116204 lines
5.6 MiB
116204 lines
5.6 MiB
$( This is the Metamath database iset.mm. $)
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$( Metamath is a formal language and associated computer program for
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archiving, verifying, and studying mathematical proofs, created by Norman
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Dwight Megill (1950--2021). For more information, visit
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http://us.metamath.org and
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https://github.com/metamath/set.mm, and feel free to ask questions at
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http://groups.google.com/group/metamath. $)
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$( New users may want to read http://us.metamath.org/ileuni/conventions.html to
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understand the label naming conventions used in iset.mm. See also "help
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verify markup" in the Metamath program for markup conventions. $)
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$( To break this file into smaller modules, in the Metamath program type "read
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iset.mm" followed by "write source iset.mm /split"; to recombine, omit
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"/split". $)
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$( The database iset.mm was created by Mario Carneiro on 21-Jan-2015 from a
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fork of the database set.mm and has been continuously enriched since then
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(list of contributors below). Many additions and updates included copying
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entire theorems and sections from set.mm in order to keep some consistency
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among these databases. $)
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$( !
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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Metamath source file for intuitionistic logic and set theory
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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~~ PUBLIC DOMAIN ~~
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This work is waived of all rights, including copyright, according to the CC0
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Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
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Currently active maintainers: See the list in the CONTRIBUTING.md file.
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Contributor list:
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DA David Abernethy
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SA Stefan Allan
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TA Thierry Arnoux
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JA Juha Arpiainen
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JB Jonathan Ben-Naim
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GB Gregory Bush
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MC Mario Carneiro
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PC Paul Chapman
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DF Drahflow
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GD Georgy Dunaev
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SF Scott Fenton
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JGH Jeff Hankins
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AH Anthony Hart
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DH David Harvey
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CH Chen-Pang He
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JH Jeff Hoffman
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SJ Szymon Jaroszewicz
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BJ Benoit Jubin
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JK Jim Kingdon
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WL Wolf Lammen
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GL Gerard Lang
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BL Brendan Leahy
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RL Raph Levien
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FL Frederic Line
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RFL Roy F. Longton
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JM Jeff Madsen
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RM Rodolfo Medina
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NM Norman Megill
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JMG Jia Ming
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MM Mykola Mostovenko
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MO Mel L. O'Cat
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SO Stefan O'Rear
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JO Jason Orendorff
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JP Josh Purinton
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SR Steve Rodriguez
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ATS Andrew Salmon
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AS Alan Sare
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ES Eric Schmidt
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GS Glauco Siliprandi
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SS Saveliy Skresanov
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JU Jarvin Udandy
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AV Alexander van der Vekens
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DAW David A. Wheeler
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JY Jonathan Yan
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FZ Fan Zheng
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Contents of this header
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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1. Naming
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2. Quick "How To"
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3. Bibliography
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4. Metamath syntax summary
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5. Other notes
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6. Acceptable shorter proofs
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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1. Naming
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Theorems which are the same as in set.mm should be named the same (that is,
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where the statement of the theorem is the same; the proof can differ without
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a new name being called for). Theorems which are different should be named
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differently (although if additional hypotheses are added in iset.mm the name
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need not be changed).
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As with set.mm, we welcome suggestions for better names (such as names which
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are more consistent with naming conventions).
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We do try to keep set.mm and iset.mm similar where we can. For example, if
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a theorem exists both places but the name in set.mm isn't great, we tend to
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keep that name for iset.mm, or change it in both files together. This is
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mainly to make it easier to copy theorems, but also to generally help people
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browse proofs, find theorems, write proofs, etc.
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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2. Quick "How To"
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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How to use this file under Windows 95/98/NT/2K/XP/Vista:
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1. Download the program metamath.exe per the instructions on the
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Metamath home page (http://us.metamath.org) and put it in the same
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directory as this file (set.mm).
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2. In Windows Explorer, double-click on metamath.exe.
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3. Type "read set.mm" and press Enter.
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4. Type "help" for a list of help topics, and "help demo" for some
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command examples.
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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3. Bibliography
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Bibliographical references are made by bracketing an identifer in a theorem's
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comment, such as [RussellWhitehead]. These refer to HTML tags on the following
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web pages:
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Logic and set theory - see http://us.metamath.org/mpegif/mmset.html#bib
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Hilbert space - see http://us.metamath.org/mpegif/mmhil.html#ref
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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4. Metamath syntax summary
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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The HELP LANGUAGE command in the Metamath program will give you a quick
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overview of Metamath. Syntax summary:
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$c ... $. - Constant declaration
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$v ... $. - Variable declaration
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$d ... $. - Disjoint (distinct) variable restriction
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<label> $f ... $. - "Floating" hypothesis (i.e. variable type declaration)
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<label> $e ... $. - "Essential" hypothesis (i.e. a logical assumption for a
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theorem or axiom)
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<label> $a ... $. - Axiom or definition or syntax construction
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<label> $p ... $= ... $. - Theorem and its proof
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${ ... $} - Block for defining the scope of the above statements
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(except $a, $p which are forever active)
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$) $( ... $)
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$( - Comments (may not be nested); see HELP LANGUAGE
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for markup features
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$[ ... $] - Include a file
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The above two-character sequences beginning with "$" are the only primitives
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built into the Metamath language. The only "logic" Metamath uses in its proof
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verification algorithm is the substitution of expressions for variables while
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checking for distinct variable violations. Everything else, including the
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axioms for logic, is defined in this database file.
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Here is some more detail about the syntax. There are two kinds of user-defined
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syntax elements, called math symbols (or just symbols) and labels. A symbol
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may contain any non-whitespace printable character except "$". A label may
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contain only alphanumeric characters and the characters "." (period), "-"
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(hyphen), and "_" (underscore). Tokens and labels are case-sensitive. All
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labels (except in proofs) must be distinct. A label may not have the same name
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as a token (to simplify the coding of certain parsers and translators).
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$c <symbollist> $.
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<symbollist> is a (whitespace-separated) list of distinct symbols that
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haven't been used before.
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$v <symbollist> $.
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<symbollist> is a list of distinct symbols that haven't been used yet
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in the current scope (see ${ ... $} below).
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$d <symbollist> $.
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<symbollist> is a (whitespace-separated) list of distinct symbols
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previously declared with $v in current scope. It means that
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substitutions into these symbols may not have variables in common.
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<label> $f <symbollist> $.
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<symbollist> is a list of 2 symbols, the first of which must be
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previously declared with $c in the current scope.
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<label> $e <symbollist> $.
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<symbollist> is a list of 2 or more symbols, the first of which must be
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previously declared with $c in the current scope.
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<label> $a <symbollist> $.
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<symbollist> is a list of 2 or more symbols, the first of which must be
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previously declared with $c in the current scope.
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<label> $p <symbollist> $= <proof> $.
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<symbollist> is a list of 2 or more symbols, the first of which must be
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previously declared with $c in the current scope. <proof> is either a
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whitespace-delimited sequence of previous labels (created by
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SAVE PROOF <label> /NORMAL) or a compressed proof (created by
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SAVE PROOF <label> /COMPRESSED). After using SAVE PROOF, use
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WRITE SOURCE to save the database file to disk.
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${ ... $}
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Block for scoping the above statements (except $a, $p which are forever
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active). Currently, $c may not occur inside of a block.
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$)
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$( <any text> $)
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$( Comment. Note: <any text> may not contain adjacent "$" and ")"
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characters.
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$[ <filename> $]
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Insert contents of <filename> at this point. If <filename> is current
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file or has been already been inserted, it will not be inserted again.
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Inside of comments, it is recommended that labels be preceded with a tilde (~)
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and math symbol tokens be enclosed in grave accents (` `). This way the LaTeX
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and HTML rendition of comments will be accurate, and (future) tools to globally
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change labels and math symbols will also change them in comments. Note that ``
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inside of grave accents is interpreted as a single ` . A special comment
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containing $ t defines LaTeX and HTML symbols. See HELP LANGUAGE and
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HELP HTML for other markup features in comments.
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The proofs in this file are in "compressed" format for storage efficiency. The
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Metamath program reads the compressed format directly. This format is
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described in an Appendix of the Metamath book. It is not intended to be read
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by humans. For viewing proofs you should use the various SHOW PROOF commands
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described in the Metamath book (or the on-line HELP).
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The Metamath program does not normally affect any content of this file (set.mm)
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other than proofs, i.e. tokens between "$=" and "$.". All other content is
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user-created. Proofs are created or modified with the PROVE command.
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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5. Other notes
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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1. It is recommended that you be familiar with chapters 2 and 4 of the
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'Metamath' book to understand the Metamath language. Chapters 2, 3 and 5
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explain how to use the program. Chapter 3 gives you an informal overview of
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what this source file is all about. Appendix A shows you the standard
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mathematical symbols corresponding to some of the ASCII tokens in this file.
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The ASCII tokens may seem cryptic at first, even if you are familiar with set
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theory, but a review of the definition summary in Chapter 3 should quickly
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enable you to see the correspondence to standard mathematical notation. To
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easily find the definition of a token, search for the first occurrences of the
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token surrounded by spaces. Some odd-looking ones include "-." for "not", and
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"C_" for "is a subset of." (HELP TEX tells you how to obtain a LaTeX output to
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see the real mathematical symbols.) Let me know if you have better suggestions
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for naming ASCII tokens.
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2. Logic and set theory provide a foundation for all of mathematics. To learn
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about them, you should study one or more of the references listed below. The
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textbooks provide a motivation for what we are doing, whereas Metamath lets you
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see in detail all hidden and implicit steps. Most standard theorems are
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accompanied by citations. Some closely followed texts include the following:
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Axioms of propositional calculus - [Margaris].
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Axioms of predicate calculus - [Megill] (System S3' in the article
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referenced).
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Theorems of propositional calculus - [WhiteheadRussell].
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Theorems of pure predicate calculus - [Margaris].
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Theorems of equality and substitution - [Monk2], [Tarski], [Megill].
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Axioms of set theory - [BellMachover].
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Development of set theory - [TakeutiZaring]. (The first part of [Quine]
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has a good explanation of the powerful device of "virtual" or
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class abstractions, which is essential to our development.)
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Construction of real and complex numbers - [Gleason]
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Theorems about real numbers - [Apostol]
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Intuitionistic logic and constructive mathematics - [Bauer] is an sampling
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of theorems and arguments which give a flavor of constructive mathematics.
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[Heyting] is a more comprehensive treatment of intuitionistic logic.
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3. Convention: All $a statements starting with "|-" have labels
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starting with "ax-" (axioms) or "df-" (definitions). "ax-" corresponds
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to what is traditionally called an axiom. "df-" introduces new symbols
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or a new relationship among symbols that can be eliminated; they always
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extend the definition of a wff or class. Metamath blindly treats $a
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statements as new given facts but does not try to justify them. The
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mmj2 program will justify the definitions as sound, except for 5 (df-bi,
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df-cleq, df-clel, df-clab, df-sbc) that require a more complex metalogical
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justification by hand.
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4. Our method of definition, the axioms for predicate calculus, and the
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development of substitution are somewhat different from those found in standard
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texts. The axioms were designed for direct derivation of standard results
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without excessive use of metatheorems. (See Theorem 9.7 of [Megill] for a
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rigorous justification.) Typically we are minimalist when introducing new
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definitions; they are introduced only when a clear advantage becomes apparent
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for reducing the number of symbols, shortening proofs, etc. We generally avoid
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the introduction of gratuitous definitions because each one requires associated
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theorems and additional elimination steps in proofs.
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5. Where possible, the notation attempts to conform to modern conventions, with
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variations due to our choice of the axiom system or to make proofs shorter.
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Listed below are some important conventions and how they correspond to textbook
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language. The notation is usually explained in more detail when first
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introduced.
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Typically, Greek letters (ph = phi, ps = psi, ch = chi, etc.) are used for
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propositional (wff) variables; x,y,z,... for individual (i.e. set)
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variables; and A,B,C,... for class variables.
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"|-", meaning "It is provable that," is the first token of all assertions
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and hypotheses that aren't syntax constructions. This is a standard
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convention in logic. For us, it also prevents any ambiguity with
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statements that are syntax constructions, such as "wff -. ph".
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"$e |- ( ph -> A. x ph ) $." should be read "Assume variable x is
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(effectively) not free in wff phi." Literally, this says "Assume it is
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provable that phi implies for all x phi."
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"|- ( -. A. x x = y -> ..." should be read "If x and y are distinct
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variables, then..." This antecedent provides us with a technical
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device (called a "distinctor" in [Megill]) to avoid the need for the
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$d statement early in our development of predicate calculus, permitting
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unrestricted substitituions as conceptually simple as those in
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propositional calculus. However, the $d eventually becomes a
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requirement, and after that this device is rarely used.
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"[ y / x ] ph" should be read "the wff that results when y is properly
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substituted for x in ph."
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"$d x y $." should be read "Assume x and y are distinct variables."
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"$d x ph $." should be read "Assume x does not occur in phi $." Sometimes
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a theorem is proved with "$e |- ( ph -> A. x ph ) $." in place of
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"$d x ph $." when a more general result is desired; ~ ax-17 can be used
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to derive the $d version. For an example of how to get from the $d
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version back to the $e version, see the proof of ~ euf from ~ df-eu .
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"$d x A $." should be read "Assume x is not a variable occurring in class A."
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"$d x A $. $d x ps $. $e |- ( x = A -> ( ph <-> ps ) ) $." is an idiom
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often used instead of explicit substitution, meaning "Assume psi results
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from the substitution of A for x in phi."
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"$e |- A e. _V $." should be read "Assume class A is a set (i.e. exists)."
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This is a convenient convention used by [Quine].
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"$d x y $. $e |- y e. A -> A. x y e. A $." should be read "Assume x is
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(effectively) not a free variable in class A."
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"`' R" should be read "converse of (relation) R" and is the same as the more
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standard notation R^{-1}.
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"( f ` x )" should be read "the value of function f at x" and is the same as
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the more familiar f(x).
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The Deduction Theorem of standard logic is never used. Instead, in set
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theory, we use other tricks to make a $e hypothesis become an antecedent.
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See the comment for theorem ~ dedth below.
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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6. Acceptable shorter proofs
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Shorter proofs are welcome, and any shorter proof I accept will be acknowledged
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in the theorem's description. However, in some cases a proof may be "shorter"
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or not depending on how it is formatted. This section provides general
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guidelines.
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Usually I will automatically accept shorter proofs that (1) shorten the set.mm
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file (with compressed proofs), (2) reduce the size of the HTML file generated
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with SHOW STATEMENT xx / HTML, (3) use only existing, unmodified theorems in
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the database (the order of theorems may be changed, though), and (4) use no
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additional axioms.
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Usually I will also automatically accept a _new_ theorem that is used to
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shorten multiple proofs, if the total size of set.mm (including the comment of
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the new theorem, not including the acknowledgment) decreases as a result.
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In borderline cases, I typically place more importance on the number of
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compressed proof steps and less on the length of the label section (since the
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names are in principal arbitrary). If two proofs have the same number of
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compressed proof steps, I will typically give preference to the one with the
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smaller number of different labels, or if these numbers are the same, the proof
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with the fewest number of characters that the proofs happen to have by chance
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when label lengths are included.
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A few theorems have a longer proof than necessary in order to avoid the use of
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certain axioms, for pedagogical purposes, and for other reasons. Usually this
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is clear from the theorem's description. For example, ~ idALT shows a proof
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directly from axioms. Shorter proofs for such cases won't be accepted, of
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course, unless the criteria described continues to be satisfied.
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$)
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$(
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###############################################################################
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FIRST ORDER LOGIC WITH EQUALITY
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###############################################################################
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$)
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$(
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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Pre-logic
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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This section includes a few "housekeeping" mechanisms before we begin
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defining the basics of logic.
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$)
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$( Declare the primitive constant symbols for propositional calculus. $)
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$c ( $. $( Left parenthesis $)
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$c ) $. $( Right parenthesis $)
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$c -> $. $( Right arrow (read: "implies") $)
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$c -. $. $( Right handle (read: "not") $)
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$c wff $. $( Well-formed formula symbol (read: "the following symbol
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sequence is a wff") $)
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$c |- $. $( Turnstile (read: "the following symbol sequence is provable" or
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'a proof exists for") $)
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$( Declare typographical constant symbols that are not directly used
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in the formalism, but *are* symbols we find useful when
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explaining the formalism. It is much easier to consistently use
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a single typographical system when generating text. $)
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$c & $. $( Ampersand (read: "and-also") $)
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$c => $. $( Big-to (read: "proves") $)
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$( wff variable sequence: ph ps ch th ta et ze si rh mu la ka $)
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$( Introduce some variable names we will use to represent well-formed
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formulas (wff's). $)
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$v ph $. $( Greek phi $)
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$v ps $. $( Greek psi $)
|
|
$v ch $. $( Greek chi $)
|
|
$v th $. $( Greek theta $)
|
|
$v ta $. $( Greek tau $)
|
|
$v et $. $( Greek eta $)
|
|
$v ze $. $( Greek zeta $)
|
|
$v si $. $( Greek sigma $)
|
|
$v rh $. $( Greek rho $)
|
|
$v mu $. $( Greek mu $)
|
|
$v la $. $( Greek lambda $)
|
|
$v ka $. $( Greek kappa $)
|
|
|
|
$( Specify some variables that we will use to represent wff's.
|
|
The fact that a variable represents a wff is relevant only to a theorem
|
|
referring to that variable, so we may use $f hypotheses. The symbol
|
|
` wff ` specifies that the variable that follows it represents a wff. $)
|
|
$( Let variable ` ph ` be a wff. $)
|
|
wph $f wff ph $.
|
|
$( Let variable ` ps ` be a wff. $)
|
|
wps $f wff ps $.
|
|
$( Let variable ` ch ` be a wff. $)
|
|
wch $f wff ch $.
|
|
$( Let variable ` th ` be a wff. $)
|
|
wth $f wff th $.
|
|
$( Let variable ` ta ` be a wff. $)
|
|
wta $f wff ta $.
|
|
$( Let variable ` et ` be a wff. $)
|
|
wet $f wff et $.
|
|
$( Let variable ` ze ` be a wff. $)
|
|
wze $f wff ze $.
|
|
$( Let variable ` si ` be a wff. $)
|
|
wsi $f wff si $.
|
|
$( Let variable ` rh ` be a wff. $)
|
|
wrh $f wff rh $.
|
|
$( Let variable ` mu ` be a wff. $)
|
|
wmu $f wff mu $.
|
|
$( Let variable ` la ` be a wff. $)
|
|
wla $f wff la $.
|
|
$( Let variable ` ka ` be a wff. $)
|
|
wka $f wff ka $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Inferences for assisting proof development
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The inference rules in this section will normally never appear in a completed
|
|
proof. They can be ignored if you are using this database to assist learning
|
|
logic - please start with the statement ~ wn instead.
|
|
|
|
$)
|
|
|
|
${
|
|
a1ii.1 $e |- ph $.
|
|
a1ii.2 $e |- ps $.
|
|
$( (_Note_: This inference rule and the next one, ~ idi , will normally
|
|
never appear in a completed proof. It can be ignored if you are using
|
|
this database to assist learning logic - please start with the statement
|
|
~ wn instead.)
|
|
|
|
This is a technical inference to assist proof development. It provides
|
|
a temporary way to add an independent subproof to a proof under
|
|
development, for later assignment to a normal proof step.
|
|
|
|
The metamath program's Proof Assistant requires proofs to be developed
|
|
backwards from the conclusion with no gaps, and it has no mechanism that
|
|
lets the user to work on isolated subproofs. This inference provides a
|
|
workaround for this limitation. It can be inserted at any point in a
|
|
proof to allow an independent subproof to be developed on the side, for
|
|
later use as part of the final proof.
|
|
|
|
_Instructions_: (1) Assign this inference to any unknown step in the
|
|
proof. Typically, the last unknown step is the most convenient, since
|
|
'assign last' can be used. This step will be replicated in hypothesis
|
|
a1ii.1, from where the development of the main proof can continue. (2)
|
|
Develop the independent subproof backwards from hypothesis a1ii.2. If
|
|
desired, use a 'let' command to pre-assign the conclusion of the
|
|
independent subproof to a1ii.2. (3) After the independent subproof is
|
|
complete, use 'improve all' to assign it automatically to an unknown
|
|
step in the main proof that matches it. (4) After the entire proof is
|
|
complete, use 'minimize *' to clean up (discard) all ~ a1ii references
|
|
automatically.
|
|
|
|
This inference was originally designed to assist importing partially
|
|
completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can
|
|
also be useful on its own. Interestingly, no axioms are required for
|
|
its proof. (Contributed by NM, 7-Feb-2006.) $)
|
|
a1ii $p |- ph $=
|
|
( ) C $.
|
|
$}
|
|
|
|
${
|
|
idi.1 $e |- ph $.
|
|
$( Inference form of ~ id . This inference rule, which requires no axioms
|
|
for its proof, is useful as a copy-paste mechanism during proof
|
|
development in mmj2. It is normally not referenced in the final version
|
|
of a proof, since it is always redundant and can be removed using the
|
|
'minimize *' command in the metamath program's Proof Assistant.
|
|
(Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
idi $p |- ph $=
|
|
( ) B $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Propositional calculus
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Recursively define primitive wffs for propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( If ` ph ` is a wff, so is ` -. ph ` or "not ` ph ` ." Part of the
|
|
recursive definition of a wff (well-formed formula). Traditionally, Greek
|
|
letters are used to represent wffs, and we follow this convention. In
|
|
propositional calculus, we define only wffs built up from other wffs, i.e.
|
|
there is no starting or "atomic" wff. Later, in predicate calculus, we
|
|
will extend the basic wff definition by including atomic wffs ( ~ weq and
|
|
~ wel ). $)
|
|
wn $a wff -. ph $.
|
|
|
|
$( If ` ph ` and ` ps ` are wff's, so is ` ( ph -> ps ) ` or " ` ph ` implies
|
|
` ps ` ." Part of the recursive definition of a wff. The left-hand wff
|
|
is called the antecedent, and the right-hand wff is called the consequent.
|
|
In the case of ` ( ph -> ( ps -> ch ) ) ` , the middle ` ps ` may be
|
|
informally called either an antecedent or part of the consequent depending
|
|
on context. $)
|
|
wi $a wff ( ph -> ps ) $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Propositional logic axioms for implication
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom _Simp_. Axiom A1 of [Margaris] p. 49. One of the axioms of
|
|
propositional calculus. This axiom is called _Simp_ or "the principle of
|
|
simplification" in _Principia Mathematica_ (Theorem *2.02 of
|
|
[WhiteheadRussell] p. 100) because "it enables us to pass from the joint
|
|
assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply."
|
|
|
|
The theorems of propositional calculus are also called _tautologies_.
|
|
Although classical propositional logic tautologies can be proved using
|
|
truth tables, there is no similarly simple system for intuitionistic
|
|
propositional logic, so proving tautologies from axioms is the preferred
|
|
approach. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-1 $a |- ( ph -> ( ps -> ph ) ) $.
|
|
|
|
$( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. This axiom "distributes" an
|
|
antecedent over two consequents. This axiom was part of Frege's original
|
|
system and is known as _Frege_ in the literature. It is also proved as
|
|
Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this
|
|
axiom also turns out to be true, as demonstrated by ~ pm5.41 .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-2 $a |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $.
|
|
|
|
$(
|
|
Postulate the modus ponens rule of inference.
|
|
$)
|
|
|
|
${
|
|
$( Minor premise for modus ponens. $)
|
|
min $e |- ph $.
|
|
$( Major premise for modus ponens. $)
|
|
maj $e |- ( ph -> ps ) $.
|
|
$( Rule of Modus Ponens. The postulated inference rule of propositional
|
|
calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if
|
|
` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be
|
|
true." This rule is sometimes called "detachment," since it detaches
|
|
the minor premise from the major premise.
|
|
|
|
Note: In some web page displays such as the Statement List, the symbols
|
|
"&" and "=>" informally indicate the relationship between the hypotheses
|
|
and the assertion (conclusion), abbreviating the English words "and" and
|
|
"implies." They are not part of the formal language. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
ax-mp $a |- ps $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical implication
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The results in this section are based on implication only, and only use
|
|
~ ax-1 , ~ ax-2 , and ~ ax-mp .
|
|
In an implication, the wff before the arrow is called the "antecedent" and
|
|
the wff after the arrow is called the "consequent."
|
|
|
|
We will use the following descriptive terms very loosely: A "closed form"
|
|
or "tautology" has no $e hypotheses. An "inference" has one or more $e
|
|
hypotheses. A "deduction" is an inference in which the hypotheses and the
|
|
conclusion share the same antecedent.
|
|
$)
|
|
|
|
${
|
|
mp2b.1 $e |- ph $.
|
|
mp2b.2 $e |- ( ph -> ps ) $.
|
|
mp2b.3 $e |- ( ps -> ch ) $.
|
|
$( A double modus ponens inference. (Contributed by Mario Carneiro,
|
|
24-Jan-2013.) $)
|
|
mp2b $p |- ch $=
|
|
( ax-mp ) BCABDEGFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ a1i . $)
|
|
a1i.1 $e |- ph $.
|
|
$( Inference derived from axiom ~ ax-1 . See ~ a1d for an explanation of
|
|
our informal use of the terms "inference" and "deduction." See also the
|
|
comment in ~ syld . (Contributed by NM, 5-Aug-1993.) $)
|
|
a1i $p |- ( ps -> ph ) $=
|
|
( wi ax-1 ax-mp ) ABADCABEF $.
|
|
$}
|
|
|
|
${
|
|
mp1i.a $e |- ph $.
|
|
mp1i.b $e |- ( ph -> ps ) $.
|
|
$( Drop and replace an antecedent. (Contributed by Stefan O'Rear,
|
|
29-Jan-2015.) $)
|
|
mp1i $p |- ( ch -> ps ) $=
|
|
( ax-mp a1i ) BCABDEFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ a2i . $)
|
|
a2i.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference derived from axiom ~ ax-2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
a2i $p |- ( ( ph -> ps ) -> ( ph -> ch ) ) $=
|
|
( wi ax-2 ax-mp ) ABCEEABEACEEDABCFG $.
|
|
$}
|
|
|
|
${
|
|
imim2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding common antecedents in an implication. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
imim2i $p |- ( ( ch -> ph ) -> ( ch -> ps ) ) $=
|
|
( wi a1i a2i ) CABABECDFG $.
|
|
$}
|
|
|
|
${
|
|
mpd.1 $e |- ( ph -> ps ) $.
|
|
mpd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A modus ponens deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
mpd $p |- ( ph -> ch ) $=
|
|
( wi a2i ax-mp ) ABFACFDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
$( First of 2 premises for ~ syl . $)
|
|
syl.1 $e |- ( ph -> ps ) $.
|
|
$( Second of 2 premises for ~ syl . $)
|
|
syl.2 $e |- ( ps -> ch ) $.
|
|
$( An inference version of the transitive laws for implication ~ imim2 and
|
|
~ imim1 , which Russell and Whitehead call "the principle of the
|
|
syllogism...because...the syllogism in Barbara is derived from them"
|
|
(quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors
|
|
call this law a "hypothetical syllogism." (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 20-Oct-2011.) (Proof shortened
|
|
by Wolf Lammen, 26-Jul-2012.) $)
|
|
syl $p |- ( ph -> ch ) $=
|
|
( wi a1i mpd ) ABCDBCFAEGH $.
|
|
$}
|
|
|
|
${
|
|
mpi.1 $e |- ps $.
|
|
mpi.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A nested modus ponens inference. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Stefan Allan, 20-Mar-2006.) $)
|
|
mpi $p |- ( ph -> ch ) $=
|
|
( a1i mpd ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
mp2.1 $e |- ph $.
|
|
mp2.2 $e |- ps $.
|
|
mp2.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
mp2 $p |- ch $=
|
|
( mpi ax-mp ) ACDABCEFGH $.
|
|
$}
|
|
|
|
${
|
|
3syl.1 $e |- ( ph -> ps ) $.
|
|
3syl.2 $e |- ( ps -> ch ) $.
|
|
3syl.3 $e |- ( ch -> th ) $.
|
|
$( Inference chaining two syllogisms. (Contributed by NM, 5-Aug-1993.) $)
|
|
3syl $p |- ( ph -> th ) $=
|
|
( syl ) ACDABCEFHGH $.
|
|
$}
|
|
|
|
${
|
|
4syl.1 $e |- ( ph -> ps ) $.
|
|
4syl.2 $e |- ( ps -> ch ) $.
|
|
4syl.3 $e |- ( ch -> th ) $.
|
|
4syl.4 $e |- ( th -> ta ) $.
|
|
$( Inference chaining three syllogisms. The use of this theorem is marked
|
|
"discouraged" because it can cause the "minimize" command to have very
|
|
long run times. However, feel free to use "minimize 4syl /override" if
|
|
you wish. (Contributed by BJ, 14-Jul-2018.)
|
|
(New usage is discouraged.) $)
|
|
4syl $p |- ( ph -> ta ) $=
|
|
( 3syl syl ) ADEABCDFGHJIK $.
|
|
$}
|
|
|
|
$( Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For
|
|
another version of the proof directly from axioms, see ~ idALT .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan,
|
|
20-Mar-2006.) $)
|
|
id $p |- ( ph -> ph ) $=
|
|
( wi ax-1 mpd ) AAABZAAACAECD $.
|
|
|
|
$( Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This
|
|
version is proved directly from the axioms for demonstration purposes.
|
|
This proof is a popular example in the literature and is identical, step
|
|
for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a)
|
|
of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of
|
|
[Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's _A
|
|
Primer for Logic and Proof_ p. 17 (PDF p. 23) at
|
|
~ http://www.mathsci.appstate.edu/~~jlh/primer/hirst.pdf . For a shorter
|
|
version of the proof that takes advantage of previously proved theorems,
|
|
see ~ id . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) Use ~ id instead.
|
|
(New usage is discouraged.) $)
|
|
idALT $p |- ( ph -> ph ) $=
|
|
( wi ax-1 ax-2 ax-mp ) AAABZBZFAACAFABBGFBAFCAFADEE $.
|
|
|
|
$( Principle of identity with antecedent. (Contributed by NM,
|
|
26-Nov-1995.) $)
|
|
idd $p |- ( ph -> ( ps -> ps ) ) $=
|
|
( wi id a1i ) BBCABDE $.
|
|
|
|
${
|
|
a1d.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing an embedded antecedent. (The proof was revised by
|
|
Stefan Allan, 20-Mar-2006.)
|
|
|
|
_Naming convention_: We often call a theorem a "deduction" and suffix
|
|
its label with "d" whenever the hypotheses and conclusion are each
|
|
prefixed with the same antecedent. This allows us to use the theorem in
|
|
places where (in traditional textbook formalizations) the standard
|
|
Deduction Theorem would be used; here ` ph ` would be replaced with a
|
|
conjunction ( ~ wa ) of the hypotheses of the would-be deduction. By
|
|
contrast, we tend to call the simpler version with no common antecedent
|
|
an "inference" and suffix its label with "i"; compare theorem ~ a1i .
|
|
Finally, a "theorem" would be the form with no hypotheses; in this case
|
|
the "theorem" form would be the original axiom ~ ax-1 . We usually show
|
|
the theorem form without a suffix on its label (e.g. ~ pm2.43 vs.
|
|
~ pm2.43i vs. ~ pm2.43d ). (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by NM, 20-Mar-2006.) $)
|
|
a1d $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( wi ax-1 syl ) ABCBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
2a1d.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing two antecedents. Two applications of ~ a1d .
|
|
Deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by BJ,
|
|
10-Aug-2020.) $)
|
|
2a1d $p |- ( ph -> ( ch -> ( th -> ps ) ) ) $=
|
|
( wi a1d ) ADBFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
a1i13.1 $e |- ( ps -> th ) $.
|
|
$( Add two antecedents to a wff. (Contributed by Jeff Hankins,
|
|
4-Aug-2009.) $)
|
|
a1i13 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi a1d a1i ) BCDFFABDCEGH $.
|
|
$}
|
|
|
|
$( A double form of ~ ax-1 . Its associated inference is ~ 2a1i . Its
|
|
associated deduction is ~ 2a1d . (Contributed by BJ, 10-Aug-2020.)
|
|
(Proof shortened by Wolf Lammen, 1-Sep-2020.) $)
|
|
2a1 $p |- ( ph -> ( ps -> ( ch -> ph ) ) ) $=
|
|
( id 2a1d ) AABCADE $.
|
|
|
|
${
|
|
a2d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Deduction distributing an embedded antecedent. (Contributed by NM,
|
|
23-Jun-1994.) $)
|
|
a2d $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $=
|
|
( wi ax-2 syl ) ABCDFFBCFBDFFEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
2a1i.1 $e |- ch $.
|
|
$( Add two antecedents to a wff. (Contributed by Jeff Hankins,
|
|
4-Aug-2009.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
2a1i $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( a1i a1d ) ACBCADEF $.
|
|
$}
|
|
|
|
${
|
|
sylcom.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylcom.2 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( Syllogism inference with commutation of antecedents. (Contributed by
|
|
NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof
|
|
shortened by Stefan Allan, 23-Feb-2006.) $)
|
|
sylcom $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a2i syl ) ABCGBDGEBCDFHI $.
|
|
$}
|
|
|
|
${
|
|
syl5com.1 $e |- ( ph -> ps ) $.
|
|
syl5com.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( Syllogism inference with commuted antecedents. (Contributed by NM,
|
|
24-May-2005.) $)
|
|
syl5com $p |- ( ph -> ( ch -> th ) ) $=
|
|
( a1d sylcom ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ com12 . See ~ pm2.04 for the theorem form. $)
|
|
com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference that swaps (commutes) antecedents in an implication.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
4-Aug-2012.) $)
|
|
com12 $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id syl5com ) BBACBEDF $.
|
|
$}
|
|
|
|
${
|
|
syl11.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl11.2 $e |- ( th -> ph ) $.
|
|
$( A syllogism inference. Commuted form of an instance of ~ syl .
|
|
(Contributed by BJ, 25-Oct-2021.) $)
|
|
syl11 $p |- ( ps -> ( th -> ch ) ) $=
|
|
( wi syl com12 ) DBCDABCGFEHI $.
|
|
$}
|
|
|
|
${
|
|
syl5.1 $e |- ( ph -> ps ) $.
|
|
syl5.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the second antecedent of the first premise. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.) $)
|
|
syl5 $p |- ( ch -> ( ph -> th ) ) $=
|
|
( syl5com com12 ) ACDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6.2 $e |- ( ch -> th ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 30-Jul-2012.) $)
|
|
syl6 $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1i sylcom ) ABCDECDGBFHI $.
|
|
$}
|
|
|
|
${
|
|
syl56.1 $e |- ( ph -> ps ) $.
|
|
syl56.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
syl56.3 $e |- ( th -> ta ) $.
|
|
$( Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.) $)
|
|
syl56 $p |- ( ch -> ( ph -> ta ) ) $=
|
|
( syl6 syl5 ) ABCEFCBDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl6com.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6com.2 $e |- ( ch -> th ) $.
|
|
$( Syllogism inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
syl6com $p |- ( ps -> ( ph -> th ) ) $=
|
|
( syl6 com12 ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
mpcom.1 $e |- ( ps -> ph ) $.
|
|
mpcom.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus ponens inference with commutation of antecedents. (Contributed by
|
|
NM, 17-Mar-1996.) $)
|
|
mpcom $p |- ( ps -> ch ) $=
|
|
( com12 mpd ) BACDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
syli.1 $e |- ( ps -> ( ph -> ch ) ) $.
|
|
syli.2 $e |- ( ch -> ( ph -> th ) ) $.
|
|
$( Syllogism inference with common nested antecedent. (Contributed by NM,
|
|
4-Nov-2004.) $)
|
|
syli $p |- ( ps -> ( ph -> th ) ) $=
|
|
( com12 sylcom ) BACDECADFGH $.
|
|
$}
|
|
|
|
${
|
|
syl2im.1 $e |- ( ph -> ps ) $.
|
|
syl2im.2 $e |- ( ch -> th ) $.
|
|
syl2im.3 $e |- ( ps -> ( th -> ta ) ) $.
|
|
$( Replace two antecedents. Implication-only version of ~ syl2an .
|
|
(Contributed by Wolf Lammen, 14-May-2013.) $)
|
|
syl2im $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wi syl5 syl ) ABCEIFCDBEGHJK $.
|
|
$}
|
|
|
|
$( This theorem, called "Assertion," can be thought of as closed form of
|
|
modus ponens ~ ax-mp . Theorem *2.27 of [WhiteheadRussell] p. 104.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wi id com12 ) ABCZABFDE $.
|
|
|
|
${
|
|
mpdd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
mpdd.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.) $)
|
|
mpdd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a2d mpd ) ABCGBDGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mpid.1 $e |- ( ph -> ch ) $.
|
|
mpid.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.) $)
|
|
mpid $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1d mpdd ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpdi.1 $e |- ( ps -> ch ) $.
|
|
mpdi.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.)
|
|
(Proof shortened by O'Cat, 15-Jan-2008.) $)
|
|
mpdi $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1i mpdd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpii.1 $e |- ch $.
|
|
mpii.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A doubly nested modus ponens inference. (Contributed by NM,
|
|
31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.) $)
|
|
mpii $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1i mpdi ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Syllogism deduction.
|
|
|
|
Notice that ~ syld has the same form as ~ syl with ` ph ` added in front
|
|
of each hypothesis and conclusion. When all theorems referenced in a
|
|
proof are converted in this way, we can replace ` ph ` with a hypothesis
|
|
of the proof, allowing the hypothesis to be eliminated with ~ id and
|
|
become an antecedent. The Deduction Theorem for propositional calculus,
|
|
e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is
|
|
always possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
|
|
syld $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1d mpdd ) ABCDEACDGBFHI $.
|
|
$( Syllogism deduction. Commuted form of ~ syld . (Contributed by BJ,
|
|
25-Oct-2021.) $)
|
|
syldc $p |- ( ps -> ( ph -> th ) ) $=
|
|
( syld com12 ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
mp2d.1 $e |- ( ph -> ps ) $.
|
|
mp2d.2 $e |- ( ph -> ch ) $.
|
|
mp2d.3 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A double modus ponens deduction. (Contributed by NM, 23-May-2013.)
|
|
(Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
mp2d $p |- ( ph -> th ) $=
|
|
( mpid mpd ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
a1dd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction introducing a nested embedded antecedent. (Contributed by NM,
|
|
17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.) $)
|
|
a1dd $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
|
|
( wi ax-1 syl6 ) ABCDCFECDGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.43i.1 $e |- ( ph -> ( ph -> ps ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43i $p |- ( ph -> ps ) $=
|
|
( id mpd ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
pm2.43d.1 $e |- ( ph -> ( ps -> ( ps -> ch ) ) ) $.
|
|
$( Deduction absorbing redundant antecedent. (Contributed by NM,
|
|
18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43d $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( id mpdi ) ABBCBEDF $.
|
|
$}
|
|
|
|
${
|
|
pm2.43a.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43a $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id mpid ) BABCBEDF $.
|
|
$}
|
|
|
|
${
|
|
pm2.43b.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
31-Oct-1995.) $)
|
|
pm2.43b $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( pm2.43a com12 ) BACABCDEF $.
|
|
$}
|
|
|
|
$( Absorption of redundant antecedent. Also called the "Contraction" or
|
|
"Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat,
|
|
15-Aug-2004.) $)
|
|
pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wi pm2.27 a2i ) AABCBABDE $.
|
|
|
|
${
|
|
imim2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding nested antecedents. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
imim2d $p |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
|
|
( wi a1d a2d ) ADBCABCFDEGH $.
|
|
$}
|
|
|
|
$( A closed form of syllogism (see ~ syl ). Theorem *2.05 of
|
|
[WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 6-Sep-2012.) $)
|
|
imim2 $p |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) $=
|
|
( wi id imim2d ) ABDZABCGEF $.
|
|
|
|
${
|
|
embantd.1 $e |- ( ph -> ps ) $.
|
|
embantd.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Deduction embedding an antecedent. (Contributed by Wolf Lammen,
|
|
4-Oct-2013.) $)
|
|
embantd $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
|
|
( wi imim2d mpid ) ABCGBDEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
3syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3syld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
3syld.3 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Triple syllogism deduction. (Contributed by Jeff Hankins,
|
|
4-Aug-2009.) $)
|
|
3syld $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( syld ) ABDEABCDFGIHI $.
|
|
$}
|
|
|
|
${
|
|
sylsyld.1 $e |- ( ph -> ps ) $.
|
|
sylsyld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
sylsyld.3 $e |- ( ps -> ( th -> ta ) ) $.
|
|
$( Virtual deduction rule. (Contributed by Alan Sare, 20-Apr-2011.) $)
|
|
sylsyld $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wi syl syld ) ACDEGABDEIFHJK $.
|
|
$}
|
|
|
|
${
|
|
imim12i.1 $e |- ( ph -> ps ) $.
|
|
imim12i.2 $e |- ( ch -> th ) $.
|
|
$( Inference joining two implications. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by O'Cat, 29-Oct-2011.) $)
|
|
imim12i $p |- ( ( ps -> ch ) -> ( ph -> th ) ) $=
|
|
( wi imim2i syl5 ) ABBCGDECDBFHI $.
|
|
$}
|
|
|
|
${
|
|
imim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding common consequents in an implication, thereby
|
|
interchanging the original antecedent and consequent. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
|
|
imim1i $p |- ( ( ps -> ch ) -> ( ph -> ch ) ) $=
|
|
( id imim12i ) ABCCDCEF $.
|
|
$}
|
|
|
|
${
|
|
imim3i.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference adding three nested antecedents. (Contributed by NM,
|
|
19-Dec-2006.) $)
|
|
imim3i $p |- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
|
|
( wi imim2i a2d ) DAFDBCABCFDEGH $.
|
|
$}
|
|
|
|
${
|
|
sylc.1 $e |- ( ph -> ps ) $.
|
|
sylc.2 $e |- ( ph -> ch ) $.
|
|
sylc.3 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
4-May-1994.) (Revised by NM, 13-Jul-2013.) $)
|
|
sylc $p |- ( ph -> th ) $=
|
|
( syl2im pm2.43i ) ADABACDEFGHI $.
|
|
$}
|
|
|
|
${
|
|
syl3c.1 $e |- ( ph -> ps ) $.
|
|
syl3c.2 $e |- ( ph -> ch ) $.
|
|
syl3c.3 $e |- ( ph -> th ) $.
|
|
syl3c.4 $e |- ( ps -> ( ch -> ( th -> ta ) ) ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by Alan
|
|
Sare, 7-Jul-2011.) $)
|
|
syl3c $p |- ( ph -> ta ) $=
|
|
( wi sylc mpd ) ADEHABCDEJFGIKL $.
|
|
$}
|
|
|
|
${
|
|
syl6mpi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6mpi.2 $e |- th $.
|
|
syl6mpi.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( ~ syl6 combined with ~ mpi . (Contributed by Alan Sare, 8-Jul-2011.)
|
|
(Proof shortened by Wolf Lammen, 13-Sep-2012.) $)
|
|
syl6mpi $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( mpi syl6 ) ABCEFCDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
mpsyl.1 $e |- ph $.
|
|
mpsyl.2 $e |- ( ps -> ch ) $.
|
|
mpsyl.3 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Modus ponens combined with a syllogism inference. (Contributed by Alan
|
|
Sare, 20-Apr-2011.) $)
|
|
mpsyl $p |- ( ps -> th ) $=
|
|
( a1i sylc ) BACDABEHFGI $.
|
|
$}
|
|
|
|
${
|
|
syl6c.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6c.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
syl6c.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( Inference combining ~ syl6 with contraction. (Contributed by Alan Sare,
|
|
2-May-2011.) $)
|
|
syl6c $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( wi syl6 mpdd ) ABDEGABCDEIFHJK $.
|
|
$}
|
|
|
|
${
|
|
syldd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syldd.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
|
|
$( Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof
|
|
shortened by Wolf Lammen, 11-May-2013.) $)
|
|
syldd $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi imim2 syl6c ) ABDEHCDHCEHGFDECIJ $.
|
|
$}
|
|
|
|
${
|
|
syl5d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl5d.2 $e |- ( ph -> ( th -> ( ch -> ta ) ) ) $.
|
|
$( A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat,
|
|
2-Feb-2006.) $)
|
|
syl5d $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
|
|
( wi a1d syldd ) ADBCEABCHDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl7.1 $e |- ( ph -> ps ) $.
|
|
syl7.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the third antecedent of the first premise. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
|
|
syl7 $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
|
|
( wi a1i syl5d ) CABDEABHCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl6d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl6d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat,
|
|
2-Feb-2006.) (Revised by NM, 3-Feb-2006.) $)
|
|
syl6d $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi a1d syldd ) ABCDEFADEHBGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl8.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl8.2 $e |- ( th -> ta ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
|
|
(Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
|
|
syl8 $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi a1i syl6d ) ABCDEFDEHAGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl9.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( A nested syllogism inference with different antecedents. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
syl9 $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
|
|
( wi a1i syl5d ) ABCDEFDCEHHAGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl9r.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( A nested syllogism inference with different antecedents. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
syl9r $p |- ( th -> ( ph -> ( ps -> ta ) ) ) $=
|
|
( wi syl9 com12 ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
imim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
imim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Deduction combining antecedents and consequents. (Contributed by NM,
|
|
7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.) $)
|
|
imim12d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) ) $=
|
|
( wi imim2d syl5d ) ABCCDHEFADECGIJ $.
|
|
$}
|
|
|
|
${
|
|
imim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 12-Sep-2012.) $)
|
|
imim1d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) ) $=
|
|
( idd imim12d ) ABCDDEADFG $.
|
|
$}
|
|
|
|
$( A closed form of syllogism (see ~ syl ). Theorem *2.06 of
|
|
[WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 25-May-2013.) $)
|
|
imim1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi id imim1d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.83 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ( ch -> th ) ) ->
|
|
( ph -> ( ps -> th ) ) ) ) $=
|
|
( wi imim1 imim3i ) BCECDEBDEABCDFG $.
|
|
|
|
${
|
|
com3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
|
|
com23 $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
|
|
( wi pm2.27 syl9 ) ABCDFCDECDGH $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com3r $p |- ( ch -> ( ph -> ( ps -> th ) ) ) $=
|
|
( wi com23 com12 ) ACBDFABCDEGH $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com13 $p |- ( ch -> ( ps -> ( ph -> th ) ) ) $=
|
|
( com3r com23 ) CABDABCDEFG $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com3l $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
|
|
( com3r ) CABDABCDEFF $.
|
|
$}
|
|
|
|
$( Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
12-Sep-2012.) $)
|
|
pm2.04 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi id com23 ) ABCDDZABCGEF $.
|
|
|
|
${
|
|
com4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
|
$( Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com34 $p |- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) ) $=
|
|
( wi pm2.04 syl6 ) ABCDEGGDCEGGFCDEHI $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.) $)
|
|
com4l $p |- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) ) $=
|
|
( wi com3l com34 ) BCADEABCDEGFHI $.
|
|
|
|
$( Commutation of antecedents. Rotate twice. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com4t $p |- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) ) $=
|
|
( com4l ) BCDAEABCDEFGG $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com4r $p |- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) $=
|
|
( com4t com4l ) CDABEABCDEFGH $.
|
|
|
|
$( Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com24 $p |- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) ) $=
|
|
( wi com4t com13 ) CDABEGABCDEFHI $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 4th. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com14 $p |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) ) $=
|
|
( wi com4l com3r ) BCDAEGABCDEFHI $.
|
|
$}
|
|
|
|
${
|
|
com5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
|
|
$( Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com45 $p |- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) ) $=
|
|
( wi pm2.04 syl8 ) ABCDEFHHEDFHHGDEFIJ $.
|
|
|
|
$( Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com35 $p |- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) ) $=
|
|
( wi com34 com45 ) ABDECFHABDCEFABCDEFHGIJI $.
|
|
|
|
$( Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com25 $p |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) ) $=
|
|
( wi com24 com45 ) ADCEBFHADCBEFABCDEFHGIJI $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins,
|
|
28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) $)
|
|
com5l $p |- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) ) $=
|
|
( wi com4l com45 ) BCDAEFABCDEFHGIJ $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen,
|
|
29-Jul-2012.) $)
|
|
com15 $p |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) ) $=
|
|
( wi com5l com4r ) BCDEAFHABCDEFGIJ $.
|
|
|
|
$( Commutation of antecedents. Rotate left twice. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com52l $p |- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) ) $=
|
|
( com5l ) BCDEAFABCDEFGHH $.
|
|
|
|
$( Commutation of antecedents. Rotate right twice. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com52r $p |- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) $=
|
|
( com52l com5l ) CDEABFABCDEFGHI $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen,
|
|
29-Jul-2012.) $)
|
|
com5r $p |- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) ) $=
|
|
( com52l ) CDEABFABCDEFGHH $.
|
|
$}
|
|
|
|
$( Elimination of a nested antecedent. (Contributed by Wolf Lammen,
|
|
9-May-2013.) $)
|
|
jarr $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
|
|
( wi ax-1 imim1i ) BABDCBAEF $.
|
|
|
|
${
|
|
pm2.86i.1 $e |- ( ( ph -> ps ) -> ( ph -> ch ) ) $.
|
|
$( Inference based on ~ pm2.86 . (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 3-Apr-2013.) $)
|
|
pm2.86i $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wi ax-1 syl com12 ) BACBABEACEBAFDGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.86d.1 $e |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $.
|
|
$( Deduction based on ~ pm2.86 . (Contributed by NM, 29-Jun-1995.) (Proof
|
|
shortened by Wolf Lammen, 3-Apr-2013.) $)
|
|
pm2.86d $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi ax-1 syl5 com23 ) ACBDCBCFABDFCBGEHI $.
|
|
$}
|
|
|
|
$( Converse of axiom ~ ax-2 . Theorem *2.86 of [WhiteheadRussell] p. 108.
|
|
(Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen,
|
|
3-Apr-2013.) $)
|
|
pm2.86 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) ->
|
|
( ph -> ( ps -> ch ) ) ) $=
|
|
( wi id pm2.86d ) ABDACDDZABCGEF $.
|
|
|
|
$( The Linearity Axiom of the infinite-valued sentential logic (L-infinity)
|
|
of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.) $)
|
|
loolin $p |- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) ) $=
|
|
( wi jarr pm2.43d ) ABCBACZCBAABFDE $.
|
|
|
|
$( An alternate for the Linearity Axiom of the infinite-valued sentential
|
|
logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, _Reports
|
|
on Mathematical Logic_ 10, 129-137 (1978). (Contributed by O'Cat,
|
|
8-Aug-2004.) $)
|
|
loowoz $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) ->
|
|
( ( ps -> ph ) -> ( ps -> ch ) ) ) $=
|
|
( wi jarr a2d ) ABDACDZDBACABGEF $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical conjunction and logical equivalence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare connectives for conjunction ('and') and biconditional. $)
|
|
$c /\ $. $( Wedge (read: 'and') $)
|
|
$c <-> $. $( Double arrow (read: 'if and only if' or
|
|
'is logically equivalent to') $)
|
|
|
|
$( Extend wff definition to include conjunction ('and'). $)
|
|
wa $a wff ( ph /\ ps ) $.
|
|
|
|
$( Extend our wff definition to include the biconditional connective. $)
|
|
wb $a wff ( ph <-> ps ) $.
|
|
|
|
$( Left 'and' elimination. One of the axioms of propositional logic. Use
|
|
its alias ~ simpl instead for naming consistency with set.mm.
|
|
(New usage is discouraged.) (Contributed by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
ax-ia1 $a |- ( ( ph /\ ps ) -> ph ) $.
|
|
|
|
$( Right 'and' elimination. One of the axioms of propositional logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ~ simpr
|
|
instead for naming consistency with set.mm.
|
|
(New usage is discouraged.) $)
|
|
ax-ia2 $a |- ( ( ph /\ ps ) -> ps ) $.
|
|
|
|
$( 'And' introduction. One of the axioms of propositional logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-ia3 $a |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $.
|
|
|
|
$( Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 13-Nov-2012.) $)
|
|
simpl $p |- ( ( ph /\ ps ) -> ph ) $=
|
|
( ax-ia1 ) ABC $.
|
|
|
|
$( Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 13-Nov-2012.) $)
|
|
simpr $p |- ( ( ph /\ ps ) -> ps ) $=
|
|
( ax-ia2 ) ABC $.
|
|
|
|
${
|
|
simpli.1 $e |- ( ph /\ ps ) $.
|
|
$( Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) $)
|
|
simpli $p |- ph $=
|
|
( wa simpl ax-mp ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
simpld.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) $)
|
|
simpld $p |- ( ph -> ps ) $=
|
|
( wa simpl syl ) ABCEBDBCFG $.
|
|
$}
|
|
|
|
${
|
|
simpri.1 $e |- ( ph /\ ps ) $.
|
|
$( Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) $)
|
|
simpri $p |- ps $=
|
|
( wa simpr ax-mp ) ABDBCABEF $.
|
|
$}
|
|
|
|
${
|
|
simprd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
simprd $p |- ( ph -> ch ) $=
|
|
( wa simpr syl ) ABCECDBCFG $.
|
|
$}
|
|
|
|
${
|
|
exp.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Exportation inference. (This theorem used to be labeled "exp" but was
|
|
changed to "ex" so as not to conflict with the math token "exp", per the
|
|
June 2006 Metamath spec change.) (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
|
|
ex $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wa ax-ia3 syl6 ) ABABECABFDG $.
|
|
|
|
$( Exportation inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
expcom $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( ex com12 ) ABCABCDEF $.
|
|
$}
|
|
|
|
$( This is our first definition, which introduces and defines the
|
|
biconditional connective ` <-> ` . We define a wff of the form
|
|
` ( ph <-> ps ) ` as an abbreviation for
|
|
` ( ( ph -> ps ) /\ ( ps -> ph ) ) ` .
|
|
|
|
Unlike most traditional developments, we have chosen not to have a
|
|
separate symbol such as "Df." to mean "is defined as." Instead, we will
|
|
later use the biconditional connective for this purpose, as it allows us
|
|
to use logic to manipulate definitions directly. For an example of such a
|
|
definition, see ~ df-3or . This greatly simplifies many proofs since it
|
|
eliminates the need for a separate mechanism for introducing and
|
|
eliminating definitions. Of course, we cannot use this mechanism to
|
|
define the biconditional itself, since it hasn't been introduced yet.
|
|
Instead, we use a more general form of definition, described as follows.
|
|
|
|
In its most general form, a definition is simply an assertion that
|
|
introduces a new symbol (or a new combination of existing symbols, as in
|
|
~ df-3an ) that is eliminable and does not strengthen the existing
|
|
language. The latter requirement means that the set of provable
|
|
statements not containing the new symbol (or new combination) should
|
|
remain exactly the same after the definition is introduced. Our
|
|
definition of the biconditional may look unusual compared to most
|
|
definitions, but it strictly satisfies these requirements.
|
|
|
|
The justification for our definition is that if we mechanically replace
|
|
` ( ph <-> ps ) ` (the definiendum i.e. the thing being defined) with
|
|
` ( ( ph -> ps ) /\ ( ps -> ph ) ) ` (the definiens i.e. the defining
|
|
expression) in the definition, the definition becomes the previously
|
|
proved theorem ~ biijust . It is impossible to use ~ df-bi to prove any
|
|
statement expressed in the original language that can't be proved from the
|
|
original axioms, because if we simply replace each instance of ~ df-bi in
|
|
the proof with the corresponding ~ biijust instance, we will end up with a
|
|
proof from the original axioms.
|
|
|
|
Note that from Metamath's point of view, a definition is just another
|
|
axiom - i.e. an assertion we claim to be true - but from our high level
|
|
point of view, we are are not strengthening the language. To indicate
|
|
this fact, we prefix definition labels with "df-" instead of "ax-". (This
|
|
prefixing is an informal convention that means nothing to the Metamath
|
|
proof verifier; it is just for human readability.)
|
|
|
|
~ df-bi itself is a conjunction of two implications (to avoid using the
|
|
biconditional in its own definition), but once we have the biconditional,
|
|
we can prove ~ dfbi2 which uses the biconditional instead.
|
|
|
|
Other definitions of the biconditional, such as ~ dfbi3dc , only hold for
|
|
decidable propositions, not all propositions. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.) $)
|
|
df-bi $a |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
|
|
/\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $.
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) (Revised by NM, 31-Jan-2015.) $)
|
|
bi1 $p |- ( ( ph <-> ps ) -> ( ph -> ps ) ) $=
|
|
( wb wi wa df-bi simpli simpld ) ABCZABDZBADZIJKEZDLIDABFGH $.
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) $)
|
|
bi3 $p |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) $=
|
|
( wi wb wa df-bi simpri ex ) ABCZBACZABDZKIJEZCLKCABFGH $.
|
|
|
|
${
|
|
biimpi.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer an implication from a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biimpi $p |- ( ph -> ps ) $=
|
|
( wb wi bi1 ax-mp ) ABDABECABFG $.
|
|
$}
|
|
|
|
${
|
|
sylbi.1 $e |- ( ph <-> ps ) $.
|
|
sylbi.2 $e |- ( ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
Useful for substituting an antecedent with a definition. (Contributed
|
|
by NM, 3-Jan-1993.) $)
|
|
sylbi $p |- ( ph -> ch ) $=
|
|
( biimpi syl ) ABCABDFEG $.
|
|
$}
|
|
|
|
${
|
|
sylib.1 $e |- ( ph -> ps ) $.
|
|
sylib.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
(Contributed by NM, 3-Jan-1993.) $)
|
|
sylib $p |- ( ph -> ch ) $=
|
|
( biimpi syl ) ABCDBCEFG $.
|
|
$}
|
|
|
|
${
|
|
sylbb.1 $e |- ( ph <-> ps ) $.
|
|
sylbb.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from two biconditionals. (Contributed by
|
|
BJ, 30-Mar-2019.) $)
|
|
sylbb $p |- ( ph -> ch ) $=
|
|
( biimpi sylbi ) ABCDBCEFG $.
|
|
$}
|
|
|
|
${
|
|
imp.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Eric Schmidt, 22-Dec-2006.) $)
|
|
imp $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa simpl simpr sylc ) ABEABCABFABGDH $.
|
|
|
|
$( Importation inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
impcom $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( com12 imp ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
impbii.1 $e |- ( ph -> ps ) $.
|
|
impbii.2 $e |- ( ps -> ph ) $.
|
|
$( Infer an equivalence from an implication and its converse. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
impbii $p |- ( ph <-> ps ) $=
|
|
( wi wb bi3 mp2 ) ABEBAEABFCDABGH $.
|
|
$}
|
|
|
|
${
|
|
impbidd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
impbidd.2 $e |- ( ph -> ( ps -> ( th -> ch ) ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by Rodolfo
|
|
Medina, 12-Oct-2010.) $)
|
|
impbidd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi wb bi3 syl6c ) ABCDGDCGCDHEFCDIJ $.
|
|
$}
|
|
|
|
${
|
|
impbid21d.1 $e |- ( ps -> ( ch -> th ) ) $.
|
|
impbid21d.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by Wolf
|
|
Lammen, 12-May-2013.) $)
|
|
impbid21d $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi a1i a1d impbidd ) ABCDBCDGGAEHADCGBFIJ $.
|
|
$}
|
|
|
|
${
|
|
impbid.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impbid.2 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) $)
|
|
impbid $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb impbid21d pm2.43i ) ABCFAABCDEGH $.
|
|
$}
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
|
|
bi2 $p |- ( ( ph <-> ps ) -> ( ps -> ph ) ) $=
|
|
( wb wi wa df-bi simpli simprd ) ABCZABDZBADZIJKEZDLIDABFGH $.
|
|
|
|
$( Commutative law for equivalence. (Contributed by Wolf Lammen,
|
|
10-Nov-2012.) $)
|
|
bicom1 $p |- ( ( ph <-> ps ) -> ( ps <-> ph ) ) $=
|
|
( wb bi2 bi1 impbid ) ABCBAABDABEF $.
|
|
|
|
${
|
|
bicomi.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference from commutative law for logical equivalence. (Contributed by
|
|
NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.) $)
|
|
bicomi $p |- ( ps <-> ph ) $=
|
|
( wb bicom1 ax-mp ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
biimpri.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer a converse implication from a logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) $)
|
|
biimpri $p |- ( ps -> ph ) $=
|
|
( bicomi biimpi ) BAABCDE $.
|
|
$}
|
|
|
|
${
|
|
sylibr.1 $e |- ( ph -> ps ) $.
|
|
sylibr.2 $e |- ( ch <-> ps ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
Useful for substituting a consequent with a definition. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sylibr $p |- ( ph -> ch ) $=
|
|
( biimpri syl ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
sylbir.1 $e |- ( ps <-> ph ) $.
|
|
sylbir.2 $e |- ( ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sylbir $p |- ( ph -> ch ) $=
|
|
( biimpri syl ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylbbr.1 $e |- ( ph <-> ps ) $.
|
|
sylbbr.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from two biconditionals.
|
|
|
|
Note on the various syllogism-like statements in set.mm. The
|
|
hypothetical syllogism ~ syl infers an implication from two implications
|
|
(and there are ~ 3syl and ~ 4syl for chaining more inferences). There
|
|
are four inferences inferring an implication from one implication and
|
|
one biconditional: ~ sylbi , ~ sylib , ~ sylbir , ~ sylibr ; four
|
|
inferences inferring an implication from two biconditionals: ~ sylbb ,
|
|
~ sylbbr , ~ sylbb1 , ~ sylbb2 ; four inferences inferring a
|
|
biconditional from two biconditionals: ~ bitri , ~ bitr2i , ~ bitr3i ,
|
|
~ bitr4i (and more for chaining more biconditionals). There are also
|
|
closed forms and deduction versions of these, like, among many others,
|
|
~ syld , ~ syl5 , ~ syl6 , ~ mpbid , ~ bitrd , ~ syl5bb , ~ syl6bb and
|
|
variants. (Contributed by BJ, 21-Apr-2019.) $)
|
|
sylbbr $p |- ( ch -> ph ) $=
|
|
( biimpri sylibr ) CBABCEFDG $.
|
|
$}
|
|
|
|
${
|
|
sylbb1.1 $e |- ( ph <-> ps ) $.
|
|
sylbb1.2 $e |- ( ph <-> ch ) $.
|
|
$( A mixed syllogism inference from two biconditionals. (Contributed by
|
|
BJ, 21-Apr-2019.) $)
|
|
sylbb1 $p |- ( ps -> ch ) $=
|
|
( biimpri sylib ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
sylbb2.1 $e |- ( ph <-> ps ) $.
|
|
sylbb2.2 $e |- ( ch <-> ps ) $.
|
|
$( A mixed syllogism inference from two biconditionals. (Contributed by
|
|
BJ, 21-Apr-2019.) $)
|
|
sylbb2 $p |- ( ph -> ch ) $=
|
|
( biimpri sylbi ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell]
|
|
p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.) $)
|
|
pm3.2 $p |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $=
|
|
( ax-ia3 ) ABC $.
|
|
|
|
$( Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell]
|
|
p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM,
|
|
11-Nov-2012.) $)
|
|
bicom $p |- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) $=
|
|
( wb bicom1 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
bicomd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Commute two sides of a biconditional in a deduction. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
bicomd $p |- ( ph -> ( ch <-> ps ) ) $=
|
|
( wb bicom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
impbid1.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impbid1.2 $e |- ( ch -> ps ) $.
|
|
$( Infer an equivalence from two implications. (Contributed by NM,
|
|
6-Mar-2007.) $)
|
|
impbid1 $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wi a1i impbid ) ABCDCBFAEGH $.
|
|
$}
|
|
|
|
${
|
|
impbid2.1 $e |- ( ps -> ch ) $.
|
|
impbid2.2 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
$( Infer an equivalence from two implications. (Contributed by NM,
|
|
6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) $)
|
|
impbid2 $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( impbid1 bicomd ) ACBACBEDFG $.
|
|
$}
|
|
|
|
${
|
|
biimpd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce an implication from a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biimpd $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wb wi bi1 syl ) ABCEBCFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
mpbi.min $e |- ph $.
|
|
mpbi.maj $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
mpbi $p |- ps $=
|
|
( biimpi ax-mp ) ABCABDEF $.
|
|
$}
|
|
|
|
${
|
|
mpbir.min $e |- ps $.
|
|
mpbir.maj $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
mpbir $p |- ph $=
|
|
( biimpri ax-mp ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mpbid.min $e |- ( ph -> ps ) $.
|
|
mpbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
mpbid $p |- ( ph -> ch ) $=
|
|
( biimpd mpd ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mpbii.min $e |- ps $.
|
|
mpbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a nested biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mpbii $p |- ( ph -> ch ) $=
|
|
( a1i mpbid ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylibd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylibd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylibd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syld ) ABCDEACDFGH $.
|
|
$}
|
|
|
|
${
|
|
sylbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylbid.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylbid $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syld ) ABCDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpbidi.min $e |- ( th -> ( ph -> ps ) ) $.
|
|
mpbidi.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 9-Aug-1994.) $)
|
|
mpbidi $p |- ( th -> ( ph -> ch ) ) $=
|
|
( biimpd sylcom ) DABCEABCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bi.1 $e |- ( ph <-> ps ) $.
|
|
syl5bi.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. Useful for substituting an embedded antecedent with a
|
|
definition. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5bi $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpi syl5 ) ABCDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5bir.1 $e |- ( ps <-> ph ) $.
|
|
syl5bir.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5bir $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpri syl5 ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5ib.1 $e |- ( ph -> ps ) $.
|
|
syl5ib.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5ib $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpd syl5 ) ABCDECBDFGH $.
|
|
|
|
$( A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) $)
|
|
syl5ibcom $p |- ( ph -> ( ch -> th ) ) $=
|
|
( syl5ib com12 ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5ibr.1 $e |- ( ph -> th ) $.
|
|
syl5ibr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised
|
|
by NM, 22-Sep-2013.) $)
|
|
syl5ibr $p |- ( ch -> ( ph -> ps ) ) $=
|
|
( bicomd syl5ib ) ADCBECBDFGH $.
|
|
|
|
$( A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) $)
|
|
syl5ibrcom $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( syl5ibr com12 ) CABABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
biimprd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce a converse implication from a logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
|
|
biimprd $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( id syl5ibr ) CBACCEDF $.
|
|
$}
|
|
|
|
${
|
|
biimpcd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce a commuted implication from a logical equivalence. (Contributed
|
|
by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
|
|
biimpcd $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id syl5ibcom ) BBACBEDF $.
|
|
|
|
$( Deduce a converse commuted implication from a logical equivalence.
|
|
(Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen,
|
|
20-Dec-2013.) $)
|
|
biimprcd $p |- ( ch -> ( ph -> ps ) ) $=
|
|
( id syl5ibrcom ) CBACCEDF $.
|
|
$}
|
|
|
|
${
|
|
syl6ib.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6ib.2 $e |- ( ch <-> th ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6ib $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpi syl6 ) ABCDECDFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6ibr.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6ibr.2 $e |- ( th <-> ch ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. Useful for substituting an embedded consequent with a
|
|
definition. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6ibr $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpri syl6 ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6bi.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bi.2 $e |- ( ch -> th ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) $)
|
|
syl6bi $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syl6 ) ABCDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6bir.1 $e |- ( ph -> ( ch <-> ps ) ) $.
|
|
syl6bir.2 $e |- ( ch -> th ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
syl6bir $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syl6 ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl7bi.1 $e |- ( ph <-> ps ) $.
|
|
syl7bi.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
|
|
$( A mixed syllogism inference from a doubly nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl7bi $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
|
|
( biimpi syl7 ) ABCDEABFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl8ib.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl8ib.2 $e |- ( th <-> ta ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
syl8ib $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( biimpi syl8 ) ABCDEFDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpbird.min $e |- ( ph -> ch ) $.
|
|
mpbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
mpbird $p |- ( ph -> ps ) $=
|
|
( biimprd mpd ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mpbiri.min $e |- ch $.
|
|
mpbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a nested biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mpbiri $p |- ( ph -> ps ) $=
|
|
( a1i mpbird ) ABCCADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylibrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylibrd.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylibrd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syld ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
sylbird.1 $e |- ( ph -> ( ch <-> ps ) ) $.
|
|
sylbird.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylbird $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syld ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
$( Principle of identity for logical equivalence. Theorem *4.2 of
|
|
[WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) $)
|
|
biid $p |- ( ph <-> ph ) $=
|
|
( id impbii ) AAABZDC $.
|
|
|
|
$( Principle of identity with antecedent. (Contributed by NM,
|
|
25-Nov-1995.) $)
|
|
biidd $p |- ( ph -> ( ps <-> ps ) ) $=
|
|
( wb biid a1i ) BBCABDE $.
|
|
|
|
$( Two propositions are equivalent if they are both true. Closed form of
|
|
~ 2th . Equivalent to a ~ bi1 -like version of the xor-connective. This
|
|
theorem stays true, no matter how you permute its operands. This is
|
|
evident from its sharper version ` ( ph <-> ( ps <-> ( ph <-> ps ) ) ) ` .
|
|
(Contributed by Wolf Lammen, 12-May-2013.) $)
|
|
pm5.1im $p |- ( ph -> ( ps -> ( ph <-> ps ) ) ) $=
|
|
( ax-1 impbid21d ) ABABBACABCD $.
|
|
|
|
${
|
|
2th.1 $e |- ph $.
|
|
2th.2 $e |- ps $.
|
|
$( Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) $)
|
|
2th $p |- ( ph <-> ps ) $=
|
|
( a1i impbii ) ABBADEABCEF $.
|
|
$}
|
|
|
|
${
|
|
2thd.1 $e |- ( ph -> ps ) $.
|
|
2thd.2 $e |- ( ph -> ch ) $.
|
|
$( Two truths are equivalent (deduction rule). (Contributed by NM,
|
|
3-Jun-2012.) (Revised by NM, 29-Jan-2013.) $)
|
|
2thd $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb pm5.1im sylc ) ABCBCFDEBCGH $.
|
|
$}
|
|
|
|
${
|
|
ibi.1 $e |- ( ph -> ( ph <-> ps ) ) $.
|
|
$( Inference that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 17-Oct-2003.) $)
|
|
ibi $p |- ( ph -> ps ) $=
|
|
( biimpd pm2.43i ) ABAABCDE $.
|
|
$}
|
|
|
|
${
|
|
ibir.1 $e |- ( ph -> ( ps <-> ph ) ) $.
|
|
$( Inference that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 22-Jul-2004.) $)
|
|
ibir $p |- ( ph -> ps ) $=
|
|
( bicomd ibi ) ABABACDE $.
|
|
$}
|
|
|
|
${
|
|
ibd.1 $e |- ( ph -> ( ps -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 26-Jun-2004.) $)
|
|
ibd $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wb bi1 syli ) BABCECDBCFG $.
|
|
$}
|
|
|
|
$( Distribution of implication over biconditional. Theorem *5.74 of
|
|
[WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 11-Apr-2013.) $)
|
|
pm5.74 $p |- ( ( ph -> ( ps <-> ch ) ) <->
|
|
( ( ph -> ps ) <-> ( ph -> ch ) ) ) $=
|
|
( wb wi bi1 imim3i bi2 impbid pm2.86d impbidd impbii ) ABCDZEZABEZACEZDZNOP
|
|
MBCABCFGMCBABCHGIQABCQABCOPFJQACBOPHJKL $.
|
|
|
|
${
|
|
pm5.74i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.74i $p |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $=
|
|
( wb wi pm5.74 mpbi ) ABCEFABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.74ri.1 $e |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $.
|
|
$( Distribution of implication over biconditional (reverse inference rule).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.74ri $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb wi pm5.74 mpbir ) ABCEFABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.74d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 21-Mar-1996.) $)
|
|
pm5.74d $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
|
|
( wb wi pm5.74 sylib ) ABCDFGBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
pm5.74rd.1 $e |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 19-Mar-1997.) $)
|
|
pm5.74rd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi wb pm5.74 sylibr ) ABCFBDFGBCDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
bitri.1 $e |- ( ph <-> ps ) $.
|
|
bitri.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) $)
|
|
bitri $p |- ( ph <-> ch ) $=
|
|
( biimpi sylib biimpri sylibr impbii ) ACABCABDFEGCBABCEHDIJ $.
|
|
$}
|
|
|
|
${
|
|
bitr2i.1 $e |- ( ph <-> ps ) $.
|
|
bitr2i.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr2i $p |- ( ch <-> ph ) $=
|
|
( bitri bicomi ) ACABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
bitr3i.1 $e |- ( ps <-> ph ) $.
|
|
bitr3i.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr3i $p |- ( ph <-> ch ) $=
|
|
( bicomi bitri ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
bitr4i.1 $e |- ( ph <-> ps ) $.
|
|
bitr4i.2 $e |- ( ch <-> ps ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr4i $p |- ( ph <-> ch ) $=
|
|
( bicomi bitri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( Deduction form of ~ bitri . (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 14-Apr-2013.) $)
|
|
bitrd $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( wi pm5.74i bitri pm5.74ri ) ABDABGACGADGABCEHACDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr2d.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( Deduction form of ~ bitr2i . (Contributed by NM, 9-Jun-2004.) $)
|
|
bitr2d $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( bitrd bicomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
$( Deduction form of ~ bitr3i . (Contributed by NM, 5-Aug-1993.) $)
|
|
bitr3d $p |- ( ph -> ( ch <-> th ) ) $=
|
|
( bicomd bitrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr4d.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
$( Deduction form of ~ bitr4i . (Contributed by NM, 5-Aug-1993.) $)
|
|
bitr4d $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( bicomd bitrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bb.1 $e |- ( ph <-> ps ) $.
|
|
syl5bb.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5bb $p |- ( ch -> ( ph <-> th ) ) $=
|
|
( wb a1i bitrd ) CABDABGCEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5rbb.1 $e |- ( ph <-> ps ) $.
|
|
syl5rbb.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5rbb $p |- ( ch -> ( th <-> ph ) ) $=
|
|
( syl5bb bicomd ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bbr.1 $e |- ( ps <-> ph ) $.
|
|
syl5bbr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5bbr $p |- ( ch -> ( ph <-> th ) ) $=
|
|
( bicomi syl5bb ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5rbbr.1 $e |- ( ps <-> ph ) $.
|
|
syl5rbbr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
syl5rbbr $p |- ( ch -> ( th <-> ph ) ) $=
|
|
( bicomi syl5rbb ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6bb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bb.2 $e |- ( ch <-> th ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6bb $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( wb a1i bitrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6rbb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6rbb.2 $e |- ( ch <-> th ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6rbb $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( syl6bb bicomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6bbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bbr.2 $e |- ( th <-> ch ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6bbr $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( bicomi syl6bb ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6rbbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6rbbr.2 $e |- ( th <-> ch ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
syl6rbbr $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( bicomi syl6rbb ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
3imtr3.1 $e |- ( ph -> ps ) $.
|
|
3imtr3.2 $e |- ( ph <-> ch ) $.
|
|
3imtr3.3 $e |- ( ps <-> th ) $.
|
|
$( A mixed syllogism inference, useful for removing a definition from both
|
|
sides of an implication. (Contributed by NM, 10-Aug-1994.) $)
|
|
3imtr3i $p |- ( ch -> th ) $=
|
|
( sylbir sylib ) CBDCABFEHGI $.
|
|
$}
|
|
|
|
${
|
|
3imtr4.1 $e |- ( ph -> ps ) $.
|
|
3imtr4.2 $e |- ( ch <-> ph ) $.
|
|
3imtr4.3 $e |- ( th <-> ps ) $.
|
|
$( A mixed syllogism inference, useful for applying a definition to both
|
|
sides of an implication. (Contributed by NM, 5-Aug-1993.) $)
|
|
3imtr4i $p |- ( ch -> th ) $=
|
|
( sylbi sylibr ) CBDCABFEHGI $.
|
|
$}
|
|
|
|
${
|
|
3imtr3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
3imtr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
|
|
$( More general version of ~ 3imtr3i . Useful for converting conditional
|
|
definitions in a formula. (Contributed by NM, 8-Apr-1996.) $)
|
|
3imtr3d $p |- ( ph -> ( th -> ta ) ) $=
|
|
( sylibd sylbird ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr4d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
|
|
3imtr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
|
|
$( More general version of ~ 3imtr4i . Useful for converting conditional
|
|
definitions in a formula. (Contributed by NM, 26-Oct-1995.) $)
|
|
3imtr4d $p |- ( ph -> ( th -> ta ) ) $=
|
|
( sylibrd sylbid ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr3g.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr3g.2 $e |- ( ps <-> th ) $.
|
|
3imtr3g.3 $e |- ( ch <-> ta ) $.
|
|
$( More general version of ~ 3imtr3i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by
|
|
Wolf Lammen, 20-Dec-2013.) $)
|
|
3imtr3g $p |- ( ph -> ( th -> ta ) ) $=
|
|
( syl5bir syl6ib ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr4g.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr4g.2 $e |- ( th <-> ps ) $.
|
|
3imtr4g.3 $e |- ( ta <-> ch ) $.
|
|
$( More general version of ~ 3imtr4i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by
|
|
Wolf Lammen, 20-Dec-2013.) $)
|
|
3imtr4g $p |- ( ph -> ( th -> ta ) ) $=
|
|
( syl5bi syl6ibr ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitri.1 $e |- ( ph <-> ps ) $.
|
|
3bitri.2 $e |- ( ps <-> ch ) $.
|
|
3bitri.3 $e |- ( ch <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3bitri $p |- ( ph <-> th ) $=
|
|
( bitri ) ABDEBCDFGHH $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitrri $p |- ( th <-> ph ) $=
|
|
( bitr2i bitr3i ) DCAGABCEFHI $.
|
|
$}
|
|
|
|
${
|
|
3bitr2i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr2i.2 $e |- ( ch <-> ps ) $.
|
|
3bitr2i.3 $e |- ( ch <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitr2i $p |- ( ph <-> th ) $=
|
|
( bitr4i bitri ) ACDABCEFHGI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitr2ri $p |- ( th <-> ph ) $=
|
|
( bitr4i bitr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3bitr3i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr3i.2 $e |- ( ph <-> ch ) $.
|
|
3bitr3i.3 $e |- ( ps <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 19-Aug-1993.) $)
|
|
3bitr3i $p |- ( ch <-> th ) $=
|
|
( bitr3i bitri ) CBDCABFEHGI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr3ri $p |- ( th <-> ch ) $=
|
|
( bitr3i ) DBCGBACEFHH $.
|
|
$}
|
|
|
|
${
|
|
3bitr4i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr4i.2 $e |- ( ch <-> ph ) $.
|
|
3bitr4i.3 $e |- ( th <-> ps ) $.
|
|
$( A chained inference from transitive law for logical equivalence. This
|
|
inference is frequently used to apply a definition to both sides of a
|
|
logical equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr4i $p |- ( ch <-> th ) $=
|
|
( bitr4i bitri ) CADFABDEGHI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 2-Sep-1995.) $)
|
|
3bitr4ri $p |- ( th <-> ch ) $=
|
|
( bitr4i bitr2i ) CADFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
3bitrd.3 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
13-Aug-1999.) $)
|
|
3bitrd $p |- ( ph -> ( ps <-> ta ) ) $=
|
|
( bitrd ) ABDEABCDFGIHI $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitrrd $p |- ( ph -> ( ta <-> ps ) ) $=
|
|
( bitr2d bitr3d ) ADEBHABCDFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr2d.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
3bitr2d.3 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr2d $p |- ( ph -> ( ps <-> ta ) ) $=
|
|
( bitr4d bitrd ) ABDEABCDFGIHJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr2rd $p |- ( ph -> ( ta <-> ps ) ) $=
|
|
( bitr4d bitr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
3bitr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. Useful for converting
|
|
conditional definitions in a formula. (Contributed by NM,
|
|
24-Apr-1996.) $)
|
|
3bitr3d $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( bitr3d bitrd ) ADCEABDCGFIHJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr3rd $p |- ( ph -> ( ta <-> th ) ) $=
|
|
( bitr3d ) ACEDHABCDFGII $.
|
|
$}
|
|
|
|
${
|
|
3bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
|
|
3bitr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
|
|
$( Deduction from transitivity of biconditional. Useful for converting
|
|
conditional definitions in a formula. (Contributed by NM,
|
|
18-Oct-1995.) $)
|
|
3bitr4d $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( bitr4d bitrd ) ADBEGABCEFHIJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr4rd $p |- ( ph -> ( ta <-> th ) ) $=
|
|
( bitr4d ) AEBDAECBHFIGI $.
|
|
$}
|
|
|
|
${
|
|
3bitr3g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr3g.2 $e |- ( ps <-> th ) $.
|
|
3bitr3g.3 $e |- ( ch <-> ta ) $.
|
|
$( More general version of ~ 3bitr3i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 4-Jun-1995.) $)
|
|
3bitr3g $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( syl5bbr syl6bb ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr4g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr4g.2 $e |- ( th <-> ps ) $.
|
|
3bitr4g.3 $e |- ( ta <-> ch ) $.
|
|
$( More general version of ~ 3bitr4i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr4g $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( syl5bb syl6bbr ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
bi3ant.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Construct a biconditional in antecedent position. (Contributed by Wolf
|
|
Lammen, 14-May-2013.) $)
|
|
bi3ant $p |- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) ->
|
|
( ( th <-> ta ) -> ch ) ) ) $=
|
|
( wi wb bi1 imim1i bi2 imim3i syl2im ) DEGZAGDEHZAGEDGZBGOBGOCGONADEIJOPB
|
|
DEKJABCOFLM $.
|
|
$}
|
|
|
|
$( Express symmetries of theorems in terms of biconditionals. (Contributed
|
|
by Wolf Lammen, 14-May-2013.) $)
|
|
bisym $p |- ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ( ps -> ph )
|
|
-> ( th -> ch ) ) -> ( ( ph <-> ps ) -> ( ch <-> th ) ) ) ) $=
|
|
( wi wb bi3 bi3ant ) CDEDCECDFABCDGH $.
|
|
|
|
$( The next three rules are useful for building up wff's around a
|
|
definition, in order to make use of the definition. $)
|
|
|
|
${
|
|
bi.a $e |- ( ph <-> ps ) $.
|
|
$( Introduce an antecedent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
6-Feb-2013.) $)
|
|
imbi2i $p |- ( ( ch -> ph ) <-> ( ch -> ps ) ) $=
|
|
( wb a1i pm5.74i ) CABABECDFG $.
|
|
$}
|
|
|
|
${
|
|
bibi.a $e |- ( ph <-> ps ) $.
|
|
$( Inference adding a biconditional to the left in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.) $)
|
|
bibi2i $p |- ( ( ch <-> ph ) <-> ( ch <-> ps ) ) $=
|
|
( wb id syl6bb syl6bbr impbii ) CAEZCBEZJCABJFDGKCBAKFDHI $.
|
|
|
|
$( Inference adding a biconditional to the right in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
bibi1i $p |- ( ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
|
|
( wb bicom bibi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.
|
|
|
|
${
|
|
bibi12.2 $e |- ( ch <-> th ) $.
|
|
$( The equivalence of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
bibi12i $p |- ( ( ph <-> ch ) <-> ( ps <-> th ) ) $=
|
|
( wb bibi2i bibi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
imbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding an antecedent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi2d $p |- ( ph -> ( ( th -> ps ) <-> ( th -> ch ) ) ) $=
|
|
( wb a1d pm5.74d ) ADBCABCFDEGH $.
|
|
|
|
$( Deduction adding a consequent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
17-Sep-2013.) $)
|
|
imbi1d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) ) $=
|
|
( wi biimprd imim1d biimpd impbid ) ABDFCDFACBDABCEGHABCDABCEIHJ $.
|
|
|
|
$( Deduction adding a biconditional to the left in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
19-May-2013.) $)
|
|
bibi2d $p |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) ) $=
|
|
( wb wi pm5.74i bibi2i pm5.74 3bitr4i pm5.74ri ) ADBFZDCFZADGZABGZFOACGZF
|
|
AMGANGPQOABCEHIADBJADCJKL $.
|
|
|
|
$( Deduction adding a biconditional to the right in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
bibi1d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) ) $=
|
|
( wb bibi2d bicom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
|
|
$}
|
|
|
|
${
|
|
imbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
imbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of implications.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi12d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> ta ) ) ) $=
|
|
( wi imbi1d imbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$( Deduction joining two equivalences to form equivalence of
|
|
biconditionals. (Contributed by NM, 5-Aug-1993.) $)
|
|
bibi12d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) ) $=
|
|
( wb bibi1d bibi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
$( Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
imbi1 $p |- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) ) $=
|
|
( wb id imbi1d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)
|
|
imbi2 $p |- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) ) $=
|
|
( wb id imbi2d ) ABDZABCGEF $.
|
|
|
|
${
|
|
imbi1i.1 $e |- ( ph <-> ps ) $.
|
|
$( Introduce a consequent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
17-Sep-2013.) $)
|
|
imbi1i $p |- ( ( ph -> ch ) <-> ( ps -> ch ) ) $=
|
|
( wb wi imbi1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
imbi12i.1 $e |- ( ph <-> ps ) $.
|
|
imbi12i.2 $e |- ( ch <-> th ) $.
|
|
$( Join two logical equivalences to form equivalence of implications.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi12i $p |- ( ( ph -> ch ) <-> ( ps -> th ) ) $=
|
|
( wi imbi2i imbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
|
|
$( Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
bibi1 $p |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) <-> ( ps <-> ch ) ) ) $=
|
|
( wb id bibi1d ) ABDZABCGEF $.
|
|
|
|
$( A wff is equivalent to itself with true antecedent. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
biimt $p |- ( ph -> ( ps <-> ( ph -> ps ) ) ) $=
|
|
( wi ax-1 pm2.27 impbid2 ) ABABCBADABEF $.
|
|
|
|
$( Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.5 $p |- ( ph -> ( ( ph -> ps ) <-> ps ) ) $=
|
|
( wi biimt bicomd ) ABABCABDE $.
|
|
|
|
${
|
|
a1bi.1 $e |- ph $.
|
|
$( Inference rule introducing a theorem as an antecedent. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
|
|
a1bi $p |- ( ps <-> ( ph -> ps ) ) $=
|
|
( wi wb biimt ax-mp ) ABABDECABFG $.
|
|
$}
|
|
|
|
$( Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 24-Jan-2013.) $)
|
|
pm5.501 $p |- ( ph -> ( ps <-> ( ph <-> ps ) ) ) $=
|
|
( wb pm5.1im bi1 com12 impbid ) ABABCZABDHABABEFG $.
|
|
|
|
$( Implication in terms of implication and biconditional. (Contributed by
|
|
NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) $)
|
|
ibib $p |- ( ( ph -> ps ) <-> ( ph -> ( ph <-> ps ) ) ) $=
|
|
( wb pm5.501 pm5.74i ) ABABCABDE $.
|
|
|
|
$( Implication in terms of implication and biconditional. (Contributed by
|
|
NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) $)
|
|
ibibr $p |- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) ) $=
|
|
( wb pm5.501 bicom syl6bb pm5.74i ) ABBACZABABCHABDABEFG $.
|
|
|
|
${
|
|
tbt.1 $e |- ph $.
|
|
$( A wff is equivalent to its equivalence with truth. (Contributed by NM,
|
|
18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
tbt $p |- ( ps <-> ( ps <-> ph ) ) $=
|
|
( wb ibibr pm5.74ri ax-mp ) ABBADZDCABHABEFG $.
|
|
$}
|
|
|
|
$( Logical equivalence of commuted antecedents. Part of Theorem *4.87 of
|
|
[WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) $)
|
|
bi2.04 $p |- ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi pm2.04 impbii ) ABCDDBACDDABCEBACEF $.
|
|
|
|
$( Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell]
|
|
p. 125. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm5.4 $p |- ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ps ) ) $=
|
|
( wi pm2.43 ax-1 impbii ) AABCZCGABDGAEF $.
|
|
|
|
$( Distributive law for implication. Compare Theorem *5.41 of
|
|
[WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) $)
|
|
imdi $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
|
|
( wi ax-2 pm2.86 impbii ) ABCDDABDACDDABCEABCFG $.
|
|
|
|
$( Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.) $)
|
|
pm5.41 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) <->
|
|
( ph -> ( ps -> ch ) ) ) $=
|
|
( wi imdi bicomi ) ABCDDABDACDDABCEF $.
|
|
|
|
$( Simplify an implication between two implications when the antecedent of
|
|
the first is a consequence of the antecedent of the second. The reverse
|
|
form is useful in producing the successor step in induction proofs.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf
|
|
Lammen, 14-Sep-2013.) $)
|
|
imim21b $p |- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <->
|
|
( ps -> ( ch -> th ) ) ) ) $=
|
|
( wi bi2.04 wb pm5.5 imbi1d imim2i pm5.74d syl5bb ) ACEZBDEEBMDEZEBAEZBCDEZ
|
|
EMBDFOBNPANPGBAMCDACHIJKL $.
|
|
|
|
${
|
|
imp3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Importation deduction. (Contributed by NM, 31-Mar-1994.) $)
|
|
impd $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( wa wi com3l imp com12 ) BCFADBCADGABCDEHIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp31 $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( wa wi imp ) ABFCDABCDGEHH $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp32 $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( wa impd imp ) ABCFDABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp3a.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Exportation deduction. (Contributed by NM, 20-Aug-1993.) $)
|
|
expd $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi wa com12 ex com3r ) BCADBCADFABCGDEHIJ $.
|
|
|
|
$( A deduction version of exportation, followed by importation.
|
|
(Contributed by NM, 6-Sep-2008.) $)
|
|
expdimp $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi expd imp ) ABCDFABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
impancom.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Mixed importation/commutation inference. (Contributed by NM,
|
|
22-Jun-2013.) $)
|
|
impancom $p |- ( ( ph /\ ch ) -> ( ps -> th ) ) $=
|
|
( wi ex com23 imp ) ACBDFABCDABCDFEGHI $.
|
|
$}
|
|
|
|
$( Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
pm3.3 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wa wi id expd ) ABDCEZABCHFG $.
|
|
|
|
$( Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
pm3.31 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wi id impd ) ABCDDZABCGEF $.
|
|
|
|
$( Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell]
|
|
p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 24-Mar-2013.) $)
|
|
impexp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wa wi pm3.3 pm3.31 impbii ) ABDCEABCEEABCFABCGH $.
|
|
|
|
$( Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell]
|
|
p. 111. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.21 $p |- ( ph -> ( ps -> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.2 com12 ) BABACBADE $.
|
|
|
|
$( Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) $)
|
|
pm3.22 $p |- ( ( ph /\ ps ) -> ( ps /\ ph ) ) $=
|
|
( wa pm3.21 imp ) ABBACABDE $.
|
|
|
|
$( Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf
|
|
Lammen, 4-Nov-2012.) $)
|
|
ancom $p |- ( ( ph /\ ps ) <-> ( ps /\ ph ) ) $=
|
|
( wa pm3.22 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
ancomd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Commutation of conjuncts in consequent. (Contributed by Jeff Hankins,
|
|
14-Aug-2009.) $)
|
|
ancomd $p |- ( ph -> ( ch /\ ps ) ) $=
|
|
( wa ancom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
ancoms.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Inference commuting conjunction in antecedent. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
ancoms $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( expcom imp ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
ancomsd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Deduction commuting conjunction in antecedent. (Contributed by NM,
|
|
12-Dec-2004.) $)
|
|
ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
|
|
( wa ancom syl5bi ) CBFBCFADCBGEH $.
|
|
$}
|
|
|
|
${
|
|
pm3.2i.1 $e |- ph $.
|
|
pm3.2i.2 $e |- ps $.
|
|
$( Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.2i $p |- ( ph /\ ps ) $=
|
|
( wa pm3.2 mp2 ) ABABECDABFG $.
|
|
$}
|
|
|
|
$( Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.43i $p |- ( ( ph -> ps ) ->
|
|
( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) ) $=
|
|
( wa pm3.2 imim3i ) BCBCDABCEF $.
|
|
|
|
${
|
|
simplbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) $)
|
|
simplbi $p |- ( ph -> ps ) $=
|
|
( wa biimpi simpld ) ABCABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
simprbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) $)
|
|
simprbi $p |- ( ph -> ch ) $=
|
|
( wa biimpi simprd ) ABCABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
adantr.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding a conjunct to the right of an antecedent. (Contributed
|
|
by NM, 30-Aug-1993.) $)
|
|
adantr $p |- ( ( ph /\ ch ) -> ps ) $=
|
|
( a1d imp ) ACBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
adantl.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding a conjunct to the left of an antecedent. (Contributed
|
|
by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
adantl $p |- ( ( ch /\ ph ) -> ps ) $=
|
|
( adantr ancoms ) ACBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
adantld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding a conjunct to the left of an antecedent. (Contributed
|
|
by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.) $)
|
|
adantld $p |- ( ph -> ( ( th /\ ps ) -> ch ) ) $=
|
|
( wa simpr syl5 ) DBFBACDBGEH $.
|
|
$}
|
|
|
|
${
|
|
adantrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding a conjunct to the right of an antecedent. (Contributed
|
|
by NM, 4-May-1994.) $)
|
|
adantrd $p |- ( ph -> ( ( ps /\ th ) -> ch ) ) $=
|
|
( wa simpl syl5 ) BDFBACBDGEH $.
|
|
$}
|
|
|
|
${
|
|
impel.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impel.2 $e |- ( th -> ps ) $.
|
|
$( An inference for implication elimination. (Contributed by Giovanni
|
|
Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen,
|
|
2-Sep-2020.) $)
|
|
impel $p |- ( ( ph /\ th ) -> ch ) $=
|
|
( syl5 imp ) ADCDBACFEGH $.
|
|
$}
|
|
|
|
${
|
|
mpan9.1 $e |- ( ph -> ps ) $.
|
|
mpan9.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( Modus ponens conjoining dissimilar antecedents. (Contributed by NM,
|
|
1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpan9 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( syl5 impcom ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syldan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
syldan.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
$( A syllogism deduction with conjoined antecents. (Contributed by NM,
|
|
24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.) $)
|
|
syldan $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa expcom adantrd mpcom ) CABGDECADBACDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan.1 $e |- ( ph -> ps ) $.
|
|
sylan.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
sylan $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( expcom mpan9 ) ABCDEBCDFGH $.
|
|
$}
|
|
|
|
${
|
|
sylanb.1 $e |- ( ph <-> ps ) $.
|
|
sylanb.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
sylanb $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( biimpi sylan ) ABCDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylanbr.1 $e |- ( ps <-> ph ) $.
|
|
sylanbr.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
sylanbr $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( biimpri sylan ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2.1 $e |- ( ph -> ch ) $.
|
|
sylan2.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
sylan2 $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( adantl syldan ) BACDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2b.1 $e |- ( ph <-> ch ) $.
|
|
sylan2b.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) $)
|
|
sylan2b $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( biimpi sylan2 ) ABCDACEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2br.1 $e |- ( ch <-> ph ) $.
|
|
sylan2br.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) $)
|
|
sylan2br $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( biimpri sylan2 ) ABCDCAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl2an.1 $e |- ( ph -> ps ) $.
|
|
syl2an.2 $e |- ( ta -> ch ) $.
|
|
syl2an.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 31-Jan-1997.) $)
|
|
syl2an $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylan sylan2 ) EACDGABCDFHIJ $.
|
|
|
|
$( A double syllogism inference. (Contributed by NM, 17-Sep-2013.) $)
|
|
syl2anr $p |- ( ( ta /\ ph ) -> th ) $=
|
|
( syl2an ancoms ) AEDABCDEFGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl2anb.1 $e |- ( ph <-> ps ) $.
|
|
syl2anb.2 $e |- ( ta <-> ch ) $.
|
|
syl2anb.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 29-Jul-1999.) $)
|
|
syl2anb $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylanb sylan2b ) EACDGABCDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl2anbr.1 $e |- ( ps <-> ph ) $.
|
|
syl2anbr.2 $e |- ( ch <-> ta ) $.
|
|
syl2anbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 29-Jul-1999.) $)
|
|
syl2anbr $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylanbr sylan2br ) EACDGABCDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
syland.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syland.2 $e |- ( ph -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
syland $p |- ( ph -> ( ( ps /\ th ) -> ta ) ) $=
|
|
( wi expd syld impd ) ABDEABCDEHFACDEGIJK $.
|
|
$}
|
|
|
|
${
|
|
sylan2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan2d.2 $e |- ( ph -> ( ( th /\ ch ) -> ta ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
sylan2d $p |- ( ph -> ( ( th /\ ps ) -> ta ) ) $=
|
|
( ancomsd syland ) ABDEABCDEFADCEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syl2and.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl2and.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
syl2and.3 $e |- ( ph -> ( ( ch /\ ta ) -> et ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
syl2and $p |- ( ph -> ( ( ps /\ th ) -> et ) ) $=
|
|
( sylan2d syland ) ABCDFGADECFHIJK $.
|
|
$}
|
|
|
|
${
|
|
biimpa.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpa $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( biimpd imp ) ABCABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpar $p |- ( ( ph /\ ch ) -> ps ) $=
|
|
( biimprd imp ) ACBABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpac $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( biimpcd imp ) BACABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimparc $p |- ( ( ch /\ ph ) -> ps ) $=
|
|
( biimprcd imp ) CABABCDEF $.
|
|
$}
|
|
|
|
$( Introduction of antecedent as conjunct. Theorem *4.73 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by
|
|
NM, 24-Mar-2013.) $)
|
|
iba $p |- ( ph -> ( ps <-> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.21 simpl impbid1 ) ABBACABDBAEF $.
|
|
|
|
$( Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
|
|
(Revised by NM, 24-Mar-2013.) $)
|
|
ibar $p |- ( ph -> ( ps <-> ( ph /\ ps ) ) ) $=
|
|
( wa pm3.2 simpr impbid1 ) ABABCABDABEF $.
|
|
|
|
${
|
|
biantru.1 $e |- ph $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biantru $p |- ( ps <-> ( ps /\ ph ) ) $=
|
|
( wa wb iba ax-mp ) ABBADECABFG $.
|
|
$}
|
|
|
|
${
|
|
biantrur.1 $e |- ph $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
biantrur $p |- ( ps <-> ( ph /\ ps ) ) $=
|
|
( wa wb ibar ax-mp ) ABABDECABFG $.
|
|
$}
|
|
|
|
${
|
|
biantrud.1 $e |- ( ph -> ps ) $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.) $)
|
|
biantrud $p |- ( ph -> ( ch <-> ( ch /\ ps ) ) ) $=
|
|
( wa wb iba syl ) ABCCBEFDBCGH $.
|
|
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
biantrurd $p |- ( ph -> ( ch <-> ( ps /\ ch ) ) ) $=
|
|
( wa wb ibar syl ) ABCBCEFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
jca.1 $e |- ( ph -> ps ) $.
|
|
jca.2 $e |- ( ph -> ch ) $.
|
|
$( Deduce conjunction of the consequents of two implications ("join
|
|
consequents with 'and'"). (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 25-Oct-2012.) $)
|
|
jca $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( wa pm3.2 sylc ) ABCBCFDEBCGH $.
|
|
$}
|
|
|
|
${
|
|
jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
$( Deduction conjoining the consequents of two implications. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
jcad $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
|
|
( wa pm3.2 syl6c ) ABCDCDGEFCDHI $.
|
|
$}
|
|
|
|
${
|
|
jca31.1 $e |- ( ph -> ps ) $.
|
|
jca31.2 $e |- ( ph -> ch ) $.
|
|
jca31.3 $e |- ( ph -> th ) $.
|
|
$( Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.) $)
|
|
jca31 $p |- ( ph -> ( ( ps /\ ch ) /\ th ) ) $=
|
|
( wa jca ) ABCHDABCEFIGI $.
|
|
|
|
$( Join three consequents. (Contributed by FL, 1-Aug-2009.) $)
|
|
jca32 $p |- ( ph -> ( ps /\ ( ch /\ th ) ) ) $=
|
|
( wa jca ) ABCDHEACDFGII $.
|
|
$}
|
|
|
|
${
|
|
jcai.1 $e |- ( ph -> ps ) $.
|
|
jcai.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction replacing implication with conjunction. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
jcai $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( mpd jca ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
jctil.1 $e |- ( ph -> ps ) $.
|
|
jctil.2 $e |- ch $.
|
|
$( Inference conjoining a theorem to left of consequent in an implication.
|
|
(Contributed by NM, 31-Dec-1993.) $)
|
|
jctil $p |- ( ph -> ( ch /\ ps ) ) $=
|
|
( a1i jca ) ACBCAEFDG $.
|
|
|
|
$( Inference conjoining a theorem to right of consequent in an implication.
|
|
(Contributed by NM, 31-Dec-1993.) $)
|
|
jctir $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( a1i jca ) ABCDCAEFG $.
|
|
$}
|
|
|
|
${
|
|
jctl.1 $e |- ps $.
|
|
$( Inference conjoining a theorem to the left of a consequent.
|
|
(Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen,
|
|
24-Oct-2012.) $)
|
|
jctl $p |- ( ph -> ( ps /\ ph ) ) $=
|
|
( id jctil ) AABADCE $.
|
|
|
|
$( Inference conjoining a theorem to the right of a consequent.
|
|
(Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
24-Oct-2012.) $)
|
|
jctr $p |- ( ph -> ( ph /\ ps ) ) $=
|
|
( id jctir ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
jctild.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jctild.2 $e |- ( ph -> th ) $.
|
|
$( Deduction conjoining a theorem to left of consequent in an implication.
|
|
(Contributed by NM, 21-Apr-2005.) $)
|
|
jctild $p |- ( ph -> ( ps -> ( th /\ ch ) ) ) $=
|
|
( a1d jcad ) ABDCADBFGEH $.
|
|
$}
|
|
|
|
${
|
|
jctird.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jctird.2 $e |- ( ph -> th ) $.
|
|
$( Deduction conjoining a theorem to right of consequent in an implication.
|
|
(Contributed by NM, 21-Apr-2005.) $)
|
|
jctird $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
|
|
( a1d jcad ) ABCDEADBFGH $.
|
|
$}
|
|
|
|
$( Conjoin antecedent to left of consequent. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
ancl $p |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) ) $=
|
|
( wa pm3.2 a2i ) ABABCABDE $.
|
|
|
|
$( Conjoin antecedent to left of consequent. Theorem *4.7 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof
|
|
shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
anclb $p |- ( ( ph -> ps ) <-> ( ph -> ( ph /\ ps ) ) ) $=
|
|
( wa ibar pm5.74i ) ABABCABDE $.
|
|
|
|
$( Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.42 $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
|
|
( wi wa ibar imbi2d pm5.74i ) ABCDBACEZDACIBACFGH $.
|
|
|
|
$( Conjoin antecedent to right of consequent. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
ancr $p |- ( ( ph -> ps ) -> ( ph -> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.21 a2i ) ABBACABDE $.
|
|
|
|
$( Conjoin antecedent to right of consequent. (Contributed by NM,
|
|
25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
ancrb $p |- ( ( ph -> ps ) <-> ( ph -> ( ps /\ ph ) ) ) $=
|
|
( wa iba pm5.74i ) ABBACABDE $.
|
|
|
|
${
|
|
ancli.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction conjoining antecedent to left of consequent. (Contributed by
|
|
NM, 12-Aug-1993.) $)
|
|
ancli $p |- ( ph -> ( ph /\ ps ) ) $=
|
|
( id jca ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
ancri.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction conjoining antecedent to right of consequent. (Contributed by
|
|
NM, 15-Aug-1994.) $)
|
|
ancri $p |- ( ph -> ( ps /\ ph ) ) $=
|
|
( id jca ) ABACADE $.
|
|
$}
|
|
|
|
${
|
|
ancld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to left of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 1-Nov-2012.) $)
|
|
ancld $p |- ( ph -> ( ps -> ( ps /\ ch ) ) ) $=
|
|
( idd jcad ) ABBCABEDF $.
|
|
$}
|
|
|
|
${
|
|
ancrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to right of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 1-Nov-2012.) $)
|
|
ancrd $p |- ( ph -> ( ps -> ( ch /\ ps ) ) ) $=
|
|
( idd jcad ) ABCBDABEF $.
|
|
$}
|
|
|
|
$( Conjoin antecedent to left of consequent in nested implication.
|
|
(Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen,
|
|
14-Jul-2013.) $)
|
|
anc2l $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
|
|
( wi wa pm5.42 biimpi ) ABCDDABACEDDABCFG $.
|
|
|
|
$( Conjoin antecedent to right of consequent in nested implication.
|
|
(Contributed by NM, 15-Aug-1994.) $)
|
|
anc2r $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ch /\ ph ) ) ) ) $=
|
|
( wi wa pm3.21 imim2d a2i ) ABCDBCAEZDACIBACFGH $.
|
|
|
|
${
|
|
anc2li.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to left of consequent in nested
|
|
implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 7-Dec-2012.) $)
|
|
anc2li $p |- ( ph -> ( ps -> ( ph /\ ch ) ) ) $=
|
|
( id jctild ) ABCADAEF $.
|
|
$}
|
|
|
|
${
|
|
anc2ri.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to right of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 7-Dec-2012.) $)
|
|
anc2ri $p |- ( ph -> ( ps -> ( ch /\ ph ) ) ) $=
|
|
( id jctird ) ABCADAEF $.
|
|
$}
|
|
|
|
$( Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.41 $p |- ( ( ph -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wa simpl imim1i ) ABDACABEF $.
|
|
|
|
$( Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.42 $p |- ( ( ps -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wa simpr imim1i ) ABDBCABEF $.
|
|
|
|
$( Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell]
|
|
p. 113. (Contributed by NM, 31-Jul-1995.) $)
|
|
pm3.4 $p |- ( ( ph /\ ps ) -> ( ph -> ps ) ) $=
|
|
( wa simpr a1d ) ABCBAABDE $.
|
|
|
|
$( Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell]
|
|
p. 119. (Contributed by NM, 17-May-1998.) $)
|
|
pm4.45im $p |- ( ph <-> ( ph /\ ( ps -> ph ) ) ) $=
|
|
( wi wa ax-1 ancli simpl impbii ) AABACZDAIABEFAIGH $.
|
|
|
|
${
|
|
anim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
anim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.) $)
|
|
anim12d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) ) $=
|
|
( wa idd syl2and ) ABCDECEHZFGAKIJ $.
|
|
$}
|
|
|
|
${
|
|
anim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Add a conjunct to right of antecedent and consequent in a deduction.
|
|
(Contributed by NM, 3-Apr-1994.) $)
|
|
anim1d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ th ) ) ) $=
|
|
( idd anim12d ) ABCDDEADFG $.
|
|
|
|
$( Add a conjunct to left of antecedent and consequent in a deduction.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
anim2d $p |- ( ph -> ( ( th /\ ps ) -> ( th /\ ch ) ) ) $=
|
|
( idd anim12d ) ADDBCADFEG $.
|
|
$}
|
|
|
|
${
|
|
anim12i.1 $e |- ( ph -> ps ) $.
|
|
anim12i.2 $e |- ( ch -> th ) $.
|
|
$( Conjoin antecedents and consequents of two premises. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.) $)
|
|
anim12i $p |- ( ( ph /\ ch ) -> ( ps /\ th ) ) $=
|
|
( wa id syl2an ) ABDBDGZCEFJHI $.
|
|
|
|
$( Variant of ~ anim12i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
anim12ci $p |- ( ( ph /\ ch ) -> ( th /\ ps ) ) $=
|
|
( wa anim12i ancoms ) CADBGCDABFEHI $.
|
|
$}
|
|
|
|
${
|
|
anim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Introduce conjunct to both sides of an implication. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anim1i $p |- ( ( ph /\ ch ) -> ( ps /\ ch ) ) $=
|
|
( id anim12i ) ABCCDCEF $.
|
|
|
|
$( Introduce conjunct to both sides of an implication. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anim2i $p |- ( ( ch /\ ph ) -> ( ch /\ ps ) ) $=
|
|
( id anim12i ) CCABCEDF $.
|
|
$}
|
|
|
|
${
|
|
anim12ii.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
anim12ii.2 $e |- ( th -> ( ps -> ta ) ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.) $)
|
|
anim12ii $p |- ( ( ph /\ th ) -> ( ps -> ( ch /\ ta ) ) ) $=
|
|
( wa wi adantr adantl jcad ) ADHBCEABCIDFJDBEIAGKL $.
|
|
$}
|
|
|
|
$( Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and
|
|
it evidently pleased him enough to call it 'praeclarum theorema' (splendid
|
|
theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 7-Apr-2013.) $)
|
|
prth $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) ->
|
|
( ( ph /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi wa simpl simpr anim12d ) ABEZCDEZFABCDJKGJKHI $.
|
|
|
|
$( Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.33 $p |- ( ( ( ph -> ps ) /\ ( ps -> ch ) ) -> ( ph -> ch ) ) $=
|
|
( wi imim1 imp ) ABDBCDACDABCEF $.
|
|
|
|
$( Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.34 $p |- ( ( ( ps -> ch ) /\ ( ph -> ps ) ) -> ( ph -> ch ) ) $=
|
|
( wi imim2 imp ) BCDABDACDBCAEF $.
|
|
|
|
$( Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112.
|
|
(Contributed by NM, 14-Dec-2002.) $)
|
|
pm3.35 $p |- ( ( ph /\ ( ph -> ps ) ) -> ps ) $=
|
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( wi pm2.27 imp ) AABCBABDE $.
|
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|
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$( Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.31 $p |- ( ( ch /\ ( ph -> ps ) ) -> ( ph -> ( ps /\ ch ) ) ) $=
|
|
( wi wa pm3.21 imim2d imp ) CABDABCEZDCBIACBFGH $.
|
|
|
|
${
|
|
imp4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
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$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4a $p |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $=
|
|
( wi wa impexp syl6ibr ) ABCDEGGCDHEGFCDEIJ $.
|
|
|
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$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4b $p |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $=
|
|
( wa wi imp4a imp ) ABCDGEHABCDEFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4c $p |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $=
|
|
( wa wi impd ) ABCGDEABCDEHFII $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4d $p |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $=
|
|
( wa imp4a impd ) ABCDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp41 $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi imp imp31 ) ABGCDEABCDEHHFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp42 $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $=
|
|
( wa wi imp32 imp ) ABCGGDEABCDEHFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp43 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $=
|
|
( wa imp4b imp ) ABGCDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp44 $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $=
|
|
( wa imp4c imp ) ABCGDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp45 $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $=
|
|
( wa imp4d imp ) ABCDGGEABCDEFHI $.
|
|
|
|
$}
|
|
|
|
${
|
|
imp5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5a $p |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) ) $=
|
|
( wi wa pm3.31 syl8 ) ABCDEFHHDEIFHGDEFJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5d $p |- ( ( ( ph /\ ps ) /\ ch ) -> ( ( th /\ ta ) -> et ) ) $=
|
|
( wa wi imp31 impd ) ABHCHDEFABCDEFIIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
|
|
( wa wi imp imp4c ) ABHCDEFABCDEFIIIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp55 $p |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) $=
|
|
( wa wi imp4a imp42 ) ABCDHEFABCDEFIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp511 $p |- ( ( ph /\ ( ( ps /\ ( ch /\ th ) ) /\ ta ) ) -> et ) $=
|
|
( wa wi imp4a imp44 ) ABCDHEFABCDEFIGJK $.
|
|
$}
|
|
|
|
${
|
|
expimpd.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Exportation followed by a deduction version of importation.
|
|
(Contributed by NM, 6-Sep-2008.) $)
|
|
expimpd $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( wi ex impd ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
exp31.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp31 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi wa ex ) ABCDFABGCDEHH $.
|
|
$}
|
|
|
|
${
|
|
exp32.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp32 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wa ex expd ) ABCDABCFDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp4a.1 $e |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4a $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa wi impexp syl6ib ) ABCDGEHCDEHHFCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
exp4b.1 $e |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
exp4b $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa wi ex exp4a ) ABCDEABCDGEHFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp4c.1 $e |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4c $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa expd ) ABCDEGABCHDEFII $.
|
|
$}
|
|
|
|
${
|
|
exp4d.1 $e |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4d $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa expd exp4a ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp41.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp41 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa ex exp31 ) ABCDEGABHCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp42.1 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp42 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa exp31 expd ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp43.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp43 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa ex exp4b ) ABCDEABGCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp44.1 $e |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp44 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa exp32 expd ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp45.1 $e |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp45 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa exp32 exp4a ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
expr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Export a wff from a right conjunct. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
expr $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi exp32 imp ) ABCDFABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp5c.1 $e |- ( ph -> ( ( ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) ) $.
|
|
$( An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
exp5c $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi wa exp4a expd ) ABCDEFHHABCIDEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp53.1 $e |- ( ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) /\ ta ) -> et ) $.
|
|
$( An exportation inference. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
exp53 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi wa ex exp43 ) ABCDEFHABICDIIEFGJK $.
|
|
$}
|
|
|
|
${
|
|
expl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Export a wff from a left conjunct. (Contributed by Jeff Hankins,
|
|
28-Aug-2009.) $)
|
|
expl $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( exp31 impd ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
impr.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Import a wff into a right conjunct. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
impr $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( wi ex imp32 ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
impl.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Export a wff from a left conjunct. (Contributed by Mario Carneiro,
|
|
9-Jul-2014.) $)
|
|
impl $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( expd imp31 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
impac.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Importation with conjunction in consequent. (Contributed by NM,
|
|
9-Aug-1994.) $)
|
|
impac $p |- ( ( ph /\ ps ) -> ( ch /\ ps ) ) $=
|
|
( wa ancrd imp ) ABCBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
exbiri.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Inference form of ~ exbir . (Contributed by Alan Sare, 31-Dec-2011.)
|
|
(Proof shortened by Wolf Lammen, 27-Jan-2013.) $)
|
|
exbiri $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
|
|
( wa biimpar exp31 ) ABDCABFCDEGH $.
|
|
$}
|
|
|
|
${
|
|
pm3.26bda.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) $)
|
|
simprbda $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa biimpa simpld ) ABFCDABCDFEGH $.
|
|
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) $)
|
|
simplbda $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa biimpa simprd ) ABFCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
pm3.26bi2.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by Alan Sare,
|
|
31-Dec-2011.) $)
|
|
simplbi2 $p |- ( ps -> ( ch -> ph ) ) $=
|
|
( wa biimpri ex ) BCAABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
simpl2im.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
simpl2im.2 $e |- ( ch -> th ) $.
|
|
$( Implication from an eliminated conjunct implied by the antecedent.
|
|
(Contributed by BJ/AV, 5-Apr-2021.) $)
|
|
simpl2im $p |- ( ph -> th ) $=
|
|
( wa simpr 3syl ) ABCGCDEBCHFI $.
|
|
$}
|
|
|
|
${
|
|
simplbiim.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
simplbiim.2 $e |- ( ch -> th ) $.
|
|
$( Implication from an eliminated conjunct equivalent to the antecedent.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
simplbiim $p |- ( ph -> th ) $=
|
|
( wa adantl sylbi ) ABCGDECDBFHI $.
|
|
$}
|
|
|
|
$( A theorem similar to the standard definition of the biconditional.
|
|
Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by NM, 31-Jan-2015.) $)
|
|
dfbi2 $p |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) $=
|
|
( wb wi wa df-bi simpli simpri impbii ) ABCZABDBADEZJKDZKJDZABFZGLMNHI $.
|
|
|
|
$( Implication in terms of biconditional and conjunction. Theorem *4.71 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 2-Dec-2012.) $)
|
|
pm4.71 $p |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) ) $=
|
|
( wa wi wb simpl biantru anclb dfbi2 3bitr4i ) AABCZDZLKADZCABDAKEMLABFGABH
|
|
AKIJ $.
|
|
|
|
$( Implication in terms of biconditional and conjunction. Theorem *4.71 of
|
|
[WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM,
|
|
25-Jul-1999.) $)
|
|
pm4.71r $p |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) ) $=
|
|
( wi wa wb pm4.71 ancom bibi2i bitri ) ABCAABDZEABADZEABFJKAABGHI $.
|
|
|
|
${
|
|
pm4.71i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference converting an implication to a biconditional with conjunction.
|
|
Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed
|
|
by NM, 4-Jan-2004.) $)
|
|
pm4.71i $p |- ( ph <-> ( ph /\ ps ) ) $=
|
|
( wi wa wb pm4.71 mpbi ) ABDAABEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
pm4.71ri.1 $e |- ( ph -> ps ) $.
|
|
$( Inference converting an implication to a biconditional with conjunction.
|
|
Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct
|
|
reversed). (Contributed by NM, 1-Dec-2003.) $)
|
|
pm4.71ri $p |- ( ph <-> ( ps /\ ph ) ) $=
|
|
( wi wa wb pm4.71r mpbi ) ABDABAEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
pm4.71rd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction converting an implication to a biconditional with conjunction.
|
|
Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed
|
|
by Mario Carneiro, 25-Dec-2016.) $)
|
|
pm4.71d $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
|
|
( wi wa wb pm4.71 sylib ) ABCEBBCFGDBCHI $.
|
|
|
|
$( Deduction converting an implication to a biconditional with conjunction.
|
|
Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed
|
|
by NM, 10-Feb-2005.) $)
|
|
pm4.71rd $p |- ( ph -> ( ps <-> ( ch /\ ps ) ) ) $=
|
|
( wi wa wb pm4.71r sylib ) ABCEBCBFGDBCHI $.
|
|
$}
|
|
|
|
$( Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 14-Mar-2014.) $)
|
|
pm4.24 $p |- ( ph <-> ( ph /\ ph ) ) $=
|
|
( id pm4.71i ) AAABC $.
|
|
|
|
$( Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 14-Mar-2014.) $)
|
|
anidm $p |- ( ( ph /\ ph ) <-> ph ) $=
|
|
( wa pm4.24 bicomi ) AAABACD $.
|
|
|
|
${
|
|
anidms.1 $e |- ( ( ph /\ ph ) -> ps ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
15-Jun-1994.) $)
|
|
anidms $p |- ( ph -> ps ) $=
|
|
( ex pm2.43i ) ABAABCDE $.
|
|
$}
|
|
|
|
$( Conjunction idempotence with antecedent. (Contributed by Roy F. Longton,
|
|
8-Aug-2005.) $)
|
|
anidmdbi $p |- ( ( ph -> ( ps /\ ps ) ) <-> ( ph -> ps ) ) $=
|
|
( wa anidm imbi2i ) BBCBABDE $.
|
|
|
|
${
|
|
anasss.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by NM, 15-Nov-2002.) $)
|
|
anasss $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( exp31 imp32 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
anassrs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by NM, 15-Nov-2002.) $)
|
|
anassrs $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( exp32 imp31 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 24-Nov-2012.) $)
|
|
anass $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
|
|
( wa id anassrs anasss impbii ) ABDCDZABCDDZABCJJEFABCIIEGH $.
|
|
|
|
${
|
|
sylanl1.1 $e |- ( ph -> ps ) $.
|
|
sylanl1.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 10-Mar-2005.) $)
|
|
sylanl1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa anim1i sylan ) ACHBCHDEABCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanl2.1 $e |- ( ph -> ch ) $.
|
|
sylanl2.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Jan-2005.) $)
|
|
sylanl2 $p |- ( ( ( ps /\ ph ) /\ th ) -> ta ) $=
|
|
( wa anim2i sylan ) BAHBCHDEACBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanr1.1 $e |- ( ph -> ch ) $.
|
|
sylanr1.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 9-Apr-2005.) $)
|
|
sylanr1 $p |- ( ( ps /\ ( ph /\ th ) ) -> ta ) $=
|
|
( wa anim1i sylan2 ) ADHBCDHEACDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanr2.1 $e |- ( ph -> th ) $.
|
|
sylanr2.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 9-Apr-2005.) $)
|
|
sylanr2 $p |- ( ( ps /\ ( ch /\ ph ) ) -> ta ) $=
|
|
( wa anim2i sylan2 ) CAHBCDHEADCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylani.1 $e |- ( ph -> ch ) $.
|
|
sylani.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 2-May-1996.) $)
|
|
sylani $p |- ( ps -> ( ( ph /\ th ) -> ta ) ) $=
|
|
( wi a1i syland ) BACDEACHBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylan2i.1 $e |- ( ph -> th ) $.
|
|
sylan2i.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Aug-1994.) $)
|
|
sylan2i $p |- ( ps -> ( ( ch /\ ph ) -> ta ) ) $=
|
|
( wi a1i sylan2d ) BADCEADHBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl2ani.1 $e |- ( ph -> ch ) $.
|
|
syl2ani.2 $e |- ( et -> th ) $.
|
|
syl2ani.3 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 3-Aug-1999.) $)
|
|
syl2ani $p |- ( ps -> ( ( ph /\ et ) -> ta ) ) $=
|
|
( sylan2i sylani ) ABCFEGFBCDEHIJK $.
|
|
$}
|
|
|
|
${
|
|
sylan9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan9.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
7-May-2011.) $)
|
|
sylan9 $p |- ( ( ph /\ th ) -> ( ps -> ta ) ) $=
|
|
( wi syl9 imp ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan9r.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sylan9r $p |- ( ( th /\ ph ) -> ( ps -> ta ) ) $=
|
|
( wi syl9r imp ) DABEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl2anc.1 $e |- ( ph -> ps ) $.
|
|
syl2anc.2 $e |- ( ph -> ch ) $.
|
|
syl2anc.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with contraction. (Contributed by NM,
|
|
16-Mar-2012.) $)
|
|
syl2anc $p |- ( ph -> th ) $=
|
|
( ex sylc ) ABCDEFBCDGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancl.1 $e |- ( ph -> ps ) $.
|
|
sylancl.2 $e |- ch $.
|
|
sylancl.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with modus ponens. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
sylancl $p |- ( ph -> th ) $=
|
|
( a1i syl2anc ) ABCDECAFHGI $.
|
|
$}
|
|
|
|
${
|
|
sylancr.1 $e |- ps $.
|
|
sylancr.2 $e |- ( ph -> ch ) $.
|
|
sylancr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with modus ponens. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
sylancr $p |- ( ph -> th ) $=
|
|
( a1i syl2anc ) ABCDBAEHFGI $.
|
|
$}
|
|
|
|
${
|
|
sylanblc.1 $e |- ( ph -> ps ) $.
|
|
sylanblc.2 $e |- ch $.
|
|
sylanblc.3 $e |- ( ( ps /\ ch ) <-> th ) $.
|
|
$( Syllogism inference combined with a biconditional. (Contributed by BJ,
|
|
25-Apr-2019.) $)
|
|
sylanblc $p |- ( ph -> th ) $=
|
|
( wa biimpi sylancl ) ABCDEFBCHDGIJ $.
|
|
$}
|
|
|
|
${
|
|
sylanblrc.1 $e |- ( ph -> ps ) $.
|
|
sylanblrc.2 $e |- ch $.
|
|
sylanblrc.3 $e |- ( th <-> ( ps /\ ch ) ) $.
|
|
$( Syllogism inference combined with a biconditional. (Contributed by BJ,
|
|
25-Apr-2019.) $)
|
|
sylanblrc $p |- ( ph -> th ) $=
|
|
( wa biimpri sylancl ) ABCDEFDBCHGIJ $.
|
|
$}
|
|
|
|
${
|
|
sylanbrc.1 $e |- ( ph -> ps ) $.
|
|
sylanbrc.2 $e |- ( ph -> ch ) $.
|
|
sylanbrc.3 $e |- ( th <-> ( ps /\ ch ) ) $.
|
|
$( Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
sylanbrc $p |- ( ph -> th ) $=
|
|
( wa jca sylibr ) ABCHDABCEFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylancb.1 $e |- ( ph <-> ps ) $.
|
|
sylancb.2 $e |- ( ph <-> ch ) $.
|
|
sylancb.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
sylancb $p |- ( ph -> th ) $=
|
|
( syl2anb anidms ) ADABCDAEFGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancbr.1 $e |- ( ps <-> ph ) $.
|
|
sylancbr.2 $e |- ( ch <-> ph ) $.
|
|
sylancbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
sylancbr $p |- ( ph -> th ) $=
|
|
( syl2anbr anidms ) ADABCDAEFGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancom.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
sylancom.2 $e |- ( ( ch /\ ps ) -> th ) $.
|
|
$( Syllogism inference with commutation of antecents. (Contributed by NM,
|
|
2-Jul-2008.) $)
|
|
sylancom $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa simpr syl2anc ) ABGCBDEABHFI $.
|
|
$}
|
|
|
|
${
|
|
mpdan.1 $e |- ( ph -> ps ) $.
|
|
mpdan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 23-May-1999.)
|
|
(Proof shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
mpdan $p |- ( ph -> ch ) $=
|
|
( id syl2anc ) AABCAFDEG $.
|
|
$}
|
|
|
|
${
|
|
mpancom.1 $e |- ( ps -> ph ) $.
|
|
mpancom.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens with commutation of antecedents.
|
|
(Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen,
|
|
7-Apr-2013.) $)
|
|
mpancom $p |- ( ps -> ch ) $=
|
|
( id syl2anc ) BABCDBFEG $.
|
|
$}
|
|
|
|
${
|
|
mpidan.1 $e |- ( ph -> ch ) $.
|
|
mpidan.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( A deduction which "stacks" a hypothesis. (Contributed by Stanislas
|
|
Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.) $)
|
|
mpidan $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa adantr mpdan ) ABGCDACBEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpan.1 $e |- ph $.
|
|
mpan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpan $p |- ( ps -> ch ) $=
|
|
( a1i mpancom ) ABCABDFEG $.
|
|
$}
|
|
|
|
${
|
|
mpan2.1 $e |- ps $.
|
|
mpan2.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpan2 $p |- ( ph -> ch ) $=
|
|
( a1i mpdan ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
mp2an.1 $e |- ph $.
|
|
mp2an.2 $e |- ps $.
|
|
mp2an.3 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Apr-1995.) $)
|
|
mp2an $p |- ch $=
|
|
( mpan ax-mp ) BCEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mp4an.1 $e |- ph $.
|
|
mp4an.2 $e |- ps $.
|
|
mp4an.3 $e |- ch $.
|
|
mp4an.4 $e |- th $.
|
|
mp4an.5 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by Jeff Madsen,
|
|
15-Jun-2011.) $)
|
|
mp4an $p |- ta $=
|
|
( wa pm3.2i mp2an ) ABKCDKEABFGLCDHILJM $.
|
|
$}
|
|
|
|
${
|
|
mpan2d.1 $e |- ( ph -> ch ) $.
|
|
mpan2d.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) $)
|
|
mpan2d $p |- ( ph -> ( ps -> th ) ) $=
|
|
( expd mpid ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpand.1 $e |- ( ph -> ps ) $.
|
|
mpand.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpand $p |- ( ph -> ( ch -> th ) ) $=
|
|
( ancomsd mpan2d ) ACBDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpani.1 $e |- ps $.
|
|
mpani.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpani $p |- ( ph -> ( ch -> th ) ) $=
|
|
( a1i mpand ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpan2i.1 $e |- ch $.
|
|
mpan2i.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpan2i $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1i mpan2d ) ABCDCAEGFH $.
|
|
$}
|
|
|
|
${
|
|
mp2ani.1 $e |- ps $.
|
|
mp2ani.2 $e |- ch $.
|
|
mp2ani.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
12-Dec-2004.) $)
|
|
mp2ani $p |- ( ph -> th ) $=
|
|
( mpani mpi ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mp2and.1 $e |- ( ph -> ps ) $.
|
|
mp2and.2 $e |- ( ph -> ch ) $.
|
|
mp2and.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) $)
|
|
mp2and $p |- ( ph -> th ) $=
|
|
( mpand mpd ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanl1.1 $e |- ph $.
|
|
mpanl1.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanl1 $p |- ( ( ps /\ ch ) -> th ) $=
|
|
( wa jctl sylan ) BABGCDBAEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanl2.1 $e |- ps $.
|
|
mpanl2.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpanl2 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( wa jctr sylan ) AABGCDABEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanl12.1 $e |- ph $.
|
|
mpanl12.2 $e |- ps $.
|
|
mpanl12.3 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
mpanl12 $p |- ( ch -> th ) $=
|
|
( mpanl1 mpan ) BCDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanr1.1 $e |- ps $.
|
|
mpanr1.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 3-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpanr1 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( anassrs mpanl2 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpanr2.1 $e |- ch $.
|
|
mpanr2.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 3-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by
|
|
Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanr2 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa jctr sylan2 ) BABCGDBCEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanr12.1 $e |- ps $.
|
|
mpanr12.2 $e |- ch $.
|
|
mpanr12.3 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Jul-2009.) $)
|
|
mpanr12 $p |- ( ph -> th ) $=
|
|
( mpanr1 mpan2 ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanlr1.1 $e |- ps $.
|
|
mpanlr1.2 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanlr1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa jctl sylanl2 ) CABCHDECBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
pm5.74da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 4-May-2007.) $)
|
|
pm5.74da $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
|
|
( wb ex pm5.74d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.) $)
|
|
imdistan $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
|
|
( wi wa pm5.42 impexp bitr4i ) ABCDDABACEZDDABEIDABCFABIGH $.
|
|
|
|
${
|
|
imdistani.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
imdistani $p |- ( ( ph /\ ps ) -> ( ph /\ ch ) ) $=
|
|
( wa anc2li imp ) ABACEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
imdistanri.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
imdistanri $p |- ( ( ps /\ ph ) -> ( ch /\ ph ) ) $=
|
|
( com12 impac ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
imdistand.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Distribution of implication with conjunction (deduction rule).
|
|
(Contributed by NM, 27-Aug-2004.) $)
|
|
imdistand $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi wa imdistan sylib ) ABCDFFBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
imdistanda.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Distribution of implication with conjunction (deduction version with
|
|
conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.) $)
|
|
imdistanda $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi ex imdistand ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.32d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.) $)
|
|
pm5.32d $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
|
|
( wa wb wi bi1 syl6 imdistand bi2 impbid ) ABCFBDFABCDABCDGZCDHECDIJKABDC
|
|
ABNDCHECDLJKM $.
|
|
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 25-Dec-2004.) $)
|
|
pm5.32rd $p |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) ) $=
|
|
( wa pm5.32d ancom 3bitr4g ) ABCFBDFCBFDBFABCDEGCBHDBHI $.
|
|
$}
|
|
|
|
${
|
|
pm5.32da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 9-Dec-2006.) $)
|
|
pm5.32da $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
|
|
( wb ex pm5.32d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
$( Distribution of implication over biconditional. Theorem *5.32 of
|
|
[WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by
|
|
NM, 31-Jan-2015.) $)
|
|
pm5.32 $p |- ( ( ph -> ( ps <-> ch ) ) <->
|
|
( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
|
|
( wb wi wa id pm5.32d ibar bibi12d biimprcd impbii ) ABCDZEZABFZACFZDZNABCN
|
|
GHAMQABOCPABIACIJKL $.
|
|
|
|
${
|
|
pm5.32i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.32i $p |- ( ( ph /\ ps ) <-> ( ph /\ ch ) ) $=
|
|
( wb wi wa pm5.32 mpbi ) ABCEFABGACGEDABCHI $.
|
|
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 12-Mar-1995.) $)
|
|
pm5.32ri $p |- ( ( ps /\ ph ) <-> ( ch /\ ph ) ) $=
|
|
( wa pm5.32i ancom 3bitr4i ) ABEACEBAECAEABCDFBAGCAGH $.
|
|
$}
|
|
|
|
${
|
|
biadan2.1 $e |- ( ph -> ps ) $.
|
|
biadan2.2 $e |- ( ps -> ( ph <-> ch ) ) $.
|
|
$( Add a conjunction to an equivalence. (Contributed by Jeff Madsen,
|
|
20-Jun-2011.) $)
|
|
biadan2 $p |- ( ph <-> ( ps /\ ch ) ) $=
|
|
( wa pm4.71ri pm5.32i bitri ) ABAFBCFABDGBACEHI $.
|
|
$}
|
|
|
|
${
|
|
bi.aa $e |- ( ph <-> ps ) $.
|
|
$( Introduce a left conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
16-Nov-2013.) $)
|
|
anbi2i $p |- ( ( ch /\ ph ) <-> ( ch /\ ps ) ) $=
|
|
( wb a1i pm5.32i ) CABABECDFG $.
|
|
|
|
$( Introduce a right conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
16-Nov-2013.) $)
|
|
anbi1i $p |- ( ( ph /\ ch ) <-> ( ps /\ ch ) ) $=
|
|
( wb a1i pm5.32ri ) CABABECDFG $.
|
|
|
|
$( Variant of ~ anbi2i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
anbi2ci $p |- ( ( ph /\ ch ) <-> ( ch /\ ps ) ) $=
|
|
( wa anbi1i ancom bitri ) ACEBCECBEABCDFBCGH $.
|
|
$}
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|
|
|
${
|
|
anbi12.1 $e |- ( ph <-> ps ) $.
|
|
anbi12.2 $e |- ( ch <-> th ) $.
|
|
$( Conjoin both sides of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anbi12i $p |- ( ( ph /\ ch ) <-> ( ps /\ th ) ) $=
|
|
( wa anbi1i anbi2i bitri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
|
|
$( Variant of ~ anbi12i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
anbi12ci $p |- ( ( ph /\ ch ) <-> ( th /\ ps ) ) $=
|
|
( wa anbi12i ancom bitri ) ACGBDGDBGABCDEFHBDIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9bb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylan9bb.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 4-Mar-1995.) $)
|
|
sylan9bb $p |- ( ( ph /\ th ) -> ( ps <-> ta ) ) $=
|
|
( wa wb adantr adantl bitrd ) ADHBCEABCIDFJDCEIAGKL $.
|
|
$}
|
|
|
|
${
|
|
sylan9bbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylan9bbr.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 4-Mar-1995.) $)
|
|
sylan9bbr $p |- ( ( th /\ ph ) -> ( ps <-> ta ) ) $=
|
|
( wb sylan9bb ancoms ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
anbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding a left conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
16-Nov-2013.) $)
|
|
anbi2d $p |- ( ph -> ( ( th /\ ps ) <-> ( th /\ ch ) ) ) $=
|
|
( wb a1d pm5.32d ) ADBCABCFDEGH $.
|
|
|
|
$( Deduction adding a right conjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 16-Nov-2013.) $)
|
|
anbi1d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) ) $=
|
|
( wb a1d pm5.32rd ) ADBCABCFDEGH $.
|
|
$}
|
|
|
|
$( Introduce a right conjunct to both sides of a logical equivalence.
|
|
Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
anbi1 $p |- ( ( ph <-> ps ) -> ( ( ph /\ ch ) <-> ( ps /\ ch ) ) ) $=
|
|
( wb id anbi1d ) ABDZABCGEF $.
|
|
|
|
$( Introduce a left conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 16-Nov-2013.) $)
|
|
anbi2 $p |- ( ( ph <-> ps ) -> ( ( ch /\ ph ) <-> ( ch /\ ps ) ) ) $=
|
|
( wb id anbi2d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
bitr $p |- ( ( ( ph <-> ps ) /\ ( ps <-> ch ) ) -> ( ph <-> ch ) ) $=
|
|
( wb bibi1 biimpar ) ABDACDBCDABCEF $.
|
|
|
|
${
|
|
anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
anbi12d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) ) $=
|
|
( wa anbi1d anbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
$( Modus ponens mixed with several conjunctions. (Contributed by Jim
|
|
Kingdon, 7-Jan-2018.) $)
|
|
mpan10 $p |- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) -> ( ps /\ ch ) ) $=
|
|
( wi wa ancom anbi2i anass 3bitr4i id imp anim1i sylbi ) ABDZCEAEZNAEZCEZBC
|
|
ENCAEZENACEZEOQRSNCAFGNCAHNACHIPBCNABNJKLM $.
|
|
|
|
$( Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
pm5.3 $p |- ( ( ( ph /\ ps ) -> ch ) <->
|
|
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
|
|
( wa wi impexp imdistan bitri ) ABDZCEABCEEIACDEABCFABCGH $.
|
|
|
|
${
|
|
adant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantll $p |- ( ( ( th /\ ph ) /\ ps ) -> ch ) $=
|
|
( wa simpr sylan ) DAFABCDAGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantlr $p |- ( ( ( ph /\ th ) /\ ps ) -> ch ) $=
|
|
( wa simpl sylan ) ADFABCADGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantrl $p |- ( ( ph /\ ( th /\ ps ) ) -> ch ) $=
|
|
( wa simpr sylan2 ) DBFABCDBGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantrr $p |- ( ( ph /\ ( ps /\ th ) ) -> ch ) $=
|
|
( wa simpl sylan2 ) BDFABCBDGEH $.
|
|
$}
|
|
|
|
${
|
|
adantl2.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) $)
|
|
adantlll $p |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th ) $=
|
|
( wa simpr sylanl1 ) EAGABCDEAHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantllr $p |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th ) $=
|
|
( wa simpl sylanl1 ) AEGABCDAEHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantlrl $p |- ( ( ( ph /\ ( ta /\ ps ) ) /\ ch ) -> th ) $=
|
|
( wa simpr sylanl2 ) EBGABCDEBHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantlrr $p |- ( ( ( ph /\ ( ps /\ ta ) ) /\ ch ) -> th ) $=
|
|
( wa simpl sylanl2 ) BEGABCDBEHFI $.
|
|
$}
|
|
|
|
${
|
|
adantr2.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrll $p |- ( ( ph /\ ( ( ta /\ ps ) /\ ch ) ) -> th ) $=
|
|
( wa simpr sylanr1 ) EBGABCDEBHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrlr $p |- ( ( ph /\ ( ( ps /\ ta ) /\ ch ) ) -> th ) $=
|
|
( wa simpl sylanr1 ) BEGABCDBEHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrrl $p |- ( ( ph /\ ( ps /\ ( ta /\ ch ) ) ) -> th ) $=
|
|
( wa simpr sylanr2 ) ECGABCDECHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrrr $p |- ( ( ph /\ ( ps /\ ( ch /\ ta ) ) ) -> th ) $=
|
|
( wa simpl sylanr2 ) CEGABCDCEHFI $.
|
|
$}
|
|
|
|
${
|
|
ad2ant.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
|
|
ad2antrr $p |- ( ( ( ph /\ ch ) /\ th ) -> ps ) $=
|
|
( adantr adantlr ) ADBCABDEFG $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
|
|
ad2antlr $p |- ( ( ( ch /\ ph ) /\ th ) -> ps ) $=
|
|
( adantr adantll ) ADBCABDEFG $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
ad2antrl $p |- ( ( ch /\ ( ph /\ th ) ) -> ps ) $=
|
|
( wa adantr adantl ) ADFBCABDEGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
ad2antll $p |- ( ( ch /\ ( th /\ ph ) ) -> ps ) $=
|
|
( wa adantl ) DAFBCABDEGG $.
|
|
|
|
$( Deduction adding three conjuncts to antecedent. (Contributed by NM,
|
|
28-Jul-2012.) $)
|
|
ad3antrrr $p |- ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa adantr ad2antrr ) ACGBDEABCFHI $.
|
|
|
|
$( Deduction adding three conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad3antlr $p |- ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa ad2antlr adantr ) CAGDGBEABCDFHI $.
|
|
|
|
$( Deduction adding 4 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad4antr $p |- ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
|
|
( wa ad3antrrr adantr ) ACHDHEHBFABCDEGIJ $.
|
|
|
|
$( Deduction adding 4 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad4antlr $p |- ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
|
|
( wa ad3antlr adantr ) CAHDHEHBFABCDEGIJ $.
|
|
|
|
$( Deduction adding 5 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad5antr $p |- ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) -> ps ) $=
|
|
( wa ad4antr adantr ) ACIDIEIFIBGABCDEFHJK $.
|
|
|
|
$( Deduction adding 5 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad5antlr $p |- ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) -> ps ) $=
|
|
( wa ad4antlr adantr ) CAIDIEIFIBGABCDEFHJK $.
|
|
|
|
$( Deduction adding 6 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad6antr $p |- ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) -> ps ) $=
|
|
( wa ad5antr adantr ) ACJDJEJFJGJBHABCDEFGIKL $.
|
|
|
|
$( Deduction adding 6 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad6antlr $p |- ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) -> ps ) $=
|
|
( wa ad5antlr adantr ) CAJDJEJFJGJBHABCDEFGIKL $.
|
|
|
|
$( Deduction adding 7 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad7antr $p |- ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa ad6antr adantr ) ACKDKEKFKGKHKBIABCDEFGHJLM $.
|
|
|
|
$( Deduction adding 7 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad7antlr $p |- ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa ad6antlr adantr ) CAKDKEKFKGKHKBIABCDEFGHJLM $.
|
|
|
|
$( Deduction adding 8 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad8antr $p |- ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa ad7antr adantr ) ACLDLELFLGLHLILBJABCDEFGHIKMN $.
|
|
|
|
$( Deduction adding 8 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad8antlr $p |- ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa ad7antlr adantr ) CALDLELFLGLHLILBJABCDEFGHIKMN $.
|
|
|
|
$( Deduction adding 9 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad9antr $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa ad8antr adantr ) ACMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.
|
|
|
|
$( Deduction adding 9 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad9antlr $p |- ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa ad8antlr adantr ) CAMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.
|
|
|
|
$( Deduction adding 10 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad10antr $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa ad9antr adantr ) ACNDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.
|
|
|
|
$( Deduction adding 10 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad10antlr $p |- ( ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa ad9antlr adantr ) CANDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.
|
|
$}
|
|
|
|
${
|
|
ad2ant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
ad2ant2l $p |- ( ( ( th /\ ph ) /\ ( ta /\ ps ) ) -> ch ) $=
|
|
( wa adantrl adantll ) AEBGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
ad2ant2r $p |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) -> ch ) $=
|
|
( wa adantrr adantlr ) ABEGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
23-Nov-2007.) $)
|
|
ad2ant2lr $p |- ( ( ( th /\ ph ) /\ ( ps /\ ta ) ) -> ch ) $=
|
|
( wa adantrr adantll ) ABEGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
24-Nov-2007.) $)
|
|
ad2ant2rl $p |- ( ( ( ph /\ th ) /\ ( ta /\ ps ) ) -> ch ) $=
|
|
( wa adantrl adantlr ) AEBGCDABCEFHI $.
|
|
$}
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.) $)
|
|
simpll $p |- ( ( ( ph /\ ps ) /\ ch ) -> ph ) $=
|
|
( id ad2antrr ) AABCADE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.) $)
|
|
simplr $p |- ( ( ( ph /\ ps ) /\ ch ) -> ps ) $=
|
|
( id ad2antlr ) BBACBDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) $)
|
|
simprl $p |- ( ( ph /\ ( ps /\ ch ) ) -> ps ) $=
|
|
( id ad2antrl ) BBACBDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) $)
|
|
simprr $p |- ( ( ph /\ ( ps /\ ch ) ) -> ch ) $=
|
|
( id ad2antll ) CCABCDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ph ) $=
|
|
( wa simpl ad2antrr ) ABEACDABFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simpllr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ps ) $=
|
|
( wa simpr ad2antrr ) ABEBCDABFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplrl $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ps ) $=
|
|
( wa simpl ad2antlr ) BCEBADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplrr $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ch ) $=
|
|
( wa simpr ad2antlr ) BCECADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprll $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( wa simpl ad2antrl ) BCEBADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprlr $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ch ) $=
|
|
( wa simpr ad2antrl ) BCECADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprrl $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ch ) $=
|
|
( wa simpl ad2antll ) CDECABCDFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprrr $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> th ) $=
|
|
( wa simpr ad2antll ) CDEDABCDFG $.
|
|
|
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$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-4l $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ph ) $=
|
|
( wa simplll adantr ) ABFCFDFAEABCDGH $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-4r $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa simpllr adantr ) ABFCFDFBEABCDGH $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-5l $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) -> ph ) $=
|
|
( wa simp-4l adantr ) ABGCGDGEGAFABCDEHI $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-5r $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) -> ps ) $=
|
|
( wa simp-4r adantr ) ABGCGDGEGBFABCDEHI $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-6l $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) -> ph ) $=
|
|
( wa simp-5l adantr ) ABHCHDHEHFHAGABCDEFIJ $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-6r $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) -> ps ) $=
|
|
( wa simp-5r adantr ) ABHCHDHEHFHBGABCDEFIJ $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-7l $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) -> ph ) $=
|
|
( wa simp-6l adantr ) ABICIDIEIFIGIAHABCDEFGJK $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-7r $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) -> ps ) $=
|
|
( wa simp-6r adantr ) ABICIDIEIFIGIBHABCDEFGJK $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-8l $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) -> ph ) $=
|
|
( wa simp-7l adantr ) ABJCJDJEJFJGJHJAIABCDEFGHKL $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-8r $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa simp-7r adantr ) ABJCJDJEJFJGJHJBIABCDEFGHKL $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-9l $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ph ) $=
|
|
( wa simp-8l adantr ) ABKCKDKEKFKGKHKIKAJABCDEFGHILM $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-9r $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa simp-8r adantr ) ABKCKDKEKFKGKHKIKBJABCDEFGHILM $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-10l $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ph ) $=
|
|
( wa simp-9l adantr ) ABLCLDLELFLGLHLILJLAKABCDEFGHIJMN $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-10r $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa simp-9r adantr ) ABLCLDLELFLGLHLILJLBKABCDEFGHIJMN $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-11l $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ph ) $=
|
|
( wa simp-10l adantr ) ABMCMDMEMFMGMHMIMJMKMALABCDEFGHIJKNO $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-11r $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa simp-10r adantr ) ABMCMDMEMFMGMHMIMJMKMBLABCDEFGHIJKNO $.
|
|
|
|
$( Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.) $)
|
|
pm4.87 $p |- ( ( ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) /\
|
|
( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) ) /\
|
|
( ( ps -> ( ph -> ch ) ) <-> ( ( ps /\ ph ) -> ch ) ) ) $=
|
|
( wa wi wb impexp bi2.04 pm3.2i bicomi ) ABDCEABCEEZFZKBACEEZFZDMBADCEZFLNA
|
|
BCGABCHIOMBACGJI $.
|
|
|
|
$( Introduce one conjunct as an antecedent to the other. "abai" stands for
|
|
"and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 7-Dec-2012.) $)
|
|
abai $p |- ( ( ph /\ ps ) <-> ( ph /\ ( ph -> ps ) ) ) $=
|
|
( wi biimt pm5.32i ) ABABCABDE $.
|
|
|
|
$( Swap two conjuncts. Note that the first digit (1) in the label refers to
|
|
the outer conjunct position, and the next digit (2) to the inner conjunct
|
|
position. (Contributed by NM, 12-Mar-1995.) $)
|
|
an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
|
|
( wa ancom anbi1i anass 3bitr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 25-Dec-2012.) $)
|
|
an32 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
|
|
( wa anass an12 ancom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof
|
|
shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
an13 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ch /\ ( ps /\ ph ) ) ) $=
|
|
( wa an12 anass ancom 3bitr2i ) ABCDDBACDDBADZCDCIDABCEBACFICGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof
|
|
shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
an31 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) ) $=
|
|
( wa an13 anass 3bitr4i ) ABCDDCBADDABDCDCBDADABCEABCFCBAFG $.
|
|
|
|
${
|
|
an12s.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Swap two conjuncts in antecedent. The label suffix "s" means that
|
|
~ an12 is combined with ~ syl (or a variant). (Contributed by NM,
|
|
13-Mar-1996.) $)
|
|
an12s $p |- ( ( ps /\ ( ph /\ ch ) ) -> th ) $=
|
|
( wa an12 sylbi ) BACFFABCFFDBACGEH $.
|
|
|
|
$( Inference commuting a nested conjunction in antecedent. (Contributed by
|
|
NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
ancom2s $p |- ( ( ph /\ ( ch /\ ps ) ) -> th ) $=
|
|
( wa pm3.22 sylan2 ) CBFABCFDCBGEH $.
|
|
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) $)
|
|
an13s $p |- ( ( ch /\ ( ps /\ ph ) ) -> th ) $=
|
|
( exp32 com13 imp32 ) CBADABCDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
an32s.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.) $)
|
|
an32s $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
|
|
( wa an32 sylbi ) ACFBFABFCFDACBGEH $.
|
|
|
|
$( Inference commuting a nested conjunction in antecedent. (Contributed by
|
|
NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
ancom1s $p |- ( ( ( ps /\ ph ) /\ ch ) -> th ) $=
|
|
( wa pm3.22 sylan ) BAFABFCDBAGEH $.
|
|
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) $)
|
|
an31s $p |- ( ( ( ch /\ ps ) /\ ph ) -> th ) $=
|
|
( exp31 com13 imp31 ) CBADABCDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
anass1rs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Commutative-associative law for conjunction in an antecedent.
|
|
(Contributed by Jeff Madsen, 19-Jun-2011.) $)
|
|
anass1rs $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
|
|
( anassrs an32s ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.)
|
|
(Proof shortened by Wolf Lammen, 16-Nov-2013.) $)
|
|
anabs1 $p |- ( ( ( ph /\ ps ) /\ ph ) <-> ( ph /\ ps ) ) $=
|
|
( wa simpl pm4.71i bicomi ) ABCZGACGAABDEF $.
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.)
|
|
(Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
|
|
anabs5 $p |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
|
|
( wa ibar bicomd pm5.32i ) AABCZBABGABDEF $.
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.)
|
|
(Proof shortened by Wolf Lammen, 17-Nov-2013.) $)
|
|
anabs7 $p |- ( ( ps /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
|
|
( wa simpr pm4.71ri bicomi ) ABCZBGCGBABDEF $.
|
|
|
|
${
|
|
anabsan.1 $e |- ( ( ( ph /\ ph ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent with conjunction. (Contributed by NM,
|
|
24-Mar-1996.) (Revised by NM, 18-Nov-2013.) $)
|
|
anabsan $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa pm4.24 sylanb ) AAAEBCAFDG $.
|
|
$}
|
|
|
|
${
|
|
anabss1.1 $e |- ( ( ( ph /\ ps ) /\ ph ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
anabss1 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( an32s anabsan ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss4.1 $e |- ( ( ( ps /\ ph ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) $)
|
|
anabss4 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabss1 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss5.1 $e |- ( ( ph /\ ( ph /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
|
|
anabss5 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anassrs anabsan ) ABCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi5.1 $e |- ( ph -> ( ( ph /\ ps ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
|
|
anabsi5 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa imp anabss5 ) ABCAABECDFG $.
|
|
$}
|
|
|
|
${
|
|
anabsi6.1 $e |- ( ph -> ( ( ps /\ ph ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
14-Aug-2000.) $)
|
|
anabsi6 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( ancomsd anabsi5 ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi7.1 $e |- ( ps -> ( ( ph /\ ps ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
|
|
anabsi7 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabsi6 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi8.1 $e |- ( ps -> ( ( ps /\ ph ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
26-Sep-1999.) $)
|
|
anabsi8 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabsi5 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss7.1 $e |- ( ( ps /\ ( ph /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.) $)
|
|
anabss7 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anassrs anabss4 ) ABCBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsan2.1 $e |- ( ( ph /\ ( ps /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent with conjunction. (Contributed by NM,
|
|
10-May-2004.) (Revised by NM, 1-Jan-2013.) $)
|
|
anabsan2 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( an12s anabss7 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss3.1 $e |- ( ( ( ph /\ ps ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
|
|
anabss3 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anasss anabsan2 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
$( Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) $)
|
|
an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
|
|
( wa an12 anbi2i anass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
|
|
LHI $.
|
|
|
|
$( Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) $)
|
|
an42 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ph /\ ch ) /\ ( th /\ ps ) ) ) $=
|
|
( wa an4 ancom anbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.
|
|
|
|
${
|
|
an4s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Inference rearranging 4 conjuncts in antecedent. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
an4s $p |- ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) -> ta ) $=
|
|
( wa an4 sylbi ) ACGBDGGABGCDGGEACBDHFI $.
|
|
$}
|
|
|
|
${
|
|
an41r3s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Inference rearranging 4 conjuncts in antecedent. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
an42s $p |- ( ( ( ph /\ ch ) /\ ( th /\ ps ) ) -> ta ) $=
|
|
( wa an4s ancom2s ) ACGBDEABCDEFHI $.
|
|
$}
|
|
|
|
$( Distribution of conjunction over conjunction. (Contributed by NM,
|
|
14-Aug-1995.) $)
|
|
anandi $p |- ( ( ph /\ ( ps /\ ch ) ) <->
|
|
( ( ph /\ ps ) /\ ( ph /\ ch ) ) ) $=
|
|
( wa anidm anbi1i an4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.
|
|
|
|
$( Distribution of conjunction over conjunction. (Contributed by NM,
|
|
24-Aug-1995.) $)
|
|
anandir $p |- ( ( ( ph /\ ps ) /\ ch ) <->
|
|
( ( ph /\ ch ) /\ ( ps /\ ch ) ) ) $=
|
|
( wa anidm anbi2i an4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.
|
|
|
|
${
|
|
anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> ta ) $.
|
|
$( Inference that undistributes conjunction in the antecedent.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
anandis $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
|
|
( wa an4s anabsan ) ABCFDABACDEGH $.
|
|
$}
|
|
|
|
${
|
|
anandirs.1 $e |- ( ( ( ph /\ ch ) /\ ( ps /\ ch ) ) -> ta ) $.
|
|
$( Inference that undistributes conjunction in the antecedent.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
anandirs $p |- ( ( ( ph /\ ps ) /\ ch ) -> ta ) $=
|
|
( wa an4s anabsan2 ) ABFCDACBCDEGH $.
|
|
$}
|
|
|
|
${
|
|
syl2an2.1 $e |- ( ph -> ps ) $.
|
|
syl2an2.2 $e |- ( ( ch /\ ph ) -> th ) $.
|
|
syl2an2.3 $e |- ( ( ps /\ th ) -> ta ) $.
|
|
$( ~ syl2an with antecedents in standard conjunction form. (Contributed by
|
|
Alan Sare, 27-Aug-2016.) $)
|
|
syl2an2 $p |- ( ( ch /\ ph ) -> ta ) $=
|
|
( wa syl2an anabss7 ) CAEABDECAIFGHJK $.
|
|
$}
|
|
|
|
${
|
|
syl2an2r.1 $e |- ( ph -> ps ) $.
|
|
syl2an2r.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
syl2an2r.3 $e |- ( ( ps /\ th ) -> ta ) $.
|
|
$( ~ syl2anr with antecedents in standard conjunction form. (Contributed
|
|
by Alan Sare, 27-Aug-2016.) $)
|
|
syl2an2r $p |- ( ( ph /\ ch ) -> ta ) $=
|
|
( wa syl2an anabss5 ) ACEABDEACIFGHJK $.
|
|
$}
|
|
|
|
${
|
|
impbida.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
impbida.2 $e |- ( ( ph /\ ch ) -> ps ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by NM,
|
|
17-Feb-2007.) $)
|
|
impbida $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( ex impbid ) ABCABCDFACBEFG $.
|
|
$}
|
|
|
|
$( Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.45 $p |- ( ( ph -> ps ) -> ( ( ph /\ ch ) -> ( ps /\ ch ) ) ) $=
|
|
( wi id anim1d ) ABDZABCGEF $.
|
|
|
|
${
|
|
im2an9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
im2an9.2 $e |- ( th -> ( ta -> et ) ) $.
|
|
$( Deduction joining nested implications to form implication of
|
|
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
|
|
im2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
|
|
( wa wi adantr adantl anim12d ) ADIBCEFABCJDGKDEFJAHLM $.
|
|
|
|
$( Deduction joining nested implications to form implication of
|
|
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
|
|
im2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
|
|
( wa wi im2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.
|
|
$}
|
|
|
|
${
|
|
anim12dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
anim12dan.2 $e |- ( ( ph /\ th ) -> ta ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by
|
|
Mario Carneiro, 12-May-2014.) $)
|
|
anim12dan $p |- ( ( ph /\ ( ps /\ th ) ) -> ( ch /\ ta ) ) $=
|
|
( wa ex anim12d imp ) ABDHCEHABCDEABCFIADEGIJK $.
|
|
$}
|
|
|
|
$( Two propositions are equivalent if they are both true. Theorem *5.1 of
|
|
[WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.) $)
|
|
pm5.1 $p |- ( ( ph /\ ps ) -> ( ph <-> ps ) ) $=
|
|
( wb pm5.501 biimpa ) ABABCABDE $.
|
|
|
|
$( Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 27-Nov-2013.) $)
|
|
pm3.43 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
|
|
( ph -> ( ps /\ ch ) ) ) $=
|
|
( wi wa pm3.43i imp ) ABDACDABCEDABCFG $.
|
|
|
|
$( Distributive law for implication over conjunction. Compare Theorem *4.76
|
|
of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 27-Nov-2013.) $)
|
|
jcab $p |- ( ( ph -> ( ps /\ ch ) ) <->
|
|
( ( ph -> ps ) /\ ( ph -> ch ) ) ) $=
|
|
( wa wi simpl imim2i simpr jca pm3.43 impbii ) ABCDZEZABEZACEZDMNOLBABCFGLC
|
|
ABCHGIABCJK $.
|
|
|
|
$( Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.76 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) <->
|
|
( ph -> ( ps /\ ch ) ) ) $=
|
|
( wa wi jcab bicomi ) ABCDEABEACEDABCFG $.
|
|
|
|
$( Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.38 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
|
|
( ( ph /\ ps ) <-> ( ch /\ th ) ) ) $=
|
|
( wb wa simpl simpr anbi12d ) ACEZBDEZFACBDJKGJKHI $.
|
|
|
|
${
|
|
bi2an9.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bi2an9.2 $e |- ( th -> ( ta <-> et ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 31-Jul-1995.) $)
|
|
bi2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
|
|
( wa anbi1d anbi2d sylan9bb ) ABEICEIDCFIABCEGJDEFCHKL $.
|
|
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 19-Feb-1996.) $)
|
|
bi2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
|
|
( wa wb bi2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.
|
|
|
|
$( Deduction joining two biconditionals with different antecedents.
|
|
(Contributed by NM, 12-May-2004.) $)
|
|
bi2bian9 $p |- ( ( ph /\ th ) -> ( ( ps <-> ta ) <-> ( ch <-> et ) ) ) $=
|
|
( wa wb adantr adantl bibi12d ) ADIBCEFABCJDGKDEFJAHLM $.
|
|
$}
|
|
|
|
$( Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.33 $p |- ( ( ph /\ ( ps -> ch ) ) <->
|
|
( ph /\ ( ( ph /\ ps ) -> ch ) ) ) $=
|
|
( wi wa ibar imbi1d pm5.32i ) ABCDABEZCDABICABFGH $.
|
|
|
|
$( Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.36 $p |- ( ( ph /\ ( ph <-> ps ) ) <-> ( ps /\ ( ph <-> ps ) ) ) $=
|
|
( wb id pm5.32ri ) ABCZABFDE $.
|
|
|
|
${
|
|
bianabs.1 $e |- ( ph -> ( ps <-> ( ph /\ ch ) ) ) $.
|
|
$( Absorb a hypothesis into the second member of a biconditional.
|
|
(Contributed by FL, 15-Feb-2007.) $)
|
|
bianabs $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wa ibar bitr4d ) ABACECDACFG $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical negation (intuitionistic)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( 'Not' introduction. One of the axioms of propositional logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-in1 $a |- ( ( ph -> -. ph ) -> -. ph ) $.
|
|
|
|
$( 'Not' elimination. One of the axioms of propositional logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-in2 $a |- ( -. ph -> ( ph -> ps ) ) $.
|
|
|
|
$( Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This
|
|
is valid intuitionistically (in the terminology of Section 1.2 of [Bauer]
|
|
p. 482 it is a proof of negation not a proof by contradiction); compare
|
|
with ~ pm2.18dc which only holds for some propositions. (Contributed by
|
|
Mario Carneiro, 12-May-2015.) $)
|
|
pm2.01 $p |- ( ( ph -> -. ph ) -> -. ph ) $=
|
|
( ax-in1 ) AB $.
|
|
|
|
$( From a wff and its negation, anything is true. Theorem *2.21 of
|
|
[WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed
|
|
by Mario Carneiro, 12-May-2015.) $)
|
|
pm2.21 $p |- ( -. ph -> ( ph -> ps ) ) $=
|
|
( ax-in2 ) ABC $.
|
|
|
|
${
|
|
pm2.01d.1 $e |- ( ph -> ( ps -> -. ps ) ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by NM,
|
|
18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.01d $p |- ( ph -> -. ps ) $=
|
|
( wn wi pm2.01 syl ) ABBDZEHCBFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.21d.1 $e |- ( ph -> -. ps ) $.
|
|
$( A contradiction implies anything. Deduction from ~ pm2.21 .
|
|
(Contributed by NM, 10-Feb-1996.) $)
|
|
pm2.21d $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wn wi pm2.21 syl ) ABEBCFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.21dd.1 $e |- ( ph -> ps ) $.
|
|
pm2.21dd.2 $e |- ( ph -> -. ps ) $.
|
|
$( A contradiction implies anything. Deduction from ~ pm2.21 .
|
|
(Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
pm2.21dd $p |- ( ph -> ch ) $=
|
|
( pm2.21d mpd ) ABCDABCEFG $.
|
|
$}
|
|
|
|
$( Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.24 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wn pm2.21 com12 ) ACABABDE $.
|
|
|
|
${
|
|
pm2.24d.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction version of ~ pm2.24 . (Contributed by NM, 30-Jan-2006.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.24d $p |- ( ph -> ( -. ps -> ch ) ) $=
|
|
( wn wi pm2.24 syl ) ABBECFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.24i.1 $e |- ph $.
|
|
$( Inference version of ~ pm2.24 . (Contributed by NM, 20-Aug-2001.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.24i $p |- ( -. ph -> ps ) $=
|
|
( wn pm2.21 mpi ) ADABCABEF $.
|
|
$}
|
|
|
|
${
|
|
con2d.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised
|
|
by NM, 12-Feb-2013.) $)
|
|
con2d $p |- ( ph -> ( ch -> -. ps ) ) $=
|
|
( wn wi ax-in2 syl6 com23 pm2.01 ) ACBBEZFKABCKABCECKFDCKGHIBJH $.
|
|
$}
|
|
|
|
${
|
|
mt2d.1 $e |- ( ph -> ch ) $.
|
|
mt2d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 4-Jul-1994.) $)
|
|
mt2d $p |- ( ph -> -. ps ) $=
|
|
( wn con2d mpd ) ACBFDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
nsyl3.1 $e |- ( ph -> -. ps ) $.
|
|
nsyl3.2 $e |- ( ch -> ps ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 1-Dec-1995.)
|
|
(Revised by NM, 13-Jun-2013.) $)
|
|
nsyl3 $p |- ( ch -> -. ph ) $=
|
|
( wn wi a1i mt2d ) CABEABFGCDHI $.
|
|
$}
|
|
|
|
${
|
|
con2i.a $e |- ( ph -> -. ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen,
|
|
13-Jun-2013.) $)
|
|
con2i $p |- ( ps -> -. ph ) $=
|
|
( id nsyl3 ) ABBCBDE $.
|
|
$}
|
|
|
|
${
|
|
nsyl.1 $e |- ( ph -> -. ps ) $.
|
|
nsyl.2 $e |- ( ch -> ps ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 31-Dec-1993.)
|
|
(Proof shortened by Wolf Lammen, 2-Mar-2013.) $)
|
|
nsyl $p |- ( ph -> -. ch ) $=
|
|
( nsyl3 con2i ) CAABCDEFG $.
|
|
$}
|
|
|
|
$( Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101.
|
|
This one holds for all propositions, but its converse only holds for
|
|
decidable propositions (see ~ notnotrdc ). (Contributed by NM,
|
|
28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) $)
|
|
notnot $p |- ( ph -> -. -. ph ) $=
|
|
( wn id con2i ) ABZAECD $.
|
|
|
|
${
|
|
notnotd.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction associated with ~ notnot and ~ notnoti . (Contributed by
|
|
Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf
|
|
Lammen, 27-Mar-2021.) $)
|
|
notnotd $p |- ( ph -> -. -. ps ) $=
|
|
( wn notnot syl ) ABBDDCBEF $.
|
|
$}
|
|
|
|
${
|
|
con3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by NM, 31-Jan-2015.) $)
|
|
con3d $p |- ( ph -> ( -. ch -> -. ps ) ) $=
|
|
( wn notnot syl6 con2d ) ABCEZABCIEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
con3i.a $e |- ( ph -> ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 20-Jun-2013.) $)
|
|
con3i $p |- ( -. ps -> -. ph ) $=
|
|
( wn id nsyl ) BDZBAGECF $.
|
|
$}
|
|
|
|
${
|
|
con3rr3.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Rotate through consequent right. (Contributed by Wolf Lammen,
|
|
3-Nov-2013.) $)
|
|
con3rr3 $p |- ( -. ch -> ( ph -> -. ps ) ) $=
|
|
( wn con3d com12 ) ACEBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
con3dimp.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Variant of ~ con3d with importation. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
con3dimp $p |- ( ( ph /\ -. ch ) -> -. ps ) $=
|
|
( wn con3d imp ) ACEBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.01da.1 $e |- ( ( ph /\ ps ) -> -. ps ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
pm2.01da $p |- ( ph -> -. ps ) $=
|
|
( wn ex pm2.01d ) ABABBDCEF $.
|
|
$}
|
|
|
|
$( In classical logic, this is just a restatement of ~ pm3.2 . In
|
|
intuitionistic logic, it still holds, but is weaker than pm3.2.
|
|
(Contributed by Mario Carneiro, 12-May-2015.) $)
|
|
pm3.2im $p |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) ) $=
|
|
( wn wi pm2.27 con2d ) AABCZDBAGEF $.
|
|
|
|
${
|
|
expi.1 $e |- ( -. ( ph -> -. ps ) -> ch ) $.
|
|
$( An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by O'Cat, 28-Nov-2008.) $)
|
|
expi $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wn wi pm3.2im syl6 ) ABABEFECABGDH $.
|
|
$}
|
|
|
|
${
|
|
pm2.65i.1 $e |- ( ph -> ps ) $.
|
|
pm2.65i.2 $e |- ( ph -> -. ps ) $.
|
|
$( Inference rule for proof by contradiction. (Contributed by NM,
|
|
18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
pm2.65i $p |- -. ph $=
|
|
( wn wi nsyl3 pm2.01 ax-mp ) AAEZFJABADCGAHI $.
|
|
$}
|
|
|
|
${
|
|
mt2.1 $e |- ps $.
|
|
mt2.2 $e |- ( ph -> -. ps ) $.
|
|
$( A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 10-Sep-2013.) $)
|
|
mt2 $p |- -. ph $=
|
|
( a1i pm2.65i ) ABBACEDF $.
|
|
$}
|
|
|
|
$( Theorem used to justify definition of intuitionistic biconditional
|
|
~ df-bi . (Contributed by NM, 24-Nov-2017.) $)
|
|
biijust $p |- ( ( ( ( ph -> ps ) /\ ( ps -> ph ) )
|
|
-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
|
|
/\ ( ( ( ph -> ps ) /\ ( ps -> ph ) )
|
|
-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) ) $=
|
|
( wi wa id pm3.2i ) ABCBACDZGCZHGEZIF $.
|
|
|
|
$( Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.) $)
|
|
con3 $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
|
|
( wi id con3d ) ABCZABFDE $.
|
|
|
|
$( Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
|
|
con2 $p |- ( ( ph -> -. ps ) -> ( ps -> -. ph ) ) $=
|
|
( wn wi id con2d ) ABCDZABGEF $.
|
|
|
|
${
|
|
mt2i.1 $e |- ch $.
|
|
mt2i.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 15-Sep-2012.) $)
|
|
mt2i $p |- ( ph -> -. ps ) $=
|
|
( a1i mt2d ) ABCCADFEG $.
|
|
$}
|
|
|
|
${
|
|
negbi.1 $e |- ph $.
|
|
$( Infer double negation. (Contributed by NM, 27-Feb-2008.) $)
|
|
notnoti $p |- -. -. ph $=
|
|
( wn notnot ax-mp ) AACCBADE $.
|
|
$}
|
|
|
|
${
|
|
pm2.21i.1 $e |- -. ph $.
|
|
$( A contradiction implies anything. Inference from ~ pm2.21 .
|
|
(Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
pm2.21i $p |- ( ph -> ps ) $=
|
|
( wn wi pm2.21 ax-mp ) ADABECABFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.24ii.1 $e |- ph $.
|
|
pm2.24ii.2 $e |- -. ph $.
|
|
$( A contradiction implies anything. Inference from ~ pm2.24 .
|
|
(Contributed by NM, 27-Feb-2008.) $)
|
|
pm2.24ii $p |- ps $=
|
|
( pm2.21i ax-mp ) ABCABDEF $.
|
|
$}
|
|
|
|
${
|
|
nsyld.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
nsyld.2 $e |- ( ph -> ( ta -> ch ) ) $.
|
|
$( A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.) $)
|
|
nsyld $p |- ( ph -> ( ps -> -. ta ) ) $=
|
|
( wn con3d syld ) ABCGDGEADCFHI $.
|
|
$}
|
|
|
|
${
|
|
nsyli.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
nsyli.2 $e |- ( th -> -. ch ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 3-May-1994.) $)
|
|
nsyli $p |- ( ph -> ( th -> -. ps ) ) $=
|
|
( wn con3d syl5 ) DCGABGFABCEHI $.
|
|
$}
|
|
|
|
$( Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
mth8 $p |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) ) $=
|
|
( wi pm2.27 con3d ) AABCBABDE $.
|
|
|
|
${
|
|
jc.1 $e |- ( ph -> ps ) $.
|
|
jc.2 $e |- ( ph -> ch ) $.
|
|
$( Inference joining the consequents of two premises. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
jc $p |- ( ph -> -. ( ps -> -. ch ) ) $=
|
|
( wn wi pm3.2im sylc ) ABCBCFGFDEBCHI $.
|
|
$}
|
|
|
|
$( Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.51 $p |- ( -. ( ph -> ps ) -> ( ph -> -. ps ) ) $=
|
|
( wi wn ax-1 con3i a1d ) ABCZDBDABHBAEFG $.
|
|
|
|
$( Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.52 $p |- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) ) $=
|
|
( wn wi pm2.21 pm2.24d com12 ) ACZABDZCBCZHIJABEFG $.
|
|
|
|
$( Exportation theorem expressed with primitive connectives. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
expt $p |- ( ( -. ( ph -> -. ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wn wi pm3.2im imim1d com12 ) AABDEDZCEBCEABICABFGH $.
|
|
|
|
$( Elimination of a nested antecedent. (Contributed by Wolf Lammen,
|
|
10-May-2013.) $)
|
|
jarl $p |- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) ) $=
|
|
( wn wi pm2.21 imim1i ) ADABECABFG $.
|
|
|
|
$( Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction.
|
|
Proofs, such as this one, which assume a proposition, here ` ph ` , derive
|
|
a contradiction, and therefore conclude ` -. ph ` , are valid
|
|
intuitionistically (and can be called "proof of negation", for example by
|
|
Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume
|
|
` -. ph ` , derive a contradiction, and conclude ` ph ` , such as
|
|
~ condandc , are only valid for decidable propositions. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.) $)
|
|
pm2.65 $p |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) ) $=
|
|
( wi wn pm2.27 con2d a2i ) ABCAABDZCZABIDAIBAHEFGF $.
|
|
|
|
${
|
|
pm2.65d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.65d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Deduction rule for proof by contradiction. (Contributed by NM,
|
|
26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) $)
|
|
pm2.65d $p |- ( ph -> -. ps ) $=
|
|
( nsyld pm2.01d ) ABABCBEDFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.65da.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
pm2.65da.2 $e |- ( ( ph /\ ps ) -> -. ch ) $.
|
|
$( Deduction rule for proof by contradiction. (Contributed by NM,
|
|
12-Jun-2014.) $)
|
|
pm2.65da $p |- ( ph -> -. ps ) $=
|
|
( ex wn pm2.65d ) ABCABCDFABCGEFH $.
|
|
$}
|
|
|
|
${
|
|
mto.1 $e |- -. ps $.
|
|
mto.2 $e |- ( ph -> ps ) $.
|
|
$( The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
mto $p |- -. ph $=
|
|
( wn a1i pm2.65i ) ABDBEACFG $.
|
|
$}
|
|
|
|
${
|
|
mtod.1 $e |- ( ph -> -. ch ) $.
|
|
mtod.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
mtod $p |- ( ph -> -. ps ) $=
|
|
( wn a1d pm2.65d ) ABCEACFBDGH $.
|
|
$}
|
|
|
|
${
|
|
mtoi.1 $e |- -. ch $.
|
|
mtoi.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof
|
|
shortened by Wolf Lammen, 15-Sep-2012.) $)
|
|
mtoi $p |- ( ph -> -. ps ) $=
|
|
( wn a1i mtod ) ABCCFADGEH $.
|
|
$}
|
|
|
|
${
|
|
mtand.1 $e |- ( ph -> -. ch ) $.
|
|
mtand.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( A modus tollens deduction. (Contributed by Jeff Hankins,
|
|
19-Aug-2009.) $)
|
|
mtand $p |- ( ph -> -. ps ) $=
|
|
( ex mtod ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction negating both sides of a logical equivalence. (Contributed by
|
|
NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
|
|
( wn biimprd con3d biimpd impbid ) ABECEACBABCDFGABCABCDHGI $.
|
|
$}
|
|
|
|
$( Contraposition. Bidirectional version of ~ con2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
con2b $p |- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) ) $=
|
|
( wn wi con2 impbii ) ABCDBACDABEBAEF $.
|
|
|
|
${
|
|
notbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Negate both sides of a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
notbii $p |- ( -. ph <-> -. ps ) $=
|
|
( wb wn id notbid ax-mp ) ABDZAEBEDCIABIFGH $.
|
|
$}
|
|
|
|
${
|
|
mtbi.1 $e |- -. ph $.
|
|
mtbi.2 $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus tollens.
|
|
(Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mtbi $p |- -. ps $=
|
|
( biimpri mto ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mtbir.1 $e |- -. ps $.
|
|
mtbir.2 $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus tollens.
|
|
(Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen,
|
|
14-Oct-2012.) $)
|
|
mtbir $p |- -. ph $=
|
|
( bicomi mtbi ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mtbid.min $e |- ( ph -> -. ps ) $.
|
|
mtbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 26-Nov-1995.) $)
|
|
mtbid $p |- ( ph -> -. ch ) $=
|
|
( biimprd mtod ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbird.min $e |- ( ph -> -. ch ) $.
|
|
mtbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 10-May-1994.) $)
|
|
mtbird $p |- ( ph -> -. ps ) $=
|
|
( biimpd mtod ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbii.min $e |- -. ps $.
|
|
mtbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 27-Nov-1995.) $)
|
|
mtbii $p |- ( ph -> -. ch ) $=
|
|
( biimprd mtoi ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbiri.min $e |- -. ch $.
|
|
mtbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 24-Aug-1995.) $)
|
|
mtbiri $p |- ( ph -> -. ps ) $=
|
|
( biimpd mtoi ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
sylnib.1 $e |- ( ph -> -. ps ) $.
|
|
sylnib.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnib $p |- ( ph -> -. ch ) $=
|
|
( wb a1i mtbid ) ABCDBCFAEGH $.
|
|
$}
|
|
|
|
${
|
|
sylnibr.1 $e |- ( ph -> -. ps ) $.
|
|
sylnibr.2 $e |- ( ch <-> ps ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
Useful for substituting an consequent with a definition. (Contributed
|
|
by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnibr $p |- ( ph -> -. ch ) $=
|
|
( bicomi sylnib ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
sylnbi.1 $e |- ( ph <-> ps ) $.
|
|
sylnbi.2 $e |- ( -. ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
Useful for substituting an antecedent with a definition. (Contributed
|
|
by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnbi $p |- ( -. ph -> ch ) $=
|
|
( wn notbii sylbi ) AFBFCABDGEH $.
|
|
$}
|
|
|
|
${
|
|
sylnbir.1 $e |- ( ps <-> ph ) $.
|
|
sylnbir.2 $e |- ( -. ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnbir $p |- ( -. ph -> ch ) $=
|
|
( bicomi sylnbi ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
xchnxbi.1 $e |- ( -. ph <-> ps ) $.
|
|
xchnxbi.2 $e |- ( ph <-> ch ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchnxbi $p |- ( -. ch <-> ps ) $=
|
|
( wn notbii bitr3i ) CFAFBACEGDH $.
|
|
$}
|
|
|
|
${
|
|
xchnxbir.1 $e |- ( -. ph <-> ps ) $.
|
|
xchnxbir.2 $e |- ( ch <-> ph ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchnxbir $p |- ( -. ch <-> ps ) $=
|
|
( bicomi xchnxbi ) ABCDCAEFG $.
|
|
$}
|
|
|
|
${
|
|
xchbinx.1 $e |- ( ph <-> -. ps ) $.
|
|
xchbinx.2 $e |- ( ps <-> ch ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchbinx $p |- ( ph <-> -. ch ) $=
|
|
( wn notbii bitri ) ABFCFDBCEGH $.
|
|
$}
|
|
|
|
${
|
|
xchbinxr.1 $e |- ( ph <-> -. ps ) $.
|
|
xchbinxr.2 $e |- ( ch <-> ps ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchbinxr $p |- ( ph <-> -. ch ) $=
|
|
( bicomi xchbinx ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
mt2bi.1 $e |- ph $.
|
|
$( A false consequent falsifies an antecedent. (Contributed by NM,
|
|
19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) $)
|
|
mt2bi $p |- ( -. ps <-> ( ps -> -. ph ) ) $=
|
|
( wn wi a1bi con2b bitri ) BDZAIEBADEAICFABGH $.
|
|
$}
|
|
|
|
$( Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by
|
|
Mario Carneiro, 31-Jan-2015.) $)
|
|
mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $=
|
|
( wn wi pm2.21 con3 com12 impbid2 ) ACZBCZBADZBAEKIJBAFGH $.
|
|
|
|
$( Two propositions are equivalent if they are both false. Theorem *5.21 of
|
|
[WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by
|
|
Mario Carneiro, 31-Jan-2015.) $)
|
|
pm5.21 $p |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) ) $=
|
|
( wn wa simpl pm2.21d simpr impbid ) ACZBCZDZABKABIJEFKBAIJGFH $.
|
|
|
|
$( Two propositions are equivalent if they are both false. Closed form of
|
|
~ 2false . Equivalent to a ~ bi2 -like version of the xor-connective.
|
|
(Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
pm5.21im $p |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) ) $=
|
|
( wn wb pm5.21 ex ) ACBCABDABEF $.
|
|
|
|
$( The negation of a wff is equivalent to the wff's equivalence to falsehood.
|
|
(Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
nbn2 $p |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) ) $=
|
|
( wn wb pm5.21im wi bi2 mtt syl5ibr impbid ) ACZBCZABDZABEMLKBAFABGABHIJ $.
|
|
|
|
$( Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.)
|
|
(Proof shortened by Wolf Lammen, 28-Jan-2013.) $)
|
|
bibif $p |- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) ) $=
|
|
( wn wb nbn2 bicom syl6rbb ) BCACBADABDBAEBAFG $.
|
|
|
|
${
|
|
nbn.1 $e |- -. ph $.
|
|
$( The negation of a wff is equivalent to the wff's equivalence to
|
|
falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 3-Oct-2013.) $)
|
|
nbn $p |- ( -. ps <-> ( ps <-> ph ) ) $=
|
|
( wb wn bibif ax-mp bicomi ) BADZBEZAEIJDCBAFGH $.
|
|
$}
|
|
|
|
${
|
|
nbn3.1 $e |- ph $.
|
|
$( Transfer falsehood via equivalence. (Contributed by NM,
|
|
11-Sep-2006.) $)
|
|
nbn3 $p |- ( -. ps <-> ( ps <-> -. ph ) ) $=
|
|
( wn notnoti nbn ) ADBACEF $.
|
|
$}
|
|
|
|
${
|
|
2false.1 $e |- -. ph $.
|
|
2false.2 $e |- -. ps $.
|
|
$( Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
2false $p |- ( ph <-> ps ) $=
|
|
( pm2.21i impbii ) ABABCEBADEF $.
|
|
$}
|
|
|
|
${
|
|
2falsed.1 $e |- ( ph -> -. ps ) $.
|
|
2falsed.2 $e |- ( ph -> -. ch ) $.
|
|
$( Two falsehoods are equivalent (deduction rule). (Contributed by NM,
|
|
11-Oct-2013.) $)
|
|
2falsed $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( pm2.21d impbid ) ABCABCDFACBEFG $.
|
|
$}
|
|
|
|
${
|
|
pm5.21ni.1 $e |- ( ph -> ps ) $.
|
|
pm5.21ni.2 $e |- ( ch -> ps ) $.
|
|
$( Two propositions implying a false one are equivalent. (Contributed by
|
|
NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)
|
|
pm5.21ni $p |- ( -. ps -> ( ph <-> ch ) ) $=
|
|
( wn con3i 2falsed ) BFACABDGCBEGH $.
|
|
|
|
pm5.21nii.3 $e |- ( ps -> ( ph <-> ch ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional.
|
|
(Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
pm5.21nii $p |- ( ph <-> ch ) $=
|
|
( wb syl ibi ibir impbii ) ACACABACGZDFHICACBLEFHJK $.
|
|
$}
|
|
|
|
${
|
|
pm5.21ndd.1 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
pm5.21ndd.2 $e |- ( ph -> ( th -> ps ) ) $.
|
|
pm5.21ndd.3 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional,
|
|
deduction version. (Contributed by Paul Chapman, 21-Nov-2012.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm5.21ndd $p |- ( ph -> ( ch <-> th ) ) $=
|
|
( wb syld ibd bicom1 syl6 impbid ) ACDACDACBCDHZEGIJADCADNDCHADBNFGICDKLJ
|
|
M $.
|
|
$}
|
|
|
|
$( Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm5.19 $p |- -. ( ph <-> -. ph ) $=
|
|
( wn wb bi1 pm2.01d id mpbird pm2.65i ) AABZCZAJAIJAAIDEZJFGKH $.
|
|
|
|
$( Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all
|
|
propositions, but compare with ~ pm4.81dc which requires a decidability
|
|
condition. (Contributed by NM, 3-Jan-2005.) $)
|
|
pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $=
|
|
( wn wi pm2.01 ax-1 impbii ) AABZCGADGAEF $.
|
|
|
|
$( Express implication in terms of conjunction. (Contributed by NM,
|
|
9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.) $)
|
|
imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
|
|
( wn wi wa pm3.2im imp con2i pm3.2 con3rr3 impbii ) ABCDZABEZCMLABLCABFGHAB
|
|
MABIJK $.
|
|
|
|
${
|
|
imnani.1 $e |- -. ( ph /\ ps ) $.
|
|
$( Express implication in terms of conjunction. (Contributed by Mario
|
|
Carneiro, 28-Sep-2015.) $)
|
|
imnani $p |- ( ph -> -. ps ) $=
|
|
( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
|
|
$}
|
|
|
|
$( Theorem to move a conjunct in and out of a negation. (Contributed by NM,
|
|
9-Nov-2003.) $)
|
|
nan $p |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) ) $=
|
|
( wa wn wi impexp imnan imbi2i bitr2i ) ABDCEZFABKFZFABCDEZFABKGLMABCHIJ $.
|
|
|
|
$( Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who
|
|
call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.)
|
|
(Revised by Mario Carneiro, 2-Feb-2015.) $)
|
|
pm3.24 $p |- -. ( ph /\ -. ph ) $=
|
|
( wn wi wa notnot imnan mpbi ) AABZBCAHDBAEAHFG $.
|
|
|
|
$( Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.) $)
|
|
notnotnot $p |- ( -. -. -. ph <-> -. ph ) $=
|
|
( wn notnot con3i impbii ) ABZBZBFAGACDFCE $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical disjunction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the connective for disjunction ('or'). $)
|
|
$c \/ $. $( Vee (read: 'or') $)
|
|
|
|
$( Extend wff definition to include disjunction ('or'). $)
|
|
wo $a wff ( ph \/ ps ) $.
|
|
|
|
$( Definition of 'or'. One of the axioms of propositional logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ~ jaob
|
|
instead. (New usage is discouraged.) $)
|
|
ax-io $a |-
|
|
( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $.
|
|
|
|
$( Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell]
|
|
p. 121. Alias of ~ ax-io . (Contributed by NM, 30-May-1994.) (Revised
|
|
by Mario Carneiro, 31-Jan-2015.) $)
|
|
jaob $p |- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $=
|
|
( ax-io ) ABCD $.
|
|
|
|
$( Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96.
|
|
(Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.) $)
|
|
olc $p |- ( ph -> ( ps \/ ph ) ) $=
|
|
( wo wi wa id jaob mpbi simpri ) BBACZDZAJDZJJDKLEJFBJAGHI $.
|
|
|
|
$( Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104.
|
|
(Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.) $)
|
|
orc $p |- ( ph -> ( ph \/ ps ) ) $=
|
|
( wo wi wa id jaob mpbi simpli ) AABCZDZBJDZJJDKLEJFAJBGHI $.
|
|
|
|
$( Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107.
|
|
(Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.) $)
|
|
pm2.67-2 $p |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) ) $=
|
|
( wo orc imim1i ) AACDBACEF $.
|
|
|
|
$( Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
pm3.44 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) ->
|
|
( ( ps \/ ch ) -> ph ) ) $=
|
|
( wo wi wa jaob biimpri ) BCDAEBAECAEFBACGH $.
|
|
|
|
${
|
|
jaoi.1 $e |- ( ph -> ps ) $.
|
|
jaoi.2 $e |- ( ch -> ps ) $.
|
|
$( Inference disjoining the antecedents of two implications. (Contributed
|
|
by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.) $)
|
|
jaoi $p |- ( ( ph \/ ch ) -> ps ) $=
|
|
( wi wo pm3.44 mp2an ) ABFCBFACGBFDEBACHI $.
|
|
$}
|
|
|
|
${
|
|
jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
$( Deduction disjoining the antecedents of two implications. (Contributed
|
|
by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.) $)
|
|
jaod $p |- ( ph -> ( ( ps \/ th ) -> ch ) ) $=
|
|
( wo wi com12 jaoi ) BDGACBACHDABCEIADCFIJI $.
|
|
|
|
jaod.3 $e |- ( ph -> ( ps \/ th ) ) $.
|
|
$( Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
mpjaod $p |- ( ph -> ch ) $=
|
|
( wo jaod mpd ) ABDHCGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
|
|
$( Inference conjoining and disjoining the antecedents of two implications.
|
|
(Contributed by NM, 30-Sep-1999.) $)
|
|
jaao $p |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) ) $=
|
|
( wa wi adantr adantl jaod ) ADHBCEABCIDFJDECIAGKL $.
|
|
|
|
$( Inference disjoining and conjoining the antecedents of two implications.
|
|
(Contributed by Stefan Allan, 1-Nov-2008.) $)
|
|
jaoa $p |- ( ( ph \/ th ) -> ( ( ps /\ ta ) -> ch ) ) $=
|
|
( wa wi adantrd adantld jaoi ) ABEHCIDABCEFJDECBGKL $.
|
|
$}
|
|
|
|
$( Theorem *2.53 of [WhiteheadRussell] p. 107. This holds
|
|
intuitionistically, although its converse does not (see ~ pm2.54dc ).
|
|
(Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.) $)
|
|
pm2.53 $p |- ( ( ph \/ ps ) -> ( -. ph -> ps ) ) $=
|
|
( wn wi pm2.24 ax-1 jaoi ) AACZBDBABEBHFG $.
|
|
|
|
${
|
|
ori.1 $e |- ( ph \/ ps ) $.
|
|
$( Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
ori $p |- ( -. ph -> ps ) $=
|
|
( wo wn wi pm2.53 ax-mp ) ABDAEBFCABGH $.
|
|
$}
|
|
|
|
${
|
|
ord.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
ord $p |- ( ph -> ( -. ps -> ch ) ) $=
|
|
( wo wn wi pm2.53 syl ) ABCEBFCGDBCHI $.
|
|
$}
|
|
|
|
$( Elimination of disjunction by denial of a disjunct. Theorem *2.55 of
|
|
[WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 21-Jul-2012.) $)
|
|
orel1 $p |- ( -. ph -> ( ( ph \/ ps ) -> ps ) ) $=
|
|
( wo wn pm2.53 com12 ) ABCADBABEF $.
|
|
|
|
$( Elimination of disjunction by denial of a disjunct. Theorem *2.56 of
|
|
[WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 5-Apr-2013.) $)
|
|
orel2 $p |- ( -. ph -> ( ( ps \/ ph ) -> ps ) ) $=
|
|
( wn idd pm2.21 jaod ) ACZBBAGBDABEF $.
|
|
|
|
$( Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
|
|
(Revised by NM, 15-Nov-2012.) $)
|
|
pm1.4 $p |- ( ( ph \/ ps ) -> ( ps \/ ph ) ) $=
|
|
( wo olc orc jaoi ) ABACBABDBAEF $.
|
|
|
|
$( Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 15-Nov-2012.) $)
|
|
orcom $p |- ( ( ph \/ ps ) <-> ( ps \/ ph ) ) $=
|
|
( wo pm1.4 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
orcomd.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Commutation of disjuncts in consequent. (Contributed by NM,
|
|
2-Dec-2010.) $)
|
|
orcomd $p |- ( ph -> ( ch \/ ps ) ) $=
|
|
( wo orcom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
orcoms.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Commutation of disjuncts in antecedent. (Contributed by NM,
|
|
2-Dec-2012.) $)
|
|
orcoms $p |- ( ( ps \/ ph ) -> ch ) $=
|
|
( wo pm1.4 syl ) BAEABECBAFDG $.
|
|
$}
|
|
|
|
${
|
|
orci.1 $e |- ph $.
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
orci $p |- ( ph \/ ps ) $=
|
|
( wo orc ax-mp ) AABDCABEF $.
|
|
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.)
|
|
(Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
olci $p |- ( ps \/ ph ) $=
|
|
( wo olc ax-mp ) ABADCABEF $.
|
|
$}
|
|
|
|
${
|
|
orcd.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.) $)
|
|
orcd $p |- ( ph -> ( ps \/ ch ) ) $=
|
|
( wo orc syl ) ABBCEDBCFG $.
|
|
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.)
|
|
(Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
olcd $p |- ( ph -> ( ch \/ ps ) ) $=
|
|
( orcd orcomd ) ABCABCDEF $.
|
|
$}
|
|
|
|
${
|
|
orcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Deduction eliminating disjunct. _Notational convention_: We sometimes
|
|
suffix with "s" the label of an inference that manipulates an
|
|
antecedent, leaving the consequent unchanged. The "s" means that the
|
|
inference eliminates the need for a syllogism ( ~ syl ) -type inference
|
|
in a proof. (Contributed by NM, 21-Jun-1994.) $)
|
|
orcs $p |- ( ph -> ch ) $=
|
|
( wo orc syl ) AABECABFDG $.
|
|
$}
|
|
|
|
${
|
|
olcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.)
|
|
(Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
olcs $p |- ( ps -> ch ) $=
|
|
( orcoms orcs ) BACABCDEF $.
|
|
$}
|
|
|
|
$( Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.07 $p |- ( ph -> ( ph \/ ph ) ) $=
|
|
( olc ) AAB $.
|
|
|
|
$( Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.45 $p |- ( -. ( ph \/ ps ) -> -. ph ) $=
|
|
( wo orc con3i ) AABCABDE $.
|
|
|
|
$( Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.46 $p |- ( -. ( ph \/ ps ) -> -. ps ) $=
|
|
( wo olc con3i ) BABCBADE $.
|
|
|
|
$( Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.47 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ ps ) ) $=
|
|
( wo wn pm2.45 orcd ) ABCDADBABEF $.
|
|
|
|
$( Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.48 $p |- ( -. ( ph \/ ps ) -> ( ph \/ -. ps ) ) $=
|
|
( wo wn pm2.46 olcd ) ABCDBDAABEF $.
|
|
|
|
$( Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.49 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ -. ps ) ) $=
|
|
( wo wn pm2.46 olcd ) ABCDBDADABEF $.
|
|
|
|
$( Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 9-Dec-2012.) $)
|
|
pm2.67 $p |- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) ) $=
|
|
( pm2.67-2 ) ABBC $.
|
|
|
|
$( A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 18-Nov-2012.) $)
|
|
biorf $p |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wn wo olc orel1 impbid2 ) ACBABDBAEABFG $.
|
|
|
|
$( A wff is equivalent to its negated disjunction with falsehood.
|
|
(Contributed by NM, 9-Jul-2012.) $)
|
|
biortn $p |- ( ph -> ( ps <-> ( -. ph \/ ps ) ) ) $=
|
|
( wn wo wb notnot biorf syl ) AACZCBIBDEAFIBGH $.
|
|
|
|
${
|
|
biorfi.1 $e |- -. ph $.
|
|
$( A wff is equivalent to its disjunction with falsehood. (Contributed by
|
|
NM, 23-Mar-1995.) $)
|
|
biorfi $p |- ( ps <-> ( ps \/ ph ) ) $=
|
|
( wn wo wb orc orel2 impbid2 ax-mp ) ADZBBAEZFCKBLBAGABHIJ $.
|
|
$}
|
|
|
|
$( Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 13-Dec-2013.) $)
|
|
pm2.621 $p |- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) ) $=
|
|
( wi id idd jaod ) ABCZABBGDGBEF $.
|
|
|
|
$( Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.) $)
|
|
pm2.62 $p |- ( ( ph \/ ps ) -> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wi wo pm2.621 com12 ) ABCABDBABEF $.
|
|
|
|
${
|
|
imorri.1 $e |- ( -. ph \/ ps ) $.
|
|
$( Infer implication from disjunction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
imorri $p |- ( ph -> ps ) $=
|
|
( wn wo wi pm2.21 ax-1 jaoi ax-mp ) ADZBEABFZCKLBABGBAHIJ $.
|
|
$}
|
|
|
|
$( Negated disjunction in terms of conjunction. This version of DeMorgan's
|
|
law is a biconditional for all propositions (not just decidable ones),
|
|
unlike ~ oranim , ~ anordc , or ~ ianordc . Compare Theorem *4.56 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Mario Carneiro, 31-Jan-2015.) $)
|
|
ioran $p |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) ) $=
|
|
( wo wn wa pm2.45 pm2.46 jca simpl con2i simpr jaoi impbii ) ABCZDZADZBDZEZ
|
|
OPQABFABGHNRARDBRAPQIJRBPQKJLJM $.
|
|
|
|
$( Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's
|
|
law). The biconditional holds for decidable propositions as seen at
|
|
~ ianordc . The converse holds for decidable propositions, as seen at
|
|
~ pm3.13dc . (Contributed by NM, 3-Jan-2005.) (Revised by Mario
|
|
Carneiro, 31-Jan-2015.) $)
|
|
pm3.14 $p |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) ) $=
|
|
( wn wa simpl con3i simpr jaoi ) ACABDZCBCIAABEFIBABGFH $.
|
|
|
|
$( Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for
|
|
decidable propositions, as seen at ~ anordc . (Contributed by NM,
|
|
3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm3.1 $p |- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) ) $=
|
|
( wn wo wa pm3.14 con2i ) ACBCDABEABFG $.
|
|
|
|
$( Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell]
|
|
p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf
|
|
Lammen, 4-Apr-2013.) $)
|
|
jao $p |- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( ph \/ ch ) -> ps ) ) ) $=
|
|
( wi wo pm3.44 ex ) ABDCBDACEBDBACFG $.
|
|
|
|
$( Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by NM, 10-Mar-2013.) $)
|
|
pm1.2 $p |- ( ( ph \/ ph ) -> ph ) $=
|
|
( id jaoi ) AAAABZDC $.
|
|
|
|
$( Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell]
|
|
p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) $)
|
|
oridm $p |- ( ( ph \/ ph ) <-> ph ) $=
|
|
( wo pm1.2 pm2.07 impbii ) AABAACADE $.
|
|
|
|
$( Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.25 $p |- ( ph <-> ( ph \/ ph ) ) $=
|
|
( wo oridm bicomi ) AABAACD $.
|
|
|
|
${
|
|
orim12i.1 $e |- ( ph -> ps ) $.
|
|
orim12i.2 $e |- ( ch -> th ) $.
|
|
$( Disjoin antecedents and consequents of two premises. (Contributed by
|
|
NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) $)
|
|
orim12i $p |- ( ( ph \/ ch ) -> ( ps \/ th ) ) $=
|
|
( wo orcd olcd jaoi ) ABDGCABDEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
orim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Introduce disjunct to both sides of an implication. (Contributed by NM,
|
|
6-Jun-1994.) $)
|
|
orim1i $p |- ( ( ph \/ ch ) -> ( ps \/ ch ) ) $=
|
|
( id orim12i ) ABCCDCEF $.
|
|
|
|
$( Introduce disjunct to both sides of an implication. (Contributed by NM,
|
|
6-Jun-1994.) $)
|
|
orim2i $p |- ( ( ch \/ ph ) -> ( ch \/ ps ) ) $=
|
|
( id orim12i ) CCABCEDF $.
|
|
$}
|
|
|
|
${
|
|
orbi2i.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding a left disjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
12-Dec-2012.) $)
|
|
orbi2i $p |- ( ( ch \/ ph ) <-> ( ch \/ ps ) ) $=
|
|
( wo biimpi orim2i biimpri impbii ) CAECBEABCABDFGBACABDHGI $.
|
|
|
|
$( Inference adding a right disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
orbi1i $p |- ( ( ph \/ ch ) <-> ( ps \/ ch ) ) $=
|
|
( wo orcom orbi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.
|
|
$}
|
|
|
|
${
|
|
orbi12i.1 $e |- ( ph <-> ps ) $.
|
|
orbi12i.2 $e |- ( ch <-> th ) $.
|
|
$( Infer the disjunction of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
orbi12i $p |- ( ( ph \/ ch ) <-> ( ps \/ th ) ) $=
|
|
( wo orbi2i orbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
|
|
$( Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm1.5 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) ) $=
|
|
( wo orc olcd olc orim2i jaoi ) ABACDZDBCDAJBACEFCJBCAGHI $.
|
|
|
|
$( Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Wolf Lammen, 14-Nov-2012.) $)
|
|
or12 $p |- ( ( ph \/ ( ps \/ ch ) ) <-> ( ps \/ ( ph \/ ch ) ) ) $=
|
|
( wo pm1.5 impbii ) ABCDDBACDDABCEBACEF $.
|
|
|
|
$( Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
orass $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( wo orcom or12 orbi2i 3bitri ) ABDZCDCIDACBDZDABCDZDICECABFJKACBEGH $.
|
|
|
|
$( Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.31 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) ) $=
|
|
( wo orass biimpri ) ABDCDABCDDABCEF $.
|
|
|
|
$( Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.32 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( wo orass biimpi ) ABDCDABCDDABCEF $.
|
|
|
|
$( A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
or32 $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ps ) ) $=
|
|
( wo orass or12 orcom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) $)
|
|
or4 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
|
|
( ( ph \/ ch ) \/ ( ps \/ th ) ) ) $=
|
|
( wo or12 orbi2i orass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
|
|
LHI $.
|
|
|
|
$( Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) $)
|
|
or42 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
|
|
( ( ph \/ ch ) \/ ( th \/ ps ) ) ) $=
|
|
( wo or4 orcom orbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.
|
|
|
|
$( Distribution of disjunction over disjunction. (Contributed by NM,
|
|
25-Feb-1995.) $)
|
|
orordi $p |- ( ( ph \/ ( ps \/ ch ) ) <->
|
|
( ( ph \/ ps ) \/ ( ph \/ ch ) ) ) $=
|
|
( wo oridm orbi1i or4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.
|
|
|
|
$( Distribution of disjunction over disjunction. (Contributed by NM,
|
|
25-Feb-1995.) $)
|
|
orordir $p |- ( ( ( ph \/ ps ) \/ ch ) <->
|
|
( ( ph \/ ch ) \/ ( ps \/ ch ) ) ) $=
|
|
( wo oridm orbi2i or4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.
|
|
|
|
$( Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.3 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) ) $=
|
|
( wo pm1.4 orim2i ) BCDCBDABCEF $.
|
|
|
|
$( Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.41 $p |- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
|
|
( wo olc id jaoi ) BABCZGBADGEF $.
|
|
|
|
$( Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.42 $p |- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wn wi pm2.21 id jaoi ) ACABDZHABEHFG $.
|
|
|
|
$( Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.4 $p |- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
|
|
( wo orc id jaoi ) AABCZGABDGEF $.
|
|
|
|
$( Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.44 $p |- ( ph <-> ( ph \/ ( ph /\ ps ) ) ) $=
|
|
( wa wo orc id simpl jaoi impbii ) AAABCZDAJEAAJAFABGHI $.
|
|
|
|
${
|
|
mtord.1 $e |- ( ph -> -. ch ) $.
|
|
mtord.2 $e |- ( ph -> -. th ) $.
|
|
mtord.3 $e |- ( ph -> ( ps -> ( ch \/ th ) ) ) $.
|
|
$( A modus tollens deduction involving disjunction. (Contributed by Jeff
|
|
Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
mtord $p |- ( ph -> -. ps ) $=
|
|
( wo wn pm2.21d jaod syld pm2.01d ) ABABCDHBIZGACNDACNEJADNFJKLM $.
|
|
$}
|
|
|
|
$( Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.45 $p |- ( ph <-> ( ph /\ ( ph \/ ps ) ) ) $=
|
|
( wo orc pm4.71i ) AABCABDE $.
|
|
|
|
$( Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM,
|
|
28-Jan-1997.) (Revised by NM, 1-Dec-2012.) $)
|
|
pm3.48 $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) ->
|
|
( ( ph \/ ch ) -> ( ps \/ th ) ) ) $=
|
|
( wi wo orc imim2i olc jaao ) ABEABDFZCDECBKABDGHDKCDBIHJ $.
|
|
|
|
${
|
|
orim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
orim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
10-May-1994.) $)
|
|
orim12d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) ) $=
|
|
( wi wo pm3.48 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
orim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
23-Apr-1995.) $)
|
|
orim1d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) ) $=
|
|
( idd orim12d ) ABCDDEADFG $.
|
|
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
23-Apr-1995.) $)
|
|
orim2d $p |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) ) $=
|
|
( idd orim12d ) ADDBCADFEG $.
|
|
$}
|
|
|
|
$( Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
orim2 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
|
|
( wi id orim2d ) BCDZBCAGEF $.
|
|
|
|
${
|
|
orbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding a left disjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
orbi2d $p |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) ) $=
|
|
( wo biimpd orim2d biimprd impbid ) ADBFDCFABCDABCEGHACBDABCEIHJ $.
|
|
|
|
$( Deduction adding a right disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
orbi1d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) ) $=
|
|
( wo orbi2d orcom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
|
|
$}
|
|
|
|
$( Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
orbi1 $p |- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) ) $=
|
|
( wb id orbi1d ) ABDZABCGEF $.
|
|
|
|
${
|
|
orbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
orbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of disjunctions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
orbi12d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) ) $=
|
|
( wo orbi1d orbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
$( Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) $)
|
|
pm5.61 $p |- ( ( ( ph \/ ps ) /\ -. ps ) <-> ( ph /\ -. ps ) ) $=
|
|
( wn wo biorf orcom syl6rbb pm5.32ri ) BCZABDZAIABADJBAEBAFGH $.
|
|
|
|
${
|
|
jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
|
|
$( Inference disjoining the antecedents of two implications. (Contributed
|
|
by NM, 23-Oct-2005.) $)
|
|
jaoian $p |- ( ( ( ph \/ th ) /\ ps ) -> ch ) $=
|
|
( wo wi ex jaoi imp ) ADGBCABCHDABCEIDBCFIJK $.
|
|
$}
|
|
|
|
${
|
|
jao1i.1 $e |- ( ps -> ( ch -> ph ) ) $.
|
|
$( Add a disjunct in the antecedent of an implication. (Contributed by
|
|
Rodolfo Medina, 24-Sep-2010.) $)
|
|
jao1i $p |- ( ( ph \/ ps ) -> ( ch -> ph ) ) $=
|
|
( wi ax-1 jaoi ) ACAEBACFDG $.
|
|
$}
|
|
|
|
${
|
|
jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
|
|
$( Deduction disjoining the antecedents of two implications. (Contributed
|
|
by NM, 14-Oct-2005.) $)
|
|
jaodan $p |- ( ( ph /\ ( ps \/ th ) ) -> ch ) $=
|
|
( wo ex jaod imp ) ABDGCABCDABCEHADCFHIJ $.
|
|
|
|
jaodan.3 $e |- ( ph -> ( ps \/ th ) ) $.
|
|
$( Eliminate a disjunction in a deduction. A translation of natural
|
|
deduction rule ` \/ ` E ( ` \/ ` elimination). (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
mpjaodan $p |- ( ph -> ch ) $=
|
|
( wo jaodan mpdan ) ABDHCGABCDEFIJ $.
|
|
$}
|
|
|
|
$( Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.77 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <->
|
|
( ( ps \/ ch ) -> ph ) ) $=
|
|
( wo wi wa jaob bicomi ) BCDAEBAECAEFBACGH $.
|
|
|
|
$( Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.63 $p |- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) ) $=
|
|
( wo wn pm2.53 idd jaod ) ABCZADBBABEHBFG $.
|
|
|
|
$( Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.64 $p |- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) ) $=
|
|
( wn wo wi ax-1 orel2 jaoi com12 ) ABCZDABDZAAKAEJAKFBAGHI $.
|
|
|
|
$( Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.53 $p |- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <->
|
|
( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) ) $=
|
|
( wo wi wa jaob anbi1i bitri ) ABEZCEDFKDFZCDFZGADFBDFGZMGKDCHLNMADBHIJ $.
|
|
|
|
$( Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.38 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) ) $=
|
|
( wi id orim1d ) BCDZBCAGEF $.
|
|
|
|
$( Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.36 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) ) $=
|
|
( wo wi pm1.4 pm2.38 syl5 ) ABDBADBCECADABFABCGH $.
|
|
|
|
$( Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.37 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo pm2.38 pm1.4 syl6 ) BCDBAECAEACEABCFCAGH $.
|
|
|
|
$( Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.73 $p |- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) ->
|
|
( ps \/ ch ) ) ) $=
|
|
( wi wo pm2.621 orim1d ) ABDABEBCABFG $.
|
|
|
|
$( Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.74 $p |- ( ( ps -> ph ) -> ( ( ( ph \/ ps ) \/ ch ) ->
|
|
( ph \/ ch ) ) ) $=
|
|
( wi wo idd id jaod orim1d ) BADZABEACJAABJAFJGHI $.
|
|
|
|
$( Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.76 $p |- ( ( ph \/ ( ps -> ch ) ) ->
|
|
( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
|
|
( wo wi orc a1d orim2 jaoi ) AABDZACDZEBCEAKJACFGABCHI $.
|
|
|
|
$( Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) $)
|
|
pm2.75 $p |- ( ( ph \/ ps ) ->
|
|
( ( ph \/ ( ps -> ch ) ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo pm2.76 com12 ) ABCDEABEACEABCFG $.
|
|
|
|
$( Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm2.8 $p |- ( ( ph \/ ps ) -> ( ( -. ps \/ ch ) -> ( ph \/ ch ) ) ) $=
|
|
( wo wn pm1.4 ord orim1d ) ABDZBEACIBAABFGH $.
|
|
|
|
$( Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.81 $p |- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) ->
|
|
( ( ph \/ ch ) -> ( ph \/ th ) ) ) ) $=
|
|
( wi wo orim2 pm2.76 syl6 ) BCDEZEABFAJFACFADFEABJGACDHI $.
|
|
|
|
$( Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.82 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ( ( ph \/ -. ch ) \/ th ) ->
|
|
( ( ph \/ ps ) \/ th ) ) ) $=
|
|
( wo wn wi ax-1 pm2.24 orim2d jaoi orim1d ) ABEZCEACFZEZMDMOMGCMOHCNBACBIJK
|
|
L $.
|
|
|
|
${
|
|
pm3.2ni.1 $e |- -. ph $.
|
|
pm3.2ni.2 $e |- -. ps $.
|
|
$( Infer negated disjunction of negated premises. (Contributed by NM,
|
|
4-Apr-1995.) $)
|
|
pm3.2ni $p |- -. ( ph \/ ps ) $=
|
|
( wo id pm2.21i jaoi mto ) ABEACAABAFBADGHI $.
|
|
$}
|
|
|
|
$( Absorption of redundant internal disjunct. Compare Theorem *4.45 of
|
|
[WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 28-Feb-2014.) $)
|
|
orabs $p |- ( ph <-> ( ( ph \/ ps ) /\ ph ) ) $=
|
|
( wo orc pm4.71ri ) AABCABDE $.
|
|
|
|
$( Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton,
|
|
23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) $)
|
|
oranabs $p |- ( ( ( ph \/ -. ps ) /\ ps ) <-> ( ph /\ ps ) ) $=
|
|
( wn wo biortn orcom syl6rbb pm5.32ri ) BABCZDZABAIADJBAEIAFGH $.
|
|
|
|
$( Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell]
|
|
p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
ordi $p |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) ) $=
|
|
( wa wo simpl orim2i simpr jca orc adantl adantr olc jaoian jaodan impbii )
|
|
ABCDZEZABEZACEZDRSTQBABCFGQCABCHGISARCARSAQJZKACRBARCUALQAMNOP $.
|
|
|
|
$( Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) $)
|
|
ordir $p |- ( ( ( ph /\ ps ) \/ ch ) <->
|
|
( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) $=
|
|
( wa wo ordi orcom anbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
|
|
ACGBCGHI $.
|
|
|
|
$( Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 5-Jan-2013.) $)
|
|
andi $p |- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) ) $=
|
|
( wo wa orc olc jaodan anim2i jaoi impbii ) ABCDZEZABEZACEZDZABPCNOFONGHNMO
|
|
BLABCFICLACBGIJK $.
|
|
|
|
$( Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) $)
|
|
andir $p |- ( ( ( ph \/ ps ) /\ ch ) <->
|
|
( ( ph /\ ch ) \/ ( ps /\ ch ) ) ) $=
|
|
( wo wa andi ancom orbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
|
|
ACGBCGHI $.
|
|
|
|
$( Double distributive law for disjunction. (Contributed by NM,
|
|
12-Aug-1994.) $)
|
|
orddi $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
|
|
( ( ( ph \/ ch ) /\ ( ph \/ th ) ) /\
|
|
( ( ps \/ ch ) /\ ( ps \/ th ) ) ) ) $=
|
|
( wa wo ordir ordi anbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
|
|
MOACDHBCDHIJ $.
|
|
|
|
$( Double distributive law for conjunction. (Contributed by NM,
|
|
12-Aug-1994.) $)
|
|
anddi $p |- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <->
|
|
( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/
|
|
( ( ps /\ ch ) \/ ( ps /\ th ) ) ) ) $=
|
|
( wo wa andir andi orbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
|
|
MOACDHBCDHIJ $.
|
|
|
|
$( Prove formula-building rules for the biconditional connective. $)
|
|
|
|
$( Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.39 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
|
|
( ( ph \/ ps ) <-> ( ch \/ th ) ) ) $=
|
|
( wb wa simpl simpr orbi12d ) ACEZBDEZFACBDJKGJKHI $.
|
|
|
|
$( Implication in terms of biconditional and disjunction. Theorem *4.72 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 30-Jan-2013.) $)
|
|
pm4.72 $p |- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wi wo wb olc pm2.621 impbid2 orc bi2 syl5 impbii ) ABCZBABDZEZMBNBAFABGHA
|
|
NOBABIBNJKL $.
|
|
|
|
$( Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) $)
|
|
pm5.16 $p |- -. ( ( ph <-> ps ) /\ ( ph <-> -. ps ) ) $=
|
|
( wb wn wa pm5.19 simpl simpr bitr3d mto ) ABCZABDZCZEZBLCBFNABLKMGKMHIJ $.
|
|
|
|
$( A wff is disjoined with truth is true. (Contributed by NM,
|
|
23-May-1999.) $)
|
|
biort $p |- ( ph -> ( ph <-> ( ph \/ ps ) ) ) $=
|
|
( wo orc ax-1 impbid2 ) AAABCZABDAGEF $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Stable propositions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
$( Declare connective for stability. $)
|
|
$c STAB $.
|
|
|
|
$( Extend wff definition to include stability. $)
|
|
wstab $a wff STAB ph $.
|
|
|
|
$( Propositions where a double-negative can be removed are called stable.
|
|
See Chapter 2 [Moschovakis] p. 2.
|
|
|
|
Our notation for stability is a connective ` STAB ` which we place before
|
|
the formula in question. For example, ` STAB x = y ` corresponds to "x =
|
|
y is stable".
|
|
|
|
(Contributed by David A. Wheeler, 13-Aug-2018.) $)
|
|
df-stab $a |- ( STAB ph <-> ( -. -. ph -> ph ) ) $.
|
|
|
|
$( Every formula of the form ` -. ph ` is stable. Uses ~ notnotnot .
|
|
(Contributed by David A. Wheeler, 13-Aug-2018.) $)
|
|
stabnot $p |- STAB -. ph $=
|
|
( wn wstab wi notnotnot biimpi df-stab mpbir ) ABZCIBBZIDJIAEFIGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Decidable propositions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
$( Declare connective for decidability. $)
|
|
$c DECID $.
|
|
|
|
$( Extend wff definition to include decidability. $)
|
|
wdc $a wff DECID ph $.
|
|
|
|
$( Propositions which are known to be true or false are called decidable.
|
|
The (classical) Law of the Excluded Middle corresponds to the principle
|
|
that all propositions are decidable, but even given intuitionistic logic,
|
|
particular kinds of propositions may be decidable (for example, the
|
|
proposition that two natural numbers are equal will be decidable under
|
|
most sets of axioms).
|
|
|
|
Our notation for decidability is a connective ` DECID ` which we place
|
|
before the formula in question. For example, ` DECID x = y ` corresponds
|
|
to "x = y is decidable".
|
|
|
|
We could transform intuitionistic logic to classical logic by adding
|
|
unconditional forms of ~ condc , ~ exmiddc , ~ peircedc , or ~ notnotrdc ,
|
|
any of which would correspond to the assertion that all propositions are
|
|
decidable.
|
|
|
|
(Contributed by Jim Kingdon, 11-Mar-2018.) $)
|
|
df-dc $a |- ( DECID ph <-> ( ph \/ -. ph ) ) $.
|
|
|
|
$( Law of excluded middle, for a decidable proposition. The law of the
|
|
excluded middle is also called the principle of _tertium non datur_.
|
|
Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is
|
|
either true or not true; there are no in-between values of truth. The key
|
|
way in which intuitionistic logic differs from classical logic is that
|
|
intuitionistic logic says that excluded middle only holds for some
|
|
propositions, and classical logic says that it holds for all propositions.
|
|
(Contributed by Jim Kingdon, 12-May-2018.) $)
|
|
exmiddc $p |- ( DECID ph -> ( ph \/ -. ph ) ) $=
|
|
( wdc wn wo df-dc biimpi ) ABAACDAEF $.
|
|
|
|
$( Commuted law of the excluded middle for a decidable proposition. Based on
|
|
theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon,
|
|
25-Mar-2018.) $)
|
|
pm2.1dc $p |- ( DECID ph -> ( -. ph \/ ph ) ) $=
|
|
( wdc wn wo df-dc orcom bitri biimpi ) ABZACZADZIAJDKAEAJFGH $.
|
|
|
|
$( A decidable proposition is decidable when negated. (Contributed by Jim
|
|
Kingdon, 25-Mar-2018.) $)
|
|
dcn $p |- ( DECID ph -> DECID -. ph ) $=
|
|
( wn wo wdc notnot orim2i orcoms df-dc 3imtr4i ) AABZCJJBZCZADJDJALAKJAEFGA
|
|
HJHI $.
|
|
|
|
${
|
|
dcbii.1 $e |- ( ph <-> ps ) $.
|
|
$( The equivalent of a decidable proposition is decidable. (Contributed by
|
|
Jim Kingdon, 28-Mar-2018.) $)
|
|
dcbii $p |- ( DECID ph <-> DECID ps ) $=
|
|
( wn wo wdc notbii orbi12i df-dc 3bitr4i ) AADZEBBDZEAFBFABKLCABCGHAIBIJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
dcbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( The equivalent of a decidable proposition is decidable. (Contributed by
|
|
Jim Kingdon, 7-Sep-2019.) $)
|
|
dcbid $p |- ( ph -> ( DECID ps <-> DECID ch ) ) $=
|
|
( wn wo wdc notbid orbi12d df-dc 3bitr4g ) ABBEZFCCEZFBGCGABCLMDABCDHIBJC
|
|
JK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Theorems of decidable propositions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Many theorems of logic hold in intuitionistic logic just as they do in
|
|
classical (non-inuitionistic) logic, for all propositions. Other theorems
|
|
only hold for decidable propositions, such as the law of the excluded
|
|
middle ( ~ df-dc ), double negation elimination ( ~ notnotrdc ), or
|
|
contraposition ( ~ condc ). Our goal is to prove all well-known
|
|
or important classical theorems, but with suitable decidability
|
|
conditions so that the proofs follow from intuitionistic axioms.
|
|
This section is focused on such proofs, given decidability conditions.
|
|
$)
|
|
|
|
$( Contraposition of a decidable proposition.
|
|
|
|
This theorem swaps or "transposes" the order of the consequents when
|
|
negation is removed. An informal example is that the statement "if there
|
|
are no clouds in the sky, it is not raining" implies the statement "if it
|
|
is raining, there are clouds in the sky." This theorem (without the
|
|
decidability condition, of course) is called _Transp_ or "the principle of
|
|
transposition" in _Principia Mathematica_ (Theorem *2.17 of
|
|
[WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will
|
|
also use the term "contraposition" for this principle, although the reader
|
|
is advised that in the field of philosophical logic, "contraposition" has
|
|
a different technical meaning.
|
|
|
|
(Contributed by Jim Kingdon, 13-Mar-2018.) $)
|
|
condc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) ) $=
|
|
( wdc wn wo wi df-dc ax-1 a1d pm2.27 ax-in2 syl6 jaoi sylbi ) ACAADZEOBDZFZ
|
|
BAFZFZAGASOARQABHIOQPROPJBAKLMN $.
|
|
|
|
$( Proof by contradiction for a decidable proposition. Based on Theorem
|
|
*2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius).
|
|
Intuitionistically it requires a decidability assumption, but compare with
|
|
~ pm2.01 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) $)
|
|
pm2.18dc $p |- ( DECID ph -> ( ( -. ph -> ph ) -> ph ) ) $=
|
|
( wdc wn wi pm2.21 a2i condc syl5 pm2.43d ) ABZACZADZALKLCZDJLADKAMAMEFALGH
|
|
I $.
|
|
|
|
$( Double negation elimination for a decidable proposition. The converse,
|
|
~ notnot , holds for all propositions, not just decidable ones. This is
|
|
Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability
|
|
condition added. (Contributed by Jim Kingdon, 11-Mar-2018.) $)
|
|
notnotrdc $p |- ( DECID ph -> ( -. -. ph -> ph ) ) $=
|
|
( wdc wn wo wi df-dc orcom bitri pm2.53 sylbi ) ABZACZADZLCAEKALDMAFALGHLAI
|
|
J $.
|
|
|
|
$( Decidability implies stability. The converse is not necessarily true.
|
|
(Contributed by David A. Wheeler, 13-Aug-2018.) $)
|
|
dcimpstab $p |- ( DECID ph -> STAB ph ) $=
|
|
( wdc wn wi wstab notnotrdc df-stab sylibr ) ABACCADAEAFAGH $.
|
|
|
|
$( Contraposition for a decidable proposition. Based on theorem *2.15 of
|
|
[WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) $)
|
|
con1dc $p |- ( DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) ) $=
|
|
( wn wi wdc notnot imim2i condc syl5 ) ACZBDJBCZCZDAEKADBLJBFGAKHI $.
|
|
|
|
${
|
|
con4biddc.1 $e |- ( ph -> ( DECID ps -> ( DECID ch ->
|
|
( -. ps <-> -. ch ) ) ) ) $.
|
|
$( A contraposition deduction. (Contributed by Jim Kingdon,
|
|
18-May-2018.) $)
|
|
con4biddc $p |- ( ph -> ( DECID ps -> ( DECID ch -> ( ps <-> ch ) ) ) ) $=
|
|
( wdc wb wa wi wn bi2 syl8 condc a2i imp31 bi1 imim2d sylcom impbid exp31
|
|
syl6 ) ABEZCEZBCFAUAGUBGBCAUAUBBCHZAUAUBCIZBIZHZHUBUCHAUAUBUEUDFZUFDUEUDJ
|
|
KUBUFUCCBLMTNAUAUBCBHZAUAUBUEUDHZHUBUHHAUAUBUGUIDUEUDOKUAUIUHUBBCLPQNRS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
impidc.1 $e |- ( DECID ch -> ( ph -> ( ps -> ch ) ) ) $.
|
|
$( An importation inference for a decidable consequent. (Contributed by
|
|
Jim Kingdon, 30-Apr-2018.) $)
|
|
impidc $p |- ( DECID ch -> ( -. ( ph -> -. ps ) -> ch ) ) $=
|
|
( wdc wn wi wa imp con3d ex com23 con1dc mpd ) CEZCFZABFZGZGRFCGOAPQOAPQG
|
|
OAHBCOABCGDIJKLCRMN $.
|
|
$}
|
|
|
|
$( Simplification given a decidable proposition. Similar to Theorem *3.27
|
|
(Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon,
|
|
30-Apr-2018.) $)
|
|
simprimdc $p |- ( DECID ps -> ( -. ( ph -> -. ps ) -> ps ) ) $=
|
|
( wi wdc idd a1i impidc ) ABBABBCCBDABEFG $.
|
|
|
|
$( Simplification for a decidable proposition. Similar to Theorem *3.26
|
|
(Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon,
|
|
29-Mar-2018.) $)
|
|
simplimdc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ph ) ) $=
|
|
( wdc wn wi pm2.21 con1dc mpi ) ACADABEZEIDAEABFAIGH $.
|
|
|
|
${
|
|
pm2.61ddc.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.61ddc.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( Deduction eliminating a decidable antecedent. (Contributed by Jim
|
|
Kingdon, 4-May-2018.) $)
|
|
pm2.61ddc $p |- ( DECID ps -> ( ph -> ch ) ) $=
|
|
( wdc wn wo wi df-dc com12 jaoi sylbi ) BFBBGZHACIZBJBONABCDKANCEKLM $.
|
|
$}
|
|
|
|
$( Case elimination for a decidable proposition. Based on theorem *2.6 of
|
|
[WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.) $)
|
|
pm2.6dc $p |- ( DECID ph ->
|
|
( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) ) $=
|
|
( wdc wn wi wo wa pm2.1dc pm3.44 syl5com expd ) ACZADZBEZABEZBLMAFNOGBAHBMA
|
|
IJK $.
|
|
|
|
${
|
|
jadc.1 $e |- ( DECID ph -> ( -. ph -> ch ) ) $.
|
|
jadc.2 $e |- ( ps -> ch ) $.
|
|
$( Inference forming an implication from the antecedents of two premises,
|
|
where a decidable antecedent is negated. (Contributed by Jim Kingdon,
|
|
25-Mar-2018.) $)
|
|
jadc $p |- ( DECID ph -> ( ( ph -> ps ) -> ch ) ) $=
|
|
( wi wdc imim2i wn pm2.6dc mpd syl5 ) ABFACFZAGZCBCAEHNAICFMCFDACJKL $.
|
|
$}
|
|
|
|
${
|
|
jaddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> th ) ) ) $.
|
|
jaddc.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Deduction forming an implication from the antecedents of two premises,
|
|
where a decidable antecedent is negated. (Contributed by Jim Kingdon,
|
|
26-Mar-2018.) $)
|
|
jaddc $p |- ( ph -> ( DECID ps -> ( ( ps -> ch ) -> th ) ) ) $=
|
|
( wi wdc imim2d wn pm2.6dc sylcom syl5d ) ABCGBDGZBHZDACDBFIAOBJDGNDGEBDK
|
|
LM $.
|
|
$}
|
|
|
|
$( Case elimination for a decidable proposition. Based on theorem *2.61 of
|
|
[WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.) $)
|
|
pm2.61dc $p |- ( DECID ph ->
|
|
( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) ) $=
|
|
( wdc wn wi pm2.6dc com23 ) ACADBEABEBABFG $.
|
|
|
|
$( Negating an implication for a decidable antecedent. Based on theorem *2.5
|
|
of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon,
|
|
29-Mar-2018.) $)
|
|
pm2.5dc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) ) $=
|
|
( wdc wi wn wa simplimdc imp pm2.24d ex ) ACZABDEZAEBDKLFABKLAABGHIJ $.
|
|
|
|
$( Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional
|
|
decidability condition. (Contributed by Jim Kingdon, 5-May-2018.) $)
|
|
pm2.521dc $p |- ( DECID ph -> ( -. ( ph -> ps ) -> ( ps -> ph ) ) ) $=
|
|
( wi wn wdc pm2.52 condc syl5 ) ABCDADBDCAEBACABFABGH $.
|
|
|
|
$( Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a
|
|
decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.) $)
|
|
con34bdc $p |- ( DECID ps -> ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) ) $=
|
|
( wdc wi wn con3 condc impbid2 ) BCABDBEAEDABFBAGH $.
|
|
|
|
$( Double negation equivalence for a decidable proposition. Like Theorem
|
|
*4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent.
|
|
The forward direction, ~ notnot , holds for all propositions, not just
|
|
decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) $)
|
|
notnotbdc $p |- ( DECID ph -> ( ph <-> -. -. ph ) ) $=
|
|
( wdc wn notnot notnotrdc impbid2 ) ABAACCADAEF $.
|
|
|
|
$( Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) $)
|
|
con1biimdc $p |- ( DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) ) $=
|
|
( wdc wn wb wi bi1 con1dc syl5 bi2 con2d a1i impbidd ) ACZADZBEZBDZAPOBFNQA
|
|
FOBGABHIPAQFFNPBAOBJKLM $.
|
|
|
|
$( Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) $)
|
|
con1bidc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) ) $=
|
|
( wdc wn wb wa wi con1biimdc adantr adantl impbid ex ) ACZBCZADBEZBDAEZEMNF
|
|
OPMOPGNABHINPOGMBAHJKL $.
|
|
|
|
$( Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) $)
|
|
con2bidc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) ) $=
|
|
( wdc wn wb wa con1bidc imp bicom 3bitr3g bicomd ex ) ACZBCZABDZEZBADZEZEMN
|
|
FZRPSQBEZOAEZRPMNTUAEABGHQBIOAIJKL $.
|
|
|
|
${
|
|
con1biddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps <-> ch ) ) ) $.
|
|
$( A contraposition deduction. (Contributed by Jim Kingdon,
|
|
4-Apr-2018.) $)
|
|
con1biddc $p |- ( ph -> ( DECID ps -> ( -. ch <-> ps ) ) ) $=
|
|
( wdc wn wb con1biimdc sylcom ) ABEBFCGCFBGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
con1biidc.1 $e |- ( DECID ph -> ( -. ph <-> ps ) ) $.
|
|
$( A contraposition inference. (Contributed by Jim Kingdon,
|
|
15-Mar-2018.) $)
|
|
con1biidc $p |- ( DECID ph -> ( -. ps <-> ph ) ) $=
|
|
( wdc wn notnotbdc notbid bitrd bicomd ) ADZABEZJAAEZEKAFJLBCGHI $.
|
|
$}
|
|
|
|
$( Contraposition. Bidirectional version of ~ con1dc . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
con1bdc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) ) $=
|
|
( wdc wn wi wb wa con1dc adantr adantl impbid ex ) ACZBCZADBEZBDAEZFMNGOPMO
|
|
PENABHINPOEMBAHJKL $.
|
|
|
|
${
|
|
con2biidc.1 $e |- ( DECID ps -> ( ph <-> -. ps ) ) $.
|
|
$( A contraposition inference. (Contributed by Jim Kingdon,
|
|
15-Mar-2018.) $)
|
|
con2biidc $p |- ( DECID ps -> ( ps <-> -. ph ) ) $=
|
|
( wdc wn bicomd con1biidc ) BDZAEBBAHABECFGF $.
|
|
$}
|
|
|
|
${
|
|
con2biddc.1 $e |- ( ph -> ( DECID ch -> ( ps <-> -. ch ) ) ) $.
|
|
$( A contraposition deduction. (Contributed by Jim Kingdon,
|
|
11-Apr-2018.) $)
|
|
con2biddc $p |- ( ph -> ( DECID ch -> ( ch <-> -. ps ) ) ) $=
|
|
( wdc wn wb bicom syl6ib con1biddc ) ACEZBFZCGCLGACBAKBCFZGMBGDBMHIJLCHI
|
|
$.
|
|
$}
|
|
|
|
${
|
|
condandc.1 $e |- ( ( ph /\ -. ps ) -> ch ) $.
|
|
condandc.2 $e |- ( ( ph /\ -. ps ) -> -. ch ) $.
|
|
$( Proof by contradiction. This only holds for decidable propositions, as
|
|
it is part of the family of theorems which assume ` -. ps ` , derive a
|
|
contradiction, and therefore conclude ` ps ` . By contrast, assuming
|
|
` ps ` , deriving a contradiction, and therefore concluding ` -. ps ` ,
|
|
as in ~ pm2.65 , is valid for all propositions. (Contributed by Jim
|
|
Kingdon, 13-May-2018.) $)
|
|
condandc $p |- ( DECID ps -> ( ph -> ps ) ) $=
|
|
( wn wdc pm2.65da notnotrdc syl5 ) ABFZFBGBAKCDEHBIJ $.
|
|
$}
|
|
|
|
${
|
|
bijadc.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
bijadc.2 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
|
|
$( Combine antecedents into a single biconditional. This inference is
|
|
reminiscent of ~ jadc . (Contributed by Jim Kingdon, 4-May-2018.) $)
|
|
bijadc $p |- ( DECID ps -> ( ( ph <-> ps ) -> ch ) ) $=
|
|
( wb bi2 syli wn bi1 con3d pm2.61ddc ) ABFZBCBMACABGDHBIMAICMABABJKEHL $.
|
|
$}
|
|
|
|
$( Relationship between an equivalence and an equivalence with some negation,
|
|
for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell]
|
|
p. 124. Given decidability, we can consider ` -. ( ph <-> -. ps ) ` to
|
|
represent "negated exclusive-or". (Contributed by Jim Kingdon,
|
|
4-Apr-2018.) $)
|
|
pm5.18dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) ) $=
|
|
( wdc wn wo wb wi df-dc wa pm5.501 a1d con1biddc imp adantr bitr2d dcn nbn2
|
|
ex syl5 jaoi sylbi ) ACAADZEBCZABFZABDZFZDZFZGZAHAUIUBAUCUHAUCIUGBUDAUCUGBF
|
|
ABUFAUEUFFUCAUEJKLMABUDFUCABJNORUBUCUHUBUCIUGUEUDUBUCUGUEFZUCUECZUBUJBPUBUE
|
|
UFUBUEDUFFUKAUEQKLSMUBUEUDFUCABQNORTUA $.
|
|
|
|
$( Definition of 'and' in terms of negation and implication, for decidable
|
|
propositions. The forward direction holds for all propositions, and can
|
|
(basically) be found at ~ pm3.2im . (Contributed by Jim Kingdon,
|
|
30-Apr-2018.) $)
|
|
dfandc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) ) $=
|
|
( wdc wa wn wi wb pm3.2im imp simplimdc adantr simprimdc adantl jca impbid2
|
|
ex ) ACZBCZABDZABEZFEZGQRDZSUAABUAABHIUBUASUBUADABUBUAAQUAAFRATJKIUBUABRUAB
|
|
FQABLMINPOP $.
|
|
|
|
$( A decidable proposition or its triple negation is true. Theorem *2.13 of
|
|
[WhiteheadRussell] p. 101 with decidability condition added. (Contributed
|
|
by Jim Kingdon, 13-May-2018.) $)
|
|
pm2.13dc $p |- ( DECID ph -> ( ph \/ -. -. -. ph ) ) $=
|
|
( wdc wn wo df-dc notnotrdc con3d orim2d syl5bi pm2.43i ) ABZAACZCZCZDZKALD
|
|
KOAEKLNAKMAAFGHIJ $.
|
|
|
|
$( Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions.
|
|
(Contributed by Jim Kingdon, 1-May-2018.) $)
|
|
pm4.63dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) ) ) $=
|
|
( wdc wn wi wa wb dfandc imp bicomd ex ) ACZBCZABDEDZABFZGLMFONLMONGABHIJK
|
|
$.
|
|
|
|
$( Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions.
|
|
(Contributed by Jim Kingdon, 1-May-2018.) $)
|
|
pm4.67dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) ) ) $=
|
|
( wdc wn wi wa wb dcn pm4.63dc syl ) ACADZCBCKBDEDKBFGEAHKBIJ $.
|
|
|
|
$( Express conjunction in terms of implication. One direction of Theorem
|
|
*4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable
|
|
propositions, as can be seen at ~ annimdc . (Contributed by Jim Kingdon,
|
|
24-Dec-2017.) $)
|
|
annimim $p |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) ) $=
|
|
( wn wi pm2.27 con3 syl imp ) ABCZABDZCZAJBDIKDABEJBFGH $.
|
|
|
|
$( One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse
|
|
holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.) $)
|
|
pm4.65r $p |- ( ( -. ph /\ -. ps ) -> -. ( -. ph -> ps ) ) $=
|
|
( wn annimim ) ACBD $.
|
|
|
|
$( An implication between two decidable propositions is decidable.
|
|
(Contributed by Jim Kingdon, 28-Mar-2018.) $)
|
|
dcim $p |- ( DECID ph -> ( DECID ps -> DECID ( ph -> ps ) ) ) $=
|
|
( wn wo wi df-dc wa anbi2i andi bitri pm3.4 annimim orim12i sylbi sylibr ex
|
|
wdc ax-in2 a1d orc syl6 jaoi ) AQAACZDBQZABEZQZEZAFAUGUCAUDUFAUDGZUEUECZDZU
|
|
FUHABGZABCZGZDZUJUHABULDZGUNUDUOABFHABULIJUKUEUMUIABKABLMNUEFZOPUCUDUEUFUCU
|
|
EUDABRSUEUJUFUEUITUPOUAUBN $.
|
|
|
|
$( Express implication in terms of conjunction. The converse only holds
|
|
given a decidability condition; see ~ imandc . (Contributed by Jim
|
|
Kingdon, 24-Dec-2017.) $)
|
|
imanim $p |- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) ) $=
|
|
( wn wa wi annimim con2i ) ABCDABEABFG $.
|
|
|
|
$( Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll]
|
|
p. 176, with an added decidability condition. The forward direction,
|
|
~ imanim , holds for all propositions, not just decidable ones.
|
|
(Contributed by Jim Kingdon, 25-Apr-2018.) $)
|
|
imandc $p |- ( DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) ) $=
|
|
( wdc wi wn wa notnotbdc imbi2d imnan syl6bb ) BCZABDABEZEZDALFEKBMABGHALIJ
|
|
$.
|
|
|
|
$( Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability
|
|
condition. (Contributed by Jim Kingdon, 24-Apr-2018.) $)
|
|
pm4.14dc $p |- ( DECID ch ->
|
|
( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) ) ) $=
|
|
( wdc wi wn wa con34bdc imbi2d impexp 3bitr4g ) CDZABCEZEACFZBFZEZEABGCEANG
|
|
OELMPABCHIABCJANOJK $.
|
|
|
|
$( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
|
|
( wa wi wn pm3.3 con3 syl6 impd ) ABDCEZACFZBFZKABCELMEABCGBCHIJ $.
|
|
|
|
$( Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) $)
|
|
pm4.15 $p |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ( ps /\ ch ) -> -. ph ) ) $=
|
|
( wa wn wi con2b nan bitr2i ) BCDZAEFAJEFABDCEFJAGABCHI $.
|
|
|
|
$( Deriving disjunction from implication for a decidable proposition. Based
|
|
on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, ~ pm2.53 ,
|
|
holds whether the proposition is decidable or not. (Contributed by Jim
|
|
Kingdon, 26-Mar-2018.) $)
|
|
pm2.54dc $p |- ( DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) ) $=
|
|
( wdc wn wi wo dcn notnotrdc orc syl6 a1d olc a1i jaddc mpd ) ACZADZCZQBEAB
|
|
FZEAGPQBSPQDZSERPTASAHABIJKBSEPBALMNO $.
|
|
|
|
$( Definition of 'or' in terms of negation and implication for a decidable
|
|
proposition. Based on definition of [Margaris] p. 49. One direction,
|
|
~ pm2.53 , holds for all propositions, not just decidable ones.
|
|
(Contributed by Jim Kingdon, 26-Mar-2018.) $)
|
|
dfordc $p |- ( DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) ) $=
|
|
( wdc wo wn wi pm2.53 pm2.54dc impbid2 ) ACABDAEBFABGABHI $.
|
|
|
|
$( Elimination of disjunction based on a disjunction, for a decidable
|
|
proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104.
|
|
(Contributed by NM, 3-Jan-2005.) $)
|
|
pm2.25dc $p |- ( DECID ph -> ( ph \/ ( ( ph \/ ps ) -> ps ) ) ) $=
|
|
( wdc wn wo wi df-dc orel1 orim2i sylbi ) ACAADZEAABEBFZEAGKLAABHIJ $.
|
|
|
|
$( Concluding disjunction from implication for a decidable proposition.
|
|
Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of ~ pm2.62
|
|
and one half of ~ dfor2dc . (Contributed by Jim Kingdon, 27-Mar-2018.) $)
|
|
pm2.68dc $p |- ( DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) ) $=
|
|
( wi wn wdc wo jarl pm2.54dc syl5 ) ABCBCADBCAEABFABBGABHI $.
|
|
|
|
$( Logical 'or' expressed in terms of implication only, for a decidable
|
|
proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124.
|
|
(Contributed by Jim Kingdon, 27-Mar-2018.) $)
|
|
dfor2dc $p |- ( DECID ph -> ( ( ph \/ ps ) <-> ( ( ph -> ps ) -> ps ) ) ) $=
|
|
( wdc wo wi pm2.62 pm2.68dc impbid2 ) ACABDABEBEABFABGH $.
|
|
|
|
$( Simplify an implication between implications, for a decidable proposition.
|
|
(Contributed by Jim Kingdon, 18-Mar-2018.) $)
|
|
imimorbdc $p |- ( DECID ps -> ( ( ( ps -> ch ) -> ( ph -> ch ) ) <->
|
|
( ph -> ( ps \/ ch ) ) ) ) $=
|
|
( wdc wo wi dfor2dc imbi2d bi2.04 syl6rbbr ) BDZABCEZFABCFZCFZFMACFFKLNABCG
|
|
HMACIJ $.
|
|
|
|
$( Implication in terms of disjunction for a decidable proposition. Based on
|
|
theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction,
|
|
~ imorr , holds for all propositions. (Contributed by Jim Kingdon,
|
|
20-Apr-2018.) $)
|
|
imordc $p |- ( DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) ) $=
|
|
( wdc wi wn wo notnotbdc imbi1d wb dcn dfordc syl bitr4d ) ACZABDAEZEZBDZOB
|
|
FZNAPBAGHNOCRQIAJOBKLM $.
|
|
|
|
$( Implication in terms of disjunction. One direction of theorem *4.6 of
|
|
[WhiteheadRussell] p. 120. The converse holds for decidable propositions,
|
|
as seen at ~ imordc . (Contributed by Jim Kingdon, 21-Jul-2018.) $)
|
|
imorr $p |- ( ( -. ph \/ ps ) -> ( ph -> ps ) ) $=
|
|
( wn wi ax-in2 ax-1 jaoi ) ACABDBABEBAFG $.
|
|
|
|
$( Implication in terms of disjunction. Like Theorem *4.62 of
|
|
[WhiteheadRussell] p. 120, but for a decidable antecedent. (Contributed
|
|
by Jim Kingdon, 21-Apr-2018.) $)
|
|
pm4.62dc $p |- ( DECID ph -> ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) ) $=
|
|
( wn imordc ) ABCD $.
|
|
|
|
$( Negated conjunction in terms of disjunction (DeMorgan's law). Theorem
|
|
*4.51 of [WhiteheadRussell] p. 120, but where one proposition is
|
|
decidable. The reverse direction, ~ pm3.14 , holds for all propositions,
|
|
but the equivalence only holds where one proposition is decidable.
|
|
(Contributed by Jim Kingdon, 21-Apr-2018.) $)
|
|
ianordc $p |- ( DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) ) $=
|
|
( wa wn wi wdc wo imnan pm4.62dc syl5bbr ) ABCDABDZEAFADKGABHABIJ $.
|
|
|
|
$( Absorption of disjunction into equivalence. (Contributed by NM,
|
|
6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) $)
|
|
oibabs $p |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) ) $=
|
|
( wo wb wi pm2.67-2 ibd olc imim1i ibibr sylibr impbid ax-1 impbii ) ABCZAB
|
|
DZEZPQABQABAPBFGQBPEBAEBOPBAHIBAJKLPOMN $.
|
|
|
|
$( Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability
|
|
condition. The reverse direction, ~ pm2.53 , holds for all propositions.
|
|
(Contributed by Jim Kingdon, 2-May-2018.) $)
|
|
pm4.64dc $p |- ( DECID ph -> ( ( -. ph -> ps ) <-> ( ph \/ ps ) ) ) $=
|
|
( wdc wo wn wi dfordc bicomd ) ACABDAEBFABGH $.
|
|
|
|
$( Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability
|
|
condition. (Contributed by Jim Kingdon, 2-May-2018.) $)
|
|
pm4.66dc $p |- ( DECID ph -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) ) $=
|
|
( wn pm4.64dc ) ABCD $.
|
|
|
|
$( One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse
|
|
also holds in classical logic. (Contributed by Jim Kingdon,
|
|
27-Jul-2018.) $)
|
|
pm4.52im $p |- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) ) $=
|
|
( wn wa wi wo annimim imorr nsyl ) ABCDABEACBFABGABHI $.
|
|
|
|
$( One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse
|
|
also holds in classical logic. (Contributed by Jim Kingdon,
|
|
27-Jul-2018.) $)
|
|
pm4.53r $p |- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) ) $=
|
|
( wn wa wo pm4.52im con2i ) ABCDACBEABFG $.
|
|
|
|
$( Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions.
|
|
One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.) $)
|
|
pm4.54dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) ) $=
|
|
( wdc wn wa wo wb wi dcn dfandc syl imp pm4.66dc adantr notbid bitrd ex ) A
|
|
CZBCZADZBEZABDZFZDZGRSEZUATUBHZDZUDRSUAUGGZRTCSUHHAITBJKLUEUFUCRUFUCGSABMNO
|
|
PQ $.
|
|
|
|
$( Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.56 $p |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) $=
|
|
( wo wn wa ioran bicomi ) ABCDADBDEABFG $.
|
|
|
|
$( Disjunction in terms of conjunction (DeMorgan's law). One direction of
|
|
Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold
|
|
intuitionistically but does hold in classical logic. (Contributed by Jim
|
|
Kingdon, 25-Jul-2018.) $)
|
|
oranim $p |- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) ) $=
|
|
( wn wa wo pm4.56 biimpi con2i ) ACBCDZABEZIJCABFGH $.
|
|
|
|
$( Implication distributes over disjunction. One direction of Theorem *4.78
|
|
of [WhiteheadRussell] p. 121. The converse holds in classical logic.
|
|
(Contributed by Jim Kingdon, 15-Jan-2018.) $)
|
|
pm4.78i $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) ->
|
|
( ph -> ( ps \/ ch ) ) ) $=
|
|
( wi wo orc imim2i olc jaoi ) ABDABCEZDACDBJABCFGCJACBHGI $.
|
|
|
|
$( Equivalence between a disjunction of two implications, and a conjunction
|
|
and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121
|
|
but with additional decidability antecedents. (Contributed by Jim
|
|
Kingdon, 28-Mar-2018.) $)
|
|
pm4.79dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ( ps -> ph ) \/ ( ch -> ph ) ) <->
|
|
( ( ps /\ ch ) -> ph ) ) ) ) $=
|
|
( wdc wi wo wa wb id jaoa simplimdc pm3.3 syl9 dcim pm2.54dc syl6 syl5d imp
|
|
wn impbid2 expcom ) BDZADZBAEZCAEZFZBCGAEZHUBUCGUFUGUDBAUECUDIUEIJUBUCUGUFE
|
|
UBUGUDSZUEEZUCUFUBUHBUGUEBAKBCALMUBUCUDDUIUFEBANUDUEOPQRTUA $.
|
|
|
|
$( Two ways of stating exclusive-or which are equivalent for a decidable
|
|
proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124.
|
|
(Contributed by Jim Kingdon, 16-Apr-2018.) $)
|
|
pm5.17dc $p |- ( DECID ps ->
|
|
( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) ) $=
|
|
( wn wb wdc wo wa bicom dfbi2 orcom dfordc syl5rbb imnan a1i anbi12d syl5bb
|
|
wi ) ABCZDRADZBEZABFZABGCZGZARHSRAQZARQZGTUCRAITUDUAUEUBUABAFTUDABJBAKLUEUB
|
|
DTABMNOPL $.
|
|
|
|
$( Reverse distribution of disjunction over implication, given decidability.
|
|
Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim
|
|
Kingdon, 1-Apr-2018.) $)
|
|
pm2.85dc $p |- ( DECID ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) ->
|
|
( ph \/ ( ps -> ch ) ) ) ) $=
|
|
( wdc wn wo wi df-dc orc a1d olc imim1i orel1 syl9r syl6 jaoi sylbi ) ADAAE
|
|
ZFABFZACFZGZABCGZFZGZAHAUDRAUCUAAUBIJRUAUBUCUABTRCBSTBAKLACMNUBAKOPQ $.
|
|
|
|
$( Disjunction distributes over implication. The forward direction,
|
|
~ pm2.76 , is valid intuitionistically. The reverse direction holds if
|
|
` ph ` is decidable, as can be seen at ~ pm2.85dc . (Contributed by Jim
|
|
Kingdon, 1-Apr-2018.) $)
|
|
orimdidc $p |- ( DECID ph -> ( ( ph \/ ( ps -> ch ) ) <->
|
|
( ( ph \/ ps ) -> ( ph \/ ch ) ) ) ) $=
|
|
( wdc wi wo pm2.76 pm2.85dc impbid2 ) ADABCEFABFACFEABCGABCHI $.
|
|
|
|
$( Decidable proposition version of theorem *2.26 of [WhiteheadRussell]
|
|
p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.) $)
|
|
pm2.26dc $p |- ( DECID ph -> ( -. ph \/ ( ( ph -> ps ) -> ps ) ) ) $=
|
|
( wdc wi wn wo pm2.27 imordc mpbii ) ACAABDBDZDAEJFABGAJHI $.
|
|
|
|
$( Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions.
|
|
This one needs a decidability condition, but compare with ~ pm4.8 which
|
|
holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.) $)
|
|
pm4.81dc $p |- ( DECID ph -> ( ( -. ph -> ph ) <-> ph ) ) $=
|
|
( wdc wn wi pm2.18dc pm2.24 impbid1 ) ABACADAAEAAFG $.
|
|
|
|
$( A decidable proposition or its negation implies a second proposition.
|
|
Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim
|
|
Kingdon, 29-Mar-2018.) $)
|
|
pm5.11dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) ) $=
|
|
( wdc wi wn wo dcim pm2.5dc pm2.54dc syl5com syld ) ACZBCABDZCZMAEBDZFZABGL
|
|
MEODNPABHMOIJK $.
|
|
|
|
$( Excluded middle with antecedents for a decidable consequent. Based on
|
|
theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon,
|
|
30-Mar-2018.) $)
|
|
pm5.12dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) ) $=
|
|
( wdc wn wo wi df-dc ax-1 orim12i sylbi ) BCBBDZEABFZAKFZEBGBLKMBAHKAHIJ $.
|
|
|
|
$( A decidable proposition is implied by or implies other propositions.
|
|
Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim
|
|
Kingdon, 30-Mar-2018.) $)
|
|
pm5.14dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ps -> ch ) ) ) $=
|
|
( wdc wn wo wi df-dc ax-1 ax-in2 orim12i sylbi ) BDBBEZFABGZBCGZFBHBNMOBAIB
|
|
CJKL $.
|
|
|
|
$( An implication holds in at least one direction, where one proposition is
|
|
decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123.
|
|
(Contributed by Jim Kingdon, 30-Mar-2018.) $)
|
|
pm5.13dc $p |- ( DECID ps -> ( ( ph -> ps ) \/ ( ps -> ph ) ) ) $=
|
|
( pm5.14dc ) ABAC $.
|
|
|
|
$( A disjunction is equivalent to one of its disjuncts, given a decidable
|
|
disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125.
|
|
(Contributed by Jim Kingdon, 30-Mar-2018.) $)
|
|
pm5.55dc $p |- ( DECID ph ->
|
|
( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) ) $=
|
|
( wdc wn wo wb df-dc biort bicomd biorf orim12i sylbi ) ACAADZEABEZAFZNBFZE
|
|
AGAOMPAANABHIMBNABJIKL $.
|
|
|
|
$( Peirce's theorem for a decidable proposition. This odd-looking theorem
|
|
can be seen as an alternative to ~ exmiddc , ~ condc , or ~ notnotrdc in
|
|
the sense of expressing the "difference" between an intuitionistic system
|
|
of propositional calculus and a classical system. In intuitionistic
|
|
logic, it only holds for decidable propositions. (Contributed by Jim
|
|
Kingdon, 3-Jul-2018.) $)
|
|
peircedc $p |- ( DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) ) $=
|
|
( wdc wn wo wi df-dc ax-1 pm2.21 imim1i com12 jaoi sylbi ) ACAADZEABFZAFZAF
|
|
ZAGAQNAPHPNANOAABIJKLM $.
|
|
|
|
$( The Inversion Axiom of the infinite-valued sentential logic (L-infinity)
|
|
of Lukasiewicz, but where one of the propositions is decidable. Using
|
|
~ dfor2dc , we can see that this expresses "disjunction commutes."
|
|
Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability
|
|
condition). (Contributed by NM, 12-Aug-2004.) $)
|
|
looinvdc $p |- ( DECID ph ->
|
|
( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) ) $=
|
|
( wi wdc imim1 peircedc syl9r ) ABCZBCBACHACADAHBAEABFG $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Testable propositions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A proposition is testable iff its negative or double-negative is true.
|
|
See Chapter 2 [Moschovakis] p. 2.
|
|
|
|
Our notation for testability is ` DECID -. ` before the formula in
|
|
question. For example, ` DECID -. x = y ` corresponds to "x = y is
|
|
testable". (Contributed by David A. Wheeler, 13-Aug-2018.) $)
|
|
dftest $p |- ( DECID -. ph <-> ( -. ph \/ -. -. ph ) ) $=
|
|
( wn df-dc ) ABC $.
|
|
|
|
$( A proposition is testable iff its negation is testable. See also ~ dcn
|
|
(which could be read as "Decidability implies testability"). (Contributed
|
|
by David A. Wheeler, 6-Dec-2018.) $)
|
|
testbitestn $p |- ( DECID -. ph <-> DECID -. -. ph ) $=
|
|
( wn wo wdc notnotnot orbi2i orcom bitri df-dc 3bitr4ri ) ABZBZLBZCZKLCZLDK
|
|
DNLKCOMKLAEFLKGHLIKIJ $.
|
|
|
|
$( "Stable and testable" is equivalent to decidable. (Contributed by David
|
|
A. Wheeler, 13-Aug-2018.) $)
|
|
stabtestimpdc $p |-
|
|
( ( STAB ph /\ DECID -. ph ) <-> DECID ph ) $=
|
|
( wstab wn wdc wa wo exmiddc adantl df-stab biimpi orim2d adantr mpd orcomd
|
|
wi df-dc sylibr dcimpstab dcn jca impbii ) ABZACZDZEZADZUEAUCFUFUEUCAUEUCUC
|
|
CZFZUCAFZUDUHUBUCGHUBUHUIOUDUBUGAUCUBUGAOAIJKLMNAPQUFUBUDARASTUA $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Miscellaneous theorems of propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
pm5.21nd.1 $e |- ( ( ph /\ ps ) -> th ) $.
|
|
pm5.21nd.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
pm5.21nd.3 $e |- ( th -> ( ps <-> ch ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional.
|
|
(Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen,
|
|
4-Nov-2013.) $)
|
|
pm5.21nd $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( ex wb wi a1i pm5.21ndd ) ADBCABDEHACDFHDBCIJAGKL $.
|
|
$}
|
|
|
|
$( Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.35 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
|
|
( ph -> ( ps <-> ch ) ) ) $=
|
|
( wi wa pm5.1 pm5.74rd ) ABDZACDZEABCHIFG $.
|
|
|
|
$( A conjunction is equivalent to one of its conjuncts, given a decidable
|
|
conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125.
|
|
(Contributed by Jim Kingdon, 30-Mar-2018.) $)
|
|
pm5.54dc $p |- ( DECID ph ->
|
|
( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) ) $=
|
|
( wdc wa wb wn df-dc simpr ax-ia3 impbid2 simpl ax-in2 orim12i sylbi orcomd
|
|
wo ) ACZABDZBEZRAEZQAAFZPSTPAGASUATARBABHABIJUARAABKARLJMNO $.
|
|
|
|
${
|
|
baib.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Move conjunction outside of biconditional. (Contributed by NM,
|
|
13-May-1999.) $)
|
|
baib $p |- ( ps -> ( ph <-> ch ) ) $=
|
|
( wa ibar syl6rbbr ) BCBCEABCFDG $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by NM,
|
|
11-Jul-1994.) $)
|
|
baibr $p |- ( ps -> ( ch <-> ph ) ) $=
|
|
( baib bicomd ) BACABCDEF $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaib $p |- ( ch -> ( ph <-> ps ) ) $=
|
|
( wa ancom bitri baib ) ACBABCECBEDBCFGH $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
|
|
( wa ancom bitri baibr ) ACBABCECBEDBCFGH $.
|
|
$}
|
|
|
|
${
|
|
baibd.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
baibd $p |- ( ( ph /\ ch ) -> ( ps <-> th ) ) $=
|
|
( wa ibar bicomd sylan9bb ) ABCDFZCDECDJCDGHI $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaibd $p |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $=
|
|
( wa iba bicomd sylan9bb ) ABCDFZDCEDCJDCGHI $.
|
|
$}
|
|
|
|
$( Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.44 $p |- ( ( ph -> ps ) -> ( ( ph -> ch ) <->
|
|
( ph -> ( ps /\ ch ) ) ) ) $=
|
|
( wa wi jcab baibr ) ABCDEABEACEABCFG $.
|
|
|
|
$( Conjunction in antecedent versus disjunction in consequent, for a
|
|
decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with
|
|
decidability condition added. The reverse implication holds for all
|
|
propositions (see ~ pm5.6r ). (Contributed by Jim Kingdon,
|
|
2-Apr-2018.) $)
|
|
pm5.6dc $p |- ( DECID ps ->
|
|
( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) ) $=
|
|
( wdc wo wi wn wa dfordc imbi2d impexp syl6rbbr ) BDZABCEZFABGZCFZFAOHCFMNP
|
|
ABCIJAOCKL $.
|
|
|
|
$( Conjunction in antecedent versus disjunction in consequent. One direction
|
|
of Theorem *5.6 of [WhiteheadRussell] p. 125. If ` ps ` is decidable, the
|
|
converse also holds (see ~ pm5.6dc ). (Contributed by Jim Kingdon,
|
|
4-Aug-2018.) $)
|
|
pm5.6r $p |- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) -> ch ) ) $=
|
|
( wo wi wn pm2.53 imim2i impd ) ABCDZEABFZCJKCEABCGHI $.
|
|
|
|
${
|
|
orcanai.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Change disjunction in consequent to conjunction in antecedent.
|
|
(Contributed by NM, 8-Jun-1994.) $)
|
|
orcanai $p |- ( ( ph /\ -. ps ) -> ch ) $=
|
|
( wn ord imp ) ABECABCDFG $.
|
|
$}
|
|
|
|
${
|
|
intnan.1 $e |- -. ph $.
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
16-Sep-1993.) $)
|
|
intnan $p |- -. ( ps /\ ph ) $=
|
|
( wa simpr mto ) BADACBAEF $.
|
|
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
3-Apr-1995.) $)
|
|
intnanr $p |- -. ( ph /\ ps ) $=
|
|
( wa simpl mto ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
intnand.1 $e |- ( ph -> -. ps ) $.
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
10-Jul-2005.) $)
|
|
intnand $p |- ( ph -> -. ( ch /\ ps ) ) $=
|
|
( wa simpr nsyl ) ABCBEDCBFG $.
|
|
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
10-Jul-2005.) $)
|
|
intnanrd $p |- ( ph -> -. ( ps /\ ch ) ) $=
|
|
( wa simpl nsyl ) ABBCEDBCFG $.
|
|
$}
|
|
|
|
$( A conjunction of two decidable propositions is decidable. (Contributed by
|
|
Jim Kingdon, 12-Apr-2018.) $)
|
|
dcan $p |- ( DECID ph -> ( DECID ps -> DECID ( ph /\ ps ) ) ) $=
|
|
( wa wn wo simpl intnanrd orim2i simpr intnand olcd jaoi df-dc anbi12i andi
|
|
wdc andir orbi1i 3bitri 3imtr4i ex ) APZBPZABCZPZUDADZBCZEZAUFEZBDZCZEZUDUD
|
|
DZEZUBUCCZUEUHUNUKUGUMUDUGABUFBFGHUKUMUDUKBAUIUJIJKLUOUIBUJEZCUIBCZUKEULUBU
|
|
IUCUPAMBMNUIBUJOUQUHUKAUFBQRSUDMTUA $.
|
|
|
|
$( A disjunction of two decidable propositions is decidable. (Contributed by
|
|
Jim Kingdon, 21-Apr-2018.) $)
|
|
dcor $p |- ( DECID ph -> ( DECID ps -> DECID ( ph \/ ps ) ) ) $=
|
|
( wdc wn wo wi df-dc orc orcd sylibr a1d wa olc adantl ioran biimpri jaodan
|
|
olcd sylan2b ex jaoi sylbi ) ACAADZEBCZABEZCZFZAGAUGUCAUFUDAUEUEDZEZUFAUEUH
|
|
ABHIUEGZJKUCUDUFUDUCBBDZEUFBGUCBUFUKUCBLZUIUFULUEUHBUEUCBAMNIUJJUCUKLZUIUFU
|
|
MUHUEUHUMABOPRUJJQSTUAUB $.
|
|
|
|
$( An equivalence of two decidable propositions is decidable. (Contributed
|
|
by Jim Kingdon, 12-Apr-2018.) $)
|
|
dcbi $p |- ( DECID ph -> ( DECID ps -> DECID ( ph <-> ps ) ) ) $=
|
|
( wdc wi wa wb dcim com12 dcan syl6c dfbi2 dcbii syl6ibr ) ACZBCZABDZBADZEZ
|
|
CZABFZCNOPCQCZSABGONUABAGHPQIJTRABKLM $.
|
|
|
|
$( Express conjunction in terms of implication. The forward direction,
|
|
~ annimim , is valid for all propositions, but as an equivalence, it
|
|
requires a decidability condition. (Contributed by Jim Kingdon,
|
|
25-Apr-2018.) $)
|
|
annimdc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) ) $=
|
|
( wdc wn wa wi wb imandc adantl dcim imp dcn dcan syl5 con2bidc sylc mpbid
|
|
ex ) ACZBCZABDZEZABFZDGZSTEZUCUBDGZUDTUFSABHIUEUCCZUBCZUFUDGSTUGABJKSTUHTUA
|
|
CSUHBLAUAMNKUCUBOPQR $.
|
|
|
|
$( Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions.
|
|
(Contributed by Jim Kingdon, 2-May-2018.) $)
|
|
pm4.55dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) ) $=
|
|
( wdc wn wa wo wb pm4.54dc imp dcn anim2i dcor syl wi dcan con2bidc biimprd
|
|
jca mpd bicomd ex ) ACZBCZADZBEZDZABDZFZGUBUCEZUHUFUIUEUHDGZUHUFGZUBUCUJABH
|
|
IUIUKUJUIUHCZUECZEUKUJGZUIULUMUIUBUGCZEULUCUOUBBJKUBUOULAUGLIMUBUCUMUBUDCUC
|
|
UMNAJUDBOMIRULUMUNUHUEPIMQSTUA $.
|
|
|
|
$( Disjunction in terms of conjunction (De Morgan's law), for decidable
|
|
propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120.
|
|
(Contributed by Jim Kingdon, 13-Dec-2021.) $)
|
|
orandc $p |- ( ( DECID ph /\ DECID ps ) ->
|
|
( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) ) $=
|
|
( wdc wa wn wo pm4.56 dcn adantr adantl dcan sylc dcor imp con2bidc mpbii
|
|
wb ) ACZBCZDZAEZBEZDZABFZEQZUDUCEQZABGTUCCZUDCZUEUFQTUACZUBCZUGRUISAHISUJRB
|
|
HJUAUBKLRSUHABMNUCUDOLP $.
|
|
|
|
${
|
|
mpbiran.1 $e |- ps $.
|
|
mpbiran.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach truth from conjunction in biconditional. (Contributed by NM,
|
|
27-Feb-1996.) (Revised by NM, 9-Jan-2015.) $)
|
|
mpbiran $p |- ( ph <-> ch ) $=
|
|
( wa biantrur bitr4i ) ABCFCEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
mpbiran2.1 $e |- ch $.
|
|
mpbiran2.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach truth from conjunction in biconditional. (Contributed by NM,
|
|
22-Feb-1996.) (Revised by NM, 9-Jan-2015.) $)
|
|
mpbiran2 $p |- ( ph <-> ps ) $=
|
|
( wa biantru bitr4i ) ABCFBECBDGH $.
|
|
$}
|
|
|
|
${
|
|
mpbir2an.1 $e |- ps $.
|
|
mpbir2an.2 $e |- ch $.
|
|
mpbiran2an.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
10-May-2005.) (Revised by NM, 9-Jan-2015.) $)
|
|
mpbir2an $p |- ph $=
|
|
( mpbiran mpbir ) ACEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpbi2and.1 $e |- ( ph -> ps ) $.
|
|
mpbi2and.2 $e |- ( ph -> ch ) $.
|
|
mpbi2and.3 $e |- ( ph -> ( ( ps /\ ch ) <-> th ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
mpbi2and $p |- ( ph -> th ) $=
|
|
( wa jca mpbid ) ABCHDABCEFIGJ $.
|
|
$}
|
|
|
|
${
|
|
mpbir2and.1 $e |- ( ph -> ch ) $.
|
|
mpbir2and.2 $e |- ( ph -> th ) $.
|
|
mpbir2and.3 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
mpbir2and $p |- ( ph -> ps ) $=
|
|
( wa jca mpbird ) ABCDHACDEFIGJ $.
|
|
$}
|
|
|
|
$( Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition.
|
|
(Contributed by Jim Kingdon, 12-May-2018.) $)
|
|
pm5.62dc $p |- ( DECID ps ->
|
|
( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) ) $=
|
|
( wdc wn wo wa wb df-dc ordir simplbi simplbi2 com12 impbid2 sylbi ) BCBBDZ
|
|
EZABFOEZAOEZGBHPQRQRPABOIZJRPQQRPSKLMN $.
|
|
|
|
$( Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition.
|
|
(Contributed by Jim Kingdon, 12-May-2018.) $)
|
|
pm5.63dc $p |- ( DECID ph ->
|
|
( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) ) $=
|
|
( wdc wo wn wa wi df-dc ordi simplbi2 sylbi simprbi impbid1 ) ACZABDZAAEZBF
|
|
DZNAPDZOQGAHQROAPBIZJKQROSLM $.
|
|
|
|
${
|
|
bianfi.1 $e |- -. ph $.
|
|
$( A wff conjoined with falsehood is false. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
|
|
bianfi $p |- ( ph <-> ( ps /\ ph ) ) $=
|
|
( wa intnan 2false ) ABADCABCEF $.
|
|
$}
|
|
|
|
${
|
|
bianfd.1 $e |- ( ph -> -. ps ) $.
|
|
$( A wff conjoined with falsehood is false. (Contributed by NM,
|
|
27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.) $)
|
|
bianfd $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
|
|
( wa intnanrd 2falsed ) ABBCEDABCDFG $.
|
|
$}
|
|
|
|
$( Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
|
|
pm4.43 $p |- ( ph <-> ( ( ph \/ ps ) /\ ( ph \/ -. ps ) ) ) $=
|
|
( wn wa wo pm3.24 biorfi ordi bitri ) AABBCZDZEABEAJEDKABFGABJHI $.
|
|
|
|
$( Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.82 $p |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph ) $=
|
|
( wi wn wa pm2.65 imp pm2.21 jca impbii ) ABCZABDZCZEADZKMNABFGNKMABHALHIJ
|
|
$.
|
|
|
|
$( Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions.
|
|
As with other case elimination theorems, like ~ pm2.61dc , it only holds
|
|
for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.) $)
|
|
pm4.83dc $p |- ( DECID ph ->
|
|
( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) ) $=
|
|
( wdc wi wn wa wo df-dc pm3.44 com12 sylbi ax-1 jca impbid1 ) ACZABDZAEZBDZ
|
|
FZBOAQGZSBDAHSTBBAQIJKBPRBALBQLMN $.
|
|
|
|
$( A transitive law of equivalence. Compare Theorem *4.22 of
|
|
[WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) $)
|
|
biantr $p |- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) ) $=
|
|
( wb id bibi2d biimparc ) CBDZACDABDHCBAHEFG $.
|
|
|
|
$( Disjunction distributes over the biconditional, for a decidable
|
|
proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On
|
|
calculational proofs" (1998),
|
|
~ http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384 .
|
|
(Contributed by Jim Kingdon, 2-Apr-2018.) $)
|
|
orbididc $p |- ( DECID ph -> ( ( ph \/ ( ps <-> ch ) ) <->
|
|
( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) ) $=
|
|
( wdc wi wo wa wb orimdidc anbi12d dfbi2 orbi2i ordi bitri 3bitr4g ) ADZABC
|
|
EZFZACBEZFZGZABFZACFZEZUCUBEZGABCHZFZUBUCHPRUDTUEABCIACBIJUGAQSGZFUAUFUHABC
|
|
KLAQSMNUBUCKO $.
|
|
|
|
$( Disjunction distributes over the biconditional, for a decidable
|
|
proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This
|
|
theorem is similar to ~ orbididc . (Contributed by Jim Kingdon,
|
|
2-Apr-2018.) $)
|
|
pm5.7dc $p |- ( DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <->
|
|
( ch \/ ( ph <-> ps ) ) ) ) $=
|
|
( wdc wb wo orbididc orcom bibi12i syl6rbb ) CDCABEFCAFZCBFZEACFZBCFZECABGK
|
|
MLNCAHCBHIJ $.
|
|
|
|
$( Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by
|
|
NM, 10-Jan-2005.) $)
|
|
bigolden $p |- ( ( ( ph /\ ps ) <-> ph ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wi wa wb wo pm4.71 pm4.72 bicom 3bitr3ri ) ABCAABDZEBABFEKAEABGABHAKIJ $.
|
|
|
|
$( Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of
|
|
[WhiteheadRussell] p. 120, but where the propositions are decidable. The
|
|
forward direction, ~ pm3.1 , holds for all propositions, but the
|
|
equivalence only holds given decidability. (Contributed by Jim Kingdon,
|
|
21-Apr-2018.) $)
|
|
anordc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) ) $=
|
|
( wdc wa wn wo wb dcan ianordc bicomd a1d con2biddc syld ) ACZBCABDZCZOAEBE
|
|
FZEGABHNQONQOEZGPNRQABIJKLM $.
|
|
|
|
$( Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable
|
|
propositions. The converse, ~ pm3.1 , holds for all propositions, not
|
|
just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.) $)
|
|
pm3.11dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) ) $=
|
|
( wdc wn wo wa wi wb anordc imp biimprd ex ) ACZBCZADBDEDZABFZGMNFPOMNPOHAB
|
|
IJKL $.
|
|
|
|
$( Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable
|
|
propositions. (Contributed by Jim Kingdon, 22-Apr-2018.) $)
|
|
pm3.12dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( -. ph \/ -. ps ) \/ ( ph /\ ps ) ) ) ) $=
|
|
( wdc wn wo wa wi pm3.11dc imp wb dcn dcor syl2im dfordc syl6 mpbird ex ) A
|
|
CZBCZADZBDZEZABFZEZRSFUDUBDUCGZRSUEABHIRSUDUEJZRSUBCZUFRTCSUACUGAKBKTUALMUB
|
|
UCNOIPQ $.
|
|
|
|
$( Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable
|
|
propositions. The converse, ~ pm3.14 , holds for all propositions.
|
|
(Contributed by Jim Kingdon, 22-Apr-2018.) $)
|
|
pm3.13dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) ) $=
|
|
( wdc wn wo wa wi dcn dcor syl2im pm3.11dc con1dc syl6c ) ACZBCZADZBDZEZCZR
|
|
DABFZGTDRGNPCOQCSAHBHPQIJABKRTLM $.
|
|
|
|
$( DN_1 for decidable propositions. Without the decidability conditions,
|
|
DN_1 can serve as a single axiom for Boolean algebra. See
|
|
~ http://www-unix.mcs.anl.gov/~~mccune/papers/basax/v12.pdf .
|
|
(Contributed by Jim Kingdon, 22-Apr-2018.) $)
|
|
dn1dc $p |- ( ( DECID ph /\ ( DECID ps /\ ( DECID ch /\ DECID th ) ) ) ->
|
|
( -. ( -. ( -. ( ph \/ ps ) \/ ch ) \/
|
|
-. ( ph \/ -. ( -. ch \/ -. ( ch \/ th ) ) ) ) <-> ch ) ) $=
|
|
( wo wn wa wdc wi pm2.45 imnan mpbi biorfi wb dcor imp anordc sylc dcn syl
|
|
orcom ordir 3bitri pm4.45 simprrl ad2antll syl5bb orbi2d anbi2d syl6 syldan
|
|
adantrr bitrd syl5rbb ) CABEZFZCEZACEZGZAHZBHZCHZDHZGZGZGZUQFACFZCDEZFZEZFZ
|
|
EZFEFZCCUPAGZEVNCEUSVNCUPAFIVNFABJUPAKLMCVNUAUPACUBUCVFUSUQVLGZVMVFURVLUQVF
|
|
CVKACCVHGZVFVKCDUDVFVBVHHZVPVKNUTVAVBVCUEZVDVQUTVAVBVCVQCDOPUFZCVHQRUGUHUIV
|
|
FUQHZVLHZVOVMNVFUPHZVBVTUTVAWBVDUTVAWBUTVAUOHWBABOUOSUJPULVRUPCORUTVEVKHZWA
|
|
VFVJHZWCVFVGHZVIHZWDVFVBWEVRCSTVFVQWFVSVHSTVGVIORVJSTUTWCWAAVKOPUKUQVLQRUMU
|
|
N $.
|
|
|
|
$( Decidable proposition version of theorem *5.71 of [WhiteheadRussell]
|
|
p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for
|
|
decidability by Jim Kingdon, 19-Apr-2018.) $)
|
|
pm5.71dc $p |- ( DECID ps ->
|
|
( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) ) $=
|
|
( wn wo wa wb wi wdc orel2 orc impbid1 anbi1d a1i pm2.21 pm5.32rd jadc ) BC
|
|
DZABEZCFACFGZBDZTHBIUASACUASABAJABKLMNRCSACSAGOPQ $.
|
|
|
|
$( Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 23-Dec-2012.) $)
|
|
pm5.75 $p |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) ->
|
|
( ( ph /\ -. ps ) <-> ch ) ) $=
|
|
( wo wb wn wa wi anbi1 anbi1i pm5.61 syl6bb pm4.71 biimpi bicomd sylan9bbr
|
|
orcom bitri ) ABCDZEZABFZGZCUAGZCUAHZCTUBSUAGZUCASUAIUECBDZUAGUCSUFUABCQJCB
|
|
KRLUDCUCUDCUCECUAMNOP $.
|
|
|
|
$( Removal of conjunct from one side of an equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
bimsc1 $p |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) )
|
|
-> ( ch <-> ph ) ) $=
|
|
( wi wa wb simpr ancr impbid2 bibi2d biimpa ) ABDZCBAEZFCAFLMACLMABAGABHIJK
|
|
$.
|
|
|
|
${
|
|
ccase.1 $e |- ( ( ph /\ ps ) -> ta ) $.
|
|
ccase.2 $e |- ( ( ch /\ ps ) -> ta ) $.
|
|
ccase.3 $e |- ( ( ph /\ th ) -> ta ) $.
|
|
ccase.4 $e |- ( ( ch /\ th ) -> ta ) $.
|
|
$( Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
|
|
(Proof shortened by Wolf Lammen, 6-Jan-2013.) $)
|
|
ccase $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
|
|
( wo jaoian jaodan ) ACJBEDABECFGKADECHIKL $.
|
|
$}
|
|
|
|
${
|
|
ccased.1 $e |- ( ph -> ( ( ps /\ ch ) -> et ) ) $.
|
|
ccased.2 $e |- ( ph -> ( ( th /\ ch ) -> et ) ) $.
|
|
ccased.3 $e |- ( ph -> ( ( ps /\ ta ) -> et ) ) $.
|
|
ccased.4 $e |- ( ph -> ( ( th /\ ta ) -> et ) ) $.
|
|
$( Deduction for combining cases. (Contributed by NM, 9-May-2004.) $)
|
|
ccased $p |- ( ph -> ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> et ) ) $=
|
|
( wo wa wi com12 ccase ) BDKCEKLAFBCDEAFMABCLFGNADCLFHNABELFINADELFJNON
|
|
$.
|
|
$}
|
|
|
|
${
|
|
ccase2.1 $e |- ( ( ph /\ ps ) -> ta ) $.
|
|
ccase2.2 $e |- ( ch -> ta ) $.
|
|
ccase2.3 $e |- ( th -> ta ) $.
|
|
$( Inference for combining cases. (Contributed by NM, 29-Jul-1999.) $)
|
|
ccase2 $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
|
|
( adantr adantl ccase ) ABCDEFCEBGIDEAHJDECHJK $.
|
|
$}
|
|
|
|
${
|
|
niabn.1 $e |- ph $.
|
|
$( Miscellaneous inference relating falsehoods. (Contributed by NM,
|
|
31-Mar-1994.) $)
|
|
niabn $p |- ( -. ps -> ( ( ch /\ ps ) <-> -. ph ) ) $=
|
|
( wa wn simpr pm2.24i pm5.21ni ) CBEBAFCBGABDHI $.
|
|
$}
|
|
|
|
$( Alternate version of ~ dedlema . (Contributed by NM, 2-Apr-1994.) (Proof
|
|
shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen,
|
|
4-Dec-2012.) $)
|
|
dedlem0a $p |- ( ph -> ( ps <-> ( ( ch -> ph ) -> ( ps /\ ph ) ) ) ) $=
|
|
( wa wi iba wb ax-1 biimt syl bitrd ) ABBADZCAEZLEZABFAMLNGACHMLIJK $.
|
|
|
|
$( Lemma for ~ iftrue . (Contributed by NM, 26-Jun-2002.) (Proof shortened
|
|
by Andrew Salmon, 7-May-2011.) $)
|
|
dedlema $p |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
|
|
( wa wn wo orc expcom wi simpl a1i pm2.24 adantld jaod impbid ) ABBADZCAEZD
|
|
ZFZBASPRGHAPBRPBIABAJKAQBCABLMNO $.
|
|
|
|
$( Lemma for ~ iffalse . (Contributed by NM, 15-May-1999.) (Proof shortened
|
|
by Andrew Salmon, 7-May-2011.) $)
|
|
dedlemb $p |- ( -. ph -> ( ch <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
|
|
( wn wa wo olc expcom pm2.21 adantld wi simpl a1i jaod impbid ) ADZCBAEZCPE
|
|
ZFZCPSRQGHPQCRPACBACIJRCKPCPLMNO $.
|
|
|
|
$( One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed
|
|
by Jim Kingdon, 4-Aug-2018.) $)
|
|
pm4.42r $p |- ( ( ( ph /\ ps ) \/ ( ph /\ -. ps ) ) -> ph ) $=
|
|
( wa wn simpl jaoi ) ABCAABDZCABEAGEF $.
|
|
|
|
${
|
|
ninba.1 $e |- ph $.
|
|
$( Miscellaneous inference relating falsehoods. (Contributed by NM,
|
|
31-Mar-1994.) $)
|
|
ninba $p |- ( -. ps -> ( -. ph <-> ( ch /\ ps ) ) ) $=
|
|
( wn wa niabn bicomd ) BECBFAEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
prlem1.1 $e |- ( ph -> ( et <-> ch ) ) $.
|
|
prlem1.2 $e |- ( ps -> -. th ) $.
|
|
$( A specialized lemma for set theory (to derive the Axiom of Pairing).
|
|
(Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon,
|
|
13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
|
|
prlem1 $p |- ( ph -> ( ps ->
|
|
( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) ) $=
|
|
( wa wo wi biimprd adantld pm2.21d adantrd jaao ex ) ABBCIZDEIZJFKARFBSAC
|
|
FBAFCGLMBDFEBDFHNOPQ $.
|
|
$}
|
|
|
|
$( A specialized lemma for set theory (to derive the Axiom of Pairing).
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
|
|
prlem2 $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
|
|
( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) ) $=
|
|
( wa wo simpl orim12i pm4.71ri ) ABEZCDEZFACFJAKCABGCDGHI $.
|
|
|
|
${
|
|
oplem1.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
oplem1.2 $e |- ( ph -> ( th \/ ta ) ) $.
|
|
oplem1.3 $e |- ( ps <-> th ) $.
|
|
oplem1.4 $e |- ( ch -> ( th <-> ta ) ) $.
|
|
$( A specialized lemma for set theory (ordered pair theorem). (Contributed
|
|
by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
|
|
(Proof shortened by Mario Carneiro, 2-Feb-2015.) $)
|
|
oplem1 $p |- ( ph -> ps ) $=
|
|
( wo idd wi ax-1 biimprcd jaoi syl syl6ibr jaod mpd ) ABCJBFABBCABKACDBAD
|
|
EJCDLZGDTEDCMCDEINOPHQRS $.
|
|
$}
|
|
|
|
$( Lemma used in construction of real numbers. (Contributed by NM,
|
|
4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rnlem $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) ) $=
|
|
( wa an4 biimpi an42 biimpri jca adantl impbii ) ABECDEEZACEBDEEZADEBCEEZEM
|
|
NOMNABCDFGOMADBCHZIJOMNOMPGKL $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Abbreviated conjunction and disjunction of three wff's
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff definition to include 3-way disjunction ('or'). $)
|
|
w3o $a wff ( ph \/ ps \/ ch ) $.
|
|
$( Extend wff definition to include 3-way conjunction ('and'). $)
|
|
w3a $a wff ( ph /\ ps /\ ch ) $.
|
|
|
|
$( These abbreviations help eliminate parentheses to aid readability. $)
|
|
|
|
$( Define disjunction ('or') of 3 wff's. Definition *2.33 of
|
|
[WhiteheadRussell] p. 105. This abbreviation reduces the number of
|
|
parentheses and emphasizes that the order of bracketing is not important
|
|
by virtue of the associative law ~ orass . (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
df-3or $a |- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) ) $.
|
|
|
|
$( Define conjunction ('and') of 3 wff.s. Definition *4.34 of
|
|
[WhiteheadRussell] p. 118. This abbreviation reduces the number of
|
|
parentheses and emphasizes that the order of bracketing is not important
|
|
by virtue of the associative law ~ anass . (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
df-3an $a |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) $.
|
|
|
|
$( Associative law for triple disjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3orass $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( w3o wo df-3or orass bitri ) ABCDABECEABCEEABCFABCGH $.
|
|
|
|
$( Associative law for triple conjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3anass $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
|
|
( w3a wa df-3an anass bitri ) ABCDABECEABCEEABCFABCGH $.
|
|
|
|
$( Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) $)
|
|
3anrot $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ch /\ ph ) ) $=
|
|
( wa w3a ancom 3anass df-3an 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.
|
|
|
|
$( Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3orrot $p |- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ch \/ ph ) ) $=
|
|
( wo w3o orcom 3orass df-3or 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.
|
|
|
|
$( Commutation law for triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3ancoma $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) ) $=
|
|
( wa w3a ancom anbi1i df-3an 3bitr4i ) ABDZCDBADZCDABCEBACEJKCABFGABCHBACHI
|
|
$.
|
|
|
|
$( Commutation law for triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3ancomb $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ch /\ ps ) ) $=
|
|
( w3a 3ancoma 3anrot bitri ) ABCDBACDACBDABCEBACFG $.
|
|
|
|
$( Commutation law for triple disjunction. (Contributed by Scott Fenton,
|
|
20-Apr-2011.) $)
|
|
3orcomb $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ch \/ ps ) ) $=
|
|
( wo w3o orcom orbi2i 3orass 3bitr4i ) ABCDZDACBDZDABCEACBEJKABCFGABCHACBHI
|
|
$.
|
|
|
|
$( Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.) $)
|
|
3anrev $p |- ( ( ph /\ ps /\ ch ) <-> ( ch /\ ps /\ ph ) ) $=
|
|
( w3a 3ancoma 3anrot bitr4i ) ABCDBACDCBADABCECBAFG $.
|
|
|
|
$( Convert triple conjunction to conjunction, then commute. (Contributed by
|
|
Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
3anan32 $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
|
|
( w3a wa df-3an an32 bitri ) ABCDABECEACEBEABCFABCGH $.
|
|
|
|
$( Convert triple conjunction to conjunction, then commute. (Contributed by
|
|
Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
3anan12 $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
|
|
( w3a wa 3ancoma 3anass bitri ) ABCDBACDBACEEABCFBACGH $.
|
|
|
|
$( Distribution of triple conjunction over conjunction. (Contributed by
|
|
David A. Wheeler, 4-Nov-2018.) $)
|
|
anandi3 $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
|
|
$=
|
|
( w3a wa 3anass anandi bitri ) ABCDABCEEABEACEEABCFABCGH $.
|
|
|
|
$( Distribution of triple conjunction over conjunction. (Contributed by
|
|
David A. Wheeler, 4-Nov-2018.) $)
|
|
anandi3r $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ch /\ ps ) ) )
|
|
$=
|
|
( w3a wa 3anan32 anandir bitri ) ABCDACEBEABECBEEABCFACBGH $.
|
|
|
|
$( Negated triple disjunction as triple conjunction. (Contributed by Scott
|
|
Fenton, 19-Apr-2011.) $)
|
|
3ioran $p |- ( -. ( ph \/ ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) ) $=
|
|
( wo wn wa w3o w3a ioran anbi1i df-3or xchnxbir df-3an 3bitr4i ) ABDZEZCEZF
|
|
ZAEZBEZFZQFABCGZESTQHPUAQABIJOCDRUBOCIABCKLSTQMN $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3simpa $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ps ) ) $=
|
|
( w3a wa df-3an simplbi ) ABCDABECABCFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3simpb $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ch ) ) $=
|
|
( w3a wa 3ancomb 3simpa sylbi ) ABCDACBDACEABCFACBGH $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|
|
(Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3simpc $p |- ( ( ph /\ ps /\ ch ) -> ( ps /\ ch ) ) $=
|
|
( w3a wa 3anrot 3simpa sylbi ) ABCDBCADBCEABCFBCAGH $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp1 $p |- ( ( ph /\ ps /\ ch ) -> ph ) $=
|
|
( w3a 3simpa simpld ) ABCDABABCEF $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp2 $p |- ( ( ph /\ ps /\ ch ) -> ps ) $=
|
|
( w3a 3simpa simprd ) ABCDABABCEF $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp3 $p |- ( ( ph /\ ps /\ ch ) -> ch ) $=
|
|
( w3a 3simpc simprd ) ABCDBCABCEF $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ph ) $=
|
|
( w3a simp1 adantr ) ABCEADABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl2 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ps ) $=
|
|
( w3a simp2 adantr ) ABCEBDABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl3 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ch ) $=
|
|
( w3a simp3 adantr ) ABCECDABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr1 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ps ) $=
|
|
( w3a simp1 adantl ) BCDEBABCDFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ch ) $=
|
|
( w3a simp2 adantl ) BCDECABCDFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr3 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> th ) $=
|
|
( w3a simp3 adantl ) BCDEDABCDFG $.
|
|
|
|
${
|
|
3simp1i.1 $e |- ( ph /\ ps /\ ch ) $.
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp1i $p |- ph $=
|
|
( w3a simp1 ax-mp ) ABCEADABCFG $.
|
|
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp2i $p |- ps $=
|
|
( w3a simp2 ax-mp ) ABCEBDABCFG $.
|
|
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp3i $p |- ch $=
|
|
( w3a simp3 ax-mp ) ABCECDABCFG $.
|
|
$}
|
|
|
|
${
|
|
3simp1d.1 $e |- ( ph -> ( ps /\ ch /\ th ) ) $.
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp1d $p |- ( ph -> ps ) $=
|
|
( w3a simp1 syl ) ABCDFBEBCDGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp2d $p |- ( ph -> ch ) $=
|
|
( w3a simp2 syl ) ABCDFCEBCDGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp3d $p |- ( ph -> th ) $=
|
|
( w3a simp3 syl ) ABCDFDEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
3simp1bi.1 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp1bi $p |- ( ph -> ps ) $=
|
|
( w3a biimpi simp1d ) ABCDABCDFEGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp2bi $p |- ( ph -> ch ) $=
|
|
( w3a biimpi simp2d ) ABCDABCDFEGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp3bi $p |- ( ph -> th ) $=
|
|
( w3a biimpi simp3d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
3adant.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant1 $p |- ( ( th /\ ph /\ ps ) -> ch ) $=
|
|
( w3a wa 3simpc syl ) DABFABGCDABHEI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant2 $p |- ( ( ph /\ th /\ ps ) -> ch ) $=
|
|
( w3a wa 3simpb syl ) ADBFABGCADBHEI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant3 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( w3a wa 3simpa syl ) ABDFABGCABDHEI $.
|
|
$}
|
|
|
|
${
|
|
3ad2ant.1 $e |- ( ph -> ch ) $.
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant1 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( adantr 3adant2 ) ADCBACDEFG $.
|
|
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant2 $p |- ( ( ps /\ ph /\ th ) -> ch ) $=
|
|
( adantr 3adant1 ) ADCBACDEFG $.
|
|
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant3 $p |- ( ( ps /\ th /\ ph ) -> ch ) $=
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( adantl 3adant1 ) DACBACDEFG $.
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$}
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp1l $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ph ) $=
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( wa simpl 3ad2ant1 ) ABECADABFG $.
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp1r $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ps ) $=
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( wa simpr 3ad2ant1 ) ABECBDABFG $.
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp2l $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ps ) $=
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( wa simpl 3ad2ant2 ) BCEABDBCFG $.
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp2r $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ch ) $=
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( wa simpr 3ad2ant2 ) BCEACDBCFG $.
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp3l $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> ch ) $=
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( wa simpl 3ad2ant3 ) CDEACBCDFG $.
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$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
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simp3r $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> th ) $=
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( wa simpr 3ad2ant3 ) CDEADBCDFG $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp11 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ph ) $=
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( w3a simp1 3ad2ant1 ) ABCFDAEABCGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp12 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ps ) $=
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( w3a simp2 3ad2ant1 ) ABCFDBEABCGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp13 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ch ) $=
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( w3a simp3 3ad2ant1 ) ABCFDCEABCGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp21 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ps ) $=
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( w3a simp1 3ad2ant2 ) BCDFABEBCDGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp22 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ch ) $=
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( w3a simp2 3ad2ant2 ) BCDFACEBCDGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp23 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> th ) $=
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( w3a simp3 3ad2ant2 ) BCDFADEBCDGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp31 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ch ) $=
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( w3a simp1 3ad2ant3 ) CDEFACBCDEGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp32 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> th ) $=
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( w3a simp2 3ad2ant3 ) CDEFADBCDEGH $.
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$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
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simp33 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ta ) $=
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( w3a simp3 3ad2ant3 ) CDEFAEBCDEGH $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simpll1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ph ) $=
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( w3a wa simpl1 adantr ) ABCFDGAEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simpll2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ps ) $=
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( w3a wa simpl2 adantr ) ABCFDGBEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simpll3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ch ) $=
|
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( w3a wa simpl3 adantr ) ABCFDGCEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simplr1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ph ) $=
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( w3a wa simpr1 adantr ) DABCFGAEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simplr2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ps ) $=
|
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( w3a wa simpr2 adantr ) DABCFGBEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simplr3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ch ) $=
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( w3a wa simpr3 adantr ) DABCFGCEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprl1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
|
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( w3a wa simpl1 adantl ) ABCFDGAEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprl2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( w3a wa simpl2 adantl ) ABCFDGBEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprl3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
|
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( w3a wa simpl3 adantl ) ABCFDGCEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprr1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
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( w3a wa simpr1 adantl ) DABCFGAEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprr2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
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( w3a wa simpr2 adantl ) DABCFGBEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simprr3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
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( w3a wa simpr3 adantl ) DABCFGCEDABCHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl1l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ph ) $=
|
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( wa w3a simp1l adantr ) ABFCDGAEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simpl1r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ps ) $=
|
|
( wa w3a simp1r adantr ) ABFCDGBEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
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simpl2l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ph ) $=
|
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( wa w3a simp2l adantr ) CABFDGAECABDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl2r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ps ) $=
|
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( wa w3a simp2r adantr ) CABFDGBECABDHI $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl3l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
|
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( wa w3a simp3l adantr ) CDABFGAECDABHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl3r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
|
|
( wa w3a simp3r adantr ) CDABFGBECDABHI $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simpr1l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
|
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( wa w3a simp1l adantl ) ABFCDGAEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simpr1r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
|
|
( wa w3a simp1r adantl ) ABFCDGBEABCDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simpr2l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
|
|
( wa w3a simp2l adantl ) CABFDGAECABDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr2r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
|
|
( wa w3a simp2r adantl ) CABFDGBECABDHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr3l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa w3a simp3l adantl ) CDABFGAECDABHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simpr3r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa w3a simp3r adantl ) CDABFGBECDABHI $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simp1ll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ph ) $=
|
|
( wa simpll 3ad2ant1 ) ABFCFDAEABCGH $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simp1lr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ps ) $=
|
|
( wa simplr 3ad2ant1 ) ABFCFDBEABCGH $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
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simp1rl $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ph ) $=
|
|
( wa simprl 3ad2ant1 ) CABFFDAECABGH $.
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1rr $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ps ) $=
|
|
( wa simprr 3ad2ant1 ) CABFFDBECABGH $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2ll $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ph ) $=
|
|
( wa simpll 3ad2ant2 ) ABFCFDAEABCGH $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2lr $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ps ) $=
|
|
( wa simplr 3ad2ant2 ) ABFCFDBEABCGH $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2rl $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
|
|
( wa simprl 3ad2ant2 ) CABFFDAECABGH $.
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2rr $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
|
|
( wa simprr 3ad2ant2 ) CABFFDBECABGH $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3ll $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ph ) $=
|
|
( wa simpll 3ad2ant3 ) ABFCFDAEABCGH $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3lr $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ps ) $=
|
|
( wa simplr 3ad2ant3 ) ABFCFDBEABCGH $.
|
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|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3rl $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa simprl 3ad2ant3 ) CABFFDAECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3rr $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa simprr 3ad2ant3 ) CABFFDBECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl11 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ph ) $=
|
|
( w3a simp11 adantr ) ABCGDEGAFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl12 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ps ) $=
|
|
( w3a simp12 adantr ) ABCGDEGBFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl13 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ch ) $=
|
|
( w3a simp13 adantr ) ABCGDEGCFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl21 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ph ) $=
|
|
( w3a simp21 adantr ) DABCGEGAFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl22 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ps ) $=
|
|
( w3a simp22 adantr ) DABCGEGBFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl23 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ch ) $=
|
|
( w3a simp23 adantr ) DABCGEGCFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl31 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
|
|
( w3a simp31 adantr ) DEABCGGAFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl32 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
|
|
( w3a simp32 adantr ) DEABCGGBFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl33 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
|
|
( w3a simp33 adantr ) DEABCGGCFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr11 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
|
|
( w3a simp11 adantl ) ABCGDEGAFABCDEHI $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr12 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
|
|
( w3a simp12 adantl ) ABCGDEGBFABCDEHI $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr13 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
|
|
( w3a simp13 adantl ) ABCGDEGCFABCDEHI $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr21 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
|
|
( w3a simp21 adantl ) DABCGEGAFDABCEHI $.
|
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|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr22 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
|
|
( w3a simp22 adantl ) DABCGEGBFDABCEHI $.
|
|
|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr23 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
|
|
( w3a simp23 adantl ) DABCGEGCFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr31 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a simp31 adantl ) DEABCGGAFDEABCHI $.
|
|
|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr32 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a simp32 adantl ) DEABCGGBFDEABCHI $.
|
|
|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr33 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a simp33 adantl ) DEABCGGCFDEABCHI $.
|
|
|
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$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant1 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant1 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant1 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant1 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant1 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant1 ) DABCGHECFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant2 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant2 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant2 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant2 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant2 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant2 ) DABCGHECFDABCIJ $.
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|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l1 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant3 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l2 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant3 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l3 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant3 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r1 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant3 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r2 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant3 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r3 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant3 ) DABCGHECFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp11l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant1 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp11r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant1 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp12l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant1 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp12r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant1 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp13l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant1 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp13r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant1 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp21l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant2 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp21r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant2 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp22l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant2 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp22r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant2 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp23l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant2 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp23r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant2 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp31l $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant3 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp31r $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant3 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp32l $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant3 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp32r $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant3 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp33l $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant3 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp33r $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant3 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp111 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp11 3ad2ant1 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp112 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp12 3ad2ant1 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp113 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp13 3ad2ant1 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp121 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp21 3ad2ant1 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp122 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp22 3ad2ant1 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp123 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp23 3ad2ant1 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp131 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp31 3ad2ant1 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp132 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp32 3ad2ant1 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp133 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp33 3ad2ant1 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp211 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ph ) $=
|
|
( w3a simp11 3ad2ant2 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp212 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ps ) $=
|
|
( w3a simp12 3ad2ant2 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp213 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ch ) $=
|
|
( w3a simp13 3ad2ant2 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp221 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ph ) $=
|
|
( w3a simp21 3ad2ant2 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp222 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ps ) $=
|
|
( w3a simp22 3ad2ant2 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp223 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ch ) $=
|
|
( w3a simp23 3ad2ant2 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp231 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ph ) $=
|
|
( w3a simp31 3ad2ant2 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp232 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ps ) $=
|
|
( w3a simp32 3ad2ant2 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp233 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ch ) $=
|
|
( w3a simp33 3ad2ant2 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp311 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
|
|
( w3a simp11 3ad2ant3 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp312 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
|
|
( w3a simp12 3ad2ant3 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp313 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
|
|
( w3a simp13 3ad2ant3 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp321 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
|
|
( w3a simp21 3ad2ant3 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp322 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
|
|
( w3a simp22 3ad2ant3 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp323 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
|
|
( w3a simp23 3ad2ant3 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp331 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a simp31 3ad2ant3 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp332 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a simp32 3ad2ant3 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp333 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a simp33 3ad2ant3 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
${
|
|
3adantl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl1 $p |- ( ( ( ta /\ ph /\ ps ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpc sylan ) EABGABHCDEABIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl2 $p |- ( ( ( ph /\ ta /\ ps ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpb sylan ) AEBGABHCDAEBIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl3 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpa sylan ) ABEGABHCDABEIFJ $.
|
|
$}
|
|
|
|
${
|
|
3adantr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
|
|
( w3a wa 3simpc sylan2 ) EBCGABCHDEBCIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( w3a wa 3simpb sylan2 ) BECGABCHDBECIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( w3a wa 3simpa sylan2 ) BCEGABCHDBCEIFJ $.
|
|
$}
|
|
|
|
${
|
|
3ad2antl.1 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl1 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
|
|
( adantlr 3adantl2 ) AECDBACDEFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl2 $p |- ( ( ( ps /\ ph /\ ta ) /\ ch ) -> th ) $=
|
|
( adantlr 3adantl1 ) AECDBACDEFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl3 $p |- ( ( ( ps /\ ta /\ ph ) /\ ch ) -> th ) $=
|
|
( adantll 3adantl1 ) EACDBACDEFGH $.
|
|
|
|
$( Deduction adding a conjuncts to antecedent. (Contributed by NM,
|
|
25-Dec-2007.) $)
|
|
3ad2antr1 $p |- ( ( ph /\ ( ch /\ ps /\ ta ) ) -> th ) $=
|
|
( adantrr 3adantr3 ) ACBDEACDBFGH $.
|
|
|
|
$( Deduction adding a conjuncts to antecedent. (Contributed by NM,
|
|
27-Dec-2007.) $)
|
|
3ad2antr2 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( adantrl 3adantr3 ) ABCDEACDBFGH $.
|
|
|
|
$( Deduction adding a conjuncts to antecedent. (Contributed by NM,
|
|
30-Dec-2007.) $)
|
|
3ad2antr3 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( adantrl 3adantr1 ) AECDBACDEFGH $.
|
|
$}
|
|
|
|
${
|
|
3anibar.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ( ch /\ ta ) ) ) $.
|
|
$( Remove a hypothesis from the second member of a biimplication.
|
|
(Contributed by FL, 22-Jul-2008.) $)
|
|
3anibar $p |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $=
|
|
( w3a wa simp3 biantrurd bitr4d ) ABCGZDCEHEFLCEABCIJK $.
|
|
$}
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix1 $p |- ( ph -> ( ph \/ ps \/ ch ) ) $=
|
|
( wo w3o orc 3orass sylibr ) AABCDZDABCEAIFABCGH $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix2 $p |- ( ph -> ( ps \/ ph \/ ch ) ) $=
|
|
( w3o 3mix1 3orrot sylibr ) AACBDBACDACBEBACFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix3 $p |- ( ph -> ( ps \/ ch \/ ph ) ) $=
|
|
( w3o 3mix1 3orrot sylib ) AABCDBCADABCEABCFG $.
|
|
|
|
${
|
|
3mixi.1 $e |- ph $.
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix1i $p |- ( ph \/ ps \/ ch ) $=
|
|
( w3o 3mix1 ax-mp ) AABCEDABCFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix2i $p |- ( ps \/ ph \/ ch ) $=
|
|
( w3o 3mix2 ax-mp ) ABACEDABCFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix3i $p |- ( ps \/ ch \/ ph ) $=
|
|
( w3o 3mix3 ax-mp ) ABCAEDABCFG $.
|
|
$}
|
|
|
|
${
|
|
3mixd.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing triple disjunction. (Contributed by Scott Fenton,
|
|
8-Jun-2011.) $)
|
|
3mix1d $p |- ( ph -> ( ps \/ ch \/ th ) ) $=
|
|
( w3o 3mix1 syl ) ABBCDFEBCDGH $.
|
|
|
|
$( Deduction introducing triple disjunction. (Contributed by Scott Fenton,
|
|
8-Jun-2011.) $)
|
|
3mix2d $p |- ( ph -> ( ch \/ ps \/ th ) ) $=
|
|
( w3o 3mix2 syl ) ABCBDFEBCDGH $.
|
|
|
|
$( Deduction introducing triple disjunction. (Contributed by Scott Fenton,
|
|
8-Jun-2011.) $)
|
|
3mix3d $p |- ( ph -> ( ch \/ th \/ ps ) ) $=
|
|
( w3o 3mix3 syl ) ABCDBFEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
3pm3.2i.1 $e |- ph $.
|
|
3pm3.2i.2 $e |- ps $.
|
|
3pm3.2i.3 $e |- ch $.
|
|
$( Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.) $)
|
|
3pm3.2i $p |- ( ph /\ ps /\ ch ) $=
|
|
( w3a wa pm3.2i df-3an mpbir2an ) ABCGABHCABDEIFABCJK $.
|
|
$}
|
|
|
|
$( ~ pm3.2 for a triple conjunction. (Contributed by Alan Sare,
|
|
24-Oct-2011.) $)
|
|
pm3.2an3 $p |- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) ) $=
|
|
( wa w3a wi pm3.2 ex df-3an bicomi syl8ib ) ABCABDZCDZABCEZABCMFLCGHNMABCIJ
|
|
K $.
|
|
|
|
${
|
|
3jca.1 $e |- ( ph -> ps ) $.
|
|
3jca.2 $e |- ( ph -> ch ) $.
|
|
3jca.3 $e |- ( ph -> th ) $.
|
|
$( Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.) $)
|
|
3jca $p |- ( ph -> ( ps /\ ch /\ th ) ) $=
|
|
( wa w3a jca31 df-3an sylibr ) ABCHDHBCDIABCDEFGJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
3jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
3jcad.3 $e |- ( ph -> ( ps -> ta ) ) $.
|
|
$( Deduction conjoining the consequents of three implications.
|
|
(Contributed by NM, 25-Sep-2005.) $)
|
|
3jcad $p |- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) ) $=
|
|
( w3a wa imp 3jca ex ) ABCDEIABJCDEABCFKABDGKABEHKLM $.
|
|
$}
|
|
|
|
${
|
|
mpbir3an.1 $e |- ps $.
|
|
mpbir3an.2 $e |- ch $.
|
|
mpbir3an.3 $e |- th $.
|
|
mpbir3an.4 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
16-Sep-2011.) (Revised by NM, 9-Jan-2015.) $)
|
|
mpbir3an $p |- ph $=
|
|
( w3a 3pm3.2i mpbir ) ABCDIBCDEFGJHK $.
|
|
$}
|
|
|
|
${
|
|
mpbir3and.1 $e |- ( ph -> ch ) $.
|
|
mpbir3and.2 $e |- ( ph -> th ) $.
|
|
mpbir3and.3 $e |- ( ph -> ta ) $.
|
|
mpbir3and.4 $e |- ( ph -> ( ps <-> ( ch /\ th /\ ta ) ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by
|
|
Mario Carneiro, 11-May-2014.) $)
|
|
mpbir3and $p |- ( ph -> ps ) $=
|
|
( w3a 3jca mpbird ) ABCDEJACDEFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl3anbrc.1 $e |- ( ph -> ps ) $.
|
|
syl3anbrc.2 $e |- ( ph -> ch ) $.
|
|
syl3anbrc.3 $e |- ( ph -> th ) $.
|
|
syl3anbrc.4 $e |- ( ta <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.) $)
|
|
syl3anbrc $p |- ( ph -> ta ) $=
|
|
( w3a 3jca sylibr ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
3anim123i.1 $e |- ( ph -> ps ) $.
|
|
3anim123i.2 $e |- ( ch -> th ) $.
|
|
3anim123i.3 $e |- ( ta -> et ) $.
|
|
$( Join antecedents and consequents with conjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3anim123i $p |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) ) $=
|
|
( w3a 3ad2ant1 3ad2ant2 3ad2ant3 3jca ) ACEJBDFACBEGKCADEHLEAFCIMN $.
|
|
$}
|
|
|
|
${
|
|
3animi.1 $e |- ( ph -> ps ) $.
|
|
$( Add two conjuncts to antecedent and consequent. (Contributed by Jeff
|
|
Hankins, 16-Aug-2009.) $)
|
|
3anim1i $p |- ( ( ph /\ ch /\ th ) -> ( ps /\ ch /\ th ) ) $=
|
|
( id 3anim123i ) ABCCDDECFDFG $.
|
|
|
|
$( Add two conjuncts to antecedent and consequent. (Contributed by AV,
|
|
21-Nov-2019.) $)
|
|
3anim2i $p |- ( ( ch /\ ph /\ th ) -> ( ch /\ ps /\ th ) ) $=
|
|
( id 3anim123i ) CCABDDCFEDFG $.
|
|
|
|
$( Add two conjuncts to antecedent and consequent. (Contributed by Jeff
|
|
Hankins, 19-Aug-2009.) $)
|
|
3anim3i $p |- ( ( ch /\ th /\ ph ) -> ( ch /\ th /\ ps ) ) $=
|
|
( id 3anim123i ) CCDDABCFDFEG $.
|
|
$}
|
|
|
|
${
|
|
bi3.1 $e |- ( ph <-> ps ) $.
|
|
bi3.2 $e |- ( ch <-> th ) $.
|
|
bi3.3 $e |- ( ta <-> et ) $.
|
|
$( Join 3 biconditionals with conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3anbi123i $p |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) ) $=
|
|
( wa w3a anbi12i df-3an 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
|
|
FMN $.
|
|
|
|
$( Join 3 biconditionals with disjunction. (Contributed by NM,
|
|
17-May-1994.) $)
|
|
3orbi123i $p |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) $=
|
|
( wo w3o orbi12i df-3or 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
|
|
FMN $.
|
|
$}
|
|
|
|
${
|
|
3anbi1i.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi1i $p |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) ) $=
|
|
( biid 3anbi123i ) ABCCDDECFDFG $.
|
|
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi2i $p |- ( ( ch /\ ph /\ th ) <-> ( ch /\ ps /\ th ) ) $=
|
|
( biid 3anbi123i ) CCABDDCFEDFG $.
|
|
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi3i $p |- ( ( ch /\ th /\ ph ) <-> ( ch /\ th /\ ps ) ) $=
|
|
( biid 3anbi123i ) CCDDABCFDFEG $.
|
|
$}
|
|
|
|
${
|
|
3imp.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Importation inference. (Contributed by NM, 8-Apr-1994.) $)
|
|
3imp $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( w3a wa df-3an imp31 sylbi ) ABCFABGCGDABCHABCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
3impa.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Importation from double to triple conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impa $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( exp31 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impb.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Importation from double to triple conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impb $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( exp32 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impia.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) $)
|
|
3impia $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( wi ex 3imp ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
3impib.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) $)
|
|
3impib $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( expd 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3exp.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Exportation inference. (Contributed by NM, 30-May-1994.) $)
|
|
3exp $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( w3a pm3.2an3 syl8 ) ABCABCFDABCGEH $.
|
|
|
|
$( Exportation from triple to double conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3expa $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( 3exp imp31 ) ABCDABCDEFG $.
|
|
|
|
$( Exportation from triple to double conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3expb $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( 3exp imp32 ) ABCDABCDEFG $.
|
|
|
|
$( Exportation from triple conjunction. (Contributed by NM,
|
|
19-May-2007.) $)
|
|
3expia $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi 3exp imp ) ABCDFABCDEGH $.
|
|
|
|
$( Exportation from triple conjunction. (Contributed by NM,
|
|
19-May-2007.) $)
|
|
3expib $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( 3exp impd ) ABCDABCDEFG $.
|
|
|
|
$( Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3com12 $p |- ( ( ps /\ ph /\ ch ) -> th ) $=
|
|
( w3a 3ancoma sylbi ) BACFABCFDBACGEH $.
|
|
|
|
$( Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3com13 $p |- ( ( ch /\ ps /\ ph ) -> th ) $=
|
|
( w3a 3anrev sylbi ) CBAFABCFDCBAGEH $.
|
|
|
|
$( Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3com23 $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
|
|
( 3exp com23 3imp ) ACBDABCDABCDEFGH $.
|
|
|
|
$( Commutation in antecedent. Rotate left. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3coml $p |- ( ( ps /\ ch /\ ph ) -> th ) $=
|
|
( 3com23 3com13 ) ACBDABCDEFG $.
|
|
|
|
$( Commutation in antecedent. Rotate right. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3comr $p |- ( ( ch /\ ph /\ ps ) -> th ) $=
|
|
( 3coml ) BCADABCDEFF $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Feb-2008.) $)
|
|
3adant3r1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
|
|
( 3expb 3adantr1 ) ABCDEABCDFGH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
17-Feb-2008.) $)
|
|
3adant3r2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( 3expb 3adantr2 ) ABCDEABCDFGH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
18-Feb-2008.) $)
|
|
3adant3r3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( 3expb 3adantr3 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
3an1rs.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Swap conjuncts. (Contributed by NM, 16-Dec-2007.) $)
|
|
3an1rs $p |- ( ( ( ph /\ ps /\ th ) /\ ch ) -> ta ) $=
|
|
( w3a wi ex 3exp com34 3imp imp ) ABDGCEABDCEHABCDEABCDEHABCGDEFIJKLM $.
|
|
$}
|
|
|
|
${
|
|
3imp1.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
|
$( Importation to left triple conjunction. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3imp1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
|
|
( w3a wi 3imp imp ) ABCGDEABCDEHFIJ $.
|
|
|
|
$( Importation deduction for triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3impd $p |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $=
|
|
( w3a wi com4l 3imp com12 ) BCDGAEBCDAEHABCDEFIJK $.
|
|
|
|
$( Importation to right triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3imp2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
|
|
( w3a 3impd imp ) ABCDGEABCDEFHI $.
|
|
$}
|
|
|
|
${
|
|
3exp1.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Exportation from left triple conjunction. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3exp1 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi w3a ex 3exp ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
3expd.1 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( Exportation deduction for triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3expd $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi w3a com12 3exp com4r ) BCDAEBCDAEGABCDHEFIJK $.
|
|
$}
|
|
|
|
${
|
|
3exp2.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( Exportation from right triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3exp2 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( w3a ex 3expd ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp5o.1 $e |- ( ( ph /\ ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp5o $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi w3a expd 3exp ) ABCDEFHHABCIDEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp516.1 $e |- ( ( ( ph /\ ( ps /\ ch /\ th ) ) /\ ta ) -> et ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp516 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi w3a exp31 3expd ) ABCDEFHABCDIEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp520.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp520 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( w3a wa ex exp5o ) ABCDEFABCHDEIFGJK $.
|
|
$}
|
|
|
|
${
|
|
3anassrs.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
3anassrs $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
|
|
( 3exp2 imp41 ) ABCDEABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
3adant1l.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant1l $p |- ( ( ( ta /\ ph ) /\ ps /\ ch ) -> th ) $=
|
|
( wa 3expb adantll 3impb ) EAGBCDABCGDEABCDFHIJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant1r $p |- ( ( ( ph /\ ta ) /\ ps /\ ch ) -> th ) $=
|
|
( wa 3expb adantlr 3impb ) AEGBCDABCGDEABCDFHIJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant2l $p |- ( ( ph /\ ( ta /\ ps ) /\ ch ) -> th ) $=
|
|
( wa 3com12 3adant1l ) EBGACDBACDEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant2r $p |- ( ( ph /\ ( ps /\ ta ) /\ ch ) -> th ) $=
|
|
( wa 3com12 3adant1r ) BEGACDBACDEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant3l $p |- ( ( ph /\ ps /\ ( ta /\ ch ) ) -> th ) $=
|
|
( wa 3com13 3adant1l ) ECGBADCBADEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant3r $p |- ( ( ph /\ ps /\ ( ch /\ ta ) ) -> th ) $=
|
|
( wa 3com13 3adant1r ) CEGBADCBADEABCDFHIH $.
|
|
$}
|
|
|
|
${
|
|
sylXanc.1 $e |- ( ph -> ps ) $.
|
|
sylXanc.2 $e |- ( ph -> ch ) $.
|
|
sylXanc.3 $e |- ( ph -> th ) $.
|
|
${
|
|
syl12anc.4 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
|
|
1-Aug-2009.) $)
|
|
syl12anc $p |- ( ph -> ta ) $=
|
|
( wa jca32 syl ) ABCDJJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl21anc.4 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
|
|
1-Aug-2009.) $)
|
|
syl21anc $p |- ( ph -> ta ) $=
|
|
( wa jca31 syl ) ABCJDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl111anc.4 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl3anc $p |- ( ph -> ta ) $=
|
|
( w3a 3jca syl ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
sylXanc.4 $e |- ( ph -> ta ) $.
|
|
${
|
|
syl22anc.5 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl22anc $p |- ( ph -> et ) $=
|
|
( wa jca syl12anc ) ABCLDEFABCGHMIJKN $.
|
|
$}
|
|
|
|
${
|
|
syl13anc.5 $e |- ( ( ps /\ ( ch /\ th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl13anc $p |- ( ph -> et ) $=
|
|
( w3a 3jca syl2anc ) ABCDELFGACDEHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl31anc.5 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl31anc $p |- ( ph -> et ) $=
|
|
( w3a 3jca syl2anc ) ABCDLEFABCDGHIMJKN $.
|
|
$}
|
|
|
|
${
|
|
syl112anc.5 $e |- ( ( ps /\ ch /\ ( th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl112anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCDELFGHADEIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl121anc.5 $e |- ( ( ps /\ ( ch /\ th ) /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl121anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCDLEFGACDHIMJKN $.
|
|
$}
|
|
|
|
${
|
|
syl211anc.5 $e |- ( ( ( ps /\ ch ) /\ th /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl211anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCLDEFABCGHMIJKN $.
|
|
$}
|
|
|
|
sylXanc.5 $e |- ( ph -> et ) $.
|
|
${
|
|
syl23anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl23anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl13anc ) ABCNDEFGABCHIOJKLMP $.
|
|
$}
|
|
|
|
${
|
|
syl32anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl32anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl31anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl122anc.6 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl122anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl121anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl212anc.6 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl212anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl211anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl221anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl221anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl211anc ) ABCDENFGHIADEJKOLMP $.
|
|
$}
|
|
|
|
${
|
|
syl113anc.6 $e |- ( ( ps /\ ch /\ ( th /\ ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl113anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDEFNGHIADEFJKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl131anc.6 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl131anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDENFGHACDEIJKOLMP $.
|
|
$}
|
|
|
|
${
|
|
syl311anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl311anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDNEFGABCDHIJOKLMP $.
|
|
$}
|
|
|
|
sylXanc.6 $e |- ( ph -> ze ) $.
|
|
${
|
|
syl33anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl33anc $p |- ( ph -> si ) $=
|
|
( w3a 3jca syl13anc ) ABCDPEFGHABCDIJKQLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl222anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl222anc $p |- ( ph -> si ) $=
|
|
( wa jca syl221anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl123anc.7 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl123anc $p |- ( ph -> si ) $=
|
|
( wa jca syl113anc ) ABCDPEFGHIACDJKQLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl132anc.7 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl132anc $p |- ( ph -> si ) $=
|
|
( wa jca syl131anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl213anc.7 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl213anc $p |- ( ph -> si ) $=
|
|
( wa jca syl113anc ) ABCPDEFGHABCIJQKLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl231anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ze )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl231anc $p |- ( ph -> si ) $=
|
|
( wa jca syl131anc ) ABCPDEFGHABCIJQKLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl312anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl312anc $p |- ( ph -> si ) $=
|
|
( wa jca syl311anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl321anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ze )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl321anc $p |- ( ph -> si ) $=
|
|
( wa jca syl311anc ) ABCDEFPGHIJKAEFLMQNOR $.
|
|
$}
|
|
|
|
sylXanc.7 $e |- ( ph -> si ) $.
|
|
${
|
|
syl133anc.8 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze /\ si ) )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl133anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl131anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl313anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze /\ si ) )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl313anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl311anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl331anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ si )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl331anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl311anc ) ABCDEFGRHIJKLAEFGMNOSPQT $.
|
|
$}
|
|
|
|
${
|
|
syl223anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl223anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl213anc ) ABCDERFGHIJKADELMSNOPQT $.
|
|
$}
|
|
|
|
${
|
|
syl232anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl232anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl231anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl322anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl322anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl321anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
|
|
$}
|
|
|
|
sylXanc.8 $e |- ( ph -> rh ) $.
|
|
${
|
|
syl233anc.9 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl233anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl133anc ) ABCTDEFGHIJABCKLUAMNOPQRSUB $.
|
|
$}
|
|
|
|
${
|
|
syl323anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl323anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl313anc ) ABCDEFTGHIJKLMAEFNOUAPQRSUB $.
|
|
$}
|
|
|
|
${
|
|
syl332anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl332anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl331anc ) ABCDEFGHITJKLMNOPAHIQRUASUB $.
|
|
$}
|
|
|
|
sylXanc.9 $e |- ( ph -> mu ) $.
|
|
${
|
|
syl333anc.10 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze )
|
|
/\ ( si /\ rh /\ mu ) ) -> la ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
10-Mar-2012.) $)
|
|
syl333anc $p |- ( ph -> la ) $=
|
|
( w3a 3jca syl331anc ) ABCDEFGHIJUBKLMNOPQAHIJRSTUCUAUD $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
syl3an1.1 $e |- ( ph -> ps ) $.
|
|
syl3an1.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( w3a 3anim1i syl ) ACDHBCDHEABCDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl3an2.1 $e |- ( ph -> ch ) $.
|
|
syl3an2.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2 $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( wi 3exp syl5 3imp ) BADEACBDEHFBCDEGIJK $.
|
|
$}
|
|
|
|
${
|
|
syl3an3.1 $e |- ( ph -> th ) $.
|
|
syl3an3.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3 $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( 3exp syl7 3imp ) BCAEADBCEFBCDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl3an1b.1 $e |- ( ph <-> ps ) $.
|
|
syl3an1b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1b $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( biimpi syl3an1 ) ABCDEABFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an2b.1 $e |- ( ph <-> ch ) $.
|
|
syl3an2b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2b $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( biimpi syl3an2 ) ABCDEACFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an3b.1 $e |- ( ph <-> th ) $.
|
|
syl3an3b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3b $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( biimpi syl3an3 ) ABCDEADFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an1br.1 $e |- ( ps <-> ph ) $.
|
|
syl3an1br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1br $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( biimpri syl3an1 ) ABCDEBAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an2br.1 $e |- ( ch <-> ph ) $.
|
|
syl3an2br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2br $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( biimpri syl3an2 ) ABCDECAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an3br.1 $e |- ( th <-> ph ) $.
|
|
syl3an3br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3br $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( biimpri syl3an3 ) ABCDEDAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an.1 $e |- ( ph -> ps ) $.
|
|
syl3an.2 $e |- ( ch -> th ) $.
|
|
syl3an.3 $e |- ( ta -> et ) $.
|
|
syl3an.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 13-May-2004.) $)
|
|
syl3an $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( w3a 3anim123i syl ) ACELBDFLGABCDEFHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl3anb.1 $e |- ( ph <-> ps ) $.
|
|
syl3anb.2 $e |- ( ch <-> th ) $.
|
|
syl3anb.3 $e |- ( ta <-> et ) $.
|
|
syl3anb.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) $)
|
|
syl3anb $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( w3a 3anbi123i sylbi ) ACELBDFLGABCDEFHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl3anbr.1 $e |- ( ps <-> ph ) $.
|
|
syl3anbr.2 $e |- ( th <-> ch ) $.
|
|
syl3anbr.3 $e |- ( et <-> ta ) $.
|
|
syl3anbr.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) $)
|
|
syl3anbr $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( bicomi syl3anb ) ABCDEFGBAHLDCILFEJLKM $.
|
|
$}
|
|
|
|
${
|
|
syld3an3.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
syld3an3.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 20-May-2007.) $)
|
|
syld3an3 $p |- ( ( ph /\ ps /\ ch ) -> ta ) $=
|
|
( w3a simp1 simp2 syl3anc ) ABCHABDEABCIABCJFGK $.
|
|
$}
|
|
|
|
${
|
|
syld3an1.1 $e |- ( ( ch /\ ps /\ th ) -> ph ) $.
|
|
syld3an1.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 7-Jul-2008.) $)
|
|
syld3an1 $p |- ( ( ch /\ ps /\ th ) -> ta ) $=
|
|
( 3com13 syld3an3 ) DBCEDBCAECBDAFHABDEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syld3an2.1 $e |- ( ( ph /\ ch /\ th ) -> ps ) $.
|
|
syld3an2.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 20-May-2007.) $)
|
|
syld3an2 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( 3com23 syld3an3 ) ADCEADCBEACDBFHABDEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syl3anl1.1 $e |- ( ph -> ps ) $.
|
|
syl3anl1.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl1 $p |- ( ( ( ph /\ ch /\ th ) /\ ta ) -> et ) $=
|
|
( w3a 3anim1i sylan ) ACDIBCDIEFABCDGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anl2.1 $e |- ( ph -> ch ) $.
|
|
syl3anl2.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl2 $p |- ( ( ( ps /\ ph /\ th ) /\ ta ) -> et ) $=
|
|
( w3a wi ex syl3an2 imp ) BADIEFABCDEFJGBCDIEFHKLM $.
|
|
$}
|
|
|
|
${
|
|
syl3anl3.1 $e |- ( ph -> th ) $.
|
|
syl3anl3.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl3 $p |- ( ( ( ps /\ ch /\ ph ) /\ ta ) -> et ) $=
|
|
( w3a 3anim3i sylan ) BCAIBCDIEFADBCGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anl.1 $e |- ( ph -> ps ) $.
|
|
syl3anl.2 $e |- ( ch -> th ) $.
|
|
syl3anl.3 $e |- ( ta -> et ) $.
|
|
syl3anl.4 $e |- ( ( ( ps /\ th /\ et ) /\ ze ) -> si ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 24-Dec-2006.) $)
|
|
syl3anl $p |- ( ( ( ph /\ ch /\ ta ) /\ ze ) -> si ) $=
|
|
( w3a 3anim123i sylan ) ACEMBDFMGHABCDEFIJKNLO $.
|
|
$}
|
|
|
|
${
|
|
syl3anr1.1 $e |- ( ph -> ps ) $.
|
|
syl3anr1.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 31-Jul-2007.) $)
|
|
syl3anr1 $p |- ( ( ch /\ ( ph /\ th /\ ta ) ) -> et ) $=
|
|
( w3a 3anim1i sylan2 ) ADEICBDEIFABDEGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anr2.1 $e |- ( ph -> th ) $.
|
|
syl3anr2.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Aug-2007.) $)
|
|
syl3anr2 $p |- ( ( ch /\ ( ps /\ ph /\ ta ) ) -> et ) $=
|
|
( w3a ancoms syl3anl2 ) BAEICFABDECFGCBDEIFHJKJ $.
|
|
$}
|
|
|
|
${
|
|
syl3anr3.1 $e |- ( ph -> ta ) $.
|
|
syl3anr3.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 23-Aug-2007.) $)
|
|
syl3anr3 $p |- ( ( ch /\ ( ps /\ th /\ ph ) ) -> et ) $=
|
|
( w3a 3anim3i sylan2 ) BDAICBDEIFAEBDGJHK $.
|
|
$}
|
|
|
|
${
|
|
3impdi.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> th ) $.
|
|
$( Importation inference (undistribute conjunction). (Contributed by NM,
|
|
14-Aug-1995.) $)
|
|
3impdi $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( anandis 3impb ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impdir.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ ps ) ) -> th ) $.
|
|
$( Importation inference (undistribute conjunction). (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impdir $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
|
|
( anandirs 3impa ) ACBDACBDEFG $.
|
|
$}
|
|
|
|
${
|
|
3anidm12.1 $e |- ( ( ph /\ ph /\ ps ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
7-Mar-2008.) $)
|
|
3anidm12 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3expib anabsi5 ) ABCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
3anidm13.1 $e |- ( ( ph /\ ps /\ ph ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
7-Mar-2008.) $)
|
|
3anidm13 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3com23 3anidm12 ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
3anidm23.1 $e |- ( ( ph /\ ps /\ ps ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
1-Feb-2007.) $)
|
|
3anidm23 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3expa anabss3 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
${
|
|
syl2an3an.1 $e |- ( ph -> ps ) $.
|
|
syl2an3an.2 $e |- ( ph -> ch ) $.
|
|
syl2an3an.3 $e |- ( th -> ta ) $.
|
|
syl2an3an.4 $e |- ( ( ps /\ ch /\ ta ) -> et ) $.
|
|
$( ~ syl3an with antecedents in standard conjunction form. (Contributed by
|
|
Alan Sare, 31-Aug-2016.) $)
|
|
syl2an3an $p |- ( ( ph /\ th ) -> et ) $=
|
|
( syl3an 3anidm12 ) ADFABACDEFGHIJKL $.
|
|
$}
|
|
|
|
${
|
|
syl2an23an.1 $e |- ( ph -> ps ) $.
|
|
syl2an23an.2 $e |- ( ph -> ch ) $.
|
|
syl2an23an.3 $e |- ( ( th /\ ph ) -> ta ) $.
|
|
syl2an23an.4 $e |- ( ( ps /\ ch /\ ta ) -> et ) $.
|
|
$( Deduction related to ~ syl3an with antecedents in standard conjunction
|
|
form. (Contributed by Alan Sare, 31-Aug-2016.) $)
|
|
syl2an23an $p |- ( ( th /\ ph ) -> et ) $=
|
|
( wa wi 3exp sylc syl5 anabsi7 ) DAFDAKEAFIABCEFLGHBCEFJMNOP $.
|
|
$}
|
|
|
|
${
|
|
3ori.1 $e |- ( ph \/ ps \/ ch ) $.
|
|
$( Infer implication from triple disjunction. (Contributed by NM,
|
|
26-Sep-2006.) $)
|
|
3ori $p |- ( ( -. ph /\ -. ps ) -> ch ) $=
|
|
( wn wa wo ioran w3o df-3or mpbi ori sylbir ) AEBEFABGZECABHNCABCINCGDABC
|
|
JKLM $.
|
|
$}
|
|
|
|
$( Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) $)
|
|
3jao $p |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ->
|
|
( ( ph \/ ch \/ th ) -> ps ) ) $=
|
|
( w3o wo wi w3a df-3or jao syl6 3imp syl5bi ) ACDEACFZDFZABGZCBGZDBGZHBACDI
|
|
PQROBGZPQNBGRSGABCJNBDJKLM $.
|
|
|
|
$( Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) $)
|
|
3jaob $p |- ( ( ( ph \/ ch \/ th ) -> ps ) <->
|
|
( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ) $=
|
|
( w3o wi w3a 3mix1 imim1i 3mix2 3mix3 3jca 3jao impbii ) ACDEZBFZABFZCBFZDB
|
|
FZGPQRSAOBACDHICOBCADJIDOBDACKILABCDMN $.
|
|
|
|
${
|
|
3jaoi.1 $e |- ( ph -> ps ) $.
|
|
3jaoi.2 $e |- ( ch -> ps ) $.
|
|
3jaoi.3 $e |- ( th -> ps ) $.
|
|
$( Disjunction of 3 antecedents (inference). (Contributed by NM,
|
|
12-Sep-1995.) $)
|
|
3jaoi $p |- ( ( ph \/ ch \/ th ) -> ps ) $=
|
|
( wi w3a w3o 3pm3.2i 3jao ax-mp ) ABHZCBHZDBHZIACDJBHNOPEFGKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
3jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
3jaod.3 $e |- ( ph -> ( ta -> ch ) ) $.
|
|
$( Disjunction of 3 antecedents (deduction). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaod $p |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) ) $=
|
|
( wi w3o 3jao syl3anc ) ABCIDCIECIBDEJCIFGHBCDEKL $.
|
|
$}
|
|
|
|
${
|
|
3jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
3jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
|
|
3jaoian.3 $e |- ( ( ta /\ ps ) -> ch ) $.
|
|
$( Disjunction of 3 antecedents (inference). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaoian $p |- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch ) $=
|
|
( w3o wi ex 3jaoi imp ) ADEIBCABCJDEABCFKDBCGKEBCHKLM $.
|
|
$}
|
|
|
|
${
|
|
3jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
3jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
|
|
3jaodan.3 $e |- ( ( ph /\ ta ) -> ch ) $.
|
|
$( Disjunction of 3 antecedents (deduction). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaodan $p |- ( ( ph /\ ( ps \/ th \/ ta ) ) -> ch ) $=
|
|
( w3o ex 3jaod imp ) ABDEICABCDEABCFJADCGJAECHJKL $.
|
|
$}
|
|
|
|
${
|
|
mpjao3dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
mpjao3dan.2 $e |- ( ( ph /\ th ) -> ch ) $.
|
|
mpjao3dan.3 $e |- ( ( ph /\ ta ) -> ch ) $.
|
|
mpjao3dan.4 $e |- ( ph -> ( ps \/ th \/ ta ) ) $.
|
|
$( Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry
|
|
Arnoux, 13-Apr-2018.) $)
|
|
mpjao3dan $p |- ( ph -> ch ) $=
|
|
( wo jaodan w3o df-3or sylib mpjaodan ) ABDJZCEABCDFGKHABDELPEJIBDEMNO $.
|
|
$}
|
|
|
|
${
|
|
3jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
|
|
3jaao.3 $e |- ( et -> ( ze -> ch ) ) $.
|
|
$( Inference conjoining and disjoining the antecedents of three
|
|
implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3jaao $p |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) ) $=
|
|
( w3a wi 3ad2ant1 3ad2ant2 3ad2ant3 3jaod ) ADFKBCEGADBCLFHMDAECLFINFAGCL
|
|
DJOP $.
|
|
$}
|
|
|
|
$( Triple disjunction implies negated triple conjunction. (Contributed by
|
|
Jim Kingdon, 23-Dec-2018.) $)
|
|
3ianorr $p |- ( ( -. ph \/ -. ps \/ -. ch ) -> -. ( ph /\ ps /\ ch ) ) $=
|
|
( wn w3a simp1 con3i simp2 simp3 3jaoi ) ADABCEZDBDCDKAABCFGKBABCHGKCABCIGJ
|
|
$.
|
|
|
|
${
|
|
syl3an9b.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl3an9b.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
syl3an9b.3 $e |- ( et -> ( ta <-> ze ) ) $.
|
|
$( Nested syllogism inference conjoining 3 dissimilar antecedents.
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
syl3an9b $p |- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) ) $=
|
|
( wb wa sylan9bb 3impa ) ADFBGKADLBEFGABCDEHIMJMN $.
|
|
$}
|
|
|
|
${
|
|
bi3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bi3d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
bi3d.3 $e |- ( ph -> ( et <-> ze ) ) $.
|
|
$( Deduction joining 3 equivalences to form equivalence of disjunctions.
|
|
(Contributed by NM, 20-Apr-1994.) $)
|
|
3orbi123d $p |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) ) $=
|
|
( wo w3o orbi12d df-3or 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
|
|
$( Deduction joining 3 equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 22-Apr-1994.) $)
|
|
3anbi123d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ ze ) ) ) $=
|
|
( wa w3a anbi12d df-3an 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
$}
|
|
|
|
${
|
|
3anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi12d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) ) $=
|
|
( biidd 3anbi123d ) ABCDEFFGHAFIJ $.
|
|
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi13d $p |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) ) $=
|
|
( biidd 3anbi123d ) ABCFFDEGAFIHJ $.
|
|
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi23d $p |- ( ph -> ( ( et /\ ps /\ th ) <-> ( et /\ ch /\ ta ) ) ) $=
|
|
( biidd 3anbi123d ) AFFBCDEAFIGHJ $.
|
|
$}
|
|
|
|
${
|
|
3anbi1d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi1d $p |- ( ph -> ( ( ps /\ th /\ ta ) <-> ( ch /\ th /\ ta ) ) ) $=
|
|
( biidd 3anbi12d ) ABCDDEFADGH $.
|
|
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi2d $p |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) ) $=
|
|
( biidd 3anbi12d ) ADDBCEADGFH $.
|
|
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi3d $p |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) ) $=
|
|
( biidd 3anbi13d ) ADDBCEADGFH $.
|
|
$}
|
|
|
|
${
|
|
3anim123d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3anim123d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
3anim123d.3 $e |- ( ph -> ( et -> ze ) ) $.
|
|
$( Deduction joining 3 implications to form implication of conjunctions.
|
|
(Contributed by NM, 24-Feb-2005.) $)
|
|
3anim123d $p |- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) ) $=
|
|
( wa w3a anim12d df-3an 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
|
|
$( Deduction joining 3 implications to form implication of disjunctions.
|
|
(Contributed by NM, 4-Apr-1997.) $)
|
|
3orim123d $p |- ( ph -> ( ( ps \/ th \/ et ) -> ( ch \/ ta \/ ze ) ) ) $=
|
|
( wo w3o orim12d df-3or 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
$}
|
|
|
|
$( Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) $)
|
|
an6 $p |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <->
|
|
( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) ) $=
|
|
( w3a wa df-3an anbi12i an4 anbi1i 3bitri bitr4i ) ABCGZDEFGZHZADHZBEHZHZCF
|
|
HZHZRSUAGQABHZCHZDEHZFHZHUCUEHZUAHUBOUDPUFABCIDEFIJUCCUEFKUGTUAABDEKLMRSUAI
|
|
N $.
|
|
|
|
$( Analog of ~ an4 for triple conjunction. (Contributed by Scott Fenton,
|
|
16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3an6 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) /\ ( ta /\ et ) ) <->
|
|
( ( ph /\ ch /\ ta ) /\ ( ps /\ th /\ et ) ) ) $=
|
|
( w3a wa an6 bicomi ) ACEGBDFGHABHCDHEFHGACEBDFIJ $.
|
|
|
|
$( Analog of ~ or4 for triple conjunction. (Contributed by Scott Fenton,
|
|
16-Mar-2011.) $)
|
|
3or6 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <->
|
|
( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) ) $=
|
|
( wo w3o or4 orbi1i bitr2i df-3or orbi12i 3bitr4i ) ABGZCDGZGZEFGZGZACGZEGZ
|
|
BDGZFGZGZOPRHACEHZBDFHZGUDTUBGZRGSTEUBFIUGQRACBDIJKOPRLUEUAUFUCACELBDFLMN
|
|
$.
|
|
|
|
${
|
|
mp3an1.1 $e |- ph $.
|
|
mp3an1.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an1 $p |- ( ( ps /\ ch ) -> th ) $=
|
|
( wa 3expb mpan ) ABCGDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an2.1 $e |- ps $.
|
|
mp3an2.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an2 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( 3expa mpanl2 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mp3an3.1 $e |- ch $.
|
|
mp3an3.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an3 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa 3expia mpi ) ABGCDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an12.1 $e |- ph $.
|
|
mp3an12.2 $e |- ps $.
|
|
mp3an12.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
mp3an12 $p |- ( ch -> th ) $=
|
|
( mp3an1 mpan ) BCDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an13.1 $e |- ph $.
|
|
mp3an13.2 $e |- ch $.
|
|
mp3an13.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-Jul-2005.) $)
|
|
mp3an13 $p |- ( ps -> th ) $=
|
|
( mp3an3 mpan ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an23.1 $e |- ps $.
|
|
mp3an23.2 $e |- ch $.
|
|
mp3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-Jul-2005.) $)
|
|
mp3an23 $p |- ( ph -> th ) $=
|
|
( mp3an3 mpan2 ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an1i.1 $e |- ps $.
|
|
mp3an1i.2 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) $)
|
|
mp3an1i $p |- ( ph -> ( ( ch /\ th ) -> ta ) ) $=
|
|
( wa wi w3a com12 mp3an1 ) CDHAEBCDAEIFABCDJEGKLK $.
|
|
$}
|
|
|
|
${
|
|
mp3anl1.1 $e |- ph $.
|
|
mp3anl1.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl1 $p |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an1 imp ) BCHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anl2.1 $e |- ps $.
|
|
mp3anl2.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl2 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an2 imp ) ACHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anl3.1 $e |- ch $.
|
|
mp3anl3.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl3 $p |- ( ( ( ph /\ ps ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an3 imp ) ABHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anr1.1 $e |- ps $.
|
|
mp3anr1.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) $)
|
|
mp3anr1 $p |- ( ( ph /\ ( ch /\ th ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl1 ) CDHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3anr2.1 $e |- ch $.
|
|
mp3anr2.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Nov-2006.) $)
|
|
mp3anr2 $p |- ( ( ph /\ ( ps /\ th ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl2 ) BDHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3anr3.1 $e |- th $.
|
|
mp3anr3.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
19-Oct-2007.) $)
|
|
mp3anr3 $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl3 ) BCHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3an.1 $e |- ph $.
|
|
mp3an.2 $e |- ps $.
|
|
mp3an.3 $e |- ch $.
|
|
mp3an.4 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
mp3an $p |- th $=
|
|
( mp3an1 mp2an ) BCDFGABCDEHIJ $.
|
|
$}
|
|
|
|
${
|
|
mpd3an3.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
mpd3an3.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) $)
|
|
mpd3an3 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa 3expa mpdan ) ABGCDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mpd3an23.1 $e |- ( ph -> ps ) $.
|
|
mpd3an23.2 $e |- ( ph -> ch ) $.
|
|
mpd3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) $)
|
|
mpd3an23 $p |- ( ph -> th ) $=
|
|
( id syl3anc ) AABCDAHEFGI $.
|
|
$}
|
|
|
|
${
|
|
mp3and.1 $e |- ( ph -> ps ) $.
|
|
mp3and.2 $e |- ( ph -> ch ) $.
|
|
mp3and.3 $e |- ( ph -> th ) $.
|
|
mp3and.4 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by Mario Carneiro,
|
|
24-Dec-2016.) $)
|
|
mp3and $p |- ( ph -> ta ) $=
|
|
( w3a 3jca mpd ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
mp3an12i.1 $e |- ph $.
|
|
mp3an12i.2 $e |- ps $.
|
|
mp3an12i.3 $e |- ( ch -> th ) $.
|
|
mp3an12i.4 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( ~ mp3an with antecedents in standard conjunction form and with one
|
|
hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) $)
|
|
mp3an12i $p |- ( ch -> ta ) $=
|
|
( mp3an12 syl ) CDEHABDEFGIJK $.
|
|
$}
|
|
|
|
${
|
|
mp3an2i.1 $e |- ph $.
|
|
mp3an2i.2 $e |- ( ps -> ch ) $.
|
|
mp3an2i.3 $e |- ( ps -> th ) $.
|
|
mp3an2i.4 $e |- ( ( ph /\ ch /\ th ) -> ta ) $.
|
|
$( ~ mp3an with antecedents in standard conjunction form and with two
|
|
hypotheses which are implications. (Contributed by Alan Sare,
|
|
28-Aug-2016.) $)
|
|
mp3an2i $p |- ( ps -> ta ) $=
|
|
( mp3an1 syl2anc ) BCDEGHACDEFIJK $.
|
|
$}
|
|
|
|
${
|
|
mp3an3an.1 $e |- ph $.
|
|
mp3an3an.2 $e |- ( ps -> ch ) $.
|
|
mp3an3an.3 $e |- ( th -> ta ) $.
|
|
mp3an3an.4 $e |- ( ( ph /\ ch /\ ta ) -> et ) $.
|
|
$( ~ mp3an with antecedents in standard conjunction form and with two
|
|
hypotheses which are implications. (Contributed by Alan Sare,
|
|
28-Aug-2016.) $)
|
|
mp3an3an $p |- ( ( ps /\ th ) -> et ) $=
|
|
( mp3an1 syl2an ) BCEFDHIACEFGJKL $.
|
|
$}
|
|
|
|
${
|
|
mp3an2ani.1 $e |- ph $.
|
|
mp3an2ani.2 $e |- ( ps -> ch ) $.
|
|
mp3an2ani.3 $e |- ( ( ps /\ th ) -> ta ) $.
|
|
mp3an2ani.4 $e |- ( ( ph /\ ch /\ ta ) -> et ) $.
|
|
$( An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) $)
|
|
mp3an2ani $p |- ( ( ps /\ th ) -> et ) $=
|
|
( wa mp3an3an anabss5 ) BDFABCBDKEFGHIJLM $.
|
|
$}
|
|
|
|
${
|
|
biimp3a.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Infer implication from a logical equivalence. Similar to ~ biimpa .
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
biimp3a $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( wa biimpa 3impa ) ABCDABFCDEGH $.
|
|
|
|
$( Infer implication from a logical equivalence. Similar to ~ biimpar .
|
|
(Contributed by NM, 2-Jan-2009.) $)
|
|
biimp3ar $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( exbiri 3imp ) ABDCABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) )
|
|
-> ta ) $.
|
|
$( Inference that undistributes a triple conjunction in the antecedent.
|
|
(Contributed by NM, 18-Apr-2007.) $)
|
|
3anandis $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
|
|
( w3a wa simpl simpr1 simpr2 simpr3 syl222anc ) ABCDGZHABACADEANIZABCDJOA
|
|
BCDKOABCDLFM $.
|
|
$}
|
|
|
|
${
|
|
3anandirs.1 $e |- ( ( ( ph /\ th ) /\ ( ps /\ th ) /\ ( ch /\ th ) )
|
|
-> ta ) $.
|
|
$( Inference that undistributes a triple conjunction in the antecedent.
|
|
(Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) $)
|
|
3anandirs $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
|
|
( w3a wa simpl1 simpr simpl2 simpl3 syl222anc ) ABCGZDHADBDCDEABCDINDJZAB
|
|
CDKOABCDLOFM $.
|
|
$}
|
|
|
|
${
|
|
ecased.1 $e |- ( ph -> -. ch ) $.
|
|
ecased.2 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon,
|
|
9-Dec-2017.) $)
|
|
ecased $p |- ( ph -> ps ) $=
|
|
( wn wo wa jca orel2 imp syl ) ACFZBCGZHBAMNDEIMNBCBJKL $.
|
|
$}
|
|
|
|
${
|
|
ecase23d.1 $e |- ( ph -> -. ch ) $.
|
|
ecase23d.2 $e |- ( ph -> -. th ) $.
|
|
ecase23d.3 $e |- ( ph -> ( ps \/ ch \/ th ) ) $.
|
|
$( Variation of ~ ecased with three disjuncts instead of two. (Contributed
|
|
by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) $)
|
|
ecase23d $p |- ( ph -> ps ) $=
|
|
( wo w3o df-3or sylib ecased ) ABCEABCHZDFABCDIMDHGBCDJKLL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
True and false constants
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Universal quantifier for use by df-tru
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
Even though it isn't ordinarily part of propositional calculus, the universal
|
|
quantifier ` A. ` is introduced here so that the soundness of definition
|
|
~ df-tru can be checked by the same algorithm that is used for predicate
|
|
calculus. Its first real use is in axiom ~ ax-5 in the predicate
|
|
calculus section below. For those who want propositional calculus to be
|
|
self-contained i.e. to use wff variables only, the alternate definition
|
|
~ dftru2 may be adopted and this subsection moved down to the start of the
|
|
subsection with ~ wex below. However, the use of ~ dftru2 as a definition
|
|
requires a more elaborate definition checking algorithm that we prefer to
|
|
avoid.
|
|
|
|
$)
|
|
|
|
$( Declare new symbols needed for predicate calculus. $)
|
|
$c A. $. $( "inverted A" universal quantifier (read: "for all") $)
|
|
$c setvar $. $( Individual variable type (read: "the following is an
|
|
individual (set) variable" $)
|
|
|
|
$( Add 'setvar' as a typecode for bound variables. $)
|
|
$( $j syntax 'setvar'; bound 'setvar'; $)
|
|
|
|
${
|
|
$v x $.
|
|
$( Let ` x ` be an individual variable (temporary declaration). $)
|
|
vx.wal $f setvar x $.
|
|
$( Extend wff definition to include the universal quantifier ('for all').
|
|
` A. x ph ` is read " ` ph ` (phi) is true for all ` x ` ." Typically,
|
|
in its final application ` ph ` would be replaced with a wff containing
|
|
a (free) occurrence of the variable ` x ` , for example ` x = y ` . In
|
|
a universe with a finite number of objects, "for all" is equivalent to a
|
|
big conjunction (AND) with one wff for each possible case of ` x ` .
|
|
When the universe is infinite (as with set theory), such a
|
|
propositional-calculus equivalent is not possible because an infinitely
|
|
long formula has no meaning, but conceptually the idea is the same. $)
|
|
wal $a wff A. x ph $.
|
|
$}
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Equality predicate for use by df-tru
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
Even though it isn't ordinarily part of propositional calculus, the equality
|
|
predicate ` = ` is introduced here so that the soundness of definition
|
|
~ df-tru can be checked by the same algorithm as is used for predicate
|
|
calculus. Its first real use is in axiom ~ ax-8 in the predicate calculus
|
|
section below. For those who want propositional calculus to be
|
|
self-contained i.e. to use wff variables only, the alternate definition
|
|
~ dftru2 may be adopted and this subsection moved down to just above ~ weq
|
|
below. However, the use of ~ dftru2 as a definition requires a more
|
|
elaborate definition checking algorithm that we prefer to avoid.
|
|
$)
|
|
|
|
$c class $.
|
|
|
|
$( Add 'class' as a typecode. $)
|
|
$( $j syntax 'class'; $)
|
|
|
|
${
|
|
$v x $.
|
|
$( Let ` x ` be an individual variable (temporary declaration). $)
|
|
vx.cv $f setvar x $.
|
|
$( This syntax construction states that a variable ` x ` , which has been
|
|
declared to be a setvar variable by $f statement vx, is also a class
|
|
expression. This can be justified informally as follows. We know that
|
|
the class builder ` { y | y e. x } ` is a class by ~ cab . Since (when
|
|
` y ` is distinct from ` x ` ) we have ` x = { y | y e. x } ` by
|
|
~ cvjust , we can argue that the syntax " ` class x ` " can be viewed as
|
|
an abbreviation for " ` class { y | y e. x } ` ". See the discussion
|
|
under the definition of class in [Jech] p. 4 showing that "Every set can
|
|
be considered to be a class."
|
|
|
|
While it is tempting and perhaps occasionally useful to view ~ cv as a
|
|
"type conversion" from a setvar variable to a class variable, keep in
|
|
mind that ~ cv is intrinsically no different from any other
|
|
class-building syntax such as ~ cab , ~ cun , or ~ c0 .
|
|
|
|
For a general discussion of the theory of classes and the role of ~ cv ,
|
|
see ~ http://us.metamath.org/mpeuni/mmset.html#class .
|
|
|
|
(The description above applies to set theory, not predicate calculus.
|
|
The purpose of introducing ` class x ` here, and not in set theory where
|
|
it belongs, is to allow us to express i.e. "prove" the ~ weq of
|
|
predicate calculus from the ~ wceq of set theory, so that we don't
|
|
overload the ` = ` connective with two syntax definitions. This is done
|
|
to prevent ambiguity that would complicate some Metamath parsers.) $)
|
|
cv $a class x $.
|
|
$}
|
|
|
|
$( Declare the equality predicate symbol. $)
|
|
$c = $. $( Equal sign (read: 'is equal to') $)
|
|
|
|
${
|
|
$v A $.
|
|
$v B $.
|
|
$( Temporary declarations of ` A ` and ` B ` . $)
|
|
cA.wceq $f class A $.
|
|
cB.wceq $f class B $.
|
|
$( Extend wff definition to include class equality.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class .
|
|
|
|
(The purpose of introducing ` wff A = B ` here, and not in set theory
|
|
where it belongs, is to allow us to express i.e. "prove" the ~ weq of
|
|
predicate calculus in terms of the ~ wceq of set theory, so that we
|
|
don't "overload" the ` = ` connective with two syntax definitions. This
|
|
is done to prevent ambiguity that would complicate some Metamath
|
|
parsers. For example, some parsers - although not the Metamath program
|
|
- stumble on the fact that the ` = ` in ` x = y ` could be the ` = ` of
|
|
either ~ weq or ~ wceq , although mathematically it makes no difference.
|
|
The class variables ` A ` and ` B ` are introduced temporarily for the
|
|
purpose of this definition but otherwise not used in predicate calculus.
|
|
See ~ df-cleq for more information on the set theory usage of
|
|
~ wceq .) $)
|
|
wceq $a wff A = B $.
|
|
$}
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Define the true and false constants
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
$c T. $.
|
|
|
|
$( ` T. ` is a wff. $)
|
|
wtru $a wff T. $.
|
|
|
|
${
|
|
$v x $.
|
|
$v y $.
|
|
$( Temporary declarations of ` x ` and ` y ` for local use by ~ df-tru .
|
|
These will be redeclared globally in the predicate calculus section. $)
|
|
vx.tru $f setvar x $.
|
|
vy.tru $f setvar y $.
|
|
$( Soundness justification theorem for ~ df-tru . (Contributed by Mario
|
|
Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) $)
|
|
trujust $p |- ( ( A. x x = x -> A. x x = x )
|
|
<-> ( A. y y = y -> A. y y = y ) ) $=
|
|
( cv wceq wal wi id 2th ) ACZIDAEZJFBCZKDBEZLFJGLGH $.
|
|
|
|
$( Definition of the truth value "true", or "verum", denoted by ` T. ` .
|
|
This is a tautology, as proved by ~ tru . In this definition, an
|
|
instance of ~ id is used as the definiens, although any tautology, such
|
|
as an axiom, can be used in its place. This particular ~ id instance
|
|
was chosen so this definition can be checked by the same algorithm that
|
|
is used for predicate calculus. This definition should be referenced
|
|
directly only by ~ tru , and other proofs should depend on ~ tru
|
|
(directly or indirectly) instead of this definition, since there are
|
|
many alternate ways to define ` T. ` . (Contributed by Anthony Hart,
|
|
13-Oct-2010.) (Revised by NM, 11-Jul-2019.)
|
|
(New usage is discouraged.) $)
|
|
df-tru $a |- ( T. <-> ( A. x x = x -> A. x x = x ) ) $.
|
|
|
|
$( The truth value ` T. ` is provable. (Contributed by Anthony Hart,
|
|
13-Oct-2010.) $)
|
|
tru $p |- T. $=
|
|
( vx.tru wtru cv wceq wal wi id df-tru mpbir ) BACZJDAEZKFKGAHI $.
|
|
$}
|
|
|
|
$c F. $.
|
|
|
|
$( ` F. ` is a wff. $)
|
|
wfal $a wff F. $.
|
|
|
|
$( Definition of the truth value "false", or "falsum", denoted by ` F. ` .
|
|
See also ~ df-tru . (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
df-fal $a |- ( F. <-> -. T. ) $.
|
|
|
|
$( The truth value ` F. ` is refutable. (Contributed by Anthony Hart,
|
|
22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) $)
|
|
fal $p |- -. F. $=
|
|
( wfal wtru wn tru notnoti df-fal mtbir ) ABCBDEFG $.
|
|
|
|
$( An alternate definition of "true". (Contributed by Anthony Hart,
|
|
13-Oct-2010.) (Revised by BJ, 12-Jul-2019.)
|
|
(New usage is discouraged.) $)
|
|
dftru2 $p |- ( T. <-> ( ph -> ph ) ) $=
|
|
( wtru wi tru id 2th ) BAACDAEF $.
|
|
|
|
${
|
|
trud.1 $e |- ( T. -> ph ) $.
|
|
$( Eliminate ` T. ` as an antecedent. A proposition implied by ` T. ` is
|
|
true. (Contributed by Mario Carneiro, 13-Mar-2014.) $)
|
|
trud $p |- ph $=
|
|
( wtru tru ax-mp ) CADBE $.
|
|
$}
|
|
|
|
$( A proposition is equivalent to itself being equivalent to ` T. ` .
|
|
(Contributed by Anthony Hart, 14-Aug-2011.) $)
|
|
tbtru $p |- ( ph <-> ( ph <-> T. ) ) $=
|
|
( wtru tru tbt ) BACD $.
|
|
|
|
$( The negation of a proposition is equivalent to itself being equivalent to
|
|
` F. ` . (Contributed by Anthony Hart, 14-Aug-2011.) $)
|
|
nbfal $p |- ( -. ph <-> ( ph <-> F. ) ) $=
|
|
( wfal fal nbn ) BACD $.
|
|
|
|
${
|
|
bitru.1 $e |- ph $.
|
|
$( A theorem is equivalent to truth. (Contributed by Mario Carneiro,
|
|
9-May-2015.) $)
|
|
bitru $p |- ( ph <-> T. ) $=
|
|
( wtru tru 2th ) ACBDE $.
|
|
$}
|
|
|
|
${
|
|
bifal.1 $e |- -. ph $.
|
|
$( A contradiction is equivalent to falsehood. (Contributed by Mario
|
|
Carneiro, 9-May-2015.) $)
|
|
bifal $p |- ( ph <-> F. ) $=
|
|
( wfal fal 2false ) ACBDE $.
|
|
$}
|
|
|
|
$( The truth value ` F. ` implies anything. Also called the principle of
|
|
explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.)
|
|
(Proof shortened by Anthony Hart, 1-Aug-2011.) $)
|
|
falim $p |- ( F. -> ph ) $=
|
|
( wfal fal pm2.21i ) BACD $.
|
|
|
|
$( The truth value ` F. ` implies anything. (Contributed by Mario Carneiro,
|
|
9-Feb-2017.) $)
|
|
falimd $p |- ( ( ph /\ F. ) -> ps ) $=
|
|
( wfal falim adantl ) CBABDE $.
|
|
|
|
$( Anything implies ` T. ` . (Contributed by FL, 20-Mar-2011.) (Proof
|
|
shortened by Anthony Hart, 1-Aug-2011.) $)
|
|
a1tru $p |- ( ph -> T. ) $=
|
|
( wtru tru a1i ) BACD $.
|
|
|
|
$( True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
|
|
(Proof shortened by Wolf Lammen, 21-Jul-2019.) $)
|
|
truan $p |- ( ( T. /\ ph ) <-> ph ) $=
|
|
( wtru wa tru biantrur bicomi ) ABACBADEF $.
|
|
|
|
$( Given falsum, we can define the negation of a wff ` ph ` as the statement
|
|
that a contradiction follows from assuming ` ph ` . (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) $)
|
|
dfnot $p |- ( -. ph <-> ( ph -> F. ) ) $=
|
|
( wfal wn wi wb fal mtt ax-mp ) BCACABDEFBAGH $.
|
|
|
|
${
|
|
inegd.1 $e |- ( ( ph /\ ps ) -> F. ) $.
|
|
$( Negation introduction rule from natural deduction. (Contributed by
|
|
Mario Carneiro, 9-Feb-2017.) $)
|
|
inegd $p |- ( ph -> -. ps ) $=
|
|
( wfal wi wn ex dfnot sylibr ) ABDEBFABDCGBHI $.
|
|
$}
|
|
|
|
${
|
|
pm2.21fal.1 $e |- ( ph -> ps ) $.
|
|
pm2.21fal.2 $e |- ( ph -> -. ps ) $.
|
|
$( If a wff and its negation are provable, then falsum is provable.
|
|
(Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
pm2.21fal $p |- ( ph -> F. ) $=
|
|
( wfal pm2.21dd ) ABECDF $.
|
|
$}
|
|
|
|
$( Negation inferred from embedded conjunct. (Contributed by NM,
|
|
20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) $)
|
|
pclem6 $p |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps ) $=
|
|
( wn wa wb wfal wi bi1 pm3.4 com12 syl9r ax-ia3 syl9 impbidd pm5.19 pm2.21i
|
|
bi2 syl6com dfnot sylibr ) ABACZDZEZBFGBCBUCAUAEZFBUCAUAUCAUBBUAAUBHUBBUABU
|
|
AIJKBUAUBUCABUALAUBQMNUDFAOPRBST $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical 'xor'
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare connective for exclusive disjunction ('xor'). $)
|
|
$c \/_ $. $( Underlined 'vee' (read: 'xor') $)
|
|
|
|
$( Extend wff definition to include exclusive disjunction ('xor'). $)
|
|
wxo $a wff ( ph \/_ ps ) $.
|
|
|
|
$( Define exclusive disjunction (logical 'xor'). Return true if either the
|
|
left or right, but not both, are true. Contrast with ` /\ ` ( ~ wa ),
|
|
` \/ ` ( ~ wo ), and ` -> ` ( ~ wi ) . (Contributed by FL, 22-Nov-2010.)
|
|
(Modified by Jim Kingdon, 1-Mar-2018.) $)
|
|
df-xor $a |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $.
|
|
|
|
$( One way of defining exclusive or. Equivalent to ~ df-xor . (Contributed
|
|
by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) $)
|
|
xoranor $p |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) ) $=
|
|
( wxo wo wn wa df-xor ax-ia3 con3d olc syl6 pm3.21 orc jaoi imdistani sylbi
|
|
wi pm3.14 anim2i sylibr impbii ) ABCZABDZAEZBEZDZFZUBUCABFZEZFZUGABGZUCUIUF
|
|
AUIUFQBAUIUEUFABUHABHIUEUDJKBUIUDUFBAUHBALIUDUEMKNOPUGUJUBUFUIUCABRSUKTUA
|
|
$.
|
|
|
|
$( This tautology shows that xor is really exclusive. (Contributed by FL,
|
|
22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) $)
|
|
excxor $p |- ( ( ph \/_ ps ) <->
|
|
( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) $=
|
|
( wxo wn wa wo xoranor andi orcom pm3.24 biorfi andir pm5.61 orbi12i 3bitri
|
|
3bitr4ri ancom orbi2i ) ABCZBADZEZABDZEZFZUCUAFUCTBEZFSABFZTUBFEUFTEZUFUBEZ
|
|
FUDABGUFTUBHUGUAUHUCUAATEZFUIUAFUAUGUAUIIUIUAAJKABTLPABMNOUAUCIUAUEUCBTQRO
|
|
$.
|
|
|
|
$( XOR implies OR. (Contributed by BJ, 19-Apr-2019.) $)
|
|
xoror $p |- ( ( ph \/_ ps ) -> ( ph \/ ps ) ) $=
|
|
( wxo wo wn xoranor simplbi ) ABCABDAEBEDABFG $.
|
|
|
|
${
|
|
xorbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction joining an equivalence and a left operand to form equivalence
|
|
of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) $)
|
|
xorbi2d $p |- ( ph -> ( ( th \/_ ps ) <-> ( th \/_ ch ) ) ) $=
|
|
( wo wa wn wxo orbi2d anbi2d notbid anbi12d df-xor 3bitr4g ) ADBFZDBGZHZG
|
|
DCFZDCGZHZGDBIDCIAPSRUAABCDEJAQTABCDEKLMDBNDCNO $.
|
|
|
|
$( Deduction joining an equivalence and a right operand to form equivalence
|
|
of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) $)
|
|
xorbi1d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) ) $=
|
|
( wo wa wn wxo orbi1d anbi1d notbid anbi12d df-xor 3bitr4g ) ABDFZBDGZHZG
|
|
CDFZCDGZHZGBDICDIAPSRUAABCDEJAQTABCDEKLMBDNCDNO $.
|
|
$}
|
|
|
|
${
|
|
xorbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
xorbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of exclusive-or.
|
|
(Contributed by Jim Kingdon, 7-Oct-2018.) $)
|
|
xorbi12d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) ) $=
|
|
( wxo xorbi1d xorbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
xorbi12.1 $e |- ( ph <-> ps ) $.
|
|
xorbi12.2 $e |- ( ch <-> th ) $.
|
|
$( Equality property for XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
xorbi12i $p |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) ) $=
|
|
( wxo wb wtru a1i xorbi12d trud ) ACGBDGHIABCDABHIEJCDHIFJKL $.
|
|
$}
|
|
|
|
$( A consequence of exclusive or. In classical logic the converse also
|
|
holds. (Contributed by Jim Kingdon, 8-Mar-2018.) $)
|
|
xorbin $p |- ( ( ph \/_ ps ) -> ( ph <-> -. ps ) ) $=
|
|
( wn wo wa wi df-xor imnan biimpri adantl sylbi pm2.53 orcoms adantr impbid
|
|
wxo ) ABPZABCZQABDZABECZEZARFZABGZTUBSUBTABHIJKQUARAFZUCSUDTBAUDBALMNKO $.
|
|
|
|
$( One direction of ~ pm5.18dc , which holds for all propositions, not just
|
|
decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) $)
|
|
pm5.18im $p |- ( ( ph <-> ps ) -> -. ( ph <-> -. ps ) ) $=
|
|
( wb wn pm5.19 bibi1 notbid mpbiri ) ABCZABDZCZDBJCZDBEIKLABJFGH $.
|
|
|
|
$( A consequence of exclusive or. For decidable propositions this is an
|
|
equivalence, as seen at ~ xornbidc . (Contributed by Jim Kingdon,
|
|
10-Mar-2018.) $)
|
|
xornbi $p |- ( ( ph \/_ ps ) -> -. ( ph <-> ps ) ) $=
|
|
( wxo wn wb xorbin pm5.18im con2i syl ) ABCABDEZABEZDABFKJABGHI $.
|
|
|
|
$( Two ways to express "exclusive or" between decidable propositions.
|
|
(Contributed by Jim Kingdon, 12-Apr-2018.) $)
|
|
xor3dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) ) $=
|
|
( wdc wb wn wa dcn dcbi syl5 imp pm5.18dc a1d con2biddc mpd bicomd ex ) ACZ
|
|
BCZABDZEZABEZDZDQRFZUBTUCUBCZUBTDQRUDRUACQUDBGAUAHIJUCSUBUCSUBEDZUDQRUEABKJ
|
|
LMNOP $.
|
|
|
|
$( ` \/_ ` is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) $)
|
|
xorcom $p |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) ) $=
|
|
( wo wa wn wxo orcom ancom notbii anbi12i df-xor 3bitr4i ) ABCZABDZEZDBACZB
|
|
ADZEZDABFBAFMPORABGNQABHIJABKBAKL $.
|
|
|
|
$( A decidable proposition is equivalent to a decidable proposition or its
|
|
negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124.
|
|
(Contributed by Jim Kingdon, 18-Apr-2018.) $)
|
|
pm5.15dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) ) ) $=
|
|
( wdc wb wn wo wa wi xor3dc imp biimpd dcbi dfordc syl mpbird ex ) ACZBCZAB
|
|
DZABEDZFZQRGZUASEZTHZUBUCTQRUCTDABIJKUBSCZUAUDDQRUEABLJSTMNOP $.
|
|
|
|
$( Two ways to express "exclusive or" between decidable propositions.
|
|
(Contributed by Jim Kingdon, 17-Apr-2018.) $)
|
|
xor2dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) ) $=
|
|
( wdc wb wn wo wa xor3dc imp pm5.17dc adantl bitr4d ex ) ACZBCZABDEZABFABGE
|
|
GZDNOGPABEDZQNOPRDABHIOQRDNABJKLM $.
|
|
|
|
$( Exclusive or is equivalent to negated biconditional for decidable
|
|
propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) $)
|
|
xornbidc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) ) $=
|
|
( wdc wxo wb wn wa wo xor2dc imp df-xor syl6rbbr ex ) ACZBCZABDZABEFZENOGQA
|
|
BHABGFGZPNOQREABIJABKLM $.
|
|
|
|
$( Two ways to express "exclusive or" between decidable propositions.
|
|
Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable
|
|
propositions. (Contributed by Jim Kingdon, 5-May-2018.) $)
|
|
xordc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) ) $=
|
|
( wdc wb wn wa wo wxo excxor ancom orbi2i bitri xornbidc imp syl5rbbr ex )
|
|
ACZBCZABDEZABEFZBAEZFZGZDUCABHZQRFSUDTUABFZGUCABIUEUBTUABJKLQRUDSDABMNOP $.
|
|
|
|
$( Exclusive or implies the left proposition is decidable. (Contributed by
|
|
Jim Kingdon, 12-Mar-2018.) $)
|
|
xordc1 $p |- ( ( ph \/_ ps ) -> DECID ph ) $=
|
|
( wo wa wn wxo wdc andir simpl imnan ancom xchbinxr pm3.35 sylan2br orim12i
|
|
wi sylbi df-xor df-dc 3imtr4i ) ABCABDZEZDZAAEZCZABFAGUCAUBDZBUBDZCUEABUBHU
|
|
FAUGUDAUBIUBBBUDPZUDUHBADUABAJABKLBUDMNOQABRAST $.
|
|
|
|
$( Move negation outside of biconditional, for decidable propositions.
|
|
Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim
|
|
Kingdon, 18-Apr-2018.) $)
|
|
nbbndc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) ) ) $=
|
|
( wdc wn wb wa xor3dc imp con2bidc bitrd bicom syl6rbb ex ) ACZBCZADZBEZABE
|
|
DZENOFZRBPEZQSRABDEZTNORUAEABGHNOUATEABIHJBPKLM $.
|
|
|
|
$( Associative law for the biconditional, for decidable propositions.
|
|
|
|
The classical version (without the decidability conditions) is an axiom of
|
|
system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure
|
|
and Applied Logic, 113:207-224, 2002,
|
|
~ http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805 , and,
|
|
interestingly, was not included in _Principia Mathematica_ but was
|
|
apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim
|
|
Kingdon, 4-May-2018.) $)
|
|
biassdc $p |- ( DECID ph -> ( DECID ps -> ( DECID ch ->
|
|
( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) ) ) $=
|
|
( wdc wb wn wo wa wi df-dc pm5.501 bibi1d bitr3d a1d nbbndc imp adantl nbn2
|
|
adantr ex jaoi sylbi expd ) ADZBDZCDZABEZCEZABCEZEZEZUDAAFZGUEUFHZUKIZAJAUN
|
|
ULAUKUMAUIUHUJABUGCABKLAUIKMNULUMUKULUMHZUIFZUHUJUOBFZCEZUPUHUMURUPEZULUEUF
|
|
USBCOPQULURUHEUMULUQUGCABRLSMULUPUJEUMAUIRSMTUAUBUC $.
|
|
|
|
$( Lukasiewicz's shortest axiom for equivalential calculus (but modified to
|
|
require decidable propositions). Storrs McCall, ed., _Polish Logic
|
|
1920-1939_ (Oxford, 1967), p. 96. (Contributed by Jim Kingdon,
|
|
5-May-2018.) $)
|
|
bilukdc $p |- ( ( ( DECID ph /\ DECID ps ) /\ DECID ch ) ->
|
|
( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) ) ) $=
|
|
( wdc wa wb bicom bibi1i biassdc imp31 syl5bb ancom1s dcbi imp adantr simpr
|
|
syl9 syl3c mpbid simplr adantlr bitr4d ) ADZBDZEZCDZEZABFZCBACFZFZFZCBFUIFZ
|
|
UGUHCFZUJFZUHUKFZUDUCUFUNUMBAFZCFZUDUCEUFEUJUHUPCABGHUDUCUFUQUJFBACIJKLUGUH
|
|
DZUFUJDZUNUOFUEURUFUCUDURABMNOUEUFPZUCUDUFUSUCUFUIDZUDUSACMZBUIMQJUHCUJIRSU
|
|
GUFUDVAULUKFUTUCUDUFTUCUFVAUDUCUFVAVBNUACBUIIRUB $.
|
|
|
|
$( An alternate definition of the biconditional for decidable propositions.
|
|
Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability
|
|
conditions. (Contributed by Jim Kingdon, 5-May-2018.) $)
|
|
dfbi3dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) ) ) $=
|
|
( wdc wb wa wn dcn xordc imp sylan2 pm5.18dc notnotbdc anbi2d ancom orbi12d
|
|
wo a1i adantl 3bitr4d ex ) ACZBCZABDZABEZAFZBFZEZPZDUAUBEAUFDFZAUFFZEZUFUEE
|
|
ZPZUCUHUBUAUFCZUIUMDZBGUAUNUOAUFHIJUAUBUCUIDABKIUBUHUMDUAUBUDUKUGULUBBUJABL
|
|
MUGULDUBUEUFNQORST $.
|
|
|
|
$( Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable
|
|
propositions. (Contributed by Jim Kingdon, 5-May-2018.) $)
|
|
pm5.24dc $p |- ( DECID ph -> ( DECID ps ->
|
|
( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <->
|
|
( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) ) $=
|
|
( wdc wa wn wo wb dfbi3dc imp notbid xordc bitr3d ex ) ACZBCZABDAEZBEZDFZEZ
|
|
AQDBPDFZGNODZABGZEZSTUAUBRNOUBRGABHIJNOUCTGABKILM $.
|
|
|
|
$( Conjunction distributes over exclusive-or, for decidable propositions.
|
|
This is one way to interpret the distributive law of multiplication over
|
|
addition in modulo 2 arithmetic. (Contributed by Jim Kingdon,
|
|
14-Jul-2018.) $)
|
|
xordidc $p |- ( DECID ph -> ( DECID ps -> ( DECID ch ->
|
|
( ( ph /\ ( ps \/_ ch ) ) <->
|
|
( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) ) ) ) $=
|
|
( wdc wxo wa wb wn dcbi imp wi annimdc pm5.32 notbii syl6bb sylan2 xornbidc
|
|
adantl anbi2d dcan adantrr adantrl sylc 3bitr4d exp32 ) ADZBDZCDZABCEZFZABF
|
|
ZACFZEZGUFUGUHFZFZABCGZHZFZUKULGZHZUJUMUNUFUPDZURUTGUGUHVABCIJUFVAFURAUPKZH
|
|
ZUTUFVAURVCGAUPLJVBUSABCMNOPUOUIUQAUNUIUQGZUFUGUHVDBCQJRSUOUKDZULDZUMUTGUFU
|
|
GVEUHUFUGVEABTJUAUFUHVFUGUFUHVFACTJUBUKULQUCUDUE $.
|
|
|
|
$( Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro
|
|
and Jim Kingdon, 7-Oct-2018.) $)
|
|
anxordi $p |- ( ( ph /\ ( ps \/_ ch ) ) <->
|
|
( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) $=
|
|
( wxo wa simpl wo wn df-xor simplbi jaoi syl ibar xorbi12d bitr3d pm5.21nii
|
|
) ABCDZEZAABEZACEZDZAQFUASTGZAUAUBSTEHSTIJSATABFACFKLAQRUAAQMABSCTABMACMNOP
|
|
$.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Truth tables: Operations on true and false constants
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
For classical logic, truth tables can be used to define propositional
|
|
logic operations, by showing the results of those operations for all
|
|
possible combinations of true ( ` T. ` ) and false ( ` F. ` ).
|
|
|
|
Although the intuitionistic logic connectives are not as simply defined,
|
|
` T. ` and ` F. ` do play similar roles as in classical logic and most
|
|
theorems from classical logic continue to hold.
|
|
|
|
Here we show that our definitions and axioms produce equivalent results for
|
|
` T. ` and ` F. ` as we would get from truth tables for
|
|
` /\ ` (conjunction aka logical 'and') ~ wa ,
|
|
` \/ ` (disjunction aka logical inclusive 'or') ~ wo ,
|
|
` -> ` (implies) ~ wi ,
|
|
` -. ` (not) ~ wn ,
|
|
` <-> ` (logical equivalence) ~ df-bi , and
|
|
` \/_ ` (exclusive or) ~ df-xor .
|
|
$)
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truantru $p |- ( ( T. /\ T. ) <-> T. ) $=
|
|
( wtru anidm ) AB $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truanfal $p |- ( ( T. /\ F. ) <-> F. ) $=
|
|
( wfal truan ) AB $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by David A. Wheeler, 23-Feb-2018.) $)
|
|
falantru $p |- ( ( F. /\ T. ) <-> F. ) $=
|
|
( wfal wtru wa simpl falim impbii ) ABCZAABDGEF $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falanfal $p |- ( ( F. /\ F. ) <-> F. ) $=
|
|
( wfal anidm ) AB $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
truortru $p |- ( ( T. \/ T. ) <-> T. ) $=
|
|
( wtru oridm ) AB $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truorfal $p |- ( ( T. \/ F. ) <-> T. ) $=
|
|
( wtru wfal wo tru orci bitru ) ABCABDEF $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falortru $p |- ( ( F. \/ T. ) <-> T. ) $=
|
|
( wfal wtru wo tru olci bitru ) ABCBADEF $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falorfal $p |- ( ( F. \/ F. ) <-> F. ) $=
|
|
( wfal oridm ) AB $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truimtru $p |- ( ( T. -> T. ) <-> T. ) $=
|
|
( wtru wi id bitru ) AABACD $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
truimfal $p |- ( ( T. -> F. ) <-> F. ) $=
|
|
( wfal wtru wi tru a1bi bicomi ) ABACBADEF $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falimtru $p |- ( ( F. -> T. ) <-> T. ) $=
|
|
( wfal wtru wi falim bitru ) ABCBDE $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falimfal $p |- ( ( F. -> F. ) <-> T. ) $=
|
|
( wfal wi id bitru ) AABACD $.
|
|
|
|
$( A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
nottru $p |- ( -. T. <-> F. ) $=
|
|
( wfal wtru wn df-fal bicomi ) ABCDE $.
|
|
|
|
$( A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
notfal $p |- ( -. F. <-> T. ) $=
|
|
( wfal wn fal bitru ) ABCD $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
trubitru $p |- ( ( T. <-> T. ) <-> T. ) $=
|
|
( wtru wb biid bitru ) AABACD $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by David A. Wheeler, 23-Feb-2018.) $)
|
|
trubifal $p |- ( ( T. <-> F. ) <-> F. ) $=
|
|
( wtru wfal wb wi wa dfbi2 truimfal falimtru anbi12i falantru 3bitri ) ABCA
|
|
BDZBADZEBAEBABFLBMAGHIJK $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falbitru $p |- ( ( F. <-> T. ) <-> F. ) $=
|
|
( wfal wtru wb bicom trubifal bitri ) ABCBACAABDEF $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falbifal $p |- ( ( F. <-> F. ) <-> T. ) $=
|
|
( wfal wb biid bitru ) AABACD $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 2-Mar-2018.) $)
|
|
truxortru $p |- ( ( T. \/_ T. ) <-> F. ) $=
|
|
( wtru wxo wo wa wn df-xor oridm nottru anidm xchnxbir anbi12i truan 3bitri
|
|
wfal ) AABAACZAADZEZDANDNAAFOAQNAGANPHAIJKNLM $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 2-Mar-2018.) $)
|
|
truxorfal $p |- ( ( T. \/_ F. ) <-> T. ) $=
|
|
( wtru wfal wxo wo wa wn df-xor truorfal notfal truan xchnxbir anidm 3bitri
|
|
anbi12i ) ABCABDZABEZFZEAAEAABGOAQAHBAPIBJKNALM $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 2-Mar-2018.) $)
|
|
falxortru $p |- ( ( F. \/_ T. ) <-> T. ) $=
|
|
( wfal wtru wo wa wn df-xor falortru notfal falantru xchnxbir anbi12i anidm
|
|
wxo 3bitri ) ABMABCZABDZEZDBBDBABFOBQBGABPHIJKBLN $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 2-Mar-2018.) $)
|
|
falxorfal $p |- ( ( F. \/_ F. ) <-> F. ) $=
|
|
( wfal wxo wo wa wn wtru df-xor oridm notfal anidm xchnxbir falantru 3bitri
|
|
anbi12i ) AABAACZAADZEZDAFDAAAGOAQFAHAFPIAJKNLM $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Stoic logic indemonstrables (Chrysippus of Soli)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The Greek Stoics developed a system of logic. The Stoic Chrysippus, in
|
|
particular, was often considered one of the greatest logicians of antiquity.
|
|
Stoic logic is different from Aristotle's system, since it focuses on
|
|
propositional logic, though later thinkers did combine the systems of the
|
|
Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with
|
|
mathematical logic it is self-evident that the Stoic dialectic is the ancient
|
|
form of modern propositional logic"
|
|
( _On the history of the logic of proposition_ by Jan Lukasiewicz (1934),
|
|
translated in: _Selected Works_ - Edited by Ludwik Borkowski -
|
|
Amsterdam, North-Holland, 1970 pp. 197-217,
|
|
referenced in "History of Logic"
|
|
~ https://www.historyoflogic.com/logic-stoics.htm ).
|
|
For more about Aristotle's system, see barbara and related theorems.
|
|
|
|
A key part of the Stoic logic system is a set of five "indemonstrables"
|
|
assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is
|
|
difficult to assign specific ideas to specific thinkers. The indemonstrables
|
|
are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and
|
|
[Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus
|
|
ponens) ~ ax-mp , modus tollendo tollens (modus tollens) ~ mto , modus
|
|
ponendo tollens I ~ mptnan , modus ponendo tollens II ~ mptxor , and modus
|
|
tollendo ponens (exclusive-or version) ~ mtpxor . The first is an axiom, the
|
|
second is already proved; in this section we prove the other three. Since we
|
|
assume or prove all of indemonstrables, the system of logic we use here is as
|
|
at least as strong as the set of Stoic indemonstrables. Note that modus
|
|
tollendo ponens ~ mtpxor originally used exclusive-or, but over time the name
|
|
modus tollendo ponens has increasingly referred to an inclusive-or variation,
|
|
which is proved in ~ mtpor . This set of indemonstrables is not the entire
|
|
system of Stoic logic.
|
|
|
|
$)
|
|
|
|
${
|
|
$( Minor premise for modus ponendo tollens 1. $)
|
|
mptnan.min $e |- ph $.
|
|
$( Major premise for modus ponendo tollens 1. $)
|
|
mptnan.maj $e |- -. ( ph /\ ps ) $.
|
|
$( Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic.
|
|
See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and
|
|
rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after
|
|
~ mptxor ) as a "safer, and these days much more common" version of
|
|
modus ponendo tollens because it avoids confusion between inclusive-or
|
|
and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) $)
|
|
mptnan $p |- -. ps $=
|
|
( wn imnani ax-mp ) ABECABDFG $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus ponendo tollens 2. $)
|
|
mptxor.min $e |- ph $.
|
|
$( Major premise for modus ponendo tollens 2. $)
|
|
mptxor.maj $e |- ( ph \/_ ps ) $.
|
|
$( Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic.
|
|
Note that this uses exclusive-or ` \/_ ` . See rule 2 on
|
|
[Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in
|
|
[Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.) $)
|
|
mptxor $p |- -. ps $=
|
|
( wo wa wn wxo df-xor mpbi simpri mptnan ) ABCABEZABFGZABHMNFDABIJKL $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus tollendo ponens (inclusive-or version). $)
|
|
mtpor.min $e |- -. ph $.
|
|
$( Major premise for modus tollendo ponens (inclusive-or version). $)
|
|
mtpor.max $e |- ( ph \/ ps ) $.
|
|
$( Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism.
|
|
This is similar to ~ mtpxor , one of the five original "indemonstrables"
|
|
in Stoic logic. However, in Stoic logic this rule used exclusive-or,
|
|
while the name modus tollendo ponens often refers to a variant of the
|
|
rule that uses inclusive-or instead. The rule says, "if ` ph ` is not
|
|
true, and ` ph ` or ` ps ` (or both) are true, then ` ps ` must be
|
|
true." An alternate phrasing is, "Once you eliminate the impossible,
|
|
whatever remains, no matter how improbable, must be the truth." --
|
|
Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four,
|
|
ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof
|
|
shortened by Wolf Lammen, 11-Nov-2017.) $)
|
|
mtpor $p |- ps $=
|
|
( wn ori ax-mp ) AEBCABDFG $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus tollendo ponens (original exclusive-or version).
|
|
$)
|
|
mtpxor.min $e |- -. ph $.
|
|
$( Major premise for modus tollendo ponens (original exclusive-or version).
|
|
$)
|
|
mtpxor.maj $e |- ( ph \/_ ps ) $.
|
|
$( Modus tollendo ponens (original exclusive-or version), aka disjunctive
|
|
syllogism, similar to ~ mtpor , one of the five "indemonstrables" in
|
|
Stoic logic. The rule says, "if ` ph ` is not true, and either ` ph `
|
|
or ` ps ` (exclusively) are true, then ` ps ` must be true." Today the
|
|
name "modus tollendo ponens" often refers to a variant, the inclusive-or
|
|
version as defined in ~ mtpor . See rule 3 on [Lopez-Astorga] p. 12
|
|
(note that the "or" is the same as ~ mptxor , that is, it is
|
|
exclusive-or ~ df-xor ), rule 3 of [Sanford] p. 39 (where it is not as
|
|
clearly stated which kind of "or" is used but it appears to be in the
|
|
same sense as ~ mptxor ), and rule A5 in [Hitchcock] p. 5 (exclusive-or
|
|
is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.)
|
|
(Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ,
|
|
19-Apr-2019.) $)
|
|
mtpxor $p |- ps $=
|
|
( wxo wo xoror ax-mp mtpor ) ABCABEABFDABGHI $.
|
|
$}
|
|
|
|
${
|
|
$( Premise 1 for Stoic logic thema 2 version a. $)
|
|
stoic2a.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Premise 2 for Stoic logic thema 2 version a. $)
|
|
stoic2a.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
$( Stoic logic Thema 2 version a.
|
|
|
|
Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic
|
|
logic thema 2 as follows: "When from two assertibles a third follows,
|
|
and from the third and one (or both) of the two another follows, then
|
|
this other follows from the first two."
|
|
|
|
Bobzien uses constructs such as ` ph ` , ` ps |- ch ` ; in Metamath we
|
|
will represent that construct as ` ph /\ ps -> ch ` .
|
|
|
|
This version a is without the phrase "or both"; see ~ stoic2b for the
|
|
version with the phrase "or both". We already have this rule as
|
|
~ syldan , so here we show the equivalence and discourage its use.
|
|
(New usage is discouraged.) (Contributed by David A. Wheeler,
|
|
17-Feb-2019.) $)
|
|
stoic2a $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( syldan ) ABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise 1 for Stoic logic thema 2 version b. $)
|
|
stoic2b.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Premise 2 for Stoic logic thema 2 version b. $)
|
|
stoic2b.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Stoic logic Thema 2 version b. See ~ stoic2a .
|
|
|
|
Version b is with the phrase "or both". We already have this rule as
|
|
~ mpd3an3 , so here we prove the equivalence and discourage its use.
|
|
(New usage is discouraged.) (Contributed by David A. Wheeler,
|
|
17-Feb-2019.) $)
|
|
stoic2b $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( mpd3an3 ) ABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise 1 for Stoic logic thema 3. $)
|
|
stoic3.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Premise 2 for Stoic logic thema 3. $)
|
|
stoic3.2 $e |- ( ( ch /\ th ) -> ta ) $.
|
|
$( Stoic logic Thema 3.
|
|
|
|
Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3.
|
|
|
|
"When from two (assemblies) a third follows, and from the one that
|
|
follows (i.e., the third) together with another, external external
|
|
assumption, another follows, then other follows from the first two and
|
|
the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by
|
|
David A. Wheeler, 17-Feb-2019.) $)
|
|
stoic3 $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
|
|
( wa sylan 3impa ) ABDEABHCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$( Premise 1 for Stoic logic thema 4a. $)
|
|
stoic4a.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Premise 2 for Stoic logic thema 4a. $)
|
|
stoic4a.2 $e |- ( ( ch /\ ph /\ th ) -> ta ) $.
|
|
$( Stoic logic Thema 4 version a.
|
|
|
|
Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic
|
|
logic thema 4: "When from two assertibles a third follows, and from the
|
|
third and one (or both) of the two and one (or more) external
|
|
assertible(s) another follows, then this other follows from the first
|
|
two and the external(s)."
|
|
|
|
We use ` th ` to represent the "external" assertibles. This is version
|
|
a, which is without the phrase "or both"; see ~ stoic4b for the version
|
|
with the phrase "or both". (Contributed by David A. Wheeler,
|
|
17-Feb-2019.) $)
|
|
stoic4a $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
|
|
( w3a 3adant3 simp1 simp3 syl3anc ) ABDHCADEABCDFIABDJABDKGL $.
|
|
$}
|
|
|
|
${
|
|
$( Premise 1 for Stoic logic thema 4b. $)
|
|
stoic4b.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Premise 2 for Stoic logic thema 4b. $)
|
|
stoic4b.2 $e |- ( ( ( ch /\ ph /\ ps ) /\ th ) -> ta ) $.
|
|
$( Stoic logic Thema 4 version b.
|
|
|
|
This is version b, which is with the phrase "or both". See ~ stoic4a
|
|
for more information. (Contributed by David A. Wheeler,
|
|
17-Feb-2019.) $)
|
|
stoic4b $p |- ( ( ph /\ ps /\ th ) -> ta ) $=
|
|
( w3a 3adant3 simp1 simp2 simp3 syl31anc ) ABDHCABDEABCDFIABDJABDKABDLGM
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical implication (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
syl6an.1 $e |- ( ph -> ps ) $.
|
|
syl6an.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
syl6an.3 $e |- ( ( ps /\ th ) -> ta ) $.
|
|
$( A syllogism deduction combined with conjoining antecedents.
|
|
(Contributed by Alan Sare, 28-Oct-2011.) $)
|
|
syl6an $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wa jctild syl6 ) ACBDIEACDBGFJHK $.
|
|
$}
|
|
|
|
${
|
|
syl10.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl10.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
|
|
syl10.3 $e |- ( ch -> ( ta -> et ) ) $.
|
|
$( A nested syllogism inference. (Contributed by Alan Sare,
|
|
17-Jul-2011.) $)
|
|
syl10 $p |- ( ph -> ( ps -> ( th -> et ) ) ) $=
|
|
( wi syl6 syldd ) ABDEFHABCEFJGIKL $.
|
|
$}
|
|
|
|
$( Exportation implication also converting head from biconditional to
|
|
conditional. (Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
exbir $p |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) ->
|
|
( ph -> ( ps -> ( th -> ch ) ) ) ) $=
|
|
( wa wb wi bi2 imim2i expd ) ABEZCDFZGABDCGZLMKCDHIJ $.
|
|
|
|
$( ~ impexp with a 3-conjunct antecedent. (Contributed by Alan Sare,
|
|
31-Dec-2011.) $)
|
|
3impexp $p |- ( ( ( ph /\ ps /\ ch ) -> th ) <->
|
|
( ph -> ( ps -> ( ch -> th ) ) ) ) $=
|
|
( w3a wi id 3expd 3impd impbii ) ABCEDFZABCDFFFZKABCDKGHLABCDLGIJ $.
|
|
|
|
$( ~ 3impexp with biconditional consequent of antecedent that is commuted in
|
|
consequent. (Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
3impexpbicom $p |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <->
|
|
( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) $=
|
|
( w3a wb wi bicom imbi2 biimpcd mpi 3expd 3impexp biimpri syl6ibr impbii )
|
|
ABCFZDEGZHZABCEDGZHHHZTABCUATSUAGZRUAHZDEIZUCTUDSUARJKLMUBRUASUDUBABCUANOUE
|
|
PQ $.
|
|
|
|
${
|
|
3impexpbicomi.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $.
|
|
$( Deduction form of ~ 3impexpbicom . (Contributed by Alan Sare,
|
|
31-Dec-2011.) $)
|
|
3impexpbicomi $p |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) $=
|
|
( wb w3a bicomd 3exp ) ABCEDGABCHDEFIJ $.
|
|
$}
|
|
|
|
$( Closed form of ~ ancoms . (Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
ancomsimp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) ) $=
|
|
( wa ancom imbi1i ) ABDBADCABEF $.
|
|
|
|
${
|
|
expcomd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Deduction form of ~ expcom . (Contributed by Alan Sare,
|
|
22-Jul-2012.) $)
|
|
expcomd $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
|
|
( expd com23 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
expdcom.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Commuted form of ~ expd . (Contributed by Alan Sare, 18-Mar-2012.) $)
|
|
expdcom $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
|
|
( expd com3l ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Implication form of ~ simplbi2com . (Contributed by Alan Sare,
|
|
22-Jul-2012.) $)
|
|
simplbi2comg $p |- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) ) $=
|
|
( wa wb bi2 expcomd ) ABCDZEBCAAHFG $.
|
|
|
|
${
|
|
simplbi2com.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( A deduction eliminating a conjunct, similar to ~ simplbi2 .
|
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf
|
|
Lammen, 10-Nov-2012.) $)
|
|
simplbi2com $p |- ( ch -> ( ps -> ph ) ) $=
|
|
( simplbi2 com12 ) BCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
syl6ci.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6ci.2 $e |- ( ph -> th ) $.
|
|
syl6ci.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by Alan
|
|
Sare, 18-Mar-2012.) $)
|
|
syl6ci $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( a1d syl6c ) ABCDEFADBGIHJ $.
|
|
$}
|
|
|
|
${
|
|
mpisyl.1 $e |- ( ph -> ps ) $.
|
|
mpisyl.2 $e |- ch $.
|
|
mpisyl.3 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( A syllogism combined with a modus ponens inference. (Contributed by
|
|
Alan Sare, 25-Jul-2011.) $)
|
|
mpisyl $p |- ( ph -> th ) $=
|
|
( mpi syl ) ABDEBCDFGHI $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Predicate calculus mostly without distinct variables
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Universal quantifier (continued)
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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The universal quantifier was introduced above in ~ wal for use by ~ df-tru .
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See the comments in that section. In this section, we continue with the
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first "real" use of it.
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$)
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$( Declare some names for individual variables. $)
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$v x $.
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$v y $.
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$v z $.
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$v w $.
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$v v $.
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$v u $.
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$v t $.
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$( Let ` x ` be an individual variable. $)
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vx $f setvar x $.
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$( Let ` y ` be an individual variable. $)
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vy $f setvar y $.
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$( Let ` z ` be an individual variable. $)
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vz $f setvar z $.
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$( Let ` w ` be an individual variable. $)
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vw $f setvar w $.
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$( Let ` v ` be an individual variable. $)
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vv $f setvar v $.
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$( Let ` u ` be an individual variable. $)
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vu $f setvar u $.
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$( Let ` t ` be an individual variable. $)
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vt $f setvar t $.
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$( Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105.
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(Contributed by NM, 5-Aug-1993.) $)
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ax-5 $a |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.
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$( Axiom of Quantifier Commutation. This axiom says universal quantifiers
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can be swapped. One of the predicate logic axioms which do not involve
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equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint).
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Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2]
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p. 113. (Contributed by NM, 5-Aug-1993.) $)
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ax-7 $a |- ( A. x A. y ph -> A. y A. x ph ) $.
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${
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ax-g.1 $e |- ph $.
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$( Rule of Generalization. The postulated inference rule of predicate
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calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if
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something is unconditionally true, then it is true for all values of a
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variable. For example, if we have proved ` x = x ` , we can conclude
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` A. x x = x ` or even ` A. y x = x ` . Theorem ~ spi shows we can go
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the other way also: in other words we can add or remove universal
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quantifiers from the beginning of any theorem as required. (Contributed
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by NM, 5-Aug-1993.) $)
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ax-gen $a |- A. x ph $.
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$}
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${
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gen2.1 $e |- ph $.
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$( Generalization applied twice. (Contributed by NM, 30-Apr-1998.) $)
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gen2 $p |- A. x A. y ph $=
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( wal ax-gen ) ACEBACDFF $.
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$}
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${
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mpg.1 $e |- ( A. x ph -> ps ) $.
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mpg.2 $e |- ph $.
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$( Modus ponens combined with generalization. (Contributed by NM,
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24-May-1994.) $)
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mpg $p |- ps $=
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( wal ax-gen ax-mp ) ACFBACEGDH $.
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$}
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${
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mpgbi.1 $e |- ( A. x ph <-> ps ) $.
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mpgbi.2 $e |- ph $.
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$( Modus ponens on biconditional combined with generalization.
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(Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan,
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28-Oct-2008.) $)
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mpgbi $p |- ps $=
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( wal ax-gen mpbi ) ACFBACEGDH $.
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$}
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${
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mpgbir.1 $e |- ( ph <-> A. x ps ) $.
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mpgbir.2 $e |- ps $.
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$( Modus ponens on biconditional combined with generalization.
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(Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan,
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28-Oct-2008.) $)
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mpgbir $p |- ph $=
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( wal ax-gen mpbir ) ABCFBCEGDH $.
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$}
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${
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a7s.1 $e |- ( A. x A. y ph -> ps ) $.
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$( Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) $)
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a7s $p |- ( A. y A. x ph -> ps ) $=
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( wal ax-7 syl ) ACFDFADFCFBADCGEH $.
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$}
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${
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alimi.1 $e |- ( ph -> ps ) $.
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$( Inference quantifying both antecedent and consequent. (Contributed by
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NM, 5-Aug-1993.) $)
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alimi $p |- ( A. x ph -> A. x ps ) $=
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( wi wal ax-5 mpg ) ABEACFBCFECABCGDH $.
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$( Inference doubly quantifying both antecedent and consequent.
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(Contributed by NM, 3-Feb-2005.) $)
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2alimi $p |- ( A. x A. y ph -> A. x A. y ps ) $=
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( wal alimi ) ADFBDFCABDEGG $.
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$}
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$( Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
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(Proof shortened by O'Cat, 30-Mar-2008.) $)
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alim $p |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
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( ax-5 ) ABCD $.
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${
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al2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
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$( Inference quantifying antecedent, nested antecedent, and consequent.
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(Contributed by NM, 5-Aug-1993.) $)
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al2imi $p |- ( A. x ph -> ( A. x ps -> A. x ch ) ) $=
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( wal wi alimi alim syl ) ADFBCGZDFBDFCDFGAKDEHBCDIJ $.
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$}
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${
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alanimi.1 $e |- ( ( ph /\ ps ) -> ch ) $.
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$( Variant of ~ al2imi with conjunctive antecedent. (Contributed by Andrew
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|
Salmon, 8-Jun-2011.) $)
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alanimi $p |- ( ( A. x ph /\ A. x ps ) -> A. x ch ) $=
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( wal ex al2imi imp ) ADFBDFCDFABCDABCEGHI $.
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$}
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$c F/ $. $( The not-free symbol. $)
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$( Extend wff definition to include the not-free predicate. $)
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wnf $a wff F/ x ph $.
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$( Define the not-free predicate for wffs. This is read " ` x ` is not free
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in ` ph ` ". Not-free means that the value of ` x ` cannot affect the
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value of ` ph ` , e.g., any occurrence of ` x ` in ` ph ` is effectively
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|
bound by a "for all" or something that expands to one (such as "there
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|
exists"). In particular, substitution for a variable not free in a wff
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|
does not affect its value ( ~ sbf ). An example of where this is used is
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~ stdpc5 . See ~ nf2 for an alternate definition which does not involve
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nested quantifiers on the same variable.
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Not-free is a commonly used constraint, so it is useful to have a notation
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for it. Surprisingly, there is no common formal notation for it, so here
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we devise one. Our definition lets us work with the not-free notion
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within the logic itself rather than as a metalogical side condition.
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To be precise, our definition really means "effectively not free," because
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it is slightly less restrictive than the usual textbook definition for
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not-free (which only considers syntactic freedom). For example, ` x ` is
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effectively not free in the bare expression ` x = x ` , even though ` x `
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|
would be considered free in the usual textbook definition, because the
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value of ` x ` in the expression ` x = x ` cannot affect the truth of the
|
|
expression (and thus substitution will not change the result).
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(Contributed by Mario Carneiro, 11-Aug-2016.) $)
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df-nf $a |- ( F/ x ph <-> A. x ( ph -> A. x ph ) ) $.
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${
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nfi.1 $e |- ( ph -> A. x ph ) $.
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$( Deduce that ` x ` is not free in ` ph ` from the definition.
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(Contributed by Mario Carneiro, 11-Aug-2016.) $)
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nfi $p |- F/ x ph $=
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( wnf wal wi df-nf mpgbir ) ABDAABEFBABGCH $.
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$}
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${
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hbth.1 $e |- ph $.
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$( No variable is (effectively) free in a theorem.
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This and later "hypothesis-building" lemmas, with labels starting
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"hb...", allow us to construct proofs of formulas of the form
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|
` |- ( ph -> A. x ph ) ` from smaller formulas of this form. These are
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|
useful for constructing hypotheses that state " ` x ` is (effectively)
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|
not free in ` ph ` ." (Contributed by NM, 5-Aug-1993.) $)
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|
hbth $p |- ( ph -> A. x ph ) $=
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( wal ax-gen a1i ) ABDAABCEF $.
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$( No variable is (effectively) free in a theorem. (Contributed by Mario
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Carneiro, 11-Aug-2016.) $)
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nfth $p |- F/ x ph $=
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( hbth nfi ) ABABCDE $.
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$}
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${
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nfnth.1 $e |- -. ph $.
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|
$( No variable is (effectively) free in a non-theorem. (Contributed by
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Mario Carneiro, 6-Dec-2016.) $)
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nfnth $p |- F/ x ph $=
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( wal pm2.21i nfi ) ABAABDCEF $.
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$}
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$( The true constant has no free variables. (This can also be proven in one
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step with ~ nfv , but this proof does not use ~ ax-17 .) (Contributed by
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|
Mario Carneiro, 6-Oct-2016.) $)
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nftru $p |- F/ x T. $=
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( wtru tru nfth ) BACD $.
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${
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alimdh.1 $e |- ( ph -> A. x ph ) $.
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alimdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
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$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM,
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4-Jan-2002.) $)
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alimdh $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
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( wal wi al2imi syl ) AADGBDGCDGHEABCDFIJ $.
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|
$}
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$( Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
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albi $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) ) $=
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( wb wal bi1 al2imi bi2 impbid ) ABDZCEACEBCEJABCABFGJBACABHGI $.
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${
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alrimih.1 $e |- ( ph -> A. x ph ) $.
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alrimih.2 $e |- ( ph -> ps ) $.
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|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
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|
5-Aug-1993.) (New usage is discouraged.) $)
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|
alrimih $p |- ( ph -> A. x ps ) $=
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|
( wal alimi syl ) AACFBCFDABCEGH $.
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|
$}
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${
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albii.1 $e |- ( ph <-> ps ) $.
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|
$( Inference adding universal quantifier to both sides of an equivalence.
|
|
(Contributed by NM, 7-Aug-1994.) $)
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|
albii $p |- ( A. x ph <-> A. x ps ) $=
|
|
( wb wal albi mpg ) ABEACFBCFECABCGDH $.
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$( Inference adding 2 universal quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 9-Mar-1997.) $)
|
|
2albii $p |- ( A. x A. y ph <-> A. x A. y ps ) $=
|
|
( wal albii ) ADFBDFCABDEGG $.
|
|
$}
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${
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|
hbxfrbi.1 $e |- ( ph <-> ps ) $.
|
|
hbxfrbi.2 $e |- ( ps -> A. x ps ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
hbxfrbi $p |- ( ph -> A. x ph ) $=
|
|
( wal albii 3imtr4i ) BBCFAACFEDABCDGH $.
|
|
$}
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|
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${
|
|
nfbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equality theorem for not-free. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfbii $p |- ( F/ x ph <-> F/ x ps ) $=
|
|
( wal wi wnf albii imbi12i df-nf 3bitr4i ) AACEZFZCEBBCEZFZCEACGBCGMOCABL
|
|
NDABCDHIHACJBCJK $.
|
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|
|
${
|
|
nfxfr.2 $e |- F/ x ps $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfxfr $p |- F/ x ph $=
|
|
( wnf nfbii mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfxfrd.2 $e |- ( ch -> F/ x ps ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
nfxfrd $p |- ( ch -> F/ x ph ) $=
|
|
( wnf nfbii sylibr ) CBDGADGFABDEHI $.
|
|
$}
|
|
$}
|
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|
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${
|
|
alcoms.1 $e |- ( A. x A. y ph -> ps ) $.
|
|
$( Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) $)
|
|
alcoms $p |- ( A. y A. x ph -> ps ) $=
|
|
( wal ax-7 syl ) ACFDFADFCFBADCGEH $.
|
|
$}
|
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|
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${
|
|
hbal.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbal $p |- ( A. y ph -> A. x A. y ph ) $=
|
|
( wal alimi ax-7 syl ) ACEZABEZCEIBEAJCDFACBGH $.
|
|
$}
|
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|
|
$( Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
alcom $p |- ( A. x A. y ph <-> A. y A. x ph ) $=
|
|
( wal ax-7 impbii ) ACDBDABDCDABCEACBEF $.
|
|
|
|
${
|
|
alrimdh.1 $e |- ( ph -> A. x ph ) $.
|
|
alrimdh.2 $e |- ( ps -> A. x ps ) $.
|
|
alrimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
alrimdh $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wal alimdh syl5 ) BBDHACDHFABCDEGIJ $.
|
|
$}
|
|
|
|
${
|
|
albidh.1 $e |- ( ph -> A. x ph ) $.
|
|
albidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
albidh $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( wb wal alrimih albi syl ) ABCGZDHBDHCDHGALDEFIBCDJK $.
|
|
$}
|
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|
|
$( Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of
|
|
[WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 4-Jul-2014.) $)
|
|
19.26 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) ) $=
|
|
( wa wal simpl alimi simpr jca id alanimi impbii ) ABDZCEZACEZBCEZDNOPMACAB
|
|
FGMBCABHGIABMCMJKL $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by
|
|
NM, 3-Feb-2005.) $)
|
|
19.26-2 $p |- ( A. x A. y ( ph /\ ps ) <->
|
|
( A. x A. y ph /\ A. x A. y ps ) ) $=
|
|
( wa wal 19.26 albii bitri ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed
|
|
by NM, 13-Sep-2011.) $)
|
|
19.26-3an $p |- ( A. x ( ph /\ ps /\ ch )
|
|
<-> ( A. x ph /\ A. x ps /\ A. x ch ) ) $=
|
|
( wa wal w3a 19.26 anbi1i bitri df-3an albii 3bitr4i ) ABEZCEZDFZADFZBDFZEZ
|
|
CDFZEZABCGZDFQRTGPNDFZTEUANCDHUCSTABDHIJUBODABCKLQRTKM $.
|
|
|
|
$( Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.33 $p |- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
|
|
( wal wo orc alimi olc jaoi ) ACDABEZCDBCDAJCABFGBJCBAHGI $.
|
|
|
|
$( Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew
|
|
Salmon, 24-May-2011.) $)
|
|
alrot3 $p |- ( A. x A. y A. z ph <-> A. y A. z A. x ph ) $=
|
|
( wal alcom albii bitri ) ADEZCEBEIBEZCEABEDEZCEIBCFJKCABDFGH $.
|
|
|
|
$( Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.)
|
|
(Proof shortened by Wolf Lammen, 28-Jun-2014.) $)
|
|
alrot4 $p |- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph ) $=
|
|
( wal alrot3 albii alcom 3bitri ) AEFDFCFZBFACFZEFZDFZBFMBFZDFLBFEFZDFKNBAC
|
|
DEGHMBDIOPDLBEIHJ $.
|
|
|
|
$( Split a biconditional and distribute quantifier. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
albiim $p |- ( A. x ( ph <-> ps ) <->
|
|
( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) ) $=
|
|
( wb wal wi wa dfbi2 albii 19.26 bitri ) ABDZCEABFZBAFZGZCEMCENCEGLOCABHIMN
|
|
CJK $.
|
|
|
|
$( Split a biconditional and distribute 2 quantifiers. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2albiim $p |- ( A. x A. y ( ph <-> ps ) <->
|
|
( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) ) $=
|
|
( wb wal wi wa albiim albii 19.26 bitri ) ABEDFZCFABGDFZBAGDFZHZCFNCFOCFHMP
|
|
CABDIJNOCKL $.
|
|
|
|
${
|
|
hband.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
hband.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hban .
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
hband $p |- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) ) $=
|
|
( wa wal anim12d 19.26 syl6ibr ) ABCGZBDHZCDHZGLDHABMCNEFIBCDJK $.
|
|
$}
|
|
|
|
${
|
|
hb3and.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
hb3and.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
hb3and.3 $e |- ( ph -> ( th -> A. x th ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hb3an .
|
|
(Contributed by NM, 17-Feb-2013.) $)
|
|
hb3and $p |- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) ) $=
|
|
( w3a wal 3anim123d 19.26-3an syl6ibr ) ABCDIZBEJZCEJZDEJZINEJABOCPDQFGHK
|
|
BCDELM $.
|
|
$}
|
|
|
|
${
|
|
hbald.1 $e |- ( ph -> A. y ph ) $.
|
|
hbald.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbal .
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
hbald $p |- ( ph -> ( A. y ps -> A. x A. y ps ) ) $=
|
|
( wal alimdh ax-7 syl6 ) ABDGZBCGZDGKCGABLDEFHBDCIJ $.
|
|
$}
|
|
|
|
$( Declare the existential quantifier symbol. $)
|
|
$c E. $. $( Backwards E (read: "there exists") $)
|
|
|
|
$( Extend wff definition to include the existential quantifier ("there
|
|
exists"). $)
|
|
wex $a wff E. x ph $.
|
|
|
|
$( ` x ` is bound in ` E. x ph ` . One of the axioms of predicate logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-ie1 $a |- ( E. x ph -> A. x E. x ph ) $.
|
|
|
|
$( Define existential quantification. ` E. x ph ` means "there exists at
|
|
least one set ` x ` such that ` ph ` is true." One of the axioms of
|
|
predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-ie2 $a |- ( A. x ( ps -> A. x ps ) ->
|
|
( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $.
|
|
|
|
$( ` x ` is not free in ` E. x ph ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbe1 $p |- ( E. x ph -> A. x E. x ph ) $=
|
|
( ax-ie1 ) ABC $.
|
|
|
|
$( ` x ` is not free in ` E. x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfe1 $p |- F/ x E. x ph $=
|
|
( wex hbe1 nfi ) ABCBABDE $.
|
|
|
|
$( Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) $)
|
|
19.23ht $p |- ( A. x ( ps -> A. x ps ) ->
|
|
( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
|
|
( ax-ie2 ) ABCD $.
|
|
|
|
${
|
|
19.23h.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 1-Feb-2015.) $)
|
|
19.23h $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( wal wi wex wb ax-gen 19.23ht ax-mp ) BBCEFZCEABFCEACGBFHLCDIABCJK $.
|
|
$}
|
|
|
|
$( Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think
|
|
of it as "if ` ph ` can be refuted for all ` x ` , then it is not possible
|
|
to find an ` x ` for which ` ph ` holds" (and likewise for the converse).
|
|
Comparing this with ~ dfexdc illustrates that statements which look
|
|
similar (to someone used to classical logic) can be different
|
|
intuitionistically due to different placement of negations. (Contributed
|
|
by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario
|
|
Carneiro, 12-May-2015.) $)
|
|
alnex $p |- ( A. x -. ph <-> -. E. x ph ) $=
|
|
( wfal wi wal wex wn fal pm2.21i 19.23h dfnot albii 3bitr4i ) ACDZBEABFZCDA
|
|
GZBEOGACBCCBEHIJPNBAKLOKM $.
|
|
|
|
${
|
|
nex.1 $e |- -. ph $.
|
|
$( Generalization rule for negated wff. (Contributed by NM,
|
|
18-May-1994.) $)
|
|
nex $p |- -. E. x ph $=
|
|
( wn wex alnex mpgbi ) ADABEDBABFCG $.
|
|
$}
|
|
|
|
$( Defining ` E. x ph ` given decidability. It is common in classical logic
|
|
to define ` E. x ph ` as ` -. A. x -. ph ` but in intuitionistic logic
|
|
without a decidability condition, that is only an implication not an
|
|
equivalence, as seen at ~ exalim . (Contributed by Jim Kingdon,
|
|
15-Mar-2018.) $)
|
|
dfexdc $p |- ( DECID E. x ph
|
|
-> ( E. x ph <-> -. A. x -. ph ) ) $=
|
|
( wn wal wex wb wdc alnex a1i con2biidc ) ACBDZABEZKLCFLGABHIJ $.
|
|
|
|
$( One direction of a classical definition of existential quantification.
|
|
One direction of Definition of [Margaris] p. 49. For a decidable
|
|
proposition, this is an equivalence, as seen as ~ dfexdc . (Contributed
|
|
by Jim Kingdon, 29-Jul-2018.) $)
|
|
exalim $p |- ( E. x ph -> -. A. x -. ph ) $=
|
|
( wn wal wex alnex biimpi con2i ) ACBDZABEZIJCABFGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equality predicate (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The equality predicate was introduced above in ~ wceq for use by ~ df-tru .
|
|
See the comments in that section. In this section, we continue with the
|
|
first "real" use of it.
|
|
$)
|
|
|
|
$( Extend wff definition to include atomic formulas using the equality
|
|
predicate.
|
|
|
|
(Instead of introducing ~ weq as an axiomatic statement, as was done in an
|
|
older version of this database, we introduce it by "proving" a special
|
|
case of set theory's more general ~ wceq . This lets us avoid overloading
|
|
the ` = ` connective, thus preventing ambiguity that would complicate
|
|
certain Metamath parsers. However, logically ~ weq is considered to be a
|
|
primitive syntax, even though here it is artificially "derived" from
|
|
~ wceq . Note: To see the proof steps of this syntax proof, type "show
|
|
proof weq /all" in the Metamath program.) (Contributed by NM,
|
|
24-Jan-2006.) $)
|
|
weq $p wff x = y $=
|
|
( cv wceq ) ACBCD $.
|
|
|
|
$( Declare the membership predicate symbol. $)
|
|
$c e. $. $( Stylized epsilon $)
|
|
|
|
$( --- Start of patch to prevent connective overloading $)
|
|
${
|
|
$v A $.
|
|
$v B $.
|
|
wcel.cA $f class A $.
|
|
wcel.cB $f class B $.
|
|
$( Extend wff definition to include the membership connective between
|
|
classes.
|
|
|
|
(The purpose of introducing ` wff A e. B ` here is to allow us to
|
|
express i.e. "prove" the ~ wel of predicate calculus in terms of the
|
|
~ wceq of set theory, so that we don't "overload" the ` e. ` connective
|
|
with two syntax definitions. This is done to prevent ambiguity that
|
|
would complicate some Metamath parsers. The class variables ` A ` and
|
|
` B ` are introduced temporarily for the purpose of this definition but
|
|
otherwise not used in predicate calculus.) $)
|
|
wcel $a wff A e. B $.
|
|
$}
|
|
|
|
$( Extend wff definition to include atomic formulas with the epsilon
|
|
(membership) predicate. This is read " ` x ` is an element of
|
|
` y ` ," " ` x ` is a member of ` y ` ," " ` x ` belongs to ` y ` ,"
|
|
or " ` y ` contains ` x ` ." Note: The phrase " ` y ` includes
|
|
` x ` " means " ` x ` is a subset of ` y ` ;" to use it also for
|
|
` x e. y ` , as some authors occasionally do, is poor form and causes
|
|
confusion, according to George Boolos (1992 lecture at MIT).
|
|
|
|
This syntactical construction introduces a binary non-logical predicate
|
|
symbol ` e. ` (epsilon) into our predicate calculus. We will eventually
|
|
use it for the membership predicate of set theory, but that is irrelevant
|
|
at this point: the predicate calculus axioms for ` e. ` apply to any
|
|
arbitrary binary predicate symbol. "Non-logical" means that the predicate
|
|
is presumed to have additional properties beyond the realm of predicate
|
|
calculus, although these additional properties are not specified by
|
|
predicate calculus itself but rather by the axioms of a theory (in our
|
|
case set theory) added to predicate calculus. "Binary" means that the
|
|
predicate has two arguments.
|
|
|
|
(Instead of introducing ~ wel as an axiomatic statement, as was done in an
|
|
older version of this database, we introduce it by "proving" a special
|
|
case of set theory's more general ~ wcel . This lets us avoid overloading
|
|
the ` e. ` connective, thus preventing ambiguity that would complicate
|
|
certain Metamath parsers. However, logically ~ wel is considered to be a
|
|
primitive syntax, even though here it is artificially "derived" from
|
|
~ wcel . Note: To see the proof steps of this syntax proof, type "show
|
|
proof wel /all" in the Metamath program.) (Contributed by NM,
|
|
24-Jan-2006.) $)
|
|
wel $p wff x e. y $=
|
|
( cv wcel ) ACBCD $.
|
|
$( --- End of patch to prevent connective overloading $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include atomic formulas with the epsilon
|
|
(membership) predicate. This is read " ` x ` is an element of ` y ` ,"
|
|
" ` x ` is a member of ` y ` ," " ` x ` belongs to ` y ` ," or " ` y `
|
|
contains ` x ` ." Note: The phrase " ` y ` includes ` x ` " means
|
|
" ` x ` is a subset of ` y ` "; to use it also for ` x e. y ` (as some
|
|
authors occasionally do) is poor form and causes confusion.
|
|
|
|
After we introduce ~ cv and ~ wcel in set theory, this syntax construction
|
|
becomes redundant, since it can be derived with the proof
|
|
"vx cv vy cv wcel". @)
|
|
wel @a wff x e. y @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Axiom of Equality. One of the equality and substitution axioms of
|
|
predicate calculus with equality. This is similar to, but not quite, a
|
|
transitive law for equality (proved later as ~ equtr ). Axiom scheme C8'
|
|
in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of
|
|
[Monk2] p. 105.
|
|
|
|
Axioms ~ ax-8 through ~ ax-16 are the axioms having to do with equality,
|
|
substitution, and logical properties of our binary predicate ` e. ` (which
|
|
later in set theory will mean "is a member of"). Note that all axioms
|
|
except ~ ax-16 and ~ ax-17 are still valid even when ` x ` , ` y ` , and
|
|
` z ` are replaced with the same variable because they do not have any
|
|
distinct variable (Metamath's $d) restrictions. Distinct variable
|
|
restrictions are required for ~ ax-16 and ~ ax-17 only. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
ax-8 $a |- ( x = y -> ( x = z -> y = z ) ) $.
|
|
|
|
$( Axiom of Quantifier Substitution. One of the equality and substitution
|
|
axioms of predicate calculus with equality. Appears as Lemma L12 in
|
|
[Megill] p. 445 (p. 12 of the preprint).
|
|
|
|
The original version of this axiom was ~ ax-10o ("o" for "old") and was
|
|
replaced with this shorter ~ ax-10 in May 2008. The old axiom is proved
|
|
from this one as theorem ~ ax10o . Conversely, this axiom is proved from
|
|
~ ax-10o as theorem ~ ax10 . (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-10 $a |- ( A. x x = y -> A. y y = x ) $.
|
|
|
|
$( Axiom of Variable Substitution. One of the 5 equality axioms of predicate
|
|
calculus. The final consequent ` A. x ( x = y -> ph ) ` is a way of
|
|
expressing " ` y ` substituted for ` x ` in wff ` ph ` " (cf. ~ sb6 ). It
|
|
is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105,
|
|
from which it can be proved by cases.
|
|
|
|
Variants of this axiom which are equivalent in classical logic but which
|
|
have not been shown to be equivalent for intuitionistic logic are
|
|
~ ax11v , ~ ax11v2 and ~ ax-11o . (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-11 $a |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $.
|
|
|
|
$( Axiom of Quantifier Introduction. One of the equality and substitution
|
|
axioms of predicate calculus with equality. Informally, it says that
|
|
whenever ` z ` is distinct from ` x ` and ` y ` , and ` x = y ` is true,
|
|
then ` x = y ` quantified with ` z ` is also true. In other words, ` z `
|
|
is irrelevant to the truth of ` x = y ` . Axiom scheme C9' in [Megill]
|
|
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
|
|
in the literature but is easily proved from textbook predicate calculus by
|
|
cases.
|
|
|
|
This axiom has been modified from the original ~ ax-12 for compatibility
|
|
with intuitionistic logic. (Contributed by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
ax-i12 $a |- ( A. z z = x \/ ( A. z z = y \/
|
|
A. z ( x = y -> A. z x = y ) ) ) $.
|
|
|
|
$( Axiom of bundling. The general idea of this axiom is that two variables
|
|
are either distinct or non-distinct. That idea could be expressed as
|
|
` A. z z = x \/ -. A. z z = x ` . However, we instead choose an axiom
|
|
which has many of the same consequences, but which is different with
|
|
respect to a universe which contains only one object. ` A. z z = x ` holds
|
|
if ` z ` and ` x ` are the same variable, likewise for ` z ` and ` y ` ,
|
|
and ` A. x A. z ( x = y -> A. z x = y ) ` holds if ` z ` is distinct from
|
|
the others (and the universe has at least two objects).
|
|
|
|
As with other statements of the form "x is decidable (either true or
|
|
false)", this does not entail the full Law of the Excluded Middle (which
|
|
is the proposition that all statements are decidable), but instead merely
|
|
the assertion that particular kinds of statements are decidable (or in
|
|
this case, an assertion similar to decidability).
|
|
|
|
This axiom implies ~ ax-i12 as can be seen at ~ axi12 . Whether ~ ax-bndl
|
|
can be proved from the remaining axioms including ~ ax-i12 is not known.
|
|
|
|
The reason we call this "bundling" is that a statement without a distinct
|
|
variable constraint "bundles" together two statements, one in which the
|
|
two variables are the same and one in which they are different.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) $)
|
|
ax-bndl $a |- ( A. z z = x \/
|
|
( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) ) $.
|
|
|
|
$( Axiom of Specialization. A quantified wff implies the wff without a
|
|
quantifier (i.e. an instance, or special case, of the generalized wff).
|
|
In other words if something is true for all ` x ` , it is true for any
|
|
specific ` x ` (that would typically occur as a free variable in the wff
|
|
substituted for ` ph ` ). (A free variable is one that does not occur in
|
|
the scope of a quantifier: ` x ` and ` y ` are both free in ` x = y ` ,
|
|
but only ` x ` is free in ` A. y x = y ` .) Axiom scheme C5' in [Megill]
|
|
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
|
|
p. 67 (under his system S2, defined in the last paragraph on p. 77).
|
|
|
|
Note that the converse of this axiom does not hold in general, but a
|
|
weaker inference form of the converse holds and is expressed as rule
|
|
~ ax-gen . Conditional forms of the converse are given by ~ ax-12 ,
|
|
~ ax-16 , and ~ ax-17 .
|
|
|
|
Unlike the more general textbook Axiom of Specialization, we cannot choose
|
|
a variable different from ` x ` for the special case. For use, that
|
|
requires the assistance of equality axioms, and we deal with it later
|
|
after we introduce the definition of proper substitution - see ~ stdpc4 .
|
|
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-4 $a |- ( A. x ph -> ph ) $.
|
|
|
|
$( Specialization. Another name for ~ ax-4 . (Contributed by NM,
|
|
21-May-2008.) $)
|
|
sp $p |- ( A. x ph -> ph ) $=
|
|
( ax-4 ) ABC $.
|
|
|
|
$( Rederive the original version of the axiom from ~ ax-i12 . (Contributed
|
|
by Mario Carneiro, 3-Feb-2015.) $)
|
|
ax-12 $p |- ( -. A. z z = x -> ( -. A. z z = y ->
|
|
( x = y -> A. z x = y ) ) ) $=
|
|
( cv wceq wal wn wi wo ax-i12 ori ord ax-4 syl6 ) CDZADZECFZGZOBDZECFZGPSEZ
|
|
UACFHZCFZUBRTUCQTUCIABCJKLUBCMN $.
|
|
|
|
$( Another name for ~ ax-i12 . (Contributed by NM, 3-Feb-2015.) $)
|
|
ax12or $p |- ( A. z z = x \/ ( A. z z = y \/
|
|
A. z ( x = y -> A. z x = y ) ) ) $=
|
|
( ax-i12 ) ABCD $.
|
|
|
|
$( Axiom of Equality. One of the equality and substitution axioms for a
|
|
non-logical predicate in our predicate calculus with equality. It
|
|
substitutes equal variables into the left-hand side of the ` e. ` binary
|
|
predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint).
|
|
It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77.
|
|
"Non-logical" means that the predicate is not a primitive of predicate
|
|
calculus proper but instead is an extension to it. "Binary" means that
|
|
the predicate has two arguments. In a system of predicate calculus with
|
|
equality, like ours, equality is not usually considered to be a
|
|
non-logical predicate. In systems of predicate calculus without equality,
|
|
it typically would be. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-13 $a |- ( x = y -> ( x e. z -> y e. z ) ) $.
|
|
|
|
$( Axiom of Equality. One of the equality and substitution axioms for a
|
|
non-logical predicate in our predicate calculus with equality. It
|
|
substitutes equal variables into the right-hand side of the ` e. ` binary
|
|
predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).
|
|
It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-14 $a |- ( x = y -> ( z e. x -> z e. y ) ) $.
|
|
|
|
$( Bound-variable hypothesis builder for ` x = x ` . This theorem tells us
|
|
that any variable, including ` x ` , is effectively not free in
|
|
` x = x ` , even though ` x ` is technically free according to the
|
|
traditional definition of free variable. (The proof uses only ~ ax-5 ,
|
|
~ ax-8 , ~ ax-12 , and ~ ax-gen . This shows that this can be proved
|
|
without ~ ax-9 , even though the theorem ~ equid cannot be. A shorter
|
|
proof using ~ ax-9 is obtainable from ~ equid and ~ hbth .) (Contributed
|
|
by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) $)
|
|
hbequid $p |- ( x = x -> A. y x = x ) $=
|
|
( cv wceq wal wi wo ax12or ax-8 pm2.43i alimi a1d ax-4 jaoi ax-mp ) BCACZDZ
|
|
BEZRPPDZSBEZFZBEZGZGUAAABHRUAUCRTSQSBQSBAAIJKLZRUAUBUDUABMNNO $.
|
|
|
|
$( Proof that ~ ax-i12 follows from ~ ax-bndl . So that we can track which
|
|
theorems rely on ~ ax-bndl , proofs should reference ~ ax-i12 rather than
|
|
this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is
|
|
discouraged). (Proof modification is discouraged.) $)
|
|
axi12 $p |- ( A. z z = x \/ ( A. z z = y \/
|
|
A. z ( x = y -> A. z x = y ) ) ) $=
|
|
( cv wceq wal wi wo ax-bndl sp orim2i ax-mp ) CDZADZECFZMBDZECFZNPEZRCFGCFZ
|
|
AFZHZHOQSHZHABCIUAUBOTSQSAJKKL $.
|
|
|
|
$( Commutation law for identical variable specifiers. The antecedent and
|
|
consequent are true when ` x ` and ` y ` are substituted with the same
|
|
variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
alequcom $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( ax-10 ) ABC $.
|
|
|
|
${
|
|
alequcoms.1 $e |- ( A. x x = y -> ph ) $.
|
|
$( A commutation rule for identical variable specifiers. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
alequcoms $p |- ( A. y y = x -> ph ) $=
|
|
( weq wal alequcom syl ) CBECFBCEBFACBGDH $.
|
|
$}
|
|
|
|
${
|
|
nalequcoms.1 $e |- ( -. A. x x = y -> ph ) $.
|
|
$( A commutation rule for distinct variable specifiers. (Contributed by
|
|
NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) $)
|
|
nalequcoms $p |- ( -. A. y y = x -> ph ) $=
|
|
( cv wceq wal wn alequcom con3i syl ) CEZBEZFCGZHMLFBGZHAONBCIJDK $.
|
|
$}
|
|
|
|
$( Consequence of the definition of not-free. (Contributed by Mario
|
|
Carneiro, 26-Sep-2016.) $)
|
|
nfr $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
|
|
( wnf wal wi df-nf sp sylbi ) ABCAABDEZBDIABFIBGH $.
|
|
|
|
${
|
|
nfri.1 $e |- F/ x ph $.
|
|
$( Consequence of the definition of not-free. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfri $p |- ( ph -> A. x ph ) $=
|
|
( wnf wal wi nfr ax-mp ) ABDAABEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
nfrd.1 $e |- ( ph -> F/ x ps ) $.
|
|
$( Consequence of the definition of not-free in a context. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
nfrd $p |- ( ph -> ( ps -> A. x ps ) ) $=
|
|
( wnf wal wi nfr syl ) ABCEBBCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
alimd.1 $e |- F/ x ph $.
|
|
alimd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alimd $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( nfri alimdh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
alrimi.1 $e |- F/ x ph $.
|
|
alrimi.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimi $p |- ( ph -> A. x ps ) $=
|
|
( nfri alrimih ) ABCACDFEG $.
|
|
$}
|
|
|
|
${
|
|
nfd.1 $e |- F/ x ph $.
|
|
nfd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduce that ` x ` is not free in ` ps ` in a context. (Contributed by
|
|
Mario Carneiro, 24-Sep-2016.) $)
|
|
nfd $p |- ( ph -> F/ x ps ) $=
|
|
( wal wi wnf nfri alrimih df-nf sylibr ) ABBCFGZCFBCHAMCACDIEJBCKL $.
|
|
$}
|
|
|
|
${
|
|
nfdh.1 $e |- ( ph -> A. x ph ) $.
|
|
nfdh.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduce that ` x ` is not free in ` ps ` in a context. (Contributed by
|
|
Mario Carneiro, 24-Sep-2016.) $)
|
|
nfdh $p |- ( ph -> F/ x ps ) $=
|
|
( nfi nfd ) ABCACDFEG $.
|
|
$}
|
|
|
|
${
|
|
nfrimi.1 $e |- F/ x ph $.
|
|
nfrimi.2 $e |- F/ x ( ph -> ps ) $.
|
|
$( Moving an antecedent outside ` F/ ` . (Contributed by Jim Kingdon,
|
|
23-Mar-2018.) $)
|
|
nfrimi $p |- ( ph -> F/ x ps ) $=
|
|
( wal wi nfri ax-5 syl2im pm2.86i nfd ) ABCDABBCFZABGZNCFAACFMNCEHACDHABC
|
|
IJKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom ax-17 - first use of the $d distinct variable statement
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Axiom to quantify a variable over a formula in which it does not occur.
|
|
Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as
|
|
Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of
|
|
[Monk2] p. 113.
|
|
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-17 $a |- ( ph -> A. x ph ) $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( ~ ax-17 with antecedent. (Contributed by NM, 1-Mar-2013.) $)
|
|
a17d $p |- ( ph -> ( ps -> A. x ps ) ) $=
|
|
( wal wi ax-17 a1i ) BBCDEABCFG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( If ` x ` is not present in ` ph ` , then ` x ` is not free in ` ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfv $p |- F/ x ph $=
|
|
( ax-17 nfi ) ABABCD $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( ~ nfv with antecedent. Useful in proofs of deduction versions of
|
|
bound-variable hypothesis builders such as ~ nfimd . (Contributed by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
nfvd $p |- ( ph -> F/ x ps ) $=
|
|
( wnf nfv a1i ) BCDABCEF $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce Axiom of Existence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Existence. One of the equality and substitution axioms of
|
|
predicate calculus with equality. One thing this axiom tells us is that
|
|
at least one thing exists (although ~ ax-4 and possibly others also tell
|
|
us that, i.e. they are not valid in the empty domain of a "free logic").
|
|
In this form (not requiring that ` x ` and ` y ` be distinct) it was used
|
|
in an axiom system of Tarski (see Axiom B7' in footnote 1 of
|
|
[KalishMontague] p. 81.) Another name for this theorem is ~ a9e , which
|
|
has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-i9 $a |- E. x x = y $.
|
|
|
|
$( Derive ~ ax-9 from ~ ax-i9 , the modified version for intuitionistic
|
|
logic. Although ~ ax-9 does hold intuistionistically, in intuitionistic
|
|
logic it is weaker than ~ ax-i9 . (Contributed by NM, 3-Feb-2015.) $)
|
|
ax-9 $p |- -. A. x -. x = y $=
|
|
( cv wceq wn wal wex ax-i9 notnoti alnex mtbir ) ACBCDZEAFLAGZEMABHILAJK $.
|
|
|
|
$( ~ equid with some quantification and negation without using ~ ax-4 or
|
|
~ ax-17 . (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf
|
|
Lammen, 27-Feb-2014.) $)
|
|
equidqe $p |- -. A. y -. x = x $=
|
|
( weq wn wal ax-9 ax-8 pm2.43i con3i alimi mto ) AACZDZBEBACZDZBEBAFMOBNLNL
|
|
BAAGHIJK $.
|
|
|
|
$( A special case of ~ ax-4 without using ~ ax-4 or ~ ax-17 . (Contributed
|
|
by NM, 13-Jan-2011.) $)
|
|
ax4sp1 $p |- ( A. y -. x = x -> -. x = x ) $=
|
|
( weq wn wal equidqe pm2.21i ) AACDZBEHABFG $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Additional intuitionistic axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . One of the axioms of predicate logic.
|
|
(Contributed by Mario Carneiro, 31-Jan-2015.) $)
|
|
ax-ial $a |- ( A. x ph -> A. x A. x ph ) $.
|
|
|
|
$( Axiom of quantifier collection. (Contributed by Mario Carneiro,
|
|
31-Jan-2015.) $)
|
|
ax-i5r $a |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) ) $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Predicate calculus including ax-4, without distinct variables
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
spi.1 $e |- A. x ph $.
|
|
$( Inference rule reversing generalization. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
spi $p |- ph $=
|
|
( wal ax-4 ax-mp ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
sps.1 $e |- ( ph -> ps ) $.
|
|
$( Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) $)
|
|
sps $p |- ( A. x ph -> ps ) $=
|
|
( wal sp syl ) ACEABACFDG $.
|
|
$}
|
|
|
|
${
|
|
spsd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) $)
|
|
spsd $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( wal sp syl5 ) BDFBACBDGEH $.
|
|
$}
|
|
|
|
${
|
|
nfbidf.1 $e |- F/ x ph $.
|
|
nfbidf.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An equality theorem for effectively not free. (Contributed by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
nfbidf $p |- ( ph -> ( F/ x ps <-> F/ x ch ) ) $=
|
|
( wal wi wnf nfri albidh imbi12d df-nf 3bitr4g ) ABBDGZHZDGCCDGZHZDGBDICD
|
|
IAPRDADEJZABCOQFABCDSFKLKBDMCDMN $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . Example in Appendix in [Megill] p. 450
|
|
(p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
hba1 $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( ax-ial ) ABC $.
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfa1 $p |- F/ x A. x ph $=
|
|
( wal hba1 nfi ) ABCBABDE $.
|
|
|
|
${
|
|
a5i.1 $e |- ( A. x ph -> ps ) $.
|
|
$( Inference generalizing a consequent. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
a5i $p |- ( A. x ph -> A. x ps ) $=
|
|
( wal wi hba1 ax-5 syl5 mpg ) ACEZBFZKBCEZFCKKCELCEMACGKBCHIDJ $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` F/ x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnf1 $p |- F/ x F/ x ph $=
|
|
( wnf wal wi df-nf nfa1 nfxfr ) ABCAABDEZBDBABFIBGH $.
|
|
|
|
${
|
|
hb.1 $e |- ( ph -> A. x ph ) $.
|
|
hb.2 $e |- ( ps -> A. x ps ) $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph -> ps ) ` . (Contributed by NM, 5-Aug-1993.) (Proof shortened
|
|
by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) $)
|
|
hbim $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wi wal ax-4 imim12i ax-i5r imim1i alimi 3syl ) ABFZACGZBCGZFOBFZCGNCGOA
|
|
BPACHEIABCJQNCAOBDKLM $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph \/ ps ) ` . (Contributed by NM, 5-Aug-1993.) (Revised by NM,
|
|
2-Feb-2015.) $)
|
|
hbor $p |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) ) $=
|
|
( wo wal orc alimi syl olc jaoi ) AABFZCGZBAACGNDAMCABHIJBBCGNEBMCBAKIJL
|
|
$.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph /\ ps ) ` . (Contributed by NM, 5-Aug-1993.) (Proof shortened
|
|
by Mario Carneiro, 2-Feb-2015.) $)
|
|
hban $p |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) ) $=
|
|
( wa wal anim12i 19.26 sylibr ) ABFZACGZBCGZFKCGALBMDEHABCIJ $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph <-> ps ) ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbbi $p |- ( ( ph <-> ps ) -> A. x ( ph <-> ps ) ) $=
|
|
( wb wi wa dfbi2 hbim hban hbxfrbi ) ABFABGZBAGZHCABIMNCABCDEJBACEDJKL $.
|
|
|
|
hb.3 $e |- ( ch -> A. x ch ) $.
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph \/ ps \/ ch ) ` . (Contributed by NM, 14-Sep-2003.) $)
|
|
hb3or $p |- ( ( ph \/ ps \/ ch ) -> A. x ( ph \/ ps \/ ch ) ) $=
|
|
( w3o wo df-3or hbor hbxfrbi ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph /\ ps /\ ch ) ` . (Contributed by NM, 14-Sep-2003.) $)
|
|
hb3an $p |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) ) $=
|
|
( w3a wa df-3an hban hbxfrbi ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
$}
|
|
|
|
$( Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) $)
|
|
hba2 $p |- ( A. y A. x ph -> A. x A. y A. x ph ) $=
|
|
( wal hba1 hbal ) ABDBCABEF $.
|
|
|
|
$( Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) $)
|
|
hbia1 $p |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> A. x ps ) ) $=
|
|
( wal hba1 hbim ) ACDBCDCACEBCEF $.
|
|
|
|
${
|
|
19.3h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( A wff may be quantified with a variable not free in it. Theorem 19.3 of
|
|
[Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM,
|
|
21-May-2007.) $)
|
|
19.3h $p |- ( A. x ph <-> ph ) $=
|
|
( wal ax-4 impbii ) ABDAABECF $.
|
|
$}
|
|
|
|
${
|
|
19.3.1 $e |- F/ x ph $.
|
|
$( A wff may be quantified with a variable not free in it. Theorem 19.3 of
|
|
[Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
19.3 $p |- ( A. x ph <-> ph ) $=
|
|
( wal sp nfri impbii ) ABDAABEABCFG $.
|
|
$}
|
|
|
|
${
|
|
19.16.1 $e |- F/ x ph $.
|
|
$( Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) $)
|
|
19.16 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> A. x ps ) ) $=
|
|
( wal wb 19.3 albi syl5bbr ) AACEABFCEBCEACDGABCHI $.
|
|
$}
|
|
|
|
${
|
|
19.17.1 $e |- F/ x ps $.
|
|
$( Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) $)
|
|
19.17 $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> ps ) ) $=
|
|
( wb wal albi 19.3 syl6bb ) ABECFACFBCFBABCGBCDHI $.
|
|
$}
|
|
|
|
${
|
|
19.21h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of
|
|
as " ` x ` is not free in ` ph ` ." New proofs should use ~ 19.21
|
|
instead. (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
19.21h $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( wi wal alim syl5 hba1 hbim ax-4 imim2i alrimih impbii ) ABEZCFZABCFZEZA
|
|
ACFPQDABCGHROCAQCDBCIJQBABCKLMN $.
|
|
$}
|
|
|
|
${
|
|
19.21bi.1 $e |- ( ph -> A. x ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.21bi $p |- ( ph -> ps ) $=
|
|
( wal ax-4 syl ) ABCEBDBCFG $.
|
|
$}
|
|
|
|
${
|
|
19.21bbi.1 $e |- ( ph -> A. x A. y ps ) $.
|
|
$( Inference removing double quantifier. (Contributed by NM,
|
|
20-Apr-1994.) $)
|
|
19.21bbi $p |- ( ph -> ps ) $=
|
|
( wal 19.21bi ) ABDABDFCEGG $.
|
|
$}
|
|
|
|
${
|
|
19.27h.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.27h $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
|
|
( wa wal 19.26 19.3h anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.27.1 $e |- F/ x ps $.
|
|
$( Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.27 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
|
|
( wa wal 19.26 19.3 anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.28h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.28h $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
|
|
( wa wal 19.26 19.3h anbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.28.1 $e |- F/ x ph $.
|
|
$( Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.28 $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
|
|
( wa wal 19.26 19.3 anbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
|
|
$}
|
|
|
|
${
|
|
nfan1.1 $e |- F/ x ph $.
|
|
nfan1.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( A closed form of ~ nfan . (Contributed by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
nfan1 $p |- F/ x ( ph /\ ps ) $=
|
|
( wa wal nfrd imdistani nfri 19.28h sylibr nfi ) ABFZCNABCGZFNCGABOABCEHI
|
|
ABCACDJKLM $.
|
|
$}
|
|
|
|
${
|
|
nfan.1 $e |- F/ x ph $.
|
|
nfan.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph /\ ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortened by Wolf Lammen, 13-Jan-2018.) $)
|
|
nfan $p |- F/ x ( ph /\ ps ) $=
|
|
( wnf a1i nfan1 ) ABCDBCFAEGH $.
|
|
|
|
nfan.3 $e |- F/ x ch $.
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph /\ ps /\ ch ) ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nf3an $p |- F/ x ( ph /\ ps /\ ch ) $=
|
|
( w3a wa df-3an nfan nfxfr ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
$}
|
|
|
|
${
|
|
nford.1 $e |- ( ph -> F/ x ps ) $.
|
|
nford.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps \/ ch ) ` . (Contributed by Jim Kingdon, 29-Oct-2019.) $)
|
|
nford $p |- ( ph -> F/ x ( ps \/ ch ) ) $=
|
|
( wo wal wi wnf wa df-nf anbi12i biimpi syl2anc 19.26 sylibr alimi imim2i
|
|
orc olc jaao syl ) ABCGZUDDHZIZDHZUDDJABBDHZIZCCDHZIZKZDHZUGAUIDHZUKDHZKZ
|
|
UMABDJZCDJZUPEFUQURKUPUQUNURUOBDLCDLMNOUIUKDPQULUFDUIBUEUKCUHUEBBUDDBCTRS
|
|
UJUECCUDDCBUARSUBRUCUDDLQ $.
|
|
$}
|
|
|
|
${
|
|
nfand.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfand.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps /\ ch ) ` . (Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfand $p |- ( ph -> F/ x ( ps /\ ch ) ) $=
|
|
( wa wal wnf jca df-nf anbi12i 19.26 bitr4i prth syl6ibr alimi sylbi syl
|
|
wi sylibr ) ABCGZUBDHZTZDHZUBDIABDIZCDIZGZUEAUFUGEFJUHBBDHZTZCCDHZTZGZDHZ
|
|
UEUHUJDHZULDHZGUNUFUOUGUPBDKCDKLUJULDMNUMUDDUMUBUIUKGUCBUICUKOBCDMPQRSUBD
|
|
KUA $.
|
|
|
|
nfand.3 $e |- ( ph -> F/ x th ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ nf3an .
|
|
(Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro,
|
|
16-Oct-2016.) $)
|
|
nf3and $p |- ( ph -> F/ x ( ps /\ ch /\ th ) ) $=
|
|
( w3a wa df-3an nfand nfxfrd ) BCDIBCJZDJAEBCDKANDEABCEFGLHLM $.
|
|
$}
|
|
|
|
${
|
|
hbim1.1 $e |- ( ph -> A. x ph ) $.
|
|
hbim1.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( A closed form of ~ hbim . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbim1 $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wi wal a2i 19.21h sylibr ) ABFZABCGZFKCGABLEHABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
nfim1.1 $e |- F/ x ph $.
|
|
nfim1.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( A closed form of ~ nfim . (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen,
|
|
2-Jan-2018.) $)
|
|
nfim1 $p |- F/ x ( ph -> ps ) $=
|
|
( wi nfri nfrd hbim1 nfi ) ABFCABCACDGABCEHIJ $.
|
|
$}
|
|
|
|
${
|
|
nfim.1 $e |- F/ x ph $.
|
|
nfim.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph -> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
nfim $p |- F/ x ( ph -> ps ) $=
|
|
( wnf a1i nfim1 ) ABCDBCFAEGH $.
|
|
$}
|
|
|
|
${
|
|
hbimd.1 $e |- ( ph -> A. x ph ) $.
|
|
hbimd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
hbimd.3 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbim .
|
|
(Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) $)
|
|
hbimd $p |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) $=
|
|
( wi wal imim2d ax-4 imim1i ax-i5r syl syl6 imim1d alimdh syld ) ABCHZBDI
|
|
ZCHZDIZSDIASBCDIZHZUBACUCBGJUDTUCHUBTBUCBDKLBCDMNOAUASDEABTCFPQR $.
|
|
$}
|
|
|
|
${
|
|
nfor.1 $e |- F/ x ph $.
|
|
nfor.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph \/ ps ) ` . (Contributed by Jim Kingdon, 11-Mar-2018.) $)
|
|
nfor $p |- F/ x ( ph \/ ps ) $=
|
|
( wo nfri hbor nfi ) ABFCABCACDGBCEGHI $.
|
|
$}
|
|
|
|
${
|
|
hbbid.1 $e |- ( ph -> A. x ph ) $.
|
|
hbbid.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
hbbid.3 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbbi .
|
|
(Contributed by NM, 1-Jan-2002.) $)
|
|
hbbid $p |- ( ph -> ( ( ps <-> ch ) -> A. x ( ps <-> ch ) ) ) $=
|
|
( wi wa wal wb hbimd anim12d dfbi2 albiim 3imtr4g ) ABCHZCBHZIQDJZRDJZIBC
|
|
KZUADJAQSRTABCDEFGLACBDEGFLMBCNBCDOP $.
|
|
$}
|
|
|
|
${
|
|
nfal.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfal $p |- F/ x A. y ph $=
|
|
( wal nfri hbal nfi ) ACEBABCABDFGH $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` F/ y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
nfnf $p |- F/ x F/ y ph $=
|
|
( wnf wal wi df-nf nfal nfim nfxfr ) ACEAACFZGZCFBACHMBCALBDABCDIJIK $.
|
|
$}
|
|
|
|
$( Closed form of ~ nfal . (Contributed by Jim Kingdon, 11-May-2018.) $)
|
|
nfalt $p |- ( A. y F/ x ph -> F/ x A. y ph ) $=
|
|
( wal wi wnf alim alcom syl6ib alimi df-nf albii bitri 3imtr4i ) AABDZEZCDZ
|
|
BDZACDZSBDZEZBDABFZCDZSBFQUABQSOCDTAOCGACBHIJUCPBDZCDRUBUDCABKLPCBHMSBKN $.
|
|
|
|
$( Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nfa2 $p |- F/ x A. y A. x ph $=
|
|
( wal nfa1 nfal ) ABDBCABEF $.
|
|
|
|
$( Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nfia1 $p |- F/ x ( A. x ph -> A. x ps ) $=
|
|
( wal nfa1 nfim ) ACDBCDCACEBCEF $.
|
|
|
|
$( Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
27-May-1997.) (New usage is discouraged.) $)
|
|
19.21ht $p |- ( A. x ( ph -> A. x ph ) ->
|
|
( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
|
|
( wal wi alim imim2d com12 sps hba1 ax-4 a1i hbimd imim2i alimi syl6 impbid
|
|
) AACDZEZCDZABEZCDZABCDZEZSUBUDECUBSUDUBRUCAABCFGHITUDUDCDUBTAUCCSCJSCKUCUC
|
|
CDETBCJLMUDUACUCBABCKNOPQ $.
|
|
|
|
$( Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
27-May-1997.) $)
|
|
19.21t $p |- ( F/ x ph ->
|
|
( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
|
|
( wnf wal wi wb df-nf 19.21ht sylbi ) ACDAACEFCEABFCEABCEFGACHABCIJ $.
|
|
|
|
${
|
|
19.21.1 $e |- F/ x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of
|
|
as " ` x ` is not free in ` ph ` ." (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.21 $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( wnf wi wal wb 19.21t ax-mp ) ACEABFCGABCGFHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
stdpc5.1 $e |- F/ x ph $.
|
|
$( An axiom scheme of standard predicate calculus that emulates Axiom 5 of
|
|
[Mendelson] p. 69. The hypothesis ` F/ x ph ` can be thought of as
|
|
emulating " ` x ` is not free in ` ph ` ." With this definition, the
|
|
meaning of "not free" is less restrictive than the usual textbook
|
|
definition; for example ` x ` would not (for us) be free in ` x = x ` by
|
|
~ nfequid . This theorem scheme can be proved as a metatheorem of
|
|
Mendelson's axiom system, even though it is slightly stronger than his
|
|
Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) $)
|
|
stdpc5 $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
|
|
( wi wal 19.21 biimpi ) ABECFABCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfimd.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfimd.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps -> ch ) ` . (Contributed by Mario Carneiro, 24-Sep-2016.)
|
|
(Proof shortened by Wolf Lammen, 30-Dec-2017.) $)
|
|
nfimd $p |- ( ph -> F/ x ( ps -> ch ) ) $=
|
|
( wnf wi wal nfnf1 nfri imim2d 19.21t biimprd syl9r alrimdh df-nf syl6ibr
|
|
nfr sylc ) ABDGZCDGZBCHZDGZEFUAUBUCUCDIZHZDIUDUAUBUFDUADBDJKUBDCDJKUBUCBC
|
|
DIZHZUAUEUBCUGBCDSLUAUEUHBCDMNOPUCDQRT $.
|
|
$}
|
|
|
|
${
|
|
aaanh.1 $e |- ( ph -> A. y ph ) $.
|
|
aaanh.2 $e |- ( ps -> A. x ps ) $.
|
|
$( Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) $)
|
|
aaanh $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
|
|
( wa wal 19.28h albii hbal 19.27h bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJ
|
|
AOCBCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
aaan.1 $e |- F/ y ph $.
|
|
aaan.2 $e |- F/ x ps $.
|
|
$( Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) $)
|
|
aaan $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
|
|
( wa wal 19.28 albii nfal 19.27 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
nfbid.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfbid.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps <-> ch ) ` . (Contributed by Mario Carneiro, 24-Sep-2016.)
|
|
(Proof shortened by Wolf Lammen, 29-Dec-2017.) $)
|
|
nfbid $p |- ( ph -> F/ x ( ps <-> ch ) ) $=
|
|
( wb wi wa dfbi2 nfimd nfand nfxfrd ) BCGBCHZCBHZIADBCJANODABCDEFKACBDFEK
|
|
LM $.
|
|
$}
|
|
|
|
${
|
|
nfbi.1 $e |- F/ x ph $.
|
|
nfbi.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph <-> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
nfbi $p |- F/ x ( ph <-> ps ) $=
|
|
( wb wnf wtru a1i nfbid trud ) ABFCGHABCACGHDIBCGHEIJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The existential quantifier
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( If a wff is true, it is true for at least one instance. Special case of
|
|
Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.8a $p |- ( ph -> E. x ph ) $=
|
|
( wex wi wal id hbe1 19.23h mpbir spi ) AABCZDZBLBEKKDKFAKBABGHIJ $.
|
|
|
|
${
|
|
19.23bi.1 $e |- ( E. x ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.23bi $p |- ( ph -> ps ) $=
|
|
( wex 19.8a syl ) AACEBACFDG $.
|
|
$}
|
|
|
|
${
|
|
exlimih.1 $e |- ( ps -> A. x ps ) $.
|
|
exlimih.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
exlimih $p |- ( E. x ph -> ps ) $=
|
|
( wi wex 19.23h mpgbi ) ABFACGBFCABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
exlimi.1 $e |- F/ x ps $.
|
|
exlimi.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
exlimi $p |- ( E. x ph -> ps ) $=
|
|
( nfri exlimih ) ABCBCDFEG $.
|
|
$}
|
|
|
|
${
|
|
exlimd2.1 $e |- ( ph -> A. x ph ) $.
|
|
exlimd2.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
exlimd2.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to ~ exlimdh
|
|
but with one slightly different hypothesis. (Contributed by Jim
|
|
Kingdon, 30-Dec-2017.) $)
|
|
exlimd2 $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( wal wi wex alrimih 19.23ht biimpd sylc ) ACCDHIZDHZBCIZDHZBDJCIZAODEFKA
|
|
QDEGKPRSBCDLMN $.
|
|
$}
|
|
|
|
${
|
|
exlimdh.1 $e |- ( ph -> A. x ph ) $.
|
|
exlimdh.2 $e |- ( ch -> A. x ch ) $.
|
|
exlimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
28-Jan-1997.) $)
|
|
exlimdh $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( wi wal wex alrimih 19.23h sylib ) ABCHZDIBDJCHANDEGKBCDFLM $.
|
|
$}
|
|
|
|
${
|
|
exlimd.1 $e |- F/ x ph $.
|
|
exlimd.2 $e |- F/ x ch $.
|
|
exlimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon,
|
|
18-Jun-2018.) $)
|
|
exlimd $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( nfri exlimdh ) ABCDADEHCDFHGI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
exlimiv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90.
|
|
|
|
This inference, along with our many variants is used to implement a
|
|
metatheorem called "Rule C" that is given in many logic textbooks. See,
|
|
for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or
|
|
Rule C in Hirst and Hirst's _A Primer for Logic and Proof_ p. 59 (PDF
|
|
p. 65) at ~ http://www.mathsci.appstate.edu/~~jlh/primer/hirst.pdf .
|
|
|
|
In informal proofs, the statement "Let C be an element such that..."
|
|
almost always means an implicit application of Rule C.
|
|
|
|
In essence, Rule C states that if we can prove that some element ` x `
|
|
exists satisfying a wff, i.e. ` E. x ph ( x ) ` where ` ph ( x ) ` has
|
|
` x ` free, then we can use ` ph ( ` C ` ) ` as a hypothesis for the
|
|
proof where C is a new (ficticious) constant not appearing previously in
|
|
the proof, nor in any axioms used, nor in the theorem to be proved. The
|
|
purpose of Rule C is to get rid of the existential quantifier.
|
|
|
|
We cannot do this in Metamath directly. Instead, we use the original
|
|
` ph ` (containing ` x ` ) as an antecedent for the main part of the
|
|
proof. We eventually arrive at ` ( ph -> ps ) ` where ` ps ` is the
|
|
theorem to be proved and does not contain ` x ` . Then we apply
|
|
~ exlimiv to arrive at ` ( E. x ph -> ps ) ` . Finally, we separately
|
|
prove ` E. x ph ` and detach it with modus ponens ~ ax-mp to arrive at
|
|
the final theorem ` ps ` . (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by NM, 25-Jul-2012.) $)
|
|
exlimiv $p |- ( E. x ph -> ps ) $=
|
|
( ax-17 exlimih ) ABCBCEDF $.
|
|
$}
|
|
|
|
$( Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 4-Jul-2014.) $)
|
|
exim $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) ) $=
|
|
( wi wal wex hba1 hbe1 19.8a imim2i sps exlimdh ) ABDZCEABCFZCMCGBCHMANDCBN
|
|
ABCIJKL $.
|
|
|
|
${
|
|
eximi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding existential quantifier to antecedent and consequent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eximi $p |- ( E. x ph -> E. x ps ) $=
|
|
( wi wex exim mpg ) ABEACFBCFECABCGDH $.
|
|
|
|
$( Inference adding 2 existential quantifiers to antecedent and consequent.
|
|
(Contributed by NM, 3-Feb-2005.) $)
|
|
2eximi $p |- ( E. x E. y ph -> E. x E. y ps ) $=
|
|
( wex eximi ) ADFBDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
eximii.1 $e |- E. x ph $.
|
|
eximii.2 $e |- ( ph -> ps ) $.
|
|
$( Inference associated with ~ eximi . (Contributed by BJ, 3-Feb-2018.) $)
|
|
eximii $p |- E. x ps $=
|
|
( wex eximi ax-mp ) ACFBCFDABCEGH $.
|
|
$}
|
|
|
|
$( A transformation of quantifiers and logical connectives. (Contributed by
|
|
NM, 19-Aug-1993.) $)
|
|
alinexa $p |- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) ) $=
|
|
( wn wi wal wa wex imnan albii alnex bitri ) ABDEZCFABGZDZCFNCHDMOCABIJNCKL
|
|
$.
|
|
|
|
$( Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
exbi $p |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) ) $=
|
|
( wb wal wex wi bi1 alimi exim syl bi2 impbid ) ABDZCEZACFZBCFZOABGZCEPQGNR
|
|
CABHIABCJKOBAGZCEQPGNSCABLIBACJKM $.
|
|
|
|
${
|
|
exbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding existential quantifier to both sides of an equivalence.
|
|
(Contributed by NM, 24-May-1994.) $)
|
|
exbii $p |- ( E. x ph <-> E. x ps ) $=
|
|
( wb wex exbi mpg ) ABEACFBCFECABCGDH $.
|
|
|
|
$( Inference adding 2 existential quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 16-Mar-1995.) $)
|
|
2exbii $p |- ( E. x E. y ph <-> E. x E. y ps ) $=
|
|
( wex exbii ) ADFBDFCABDEGG $.
|
|
|
|
$( Inference adding 3 existential quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 2-May-1995.) $)
|
|
3exbii $p |- ( E. x E. y E. z ph <-> E. x E. y E. z ps ) $=
|
|
( wex exbii 2exbii ) AEGBEGCDABEFHI $.
|
|
$}
|
|
|
|
$( Commutation of conjunction inside an existential quantifier. (Contributed
|
|
by NM, 18-Aug-1993.) $)
|
|
exancom $p |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) ) $=
|
|
( wa ancom exbii ) ABDBADCABEF $.
|
|
|
|
${
|
|
alrimdd.1 $e |- F/ x ph $.
|
|
alrimdd.2 $e |- ( ph -> F/ x ps ) $.
|
|
alrimdd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimdd $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wal nfrd alimd syld ) ABBDHCDHABDFIABCDEGJK $.
|
|
$}
|
|
|
|
${
|
|
alrimd.1 $e |- F/ x ph $.
|
|
alrimd.2 $e |- F/ x ps $.
|
|
alrimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimd $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wnf a1i alrimdd ) ABCDEBDHAFIGJ $.
|
|
$}
|
|
|
|
${
|
|
eximdh.1 $e |- ( ph -> A. x ph ) $.
|
|
eximdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
20-May-1996.) $)
|
|
eximdh $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( wi wal wex alrimih exim syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
eximd.1 $e |- F/ x ph $.
|
|
eximd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
eximd $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( nfri eximdh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
nexd.1 $e |- ( ph -> A. x ph ) $.
|
|
nexd.2 $e |- ( ph -> -. ps ) $.
|
|
$( Deduction for generalization rule for negated wff. (Contributed by NM,
|
|
2-Jan-2002.) $)
|
|
nexd $p |- ( ph -> -. E. x ps ) $=
|
|
( wn wal wex alrimih alnex sylib ) ABFZCGBCHFALCDEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
exbidh.1 $e |- ( ph -> A. x ph ) $.
|
|
exbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
exbidh $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( wb wal wex alrimih exbi syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
albid.1 $e |- F/ x ph $.
|
|
albid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
albid $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( nfri albidh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
exbid.1 $e |- F/ x ph $.
|
|
exbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
exbid $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( nfri exbidh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
$( Simplification of an existentially quantified conjunction. (Contributed
|
|
by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
exsimpl $p |- ( E. x ( ph /\ ps ) -> E. x ph ) $=
|
|
( wa simpl eximi ) ABDACABEF $.
|
|
|
|
$( Simplification of an existentially quantified conjunction. (Contributed
|
|
by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
exsimpr $p |- ( E. x ( ph /\ ps ) -> E. x ps ) $=
|
|
( wa simpr eximi ) ABDBCABEF $.
|
|
|
|
$( Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The
|
|
forward direction holds for all propositions, as seen at ~ alexim .
|
|
(Contributed by Jim Kingdon, 2-Jun-2018.) $)
|
|
alexdc $p |- ( A. x DECID ph -> ( A. x ph <-> -. E. x -. ph ) ) $=
|
|
( wdc wal wn wex nfa1 wb notnotbdc sps albid alnex syl6bb ) ACZBDZABDAEZEZB
|
|
DPBFEOAQBNBGNAQHBAIJKPBLM $.
|
|
|
|
$( Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
19.29 $p |- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) $=
|
|
( wal wex wa wi pm3.2 alimi exim syl imp ) ACDZBCEZABFZCEZMBOGZCDNPGAQCABHI
|
|
BOCJKL $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
19.29r $p |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) ) $=
|
|
( wal wex wa 19.29 ancom exancom 3imtr4i ) BCDZACEZFBAFCELKFABFCEBACGLKHABC
|
|
IJ $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification.
|
|
(Contributed by NM, 3-Feb-2005.) $)
|
|
19.29r2 $p |- ( ( E. x E. y ph /\ A. x A. y ps ) ->
|
|
E. x E. y ( ph /\ ps ) ) $=
|
|
( wex wal wa 19.29r eximi syl ) ADEZCEBDFZCFGKLGZCEABGDEZCEKLCHMNCABDHIJ $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification.
|
|
(Contributed by NM, 11-Feb-2005.) $)
|
|
19.29x $p |- ( ( E. x A. y ph /\ A. x E. y ps ) ->
|
|
E. x E. y ( ph /\ ps ) ) $=
|
|
( wal wex wa 19.29r 19.29 eximi syl ) ADEZCFBDFZCEGLMGZCFABGDFZCFLMCHNOCABD
|
|
IJK $.
|
|
|
|
$( Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse
|
|
holds for classical logic but not (for all propositions) in intuitionistic
|
|
logic (Contributed by Mario Carneiro, 2-Feb-2015.) $)
|
|
19.35-1 $p |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) ) $=
|
|
( wal wi wex wa 19.29 pm3.35 eximi syl expcom ) ACDZABEZCFZBCFZMOGANGZCFPAN
|
|
CHQBCABIJKL $.
|
|
|
|
${
|
|
19.35i.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by NM, 2-Feb-2015.) $)
|
|
19.35i $p |- ( A. x ph -> E. x ps ) $=
|
|
( wi wex wal 19.35-1 ax-mp ) ABECFACGBCFEDABCHI $.
|
|
$}
|
|
|
|
$( Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by NM, 2-Feb-2015.) $)
|
|
19.25 $p |- ( A. y E. x ( ph -> ps ) ->
|
|
( E. y A. x ph -> E. y E. x ps ) ) $=
|
|
( wi wex wal 19.35-1 alimi exim syl ) ABECFZDGACGZBCFZEZDGMDFNDFELODABCHIMN
|
|
DJK $.
|
|
|
|
$( Theorem 19.30 of [Margaris] p. 90, with an additional decidability
|
|
condition. (Contributed by Jim Kingdon, 21-Jul-2018.) $)
|
|
19.30dc $p |- ( DECID E. x ps ->
|
|
( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) ) $=
|
|
( wex wdc wn wo wal df-dc olc a1d alnex orel2 al2imi sylbir syl6 jaoi sylbi
|
|
wi orc ) BCDZEUAUAFZGABGZCHZACHZUAGZSZUAIUAUGUBUAUFUDUAUEJKUBUDUEUFUBBFZCHU
|
|
DUESBCLUHUCACBAMNOUEUATPQR $.
|
|
|
|
$( Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Mario Carneiro, 2-Feb-2015.) $)
|
|
19.43 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
|
|
( wo wex hbe1 hbor 19.8a orim12i exlimih orc eximi olc jaoi impbii ) ABDZCE
|
|
ZACEZBCEZDZPTCRSCACFBCFGARBSACHBCHIJRQSAPCABKLBPCBAMLNO $.
|
|
|
|
$( The antecedent provides a condition implying the converse of ~ 19.33 .
|
|
Compare Theorem 19.33 of [Margaris] p. 90. This variation of ~ 19.33bdc
|
|
is intuitionistically valid without a decidability condition.
|
|
(Contributed by Mario Carneiro, 2-Feb-2015.) $)
|
|
19.33b2 $p |- ( ( -. E. x ph \/ -. E. x ps ) ->
|
|
( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) $=
|
|
( wex wn wal orcom alnex orbi12i bitr4i pm2.53 orcoms al2imi orim12d syl5bi
|
|
wo wi com12 19.33 impbid1 ) ACDEZBCDEZPZABPZCFZACFZBCFZPZUEUCUHUCBEZCFZAEZC
|
|
FZPZUEUHUCUBUAPUMUAUBGUJUBULUABCHACHIJUEUJUFULUGUDUIACBAUIAQBAKLMUDUKBCABKM
|
|
NORABCST $.
|
|
|
|
$( Converse of ~ 19.33 given ` -. ( E. x ph /\ E. x ps ) ` and a decidability
|
|
condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version
|
|
which does not require a decidability condition, see ~ 19.33b2
|
|
(Contributed by Jim Kingdon, 23-Apr-2018.) $)
|
|
19.33bdc $p |- ( DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) ->
|
|
( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) ) $=
|
|
( wex wdc wa wn wo wal wb ianordc 19.33b2 syl6bi ) ACDZENBCDZFGNGOGHABHCIAC
|
|
IBCIHJNOKABCLM $.
|
|
|
|
$( Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.40 $p |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) ) $=
|
|
( wa wex exsimpl simpr eximi jca ) ABDZCEACEBCEABCFJBCABGHI $.
|
|
|
|
$( Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris]
|
|
p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) $)
|
|
19.40-2 $p |- ( E. x E. y ( ph /\ ps ) ->
|
|
( E. x E. y ph /\ E. x E. y ps ) ) $=
|
|
( wa wex 19.40 eximi syl ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.
|
|
|
|
$( Add/remove a conjunct in the scope of an existential quantifier.
|
|
(Contributed by Raph Levien, 3-Jul-2006.) $)
|
|
exintrbi $p |- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) ) $=
|
|
( wi wal wa wb wex pm4.71 albii exbi sylbi ) ABDZCEAABFZGZCEACHNCHGMOCABIJA
|
|
NCKL $.
|
|
|
|
$( Introduce a conjunct in the scope of an existential quantifier.
|
|
(Contributed by NM, 11-Aug-1993.) $)
|
|
exintr $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) ) $=
|
|
( wi wal wex wa exintrbi biimpd ) ABDCEACFABGCFABCHI $.
|
|
|
|
$( Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew
|
|
Salmon, 8-Jun-2011.) $)
|
|
alsyl $p |- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ch ) ) ->
|
|
A. x ( ph -> ch ) ) $=
|
|
( wi pm3.33 alanimi ) ABEBCEACEDABCFG $.
|
|
|
|
${
|
|
hbex.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) $)
|
|
hbex $p |- ( E. y ph -> A. x E. y ph ) $=
|
|
( wex wal hbe1 hbal 19.8a alrimih exlimih ) AACEZBFCLCBACGHALBDACIJK $.
|
|
$}
|
|
|
|
${
|
|
nfex.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
nfex $p |- F/ x E. y ph $=
|
|
( wex nfri hbex nfi ) ACEBABCABDFGH $.
|
|
$}
|
|
|
|
$( Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables.
|
|
(Contributed by O'Cat, 31-Mar-2008.) $)
|
|
19.2 $p |- ( A. x ph -> E. y ph ) $=
|
|
( wex 19.8a sps ) AACDBACEF $.
|
|
|
|
${
|
|
i19.24.1 $e |- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $.
|
|
$( Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The
|
|
hypothesis is the converse of ~ 19.35-1 , and is a theorem of classical
|
|
logic, but in intuitionistic logic it will only be provable for some
|
|
propositions. (Contributed by Jim Kingdon, 22-Jul-2018.) $)
|
|
i19.24 $p |- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) ) $=
|
|
( wal wi wex 19.2 imim2i syl ) ACEZBCEZFKBCGZFABFCGLMKBCCHIDJ $.
|
|
|
|
$( Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The
|
|
hypothesis is the converse of ~ 19.35-1 , and is a theorem of classical
|
|
logic, but in intuitionistic logic it will only be provable for some
|
|
propositions. (Contributed by Jim Kingdon, 22-Jul-2018.) $)
|
|
i19.39 $p |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $=
|
|
( wex wi wal 19.2 imim1i syl ) ACEZBCEZFACGZLFABFCEMKLACCHIDJ $.
|
|
$}
|
|
|
|
$( A closed version of one direction of ~ 19.9 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.9ht $p |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) ) $=
|
|
( wal wi wex id ax-gen 19.23ht mpbii ) AABCDBCAADZBCABEADJBAFGAABHI $.
|
|
|
|
$( A closed version of ~ 19.9 . (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen,
|
|
30-Dec-2017.) $)
|
|
19.9t $p |- ( F/ x ph -> ( E. x ph <-> ph ) ) $=
|
|
( wnf wex wal wi df-nf 19.9ht sylbi 19.8a impbid1 ) ABCZABDZALAABEFBEMAFABG
|
|
ABHIABJK $.
|
|
|
|
${
|
|
19.9h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( A wff may be existentially quantified with a variable not free in it.
|
|
Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) $)
|
|
19.9h $p |- ( E. x ph <-> ph ) $=
|
|
( wex wal wi 19.9ht mpg 19.8a impbii ) ABDZAAABEFKAFBABGCHABIJ $.
|
|
$}
|
|
|
|
${
|
|
19.9.1 $e |- F/ x ph $.
|
|
$( A wff may be existentially quantified with a variable not free in it.
|
|
Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
19.9 $p |- ( E. x ph <-> ph ) $=
|
|
( nfri 19.9h ) ABABCDE $.
|
|
$}
|
|
|
|
$( One direction of theorem 19.6 of [Margaris] p. 89. The converse holds
|
|
given a decidability condition, as seen at ~ alexdc . (Contributed by Jim
|
|
Kingdon, 2-Jul-2018.) $)
|
|
alexim $p |- ( A. x ph -> -. E. x -. ph ) $=
|
|
( wal wn wex wfal wi pm2.24 alimi exim syl nfv 19.9 syl6ib dfnot sylibr ) A
|
|
BCZADZBEZFGSDQSFBEZFQRFGZBCSTGAUABAFHIRFBJKFBFBLMNSOP $.
|
|
|
|
$( One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic
|
|
the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) $)
|
|
exnalim $p |- ( E. x -. ph -> -. A. x ph ) $=
|
|
( wal wn wex alexim con2i ) ABCADBEABFG $.
|
|
|
|
$( A transformation of quantifiers and logical connectives. In classical
|
|
logic the converse also holds. (Contributed by Jim Kingdon,
|
|
15-Jul-2018.) $)
|
|
exanaliim $p |- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) ) $=
|
|
( wn wa wex wi wal annimim eximi exnalim syl ) ABDEZCFABGZDZCFNCHDMOCABIJNC
|
|
KL $.
|
|
|
|
$( A relationship between two quantifiers and negation. (Contributed by Jim
|
|
Kingdon, 27-Aug-2018.) $)
|
|
alexnim $p |- ( A. x E. y -. ph -> -. E. x A. y ph ) $=
|
|
( wn wex wal exnalim alimi alnex sylib ) ADCEZBFACFZDZBFLBEDKMBACGHLBIJ $.
|
|
|
|
${
|
|
ax6blem.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` -. ph ` . This
|
|
theorem doesn't use ~ ax6b compared to ~ hbnt . (Contributed by GD,
|
|
27-Jan-2018.) $)
|
|
ax6blem $p |- ( -. ph -> A. x -. ph ) $=
|
|
( wn wex wal id exlimih con3i alnex sylibr ) ADZABEZDLBFMAAABCAGHIABJK $.
|
|
$}
|
|
|
|
$( Quantified Negation. Axiom C5-2 of [Monk2] p. 113.
|
|
|
|
(Contributed by GD, 27-Jan-2018.) $)
|
|
ax6b $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( wal ax-ial ax6blem ) ABCBABDE $.
|
|
|
|
$( ` x ` is not free in ` -. A. x ph ` . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 18-Aug-2014.) $)
|
|
hbn1 $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( ax6b ) ABC $.
|
|
|
|
$( Closed theorem version of bound-variable hypothesis builder ~ hbn .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) $)
|
|
hbnt $p |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) ) $=
|
|
( wn wal wi ax-4 con3i ax6b syl con3 al2imi syl5 ) ACZABDZCZBDZANEZBDMBDMOP
|
|
NAABFGABHIQOMBANJKL $.
|
|
|
|
${
|
|
hbn.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` -. ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbn $p |- ( -. ph -> A. x -. ph ) $=
|
|
( wal wi wn hbnt mpg ) AABDEAFZIBDEBABGCH $.
|
|
$}
|
|
|
|
${
|
|
hbnd.1 $e |- ( ph -> A. x ph ) $.
|
|
hbnd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbn .
|
|
(Contributed by NM, 3-Jan-2002.) $)
|
|
hbnd $p |- ( ph -> ( -. ps -> A. x -. ps ) ) $=
|
|
( wal wi wn alrimih hbnt syl ) ABBCFGZCFBHZMCFGALCDEIBCJK $.
|
|
$}
|
|
|
|
$( If ` x ` is not free in ` ph ` , then it is not free in ` -. ph ` .
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) $)
|
|
nfnt $p |- ( F/ x ph -> F/ x -. ph ) $=
|
|
( wnf wn nfnf1 wal wi df-nf hbnt sylbi nfd ) ABCZADZBABELAABFGBFMMBFGABHABI
|
|
JK $.
|
|
|
|
${
|
|
nfnd.1 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction associated with ~ nfnt . (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nfnd $p |- ( ph -> F/ x -. ps ) $=
|
|
( wnf wn nfnt syl ) ABCEBFCEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
nfn.1 $e |- F/ x ph $.
|
|
$( Inference associated with ~ nfnt . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfn $p |- F/ x -. ph $=
|
|
( wnf wn nfnt ax-mp ) ABDAEBDCABFG $.
|
|
$}
|
|
|
|
${
|
|
nfdc.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` DECID ph ` .
|
|
(Contributed by Jim Kingdon, 11-Mar-2018.) $)
|
|
nfdc $p |- F/ x DECID ph $=
|
|
( wdc wn wo df-dc nfn nfor nfxfr ) ADAAEZFBAGAKBCABCHIJ $.
|
|
$}
|
|
|
|
$( The analog in our predicate calculus of axiom 5 of modal logic S5.
|
|
(Contributed by NM, 5-Oct-2005.) $)
|
|
modal-5 $p |- ( -. A. x -. ph -> A. x -. A. x -. ph ) $=
|
|
( wn hbn1 ) ACBD $.
|
|
|
|
${
|
|
19.9d.1 $e |- ( ps -> F/ x ph ) $.
|
|
$( A deduction version of one direction of ~ 19.9 . (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.9d $p |- ( ps -> ( E. x ph -> ph ) ) $=
|
|
( wex wnf wb 19.9t syl biimpd ) BACEZABACFKAGDACHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.9hd.1 $e |- ( ps -> A. x ps ) $.
|
|
19.9hd.2 $e |- ( ps -> ( ph -> A. x ph ) ) $.
|
|
$( A deduction version of one direction of ~ 19.9 . This is an older
|
|
variation of this theorem; new proofs should use ~ 19.9d . (Contributed
|
|
by NM, 5-Aug-1993.) (New usage is discouraged.) $)
|
|
19.9hd $p |- ( ps -> ( E. x ph -> ph ) ) $=
|
|
( wal wi wex alimi 19.9ht 3syl ) BBCFAACFGZCFACHAGDBLCEIACJK $.
|
|
$}
|
|
|
|
$( One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
excomim $p |- ( E. x E. y ph -> E. y E. x ph ) $=
|
|
( wex 19.8a 2eximi hbe1 hbex 19.9h sylib ) ACDBDABDZCDZBDLAKBCABEFLBKBCABGH
|
|
IJ $.
|
|
|
|
$( Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
excom $p |- ( E. x E. y ph <-> E. y E. x ph ) $=
|
|
( wex excomim impbii ) ACDBDABDCDABCEACBEF $.
|
|
|
|
$( Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake
|
|
sometimes made by beginners! (Contributed by NM, 5-Aug-1993.) $)
|
|
19.12 $p |- ( E. x A. y ph -> A. y E. x ph ) $=
|
|
( wal wex hba1 hbex ax-4 eximi alrimih ) ACDZBEABECKCBACFGKABACHIJ $.
|
|
|
|
${
|
|
19.19.1 $e |- F/ x ph $.
|
|
$( Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) $)
|
|
19.19 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) ) $=
|
|
( wex wb wal 19.9 exbi syl5bbr ) AACEABFCGBCEACDHABCIJ $.
|
|
$}
|
|
|
|
${
|
|
19.21-2.1 $e |- F/ x ph $.
|
|
19.21-2.2 $e |- F/ y ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed
|
|
by NM, 4-Feb-2005.) $)
|
|
19.21-2 $p |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) ) $=
|
|
( wi wal 19.21 albii bitri ) ABGDHZCHABDHZGZCHAMCHGLNCABDFIJAMCEIK $.
|
|
$}
|
|
|
|
$( An alternate definition of ~ df-nf , which does not involve nested
|
|
quantifiers on the same variable. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nf2 $p |- ( F/ x ph <-> ( E. x ph -> A. x ph ) ) $=
|
|
( wnf wal wi wex df-nf nfa1 nfri 19.23h bitri ) ABCAABDZEBDABFLEABGALBLBABH
|
|
IJK $.
|
|
|
|
$( An alternate definition of ~ df-nf . (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nf3 $p |- ( F/ x ph <-> A. x ( E. x ph -> ph ) ) $=
|
|
( wnf wex wal wi nf2 nfe1 nfri 19.21h bitr4i ) ABCABDZABEFLAFBEABGLABLBABHI
|
|
JK $.
|
|
|
|
$( Variable ` x ` is effectively not free in ` ph ` iff ` ph ` is always true
|
|
or always false, given a decidability condition. The reverse direction,
|
|
~ nf4r , holds for all propositions. (Contributed by Jim Kingdon,
|
|
21-Jul-2018.) $)
|
|
nf4dc $p |- ( DECID E. x ph -> ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) ) $=
|
|
( wex wdc wnf wn wal wo nf2 imordc syl5bb orcom alnex orbi2i bitr4i syl6bb
|
|
wi ) ABCZDZABEZRFZABGZHZUBAFBGZHZTRUBQSUCABIRUBJKUCUBUAHUEUAUBLUDUAUBABMNOP
|
|
$.
|
|
|
|
$( If ` ph ` is always true or always false, then variable ` x ` is
|
|
effectively not free in ` ph ` . The converse holds given a decidability
|
|
condition, as seen at ~ nf4dc . (Contributed by Jim Kingdon,
|
|
21-Jul-2018.) $)
|
|
nf4r $p |- ( ( A. x ph \/ A. x -. ph ) -> F/ x ph ) $=
|
|
( wal wn wo wex wnf orcom alnex orbi2i bitr4i wi imorr nf2 sylibr sylbir )
|
|
ABCZADBCZEZABFZDZQEZABGZUBQUAESUAQHRUAQABIJKUBTQLUCTQMABNOP $.
|
|
|
|
${
|
|
19.36i.1 $e |- F/ x ps $.
|
|
19.36i.2 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by NM, 2-Feb-2015.) $)
|
|
19.36i $p |- ( A. x ph -> ps ) $=
|
|
( wal wex 19.35i id exlimi syl ) ACFBCGBABCEHBBCDBIJK $.
|
|
$}
|
|
|
|
${
|
|
19.36-1.1 $e |- F/ x ps $.
|
|
$( Closed form of ~ 19.36i . One direction of Theorem 19.36 of [Margaris]
|
|
p. 90. The converse holds in classical logic, but does not hold (for
|
|
all propositions) in intuitionistic logic. (Contributed by Jim Kingdon,
|
|
20-Jun-2018.) $)
|
|
19.36-1 $p |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) ) $=
|
|
( wi wex wal 19.35-1 19.9 syl6ib ) ABECFACGBCFBABCHBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
19.37-1.1 $e |- F/ x ph $.
|
|
$( One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds
|
|
in classical logic but not, in general, here. (Contributed by Jim
|
|
Kingdon, 21-Jun-2018.) $)
|
|
19.37-1 $p |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) $=
|
|
( wal wi wex 19.3 19.35-1 syl5bir ) AACEABFCGBCGACDHABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
19.37aiv.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.37aiv $p |- ( ph -> E. x ps ) $=
|
|
( wi wex nfv 19.37-1 ax-mp ) ABECFABCFEDABCACGHI $.
|
|
$}
|
|
|
|
$( Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.38 $p |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wex wal wi hbe1 hba1 hbim 19.8a ax-4 imim12i alrimih ) ACDZBCEZFABFCNOCAC
|
|
GBCHIANOBACJBCKLM $.
|
|
|
|
$( Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
19.23t $p |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
|
|
( wnf wi wal wex exim 19.9t biimpd syl9r nfr imim2d 19.38 syl6 impbid ) BCD
|
|
ZABECFZACGZBEZRSBCGZQBABCHQUABBCIJKQTSBCFZERQBUBSBCLMABCNOP $.
|
|
|
|
${
|
|
19.23.1 $e |- F/ x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.23 $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( wnf wi wal wex wb 19.23t ax-mp ) BCEABFCGACHBFIDABCJK $.
|
|
$}
|
|
|
|
${
|
|
19.32dc.1 $e |- F/ x ph $.
|
|
$( Theorem 19.32 of [Margaris] p. 90, where ` ph ` is decidable.
|
|
(Contributed by Jim Kingdon, 4-Jun-2018.) $)
|
|
19.32dc $p |- ( DECID ph ->
|
|
( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) ) $=
|
|
( wdc wn wi wal wo wb nfn 19.21 a1i nfdc dfordc albid 3bitr4d ) AEZAFZBGZ
|
|
CHZSBCHZGZABIZCHAUBIUAUCJRSBCACDKLMRUDTCACDNABOPAUBOQ $.
|
|
$}
|
|
|
|
${
|
|
19.32r.1 $e |- F/ x ph $.
|
|
$( One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds
|
|
if ` ph ` is decidable, as seen at ~ 19.32dc . (Contributed by Jim
|
|
Kingdon, 28-Jul-2018.) $)
|
|
19.32r $p |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
|
|
( wo wal orc alrimi olc alimi jaoi ) AABEZCFBCFALCDABGHBLCBAIJK $.
|
|
$}
|
|
|
|
${
|
|
19.31r.1 $e |- F/ x ps $.
|
|
$( One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds
|
|
in classical logic, but not intuitionistic logic. (Contributed by Jim
|
|
Kingdon, 28-Jul-2018.) $)
|
|
19.31r $p |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) ) $=
|
|
( wal wo 19.32r orcom albii 3imtr4i ) BACEZFBAFZCEKBFABFZCEBACDGKBHMLCABH
|
|
IJ $.
|
|
$}
|
|
|
|
${
|
|
19.44.1 $e |- F/ x ps $.
|
|
$( Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) $)
|
|
19.44 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) ) $=
|
|
( wo wex 19.43 19.9 orbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.45.1 $e |- F/ x ph $.
|
|
$( Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) $)
|
|
19.45 $p |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) ) $=
|
|
( wo wex 19.43 19.9 orbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
|
|
$}
|
|
|
|
$( Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.34 $p |- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) ) $=
|
|
( wal wex wo 19.2 orim1i 19.43 sylibr ) ACDZBCEZFACEZLFABFCEKMLACCGHABCIJ
|
|
$.
|
|
|
|
${
|
|
19.41h.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Theorem 19.41 of [Margaris] p. 90. New proofs should use ~ 19.41
|
|
instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 25-May-2011.) (New usage is discouraged.) $)
|
|
19.41h $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
|
|
( wa wex 19.40 id exlimih anim2i syl pm3.21 eximdh impcom impbii ) ABEZCF
|
|
ZACFZBEZQRBCFZESABCGTBRBBCDBHIJKBRQBAPCDBALMNO $.
|
|
$}
|
|
|
|
${
|
|
19.41.1 $e |- F/ x ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by
|
|
Wolf Lammen, 12-Jan-2018.) $)
|
|
19.41 $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
|
|
( wa wex 19.40 19.9 anbi2i sylib pm3.21 eximd impcom impbii ) ABEZCFZACFZ
|
|
BEZPQBCFZERABCGSBQBCDHIJBQPBAOCDBAKLMN $.
|
|
$}
|
|
|
|
${
|
|
19.42h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Theorem 19.42 of [Margaris] p. 90. New proofs should use ~ 19.42
|
|
instead. (Contributed by NM, 18-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
19.42h $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
|
|
( wa wex 19.41h exancom ancom 3bitr4i ) BAECFBCFZAEABECFAKEBACDGABCHAKIJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
19.42.1 $e |- F/ x ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) $)
|
|
19.42 $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
|
|
( wa wex 19.41 exancom ancom 3bitr4i ) BAECFBCFZAEABECFAKEBACDGABCHAKIJ
|
|
$.
|
|
$}
|
|
|
|
$( Swap 1st and 3rd existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
excom13 $p |- ( E. x E. y E. z ph <-> E. z E. y E. x ph ) $=
|
|
( wex excom exbii 3bitri ) ADEZCEBEIBEZCEABEZDEZCEKCEDEIBCFJLCABDFGKCDFH $.
|
|
|
|
$( Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) $)
|
|
exrot3 $p |- ( E. x E. y E. z ph <-> E. y E. z E. x ph ) $=
|
|
( wex excom13 excom bitri ) ADECEBEABEZCEDEIDECEABCDFIDCGH $.
|
|
|
|
$( Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) $)
|
|
exrot4 $p |- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph ) $=
|
|
( wex excom13 exbii bitri ) AEFDFCFZBFACFZDFEFZBFKBFEFDFJLBACDEGHKBEDGI $.
|
|
|
|
${
|
|
nexr.1 $e |- -. E. x ph $.
|
|
$( Inference from ~ 19.8a . (Contributed by Jeff Hankins, 26-Jul-2009.) $)
|
|
nexr $p |- -. ph $=
|
|
( wex 19.8a mto ) AABDCABEF $.
|
|
$}
|
|
|
|
${
|
|
exan.1 $e |- ( E. x ph /\ ps ) $.
|
|
$( Place a conjunct in the scope of an existential quantifier.
|
|
(Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
exan $p |- E. x ( ph /\ ps ) $=
|
|
( wex wal wa hbe1 19.28h mpgbi 19.29r ax-mp ) ACEZBCFGZABGCEMBGNCMBCACHID
|
|
JABCKL $.
|
|
$}
|
|
|
|
${
|
|
hbexd.1 $e |- ( ph -> A. y ph ) $.
|
|
hbexd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbex .
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
hbexd $p |- ( ph -> ( E. y ps -> A. x E. y ps ) ) $=
|
|
( wex wal eximdh 19.12 syl6 ) ABDGZBCHZDGLCHABMDEFIBDCJK $.
|
|
$}
|
|
|
|
${
|
|
eeor.1 $e |- F/ y ph $.
|
|
eeor.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) $)
|
|
eeor $p |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) ) $=
|
|
( wo wex 19.45 exbii nfex 19.44 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equality theorems without distinct variables
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( At least one individual exists. This is not a theorem of free logic,
|
|
which is sound in empty domains. For such a logic, we would add this
|
|
theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the
|
|
system consisting of ~ ax-5 through ~ ax-14 and ~ ax-17 , all axioms other
|
|
than ~ ax-9 are believed to be theorems of free logic, although the system
|
|
without ~ ax-9 is probably not complete in free logic. (Contributed by
|
|
NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) $)
|
|
a9e $p |- E. x x = y $=
|
|
( ax-i9 ) ABC $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( At least one individual exists. Weaker version of ~ a9e . (Contributed
|
|
by NM, 3-Aug-2017.) $)
|
|
a9ev $p |- E. x x = y $=
|
|
( ax-i9 ) ABC $.
|
|
$}
|
|
|
|
$( An implication related to substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by NM, 3-Feb-2015.) $)
|
|
ax9o $p |- ( A. x ( x = y -> A. x ph ) -> ph ) $=
|
|
( cv wceq wex wal wi a9e wa 19.29r hba1 pm3.35 exlimih ax-4 syl mpan ) BDCD
|
|
EZBFZRABGZHZBGZABCISUBJRUAJZBFZARUABKUDTAUCTBABLRTMNABOPPQ $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68.
|
|
This is often an axiom of equality in textbook systems, but we don't
|
|
need it as an axiom since it can be proved from our other axioms.
|
|
|
|
This proof is similar to Tarski's and makes use of a dummy variable
|
|
` y ` . It also works in intuitionistic logic, unlike some other
|
|
possible ways of proving this theorem. (Contributed by NM,
|
|
1-Apr-2005.) $)
|
|
equid $p |- x = x $=
|
|
( vy weq wex a9e ax-17 ax-8 pm2.43i exlimih ax-mp ) BACZBDAACZBAEKLBLBFKL
|
|
BAAGHIJ $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for ` x = x ` . This theorem tells us
|
|
that any variable, including ` x ` , is effectively not free in
|
|
` x = x ` , even though ` x ` is technically free according to the
|
|
traditional definition of free variable. (Contributed by NM,
|
|
13-Jan-2011.) (Revised by NM, 21-Aug-2017.) $)
|
|
nfequid $p |- F/ y x = x $=
|
|
( weq equid nfth ) AACBADE $.
|
|
|
|
$( One of the two equality axioms of standard predicate calculus, called
|
|
reflexivity of equality. (The other one is ~ stdpc7 .) Axiom 6 of
|
|
[Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant
|
|
quantifier, but it was probably to be compatible with free logic (which is
|
|
valid in the empty domain). (Contributed by NM, 16-Feb-2005.) $)
|
|
stdpc6 $p |- A. x x = x $=
|
|
( weq equid ax-gen ) AABAACD $.
|
|
|
|
$( Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
equcomi $p |- ( x = y -> y = x ) $=
|
|
( weq equid ax-8 mpi ) ABCAACBACADABAEF $.
|
|
|
|
$( Commutative law for equality. (Contributed by NM, 20-Aug-1993.) $)
|
|
equcom $p |- ( x = y <-> y = x ) $=
|
|
( weq equcomi impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
equcoms.1 $e |- ( x = y -> ph ) $.
|
|
$( An inference commuting equality in antecedent. Used to eliminate the
|
|
need for a syllogism. (Contributed by NM, 5-Aug-1993.) $)
|
|
equcoms $p |- ( y = x -> ph ) $=
|
|
( weq equcomi syl ) CBEBCEACBFDG $.
|
|
$}
|
|
|
|
$( A transitive law for equality. (Contributed by NM, 23-Aug-1993.) $)
|
|
equtr $p |- ( x = y -> ( y = z -> x = z ) ) $=
|
|
( weq wi ax-8 equcoms ) BCDACDEBABACFG $.
|
|
|
|
$( A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the
|
|
preprint). (Contributed by NM, 23-Aug-1993.) $)
|
|
equtrr $p |- ( x = y -> ( z = x -> z = y ) ) $=
|
|
( weq equtr com12 ) CADABDCBDCABEF $.
|
|
|
|
$( A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
equtr2 $p |- ( ( x = z /\ y = z ) -> x = y ) $=
|
|
( weq wi equtrr equcoms impcom ) BCDACDZABDZIJECBCBAFGH $.
|
|
|
|
$( An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) $)
|
|
equequ1 $p |- ( x = y -> ( x = z <-> y = z ) ) $=
|
|
( weq ax-8 equtr impbid ) ABDACDBCDABCEABCFG $.
|
|
|
|
$( An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) $)
|
|
equequ2 $p |- ( x = y -> ( z = x <-> z = y ) ) $=
|
|
( weq equtrr wi equcoms impbid ) ABDCADZCBDZABCEJIFBABACEGH $.
|
|
|
|
$( An identity law for the non-logical predicate. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
elequ1 $p |- ( x = y -> ( x e. z <-> y e. z ) ) $=
|
|
( weq wel ax-13 wi equcoms impbid ) ABDACEZBCEZABCFKJGBABACFHI $.
|
|
|
|
$( An identity law for the non-logical predicate. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
elequ2 $p |- ( x = y -> ( z e. x <-> z e. y ) ) $=
|
|
( weq wel ax-14 wi equcoms impbid ) ABDCAEZCBEZABCFKJGBABACFHI $.
|
|
|
|
${
|
|
ax11i.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
ax11i.2 $e |- ( ps -> A. x ps ) $.
|
|
$( Inference that has ~ ax-11 (without ` A. y ` ) as its conclusion and
|
|
doesn't require ~ ax-10 , ~ ax-11 , or ~ ax-12 for its proof. The
|
|
hypotheses may be eliminable without one or more of these axioms in
|
|
special cases. Proof similar to Lemma 16 of [Tarski] p. 70.
|
|
(Contributed by NM, 20-May-2008.) $)
|
|
ax11i $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wi wal biimprcd alrimih syl6bi ) CDGZABMAHZCIEBNCFMABEJKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axioms ax-10 and ax-11
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Show that ~ ax-10o can be derived from ~ ax-10 . An open problem is
|
|
whether this theorem can be derived from ~ ax-10 and the others when
|
|
~ ax-11 is replaced with ~ ax-11o . See theorem ~ ax10 for the
|
|
rederivation of ~ ax-10 from ~ ax10o .
|
|
|
|
Normally, ~ ax10o should be used rather than ~ ax-10o , except by theorems
|
|
specifically studying the latter's properties. (Contributed by NM,
|
|
16-May-2008.) $)
|
|
ax10o $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
|
|
( weq wal wi ax-10 ax-11 equcoms sps pm2.27 al2imi sylsyld ) BCDZBECBDZCEAB
|
|
EZOAFZCEZACEBCGNPRFZBSCBACBHIJOQACOAKLM $.
|
|
|
|
$( Axiom ~ ax-10o ("o" for "old") was the original version of ~ ax-10 ,
|
|
before it was discovered (in May 2008) that the shorter ~ ax-10 could
|
|
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
|
|
the preprint).
|
|
|
|
This axiom is redundant, as shown by theorem ~ ax10o .
|
|
|
|
Normally, ~ ax10o should be used rather than ~ ax-10o , except by theorems
|
|
specifically studying the latter's properties. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
ax-10o $a |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $.
|
|
|
|
$( Rederivation of ~ ax-10 from original version ~ ax-10o . See theorem
|
|
~ ax10o for the derivation of ~ ax-10o from ~ ax-10 .
|
|
|
|
This theorem should not be referenced in any proof. Instead, use ~ ax-10
|
|
above so that uses of ~ ax-10 can be more easily identified. (Contributed
|
|
by NM, 16-May-2008.) (New usage is discouraged.) $)
|
|
ax10 $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( weq wal ax-10o pm2.43i equcomi alimi syl ) ABCZADZJBDZBACZBDKLJABEFJMBABG
|
|
HI $.
|
|
|
|
$( All variables are effectively bound in an identical variable specifier.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) $)
|
|
hbae $p |- ( A. x x = y -> A. z A. x x = y ) $=
|
|
( cv wceq wal wi ax12or ax10o alequcoms pm2.43i syl5 ax-4 imim1i jaoi ax-mp
|
|
wo sps a5i ax-7 syl ) ADZBDZEZAFZUDCFZAFUECFUDUFACDZUBECFZUGUCECFZUDUFGZCFZ
|
|
QZQUEUFGZABCHUHUMULUMACUDACIJUIUMUKUMBCUEUDBFZUCUGEBFUFUEUNUDABIKUDBCILJUJU
|
|
MCUEUDUFUDAMNROOPSUDACTUA $.
|
|
|
|
$( All variables are effectively bound in an identical variable specifier.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfae $p |- F/ z A. x x = y $=
|
|
( weq wal hbae nfi ) ABDAECABCFG $.
|
|
|
|
${
|
|
hbalequs.1 $e |- ( A. z A. x x = y -> ph ) $.
|
|
$( Rule that applies ~ hbae to antecedent. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbaes $p |- ( A. x x = y -> ph ) $=
|
|
( weq wal hbae syl ) BCFBGZJDGABCDHEI $.
|
|
$}
|
|
|
|
$( All variables are effectively bound in a distinct variable specifier.
|
|
Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbnae $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
|
|
( weq wal hbae hbn ) ABDAECABCFG $.
|
|
|
|
$( All variables are effectively bound in a distinct variable specifier.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfnae $p |- F/ z -. A. x x = y $=
|
|
( weq wal nfae nfn ) ABDAECABCFG $.
|
|
|
|
${
|
|
hbnalequs.1 $e |- ( A. z -. A. x x = y -> ph ) $.
|
|
$( Rule that applies ~ hbnae to antecedent. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbnaes $p |- ( -. A. x x = y -> ph ) $=
|
|
( weq wal wn hbnae syl ) BCFBGHZKDGABCDIEJ $.
|
|
$}
|
|
|
|
${
|
|
naecoms.1 $e |- ( -. A. x x = y -> ph ) $.
|
|
$( A commutation rule for distinct variable specifiers. (Contributed by
|
|
NM, 2-Jan-2002.) $)
|
|
naecoms $p |- ( -. A. y y = x -> ph ) $=
|
|
( cv wceq wal wn ax-10 con3i syl ) CEZBEZFCGZHMLFBGZHAONBCIJDK $.
|
|
$}
|
|
|
|
$( Lemma used in proofs of substitution properties. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) $)
|
|
equs4 $p |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) $=
|
|
( cv wceq wi wal wa wex a9e 19.29 mpan2 ancl imp eximi syl ) BDCDEZAFZBGZRQ
|
|
HZBIZQAHZBISQBIUABCJRQBKLTUBBRQUBQAMNOP $.
|
|
|
|
${
|
|
equsalh.1 $e |- ( ps -> A. x ps ) $.
|
|
equsalh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. New proofs should use
|
|
~ equsal instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) $)
|
|
equsalh $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( weq wi wal 19.3h syl6bbr pm5.74i albii a1d alrimih ax9o impbii bitr4i )
|
|
CDGZAHZCISBCIZHZCIZBTUBCSAUASABUAFBCEJKLMBUCBUBCEBUASENOBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
equsal.1 $e |- F/ x ps $.
|
|
equsal.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised
|
|
by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen,
|
|
5-Feb-2018.) $)
|
|
equsal $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( weq wi wal wex 19.23 pm5.74i albii a9e a1bi 3bitr4i ) CDGZBHZCIQCJZBHQA
|
|
HZCIBQBCEKTRCQABFLMSBCDNOP $.
|
|
$}
|
|
|
|
${
|
|
equsex.1 $e |- ( ps -> A. x ps ) $.
|
|
equsex.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by NM, 3-Feb-2015.) $)
|
|
equsex $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
|
|
( cv wceq wa wex biimpa exlimih a9e idd biimprcd jcad eximdh mpi impbii )
|
|
CGDGHZAIZCJZBUABCETABFKLBTCJUBCDMBTUACEBTTABTNTABFOPQRS $.
|
|
$}
|
|
|
|
${
|
|
equsexd.1 $e |- ( ph -> A. x ph ) $.
|
|
equsexd.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
equsexd.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction form of ~ equsex . (Contributed by Jim Kingdon,
|
|
29-Dec-2017.) $)
|
|
equsexd $p |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) ) $=
|
|
( cv wceq wa wex wb wi bi1 imim2i 3syl wal a1i imp pm3.31 exlimd2 a9e jca
|
|
prth syl 19.26 syl6ibr anabsi5 idd bi2 pm2.04 jcad eximdh mpi ex impbid )
|
|
ADIEIJZBKZDLZCAUSCDFGAURBCMZNZURBCNZNUSCNHVAVCURBCOPURBCUAQUBACUTACKZURDL
|
|
UTDEUCVDURUSDACVDDRZAVDADRZCDRZKZVEAAVFNZCVGNZKVDVHNAVIVJVIAFSGUDAVFCVGUE
|
|
UFACDUGUHUIVDURURBACURURNZCVKNACURUJSTACURBNZAVBURCBNZNCVLNHVAVMURBCUKPUR
|
|
CBULQTUMUNUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
dral1.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 24-Nov-1994.) $)
|
|
dral1 $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
|
|
( weq wal hbae biimpd alimdh ax10o syld biimprd wi alequcoms impbid ) CDF
|
|
CGZACGZBDGZQRBCGSQABCCDCHQABEIJBCDKLQSADGZRQBADCDDHQABEMJTRNDCADCKOLP $.
|
|
$}
|
|
|
|
${
|
|
dral2.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) $)
|
|
dral2 $p |- ( A. x x = y -> ( A. z ph <-> A. z ps ) ) $=
|
|
( weq wal hbae albidh ) CDGCHABECDEIFJ $.
|
|
$}
|
|
|
|
${
|
|
drex2.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) $)
|
|
drex2 $p |- ( A. x x = y -> ( E. z ph <-> E. z ps ) ) $=
|
|
( weq wal hbae exbidh ) CDGCHABECDEIFJ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 4-Oct-2016.) $)
|
|
drnf1 $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
|
|
( weq wal wi wnf dral1 imbi12d df-nf 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGAC
|
|
IBDIPRCDNABOQEABCDEJKJACLBDLM $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 4-Oct-2016.) $)
|
|
drnf2 $p |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) ) $=
|
|
( weq wal wi wnf dral2 imbi12d df-nf 3bitr4g ) CDGCHZAAEHZIZEHBBEHZIZEHAE
|
|
JBEJQSCDEOABPRFABCDEFKLKAEMBEMN $.
|
|
$}
|
|
|
|
$( Closed theorem form of ~ spim . (Contributed by NM, 15-Jan-2008.)
|
|
(New usage is discouraged.) $)
|
|
spimth $p |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) ->
|
|
( A. x ph -> ps ) ) $=
|
|
( wal wi weq wa imim2 imim2d imp com23 al2imi ax9o syl6 ) BBCEZFZCDGZABFZFZ
|
|
HZCEACERPFZCEBUAAUBCUARAPQTRAPFZFQSUCRBPAIJKLMBCDNO $.
|
|
|
|
$( Closed theorem form of ~ spim . (Contributed by NM, 15-Jan-2008.)
|
|
(Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf
|
|
Lammen, 24-Feb-2018.) $)
|
|
spimt $p |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) ->
|
|
( A. x ph -> ps ) ) $=
|
|
( cv wceq wi wal wex wnf a9e exim mpi 19.35-1 syl 19.9t biimpd sylan9r ) CE
|
|
DEFZABGZGCHZACHZBCIZBCJZBUATCIZUBUCGUASCIUECDKSTCLMABCNOUDUCBBCPQR $.
|
|
|
|
${
|
|
spimh.1 $e |- ( ps -> A. x ps ) $.
|
|
spimh.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization, using implicit substitition. Compare Lemma 14 of
|
|
[Tarski] p. 70. The ~ spim series of theorems requires that only one
|
|
direction of the substitution hypothesis hold. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by NM, 8-May-2008.)
|
|
(New usage is discouraged.) $)
|
|
spimh $p |- ( A. x ph -> ps ) $=
|
|
( wal weq wi syl6com alimi ax9o syl ) ACGCDHZBCGZIZCGBAPCNABOFEJKBCDLM $.
|
|
$}
|
|
|
|
${
|
|
spim.1 $e |- F/ x ps $.
|
|
spim.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization, using implicit substitution. Compare Lemma 14 of
|
|
[Tarski] p. 70. The ~ spim series of theorems requires that only one
|
|
direction of the substitution hypothesis hold. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten
|
|
by Jim Kingdon, 10-Jun-2018.) $)
|
|
spim $p |- ( A. x ph -> ps ) $=
|
|
( nfri spimh ) ABCDBCEGFH $.
|
|
$}
|
|
|
|
${
|
|
spimeh.1 $e |- ( ph -> A. x ph ) $.
|
|
spimeh.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Existential introduction, using implicit substitition. Compare Lemma 14
|
|
of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM,
|
|
3-Feb-2015.) (New usage is discouraged.) $)
|
|
spimeh $p |- ( ph -> E. x ps ) $=
|
|
( cv wceq wex a9e com12 eximdh mpi ) ACGDGHZCIBCICDJANBCENABFKLM $.
|
|
$}
|
|
|
|
${
|
|
spimed.1 $e |- ( ch -> F/ x ph ) $.
|
|
spimed.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Deduction version of ~ spime . (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf
|
|
Lammen, 19-Feb-2018.) $)
|
|
spimed $p |- ( ch -> ( ph -> E. x ps ) ) $=
|
|
( wal wex nfrd weq wi a9e eximii 19.35i syl6 ) CAADHBDICADFJABDDEKABLDDEM
|
|
GNOP $.
|
|
$}
|
|
|
|
${
|
|
spime.1 $e |- F/ x ph $.
|
|
spime.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Existential introduction, using implicit substitution. Compare Lemma 14
|
|
of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario
|
|
Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) $)
|
|
spime $p |- ( ph -> E. x ps ) $=
|
|
( wex wi wtru wnf a1i spimed trud ) ABCGHABICDACJIEKFLM $.
|
|
$}
|
|
|
|
${
|
|
cbv3.1 $e |- F/ y ph $.
|
|
cbv3.2 $e |- F/ x ps $.
|
|
cbv3.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
12-May-2018.) $)
|
|
cbv3 $p |- ( A. x ph -> A. y ps ) $=
|
|
( wal nfal spim alrimi ) ACHBDADCEIABCDFGJK $.
|
|
$}
|
|
|
|
${
|
|
cbv3h.1 $e |- ( ph -> A. y ph ) $.
|
|
cbv3h.2 $e |- ( ps -> A. x ps ) $.
|
|
cbv3h.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) $)
|
|
cbv3h $p |- ( A. x ph -> A. y ps ) $=
|
|
( nfi cbv3 ) ABCDADEHBCFHGI $.
|
|
$}
|
|
|
|
${
|
|
cbv1.1 $e |- F/ x ph $.
|
|
cbv1.2 $e |- F/ y ph $.
|
|
cbv1.3 $e |- ( ph -> F/ y ps ) $.
|
|
cbv1.4 $e |- ( ph -> F/ x ch ) $.
|
|
cbv1.5 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
Revised to format hypotheses to common style. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf
|
|
Lammen, 13-May-2018.) $)
|
|
cbv1 $p |- ( ph -> ( A. x ps -> A. y ch ) ) $=
|
|
( wal wi nfim1 weq com12 a2d cbv3 19.21 3imtr3i pm2.86i ) ABDKZCEKZABLZDK
|
|
ACLZEKAUALAUBLUCUDDEABEGHMACDFIMDENZABCAUEBCLJOPQABDFRACEGRST $.
|
|
$}
|
|
|
|
${
|
|
cbv1h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
|
|
cbv1h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
cbv1h.3 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
13-May-2018.) $)
|
|
cbv1h $p |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) ) $=
|
|
( wal nfa1 nfa2 wi sp sps syl nfd weq cbv1 ) AEIZDIZBCDESDJZAEDKZTBEUBTAB
|
|
BEILSADAEMNZFOPTCDUATACCDILUCGOPTADEQBCLLUCHOR $.
|
|
$}
|
|
|
|
${
|
|
cbv2h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
|
|
cbv2h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
cbv2h.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbv2h $p |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( wal weq wb wi bi1 syl6 cbv1h equcomi bi2 syl56 a7s impbid ) AEIDIBDIZCE
|
|
IZABCDEFGADEJZBCKZBCLHBCMNOAUBUALEDACBEDGFEDJUCAUDCBLEDPHBCQROST $.
|
|
$}
|
|
|
|
${
|
|
cbv2.1 $e |- F/ x ph $.
|
|
cbv2.2 $e |- F/ y ph $.
|
|
cbv2.3 $e |- ( ph -> F/ y ps ) $.
|
|
cbv2.4 $e |- ( ph -> F/ x ch ) $.
|
|
cbv2.5 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
Revised to align format of hypotheses to common style. (Contributed by
|
|
NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by
|
|
Wolf Lammen, 13-May-2018.) $)
|
|
cbv2 $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( wal wb nfri nfal syl nfrd cbv2h ) AAEKZDKZBDKCEKLARSAEGMRDADEFNMOABCDEA
|
|
BEHPACDIPJQO $.
|
|
$}
|
|
|
|
${
|
|
cbvalh.1 $e |- ( ph -> A. y ph ) $.
|
|
cbvalh.2 $e |- ( ps -> A. x ps ) $.
|
|
cbvalh.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitition.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
cbvalh $p |- ( A. x ph <-> A. y ps ) $=
|
|
( wal weq biimpd cbv3h wb equcoms biimprd impbii ) ACHBDHABCDEFCDIABGJKBA
|
|
DCFEDCIABABLCDGMNKO $.
|
|
$}
|
|
|
|
${
|
|
cbval.1 $e |- F/ y ph $.
|
|
cbval.2 $e |- F/ x ps $.
|
|
cbval.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
cbval $p |- ( A. x ph <-> A. y ps ) $=
|
|
( nfri cbvalh ) ABCDADEHBCFHGI $.
|
|
$}
|
|
|
|
${
|
|
cbvexh.1 $e |- ( ph -> A. y ph ) $.
|
|
cbvexh.2 $e |- ( ps -> A. x ps ) $.
|
|
cbvexh.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitition.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
3-Feb-2015.) $)
|
|
cbvexh $p |- ( E. x ph <-> E. y ps ) $=
|
|
( wex hbex cv wceq wa wb bicomd equcoms equsex simpr eximi sylbir exlimih
|
|
impbii ) ACHZBDHZAUCCBCDFIADJZCJZKZBLZDHUCBADCEBAMCDUEUDKZABGNOPUGBDUFBQR
|
|
STBUBDADCEIBUHALZCHUBABCDFGPUIACUHAQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
cbvex.1 $e |- F/ y ph $.
|
|
cbvex.2 $e |- F/ x ps $.
|
|
cbvex.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvex $p |- ( E. x ph <-> E. y ps ) $=
|
|
( nfri cbvexh ) ABCDADEHBCFHGI $.
|
|
$}
|
|
|
|
${
|
|
chvar.1 $e |- F/ x ps $.
|
|
chvar.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
chvar.3 $e |- ph $.
|
|
$( Implicit substitution of ` y ` for ` x ` into a theorem. (Contributed
|
|
by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
chvar $p |- ps $=
|
|
( weq biimpd spim mpg ) ABCABCDECDHABFIJGK $.
|
|
$}
|
|
|
|
$( A variable introduction law for equality. Lemma 15 of [Monk2] p. 109,
|
|
however we do not require ` z ` to be distinct from ` x ` and ` y `
|
|
(making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
equvini $p |- ( x = y -> E. z ( x = z /\ z = y ) ) $=
|
|
( cv wceq wal wi wo wex ax12or equcomi alimi a9e jctir a1d 19.29 syl6 ax-mp
|
|
wa jaoi eximi 2a1i anc2ri 19.29r ax-8 anc2li equcoms com12 exim syl mpi sps
|
|
imim2i ) CDZADZEZCFZUNBDZEZCFZUOUREZVACFZGZCFZHZHVAUOUNEZUSSZCIZGZABCJUQVIV
|
|
EUQVAVFCFZUSCIZSZVHUQVLVAUQVJVKUPVFCCAKZLCBMNOVFUSCPQUTVIVDUTVAVFCIZUTSVHUT
|
|
VAVNUTVAVNUPCIZVNCAMZUPVFCVMUARUBUCVFUSCUDQVCVICVBVHVAVBVOVHVPVBUPVGGZCFVOV
|
|
HGVAVQCUPVAVGVAVGGACVFVAUSACBUEUFUGUHLUPVGCUIUJUKUMULTTR $.
|
|
|
|
$( A variable elimination law for equality with no distinct variable
|
|
requirements. (Compare ~ equvini .) (Contributed by NM, 1-Mar-2013.)
|
|
(Revised by NM, 3-Feb-2015.) $)
|
|
equveli $p |- ( A. z ( z = x <-> z = y ) -> x = y ) $=
|
|
( cv wb wal wi wa albiim wo ax12or equequ1 imbi12d sps equid syl6bi adantrd
|
|
wceq ax-4 jaoi dral2 a1bi biimpri dral1 mpi equcomi syl adantld hbequid a1i
|
|
hba1 hbimd equtr ax-8 imim12d ax-gen 19.26 spimth sylbir sylancl mpii ax-mp
|
|
a5i sylbi ) CDZADZRZVEBDZRZECFVGVIGZCFZVIVGGZCFZHZVFVHRZVGVICIVGCFZVICFZVOV
|
|
OCFGZCFZJZJVNVOGZABCKVPWAVTVPVKVOVMVPVKVFVFRZVOGZCFZVOVJWCCACVGVJWCECVGVGWB
|
|
VIVOCAALCABLMNUAWCVOCVOWCWBVOAOZUBUCNPQVQWAVSVQVMVOVKVQVMVHVHRZVHVFRZGZBFZV
|
|
OVLWHCBVIVLWHECVIVIWFVGWGCBBLCBALMNUDWIWGVOWIWFWGBOWHBSUEBAUFUGPUHVSVKVOVMV
|
|
SVKWBVOWEVSWCWDGZCFZVGVJWCGGZCFZVKWCGZVRWJCVSWBVOCVRCUKWBWBCFGVSACUIUJVRCSU
|
|
LVCWLCVGWBVGVIVOCAAUMCABUNUOUPWKWMHWJWLHCFWNWJWLCUQVJWCCAURUSUTVAQTTVBVD $.
|
|
|
|
${
|
|
nfald.1 $e |- F/ y ph $.
|
|
nfald.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 6-Jan-2018.) $)
|
|
nfald $p |- ( ph -> F/ x A. y ps ) $=
|
|
( wnf wal nfri alrimih nfnf1 nfal hba1 sp nfrd hbald nfd syl ) ABCGZDHZBD
|
|
HZCGASDADEIFJTUACSCDBCKLTBCDSDMTBCSDNOPQR $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim
|
|
Kingdon, 7-Feb-2018.) $)
|
|
nfexd $p |- ( ph -> F/ x E. y ps ) $=
|
|
( wex wal wnf nfri df-nf sylib alrimih alcom exim alimi syl 19.12 imim2i
|
|
wi sylibr ) ABDGZUBCHZTZCHZUBCIAUBBCHZDGZTZCHZUEABUFTZDHZCHZUIAUJCHZDHULA
|
|
UMDADEJABCIUMFBCKLMUJDCNLUKUHCBUFDOPQUHUDCUGUCUBBDCRSPQUBCKUA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Substitution (without distinct variables)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c [ $. $( Left bracket $)
|
|
$c / $. $( Division. $)
|
|
$c ] $. $( Right bracket $)
|
|
|
|
$( Extend wff definition to include proper substitution (read "the wff that
|
|
results when ` y ` is properly substituted for ` x ` in wff ` ph ` ").
|
|
(Contributed by NM, 24-Jan-2006.) $)
|
|
wsb $a wff [ y / x ] ph $.
|
|
|
|
$( Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
|
|
preprint). For our notation, we use ` [ y / x ] ph ` to mean "the wff
|
|
that results when ` y ` is properly substituted for ` x ` in the wff
|
|
` ph ` ." We can also use ` [ y / x ] ph ` in place of the "free for"
|
|
side condition used in traditional predicate calculus; see, for example,
|
|
~ stdpc4 .
|
|
|
|
Our notation was introduced in Haskell B. Curry's _Foundations of
|
|
Mathematical Logic_ (1977), p. 316 and is frequently used in textbooks of
|
|
lambda calculus and combinatory logic. This notation improves the common
|
|
but ambiguous notation, " ` ph ( y ) ` is the wff that results when ` y `
|
|
is properly substituted for ` x ` in ` ph ( x ) ` ." For example, if the
|
|
original ` ph ( x ) ` is ` x = y ` , then ` ph ( y ) ` is ` y = y ` , from
|
|
which we obtain that ` ph ( x ) ` is ` x = x ` . So what exactly does
|
|
` ph ( x ) ` mean? Curry's notation solves this problem.
|
|
|
|
In most books, proper substitution has a somewhat complicated recursive
|
|
definition with multiple cases based on the occurrences of free and bound
|
|
variables in the wff. Instead, we use a single formula that is exactly
|
|
equivalent and gives us a direct definition. We later prove that our
|
|
definition has the properties we expect of proper substitution (see
|
|
theorems ~ sbequ , ~ sbcom2 and ~ sbid2v ).
|
|
|
|
Note that our definition is valid even when ` x ` and ` y ` are replaced
|
|
with the same variable, as ~ sbid shows. We achieve this by having ` x `
|
|
free in the first conjunct and bound in the second. We can also achieve
|
|
this by using a dummy variable, as the alternate definition ~ dfsb7 shows
|
|
(which some logicians may prefer because it doesn't mix free and bound
|
|
variables). Another alternate definition which uses a dummy variable is
|
|
~ dfsb7a .
|
|
|
|
When ` x ` and ` y ` are distinct, we can express proper substitution with
|
|
the simpler expressions of ~ sb5 and ~ sb6 .
|
|
|
|
In classical logic, another possible definition is
|
|
` ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ` but we do not have an
|
|
intuitionistic proof that this is equivalent.
|
|
|
|
There are no restrictions on any of the variables, including what
|
|
variables may occur in wff ` ph ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-sb $a |- ( [ y / x ] ph <->
|
|
( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) $.
|
|
|
|
${
|
|
sbimi.1 $e |- ( ph -> ps ) $.
|
|
$( Infer substitution into antecedent and consequent of an implication.
|
|
(Contributed by NM, 25-Jun-1998.) $)
|
|
sbimi $p |- ( [ y / x ] ph -> [ y / x ] ps ) $=
|
|
( weq wi wa wex wsb imim2i anim2i eximi anim12i df-sb 3imtr4i ) CDFZAGZQA
|
|
HZCIZHQBGZQBHZCIZHACDJBCDJRUATUCABQEKSUBCABQELMNACDOBCDOP $.
|
|
$}
|
|
|
|
${
|
|
sbbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer substitution into both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbbii $p |- ( [ y / x ] ph <-> [ y / x ] ps ) $=
|
|
( wsb biimpi sbimi biimpri impbii ) ACDFBCDFABCDABEGHBACDABEIHJ $.
|
|
$}
|
|
|
|
$( One direction of a simplified definition of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb1 $p |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wa wex df-sb simprbi ) ABCDBCEZAFKAGBHABCIJ $.
|
|
|
|
$( One direction of a simplified definition of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb2 $p |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) $=
|
|
( weq wi wal wa wex wsb ax-4 equs4 df-sb sylanbrc ) BCDZAEZBFONAGBHABCIOBJA
|
|
BCKABCLM $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ1 $p |- ( x = y -> ( ph -> [ y / x ] ph ) ) $=
|
|
( weq wsb wa wi wex pm3.4 19.8a df-sb sylanbrc ex ) BCDZAABCEZNAFZNAGPBHONA
|
|
IPBJABCKLM $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ2 $p |- ( x = y -> ( [ y / x ] ph -> ph ) ) $=
|
|
( wsb weq wi wa wex df-sb simpl com12 syl5bi ) ABCDBCEZAFZMAGBHZGZMAABCIPMA
|
|
NOJKL $.
|
|
|
|
$( One of the two equality axioms of standard predicate calculus, called
|
|
substitutivity of equality. (The other one is ~ stdpc6 .) Translated to
|
|
traditional notation, it can be read: " ` x = y -> ( ph ( x ` ,
|
|
` x ) -> ph ( x ` , ` y ) ) ` , provided that ` y ` is free for ` x ` in
|
|
` ph ( x ` , ` y ) ` ." Axiom 7 of [Mendelson] p. 95. (Contributed by
|
|
NM, 15-Feb-2005.) $)
|
|
stdpc7 $p |- ( x = y -> ( [ x / y ] ph -> ph ) ) $=
|
|
( wsb wi sbequ2 equcoms ) ACBDAECBACBFG $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ12 $p |- ( x = y -> ( ph <-> [ y / x ] ph ) ) $=
|
|
( weq wsb sbequ1 sbequ2 impbid ) BCDAABCEABCFABCGH $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.)
|
|
(Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
sbequ12r $p |- ( x = y -> ( [ x / y ] ph <-> ph ) ) $=
|
|
( wsb wb weq sbequ12 bicomd equcoms ) ACBDZAECBCBFAJACBGHI $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ12a $p |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) ) $=
|
|
( weq wsb sbequ12 wb equcoms bitr3d ) BCDAABCEACBEZABCFAJGCBACBFHI $.
|
|
|
|
$( An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p.
|
|
15 of the preprint). (Contributed by NM, 5-Aug-1993.) $)
|
|
sbid $p |- ( [ x / x ] ph <-> ph ) $=
|
|
( wsb weq wb equid sbequ12 ax-mp bicomi ) AABBCZBBDAJEBFABBGHI $.
|
|
|
|
$( The specialization axiom of standard predicate calculus. It states that
|
|
if a statement ` ph ` holds for all ` x ` , then it also holds for the
|
|
specific case of ` y ` (properly) substituted for ` x ` . Translated to
|
|
traditional notation, it can be read: " ` A. x ph ( x ) -> ph ( y ) ` ,
|
|
provided that ` y ` is free for ` x ` in ` ph ( x ) ` ." Axiom 4 of
|
|
[Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.) $)
|
|
stdpc4 $p |- ( A. x ph -> [ y / x ] ph ) $=
|
|
( wal weq wi wsb ax-1 alimi sb2 syl ) ABDBCEZAFZBDABCGAMBALHIABCJK $.
|
|
|
|
${
|
|
sbh.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) $)
|
|
sbh $p |- ( [ y / x ] ph <-> ph ) $=
|
|
( wsb weq wex wa sb1 19.41h sylib simprd wal stdpc4 syl impbii ) ABCEZAQB
|
|
CFZBGZAQRAHBGSAHABCIRABDJKLAABMQDABCNOP $.
|
|
$}
|
|
|
|
${
|
|
sbf.1 $e |- F/ x ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sbf $p |- ( [ y / x ] ph <-> ph ) $=
|
|
( nfri sbh ) ABCABDEF $.
|
|
$}
|
|
|
|
$( Substitution has no effect on a bound variable. (Contributed by NM,
|
|
1-Jul-2005.) $)
|
|
sbf2 $p |- ( [ y / x ] A. x ph <-> A. x ph ) $=
|
|
( wal nfa1 sbf ) ABDBCABEF $.
|
|
|
|
${
|
|
sb6x.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Equivalence involving substitution for a variable not free.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
12-Aug-2011.) $)
|
|
sb6x $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( wsb weq wi wal sbh biidd equsalh bitr4i ) ABCEABCFZAGBHABCDIAABCDMAJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfs1f.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1f $p |- F/ x [ y / x ] ph $=
|
|
( wsb nfri sbh nfxfr ) ABCEABABCABDFGDH $.
|
|
$}
|
|
|
|
${
|
|
hbs1f.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
hbs1f $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
|
|
( wsb sbh hbxfrbi ) ABCEABABCDFDG $.
|
|
$}
|
|
|
|
$( Substitution does not change an identical variable specifier.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) $)
|
|
sbequ5 $p |- ( [ w / z ] A. x x = y <-> A. x x = y ) $=
|
|
( weq wal nfae sbf ) ABEAFCDABCGH $.
|
|
|
|
$( Substitution does not change a distinctor. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by NM, 14-May-2005.) $)
|
|
sbequ6 $p |- ( [ w / z ] -. A. x x = y <-> -. A. x x = y ) $=
|
|
( weq wal wn nfnae sbf ) ABEAFGCDABCHI $.
|
|
|
|
${
|
|
sbt.1 $e |- ph $.
|
|
$( A substitution into a theorem remains true. (See ~ chvar and ~ chvarv
|
|
for versions using implicit substitition.) (Contributed by NM,
|
|
21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sbt $p |- [ y / x ] ph $=
|
|
( wsb nfth sbf mpbir ) ABCEADABCABDFGH $.
|
|
$}
|
|
|
|
$( Substitution applied to an atomic wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsb1 $p |- [ y / x ] x = y $=
|
|
( weq wi wsb sb2 id mpg ) ABCZIDIABEAIABFIGH $.
|
|
|
|
$( Substitution applied to an atomic wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsb2 $p |- [ y / x ] y = x $=
|
|
( weq wi wsb sb2 equcomi mpg ) ABCBACZDIABEAIABFABGH $.
|
|
|
|
${
|
|
sbiedh.1 $e |- ( ph -> A. x ph ) $.
|
|
sbiedh.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
sbiedh.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution (deduction
|
|
version of ~ sbieh ). New proofs should use ~ sbied instead.
|
|
(Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) (New usage is discouraged.) $)
|
|
sbiedh $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
|
|
( wsb wex weq wa sb1 wb wi bi1 syl6 impd syld wal eximdh 19.9hd bi2 com23
|
|
syl5 alimdh sb2 impbid ) ABDEIZCAUICDJZCUIDEKZBLZDJAUJBDEMAULCDFAUKBCAUKB
|
|
CNZBCOHBCPQRUAUECADFGUBSACCDTZUIGAUNUKBOZDTUIACUODFAUKCBAUKUMCBOHBCUCQUDU
|
|
FBDEUGQSUH $.
|
|
$}
|
|
|
|
${
|
|
sbied.1 $e |- F/ x ph $.
|
|
sbied.2 $e |- ( ph -> F/ x ch ) $.
|
|
sbied.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution (deduction
|
|
version of ~ sbie ). (Contributed by NM, 30-Jun-1994.) (Revised by
|
|
Mario Carneiro, 4-Oct-2016.) $)
|
|
sbied $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
|
|
( nfri nfrd sbiedh ) ABCDEADFIACDGJHK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
sbiedv.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution (deduction
|
|
version of ~ sbie ). (Contributed by NM, 7-Jan-2017.) $)
|
|
sbiedv $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
|
|
( nfv nfvd weq wb ex sbied ) ABCDEADGACDHADEIBCJFKL $.
|
|
$}
|
|
|
|
${
|
|
sbieh.1 $e |- ( ps -> A. x ps ) $.
|
|
sbieh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution. New
|
|
proofs should use ~ sbie instead. (Contributed by NM, 30-Jun-1994.)
|
|
(New usage is discouraged.) $)
|
|
sbieh $p |- ( [ y / x ] ph <-> ps ) $=
|
|
( wi wsb wb id hbth wal a1i weq sbiedh ax-mp ) AAGZACDHBIAJZQABCDQCRKBBCL
|
|
GQEMCDNABIGQFMOP $.
|
|
$}
|
|
|
|
${
|
|
sbie.1 $e |- F/ x ps $.
|
|
sbie.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution.
|
|
(Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.) $)
|
|
sbie $p |- ( [ y / x ] ph <-> ps ) $=
|
|
( nfri sbieh ) ABCDBCEGFH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Theorems using axiom ax-11
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A property related to substitution that unlike ~ equs5 doesn't require a
|
|
distinctor antecedent. (Contributed by NM, 2-Feb-2007.) $)
|
|
equs5a $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
|
|
( weq wal wa wi hba1 ax-11 imp exlimih ) BCDZACEZFLAGZBEZBNBHLMOABCIJK $.
|
|
|
|
$( A property related to substitution that unlike ~ equs5 doesn't require a
|
|
distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM,
|
|
3-Feb-2015.) $)
|
|
equs5e $p |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) ) $=
|
|
( cv wceq wa wex wal wi 19.8a hbe1 syl anim2i eximi equs5a ) BDCDEZAFZBGPAC
|
|
GZCHZFZBGPRIBHQTBASPARSACJACKLMNRBCOL $.
|
|
|
|
$( Analogue to ~ ax-11 but for existential quantification. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro,
|
|
9-Feb-2018.) $)
|
|
ax11e $p |- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) ) $=
|
|
( cv wceq wa wex wi equs5e 19.21bi com12 ) BDCDEZAFBGZLACGZMLNHBABCIJK $.
|
|
|
|
$( Quantifier Substitution for existential quantifiers. Analogue to ~ ax10o
|
|
but for ` E. ` rather than ` A. ` . (Contributed by Jim Kingdon,
|
|
21-Dec-2017.) $)
|
|
ax10oe $p |- ( A. x x = y -> ( E. x ps -> E. y ps ) ) $=
|
|
( cv wceq wal wex wa wi ax-ia3 alimi exim syl ax11e sps syld ) BDCDEZBFZABG
|
|
ZQAHZBGZACGZRATIZBFSUAIQUCBQAJKATBLMQUAUBIBABCNOP $.
|
|
|
|
${
|
|
drex1.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.) $)
|
|
drex1 $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $=
|
|
( cv wceq wal wex wa hbae ax-4 biantrurd bitr2d exbidh wi sylbird equcomi
|
|
ax11e sps bitr3d alequcoms impbid ) CFZDFZGZCHZACIZBDIZUGUHUFBJZCIZUIUGUJ
|
|
ACCDCKUGABUJEUGUFBUFCLMNOUFUKUIPCBCDSTQUGUIUEUDGZAJZDIZUHUGUMBDCDDKUGAUMB
|
|
UGULAUFULCCDRTMEUAOUNUHPZDCULUODADCSTUBQUC $.
|
|
$}
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
drsb1 $p |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) ) $=
|
|
( weq wal wi wa wex wsb wb equequ1 sps imbi1d anbi1d drex1 anbi12d 3bitr4g
|
|
df-sb ) BCEZBFZBDEZAGZUBAHZBIZHCDEZAGZUFAHZCIZHABDJACDJUAUCUGUEUIUAUBUFATUB
|
|
UFKBBCDLMZNUDUHBCUAUBUFAUJOPQABDSACDSR $.
|
|
|
|
${
|
|
exdistrfor.1 $e |- ( A. x x = y \/ A. x F/ y ph ) $.
|
|
$( Distribution of existential quantifiers, with a bound-variable
|
|
hypothesis saying that ` y ` is not free in ` ph ` , but ` x ` can be
|
|
free in ` ph ` (and there is no distinct variable condition on ` x ` and
|
|
` y ` ). (Contributed by Jim Kingdon, 25-Feb-2018.) $)
|
|
exdistrfor $p |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) $=
|
|
( weq wal wnf wo wa wex biidd drex1 drex2 hbe1 19.9h 19.8a anim2i eximi
|
|
wi sylbi syl6bir ax-ial 19.40 19.9t biimpd anim1d syl5 eximdh jaoi ax-mp
|
|
sps ) CDFCGZADHZCGZIABJZDKZCKZABDKZJZCKZTZEUMVBUOUMURUPCKZCKZVAVCUQCDCUPU
|
|
PCDUMUPLMNVDVCVAVCCUPCOPUPUTCBUSABDQRSUAUBUOUQUTCUNCUCUNUQUTTCUQADKZUSJUN
|
|
UTABDUDUNVEAUSUNVEAADUEUFUGUHULUIUJUK $.
|
|
$}
|
|
|
|
$( A version of ~ sb4 that doesn't require a distinctor antecedent.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
sb4a $p |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) ) $=
|
|
( wal wsb weq wa wex wi sb1 equs5a syl ) ACDZBCEBCFZMGBHNAIBDMBCJABCKL $.
|
|
|
|
${
|
|
equs45f.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Two ways of expressing substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 25-Apr-2008.) $)
|
|
equs45f $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wa wex wi wal anim2i eximi equs5a syl equs4 impbii ) BCEZAFZBGZPAHB
|
|
IZRPACIZFZBGSQUABATPDJKABCLMABCNO $.
|
|
|
|
$( Equivalence for substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) $)
|
|
sb6f $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( wsb weq wi wal sbimi sb4a syl sb2 impbii ) ABCEZBCFAGBHZNACHZBCEOAPBCDI
|
|
ABCJKABCLM $.
|
|
|
|
$( Equivalence for substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) $)
|
|
sb5f $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wal wa wex sb6f equs45f bitr4i ) ABCEBCFZAGBHNAIBJABCDKABCDL
|
|
M $.
|
|
$}
|
|
|
|
$( One direction of a simplified definition of substitution that unlike ~ sb4
|
|
doesn't require a distinctor antecedent. (Contributed by NM,
|
|
2-Feb-2007.) $)
|
|
sb4e $p |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) ) $=
|
|
( wsb weq wa wex wi wal sb1 equs5e syl ) ABCDBCEZAFBGMACGHBIABCJABCKL $.
|
|
|
|
$( Special case of a bound-variable hypothesis builder for substitution.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
hbsb2a $p |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph ) $=
|
|
( wal wsb weq wi sb4a sb2 a5i syl ) ACDBCEBCFAGZBDABCEZBDABCHLMBABCIJK $.
|
|
|
|
$( Special case of a bound-variable hypothesis builder for substitution.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
hbsb2e $p |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph ) $=
|
|
( wsb weq wex wi wal sb4e sb2 a5i syl ) ABCDBCEACFZGZBHMBCDZBHABCINOBMBCJKL
|
|
$.
|
|
|
|
${
|
|
hbsb3.1 $e |- ( ph -> A. y ph ) $.
|
|
$( If ` y ` is not free in ` ph ` , ` x ` is not free in ` [ y / x ] ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbsb3 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
|
|
( wsb wal sbimi hbsb2a syl ) ABCEZACFZBCEJBFAKBCDGABCHI $.
|
|
$}
|
|
|
|
${
|
|
nfs1.1 $e |- F/ y ph $.
|
|
$( If ` y ` is not free in ` ph ` , ` x ` is not free in ` [ y / x ] ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1 $p |- F/ x [ y / x ] ph $=
|
|
( wsb nfri hbsb3 nfi ) ABCEBABCACDFGH $.
|
|
$}
|
|
|
|
${
|
|
sbcof2.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Version of ~ sbco where ` x ` is not free in ` ph ` . (Contributed by
|
|
Jim Kingdon, 28-Dec-2017.) $)
|
|
sbcof2 $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
|
|
( wsb weq wi wal hbsb3 sb6f imbi2i albii bitri alimi wex 3syl wa jca sb5f
|
|
eximi ax-11 equcomi imim1i imim2i pm2.43d syl6 a2i sylbi ax-i9 mpi ax-ial
|
|
exim 19.9h sylib sb2 simpl 19.8a anim1i ax11e syl5 imdistani anbi2i exbii
|
|
sb1 sylibr impbii ) ACBEZBCEZABCEZVHBCFZVJAGZBHZGZBHZVLVIVHVJCBFZAGZCHZGZ
|
|
BHZVNVHVJVGGZBHVSVGBCACBDIZJVTVRBVGVQVJACBDJKLMVRVMBVJVQVLVJVQVJVPGZBHVLV
|
|
PBCUAWBVKBWBVJAVPVKVJVJVOABCUBZUCUDUENUFUGNUHVNVLBOZVLVNVJBOWDBCUIVJVLBUL
|
|
UJVLBVKBUKUMUNABCUOPVIVJVOAQZCOZQZBOZVHVIVJAQZBOZVJWJQZBOWHABCVDWIWKBWIVJ
|
|
WJVJAUPZWIBUQRTWKWGBVJWJWFWJVJWEQZBOVJWFWIWMBWIVJWEWLVJVOAWCURRTWEBCUSUTV
|
|
ATPVHVJVGQZBOWHVGBCWASWNWGBVGWFVJACBDSVBVCMVEVF $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Predicate calculus with distinct variables
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the axiom of distinct variables ax-16
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x ps $.
|
|
spimv.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( A version of ~ spim with a distinct variable requirement instead of a
|
|
bound variable hypothesis. (Contributed by NM, 5-Aug-1993.) $)
|
|
spimv $p |- ( A. x ph -> ps ) $=
|
|
( nfv spim ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$v f $.
|
|
$( Define a temporary individual variable. $)
|
|
aev.vf $f setvar f $.
|
|
$d f u v $. $d f u x y $. $d u w $.
|
|
$( A "distinctor elimination" lemma with no restrictions on variables in
|
|
the consequent, proved without using ~ ax-16 . (Contributed by NM,
|
|
8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
aev $p |- ( A. x x = y -> A. z w = v ) $=
|
|
( aev.vf vu weq wal hbae ax-8 alrimih equcomi syl6 alequcoms a5i alequcom
|
|
spimv 3syl ) ABHZAIZDEHZCABCJUAFBHZFIZGEHZGIZUBUAUCFABFJTUCAFAFBKRLUDFGHZ
|
|
FIZEGHZEIUFUCUGFUGBFBFHZUGBGBGHUJGFHUGBGFKGFMNROPUHUIEFGEJUGUIFEFEGKRLEGQ
|
|
SUEUBGDGDEKRSL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d z ph $.
|
|
$( Theorem showing that ~ ax-16 is redundant if ~ ax-17 is included in the
|
|
axiom system. The important part of the proof is provided by ~ aev .
|
|
|
|
See ~ ax16ALT for an alternate proof that does not require ~ ax-10 or
|
|
~ ax-12 .
|
|
|
|
This theorem should not be referenced in any proof. Instead, use
|
|
~ ax-16 below so that theorems needing ~ ax-16 can be more easily
|
|
identified. (Contributed by NM, 8-Nov-2006.) $)
|
|
ax16 $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( vz weq wal wi aev wsb ax-17 sbequ12 biimpcd alimdh hbsb3 stdpc7 syl6com
|
|
cbv3h syl ) BCEBFBDEZDFZAABFZGBCDBDHATABDIZDFUAASUBDADJZSAUBABDKLMUBADBAB
|
|
DUCNUCADBOQPR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of Distinct Variables. The only axiom of predicate calculus
|
|
requiring that variables be distinct (if we consider ~ ax-17 to be a
|
|
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
|
|
16 of the preprint). It apparently does not otherwise appear in the
|
|
literature but is easily proved from textbook predicate calculus by
|
|
cases. It is a somewhat bizarre axiom since the antecedent is always
|
|
false in set theory, but nonetheless it is technically necessary as you
|
|
can see from its uses.
|
|
|
|
This axiom is redundant if we include ~ ax-17 ; see theorem ~ ax16 .
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax16 . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-16 $a |- ( A. x x = y -> ( ph -> A. x ph ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d z x $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveeq2 $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
|
|
( weq wal wn wi wo ax-i12 orcom orbi2i mpbi orass mpbir orel2 mpi ax16 sp
|
|
jaoi syl ) ABDAEZFZACDAEZCBDZUDAEGZAEZHZUEUBUGUAHZUGUHUCUFUAHZHZUCUAUFHZH
|
|
UJCBAIUKUIUCUAUFJKLUCUFUAMNUAUGOPUCUEUFUDACQUEARST $.
|
|
$}
|
|
|
|
${
|
|
$d z x $.
|
|
$( Quantifier introduction when one pair of variables is distinct. Like
|
|
~ dveeq2 but connecting ` A. x x = y ` by a disjunction rather than
|
|
negation and implication makes the theorem stronger in intuitionistic
|
|
logic. (Contributed by Jim Kingdon, 1-Feb-2018.) $)
|
|
dveeq2or $p |- ( A. x x = y \/ F/ x z = y ) $=
|
|
( weq wal wi wnf ax-i12 orass mpbir pm1.4 orim1i ax-mp mpbi ax16 a5i jaoi
|
|
wo id orim2i df-nf biimpri ) ABDAEZCBDZUDAEFZAEZRZUCUDAGZRUCACDZAEZUFRZRZ
|
|
UGUCUJRZUFRZULUJUCRZUFRZUNUPUJUGRCBAHUJUCUFIJUOUMUFUJUCKLMUCUJUFINUKUFUCU
|
|
JUFUFUIUEAUDACOPUFSQTMUFUHUCUHUFUDAUAUBTM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
dvelimfALT2.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimfALT2.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimfALT2.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
dvelimfALT2.4 $e |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $.
|
|
$( Proof of ~ dvelimf using ~ dveeq2 (shown as the last hypothesis) instead
|
|
of ~ ax-12 . This shows that ~ ax-12 could be replaced by ~ dveeq2 (the
|
|
last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) $)
|
|
dvelimfALT2 $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( cv wceq wal wn wi ax-17 hbn1 a1i hbimd hbald equsalh albii 3imtr3g ) CJ
|
|
DJZKZCLMZEJUCKZANZELZUHCLBBCLUEUGCEUEEOUEUFACUDCPIAACLNUEFQRSABEDGHTZUHBC
|
|
UIUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d z x $.
|
|
$( A lemma for proving conditionless ZFC axioms. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
nd5 $p |- ( -. A. y y = x -> ( z = y -> A. x z = y ) ) $=
|
|
( cv wceq wal wi dveeq2 nalequcoms ) CDBDEZJAFGABABCHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $.
|
|
exlimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
exlimdv $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( ax-17 exlimdh ) ABCDADFCDFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11v2.1 $e |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Recovery of ~ ax11o from ~ ax11v without using ~ ax-11 . The hypothesis
|
|
is even weaker than ~ ax11v , with ` z ` both distinct from ` x ` _and_
|
|
not occurring in ` ph ` . Thus the hypothesis provides an alternate
|
|
axiom that can be used in place of ~ ax11o . (Contributed by NM,
|
|
2-Feb-2007.) $)
|
|
ax11v2 $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( weq wal wn wex wi a9e wa wb equequ2 adantl dveeq2 imp hba1 imbi1d sps
|
|
albidh syl imbi2d imbi12d mpbii ex exlimdv mpi ) BCFZBGHZDCFZDIUIAUIAJZBG
|
|
ZJZJZDCKUJUKUODUJUKUOUJUKLZBDFZAUQAJZBGZJZJUOEUPUQUIUTUNUKUQUIMUJDCBNZOUP
|
|
USUMAUPUKBGZUSUMMUJUKVBBCDPQVBURULBUKBRUKURULMBUKUQUIAVASTUAUBUCUDUEUFUGU
|
|
H $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11a2.1 $e |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Derive ~ ax-11o from a hypothesis in the form of ~ ax-11 . The
|
|
hypothesis is even weaker than ~ ax-11 , with ` z ` both distinct from
|
|
` x ` and not occurring in ` ph ` . Thus the hypothesis provides an
|
|
alternate axiom that can be used in place of ~ ax11o . (Contributed by
|
|
NM, 2-Feb-2007.) $)
|
|
ax11a2 $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( wal weq wi ax-17 syl5 ax11v2 ) ABCDAADFBDGZLAHBFADIEJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the obsolete axiom of variable substitution ax-11o
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( Derivation of set.mm's original ~ ax-11o from the shorter ~ ax-11 that
|
|
has replaced it.
|
|
|
|
An open problem is whether this theorem can be proved without relying on
|
|
~ ax-16 or ~ ax-17 .
|
|
|
|
Normally, ~ ax11o should be used rather than ~ ax-11o , except by
|
|
theorems specifically studying the latter's properties. (Contributed by
|
|
NM, 3-Feb-2007.) $)
|
|
ax11o $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( vz ax-11 ax11a2 ) ABCDABDEF $.
|
|
$}
|
|
|
|
$( Axiom ~ ax-11o ("o" for "old") was the original version of ~ ax-11 ,
|
|
before it was discovered (in Jan. 2007) that the shorter ~ ax-11 could
|
|
replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
|
|
the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
|
|
[Monk2] p. 105, from which it can be proved by cases. To understand this
|
|
theorem more easily, think of " ` -. A. x x = y -> ` ..." as informally
|
|
meaning "if ` x ` and ` y ` are distinct variables then..." The
|
|
antecedent becomes false if the same variable is substituted for ` x ` and
|
|
` y ` , ensuring the theorem is sound whenever this is the case. In some
|
|
later theorems, we call an antecedent of the form ` -. A. x x = y ` a
|
|
"distinctor."
|
|
|
|
This axiom is redundant, as shown by theorem ~ ax11o .
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax11o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-11o $a |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
More theorems related to ax-11 and substitution
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x ph $.
|
|
albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
albidv $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( ax-17 albidh ) ABCDADFEG $.
|
|
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
exbidv $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( ax-17 exbidh ) ABCDADFEG $.
|
|
$}
|
|
|
|
$( A bidirectional version of ~ ax-11o . (Contributed by NM,
|
|
30-Jun-2006.) $)
|
|
ax11b $p |- ( ( -. A. x x = y /\ x = y ) ->
|
|
( ph <-> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wa wi ax11o imp ax-4 com12 adantl impbid ) BCDZBEFZOGAOAHZBEZP
|
|
OARHABCIJORAHPROAQBKLMN $.
|
|
|
|
${
|
|
$d x y $. $d x z $. $d y z $. $d ph z $.
|
|
$( This is a version of ~ ax-11o when the variables are distinct. Axiom
|
|
(C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Jim Kingdon, 15-Dec-2017.) $)
|
|
ax11v $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( vz cv wceq wex wi wal a9e ax-17 ax-11 syl5 equequ2 imbi1d albidv imbi2d
|
|
imbi12d mpbii exlimiv ax-mp ) DEZCEZFZDGBEZUCFZAUFAHZBIZHZHZDCJUDUJDUDUEU
|
|
BFZAUKAHZBIZHZHUJAADIUKUMADKABDLMUDUKUFUNUIDCBNZUDUMUHAUDULUGBUDUKUFAUOOP
|
|
QRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $. $d y z $. $d ph z $.
|
|
$( Analogue to ~ ax11v for existential quantification. (Contributed by Jim
|
|
Kingdon, 9-Jan-2018.) $)
|
|
ax11ev $p |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) ) $=
|
|
( vz cv wceq wex wa wi a9e ax11e ax-17 19.9h syl6ib equequ2 anbi1d exbidv
|
|
imbi1d imbi12d mpbii exlimiv ax-mp ) DEZCEZFZDGBEZUDFZUGAHZBGZAIZIZDCJUEU
|
|
KDUEUFUCFZULAHZBGZAIZIUKULUNADGAABDKADADLMNUEULUGUOUJDCBOZUEUNUIAUEUMUHBU
|
|
EULUGAUPPQRSTUAUB $.
|
|
$}
|
|
|
|
$( Lemma used in proofs of substitution properties. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equs5 $p |- ( -. A. x x = y ->
|
|
( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wa wi hbnae hba1 ax11o impd exlimdh ) BCDZBEFZNAGNAHZBEZBBCBIP
|
|
BJONAQABCKLM $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $.
|
|
|
|
$( Lemma used in proofs of substitution properties. Like ~ equs5 but, in
|
|
intuitionistic logic, replacing negation and implication with
|
|
disjunction makes this a stronger result. (Contributed by Jim Kingdon,
|
|
2-Feb-2018.) $)
|
|
equs5or $p |- ( A. x x = y \/
|
|
( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) $=
|
|
( vz weq wex wal wa wi wo a9e wnf dveeq2or nfnf1 nfri ax11v equequ2 ax-mp
|
|
wb hba1 adantl nfr imp imbi1d sps albidh syl imbi2d imbi12d mpbii alrimih
|
|
ex imp4a 19.21t mpbid 19.23h syl6ib orim2i pm2.76 olcs exlimiv ) DCEZDFBC
|
|
EZBGZVCAHZBFVCAIZBGZIZJZDCKVBVIDVDVBVIVDVBVHIZJZVDVBJVIIVDVBBLZJVKBCDMVLV
|
|
JVDVLVBVEVGIZBGZVHVLVBVMIZBGVBVNIVLVOBVLBVBBNOVLVBVCAVGVLVBVCAVGIZIZVLVBH
|
|
ZBDEZAVSAIZBGZIZIVQABDPVRVSVCWBVPVBVSVCSVLDCBQZUAVRWAVGAVRVBBGZWAVGSVLVBW
|
|
DVBBUBUCWDVTVFBVBBTVBVTVFSBVBVSVCAWCUDUEUFUGUHUIUJULUMUKVBVMBUNUOVEVGBVFB
|
|
TUPUQURRVDVBVHUSRUTVAR $.
|
|
$}
|
|
|
|
$( One direction of a simplified definition of substitution when variables
|
|
are distinct. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb3 $p |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) ) $=
|
|
( weq wal wn wa wex wi wsb equs5 sb2 syl6 ) BCDZBEFNAGBHNAIBEABCJABCKABCLM
|
|
$.
|
|
|
|
$( One direction of a simplified definition of substitution when variables
|
|
are distinct. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb4 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( wsb weq wa wex wal wn wi sb1 equs5 syl5 ) ABCDBCEZAFBGNBHINAJBHABCKABCLM
|
|
$.
|
|
|
|
$( One direction of a simplified definition of substitution when variables
|
|
are distinct. Similar to ~ sb4 but stronger in intuitionistic logic.
|
|
(Contributed by Jim Kingdon, 2-Feb-2018.) $)
|
|
sb4or $p |- ( A. x x = y \/
|
|
A. x ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wa wex wi wo wsb equs5or nfe1 nfa1 nfim sb1 imim1i alrimih orim2i
|
|
nfri ax-mp ) BCDZBEZUAAFZBGZUAAHZBEZHZIUBABCJZUFHZBEZIABCKUGUJUBUGUIBUGBUDU
|
|
FBUCBLUEBMNSUHUDUFABCOPQRT $.
|
|
|
|
$( Simplified definition of substitution when variables are distinct.
|
|
(Contributed by NM, 27-May-1997.) $)
|
|
sb4b $p |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wsb wi sb4 sb2 impbid1 ) BCDZBEFABCGLAHBEABCIABCJK $.
|
|
|
|
$( Simplified definition of substitution when variables are distinct,
|
|
expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) $)
|
|
sb4bor $p |- ( A. x x = y \/
|
|
A. x ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wsb wi wo wb sb4or sb2 wa df-bi simpri mpan2 alimi orim2i ax-mp )
|
|
BCDZBEZABCFZSAGBEZGZBEZHTUAUBIZBEZHABCJUDUFTUCUEBUCUBUAGZUEABCKUEUCUGLZGUHU
|
|
EGUAUBMNOPQR $.
|
|
|
|
$( Bound-variable hypothesis builder for substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbsb2 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) ) $=
|
|
( weq wal wn wsb wi sb4 sb2 a5i syl6 ) BCDZBEFABCGZMAHZBENBEABCIONBABCJKL
|
|
$.
|
|
|
|
$( Bound-variable hypothesis builder for substitution. Similar to ~ hbsb2
|
|
but in intuitionistic logic a disjunction is stronger than an implication.
|
|
(Contributed by Jim Kingdon, 2-Feb-2018.) $)
|
|
nfsb2or $p |- ( A. x x = y \/ F/ x [ y / x ] ph ) $=
|
|
( weq wal wsb wi wnf sb4or sb2 a5i imim2i alimi df-nf sylibr orim2i ax-mp
|
|
wo ) BCDZBEZABCFZSAGZBEZGZBEZRTUABHZRABCIUEUFTUEUAUABEZGZBEUFUDUHBUCUGUAUBU
|
|
ABABCJKLMUABNOPQ $.
|
|
|
|
${
|
|
sbequilem.1 $e |- ( ph \/ ( ps -> ( ch -> th ) ) ) $.
|
|
sbequilem.2 $e |- ( ta \/ ( ps -> ( th -> et ) ) ) $.
|
|
$( Propositional logic lemma used in the ~ sbequi proof. (Contributed by
|
|
Jim Kingdon, 1-Feb-2018.) $)
|
|
sbequilem $p |- ( ph \/ ( ta \/ ( ps -> ( ch -> et ) ) ) ) $=
|
|
( wo wi wa pm3.2i andi mpbi andir orbi12i orim2i ax-mp orim1i orass simpr
|
|
pm3.43 pm3.33 syl6 sylbir simpl mpbir orcom orbi1i ) AEIZBCFJZJZIZAEULIIE
|
|
AIZULIZUMUOEAULIZIZEABDFJZJZKZULIZIZUQAEKBCDJZJZEKIZVAIZVBVEUTVDUSKZIZIZV
|
|
FAVDIZEKZVJUSKZIZVIVJEUSIZKVMVJVNGHLVJEUSMNVKVEVLVHAVDEOZAVDUSOPNVHVAVEVG
|
|
ULUTVGBVCURKUKBVCURUBCDFUCUDQQRVEEVAVEVKEVOVJEUAUESRVAUPEUTAULAUSUFSQREAU
|
|
LTUGUNUJULEAUHUINAEULTN $.
|
|
$}
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modified by Jim Kingdon, 1-Feb-2018.) $)
|
|
sbequi $p |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) $=
|
|
( weq wal wsb wi wo wex nfsb2or wa stdpc7 sbequ1 sylan9 orim2i ax-mp biimpd
|
|
wnf sps equvini eximi 19.35-1 3syl syl9 19.9t sbequilem sbequ2 adantr drsb1
|
|
nfr ax-1 alequcoms sylan9r syld ex orim1i pm1.2 syl jaoi ) DBEZDFZDCEZDFZBC
|
|
EZADBGZADCGZHZHZIZIVIVBVEVFVGDJZVDVGVBVFDSZIVBVEVFVKHHZIADBKVLVMVBVLVFVFDFZ
|
|
VEVKVFDUKVEBDEZVCLZDJVHDJVNVKHBCDUAVPVHDVOVFAVCVGABDMADCNZOUBVFVGDUCUDUEPQV
|
|
DVKVGHZIZVDVEVRHZIVDVGDSZIVSADCKWAVRVDWAVKVGVGDUFRPQVRVTVDVRVEULPQUGVBVIVJV
|
|
BVEVHVBVELVFAVGVBVFAHZVEVAWBDADBUHTUIVEAABCGZVBVGABCNWCVGHBDVOBFWCVGABDCUJR
|
|
UMUNUOUPVJVIVIIVIVDVIVIVDVEVHVDVELVFAVGVDVFACBGZVEAVDVFWDADCBUJRABCMOVDAVGH
|
|
ZVEVCWEDVQTUIUOUPUQVIURUSUTQ $.
|
|
|
|
$( An equality theorem for substitution. Used in proof of Theorem 9.7 in
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbequ $p |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
|
|
( weq wsb sbequi wi equcoms impbid ) BCEADBFZADCFZABCDGLKHCBACBDGIJ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed
|
|
by NM, 27-Feb-2005.) $)
|
|
drsb2 $p |- ( A. x x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
|
|
( weq wsb wb sbequ sps ) BCEADBFADCFGBABCDHI $.
|
|
|
|
$( A specialization theorem, mostly the same as Theorem 19.8 of [Margaris]
|
|
p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
|
|
29-Dec-2017.) $)
|
|
spsbe $p |- ( [ y / x ] ph -> E. x ph ) $=
|
|
( wsb weq wa wex sb1 simpr eximi syl ) ABCDBCEZAFZBGABGABCHMABLAIJK $.
|
|
|
|
$( Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
rewritten by Jim Kingdon, 21-Jan-2018.) $)
|
|
spsbim $p |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( wi wal weq wa wex wsb imim2 sps id anim2d alimi syl anim12d df-sb 3imtr4g
|
|
exim ) ABEZCFZCDGZAEZUCAHZCIZHUCBEZUCBHZCIZHACDJBCDJUBUDUGUFUIUAUDUGECABUCK
|
|
LUBUEUHEZCFUFUIEUAUJCUAABUCUAMNOUEUHCTPQACDRBCDRS $.
|
|
|
|
$( Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
rewritten by Jim Kingdon, 21-Jan-2018.) $)
|
|
spsbbi $p |- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
|
|
( wi wal wa wsb wb spsbim anim12i albiim dfbi2 3imtr4i ) ABECFZBAECFZGACDHZ
|
|
BCDHZEZRQEZGABICFQRIOSPTABCDJBACDJKABCLQRMN $.
|
|
|
|
${
|
|
sbbidh.1 $e |- ( ph -> A. x ph ) $.
|
|
sbbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction substituting both sides of a biconditional. New proofs should
|
|
use ~ sbbid instead. (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
sbbidh $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
|
|
( wb wal wsb alrimih spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.
|
|
$}
|
|
|
|
${
|
|
sbbid.1 $e |- F/ x ph $.
|
|
sbbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction substituting both sides of a biconditional. (Contributed by
|
|
NM, 30-Jun-1993.) $)
|
|
sbbid $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
|
|
( wb wal wsb alrimi spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.
|
|
$}
|
|
|
|
$( Elimination of equality from antecedent after substitution. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) $)
|
|
sbequ8 $p |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) ) $=
|
|
( weq wi wa wex wsb pm5.4 simpl pm3.35 jca pm3.4 impbii exbii anbi12i df-sb
|
|
3bitr4ri ) BCDZSAEZEZSTFZBGZFTSAFZBGZFTBCHABCHUATUCUESAIUBUDBUBUDUBSASTJSAK
|
|
LUDSTSAJSAMLNOPTBCQABCQR $.
|
|
|
|
$( Substitution has no effect on a non-free variable. (Contributed by NM,
|
|
30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened
|
|
by Wolf Lammen, 3-May-2018.) $)
|
|
sbft $p |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) $=
|
|
( wnf wsb wex spsbe 19.9t syl5ib wal nfr stdpc4 syl6 impbid ) ABDZABCEZAPAB
|
|
FOAABCGABHIOAABJPABKABCLMN $.
|
|
|
|
${
|
|
sbid2h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( An identity law for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbid2h $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
|
|
( wsb sbcof2 sbh bitri ) ACBEBCEABCEAABCDFABCDGH $.
|
|
$}
|
|
|
|
${
|
|
sbid2.1 $e |- F/ x ph $.
|
|
$( An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sbid2 $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
|
|
( nfri sbid2h ) ABCABDEF $.
|
|
$}
|
|
|
|
$( An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.)
|
|
(Proof rewritten by Jim Kingdon, 21-Jan-2018.) $)
|
|
sbidm $p |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph ) $=
|
|
( wsb weq wi wa df-sb simplbi sbimi sbequ8 sylibr ax-1 pm4.24 ax-ie1 19.41h
|
|
wex sb1 bitr4i exbii anim2i anim1i eximi sylbi anass anbi2i sylanbrc impbii
|
|
sylib syl ) ABCDZBCDZUKULBCEZAFZBCDUKUKUNBCUKUNUMAGZBQZABCHZIJABCKLUKUMUKFU
|
|
MUKGZBQZULUKUMMUKUPUSABCRUPUMUNUPGZGZBQZUSUPUMUNGZUPGZBQZVBUPUOUPGZBQZVEUPU
|
|
PUPGVGUPNUOUPBUOBOPSVFVDBUOVCUPAUNUMAUMMUAUBUCUDVDVABUMUNUPUETUIURVABUKUTUM
|
|
UQUFTLUJUKBCHUGUH $.
|
|
|
|
${
|
|
sb5rf.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sb5rf $p |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) ) $=
|
|
( weq wsb wa wex sbid2h sb1 sylbir stdpc7 imp exlimih impbii ) ACBEZABCFZ
|
|
GZCHZAQCBFSACBDIQCBJKRACDPQAACBLMNO $.
|
|
|
|
$( Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sb6rf $p |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) ) $=
|
|
( weq wsb wi wal sbequ1 equcoms com12 alrimih sb2 sbid2h sylib impbii ) A
|
|
CBEZABCFZGZCHZASCDQARARGBCABCIJKLTRCBFARCBMACBDNOP $.
|
|
$}
|
|
|
|
${
|
|
sb8h.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Substitution of variable in universal quantifier. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof
|
|
shortened by Jim Kingdon, 15-Jan-2018.) $)
|
|
sb8h $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
|
|
( wsb hbsb3 sbequ12 cbvalh ) AABCEBCDABCDFABCGH $.
|
|
$}
|
|
|
|
${
|
|
sb8eh.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Substitution of variable in existential quantifier. (Contributed by NM,
|
|
12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) $)
|
|
sb8eh $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
|
|
( wsb hbsb3 sbequ12 cbvexh ) AABCEBCDABCDFABCGH $.
|
|
$}
|
|
|
|
${
|
|
sb8e.1 $e |- F/ y ph $.
|
|
$( Substitution of variable in universal quantifier. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened
|
|
by Jim Kingdon, 15-Jan-2018.) $)
|
|
sb8 $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
|
|
( wsb nfs1 sbequ12 cbval ) AABCEBCDABCDFABCGH $.
|
|
|
|
$( Substitution of variable in existential quantifier. (Contributed by NM,
|
|
12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof
|
|
shortened by Jim Kingdon, 15-Jan-2018.) $)
|
|
sb8e $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
|
|
( wsb nfs1 sbequ12 cbvex ) AABCEBCDABCDFABCGH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Predicate calculus with distinct variables (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
ax16i.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
ax16i.2 $e |- ( ps -> A. x ps ) $.
|
|
$( Inference with ~ ax-16 as its conclusion, that doesn't require ~ ax-10 ,
|
|
~ ax-11 , or ~ ax-12 for its proof. The hypotheses may be eliminable
|
|
without one or more of these axioms in special cases. (Contributed by
|
|
NM, 20-May-2008.) $)
|
|
ax16i $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( weq wal wi ax-17 ax-8 cbv3h spimv equid mpi syl syl5com alimdh mpcom
|
|
alimi biimpcd biimprd syl6com ) CDHZCIZCEHZEIZAACIZJUFEDHZEIZUHUEUJCEUEEK
|
|
ZUJCKCEDLMUKECHZEIZUHUEUKUNUJUEECECDLNUEUJUMEULUEDCHZUJUMUECCHUOCOCDCLPUJ
|
|
DEHZUOUMJUJEEHZUPEOZEDELPDECLQRSTUMUGEUMUQUGURECELPZUAQQAUHBEIUIAUGBEAEKZ
|
|
UGABFUBSBAECGUTUMUGBAJUSUGABFUCQMUDQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
$( Version of ~ ax16 that doesn't require ~ ax-10 or ~ ax-12 for its proof.
|
|
(Contributed by NM, 17-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax16ALT $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( vz wsb sbequ12 ax-17 hbsb3 ax16i ) AABDEBCDABDFABDADGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
spv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Specialization, using implicit substitition. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
spv $p |- ( A. x ph -> ps ) $=
|
|
( weq biimpd spimv ) ABCDCDFABEGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
spimev.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Distinct-variable version of ~ spime . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
spimev $p |- ( ph -> E. x ps ) $=
|
|
( nfv spime ) ABCDACFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
speiv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
speiv.2 $e |- ps $.
|
|
$( Inference from existential specialization, using implicit substitition.
|
|
(Contributed by NM, 19-Aug-1993.) $)
|
|
speiv $p |- E. x ph $=
|
|
( wex weq biimprd spimev ax-mp ) BACGFBACDCDHABEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( A variable introduction law for equality. Lemma 15 of [Monk2] p. 109.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
equvin $p |- ( x = y <-> E. z ( x = z /\ z = y ) ) $=
|
|
( weq wa wex equvini ax-17 equtr imp exlimih impbii ) ABDZACDZCBDZEZCFABC
|
|
GPMCMCHNOMACBIJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A generalization of axiom ~ ax-16 . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
a16g $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
|
|
( weq wal aev ax16 biidd dral1 biimprd sylsyld ) BCEBFDBEDFZAABFZADFZBCDD
|
|
BGABCHMONAADBMAIJKL $.
|
|
|
|
$( A generalization of axiom ~ ax-16 . (Contributed by NM, 5-Aug-1993.) $)
|
|
a16gb $p |- ( A. x x = y -> ( ph <-> A. z ph ) ) $=
|
|
( weq wal a16g ax-4 impbid1 ) BCEBFAADFABCDGADHI $.
|
|
|
|
$( If there is only one element in the universe, then everything satisfies
|
|
` F/ ` . (Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
a16nf $p |- ( A. x x = y -> F/ z ph ) $=
|
|
( weq wal nfae a16g nfd ) BCEBFADBCDGABCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 2 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 4-Mar-1997.) $)
|
|
2albidv $p |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) ) $=
|
|
( wal albidv ) ABEGCEGDABCEFHH $.
|
|
|
|
$( Formula-building rule for 2 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
2exbidv $p |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) ) $=
|
|
( wex exbidv ) ABEGCEGDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $.
|
|
3exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 3 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
3exbidv $p |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) ) $=
|
|
( wex exbidv 2exbidv ) ABFHCFHDEABCFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $. $d w ph $.
|
|
4exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 4 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 3-Aug-1995.) $)
|
|
4exbidv $p |- ( ph ->
|
|
( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) ) $=
|
|
( wex 2exbidv ) ABGIFICGIFIDEABCFGHJJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM,
|
|
28-May-1995.) (Revised by NM, 21-May-2007.) $)
|
|
19.9v $p |- ( E. x ph <-> ph ) $=
|
|
( ax-17 19.9h ) ABABCD $.
|
|
$}
|
|
|
|
${
|
|
exlimdd.1 $e |- F/ x ph $.
|
|
exlimdd.2 $e |- F/ x ch $.
|
|
exlimdd.3 $e |- ( ph -> E. x ps ) $.
|
|
exlimdd.4 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Existential elimination rule of natural deduction. (Contributed by
|
|
Mario Carneiro, 9-Feb-2017.) $)
|
|
exlimdd $p |- ( ph -> ch ) $=
|
|
( wex ex exlimd mpd ) ABDICGABCDEFABCHJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.21 of [Margaris] p. 90. _Notational
|
|
convention_: We sometimes suffix with "v" the label of a theorem
|
|
eliminating a hypothesis such as ` ( ph -> A. x ph ) ` in ~ 19.21 via
|
|
the use of distinct variable conditions combined with ~ ax-17 .
|
|
Conversely, we sometimes suffix with "f" the label of a theorem
|
|
introducing such a hypothesis to eliminate the need for the distinct
|
|
variable condition; e.g. ~ euf derived from ~ df-eu . The "f" stands
|
|
for "not free in" which is less restrictive than "does not occur in."
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
19.21v $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( ax-17 19.21h ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
alrimiv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
alrimiv $p |- ( ph -> A. x ps ) $=
|
|
( ax-17 alrimih ) ABCACEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
alrimivv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
31-Jul-1995.) $)
|
|
alrimivv $p |- ( ph -> A. x A. y ps ) $=
|
|
( wal alrimiv ) ABDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
alrimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
alrimdv $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( ax-17 alrimdh ) ABCDADFBDFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nfdv.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Apply the definition of not-free in a context. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfdv $p |- ( ph -> F/ x ps ) $=
|
|
( wal wi wnf alrimiv df-nf sylibr ) ABBCEFZCEBCGAKCDHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
$( Quantification of two variables over a formula in which they do not
|
|
occur. (Contributed by Alan Sare, 12-Apr-2011.) $)
|
|
2ax17 $p |- ( ph -> A. x A. y ph ) $=
|
|
( id alrimivv ) AABCADE $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM,
|
|
3-Apr-1994.) $)
|
|
alimdv $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( ax-17 alimdh ) ABCDADFEG $.
|
|
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
eximdv $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( ax-17 eximdh ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-2004.) $)
|
|
2alimdv $p |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) ) $=
|
|
( wal alimdv ) ABEGCEGDABCEFHH $.
|
|
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
3-Aug-1995.) $)
|
|
2eximdv $p |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) ) $=
|
|
( wex eximdv ) ABEGCEGDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
19.23v $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( ax-17 19.23h ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 extended to two variables.
|
|
(Contributed by NM, 10-Aug-2004.) $)
|
|
19.23vv $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) ) $=
|
|
( wi wal wex 19.23v albii bitri ) ABEDFZCFADGZBEZCFLCGBEKMCABDHILBCHJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Two equivalent ways of expressing the proper substitution of ` y ` for
|
|
` x ` in ` ph ` , when ` x ` and ` y ` are distinct. Theorem 6.2 of
|
|
[Quine] p. 40. The proof does not involve ~ df-sb . (Contributed by
|
|
NM, 14-Apr-2008.) $)
|
|
sb56 $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wi wal hba1 ax11v ax-4 com12 impbid equsex ) ABCDZAEZBFZBCNBGMAOABC
|
|
HOMANBIJKL $.
|
|
|
|
$( Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40.
|
|
Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM,
|
|
18-Aug-1993.) (Revised by NM, 14-Apr-2008.) $)
|
|
sb6 $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wi wa wex wal wsb sb56 anbi2i df-sb ax-4 pm4.71ri 3bitr4i ) BCDZAEZ
|
|
PAFBGZFQQBHZFABCISRSQABCJKABCLSQQBMNO $.
|
|
|
|
$( Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40.
|
|
(Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) $)
|
|
sb5 $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wal wa wex sb6 sb56 bitr4i ) ABCDBCEZAFBGMAHBIABCJABCKL $.
|
|
|
|
$( Version of ~ sbn where ` x ` and ` y ` are distinct. (Contributed by
|
|
Jim Kingdon, 18-Dec-2017.) $)
|
|
sbnv $p |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) $=
|
|
( wn wsb weq wa wex wi wal sb6 alinexa bitri sb5 xchbinxr ) ADZBCEZBCFZAG
|
|
BHZABCEQRPIBJSDPBCKRABLMABCNO $.
|
|
|
|
$( Version of ~ sban where ` x ` and ` y ` are distinct. (Contributed by
|
|
Jim Kingdon, 24-Dec-2017.) $)
|
|
sbanv $p |- ( [ y / x ] ( ph /\ ps ) <->
|
|
( [ y / x ] ph /\ [ y / x ] ps ) ) $=
|
|
( wa wsb weq wi wal sb6 anbi12i 19.26 pm4.76 albii 3bitr2i bitr4i ) ABEZC
|
|
DFCDGZQHZCIZACDFZBCDFZEZQCDJUCRAHZCIZRBHZCIZEUDUFEZCITUAUEUBUGACDJBCDJKUD
|
|
UFCLUHSCRABMNOP $.
|
|
|
|
$( Version of ~ sbor where ` x ` and ` y ` are distinct. (Contributed by
|
|
Jim Kingdon, 3-Feb-2018.) $)
|
|
sborv $p |- ( [ y / x ] ( ph \/ ps ) <->
|
|
( [ y / x ] ph \/ [ y / x ] ps ) ) $=
|
|
( wo wsb weq wa wex sb5 andi exbii 19.43 3bitri orbi12i bitr4i ) ABEZCDFZ
|
|
CDGZAHZCIZSBHZCIZEZACDFZBCDFZERSQHZCITUBEZCIUDQCDJUGUHCSABKLTUBCMNUEUAUFU
|
|
CACDJBCDJOP $.
|
|
|
|
$( Forward direction of ~ sbimv . (Contributed by Jim Kingdon,
|
|
25-Dec-2017.) $)
|
|
sbi1v $p |- ( [ y / x ] ( ph -> ps )
|
|
-> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( wsb weq wi wal sb6 ax-2 al2imi sb2 syl6 sylbi syl5bi ) ACDECDFZAGZCHZAB
|
|
GZCDEZBCDEZACDITPSGZCHZRUAGSCDIUCRPBGZCHUAUBQUDCPABJKBCDLMNO $.
|
|
|
|
$( Reverse direction of ~ sbimv . (Contributed by Jim Kingdon,
|
|
18-Jan-2018.) $)
|
|
sbi2v $p |- ( ( [ y / x ] ph -> [ y / x ] ps )
|
|
-> [ y / x ] ( ph -> ps ) ) $=
|
|
( weq wa wex wi wal wsb 19.38 pm3.3 pm2.04 syli alimi syl sb5 sb6 imbi12i
|
|
3imtr4i ) CDEZAFZCGZUABHZCIZHZUAABHZHZCIZACDJZBCDJZHUGCDJUFUBUDHZCIUIUBUD
|
|
CKULUHCUAULAUDHUGUAAUDLAUABMNOPUJUCUKUEACDQBCDRSUGCDRT $.
|
|
|
|
$( Intuitionistic proof of ~ sbim where ` x ` and ` y ` are distinct.
|
|
(Contributed by Jim Kingdon, 18-Jan-2018.) $)
|
|
sbimv $p |- ( [ y / x ] ( ph -> ps )
|
|
<-> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( wi wsb sbi1v sbi2v impbii ) ABECDFACDFBCDFEABCDGABCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
sblimv.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Version of ~ sblim where ` x ` and ` y ` are distinct. (Contributed by
|
|
Jim Kingdon, 19-Jan-2018.) $)
|
|
sblimv $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) $=
|
|
( wi wsb sbimv sbh imbi2i bitri ) ABFCDGACDGZBCDGZFLBFABCDHMBLBCDEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps x $.
|
|
$( Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew
|
|
Salmon, 24-May-2011.) $)
|
|
pm11.53 $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) $=
|
|
( wi wal wex 19.21v albii ax-17 hbal 19.23h bitri ) ABEDFZCFABDFZEZCFACGO
|
|
ENPCABDHIAOCBCDBCJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
exlimivv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
1-Aug-1995.) $)
|
|
exlimivv $p |- ( E. x E. y ph -> ps ) $=
|
|
( wex exlimiv ) ADFBCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $. $d y ch $. $d y ph $.
|
|
exlimdvv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
31-Jul-1995.) $)
|
|
exlimdvv $p |- ( ph -> ( E. x E. y ps -> ch ) ) $=
|
|
( wex exlimdv ) ABEGCDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $.
|
|
exlimddv.1 $e |- ( ph -> E. x ps ) $.
|
|
exlimddv.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Existential elimination rule of natural deduction. (Contributed by
|
|
Mario Carneiro, 15-Jun-2016.) $)
|
|
exlimddv $p |- ( ph -> ch ) $=
|
|
( wex ex exlimdv mpd ) ABDGCEABCDABCFHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.) $)
|
|
19.27v $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
|
|
( ax-17 19.27h ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.) $)
|
|
19.28v $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
|
|
( ax-17 19.28h ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
19.36aiv.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.36aiv $p |- ( A. x ph -> ps ) $=
|
|
( nfv 19.36i ) ABCBCEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.41v $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
|
|
( ax-17 19.41h ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by
|
|
NM, 30-Apr-1995.) $)
|
|
19.41vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) ) $=
|
|
( wa wex 19.41v exbii bitri ) ABEDFZCFADFZBEZCFKCFBEJLCABDGHKBCGI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $. $d z ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by
|
|
NM, 30-Apr-1995.) $)
|
|
19.41vvv $p |- ( E. x E. y E. z ( ph /\ ps ) <->
|
|
( E. x E. y E. z ph /\ ps ) ) $=
|
|
( wa wex 19.41vv exbii 19.41v bitri ) ABFEGDGZCGAEGDGZBFZCGMCGBFLNCABDEHI
|
|
MBCJK $.
|
|
$}
|
|
|
|
${
|
|
$d w ps $. $d x ps $. $d y ps $. $d z ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by
|
|
FL, 14-Jul-2007.) $)
|
|
19.41vvvv $p |- ( E. w E. x E. y E. z ( ph /\ ps ) <->
|
|
( E. w E. x E. y E. z ph /\ ps ) ) $=
|
|
( wa wex 19.41vvv exbii 19.41v bitri ) ABGEHDHCHZFHAEHDHCHZBGZFHNFHBGMOFA
|
|
BCDEIJNBFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.42v $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
|
|
( ax-17 19.42h ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
exdistr $p |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) ) $=
|
|
( wa wex 19.42v exbii ) ABEDFABDFECABDGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by
|
|
NM, 16-Mar-1995.) $)
|
|
19.42vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) ) $=
|
|
( wa wex exdistr 19.42v bitri ) ABEDFCFABDFZECFAJCFEABCDGAJCHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by
|
|
NM, 21-Sep-2011.) $)
|
|
19.42vvv $p |- ( E. x E. y E. z ( ph /\ ps )
|
|
<-> ( ph /\ E. x E. y E. z ps ) ) $=
|
|
( wa wex 19.42vv exbii 19.42v bitri ) ABFEGDGZCGABEGDGZFZCGAMCGFLNCABDEHI
|
|
AMCJK $.
|
|
$}
|
|
|
|
${
|
|
$d w ph $. $d x ph $. $d y ph $. $d z ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by
|
|
Jim Kingdon, 23-Nov-2019.) $)
|
|
19.42vvvv $p |- ( E. w E. x E. y E. z ( ph /\ ps ) <->
|
|
( ph /\ E. w E. x E. y E. z ps ) ) $=
|
|
( wa wex 19.42vv 2exbii bitri ) ABGEHDHZCHFHABEHDHZGZCHFHAMCHFHGLNFCABDEI
|
|
JAMFCIK $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
17-Mar-1995.) $)
|
|
exdistr2 $p |- ( E. x E. y E. z ( ph /\ ps ) <->
|
|
E. x ( ph /\ E. y E. z ps ) ) $=
|
|
( wa wex 19.42vv exbii ) ABFEGDGABEGDGFCABDEHI $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d z ps $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3exdistr $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
|
|
E. x ( ph /\ E. y ( ps /\ E. z ch ) ) ) $=
|
|
( w3a wex wa 3anass 2exbii 19.42vv exdistr anbi2i 3bitri exbii ) ABCGZFHE
|
|
HZABCFHIEHZIZDRABCIZIZFHEHAUAFHEHZITQUBEFABCJKAUAEFLUCSABCEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d w ph $. $d z ps $. $d w ps $. $d w ch $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
4exdistr $p |- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) ) $=
|
|
( wa wex anass exbii 19.42v anbi2i 3bitri bitri ) ABICDIZIZHJZGJZFJZABCDH
|
|
JIZGJIZFJIZEUAAUCIZFJUDTUEFTABUBIZIZGJAUFGJZIUESUGGSABQIZIZHJZUGRUJHABQKL
|
|
UKAUIHJZIABQHJZIZIUGAUIHMULUNABQHMNUNUFAUMUBBCDHMNNOPLAUFGMUHUCABUBGMNOLA
|
|
UCFMPL $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvalv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitition.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvalv $p |- ( A. x ph <-> A. y ps ) $=
|
|
( ax-17 cbvalh ) ABCDADFBCFEG $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitition.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvexv $p |- ( E. x ph <-> E. y ps ) $=
|
|
( ax-17 cbvexh ) ABCDADFBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y z $. $d w x $. $d w z $.
|
|
cbval2.1 $e |- F/ z ph $.
|
|
cbval2.2 $e |- F/ w ph $.
|
|
cbval2.3 $e |- F/ x ps $.
|
|
cbval2.4 $e |- F/ y ps $.
|
|
cbval2.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) $)
|
|
cbval2 $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
|
|
( wal nfal weq wi nfv nfim wb cbval 19.21v pm5.74d 3bitr3i pm5.74ri
|
|
expcom ) ADLZBFLZCEAEDGMBCFIMCENZUEUFUGAOZDLUGBOZFLUGUEOUGUFOUHUIDFUGAFUG
|
|
FPHQUGBDUGDPJQDFNZUGABUGUJABRKUDUASUGADTUGBFTUBUCS $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) $)
|
|
cbvex2 $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
|
|
( wex nfex weq wb wa nfv nfan cbvex 19.42v wi expcom pm5.32d pm5.32 mpbir
|
|
3bitr3i ) ADLZBFLZCEAEDGMBCFIMCENZUGUHOUAUIUGPZUIUHPZOUIAPZDLUIBPZFLUJUKU
|
|
LUMDFUIAFUIFQHRUIBDUIDQJRDFNZUIABUIUNABOKUBUCSUIADTUIBFTUFUIUGUHUDUES $.
|
|
$}
|
|
|
|
${
|
|
$d z w ph $. $d x y ps $. $d x w $. $d z y $.
|
|
cbval2v.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 4-Feb-2005.) $)
|
|
cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
|
|
( nfv cbval2 ) ABCDEFAEHAFHBCHBDHGI $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 26-Jul-1995.) $)
|
|
cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
|
|
( nfv cbvex2 ) ABCDEFAEHAFHBCHBDHGI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
cbvald.1 $e |- F/ y ph $.
|
|
cbvald.2 $e |- ( ph -> F/ y ps ) $.
|
|
cbvald.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction used to change bound variables, using implicit substitution,
|
|
particularly useful in conjunction with ~ dvelim . (Contributed by NM,
|
|
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf
|
|
Lammen, 13-May-2018.) $)
|
|
cbvald $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( nfv wnf a1i cbv2 ) ABCDEADIFGCDJACDIKHL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
cbvexdh.1 $e |- ( ph -> A. y ph ) $.
|
|
cbvexdh.2 $e |- ( ph -> ( ps -> A. y ps ) ) $.
|
|
cbvexdh.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction used to change bound variables, using implicit substitition,
|
|
particularly useful in conjunction with ~ dvelim . (Contributed by NM,
|
|
2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.) $)
|
|
cbvexdh $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
|
|
( wex ax-17 cv wceq wa wb wi equsexd simpr eximi syl6bir wal hbex equcomi
|
|
bicom1 imim12i syl exlimdh eximdh 19.12 syl6 a1i exlimd2 impbid ) ABDIZCE
|
|
IZABUNDADJZCDECDJZUAABEKZDKZLZCMZEIUNACBEDFGAURUQLZBCNZOUSCBNZOHUSVAVBVCE
|
|
DUBBCUCUDUEPUTCEUSCQRSUFACUMEFAUMBETZDIUMETABVDDUOGUGBDEUHUIACVABMZDIUMAB
|
|
CDEUOCCDTOAUPUJHPVEBDVABQRSUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
cbvexd.1 $e |- F/ y ph $.
|
|
cbvexd.2 $e |- ( ph -> F/ y ps ) $.
|
|
cbvexd.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction used to change bound variables, using implicit substitution,
|
|
particularly useful in conjunction with ~ dvelim . (Contributed by NM,
|
|
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten
|
|
by Jim Kingdon, 10-Jun-2018.) $)
|
|
cbvexd $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
|
|
( nfri nfrd cbvexdh ) ABCDEAEFIABEGJHK $.
|
|
$}
|
|
|
|
${
|
|
$d ps y $. $d ch x $. $d ph x $. $d ph y $.
|
|
cbvaldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change the bound variable in a universal quantifier with
|
|
implicit substitution. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
cbvaldva $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( nfv nfvd weq wb ex cbvald ) ABCDEAEGABEHADEIBCJFKL $.
|
|
|
|
$( Rule used to change the bound variable in an existential quantifier with
|
|
implicit substitution. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
cbvexdva $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
|
|
( nfv nfvd weq wb ex cbvexd ) ABCDEAEGABEHADEIBCJFKL $.
|
|
$}
|
|
|
|
${
|
|
$v f $.
|
|
$v g $.
|
|
$( Define temporary individual variables. $)
|
|
cbvex4v.vf $f setvar f $.
|
|
cbvex4v.vg $f setvar g $.
|
|
$d w z ch $. $d u v ph $. $d x y ps $. $d f g ps $. $d f w $.
|
|
$d g z $. $d u v w z $. $d u w x z $. $d v w y z $. $d w x y z $.
|
|
cbvex4v.1 $e |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) ) $.
|
|
cbvex4v.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitition.
|
|
(Contributed by NM, 26-Jul-1995.) $)
|
|
cbvex4v $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $=
|
|
( wex weq wa 2exbidv cbvex2v 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZINH
|
|
NUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $.
|
|
$}
|
|
|
|
${
|
|
eean.1 $e |- F/ y ph $.
|
|
eean.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
eean $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
|
|
( wa wex 19.42 exbii nfex 19.41 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) $)
|
|
eeanv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
|
|
( nfv eean ) ABCDADEBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d x z ps $. $d x y ch $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
eeeanv $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
|
|
( E. x ph /\ E. y ps /\ E. z ch ) ) $=
|
|
( w3a wex wa df-3an 3exbii eeanv exbii anbi1i 19.41v 3bitr4i 3bitri ) ABC
|
|
GZFHEHDHABIZCIZFHEHZDHSEHZCFHZIZDHZADHZBEHZUCGZRTDEFABCJKUAUDDSCEFLMUBDHZ
|
|
UCIUFUGIZUCIUEUHUIUJUCABDELNUBUCDOUFUGUCJPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z ph $. $d w ph $. $d x ps $. $d y ps $. $d y z $. $d w x $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) $)
|
|
ee4anv $p |- ( E. x E. y E. z E. w ( ph /\ ps ) <->
|
|
( E. x E. y ph /\ E. z E. w ps ) ) $=
|
|
( wa wex excom exbii eeanv 2exbii 3bitri ) ABGFHZEHDHZCHNDHZEHZCHADHZBFHZ
|
|
GZEHCHRCHSEHGOQCNDEIJPTCEABDFKLRSCEKM $.
|
|
$}
|
|
|
|
${
|
|
$v s $.
|
|
$( Define a temporary individual variable. $)
|
|
ee8anv.vs $f setvar s $.
|
|
|
|
$d v ph $. $d u ph $. $d t ph $. $d s ph $. $d x ps $. $d y ps $.
|
|
$d z ps $. $d w ps $. $d s x $. $d s y $. $d s z $. $d t w $.
|
|
$d t x $. $d t y $. $d u w $. $d u x $. $d u z $. $d v w $. $d v y $.
|
|
$d v z $.
|
|
$( Rearrange existential quantifiers. (Contributed by Jim Kingdon,
|
|
23-Nov-2019.) $)
|
|
ee8anv $p |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ph /\ ps ) <->
|
|
( E. x E. y E. z E. w ph /\ E. v E. u E. t E. s ps ) ) $=
|
|
( wa wex exrot4 2exbii ee4anv 3bitri ) ABKJLILZHLGLFLELZDLCLQFLELZHLGLZDL
|
|
CLAFLELZBJLILZKZHLGLZDLCLUADLCLUBHLGLKRTCDQEFGHMNTUDCDSUCGHABEFIJONNUAUBC
|
|
DGHOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nexdv.1 $e |- ( ph -> -. ps ) $.
|
|
$( Deduction for generalization rule for negated wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
nexdv $p |- ( ph -> -. E. x ps ) $=
|
|
( ax-17 nexd ) ABCACEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
chv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
chv.2 $e |- ph $.
|
|
$( Implicit substitution of ` y ` for ` x ` into a theorem. (Contributed
|
|
by NM, 20-Apr-1994.) $)
|
|
chvarv $p |- ps $=
|
|
( spv mpg ) ABCABCDEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( When the class variables of set theory are replaced with setvar
|
|
variables, this theorem of predicate calculus is the result. This
|
|
theorem provides part of the justification for the consistency of that
|
|
definition, which "overloads" the setvar variables in ~ wel with the
|
|
class variables in ~ wcel . (Contributed by NM, 28-Jan-2004.) $)
|
|
cleljust $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
|
|
( weq wel wa wex ax-17 elequ1 equsex bicomi ) CADCBEZFCGABEZLMCAMCHCABIJK
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
More substitution theorems
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The theorems in this section make use of the $d statement. $)
|
|
|
|
${
|
|
$d x y $.
|
|
$( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are distinct.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.)
|
|
(New usage is discouraged.) $)
|
|
hbs1 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
|
|
( wsb weq wi wal sb6 ax-ial hbxfrbi ) ABCDBCEAFZBGBABCHKBIJ $.
|
|
|
|
$( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are distinct.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1v $p |- F/ x [ y / x ] ph $=
|
|
( wsb hbs1 nfi ) ABCDBABCEF $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
|
|
(Contributed by NM, 29-May-2009.) $)
|
|
sbhb $p |- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) $=
|
|
( wal wi wsb ax-17 sb8h imbi2i 19.21v bitr4i ) AABDZEAABCFZCDZEAMECDLNAAB
|
|
CACGHIAMCJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
hbsbv.1 $e |- ( ph -> A. z ph ) $.
|
|
$( This is a version of ~ hbsb with an extra distinct variable constraint,
|
|
on ` z ` and ` x ` . (Contributed by Jim Kingdon, 25-Dec-2017.) $)
|
|
hbsbv $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
|
|
( wsb weq wi wa wex df-sb ax-17 hbim hban hbex hbxfrbi ) ABCFBCGZAHZQAIZB
|
|
JZIDABCKRTDQADQDLZEMSDBQADUAENONP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $.
|
|
nfsbxy.1 $e |- F/ z ph $.
|
|
$( Similar to ~ hbsb but with an extra distinct variable constraint, on
|
|
` x ` and ` y ` . (Contributed by Jim Kingdon, 19-Mar-2018.) $)
|
|
nfsbxy $p |- F/ z [ y / x ] ph $=
|
|
( weq wal wi wo wsb wnf ax-bndl nfs1v drsb1 drnf2 mpbii a16nf df-nf albii
|
|
jaoi wa wex sb5 nfa1 sp a1i nfand nfexd nfxfrd sylbir ax-mp ) DBFDGZDCFDG
|
|
ZBCFZUNDGHDGZBGZIZIABCJZDKZBCDLULUSUQULADCJZDKUSADCMUTURDBDADBCNOPUMUSUPU
|
|
RDCDQUPUNDKZBGZUSVAUOBUNDRSURUNAUAZBUBVBDABCUCVBVCDBVABUDVBUNADVABUEADKVB
|
|
EUFUGUHUIUJTTUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $.
|
|
$( Closed form of ~ nfsbxy . (Contributed by Jim Kingdon, 9-May-2018.) $)
|
|
nfsbxyt $p |- ( A. x F/ z ph -> F/ z [ y / x ] ph ) $=
|
|
( weq wal wi wo wnf wsb ax-bndl nfs1v drsb1 drnf2 mpbii a1d wa nfa1 jaoi
|
|
sp a16nf df-nf albii wex sb5 nfan adantr adantl nfand nfexd nfxfrd sylbir
|
|
ex ax-mp ) DBEDFZDCEDFZBCEZUQDFGDFZBFZHZHADIZBFZABCJZDIZGZBCDKUOVEUTUOVDV
|
|
BUOADCJZDIVDADCLVFVCDBDADBCMNOPUPVEUSUPVDVBVCDCDUAPUSUQDIZBFZVEVGURBUQDUB
|
|
UCVHVBVDVCUQAQZBUDVHVBQZDABCUEVJVIDBVHVBBVGBRVABRUFVJUQADVHVGVBVGBTUGVBVA
|
|
VHVABTUHUIUJUKUMULSSUN $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sbco2vlem.1 $e |- ( ph -> A. z ph ) $.
|
|
$( This is a version of ~ sbco2 where ` z ` is distinct from ` x ` and from
|
|
` y ` . It is a lemma on the way to proving ~ sbco2v which only
|
|
requires that ` z ` and ` x ` be distinct. (Contributed by Jim Kingdon,
|
|
25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon,
|
|
3-Feb-2018.) $)
|
|
sbco2vlem $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( wsb hbsbv sbequ sbieh ) ABDFABCFDCABCDEGADCBHI $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y w $. $d z w $. $d ph w $.
|
|
sbco2v.1 $e |- ( ph -> A. z ph ) $.
|
|
$( This is a version of ~ sbco2 where ` z ` is distinct from ` x ` .
|
|
(Contributed by Jim Kingdon, 12-Feb-2018.) $)
|
|
sbco2v $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( vw wsb sbco2vlem sbbii ax-17 3bitr3i ) ABDGZDFGZFCGABFGZFCGLDCGABCGMNFC
|
|
ABFDEHILDCFLFJHABCFAFJHK $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w ph $. $d w x $.
|
|
nfsb.1 $e |- F/ z ph $.
|
|
$( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
|
|
` y ` and ` z ` are distinct. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) $)
|
|
nfsb $p |- F/ z [ y / x ] ph $=
|
|
( vw wsb wnf nfsbxy ax-17 sbco2v nfbii mpbi ) ABFGZFCGZDHABCGZDHNFCDABFDE
|
|
IIOPDABCFAFJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
hbsb.1 $e |- ( ph -> A. z ph ) $.
|
|
$( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
|
|
` y ` and ` z ` are distinct. (Contributed by NM, 12-Aug-1993.) (Proof
|
|
rewritten by Jim Kingdon, 22-Mar-2018.) $)
|
|
hbsb $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
|
|
( wsb nfi nfsb nfri ) ABCFDABCDADEGHI $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x y $.
|
|
$( Lemma for ~ equsb3 . (Contributed by NM, 4-Dec-2005.) (Proof shortened
|
|
by Andrew Salmon, 14-Jun-2011.) $)
|
|
equsb3lem $p |- ( [ x / y ] y = z <-> x = z ) $=
|
|
( cv wceq ax-17 equequ1 sbieh ) BDCDZEADIEZBAJBFBACGH $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic wff. (Contributed by Raph Levien and
|
|
FL, 4-Dec-2005.) $)
|
|
equsb3 $p |- ( [ x / y ] y = z <-> x = z ) $=
|
|
( vw weq wsb equsb3lem sbbii ax-17 sbco2v 3bitr3i ) BCEZBDFZDAFDCEZDAFLBA
|
|
FACEMNDADBCGHLBADLDIJADCGK $.
|
|
$}
|
|
|
|
${
|
|
$d z ph $. $d z ps $. $d z x $. $d z y $.
|
|
|
|
$( Negation inside and outside of substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
|
|
3-Feb-2018.) $)
|
|
sbn $p |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) $=
|
|
( vz wn wsb sbnv sbbii bitri ax-17 hbn sbco2v notbii 3bitr3i ) AEZBDFZDCF
|
|
ZABDFZDCFZEZOBCFABCFZEQREZDCFTPUBDCABDGHRDCGIOBCDADADJZKLSUAABCDUCLMN $.
|
|
|
|
$( Implication inside and outside of substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
|
|
3-Feb-2018.) $)
|
|
sbim $p |- ( [ y / x ] ( ph -> ps ) <->
|
|
( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( vz wi wsb sbimv sbbii bitri ax-17 sbco2v imbi12i 3bitr3i ) ABFZCEGZEDGZ
|
|
ACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSUC
|
|
UAUDACDEAEKLBCDEBEKLMN $.
|
|
|
|
$( Logical OR inside and outside of substitution are equivalent.
|
|
(Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon,
|
|
3-Feb-2018.) $)
|
|
sbor $p |- ( [ y / x ] ( ph \/ ps ) <->
|
|
( [ y / x ] ph \/ [ y / x ] ps ) ) $=
|
|
( vz wo wsb sborv sbbii bitri ax-17 sbco2v orbi12i 3bitr3i ) ABFZCEGZEDGZ
|
|
ACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSUC
|
|
UAUDACDEAEKLBCDEBEKLMN $.
|
|
|
|
$( Conjunction inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon,
|
|
3-Feb-2018.) $)
|
|
sban $p |- ( [ y / x ] ( ph /\ ps ) <->
|
|
( [ y / x ] ph /\ [ y / x ] ps ) ) $=
|
|
( vz wa wsb sbanv sbbii bitri ax-17 sbco2v anbi12i 3bitr3i ) ABFZCEGZEDGZ
|
|
ACEGZEDGZBCEGZEDGZFZOCDGACDGZBCDGZFQRTFZEDGUBPUEEDABCEHIRTEDHJOCDEOEKLSUC
|
|
UAUDACDEAEKLBCDEBEKLMN $.
|
|
$}
|
|
|
|
${
|
|
sbrim.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Substitution with a variable not free in antecedent affects only the
|
|
consequent. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbrim $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
|
|
( wi wsb sbim sbh imbi1i bitri ) ABFCDGACDGZBCDGZFAMFABCDHLAMACDEIJK $.
|
|
$}
|
|
|
|
${
|
|
sblim.1 $e |- F/ x ps $.
|
|
$( Substitution with a variable not free in consequent affects only the
|
|
antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
sblim $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) $=
|
|
( wi wsb sbim sbf imbi2i bitri ) ABFCDGACDGZBCDGZFLBFABCDHMBLBCDEIJK $.
|
|
$}
|
|
|
|
$( Conjunction inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 14-Dec-2006.) $)
|
|
sb3an $p |- ( [ y / x ] ( ph /\ ps /\ ch ) <->
|
|
( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) ) $=
|
|
( w3a wsb wa df-3an sbbii sban anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDEGPDE
|
|
GZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.
|
|
|
|
$( Equivalence inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbbi $p |- ( [ y / x ] ( ph <-> ps )
|
|
<-> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
|
|
( wb wsb wi wa dfbi2 sbbii sbim anbi12i sban 3bitr4i bitri ) ABEZCDFABGZBAG
|
|
ZHZCDFZACDFZBCDFZEZPSCDABIJQCDFZRCDFZHUAUBGZUBUAGZHTUCUDUFUEUGABCDKBACDKLQR
|
|
CDMUAUBINO $.
|
|
|
|
${
|
|
sblbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce left biconditional inside of a substitution. (Contributed by
|
|
NM, 19-Aug-1993.) $)
|
|
sblbis $p |- ( [ y / x ] ( ch <-> ph ) <-> ( [ y / x ] ch <-> ps ) ) $=
|
|
( wb wsb sbbi bibi2i bitri ) CAGDEHCDEHZADEHZGLBGCADEIMBLFJK $.
|
|
$}
|
|
|
|
${
|
|
sbrbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce right biconditional inside of a substitution. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
sbrbis $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) ) $=
|
|
( wb wsb sbbi bibi1i bitri ) ACGDEHADEHZCDEHZGBMGACDEILBMFJK $.
|
|
$}
|
|
|
|
${
|
|
sbrbif.1 $e |- ( ch -> A. x ch ) $.
|
|
sbrbif.2 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce right biconditional inside of a substitution. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
sbrbif $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
|
|
( wb wsb sbrbis sbh bibi2i bitri ) ACHDEIBCDEIZHBCHABCDEGJNCBCDEFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
sbco2yz.1 $e |- F/ z ph $.
|
|
$( This is a version of ~ sbco2 where ` z ` is distinct from ` y ` . It is
|
|
a lemma on the way to proving ~ sbco2 which has no distinct variable
|
|
constraints. (Contributed by Jim Kingdon, 19-Mar-2018.) $)
|
|
sbco2yz $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( wsb nfsb nfri sbequ sbieh ) ABDFABCFZDCKDABCDEGHADCBIJ $.
|
|
$}
|
|
|
|
${
|
|
$d w z $. $d w x $. $d w y $. $d ph w $.
|
|
sbco2h.1 $e |- ( ph -> A. z ph ) $.
|
|
$( A composition law for substitution. (Contributed by NM, 30-Jun-1994.)
|
|
(Proof rewritten by Jim Kingdon, 19-Mar-2018.) $)
|
|
sbco2h $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( vw wsb nfi sbco2yz sbbii nfv 3bitr3i ) ABDGZDFGZFCGABFGZFCGMDCGABCGNOFC
|
|
ABFDADEHIJMDCFMFKIABCFAFKIL $.
|
|
$}
|
|
|
|
${
|
|
sbco2.1 $e |- F/ z ph $.
|
|
$( A composition law for substitution. (Contributed by NM, 30-Jun-1994.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sbco2 $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( nfri sbco2h ) ABCDADEFG $.
|
|
$}
|
|
|
|
${
|
|
sbco2d.1 $e |- ( ph -> A. x ph ) $.
|
|
sbco2d.2 $e |- ( ph -> A. z ph ) $.
|
|
sbco2d.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbco2d $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
|
|
( wsb wi hbim1 sbco2h sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCDI
|
|
ZABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCDF
|
|
MPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
sbco2vd.1 $e |- ( ph -> A. x ph ) $.
|
|
sbco2vd.2 $e |- ( ph -> A. z ph ) $.
|
|
sbco2vd.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
|
|
$( Version of ~ sbco2d with a distinct variable constraint between ` x `
|
|
and ` z ` . (Contributed by Jim Kingdon, 19-Feb-2018.) $)
|
|
sbco2vd $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
|
|
( wsb wi hbim1 sbco2v sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCDI
|
|
ZABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCDF
|
|
MPQ $.
|
|
$}
|
|
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbco $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
|
|
( wsb wb weq equsb2 sbequ12 bicomd sbimi ax-mp sbbi mpbi ) ACBDZAEZBCDZNBCD
|
|
ABCDECBFZBCDPBCGQOBCQANACBHIJKNABCLM $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Version of ~ sbco3 with a distinct variable constraint between ` x ` and
|
|
` y ` . (Contributed by Jim Kingdon, 19-Feb-2018.) $)
|
|
sbco3v $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
|
|
( wsb nfs1v nfri sbco2v sbco sbbii bitr3i ) ABCEZCDELCBEZBDEACBEZBDELCDBL
|
|
BABCFGHMNBDACBIJK $.
|
|
$}
|
|
|
|
$( Relationship between composition and commutativity for substitution.
|
|
(Contributed by Jim Kingdon, 28-Feb-2018.) $)
|
|
sbcocom $p |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) $=
|
|
( wsb wb weq equsb1 sbequ sbimi ax-mp sbbi mpbi ) ABCEZABDEZFZCDEZNCDEOCDEF
|
|
CDGZCDEQCDHRPCDACDBIJKNOCDLM $.
|
|
|
|
${
|
|
$d x z $.
|
|
$( Version of ~ sbcom with a distinct variable constraint between ` x ` and
|
|
` z ` . (Contributed by Jim Kingdon, 28-Feb-2018.) $)
|
|
sbcomv $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
|
|
( wsb sbco3v sbcocom 3bitr3i ) ABDEDCEADBEBCEABCEDCEADCEBCEABDCFABDCGADBC
|
|
GH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $.
|
|
$( Version of ~ sbcom with distinct variable constraints between ` x ` and
|
|
` y ` , and ` y ` and ` z ` . (Contributed by Jim Kingdon,
|
|
21-Mar-2018.) $)
|
|
sbcomxyyz $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
|
|
( weq wal wi wo wsb ax-ial drsb1 sbbidh bitr3d sbequ12 sps wnf nfs1v nfrd
|
|
wb a1i ax-bndl hbae df-nf albii nfsb nfr wa nfnf1 nfa1 nfan nfri sylan9bb
|
|
ex adantl sbiedh syld bicomd sylbir jaoi ax-mp ) DBEZDFZDCEZDFZBCEZVEDFZG
|
|
DFZBFZHZHABCIZDCIZADCIZBCIZSZBCDUAVBVNVIVBVLDCIVKVMVBVLVJDCVADJADBCKLVLDB
|
|
CKMVDVNVHVDVJVKVMVCVJVKSDVJDCNZOVDAVLBCDCBUBVCAVLSDADCNOLMVHVEDPZBFZVNVPV
|
|
GBVEDUCUDVQVMVKVQVLVKBCVPBJVQVKBVKBPVQVJDCBABCQUETRVPVEVLVKSZGBVPVEVFVRVE
|
|
DUFVPVFVRVPVFUGZAVKDCVSDVPVFDVEDUHVEDUIUJUKVSVKDVKDPVSVJDCQTRVFVCAVKSZGZV
|
|
PVEWADVEVCVTVEAVJVCVKABCNVOULUMOUNUOUMUPOUOUQURUSUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Version of ~ sbco3 with distinct variable constraints between ` x ` and
|
|
` z ` , and ` y ` and ` z ` . Lemma for proving ~ sbco3 . (Contributed
|
|
by Jim Kingdon, 22-Mar-2018.) $)
|
|
sbco3xzyz $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
|
|
( wsb sbcomxyyz sbcocom 3bitr4i ) ABDECDEACDEBDEABCECDEACBEBDEABDCFABCDGA
|
|
CBDGH $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w ph $.
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof rewritten by Jim Kingdon, 22-Mar-2018.) $)
|
|
sbco3 $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
|
|
( vw wsb sbco3xzyz sbbii ax-17 sbco2h 3bitr3i ) ABCFZCEFZEDFACBFZBEFZEDFL
|
|
CDFNBDFMOEDABCEGHLCDELEIJNBDENEIJK $.
|
|
$}
|
|
|
|
$( A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
|
|
(Proof rewritten by Jim Kingdon, 22-Mar-2018.) $)
|
|
sbcom $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
|
|
( wsb sbco3 sbcocom 3bitr3i ) ABDEDCEADBEBCEABCEDCEADCEBCEABDCFABDCGADBCGH
|
|
$.
|
|
|
|
${
|
|
$d w y z $. $d w ph $. $d w x $.
|
|
$( Closed form of ~ nfsb . (Contributed by Jim Kingdon, 9-May-2018.) $)
|
|
nfsbt $p |- ( A. x F/ z ph -> F/ z [ y / x ] ph ) $=
|
|
( vw wnf wal wsb ax-17 nfsbxyt alimi syl nfv sbco2 nfbii sylib ) ADFBGZQE
|
|
GZABCHZDFZQEIRABEHZECHZDFZTRUADFZEGUCQUDEABEDJKUAECDJLUBSDABCEAEMNOPL $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
nfsbd.1 $e |- F/ x ph $.
|
|
nfsbd.2 $e |- ( ph -> F/ z ps ) $.
|
|
$( Deduction version of ~ nfsb . (Contributed by NM, 15-Feb-2013.) $)
|
|
nfsbd $p |- ( ph -> F/ z [ y / x ] ps ) $=
|
|
( wal wnf wsb nfri alimi nfsbt 3syl ) AACHBEIZCHBCDJEIACFKAOCGLBCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic membership wff. (Contributed by NM,
|
|
7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
elsb3 $p |- ( [ x / y ] y e. z <-> x e. z ) $=
|
|
( vw wel wsb ax-17 elequ1 sbieh sbbii sbco2h bitr3i wb equsb1 sbimi ax-mp
|
|
weq sbbi mpbi sbh 3bitri ) BCEZBAFZDCEZDAFZACEZDAFZUFUCUDDBFZBAFUEUHUBBAU
|
|
DUBDBUBDGDBCHIJUDDABUDBGKLUDUFMZDAFZUEUGMDAQZDAFUJDANUKUIDADACHOPUDUFDARS
|
|
UFDAUFDGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic membership wff. (Contributed by
|
|
Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
elsb4 $p |- ( [ x / y ] z e. y <-> z e. x ) $=
|
|
( vw wel wsb ax-17 elequ2 sbieh sbbii sbco2h bitr3i wb equsb1 sbimi ax-mp
|
|
weq sbbi mpbi sbh 3bitri ) CBEZBAFZCDEZDAFZCAEZDAFZUFUCUDDBFZBAFUEUHUBBAU
|
|
DUBDBUBDGDBCHIJUDDABUDBGKLUDUFMZDAFZUEUGMDAQZDAFUJDANUKUIDADACHOPUDUFDARS
|
|
UFDAUFDGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Like ~ sb9 but with a distinct variable constraint between ` x ` and
|
|
` y ` . (Contributed by Jim Kingdon, 28-Feb-2018.) $)
|
|
sb9v $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
|
|
( wsb hbs1 weq wb sbequ12 equcoms bitr3d cbvalh ) ACBDZABCDZBCACBEABCEBCF
|
|
ALMALGCBACBHIABCHJK $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w ph $.
|
|
$( Commutation of quantification and substitution variables. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) $)
|
|
sb9 $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
|
|
( vw wsb wal sb9v sbcom albii 3bitri ax-17 sbco2h 3bitr3ri ) ABDEZDCEZCFZ
|
|
ACDEZDBEZBFZABCEZCFACBEZBFPNCDEZDFQBDEZDFSNCDGUBUCDABDCHIQDBGJOTCABCDADKZ
|
|
LIRUABACBDUDLIM $.
|
|
$}
|
|
|
|
$( Commutation of quantification and substitution variables. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) $)
|
|
sb9i $p |- ( A. x [ x / y ] ph -> A. y [ y / x ] ph ) $=
|
|
( wsb wal sb9 biimpi ) ACBDBEABCDCEABCFG $.
|
|
|
|
${
|
|
$d x y z $. $d y z ph $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
|
|
(Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario
|
|
Carneiro, 6-Oct-2016.) $)
|
|
sbnf2 $p |- ( F/ x ph
|
|
<-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) ) $=
|
|
( wsb wb wal wi wnf 2albiim sbhb albii alcom 3bitri nfv nfs1v sblim bitri
|
|
wa sb8 df-nf anbi12i anidm 3bitr2ri ) ABCEZABDEZFDGCGUEUFHZDGCGZUFUEHZDGZ
|
|
CGZSABIZULSULUEUFCDJULUHULUKULAUFHZBGZDGZUGCGZDGUHULAABGHZBGZUMDGZBGUOABU
|
|
AZUQUSBABDKLUMBDMNUNUPDUNUMBCEZCGUPUMBCUMCOTVAUGCAUFBCABDPQLRLUGDCMNULAUE
|
|
HZBGZCGZUKULURVBCGZBGVDUTUQVEBABCKLVBBCMNVCUJCVCVBBDEZDGUJVBBDVBDOTVFUIDA
|
|
UEBDABCPQLRLRUBULUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
hbsbd.1 $e |- ( ph -> A. x ph ) $.
|
|
hbsbd.2 $e |- ( ph -> A. z ph ) $.
|
|
hbsbd.3 $e |- ( ph -> ( ps -> A. z ps ) ) $.
|
|
$( Deduction version of ~ hbsb . (Contributed by NM, 15-Feb-2013.) (Proof
|
|
rewritten by Jim Kingdon, 23-Mar-2018.) $)
|
|
hbsbd $p |- ( ph -> ( [ y / x ] ps -> A. z [ y / x ] ps ) ) $=
|
|
( wsb nfi wi wnf nfdh nfim1 nfsb sbrim nfbii mpbi nfrimi nfrd ) ABCDIZEAU
|
|
AEAEGJZABKZCDIZELAUAKZELUCCDEABEUBABEGHMNOUDUEEABCDFPQRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d w y $.
|
|
$( Equivalence for double substitution. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2sb5 $p |- ( [ z / x ] [ w / y ] ph <->
|
|
E. x E. y ( ( x = z /\ y = w ) /\ ph ) ) $=
|
|
( wsb weq wa wex sb5 19.42v anass exbii anbi2i 3bitr4ri bitri ) ACEFZBDFB
|
|
DGZQHZBIRCEGZHAHZCIZBIQBDJSUBBRTAHZHZCIRUCCIZHUBSRUCCKUAUDCRTALMQUERACEJN
|
|
OMP $.
|
|
|
|
$( Equivalence for double substitution. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2sb6 $p |- ( [ z / x ] [ w / y ] ph <->
|
|
A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
|
|
( wsb weq wi wal wa sb6 19.21v impexp albii imbi2i 3bitr4ri bitri ) ACEFZ
|
|
BDFBDGZRHZBISCEGZJAHZCIZBIRBDKTUCBSUAAHZHZCISUDCIZHUCTSUDCLUBUECSUAAMNRUF
|
|
SACEKOPNQ $.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d x y z $.
|
|
$( Lemma for proving ~ sbcom2 . It is the same as ~ sbcom2 but with
|
|
additional distinct variable constraints on ` x ` and ` y ` , and on
|
|
` w ` and ` z ` . (Contributed by Jim Kingdon, 19-Feb-2018.) $)
|
|
sbcom2v $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
|
|
( weq wal wsb alcom bi2.04 albii 19.21v bitri 3bitr3i sb6 imbi2i 3bitr4i
|
|
wi ) DEFZBCFZARZBGZRZDGZTSARZDGZRZBGZABCHZDEHZADEHZBCHZTUERZBGZDGUMDGZBGU
|
|
DUHUMDBIUNUCDUNSUARZBGUCUMUPBTSAJKSUABLMKUOUGBTUEDLKNUJSUIRZDGUDUIDEOUQUC
|
|
DUIUBSABCOPKMULTUKRZBGUHUKBCOURUGBUKUFTADEOPKMQ $.
|
|
$}
|
|
|
|
${
|
|
$d v w x z $. $d v y z $. $d v ph $.
|
|
$( Lemma for proving ~ sbcom2 . It is the same as ~ sbcom2v but removes
|
|
the distinct variable constraint on ` x ` and ` y ` . (Contributed by
|
|
Jim Kingdon, 19-Feb-2018.) $)
|
|
sbcom2v2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
|
|
( vv wsb sbcom2v sbbii bitri ax-17 sbco2v 3bitr3i ) ABFGZFCGZDEGZADEGZBFG
|
|
ZFCGZABCGZDEGQBCGPNDEGZFCGSNFCDEHUARFCABFDEHIJOTDEABCFAFKLIQBCFQFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d v x w $. $d v y z $. $d v ph $.
|
|
$( Commutativity law for substitution. Used in proof of Theorem 9.7 of
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon,
|
|
19-Feb-2018.) $)
|
|
sbcom2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
|
|
( vv wsb sbcom2v2 sbbii bitri ax-17 sbco2v 3bitr3i ) ABCGZDFGZFEGZADFGZFE
|
|
GZBCGZNDEGADEGZBCGPQBCGZFEGSOUAFEABCDFHIQBCFEHJNDEFNFKLRTBCADEFAFKLIM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb6a $p |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) ) $=
|
|
( wsb weq wi wal sb6 wb sbequ12 equcoms pm5.74i albii bitri ) ABCDBCEZAFZ
|
|
BGOACBDZFZBGABCHPRBOAQAQICBACBJKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x w $. $d y z $. $d z w $.
|
|
2sb5rf.1 $e |- ( ph -> A. z ph ) $.
|
|
2sb5rf.2 $e |- ( ph -> A. w ph ) $.
|
|
$( Reversed double substitution. (Contributed by NM, 3-Feb-2005.) $)
|
|
2sb5rf $p |- ( ph <->
|
|
E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) ) $=
|
|
( weq wsb wex sb5rf 19.42v sbcom2 anbi2i anass bitri exbii hbsbv 3bitr4ri
|
|
wa ) ADBHZABDIZTZDJUAECHZTZACEIBDIZTZEJZDJABDFKUCUHDUAUDUBCEIZTZTZEJUAUJE
|
|
JZTUHUCUAUJELUGUKEUGUEUITUKUFUIUEACEBDMNUAUDUIOPQUBULUAUBCEABDEGRKNSQP $.
|
|
|
|
$( Reversed double substitution. (Contributed by NM, 3-Feb-2005.) $)
|
|
2sb6rf $p |- ( ph <->
|
|
A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) ) $=
|
|
( weq wsb wi wal wa sb6rf 19.21v sbcom2 imbi2i impexp bitri albii hbsbv
|
|
3bitr4ri ) ADBHZABDIZJZDKUBECHZLZACEIBDIZJZEKZDKABDFMUDUIDUBUEUCCEIZJZJZE
|
|
KUBUKEKZJUIUDUBUKENUHULEUHUFUJJULUGUJUFACEBDOPUBUEUJQRSUCUMUBUCCEABDEGTMP
|
|
UASR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( An alternate definition of proper substitution ~ df-sb . By introducing
|
|
a dummy variable ` z ` in the definiens, we are able to eliminate any
|
|
distinct variable restrictions among the variables ` x ` , ` y ` , and
|
|
` ph ` of the definiendum. No distinct variable conflicts arise because
|
|
` z ` effectively insulates ` x ` from ` y ` . To achieve this, we use
|
|
a chain of two substitutions in the form of ~ sb5 , first ` z ` for
|
|
` x ` then ` y ` for ` z ` . Compare Definition 2.1'' of [Quine] p. 17.
|
|
Theorem ~ sb7f provides a version where ` ph ` and ` z ` don't have to
|
|
be distinct. (Contributed by NM, 28-Jan-2004.) $)
|
|
dfsb7 $p |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
|
|
( wsb weq wa wex sb5 sbbii ax-17 sbco2v 3bitr3i ) ABDEZDCEBDFAGBHZDCEABCE
|
|
DCFOGDHNODCABDIJABCDADKLODCIM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sb7f.1 $e |- ( ph -> A. z ph ) $.
|
|
$( This version of ~ dfsb7 does not require that ` ph ` and ` z ` be
|
|
distinct. This permits it to be used as a definition for substitution
|
|
in a formalization that omits the logically redundant axiom ~ ax-17 i.e.
|
|
that doesn't have the concept of a variable not occurring in a wff.
|
|
( ~ df-sb is also suitable, but its mixing of free and bound variables
|
|
is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sb7f $p |- ( [ y / x ] ph <->
|
|
E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
|
|
( wsb weq wa wex sb5 sbbii sbco2v 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDCGOHD
|
|
INODCABDJKABCDELODCJM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sb7af.1 $e |- F/ z ph $.
|
|
$( An alternate definition of proper substitution ~ df-sb . Similar to
|
|
~ dfsb7a but does not require that ` ph ` and ` z ` be distinct.
|
|
Similar to ~ sb7f in that it involves a dummy variable ` z ` , but
|
|
expressed in terms of ` A. ` rather than ` E. ` . (Contributed by Jim
|
|
Kingdon, 5-Feb-2018.) $)
|
|
sb7af $p |- ( [ y / x ] ph
|
|
<-> A. z
|
|
( z = y
|
|
-> A. x ( x = z -> ph ) ) ) $=
|
|
( wsb weq wi wal sb6 sbbii sbco2 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDCGOHDI
|
|
NODCABDJKABCDELODCJM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( An alternate definition of proper substitution ~ df-sb . Similar to
|
|
~ dfsb7 in that it involves a dummy variable ` z ` , but expressed in
|
|
terms of ` A. ` rather than ` E. ` . For a version which only requires
|
|
` F/ z ph ` rather than ` z ` and ` ph ` being distinct, see ~ sb7af .
|
|
(Contributed by Jim Kingdon, 5-Feb-2018.) $)
|
|
dfsb7a $p |- ( [ y / x ] ph
|
|
<-> A. z
|
|
( z = y
|
|
-> A. x ( x = z -> ph ) ) ) $=
|
|
( nfv sb7af ) ABCDADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
sb10f.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Hao Wang's identity axiom P6 in Irving Copi, _Symbolic Logic_ (5th ed.,
|
|
1979), p. 328. In traditional predicate calculus, this is a sole axiom
|
|
for identity from which the usual ones can be derived. (Contributed by
|
|
NM, 9-May-2005.) $)
|
|
sb10f $p |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) ) $=
|
|
( weq wsb wa wex hbsb sbequ equsex bicomi ) BCFADBGZHBIADCGZNOBCADCBEJABC
|
|
DKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( An identity law for substitution. Used in proof of Theorem 9.7 of
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbid2v $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
|
|
( ax-17 sbid2h ) ABCABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ph $.
|
|
$( Elimination of substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbelx $p |- ( ph <-> E. x ( x = y /\ [ x / y ] ph ) ) $=
|
|
( ax-17 sb5rf ) ACBABDE $.
|
|
$}
|
|
|
|
${
|
|
$( Note: A more general case could also be proved with
|
|
"$d x z $. $d y w $. $d x ph $. $d y ph $.", but with more
|
|
difficulty. $)
|
|
$d x y z $. $d w y $. $d x y ph $.
|
|
$( Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbel2x $p |- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\
|
|
[ y / w ] [ x / z ] ph ) ) $=
|
|
( weq wsb wa wex sbelx anbi2i exbii exdistr 3bitr4i anass 2exbii bitr4i )
|
|
ABDFZCEFZADBGZECGZHZHZCIBIZRSHUAHZCIBIRTHZBIRUBCIZHZBIAUDUFUHBTUGRTCEJKLA
|
|
BDJRUBBCMNUEUCBCRSUAOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Move universal quantifier in and out of substitution. Identical to
|
|
~ sbal except that it has an additional distinct variable constraint on
|
|
` y ` and ` z ` . (Contributed by Jim Kingdon, 29-Dec-2017.) $)
|
|
sbalyz $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
|
|
( wal wsb nfa1 nfsbxy ax-4 sbimi alrimi weq wi sb6 albii alcom nfv stdpc5
|
|
bitri alimi sb2 syl sylbi impbii ) ABEZCDFZACDFZBEZUFUGBUECDBABGHUEACDABI
|
|
JKUHCDLZAMZBEZCEZUFUHUJCEZBEULUGUMBACDNOUJBCPSULUIUEMZCEUFUKUNCUIABUIBQRT
|
|
UECDUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y w $. $d x z w $. $d w ph $.
|
|
$( Move universal quantifier in and out of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) $)
|
|
sbal $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
|
|
( vw wal wsb sbalyz sbbii bitri ax-17 sbco2v albii 3bitr3i ) ABFZCEGZEDGZ
|
|
ACEGZEDGZBFZOCDGACDGZBFQRBFZEDGTPUBEDABCEHIRBEDHJOCDEOEKLSUABACDEAEKLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $.
|
|
$( Lemma for proving ~ sbal1 . Same as ~ sbal1 but with an additional
|
|
distinct variable constraint on ` y ` and ` z ` . (Contributed by Jim
|
|
Kingdon, 23-Feb-2018.) $)
|
|
sbal1yz $p |- ( -. A. x x = z ->
|
|
( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
|
|
( weq wal wn wsb wi wb wo dveeq2or equcom nfbii 19.21t sylbi imbi1i albii
|
|
wnf sb6 orim2i ax-mp ori albidv alcom bitri bitr4i bitr2i 3bitr3g bicomd
|
|
) BDEBFZGZACDHZBFZABFZCDHZULDCEZAIZBFZCFZUQUOIZCFZUNUPULUSVACUKUSVAJZUKCD
|
|
EZBSZKUKVCKBDCLVEVCUKVEUQBSVCVDUQBCDMZNUQABOPUAUBUCUDUTURCFZBFUNURCBUEUMV
|
|
GBUMVDAIZCFVGACDTVHURCVDUQAVFQRUFRUGUPVDUOIZCFVBUOCDTVIVACVDUQUOVFQRUHUIU
|
|
J $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d w x $. $d w y $. $d w z $. $d ph w $.
|
|
|
|
$( A theorem used in elimination of disjoint variable restriction on ` x `
|
|
and ` y ` by replacing it with a distinctor ` -. A. x x = z ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon,
|
|
24-Feb-2018.) $)
|
|
sbal1 $p |- ( -. A. x x = z ->
|
|
( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
|
|
( vw weq wal wn wsb sbal sbbii sbal1yz syl5bb ax-17 sbco2v albii 3bitr3g
|
|
) BDFBGHZABGZCEIZEDIZACEIZEDIZBGZSCDIACDIZBGUAUBBGZEDIRUDTUFEDABCEJKUBBED
|
|
LMSCDESENOUCUEBACDEAENOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Move existential quantifier in and out of substitution. Identical to
|
|
~ sbex except that it has an additional distinct variable constraint on
|
|
` y ` and ` z ` . (Contributed by Jim Kingdon, 29-Dec-2017.) $)
|
|
sbexyz $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
|
|
( wex wsb weq wa sb5 exdistr excom 3bitr2i exbii bitr4i ) ABEZCDFZCDGZAHZ
|
|
CEZBEZACDFZBEPQOHCERBECETOCDIQACBJRCBKLUASBACDIMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y w $. $d x z w $. $d w ph $.
|
|
$( Move existential quantifier in and out of substitution. (Contributed by
|
|
NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) $)
|
|
sbex $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
|
|
( vw wex wsb sbexyz sbbii bitri ax-17 sbco2v exbii 3bitr3i ) ABFZCEGZEDGZ
|
|
ACEGZEDGZBFZOCDGACDGZBFQRBFZEDGTPUBEDABCEHIRBEDHJOCDEOEKLSUABACDEAEKLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sbalv.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Quantify with new variable inside substitution. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
sbalv $p |- ( [ y / x ] A. z ph <-> A. z ps ) $=
|
|
( wal wsb sbal albii bitri ) AEGCDHACDHZEGBEGAECDILBEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d v w ph $. $d v w x $. $d v w y $.
|
|
$( Lemma for ~ sbco4 . It replaces the temporary variable ` v ` with
|
|
another temporary variable ` w ` . (Contributed by Jim Kingdon,
|
|
26-Sep-2018.) $)
|
|
sbco4lem $p |- ( [ x / v ] [ y / x ] [ v / y ] ph <->
|
|
[ x / w ] [ y / x ] [ w / y ] ph ) $=
|
|
( wsb sbcom2 sbbii nfv sbco2 bitri sbid2 3bitr3i ) ACDFZDEFZBCFZEDFZDBFZO
|
|
EDFZBCFZDBFACEFZBCFZEBFZNBCFZDBFQTDBOBCEDGHRUBEDFZDBFUCQUEDBPUBEDOUABCACE
|
|
DADIJHHHUBEBDUBDIJKTUDDBSNBCNEDNEILHHM $.
|
|
$}
|
|
|
|
${
|
|
$d t u v ph $. $d t u v x $. $d t u v y $. $d w ph $. $d w x $.
|
|
$d w y $. $d t w $.
|
|
$( Two ways of exchanging two variables. Both sides of the biconditional
|
|
exchange ` x ` and ` y ` , either via two temporary variables ` u ` and
|
|
` v ` , or a single temporary ` w ` . (Contributed by Jim Kingdon,
|
|
25-Sep-2018.) $)
|
|
sbco4 $p |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <->
|
|
[ x / w ] [ y / x ] [ w / y ] ph ) $=
|
|
( vt wsb sbcom2 nfv sbco2 sbbii bitr3i sbco4lem 3bitri ) ACEHZBFHZEBHFCHZ
|
|
PBCHZEBHZACGHBCHGBHACDHBCHDBHRQFCHZEBHTQFCEBIUASEBPBCFPFJKLMABCGENABCDGNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( An equivalent expression for existence. (Contributed by NM,
|
|
2-Feb-2005.) $)
|
|
exsb $p |- ( E. x ph <-> E. y A. x ( x = y -> ph ) ) $=
|
|
( wex wsb weq wi wal ax-17 sb8eh sb6 exbii bitri ) ABDABCEZCDBCFAGBHZCDAB
|
|
CACIJNOCABCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y w $. $d z w ph $.
|
|
$( An equivalent expression for double existence. (Contributed by NM,
|
|
2-Feb-2005.) $)
|
|
2exsb $p |- ( E. x E. y ph <->
|
|
E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
|
|
( wex weq wi wal exsb exbii excom bitri impexp albii 19.21v bitr2i 3bitri
|
|
wa ) ACFZBFZCEGZAHZCIZBFZEFZBDGZUBSAHZCIZBIZDFZEFUJEFDFUAUDEFZBFUFTULBACE
|
|
JKUDBELMUEUKEUEUGUDHZBIZDFUKUDBDJUNUJDUMUIBUIUGUCHZCIUMUHUOCUGUBANOUGUCCP
|
|
QOKMKUJEDLR $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d x z $. $d y z $.
|
|
dvelimALT.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimALT.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelim that doesn't use ~ ax-10 . Because it has different
|
|
distinct variable constraints than ~ dvelim and is used in important
|
|
proofs, it would be better if it had a name which does not end in ALT
|
|
(ideally more close to set.mm naming). (Contributed by NM,
|
|
17-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
dvelimALT $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi wnf nfv wo ax-i12 orcom orbi2i mpbi a1i nfimd orass mpbir
|
|
nfa1 ax16ALT nfd nfi df-nf id sylbir orim1i ax-mp ori nfald ax-17 equsalh
|
|
jaoi nfbii sylib nfrd ) CDHCIZJZBCVAEDHZAKZEIZCLBCLVAVCCEVAEMUTVCCLZVEUTN
|
|
ZUTVENCEHZCIZVBVBCIKCIZNZUTNZVFVKVHVIUTNZNZVHUTVINZNVMEDCOVNVLVHUTVIPQRVH
|
|
VIUTUAUBVJVEUTVHVEVIVHVBACVHVBCVGCUCVBCEUDUEACLZVHACFUFZSTVIVBCLZVEVBCUGV
|
|
QVBACVQUHVOVQVPSTUIUPUJUKVEUTPRULUMVDBCABEDBEUNGUOUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
dvelimfv.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimfv.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimfv.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Like ~ dvelimf but with a distinct variable constraint on ` x ` and
|
|
` z ` . (Contributed by Jim Kingdon, 6-Mar-2018.) $)
|
|
dvelimfv $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi wnf nfnae wo ax-i12 orcom mpbi a1i nfimd orass mpbir nfae
|
|
orbi2i ax16ALT nfd nfi df-nf sylbir jaoi orim1i ax-mp nfald equsalh nfbii
|
|
id ori sylib nfrd ) CDICJZKZBCVAEDIZALZEJZCMBCMVAVCCECDENUTVCCMZVEUTOZUTV
|
|
EOCEICJZVBVBCJLCJZOZUTOZVFVJVGVHUTOZOZVGUTVHOZOVLEDCPVMVKVGUTVHQUDRVGVHUT
|
|
UAUBVIVEUTVGVEVHVGVBACVGVBCCECUCVBCEUEUFACMZVGACFUGZSTVHVBCMZVEVBCUHVPVBA
|
|
CVPUPVNVPVOSTUIUJUKULVEUTQRUQUMVDBCABEDGHUNUOURUS $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d ph w $.
|
|
hbsb4.1 $e |- ( ph -> A. z ph ) $.
|
|
$( A variable not free remains so after substitution with a distinct
|
|
variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim
|
|
Kingdon, 23-Mar-2018.) $)
|
|
hbsb4 $p |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) $=
|
|
( vw wsb hbsb sbequ dvelimALT ) ABFGABCGDCFABFDEHAFCBIJ $.
|
|
$}
|
|
|
|
$( A variable not free remains so after substitution with a distinct variable
|
|
(closed form of ~ hbsb4 ). (Contributed by NM, 7-Apr-2004.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
hbsb4t $p |- ( A. x A. z ( ph -> A. z ph ) ->
|
|
( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) ) $=
|
|
( weq wal wn wsb wi hba1 hbsb4 spsbim sps ax-4 sbimi alimi a1i imim12d syl5
|
|
a7s ) DCEDFGADFZBCHZUBDFZIZAUAIZDFBFABCHZUFDFZIZUABCDADJKUEUDUHIDBUEBFZDFZU
|
|
FUBUCUGUIUFUBIDAUABCLMUCUGIUJUBUFDUAABCADNOPQRTS $.
|
|
|
|
$( A variable not free remains so after substitution with a distinct variable
|
|
(closed form of ~ hbsb4 ). (Contributed by NM, 7-Apr-2004.) (Revised by
|
|
Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon,
|
|
9-May-2018.) $)
|
|
nfsb4t $p |- ( A. x F/ z ph ->
|
|
( -. A. z z = y -> F/ z [ y / x ] ph ) ) $=
|
|
( wnf wal weq wn wsb wa nfnf1 nfal nfnae nfan df-nf albii hbsb4t sylbi imp
|
|
wi nfd ex ) ADEZBFZDCGDFHZABCIZDEUDUEJUFDUDUEDUCDBADKLDCDMNUDUEUFUFDFTZUDAA
|
|
DFTDFZBFUEUGTUCUHBADOPABCDQRSUAUB $.
|
|
|
|
${
|
|
dvelimf.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimf.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimf.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelim without any variable restrictions. (Contributed by
|
|
NM, 1-Oct-2002.) $)
|
|
dvelimf $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wsb hbsb4 sbieh albii 3imtr3g ) CDICJKAEDLZQCJBBCJAEDCFMABED
|
|
GHNZQBCROP $.
|
|
$}
|
|
|
|
${
|
|
dvelimdf.1 $e |- F/ x ph $.
|
|
dvelimdf.2 $e |- F/ z ph $.
|
|
dvelimdf.3 $e |- ( ph -> F/ x ps ) $.
|
|
dvelimdf.4 $e |- ( ph -> F/ z ch ) $.
|
|
dvelimdf.5 $e |- ( ph -> ( z = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction form of ~ dvelimf . This version may be useful if we want to
|
|
avoid ~ ax-17 and use ~ ax-16 instead. (Contributed by NM, 7-Apr-2004.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf
|
|
Lammen, 11-May-2018.) $)
|
|
dvelimdf $p |- ( ph -> ( -. A. x x = y -> F/ x ch ) ) $=
|
|
( weq wal wn wsb wnf wi alrimi nfsb4t syl sbied nfbidf sylibd ) ADELDMNZB
|
|
FEOZDPZCDPABDPZFMUDUFQAUGFHIRBFEDSTAUECDGABCFEHJKUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $.
|
|
dvelim.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelim.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( This theorem can be used to eliminate a distinct variable restriction on
|
|
` x ` and ` z ` and replace it with the "distinctor" ` -. A. x x = y `
|
|
as an antecedent. ` ph ` normally has ` z ` free and can be read
|
|
` ph ( z ) ` , and ` ps ` substitutes ` y ` for ` z ` and can be read
|
|
` ph ( y ) ` . We don't require that ` x ` and ` y ` be distinct: if
|
|
they aren't, the distinctor will become false (in multiple-element
|
|
domains of discourse) and "protect" the consequent.
|
|
|
|
To obtain a closed-theorem form of this inference, prefix the hypotheses
|
|
with ` A. x A. z ` , conjoin them, and apply ~ dvelimdf .
|
|
|
|
Other variants of this theorem are ~ dvelimf (with no distinct variable
|
|
restrictions) and ~ dvelimALT (that avoids ~ ax-10 ). (Contributed by
|
|
NM, 23-Nov-1994.) $)
|
|
dvelim $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( ax-17 dvelimf ) ABCDEFBEHGI $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d x z $.
|
|
dvelimor.1 $e |- F/ x ph $.
|
|
dvelimor.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Disjunctive distinct variable constraint elimination. A user of this
|
|
theorem starts with a formula ` ph ` (containing ` z ` ) and a distinct
|
|
variable constraint between ` x ` and ` z ` . The theorem makes it
|
|
possible to replace the distinct variable constraint with the disjunct
|
|
` A. x x = y ` ( ` ps ` is just a version of ` ph ` with ` y `
|
|
substituted for ` z ` ). (Contributed by Jim Kingdon, 11-May-2018.) $)
|
|
dvelimor $p |- ( A. x x = y \/ F/ x ps ) $=
|
|
( weq wal wi wnf wo ax-bndl orcom orbi2i mpbi orass mpbir nfae ax-mp jaoi
|
|
a16nf alrimi df-nf id nfimd sylbir alimi orim1i nfalt ax-17 equsalh nfbii
|
|
a1i sylib orim2i ) CDHCIZEDHZAJZCKZEIZLZUQBCKZLVAUQLZVBCEHCIZURURCIJCIZEI
|
|
ZLZUQLZVDVIVEVGUQLZLZVEUQVGLZLVKEDCMVLVJVEUQVGNOPVEVGUQQRVHVAUQVEVAVGVEUT
|
|
ECEESUSCECUBUCVFUTEVFURCKZUTURCUDVMURACVMUEACKVMFUNUFUGUHUAUITVAUQNPVAVCU
|
|
QVAUSEIZCKVCUSCEUJVNBCABEDBEUKGULUMUOUPT $.
|
|
$}
|
|
|
|
${
|
|
$d z x $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon,
|
|
19-Feb-2018.) $)
|
|
dveeq1 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( weq wal wn dveeq2 equcom albii 3imtr3g ) ABDAEFCBDZKAEBCDZLAEABCGCBHZKL
|
|
AMIJ $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveel1 $p |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) ) $=
|
|
( vw wel ax-17 elequ1 dvelimf ) DCEZBCEZABDIAFJDFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveel2 $p |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) ) $=
|
|
( vw wel ax-17 elequ2 dvelimf ) CDEZCBEZABDIAFJDFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d z y $. $d z x $.
|
|
$( Move quantifier in and out of substitution. (Contributed by NM,
|
|
2-Jan-2002.) $)
|
|
sbal2 $p |- ( -. A. x x = y ->
|
|
( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
|
|
( weq wal wn wi wsb alcom hbnae wb dveeq1 alimi hbnaes 19.21ht syl albidh
|
|
syl5rbbr sb6 albii 3bitr4g ) BCEBFGZCDEZABFZHZCFZUDAHZCFZBFZUECDIACDIZBFU
|
|
JUHBFZCFUCUGUHCBJUCULUFCBCCKUCUDUDBFHZBFZULUFLUNBCBUCUMBBCDMNOUDABPQRSUEC
|
|
DTUKUIBACDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d ph w $.
|
|
nfsb4or.1 $e |- F/ z ph $.
|
|
$( A variable not free remains so after substitution with a distinct
|
|
variable. (Contributed by Jim Kingdon, 11-May-2018.) $)
|
|
nfsb4or $p |- ( A. z z = y \/ F/ z [ y / x ] ph ) $=
|
|
( vw wsb nfsb sbequ dvelimor ) ABFGABCGDCFABFDEHAFCBIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Existential uniqueness
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols needed for uniqueness notation. $)
|
|
$c E! $. $( Backwards E exclamation point. $)
|
|
$c E* $. $( Backwards E superscript *. $)
|
|
|
|
$( Extend wff definition to include existential uniqueness ("there exists a
|
|
unique ` x ` such that ` ph ` "). $)
|
|
weu $a wff E! x ph $.
|
|
|
|
$( Extend wff definition to include uniqueness ("there exists at most one
|
|
` x ` such that ` ph ` "). $)
|
|
wmo $a wff E* x ph $.
|
|
|
|
${
|
|
$d w x y $. $d x z $. $d y ph $. $d w z ph $.
|
|
$( A soundness justification theorem for ~ df-eu , showing that the
|
|
definition is equivalent to itself with its dummy variable renamed.
|
|
Note that ` y ` and ` z ` needn't be distinct variables. (Contributed
|
|
by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
eujust $p |- ( E. y A. x ( ph <-> x = y )
|
|
<-> E. z A. x ( ph <-> x = z ) ) $=
|
|
( vw cv wceq wb wal wex equequ2 bibi2d albidv cbvexv bitri ) ABFZCFZGZHZB
|
|
IZCJAPEFZGZHZBIZEJAPDFZGZHZBIZDJTUDCEQUAGZSUCBUIRUBACEBKLMNUDUHEDUAUEGZUC
|
|
UGBUJUBUFAEDBKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Define existential uniqueness, i.e. "there exists exactly one ` x `
|
|
such that ` ph ` ." Definition 10.1 of [BellMachover] p. 97; also
|
|
Definition *14.02 of [WhiteheadRussell] p. 175. Other possible
|
|
definitions are given by ~ eu1 , ~ eu2 , ~ eu3 , and ~ eu5 (which in
|
|
some cases we show with a hypothesis ` ph -> A. y ph ` in place of a
|
|
distinct variable condition on ` y ` and ` ph ` ). Double uniqueness is
|
|
tricky: ` E! x E! y ph ` does not mean "exactly one ` x ` and one
|
|
` y ` " (see ~ 2eu4 ). (Contributed by NM, 5-Aug-1993.) $)
|
|
df-eu $a |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $.
|
|
$}
|
|
|
|
$( Define "there exists at most one ` x ` such that ` ph ` ." Here we define
|
|
it in terms of existential uniqueness. Notation of [BellMachover] p. 460,
|
|
whose definition we show as ~ mo3 . For another possible definition see
|
|
~ mo4 . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-mo $a |- ( E* x ph <-> ( E. x ph -> E! x ph ) ) $.
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
euf.1 $e |- ( ph -> A. y ph ) $.
|
|
$( A version of the existential uniqueness definition with a hypothesis
|
|
instead of a distinct variable condition. (Contributed by NM,
|
|
12-Aug-1993.) $)
|
|
euf $p |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $=
|
|
( vz weu weq wb wal wex df-eu ax-17 hbbi hbal equequ2 bibi2d albidv bitri
|
|
cbvexh ) ABFABEGZHZBIZEJABCGZHZBIZCJABEKUBUEECUACBATCDTCLMNUEELECGZUAUDBU
|
|
FTUCAECBOPQSR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $. $d y ch $.
|
|
eubidh.1 $e |- ( ph -> A. x ph ) $.
|
|
eubidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for uniqueness quantifier (deduction rule).
|
|
(Contributed by NM, 9-Jul-1994.) $)
|
|
eubidh $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
|
|
( vy weq wb wal wex weu bibi1d albidh exbidv df-eu 3bitr4g ) ABDGHZIZDJZG
|
|
KCRIZDJZGKBDLCDLATUBGASUADEABCRFMNOBDGPCDGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $. $d y ch $.
|
|
eubid.1 $e |- F/ x ph $.
|
|
eubid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for uniqueness quantifier (deduction rule).
|
|
(Contributed by NM, 9-Jul-1994.) $)
|
|
eubid $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
|
|
( vy weq wb wal wex weu bibi1d albid exbidv df-eu 3bitr4g ) ABDGHZIZDJZGK
|
|
CRIZDJZGKBDLCDLATUBGASUADEABCRFMNOBDGPCDGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
eubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for uniqueness quantifier (deduction rule).
|
|
(Contributed by NM, 9-Jul-1994.) $)
|
|
eubidv $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
|
|
( nfv eubid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
eubii.1 $e |- ( ph <-> ps ) $.
|
|
$( Introduce uniqueness quantifier to both sides of an equivalence.
|
|
(Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) $)
|
|
eubii $p |- ( E! x ph <-> E! x ps ) $=
|
|
( weu wb wtru a1i eubidv trud ) ACEBCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for uniqueness. (Contributed by NM,
|
|
9-Jul-1994.) $)
|
|
hbeu1 $p |- ( E! x ph -> A. x E! x ph ) $=
|
|
( vy weu weq wb wal wex df-eu hba1 hbex hbxfrbi ) ABDABCEFZBGZCHBABCINBCM
|
|
BJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for uniqueness. (Contributed by NM,
|
|
9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfeu1 $p |- F/ x E! x ph $=
|
|
( vy weu weq wb wal wex df-eu nfa1 nfex nfxfr ) ABDABCEFZBGZCHBABCINBCMBJ
|
|
KL $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by NM,
|
|
8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfmo1 $p |- F/ x E* x ph $=
|
|
( wmo wex weu wi df-mo nfe1 nfeu1 nfim nfxfr ) ABCABDZABEZFBABGLMBABHABIJK
|
|
$.
|
|
|
|
${
|
|
$d w y z $. $d ph z w $. $d w x z $.
|
|
sb8eu.1 $e |- F/ y ph $.
|
|
$( Variable substitution in uniqueness quantifier. (Contributed by NM,
|
|
7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
sb8eu $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
|
|
( vz vw weq wb wal wex wsb weu nfv sb8 sbbi nfsb equsb3 nfxfr nfbi df-eu
|
|
sbequ cbval sblbis albii 3bitri exbii 3bitr4i ) ABEGZHZBIZEJABCKZCEGZHZCI
|
|
ZEJABLUKCLUJUNEUJUIBFKZFIUIBCKZCIUNUIBFUIFMNUOUPFCUOABFKZUHBFKZHCAUHBFOUQ
|
|
URCABFCDPURFEGZCFBEQUSCMRSRUPFMUIFCBUAUBUPUMCUHULABCCBEQUCUDUEUFABETUKCET
|
|
UG $.
|
|
|
|
$( Variable substitution for "at most one." (Contributed by Alexander van
|
|
der Vekens, 17-Jun-2017.) $)
|
|
sb8mo $p |- ( E* x ph <-> E* y [ y / x ] ph ) $=
|
|
( wex weu wi wsb wmo sb8e sb8eu imbi12i df-mo 3bitr4i ) ABEZABFZGABCHZCEZ
|
|
QCFZGABIQCIORPSABCDJABCDKLABMQCMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $. $d z ps $.
|
|
nfeudv.1 $e |- F/ y ph $.
|
|
nfeudv.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfeu . Similar to ~ nfeud but has the additional
|
|
constraint that ` x ` and ` y ` must be distinct. (Contributed by Jim
|
|
Kingdon, 25-May-2018.) $)
|
|
nfeudv $p |- ( ph -> F/ x E! y ps ) $=
|
|
( vz cv wceq wb wal wex wnf weu nfv a1i nfbid nfald nfexd df-eu sylibr
|
|
nfbii ) ABDHGHIZJZDKZGLZCMBDNZCMAUECGAGOAUDCDEABUCCFUCCMAUCCOPQRSUGUFCBDG
|
|
TUBUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $.
|
|
nfeud.1 $e |- F/ y ph $.
|
|
nfeud.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfeu . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim
|
|
Kingdon, 25-May-2018.) $)
|
|
nfeud $p |- ( ph -> F/ x E! y ps ) $=
|
|
( vz weu wsb nfv sb8eu nfsbd nfeudv nfxfrd ) BDHBDGIZGHACBDGBGJKAOCGAGJAB
|
|
DGCEFLMN $.
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by
|
|
Mario Carneiro, 14-Nov-2016.) $)
|
|
nfmod $p |- ( ph -> F/ x E* y ps ) $=
|
|
( wmo wex weu wi df-mo nfexd nfeud nfimd nfxfrd ) BDGBDHZBDIZJACBDKAPQCAB
|
|
CDEFLABCDEFMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
nfeuv.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for existential uniqueness. This is
|
|
similar to ~ nfeu but has the additional constraint that ` x ` and ` y `
|
|
must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) $)
|
|
nfeuv $p |- F/ x E! y ph $=
|
|
( vz weu wnf weq wb wal wex nfv nfbi nfal nfex df-eu nfbii mpbir ) ACFZBG
|
|
ACEHZIZCJZEKZBGUBBEUABCATBDTBLMNOSUCBACEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x z $. $d z ph $.
|
|
nfeu.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for existential uniqueness. Note that
|
|
` x ` and ` y ` needn't be distinct. (Contributed by NM, 8-Mar-1995.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim
|
|
Kingdon, 23-May-2018.) $)
|
|
nfeu $p |- F/ x E! y ph $=
|
|
( vz weu wsb nfv sb8eu nfsb nfeuv nfxfr ) ACFACEGZEFBACEAEHIMBEACEBDJKL
|
|
$.
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by
|
|
NM, 9-Mar-1995.) $)
|
|
nfmo $p |- F/ x E* y ph $=
|
|
( wmo wnf wtru nftru a1i nfmod trud ) ACEBFGABCCHABFGDIJK $.
|
|
$}
|
|
|
|
${
|
|
hbeu.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for uniqueness. Note that ` x ` and
|
|
` y ` needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof
|
|
rewritten by Jim Kingdon, 24-May-2018.) $)
|
|
hbeu $p |- ( E! y ph -> A. x E! y ph ) $=
|
|
( weu nfi nfeu nfri ) ACEBABCABDFGH $.
|
|
$}
|
|
|
|
${
|
|
hbeud.1 $e |- ( ph -> A. x ph ) $.
|
|
hbeud.2 $e |- ( ph -> A. y ph ) $.
|
|
hbeud.3 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction version of ~ hbeu . (Contributed by NM, 15-Feb-2013.) (Proof
|
|
rewritten by Jim Kingdon, 25-May-2018.) $)
|
|
hbeud $p |- ( ph -> ( E! y ps -> A. x E! y ps ) ) $=
|
|
( weu nfi nfd nfeud nfrd ) ABDHCABCDADFIABCACEIGJKL $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d ph z w $. $d w x z $.
|
|
sb8euh.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Variable substitution in uniqueness quantifier. (Contributed by NM,
|
|
7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) $)
|
|
sb8euh $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
|
|
( vz vw weq wb wal wex wsb ax-17 sb8h sbbi hbsb equsb3 hbxfrbi hbbi df-eu
|
|
weu sbequ cbvalh sblbis albii 3bitri exbii 3bitr4i ) ABEGZHZBIZEJABCKZCEG
|
|
ZHZCIZEJABTUKCTUJUNEUJUIBFKZFIUIBCKZCIUNUIBFUIFLMUOUPFCUOABFKZUHBFKZHCAUH
|
|
BFNUQURCABFCDOURFEGZCFBEPUSCLQRQUPFLUIFCBUAUBUPUMCUHULABCCBEPUCUDUEUFABES
|
|
UKCESUG $.
|
|
$}
|
|
|
|
${
|
|
cbveu.1 $e |- F/ y ph $.
|
|
cbveu.2 $e |- F/ x ps $.
|
|
cbveu.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
cbveu $p |- ( E! x ph <-> E! y ps ) $=
|
|
( weu wsb sb8eu sbie eubii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu1.1 $e |- ( ph -> A. y ph ) $.
|
|
$( An alternate way to express uniqueness used by some authors. Exercise
|
|
2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) $)
|
|
eu1 $p |- ( E! x ph <->
|
|
E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) ) $=
|
|
( wsb weu weq wb wal wex wi wa hbs1 euf sb8euh equcom albii sb6rf 3bitr4i
|
|
imbi2i anbi12i ancom albiim exbii ) ABCEZCFUECBGZHCIZBJABFAUEBCGZKZCIZLZB
|
|
JUECBABCMNABCDOUKUGBUJALUEUFKZCIZUFUEKCIZLUKUGUJUMAUNUIULCUHUFUEBCPTQABCD
|
|
RUAAUJUBUEUFCUCSUDS $.
|
|
$}
|
|
|
|
${
|
|
euor.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Introduce a disjunct into a uniqueness quantifier. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
euor $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
|
|
( wn weu wo hbn biorf eubidh biimpa ) AEZBCFABGZCFLBMCACDHABIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduce a disjunct into a uniqueness quantifier. (Contributed by NM,
|
|
23-Mar-1995.) $)
|
|
euorv $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
|
|
( ax-17 euor ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mon.1 $e |- F/ y ph $.
|
|
$( There is at most one of something which does not exist. (Contributed by
|
|
Jim Kingdon, 2-Jul-2018.) $)
|
|
mo2n $p |- ( -. E. x ph -> E. y A. x ( ph -> x = y ) ) $=
|
|
( wex wsb weq wi wal sb8e wn alnex nfs1v sbequ1 equcoms con3d cbv3 pm2.21
|
|
nfn alimi 19.8a 3syl sylbir sylnbi ) ABEABCFZCEZABCGZHZBIZCEZABCDJUFKUEKZ
|
|
CIZUJUECLULAKZBIUIUJUKUMCBUEBABCMSACDSCBGAUEAUEHBCABCNOPQUMUHBAUGRTUICUAU
|
|
BUCUD $.
|
|
$}
|
|
|
|
$( There is at most one of something which does not exist. (Contributed by
|
|
Jim Kingdon, 5-Jul-2018.) $)
|
|
mon $p |- ( -. E. x ph -> E* x ph ) $=
|
|
( wex wn weu wi wmo ax-in2 df-mo sylibr ) ABCZDKABEZFABGKLHABIJ $.
|
|
|
|
${
|
|
$d x y $. $d ph y $.
|
|
$( Existential uniqueness implies existence. (Contributed by NM,
|
|
15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
euex $p |- ( E! x ph -> E. x ph ) $=
|
|
( vy weu wsb weq wi wal wa wex ax-17 eu1 exsimpl sylbi ) ABDAABCEBCFGCHZI
|
|
BJABJABCACKLAOBMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eumo0.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Existential uniqueness implies "at most one." (Contributed by NM,
|
|
8-Jul-1994.) $)
|
|
eumo0 $p |- ( E! x ph -> E. y A. x ( ph -> x = y ) ) $=
|
|
( weu weq wb wal wex wi euf bi1 alimi eximi sylbi ) ABEABCFZGZBHZCIAPJZBH
|
|
ZCIABCDKRTCQSBAPLMNO $.
|
|
$}
|
|
|
|
$( Existential uniqueness implies "at most one." (Contributed by NM,
|
|
23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) $)
|
|
eumo $p |- ( E! x ph -> E* x ph ) $=
|
|
( weu wex wi wmo ax-1 df-mo sylibr ) ABCZABDZJEABFJKGABHI $.
|
|
|
|
${
|
|
eumoi.1 $e |- E! x ph $.
|
|
$( "At most one" inferred from existential uniqueness. (Contributed by NM,
|
|
5-Apr-1995.) $)
|
|
eumoi $p |- E* x ph $=
|
|
( weu wmo eumo ax-mp ) ABDABECABFG $.
|
|
$}
|
|
|
|
${
|
|
mobidh.1 $e |- ( ph -> A. x ph ) $.
|
|
mobidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for "at most one" quantifier (deduction rule).
|
|
(Contributed by NM, 8-Mar-1995.) $)
|
|
mobidh $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
|
|
( wex weu wi wmo exbidh eubidh imbi12d df-mo 3bitr4g ) ABDGZBDHZICDGZCDHZ
|
|
IBDJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
|
|
$}
|
|
|
|
${
|
|
mobid.1 $e |- F/ x ph $.
|
|
mobid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for "at most one" quantifier (deduction rule).
|
|
(Contributed by NM, 8-Mar-1995.) $)
|
|
mobid $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
|
|
( wex weu wi wmo exbid eubid imbi12d df-mo 3bitr4g ) ABDGZBDHZICDGZCDHZIB
|
|
DJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for "at most one" quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
mobidv $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
|
|
( nfv mobid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
mobii.1 $e |- ( ps <-> ch ) $.
|
|
$( Formula-building rule for "at most one" quantifier (inference rule).
|
|
(Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro,
|
|
17-Oct-2016.) $)
|
|
mobii $p |- ( E* x ps <-> E* x ch ) $=
|
|
( wmo wb wtru a1i mobidv trud ) ACEBCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
hbmo1 $p |- ( E* x ph -> A. x E* x ph ) $=
|
|
( wmo wex weu wi df-mo hbe1 hbeu1 hbim hbxfrbi ) ABCABDZABEZFBABGLMBABHABIJ
|
|
K $.
|
|
|
|
${
|
|
hbmo.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by
|
|
NM, 9-Mar-1995.) $)
|
|
hbmo $p |- ( E* y ph -> A. x E* y ph ) $=
|
|
( wmo wex weu wi df-mo hbex hbeu hbim hbxfrbi ) ACEACFZACGZHBACINOBABCDJA
|
|
BCDKLM $.
|
|
$}
|
|
|
|
${
|
|
cbvmo.1 $e |- F/ y ph $.
|
|
cbvmo.2 $e |- F/ x ps $.
|
|
cbvmo.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvmo $p |- ( E* x ph <-> E* y ps ) $=
|
|
( wex weu wi wmo cbvex cbveu imbi12i df-mo 3bitr4i ) ACHZACIZJBDHZBDIZJAC
|
|
KBDKQSRTABCDEFGLABCDEFGMNACOBDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
mo23.1 $e |- F/ y ph $.
|
|
$( An implication between two definitions of "there exists at most one."
|
|
(Contributed by Jim Kingdon, 25-Jun-2018.) $)
|
|
mo23 $p |- ( E. y A. x ( ph -> x = y ) ->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( vz weq wi wal wex wsb wa nfv nfim nfal equequ2 imbi2d albidv cbvex nfri
|
|
nfs1v sbequ2 ax-8 imim12d cbv3 ancli aaanh sylibr prth equtr2 syl6 2alimi
|
|
syl exlimiv sylbir ) ABCFZGZBHZCIABEFZGZBHZEIAABCJZKZUOGZCHBHZUTUQECUSCBA
|
|
URCDURCLMZNUQELECFZUSUPBVFURUOAECBOPQRUTVDEUTUSVACEFZGZKZCHBHZVDUTUTVHCHZ
|
|
KVJUTVKUSVHBCVEVAVGBABCTVGBLMZUOVAAURVGABCUABCEUBUCUDUEUSVHBCUSCVESVHBVLS
|
|
UFUGVIVCBCVIVBURVGKUOAURVAVGUHBCEUIUJUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mor.1 $e |- F/ y ph $.
|
|
$( Converse of ~ mo23 with an additional ` E. x ph ` condition.
|
|
(Contributed by Jim Kingdon, 25-Jun-2018.) $)
|
|
mor $p |- ( E. x ph ->
|
|
( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ->
|
|
E. y A. x ( ph -> x = y ) ) ) $=
|
|
( wex wsb wa weq wi sb8e impexp bi2.04 bitri 2albii nfs1v nfri eximi alim
|
|
wal alimi a7s exim syl syl5com syl5bi sylbi ) ABEABCFZCEZAUGGBCHZIZCSBSZA
|
|
UIIZBSZCEZIABCDJUKUGULIZCSBSZUHUNUJUOBCUJAUGUIIIUOAUGUIKAUGUILMNUHUGBSZCE
|
|
ZUPUNUGUQCUGBABCOPQUPUQUMIZCSZURUNIUOUTCBUOBSUSCUGULBRTUAUQUMCUBUCUDUEUF
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
modc.1 $e |- F/ y ph $.
|
|
$( Equivalent definitions of "there exists at most one," given decidable
|
|
existence. (Contributed by Jim Kingdon, 1-Jul-2018.) $)
|
|
modc $p |- ( DECID E. x ph ->
|
|
( E. y A. x ( ph -> x = y ) <->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
|
|
( wex wdc weq wi wal wsb wa mo23 wn wo exmiddc mor mo2n a1d jaoi syl
|
|
impbid2 ) ABEZFZABCGZHBICEZAABCJKUDHCIBIZABCDLUCUBUBMZNUFUEHZUBOUBUHUGABC
|
|
DPUGUEUFABCDQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu2.1 $e |- F/ y ph $.
|
|
$( An alternate way of defining existential uniqueness. Definition 6.10 of
|
|
[TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) $)
|
|
eu2 $p |- ( E! x ph <->
|
|
( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
|
|
( weu wex wsb wa weq wal euex nfri eumo0 mo23 syl jca 19.29r impexp albii
|
|
wi 19.21 bitri anbi2i abai bitr4i exbii sylib eu1 sylibr impbii ) ABEZABF
|
|
ZAABCGZHBCIZTZCJZBJZHZUKULUQABKUKAUNTBJCFUQABCACDLZMABCDNOPURAUMUNTZCJZHZ
|
|
BFZUKURAUPHZBFVCAUPBQVDVBBVDAAVATZHVBUPVEAUPAUTTZCJVEUOVFCAUMUNRSAUTCDUAU
|
|
BUCAVAUDUEUFUGABCUSUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu3h.1 $e |- ( ph -> A. y ph ) $.
|
|
$( An alternate way to express existential uniqueness. (Contributed by NM,
|
|
8-Jul-1994.) (New usage is discouraged.) $)
|
|
eu3h $p |- ( E! x ph <->
|
|
( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
|
|
( weu wex weq wi wal euex eumo0 jca wsb nfi mo23 anim2i eu2 sylibr impbii
|
|
wa ) ABEZABFZABCGZHBICFZTZUAUBUDABJABCDKLUEUBAABCMTUCHCIBIZTUAUDUFUBABCAC
|
|
DNZOPABCUGQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu3.1 $e |- F/ y ph $.
|
|
$( An alternate way to express existential uniqueness. (Contributed by NM,
|
|
8-Jul-1994.) $)
|
|
eu3 $p |- ( E! x ph <->
|
|
( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
|
|
( nfri eu3h ) ABCACDEF $.
|
|
$}
|
|
|
|
$( Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)
|
|
(Proof rewritten by Jim Kingdon, 27-May-2018.) $)
|
|
eu5 $p |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $=
|
|
( weu wex wmo wa euex eumo jca wi df-mo biimpi imp ancoms impbii ) ABCZABDZ
|
|
ABEZFPQRABGABHIRQPRQPRQPJABKLMNO $.
|
|
|
|
$( Existence implies "at most one" is equivalent to uniqueness. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
exmoeu2 $p |- ( E. x ph -> ( E* x ph <-> E! x ph ) ) $=
|
|
( weu wex wmo eu5 baibr ) ABCABDABEABFG $.
|
|
|
|
$( Absorption of existence condition by "at most one." (Contributed by NM,
|
|
4-Nov-2002.) $)
|
|
moabs $p |- ( E* x ph <-> ( E. x ph -> E* x ph ) ) $=
|
|
( wex weu wi wmo pm5.4 df-mo imbi2i 3bitr4ri ) ABCZKABDZEZEMKABFZENKLGNMKAB
|
|
HZIOJ $.
|
|
|
|
$( If existence is decidable, something exists or at most one exists.
|
|
(Contributed by Jim Kingdon, 30-Jun-2018.) $)
|
|
exmodc $p |- ( DECID E. x ph -> ( E. x ph \/ E* x ph ) ) $=
|
|
( wex wdc wn wo wmo df-dc weu wi pm2.21 df-mo sylibr orim2i sylbi ) ABCZDPP
|
|
EZFPABGZFPHQRPQPABIZJRPSKABLMNO $.
|
|
|
|
$( There is at most one of something which does not exist. Unlike ~ exmodc
|
|
there is no decidability condition. (Contributed by Jim Kingdon,
|
|
22-Sep-2018.) $)
|
|
exmonim $p |- ( -. E. x ph -> E* x ph ) $=
|
|
( wex wn weu wi wmo pm2.21 df-mo sylibr ) ABCZDKABEZFABGKLHABIJ $.
|
|
|
|
${
|
|
$d x y $.
|
|
mo2r.1 $e |- F/ y ph $.
|
|
$( A condition which implies "at most one." (Contributed by Jim Kingdon,
|
|
2-Jul-2018.) $)
|
|
mo2r $p |- ( E. y A. x ( ph -> x = y ) -> E* x ph ) $=
|
|
( weq wi wal wex weu wmo nfri eu3h simplbi2com df-mo sylibr ) ABCEFBGCHZA
|
|
BHZABIZFABJRQPABCACDKLMABNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mo3h.1 $e |- ( ph -> A. y ph ) $.
|
|
$( Alternate definition of "at most one." Definition of [BellMachover]
|
|
p. 460, except that definition has the side condition that ` y ` not
|
|
occur in ` ph ` in place of our hypothesis. (Contributed by NM,
|
|
8-Mar-1995.) (New usage is discouraged.) $)
|
|
mo3h $p |- ( E* x ph <->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( wmo wsb wa weq wi wal wex weu nfi eu2 imbi2i df-mo anclb 3bitr4i sylibr
|
|
19.38 19.21 albii anabs5 pm3.31 syl5bir 2alimi sylbi simplbi2com impbii
|
|
syl ) ABEZAABCFZGZBCHZIZCJZBJZUKABKZUQIZUQURABLZIZURURUQGZIUKUSUTVBURABCA
|
|
CDMZNZOABPZURUQQRUSAUOIZCJZBJZUQUSAUPIZBJVHAUPBTVGVIBAUOCVCUAUBSVFUOBCUMA
|
|
UMGVFUNAULUCAUMUNUDUEUFUJUGUQVAUKUTURUQVDUHVESUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mo3.1 $e |- F/ y ph $.
|
|
$( Alternate definition of "at most one." Definition of [BellMachover]
|
|
p. 460, except that definition has the side condition that ` y ` not
|
|
occur in ` ph ` in place of our hypothesis. (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
mo3 $p |- ( E* x ph <->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( nfri mo3h ) ABCACDEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mo2dc.1 $e |- F/ y ph $.
|
|
$( Alternate definition of "at most one" where existence is decidable.
|
|
(Contributed by Jim Kingdon, 2-Jul-2018.) $)
|
|
mo2dc $p |- ( DECID E. x ph ->
|
|
( E* x ph <-> E. y A. x ( ph -> x = y ) ) ) $=
|
|
( wex wdc weq wi wal wsb wa wmo modc nfri mo3h syl6rbbr ) ABEFABCGZHBICEA
|
|
ABCJKQHCIBIABLABCDMABCACDNOP $.
|
|
$}
|
|
|
|
${
|
|
euan.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Introduction of a conjunct into uniqueness quantifier. (Contributed by
|
|
NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
euan $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
|
|
( wa weu wex wmo simpl exlimih adantr simpr eximi hbe1 a1d impbid2 mobidh
|
|
ancrd biimpa eu5 jca32 anbi2i 3imtr4i ibar eubidh impbii ) ABEZCFZABCFZEZ
|
|
UGCGZUGCHZEZABCGZBCHZEZEUHUJUMAUNUOUKAULUGACDABIJZKUKUNULUGBCABLZMKUKULUO
|
|
UKUGBCUGCNUKUGBURUKBAUKABUQORPQSUAUGCTUIUPABCTUBUCAUIUHABUGCDABUDUESUF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduction of a conjunct into uniqueness quantifier. (Contributed by
|
|
NM, 23-Mar-1995.) $)
|
|
euanv $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
|
|
( ax-17 euan ) ABCACDE $.
|
|
$}
|
|
|
|
$( Introduce or eliminate a disjunct in a uniqueness quantifier.
|
|
(Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
euor2 $p |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) ) $=
|
|
( wex wn wo hbe1 hbn wb 19.8a con3i orel1 olc impbid1 syl eubidh ) ACDZEZAB
|
|
FZBCQCACGHRAEZSBIAQACJKTSBABLBAMNOP $.
|
|
|
|
${
|
|
$d w x z $. $d w y z $. $d w ph $.
|
|
$( Substitution into "at most one". (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
sbmo $p |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph ) $=
|
|
( vw wsb wa cv wceq wal wmo nfv sblim sban imbi1i sbcom2 anbi2i sbalv mo3
|
|
wi 3bitri sbbii 3bitr4i ) AADEFZGZDHEHIZTZEJZDJZBCFABCFZUJDEFZGZUFTZEJZDJ
|
|
ADKZBCFUJDKUHUNBCDUGUMBCEUGBCFUEBCFZUFTUJUDBCFZGZUFTUMUEUFBCUFBLMUPURUFAU
|
|
DBCNOURULUFUQUKUJADEBCPQOUARRUOUIBCADEAELSUBUJDEUJELSUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
mo4f.1 $e |- F/ x ps $.
|
|
mo4f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( "At most one" expressed using implicit substitution. (Contributed by
|
|
NM, 10-Apr-2004.) $)
|
|
mo4f $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( wmo wsb wa weq wi wal ax-17 mo3h sbie anbi2i imbi1i 2albii bitri ) ACGA
|
|
ACDHZIZCDJZKZDLCLABIZUBKZDLCLACDADMNUCUECDUAUDUBTBAABCDEFOPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d x ps $.
|
|
mo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( "At most one" expressed using implicit substitution. (Contributed by
|
|
NM, 26-Jul-1995.) $)
|
|
mo4 $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( nfv mo4f ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d x ps $.
|
|
eu4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Uniqueness using implicit substitution. (Contributed by NM,
|
|
26-Jul-1995.) $)
|
|
eu4 $p |- ( E! x ph <-> ( E. x ph /\
|
|
A. x A. y ( ( ph /\ ps ) -> x = y ) ) ) $=
|
|
( weu wex wmo wa weq wi wal eu5 mo4 anbi2i bitri ) ACFACGZACHZIQABICDJKDL
|
|
CLZIACMRSQABCDENOP $.
|
|
$}
|
|
|
|
$( Existence in terms of "at most one" and uniqueness. (Contributed by Jim
|
|
Kingdon, 3-Jul-2018.) $)
|
|
exmoeudc $p |- ( DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) ) $=
|
|
( wex wdc wmo weu wi df-mo com12 biimpri euex imim12i peircedc syl5 impbid2
|
|
biimpi ) ABCZDZQABEZABFZGZSQTSQTGZABHZPIUAUBQGRQUBSTQSUBUCJABKLQTMNO $.
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
$( "At most one" is preserved through implication (notice wff reversal).
|
|
(Contributed by NM, 22-Apr-1995.) $)
|
|
moim $p |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) ) $=
|
|
( vy wi wal wsb wa weq nfa1 ax-4 spsbim anim12d imim1d alimdv alimd ax-17
|
|
wmo mo3h 3imtr4g ) ABEZCFZBBCDGZHZCDIZEZDFZCFAACDGZHZUEEZDFZCFBCRACRUBUGU
|
|
KCUACJUBUFUJDUBUIUDUEUBABUHUCUACKABCDLMNOPBCDBDQSACDADQST $.
|
|
$}
|
|
|
|
${
|
|
moimi.1 $e |- ( ph -> ps ) $.
|
|
$( "At most one" is preserved through implication (notice wff reversal).
|
|
(Contributed by NM, 15-Feb-2006.) $)
|
|
moimi $p |- ( E* x ps -> E* x ph ) $=
|
|
( wi wmo moim mpg ) ABEBCFACFECABCGDH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x y ph $. $d y ps $.
|
|
$( Move antecedent outside of "at most one." (Contributed by NM,
|
|
28-Jul-1995.) $)
|
|
moimv $p |- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) ) $=
|
|
( vy wi wmo wsb weq wal ax-1 a1i sbimi nfv sbf sbim 3imtr3i anim12d ax-17
|
|
wa mo3h imim1d 2alimdv 3imtr4g com12 ) AABEZCFZBCFZAUEUECDGZSZCDHZEZDICIB
|
|
BCDGZSZUJEZDICIUFUGAUKUNCDAUMUIUJABUEULUHBUEEZABAJKZACDGUOCDGAULUHEAUOCDU
|
|
PLACDACMNBUECDOPQUAUBUECDUEDRTBCDBDRTUCUD $.
|
|
$}
|
|
|
|
$( Uniqueness implies "at most one" through implication. (Contributed by NM,
|
|
22-Apr-1995.) $)
|
|
euimmo $p |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) ) $=
|
|
( weu wmo wi wal eumo moim syl5 ) BCDBCEABFCGACEBCHABCIJ $.
|
|
|
|
$( Add existential uniqueness quantifiers to an implication. Note the
|
|
reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
euim $p |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) ) $=
|
|
( wex wi wal wa weu wmo ax-1 euimmo anim12ii eu5 syl6ibr ) ACDZABECFZGBCHZO
|
|
ACIZGACHOQOPROQJABCKLACMN $.
|
|
|
|
$( "At most one" is still the case when a conjunct is added. (Contributed by
|
|
NM, 22-Apr-1995.) $)
|
|
moan $p |- ( E* x ph -> E* x ( ps /\ ph ) ) $=
|
|
( wa simpr moimi ) BADACBAEF $.
|
|
|
|
${
|
|
moani.1 $e |- E* x ph $.
|
|
$( "At most one" is still true when a conjunct is added. (Contributed by
|
|
NM, 9-Mar-1995.) $)
|
|
moani $p |- E* x ( ps /\ ph ) $=
|
|
( wmo wa moan ax-mp ) ACEBAFCEDABCGH $.
|
|
$}
|
|
|
|
$( "At most one" is still the case when a disjunct is removed. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
moor $p |- ( E* x ( ph \/ ps ) -> E* x ph ) $=
|
|
( wo orc moimi ) AABDCABEF $.
|
|
|
|
$( "At most one" imports disjunction to conjunction. (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mooran1 $p |- ( ( E* x ph \/ E* x ps ) -> E* x ( ph /\ ps ) ) $=
|
|
( wmo wa simpl moimi moan jaoi ) ACDABEZCDBCDJACABFGBACHI $.
|
|
|
|
$( "At most one" exports disjunction to conjunction. (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mooran2 $p |- ( E* x ( ph \/ ps ) -> ( E* x ph /\ E* x ps ) ) $=
|
|
( wo wmo moor olc moimi jca ) ABDZCEACEBCEABCFBJCBAGHI $.
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
moanim.1 $e |- F/ x ph $.
|
|
$( Introduction of a conjunct into "at most one" quantifier. (Contributed
|
|
by NM, 3-Dec-2001.) $)
|
|
moanim $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
|
|
( vy wsb wa weq wi wal wmo anandi imbi1i impexp sbf anbi1i ax-17 3bitr4ri
|
|
sban mo3h bitr2i anbi2i 3bitr3i 2albii 19.21 19.21v albii imbi2i ) ABBCEF
|
|
ZGZCEHZIZIZEJZCJZABGZUPCEFZGZUKIZEJCJABCKZIZUPCKUMUSCEAUJGZUKIUPAUIGZGZUK
|
|
IUMUSVBVDUKABUILMAUJUKNVDURUKVCUQUPUQACEFZUIGVCABCESVEAUIACEDOPUAUBMUCUDA
|
|
ULEJZIZCJAVFCJZIUOVAAVFCDUEUNVGCAULEUFUGUTVHABCEBEQTUHRUPCEUPEQTR $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduction of a conjunct into "at most one" quantifier. (Contributed
|
|
by NM, 23-Mar-1995.) $)
|
|
moanimv $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
|
|
( nfv moanim ) ABCACDE $.
|
|
$}
|
|
|
|
$( Nested "at most one" and uniqueness quantifiers. (Contributed by NM,
|
|
25-Jan-2006.) $)
|
|
moaneu $p |- E* x ( ph /\ E! x ph ) $=
|
|
( weu wa wmo wi eumo nfeu1 moanim mpbir ancom mobii ) AABCZDZBEMADZBEZPMABE
|
|
FABGMABABHIJNOBAMKLJ $.
|
|
|
|
$( Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) $)
|
|
moanmo $p |- E* x ( ph /\ E* x ph ) $=
|
|
( wmo wa wi id nfmo1 moanim mpbir ancom mobii ) AABCZDZBCLADZBCZOLLELFLABAB
|
|
GHIMNBALJKI $.
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
$( "At most one" picks a variable value, eliminating an existential
|
|
quantifier. (Contributed by NM, 27-Jan-1997.) $)
|
|
mopick $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
|
|
( vy wa wex wmo wi wsb hbs1 hban weq sbequ12 anbi12d cbvexh wal mo3h ax-4
|
|
ax-17 sylbi sps sbequ2 imim2i expd com4t imp syl5 exlimiv impcom ) ABEZCF
|
|
ZACGZABHZUKACDIZBCDIZEZDFULUMHZUJUPCDUJDSUNUOCACDJBCDJKCDLZAUNBUOACDMBCDM
|
|
NOUPUQDULAUNEZURHZUPUMULUTDPZCPUTACDADSQVAUTCUTDRUATUNUOUTUMHUTAUNUOBUTAU
|
|
NUOBHZURVBUSBCDUBUCUDUEUFUGUHTUI $.
|
|
$}
|
|
|
|
$( Existential uniqueness "picks" a variable value for which another wff is
|
|
true. If there is only one thing ` x ` such that ` ph ` is true, and
|
|
there is also an ` x ` (actually the same one) such that ` ph ` and ` ps `
|
|
are both true, then ` ph ` implies ` ps ` regardless of ` x ` . This
|
|
theorem can be useful for eliminating existential quantifiers in a
|
|
hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192.
|
|
(Contributed by NM, 10-Jul-1994.) $)
|
|
eupick $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
|
|
( weu wmo wa wex wi eumo mopick sylan ) ACDACEABFCGABHACIABCJK $.
|
|
|
|
$( Version of ~ eupick with closed formulas. (Contributed by NM,
|
|
6-Sep-2008.) $)
|
|
eupicka $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) ) $=
|
|
( weu wa wex wi hbeu1 hbe1 hban eupick alrimih ) ACDZABEZCFZEABGCMOCACHNCIJ
|
|
ABCKL $.
|
|
|
|
$( Existential uniqueness "pick" showing wff equivalence. (Contributed by
|
|
NM, 25-Nov-1994.) $)
|
|
eupickb $p |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) ->
|
|
( ph <-> ps ) ) $=
|
|
( weu wa wex w3a wi eupick 3adant2 3simpc pm3.22 eximi anim2i 3syl impbid )
|
|
ACDZBCDZABEZCFZGZABQTABHRABCIJUARTERBAEZCFZEBAHQRTKTUCRSUBCABLMNBACIOP $.
|
|
|
|
$( Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) $)
|
|
eupickbi $p |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) ) $=
|
|
( weu wa wex wi wal eupicka ex hba1 wb ancl simpl impbid1 sps eubidh syl6bi
|
|
euex com12 impbid ) ACDZABEZCFZABGZCHZUBUDUFABCIJUFUBUDUFUBUCCDUDUFAUCCUECK
|
|
UEAUCLCUEAUCABMABNOPQUCCSRTUA $.
|
|
|
|
$( "At most one" can show the existence of a common value. In this case we
|
|
can infer existence of conjunction from a conjunction of existence, and it
|
|
is one way to achieve the converse of ~ 19.40 . (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mopick2 $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) ->
|
|
E. x ( ph /\ ps /\ ch ) ) $=
|
|
( wmo wa wex w3a hbmo1 hbe1 hban mopick anim1d df-3an syl6ibr eximdh 3impia
|
|
ancld ) ADEZABFZDGZACFZDGABCHZDGSUAFZUBUCDSUADADITDJKUDUBTCFUCUDATCUDABABDL
|
|
RMABCNOPQ $.
|
|
|
|
${
|
|
moexexdc.1 $e |- F/ y ph $.
|
|
$( "At most one" double quantification. (Contributed by Jim Kingdon,
|
|
5-Jul-2018.) $)
|
|
moexexdc $p |- ( DECID E. x ph ->
|
|
( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) ) $=
|
|
( wex wdc wn wo wmo wal wa wi df-dc hbmo1 hba1 hbmo hbim exlimih a1d hbe1
|
|
nfri mopick com3r alrimdh moim spsd syl6 hbex exsimpl con3i mon jaoi impd
|
|
ex syl sylbi ) ACFZGURURHZIZACJZBDJZCKZLABLZCFZDJZMURNUTVAVCVFURVAVCVFMZM
|
|
ZUSAVHCVAVGCACOVCVFCVBCPVECDVDCUAQRRAVAVEBMZDKZVGAVAVIDADEUBZADCVKQVAVEAB
|
|
VAVEABMABCUCUOUDUEVJVBVFCVEBDUFUGUHSUSVGVAUSVFVCUSVEDFZHVFVLURVEURDADCVKU
|
|
IABCUJSUKVEDULUPTTUMUNUQ $.
|
|
$}
|
|
|
|
${
|
|
euexex.1 $e |- F/ y ph $.
|
|
$( Existential uniqueness and "at most one" double quantification.
|
|
(Contributed by Jim Kingdon, 28-Dec-2018.) $)
|
|
euexex $p |-
|
|
( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
|
|
( weu wmo wal wa wex wi eu5 nfmo1 nfa1 nfe1 nfmo nfim mopick ex imp com3r
|
|
alrimd moim spsd syl6 exlimi sylbi ) ACFZBDGZCHZABIZCJZDGZUHACJZACGZIUJUM
|
|
KZACLUNUOUPAUOUPKCUOUPCACMUJUMCUICNULCDUKCOPQQAUOULBKZDHZUPAUOUQDEADCEPUO
|
|
ULABUOULABKABCRSUAUBURUIUMCULBDUCUDUEUFTUGT $.
|
|
$}
|
|
|
|
$( Double quantification with "at most one." (Contributed by NM,
|
|
3-Dec-2001.) $)
|
|
2moex $p |- ( E* x E. y ph -> A. y E* x ph ) $=
|
|
( wex wmo hbe1 hbmo 19.8a moimi alrimih ) ACDZBEABECKCBACFGAKBACHIJ $.
|
|
|
|
$( Double quantification with existential uniqueness. (Contributed by NM,
|
|
3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
2euex $p |- ( E! x E. y ph -> E. y E! x ph ) $=
|
|
( wex weu wmo wa eu5 excom hbe1 hbmo 19.8a moimi df-mo eximdh syl5bi impcom
|
|
wi sylib sylbi ) ACDZBEUABDZUABFZGABEZCDZUABHUCUBUEUBABDZCDUCUEABCIUCUFUDCU
|
|
ACBACJKUCABFUFUDRAUABACLMABNSOPQT $.
|
|
|
|
$( Double quantification with existential uniqueness and "at most one."
|
|
(Contributed by NM, 3-Dec-2001.) $)
|
|
2eumo $p |- ( E! x E* y ph -> E* x E! y ph ) $=
|
|
( weu wmo wi euimmo eumo mpg ) ACDZACEZFKBDJBEFBJKBGACHI $.
|
|
|
|
$( Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) $)
|
|
2eu2ex $p |- ( E! x E! y ph -> E. x E. y ph ) $=
|
|
( weu wex euex eximi syl ) ACDZBDIBEACEZBEIBFIJBACFGH $.
|
|
|
|
$( A condition allowing swap of "at most one" and existential quantifiers.
|
|
(Contributed by Jim Kingdon, 6-Jul-2018.) $)
|
|
2moswapdc $p |- ( DECID E. x E. y ph -> ( A. x E* y ph ->
|
|
( E* x E. y ph -> E* y E. x ph ) ) ) $=
|
|
( wex wdc wmo wa wi nfe1 moexexdc expcomd 19.8a pm4.71ri exbii mobii imbi2i
|
|
wal syl6ibr ) ACDZBDEZACFBQZSBFZSAGZBDZCFZHUBABDZCFZHTUBUAUESABCACIJKUGUEUB
|
|
UFUDCAUCBASACLMNOPR $.
|
|
|
|
$( A condition allowing swap of uniqueness and existential quantifiers.
|
|
(Contributed by Jim Kingdon, 7-Jul-2018.) $)
|
|
2euswapdc $p |- ( DECID E. x E. y ph -> ( A. x E* y ph ->
|
|
( E! x E. y ph -> E! y E. x ph ) ) ) $=
|
|
( wex wdc wmo wal weu wi wa excomim a1i 2moswapdc imp anim12d eu5 3imtr4g
|
|
ex ) ACDZBDZEZACFBGZSBHZABDZCHZIUAUBJZTSBFZJUDCDZUDCFZJUCUEUFTUHUGUITUHIUFA
|
|
BCKLUAUBUGUIIABCMNOSBPUDCPQR $.
|
|
|
|
$( Double existential uniqueness implies double uniqueness quantification.
|
|
(Contributed by NM, 3-Dec-2001.) $)
|
|
2exeu $p |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) $=
|
|
( wex wmo weu excom hbe1 hbmo 19.41h 19.8a moimi anim2i eximi sylbir sylanb
|
|
wa simpl eu5 anbi12i adantl anim12i ancoms exbii mobii bitri 3imtr4i ) ACDZ
|
|
BDZUHBEZQZABDZCDZULCEZQZQUHACEZQZBDZUQBEZQZUHBFZULCFZQACFZBFZUOUKUTUOURUKUS
|
|
UMUIUNURACBGUIUNQUHUNQZBDURUHUNBULBCABHIJVEUQBUNUPUHAULCABKLMNOPUJUSUIUQUHB
|
|
UHUPRLUAUBUCVAUKVBUOUHBSULCSTVDVCBDZVCBEZQUTVCBSVFURVGUSVCUQBACSZUDVCUQBVHU
|
|
ETUFUG $.
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( This theorem provides us with a definition of double existential
|
|
uniqueness ("exactly one ` x ` and exactly one ` y ` "). Naively one
|
|
might think (incorrectly) that it could be defined by ` E! x E! y ph ` .
|
|
See ~ 2exeu for a one-way implication. (Contributed by NM,
|
|
3-Dec-2001.) $)
|
|
2eu4 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
|
|
( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) ) $=
|
|
( wex weu wa weq wi wal ax-17 eu3h anbi12i anbi2i bitri 19.3h 19.26 albii
|
|
hba1 excom anidm jcab 3bitr4ri alcom bitr4i 19.23v 2albii hbe1 hbim aaanh
|
|
an4 3bitri 2exbii eeanv bitr2i ) ACFZBGZABFZCGZHUQBFZUQBDIZJZBKZDFZHZUSCF
|
|
ZUSCEIZJZCKZEFZHZHVAVGHZVEVKHZHVAAVBVHHJZCKZBKZEFDFZHURVFUTVLUQBDUQDLMUSC
|
|
EUSELMNVAVEVGVKULVMVAVNVRVMVAVAHVAVGVAVAACBUAOVAUBPVRVDVJHZEFDFVNVQVSDEVQ
|
|
AVBJZCKZAVHJZBKZHZCKZBKZVCVIHZCKBKVSVQWAWBCKZBKZHZBKZWFWABKZWIBKZHWLWIHZW
|
|
KVQWMWIWLWIBWHBTQOWAWIBRVQWAWHHZBKWNVPWOBVPVTWBHZCKWOVOWPCAVBVHUCSVTWBCRP
|
|
SWAWHBRPUDWEWJBWEWACKZWCCKZHWJWAWCCRWQWAWRWIWACVTCTQWBCBUENPSUFWDWGBCWAVC
|
|
WCVIAVBCUGAVHBUGNUHVCVIBCUQVBCACUIVBCLUJUSVHBABUIVHBLUJUKUMUNVDVJDEUOUPNU
|
|
M $.
|
|
$}
|
|
|
|
$( Two equivalent expressions for double existential uniqueness.
|
|
(Contributed by NM, 19-Feb-2005.) $)
|
|
2eu7 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
|
|
E! x E! y ( E. x ph /\ E. y ph ) ) $=
|
|
( wex weu wa hbe1 hbeu euan ancom eubii 3bitri 3bitr4ri ) ABDZCEZACDZFZBEOP
|
|
BEZFNPFZCEZBEROFOPBNBCABGHITQBTPNFZCEPOFQSUACNPJKPNCACGIPOJLKROJM $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Equality has existential uniqueness. (Contributed by Stefan Allan,
|
|
4-Dec-2008.) $)
|
|
euequ1 $p |- E! x x = y $=
|
|
( vz weq weu wex wa wi wal a9e equtr2 gen2 equequ1 eu4 mpbir2an ) ABDZAEP
|
|
AFPCBDZGACDHZCIAIABJRACACBKLPQACACBMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Two ways to express "only one thing exists." The left-hand side
|
|
requires only one variable to express this. Both sides are false in set
|
|
theory. (Contributed by NM, 5-Apr-2004.) $)
|
|
exists1 $p |- ( E! x x = x <-> A. x x = y ) $=
|
|
( weq weu wb wal wex df-eu equid tbt bicom bitri albii exbii hbae 3bitr2i
|
|
19.9h ) AACZADRABCZEZAFZBGSAFZBGUBRABHUBUABSTASSRETRSAIJSRKLMNUBBABBOQP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A condition implying that at least two things exist. (Contributed by
|
|
NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
exists2 $p |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x ) $=
|
|
( vy wex wn cv wceq weu wal hbeu1 hba1 wi exists1 ax16 sylbi com12 alexim
|
|
exlimdh syl6 con2d imp ) ABDZAEBDZBFZUDGZBHZEUBUFUCUBUFABIZUCEUFUBUGUFAUG
|
|
BUEBJABKUFUDCFGBIAUGLBCMABCNORPABQSTUA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Aristotelian logic: Assertic syllogisms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Model the Aristotelian assertic syllogisms using modern notation.
|
|
This section shows that the Aristotelian assertic syllogisms can be proven
|
|
with our axioms of logic, and also provides generally useful theorems.
|
|
|
|
In antiquity Aristotelian logic and Stoic logic
|
|
(see ~ mptnan ) were the leading logical systems.
|
|
Aristotelian logic became the leading system in medieval Europe;
|
|
this section models this system (including later refinements to it).
|
|
Aristotle defined syllogisms very generally
|
|
("a discourse in which certain (specific) things having been supposed,
|
|
something different from the things supposed results of necessity
|
|
because these things are so")
|
|
Aristotle, _Prior Analytics_ 24b18-20.
|
|
However, in _Prior Analytics_ he limits himself to
|
|
categorical syllogisms that consist of three categorical propositions
|
|
with specific structures. The syllogisms are the valid subset of
|
|
the possible combinations of these structures.
|
|
The medieval schools used vowels to identify the types of terms
|
|
(a=all, e=none, i=some, and o=some are not), and named the different
|
|
syllogisms with Latin words that had the vowels in the intended order.
|
|
|
|
"There is a surprising amount of scholarly debate
|
|
about how best to formalize Aristotle's syllogisms..." according to
|
|
_Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic_,
|
|
Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28.
|
|
For example, Lukasiewicz believes it is important to note that
|
|
"Aristotle does not introduce singular terms or premisses into his system".
|
|
Lukasiewicz also believes that Aristotelian syllogisms are
|
|
predicates (having a true/false value), not inference rules:
|
|
"The characteristic sign of an inference is the word 'therefore'...
|
|
no syllogism is formulated by Aristotle primarily as an inference,
|
|
but they are all implications."
|
|
Jan Lukasiewicz, _Aristotle's Syllogistic from the Standpoint of
|
|
Modern Formal Logic_, Second edition, Oxford, 1957, page 1-2.
|
|
Lukasiewicz devised a specialized prefix notation for representing
|
|
Aristotelian syllogisms instead of using standard predicate logic notation.
|
|
|
|
We instead translate each Aristotelian syllogism into an inference rule,
|
|
and each rule is defined using standard predicate logic notation and
|
|
predicates. The predicates are represented by wff variables
|
|
that may depend on the quantified variable ` x ` .
|
|
Our translation is essentially identical to the one
|
|
use in Rini page 18, Table 2 "Non-Modal Syllogisms in
|
|
Lower Predicate Calculus (LPC)", which uses
|
|
standard predicate logic with predicates. Rini states,
|
|
"the crucial point is that we capture the meaning Aristotle intends,
|
|
and the method by which we represent that meaning is less important."
|
|
There are two differences: we make the existence criteria explicit, and
|
|
we use ` ph ` , ` ps ` , and ` ch ` in the order they appear
|
|
(a common Metamath convention).
|
|
Patzig also uses standard predicate logic notation and predicates
|
|
(though he interprets them as conditional propositions, not as
|
|
inference rules); see
|
|
Gunther Patzig, _Aristotle's Theory of the Syllogism_ second edition, 1963,
|
|
English translation by Jonathan Barnes, 1968, page 38.
|
|
Terms such as "all" and "some" are translated into predicate logic
|
|
using the aproach devised by Frege and Russell.
|
|
"Frege (and Russell) devised an ingenious procedure for regimenting
|
|
binary quantifiers like "every" and "some" in terms of unary quantifiers
|
|
like "everything" and "something": they formalized sentences of the form
|
|
"Some A is B" and "Every A is B" as
|
|
exists x (Ax and Bx) and all x (Ax implies Bx), respectively."
|
|
"Quantifiers and Quantification", _Stanford Encyclopedia of Philosophy_,
|
|
~ http://plato.stanford.edu/entries/quantification/ .
|
|
See _Principia Mathematica_ page 22 and *10 for more information
|
|
(especially *10.3 and *10.26).
|
|
|
|
Expressions of the form "no ` ph ` is ` ps ` " are consistently translated as
|
|
` A. x ( ph -> -. ps ) ` . These can also be expressed as
|
|
` -. E. x ( ph /\ ps ) ` , per ~ alinexa .
|
|
We translate "all ` ph ` is ` ps ` " to ` A. x ( ph -> ps ) ` ,
|
|
"some ` ph ` is ` ps ` " to ` E. x ( ph /\ ps ) ` , and
|
|
"some ` ph ` is not ` ps ` " to ` E. x ( ph /\ -. ps ) ` .
|
|
It is traditional to use the singular verb "is", not the plural
|
|
verb "are", in the generic expressions.
|
|
By convention the major premise is listed first.
|
|
|
|
In traditional Aristotelian syllogisms the predicates
|
|
have a restricted form ("x is a ..."); those predicates
|
|
could be modeled in modern notation by constructs such as
|
|
` x = A ` , ` x e. A ` , or ` x C_ A ` .
|
|
Here we use wff variables instead of specialized restricted forms.
|
|
This generalization makes the syllogisms more useful
|
|
in more circumstances. In addition, these expressions make
|
|
it clearer that the syllogisms of Aristolean logic are the
|
|
forerunners of predicate calculus. If we used restricted forms
|
|
like ` x e. A ` instead, we would not only unnecessarily limit
|
|
their use, but we would also need to use set and class axioms,
|
|
making their relationship to predicate calculus less clear.
|
|
|
|
There are some widespread misconceptions about the existential
|
|
assumptions made by Aristotle (aka "existential import").
|
|
Aristotle was not trying to develop something exactly corresponding
|
|
to modern logic. Aristotle devised "a companion-logic for science.
|
|
He relegates fictions like fairy godmothers and mermaids and unicorns to
|
|
the realms of poetry and literature. In his mind, they exist outside the
|
|
ambit of science. This is why he leaves no room for such non-existent
|
|
entities in his logic. This is a thoughtful choice, not an inadvertent
|
|
omission. Technically, Aristotelian science is a search for definitions,
|
|
where a definition is "a phrase signifying a thing's essence."
|
|
(Topics, I.5.102a37, Pickard-Cambridge.)...
|
|
Because non-existent entities cannot be anything, they do not, in
|
|
Aristotle's mind, possess an essence... This is why he leaves
|
|
no place for fictional entities like goat-stags (or unicorns)."
|
|
Source: Louis F. Groarke, "Aristotle: Logic",
|
|
section 7. (Existential Assumptions),
|
|
_Internet Encyclopedia of Philosophy_ (A Peer-Reviewed Academic Resource),
|
|
~ http://www.iep.utm.edu/aris-log/ .
|
|
Thus, some syllogisms have "extra" existence
|
|
hypotheses that do not directly appear in Aristotle's original materials
|
|
(since they were always assumed); they are added where they are needed.
|
|
This affects ~ barbari , ~ celaront , ~ cesaro , ~ camestros , ~ felapton ,
|
|
~ darapti , ~ calemos , ~ fesapo , and ~ bamalip .
|
|
|
|
These are only the _assertic_ syllogisms.
|
|
Aristotle also defined modal syllogisms that deal with modal
|
|
qualifiers such as "necessarily" and "possibly".
|
|
Historically Aristotelian modal syllogisms were not as widely used.
|
|
For more about modal syllogisms in a modern context, see Rini as well as
|
|
_Aristotle's Modal Syllogistic_ by Marko Malink, Harvard
|
|
University Press, November 2013. We do not treat them further here.
|
|
|
|
Aristotelean logic is essentially the forerunner of predicate calculus
|
|
(as well as set theory since it discusses membership in groups),
|
|
while Stoic logic is essentially the forerunner of propositional calculus.
|
|
$)
|
|
|
|
$( Figure 1. Aristotelian syllogisms are grouped by "figures",
|
|
which doesn't matter for our purposes but is a reasonable way
|
|
to order them. $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Barbara", e.g.,
|
|
"All men are mortal". By convention, the major premise is first. $)
|
|
barbara.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Barbara, e.g., "Socrates is a man". $)
|
|
barbara.min $e |- A. x ( ch -> ph ) $.
|
|
$( "Barbara", one of the fundamental syllogisms of Aristotelian logic. All
|
|
` ph ` is ` ps ` , and all ` ch ` is ` ph ` , therefore all ` ch ` is
|
|
` ps ` . (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.)
|
|
For example, given "All men are mortal" and "Socrates is a man", we can
|
|
prove "Socrates is mortal". If H is the set of men, M is the set of
|
|
mortal beings, and S is Socrates, these word phrases can be represented
|
|
as ` A. x ( x e. H -> x e. M ) ` (all men are mortal) and
|
|
` A. x ( x = S -> x e. H ) ` (Socrates is a man) therefore
|
|
` A. x ( x = S -> x e. M ) ` (Socrates is mortal). Russell and
|
|
Whitehead note that the "syllogism in Barbara is derived..." from
|
|
~ syl . (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most
|
|
of the proof is in ~ alsyl . There are a legion of sources for Barbara,
|
|
including ~ http://www.friesian.com/aristotl.htm ,
|
|
~ http://plato.stanford.edu/entries/aristotle-logic/ , and
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) $)
|
|
barbara $p |- A. x ( ch -> ps ) $=
|
|
( wi wal alsyl mp2an ) CAGDHABGDHCBGDHFECABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Celarent", e.g.,
|
|
"No reptiles have fur". $)
|
|
celarent.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Celarent, e.g., "All snakes are reptiles". $)
|
|
celarent.min $e |- A. x ( ch -> ph ) $.
|
|
$( "Celarent", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and all ` ch ` is ` ph ` , therefore no ` ch ` is ` ps ` . (In
|
|
Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example,
|
|
given the "No reptiles have fur" and "All snakes are reptiles",
|
|
therefore "No snakes have fur". Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
celarent $p |- A. x ( ch -> -. ps ) $=
|
|
( wn barbara ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Darii", e.g.,
|
|
"All rabbits have fur". $)
|
|
darii.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Darii, e.g., "Some pets are rabbits." $)
|
|
darii.min $e |- E. x ( ch /\ ph ) $.
|
|
$( "Darii", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For
|
|
example, given "All rabbits have fur" and "Some pets are rabbits",
|
|
therefore "Some pets have fur". Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) $)
|
|
darii $p |- E. x ( ch /\ ps ) $=
|
|
( wa wi spi anim2i eximii ) CAGCBGDFABCABHDEIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Ferio" ("Ferioque"),
|
|
e.g., "No homework is fun". $)
|
|
ferio.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Ferio, e.g., "Some reading is homework." $)
|
|
ferio.min $e |- E. x ( ch /\ ph ) $.
|
|
$( "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No
|
|
` ph ` is ` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is
|
|
not ` ps ` . (In Aristotelian notation, EIO-1: MeP and SiM therefore
|
|
SoP.) For example, given "No homework is fun" and "Some reading is
|
|
homework", therefore "Some reading is not fun". This is essentially a
|
|
logical axiom in Aristotelian logic. Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
ferio $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn darii ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Barbari", e.g.,
|
|
e.g., "All men are mortal". $)
|
|
barbari.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Barbari, e.g., "All Greeks are men." $)
|
|
barbari.min $e |- A. x ( ch -> ph ) $.
|
|
$( Existence premise for Barbari, e.g., "Greeks exist." $)
|
|
barbari.e $e |- E. x ch $.
|
|
$( "Barbari", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
|
|
` ch ` is ` ps ` . (In Aristotelian notation, AAI-1: MaP and SaM
|
|
therefore SiP.) For example, given "All men are mortal", "All Greeks are
|
|
men", and "Greeks exist", therefore "Some Greeks are mortal". Note the
|
|
existence hypothesis (to prove the "some" in the conclusion). Example
|
|
from ~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David
|
|
A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler,
|
|
30-Aug-2016.) $)
|
|
barbari $p |- E. x ( ch /\ ps ) $=
|
|
( wa wi barbara spi ancli eximii ) CCBHDGCBCBIDABCDEFJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Celaront", e.g.,
|
|
e.g., "No reptiles have fur". $)
|
|
celaront.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Celaront, e.g., "All Snakes are reptiles." $)
|
|
celaront.min $e |- A. x ( ch -> ph ) $.
|
|
$( Existence premise for Celaront, e.g., "Snakes exist." $)
|
|
celaront.e $e |- E. x ch $.
|
|
$( "Celaront", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
|
|
` ch ` is not ` ps ` . (In Aristotelian notation, EAO-1: MeP and SaM
|
|
therefore SoP.) For example, given "No reptiles have fur", "All snakes
|
|
are reptiles.", and "Snakes exist.", prove "Some snakes have no fur".
|
|
Note the existence hypothesis. Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
celaront $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn barbari ) ABHCDEFGI $.
|
|
$}
|
|
|
|
$( Figure 2 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Cesare" $)
|
|
cesare.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Cesare $)
|
|
cesare.min $e |- A. x ( ch -> ps ) $.
|
|
$( "Cesare", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and all ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to
|
|
~ celarent . (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised
|
|
by David A. Wheeler, 13-Nov-2016.) $)
|
|
cesare $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi nsyl3 ax-gen ) CAGHDABCABGHDEICBHDFIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Camestres" $)
|
|
camestres.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Camestres $)
|
|
camestres.min $e |- A. x ( ch -> -. ps ) $.
|
|
$( "Camestres", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and no ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed
|
|
by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
camestres $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi nsyl ax-gen ) CAGHDCBACBGHDFIABHDEIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Festino" $)
|
|
festino.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Festino $)
|
|
festino.min $e |- E. x ( ch /\ ps ) $.
|
|
$( "Festino", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and some ` ch ` is ` ps ` , therefore some ` ch ` is not
|
|
` ph ` . (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.)
|
|
(Contributed by David A. Wheeler, 25-Nov-2016.) $)
|
|
festino $p |- E. x ( ch /\ -. ph ) $=
|
|
( wa wn wi spi con2i anim2i eximii ) CBGCAHZGDFBNCABABHIDEJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Baroco" $)
|
|
baroco.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Baroco $)
|
|
baroco.min $e |- E. x ( ch /\ -. ps ) $.
|
|
$( "Baroco", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ch ` is not ` ps ` , therefore some ` ch ` is not
|
|
` ph ` . (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.)
|
|
For example, "All informative things are useful", "Some websites are not
|
|
useful", therefore "Some websites are not informative." (Contributed by
|
|
David A. Wheeler, 28-Aug-2016.) $)
|
|
baroco $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wi spi con3i anim2i eximii ) CBGZHCAGZHDFNOCABABIDEJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Cesaro" $)
|
|
cesaro.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Cesaro $)
|
|
cesaro.min $e |- A. x ( ch -> ps ) $.
|
|
$( Existence premise for Cesaro $)
|
|
cesaro.e $e |- E. x ch $.
|
|
$( "Cesaro", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, EAO-2: PeM and SaM
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
cesaro $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wi spi nsyl3 ancli eximii ) CCAHZIDGCOABCABHJDEKCBJDFKLMN $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Camestros" $)
|
|
camestros.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
camestros.min $e |- A. x ( ch -> -. ps ) $.
|
|
$( Existence premise for Camestros $)
|
|
camestros.e $e |- E. x ch $.
|
|
$( "Camestros", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , no ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, AEO-2: PaM and SeM
|
|
therefore SoP.) For example, "All horses have hooves", "No humans have
|
|
hooves", and humans exist, therefore "Some humans are not horses".
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A.
|
|
Wheeler, 2-Sep-2016.) $)
|
|
camestros $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wi spi nsyl ancli eximii ) CCAHZIDGCOCBACBHJDFKABJDEKLMN $.
|
|
$}
|
|
|
|
$( Figure 3 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Datisi" $)
|
|
datisi.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
datisi.min $e |- E. x ( ph /\ ch ) $.
|
|
$( "Datisi", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, AII-3: MaP and MiS therefore SiP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
datisi $p |- E. x ( ch /\ ps ) $=
|
|
( wa simpr wi spi adantr jca eximii ) ACGZCBGDFNCBACHABCABIDEJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Disamis" $)
|
|
disamis.maj $e |- E. x ( ph /\ ps ) $.
|
|
$( Minor premise for $)
|
|
disamis.min $e |- A. x ( ph -> ch ) $.
|
|
$( "Disamis", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
disamis $p |- E. x ( ch /\ ps ) $=
|
|
( wa wi spi anim1i eximii ) ABGCBGDEACBACHDFIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Ferison" $)
|
|
ferison.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
ferison.min $e |- E. x ( ph /\ ch ) $.
|
|
$( "Ferison", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is not
|
|
` ps ` . (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A.
|
|
Wheeler, 2-Sep-2016.) $)
|
|
ferison $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn datisi ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Bocardo" $)
|
|
bocardo.maj $e |- E. x ( ph /\ -. ps ) $.
|
|
$( Minor premise for $)
|
|
bocardo.min $e |- A. x ( ph -> ch ) $.
|
|
$( "Bocardo", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
not ` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is not
|
|
` ps ` . (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.)
|
|
For example, "Some cats have no tails", "All cats are mammals",
|
|
therefore "Some mammals have no tails". A reorder of ~ disamis ; prefer
|
|
using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(New usage is discouraged.) $)
|
|
bocardo $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn disamis ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Felapton" $)
|
|
felapton.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
felapton.min $e |- A. x ( ph -> ch ) $.
|
|
$( Existence premise for Felapton $)
|
|
felapton.e $e |- E. x ph $.
|
|
$( "Felapton", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
|
|
` ch ` is not ` ps ` . (In Aristotelian notation, EAO-3: MeP and MaS
|
|
therefore SoP.) For example, "No flowers are animals" and "All flowers
|
|
are plants", therefore "Some plants are not animals". (Contributed by
|
|
David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
felapton $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn wa wi spi jca eximii ) ACBHZIDGACNACJDFKANJDEKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Darapti" $)
|
|
darapti.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
darapti.min $e |- A. x ( ph -> ch ) $.
|
|
$( Existence premise for Darapti $)
|
|
darapti.e $e |- E. x ph $.
|
|
$( "Darapti", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
|
|
` ch ` is ` ps ` . (In Aristotelian notation, AAI-3: MaP and MaS
|
|
therefore SiP.) For example, "All squares are rectangles" and "All
|
|
squares are rhombuses", therefore "Some rhombuses are rectangles".
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
darapti $p |- E. x ( ch /\ ps ) $=
|
|
( wa wi spi jca eximii ) ACBHDGACBACIDFJABIDEJKL $.
|
|
$}
|
|
|
|
$( Figure 4 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Calemes" $)
|
|
calemes.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
calemes.min $e |- A. x ( ps -> -. ch ) $.
|
|
$( "Calemes", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and no ` ps ` is ` ch ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed
|
|
by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
calemes $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi con2i nsyl ax-gen ) CAGHDCBABCBCGHDFIJABHDEIKL $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Dimatis" $)
|
|
dimatis.maj $e |- E. x ( ph /\ ps ) $.
|
|
$( Minor premise for $)
|
|
dimatis.min $e |- A. x ( ps -> ch ) $.
|
|
$( "Dimatis", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
` ps ` , and all ` ps ` is ` ch ` , therefore some ` ch ` is ` ph ` .
|
|
(In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For
|
|
example, "Some pets are rabbits.", "All rabbits have fur", therefore
|
|
"Some fur bearing animals are pets". Like ~ darii with positions
|
|
interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
dimatis $p |- E. x ( ch /\ ph ) $=
|
|
( wa wi spi adantl simpl jca eximii ) ABGZCAGDENCABCABCHDFIJABKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Fresison" $)
|
|
fresison.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
fresison.min $e |- E. x ( ps /\ ch ) $.
|
|
$( "Fresison", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` (PeM), and some ` ps ` is ` ch ` (MiS), therefore some ` ch ` is
|
|
not ` ph ` (SoP). (In Aristotelian notation, EIO-4: PeM and MiS
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
fresison $p |- E. x ( ch /\ -. ph ) $=
|
|
( wa wn simpr wi spi con2i adantr jca eximii ) BCGZCAHZGDFPCQBCIBQCABABHJ
|
|
DEKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Calemos" $)
|
|
calemos.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
calemos.min $e |- A. x ( ps -> -. ch ) $.
|
|
$( Existence premise for Calemos $)
|
|
calemos.e $e |- E. x ch $.
|
|
$( "Calemos", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` (PaM), no ` ps ` is ` ch ` (MeS), and ` ch ` exist, therefore
|
|
some ` ch ` is not ` ph ` (SoP). (In Aristotelian notation, AEO-4: PaM
|
|
and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
calemos $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wi spi con2i nsyl ancli eximii ) CCAHZIDGCPCBABCBCHJDFKLABJDEKMNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Fesapo" $)
|
|
fesapo.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
fesapo.min $e |- A. x ( ps -> ch ) $.
|
|
$( Existence premise for Fesapo $)
|
|
fesapo.e $e |- E. x ps $.
|
|
$( "Fesapo", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ps ` is ` ch ` , and ` ps ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, EAO-4: PeM and MaS
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
fesapo $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wi spi con2i jca eximii ) BCAHZIDGBCOBCJDFKABABHJDEKLMN $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Bamalip" $)
|
|
bamalip.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
bamalip.min $e |- A. x ( ps -> ch ) $.
|
|
$( Existence premise for Bamalip $)
|
|
bamalip.e $e |- E. x ph $.
|
|
$( "Bamalip", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ps ` is ` ch ` , and ` ph ` exist, therefore some ` ch `
|
|
is ` ph ` . (In Aristotelian notation, AAI-4: PaM and MaS therefore
|
|
SiP.) Like ~ barbari . (Contributed by David A. Wheeler,
|
|
28-Aug-2016.) $)
|
|
bamalip $p |- E. x ( ch /\ ph ) $=
|
|
( wa wi spi syl ancri eximii ) ACAHDGACABCABIDEJBCIDFJKLM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
SET THEORY
|
|
###############################################################################
|
|
|
|
Set theory uses the formalism of propositional and predicate calculus to
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assert properties of arbitrary mathematical objects called "sets." A set can
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be an element of another set, and this relationship is indicated by the
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` e. `
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symbol. Starting with the simplest mathematical object, called the empty
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set, set theory builds up more and more complex structures whose existence
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follows from the axioms, eventually resulting in extremely complicated sets
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that we identify with the real numbers and other familiar mathematical
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objects.
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Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel
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(IZF) system, mostly following the IZF axioms as laid out in [Crosilla].
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Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not
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as easy to formalize in Metamath because the statement of some of its axioms
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uses the notion of "bounded formula". Since Metamath has, purposefully, a
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very weak metalogic, that notion must be developed in the logic itself. This
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is similar to our treatment of substitution ( ~ df-sb ) and our definition of
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the nonfreeness predicate ( ~ df-nf ), whereas substitution and bound and
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free variables are ordinarily defined in the metalogic. The development of
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CZF has begun in BJ's mathbox, see ~ wbd .
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$)
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$(
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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IZF Set Theory - start with the Axiom of Extensionality
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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$)
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Introduce the Axiom of Extensionality
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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${
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$d x y z $.
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$( Axiom of Extensionality. It states that two sets are identical if they
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contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and
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IZF" (with unnecessary quantifiers removed).
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Set theory can also be formulated with a _single_ primitive predicate
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` e. ` on top of traditional predicate calculus _without_ equality. In
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that case the Axiom of Extensionality becomes
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` ( A. w ( w e. x <-> w e. y ) -> ( x e. z -> y e. z ) ) ` , and
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equality ` x = y ` is _defined_ as ` A. w ( w e. x <-> w e. y ) ` . All
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of the usual axioms of equality then become theorems of set theory.
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See, for example, Axiom 1 of [TakeutiZaring] p. 8.
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To use the above "equality-free" version of Extensionality with
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Metamath's logical axioms, we would rewrite ~ ax-8 through ~ ax-16 with
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equality expanded according to the above definition. Some of those
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axioms could be proved from set theory and would be redundant. Not all
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of them are redundant, since our axioms of predicate calculus make
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essential use of equality for the proper substitution that is a
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primitive notion in traditional predicate calculus. A study of such an
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axiomatization would be an interesting project for someone exploring the
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foundations of logic.
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It is important to understand that strictly speaking, all of our set
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theory axioms are really schemes that represent an infinite number of
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actual axioms. This is inherent in the design of Metamath
|
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("metavariable math"), which manipulates only metavariables. For
|
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example, the metavariable ` x ` in ~ ax-ext can represent any actual
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variable _v1_, _v2_, _v3_,... . Distinct variable restrictions ($d)
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|
prevent us from substituting say _v1_ for both ` x ` and ` z ` . This
|
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is in contrast to typical textbook presentations that present actual
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axioms (except for axioms which involve wff metavariables). In
|
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practice, though, the theorems and proofs are essentially the same. The
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$d restrictions make each of the infinite axioms generated by the
|
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~ ax-ext scheme exactly logically equivalent to each other and in
|
|
particular to the actual axiom of the textbook version. (Contributed by
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NM, 5-Aug-1993.) $)
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ax-ext $a |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $.
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$}
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${
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$d z x w $. $d z y w $.
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$( A generalization of the Axiom of Extensionality in which ` x ` and ` y `
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need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof
|
|
shortened by Andrew Salmon, 12-Aug-2011.) $)
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axext3 $p |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $=
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( vw wel wb wal weq wi elequ2 bibi1d albidv equequ1 imbi12d ax-ext chvarv
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) CDEZCBEZFZCGZDBHZICAEZRFZCGZABHZIDADAHZTUDUAUEUFSUCCUFQUBRDACJKLDABMNDB
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COP $.
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$( A bidirectional version of Extensionality. Although this theorem
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"looks" like it is just a definition of equality, it requires the Axiom
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of Extensionality for its proof under our axiomatization. See the
|
|
comments for ~ ax-ext . (Contributed by NM, 14-Nov-2008.) $)
|
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axext4 $p |- ( x = y <-> A. z ( z e. x <-> z e. y ) ) $=
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( weq wel wb wal elequ2 alrimiv axext3 impbii ) ABDZCAECBEFZCGLMCABCHIABC
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JK $.
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$}
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${
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$d x y z $. $d ph z $.
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|
bm1.1.1 $e |- F/ x ph $.
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|
$( Any set defined by a property is the only set defined by that property.
|
|
Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM,
|
|
30-Jun-1994.) $)
|
|
bm1.1 $p |- ( E. x A. y ( y e. x <-> ph ) ->
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E! x A. y ( y e. x <-> ph ) ) $=
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( vz wel wb wal wex wsb wa weq wi weu nfv nfbi nfal elequ2 bibi1d albidv
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sbie 19.26 biantr alimi ax-ext syl sylbir sylan2b gen2 jctr eu2 sylibr )
|
|
CBFZAGZCHZBIZUPUOUOBEJZKBELZMZEHBHZKUOBNUPUTUSBEUQUOCEFZAGZCHZURUOVCBEVBB
|
|
CVAABVABODPQURUNVBCURUMVAABECRSTUAUOVCKUNVBKZCHZURUNVBCUBVEUMVAGZCHURVDVF
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CUMAVAUCUDBECUEUFUGUHUIUJUOBEUOEOUKUL $.
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$}
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Class abstractions (a.k.a. class builders)
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
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|
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$( Declare new constants use in class definition. $)
|
|
$c { $. $( Left brace $)
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|
$c | $. $( Vertical bar $)
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|
$c } $. $( Right brace $)
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|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
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|
@c class @. @( Class variable type @)
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|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
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|
|
|
$( Declare symbols as variables $)
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|
$v ./\ $.
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|
$v .\/ $.
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|
$v .<_ $.
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|
$v .< $.
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|
$v .+ $.
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$v .- $.
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$v .X. $.
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$v ./ $.
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$v .^ $.
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$v .0. $.
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$v .1. $.
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$v .|| $.
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$v .~ $.
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$v ._|_ $.
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$v .+^ $.
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$v .+b $.
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$v .(+) $.
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$v .* $.
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$v .x. $.
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$v .xb $.
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|
$v ., $.
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|
$v .(x) $.
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$v .0b $.
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|
|
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$( Declare variable symbols that will be used to represent classes. Note
|
|
that later on ` R ` , ` S ` , ` F ` and ` G ` denote relations and
|
|
functions, but these letters serve as mnemonics only and in fact behave
|
|
no differently from the variables ` A ` through ` D ` . $)
|
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$v A $.
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$v B $.
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$v C $.
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$v D $.
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$v P $.
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|
$v Q $.
|
|
$v R $.
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$v S $.
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$v T $.
|
|
$v U $.
|
|
|
|
$( Introduce the class builder or class abstraction notation ("the class of
|
|
sets ` x ` such that ` ph ` is true"). Our class variables ` A ` ,
|
|
` B ` , etc. range over class builders (sometimes implicitly). Note that
|
|
a setvar variable can be expressed as a class builder per theorem
|
|
~ cvjust , justifying the assignment of setvar variables to class
|
|
variables via the use of ~ cv . $)
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|
cab $a class { x | ph } $.
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$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( A setvar variable is a class expression. The syntax " ` class x ` " can
|
|
be
|
|
viewed as an abbreviation for " ` class { y | y e. x } ` " (a special case
|
|
of ~ cab ), where ` y ` is distinct from ` x ` . See the discussion under
|
|
the definition of class in [Jech] p. 4. Note that ` { y | y e. x } = x `
|
|
by ~ cvjust . @)
|
|
cv @a class x @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
$( $j primitive 'cv' 'wceq' 'wcel' 'cab'; $)
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|
|
|
$( Let ` A ` be a class variable. $)
|
|
cA $f class A $.
|
|
$( Let ` B ` be a class variable. $)
|
|
cB $f class B $.
|
|
$( Let ` C ` be a class variable. $)
|
|
cC $f class C $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pa $f class .|| $.
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|
|
|
$( Let ` D ` be a class variable. $)
|
|
cD $f class D $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.dv $f class ./ $.
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|
|
|
$( Let ` P ` be a class variable. $)
|
|
cP $f class P $.
|
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|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pl $f class .+ $.
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|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pd $f class .+^ $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pb $f class .+b $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.po $f class .(+) $.
|
|
|
|
$( Let ` Q ` be a class variable. $)
|
|
cQ $f class Q $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.sm $f class .~ $.
|
|
|
|
$( Let ` R ` be a class variable. $)
|
|
cR $f class R $.
|
|
$( Let ` S ` be a class variable. $)
|
|
cS $f class S $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.lt $f class .< $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xb $f class .xb $.
|
|
|
|
$( Let ` T ` be a class variable. $)
|
|
cT $f class T $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.x $f class .x. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xp $f class .X. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xo $f class .(x) $.
|
|
|
|
$( Let ` U ` be a class variable. $)
|
|
cU $f class U $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.1 $f class .1. $.
|
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|
|
$v e $.
|
|
$v f $.
|
|
$v g $.
|
|
$v h $.
|
|
$v i $.
|
|
$v j $.
|
|
$v k $.
|
|
$v m $.
|
|
$v n $.
|
|
$v o $.
|
|
$v E $.
|
|
$v F $.
|
|
$v G $.
|
|
$v H $.
|
|
$v I $.
|
|
$v J $.
|
|
$v K $.
|
|
$v L $.
|
|
$v M $.
|
|
$v N $.
|
|
$v V $.
|
|
$v W $.
|
|
$v X $.
|
|
$v Y $.
|
|
$v Z $.
|
|
$v O $.
|
|
$v s $.
|
|
$v r $.
|
|
$v q $.
|
|
$v p $.
|
|
$v a $.
|
|
$v b $.
|
|
$v c $.
|
|
$v d $.
|
|
$v l $.
|
|
|
|
$( Let ` e ` be an individual variable. $)
|
|
ve $f setvar e $.
|
|
$( Let ` f ` be an individual variable. $)
|
|
vf $f setvar f $.
|
|
$( Let ` g ` be an individual variable. $)
|
|
vg $f setvar g $.
|
|
$( Let ` h ` be an individual variable. $)
|
|
vh $f setvar h $.
|
|
$( Let ` i ` be an individual variable. $)
|
|
vi $f setvar i $.
|
|
$( Let ` j ` be an individual variable. $)
|
|
vj $f setvar j $.
|
|
$( Let ` k ` be an individual variable. $)
|
|
vk $f setvar k $.
|
|
$( Let ` m ` be an individual variable. $)
|
|
vm $f setvar m $.
|
|
$( Let ` n ` be an individual variable. $)
|
|
vn $f setvar n $.
|
|
$( Let ` o ` be an individual variable. $)
|
|
vo $f setvar o $.
|
|
$( Let ` E ` be a class variable. $)
|
|
cE $f class E $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.ex $f class .^ $.
|
|
|
|
$( Let ` F ` be a class variable. $)
|
|
cF $f class F $.
|
|
$( Let ` G ` be a class variable. $)
|
|
cG $f class G $.
|
|
$( Let ` H ` be a class variable. $)
|
|
cH $f class H $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xi $f class ., $.
|
|
|
|
$( Let ` I ` be a class variable. $)
|
|
cI $f class I $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.as $f class .* $.
|
|
|
|
$( Let ` J ` be a class variable. $)
|
|
cJ $f class J $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.or $f class .\/ $.
|
|
|
|
$( Let ` K ` be a class variable. $)
|
|
cK $f class K $.
|
|
$( Let ` L ` be a class variable. $)
|
|
cL $f class L $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.le $f class .<_ $.
|
|
|
|
$( Let ` M ` be a class variable. $)
|
|
cM $f class M $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.an $f class ./\ $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.mi $f class .- $.
|
|
|
|
$( Let ` N ` be a class variable. $)
|
|
cN $f class N $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pe $f class ._|_ $.
|
|
|
|
$( Let ` O ` be a class variable. $)
|
|
cO $f class O $.
|
|
$( Let ` V ` be a class variable. $)
|
|
cV $f class V $.
|
|
$( Let ` W ` be a class variable. $)
|
|
cW $f class W $.
|
|
$( Let ` X ` be a class variable. $)
|
|
cX $f class X $.
|
|
$( Let ` Y ` be a class variable. $)
|
|
cY $f class Y $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.0 $f class .0. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.0b $f class .0b $.
|
|
|
|
$( Let ` Z ` be a class variable. $)
|
|
cZ $f class Z $.
|
|
$( Let ` s ` be an individual variable. $)
|
|
vs $f setvar s $.
|
|
$( Let ` r ` be an individual variable. $)
|
|
vr $f setvar r $.
|
|
$( Let ` q ` be an individual variable. $)
|
|
vq $f setvar q $.
|
|
$( Let ` p ` be an individual variable. $)
|
|
vp $f setvar p $.
|
|
$( Let ` a ` be an individual variable. $)
|
|
va $f setvar a $.
|
|
$( Let ` b ` be an individual variable. $)
|
|
vb $f setvar b $.
|
|
$( Let ` c ` be an individual variable. $)
|
|
vc $f setvar c $.
|
|
$( Let ` d ` be an individual variable. $)
|
|
vd $f setvar d $.
|
|
$( Let ` l ` be an individual variable. $)
|
|
vl $f setvar l $.
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include class equality. @)
|
|
wceq @a wff A = B @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include the membership connective between
|
|
classes. @)
|
|
wcel @a wff A e. B @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Define class abstraction notation (so-called by Quine), also called a
|
|
"class builder" in the literature. ` x ` and ` y ` need not be distinct.
|
|
Definition 2.1 of [Quine] p. 16. Typically, ` ph ` will have ` y ` as a
|
|
free variable, and " ` { y | ph } ` " is read "the class of all sets ` y `
|
|
such that ` ph ( y ) ` is true." We do not define ` { y | ph } ` in
|
|
isolation but only as part of an expression that extends or "overloads"
|
|
the ` e. ` relationship.
|
|
|
|
This is our first use of the ` e. ` symbol to connect classes instead of
|
|
sets. The syntax definition ~ wcel , which extends or "overloads" the
|
|
~ wel definition connecting setvar variables, requires that both sides of
|
|
` e. ` be a class. In ~ df-cleq and ~ df-clel , we introduce a new kind
|
|
of variable (class variable) that can substituted with expressions such as
|
|
` { y | ph } ` . In the present definition, the ` x ` on the left-hand
|
|
side is a setvar variable. Syntax definition ~ cv allows us to substitute
|
|
a setvar variable ` x ` for a class variable: all sets are classes by
|
|
~ cvjust (but not necessarily vice-versa). For a full description of how
|
|
classes are introduced and how to recover the primitive language, see the
|
|
discussion in Quine (and under ~ abeq2 for a quick overview).
|
|
|
|
Because class variables can be substituted with compound expressions and
|
|
setvar variables cannot, it is often useful to convert a theorem
|
|
containing a free setvar variable to a more general version with a class
|
|
variable.
|
|
|
|
This is called the "axiom of class comprehension" by [Levy] p. 338, who
|
|
treats the theory of classes as an extralogical extension to our logic and
|
|
set theory axioms. He calls the construction ` { y | ph } ` a "class
|
|
term".
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-clab $a |- ( x e. { y | ph } <-> [ x / y ] ph ) $.
|
|
|
|
$( Simplification of class abstraction notation when the free and bound
|
|
variables are identical. (Contributed by NM, 5-Aug-1993.) $)
|
|
abid $p |- ( x e. { x | ph } <-> ph ) $=
|
|
( cv cab wcel wsb df-clab sbid bitri ) BCABDEABBFAABBGABHI $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
hbab1 $p |- ( y e. { x | ph } -> A. x y e. { x | ph } ) $=
|
|
( cv cab wcel wsb df-clab hbs1 hbxfrbi ) CDABEFABCGBACBHABCIJ $.
|
|
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfsab1 $p |- F/ x y e. { x | ph } $=
|
|
( cv cab wcel hbab1 nfi ) CDABEFBABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
hbab.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by NM, 1-Mar-1995.) $)
|
|
hbab $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $=
|
|
( cv cab wcel wsb df-clab hbsb hbxfrbi ) DFACGHACDIBADCJACDBEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
nfsab.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfsab $p |- F/ x z e. { y | ph } $=
|
|
( cv cab wcel nfri hbab nfi ) DFACGHBABCDABEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x y z $.
|
|
df-cleq.1 $e |- ( A. x ( x e. y <-> x e. z ) -> y = z ) $.
|
|
$( Define the equality connective between classes. Definition 2.7 of
|
|
[Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4
|
|
provides its justification and methods for eliminating it. Note that
|
|
its elimination will not necessarily result in a single wff in the
|
|
original language but possibly a "scheme" of wffs.
|
|
|
|
This is an example of a somewhat "risky" definition, meaning that it has
|
|
a more complex than usual soundness justification (outside of Metamath),
|
|
because it "overloads" or reuses the existing equality symbol rather
|
|
than introducing a new symbol. This allows us to make statements that
|
|
may not hold for the original symbol. For example, it permits us to
|
|
deduce ` y = z <-> A. x ( x e. y <-> x e. z ) ` , which is not a theorem
|
|
of logic but rather presupposes the Axiom of Extensionality (see theorem
|
|
~ axext4 ). We therefore include this axiom as a hypothesis, so that
|
|
the use of Extensionality is properly indicated.
|
|
|
|
We could avoid this complication by introducing a new symbol, say =_2,
|
|
in place of ` = ` . This would also have the advantage of making
|
|
elimination of the definition straightforward, so that we could
|
|
eliminate Extensionality as a hypothesis. We would then also have the
|
|
advantage of being able to identify in various proofs exactly where
|
|
Extensionality truly comes into play rather than just being an artifact
|
|
of a definition. One of our theorems would then be ` x ` =_2
|
|
` y <-> x = y ` by invoking Extensionality.
|
|
|
|
However, to conform to literature usage, we retain this overloaded
|
|
definition. This also makes some proofs shorter and probably easier to
|
|
read, without the constant switching between two kinds of equality.
|
|
|
|
See also comments under ~ df-clab , ~ df-clel , and ~ abeq2 .
|
|
|
|
In the form of ~ dfcleq , this is called the "axiom of extensionality"
|
|
by [Levy] p. 338, who treats the theory of classes as an extralogical
|
|
extension to our logic and set theory axioms.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
15-Sep-1993.) $)
|
|
df-cleq $a |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x y z $.
|
|
$( The same as ~ df-cleq with the hypothesis removed using the Axiom of
|
|
Extensionality ~ ax-ext . (Contributed by NM, 15-Sep-1993.) $)
|
|
dfcleq $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( vy vz ax-ext df-cleq ) ADEBCDEAFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This
|
|
theorem shows that a setvar variable can be expressed as a class
|
|
abstraction. This provides a motivation for the class syntax
|
|
construction ~ cv , which allows us to substitute a setvar variable for
|
|
a class variable. See also ~ cab and ~ df-clab . Note that this is not
|
|
a rigorous justification, because ~ cv is used as part of the proof of
|
|
this theorem, but a careful argument can be made outside of the
|
|
formalism of Metamath, for example as is done in Chapter 4 of Takeuti
|
|
and Zaring. See also the discussion under the definition of class in
|
|
[Jech] p. 4 showing that "Every set can be considered to be a class."
|
|
(Contributed by NM, 7-Nov-2006.) $)
|
|
cvjust $p |- x = { y | y e. x } $=
|
|
( vz cv wcel cab wceq wb dfcleq wsb df-clab elsb3 bitr2i mpgbir ) ADZBDOE
|
|
ZBFZGCDZOEZRQEZHCCOQITPBCJSPCBKCBALMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Define the membership connective between classes. Theorem 6.3 of
|
|
[Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
|
|
adopt as a definition. See these references for its metalogical
|
|
justification. Note that like ~ df-cleq it extends or "overloads" the
|
|
use of the existing membership symbol, but unlike ~ df-cleq it does not
|
|
strengthen the set of valid wffs of logic when the class variables are
|
|
replaced with setvar variables (see ~ cleljust ), so we don't include
|
|
any set theory axiom as a hypothesis. See also comments about the
|
|
syntax under ~ df-clab .
|
|
|
|
This is called the "axiom of membership" by [Levy] p. 338, who treats
|
|
the theory of classes as an extralogical extension to our logic and set
|
|
theory axioms.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-clel $a |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
eqriv.1 $e |- ( x e. A <-> x e. B ) $.
|
|
$( Infer equality of classes from equivalence of membership. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
eqriv $p |- A = B $=
|
|
( wceq cv wcel wb dfcleq mpgbir ) BCEAFZBGKCGHAABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
eqrdv.1 $e |- ( ph -> ( x e. A <-> x e. B ) ) $.
|
|
$( Deduce equality of classes from equivalence of membership. (Contributed
|
|
by NM, 17-Mar-1996.) $)
|
|
eqrdv $p |- ( ph -> A = B ) $=
|
|
( cv wcel wb wal wceq alrimiv dfcleq sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
eqrdav.1 $e |- ( ( ph /\ x e. A ) -> x e. C ) $.
|
|
eqrdav.2 $e |- ( ( ph /\ x e. B ) -> x e. C ) $.
|
|
eqrdav.3 $e |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) ) $.
|
|
$( Deduce equality of classes from an equivalence of membership that
|
|
depends on the membership variable. (Contributed by NM, 7-Nov-2008.) $)
|
|
eqrdav $p |- ( ph -> A = B ) $=
|
|
( cv wcel wa biimpd impancom mpd wi exbiri com23 imp impbida eqrdv ) ABCD
|
|
ABIZCJZUADJZAUBKUAEJZUCFAUDUBUCAUDKUBUCHLMNAUCKUDUBGAUCUDUBOAUDUCUBAUDUBU
|
|
CHPQRNST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
|
|
p. 41.
|
|
|
|
This law is thought to have originated with Aristotle (_Metaphysics_,
|
|
Zeta, 17, 1041 a, 10-20). (Thanks to Stefan Allan and BJ for this
|
|
information.) (Contributed by NM, 5-Aug-1993.) (Revised by BJ,
|
|
14-Oct-2017.) $)
|
|
eqid $p |- A = A $=
|
|
( vx cv wcel biid eqriv ) BAABCADEF $.
|
|
$}
|
|
|
|
$( Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) $)
|
|
eqidd $p |- ( ph -> A = A ) $=
|
|
( wceq eqid a1i ) BBCABDE $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eqcom $p |- ( A = B <-> B = A ) $=
|
|
( vx cv wcel wb wal wceq bicom albii dfcleq 3bitr4i ) CDZAEZMBEZFZCGONFZC
|
|
GABHBAHPQCNOIJCABKCBAKL $.
|
|
$}
|
|
|
|
${
|
|
eqcoms.1 $e |- ( A = B -> ph ) $.
|
|
$( Inference applying commutative law for class equality to an antecedent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eqcoms $p |- ( B = A -> ph ) $=
|
|
( wceq eqcom sylbi ) CBEBCEACBFDG $.
|
|
$}
|
|
|
|
${
|
|
eqcomi.1 $e |- A = B $.
|
|
$( Inference from commutative law for class equality. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqcomi $p |- B = A $=
|
|
( wceq eqcom mpbi ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
eqcomd.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from commutative law for class equality. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
eqcomd $p |- ( ph -> B = A ) $=
|
|
( wceq eqcom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality implies equivalence of equalities. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqeq1 $p |- ( A = B -> ( A = C <-> B = C ) ) $=
|
|
( vx wceq cv wcel wb wal dfcleq biimpi 19.21bi bibi1d albidv 3bitr4g ) AB
|
|
EZDFZAGZQCGZHZDIQBGZSHZDIACEBCEPTUBDPRUASPRUAHZDPUCDIDABJKLMNDACJDBCJO $.
|
|
$}
|
|
|
|
${
|
|
eqeq1i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of equalities. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeq1i $p |- ( A = C <-> B = C ) $=
|
|
( wceq wb eqeq1 ax-mp ) ABEACEBCEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
eqeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of equalities. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eqeq1d $p |- ( ph -> ( A = C <-> B = C ) ) $=
|
|
( wceq wb eqeq1 syl ) ABCFBDFCDFGEBCDHI $.
|
|
$}
|
|
|
|
$( Equality implies equivalence of equalities. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqeq2 $p |- ( A = B -> ( C = A <-> C = B ) ) $=
|
|
( wceq eqeq1 eqcom 3bitr4g ) ABDACDBCDCADCBDABCECAFCBFG $.
|
|
|
|
${
|
|
eqeq2i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of equalities. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeq2i $p |- ( C = A <-> C = B ) $=
|
|
( wceq wb eqeq2 ax-mp ) ABECAECBEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
eqeq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of equalities. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eqeq2d $p |- ( ph -> ( C = A <-> C = B ) ) $=
|
|
( wceq wb eqeq2 syl ) ABCFDBFDCFGEBCDHI $.
|
|
$}
|
|
|
|
$( Equality relationship among 4 classes. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
eqeq12 $p |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq eqeq1 eqeq2 sylan9bb ) ABEACEBCECDEBDEABCFCDBGH $.
|
|
|
|
${
|
|
eqeq12i.1 $e |- A = B $.
|
|
eqeq12i.2 $e |- C = D $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeq12i $p |- ( A = C <-> B = D ) $=
|
|
( wceq wb eqeq12 mp2an ) ABGCDGACGBDGHEFABCDIJ $.
|
|
|
|
$( Theorem eqeq12i is the congruence law for equality. $)
|
|
$( $j congruence 'eqeq12i'; $)
|
|
$}
|
|
|
|
${
|
|
eqeq12d.1 $e |- ( ph -> A = B ) $.
|
|
eqeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeq12d $p |- ( ph -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeq12 syl2anc ) ABCHDEHBDHCEHIFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
eqeqan12d.1 $e |- ( ph -> A = B ) $.
|
|
eqeqan12d.2 $e |- ( ps -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeqan12d $p |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeq12 syl2an ) ACDIEFICEIDFIJBGHCDEFKL $.
|
|
$}
|
|
|
|
${
|
|
eqeqan12rd.1 $e |- ( ph -> A = B ) $.
|
|
eqeqan12rd.2 $e |- ( ps -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 9-Aug-1994.) $)
|
|
eqeqan12rd $p |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeqan12d ancoms ) ABCEIDFIJABCDEFGHKL $.
|
|
$}
|
|
|
|
$( Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring]
|
|
p. 13. (Contributed by NM, 25-Jan-2004.) $)
|
|
eqtr $p |- ( ( A = B /\ B = C ) -> A = C ) $=
|
|
( wceq eqeq1 biimpar ) ABDACDBCDABCEF $.
|
|
|
|
$( A transitive law for class equality. (Contributed by NM, 20-May-2005.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
eqtr2 $p |- ( ( A = B /\ A = C ) -> B = C ) $=
|
|
( wceq eqcom eqtr sylanb ) ABDBADACDBCDABEBACFG $.
|
|
|
|
$( A transitive law for class equality. (Contributed by NM, 20-May-2005.) $)
|
|
eqtr3 $p |- ( ( A = C /\ B = C ) -> A = B ) $=
|
|
( wceq eqcom eqtr sylan2b ) BCDACDCBDABDBCEACBFG $.
|
|
|
|
${
|
|
eqtri.1 $e |- A = B $.
|
|
eqtri.2 $e |- B = C $.
|
|
$( An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtri $p |- A = C $=
|
|
( wceq eqeq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
eqtr2i.1 $e |- A = B $.
|
|
eqtr2i.2 $e |- B = C $.
|
|
$( An equality transitivity inference. (Contributed by NM,
|
|
21-Feb-1995.) $)
|
|
eqtr2i $p |- C = A $=
|
|
( eqtri eqcomi ) ACABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
eqtr3i.1 $e |- A = B $.
|
|
eqtr3i.2 $e |- A = C $.
|
|
$( An equality transitivity inference. (Contributed by NM, 6-May-1994.) $)
|
|
eqtr3i $p |- B = C $=
|
|
( eqcomi eqtri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
eqtr4i.1 $e |- A = B $.
|
|
eqtr4i.2 $e |- C = B $.
|
|
$( An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtr4i $p |- A = C $=
|
|
( eqcomi eqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( Register '=' as an equality for its type (class). $)
|
|
$( $j equality 'wceq' from 'eqid' 'eqcomi' 'eqtri'; $)
|
|
|
|
${
|
|
3eqtri.1 $e |- A = B $.
|
|
3eqtri.2 $e |- B = C $.
|
|
3eqtri.3 $e |- C = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
29-Aug-1993.) $)
|
|
3eqtri $p |- A = D $=
|
|
( eqtri ) ABDEBCDFGHH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtrri $p |- D = A $=
|
|
( eqtri eqtr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3eqtr2i.1 $e |- A = B $.
|
|
3eqtr2i.2 $e |- C = B $.
|
|
3eqtr2i.3 $e |- C = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) $)
|
|
3eqtr2i $p |- A = D $=
|
|
( eqtr4i eqtri ) ACDABCEFHGI $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr2ri $p |- D = A $=
|
|
( eqtr4i eqtr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3i.1 $e |- A = B $.
|
|
3eqtr3i.2 $e |- A = C $.
|
|
3eqtr3i.3 $e |- B = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr3i $p |- C = D $=
|
|
( eqtr3i ) BCDABCEFHGH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
15-Aug-2004.) $)
|
|
3eqtr3ri $p |- D = C $=
|
|
( eqtr3i ) BDCGABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4i.1 $e |- A = B $.
|
|
3eqtr4i.2 $e |- C = A $.
|
|
3eqtr4i.3 $e |- D = B $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4i $p |- C = D $=
|
|
( eqtr4i ) CADFDBAGEHH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4ri $p |- D = C $=
|
|
( eqtr4i ) DACDBAGEHFH $.
|
|
$}
|
|
|
|
${
|
|
eqtrd.1 $e |- ( ph -> A = B ) $.
|
|
eqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtrd $p |- ( ph -> A = C ) $=
|
|
( wceq eqeq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
eqtr2d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr2d.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
18-Oct-1999.) $)
|
|
eqtr2d $p |- ( ph -> C = A ) $=
|
|
( eqtrd eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
eqtr3d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr3d.2 $e |- ( ph -> A = C ) $.
|
|
$( An equality transitivity equality deduction. (Contributed by NM,
|
|
18-Jul-1995.) $)
|
|
eqtr3d $p |- ( ph -> B = C ) $=
|
|
( eqcomd eqtrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
eqtr4d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr4d.2 $e |- ( ph -> C = B ) $.
|
|
$( An equality transitivity equality deduction. (Contributed by NM,
|
|
18-Jul-1995.) $)
|
|
eqtr4d $p |- ( ph -> A = C ) $=
|
|
( eqcomd eqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3eqtrd.1 $e |- ( ph -> A = B ) $.
|
|
3eqtrd.2 $e |- ( ph -> B = C ) $.
|
|
3eqtrd.3 $e |- ( ph -> C = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
29-Oct-1995.) $)
|
|
3eqtrd $p |- ( ph -> A = D ) $=
|
|
( eqtrd ) ABCEFACDEGHII $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtrrd $p |- ( ph -> D = A ) $=
|
|
( eqtrd eqtr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr2d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr2d.2 $e |- ( ph -> C = B ) $.
|
|
3eqtr2d.3 $e |- ( ph -> C = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3eqtr2d $p |- ( ph -> A = D ) $=
|
|
( eqtr4d eqtrd ) ABDEABCDFGIHJ $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3eqtr2rd $p |- ( ph -> D = A ) $=
|
|
( eqtr4d eqtr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr3d.2 $e |- ( ph -> A = C ) $.
|
|
3eqtr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr3d $p |- ( ph -> C = D ) $=
|
|
( eqtr3d ) ACDEABCDFGIHI $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
14-Jan-2006.) $)
|
|
3eqtr3rd $p |- ( ph -> D = C ) $=
|
|
( eqtr3d ) ACEDHABCDFGII $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr4d.2 $e |- ( ph -> C = A ) $.
|
|
3eqtr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4d $p |- ( ph -> C = D ) $=
|
|
( eqtr4d ) ADBEGAECBHFII $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
21-Sep-1995.) $)
|
|
3eqtr4rd $p |- ( ph -> D = C ) $=
|
|
( eqtr4d ) AEBDAECBHFIGI $.
|
|
$}
|
|
|
|
${
|
|
syl5eq.1 $e |- A = B $.
|
|
syl5eq.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5eq $p |- ( ph -> A = C ) $=
|
|
( wceq a1i eqtrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5req.1 $e |- A = B $.
|
|
syl5req.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl5req $p |- ( ph -> C = A ) $=
|
|
( syl5eq eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5eqr.1 $e |- B = A $.
|
|
syl5eqr.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5eqr $p |- ( ph -> A = C ) $=
|
|
( eqcomi syl5eq ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5reqr.1 $e |- B = A $.
|
|
syl5reqr.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl5reqr $p |- ( ph -> C = A ) $=
|
|
( eqcomi syl5req ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6eq.1 $e |- ( ph -> A = B ) $.
|
|
syl6eq.2 $e |- B = C $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6eq $p |- ( ph -> A = C ) $=
|
|
( wceq a1i eqtrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6req.1 $e |- ( ph -> A = B ) $.
|
|
syl6req.2 $e |- B = C $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl6req $p |- ( ph -> C = A ) $=
|
|
( syl6eq eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6eqr.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqr.2 $e |- C = B $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6eqr $p |- ( ph -> A = C ) $=
|
|
( eqcomi syl6eq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6reqr.1 $e |- ( ph -> A = B ) $.
|
|
syl6reqr.2 $e |- C = B $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl6reqr $p |- ( ph -> C = A ) $=
|
|
( eqcomi syl6req ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
sylan9eq.1 $e |- ( ph -> A = B ) $.
|
|
sylan9eq.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sylan9eq $p |- ( ( ph /\ ps ) -> A = C ) $=
|
|
( wceq eqtr syl2an ) ACDHDEHCEHBFGCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9req.1 $e |- ( ph -> B = A ) $.
|
|
sylan9req.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
23-Jun-2007.) $)
|
|
sylan9req $p |- ( ( ph /\ ps ) -> A = C ) $=
|
|
( eqcomd sylan9eq ) ABCDEADCFHGI $.
|
|
$}
|
|
|
|
${
|
|
sylan9eqr.1 $e |- ( ph -> A = B ) $.
|
|
sylan9eqr.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 8-May-1994.) $)
|
|
sylan9eqr $p |- ( ( ps /\ ph ) -> A = C ) $=
|
|
( wceq sylan9eq ancoms ) ABCEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3g.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr3g.2 $e |- A = C $.
|
|
3eqtr3g.3 $e |- B = D $.
|
|
$( A chained equality inference, useful for converting from definitions.
|
|
(Contributed by NM, 15-Nov-1994.) $)
|
|
3eqtr3g $p |- ( ph -> C = D ) $=
|
|
( syl5eqr syl6eq ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3a.1 $e |- A = B $.
|
|
3eqtr3a.2 $e |- ( ph -> A = C ) $.
|
|
3eqtr3a.3 $e |- ( ph -> B = D ) $.
|
|
$( A chained equality inference, useful for converting from definitions.
|
|
(Contributed by Mario Carneiro, 6-Nov-2015.) $)
|
|
3eqtr3a $p |- ( ph -> C = D ) $=
|
|
( syl5eq eqtr3d ) ABDEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4g.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr4g.2 $e |- C = A $.
|
|
3eqtr4g.3 $e |- D = B $.
|
|
$( A chained equality inference, useful for converting to definitions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3eqtr4g $p |- ( ph -> C = D ) $=
|
|
( syl5eq syl6eqr ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4a.1 $e |- A = B $.
|
|
3eqtr4a.2 $e |- ( ph -> C = A ) $.
|
|
3eqtr4a.3 $e |- ( ph -> D = B ) $.
|
|
$( A chained equality inference, useful for converting to definitions.
|
|
(Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
3eqtr4a $p |- ( ph -> C = D ) $=
|
|
( syl6eq eqtr4d ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
eq2tr.1 $e |- ( A = C -> D = F ) $.
|
|
eq2tr.2 $e |- ( B = D -> C = G ) $.
|
|
$( A compound transitive inference for class equality. (Contributed by NM,
|
|
22-Jan-2004.) $)
|
|
eq2tri $p |- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) ) $=
|
|
( wceq wa ancom eqeq2d pm5.32i 3bitr3i ) ACIZBDIZJPOJOBEIZJPAFIZJOPKOPQOD
|
|
EBGLMPORPCFAHLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eleq1 $p |- ( A = B -> ( A e. C <-> B e. C ) ) $=
|
|
( vx wceq cv wcel wa wex eqeq2 anbi1d exbidv df-clel 3bitr4g ) ABEZDFZAEZ
|
|
PCGZHZDIPBEZRHZDIACGBCGOSUADOQTRABPJKLDACMDBCMN $.
|
|
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eleq2 $p |- ( A = B -> ( C e. A <-> C e. B ) ) $=
|
|
( vx wceq cv wcel wa wex wal dfcleq biimpi 19.21bi anbi2d df-clel 3bitr4g
|
|
wb exbidv ) ABEZDFZCEZTAGZHZDIUATBGZHZDICAGCBGSUCUEDSUBUDUASUBUDQZDSUFDJD
|
|
ABKLMNRDCAODCBOP $.
|
|
$}
|
|
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
eleq12 $p |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) ) $=
|
|
( wceq wcel eleq1 eleq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.
|
|
|
|
${
|
|
eleq1i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq1i $p |- ( A e. C <-> B e. C ) $=
|
|
( wceq wcel wb eleq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq2i $p |- ( C e. A <-> C e. B ) $=
|
|
( wceq wcel wb eleq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
eleq12i.2 $e |- C = D $.
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 31-May-1994.) $)
|
|
eleq12i $p |- ( A e. C <-> B e. D ) $=
|
|
( wcel eleq2i eleq1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
|
|
$( Theorem eleq12i is the congruence law for elementhood. $)
|
|
$( $j congruence 'eleq12i'; $)
|
|
$}
|
|
$}
|
|
|
|
${
|
|
eleq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq1d $p |- ( ph -> ( A e. C <-> B e. C ) ) $=
|
|
( wceq wcel wb eleq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eleq2d $p |- ( ph -> ( C e. A <-> C e. B ) ) $=
|
|
( wceq wcel wb eleq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
${
|
|
eleq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 31-May-1994.) $)
|
|
eleq12d $p |- ( ph -> ( A e. C <-> B e. D ) ) $=
|
|
( wcel eleq2d eleq1d bitrd ) ABDHBEHCEHADEBGIABCEFJK $.
|
|
$}
|
|
$}
|
|
|
|
$( A transitive-type law relating membership and equality. (Contributed by
|
|
NM, 9-Apr-1994.) $)
|
|
eleq1a $p |- ( A e. B -> ( C = A -> C e. B ) ) $=
|
|
( wceq wcel eleq1 biimprcd ) CADCBEABECABFG $.
|
|
|
|
${
|
|
eqeltr.1 $e |- A = B $.
|
|
eqeltr.2 $e |- B e. C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeltri $p |- A e. C $=
|
|
( wcel eleq1i mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
eqeltrr.1 $e |- A = B $.
|
|
eqeltrr.2 $e |- A e. C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeltrri $p |- B e. C $=
|
|
( eqcomi eqeltri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
eleqtr.1 $e |- A e. B $.
|
|
eleqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleqtri $p |- A e. C $=
|
|
( wcel eleq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
eleqtrr.1 $e |- A e. B $.
|
|
eleqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleqtrri $p |- A e. C $=
|
|
( eqcomi eleqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
eqeltrd.1 $e |- ( ph -> A = B ) $.
|
|
eqeltrd.2 $e |- ( ph -> B e. C ) $.
|
|
$( Substitution of equal classes into membership relation, deduction form.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
eqeltrd $p |- ( ph -> A e. C ) $=
|
|
( wcel eleq1d mpbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqeltrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqeltrrd.2 $e |- ( ph -> A e. C ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eqeltrrd $p |- ( ph -> B e. C ) $=
|
|
( eqcomd eqeltrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
eleqtrd.1 $e |- ( ph -> A e. B ) $.
|
|
eleqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eleqtrd $p |- ( ph -> A e. C ) $=
|
|
( wcel eleq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
eleqtrrd.1 $e |- ( ph -> A e. B ) $.
|
|
eleqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eleqtrrd $p |- ( ph -> A e. C ) $=
|
|
( eqcomd eleqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3eltr3.1 $e |- A e. B $.
|
|
3eltr3.2 $e |- A = C $.
|
|
3eltr3.3 $e |- B = D $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3i $p |- C e. D $=
|
|
( eleqtri eqeltrri ) ACDFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3eltr4.1 $e |- A e. B $.
|
|
3eltr4.2 $e |- C = A $.
|
|
3eltr4.3 $e |- D = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4i $p |- C e. D $=
|
|
( eleqtrri eqeltri ) CADFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3eltr3d.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr3d.2 $e |- ( ph -> A = C ) $.
|
|
3eltr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3d $p |- ( ph -> C e. D ) $=
|
|
( eleqtrd eqeltrrd ) ABDEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eltr4d.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr4d.2 $e |- ( ph -> C = A ) $.
|
|
3eltr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4d $p |- ( ph -> C e. D ) $=
|
|
( eleqtrrd eqeltrd ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eltr3g.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr3g.2 $e |- A = C $.
|
|
3eltr3g.3 $e |- B = D $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3g $p |- ( ph -> C e. D ) $=
|
|
( wcel eleq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3eltr4g.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr4g.2 $e |- C = A $.
|
|
3eltr4g.3 $e |- D = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4g $p |- ( ph -> C e. D ) $=
|
|
( wcel eleq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
syl5eqel.1 $e |- A = B $.
|
|
syl5eqel.2 $e |- ( ph -> B e. C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eqel $p |- ( ph -> A e. C ) $=
|
|
( wceq a1i eqeltrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqelr.1 $e |- B = A $.
|
|
syl5eqelr.2 $e |- ( ph -> B e. C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eqelr $p |- ( ph -> A e. C ) $=
|
|
( eqcomi syl5eqel ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5eleq.1 $e |- A e. B $.
|
|
syl5eleq.2 $e |- ( ph -> B = C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eleq $p |- ( ph -> A e. C ) $=
|
|
( wcel a1i eleqtrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5eleqr.1 $e |- A e. B $.
|
|
syl5eleqr.2 $e |- ( ph -> C = B ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eleqr $p |- ( ph -> A e. C ) $=
|
|
( eqcomd syl5eleq ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6eqel.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqel.2 $e |- B e. C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqel $p |- ( ph -> A e. C ) $=
|
|
( wcel a1i eqeltrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqelr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqelr.2 $e |- B e. C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqelr $p |- ( ph -> A e. C ) $=
|
|
( eqcomd syl6eqel ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6eleq.1 $e |- ( ph -> A e. B ) $.
|
|
syl6eleq.2 $e |- B = C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eleq $p |- ( ph -> A e. C ) $=
|
|
( wceq a1i eleqtrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eleqr.1 $e |- ( ph -> A e. B ) $.
|
|
syl6eleqr.2 $e |- C = B $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl6eleqr $p |- ( ph -> A e. C ) $=
|
|
( eqcomi syl6eleq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
eleq2s.1 $e |- ( A e. B -> ph ) $.
|
|
eleq2s.2 $e |- C = B $.
|
|
$( Substitution of equal classes into a membership antecedent.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
eleq2s $p |- ( A e. C -> ph ) $=
|
|
( wcel eleq2i sylbi ) BDGBCGADCBFHEI $.
|
|
$}
|
|
|
|
${
|
|
eqneltrd.1 $e |- ( ph -> A = B ) $.
|
|
eqneltrd.2 $e |- ( ph -> -. B e. C ) $.
|
|
$( If a class is not an element of another class, an equal class is also
|
|
not an element. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
eqneltrd $p |- ( ph -> -. A e. C ) $=
|
|
( wcel eleq1d mtbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqneltrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqneltrrd.2 $e |- ( ph -> -. A e. C ) $.
|
|
$( If a class is not an element of another class, an equal class is also
|
|
not an element. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
eqneltrrd $p |- ( ph -> -. B e. C ) $=
|
|
( wcel eleq1d mtbid ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
neleqtrd.1 $e |- ( ph -> -. C e. A ) $.
|
|
neleqtrd.2 $e |- ( ph -> A = B ) $.
|
|
$( If a class is not an element of another class, it is also not an element
|
|
of an equal class. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neleqtrd $p |- ( ph -> -. C e. B ) $=
|
|
( wcel eleq2d mtbid ) ADBGDCGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
neleqtrrd.1 $e |- ( ph -> -. C e. B ) $.
|
|
neleqtrrd.2 $e |- ( ph -> A = B ) $.
|
|
$( If a class is not an element of another class, it is also not an element
|
|
of an equal class. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neleqtrrd $p |- ( ph -> -. C e. A ) $=
|
|
( wcel eleq2d mtbird ) ADBGDCGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
cleqh.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
cleqh.2 $e |- ( y e. B -> A. x y e. B ) $.
|
|
$( Establish equality between classes, using bound-variable hypotheses
|
|
instead of distinct variable conditions. See also ~ cleqf .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cleqh $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( wceq cv wcel wb wal dfcleq ax-17 wi wa dfbi2 hbim weq eleq1 cbv3h hban
|
|
hbxfrbi bibi12d biimpd equcoms biimprd impbii bitr4i ) CDGBHZCIZUIDIZJZBK
|
|
ZAHZCIZUNDIZJZAKZBCDLURUMUQULABUQBMZULUJUKNZUKUJNZOAUJUKPUTVAAUJUKAEFQUKU
|
|
JAFEQUAUBZABRZUQULVCUOUJUPUKUNUICSUNUIDSUCZUDTULUQBAVBUSBARUQULUQULJABVDU
|
|
EUFTUGUH $.
|
|
$}
|
|
|
|
$( A way of showing two classes are not equal. (Contributed by NM,
|
|
1-Apr-1997.) $)
|
|
nelneq $p |- ( ( A e. C /\ -. B e. C ) -> -. A = B ) $=
|
|
( wcel wceq eleq1 biimpcd con3dimp ) ACDZABEZBCDZJIKABCFGH $.
|
|
|
|
$( A way of showing two classes are not equal. (Contributed by NM,
|
|
12-Jan-2002.) $)
|
|
nelneq2 $p |- ( ( A e. B /\ -. A e. C ) -> -. B = C ) $=
|
|
( wcel wceq eleq2 biimpcd con3dimp ) ABDZBCEZACDZJIKBCAFGH $.
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Lemma for ~ eqsb3 . (Contributed by Rodolfo Medina, 28-Apr-2010.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
eqsb3lem $p |- ( [ x / y ] y = A <-> x = A ) $=
|
|
( cv wceq nfv eqeq1 sbie ) BDZCEADZCEZBAKBFIJCGH $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d w y $. $d w A $. $d x w $.
|
|
$( Substitution applied to an atomic wff (class version of ~ equsb3 ).
|
|
(Contributed by Rodolfo Medina, 28-Apr-2010.) $)
|
|
eqsb3 $p |- ( [ x / y ] y = A <-> x = A ) $=
|
|
( vw cv wceq wsb eqsb3lem sbbii nfv sbco2 3bitr3i ) BECFZBDGZDAGDECFZDAGM
|
|
BAGAECFNODADBCHIMBADMDJKADCHL $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d w y $. $d w A $. $d w x $.
|
|
$( Substitution applied to an atomic wff (class version of ~ elsb3 ).
|
|
(Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by
|
|
Andrew Salmon, 14-Jun-2011.) $)
|
|
clelsb3 $p |- ( [ x / y ] y e. A <-> x e. A ) $=
|
|
( vw cv wcel wsb nfv sbco2 eleq1 sbie sbbii 3bitr3i ) DEZCFZDBGZBAGODAGBE
|
|
ZCFZBAGAEZCFZODABOBHIPRBAORDBRDHNQCJKLOTDATDHNSCJKM $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d w y $. $d w A $. $d w x $.
|
|
$( Substitution applied to an atomic wff (class version of ~ elsb4 ).
|
|
(Contributed by Jim Kingdon, 22-Nov-2018.) $)
|
|
clelsb4 $p |- ( [ x / y ] A e. y <-> A e. x ) $=
|
|
( vw cv wcel wsb nfv sbco2 eleq2 sbie sbbii 3bitr3i ) CDEZFZDBGZBAGODAGCB
|
|
EZFZBAGCAEZFZODABOBHIPRBAORDBRDHNQCJKLOTDATDHNSCJKM $.
|
|
$}
|
|
|
|
${
|
|
hbxfr.1 $e |- A = B $.
|
|
hbxfr.2 $e |- ( y e. B -> A. x y e. B ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. See ~ hbxfrbi for equivalence version. (Contributed by NM,
|
|
21-Aug-2007.) $)
|
|
hbxfreq $p |- ( y e. A -> A. x y e. A ) $=
|
|
( cv wcel eleq2i hbxfrbi ) BGZCHKDHACDKEIFJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x z $.
|
|
hblem.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
$( Change the free variable of a hypothesis builder. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
hblem $p |- ( z e. A -> A. x z e. A ) $=
|
|
( cv wcel wsb wal hbsb clelsb3 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAEJ
|
|
CBDKZOPAQLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Equality of a class variable and a class abstraction (also called a
|
|
class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the
|
|
relationship between expressions with class abstractions and expressions
|
|
with class variables. Note that ~ abbi and its relatives are among
|
|
those useful for converting theorems with class variables to equivalent
|
|
theorems with wff variables, by first substituting a class abstraction
|
|
for each class variable.
|
|
|
|
Class variables can always be eliminated from a theorem to result in an
|
|
equivalent theorem with wff variables, and vice-versa. The idea is
|
|
roughly as follows. To convert a theorem with a wff variable ` ph `
|
|
(that has a free variable ` x ` ) to a theorem with a class variable
|
|
` A ` , we substitute ` x e. A ` for ` ph ` throughout and simplify,
|
|
where ` A ` is a new class variable not already in the wff. Conversely,
|
|
to convert a theorem with a class variable ` A ` to one with ` ph ` , we
|
|
substitute ` { x | ph } ` for ` A ` throughout and simplify, where ` x `
|
|
and ` ph ` are new set and wff variables not already in the wff. For
|
|
more information on class variables, see Quine pp. 15-21 and/or Takeuti
|
|
and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.) $)
|
|
abeq2 $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
|
|
( vy cab wceq cv wcel wb wal ax-17 hbab1 cleqh abid bibi2i albii bitri )
|
|
CABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Equality of a class variable and a class abstraction. (Contributed by
|
|
NM, 20-Aug-1993.) $)
|
|
abeq1 $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
|
|
( cab wceq cv wcel wb wal abeq2 eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
|
|
ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
|
|
$}
|
|
|
|
${
|
|
abeqi.1 $e |- A = { x | ph } $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 3-Apr-1996.) $)
|
|
abeq2i $p |- ( x e. A <-> ph ) $=
|
|
( cv wcel cab eleq2i abid bitri ) BEZCFKABGZFACLKDHABIJ $.
|
|
$}
|
|
|
|
${
|
|
abeqri.1 $e |- { x | ph } = A $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 31-Jul-1994.) $)
|
|
abeq1i $p |- ( ph <-> x e. A ) $=
|
|
( cv cab wcel abid eleq2i bitr3i ) ABEZABFZGKCGABHLCKDIJ $.
|
|
$}
|
|
|
|
${
|
|
abeqd.1 $e |- ( ph -> A = { x | ps } ) $.
|
|
$( Equality of a class variable and a class abstraction (deduction).
|
|
(Contributed by NM, 16-Nov-1995.) $)
|
|
abeq2d $p |- ( ph -> ( x e. A <-> ps ) ) $=
|
|
( cv wcel cab eleq2d abid syl6bb ) ACFZDGLBCHZGBADMLEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps y $. $d x y $.
|
|
$( Equivalent wff's correspond to equal class abstractions. (Contributed
|
|
by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) $)
|
|
abbi $p |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } ) $=
|
|
( vy cab wceq wcel wal dfcleq nfsab1 nfbi nfv wsb df-clab sbequ12r syl5bb
|
|
cv wb bibi12d cbval bitr2i ) ACEZBCEZFDQZUBGZUDUCGZRZDHABRZCHDUBUCIUGUHDC
|
|
UEUFCACDJBCDJKUHDLUDCQFZUEAUFBUEACDMUIAADCNADCOPUFBCDMUIBBDCNBDCOPSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
abbiri.1 $e |- ( x e. A <-> ph ) $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
abbi2i $p |- A = { x | ph } $=
|
|
( cab wceq cv wcel wb abeq2 mpgbir ) CABEFBGCHAIBABCJDK $.
|
|
$}
|
|
|
|
${
|
|
abbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal class abstractions (inference rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
abbii $p |- { x | ph } = { x | ps } $=
|
|
( wb cab wceq abbi mpgbi ) ABEACFBCFGCABCHDI $.
|
|
|
|
$( Theorem abbii is the congruence law for class abstraction. $)
|
|
$( $j congruence 'abbii'; $)
|
|
$}
|
|
|
|
${
|
|
abbid.1 $e |- F/ x ph $.
|
|
abbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal class abstractions (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
abbid $p |- ( ph -> { x | ps } = { x | ch } ) $=
|
|
( wb wal cab wceq alrimi abbi sylib ) ABCGZDHBDICDIJANDEFKBCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
abbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal class abstractions (deduction rule).
|
|
(Contributed by NM, 10-Aug-1993.) $)
|
|
abbidv $p |- ( ph -> { x | ps } = { x | ch } ) $=
|
|
( nfv abbid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d ph x $.
|
|
abbirdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $.
|
|
$( Deduction from a wff to a class abstraction. (Contributed by NM,
|
|
9-Jul-1994.) $)
|
|
abbi2dv $p |- ( ph -> A = { x | ps } ) $=
|
|
( cv wcel wb wal cab wceq alrimiv abeq2 sylibr ) ACFDGBHZCIDBCJKAOCELBCDM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d ph x $.
|
|
abbildv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $.
|
|
$( Deduction from a wff to a class abstraction. (Contributed by NM,
|
|
9-Jul-1994.) $)
|
|
abbi1dv $p |- ( ph -> { x | ps } = A ) $=
|
|
( cv wcel wb wal cab wceq alrimiv abeq1 sylibr ) ABCFDGHZCIBCJDKAOCELBCDM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|
|
(Contributed by NM, 26-Dec-1993.) $)
|
|
abid2 $p |- { x | x e. A } = A $=
|
|
( cv wcel cab biid abbi2i eqcomi ) BACBDZAEIABIFGH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $.
|
|
sb8ab.1 $e |- F/ y ph $.
|
|
$( Substitution of variable in class abstraction. (Contributed by Jim
|
|
Kingdon, 27-Sep-2018.) $)
|
|
sb8ab $p |- { x | ph } = { y | [ y / x ] ph } $=
|
|
( vz cab wsb cv wcel sbco2 df-clab 3bitr4ri eqriv ) EABFZABCGZCFZOCEGABEG
|
|
EHZPIQNIABECDJOECKAEBKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $. $d ps z $.
|
|
cbvab.1 $e |- F/ y ph $.
|
|
cbvab.2 $e |- F/ x ps $.
|
|
cbvab.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvab $p |- { x | ph } = { y | ps } $=
|
|
( vz cab wsb cv wcel nfsb weq wb equcoms bicomd sbie sbequ df-clab eqriv
|
|
syl5bbr 3bitr4i ) HACIZBDIZACHJBDHJZHKZUDLUGUELAUFCHBDHCFMABDCJCHNUFBADCE
|
|
DCNABABOCDGPQRBCHDSUBRAHCTBHDTUCUA $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 26-May-1999.) $)
|
|
cbvabv $p |- { x | ph } = { y | ps } $=
|
|
( nfv cbvab ) ABCDADFBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Membership of a class variable in a class abstraction. (Contributed by
|
|
NM, 23-Dec-1993.) $)
|
|
clelab $p |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy cv wceq cab wcel wex wsb df-clab anbi2i exbii df-clel nfv nfs1v nfan
|
|
wa eqeq1 sbequ12 anbi12d cbvex 3bitr4i ) DEZCFZUDABGZHZRZDIUEABDJZRZDICUF
|
|
HBEZCFZARZBIUHUJDUGUIUEADBKLMDCUFNUMUJBDUMDOUEUIBUEBOABDPQUKUDFULUEAUIUKU
|
|
DCSABDTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( Membership of a class abstraction in another class. (Contributed by NM,
|
|
17-Jan-2006.) $)
|
|
clabel $p |- ( { x | ph } e. A <->
|
|
E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) ) $=
|
|
( cab wcel cv wceq wa wex wb wal df-clel abeq2 anbi2ci exbii bitri ) ABEZ
|
|
DFCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z x $. $d z y $.
|
|
$( The right-hand side of the second equality is a way of representing
|
|
proper substitution of ` y ` for ` x ` into a class variable.
|
|
(Contributed by NM, 14-Sep-2003.) $)
|
|
sbab $p |- ( x = y -> A = { z | [ y / x ] z e. A } ) $=
|
|
( weq cv wcel wsb sbequ12 abbi2dv ) ABECFDGZABHCDKABIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Class form not-free predicate
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c F/_ $. $( Underlined not-free symbol. $)
|
|
|
|
$( Extend wff definition to include the not-free predicate for classes. $)
|
|
wnfc $a wff F/_ x A $.
|
|
|
|
${
|
|
$d x y z $. $d y z A $.
|
|
$( Justification theorem for ~ df-nfc . (Contributed by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
nfcjust $p |- ( A. y F/ x y e. A <-> A. z F/ x z e. A ) $=
|
|
( cv wcel wnf weq nfv eleq1 nfbidf cbvalv ) BEZDFZAGCEZDFZAGBCBCHZNPAQAIM
|
|
ODJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Define the not-free predicate for classes. This is read " ` x ` is not
|
|
free in ` A ` ". Not-free means that the value of ` x ` cannot affect
|
|
the value of ` A ` , e.g., any occurrence of ` x ` in ` A ` is
|
|
effectively bound by a quantifier or something that expands to one (such
|
|
as "there exists at most one"). It is defined in terms of the not-free
|
|
predicate ~ df-nf for wffs; see that definition for more information.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
df-nfc $a |- ( F/_ x A <-> A. y F/ x y e. A ) $.
|
|
|
|
${
|
|
nfci.1 $e |- F/ x y e. A $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfci $p |- F/_ x A $=
|
|
( wnfc cv wcel wnf df-nfc mpgbir ) ACEBFCGAHBABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
nfcii.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcii $p |- F/_ x A $=
|
|
( cv wcel nfi nfci ) ABCBECFADGH $.
|
|
$}
|
|
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcr $p |- ( F/_ x A -> F/ x y e. A ) $=
|
|
( wnfc cv wcel wnf wal df-nfc sp sylbi ) ACDBECFAGZBHLABCILBJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z A $.
|
|
nfcri.1 $e |- F/_ x A $.
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcrii $p |- ( y e. A -> A. x y e. A ) $=
|
|
( vz cv wcel wnfc wnf nfcr ax-mp nfri hblem ) AEBCEFCGZAACHNAIDAECJKLM $.
|
|
|
|
$( Consequence of the not-free predicate. (Note that unlike ~ nfcr , this
|
|
does not require ` y ` and ` A ` to be disjoint.) (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfcri $p |- F/ x y e. A $=
|
|
( cv wcel nfcrii nfi ) BECFAABCDGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
nfcd.1 $e |- F/ y ph $.
|
|
nfcd.2 $e |- ( ph -> F/ x y e. A ) $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcd $p |- ( ph -> F/_ x A ) $=
|
|
( cv wcel wnf wal wnfc alrimi df-nfc sylibr ) ACGDHBIZCJBDKAOCEFLBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfceqi.1 $e |- A = B $.
|
|
$( Equality theorem for class not-free. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfceqi $p |- ( F/_ x A <-> F/_ x B ) $=
|
|
( vy cv wcel wnf wal wnfc eleq2i nfbii albii df-nfc 3bitr4i ) EFZBGZAHZEI
|
|
PCGZAHZEIABJACJRTEQSABCPDKLMAEBNAECNO $.
|
|
|
|
${
|
|
nfcxfr.2 $e |- F/_ x B $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcxfr $p |- F/_ x A $=
|
|
( wnfc nfceqi mpbir ) ABFACFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfcxfrd.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcxfrd $p |- ( ph -> F/_ x A ) $=
|
|
( wnfc nfceqi sylibr ) ABDGBCGFBCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $. $d B y $. $d ph y $.
|
|
nfceqdf.1 $e |- F/ x ph $.
|
|
nfceqdf.2 $e |- ( ph -> A = B ) $.
|
|
$( An equality theorem for effectively not free. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
nfceqdf $p |- ( ph -> ( F/_ x A <-> F/_ x B ) ) $=
|
|
( vy cv wcel wnf wal wnfc eleq2d nfbidf albidv df-nfc 3bitr4g ) AGHZCIZBJ
|
|
ZGKRDIZBJZGKBCLBDLATUBGASUABEACDRFMNOBGCPBGDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcv $p |- F/_ x A $=
|
|
( vy cv wcel nfv nfci ) ACBCDBEAFG $.
|
|
|
|
$( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
|
|
(Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfcvd $p |- ( ph -> F/_ x A ) $=
|
|
( wnfc nfcv a1i ) BCDABCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfab1 $p |- F/_ x { x | ph } $=
|
|
( vy cab nfsab1 nfci ) BCABDABCEF $.
|
|
|
|
$( ` x ` is bound in ` F/_ x A ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnfc1 $p |- F/ x F/_ x A $=
|
|
( vy wnfc cv wcel wnf wal df-nfc nfnf1 nfal nfxfr ) ABDCEBFZAGZCHAACBINAC
|
|
MAJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
nfab.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfab $p |- F/_ x { y | ph } $=
|
|
( vz cab nfsab nfci ) BEACFABCEDGH $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfaba1 $p |- F/_ x { y | A. x ph } $=
|
|
( wal nfa1 nfab ) ABDBCABEF $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $.
|
|
nfnfc.1 $e |- F/_ x A $.
|
|
$( Hypothesis builder for ` F/_ y A ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnfc $p |- F/ x F/_ y A $=
|
|
( vz wnfc cv wcel wnf wal df-nfc nfcri nfnf nfal nfxfr ) BCFEGCHZBIZEJABE
|
|
CKQAEPABAECDLMNO $.
|
|
|
|
nfeq.2 $e |- F/_ x B $.
|
|
$( Hypothesis builder for equality. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfeq $p |- F/ x A = B $=
|
|
( vz wceq cv wcel wb wal dfcleq nfcri nfbi nfal nfxfr ) BCGFHZBIZQCIZJZFK
|
|
AFBCLTAFRSAAFBDMAFCEMNOP $.
|
|
|
|
$( Hypothesis builder for elementhood. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfel $p |- F/ x A e. B $=
|
|
( vz wcel cv wceq wa wex df-clel nfcv nfeq nfcri nfan nfex nfxfr ) BCGFHZ
|
|
BIZSCGZJZFKAFBCLUBAFTUAAASBASMDNAFCEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
nfeq1.1 $e |- F/_ x A $.
|
|
$( Hypothesis builder for equality, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfeq1 $p |- F/ x A = B $=
|
|
( nfcv nfeq ) ABCDACEF $.
|
|
|
|
$( Hypothesis builder for elementhood, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfel1 $p |- F/ x A e. B $=
|
|
( nfcv nfel ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
nfeq2.1 $e |- F/_ x B $.
|
|
$( Hypothesis builder for equality, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfeq2 $p |- F/ x A = B $=
|
|
( nfcv nfeq ) ABCABEDF $.
|
|
|
|
$( Hypothesis builder for elementhood, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfel2 $p |- F/ x A e. B $=
|
|
( nfcv nfel ) ABCABEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfeqd.1 $e |- ( ph -> F/_ x A ) $.
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcrd $p |- ( ph -> F/ x y e. A ) $=
|
|
( wnfc cv wcel wnf nfcr syl ) ABDFCGDHBIEBCDJK $.
|
|
|
|
$d y ph $.
|
|
nfeqd.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Hypothesis builder for equality. (Contributed by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfeqd $p |- ( ph -> F/ x A = B ) $=
|
|
( vy wceq cv wcel wb wal dfcleq nfv nfcrd nfbid nfald nfxfrd ) CDHGIZCJZS
|
|
DJZKZGLABGCDMAUBBGAGNATUABABGCEOABGDFOPQR $.
|
|
|
|
$( Hypothesis builder for elementhood. (Contributed by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfeld $p |- ( ph -> F/ x A e. B ) $=
|
|
( vy wcel cv wceq wa wex df-clel nfv nfcvd nfeqd nfcrd nfand nfexd nfxfrd
|
|
) CDHGIZCJZUADHZKZGLABGCDMAUDBGAGNAUBUCBABUACABUAOEPABGDFQRST $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w A $. $d w B $.
|
|
drnfc1.1 $e |- ( A. x x = y -> A = B ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
drnfc1 $p |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) $=
|
|
( vw weq wal cv wcel wnf wnfc eleq2d drnf1 dral2 df-nfc 3bitr4g ) ABGAHZF
|
|
IZCJZAKZFHSDJZBKZFHACLBDLUAUCABFTUBABRCDSEMNOAFCPBFDPQ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
drnfc2 $p |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) $=
|
|
( vw weq wal cv wcel wnf wnfc eleq2d drnf2 dral2 df-nfc 3bitr4g ) ABHAIZG
|
|
JZDKZCLZGITEKZCLZGICDMCEMUBUDABGUAUCABCSDETFNOPCGDQCGEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $.
|
|
nfabd.1 $e |- F/ y ph $.
|
|
nfabd.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfabd $p |- ( ph -> F/_ x { y | ps } ) $=
|
|
( vz cab nfv cv wcel wsb df-clab nfsbd nfxfrd nfcd ) ACGBDHZAGIGJQKBDGLAC
|
|
BGDMABDGCEFNOP $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w A $. $d w B $. $d w ph $.
|
|
dvelimdc.1 $e |- F/ x ph $.
|
|
dvelimdc.2 $e |- F/ z ph $.
|
|
dvelimdc.3 $e |- ( ph -> F/_ x A ) $.
|
|
dvelimdc.4 $e |- ( ph -> F/_ z B ) $.
|
|
dvelimdc.5 $e |- ( ph -> ( z = y -> A = B ) ) $.
|
|
$( Deduction form of ~ dvelimc . (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
dvelimdc $p |- ( ph -> ( -. A. x x = y -> F/_ x B ) ) $=
|
|
( vw weq wal wn wnfc wa nfv wcel nfcrd cv wnf wceq wb eleq2 syl6 dvelimdf
|
|
imp nfcd ex ) ABCMBNOZBFPAUKQZBLFULLRAUKLUAZFSZBUBAUMESZUNBCDGHABLEITADLF
|
|
JTADCMEFUCUOUNUDKEFUMUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
dvelimc.1 $e |- F/_ x A $.
|
|
dvelimc.2 $e |- F/_ z B $.
|
|
dvelimc.3 $e |- ( z = y -> A = B ) $.
|
|
$( Version of ~ dvelim for classes. (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
dvelimc $p |- ( -. A. x x = y -> F/_ x B ) $=
|
|
( weq wal wn wnfc wi wtru nftru a1i wceq dvelimdc trud ) ABIAJKAELMNABCDE
|
|
AOCOADLNFPCELNGPCBIDEQMNHPRS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( If ` x ` and ` y ` are distinct, then ` x ` is not free in ` y ` .
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfcvf $p |- ( -. A. x x = y -> F/_ x y ) $=
|
|
( vz cv nfcv weq id dvelimc ) ABCCDZBDZAIECJECBFGH $.
|
|
|
|
$( If ` x ` and ` y ` are distinct, then ` y ` is not free in ` x ` .
|
|
(Contributed by Mario Carneiro, 5-Dec-2016.) $)
|
|
nfcvf2 $p |- ( -. A. x x = y -> F/_ y x ) $=
|
|
( cv wnfc nfcvf naecoms ) BACDBABAEF $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
cleqf.1 $e |- F/_ x A $.
|
|
cleqf.2 $e |- F/_ x B $.
|
|
$( Establish equality between classes, using bound-variable hypotheses
|
|
instead of distinct variable conditions. See also ~ cleqh .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
cleqf $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( vy wceq cv wcel wb wal dfcleq nfv nfcri nfbi eleq1 bibi12d cbval bitr4i
|
|
) BCGFHZBIZTCIZJZFKAHZBIZUDCIZJZAKFBCLUGUCAFUGFMUAUBAAFBDNAFCENOUDTGUEUAU
|
|
FUBUDTBPUDTCPQRS $.
|
|
$}
|
|
|
|
${
|
|
abid2f.1 $e |- F/_ x A $.
|
|
$( A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|
|
(Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
abid2f $p |- { x | x e. A } = A $=
|
|
( cv wcel cab wceq wb wal nfab1 cleqf abid bibi2i albii bitri biid mpgbir
|
|
eqcomi ) BADZBEZAFZBUAGZTTHZAUBTSUAEZHZAIUCAIABUACTAJKUEUCAUDTTTALMNOTPQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d v A $. $d x z v $. $d y z v $. $d v ph $.
|
|
sbabel.1 $e |- F/_ x A $.
|
|
$( Theorem to move a substitution in and out of a class abstraction.
|
|
(Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
sbabel $p |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A ) $=
|
|
( vv cv cab wceq wcel wa wex wsb wb wal sbf abeq2 sbbii 3bitr4i sbex sban
|
|
nfv sbrbis sbalv nfcri anbi12i bitri exbii df-clel ) GHZADIZJZUKEKZLZGMZB
|
|
CNZUKABCNZDIZJZUNLZGMZULEKZBCNUSEKUQUOBCNZGMVBUOGBCUAVDVAGVDUMBCNZUNBCNZL
|
|
VAUMUNBCUBVEUTVFUNDHUKKZAOZDPZBCNVGUROZDPVEUTVHVJBCDVGVGABCVGBCVGBUCQUDUE
|
|
UMVIBCADUKRSURDUKRTUNBCBGEFUFQUGUHUIUHVCUPBCGULEUJSGUSEUJT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Negated equality and membership
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Negated equality
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
$c =/= $. $( Not equal to (equal sign with slash through it). $)
|
|
|
|
$( Extend wff notation to include inequality. $)
|
|
wne $a wff A =/= B $.
|
|
|
|
$( Define inequality. (Contributed by NM, 5-Aug-1993.) $)
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df-ne $a |- ( A =/= B <-> -. A = B ) $.
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${
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neii.1 $e |- A =/= B $.
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$( Inference associated with ~ df-ne . (Contributed by BJ, 7-Jul-2018.) $)
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neii $p |- -. A = B $=
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( wne wceq wn df-ne mpbi ) ABDABEFCABGH $.
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$}
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${
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neir.1 $e |- -. A = B $.
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$( Inference associated with ~ df-ne . (Contributed by BJ, 7-Jul-2018.) $)
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neir $p |- A =/= B $=
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( wne wceq wn df-ne mpbir ) ABDABEFCABGH $.
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$}
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$( Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.) $)
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nner $p |- ( A = B -> -. A =/= B ) $=
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( wne wceq wn df-ne biimpi con2i ) ABCZABDZIJEABFGH $.
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$( Negation of inequality where equality is decidable. (Contributed by Jim
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Kingdon, 15-May-2018.) $)
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nnedc $p |- ( DECID A = B -> ( -. A =/= B <-> A = B ) ) $=
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( wceq wdc wne wn wb df-ne a1i con2biidc bicomd ) ABCZDZLABEZFNLNLFGMABHIJK
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$.
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${
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dcned.eq $e |- ( ph -> DECID A = B ) $.
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$( Decidable equality implies decidable negated equality. (Contributed by
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Jim Kingdon, 3-May-2020.) $)
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dcned $p |- ( ph -> DECID A =/= B ) $=
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( wceq wn wdc wne dcn syl df-ne dcbii sylibr ) ABCEZFZGZBCHZGANGPDNIJQOBC
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KLM $.
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$}
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${
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neqned.1 $e |- ( ph -> -. A = B ) $.
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$( If it is not the case that two classes are equal, they are unequal.
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Converse of ~ neneqd . One-way deduction form of ~ df-ne .
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(Contributed by David Moews, 28-Feb-2017.) Allow a shortening of
|
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~ necon3bi . (Revised by Wolf Lammen, 22-Nov-2019.) $)
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neqned $p |- ( ph -> A =/= B ) $=
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( wceq wn wne df-ne sylibr ) ABCEFBCGDBCHI $.
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$}
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$( From non equality to inequality. (Contributed by Glauco Siliprandi,
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11-Dec-2019.) $)
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neqne $p |- ( -. A = B -> A =/= B ) $=
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( wceq wn id neqned ) ABCDZABGEF $.
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$( No class is unequal to itself. (Contributed by Stefan O'Rear,
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|
1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
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neirr $p |- -. A =/= A $=
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( wne wn wceq eqid notnoti df-ne notbii mpbir ) AABZCAADZCZCKAEFJLAAGHI $.
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$( A contradiction concerning equality implies anything. (Contributed by
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Alexander van der Vekens, 25-Jan-2018.) $)
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eqneqall $p |- ( A = B -> ( A =/= B -> ph ) ) $=
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( wne wceq wn df-ne pm2.24 syl5bi ) BCDBCEZFJABCGJAHI $.
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$( Decidable equality expressed in terms of ` =/= ` . Basically the same as
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~ df-dc . (Contributed by Jim Kingdon, 14-Mar-2020.) $)
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dcne $p |- ( DECID A = B <-> ( A = B \/ A =/= B ) ) $=
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( wceq wdc wn wo wne df-dc df-ne orbi2i bitr4i ) ABCZDLLEZFLABGZFLHNMLABIJK
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$.
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$( Law of noncontradiction with equality and inequality. (Contributed by NM,
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|
3-Feb-2012.) $)
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nonconne $p |- -. ( A = B /\ A =/= B ) $=
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( wceq wne wa wn pm3.24 df-ne anbi2i mtbir ) ABCZABDZEKKFZEKGLMKABHIJ $.
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$( Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) $)
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neeq1 $p |- ( A = B -> ( A =/= C <-> B =/= C ) ) $=
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|
( wceq wn wne eqeq1 notbid df-ne 3bitr4g ) ABDZACDZEBCDZEACFBCFKLMABCGHACIB
|
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CIJ $.
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$( Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) $)
|
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neeq2 $p |- ( A = B -> ( C =/= A <-> C =/= B ) ) $=
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|
( wceq wn wne eqeq2 notbid df-ne 3bitr4g ) ABDZCADZECBDZECAFCBFKLMABCGHCAIC
|
|
BIJ $.
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${
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|
neeq1i.1 $e |- A = B $.
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$( Inference for inequality. (Contributed by NM, 29-Apr-2005.) $)
|
|
neeq1i $p |- ( A =/= C <-> B =/= C ) $=
|
|
( wceq wne wb neeq1 ax-mp ) ABEACFBCFGDABCHI $.
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$( Inference for inequality. (Contributed by NM, 29-Apr-2005.) $)
|
|
neeq2i $p |- ( C =/= A <-> C =/= B ) $=
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( wceq wne wb neeq2 ax-mp ) ABECAFCBFGDABCHI $.
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neeq12i.2 $e |- C = D $.
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$( Inference for inequality. (Contributed by NM, 24-Jul-2012.) $)
|
|
neeq12i $p |- ( A =/= C <-> B =/= D ) $=
|
|
( wne neeq2i neeq1i bitri ) ACGADGBDGCDAFHABDEIJ $.
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|
$}
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${
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|
neeq1d.1 $e |- ( ph -> A = B ) $.
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|
$( Deduction for inequality. (Contributed by NM, 25-Oct-1999.) $)
|
|
neeq1d $p |- ( ph -> ( A =/= C <-> B =/= C ) ) $=
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|
( wceq wne wb neeq1 syl ) ABCFBDGCDGHEBCDIJ $.
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$( Deduction for inequality. (Contributed by NM, 25-Oct-1999.) $)
|
|
neeq2d $p |- ( ph -> ( C =/= A <-> C =/= B ) ) $=
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|
( wceq wne wb neeq2 syl ) ABCFDBGDCGHEBCDIJ $.
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neeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Deduction for inequality. (Contributed by NM, 24-Jul-2012.) $)
|
|
neeq12d $p |- ( ph -> ( A =/= C <-> B =/= D ) ) $=
|
|
( wne neeq1d neeq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
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|
$}
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${
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|
neneqd.1 $e |- ( ph -> A =/= B ) $.
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|
$( Deduction eliminating inequality definition. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
neneqd $p |- ( ph -> -. A = B ) $=
|
|
( wne wceq wn df-ne sylib ) ABCEBCFGDBCHI $.
|
|
$}
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|
$( From inequality to non equality. (Contributed by Glauco Siliprandi,
|
|
11-Dec-2019.) $)
|
|
neneq $p |- ( A =/= B -> -. A = B ) $=
|
|
( wne id neneqd ) ABCZABFDE $.
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${
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|
eqnetr.1 $e |- A = B $.
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|
eqnetr.2 $e |- B =/= C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetri $p |- A =/= C $=
|
|
( wne neeq1i mpbir ) ACFBCFEABCDGH $.
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|
$}
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${
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|
eqnetrd.1 $e |- ( ph -> A = B ) $.
|
|
eqnetrd.2 $e |- ( ph -> B =/= C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrd $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq1d mpbird ) ABDGCDGFABCDEHI $.
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|
$}
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${
|
|
eqnetrr.1 $e |- A = B $.
|
|
eqnetrr.2 $e |- A =/= C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrri $p |- B =/= C $=
|
|
( eqcomi eqnetri ) BACABDFEG $.
|
|
$}
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|
|
${
|
|
eqnetrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqnetrrd.2 $e |- ( ph -> A =/= C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrrd $p |- ( ph -> B =/= C ) $=
|
|
( eqcomd eqnetrd ) ACBDABCEGFH $.
|
|
$}
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|
|
${
|
|
neeqtr.1 $e |- A =/= B $.
|
|
neeqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtri $p |- A =/= C $=
|
|
( wne neeq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
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|
|
${
|
|
neeqtrd.1 $e |- ( ph -> A =/= B ) $.
|
|
neeqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrd $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
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|
|
${
|
|
neeqtrr.1 $e |- A =/= B $.
|
|
neeqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrri $p |- A =/= C $=
|
|
( eqcomi neeqtri ) ABCDCBEFG $.
|
|
$}
|
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|
|
${
|
|
neeqtrrd.1 $e |- ( ph -> A =/= B ) $.
|
|
neeqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrrd $p |- ( ph -> A =/= C ) $=
|
|
( eqcomd neeqtrd ) ABCDEADCFGH $.
|
|
$}
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|
|
${
|
|
syl5eqner.1 $e |- B = A $.
|
|
syl5eqner.2 $e |- ( ph -> B =/= C ) $.
|
|
$( B chained equality inference for inequality. (Contributed by NM,
|
|
6-Jun-2012.) $)
|
|
syl5eqner $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq1i sylib ) ACDGBDGFCBDEHI $.
|
|
$}
|
|
|
|
${
|
|
3netr3d.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr3d.2 $e |- ( ph -> A = C ) $.
|
|
3netr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr3d $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12d mpbid ) ABCIDEIFABDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr4d.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr4d.2 $e |- ( ph -> C = A ) $.
|
|
3netr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr4d $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12d mpbird ) ADEIBCIFADBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr3g.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr3g.2 $e |- A = C $.
|
|
3netr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr3g $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr4g.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr4g.2 $e |- C = A $.
|
|
3netr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 14-Jun-2012.) $)
|
|
3netr4g $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
necon3abii.1 $e |- ( A = B <-> ph ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
9-Nov-2007.) $)
|
|
necon3abii $p |- ( A =/= B <-> -. ph ) $=
|
|
( wne wceq df-ne xchbinx ) BCEBCFABCGDH $.
|
|
$}
|
|
|
|
${
|
|
necon3bbii.1 $e |- ( ph <-> A = B ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon3bbii $p |- ( -. ph <-> A =/= B ) $=
|
|
( wne wn wceq bicomi necon3abii ) BCEAFABCABCGDHIH $.
|
|
$}
|
|
|
|
${
|
|
necon3bii.1 $e |- ( A = B <-> C = D ) $.
|
|
$( Inference from equality to inequality. (Contributed by NM,
|
|
23-Feb-2005.) $)
|
|
necon3bii $p |- ( A =/= B <-> C =/= D ) $=
|
|
( wne wceq wn necon3abii df-ne bitr4i ) ABFCDGZHCDFLABEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon3abid.1 $e |- ( ph -> ( A = B <-> ps ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
21-Mar-2007.) $)
|
|
necon3abid $p |- ( ph -> ( A =/= B <-> -. ps ) ) $=
|
|
( wne wceq wn df-ne notbid syl5bb ) CDFCDGZHABHCDIALBEJK $.
|
|
$}
|
|
|
|
${
|
|
necon3bbid.1 $e |- ( ph -> ( ps <-> A = B ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
2-Jun-2007.) $)
|
|
necon3bbid $p |- ( ph -> ( -. ps <-> A =/= B ) ) $=
|
|
( wne wn wceq bicomd necon3abid ) ACDFBGABCDABCDHEIJI $.
|
|
$}
|
|
|
|
${
|
|
necon3bid.1 $e |- ( ph -> ( A = B <-> C = D ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3bid $p |- ( ph -> ( A =/= B <-> C =/= D ) ) $=
|
|
( wne wceq wn df-ne necon3bbid syl5bb ) BCGBCHZIADEGBCJAMDEFKL $.
|
|
$}
|
|
|
|
${
|
|
necon3ad.1 $e |- ( ph -> ( ps -> A = B ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
|
|
necon3ad $p |- ( ph -> ( A =/= B -> -. ps ) ) $=
|
|
( wne wceq wn df-ne con3d syl5bi ) CDFCDGZHABHCDIABLEJK $.
|
|
$}
|
|
|
|
${
|
|
necon3bd.1 $e |- ( ph -> ( A = B -> ps ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
|
|
necon3bd $p |- ( ph -> ( -. ps -> A =/= B ) ) $=
|
|
( wn wceq wne con3d df-ne syl6ibr ) ABFCDGZFCDHALBEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon3d.1 $e |- ( ph -> ( A = B -> C = D ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
10-Jun-2006.) $)
|
|
necon3d $p |- ( ph -> ( C =/= D -> A =/= B ) ) $=
|
|
( wne wceq wn necon3ad df-ne syl6ibr ) ADEGBCHZIBCGAMDEFJBCKL $.
|
|
$}
|
|
|
|
$( Characterization of inequality in terms of reversed equality (see
|
|
~ bicom ). (Contributed by BJ, 7-Jul-2018.) $)
|
|
nesym $p |- ( A =/= B <-> -. B = A ) $=
|
|
( wceq eqcom necon3abii ) BACABABDE $.
|
|
|
|
${
|
|
nesymi.1 $e |- A =/= B $.
|
|
$( Inference associated with ~ nesym . (Contributed by BJ, 7-Jul-2018.) $)
|
|
nesymi $p |- -. B = A $=
|
|
( wne wceq wn nesym mpbi ) ABDBAEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
nesymir.1 $e |- -. A = B $.
|
|
$( Inference associated with ~ nesym . (Contributed by BJ, 7-Jul-2018.) $)
|
|
nesymir $p |- B =/= A $=
|
|
( wne wceq wn nesym mpbir ) BADABEFCBAGH $.
|
|
$}
|
|
|
|
${
|
|
necon3i.1 $e |- ( A = B -> C = D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
9-Aug-2006.) $)
|
|
necon3i $p |- ( C =/= D -> A =/= B ) $=
|
|
( wceq wi wne id necon3d ax-mp ) ABFCDFGZCDHABHGELABCDLIJK $.
|
|
$}
|
|
|
|
${
|
|
necon3ai.1 $e |- ( ph -> A = B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
|
|
necon3ai $p |- ( A =/= B -> -. ph ) $=
|
|
( wne wceq wn df-ne con3i sylbi ) BCEBCFZGAGBCHAKDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon3bi.1 $e |- ( A = B -> ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $)
|
|
necon3bi $p |- ( -. ph -> A =/= B ) $=
|
|
( wn wceq wne con3i df-ne sylibr ) AEBCFZEBCGKADHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1aidc.1 $e |- ( DECID ph -> ( -. ph -> A = B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
15-May-2018.) $)
|
|
necon1aidc $p |- ( DECID ph -> ( A =/= B -> ph ) ) $=
|
|
( wne wceq wn wdc df-ne wi con1dc mpd syl5bi ) BCEBCFZGZAHZABCIPAGNJOAJDA
|
|
NKLM $.
|
|
$}
|
|
|
|
${
|
|
necon1bidc.1 $e |- ( DECID A = B -> ( A =/= B -> ph ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
15-May-2018.) $)
|
|
necon1bidc $p |- ( DECID A = B -> ( -. ph -> A = B ) ) $=
|
|
( wceq wdc wn wi wne df-ne syl5bir con1dc mpd ) BCEZFZNGZAHAGNHPBCIOABCJD
|
|
KNALM $.
|
|
$}
|
|
|
|
${
|
|
necon1idc.1 $e |- ( A =/= B -> C = D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon1idc $p |- ( DECID A = B -> ( C =/= D -> A = B ) ) $=
|
|
( wceq wn wi wdc wne df-ne sylbir a1i necon1aidc ) ABFZCDOGZCDFZHOIPABJQA
|
|
BKELMN $.
|
|
$}
|
|
|
|
${
|
|
necon2ai.1 $e |- ( A = B -> -. ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.) $)
|
|
necon2ai $p |- ( ph -> A =/= B ) $=
|
|
( wceq wn wne con2i df-ne sylibr ) ABCEZFBCGKADHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
necon2bi.1 $e |- ( ph -> A =/= B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
1-Apr-2007.) $)
|
|
necon2bi $p |- ( A = B -> -. ph ) $=
|
|
( wceq neneqd con2i ) ABCEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
necon2i.1 $e |- ( A = B -> C =/= D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
18-Mar-2007.) $)
|
|
necon2i $p |- ( C = D -> A =/= B ) $=
|
|
( wceq neneqd necon2ai ) CDFABABFCDEGH $.
|
|
$}
|
|
|
|
${
|
|
necon2ad.1 $e |- ( ph -> ( A = B -> -. ps ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.) $)
|
|
necon2ad $p |- ( ph -> ( ps -> A =/= B ) ) $=
|
|
( wceq wn wne con2d df-ne syl6ibr ) ABCDFZGCDHALBEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon2bd.1 $e |- ( ph -> ( ps -> A =/= B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon2bd $p |- ( ph -> ( A = B -> -. ps ) ) $=
|
|
( wceq wne wn df-ne syl6ib con2d ) ABCDFZABCDGLHECDIJK $.
|
|
$}
|
|
|
|
${
|
|
necon2d.1 $e |- ( ph -> ( A = B -> C =/= D ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
28-Dec-2008.) $)
|
|
necon2d $p |- ( ph -> ( C = D -> A =/= B ) ) $=
|
|
( wceq wne wn df-ne syl6ib necon2ad ) ADEGZBCABCGDEHMIFDEJKL $.
|
|
$}
|
|
|
|
${
|
|
necon1abiidc.1 $e |- ( DECID ph -> ( -. ph <-> A = B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon1abiidc $p |- ( DECID ph -> ( A =/= B <-> ph ) ) $=
|
|
( wne wceq wn wdc df-ne con1biidc syl5bb ) BCEBCFZGAHABCIALDJK $.
|
|
$}
|
|
|
|
${
|
|
necon1bbiidc.1 $e |- ( DECID A = B -> ( A =/= B <-> ph ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon1bbiidc $p |- ( DECID A = B -> ( -. ph <-> A = B ) ) $=
|
|
( wceq wn wne wdc df-ne syl5bbr con1biidc ) BCEZALFBCGLHABCIDJK $.
|
|
$}
|
|
|
|
${
|
|
necon1abiddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps <-> A = B ) ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon1abiddc $p |- ( ph -> ( DECID ps -> ( A =/= B <-> ps ) ) ) $=
|
|
( wdc wceq wn wb wne con1biddc df-ne bibi1i syl6ibr ) ABFCDGZHZBICDJZBIAB
|
|
OEKQPBCDLMN $.
|
|
$}
|
|
|
|
${
|
|
necon1bbiddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B <-> ps ) ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon1bbiddc $p |- ( ph -> ( DECID A = B -> ( -. ps <-> A = B ) ) ) $=
|
|
( wceq wdc wne wb wn df-ne bibi1i syl6ib con1biddc ) ACDFZBAOGCDHZBIOJZBI
|
|
EPQBCDKLMN $.
|
|
$}
|
|
|
|
${
|
|
necon2abiidc.1 $e |- ( DECID ph -> ( A = B <-> -. ph ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon2abiidc $p |- ( DECID ph -> ( ph <-> A =/= B ) ) $=
|
|
( wdc wne wceq wn bicomd necon1abiidc ) AEZBCFAABCKBCGAHDIJI $.
|
|
$}
|
|
|
|
${
|
|
necon2bbii.1 $e |- ( DECID A = B -> ( ph <-> A =/= B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon2bbii $p |- ( DECID A = B -> ( A = B <-> -. ph ) ) $=
|
|
( wceq wdc wn wne bicomd necon1bbiidc ) BCEZFZAGKABCLABCHDIJI $.
|
|
$}
|
|
|
|
${
|
|
necon2abiddc.1 $e |- ( ph -> ( DECID ps -> ( A = B <-> -. ps ) ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon2abiddc $p |- ( ph -> ( DECID ps -> ( ps <-> A =/= B ) ) ) $=
|
|
( wdc wne wb wceq wn bicom syl6ib necon1abiddc ) ABFZCDGZBHBOHABCDANCDIZB
|
|
JZHQPHEPQKLMOBKL $.
|
|
$}
|
|
|
|
${
|
|
necon2bbiddc.1 $e |- ( ph -> ( DECID A = B -> ( ps <-> A =/= B ) ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon2bbiddc $p |- ( ph -> ( DECID A = B -> ( A = B <-> -. ps ) ) ) $=
|
|
( wceq wdc wn wb wne bicom syl6ib necon1bbiddc ) ACDFZGZBHZNINPIABCDAOBCD
|
|
JZIQBIEBQKLMPNKL $.
|
|
$}
|
|
|
|
${
|
|
necon4aidc.1 $e |- ( DECID A = B -> ( A =/= B -> -. ph ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon4aidc $p |- ( DECID A = B -> ( ph -> A = B ) ) $=
|
|
( wceq wdc wn wi wne df-ne syl5bir condc mpd ) BCEZFZNGZAGZHANHPBCIOQBCJD
|
|
KNALM $.
|
|
$}
|
|
|
|
${
|
|
necon4idc.1 $e |- ( DECID A = B -> ( A =/= B -> C =/= D ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
16-May-2018.) $)
|
|
necon4idc $p |- ( DECID A = B -> ( C = D -> A = B ) ) $=
|
|
( wceq wdc wne wn df-ne syl6ib necon4aidc ) CDFZABABFGABHCDHMIECDJKL $.
|
|
$}
|
|
|
|
${
|
|
necon4addc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> -. ps ) ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
17-May-2018.) $)
|
|
necon4addc $p |- ( ph -> ( DECID A = B -> ( ps -> A = B ) ) ) $=
|
|
( wceq wdc wne wn wi df-ne imbi1i condc syl5bi sylcom ) ACDFZGZCDHZBIZJZB
|
|
PJZETPIZSJQUARUBSCDKLPBMNO $.
|
|
$}
|
|
|
|
${
|
|
necon4bddc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> A =/= B ) ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
17-May-2018.) $)
|
|
necon4bddc $p |- ( ph -> ( DECID ps -> ( A = B -> ps ) ) ) $=
|
|
( wdc wn wceq wi wne df-ne syl8ib condc sylcom ) ABFZBGZCDHZGZIQBIAOPCDJR
|
|
ECDKLBQMN $.
|
|
$}
|
|
|
|
${
|
|
necon4ddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> C =/= D ) ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by Jim Kingdon,
|
|
17-May-2018.) $)
|
|
necon4ddc $p |- ( ph -> ( DECID A = B -> ( C = D -> A = B ) ) ) $=
|
|
( wceq wdc wn wi wne df-ne imbi12i syl6ib condc sylcom ) ABCGZHZQIZDEGZIZ
|
|
JZTQJARBCKZDEKZJUBFUCSUDUABCLDELMNQTOP $.
|
|
$}
|
|
|
|
${
|
|
necon4abiddc.1 $e |- ( ph -> ( DECID A = B -> ( DECID ps ->
|
|
( A =/= B <-> -. ps ) ) ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by Jim
|
|
Kingdon, 18-May-2018.) $)
|
|
necon4abiddc $p |- ( ph -> ( DECID A = B -> ( DECID ps ->
|
|
( A = B <-> ps ) ) ) ) $=
|
|
( wceq wdc wne wn wb df-ne bibi1i syl8ib con4biddc ) ACDFZBAOGBGCDHZBIZJO
|
|
IZQJEPRQCDKLMN $.
|
|
$}
|
|
|
|
${
|
|
necon4bbiddc.1 $e |- ( ph -> ( DECID ps -> ( DECID A = B ->
|
|
( -. ps <-> A =/= B ) ) ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by Jim
|
|
Kingdon, 19-May-2018.) $)
|
|
necon4bbiddc $p |- ( ph -> ( DECID ps -> ( DECID A = B ->
|
|
( ps <-> A = B ) ) ) ) $=
|
|
( wdc wceq wb wne wn bicom syl8ib com23 necon4abiddc ) ABFZCDGZFZPBHZBPHA
|
|
QORABCDAOQCDIZBJZHZAOQTSHUAETSKLMNMPBKL $.
|
|
$}
|
|
|
|
${
|
|
necon4biddc.1 $e |- ( ph -> ( DECID A = B -> ( DECID C = D ->
|
|
( A =/= B <-> C =/= D ) ) ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by Jim
|
|
Kingdon, 19-May-2018.) $)
|
|
necon4biddc $p |- ( ph -> ( DECID A = B -> ( DECID C = D ->
|
|
( A = B <-> C = D ) ) ) ) $=
|
|
( wceq wdc wne wb wn df-ne bibi2i syl8ib necon4abiddc ) ADEGZBCABCGHPHBCI
|
|
ZDEIZJQPKZJFRSQDELMNO $.
|
|
$}
|
|
|
|
${
|
|
necon1addc.1 $e |- ( ph -> ( DECID ps -> ( -. ps -> A = B ) ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
|
|
19-May-2018.) $)
|
|
necon1addc $p |- ( ph -> ( DECID ps -> ( A =/= B -> ps ) ) ) $=
|
|
( wne wceq wn wdc df-ne wi con1dc sylcom syl7bi ) CDFCDGZHZABIZBCDJAQBHOK
|
|
PBKEBOLMN $.
|
|
$}
|
|
|
|
${
|
|
necon1bddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> ps ) ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by Jim Kingdon,
|
|
19-May-2018.) $)
|
|
necon1bddc $p |- ( ph -> ( DECID A = B -> ( -. ps -> A = B ) ) ) $=
|
|
( wceq wdc wn wi wne df-ne imbi1i syl6ib con1dc sylcom ) ACDFZGZPHZBIZBHP
|
|
IAQCDJZBISETRBCDKLMPBNO $.
|
|
$}
|
|
|
|
${
|
|
necon1ddc.1 $e |- ( ph -> ( DECID A = B -> ( A =/= B -> C = D ) ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by Jim
|
|
Kingdon, 19-May-2018.) $)
|
|
necon1ddc $p |- ( ph -> ( DECID A = B -> ( C =/= D -> A = B ) ) ) $=
|
|
( wne wceq wn wdc df-ne necon1bddc syl7bi ) DEGDEHZIABCHZJODEKANBCFLM $.
|
|
$}
|
|
|
|
${
|
|
neneqad.1 $e |- ( ph -> -. A = B ) $.
|
|
$( If it is not the case that two classes are equal, they are unequal.
|
|
Converse of ~ neneqd . One-way deduction form of ~ df-ne .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
neneqad $p |- ( ph -> A =/= B ) $=
|
|
( wceq con2i necon2ai ) ABCABCEDFG $.
|
|
$}
|
|
|
|
$( Contraposition law for inequality. (Contributed by Jim Kingdon,
|
|
19-May-2018.) $)
|
|
nebidc $p |- ( DECID A = B -> ( DECID C = D ->
|
|
( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) ) ) $=
|
|
( wceq wdc wb wne wa id necon3bid wi a1d necon4biddc com3l imp impbid2 ex )
|
|
ABEZFZCDEZFZSUAGZABHCDHGZGTUBIUCUDUCABCDUCJKTUBUDUCLUDTUBUCUDABCDUDUBUDLTUD
|
|
UDUBUDJMMNOPQR $.
|
|
|
|
$( Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew
|
|
Salmon, 3-Jun-2011.) $)
|
|
pm13.18 $p |- ( ( A = B /\ A =/= C ) -> B =/= C ) $=
|
|
( wceq wne eqeq1 biimprd necon3d imp ) ABDZACEBCEJBCACJACDBCDABCFGHI $.
|
|
|
|
$( Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew
|
|
Salmon, 3-Jun-2011.) $)
|
|
pm13.181 $p |- ( ( A = B /\ B =/= C ) -> A =/= C ) $=
|
|
( wceq wne eqcom pm13.18 sylanb ) ABDBADBCEACEABFBACGH $.
|
|
|
|
${
|
|
pm2.21ddne.1 $e |- ( ph -> A = B ) $.
|
|
pm2.21ddne.2 $e |- ( ph -> A =/= B ) $.
|
|
$( A contradiction implies anything. Equality/inequality deduction form.
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
pm2.21ddne $p |- ( ph -> ps ) $=
|
|
( wceq neneqd pm2.21dd ) ACDGBEACDFHI $.
|
|
$}
|
|
|
|
$( Commutation of inequality. (Contributed by NM, 14-May-1999.) $)
|
|
necom $p |- ( A =/= B <-> B =/= A ) $=
|
|
( eqcom necon3bii ) ABBAABCD $.
|
|
|
|
${
|
|
necomi.1 $e |- A =/= B $.
|
|
$( Inference from commutative law for inequality. (Contributed by NM,
|
|
17-Oct-2012.) $)
|
|
necomi $p |- B =/= A $=
|
|
( wne necom mpbi ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
necomd.1 $e |- ( ph -> A =/= B ) $.
|
|
$( Deduction from commutative law for inequality. (Contributed by NM,
|
|
12-Feb-2008.) $)
|
|
necomd $p |- ( ph -> B =/= A ) $=
|
|
( wne necom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
$( A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.) $)
|
|
neanior $p |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) ) $=
|
|
( wne wa wceq wn wo df-ne anbi12i pm4.56 bitri ) ABEZCDEZFABGZHZCDGZHZFPRIH
|
|
NQOSABJCDJKPRLM $.
|
|
|
|
$( A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
|
|
(Proof rewritten by Jim Kingdon, 19-May-2018.) $)
|
|
ne3anior $p |- ( ( A =/= B /\ C =/= D /\ E =/= F )
|
|
<-> -. ( A = B \/ C = D \/ E = F ) ) $=
|
|
( wne w3a wceq wn w3o df-ne 3anbi123i 3ioran bitr4i ) ABGZCDGZEFGZHABIZJZCD
|
|
IZJZEFIZJZHSUAUCKJPTQUBRUDABLCDLEFLMSUAUCNO $.
|
|
|
|
${
|
|
nemtbir.1 $e |- A =/= B $.
|
|
nemtbir.2 $e |- ( ph <-> A = B ) $.
|
|
$( An inference from an inequality, related to modus tollens. (Contributed
|
|
by NM, 13-Apr-2007.) $)
|
|
nemtbir $p |- -. ph $=
|
|
( wceq wne wn df-ne mpbi mtbir ) ABCFZBCGLHDBCIJEK $.
|
|
$}
|
|
|
|
$( Two classes are different if they don't contain the same element.
|
|
(Contributed by NM, 3-Feb-2012.) $)
|
|
nelne1 $p |- ( ( A e. B /\ -. A e. C ) -> B =/= C ) $=
|
|
( wcel wn wne wceq eleq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJ
|
|
K $.
|
|
|
|
$( Two classes are different if they don't belong to the same class.
|
|
(Contributed by NM, 25-Jun-2012.) $)
|
|
nelne2 $p |- ( ( A e. C /\ -. B e. C ) -> A =/= B ) $=
|
|
( wcel wn wne wceq eleq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJ
|
|
K $.
|
|
|
|
${
|
|
nfne.1 $e |- F/_ x A $.
|
|
nfne.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by NM,
|
|
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfne $p |- F/ x A =/= B $=
|
|
( wne wceq wn df-ne nfeq nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfned.1 $e |- ( ph -> F/_ x A ) $.
|
|
nfned.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by NM,
|
|
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfned $p |- ( ph -> F/ x A =/= B ) $=
|
|
( wne wceq wn df-ne nfeqd nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Negated membership
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
$c e/ $. $( Not an element of (epsilon with slash through it). $)
|
|
|
|
$( Extend wff notation to include negated membership. $)
|
|
wnel $a wff A e/ B $.
|
|
|
|
$( Define negated membership. (Contributed by NM, 7-Aug-1994.) $)
|
|
df-nel $a |- ( A e/ B <-> -. A e. B ) $.
|
|
|
|
${
|
|
neli.1 $e |- A e/ B $.
|
|
$( Inference associated with ~ df-nel . (Contributed by BJ,
|
|
7-Jul-2018.) $)
|
|
neli $p |- -. A e. B $=
|
|
( wnel wcel wn df-nel mpbi ) ABDABEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
nelir.1 $e |- -. A e. B $.
|
|
$( Inference associated with ~ df-nel . (Contributed by BJ,
|
|
7-Jul-2018.) $)
|
|
nelir $p |- A e/ B $=
|
|
( wnel wcel wn df-nel mpbir ) ABDABEFCABGH $.
|
|
$}
|
|
|
|
$( Equality theorem for negated membership. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
neleq1 $p |- ( A = B -> ( A e/ C <-> B e/ C ) ) $=
|
|
( wceq wcel wn wnel eleq1 notbid df-nel 3bitr4g ) ABDZACEZFBCEZFACGBCGLMNAB
|
|
CHIACJBCJK $.
|
|
|
|
$( Equality theorem for negated membership. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
neleq2 $p |- ( A = B -> ( C e/ A <-> C e/ B ) ) $=
|
|
( wceq wcel wn wnel eleq2 notbid df-nel 3bitr4g ) ABDZCAEZFCBEZFCAGCBGLMNAB
|
|
CHICAJCBJK $.
|
|
|
|
${
|
|
neleq12d.1 $e |- ( ph -> A = B ) $.
|
|
neleq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality theorem for negated membership. (Contributed by FL,
|
|
10-Aug-2016.) $)
|
|
neleq12d $p |- ( ph -> ( A e/ C <-> B e/ D ) ) $=
|
|
( wnel wceq wb neleq1 syl neleq2 bitrd ) ABDHZCDHZCEHZABCIOPJFBCDKLADEIPQ
|
|
JGDECMLN $.
|
|
$}
|
|
|
|
${
|
|
nfnel.1 $e |- F/_ x A $.
|
|
nfnel.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for negated membership. (Contributed
|
|
by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfnel $p |- F/ x A e/ B $=
|
|
( wnel wcel wn df-nel nfel nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfneld.1 $e |- ( ph -> F/_ x A ) $.
|
|
nfneld.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Bound-variable hypothesis builder for negated membership. (Contributed
|
|
by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfneld $p |- ( ph -> F/ x A e/ B ) $=
|
|
( wnel wcel wn df-nel nfeld nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Restricted quantification
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff notation to include restricted universal quantification. $)
|
|
wral $a wff A. x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted existential quantification. $)
|
|
wrex $a wff E. x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted existential uniqueness. $)
|
|
wreu $a wff E! x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted "at most one." $)
|
|
wrmo $a wff E* x e. A ph $.
|
|
|
|
$( Extend class notation to include the restricted class abstraction (class
|
|
builder). $)
|
|
crab $a class { x e. A | ph } $.
|
|
|
|
$( Define restricted universal quantification. Special case of Definition
|
|
4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.) $)
|
|
df-ral $a |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) ) $.
|
|
|
|
$( Define restricted existential quantification. Special case of Definition
|
|
4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.) $)
|
|
df-rex $a |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define restricted existential uniqueness. (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define restricted "at most one". (Contributed by NM, 16-Jun-2017.) $)
|
|
df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define a restricted class abstraction (class builder), which is the class
|
|
of all ` x ` in ` A ` such that ` ph ` is true. Definition of
|
|
[TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.) $)
|
|
df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
ralnex $p |- ( A. x e. A -. ph <-> -. E. x e. A ph ) $=
|
|
( wn wral cv wcel wi wal wrex df-ral wa wex alinexa df-rex xchbinxr bitri )
|
|
ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers. In
|
|
classical logic this would be a biconditional. (Contributed by Jim
|
|
Kingdon, 17-Aug-2018.) $)
|
|
rexnalim $p |- ( E. x e. A -. ph -> -. A. x e. A ph ) $=
|
|
( wn wrex cv wcel wa wex wral df-rex wi wal exanaliim df-ral sylnibr sylbi
|
|
) ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by Jim Kingdon, 17-Aug-2018.) $)
|
|
ralexim $p |- ( A. x e. A ph -> -. E. x e. A -. ph ) $=
|
|
( wn wrex wral rexnalim con2i ) ADBCEABCFABCGH $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by Jim Kingdon, 17-Aug-2018.) $)
|
|
rexalim $p |- ( E. x e. A ph -> -. A. x e. A -. ph ) $=
|
|
( wn wral wrex ralnex biimpi con2i ) ADBCEZABCFZJKDABCGHI $.
|
|
|
|
${
|
|
ralbida.1 $e |- F/ x ph $.
|
|
ralbida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 6-Oct-2003.) $)
|
|
ralbida $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral pm5.74da albid df-ral 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 6-Oct-2003.) $)
|
|
rexbida $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( cv wcel wa wex wrex pm5.32da exbid df-rex 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 4-Mar-1997.) $)
|
|
ralbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( nfv ralbida ) ABCDEADGFH $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 9-Mar-1997.) $)
|
|
rexbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( nfv rexbida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
ralbid.1 $e |- F/ x ph $.
|
|
ralbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 27-Jun-1998.) $)
|
|
ralbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( wb cv wcel adantr ralbida ) ABCDEFABCHDIEJGKL $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 27-Jun-1998.) $)
|
|
rexbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( wb cv wcel adantr rexbida ) ABCDEFABCHDIEJGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 20-Nov-1994.) $)
|
|
ralbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( nfv ralbid ) ABCDEADGFH $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 20-Nov-1994.) $)
|
|
rexbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( nfv rexbid ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidv2.1 $e |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 6-Apr-1997.) $)
|
|
ralbidv2 $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( cv wcel wi wal wral albidv df-ral 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rexbidv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 22-May-1999.) $)
|
|
rexbidv2 $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( cv wcel wa wex wrex exbidv df-rex 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
ralbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding restricted universal quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
|
|
Carneiro, 17-Oct-2016.) $)
|
|
ralbii $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
|
|
( wral wb wtru a1i ralbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
|
|
$( Inference adding restricted existential quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
|
|
Carneiro, 17-Oct-2016.) $)
|
|
rexbii $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
|
|
( wrex wb wtru a1i rexbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
|
|
$( Inference adding two restricted universal quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 1-Aug-2004.) $)
|
|
2ralbii $p |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps ) $=
|
|
( wral ralbii ) ADFHBDFHCEABDFGII $.
|
|
|
|
$( Inference adding two restricted existential quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 11-Nov-1995.) $)
|
|
2rexbii $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
|
|
( wrex rexbii ) ADFHBDFHCEABDFGII $.
|
|
$}
|
|
|
|
${
|
|
ralbii2.1 $e |- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) ) $.
|
|
$( Inference adding different restricted universal quantifiers to each side
|
|
of an equivalence. (Contributed by NM, 15-Aug-2005.) $)
|
|
ralbii2 $p |- ( A. x e. A ph <-> A. x e. B ps ) $=
|
|
( cv wcel wi wal wral albii df-ral 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
rexbii2.1 $e |- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) ) $.
|
|
$( Inference adding different restricted existential quantifiers to each
|
|
side of an equivalence. (Contributed by NM, 4-Feb-2004.) $)
|
|
rexbii2 $p |- ( E. x e. A ph <-> E. x e. B ps ) $=
|
|
( cv wcel wa wex wrex exbii df-rex 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
raleqbii.1 $e |- A = B $.
|
|
raleqbii.2 $e |- ( ps <-> ch ) $.
|
|
$( Equality deduction for restricted universal quantifier, changing both
|
|
formula and quantifier domain. Inference form. (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
raleqbii $p |- ( A. x e. A ps <-> A. x e. B ch ) $=
|
|
( cv wcel eleq2i imbi12i ralbii2 ) ABCDECHZDIMEIABDEMFJGKL $.
|
|
|
|
$( Equality deduction for restricted existential quantifier, changing both
|
|
formula and quantifier domain. Inference form. (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
rexeqbii $p |- ( E. x e. A ps <-> E. x e. B ch ) $=
|
|
( cv wcel eleq2i anbi12i rexbii2 ) ABCDECHZDIMEIABDEMFJGKL $.
|
|
$}
|
|
|
|
${
|
|
ralbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Inference adding restricted universal quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 26-Nov-2000.) $)
|
|
ralbiia $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
|
|
( cv wcel pm5.74i ralbii2 ) ABCDDCFDGABEHI $.
|
|
|
|
$( Inference adding restricted existential quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 26-Oct-1999.) $)
|
|
rexbiia $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
|
|
( cv wcel pm5.32i rexbii2 ) ABCDDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
2rexbiia.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph <-> ps ) ) $.
|
|
$( Inference adding two restricted existential quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 1-Aug-2004.) $)
|
|
2rexbiia $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
|
|
( wrex cv wcel rexbidva rexbiia ) ADFHBDFHCECIEJABDFGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
r2alf.1 $e |- F/_ y A $.
|
|
$( Double restricted universal quantification. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
r2alf $p |- ( A. x e. A A. y e. B ph <->
|
|
A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( wral cv wi wal wa df-ral nfcri 19.21 impexp albii imbi2i 3bitr4i bitr4i
|
|
wcel ) ACEGZBDGBHDTZUAIZBJUBCHETZKAIZCJZBJUABDLUFUCBUBUDAIZIZCJUBUGCJZIUF
|
|
UCUBUGCCBDFMNUEUHCUBUDAOPUAUIUBACELQRPS $.
|
|
|
|
$( Double restricted existential quantification. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
r2exf $p |- ( E. x e. A E. y e. B ph <->
|
|
E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
|
|
( wrex cv wcel wex df-rex nfcri 19.42 anass exbii anbi2i 3bitr4i bitr4i
|
|
wa ) ACEGZBDGBHDIZTSZBJUACHEIZSASZCJZBJTBDKUEUBBUAUCASZSZCJUAUFCJZSUEUBUA
|
|
UFCCBDFLMUDUGCUAUCANOTUHUAACEKPQOR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Double restricted universal quantification. (Contributed by NM,
|
|
19-Nov-1995.) $)
|
|
r2al $p |- ( A. x e. A A. y e. B ph <->
|
|
A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( nfcv r2alf ) ABCDECDFG $.
|
|
|
|
$( Double restricted existential quantification. (Contributed by NM,
|
|
11-Nov-1995.) $)
|
|
r2ex $p |- ( E. x e. A E. y e. B ph <->
|
|
E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
|
|
( nfcv r2exf ) ABCDECDFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
2ralbida.1 $e |- F/ x ph $.
|
|
2ralbida.2 $e |- F/ y ph $.
|
|
2ralbida.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 24-Feb-2004.) $)
|
|
2ralbida $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( wral cv wcel wa nfv nfan wb anassrs ralbida ) ABEGKCEGKDFHADLFMZNBCEGAT
|
|
EITEOPATELGMBCQJRSS $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $.
|
|
2ralbidva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifiers (deduction
|
|
rule). (Contributed by NM, 4-Mar-1997.) $)
|
|
2ralbidva $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( nfv 2ralbida ) ABCDEFGADIAEIHJ $.
|
|
|
|
$( Formula-building rule for restricted existential quantifiers (deduction
|
|
rule). (Contributed by NM, 15-Dec-2004.) $)
|
|
2rexbidva $p |- ( ph ->
|
|
( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
|
|
( wrex cv wcel wa wb anassrs rexbidva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
|
|
OO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifiers (deduction
|
|
rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon
|
|
Jaroszewicz, 16-Mar-2007.) $)
|
|
2ralbidv $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( wral ralbidv ) ABEGICEGIDFABCEGHJJ $.
|
|
|
|
$( Formula-building rule for restricted existential quantifiers (deduction
|
|
rule). (Contributed by NM, 28-Jan-2006.) $)
|
|
2rexbidv $p |- ( ph ->
|
|
( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
|
|
( wrex rexbidv ) ABEGICEGIDFABCEGHJJ $.
|
|
|
|
$( Formula-building rule for restricted quantifiers (deduction rule).
|
|
(Contributed by NM, 28-Jan-2006.) $)
|
|
rexralbidv $p |- ( ph ->
|
|
( E. x e. A A. y e. B ps <-> E. x e. A A. y e. B ch ) ) $=
|
|
( wral ralbidv rexbidv ) ABEGICEGIDFABCEGHJK $.
|
|
$}
|
|
|
|
$( A transformation of restricted quantifiers and logical connectives.
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
ralinexa $p |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) ) $=
|
|
( wn wi wral wa wrex imnan ralbii ralnex bitri ) ABEFZCDGABHZEZCDGOCDIENPCD
|
|
ABJKOCDLM $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Two ways to say " ` A ` belongs to ` B ` ." (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
risset $p |- ( A e. B <-> E. x e. B x = A ) $=
|
|
( cv wcel wceq wa wex wrex exancom df-rex df-clel 3bitr4ri ) ADZCEZNBFZGA
|
|
HPOGAHPACIBCEOPAJPACKABCLM $.
|
|
$}
|
|
|
|
${
|
|
hbral.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
hbral.2 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for restricted quantification.
|
|
(Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy,
|
|
13-Dec-2009.) $)
|
|
hbral $p |- ( A. y e. A ph -> A. x A. y e. A ph ) $=
|
|
( wral cv wcel wi wal df-ral hbim hbal hbxfrbi ) ACDGCHDIZAJZCKBACDLQBCPA
|
|
BEFMNO $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` A. x e. A ph ` . (Contributed by NM,
|
|
18-Oct-1996.) $)
|
|
hbra1 $p |- ( A. x e. A ph -> A. x A. x e. A ph ) $=
|
|
( wral cv wcel wi wal df-ral hba1 hbxfrbi ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
$( ` x ` is not free in ` A. x e. A ph ` . (Contributed by NM, 18-Oct-1996.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfra1 $p |- F/ x A. x e. A ph $=
|
|
( wral cv wcel wi wal df-ral nfa1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
$d x y $.
|
|
nfraldxy.2 $e |- F/ y ph $.
|
|
nfraldxy.3 $e |- ( ph -> F/_ x A ) $.
|
|
nfraldxy.4 $e |- ( ph -> F/ x ps ) $.
|
|
$( Not-free for restricted universal quantification where ` x ` and ` y `
|
|
are distinct. See ~ nfraldya for a version with ` y ` and ` A `
|
|
distinct instead. (Contributed by Jim Kingdon, 29-May-2018.) $)
|
|
nfraldxy $p |- ( ph -> F/ x A. y e. A ps ) $=
|
|
( wral cv wcel wi wal df-ral wnfc nfcv a1i nfeld nfimd nfald nfxfrd ) BDE
|
|
IDJZEKZBLZDMACBDENAUDCDFAUCBCACUBECUBOACUBPQGRHSTUA $.
|
|
|
|
$( Not-free for restricted existential quantification where ` x ` and ` y `
|
|
are distinct. See ~ nfrexdya for a version with ` y ` and ` A `
|
|
distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) $)
|
|
nfrexdxy $p |- ( ph -> F/ x E. y e. A ps ) $=
|
|
( wrex cv wcel wa wex df-rex wnfc nfcv a1i nfeld nfand nfexd nfxfrd ) BDE
|
|
IDJZEKZBLZDMACBDENAUDCDFAUCBCACUBECUBOACUBPQGRHSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x z $. $d y z $. $d A z $. $d ph z $. $d ps z $.
|
|
nfraldya.2 $e |- F/ y ph $.
|
|
nfraldya.3 $e |- ( ph -> F/_ x A ) $.
|
|
nfraldya.4 $e |- ( ph -> F/ x ps ) $.
|
|
$( Not-free for restricted universal quantification where ` y ` and ` A `
|
|
are distinct. See ~ nfraldxy for a version with ` x ` and ` y `
|
|
distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) $)
|
|
nfraldya $p |- ( ph -> F/ x A. y e. A ps ) $=
|
|
( vz wral cv wcel wi wal df-ral wsb sbim clelsb3 nfv nfxfrd bitri 3bitr4i
|
|
imbi1i albii sb8 nfsbd nfraldxy ) BDEJDKELZBMZDNZACBDEOUJBDIPZIEJZACUIDIP
|
|
ZINIKELZUKMZINUJULUMUOIUMUHDIPZUKMUOUHBDIQUPUNUKIDERUCUAUDUIDIUIISUEUKIEO
|
|
UBAUKCIEAISGABDICFHUFUGTT $.
|
|
|
|
$( Not-free for restricted existential quantification where ` y ` and ` A `
|
|
are distinct. See ~ nfrexdxy for a version with ` x ` and ` y `
|
|
distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) $)
|
|
nfrexdya $p |- ( ph -> F/ x E. y e. A ps ) $=
|
|
( vz wrex cv wcel wa wex df-rex wsb sban clelsb3 nfv nfxfrd bitri 3bitr4i
|
|
anbi1i exbii sb8e nfsbd nfrexdxy ) BDEJDKELZBMZDNZACBDEOUJBDIPZIEJZACUIDI
|
|
PZINIKELZUKMZINUJULUMUOIUMUHDIPZUKMUOUHBDIQUPUNUKIDERUCUAUDUIDIUIISUEUKIE
|
|
OUBAUKCIEAISGABDICFHUFUGTT $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nfralxy.1 $e |- F/_ x A $.
|
|
nfralxy.2 $e |- F/ x ph $.
|
|
$( Not-free for restricted universal quantification where ` x ` and ` y `
|
|
are distinct. See ~ nfralya for a version with ` y ` and ` A ` distinct
|
|
instead. (Contributed by Jim Kingdon, 30-May-2018.) $)
|
|
nfralxy $p |- F/ x A. y e. A ph $=
|
|
( wral wnf wtru nftru wnfc a1i nfraldxy trud ) ACDGBHIABCDCJBDKIELABHIFLM
|
|
N $.
|
|
|
|
$( Not-free for restricted existential quantification where ` x ` and ` y `
|
|
are distinct. See ~ nfrexya for a version with ` y ` and ` A ` distinct
|
|
instead. (Contributed by Jim Kingdon, 30-May-2018.) $)
|
|
nfrexxy $p |- F/ x E. y e. A ph $=
|
|
( wrex wnf wtru nftru wnfc a1i nfrexdxy trud ) ACDGBHIABCDCJBDKIELABHIFLM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d y A $.
|
|
nfralya.1 $e |- F/_ x A $.
|
|
nfralya.2 $e |- F/ x ph $.
|
|
$( Not-free for restricted universal quantification where ` y ` and ` A `
|
|
are distinct. See ~ nfralxy for a version with ` x ` and ` y ` distinct
|
|
instead. (Contributed by Jim Kingdon, 3-Jun-2018.) $)
|
|
nfralya $p |- F/ x A. y e. A ph $=
|
|
( wral wnf wtru nftru wnfc a1i nfraldya trud ) ACDGBHIABCDCJBDKIELABHIFLM
|
|
N $.
|
|
|
|
$( Not-free for restricted existential quantification where ` y ` and ` A `
|
|
are distinct. See ~ nfrexxy for a version with ` x ` and ` y ` distinct
|
|
instead. (Contributed by Jim Kingdon, 3-Jun-2018.) $)
|
|
nfrexya $p |- F/ x E. y e. A ph $=
|
|
( wrex wnf wtru nftru wnfc a1i nfrexdya trud ) ACDGBHIABCDCJBDKIELABHIFLM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $.
|
|
$( Not-free given two restricted quantifiers. (Contributed by Jim Kingdon,
|
|
20-Aug-2018.) $)
|
|
nfra2xy $p |- F/ y A. x e. A A. y e. B ph $=
|
|
( wral nfcv nfra1 nfralxy ) ACEFCBDCDGACEHI $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` E. x e. A ph ` . (Contributed by NM, 19-Mar-1997.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfre1 $p |- F/ x E. x e. A ph $=
|
|
( wrex cv wcel wa wex df-rex nfe1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d z B $.
|
|
$( Triple restricted universal quantification. (Contributed by NM,
|
|
19-Nov-1995.) $)
|
|
r3al $p |- ( A. x e. A A. y e. B A. z e. C ph <->
|
|
A. x A. y A. z ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) ) $=
|
|
( cv wcel wa wi wal wral w3a df-ral r2al ralbii bitri albii 19.21v 3anass
|
|
imbi1i impexp 3bitr4i ) CHFIZDHGIZJZAKZDLZCLZBEMBHEIZUJKZBLADGMCFMZBEMUKU
|
|
EUFNZAKZDLZCLZBLUJBEOUMUJBEACDFGPQUQULBUQUKUIKZCLULUPURCUPUKUHKZDLURUOUSD
|
|
UOUKUGJZAKUSUNUTAUKUEUFUAUBUKUGAUCRSUKUHDTRSUKUICTRSUD $.
|
|
$}
|
|
|
|
$( Universal quantification implies restricted quantification. (Contributed
|
|
by NM, 20-Oct-2006.) $)
|
|
alral $p |- ( A. x ph -> A. x e. A ph ) $=
|
|
( wal cv wcel wi wral ax-1 alimi df-ral sylibr ) ABDBECFZAGZBDABCHANBAMIJAB
|
|
CKL $.
|
|
|
|
$( Restricted existence implies existence. (Contributed by NM,
|
|
11-Nov-1995.) $)
|
|
rexex $p |- ( E. x e. A ph -> E. x ph ) $=
|
|
( wrex cv wcel wa wex df-rex simpr eximi sylbi ) ABCDBECFZAGZBHABHABCINABMA
|
|
JKL $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 17-Oct-1996.) $)
|
|
rsp $p |- ( A. x e. A ph -> ( x e. A -> ph ) ) $=
|
|
( wral cv wcel wi wal df-ral sp sylbi ) ABCDBECFAGZBHLABCILBJK $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 12-Oct-1999.) $)
|
|
rspe $p |- ( ( x e. A /\ ph ) -> E. x e. A ph ) $=
|
|
( cv wcel wa wex wrex 19.8a df-rex sylibr ) BDCEAFZLBGABCHLBIABCJK $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 11-Feb-1997.) $)
|
|
rsp2 $p |- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( wral cv wcel wi rsp syl6 impd ) ACEFZBDFZBGDHZCGEHZANOMPAIMBDJACEJKL $.
|
|
|
|
$( Restricted specialization. (Contributed by FL, 4-Jun-2012.) $)
|
|
rsp2e $p |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph ) $=
|
|
( cv wcel w3a wrex wa wex simp1 rspe 3adant1 19.8a syl2anc df-rex sylibr )
|
|
BFDGZCFEGZAHZSACEIZJZBKZUBBDIUASUBUDSTALTAUBSACEMNUCBOPUBBDQR $.
|
|
|
|
${
|
|
rspec.1 $e |- A. x e. A ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
19-Nov-1994.) $)
|
|
rspec $p |- ( x e. A -> ph ) $=
|
|
( wral cv wcel wi rsp ax-mp ) ABCEBFCGAHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
rgen.1 $e |- ( x e. A -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
19-Nov-1994.) $)
|
|
rgen $p |- A. x e. A ph $=
|
|
( wral cv wcel wi df-ral mpgbir ) ABCEBFCGAHBABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z $.
|
|
rgen2a.1 $e |- ( ( x e. A /\ y e. A ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. Note that ` x ` and
|
|
` y ` needn't be distinct (and illustrates the use of ~ dvelimor ).
|
|
(Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon,
|
|
1-Jun-2018.) $)
|
|
rgen2a $p |- A. x e. A A. y e. A ph $=
|
|
( vz wral cv wcel wi wal wceq wnf wo nfv eleq1 dvelimor ex syl6bi alimi
|
|
pm2.43d a1d nfr syl6 jaoi ax-mp df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZA
|
|
JZCKZUJUMUKLZCKZULCMZNULUPJZFHZDIZULCBFVBCOVAUKDPQURUTUSURUPULUQUOCUQUNAU
|
|
QUNULUOUMUKDPULUNAERZSUATUBUSULULCKUPULCUCULUOCVCTUDUEUFACDUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
rgenw.1 $e |- ph $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
18-Jun-2014.) $)
|
|
rgenw $p |- A. x e. A ph $=
|
|
( cv wcel a1i rgen ) ABCABECFDGH $.
|
|
|
|
$( Generalization rule for restricted quantification. Note that ` x ` and
|
|
` y ` needn't be distinct. (Contributed by NM, 18-Jun-2014.) $)
|
|
rgen2w $p |- A. x e. A A. y e. B ph $=
|
|
( wral rgenw ) ACEGBDACEFHH $.
|
|
$}
|
|
|
|
${
|
|
mprg.1 $e |- ( A. x e. A ph -> ps ) $.
|
|
mprg.2 $e |- ( x e. A -> ph ) $.
|
|
$( Modus ponens combined with restricted generalization. (Contributed by
|
|
NM, 10-Aug-2004.) $)
|
|
mprg $p |- ps $=
|
|
( wral rgen ax-mp ) ACDGBACDFHEI $.
|
|
$}
|
|
|
|
${
|
|
mprgbir.1 $e |- ( ph <-> A. x e. A ps ) $.
|
|
mprgbir.2 $e |- ( x e. A -> ps ) $.
|
|
$( Modus ponens on biconditional combined with restricted generalization.
|
|
(Contributed by NM, 21-Mar-2004.) $)
|
|
mprgbir $p |- ph $=
|
|
( wral rgen mpbir ) ABCDGBCDFHEI $.
|
|
$}
|
|
|
|
$( Distribution of restricted quantification over implication. (Contributed
|
|
by NM, 9-Feb-1997.) $)
|
|
ralim $p |- ( A. x e. A ( ph -> ps ) ->
|
|
( A. x e. A ph -> A. x e. A ps ) ) $=
|
|
( wi wral cv wcel wal df-ral ax-2 al2imi sylbi 3imtr4g ) ABEZCDFZCGDHZAEZCI
|
|
ZQBEZCIZACDFBCDFPQOEZCISUAEOCDJUBRTCQABKLMACDJBCDJN $.
|
|
|
|
${
|
|
ralimi2.1 $e |- ( ( x e. A -> ph ) -> ( x e. B -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 22-Feb-2004.) $)
|
|
ralimi2 $p |- ( A. x e. A ph -> A. x e. B ps ) $=
|
|
( cv wcel wi wal wral alimi df-ral 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
ralimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 19-Jul-1996.) $)
|
|
ralimia $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( cv wcel a2i ralimi2 ) ABCDDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
ralimiaa.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 4-Aug-2007.) $)
|
|
ralimiaa $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( cv wcel ex ralimia ) ABCDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
ralimi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent, with strong
|
|
hypothesis. (Contributed by NM, 4-Mar-1997.) $)
|
|
ralimi $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( wi cv wcel a1i ralimia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
ral2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference quantifying antecedent, nested antecedent, and consequent,
|
|
with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) $)
|
|
ral2imi $p |- ( A. x e. A ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( wral wi ralimi ralim syl ) ADEGBCHZDEGBDEGCDEGHALDEFIBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ralimdaa.1 $e |- F/ x ph $.
|
|
ralimdaa.2 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) $)
|
|
ralimdaa $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral ex a2d alimd df-ral 3imtr4g ) ADHEIZBJZDKRCJZDKBDEL
|
|
CDELASTDFARBCARBCJGMNOBDEPCDEPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) $)
|
|
ralimdva $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( nfv ralimdaa ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.) $)
|
|
ralimdv $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( wi cv wcel adantr ralimdva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdv2.1 $e |- ( ph -> ( ( x e. A -> ps ) -> ( x e. B -> ch ) ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 1-Feb-2005.) $)
|
|
ralimdv2 $p |- ( ph -> ( A. x e. A ps -> A. x e. B ch ) ) $=
|
|
( cv wcel wi wal wral alimdv df-ral 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
ralrimi.1 $e |- F/ x ph $.
|
|
ralrimi.2 $e |- ( ph -> ( x e. A -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 10-Oct-1999.) $)
|
|
ralrimi $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel wi wal wral alrimi df-ral sylibr ) ACGDHBIZCJBCDKAOCEFLBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimiv.1 $e |- ( ph -> ( x e. A -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 22-Nov-1994.) $)
|
|
ralrimiv $p |- ( ph -> A. x e. A ps ) $=
|
|
( nfv ralrimi ) ABCDACFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimiva.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 2-Jan-2006.) $)
|
|
ralrimiva $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel ex ralrimiv ) ABCDACFDGBEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimivw.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 18-Jun-2014.) $)
|
|
ralrimivw $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel a1d ralrimiv ) ABCDABCFDGEHI $.
|
|
$}
|
|
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed
|
|
theorem version). (Contributed by NM, 1-Mar-2008.) $)
|
|
r19.21t $p |- ( F/ x ph ->
|
|
( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) ) $=
|
|
( wnf cv wcel wi wal wral bi2.04 albii 19.21t syl5bb df-ral imbi2i 3bitr4g
|
|
) ACEZCFDGZABHZHZCIZASBHZCIZHZTCDJABCDJZHUBAUCHZCIRUEUAUGCSABKLAUCCMNTCDOUF
|
|
UDABCDOPQ $.
|
|
|
|
${
|
|
r19.21.1 $e |- F/ x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by Scott Fenton, 30-Mar-2011.) $)
|
|
r19.21 $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
|
|
( wnf wi wral wb r19.21t ax-mp ) ACFABGCDHABCDHGIEABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.21v $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
|
|
( nfv r19.21 ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
ralrimd.1 $e |- F/ x ph $.
|
|
ralrimd.2 $e |- F/ x ps $.
|
|
ralrimd.3 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 16-Feb-2004.) $)
|
|
ralrimd $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral alrimd df-ral syl6ibr ) ABDIEJCKZDLCDEMABQDFGHNCDEO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
ralrimdv.1 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 27-May-1998.) $)
|
|
ralrimdv $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( nfv ralrimd ) ABCDEADGBDGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
ralrimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 2-Feb-2008.) $)
|
|
ralrimdva $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi ex com23 ralrimdv ) ABCDEADGEHZBCAMBCIFJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $.
|
|
ralrimivv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
24-Jul-2004.) $)
|
|
ralrimivv $p |- ( ph -> A. x e. A A. y e. B ps ) $=
|
|
( wral cv wcel expd ralrimdv ralrimiv ) ABDFHCEACIEJZBDFANDIFJBGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y $. $d A y $.
|
|
ralrimivva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by Jeff
|
|
Madsen, 19-Jun-2011.) $)
|
|
ralrimivva $p |- ( ph -> A. x e. A A. y e. B ps ) $=
|
|
( cv wcel wa ex ralrimivv ) ABCDEFACHEIDHFIJBGKL $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $. $d A y z $. $d B z $.
|
|
ralrimivvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with triple quantification.) (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ralrimivvva $p |- ( ph -> A. x e. A A. y e. B A. z e. C ps ) $=
|
|
( wral cv wcel wa 3anassrs ralrimiva ) ABEHJZDGJCFACKFLZMZPDGRDKGLZMBEHAQ
|
|
SEKHLBINOOO $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ps $. $d y A $.
|
|
ralrimdvv.1 $e |- ( ph -> ( ps -> ( ( x e. A /\ y e. B ) -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
1-Jun-2005.) $)
|
|
ralrimdvv $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
|
|
( wral wa cv wcel wi imp ralrimivv ex ) ABCEGIDFIABJCDEFGABDKFLEKGLJCMHNO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ps $. $d y A $.
|
|
ralrimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
2-Feb-2008.) $)
|
|
ralrimdvva $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
|
|
( cv wcel wa wi ex com23 ralrimdvv ) ABCDEFGADIFJEIGJKZBCAPBCLHMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
rgen2.1 $e |- ( ( x e. A /\ y e. B ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
30-May-1999.) $)
|
|
rgen2 $p |- A. x e. A A. y e. B ph $=
|
|
( wral cv wcel ralrimiva rgen ) ACEGBDBHDIACEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d z B $. $d x y z $.
|
|
rgen3.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
12-Jan-2008.) $)
|
|
rgen3 $p |- A. x e. A A. y e. B A. z e. C ph $=
|
|
( wral cv wcel wa 3expa ralrimiva rgen2 ) ADGIBCEFBJEKZCJFKZLADGPQDJGKAHM
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
r19.21bi.1 $e |- ( ph -> A. x e. A ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 20-Nov-1994.) $)
|
|
r19.21bi $p |- ( ( ph /\ x e. A ) -> ps ) $=
|
|
( cv wcel wi wral wal df-ral sylib 19.21bi imp ) ACFDGZBAOBHZCABCDIPCJEBC
|
|
DKLMN $.
|
|
$}
|
|
|
|
${
|
|
rspec2.1 $e |- A. x e. A A. y e. B ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
rspec2 $p |- ( ( x e. A /\ y e. B ) -> ph ) $=
|
|
( cv wcel wral rspec r19.21bi ) BGDHACEACEIBDFJK $.
|
|
$}
|
|
|
|
${
|
|
rspec3.1 $e |- A. x e. A A. y e. B A. z e. C ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
rspec3 $p |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $=
|
|
( cv wcel wa wral rspec2 r19.21bi 3impa ) BIEJZCIFJZDIGJAPQKADGADGLBCEFHM
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
r19.21be.1 $e |- ( ph -> A. x e. A ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 21-Nov-1994.) $)
|
|
r19.21be $p |- A. x e. A ( ph -> ps ) $=
|
|
( wi cv wcel r19.21bi expcom rgen ) ABFCDACGDHBABCDEIJK $.
|
|
$}
|
|
|
|
${
|
|
nrex.1 $e |- ( x e. A -> -. ps ) $.
|
|
$( Inference adding restricted existential quantifier to negated wff.
|
|
(Contributed by NM, 16-Oct-2003.) $)
|
|
nrex $p |- -. E. x e. A ps $=
|
|
( wn wral wrex rgen ralnex mpbi ) AEZBCFABCGEKBCDHABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nrexdv.1 $e |- ( ( ph /\ x e. A ) -> -. ps ) $.
|
|
$( Deduction adding restricted existential quantifier to negated wff.
|
|
(Contributed by NM, 16-Oct-2003.) $)
|
|
nrexdv $p |- ( ph -> -. E. x e. A ps ) $=
|
|
( wn wral wrex ralrimiva ralnex sylib ) ABFZCDGBCDHFALCDEIBCDJK $.
|
|
$}
|
|
|
|
$( Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.)
|
|
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
rexim $p |- ( A. x e. A ( ph -> ps ) ->
|
|
( E. x e. A ph -> E. x e. A ps ) ) $=
|
|
( wi wral cv wcel wa wex wrex wal df-ral simpl a1i pm3.31 jcad alimi df-rex
|
|
sylbi exim syl 3imtr4g ) ABEZCDFZCGDHZAIZCJZUFBIZCJZACDKBCDKUEUGUIEZCLZUHUJ
|
|
EUEUFUDEZCLULUDCDMUMUKCUMUGUFBUGUFEUMUFANOUFABPQRTUGUICUAUBACDSBCDSUC $.
|
|
|
|
${
|
|
reximia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 10-Feb-1997.) $)
|
|
reximia $p |- ( E. x e. A ph -> E. x e. A ps ) $=
|
|
( wi wrex rexim mprg ) ABFACDGBCDGFCDABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
reximi2.1 $e |- ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.) $)
|
|
reximi2 $p |- ( E. x e. A ph -> E. x e. B ps ) $=
|
|
( cv wcel wa wex wrex eximi df-rex 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
reximi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 18-Oct-1996.) $)
|
|
reximi $p |- ( E. x e. A ph -> E. x e. A ps ) $=
|
|
( wi cv wcel a1i reximia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
reximdai.1 $e |- F/ x ph $.
|
|
reximdai.2 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 31-Aug-1999.) $)
|
|
reximdai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( wi wral wrex ralrimi rexim syl ) ABCHZDEIBDEJCDEJHANDEFGKBCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.) $)
|
|
reximdv2 $p |- ( ph -> ( E. x e. A ps -> E. x e. B ch ) ) $=
|
|
( cv wcel wa wex wrex eximdv df-rex 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdvai.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) $)
|
|
reximdvai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( nfv reximdai ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted
|
|
quantifier version with strong hypothesis.) (Contributed by NM,
|
|
24-Jun-1998.) $)
|
|
reximdv $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( wi cv wcel a1d reximdvai ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) $)
|
|
reximdva $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( cv wcel wi ex reximdvai ) ABCDEADGEHBCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximddva.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
|
|
reximddva.2 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by
|
|
Thierry Arnoux, 7-Dec-2016.) $)
|
|
reximddv $p |- ( ph -> E. x e. A ch ) $=
|
|
( wrex cv wcel expr reximdva mpd ) ABDEHCDEHGABCDEADIEJBCFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ph x y $.
|
|
reximddv2.1 $e |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch ) $.
|
|
reximddv2.2 $e |- ( ph -> E. x e. A E. y e. B ps ) $.
|
|
$( Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed
|
|
by Thierry Arnoux, 15-Dec-2019.) $)
|
|
reximddv2 $p |- ( ph -> E. x e. A E. y e. B ch ) $=
|
|
( wrex cv wcel wa ex reximdva impr reximddv ) ABEGJZCEGJZDFADKFLZRSATMZBC
|
|
EGUAEKGLMBCHNOPIQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d x B $.
|
|
$( Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.12 $p |- ( E. x e. A A. y e. B ph -> A. y e. B E. x e. A ph ) $=
|
|
( wral wrex nfcv nfra1 nfrexxy cv wcel ax-1 ralrimi com12 reximdv ralimia
|
|
rsp syl ) ACEFZBDGZUACEFABDGZCEFUAUACETCBDCDHACEIJUACKELZMNUAUBCEUCTABDTU
|
|
CAACEROPQS $.
|
|
$}
|
|
|
|
$( Closed theorem form of ~ r19.23 . (Contributed by NM, 4-Mar-2013.)
|
|
(Revised by Mario Carneiro, 8-Oct-2016.) $)
|
|
r19.23t $p |- ( F/ x ps ->
|
|
( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) ) $=
|
|
( wnf cv wcel wa wal wex wral wrex 19.23t df-ral impexp albii bitr4i df-rex
|
|
wi imbi1i 3bitr4g ) BCECFDGZAHZBSZCIZUCCJZBSABSZCDKZACDLZBSUCBCMUHUBUGSZCIU
|
|
EUGCDNUDUJCUBABOPQUIUFBACDRTUA $.
|
|
|
|
${
|
|
r19.23.1 $e |- F/ x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
r19.23 $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
|
|
( wnf wi wral wrex wb r19.23t ax-mp ) BCFABGCDHACDIBGJEABCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 31-Aug-1999.) $)
|
|
r19.23v $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
|
|
( nfv r19.23 ) ABCDBCEF $.
|
|
$}
|
|
|
|
${
|
|
rexlimi.1 $e |- F/ x ps $.
|
|
rexlimi.2 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof
|
|
shortened by Andrew Salmon, 30-May-2011.) $)
|
|
rexlimi $p |- ( E. x e. A ph -> ps ) $=
|
|
( wi wral wrex rgen r19.23 mpbi ) ABGZCDHACDIBGMCDFJABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
rexlimiv.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 20-Nov-1994.) $)
|
|
rexlimiv $p |- ( E. x e. A ph -> ps ) $=
|
|
( nfv rexlimi ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
rexlimiva.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 18-Dec-2006.) $)
|
|
rexlimiva $p |- ( E. x e. A ph -> ps ) $=
|
|
( cv wcel ex rexlimiv ) ABCDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
$d ps x $.
|
|
rexlimivw.1 $e |- ( ph -> ps ) $.
|
|
$( Weaker version of ~ rexlimiv . (Contributed by FL, 19-Sep-2011.) $)
|
|
rexlimivw $p |- ( E. x e. A ph -> ps ) $=
|
|
( wi cv wcel a1i rexlimiv ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
rexlimd.1 $e |- F/ x ph $.
|
|
rexlimd.2 $e |- F/ x ch $.
|
|
rexlimd.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew
|
|
Salmon, 30-May-2011.) $)
|
|
rexlimd $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi wral wrex ralrimi r19.23 sylib ) ABCIZDEJBDEKCIAODEFHLBCDEGMN $.
|
|
$}
|
|
|
|
${
|
|
rexlimd2.1 $e |- F/ x ph $.
|
|
rexlimd2.2 $e |- ( ph -> F/ x ch ) $.
|
|
rexlimd2.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Version of ~ rexlimd with deduction version of second hypothesis.
|
|
(Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
rexlimd2 $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi wral wrex ralrimi wnf wb r19.23t syl mpbid ) ABCIZDEJZBDEKCIZARDEFHL
|
|
ACDMSTNGBCDEOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdv.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric
|
|
Schmidt, 22-Dec-2006.) $)
|
|
rexlimdv $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( nfv rexlimd ) ABCDEADGCDGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 20-Jan-2007.) $)
|
|
rexlimdva $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel wi ex rexlimdv ) ABCDEADGEHBCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdvaa.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by Mario Carneiro, 15-Jun-2016.) $)
|
|
rexlimdvaa $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel expr rexlimdva ) ABCDEADGEHBCFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdv3a.1 $e |- ( ( ph /\ x e. A /\ ps ) -> ch ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). Frequently-used variant of ~ rexlimdv . (Contributed by NM,
|
|
7-Jun-2015.) $)
|
|
rexlimdv3a $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel 3exp rexlimdv ) ABCDEADGEHBCFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdvw.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 18-Jun-2014.) $)
|
|
rexlimdvw $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi cv wcel a1d rexlimdv ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimddv.1 $e |- ( ph -> E. x e. A ps ) $.
|
|
rexlimddv.2 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
|
|
$( Restricted existential elimination rule of natural deduction.
|
|
(Contributed by Mario Carneiro, 15-Jun-2016.) $)
|
|
rexlimddv $p |- ( ph -> ch ) $=
|
|
( wrex rexlimdvaa mpd ) ABDEHCFABCDEGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d y A $.
|
|
rexlimivv.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 17-Feb-2004.) $)
|
|
rexlimivv $p |- ( E. x e. A E. y e. B ph -> ps ) $=
|
|
( wrex cv wcel rexlimdva rexlimiv ) ADFHBCECIEJABDFGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ch $. $d y A $.
|
|
rexlimdvv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 22-Jul-2004.) $)
|
|
rexlimdvv $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
|
|
( wrex cv wcel wa wi expdimp rexlimdv rexlimdva ) ABEGICDFADJFKZLBCEGAQEJ
|
|
GKBCMHNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ch $. $d y A $.
|
|
rexlimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 18-Jun-2014.) $)
|
|
rexlimdvva $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
|
|
( cv wcel wa wi ex rexlimdvv ) ABCDEFGADIFJEIGJKBCLHMN $.
|
|
$}
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.26 $p |- ( A. x e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. x e. A ps ) ) $=
|
|
( wa wral simpl ralimi simpr jca pm3.2 ral2imi imp impbii ) ABEZCDFZACDFZBC
|
|
DFZEPQROACDABGHOBCDABIHJQRPABOCDABKLMN $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers.
|
|
(Contributed by NM, 10-Aug-2004.) $)
|
|
r19.26-2 $p |- ( A. x e. A A. y e. B ( ph /\ ps ) <->
|
|
( A. x e. A A. y e. B ph /\ A. x e. A A. y e. B ps ) ) $=
|
|
( wa wral r19.26 ralbii bitri ) ABGDFHZCEHADFHZBDFHZGZCEHMCEHNCEHGLOCEABDFI
|
|
JMNCEIK $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers.
|
|
(Contributed by FL, 22-Nov-2010.) $)
|
|
r19.26-3 $p |- ( A. x e. A ( ph /\ ps /\ ch ) <->
|
|
( A. x e. A ph /\ A. x e. A ps /\ A. x e. A ch ) ) $=
|
|
( w3a wral wa df-3an ralbii r19.26 anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDE
|
|
GPDEGZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by
|
|
NM, 22-Feb-2004.) $)
|
|
r19.26m $p |- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <->
|
|
( A. x e. A ph /\ A. x e. B ps ) ) $=
|
|
( cv wcel wi wa wal wral 19.26 df-ral anbi12i bitr4i ) CFZDGAHZPEGBHZICJQCJ
|
|
ZRCJZIACDKZBCEKZIQRCLUASUBTACDMBCEMNO $.
|
|
|
|
$( Distribute a restricted universal quantifier over a biconditional.
|
|
Theorem 19.15 of [Margaris] p. 90 with restricted quantification.
|
|
(Contributed by NM, 6-Oct-2003.) $)
|
|
ralbi $p |- ( A. x e. A ( ph <-> ps ) ->
|
|
( A. x e. A ph <-> A. x e. A ps ) ) $=
|
|
( wb wral nfra1 cv wcel rsp imp ralbida ) ABEZCDFZABCDMCDGNCHDIMMCDJKL $.
|
|
|
|
$( Distribute a restricted existential quantifier over a biconditional.
|
|
Theorem 19.18 of [Margaris] p. 90 with restricted quantification.
|
|
(Contributed by Jim Kingdon, 21-Jan-2019.) $)
|
|
rexbi $p |- ( A. x e. A ( ph <-> ps ) ->
|
|
( E. x e. A ph <-> E. x e. A ps ) ) $=
|
|
( wb wral nfra1 cv wcel rsp imp rexbida ) ABEZCDFZABCDMCDGNCHDIMMCDJKL $.
|
|
|
|
$( Split a biconditional and distribute quantifier. (Contributed by NM,
|
|
3-Jun-2012.) $)
|
|
ralbiim $p |- ( A. x e. A ( ph <-> ps ) <->
|
|
( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) ) $=
|
|
( wb wral wi wa dfbi2 ralbii r19.26 bitri ) ABEZCDFABGZBAGZHZCDFNCDFOCDFHMP
|
|
CDABIJNOCDKL $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Restricted version of one direction of Theorem 19.27 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.27av $p |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) ) $=
|
|
( wral wa cv wcel ax-1 ralrimiv anim2i r19.26 sylibr ) ACDEZBFNBCDEZFABFC
|
|
DEBONBBCDBCGDHIJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.28 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.28av $p |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) $=
|
|
( wral wa r19.27av ancom ralbii 3imtr4i ) BCDEZAFBAFZCDEAKFABFZCDEBACDGAK
|
|
HMLCDABHIJ $.
|
|
$}
|
|
|
|
$( Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.29 $p |- ( ( A. x e. A ph /\ E. x e. A ps ) ->
|
|
E. x e. A ( ph /\ ps ) ) $=
|
|
( wral wrex wa wi pm3.2 ralimi rexim syl imp ) ACDEZBCDFZABGZCDFZNBPHZCDEOQ
|
|
HARCDABIJBPCDKLM $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with restricted
|
|
quantifiers. (Contributed by NM, 31-Aug-1999.) $)
|
|
r19.29r $p |- ( ( E. x e. A ph /\ A. x e. A ps ) ->
|
|
E. x e. A ( ph /\ ps ) ) $=
|
|
( wral wrex wa r19.29 ancom rexbii 3imtr4i ) BCDEZACDFZGBAGZCDFMLGABGZCDFBA
|
|
CDHMLIONCDABIJK $.
|
|
|
|
${
|
|
r19.29af2.p $e |- F/ x ph $.
|
|
r19.29af2.c $e |- F/ x ch $.
|
|
r19.29af2.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
|
|
r19.29af2.2 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( A commonly used pattern based on ~ r19.29 (Contributed by Thierry
|
|
Arnoux, 17-Dec-2017.) $)
|
|
r19.29af2 $p |- ( ph -> ch ) $=
|
|
( wrex wi wral wa cv wcel exp31 ralrimi jca r19.29r pm3.35 rexlimi 3syl
|
|
a1i ) ABDEJZBCKZDELZMBUEMZDEJCAUDUFIAUEDEFADNEOZBCHPQRBUEDESUGCDEGUGCKUHB
|
|
CTUCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $.
|
|
r19.29af.0 $e |- F/ x ph $.
|
|
r19.29af.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
|
|
r19.29af.2 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( A commonly used pattern based on ~ r19.29 (Contributed by Thierry
|
|
Arnoux, 29-Nov-2017.) $)
|
|
r19.29af $p |- ( ph -> ch ) $=
|
|
( nfv r19.29af2 ) ABCDEFCDIGHJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $.
|
|
r19.29a.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
|
|
r19.29a.2 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( A commonly used pattern based on ~ r19.29 (Contributed by Thierry
|
|
Arnoux, 22-Nov-2017.) $)
|
|
r19.29a $p |- ( ph -> ch ) $=
|
|
( nfv r19.29af ) ABCDEADHFGI $.
|
|
$}
|
|
|
|
${
|
|
r19.29d2r.1 $e |- ( ph -> A. x e. A A. y e. B ps ) $.
|
|
r19.29d2r.2 $e |- ( ph -> E. x e. A E. y e. B ch ) $.
|
|
$( Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers,
|
|
deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.) $)
|
|
r19.29d2r $p |- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) ) $=
|
|
( wral wrex wa r19.29 syl2anc reximi syl ) ABEGJZCEGKZLZDFKZBCLEGKZDFKAQD
|
|
FJRDFKTHIQRDFMNSUADFBCEGMOP $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y ch $. $d x y ph $.
|
|
r19.29vva.1 $e |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch ) $.
|
|
r19.29vva.2 $e |- ( ph -> E. x e. A E. y e. B ps ) $.
|
|
$( A commonly used pattern based on ~ r19.29 , version with two restricted
|
|
quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) $)
|
|
r19.29vva $p |- ( ph -> ch ) $=
|
|
( wi wa wrex wral cv wcel ex ralrimiva r19.29d2r pm3.35 rexlimivw ancoms
|
|
syl ) ABCJZBKZEGLZDFLCAUCBDEFGAUCEGMDFADNFOKZUCEGUFENGOKBCHPQQIRUECDFUDCE
|
|
GBUCCBCSUATTUB $.
|
|
$}
|
|
|
|
${
|
|
r19.32r.1 $e |- F/ x ph $.
|
|
$( One direction of Theorem 19.32 of [Margaris] p. 90 with restricted
|
|
quantifiers. For decidable propositions this is an equivalence.
|
|
(Contributed by Jim Kingdon, 19-Aug-2018.) $)
|
|
r19.32r $p |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) ) $=
|
|
( wral wo cv wcel wal orc a1d alrimi df-ral olc imim2i alimi sylbi jaoi
|
|
wi sylibr ) ABCDFZGCHDIZABGZTZCJZUDCDFAUFUBAUECEAUDUCABKLMUBUCBTZCJUFBCDN
|
|
UGUECBUDUCBAOPQRSUDCDNUA $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( One direction of Theorem 19.32 of [Margaris] p. 90 with restricted
|
|
quantifiers. For decidable propositions this is an equivalence, as seen
|
|
at ~ r19.32vdc . (Contributed by Jim Kingdon, 19-Aug-2018.) $)
|
|
r19.32vr $p |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) ) $=
|
|
( nfv r19.32r ) ABCDACEF $.
|
|
|
|
$( Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where
|
|
` ph ` is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.) $)
|
|
r19.32vdc $p |- ( DECID ph ->
|
|
( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) ) $=
|
|
( wdc wn wi wral wo wb r19.21v a1i dfordc ralbidv 3bitr4d ) AEZAFZBGZCDHZ
|
|
QBCDHZGZABIZCDHATISUAJPQBCDKLPUBRCDABMNATMO $.
|
|
$}
|
|
|
|
$( Restricted quantifier version of ~ 19.35-1 . (Contributed by Jim Kingdon,
|
|
4-Jun-2018.) $)
|
|
r19.35-1 $p |- ( E. x e. A ( ph -> ps ) ->
|
|
( A. x e. A ph -> E. x e. A ps ) ) $=
|
|
( wral wi wrex wa r19.29 pm3.35 reximi syl expcom ) ACDEZABFZCDGZBCDGZNPHAO
|
|
HZCDGQAOCDIRBCDABJKLM $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( One direction of a restricted quantifier version of Theorem 19.36 of
|
|
[Margaris] p. 90. In classical logic, the converse would hold if ` A `
|
|
has at least one element, but in intuitionistic logic, that is not a
|
|
sufficient condition. (Contributed by NM, 22-Oct-2003.) $)
|
|
r19.36av $p |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) ) $=
|
|
( wi wrex wral r19.35-1 cv wcel idd rexlimiv imim2i syl ) ABECDFACDGZBCDF
|
|
ZEOBEABCDHPBOBBCDCIDJBKLMN $.
|
|
$}
|
|
|
|
${
|
|
r19.37.1 $e |- F/ x ph $.
|
|
$( Restricted version of one direction of Theorem 19.37 of [Margaris]
|
|
p. 90. In classical logic the converse would hold if ` A ` has at least
|
|
one element, but that is not sufficient in intuitionistic logic.
|
|
(Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
r19.37 $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
|
|
( wral wi wrex cv wcel ax-1 ralrimi r19.35-1 syl5 ) AACDFABGCDHBCDHAACDEA
|
|
CIDJKLABCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.37 of [Margaris]
|
|
p. 90. (Contributed by NM, 2-Apr-2004.) $)
|
|
r19.37av $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
|
|
( nfv r19.37 ) ABCDACEF $.
|
|
$}
|
|
|
|
$( Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90.
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.40 $p |- ( E. x e. A ( ph /\ ps ) ->
|
|
( E. x e. A ph /\ E. x e. A ps ) ) $=
|
|
( wa wrex simpl reximi simpr jca ) ABEZCDFACDFBCDFKACDABGHKBCDABIHJ $.
|
|
|
|
${
|
|
r19.41.1 $e |- F/ x ps $.
|
|
$( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
|
|
(Contributed by NM, 1-Nov-2010.) $)
|
|
r19.41 $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
|
|
( cv wcel wa wex wrex anass exbii 19.41 bitr3i df-rex anbi1i 3bitr4i ) CF
|
|
DGZABHZHZCIZRAHZCIZBHZSCDJACDJZBHUAUBBHZCIUDUFTCRABKLUBBCEMNSCDOUEUCBACDO
|
|
PQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
|
|
(Contributed by NM, 17-Dec-2003.) $)
|
|
r19.41v $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
|
|
( nfv r19.41 ) ABCDBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed
|
|
by NM, 27-May-1998.) $)
|
|
r19.42v $p |- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) ) $=
|
|
( wa wrex r19.41v ancom rexbii 3bitr4i ) BAEZCDFBCDFZAEABEZCDFALEBACDGMKC
|
|
DABHIALHJ $.
|
|
$}
|
|
|
|
$( Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by
|
|
NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.) $)
|
|
r19.43 $p |- ( E. x e. A ( ph \/ ps ) <->
|
|
( E. x e. A ph \/ E. x e. A ps ) ) $=
|
|
( wo wrex cv wcel wa wex df-rex andi exbii bitri 19.43 orbi12i bitr4i ) ABE
|
|
ZCDFZCGDHZAIZCJZTBIZCJZEZACDFZBCDFZESUAUCEZCJZUESTRIZCJUIRCDKUJUHCTABLMNUAU
|
|
CCONUFUBUGUDACDKBCDKPQ $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( One direction of a restricted quantifier version of Theorem 19.44 of
|
|
[Margaris] p. 90. The other direction doesn't hold when ` A ` is empty.
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.44av $p |- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) ) $=
|
|
( wo wrex r19.43 cv wcel idd rexlimiv orim2i sylbi ) ABECDFACDFZBCDFZENBE
|
|
ABCDGOBNBBCDCHDIBJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.45 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.45av $p |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) ) $=
|
|
( wo wrex r19.43 cv wcel idd rexlimiv orim1i sylbi ) ABECDFACDFZBCDFZEAOE
|
|
ABCDGNAOAACDCHDIAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
ralcomf.1 $e |- F/_ y A $.
|
|
ralcomf.2 $e |- F/_ x B $.
|
|
$( Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
ralcomf $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
|
|
( cv wcel wa wi wal wral ancomsimp 2albii alcom bitri r2alf 3bitr4i ) BHD
|
|
IZCHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABC
|
|
DEFRACBEDGRS $.
|
|
|
|
$( Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
rexcomf $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
|
|
( cv wcel wa wex wrex ancom anbi1i 2exbii excom bitri r2exf 3bitr4i ) BHD
|
|
IZCHEIZJZAJZCKBKZUATJZAJZBKCKZACELBDLABDLCELUDUFCKBKUGUCUFBCUBUEATUAMNOUF
|
|
BCPQABCDEFRACBEDGRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x B $. $d y A $.
|
|
$( Commutation of restricted quantifiers. (Contributed by NM,
|
|
13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
ralcom $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
|
|
( nfcv ralcomf ) ABCDECDFBEFG $.
|
|
|
|
$( Commutation of restricted quantifiers. (Contributed by NM,
|
|
19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
rexcom $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
|
|
( nfcv rexcomf ) ABCDECDFBEFG $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d x y C $.
|
|
$( Swap 1st and 3rd restricted existential quantifiers. (Contributed by
|
|
NM, 8-Apr-2015.) $)
|
|
rexcom13 $p |- ( E. x e. A E. y e. B E. z e. C ph
|
|
<-> E. z e. C E. y e. B E. x e. A ph ) $=
|
|
( wrex rexcom rexbii 3bitri ) ADGHZCFHBEHLBEHZCFHABEHZDGHZCFHNCFHDGHLBCEF
|
|
IMOCFABDEGIJNCDFGIK $.
|
|
$}
|
|
|
|
${
|
|
$d w z A $. $d w z B $. $d w x y C $. $d x y z D $.
|
|
$( Rotate existential restricted quantifiers twice. (Contributed by NM,
|
|
8-Apr-2015.) $)
|
|
rexrot4 $p |- ( E. x e. A E. y e. B E. z e. C E. w e. D ph
|
|
<-> E. z e. C E. w e. D E. x e. A E. y e. B ph ) $=
|
|
( wrex rexcom13 rexbii bitri ) AEIJDHJCGJZBFJACGJZDHJEIJZBFJOBFJEIJDHJNPB
|
|
FACDEGHIKLOBEDFIHKM $.
|
|
$}
|
|
|
|
$( A commutative law for restricted quantifiers that swaps the domain of the
|
|
restriction. (Contributed by NM, 22-Feb-2004.) $)
|
|
ralcom3 $p |- ( A. x e. A ( x e. B -> ph ) <->
|
|
A. x e. B ( x e. A -> ph ) ) $=
|
|
( cv wcel wi wral pm2.04 ralimi2 impbii ) BEZDFZAGZBCHLCFZAGZBDHNPBCDOMAIJP
|
|
NBDCMOAIJK $.
|
|
|
|
${
|
|
$d y A $. $d x B $. $d x y $.
|
|
reean.1 $e |- F/ y ph $.
|
|
reean.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
|
|
(Proof shortened by Andrew Salmon, 30-May-2011.) $)
|
|
reean $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
|
|
( E. x e. A ph /\ E. y e. B ps ) ) $=
|
|
( cv wcel wa wex wrex an4 2exbii nfv nfan eean bitri df-rex r2ex anbi12i
|
|
3bitr4i ) CIEJZDIFJZKABKZKZDLCLZUDAKZCLZUEBKZDLZKZUFDFMCEMACEMZBDFMZKUHUI
|
|
UKKZDLCLUMUGUPCDUDUEABNOUIUKCDUDADUDDPGQUEBCUECPHQRSUFCDEFUAUNUJUOULACETB
|
|
DFTUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y $. $d y A $. $d x B $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.) $)
|
|
reeanv $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
|
|
( E. x e. A ph /\ E. y e. B ps ) ) $=
|
|
( nfv reean ) ABCDEFADGBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d ph y z $. $d ps x z $. $d ch x y $. $d A y $. $d B x z $.
|
|
$d C x y $.
|
|
$( Rearrange three existential quantifiers. (Contributed by Jeff Madsen,
|
|
11-Jun-2010.) $)
|
|
3reeanv $p |- ( E. x e. A E. y e. B E. z e. C ( ph /\ ps /\ ch )
|
|
<-> ( E. x e. A ph /\ E. y e. B ps /\ E. z e. C ch ) ) $=
|
|
( wa wrex w3a r19.41v reeanv anbi1i bitri df-3an 2rexbii rexbii 3bitr4i )
|
|
ABJZEHKZCFIKZJZDGKZADGKZBEHKZJZUCJZABCLZFIKEHKZDGKUFUGUCLUEUBDGKZUCJUIUBU
|
|
CDGMULUHUCABDEGHNOPUKUDDGUKUACJZFIKEHKUDUJUMEFHIABCQRUACEFHINPSUFUGUCQT
|
|
$.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` E! x e. A ph ` . (Contributed by NM,
|
|
19-Mar-1997.) $)
|
|
nfreu1 $p |- F/ x E! x e. A ph $=
|
|
( wreu cv wcel wa weu df-reu nfeu1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
$( ` x ` is not free in ` E* x e. A ph ` . (Contributed by NM,
|
|
16-Jun-2017.) $)
|
|
nfrmo1 $p |- F/ x E* x e. A ph $=
|
|
( wrmo cv wcel wa wmo df-rmo nfmo1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
$d x y $.
|
|
nfreudxy.1 $e |- F/ y ph $.
|
|
nfreudxy.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfreudxy.3 $e |- ( ph -> F/ x ps ) $.
|
|
$( Not-free deduction for restricted uniqueness. This is a version where
|
|
` x ` and ` y ` are distinct. (Contributed by Jim Kingdon,
|
|
6-Jun-2018.) $)
|
|
nfreudxy $p |- ( ph -> F/ x E! y e. A ps ) $=
|
|
( cv wcel wa weu wnf wreu wnfc nfcv a1i nfeld nfand nfeud df-reu sylibr
|
|
nfbii ) ADIZEJZBKZDLZCMBDENZCMAUFCDFAUEBCACUDECUDOACUDPQGRHSTUHUGCBDEUAUC
|
|
UB $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nfreuxy.1 $e |- F/_ x A $.
|
|
nfreuxy.2 $e |- F/ x ph $.
|
|
$( Not-free for restricted uniqueness. This is a version where ` x ` and
|
|
` y ` are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.) $)
|
|
nfreuxy $p |- F/ x E! y e. A ph $=
|
|
( wreu wnf wtru nftru wnfc a1i nfreudxy trud ) ACDGBHIABCDCJBDKIELABHIFLM
|
|
N $.
|
|
$}
|
|
|
|
$( An "identity" law of concretion for restricted abstraction. Special case
|
|
of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $)
|
|
rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $=
|
|
( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( An "identity" law for restricted class abstraction. (Contributed by NM,
|
|
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) $)
|
|
rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
|
|
( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i
|
|
wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
|
|
CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
|
|
$}
|
|
|
|
$( Equivalent wff's correspond to equal restricted class abstractions.
|
|
Closed theorem form of ~ rabbidva . (Contributed by NM, 25-Nov-2013.) $)
|
|
rabbi $p |- ( A. x e. A ( ps <-> ch )
|
|
<-> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( cv wcel wa wb wal wceq wral crab abbi wi df-ral pm5.32 albii bitri df-rab
|
|
cab eqeq12i 3bitr4i ) CEDFZAGZUCBGZHZCIZUDCTZUECTZJABHZCDKZACDLZBCDLZJUDUEC
|
|
MUKUCUJNZCIUGUJCDOUNUFCUCABPQRULUHUMUIACDSBCDSUAUB $.
|
|
|
|
$( Swap with a membership relation in a restricted class abstraction.
|
|
(Contributed by NM, 4-Jul-2005.) $)
|
|
rabswap $p |- { x e. A | x e. B } = { x e. B | x e. A } $=
|
|
( cv wcel wa cab crab ancom abbii df-rab 3eqtr4i ) ADZBEZMCEZFZAGONFZAGOABH
|
|
NACHPQANOIJOABKNACKL $.
|
|
|
|
$( The abstraction variable in a restricted class abstraction isn't free.
|
|
(Contributed by NM, 19-Mar-1997.) $)
|
|
nfrab1 $p |- F/_ x { x e. A | ph } $=
|
|
( crab cv wcel wa cab df-rab nfab1 nfcxfr ) BABCDBECFAGZBHABCILBJK $.
|
|
|
|
${
|
|
$d x y $.
|
|
nfrabxy.1 $e |- F/ x ph $.
|
|
nfrabxy.2 $e |- F/_ x A $.
|
|
$( A variable not free in a wff remains so in a restricted class
|
|
abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.) $)
|
|
nfrabxy $p |- F/_ x { y e. A | ph } $=
|
|
( crab cv wcel wa cab df-rab nfcri nfan nfab nfcxfr ) BACDGCHDIZAJZCKACDL
|
|
RBCQABBCDFMENOP $.
|
|
$}
|
|
|
|
${
|
|
reubida.1 $e |- F/ x ph $.
|
|
reubida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by Mario Carneiro, 19-Nov-2016.) $)
|
|
reubida $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( cv wcel wa weu wreu pm5.32da eubid df-reu 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reubidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 13-Nov-2004.) $)
|
|
reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( nfv reubida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 17-Oct-1996.) $)
|
|
reubidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( wb cv wcel adantr reubidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
reubiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 14-Nov-2004.) $)
|
|
reubiia $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
|
|
( cv wcel wa weu wreu pm5.32i eubii df-reu 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
reubii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 22-Oct-1999.) $)
|
|
reubii $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
|
|
( wb cv wcel a1i reubiia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
rmobida.1 $e |- F/ x ph $.
|
|
rmobida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobida $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( cv wcel wa wmo wrmo pm5.32da mobid df-rmo 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( nfv rmobida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobidv $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( wb cv wcel adantr rmobidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobiia $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
|
|
( cv wcel wa wmo wrmo pm5.32i mobii df-rmo 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
rmobii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobii $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
|
|
( wb cv wcel a1i rmobiia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
raleq1f.1 $e |- F/_ x A $.
|
|
raleq1f.2 $e |- F/_ x B $.
|
|
$( Equality theorem for restricted universal quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
raleqf $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
|
|
( wceq cv wcel wi wal wral nfeq eleq2 imbi1d albid df-ral 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted existential quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
|
|
( wceq cv wcel wa wex wrex nfeq eleq2 anbi1d exbid df-rex 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
reueq1f $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
|
|
( wceq cv wcel wa weu wreu nfeq eleq2 anbi1d eubid df-reu 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeq1f $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
|
|
( wceq cv wcel wa wmo wrmo nfeq eleq2 anbi1d mobid df-rmo 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for restricted universal quantifier. (Contributed by
|
|
NM, 16-Nov-1995.) $)
|
|
raleq $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
|
|
( nfcv raleqf ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted existential quantifier. (Contributed by
|
|
NM, 29-Oct-1995.) $)
|
|
rexeq $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
|
|
( nfcv rexeqf ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier. (Contributed by
|
|
NM, 5-Apr-2004.) $)
|
|
reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
|
|
( nfcv reueq1f ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier. (Contributed by
|
|
Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
|
|
( nfcv rmoeq1f ) ABCDBCEBDEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
raleq1i.1 $e |- A = B $.
|
|
$( Equality inference for restricted universal qualifier. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
raleqi $p |- ( A. x e. A ph <-> A. x e. B ph ) $=
|
|
( wceq wral wb raleq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
|
|
$( Equality inference for restricted existential qualifier. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rexeqi $p |- ( E. x e. A ph <-> E. x e. B ph ) $=
|
|
( wceq wrex wb rexeq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
raleq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 13-Nov-2005.) $)
|
|
raleqdv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) ) $=
|
|
( wceq wral wb raleq syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
|
|
$( Equality deduction for restricted existential quantifier. (Contributed
|
|
by NM, 14-Jan-2007.) $)
|
|
rexeqdv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) ) $=
|
|
( wceq wrex wb rexeq syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
raleqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 16-Nov-1995.) $)
|
|
raleqbi1dv $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) ) $=
|
|
( wceq wral raleq ralbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted existential quantifier. (Contributed
|
|
by NM, 18-Mar-1997.) $)
|
|
rexeqbi1dv $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) ) $=
|
|
( wceq wrex rexeq rexbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted uniqueness quantifier. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
reueqd $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) ) $=
|
|
( wceq wreu reueq1 reubidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted uniqueness quantifier. (Contributed
|
|
by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeqd $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) ) $=
|
|
( wceq wrmo rmoeq1 rmobidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
raleqbidv.1 $e |- ( ph -> A = B ) $.
|
|
raleqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 6-Nov-2007.) $)
|
|
raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( wral raleqdv ralbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 6-Nov-2007.) $)
|
|
rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( wrex rexeqdv rexbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
raleqbidva.1 $e |- ( ph -> A = B ) $.
|
|
raleqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
Mario Carneiro, 5-Jan-2017.) $)
|
|
raleqbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( wral ralbidva raleqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
|
|
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
Mario Carneiro, 5-Jan-2017.) $)
|
|
rexeqbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( wrex rexbidva rexeqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
|
|
$}
|
|
|
|
$( Unrestricted "at most one" implies restricted "at most one". (Contributed
|
|
by NM, 16-Jun-2017.) $)
|
|
mormo $p |- ( E* x ph -> E* x e. A ph ) $=
|
|
( wmo cv wcel wa wrmo moan df-rmo sylibr ) ABDBECFZAGBDABCHALBIABCJK $.
|
|
|
|
$( Restricted uniqueness in terms of "at most one." (Contributed by NM,
|
|
23-May-1999.) (Revised by NM, 16-Jun-2017.) $)
|
|
reu5 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) $=
|
|
( cv wcel wa weu wex wmo wreu wrex eu5 df-reu df-rex df-rmo anbi12i 3bitr4i
|
|
wrmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCRZFSBLABCMUBTUCUAABCNABCOPQ $.
|
|
|
|
$( Restricted unique existence implies restricted existence. (Contributed by
|
|
NM, 19-Aug-1999.) $)
|
|
reurex $p |- ( E! x e. A ph -> E. x e. A ph ) $=
|
|
( wreu wrex wrmo reu5 simplbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( Restricted existential uniqueness implies restricted "at most one."
|
|
(Contributed by NM, 16-Jun-2017.) $)
|
|
reurmo $p |- ( E! x e. A ph -> E* x e. A ph ) $=
|
|
( wreu wrex wrmo reu5 simprbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( Restricted "at most one" in term of uniqueness. (Contributed by NM,
|
|
16-Jun-2017.) $)
|
|
rmo5 $p |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) ) $=
|
|
( cv wcel wa wmo wex weu wrmo wrex wreu df-mo df-rmo df-rex imbi12i 3bitr4i
|
|
wi df-reu ) BDCEAFZBGTBHZTBIZRABCJABCKZABCLZRTBMABCNUCUAUDUBABCOABCSPQ $.
|
|
|
|
$( Nonexistence implies restricted "at most one". (Contributed by NM,
|
|
17-Jun-2017.) $)
|
|
nrexrmo $p |- ( -. E. x e. A ph -> E* x e. A ph ) $=
|
|
( wrex wn wreu wi wrmo pm2.21 rmo5 sylibr ) ABCDZELABCFZGABCHLMIABCJK $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z ps $. $d z ph $.
|
|
cbvralf.1 $e |- F/_ x A $.
|
|
cbvralf.2 $e |- F/_ y A $.
|
|
cbvralf.3 $e |- F/ y ph $.
|
|
cbvralf.4 $e |- F/ x ps $.
|
|
cbvralf.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro,
|
|
9-Oct-2016.) $)
|
|
cbvralf $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v weq eleq1 sbequ12 cbval
|
|
imbi12d nfsb sbequ sbie syl6bb bitri df-ral 3bitr4i ) CLZEMZANZCOZDLZEMZB
|
|
NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
|
|
EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
|
|
HABCDIJUIUJUFUEUKACEULBDEULUM $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro,
|
|
9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) $)
|
|
cbvrexf $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( vz cv wcel wa wex wrex wsb nfv nfcri nfan nfs1v weq eleq1 sbequ12 cbvex
|
|
anbi12d nfsb sbequ sbie syl6bb bitri df-rex 3bitr4i ) CLZEMZANZCOZDLZEMZB
|
|
NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
|
|
EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
|
|
HABCDIJUIUJUFUEUKACEULBDEULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $. $d z ps $.
|
|
cbvral.1 $e |- F/ y ph $.
|
|
cbvral.2 $e |- F/ x ps $.
|
|
cbvral.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 31-Jul-2003.) $)
|
|
cbvral $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( nfcv cbvralf ) ABCDECEIDEIFGHJ $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvrex $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( nfcv cbvrexf ) ABCDECEIDEIFGHJ $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvreu $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
|
|
( vz cv wcel wa weu wreu wsb nfv sb8eu sban eubii df-reu anbi1i nfsb nfan
|
|
clelsb3 weq eleq1 sbequ sbie syl6bb anbi12d cbveu bitri 3bitri 3bitr4i )
|
|
CJEKZALZCMZDJZEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZVAUPCIUPIPQVBVEI
|
|
UOACIRSVFIJZEKZVDLZIMVAVEVIIVCVHVDICEUDUASVIUTIDVHVDDVHDPACIDFUBUCUTIPIDU
|
|
EZVHUSVDBVGUREUFVJVDACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $.
|
|
|
|
$( Change the bound variable of restricted "at most one" using implicit
|
|
substitution. (Contributed by NM, 16-Jun-2017.) $)
|
|
cbvrmo $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
|
|
( wrex wreu wi wrmo cbvrex cbvreu imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDEIZ
|
|
BDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y ph $. $d x ps $.
|
|
cbvralv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change the bound variable of a restricted universal quantifier using
|
|
implicit substitution. (Contributed by NM, 28-Jan-1997.) $)
|
|
cbvralv $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( nfv cbvral ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted existential quantifier using
|
|
implicit substitution. (Contributed by NM, 2-Jun-1998.) $)
|
|
cbvrexv $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( nfv cbvrex ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
|
|
( nfv cbvreu ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by Alexander van der Vekens,
|
|
17-Jun-2017.) $)
|
|
cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
|
|
( nfv cbvrmo ) ABCDEADGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ps y $. $d B x $. $d ch x $. $d x ph y $.
|
|
cbvraldva2.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
cbvraldva2.2 $e |- ( ( ph /\ x = y ) -> A = B ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvraldva2 $p |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) ) $=
|
|
( cv wcel wi wal wral weq wa simpr eleq12d imbi12d df-ral cbvaldva
|
|
3bitr4g ) ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUEUHDEADEOZPZUDUGBCUJUCUFFGAUIQ
|
|
IRHSUABDFTCEGTUB $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $=
|
|
( cv wcel wa wex wrex weq simpr eleq12d anbi12d cbvexdva df-rex 3bitr4g )
|
|
ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUDUGDEADEOZLZUCUFBCUIUBUEFGAUHPIQHRSBDFTC
|
|
EGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d ps y $. $d ch x $. $d A x y $. $d x ph y $.
|
|
cbvraldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution. Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
|
|
( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution. Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $=
|
|
( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d z A $. $d x y B $. $d z y B $. $d w B $. $d z ph $.
|
|
$d y ps $. $d x ch $. $d w ch $.
|
|
cbvral2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
|
|
cbvral2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
|
|
$( Change bound variables of double restricted universal quantification,
|
|
using implicit substitution. (Contributed by NM, 10-Aug-2004.) $)
|
|
cbvral2v $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $=
|
|
( wral weq ralbidv cbvralv ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF
|
|
MACEIJNOSTFHCBEGIKOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A z $. $d B w $. $d B x y $. $d B z y $. $d ch w $.
|
|
$d ch x $. $d ph z $. $d ps y $.
|
|
cbvrex2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
|
|
cbvrex2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
|
|
$( Change bound variables of double restricted universal quantification,
|
|
using implicit substitution. (Contributed by FL, 2-Jul-2012.) $)
|
|
cbvrex2v $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $=
|
|
( wrex weq rexbidv cbvrexv rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF
|
|
MACEIJNOSTFHCBEGIKOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d w ph $. $d z ps $. $d x ch $. $d v ch $. $d y u th $. $d x A $.
|
|
$d w A $. $d x y B $. $d w y B $. $d v B $. $d x y z C $.
|
|
$d w y z C $. $d v z C $. $d z y C $. $d z C $. $d u C $.
|
|
cbvral3v.1 $e |- ( x = w -> ( ph <-> ch ) ) $.
|
|
cbvral3v.2 $e |- ( y = v -> ( ch <-> th ) ) $.
|
|
cbvral3v.3 $e |- ( z = u -> ( th <-> ps ) ) $.
|
|
$( Change bound variables of triple restricted universal quantification,
|
|
using implicit substitution. (Contributed by NM, 10-May-2005.) $)
|
|
cbvral3v $p |- ( A. x e. A A. y e. B A. z e. C ph <->
|
|
A. w e. A A. v e. B A. u e. C ps ) $=
|
|
( wral weq 2ralbidv cbvralv cbvral2v ralbii bitri ) AGMQFLQZEKQCGMQFLQZHK
|
|
QBJMQILQZHKQUDUEEHKEHRACFGLMNSTUEUFHKCBDFGIJLMOPUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d z x A $. $d y A $. $d z y ph $.
|
|
$( Change bound variable by using a substitution. (Contributed by NM,
|
|
20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvralsv $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
|
|
( vz wral wsb nfv nfs1v sbequ12 cbvral nfsb sbequ bitri ) ABDFABEGZEDFABC
|
|
GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
|
|
$}
|
|
|
|
${
|
|
$d z x A $. $d y z ph $. $d y A $.
|
|
$( Change bound variable by using a substitution. (Contributed by NM,
|
|
2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvrexsv $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $=
|
|
( vz wrex wsb nfv nfs1v sbequ12 cbvrex nfsb sbequ bitri ) ABDFABEGZEDFABC
|
|
GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z ph $. $d x z ps $.
|
|
sbralie.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Implicit to explicit substitution that swaps variables in a quantified
|
|
expression. (Contributed by NM, 5-Sep-2004.) $)
|
|
sbralie $p |- ( [ x / y ] A. x e. y ph <-> A. y e. x ps ) $=
|
|
( vz cv wral wsb cbvralsv sbbii nfv raleq sbie bitri sbco2 ralbii ) ACDGZ
|
|
HZDCIZACFIZFCGZHZBDUBHZTUAFRHZDCIUCSUEDCACFRJKUEUCDCUCDLUAFRUBMNOUCUAFDIZ
|
|
DUBHUDUAFDUBJUFBDUBUFACDIBACDFAFLPABCDBCLENOQOO $.
|
|
$}
|
|
|
|
${
|
|
rabbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (inference
|
|
rule). (Contributed by NM, 22-May-1999.) $)
|
|
rabbiia $p |- { x e. A | ph } = { x e. A | ps } $=
|
|
( cv wcel wa cab crab pm5.32i abbii df-rab 3eqtr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rabbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (deduction
|
|
rule). (Contributed by NM, 28-Nov-2003.) $)
|
|
rabbidva $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( wb wral crab wceq ralrimiva rabbi sylib ) ABCGZDEHBDEICDEIJANDEFKBCDELM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (deduction
|
|
rule). (Contributed by NM, 10-Feb-1995.) $)
|
|
rabbidv $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( wb cv wcel adantr rabbidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
rabeqf.1 $e |- F/_ x A $.
|
|
rabeqf.2 $e |- F/_ x B $.
|
|
$( Equality theorem for restricted class abstractions, with bound-variable
|
|
hypotheses instead of distinct variable restrictions. (Contributed by
|
|
NM, 7-Mar-2004.) $)
|
|
rabeqf $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
|
|
( wceq cv wcel wa cab crab nfeq eleq2 anbi1d abbid df-rab 3eqtr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for restricted class abstractions. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
rabeq $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
|
|
( nfcv rabeqf ) ABCDBCEBDEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $.
|
|
rabeqbidv.1 $e |- ( ph -> A = B ) $.
|
|
rabeqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equality of restricted class abstractions. (Contributed by Jeff Madsen,
|
|
1-Dec-2009.) $)
|
|
rabeqbidv $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
|
|
( crab wceq rabeq syl rabbidv eqtrd ) ABDEIZBDFIZCDFIAEFJOPJGBDEFKLABCDFH
|
|
MN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $.
|
|
rabeqbidva.1 $e |- ( ph -> A = B ) $.
|
|
rabeqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equality of restricted class abstractions. (Contributed by Mario
|
|
Carneiro, 26-Jan-2017.) $)
|
|
rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
|
|
( crab rabbidva wceq rabeq syl eqtrd ) ABDEICDEIZCDFIZABCDEHJAEFKOPKGCDEF
|
|
LMN $.
|
|
$}
|
|
|
|
${
|
|
rabeqi.1 $e |- A = { x e. B | ph } $.
|
|
$( Inference rule from equality of a class variable and a restricted class
|
|
abstraction. (Contributed by NM, 16-Feb-2004.) $)
|
|
rabeq2i $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
|
|
( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d A z $. $d ph z $. $d ps z $.
|
|
cbvrab.1 $e |- F/_ x A $.
|
|
cbvrab.2 $e |- F/_ y A $.
|
|
cbvrab.3 $e |- F/ y ph $.
|
|
cbvrab.4 $e |- F/ x ps $.
|
|
cbvrab.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule to change the bound variable in a restricted class abstraction,
|
|
using implicit substitution. This version has bound-variable hypotheses
|
|
in place of distinct variable conditions. (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) $)
|
|
cbvrab $p |- { x e. A | ph } = { y e. A | ps } $=
|
|
( vz cv wcel wa cab crab wsb nfv nfcri nfan nfs1v weq eleq1 sbequ12 cbvab
|
|
anbi12d nfsb sbequ sbie syl6bb eqtri df-rab 3eqtr4i ) CLZEMZANZCOZDLZEMZB
|
|
NZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATCKUBUOVCAVDUNVB
|
|
EUCACKUDUFUEVEUTKDVCVDDDKEGSACKDHUGTUTKRKDUBZVCUSVDBVBUREUCVFVDACDQBAKDCU
|
|
HABCDIJUIUJUFUEUKACEULBDEULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $. $d x ps $.
|
|
cbvrabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule to change the bound variable in a restricted class abstraction,
|
|
using implicit substitution. (Contributed by NM, 26-May-1999.) $)
|
|
cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $=
|
|
( nfcv nfv cbvrab ) ABCDECEGDEGADHBCHFI $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The universal class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for the universal class. $)
|
|
$c _V $. $( Letter V (for the universal class) $)
|
|
|
|
$( Extend class notation to include the universal class symbol. $)
|
|
cvv $a class _V $.
|
|
|
|
${
|
|
$d z x $. $d z y $.
|
|
$( Soundness justification theorem for ~ df-v . (Contributed by Rodolfo
|
|
Medina, 27-Apr-2010.) $)
|
|
vjust $p |- { x | x = x } = { y | y = y } $=
|
|
( vz weq cab wsb cv wcel equid sbt 2th df-clab 3bitr4i eqriv ) CAADZAEZBB
|
|
DZBEZOACFZQBCFZCGZPHUARHSTOACAIJQBCBIJKOCALQCBLMN $.
|
|
$}
|
|
|
|
$( Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21.
|
|
Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.) $)
|
|
df-v $a |- _V = { x | x = x } $.
|
|
|
|
$( All setvar variables are sets (see ~ isset ). Theorem 6.8 of [Quine]
|
|
p. 43. (Contributed by NM, 5-Aug-1993.) $)
|
|
vex $p |- x e. _V $=
|
|
( cv cvv wcel weq equid df-v abeq2i mpbir ) ABCDAAEZAFJACAGHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to say " ` A ` is a set": A class ` A ` is a member of the
|
|
universal class ` _V ` (see ~ df-v ) if and only if the class ` A `
|
|
exists (i.e. there exists some set ` x ` equal to class ` A ` ).
|
|
Theorem 6.9 of [Quine] p. 43. _Notational convention_: We will use the
|
|
notational device " ` A e. _V ` " to mean " ` A ` is a set" very
|
|
frequently, for example in ~ uniex . Note the when ` A ` is not a set,
|
|
it is called a proper class. In some theorems, such as ~ uniexg , in
|
|
order to shorten certain proofs we use the more general antecedent
|
|
` A e. V ` instead of ` A e. _V ` to mean " ` A ` is a set."
|
|
|
|
Note that a constant is implicitly considered distinct from all
|
|
variables. This is why ` _V ` is not included in the distinct variable
|
|
list, even though ~ df-clel requires that the expression substituted for
|
|
` B ` not contain ` x ` . (Also, the Metamath spec does not allow
|
|
constants in the distinct variable list.) (Contributed by NM,
|
|
26-May-1993.) $)
|
|
isset $p |- ( A e. _V <-> E. x x = A ) $=
|
|
( cvv wcel cv wceq wa wex df-clel vex biantru exbii bitr4i ) BCDAEZBFZNCD
|
|
ZGZAHOAHABCIOQAPOAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d x y $.
|
|
issetf.1 $e |- F/_ x A $.
|
|
$( A version of isset that does not require x and A to be distinct.
|
|
(Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
issetf $p |- ( A e. _V <-> E. x x = A ) $=
|
|
( vy cvv wcel cv wceq wex isset nfeq2 nfv eqeq1 cbvex bitri ) BEFDGZBHZDI
|
|
AGZBHZAIDBJQSDAAPBCKSDLPRBMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
isseti.1 $e |- A e. _V $.
|
|
$( A way to say " ` A ` is a set" (inference rule). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
isseti $p |- E. x x = A $=
|
|
( cvv wcel cv wceq wex isset mpbi ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
issetri.1 $e |- E. x x = A $.
|
|
$( A way to say " ` A ` is a set" (inference rule). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
issetri $p |- A e. _V $=
|
|
( cvv wcel cv wceq wex isset mpbir ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
$( A class equal to a variable is a set. Note the absence of dv condition,
|
|
contrary to ~ isset and ~ issetri . (Contributed by BJ, 27-Apr-2019.) $)
|
|
eqvisset $p |- ( x = A -> A e. _V ) $=
|
|
( cv wceq cvv wcel vex eleq1 mpbii ) ACZBDJEFBEFAGJBEHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( If a class is a member of another class, it is a set. Theorem 6.12 of
|
|
[Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Andrew Salmon, 8-Jun-2011.) $)
|
|
elex $p |- ( A e. B -> A e. _V ) $=
|
|
( vx cv wceq wcel wa wex cvv exsimpl df-clel isset 3imtr4i ) CDZAEZNBFZGC
|
|
HOCHABFAIFOPCJCABKCALM $.
|
|
$}
|
|
|
|
${
|
|
elisseti.1 $e |- A e. B $.
|
|
$( If a class is a member of another class, it is a set. (Contributed by
|
|
NM, 11-Jun-1994.) $)
|
|
elexi $p |- A e. _V $=
|
|
( wcel cvv elex ax-mp ) ABDAEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
elexd.1 $e |- ( ph -> A e. V ) $.
|
|
$( If a class is a member of another class, it is a set. (Contributed by
|
|
Glauco Siliprandi, 11-Oct-2020.) $)
|
|
elexd $p |- ( ph -> A e. _V ) $=
|
|
( wcel cvv elex syl ) ABCEBFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An element of a class exists. (Contributed by NM, 1-May-1995.) $)
|
|
elisset $p |- ( A e. V -> E. x x = A ) $=
|
|
( wcel cvv cv wceq wex elex isset sylib ) BCDBEDAFBGAHBCIABJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( If two classes each contain another class, then both contain some set.
|
|
(Contributed by Alan Sare, 24-Oct-2011.) $)
|
|
elex22 $p |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) ) $=
|
|
( wcel wa cv wceq wi wal eleq1a anim12ii alrimiv elisset adantr exim sylc
|
|
wex ) BCEZBDEZFZAGZBHZUBCEZUBDEZFZIZAJUCARZUFARUAUGASUCUDTUEBCUBKBDUBKLMS
|
|
UHTABCNOUCUFAPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( If a class contains another class, then it contains some set.
|
|
(Contributed by Alan Sare, 25-Sep-2011.) $)
|
|
elex2 $p |- ( A e. B -> E. x x e. B ) $=
|
|
( wcel cv wceq wi wal wex eleq1a alrimiv elisset exim sylc ) BCDZAEZBFZPC
|
|
DZGZAHQAIRAIOSABCPJKABCLQRAMN $.
|
|
$}
|
|
|
|
$( A universal quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 26-Mar-2004.) $)
|
|
ralv $p |- ( A. x e. _V ph <-> A. x ph ) $=
|
|
( cvv wral cv wcel wi wal df-ral vex a1bi albii bitr4i ) ABCDBECFZAGZBHABHA
|
|
BCIAOBNABJKLM $.
|
|
|
|
$( An existential quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 26-Mar-2004.) $)
|
|
rexv $p |- ( E. x e. _V ph <-> E. x ph ) $=
|
|
( cvv wrex cv wcel wa wex df-rex vex biantrur exbii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A uniqueness quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 1-Nov-2010.) $)
|
|
reuv $p |- ( E! x e. _V ph <-> E! x ph ) $=
|
|
( cvv wreu cv wcel wa weu df-reu vex biantrur eubii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A uniqueness quantifier restricted to the universe is unrestricted.
|
|
(Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmov $p |- ( E* x e. _V ph <-> E* x ph ) $=
|
|
( cvv wrmo cv wcel wa wmo df-rmo vex biantrur mobii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A class abstraction restricted to the universe is unrestricted.
|
|
(Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
rabab $p |- { x e. _V | ph } = { x | ph } $=
|
|
( cvv crab cv wcel wa cab df-rab vex biantrur abbii eqtr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Commutation of restricted and unrestricted universal quantifiers.
|
|
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
ralcom4 $p |- ( A. x e. A A. y ph <-> A. y A. x e. A ph ) $=
|
|
( cvv wral wal ralcom ralv ralbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
|
|
DEHLNBDACIJMCIK $.
|
|
|
|
$( Commutation of restricted and unrestricted existential quantifiers.
|
|
(Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
rexcom4 $p |- ( E. x e. A E. y ph <-> E. y E. x e. A ph ) $=
|
|
( cvv wrex wex rexcom rexv rexbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
|
|
DEHLNBDACIJMCIK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d x y $. $d ph x $.
|
|
$( Specialized existential commutation lemma. (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) $)
|
|
rexcom4a $p |- ( E. x E. y e. A ( ph /\ ps )
|
|
<-> E. y e. A ( ph /\ E. x ps ) ) $=
|
|
( wa wrex wex rexcom4 19.42v rexbii bitr3i ) ABFZDEGCHMCHZDEGABCHFZDEGMDC
|
|
EINODEABCJKL $.
|
|
|
|
$d B x $.
|
|
rexcom4b.1 $e |- B e. _V $.
|
|
$( Specialized existential commutation lemma. (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) $)
|
|
rexcom4b $p |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph ) $=
|
|
( cv wceq wa wrex wex rexcom4a isseti biantru rexbii bitr4i ) ABGEHZICDJB
|
|
KAQBKZIZCDJACDJAQBCDLASCDRABEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Closed theorem version of ~ ceqsalg . (Contributed by NM, 28-Feb-2013.)
|
|
(Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
ceqsalt $p |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V )
|
|
-> ( A. x ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel w3a wex elisset 3ad2ant3 bi1 imim3i 3ad2ant2
|
|
3ad2ant1 al2imi 19.23t sylibd mpid imim2i com23 alimi 19.21t mpbid impbid
|
|
bi2 ) BCFZCGDHZABIZJZCKZDELZMZUMAJZCKZBURUTUMCNZBUQULVAUPCDEOPURUTUMBJZCK
|
|
ZVABJZUPULUTVCJUQUOUSVBCUNABUMABQRUASULUPVCVDIUQUMBCUBTUCUDURBUSJZCKZBUTJ
|
|
ZUPULVFUQUOVECUOUMBAUNBAJUMABUKUEUFUGSULUPVFVGIUQBUSCUHTUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Restricted quantifier version of ~ ceqsalt . (Contributed by NM,
|
|
28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
ceqsralt $p |- ( ( F/ x ps
|
|
/\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
|
|
-> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel w3a wral df-ral eleq1 pm5.32ri imbi1i impexp
|
|
wa 3bitr3i albii a1i syl5bb 19.21v syl6bb biimt 3ad2ant3 ceqsalt 3bitr2d
|
|
) BCFZCGZDHZABIJCKZDELZMZUMAJZCENZUOUQCKZJZUSBUPURUOUQJZCKZUTURULELZUQJZC
|
|
KZUPVBUQCEOVEVBIUPVDVACVCUMTZAJUOUMTZAJVDVAVFVGAUMVCUOULDEPQRVCUMASUOUMAS
|
|
UAUBUCUDUOUQCUEUFUOUKUSUTIUNUOUSUGUHABCDEUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsalg.1 $e |- F/ x ps $.
|
|
ceqsalg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
ceqsalg $p |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) ) $=
|
|
( wcel cv wceq wi wal wex elisset nfa1 biimpd a2i sps exlimd syl5com
|
|
biimprcd alrimi impbid1 ) DEHZCIDJZAKZCLZBUDUECMUGBCDENUGUEBCUFCOFUFUEBKC
|
|
UEABUEABGPQRSTBUFCFUEABGUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsal.1 $e |- F/ x ps $.
|
|
ceqsal.2 $e |- A e. _V $.
|
|
ceqsal.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
ceqsal $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
|
|
( cvv wcel cv wceq wi wal wb ceqsalg ax-mp ) DHICJDKALCMBNFABCDHEGOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsalv.1 $e |- A e. _V $.
|
|
ceqsalv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
ceqsalv $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
|
|
( nfv ceqsal ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
ceqsralv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted quantifier version of ~ ceqsalv . (Contributed by NM,
|
|
21-Jun-2013.) $)
|
|
ceqsralv $p |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel wral nfv ax-gen ceqsralt mp3an12 ) BCGCHDIZA
|
|
BJKZCLDEMSAKCENBJBCOTCFPABCDEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
gencl.1 $e |- ( th <-> E. x ( ch /\ A = B ) ) $.
|
|
gencl.2 $e |- ( A = B -> ( ph <-> ps ) ) $.
|
|
gencl.3 $e |- ( ch -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
gencl $p |- ( th -> ps ) $=
|
|
( wceq wa wex syl5ib impcom exlimiv sylbi ) DCFGKZLZEMBHSBERCBCARBJINOPQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x R $. $d x ps $. $d y C $. $d y S $. $d y ch $.
|
|
2gencl.1 $e |- ( C e. S <-> E. x e. R A = C ) $.
|
|
2gencl.2 $e |- ( D e. S <-> E. y e. R B = D ) $.
|
|
2gencl.3 $e |- ( A = C -> ( ph <-> ps ) ) $.
|
|
2gencl.4 $e |- ( B = D -> ( ps <-> ch ) ) $.
|
|
2gencl.5 $e |- ( ( x e. R /\ y e. R ) -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
2gencl $p |- ( ( C e. S /\ D e. S ) -> ch ) $=
|
|
( wcel wi cv wceq wrex wa wex df-rex bitri imbi2d ex gencl com12 impcom )
|
|
IKQZHKQZCULBRULCRESJQZUKEGIUKGITZEJUAUMUNUBEUCMUNEJUDUEUNBCULOUFULUMBUMAR
|
|
UMBRDSJQZULDFHULFHTZDJUAUOUPUBDUCLUPDJUDUEUPABUMNUFUOUMAPUGUHUIUHUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z D $. $d z F $. $d x y R $. $d y z S $. $d x ps $.
|
|
$d y ch $. $d z th $.
|
|
3gencl.1 $e |- ( D e. S <-> E. x e. R A = D ) $.
|
|
3gencl.2 $e |- ( F e. S <-> E. y e. R B = F ) $.
|
|
3gencl.3 $e |- ( G e. S <-> E. z e. R C = G ) $.
|
|
3gencl.4 $e |- ( A = D -> ( ph <-> ps ) ) $.
|
|
3gencl.5 $e |- ( B = F -> ( ps <-> ch ) ) $.
|
|
3gencl.6 $e |- ( C = G -> ( ch <-> th ) ) $.
|
|
3gencl.7 $e |- ( ( x e. R /\ y e. R /\ z e. R ) -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
3gencl $p |- ( ( D e. S /\ F e. S /\ G e. S ) -> th ) $=
|
|
( wcel wa wi wceq wrex wex df-rex bitri imbi2d 3expia 2gencl com12 3impia
|
|
cv gencl ) KMUCZNMUCZOMUCZDUTURUSUDZDVACUEVADUEGUPLUCZUTGJOUTJOUFZGLUGVBV
|
|
CUDGUHRVCGLUIUJVCCDVAUAUKVAVBCVBAUEVBBUEVBCUEEFHIKNLMPQHKUFABVBSUKINUFBCV
|
|
BTUKEUPLUCFUPLUCVBAUBULUMUNUQUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
cgsexg.1 $e |- ( x = A -> ch ) $.
|
|
cgsexg.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for general classes. (Contributed by
|
|
NM, 26-Aug-2007.) $)
|
|
cgsexg $p |- ( A e. V ->
|
|
( E. x ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex biimpa exlimiv cv wceq elisset eximi syl biimprcd ancld
|
|
eximdv syl5com impbid2 ) EFIZCAJZDKZBUEBDCABHLMUDCDKZBUFUDDNEOZDKUGDEFPUH
|
|
CDGQRBCUEDBCACABHSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
cgsex2g.1 $e |- ( ( x = A /\ y = B ) -> ch ) $.
|
|
cgsex2g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for general classes. (Contributed by
|
|
NM, 26-Jul-1995.) $)
|
|
cgsex2g $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x E. y ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex biimpa exlimivv cv wceq elisset anim12i eeanv sylibr 2eximi
|
|
syl biimprcd ancld 2eximdv syl5com impbid2 ) FHLZGILZMZCAMZENDNZBUMBDECAB
|
|
KOPULCENDNZBUNULDQFRZEQGRZMZENDNZUOULUPDNZUQENZMUSUJUTUKVADFHSEGISTUPUQDE
|
|
UAUBURCDEJUCUDBCUMDEBCACABKUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y z w ps $.
|
|
cgsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch ) $.
|
|
cgsex4g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( An implicit substitution inference for 4 general classes. (Contributed
|
|
by NM, 5-Aug-1995.) $)
|
|
cgsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
|
|
( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex cv wceq biimpa exlimivv elisset anim12i eeanv sylibr ee4anv
|
|
2eximi syl biimprcd ancld 2eximdv syl5com impbid2 ) HLPZIMPZQZJLPZKMPZQZQ
|
|
ZCAQZGRFRZERDRZBVCBDEVBBFGCABOUAUBUBVACGRFRZERDRZBVDVADSHTZESITZQZFSJTZGS
|
|
KTZQZQZGRFRZERDRZVFVAVIERDRZVLGRFRZQVOUQVPUTVQUQVGDRZVHERZQVPUOVRUPVSDHLU
|
|
CEIMUCUDVGVHDEUEUFUTVJFRZVKGRZQVQURVTUSWAFJLUCGKMUCUDVJVKFGUEUFUDVIVLDEFG
|
|
UGUFVNVEDEVMCFGNUHUHUIBVEVCDEBCVBFGBCACABOUJUKULULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsex.1 $e |- F/ x ps $.
|
|
ceqsex.2 $e |- A e. _V $.
|
|
ceqsex.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
ceqsex $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
|
|
( cv wceq wa wex biimpa exlimi wal biimprcd alrimi isseti exintr mpisyl
|
|
wi impbii ) CHDIZAJZCKZBUCBCEUBABGLMBUBATZCNUBCKUDBUECEUBABGOPCDFQUBACRSU
|
|
A $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsexv.1 $e |- A e. _V $.
|
|
ceqsexv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 2-Mar-1995.) $)
|
|
ceqsexv $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
|
|
( nfv ceqsex ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
ceqsex2.1 $e |- F/ x ps $.
|
|
ceqsex2.2 $e |- F/ y ch $.
|
|
ceqsex2.3 $e |- A e. _V $.
|
|
ceqsex2.4 $e |- B e. _V $.
|
|
ceqsex2.5 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex2.6 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of two existential quantifiers, using implicit substitution.
|
|
(Contributed by Scott Fenton, 7-Jun-2006.) $)
|
|
ceqsex2 $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
|
|
( cv wceq w3a wex wa exbii ceqsex 3anass 19.42v nfan anbi2d exbidv 3bitri
|
|
bitri nfv nfex ) DNFOZENGOZAPZEQZDQUJUKARZEQZRZDQUKBRZEQZCUMUPDUMUJUNRZEQ
|
|
UPULUSEUJUKAUASUJUNEUBUGSUOURDFUQDEUKBDUKDUHHUCUIJUJUNUQEUJABUKLUDUETBCEG
|
|
IKMTUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x ps $. $d y ch $.
|
|
ceqsex2v.1 $e |- A e. _V $.
|
|
ceqsex2v.2 $e |- B e. _V $.
|
|
ceqsex2v.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex2v.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of two existential quantifiers, using implicit substitution.
|
|
(Contributed by Scott Fenton, 7-Jun-2006.) $)
|
|
ceqsex2v $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
|
|
( nfv ceqsex2 ) ABCDEFGBDLCELHIJKM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x ps $. $d y ch $.
|
|
$d z th $.
|
|
ceqsex3v.1 $e |- A e. _V $.
|
|
ceqsex3v.2 $e |- B e. _V $.
|
|
ceqsex3v.3 $e |- C e. _V $.
|
|
ceqsex3v.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex3v.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex3v.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
$( Elimination of three existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 16-Aug-2011.) $)
|
|
ceqsex3v $p |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph )
|
|
<-> th ) $=
|
|
( cv wceq wa wex anass 3anass anbi1i df-3an anbi2i 3bitr4i 2exbii 19.42vv
|
|
w3a bitri exbii 3anbi3d 2exbidv ceqsexv ceqsex2v ) EQHRZFQIRZGQJRZUIZASZG
|
|
TFTZETUPUQURAUIZGTFTZSZETZDVAVDEVAUPVBSZGTFTVDUTVFFGUPUQURSZSZASUPVGASZSU
|
|
TVFUPVGAUAUSVHAUPUQURUBUCVBVIUPUQURAUDUEUFUGUPVBFGUHUJUKVEUQURBUIZGTFTZDV
|
|
CVKEHKUPVBVJFGUPABUQURNULUMUNBCDFGIJLMOPUOUJUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x ps $. $d y ch $. $d z th $. $d w ta $.
|
|
ceqsex4v.1 $e |- A e. _V $.
|
|
ceqsex4v.2 $e |- B e. _V $.
|
|
ceqsex4v.3 $e |- C e. _V $.
|
|
ceqsex4v.4 $e |- D e. _V $.
|
|
ceqsex4v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex4v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex4v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex4v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
$( Elimination of four existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 23-Sep-2011.) $)
|
|
ceqsex4v $p |- ( E. x E. y E. z E. w
|
|
( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta ) $=
|
|
( wceq w3a wex 19.42vv 3anass df-3an anbi2i bitr4i 2exbii 3bitr4i 3anbi3d
|
|
cv wa 2exbidv ceqsex2v 3bitri ) FUMJUBZGUMKUBZUNZHUMLUBZIUMMUBZUNZAUCZIUD
|
|
HUDZGUDFUDURUSVAVBAUCZIUDHUDZUCZGUDFUDVAVBCUCZIUDHUDZEVEVHFGUTVFUNZIUDHUD
|
|
UTVGUNVEVHUTVFHIUEVDVKHIVDUTVCAUNZUNVKUTVCAUFVFVLUTVAVBAUGUHUIUJURUSVGUGU
|
|
KUJVGVAVBBUCZIUDHUDVJFGJKNOURVFVMHIURABVAVBRULUOUSVMVIHIUSBCVAVBSULUOUPCD
|
|
EHILMPQTUAUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $. $d x y z w v u E $. $d x y z w v u F $. $d x ps $.
|
|
$d y ch $. $d z th $. $d w ta $. $d v et $. $d u ze $.
|
|
ceqsex6v.1 $e |- A e. _V $.
|
|
ceqsex6v.2 $e |- B e. _V $.
|
|
ceqsex6v.3 $e |- C e. _V $.
|
|
ceqsex6v.4 $e |- D e. _V $.
|
|
ceqsex6v.5 $e |- E e. _V $.
|
|
ceqsex6v.6 $e |- F e. _V $.
|
|
ceqsex6v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex6v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex6v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex6v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
ceqsex6v.11 $e |- ( v = E -> ( ta <-> et ) ) $.
|
|
ceqsex6v.12 $e |- ( u = F -> ( et <-> ze ) ) $.
|
|
$( Elimination of six existential quantifiers, using implicit substitution.
|
|
(Contributed by NM, 21-Sep-2011.) $)
|
|
ceqsex6v $p |- ( E. x E. y E. z E. w E. v E. u
|
|
( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph )
|
|
<-> ze ) $=
|
|
( cv wceq w3a wex wa 3anass 3exbii 19.42vvv bitri anbi2d 3exbidv ceqsex3v
|
|
) HULNUMZIULOUMZJULPUMZUNZKULQUMLULRUMMULSUMUNZAUNZMUOLUOKUOZJUOIUOHUOVGV
|
|
HAUPZMUOLUOKUOZUPZJUOIUOHUOZGVJVMHIJVJVGVKUPZMUOLUOKUOVMVIVOKLMVGVHAUQURV
|
|
GVKKLMUSUTURVNVHDUPZMUOLUOKUOZGVLVHBUPZMUOLUOKUOVHCUPZMUOLUOKUOVQHIJNOPTU
|
|
AUBVDVKVRKLMVDABVHUFVAVBVEVRVSKLMVEBCVHUGVAVBVFVSVPKLMVFCDVHUHVAVBVCDEFGK
|
|
LMQRSUCUDUEUIUJUKVCUTUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u t s A $. $d x y z w v u t s B $. $d x y z w v u t s C $.
|
|
$d x y z w v u t s D $. $d x y z w v u t s E $. $d x y z w v u t s F $.
|
|
$d x y z w v u t s G $. $d x y z w v u t s H $. $d x ps $. $d y ch $.
|
|
$d z th $. $d w ta $. $d v et $. $d u ze $. $d t si $. $d s rh $.
|
|
ceqsex8v.1 $e |- A e. _V $.
|
|
ceqsex8v.2 $e |- B e. _V $.
|
|
ceqsex8v.3 $e |- C e. _V $.
|
|
ceqsex8v.4 $e |- D e. _V $.
|
|
ceqsex8v.5 $e |- E e. _V $.
|
|
ceqsex8v.6 $e |- F e. _V $.
|
|
ceqsex8v.7 $e |- G e. _V $.
|
|
ceqsex8v.8 $e |- H e. _V $.
|
|
ceqsex8v.9 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex8v.10 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex8v.11 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex8v.12 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
ceqsex8v.13 $e |- ( v = E -> ( ta <-> et ) ) $.
|
|
ceqsex8v.14 $e |- ( u = F -> ( et <-> ze ) ) $.
|
|
ceqsex8v.15 $e |- ( t = G -> ( ze <-> si ) ) $.
|
|
ceqsex8v.16 $e |- ( s = H -> ( si <-> rh ) ) $.
|
|
$( Elimination of eight existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 23-Sep-2011.) $)
|
|
ceqsex8v $p |- ( E. x E. y E. z E. w E. v E. u E. t E. s
|
|
( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) )
|
|
/\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh ) $=
|
|
( cv wceq wa w3a wex 19.42vvvv 3anass df-3an anbi2i bitr4i 2exbii 3bitr4i
|
|
3anbi3d 4exbidv ceqsex4v bitri ) JVBQVCZKVBRVCZVDZLVBSVCZMVBTVCZVDZVDZNVB
|
|
UAVCOVBUBVCVDZPVBUCVCUEVBUDVCVDZVDZAVEZUEVFPVFZOVFNVFZMVFLVFZKVFJVFVTWCWE
|
|
WFAVEZUEVFPVFOVFNVFZVEZMVFLVFZKVFJVFZIWKWOJKWJWNLMWDWLVDZUEVFPVFZOVFNVFWD
|
|
WMVDWJWNWDWLOPUENVGWIWRNOWHWQPUEWHWDWGAVDZVDWQWDWGAVHWLWSWDWEWFAVIVJVKVLV
|
|
LVTWCWMVIVMVLVLWPWEWFEVEZUEVFPVFOVFNVFZIWMWEWFBVEZUEVFPVFOVFNVFWEWFCVEZUE
|
|
VFPVFOVFNVFWEWFDVEZUEVFPVFOVFNVFXAJKLMQRSTUFUGUHUIVRWLXBNOPUEVRABWEWFUNVN
|
|
VOVSXBXCNOPUEVSBCWEWFUOVNVOWAXCXDNOPUEWACDWEWFUPVNVOWBXDWTNOPUEWBDEWEWFUQ
|
|
VNVOVPEFGHINOPUEUAUBUCUDUJUKULUMURUSUTVAVPVQVQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbvex.1 $e |- A e. _V $.
|
|
gencbvex.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbvex.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbvex.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
|
|
$( Change of bound variable using implicit substitution. (Contributed by
|
|
NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
gencbvex $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
|
|
( cv wceq wa wex excom wb anbi12d bicomd exbii eqcoms ceqsexv simpr eqcom
|
|
19.41v biimpi adantl eximi sylbi adantr ancri impbii bitri 3bitr3i ) FLZG
|
|
MZDBNZNZFOZEOUREOZFOCANZEOUQFOUREFPUSVAEUQVAFGHUQVAQGUOGUOMZVAUQVBCDABJIR
|
|
SUAUBTUTUQFUTUPEOZUQNZUQUPUQEUEVDUQVCUQUCUQVCDVCBDCVBNZEOVCKVEUPEVBUPCVBU
|
|
PGUOUDUFUGUHUIUJUKULUMTUN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbvex2.1 $e |- A e. _V $.
|
|
gencbvex2.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbvex2.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbvex2.4 $e |- ( th -> E. x ( ch /\ A = y ) ) $.
|
|
$( Restatement of ~ gencbvex with weaker hypotheses. (Contributed by Jeff
|
|
Hankins, 6-Dec-2006.) $)
|
|
gencbvex2 $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
|
|
( cv wceq wa wex biimpac exlimiv impbii gencbvex ) ABCDEFGHIJDCGFLMZNZEOK
|
|
UADETCDJPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbval.1 $e |- A e. _V $.
|
|
gencbval.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbval.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbval.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
|
|
$( Change of bound variable using implicit substitution. (Contributed by
|
|
NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.) $)
|
|
gencbval $p |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) ) $=
|
|
( cv wceq wi wal alcom wb imbi12d albii wex bicomd eqcoms 19.23v wa eqcom
|
|
ceqsalv biimpi adantl eximi sylbi pm2.04 mpdi ax-1 impbii bitri 3bitr3i )
|
|
FLZGMZDBNZNZFOZEOUTEOZFOCANZEOUSFOUTEFPVAVCEUSVCFGHUSVCQGUQGUQMZVCUSVDCDA
|
|
BJIRUAUBUFSVBUSFVBURETZUSNZUSURUSEUCVFUSVFDVEBDCVDUDZETVEKVGUREVDURCVDURG
|
|
UQUEUGUHUIUJVEDBUKULUSVEUMUNUOSUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d x y $.
|
|
sbhypf.1 $e |- F/ x ps $.
|
|
sbhypf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Introduce an explicit substitution into an implicit substitution
|
|
hypothesis. See also csbhypf . (Contributed by Raph Levien,
|
|
10-Apr-2004.) $)
|
|
sbhypf $p |- ( y = A -> ( [ y / x ] ph <-> ps ) ) $=
|
|
( cv wceq wa wex wsb wb vex eqeq1 ceqsexv nfs1v nfbi sbequ12 bicomd
|
|
sylan9bb exlimi sylbir ) DHZEIZCHZUDIZUFEIZJZCKACDLZBMZUHUECUDDNUFUDEOPUI
|
|
UKCUJBCACDQFRUGUJAUHBUGAUJACDSTGUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d x z $.
|
|
$( Closed theorem form of ~ vtoclgf . (Contributed by NM, 17-Feb-2013.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
vtoclgft $p |- ( ( ( F/_ x A /\ F/ x ps )
|
|
/\ ( A. x ( x = A -> ( ph <-> ps ) )
|
|
/\ A. x ph ) /\ A e. V ) -> ps ) $=
|
|
( vz wcel wnfc wnf wa cv wceq wb wi wal cvv elex w3a wex mpbid elisset id
|
|
3ad2ant3 nfnfc1 nfcvd nfeqd eqeq1 a1i cbvexd 3adant3 bi1 imim2i com23 imp
|
|
ad2antrr alanimi 3ad2ant2 simp1r 19.23t syl mpd syl3an3 ) DEGCDHZBCIZJZCK
|
|
ZDLZABMZNZCOACOJZDPGZBDEQVEVJVKRZVGCSZBVLFKZDLZFSZVMVKVEVPVJFDPUAUCVEVJVP
|
|
VMMZVKVCVQVDVJVCVOVGFCCDUDVCCVNDVCCVNUEVCUBUFVNVFLVOVGMNVCVNVFDUGUHUIUOUJ
|
|
TVLVGBNZCOZVMBNZVJVEVSVKVIAVRCVIAVRVIVGABVHABNVGABUKULUMUNUPUQVLVDVSVTMVC
|
|
VDVJVKURVGBCUSUTTVAVB $.
|
|
$}
|
|
|
|
${
|
|
vtocld.1 $e |- ( ph -> A e. V ) $.
|
|
vtocld.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
vtocld.3 $e |- ( ph -> ps ) $.
|
|
${
|
|
vtocldf.4 $e |- F/ x ph $.
|
|
vtocldf.5 $e |- ( ph -> F/_ x A ) $.
|
|
vtocldf.6 $e |- ( ph -> F/ x ch ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed
|
|
by Mario Carneiro, 15-Oct-2016.) $)
|
|
vtocldf $p |- ( ph -> ch ) $=
|
|
( wnfc wnf cv wceq wb wi wal alrimi wcel ex vtoclgft syl221anc ) ADEMCD
|
|
NDOEPZBCQZRZDSBDSEFUACKLAUGDJAUEUFHUBTABDJITGBCDEFUCUD $.
|
|
$}
|
|
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
vtocld $p |- ( ph -> ch ) $=
|
|
( nfv nfcvd nfvd vtocldf ) ABCDEFGHIADJADEKACDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
vtoclf.1 $e |- F/ x ps $.
|
|
vtoclf.2 $e |- A e. _V $.
|
|
vtoclf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclf.4 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. This is a
|
|
generalization of ~ chvar . (Contributed by NM, 30-Aug-1993.) $)
|
|
vtoclf $p |- ps $=
|
|
( cv wceq wi isseti biimpd eximii 19.36i mpg ) ABCABCECIDJZABKCCDFLQABGMN
|
|
OHP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
vtocl.1 $e |- A e. _V $.
|
|
vtocl.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl.3 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
vtocl $p |- ps $=
|
|
( nfv vtoclf ) ABCDBCHEFGI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
vtocl2.1 $e |- A e. _V $.
|
|
vtocl2.2 $e |- B e. _V $.
|
|
vtocl2.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
vtocl2.4 $e |- ph $.
|
|
$( Implicit substitution of classes for setvar variables. (Contributed by
|
|
NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
vtocl2 $p |- ps $=
|
|
( wal wi wex cv wceq isseti wa eeanv biimpd 2eximi mp2an 19.36-1 19.36aiv
|
|
sylbir nfv eximii ax-gen mpg ) ADKZBCUIBCABLZDMZUIBLCCNEOZCMZDNFOZDMZUKCM
|
|
ZCEGPDFHPUMUOQULUNQZDMCMUPULUNCDRUQUJCDUQABISTUDUAABDBDUEUBUFUCADJUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
|
|
vtocl3.1 $e |- A e. _V $.
|
|
vtocl3.2 $e |- B e. _V $.
|
|
vtocl3.3 $e |- C e. _V $.
|
|
vtocl3.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
vtocl3.5 $e |- ph $.
|
|
$( Implicit substitution of classes for setvar variables. (Contributed by
|
|
NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
vtocl3 $p |- ps $=
|
|
( wal wi wex cv wceq isseti w3a eeeanv biimpd eximi 2eximi sylbir 19.36-1
|
|
mp3an nfv ax-mp eximii 19.36aiv gen2 mpg ) AENZDNZBCUOBCUNBOZDPZUOBOCABOZ
|
|
EPZDPCPZUQCPCQFRZCPZDQGRZDPZEQHRZEPZUTCFISDGJSEHKSVBVDVFTVAVCVETZEPZDPCPU
|
|
TVAVCVECDEUAVHUSCDVGUREVGABLUBUCUDUEUGUSUPCDABEBEUHUFUDUIUNBDBDUHUFUJUKAD
|
|
EMULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ch $. $d x th $.
|
|
vtoclb.1 $e |- A e. _V $.
|
|
vtoclb.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
vtoclb.3 $e |- ( x = A -> ( ps <-> th ) ) $.
|
|
vtoclb.4 $e |- ( ph <-> ps ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 23-Dec-1993.) $)
|
|
vtoclb $p |- ( ch <-> th ) $=
|
|
( wb cv wceq bibi12d vtocl ) ABKCDKEFGELFMACBDHINJO $.
|
|
$}
|
|
|
|
${
|
|
vtoclgf.1 $e |- F/_ x A $.
|
|
vtoclgf.2 $e |- F/ x ps $.
|
|
vtoclgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclgf.4 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable, with
|
|
bound-variable hypotheses in place of distinct variable restrictions.
|
|
(Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
vtoclgf $p |- ( A e. V -> ps ) $=
|
|
( wcel cvv elex cv wceq wex issetf mpbii exlimi sylbi syl ) DEJDKJZBDELUA
|
|
CMDNZCOBCDFPUBBCGUBABIHQRST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
vtoclg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclg.2 $e |- ph $.
|
|
$( Implicit substitution of a class expression for a setvar variable.
|
|
(Contributed by NM, 17-Apr-1995.) $)
|
|
vtoclg $p |- ( A e. V -> ps ) $=
|
|
( nfcv nfv vtoclgf ) ABCDECDHBCIFGJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ch $. $d x th $.
|
|
vtoclbg.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
vtoclbg.2 $e |- ( x = A -> ( ps <-> th ) ) $.
|
|
vtoclbg.3 $e |- ( ph <-> ps ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 29-Apr-1994.) $)
|
|
vtoclbg $p |- ( A e. V -> ( ch <-> th ) ) $=
|
|
( wb cv wceq bibi12d vtoclg ) ABKCDKEFGELFMACBDHINJO $.
|
|
$}
|
|
|
|
${
|
|
vtocl2gf.1 $e |- F/_ x A $.
|
|
vtocl2gf.2 $e |- F/_ y A $.
|
|
vtocl2gf.3 $e |- F/_ y B $.
|
|
vtocl2gf.4 $e |- F/ x ps $.
|
|
vtocl2gf.5 $e |- F/ y ch $.
|
|
vtocl2gf.6 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2gf.7 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2gf.8 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 25-Apr-1995.) $)
|
|
vtocl2gf $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
|
|
( wcel cvv wi elex nfel1 nfim cv wceq imbi2d vtoclgf mpan9 ) FHRFSRZGIRCF
|
|
HUAUIBTUICTEGILUICEEFSKUBNUCEUDGUEBCUIPUFABDFSJMOQUGUGUH $.
|
|
$}
|
|
|
|
${
|
|
vtocl3gf.a $e |- F/_ x A $.
|
|
vtocl3gf.b $e |- F/_ y A $.
|
|
vtocl3gf.c $e |- F/_ z A $.
|
|
vtocl3gf.d $e |- F/_ y B $.
|
|
vtocl3gf.e $e |- F/_ z B $.
|
|
vtocl3gf.f $e |- F/_ z C $.
|
|
vtocl3gf.1 $e |- F/ x ps $.
|
|
vtocl3gf.2 $e |- F/ y ch $.
|
|
vtocl3gf.3 $e |- F/ z th $.
|
|
vtocl3gf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3gf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3gf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3gf.7 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
vtocl3gf $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> th ) $=
|
|
( wcel cvv wa elex wi nfel1 nfim wceq imbi2d vtoclgf vtocl2gf mpan9 3impb
|
|
cv ) HKUGZILUGZJMUGZDVAHUHUGZVBVCUIDHKUJVDBUKVDCUKVDDUKFGIJLMQRSVDCFFHUHO
|
|
ULUAUMVDDGGHUHPULUBUMFUTIUNBCVDUDUOGUTJUNCDVDUEUOABEHUHNTUCUFUPUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y B $. $d x ps $. $d y ch $.
|
|
vtocl2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2g.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2g.3 $e |- ph $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 25-Apr-1995.) $)
|
|
vtocl2g $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
|
|
( nfcv nfv vtocl2gf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
vtoclgaf.1 $e |- F/_ x A $.
|
|
vtoclgaf.2 $e |- F/ x ps $.
|
|
vtoclgaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclgaf.4 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
vtoclgaf $p |- ( A e. B -> ps ) $=
|
|
( wcel cv wi nfel1 nfim wceq eleq1 imbi12d vtoclgf pm2.43i ) DEJZBCKZEJZA
|
|
LTBLCDEFTBCCDEFMGNUADOUBTABUADEPHQIRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
vtoclga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclga.2 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 20-Aug-1995.) $)
|
|
vtoclga $p |- ( A e. B -> ps ) $=
|
|
( nfcv nfv vtoclgaf ) ABCDECDHBCIFGJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d x y D $.
|
|
vtocl2gaf.a $e |- F/_ x A $.
|
|
vtocl2gaf.b $e |- F/_ y A $.
|
|
vtocl2gaf.c $e |- F/_ y B $.
|
|
vtocl2gaf.1 $e |- F/ x ps $.
|
|
vtocl2gaf.2 $e |- F/ y ch $.
|
|
vtocl2gaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2gaf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2gaf.5 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 10-Aug-2013.) $)
|
|
vtocl2gaf $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
|
|
( wcel wa wi cv nfel1 nfan nfim wceq eleq1 anbi1d imbi12d anbi2d vtocl2gf
|
|
nfv pm2.43i ) FHRZGIRZSZCDUAZHRZEUAZIRZSZATUMUSSZBTUOCTDEFGHIJKLVABDUMUSD
|
|
DFHJUBUSDUKUCMUDUOCEUMUNEEFHKUBEGILUBUCNUDUPFUEZUTVAABVBUQUMUSUPFHUFUGOUH
|
|
URGUEZVAUOBCVCUSUNUMURGIUFUIPUHQUJUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $. $d x ps $. $d y ch $.
|
|
vtocl2ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2ga.3 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 20-Aug-1995.) $)
|
|
vtocl2ga $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
|
|
( nfcv nfv vtocl2gaf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z S $. $d x y z T $.
|
|
vtocl3gaf.a $e |- F/_ x A $.
|
|
vtocl3gaf.b $e |- F/_ y A $.
|
|
vtocl3gaf.c $e |- F/_ z A $.
|
|
vtocl3gaf.d $e |- F/_ y B $.
|
|
vtocl3gaf.e $e |- F/_ z B $.
|
|
vtocl3gaf.f $e |- F/_ z C $.
|
|
vtocl3gaf.1 $e |- F/ x ps $.
|
|
vtocl3gaf.2 $e |- F/ y ch $.
|
|
vtocl3gaf.3 $e |- F/ z th $.
|
|
vtocl3gaf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3gaf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3gaf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3gaf.7 $e |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph ) $.
|
|
$( Implicit substitution of 3 classes for 3 setvar variables. (Contributed
|
|
by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) $)
|
|
vtocl3gaf $p |- ( ( A e. R /\ B e. S /\ C e. T ) -> th ) $=
|
|
( wcel w3a cv nfel1 nf3an nfim wceq eleq1 3anbi1d imbi12d 3anbi2d 3anbi3d
|
|
wi nfv vtocl3gf pm2.43i ) HKUGZILUGZJMUGZUHZDEUIZKUGZFUIZLUGZGUIZMUGZUHZA
|
|
USVCVJVLUHZBUSVCVDVLUHZCUSVFDUSEFGHIJKLMNOPQRSVNBEVCVJVLEEHKNUJVJEUTVLEUT
|
|
UKTULVOCFVCVDVLFFHKOUJFILQUJVLFUTUKUAULVFDGVCVDVEGGHKPUJGILRUJGJMSUJUKUBU
|
|
LVGHUMZVMVNABVPVHVCVJVLVGHKUNUOUCUPVIIUMZVNVOBCVQVJVDVCVLVIILUNUQUDUPVKJU
|
|
MZVOVFCDVRVLVEVCVDVKJMUNURUEUPUFVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $. $d x y z D $. $d x y z R $.
|
|
$d x y z S $. $d x ps $. $d y ch $. $d z th $.
|
|
vtocl3ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3ga.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3ga.4 $e |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph ) $.
|
|
$( Implicit substitution of 3 classes for 3 setvar variables. (Contributed
|
|
by NM, 20-Aug-1995.) $)
|
|
vtocl3ga $p |- ( ( A e. D /\ B e. R /\ C e. S ) -> th ) $=
|
|
( nfcv nfv vtocl3gaf ) ABCDEFGHIJKLMEHRFHRGHRFIRGIRGJRBESCFSDGSNOPQT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
vtocleg.1 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 10-Jan-2004.) $)
|
|
vtocleg $p |- ( A e. V -> ph ) $=
|
|
( wcel cv wceq wex elisset exlimiv syl ) CDFBGCHZBIABCDJMABEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Implicit substitution of a class for a setvar variable. (Closed theorem
|
|
version of ~ vtoclef .) (Contributed by NM, 7-Nov-2005.) (Revised by
|
|
Mario Carneiro, 11-Oct-2016.) $)
|
|
vtoclegft $p |- ( ( A e. B /\ F/ x ph /\
|
|
A. x ( x = A -> ph ) ) -> ph ) $=
|
|
( wcel wnf cv wceq wi wal w3a wex elisset mpan9 3adant2 wb 19.9t 3ad2ant2
|
|
exim mpbid ) CDEZABFZBGCHZAIBJZKABLZAUAUDUEUBUAUCBLUDUEBCDMUCABSNOUBUAUEA
|
|
PUDABQRT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
vtoclef.1 $e |- F/ x ph $.
|
|
vtoclef.2 $e |- A e. _V $.
|
|
vtoclef.3 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
vtoclef $p |- ph $=
|
|
( cv wceq wex isseti exlimi ax-mp ) BGCHZBIABCEJMABDFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
vtocle.1 $e |- A e. _V $.
|
|
vtocle.2 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 9-Sep-1993.) $)
|
|
vtocle $p |- ph $=
|
|
( cvv wcel vtocleg ax-mp ) CFGADABCFEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
vtoclri.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclri.2 $e |- A. x e. B ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 21-Nov-1994.) $)
|
|
vtoclri $p |- ( A e. B -> ps ) $=
|
|
( rspec vtoclga ) ABCDEFACEGHI $.
|
|
$}
|
|
|
|
${
|
|
spcimgft.1 $e |- F/ x ps $.
|
|
spcimgft.2 $e |- F/_ x A $.
|
|
$( A closed version of ~ spcimgf . (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
spcimgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B ->
|
|
( A. x ph -> ps ) ) ) $=
|
|
( wcel cvv cv wceq wi wal elex wex issetf exim syl5bi 19.36-1 syl6 syl5 )
|
|
DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFSTUA $.
|
|
|
|
$( A closed version of ~ spcgf . (Contributed by Andrew Salmon,
|
|
6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcgft $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
|
|
( A. x ph -> ps ) ) ) $=
|
|
( cv wceq wb wi wal wcel bi1 imim2i alimi spcimgft syl ) CHDIZABJZKZCLSAB
|
|
KZKZCLDEMACLBKKUAUCCTUBSABNOPABCDEFGQR $.
|
|
|
|
$( A closed version of ~ spcimegf . (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
spcimegft $p |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B ->
|
|
( ps -> E. x ph ) ) ) $=
|
|
( wcel cvv cv wceq wi wal wex elex issetf exim syl5bi 19.37-1 syl6 syl5 )
|
|
DEHDIHZCJDKZBALZLCMZBACNLZDEOUEUBUDCNZUFUBUCCNUEUGCDGPUCUDCQRBACFSTUA $.
|
|
|
|
$( A closed version of ~ spcegf . (Contributed by Jim Kingdon,
|
|
22-Jun-2018.) $)
|
|
spcegft $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
|
|
( ps -> E. x ph ) ) ) $=
|
|
( cv wceq wb wi wal wcel wex bi2 imim2i alimi spcimegft syl ) CHDIZABJZKZ
|
|
CLTBAKZKZCLDEMBACNKKUBUDCUAUCTABOPQABCDEFGRS $.
|
|
$}
|
|
|
|
${
|
|
spcimgf.1 $e |- F/_ x A $.
|
|
spcimgf.2 $e |- F/ x ps $.
|
|
${
|
|
spcimgf.3 $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( cv wceq wi wcel wal spcimgft mpg ) CIDJABKKDELACMBKKCABCDEGFNHO $.
|
|
$}
|
|
|
|
spcimegf.3 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( cv wceq wi wcel wex spcimegft mpg ) CIDJBAKKDELBACMKKCABCDEGFNHO $.
|
|
$}
|
|
|
|
${
|
|
spcgf.1 $e |- F/_ x A $.
|
|
spcgf.2 $e |- F/ x ps $.
|
|
spcgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by
|
|
Andrew Salmon, 12-Aug-2011.) $)
|
|
spcgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( cv wceq wb wi wcel wal spcgft mpg ) CIDJABKLDEMACNBLLCABCDEGFOHP $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 2-Feb-1997.) $)
|
|
spcegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( cv wceq wb wi wcel wex spcegft mpg ) CIDJABKLDEMBACNLLCABCDEGFOHP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
spcimdv.1 $e |- ( ph -> A e. B ) $.
|
|
${
|
|
spcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( cv wceq wi wal wcel ex alrimiv nfv nfcv spcimgft sylc ) ADIEJZBCKZKZD
|
|
LEFMBDLCKAUBDATUAHNOGBCDEFCDPDEQRS $.
|
|
$}
|
|
|
|
${
|
|
spcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Analogous to
|
|
~ rspcdv . (Contributed by David Moews, 1-May-2017.) $)
|
|
spcdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( cv wceq wa biimpd spcimdv ) ABCDEFGADIEJKBCHLM $.
|
|
$}
|
|
|
|
spcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimedv $p |- ( ph -> ( ch -> E. x ps ) ) $=
|
|
( cv wceq wi wal wcel wex ex alrimiv nfv nfcv spcimegft sylc ) ADIEJZCBKZ
|
|
KZDLEFMCBDNKAUCDAUAUBHOPGBCDEFCDQDERST $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
spcgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) $)
|
|
spcgv $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( nfcv nfv spcgf ) ABCDECDGBCHFI $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 14-Aug-1994.) $)
|
|
spcegv $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( nfcv nfv spcegf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
spc2egv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization with 2 quantifiers, using implicit
|
|
substitution. (Contributed by NM, 3-Aug-1995.) $)
|
|
spc2egv $p |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) ) $=
|
|
( wcel wa cv wceq wex elisset anim12i eeanv sylibr biimprcd 2eximdv
|
|
syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNCNZBADNCNUDUECNZUFDNZKUHUBUIUCUJCEGODFH
|
|
OPUEUFCDQRBUGACDUGABISTUA $.
|
|
|
|
$( Specialization with 2 quantifiers, using implicit substitution.
|
|
(Contributed by NM, 27-Apr-2004.) $)
|
|
spc2gv $p |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) ) $=
|
|
( wcel wa cv wceq wex wal elisset anim12i wi exim 19.9v eeanv sylibr 3syl
|
|
biimpcd 2alimi alimi bitri syl6ib syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNZCNZA
|
|
DOCOZBULUMCNZUNDNZKUQUJUSUKUTCEGPDFHPQUMUNCDUAUBURUQBDNZCNZBURUOBRZDOZCOU
|
|
PVARZCOUQVBRAVCCDUOABIUDUEVDVECUOBDSUFUPVACSUCVBVABVACTBDTUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
|
|
spc3egv.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization with 3 quantifiers, using implicit
|
|
substitution. (Contributed by NM, 12-May-2008.) $)
|
|
spc3egv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( ps -> E. x E. y E. z ph ) ) $=
|
|
( wcel w3a cv wceq wex elisset 3anim123i eeeanv biimprcd 2eximdv syl5com
|
|
sylibr eximdv ) FIMZGJMZHKMZNZCOFPZDOGPZEOHPZNZEQZDQCQZBAEQZDQCQUIUJCQZUK
|
|
DQZULEQZNUOUFUQUGURUHUSCFIRDGJREHKRSUJUKULCDETUDBUNUPCDBUMAEUMABLUAUEUBUC
|
|
$.
|
|
|
|
$( Specialization with 3 quantifiers, using implicit substitution.
|
|
(Contributed by NM, 12-May-2008.) $)
|
|
spc3gv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( A. x A. y A. z ph -> ps ) ) $=
|
|
( wcel w3a cv wceq wex wal elisset wi 3anim123i eeeanv sylibr 2alimi exim
|
|
biimpcd alimi syl 3syl 19.9v 3bitri syl6ib syl5com ) FIMZGJMZHKMZNZCOFPZD
|
|
OGPZEOHPZNZEQZDQZCQZAERDRZCRZBUQURCQZUSDQZUTEQZNVDUNVGUOVHUPVICFISDGJSEHK
|
|
SUAURUSUTCDEUBUCVFVDBEQZDQZCQZBVFVBVJTZDRZCRZVCVKTZCRVDVLTVFVABTZERZDRZCR
|
|
VOVEVSCAVQDEVAABLUFUDUGVRVMCDVABEUEUDUHVNVPCVBVJDUEUGVCVKCUEUIVLVKVJBVKCU
|
|
JVJDUJBEUJUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
spcv.1 $e |- A e. _V $.
|
|
spcv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. (Contributed by
|
|
NM, 22-Jun-1994.) $)
|
|
spcv $p |- ( A. x ph -> ps ) $=
|
|
( cvv wcel wal wi spcgv ax-mp ) DGHACIBJEABCDGFKL $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
|
|
spcev $p |- ( ps -> E. x ph ) $=
|
|
( cvv wcel wex wi spcegv ax-mp ) DGHBACIJEABCDGFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
spc2ev.1 $e |- A e. _V $.
|
|
spc2ev.2 $e |- B e. _V $.
|
|
spc2ev.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 3-Aug-1995.) $)
|
|
spc2ev $p |- ( ps -> E. x E. y ph ) $=
|
|
( cvv wcel wex wi spc2egv mp2an ) EJKFJKBADLCLMGHABCDEFJJINO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
rspct.1 $e |- F/ x ps $.
|
|
$( A closed version of ~ rspc . (Contributed by Andrew Salmon,
|
|
6-Jun-2011.) $)
|
|
rspct $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
|
|
( A. x e. B ph -> ps ) ) ) $=
|
|
( cv wceq wb wi wal wcel wral df-ral wa eleq1 adantr simpr imbi12d ex a2i
|
|
alimi nfv nfim nfcv spcgft syl syl7bi com34 pm2.43d ) CGZDHZABIZJZCKZDELZ
|
|
ACEMZBJUOUPUQUPBUQUKELZAJZCKZUOUPUPBJZACENUOULUSVAIZJZCKUPUTVAJJUNVCCULUM
|
|
VBULUMVBULUMOURUPABULURUPIUMUKDEPQULUMRSTUAUBUSVACDEUPBCUPCUCFUDCDUEUFUGU
|
|
HUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
rspc.1 $e |- F/ x ps $.
|
|
rspc.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) $)
|
|
rspc $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
|
|
( wral cv wcel wi wal df-ral nfcv nfv nfim wceq eleq1 imbi12d spcgf
|
|
pm2.43a syl5bi ) ACEHCIZEJZAKZCLZDEJZBACEMUFUGBUEUGBKCDECDNUGBCUGCOFPUCDQ
|
|
UDUGABUCDERGSTUAUB $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro,
|
|
11-Oct-2016.) $)
|
|
rspce $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
|
|
( wcel wa cv wex wrex nfcv nfv nfan wceq eleq1 anbi12d spcegf anabsi5
|
|
df-rex sylibr ) DEHZBIZCJZEHZAIZCKZACELUCBUHUGUDCDECDMUCBCUCCNFOUEDPUFUCA
|
|
BUEDEQGRSTACEUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
rspcv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 26-May-1998.) $)
|
|
rspcv $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
|
|
( nfv rspc ) ABCDEBCGFH $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 2-Feb-2006.) $)
|
|
rspccv $p |- ( A. x e. B ph -> ( A e. B -> ps ) ) $=
|
|
( wcel wral rspcv com12 ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 13-Sep-2005.) $)
|
|
rspcva $p |- ( ( A e. B /\ A. x e. B ph ) -> ps ) $=
|
|
( wcel wral rspcv imp ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
rspccva $p |- ( ( A. x e. B ph /\ A e. B ) -> ps ) $=
|
|
( wcel wral rspcv impcom ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by NM, 26-May-1998.) $)
|
|
rspcev $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
|
|
( nfv rspce ) ABCDEBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $. $d x ch $.
|
|
rspcimdv.1 $e |- ( ph -> A e. B ) $.
|
|
${
|
|
rspcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcimdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
|
|
( wral cv wcel wi wal df-ral wceq wa simpr eleq1d biimprd imim12d mpid
|
|
spcimdv syl5bi ) BDFIDJZFKZBLZDMZACBDFNAUGEFKZCGAUFUHCLDEFGAUDEOZPZUHUE
|
|
BCUJUEUHUJUDEFAUIQRSHTUBUAUC $.
|
|
$}
|
|
|
|
rspcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcimedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
|
|
( cv wcel wa wex wrex wceq simpr eleq1d biimprd anim12d spcimedv mpand
|
|
df-rex syl6ibr ) ACDIZFJZBKZDLZBDFMAEFJZCUFGAUEUGCKDEFGAUCENZKZUGUDCBUIUD
|
|
UGUIUCEFAUHOPQHRSTBDFUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $. $d x ch $.
|
|
rspcdv.1 $e |- ( ph -> A e. B ) $.
|
|
rspcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
|
|
( cv wceq wa biimpd rspcimdv ) ABCDEFGADIEJKBCHLM $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
rspcedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
|
|
( cv wceq wa biimprd rspcimedv ) ABCDEFGADIEJKBCHLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d C x $. $d ch x $.
|
|
rspcdva.1 $e |- ( x = C -> ( ps <-> ch ) ) $.
|
|
rspcdva.2 $e |- ( ph -> A. x e. A ps ) $.
|
|
rspcdva.3 $e |- ( ph -> C e. A ) $.
|
|
${
|
|
rspcda.1 $e |- F/ x ph $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed
|
|
by Thierry Arnoux, 29-Jun-2020.) $)
|
|
rspcda $p |- ( ph -> ch ) $=
|
|
( wcel wral rspcv sylc ) AFEKBDELCIHBCDFEGMN $.
|
|
$}
|
|
|
|
$d ph x $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
Thierry Arnoux, 21-Jun-2020.) $)
|
|
rspcdva $p |- ( ph -> ch ) $=
|
|
( wral cv wceq wb adantl rspcdv mpd ) ABDEJCHABCDFEIDKFLBCMAGNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $. $d x ch $.
|
|
rspcedvd.1 $e |- ( ph -> A e. B ) $.
|
|
rspcedvd.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
rspcedvd.3 $e |- ( ph -> ch ) $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
Variant of ~ rspcedv . (Contributed by AV, 27-Nov-2019.) $)
|
|
rspcedvd $p |- ( ph -> E. x e. B ps ) $=
|
|
( wrex rspcedv mpd ) ACBDFJIABCDEFGHKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
rspcedeqvd.1 $e |- ( ph -> A e. B ) $.
|
|
rspcedeqvd.2 $e |- ( ( ph /\ x = A ) -> C = D ) $.
|
|
${
|
|
$d x D $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
Variant of ~ rspcedvd for equations, in which the left hand side
|
|
depends on the quantified variable. (Contributed by AV,
|
|
24-Dec-2019.) $)
|
|
rspcedeq1vd $p |- ( ph -> E. x e. B C = D ) $=
|
|
( wceq cv wa eqeq1d eqidd rspcedvd ) AEFIFFIBCDGABJCIKEFFHLAFMN $.
|
|
$}
|
|
|
|
$d x C $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
Variant of ~ rspcedvd for equations, in which the right hand side
|
|
depends on the quantified variable. (Contributed by AV,
|
|
24-Dec-2019.) $)
|
|
rspcedeq2vd $p |- ( ph -> E. x e. B C = D ) $=
|
|
( wceq cv wa eqcomd eqeq2d eqidd rspcedvd ) AEFIEEIBCDGABJCIKZFEEPEFHLMAE
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C $. $d x y D $.
|
|
rspc2.1 $e |- F/ x ch $.
|
|
rspc2.2 $e |- F/ y ps $.
|
|
rspc2.3 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc2.4 $e |- ( y = B -> ( ch <-> ps ) ) $.
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 9-Nov-2012.) $)
|
|
rspc2 $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
|
|
ps ) ) $=
|
|
( wcel wral nfcv nfralxy cv wceq rspc ralbidv sylan9 ) FHNAEIOZDHOCEIOZGI
|
|
NBUCUDDFHCDEIDIPJQDRFSACEILUATCBEGIKMTUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y V $. $d x y W $. $d x y ps $.
|
|
rspc2gv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Restricted specialization with two quantifiers, using implicit
|
|
substitution. (Contributed by BJ, 2-Dec-2021.) $)
|
|
rspc2gv $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A. x e. V A. y e. W ph -> ps ) ) $=
|
|
( wral cv wcel wi wal wa df-ral albii wceq eleq1 syl5bi bi2anan9 imbi12d
|
|
imbi2i 19.21v bicomi impexp syl5bbr spc2gv pm2.43a ) ADHJZCGJCKZGLZUJMZCN
|
|
ZEGLZFHLZOZBUJCGPUNULDKZHLZAMZDNZMZCNZUQBUMVBCUJVAULADHPUCQVCULUTMZDNZCNZ
|
|
UQBVBVECVEVBULUTDUDUEQVFUQBVDUQBMZCDEFGHVDULUSOZAMUKERZURFRZOZVGULUSAUFVK
|
|
VHUQABVIULUOVJUSUPUKEGSURFHSUAIUBUGUHUITTT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C $. $d x y D $. $d x ch $. $d y ps $.
|
|
rspc2v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc2v.2 $e |- ( y = B -> ( ch <-> ps ) ) $.
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1999.) $)
|
|
rspc2v $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
|
|
ps ) ) $=
|
|
( nfv rspc2 ) ABCDEFGHICDLBELJKM $.
|
|
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 18-Jun-2014.) $)
|
|
rspc2va $p |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) ->
|
|
ps ) $=
|
|
( wcel wa wral rspc2v imp ) FHLGILMAEINDHNBABCDEFGHIJKOP $.
|
|
|
|
$( 2-variable restricted existential specialization, using implicit
|
|
substitution. (Contributed by NM, 16-Oct-1999.) $)
|
|
rspc2ev $p |- ( ( A e. C /\ B e. D /\ ps ) -> E. x e. C E. y e. D ph ) $=
|
|
( wcel w3a wrex wa rspcev anim2i 3impb cv wceq rexbidv syl ) FHLZGILZBMUC
|
|
CEINZOZAEINZDHNUCUDBUFUDBOUEUCCBEGIKPQRUGUEDFHDSFTACEIJUAPUB $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d x ch $. $d y th $. $d x y z A $. $d y z B $. $d z C $.
|
|
$d x R $. $d x y S $. $d x y z T $.
|
|
rspc3v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc3v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
|
|
rspc3v.3 $e |- ( z = C -> ( th <-> ps ) ) $.
|
|
$( 3-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 10-May-2005.) $)
|
|
rspc3v $p |- ( ( A e. R /\ B e. S /\ C e. T ) ->
|
|
( A. x e. R A. y e. S A. z e. T ph -> ps ) ) $=
|
|
( wcel wral cv wceq wi wa ralbidv rspc2v rspcv sylan9 3impa ) HKQZILQZJMQ
|
|
ZAGMRZFLREKRZBUAUHUIUBULDGMRZUJBUKUMCGMREFHIKLESHTACGMNUCFSITCDGMOUCUDDBG
|
|
JMPUEUFUG $.
|
|
|
|
$( 3-variable restricted existentional specialization, using implicit
|
|
substitution. (Contributed by NM, 25-Jul-2012.) $)
|
|
rspc3ev $p |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) ->
|
|
E. x e. R E. y e. S E. z e. T ph ) $=
|
|
( wcel wrex cv wceq w3a wa simpl1 simpl2 rspcev 3ad2antl3 rexbidv rspc2ev
|
|
syl3anc ) HKQZILQZJMQZUABUBUJUKDGMRZAGMRZFLREKRUJUKULBUCUJUKULBUDULUJBUMU
|
|
KDBGJMPUEUFUNUMCGMREFHIKLESHTACGMNUGFSITCDGMOUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
eqvinc.1 $e |- A e. _V $.
|
|
$( A variable introduction law for class equality. (Contributed by NM,
|
|
14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
eqvinc $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
|
|
( wceq cv wa wex wi isseti ax-1 eqtr jca eximi pm3.43 mp2b 19.37aiv eqtr2
|
|
ex exlimiv impbii ) BCEZAFZBEZUCCEZGZAHUBUFAUDAHUBUDIZUBUEIZGZAHUBUFIZAHA
|
|
BDJUDUIAUDUGUHUDUBKUDUBUEUCBCLSMNUIUJAUBUDUEONPQUFUBAUCBCRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A variable introduction law for class equality, deduction version.
|
|
(Contributed by Thierry Arnoux, 2-Mar-2017.) $)
|
|
eqvincg $p |- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) ) $=
|
|
( wcel wceq cv wa wex wi elisset ax-1 eqtr ex jca pm3.43 3syl nfv 19.37-1
|
|
eximi syl eqtr2 exlimiv impbid1 ) BDEZBCFZAGZBFZUGCFZHZAIZUEUFUJJZAIZUFUK
|
|
JUEUHAIUFUHJZUFUIJZHZAIUMABDKUHUPAUHUNUOUHUFLUHUFUIUGBCMNOTUPULAUFUHUIPTQ
|
|
UFUJAUFARSUAUJUFAUGBCUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
eqvincf.1 $e |- F/_ x A $.
|
|
eqvincf.2 $e |- F/_ x B $.
|
|
eqvincf.3 $e |- A e. _V $.
|
|
$( A variable introduction law for class equality, using bound-variable
|
|
hypotheses instead of distinct variable conditions. (Contributed by NM,
|
|
14-Sep-2003.) $)
|
|
eqvincf $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
|
|
( vy wceq cv wa wex eqvinc nfeq2 nfan nfv eqeq1 anbi12d cbvex bitri ) BCH
|
|
GIZBHZTCHZJZGKAIZBHZUDCHZJZAKGBCFLUCUGGAUAUBAATBDMATCEMNUGGOTUDHUAUEUBUFT
|
|
UDBPTUDCPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
alexeq.1 $e |- A e. _V $.
|
|
$( Two ways to express substitution of ` A ` for ` x ` in ` ph ` .
|
|
(Contributed by NM, 2-Mar-1995.) $)
|
|
alexeq $p |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy cv wceq wa wex wi wal anbi1d exbidv imbi1d albidv sb56 vtoclb bicomi
|
|
eqeq2 ) BFZCGZAHZBIZUAAJZBKZTEFZGZAHZBIUGAJZBKUCUEECDUFCGZUHUBBUJUGUAAUFC
|
|
TSZLMUJUIUDBUJUGUAAUKNOABEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Equality implies equivalence with substitution. (Contributed by NM,
|
|
2-Mar-1995.) $)
|
|
ceqex $p |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) $=
|
|
( vy cvv wcel cv wceq wa wex wb 19.8a isset sylibr wi eqeq2 anbi1d exbidv
|
|
bibi2d imbi12d ex wal vex alexeq sp com12 syl5bir impbid vtoclg mpcom ) C
|
|
EFZBGZCHZAUMAIZBJZKZUMUMBJUKUMBLBCMNULDGZHZAURAIZBJZKZOUMUPODCEUQCHZURUMV
|
|
AUPUQCULPZVBUTUOAVBUSUNBVBURUMAVCQRSTURAUTURAUTUSBLUAUTURAOZBUBZURAABUQDU
|
|
CUDVEURAVDBUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsexg.1 $e |- F/ x ps $.
|
|
ceqsexg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
11-Oct-2004.) $)
|
|
ceqsexg $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
|
|
( wb cv wceq wa wex nfcv nfe1 nfbi ceqex bibi12d biid vtoclgf ) AAHCIDJZA
|
|
KZCLZBHCDECDMUBBCUACNFOTAUBABACDPGQARS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsexgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 29-Dec-1996.) $)
|
|
ceqsexgv $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
|
|
( nfv ceqsexg ) ABCDEBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
ceqsrexv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by NM, 30-Apr-2004.) $)
|
|
ceqsrexv $p |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) ) $=
|
|
( cv wceq wa wrex wcel wex df-rex an12 exbii bitr4i eleq1 anbi12d bianabs
|
|
ceqsexgv syl5bb ) CGZDHZAIZCEJZUCUBEKZAIZIZCLZDEKZBUEUFUDIZCLUIUDCEMUHUKC
|
|
UCUFANOPUJUIBUGUJBICDEUCUFUJABUBDEQFRTSUA $.
|
|
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) $)
|
|
ceqsrexbv $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $=
|
|
( wcel cv wceq wa wrex r19.42v eleq1 adantr pm5.32ri bicomi baib ceqsrexv
|
|
wb rexbiia pm5.32i 3bitr3i ) DEGZCHZDIZAJZJZCEKUCUFCEKZJUHUCBJUCUFCELUGUF
|
|
CEUGUDEGZUFUIUFJUGUFUIUCUEUIUCSAUDDEMNOPQTUCUHBABCDEFRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x C $. $d x y D $. $d x ps $. $d y ch $.
|
|
ceqsrex2v.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsrex2v.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by NM, 29-Oct-2005.) $)
|
|
ceqsrex2v $p |- ( ( A e. C /\ B e. D ) ->
|
|
( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) ) $=
|
|
( wcel cv wceq wa wrex anass rexbii r19.42v ceqsrexv bitri anbi2d rexbidv
|
|
syl5bb sylan9bb ) FHLZDMFNZEMGNZOAOZEIPZDHPZUHBOZEIPZGILCUKUGUHAOZEIPZOZD
|
|
HPUFUMUJUPDHUJUGUNOZEIPUPUIUQEIUGUHAQRUGUNEISUARUOUMDFHUGUNULEIUGABUHJUBU
|
|
CTUDBCEGIKTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel2.1 $e |- A e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel2 $p |- ( A e. B <-> A. x ( x = A -> x e. B ) ) $=
|
|
( cv wceq wcel wi wal eleq1 ceqsalv bicomi ) AEZBFMCGZHAIBCGZNOABDMBCJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 13-Aug-2005.) $)
|
|
clel3g $p |- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) ) $=
|
|
( wcel cv wceq wa wex eleq2 ceqsexgv bicomd ) CDEAFZCGBMEZHAIBCEZNOACDMCB
|
|
JKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel3.1 $e |- B e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel3 $p |- ( A e. B <-> E. x ( x = B /\ A e. x ) ) $=
|
|
( cvv wcel cv wceq wa wex wb clel3g ax-mp ) CEFBCFAGZCHBNFIAJKDABCELM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel4.1 $e |- B e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel4 $p |- ( A e. B <-> A. x ( x = B -> A e. x ) ) $=
|
|
( cv wceq wcel wi wal eleq2 ceqsalv bicomi ) AEZCFBMGZHAIBCGZNOACDMCBJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d y A z $. $d y B z $.
|
|
$( Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only ` A ` is
|
|
required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) $)
|
|
pm13.183 $p |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) ) $=
|
|
( vy cv wceq wal eqeq1 eqeq2 bibi1d albidv alrimiv wsb stdpc4 sbbi bibi2i
|
|
wb eqsb3 sylbi equsb1 bi1 mpi syl impbii vtoclbg ) EFZCGZAFZUGGZUICGZRZAH
|
|
ZBCGUIBGZUKRZAHEBDUGBCIUGBGZULUOAUPUJUNUKUGBUIJKLUHUMUHULAUGCUIJMUMULAENZ
|
|
UHULAEOUQUJAENZUKAENZRZUHUJUKAEPUTURUHRZUHUSUHUREACSQVAURUHAEUAURUHUBUCTT
|
|
UDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $. $d y ph $.
|
|
$( Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89.
|
|
(Contributed by NM, 25-Oct-2012.) $)
|
|
rr19.3v $p |- ( A. x e. A A. y e. A ph <-> A. x e. A ph ) $=
|
|
( wral cv weq biidd rspcv ralimia wcel ax-1 ralrimiv ralimi impbii ) ACDE
|
|
ZBDEABDEPABDAACBFDCBGAHIJAPBDAACDACFDKLMNO $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.
|
|
(Contributed by NM, 29-Oct-2012.) $)
|
|
rr19.28v $p |- ( A. x e. A A. y e. A ( ph /\ ps )
|
|
<-> A. x e. A ( ph /\ A. y e. A ps ) ) $=
|
|
( wa wral cv wcel simpl ralimi weq biidd rspcv syl5 wi simpr jcad ralimia
|
|
a1i r19.28av impbii ) ABFZDEGZCEGABDEGZFZCEGUDUFCECHZEIZUDAUEUDADEGUHAUCA
|
|
DEABJKAADUGEDCLAMNOUDUEPUHUCBDEABQKTRSUFUDCEABDEUAKUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
$( Membership in a class abstraction, using implicit substitution. (Closed
|
|
theorem version of ~ elabg .) (Contributed by NM, 7-Nov-2005.) (Proof
|
|
shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
elabgt $p |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) ->
|
|
( A e. { x | ph } <-> ps ) ) $=
|
|
( cv wceq wb wal wcel cab abid eleq1 syl5bbr bibi1d biimpd a2i alimi nfcv
|
|
wi nfab1 nfel2 nfv nfbi pm5.5 spcgf imp sylan2 ) CFZDGZABHZTZCIDEJZUJDACK
|
|
ZJZBHZTZCIZUPULUQCUJUKUPUJUKUPUJAUOBAUIUNJUJUOACLUIDUNMNOPQRUMURUPUQUPCDE
|
|
CDSUOBCCDUNACUAUBBCUCUDUJUPUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
elabgf.1 $e |- F/_ x A $.
|
|
elabgf.2 $e |- F/ x ps $.
|
|
elabgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. This version has bound-variable
|
|
hypotheses in place of distinct variable restrictions. (Contributed by
|
|
NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
elabgf $p |- ( A e. B -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( cv cab wcel wb nfab1 nfel nfbi wceq eleq1 bibi12d abid vtoclgf ) CIZACJ
|
|
ZKZALDUBKZBLCDEFUDBCCDUBFACMNGOUADPUCUDABUADUBQHRACST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
elabf.1 $e |- F/ x ps $.
|
|
elabf.2 $e |- A e. _V $.
|
|
elabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) $)
|
|
elabf $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( cvv wcel cab wb nfcv elabgf ax-mp ) DHIDACJIBKFABCDHCDLEGMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab.1 $e |- A e. _V $.
|
|
elab.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) $)
|
|
elab $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( nfv elabf ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) $)
|
|
elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( nfcv nfv elabgf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab2g.2 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1995.) $)
|
|
elab2g $p |- ( A e. V -> ( A e. B <-> ps ) ) $=
|
|
( wcel cab eleq2i elabg syl5bb ) DEIDACJZIDFIBENDHKABCDFGLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab2.1 $e |- A e. _V $.
|
|
elab2.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab2.3 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1995.) $)
|
|
elab2 $p |- ( A e. B <-> ps ) $=
|
|
( cvv wcel wb elab2g ax-mp ) DIJDEJBKFABCDEIGHLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab4g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab4g.2 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 17-Oct-2012.) $)
|
|
elab4g $p |- ( A e. B <-> ( A e. _V /\ ps ) ) $=
|
|
( wcel cvv elex elab2g biadan2 ) DEHDIHBDEJABCDEIFGKL $.
|
|
$}
|
|
|
|
${
|
|
elab3gf.1 $e |- F/_ x A $.
|
|
elab3gf.2 $e |- F/ x ps $.
|
|
elab3gf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, with a weaker antecedent than
|
|
~ elabgf . (Contributed by NM, 6-Sep-2011.) $)
|
|
elab3gf $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( wcel wi cab elabgf ibi wb imim2i bi2 syli impbid2 ) BDEIZJZDACKZIZBUBBA
|
|
BCDUAFGHLMBTUBBNZUBSUCBABCDEFGHLOUBBPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab3g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, with a weaker antecedent than
|
|
~ elabg . (Contributed by NM, 29-Aug-2006.) $)
|
|
elab3g $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( nfcv nfv elab3gf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab3.1 $e |- ( ps -> A e. _V ) $.
|
|
elab3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction using implicit substitution.
|
|
(Contributed by NM, 10-Nov-2000.) $)
|
|
elab3 $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( cvv wcel wi cab wb elab3g ax-mp ) BDGHIDACJHBKEABCDGFLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d V x $.
|
|
$( Implication for the membership in a restricted class abstraction.
|
|
(Contributed by Alexander van der Vekens, 31-Dec-2017.) $)
|
|
elrabi $p |- ( A e. { x e. V | ph } -> A e. V ) $=
|
|
( wcel cv wa cab crab wceq wex clelab eleq1 anbi1d simprbda exlimiv sylbi
|
|
df-rab eleq2s ) CDEZCBFZDEZAGZBHZABDICUDEUACJZUCGZBKTUCBCLUFTBUEUCTAUEUBT
|
|
AUACDMNOPQABDRS $.
|
|
$}
|
|
|
|
${
|
|
elrabf.1 $e |- F/_ x A $.
|
|
elrabf.2 $e |- F/_ x B $.
|
|
elrabf.3 $e |- F/ x ps $.
|
|
elrabf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. This version has bound-variable hypotheses in place of
|
|
distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) $)
|
|
elrabf $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
|
|
( crab wcel cvv wa elex adantr cv cab df-rab eleq2i nfel nfan wceq elabgf
|
|
eleq1 anbi12d syl5bb pm5.21nii ) DACEJZKZDLKZDEKZBMZDUHNUKUJBDENOUIDCPZEK
|
|
ZAMZCQZKUJULUHUPDACERSUOULCDLFUKBCCDEFGTHUAUMDUBUNUKABUMDEUDIUEUCUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. (Closed theorem version of ~ elrab3 .) (Contributed by
|
|
Thierry Arnoux, 31-Aug-2017.) $)
|
|
elrab3t $p |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) ->
|
|
( A e. { x e. B | ph } <-> ps ) ) $=
|
|
( cv wceq wb wi wal wcel wa cab crab simpr nfa1 nfv nfan simpl 19.21bi
|
|
eleq1 biimparc biantrurd bibi1d adantl mpbid alrimi elabgt syl2anc df-rab
|
|
pm5.74da eleq2i bibi1i sylibr ) CFZDGZABHZIZCJZDEKZLZDUOEKZALZCMZKZBHZDAC
|
|
ENZKZBHVAUTUPVCBHZIZCJVFUSUTOVAVJCUSUTCURCPUTCQRVAURVJVAURCUSUTSTUTURVJHU
|
|
SUTUPUQVIUTUPLZAVCBVKVBAUPVBUTUODEUAUBUCUDUKUEUFUGVCBCDEUHUIVHVEBVGVDDACE
|
|
UJULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $. $d x B $.
|
|
elrab.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. (Contributed by NM, 21-May-1999.) $)
|
|
elrab $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
|
|
( nfcv nfv elrabf ) ABCDECDGCEGBCHFI $.
|
|
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. (Contributed by NM, 5-Oct-2006.) $)
|
|
elrab3 $p |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) ) $=
|
|
( crab wcel elrab baib ) DACEGHDEHBABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $. $d x B $.
|
|
elrab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elrab2.2 $e |- C = { x e. B | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 2-Nov-2006.) $)
|
|
elrab2 $p |- ( A e. C <-> ( A e. B /\ ps ) ) $=
|
|
( wcel crab wa eleq2i elrab bitri ) DFIDACEJZIDEIBKFODHLABCDEGMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ps $.
|
|
ralab.1 $e |- ( y = x -> ( ph <-> ps ) ) $.
|
|
$( Universal quantification over a class abstraction. (Contributed by Jeff
|
|
Madsen, 10-Jun-2010.) $)
|
|
ralab $p |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) ) $=
|
|
( cab wral cv wcel wi wal df-ral vex elab imbi1i albii bitri ) CDAEGZHDIZ
|
|
SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.
|
|
|
|
$( Universal quantification over a restricted class abstraction.
|
|
(Contributed by Jeff Madsen, 10-Jun-2010.) $)
|
|
ralrab $p |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) ) $=
|
|
( wi crab cv wcel wa elrab imbi1i impexp bitri ralbii2 ) CBCHZDAEFIZFDJZS
|
|
KZCHTFKZBLZCHUBRHUAUCCABETFGMNUBBCOPQ $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro,
|
|
3-Sep-2015.) $)
|
|
rexab $p |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) ) $=
|
|
( cab wrex cv wcel wa wex df-rex vex elab anbi1i exbii bitri ) CDAEGZHDIZ
|
|
SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) $)
|
|
rexrab $p |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) ) $=
|
|
( wa crab cv wcel elrab anbi1i anass bitri rexbii2 ) CBCHZDAEFIZFDJZRKZCH
|
|
SFKZBHZCHUAQHTUBCABESFGLMUABCNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ch $. $d x ph $. $d y ps $.
|
|
ralab2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
$( Universal quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
ralab2 $p |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) ) $=
|
|
( cab wral cv wcel wi wal df-ral nfsab1 nfv nfim weq eleq1 abid syl6bb
|
|
imbi12d cbval bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
|
|
ODEQZUFABCUIUFEIZUDJAUEUJUDRAESTFUAUBUC $.
|
|
|
|
$( Universal quantification over a restricted class abstraction.
|
|
(Contributed by Mario Carneiro, 3-Sep-2015.) $)
|
|
ralrab2 $p |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) ) $=
|
|
( crab wral cv wcel wa cab wi wal df-rab raleqi ralab2 impexp albii
|
|
df-ral bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUFCNZEOZACNZEFIZBDUDUGAEFPQU
|
|
FBCDEGRUIUEUJNZEOUKUHULEUEACSTUJEFUAUBUC $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
rexab2 $p |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) ) $=
|
|
( cab wrex cv wcel wa wex df-rex nfsab1 nfv nfan weq eleq1 abid syl6bb
|
|
anbi12d cbvex bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
|
|
ODEQZUFABCUIUFEIZUDJAUEUJUDRAESTFUAUBUC $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
rexrab2 $p |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) ) $=
|
|
( crab wrex cv wcel wa cab wex df-rab rexeqi rexab2 anass exbii df-rex
|
|
bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUECLZENZACLZEFIZBDUCUFAEFOPUEBCDEGQ
|
|
UHUDUILZENUJUGUKEUDACRSUIEFTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d z A $.
|
|
$( Identity used to create closed-form versions of bound-variable
|
|
hypothesis builders for class expressions. (Contributed by NM,
|
|
10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) $)
|
|
abidnf $p |- ( F/_ x A -> { z | A. x z e. A } = A ) $=
|
|
( wnfc cv wcel wal sp nfcr nfrd impbid2 abbi1dv ) ACDZBECFZAGZBCMONNAHMNA
|
|
ABCIJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d z A $.
|
|
dedhb.1 $e |- ( A = { z | A. x z e. A } -> ( ph <-> ps ) ) $.
|
|
dedhb.2 $e |- ps $.
|
|
$( A deduction theorem for converting the inference ` |- F/_ x A ` =>
|
|
` |- ph ` into a closed theorem. Use ~ nfa1 and ~ nfab to eliminate the
|
|
hypothesis of the substitution instance ` ps ` of the inference. For
|
|
converting the inference form into a deduction form, ~ abidnf is useful.
|
|
(Contributed by NM, 8-Dec-2006.) $)
|
|
dedhb $p |- ( F/_ x A -> ph ) $=
|
|
( wnfc cv wcel wal cab wceq wb abidnf eqcomd syl mpbiri ) CEHZABGSEDIEJCK
|
|
DLZMABNSTECDEOPFQR $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x y ps $. $d x y A $.
|
|
eqeu.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A condition which implies existential uniqueness. (Contributed by Jeff
|
|
Hankins, 8-Sep-2009.) $)
|
|
eqeu $p |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph ) $=
|
|
( vy wcel cv wceq wi wal w3a wex weu spcegv imp 3adant3 eqeq2 imbi2d nfv
|
|
albidv 3adant2 eu3 sylanbrc ) DEHZBACIZDJZKZCLZMACNZAUGGIZJZKZCLZGNZACOUF
|
|
BUKUJUFBUKABCDEFPQRUFUJUPBUFUJUPUOUJGDEULDJZUNUICUQUMUHAULDUGSTUBPQUCACGA
|
|
GUAUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Equality has existential uniqueness. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
eueq $p |- ( A e. _V <-> E! x x = A ) $=
|
|
( vy cv wceq wex wa wi wal cvv wcel weu eqtr3 biantru isset eqeq1 3bitr4i
|
|
gen2 eu4 ) ADZBEZAFZUBUACDZBEZGTUCEHZCIAIZGBJKUAALUFUBUEACTUCBMRNABOUAUDA
|
|
CTUCBPSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
eueq1.1 $e |- A e. _V $.
|
|
$( Equality has existential uniqueness. (Contributed by NM,
|
|
5-Apr-1995.) $)
|
|
eueq1 $p |- E! x x = A $=
|
|
( cvv wcel cv wceq weu eueq mpbi ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
eueq2dc.1 $e |- A e. _V $.
|
|
eueq2dc.2 $e |- B e. _V $.
|
|
$( Equality has existential uniqueness (split into 2 cases). (Contributed
|
|
by NM, 5-Apr-1995.) $)
|
|
eueq2dc $p |- ( DECID ph ->
|
|
E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) $=
|
|
( wdc wn wo wceq weu eueq1 euanv biimpri mpan2 euorv bianfd eubidv mpbid
|
|
wa cv df-dc notnot syl2anc orcom orbi2d syl5bb mpdan id orbi1d jaoi sylbi
|
|
) AGAAHZIABUAZCJZTZUMUNDJZTZIZBKZAUBAUTUMAUMUPIZBKZUTAUMHUPBKZVBAUCZAUOBK
|
|
ZVCBCELVCAVETAUOBMNOUMUPBPUDAVAUSBVAUPUMIAUSUMUPUEAUMURUPAUMUQVDQUFUGRSUM
|
|
AURIZBKZUTUMURBKZVGUMUQBKZVHBDFLVHUMVITUMUQBMNOAURBPUHUMVFUSBUMAUPURUMAUO
|
|
UMUIQUJRSUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $. $d x A $. $d x B $. $d x C $.
|
|
eueq3dc.1 $e |- A e. _V $.
|
|
eueq3dc.2 $e |- B e. _V $.
|
|
eueq3dc.3 $e |- C e. _V $.
|
|
eueq3dc.4 $e |- -. ( ph /\ ps ) $.
|
|
$( Equality has existential uniqueness (split into 3 cases). (Contributed
|
|
by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro,
|
|
28-Sep-2015.) $)
|
|
eueq3dc $p |- ( DECID ph -> ( DECID ps ->
|
|
E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
|
|
\/ ( ps /\ x = C ) ) ) ) $=
|
|
( wdc wo wceq wa wn w3o weu eueq1 con2i jaoi cv dcor df-dc ibar wb pm2.45
|
|
imnani bianfd orbi12d mtbid biorf bitrd 3orrot df-3or bitri syl6bbr mpbii
|
|
syl eubidv adantr pm2.46 simpl orim12i con3i 3orcomb sylbi syl6 ) AKBKABL
|
|
ZKZACUAZDMZNZVHOZVJEMZNZBVJFMZNZPZCQZABUBVIVHVMLVSVHUCVHVSVMAVSBAVKCQVSCD
|
|
GRAVKVRCAVKVOVQLZVLLZVRAVKVLWAAVKUDAVTOVLWAUEAVMBLZVTWBAVMAOBABUFZABABJUG
|
|
ZSTSAVMVOBVQAVMVNVMAWCSUHABVPWDUHUIUJVTVLUKURULVRVOVQVLPWAVLVOVQUMVOVQVLU
|
|
NUOUPUSUQBVPCQVSCFIRBVPVRCBVPVLVOLZVQLZVRBVPVQWFBVPUDBWEOVQWFUEWEBVLBOZVO
|
|
AWGVKWDUTVMWGVNABVAUTTSWEVQUKURULVLVOVQUNUPUSUQTVMVNCQVSCEHRVMVNVRCVMVNVL
|
|
VQLZVOLZVRVMVNVOWIVMVNUDVMWHOVOWIUEWHVHVLAVQBAVKVBBVPVBVCVDWHVOUKURULVRVL
|
|
VQVOPWIVLVOVQVEVLVQVOUNUOUPUSUQTVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( There is at most one set equal to a class. (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
moeq $p |- E* x x = A $=
|
|
( cv wceq wmo wex weu wi cvv wcel isset eueq bitr3i biimpi df-mo mpbir )
|
|
ACBDZAEQAFZQAGZHRSRBIJSABKABLMNQAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $. $d x A $. $d x B $. $d x C $.
|
|
moeq3dc.1 $e |- A e. _V $.
|
|
moeq3dc.2 $e |- B e. _V $.
|
|
moeq3dc.3 $e |- C e. _V $.
|
|
moeq3dc.4 $e |- -. ( ph /\ ps ) $.
|
|
$( "At most one" property of equality (split into 3 cases). (Contributed
|
|
by Jim Kingdon, 7-Jul-2018.) $)
|
|
moeq3dc $p |- ( DECID ph -> ( DECID ps ->
|
|
E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
|
|
\/ ( ps /\ x = C ) ) ) ) $=
|
|
( wdc cv wceq wa wo wn w3o weu wmo eueq3dc eumo syl6 ) AKBKACLZDMNABOPUCE
|
|
MNBUCFMNQZCRUDCSABCDEFGHIJTUDCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( "At most one" remains true after substitution. (Contributed by Jim
|
|
Kingdon, 18-Jan-2019.) $)
|
|
mosubt $p |- ( A. y E* x ph -> E* x E. y ( y = A /\ ph ) ) $=
|
|
( wmo wal cv wceq wex wa wi weu cvv wcel eueq isset bitr3i euexex sylanbr
|
|
nfv expcom moanimv sylibr simpl eximi ancri moimi syl ) ABECFZCGDHZCIZUJA
|
|
JZCIZJZBEZUMBEZUIUKUPKUOUKUIUPUKUJCLZUIUPUQDMNUKCDOCDPQUJACBUJBTRSUAUKUMB
|
|
UBUCUMUNBUMUKULUJCUJAUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
mosub.1 $e |- E* x ph $.
|
|
$( "At most one" remains true after substitution. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
mosub $p |- E* x E. y ( y = A /\ ph ) $=
|
|
( wmo cv wceq wa wex mosubt mpg ) ABFCGDHAICJBFCABCDKEL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Theorem for inferring "at most one." (Contributed by NM,
|
|
17-Oct-1996.) $)
|
|
mo2icl $p |- ( A. x ( ph -> x = A ) -> E* x ph ) $=
|
|
( vy cv wceq wal wex wmo cvv wcel nfa1 vex eleq1 mpbii imim2i eximd 19.9v
|
|
wi sps syl6ib eqeq2 imbi2d albidv imbi1d nfv mo2r vtoclg com12 syld moabs
|
|
19.23bi sylibr ) ABEZCFZSZBGZABHZABIZSUSUQURCJKZUSUQURUTBHUTUQAUTBUPBLUPA
|
|
UTSBUOUTAUOUNJKUTBMUNCJNOPTQUTBRUAUTUQUSAUNDEZFZSZBGZUSSUQUSSDCJVACFZVDUQ
|
|
USVEVCUPBVEVBUOAVACUNUBUCUDUEVDUSDABDADUFUGULUHUIUJABUKUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $. $d x y ps $.
|
|
moi2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) $)
|
|
mob2 $p |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) ) $=
|
|
( vy wcel wmo w3a cv wceq simp3 syl5ibcom wi wa wsb wal nfs1v sbequ12 nfv
|
|
mo4f sylbi sbhypf anbi2d eqeq2 imbi12d spcgv syl5 imp expd 3impia impbid
|
|
sp ) DEHZACIZAJZCKZDLZBUQAUSBUOUPAMFNUOUPABUSOUOUPPABUSUOUPABPZUSOZUPAACG
|
|
QZPZURGKZLZOZGRZUOVAUPVGCRVGAVBCGACGSACGTUBVGCUNUCVFVAGDEVDDLZVCUTVEUSVHV
|
|
BBAABCGDBCUAFUDUEVDDURUFUGUHUIUJUKULUM $.
|
|
|
|
$( Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) $)
|
|
moi2 $p |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A ) $=
|
|
( wcel wmo wa cv wceq wb mob2 3expa biimprd impr ) DEGZACHZIZABCJDKZSAITB
|
|
QRATBLABCDEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ch $. $d x ps $.
|
|
moi.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
moi.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) $)
|
|
mob $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) ->
|
|
( A = B <-> ch ) ) $=
|
|
( wcel wa wmo wceq wb wi cvv elex w3a nfv cv nfcv nfmo1 nf3an nfim bibi1d
|
|
3anbi3d eqeq1 imbi12d mob2 vtoclgf com12 3expib syl com3r imp 3impib ) EG
|
|
KZFHKZLADMZBEFNZCOZURUSUTBLZVBPUSVCURVBUSFQKZVCURVBPZPFHRVDUTBVEURVDUTBSZ
|
|
VBVDUTASZDUAZFNZCOZPVFVBPDEGDEUBVFVBDVDUTBDVDDTADUCBDTUDVBDTUEVHENZVGVFVJ
|
|
VBVKABVDUTIUGVKVIVACVHEFUHUFUIACDFQJUJUKULUMUNUOUPUQ $.
|
|
|
|
$( Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) $)
|
|
moi $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) ->
|
|
A = B ) $=
|
|
( wcel wa wmo wceq wi w3a mob biimprd 3expia impd 3impia ) EGKFHKLZADMZBC
|
|
LEFNZUBUCLBCUDUBUCBCUDOUBUCBPUDCABCDEFGHIJQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d B x $. $d A x $. $d ps x $.
|
|
morex.1 $e |- B e. _V $.
|
|
morex.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Derive membership from uniqueness. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
morex $p |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) ) $=
|
|
( wmo wrex wcel wi cv wa wex df-rex exancom bitri wal nfmo1 nfe1 nfan syl
|
|
mopick alrimi wceq eleq1 imbi12d spcv sylan2b ancoms ) ACHZACDIZBEDJZKZUL
|
|
UKACLZDJZMZCNZUNULUPAMCNURACDOUPACPQUKURMZAUPKZCRUNUSUTCUKURCACSUQCTUAAUP
|
|
CUCUDUTUNCEFUOEUEABUPUMGUOEDUFUGUHUBUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
euxfr2dc.1 $e |- A e. _V $.
|
|
euxfr2dc.2 $e |- E* y x = A $.
|
|
$( Transfer existential uniqueness from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
14-Nov-2004.) $)
|
|
euxfr2dc $p |- ( DECID E. y E. x ( x = A /\ ph ) ->
|
|
( E! x E. y ( x = A /\ ph ) <-> E! y ph ) ) $=
|
|
( cv wa wex wdc weu wmo wal wi moani mobii mpbi ax-gen 2euswapdc mpi wceq
|
|
ancom excom dcbii sylbi moeq impbid biidd ceqsexv eubii syl6bb ) BGDUAZAH
|
|
ZBIZCIZJZUMCIZBKZUNCKZACKUPURUSUPUMCLZBMZURUSNZUTBAULHZCLUTULACFOVCUMCAUL
|
|
UBZPQRUPUQBIZJVAVBNUOVEUMCBUCUDUMBCSUETUPUMBLZCMUSURNVFCVCBLVFULABBDUFOVC
|
|
UMBVDPQRUMCBSTUGUNACAABDEULAUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x A $.
|
|
euxfrdc.1 $e |- A e. _V $.
|
|
euxfrdc.2 $e |- E! y x = A $.
|
|
euxfrdc.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Transfer existential uniqueness from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
14-Nov-2004.) $)
|
|
euxfrdc $p |- ( DECID E. y E. x ( x = A /\ ps ) ->
|
|
( E! x ph <-> E! y ps ) ) $=
|
|
( weu cv wceq wa wex wdc euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i
|
|
eubii eumoi euxfr2dc syl5bb ) ACICJEKZBLZDMZCIUGCMDMNBDIAUHCAUFDMZALUFALZ
|
|
DMUHUIAUFDIUIGUFDOPQUFADRUJUGDUFABHSTUAUBBCDEFUFDGUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d y z w ph $. $d x z ps $. $d y z w A $. $d x z B $. $d x y w $.
|
|
euind.1 $e |- B e. _V $.
|
|
euind.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
euind.3 $e |- ( x = y -> A = B ) $.
|
|
$( Existential uniqueness via an indirect equality. (Contributed by NM,
|
|
11-Oct-2010.) $)
|
|
euind $p |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph )
|
|
-> E! z A. x ( ph -> z = A ) ) $=
|
|
( vw wa wceq wi wal wex cv exbii bitri imim2i weu cbvexv biantrur 3bitr2i
|
|
isseti 19.41v excom eqeq2 bi2 an31 imbi1i impexp 3bitr3i sylib syl 2alimi
|
|
wb 19.23v albii 19.21v eximdv syl5bi imp pm4.24 biimpi prth eqtr3 alanimi
|
|
syl56 com12 syl5 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc ) ABLZF
|
|
GMZNZDOCOZACPZLAEQZFMZNZCOZEPZWGAKQZFMZNZCOZLZWDWIMZNZKOEOZWGEUAWBWCWHWCW
|
|
DGMZBLZDPZEPZWBWHWCBDPZWTABCDIUBXAWQEPZBLZDPWREPZDPWTBXCDXBBEGHUEUCRXDXCD
|
|
WQBEUFRWRDEUGUDSWBWSWGEWBWRWFNZDOZCOZWSWGNZWAXECDWAVSWEWQUQZNZXEVTXIVSFGW
|
|
DUHTXJVSWQWENZNZXEXIXKVSWEWQUITVSWQLZWENWRALZWENXLXEXMXNWEABWQUJUKVSWQWEU
|
|
LWRAWEULUMUNUOUPXGWSWFNZCOXHXFXOCWRWFDURUSWSWFCUTSUNVAVBVCWCWPWBWCWOEKWMA
|
|
WNNZCOZWCWNWFWKXPCAAALZWFWKLWEWJLWNAXRAVDVEAWEAWJVFWDWIFVGVIVHXQWCWNXQWCW
|
|
NNAWNCURVEVJVKVLVMWGWLEKWNWFWKCWNWEWJAWDWIFVNVOVPVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y ph $. $d x ps $.
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
reu2 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
|
|
( cv wcel wa weu wex wsb weq wi wal wreu wral df-ral impexp albii 3bitr4i
|
|
nfv wrex eu2 df-reu df-rex 19.21v nfs1v nfan eleq1 sbequ12 anbi12d anbi2i
|
|
sbie an4 bitri imbi1i 3bitri imbi2i bitr4i anbi12i ) BEZDFZAGZBHVBBIZVBVB
|
|
BCJZGZBCKZLZCMZBMZGABDNABDUAZAABCJZGZVFLZCDOZBDOZGVBBCVBCTUBABDUCVJVCVOVI
|
|
ABDUDVOVAVNLZBMVIVNBDPVHVPBVACEZDFZVMLZLZCMVAVSCMZLVHVPVAVSCUEVGVTCVGVAVR
|
|
GZVLGZVFLWBVMLVTVEWCVFVEVBVRVKGZGWCVDWDVBVBWDBCVRVKBVRBTABCUFUGVFVAVRAVKU
|
|
TVQDUHABCUIUJULUKVAAVRVKUMUNUOWBVLVFQVAVRVMQUPRVNWAVAVMCDPUQSRURUSS $.
|
|
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
20-Oct-2006.) $)
|
|
reu6 $p |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) ) $=
|
|
( cv wcel wa weq wb wal wex wi bi1 bi2 adantr impbid imim2i imp 3bitr4i
|
|
ex wreu weu wral wrex df-reu 19.28v eleq1 sbequ12 anbi12d equequ1 bibi12d
|
|
wsb equid tbt simpl sylbir syl6bi spimv expdimp simpr syl6 sps jca adantl
|
|
a5i impd eleq1a com23 adantll jcai alimi impbii df-ral anbi2i exbii df-eu
|
|
df-rex bitri ) ABDUABEZDFZAGZBUBZABCHZIZBDUCZCDUDZABDUEWAWCIZBJZCKCEZDFZW
|
|
EGZCKWBWFWHWKCWJVTWDLZGZBJZWJWLBJZGWHWKWJWLBUFWHWNWGWMBWHWJWLWGWJBCWCWGWJ
|
|
ABCULZGZCCHZIZWJWCWAWQWCWRWCVTWJAWPVSWIDUGABCUHUIBCCUJUKWSWQWJWRWQCUMUNWJ
|
|
WPUOUPUQURWGWLBWGVTWDWGVTGAWCWGVTAWCWAWCMUSWGWCALZVTWGWCWAAWAWCNVTAUTVAOP
|
|
TVBVCVEWMWGBWMWAWCWLWAWCLWJWLVTAWCWDAWCLVTAWCMQVFVDWMWCWAWMWCGVTAWMWCVTWJ
|
|
WCVTLWLWIDVSVGORWLWCVTALZWJWLWCXAWLVTWCAWDWTVTAWCNQVHRVIVJTPVKVLWEWOWJWDB
|
|
DVMVNSVOWABCVPWECDVQSVR $.
|
|
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu3 $p |- ( E! x e. A ph <->
|
|
( E. x e. A ph /\ E. y e. A A. x e. A ( ph -> x = y ) ) ) $=
|
|
( wreu wrex weq wi wral wa reurex wb reu6 bi1 ralimi reximi sylbi jca wex
|
|
wal rexex anim2i cv wcel weu nfv df-reu df-rex df-ral impexp albii bitr4i
|
|
eu3 exbii anbi12i 3bitr4i sylibr impbii ) ABDEZABDFZABCGZHZBDIZCDFZJZUSUT
|
|
VDABDKUSAVALZBDIZCDFVDABCDMVGVCCDVFVBBDAVANOPQRVEUTVCCSZJZUSVDVHUTVCCDUAU
|
|
BBUCDUDZAJZBUEVKBSZVKVAHZBTZCSZJUSVIVKBCVKCUFUMABDUGUTVLVHVOABDUHVCVNCVCV
|
|
JVBHZBTVNVBBDUIVMVPBVJAVAUJUKULUNUOUPUQUR $.
|
|
|
|
$( A condition which implies existential uniqueness. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
reu6i $p |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph ) $=
|
|
( vy wcel cv wceq wb wral wa wrex wreu eqeq2 bibi2d ralbidv rspcev sylibr
|
|
reu6 ) DCFABGZDHZIZBCJZKATEGZHZIZBCJZECLABCMUGUCEDCUDDHZUFUBBCUHUEUAAUDDT
|
|
NOPQABECSR $.
|
|
|
|
eqreu.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( A condition which implies existential uniqueness. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
eqreu $p |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) ->
|
|
E! x e. A ph ) $=
|
|
( wcel cv wceq wi wral wreu wa wb ralbiim ceqsralv anbi2d syl5bb reu6i ex
|
|
sylbird 3impib 3com23 ) EDGZACHEIZJCDKZBACDLZUDUFBUGUDUFBMZAUENCDKZUGUIUF
|
|
UEAJCDKZMUDUHAUECDOUDUJBUFABCEDFPQRUDUIUGACDESTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z ph $. $d x z ps $.
|
|
rmo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Restricted "at most one" using implicit substitution. (Contributed by
|
|
NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmo4 $p |- ( E* x e. A ph <->
|
|
A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( wrmo cv wcel wa wmo weq wi wral df-rmo wal bitri impexp albii df-ral
|
|
an4 ancom anbi1i imbi1i 3bitri r19.21v 3bitr2i eleq1 anbi12d mo4 3bitr4i
|
|
) ACEGCHZEIZAJZCKZABJZCDLZMZDENZCENZACEOUNDHZEIZBJZJZUQMZDPZCPUMUSMZCPUOU
|
|
TVFVGCVFVBUMURMZMZDPVHDENVGVEVIDVEVBUMJZUPJZUQMVJURMVIVDVKUQVDUMVBJZUPJVK
|
|
UMAVBBUAVLVJUPUMVBUBUCQUDVJUPUQRVBUMURRUESVHDETUMURDEUFUGSUNVCCDUQUMVBABU
|
|
LVAEUHFUIUJUSCETUKQ $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
reu4 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) ) $=
|
|
( wreu wrex wrmo wa weq wi wral reu5 rmo4 anbi2i bitri ) ACEGACEHZACEIZJR
|
|
ABJCDKLDEMCEMZJACENSTRABCDEFOPQ $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu7 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
E. x e. A A. y e. A ( ps -> x = y ) ) ) $=
|
|
( vz wreu wrex weq wi wral wa equequ1 equcom syl6bb imbi12d cbvralv bitri
|
|
reu3 rexbii imbi2d ralbidv cbvrexv anbi2i ) ACEHACEIZACGJZKZCELZGEIZMUFBC
|
|
DJZKZDELZCEIZMACGETUJUNUFUJBGDJZKZDELZGEIUNUIUQGEUHUPCDEUKABUGUOFUKUGDGJU
|
|
OCDGNDGOPQRUAUQUMGCEGCJZUPULDEURUOUKBGCDNUBUCUDSUES $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu8 $p |- ( E! x e. A ph <-> E. x e. A ( ph /\
|
|
A. y e. A ( ps -> x = y ) ) ) $=
|
|
( wreu weq wb wral wrex wi wa cbvreuv reu6 cv wcel ralbii wal syl5bb a1i
|
|
dfbi2 ancom equcom imbi2i biimt df-ral bi2.04 albii eleq1 imbi12d equcoms
|
|
vex bicomd ceqsalv 3bitrri syl6bb anbi12d r19.26 syl6rbbr rexbiia 3bitri
|
|
) ACEGBDEGBDCHZIZDEJZCEKABCDHZLZDEJZMZCEKABCDEFNBDCEOVEVICEVEBVCLZVCBLZMZ
|
|
DEJZCPZEQZVIVDVLDEBVCUBRVOVIVJDEJZVKDEJZMZVMVIVHAMVOVRAVHUCVOVHVPAVQVHVPI
|
|
VOVGVJDEVFVCBCDUDUERUAVOAVOALZVQVOAUFVQDPZEQZVKLZDSVCWABLZLZDSVSVKDEUGWBW
|
|
DDWAVCBUHUIWCVSDVNCUMWCVSICDVFVSWCVFVOWAABVNVTEUJFUKUNULUOUPUQURTVJVKDEUS
|
|
UTTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality has existential uniqueness. (Contributed by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
reueq $p |- ( B e. A <-> E! x e. A x = B ) $=
|
|
( wcel cv wceq wrex wreu risset wrmo wmo moeq mormo ax-mp mpbiran2 bitr4i
|
|
reu5 ) CBDAECFZABGZRABHZACBITSRABJZRAKUAACLRABMNRABQOP $.
|
|
$}
|
|
|
|
$( Restricted "at most one" still holds when a conjunct is added.
|
|
(Contributed by NM, 16-Jun-2017.) $)
|
|
rmoan $p |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) ) $=
|
|
( cv wcel wa wmo wrmo moan an12 mobii sylib df-rmo 3imtr4i ) CEDFZAGZCHZPBA
|
|
GZGZCHZACDISCDIRBQGZCHUAQBCJUBTCBPAKLMACDNSCDNO $.
|
|
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoim $p |- ( A. x e. A ( ph -> ps )
|
|
-> ( E* x e. A ps -> E* x e. A ph ) ) $=
|
|
( wi wral cv wcel wa wal wrmo df-ral imdistan albii wmo moim df-rmo 3imtr4g
|
|
bitri sylbi ) ABEZCDFZCGDHZAIZUCBIZEZCJZBCDKZACDKZEUBUCUAEZCJUGUACDLUJUFCUC
|
|
ABMNSUGUECOUDCOUHUIUDUECPBCDQACDQRT $.
|
|
|
|
${
|
|
rmoimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoimia $p |- ( E* x e. A ps -> E* x e. A ph ) $=
|
|
( wi wrmo rmoim mprg ) ABFBCDGACDGFCDABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
rmoimi2.1 $e |- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoimi2 $p |- ( E* x e. B ps -> E* x e. A ph ) $=
|
|
( cv wcel wa wmo wrmo wi wal moim ax-mp df-rmo 3imtr4i ) CGZEHBIZCJZRDHAI
|
|
ZCJZBCEKACDKUASLCMTUBLFUASCNOBCEPACDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
$( A condition allowing swap of uniqueness and existential quantifiers.
|
|
(Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM,
|
|
16-Jun-2017.) $)
|
|
2reuswapdc $p |- ( DECID E. x E. y ( x e. A /\ ( y e. B /\ ph ) ) ->
|
|
( A. x e. A E* y e. B ph ->
|
|
( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) ) $=
|
|
( wral cv wcel wa wmo wex wrex wreu wal weu df-reu r19.42v df-rex bitri
|
|
wi wrmo df-rmo ralbii df-ral moanimv albii bitr4i bitr3i an12 exbii eubii
|
|
wdc 2euswapdc syl7bi imbi2i syl6ibr syl5bi ) ACEUAZBDFCGEHZAIZCJZBDFZBGDH
|
|
ZUTIZCKZBKULZACELZBDMZABDLZCEMZTZURVABDACEUBUCVBVDCJZBNZVFVKVBVCVATZBNVMV
|
|
ABDUDVLVNBVCUTCUEUFUGVFVMVHVDBKZCOZTVKVHVEBOZVFVMVPVHVCVGIZBOVQVGBDPVRVEB
|
|
VRUSVCAIZIZCKZVEVRVSCELWAVCACEQVSCERUHVTVDCUSVCAUIUJSUKSVDBCUMUNVJVPVHVJU
|
|
SVIIZCOVPVICEPWBVOCWBUTBDLVOUSABDQUTBDRUHUKSUOUPUQUQ $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d x z B $. $d w x y z C $. $d w y z ph $. $d x z ps $.
|
|
reuind.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
reuind.2 $e |- ( x = y -> A = B ) $.
|
|
$( Existential uniqueness via an indirect equality. (Contributed by NM,
|
|
16-Oct-2010.) $)
|
|
reuind $p |- ( ( A. x A. y ( ( ( A e. C /\ ph ) /\ ( B e. C /\ ps ) )
|
|
-> A = B ) /\ E. x ( A e. C /\ ph ) )
|
|
-> E! z e. C A. x ( ( A e. C /\ ph ) -> z = A ) ) $=
|
|
( vw wcel wa wceq wi wal wex cv wrex bitri wral wreu eleq1d anbi12d exbii
|
|
cbvexv r19.41v rexcom4 risset anbi1i 3bitr4ri wb eqeq2 imim2i an31 imbi1i
|
|
bi2 impexp 3bitr3i sylib 2alimi 19.23v an12 adantr pm5.32ri bitr4i 19.42v
|
|
syl eleq1 albii 19.21v expd reximdvai syl5bi imp pm4.24 biimpi prth eqtr3
|
|
syl56 alanimi com12 syl5 a1d ralrimivv adantl imbi2d albidv reu4 sylanbrc
|
|
eqeq1 ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWMCQZMWMERZFNZOZCPZEHSZXDWMKRZFNZOZCPZ
|
|
MZXAXFNZOZKHUAEHUAZXDEHUBWSWTXEWTXAGNZBMZDQZEHSZWSXEWTWODQZXQWMWOCDCRDRNZ
|
|
WLWNABXSFGHJUCIUDUFXOEHSZDQXNEHSZBMZDQXQXRXTYBDXNBEHUGUEXOEDHUHWOYBDWNYAB
|
|
EGHUIUJUEUKTWSXPXDEHWSXAHLZXPXDWSXNWOMZXCOZDPZCPZYCXPMZXDOZWRYECDWRWPXBXN
|
|
ULZOZYEWQYJWPFGXAUMUNYKWPXNXBOZOZYEYJYLWPXBXNUQUNWPXNMZXBOYDWMMZXBOYMYEYN
|
|
YOXBWMWOXNUOUPWPXNXBURYDWMXBURUSUTVHVAYGYHXCOZCPYIYFYPCYFYDDQZXCOYPYDXCDV
|
|
BYQYHXCYQYCXOMZDQYHYDYRDYDWNXOMYRXNWNBVCXOYCWNXNYCWNULBXAGHVIVDVEVFUEYCXO
|
|
DVGTUPTVJYHXCCVKTUTVLVMVNVOWTXMWSWTXLEKHHWTXLYCXFHLMXJWMXKOZCPZWTXKXCXHYS
|
|
CWMWMWMMZXCXHMXBXGMXKWMUUAWMVPVQWMXBWMXGVRXAXFFVSVTWAYTWTXKYTWTXKOWMXKCVB
|
|
VQWBWCWDWEWFXDXIEKHXKXCXHCXKXBXGWMXAXFFWKWGWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x B $. $d x y $.
|
|
$( Double restricted quantification with "at most one," analogous to
|
|
~ 2moex . (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
2rmorex $p |- ( E* x e. A E. y e. B ph -> A. y e. B E* x e. A ph ) $=
|
|
( wrex wrmo cv wcel wa wmo wal wral wex df-rex mobii df-rmo albii sylib
|
|
wi anbi2i 19.42v 3bitr4i 2moex sylbi moanimv df-ral imbi2i bitri sylibr
|
|
an12 ) ACEFZBDGZCHEIZBHDIZAJZBKZTZCLZABDGZCEMZUMUNUPJZBKZCLZUSUMUOUNAJZJZ
|
|
BKZCLZVDUMVFCNZBKZVHUOULJZBKUOVECNZJZBKUMVJVKVMBULVLUOACEOUAPULBDQVIVMBUO
|
|
VECUBPUCVFBCUDUEVGVCCVFVBBUOUNAUKPRSVCURCUNUPBUFRSVAUNUTTZCLUSUTCEUGVNURC
|
|
UTUQUNABDQUHRUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
nelrdva.1 $e |- ( ( ph /\ x e. A ) -> x =/= B ) $.
|
|
$( Deduce negative membership from an implication. (Contributed by Thierry
|
|
Arnoux, 27-Nov-2017.) $)
|
|
nelrdva $p |- ( ph -> -. B e. A ) $=
|
|
( wcel wceq wa eqidd wne cv wi eleq1 anbi2d imbi12d vtoclg anabsi7 neneqd
|
|
neeq1 pm2.65da ) ADCFZDDGAUAHZDIUBDDAUADDJZABKZCFZHZUDDJZLUBUCLBDCUDDGZUF
|
|
UBUGUCUHUEUAAUDDCMNUDDDSOEPQRT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Conditional equality (experimental)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
This is a very useless definition, which "abbreviates" ` ( x = y -> ph ) ` as
|
|
` CondEq ( x = y -> ph ) ` . What this display hides, though, is that the
|
|
first expression, even though it has a shorter constant string, is actually
|
|
much more complicated in its parse tree: it is parsed as
|
|
(wi (wceq (cv vx) (cv vy)) wph), while the ` CondEq ` version is parsed as
|
|
(wcdeq vx vy wph). It also allows us to give a name to the specific ternary
|
|
operation ` ( x = y -> ph ) ` .
|
|
|
|
This is all used as part of a metatheorem: we want to say that
|
|
` |- ( x = y -> ( ph ( x ) <-> ph ( y ) ) ) ` and
|
|
` |- ( x = y -> A ( x ) = A ( y ) ) ` are provable, for any expressions
|
|
` ph ( x ) ` or ` A ( x ) ` in the language. The proof is by induction, so
|
|
the base case is each of the primitives, which is why you will see a theorem
|
|
for each of the set.mm primitive operations.
|
|
|
|
The metatheorem comes with a disjoint variables assumption: every variable in
|
|
` ph ( x ) ` is assumed disjoint from ` x ` except ` x ` itself. For such a
|
|
proof by induction, we must consider each of the possible forms of
|
|
` ph ( x ) ` . If it is a variable other than ` x ` , then we have
|
|
` CondEq ( x = y -> A = A ) ` or ` CondEq ( x = y -> ( ph <-> ph ) ) ` ,
|
|
which is provable by ~ cdeqth and reflexivity. Since we are only working
|
|
with class and wff expressions, it can't be ` x ` itself in set.mm, but if it
|
|
was we'd have to also prove ` CondEq ( x = y -> x = y ) ` (where _set_
|
|
equality is being used on the right).
|
|
|
|
Otherwise, it is a primitive operation applied to smaller expressions. In
|
|
these cases, for each setvar variable parameter to the operation, we must
|
|
consider if it is equal to ` x ` or not, which yields 2^n proof obligations.
|
|
Luckily, all primitive operations in set.mm have either zero or one set
|
|
variable, so we only need to prove one statement for the non-set constructors
|
|
(like implication) and two for the constructors taking a set (the forall and
|
|
the class builder).
|
|
|
|
In each of the primitive proofs, we are allowed to assume that ` y ` is
|
|
disjoint from ` ph ( x ) ` and vice versa, because this is maintained through
|
|
the induction. This is how we satisfy the DV assumptions of ~ cdeqab1 and
|
|
~ cdeqab .
|
|
|
|
$)
|
|
|
|
$c CondEq $. $( conditional equality $)
|
|
|
|
$( Extend wff notation to include conditional equality. This is a technical
|
|
device used in the proof that ` F/ ` is the not-free predicate, and that
|
|
definitions are conservative as a result. $)
|
|
wcdeq $a wff CondEq ( x = y -> ph ) $.
|
|
|
|
$( Define conditional equality. All the notation to the left of the ` <-> `
|
|
is fake; the parentheses and arrows are all part of the notation, which
|
|
could equally well be written ` CondEq x y ph ` . On the right side is
|
|
the actual implication arrow. The reason for this definition is to
|
|
"flatten" the structure on the right side (whose tree structure is
|
|
something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy
|
|
wph). (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
df-cdeq $a |- ( CondEq ( x = y -> ph ) <-> ( x = y -> ph ) ) $.
|
|
|
|
${
|
|
cdeqi.1 $e |- ( x = y -> ph ) $.
|
|
$( Deduce conditional equality. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
cdeqi $p |- CondEq ( x = y -> ph ) $=
|
|
( wcdeq weq wi df-cdeq mpbir ) ABCEBCFAGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
cdeqri.1 $e |- CondEq ( x = y -> ph ) $.
|
|
$( Property of conditional equality. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
cdeqri $p |- ( x = y -> ph ) $=
|
|
( wcdeq weq wi df-cdeq mpbi ) ABCEBCFAGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
cdeqth.1 $e |- ph $.
|
|
$( Deduce conditional equality from a theorem. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqth $p |- CondEq ( x = y -> ph ) $=
|
|
( weq a1i cdeqi ) ABCABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
cdeqnot.1 $e |- CondEq ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Distribute conditional equality over negation. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqnot $p |- CondEq ( x = y -> ( -. ph <-> -. ps ) ) $=
|
|
( wn wb weq cdeqri notbid cdeqi ) AFBFGCDCDHABABGCDEIJK $.
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Distribute conditional equality over quantification. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
cdeqal $p |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) ) $=
|
|
( wal wb weq cdeqri albidv cdeqi ) AEGBEGHCDCDIABEABHCDFJKL $.
|
|
|
|
$( Distribute conditional equality over abstraction. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
cdeqab $p |- CondEq ( x = y -> { z | ph } = { z | ps } ) $=
|
|
( cab wceq weq wb cdeqri abbidv cdeqi ) AEGBEGHCDCDIABEABJCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $.
|
|
$( Distribute conditional equality over quantification. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
cdeqal1 $p |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) ) $=
|
|
( wal wb cdeqri cbvalv cdeqth ) ACFBDFGCDABCDABGCDEHIJ $.
|
|
|
|
$( Distribute conditional equality over abstraction. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
cdeqab1 $p |- CondEq ( x = y -> { x | ph } = { y | ps } ) $=
|
|
( cab wceq wb cdeqri cbvabv cdeqth ) ACFBDFGCDABCDABHCDEIJK $.
|
|
$}
|
|
|
|
cdeqim.1 $e |- CondEq ( x = y -> ( ch <-> th ) ) $.
|
|
$( Distribute conditional equality over implication. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqim $p |- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) $=
|
|
( wi wb weq cdeqri imbi12d cdeqi ) ACIBDIJEFEFKABCDABJEFGLCDJEFHLMN $.
|
|
$}
|
|
|
|
$( Conditional equality for set-to-class promotion. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqcv $p |- CondEq ( x = y -> x = y ) $=
|
|
( weq id cdeqi ) ABCZABFDE $.
|
|
|
|
${
|
|
cdeqeq.1 $e |- CondEq ( x = y -> A = B ) $.
|
|
cdeqeq.2 $e |- CondEq ( x = y -> C = D ) $.
|
|
$( Distribute conditional equality over equality. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqeq $p |- CondEq ( x = y -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb weq cdeqri eqeq12d cdeqi ) CEIDFIJABABKCDEFCDIABGLEFIABHLMN $.
|
|
|
|
$( Distribute conditional equality over elementhood. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
cdeqel $p |- CondEq ( x = y -> ( A e. C <-> B e. D ) ) $=
|
|
( wcel wb weq wceq cdeqri eleq12d cdeqi ) CEIDFIJABABKCDEFCDLABGMEFLABHMN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $.
|
|
nfcdeq.1 $e |- F/ x ph $.
|
|
nfcdeq.2 $e |- CondEq ( x = y -> ( ph <-> ps ) ) $.
|
|
$( If we have a conditional equality proof, where ` ph ` is ` ph ( x ) `
|
|
and ` ps ` is ` ph ( y ) ` , and ` ph ( x ) ` in fact does not have
|
|
` x ` free in it according to ` F/ ` , then ` ph ( x ) <-> ph ( y ) `
|
|
unconditionally. This proves that ` F/ x ph ` is actually a not-free
|
|
predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcdeq $p |- ( ph <-> ps ) $=
|
|
( wsb sbf nfv wb cdeqri sbie bitr3i ) AACDGBACDEHABCDBCIABJCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z B $. $d y z A $.
|
|
nfccdeq.1 $e |- F/_ x A $.
|
|
nfccdeq.2 $e |- CondEq ( x = y -> A = B ) $.
|
|
$( Variation of ~ nfcdeq for classes. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfccdeq $p |- A = B $=
|
|
( vz cv wcel nfcri weq equid cdeqth cdeqel nfcdeq eqriv ) GCDGHZCIQDIABAG
|
|
CEJABQQCDGGKABGLMFNOP $.
|
|
$}
|
|
|
|
$( Let the computer know the theorems to look for to prove the metatheorem $)
|
|
$( $j
|
|
condequality 'wcdeq' from 'cdeqth';
|
|
condcongruence 'cdeqnot' 'cdeqim' 'cdeqal1' 'cdeqal' 'cdeqcv' 'cdeqeq'
|
|
'cdeqel' 'cdeqab1' 'cdeqab';
|
|
notfree 'wnf' from 'nfcdeq';
|
|
notfree 'wnfc' from 'nfccdeq';
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Russell's Paradox
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y $.
|
|
$( Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
|
|
|
|
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension,
|
|
expressed in our notation as ` A e. _V ` , asserted that any collection
|
|
of sets ` A ` is a set i.e. belongs to the universe ` _V ` of all sets.
|
|
In particular, by substituting ` { x | x e/ x } ` (the "Russell class")
|
|
for ` A ` , it asserted ` { x | x e/ x } e. _V ` , meaning that the
|
|
"collection of all sets which are not members of themselves" is a set.
|
|
However, here we prove ` { x | x e/ x } e/ _V ` . This contradiction
|
|
was discovered by Russell in 1901 (published in 1903), invalidating the
|
|
Comprehension Axiom and leading to the collapse of Frege's system.
|
|
|
|
In 1908, Zermelo rectified this fatal flaw by replacing Comprehension
|
|
with a weaker Subset (or Separation) Axiom asserting that ` A ` is a set
|
|
only when it is smaller than some other set ` B ` . The intuitionistic
|
|
set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla]
|
|
p. "Axioms of CZF and IZF", which we include as ~ ax-sep .
|
|
(Contributed by NM, 7-Aug-1994.) $)
|
|
ru $p |- { x | x e/ x } e/ _V $=
|
|
( vy cv wnel cab cvv wcel wn wceq wex wel wal pm5.19 eleq1 df-nel eleq12d
|
|
wb id notbid mtbir syl5bb bibi12d spv mto abeq2 nex isset mpbir ) ACZUIDZ
|
|
AEZFDUKFGZHULBCZUKIZBJUNBUNABKZUJQZALZUQBBKZURHZQZURMUPUTABUIUMIZUOURUJUS
|
|
UIUMUMNUJAAKZHVAUSUIUIOVAVBURVAUIUMUIUMVARZVCPSUAUBUCUDUJAUMUETUFBUKUGTUK
|
|
FOUH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Proper substitution of classes for sets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c [. $.
|
|
$c ]. $.
|
|
|
|
$( Extend wff notation to include the proper substitution of a class for a
|
|
set. Read this notation as "the proper substitution of class ` A ` for
|
|
setvar variable ` x ` in wff ` ph ` ." $)
|
|
wsbc $a wff [. A / x ]. ph $.
|
|
|
|
$( Define the proper substitution of a class for a set.
|
|
|
|
When ` A ` is a proper class, our definition evaluates to false. This is
|
|
somewhat arbitrary: we could have, instead, chosen the conclusion of
|
|
~ sbc6 for our definition, which always evaluates to true for proper
|
|
classes.
|
|
|
|
Our definition also does not produce the same results as discussed in the
|
|
proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does
|
|
hold, as shown by ~ dfsbcq below). Unfortunately, Quine's definition
|
|
requires a recursive syntactical breakdown of ` ph ` , and it does not
|
|
seem possible to express it with a single closed formula.
|
|
|
|
If we did not want to commit to any specific proper class behavior, we
|
|
could use this definition _only_ to prove theorem ~ dfsbcq , which holds
|
|
for both our definition and Quine's, and from which we can derive a weaker
|
|
version of ~ df-sbc in the form of ~ sbc8g . However, the behavior of
|
|
Quine's definition at proper classes is similarly arbitrary, and for
|
|
practical reasons (to avoid having to prove sethood of ` A ` in every use
|
|
of this definition) we allow direct reference to ~ df-sbc and assert that
|
|
` [. A / x ]. ph ` is always false when ` A ` is a proper class.
|
|
|
|
The related definition df-csb defines proper substitution into a class
|
|
variable (as opposed to a wff variable). (Contributed by NM,
|
|
14-Apr-1995.) (Revised by NM, 25-Dec-2016.) $)
|
|
df-sbc $a |- ( [. A / x ]. ph <-> A e. { x | ph } ) $.
|
|
|
|
$( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
|
|
under both our definition and Quine's, provides us with a weak definition
|
|
of the proper substitution of a class for a set. Since our ~ df-sbc does
|
|
not result in the same behavior as Quine's for proper classes, if we
|
|
wished to avoid conflict with Quine's definition we could start with this
|
|
theorem and ~ dfsbcq2 instead of ~ df-sbc . ( ~ dfsbcq2 is needed because
|
|
unlike Quine we do not overload the ~ df-sb syntax.) As a consequence of
|
|
these theorems, we can derive ~ sbc8g , which is a weaker version of
|
|
~ df-sbc that leaves substitution undefined when ` A ` is a proper class.
|
|
|
|
However, it is often a nuisance to have to prove the sethood hypothesis of
|
|
~ sbc8g , so we will allow direct use of ~ df-sbc . Proper substiution
|
|
with a proper class is rarely needed, and when it is, we can simply use
|
|
the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) $)
|
|
dfsbcq $p |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) ) $=
|
|
( wceq cab wcel wsbc eleq1 df-sbc 3bitr4g ) CDECABFZGDLGABCHABDHCDLIABCJABD
|
|
JK $.
|
|
|
|
$( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
|
|
under both our definition and Quine's, relates logic substitution ~ df-sb
|
|
and substitution for class variables ~ df-sbc . Unlike Quine, we use a
|
|
different syntax for each in order to avoid overloading it. See remarks
|
|
in ~ dfsbcq . (Contributed by NM, 31-Dec-2016.) $)
|
|
dfsbcq2 $p |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) ) $=
|
|
( cv wceq cab wcel wsb wsbc eleq1 df-clab df-sbc bicomi 3bitr3g ) CEZDFPABG
|
|
ZHDQHZABCIABDJZPDQKACBLSRABDMNO $.
|
|
|
|
$( Show that ~ df-sb and ~ df-sbc are equivalent when the class term ` A ` in
|
|
~ df-sbc is a setvar variable. This theorem lets us reuse theorems based
|
|
on ~ df-sb for proofs involving ~ df-sbc . (Contributed by NM,
|
|
31-Dec-2016.) (Proof modification is discouraged.) $)
|
|
sbsbc $p |- ( [ y / x ] ph <-> [. y / x ]. ph ) $=
|
|
( weq wsb cv wsbc wb eqid dfsbcq2 ax-mp ) CCDABCEABCFZGHLIABCLJK $.
|
|
|
|
${
|
|
sbceq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for class substitution. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) $)
|
|
sbceq1d $p |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) ) $=
|
|
( wceq wsbc wb dfsbcq syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
|
|
sbceq1dd.2 $e |- ( ph -> [. A / x ]. ps ) $.
|
|
$( Equality theorem for class substitution. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) $)
|
|
sbceq1dd $p |- ( ph -> [. B / x ]. ps ) $=
|
|
( wsbc sbceq1d mpbid ) ABCDHBCEHGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( This is the closest we can get to ~ df-sbc if we start from ~ dfsbcq
|
|
(see its comments) and ~ dfsbcq2 . (Contributed by NM, 18-Nov-2008.)
|
|
(Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
|
(Proof modification is discouraged.) $)
|
|
sbc8g $p |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) ) $=
|
|
( vy cv wsbc cab wcel dfsbcq eleq1 wsb df-clab weq wb equid dfsbcq2 ax-mp
|
|
bitr2i vtoclbg ) ABEFZGZUAABHZIZABCGCUCIECDABUACJUACUCKUDABELZUBAEBMEENUE
|
|
UBOEPABEUAQRST $.
|
|
$}
|
|
|
|
$( By our definition of proper substitution, it can only be true if the
|
|
substituted expression is a set. (Contributed by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbcex $p |- ( [. A / x ]. ph -> A e. _V ) $=
|
|
( wsbc cab wcel cvv df-sbc elex sylbi ) ABCDCABEZFCGFABCHCKIJ $.
|
|
|
|
$( Equality theorem for class substitution. Class version of ~ sbequ12 .
|
|
(Contributed by NM, 26-Sep-2003.) $)
|
|
sbceq1a $p |- ( x = A -> ( ph <-> [. A / x ]. ph ) ) $=
|
|
( wsb cv wceq wsbc sbid dfsbcq2 syl5bbr ) AABBDBECFABCGABHABBCIJ $.
|
|
|
|
$( Equality theorem for class substitution. Class version of ~ sbequ12r .
|
|
(Contributed by NM, 4-Jan-2017.) $)
|
|
sbceq2a $p |- ( A = x -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( cv wceq wsbc wb sbceq1a eqcoms bicomd ) CBDZEAABCFZALGKCABCHIJ $.
|
|
|
|
${
|
|
$d ph y $. $d A y $. $d x y $.
|
|
$( Specialization: if a formula is true for all sets, it is true for any
|
|
class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See
|
|
also ~ stdpc4 and ~ rspsbc . (Contributed by NM, 16-Jan-2004.) $)
|
|
spsbc $p |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) ) $=
|
|
( vy wal wsbc wi cv wceq wsb stdpc4 sbsbc sylib dfsbcq syl5ib vtocleg ) A
|
|
BFZABCGZHECDRABEIZGZTCJSRABEKUAABELABEMNABTCOPQ $.
|
|
|
|
spsbcd.1 $e |- ( ph -> A e. V ) $.
|
|
spsbcd.2 $e |- ( ph -> A. x ps ) $.
|
|
$( Specialization: if a formula is true for all sets, it is true for any
|
|
class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See
|
|
also ~ stdpc4 and ~ rspsbc . (Contributed by Mario Carneiro,
|
|
9-Feb-2017.) $)
|
|
spsbcd $p |- ( ph -> [. A / x ]. ps ) $=
|
|
( wcel wal wsbc spsbc sylc ) ADEHBCIBCDJFGBCDEKL $.
|
|
$}
|
|
|
|
${
|
|
sbcth.1 $e |- ph $.
|
|
$( A substitution into a theorem remains true (when ` A ` is a set).
|
|
(Contributed by NM, 5-Nov-2005.) $)
|
|
sbcth $p |- ( A e. V -> [. A / x ]. ph ) $=
|
|
( wcel wal wsbc ax-gen spsbc mpi ) CDFABGABCHABEIABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcthdv.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction version of ~ sbcth . (Contributed by NM, 30-Nov-2005.)
|
|
(Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbcthdv $p |- ( ( ph /\ A e. V ) -> [. A / x ]. ps ) $=
|
|
( wal wcel wsbc alrimiv spsbc mpan9 ) ABCGDEHBCDIABCFJBCDEKL $.
|
|
$}
|
|
|
|
$( An identity theorem for substitution. See ~ sbid . (Contributed by Mario
|
|
Carneiro, 18-Feb-2017.) $)
|
|
sbcid $p |- ( [. x / x ]. ph <-> ph )
|
|
$=
|
|
( cv wsbc wsb sbsbc sbid bitr3i ) ABBCDABBEAABBFABGH $.
|
|
|
|
${
|
|
nfsbc1d.2 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfsbc1 . (Contributed by NM, 23-May-2006.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1d $p |- ( ph -> F/ x [. A / x ]. ps ) $=
|
|
( wsbc cab wcel df-sbc wnfc nfab1 a1i nfeld nfxfrd ) BCDFDBCGZHACBCDIACDO
|
|
ECOJABCKLMN $.
|
|
$}
|
|
|
|
${
|
|
nfsbc1.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1 $p |- F/ x [. A / x ]. ph $=
|
|
( wsbc wnf wtru wnfc a1i nfsbc1d trud ) ABCEBFGABCBCHGDIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1v $p |- F/ x [. A / x ]. ph $=
|
|
( nfcv nfsbc1 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
nfsbcd.1 $e |- F/ y ph $.
|
|
nfsbcd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfsbcd.3 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfsbc . (Contributed by NM, 23-Nov-2005.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbcd $p |- ( ph -> F/ x [. A / y ]. ps ) $=
|
|
( wsbc cab wcel df-sbc nfabd nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHMN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
nfsbc.1 $e |- F/_ x A $.
|
|
nfsbc.2 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc $p |- F/ x [. A / y ]. ph $=
|
|
( wsbc wnf wtru nftru wnfc a1i nfsbcd trud ) ACDGBHIABCDCJBDKIELABHIFLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d z A $. $d y z ph $.
|
|
$( A composition law for class substitution. (Contributed by NM,
|
|
26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbcco $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $=
|
|
( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2 3bitr3ri vtoclbg
|
|
nfv bitri pm5.21nii ) ABCFGZCDGZDHIABDGZUACDJABDJUACEFZGZABUDGZUBUCEDHUAC
|
|
UDDKABUDDKUEABELZUFABCLZCELUACELUGUEUHUACEABCMNABECACROUACEMPABEMSQT $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d A y $.
|
|
sbcco2.1 $e |- ( x = y -> A = B ) $.
|
|
$( A composition law for class substitution. Importantly, ` x ` may occur
|
|
free in the class expression substituted for ` A ` . (Contributed by
|
|
NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbcco2 $p |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph ) $=
|
|
( wsbc cv wsb sbsbc nfv weq wceq wb equcoms dfsbcq bicomd syl sbie bitr3i
|
|
) ABEGZCBHGUACBIABDGZUACBJUAUBCBUBCKCBLDEMZUAUBNUCBCFOUCUBUAABDEPQRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
sbc5 $p |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy wsbc cvv wcel cv wceq wex sbcex exsimpl isset sylibr wsb weq dfsbcq2
|
|
wa eqeq2 anbi1d exbidv sb5 vtoclbg pm5.21nii ) ABCEZCFGZBHZCIZARZBJZABCKU
|
|
JUHBJUFUHABLBCMNABDOBDPZARZBJUEUJDCFABDCQDHZCIZULUIBUNUKUHAUMCUGSTUAABDUB
|
|
UCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbc6g $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
|
|
( wcel cv wceq wi wal wa wex wsbc nfe1 ceqex ceqsalg sbc5 syl6rbbr ) CDEB
|
|
FCGZAHBIRAJZBKZABCLATBCDSBMABCNOABCPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbc6.1 $e |- A e. _V $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
|
|
sbc6 $p |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) $=
|
|
( cvv wcel wsbc cv wceq wi wal wb sbc6g ax-mp ) CEFABCGBHCIAJBKLDABCEMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( An equivalence for class substitution in the spirit of ~ df-clab . Note
|
|
that ` x ` and ` A ` don't have to be distinct. (Contributed by NM,
|
|
18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbc7 $p |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) ) $=
|
|
( wsbc cv wceq wa wex sbcco sbc5 bitr3i ) ABDEABCFZEZCDEMDGNHCIABCDJNCDKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
cbvsbc.1 $e |- F/ y ph $.
|
|
cbvsbc.2 $e |- F/ x ps $.
|
|
cbvsbc.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variables in a wff substitution. (Contributed by Jeff
|
|
Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvsbc $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
|
|
( cab wcel wsbc cbvab eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABCD
|
|
FGHLMACENBDENO $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvsbcv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change the bound variable of a class substitution using implicit
|
|
substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario
|
|
Carneiro, 13-Oct-2016.) $)
|
|
cbvsbcv $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
|
|
( nfv cbvsbc ) ABCDEADGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
using a bound-variable hypothesis instead of distinct variables.
|
|
(Closed theorem version of ~ sbciegf .) (Contributed by NM,
|
|
10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbciegft $p |- ( ( A e. V /\ F/ x ps /\
|
|
A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( wcel wnf cv wceq wb wi wal w3a wsbc imim2i alimi biimpa sylan2 3adant1
|
|
wa wex sbc5 bi1 impd 19.23t syl5bi bi2 com23 19.21t sbc6g 3ad2ant1 impbid
|
|
sylibrd ) DEFZBCGZCHDIZABJZKZCLZMZACDNZBVAUPATZCUAZUTBACDUBUOUSVCBKZUNUSU
|
|
OVBBKZCLZVDURVECURUPABUQABKUPABUCOUDPUOVFVDVBBCUEQRSUFUTBUPAKZCLZVAUOUSBV
|
|
HKZUNUSUOBVGKZCLZVIURVJCURUPBAUQBAKUPABUGOUHPUOVKVIBVGCUIQRSUNUOVAVHJUSAC
|
|
DEUJUKUMUL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbciegf.1 $e |- F/ x ps $.
|
|
sbciegf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbciegf $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( wcel wnf cv wceq wb wi wal wsbc ax-gen sbciegft mp3an23 ) DEHBCICJDKABL
|
|
MZCNACDOBLFSCGPABCDEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
sbcieg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 10-Nov-2005.) $)
|
|
sbcieg $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( nfv sbciegf ) ABCDEBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $. $d ch y $. $d ph y $. $d ps x $.
|
|
sbcie2g.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
sbcie2g.2 $e |- ( y = A -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
This version of ~ sbcie avoids a disjointness condition on ` x ` and
|
|
` A ` by substituting twice. (Contributed by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
sbcie2g $p |- ( A e. V -> ( [. A / x ]. ph <-> ch ) ) $=
|
|
( cv wsbc dfsbcq wsb sbsbc nfv sbie bitr3i vtoclbg ) ADEJZKZBADFKCEFGADSF
|
|
LITADEMBADENABDEBDOHPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
sbcie.1 $e |- A e. _V $.
|
|
sbcie.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 4-Sep-2004.) $)
|
|
sbcie $p |- ( [. A / x ]. ph <-> ps ) $=
|
|
( cvv wcel wsbc wb sbcieg ax-mp ) DGHACDIBJEABCDGFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbcied.1 $e |- ( ph -> A e. V ) $.
|
|
sbcied.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
${
|
|
sbciedf.3 $e |- F/ x ph $.
|
|
sbciedf.4 $e |- ( ph -> F/ x ch ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 29-Dec-2014.) $)
|
|
sbciedf $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( wcel wnf cv wceq wb wi wal wsbc ex alrimi sbciegft syl3anc ) AEFKCDLD
|
|
MENZBCOZPZDQBDERCOGJAUEDIAUCUDHSTBCDEFUAUB $.
|
|
$}
|
|
|
|
$d x ph $. $d x ch $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 13-Dec-2014.) $)
|
|
sbcied $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( nfv nfvd sbciedf ) ABCDEFGHADIACDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
sbcied2.1 $e |- ( ph -> A e. V ) $.
|
|
sbcied2.2 $e |- ( ph -> A = B ) $.
|
|
sbcied2.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 13-Dec-2014.) $)
|
|
sbcied2 $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( cv wceq wb id sylan9eqr syldan sbcied ) ABCDEGHADKZELZRFLBCMSAREFSNIOJP
|
|
Q $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y ph $. $d x y $.
|
|
elrabsf.1 $e |- F/_ x B $.
|
|
$( Membership in a restricted class abstraction, expressed with explicit
|
|
class substitution. (The variation ~ elrabf has implicit substitution).
|
|
The hypothesis specifies that ` x ` must not be a free variable in
|
|
` B ` . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario
|
|
Carneiro, 13-Oct-2016.) $)
|
|
elrabsf $p |- ( A e. { x e. B | ph }
|
|
<-> ( A e. B /\ [. A / x ]. ph ) ) $=
|
|
( vy cv wsbc crab dfsbcq nfcv nfv nfsbc1v sbceq1a cbvrab elrab2 ) ABFGZHZ
|
|
ABCHFCDABDIABQCJARBFDEFDKAFLABQMABQNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y B $. $d y A $.
|
|
$( Substitution applied to an atomic wff. Set theory version of ~ eqsb3 .
|
|
(Contributed by Andrew Salmon, 29-Jun-2011.) $)
|
|
eqsbc3 $p |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) ) $=
|
|
( vy cv wceq wsbc dfsbcq eqeq1 wsb sbsbc eqsb3 bitr3i vtoclbg ) AFCGZAEFZ
|
|
HZQCGZPABHBCGEBDPAQBIQBCJRPAEKSPAELEACMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $. $d y ps $.
|
|
$( Move negation in and out of class substitution. (Contributed by NM,
|
|
16-Jan-2004.) $)
|
|
sbcng $p |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) ) $=
|
|
( vy wn wsb wsbc dfsbcq2 cv wceq notbid sbn vtoclbg ) AFZBEGABEGZFOBCHABC
|
|
HZFECDOBECIEJCKPQABECILABEMN $.
|
|
|
|
$( Distribution of class substitution over implication. (Contributed by
|
|
NM, 16-Jan-2004.) $)
|
|
sbcimg $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) ) $=
|
|
( vy wi wsb wsbc dfsbcq2 cv wceq imbi12d sbim vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over conjunction. (Contributed by
|
|
NM, 31-Dec-2016.) $)
|
|
sbcan $p |- ( [. A / x ]. ( ph /\ ps )
|
|
<-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) $=
|
|
( vy wa wsbc cvv wcel sbcex adantl dfsbcq2 cv wceq anbi12d sban pm5.21nii
|
|
wsb vtoclbg ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUDUBUCBCDJKTCERACERZBCERZFUAUEE
|
|
DHTCEDLEMDNUFUCUGUDACEDLBCEDLOABCEPSQ $.
|
|
|
|
$( Distribution of class substitution over conjunction. (Contributed by
|
|
NM, 21-May-2004.) $)
|
|
sbcang $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) ) $=
|
|
( vy wa wsb wsbc dfsbcq2 cv wceq anbi12d sban vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over disjunction. (Contributed by
|
|
NM, 31-Dec-2016.) $)
|
|
sbcor $p |- ( [. A / x ]. ( ph \/ ps )
|
|
<-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) $=
|
|
( vy wo wsbc cvv wcel sbcex jaoi wsb dfsbcq2 cv wceq orbi12d sbor vtoclbg
|
|
pm5.21nii ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUCUBUDACDJBCDJKTCELACELZBCELZFUAU
|
|
EEDHTCEDMENDOUFUCUGUDACEDMBCEDMPABCEQRS $.
|
|
|
|
$( Distribution of class substitution over disjunction. (Contributed by
|
|
NM, 21-May-2004.) $)
|
|
sbcorg $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ) $=
|
|
( vy wo wsb wsbc dfsbcq2 cv wceq orbi12d sbor vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over biconditional. (Contributed by
|
|
Raph Levien, 10-Apr-2004.) $)
|
|
sbcbig $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) ) $=
|
|
( vy wb wsb wsbc dfsbcq2 cv wceq bibi12d sbbi vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
$}
|
|
|
|
$( Move negation in and out of class substitution. One direction of ~ sbcng
|
|
that holds for proper classes. (Contributed by NM, 17-Aug-2018.) $)
|
|
sbcn1 $p |- ( [. A / x ]. -. ph -> -. [. A / x ]. ph ) $=
|
|
( cvv wcel wn wsbc sbcex sbcng biimpd mpcom ) CDEZAFZBCGZABCGFZMBCHLNOABCDI
|
|
JK $.
|
|
|
|
$( Distribution of class substitution over implication. One direction of
|
|
~ sbcimg that holds for proper classes. (Contributed by NM,
|
|
17-Aug-2018.) $)
|
|
sbcim1 $p |- ( [. A / x ]. ( ph -> ps )
|
|
-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) $=
|
|
( cvv wcel wi wsbc sbcex sbcimg biimpd mpcom ) DEFZABGZCDHZACDHBCDHGZNCDIMO
|
|
PABCDEJKL $.
|
|
|
|
$( Distribution of class substitution over biconditional. One direction of
|
|
~ sbcbig that holds for proper classes. (Contributed by NM,
|
|
17-Aug-2018.) $)
|
|
sbcbi1 $p |- ( [. A / x ]. ( ph <-> ps )
|
|
-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
|
|
( cvv wcel wb wsbc sbcex sbcbig biimpd mpcom ) DEFZABGZCDHZACDHBCDHGZNCDIMO
|
|
PABCDEJKL $.
|
|
|
|
$( Substituting into equivalent wff's gives equivalent results. (Contributed
|
|
by Giovanni Mascellani, 9-Apr-2018.) $)
|
|
sbcbi2 $p |- ( A. x ( ph <-> ps ) ->
|
|
( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
|
|
( wb wal cab wcel wsbc wceq abbi eleq2 sylbi df-sbc 3bitr4g ) ABECFZDACGZHZ
|
|
DBCGZHZACDIBCDIPQSJRTEABCKQSDLMACDNBCDNO $.
|
|
|
|
${
|
|
$d x z A $. $d x y z $. $d z ph $.
|
|
$( Move universal quantifier in and out of class substitution.
|
|
(Contributed by NM, 31-Dec-2016.) $)
|
|
sbcal $p |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) $=
|
|
( vz wal wsbc cvv wcel sbcex sps dfsbcq2 cv wceq albidv vtoclbg pm5.21nii
|
|
wsb sbal ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCERACERZBFUAUDEDHTCEDLEMDNU
|
|
EUCBACEDLOABCESPQ $.
|
|
|
|
$( Move universal quantifier in and out of class substitution.
|
|
(Contributed by NM, 16-Jan-2004.) $)
|
|
sbcalg $p |- ( A e. V
|
|
-> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) ) $=
|
|
( vz wal wsb wsbc dfsbcq2 cv wceq albidv sbal vtoclbg ) ABGZCFHACFHZBGPCD
|
|
IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.
|
|
|
|
$( Move existential quantifier in and out of class substitution.
|
|
(Contributed by NM, 21-May-2004.) $)
|
|
sbcex2 $p |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) $=
|
|
( vz wex wsbc cvv wcel sbcex exlimiv wsb dfsbcq2 wceq exbidv sbex vtoclbg
|
|
cv pm5.21nii ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCELACELZBFUAUDEDHTCEDME
|
|
RDNUEUCBACEDMOABCEPQS $.
|
|
|
|
$( Move existential quantifier in and out of class substitution.
|
|
(Contributed by NM, 21-May-2004.) $)
|
|
sbcexg $p |- ( A e. V
|
|
-> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) ) $=
|
|
( vz wex wsb wsbc dfsbcq2 cv wceq exbidv sbex vtoclbg ) ABGZCFHACFHZBGPCD
|
|
IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x A $.
|
|
$( A variation of extensionality for classes. (Contributed by Andrew
|
|
Salmon, 28-Jun-2011.) $)
|
|
sbceqal $p |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) ) $=
|
|
( wcel cv wceq wi wal wsbc spsbc sbcimg wb eqsbc3 mpbiri pm5.5 syl 3bitrd
|
|
eqid sylibd ) BDEZAFZBGZUBCGZHZAIUEABJZBCGZUEABDKUAUFUCABJZUDABJZHZUIUGUC
|
|
UDABDLUAUHUJUIMUAUHBBGBSABBDNOUHUIPQABCDNRT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew
|
|
Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) $)
|
|
sbeqalb $p |- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <->
|
|
x = B ) ) -> A = B ) ) $=
|
|
( cv wceq wb wal wa wi wcel bibi1 biimpa biimpd alanimi sbceqal syl5 ) AB
|
|
FZCGZHZBIASDGZHZBIJTUBKZBICELCDGUAUCUDBUAUCJTUBUAUCTUBHATUBMNOPBCDEQR $.
|
|
$}
|
|
|
|
${
|
|
sbcbid.1 $e |- F/ x ph $.
|
|
sbcbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 29-Dec-2014.) $)
|
|
sbcbid $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
|
|
( cab wcel wsbc abbid eleq2d df-sbc 3bitr4g ) AEBDHZIECDHZIBDEJCDEJAOPEAB
|
|
CDFGKLBDEMCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 29-Dec-2014.) $)
|
|
sbcbidv $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
|
|
( nfv sbcbid ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
sbcbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building inference rule for class substitution. (Contributed by
|
|
NM, 11-Nov-2005.) $)
|
|
sbcbii $p |- ( [. A / x ]. ph <-> [. A / x ]. ps ) $=
|
|
( wsbc wb wtru a1i sbcbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
$( ~ eqsbc3 with setvar variable on right side of equals sign.
|
|
(Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ,
|
|
7-Jul-2021.) $)
|
|
eqsbc3r $p |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) ) $=
|
|
( wcel cv wceq wsbc eqsbc3 eqcom sbcbii 3bitr4g ) BDEAFZCGZABHBCGCMGZABHC
|
|
BGABCDIONABCMJKCBJL $.
|
|
$}
|
|
|
|
$( Distribution of class substitution over triple conjunction. (Contributed
|
|
by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) $)
|
|
sbc3an $p |- ( [. A / x ]. ( ph /\ ps /\ ch ) <->
|
|
( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) $=
|
|
( w3a wsbc wa df-3an sbcbii sbcan anbi1i 3bitri bitr4i ) ABCFZDEGZADEGZBDEG
|
|
ZHZCDEGZHZQRTFPABHZCHZDEGUBDEGZTHUAOUCDEABCIJUBCDEKUDSTABDEKLMQRTIN $.
|
|
|
|
${
|
|
$d y A $. $d x y B $.
|
|
$( Class substitution into a membership relation. (Contributed by NM,
|
|
17-Aug-2018.) $)
|
|
sbcel1v $p |- ( [. A / x ]. x e. B <-> A e. B ) $=
|
|
( vy wcel wsbc cvv sbcex elex wsb dfsbcq2 eleq1 clelsb3 vtoclbg pm5.21nii
|
|
cv ) APCEZABFZBGEBCEZQABHBCIQADJDPZCERSDBGQADBKTBCLDACMNO $.
|
|
$}
|
|
|
|
${
|
|
$d y B $. $d x y A $.
|
|
$( Class substitution into a membership relation. (Contributed by NM,
|
|
17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
sbcel2gv $p |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) ) $=
|
|
( vy cv wcel eleq2 sbcie2g ) BAFZGBEFZGBCGAECDJKBHKCBHI $.
|
|
|
|
$( Class substitution into a membership relation. One direction of
|
|
~ sbcel2gv that holds for proper classes. (Contributed by NM,
|
|
17-Aug-2018.) $)
|
|
sbcel21v $p |- ( [. B / x ]. A e. x -> A e. B ) $=
|
|
( cvv wcel cv wsbc sbcex sbcel2gv biimpd mpcom ) CDEZBAFEZACGZBCEZMACHLNO
|
|
ABCDIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Substitution analogue of Theorem 19.20 of [Margaris] p. 90 ( ~ alim ).
|
|
(Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof
|
|
shortened by JJ, 7-Jul-2021.) $)
|
|
sbcimdv $p |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) $=
|
|
( wsbc cvv wcel sbcex wi wal alrimiv spsbc sbcim1 syl56 com3l mpdi ) ABDE
|
|
GZEHIZCDEGZBDEJTASUAABCKZDLTUBDEGSUAKAUBDFMUBDEHNBCDEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
sbctt $p |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( vy wcel wnf wsbc wb wsb wi wceq dfsbcq2 bibi1d imbi2d sbft vtoclg imp
|
|
cv ) CDFABGZABCHZAIZTABEJZAIZKTUBKECDESCLZUDUBTUEUCUAAABECMNOABEPQR $.
|
|
$}
|
|
|
|
${
|
|
sbcgf.1 $e |- F/ x ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcgf $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( wcel wnf wsbc wb sbctt mpan2 ) CDFABGABCHAIEABCDJK $.
|
|
|
|
$( Substitution for a variable not free in antecedent affects only the
|
|
consequent. (Contributed by NM, 11-Oct-2004.) $)
|
|
sbc19.21g $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) ) $=
|
|
( wcel wi wsbc sbcimg sbcgf imbi1d bitrd ) DEGZABHCDIACDIZBCDIZHAPHABCDEJ
|
|
NOAPACDEFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Substitution for a variable not occurring in a wff does not affect it.
|
|
Distinct variable form of ~ sbcgf . (Contributed by Alan Sare,
|
|
10-Nov-2012.) $)
|
|
sbcg $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( nfv sbcgf ) ABCDABEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x V $. $d y W $.
|
|
sbc2iegf.1 $e |- F/ x ps $.
|
|
sbc2iegf.2 $e |- F/ y ps $.
|
|
sbc2iegf.3 $e |- F/ x B e. W $.
|
|
sbc2iegf.4 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
sbc2iegf $p |- ( ( A e. V /\ B e. W ) ->
|
|
( [. A / x ]. [. B / y ]. ph <-> ps ) ) $=
|
|
( wcel wa simpl cv wceq wb adantll nfv wsbc wnf a1i sbciedf nfan ) EGMZFH
|
|
MZNZADFUAZBCEGUFUGOUGCPEQZUIBRUFUGUJNZABDFHUGUJOUJDPFQABRUGLSUKDTBDUBUKJU
|
|
CUDSUFUGCUFCTKUEBCUBUHIUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ps $.
|
|
sbc2ie.1 $e |- A e. _V $.
|
|
sbc2ie.2 $e |- B e. _V $.
|
|
sbc2ie.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro,
|
|
19-Dec-2013.) $)
|
|
sbc2ie $p |- ( [. A / x ]. [. B / y ]. ph <-> ps ) $=
|
|
( cvv wcel wsbc wb nfv nfth sbc2iegf mp2an ) EJKFJKZADFLCELBMGHABCDEFJJBC
|
|
NBDNRCHOIPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ph $. $d x y ch $.
|
|
sbc2iedv.1 $e |- A e. _V $.
|
|
sbc2iedv.2 $e |- B e. _V $.
|
|
sbc2iedv.3 $e |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro,
|
|
18-Oct-2016.) $)
|
|
sbc2iedv $p |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) ) $=
|
|
( wsbc cvv wcel a1i cv wceq wa wb impl sbcied ) ABEGKCDFLFLMAHNADOFPZQZBC
|
|
EGLGLMUBINAUAEOGPBCRJSTT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $. $d x y z ps $.
|
|
sbc3ie.1 $e |- A e. _V $.
|
|
sbc3ie.2 $e |- B e. _V $.
|
|
sbc3ie.3 $e |- C e. _V $.
|
|
sbc3ie.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario
|
|
Carneiro, 29-Dec-2014.) $)
|
|
sbc3ie $p |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps ) $=
|
|
( wsbc cv wceq wa cvv wcel a1i wb 3expa sbcied sbc2ie ) AEHMBCDFGIJCNFOZD
|
|
NGOZPZABEHQHQRUFKSUDUEENHOABTLUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Lemma for ~ sbccom . (Contributed by NM, 14-Nov-2005.) (Revised by
|
|
Mario Carneiro, 18-Oct-2016.) $)
|
|
sbccomlem $p |- ( [. A / x ]. [. B / y ]. ph
|
|
<-> [. B / y ]. [. A / x ]. ph ) $=
|
|
( cv wceq wa wex wsbc excom exdistr an12 exbii bitri 3bitr3i sbc5 3bitr4i
|
|
19.42v sbcbii ) CFEGZAHZCIZBDJZBFDGZAHZBIZCEJZACEJZBDJABDJZCEJUEUCHBIZUAU
|
|
GHZCIZUDUHUEUBHZCIBIUNBIZCIUKUMUNBCKUEUBBCLUOULCUOUAUFHZBIULUNUPBUEUAAMNU
|
|
AUFBSONPUCBDQUGCEQRUIUCBDACEQTUJUGCEABDQTR $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w x z B $. $d w z ph $. $d x y $.
|
|
$( Commutative law for double class substitution. (Contributed by NM,
|
|
15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) $)
|
|
sbccom $p |- ( [. A / x ]. [. B / y ]. ph
|
|
<-> [. B / y ]. [. A / x ]. ph ) $=
|
|
( vw vz cv wsbc sbccomlem sbcbii bitri 3bitr3i sbcco ) ACFHZIZFEIZBDIZABG
|
|
HZIZGDIZCEIZACEIZBDIABDIZCEIQBSIZGDIZUACOIZFEIZRUBTCOIZFEIZGDIUIGDIZFEIUF
|
|
UHUIGFDEJUJUEGDUJPBSIZFEIUEUIULFEACBOSJKPFBESJLKUKUGFETGCDOJKMQBGDNUACFEN
|
|
MQUCBDACFENKUAUDCEABGDNKM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A z $. $d B x z $. $d V z $. $d ph z $.
|
|
$( Interchange class substitution and restricted quantifier. (Contributed
|
|
by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) $)
|
|
sbcralt $p |- ( ( A e. V /\ F/_ y A ) ->
|
|
( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wral wsbc cv wcel wnfc wa sbcco simpl wsb wceq sbsbc nfcv wb nfralxy
|
|
nfs1v weq sbequ12 ralbidv sbie bitr3i nfnfc1 nfcvd id nfeqd nfan1 dfsbcq2
|
|
adantl ralbid adantll syl5bb sbcied syl5bbr ) ACEHZBDIUTBGJZIZGDIDFKZCDLZ
|
|
MZABDIZCEHZUTBGDNVEVBVGGDFVCVDOVBABGPZCEHZVEVADQZMVGVBUTBGPVIUTBGRUTVIBGV
|
|
HBCEBESABGUBUABGUCAVHCEABGUDUEUFUGVDVJVIVGTVCVDVJMVHVFCEVDVJCCDUHVDCVADVD
|
|
CVAUIVDUJUKULVJVHVFTVDABGDUMUNUOUPUQURUS $.
|
|
|
|
$( Interchange class substitution and restricted existential quantifier.
|
|
(Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbcrext $p |- ( F/_ y A ->
|
|
( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wnfc cvv wcel wrex wsbc wi sbcex a1i nfcvd cv wb wa wsb wceq id 2a1i
|
|
nfnfc1 nfeld rexlimd2 sbcco simpl sbsbc nfcv nfrexxy sbequ12 rexbidv sbie
|
|
nfs1v bitr3i nfeqd dfsbcq2 adantl rexbid adantll syl5bb syl5bbr pm5.21ndd
|
|
nfan1 sbcied expcom ) CDGZDHIZACEJZBDKZABDKZCEJZVJVHLVGVIBDMNVGVKVHCECDUC
|
|
ZVGCDHVGUAZVGCHOUDVGCPEIVKVHLABDMUBUEVHVGVJVLQVJVIBFPZKZFDKVHVGRZVLVIBFDU
|
|
FVQVPVLFDHVHVGUGVPABFSZCEJZVQVODTZRVLVPVIBFSVSVIBFUHVIVSBFVRBCEBEUIABFUNU
|
|
JBPVOTAVRCEABFUKULUMUOVGVTVSVLQVHVGVTRVRVKCEVGVTCVMVGCVODVGCVOOVNUPVDVTVR
|
|
VKQVGABFDUQURUSUTVAVEVBVFVC $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x B $. $d x y z $. $d ph z $. $d B z $.
|
|
$( Interchange class substitution and restricted quantifier. (Contributed
|
|
by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcralg $p |- ( A e. V ->
|
|
( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wral wsb wsbc dfsbcq2 cv wceq ralbidv nfcv nfs1v nfralxy weq sbequ12
|
|
sbie vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGLDMUCUECEABGDKNUBU
|
|
DBGUCBCEBEOABGPQBGRAUCCEABGSNTUA $.
|
|
|
|
$( Interchange class substitution and restricted existential quantifier.
|
|
(Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) $)
|
|
sbcrex $p |- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) $=
|
|
( wnfc wrex wsbc wb nfcv sbcrext ax-mp ) CDFACEGBDHABDHCEGICDJABCDEKL $.
|
|
|
|
$( Interchange class substitution and restricted uniqueness quantifier.
|
|
(Contributed by NM, 24-Feb-2013.) $)
|
|
sbcreug $p |- ( A e. V ->
|
|
( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) ) $=
|
|
( vz wreu wsb wsbc dfsbcq2 cv wceq reubidv nfcv nfs1v nfreuxy weq sbequ12
|
|
sbie vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGLDMUCUECEABGDKNUBU
|
|
DBGUCBCEBEOABGPQBGRAUCCEABGSNTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y w A $. $d w B $. $d w ph $. $d x y $. $d w x $.
|
|
sbcabel.1 $e |- F/_ x B $.
|
|
$( Interchange class substitution and class abstraction. (Contributed by
|
|
NM, 5-Nov-2005.) $)
|
|
sbcabel $p |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <->
|
|
{ y | [. A / x ]. ph } e. B ) ) $=
|
|
( vw wcel cvv cab wsbc wb cv wceq wa wex wal bitrd abeq2 elex sbcexg sbcg
|
|
sbcang sbcalg sbcbig bibi1d albidv sbcbii 3bitr4g nfcri sbcgf anbi12d syl
|
|
exbidv df-clel ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAUQHNZUROZVDEIZPZHQZBDL
|
|
ZVDVBOZVFPZHQZUTVCUQVIVGBDLZHQVLVGHBDJUBUQVMVKHUQVMVEBDLZVFBDLZPVKVEVFBDJ
|
|
UDUQVNVJVOVFUQCNVDIZAMZCRZBDLZVPVAMZCRZVNVJUQVSVQBDLZCRWAVQCBDJUEUQWBVTCU
|
|
QWBVPBDLZVAMVTVPABDJUFUQWCVPVAVPBDJUCUGSUHSVEVRBDACVDTUIVACVDTUJVFBDJBHEG
|
|
UKULUMSUOSUSVHBDHUREUPUIHVBEUPUJUN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $. $d y ph $.
|
|
$( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This
|
|
provides an axiom for a predicate calculus for a restricted domain.
|
|
This theorem generalizes the unrestricted ~ stdpc4 and ~ spsbc . See
|
|
also ~ rspsbca and rspcsbela . (Contributed by NM, 17-Nov-2006.)
|
|
(Proof shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
rspsbc $p |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) ) $=
|
|
( vy wral wsb wcel wsbc cbvralsv dfsbcq2 rspcv syl5bi ) ABDFABEGZEDFCDHAB
|
|
CIZABEDJNOECDABECKLM $.
|
|
|
|
$( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69.
|
|
(Contributed by NM, 14-Dec-2005.) $)
|
|
rspsbca $p |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph ) $=
|
|
( wcel wral wsbc rspsbc imp ) CDEABDFABCGABCDHI $.
|
|
|
|
$( Existence form of ~ rspsbca . (Contributed by NM, 29-Feb-2008.) (Proof
|
|
shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
rspesbca $p |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph ) $=
|
|
( vy wcel wsbc wa wsb wrex dfsbcq2 rspcev cbvrexsv sylibr ) CDFABCGZHABEI
|
|
ZEDJABDJPOECDABECKLABEDMN $.
|
|
|
|
$( Existence form of ~ spsbc . (Contributed by Mario Carneiro,
|
|
18-Nov-2016.) $)
|
|
spesbc $p |- ( [. A / x ]. ph -> E. x ph ) $=
|
|
( wsbc cvv wrex wex wcel sbcex rspesbca mpancom rexv sylib ) ABCDZABEFZAB
|
|
GCEHNOABCIABCEJKABLM $.
|
|
|
|
spesbcd.1 $e |- ( ph -> [. A / x ]. ps ) $.
|
|
$( form of ~ spsbc . (Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
spesbcd $p |- ( ph -> E. x ps ) $=
|
|
( wsbc wex spesbc syl ) ABCDFBCGEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
sbcth2.1 $e |- ( x e. B -> ph ) $.
|
|
$( A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof
|
|
shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbcth2 $p |- ( A e. B -> [. A / x ]. ph ) $=
|
|
( wcel wral wsbc rgen rspsbc mpi ) CDFABDGABCHABDEIABCDJK $.
|
|
$}
|
|
|
|
${
|
|
ra5.1 $e |- F/ x ph $.
|
|
$( Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is
|
|
an axiom of a predicate calculus for a restricted domain. Compare the
|
|
unrestricted ~ stdpc5 . (Contributed by NM, 16-Jan-2004.) $)
|
|
ra5 $p |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) ) $=
|
|
( wi wral cv wcel wal df-ral bi2.04 albii bitri stdpc5 sylbi syl6ibr ) AB
|
|
FZCDGZACHDIZBFZCJZBCDGSAUAFZCJZAUBFSTRFZCJUDRCDKUEUCCTABLMNAUACEOPBCDKQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
rmo2.1 $e |- F/ y ph $.
|
|
$( Condition implying restricted "at most one." (Contributed by Jim
|
|
Kingdon, 14-Jul-2018.) $)
|
|
rmo2ilem $p |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
|
|
( cv wceq wi wral wex wcel wal wrmo impexp albii df-ral bitr4i exbii wmo
|
|
wa nfv nfan mo2r df-rmo sylibr sylbir ) ABFZCFGZHZBDIZCJUGDKZATZUHHZBLZCJ
|
|
ZABDMZUNUJCUNUKUIHZBLUJUMUQBUKAUHNOUIBDPQRUOULBSUPULBCUKACUKCUAEUBUCABDUD
|
|
UEUF $.
|
|
|
|
$( Condition implying restricted "at most one." (Contributed by NM,
|
|
17-Jun-2017.) $)
|
|
rmo2i $p |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
|
|
( cv wceq wi wral wrex wex wrmo rexex rmo2ilem syl ) ABFCFGHBDIZCDJPCKABD
|
|
LPCDMABCDENO $.
|
|
|
|
$( Restricted "at most one" using explicit substitution. (Contributed by
|
|
NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmo3 $p |- ( E* x e. A ph <->
|
|
A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( wrmo cv wcel wa wmo wsb wi wral anbi1i bitri 3bitri impexp albii df-ral
|
|
wal weq df-rmo sban clelsb3 anbi2i an4 ancom r19.21v 3bitr2i nfv nfan mo3
|
|
imbi1i 3bitr4i ) ABDFBGDHZAIZBJZAABCKZIZBCUAZLZCDMZBDMZABDUBUPUPBCKZIZUTL
|
|
ZCTZBTUOVBLZBTUQVCVGVHBVGCGDHZUOVALZLZCTVJCDMVHVFVKCVFVIUOIZUSIZUTLVLVALV
|
|
KVEVMUTVEUPVIURIZIUOVIIZUSIVMVDVNUPVDUOBCKZURIVNUOABCUCVPVIURCBDUDNOUEUOA
|
|
VIURUFVOVLUSUOVIUGNPUMVLUSUTQVIUOVAQPRVJCDSUOVACDUHUIRUPBCUOACUOCUJEUKULV
|
|
BBDSUNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $.
|
|
rmoi.b $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
rmoi.c $e |- ( x = C -> ( ph <-> ch ) ) $.
|
|
$( Consequence of "at most one", using implicit substitution. (Contributed
|
|
by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmob $p |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) ->
|
|
( B = C <-> ( C e. A /\ ch ) ) ) $=
|
|
( wrmo cv wcel wa wmo wceq wb df-rmo simprl eleq1 anbi12d syl5ibcom simpl
|
|
wi a1i simplrl simpr simpll simplrr mob syl212anc ex pm5.21ndd sylanb ) A
|
|
DEJDKZELZAMZDNZFELZBMZFGOZGELZCMZPZADEQUQUSMZVAUTVBVDURUTVAUQURBRFGESUAVB
|
|
VAUCVDVACUBUDVDVAVCVDVAMURVAUQURBVCUQURBVAUEZVDVAUFUQUSVAUGVEUQURBVAUHUPU
|
|
SVBDFGEEUNFOUOURABUNFESHTUNGOUOVAACUNGESITUIUJUKULUM $.
|
|
|
|
$( Consequence of "at most one", using implicit substitution. (Contributed
|
|
by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmoi $p |- ( ( E* x e. A ph
|
|
/\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C ) $=
|
|
( wrmo wcel wa wceq rmob biimp3ar ) ADEJFEKBLFGMGEKCLABCDEFGHINO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Proper substitution of classes for sets into classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c [_ $. $( Underlined left bracket $)
|
|
$c ]_ $. $( Underlined right bracket $)
|
|
|
|
$( Extend class notation to include the proper substitution of a class for a
|
|
set into another class. $)
|
|
csb $a class [_ A / x ]_ B $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Define the proper substitution of a class for a set into another class.
|
|
The underlined brackets distinguish it from the substitution into a wff,
|
|
~ wsbc , to prevent ambiguity. Theorem ~ sbcel1g shows an example of
|
|
how ambiguity could arise if we didn't use distinguished brackets.
|
|
Theorem ~ sbccsbg recreates substitution into a wff from this
|
|
definition. (Contributed by NM, 10-Nov-2005.) $)
|
|
df-csb $a |- [_ A / x ]_ B = { y | [. A / x ]. y e. B } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y $.
|
|
$( Alternate expression for the proper substitution into a class, without
|
|
referencing substitution into a wff. Note that ` x ` can be free in
|
|
` B ` but cannot occur in ` A ` . (Contributed by NM, 2-Dec-2013.) $)
|
|
csb2 $p |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) } $=
|
|
( csb cv wcel wsbc cab wceq wa wex df-csb sbc5 abbii eqtri ) ACDEBFDGZACH
|
|
ZBIAFCJQKALZBIABCDMRSBQACNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Analog of ~ dfsbcq for proper substitution into a class. (Contributed
|
|
by NM, 10-Nov-2005.) $)
|
|
csbeq1 $p |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
|
|
( vy wceq cv wcel wsbc cab csb dfsbcq abbidv df-csb 3eqtr4g ) BCFZEGDHZAB
|
|
IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z C $. $d z D $.
|
|
cbvcsb.1 $e |- F/_ y C $.
|
|
cbvcsb.2 $e |- F/_ x D $.
|
|
cbvcsb.3 $e |- ( x = y -> C = D ) $.
|
|
$( Change bound variables in a class substitution. Interestingly, this
|
|
does not require any bound variable conditions on ` A ` . (Contributed
|
|
by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
cbvcsb $p |- [_ A / x ]_ C = [_ A / y ]_ D $=
|
|
( vz cv wcel wsbc cab csb nfcri weq eleq2d cbvsbc abbii df-csb 3eqtr4i )
|
|
IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
|
|
TUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y B $. $d x C $.
|
|
cbvcsbv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Change the bound variable of a proper substitution into a class using
|
|
implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by
|
|
Mario Carneiro, 13-Oct-2016.) $)
|
|
cbvcsbv $p |- [_ A / x ]_ B = [_ A / y ]_ C $=
|
|
( nfcv cbvcsb ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
csbeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for proper substitution into a class. (Contributed
|
|
by NM, 3-Dec-2005.) $)
|
|
csbeq1d $p |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
|
|
( wceq csb csbeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Analog of ~ sbid for proper substitution into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbid $p |- [_ x / x ]_ A = A $=
|
|
( vy cv csb wcel wsbc cab df-csb sbcid abbii abid2 3eqtri ) AADZBECDBFZAN
|
|
GZCHOCHBACNBIPOCOAJKCBLM $.
|
|
$}
|
|
|
|
$( Equality theorem for proper substitution into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbeq1a $p |- ( x = A -> B = [_ A / x ]_ B ) $=
|
|
( cv wceq csb csbid csbeq1 syl5eqr ) ADZBECAJCFABCFACGAJBCHI $.
|
|
|
|
${
|
|
$d z A $. $d y z B $. $d x z $.
|
|
$( Composition law for chained substitutions into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbco $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
|
|
( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbcco bitri abbii 3eqtr4i
|
|
) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
|
|
UCABCNOPBECTKAECDKQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y V $. $d x y $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). (Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
csbtt $p |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B ) $=
|
|
( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 abbi1dv
|
|
syl5eq ) BDFZACGZHZABCIEJCFZABKZELCAEBCMUCUEECUBUAUDANUEUDOAECPUDABDQRST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
csbconstgf.1 $e |- F/_ x B $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). (Contributed by NM, 10-Nov-2005.) $)
|
|
csbconstgf $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
|
|
( wcel wnfc csb wceq csbtt mpan2 ) BDFACGABCHCIEABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d B x $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). ~ csbconstgf with distinct variable requirement. (Contributed by
|
|
Alan Sare, 22-Jul-2012.) $)
|
|
csbconstg $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
|
|
( nfcv csbconstgf ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d y z B $. $d y z C $.
|
|
$( Distribute proper substitution through a membership relation.
|
|
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcel12g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
[_ A / x ]_ B e. [_ A / x ]_ C ) ) $=
|
|
( vy vz wcel wsbc cv cab csb wsb dfsbcq2 abbidv eleq12d nfs1v nfab df-csb
|
|
sbab wceq nfel weq sbie vtoclbg eleq12i syl6bbr ) BEHCDHZABIZFJZCHZABIZFK
|
|
ZUJDHZABIZFKZHZABCLZABDLZHUHAGMUKAGMZFKZUNAGMZFKZHZUIUQGBEUHAGBNGJBUAZVAU
|
|
MVCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUBAGUCCV
|
|
ADVCAGFCTAGFDTPUDUEURUMUSUPAFBCSAFBDSUFUG $.
|
|
|
|
$( Distribute proper substitution through an equality relation.
|
|
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbceqg $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
[_ A / x ]_ B = [_ A / x ]_ C ) ) $=
|
|
( vy vz wcel wceq wsbc cab csb wsb dfsbcq2 abbidv eqeq12d nfs1v nfab sbab
|
|
cv nfeq weq sbie vtoclbg df-csb eqeq12i syl6bbr ) BEHCDIZABJZFTZCHZABJZFK
|
|
ZUJDHZABJZFKZIZABCLZABDLZIUHAGMUKAGMZFKZUNAGMZFKZIZUIUQGBEUHAGBNGTBIZVAUM
|
|
VCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUAAGUBCVA
|
|
DVCAGFCSAGFDSPUCUDURUMUSUPAFBCUEAFBDUEUFUG $.
|
|
$}
|
|
|
|
$( Distribute proper substitution through negated membership. (Contributed
|
|
by Andrew Salmon, 18-Jun-2011.) $)
|
|
sbcnel12g $p |- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/
|
|
[_ A / x ]_ C ) ) $=
|
|
( wcel wnel wsbc wn csb wb df-nel sbcbii a1i sbcel12g notbid syl6bbr 3bitrd
|
|
sbcng ) BEFZCDGZABHZCDFZIZABHZUCABHZIZABCJZABDJZGZUBUEKTUAUDABCDLMNUCABESTU
|
|
GUHUIFZIUJTUFUKABCDEOPUHUILQR $.
|
|
|
|
$( Distribute proper substitution through an inequality. (Contributed by
|
|
Andrew Salmon, 18-Jun-2011.) $)
|
|
sbcne12g $p |- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/=
|
|
[_ A / x ]_ C ) ) $=
|
|
( wcel wne wsbc csb wb wceq sbceqg notbid df-ne sbcbii sbcng syl5bb bibi12d
|
|
wn a1i mpbird ) BEFZCDGZABHZABCIZABDIZGZJCDKZABHZSZUEUFKZSZJUBUIUKABCDELMUB
|
|
UDUJUGULUDUHSZABHUBUJUCUMABCDNOUHABEPQUGULJUBUEUFNTRUA $.
|
|
|
|
${
|
|
$d x C $.
|
|
$( Move proper substitution in and out of a membership relation. Note that
|
|
the scope of ` [. A / x ]. ` is the wff ` B e. C ` , whereas the scope
|
|
of ` [_ A / x ]_ ` is the class ` B ` . (Contributed by NM,
|
|
10-Nov-2005.) $)
|
|
sbcel1g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
[_ A / x ]_ B e. C ) ) $=
|
|
( wcel wsbc csb sbcel12g csbconstg eleq2d bitrd ) BEFZCDFABGABCHZABDHZFND
|
|
FABCDEIMODNABDEJKL $.
|
|
|
|
$( Move proper substitution to first argument of an equality. (Contributed
|
|
by NM, 30-Nov-2005.) $)
|
|
sbceq1g $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
[_ A / x ]_ B = C ) ) $=
|
|
( wcel wceq wsbc csb sbceqg csbconstg eqeq2d bitrd ) BEFZCDGABHABCIZABDIZ
|
|
GODGABCDEJNPDOABDEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
$( Move proper substitution in and out of a membership relation.
|
|
(Contributed by NM, 14-Nov-2005.) $)
|
|
sbcel2g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
B e. [_ A / x ]_ C ) ) $=
|
|
( wcel wsbc csb sbcel12g csbconstg eleq1d bitrd ) BEFZCDFABGABCHZABDHZFCO
|
|
FABCDEIMNCOABCEJKL $.
|
|
|
|
$( Move proper substitution to second argument of an equality.
|
|
(Contributed by NM, 30-Nov-2005.) $)
|
|
sbceq2g $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
B = [_ A / x ]_ C ) ) $=
|
|
( wcel wceq wsbc csb sbceqg csbconstg eqeq1d bitrd ) BEFZCDGABHABCIZABDIZ
|
|
GCPGABCDEJNOCPABCEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d z C $. $d x y $.
|
|
$( Commutative law for double substitution into a class. (Contributed by
|
|
NM, 14-Nov-2005.) $)
|
|
csbcomg $p |- ( ( A e. V /\ B e. W ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C ) $=
|
|
( vz wcel cvv csb wceq elex wsbc wb sbcel2g sbcbidv adantl adantr 3bitr3d
|
|
wa cv sbccom a1i eqrdv syl2an ) CFICJIZDJIZACBDEKZKZBDACEKZKZLDGICFMDGMUG
|
|
UHUAZHUJULUMHUBZUIIZACNZUNUKIZBDNZUNUJIZUNULIZUMUNEIZBDNZACNZVAACNZBDNZUP
|
|
URVCVEOUMVAABCDUCUDUHVCUPOUGUHVBUOACBDUNEJPQRUGVEUROUHUGVDUQBDACUNEJPQSTU
|
|
GUPUSOUHACUNUIJPSUHURUTOUGBDUNUKJPRTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $. $d y ph $.
|
|
csbeq2d.1 $e |- F/ x ph $.
|
|
csbeq2d.2 $e |- ( ph -> B = C ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2d $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
|
|
( vy cv wcel wsbc cab csb eleq2d sbcbid abbidv df-csb 3eqtr4g ) AHIZDJZBC
|
|
KZHLSEJZBCKZHLBCDMBCEMAUAUCHATUBBCFADESGNOPBHCDQBHCEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
csbeq2dv.1 $e |- ( ph -> B = C ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2dv $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
|
|
( nfv csbeq2d ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
csbeq2i.1 $e |- B = C $.
|
|
$( Formula-building inference rule for class substitution. (Contributed by
|
|
NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2i $p |- [_ A / x ]_ B = [_ A / x ]_ C $=
|
|
( csb wceq wtru a1i csbeq2dv trud ) ABCFABDFGHABCDCDGHEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
$( The proper substitution of a class for setvar variable results in the
|
|
class (if the class exists). (Contributed by NM, 10-Nov-2005.) $)
|
|
csbvarg $p |- ( A e. V -> [_ A / x ]_ x = A ) $=
|
|
( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv abbi1dv syl5eq
|
|
vex ax-mp csbeq2i csbco 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUDUFDHZEHZFEBLZD
|
|
MZBEBAUHUEIZIEBUHIUFUJEBUKUHUHGFZUKUHJERULUKUGUEFAUHLZDMUHADUHUENULUMDUHA
|
|
UGUHGOPQSTAEBUEUAEDBUHNUBUDUIDBEUGBGOPQUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Substitution into a wff expressed in terms of substitution into a class.
|
|
(Contributed by NM, 15-Aug-2007.) $)
|
|
sbccsbg $p |- ( A e. V ->
|
|
( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) ) $=
|
|
( wsbc cv cab wcel csb abid sbcbii sbcel2g syl5bbr ) ABDFCGZACHZIZBDFDEIO
|
|
BDPJIQABDACKLBDOPEMN $.
|
|
$}
|
|
|
|
$( Substitution into a wff expressed in using substitution into a class.
|
|
(Contributed by NM, 27-Nov-2005.) $)
|
|
sbccsb2g $p |- ( A e. V ->
|
|
( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) ) $=
|
|
( wsbc cv cab wcel csb abid sbcbii sbcel12g csbvarg eleq1d bitrd syl5bbr )
|
|
ABCEBFZABGZHZBCEZCDHZCBCRIZHZSABCABJKUATBCQIZUBHUCBCQRDLUAUDCUBBCDMNOP $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y ph $.
|
|
nfcsb1d.1 $e |- ( ph -> F/_ x A ) $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb1d $p |- ( ph -> F/_ x [_ A / x ]_ B ) $=
|
|
( vy csb cv wcel wsbc cab df-csb nfv nfsbc1d nfabd nfcxfrd ) ABBCDGFHDIZB
|
|
CJZFKBFCDLARBFAFMAQBCENOP $.
|
|
$}
|
|
|
|
${
|
|
nfcsb1.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb1 $p |- F/_ x [_ A / x ]_ B $=
|
|
( csb wnfc wtru a1i nfcsb1d trud ) AABCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) $)
|
|
nfcsb1v $p |- F/_ x [_ A / x ]_ B $=
|
|
( nfcv nfcsb1 ) ABCABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z ph $.
|
|
nfcsbd.1 $e |- F/ y ph $.
|
|
nfcsbd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfcsbd.3 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of ~ nfcsb . (Contributed by NM, 21-Nov-2005.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsbd $p |- ( ph -> F/_ x [_ A / y ]_ B ) $=
|
|
( vz csb cv wcel wsbc cab df-csb nfv nfcrd nfsbcd nfabd nfcxfrd ) ABCDEJI
|
|
KELZCDMZINCIDEOAUBBIAIPAUABCDFGABIEHQRST $.
|
|
$}
|
|
|
|
${
|
|
nfcsb.1 $e |- F/_ x A $.
|
|
nfcsb.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb $p |- F/_ x [_ A / y ]_ B $=
|
|
( csb wnfc wtru nftru a1i nfcsbd trud ) ABCDGHIABCDBJACHIEKADHIFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
csbhypf.1 $e |- F/_ x A $.
|
|
csbhypf.2 $e |- F/_ x C $.
|
|
csbhypf.3 $e |- ( x = A -> B = C ) $.
|
|
$( Introduce an explicit substitution into an implicit substitution
|
|
hypothesis. See ~ sbhypf for class substitution version. (Contributed
|
|
by NM, 19-Dec-2008.) $)
|
|
csbhypf $p |- ( y = A -> [_ y / x ]_ B = C ) $=
|
|
( cv wceq wi csb nfeq2 nfcsb1v nfeq nfim eqeq1 csbeq1a eqeq1d imbi12d
|
|
chvar ) AIZCJZDEJZKBIZCJZAUEDLZEJZKABUFUHAAUECFMAUGEAUEDNGOPUBUEJZUCUFUDU
|
|
HUBUECQUIDUGEAUEDRSTHUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Closed theorem version of ~ csbiegf .) (Contributed by NM,
|
|
11-Nov-2005.) $)
|
|
csbiebt $p |- ( ( A e. V /\ F/_ x C ) ->
|
|
( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
|
|
( wcel cvv wnfc cv wceq wi wal csb wb elex wa wsbc adantl a1i nfeqd spsbc
|
|
adantr simpl biimt csbeq1a eqeq1d bitr3d nfv nfnfc1 nfcsb1v simpr sbciedf
|
|
nfan sylibd id nfan1 biimprcd alrimi ex impbid sylan ) BEFBGFZADHZAIBJZCD
|
|
JZKZALZABCMZDJZNBEOVBVCPZVGVIVJVGVFABQZVIVBVGVKKVCVFABGUAUBVJVFVIABGVBVCU
|
|
CVDVFVINVJVDVEVFVIVDVEUDVDCVHDABCUEUFZUGRVBVCAVBAUHADUIZUMVJAVHDAVHHZVJAB
|
|
CUJZSVBVCUKTULUNVCVIVGKVBVCVIVGVCVIPVFAVCVIAVMVCAVHDVNVCVOSVCUOTUPVIVFVCV
|
|
DVEVIVLUQRURUSRUTVA $.
|
|
|
|
csbiedf.1 $e |- F/ x ph $.
|
|
csbiedf.2 $e |- ( ph -> F/_ x C ) $.
|
|
csbiedf.3 $e |- ( ph -> A e. V ) $.
|
|
csbiedf.4 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by Mario Carneiro, 13-Oct-2016.) $)
|
|
csbiedf $p |- ( ph -> [_ A / x ]_ B = C ) $=
|
|
( cv wceq wi wal csb ex alrimi wcel wnfc wb csbiebt syl2anc mpbid ) ABKCL
|
|
ZDELZMZBNZBCDOELZAUFBGAUDUEJPQACFRBESUGUHTIHBCDEFUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
csbieb.1 $e |- A e. _V $.
|
|
csbieb.2 $e |- F/_ x C $.
|
|
$( Bidirectional conversion between an implicit class substitution
|
|
hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
|
|
(Contributed by NM, 2-Mar-2008.) $)
|
|
csbieb $p |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) $=
|
|
( cvv wcel wnfc cv wceq wi wal csb wb csbiebt mp2an ) BGHADIAJBKCDKLAMABC
|
|
NDKOEFABCDGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d a x A $. $d a B $. $d a C $.
|
|
csbiebg.2 $e |- F/_ x C $.
|
|
$( Bidirectional conversion between an implicit class substitution
|
|
hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
|
|
(Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
csbiebg $p |- ( A e. V ->
|
|
( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
|
|
( va cv wceq wal csb eqeq2 imbi1d albidv csbeq1 eqeq1d vex csbieb vtoclbg
|
|
wi ) AHZGHZIZCDIZTZAJAUBCKZDIUABIZUDTZAJABCKZDIGBEUBBIZUEUHAUJUCUGUDUBBUA
|
|
LMNUJUFUIDAUBBCOPAUBCDGQFRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x V $.
|
|
csbiegf.1 $e |- ( A e. V -> F/_ x C ) $.
|
|
csbiegf.2 $e |- ( x = A -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
csbiegf $p |- ( A e. V -> [_ A / x ]_ B = C ) $=
|
|
( wcel cv wceq wi wal csb ax-gen wnfc wb csbiebt mpdan mpbii ) BEHZAIBJCD
|
|
JKZALZABCMDJZUAAGNTADOUBUCPFABCDEQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
csbief.1 $e |- A e. _V $.
|
|
csbief.2 $e |- F/_ x C $.
|
|
csbief.3 $e |- ( x = A -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
csbief $p |- [_ A / x ]_ B = C $=
|
|
( cvv wcel csb wceq wnfc a1i csbiegf ax-mp ) BHIZABCJDKEABCDHADLPFMGNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x ph $.
|
|
csbied.1 $e |- ( ph -> A e. V ) $.
|
|
csbied.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario
|
|
Carneiro, 13-Oct-2016.) $)
|
|
csbied $p |- ( ph -> [_ A / x ]_ B = C ) $=
|
|
( nfv nfcvd csbiedf ) ABCDEFABIABEJGHK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x D $.
|
|
csbied2.1 $e |- ( ph -> A e. V ) $.
|
|
csbied2.2 $e |- ( ph -> A = B ) $.
|
|
csbied2.3 $e |- ( ( ph /\ x = B ) -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) $)
|
|
csbied2 $p |- ( ph -> [_ A / x ]_ C = D ) $=
|
|
( cv wceq id sylan9eqr syldan csbied ) ABCEFGHABKZCLZQDLEFLRAQCDRMINJOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y D $.
|
|
csbie2t.1 $e |- A e. _V $.
|
|
csbie2t.2 $e |- B e. _V $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class (closed form of ~ csbie2 ). (Contributed by NM, 3-Sep-2007.)
|
|
(Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
csbie2t $p |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = D ) $=
|
|
( cv wceq wa wi wal csb cvv nfa1 nfcvd wcel a1i csbiedf nfa2 nfv nfan sps
|
|
sp impl ) AICJZBIDJZKEFJZLZBMZAMZACBDENFOUKAPULAFQCORULGSULUGKZBDEFOULUGB
|
|
UJBAUAUGBUBUCUMBFQDORUMHSULUGUHUIUKUJAUJBUEUDUFTT $.
|
|
|
|
csbie2.3 $e |- ( ( x = A /\ y = B ) -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 27-Aug-2007.) $)
|
|
csbie2 $p |- [_ A / x ]_ [_ B / y ]_ C = D $=
|
|
( cv wceq wa wi wal csb gen2 csbie2t ax-mp ) AJCKBJDKLEFKMZBNANACBDEOOFKS
|
|
ABIPABCDEFGHQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A y z $. $d B y z $. $d C x $. $d D y z $. $d V z $.
|
|
csbie2g.1 $e |- ( x = y -> B = C ) $.
|
|
csbie2g.2 $e |- ( y = A -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
This version of ~ sbcie avoids a disjointness condition on ` x ` and
|
|
` A ` by substituting twice. (Contributed by Mario Carneiro,
|
|
11-Nov-2016.) $)
|
|
csbie2g $p |- ( A e. V -> [_ A / x ]_ B = D ) $=
|
|
( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g abbi1dv syl5eq ) CGK
|
|
ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z C $. $d z ph $.
|
|
$( Nest the composition of two substitutions. (Contributed by Mario
|
|
Carneiro, 11-Nov-2016.) $)
|
|
sbcnestgf $p |- ( ( A e. V /\ A. y F/ x ph ) ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
|
|
( vz wcel wnf wal wsbc csb wb cv wi wceq dfsbcq syl cvv a1i csbeq1 imbi2d
|
|
bibi12d vex csbeq1a adantl nfnf1 nfal nfa1 wnfc nfcsb1v sp nfsbcd sbciedf
|
|
vtoclg imp ) DFHABIZCJZACEKZBDKZACBDELZKZMZURUSBGNZKZACBVDELZKZMZOURVCOGD
|
|
FVDDPZVHVCURVIVEUTVGVBUSBVDDQVIVFVAPVGVBMBVDDEUAACVFVAQRUCUBURUSVGBVDSVDS
|
|
HURGUDTBNVDPZUSVGMZURVJEVFPVKBVDEUEACEVFQRUFUQBCABUGUHURABCVFUQCUIBVFUJUR
|
|
BVDEUKTUQCULUMUNUOUP $.
|
|
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
|
|
csbnestgf $p |- ( ( A e. V /\ A. y F/_ x C ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
|
|
( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
|
|
sbcbii wnf alimi sbcnestgf sylan2 syl5bb abbidv sylan 3eqtr4g ) CFHZAEIZB
|
|
JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
|
|
URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
|
|
FUGUHUIUJAGCUPSBGVAESUK $.
|
|
|
|
$d x ph $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
|
|
sbcnestg $p |- ( A e. V ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
|
|
( wcel wnf wal wsbc csb wb nfv ax-gen sbcnestgf mpan2 ) DFGABHZCIACEJBDJA
|
|
CBDEKJLQCABMNABCDEFOP $.
|
|
|
|
$d x C $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
|
|
csbnestg $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
|
|
( wcel wnfc wal csb wceq nfcv ax-gen csbnestgf mpan2 ) CFGAEHZBIACBDEJJBA
|
|
CDJEJKPBAELMABCDEFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y C $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
|
|
csbnest1g $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C ) $=
|
|
( vy wcel cv csb wnfc wceq nfcsb1v ax-gen csbnestgf mpan2 csbeq2i 3eqtr3g
|
|
wal csbco ) BEGZABFCAFHZDIZIZIZFABCIZUBIZABACDIZIAUEDITAUBJZFRUDUFKUHFAUA
|
|
DLMAFBCUBENOABUCUGAFCDSPAFUEDSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for class substitutions. (Contributed by NM,
|
|
1-Mar-2008.) $)
|
|
csbidmg $p |- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B ) $=
|
|
( wcel cvv csb wceq elex csbnest1g csbconstg csbeq1d eqtrd syl ) BDEBFEZA
|
|
BABCGZGZPHBDIOQAABBGZCGPABBCFJOARBCABBFKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x C $. $d x D $.
|
|
sbcco3g.1 $e |- ( x = A -> B = C ) $.
|
|
$( Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
|
|
(Revised by Mario Carneiro, 11-Nov-2016.) $)
|
|
sbcco3g $p |- ( A e. V ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $=
|
|
( wcel wsbc csb sbcnestg cvv wceq wb elex nfcvd csbiegf dfsbcq 3syl bitrd
|
|
) DGIZACEJBDJACBDEKZJZACFJZABCDEGLUBDMIZUCFNUDUEODGPBDEFMUFBFQHRACUCFSTUA
|
|
$.
|
|
|
|
$( Composition of two class substitutions. (Contributed by NM,
|
|
27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) $)
|
|
csbco3g $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D ) $=
|
|
( wcel csb csbnestg cvv wceq elex nfcvd csbiegf syl csbeq1d eqtrd ) CGIZA
|
|
CBDFJJBACDJZFJBEFJABCDFGKTBUAEFTCLIZUAEMCGNACDELUBAEOHPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x D $.
|
|
$( Special case related to ~ rspsbc . (Contributed by NM, 10-Dec-2005.)
|
|
(Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
|
|
rspcsbela $p |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D ) $=
|
|
( wcel wral csb wsbc rspsbc sbcel1g sylibd imp ) BCFZDEFZACGZABDHEFZNPOAB
|
|
IQOABCJABDECKLM $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d w y z A $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` A ` ."
|
|
(Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
sbnfc2 $p |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A ) $=
|
|
( vw cv csb wceq wal cvv wcel vex csbtt mpan wsb wsbc sbsbc sbcel2g ax-mp
|
|
wb eqtr4d alrimivv nfv wnf eleq2 bitri 3bitr4g 2alimi sbnf2 sylibr impbii
|
|
wnfc nfcd ) ADULZABFZDGZACFZDGZHZCIBIZUNUSBCUNUPDURUOJKZUNUPDHBLZAUODJMNU
|
|
QJKZUNURDHCLZAUQDJMNUAUBUTAEDUTEUCUTEFZDKZABOZVFACOZTZCIBIVFAUDUSVIBCUSVE
|
|
UPKZVEURKZVGVHUPURVEUEVGVFAUOPZVJVFABQVAVLVJTVBAUOVEDJRSUFVHVFAUQPZVKVFAC
|
|
QVCVMVKTVDAUQVEDJRSUFUGUHVFABCUIUJUMUK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d z ph $. $d x y z $. $d V z $.
|
|
$( Move substitution into a class abstraction. (Contributed by NM,
|
|
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
csbabg $p |- ( A e. V ->
|
|
[_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } ) $=
|
|
( vz wcel cab csb wsbc cv sbccom wsb df-clab sbsbc sbcbii 3bitr4i sbcel2g
|
|
bitri syl5rbb eqrdv ) DEGZFBDACHZIZABDJZCHZFKZUFGZUGUCGZBDJZUBUGUDGUECUGJ
|
|
ZACUGJZBDJUHUJACBUGDLUHUECFMUKUEFCNUECFOSUIULBDUIACFMULAFCNACFOSPQBDUGUCE
|
|
RTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x v z $. $d y v z $. $d A v z $. $d B v z $. $d ph v z $.
|
|
$d ps v z $.
|
|
cbvralcsf.1 $e |- F/_ y A $.
|
|
cbvralcsf.2 $e |- F/_ x B $.
|
|
cbvralcsf.3 $e |- F/ y ph $.
|
|
cbvralcsf.4 $e |- F/ x ps $.
|
|
cbvralcsf.5 $e |- ( x = y -> A = B ) $.
|
|
cbvralcsf.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A more general version of ~ cbvralf that doesn't require ` A ` and ` B `
|
|
to be distinct from ` x ` or ` y ` . Changes bound variables using
|
|
implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvralcsf $p |- ( A. x e. A ph <-> A. y e. B ps ) $=
|
|
( vz vv cv wcel wi wal wsbc nfcri wral csb nfv nfcsb1v nfsbc1v id csbeq1a
|
|
nfim weq eleq12d sbceq1a imbi12d cbval nfcv nfcsb nfsbc csbeq1 cab df-csb
|
|
eleq2d sbie sbsbc bitr3i abbi2i eqtr4i syl6eq dfsbcq syl6bb bitri 3bitr4i
|
|
wsb df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRVT
|
|
VOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWAU
|
|
KULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUPUHVSMUCMDUIZWCVRWDBWHWAVQWBF
|
|
WHUFWHWBCVQEUBZFCWAVQEUQWINOZEPZCVQSZNURFCNVQEUSWLNFWJFPZWKCDVKWLWKWMCDCN
|
|
FHTCDUIEFWJKUTVAWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDVKBACDVBABCDJLVAV
|
|
CVHULUMVIACEVLBDFVLVJ $.
|
|
|
|
$( A more general version of ~ cbvrexf that has no distinct variable
|
|
restrictions. Changes bound variables using implicit substitution.
|
|
(Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario
|
|
Carneiro, 7-Dec-2014.) $)
|
|
cbvrexcsf $p |- ( E. x e. A ph <-> E. y e. B ps ) $=
|
|
( vz vv cv wcel wa wex wsbc nfcri wrex csb nfv nfcsb1v nfsbc1v id csbeq1a
|
|
nfan weq eleq12d sbceq1a anbi12d cbvex nfcv nfcsb nfsbc csbeq1 cab df-csb
|
|
eleq2d sbie sbsbc bitr3i abbi2i eqtr4i syl6eq dfsbcq syl6bb bitri 3bitr4i
|
|
wsb df-rex ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRVT
|
|
VOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWAU
|
|
KULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUPUHVSMUCMDUIZWCVRWDBWHWAVQWBF
|
|
WHUFWHWBCVQEUBZFCWAVQEUQWINOZEPZCVQSZNURFCNVQEUSWLNFWJFPZWKCDVKWLWKWMCDCN
|
|
FHTCDUIEFWJKUTVAWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDVKBACDVBABCDJLVAV
|
|
CVHULUMVIACEVLBDFVLVJ $.
|
|
|
|
$( A more general version of ~ cbvreuv that has no distinct variable
|
|
rextrictions. Changes bound variables using implicit substitution.
|
|
(Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvreucsf $p |- ( E! x e. A ph <-> E! y e. B ps ) $=
|
|
( vz vv cv wcel wa weu wsb nfcri wreu csb nfv nfcsb1v nfan weq id csbeq1a
|
|
nfs1v eleq12d sbequ12 anbi12d cbveu nfcv nfcsb nfsb csbeq1 cab wsbc sbsbc
|
|
abbii eleq2d sbie bicomi abbi2i df-csb 3eqtr4ri syl6eq sbequ syl6bb bitri
|
|
df-reu 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZM
|
|
RWAVPWFCMVPMUCWDWECCMWCCWBEUDTACMUIUECMUFZVOWDAWEWGVNWBEWCWGUGCWBEUHUJACM
|
|
UKULUMWFVTMDWDWEDDMWCDCWBEDWBUNGUOTACMDIUPUEVTMUCMDUFZWDVSWEBWHWBVRWCFWHU
|
|
GWHWCCVREUBZFCWBVREUQNOZEPZCDSZNURWKCVRUSZNURFWIWLWMNWKCDUTVAWLNFWLWJFPZW
|
|
KWNCDCNFHTCDUFEFWJKVBVCVDVECNVREVFVGVHUJWHWEACDSBAMDCVIABCDJLVCVJULUMVKAC
|
|
EVLBDFVLVM $.
|
|
|
|
$( A more general version of ~ cbvrab with no distinct variable
|
|
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvrabcsf $p |- { x e. A | ph } = { y e. B | ps } $=
|
|
( vz vv cv wcel wa cab wsb nfcri crab csb nfv nfcsb1v nfan weq id csbeq1a
|
|
nfs1v eleq12d sbequ12 anbi12d cbvab nfcv nfcsb csbeq1 df-csb eleq2d sbsbc
|
|
nfsb wsbc bitr3i abbi2i eqtr4i syl6eq sbequ syl6bb eqtri df-rab 3eqtr4i
|
|
sbie ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVOMOZCVTEUBZPZACMSZQZMRVSVNWDCMV
|
|
NMUCWBWCCCMWACVTEUDTACMUIUECMUFZVMWBAWCWEVLVTEWAWEUGCVTEUHUJACMUKULUMWDVR
|
|
MDWBWCDDMWADCVTEDVTUNGUOTACMDIUTUEVRMUCMDUFZWBVQWCBWFVTVPWAFWFUGWFWACVPEU
|
|
BZFCVTVPEUPWGNOZEPZCVPVAZNRFCNVPEUQWJNFWHFPZWICDSWJWIWKCDCNFHTCDUFEFWHKUR
|
|
VKWICDUSVBVCVDVEUJWFWCACDSBAMDCVFABCDJLVKVGULUMVHACEVIBDFVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ps y $. $d B x $. $d ch x $.
|
|
cbvralv2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
cbvralv2.2 $e |- ( x = y -> A = B ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvralv2 $p |- ( A. x e. A ps <-> A. y e. B ch ) $=
|
|
( nfcv nfv cbvralcsf ) ABCDEFDEICFIADJBCJHGK $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvrexv2 $p |- ( E. x e. A ps <-> E. y e. B ch ) $=
|
|
( nfcv nfv cbvrexcsf ) ABCDEFDEICFIADJBCJHGK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Define basic set operations and relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols. $)
|
|
$c \ $. $( Backslash (difference) $)
|
|
$c u. $. $( Cup (union) $)
|
|
$c i^i $. $( Cap (intersection) $)
|
|
$c C_ $. $( Subclass or subset symbol $)
|
|
|
|
$( Extend class notation to include class difference (read: " ` A ` minus
|
|
` B ` "). $)
|
|
cdif $a class ( A \ B ) $.
|
|
|
|
$( Extend class notation to include union of two classes (read: " ` A `
|
|
union ` B ` "). $)
|
|
cun $a class ( A u. B ) $.
|
|
|
|
$( Extend class notation to include the intersection of two classes (read:
|
|
" ` A ` intersect ` B ` "). $)
|
|
cin $a class ( A i^i B ) $.
|
|
|
|
$( Extend wff notation to include the subclass relation. This is
|
|
read " ` A ` is a subclass of ` B ` " or " ` B ` includes ` A ` ." When
|
|
` A ` exists as a set, it is also read " ` A ` is a subset of ` B ` ." $)
|
|
wss $a wff A C_ B $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d y A $. $d y B $. $d z x $. $d z y $. $d z A $.
|
|
$d z B $.
|
|
$( Soundness justification theorem for ~ df-dif . (Contributed by Rodolfo
|
|
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
difjust $p |- { x | ( x e. A /\ -. x e. B ) }
|
|
= { y | ( y e. A /\ -. y e. B ) } $=
|
|
( vz cv wcel wn wa cab weq eleq1 notbid anbi12d cbvabv eqtri ) AFZCGZQDGZ
|
|
HZIZAJEFZCGZUBDGZHZIZEJBFZCGZUGDGZHZIZBJUAUFAEAEKZRUCTUEQUBCLULSUDQUBDLMN
|
|
OUFUKEBEBKZUCUHUEUJUBUGCLUMUDUIUBUGDLMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Define class difference, also called relative complement. Definition
|
|
5.12 of [TakeutiZaring] p. 20. Contrast this operation with union
|
|
` ( A u. B ) ` ( ~ df-un ) and intersection ` ( A i^i B ) ` ( ~ df-in ).
|
|
Several notations are used in the literature; we chose the ` \ `
|
|
convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the
|
|
more common minus sign to reserve the latter for later use in, e.g.,
|
|
arithmetic. We will use the terminology " ` A ` excludes ` B ` " to
|
|
mean ` A \ B ` . We will use " ` B ` is removed from ` A ` " to mean
|
|
` A \ { B } ` i.e. the removal of an element or equivalently the
|
|
exclusion of a singleton. (Contributed by NM, 29-Apr-1994.) $)
|
|
df-dif $a |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d y A $. $d y B $. $d z x $. $d z y $. $d z A $.
|
|
$d z B $.
|
|
$( Soundness justification theorem for ~ df-un . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
unjust $p |- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) } $=
|
|
( vz cv wcel wo cab weq eleq1 orbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCGZ
|
|
SDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAOSCKOSDKLMUBUFEBEBJTUDUAUESUCCKSUCDKL
|
|
MN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Define the union of two classes. Definition 5.6 of [TakeutiZaring]
|
|
p. 16. Contrast this operation with difference ` ( A \ B ) `
|
|
( ~ df-dif ) and intersection ` ( A i^i B ) ` ( ~ df-in ). (Contributed
|
|
by NM, 23-Aug-1993.) $)
|
|
df-un $a |- ( A u. B ) = { x | ( x e. A \/ x e. B ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d y A $. $d y B $. $d z x $. $d z y $. $d z A $.
|
|
$d z B $.
|
|
$( Soundness justification theorem for ~ df-in . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
injust $p |- { x | ( x e. A /\ x e. B ) }
|
|
= { y | ( y e. A /\ y e. B ) } $=
|
|
( vz cv wcel wa cab weq eleq1 anbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCGZ
|
|
SDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAOSCKOSDKLMUBUFEBEBJTUDUAUESUCCKSUCDKL
|
|
MN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Define the intersection of two classes. Definition 5.6 of
|
|
[TakeutiZaring] p. 16. Contrast this operation with union
|
|
` ( A u. B ) ` ( ~ df-un ) and difference ` ( A \ B ) ` ( ~ df-dif ).
|
|
(Contributed by NM, 29-Apr-1994.) $)
|
|
df-in $a |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) } $.
|
|
|
|
$( Alternate definition for the intersection of two classes. (Contributed
|
|
by NM, 6-Jul-2005.) $)
|
|
dfin5 $p |- ( A i^i B ) = { x e. A | x e. B } $=
|
|
( cin cv wcel wa cab crab df-in df-rab eqtr4i ) BCDAEZBFMCFZGAHNABIABCJNA
|
|
BKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of class difference. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
dfdif2 $p |- ( A \ B ) = { x e. A | -. x e. B } $=
|
|
( cdif cv wcel wn wa cab crab df-dif df-rab eqtr4i ) BCDAEZBFNCFGZHAIOABJ
|
|
ABCKOABLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Expansion of membership in a class difference. (Contributed by NM,
|
|
29-Apr-1994.) $)
|
|
eldif $p |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) ) $=
|
|
( vx cdif wcel cvv wn wa elex adantr cv wceq notbid anbi12d df-dif elab2g
|
|
eleq1 pm5.21nii ) ABCEZFAGFZABFZACFZHZIZATJUBUAUDABJKDLZBFZUFCFZHZIUEDATG
|
|
UFAMZUGUBUIUDUFABRUJUHUCUFACRNODBCPQS $.
|
|
$}
|
|
|
|
${
|
|
eldifd.1 $e |- ( ph -> A e. B ) $.
|
|
eldifd.2 $e |- ( ph -> -. A e. C ) $.
|
|
$( If a class is in one class and not another, it is also in their
|
|
difference. One-way deduction form of ~ eldif . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
eldifd $p |- ( ph -> A e. ( B \ C ) ) $=
|
|
( wcel wn cdif eldif sylanbrc ) ABCGBDGHBCDIGEFBCDJK $.
|
|
$}
|
|
|
|
${
|
|
eldifad.1 $e |- ( ph -> A e. ( B \ C ) ) $.
|
|
$( If a class is in the difference of two classes, it is also in the
|
|
minuend. One-way deduction form of ~ eldif . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
eldifad $p |- ( ph -> A e. B ) $=
|
|
( wcel wn cdif wa eldif sylib simpld ) ABCFZBDFGZABCDHFMNIEBCDJKL $.
|
|
$}
|
|
|
|
${
|
|
eldifbd.1 $e |- ( ph -> A e. ( B \ C ) ) $.
|
|
$( If a class is in the difference of two classes, it is not in the
|
|
subtrahend. One-way deduction form of ~ eldif . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
eldifbd $p |- ( ph -> -. A e. C ) $=
|
|
( wcel wn cdif wa eldif sylib simprd ) ABCFZBDFGZABCDHFMNIEBCDJKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Subclasses and subsets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
|
|
Note that ` A C_ A ` (proved in ~ ssid ). For a more traditional
|
|
definition, but requiring a dummy variable, see ~ dfss2 . Other possible
|
|
definitions are given by ~ dfss3 , ~ ssequn1 , ~ ssequn2 , and ~ sseqin2 .
|
|
(Contributed by NM, 27-Apr-1994.) $)
|
|
df-ss $a |- ( A C_ B <-> ( A i^i B ) = A ) $.
|
|
|
|
$( Variant of subclass definition ~ df-ss . (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
dfss $p |- ( A C_ B <-> A = ( A i^i B ) ) $=
|
|
( wss cin wceq df-ss eqcom bitri ) ABCABDZAEAIEABFIAGH $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of the subclass relationship between two classes.
|
|
Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
dfss2 $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
|
|
( wss cv wcel wa wb wal wi cin wceq dfss df-in eqeq2i abeq2 3bitri pm4.71
|
|
cab albii bitr4i ) BCDZAEZBFZUDUCCFZGZHZAIZUDUEJZAIUBBBCKZLBUFASZLUHBCMUJ
|
|
UKBABCNOUFABPQUIUGAUDUERTUA $.
|
|
|
|
$( Alternate definition of subclass relationship. (Contributed by NM,
|
|
14-Oct-1999.) $)
|
|
dfss3 $p |- ( A C_ B <-> A. x e. A x e. B ) $=
|
|
( wss cv wcel wi wal wral dfss2 df-ral bitr4i ) BCDAEZBFMCFZGAHNABIABCJNA
|
|
BKL $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x z $.
|
|
dfss2f.1 $e |- F/_ x A $.
|
|
dfss2f.2 $e |- F/_ x B $.
|
|
$( Equivalence for subclass relation, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) $)
|
|
dfss2f $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
|
|
( vz wss cv wcel wal dfss2 nfcri nfim nfv weq eleq1 imbi12d cbval bitri
|
|
wi ) BCGFHZBIZUACIZTZFJAHZBIZUECIZTZAJFBCKUDUHFAUBUCAAFBDLAFCELMUHFNFAOUB
|
|
UFUCUGUAUEBPUAUECPQRS $.
|
|
|
|
$( Equivalence for subclass relation, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
20-Mar-2004.) $)
|
|
dfss3f $p |- ( A C_ B <-> A. x e. A x e. B ) $=
|
|
( wss cv wcel wi wal wral dfss2f df-ral bitr4i ) BCFAGZBHOCHZIAJPABKABCDE
|
|
LPABMN $.
|
|
|
|
$( If ` x ` is not free in ` A ` and ` B ` , it is not free in ` A C_ B ` .
|
|
(Contributed by NM, 27-Dec-1996.) $)
|
|
nfss $p |- F/ x A C_ B $=
|
|
( wss cv wcel wral dfss3f nfra1 nfxfr ) BCFAGCHZABIAABCDEJMABKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership relationships follow from a subclass relationship.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ssel $p |- ( A C_ B -> ( C e. A -> C e. B ) ) $=
|
|
( vx wss cv wceq wa wex wi wal dfss2 biimpi 19.21bi anim2d eximdv df-clel
|
|
wcel 3imtr4g ) ABEZDFZCGZUAARZHZDIUBUABRZHZDICARCBRTUDUFDTUCUEUBTUCUEJZDT
|
|
UGDKDABLMNOPDCAQDCBQS $.
|
|
$}
|
|
|
|
$( Membership relationships follow from a subclass relationship.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
ssel2 $p |- ( ( A C_ B /\ C e. A ) -> C e. B ) $=
|
|
( wss wcel ssel imp ) ABDCAECBEABCFG $.
|
|
|
|
${
|
|
sseli.1 $e |- A C_ B $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sseli $p |- ( C e. A -> C e. B ) $=
|
|
( wss wcel wi ssel ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
sselii.2 $e |- C e. A $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
sselii $p |- C e. B $=
|
|
( wcel sseli ax-mp ) CAFCBFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
sseldi.2 $e |- ( ph -> C e. A ) $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
25-Jun-2014.) $)
|
|
sseldi $p |- ( ph -> C e. B ) $=
|
|
( wcel sseli syl ) ADBGDCGFBCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
sseld.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Membership deduction from subclass relationship. (Contributed by NM,
|
|
15-Nov-1995.) $)
|
|
sseld $p |- ( ph -> ( C e. A -> C e. B ) ) $=
|
|
( wss wcel wi ssel syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
$( Membership deduction from subclass relationship. (Contributed by NM,
|
|
26-Jun-2014.) $)
|
|
sselda $p |- ( ( ph /\ C e. A ) -> C e. B ) $=
|
|
( wcel sseld imp ) ADBFDCFABCDEGH $.
|
|
|
|
${
|
|
sseldd.2 $e |- ( ph -> C e. A ) $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
14-Dec-2004.) $)
|
|
sseldd $p |- ( ph -> C e. B ) $=
|
|
( wcel sseld mpd ) ADBGDCGFABCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ssneld.1 $e |- ( ph -> A C_ B ) $.
|
|
$( If a class is not in another class, it is also not in a subclass of that
|
|
class. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
ssneld $p |- ( ph -> ( -. C e. B -> -. C e. A ) ) $=
|
|
( wcel sseld con3d ) ADBFDCFABCDEGH $.
|
|
|
|
ssneldd.2 $e |- ( ph -> -. C e. B ) $.
|
|
$( If an element is not in a class, it is also not in a subclass of that
|
|
class. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
ssneldd $p |- ( ph -> -. C e. A ) $=
|
|
( wcel wn ssneld mpd ) ADCGHDBGHFABCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
ssriv.1 $e |- ( x e. A -> x e. B ) $.
|
|
$( Inference rule based on subclass definition. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ssriv $p |- A C_ B $=
|
|
( wss cv wcel wi dfss2 mpgbir ) BCEAFZBGKCGHAABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
ssrd.0 $e |- F/ x ph $.
|
|
ssrd.1 $e |- F/_ x A $.
|
|
ssrd.2 $e |- F/_ x B $.
|
|
ssrd.3 $e |- ( ph -> ( x e. A -> x e. B ) ) $.
|
|
$( Deduction rule based on subclass definition. (Contributed by Thierry
|
|
Arnoux, 8-Mar-2017.) $)
|
|
ssrd $p |- ( ph -> A C_ B ) $=
|
|
( cv wcel wi wal wss alrimi dfss2f sylibr ) ABIZCJQDJKZBLCDMARBEHNBCDFGOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
ssrdv.1 $e |- ( ph -> ( x e. A -> x e. B ) ) $.
|
|
$( Deduction rule based on subclass definition. (Contributed by NM,
|
|
15-Nov-1995.) $)
|
|
ssrdv $p |- ( ph -> A C_ B ) $=
|
|
( cv wcel wi wal wss alrimiv dfss2 sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
sstr2 $p |- ( A C_ B -> ( B C_ C -> A C_ C ) ) $=
|
|
( vx wss cv wcel wi wal ssel imim1d alimdv dfss2 3imtr4g ) ABEZDFZBGZPCGZ
|
|
HZDIPAGZRHZDIBCEACEOSUADOTQRABPJKLDBCMDACMN $.
|
|
$}
|
|
|
|
$( Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
|
|
NM, 5-Sep-2003.) $)
|
|
sstr $p |- ( ( A C_ B /\ B C_ C ) -> A C_ C ) $=
|
|
( wss sstr2 imp ) ABDBCDACDABCEF $.
|
|
|
|
${
|
|
sstri.1 $e |- A C_ B $.
|
|
sstri.2 $e |- B C_ C $.
|
|
$( Subclass transitivity inference. (Contributed by NM, 5-May-2000.) $)
|
|
sstri $p |- A C_ C $=
|
|
( wss sstr2 mp2 ) ABFBCFACFDEABCGH $.
|
|
$}
|
|
|
|
${
|
|
sstrd.1 $e |- ( ph -> A C_ B ) $.
|
|
sstrd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.) $)
|
|
sstrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sstr syl2anc ) ABCGCDGBDGEFBCDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5ss.1 $e |- A C_ B $.
|
|
syl5ss.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.) $)
|
|
syl5ss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i sstrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl6ss.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6ss.2 $e |- B C_ C $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl6ss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i sstrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
sylan9ss.1 $e |- ( ph -> A C_ B ) $.
|
|
sylan9ss.2 $e |- ( ps -> B C_ C ) $.
|
|
$( A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
sylan9ss $p |- ( ( ph /\ ps ) -> A C_ C ) $=
|
|
( wss sstr syl2an ) ACDHDEHCEHBFGCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9ssr.1 $e |- ( ph -> A C_ B ) $.
|
|
sylan9ssr.2 $e |- ( ps -> B C_ C ) $.
|
|
$( A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) $)
|
|
sylan9ssr $p |- ( ( ps /\ ph ) -> A C_ C ) $=
|
|
( wss sylan9ss ancoms ) ABCEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The subclass relationship is antisymmetric. Compare Theorem 4 of
|
|
[Suppes] p. 22. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqss $p |- ( A = B <-> ( A C_ B /\ B C_ A ) ) $=
|
|
( vx cv wcel wb wal wi wa wceq wss albiim dfcleq dfss2 anbi12i 3bitr4i )
|
|
CDZAEZQBEZFCGRSHCGZSRHCGZIABJABKZBAKZIRSCLCABMUBTUCUACABNCBANOP $.
|
|
$}
|
|
|
|
${
|
|
eqssi.1 $e |- A C_ B $.
|
|
eqssi.2 $e |- B C_ A $.
|
|
$( Infer equality from two subclass relationships. Compare Theorem 4 of
|
|
[Suppes] p. 22. (Contributed by NM, 9-Sep-1993.) $)
|
|
eqssi $p |- A = B $=
|
|
( wceq wss eqss mpbir2an ) ABEABFBAFCDABGH $.
|
|
$}
|
|
|
|
${
|
|
eqssd.1 $e |- ( ph -> A C_ B ) $.
|
|
eqssd.2 $e |- ( ph -> B C_ A ) $.
|
|
$( Equality deduction from two subclass relationships. Compare Theorem 4
|
|
of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.) $)
|
|
eqssd $p |- ( ph -> A = B ) $=
|
|
( wss wceq eqss sylanbrc ) ABCFCBFBCGDEBCHI $.
|
|
$}
|
|
|
|
${
|
|
eqrd.0 $e |- F/ x ph $.
|
|
eqrd.1 $e |- F/_ x A $.
|
|
eqrd.2 $e |- F/_ x B $.
|
|
eqrd.3 $e |- ( ph -> ( x e. A <-> x e. B ) ) $.
|
|
$( Deduce equality of classes from equivalence of membership. (Contributed
|
|
by Thierry Arnoux, 21-Mar-2017.) $)
|
|
eqrd $p |- ( ph -> A = B ) $=
|
|
( cv wcel biimpd ssrd biimprd eqssd ) ACDABCDEFGABIZCJZODJZHKLABDCEGFAPQH
|
|
MLN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Any class is a subclass of itself. Exercise 10 of [TakeutiZaring]
|
|
p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 14-Jun-2011.) $)
|
|
ssid $p |- A C_ A $=
|
|
( vx cv wcel id ssriv ) BAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Any class is a subclass of the universal class. (Contributed by NM,
|
|
31-Oct-1995.) $)
|
|
ssv $p |- A C_ _V $=
|
|
( vx cvv cv elex ssriv ) BACBDAEF $.
|
|
$}
|
|
|
|
$( Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
sseq1 $p |- ( A = B -> ( A C_ C <-> B C_ C ) ) $=
|
|
( wceq wss wa wb eqss wi sstr2 adantl adantr impbid sylbi ) ABDABEZBAEZFZAC
|
|
EZBCEZGABHQRSPRSIOBACJKOSRIPABCJLMN $.
|
|
|
|
$( Equality theorem for the subclass relationship. (Contributed by NM,
|
|
25-Jun-1998.) $)
|
|
sseq2 $p |- ( A = B -> ( C C_ A <-> C C_ B ) ) $=
|
|
( wss wa wi wceq wb sstr2 com12 anim12i eqss dfbi2 3imtr4i ) ABDZBADZECADZC
|
|
BDZFZRQFZEABGQRHOSPTQORCABIJRPQCBAIJKABLQRMN $.
|
|
|
|
$( Equality theorem for the subclass relationship. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
sseq12 $p |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) ) $=
|
|
( wceq wss sseq1 sseq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.
|
|
|
|
${
|
|
sseq1i.1 $e |- A = B $.
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
sseq1i $p |- ( A C_ C <-> B C_ C ) $=
|
|
( wceq wss wb sseq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
sseq2i $p |- ( C C_ A <-> C C_ B ) $=
|
|
( wceq wss wb sseq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
sseq12i.2 $e |- C = D $.
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
sseq12i $p |- ( A C_ C <-> B C_ D ) $=
|
|
( wceq wss wb sseq12 mp2an ) ABGCDGACHBDHIEFABCDJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
sseq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 14-Aug-1994.) $)
|
|
sseq1d $p |- ( ph -> ( A C_ C <-> B C_ C ) ) $=
|
|
( wceq wss wb sseq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 14-Aug-1994.) $)
|
|
sseq2d $p |- ( ph -> ( C C_ A <-> C C_ B ) ) $=
|
|
( wceq wss wb sseq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
${
|
|
sseq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 31-May-1999.) $)
|
|
sseq12d $p |- ( ph -> ( A C_ C <-> B C_ D ) ) $=
|
|
( wss sseq1d sseq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
eqsstr.1 $e |- A = B $.
|
|
eqsstr.2 $e |- B C_ C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 16-Jul-1995.) $)
|
|
eqsstri $p |- A C_ C $=
|
|
( wss sseq1i mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
eqsstr3.1 $e |- B = A $.
|
|
eqsstr3.2 $e |- B C_ C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 19-Oct-1999.) $)
|
|
eqsstr3i $p |- A C_ C $=
|
|
( eqcomi eqsstri ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sseqtr.1 $e |- A C_ B $.
|
|
sseqtr.2 $e |- B = C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 28-Jul-1995.) $)
|
|
sseqtri $p |- A C_ C $=
|
|
( wss sseq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
sseqtr4.1 $e |- A C_ B $.
|
|
sseqtr4.2 $e |- C = B $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 4-Apr-1995.) $)
|
|
sseqtr4i $p |- A C_ C $=
|
|
( eqcomi sseqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
eqsstrd.1 $e |- ( ph -> A = B ) $.
|
|
eqsstrd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
eqsstrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq1d mpbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqsstr3d.1 $e |- ( ph -> B = A ) $.
|
|
eqsstr3d.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
eqsstr3d $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd eqsstrd ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
sseqtrd.1 $e |- ( ph -> A C_ B ) $.
|
|
sseqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
sseqtrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
sseqtr4d.1 $e |- ( ph -> A C_ B ) $.
|
|
sseqtr4d.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
sseqtr4d $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd sseqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3sstr3.1 $e |- A C_ B $.
|
|
3sstr3.2 $e |- A = C $.
|
|
3sstr3.3 $e |- B = D $.
|
|
$( Substitution of equality in both sides of a subclass relationship.
|
|
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr3i $p |- C C_ D $=
|
|
( wss sseq12i mpbi ) ABHCDHEACBDFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3sstr4.1 $e |- A C_ B $.
|
|
3sstr4.2 $e |- C = A $.
|
|
3sstr4.3 $e |- D = B $.
|
|
$( Substitution of equality in both sides of a subclass relationship.
|
|
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4i $p |- C C_ D $=
|
|
( wss sseq12i mpbir ) CDHABHECADBFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3sstr3g.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr3g.2 $e |- A = C $.
|
|
3sstr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 1-Oct-2000.) $)
|
|
3sstr3g $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr4g.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr4g.2 $e |- C = A $.
|
|
3sstr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4g $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr3d.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr3d.2 $e |- ( ph -> A = C ) $.
|
|
3sstr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 1-Oct-2000.) $)
|
|
3sstr3d $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12d mpbid ) ABCIDEIFABDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr4d.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr4d.2 $e |- ( ph -> C = A ) $.
|
|
3sstr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4d $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12d mpbird ) ADEIBCIFADBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
syl5eqss.1 $e |- A = B $.
|
|
syl5eqss.2 $e |- ( ph -> B C_ C ) $.
|
|
$( B chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl5eqss $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq1i sylibr ) ACDGBDGFBCDEHI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqssr.1 $e |- B = A $.
|
|
syl5eqssr.2 $e |- ( ph -> B C_ C ) $.
|
|
$( B chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl5eqssr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomi syl5eqss ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6sseq.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6sseq.2 $e |- B = C $.
|
|
$( A chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl6sseq $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq2i sylib ) ABCGBDGECDBFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6ssr.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6ssr.2 $e |- C = B $.
|
|
$( A chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl6sseqr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomi syl6sseq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5sseq.1 $e |- B C_ A $.
|
|
syl5sseq.2 $e |- ( ph -> A = C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl5sseq $p |- ( ph -> B C_ C ) $=
|
|
( wceq wss sseq2 biimpa sylancl ) ABDGZCBHZCDHZFELMNBDCIJK $.
|
|
$}
|
|
|
|
${
|
|
syl5sseqr.1 $e |- B C_ A $.
|
|
syl5sseqr.2 $e |- ( ph -> C = A ) $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl5sseqr $p |- ( ph -> B C_ C ) $=
|
|
( wss a1i sseqtr4d ) ACBDCBGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqss.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqss.2 $e |- B C_ C $.
|
|
$( A chained subclass and equality deduction. (Contributed by Mario
|
|
Carneiro, 2-Jan-2017.) $)
|
|
syl6eqss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i eqsstrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqssr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqssr.2 $e |- B C_ C $.
|
|
$( A chained subclass and equality deduction. (Contributed by Mario
|
|
Carneiro, 2-Jan-2017.) $)
|
|
syl6eqssr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd syl6eqss ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
$( Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
eqimss $p |- ( A = B -> A C_ B ) $=
|
|
( wceq wss eqss simplbi ) ABCABDBADABEF $.
|
|
|
|
$( Equality implies the subclass relation. (Contributed by NM,
|
|
23-Nov-2003.) $)
|
|
eqimss2 $p |- ( B = A -> A C_ B ) $=
|
|
( wss eqimss eqcoms ) ABCABABDE $.
|
|
|
|
${
|
|
eqimssi.1 $e |- A = B $.
|
|
$( Infer subclass relationship from equality. (Contributed by NM,
|
|
6-Jan-2007.) $)
|
|
eqimssi $p |- A C_ B $=
|
|
( ssid sseqtri ) AABADCE $.
|
|
|
|
$( Infer subclass relationship from equality. (Contributed by NM,
|
|
7-Jan-2007.) $)
|
|
eqimss2i $p |- B C_ A $=
|
|
( ssid sseqtr4i ) BBABDCE $.
|
|
$}
|
|
|
|
$( Two classes are different if they don't include the same class.
|
|
(Contributed by NM, 23-Apr-2015.) $)
|
|
nssne1 $p |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C ) $=
|
|
( wss wn wne wceq sseq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJK
|
|
$.
|
|
|
|
$( Two classes are different if they are not subclasses of the same class.
|
|
(Contributed by NM, 23-Apr-2015.) $)
|
|
nssne2 $p |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B ) $=
|
|
( wss wn wne wceq sseq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJK
|
|
$.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Negation of subclass relationship. One direction of Exercise 13 of
|
|
[TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.) $)
|
|
nssr $p |- ( E. x ( x e. A /\ -. x e. B ) -> -. A C_ B ) $=
|
|
( cv wcel wn wa wex wi wal wss exanaliim dfss2 sylnibr ) ADZBEZOCEZFGAHPQ
|
|
IAJBCKPQALABCMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Quantification restricted to a subclass. (Contributed by NM,
|
|
11-Mar-2006.) $)
|
|
ssralv $p |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) ) $=
|
|
( wss cv wcel ssel imim1d ralimdv2 ) CDEZAABDCKBFZCGLDGACDLHIJ $.
|
|
|
|
$( Existential quantification restricted to a subclass. (Contributed by
|
|
NM, 11-Jan-2007.) $)
|
|
ssrexv $p |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) ) $=
|
|
( wss cv wcel ssel anim1d reximdv2 ) CDEZAABCDKBFZCGLDGACDLHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Restricted universal quantification on a subset in terms of superset.
|
|
(Contributed by Stefan O'Rear, 3-Apr-2015.) $)
|
|
ralss $p |- ( A C_ B -> ( A. x e. A ph <->
|
|
A. x e. B ( x e. A -> ph ) ) ) $=
|
|
( wss cv wcel wi wa ssel pm4.71rd imbi1d impexp syl6bb ralbidv2 ) CDEZABF
|
|
ZCGZAHZBCDPSQDGZRIZAHTSHPRUAAPRTCDQJKLTRAMNO $.
|
|
|
|
$( Restricted existential quantification on a subset in terms of superset.
|
|
(Contributed by Stefan O'Rear, 3-Apr-2015.) $)
|
|
rexss $p |- ( A C_ B -> ( E. x e. A ph <->
|
|
E. x e. B ( x e. A /\ ph ) ) ) $=
|
|
( wss cv wcel wa ssel pm4.71rd anbi1d anass syl6bb rexbidv2 ) CDEZABFZCGZ
|
|
AHZBCDORPDGZQHZAHSRHOQTAOQSCDPIJKSQALMN $.
|
|
$}
|
|
|
|
$( Class abstractions in a subclass relationship. (Contributed by NM,
|
|
3-Jul-1994.) $)
|
|
ss2ab $p |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) ) $=
|
|
( cab wss cv wcel wi wal nfab1 dfss2f abid imbi12i albii bitri ) ACDZBCDZEC
|
|
FZPGZRQGZHZCIABHZCICPQACJBCJKUAUBCSATBACLBCLMNO $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Class abstraction in a subclass relationship. (Contributed by NM,
|
|
16-Aug-2006.) $)
|
|
abss $p |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) ) $=
|
|
( cab wss cv wcel wi wal abid2 sseq2i ss2ab bitr3i ) ABDZCENBFCGZBDZEAOHB
|
|
IPCNBCJKAOBLM $.
|
|
|
|
$( Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.) $)
|
|
ssab $p |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) ) $=
|
|
( cab wss cv wcel wi wal abid2 sseq1i ss2ab bitr3i ) CABDZEBFCGZBDZNEOAHB
|
|
IPCNBCJKOABLM $.
|
|
|
|
$( The relation for a subclass of a class abstraction is equivalent to
|
|
restricted quantification. (Contributed by NM, 6-Sep-2006.) $)
|
|
ssabral $p |- ( A C_ { x | ph } <-> A. x e. A ph ) $=
|
|
( cab wss cv wcel wi wal wral ssab df-ral bitr4i ) CABDEBFCGAHBIABCJABCKA
|
|
BCLM $.
|
|
$}
|
|
|
|
${
|
|
ss2abi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of abstraction subclass from implication. (Contributed by NM,
|
|
31-Mar-1995.) $)
|
|
ss2abi $p |- { x | ph } C_ { x | ps } $=
|
|
( cab wss wi ss2ab mpgbir ) ACEBCEFABGCABCHDI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ss2abdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction of abstraction subclass from implication. (Contributed by NM,
|
|
29-Jul-2011.) $)
|
|
ss2abdv $p |- ( ph -> { x | ps } C_ { x | ch } ) $=
|
|
( wi wal cab wss alrimiv ss2ab sylibr ) ABCFZDGBDHCDHIAMDEJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
abssdv.1 $e |- ( ph -> ( ps -> x e. A ) ) $.
|
|
$( Deduction of abstraction subclass from implication. (Contributed by NM,
|
|
20-Jan-2006.) $)
|
|
abssdv $p |- ( ph -> { x | ps } C_ A ) $=
|
|
( cv wcel wi wal cab wss alrimiv abss sylibr ) ABCFDGHZCIBCJDKAOCELBCDMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
abssi.1 $e |- ( ph -> x e. A ) $.
|
|
$( Inference of abstraction subclass from implication. (Contributed by NM,
|
|
20-Jan-2006.) $)
|
|
abssi $p |- { x | ph } C_ A $=
|
|
( cab cv wcel ss2abi abid2 sseqtri ) ABEBFCGZBECAKBDHBCIJ $.
|
|
$}
|
|
|
|
$( Restricted abstraction classes in a subclass relationship. (Contributed
|
|
by NM, 30-May-1999.) $)
|
|
ss2rab $p |- ( { x e. A | ph } C_ { x e. A | ps } <->
|
|
A. x e. A ( ph -> ps ) ) $=
|
|
( crab wss cv wcel wa cab wi wal df-rab sseq12i ss2ab df-ral imdistan albii
|
|
wral bitr2i 3bitri ) ACDEZBCDEZFCGDHZAIZCJZUDBIZCJZFUEUGKZCLZABKZCDSZUBUFUC
|
|
UHACDMBCDMNUEUGCOULUDUKKZCLUJUKCDPUMUICUDABQRTUA $.
|
|
|
|
${
|
|
$d x B $.
|
|
$( Restricted class abstraction in a subclass relationship. (Contributed
|
|
by NM, 16-Aug-2006.) $)
|
|
rabss $p |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) ) $=
|
|
( crab wss cv wcel wa cab wal wral df-rab sseq1i abss impexp albii df-ral
|
|
wi bitr4i 3bitri ) ABCEZDFBGZCHZAIZBJZDFUEUCDHZSZBKZAUGSZBCLZUBUFDABCMNUE
|
|
BDOUIUDUJSZBKUKUHULBUDAUGPQUJBCRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass of a restricted class abstraction. (Contributed by NM,
|
|
16-Aug-2006.) $)
|
|
ssrab $p |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) ) $=
|
|
( crab wss cv wcel wa cab wal wral df-rab sseq2i ssab dfss3 anbi1i r19.26
|
|
wi df-ral 3bitr2ri 3bitri ) DABCEZFDBGZCHZAIZBJZFUDDHUFSBKZDCFZABDLZIZUCU
|
|
GDABCMNUFBDOUKUEBDLZUJIUFBDLUHUIULUJBDCPQUEABDRUFBDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
ssrabdv.1 $e |- ( ph -> B C_ A ) $.
|
|
ssrabdv.2 $e |- ( ( ph /\ x e. B ) -> ps ) $.
|
|
$( Subclass of a restricted class abstraction (deduction rule).
|
|
(Contributed by NM, 31-Aug-2006.) $)
|
|
ssrabdv $p |- ( ph -> B C_ { x e. A | ps } ) $=
|
|
( wss wral crab ralrimiva ssrab sylanbrc ) AEDHBCEIEBCDJHFABCEGKBCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x ph $.
|
|
rabssdv.1 $e |- ( ( ph /\ x e. A /\ ps ) -> x e. B ) $.
|
|
$( Subclass of a restricted class abstraction (deduction rule).
|
|
(Contributed by NM, 2-Feb-2015.) $)
|
|
rabssdv $p |- ( ph -> { x e. A | ps } C_ B ) $=
|
|
( cv wcel wi wral crab wss 3exp ralrimiv rabss sylibr ) ABCGZEHZIZCDJBCDK
|
|
ELASCDAQDHBRFMNBCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ss2rabdv.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction of restricted abstraction subclass from implication.
|
|
(Contributed by NM, 30-May-2006.) $)
|
|
ss2rabdv $p |- ( ph -> { x e. A | ps } C_ { x e. A | ch } ) $=
|
|
( wi wral crab wss ralrimiva ss2rab sylibr ) ABCGZDEHBDEICDEIJANDEFKBCDEL
|
|
M $.
|
|
$}
|
|
|
|
${
|
|
ss2rabi.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference of restricted abstraction subclass from implication.
|
|
(Contributed by NM, 14-Oct-1999.) $)
|
|
ss2rabi $p |- { x e. A | ph } C_ { x e. A | ps } $=
|
|
( crab wss wi ss2rab mprgbir ) ACDFBCDFGABHCDABCDIEJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass law for restricted abstraction. (Contributed by NM,
|
|
18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rabss2 $p |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } ) $=
|
|
( wss cv wcel wa cab crab wi wal pm3.45 alimi dfss2 ss2ab 3imtr4i 3sstr4g
|
|
df-rab ) CDEZBFZCGZAHZBIZUADGZAHZBIZABCJABDJUBUEKZBLUCUFKZBLTUDUGEUHUIBUB
|
|
UEAMNBCDOUCUFBPQABCSABDSR $.
|
|
|
|
$( Subclass relation for the restriction of a class abstraction.
|
|
(Contributed by NM, 31-Mar-1995.) $)
|
|
ssab2 $p |- { x | ( x e. A /\ ph ) } C_ A $=
|
|
( cv wcel wa simpl abssi ) BDCEZAFBCIAGH $.
|
|
|
|
$( Subclass relation for a restricted class. (Contributed by NM,
|
|
19-Mar-1997.) $)
|
|
ssrab2 $p |- { x e. A | ph } C_ A $=
|
|
( crab cv wcel wa cab df-rab ssab2 eqsstri ) ABCDBECFAGBHCABCIABCJK $.
|
|
$}
|
|
|
|
${
|
|
$d V x $.
|
|
$( If the restricting class of a restricted class abstraction is a subset
|
|
of this restricted class abstraction, it is equal to this restricted
|
|
class abstraction. (Contributed by Alexander van der Vekens,
|
|
31-Dec-2017.) $)
|
|
ssrabeq $p |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } ) $=
|
|
( crab wss wa wceq ssrab2 biantru eqss bitr4i ) CABCDZEZMLCEZFCLGNMABCHIC
|
|
LJK $.
|
|
$}
|
|
|
|
$( A restricted class is a subclass of the corresponding unrestricted class.
|
|
(Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
rabssab $p |- { x e. A | ph } C_ { x | ph } $=
|
|
( crab cv wcel wa cab df-rab simpr ss2abi eqsstri ) ABCDBECFZAGZBHABHABCINA
|
|
BMAJKL $.
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d y z B $. $d x z C $.
|
|
$( A subset relationship useful for converting union to indexed union using
|
|
dfiun2 or dfiun2g and intersection to indexed intersection using
|
|
dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario
|
|
Carneiro, 26-Sep-2015.) $)
|
|
uniiunlem $p |- ( A. x e. A B e. D ->
|
|
( A. x e. A B e. C <-> { y | E. x e. A y = B } C_ C ) ) $=
|
|
( vz cv wceq wrex cab wss wcel wi wal wral eqeq1 rexbidv cbvabv wb sseq1i
|
|
r19.23v albii ralcom4 abss 3bitr4i bitr4i nfv eleq1 ceqsalg ralbi syl5rbb
|
|
ralimi syl ) BHZDIZACJZBKZELZGHZDIZUTEMZNZGOZACPZDFMZACPZDEMZACPZUSVAACJZ
|
|
GKZELZVEURVKEUQVJBGUOUTIUPVAACUOUTDQRSUAVCACPZGOVJVBNZGOVEVLVMVNGVAVBACUB
|
|
UCVCAGCUDVJGEUEUFUGVGVDVHTZACPVEVITVFVOACVBVHGDFVHGUHUTDEUIUJUMVDVHACUKUN
|
|
UL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The difference, union, and intersection of two classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
The difference of two classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for class difference. (Contributed by NM,
|
|
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeq1 $p |- ( A = B -> ( A \ C ) = ( B \ C ) ) $=
|
|
( vx wceq cv wcel wn crab cdif rabeq dfdif2 3eqtr4g ) ABEDFCGHZDAINDBIACJ
|
|
BCJNDABKDACLDBCLM $.
|
|
|
|
$( Equality theorem for class difference. (Contributed by NM,
|
|
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeq2 $p |- ( A = B -> ( C \ A ) = ( C \ B ) ) $=
|
|
( vx wceq cv wcel wn crab cdif eleq2 notbid rabbidv dfdif2 3eqtr4g ) ABEZ
|
|
DFZAGZHZDCIQBGZHZDCICAJCBJPSUADCPRTABQKLMDCANDCBNO $.
|
|
$}
|
|
|
|
$( Equality theorem for class difference. (Contributed by FL,
|
|
31-Aug-2009.) $)
|
|
difeq12 $p |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) ) $=
|
|
( wceq cdif difeq1 difeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
difeq1i.1 $e |- A = B $.
|
|
$( Inference adding difference to the right in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq1i $p |- ( A \ C ) = ( B \ C ) $=
|
|
( wceq cdif difeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Inference adding difference to the left in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq2i $p |- ( C \ A ) = ( C \ B ) $=
|
|
( wceq cdif difeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
difeq12i.2 $e |- C = D $.
|
|
$( Equality inference for class difference. (Contributed by NM,
|
|
29-Aug-2004.) $)
|
|
difeq12i $p |- ( A \ C ) = ( B \ D ) $=
|
|
( cdif difeq1i difeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
difeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction adding difference to the right in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq1d $p |- ( ph -> ( A \ C ) = ( B \ C ) ) $=
|
|
( wceq cdif difeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Deduction adding difference to the left in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq2d $p |- ( ph -> ( C \ A ) = ( C \ B ) ) $=
|
|
( wceq cdif difeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
difeq12d.1 $e |- ( ph -> A = B ) $.
|
|
difeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for class difference. (Contributed by FL,
|
|
29-May-2014.) $)
|
|
difeq12d $p |- ( ph -> ( A \ C ) = ( B \ D ) ) $=
|
|
( cdif difeq1d difeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
difeqri.1 $e |- ( ( x e. A /\ -. x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to difference. (Contributed by NM,
|
|
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeqri $p |- ( A \ B ) = C $=
|
|
( cdif cv wcel wn wa eldif bitri eqriv ) ABCFZDAGZNHOBHOCHIJODHOBCKELM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfdif.1 $e |- F/_ x A $.
|
|
nfdif.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for class difference. (Contributed by
|
|
NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
nfdif $p |- F/_ x ( A \ B ) $=
|
|
( vy cdif cv wcel wn crab dfdif2 nfcri nfn nfrabxy nfcxfr ) ABCGFHCIZJZFB
|
|
KFBCLRAFBQAAFCEMNDOP $.
|
|
$}
|
|
|
|
$( Implication of membership in a class difference. (Contributed by NM,
|
|
29-Apr-1994.) $)
|
|
eldifi $p |- ( A e. ( B \ C ) -> A e. B ) $=
|
|
( cdif wcel wn eldif simplbi ) ABCDEABEACEFABCGH $.
|
|
|
|
$( Implication of membership in a class difference. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
eldifn $p |- ( A e. ( B \ C ) -> -. A e. C ) $=
|
|
( cdif wcel wn eldif simprbi ) ABCDEABEACEFABCGH $.
|
|
|
|
$( A set does not belong to a class excluding it. (Contributed by NM,
|
|
27-Jun-1994.) $)
|
|
elndif $p |- ( A e. B -> -. A e. ( C \ B ) ) $=
|
|
( cdif wcel eldifn con2i ) ACBDEABEACBFG $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
|
|
(Contributed by NM, 17-May-1998.) $)
|
|
difdif $p |- ( A \ ( B \ A ) ) = A $=
|
|
( vx cdif cv wcel wn wa simpl pm4.45im imanim eldif sylnibr anim2i impbii
|
|
wi sylbi difeqri ) CABADZACEZAFZTSFZGZHZUAUAUCIUAUATBFZUAPZHUDUAUEJUFUCUA
|
|
UFUEUAGHUBUEUAKTBALMNQOR $.
|
|
|
|
$( Subclass relationship for class difference. Exercise 14 of
|
|
[TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) $)
|
|
difss $p |- ( A \ B ) C_ A $=
|
|
( vx cdif cv eldifi ssriv ) CABDACEABFG $.
|
|
$}
|
|
|
|
$( A difference of two classes is contained in the minuend. Deduction form
|
|
of ~ difss . (Contributed by David Moews, 1-May-2017.) $)
|
|
difssd $p |- ( ph -> ( A \ B ) C_ A ) $=
|
|
( cdif wss difss a1i ) BCDBEABCFG $.
|
|
|
|
$( If a class is contained in a difference, it is contained in the minuend.
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
difss2 $p |- ( A C_ ( B \ C ) -> A C_ B ) $=
|
|
( cdif wss id difss syl6ss ) ABCDZEZAIBJFBCGH $.
|
|
|
|
${
|
|
difss2d.1 $e |- ( ph -> A C_ ( B \ C ) ) $.
|
|
$( If a class is contained in a difference, it is contained in the minuend.
|
|
Deduction form of ~ difss2 . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
difss2d $p |- ( ph -> A C_ B ) $=
|
|
( cdif wss difss2 syl ) ABCDFGBCGEBCDHI $.
|
|
$}
|
|
|
|
$( Preservation of a subclass relationship by class difference. (Contributed
|
|
by NM, 15-Feb-2007.) $)
|
|
ssdifss $p |- ( A C_ B -> ( A \ C ) C_ B ) $=
|
|
( cdif wss difss sstr mpan ) ACDZAEABEIBEACFIABGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
ddifnel.1 $e |- ( -. x e. ( _V \ A ) -> x e. A ) $.
|
|
$( Double complement under universal class. The hypothesis corresponds to
|
|
stability of membership in ` A ` , which is weaker than decidability
|
|
(see ~ dcimpstab ). Actually, the conclusion is a characterization of
|
|
stability of membership in a class (see ~ ddifstab ) . Exercise 4.10(s)
|
|
of [Mendelson] p. 231, but with an additional hypothesis. For a version
|
|
without a hypothesis, but which only states that ` A ` is a subset of
|
|
` _V \ ( _V \ A ) ` , see ~ ddifss . (Contributed by Jim Kingdon,
|
|
21-Jul-2018.) $)
|
|
ddifnel $p |- ( _V \ ( _V \ A ) ) = A $=
|
|
( cvv cdif cv wcel wn wa adantl elndif vex jctil impbii difeqri ) ADDBEZB
|
|
AFZDGZQPGHZIQBGZSTRCJTSRQBDKALMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A class is equal to its double complement if and only if it is stable
|
|
(that is, membership in it is a stable property). (Contributed by BJ,
|
|
12-Dec-2021.) $)
|
|
ddifstab $p |- ( ( _V \ ( _V \ A ) ) = A <-> A. x STAB x e. A ) $=
|
|
( cvv cdif wceq cv wb wal wstab dfcleq wn wi wa eldif vex biantrur bitr4i
|
|
wcel notbii bitri 3bitr2i bibi1i bi1 notnot impbid1 impbii df-stab albii
|
|
id ) CCBDZDZBEAFZUKRZULBRZGZAHUNIZAHAUKBJUOUPAUOUNKZKZUNLZUPUOURUNGZUSUMU
|
|
RUNUMULCRZULUJRZKZMVCURULCUJNVAVCAOZPVBUQVBVAUQMUQULCBNVAUQVDPQSUAUBUTUSU
|
|
RUNUCUSURUNUSUIUNUDUEUFTUNUGQUHT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) $)
|
|
ssconb $p |- ( ( A C_ C /\ B C_ C ) ->
|
|
( A C_ ( C \ B ) <-> B C_ ( C \ A ) ) ) $=
|
|
( vx wss wa cv wcel cdif wi wal wn wb ssel pm5.1 jcab 3bitr4g eldif dfss2
|
|
imbi2i syl2an con2b a1i anbi12d albidv ) ACEZBCEZFZDGZAHZUICBIZHZJZDKUIBH
|
|
ZUICAIZHZJZDKAUKEBUOEUHUMUQDUHUJUICHZUNLZFZJZUNURUJLZFZJZUMUQUHUJURJZUJUS
|
|
JZFUNURJZUNVBJZFVAVDUHVEVGVFVHUFVEVGVEVGMUGACUINBCUINVEVGOUAVFVHMUHUJUNUB
|
|
UCUDUJURUSPUNURVBPQULUTUJUICBRTUPVCUNUICARTQUEDAUKSDBUOSQ $.
|
|
|
|
$( Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
|
|
(Contributed by NM, 22-Mar-1998.) $)
|
|
sscon $p |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) ) $=
|
|
( vx wss cdif cv wcel wn wa ssel con3d anim2d eldif 3imtr4g ssrdv ) ABEZD
|
|
CBFZCAFZQDGZCHZTBHZIZJUATAHZIZJTRHTSHQUCUEUAQUDUBABTKLMTCBNTCANOP $.
|
|
|
|
$( Difference law for subsets. (Contributed by NM, 28-May-1998.) $)
|
|
ssdif $p |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) ) $=
|
|
( vx wss cdif cv wcel wn wa ssel anim1d eldif 3imtr4g ssrdv ) ABEZDACFZBC
|
|
FZPDGZAHZSCHIZJSBHZUAJSQHSRHPTUBUAABSKLSACMSBCMNO $.
|
|
$}
|
|
|
|
${
|
|
ssdifd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is contained in
|
|
` ( B \ C ) ` . Deduction form of ~ ssdif . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
ssdifd $p |- ( ph -> ( A \ C ) C_ ( B \ C ) ) $=
|
|
( wss cdif ssdif syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( If ` A ` is contained in ` B ` , then ` ( C \ B ) ` is contained in
|
|
` ( C \ A ) ` . Deduction form of ~ sscon . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
sscond $p |- ( ph -> ( C \ B ) C_ ( C \ A ) ) $=
|
|
( wss cdif sscon syl ) ABCFDCGDBGFEBCDHI $.
|
|
|
|
$( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is also contained in
|
|
` B ` . Deduction form of ~ ssdifss . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
ssdifssd $p |- ( ph -> ( A \ C ) C_ B ) $=
|
|
( wss cdif ssdifss syl ) ABCFBDGCFEBCDHI $.
|
|
|
|
ssdif2d.2 $e |- ( ph -> C C_ D ) $.
|
|
$( If ` A ` is contained in ` B ` and ` C ` is contained in ` D ` , then
|
|
` ( A \ D ) ` is contained in ` ( B \ C ) ` . Deduction form.
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
ssdif2d $p |- ( ph -> ( A \ D ) C_ ( B \ C ) ) $=
|
|
( cdif sscond ssdifd sstrd ) ABEHBDHCDHADEBGIABCDFJK $.
|
|
$}
|
|
|
|
$( Restricted universal quantification on a class difference in terms of an
|
|
implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.) $)
|
|
raldifb $p |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph ) $=
|
|
( cv wnel wi cdif wcel wa impexp bicomi wn df-nel anbi2i eldif bitri imbi1i
|
|
ralbii2 ) BEZDFZAGZABCCDHZTCIZUBGZUDUAJZAGZTUCIZAGUGUEUDUAAKLUFUHAUFUDTDIMZ
|
|
JZUHUAUIUDTDNOUHUJTCDPLQRQS $.
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
The union of two classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
|
|
(Contributed by NM, 7-Aug-1994.) $)
|
|
elun $p |- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) ) $=
|
|
( vx cun wcel cvv wo elex jaoi wceq eleq1 orbi12d df-un elab2g pm5.21nii
|
|
cv ) ABCEZFAGFZABFZACFZHZARITSUAABIACIJDQZBFZUCCFZHUBDARGUCAKUDTUEUAUCABL
|
|
UCACLMDBCNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
uneqri.1 $e |- ( ( x e. A \/ x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to union. (Contributed by NM, 5-Aug-1993.) $)
|
|
uneqri $p |- ( A u. B ) = C $=
|
|
( cun cv wcel wo elun bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
unidm $p |- ( A u. A ) = A $=
|
|
( vx cv wcel oridm uneqri ) BAAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Commutative law for union of classes. Exercise 6 of [TakeutiZaring]
|
|
p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
uncom $p |- ( A u. B ) = ( B u. A ) $=
|
|
( vx cun cv wcel wo orcom elun bitr4i uneqri ) CABBADZCEZAFZMBFZGONGMLFNO
|
|
HMBAIJK $.
|
|
$}
|
|
|
|
$( If a class equals the union of two other classes, then it equals the union
|
|
of those two classes commuted. (Contributed by Alan Sare,
|
|
18-Feb-2012.) $)
|
|
equncom $p |- ( A = ( B u. C ) <-> A = ( C u. B ) ) $=
|
|
( cun uncom eqeq2i ) BCDCBDABCEF $.
|
|
|
|
${
|
|
equncomi.1 $e |- A = ( B u. C ) $.
|
|
$( Inference form of ~ equncom . (Contributed by Alan Sare,
|
|
18-Feb-2012.) $)
|
|
equncomi $p |- A = ( C u. B ) $=
|
|
( cun wceq equncom mpbi ) ABCEFACBEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for union of two classes. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
uneq1 $p |- ( A = B -> ( A u. C ) = ( B u. C ) ) $=
|
|
( vx wceq cun cv wcel wo eleq2 orbi1d elun 3bitr4g eqrdv ) ABEZDACFZBCFZO
|
|
DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
$}
|
|
|
|
$( Equality theorem for the union of two classes. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
uneq2 $p |- ( A = B -> ( C u. A ) = ( C u. B ) ) $=
|
|
( wceq cun uneq1 uncom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for union of two classes. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
uneq12 $p |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) ) $=
|
|
( wceq cun uneq1 uneq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
uneq1i.1 $e |- A = B $.
|
|
$( Inference adding union to the right in a class equality. (Contributed
|
|
by NM, 30-Aug-1993.) $)
|
|
uneq1i $p |- ( A u. C ) = ( B u. C ) $=
|
|
( wceq cun uneq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Inference adding union to the left in a class equality. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
uneq2i $p |- ( C u. A ) = ( C u. B ) $=
|
|
( wceq cun uneq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
uneq12i.2 $e |- C = D $.
|
|
$( Equality inference for union of two classes. (Contributed by NM,
|
|
12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
uneq12i $p |- ( A u. C ) = ( B u. D ) $=
|
|
( wceq cun uneq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
uneq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction adding union to the right in a class equality. (Contributed
|
|
by NM, 29-Mar-1998.) $)
|
|
uneq1d $p |- ( ph -> ( A u. C ) = ( B u. C ) ) $=
|
|
( wceq cun uneq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Deduction adding union to the left in a class equality. (Contributed by
|
|
NM, 29-Mar-1998.) $)
|
|
uneq2d $p |- ( ph -> ( C u. A ) = ( C u. B ) ) $=
|
|
( wceq cun uneq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
uneq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for union of two classes. (Contributed by NM,
|
|
29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
uneq12d $p |- ( ph -> ( A u. C ) = ( B u. D ) ) $=
|
|
( wceq cun uneq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfun.1 $e |- F/_ x A $.
|
|
nfun.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for the union of classes.
|
|
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfun $p |- F/_ x ( A u. B ) $=
|
|
( vy cun cv wcel wo cab df-un nfcri nfor nfab nfcxfr ) ABCGFHZBIZQCIZJZFK
|
|
FBCLTAFRSAAFBDMAFCEMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Associative law for union of classes. Exercise 8 of [TakeutiZaring]
|
|
p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
unass $p |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) ) $=
|
|
( vx cun cv wcel wo elun orbi2i orbi1i orass bitr2i 3bitrri uneqri ) DABE
|
|
ZCABCEZEZDFZRGSAGZSQGZHTSBGZSCGZHZHZSPGZUCHZSAQIUAUDTSBCIJUGTUBHZUCHUEUFU
|
|
HUCSABIKTUBUCLMNO $.
|
|
$}
|
|
|
|
$( A rearrangement of union. (Contributed by NM, 12-Aug-2004.) $)
|
|
un12 $p |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) ) $=
|
|
( cun uncom uneq1i unass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
|
|
$.
|
|
|
|
$( A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
un23 $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B ) $=
|
|
( cun unass un12 uncom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of the union of 4 classes. (Contributed by NM,
|
|
12-Aug-2004.) $)
|
|
un4 $p |- ( ( A u. B ) u. ( C u. D ) ) =
|
|
( ( A u. C ) u. ( B u. D ) ) $=
|
|
( cun un12 uneq2i unass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
|
|
CLHI $.
|
|
|
|
$( Union distributes over itself. (Contributed by NM, 17-Aug-2004.) $)
|
|
unundi $p |- ( A u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) ) $=
|
|
( cun unidm uneq1i un4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.
|
|
|
|
$( Union distributes over itself. (Contributed by NM, 17-Aug-2004.) $)
|
|
unundir $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) ) $=
|
|
( cun unidm uneq2i un4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for union of classes. Theorem 25 of [Suppes]
|
|
p. 27. (Contributed by NM, 5-Aug-1993.) $)
|
|
ssun1 $p |- A C_ ( A u. B ) $=
|
|
( vx cun cv wcel wo orc elun sylibr ssriv ) CAABDZCEZAFZNMBFZGMLFNOHMABIJ
|
|
K $.
|
|
$}
|
|
|
|
$( Subclass relationship for union of classes. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
ssun2 $p |- A C_ ( B u. A ) $=
|
|
( cun ssun1 uncom sseqtri ) AABCBACABDABEF $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) $)
|
|
ssun3 $p |- ( A C_ B -> A C_ ( B u. C ) ) $=
|
|
( wss cun ssun1 sstr2 mpi ) ABDBBCEZDAIDBCFABIGH $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) $)
|
|
ssun4 $p |- ( A C_ B -> A C_ ( C u. B ) ) $=
|
|
( wss cun ssun2 sstr2 mpi ) ABDBCBEZDAIDBCFABIGH $.
|
|
|
|
$( Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) $)
|
|
elun1 $p |- ( A e. B -> A e. ( B u. C ) ) $=
|
|
( cun ssun1 sseli ) BBCDABCEF $.
|
|
|
|
$( Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) $)
|
|
elun2 $p |- ( A e. B -> A e. ( C u. B ) ) $=
|
|
( cun ssun2 sseli ) BCBDABCEF $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
unss1 $p |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) ) $=
|
|
( vx wss cun cv wcel wo ssel orim1d elun 3imtr4g ssrdv ) ABEZDACFZBCFZODG
|
|
ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
|
|
$( A relationship between subclass and union. Theorem 26 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
ssequn1 $p |- ( A C_ B <-> ( A u. B ) = B ) $=
|
|
( vx cv wcel wi wal cun wb wceq wo bicom pm4.72 elun bibi1i 3bitr4i albii
|
|
wss dfss2 dfcleq ) CDZAEZUABEZFZCGUAABHZEZUCIZCGABRUEBJUDUGCUCUBUCKZIUHUC
|
|
IUDUGUCUHLUBUCMUFUHUCUAABNOPQCABSCUEBTP $.
|
|
$}
|
|
|
|
$( Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
|
|
(Contributed by NM, 14-Oct-1999.) $)
|
|
unss2 $p |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) ) $=
|
|
( wss cun unss1 uncom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) $)
|
|
unss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) ) $=
|
|
( wss cun unss1 unss2 sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
$( A relationship between subclass and union. (Contributed by NM,
|
|
13-Jun-1994.) $)
|
|
ssequn2 $p |- ( A C_ B <-> ( B u. A ) = B ) $=
|
|
( wss cun wceq ssequn1 uncom eqeq1i bitri ) ABCABDZBEBADZBEABFJKBABGHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
|
|
and its converse. (Contributed by NM, 11-Jun-2004.) $)
|
|
unss $p |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C ) $=
|
|
( vx cun wss cv wcel wi wal wa dfss2 19.26 elun imbi1i jaob bitri anbi12i
|
|
wo albii 3bitr4i bitr2i ) ABEZCFDGZUCHZUDCHZIZDJZACFZBCFZKZDUCCLUDAHZUFIZ
|
|
UDBHZUFIZKZDJUMDJZUODJZKUHUKUMUODMUGUPDUGULUNSZUFIUPUEUSUFUDABNOULUFUNPQT
|
|
UIUQUJURDACLDBCLRUAUB $.
|
|
$}
|
|
|
|
${
|
|
unssi.1 $e |- A C_ C $.
|
|
unssi.2 $e |- B C_ C $.
|
|
$( An inference showing the union of two subclasses is a subclass.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
unssi $p |- ( A u. B ) C_ C $=
|
|
( wss wa cun pm3.2i unss mpbi ) ACFZBCFZGABHCFLMDEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
unssd.1 $e |- ( ph -> A C_ C ) $.
|
|
unssd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( A deduction showing the union of two subclasses is a subclass.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
unssd $p |- ( ph -> ( A u. B ) C_ C ) $=
|
|
( wss cun wa unss biimpi syl2anc ) ABDGZCDGZBCHDGZEFMNIOBCDJKL $.
|
|
$}
|
|
|
|
${
|
|
unssad.1 $e |- ( ph -> ( A u. B ) C_ C ) $.
|
|
$( If ` ( A u. B ) ` is contained in ` C ` , so is ` A ` . One-way
|
|
deduction form of ~ unss . Partial converse of ~ unssd . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
unssad $p |- ( ph -> A C_ C ) $=
|
|
( wss cun wa unss sylibr simpld ) ABDFZCDFZABCGDFLMHEBCDIJK $.
|
|
|
|
$( If ` ( A u. B ) ` is contained in ` C ` , so is ` B ` . One-way
|
|
deduction form of ~ unss . Partial converse of ~ unssd . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
unssbd $p |- ( ph -> B C_ C ) $=
|
|
( wss cun wa unss sylibr simprd ) ABDFZCDFZABCGDFLMHEBCDIJK $.
|
|
$}
|
|
|
|
$( A condition that implies inclusion in the union of two classes.
|
|
(Contributed by NM, 23-Nov-2003.) $)
|
|
ssun $p |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) ) $=
|
|
( wss cun ssun3 ssun4 jaoi ) ABDABCEDACDABCFACBGH $.
|
|
|
|
$( Restricted existential quantification over union. (Contributed by Jeff
|
|
Madsen, 5-Jan-2011.) $)
|
|
rexun $p |- ( E. x e. ( A u. B ) ph <->
|
|
( E. x e. A ph \/ E. x e. B ph ) ) $=
|
|
( cun wrex cv wcel wa wo df-rex 19.43 elun anbi1i andir bitri exbii orbi12i
|
|
wex 3bitr4i ) ABCDEZFBGZUAHZAIZBSZABCFZABDFZJZABUAKUBCHZAIZUBDHZAIZJZBSUJBS
|
|
ZULBSZJUEUHUJULBLUDUMBUDUIUKJZAIUMUCUPAUBCDMNUIUKAOPQUFUNUGUOABCKABDKRTP $.
|
|
|
|
$( Restricted quantification over a union. (Contributed by Scott Fenton,
|
|
12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
ralunb $p |- ( A. x e. ( A u. B ) ph <->
|
|
( A. x e. A ph /\ A. x e. B ph ) ) $=
|
|
( cv cun wcel wi wal wa wral wo elun imbi1i jaob bitri albii df-ral anbi12i
|
|
19.26 3bitr4i ) BEZCDFZGZAHZBIZUBCGZAHZBIZUBDGZAHZBIZJZABUCKABCKZABDKZJUFUH
|
|
UKJZBIUMUEUPBUEUGUJLZAHUPUDUQAUBCDMNUGAUJOPQUHUKBTPABUCRUNUIUOULABCRABDRSUA
|
|
$.
|
|
|
|
$( Restricted quantification over union. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
ralun $p |- ( ( A. x e. A ph /\ A. x e. B ph ) -> A. x e. ( A u. B ) ph ) $=
|
|
( cun wral wa ralunb biimpri ) ABCDEFABCFABDFGABCDHI $.
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
The intersection of two classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Expansion of membership in an intersection of two classes. Theorem 12
|
|
of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.) $)
|
|
elin $p |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) ) $=
|
|
( vx cin wcel wa elex adantl cv wceq eleq1 anbi12d df-in elab2g pm5.21nii
|
|
cvv ) ABCEZFAQFZABFZACFZGZARHUASTACHIDJZBFZUCCFZGUBDARQUCAKUDTUEUAUCABLUC
|
|
ACLMDBCNOP $.
|
|
$}
|
|
|
|
${
|
|
elin2.x $e |- X = ( B i^i C ) $.
|
|
$( Membership in a class defined as an intersection. (Contributed by
|
|
Stefan O'Rear, 29-Mar-2015.) $)
|
|
elin2 $p |- ( A e. X <-> ( A e. B /\ A e. C ) ) $=
|
|
( wcel cin wa eleq2i elin bitri ) ADFABCGZFABFACFHDLAEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
elin3.x $e |- X = ( ( B i^i C ) i^i D ) $.
|
|
$( Membership in a class defined as a ternary intersection. (Contributed
|
|
by Stefan O'Rear, 29-Mar-2015.) $)
|
|
elin3 $p |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) ) $=
|
|
( cin wcel wa w3a elin anbi1i elin2 df-3an 3bitr4i ) ABCGZHZADHZIABHZACHZ
|
|
IZRIAEHSTRJQUARABCKLAPDEFMSTRNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Commutative law for intersection of classes. Exercise 7 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) $)
|
|
incom $p |- ( A i^i B ) = ( B i^i A ) $=
|
|
( vx cin cv wcel wa ancom elin 3bitr4i eqriv ) CABDZBADZCEZAFZNBFZGPOGNLF
|
|
NMFOPHNABINBAIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
ineqri.1 $e |- ( ( x e. A /\ x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to intersection. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ineqri $p |- ( A i^i B ) = C $=
|
|
( cin cv wcel wa elin bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
14-Dec-1993.) $)
|
|
ineq1 $p |- ( A = B -> ( A i^i C ) = ( B i^i C ) ) $=
|
|
( vx wceq cin cv wcel wa eleq2 anbi1d elin 3bitr4g eqrdv ) ABEZDACFZBCFZO
|
|
DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
$}
|
|
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq2 $p |- ( A = B -> ( C i^i A ) = ( C i^i B ) ) $=
|
|
( wceq cin ineq1 incom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
8-May-1994.) $)
|
|
ineq12 $p |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq1 ineq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
ineq1i.1 $e |- A = B $.
|
|
$( Equality inference for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq1i $p |- ( A i^i C ) = ( B i^i C ) $=
|
|
( wceq cin ineq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq2i $p |- ( C i^i A ) = ( C i^i B ) $=
|
|
( wceq cin ineq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
ineq12i.2 $e |- C = D $.
|
|
$( Equality inference for intersection of two classes. (Contributed by
|
|
NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
ineq12i $p |- ( A i^i C ) = ( B i^i D ) $=
|
|
( wceq cin ineq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ineq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
ineq1d $p |- ( ph -> ( A i^i C ) = ( B i^i C ) ) $=
|
|
( wceq cin ineq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for intersection of two classes. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
ineq2d $p |- ( ph -> ( C i^i A ) = ( C i^i B ) ) $=
|
|
( wceq cin ineq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
ineq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by
|
|
NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ineq12d $p |- ( ph -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ineqan12d.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by
|
|
NM, 7-Feb-2007.) $)
|
|
ineqan12d $p |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq12 syl2an ) ACDIEFICEJDFJIBGHCDEFKL $.
|
|
$}
|
|
$}
|
|
|
|
$( A frequently-used variant of subclass definition ~ df-ss . (Contributed
|
|
by NM, 10-Jan-2015.) $)
|
|
dfss1 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
|
|
( wss cin wceq df-ss incom eqeq1i bitri ) ABCABDZAEBADZAEABFJKAABGHI $.
|
|
|
|
$( Another definition of subclasshood. Similar to ~ df-ss , ~ dfss , and
|
|
~ dfss1 . (Contributed by David Moews, 1-May-2017.) $)
|
|
dfss5 $p |- ( A C_ B <-> A = ( B i^i A ) ) $=
|
|
( wss cin wceq dfss1 eqcom bitri ) ABCBADZAEAIEABFIAGH $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfin.1 $e |- F/_ x A $.
|
|
nfin.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for the intersection of classes.
|
|
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfin $p |- F/_ x ( A i^i B ) $=
|
|
( vy cin cv wcel crab dfin5 nfcri nfrabxy nfcxfr ) ABCGFHCIZFBJFBCKOAFBAF
|
|
CELDMN $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d C y $. $d D y $. $d x y $.
|
|
$( Distribute proper substitution through an intersection relation.
|
|
(Contributed by Alan Sare, 22-Jul-2012.) $)
|
|
csbing $p |- ( A e. B -> [_ A / x ]_ ( C i^i D ) =
|
|
( [_ A / x ]_ C i^i [_ A / x ]_ D ) ) $=
|
|
( vy cv cin csb wceq csbeq1 ineq12d eqeq12d nfcsb1v csbeq1a csbief vtoclg
|
|
vex nfin ) AFGZDEHZIZATDIZATEIZHZJABUAIZABDIZABEIZHZJFBCTBJZUBUFUEUIATBUA
|
|
KUJUCUGUDUHATBDKATBEKLMATUAUEFRAUCUDATDNATENSAGTJDUCEUDATDOATEOLPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
rabbi2dva.1 $e |- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) ) $.
|
|
$( Deduction from a wff to a restricted class abstraction. (Contributed by
|
|
NM, 14-Jan-2014.) $)
|
|
rabbi2dva $p |- ( ph -> ( A i^i B ) = { x e. A | ps } ) $=
|
|
( cin cv wcel crab dfin5 rabbidva syl5eq ) ADEGCHEIZCDJBCDJCDEKANBCDFLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for intersection of classes. Theorem 15 of [Suppes]
|
|
p. 26. (Contributed by NM, 5-Aug-1993.) $)
|
|
inidm $p |- ( A i^i A ) = A $=
|
|
( vx cv wcel anidm ineqri ) BAAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Associative law for intersection of classes. Exercise 9 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) $)
|
|
inass $p |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) ) $=
|
|
( vx cin cv wcel wa anass elin anbi2i bitr4i anbi1i 3bitr4i ineqri ) DABE
|
|
ZCABCEZEZDFZAGZSBGZHZSCGZHZTSQGZHZSPGZUCHSRGUDTUAUCHZHUFTUAUCIUEUHTSBCJKL
|
|
UGUBUCSABJMSAQJNO $.
|
|
$}
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) $)
|
|
in12 $p |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) ) $=
|
|
( cin incom ineq1i inass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
|
|
$.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
in32 $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B ) $=
|
|
( cin inass in12 incom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) $)
|
|
in13 $p |- ( A i^i ( B i^i C ) ) = ( C i^i ( B i^i A ) ) $=
|
|
( cin in32 incom 3eqtr4i ) BCDZADBADZCDAHDCIDBCAEAHFCIFG $.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) $)
|
|
in31 $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A ) $=
|
|
( cin in12 incom 3eqtr4i ) CABDZDACBDZDHCDIADCABEHCFIAFG $.
|
|
|
|
$( Rotate the intersection of 3 classes. (Contributed by NM,
|
|
27-Aug-2012.) $)
|
|
inrot $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i A ) i^i B ) $=
|
|
( cin in31 in32 eqtri ) ABDCDCBDADCADBDABCECBAFG $.
|
|
|
|
$( Rearrangement of intersection of 4 classes. (Contributed by NM,
|
|
21-Apr-2001.) $)
|
|
in4 $p |- ( ( A i^i B ) i^i ( C i^i D ) ) =
|
|
( ( A i^i C ) i^i ( B i^i D ) ) $=
|
|
( cin in12 ineq2i inass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
|
|
CLHI $.
|
|
|
|
$( Intersection distributes over itself. (Contributed by NM, 6-May-1994.) $)
|
|
inindi $p |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) ) $=
|
|
( cin inidm ineq1i in4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.
|
|
|
|
$( Intersection distributes over itself. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
inindir $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) ) $=
|
|
( cin inidm ineq2i in4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.
|
|
|
|
$( A relationship between subclass and intersection. Similar to Exercise 9
|
|
of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.) $)
|
|
sseqin2 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
|
|
( dfss1 ) ABC $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The intersection of two classes is a subset of one of them. Part of
|
|
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
inss1 $p |- ( A i^i B ) C_ A $=
|
|
( vx cin cv wcel elin simplbi ssriv ) CABDZACEZJFKAFKBFKABGHI $.
|
|
$}
|
|
|
|
$( The intersection of two classes is a subset of one of them. Part of
|
|
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
inss2 $p |- ( A i^i B ) C_ B $=
|
|
( cin incom inss1 eqsstr3i ) ABCBACBBADBAEF $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
|
|
(Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
ssin $p |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) ) $=
|
|
( vx cv wcel wi wal wa cin wss elin imbi2i albii jcab 19.26 3bitrri dfss2
|
|
anbi12i 3bitr4i ) DEZAFZUABFZGZDHZUBUACFZGZDHZIZUBUABCJZFZGZDHZABKZACKZIA
|
|
UJKUMUBUCUFIZGZDHUDUGIZDHUIULUQDUKUPUBUABCLMNUQURDUBUCUFONUDUGDPQUNUEUOUH
|
|
DABRDACRSDAUJRT $.
|
|
$}
|
|
|
|
${
|
|
ssini.1 $e |- A C_ B $.
|
|
ssini.2 $e |- A C_ C $.
|
|
$( An inference showing that a subclass of two classes is a subclass of
|
|
their intersection. (Contributed by NM, 24-Nov-2003.) $)
|
|
ssini $p |- A C_ ( B i^i C ) $=
|
|
( wss wa cin pm3.2i ssin mpbi ) ABFZACFZGABCHFLMDEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
ssind.1 $e |- ( ph -> A C_ B ) $.
|
|
ssind.2 $e |- ( ph -> A C_ C ) $.
|
|
$( A deduction showing that a subclass of two classes is a subclass of
|
|
their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
ssind $p |- ( ph -> A C_ ( B i^i C ) ) $=
|
|
( wss cin wa ssin biimpi syl2anc ) ABCGZBDGZBCDHGZEFMNIOBCDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Add right intersection to subclass relation. (Contributed by NM,
|
|
16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ssrin $p |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) ) $=
|
|
( vx wss cin cv wcel wa ssel anim1d elin 3imtr4g ssrdv ) ABEZDACFZBCFZODG
|
|
ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
|
|
$( Add left intersection to subclass relation. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
sslin $p |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) ) $=
|
|
( wss cin ssrin incom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
$}
|
|
|
|
$( Intersection of subclasses. (Contributed by NM, 5-May-2000.) $)
|
|
ss2in $p |- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) ) $=
|
|
( wss cin ssrin sslin sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
$( Intersection preserves subclass relationship. (Contributed by NM,
|
|
14-Sep-1999.) $)
|
|
ssinss1 $p |- ( A C_ C -> ( A i^i B ) C_ C ) $=
|
|
( cin wss wi inss1 sstr2 ax-mp ) ABDZAEACEJCEFABGJACHI $.
|
|
|
|
$( Inclusion of an intersection of two classes. (Contributed by NM,
|
|
30-Oct-2014.) $)
|
|
inss $p |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C ) $=
|
|
( wss cin ssinss1 incom syl5eqss jaoi ) ACDABEZCDBCDZABCFKJBAECABGBACFHI $.
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Combinations of difference, union, and intersection of two classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
$( Absorption law for union. (Contributed by NM, 16-Apr-2006.) $)
|
|
unabs $p |- ( A u. ( A i^i B ) ) = A $=
|
|
( cin wss cun wceq inss1 ssequn2 mpbi ) ABCZADAJEAFABGJAHI $.
|
|
|
|
$( Absorption law for intersection. (Contributed by NM, 16-Apr-2006.) $)
|
|
inabs $p |- ( A i^i ( A u. B ) ) = A $=
|
|
( cun wss cin wceq ssun1 df-ss mpbi ) AABCZDAJEAFABGAJHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Double complement and subset. Similar to ~ ddifss but inside a class
|
|
` B ` instead of the universal class ` _V ` . In classical logic the
|
|
subset operation on the right hand side could be an equality (that is,
|
|
` A C_ B <-> ( B \ ( B \ A ) ) = A ` ). (Contributed by Jim Kingdon,
|
|
24-Jul-2018.) $)
|
|
ssddif $p |- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) ) $=
|
|
( vx cv wcel wi wal cdif wss wa wn simpr con2i anim2i eldif notbii anbi2i
|
|
ancr bitri dfss2 sylibr syl6 eldifi imim2i impbii albii 3bitr4i ) CDZAEZU
|
|
HBEZFZCGUIUHBBAHZHZEZFZCGABIAUMIUKUOCUKUOUKUIUJUIJZUNUIUJRUPUJUJUIKZJZKZJ
|
|
ZUNUIUSUJURUIUJUQLMNUNUJUHULEZKZJUTUHBULOVBUSUJVAURUHBAOPQSUAUBUNUJUIUHBU
|
|
LUCUDUEUFCABTCAUMTUG $.
|
|
|
|
$( Union of two classes and class difference. In classical logic this
|
|
would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.) $)
|
|
unssdif $p |- ( A u. B ) C_ ( _V \ ( ( _V \ A ) \ B ) ) $=
|
|
( vx cun cvv cdif cv wcel wo wn wa vex eldif mpbiran anbi1i ioran 3bitr4i
|
|
biimpi con2i elun 3imtr4i ssriv ) CABDZEEAFZBFZFZCGZAHZUGBHZIZUGUEHZJZUGU
|
|
CHUGUFHZUKUJUKUJJZUGUDHZUIJZKUHJZUPKUKUNUOUQUPUOUGEHZUQCLZUGEAMNOUGUDBMUH
|
|
UIPQRSUGABTUMURULUSUGEUEMNUAUB $.
|
|
|
|
$( Intersection of two classes and class difference. In classical logic
|
|
this would be an equality. (Contributed by Jim Kingdon,
|
|
24-Jul-2018.) $)
|
|
inssdif $p |- ( A i^i B ) C_ ( A \ ( _V \ B ) ) $=
|
|
( vx cin cvv cdif cv wcel wa wn elndif anim2i elin eldif 3imtr4i ssriv )
|
|
CABDZAEBFZFZCGZAHZTBHZIUATRHJZITQHTSHUBUCUATBEKLTABMTARNOP $.
|
|
|
|
$( Difference with intersection. Theorem 33 of [Suppes] p. 29.
|
|
(Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
difin $p |- ( A \ ( A i^i B ) ) = ( A \ B ) $=
|
|
( vx cdif cv wcel wa wn wfal wi ax-in2 expd dfnot syl6ibr com12 imdistani
|
|
cin simpr con3i eldif anim2i impbii notbii anbi2i bitri 3bitr4i eqriv
|
|
elin ) CAABQZDZABDZCEZAFZUMULBFZGZHZGZUMUNHZGZULUJFZULUKFUQUSUMUPURUPUMUR
|
|
UPUMUNIJURUPUMUNIUOIKLUNMNOPURUPUMUOUNUMUNRSUAUBUTUMULUIFZHZGUQULAUITVBUP
|
|
UMVAUOULABUHUCUDUEULABTUFUG $.
|
|
$}
|
|
|
|
$( Double complement under universal class. In classical logic (or given an
|
|
additional hypothesis, as in ~ ddifnel ), this is equality rather than
|
|
subset. (Contributed by Jim Kingdon, 24-Jul-2018.) $)
|
|
ddifss $p |- A C_ ( _V \ ( _V \ A ) ) $=
|
|
( cvv wss cdif ssv ssddif mpbi ) ABCABBADDCAEABFG $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Union as a subset of class complement and intersection (De Morgan's
|
|
law). One direction of the definition of union in [Mendelson] p. 231.
|
|
This would be an equality, rather than subset, in classical logic.
|
|
(Contributed by Jim Kingdon, 25-Jul-2018.) $)
|
|
unssin $p |- ( A u. B ) C_ ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) ) $=
|
|
( vx cun cvv cdif cin cv wcel wo wn wa oranim eldifn anim12i nsyl sylnibr
|
|
elin elun vex eldif mpbiran 3imtr4i ssriv ) CABDZEEAFZEBFZGZFZCHZAIZUJBIZ
|
|
JZUJUHIZKZUJUEIUJUIIZUMUJUFIZUJUGIZLZUNUMUKKZULKZLUSUKULMUQUTURVAUJEANUJE
|
|
BNOPUJUFUGRQUJABSUPUJEIUOCTUJEUHUAUBUCUD $.
|
|
|
|
$( Intersection in terms of class difference and union (De Morgan's law).
|
|
Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an
|
|
equality, rather than subset, in classical logic. (Contributed by Jim
|
|
Kingdon, 25-Jul-2018.) $)
|
|
inssun $p |- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) $=
|
|
( vx cin cvv cdif cv wcel wa wn wo pm3.1 eldifn orim12i nsyl elun sylnibr
|
|
cun elin vex eldif mpbiran 3imtr4i ssriv ) CABDZEEAFZEBFZRZFZCGZAHZUJBHZI
|
|
ZUJUHHZJZUJUEHUJUIHZUMUJUFHZUJUGHZKZUNUMUKJZULJZKUSUKULLUQUTURVAUJEAMUJEB
|
|
MNOUJUFUGPQUJABSUPUJEHUOCTUJEUHUAUBUCUD $.
|
|
$}
|
|
|
|
$( Intersection of two classes and class difference. In classical logic,
|
|
such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather
|
|
than subset. (Contributed by Jim Kingdon, 26-Jul-2018.) $)
|
|
inssddif $p |- ( A i^i B ) C_ ( A \ ( A \ B ) ) $=
|
|
( cin cdif wss inss1 ssddif mpbi difin difeq2i sseqtri ) ABCZAALDZDZAABDZDL
|
|
AELNEABFLAGHMOAABIJK $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Intersection with universal complement. Remark in [Stoll] p. 20.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
invdif $p |- ( A i^i ( _V \ B ) ) = ( A \ B ) $=
|
|
( vx cvv cdif cin cv wcel wa vex eldif mpbiran anbi2i elin 3bitr4i eqriv
|
|
wn ) CADBEZFZABEZCGZAHZUARHZIUBUABHQZIUASHUATHUCUDUBUCUADHUDCJUADBKLMUAAR
|
|
NUAABKOP $.
|
|
|
|
$( Intersection with class difference. Theorem 34 of [Suppes] p. 29.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
indif $p |- ( A i^i ( A \ B ) ) = ( A \ B ) $=
|
|
( vx cdif cin cv wcel wn wa anabs5 elin eldif anbi2i bitri 3bitr4i eqriv
|
|
) CAABDZEZQCFZAGZTSBGHZIZIZUBSRGZSQGZTUAJUDTUEIUCSAQKUEUBTSABLZMNUFOP $.
|
|
$}
|
|
|
|
$( Bring an intersection in and out of a class difference. (Contributed by
|
|
Jeff Hankins, 15-Jul-2009.) $)
|
|
indif2 $p |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C ) $=
|
|
( cin cvv cdif inass invdif ineq2i 3eqtr3ri ) ABDZECFZDABLDZDKCFABCFZDABLGK
|
|
CHMNABCHIJ $.
|
|
|
|
$( Bring an intersection in and out of a class difference. (Contributed by
|
|
Mario Carneiro, 15-May-2015.) $)
|
|
indif1 $p |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C ) $=
|
|
( cdif cin indif2 incom difeq1i 3eqtr3i ) BACDZEBAEZCDJBEABEZCDBACFBJGKLCBA
|
|
GHI $.
|
|
|
|
$( Commutation law for intersection and difference. (Contributed by Scott
|
|
Fenton, 18-Feb-2013.) $)
|
|
indifcom $p |- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) ) $=
|
|
( cin cdif incom difeq1i indif2 3eqtr4i ) ABDZCEBADZCEABCEDBACEDJKCABFGABCH
|
|
BACHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distributive law for intersection over union. Exercise 10 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
indi $p |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) ) $=
|
|
( vx cun cin cv wcel wo wa andi elin orbi12i bitr4i anbi2i 3bitr4i ineqri
|
|
elun ) DABCEZABFZACFZEZDGZAHZUCBHZUCCHZIZJZUCTHZUCUAHZIZUDUCSHZJUCUBHUHUD
|
|
UEJZUDUFJZIUKUDUEUFKUIUMUJUNUCABLUCACLMNULUGUDUCBCROUCTUARPQ $.
|
|
|
|
$( Distributive law for union over intersection. Exercise 11 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
undi $p |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) ) $=
|
|
( vx cin cv wcel wo wa elin orbi2i ordi elun anbi12i bitr2i 3bitri uneqri
|
|
cun ) DABCEZABRZACRZEZDFZAGZUCSGZHUDUCBGZUCCGZIZHUDUFHZUDUGHZIZUCUBGZUEUH
|
|
UDUCBCJKUDUFUGLULUCTGZUCUAGZIUKUCTUAJUMUIUNUJUCABMUCACMNOPQ $.
|
|
$}
|
|
|
|
$( Distributive law for intersection over union. Theorem 28 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Sep-2002.) $)
|
|
indir $p |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) ) $=
|
|
( cun cin indi incom uneq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
|
|
NLACGBCGHI $.
|
|
|
|
$( Distributive law for union over intersection. Theorem 29 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Sep-2002.) $)
|
|
undir $p |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) ) $=
|
|
( cin cun undi uncom ineq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
|
|
NLACGBCGHI $.
|
|
|
|
$( Equality of union and intersection implies equality of their arguments.
|
|
(Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
uneqin $p |- ( ( A u. B ) = ( A i^i B ) <-> A = B ) $=
|
|
( cun cin wceq wss wa eqimss unss ssin sstr sylbir simpl anim12i syl sylibr
|
|
eqss unidm inidm eqtr4i uneq2 ineq2 3eqtr3a impbii ) ABCZABDZEZABEZUGABFZBA
|
|
FZGZUHUGUEUFFZUKUEUFHULAUFFZBUFFZGUKABUFIUMUIUNUJUMAAFUIGUIAABJAABKLUNUJBBF
|
|
ZGUJBABJUJUOMLNLOABQPUHAACZAADZUEUFUPAUQARASTABAUAABAUBUCUD $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
difundi $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) i^i ( A \ C ) ) $=
|
|
( vx cun cdif cin cv wcel wa wn eldif anbi12i elin wo notbii anbi2i bitri
|
|
elun ioran anandi 3bitr4ri eqriv ) DABCEZFZABFZACFZGZDHZUFIZUIUGIZJUIAIZU
|
|
IBIZKZJZULUICIZKZJZJZUIUHIUIUEIZUJUOUKURUIABLUIACLMUIUFUGNUTULUNUQJZJZUSU
|
|
TULUMUPOZKZJZVBUTULUIUDIZKZJVEUIAUDLVGVDULVFVCUIBCSPQRVDVAULUMUPTQRULUNUQ
|
|
UARUBUC $.
|
|
$}
|
|
|
|
$( Distributive law for class difference. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
difundir $p |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) ) $=
|
|
( cun cvv cdif cin indir invdif uneq12i 3eqtr3i ) ABDZECFZGAMGZBMGZDLCFACFZ
|
|
BCFZDABMHLCINPOQACIBCIJK $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distributive law for class difference. In classical logic, for example,
|
|
theorem 40 of [Suppes] p. 29, this is an equality instead of subset.
|
|
(Contributed by Jim Kingdon, 26-Jul-2018.) $)
|
|
difindiss $p |- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) ) $=
|
|
( vx cdif cun cin cv wcel wo elun wa wn orbi12i andi bitr4i pm3.14 anim2i
|
|
eldif sylbi elin notbii anbi2i bitr2i sylib ssriv ) DABEZACEZFZABCGZEZDHZ
|
|
UIIULUGIZULUHIZJZULUKIZULUGUHKUOULAIZULBIZULCIZLZMZLZUPUOUQURMZUSMZJZLZVB
|
|
UOUQVCLZUQVDLZJVFUMVGUNVHULABSULACSNUQVCVDOPVEVAUQURUSQRTUPUQULUJIZMZLVBU
|
|
LAUJSVJVAUQVIUTULBCUAUBUCUDUETUF $.
|
|
$}
|
|
|
|
$( Distributive law for class difference. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
difindir $p |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) ) $=
|
|
( cin cvv cdif inindir invdif ineq12i 3eqtr3i ) ABDZECFZDALDZBLDZDKCFACFZBC
|
|
FZDABLGKCHMONPACHBCHIJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Distribute intersection over difference. (Contributed by Scott Fenton,
|
|
14-Apr-2011.) $)
|
|
indifdir $p |- ( ( A \ B ) i^i C ) = ( ( A i^i C ) \ ( B i^i C ) ) $=
|
|
( vx cdif cv wcel wn wa elin notbii anbi12i eldif anbi1i bitri an32 simpl
|
|
cin con3i wfal anim2i ax-in2 expcomd impcom dfnot sylibr adantll 3bitr4ri
|
|
wi jca impbii eqriv ) DABEZCRZACRZBCRZEZDFZUOGZURUPGZHZIURAGZURCGZIZURBGZ
|
|
VCIZHZIZURUQGURUNGZUSVDVAVGURACJUTVFURBCJKLURUOUPMVIVBVEHZIZVCIZVHVIURUMG
|
|
ZVCIVLURUMCJVMVKVCURABMNOVLVDVJIZVHVBVJVCPVNVHVJVGVDVFVEVEVCQSUAVHVDVJVDV
|
|
GQVCVGVJVBVCVGIVETUIZVJVGVCVOVGVEVCTVFTUBUCUDVEUEUFUGUJUKOOUHUL $.
|
|
$}
|
|
|
|
$( Set difference with a set difference. In classical logic this would be
|
|
equality rather than subset. (Contributed by Jim Kingdon,
|
|
27-Jul-2018.) $)
|
|
difdif2ss $p |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) ) $=
|
|
( cin cun cvv wss inssdif unss2 ax-mp difindiss sstri invdif eqcomi difeq2i
|
|
cdif sseqtr4i ) ABPZACDZEZABFCPZDZPZABCPZPTRAUAPZEZUCSUEGTUFGACHSUERIJABUAK
|
|
LUDUBAUBUDBCMNOQ $.
|
|
|
|
$( De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
|
|
(Contributed by NM, 18-Aug-2004.) $)
|
|
undm $p |- ( _V \ ( A u. B ) ) = ( ( _V \ A ) i^i ( _V \ B ) ) $=
|
|
( cvv difundi ) CABD $.
|
|
|
|
$( De Morgan's law for intersection. In classical logic, this would be
|
|
equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19.
|
|
(Contributed by Jim Kingdon, 27-Jul-2018.) $)
|
|
indmss $p |- ( ( _V \ A ) u. ( _V \ B ) ) C_ ( _V \ ( A i^i B ) ) $=
|
|
( cvv difindiss ) CABD $.
|
|
|
|
$( A relationship involving double difference and union. (Contributed by NM,
|
|
29-Aug-2004.) $)
|
|
difun1 $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C ) $=
|
|
( cvv cdif cin cun inass invdif eqtr3i undm ineq2i difeq1i ) ADBEZFZCEZABCG
|
|
ZEZABEZCEANDCEZFZFZPROTFUBPANTHOCIJADQEZFUBRUCUAABCKLAQIJJOSCABIMJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( A subset relationship involving class union and class difference. In
|
|
classical logic, this would be equality rather than subset, as in the
|
|
first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by
|
|
Jim Kingdon, 28-Jul-2018.) $)
|
|
undif3ss $p |- ( A u. ( B \ C ) ) C_ ( ( A u. B ) \ ( C \ A ) ) $=
|
|
( vx cdif cun cv wcel wo wn wa elun eldif orbi2i orc olc jca anim12i jaoi
|
|
simpl orcd adantr adantl ccase impbii 3bitri biimpri pm4.53r sylibr sylbi
|
|
sylnibr ssriv ) DABCEZFZABFZCAEZEZDGZUNHZURAHZURBHZIZURCHZJZUTIZKZURUQHZU
|
|
SUTURUMHZIUTVAVDKZIZVFURAUMLVHVIUTURBCMNVJVFUTVFVIUTVBVEUTVAOUTVDPQVAVBVD
|
|
VEVAUTPVDUTORSUTVDVAUTVJUTVDKUTVIUTVDTUAVIUTPUTVJUTUTVIOZUBUTVJVAVKUCUDUE
|
|
UFVFURUOHZURUPHZJZKVGVBVLVEVNVLVBURABLUGVEVCUTJKVMVCUTUHURCAMUKRURUOUPMUI
|
|
UJUL $.
|
|
|
|
$( Represent a set difference as an intersection with a larger difference.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
difin2 $p |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) ) $=
|
|
( vx wss cdif cin cv wcel wn ssel pm4.71d anbi1d eldif anbi1i ancom anass
|
|
wa elin bitr4i 3bitri 3bitr4g eqrdv ) ACEZDABFZCBFZAGZUDDHZAIZUHBIJZRUIUH
|
|
CIZRZUJRZUHUEIUHUGIZUDUIULUJUDUIUKACUHKLMUHABNUNUHUFIZUIRUKUJRZUIRZUMUHUF
|
|
ASUOUPUIUHCBNOUQUIUPRUMUPUIPUIUKUJQTUAUBUC $.
|
|
$}
|
|
|
|
$( Swap second and third argument of double difference. (Contributed by NM,
|
|
18-Aug-2004.) $)
|
|
dif32 $p |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B ) $=
|
|
( cun cdif uncom difeq2i difun1 3eqtr3i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
|
|
ACBHI $.
|
|
|
|
$( Absorption-like law for class difference: you can remove a class only
|
|
once. (Contributed by FL, 2-Aug-2009.) $)
|
|
difabs $p |- ( ( A \ B ) \ B ) = ( A \ B ) $=
|
|
( cun cdif difun1 unidm difeq2i eqtr3i ) ABBCZDABDZBDJABBEIBABFGH $.
|
|
|
|
$( Two ways to express symmetric difference. This theorem shows the
|
|
equivalence of the definition of symmetric difference in [Stoll] p. 13 and
|
|
the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by
|
|
NM, 17-Aug-2004.) $)
|
|
symdif1 $p |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) ) $=
|
|
( cun cin cdif difundir difin incom difeq2i eqtri uneq12i eqtr2i ) ABCABDZE
|
|
AMEZBMEZCABEZBAEZCABMFNPOQABGOBBADZEQMRBABHIBAGJKL $.
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Class abstractions with difference, union, and intersection of two classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Expressing symmetric difference with exclusive-or or two differences.
|
|
(Contributed by Jim Kingdon, 28-Jul-2018.) $)
|
|
symdifxor $p |- ( ( A \ B ) u. ( B \ A ) ) =
|
|
{ x | ( x e. A \/_ x e. B ) } $=
|
|
( cv wcel wxo cdif cun wo wn eldif orbi12i elun excxor ancom orbi2i bitri
|
|
wa 3bitr4i abbi2i ) ADZBEZUACEZFZABCGZCBGZHZUAUEEZUAUFEZIUBUCJRZUCUBJZRZI
|
|
ZUAUGEUDUHUJUIULUABCKUACBKLUAUEUFMUDUJUKUCRZIUMUBUCNUNULUJUKUCOPQST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d ph y $. $d ps y $.
|
|
$( Union of two class abstractions. (Contributed by NM, 29-Sep-2002.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
unab $p |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) } $=
|
|
( vy cab wo wsb cv wcel sbor df-clab orbi12i 3bitr4ri uneqri ) DACEZBCEZA
|
|
BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.
|
|
|
|
$( Intersection of two class abstractions. (Contributed by NM,
|
|
29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
inab $p |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) } $=
|
|
( vy cab wa wsb cv wcel sban df-clab anbi12i 3bitr4ri ineqri ) DACEZBCEZA
|
|
BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.
|
|
|
|
$( Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difab $p |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) } $=
|
|
( vy cab wn wcel wsb df-clab sban bicomi xchbinxr anbi12i 3bitrri difeqri
|
|
wa cv sbn ) DACEZBCEZABFZPZCEZDQZUCGUBCDHACDHZUACDHZPUDSGZUDTGZFZPUBDCIAU
|
|
ACDJUEUGUFUIUGUEADCIKUFBCDHUHBCDRBDCILMNO $.
|
|
$}
|
|
|
|
$( A class builder defined by a negation. (Contributed by FL,
|
|
18-Sep-2010.) $)
|
|
notab $p |- { x | -. ph } = ( _V \ { x | ph } ) $=
|
|
( cv cvv wcel wn wa cab cdif crab df-rab rabab eqtr3i difab abid2 difeq1i )
|
|
BCDEZAFZGBHZRBHZDABHZIZRBDJSTRBDKRBLMQBHZUAISUBQABNUCDUABDOPMM $.
|
|
|
|
$( Union of two restricted class abstractions. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
unrab $p |- ( { x e. A | ph } u. { x e. A | ps } ) =
|
|
{ x e. A | ( ph \/ ps ) } $=
|
|
( crab cun cv wcel wa cab wo df-rab uneq12i unab andi abbii eqtr4i ) ACDEZB
|
|
CDEZFCGDHZAIZCJZTBIZCJZFZABKZCDEZRUBSUDACDLBCDLMUGTUFIZCJZUEUFCDLUEUAUCKZCJ
|
|
UIUAUCCNUHUJCTABOPQQQ $.
|
|
|
|
$( Intersection of two restricted class abstractions. (Contributed by NM,
|
|
1-Sep-2006.) $)
|
|
inrab $p |- ( { x e. A | ph } i^i { x e. A | ps } ) =
|
|
{ x e. A | ( ph /\ ps ) } $=
|
|
( crab cin cv wcel wa cab df-rab ineq12i inab anandi abbii eqtr4i ) ACDEZBC
|
|
DEZFCGDHZAIZCJZSBIZCJZFZABIZCDEZQUARUCACDKBCDKLUFSUEIZCJZUDUECDKUDTUBIZCJUH
|
|
TUBCMUGUICSABNOPPP $.
|
|
|
|
${
|
|
$d x B $.
|
|
$( Intersection with a restricted class abstraction. (Contributed by NM,
|
|
19-Nov-2007.) $)
|
|
inrab2 $p |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph } $=
|
|
( crab cin cv wcel cab df-rab abid2 eqcomi ineq12i inab elin anbi1i bitri
|
|
wa an32 eqtr4i abbii ) ABCEZDFBGZCHZARZBIZUCDHZBIZFZABCDFZEZUBUFDUHABCJUH
|
|
DBDKLMUKUCUJHZARZBIZUIABUJJUIUEUGRZBIUNUEUGBNUMUOBUMUDUGRZARUOULUPAUCCDOP
|
|
UDUGASQUATTT $.
|
|
$}
|
|
|
|
$( Difference of two restricted class abstractions. (Contributed by NM,
|
|
23-Oct-2004.) $)
|
|
difrab $p |- ( { x e. A | ph } \ { x e. A | ps } ) =
|
|
{ x e. A | ( ph /\ -. ps ) } $=
|
|
( crab cdif cv wcel wa wn df-rab difeq12i difab anass simpr con3i anim2i wi
|
|
cab eqtr4i pm3.2 adantr con3d imdistani impbii bitr3i abbii ) ACDEZBCDEZFCG
|
|
DHZAIZCSZUJBIZCSZFZABJZIZCDEZUHULUIUNACDKBCDKLURUJUQIZCSZUOUQCDKUOUKUMJZIZC
|
|
SUTUKUMCMUSVBCUSUKUPIZVBUJAUPNVCVBUPVAUKUMBUJBOPQUKVAUPUKBUMUJBUMRAUJBUAUBU
|
|
CUDUEUFUGTTT $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of restricted class abstraction. (Contributed by
|
|
NM, 20-Sep-2003.) $)
|
|
dfrab2 $p |- { x e. A | ph } = ( { x | ph } i^i A ) $=
|
|
( crab cv wcel wa cab cin df-rab inab abid2 ineq1i eqtr3i incom 3eqtri )
|
|
ABCDBECFZAGBHZCABHZIZSCIABCJQBHZSIRTQABKUACSBCLMNCSOP $.
|
|
|
|
$( Alternate definition of restricted class abstraction. (Contributed by
|
|
Mario Carneiro, 8-Sep-2013.) $)
|
|
dfrab3 $p |- { x e. A | ph } = ( A i^i { x | ph } ) $=
|
|
( crab cv wcel wa cab cin df-rab inab abid2 ineq1i 3eqtr2i ) ABCDBECFZAGB
|
|
HOBHZABHZICQIABCJOABKPCQBCLMN $.
|
|
|
|
$( Complementation of restricted class abstractions. (Contributed by Mario
|
|
Carneiro, 3-Sep-2015.) $)
|
|
notrab $p |- ( A \ { x e. A | ph } ) = { x e. A | -. ph } $=
|
|
( cv wcel cab cdif wn crab difab cin difin dfrab3 difeq2i difeq1i 3eqtr4i
|
|
wa abid2 df-rab ) BDCEZBFZABFZGZTAHZQBFCABCIZGZUDBCITABJCCUBKZGCUBGUFUCCU
|
|
BLUEUGCABCMNUACUBBCROPUDBCSP $.
|
|
|
|
$( Restricted class abstraction with a common superset. (Contributed by
|
|
Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro,
|
|
8-Nov-2015.) $)
|
|
dfrab3ss $p |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) ) $=
|
|
( wss cab crab wceq df-ss ineq1 eqcomd sylbi dfrab3 ineq2i eqtr4i 3eqtr4g
|
|
cin inass ) CDEZCABFZQZCDQZTQZABCGCABDGZQZSUBCHZUAUCHCDIUFUCUAUBCTJKLABCM
|
|
UECDTQZQUCUDUGCABDMNCDTROP $.
|
|
$}
|
|
|
|
$( Abstraction restricted to a union. (Contributed by Stefan O'Rear,
|
|
5-Feb-2015.) $)
|
|
rabun2 $p |- { x e. ( A u. B ) | ph } =
|
|
( { x e. A | ph } u. { x e. B | ph } ) $=
|
|
( cun crab cv wcel wa cab df-rab uneq12i elun anbi1i andir bitri abbii unab
|
|
wo eqtr4i ) ABCDEZFBGZUAHZAIZBJZABCFZABDFZEZABUAKUHUBCHZAIZBJZUBDHZAIZBJZEZ
|
|
UEUFUKUGUNABCKABDKLUEUJUMSZBJUOUDUPBUDUIULSZAIUPUCUQAUBCDMNUIULAOPQUJUMBRTT
|
|
T $.
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Restricted uniqueness with difference, union, and intersection
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Transfer uniqueness to a smaller subclass. (Contributed by NM,
|
|
20-Oct-2005.) $)
|
|
reuss2 $p |- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\
|
|
( E. x e. A ph /\ E! x e. B ps ) ) -> E! x e. A ph ) $=
|
|
( wrex wreu wa wss wi wral wcel wex weu df-rex df-reu anbi12i wal sylan2b
|
|
cv wmo df-ral ssel prth sylan exp4b com23 a2d imp4a alimdv imp euimmo syl
|
|
eu5 simplbi2 syl9 imp32 sylibr ) ACDFZBCEGZHDEIZABJZCDKZHZCTZDLZAHZCMZVEE
|
|
LZBHZCNZHZACDGZUSVHUTVKACDOBCEPQVDVLHVGCNZVMVDVHVKVNVDVKVGCUAZVHVNVDVGVJJ
|
|
ZCRZVKVOJVCVAVFVBJZCRZVQVBCDUBVAVSVQVAVRVPCVAVRVFAVJVAVFVBAVJJZVAVBVFVTVA
|
|
VBVFAVJVAVFVIJVBVPDEVEUCVFVIABUDUEUFUGUHUIUJUKSVGVJCULUMVNVHVOVGCUNUOUPUQ
|
|
ACDPURS $.
|
|
|
|
$( Transfer uniqueness to a smaller subclass. (Contributed by NM,
|
|
21-Aug-1999.) $)
|
|
reuss $p |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) ->
|
|
E! x e. A ph ) $=
|
|
( wss wrex wreu wi wral wa cv wcel idd rgen reuss2 mpanl2 3impb ) CDEZABC
|
|
FZABDGZABCGZRAAHZBCISTJUAUBBCBKCLAMNAABCDOPQ $.
|
|
|
|
$( Transfer uniqueness to a smaller class. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
reuun1 $p |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) )
|
|
-> E! x e. A ph ) $=
|
|
( cun wss wo wi wral wrex wreu wa ssun1 orc rgenw reuss2 mpanl12 ) DDEFZG
|
|
AABHZIZCDJACDKTCSLMACDLDENUACDABOPATCDSQR $.
|
|
|
|
$( Transfer uniqueness to a smaller or larger class. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
reuun2 $p |- ( -. E. x e. B ph ->
|
|
( E! x e. ( A u. B ) ph <-> E! x e. A ph ) ) $=
|
|
( wrex wn cv wcel wa wo weu cun wreu wex df-rex euor2 sylnbi df-reu bitri
|
|
wb elun anbi1i andir orcom eubii 3bitr4g ) ABDEZFBGZDHZAIZUHCHZAIZJZBKZUL
|
|
BKZABCDLZMZABCMUGUJBNUNUOTABDOUJULBPQUQUHUPHZAIZBKUNABUPRUSUMBUSUKUIJZAIZ
|
|
UMURUTAUHCDUAUBVAULUJJUMUKUIAUCULUJUDSSUESABCRUF $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
NM, 21-Aug-1999.) $)
|
|
reupick $p |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) ->
|
|
( x e. A <-> x e. B ) ) $=
|
|
( wss wrex wreu wa cv wcel wi ssel ad2antrr wex weu df-rex df-reu anbi12i
|
|
ancrd anim1d an32 syl6ib eximdv eupick ex com23 imp32 sylan2b expcomd imp
|
|
syl9 impbid ) CDEZABCFZABDGZHZHZAHBIZCJZURDJZUMUSUTKUPACDURLZMUQAUTUSKUQU
|
|
TAUSUPUMUSAHZBNZUTAHZBOZHVDUSKZUNVCUOVEABCPABDQRUMVCVEVFUMVEVCVFUMVCVDUSH
|
|
ZBNZVEVFUMVBVGBUMVBUTUSHZAHVGUMUSVIAUMUSUTVASTUTUSAUAUBUCVEVHVFVDUSBUDUEU
|
|
KUFUGUHUIUJUL $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
Mario Carneiro, 19-Nov-2016.) $)
|
|
reupick3 $p |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) ->
|
|
( ph -> ps ) ) $=
|
|
( wreu wa wrex cv wcel wi weu wex df-reu df-rex anass exbii bitr4i eupick
|
|
syl2anb expd 3impia ) ACDEZABFZCDGZCHDIZABJUBUDFUEABUBUEAFZCKUFBFZCLZUFBJ
|
|
UDACDMUDUEUCFZCLUHUCCDNUGUICUEABOPQUFBCRSTUA $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro,
|
|
19-Nov-2016.) $)
|
|
reupick2 $p |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\
|
|
E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) ) $=
|
|
( wi wral wrex wreu w3a cv wcel ancr ralimi rexim syl reupick3 3exp com12
|
|
wa syl6 3imp1 rsp 3ad2ant1 imp impbid ) BAEZCDFZBCDGZACDHZIZCJDKZSABUGUHU
|
|
IUKABEZUGUHABSZCDGZUIUKULEZEUGBUMEZCDFUHUNEUFUPCDBALMBUMCDNOUIUNUOUIUNUKU
|
|
LABCDPQRTUAUJUKUFUGUHUKUFEUIUFCDUBUCUDUE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The empty set
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for the empty or null set. $)
|
|
$c (/) $. $( null set $)
|
|
|
|
$( Extend class notation to include the empty set. $)
|
|
c0 $a class (/) $.
|
|
|
|
$( Define the empty set. Special case of Exercise 4.10(o) of [Mendelson]
|
|
p. 231. For a more traditional definition, but requiring a dummy
|
|
variable, see ~ dfnul2 . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-nul $a |- (/) = ( _V \ _V ) $.
|
|
|
|
$( Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring]
|
|
p. 20. (Contributed by NM, 26-Dec-1996.) $)
|
|
dfnul2 $p |- (/) = { x | -. x = x } $=
|
|
( cv wceq wn c0 wcel cdif wa df-nul eleq2i eldif pm3.24 eqid notnoti 2false
|
|
cvv 3bitri abbi2i ) ABZSCZDZAESEFSPPGZFSPFZUCDHZUAEUBSIJSPPKUDUAUCLTSMNOQR
|
|
$.
|
|
|
|
$( Alternate definition of the empty set. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
dfnul3 $p |- (/) = { x e. A | -. x e. A } $=
|
|
( weq wn cv wcel wa c0 crab equid notnoti pm3.24 2false abbii dfnul2 df-rab
|
|
cab 3eqtr4i ) AACZDZAQAEBFZUADZGZAQHUBABITUCATUCSAJKUALMNAOUBABPR $.
|
|
|
|
$( The empty set has no elements. Theorem 6.14 of [Quine] p. 44.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
noel $p |- -. A e. (/) $=
|
|
( c0 wcel cvv cdif eldifi eldifn pm2.65i df-nul eleq2i mtbir ) ABCADDEZCZMA
|
|
DCADDFADDGHBLAIJK $.
|
|
|
|
$( If a set has elements, it is not empty. A set with elements is also
|
|
inhabited, see ~ elex2 . (Contributed by NM, 31-Dec-1993.) $)
|
|
n0i $p |- ( B e. A -> -. A = (/) ) $=
|
|
( c0 wceq wcel noel eleq2 mtbiri con2i ) ACDZBAEZJKBCEBFACBGHI $.
|
|
|
|
$( If a set has elements, it is not empty. A set with elements is also
|
|
inhabited, see ~ elex2 . (Contributed by NM, 31-Dec-1993.) $)
|
|
ne0i $p |- ( B e. A -> A =/= (/) ) $=
|
|
( wcel c0 n0i neneqad ) BACADABEF $.
|
|
|
|
$( The universal class is not equal to the empty set. (Contributed by NM,
|
|
11-Sep-2008.) $)
|
|
vn0 $p |- _V =/= (/) $=
|
|
( vx cv cvv wcel c0 wne vex ne0i ax-mp ) ABZCDCEFAGCJHI $.
|
|
|
|
$( The universal class is inhabited. (Contributed by Jim Kingdon,
|
|
17-Dec-2018.) $)
|
|
vn0m $p |- E. x x e. _V $=
|
|
( cv cvv wcel wex vex 19.8a ax-mp ) ABCDZIAEAFIAGH $.
|
|
|
|
${
|
|
n0rf.1 $e |- F/_ x A $.
|
|
$( An inhabited class is nonempty. Following the Definition of [Bauer],
|
|
p. 483, we call a class ` A ` nonempty if ` A =/= (/) ` and inhabited if
|
|
it has at least one element. In classical logic these two concepts are
|
|
equivalent, for example see Proposition 5.17(1) of [TakeutiZaring]
|
|
p. 20. This version of ~ n0r requires only that ` x ` not be free in,
|
|
rather than not occur in, ` A ` . (Contributed by Jim Kingdon,
|
|
31-Jul-2018.) $)
|
|
n0rf $p |- ( E. x x e. A -> A =/= (/) ) $=
|
|
( cv wcel wex wn wal c0 wne exalim wceq wb nfcv cleqf noel nbn necon3abii
|
|
albii bitr4i sylibr ) ADZBEZAFUCGZAHZGBIJUCAKUEBIBILUCUBIEZMZAHUEABICAINO
|
|
UDUGAUFUCUBPQSTRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An inhabited class is nonempty. See ~ n0rf for more discussion.
|
|
(Contributed by Jim Kingdon, 31-Jul-2018.) $)
|
|
n0r $p |- ( E. x x e. A -> A =/= (/) ) $=
|
|
( nfcv n0rf ) ABABCD $.
|
|
|
|
$( An inhabited class is nonempty. See ~ n0rf for more discussion.
|
|
(Contributed by Jim Kingdon, 31-Jul-2018.) $)
|
|
neq0r $p |- ( E. x x e. A -> -. A = (/) ) $=
|
|
( cv wcel wex c0 n0r neneqd ) ACBDAEBFABGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
reximdva0m.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
|
|
$( Restricted existence deduced from inhabited class. (Contributed by Jim
|
|
Kingdon, 31-Jul-2018.) $)
|
|
reximdva0m $p |- ( ( ph /\ E. x x e. A ) -> E. x e. A ps ) $=
|
|
( cv wcel wex wa wrex ex ancld eximdv imp df-rex sylibr ) ACFDGZCHZIQBIZC
|
|
HZBCDJARTAQSCAQBAQBEKLMNBCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A case of equivalence of "at most one" and "only one". If a class is
|
|
inhabited, that class having at most one element is equivalent to it
|
|
having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.) $)
|
|
n0mmoeu $p |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) ) $=
|
|
( cv wcel exmoeu2 ) ACBDAE $.
|
|
$}
|
|
|
|
$( Vacuous existential quantification is false. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
rex0 $p |- -. E. x e. (/) ph $=
|
|
( c0 cv wcel wn noel pm2.21i nrex ) ABCBDZCEAFJGHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The empty set has no elements. Theorem 2 of [Suppes] p. 22.
|
|
(Contributed by NM, 29-Aug-1993.) $)
|
|
eq0 $p |- ( A = (/) <-> A. x -. x e. A ) $=
|
|
( c0 wceq cv wcel wb wal wn nfcv cleqf noel nbn albii bitr4i ) BCDAEZBFZP
|
|
CFZGZAHQIZAHABCABJACJKTSARQPLMNO $.
|
|
|
|
$( The universe contains every set. (Contributed by NM, 11-Sep-2006.) $)
|
|
eqv $p |- ( A = _V <-> A. x x e. A ) $=
|
|
( cvv wceq cv wcel wb wal dfcleq vex tbt albii bitr4i ) BCDAEZBFZNCFZGZAH
|
|
OAHABCIOQAPOAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y $.
|
|
$( Membership of the empty set in another class. (Contributed by NM,
|
|
29-Jun-2004.) $)
|
|
0el $p |- ( (/) e. A <-> E. x e. A A. y -. y e. x ) $=
|
|
( c0 wcel cv wceq wrex wn wal risset eq0 rexbii bitri ) DCEAFZDGZACHBFOEI
|
|
BJZACHADCKPQACBOLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( The class builder of a decidable proposition not containing the
|
|
abstraction variable is either the universal class or the empty set.
|
|
(Contributed by Jim Kingdon, 1-Aug-2018.) $)
|
|
abvor0dc $p |- ( DECID ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) ) $=
|
|
( wdc wn wo cab cvv wceq c0 df-dc cv wcel id vex a1i 2thd abbi1dv 2falsed
|
|
noel orim12i sylbi ) ACAADZEABFZGHZUCIHZEAJAUDUBUEAABGAABKZGLZAMUGABNOPQU
|
|
BABIUBAUFILZUBMUHDUBUFSORQTUA $.
|
|
$}
|
|
|
|
$( Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.) $)
|
|
abn0r $p |- ( E. x ph -> { x | ph } =/= (/) ) $=
|
|
( wex cv cab wcel c0 wne abid exbii nfab1 n0rf sylbir ) ABCBDABEZFZBCNGHOAB
|
|
ABIJBNABKLM $.
|
|
|
|
$( Non-empty restricted class abstraction. (Contributed by Jim Kingdon,
|
|
1-Aug-2018.) $)
|
|
rabn0r $p |- ( E. x e. A ph -> { x e. A | ph } =/= (/) ) $=
|
|
( cv wcel wa wex cab c0 wne wrex crab abn0r df-rex df-rab neeq1i 3imtr4i )
|
|
BDCEAFZBGRBHZIJABCKABCLZIJRBMABCNTSIABCOPQ $.
|
|
|
|
${
|
|
$d x y $. $d A y $. $d ph y $.
|
|
$( Inhabited restricted class abstraction. (Contributed by Jim Kingdon,
|
|
18-Sep-2018.) $)
|
|
rabn0m $p |- ( E. y y e. { x e. A | ph } <-> E. x e. A ph ) $=
|
|
( wrex cv wcel wa wex crab df-rex rabid exbii nfv cab df-rab eleq2i nfxfr
|
|
nfsab1 eleq1 cbvex 3bitr2ri ) ABDEBFZDGAHZBIUCABDJZGZBICFZUEGZCIABDKUFUDB
|
|
ABDLMUFUHBCUFCNUHUGUDBOZGBUEUIUGABDPQUDBCSRUCUGUETUAUB $.
|
|
$}
|
|
|
|
$( Any restricted class abstraction restricted to the empty set is empty.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
rab0 $p |- { x e. (/) | ph } = (/) $=
|
|
( cv c0 wcel wa wceq wn crab noel intnanr equid notnoti 2false abbii df-rab
|
|
cab dfnul2 3eqtr4i ) BCZDEZAFZBQTTGZHZBQABDIDUBUDBUBUDUAATJKUCBLMNOABDPBRS
|
|
$.
|
|
|
|
${
|
|
$d x y $. $d A y $. $d ph y $.
|
|
|
|
$( Condition for a restricted class abstraction to be empty. (Contributed
|
|
by Jeff Madsen, 7-Jun-2010.) $)
|
|
rabeq0 $p |- ( { x e. A | ph } = (/) <-> A. x e. A -. ph ) $=
|
|
( vy cv wcel wn wi wa wral crab c0 wceq imnan albii df-ral bitri 3bitr4ri
|
|
wal wsb sbn nfv sb8 eq0 cab df-rab eleq2i df-clab notbii ) BECFZAGZHZBSUJ
|
|
AIZGZBSZUKBCJABCKZLMZULUNBUJANOUKBCPUNBDTZDSUMBDTZGZDSZUOUQURUTDUMBDUAOUN
|
|
BDUNDUBUCUQDEZUPFZGZDSVADUPUDVDUTDVCUSVCVBUMBUEZFUSUPVEVBABCUFUGUMDBUHQUI
|
|
OQRR $.
|
|
|
|
$( Condition for a class abstraction to be empty. (Contributed by Jim
|
|
Kingdon, 12-Aug-2018.) $)
|
|
abeq0 $p |- ( { x | ph } = (/) <-> A. x -. ph ) $=
|
|
( vy wn wsb wal cab c0 wceq sbn albii nfv sb8 cv eq0 df-clab notbii bitri
|
|
wcel 3bitr4ri ) ADZBCEZCFABCEZDZCFZUABFABGZHIZUBUDCABCJKUABCUACLMUGCNUFSZ
|
|
DZCFUECUFOUIUDCUHUCACBPQKRT $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Law of excluded middle given decidability, in terms of restricted class
|
|
abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.) $)
|
|
rabxmdc $p |- ( A. x DECID ph ->
|
|
A = ( { x e. A | ph } u. { x e. A | -. ph } ) ) $=
|
|
( wdc wal wn wo crab cun wral wceq cv wcel wi exmiddc alimi df-ral sylibr
|
|
a1d rabid2 unrab syl6eqr ) ADZBEZCAAFZGZBCHZABCHUEBCHIUDUFBCJZCUGKUDBLCMZ
|
|
UFNZBEUHUCUJBUCUFUIAOSPUFBCQRUFBCTRAUEBCUAUB $.
|
|
|
|
$( Law of noncontradiction, in terms of restricted class abstractions.
|
|
(Contributed by Jeff Madsen, 20-Jun-2011.) $)
|
|
rabnc $p |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/) $=
|
|
( crab wn cin wa c0 inrab wceq rabeq0 cv wcel pm3.24 a1i mprgbir eqtri )
|
|
ABCDAEZBCDFARGZBCDZHARBCITHJSEZBCSBCKUABLCMANOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The union of a class with the empty set is itself. Theorem 24 of
|
|
[Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) $)
|
|
un0 $p |- ( A u. (/) ) = A $=
|
|
( vx c0 cv wcel wo noel biorfi bicomi uneqri ) BACABDZAEZLKCEZFMLKGHIJ $.
|
|
|
|
$( The intersection of a class with the empty set is the empty set.
|
|
Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.) $)
|
|
in0 $p |- ( A i^i (/) ) = (/) $=
|
|
( vx c0 cv wcel wa noel bianfi bicomi ineqri ) BACCBDZCEZKAEZLFLMKGHIJ $.
|
|
$}
|
|
|
|
$( The intersection of a class with the universal class is itself. Exercise
|
|
4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.) $)
|
|
inv1 $p |- ( A i^i _V ) = A $=
|
|
( cvv cin inss1 ssid ssv ssini eqssi ) ABCAABDAABAEAFGH $.
|
|
|
|
$( The union of a class with the universal class is the universal class.
|
|
Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM,
|
|
17-May-1998.) $)
|
|
unv $p |- ( A u. _V ) = _V $=
|
|
( cvv cun ssv ssun2 eqssi ) ABCZBGDBAEF $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( The null set is a subset of any class. Part of Exercise 1 of
|
|
[TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) $)
|
|
0ss $p |- (/) C_ A $=
|
|
( vx c0 cv wcel noel pm2.21i ssriv ) BCABDZCEIAEIFGH $.
|
|
$}
|
|
|
|
$( Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its
|
|
converse. (Contributed by NM, 17-Sep-2003.) $)
|
|
ss0b $p |- ( A C_ (/) <-> A = (/) ) $=
|
|
( c0 wceq wss 0ss eqss mpbiran2 bicomi ) ABCZABDZIJBADAEABFGH $.
|
|
|
|
$( Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23.
|
|
(Contributed by NM, 13-Aug-1994.) $)
|
|
ss0 $p |- ( A C_ (/) -> A = (/) ) $=
|
|
( c0 wss wceq ss0b biimpi ) ABCABDAEF $.
|
|
|
|
$( A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
sseq0 $p |- ( ( A C_ B /\ B = (/) ) -> A = (/) ) $=
|
|
( c0 wceq wss sseq2 ss0 syl6bi impcom ) BCDZABEZACDZJKACELBCAFAGHI $.
|
|
|
|
$( A class with a nonempty subclass is nonempty. (Contributed by NM,
|
|
17-Feb-2007.) $)
|
|
ssn0 $p |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) ) $=
|
|
( wss c0 wne wceq sseq0 ex necon3d imp ) ABCZADEBDEKBDADKBDFADFABGHIJ $.
|
|
|
|
${
|
|
abf.1 $e |- -. ph $.
|
|
$( A class builder with a false argument is empty. (Contributed by NM,
|
|
20-Jan-2012.) $)
|
|
abf $p |- { x | ph } = (/) $=
|
|
( cab c0 wss wceq cv wcel pm2.21i abssi ss0 ax-mp ) ABDZEFNEGABEABHEICJKN
|
|
LM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
eq0rdv.1 $e |- ( ph -> -. x e. A ) $.
|
|
$( Deduction rule for equality to the empty set. (Contributed by NM,
|
|
11-Jul-2014.) $)
|
|
eq0rdv $p |- ( ph -> A = (/) ) $=
|
|
( c0 wss wceq cv wcel pm2.21d ssrdv ss0 syl ) ACEFCEGABCEABHZCINEIDJKCLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( The proper substitution of a proper class for a set into a class results
|
|
in the empty set. (Contributed by NM, 17-Aug-2018.) $)
|
|
csbprc $p |- ( -. A e. _V -> [_ A / x ]_ B = (/) ) $=
|
|
( vy cvv wcel wn csb cv wsbc cab c0 df-csb wfal sbcex con3i pm2.21d falim
|
|
impbid1 abbidv fal abf syl6eq syl5eq ) BEFZGZABCHDICFZABJZDKZLADBCMUFUIND
|
|
KLUFUHNDUFUHNUFUHNUHUEUGABOPQUHRSTNDUAUBUCUD $.
|
|
$}
|
|
|
|
$( Two classes are empty iff their union is empty. (Contributed by NM,
|
|
11-Aug-2004.) $)
|
|
un00 $p |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) ) $=
|
|
( c0 wceq wa cun uneq12 un0 syl6eq wss ssun1 sseq2 mpbii sylib ssun2 impbii
|
|
ss0b jca ) ACDZBCDZEZABFZCDZUAUBCCFCACBCGCHIUCSTUCACJZSUCAUBJUDABKUBCALMAQN
|
|
UCBCJZTUCBUBJUEBAOUBCBLMBQNRP $.
|
|
|
|
$( Only the universal class has the universal class as a subclass.
|
|
(Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
vss $p |- ( _V C_ A <-> A = _V ) $=
|
|
( cvv wss wa wceq ssv biantrur eqss bitr4i ) BACZABCZJDABEKJAFGABHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Two ways of saying that two classes are disjoint (have no members in
|
|
common). (Contributed by NM, 17-Feb-2004.) $)
|
|
disj $p |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B ) $=
|
|
( cin c0 wceq cv wcel wn wi wal wral wa cab df-in eqeq1i abeq1 imnan noel
|
|
wb nbn bitr2i albii 3bitri df-ral bitr4i ) BCDZEFZAGZBHZUICHZIZJZAKZULABL
|
|
UHUJUKMZANZEFUOUIEHZTZAKUNUGUPEABCOPUOAEQURUMAUMUOIURUJUKRUQUOUISUAUBUCUD
|
|
ULABUEUF $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by Jeff
|
|
Madsen, 19-Jun-2011.) $)
|
|
disjr $p |- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A ) $=
|
|
( cin c0 wceq cv wcel wn wral incom eqeq1i disj bitri ) BCDZEFCBDZEFAGBHI
|
|
ACJOPEBCKLACBMN $.
|
|
|
|
$( Two ways of saying that two classes are disjoint (have no members in
|
|
common). (Contributed by NM, 19-Aug-1993.) $)
|
|
disj1 $p |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) ) $=
|
|
( cin c0 wceq cv wcel wn wral wi wal disj df-ral bitri ) BCDEFAGZCHIZABJP
|
|
BHQKALABCMQABNO $.
|
|
|
|
$( Two ways of saying that two classes are disjoint, using the complement
|
|
of ` B ` relative to a universe ` C ` . (Contributed by NM,
|
|
15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
reldisj $p |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) ) $=
|
|
( vx wss cv wcel wn wi wal cdif cin c0 wceq wb dfss2 pm5.44 eldif imbi2i
|
|
wa syl6bbr sps sylbi albidv disj1 3bitr4g ) ACEZDFZAGZUHBGHZIZDJUIUHCBKZG
|
|
ZIZDJABLMNAULEUGUKUNDUGUIUHCGZIZDJUKUNOZDACPUPUQDUPUKUIUOUJTZIUNUIUOUJQUM
|
|
URUIUHCBRSUAUBUCUDDABUEDAULPUF $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by NM,
|
|
19-May-1998.) $)
|
|
disj3 $p |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) ) $=
|
|
( vx cv wcel wn wi wal cdif wb cin c0 wa pm4.71 eldif bibi2i bitr4i albii
|
|
wceq disj1 dfcleq 3bitr4i ) CDZAEZUCBEFZGZCHUDUCABIZEZJZCHABKLSAUGSUFUICU
|
|
FUDUDUEMZJUIUDUENUHUJUDUCABOPQRCABTCAUGUAUB $.
|
|
|
|
$( Members of disjoint sets are not equal. (Contributed by NM,
|
|
28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
disjne $p |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D ) $=
|
|
( vx cin c0 wceq wcel wne cv wn wral wi disj eleq1 notbid rspccva eleq1a
|
|
wa necon3bd syl5com sylanb 3impia ) ABFGHZCAIZDBIZCDJZUEEKZBIZLZEAMZUFUGU
|
|
HNEABOULUFTCBIZLZUGUHUKUNECAUICHUJUMUICBPQRUGUMCDDBCSUAUBUCUD $.
|
|
$}
|
|
|
|
$( A set can't belong to both members of disjoint classes. (Contributed by
|
|
NM, 28-Feb-2015.) $)
|
|
disjel $p |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B ) $=
|
|
( cin c0 wceq wcel wn cdif wi disj3 eleq2 eldifn syl6bi sylbi imp ) ABDEFZC
|
|
AGZCBGHZQAABIZFZRSJABKUARCTGSATCLCABMNOP $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by NM,
|
|
17-May-1998.) $)
|
|
disj2 $p |- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) ) $=
|
|
( cvv wss cin c0 wceq cdif wb ssv reldisj ax-mp ) ACDABEFGACBHDIAJABCKL $.
|
|
|
|
$( Intersection with a subclass of a disjoint class. (Contributed by FL,
|
|
24-Jan-2007.) $)
|
|
ssdisj $p |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) ) $=
|
|
( wss cin c0 wceq wa ss0b wi ssrin sstr2 syl syl5bir imp ss0 ) ABDZBCEZFGZH
|
|
ACEZFDZTFGQSUASRFDZQUARIQTRDUBUAJABCKTRFLMNOTPM $.
|
|
|
|
$( The union of disjoint classes is disjoint. (Contributed by NM,
|
|
26-Sep-2004.) $)
|
|
undisj1 $p |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <->
|
|
( ( A u. B ) i^i C ) = (/) ) $=
|
|
( cin c0 wceq wa cun un00 indir eqeq1i bitr4i ) ACDZEFBCDZEFGMNHZEFABHCDZEF
|
|
MNIPOEABCJKL $.
|
|
|
|
$( The union of disjoint classes is disjoint. (Contributed by NM,
|
|
13-Sep-2004.) $)
|
|
undisj2 $p |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <->
|
|
( A i^i ( B u. C ) ) = (/) ) $=
|
|
( cin c0 wceq wa cun un00 indi eqeq1i bitr4i ) ABDZEFACDZEFGMNHZEFABCHDZEFM
|
|
NIPOEABCJKL $.
|
|
|
|
$( Subclass implies empty intersection with difference from the universal
|
|
class. (Contributed by NM, 17-Sep-2003.) $)
|
|
ssindif0im $p |- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) ) $=
|
|
( wss cvv cdif cin c0 wceq ddifss sstr mpan2 disj2 sylibr ) ABCZADDBEZEZCZA
|
|
OFGHNBPCQBIABPJKAOLM $.
|
|
|
|
$( The intersection of classes with a common member is nonempty.
|
|
(Contributed by NM, 7-Apr-1994.) $)
|
|
inelcm $p |- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) ) $=
|
|
( wcel wa cin c0 wne elin ne0i sylbir ) ABDACDEABCFZDLGHABCILAJK $.
|
|
|
|
$( A minimum element of a class has no elements in common with the class.
|
|
(Contributed by NM, 22-Jun-1994.) $)
|
|
minel $p |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C ) $=
|
|
( cin c0 wceq wcel wn wa wi inelcm necon2bi imnan sylibr con2d impcom ) CBD
|
|
ZEFZABGZACGZHRTSRTSIZHTSHJUAQEACBKLTSMNOP $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distribute union over difference. (Contributed by NM, 17-May-1998.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
undif4 $p |- ( ( A i^i C ) = (/) ->
|
|
( A u. ( B \ C ) ) = ( ( A u. B ) \ C ) ) $=
|
|
( vx cv wcel wn wi wal cdif cun wb cin c0 wceq wo wa eldif elun 3bitr4g
|
|
pm2.621 olc impbid1 anbi2d orbi2i bitri anbi1i alimi disj1 dfcleq 3imtr4i
|
|
ordi ) DEZAFZUMCFGZHZDIUMABCJZKZFZUMABKZCJZFZLZDIACMNOURVAOUPVCDUPUNUMUQF
|
|
ZPZUMUTFZUOQZUSVBUPUNUMBFZPZUNUOPZQZVIUOQVEVGUPVJUOVIUPVJUOUNUOUAUOUNUBUC
|
|
UDVEUNVHUOQZPVKVDVLUNUMBCRUEUNVHUOULUFVFVIUOUMABSUGTUMAUQSUMUTCRTUHDACUID
|
|
URVAUJUK $.
|
|
|
|
$( Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
disjssun $p |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) ) $=
|
|
( cin c0 wceq cun wss indi equncomi uneq2 un0 syl6eq syl5eq df-ss 3bitr4g
|
|
eqeq1d ) ABDZEFZABCGZDZAFACDZAFATHACHSUAUBASUAUBRGZUBUARUBABCIJSUCUBEGUBR
|
|
EUBKUBLMNQATOACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass implies empty difference. One direction of Exercise 7 of
|
|
[TakeutiZaring] p. 22. In classical logic this would be an equivalence.
|
|
(Contributed by Jim Kingdon, 2-Aug-2018.) $)
|
|
ssdif0im $p |- ( A C_ B -> ( A \ B ) = (/) ) $=
|
|
( vx cv wcel wi wal cdif wn wss c0 wceq wa imanim eldif sylnibr alimi eq0
|
|
dfss2 3imtr4i ) CDZAEZUABEZFZCGUAABHZEZIZCGABJUEKLUDUGCUDUBUCIMUFUBUCNUAA
|
|
BOPQCABSCUERT $.
|
|
$}
|
|
|
|
$( Universal class equality in terms of empty difference. (Contributed by
|
|
Jim Kingdon, 3-Aug-2018.) $)
|
|
vdif0im $p |- ( A = _V -> ( _V \ A ) = (/) ) $=
|
|
( cvv wceq wss cdif c0 vss ssdif0im sylbir ) ABCBADBAEFCAGBAHI $.
|
|
|
|
${
|
|
$d V x $.
|
|
$( If the difference between the restricting class of a restricted class
|
|
abstraction and the restricted class abstraction is empty, the
|
|
restricting class is equal to this restricted class abstraction.
|
|
(Contributed by Jim Kingdon, 3-Aug-2018.) $)
|
|
difrab0eqim $p |-
|
|
( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) ) $=
|
|
( crab wceq wss cdif c0 ssrabeq ssdif0im sylbir ) CABCDZECLFCLGHEABCICLJK
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Intersection, subclass, and difference relationship. In classical logic
|
|
the converse would also hold. (Contributed by Jim Kingdon,
|
|
3-Aug-2018.) $)
|
|
inssdif0im $p |- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) ) $=
|
|
( vx cv cin wcel wi wal cdif wn c0 wceq wa elin imbi1i imanim sylbi eldif
|
|
wss anbi2i anass 3bitr4ri sylnib alimi dfss2 eq0 3imtr4i ) DEZABFZGZUICGZ
|
|
HZDIUIABCJZFZGZKZDIUJCTUOLMUMUQDUMUIAGZUIBGZNZULKZNZUPUMUTULHVBKUKUTULUIA
|
|
BOPUTULQRURUIUNGZNURUSVANZNUPVBVCVDURUIBCSUAUIAUNOURUSVAUBUCUDUEDUJCUFDUO
|
|
UGUH $.
|
|
$}
|
|
|
|
$( The difference between a class and itself is the empty set. Proposition
|
|
5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.
|
|
(Contributed by NM, 22-Apr-2004.) $)
|
|
difid $p |- ( A \ A ) = (/) $=
|
|
( wss cdif c0 wceq ssid ssdif0im ax-mp ) AABAACDEAFAAGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The difference between a class and itself is the empty set. Proposition
|
|
5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.
|
|
Alternate proof of ~ difid . (Contributed by David Abernethy,
|
|
17-Jun-2012.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
difidALT $p |- ( A \ A ) = (/) $=
|
|
( vx cdif cv wcel wn crab c0 dfdif2 dfnul3 eqtr4i ) AACBDAEFBAGHBAAIBAJK
|
|
$.
|
|
$}
|
|
|
|
$( The difference between a class and the empty set. Part of Exercise 4.4 of
|
|
[Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) $)
|
|
dif0 $p |- ( A \ (/) ) = A $=
|
|
( cdif c0 difid difeq2i difdif eqtr3i ) AAABZBACBAHCAADEAAFG $.
|
|
|
|
$( The difference between the empty set and a class. Part of Exercise 4.4 of
|
|
[Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) $)
|
|
0dif $p |- ( (/) \ A ) = (/) $=
|
|
( c0 cdif wss wceq difss ss0 ax-mp ) BACZBDIBEBAFIGH $.
|
|
|
|
$( A class and its relative complement are disjoint. Theorem 38 of [Suppes]
|
|
p. 29. (Contributed by NM, 24-Mar-1998.) $)
|
|
disjdif $p |- ( A i^i ( B \ A ) ) = (/) $=
|
|
( cin wss cdif c0 wceq inss1 inssdif0im ax-mp ) ABCADABAECFGABHABAIJ $.
|
|
|
|
$( The difference of a class from its intersection is empty. Theorem 37 of
|
|
[Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by
|
|
Andrew Salmon, 26-Jun-2011.) $)
|
|
difin0 $p |- ( ( A i^i B ) \ B ) = (/) $=
|
|
( cin wss cdif c0 wceq inss2 ssdif0im ax-mp ) ABCZBDKBEFGABHKBIJ $.
|
|
|
|
$( Absorption of difference by union. In classical logic, as Theorem 35 of
|
|
[Suppes] p. 29, this would be equality rather than subset. (Contributed
|
|
by Jim Kingdon, 4-Aug-2018.) $)
|
|
undif1ss $p |- ( ( A \ B ) u. B ) C_ ( A u. B ) $=
|
|
( cdif wss cun difss unss1 ax-mp ) ABCZADIBEABEDABFIABGH $.
|
|
|
|
$( Absorption of difference by union. In classical logic, as in Part of
|
|
proof of Corollary 6K of [Enderton] p. 144, this would be equality rather
|
|
than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) $)
|
|
undif2ss $p |- ( A u. ( B \ A ) ) C_ ( A u. B ) $=
|
|
( cdif cun undif1ss uncom 3sstr4i ) BACZADBADAHDABDBAEAHFABFG $.
|
|
|
|
$( Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) $)
|
|
undifabs $p |- ( A u. ( A \ B ) ) = A $=
|
|
( cdif cun ssid difss unssi ssun1 eqssi ) AABCZDAAJAAEABFGAJHI $.
|
|
|
|
$( The intersection and class difference of a class with another class are
|
|
contained in the original class. In classical logic we'd be able to make
|
|
a stronger statement: that everything in the original class is in the
|
|
intersection or the difference (that is, this theorem would be equality
|
|
rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.) $)
|
|
inundifss $p |- ( ( A i^i B ) u. ( A \ B ) ) C_ A $=
|
|
( cin cdif inss1 difss unssi ) ABCABDAABEABFG $.
|
|
|
|
$( Absorption of union by difference. Theorem 36 of [Suppes] p. 29.
|
|
(Contributed by NM, 19-May-1998.) $)
|
|
difun2 $p |- ( ( A u. B ) \ B ) = ( A \ B ) $=
|
|
( cun cdif c0 difundir difid uneq2i un0 3eqtri ) ABCBDABDZBBDZCKECKABBFLEKB
|
|
GHKIJ $.
|
|
|
|
$( Union of complementary parts into whole. (Contributed by Jim Kingdon,
|
|
4-Aug-2018.) $)
|
|
undifss $p |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B ) $=
|
|
( wss cdif cun wa difss jctr unss sylib ssun1 sstr mpan impbii ) ABCZABADZE
|
|
ZBCZOOPBCZFROSBAGHAPBIJAQCROAPKAQBLMN $.
|
|
|
|
$( A subset of a difference does not intersect the subtrahend. (Contributed
|
|
by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro,
|
|
24-Aug-2015.) $)
|
|
ssdifin0 $p |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) ) $=
|
|
( cdif wss cin c0 wceq ssrin incom disjdif eqtri sseq0 sylancl ) ABCDZEACFZ
|
|
OCFZEQGHPGHAOCIQCOFGOCJCBKLPQMN $.
|
|
|
|
$( A class is a subclass of itself subtracted from another iff it is the
|
|
empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) $)
|
|
ssdifeq0 $p |- ( A C_ ( B \ A ) <-> A = (/) ) $=
|
|
( cdif wss wceq cin inidm ssdifin0 syl5eqr 0ss difeq2 sseq12d mpbiri impbii
|
|
c0 id ) ABACZDZAOEZRAAAFOAGABAHISROBOCZDTJSAOQTSPAOBKLMN $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( A consequence of inclusion in the union of two classes. In classical
|
|
logic this would be a biconditional. (Contributed by Jim Kingdon,
|
|
4-Aug-2018.) $)
|
|
ssundifim $p |- ( A C_ ( B u. C ) -> ( A \ B ) C_ C ) $=
|
|
( vx cv wcel cun wi wal cdif wss wo wn wa pm5.6r elun eldif 3imtr4i dfss2
|
|
imbi2i imbi1i alimi ) DEZAFZUCBCGZFZHZDIUCABJZFZUCCFZHZDIAUEKUHCKUGUKDUDU
|
|
CBFZUJLZHUDULMNZUJHUGUKUDULUJOUFUMUDUCBCPTUIUNUJUCABQUARUBDAUESDUHCSR $.
|
|
$}
|
|
|
|
$( Distributive law for class difference. In classical logic, as in Exercise
|
|
4.8 of [Stoll] p. 16, this would be equality rather than subset.
|
|
(Contributed by Jim Kingdon, 4-Aug-2018.) $)
|
|
difdifdirss $p |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) ) $=
|
|
( cdif cvv cun cin dif32 invdif eqtr4i un0 indi disjdif incom eqtr3i uneq2i
|
|
c0 wss ax-mp sseqtri ddifss unss2 indmss difeq2i sstri sslin eqsstri ) ABDC
|
|
DZACDZEBDZCFZGZUIBCDZDZUHUIUJGZQFZULUHUOUPUHUIBDUOABCHUIBIJUOKJULUOUICGZFUP
|
|
UIUJCLQUQUOCUIGQUQCAMCUINOPJJULUIEUMDZGZUNUKURRULUSRUKUJEECDZDZFZURCVARUKVB
|
|
RCUACVAUJUBSVBEBUTGZDURBUTUCVCUMEBCIUDTUEUKURUIUFSUIUMITUG $.
|
|
|
|
$( Two ways that ` A ` and ` B ` can "partition" ` C ` (when ` A ` and ` B `
|
|
don't overlap and ` A ` is a part of ` C ` ). In classical logic, the
|
|
second implication would be a biconditional. (Contributed by Jim Kingdon,
|
|
4-Aug-2018.) $)
|
|
uneqdifeqim $p |- ( ( A C_ C /\ ( A i^i B ) = (/) )
|
|
-> ( ( A u. B ) = C -> ( C \ A ) = B ) ) $=
|
|
( cin c0 wceq cun cdif wi wss uncom eqcomd difeq1 difun2 incom eqeq1i disj3
|
|
wa eqtr bitri expcom eqcoms sylbi syl5com sylancl syl mpan com12 adantl ) A
|
|
BDZEFZABGZCFZCAHZBFZIACJUMUKUOBAGZULFZUMUKUOIZBAKUQUMRZCUPFZURUSUPCUPULCSLU
|
|
TUNUPAHZFZVABAHZFZURCUPAMBANVBVDRUNVCFZUKUOUNVAVCSUKBVCFZVEUOIZUKBADZEFVFUJ
|
|
VHEABOPBAQTVGVCBVEVCBFUOUNVCBSUAUBUCUDUEUFUGUHUI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare
|
|
~ 19.2 ). The restricted version is valid only when the domain of
|
|
quantification is inhabited. (Contributed by Jim Kingdon,
|
|
5-Aug-2018.) $)
|
|
r19.2m $p |- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph ) $=
|
|
( wral cv wcel wex wrex wa wal df-ral exintr sylbi df-rex syl6ibr impcom
|
|
wi ) ABCDZBECFZBGZABCHZRTSAIBGZUARSAQBJTUBQABCKSABLMABCNOP $.
|
|
$}
|
|
|
|
${
|
|
$d a x A $. $d a y A $.
|
|
r19.3rm.1 $e |- F/ x ph $.
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by Jim Kingdon, 19-Dec-2018.) $)
|
|
r19.3rm $p |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) ) $=
|
|
( va cv wcel wex wral wb eleq1 cbvexv wi biimt df-ral 19.23 bitri syl6bbr
|
|
wal sylbi sylbir ) CGZDHZCIFGZDHZFIZAABDJZKZUFUDFCUEUCDLMUGBGZDHZBIZUIUFU
|
|
KFBUEUJDLMULAULANZUHULAOUHUKANBTUMABDPUKABEQRSUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
r19.28m.1 $e |- F/ x ph $.
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is inhabited.
|
|
(Contributed by Jim Kingdon, 5-Aug-2018.) $)
|
|
r19.28m $p |- ( E. x x e. A ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
|
|
( cv wcel wex wral wa r19.3rm anbi1d r19.26 syl6rbbr ) CFDGCHZABCDIZJACDI
|
|
ZPJABJCDIOAQPACCDEKLABCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d x ph $.
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by Jim Kingdon, 6-Aug-2018.) $)
|
|
r19.3rmv $p |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) ) $=
|
|
( nfv r19.3rm ) ABCDABEF $.
|
|
$}
|
|
|
|
${
|
|
$d x a A $. $d y a A $. $d x ph $.
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by Jim Kingdon, 5-Aug-2018.) $)
|
|
r19.9rmv $p |- ( E. y y e. A -> ( ph <-> E. x e. A ph ) ) $=
|
|
( va cv wcel wex wrex wb eleq1 cbvexv wa df-rex 19.41v bitri baibr sylbir
|
|
sylbi ) CFZDGZCHEFZDGZEHZAABDIZJZUCUAECUBTDKLUDBFZDGZBHZUFUCUHEBUBUGDKLUE
|
|
UIAUEUHAMBHUIAMABDNUHABOPQSR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is inhabited.
|
|
(Contributed by Jim Kingdon, 6-Aug-2018.) $)
|
|
r19.28mv $p |- ( E. x x e. A ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
|
|
( nfv r19.28m ) ABCDACEF $.
|
|
|
|
$( Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed
|
|
by NM, 27-May-1998.) $)
|
|
r19.45mv $p |- ( E. x x e. A ->
|
|
( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) ) $=
|
|
( cv wcel wex wrex wo r19.9rmv orbi1d r19.43 syl6rbbr ) CEDFCGZABCDHZIACD
|
|
HZOIABICDHNAPOACCDJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d ps x $.
|
|
$( Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed
|
|
by NM, 27-May-1998.) $)
|
|
r19.44mv $p |- ( E. y y e. A ->
|
|
( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) ) $=
|
|
( cv wcel wex wrex wo r19.9rmv orbi2d r19.43 syl6rbbr ) DFEGDHZACEIZBJPBC
|
|
EIZJABJCEIOBQPBCDEKLABCEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
r19.27m.1 $e |- F/ x ps $.
|
|
$( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is inhabited.
|
|
(Contributed by Jim Kingdon, 5-Aug-2018.) $)
|
|
r19.27m $p |- ( E. x x e. A ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
|
|
( cv wcel wex wral wa r19.3rm anbi2d r19.26 syl6rbbr ) CFDGCHZACDIZBJPBCD
|
|
IZJABJCDIOBQPBCCDEKLABCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
$( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is inhabited.
|
|
(Contributed by Jim Kingdon, 5-Aug-2018.) $)
|
|
r19.27mv $p |- ( E. x x e. A ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
|
|
( nfv r19.27m ) ABCDBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Vacuous quantification is always true. (Contributed by NM,
|
|
11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rzal $p |- ( A = (/) -> A. x e. A ph ) $=
|
|
( c0 wceq cv wcel ne0i necon2bi pm2.21d ralrimiv ) CDEZABCLBFZCGZANCDCMHI
|
|
JK $.
|
|
|
|
$( Restricted existential quantification implies its restriction is
|
|
nonempty (it is also inhabited as shown in ~ rexm ). (Contributed by
|
|
Szymon Jaroszewicz, 3-Apr-2007.) $)
|
|
rexn0 $p |- ( E. x e. A ph -> A =/= (/) ) $=
|
|
( c0 wne cv wcel ne0i a1d rexlimiv ) ACDEZBCBFZCGKACLHIJ $.
|
|
|
|
$( Restricted existential quantification implies its restriction is
|
|
inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) $)
|
|
rexm $p |- ( E. x e. A ph -> E. x x e. A ) $=
|
|
( wrex cv wcel wa wex df-rex simpl eximi sylbi ) ABCDBECFZAGZBHMBHABCINMB
|
|
MAJKL $.
|
|
|
|
$( Idempotent law for restricted quantifier. (Contributed by NM,
|
|
28-Mar-1997.) $)
|
|
ralidm $p |- ( A. x e. A A. x e. A ph <-> A. x e. A ph ) $=
|
|
( wral nfra1 cv wcel wa anidm rsp2 syl5bir ralrimi wi wal wex ax-1 sylibr
|
|
19.23 df-ral impbii ) ABCDZBCDZUAUBABCUABCEBFCGZUCUCHUBAUCIABBCCJKLUAUCUA
|
|
MBNZUBUAUCBOZUAMUDUAUEPUCUABABCERQUABCSQT $.
|
|
$}
|
|
|
|
$( Vacuous universal quantification is always true. (Contributed by NM,
|
|
20-Oct-2005.) $)
|
|
ral0 $p |- A. x e. (/) ph $=
|
|
( c0 cv wcel noel pm2.21i rgen ) ABCBDZCEAIFGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
rgenm.1 $e |- ( ( E. x x e. A /\ x e. A ) -> ph ) $.
|
|
$( Generalization rule that eliminates an inhabited class requirement.
|
|
(Contributed by Jim Kingdon, 5-Aug-2018.) $)
|
|
rgenm $p |- A. x e. A ph $=
|
|
( wral cv wcel wi wal wex nfe1 alrimi 19.38 ax-mp pm5.4 albii mpbi df-ral
|
|
ex mpbir ) ABCEBFCGZAHZBIZUAUBHZBIZUCUABJZUCHUEUFUBBUABKUFUAADSLUAUBBMNUD
|
|
UBBUAAOPQABCRT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ralf0.1 $e |- -. ph $.
|
|
$( The quantification of a falsehood is vacuous when true. (Contributed by
|
|
NM, 26-Nov-2005.) $)
|
|
ralf0 $p |- ( A. x e. A ph <-> A = (/) ) $=
|
|
( wral c0 wceq cv wcel wi wal wn con3 mpi alimi df-ral eq0 3imtr4i impbii
|
|
rzal ) ABCEZCFGZBHCIZAJZBKUCLZBKUAUBUDUEBUDALUEDUCAMNOABCPBCQRABCTS $.
|
|
$}
|
|
|
|
$( Inhabited classes and restricted quantification. (Contributed by Jim
|
|
Kingdon, 6-Aug-2018.) $)
|
|
ralm $p |- ( ( E. x x e. A -> A. x e. A ph ) <-> A. x e. A ph ) $=
|
|
( cv wcel wex wral wi wal df-ral imbi2i 19.38 sylbi pm2.43 alimi syl sylibr
|
|
ax-1 impbii ) BDCEZBFZABCGZHZUBUCTAHZBIZUBUCTUDHZBIZUEUCUAUEHUGUBUEUAABCJZK
|
|
TUDBLMUFUDBTANOPUHQUBUARS $.
|
|
|
|
${
|
|
$d x y A $.
|
|
raaan.1 $e |- F/ y ph $.
|
|
raaan.2 $e |- F/ x ps $.
|
|
$( Special case of ~ raaan where ` A ` is inhabited. (Contributed by Jim
|
|
Kingdon, 6-Aug-2018.) $)
|
|
raaanlem $p |- ( E. x x e. A ->
|
|
( A. x e. A A. y e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. y e. A ps ) ) ) $=
|
|
( cv wcel wex wa wral wb eleq1 cbvexv r19.28m ralbidv sylbi nfcv nfralxy
|
|
r19.27m bitrd ) CHZEIZCJZABKDELZCELZABDELZKZCELZACELUHKUEDHZEIZDJZUGUJMUD
|
|
ULCDUCUKENOUMUFUICEABDEFPQRAUHCEBCDECESGTUAUB $.
|
|
|
|
$( Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) $)
|
|
raaan $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. y e. A ps ) ) $=
|
|
( cv wcel wex wa wral wi raaanlem pm5.74i ralm jcab eleq1 cbvexv bitri
|
|
imbi1i anbi12i 3bitr3i ) CHZEIZCJZABKDELZCELZMUFACELZBDELZKZMZUHUKUFUHUKA
|
|
BCDEFGNOUGCEPULUFUIMZUFUJMZKUKUFUIUJQUMUIUNUJACEPUNDHZEIZDJZUJMUJUFUQUJUE
|
|
UPCDUDUOERSUABDEPTUBTUC $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y A $.
|
|
$( Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) $)
|
|
raaanv $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. y e. A ps ) ) $=
|
|
( nfv raaan ) ABCDEADFBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d z y $. $d z x A $.
|
|
$( Set substitution into the first argument of a subset relation.
|
|
(Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
sbss $p |- ( [ y / x ] x C_ A <-> y C_ A ) $=
|
|
( vz cv wss wsb vex sbequ sseq1 nfv sbie vtoclb ) AEZCFZADGDEZCFZOABGBEZC
|
|
FDRBHODBAIPRCJOQADQAKNPCJLM $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $. $d V y $. $d x y $.
|
|
$( Distribute proper substitution through a subclass relation.
|
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander
|
|
van der Vekens, 23-Jul-2017.) $)
|
|
sbcssg $p |- ( A e. V -> ( [. A / x ]. B C_ C <->
|
|
[_ A / x ]_ B C_ [_ A / x ]_ C ) ) $=
|
|
( vy wcel cv wi wal wsbc csb wss sbcalg sbcimg sbcel2g bitrd albidv dfss2
|
|
imbi12d sbcbii 3bitr4g ) BEGZFHZCGZUDDGZIZFJZABKZUDABCLZGZUDABDLZGZIZFJZC
|
|
DMZABKUJULMUCUIUGABKZFJUOUGFABENUCUQUNFUCUQUEABKZUFABKZIUNUEUFABEOUCURUKU
|
|
SUMABUDCEPABUDDEPTQRQUPUHABFCDSUAFUJULSUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Conditional operator
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c , $. $( Comma (also used for unordered pair notation later) $)
|
|
$c if $. $( Conditional operator (was "ded" for "deduction class"). $)
|
|
|
|
$( Extend class notation to include the conditional operator. See ~ df-if
|
|
for a description. (In older databases this was denoted "ded".) $)
|
|
cif $a class if ( ph , A , B ) $.
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
$( Define the conditional operator. Read ` if ( ph , A , B ) ` as "if
|
|
` ph ` then ` A ` else ` B ` ." See ~ iftrue and ~ iffalse for its
|
|
values. In mathematical literature, this operator is rarely defined
|
|
formally but is implicit in informal definitions such as "let f(x)=0 if
|
|
x=0 and 1/x otherwise."
|
|
|
|
In the absence of excluded middle, this will tend to be useful where
|
|
` ph ` is decidable (in the sense of ~ df-dc ). (Contributed by NM,
|
|
15-May-1999.) $)
|
|
df-if $a |- if ( ph , A , B ) =
|
|
{ x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $. $d x C $.
|
|
$( An alternate definition of the conditional operator ~ df-if as a simple
|
|
class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) $)
|
|
dfif6 $p |- if ( ph , A , B ) =
|
|
( { x e. A | ph } u. { x e. B | -. ph } ) $=
|
|
( cv wcel wa cab wn cun wo crab cif unab df-rab uneq12i df-if 3eqtr4ri )
|
|
BEZCFAGZBHZSDFAIZGZBHZJTUCKBHABCLZUBBDLZJACDMTUCBNUEUAUFUDABCOUBBDOPABCDQ
|
|
R $.
|
|
|
|
$( Equality theorem for conditional operator. (Contributed by NM,
|
|
1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
ifeq1 $p |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) ) $=
|
|
( vx wceq crab wn cun cif rabeq uneq1d dfif6 3eqtr4g ) BCFZAEBGZAHEDGZIAE
|
|
CGZQIABDJACDJOPRQAEBCKLAEBDMAECDMN $.
|
|
|
|
$( Equality theorem for conditional operator. (Contributed by NM,
|
|
1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
ifeq2 $p |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) ) $=
|
|
( vx wceq crab wn cun cif rabeq uneq2d dfif6 3eqtr4g ) BCFZAEDGZAHZEBGZIP
|
|
QECGZIADBJADCJORSPQEBCKLAEDBMAEDCMN $.
|
|
|
|
$( Value of the conditional operator when its first argument is true.
|
|
(Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
iftrue $p |- ( ph -> if ( ph , A , B ) = A ) $=
|
|
( vx cv wcel wa wn wo cab cif dedlema abbi2dv df-if syl6reqr ) ABDEZBFZAG
|
|
PCFZAHGIZDJABCKASDBAQRLMADBCNO $.
|
|
$}
|
|
|
|
${
|
|
iftruei.1 $e |- ph $.
|
|
$( Inference associated with ~ iftrue . (Contributed by BJ,
|
|
7-Oct-2018.) $)
|
|
iftruei $p |- if ( ph , A , B ) = A $=
|
|
( cif wceq iftrue ax-mp ) AABCEBFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
iftrued.1 $e |- ( ph -> ch ) $.
|
|
$( Value of the conditional operator when its first argument is true.
|
|
(Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
|
|
iftrued $p |- ( ph -> if ( ch , A , B ) = A ) $=
|
|
( cif wceq iftrue syl ) ABBCDFCGEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
$( Value of the conditional operator when its first argument is false.
|
|
(Contributed by NM, 14-Aug-1999.) $)
|
|
iffalse $p |- ( -. ph -> if ( ph , A , B ) = B ) $=
|
|
( vx wn cv wcel wa wo cab cif dedlemb abbi2dv df-if syl6reqr ) AEZCDFZBGZ
|
|
AHQCGZPHIZDJABCKPTDCARSLMADBCNO $.
|
|
$}
|
|
|
|
${
|
|
iffalsei.1 $e |- -. ph $.
|
|
$( Inference associated with ~ iffalse . (Contributed by BJ,
|
|
7-Oct-2018.) $)
|
|
iffalsei $p |- if ( ph , A , B ) = B $=
|
|
( wn cif wceq iffalse ax-mp ) AEABCFCGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
iffalsed.1 $e |- ( ph -> -. ch ) $.
|
|
$( Value of the conditional operator when its first argument is false.
|
|
(Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
|
|
iffalsed $p |- ( ph -> if ( ch , A , B ) = B ) $=
|
|
( wn cif wceq iffalse syl ) ABFBCDGDHEBCDIJ $.
|
|
$}
|
|
|
|
$( When values are unequal, but an "if" condition checks if they are equal,
|
|
then the "false" branch results. This is a simple utility to provide a
|
|
slight shortening and simplification of proofs vs. applying ~ iffalse
|
|
directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) $)
|
|
ifnefalse $p |- ( A =/= B -> if ( A = B , C , D ) = D ) $=
|
|
( wne wceq wn cif df-ne iffalse sylbi ) ABEABFZGLCDHDFABILCDJK $.
|
|
|
|
${
|
|
ifsbdc.1 $e |- ( if ( ph , A , B ) = A -> C = D ) $.
|
|
ifsbdc.2 $e |- ( if ( ph , A , B ) = B -> C = E ) $.
|
|
$( Distribute a function over an if-clause. (Contributed by Jim Kingdon,
|
|
1-Jan-2022.) $)
|
|
ifsbdc $p |- ( DECID ph -> C = if ( ph , D , E ) ) $=
|
|
( wdc wn wo cif wceq exmiddc iftrue syl eqtr4d iffalse jaoi ) AIAAJZKDAEF
|
|
LZMZANAUBTADEUAAABCLZBMDEMABCOGPAEFOQTDFUATUCCMDFMABCRHPAEFRQSP $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y ph $.
|
|
dfif3.1 $e |- C = { x | ph } $.
|
|
$( Alternate definition of the conditional operator ~ df-if . Note that
|
|
` ph ` is independent of ` x ` i.e. a constant true or false.
|
|
(Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
dfif3 $p |- if ( ph , A , B )
|
|
= ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) $=
|
|
( vy cif crab wn cun cin cvv cdif dfif6 cab weq ineq2i dfrab3 eqtr4i
|
|
biidd cbvabv eqtri notab difeq2i eqtr2i uneq12i ) ACDHAGCIZAJZGDIZKCELZDM
|
|
ENZLZKAGCDOUKUHUMUJUKCAGPZLUHEUNCEABPUNFAABGBGQAUAUBUCZRAGCSTUJDUIGPZLUMU
|
|
IGDSUPULDUPMUNNULAGUDEUNMUOUETRUFUGT $.
|
|
$}
|
|
|
|
$( Equality theorem for conditional operators. (Contributed by NM,
|
|
1-Sep-2004.) $)
|
|
ifeq12 $p |- ( ( A = B /\ C = D ) ->
|
|
if ( ph , A , C ) = if ( ph , B , D ) ) $=
|
|
( wceq cif ifeq1 ifeq2 sylan9eq ) BCFDEFABDGACDGACEGABCDHADECIJ $.
|
|
|
|
${
|
|
ifeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
16-Feb-2005.) $)
|
|
ifeq1d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
|
|
( wceq cif ifeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
16-Feb-2005.) $)
|
|
ifeq2d $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
|
|
( wceq cif ifeq2 syl ) ACDGBECHBEDHGFBCDEIJ $.
|
|
|
|
ifeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
24-Mar-2015.) $)
|
|
ifeq12d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) ) $=
|
|
( cif ifeq1d ifeq2d eqtrd ) ABCEIBDEIBDFIABCDEGJABEFDHKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $. $d ps x $.
|
|
$( Equivalence theorem for conditional operators. (Contributed by Raph
|
|
Levien, 15-Jan-2004.) $)
|
|
ifbi $p |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) ) $=
|
|
( vx wb cv wcel wa wn wo cab cif anbi2 notbid anbi2d orbi12d abbidv df-if
|
|
id 3eqtr4g ) ABFZEGZCHZAIZUCDHZAJZIZKZELUDBIZUFBJZIZKZELACDMBCDMUBUIUMEUB
|
|
UEUJUHULABUDNUBUGUKUFUBABUBTOPQRAECDSBECDSUA $.
|
|
$}
|
|
|
|
${
|
|
ifbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by NM,
|
|
18-Apr-2005.) $)
|
|
ifbid $p |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) ) $=
|
|
( wb cif wceq ifbi syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ifbieq1d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
ifbieq1d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equivalence/equality deduction for conditional operators. (Contributed
|
|
by JJ, 25-Sep-2018.) $)
|
|
ifbieq1d $p |- ( ph -> if ( ps , A , C ) = if ( ch , B , C ) ) $=
|
|
( cif ifbid ifeq1d eqtrd ) ABDFICDFICEFIABCDFGJACDEFHKL $.
|
|
$}
|
|
|
|
${
|
|
ifbieq2i.1 $e |- ( ph <-> ps ) $.
|
|
ifbieq2i.2 $e |- A = B $.
|
|
$( Equivalence/equality inference for conditional operators. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
ifbieq2i $p |- if ( ph , C , A ) = if ( ps , C , B ) $=
|
|
( cif wb wceq ifbi ax-mp ifeq2 eqtri ) AECHZBECHZBEDHZABIOPJFABECKLCDJPQJ
|
|
GBCDEMLN $.
|
|
$}
|
|
|
|
${
|
|
ifbieq2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
ifbieq2d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equivalence/equality deduction for conditional operators. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
ifbieq2d $p |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) ) $=
|
|
( cif ifbid ifeq2d eqtrd ) ABFDICFDICFEIABCFDGJACDEFHKL $.
|
|
$}
|
|
|
|
${
|
|
ifbieq12i.1 $e |- ( ph <-> ps ) $.
|
|
ifbieq12i.2 $e |- A = C $.
|
|
ifbieq12i.3 $e |- B = D $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by NM,
|
|
18-Mar-2013.) $)
|
|
ifbieq12i $p |- if ( ph , A , B ) = if ( ps , C , D ) $=
|
|
( cif wceq ifeq1 ax-mp ifbieq2i eqtri ) ACDJZAEDJZBEFJCEKPQKHACEDLMABDFEG
|
|
INO $.
|
|
$}
|
|
|
|
${
|
|
ifbieq12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
ifbieq12d.2 $e |- ( ph -> A = C ) $.
|
|
ifbieq12d.3 $e |- ( ph -> B = D ) $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
ifbieq12d $p |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) ) $=
|
|
( cif ifbid ifeq12d eqtrd ) ABDEKCDEKCFGKABCDEHLACDFEGIJMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y ph $. $d y ps $.
|
|
nfifd.2 $e |- ( ph -> F/ x ps ) $.
|
|
nfifd.3 $e |- ( ph -> F/_ x A ) $.
|
|
nfifd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of ~ nfif . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
nfifd $p |- ( ph -> F/_ x if ( ps , A , B ) ) $=
|
|
( vy cif cv wcel wa wn wo cab df-if nfv nfcrd nfand nford nfabd nfcxfrd
|
|
nfnd ) ACBDEJIKZDLZBMZUEELZBNZMZOZIPBIDEQAUKCIAIRAUGUJCAUFBCACIDGSFTAUHUI
|
|
CACIEHSABCFUDTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
nfif.1 $e |- F/ x ph $.
|
|
nfif.2 $e |- F/_ x A $.
|
|
nfif.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a conditional operator.
|
|
(Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
nfif $p |- F/_ x if ( ph , A , B ) $=
|
|
( cif wnfc wtru wnf a1i nfifd trud ) BACDHIJABCDABKJELBCIJFLBDIJGLMN $.
|
|
$}
|
|
|
|
${
|
|
ifcldadc.1 $e |- ( ( ph /\ ps ) -> A e. C ) $.
|
|
ifcldadc.2 $e |- ( ( ph /\ -. ps ) -> B e. C ) $.
|
|
ifcldadc.dc $e |- ( ph -> DECID ps ) $.
|
|
$( Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) $)
|
|
ifcldadc $p |- ( ph -> if ( ps , A , B ) e. C ) $=
|
|
( cif wcel wn wa wceq iftrue adantl eqeltrd iffalse wdc wo exmiddc syl
|
|
mpjaodan ) ABBCDIZEJBKZABLUCCEBUCCMABCDNOFPAUDLUCDEUDUCDMABCDQOGPABRBUDSH
|
|
BTUAUB $.
|
|
$}
|
|
|
|
${
|
|
ifeq1dadc.1 $e |- ( ( ph /\ ps ) -> A = B ) $.
|
|
ifeq1dadc.dc $e |- ( ph -> DECID ps ) $.
|
|
$( Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) $)
|
|
ifeq1dadc $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
|
|
( cif wceq wn wa ifeq1d iffalse eqtr4d adantl wdc wo exmiddc syl mpjaodan
|
|
) ABBCEHZBDEHZIZBJZABKBCDEFLUDUCAUDUAEUBBCEMBDEMNOABPBUDQGBRST $.
|
|
$}
|
|
|
|
${
|
|
ifbothdc.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
|
|
ifbothdc.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.
|
|
$( A wff ` th ` containing a conditional operator is true when both of its
|
|
cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.) $)
|
|
ifbothdc $p |- ( ( ps /\ ch /\ DECID ph ) -> th ) $=
|
|
( wdc w3a wn wi cif wceq wb iftrue eqcomd syl biimpcd 3ad2ant1 iffalse wo
|
|
3ad2ant2 exmiddc 3ad2ant3 mpjaod ) BCAIZJADAKZBCADLUGABDAEAEFMZNBDOAUIEAE
|
|
FPQGRSTCBUHDLUGUHCDUHFUINCDOUHUIFAEFUAQHRSUCUGBAUHUBCAUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
ifcldcd.a $e |- ( ph -> A e. C ) $.
|
|
ifcldcd.b $e |- ( ph -> B e. C ) $.
|
|
ifcldcd.dc $e |- ( ph -> DECID ps ) $.
|
|
$( Membership (closure) of a conditional operator, deduction form.
|
|
(Contributed by Jim Kingdon, 8-Aug-2021.) $)
|
|
ifcldcd $p |- ( ph -> if ( ps , A , B ) e. C ) $=
|
|
( cif wcel wn wa wceq iftrue adantl adantr eqeltrd iffalse wdc wo df-dc
|
|
sylib mpjaodan ) ABBCDIZEJBKZABLUDCEBUDCMABCDNOACEJBFPQAUELUDDEUEUDDMABCD
|
|
ROADEJUEGPQABSBUETHBUAUBUC $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Power classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for power class. $)
|
|
$c ~P $. $( Calligraphic P $)
|
|
|
|
$( Extend class notation to include power class. (The tilde in the Metamath
|
|
token is meant to suggest the calligraphic font of the P.) $)
|
|
cpw $a class ~P A $.
|
|
|
|
${
|
|
$d x A $. $d y A $. $d z A $. $d z x $. $d z y $.
|
|
$( Soundness justification theorem for ~ df-pw . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
pwjust $p |- { x | x C_ A } = { y | y C_ A } $=
|
|
( vz cv wss cab sseq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINPD
|
|
BMOCHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we
|
|
also let it apply to proper classes, i.e. those that are not members of
|
|
` _V ` . When applied to a set, this produces its power set. A power
|
|
set of S is the set of all subsets of S, including the empty set and S
|
|
itself. For example, if ` A ` is { 3 , 5 , 7 }, then ` ~P A ` is {
|
|
(/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 ,
|
|
5 , 7 } }. We will later introduce the Axiom of Power Sets. Still
|
|
later we will prove that the size of the power set of a finite set is 2
|
|
raised to the power of the size of the set. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-pw $a |- ~P A = { x | x C_ A } $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) $)
|
|
pweq $p |- ( A = B -> ~P A = ~P B ) $=
|
|
( vx wceq cv wss cab cpw sseq2 abbidv df-pw 3eqtr4g ) ABDZCEZAFZCGNBFZCGA
|
|
HBHMOPCABNIJCAKCBKL $.
|
|
$}
|
|
|
|
${
|
|
pweqi.1 $e |- A = B $.
|
|
$( Equality inference for power class. (Contributed by NM,
|
|
27-Nov-2013.) $)
|
|
pweqi $p |- ~P A = ~P B $=
|
|
( wceq cpw pweq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
pweqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for power class. (Contributed by NM,
|
|
27-Nov-2013.) $)
|
|
pweqd $p |- ( ph -> ~P A = ~P B ) $=
|
|
( wceq cpw pweq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
${
|
|
elpw.1 $e |- A e. _V $.
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47.
|
|
(Contributed by NM, 31-Dec-1993.) $)
|
|
elpw $p |- ( A e. ~P B <-> A C_ B ) $=
|
|
( vx cv wss cpw sseq1 df-pw elab2 ) DEZBFABFDABGCKABHDBIJ $.
|
|
$}
|
|
|
|
$( Setvar variable membership in a power class (common case). See ~ elpw .
|
|
(Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
selpw $p |- ( x e. ~P A <-> x C_ A ) $=
|
|
( cv vex elpw ) ACBADE $.
|
|
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47.
|
|
(Contributed by NM, 6-Aug-2000.) $)
|
|
elpwg $p |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) $=
|
|
( vx cv cpw wcel wss eleq1 sseq1 vex elpw vtoclbg ) DEZBFZGNBHAOGABHDACNA
|
|
OINABJNBDKLM $.
|
|
$}
|
|
|
|
$( Subset relation implied by membership in a power class. (Contributed by
|
|
NM, 17-Feb-2007.) $)
|
|
elpwi $p |- ( A e. ~P B -> A C_ B ) $=
|
|
( cpw wcel wss elpwg ibi ) ABCZDABEABHFG $.
|
|
|
|
${
|
|
elpwid.1 $e |- ( ph -> A e. ~P B ) $.
|
|
$( An element of a power class is a subclass. Deduction form of ~ elpwi .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
elpwid $p |- ( ph -> A C_ B ) $=
|
|
( cpw wcel wss elpwi syl ) ABCEFBCGDBCHI $.
|
|
$}
|
|
|
|
$( If ` A ` belongs to a part of ` C ` then ` A ` belongs to ` C ` .
|
|
(Contributed by FL, 3-Aug-2009.) $)
|
|
elelpwi $p |- ( ( A e. B /\ B e. ~P C ) -> A e. C ) $=
|
|
( cpw wcel elpwi sseld impcom ) BCDEZABEACEIBCABCFGH $.
|
|
|
|
${
|
|
$d y A $. $d x y $.
|
|
nfpw.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for power class. (Contributed by NM,
|
|
28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
nfpw $p |- F/_ x ~P A $=
|
|
( vy cpw cv wss cab df-pw nfcv nfss nfab nfcxfr ) ABEDFZBGZDHDBIOADANBANJ
|
|
CKLM $.
|
|
$}
|
|
|
|
$( Membership of the original in a power set. (Contributed by Stefan O'Rear,
|
|
1-Feb-2015.) $)
|
|
pwidg $p |- ( A e. V -> A e. ~P A ) $=
|
|
( wcel cpw wss ssid elpwg mpbiri ) ABCAADCAAEAFAABGH $.
|
|
|
|
${
|
|
pwid.1 $e |- A e. _V $.
|
|
$( A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
pwid $p |- A e. ~P A $=
|
|
( cvv wcel cpw pwidg ax-mp ) ACDAAEDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for power class. (Contributed by NM,
|
|
21-Jun-2009.) $)
|
|
pwss $p |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) ) $=
|
|
( cpw wss cv wcel wi wal dfss2 df-pw abeq2i imbi1i albii bitri ) BDZCEAFZ
|
|
PGZQCGZHZAIQBEZSHZAIAPCJTUBARUASUAAPABKLMNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Unordered and ordered pairs
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols needed. $)
|
|
$c <. $. $( Bracket (the period distinguishes it from 'less than') $)
|
|
$c >. $. $( Bracket (the period distinguishes it from 'greater than') $)
|
|
|
|
$( Extend class notation to include singleton. $)
|
|
csn $a class { A } $.
|
|
$( Extend class notation to include unordered pair. $)
|
|
cpr $a class { A , B } $.
|
|
$( Extend class notation to include unordered triplet. $)
|
|
ctp $a class { A , B , C } $.
|
|
$( Extend class notation to include ordered pair. $)
|
|
cop $a class <. A , B >. $.
|
|
$( Extend class notation to include ordered triple. $)
|
|
cotp $a class <. A , B , C >. $.
|
|
|
|
${
|
|
$d x A $. $d y A $. $d z x $. $d z y $. $d z A $.
|
|
$( Soundness justification theorem for ~ df-sn . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
snjust $p |- { x | x = A } = { y | y = A } $=
|
|
( vz cv wceq cab eqeq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINP
|
|
DBMOCHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For
|
|
convenience, it is well-defined for proper classes, i.e., those that are
|
|
not elements of ` _V ` , although it is not very meaningful in this
|
|
case. For an alternate definition see ~ dfsn2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-sn $a |- { A } = { x | x = A } $.
|
|
$}
|
|
|
|
$( Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They
|
|
are unordered, so ` { A , B } = { B , A } ` as proven by ~ prcom . For a
|
|
more traditional definition, but requiring a dummy variable, see ~ dfpr2 .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
df-pr $a |- { A , B } = ( { A } u. { B } ) $.
|
|
|
|
$( Define unordered triple of classes. Definition of [Enderton] p. 19.
|
|
(Contributed by NM, 9-Apr-1994.) $)
|
|
df-tp $a |- { A , B , C } = ( { A , B } u. { C } ) $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Definition of an ordered pair, equivalent to Kuratowski's definition
|
|
` { { A } , { A , B } } ` when the arguments are sets. Since the
|
|
behavior of Kuratowski definition is not very useful for proper classes,
|
|
we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For
|
|
Kuratowski's actual definition when the arguments are sets, see ~ dfop .
|
|
|
|
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally
|
|
as ` <. A , B >. = { { A } , { A , B } } ` , which has different
|
|
behavior from our ~ df-op when the arguments are proper classes.
|
|
Ordinarily this difference is not important, since neither definition is
|
|
meaningful in that case. Our ~ df-op was chosen because it often makes
|
|
proofs shorter by eliminating unnecessary sethood hypotheses.
|
|
|
|
There are other ways to define ordered pairs. The basic requirement is
|
|
that two ordered pairs are equal iff their respective members are equal.
|
|
In 1914 Norbert Wiener gave the first successful definition
|
|
` <. A , B >. ` _2 ` = { { { A } , (/) } , { { B } } } ` . This was
|
|
simplified by Kazimierz Kuratowski in 1921 to our present definition.
|
|
An even simpler definition is ` <. A , B >. ` _3
|
|
` = { A , { A , B } } ` , but it requires the Axiom of Regularity for
|
|
its justification and is not commonly used. Finally, an ordered pair of
|
|
real numbers can be represented by a complex number. (Contributed by
|
|
NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
df-op $a |- <. A , B >. = { x |
|
|
( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } $.
|
|
$}
|
|
|
|
$( Define ordered triple of classes. Definition of ordered triple in [Stoll]
|
|
p. 25. (Contributed by NM, 3-Apr-2015.) $)
|
|
df-ot $a |- <. A , B , C >. = <. <. A , B >. , C >. $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 5-Aug-1993.) $)
|
|
sneq $p |- ( A = B -> { A } = { B } ) $=
|
|
( vx wceq cv cab csn eqeq2 abbidv df-sn 3eqtr4g ) ABDZCEZADZCFMBDZCFAGBGL
|
|
NOCABMHICAJCBJK $.
|
|
$}
|
|
|
|
${
|
|
sneqi.1 $e |- A = B $.
|
|
$( Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) $)
|
|
sneqi $p |- { A } = { B } $=
|
|
( wceq csn sneq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
sneqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) $)
|
|
sneqd $p |- ( ph -> { A } = { B } ) $=
|
|
( wceq csn sneq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Alternate definition of singleton. Definition 5.1 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 24-Apr-1994.) $)
|
|
dfsn2 $p |- { A } = { A , A } $=
|
|
( cpr csn cun df-pr unidm eqtr2i ) AABACZHDHAAEHFG $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( There is exactly one element in a singleton. Exercise 2 of
|
|
[TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.)
|
|
(Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
elsng $p |- ( A e. V -> ( A e. { B } <-> A = B ) ) $=
|
|
( vx cv wceq csn eqeq1 df-sn elab2g ) DEZBFABFDABGCKABHDBIJ $.
|
|
$}
|
|
|
|
${
|
|
elsn.1 $e |- A e. _V $.
|
|
$( There is exactly one element in a singleton. Exercise 2 of
|
|
[TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) $)
|
|
elsn $p |- ( A e. { B } <-> A = B ) $=
|
|
( cvv wcel csn wceq wb elsng ax-mp ) ADEABFEABGHCABDIJ $.
|
|
$}
|
|
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 21-Jun-1993.) $)
|
|
velsn $p |- ( x e. { A } <-> x = A ) $=
|
|
( cv vex elsn ) ACBADE $.
|
|
|
|
$( There is only one element in a singleton. (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
elsni $p |- ( A e. { B } -> A = B ) $=
|
|
( csn wcel wceq elsng ibi ) ABCZDABEABHFG $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of unordered pair. Definition 5.1 of
|
|
[TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) $)
|
|
dfpr2 $p |- { A , B } = { x | ( x = A \/ x = B ) } $=
|
|
( cpr csn cun cv wceq wo df-pr wcel elun velsn orbi12i bitri abbi2i eqtri
|
|
cab ) BCDBEZCEZFZAGZBHZUBCHZIZARBCJUEAUAUBUAKUBSKZUBTKZIUEUBSTLUFUCUGUDAB
|
|
MACMNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM,
|
|
13-Sep-1995.) $)
|
|
elprg $p |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) ) $=
|
|
( vx cv wceq wo cpr eqeq1 orbi12d dfpr2 elab2g ) EFZBGZNCGZHABGZACGZHEABC
|
|
IDNAGOQPRNABJNACJKEBCLM $.
|
|
$}
|
|
|
|
${
|
|
elpr.1 $e |- A e. _V $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
13-Sep-1995.) $)
|
|
elpr $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
|
|
( cvv wcel cpr wceq wo wb elprg ax-mp ) AEFABCGFABHACHIJDABCEKL $.
|
|
$}
|
|
|
|
${
|
|
elpr2.1 $e |- B e. _V $.
|
|
elpr2.2 $e |- C e. _V $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
elpr2 $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
|
|
( cpr wcel wceq wo elprg ibi cvv wb eleq1 mpbiri jaoi syl ibir impbii ) A
|
|
BCFZGZABHZACHZIZUAUDABCTJKUDUAUDALGZUAUDMUBUEUCUBUEBLGDABLNOUCUECLGEACLNO
|
|
PABCLJQRS $.
|
|
$}
|
|
|
|
$( If a class is an element of a pair, then it is one of the two paired
|
|
elements. (Contributed by Scott Fenton, 1-Apr-2011.) $)
|
|
elpri $p |- ( A e. { B , C } -> ( A = B \/ A = C ) ) $=
|
|
( cpr wcel wceq wo elprg ibi ) ABCDZEABFACFGABCJHI $.
|
|
|
|
${
|
|
nelpri.1 $e |- A =/= B $.
|
|
nelpri.2 $e |- A =/= C $.
|
|
$( If an element doesn't match the items in an unordered pair, it is not in
|
|
the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
nelpri $p |- -. A e. { B , C } $=
|
|
( wne cpr wcel wn wa wceq wo neanior elpri con3i sylbi mp2an ) ABFZACFZAB
|
|
CGHZIZDERSJABKACKLZIUAABACMTUBABCNOPQ $.
|
|
$}
|
|
|
|
$( A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|
|
(Contributed by NM, 28-Oct-2003.) $)
|
|
snidg $p |- ( A e. V -> A e. { A } ) $=
|
|
( wcel csn wceq eqid elsng mpbiri ) ABCAADCAAEAFAABGH $.
|
|
|
|
$( A class is a set iff it is a member of its singleton. (Contributed by NM,
|
|
5-Apr-2004.) $)
|
|
snidb $p |- ( A e. _V <-> A e. { A } ) $=
|
|
( cvv wcel csn snidg elex impbii ) ABCAADZCABEAHFG $.
|
|
|
|
${
|
|
snid.1 $e |- A e. _V $.
|
|
$( A set is a member of its singleton. Part of Theorem 7.6 of [Quine]
|
|
p. 49. (Contributed by NM, 31-Dec-1993.) $)
|
|
snid $p |- A e. { A } $=
|
|
( cvv wcel csn snidb mpbi ) ACDAAEDBAFG $.
|
|
$}
|
|
|
|
$( A setvar variable is a member of its singleton (common case).
|
|
(Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
vsnid $p |- x e. { x } $=
|
|
( cv vex snid ) ABACD $.
|
|
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. This variation requires only that ` B ` , rather than ` A ` , be a
|
|
set. (Contributed by NM, 28-Oct-2003.) $)
|
|
elsn2g $p |- ( B e. V -> ( A e. { B } <-> A = B ) ) $=
|
|
( wcel csn wceq elsni snidg eleq1 syl5ibrcom impbid2 ) BCDZABEZDZABFZABGLNO
|
|
BMDBCHABMIJK $.
|
|
|
|
${
|
|
elsn2.1 $e |- B e. _V $.
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. This variation requires only that ` B ` , rather than ` A ` , be
|
|
a set. (Contributed by NM, 12-Jun-1994.) $)
|
|
elsn2 $p |- ( A e. { B } <-> A = B ) $=
|
|
( cvv wcel csn wceq wb elsn2g ax-mp ) BDEABFEABGHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A singleton has at most one element. This works whether ` A ` is a
|
|
proper class or not, and in that sense can be seen as encompassing both
|
|
~ snmg and ~ snprc . (Contributed by Jim Kingdon, 30-Aug-2018.) $)
|
|
mosn $p |- E* x x e. { A } $=
|
|
( cv csn wcel wmo wceq moeq velsn mobii mpbir ) ACZBDEZAFLBGZAFABHMNAABIJ
|
|
K $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
ralsnsg $p |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) ) $=
|
|
( wcel wsbc cv wceq wi wal csn wral sbc6g df-ral velsn imbi1i albii bitri
|
|
syl6rbbr ) CDEABCFBGZCHZAIZBJZABCKZLZABCDMUETUDEZAIZBJUCABUDNUGUBBUFUAABC
|
|
OPQRS $.
|
|
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralsns $p |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) ) $=
|
|
( wcel wsbc cv wceq wi wal csn wral sbc6g df-ral velsn imbi1i albii bitri
|
|
syl6rbbr ) CDEABCFBGZCHZAIZBJZABCKZLZABCDMUETUDEZAIZBJUCABUDNUGUBBUFUAABC
|
|
OPQRS $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) $)
|
|
rexsns $p |- ( E. x e. { A } ph <-> [. A / x ]. ph ) $=
|
|
( cv csn wcel wex wceq wrex wsbc velsn anbi1i exbii df-rex sbc5 3bitr4i
|
|
wa ) BDZCEZFZAQZBGRCHZAQZBGABSIABCJUAUCBTUBABCKLMABSNABCOP $.
|
|
|
|
ralsng.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
ralsng $p |- ( A e. V -> ( A. x e. { A } ph <-> ps ) ) $=
|
|
( wcel csn wral wsbc ralsns sbcieg bitrd ) DEGACDHIACDJBACDEKABCDEFLM $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
NM, 29-Jan-2012.) $)
|
|
rexsng $p |- ( A e. V -> ( E. x e. { A } ph <-> ps ) ) $=
|
|
( csn wrex wsbc wcel rexsns sbcieg syl5bb ) ACDGHACDIDEJBACDKABCDEFLM $.
|
|
$}
|
|
|
|
$( There is a set being the element of a singleton if and only if there is an
|
|
element of the singleton. (Contributed by Alexander van der Vekens,
|
|
1-Jan-2018.) $)
|
|
exsnrex $p |- ( E. x M = { x } <-> E. x e. M M = { x } ) $=
|
|
( cv csn wceq wex wcel wa wrex vex snid eleq2 mpbiri pm4.71ri df-rex bitr4i
|
|
exbii ) BACZDZEZAFRBGZTHZAFTABITUBATUATUARSGRAJKBSRLMNQTABOP $.
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
ralsn.1 $e |- A e. _V $.
|
|
ralsn.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Convert a quantification over a singleton to a substitution.
|
|
(Contributed by NM, 27-Apr-2009.) $)
|
|
ralsn $p |- ( A. x e. { A } ph <-> ps ) $=
|
|
( cvv wcel csn wral wb ralsng ax-mp ) DGHACDIJBKEABCDGFLM $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
Jeff Madsen, 5-Jan-2011.) $)
|
|
rexsn $p |- ( E. x e. { A } ph <-> ps ) $=
|
|
( cvv wcel csn wrex wb rexsng ax-mp ) DGHACDIJBKEABCDGFLM $.
|
|
$}
|
|
|
|
$( Members of an unordered triple of classes. (Contributed by FL,
|
|
2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltpg $p |- ( A e. V -> ( A e. { B , C , D } <->
|
|
( A = B \/ A = C \/ A = D ) ) ) $=
|
|
( wcel cpr csn wceq ctp w3o elprg elsng orbi12d cun df-tp eleq2i elun bitri
|
|
wo df-3or 3bitr4g ) AEFZABCGZFZADHZFZTZABIZACIZTZADIZTABCDJZFZUIUJULKUCUEUK
|
|
UGULABCELADEMNUNAUDUFOZFUHUMUOABCDPQAUDUFRSUIUJULUAUB $.
|
|
|
|
$( A member of an unordered triple of classes is one of them. (Contributed
|
|
by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltpi $p |- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) ) $=
|
|
( ctp wcel wceq w3o eltpg ibi ) ABCDEZFABGACGADGHABCDKIJ $.
|
|
|
|
${
|
|
eltp.1 $e |- A e. _V $.
|
|
$( A member of an unordered triple of classes is one of them. Special case
|
|
of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM,
|
|
8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltp $p |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) $=
|
|
( cvv wcel ctp wceq w3o wb eltpg ax-mp ) AFGABCDHGABIACIADIJKEABCDFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Alternate definition of unordered triple of classes. Special case of
|
|
Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
dftp2 $p |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) } $=
|
|
( cv wceq w3o ctp vex eltp abbi2i ) AEZBFLCFLDFGABCDHLBCDAIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
nfpr.1 $e |- F/_ x A $.
|
|
nfpr.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for unordered pairs. (Contributed by
|
|
NM, 14-Nov-1995.) $)
|
|
nfpr $p |- F/_ x { A , B } $=
|
|
( vy cpr cv wceq wo cab dfpr2 nfeq2 nfor nfab nfcxfr ) ABCGFHZBIZQCIZJZFK
|
|
FBCLTAFRSAAQBDMAQCEMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $. $d x th $.
|
|
ralprg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ralprg.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Convert a quantification over a pair to a conjunction. (Contributed by
|
|
NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralprg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A. x e. { A , B } ph <-> ( ps /\ ch ) ) ) $=
|
|
( cpr wral csn wa wcel cun df-pr raleqi ralunb ralsng bi2anan9 syl5bb
|
|
bitri ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZNBCNUEADUFUHPZLUJADUDUMEFQRADUFUH
|
|
SUCUKUGBULUICABDEGITACDFHJTUAUB $.
|
|
|
|
$( Convert a quantification over a pair to a disjunction. (Contributed by
|
|
NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
rexprg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x e. { A , B } ph <-> ( ps \/ ch ) ) ) $=
|
|
( cpr wrex csn wo wcel wa cun df-pr rexeqi rexsng orbi1d orbi2d sylan9bb
|
|
rexun bitri syl5bb ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZPBCNZUHADUIUKQZLUMAD
|
|
UGUQEFRSADUIUKUDUEUNUMBULNUOUPUNUJBULABDEGITUAUOULCBACDFHJTUBUCUF $.
|
|
|
|
raltpg.3 $e |- ( x = C -> ( ph <-> th ) ) $.
|
|
$( Convert a quantification over a triple to a conjunction. (Contributed
|
|
by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
raltpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) ) $=
|
|
( wcel w3a cpr wral csn wa ctp wb ralprg ralsng bi2anan9 3impa cun raleqi
|
|
df-tp ralunb bitri df-3an 3bitr4g ) FIOZGJOZHKOZPAEFGQZRZAEHSZRZTZBCTZDTZ
|
|
AEFGHUAZRZBCDPUNUOUPVAVCUBUNUOTURVBUPUTDABCEFGIJLMUCADEHKNUDUEUFVEAEUQUSU
|
|
GZRVAAEVDVFFGHUIUHAEUQUSUJUKBCDULUM $.
|
|
|
|
$( Convert a quantification over a triple to a disjunction. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rextpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) ) $=
|
|
( wcel w3a cpr wrex csn wo ctp wb wa rexprg orbi1d rexsng orbi2d sylan9bb
|
|
w3o 3impa cun df-tp rexeqi rexun bitri df-3or 3bitr4g ) FIOZGJOZHKOZPAEFG
|
|
QZRZAEHSZRZTZBCTZDTZAEFGHUAZRZBCDUIURUSUTVEVGUBURUSUCZVEVFVDTUTVGVJVBVFVD
|
|
ABCEFGIJLMUDUEUTVDDVFADEHKNUFUGUHUJVIAEVAVCUKZRVEAEVHVKFGHULUMAEVAVCUNUOB
|
|
CDUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $. $d x ch $.
|
|
ralpr.1 $e |- A e. _V $.
|
|
ralpr.2 $e |- B e. _V $.
|
|
ralpr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ralpr.4 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Convert a quantification over a pair to a conjunction. (Contributed by
|
|
NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralpr $p |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) $=
|
|
( cvv wcel cpr wral wa wb ralprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
|
|
$.
|
|
|
|
$( Convert an existential quantification over a pair to a disjunction.
|
|
(Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
rexpr $p |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) $=
|
|
( cvv wcel cpr wrex wo wb rexprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $. $d x th $.
|
|
raltp.1 $e |- A e. _V $.
|
|
raltp.2 $e |- B e. _V $.
|
|
raltp.3 $e |- C e. _V $.
|
|
raltp.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
raltp.5 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
raltp.6 $e |- ( x = C -> ( ph <-> th ) ) $.
|
|
$( Convert a quantification over a triple to a conjunction. (Contributed
|
|
by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
raltp $p |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) $=
|
|
( cvv wcel ctp wral w3a wb raltpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
|
|
HOOOLMNUAUB $.
|
|
|
|
$( Convert a quantification over a triple to a disjunction. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rextp $p |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) $=
|
|
( cvv wcel ctp wrex w3o wb rextpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
|
|
HOOOLMNUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( TODO - make obsolete; use ralsnsSBC instead - also,
|
|
shorten posn w/ ralsn or ralsng $)
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
sbcsng $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x e. { A } ph ) ) $=
|
|
( wcel csn wral wsbc ralsns bicomd ) CDEABCFGABCHABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
nfsn.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for singletons. (Contributed by NM,
|
|
14-Nov-1995.) $)
|
|
nfsn $p |- F/_ x { A } $=
|
|
( csn cpr dfsn2 nfpr nfcxfr ) ABDBBEBFABBCCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d V y $. $d x y $.
|
|
$( Distribute proper substitution through the singleton of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbsng $p |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } ) $=
|
|
( vy wcel wceq cab csb csn wsbc csbabg sbceq2g abbidv eqtrd df-sn csbeq2i
|
|
cv 3eqtr4g ) BDFZABERZCGZEHZIZUAABCIZGZEHZABCJZIUEJTUDUBABKZEHUGUBAEBDLTU
|
|
IUFEABUACDMNOABUHUCECPQEUEPS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Intersection with the singleton of a non-member is disjoint.
|
|
(Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
|
|
disjsn $p |- ( ( A i^i { B } ) = (/) <-> -. B e. A ) $=
|
|
( vx csn cin c0 wceq cv wcel wn wi wal wa disj1 con2b velsn imbi1i 3bitri
|
|
imnan albii wex alnex df-clel xchbinxr ) ABDZEFGCHZAIZUFUEIZJKZCLUFBGZUGM
|
|
ZJZCLZBAIZJCAUENUIULCUIUHUGJZKUJUOKULUGUHOUHUJUOCBPQUJUGSRTUMUKCUAUNUKCUB
|
|
CBAUCUDR $.
|
|
$}
|
|
|
|
$( Intersection of distinct singletons is disjoint. (Contributed by NM,
|
|
25-May-1998.) $)
|
|
disjsn2 $p |- ( A =/= B -> ( { A } i^i { B } ) = (/) ) $=
|
|
( wne csn wcel wn cin c0 wceq elsni eqcomd necon3ai disjsn sylibr ) ABCBADZ
|
|
EZFOBDGHIPABPBABAJKLOBMN $.
|
|
|
|
$( The intersection of distinct unordered pairs is disjoint. (Contributed by
|
|
Alexander van der Vekens, 11-Nov-2017.) $)
|
|
disjpr2 $p |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) )
|
|
-> ( { A , B } i^i { C , D } ) = (/) ) $=
|
|
( wne wa cpr cin csn c0 wceq df-pr ineq1i indir eqtri disjsn2 adantr adantl
|
|
cun syl5eq a1i ineq2d indi jca un00 sylib uneq12d un0 syl6eq eqtrd ) ACEZBC
|
|
EZFZADEZBDEZFZFZABGZCDGZHURCIZDIZSZHZJUQUSVBURUSVBKUQCDLUAUBUQVCURUTHZURVAH
|
|
ZSZJURUTVAUCUQVFJJSJUQVDJVEJUQVDAIZUTHZBIZUTHZSZJVDVGVISZUTHVKURVLUTABLZMVG
|
|
VIUTNOUQVHJKZVJJKZFVKJKUQVNVOUMVNUPUKVNULACPQQUMVOUPULVOUKBCPRQUDVHVJUEUFTU
|
|
QVEVGVAHZVIVAHZSZJVEVLVAHVRURVLVAVMMVGVIVANOUQVPJKZVQJKZFVRJKUQVSVTUPVSUMUN
|
|
VSUOADPQRUPVTUMUOVTUNBDPRRUDVPVQUEUFTUGJUHUITUJ $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The singleton of a proper class (one that doesn't exist) is the empty
|
|
set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.) $)
|
|
snprc $p |- ( -. A e. _V <-> { A } = (/) ) $=
|
|
( vx cv csn wcel wex wn wceq cvv velsn exbii notbii wal alnex bitri isset
|
|
c0 eq0 3bitr4ri ) BCZADZEZBFZGZTAHZBFZGUAQHZAIEZGUCUFUBUEBBAJKLUGUBGBMUDB
|
|
UARUBBNOUHUFBAPLS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
$( Special case of ~ r19.12 where its converse holds. (Contributed by NM,
|
|
19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by
|
|
BJ, 20-Dec-2021.) $)
|
|
r19.12sn $p |- ( A e. V ->
|
|
( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) ) $=
|
|
( wcel wral wsbc csn wrex sbcralg rexsns ralbii 3bitr4g ) DFGACEHZBDIABDI
|
|
ZCEHPBDJZKABRKZCEHABCDEFLPBDMSQCEABDMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Condition where a restricted class abstraction is a singleton.
|
|
(Contributed by NM, 28-May-2006.) $)
|
|
rabsn $p |- ( B e. A -> { x e. A | x = B } = { B } ) $=
|
|
( wcel cv wceq wa cab crab csn eleq1 pm5.32ri abbidv df-rab df-sn 3eqtr4g
|
|
baib ) CBDZAEZBDZSCFZGZAHUAAHUAABICJRUBUAAUBRUAUATRSCBKLQMUAABNACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
rabrsndc.1 $e |- A e. _V $.
|
|
rabrsndc.2 $e |- DECID ph $.
|
|
$( A class abstraction over a decidable proposition restricted to a
|
|
singleton is either the empty set or the singleton itself. (Contributed
|
|
by Jim Kingdon, 8-Aug-2018.) $)
|
|
rabrsndc $p |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) ) $=
|
|
( csn crab wceq c0 wo wral wsbc cvv ax-mp wb ralsns orbi12i mpbir eqeq1
|
|
wn wcel pm2.1dc sbcth sbcor mpbi rabeq0 eqcom rabid2 bitri orbi12d mpbiri
|
|
wdc ) DABCGZHZIZDJIZDUNIZKUOJIZUOUNIZKZVAAUAZBUNLZABUNLZKZVEVBBCMZABCMZKZ
|
|
VBAKZBCMZVHCNUBZVJEVIBCNAUMVIFAUCOUDOVBABCUEUFVCVFVDVGVKVCVFPEVBBCNQOVKVD
|
|
VGPEABCNQORSUSVCUTVDABUNUGUTUNUOIVDUOUNUHABUNUIUJRSUPUQUSURUTDUOJTDUOUNTU
|
|
KUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y A $.
|
|
$( Another way to express existential uniqueness of a wff: its class
|
|
abstraction is a singleton. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
euabsn2 $p |- ( E! x ph <-> E. y { x | ph } = { y } ) $=
|
|
( weu cv wceq wb wal wex cab csn df-eu wcel abeq1 velsn albii bitri exbii
|
|
bibi2i bitr4i ) ABDABEZCEZFZGZBHZCIABJUBKZFZCIABCLUGUECUGAUAUFMZGZBHUEABU
|
|
FNUIUDBUHUCABUBOSPQRT $.
|
|
|
|
$( Another way to express existential uniqueness of a wff: its class
|
|
abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) $)
|
|
euabsn $p |- ( E! x ph <-> E. x { x | ph } = { x } ) $=
|
|
( vy weu cab cv csn wceq wex euabsn2 nfab1 nfeq1 sneq eqeq2d cbvex bitr4i
|
|
nfv ) ABDABEZCFZGZHZCIRBFZGZHZBIABCJUDUABCUDCQBRTABKLUBSHUCTRUBSMNOP $.
|
|
|
|
$( A way to express restricted existential uniqueness of a wff: its
|
|
restricted class abstraction is a singleton. (Contributed by NM,
|
|
30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
|
|
reusn $p |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } ) $=
|
|
( cv wcel wa weu cab csn wceq wex wreu euabsn2 df-reu df-rab eqeq1i exbii
|
|
crab 3bitr4i ) BEDFAGZBHUABIZCEJZKZCLABDMABDSZUCKZCLUABCNABDOUFUDCUEUBUCA
|
|
BDPQRT $.
|
|
|
|
$( Restricted existential uniqueness determined by a singleton.
|
|
(Contributed by NM, 29-May-2006.) $)
|
|
absneu $p |- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph ) $=
|
|
( vy wcel cab csn wceq wa wex weu sneq eqeq2d spcegv imp euabsn2 sylibr
|
|
cv ) CDFZABGZCHZIZJUAESZHZIZEKZABLTUCUGUFUCECDUDCIUEUBUAUDCMNOPABEQR $.
|
|
|
|
$( Restricted existential uniqueness determined by a singleton.
|
|
(Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
rabsneu $p |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph ) $=
|
|
( wcel crab csn wceq wa cv weu df-rab eqeq1i absneu sylan2b df-reu sylibr
|
|
wreu cab ) CEFZABDGZCHZIZJBKDFAJZBLZABDSUDUAUEBTZUCIUFUBUGUCABDMNUEBCEOPA
|
|
BDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to express " ` A ` is a singleton." (Contributed by NM,
|
|
30-Oct-2010.) $)
|
|
eusn $p |- ( E! x x e. A <-> E. x A = { x } ) $=
|
|
( cv wcel weu cab csn wceq wex euabsn abid2 eqeq1i exbii bitri ) ACZBDZAE
|
|
PAFZOGZHZAIBRHZAIPAJSTAQBRABKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
rabsnt.1 $e |- B e. _V $.
|
|
rabsnt.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Truth implied by equality of a restricted class abstraction and a
|
|
singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
rabsnt $p |- ( { x e. A | ph } = { B } -> ps ) $=
|
|
( crab csn wceq wcel snid id syl5eleqr elrab simprbi syl ) ACDHZEIZJZERKZ
|
|
BTESREFLTMNUAEDKBABCEDGOPQ $.
|
|
$}
|
|
|
|
$( Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) $)
|
|
prcom $p |- { A , B } = { B , A } $=
|
|
( csn cun cpr uncom df-pr 3eqtr4i ) ACZBCZDJIDABEBAEIJFABGBAGH $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
preq1 $p |- ( A = B -> { A , C } = { B , C } ) $=
|
|
( wceq csn cun cpr sneq uneq1d df-pr 3eqtr4g ) ABDZAEZCEZFBEZNFACGBCGLMONAB
|
|
HIACJBCJK $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) $)
|
|
preq2 $p |- ( A = B -> { C , A } = { C , B } ) $=
|
|
( wceq cpr preq1 prcom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12 $p |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } ) $=
|
|
( wceq cpr preq1 preq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.
|
|
|
|
${
|
|
preq1i.1 $e |- A = B $.
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq1i $p |- { A , C } = { B , C } $=
|
|
( wceq cpr preq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq2i $p |- { C , A } = { C , B } $=
|
|
( wceq cpr preq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
preq12i.2 $e |- C = D $.
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12i $p |- { A , C } = { B , D } $=
|
|
( wceq cpr preq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
preq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq1d $p |- ( ph -> { A , C } = { B , C } ) $=
|
|
( wceq cpr preq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq2d $p |- ( ph -> { C , A } = { C , B } ) $=
|
|
( wceq cpr preq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
preq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12d $p |- ( ph -> { A , C } = { B , D } ) $=
|
|
( wceq cpr preq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
|
|
$}
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq1 $p |- ( A = B -> { A , C , D } = { B , C , D } ) $=
|
|
( wceq cpr csn cun ctp preq1 uneq1d df-tp 3eqtr4g ) ABEZACFZDGZHBCFZPHACDIB
|
|
CDINOQPABCJKACDLBCDLM $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq2 $p |- ( A = B -> { C , A , D } = { C , B , D } ) $=
|
|
( wceq cpr csn cun ctp preq2 uneq1d df-tp 3eqtr4g ) ABEZCAFZDGZHCBFZPHCADIC
|
|
BDINOQPABCJKCADLCBDLM $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq3 $p |- ( A = B -> { C , D , A } = { C , D , B } ) $=
|
|
( wceq cpr csn cun ctp sneq uneq2d df-tp 3eqtr4g ) ABEZCDFZAGZHOBGZHCDAICDB
|
|
INPQOABJKCDALCDBLM $.
|
|
|
|
${
|
|
tpeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq1d $p |- ( ph -> { A , C , D } = { B , C , D } ) $=
|
|
( wceq ctp tpeq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq2d $p |- ( ph -> { C , A , D } = { C , B , D } ) $=
|
|
( wceq ctp tpeq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq3d $p |- ( ph -> { C , D , A } = { C , D , B } ) $=
|
|
( wceq ctp tpeq3 syl ) ABCGDEBHDECHGFBCDEIJ $.
|
|
|
|
tpeq123d.2 $e |- ( ph -> C = D ) $.
|
|
tpeq123d.3 $e |- ( ph -> E = F ) $.
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq123d $p |- ( ph -> { A , C , E } = { B , D , F } ) $=
|
|
( ctp tpeq1d tpeq2d tpeq3d 3eqtrd ) ABDFKCDFKCEFKCEGKABCDFHLADECFIMAFGCEJ
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Rotation of the elements of an unordered triple. (Contributed by Alan
|
|
Sare, 24-Oct-2011.) $)
|
|
tprot $p |- { A , B , C } = { B , C , A } $=
|
|
( vx cv wceq w3o cab ctp 3orrot abbii dftp2 3eqtr4i ) DEZAFZNBFZNCFZGZDHP
|
|
QOGZDHABCIBCAIRSDOPQJKDABCLDBCALM $.
|
|
$}
|
|
|
|
$( Swap 1st and 2nd members of an undordered triple. (Contributed by NM,
|
|
22-May-2015.) $)
|
|
tpcoma $p |- { A , B , C } = { B , A , C } $=
|
|
( cpr csn cun ctp prcom uneq1i df-tp 3eqtr4i ) ABDZCEZFBADZMFABCGBACGLNMABH
|
|
IABCJBACJK $.
|
|
|
|
$( Swap 2nd and 3rd members of an undordered triple. (Contributed by NM,
|
|
22-May-2015.) $)
|
|
tpcomb $p |- { A , B , C } = { A , C , B } $=
|
|
( ctp tpcoma tprot 3eqtr4i ) BCADCBADABCDACBDBCAEABCFACBFG $.
|
|
|
|
$( Split off the first element of an unordered triple. (Contributed by Mario
|
|
Carneiro, 5-Jan-2016.) $)
|
|
tpass $p |- { A , B , C } = ( { A } u. { B , C } ) $=
|
|
( ctp cpr csn cun df-tp tprot uncom 3eqtr4i ) BCADBCEZAFZGABCDMLGBCAHABCIML
|
|
JK $.
|
|
|
|
$( Two ways to write an unordered quadruple. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
qdass $p |- ( { A , B } u. { C , D } ) = ( { A , B , C } u. { D } ) $=
|
|
( cpr csn cun ctp unass df-tp uneq1i df-pr uneq2i 3eqtr4ri ) ABEZCFZGZDFZGO
|
|
PRGZGABCHZRGOCDEZGOPRITQRABCJKUASOCDLMN $.
|
|
|
|
$( Two ways to write an unordered quadruple. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
qdassr $p |- ( { A , B } u. { C , D } ) = ( { A } u. { B , C , D } ) $=
|
|
( csn cun cpr ctp unass df-pr uneq1i tpass uneq2i 3eqtr4i ) AEZBEZFZCDGZFOP
|
|
RFZFABGZRFOBCDHZFOPRITQRABJKUASOBCDLMN $.
|
|
|
|
$( Unordered triple ` { A , A , B } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm12 $p |- { A , A , B } = { A , B } $=
|
|
( csn cun cpr ctp dfsn2 uneq1i df-pr df-tp 3eqtr4ri ) ACZBCZDAAEZMDABEAABFL
|
|
NMAGHABIAABJK $.
|
|
|
|
$( Unordered triple ` { A , B , A } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm13 $p |- { A , B , A } = { A , B } $=
|
|
( ctp cpr tprot tpidm12 eqtr3i ) AABCABACABDAABEABFG $.
|
|
|
|
$( Unordered triple ` { A , B , B } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm23 $p |- { A , B , B } = { A , B } $=
|
|
( ctp cpr tprot tpidm12 prcom 3eqtri ) ABBCBBACBADABDABBEBAFBAGH $.
|
|
|
|
$( Unordered triple ` { A , A , A } ` is just an overlong way to write
|
|
` { A } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm $p |- { A , A , A } = { A } $=
|
|
( ctp cpr csn tpidm12 dfsn2 eqtr4i ) AAABAACADAAEAFG $.
|
|
|
|
$( An unordered triple is an unordered pair if one of its elements is
|
|
identical with another element. (Contributed by Alexander van der Vekens,
|
|
6-Oct-2017.) $)
|
|
tppreq3 $p |- ( B = C -> { A , B , C } = { A , B } ) $=
|
|
( wceq ctp cpr tpeq3 eqcoms tpidm23 syl6eq ) BCDABCEZABBEZABFKLDCBCBABGHABI
|
|
J $.
|
|
|
|
$( An unordered pair contains its first member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) $)
|
|
prid1g $p |- ( A e. V -> A e. { A , B } ) $=
|
|
( wcel cpr wceq wo eqid orci elprg mpbiri ) ACDAABEDAAFZABFZGLMAHIAABCJK $.
|
|
|
|
$( An unordered pair contains its second member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) $)
|
|
prid2g $p |- ( B e. V -> B e. { A , B } ) $=
|
|
( wcel cpr prid1g prcom syl6eleq ) BCDBBAEABEBACFBAGH $.
|
|
|
|
${
|
|
prid1.1 $e |- A e. _V $.
|
|
$( An unordered pair contains its first member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
prid1 $p |- A e. { A , B } $=
|
|
( cvv wcel cpr prid1g ax-mp ) ADEAABFECABDGH $.
|
|
$}
|
|
|
|
${
|
|
prid2.1 $e |- B e. _V $.
|
|
$( An unordered pair contains its second member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
prid2 $p |- B e. { A , B } $=
|
|
( cpr prid1 prcom eleqtri ) BBADABDBACEBAFG $.
|
|
$}
|
|
|
|
$( A proper class vanishes in an unordered pair. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
prprc1 $p |- ( -. A e. _V -> { A , B } = { B } ) $=
|
|
( cvv wcel wn csn c0 wceq cpr snprc cun uneq1 df-pr uncom un0 3eqtr4g sylbi
|
|
eqtr2i ) ACDEAFZGHZABIZBFZHAJTSUBKGUBKZUAUBSGUBLABMUCUBGKUBGUBNUBORPQ $.
|
|
|
|
$( A proper class vanishes in an unordered pair. (Contributed by NM,
|
|
22-Mar-2006.) $)
|
|
prprc2 $p |- ( -. B e. _V -> { A , B } = { A } ) $=
|
|
( cvv wcel wn cpr csn prcom prprc1 syl5eq ) BCDEABFBAFAGABHBAIJ $.
|
|
|
|
$( An unordered pair containing two proper classes is the empty set.
|
|
(Contributed by NM, 22-Mar-2006.) $)
|
|
prprc $p |- ( ( -. A e. _V /\ -. B e. _V ) -> { A , B } = (/) ) $=
|
|
( cvv wcel wn cpr csn c0 prprc1 wceq snprc biimpi sylan9eq ) ACDEBCDEZABFBG
|
|
ZHABINOHJBKLM $.
|
|
|
|
${
|
|
tpid1.1 $e |- A e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid1 $p |- A e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix1i eltp mpbir ) AABCEFAAGZABGZACGZHMNOAIJAAB
|
|
CDKL $.
|
|
$}
|
|
|
|
${
|
|
tpid2.1 $e |- B e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid2 $p |- B e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix2i eltp mpbir ) BABCEFBAGZBBGZBCGZHNMOBIJBAB
|
|
CDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
$( Closed theorem form of ~ tpid3 . (Contributed by Alan Sare,
|
|
24-Oct-2011.) $)
|
|
tpid3g $p |- ( A e. B -> A e. { C , D , A } ) $=
|
|
( vx wcel cv wceq wex ctp elisset w3o cab wi 3mix3 a1i abid syl6ibr dftp2
|
|
eleq2i eleq1 mpbidi exlimdv mpd ) ABFZEGZAHZEIACDAJZFZEABKUEUGUIEUGUFUHFZ
|
|
UIUEUEUGUFUFCHZUFDHZUGLZEMZFZUJUEUGUMUOUGUMNUEUGUKULOPUMEQRUHUNUFECDASTRU
|
|
FAUHUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
tpid3.1 $e |- C e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid3 $p |- C e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix3i eltp mpbir ) CABCEFCAGZCBGZCCGZHOMNCIJCAB
|
|
CDKL $.
|
|
$}
|
|
|
|
$( The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.) $)
|
|
snnzg $p |- ( A e. V -> { A } =/= (/) ) $=
|
|
( wcel csn c0 wne snidg ne0i syl ) ABCAADZCJEFABGJAHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The singleton of a set is inhabited. (Contributed by Jim Kingdon,
|
|
11-Aug-2018.) $)
|
|
snmg $p |- ( A e. V -> E. x x e. { A } ) $=
|
|
( wcel csn cv wex snidg elex2 syl ) BCDBBEZDAFKDAGBCHABKIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
snnz.1 $e |- A e. _V $.
|
|
$( The singleton of a set is not empty. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
snnz $p |- { A } =/= (/) $=
|
|
( cvv wcel csn c0 wne snnzg ax-mp ) ACDAEFGBACHI $.
|
|
|
|
$( The singleton of a set is inhabited. (Contributed by Jim Kingdon,
|
|
11-Aug-2018.) $)
|
|
snm $p |- E. x x e. { A } $=
|
|
( cvv wcel cv csn wex snmg ax-mp ) BDEAFBGEAHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A pair containing a set is inhabited. (Contributed by Jim Kingdon,
|
|
21-Sep-2018.) $)
|
|
prmg $p |- ( A e. V -> E. x x e. { A , B } ) $=
|
|
( wcel cv csn wex cpr snmg wceq wo orc velsn vex elpr 3imtr4i eximi syl )
|
|
BDEAFZBGEZAHTBCIEZAHABDJUAUBATBKZUCTCKZLUAUBUCUDMABNTBCAOPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
prnz.1 $e |- A e. _V $.
|
|
$( A pair containing a set is not empty. (Contributed by NM,
|
|
9-Apr-1994.) $)
|
|
prnz $p |- { A , B } =/= (/) $=
|
|
( cpr wcel c0 wne prid1 ne0i ax-mp ) AABDZEKFGABCHKAIJ $.
|
|
|
|
$( A pair containing a set is inhabited. (Contributed by Jim Kingdon,
|
|
21-Sep-2018.) $)
|
|
prm $p |- E. x x e. { A , B } $=
|
|
( cvv wcel cv cpr wex prmg ax-mp ) BEFAGBCHFAIDABCEJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A pair containing a set is not empty. (Contributed by FL,
|
|
19-Sep-2011.) $)
|
|
prnzg $p |- ( A e. V -> { A , B } =/= (/) ) $=
|
|
( vx cv cpr c0 wne wceq preq1 neeq1d vex prnz vtoclg ) DEZBFZGHABFZGHDACO
|
|
AIPQGOABJKOBDLMN $.
|
|
$}
|
|
|
|
${
|
|
tpnz.1 $e |- A e. _V $.
|
|
$( A triplet containing a set is not empty. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
tpnz $p |- { A , B , C } =/= (/) $=
|
|
( ctp wcel c0 wne tpid1 ne0i ax-mp ) AABCEZFLGHABCDILAJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
snss.1 $e |- A e. _V $.
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
snss $p |- ( A e. B <-> { A } C_ B ) $=
|
|
( vx cv csn wcel wi wal wceq wss velsn imbi1i albii dfss2 clel2 3bitr4ri
|
|
) DEZAFZGZRBGZHZDIRAJZUAHZDISBKABGUBUDDTUCUADALMNDSBODABCPQ $.
|
|
$}
|
|
|
|
$( Membership in a set with an element removed. (Contributed by NM,
|
|
10-Oct-2007.) $)
|
|
eldifsn $p |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) ) $=
|
|
( csn cdif wcel wn wa wne eldif elsng necon3bbid pm5.32i bitri ) ABCDZEFABF
|
|
ZAOFZGZHPACIZHABOJPRSPQACACBKLMN $.
|
|
|
|
$( Membership in a set with an element removed. (Contributed by NM,
|
|
10-Mar-2015.) $)
|
|
eldifsni $p |- ( A e. ( B \ { C } ) -> A =/= C ) $=
|
|
( csn cdif wcel wne eldifsn simprbi ) ABCDEFABFACGABCHI $.
|
|
|
|
$( ` A ` is not in ` ( B \ { A } ) ` . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neldifsn $p |- -. A e. ( B \ { A } ) $=
|
|
( csn cdif wcel wne neirr eldifsni mto ) ABACDEAAFAGABAHI $.
|
|
|
|
$( ` A ` is not in ` ( B \ { A } ) ` . Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
neldifsnd $p |- ( ph -> -. A e. ( B \ { A } ) ) $=
|
|
( csn cdif wcel wn neldifsn a1i ) BCBDEFGABCHI $.
|
|
|
|
$( Restricted existential quantification over a set with an element removed.
|
|
(Contributed by NM, 4-Feb-2015.) $)
|
|
rexdifsn $p |- ( E. x e. ( A \ { B } ) ph
|
|
<-> E. x e. A ( x =/= B /\ ph ) ) $=
|
|
( cv wne wa csn cdif wcel eldifsn anbi1i anass bitri rexbii2 ) ABEZDFZAGZBC
|
|
DHIZCPSJZAGPCJZQGZAGUARGTUBAPCDKLUAQAMNO $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) $)
|
|
snssg $p |- ( A e. V -> ( A e. B <-> { A } C_ B ) ) $=
|
|
( vx cv wcel csn wss eleq1 wceq sneq sseq1d vex snss vtoclbg ) DEZBFPGZBH
|
|
ABFAGZBHDACPABIPAJQRBPAKLPBDMNO $.
|
|
|
|
$( An element not in a set can be removed without affecting the set.
|
|
(Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
difsn $p |- ( -. A e. B -> ( B \ { A } ) = B ) $=
|
|
( vx wcel wn csn cdif cv wne wa eldifsn simpl wceq eleq1 biimpcd necon3bd
|
|
com12 ancld impbid2 syl5bb eqrdv ) ABDZEZCBAFGZBCHZUDDUEBDZUEAIZJZUCUFUEB
|
|
AKUCUHUFUFUGLUCUFUGUFUCUGUFUBUEAUEAMUFUBUEABNOPQRSTUA $.
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by NM,
|
|
16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
difprsnss $p |- ( { A , B } \ { A } ) C_ { B } $=
|
|
( vx cpr csn cdif cv wcel wn wa wceq vex elpr velsn notbii biorf biimparc
|
|
wo syl2anb eldif 3imtr4i ssriv ) CABDZAEZFZBEZCGZUCHZUGUDHZIZJUGBKZUGUEHU
|
|
GUFHUHUGAKZUKRZULIZUKUJUGABCLMUIULCANOUNUKUMULUKPQSUGUCUDTCBNUAUB $.
|
|
$}
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by Thierry
|
|
Arnoux, 4-Feb-2017.) $)
|
|
difprsn1 $p |- ( A =/= B -> ( { A , B } \ { A } ) = { B } ) $=
|
|
( wne cpr csn cdif wceq necom cin c0 disjsn2 disj3 sylib cun df-pr equncomi
|
|
difeq1i difun2 eqtri syl6reqr sylbir ) ABCBACZABDZAEZFZBEZGBAHUBUFUFUDFZUEU
|
|
BUFUDIJGUFUGGBAKUFUDLMUEUFUDNZUDFUGUCUHUDUCUDUFABOPQUFUDRSTUA $.
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by Alexander
|
|
van der Vekens, 5-Oct-2017.) $)
|
|
difprsn2 $p |- ( A =/= B -> ( { A , B } \ { B } ) = { A } ) $=
|
|
( wne cpr csn cdif prcom difeq1i wceq necom difprsn1 sylbi syl5eq ) ABCZABD
|
|
ZBEZFBADZPFZAEZOQPABGHNBACRSIABJBAKLM $.
|
|
|
|
$( Removal of a singleton from an unordered triple. (Contributed by
|
|
Alexander van der Vekens, 5-Oct-2017.) $)
|
|
diftpsn3 $p |- ( ( A =/= C /\ B =/= C )
|
|
-> ( { A , B , C } \ { C } ) = { A , B } ) $=
|
|
( wne wa ctp csn cdif cpr cun a1i c0 cin necom disjsn2 sylbi uneq12d syl6eq
|
|
wceq 3eqtrd df-tp difeq1d df-pr ineq1d incom indi eqtri adantr adantl unidm
|
|
difundir disj3 sylib eqcomd difid un0 ) ACDZBCDZEZABCFZCGZHABIZVAJZVAHZVBVA
|
|
HZVAVAHZJZVBUSUTVCVAUTVCSUSABCUAKUBVDVGSUSVBVAVAUKKUSVGVBLJVBUSVEVBVFLUSVBV
|
|
EUSVBVAMZLSVBVESUSVHAGZBGZJZVAMZVAVIMZVAVJMZJZLUSVBVKVAVBVKSUSABUCKUDVLVOSU
|
|
SVLVAVKMVOVKVAUEVAVIVJUFUGKUSVOLLJLUSVMLVNLUQVMLSZURUQCADVPACNCAOPUHURVNLSZ
|
|
UQURCBDVQBCNCBOPUIQLUJRTVBVAULUMUNVFLSUSVAUOKQVBUPRT $.
|
|
|
|
$( ` ( B \ { A } ) ` equals ` B ` if and only if ` A ` is not a member of
|
|
` B ` . Generalization of ~ difsn . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
difsnb $p |- ( -. A e. B <-> ( B \ { A } ) = B ) $=
|
|
( wcel wn csn cdif wceq difsn neldifsnd nelne1 mpdan necomd necon2bi impbii
|
|
wne ) ABCZDBAEFZBGABHPQBPBQPAQCDBQOPABIABQJKLMN $.
|
|
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
(Contributed by NM, 6-Jun-1994.) $)
|
|
snssi $p |- ( A e. B -> { A } C_ B ) $=
|
|
( wcel csn wss snssg ibi ) ABCADBEABBFG $.
|
|
|
|
${
|
|
snssd.1 $e |- ( ph -> A e. B ) $.
|
|
$( The singleton of an element of a class is a subset of the class
|
|
(deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
snssd $p |- ( ph -> { A } C_ B ) $=
|
|
( wcel csn wss wb snssg syl mpbid ) ABCEZBFCGZDALLMHDBCCIJK $.
|
|
$}
|
|
|
|
$( If we remove a single element from a class then put it back in, we end up
|
|
with a subset of the original class. If equality is decidable, we can
|
|
replace subset with equality as seen in ~ nndifsnid . (Contributed by Jim
|
|
Kingdon, 10-Aug-2018.) $)
|
|
difsnss $p |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A ) $=
|
|
( wcel csn cdif cun uncom wss snssi undifss sylib syl5eqss ) BACZABDZEZNFNO
|
|
FZAONGMNAHPAHBAINAJKL $.
|
|
|
|
$( Note that ` x ` is a dummy variable in the proof below. $)
|
|
$( Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
pw0 $p |- ~P (/) = { (/) } $=
|
|
( vx cv c0 wss cab wceq cpw csn ss0b abbii df-pw df-sn 3eqtr4i ) ABZCDZAENC
|
|
FZAECGCHOPANIJACKACLM $.
|
|
|
|
$( A singleton is a subset of an unordered pair containing its member.
|
|
(Contributed by NM, 27-Aug-2004.) $)
|
|
snsspr1 $p |- { A } C_ { A , B } $=
|
|
( csn cun cpr ssun1 df-pr sseqtr4i ) ACZIBCZDABEIJFABGH $.
|
|
|
|
$( A singleton is a subset of an unordered pair containing its member.
|
|
(Contributed by NM, 2-May-2009.) $)
|
|
snsspr2 $p |- { B } C_ { A , B } $=
|
|
( csn cun cpr ssun2 df-pr sseqtr4i ) BCZACZIDABEIJFABGH $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp1 $p |- { A } C_ { A , B , C } $=
|
|
( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtr4i ) ADZABEZCDZFZABCGMNPA
|
|
BHNOIJABCKL $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp2 $p |- { B } C_ { A , B , C } $=
|
|
( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtr4i ) BDZABEZCDZFZABCGMNPA
|
|
BHNOIJABCKL $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp3 $p |- { C } C_ { A , B , C } $=
|
|
( csn cpr cun ctp ssun2 df-tp sseqtr4i ) CDZABEZKFABCGKLHABCIJ $.
|
|
|
|
$( A pair is a subset of an unordered triple containing its members.
|
|
(Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
prsstp12 $p |- { A , B } C_ { A , B , C } $=
|
|
( cpr csn cun ctp ssun1 df-tp sseqtr4i ) ABDZKCEZFABCGKLHABCIJ $.
|
|
|
|
$( A pair is a subset of an unordered triple containing its members.
|
|
(Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
prsstp13 $p |- { A , C } C_ { A , B , C } $=
|
|
( cpr ctp prsstp12 tpcomb sseqtr4i ) ACDACBEABCEACBFABCGH $.
|
|
|
|
$( A pair is a subset of an unordered triple containing its members.
|
|
(Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
prsstp23 $p |- { B , C } C_ { A , B , C } $=
|
|
( cpr ctp prsstp12 tprot sseqtr4i ) BCDBCAEABCEBCAFABCGH $.
|
|
|
|
${
|
|
prss.1 $e |- A e. _V $.
|
|
prss.2 $e |- B e. _V $.
|
|
$( A pair of elements of a class is a subset of the class. Theorem 7.5 of
|
|
[Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by
|
|
Andrew Salmon, 29-Jun-2011.) $)
|
|
prss $p |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) $=
|
|
( csn wss wa cun wcel cpr unss snss anbi12i df-pr sseq1i 3bitr4i ) AFZCGZ
|
|
BFZCGZHRTIZCGACJZBCJZHABKZCGRTCLUCSUDUAACDMBCEMNUEUBCABOPQ $.
|
|
$}
|
|
|
|
$( A pair of elements of a class is a subset of the class. Theorem 7.5 of
|
|
[Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by
|
|
Andrew Salmon, 29-Jun-2011.) $)
|
|
prssg $p |- ( ( A e. V /\ B e. W )
|
|
-> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) ) $=
|
|
( wcel wa csn wss cpr snssg bi2anan9 cun unss df-pr sseq1i bitr4i syl6bb )
|
|
ADFZBEFZGACFZBCFZGAHZCIZBHZCIZGZABJZCIZSUAUDTUBUFACDKBCEKLUGUCUEMZCIUIUCUEC
|
|
NUHUJCABOPQR $.
|
|
|
|
$( A pair of elements of a class is a subset of the class. (Contributed by
|
|
NM, 16-Jan-2015.) $)
|
|
prssi $p |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C ) $=
|
|
( wcel wa cpr wss prssg ibi ) ACDBCDEABFCGABCCCHI $.
|
|
|
|
$( An unordered pair belongs to the power class of a class iff each member
|
|
belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.)
|
|
(Revised by NM, 18-Jan-2018.) $)
|
|
prsspwg $p |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C
|
|
<-> ( A C_ C /\ B C_ C ) ) ) $=
|
|
( wcel wa cpw cpr wss prssg elpwg bi2anan9 bitr3d ) ADFZBEFZGACHZFZBQFZGABI
|
|
QJACJZBCJZGABQDEKORTPSUAACDLBCELMN $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Empty set and the singleton itself are subsets of a singleton.
|
|
(Contributed by Jim Kingdon, 10-Aug-2018.) $)
|
|
sssnr $p |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } ) $=
|
|
( c0 wceq csn wss 0ss sseq1 mpbiri eqimss jaoi ) ACDZABEZFZAMDLNCMFMGACMH
|
|
IAMJK $.
|
|
|
|
$( The inhabited subset of a singleton. (Contributed by Jim Kingdon,
|
|
10-Aug-2018.) $)
|
|
sssnm $p |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) ) $=
|
|
( cv wcel wex csn wss wceq wa wb ssel elsni syl6 eleq1 ibd exlimdv anc2li
|
|
snssi eqss syl6ibr com12 eqimss impbid1 ) ADZBEZAFZBCGZHZBUHIZUIUGUJUIUGU
|
|
IUHBHZJUJUIUGUKUIUGCBEZUKUIUFULAUIUFULUIUFUECIZUFULKUIUFUEUHEUMBUHUELUECM
|
|
NUECBONPQCBSNRBUHTUAUBBUHUCUD $.
|
|
|
|
$( Two ways to express that an inhabited set equals a singleton.
|
|
(Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
eqsnm $p |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) ) $=
|
|
( cv wceq wral csn wss wcel wex dfss3 velsn ralbii bitri sssnm syl5rbbr )
|
|
ADZCEZABFZBCGZHZQBIAJBTEUAQTIZABFSABTKUBRABACLMNABCOP $.
|
|
$}
|
|
|
|
$( The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
ssprr $p |- (
|
|
( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) )
|
|
-> A C_ { B , C } ) $=
|
|
( c0 wceq csn wo cpr wss 0ss sseq1 mpbiri snsspr1 jaoi snsspr2 eqimss ) ADE
|
|
ZABFZEZGABCHZIZACFZEZATEZGQUASQUADTITJADTKLSUARTIBCMARTKLNUCUAUDUCUAUBTIBCO
|
|
AUBTKLATPNN $.
|
|
|
|
$( The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.) $)
|
|
sstpr $p |- (
|
|
( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/
|
|
( ( A = { D } \/ A = { B , D } ) \/
|
|
( A = { C , D } \/ A = { B , C , D } ) ) )
|
|
-> A C_ { B , C , D } ) $=
|
|
( c0 csn wo cpr ctp wss ssprr prsstp12 syl6ss snsstp3 sseq1 mpbiri prsstp13
|
|
wceq jaoi prsstp23 eqimss ) AERABFRGACFRABCHZRGGZABCDIZJZADFZRZABDHZRZGZACD
|
|
HZRZAUDRZGZGUCAUBUDABCKBCDLMUJUEUNUGUEUIUGUEUFUDJBCDNAUFUDOPUIUEUHUDJBCDQAU
|
|
HUDOPSULUEUMULUEUKUDJBCDTAUKUDOPAUDUASSS $.
|
|
|
|
${
|
|
tpss.1 $e |- A e. _V $.
|
|
tpss.2 $e |- B e. _V $.
|
|
tpss.3 $e |- C e. _V $.
|
|
$( A triplet of elements of a class is a subset of the class. (Contributed
|
|
by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpss $p |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D ) $=
|
|
( cpr wss csn wa cun wcel w3a ctp unss df-3an prss snss anbi12i 3bitr4i
|
|
bitri df-tp sseq1i ) ABHZDIZCJZDIZKZUEUGLZDIADMZBDMZCDMZNZABCOZDIUEUGDPUN
|
|
UKULKZUMKUIUKULUMQUPUFUMUHABDEFRCDGSTUBUOUJDABCUCUDUA $.
|
|
$}
|
|
|
|
$( A triple of elements of a class is a subset of the class. (Contributed by
|
|
Alexander van der Vekens, 1-Feb-2018.) $)
|
|
tpssi $p |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D ) $=
|
|
( w3a ctp cpr csn cun df-tp wss prssi 3adant3 snssi 3ad2ant3 unssd syl5eqss
|
|
wcel ) ADRZBDRZCDRZEZABCFABGZCHZIDABCJUBUCUDDSTUCDKUAABDLMUASUDDKTCDNOPQ $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
sneqr.1 $e |- A e. _V $.
|
|
$( If the singletons of two sets are equal, the two sets are equal. Part
|
|
of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
27-Aug-1993.) $)
|
|
sneqr $p |- ( { A } = { B } -> A = B ) $=
|
|
( csn wceq wcel snid eleq2 mpbii elsn sylib ) ADZBDZEZAMFZABENALFOACGLMAH
|
|
IABCJK $.
|
|
|
|
$( If a singleton is a subset of another, their members are equal.
|
|
(Contributed by NM, 28-May-2006.) $)
|
|
snsssn $p |- ( { A } C_ { B } -> A = B ) $=
|
|
( vx csn wss cv wceq wal wcel dfss2 velsn imbi12i albii bitri cvv sbceqal
|
|
wi ax-mp sylbi ) AEZBEZFZDGZAHZUDBHZRZDIZABHZUCUDUAJZUDUBJZRZDIUHDUAUBKUL
|
|
UGDUJUEUKUFDALDBLMNOAPJUHUIRCDABPQST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Closed form of ~ sneqr . (Contributed by Scott Fenton, 1-Apr-2011.) $)
|
|
sneqrg $p |- ( A e. V -> ( { A } = { B } -> A = B ) ) $=
|
|
( vx cv csn wceq wi sneq eqeq1d eqeq1 imbi12d vex sneqr vtoclg ) DEZFZBFZ
|
|
GZPBGZHAFZRGZABGZHDACPAGZSUBTUCUDQUARPAIJPABKLPBDMNO $.
|
|
|
|
$}
|
|
|
|
$( Two singletons of sets are equal iff their elements are equal.
|
|
(Contributed by Scott Fenton, 16-Apr-2012.) $)
|
|
sneqbg $p |- ( A e. V -> ( { A } = { B } <-> A = B ) ) $=
|
|
( wcel csn wceq sneqrg sneq impbid1 ) ACDAEBEFABFABCGABHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The singleton of a class is a subset of its power class. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
snsspw $p |- { A } C_ ~P A $=
|
|
( vx csn cpw cv wceq wss wcel eqimss velsn df-pw abeq2i 3imtr4i ssriv ) B
|
|
ACZADZBEZAFQAGZQOHQPHQAIBAJRBPBAKLMN $.
|
|
$}
|
|
|
|
${
|
|
prsspw.1 $e |- A e. _V $.
|
|
prsspw.2 $e |- B e. _V $.
|
|
$( An unordered pair belongs to the power class of a class iff each member
|
|
belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
prsspw $p |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) $=
|
|
( cpr cpw wss wcel wa prss elpw anbi12i bitr3i ) ABFCGZHAOIZBOIZJACHZBCHZ
|
|
JABODEKPRQSACDLBCELMN $.
|
|
$}
|
|
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same second element, the first elements are equal. Closed form of
|
|
~ preqr1 . (Contributed by Jim Kingdon, 21-Sep-2018.) $)
|
|
preqr1g $p |- ( ( A e. _V /\ B e. _V ) ->
|
|
( { A , C } = { B , C } -> A = B ) ) $=
|
|
( cvv wcel wa cpr wceq wo wi prid1g eleq2 syl5ibcom elprg sylibd adantr imp
|
|
syl5ibrcom adantl eqcom eqeq2 oplem1 ex ) ADEZBDEZFZACGZBCGZHZABHZUFUIFUJAC
|
|
HZBAHZBCHZUFUIUJUKIZUDUIUNJUEUDUIAUHEZUNUDAUGEUIUOACDKUGUHALMABCDNOPQUFUIUL
|
|
UMIZUEUIUPJUDUEUIBUGEZUPUEUQUIBUHEBCDKUGUHBLRBACDNOSQABTACBUAUBUC $.
|
|
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same second element, the second elements are equal. Closed form of
|
|
~ preqr2 . (Contributed by Jim Kingdon, 21-Sep-2018.) $)
|
|
preqr2g $p |- ( ( A e. _V /\ B e. _V ) ->
|
|
( { C , A } = { C , B } -> A = B ) ) $=
|
|
( cpr wceq cvv wcel wa prcom eqeq12i preqr1g syl5bi ) CADZCBDZEACDZBCDZEAFG
|
|
BFGHABEMONPCAICBIJABCKL $.
|
|
|
|
${
|
|
preqr1.1 $e |- A e. _V $.
|
|
preqr1.2 $e |- B e. _V $.
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same second element, the first elements are equal. (Contributed by
|
|
NM, 18-Oct-1995.) $)
|
|
preqr1 $p |- ( { A , C } = { B , C } -> A = B ) $=
|
|
( cpr wceq wcel wo prid1 eleq2 mpbii elpr sylib mpbiri eqcom eqeq2 oplem1
|
|
) ACFZBCFZGZABGZACGZBAGZBCGZUAATHZUBUCIUAASHUFACDJSTAKLABCDMNUABSHZUDUEIU
|
|
AUGBTHBCEJSTBKOBACEMNABPACBQR $.
|
|
$}
|
|
|
|
${
|
|
preqr2.1 $e |- A e. _V $.
|
|
preqr2.2 $e |- B e. _V $.
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same first element, the second elements are equal. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
preqr2 $p |- ( { C , A } = { C , B } -> A = B ) $=
|
|
( cpr wceq prcom eqeq12i preqr1 sylbi ) CAFZCBFZGACFZBCFZGABGLNMOCAHCBHIA
|
|
BCDEJK $.
|
|
$}
|
|
|
|
${
|
|
preq12b.1 $e |- A e. _V $.
|
|
preq12b.2 $e |- B e. _V $.
|
|
preq12b.3 $e |- C e. _V $.
|
|
preq12b.4 $e |- D e. _V $.
|
|
$( Equality relationship for two unordered pairs. (Contributed by NM,
|
|
17-Oct-1996.) $)
|
|
preq12b $p |- ( { A , B } = { C , D } <->
|
|
( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) $=
|
|
( cpr wceq wa wo wcel preq1 eqeq1d preqr2 syl6bi com12 ancld prcom eqeq2i
|
|
prid1 eleq2 mpbii elpr sylib wi sylbi orim12d preq12 syl6eq sylan9eq jaoi
|
|
mpd impbii ) ABIZCDIZJZACJZBDJZKZADJZBCJZKZLZURUSVBLZVEURAUQMZVFURAUPMVGA
|
|
BEUBUPUQAUCUDACDEUEUFURUSVAVBVDURUSUTUSURUTUSURCBIZUQJUTUSUPVHUQACBNOBDCF
|
|
HPQRSURVBVCURUPDCIZJZVBVCUGUQVIUPCDTUAVBVJVCVBVJDBIZVIJVCVBUPVKVIADBNZOBC
|
|
DFGPQRUHSUIUNVAURVDABCDUJVBVCUPBDIZUQVBUPVKVMVLDBTUKBCDNULUMUO $.
|
|
|
|
$( Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) $)
|
|
prel12 $p |- ( -. A = B -> ( { A , B } = { C , D } <->
|
|
( A e. { C , D } /\ B e. { C , D } ) ) ) $=
|
|
( wceq wn cpr wcel wa eleq2 mpbii wo wi elpr eqeq2 notbid prid1 prid2 jca
|
|
orel2 syl6bi com3l imp ancrd orel1 orim12d orcom bitri preq12b 3imtr4g ex
|
|
syl5bi impd impbid2 ) ABIZJZABKZCDKZIZAVBLZBVBLZMVCVDVEVCAVALVDABEUAVAVBA
|
|
NOVCBVALVEABFUBVAVBBNOUCUTVDVEVCVDACIZADIZPZUTVEVCQZACDERUTVHVIUTVHMZBDIZ
|
|
BCIZPZVFVKMZVGVLMZPVEVCVJVKVNVLVOVJVKVFUTVHVKVFQVKUTVHVFVKUTVGJVHVFQVKUSV
|
|
GBDASTVGVFUDUEUFUGUHVJVLVGUTVHVLVGQVLUTVHVGVLUTVFJVHVGQVLUSVFBCASTVFVGUIU
|
|
EUFUGUHUJVEVLVKPVMBCDFRVLVKUKULABCDEFGHUMUNUOUPUQUR $.
|
|
|
|
$( A way to represent ordered pairs using unordered pairs with distinct
|
|
members. (Contributed by NM, 27-Mar-2007.) $)
|
|
opthpr $p |- ( A =/= D ->
|
|
( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) ) $=
|
|
( cpr wceq wa wo wne preq12b idd wn wi df-ne pm2.21 sylbi impd orc syl5bb
|
|
jaod impbid1 ) ABICDIJACJBDJKZADJZBCJZKZLZADMZUFABCDEFGHNUKUJUFUKUFUFUIUK
|
|
UFOUKUGUHUFUKUGPUGUHUFQZQADRUGULSTUAUDUFUIUBUEUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d C x y z w $. $d D x y z w $.
|
|
$d V x y z w $. $d W x y z w $. $d X x y z w $. $d Y x y z w $.
|
|
$( Closed form of ~ preq12b . (Contributed by Scott Fenton,
|
|
28-Mar-2014.) $)
|
|
preq12bg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) ->
|
|
( { A , B } = { C , D } <->
|
|
( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) $=
|
|
( vx vy vz vw wcel wa cpr wceq wo wb wi cv weq preq1 eqeq1d eqeq1 orbi12d
|
|
anbi1d bibi12d imbi2d preq2 anbi2d eqeq2 w3a vex preq12b vtoclbg vtocl3ga
|
|
eqeq2d a1i 3expa impr ) AEMZBFMZNCGMZDHMZABOZCDOZPZACPZBDPZNZADPZBCPZNZQZ
|
|
RZVAVBVCVDVOSZVDITZJTZOZKTZDOZPZIKUAZVRDPZNZVQDPZJKUAZNZQZRZSZVDAVROZWAPZ
|
|
AVTPZWDNZVKWGNZQZRZSVDVEWAPZWNVINZVKBVTPZNZQZRZSVPIJKABCEFGVQAPZWJWRVDXEW
|
|
BWMWIWQXEVSWLWAVQAVRUBUCXEWEWOWHWPXEWCWNWDVQAVTUDUFXEWFVKWGVQADUDUFUEUGUH
|
|
VRBPZWRXDVDXFWMWSWQXCXFWLVEWAVRBAUIUCXFWOWTWPXBXFWDVIWNVRBDUDUJXFWGXAVKVR
|
|
BVTUDUJUEUGUHVTCPZXDVOVDXGWSVGXCVNXGWAVFVEVTCDUBUQXGWTVJXBVMXGWNVHVIVTCAU
|
|
KUFXGXAVLVKVTCBUKUJUEUGUHWKVQEMVRFMVTGMULVSVTLTZOZPWCJLUAZNZILUAZWGNZQWBW
|
|
ILDHXHDPZXIWAVSXHDVTUIUQXNXKWEXMWHXNXJWDWCXHDVRUKUJXNXLWFWGXHDVQUKUFUEVQV
|
|
RVTXHIUMJUMKUMLUMUNUOURUPUSUT $.
|
|
$}
|
|
|
|
$( Two pairs are not equal if at least one element of the first pair is not
|
|
contained in the second pair. (Contributed by Alexander van der Vekens,
|
|
13-Aug-2017.) $)
|
|
prneimg $p |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) )
|
|
-> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) )
|
|
-> { A , B } =/= { C , D } ) ) $=
|
|
( wcel wa wne wo cpr wceq wn preq12bg oranim df-ne anbi12i sylnibr ad2antll
|
|
orddi simpll pm1.4 jca sylbi syl6bi anim12i syl6 pm4.56 syl6ib necon2ad ) A
|
|
EIBFIJCGIDHIJJZACKZADKZJZBCKZBDKZJZLZABMZCDMZUMVAVBNZUPOZUSOZJZUTOUMVCACNZA
|
|
DNZLZBCNZBDNZLZJZVFUMVCVGVKJVHVJJLZVMABCDEFGHPVNVIVGVJLZJZVKVHLZVKVJLZJZJZV
|
|
MVGVKVHVJUBVTVIVLVIVOVSUCVRVLVPVQVKVJUDUAUEUFUGVIVDVLVEVIVGOZVHOZJUPVGVHQUN
|
|
WAUOWBACRADRSTVLVJOZVKOZJUSVJVKQUQWCURWDBCRBDRSTUHUIUPUSUJUKUL $.
|
|
|
|
${
|
|
preqsn.1 $e |- A e. _V $.
|
|
preqsn.2 $e |- B e. _V $.
|
|
preqsn.3 $e |- C e. _V $.
|
|
$( Equivalence for a pair equal to a singleton. (Contributed by NM,
|
|
3-Jun-2008.) $)
|
|
preqsn $p |- ( { A , B } = { C } <-> ( A = B /\ B = C ) ) $=
|
|
( cpr csn wceq wa dfsn2 eqeq2i preq12b oridm eqtr3 simpr jca eqtr bitri
|
|
wo impbii ) ABGZCHZIUBCCGZIZABIZBCIZJZUCUDUBCKLUEACIZUGJZUJTZUHABCCDEFFMU
|
|
KUJUHUJNUJUHUJUFUGABCOUIUGPQUHUIUGABCRUFUGPQUASSS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Value of the ordered pair when the arguments are sets. (Contributed by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
dfopg $p |- ( ( A e. V /\ B e. W ) ->
|
|
<. A , B >. = { { A } , { A , B } } ) $=
|
|
( vx wcel cvv cop csn cpr wceq elex wa cab w3a df-3an baibr abbidv abid2
|
|
cv df-op eqcomi 3eqtr3g eqcomd syl2an ) ACFAGFZBGFZABHZAIABJJZKBDFACLBDLU
|
|
FUGMZUIUHUJETUIFZENUFUGUKOZENZUIUHUJUKULEULUJUKUFUGUKPQREUISUHUMEABUAUBUC
|
|
UDUE $.
|
|
$}
|
|
|
|
${
|
|
dfop.1 $e |- A e. _V $.
|
|
dfop.2 $e |- B e. _V $.
|
|
$( Value of an ordered pair when the arguments are sets, with the
|
|
conclusion corresponding to Kuratowski's original definition.
|
|
(Contributed by NM, 25-Jun-1998.) $)
|
|
dfop $p |- <. A , B >. = { { A } , { A , B } } $=
|
|
( cvv wcel cop csn cpr wceq dfopg mp2an ) AEFBEFABGAHABIIJCDABEEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opeq1 $p |- ( A = B -> <. A , C >. = <. B , C >. ) $=
|
|
( vx wceq cvv wcel cv csn cpr w3a cab cop eleq1 anbi1d preq1 df-3an df-op
|
|
wa sneq preq12d eleq2d anbi12d 3bitr4g abbidv 3eqtr4g ) ABEZAFGZCFGZDHZAI
|
|
ZACJZJZGZKZDLBFGZUIUJBIZBCJZJZGZKZDLACMBCMUGUOVADUGUHUISZUNSUPUISZUTSUOVA
|
|
UGVBVCUNUTUGUHUPUIABFNOUGUMUSUJUGUKUQULURABTABCPUAUBUCUHUIUNQUPUIUTQUDUED
|
|
ACRDBCRUF $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opeq2 $p |- ( A = B -> <. C , A >. = <. C , B >. ) $=
|
|
( vx wceq cvv wcel cv csn cpr w3a cab cop eleq1 anbi2d eqidd preq2 df-3an
|
|
wa df-op preq12d eleq2d anbi12d 3bitr4g abbidv 3eqtr4g ) ABEZCFGZAFGZDHZC
|
|
IZCAJZJZGZKZDLUHBFGZUJUKCBJZJZGZKZDLCAMCBMUGUOUTDUGUHUISZUNSUHUPSZUSSUOUT
|
|
UGVAVBUNUSUGUIUPUHABFNOUGUMURUJUGUKUKULUQUGUKPABCQUAUBUCUHUIUNRUHUPUSRUDU
|
|
EDCATDCBTUF $.
|
|
$}
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) $)
|
|
opeq12 $p |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. ) $=
|
|
( wceq cop opeq1 opeq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.
|
|
|
|
${
|
|
opeq1i.1 $e |- A = B $.
|
|
$( Equality inference for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opeq1i $p |- <. A , C >. = <. B , C >. $=
|
|
( wceq cop opeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opeq2i $p |- <. C , A >. = <. C , B >. $=
|
|
( wceq cop opeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
opeq12i.2 $e |- C = D $.
|
|
$( Equality inference for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
|
|
opeq12i $p |- <. A , C >. = <. B , D >. $=
|
|
( wceq cop opeq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
opeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opeq1d $p |- ( ph -> <. A , C >. = <. B , C >. ) $=
|
|
( wceq cop opeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opeq2d $p |- ( ph -> <. C , A >. = <. C , B >. ) $=
|
|
( wceq cop opeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
opeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|
|
(Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
opeq12d $p |- ( ph -> <. A , C >. = <. B , D >. ) $=
|
|
( wceq cop opeq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
|
|
$}
|
|
|
|
$( Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) $)
|
|
oteq1 $p |- ( A = B -> <. A , C , D >. = <. B , C , D >. ) $=
|
|
( wceq cop cotp opeq1 opeq1d df-ot 3eqtr4g ) ABEZACFZDFBCFZDFACDGBCDGLMNDAB
|
|
CHIACDJBCDJK $.
|
|
|
|
$( Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) $)
|
|
oteq2 $p |- ( A = B -> <. C , A , D >. = <. C , B , D >. ) $=
|
|
( wceq cop cotp opeq2 opeq1d df-ot 3eqtr4g ) ABEZCAFZDFCBFZDFCADGCBDGLMNDAB
|
|
CHICADJCBDJK $.
|
|
|
|
$( Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) $)
|
|
oteq3 $p |- ( A = B -> <. C , D , A >. = <. C , D , B >. ) $=
|
|
( wceq cop cotp opeq2 df-ot 3eqtr4g ) ABECDFZAFKBFCDAGCDBGABKHCDAICDBIJ $.
|
|
|
|
${
|
|
oteq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for ordered triples. (Contributed by Mario Carneiro,
|
|
11-Jan-2017.) $)
|
|
oteq1d $p |- ( ph -> <. A , C , D >. = <. B , C , D >. ) $=
|
|
( wceq cotp oteq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for ordered triples. (Contributed by Mario Carneiro,
|
|
11-Jan-2017.) $)
|
|
oteq2d $p |- ( ph -> <. C , A , D >. = <. C , B , D >. ) $=
|
|
( wceq cotp oteq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for ordered triples. (Contributed by Mario Carneiro,
|
|
11-Jan-2017.) $)
|
|
oteq3d $p |- ( ph -> <. C , D , A >. = <. C , D , B >. ) $=
|
|
( wceq cotp oteq3 syl ) ABCGDEBHDECHGFBCDEIJ $.
|
|
|
|
oteq123d.2 $e |- ( ph -> C = D ) $.
|
|
oteq123d.3 $e |- ( ph -> E = F ) $.
|
|
$( Equality deduction for ordered triples. (Contributed by Mario Carneiro,
|
|
11-Jan-2017.) $)
|
|
oteq123d $p |- ( ph -> <. A , C , E >. = <. B , D , F >. ) $=
|
|
( cotp oteq1d oteq2d oteq3d 3eqtrd ) ABDFKCDFKCEFKCEGKABCDFHLADECFIMAFGCE
|
|
JNO $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y x $.
|
|
nfop.1 $e |- F/_ x A $.
|
|
nfop.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for ordered pairs. (Contributed by
|
|
NM, 14-Nov-1995.) $)
|
|
nfop $p |- F/_ x <. A , B >. $=
|
|
( vy cop cvv wcel cv csn cpr w3a cab df-op nfel1 nfsn nfpr nfcri nf3an
|
|
nfab nfcxfr ) ABCGBHIZCHIZFJBKZBCLZLZIZMZFNFBCOUIAFUCUDUHAABHDPACHEPAFUGA
|
|
UEUFABDQABCDERRSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d z B $. $d z A $. $d x z $.
|
|
nfopd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfopd.3 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfop . This
|
|
shows how the deduction version of a not-free theorem such as ~ nfop can
|
|
be created from the corresponding not-free inference theorem.
|
|
(Contributed by NM, 4-Feb-2008.) $)
|
|
nfopd $p |- ( ph -> F/_ x <. A , B >. ) $=
|
|
( vz cv wcel wal cab cop wnfc nfaba1 nfop wb wa nfnfc1 wceq abidnf adantr
|
|
nfan adantl opeq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
|
|
DLZMZBUJULUIBGNUKBGNOABCMZBDMZUNUPPEFUQURQZBUMUOUQURBBCRBDRUBUSUJCULDUQUJ
|
|
CSURBGCTUAURULDSUQBGDTUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
opid.1 $e |- A e. _V $.
|
|
$( The ordered pair ` <. A , A >. ` in Kuratowski's representation.
|
|
(Contributed by FL, 28-Dec-2011.) $)
|
|
opid $p |- <. A , A >. = { { A } } $=
|
|
( csn cpr cop dfsn2 eqcomi preq2i dfop 3eqtr4i ) ACZAADZDKKDAAEKCLKKKLAFG
|
|
HAABBIKFJ $.
|
|
$}
|
|
|
|
${
|
|
$d B x $. $d ps x $.
|
|
ralunsn.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Restricted quantification over the union of a set and a singleton, using
|
|
implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
|
|
(Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralunsn $p |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <->
|
|
( A. x e. A ph /\ ps ) ) ) $=
|
|
( csn cun wral wa wcel ralunb ralsng anbi2d syl5bb ) ACDEHZIJACDJZACQJZKE
|
|
FLZRBKACDQMTSBRABCEFGNOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x y $. $d C x $. $d ch x $. $d ps y $. $d th x $.
|
|
2ralunsn.1 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
2ralunsn.2 $e |- ( y = B -> ( ph <-> ps ) ) $.
|
|
2ralunsn.3 $e |- ( x = B -> ( ps <-> th ) ) $.
|
|
$( Double restricted quantification over the union of a set and a
|
|
singleton, using implicit substitution. (Contributed by Paul Chapman,
|
|
17-Nov-2012.) $)
|
|
2ralunsn $p |- ( B e. C ->
|
|
( A. x e. ( A u. { B } ) A. y e. ( A u. { B } ) ph <->
|
|
( ( A. x e. A A. y e. A ph /\ A. x e. A ps ) /\
|
|
( A. y e. A ch /\ th ) ) ) ) $=
|
|
( wcel csn cun wral wa ralunsn ralbidv cv wceq r19.26 anbi1i syl6bb bitrd
|
|
anbi12d ) HIMZAFGHNOZPZEUHPAFGPZBQZEUHPZUJEGPBEGPQZCFGPZDQZQZUGUIUKEUHABF
|
|
GHIKRSUGULUKEGPZUOQUPUKUOEGHIETHUAZUJUNBDURACFGJSLUFRUQUMUOUJBEGUBUCUDUE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Expansion of an ordered pair when either member is a proper class.
|
|
(Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
opprc $p |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) ) $=
|
|
( vx cvv wcel wa wn cop cv csn cpr w3a cab c0 df-op wal wceq 3simpa con3i
|
|
alrimiv abeq0 sylibr syl5eq ) ADEZBDEZFZGZABHUDUECIAJABKKEZLZCMZNCABOUGUI
|
|
GZCPUJNQUGUKCUIUFUDUEUHRSTUICUAUBUC $.
|
|
$}
|
|
|
|
$( Expansion of an ordered pair when the first member is a proper class. See
|
|
also ~ opprc . (Contributed by NM, 10-Apr-2004.) (Revised by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
opprc1 $p |- ( -. A e. _V -> <. A , B >. = (/) ) $=
|
|
( cvv wcel wn wa cop c0 wceq simpl con3i opprc syl ) ACDZENBCDZFZEABGHIPNNO
|
|
JKABLM $.
|
|
|
|
$( Expansion of an ordered pair when the second member is a proper class.
|
|
See also ~ opprc . (Contributed by NM, 15-Nov-1994.) (Revised by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
opprc2 $p |- ( -. B e. _V -> <. A , B >. = (/) ) $=
|
|
( cvv wcel wn wa cop c0 wceq simpr con3i opprc syl ) BCDZEACDZNFZEABGHIPNON
|
|
JKABLM $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d C y $.
|
|
$( If an ordered pair has an element, then its arguments are sets.
|
|
(Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
oprcl $p |- ( C e. <. A , B >. -> ( A e. _V /\ B e. _V ) ) $=
|
|
( vy vx cop wcel cv wex cvv wa elex2 wsb csn cpr w3a df-op eleq2i df-clab
|
|
cab bitri 3simpa sbimi sylbi nfv sbf sylib exlimiv syl ) CABFZGDHZUJGZDIA
|
|
JGZBJGZKZDCUJLULUODULUOEDMZUOULUMUNEHANABOOGZPZEDMZUPULUKURETZGUSUJUTUKEA
|
|
BQRURDESUAURUOEDUMUNUQUBUCUDUOEDUOEUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( The power set of a singleton. (Contributed by Jim Kingdon,
|
|
12-Aug-2018.) $)
|
|
pwsnss $p |- { (/) , { A } } C_ ~P { A } $=
|
|
( vx cv c0 wceq csn wo cab wss cpr cpw sssnr ss2abi dfpr2 df-pw 3sstr4i )
|
|
BCZDEQAFZEGZBHQRIZBHDRJRKSTBQALMBDRNBROP $.
|
|
|
|
$( Compute the power set of the power set of the empty set. (See ~ pw0 for
|
|
the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with
|
|
subset in place of equality). (Contributed by Jim Kingdon,
|
|
12-Aug-2018.) $)
|
|
pwpw0ss $p |- { (/) , { (/) } } C_ ~P { (/) } $=
|
|
( c0 pwsnss ) AB $.
|
|
|
|
$( The power set of an unordered pair. (Contributed by Jim Kingdon,
|
|
13-Aug-2018.) $)
|
|
pwprss $p |- ( { (/) , { A } } u. { { B } , { A , B } } ) C_
|
|
~P { A , B } $=
|
|
( vx c0 csn cpr cun cpw cv wcel wo wss wceq elpr orbi12i ssprr sylbi elun
|
|
vex elpw 3imtr4i ssriv ) CDAEZFZBEZABFZFZGZUFHZCIZUDJZUJUGJZKZUJUFLZUJUHJ
|
|
UJUIJUMUJDMUJUCMKZUJUEMUJUFMKZKUNUKUOULUPUJDUCCSZNUJUEUFUQNOUJABPQUJUDUGR
|
|
UJUFUQTUAUB $.
|
|
|
|
$( The power set of an unordered triple. (Contributed by Jim Kingdon,
|
|
13-Aug-2018.) $)
|
|
pwtpss $p |-
|
|
( ( { (/) , { A } } u. { { B } , { A , B } } ) u.
|
|
( { { C } , { A , C } } u. { { B , C } , { A , B , C } } ) )
|
|
C_ ~P { A , B , C } $=
|
|
( vx c0 csn cpr cun ctp cpw cv wceq wo wcel sstpr elun elpr orbi12i bitri
|
|
wss vex elpw 3imtr4i ssriv ) DEAFZGZBFZABGZGZHZCFZACGZGZBCGZABCIZGZHZHZUO
|
|
JZDKZELUTUELMZUTUGLUTUHLMZMZUTUKLUTULLMZUTUNLUTUOLMZMZMZUTUOTUTURNZUTUSNU
|
|
TABCOVHUTUJNZUTUQNZMVGUTUJUQPVIVCVJVFVIUTUFNZUTUINZMVCUTUFUIPVKVAVLVBUTEU
|
|
EDUAZQUTUGUHVMQRSVJUTUMNZUTUPNZMVFUTUMUPPVNVDVOVEUTUKULVMQUTUNUOVMQRSRSUT
|
|
UOVMUBUCUD $.
|
|
$}
|
|
|
|
$( Compute the power set of the power set of the power set of the empty set.
|
|
(See also ~ pw0 and ~ pwpw0ss .) (Contributed by Jim Kingdon,
|
|
13-Aug-2018.) $)
|
|
pwpwpw0ss $p |-
|
|
( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } )
|
|
C_ ~P { (/) , { (/) } } $=
|
|
( c0 csn pwprss ) AABC $.
|
|
|
|
$( The power class of the universe is the universe. Exercise 4.12(d) of
|
|
[Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) $)
|
|
pwv $p |- ~P _V = _V $=
|
|
( vx cvv cpw cv wcel wss ssv vex elpw mpbir 2th eqriv ) ABCZBADZMEZNBEONBFN
|
|
GNBAHZIJPKL $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The union of a class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare class union symbol. $)
|
|
$c U. $. $( Big cup $)
|
|
|
|
$( Extend class notation to include the union of a class (read: 'union
|
|
` A ` ') $)
|
|
cuni $a class U. A $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Define the union of a class i.e. the collection of all members of the
|
|
members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For
|
|
example, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to
|
|
the union of two classes ~ df-un . (Contributed by NM, 23-Aug-1993.) $)
|
|
df-uni $a |- U. A = { x | E. y ( x e. y /\ y e. A ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of class union. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
dfuni2 $p |- U. A = { x | E. y e. A x e. y } $=
|
|
( cuni cv wcel wa wex cab wrex df-uni exancom df-rex bitr4i abbii eqtri )
|
|
CDAEBEZFZQCFZGBHZAIRBCJZAIABCKTUAATSRGBHUARSBLRBCMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d x B y $.
|
|
$( Membership in class union. (Contributed by NM, 22-May-1994.) $)
|
|
eluni $p |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) $=
|
|
( vy cuni wcel cvv cv wa wex elex adantr exlimiv wceq eleq1 anbi1d exbidv
|
|
df-uni elab2g pm5.21nii ) BCEZFBGFZBAHZFZUCCFZIZAJZBUAKUFUBAUDUBUEBUCKLMD
|
|
HZUCFZUEIZAJUGDBUAGUHBNZUJUFAUKUIUDUEUHBUCOPQDACRST $.
|
|
|
|
$( Membership in class union. Restricted quantifier version. (Contributed
|
|
by NM, 31-Aug-1999.) $)
|
|
eluni2 $p |- ( A e. U. B <-> E. x e. B A e. x ) $=
|
|
( cv wcel wa wex cuni wrex exancom eluni df-rex 3bitr4i ) BADZEZNCEZFAGPO
|
|
FAGBCHEOACIOPAJABCKOACLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership in class union. (Contributed by NM, 24-Mar-1995.) $)
|
|
elunii $p |- ( ( A e. B /\ B e. C ) -> A e. U. C ) $=
|
|
( vx wcel wa cv cuni wceq eleq2 eleq1 anbi12d spcegv anabsi7 eluni sylibr
|
|
wex ) ABEZBCEZFZADGZEZUACEZFZDQZACHERSUEUDTDBCUABIUBRUCSUABAJUABCKLMNDACO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfuni.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for union. (Contributed by NM,
|
|
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
nfuni $p |- F/_ x U. A $=
|
|
( vy vz cuni wel wrex cab dfuni2 nfv nfrexxy nfab nfcxfr ) ABFDEGZEBHZDID
|
|
EBJPADOAEBCOAKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $. $d y z ph $.
|
|
nfunid.3 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfuni . (Contributed by NM, 18-Feb-2013.) $)
|
|
nfunid $p |- ( ph -> F/_ x U. A ) $=
|
|
( vy vz cuni wel wrex cab dfuni2 nfv nfvd nfrexdxy nfabd nfcxfrd ) ABCGEF
|
|
HZFCIZEJEFCKARBEAELAQBFCAFLDAQBMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B y z $. $d V y z $. $d x y z $.
|
|
$( Distribute proper substitution through the union of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbunig $p |- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B ) $=
|
|
( vz vy wcel cv wa wex cab csb cuni wsbc csbabg sbcexg sbcang sbcg df-uni
|
|
bitrd sbcel2g anbi12d exbidv abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHZGZ
|
|
UICGZIZFJZEKZLZUJUIABCLZGZIZFJZEKZABCMZLUPMUHUOUMABNZEKUTUMAEBDOUHVBUSEUH
|
|
VBULABNZFJUSULFABDPUHVCURFUHVCUJABNZUKABNZIURUJUKABDQUHVDUJVEUQUJABDRABUI
|
|
CDUAUBTUCTUDUEABVAUNEFCSUFEFUPSUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
|
|
(Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
unieq $p |- ( A = B -> U. A = U. B ) $=
|
|
( vy vx wceq wel wrex cab cuni rexeq abbidv dfuni2 3eqtr4g ) ABEZCDFZDAGZ
|
|
CHODBGZCHAIBINPQCODABJKCDALCDBLM $.
|
|
$}
|
|
|
|
${
|
|
unieqi.1 $e |- A = B $.
|
|
$( Inference of equality of two class unions. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
unieqi $p |- U. A = U. B $=
|
|
( wceq cuni unieq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
unieqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction of equality of two class unions. (Contributed by NM,
|
|
21-Apr-1995.) $)
|
|
unieqd $p |- ( ph -> U. A = U. B ) $=
|
|
( wceq cuni unieq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Membership in union of a class abstraction. (Contributed by NM,
|
|
11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) $)
|
|
eluniab $p |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) ) $=
|
|
( vy cab cuni wcel cv wa wex eluni nfv nfsab1 nfan weq eleq2 eleq1 syl6bb
|
|
abid anbi12d cbvex bitri ) CABEZFGCDHZGZUDUCGZIZDJCBHZGZAIZBJDCUCKUGUJDBU
|
|
EUFBUEBLABDMNUJDLDBOZUEUIUFAUDUHCPUKUFUHUCGAUDUHUCQABSRTUAUB $.
|
|
|
|
$( Membership in union of a class abstraction. (Contributed by NM,
|
|
4-Oct-2006.) $)
|
|
elunirab $p |- ( A e. U. { x e. B | ph } <->
|
|
E. x e. B ( A e. x /\ ph ) ) $=
|
|
( cv wcel cab cuni wex crab wrex eluniab df-rab unieqi eleq2i df-rex an12
|
|
wa exbii bitri 3bitr4i ) CBEZDFZARZBGZHZFCUBFZUDRZBIZCABDJZHZFUGARZBDKZUD
|
|
BCLUKUFCUJUEABDMNOUMUCULRZBIUIULBDPUNUHBUCUGAQSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
unipr.1 $e |- A e. _V $.
|
|
unipr.2 $e |- B e. _V $.
|
|
$( The union of a pair is the union of its members. Proposition 5.7 of
|
|
[TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) $)
|
|
unipr $p |- U. { A , B } = ( A u. B ) $=
|
|
( vx vy cv wcel wo cab cpr wa wex cun cuni wceq 19.43 bitri clel3 exancom
|
|
vex elpr anbi2i andi exbii orbi12i 3bitr4ri abbii df-un df-uni 3eqtr4ri )
|
|
EGZAHZULBHZIZEJULFGZHZUPABKZHZLZFMZEJABNUROUOVAEUQUPAPZLZUQUPBPZLZIZFMVCF
|
|
MZVEFMZIVAUOVCVEFQUTVFFUTUQVBVDIZLVFUSVIUQUPABFUAUBUCUQVBVDUDRUEUMVGUNVHU
|
|
MVBUQLFMVGFULACSVBUQFTRUNVDUQLFMVHFULBDSVDUQFTRUFUGUHEABUIEFURUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The union of a pair is the union of its members. Proposition 5.7 of
|
|
[TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) $)
|
|
uniprg $p |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) ) $=
|
|
( vx vy cv cpr cuni cun wceq preq1 unieqd uneq1 eqeq12d preq2 uneq2 unipr
|
|
vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
|
|
HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
|
|
$}
|
|
|
|
${
|
|
unisn.1 $e |- A e. _V $.
|
|
$( A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|
|
(Contributed by NM, 30-Aug-1993.) $)
|
|
unisn $p |- U. { A } = A $=
|
|
( csn cuni cpr cun dfsn2 unieqi unipr unidm 3eqtri ) ACZDAAEZDAAFALMAGHAA
|
|
BBIAJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|
|
(Contributed by NM, 13-Aug-2002.) $)
|
|
unisng $p |- ( A e. V -> U. { A } = A ) $=
|
|
( vx cv csn cuni wceq sneq unieqd id eqeq12d vex unisn vtoclg ) CDZEZFZOG
|
|
AEZFZAGCABOAGZQSOATPROAHITJKOCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( An alternate statement of the effective freeness of a class ` A ` , when
|
|
it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
dfnfc2 $p |- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) ) $=
|
|
( wcel wal wnfc cv wceq wnf nfcvd id nfeqd alrimiv csn simpr df-nfc velsn
|
|
wa cuni nfbii albii bitri sylibr nfunid nfa1 nfnf1 nfal unisng sps adantr
|
|
nfan nfceqdf mpbid ex impbid2 ) CDEZAFZACGZBHZCIZAJZBFZUSVBBUSAUTCUSAUTKU
|
|
SLMNURVCUSURVCSZACOZTZGUSVDAVEVDVCAVEGZURVCPVGUTVEEZAJZBFVCABVEQVIVBBVHVA
|
|
ABCRUAUBUCUDUEVDAVFCURVCAUQAUFVBABVAAUGUHULURVFCIZVCUQVJACDUIUJUKUMUNUOUP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The class union of the union of two classes. Theorem 8.3 of [Quine]
|
|
p. 53. (Contributed by NM, 20-Aug-1993.) $)
|
|
uniun $p |- U. ( A u. B ) = ( U. A u. U. B ) $=
|
|
( vx vy cun cuni cv wcel wa wo 19.43 elun anbi2i andi bitri exbii orbi12i
|
|
wex eluni 3bitr4i eqriv ) CABEZFZAFZBFZEZCGZDGZHZUHUBHZIZDRZUGUDHZUGUEHZJ
|
|
ZUGUCHUGUFHUIUHAHZIZUIUHBHZIZJZDRUQDRZUSDRZJULUOUQUSDKUKUTDUKUIUPURJZIUTU
|
|
JVCUIUHABLMUIUPURNOPUMVAUNVBDUGASDUGBSQTDUGUBSUGUDUELTUA $.
|
|
|
|
$( The class union of the intersection of two classes. Exercise 4.12(n) of
|
|
[Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened
|
|
by Andrew Salmon, 29-Jun-2011.) $)
|
|
uniin $p |- U. ( A i^i B ) C_ ( U. A i^i U. B ) $=
|
|
( vx vy cin cuni cv wcel wex 19.40 elin anbi2i anandi bitri exbii anbi12i
|
|
wa eluni 3imtr4i ssriv ) CABEZFZAFZBFZEZCGZDGZHZUGUAHZQZDIZUFUCHZUFUDHZQZ
|
|
UFUBHUFUEHUHUGAHZQZUHUGBHZQZQZDIUPDIZURDIZQUKUNUPURDJUJUSDUJUHUOUQQZQUSUI
|
|
VBUHUGABKLUHUOUQMNOULUTUMVADUFARDUFBRPSDUFUARUFUCUDKST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
|
|
(Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
uniss $p |- ( A C_ B -> U. A C_ U. B ) $=
|
|
( vx vy wss cuni cv wcel wa wex ssel anim2d eximdv eluni 3imtr4g ssrdv )
|
|
ABEZCAFZBFZQCGZDGZHZUAAHZIZDJUBUABHZIZDJTRHTSHQUDUFDQUCUEUBABUAKLMDTANDTB
|
|
NOP $.
|
|
|
|
$( Subclass relationship for class union. (Contributed by NM,
|
|
24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
ssuni $p |- ( ( A C_ B /\ B e. C ) -> A C_ U. C ) $=
|
|
( vy vx wcel wss cuni cv wi wal eleq2 imbi1d elunii expcom vtoclga imim2d
|
|
wceq alimdv dfss2 3imtr4g impcom ) BCFZABGZACHZGZUCDIZAFZUGBFZJZDKUHUGUEF
|
|
ZJZDKUDUFUCUJULDUCUIUKUHUGEIZFZUKJUIUKJEBCUMBRUNUIUKUMBUGLMUNUMCFUKUGUMCN
|
|
OPQSDABTDAUETUAUB $.
|
|
$}
|
|
|
|
${
|
|
unissi.1 $e |- A C_ B $.
|
|
$( Subclass relationship for subclass union. Inference form of ~ uniss .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
unissi $p |- U. A C_ U. B $=
|
|
( wss cuni uniss ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
unissd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Subclass relationship for subclass union. Deduction form of ~ uniss .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
unissd $p |- ( ph -> U. A C_ U. B ) $=
|
|
( wss cuni uniss syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The union of a set is empty iff the set is included in the singleton of
|
|
the empty set. (Contributed by NM, 12-Sep-2004.) $)
|
|
uni0b $p |- ( U. A = (/) <-> A C_ { (/) } ) $=
|
|
( vx vy cv c0 wceq wral wcel wn wal csn wss cuni eq0 ralbii ralcom4 bitri
|
|
dfss3 velsn albii wrex eluni2 notbii ralnex 3bitr4i 3bitr4ri ) BDZEFZBAGZ
|
|
CDZUGHZIZBAGZCJZAEKZLZAMZEFZUIULCJZBAGUNUHUSBACUGNOULBCAPQUPUGUOHZBAGUIBA
|
|
UORUTUHBABESOQUJUQHZIZCJUKBAUAZIZCJURUNVBVDCVAVCBUJAUBUCTCUQNUMVDCUKBAUDT
|
|
UEUF $.
|
|
|
|
$( The union of a set is empty iff all of its members are empty.
|
|
(Contributed by NM, 16-Aug-2006.) $)
|
|
uni0c $p |- ( U. A = (/) <-> A. x e. A x = (/) ) $=
|
|
( cuni c0 wceq csn wss cv wcel wral uni0b dfss3 velsn ralbii 3bitri ) BCD
|
|
EBDFZGAHZPIZABJQDEZABJBKABPLRSABADMNO $.
|
|
$}
|
|
|
|
$( The union of the empty set is the empty set. Theorem 8.7 of [Quine]
|
|
p. 54. (Reproved without relying on ax-nul by Eric Schmidt.)
|
|
(Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt,
|
|
4-Apr-2007.) $)
|
|
uni0 $p |- U. (/) = (/) $=
|
|
( c0 cuni wceq csn wss 0ss uni0b mpbir ) ABACAADZEIFAGH $.
|
|
|
|
$( An element of a class is a subclass of its union. Theorem 8.6 of [Quine]
|
|
p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40.
|
|
(Contributed by NM, 6-Jun-1994.) $)
|
|
elssuni $p |- ( A e. B -> A C_ U. B ) $=
|
|
( wss wcel cuni ssid ssuni mpan ) AACABDABECAFAABGH $.
|
|
|
|
$( Condition turning a subclass relationship for union into an equality.
|
|
(Contributed by NM, 18-Jul-2006.) $)
|
|
unissel $p |- ( ( U. A C_ B /\ B e. A ) -> U. A = B ) $=
|
|
( cuni wss wcel wa simpl elssuni adantl eqssd ) ACZBDZBAEZFKBLMGMBKDLBAHIJ
|
|
$.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Relationship involving membership, subset, and union. Exercise 5 of
|
|
[Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) $)
|
|
unissb $p |- ( U. A C_ B <-> A. x e. A x C_ B ) $=
|
|
( vy cv cuni wcel wi wal wss wa wex eluni imbi1i 19.23v albii bitri dfss2
|
|
wral 3bitr4i bitr4i alcom 19.21v impexp bi2.04 imbi2i df-ral ) DEZBFZGZUH
|
|
CGZHZDIZAEZBGZUNCJZHZAIZUICJUPABSUMUHUNGZUOKZUKHZAIZDIZURULVBDULUTALZUKHV
|
|
BUJVDUKAUHBMNUTUKAOUAPVCVADIZAIURVADAUBVEUQAUOUSUKHZHZDIUOVFDIZHVEUQUOVFD
|
|
UCVAVGDVAUSUOUKHHVGUSUOUKUDUSUOUKUEQPUPVHUODUNCRUFTPQQDUICRUPABUGT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $.
|
|
$( A subclass condition on the members of two classes that implies a
|
|
subclass relation on their unions. Proposition 8.6 of [TakeutiZaring]
|
|
p. 59. (Contributed by NM, 22-Mar-2004.) $)
|
|
uniss2 $p |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B ) $=
|
|
( cv wss wrex wral cuni wcel ssuni expcom rexlimiv ralimi unissb sylibr )
|
|
AEZBEZFZBDGZACHQDIZFZACHCIUAFTUBACSUBBDSRDJUBQRDKLMNACUAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( If the difference ` A \ B ` contains the largest members of ` A ` , then
|
|
the union of the difference is the union of ` A ` . (Contributed by NM,
|
|
22-Mar-2004.) $)
|
|
unidif $p |- ( A. x e. A E. y e. ( A \ B ) x C_ y ->
|
|
U. ( A \ B ) = U. A ) $=
|
|
( cv wss cdif wrex wral cuni wceq uniss2 difss unissi jctil eqss sylibr
|
|
wa ) AEBEFBCDGZHACIZSJZCJZFZUBUAFZRUAUBKTUDUCABCSLSCCDMNOUAUBPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Relationship implying union. (Contributed by NM, 10-Nov-1999.) $)
|
|
ssunieq $p |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B ) $=
|
|
( wcel cv wss wral cuni wceq elssuni unissb biimpri anim12i eqss sylibr
|
|
wa ) BCDZAEBFACGZPBCHZFZSBFZPBSIQTRUABCJUARACBKLMBSNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Any member of a class is the largest of those members that it includes.
|
|
(Contributed by NM, 13-Aug-2002.) $)
|
|
unimax $p |- ( A e. B -> U. { x e. B | x C_ A } = A ) $=
|
|
( vy wcel cv wss crab wral cuni wceq ssid sseq1 elrab3 elrab simprbi rgen
|
|
mpbiri wa ssunieq eqcomd sylancl ) BCEZBAFZBGZACHZEZDFZBGZDUFIZUFJZBKUCUG
|
|
BBGZBLUEULABCUDBBMNRUIDUFUHUFEUHCEUIUEUIAUHCUDUHBMOPQUGUJSBUKDBUFTUAUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The intersection of a class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare class intersection symbol. $)
|
|
$c |^| $. $( Big cap $)
|
|
|
|
$( Extend class notation to include the intersection of a class (read:
|
|
'intersect ` A ` '). $)
|
|
cint $a class |^| A $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Define the intersection of a class. Definition 7.35 of [TakeutiZaring]
|
|
p. 44. For example, ` |^| ` { { 1 , 3 } , { 1 , 8 } } = { 1 } .
|
|
Compare this with the intersection of two classes, ~ df-in .
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
df-int $a |- |^| A = { x | A. y ( y e. A -> x e. y ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of class intersection. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
dfint2 $p |- |^| A = { x | A. y e. A x e. y } $=
|
|
( cint cv wcel wi wal cab wral df-int df-ral abbii eqtr4i ) CDBEZCFAEOFZG
|
|
BHZAIPBCJZAIABCKRQAPBCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Equality law for intersection. (Contributed by NM, 13-Sep-1999.) $)
|
|
inteq $p |- ( A = B -> |^| A = |^| B ) $=
|
|
( vx vy wceq wel wral cab cint raleq abbidv dfint2 3eqtr4g ) ABEZCDFZDAGZ
|
|
CHODBGZCHAIBINPQCODABJKCDALCDBLM $.
|
|
$}
|
|
|
|
${
|
|
inteqi.1 $e |- A = B $.
|
|
$( Equality inference for class intersection. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
inteqi $p |- |^| A = |^| B $=
|
|
( wceq cint inteq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
inteqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for class intersection. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
inteqd $p |- ( ph -> |^| A = |^| B ) $=
|
|
( wceq cint inteq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d x B y $.
|
|
elint.1 $e |- A e. _V $.
|
|
$( Membership in class intersection. (Contributed by NM, 21-May-1994.) $)
|
|
elint $p |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) ) $=
|
|
( vy cv wcel wi wal cint wceq eleq1 imbi2d albidv df-int elab2 ) AFZCGZEF
|
|
ZQGZHZAIRBQGZHZAIEBCJDSBKZUAUCAUDTUBRSBQLMNEACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
elint2.1 $e |- A e. _V $.
|
|
$( Membership in class intersection. (Contributed by NM, 14-Oct-1999.) $)
|
|
elint2 $p |- ( A e. |^| B <-> A. x e. B A e. x ) $=
|
|
( cint wcel cv wi wal wral elint df-ral bitr4i ) BCEFAGZCFBNFZHAIOACJABCD
|
|
KOACLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Membership in class intersection, with the sethood requirement expressed
|
|
as an antecedent. (Contributed by NM, 20-Nov-2003.) $)
|
|
elintg $p |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) ) $=
|
|
( vy cv cint wcel wral eleq1 wceq ralbidv vex elint2 vtoclbg ) EFZCGZHPAF
|
|
ZHZACIBQHBRHZACIEBDPBQJPBKSTACPBRJLAPCEMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
|
(Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
elinti $p |- ( A e. |^| B -> ( C e. B -> A e. C ) ) $=
|
|
( vx cint wcel wi cv wral elintg eleq2 rspccv syl6bi pm2.43i ) ABEZFZCBFA
|
|
CFZGZPPADHZFZDBIRDABOJTQDCBSCAKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfint.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for intersection. (Contributed by NM,
|
|
2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) $)
|
|
nfint $p |- F/_ x |^| A $=
|
|
( vy vz cint wel wral cab dfint2 nfv nfralxy nfab nfcxfr ) ABFDEGZEBHZDID
|
|
EBJPADOAEBCOAKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d ph y $.
|
|
inteqab.1 $e |- A e. _V $.
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
elintab $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $=
|
|
( vy cab cint wcel cv wi wal elint nfsab1 nfv nfim weq eleq1 syl6bb eleq2
|
|
abid imbi12d cbval bitri ) CABFZGHEIZUDHZCUEHZJZEKACBIZHZJZBKECUDDLUHUKEB
|
|
UFUGBABEMUGBNOUKENEBPZUFAUGUJULUFUIUDHAUEUIUDQABTRUEUICSUAUBUC $.
|
|
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 17-Oct-1999.) $)
|
|
elintrab $p |- ( A e. |^| { x e. B | ph } <->
|
|
A. x e. B ( ph -> A e. x ) ) $=
|
|
( cv wcel wa cab cint wi wal crab wral elintab impexp albii df-rab inteqi
|
|
bitri eleq2i df-ral 3bitr4i ) CBFZDGZAHZBIZJZGZUEACUDGZKZKZBLZCABDMZJZGUK
|
|
BDNUIUFUJKZBLUMUFBCEOUPULBUEAUJPQTUOUHCUNUGABDRSUAUKBDUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y ph $.
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 17-Feb-2007.) $)
|
|
elintrabg $p |- ( A e. V -> ( A e. |^| { x e. B | ph } <->
|
|
A. x e. B ( ph -> A e. x ) ) ) $=
|
|
( vy cv crab cint wcel wi wral eleq1 wceq imbi2d ralbidv elintrab vtoclbg
|
|
vex ) FGZABDHIZJATBGZJZKZBDLCUAJACUBJZKZBDLFCETCUAMTCNZUDUFBDUGUCUEATCUBM
|
|
OPABTDFSQR $.
|
|
|
|
$( The intersection of the empty set is the universal class. Exercise 2 of
|
|
[TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) $)
|
|
int0 $p |- |^| (/) = _V $=
|
|
( vy vx cv c0 wcel wi wal cab weq cint cvv pm2.21i ax-gen equid 2th abbii
|
|
noel df-int df-v 3eqtr4i ) ACZDEZBCUAEZFZAGZBHBBIZBHDJKUEUFBUEUFUDAUBUCUA
|
|
QLMBNOPBADRBST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y ph $.
|
|
$( An element of a class includes the intersection of the class. Exercise
|
|
4 of [TakeutiZaring] p. 44 (with correction), generalized to classes.
|
|
(Contributed by NM, 18-Nov-1995.) $)
|
|
intss1 $p |- ( A e. B -> |^| B C_ A ) $=
|
|
( vx vy wcel cint cv wal vex elint wceq eleq1 eleq2 imbi12d spcgv pm2.43a
|
|
wi syl5bi ssrdv ) ABEZCBFZACGZUAEDGZBEZUBUCEZQZDHZTUBAEZDUBBCIJUGTUHUFTUH
|
|
QDABUCAKUDTUEUHUCABLUCAUBMNOPRS $.
|
|
|
|
$( Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
|
|
and its converse. (Contributed by NM, 14-Oct-1999.) $)
|
|
ssint $p |- ( A C_ |^| B <-> A. x e. B A C_ x ) $=
|
|
( vy cint wss cv wcel wral dfss3 vex elint2 ralbii ralcom bitr4i 3bitri )
|
|
BCEZFDGZQHZDBIRAGZHZACIZDBIZBTFZACIZDBQJSUBDBARCDKLMUCUADBIZACIUEUADABCNU
|
|
DUFACDBTJMOP $.
|
|
|
|
$( Subclass of the intersection of a class abstraction. (Contributed by
|
|
NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
ssintab $p |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) ) $=
|
|
( vy cab cint wss cv wral wi wal ssint sseq2 ralab2 bitri ) CABEZFGCDHZGZ
|
|
DPIACBHZGZJBKDCPLARTDBQSCMNO $.
|
|
|
|
$( Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.) $)
|
|
ssintub $p |- A C_ |^| { x e. B | A C_ x } $=
|
|
( vy cv wss crab cint ssint wcel sseq2 elrab simprbi mprgbir ) BBAEZFZACG
|
|
ZHFBDEZFZDQDBQIRQJRCJSPSARCORBKLMN $.
|
|
|
|
$( Subclass of the minimum value of class of supersets. (Contributed by
|
|
NM, 10-Aug-2006.) $)
|
|
ssmin $p |- A C_ |^| { x | ( A C_ x /\ ph ) } $=
|
|
( cv wss wa cab cint wi ssintab simpl mpgbir ) CCBDEZAFZBGHENMIBNBCJMAKL
|
|
$.
|
|
|
|
$( Any member of a class is the smallest of those members that include it.
|
|
(Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
intmin $p |- ( A e. B -> |^| { x e. B | A C_ x } = A ) $=
|
|
( vy wcel cv wss crab cint wi wral elintrab ssid wceq sseq2 eleq2 imbi12d
|
|
vex rspcv mpii syl5bi ssrdv ssintub a1i eqssd ) BCEZBAFZGZACHIZBUFDUIBDFZ
|
|
UIEUHUJUGEZJZACKZUFUJBEZUHAUJCDRLUFUMBBGZUNBMULUOUNJABCUGBNUHUOUKUNUGBBOU
|
|
GBUJPQSTUAUBBUIGUFABCUCUDUE $.
|
|
|
|
$( Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) $)
|
|
intss $p |- ( A C_ B -> |^| B C_ |^| A ) $=
|
|
( vy vx cv wcel wal cint wss imim1 al2imi vex elint 3imtr4g alrimiv dfss2
|
|
wi 3imtr4i ) CEZAFZSBFZQZCGZDEZBHZFZUDAHZFZQZDGABIUEUGIUCUIDUCUAUDSFZQZCG
|
|
TUJQZCGUFUHUBUKULCTUAUJJKCUDBDLZMCUDAUMMNOCABPDUEUGPR $.
|
|
|
|
$( The intersection of an inhabited set is a subclass of its union.
|
|
(Contributed by NM, 29-Jul-2006.) $)
|
|
intssunim $p |- ( E. x x e. A -> |^| A C_ U. A ) $=
|
|
( vy cv wcel wex cint cuni wral r19.2m ex vex elint2 eluni2 3imtr4g ssrdv
|
|
wrex ) ADZBEAFZCBGZBHZSCDZREZABIZUCABQZUBTEUBUAESUDUEUCABJKAUBBCLMAUBBNOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Subclass of the intersection of a restricted class builder.
|
|
(Contributed by NM, 30-Jan-2015.) $)
|
|
ssintrab $p |- ( A C_ |^| { x e. B | ph }
|
|
<-> A. x e. B ( ph -> A C_ x ) ) $=
|
|
( crab cint wss cv wcel wa cab wral df-rab inteqi sseq2i wal impexp albii
|
|
wi ssintab df-ral 3bitr4i bitri ) CABDEZFZGCBHZDIZAJZBKZFZGZACUFGZSZBDLZU
|
|
EUJCUDUIABDMNOUHULSZBPUGUMSZBPUKUNUOUPBUGAULQRUHBCTUMBDUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Subclass relationship for intersection and union. (Contributed by Jim
|
|
Kingdon, 14-Aug-2018.) $)
|
|
intssuni2m $p |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B ) $=
|
|
( cv wcel wex wss cint cuni intssunim uniss sylan9ssr ) ADBEAFBCGBHBICIAB
|
|
JBCKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
intminss.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Under subset ordering, the intersection of a restricted class
|
|
abstraction is less than or equal to any of its members. (Contributed
|
|
by NM, 7-Sep-2013.) $)
|
|
intminss $p |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A ) $=
|
|
( wcel wa crab cint wss elrab intss1 sylbir ) DEGBHDACEIZGOJDKABCDEFLDOMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
intmin2.1 $e |- A e. _V $.
|
|
$( Any set is the smallest of all sets that include it. (Contributed by
|
|
NM, 20-Sep-2003.) $)
|
|
intmin2 $p |- |^| { x | A C_ x } = A $=
|
|
( cv wss cvv crab cint cab rabab inteqi wcel wceq intmin ax-mp eqtr3i ) B
|
|
ADEZAFGZHZQAIZHBRTQAJKBFLSBMCABFNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
intmin3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
intmin3.3 $e |- ps $.
|
|
$( Under subset ordering, the intersection of a class abstraction is less
|
|
than or equal to any of its members. (Contributed by NM,
|
|
3-Jul-2005.) $)
|
|
intmin3 $p |- ( A e. V -> |^| { x | ph } C_ A ) $=
|
|
( wcel cab cint wss elabg mpbiri intss1 syl ) DEHZDACIZHZQJDKPRBGABCDEFLM
|
|
DQNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Elimination of a conjunct in a class intersection. (Contributed by NM,
|
|
31-Jul-2006.) $)
|
|
intmin4 $p |- ( A C_ |^| { x | ph } ->
|
|
|^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } ) $=
|
|
( vy cab cint wss cv wa wcel wi wal wb ssintab simpr impbid2 imbi1d alimi
|
|
ancr elintab albi syl sylbi vex 3bitr4g eqrdv ) CABEFZGZDCBHZGZAIZBEFZUGU
|
|
HUKDHZUIJZKZBLZAUNKZBLZUMULJUMUGJUHAUJKZBLZUPURMZABCNUTUOUQMZBLVAUSVBBUSU
|
|
KAUNUSUKAUJAOAUJSPQRUOUQBUAUBUCUKBUMDUDZTABUMVCTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d x z ph $. $d x y z $.
|
|
intab.1 $e |- A e. _V $.
|
|
intab.2 $e |- { x | E. y ( ph /\ x = A ) } e. _V $.
|
|
$( The intersection of a special case of a class abstraction. ` y ` may be
|
|
free in ` ph ` and ` A ` , which can be thought of a ` ph ( y ) ` and
|
|
` A ( y ) ` . (Contributed by NM, 28-Jul-2006.) (Proof shortened by
|
|
Mario Carneiro, 14-Nov-2016.) $)
|
|
intab $p |- |^| { x | A. y ( ph -> A e. x ) } =
|
|
{ x | E. y ( ph /\ x = A ) } $=
|
|
( vz cv wcel wi wal cab cint wceq wa wex wss ex alrimiv sylibr cvv anbi2d
|
|
eqeq1 exbidv cbvabv eqeltri nfe1 nfab nfeq2 eleq2 imbi2d albid elab 19.8a
|
|
wsbc sbc6 df-sbc sylib mpgbir intss1 19.29r simplr pm3.35 adantlr eqeltrd
|
|
ax-mp exlimiv syl vex elintab abssi eqssi eqtri ) ADBHZIZJZCKZBLZMZAGHZDN
|
|
ZOZCPZGLZAVNDNZOZCPZBLZVSWDWDVRIZVSWDQWIADWDIZJZCVQWKCKBWDWDWHUAWCWGGBVTV
|
|
NNZWBWFCWLWAWEAVTVNDUCUBUDUEZFUFVNWDNZVPWKCCVNWDWCCGWBCUGUHUIWNVOWJAVNWDD
|
|
UJUKULUMAWCGDUOZWJAWAWCJZGKWOAWPGAWAWCWBCUNRSWCGDEUPTWCGDUQURUSWDVRUTVFWC
|
|
GVSWCVQVTVNIZJZBKVTVSIWCWRBWCVQWQWCVQOWBVPOZCPWQWBVPCVAWSWQCWSVTDVNAWAVPV
|
|
BAVPVOWAAVOVCVDVEVGVHRSVQBVTGVIVJTVKVLWMVM $.
|
|
$}
|
|
|
|
$( The intersection of a class containing the empty set is empty.
|
|
(Contributed by NM, 24-Apr-2004.) $)
|
|
int0el $p |- ( (/) e. A -> |^| A = (/) ) $=
|
|
( c0 wcel cint intss1 wss 0ss a1i eqssd ) BACZADZBBAEBKFJKGHI $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The class intersection of the union of two classes. Theorem 78 of
|
|
[Suppes] p. 42. (Contributed by NM, 22-Sep-2002.) $)
|
|
intun $p |- |^| ( A u. B ) = ( |^| A i^i |^| B ) $=
|
|
( vx vy cun cint cin cv wcel wi wal wa 19.26 elun imbi1i jaob bitri elint
|
|
wo 3bitr4i albii vex anbi12i elin eqriv ) CABEZFZAFZBFZGZDHZUFIZCHZUKIZJZ
|
|
DKZUMUHIZUMUIIZLZUMUGIUMUJIUKAIZUNJZUKBIZUNJZLZDKVADKZVCDKZLUPUSVAVCDMUOV
|
|
DDUOUTVBSZUNJVDULVGUNUKABNOUTUNVBPQUAUQVEURVFDUMACUBZRDUMBVHRUCTDUMUFVHRU
|
|
MUHUIUDTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
intpr.1 $e |- A e. _V $.
|
|
intpr.2 $e |- B e. _V $.
|
|
$( The intersection of a pair is the intersection of its members. Theorem
|
|
71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) $)
|
|
intpr $p |- |^| { A , B } = ( A i^i B ) $=
|
|
( vx vy cpr cint cin cv wcel wi wal wa wceq 19.26 wo vex clel4 3bitr4i
|
|
elpr imbi1i jaob bitri albii anbi12i elint elin eqriv ) EABGZHZABIZFJZUJK
|
|
ZEJZUMKZLZFMZUOAKZUOBKZNZUOUKKUOULKUMAOZUPLZUMBOZUPLZNZFMVCFMZVEFMZNURVAV
|
|
CVEFPUQVFFUQVBVDQZUPLVFUNVIUPUMABFRUAUBVBUPVDUCUDUEUSVGUTVHFUOACSFUOBDSUF
|
|
TFUOUJERUGUOABUHTUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The intersection of a pair is the intersection of its members. Closed
|
|
form of ~ intpr . Theorem 71 of [Suppes] p. 42. (Contributed by FL,
|
|
27-Apr-2008.) $)
|
|
intprg $p |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) ) $=
|
|
( vx vy cv cpr cint cin wceq preq1 inteqd ineq1 eqeq12d preq2 ineq2 intpr
|
|
vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
|
|
HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
|
|
$}
|
|
|
|
$( Intersection of a singleton. (Contributed by Stefan O'Rear,
|
|
22-Feb-2015.) $)
|
|
intsng $p |- ( A e. V -> |^| { A } = A ) $=
|
|
( wcel csn cint cpr dfsn2 inteqi cin wceq intprg anidms inidm syl6eq syl5eq
|
|
) ABCZADZEAAFZEZAQRAGHPSAAIZAPSTJAABBKLAMNO $.
|
|
|
|
${
|
|
intsn.1 $e |- A e. _V $.
|
|
$( The intersection of a singleton is its member. Theorem 70 of [Suppes]
|
|
p. 41. (Contributed by NM, 29-Sep-2002.) $)
|
|
intsn $p |- |^| { A } = A $=
|
|
( cvv wcel csn cint wceq intsng ax-mp ) ACDAEFAGBACHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( The union and intersection of a singleton are equal. See also ~ eusn .
|
|
(Contributed by Jim Kingdon, 14-Aug-2018.) $)
|
|
uniintsnr $p |- ( E. x A = { x } -> U. A = |^| A ) $=
|
|
( csn wceq cuni cint vex unisn unieq inteq intsn syl6eq 3eqtr4a exlimiv
|
|
cv ) BAOZCZDZBEZBFZDARQEPSTPAGZHBQIRTQFPBQJPUAKLMN $.
|
|
|
|
$( The union and the intersection of a class abstraction are equal if there
|
|
is a unique satisfying value of ` ph ( x ) ` . (Contributed by Jim
|
|
Kingdon, 14-Aug-2018.) $)
|
|
uniintabim $p |- ( E! x ph -> U. { x | ph } = |^| { x | ph } ) $=
|
|
( vy weu cab cv csn wceq wex cuni cint euabsn2 uniintsnr sylbi ) ABDABEZC
|
|
FGHCIOJOKHABCLCOMN $.
|
|
$}
|
|
|
|
${
|
|
intunsn.1 $e |- B e. _V $.
|
|
$( Theorem joining a singleton to an intersection. (Contributed by NM,
|
|
29-Sep-2002.) $)
|
|
intunsn $p |- |^| ( A u. { B } ) = ( |^| A i^i B ) $=
|
|
( csn cun cint cin intun intsn ineq2i eqtri ) ABDZEFAFZLFZGMBGALHNBMBCIJK
|
|
$.
|
|
$}
|
|
|
|
$( Relative intersection of an empty set. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
rint0 $p |- ( X = (/) -> ( A i^i |^| X ) = A ) $=
|
|
( c0 wceq cint cin inteq ineq2d cvv int0 ineq2i inv1 eqtri syl6eq ) BCDZABE
|
|
ZFACEZFZAOPQABCGHRAIFAQIAJKALMN $.
|
|
|
|
${
|
|
$d B y $. $d X y $.
|
|
$( Membership in a restricted intersection. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
elrint $p |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) ) $=
|
|
( cint cin wcel wa cv wral elin elintg pm5.32i bitri ) DBCEZFGDBGZDOGZHPD
|
|
AIGACJZHDBOKPQRADCBLMN $.
|
|
|
|
$( Membership in a restricted intersection. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
elrint2 $p |- ( X e. A -> ( X e. ( A i^i |^| B ) <->
|
|
A. y e. B X e. y ) ) $=
|
|
( cint cin wcel cv wral elrint baib ) DBCEFGDBGDAHGACIABCDJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Indexed union and intersection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c U_ $. $( Underlined big cup. $)
|
|
$c |^|_ $. $( Underlined big cap. $)
|
|
|
|
$( Extend class notation to include indexed union. Note: Historically
|
|
(prior to 21-Oct-2005), set.mm used the notation ` U. x e. A B ` , with
|
|
the same union symbol as ~ cuni . While that syntax was unambiguous, it
|
|
did not allow for LALR parsing of the syntax constructions in set.mm. The
|
|
new syntax uses as distinguished symbol ` U_ ` instead of ` U. ` and does
|
|
allow LALR parsing. Thanks to Peter Backes for suggesting this change. $)
|
|
ciun $a class U_ x e. A B $.
|
|
|
|
$( Extend class notation to include indexed intersection. Note:
|
|
Historically (prior to 21-Oct-2005), set.mm used the notation
|
|
` |^| x e. A B ` , with the same intersection symbol as ~ cint . Although
|
|
that syntax was unambiguous, it did not allow for LALR parsing of the
|
|
syntax constructions in set.mm. The new syntax uses a distinguished
|
|
symbol ` |^|_ ` instead of ` |^| ` and does allow LALR parsing. Thanks to
|
|
Peter Backes for suggesting this change. $)
|
|
ciin $a class |^|_ x e. A B $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( Define indexed union. Definition indexed union in [Stoll] p. 45. In
|
|
most applications, ` A ` is independent of ` x ` (although this is not
|
|
required by the definition), and ` B ` depends on ` x ` i.e. can be read
|
|
informally as ` B ( x ) ` . We call ` x ` the index, ` A ` the index
|
|
set, and ` B ` the indexed set. In most books, ` x e. A ` is written as
|
|
a subscript or underneath a union symbol ` U. ` . We use a special
|
|
union symbol ` U_ ` to make it easier to distinguish from plain class
|
|
union. In many theorems, you will see that ` x ` and ` A ` are in the
|
|
same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and
|
|
that ` B ` and ` x ` do not share a distinct variable group (meaning
|
|
that can be thought of as ` B ( x ) ` i.e. can be substituted with a
|
|
class expression containing ` x ` ). An alternate definition tying
|
|
indexed union to ordinary union is ~ dfiun2 . Theorem ~ uniiun provides
|
|
a definition of ordinary union in terms of indexed union. (Contributed
|
|
by NM, 27-Jun-1998.) $)
|
|
df-iun $a |- U_ x e. A B = { y | E. x e. A y e. B } $.
|
|
|
|
$( Define indexed intersection. Definition of [Stoll] p. 45. See the
|
|
remarks for its sibling operation of indexed union ~ df-iun . An
|
|
alternate definition tying indexed intersection to ordinary intersection
|
|
is ~ dfiin2 . Theorem ~ intiin provides a definition of ordinary
|
|
intersection in terms of indexed intersection. (Contributed by NM,
|
|
27-Jun-1998.) $)
|
|
df-iin $a |- |^|_ x e. A B = { y | A. x e. A y e. B } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y C $.
|
|
$( Membership in indexed union. (Contributed by NM, 3-Sep-2003.) $)
|
|
eliun $p |- ( A e. U_ x e. B C <-> E. x e. B A e. C ) $=
|
|
( vy ciun wcel wrex elex rexlimivw cv wceq eleq1 rexbidv df-iun pm5.21nii
|
|
cvv elab2g ) BACDFZGBQGZBDGZACHZBSIUATACBDIJEKZDGZACHUBEBSQUCBLUDUAACUCBD
|
|
MNAECDORP $.
|
|
|
|
$( Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.) $)
|
|
eliin $p |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) ) $=
|
|
( vy cv wcel wral ciin wceq eleq1 ralbidv df-iin elab2g ) FGZDHZACIBDHZAC
|
|
IFBACDJEPBKQRACPBDLMAFCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d z C $. $d x y $.
|
|
$( Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.) $)
|
|
iuncom $p |- U_ x e. A U_ y e. B C = U_ y e. B U_ x e. A C $=
|
|
( vz ciun cv wcel wrex rexcom eliun rexbii 3bitr4i eqriv ) FACBDEGZGZBDAC
|
|
EGZGZFHZPIZACJZTRIZBDJZTQITSITEIZBDJZACJUEACJZBDJUBUDUEABCDKUAUFACBTDELMU
|
|
CUGBDATCELMNATCPLBTDRLNO $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
$( Commutation of union with indexed union. (Contributed by Mario
|
|
Carneiro, 18-Jan-2014.) $)
|
|
iuncom4 $p |- U_ x e. A U. B = U. U_ x e. A B $=
|
|
( vy vz cuni ciun cv wcel wa wex df-rex rexbii rexcom4 bitri exbii eluni2
|
|
wrex eliun 3bitr4i r19.41v anbi1i eqriv ) DABCFZGZABCGZFZDHZUDIZABRZUHEHZ
|
|
IZEUFRZUHUEIUHUGIULECRZABRZUKCIZABRZULJZEKZUJUMUOUPULJZABRZEKZUSUOUTEKZAB
|
|
RVBUNVCABULECLMUTAEBNOVAUREUPULABUAPOUIUNABEUHCQMUMUKUFIZULJZEKUSULEUFLVE
|
|
UREVDUQULAUKBCSUBPOTAUHBUDSEUHUFQTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Indexed union of a constant class, i.e. where ` B ` does not depend on
|
|
` x ` . (Contributed by Jim Kingdon, 15-Aug-2018.) $)
|
|
iunconstm $p |- ( E. x x e. A -> U_ x e. A B = B ) $=
|
|
( vy cv wcel wex ciun wrex r19.9rmv eliun syl6rbbr eqrdv ) AEBFAGZDABCHZC
|
|
NDEZCFZQABIPOFQAABJAPBCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d x z B $. $d y z A $.
|
|
$( Indexed intersection of a constant class, i.e. where ` B ` does not
|
|
depend on ` x ` . (Contributed by Jim Kingdon, 19-Dec-2018.) $)
|
|
iinconstm $p |- ( E. y y e. A -> |^|_ x e. A B = B ) $=
|
|
( vz cv wcel wex ciin wral r19.3rmv cvv wb vex eliin ax-mp syl6rbbr eqrdv
|
|
) BFCGBHZEACDIZDSEFZDGZUBACJZUATGZUBABCKUALGUDUCMENAUACDLOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d x z B $. $d z C $.
|
|
$( Law combining indexed union with indexed intersection. Eq. 14 in
|
|
[KuratowskiMostowski] p. 109. This theorem also appears as the last
|
|
example at ~ http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29 .
|
|
(Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon,
|
|
25-Jul-2011.) $)
|
|
iuniin $p |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C $=
|
|
( vz ciin ciun cv wcel wrex wral r19.12 cvv vex eliin ax-mp eliun 3imtr4i
|
|
wb rexbii ralbii ssriv ) FACBDEGZHZBDACEHZGZFIZUDJZACKZUHUFJZBDLZUHUEJUHU
|
|
GJZUHEJZBDLZACKUNACKZBDLUJULUNABCDMUIUOACUHNJZUIUOTFOZBUHDENPQUAUKUPBDAUH
|
|
CERUBSAUHCUDRUQUMULTURBUHDUFNPQSUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iunss1 $p |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C ) $=
|
|
( vy wss ciun cv wcel wrex ssrexv eliun 3imtr4g ssrdv ) BCFZEABDGZACDGZOE
|
|
HZDIZABJSACJRPIRQISABCKARBDLARCDLMN $.
|
|
|
|
$( Subclass theorem for indexed union. (Contributed by NM,
|
|
24-Jan-2012.) $)
|
|
iinss1 $p |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C ) $=
|
|
( vy wss ciin cv wcel wral ssralv cvv wb vex eliin ax-mp 3imtr4g ssrdv )
|
|
BCFZEACDGZABDGZSEHZDIZACJZUCABJZUBTIZUBUAIZUCABCKUBLIZUFUDMENZAUBCDLOPUHU
|
|
GUEMUIAUBBDLOPQR $.
|
|
|
|
$( Equality theorem for indexed union. (Contributed by NM,
|
|
27-Jun-1998.) $)
|
|
iuneq1 $p |- ( A = B -> U_ x e. A C = U_ x e. B C ) $=
|
|
( wss wa ciun wceq iunss1 anim12i eqss 3imtr4i ) BCEZCBEZFABDGZACDGZEZPOE
|
|
ZFBCHOPHMQNRABCDIACBDIJBCKOPKL $.
|
|
|
|
$( Equality theorem for restricted existential quantifier. (Contributed by
|
|
NM, 27-Jun-1998.) $)
|
|
iineq1 $p |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C ) $=
|
|
( vy wceq cv wcel wral cab ciin raleq abbidv df-iin 3eqtr4g ) BCFZEGDHZAB
|
|
IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ss2iun $p |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C ) $=
|
|
( vy wss wral ciun cv wcel wrex ssel ralimi rexim syl eliun 3imtr4g ssrdv
|
|
wi ) CDFZABGZEABCHZABDHZUAEIZCJZABKZUDDJZABKZUDUBJUDUCJUAUEUGSZABGUFUHSTU
|
|
IABCDUDLMUEUGABNOAUDBCPAUDBDPQR $.
|
|
|
|
$( Equality theorem for indexed union. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iuneq2 $p |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wss wral wa ciun wceq ss2iun anim12i eqss ralbii r19.26 bitri 3imtr4i )
|
|
CDEZABFZDCEZABFZGZABCHZABDHZEZUCUBEZGCDIZABFZUBUCIRUDTUEABCDJABDCJKUGQSGZ
|
|
ABFUAUFUHABCDLMQSABNOUBUCLP $.
|
|
|
|
$( Equality theorem for indexed intersection. (Contributed by NM,
|
|
22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iineq2 $p |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( vy wceq wral cv wcel cab ciin wb eleq2 ralimi syl abbidv df-iin 3eqtr4g
|
|
ralbi ) CDFZABGZEHZCIZABGZEJUBDIZABGZEJABCKABDKUAUDUFEUAUCUELZABGUDUFLTUG
|
|
ABCDUBMNUCUEABSOPAEBCQAEBDQR $.
|
|
$}
|
|
|
|
${
|
|
iuneq2i.1 $e |- ( x e. A -> B = C ) $.
|
|
$( Equality inference for indexed union. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iuneq2i $p |- U_ x e. A B = U_ x e. A C $=
|
|
( wceq ciun iuneq2 mprg ) CDFABCGABDGFABABCDHEI $.
|
|
|
|
$( Equality inference for indexed intersection. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iineq2i $p |- |^|_ x e. A B = |^|_ x e. A C $=
|
|
( wceq ciin iineq2 mprg ) CDFABCGABDGFABABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
iineq2d.1 $e |- F/ x ph $.
|
|
iineq2d.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Equality deduction for indexed intersection. (Contributed by NM,
|
|
7-Dec-2011.) $)
|
|
iineq2d $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( wceq wral ciin cv wcel ex ralrimi iineq2 syl ) ADEHZBCIBCDJBCEJHAQBCFAB
|
|
KCLQGMNBCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
iuneq2dv.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Equality deduction for indexed union. (Contributed by NM,
|
|
3-Aug-2004.) $)
|
|
iuneq2dv $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wceq wral ciun ralrimiva iuneq2 syl ) ADEGZBCHBCDIBCEIGAMBCFJBCDEKL $.
|
|
|
|
$( Equality deduction for indexed intersection. (Contributed by NM,
|
|
3-Aug-2004.) $)
|
|
iineq2dv $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( nfv iineq2d ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
iuneq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for indexed union, deduction version. (Contributed by
|
|
Drahflow, 22-Oct-2015.) $)
|
|
iuneq1d $p |- ( ph -> U_ x e. A C = U_ x e. B C ) $=
|
|
( wceq ciun iuneq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
|
|
${
|
|
$d x ph $.
|
|
iuneq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for indexed union, deduction version. (Contributed
|
|
by Drahflow, 22-Oct-2015.) $)
|
|
iuneq12d $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
|
|
( ciun iuneq1d wceq cv wcel adantr iuneq2dv eqtrd ) ABCEIBDEIBDFIABCDEG
|
|
JABDEFAEFKBLDMHNOP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
iuneq2d.2 $e |- ( ph -> B = C ) $.
|
|
$( Equality deduction for indexed union. (Contributed by Drahflow,
|
|
22-Oct-2015.) $)
|
|
iuneq2d $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wceq cv wcel adantr iuneq2dv ) ABCDEADEGBHCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d z A $. $d z B $. $d x z $. $d y z $.
|
|
nfiunxy.1 $e |- F/_ y A $.
|
|
nfiunxy.2 $e |- F/_ y B $.
|
|
$( Bound-variable hypothesis builder for indexed union. (Contributed by
|
|
Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiunxy $p |- F/_ y U_ x e. A B $=
|
|
( vz ciun cv wcel wrex cab df-iun nfcri nfrexxy nfab nfcxfr ) BACDHGIDJZA
|
|
CKZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
|
|
$( Bound-variable hypothesis builder for indexed intersection.
|
|
(Contributed by Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiinxy $p |- F/_ y |^|_ x e. A B $=
|
|
( vz ciin cv wcel wral cab df-iin nfcri nfralxy nfab nfcxfr ) BACDHGIDJZA
|
|
CKZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d z A $. $d z B $. $d x z $. $d y z $.
|
|
nfiunya.1 $e |- F/_ y A $.
|
|
nfiunya.2 $e |- F/_ y B $.
|
|
$( Bound-variable hypothesis builder for indexed union. (Contributed by
|
|
Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiunya $p |- F/_ y U_ x e. A B $=
|
|
( vz ciun cv wcel wrex cab df-iun nfcri nfrexya nfab nfcxfr ) BACDHGIDJZA
|
|
CKZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
|
|
$( Bound-variable hypothesis builder for indexed intersection.
|
|
(Contributed by Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiinya $p |- F/_ y |^|_ x e. A B $=
|
|
( vz ciin cv wcel wral cab df-iin nfcri nfralya nfab nfcxfr ) BACDHGIDJZA
|
|
CKZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Bound-variable hypothesis builder for indexed union. (Contributed by
|
|
NM, 12-Oct-2003.) $)
|
|
nfiu1 $p |- F/_ x U_ x e. A B $=
|
|
( vy ciun cv wcel wrex cab df-iun nfre1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
|
|
CJOADNABKLM $.
|
|
|
|
$( Bound-variable hypothesis builder for indexed intersection.
|
|
(Contributed by NM, 15-Oct-2003.) $)
|
|
nfii1 $p |- F/_ x |^|_ x e. A B $=
|
|
( vy ciin cv wcel wral cab df-iin nfra1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
|
|
CJOADNABKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z w A $. $d y z w B $. $d w C z $. $d w x y z $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. Definition
|
|
15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof
|
|
shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
dfiun2g $p |- ( A. x e. A B e. C ->
|
|
U_ x e. A B = U. { y | E. x e. A y = B } ) $=
|
|
( vz wcel wral ciun cv wceq wrex cab cuni wa wex nfra1 wb rsp syl6bb syl6
|
|
clel3g rexbida rexcom4 r19.41v exbii exancom bitri eliun eluniab 3bitr4g
|
|
imp eqrdv ) DEGZACHZFACDIZBJZDKZACLZBMNZUOFJZDGZACLZVAUQGZUSOBPZVAUPGVAUT
|
|
GUOVCURVDOZACLZBPZVEUOVCVFBPZACLVHUOVBVIACUNACQUOAJCGZVBVIRZUOVJUNVKUNACS
|
|
BVADEUBUAULUCVFABCUDTVHUSVDOZBPVEVGVLBURVDACUEUFUSVDBUGUHTAVACDUIUSBVAUJU
|
|
KUM $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
(Contributed by Jeff Hankins, 27-Aug-2009.) $)
|
|
dfiin2g $p |- ( A. x e. A B e. C
|
|
-> |^|_ x e. A B = |^| { y | E. x e. A y = B } ) $=
|
|
( vw vz wcel wral cv cab wceq wrex wi wal df-ral wb eqeq1 albii bitr4i
|
|
ciin cint eleq2 biimprcd alrimiv eqid imbi12d mpii impbid2 imim2i pm5.74d
|
|
spcgv alimi albi syl sylbi alcom r19.23v vex rexbidv elab imbi1i 3bitr3ri
|
|
19.21v syl6bb syl5bb abbidv df-iin df-int 3eqtr4g ) DEHZACIZFJZDHZACIZFKG
|
|
JZBJZDLZACMZBKZHZVMVPHZNZGOZFKACDUAVTUBVLVOWDFVOAJCHZVNNZAOZVLWDVNACPVLWG
|
|
WEVPDLZWBNZGOZNZAOZWDVLWEVKNZAOZWGWLQZVKACPWNWFWKQZAOWOWMWPAWMWEVNWJVKVNW
|
|
JQWEVKVNWJVNWIGWHWBVNVPDVMUCZUDUEVKWJDDLZVNDUFWIWRVNNGDEWHWHWRWBVNVPDDRWQ
|
|
UGULUHUIUJUKUMWFWKAUNUOUPWIACIZGOZWEWINZGOZAOZWDWLWTXAAOZGOXCWSXDGWIACPSX
|
|
AAGUQTWSWCGWSWHACMZWBNWCWHWBACURWAXEWBVSXEBVPGUSVQVPLVRWHACVQVPDRUTVAVBTS
|
|
XBWKAWEWIGVDSVCVEVFVGAFCDVHFGVTVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
dfiun2.1 $e |- B e. _V $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. Definition
|
|
15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by
|
|
David Abernethy, 19-Jun-2012.) $)
|
|
dfiun2 $p |- U_ x e. A B = U. { y | E. x e. A y = B } $=
|
|
( cvv wcel ciun cv wceq wrex cab cuni dfiun2g a1i mprg ) DFGZACDHBIDJACKB
|
|
LMJACABCDFNQAICGEOP $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
dfiin2 $p |- |^|_ x e. A B = |^| { y | E. x e. A y = B } $=
|
|
( cvv wcel ciin cv wceq wrex cab cint dfiin2g a1i mprg ) DFGZACDHBIDJACKB
|
|
LMJACABCDFNQAICGEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z w $. $d z A $. $d z w B $. $d z w C $.
|
|
$( Define double indexed union. (Contributed by FL, 6-Nov-2013.) $)
|
|
dfiunv2 $p |- U_ x e. A U_ y e. B C =
|
|
{ z | E. x e. A E. y e. B z e. C } $=
|
|
( vw ciun cv wcel wrex cab wceq df-iun a1i iuneq2i vex weq eleq1 rexbidv
|
|
elab rexbii abbii 3eqtri ) ADBEFHZHADGIZFJZBEKZGLZHCIZUIJZADKZCLUJFJZBEKZ
|
|
ADKZCLADUEUIUEUIMAIDJBGEFNOPACDUINULUOCUKUNADUHUNGUJCQGCRUGUMBEUFUJFSTUAU
|
|
BUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d z y A $. $d z x A $. $d z B $. $d z C $.
|
|
cbviun.1 $e |- F/_ y B $.
|
|
cbviun.2 $e |- F/_ x C $.
|
|
cbviun.3 $e |- ( x = y -> B = C ) $.
|
|
$( Rule used to change the bound variables in an indexed union, with the
|
|
substitution specified implicitly by the hypothesis. (Contributed by
|
|
NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) $)
|
|
cbviun $p |- U_ x e. A B = U_ y e. A C $=
|
|
( vz cv wcel wrex cab ciun nfcri weq eleq2d cbvrex abbii df-iun 3eqtr4i )
|
|
IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
|
|
TUA $.
|
|
|
|
$( Change bound variables in an indexed intersection. (Contributed by Jeff
|
|
Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
cbviin $p |- |^|_ x e. A B = |^|_ y e. A C $=
|
|
( vz cv wcel wral cab ciin nfcri weq eleq2d cbvral abbii df-iin 3eqtr4i )
|
|
IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE
|
|
TUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y B $. $d x C $.
|
|
cbviunv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Rule used to change the bound variables in an indexed union, with the
|
|
substitution specified implicitly by the hypothesis. (Contributed by
|
|
NM, 15-Sep-2003.) $)
|
|
cbviunv $p |- U_ x e. A B = U_ y e. A C $=
|
|
( nfcv cbviun ) ABCDEBDGAEGFH $.
|
|
|
|
$( Change bound variables in an indexed intersection. (Contributed by Jeff
|
|
Hankins, 26-Aug-2009.) $)
|
|
cbviinv $p |- |^|_ x e. A B = |^|_ y e. A C $=
|
|
( nfcv cbviin ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iunss $p |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C ) $=
|
|
( vy ciun wss cv wcel wrex cab wal wral df-iun sseq1i abss ralbii ralcom4
|
|
wi dfss2 r19.23v albii 3bitrri 3bitri ) ABCFZDGEHZCIZABJZEKZDGUHUFDIZSZEL
|
|
ZCDGZABMZUEUIDAEBCNOUHEDPUNUGUJSZELZABMUOABMZELULUMUPABECDTQUOAEBRUQUKEUG
|
|
UJABUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset implication for an indexed union. (Contributed by NM,
|
|
3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ssiun $p |- ( E. x e. A C C_ B -> C C_ U_ x e. A B ) $=
|
|
( vy wss wrex ciun cv wcel ssel reximi r19.37av syl eliun syl6ibr ssrdv
|
|
wi ) DCFZABGZEDABCHZTEIZDJZUBCJZABGZUBUAJTUCUDRZABGUCUERSUFABDCUBKLUCUDAB
|
|
MNAUBBCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Identity law for subset of an indexed union. (Contributed by NM,
|
|
12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ssiun2 $p |- ( x e. A -> B C_ U_ x e. A B ) $=
|
|
( vy cv wcel ciun wrex rspe ex eliun syl6ibr ssrdv ) AEBFZDCABCGZNDEZCFZQ
|
|
ABHZPOFNQRQABIJAPBCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $.
|
|
ssiun2s.1 $e |- ( x = C -> B = D ) $.
|
|
$( Subset relationship for an indexed union. (Contributed by NM,
|
|
26-Oct-2003.) $)
|
|
ssiun2s $p |- ( C e. A -> D C_ U_ x e. A B ) $=
|
|
( ciun wss nfcv nfiu1 nfss cv wceq sseq1d ssiun2 vtoclgaf ) CABCGZHEQHADB
|
|
ADIAEQAEIABCJKALDMCEQFNABCOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x B $. $d y C $. $d x D $.
|
|
$( A subclass condition on the members of two indexed classes ` C ( x ) `
|
|
and ` D ( y ) ` that implies a subclass relation on their indexed
|
|
unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59.
|
|
Compare ~ uniss2 . (Contributed by NM, 9-Dec-2004.) $)
|
|
iunss2 $p |- ( A. x e. A E. y e. B C C_ D ->
|
|
U_ x e. A C C_ U_ y e. B D ) $=
|
|
( wss wrex wral ciun ssiun ralimi iunss sylibr ) EFGBDHZACIEBDFJZGZACIACE
|
|
JPGOQACBDFEKLACEPMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $. $d x B $.
|
|
$( The indexed union of a class abstraction. (Contributed by NM,
|
|
27-Dec-2004.) $)
|
|
iunab $p |- U_ x e. A { y | ph } = { y | E. x e. A ph } $=
|
|
( cab ciun wrex wceq cv wcel nfcv nfab1 nfiunxy cleqf abid rexbii 3bitr4i
|
|
wb eliun mpgbir ) BDACEZFZABDGZCEZHCIZUBJZUEUDJZRCCUBUDBCDUACDKACLMUCCLNU
|
|
EUAJZBDGUCUFUGUHABDACOPBUEDUASUCCOQT $.
|
|
|
|
$( The indexed union of a restricted class abstraction. (Contributed by
|
|
NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
|
|
iunrab $p |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph } $=
|
|
( cv wcel cab ciun wrex crab iunab wceq df-rab a1i iuneq2i r19.42v eqtr4i
|
|
wa abbii 3eqtr4i ) BDCFEGZASZCHZIUCBDJZCHZBDACEKZIABDJZCEKZUCBCDLBDUGUDUG
|
|
UDMBFDGACENOPUIUBUHSZCHUFUHCENUEUJCUBABDQTRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $.
|
|
iunxdif2.1 $e |- ( x = y -> C = D ) $.
|
|
$( Indexed union with a class difference as its index. (Contributed by NM,
|
|
10-Dec-2004.) $)
|
|
iunxdif2 $p |- ( A. x e. A E. y e. ( A \ B ) C C_ D ->
|
|
U_ y e. ( A \ B ) D = U_ x e. A C ) $=
|
|
( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtr4i
|
|
wa jctil eqss sylibr ) EFHBCDIZJACKZBUDFLZACELZHZUGUFHZTUFUGMUEUIUHABCUDE
|
|
FNUFBCFLZUGUDCHUFUJHCDOBUDCFPQABCEFGRSUAUFUGUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d x y $.
|
|
ssiinf.1 $e |- F/_ x C $.
|
|
$( Subset theorem for an indexed intersection. (Contributed by FL,
|
|
15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.) $)
|
|
ssiinf $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
|
|
( vy cv ciin wcel wral wss cvv vex eliin ax-mp ralbii nfcv ralcomf dfss3
|
|
wb bitri 3bitr4i ) FGZABCHZIZFDJZUCCIZFDJZABJZDUDKDCKZABJUFUGABJZFDJUIUEU
|
|
KFDUCLIUEUKTFMAUCBCLNOPUGFADBEFBQRUAFDUDSUJUHABFDCSPUB $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Subset theorem for an indexed intersection. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
ssiin $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
|
|
( nfcv ssiinf ) ABCDADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset implication for an indexed intersection. (Contributed by NM,
|
|
15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iinss $p |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C ) $=
|
|
( vy wss wrex ciin cv wcel wral cvv wb vex eliin ax-mp wi reximi r19.36av
|
|
ssel syl syl5bi ssrdv ) CDFZABGZEABCHZDEIZUFJZUGCJZABKZUEUGDJZUGLJUHUJMEN
|
|
AUGBCLOPUEUIUKQZABGUJUKQUDULABCDUGTRUIUKABSUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
$( An indexed intersection is included in any of its members. (Contributed
|
|
by FL, 15-Oct-2012.) $)
|
|
iinss2 $p |- ( x e. A -> |^|_ x e. A B C_ B ) $=
|
|
( vy cv wcel ciin wral wi cvv wb vex eliin ax-mp rsp sylbi com12 ssrdv )
|
|
AEBFZDABCGZCDEZTFZSUACFZUBUCABHZSUCIUAJFUBUDKDLAUABCJMNUCABOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Class union in terms of indexed union. Definition in [Stoll] p. 43.
|
|
(Contributed by NM, 28-Jun-1998.) $)
|
|
uniiun $p |- U. A = U_ x e. A x $=
|
|
( vy cuni wel wrex cab cv ciun dfuni2 df-iun eqtr4i ) BDCAEABFCGABAHZICAB
|
|
JACBMKL $.
|
|
|
|
$( Class intersection in terms of indexed intersection. Definition in
|
|
[Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) $)
|
|
intiin $p |- |^| A = |^|_ x e. A x $=
|
|
( vy cint wel wral cab cv ciin dfint2 df-iin eqtr4i ) BDCAEABFCGABAHZICAB
|
|
JACBMKL $.
|
|
|
|
$( An indexed union of singletons recovers the index set. (Contributed by
|
|
NM, 6-Sep-2005.) $)
|
|
iunid $p |- U_ x e. A { x } = A $=
|
|
( vy cv csn ciun wceq cab wcel df-sn equcom abbii eqtri a1i iuneq2i iunab
|
|
wrex risset abid2 3eqtr2i ) ABADZEZFABUACDZGZCHZFZBABUBUEUBUEGUABIUBUCUAG
|
|
ZCHUECUAJUGUDCCAKLMNOUFUDABQZCHUCBIZCHBUDACBPUIUHCAUCBRLCBSTM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( An indexed union of the empty set is empty. (Contributed by NM,
|
|
26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iun0 $p |- U_ x e. A (/) = (/) $=
|
|
( vy c0 ciun cv wcel wrex wn noel a1i nrex eliun mtbir 2false eqriv ) CAB
|
|
DEZDCFZQGZRDGZSTABHTABTIAFBGRJZKLARBDMNUAOP $.
|
|
|
|
$( An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
0iun $p |- U_ x e. (/) A = (/) $=
|
|
( vy c0 ciun cv wcel wrex rex0 eliun mtbir noel 2false eqriv ) CADBEZDCFZ
|
|
OGZPDGQPBGZADHRAIAPDBJKPLMN $.
|
|
|
|
$( An empty indexed intersection is the universal class. (Contributed by
|
|
NM, 20-Oct-2005.) $)
|
|
0iin $p |- |^|_ x e. (/) A = _V $=
|
|
( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th abbi2i eqtr4i ) ADB
|
|
ECFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $.
|
|
|
|
$( Indexed intersection with a universal index class. (Contributed by NM,
|
|
11-Sep-2008.) $)
|
|
viin $p |- |^|_ x e. _V A = { y | A. x y e. A } $=
|
|
( cvv ciin cv wcel wral cab wal df-iin ralv abbii eqtri ) ADCEBFCGZADHZBI
|
|
OAJZBIABDCKPQBOALMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( There is an inhabited class in an indexed collection ` B ( x ) ` iff the
|
|
indexed union of them is inhabited. (Contributed by Jim Kingdon,
|
|
16-Aug-2018.) $)
|
|
iunn0m $p |- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B ) $=
|
|
( cv wcel wex wrex ciun rexcom4 eliun exbii bitr4i ) BEZDFZBGACHOACHZBGNA
|
|
CDIFZBGOABCJQPBANCDKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $.
|
|
$( Indexed intersection of a class builder. (Contributed by NM,
|
|
6-Dec-2011.) $)
|
|
iinab $p |- |^|_ x e. A { y | ph } = { y | A. x e. A ph } $=
|
|
( cab ciin wral wceq cv wcel nfcv nfab1 nfiinxy cleqf abid ralbii cvv vex
|
|
wb eliin ax-mp 3bitr4i mpgbir ) BDACEZFZABDGZCEZHCIZUEJZUHUGJZSCCUEUGBCDU
|
|
DCDKACLMUFCLNUHUDJZBDGZUFUIUJUKABDACOPUHQJUIULSCRBUHDUDQTUAUFCOUBUC $.
|
|
|
|
$d x A $. $d x B $.
|
|
$( Indexed intersection of a restricted class builder. (Contributed by Jim
|
|
Kingdon, 16-Aug-2018.) $)
|
|
iinrabm $p |- ( E. x x e. A
|
|
-> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } ) $=
|
|
( cv wcel wex wa wral cab crab ciin r19.28mv abbidv wceq df-rab a1i iinab
|
|
iineq2i eqtri 3eqtr4g ) BFDGZBHZCFEGZAIZBDJZCKZUEABDJZIZCKBDACELZMZUICELU
|
|
DUGUJCUEABDNOULBDUFCKZMUHBDUKUMUKUMPUCACEQRTUFBCDSUAUICEQUB $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $. $d y C $.
|
|
$( Indexed union of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ uniiun to recover
|
|
Enderton's theorem. (Contributed by NM, 26-Mar-2004.) $)
|
|
iunin2 $p |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C ) $=
|
|
( vy cin ciun cv wcel wrex r19.42v elin rexbii eliun anbi2i 3bitr4i eqriv
|
|
wa ) EABCDFZGZCABDGZFZEHZSIZABJZUCCIZUCUAIZRZUCTIUCUBIUFUCDIZRZABJUFUIABJ
|
|
ZRUEUHUFUIABKUDUJABUCCDLMUGUKUFAUCBDNOPAUCBSNUCCUALPQ $.
|
|
|
|
$( Indexed union of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ uniiun to recover
|
|
Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.) $)
|
|
iunin1 $p |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B ) $=
|
|
( cin ciun iunin2 wceq cv wcel incom a1i iuneq2i 3eqtr4i ) ABCDEZFCABDFZE
|
|
ABDCEZFPCEABCDGABQOQOHAIBJDCKLMPCKN $.
|
|
|
|
$( Indexed union of class difference. Compare to theorem "De Morgan's
|
|
laws" in [Enderton] p. 31. (Contributed by Jim Kingdon,
|
|
17-Aug-2018.) $)
|
|
iundif2ss $p |- U_ x e. A ( B \ C ) C_ ( B \ |^|_ x e. A C ) $=
|
|
( vy cdif ciun ciin cv wcel wrex wn wa eldif rexbii r19.42v wral rexnalim
|
|
bitri cvv wb vex eliin ax-mp sylnibr anim2i sylbi eliun 3imtr4i ssriv ) E
|
|
ABCDFZGZCABDHZFZEIZUKJZABKZUOCJZUOUMJZLZMZUOULJUOUNJUQURUODJZLZABKZMZVAUQ
|
|
URVCMZABKVEUPVFABUOCDNOURVCABPSVDUTURVDVBABQZUSVBABRUOTJUSVGUAEUBAUOBDTUC
|
|
UDUEUFUGAUOBUKUHUOCUMNUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d y C $. $d x D $. $d x y $.
|
|
$( Rearrange indexed unions over intersection. (Contributed by NM,
|
|
18-Dec-2008.) $)
|
|
2iunin $p |- U_ x e. A U_ y e. B ( C i^i D )
|
|
= ( U_ x e. A C i^i U_ y e. B D ) $=
|
|
( cin ciun wceq cv wcel iunin2 a1i iuneq2i iunin1 eqtri ) ACBDEFGHZHACEBD
|
|
FHZGZHACEHRGACQSQSIAJCKBDEFLMNACREOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( Indexed intersection of class difference. Compare to Theorem "De
|
|
Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon,
|
|
17-Aug-2018.) $)
|
|
iindif2m $p |- ( E. x x e. A ->
|
|
|^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) ) $=
|
|
( vy cv wcel wex cdif ciin ciun wral wn r19.28mv eldif bicomi ralbii wrex
|
|
wa cvv ralnex eliun xchbinxr anbi2i 3bitr3g vex eliin ax-mp 3bitr4g eqrdv
|
|
wb ) AFBGAHZEABCDIZJZCABDKZIZULEFZUMGZABLZUQCGZUQUOGZMZSZUQUNGZUQUPGULUTU
|
|
QDGZMZSZABLUTVFABLZSUSVCUTVFABNVGURABURVGUQCDOPQVHVBUTVHVEABRVAVEABUAAUQB
|
|
DUBUCUDUEUQTGVDUSUKEUFAUQBUMTUGUHUQCUOOUIUJ $.
|
|
|
|
$( Indexed intersection of intersection. Compare to Theorem "Distributive
|
|
laws" in [Enderton] p. 30. (Contributed by Jim Kingdon,
|
|
17-Aug-2018.) $)
|
|
iinin2m $p |- ( E. x x e. A ->
|
|
|^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) ) $=
|
|
( vy cv wcel wex cin ciin wral wa r19.28mv elin ralbii cvv wb eliin ax-mp
|
|
3bitr4g vex anbi2i eqrdv ) AFBGAHZEABCDIZJZCABDJZIZUDEFZUEGZABKZUICGZUIUG
|
|
GZLZUIUFGZUIUHGUDULUIDGZLZABKULUPABKZLUKUNULUPABMUJUQABUICDNOUMURULUIPGZU
|
|
MURQEUAZAUIBDPRSUBTUSUOUKQUTAUIBUEPRSUICUGNTUC $.
|
|
|
|
$( Indexed intersection of intersection. Compare to Theorem "Distributive
|
|
laws" in [Enderton] p. 30. (Contributed by Jim Kingdon,
|
|
17-Aug-2018.) $)
|
|
iinin1m $p |- ( E. x x e. A ->
|
|
|^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) ) $=
|
|
( cv wcel wex cin ciin iinin2m wceq incom a1i iineq2i 3eqtr4g ) AEBFZAGAB
|
|
CDHZICABDIZHABDCHZIRCHABCDJABSQSQKPDCLMNRCLO $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d X x $. $d B x $.
|
|
$( Elementhood in a relative intersection. (Contributed by Mario Carneiro,
|
|
30-Dec-2016.) $)
|
|
elriin $p |- ( B e. ( A i^i |^|_ x e. X S ) <->
|
|
( B e. A /\ A. x e. X B e. S ) ) $=
|
|
( ciin cin wcel wa wral elin eliin pm5.32i bitri ) CBAEDFZGHCBHZCOHZIPCDH
|
|
AEJZICBOKPQRACEDBLMN $.
|
|
|
|
$( Relative intersection of an empty family. (Contributed by Stefan
|
|
O'Rear, 3-Apr-2015.) $)
|
|
riin0 $p |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A ) $=
|
|
( c0 wceq ciin cin iineq1 ineq2d cvv 0iin ineq2i inv1 eqtri syl6eq ) DEFZ
|
|
BADCGZHBAECGZHZBQRSBADECIJTBKHBSKBACLMBNOP $.
|
|
|
|
$( Relative intersection of an inhabited family. (Contributed by Jim
|
|
Kingdon, 19-Aug-2018.) $)
|
|
riinm $p |- ( ( A. x e. X S C_ A /\ E. x x e. X ) ->
|
|
( A i^i |^|_ x e. X S ) = |^|_ x e. X S ) $=
|
|
( wss wral cv wcel wex ciin cin incom wceq wrex r19.2m ancoms iinss df-ss
|
|
wa syl sylib syl5eq ) CBEZADFZAGDHAIZSZBADCJZKUGBKZUGBUGLUFUGBEZUHUGMUFUC
|
|
ADNZUIUEUDUJUCADOPADCBQTUGBRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d y V $.
|
|
iinxsng.1 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed intersection's
|
|
argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario
|
|
Carneiro, 17-Nov-2016.) $)
|
|
iinxsng $p |- ( A e. V -> |^|_ x e. { A } B = C ) $=
|
|
( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng abbi1dv syl5eq )
|
|
BEHZABIZCJGKZCHZAUALZGMDAGUACNTUDGDUCUBDHABEAKBOCDUBFPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x y D $. $d x y E $. $d y V $.
|
|
$d y W $.
|
|
iinxprg.1 $e |- ( x = A -> C = D ) $.
|
|
iinxprg.2 $e |- ( x = B -> C = E ) $.
|
|
$( Indexed intersection with an unordered pair index. (Contributed by NM,
|
|
25-Jan-2012.) $)
|
|
iinxprg $p |- ( ( A e. V /\ B e. W )
|
|
-> |^|_ x e. { A , B } C = ( D i^i E ) ) $=
|
|
( vy wcel wa cv cpr wral cab ciin wceq eleq2d ralprg abbidv df-in 3eqtr4g
|
|
cin df-iin ) BGLCHLMZKNZDLZABCOZPZKQUHELZUHFLZMZKQAUJDREFUEUGUKUNKUIULUMA
|
|
BCGHANZBSDEUHITUOCSDFUHJTUAUBAKUJDUFKEFUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d y V $.
|
|
iunxsng.1 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed union's argument.
|
|
(Contributed by Mario Carneiro, 25-Jun-2016.) $)
|
|
iunxsng $p |- ( A e. V -> U_ x e. { A } B = C ) $=
|
|
( vy wcel csn ciun cv wrex eliun wceq eleq2d rexsng syl5bb eqrdv ) BEHZGA
|
|
BIZCJZDGKZUAHUBCHZATLSUBDHZAUBTCMUCUDABEAKBNCDUBFOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
iunxsn.1 $e |- A e. _V $.
|
|
iunxsn.2 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed union's argument.
|
|
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro,
|
|
25-Jun-2016.) $)
|
|
iunxsn $p |- U_ x e. { A } B = C $=
|
|
( cvv wcel csn ciun wceq iunxsng ax-mp ) BGHABICJDKEABCDGFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.)
|
|
(Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
iunun $p |- U_ x e. A ( B u. C ) = ( U_ x e. A B u. U_ x e. A C ) $=
|
|
( vy cun ciun cv wcel wrex r19.43 elun rexbii eliun orbi12i 3bitr4i eqriv
|
|
wo ) EABCDFZGZABCGZABDGZFZEHZSIZABJZUDUAIZUDUBIZRZUDTIUDUCIUDCIZUDDIZRZAB
|
|
JUJABJZUKABJZRUFUIUJUKABKUEULABUDCDLMUGUMUHUNAUDBCNAUDBDNOPAUDBSNUDUAUBLP
|
|
Q $.
|
|
|
|
$( Separate a union in the index of an indexed union. (Contributed by NM,
|
|
26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
iunxun $p |- U_ x e. ( A u. B ) C = ( U_ x e. A C u. U_ x e. B C ) $=
|
|
( vy cun ciun cv wcel wrex rexun eliun orbi12i bitr4i elun 3bitr4i eqriv
|
|
wo ) EABCFZDGZABDGZACDGZFZEHZDIZASJZUDUAIZUDUBIZRZUDTIUDUCIUFUEABJZUEACJZ
|
|
RUIUEABCKUGUJUHUKAUDBDLAUDCDLMNAUDSDLUDUAUBOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d x z A $. $d z B $. $d y z C $.
|
|
$( Separate an indexed union in the index of an indexed union.
|
|
(Contributed by Mario Carneiro, 5-Dec-2016.) $)
|
|
iunxiun $p |- U_ x e. U_ y e. A B C = U_ y e. A U_ x e. B C $=
|
|
( vz ciun cv wcel wa wex eliun anbi1i r19.41v bitr4i exbii rexcom4 df-rex
|
|
wrex 3bitr4i bitri rexbii eqriv ) FABCDGZEGZBCADEGZGZFHZEIZAUDSZUHUFIZBCS
|
|
ZUHUEIUHUGIAHZUDIZUIJZAKZUMDIZUIJZAKZBCSZUJULUPURBCSZAKUTUOVAAUOUQBCSZUIJ
|
|
VAUNVBUIBUMCDLMUQUIBCNOPURBACQOUIAUDRUKUSBCUKUIADSUSAUHDELUIADRUAUBTAUHUD
|
|
ELBUHCUFLTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( A relationship involving union and indexed intersection. Exercise 23 of
|
|
[Enderton] p. 33 but with equality changed to subset. (Contributed by
|
|
Jim Kingdon, 19-Aug-2018.) $)
|
|
iinuniss $p |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x ) $=
|
|
( vy cv wcel cint wo cab cun wral ciin r19.32vr elint2 orbi2i elun ralbii
|
|
vex 3imtr4i ss2abi df-un df-iin 3sstr4i ) DEZBFZUDCGZFZHZDIUDBAEZJZFZACKZ
|
|
DIBUFJACUJLUHULDUEUDUIFZACKZHUEUMHZACKUHULUEUMACMUGUNUEAUDCDRNOUKUOACUDBU
|
|
IPQSTDBUFUAADCUJUBUC $.
|
|
|
|
$( A relationship involving union and indexed union. Exercise 25 of
|
|
[Enderton] p. 33 but with biconditional changed to implication.
|
|
(Contributed by Jim Kingdon, 19-Aug-2018.) $)
|
|
iununir $p |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) ->
|
|
( B = (/) -> A = (/) ) ) $=
|
|
( c0 wceq cuni cun cv ciun unieq syl6eq uneq2d un0 iuneq1 eqeq12d biimpcd
|
|
uni0 0iun ) CDEZBCFZGZACBAHGZIZEBDESUABUCDSUABDGBSTDBSTDFDCDJQKLBMKSUCADU
|
|
BIDACDUBNAUBRKOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for power class and union. (Contributed by NM,
|
|
18-Jul-2006.) $)
|
|
sspwuni $p |- ( A C_ ~P B <-> U. A C_ B ) $=
|
|
( vx cv cpw wcel wral wss cuni vex elpw ralbii dfss3 unissb 3bitr4i ) CDZ
|
|
BEZFZCAGPBHZCAGAQHAIBHRSCAPBCJKLCAQMCABNO $.
|
|
|
|
$( Two ways to express a collection of subclasses. (Contributed by NM,
|
|
19-Jul-2006.) $)
|
|
pwssb $p |- ( A C_ ~P B <-> A. x e. A x C_ B ) $=
|
|
( cpw wss cuni cv wral sspwuni unissb bitri ) BCDEBFCEAGCEABHBCIABCJK $.
|
|
$}
|
|
|
|
$( Relationship for power class and union. (Contributed by NM,
|
|
18-Jul-2006.) $)
|
|
elpwuni $p |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) ) $=
|
|
( cpw wss cuni wcel wceq sspwuni unissel expcom eqimss impbid1 syl5bb ) ABC
|
|
DAEZBDZBAFZNBGZABHPOQOPQABIJNBKLM $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The power class of an intersection in terms of indexed intersection.
|
|
Exercise 24(a) of [Enderton] p. 33. (Contributed by NM,
|
|
29-Nov-2003.) $)
|
|
iinpw $p |- ~P |^| A = |^|_ x e. A ~P x $=
|
|
( vy cint cpw cv ciin wss wcel wral ssint vex elpw ralbii bitr4i wb eliin
|
|
cvv ax-mp 3bitr4i eqriv ) CBDZEZABAFZEZGZCFZUBHZUGUEIZABJZUGUCIUGUFIZUHUG
|
|
UDHZABJUJAUGBKUIULABUGUDCLZMNOUGUBUMMUGRIUKUJPUMAUGBUERQSTUA $.
|
|
|
|
$( Inclusion of an indexed union of a power class in the power class of the
|
|
union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
|
|
(Contributed by NM, 25-Nov-2003.) $)
|
|
iunpwss $p |- U_ x e. A ~P x C_ ~P U. A $=
|
|
( vy cpw ciun cuni wss wrex wcel ssiun eliun vex elpw rexbii bitri uniiun
|
|
cv sseq2i 3imtr4i ssriv ) CABAQZDZEZBFZDZCQZUAGZABHZUFABUAEZGZUFUCIZUFUEI
|
|
ZABUAUFJUKUFUBIZABHUHAUFBUBKUMUGABUFUACLZMNOULUFUDGUJUFUDUNMUDUIUFABPROST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x X $.
|
|
$( Relative intersection of an inhabited class. (Contributed by Jim
|
|
Kingdon, 19-Aug-2018.) $)
|
|
rintm $p |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X ) $=
|
|
( cpw wss cv wcel wex wa cint cin incom wceq cuni intssuni2m ssid sspwuni
|
|
mpbi syl6ss df-ss sylib syl5eq ) CBDZEAFCGAHIZBCJZKUEBKZUEBUELUDUEBEUFUEM
|
|
UDUEUCNZBACUCOUCUCEUGBEUCPUCBQRSUEBTUAUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Disjointness
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c Disj_ $.
|
|
|
|
$( Extend wff notation to include the statement that a family of classes
|
|
` B ( x ) ` , for ` x e. A ` , is a disjoint family. $)
|
|
wdisj $a wff Disj_ x e. A B $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( A collection of classes ` B ( x ) ` is disjoint when for each element
|
|
` y ` , it is in ` B ( x ) ` for at most one ` x ` . (Contributed by
|
|
Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.) $)
|
|
df-disj $a |- ( Disj_ x e. A B <-> A. y E* x e. A y e. B ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( Alternate definition for disjoint classes. (Contributed by NM,
|
|
17-Jun-2017.) $)
|
|
dfdisj2 $p |- ( Disj_ x e. A B <-> A. y E* x ( x e. A /\ y e. B ) ) $=
|
|
( wdisj cv wcel wrmo wal wa wmo df-disj df-rmo albii bitri ) ACDEBFDGZACH
|
|
ZBIAFCGPJAKZBIABCDLQRBPACMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( If each element of a collection is contained in a disjoint collection,
|
|
the original collection is also disjoint. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjss2 $p |- ( A. x e. A B C_ C ->
|
|
( Disj_ x e. A C -> Disj_ x e. A B ) ) $=
|
|
( vy wss wral cv wcel wrmo wal wdisj ssel ralimi rmoim syl alimdv df-disj
|
|
wi 3imtr4g ) CDFZABGZEHZDIZABJZEKUCCIZABJZEKABDLABCLUBUEUGEUBUFUDSZABGUEU
|
|
GSUAUHABCDUCMNUFUDABOPQAEBDRAEBCRT $.
|
|
$}
|
|
|
|
$( Equality theorem for disjoint collection. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
disjeq2 $p |- ( A. x e. A B = C ->
|
|
( Disj_ x e. A B <-> Disj_ x e. A C ) ) $=
|
|
( wceq wral wdisj wss wi eqimss2 ralimi disjss2 syl eqimss impbid ) CDEZABF
|
|
ZABCGZABDGZQDCHZABFRSIPTABDCJKABDCLMQCDHZABFSRIPUAABCDNKABCDLMO $.
|
|
|
|
${
|
|
$d x ph $.
|
|
disjeq2dv.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Equality deduction for disjoint collection. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjeq2dv $p |- ( ph -> ( Disj_ x e. A B <-> Disj_ x e. A C ) ) $=
|
|
( wceq wral wdisj wb ralrimiva disjeq2 syl ) ADEGZBCHBCDIBCEIJANBCFKBCDEL
|
|
M $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( A subset of a disjoint collection is disjoint. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjss1 $p |- ( A C_ B -> ( Disj_ x e. B C -> Disj_ x e. A C ) ) $=
|
|
( vy wss cv wcel wa wmo wal wdisj wi ssel anim1d alrimiv moim syl dfdisj2
|
|
alimdv 3imtr4g ) BCFZAGZCHZEGDHZIZAJZEKUCBHZUEIZAJZEKACDLABDLUBUGUJEUBUIU
|
|
FMZAKUGUJMUBUKAUBUHUDUEBCUCNOPUIUFAQRTAECDSAEBDSUA $.
|
|
|
|
$( Equality theorem for disjoint collection. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjeq1 $p |- ( A = B -> ( Disj_ x e. A C <-> Disj_ x e. B C ) ) $=
|
|
( wceq wdisj wss wi eqimss2 disjss1 syl eqimss impbid ) BCEZABDFZACDFZNCB
|
|
GOPHCBIACBDJKNBCGPOHBCLABCDJKM $.
|
|
|
|
disjeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for disjoint collection. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjeq1d $p |- ( ph -> ( Disj_ x e. A C <-> Disj_ x e. B C ) ) $=
|
|
( wceq wdisj wb disjeq1 syl ) ACDGBCEHBDEHIFBCDEJK $.
|
|
|
|
$d x ph $.
|
|
disjeq12d.1 $e |- ( ph -> C = D ) $.
|
|
$( Equality theorem for disjoint collection. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
disjeq12d $p |- ( ph -> ( Disj_ x e. A C <-> Disj_ x e. B D ) ) $=
|
|
( wdisj disjeq1d wceq cv wcel adantr disjeq2dv bitrd ) ABCEIBDEIBDFIABCDE
|
|
GJABDEFAEFKBLDMHNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d z B $. $d z C $.
|
|
cbvdisj.1 $e |- F/_ y B $.
|
|
cbvdisj.2 $e |- F/_ x C $.
|
|
cbvdisj.3 $e |- ( x = y -> B = C ) $.
|
|
$( Change bound variables in a disjoint collection. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
cbvdisj $p |- ( Disj_ x e. A B <-> Disj_ y e. A C ) $=
|
|
( vz cv wcel wrmo wal wdisj nfcri weq eleq2d cbvrmo albii df-disj 3bitr4i
|
|
) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBI
|
|
CETUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C $.
|
|
cbvdisjv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Change bound variables in a disjoint collection. (Contributed by Mario
|
|
Carneiro, 11-Dec-2016.) $)
|
|
cbvdisjv $p |- ( Disj_ x e. A B <-> Disj_ y e. A C ) $=
|
|
( nfcv cbvdisj ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x y z $.
|
|
nfdisjv.1 $e |- F/_ y A $.
|
|
nfdisjv.2 $e |- F/_ y B $.
|
|
$( Bound-variable hypothesis builder for disjoint collection. (Contributed
|
|
by Jim Kingdon, 19-Aug-2018.) $)
|
|
nfdisjv $p |- F/ y Disj_ x e. A B $=
|
|
( vz wdisj cv wcel wa wmo wal dfdisj2 nfcv nfel nfcri nfan nfmo nfal
|
|
nfxfr ) ACDHAIZCJZGIDJZKZALZGMBAGCDNUFBGUEBAUCUDBBUBCBUBOEPBGDFQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Bound-variable hypothesis builder for disjoint collection. (Contributed
|
|
by Mario Carneiro, 14-Nov-2016.) $)
|
|
nfdisj1 $p |- F/ x Disj_ x e. A B $=
|
|
( vy wdisj cv wcel wrmo wal df-disj nfrmo1 nfal nfxfr ) ABCEDFCGZABHZDIAA
|
|
DBCJOADNABKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d x C $.
|
|
$( If there is a function ` C ( y ) ` such that ` C ( y ) = x ` for all
|
|
` y e. B ( x ) ` , then the sets ` B ( x ) ` for distinct ` x e. A ` are
|
|
disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.) $)
|
|
invdisj $p |- ( A. x e. A A. y e. B C = x -> Disj_ x e. A B ) $=
|
|
( cv wceq wral wa wmo wal wdisj nfra2xy wi df-ral rsp eqcom syl6ib imim2i
|
|
wcel impd alimi sylbi mo2icl syl alrimi dfdisj2 sylibr ) EAFZGZBDHZACHZUI
|
|
CTZBFDTZIZAJZBKACDLULUPBUJABCDMULUOUIEGZNZAKZUPULUMUKNZAKUSUKACOUTURAUTUM
|
|
UNUQUKUNUQNUMUKUNUJUQUJBDPEUIQRSUAUBUCUOAEUDUEUFABCDUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Any collection of singletons is disjoint. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
sndisj $p |- Disj_ x e. A { x } $=
|
|
( vy cv csn wdisj wcel wa wmo dfdisj2 moeq simpr velsn sylib eqcomd moimi
|
|
weq ax-mp mpgbir ) ABADZEZFTBGZCDZUAGZHZAIZCACBUAJACQZAIUFAUCKUEUGAUEUCTU
|
|
EUDCAQUBUDLCTMNOPRS $.
|
|
$}
|
|
|
|
$( Any collection of empty sets is disjoint. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
0disj $p |- Disj_ x e. A (/) $=
|
|
( c0 cv csn wss wral wdisj 0ss rgenw sndisj disjss2 mp2 ) CADEZFZABGABNHABC
|
|
HOABNIJABKABCNLM $.
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( A singleton collection is disjoint. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
disjxsn $p |- Disj_ x e. { A } B $=
|
|
( vy csn wdisj cv wcel wa wmo dfdisj2 wceq moeq elsni adantr moimi mpgbir
|
|
ax-mp ) ABEZCFAGZSHZDGCHZIZAJZDADSCKTBLZAJUDABMUCUEAUAUEUBTBNOPRQ $.
|
|
|
|
$( An empty collection is disjoint. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
disjx0 $p |- Disj_ x e. (/) B $=
|
|
( c0 csn wss wdisj 0ss disjxsn disjss1 mp2 ) CCDZEAKBFACBFKGACBHACKBIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Binary relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff notation to include the general binary relation predicate.
|
|
Note that the syntax is simply three class symbols in a row. Since binary
|
|
relations are the only possible wff expressions consisting of three class
|
|
expressions in a row, the syntax is unambiguous. $)
|
|
wbr $a wff A R B $.
|
|
|
|
$( Define a general binary relation. Note that the syntax is simply three
|
|
class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29
|
|
generalized to arbitrary classes. This definition of relations is
|
|
well-defined, although not very meaningful, when classes ` A ` and/or
|
|
` B ` are proper classes (i.e. are not sets). On the other hand, we often
|
|
find uses for this definition when ` R ` is a proper class (see for
|
|
example ~ iprc ). (Contributed by NM, 31-Dec-1993.) $)
|
|
df-br $a |- ( A R B <-> <. A , B >. e. R ) $.
|
|
|
|
$( Equality theorem for binary relations. (Contributed by NM,
|
|
4-Jun-1995.) $)
|
|
breq $p |- ( R = S -> ( A R B <-> A S B ) ) $=
|
|
( wceq cop wcel wbr eleq2 df-br 3bitr4g ) CDEABFZCGLDGABCHABDHCDLIABCJABDJK
|
|
$.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
breq1 $p |- ( A = B -> ( A R C <-> B R C ) ) $=
|
|
( wceq cop wcel wbr opeq1 eleq1d df-br 3bitr4g ) ABEZACFZDGBCFZDGACDHBCDHMN
|
|
ODABCIJACDKBCDKL $.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
breq2 $p |- ( A = B -> ( C R A <-> C R B ) ) $=
|
|
( wceq cop wcel wbr opeq2 eleq1d df-br 3bitr4g ) ABEZCAFZDGCBFZDGCADHCBDHMN
|
|
ODABCIJCADKCBDKL $.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq12 $p |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr breq1 breq2 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
breqi.1 $e |- R = S $.
|
|
$( Equality inference for binary relations. (Contributed by NM,
|
|
19-Feb-2005.) $)
|
|
breqi $p |- ( A R B <-> A S B ) $=
|
|
( wceq wbr wb breq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
breq1i.1 $e |- A = B $.
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq1i $p |- ( A R C <-> B R C ) $=
|
|
( wceq wbr wb breq1 ax-mp ) ABFACDGBCDGHEABCDIJ $.
|
|
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq2i $p |- ( C R A <-> C R B ) $=
|
|
( wceq wbr wb breq2 ax-mp ) ABFCADGCBDGHEABCDIJ $.
|
|
|
|
${
|
|
breq12i.2 $e |- C = D $.
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
|
|
breq12i $p |- ( A R C <-> B R D ) $=
|
|
( wceq wbr wb breq12 mp2an ) ABHCDHACEIBDEIJFGABCDEKL $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
breq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq1d $p |- ( ph -> ( A R C <-> B R C ) ) $=
|
|
( wceq wbr wb breq1 syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
29-Oct-2011.) $)
|
|
breqd $p |- ( ph -> ( C A D <-> C B D ) ) $=
|
|
( wceq wbr wb breq syl ) ABCGDEBHDECHIFDEBCJK $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq2d $p |- ( ph -> ( C R A <-> C R B ) ) $=
|
|
( wceq wbr wb breq2 syl ) ABCGDBEHDCEHIFBCDEJK $.
|
|
|
|
${
|
|
breq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
breq12d $p |- ( ph -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr wb breq12 syl2anc ) ABCIDEIBDFJCEFJKGHBCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
breq123d.2 $e |- ( ph -> R = S ) $.
|
|
breq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
29-Oct-2011.) $)
|
|
breq123d $p |- ( ph -> ( A R C <-> B S D ) ) $=
|
|
( wbr breq12d breqd bitrd ) ABDFKCEFKCEGKABCDEFHJLAFGCEIMN $.
|
|
$}
|
|
|
|
${
|
|
breqan12i.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breqan12d $p |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr wb breq12 syl2an ) ACDJEFJCEGKDFGKLBHICDEFGMN $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breqan12rd $p |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) ) $=
|
|
( wbr wb breqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
|
|
$}
|
|
$}
|
|
|
|
$( Two classes are different if they don't have the same relationship to a
|
|
third class. (Contributed by NM, 3-Jun-2012.) $)
|
|
nbrne1 $p |- ( ( A R B /\ -. A R C ) -> B =/= C ) $=
|
|
( wbr wn wne wceq breq2 biimpcd necon3bd imp ) ABDEZACDEZFBCGMNBCBCHMNBCADI
|
|
JKL $.
|
|
|
|
$( Two classes are different if they don't have the same relationship to a
|
|
third class. (Contributed by NM, 3-Jun-2012.) $)
|
|
nbrne2 $p |- ( ( A R C /\ -. B R C ) -> A =/= B ) $=
|
|
( wbr wn wne wceq breq1 biimpcd necon3bd imp ) ACDEZBCDEZFABGMNABABHMNABCDI
|
|
JKL $.
|
|
|
|
${
|
|
eqbrtr.1 $e |- A = B $.
|
|
eqbrtr.2 $e |- B R C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqbrtri $p |- A R C $=
|
|
( wbr breq1i mpbir ) ACDGBCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrd.1 $e |- ( ph -> A = B ) $.
|
|
eqbrtrd.2 $e |- ( ph -> B R C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 8-Oct-1999.) $)
|
|
eqbrtrd $p |- ( ph -> A R C ) $=
|
|
( wbr breq1d mpbird ) ABDEHCDEHGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrr.1 $e |- A = B $.
|
|
eqbrtrr.2 $e |- A R C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqbrtrri $p |- B R C $=
|
|
( eqcomi eqbrtri ) BACDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqbrtrrd.2 $e |- ( ph -> A R C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
eqbrtrrd $p |- ( ph -> B R C ) $=
|
|
( eqcomd eqbrtrd ) ACBDEABCFHGI $.
|
|
$}
|
|
|
|
${
|
|
breqtr.1 $e |- A R B $.
|
|
breqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
breqtri $p |- A R C $=
|
|
( wbr breq2i mpbi ) ABDGACDGEBCADFHI $.
|
|
$}
|
|
|
|
${
|
|
breqtrd.1 $e |- ( ph -> A R B ) $.
|
|
breqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
breqtrd $p |- ( ph -> A R C ) $=
|
|
( wbr breq2d mpbid ) ABCEHBDEHFACDBEGIJ $.
|
|
$}
|
|
|
|
${
|
|
breqtrr.1 $e |- A R B $.
|
|
breqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
breqtrri $p |- A R C $=
|
|
( eqcomi breqtri ) ABCDECBFGH $.
|
|
$}
|
|
|
|
${
|
|
breqtrrd.1 $e |- ( ph -> A R B ) $.
|
|
breqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
breqtrrd $p |- ( ph -> A R C ) $=
|
|
( eqcomd breqtrd ) ABCDEFADCGHI $.
|
|
$}
|
|
|
|
${
|
|
3brtr3.1 $e |- A R B $.
|
|
3brtr3.2 $e |- A = C $.
|
|
3brtr3.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 11-Aug-1999.) $)
|
|
3brtr3i $p |- C R D $=
|
|
( eqbrtrri breqtri ) CBDEACBEGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3brtr4.1 $e |- A R B $.
|
|
3brtr4.2 $e |- C = A $.
|
|
3brtr4.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 11-Aug-1999.) $)
|
|
3brtr4i $p |- C R D $=
|
|
( eqbrtri breqtrri ) CBDECABEGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3brtr3d.1 $e |- ( ph -> A R B ) $.
|
|
3brtr3d.2 $e |- ( ph -> A = C ) $.
|
|
3brtr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 18-Oct-1999.) $)
|
|
3brtr3d $p |- ( ph -> C R D ) $=
|
|
( wbr breq12d mpbid ) ABCFJDEFJGABDCEFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr4d.1 $e |- ( ph -> A R B ) $.
|
|
3brtr4d.2 $e |- ( ph -> C = A ) $.
|
|
3brtr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 21-Feb-2005.) $)
|
|
3brtr4d $p |- ( ph -> C R D ) $=
|
|
( wbr breq12d mpbird ) ADEFJBCFJGADBECFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr3g.1 $e |- ( ph -> A R B ) $.
|
|
3brtr3g.2 $e |- A = C $.
|
|
3brtr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 16-Jan-1997.) $)
|
|
3brtr3g $p |- ( ph -> C R D ) $=
|
|
( wbr breq12i sylib ) ABCFJDEFJGBDCEFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr4g.1 $e |- ( ph -> A R B ) $.
|
|
3brtr4g.2 $e |- C = A $.
|
|
3brtr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 16-Jan-1997.) $)
|
|
3brtr4g $p |- ( ph -> C R D ) $=
|
|
( wbr breq12i sylibr ) ABCFJDEFJGDBECFHIKL $.
|
|
$}
|
|
|
|
${
|
|
syl5eqbr.1 $e |- A = B $.
|
|
syl5eqbr.2 $e |- ( ph -> B R C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl5eqbr $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr4g ) ACDBDEGFDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqbrr.1 $e |- B = A $.
|
|
syl5eqbrr.2 $e |- ( ph -> B R C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
17-Sep-2004.) $)
|
|
syl5eqbrr $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr3g ) ACDBDEGFDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5breq.1 $e |- A R B $.
|
|
syl5breq.2 $e |- ( ph -> B = C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl5breq $p |- ( ph -> A R C ) $=
|
|
( wbr a1i breqtrd ) ABCDEBCEHAFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl5breqr.1 $e |- A R B $.
|
|
syl5breqr.2 $e |- ( ph -> C = B ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl5breqr $p |- ( ph -> A R C ) $=
|
|
( eqcomd syl5breq ) ABCDEFADCGHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqbr.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqbr.2 $e |- B R C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
12-Oct-1999.) $)
|
|
syl6eqbr $p |- ( ph -> A R C ) $=
|
|
( wbr breq1d mpbiri ) ABDEHCDEHGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
syl6eqbrr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqbrr.2 $e |- B R C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqbrr $p |- ( ph -> A R C ) $=
|
|
( eqcomd syl6eqbr ) ABCDEACBFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl6breq.1 $e |- ( ph -> A R B ) $.
|
|
syl6breq.2 $e |- B = C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl6breq $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr3g ) ABCBDEFBHGI $.
|
|
$}
|
|
|
|
${
|
|
syl6breqr.1 $e |- ( ph -> A R B ) $.
|
|
syl6breqr.2 $e |- C = B $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl6breqr $p |- ( ph -> A R C ) $=
|
|
( eqcomi syl6breq ) ABCDEFDCGHI $.
|
|
$}
|
|
|
|
${
|
|
ssbrd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Deduction from a subclass relationship of binary relations.
|
|
(Contributed by NM, 30-Apr-2004.) $)
|
|
ssbrd $p |- ( ph -> ( C A D -> C B D ) ) $=
|
|
( cop wcel wbr sseld df-br 3imtr4g ) ADEGZBHMCHDEBIDECIABCMFJDEBKDECKL $.
|
|
$}
|
|
|
|
${
|
|
ssbri.1 $e |- A C_ B $.
|
|
$( Inference from a subclass relationship of binary relations.
|
|
(Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro,
|
|
8-Feb-2015.) $)
|
|
ssbri $p |- ( C A D -> C B D ) $=
|
|
( wbr wi wtru wss a1i ssbrd trud ) CDAFCDBFGHABCDABIHEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfbrd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfbrd.3 $e |- ( ph -> F/_ x R ) $.
|
|
nfbrd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfbr .
|
|
(Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfbrd $p |- ( ph -> F/ x A R B ) $=
|
|
( wbr cop wcel df-br nfopd nfeld nfxfrd ) CDEICDJZEKABCDELABPEABCDFHMGNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfbr.1 $e |- F/_ x A $.
|
|
nfbr.2 $e |- F/_ x R $.
|
|
nfbr.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for binary relation. (Contributed by
|
|
NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfbr $p |- F/ x A R B $=
|
|
( wbr wnf wtru wnfc a1i nfbrd trud ) BCDHAIJABCDABKJELADKJFLACKJGLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d y z R $.
|
|
$( Relationship between a binary relation and a class abstraction.
|
|
(Contributed by Andrew Salmon, 8-Jul-2011.) $)
|
|
brab1 $p |- ( x R A <-> x e. { z | z R A } ) $=
|
|
( vy cv wbr wsbc cab wcel cvv wb vex breq1 sbcie2g ax-mp df-sbc bitr3i )
|
|
AFZCDGZBFZCDGZBSHZSUBBIJSKJUCTLAMUBEFZCDGTBESKUAUDCDNUDSCDNOPUBBSQR $.
|
|
$}
|
|
|
|
$( The union of two binary relations. (Contributed by NM, 21-Dec-2008.) $)
|
|
brun $p |- ( A ( R u. S ) B <-> ( A R B \/ A S B ) ) $=
|
|
( cop cun wcel wo wbr elun df-br orbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
|
|
CIZABDIZHNCDJABOKRPSQABCKABDKLM $.
|
|
|
|
$( The intersection of two relations. (Contributed by FL, 7-Oct-2008.) $)
|
|
brin $p |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) ) $=
|
|
( cop cin wcel wa wbr elin df-br anbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
|
|
CIZABDIZHNCDJABOKRPSQABCKABDKLM $.
|
|
|
|
$( The difference of two binary relations. (Contributed by Scott Fenton,
|
|
11-Apr-2011.) $)
|
|
brdif $p |- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) ) $=
|
|
( cop cdif wcel wn wa wbr eldif df-br notbii anbi12i 3bitr4i ) ABEZCDFZGPCG
|
|
ZPDGZHZIABQJABCJZABDJZHZIPCDKABQLUARUCTABCLUBSABDLMNO $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y D $. $d y R $. $d x y $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
sbcbrg $p |- ( A e. D -> ( [. A / x ]. B R C <->
|
|
[_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) ) $=
|
|
( vy wbr wsb cv csb wsbc dfsbcq2 wceq csbeq1 breq123d nfcsb1v weq csbeq1a
|
|
nfbr sbie vtoclbg ) CDFHZAGIAGJZCKZAUDDKZAUDFKZHZUCABLABCKZABDKZABFKZHGBE
|
|
UCAGBMUDBNUEUIUFUJUGUKAUDBCOAUDBFOAUDBDOPUCUHAGAUEUFUGAUDCQAUDFQAUDDQTAGR
|
|
CUEDUFFUGAUDCSAUDFSAUDDSPUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr12g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) ) $=
|
|
( wcel wbr wsbc csb sbcbrg csbconstg breqd bitrd ) BEGZCDFHABIABCJZABDJZA
|
|
BFJZHPQFHABCDEFKORFPQABFELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x C $. $d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr1g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) ) $=
|
|
( wcel wbr wsbc csb sbcbr12g csbconstg breq2d bitrd ) BEGZCDFHABIABCJZABD
|
|
JZFHPDFHABCDEFKOQDPFABDELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr2g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) ) $=
|
|
( wcel wbr wsbc csb sbcbr12g csbconstg breq1d bitrd ) BEGZCDFHABIABCJZABD
|
|
JZFHCQFHABCDEFKOPCQFABCELMN $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered-pair class abstractions (class builders)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c |-> $. $( Maps-to symbol $)
|
|
|
|
$( Extend class notation to include ordered-pair class abstraction (class
|
|
builder). $)
|
|
copab $a class { <. x , y >. | ph } $.
|
|
|
|
$( Extend the definition of a class to include maps-to notation for defining
|
|
a function via a rule. $)
|
|
cmpt $a class ( x e. A |-> B ) $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( Define the class abstraction of a collection of ordered pairs.
|
|
Definition 3.3 of [Monk1] p. 34. Usually ` x ` and ` y ` are distinct,
|
|
although the definition doesn't strictly require it. The brace notation
|
|
is called "class abstraction" by Quine; it is also (more commonly)
|
|
called a "class builder" in the literature. (Contributed by NM,
|
|
4-Jul-1994.) $)
|
|
df-opab $a |- { <. x , y >. | ph } =
|
|
{ z | E. x E. y ( z = <. x , y >. /\ ph ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( Define maps-to notation for defining a function via a rule. Read as
|
|
"the function defined by the map from ` x ` (in ` A ` ) to
|
|
` B ( x ) ` ." The class expression ` B ` is the value of the function
|
|
at ` x ` and normally contains the variable ` x ` . Similar to the
|
|
definition of mapping in [ChoquetDD] p. 2. (Contributed by NM,
|
|
17-Feb-2008.) $)
|
|
df-mpt $a |- ( x e. A |-> B ) =
|
|
{ <. x , y >. | ( x e. A /\ y = B ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x z R $. $d y z R $.
|
|
$( The collection of ordered pairs in a class is a subclass of it.
|
|
(Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
opabss $p |- { <. x , y >. | x R y } C_ R $=
|
|
( vz cv wbr copab cop wceq wa wex cab df-opab df-br eleq1 biimpar sylan2b
|
|
wcel exlimivv abssi eqsstri ) AEZBEZCFZABGDEZUBUCHZIZUDJZBKAKZDLCUDABDMUI
|
|
DCUHUECRZABUDUGUFCRZUJUBUCCNUGUJUKUEUFCOPQSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $. $d z ch $.
|
|
opabbid.1 $e |- F/ x ph $.
|
|
opabbid.2 $e |- F/ y ph $.
|
|
opabbid.3 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal ordered-pair class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew
|
|
Salmon, 9-Jul-2011.) $)
|
|
opabbid $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
|
|
( vz cv cop wceq wa wex cab copab anbi2d exbid abbidv df-opab 3eqtr4g ) A
|
|
IJDJEJKLZBMZENZDNZIOUBCMZENZDNZIOBDEPCDEPAUEUHIAUDUGDFAUCUFEGABCUBHQRRSBD
|
|
EITCDEITUA $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
opabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal ordered-pair class abstractions (deduction
|
|
rule). (Contributed by NM, 15-May-1995.) $)
|
|
opabbidv $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
|
|
( nfv opabbid ) ABCDEADGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $.
|
|
opabbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal class abstractions. (Contributed by NM,
|
|
15-May-1995.) $)
|
|
opabbii $p |- { <. x , y >. | ph } = { <. x , y >. | ps } $=
|
|
( vz cv wceq copab eqid wb a1i opabbidv ax-mp ) FGZOHZACDIBCDIHOJPABCDABK
|
|
PELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d ph w $.
|
|
nfopab.1 $e |- F/ z ph $.
|
|
$( Bound-variable hypothesis builder for class abstraction. (Contributed
|
|
by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were
|
|
removed by Andrew Salmon, 11-Jul-2011.) $)
|
|
nfopab $p |- F/_ z { <. x , y >. | ph } $=
|
|
( vw copab cv cop wceq wa wex cab df-opab nfv nfan nfex nfab nfcxfr ) DAB
|
|
CGFHBHCHIJZAKZCLZBLZFMABCFNUCDFUBDBUADCTADTDOEPQQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( The first abstraction variable in an ordered-pair class abstraction
|
|
(class builder) is effectively not free. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfopab1 $p |- F/_ x { <. x , y >. | ph } $=
|
|
( vz copab cv cop wceq wa wex cab df-opab nfe1 nfab nfcxfr ) BABCEDFBFCFG
|
|
HAICJZBJZDKABCDLQBDPBMNO $.
|
|
|
|
$( The second abstraction variable in an ordered-pair class abstraction
|
|
(class builder) is effectively not free. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfopab2 $p |- F/_ y { <. x , y >. | ph } $=
|
|
( vz copab cv cop wceq wa wex cab df-opab nfe1 nfex nfab nfcxfr ) CABCEDF
|
|
BFCFGHAIZCJZBJZDKABCDLSCDRCBQCMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d v ph $. $d v ps $.
|
|
cbvopab.1 $e |- F/ z ph $.
|
|
cbvopab.2 $e |- F/ w ph $.
|
|
cbvopab.3 $e |- F/ x ps $.
|
|
cbvopab.4 $e |- F/ y ps $.
|
|
cbvopab.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables in an ordered-pair class
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
14-Sep-2003.) $)
|
|
cbvopab $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
|
|
( vv cv cop wceq wa wex cab nfv nfan opeq12 eqeq2d anbi12d cbvex2 df-opab
|
|
copab abbii 3eqtr4i ) LMZCMZDMZNZOZAPZDQCQZLRUIEMZFMZNZOZBPZFQEQZLRACDUFB
|
|
EFUFUOVALUNUTCDEFUMAEUMESGTUMAFUMFSHTUSBCUSCSITUSBDUSDSJTUJUPOUKUQOPZUMUS
|
|
ABVBULURUIUJUKUPUQUAUBKUCUDUGACDLUEBEFLUEUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $. $d x y ps $.
|
|
cbvopabv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables in an ordered-pair class
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
15-Oct-1996.) $)
|
|
cbvopabv $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
|
|
( nfv cbvopab ) ABCDEFAEHAFHBCHBDHGI $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y $. $d v w y z $. $d v w ph $. $d v w ps $.
|
|
cbvopab1.1 $e |- F/ z ph $.
|
|
cbvopab1.2 $e |- F/ x ps $.
|
|
cbvopab1.3 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
$( Change first bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by
|
|
Mario Carneiro, 14-Oct-2016.) $)
|
|
cbvopab1 $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
|
|
( vw vv cv cop wceq wa wex cab copab wsb nfv nfan nfs1v nfex opeq1 eqeq2d
|
|
sbequ12 anbi12d exbidv cbvex nfsb sbequ sbie syl6bb bitri df-opab 3eqtr4i
|
|
abbii ) IKZCKZDKZLZMZANZDOZCOZIPUQEKZUSLZMZBNZDOZEOZIPACDQBEDQVDVJIVDUQJK
|
|
ZUSLZMZACJRZNZDOZJOVJVCVPCJVCJSVOCDVMVNCVMCSACJUATUBURVKMZVBVODVQVAVMAVNV
|
|
QUTVLUQURVKUSUCUDACJUEUFUGUHVPVIJEVOEDVMVNEVMESACJEFUITUBVIJSVKVEMZVOVHDV
|
|
RVMVGVNBVRVLVFUQVKVEUSUCUDVRVNACERBAJECUJABCEGHUKULUFUGUHUMUPACDIUNBEDIUN
|
|
UO $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d w ph $. $d w ps $.
|
|
cbvopab2.1 $e |- F/ z ph $.
|
|
cbvopab2.2 $e |- F/ y ps $.
|
|
cbvopab2.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Change second bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 22-Aug-2013.) $)
|
|
cbvopab2 $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
|
|
( vw cv cop wceq wa wex cab copab nfv nfan opeq2 df-opab anbi12d 3eqtr4i
|
|
eqeq2d cbvex exbii abbii ) IJZCJZDJZKZLZAMZDNZCNZIOUGUHEJZKZLZBMZENZCNZIO
|
|
ACDPBCEPUNUTIUMUSCULURDEUKAEUKEQFRUQBDUQDQGRUIUOLZUKUQABVAUJUPUGUIUOUHSUC
|
|
HUAUDUEUFACDITBCEITUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( Change first bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 31-Jul-2003.) $)
|
|
cbvopab1s $p |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph } $=
|
|
( vw cv cop wceq wa wex cab wsb copab nfv nfs1v nfan opeq1 eqeq2d df-opab
|
|
nfex sbequ12 anbi12d exbidv cbvex abbii 3eqtr4i ) EFZBFZCFZGZHZAIZCJZBJZE
|
|
KUGDFZUIGZHZABDLZIZCJZDJZEKABCMURDCMUNVAEUMUTBDUMDNUSBCUQURBUQBNABDOPTUHU
|
|
OHZULUSCVBUKUQAURVBUJUPUGUHUOUIQRABDUAUBUCUDUEABCESURDCESUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $. $d z ph $. $d x ps $.
|
|
cbvopab1v.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the first bound variable in an ordered pair
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
|
|
cbvopab1v $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
|
|
( nfv cbvopab1 ) ABCDEAEGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $. $d y w ps $.
|
|
cbvopab2v.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the second bound variable in an ordered pair
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
2-Sep-1999.) $)
|
|
cbvopab2v $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
|
|
( vw cv cop wceq wex cab copab opeq2 eqeq2d anbi12d cbvexv exbii df-opab
|
|
wa abbii 3eqtr4i ) GHZCHZDHZIZJZATZDKZCKZGLUCUDEHZIZJZBTZEKZCKZGLACDMBCEM
|
|
UJUPGUIUOCUHUNDEUEUKJZUGUMABUQUFULUCUEUKUDNOFPQRUAACDGSBCEGSUB $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w ph $. $d w x y z $.
|
|
$( Move substitution into a class abstraction. (Contributed by NM,
|
|
6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
csbopabg $p |- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } =
|
|
{ <. y , z >. | [. A / x ]. ph } ) $=
|
|
( vw cv copab csb wsb wceq wsbc csbeq1 dfsbcq2 opabbidv eqeq12d vex nfs1v
|
|
nfopab sbequ12 csbief vtoclg ) BGHZACDIZJZABGKZCDIZLBEUEJZABEMZCDIZLGEFUD
|
|
ELZUFUIUHUKBUDEUENULUGUJCDABGEOPQBUDUEUHGRUGCDBABGSTBHUDLAUGCDABGUAPUBUC
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $. $d ps z $.
|
|
$( Union of two ordered pair class abstractions. (Contributed by NM,
|
|
30-Sep-2002.) $)
|
|
unopab $p |- ( { <. x , y >. | ph } u. { <. x , y >. | ps } ) =
|
|
{ <. x , y >. | ( ph \/ ps ) } $=
|
|
( vz cv cop wceq wa wex cab wo copab unab 19.43 andi exbii bitr2i df-opab
|
|
cun bitr3i abbii eqtri uneq12i 3eqtr4i ) EFCFDFGHZAIZDJZCJZEKZUFBIZDJZCJZ
|
|
EKZTZUFABLZIZDJZCJZEKZACDMZBCDMZTUPCDMUOUIUMLZEKUTUIUMENVCUSEVCUHULLZCJUS
|
|
UHULCOVDURCURUGUKLZDJVDUQVEDUFABPQUGUKDORQUAUBUCVAUJVBUNACDESBCDESUDUPCDE
|
|
SUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $. $d y B $. $d y C $. $d y D $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq12f $p |- ( ( A. x A = C /\ A. x e. A B = D ) ->
|
|
( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( vy wceq wal wral wa cv wcel copab cmpt nfa1 nfra1 nfan nfv rsp df-mpt
|
|
imp eqeq2d pm5.32da sp eleq2d anbi1d sylan9bbr opabbid 3eqtr4g ) BDGZAHZC
|
|
EGZABIZJZAKZBLZFKZCGZJZAFMUODLZUQEGZJZAFMABCNADENUNUSVBAFUKUMAUJAOULABPQU
|
|
NFRUMUSUPVAJUKVBUMUPURVAUMUPJCEUQUMUPULULABSUAUBUCUKUPUTVAUKBDUOUJAUDUEUF
|
|
UGUHAFBCTAFDETUI $.
|
|
|
|
mpteq12dv.1 $e |- ( ph -> A = C ) $.
|
|
${
|
|
mpteq12dva.2 $e |- ( ( ph /\ x e. A ) -> B = D ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 26-Jan-2017.) $)
|
|
mpteq12dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( wceq wal wral cmpt alrimiv ralrimiva mpteq12f syl2anc ) ACEIZBJDFIZBC
|
|
KBCDLBEFLIAQBGMARBCHNBCDEFOP $.
|
|
$}
|
|
|
|
mpteq12dv.2 $e |- ( ph -> B = D ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by NM,
|
|
24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) $)
|
|
mpteq12dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( wceq cv wcel adantr mpteq12dva ) ABCDEFGADFIBJCKHLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
$( An equality theorem for the maps to notation. (Contributed by NM,
|
|
16-Dec-2013.) $)
|
|
mpteq12 $p |- ( ( A = C /\ A. x e. A B = D ) ->
|
|
( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( wceq wal wral cmpt ax-17 mpteq12f sylan ) BDFZMAGCEFABHABCIADEIFMAJABCD
|
|
EKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq1 $p |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
|
|
( wceq wral cmpt cv wcel eqidd rgen mpteq12 mpan2 ) BCEDDEZABFABDGACDGENA
|
|
BAHBIDJKABDCDLM $.
|
|
|
|
mpteq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 11-Jun-2016.) $)
|
|
mpteq1d $p |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
|
|
( wceq cmpt mpteq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
mpteq2ia.1 $e |- ( x e. A -> B = C ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq2ia $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
|
|
( wceq wal wral cmpt eqid ax-gen rgen mpteq12f mp2an ) BBFZAGCDFZABHABCIA
|
|
BDIFOABJKPABELABCBDMN $.
|
|
$}
|
|
|
|
${
|
|
mpteq2i.1 $e |- B = C $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq2i $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
|
|
( wceq cv wcel a1i mpteq2ia ) ABCDCDFAGBHEIJ $.
|
|
$}
|
|
|
|
${
|
|
mpteq12i.1 $e |- A = C $.
|
|
mpteq12i.2 $e |- B = D $.
|
|
$( An equality inference for the maps to notation. (Contributed by Scott
|
|
Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.) $)
|
|
mpteq12i $p |- ( x e. A |-> B ) = ( x e. C |-> D ) $=
|
|
( cmpt wceq wtru a1i mpteq12dv trud ) ABCHADEHIJABCDEBDIJFKCEIJGKLM $.
|
|
$}
|
|
|
|
${
|
|
mpteq2da.1 $e |- F/ x ph $.
|
|
mpteq2da.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro,
|
|
16-Dec-2013.) $)
|
|
mpteq2da $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( wceq wal wral cmpt eqid ax-gen cv wcel ex ralrimi mpteq12f sylancr ) AC
|
|
CHZBIDEHZBCJBCDKBCEKHTBCLMAUABCFABNCOUAGPQBCDCERS $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mpteq2dva.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by Scott Fenton, 25-Apr-2012.) $)
|
|
mpteq2dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( nfv mpteq2da ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mpteq2dv.1 $e |- ( ph -> B = C ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 23-Aug-2014.) $)
|
|
mpteq2dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( wceq cv wcel adantr mpteq2dva ) ABCDEADEGBHCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x y z $.
|
|
nfmpt.1 $e |- F/_ x A $.
|
|
nfmpt.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by NM, 20-Feb-2013.) $)
|
|
nfmpt $p |- F/_ x ( y e. A |-> B ) $=
|
|
( vz cmpt cv wcel wceq wa copab df-mpt nfcri nfeq2 nfan nfopab nfcxfr ) A
|
|
BCDHBICJZGIZDKZLZBGMBGCDNUCBGATUBAABCEOAUADFPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d A z $. $d B z $. $d x z $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by FL, 17-Feb-2008.) $)
|
|
nfmpt1 $p |- F/_ x ( x e. A |-> B ) $=
|
|
( vz cmpt cv wcel wceq wa copab df-mpt nfopab1 nfcxfr ) AABCEAFBGDFCHIZAD
|
|
JADBCKNADLM $.
|
|
$}
|
|
|
|
${
|
|
$d w z x A $. $d w z y A $. $d w z B $. $d w z C $.
|
|
cbvmpt.1 $e |- F/_ y B $.
|
|
cbvmpt.2 $e |- F/_ x C $.
|
|
cbvmpt.3 $e |- ( x = y -> B = C ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. This version has bound-variable hypotheses in place of
|
|
distinct variable conditions. (Contributed by NM, 11-Sep-2011.) $)
|
|
cbvmpt $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
|
|
( vz vw cv wcel wceq wa copab cmpt wsb nfv nfan weq nfs1v sbequ12 anbi12d
|
|
eleq1 cbvopab1 nfeq2 nfsb sbequ eqeq2d sbie syl6bb eqtri df-mpt 3eqtr4i )
|
|
AKZCLZIKZDMZNZAIOZBKZCLZUQEMZNZBIOZACDPBCEPUTJKZCLZURAJQZNZJIOVEUSVIAIJUS
|
|
JRVGVHAVGARURAJUASAJTUPVGURVHUOVFCUDURAJUBUCUEVIVDJIBVGVHBVGBRURAJBBUQDFU
|
|
FUGSVDJRJBTZVGVBVHVCVFVACUDVJVHURABQVCURJBAUHURVCABAUQEGUFABTDEUQHUIUJUKU
|
|
CUEULAICDUMBICEUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B y $. $d C x $.
|
|
cbvmptv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) $)
|
|
cbvmptv $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
|
|
( nfcv cbvmpt ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y B $.
|
|
$( Function with universal domain in maps-to notation. (Contributed by NM,
|
|
16-Aug-2013.) $)
|
|
mptv $p |- ( x e. _V |-> B ) = { <. x , y >. | y = B } $=
|
|
( cvv cmpt cv wcel wceq wa copab df-mpt vex biantrur opabbii eqtr4i ) ADC
|
|
EAFDGZBFCHZIZABJQABJABDCKQRABPQALMNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Transitive classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare a new symbol. $)
|
|
$c Tr $. $( Transitive predicate (read: "the following class is
|
|
transitive") $)
|
|
|
|
$( Extend wff notation to include transitive classes. Notation from
|
|
[TakeutiZaring] p. 35. $)
|
|
wtr $a wff Tr A $.
|
|
|
|
$( Define the transitive class predicate. Definition of [Enderton] p. 71
|
|
extended to arbitrary classes. For alternate definitions, see ~ dftr2
|
|
(which is suggestive of the word "transitive"), ~ dftr3 , ~ dftr4 , and
|
|
~ dftr5 . The term "complete" is used instead of "transitive" in
|
|
Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.) $)
|
|
df-tr $a |- ( Tr A <-> U. A C_ A ) $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( An alternate way of defining a transitive class. Exercise 7 of
|
|
[TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.) $)
|
|
dftr2 $p |- ( Tr A <-> A. x A. y ( ( x e. y /\ y e. A ) -> x e. A ) ) $=
|
|
( cuni wss cv wcel wi wal wa dfss2 df-tr 19.23v eluni imbi1i bitr4i albii
|
|
wtr wex 3bitr4i ) CDZCEAFZUAGZUBCGZHZAICRUBBFZGUFCGJZUDHBIZAIAUACKCLUHUEA
|
|
UHUGBSZUDHUEUGUDBMUCUIUDBUBCNOPQT $.
|
|
|
|
$( An alternate way of defining a transitive class. (Contributed by NM,
|
|
20-Mar-2004.) $)
|
|
dftr5 $p |- ( Tr A <-> A. x e. A A. y e. x y e. A ) $=
|
|
( wtr cv wcel wa wi wal wral dftr2 alcom impexp albii df-ral bitr4i bitri
|
|
r19.21v ) CDBEZAEZFZTCFZGSCFZHZAIBIZUCBTJZACJZBACKUEUDBIZAIZUGUDBALUIUBUF
|
|
HZAIUGUHUJAUHUBUCHZBTJZUJUHUAUKHZBIULUDUMBUAUBUCMNUKBTOPUBUCBTRQNUFACOPQQ
|
|
$.
|
|
|
|
$( An alternate way of defining a transitive class. Definition 7.1 of
|
|
[TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.) $)
|
|
dftr3 $p |- ( Tr A <-> A. x e. A x C_ A ) $=
|
|
( vy wtr cv wcel wral wss dftr5 dfss3 ralbii bitr4i ) BDCEBFCAEZGZABGMBHZ
|
|
ABGACBIONABCMBJKL $.
|
|
$}
|
|
|
|
$( An alternate way of defining a transitive class. Definition of [Enderton]
|
|
p. 71. (Contributed by NM, 29-Aug-1993.) $)
|
|
dftr4 $p |- ( Tr A <-> A C_ ~P A ) $=
|
|
( wtr cuni wss cpw df-tr sspwuni bitr4i ) ABACADAAEDAFAAGH $.
|
|
|
|
$( Equality theorem for the transitive class predicate. (Contributed by NM,
|
|
17-Sep-1993.) $)
|
|
treq $p |- ( A = B -> ( Tr A <-> Tr B ) ) $=
|
|
( wceq cuni wss wtr unieq sseq1d sseq2 bitrd df-tr 3bitr4g ) ABCZADZAEZBDZB
|
|
EZAFBFMOPAEQMNPAABGHABPIJAKBKL $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( In a transitive class, the membership relation is transitive.
|
|
(Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
trel $p |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) ) $=
|
|
( vy vx wtr cv wcel wa wi wal dftr2 eleq12 wb eleq1 adantl anbi12d adantr
|
|
wceq imbi12d spc2gv pm2.43b sylbi ) AFDGZEGZHZUEAHZIZUDAHZJZEKDKZBCHZCAHZ
|
|
IZBAHZJZDEALUKUNUOUJUPDEBCCAUDBSZUECSZIZUHUNUIUOUSUFULUGUMUDBUECMURUGUMNU
|
|
QUECAOPQUQUIUONURUDBAORTUAUBUC $.
|
|
$}
|
|
|
|
$( In a transitive class, the membership relation is transitive.
|
|
(Contributed by NM, 19-Apr-1994.) $)
|
|
trel3 $p |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) ) $=
|
|
( wtr wcel w3a wa 3anass trel anim2d syl5bi syld ) AEZBCFZCDFZDAFZGZOCAFZHZ
|
|
BAFROPQHZHNTOPQINUASOACDJKLABCJM $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An element of a transitive class is a subset of the class. (Contributed
|
|
by NM, 7-Aug-1994.) $)
|
|
trss $p |- ( Tr A -> ( B e. A -> B C_ A ) ) $=
|
|
( vx wtr wcel wss cv wceq eleq1 sseq1 imbi12d imbi2d wral dftr3 rsp sylbi
|
|
wi vtoclg pm2.43b ) ADZBAEZBAFZTCGZAEZUCAFZQZQTUAUBQZQCBAUCBHZUFUGTUHUDUA
|
|
UEUBUCBAIUCBAJKLTUECAMUFCANUECAOPRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The intersection of transitive classes is transitive. (Contributed by
|
|
NM, 9-May-1994.) $)
|
|
trin $p |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) ) $=
|
|
( vx wtr cin wss wral wcel elin trss im2anan9 syl5bi ssin syl6ib ralrimiv
|
|
wa cv dftr3 sylibr ) ADZBDZPZCQZABEZFZCUDGUDDUBUECUDUBUCUDHZUCAFZUCBFZPZU
|
|
EUFUCAHZUCBHZPUBUIUCABITUJUGUAUKUHAUCJBUCJKLUCABMNOCUDRS $.
|
|
$}
|
|
|
|
$( The empty set is transitive. (Contributed by NM, 16-Sep-1993.) $)
|
|
tr0 $p |- Tr (/) $=
|
|
( c0 wtr cpw wss 0ss dftr4 mpbir ) ABAACZDHEAFG $.
|
|
|
|
$( The universe is transitive. (Contributed by NM, 14-Sep-2003.) $)
|
|
trv $p |- Tr _V $=
|
|
( cvv wtr cuni wss ssv df-tr mpbir ) ABACZADHEAFG $.
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The indexed union of a class of transitive sets is transitive.
|
|
(Contributed by Mario Carneiro, 16-Nov-2014.) $)
|
|
triun $p |- ( A. x e. A Tr B -> Tr U_ x e. A B ) $=
|
|
( vy wtr wral cv ciun wss wcel wrex eliun r19.29 nfcv nfiu1 nfss trss imp
|
|
wa ssiun2 sstr2 syl5com syl5 rexlimi syl sylan2b ralrimiva dftr3 sylibr )
|
|
CEZABFZDGZABCHZIZDUMFUMEUKUNDUMULUMJUKULCJZABKZUNAULBCLUKUPSUJUOSZABKUNUJ
|
|
UOABMUQUNABAULUMAULNABCOPUQULCIZAGBJZUNUJUOURCULQRUSCUMIURUNABCTULCUMUAUB
|
|
UCUDUEUFUGDUMUHUI $.
|
|
|
|
$( The union of a class of transitive sets is transitive. Exercise 5(a) of
|
|
[Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof
|
|
shortened by Mario Carneiro, 26-Apr-2014.) $)
|
|
truni $p |- ( A. x e. A Tr x -> Tr U. A ) $=
|
|
( cv wtr wral ciun cuni triun wceq wb uniiun treq ax-mp sylibr ) ACZDABEA
|
|
BOFZDZBGZDZABOHRPISQJABKRPLMN $.
|
|
|
|
$( The intersection of a class of transitive sets is transitive. Exercise
|
|
5(b) of [Enderton] p. 73. (Contributed by Scott Fenton,
|
|
25-Feb-2011.) $)
|
|
trint $p |- ( A. x e. A Tr x -> Tr |^| A ) $=
|
|
( vy cv wtr wral wel wss wi cint dftr3 ralbii biimpi df-ral ralcom4 bitri
|
|
wal sylib ralim alimi syl wcel vex elint2 ssint imbi12i albii sylibr ) AD
|
|
ZEZABFZCAGZABFZCDZUIHZABFZIZCQZBJZEZUKULUOIZABFZCQZURUKUOCUIFZABFZVCUKVEU
|
|
JVDABCUIKLMVEVACQZABFVCVDVFABUOCUINLVAACBOPRVBUQCULUOABSTUAUTUNUSHZCUSFZU
|
|
RCUSKVHUNUSUBZVGIZCQURVGCUSNVJUQCVIUMVGUPAUNBCUCUDAUNBUEUFUGPPUH $.
|
|
|
|
$( Any inhabited transitive class includes its intersection. Similar to
|
|
Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include
|
|
the inhabitedness hypothesis). (Contributed by Jim Kingdon,
|
|
22-Aug-2018.) $)
|
|
trintssm $p |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A ) $=
|
|
( cv wcel wex wtr cint wss intss1 trss com12 sstr2 sylsyld exlimiv impcom
|
|
wi ) ACZBDZAEBFZBGZBHZRSUAPARTQHSQBHZUAQBISRUBBQJKTQBLMNO $.
|
|
|
|
$( Obsolete version of ~ trintssm as of 30-Oct-2021. (Contributed by Jim
|
|
Kingdon, 22-Aug-2018.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
trintssmOLD $p |- ( ( E. x x e. A /\ Tr A ) -> |^| A C_ A ) $=
|
|
( vy cv wcel wex wtr wa cint wral vex elint2 wrex r19.2m ex trel rexlimdv
|
|
expcomd sylan9 syl5bi ssrdv ) ADZBEZAFZBGZHZCBIZBCDZUGEUHUBEZABJZUFUHBEZA
|
|
UHBCKLUDUJUIABMZUEUKUDUJULUIABNOUEUIUKABUEUIUCUKBUHUBPRQSTUA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
IZF Set Theory - add the Axioms of Collection and Separation
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Collection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y a b $.
|
|
ax-coll.1 $e |- F/ b ph $.
|
|
$( Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF"
|
|
(with unnecessary quantifier removed). It is similar to ~ bnd but uses
|
|
a freeness hypothesis in place of one of the distinct variable
|
|
constraints. (Contributed by Jim Kingdon, 23-Aug-2018.) $)
|
|
ax-coll $a |- ( A. x e. a E. y ph -> E. b A. x e. a E. y e. b ph ) $.
|
|
|
|
$( Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and
|
|
IZF" (with unnecessary quantifier removed). In our context this is not
|
|
an axiom, but a theorem proved from ~ ax-coll . It is identical to
|
|
~ zfrep6 except for the choice of a freeness hypothesis rather than a
|
|
distinct variable constraint between ` b ` and ` ph ` . (Contributed by
|
|
Jim Kingdon, 23-Aug-2018.) $)
|
|
repizf $p |- ( A. x e. a E! y ph -> E. b A. x e. a E. y e. b ph ) $=
|
|
( weu cv wral wex wrex euex ralimi ax-coll syl ) ACGZBDHZIACJZBQIACEHKBQI
|
|
EJPRBQACLMABCDEFNO $.
|
|
$}
|
|
|
|
${
|
|
$d ph w $. $d x y z w $.
|
|
$( A version of the Axiom of Replacement. Normally ` ph ` would have free
|
|
variables ` x ` and ` y ` . Axiom 6 of [Kunen] p. 12. The Separation
|
|
Scheme ~ ax-sep cannot be derived from this version and must be stated
|
|
as a separate axiom in an axiom system (such as Kunen's) that uses this
|
|
version. (Contributed by NM, 10-Oct-2003.) $)
|
|
zfrep6 $p |- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph ) $=
|
|
( nfv repizf ) ABCDEAEFG $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Separation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $. $d y z ph $.
|
|
$( The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p.
|
|
"Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a
|
|
` F/ y ph ` condition replaced by a distinct variable constraint between
|
|
` y ` and ` ph ` ).
|
|
|
|
The Separation Scheme is a weak form of Frege's Axiom of Comprehension,
|
|
conditioning it (with ` x e. z ` ) so that it asserts the existence of a
|
|
collection only if it is smaller than some other collection ` z ` that
|
|
already exists. This prevents Russell's paradox ~ ru . In some texts,
|
|
this scheme is called "Aussonderung" or the Subset Axiom.
|
|
|
|
(Contributed by NM, 11-Sep-2006.) $)
|
|
ax-sep $a |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d y ph w $. $d z w $.
|
|
$( A less restrictive version of the Separation Scheme ~ ax-sep , where
|
|
variables ` x ` and ` z ` can both appear free in the wff ` ph ` , which
|
|
can therefore be thought of as ` ph ( x , z ) ` . This version was
|
|
derived from the more restrictive ~ ax-sep with no additional set theory
|
|
axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario
|
|
Carneiro, 17-Nov-2016.) $)
|
|
axsep2 $p |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $=
|
|
( vw wel wa wb wal wex cv eleq2 anbi1d anabs5 syl6bb bibi2d albidv exbidv
|
|
weq ax-sep chvarv ) BCFZBEFZBDFZAGZGZHZBIZCJUBUEHZBIZCJEDEDSZUHUJCUKUGUIB
|
|
UKUFUEUBUKUFUDUEGUEUKUCUDUEEKDKBKLMUDANOPQRUEBCETUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A z $. $d y ph z $.
|
|
zfauscl.1 $e |- A e. _V $.
|
|
$( Separation Scheme (Aussonderung) using a class variable. To derive this
|
|
from ~ ax-sep , we invoke the Axiom of Extensionality (indirectly via
|
|
~ vtocl ), which is needed for the justification of class variable
|
|
notation. (Contributed by NM, 5-Aug-1993.) $)
|
|
zfauscl $p |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
|
|
( vz cv wcel wa wb wal wex eleq2 anbi1d bibi2d albidv exbidv ax-sep vtocl
|
|
wceq ) BGZCGHZUAFGZHZAIZJZBKZCLUBUADHZAIZJZBKZCLFDEUCDTZUGUKCULUFUJBULUEU
|
|
IUBULUDUHAUCDUAMNOPQABCFRS $.
|
|
$}
|
|
|
|
${
|
|
$d x ph z $. $d x y z $.
|
|
bm1.3ii.1 $e |- E. x A. y ( ph -> y e. x ) $.
|
|
$( Convert implication to equivalence using the Separation Scheme
|
|
(Aussonderung) ~ ax-sep . Similar to Theorem 1.3ii of [BellMachover]
|
|
p. 463. (Contributed by NM, 5-Aug-1993.) $)
|
|
bm1.3ii $p |- E. x A. y ( y e. x <-> ph ) $=
|
|
( vz wel wi wal wa wex weq elequ2 imbi2d albidv cbvexv mpbi ax-sep pm3.2i
|
|
wb exan 19.42v bimsc1 alanimi eximi sylbir exlimiv ax-mp ) ACEFZGZCHZCBFZ
|
|
UHAISZCHZBJZIZEJUKASZCHZBJZUJUNEUJEJZUNAUKGZCHZBJUSDVAUJBEBEKZUTUICVBUKUH
|
|
ABECLMNOPACBEQRTUOUREUOUJUMIZBJURUJUMBUAVCUQBUIULUPCAUHUKUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Derive a weakened version of ~ ax-i9 , where ` x ` and ` y ` must be
|
|
distinct, from Separation ~ ax-sep and Extensionality ~ ax-ext . The
|
|
theorem ` -. A. x -. x = y ` also holds ( ~ ax9vsep ), but in
|
|
intuitionistic logic ` E. x x = y ` is stronger. (Contributed by Jim
|
|
Kingdon, 25-Aug-2018.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
a9evsep $p |- E. x x = y $=
|
|
( vz cv wcel wceq wi wa wb wal wex ax-sep id biantru bibi2i biimpri alimi
|
|
ax-ext syl eximi ax-mp ) CDZADZEZUBBDZEZUBUBFZUGGZHZIZCJZAKUCUEFZAKUHCABL
|
|
UKULAUKUDUFIZCJULUJUMCUMUJUFUIUDUHUFUGMNOPQABCRSTUA $.
|
|
|
|
$( Derive a weakened version of ~ ax-9 , where ` x ` and ` y ` must be
|
|
distinct, from Separation ~ ax-sep and Extensionality ~ ax-ext . In
|
|
intuitionistic logic ~ a9evsep is stronger and also holds. (Contributed
|
|
by NM, 12-Nov-2013.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax9vsep $p |- -. A. x -. x = y $=
|
|
( cv wceq wex wn wal a9evsep exalim ax-mp ) ACBCDZAEKFAGFABHKAIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Null Set Axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y $.
|
|
zfnuleu.1 $e |- E. x A. y -. y e. x $.
|
|
$( Show the uniqueness of the empty set (using the Axiom of Extensionality
|
|
via ~ bm1.1 to strengthen the hypothesis in the form of ~ axnul ).
|
|
(Contributed by NM, 22-Dec-2007.) $)
|
|
zfnuleu $p |- E! x A. y -. y e. x $=
|
|
( wel wn wal weu wfal wb wex nbfal albii exbii mpbi nfv bm1.1 ax-mp eubii
|
|
mpbir ) BADZEZBFZAGTHIZBFZAGZUDAJZUEUBAJUFCUBUDAUAUCBTKLZMNHABHAOPQUBUDAU
|
|
GRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The Null Set Axiom of ZF set theory: there exists a set with no
|
|
elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks,
|
|
this is presented as a separate axiom; here we show it can be derived
|
|
from Separation ~ ax-sep . This version of the Null Set Axiom tells us
|
|
that at least one empty set exists, but does not tell us that it is
|
|
unique - we need the Axiom of Extensionality to do that (see
|
|
~ zfnuleu ).
|
|
|
|
This theorem should not be referenced by any proof. Instead, use
|
|
~ ax-nul below so that the uses of the Null Set Axiom can be more easily
|
|
identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM,
|
|
4-Feb-2008.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
axnul $p |- E. x A. y -. y e. x $=
|
|
( vz cv wn wa wb wal wex ax-sep pm3.24 intnan id mtbiri alimi eximi ax-mp
|
|
wcel ) BDZADRZSCDRZSSRZUBEFZFZGZBHZAITEZBHZAIUCBACJUFUHAUEUGBUETUDUCUAUBK
|
|
LUEMNOPQ $.
|
|
|
|
$( The Null Set Axiom of IZF set theory. It was derived as ~ axnul above
|
|
and is therefore redundant, but we state it as a separate axiom here so
|
|
that its uses can be identified more easily. Axiom 4 of [Crosilla] p.
|
|
"Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.) $)
|
|
ax-nul $a |- E. x A. y -. y e. x $.
|
|
|
|
$( The Null Set Axiom of ZF set theory: the empty set exists. Corollary
|
|
5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see
|
|
~ ax-nul . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 9-Jul-2011.) $)
|
|
0ex $p |- (/) e. _V $=
|
|
( vx vy c0 cv wceq wex wel wn wal ax-nul eq0 exbii mpbir issetri ) ACADZC
|
|
EZAFBAGHBIZAFABJPQABOKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( The existence of proper substitution into a class. (Contributed by NM,
|
|
10-Nov-2005.) $)
|
|
csbexga $p |- ( ( A e. V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V ) $=
|
|
( vy wcel wal wa csb wsbc cab cvv df-csb abid2 elex syl5eqel alimi spsbc
|
|
cv syl5 imp wb nfcv sbcabel adantr mpbid ) BDGZCEGZAHZIZABCJFTCGZABKFLZMA
|
|
FBCNUKULFLZMGZABKZUMMGZUHUJUPUJUOAHUHUPUIUOAUIUNCMFCOCEPQRUOABDSUAUBUHUPU
|
|
QUCUJULAFBMDAMUDUEUFUGQ $.
|
|
$}
|
|
|
|
${
|
|
csbexa.1 $e |- A e. _V $.
|
|
csbexa.2 $e |- B e. _V $.
|
|
$( The existence of proper substitution into a class. (Contributed by NM,
|
|
7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
csbexa $p |- [_ A / x ]_ B e. _V $=
|
|
( cvv wcel csb wal csbexga mpan mpg ) CFGZABCHFGZABFGMAINDABCFFJKEL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Theorems requiring subset and intersection existence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $.
|
|
$( No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed
|
|
by NM, 23-Aug-1993.) $)
|
|
nalset $p |- -. E. x A. y y e. x $=
|
|
( vz wel wn wex wal alexnim wa wb ax-sep weq elequ1 elequ2 notbid anbi12d
|
|
bitrd bibi12d spv pclem6 syl eximii mpg ) BADZEZBFUDBGAFEAUDABHCBDZCADZCC
|
|
DZEZIZJZCGZUEBUICBAKULBBDZUDUMEZIZJZUEUKUPCBCBLZUFUMUJUOCBBMUQUGUDUIUNCBA
|
|
MUQUHUMUQUHBCDUMCBCMCBBNQOPRSUMUDTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The universal class is not a member of itself (and thus is not a set).
|
|
Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not
|
|
depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.) $)
|
|
vprc $p |- -. _V e. _V $=
|
|
( vx vy cvv wcel cv wceq wex wal nalset vex tbt albii dfcleq bitr4i exbii
|
|
wb mtbi isset mtbir ) CCDAEZCFZAGZBEZTDZBHZAGUBABIUEUAAUEUDUCCDZPZBHUAUDU
|
|
GBUFUDBJKLBTCMNOQACRS $.
|
|
$}
|
|
|
|
$( The universal class doesn't belong to any class. (Contributed by FL,
|
|
31-Dec-2006.) $)
|
|
nvel $p |- -. _V e. A $=
|
|
( cvv wcel vprc elex mto ) BACBBCDBAEF $.
|
|
|
|
$( The universal class does not exist. (Contributed by NM, 4-Jul-2005.) $)
|
|
vnex $p |- -. E. x x = _V $=
|
|
( cvv wcel cv wceq wex vprc isset mtbi ) BBCADBEAFGABHI $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
inex1.1 $e |- A e. _V $.
|
|
$( Separation Scheme (Aussonderung) using class notation. Compare Exercise
|
|
4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) $)
|
|
inex1 $p |- ( A i^i B ) e. _V $=
|
|
( vx vy cin cv wceq wex wcel wa wb zfauscl dfcleq elin bibi2i albii bitri
|
|
wal exbii mpbir issetri ) DABFZDGZUCHZDIEGZUDJZUFAJUFBJZKZLZESZDIUHEDACMU
|
|
EUKDUEUGUFUCJZLZESUKEUDUCNUMUJEULUIUGUFABOPQRTUAUB $.
|
|
$}
|
|
|
|
${
|
|
inex2.1 $e |- A e. _V $.
|
|
$( Separation Scheme (Aussonderung) using class notation. (Contributed by
|
|
NM, 27-Apr-1994.) $)
|
|
inex2 $p |- ( B i^i A ) e. _V $=
|
|
( cin cvv incom inex1 eqeltri ) BADABDEBAFABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Closed-form, generalized Separation Scheme. (Contributed by NM,
|
|
7-Apr-1995.) $)
|
|
inex1g $p |- ( A e. V -> ( A i^i B ) e. _V ) $=
|
|
( vx cv cin cvv wcel wceq ineq1 eleq1d vex inex1 vtoclg ) DEZBFZGHABFZGHD
|
|
ACOAIPQGOABJKOBDLMN $.
|
|
$}
|
|
|
|
${
|
|
ssex.1 $e |- B e. _V $.
|
|
$( The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22.
|
|
This is one way to express the Axiom of Separation ~ ax-sep (a.k.a.
|
|
Subset Axiom). (Contributed by NM, 27-Apr-1994.) $)
|
|
ssex $p |- ( A C_ B -> A e. _V ) $=
|
|
( wss cin wceq cvv wcel df-ss inex2 eleq1 mpbii sylbi ) ABDABEZAFZAGHZABI
|
|
ONGHPBACJNAGKLM $.
|
|
$}
|
|
|
|
${
|
|
ssexi.1 $e |- B e. _V $.
|
|
ssexi.2 $e |- A C_ B $.
|
|
$( The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) $)
|
|
ssexi $p |- A e. _V $=
|
|
( wss cvv wcel ssex ax-mp ) ABEAFGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22
|
|
(generalized). (Contributed by NM, 14-Aug-1994.) $)
|
|
ssexg $p |- ( ( A C_ B /\ B e. C ) -> A e. _V ) $=
|
|
( vx wcel wss cvv cv wi wceq sseq2 imbi1d vex ssex vtoclg impcom ) BCEABF
|
|
ZAGEZADHZFZRIQRIDBCSBJTQRSBAKLASDMNOP $.
|
|
$}
|
|
|
|
${
|
|
ssexd.1 $e |- ( ph -> B e. C ) $.
|
|
ssexd.2 $e |- ( ph -> A C_ B ) $.
|
|
$( A subclass of a set is a set. Deduction form of ~ ssexg . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
ssexd $p |- ( ph -> A e. _V ) $=
|
|
( wss wcel cvv ssexg syl2anc ) ABCGCDHBIHFEBCDJK $.
|
|
$}
|
|
|
|
$( Existence of a difference. (Contributed by NM, 26-May-1998.) $)
|
|
difexg $p |- ( A e. V -> ( A \ B ) e. _V ) $=
|
|
( cdif wss wcel cvv difss ssexg mpan ) ABDZAEACFKGFABHKACIJ $.
|
|
|
|
${
|
|
$d x A $.
|
|
zfausab.1 $e |- A e. _V $.
|
|
$( Separation Scheme (Aussonderung) in terms of a class abstraction.
|
|
(Contributed by NM, 8-Jun-1994.) $)
|
|
zfausab $p |- { x | ( x e. A /\ ph ) } e. _V $=
|
|
( cv wcel wa cab ssab2 ssexi ) BECFAGBHCDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Separation Scheme in terms of a restricted class abstraction.
|
|
(Contributed by NM, 23-Oct-1999.) $)
|
|
rabexg $p |- ( A e. V -> { x e. A | ph } e. _V ) $=
|
|
( crab wss wcel cvv ssrab2 ssexg mpan ) ABCEZCFCDGLHGABCILCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
rabex.1 $e |- A e. _V $.
|
|
$( Separation Scheme in terms of a restricted class abstraction.
|
|
(Contributed by NM, 19-Jul-1996.) $)
|
|
rabex $p |- { x e. A | ph } e. _V $=
|
|
( cvv wcel crab rabexg ax-mp ) CEFABCGEFDABCEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
elssabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction involving a subset. Unlike ~ elabg ,
|
|
` A ` does not have to be a set. (Contributed by NM, 29-Aug-2006.) $)
|
|
elssabg $p |- ( B e. V ->
|
|
( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) ) $=
|
|
( wcel wss wa cvv wi cv cab wb ssexg expcom adantrd wceq sseq1 elab3g syl
|
|
anbi12d ) EFHZDEIZBJZDKHZLDCMZEIZAJZCNHUFOUDUEUGBUEUDUGDEFPQRUJUFCDKUHDSU
|
|
IUEABUHDETGUCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The intersection of an inhabited class exists. (Contributed by Jim
|
|
Kingdon, 27-Aug-2018.) $)
|
|
inteximm $p |- ( E. x x e. A -> |^| A e. _V ) $=
|
|
( cv wcel cint cvv wss intss1 vex ssex syl exlimiv ) ACZBDZBEZFDZANOMGPMB
|
|
HOMAIJKL $.
|
|
$}
|
|
|
|
$( If the intersection of a class exists, the class is non-empty.
|
|
(Contributed by Jim Kingdon, 27-Aug-2018.) $)
|
|
intexr $p |- ( |^| A e. _V -> A =/= (/) ) $=
|
|
( cint cvv wcel c0 wceq vprc inteq int0 syl6eq eleq1d mtbiri necon2ai ) ABZ
|
|
CDZAEAEFZOCCDGPNCCPNEBCAEHIJKLM $.
|
|
|
|
$( If a class intersection is the universe, it is not a set. In classical
|
|
logic this would be an equivalence. (Contributed by Jim Kingdon,
|
|
27-Aug-2018.) $)
|
|
intnexr $p |- ( |^| A = _V -> -. |^| A e. _V ) $=
|
|
( cint cvv wceq wcel vprc eleq1 mtbiri ) ABZCDICECCEFICCGH $.
|
|
|
|
${
|
|
$d x y $. $d ph y $.
|
|
$( The intersection of an inhabited class abstraction exists. (Contributed
|
|
by Jim Kingdon, 27-Aug-2018.) $)
|
|
intexabim $p |- ( E. x ph -> |^| { x | ph } e. _V ) $=
|
|
( vy wex cv cab wcel cint cvv abid exbii nfsab1 nfv eleq1 inteximm sylbir
|
|
cbvex ) ABDBEZABFZGZBDZSHIGZTABABJKUACEZSGZCDUBUDTCBABCLTCMUCRSNQCSOPP $.
|
|
$}
|
|
|
|
$( The intersection of an inhabited restricted class abstraction exists.
|
|
(Contributed by Jim Kingdon, 27-Aug-2018.) $)
|
|
intexrabim $p |- ( E. x e. A ph -> |^| { x e. A | ph } e. _V ) $=
|
|
( cv wcel wex cab cint cvv wrex crab intexabim df-rex df-rab inteqi 3imtr4i
|
|
wa eleq1i ) BDCEAQZBFSBGZHZIEABCJABCKZHZIESBLABCMUCUAIUBTABCNORP $.
|
|
|
|
${
|
|
$d A x y $. $d B y $. $d x y z $. $d A z $. $d B z $.
|
|
$( The existence of an indexed union. ` x ` is normally a free-variable
|
|
parameter in ` B ` , which should be read ` B ( x ) ` . (Contributed by
|
|
Jim Kingdon, 28-Aug-2018.) $)
|
|
iinexgm $p |- ( ( E. x x e. A /\ A. x e. A B e. C ) ->
|
|
|^|_ x e. A B e. _V ) $=
|
|
( vy vz cv wcel wex wral wa ciin wceq wrex cab cint cvv wi syl sylib abid
|
|
dfiin2g adantl elisset rgenw r19.2m mpan2 r19.35-1 imp rexcom4 sylibr nfv
|
|
exbii nfsab1 eleq1 cbvex inteximm eqeltrd ) AGBHAIZCDHZABJZKZABCLZEGZCMZA
|
|
BNZEOZPZQVAVCVHMUSAEBCDUBUCVBFGZVGHZFIZVHQHVBVDVGHZEIZVKVBVFEIZVMVBVEEIZA
|
|
BNZVNUSVAVPUSUTVORZABNZVAVPRUSVQABJVRVQABECDUDUEVQABUFUGUTVOABUHSUIVEAEBU
|
|
JTVLVFEVFEUAUMUKVLVJEFVLFULVFEFUNVDVIVGUOUPTFVGUQSUR $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( The intersection of a union ` U. A ` with a class ` B ` is equal to the
|
|
union of the intersections of each element of ` A ` with ` B ` .
|
|
(Contributed by FL, 24-Mar-2007.) $)
|
|
inuni $p |- ( U. A i^i B ) = U. { x | E. y e. A x = ( y i^i B ) } $=
|
|
( vz cuni cin cv wceq wrex cab wcel wex eluni2 anbi1i elin r19.41v bitr4i
|
|
wa bitri ancom exbii rexcom4 vex inex1 eleq2 ceqsexv rexbii 3bitr4i eqriv
|
|
eluniab ) ECFZDGZAHZBHZDGZIZBCJZAKFZEHZUMLZUTUNLZURSZAMZUTUSLUTULLZUTDLZS
|
|
UTUOLZBCJZVFSZVAVDVEVHVFBUTCNOUTULDPVDUQVBSZAMZBCJZVIVDVJBCJZAMVLVCVMAVCU
|
|
RVBSVMVBURUAUQVBBCQRUBVJBACUCRVLVGVFSZBCJVIVKVNBCVKUTUPLZVNVBVOAUPUODBUDU
|
|
EUNUPUTUFUGUTUODPTUHVGVFBCQTTUIURAUTUKRUJ $.
|
|
$}
|
|
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed
|
|
by NM, 7-Aug-2000.) $)
|
|
elpw2g $p |- ( B e. V -> ( A e. ~P B <-> A C_ B ) ) $=
|
|
( wcel cpw wss elpwi cvv ssexg elpwg biimparc syldan expcom impbid2 ) BCDZA
|
|
BEDZABFZABGQOPQOAHDZPABCIRPQABHJKLMN $.
|
|
|
|
${
|
|
elpw2.1 $e |- B e. _V $.
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47.
|
|
(Contributed by NM, 11-Oct-2007.) $)
|
|
elpw2 $p |- ( A e. ~P B <-> A C_ B ) $=
|
|
( cvv wcel cpw wss wb elpw2g ax-mp ) BDEABFEABGHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d V x y $.
|
|
$( The power set of a set is never a subset. (Contributed by Stefan
|
|
O'Rear, 22-Feb-2015.) $)
|
|
pwnss $p |- ( A e. V -> -. ~P A C_ A ) $=
|
|
( vx vy cpw cv wnel crab wcel wn wa wb eleq12 anidms notbid df-nel syl5bb
|
|
wss wceq cbvrabv elrab2 pclem6 ax-mp ssel mtoi ssrab2 elpw2g mpbiri nsyl3
|
|
) AEZARZCFZULGZCAHZUJIZABIZUKUOUNAIZUNUNIZUQURJZKLUQJDFZUTIZJZUSDUNAUNUTU
|
|
NSZVAURVCVAURLUTUNUTUNMNOUMVBCDAUMULULIZJULUTSZVBULULPVEVDVAVEVDVALULUTUL
|
|
UTMNOQTUAURUQUBUCUJAUNUDUEUPUOUNARUMCAUFUNABUGUHUI $.
|
|
$}
|
|
|
|
$( No set equals its power set. The sethood antecedent is necessary; compare
|
|
~ pwv . (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
pwne $p |- ( A e. V -> ~P A =/= A ) $=
|
|
( wcel cpw wss wn wne pwnss eqimss necon3bi syl ) ABCADZAEZFLAGABHMLALAIJK
|
|
$.
|
|
|
|
$( Lemma for ~ repizf2 . If we have a function-like proposition which
|
|
provides at most one value of ` y ` for each ` x ` in a set ` w ` , we can
|
|
change "at most one" to "exactly one" by restricting the values of ` x `
|
|
to those values for which the proposition provides a value of ` y ` .
|
|
(Contributed by Jim Kingdon, 7-Sep-2018.) $)
|
|
repizf2lem $p |- ( A. x e. w E* y ph <->
|
|
A. x e. { x e. w | E. y ph } E! y ph ) $=
|
|
( cv wcel wmo wi wal wex wral crab df-mo imbi2i albii df-ral wa rabid bitri
|
|
weu imbi1i impexp 3bitr4i ) BEZDEZFZACGZHZBIUFACJZACTZHZHZBIZUGBUEKUJBUIBUE
|
|
LZKZUHULBUGUKUFACMNOUGBUEPUOUDUNFZUJHZBIUMUJBUNPUQULBUQUFUIQZUJHULUPURUJUIB
|
|
UERUAUFUIUJUBSOSUC $.
|
|
|
|
${
|
|
$d v x y z w $. $d v ph $.
|
|
repizf2.1 $e |- F/ z ph $.
|
|
$( Replacement. This version of replacement is stronger than ~ repizf in
|
|
the sense that ` ph ` does not need to map all values of ` x ` in ` w `
|
|
to a value of ` y ` . The resulting set contains those elements for
|
|
which there is a value of ` y ` and in that sense, this theorem combines
|
|
~ repizf with ~ ax-sep . Another variation would be
|
|
` A. x e. w E* y ph -> { y | E. x ( x e. w /\ ph ) } e. _V ` but we
|
|
don't have a proof of that yet. (Contributed by Jim Kingdon,
|
|
7-Sep-2018.) $)
|
|
repizf2 $p |- ( A. x e. w E* y ph ->
|
|
E. z A. x e. { x e. w | E. y ph } E. y e. z ph ) $=
|
|
( vv wmo cv wral wrex wex crab wi vex rabex wceq weu repizf2lem raleqf wa
|
|
nfcv nfrab1 repizf syl6bir syl5bi wel cab df-rab nfv nfex nfan nfab nfeq2
|
|
nfcxfr exbid sylibd vtocle ) ACHBEIZJZACDIKZBACLZBUSMZJZDLZNGVCVBBUSEOPGI
|
|
ZVCQZUTVABVFJZDLZVEUTACRZBVCJZVGVIABCESVGVKVJBVFJVIVJBVFVCBVFUBZVBBUSUCZT
|
|
ABCGDFUDUEUFVGVHVDDDVFVCDVCBEUGZVBUAZBUHVBBUSUIVODBVNVBDVNDUJADCFUKULUMUO
|
|
UNVABVFVCVLVMTUPUQUR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Theorems requiring empty set existence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x A $.
|
|
$( Equality theorem for classes and sets . (Contributed by NM,
|
|
13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) $)
|
|
class2seteq $p |- ( A e. V -> { x e. A | A e. _V } = A ) $=
|
|
( wcel cvv crab wceq elex wral cv ax-1 ralrimiv rabid2 sylibr eqcomd syl
|
|
) BCDBEDZQABFZBGBCHQBRQQABIBRGQQABQAJBDKLQABMNOP $.
|
|
$}
|
|
|
|
$( Every power class contains the empty set. (Contributed by NM,
|
|
25-Oct-2007.) $)
|
|
0elpw $p |- (/) e. ~P A $=
|
|
( c0 cpw wcel wss 0ss 0ex elpw mpbir ) BACDBAEAFBAGHI $.
|
|
|
|
$( The empty set and its power set are not equal. (Contributed by NM,
|
|
23-Dec-1993.) $)
|
|
0nep0 $p |- (/) =/= { (/) } $=
|
|
( c0 csn 0ex snnz necomi ) ABAACDE $.
|
|
|
|
$( Something cannot be equal to both the null set and the power set of the
|
|
null set. (Contributed by NM, 30-Sep-2003.) $)
|
|
0inp0 $p |- ( A = (/) -> -. A = { (/) } ) $=
|
|
( c0 wceq csn wne 0nep0 neeq1 mpbiri neneqd ) ABCZABDZJAKEBKEFABKGHI $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The removal of the empty set from a class does not affect its union.
|
|
(Contributed by NM, 22-Mar-2004.) $)
|
|
unidif0 $p |- U. ( A \ { (/) } ) = U. A $=
|
|
( vx vy c0 csn cdif cuni cv wcel wn wa wex n0i pm4.71i anbi1i exbii eluni
|
|
wceq anbi2i bitri an32 anass 3bitr2ri eldif velsn notbii 3bitr4i eqriv )
|
|
BADEZFZGZAGZBHZCHZIZUNAIZUNDRZJZKZKZCLZUOUPKZCLUMUKIZUMULIUTVBCVBUOURKZUP
|
|
KVBURKUTUOVDUPUOURUNUMMNOUOUPURUAUOUPURUBUCPVCUOUNUJIZKZCLVACUMUJQVFUTCVE
|
|
USUOVEUPUNUIIZJZKUSUNAUIUDVHURUPVGUQCDUEUFSTSPTCUMAQUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x A $.
|
|
$( An indexed intersection of the empty set, with an inhabited index set,
|
|
is empty. (Contributed by Jim Kingdon, 29-Aug-2018.) $)
|
|
iin0imm $p |- ( E. y y e. A -> |^|_ x e. A (/) = (/) ) $=
|
|
( c0 iinconstm ) ABCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( If an indexed intersection of the empty set is empty, the index set is
|
|
non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.) $)
|
|
iin0r $p |- ( |^|_ x e. A (/) = (/) -> A =/= (/) ) $=
|
|
( c0 ciin wceq cvv wcel wn 0ex n0i ax-mp 0iin eqeq1i iineq1 eqeq1d mtbiri
|
|
mtbir necon2ai ) ABCDZCEZBCBCEZTACCDZCEZUCFCEZCFGUDHIFCJKUBFCACLMQUASUBCA
|
|
BCCNOPR $.
|
|
$}
|
|
|
|
$( The intersection of the universal class is empty. (Contributed by NM,
|
|
11-Sep-2008.) $)
|
|
intv $p |- |^| _V = (/) $=
|
|
( c0 cvv wcel cint wceq 0ex int0el ax-mp ) ABCBDAEFBGH $.
|
|
|
|
${
|
|
$d x y z A $.
|
|
axpweq.1 $e |- A e. _V $.
|
|
$( Two equivalent ways to express the Power Set Axiom. Note that ~ ax-pow
|
|
is not used by the proof. (Contributed by NM, 22-Jun-2009.) $)
|
|
axpweq $p |- ( ~P A e. _V
|
|
<-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) ) $=
|
|
( cpw cvv wcel cv wex wel wi wal pwidg wceq pweq eleq2d spcegv wss bitri
|
|
mpd elex exlimiv impbii vex elpw2 pwss dfss2 imbi1i albii exbii ) DFZGHZU
|
|
LAIZFZHZAJZCBKCIDHLCMZBAKZLZBMZAJUMUQUMULULFZHZUQULGNUPVCAULGUNULOUOVBULU
|
|
NULPQRUAUPUMAULUOUBUCUDUPVAAUPULUNSZVAULUNAUEUFVDBIZDSZUSLZBMVABDUNUGVGUT
|
|
BVFURUSCVEDUHUIUJTTUKT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Collection principle
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d ph z w $. $d x y z w $.
|
|
$( A very strong generalization of the Axiom of Replacement (compare
|
|
~ zfrep6 ). Its strength lies in the rather profound fact that
|
|
` ph ( x , y ) ` does not have to be a "function-like" wff, as it does
|
|
in the standard Axiom of Replacement. This theorem is sometimes called
|
|
the Boundedness Axiom. In the context of IZF, it is just a slight
|
|
variation of ~ ax-coll . (Contributed by NM, 17-Oct-2004.) $)
|
|
bnd $p |- ( A. x e. z E. y ph -> E. w A. x e. z E. y e. w ph ) $=
|
|
( nfv ax-coll ) ABCDEAEFG $.
|
|
$}
|
|
|
|
${
|
|
$d ph z w v $. $d x z w v A $. $d x y z w v B $.
|
|
bnd2.1 $e |- A e. _V $.
|
|
$( A variant of the Boundedness Axiom ~ bnd that picks a subset ` z ` out
|
|
of a possibly proper class ` B ` in which a property is true.
|
|
(Contributed by NM, 4-Feb-2004.) $)
|
|
bnd2 $p |- ( A. x e. A E. y e. B ph ->
|
|
E. z ( z C_ B /\ A. x e. A E. y e. z ph ) ) $=
|
|
( vw vv wrex wral cv wcel wa wex wss df-rex wi wceq raleq bnd vtocl sylbi
|
|
ralbii exbidv imbi12d cin inex1 inss2 sseq1 mpbiri biantrurd rexeq anbi1i
|
|
vex elin anass bitri rexbii2 syl6bb ralbidv bitr3d spcev exlimiv syl ) AC
|
|
FJZBEKZCLZFMZANZCHLZJZBEKZHOZDLZFPZACVOJZBEKZNZDOZVGVJCOZBEKZVNVFWABEACFQ
|
|
UDWABILZKZVLBWCKZHOZRWBVNRIEGWCESZWDWBWFVNWABWCETWGWEVMHVLBWCETUEUFVJBCIH
|
|
UAUBUCVMVTHVSVMDVKFUGZVKFHUOUHVOWHSZVRVSVMWIVPVRWIVPWHFPVKFUIVOWHFUJUKULW
|
|
IVQVLBEWIVQACWHJVLACVOWHUMAVJCWHVKVHWHMZANVHVKMZVINZANWKVJNWJWLAVHVKFUPUN
|
|
WKVIAUQURUSUTVAVBVCVDVE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
IZF Set Theory - add the Axioms of Power Sets and Pairing
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Power Sets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set
|
|
theory. It states that a set ` y ` exists that includes the power set
|
|
of a given set ` x ` i.e. contains every subset of ` x ` . This is
|
|
Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary
|
|
quantifiers are removed, and (b) Crosilla has a biconditional rather
|
|
than an implication (but the two are equivalent by ~ bm1.3ii ).
|
|
|
|
The variant ~ axpow2 uses explicit subset notation. A version using
|
|
class notation is ~ pwex . (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-pow $a |- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y ) $.
|
|
|
|
$( Axiom of Power Sets expressed with the fewest number of different
|
|
variables. (Contributed by NM, 14-Aug-2003.) $)
|
|
zfpow $p |- E. x A. y ( A. x ( x e. y -> x e. z ) -> y e. x ) $=
|
|
( vw wel wal wex ax-pow weq elequ1 imbi12d cbvalv imbi1i albii exbii mpbi
|
|
wi ) DBEZDCEZQZDFZBAEZQZBFZAGABEZACEZQZAFZUBQZBFZAGCABDHUDUJAUCUIBUAUHUBT
|
|
UGDADAIRUESUFDABJDACJKLMNOP $.
|
|
|
|
$( A variant of the Axiom of Power Sets ~ ax-pow using subset notation.
|
|
Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) $)
|
|
axpow2 $p |- E. y A. z ( z C_ x -> z e. y ) $=
|
|
( vw cv wss wel wi wal wex ax-pow dfss2 imbi1i albii exbii mpbir ) CEZAEZ
|
|
FZCBGZHZCIZBJDCGDAGHDIZTHZCIZBJABCDKUBUEBUAUDCSUCTDQRLMNOP $.
|
|
|
|
$( A variant of the Axiom of Power Sets ~ ax-pow . For any set ` x ` ,
|
|
there exists a set ` y ` whose members are exactly the subsets of ` x `
|
|
i.e. the power set of ` x ` . Axiom Pow of [BellMachover] p. 466.
|
|
(Contributed by NM, 4-Jun-2006.) $)
|
|
axpow3 $p |- E. y A. z ( z C_ x <-> z e. y ) $=
|
|
( cv wss wel wb wal wex axpow2 bm1.3ii bicom albii exbii mpbir ) CDADEZCB
|
|
FZGZCHZBIQPGZCHZBIPBCABCJKSUABRTCPQLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Every set is an element of some other set. (Contributed by NM,
|
|
4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
el $p |- E. y x e. y $=
|
|
( vz wel wi wal zfpow weq ax-14 alrimiv ax-13 embantd spimv eximii ) BCDB
|
|
ADEZBFZCBDZEZCFABDZBBCAGRSCACAHZPQSTOBCABIJCABKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
zfpowcl.1 $e |- A e. _V $.
|
|
$( Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring]
|
|
p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 25-Jul-2011.) $)
|
|
pwex $p |- ~P A e. _V $=
|
|
( vz vy vx cv cpw cvv wcel wceq pweq eleq1d wss cab df-pw wex wal bm1.3ii
|
|
wb axpow2 abeq2 exbii mpbir issetri eqeltri vtocl ) CFZGZHIAGZHICABUGAJUH
|
|
UIHUGAKLUHDFZUGMZDNZHDUGOEULEFZULJZEPUJUMIUKSDQZEPUKEDCEDTRUNUOEUKDUMUAUB
|
|
UCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Power set axiom expressed in class notation, with the sethood
|
|
requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17.
|
|
(Contributed by NM, 30-Oct-2003.) $)
|
|
pwexg $p |- ( A e. V -> ~P A e. _V ) $=
|
|
( vx cv cpw cvv wcel wceq pweq eleq1d vex pwex vtoclg ) CDZEZFGAEZFGCABNA
|
|
HOPFNAIJNCKLM $.
|
|
|
|
$( Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
abssexg $p |- ( A e. V -> { x | ( x C_ A /\ ph ) } e. _V ) $=
|
|
( wcel cpw cvv cv wss wa pwexg df-pw eleq1i simpl ss2abi ssexg mpan sylbi
|
|
cab syl ) CDECFZGEZBHCIZAJZBSZGEZCDKUBUCBSZGEZUFUAUGGBCLMUEUGIUHUFUDUCBUC
|
|
ANOUEUGGPQRT $.
|
|
$}
|
|
|
|
$( A singleton whose element exists is a set. The ` A e. _V ` case of
|
|
Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power
|
|
Set, and Separation. Replacement is not needed. (Contributed by Jim
|
|
Kingdon, 1-Sep-2018.) $)
|
|
snexg $p |- ( A e. V -> { A } e. _V ) $=
|
|
( wcel cpw cvv csn pwexg wss snsspw ssexg mpan syl ) ABCADZECZAFZECZABGOMHN
|
|
PAIOMEJKL $.
|
|
|
|
${
|
|
snex.1 $e |- A e. _V $.
|
|
$( A singleton whose element exists is a set. (Contributed by NM,
|
|
7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) $)
|
|
snex $p |- { A } e. _V $=
|
|
( cvv wcel csn snexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( A singleton whose element is a proper class is a set. The ` -. A e. _V `
|
|
case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality,
|
|
Power Set, and Separation. Replacement is not needed. (Contributed by
|
|
Jim Kingdon, 1-Sep-2018.) $)
|
|
snexprc $p |- ( -. A e. _V -> { A } e. _V ) $=
|
|
( cvv wcel wn csn c0 wceq snprc biimpi 0ex syl6eqel ) ABCDZAEZFBLMFGAHIJK
|
|
$.
|
|
|
|
$( The power set of the empty set (the ordinal 1) is a set. (Contributed by
|
|
NM, 23-Dec-1993.) $)
|
|
p0ex $p |- { (/) } e. _V $=
|
|
( c0 cpw csn cvv pw0 0ex pwex eqeltrri ) ABACDEAFGH $.
|
|
|
|
$( ` { (/) , { (/) } } ` (the ordinal 2) is a set. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
pp0ex $p |- { (/) , { (/) } } e. _V $=
|
|
( c0 csn cpr cpw p0ex pwex pwpw0ss ssexi ) AABZCIDIEFGH $.
|
|
|
|
$( The ordinal number 3 is a set, proved without the Axiom of Union.
|
|
(Contributed by NM, 2-May-2009.) $)
|
|
ord3ex $p |- { (/) , { (/) } , { (/) , { (/) } } } e. _V $=
|
|
( csn cpr ctp cun cvv df-tp cpw pp0ex pwex pwprss ssexi snsspr2 unss2 ax-mp
|
|
c0 wss eqeltri ) OOAZORBZCSSAZDZEORSFUASRAZSBZDZUDSGSHIORJKTUCPUAUDPUBSLTUC
|
|
SMNKQ $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( At least two sets exist (or in terms of first-order logic, the universe
|
|
of discourse has two or more objects). This theorem asserts the
|
|
existence of two sets which do not equal each other; compare with
|
|
~ dtruex in which we are given a set ` y ` and go from there to a set
|
|
` x ` which is not equal to it. (Contributed by Jim Kingdon,
|
|
2-Sep-2018.) $)
|
|
dtruarb $p |- E. x E. y -. x = y $=
|
|
( vz wel wn wa wex weq el wal ax-nul eximii eeanv mpbir2an nelneq2 2eximi
|
|
sp cv ax-mp ) CADZCBDEZFZBGAGZABHEZBGAGUCTAGUABGCAIUACJUABBCKUACQLTUAABMN
|
|
UBUDABCRARBROPS $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A class is a subclass of the power class of its union. Exercise 6(b) of
|
|
[Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) $)
|
|
pwuni $p |- A C_ ~P U. A $=
|
|
( vx cuni cpw cv wcel wss elssuni vex elpw sylibr ssriv ) BAACZDZBEZAFOMG
|
|
ONFOAHOMBIJKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom of Pairing
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x z w $. $d y z w $.
|
|
$( The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p.
|
|
"Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed,
|
|
and (b) Crosilla has a biconditional rather than an implication (but the
|
|
two are equivalent by ~ bm1.3ii ). (Contributed by NM, 14-Nov-2006.) $)
|
|
ax-pr $a |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $.
|
|
|
|
$( Derive the abbreviated version of the Axiom of Pairing from ~ ax-pr .
|
|
(Contributed by NM, 14-Nov-2006.) $)
|
|
zfpair2 $p |- { x , y } e. _V $=
|
|
( vz vw cv cpr wceq wex wcel wo wal ax-pr bm1.3ii dfcleq vex bibi2i albii
|
|
wb elpr bitri exbii mpbir issetri ) CAEZBEZFZCEZUFGZCHDEZUGIZUIUDGUIUEGJZ
|
|
RZDKZCHUKCDABCDLMUHUMCUHUJUIUFIZRZDKUMDUGUFNUOULDUNUKUJUIUDUEDOSPQTUAUBUC
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x W $.
|
|
$( The Axiom of Pairing using class variables. Theorem 7.13 of [Quine]
|
|
p. 51, but restricted to classes which exist. For proper classes, see
|
|
~ prprc , ~ prprc1 , and ~ prprc2 . (Contributed by Jim Kingdon,
|
|
16-Sep-2018.) $)
|
|
prexg $p |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V ) $=
|
|
( vx vy wcel cpr cvv wi cv wceq preq2 eleq1d zfpair2 vtoclg preq1 vtocleg
|
|
syl5ib imp ) ACGBDGZABHZIGZUAUCJEACUAEKZBHZIGZUDALZUCUDFKZHZIGUFFBDUHBLUI
|
|
UEIUHBUDMNEFOPUGUEUBIUDABQNSRT $.
|
|
$}
|
|
|
|
$( A singleton of a set belongs to the power class of a class containing the
|
|
set. (Contributed by Alan Sare, 25-Aug-2011.) $)
|
|
snelpwi $p |- ( A e. B -> { A } e. ~P B ) $=
|
|
( wcel csn cpw wss snssi cvv wb elex snexg elpwg 3syl mpbird ) ABCZADZBECZP
|
|
BFZABGOAHCPHCQRIABJAHKPBHLMN $.
|
|
|
|
${
|
|
snelpw.1 $e |- A e. _V $.
|
|
$( A singleton of a set belongs to the power class of a class containing
|
|
the set. (Contributed by NM, 1-Apr-1998.) $)
|
|
snelpw $p |- ( A e. B <-> { A } e. ~P B ) $=
|
|
( wcel csn wss cpw snss snex elpw bitr4i ) ABDAEZBFLBGDABCHLBACIJK $.
|
|
$}
|
|
|
|
$( A pair of two sets belongs to the power class of a class containing those
|
|
two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) $)
|
|
prelpwi $p |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C ) $=
|
|
( wcel wa cpr cpw wss prssi cvv wb prexg elpwg syl mpbird ) ACDBCDEZABFZCGD
|
|
ZQCHZABCIPQJDRSKABCCLQCJMNO $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( A theorem similar to extensionality, requiring the existence of a
|
|
singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM,
|
|
10-Aug-1993.) $)
|
|
rext $p |- ( A. z ( x e. z -> y e. z ) -> x = y ) $=
|
|
( cv wcel wi wal csn wceq vsnid vex snex eleq2 imbi12d spcv velsn equcomi
|
|
mpi sylbi syl ) ADZCDZEZBDZUBEZFZCGZUDUAHZEZUAUDIZUGUAUHEZUIAJUFUKUIFCUHU
|
|
AAKLUBUHIUCUKUEUIUBUHUAMUBUHUDMNORUIUDUAIUJBUAPBAQST $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Classes are subclasses if and only if their power classes are
|
|
subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM,
|
|
13-Oct-1996.) $)
|
|
sspwb $p |- ( A C_ B <-> ~P A C_ ~P B ) $=
|
|
( vx wss cpw wcel sstr2 com12 vex elpw 3imtr4g ssrdv csn ssel snex bitr4i
|
|
cv snss 3imtr3g impbii ) ABDZAEZBEZDZUACUBUCUACQZADZUEBDZUEUBFUEUCFUFUAUG
|
|
UEABGHUEACIZJUEBUHJKLUDCABUDUEMZUBFZUIUCFZUEAFZUEBFZUBUCUINUJUIADULUIAUEU
|
|
HOZJUEAUHRPUKUIBDUMUIBUNJUEBUHRPSLT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( A class equals the union of its power class. Exercise 6(a) of
|
|
[Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened
|
|
by Alan Sare, 28-Dec-2008.) $)
|
|
unipw $p |- U. ~P A = A $=
|
|
( vx vy cpw cuni cv wcel wex eluni elelpwi exlimiv sylbi csn snid snelpwi
|
|
wa vex elunii sylancr impbii eqriv ) BADZEZABFZUCGZUDAGZUEUDCFZGUGUBGPZCH
|
|
UFCUDUBIUHUFCUDUGAJKLUFUDUDMZGUIUBGUEUDBQNUDAOUDUIUBRSTUA $.
|
|
$}
|
|
|
|
$( Membership of a power class. Exercise 10 of [Enderton] p. 26.
|
|
(Contributed by NM, 13-Jan-2007.) $)
|
|
pwel $p |- ( A e. B -> ~P A e. ~P ~P U. B ) $=
|
|
( wcel cpw cuni wss elssuni sspwb sylib cvv wb pwexg elpwg syl mpbird ) ABC
|
|
ZADZBEZDZDCZQSFZPARFUAABGARHIPQJCTUAKABLQSJMNO $.
|
|
|
|
$( A class is transitive iff its power class is transitive. (Contributed by
|
|
Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.) $)
|
|
pwtr $p |- ( Tr A <-> Tr ~P A ) $=
|
|
( cpw cuni wss wtr unipw sseq1i df-tr dftr4 3bitr4ri ) ABZCZKDAKDKEAELAKAFG
|
|
KHAIJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( An extensionality-like principle defining subclass in terms of subsets.
|
|
(Contributed by NM, 30-Jun-2004.) $)
|
|
ssextss $p |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) ) $=
|
|
( wss cpw cv wcel wi wal sspwb dfss2 vex elpw imbi12i albii 3bitri ) BCDB
|
|
EZCEZDAFZQGZSRGZHZAISBDZSCDZHZAIBCJAQRKUBUEATUCUAUDSBALZMSCUFMNOP $.
|
|
|
|
$( An extensionality-like principle that uses the subset instead of the
|
|
membership relation: two classes are equal iff they have the same
|
|
subsets. (Contributed by NM, 30-Jun-2004.) $)
|
|
ssext $p |- ( A = B <-> A. x ( x C_ A <-> x C_ B ) ) $=
|
|
( wss wa cv wi wal wceq wb ssextss anbi12i eqss albiim 3bitr4i ) BCDZCBDZ
|
|
EAFZBDZRCDZGAHZTSGAHZEBCISTJAHPUAQUBABCKACBKLBCMSTANO $.
|
|
|
|
$( Negation of subclass relationship. Compare ~ nssr . (Contributed by
|
|
Jim Kingdon, 17-Sep-2018.) $)
|
|
nssssr $p |- ( E. x ( x C_ A /\ -. x C_ B ) -> -. A C_ B ) $=
|
|
( cv wss wn wa wex wi wal exanaliim ssextss sylnibr ) ADZBEZNCEZFGAHOPIAJ
|
|
BCEOPAKABCLM $.
|
|
$}
|
|
|
|
$( Classes are equal if and only if their power classes are equal. Exercise
|
|
19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) $)
|
|
pweqb $p |- ( A = B <-> ~P A = ~P B ) $=
|
|
( wss wa cpw wceq sspwb anbi12i eqss 3bitr4i ) ABCZBACZDAEZBEZCZNMCZDABFMNF
|
|
KOLPABGBAGHABIMNIJ $.
|
|
|
|
${
|
|
$d x A $.
|
|
intid.1 $e |- A e. _V $.
|
|
$( The intersection of all sets to which a set belongs is the singleton of
|
|
that set. (Contributed by NM, 5-Jun-2009.) $)
|
|
intid $p |- |^| { x | A e. x } = { A } $=
|
|
( cv wcel cab cint csn cvv wss snex eleq2 intmin3 ax-mp wi elintab mpgbir
|
|
snid id snssi eqssi ) BADZEZAFGZBHZUEIEUDUEJBCKUCBUEEAUEIUBUEBLBCRMNBUDEZ
|
|
UEUDJUFUCUCOAUCABCPUCSQBUDTNUA $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
$( The abstraction of a wff with existential uniqueness exists.
|
|
(Contributed by NM, 25-Nov-1994.) $)
|
|
euabex $p |- ( E! x ph -> { x | ph } e. _V ) $=
|
|
( vy weu cab cv csn wceq wex cvv wcel euabsn2 vex snex eleq1 mpbiri sylbi
|
|
exlimiv ) ABDABEZCFZGZHZCISJKZABCLUBUCCUBUCUAJKTCMNSUAJOPRQ $.
|
|
$}
|
|
|
|
${
|
|
$d w x y $. $d w x z $. $d x y A $.
|
|
$( An inhabited class (even if proper) has an inhabited subset.
|
|
(Contributed by Jim Kingdon, 17-Sep-2018.) $)
|
|
mss $p |- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) ) $=
|
|
( vw cv wcel wex wss wa csn vex snss snex wceq sseq1 eleq2 exbidv anbi12d
|
|
snm spcev mpan2 sylbi exlimiv elequ1 cbvexv anbi2i exbii sylibr ) BFZDGZB
|
|
HAFZDIZEFZULGZEHZJZAHZUMCFULGZCHZJZAHUKURBUKUJKZDIZURUJDBLZMVCUNVBGZEHZUR
|
|
EUJVDTUQVCVFJAVBUJVDNULVBOZUMVCUPVFULVBDPVGUOVEEULVBUNQRSUAUBUCUDVAUQAUTU
|
|
PUMUSUOCECEAUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A w $. $d y z ph w $.
|
|
$( Restricted existence in a class (even if proper) implies restricted
|
|
existence in a subset. (Contributed by NM, 23-Aug-2003.) $)
|
|
exss $p |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) ) $=
|
|
( vz vw wrex cv wcel wa cab wex wss crab rabn0m df-rab eleq2i sylbi bitri
|
|
wsb exbii bitr3i csn snss ssab2 sstr2 simpr wceq equsb1 velsn sbbii mpbir
|
|
vex mpi jctil df-clab sban 3imtr4i elex2 syl sylib snex sseq1 rexeq spcev
|
|
anbi12d syl2anc exlimiv ) ABDGZEHZBHZDIZAJZBKZIZELZCHZDMZABVQGZJZCLZVIVJA
|
|
BDNZIZELVPABEDOWCVOEWBVNVJABDPQUAUBVOWAEVOVJUCZDMZABWDGZWAVOWDVNMZWEVJVNE
|
|
UMZUDWGVNDMWEABDUEWDVNDUFUNRVOFHABWDNZIFLZWFVOVJWIIZWJVLBETZABETZJZVKWDIZ
|
|
BETZWMJZVOWKWNWMWPWLWMUGWPVKVJUHZBETBEUIWOWRBEBVJUJUKULUOVOVMBETWNVMEBUPV
|
|
LABEUQSWKVJWOAJZBKZIZWQWIWTVJABWDPQXAWSBETWQWSEBUPWOABEUQSSURFVJWIUSUTABF
|
|
WDOVAVTWEWFJCWDVJWHVBVQWDUHVRWEVSWFVQWDDVCABVQWDVDVFVEVGVHR $.
|
|
$}
|
|
|
|
$( An ordered pair of sets is a set. (Contributed by Jim Kingdon,
|
|
11-Jan-2019.) $)
|
|
opexg $p |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V ) $=
|
|
( wcel wa cop csn cpr cvv dfopg elex snexg syl adantr prexg syl2anc eqeltrd
|
|
syl2an ) ACEZBDEZFZABGAHZABIZIZJABCDKUBUCJEZUDJEZUEJETUFUATAJEZUFACLZAJMNOT
|
|
UHBJEUGUAUIBDLABJJPSUCUDJJPQR $.
|
|
|
|
${
|
|
opex.1 $e |- A e. _V $.
|
|
opex.2 $e |- B e. _V $.
|
|
$( An ordered pair of sets is a set. (Contributed by Jim Kingdon,
|
|
24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.) $)
|
|
opex $p |- <. A , B >. e. _V $=
|
|
( cvv wcel cop opexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( An ordered triple of sets is a set. (Contributed by Jim Kingdon,
|
|
19-Sep-2018.) $)
|
|
otexg $p |- ( ( A e. U /\ B e. V /\ C e. W ) -> <. A , B , C >. e. _V ) $=
|
|
( wcel cotp cvv wa cop df-ot opexg sylan syl5eqel 3impa ) ADGZBEGZCFGZABCHZ
|
|
IGQRJZSJTABKZCKZIABCLUAUBIGSUCIGABDEMUBCIFMNOP $.
|
|
|
|
${
|
|
elop.1 $e |- A e. _V $.
|
|
elop.2 $e |- B e. _V $.
|
|
elop.3 $e |- C e. _V $.
|
|
$( An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
elop $p |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) ) $=
|
|
( cop wcel csn cpr wceq wo dfop eleq2i elpr bitri ) ABCGZHABIZBCJZJZHARKA
|
|
SKLQTABCEFMNARSDOP $.
|
|
$}
|
|
|
|
${
|
|
opi1.1 $e |- A e. _V $.
|
|
opi1.2 $e |- B e. _V $.
|
|
$( One of the two elements in an ordered pair. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opi1 $p |- { A } e. <. A , B >. $=
|
|
( csn cpr cop snex prid1 dfop eleqtrri ) AEZLABFZFABGLMACHIABCDJK $.
|
|
|
|
$( One of the two elements of an ordered pair. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opi2 $p |- { A , B } e. <. A , B >. $=
|
|
( cpr csn cop cvv wcel prexg mp2an prid2 dfop eleqtrri ) ABEZAFZOEABGPOAH
|
|
IBHIOHICDABHHJKLABCDMN $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered pair theorem
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
|
|
$( An ordered pair is inhabited iff the arguments are sets. (Contributed
|
|
by Jim Kingdon, 21-Sep-2018.) $)
|
|
opm $p |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) ) $=
|
|
( cv cop wcel wex cvv csn cpr w3a wa df-op eleq2i exbii abid bitri 19.42v
|
|
cab df-3an 3bitr4ri 3simpa snexg adantr prmg syl sylanbrc impbii 3bitr2i
|
|
id ) ADZBCEZFZAGZBHFZCHFZUKBIZBCJZJFZKZAGZUOUPUSAGZKZUOUPLZUNUKUTASZFZAGV
|
|
AUMVFAULVEUKABCMNOVFUTAUTAPOQVDUSLZAGVDVBLVAVCVDUSARUTVGAUOUPUSTOUOUPVBTZ
|
|
UAVCVDUOUPVBUBVDVDVBVCVDUJVDUQHFZVBUOVIUPBHUCUDAUQURHUEUFVHUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
opth1.1 $e |- A e. _V $.
|
|
opth1.2 $e |- B e. _V $.
|
|
$( An ordered pair is nonempty if the arguments are sets (it is also
|
|
inhabited; see ~ opm ). (Contributed by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
opnzi $p |- <. A , B >. =/= (/) $=
|
|
( vx cv cop wcel wex c0 wne cvv opm mpbir2an n0r ax-mp ) EFABGZHEIZQJKRAL
|
|
HBLHCDEABMNEQOP $.
|
|
|
|
$( Equality of the first members of equal ordered pairs. (Contributed by
|
|
NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opth1 $p |- ( <. A , B >. = <. C , D >. -> A = C ) $=
|
|
( cop wceq csn cpr wi sneqr a1i wcel cvv wa opi1 id syl5eleq syl oprcl wo
|
|
simpld prid1g eleq2 syl5ibrcom elsni eqcomd syl6 dfopg 3syl eleqtrd elpri
|
|
mpjaod ) ABGZCDGZHZAIZCIZHZACHZURCDJZHZUTVAKUQACELMUQVCCURNZVAUQVDVCCVBNZ
|
|
UQCONZVEUQVFDONZUQURUPNZVFVGPZUQURUOUPABEFQUQRSZCDURUAZTUCCDOUDTURVBCUEUF
|
|
VDCACAUGUHUIUQURUSVBJZNUTVCUBUQURUPVLVJUQVHVIUPVLHVJVKCDOOUJUKULURUSVBUMT
|
|
UN $.
|
|
|
|
$( The ordered pair theorem. If two ordered pairs are equal, their first
|
|
elements are equal and their second elements are equal. Exercise 6 of
|
|
[TakeutiZaring] p. 16. Note that ` C ` and ` D ` are not required to be
|
|
sets due our specific ordered pair definition. (Contributed by NM,
|
|
28-May-1995.) $)
|
|
opth $p |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) $=
|
|
( vx cop wceq wa cvv wcel cpr csn syl eqtr3d dfopg sylancl wi prexg opth1
|
|
opi1 id syl5eleq oprcl simprd opeq1d simpld preqr2g syl2anc mpd cv eqeq2d
|
|
preq2 eqeq2 imbi12d vex preqr2 vtoclg sylc jca opeq12 impbii ) ABHZCDHZIZ
|
|
ACIZBDIZJVFVGVHABCDEFUAZVFDKLZCBMZCDMZIZVHVFCKLZVJVFANZVELVNVJJZVFVOVDVEA
|
|
BEFUBVFUCZUDCDVOUEOZUFVFCNZVKMZVSVLMZIZVMVFVEVTWAVFCBHZVEVTVFVDWCVEVFACBV
|
|
IUGVQPVFVNBKLZWCVTIVFVNVJVRUHZFCBKKQRPVFVPVEWAIVRCDKKQOPVFVKKLZVLKLZWBVMS
|
|
VFVNWDWFWEFCBKKTRVFVPWGVRCDKKTOVKVLVSUIUJUKVKCGULZMZIZBWHIZSVMVHSGDKWHDIZ
|
|
WJVMWKVHWLWIVLVKWHDCUNUMWHDBUOUPBWHCFGUQURUSUTVAABCDVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $.
|
|
$( Ordered pair theorem. ` C ` and ` D ` are not required to be sets under
|
|
our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opthg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) $=
|
|
( vx vy cv cop wceq wa wb opeq1 eqeq1d eqeq1 anbi1d bibi12d opeq2 vex
|
|
anbi2d opth vtocl2g ) GIZHIZJZCDJZKZUDCKZUEDKZLZMAUEJZUGKZACKZUJLZMABJZUG
|
|
KZUNBDKZLZMGHABEFUDAKZUHUMUKUOUTUFULUGUDAUENOUTUIUNUJUDACPQRUEBKZUMUQUOUS
|
|
VAULUPUGUEBASOVAUJURUNUEBDPUARUDUECDGTHTUBUC $.
|
|
$}
|
|
|
|
$( Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
opthg2 $p |- ( ( C e. V /\ D e. W ) ->
|
|
( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) ) $=
|
|
( wcel wa cop wceq opthg eqcom anbi12i 3bitr4g ) CEGDFGHCDIZABIZJCAJZDBJZHP
|
|
OJACJZBDJZHCDABEFKPOLSQTRACLBDLMN $.
|
|
|
|
${
|
|
opth2.1 $e |- C e. _V $.
|
|
opth2.2 $e |- D e. _V $.
|
|
$( Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) $)
|
|
opth2 $p |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) $=
|
|
( cvv wcel cop wceq wa wb opthg2 mp2an ) CGHDGHABICDIJACJBDJKLEFABCDGGMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
otth.1 $e |- A e. _V $.
|
|
otth.2 $e |- B e. _V $.
|
|
otth.3 $e |- R e. _V $.
|
|
$( Ordered triple theorem, with triple express with ordered pairs.
|
|
(Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
otth2 $p |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <->
|
|
( A = C /\ B = D /\ R = S ) ) $=
|
|
( cop wceq wa w3a opth anbi1i opex df-3an 3bitr4i ) ABJZCDJZKZEFKZLACKZBD
|
|
KZLZUBLSEJTFJKUCUDUBMUAUEUBABCDGHNOSETFABGHPINUCUDUBQR $.
|
|
|
|
$( Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
otth $p |- ( <. A , B , R >. = <. C , D , S >. <->
|
|
( A = C /\ B = D /\ R = S ) ) $=
|
|
( cotp wceq cop w3a df-ot eqeq12i otth2 bitri ) ABEJZCDFJZKABLELZCDLFLZKA
|
|
CKBDKEFKMRTSUAABENCDFNOABCDEFGHIPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
eqvinop.1 $e |- B e. _V $.
|
|
eqvinop.2 $e |- C e. _V $.
|
|
$( A variable introduction law for ordered pairs. Analog of Lemma 15 of
|
|
[Monk2] p. 109. (Contributed by NM, 28-May-1995.) $)
|
|
eqvinop $p |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\
|
|
<. x , y >. = <. B , C >. ) ) $=
|
|
( cv cop wceq wa wex opth2 anbi2i ancom anass 3bitri exbii eqeq2d ceqsexv
|
|
19.42v opeq2 opeq1 bitr2i ) CAHZBHZIZJZUGDEIZJZKZBLZALUEDJZCUEEIZJZKZALCU
|
|
IJZULUPAULUMUFEJZUHKZKZBLUMUSBLZKUPUKUTBUKUHUMURKZKVBUHKUTUJVBUHUEUFDEFGM
|
|
NUHVBOUMURUHPQRUMUSBUAVAUOUMUHUOBEGURUGUNCUFEUEUBSTNQRUOUQADFUMUNUICUEDEU
|
|
CSTUD $.
|
|
$}
|
|
|
|
${
|
|
$d x z w A $. $d y z w A $. $d z w ph $.
|
|
$( Substitution of class ` A ` for ordered pair ` <. x , y >. ` .
|
|
(Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
copsexg $p |- ( A = <. x , y >. ->
|
|
( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) $=
|
|
( vz vw cv cop wceq wa wex wb wi vex 19.8a weq syl5bi syl5 weu euequ1 wal
|
|
eqvinop 19.23bi ex opth anbi1i 2exbii nfe1 wnf wo dveeq2or nfae anass a1i
|
|
anim2d eximd biidd drex1 sylibd exbii 19.40 19.9t biimpd anim1d syl6 jaoi
|
|
ax-mp exlimi equcom eubii eupick com12 sylan9 sylbi impbid anbi1d 2exbidv
|
|
mpbi mpan eqeq1 bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZCGZHZ
|
|
IZAWJAJZCKBKZLZWJDEGZFGZHZIZWPWIIZJZFKEKWJWMMZEFDWGWHBNCNUBWSWTEFWQWTWRWQ
|
|
WTWRAWRAJZCKZBKZLZMWRAXCWRAXCXAXCCXBBOUCUDWREBPZFCPZJZXCAMWNWOWGWHENFNUEZ
|
|
XCXGAJZCKZBKZXGAXAXIBCWRXGAXHUFUGXKXEXFAJZCKZJZBKZXGAXJXOBXNBUHCBPCUAZXEC
|
|
UIZUJXJXOMZCBEUKXPXRXQXPXJXNCKXOXPXIXNCCBCULXIXEXLJZXPXNXEXFAUMZXPXLXMXEX
|
|
LXMMXPXLCOUNUOQUPXNXNCBXPXNUQURUSXQXJXNXOXJXSCKZXQXNXIXSCXTUTYAXECKZXMJXQ
|
|
XNXEXLCVAXQYBXEXMXQYBXEXECVBVCVDRQXNBOVEVFVGVHXEXOXMXFAXOXEXMXEBSZXOXEXMM
|
|
BEPZBSYCBETYDXEBBEVIVJVRXEXMBVKVSVLXMXFAXFCSZXMXFAMCFPZCSYECFTYFXFCCFVIVJ
|
|
VRXFACVKVSVLVMRQVNVOWQWJWRWMXDDWPWIVTZWQWLXCAWQWKXABCWQWJWRAYGVPVQWAWBWCW
|
|
DWEVNWF $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
$( Closed theorem form of ~ copsex2g . (Contributed by NM,
|
|
17-Feb-2013.) $)
|
|
copsex2t $p |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
|
|
/\ ( A e. V /\ B e. W ) ) ->
|
|
( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
|
|
( wcel wa cv wceq wb wal wex cop elisset nfe1 nfv nfbi anim12i eeanv nfa1
|
|
wi sylibr nfa2 nfex opeq12 copsexg eqcoms syl adantl sp 19.21bi bitr3d ex
|
|
imp exlimd sylan2 ) EGIZFHIZJZCKZELZDKZFLZJZABMZUDZDNZCNZVGDOZCOZEFPZVCVE
|
|
PZLAJZDOZCOZBMZVBVDCOZVFDOZJVMUTVTVAWACEGQDFHQUAVDVFCDUBUEVKVMVSVKVLVSCVJ
|
|
CUCVRBCVQCRBCSTVKVGVSDVIDCUFVRBDVQDCVPDRUGBDSTVKVGVSVKVGJAVRBVGAVRMZVKVGV
|
|
OVNLWBVCVEEFUHWBVNVOACDVNUIUJUKULVKVGVHVKVIDVJCUMUNUQUOUPURURUQUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
copsex2g.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for ordered pairs. (Contributed by NM,
|
|
28-May-1995.) $)
|
|
copsex2g $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
|
|
( wcel cv wceq wex cop wa wb elisset nfe1 nfv nfbi eeanv nfex copsexg syl
|
|
opeq12 eqcoms bitr3d exlimi sylbir syl2an ) EGJCKZELZCMZDKZFLZDMZEFNZUKUN
|
|
NZLAOZDMZCMZBPZFHJCEGQDFHQUMUPOULUOOZDMZCMVBULUOCDUAVDVBCVABCUTCRBCSTVCVB
|
|
DVABDUTDCUSDRUBBDSTVCAVABVCURUQLAVAPZUKUNEFUEVEUQURACDUQUCUFUDIUGUHUHUIUJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y z w ps $. $d x y z w R $. $d x y z w S $.
|
|
copsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) ->
|
|
( ph <-> ps ) ) $.
|
|
$( An implicit substitution inference for 2 ordered pairs. (Contributed by
|
|
NM, 5-Aug-1995.) $)
|
|
copsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
|
|
( E. x E. y E. z E. w ( ( <. A , B >. = <. x , y >. /\
|
|
<. C , D >. = <. z , w >. ) /\ ph ) <-> ps ) ) $=
|
|
( wcel wa cop cv wceq wex vex eqcom opth bitri anbi12i anbi1i a1i 4exbidv
|
|
wb id cgsex4g bitrd ) GKNHLNOIKNJLNOOZGHPZCQZDQZPZRZIJPZEQZFQZPZRZOZAOZFS
|
|
ESDSCSUNGRUOHROZUSIRUTJROZOZAOZFSESDSCSBULVDVHCDEFVDVHUHULVCVGAUQVEVBVFUQ
|
|
UPUMRVEUMUPUAUNUOGHCTDTUBUCVBVAURRVFURVAUAUSUTIJETFTUBUCUDUEUFUGABVGCDEFG
|
|
HIJKLVGUIMUJUK $.
|
|
$}
|
|
|
|
$( A property of ordered pairs. (Contributed by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
0nelop $p |- -. (/) e. <. A , B >. $=
|
|
( c0 cop wcel csn cpr wo id cvv wa oprcl dfopg syl eleqtrd elpri wne necomd
|
|
wceq wn simpld snnzg prnzg jca neanior sylib pm2.65i ) CABDZEZCAFZSCABGZSHZ
|
|
UICUJUKGZEULUICUHUMUIIUIAJEZBJEZKUHUMSABCLZABJJMNOCUJUKPNUICUJQZCUKQZKULTUI
|
|
UQURUIUJCUIUNUJCQUIUNUOUPUAZAJUBNRUIUKCUIUNUKCQUSABJUCNRUDCUJCUKUEUFUG $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d D x $.
|
|
|
|
$( Equivalence of existence implied by equality of ordered pairs.
|
|
(Contributed by NM, 28-May-2008.) $)
|
|
opeqex $p |- ( <. A , B >. = <. C , D >. ->
|
|
( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) ) $=
|
|
( vx cop wceq cv wcel wex cvv wa eleq2 exbidv opm 3bitr3g ) ABFZCDFZGZEHZ
|
|
QIZEJTRIZEJAKIBKILCKIDKILSUAUBEQRTMNEABOECDOP $.
|
|
$}
|
|
|
|
${
|
|
opcom.1 $e |- A e. _V $.
|
|
opcom.2 $e |- B e. _V $.
|
|
$( An ordered pair commutes iff its members are equal. (Contributed by NM,
|
|
28-May-2009.) $)
|
|
opcom $p |- ( <. A , B >. = <. B , A >. <-> A = B ) $=
|
|
( cop wceq wa opth eqcom anbi2i anidm 3bitri ) ABEBAEFABFZBAFZGMMGMABBACD
|
|
HNMMBAIJMKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
moop2.1 $e |- B e. _V $.
|
|
$( "At most one" property of an ordered pair. (Contributed by NM,
|
|
11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
moop2 $p |- E* x A = <. B , x >. $=
|
|
( vy cv cop wceq wmo csb wa wal eqtr2 vex opth simprbi syl gen2 nfcsb1v
|
|
wi nfcv nfop nfeq2 csbeq1a id opeq12d eqeq2d mo4f mpbir ) BCAFZGZHZAIULBA
|
|
EFZCJZUMGZHZKZUJUMHZTZELALUSAEUQUKUOHZURBUKUOMUTCUNHURCUJUNUMDANOPQRULUPA
|
|
EABUOAUNUMAUMCSAUMUAUBUCURUKUOBURCUNUJUMAUMCUDURUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
opeqsn.1 $e |- A e. _V $.
|
|
opeqsn.2 $e |- B e. _V $.
|
|
opeqsn.3 $e |- C e. _V $.
|
|
$( Equivalence for an ordered pair equal to a singleton. (Contributed by
|
|
NM, 3-Jun-2008.) $)
|
|
opeqsn $p |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) ) $=
|
|
( cop csn wceq cpr wa dfop eqeq1i snex cvv wcel preqsn eqcom bitri 3bitri
|
|
prexg mp2an anbi2i anidm anbi1i dfsn2 preq2 syl5req eqeq1d syl6bb pm5.32i
|
|
) ABGZCHZIAHZABJZJZUMIUNUOIZUOCIZKZABIZCUNIZKZULUPUMABDELMUNUOCADNAOPBOPU
|
|
OOPDEABOOUAUBFQUSUTURKVBUQUTURUQUOUNIUTBAIZKZUTUNUORABADEDQVDUTUTKUTVCUTU
|
|
TBARUCUTUDSTUEUTURVAUTURUNCIVAUTUOUNCUTUNAAJUOAUFABAUGUHUIUNCRUJUKST $.
|
|
$}
|
|
|
|
${
|
|
opeqpr.1 $e |- A e. _V $.
|
|
opeqpr.2 $e |- B e. _V $.
|
|
opeqpr.3 $e |- C e. _V $.
|
|
opeqpr.4 $e |- D e. _V $.
|
|
$( Equivalence for an ordered pair equal to an unordered pair.
|
|
(Contributed by NM, 3-Jun-2008.) $)
|
|
opeqpr $p |- ( <. A , B >. = { C , D }
|
|
<-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) ) $=
|
|
( cop cpr wceq csn wa wo eqcom dfop eqeq2i snex cvv wcel preq12b 3bitri
|
|
prexg mp2an ) ABIZCDJZKUFUEKUFALZABJZJZKCUGKDUHKMCUHKDUGKMNUEUFOUEUIUFABE
|
|
FPQCDUGUHGHAERASTBSTUHSTEFABSSUCUDUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d a b c x y A $. $d a b c x y B $. $d a b c x y C $. $d a b c x ph $.
|
|
$d y ps $.
|
|
euotd.1 $e |- ( ph -> A e. _V ) $.
|
|
euotd.2 $e |- ( ph -> B e. _V ) $.
|
|
euotd.3 $e |- ( ph -> C e. _V ) $.
|
|
euotd.4 $e |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) ) $.
|
|
$( Prove existential uniqueness for an ordered triple. (Contributed by
|
|
Mario Carneiro, 20-May-2015.) $)
|
|
euotd $p |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) $=
|
|
( vy cv wceq wa wex cvv wtru cotp wb wal weu otexg syl3anc w3a biimpa vex
|
|
wcel otth sylibr eqeq2d biimpd impancom expimpd exlimdv exlimdvv wsbc tru
|
|
adantr ad2antrr simpr eqcomd biimpar jca a1tru 2thd sbcied mpbiri spesbcd
|
|
3anassrs nfcv nfsbc1v nfex sbceq1a exbidv spcegf sylc 2exbidv sylib eqeq1
|
|
excom13 anbi1d 3exbidv syl5ibrcom impbid eqeq2 bibi2d albidv spcegv df-eu
|
|
alrimiv ) ACOZGOZHOZIOZUAZPZBQZIRZHRGRZWNNOZPZUBZCUCZNRZXBCUDADEFUAZSUJZX
|
|
BWNXHPZUBZCUCZXGADSUJESUJZFSUJZXIJKLDEFSSSUEUFAXKCAXBXJAXAXJGHAWTXJIAWSBX
|
|
JABWSXJABQZWSXJXOWRXHWNXOWODPZWPEPZWQFPZUGZWRXHPZABXSMUHWOWPDEWQFGUIHUIIU
|
|
IUKZULUMUNUOUPUQURAXBXJXHWRPZBQZIRHRGRZAYCGRHRZIRZYDAXNYCIFUSZGRZHRZYFLAX
|
|
MYGHEUSZGRZYIKAYJGDAYJGDUSTUTAYJTGDSJAXPQZYGTHESAXMXPKVAYLXQQYCTIFSAXNXPX
|
|
QLVBAXPXQXRYCTUBAXSQZYCTYMYBBYMWRXHYMXSXTAXSVCYAULVDABXSMVEVFYMVGVHVLVIVI
|
|
VIVJVKYHYKHESHEVMYJHGYGHEVNVOXQYGYJGYGHEVPVQVRVSYEYIIFSIFVMYHIHYGIGYCIFVN
|
|
VOVOXRYCYGHGYCIFVPVTVRVSYCIHGWCWAXJWTYCGHIXJWSYBBWNXHWRWBWDWEWFWGWMXFXLNX
|
|
HSXCXHPZXEXKCYNXDXJXBXCXHWNWHWIWJWKVSXBCNWLUL $.
|
|
$}
|
|
|
|
${
|
|
opthw.1 $e |- A e. _V $.
|
|
opthw.2 $e |- B e. _V $.
|
|
$( The union of an ordered pair. Theorem 65 of [Suppes] p. 39.
|
|
(Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
uniop $p |- U. <. A , B >. = { A , B } $=
|
|
( cop cuni csn cpr cun dfop unieqi snex cvv prexg mp2an unipr wss snsspr1
|
|
wcel wceq ssequn1 mpbi 3eqtri ) ABEZFAGZABHZHZFUEUFIZUFUDUGABCDJKUEUFACLA
|
|
MSBMSUFMSCDABMMNOPUEUFQUHUFTABRUEUFUAUBUC $.
|
|
|
|
$( Ordered pair membership is inherited by class union. (Contributed by
|
|
NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
uniopel $p |- ( <. A , B >. e. C -> U. <. A , B >. e. U. C ) $=
|
|
( cop wcel cuni cpr uniop opi2 eqeltri elssuni sseld mpi ) ABFZCGZPHZPGRC
|
|
HZGRABIPABDEJABDEKLQPSRPCMNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered-pair class abstractions (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $.
|
|
$( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.
|
|
(Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon,
|
|
25-Jul-2011.) $)
|
|
opabid $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
|
|
( vz cv cop wceq wa wex copab vex opex copsexg bicomd df-opab elab2 ) DEZ
|
|
BEZCEZFZGZAHCIBIZADTABCJRSBKCKLUAAUBABCQMNABCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $.
|
|
$( Membership in a class abstraction of pairs. (Contributed by NM,
|
|
24-Mar-1998.) $)
|
|
elopab $p |- ( A e. { <. x , y >. | ph } <->
|
|
E. x E. y ( A = <. x , y >. /\ ph ) ) $=
|
|
( vz copab wcel cvv cv cop wceq wex elex vex eleq1 mpbiri adantr exlimivv
|
|
wa opex eqeq1 anbi1d 2exbidv df-opab elab2g pm5.21nii ) DABCFZGDHGZDBIZCI
|
|
ZJZKZASZCLBLZDUGMUMUHBCULUHAULUHUKHGUIUJBNCNTDUKHOPQREIZUKKZASZCLBLUNEDUG
|
|
HUODKZUQUMBCURUPULAUODUKUAUBUCABCEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d x y w $.
|
|
$( The law of concretion in terms of substitutions. Less general than
|
|
~ opelopabsb , but having a much shorter proof. (Contributed by NM,
|
|
30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
opelopabsbALT $p |- ( <. z , w >. e. { <. x , y >. | ph }
|
|
<-> [ w / y ] [ z / x ] ph ) $=
|
|
( cv cop wceq wa wex copab wcel wsb excom vex opth equcom anbi12ci anbi1i
|
|
bitri 2exbii elopab 2sb5 3bitr4i ) DFZEFZGZBFZCFZGHZAIZCJBJZUIUFHZUHUEHZI
|
|
ZAIZBJCJZUGABCKLABDMCEMULUKBJCJUQUKBCNUKUPCBUJUOAUJUEUHHZUFUIHZIUOUEUFUHU
|
|
IDOEOPURUNUSUMDBQECQRTSUATABCUGUBACBEDUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d u v x y z w $. $d u v w z A $. $d u v w x B $. $d u v w z ph $.
|
|
$( The law of concretion in terms of substitutions. (Contributed by NM,
|
|
30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) $)
|
|
opelopabsb $p |- ( <. A , B >. e. { <. x , y >. | ph }
|
|
<-> [. A / x ]. [. B / y ]. ph ) $=
|
|
( vv vu vz vw cop wcel cvv wa wsb cv wceq wex nfs1v weq wb copab wsbc vex
|
|
elopab simpl eqcomd opth sylib 2eximi eeanv isset bitr4i sylbi nfv nfsbxy
|
|
anbi12i sbequ12 sylan9bbr cbvopab eleq2s sbcex spesbc exlimiv syl dfsbcq2
|
|
jca opeq1 eleq1d bibi12d opeq2 sbcbidv nfopab1 nfel2 nfopab2 opabid chvar
|
|
nfbi vtocl2g pm5.21nii ) DEJZABCUAZKZDLKZELKZMZACEUBZBDUBZWEVTACFNZBGNZGF
|
|
UAZWAVTWJKVTGOZFOZJZPZWIMZFQGQZWEWIGFVTUDWPWKDPZWLEPZMZFQGQZWEWOWSGFWOWMV
|
|
TPWSWOVTWMWNWIUEUFWKWLDEGUCFUCUGUHUIWTWQGQZWRFQZMWEWQWRGFUJWCXAWDXBGDUKFE
|
|
UKUPULUHUMAWIBCGFAGUNAFUNWHBGRWHBGCACFRUOCFSAWHBGSWIACFUQWHBGUQURUSUTWGWC
|
|
WDWFBDVAWGWFBQWDWFBDVBWFWDBACEVAVCVDVFHOZIOZJZWAKZACINZBHNZTZDXDJZWAKZXGB
|
|
DUBZTWBWGTHIDELLXCDPZXFXKXHXLXMXEXJWAXCDXDVGVHXGBHDVEVIXDEPZXKWBXLWGXNXJV
|
|
TWAXDEDVJVHXNXGWFBDACIEVEVKVIBOZXDJZWAKZXGTZXIBHXFXHBBXEWAABCVLVMXGBHRVQB
|
|
HSZXQXFXGXHXSXPXEWAXOXCXDVGVHXGBHUQVIXOCOZJZWAKZATXRCIXQXGCCXPWAABCVNVMAC
|
|
IRVQCISZYBXQAXGYCYAXPWAXTXDXOVJVHACIUQVIABCVOVPVPVRVS $.
|
|
|
|
brabsb.1 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion in terms of substitutions. (Contributed by NM,
|
|
17-Mar-2008.) $)
|
|
brabsb $p |- ( A R B <-> [. A / x ]. [. B / y ]. ph ) $=
|
|
( wbr cop wcel copab wsbc df-br eleq2i opelopabsb 3bitri ) DEFHDEIZFJQABC
|
|
KZJACELBDLDEFMFRQGNABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
$( Closed theorem form of ~ opelopab . (Contributed by NM,
|
|
19-Feb-2013.) $)
|
|
opelopabt $p |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) )
|
|
/\ A. x A. y ( y = B -> ( ps <-> ch ) )
|
|
/\ ( A e. V /\ B e. W ) ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
|
|
( cop copab wcel cv wceq wa wex wb wi wal w3a elopab prth 2alimi copsex2t
|
|
19.26-2 bitr syl6 sylbir sylan 3impa syl5bb ) FGJZADEKLULDMZEMZJNAOEPDPZU
|
|
MFNZABQZRZESDSZUNGNZBCQZRZESDSZFHLGILOZTCADEULUAUSVCVDUOCQZUSVCOZUPUTOZAC
|
|
QZRZESDSZVDVEVFURVBOZESDSVJURVBDEUEVKVIDEVKVGUQVAOVHUPUQUTVAUBABCUFUGUCUH
|
|
ACDEFGHIUDUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
opelopabga.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopabga $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ps ) ) $=
|
|
( cop copab wcel cv wceq wa wex elopab copsex2g syl5bb ) EFJZACDKLTCMDMJN
|
|
AODPCPEGLFHLOBACDTQABCDEFGHIRS $.
|
|
|
|
${
|
|
brabga.2 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by Mario
|
|
Carneiro, 19-Dec-2013.) $)
|
|
brabga $p |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) ) $=
|
|
( wbr cop copab wcel wa df-br eleq2i bitri opelopabga syl5bb ) EFGLZEFM
|
|
ZACDNZOZEHOFIOPBUBUCGOUEEFGQGUDUCKRSABCDEFHIJTUA $.
|
|
$}
|
|
|
|
$d x y C $. $d x y D $.
|
|
$( Ordered pair membership in an ordered pair class abstraction.
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopab2a $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
|
|
{ <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) ) $=
|
|
( wcel wa cop cv copab wceq eleq1 bi2anan9 anbi12d opelopabga bianabs ) E
|
|
GJZFHJZKZEFLCMZGJZDMZHJZKZAKZCDNJBUIUCBKCDEFGHUDEOZUFFOZKUHUCABUJUEUAUKUG
|
|
UBUDEGPUFFHPQIRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
opelopaba.1 $e |- A e. _V $.
|
|
opelopaba.2 $e |- B e. _V $.
|
|
opelopaba.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopaba $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
|
|
( cvv wcel cop copab wb opelopabga mp2an ) EJKFJKEFLACDMKBNGHABCDEFJJIOP
|
|
$.
|
|
|
|
${
|
|
braba.4 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
19-Dec-2013.) $)
|
|
braba $p |- ( A R B <-> ps ) $=
|
|
( cvv wcel wbr wb brabga mp2an ) ELMFLMEFGNBOHIABCDEFGLLJKPQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
opelopabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopabg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
|
|
( cv wceq sylan9bb opelopabga ) ACDEFGHIDLFMABELGMCJKNO $.
|
|
|
|
${
|
|
brabg.5 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
brabg $p |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) ) $=
|
|
( cv wceq sylan9bb brabga ) ACDEFGJHIDNFOABENGOCKLPMQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y ch $.
|
|
opelopab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopab2.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Ordered pair membership in an ordered pair class abstraction.
|
|
(Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro,
|
|
19-Dec-2013.) $)
|
|
opelopab2 $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
|
|
{ <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) ) $=
|
|
( cv wceq sylan9bb opelopab2a ) ACDEFGHIDLFMABELGMCJKNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
opelopab.1 $e |- A e. _V $.
|
|
opelopab.2 $e |- B e. _V $.
|
|
opelopab.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopab.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
NM, 16-May-1995.) $)
|
|
opelopab $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
|
|
( cvv wcel cop copab wb opelopabg mp2an ) FLMGLMFGNADEOMCPHIABCDEFGLLJKQR
|
|
$.
|
|
|
|
${
|
|
brab.5 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
16-Aug-1999.) $)
|
|
brab $p |- ( A R B <-> ch ) $=
|
|
( cvv wcel wbr wb brabg mp2an ) FNOGNOFGHPCQIJABCDEFGNNHKLMRS $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
opelopabaf.x $e |- F/ x ps $.
|
|
opelopabaf.y $e |- F/ y ps $.
|
|
opelopabaf.1 $e |- A e. _V $.
|
|
opelopabaf.2 $e |- B e. _V $.
|
|
opelopabaf.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of
|
|
~ opelopab uses bound-variable hypotheses in place of distinct variable
|
|
conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof
|
|
shortened by Mario Carneiro, 18-Nov-2016.) $)
|
|
opelopabaf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
|
|
( cop copab wcel wsbc opelopabsb cvv wb nfv sbc2iegf mp2an bitri ) EFLACD
|
|
MNADFOCEOZBACDEFPEQNFQNZUCBRIJABCDEFQQGHUDCSKTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
opelopabf.x $e |- F/ x ps $.
|
|
opelopabf.y $e |- F/ y ch $.
|
|
opelopabf.1 $e |- A e. _V $.
|
|
opelopabf.2 $e |- B e. _V $.
|
|
opelopabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopabf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of
|
|
~ opelopab uses bound-variable hypotheses in place of distinct variable
|
|
conditions." (Contributed by NM, 19-Dec-2008.) $)
|
|
opelopabf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
|
|
( cop wcel wsbc cvv wb sbciegf ax-mp copab opelopabsb nfcv cv wceq 3bitri
|
|
nfsbc sbcbidv ) FGNADEUAOAEGPZDFPZBEGPZCADEFGUBFQOUJUKRJUIUKDFQBDEGDGUCHU
|
|
GDUDFUEABEGLUHSTGQOUKCRKBCEGQIMSTUF $.
|
|
$}
|
|
|
|
${
|
|
$d ph z $. $d ps z $. $d x z $. $d y z $.
|
|
$( Equivalence of ordered pair abstraction subclass and implication.
|
|
(Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro,
|
|
19-May-2013.) $)
|
|
ssopab2 $p |- ( A. x A. y ( ph -> ps ) ->
|
|
{ <. x , y >. | ph } C_ { <. x , y >. | ps } ) $=
|
|
( vz wi wal cv cop wceq wa wex cab copab nfa1 sp anim2d eximd sps df-opab
|
|
ss2abdv 3sstr4g ) ABFZDGZCGZEHCHDHIJZAKZDLZCLZEMUFBKZDLZCLZEMACDNBCDNUEUI
|
|
ULEUEUHUKCUDCOUDUHUKFCUDUGUJDUCDOUDABUFUCDPQRSRUAACDETBCDETUB $.
|
|
$}
|
|
|
|
$( Equivalence of ordered pair abstraction subclass and implication.
|
|
(Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro,
|
|
18-Nov-2016.) $)
|
|
ssopab2b $p |- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <->
|
|
A. x A. y ( ph -> ps ) ) $=
|
|
( copab wss wi wal nfopab1 nfss nfopab2 cop wcel ssel opabid 3imtr3g alrimi
|
|
cv ssopab2 impbii ) ACDEZBCDEZFZABGZDHZCHUCUECCUAUBACDIBCDIJUCUDDDUAUBACDKB
|
|
CDKJUCCRDRLZUAMUFUBMABUAUBUFNACDOBCDOPQQABCDST $.
|
|
|
|
${
|
|
ssopab2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of ordered pair abstraction subclass from implication.
|
|
(Contributed by NM, 5-Apr-1995.) $)
|
|
ssopab2i $p |- { <. x , y >. | ph } C_ { <. x , y >. | ps } $=
|
|
( wi wal copab wss ssopab2 ax-gen mpg ) ABFZDGACDHBCDHICABCDJMDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
ssopab2dv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference of ordered pair abstraction subclass from implication.
|
|
(Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro,
|
|
24-Jun-2014.) $)
|
|
ssopab2dv $p |- ( ph -> { <. x , y >. | ps } C_ { <. x , y >. | ch } ) $=
|
|
( wi wal copab wss alrimivv ssopab2 syl ) ABCGZEHDHBDEICDEIJANDEFKBCDELM
|
|
$.
|
|
$}
|
|
|
|
$( Equivalence of ordered pair abstraction equality and biconditional.
|
|
(Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
eqopab2b $p |- ( { <. x , y >. | ph } = { <. x , y >. | ps }
|
|
<-> A. x A. y ( ph <-> ps ) ) $=
|
|
( copab wss wa wi wal wceq wb ssopab2b anbi12i eqss 2albiim 3bitr4i ) ACDEZ
|
|
BCDEZFZRQFZGABHDICIZBAHDICIZGQRJABKDICISUATUBABCDLBACDLMQRNABCDOP $.
|
|
|
|
${
|
|
$d z ph $. $d z x $. $d z y $.
|
|
$( Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon,
|
|
29-Sep-2018.) $)
|
|
opabm $p |- ( E. z z e. { <. x , y >. | ph } <-> E. x E. y ph ) $=
|
|
( cv copab wcel wex cop wceq wa elopab exbii exrot3 isseti 19.41v mpbiran
|
|
vex opex 2exbii 3bitri ) DEZABCFGZDHUBBEZCEZIZJZAKZCHBHZDHUHDHZCHBHACHBHU
|
|
CUIDABCUBLMUHDBCNUJABCUJUGDHADUFUDUEBRCRSOUGADPQTUA $.
|
|
$}
|
|
|
|
${
|
|
$d ph w $. $d A w x $. $d A y $. $d w y z $. $d x z $.
|
|
$( Move indexed union inside an ordered-pair abstraction. (Contributed by
|
|
Stefan O'Rear, 20-Feb-2015.) $)
|
|
iunopab $p |- U_ z e. A { <. x , y >. | ph } =
|
|
{ <. x , y >. | E. z e. A ph } $=
|
|
( vw cv copab wcel wrex cab cop wceq wex ciun elopab rexcom4 exbii bitri
|
|
wa rexbii r19.42v abbii df-iun df-opab 3eqtr4i ) FGZABCHZIZDEJZFKUGBGCGLM
|
|
ZADEJZTZCNZBNZFKDEUHOULBCHUJUOFUJUKATZCNZBNZDEJZUOUIURDEABCUGPUAUSUQDEJZB
|
|
NUOUQDBEQUTUNBUTUPDEJZCNUNUPDCEQVAUMCUKADEUBRSRSSUCDFEUHUDULBCFUEUF $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Power class of union and intersection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( The power class of the intersection of two classes is the intersection
|
|
of their power classes. Exercise 4.12(j) of [Mendelson] p. 235.
|
|
(Contributed by NM, 23-Nov-2003.) $)
|
|
pwin $p |- ~P ( A i^i B ) = ( ~P A i^i ~P B ) $=
|
|
( vx cpw cin cv wss wa wcel ssin vex elpw anbi12i 3bitr4i ineqri eqcomi )
|
|
ADZBDZEABEZDZCQRTCFZAGZUABGZHUASGUAQIZUARIZHUATIUAABJUDUBUEUCUAACKZLUABUF
|
|
LMUASUFLNOP $.
|
|
|
|
$( The power class of the union of two classes includes the union of their
|
|
power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by
|
|
NM, 23-Nov-2003.) $)
|
|
pwunss $p |- ( ~P A u. ~P B ) C_ ~P ( A u. B ) $=
|
|
( vx cpw cun cv wss wcel ssun elun vex elpw orbi12i bitri 3imtr4i ssriv
|
|
wo ) CADZBDZEZABEZDZCFZAGZUCBGZQZUCUAGUCTHZUCUBHUCABIUGUCRHZUCSHZQUFUCRSJ
|
|
UHUDUIUEUCACKZLUCBUJLMNUCUAUJLOP $.
|
|
|
|
$( The power class of the union of two classes is a subset of the union of
|
|
their power classes, if one class is a subclass of the other. One
|
|
direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by
|
|
Jim Kingdon, 30-Sep-2018.) $)
|
|
pwssunim $p |- ( ( A C_ B \/ B C_ A ) ->
|
|
~P ( A u. B ) C_ ( ~P A u. ~P B ) ) $=
|
|
( wss wo cun cpw wceq ssequn2 pweq eqimss syl ssequn1 orim12i orcoms ssun
|
|
sylbi ) ABCZBACZDABEZFZAFZCZTBFZCZDZTUAUCECRQUERUBQUDRSAGZUBBAHUFTUAGUBSA
|
|
ITUAJKPQSBGZUDABLUGTUCGUDSBITUCJKPMNTUAUCOK $.
|
|
$}
|
|
|
|
$( Break up the power class of a union into a union of smaller classes.
|
|
(Contributed by Jim Kingdon, 30-Sep-2018.) $)
|
|
pwundifss $p |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_
|
|
~P ( A u. B ) $=
|
|
( cun cpw cdif undif1ss wss wceq wa pwunss unss simpli ssequn2 mpbi sseqtri
|
|
mpbir ) ABCDZADZERCQRCZQQRFRQGZSQHTBDZQGZTUBIRUACQGABJRUAQKPLRQMNO $.
|
|
|
|
$( The power class of the union of two classes equals the union of their
|
|
power classes, iff one class is a subclass of the other. Part of Exercise
|
|
7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.) $)
|
|
pwunim $p |- ( ( A C_ B \/ B C_ A ) ->
|
|
~P ( A u. B ) = ( ~P A u. ~P B ) ) $=
|
|
( wss wo cun cpw wa wceq pwssunim pwunss biantru sylib eqss sylibr ) ABCBAC
|
|
DZABEFZAFBFEZCZQPCZGZPQHORTABISRABJKLPQMN $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Epsilon and identity relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c _E $. $( Letter E (for epsilon relation) $)
|
|
$c _I $. $( Letter I (for identity relation) $)
|
|
|
|
$( Extend class notation to include the epsilon relation. $)
|
|
cep $a class _E $.
|
|
|
|
$( Extend the definition of a class to include identity relation. $)
|
|
cid $a class _I $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the epsilon relation. Similar to Definition 6.22 of
|
|
[TakeutiZaring] p. 30. The epsilon relation and set membership are the
|
|
same, that is, ` ( A _E B <-> A e. B ) ` when ` B ` is a set by
|
|
~ epelg . Thus, 5 ` _E ` { 1 , 5 }. (Contributed by NM,
|
|
13-Aug-1995.) $)
|
|
df-eprel $a |- _E = { <. x , y >. | x e. y } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The epsilon relation and membership are the same. General version of
|
|
~ epel . (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
epelg $p |- ( B e. V -> ( A _E B <-> A e. B ) ) $=
|
|
( vx vy wcel cvv cep wbr wi cop df-br cv copab wceq wa wex vex sylbi a1i
|
|
elopab pm3.2i opeqex mpbiri simpld adantr exlimivv df-eprel eleq2s eleq12
|
|
elex wb brabga expcom pm5.21ndd ) BCFZAGFZABHIZABFZURUQJUPURABKZHFUQABHLU
|
|
QUTDMZEMZFZDENZHUTVDFUTVAVBKOZVCPZEQDQUQVCDEUTUAVFUQDEVEUQVCVEUQBGFZVEUQV
|
|
GPVAGFZVBGFZPVHVIDRERUBABVAVBUCUDUEUFUGSDEUHZUISTUSUQJUPABUKTUQUPURUSULVC
|
|
USDEABHGCVAAVBBUJVJUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
epelc.1 $e |- B e. _V $.
|
|
$( The epsilon relationship and the membership relation are the same.
|
|
(Contributed by Scott Fenton, 11-Apr-2012.) $)
|
|
epelc $p |- ( A _E B <-> A e. B ) $=
|
|
( cvv wcel cep wbr wb epelg ax-mp ) BDEABFGABEHCABDIJ $.
|
|
$}
|
|
|
|
$( The epsilon relation and the membership relation are the same.
|
|
(Contributed by NM, 13-Aug-1995.) $)
|
|
epel $p |- ( x _E y <-> x e. y ) $=
|
|
( cv vex epelc ) ACBCBDE $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the identity relation. Definition 9.15 of [Quine] p. 64. For
|
|
example, 5 ` _I ` 5 and ` -. ` 4 ` _I ` 5. (Contributed by NM,
|
|
13-Aug-1995.) $)
|
|
df-id $a |- _I = { <. x , y >. | x = y } $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Partial and complete ordering
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$(
|
|
We have not yet defined relations ( ~ df-rel ), but here we introduce a
|
|
few related notions we will use to develop ordinals. The class variable
|
|
` R ` is no different from other class variables, but it reminds us that
|
|
normally it represents what we will later call a "relation."
|
|
$)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c Po $. $( Partial ordering predicate symbol (read: 'partial ordering'). $)
|
|
$c Or $. $( Strict linear ordering predicate symbol (read: 'orders'). $)
|
|
|
|
$( Extend wff notation to include the strict partial ordering predicate.
|
|
Read: ' ` R ` is a partial order on ` A ` .' $)
|
|
wpo $a wff R Po A $.
|
|
|
|
$( Extend wff notation to include the strict linear ordering predicate.
|
|
Read: ' ` R ` orders ` A ` .' $)
|
|
wor $a wff R Or A $.
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $.
|
|
$( Define the strict partial order predicate. Definition of [Enderton]
|
|
p. 168. The expression ` R Po A ` means ` R ` is a partial order on
|
|
` A ` . (Contributed by NM, 16-Mar-1997.) $)
|
|
df-po $a |- ( R Po A <-> A. x e. A A. y e. A A. z e. A
|
|
( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) $.
|
|
|
|
$( Define the strict linear order predicate. The expression ` R Or A ` is
|
|
true if relationship ` R ` orders ` A ` . The property
|
|
` x R y -> ( x R z \/ z R y ) ` is called weak linearity by Proposition
|
|
11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would
|
|
be equivalent to trichotomy, ` x R y \/ x = y \/ y R x ` . (Contributed
|
|
by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.) $)
|
|
df-iso $a |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A
|
|
( x R y -> ( x R z \/ z R y ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z B $.
|
|
$( Subset theorem for the partial ordering predicate. (Contributed by NM,
|
|
27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) $)
|
|
poss $p |- ( A C_ B -> ( R Po B -> R Po A ) ) $=
|
|
( vx vy vz wss cv wbr wn wa wi wral wpo ssralv ralimdv syld df-po 3imtr4g
|
|
) ABGZDHZUACIJUAEHZCIUBFHZCIKUAUCCILKZFBMZEBMZDBMZUDFAMZEAMZDAMZBCNACNTUG
|
|
UFDAMUJUFDABOTUFUIDATUFUEEAMUIUEEABOTUEUHEAUDFABOPQPQDEFBCRDEFACRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z S $. $d x y z A $.
|
|
$( Equality theorem for partial ordering predicate. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
poeq1 $p |- ( R = S -> ( R Po A <-> S Po A ) ) $=
|
|
( vx vy vz wceq cv wbr wn wa wral wpo breq notbid anbi12d imbi12d ralbidv
|
|
wi df-po 2ralbidv 3bitr4g ) BCGZDHZUDBIZJZUDEHZBIZUGFHZBIZKZUDUIBIZSZKZFA
|
|
LZEALDALUDUDCIZJZUDUGCIZUGUICIZKZUDUICIZSZKZFALZEALDALABMACMUCUOVDDEAAUCU
|
|
NVCFAUCUFUQUMVBUCUEUPUDUDBCNOUCUKUTULVAUCUHURUJUSUDUGBCNUGUIBCNPUDUIBCNQP
|
|
RUADEFABTDEFACTUB $.
|
|
$}
|
|
|
|
$( Equality theorem for partial ordering predicate. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
poeq2 $p |- ( A = B -> ( R Po A <-> R Po B ) ) $=
|
|
( wceq wpo wss wi eqimss2 poss syl eqimss impbid ) ABDZACEZBCEZMBAFNOGBAHBA
|
|
CIJMABFONGABKABCIJL $.
|
|
|
|
${
|
|
$d R a b c $. $d A a b c $. $d x a b c $.
|
|
nfpo.r $e |- F/_ x R $.
|
|
nfpo.a $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for partial orders. (Contributed by
|
|
Stefan O'Rear, 20-Jan-2015.) $)
|
|
nfpo $p |- F/ x R Po A $=
|
|
( va vb vc wpo cv wbr wn wa wi wral df-po nfcv nfbr nfan nfralxy nfn nfim
|
|
nfxfr ) BCIFJZUDCKZLZUDGJZCKZUGHJZCKZMZUDUICKZNZMZHBOZGBOZFBOAFGHBCPUPAFB
|
|
EUOAGBEUNAHBEUFUMAUEAAUDUDCAUDQZDUQRUAUKULAUHUJAAUDUGCUQDAUGQZRAUGUICURDA
|
|
UIQZRSAUDUICUQDUSRUBSTTTUC $.
|
|
|
|
$( Bound-variable hypothesis builder for total orders. (Contributed by
|
|
Stefan O'Rear, 20-Jan-2015.) $)
|
|
nfso $p |- F/ x R Or A $=
|
|
( va vb vc wor wpo cv wbr wo wi wral wa df-iso nfcv nfbr nfralxy nfxfr
|
|
nfpo nfor nfim nfan ) BCIBCJZFKZGKZCLZUGHKZCLZUJUHCLZMZNZHBOZGBOZFBOZPAFG
|
|
HBCQUFUQAABCDEUBUPAFBEUOAGBEUNAHBEUIUMAAUGUHCAUGRZDAUHRZSUKULAAUGUJCURDAU
|
|
JRZSAUJUHCUTDUSSUCUDTTTUEUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $.
|
|
$( Properties of partial order relation in class notation. (Contributed by
|
|
NM, 27-Mar-1997.) $)
|
|
pocl $p |- ( R Po A -> ( ( B e. A /\ C e. A /\ D e. A ) ->
|
|
( -. B R B /\ ( ( B R C /\ C R D ) -> B R D ) ) ) ) $=
|
|
( vx vy vz wcel wbr wa wi cv wceq breq1 imbi2d breq2 anbi2d wal wral r3al
|
|
w3a wn id breq12d notbid anbi1d imbi12d anbi12d imbi1d df-po bitri biimpi
|
|
wpo 19.21bbi 19.21bi com12 vtocl3ga ) BAICAIDAIUBAEUNZBBEJZUCZBCEJZCDEJZK
|
|
ZBDEJZLZKZUSFMZVHEJZUCZVHGMZEJZVKHMZEJZKZVHVMEJZLZKZLUSVABVKEJZVNKZBVMEJZ
|
|
LZKZLUSVAVBCVMEJZKZWALZKZLUSVGLFGHBCDAAAVHBNZVRWCUSWHVJVAVQWBWHVIUTWHVHBV
|
|
HBEWHUDZWIUEUFWHVOVTVPWAWHVLVSVNVHBVKEOUGVHBVMEOUHUIPVKCNZWCWGUSWJWBWFVAW
|
|
JVTWEWAWJVSVBVNWDVKCBEQVKCVMEOUIUJRPVMDNZWGVGUSWKWFVFVAWKWEVDWAVEWKWDVCVB
|
|
VMDCEQRVMDBEQUHRPUSVHAIVKAIVMAIUBZVRUSWLVRLZHUSWMHSZFGUSWNGSFSZUSVRHATGAT
|
|
FATWOFGHAEUKVRFGHAAAUAULUMUOUPUQURUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z R $. $d x y z ph $.
|
|
ispod.1 $e |- ( ( ph /\ x e. A ) -> -. x R x ) $.
|
|
ispod.2 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
|
|
( ( x R y /\ y R z ) -> x R z ) ) $.
|
|
$( Sufficient conditions for a partial order. (Contributed by NM,
|
|
9-Jul-2014.) $)
|
|
ispod $p |- ( ph -> R Po A ) $=
|
|
( cv wbr wn wa wi wral wpo wcel w3a 3ad2antr1 jca ralrimivvva sylibr
|
|
df-po ) ABIZUCFJKZUCCIZFJUEDIZFJLUCUFFJMZLZDENCENBENEFOAUHBCDEEEAUCEPZUEE
|
|
PZUFEPZQLUDUGAUJUIUDUKGRHSTBCDEFUBUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z ph $. $d x y z R $. $d x y z X $. $d y z Y $.
|
|
$d z Z $.
|
|
swopolem.1 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
|
|
( x R y -> ( x R z \/ z R y ) ) ) $.
|
|
$( Perform the substitutions into the strict weak ordering law.
|
|
(Contributed by Mario Carneiro, 31-Dec-2014.) $)
|
|
swopolem $p |- ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) ->
|
|
( X R Y -> ( X R Z \/ Z R Y ) ) ) $=
|
|
( cv wbr wo wi wral wcel wceq breq1 imbi12d breq2 w3a ralrimivvva orbi12d
|
|
orbi1d orbi2d imbi2d rspc3v mpan9 ) ABKZCKZFLZUIDKZFLZULUJFLZMZNZDEOCEOBE
|
|
OGEPHEPIEPUAGHFLZGIFLZIHFLZMZNZAUPBCDEEEJUBUPVAGUJFLZGULFLZUNMZNUQVCULHFL
|
|
ZMZNBCDGHIEEEUIGQZUKVBUOVDUIGUJFRVGUMVCUNUIGULFRUDSUJHQZVBUQVDVFUJHGFTVHU
|
|
NVEVCUJHULFTUESULIQZVFUTUQVIVCURVEUSULIGFTULIHFRUCUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z R $. $d x y z ph $.
|
|
swopo.1 $e |- ( ( ph /\ ( y e. A /\ z e. A ) ) ->
|
|
( y R z -> -. z R y ) ) $.
|
|
swopo.2 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
|
|
( x R y -> ( x R z \/ z R y ) ) ) $.
|
|
$( A strict weak order is a partial order. (Contributed by Mario Carneiro,
|
|
9-Jul-2014.) $)
|
|
swopo $p |- ( ph -> R Po A ) $=
|
|
( cv wcel wa wbr wn wi wral weq breq1 breq2 notbid imbi12d ralrimivva w3a
|
|
id ancli rspc2va syl2anr pm2.01d 3adantr1 wo imp orcomd ord expimpd ispod
|
|
sylan2d ) ABCDEFABIZEJZKUPUPFLZUQUQUQKCIZDIZFLZUTUSFLZMZNZDEOCEOURURMZNZA
|
|
UQUQUQUCUDAVDCDEEGUAVDVFUPUTFLZUTUPFLZMZNCDUPUPEECBPZVAVGVCVIUSUPUTFQVJVB
|
|
VHUSUPUTFRSTDBPZVGURVIVEUTUPUPFRVKVHURUTUPUPFQSTUEUFUGAUQUSEJZUTEJZUBKZVA
|
|
VCUPUSFLZVGAVLVMVDUQGUHVNVOVCVGVNVOKZVBVGVPVGVBVNVOVGVBUIHUJUKULUMUOUN $.
|
|
$}
|
|
|
|
$( A partial order relation is irreflexive. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
poirr $p |- ( ( R Po A /\ B e. A ) -> -. B R B ) $=
|
|
( wcel wpo w3a wbr wn wa df-3an anabs1 anidm 3bitrri wi pocl simpld sylan2b
|
|
imp ) BADZACEZSSSFZBBCGZHZUASSIZSIUDSSSSJSSKSLMTUAIUCUBUBIUBNZTUAUCUEIABBBC
|
|
ORPQ $.
|
|
|
|
$( A partial order relation is a transitive relation. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
potr $p |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
|
|
( ( B R C /\ C R D ) -> B R D ) ) $=
|
|
( wpo wcel w3a wa wbr wn wi pocl imp simprd ) AEFZBAGCAGDAGHZIBBEJKZBCEJCDE
|
|
JIBDEJLZPQRSIABCDEMNO $.
|
|
|
|
$( A partial order relation has no 2-cycle loops. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
po2nr $p |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) ->
|
|
-. ( B R C /\ C R B ) ) $=
|
|
( wpo wcel wa wbr wn poirr adantrr wi potr 3exp2 com34 pm2.43d imp32 mtod )
|
|
ADEZBAFZCAFZGGBCDHCBDHGZBBDHZSTUCIUAABDJKSTUAUBUCLZSTUAUDLSTUATUDSTUATUDABC
|
|
BDMNOPQR $.
|
|
|
|
$( A partial order relation has no 3-cycle loops. (Contributed by NM,
|
|
27-Mar-1997.) $)
|
|
po3nr $p |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
|
|
-. ( B R C /\ C R D /\ D R B ) ) $=
|
|
( wpo wcel w3a wa wbr wn po2nr 3adantr2 df-3an potr anim1d syl5bi mtod ) AE
|
|
FZBAGZCAGZDAGZHIZBCEJZCDEJZDBEJZHZBDEJZUFIZSTUBUIKUAABDELMUGUDUEIZUFIUCUIUD
|
|
UEUFNUCUJUHUFABCDEOPQR $.
|
|
|
|
${
|
|
$d x y z R $.
|
|
$( Any relation is a partial ordering of the empty set. (Contributed by
|
|
NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
po0 $p |- R Po (/) $=
|
|
( vx vy vz c0 wpo cv wbr wn wa wi wral ral0 df-po mpbir ) EAFBGZPAHIPCGZA
|
|
HQDGZAHJPRAHKJDELCELZBELSBMBCDEANO $.
|
|
$}
|
|
|
|
${
|
|
$d R v w x y z $. $d S v w z $. $d X v w y z $. $d Y x z $.
|
|
$d A v w x z $. $d B v w x z $.
|
|
pofun.1 $e |- S = { <. x , y >. | X R Y } $.
|
|
pofun.2 $e |- ( x = y -> X = Y ) $.
|
|
$( A function preserves a partial order relation. (Contributed by Jeff
|
|
Madsen, 18-Jun-2011.) $)
|
|
pofun $p |- ( ( R Po B /\ A. x e. A X e. B ) -> S Po A ) $=
|
|
( vv vw vz wcel wa cv wbr csb weq cop wpo wn nfcsb1v nfel1 csbeq1a eleq1d
|
|
wral rspc impcom poirr copab df-br eleq2i nfcv nfbr nfv vex breq1d csbief
|
|
csbeq1 syl5eqr breq2d opelopabf 3bitri sylnibr sylan2 w3a com12 3anim123d
|
|
anassrs wi imp adantll potr anbi12i 3imtr4g adantlr syldan ispod ) DEUAZG
|
|
DNZACUGZOZKLMCFVTWBKPZCNZWDWDFQZUBZWBWEOVTAWDGRZDNZWGWEWBWIWAWIAWDCAWHDAW
|
|
DGUCZUDAKSZGWHDAWDGUEZUFUHZUIVTWIOWHWHEQZWFDWHEUJWFWDWDTZFNWOGHEQZABUKZNW
|
|
NWDWDFULFWQWOIUMWPWHHEQZWNABWDWDAWHHEWJAEUNZAHUNZUOZWNBUPKUQZXBWKGWHHEWLU
|
|
RZBKSZHWHWHEXDHABPZGRZWHAXEGHBUQWTJUSZAXEWDGUTVAVBVCVDVEVFVJWCWELPZCNZMPZ
|
|
CNZVGZWIAXHGRZDNZAXJGRZDNZVGZWDXHFQZXHXJFQZOZWDXJFQZVKZWBXLXQVTWBXLXQWBWE
|
|
WIXIXNXKXPWEWBWIWMVHXIWBXNWAXNAXHCAXMDAXHGUCZUDALSZGXMDAXHGUEZUFUHVHXKWBX
|
|
PWAXPAXJCAXODAXJGUCUDAMSGXODAXJGUEUFUHVHVIVLVMVTXQYBWBVTXQOWHXMEQZXMXOEQZ
|
|
OWHXOEQZXTYADWHXMXOEVNXRYFXSYGXRWDXHTZFNYIWQNYFWDXHFULFWQYIIUMWPWRYFABWDX
|
|
HXAYFBUPXBLUQZXCBLSZHXMWHEYKHXFXMXGAXEXHGUTVAVBVCVDXSXHXJTZFNYLWQNYGXHXJF
|
|
ULFWQYLIUMWPXMHEQYGABXHXJAXMHEYCWSWTUOYGBUPYJMUQZYDGXMHEYEURBMSZHXOXMEYNH
|
|
XFXOXGAXEXJGUTVAZVBVCVDVOYAWDXJTZFNYPWQNYHWDXJFULFWQYPIUMWPWRYHABWDXJXAYH
|
|
BUPXBYMXCYNHXOWHEYOVBVCVDVPVQVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $.
|
|
$( A strict linear order is a strict partial order. (Contributed by NM,
|
|
28-Mar-1997.) $)
|
|
sopo $p |- ( R Or A -> R Po A ) $=
|
|
( vx vy vz wor wpo cv wbr wo wi wral df-iso simplbi ) ABFABGCHZDHZBIOEHZB
|
|
IQPBIJKEALDALCALCDEABMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z B $.
|
|
$( Subset theorem for the strict ordering predicate. (Contributed by NM,
|
|
16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
soss $p |- ( A C_ B -> ( R Or B -> R Or A ) ) $=
|
|
( vx vy vz wpo cv wbr wi wral wor wcel w3a wal ssel r3al 3imtr4g df-iso
|
|
wa wss wo poss 3anim123d imim1d 2alimdv alimdv anim12d ) ABUAZBCGZDHZEHZC
|
|
IUKFHZCIUMULCIUBJZFBKEBKDBKZTACGZUNFAKEAKDAKZTBCLACLUIUJUPUOUQABCUCUIUKBM
|
|
ZULBMZUMBMZNZUNJZFOEOZDOUKAMZULAMZUMAMZNZUNJZFOEOZDOUOUQUIVCVIDUIVBVHEFUI
|
|
VGVAUNUIVDURVEUSVFUTABUKPABULPABUMPUDUEUFUGUNDEFBBBQUNDEFAAAQRUHDEFBCSDEF
|
|
ACSR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z S $. $d x y z A $.
|
|
$( Equality theorem for the strict ordering predicate. (Contributed by NM,
|
|
16-Mar-1997.) $)
|
|
soeq1 $p |- ( R = S -> ( R Or A <-> S Or A ) ) $=
|
|
( vx vy vz wceq wpo cv wbr wo wi wral wa wor poeq1 orbi12d imbi12d df-iso
|
|
breq 2ralbidv ralbidv anbi12d 3bitr4g ) BCGZABHZDIZEIZBJZUGFIZBJZUJUHBJZK
|
|
ZLZFAMEAMZDAMZNACHZUGUHCJZUGUJCJZUJUHCJZKZLZFAMEAMZDAMZNABOACOUEUFUQUPVDA
|
|
BCPUEUOVCDAUEUNVBEFAAUEUIURUMVAUGUHBCTUEUKUSULUTUGUJBCTUJUHBCTQRUAUBUCDEF
|
|
ABSDEFACSUD $.
|
|
$}
|
|
|
|
$( Equality theorem for the strict ordering predicate. (Contributed by NM,
|
|
16-Mar-1997.) $)
|
|
soeq2 $p |- ( A = B -> ( R Or A <-> R Or B ) ) $=
|
|
( wceq wor wss wa wi wb soss anim12i eqss dfbi2 3imtr4i bicomd ) ABDZBCEZAC
|
|
EZABFZBAFZGQRHZRQHZGPQRISUATUBABCJBACJKABLQRMNO $.
|
|
|
|
$( A strict order relation is irreflexive. (Contributed by NM,
|
|
24-Nov-1995.) $)
|
|
sonr $p |- ( ( R Or A /\ B e. A ) -> -. B R B ) $=
|
|
( wor wpo wcel wbr wn sopo poirr sylan ) ACDACEBAFBBCGHACIABCJK $.
|
|
|
|
$( A strict order relation is a transitive relation. (Contributed by NM,
|
|
21-Jan-1996.) $)
|
|
sotr $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
|
|
( ( B R C /\ C R D ) -> B R D ) ) $=
|
|
( wor wpo wcel w3a wbr wa wi sopo potr sylan ) AEFAEGBAHCAHDAHIBCEJCDEJKBDE
|
|
JLAEMABCDENO $.
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z ph $.
|
|
issod.1 $e |- ( ph -> R Po A ) $.
|
|
issod.2 $e |- ( ( ph /\ ( x e. A /\ y e. A ) ) ->
|
|
( x R y \/ x = y \/ y R x ) ) $.
|
|
$( An irreflexive, transitive, trichotomous relation is a linear ordering
|
|
(in the sense of ~ df-iso ). (Contributed by NM, 21-Jan-1996.)
|
|
(Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
issod $p |- ( ph -> R Or A ) $=
|
|
( vz wpo cv wbr wo wi wral wor wcel wa w3a syl6 ralrimiva weq w3o 3adant3
|
|
orc a1i simp3r breq1 syl5ibcom simp1 simp2r simp2l simp3l 3jca potr sylan
|
|
olc expcomd imp syl21anc 3jaod mpd 3expa expr anassrs ralcom sylib df-iso
|
|
sylanbrc ) ADEIZBJZHJZEKZVJCJZEKZVMVKEKZLZMZCDNHDNZBDNDEOFAVRBDAVJDPZQZVQ
|
|
HDNZCDNVRVTWACDAVSVMDPZWAAVSWBQZQZVQHDWDVKDPZVLVPAWCWEVLQZVPAWCWFRZVNBCUA
|
|
ZVMVJEKZUBZVPAWCWJWFGUCWGVNVPWHWIVNVPMWGVNVOUDUEWGWHVOVPWGVLWHVOAWCWEVLUF
|
|
ZVJVMVKEUGUHVOVNUPZSWGWIVOVPWGAWBVSWERZVLWIVOMZAWCWFUIWGWBVSWEAVSWBWFUJAV
|
|
SWBWFUKAWCWEVLULUMWKAWMQZVLWNWOWIVLVOAVIWMWIVLQVOMFDVMVJVKEUNUOUQURUSWLSU
|
|
TVAVBVCTVDTVQCHDDVEVFTBHCDEVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z R $.
|
|
$( A strict order relation satisfies weak linearity. (Contributed by Jim
|
|
Kingdon, 6-Oct-2018.) $)
|
|
sowlin $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
|
|
( B R C -> ( B R D \/ D R C ) ) ) $=
|
|
( vx vy vz wcel w3a wbr wo wi cv wceq breq1 imbi2d breq2 wral wa wor rsp2
|
|
orbi1d imbi12d orbi2d orbi12d df-iso 3anass syl6 impd syl5bi adantl sylbi
|
|
wpo rsp com12 vtocl3ga impcom ) BAICAIDAIJAEUAZBCEKZBDEKZDCEKZLZMZUSFNZGN
|
|
ZEKZVEHNZEKZVHVFEKZLZMZMUSBVFEKZBVHEKZVJLZMZMUSUTVNVHCEKZLZMZMUSVDMFGHBCD
|
|
AAAVEBOZVLVPUSVTVGVMVKVOVEBVFEPVTVIVNVJVEBVHEPUCUDQVFCOZVPVSUSWAVMUTVOVRV
|
|
FCBERWAVJVQVNVFCVHERUEUDQVHDOZVSVDUSWBVRVCUTWBVNVAVQVBVHDBERVHDCEPUFQQUSV
|
|
EAIZVFAIZVHAIZJZVLUSAEUNZVLHASGASZFASZTWFVLMZFGHAEUGWIWJWGWFWCWDWETZTWIVL
|
|
WCWDWEUHWIWCWKVLWIWCWHWKVLMWHFAUOVLGHAAUBUIUJUKULUMUPUQUR $.
|
|
$}
|
|
|
|
$( A strict order relation has no 2-cycle loops. (Contributed by NM,
|
|
21-Jan-1996.) $)
|
|
so2nr $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
|
|
-. ( B R C /\ C R B ) ) $=
|
|
( wor wpo wcel wa wbr wn sopo po2nr sylan ) ADEADFBAGCAGHBCDICBDIHJADKABCDL
|
|
M $.
|
|
|
|
$( A strict order relation has no 3-cycle loops. (Contributed by NM,
|
|
21-Jan-1996.) $)
|
|
so3nr $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
|
|
-. ( B R C /\ C R D /\ D R B ) ) $=
|
|
( wor wpo wcel w3a wbr wn sopo po3nr sylan ) AEFAEGBAHCAHDAHIBCEJCDEJDBEJIK
|
|
AELABCDEMN $.
|
|
|
|
$( One direction of ~ sotritric holds for all weakly linear orders.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
sotricim $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
|
|
( B R C -> -. ( B = C \/ C R B ) ) ) $=
|
|
( wor wcel wa wbr wceq wn wo w3a adantrr 3adant3 wi breq2 biimprcd 3ad2ant3
|
|
sonr mtod 3expia so2nr imnan sylibr jcad ioran syl6ibr ) ADEZBAFZCAFZGZGZBC
|
|
DHZBCIZJZCBDHZJZGUNUPKJULUMUOUQUHUKUMUOUHUKUMLUNBBDHZUHUKURJZUMUHUIUSUJABDS
|
|
MNUMUHUNUROUKUNURUMBCBDPQRTUAULUMUPGJUMUQOABCDUBUMUPUCUDUEUNUPUFUG $.
|
|
|
|
${
|
|
sotritric.or $e |- R Or A $.
|
|
sotritric.tri $e |- ( ( B e. A /\ C e. A ) ->
|
|
( B R C \/ B = C \/ C R B ) ) $.
|
|
$( A trichotomy relationship, given a trichotomous order. (Contributed by
|
|
Jim Kingdon, 28-Sep-2019.) $)
|
|
sotritric $p |- ( ( B e. A /\ C e. A ) ->
|
|
( B R C <-> -. ( B = C \/ C R B ) ) ) $=
|
|
( wcel wa wbr wceq wo wn wor wi sotricim mpan w3o 3orass ax-1 pm2.24 jaoi
|
|
sylbi syl impbid ) BAGCAGHZBCDIZBCJZCBDIZKZLZADMUEUFUJNEABCDOPUEUFUGUHQZU
|
|
JUFNZFUKUFUIKULUFUGUHRUFULUIUFUJSUIUFTUAUBUCUD $.
|
|
|
|
$( A trichotomy relationship, given a trichotomous order. (Contributed by
|
|
Jim Kingdon, 13-Dec-2019.) $)
|
|
sotritrieq $p |- ( ( B e. A /\ C e. A ) ->
|
|
( B = C <-> -. ( B R C \/ C R B ) ) ) $=
|
|
( wcel wa wceq wbr wo wn wi wor sonr mpan breq2 notbid syl5ibcom w3o jcad
|
|
breq1 ioran syl6ibr adantr 3orrot 3orcomb 3orass 3bitri biimpi orcomd ord
|
|
syl impbid ) BAGZCAGZHZBCIZBCDJZCBDJZKZLZUOURVBMUPUOURUSLZUTLZHVBUOURVCVD
|
|
UOBBDJZLZURVCADNUOVFEABDOPZURVEUSBCBDQRSUOVFURVDVGURVEUTBCBDUBRSUAUSUTUCU
|
|
DUEUQUSURUTTZVBURMFVHVAURVHURVAVHURVAKZVHURUTUSTURUSUTTVIUSURUTUFURUTUSUG
|
|
URUSUTUHUIUJUKULUMUN $.
|
|
$}
|
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|
|
${
|
|
$d x y z R $.
|
|
$( Any relation is a strict ordering of the empty set. (Contributed by NM,
|
|
16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
so0 $p |- R Or (/) $=
|
|
( vx vy vz c0 wor wpo cv wbr wo wi wral po0 ral0 df-iso mpbir2an ) EAFEAG
|
|
BHZCHZAIQDHZAISRAIJKDELCELZBELAMTBNBCDEAOP $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Founded and set-like relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c FrFor $. $( Well-founded predicate symbol (read: 'well-founded'). $)
|
|
$c Fr $. $( Well-founded predicate symbol (read: 'well-founded'). $)
|
|
$c Se $. $( Set-like predicate symbol (read: 'set-like'). $)
|
|
$c We $. $( Well-ordering predicate symbol (read: 'well-orders') $)
|
|
|
|
$( Extend wff notation to include the well-founded predicate. $)
|
|
wfrfor $a wff FrFor R A S $.
|
|
|
|
$( Extend wff notation to include the well-founded predicate. Read: ' ` R `
|
|
is a well-founded relation on ` A ` .' $)
|
|
wfr $a wff R Fr A $.
|
|
|
|
$( Extend wff notation to include the set-like predicate. Read: ' ` R ` is
|
|
set-like on ` A ` .' $)
|
|
wse $a wff R Se A $.
|
|
|
|
$( Extend wff notation to include the well-ordering predicate. Read:
|
|
' ` R ` well-orders ` A ` .' $)
|
|
wwe $a wff R We A $.
|
|
|
|
${
|
|
$d x y R $. $d x y A $. $d x y S $.
|
|
$( Define the well-founded relation predicate where ` A ` might be a proper
|
|
class. By passing in ` S ` we allow it potentially to be a proper class
|
|
rather than a set. (Contributed by Jim Kingdon and Mario Carneiro,
|
|
22-Sep-2021.) $)
|
|
df-frfor $a |- ( FrFor R A S <->
|
|
( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S )
|
|
-> A C_ S ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d s R $. $d s A $.
|
|
$( Define the well-founded relation predicate. In the presence of excluded
|
|
middle, there are a variety of equivalent ways to define this. In our
|
|
case, this definition, in terms of an inductive principle, works better
|
|
than one along the lines of "there is an element which is minimal when A
|
|
is ordered by R". Because ` s ` is constrained to be a set (not a
|
|
proper class) here, sometimes it may be necessary to use ` FrFor `
|
|
directly rather than via ` Fr ` . (Contributed by Jim Kingdon and Mario
|
|
Carneiro, 21-Sep-2021.) $)
|
|
df-frind $a |- ( R Fr A <-> A. s FrFor R A s ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y R $. $d x y A $.
|
|
$( Define the set-like predicate. (Contributed by Mario Carneiro,
|
|
19-Nov-2014.) $)
|
|
df-se $a |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V ) $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d R x y z $.
|
|
$( Define the well-ordering predicate. It is unusual to define
|
|
"well-ordering" in the absence of excluded middle, but we mean an
|
|
ordering which is like the ordering which we have for ordinals (for
|
|
example, it does not entail trichotomy because ordinals don't have that
|
|
as seen at ~ ordtriexmid ). Given excluded middle, well-ordering is
|
|
usually defined to require trichotomy (and the defintion of ` Fr ` is
|
|
typically also different). (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 23-Sep-2021.) $)
|
|
df-wetr $a |- ( R We A <-> ( R Fr A /\
|
|
A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y V $.
|
|
$( The ` R ` -preimage of an element of the base set in a set-like relation
|
|
is a set. (Contributed by Mario Carneiro, 19-Nov-2014.) $)
|
|
seex $p |- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V ) $=
|
|
( vy wse cv wbr crab cvv wcel wral df-se wceq breq2 rabbidv eleq1d sylanb
|
|
rspccva ) BDFAGZEGZDHZABIZJKZEBLCBKTCDHZABIZJKZEABDMUDUGECBUACNZUCUFJUHUB
|
|
UEABUACTDOPQSR $.
|
|
|
|
$( Any relation on a set is set-like on it. (Contributed by Mario
|
|
Carneiro, 22-Jun-2015.) $)
|
|
exse $p |- ( A e. V -> R Se A ) $=
|
|
( vy vx wcel cv wbr crab cvv wral wse rabexg ralrimivw df-se sylibr ) ACF
|
|
ZDGEGBHZDAIJFZEAKABLQSEARDACMNEDABOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $.
|
|
$( Subset theorem for the set-like predicate. (Contributed by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
sess1 $p |- ( R C_ S -> ( S Se A -> R Se A ) ) $=
|
|
( vy vx wss cv wbr crab cvv wcel wral wi simpl ssbrd ss2rabdv ssexg df-se
|
|
wse wa ex syl ralimdv 3imtr4g ) BCFZDGZEGZCHZDAIZJKZEALUFUGBHZDAIZJKZEALA
|
|
CSABSUEUJUMEAUEULUIFZUJUMMUEUKUHDAUEUFAKZTBCUFUGUEUONOPUNUJUMULUIJQUAUBUC
|
|
EDACREDABRUD $.
|
|
|
|
$( Subset theorem for the set-like predicate. (Contributed by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
sess2 $p |- ( A C_ B -> ( R Se B -> R Se A ) ) $=
|
|
( vy vx wss cv wbr crab cvv wcel wral wse ssralv wi rabss2 ssexg ex df-se
|
|
syl ralimdv syld 3imtr4g ) ABFZDGEGCHZDBIZJKZEBLZUEDAIZJKZEALZBCMACMUDUHU
|
|
GEALUKUGEABNUDUGUJEAUDUIUFFZUGUJOUEDABPULUGUJUIUFJQRTUAUBEDBCSEDACSUC $.
|
|
$}
|
|
|
|
$( Equality theorem for the set-like predicate. (Contributed by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
seeq1 $p |- ( R = S -> ( R Se A <-> S Se A ) ) $=
|
|
( wceq wse wss wi eqimss2 sess1 syl eqimss impbid ) BCDZABEZACEZMCBFNOGCBHA
|
|
CBIJMBCFONGBCKABCIJL $.
|
|
|
|
$( Equality theorem for the set-like predicate. (Contributed by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
seeq2 $p |- ( A = B -> ( R Se A <-> R Se B ) ) $=
|
|
( wceq wse wss wi eqimss2 sess2 syl eqimss impbid ) ABDZACEZBCEZMBAFNOGBAHB
|
|
ACIJMABFONGABKABCIJL $.
|
|
|
|
${
|
|
$d R a b $. $d A a b $. $d x a b $.
|
|
nfse.r $e |- F/_ x R $.
|
|
nfse.a $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for set-like relations. (Contributed
|
|
by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfse $p |- F/ x R Se A $=
|
|
( va vb wse wbr crab cvv wcel wral df-se nfcv nfbr nfrabxy nfel1 nfralxy
|
|
cv nfxfr ) BCHFTZGTZCIZFBJZKLZGBMAGFBCNUFAGBEAUEKUDAFBAUBUCCAUBODAUCOPEQR
|
|
SUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The epsilon relation is set-like on any class. (This is the origin of
|
|
the term "set-like": a set-like relation "acts like" the epsilon
|
|
relation of sets and their elements.) (Contributed by Mario Carneiro,
|
|
22-Jun-2015.) $)
|
|
epse $p |- _E Se A $=
|
|
( vy vx cep wse cv wbr crab cvv wcel wral cab epel bicomi abbi2i eqeltrri
|
|
vex rabssab ssexi rgenw df-se mpbir ) ADEBFZCFZDGZBAHZIJZCAKUGCAUFUEBLZUD
|
|
UHIUEBUDUEUCUDJBCMNOCQPUEBARSTCBADUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d R x y $. $d S x y $. $d T x y $.
|
|
$( Equality theorem for the well-founded predicate. (Contributed by Jim
|
|
Kingdon, 22-Sep-2021.) $)
|
|
frforeq1 $p |- ( R = S -> ( FrFor R A T <-> FrFor S A T ) ) $=
|
|
( vy vx wceq cv wbr wcel wi wral wss wfrfor breq ralbidv df-frfor 3bitr4g
|
|
imbi1d ) BCGZEHZFHZBIZUADJZKZEALZUBDJZKZFALZADMZKUAUBCIZUDKZEALZUGKZFALZU
|
|
JKABDNACDNTUIUOUJTUHUNFATUFUMUGTUEULEATUCUKUDUAUBBCOSPSPSFEABDQFEACDQR $.
|
|
$}
|
|
|
|
${
|
|
$d s R $. $d s S $. $d s A $.
|
|
$( Equality theorem for the well-founded predicate. (Contributed by NM,
|
|
9-Mar-1997.) $)
|
|
freq1 $p |- ( R = S -> ( R Fr A <-> S Fr A ) ) $=
|
|
( vs wceq cv wfrfor wal wfr frforeq1 albidv df-frind 3bitr4g ) BCEZABDFZG
|
|
ZDHACOGZDHABIACINPQDABCOJKABDLACDLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d R x y $. $d T x y $.
|
|
$( Equality theorem for the well-founded predicate. (Contributed by Jim
|
|
Kingdon, 22-Sep-2021.) $)
|
|
frforeq2 $p |- ( A = B -> ( FrFor R A T <-> FrFor R B T ) ) $=
|
|
( vy vx wceq cv wbr wcel wi wral wss wfrfor raleq imbi1d raleqbi1dv sseq1
|
|
imbi12d df-frfor 3bitr4g ) ABGZEHZFHZCIUCDJKZEALZUDDJZKZFALZADMZKUEEBLZUG
|
|
KZFBLZBDMZKACDNBCDNUBUIUMUJUNUHULFABUBUFUKUGUEEABOPQABDRSFEACDTFEBCDTUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A s $. $d B s $. $d R s $.
|
|
$( Equality theorem for the well-founded predicate. (Contributed by NM,
|
|
3-Apr-1994.) $)
|
|
freq2 $p |- ( A = B -> ( R Fr A <-> R Fr B ) ) $=
|
|
( vs wceq cv wfrfor wal wfr frforeq2 albidv df-frind 3bitr4g ) ABEZACDFZG
|
|
ZDHBCOGZDHACIBCINPQDABCOJKACDLBCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d R x y $. $d S x y $. $d T x y $.
|
|
$( Equality theorem for the well-founded predicate. (Contributed by Jim
|
|
Kingdon, 22-Sep-2021.) $)
|
|
frforeq3 $p |- ( S = T -> ( FrFor R A S <-> FrFor R A T ) ) $=
|
|
( vy vx wceq cv wbr wcel wi wss wfrfor eleq2 imbi2d ralbidv imbi12d sseq2
|
|
wral df-frfor 3bitr4g ) CDGZEHZFHZBIZUCCJZKZEASZUDCJZKZFASZACLZKUEUCDJZKZ
|
|
EASZUDDJZKZFASZADLZKABCMABDMUBUKURULUSUBUJUQFAUBUHUOUIUPUBUGUNEAUBUFUMUEC
|
|
DUCNOPCDUDNQPCDARQFEABCTFEABDTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A u v $. $d R u v $. $d S u v $. $d u v x $.
|
|
nffrfor.r $e |- F/_ x R $.
|
|
nffrfor.a $e |- F/_ x A $.
|
|
nffrfor.s $e |- F/_ x S $.
|
|
$( Bound-variable hypothesis builder for well-founded relations.
|
|
(Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
nffrfor $p |- F/ x FrFor R A S $=
|
|
( vv vu wfrfor cv wbr wcel wi wral wss nfcv nfcri nfim nfralxy nfbr nfxfr
|
|
df-frfor nfss ) BCDJHKZIKZCLZUEDMZNZHBOZUFDMZNZIBOZBDPZNAIHBCDUCUMUNAULAI
|
|
BFUJUKAUIAHBFUGUHAAUEUFCAUEQEAUFQUAAHDGRSTAIDGRSTABDFGUDSUB $.
|
|
$}
|
|
|
|
${
|
|
$d A s $. $d R s $. $d s x $.
|
|
nffr.r $e |- F/_ x R $.
|
|
nffr.a $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for well-founded relations.
|
|
(Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
nffr $p |- F/ x R Fr A $=
|
|
( vs wfr cv wfrfor wal df-frind nfcv nffrfor nfal nfxfr ) BCGBCFHZIZFJABC
|
|
FKQAFABCPDEAPLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A s x y $. $d B s x y $. $d R s x y $. $d V x $.
|
|
$( A well-founded relation is irreflexive. This is the case where ` A `
|
|
exists. (Contributed by Jim Kingdon, 21-Sep-2021.) $)
|
|
frirrg $p |- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B ) $=
|
|
( vy vx vs wcel wbr wa simpr neldifsnd pm2.65da cv wi wral simplr imbi12d
|
|
wss wceq wfr w3a cdif simpl3 sseldd simpll3 ad2antrr breqtrrd breq1 eleq1
|
|
csn rspcv syl3c velsn sylnibr eldifd ex ralrimiva df-frind df-frfor albii
|
|
wal wfrfor bitri biimpi 3ad2ant1 cvv difexg eleq2 ralbidv sseq2 spcgv syl
|
|
imbi2d 3ad2ant2 mpd adantr mtand ) ACUAZADHZBAHZUBZBBCIZAABUKZUCZSZWBWFBW
|
|
EHZWBWFJZAWEBWBWFKVSVTWAWFUDUEWHBALMWBWCJZENZFNZCIZWJWEHZOZEAPZWKWEHZOZFA
|
|
PZWFWIWQFAWIWKAHZJZWOWPWTWOJZWKAWDWIWSWOQXAWKBTZWKWDHXAXBWGXAXBJZWAWOBWKC
|
|
IZWGWTWAWOXBVSVTWAWCWSUFUGWTWOXBQXCBBWKCWTWCWOXBWBWCWSQUGXAXBKUHWNXDWGOEB
|
|
AWJBTWLXDWMWGWJBWKCUIWJBWEUJRULUMXCBALMFBUNUOUPUQURWBWRWFOZWCWBWLWJGNZHZO
|
|
ZEAPZWKXFHZOZFAPZAXFSZOZGVBZXEVSVTXOWAVSXOVSACXFVCZGVBXOACGUSXPXNGFEACXFU
|
|
TVAVDVEVFVTVSXOXEOZWAVTWEVGHXQAWDDVHXNXEGWEVGXFWETZXLWRXMWFXRXKWQFAXRXIWO
|
|
XJWPXRXHWNEAXRXGWMWLXFWEWJVIVNVJXFWEWKVIRVJXFWEAVKRVLVMVOVPVQVPVR $.
|
|
$}
|
|
|
|
${
|
|
$d R s x y $.
|
|
$( Any relation is well-founded on the empty set. (Contributed by NM,
|
|
17-Sep-1993.) $)
|
|
fr0 $p |- R Fr (/) $=
|
|
( vs vy vx c0 wfr cv wfrfor df-frind wbr wel wi wral wss 0ss a1i df-frfor
|
|
mpbir mpgbir ) EAFEABGZHZBEABIUACGDGAJCBKLCEMDBKLDEMZETNZLUCUBTOPDCEATQRS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y s z $. $d R x y s z $. $d ch x $. $d ph y s z $. $d ps x z $.
|
|
frind.sb $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
frind.ind $e |- ( ( ch /\ x e. A ) ->
|
|
( A. y e. A ( y R x -> ps ) -> ph ) ) $.
|
|
frind.fr $e |- ( ch -> R Fr A ) $.
|
|
frind.a $e |- ( ch -> A e. V ) $.
|
|
$( Induction over a well-founded set. (Contributed by Jim Kingdon,
|
|
28-Sep-2021.) $)
|
|
frind $p |- ( ( ch /\ x e. A ) -> ph ) $=
|
|
( vz vs wss wral cv wcel wi sylib crab wa wbr wsb ralrimiva nfv nfs1v weq
|
|
nfim breq2 imbi1d ralbidv sbequ12 imbi12d cbvral wb elrab3 imbi2d ralbiia
|
|
a1i nfcv elrabf baib sylibr wfrfor wal wfr df-frind rabexg frforeq3 spcgv
|
|
cvv 3syl mpd df-frfor ssrab simprd r19.21bi ) CADFCFFOZADFPZCFADFUAZOZVSV
|
|
TUBCEQZMQZGUCZWCWARZSZEFPZWDWARZSZMFPZWBCWEBSZEFPZADMUDZSZMFPZWKCWCDQZGUC
|
|
ZBSZEFPZASZDFPWPCXADFJUEXAWODMFXAMUFWMWNDWMDUFADMUGZUIDMUHZWTWMAWNXCWSWLE
|
|
FXCWRWEBWQWDWCGUJUKULADMUMZUNUOTWJWOMFWDFRZWHWMWIWNWHWMUPXEWGWLEFWCFRWFBW
|
|
EABDWCFIUQURUSUTWIXEWNAWNDWDFDWDVADFVAXBXDVBVCUNUSVDCFGWAVEZWKWBSCFGNQZVE
|
|
ZNVFZXFCFGVGXIKFGNVHTCFHRWAVLRXIXFSLADFHVIXHXFNWAVLFGXGWAVJVKVMVNMEFGWAVO
|
|
TVNADFFVPTVQVR $.
|
|
$}
|
|
|
|
$( Irreflexivity of the epsilon relation: a class founded by epsilon is not a
|
|
member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario
|
|
Carneiro, 22-Jun-2015.) $)
|
|
efrirr $p |- ( _E Fr A -> -. A e. A ) $=
|
|
( cep wfr wcel wa wbr wn frirrg 3anidm23 wb epelg adantl mtbid pm2.01da ) A
|
|
BCZAADZOPEAABFZPOPQGAABAHIPQPJOAAAKLMN $.
|
|
|
|
$( Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom
|
|
of Regularity is not required due to antecedent ` _E Fr A ` .
|
|
(Contributed by NM, 4-May-1994.) $)
|
|
tz7.2 $p |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) ) $=
|
|
( wtr cep wfr wcel wss wne wa trss wn wceq efrirr eleq1 syl5ibrcom necon2ad
|
|
notbid anim12ii 3impia ) ACZADEZBAFZBAGZBAHZITUBUCUAUDABJUAUBBAUAUBKBALZAAF
|
|
ZKAMUEUBUFBAANQOPRS $.
|
|
|
|
${
|
|
$d a b c x $. $d a b c A $. $d a b c R $.
|
|
nfwe.r $e |- F/_ x R $.
|
|
nfwe.a $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for well-orderings. (Contributed by
|
|
Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfwe $p |- F/ x R We A $=
|
|
( va vb vc wwe wfr cv wbr wa wi wral df-wetr nfcv nfbr nfan nfralxy nfxfr
|
|
nffr nfim ) BCIBCJZFKZGKZCLZUFHKZCLZMZUEUHCLZNZHBOZGBOZFBOZMAFGHBCPUDUOAA
|
|
BCDEUBUNAFBEUMAGBEULAHBEUJUKAUGUIAAUEUFCAUEQZDAUFQZRAUFUHCUQDAUHQZRSAUEUH
|
|
CUPDURRUCTTTSUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d R x y z $. $d S x y z $.
|
|
$( Equality theorem for the well-ordering predicate. (Contributed by NM,
|
|
9-Mar-1997.) $)
|
|
weeq1 $p |- ( R = S -> ( R We A <-> S We A ) ) $=
|
|
( vx vy vz wceq wfr cv wbr wa wral wwe freq1 breq anbi12d imbi12d ralbidv
|
|
wi df-wetr 3bitr4g ) BCGZABHZDIZEIZBJZUEFIZBJZKZUDUGBJZSZFALZEALZDALZKACH
|
|
ZUDUECJZUEUGCJZKZUDUGCJZSZFALZEALZDALZKABMACMUBUCUOUNVCABCNUBUMVBDAUBULVA
|
|
EAUBUKUTFAUBUIURUJUSUBUFUPUHUQUDUEBCOUEUGBCOPUDUGBCOQRRRPDEFABTDEFACTUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d R x y z $.
|
|
$( Equality theorem for the well-ordering predicate. (Contributed by NM,
|
|
3-Apr-1994.) $)
|
|
weeq2 $p |- ( A = B -> ( R We A <-> R We B ) ) $=
|
|
( vx vy vz wceq wfr cv wbr wa wral freq2 raleq raleqbi1dv anbi12d df-wetr
|
|
wi wwe 3bitr4g ) ABGZACHZDIZEIZCJUDFIZCJKUCUECJRZFALZEALZDALZKBCHZUFFBLZE
|
|
BLZDBLZKACSBCSUAUBUJUIUMABCMUHULDABUGUKEABUFFABNOOPDEFACQDEFBCQT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d R x y z $.
|
|
$( A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) $)
|
|
wefr $p |- ( R We A -> R Fr A ) $=
|
|
( vx vy vz wwe wfr cv wbr wa wi wral df-wetr simplbi ) ABFABGCHZDHZBIPEHZ
|
|
BIJOQBIKEALDALCALCDEABMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d R x y z $. $d V x y z $.
|
|
$( A well-ordering is a partial ordering. (Contributed by Jim Kingdon,
|
|
23-Sep-2021.) $)
|
|
wepo $p |- ( ( R We A /\ A e. V ) -> R Po A ) $=
|
|
( vx vy vz wwe wcel wa cv wbr wfr wefr frirrg syl3an1 3expa wral r19.21bi
|
|
wn anasss w3a wi df-3an df-wetr simprbi adantr sylan2b ispod ) ABGZACHZIZ
|
|
DEFABUIUJDJZAHZULULBKSZUIABLZUJUMUNABMAULBCNOPUMEJZAHZFJZAHZUAUKUMUQIZUSI
|
|
ULUPBKUPURBKIULURBKUBZUMUQUSUCUKUTUSVAUKUTIVAFAUKUMUQVAFAQZUKUMIVBEAUKVBE
|
|
AQZDAUIVCDAQZUJUIUOVDDEFABUDUEUFRRTRTUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( An epsilon well-ordering is a transitive relation. (Contributed by NM,
|
|
22-Apr-1994.) $)
|
|
wetrep $p |- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) ->
|
|
( ( x e. y /\ y e. z ) -> x e. z ) ) $=
|
|
( cep wwe cv wcel w3a wa wbr wel wi df-3an wral wfr df-wetr r19.21bi epel
|
|
anasss simprbi sylan2b anbi12i 3imtr3g ) DEFZAGZDHZBGZDHZCGZDHZIZJUFUHEKZ
|
|
UHUJEKZJZUFUJEKZABLZBCLZJACLULUEUGUIJZUKJUOUPMZUGUIUKNUEUSUKUTUEUSJUTCDUE
|
|
UGUIUTCDOZUEUGJVABDUEVABDOZADUEDEPVBADOABCDEQUARRTRTUBUMUQUNURABSBCSUCACS
|
|
UD $.
|
|
$}
|
|
|
|
${
|
|
$d R x y z $.
|
|
$( Any relation is a well-ordering of the empty set. (Contributed by NM,
|
|
16-Mar-1997.) $)
|
|
we0 $p |- R We (/) $=
|
|
( vx vy vz c0 wwe wfr cv wbr wa wi wral fr0 ral0 df-wetr mpbir2an ) EAFEA
|
|
GBHZCHZAIRDHZAIJQSAIKDELCELZBELAMTBNBCDEAOP $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordinals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c Ord $. $( Ordinal predicate $)
|
|
$c On $. $( The class of ordinal numbers $)
|
|
$c Lim $. $( Limit ordinal predicate $)
|
|
$c suc $. $( Successor function (read: 'successor of') $)
|
|
|
|
$( Extend the definition of a wff to include the ordinal predicate. $)
|
|
word $a wff Ord A $.
|
|
|
|
$( Extend the definition of a class to include the class of all ordinal
|
|
numbers. (The 0 in the name prevents creating a file called con.html,
|
|
which causes problems in Windows.) $)
|
|
con0 $a class On $.
|
|
|
|
$( Extend the definition of a wff to include the limit ordinal predicate. $)
|
|
wlim $a wff Lim A $.
|
|
|
|
$( Extend class notation to include the successor function. $)
|
|
csuc $a class suc A $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define the ordinal predicate, which is true for a class that is
|
|
transitive and whose elements are transitive. Definition of ordinal in
|
|
[Crosilla], p. "Set-theoretic principles incompatible with
|
|
intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use
|
|
its alias ~ dford3 instead for naming consistency with set.mm.
|
|
(New usage is discouraged.) $)
|
|
df-iord $a |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Alias for ~ df-iord . Use it instead of ~ df-iord for naming
|
|
consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.) $)
|
|
dford3 $p |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) ) $=
|
|
( df-iord ) ABC $.
|
|
$}
|
|
|
|
$( Define the class of all ordinal numbers. Definition 7.11 of
|
|
[TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.) $)
|
|
df-on $a |- On = { x | Ord x } $.
|
|
|
|
$( Define the limit ordinal predicate, which is true for an ordinal that has
|
|
the empty set as an element and is not a successor (i.e. that is the union
|
|
of itself). Our definition combines the definition of Lim of
|
|
[BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then
|
|
changes ` A =/= (/) ` to ` (/) e. A ` (which would be equivalent given the
|
|
law of the excluded middle, but which is not for us). (Contributed by Jim
|
|
Kingdon, 11-Nov-2018.) Use its alias ~ dflim2 instead for naming
|
|
consistency with set.mm. (New usage is discouraged.) $)
|
|
df-ilim $a |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) ) $.
|
|
|
|
$( Alias for ~ df-ilim . Use it instead of ~ df-ilim for naming consistency
|
|
with set.mm. (Contributed by NM, 4-Nov-2004.) $)
|
|
dflim2 $p |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) ) $=
|
|
( df-ilim ) AB $.
|
|
|
|
$( Define the successor of a class. When applied to an ordinal number, the
|
|
successor means the same thing as "plus 1". Definition 7.22 of
|
|
[TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our
|
|
definition is a generalization to classes. Although it is not
|
|
conventional to use it with proper classes, it has no effect on a proper
|
|
class ( ~ sucprc ). Some authors denote the successor operation with a
|
|
prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134
|
|
and the definition of successor in [Mendelson] p. 246 (who uses the symbol
|
|
"Suc" as a predicate to mean "is a successor ordinal"). The definition of
|
|
successor of [Enderton] p. 68 denotes the operation with a plus-sign
|
|
superscript. (Contributed by NM, 30-Aug-1993.) $)
|
|
df-suc $a |- suc A = ( A u. { A } ) $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for the ordinal predicate. (Contributed by NM,
|
|
17-Sep-1993.) $)
|
|
ordeq $p |- ( A = B -> ( Ord A <-> Ord B ) ) $=
|
|
( vx wceq wtr cv wral wa word treq raleq anbi12d dford3 3bitr4g ) ABDZAEZ
|
|
CFEZCAGZHBEZQCBGZHAIBIOPSRTABJQCABKLCAMCBMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An ordinal number is an ordinal set. (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
elong $p |- ( A e. V -> ( A e. On <-> Ord A ) ) $=
|
|
( vx cv word con0 ordeq df-on elab2g ) CDZEAECAFBJAGCHI $.
|
|
$}
|
|
|
|
${
|
|
elon.1 $e |- A e. _V $.
|
|
$( An ordinal number is an ordinal set. (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
elon $p |- ( A e. On <-> Ord A ) $=
|
|
( cvv wcel con0 word wb elong ax-mp ) ACDAEDAFGBACHI $.
|
|
$}
|
|
|
|
$( An ordinal number has the ordinal property. (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
eloni $p |- ( A e. On -> Ord A ) $=
|
|
( con0 wcel word elong ibi ) ABCADABEF $.
|
|
|
|
$( An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) $)
|
|
elon2 $p |- ( A e. On <-> ( Ord A /\ A e. _V ) ) $=
|
|
( con0 wcel word cvv wa eloni elex jca elong biimparc impbii ) ABCZADZAECZF
|
|
MNOAGABHIOMNAEJKL $.
|
|
|
|
$( Equality theorem for the limit predicate. (Contributed by NM,
|
|
22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
limeq $p |- ( A = B -> ( Lim A <-> Lim B ) ) $=
|
|
( wceq word c0 wcel cuni wlim ordeq eleq2 id unieq eqeq12d 3anbi123d dflim2
|
|
w3a 3bitr4g ) ABCZADZEAFZAAGZCZPBDZEBFZBBGZCZPAHBHRSUCTUDUBUFABIABEJRABUAUE
|
|
RKABLMNAOBOQ $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) $)
|
|
ordtr $p |- ( Ord A -> Tr A ) $=
|
|
( vx word wtr cv wral dford3 simplbi ) ACADBEDBAFBAGH $.
|
|
$}
|
|
|
|
$( An element of an ordinal class is a subset of it. (Contributed by NM,
|
|
30-May-1994.) $)
|
|
ordelss $p |- ( ( Ord A /\ B e. A ) -> B C_ A ) $=
|
|
( word wtr wcel wss ordtr trss imp sylan ) ACADZBAEZBAFZAGKLMABHIJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( A transitive subclass of an ordinal class is ordinal. (Contributed by
|
|
NM, 29-May-1994.) $)
|
|
trssord $p |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A ) $=
|
|
( vx wtr wss word w3a cv wral dford3 simprbi ssralv syl5 imp anim2i 3impb
|
|
wa sylibr ) ADZABEZBFZGSCHDZCAIZQZAFSTUAUDTUAQUCSTUAUCUAUBCBIZTUCUABDUECB
|
|
JKUBCABLMNOPCAJR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An element of an ordinal class is ordinal. Proposition 7.6 of
|
|
[TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) $)
|
|
ordelord $p |- ( ( Ord A /\ B e. A ) -> Ord B ) $=
|
|
( vx word wcel cv wa wceq eleq1 anbi2d ordeq imbi12d wtr wss wral simprbi
|
|
wi dford3 r19.21bi ordelss simpl trssord syl3anc vtoclg anabsi7 ) ADZBAEZ
|
|
BDZUFCFZAEZGZUIDZQUFUGGZUHQCBAUIBHZUKUMULUHUNUJUGUFUIBAIJUIBKLUKUIMZUIANU
|
|
FULUFUOCAUFAMUOCAOCARPSAUITUFUJUAUIAUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The class of all ordinal numbers is transitive. (Contributed by NM,
|
|
4-May-2009.) $)
|
|
tron $p |- Tr On $=
|
|
( vx vy con0 wtr cv wss dftr3 wcel word vex elon ordelord sylanb ex ssrdv
|
|
syl6ibr mprgbir ) CDAEZCFACACGRCHZBRCSBEZRHZTIZTCHSUAUBSRIUAUBRAJKRTLMNTB
|
|
JKPOQ $.
|
|
$}
|
|
|
|
$( An element of an ordinal class is an ordinal number. (Contributed by NM,
|
|
26-Oct-2003.) $)
|
|
ordelon $p |- ( ( Ord A /\ B e. A ) -> B e. On ) $=
|
|
( word wcel wa con0 ordelord wb elong adantl mpbird ) ACZBADZEBFDZBCZABGMNO
|
|
HLBAIJK $.
|
|
|
|
$( An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of
|
|
[BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) $)
|
|
onelon $p |- ( ( A e. On /\ B e. A ) -> B e. On ) $=
|
|
( con0 wcel word eloni ordelon sylan ) ACDAEBADBCDAFABGH $.
|
|
|
|
$( The intersection of two ordinal classes is ordinal. Proposition 7.9 of
|
|
[TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.) $)
|
|
ordin $p |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) ) $=
|
|
( word cin wtr ordtr trin syl2an wss inss2 trssord mp3an2 sylancom ) ACZBCZ
|
|
ABDZEZPCZNAEBEQOAFBFABGHQPBIORABJPBKLM $.
|
|
|
|
$( The intersection of two ordinal numbers is an ordinal number.
|
|
(Contributed by NM, 7-Apr-1995.) $)
|
|
onin $p |- ( ( A e. On /\ B e. On ) -> ( A i^i B ) e. On ) $=
|
|
( con0 wcel wa cin word eloni ordin syl2an cvv wb simpl inex1g elong mpbird
|
|
3syl ) ACDZBCDZEZABFZCDZUAGZRAGBGUCSAHBHABIJTRUAKDUBUCLRSMABCNUAKOQP $.
|
|
|
|
$( An element of an ordinal number is a subset of the number. (Contributed
|
|
by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
onelss $p |- ( A e. On -> ( B e. A -> B C_ A ) ) $=
|
|
( con0 wcel word wss wi eloni ordelss ex syl ) ACDAEZBADZBAFZGAHLMNABIJK $.
|
|
|
|
$( Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) $)
|
|
ordtr1 $p |- ( Ord C -> ( ( A e. B /\ B e. C ) -> A e. C ) ) $=
|
|
( word wtr wcel wa wi ordtr trel syl ) CDCEABFBCFGACFHCICABJK $.
|
|
|
|
$( Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192.
|
|
(Contributed by NM, 11-Aug-1994.) $)
|
|
ontr1 $p |- ( C e. On -> ( ( A e. B /\ B e. C ) -> A e. C ) ) $=
|
|
( con0 wcel word wa wi eloni ordtr1 syl ) CDECFABEBCEGACEHCIABCJK $.
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
onintss.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( If a property is true for an ordinal number, then the minimum ordinal
|
|
number for which it is true is smaller or equal. Theorem Schema 61 of
|
|
[Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) $)
|
|
onintss $p |- ( A e. On -> ( ps -> |^| { x e. On | ph } C_ A ) ) $=
|
|
( con0 wcel crab cint wss intminss ex ) DFGBACFHIDJABCDFEKL $.
|
|
$}
|
|
|
|
$( The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) $)
|
|
ord0 $p |- Ord (/) $=
|
|
( vx c0 word wtr cv wral tr0 ral0 dford3 mpbir2an ) BCBDAEDZABFGKAHABIJ $.
|
|
|
|
$( The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193.
|
|
(Contributed by NM, 17-Sep-1993.) $)
|
|
0elon $p |- (/) e. On $=
|
|
( c0 con0 wcel word ord0 0ex elon mpbir ) ABCADEAFGH $.
|
|
|
|
$( The intersection of the class of ordinal numbers is the empty set.
|
|
(Contributed by NM, 20-Oct-2003.) $)
|
|
inton $p |- |^| On = (/) $=
|
|
( c0 con0 wcel cint wceq 0elon int0el ax-mp ) ABCBDAEFBGH $.
|
|
|
|
$( The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
nlim0 $p |- -. Lim (/) $=
|
|
( c0 wlim word wcel cuni wceq w3a noel simp2 mto dflim2 mtbir ) ABACZAADZAA
|
|
EFZGZPNAHMNOIJAKL $.
|
|
|
|
$( A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) $)
|
|
limord $p |- ( Lim A -> Ord A ) $=
|
|
( wlim word c0 wcel cuni wceq dflim2 simp1bi ) ABACDAEAAFGAHI $.
|
|
|
|
$( A limit ordinal is its own supremum (union). (Contributed by NM,
|
|
4-May-1995.) $)
|
|
limuni $p |- ( Lim A -> A = U. A ) $=
|
|
( wlim word c0 wcel cuni wceq dflim2 simp3bi ) ABACDAEAAFGAHI $.
|
|
|
|
$( The union of a limit ordinal is a limit ordinal. (Contributed by NM,
|
|
19-Sep-2006.) $)
|
|
limuni2 $p |- ( Lim A -> Lim U. A ) $=
|
|
( wlim cuni wceq wb limuni limeq syl ibi ) ABZACZBZJAKDJLEAFAKGHI $.
|
|
|
|
$( A limit ordinal contains the empty set. (Contributed by NM,
|
|
15-May-1994.) $)
|
|
0ellim $p |- ( Lim A -> (/) e. A ) $=
|
|
( wlim word c0 wcel cuni wceq dflim2 simp2bi ) ABACDAEAAFGAHI $.
|
|
|
|
$( A limit ordinal class that is also a set is an ordinal number.
|
|
(Contributed by NM, 26-Apr-2004.) $)
|
|
limelon $p |- ( ( A e. B /\ Lim A ) -> A e. On ) $=
|
|
( wcel wlim con0 word limord elong syl5ibr imp ) ABCZADZAECZLMKAFAGABHIJ $.
|
|
|
|
$( The class of all ordinal numbers is not empty. (Contributed by NM,
|
|
17-Sep-1995.) $)
|
|
onn0 $p |- On =/= (/) $=
|
|
( c0 con0 wcel wne 0elon ne0i ax-mp ) ABCBADEBAFG $.
|
|
|
|
$( The class of all ordinal numbers is inhabited. (Contributed by Jim
|
|
Kingdon, 6-Mar-2019.) $)
|
|
onm $p |- E. x x e. On $=
|
|
( cv c0 wceq con0 wcel wa wex 0elon 0ex eleq1 ceqsexv mpbir exsimpr ax-mp )
|
|
ABZCDZPEFZGAHZRAHSCEFZIRTACJPCEKLMQRANO $.
|
|
|
|
$( Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
suceq $p |- ( A = B -> suc A = suc B ) $=
|
|
( wceq csn cun csuc id sneq uneq12d df-suc 3eqtr4g ) ABCZAADZEBBDZEAFBFLABM
|
|
NLGABHIAJBJK $.
|
|
|
|
$( Membership in a successor. This one-way implication does not require that
|
|
either ` A ` or ` B ` be sets. (Contributed by NM, 6-Jun-1994.) $)
|
|
elsuci $p |- ( A e. suc B -> ( A e. B \/ A = B ) ) $=
|
|
( csuc wcel csn wo wceq cun df-suc eleq2i elun bitri elsni orim2i sylbi ) A
|
|
BCZDZABDZABEZDZFZRABGZFQABSHZDUAPUCABIJABSKLTUBRABMNO $.
|
|
|
|
$( Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.
|
|
(Contributed by NM, 15-Sep-1995.) $)
|
|
elsucg $p |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) ) $=
|
|
( csuc wcel csn wo wceq cun df-suc eleq2i elun bitri elsng orbi2d syl5bb )
|
|
ABDZEZABEZABFZEZGZACEZSABHZGRABTIZEUBQUEABJKABTLMUCUAUDSABCNOP $.
|
|
|
|
$( Variant of membership in a successor, requiring that ` B ` rather than
|
|
` A ` be a set. (Contributed by NM, 28-Oct-2003.) $)
|
|
elsuc2g $p |- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) ) $=
|
|
( csuc wcel csn cun wceq wo df-suc eleq2i elun elsn2g orbi2d syl5bb ) ABDZE
|
|
ABBFZGZEZBCEZABEZABHZIZPRABJKSUAAQEZITUCABQLTUDUBUAABCMNOO $.
|
|
|
|
${
|
|
elsuc.1 $e |- A e. _V $.
|
|
$( Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.
|
|
(Contributed by NM, 15-Sep-2003.) $)
|
|
elsuc $p |- ( A e. suc B <-> ( A e. B \/ A = B ) ) $=
|
|
( cvv wcel csuc wceq wo wb elsucg ax-mp ) ADEABFEABEABGHICABDJK $.
|
|
|
|
$( Membership in a successor. (Contributed by NM, 15-Sep-2003.) $)
|
|
elsuc2 $p |- ( B e. suc A <-> ( B e. A \/ B = A ) ) $=
|
|
( cvv wcel csuc wceq wo wb elsuc2g ax-mp ) ADEBAFEBAEBAGHICBADJK $.
|
|
$}
|
|
|
|
${
|
|
nfsuc.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for successor. (Contributed by NM,
|
|
15-Sep-2003.) $)
|
|
nfsuc $p |- F/_ x suc A $=
|
|
( csuc csn cun df-suc nfsn nfun nfcxfr ) ABDBBEZFBGABKCABCHIJ $.
|
|
$}
|
|
|
|
$( Membership in a successor. (Contributed by NM, 20-Jun-1998.) $)
|
|
elelsuc $p |- ( A e. B -> A e. suc B ) $=
|
|
( wcel csuc wceq wo orc elsucg mpbird ) ABCZABDCJABEZFJKGABBHI $.
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
$( Membership of a successor in another class. (Contributed by NM,
|
|
29-Jun-2004.) $)
|
|
sucel $p |- ( suc A e. B <->
|
|
E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) ) $=
|
|
( csuc wcel cv wceq wrex wo wb wal risset dfcleq elsuc bibi2i albii bitri
|
|
vex rexbii ) CEZDFAGZUAHZADIBGZUBFZUDCFUDCHJZKZBLZADIAUADMUCUHADUCUEUDUAF
|
|
ZKZBLUHBUBUANUJUGBUIUFUEUDCBSOPQRTR $.
|
|
$}
|
|
|
|
$( The successor of the empty set. (Contributed by NM, 1-Feb-2005.) $)
|
|
suc0 $p |- suc (/) = { (/) } $=
|
|
( c0 csuc csn cun df-suc uncom un0 3eqtri ) ABAACZDIADIAEAIFIGH $.
|
|
|
|
$( A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) $)
|
|
sucprc $p |- ( -. A e. _V -> suc A = A ) $=
|
|
( cvv wcel wn csuc cun csn df-suc wceq snprc uneq2 sylbi syl5eq un0 syl6eq
|
|
c0 ) ABCDZAEZAPFZAQRAAGZFZSAHQTPIUASIAJTPAKLMANO $.
|
|
|
|
${
|
|
unisuc.1 $e |- A e. _V $.
|
|
$( A transitive class is equal to the union of its successor. Combines
|
|
Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.
|
|
(Contributed by NM, 30-Aug-1993.) $)
|
|
unisuc $p |- ( Tr A <-> U. suc A = A ) $=
|
|
( cuni wss cun wceq wtr csuc ssequn1 df-tr csn df-suc unieqi uniun uneq2i
|
|
unisn 3eqtri eqeq1i 3bitr4i ) ACZADTAEZAFAGAHZCZAFTAIAJUCUAAUCAAKZEZCTUDC
|
|
ZEUAUBUEALMAUDNUFATABPOQRS $.
|
|
$}
|
|
|
|
$( A transitive class is equal to the union of its successor. Combines
|
|
Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.
|
|
(Contributed by Jim Kingdon, 18-Aug-2019.) $)
|
|
unisucg $p |- ( A e. V -> ( Tr A <-> U. suc A = A ) ) $=
|
|
( wcel csuc cuni cun wtr csn df-suc unieqi uniun eqtri unisng uneq2d syl5eq
|
|
wceq eqeq1d wss df-tr ssequn1 bitri syl6rbbr ) ABCZADZEZAPAEZAFZAPZAGZUCUEU
|
|
GAUCUEUFAHZEZFZUGUEAUJFZEULUDUMAIJAUJKLUCUKAUFABMNOQUIUFARUHASUFATUAUB $.
|
|
|
|
$( A class is included in its own successor. Part of Proposition 7.23 of
|
|
[TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by
|
|
NM, 31-May-1994.) $)
|
|
sssucid $p |- A C_ suc A $=
|
|
( csn cun csuc ssun1 df-suc sseqtr4i ) AAABZCADAHEAFG $.
|
|
|
|
$( Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized).
|
|
(Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton,
|
|
20-Feb-2012.) $)
|
|
sucidg $p |- ( A e. V -> A e. suc A ) $=
|
|
( wcel csuc wceq wo eqid olci elsucg mpbiri ) ABCAADCAACZAAEZFLKAGHAABIJ $.
|
|
|
|
${
|
|
sucid.1 $e |- A e. _V $.
|
|
$( A set belongs to its successor. (Contributed by NM, 22-Jun-1994.)
|
|
(Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott
|
|
Fenton, 20-Feb-2012.) $)
|
|
sucid $p |- A e. suc A $=
|
|
( cvv wcel csuc sucidg ax-mp ) ACDAAEDBACFG $.
|
|
$}
|
|
|
|
$( No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) $)
|
|
nsuceq0g $p |- ( A e. V -> suc A =/= (/) ) $=
|
|
( wcel csuc c0 wceq noel sucidg eleq2 syl5ibcom mtoi neneqad ) ABCZADZEMNEF
|
|
ZAECZAGMANCOPABHNEAIJKL $.
|
|
|
|
${
|
|
eqelsuc.1 $e |- A e. _V $.
|
|
$( A set belongs to the successor of an equal set. (Contributed by NM,
|
|
18-Aug-1994.) $)
|
|
eqelsuc $p |- ( A = B -> A e. suc B ) $=
|
|
( wceq csuc sucid suceq syl5eleq ) ABDAAEBEACFABGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d C x $.
|
|
iunsuc.1 $e |- A e. _V $.
|
|
iunsuc.2 $e |- ( x = A -> B = C ) $.
|
|
$( Inductive definition for the indexed union at a successor. (Contributed
|
|
by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro,
|
|
18-Nov-2016.) $)
|
|
iunsuc $p |- U_ x e. suc A B = ( U_ x e. A B u. C ) $=
|
|
( csuc ciun csn cun wceq df-suc iuneq1 ax-mp iunxun iunxsn uneq2i 3eqtri
|
|
) ABGZCHZABBIZJZCHZABCHZAUACHZJUDDJSUBKTUCKBLASUBCMNABUACOUEDUDABCDEFPQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d y z A $.
|
|
$( The successor of a transitive class is transitive. (Contributed by Alan
|
|
Sare, 11-Apr-2009.) $)
|
|
suctr $p |- ( Tr A -> Tr suc A ) $=
|
|
( vz vy wtr cv wcel csuc wa wi wal wceq simpr vex elsuc sylib simpl eleq2
|
|
wo syl6 mpdi syl5ibcom elelsuc trel expd adantrd syl8 jao alrimivv sylibr
|
|
dftr2 ) ADZBEZCEZFZUMAGZFZHZULUOFZIZCJBJUODUKUSBCUKUQUMAFZUMAKZRZURUQUPVB
|
|
UNUPLUMACMNOUKUQVAURIZVBURIZUQVAULAFZURUQUNVAVEUNUPPUMAULQUAULAUBZSUKUQUT
|
|
URIVCVDIUKUQUTVEURUKUNUTVEIUPUKUNUTVEAULUMUCUDUEVFUFUTURVAUGSTTUHBCUOUJUI
|
|
$.
|
|
$}
|
|
|
|
$( A set whose successor belongs to a transitive class also belongs.
|
|
(Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon,
|
|
12-Aug-2011.) $)
|
|
trsuc $p |- ( ( Tr A /\ suc B e. A ) -> B e. A ) $=
|
|
( wtr csuc wcel cvv wss sssucid ssexg mpan sucidg syl ancri trel syl5 imp
|
|
wa ) ACZBDZAEZBAEZTBSEZTQRUATUBTBFEZUBBSGTUCBHBSAIJBFKLMABSNOP $.
|
|
|
|
$( A member of the successor of a transitive class is a subclass of it.
|
|
(Contributed by NM, 4-Oct-2003.) $)
|
|
trsucss $p |- ( Tr A -> ( B e. suc A -> B C_ A ) ) $=
|
|
( csuc wcel wceq wo wtr wss elsuci trss wi eqimss a1i jaod syl5 ) BACDBADZB
|
|
AEZFAGZBAHZBAIRPSQABJQSKRBALMNO $.
|
|
|
|
$( A set whose successor is a subset of another class is a member of that
|
|
class. (Contributed by NM, 16-Sep-1995.) $)
|
|
sucssel $p |- ( A e. V -> ( suc A C_ B -> A e. B ) ) $=
|
|
( wcel csuc wss sucidg ssel syl5com ) ACDAAEZDJBFABDACGJBAHI $.
|
|
|
|
$( An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) $)
|
|
orduniss $p |- ( Ord A -> U. A C_ A ) $=
|
|
( word wtr cuni wss ordtr df-tr sylib ) ABACADAEAFAGH $.
|
|
|
|
${
|
|
on.1 $e |- A e. On $.
|
|
$( An ordinal number is an ordinal class. (Contributed by NM,
|
|
11-Jun-1994.) $)
|
|
onordi $p |- Ord A $=
|
|
( con0 wcel word eloni ax-mp ) ACDAEBAFG $.
|
|
|
|
$( An ordinal number is a transitive class. (Contributed by NM,
|
|
11-Jun-1994.) $)
|
|
ontrci $p |- Tr A $=
|
|
( word wtr onordi ordtr ax-mp ) ACADABEAFG $.
|
|
|
|
$( A member of an ordinal number is an ordinal number. Theorem 7M(a) of
|
|
[Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) $)
|
|
oneli $p |- ( B e. A -> B e. On ) $=
|
|
( con0 wcel onelon mpan ) ADEBAEBDECABFG $.
|
|
|
|
$( A member of an ordinal number is a subset of it. (Contributed by NM,
|
|
11-Aug-1994.) $)
|
|
onelssi $p |- ( B e. A -> B C_ A ) $=
|
|
( con0 wcel wss wi onelss ax-mp ) ADEBAEBAFGCABHI $.
|
|
|
|
$( An element of an ordinal number equals the intersection with it.
|
|
(Contributed by NM, 11-Jun-1994.) $)
|
|
onelini $p |- ( B e. A -> B = ( B i^i A ) ) $=
|
|
( wcel wss cin wceq onelssi dfss sylib ) BADBAEBBAFGABCHBAIJ $.
|
|
|
|
$( An ordinal number equals its union with any element. (Contributed by
|
|
NM, 13-Jun-1994.) $)
|
|
oneluni $p |- ( B e. A -> ( A u. B ) = A ) $=
|
|
( wcel wss cun wceq onelssi ssequn2 sylib ) BADBAEABFAGABCHBAIJ $.
|
|
|
|
$( An ordinal number is equal to the union of its successor. (Contributed
|
|
by NM, 12-Jun-1994.) $)
|
|
onunisuci $p |- U. suc A = A $=
|
|
( wtr csuc cuni wceq ontrci con0 elexi unisuc mpbi ) ACADEAFABGAAHBIJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
IZF Set Theory - add the Axiom of Union
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Union
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d w x y z $.
|
|
$( Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory.
|
|
It states that a set ` y ` exists that includes the union of a given set
|
|
` x ` i.e. the collection of all members of the members of ` x ` . The
|
|
variant ~ axun2 states that the union itself exists. A version with the
|
|
standard abbreviation for union is ~ uniex2 . A version using class
|
|
notation is ~ uniex .
|
|
|
|
This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a)
|
|
unnecessary quantifiers are removed, (b) Crosilla has a biconditional
|
|
rather than an implication (but the two are equivalent by ~ bm1.3ii ),
|
|
and (c) the order of the conjuncts is swapped (which is equivalent by
|
|
~ ancom ).
|
|
|
|
The union of a class ~ df-uni should not be confused with the union of
|
|
two classes ~ df-un . Their relationship is shown in ~ unipr .
|
|
(Contributed by NM, 23-Dec-1993.) $)
|
|
ax-un $a |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y ) $.
|
|
|
|
$( Axiom of Union expressed with the fewest number of different variables.
|
|
(Contributed by NM, 14-Aug-2003.) $)
|
|
zfun $p |- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x ) $=
|
|
( vw wel wa wex wi wal ax-un weq elequ2 elequ1 anbi12d cbvexv albii exbii
|
|
imbi1i mpbi ) BDEZDCEZFZDGZBAEZHZBIZAGUDACEZFZAGZUDHZBIZAGCABDJUFUKAUEUJB
|
|
UCUIUDUBUHDADAKTUDUAUGDABLDACMNORPQS $.
|
|
|
|
$( A variant of the Axiom of Union ~ ax-un . For any set ` x ` , there
|
|
exists a set ` y ` whose members are exactly the members of the members
|
|
of ` x ` i.e. the union of ` x ` . Axiom Union of [BellMachover]
|
|
p. 466. (Contributed by NM, 4-Jun-2006.) $)
|
|
axun2 $p |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) ) $=
|
|
( wel wa wex ax-un bm1.3ii ) CDEDAEFDGBCABCDHI $.
|
|
|
|
$( The Axiom of Union using the standard abbreviation for union. Given any
|
|
set ` x ` , its union ` y ` exists. (Contributed by NM, 4-Jun-2006.) $)
|
|
uniex2 $p |- E. y y = U. x $=
|
|
( vz cv cuni wceq wex wcel wb wi wa zfun eluni imbi1i albii exbii bm1.3ii
|
|
wal mpbir dfcleq ) BDZADZEZFZBGCDZUAHZUEUCHZICRZBGUGBCUGUFJZCRZBGUFUAUBHK
|
|
BGZUFJZCRZBGBCALUJUMBUIULCUGUKUFBUEUBMNOPSQUDUHBCUAUCTPS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
uniex.1 $e |- A e. _V $.
|
|
$( The Axiom of Union in class notation. This says that if ` A ` is a set
|
|
i.e. ` A e. _V ` (see ~ isset ), then the union of ` A ` is also a set.
|
|
Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM,
|
|
11-Aug-1993.) $)
|
|
uniex $p |- U. A e. _V $=
|
|
( vx vy cv cuni cvv wcel wceq unieq eleq1d uniex2 issetri vtocl ) CEZFZGH
|
|
AFZGHCABOAIPQGOAJKDPCDLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The ZF Axiom of Union in class notation, in the form of a theorem
|
|
instead of an inference. We use the antecedent ` A e. V ` instead of
|
|
` A e. _V ` to make the theorem more general and thus shorten some
|
|
proofs; obviously the universal class constant ` _V ` is one possible
|
|
substitution for class variable ` V ` . (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
uniexg $p |- ( A e. V -> U. A e. _V ) $=
|
|
( vx cv cuni cvv wcel wceq unieq eleq1d vex uniex vtoclg ) CDZEZFGAEZFGCA
|
|
BNAHOPFNAIJNCKLM $.
|
|
$}
|
|
|
|
${
|
|
unex.1 $e |- A e. _V $.
|
|
unex.2 $e |- B e. _V $.
|
|
$( The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16.
|
|
(Contributed by NM, 1-Jul-1994.) $)
|
|
unex $p |- ( A u. B ) e. _V $=
|
|
( cpr cuni cun cvv unipr wcel prexg mp2an uniex eqeltrri ) ABEZFABGHABCDI
|
|
OAHJBHJOHJCDABHHKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Existence of union is equivalent to existence of its components.
|
|
(Contributed by NM, 11-Jun-1998.) $)
|
|
unexb $p |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V ) $=
|
|
( vx vy cvv wcel wa cun cv wceq uneq1 eleq1d uneq2 vex unex vtocl2g ssun1
|
|
wss ssexg mpan ssun2 jca impbii ) AEFZBEFZGABHZEFZCIZDIZHZEFAUIHZEFUGCDAB
|
|
EEUHAJUJUKEUHAUIKLUIBJUKUFEUIBAMLUHUICNDNOPUGUDUEAUFRUGUDABQAUFESTBUFRUGU
|
|
EBAUABUFESTUBUC $.
|
|
$}
|
|
|
|
$( A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16.
|
|
(Contributed by NM, 18-Sep-2006.) $)
|
|
unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
|
|
( wcel cvv cun elex wa unexb biimpi syl2an ) ACEAFEZBFEZABGFEZBDEACHBDHMNIO
|
|
ABJKL $.
|
|
|
|
$( An unordered triple of classes exists. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
tpexg $p |- ( ( A e. U /\ B e. V /\ C e. W ) -> { A , B , C } e. _V ) $=
|
|
( wcel w3a ctp cpr csn cun cvv df-tp wa prexg snexg anim12i 3impa unexg syl
|
|
syl5eqel ) ADGZBEGZCFGZHZABCIABJZCKZLZMABCNUFUGMGZUHMGZOZUIMGUCUDUEULUCUDOU
|
|
JUEUKABDEPCFQRSUGUHMMTUAUB $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Union of a singleton in the form of a restricted class abstraction.
|
|
(Contributed by NM, 3-Jul-2008.) $)
|
|
unisn3 $p |- ( A e. B -> U. { x e. B | x = A } = A ) $=
|
|
( wcel cv wceq crab cuni csn rabsn unieqd unisng eqtrd ) BCDZAEBFACGZHBIZ
|
|
HBNOPACBJKBCLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The class of all singletons is a proper class. (Contributed by NM,
|
|
10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) $)
|
|
snnex $p |- { x | E. y x = { y } } e/ _V $=
|
|
( vz cv csn wceq wex cab wcel cuni vprc wa vsnid a9ev sneq equcoms eximii
|
|
cvv vex snex eleq2 eqeq1 exbidv anbi12d spcev mp2an mpbir 2th eqriv mtbir
|
|
eluniab eleq1i uniexg mto nelir ) ADZBDZEZFZBGZAHZRVARIVAJZRIZVCRRIKVBRRC
|
|
VBRCDZVBIZVDRIVEVDUPIZUTLZAGZVDVDEZIZVIURFZBGZVHCMUQVDFVKBBCNVKCBVDUQOPQV
|
|
GVJVLLAVIVDCSZTUPVIFZVFVJUTVLUPVIVDUAVNUSVKBUPVIURUBUCUDUEUFUTAVDUKUGVMUH
|
|
UIULUJVARUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
opeluu.1 $e |- A e. _V $.
|
|
opeluu.2 $e |- B e. _V $.
|
|
$( Each member of an ordered pair belongs to the union of the union of a
|
|
class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41.
|
|
(Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro,
|
|
27-Feb-2016.) $)
|
|
opeluu $p |- ( <. A , B >. e. C ->
|
|
( A e. U. U. C /\ B e. U. U. C ) ) $=
|
|
( cop wcel cuni cpr prid1 opi2 elunii mpan sylancr prid2 jca ) ABFZCGZACH
|
|
ZHZGZBTGZRAABIZGUCSGZUAABDJUCQGRUDABDEKUCQCLMZAUCSLNRBUCGUDUBABEOUEBUCSLN
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d A x y v z $. $d A x y u z $.
|
|
$( Expression for double union that moves union into a class builder.
|
|
(Contributed by FL, 28-May-2007.) $)
|
|
uniuni $p |- U. U. A = U. { x | E. y ( x = U. y /\ y e. A ) } $=
|
|
( vz vu vv wel cv cuni wcel wex cab wceq eluni anbi2i exbii 19.42v bicomi
|
|
wa 3bitri excom anass ancom bitr3i 2exbii exdistr vex uniex eleq2 ceqsexv
|
|
exancom bitri 3bitr2i eqeq1 anbi1d exbidv elab abbii df-uni 3eqtr4i ) DEG
|
|
ZEHZCIZJZSZEKZDLDFGZFHZAHZBHZIZMZVJCJZSZBKZALZJZSZFKZDLVCIVPIVFVSDVFVAEBG
|
|
ZVMSZBKZSZEKZVMDHZVKJZSZBKZVSVEWCEVDWBVABVBCNOPWDVAWASZBKZEKZVMVAVTSZEKZS
|
|
ZBKZWHWCWJEWJWCVAWABQRPWKWIEKBKVMWLSZEKBKWOWIEBUAWIWPBEWIWLVMSWPVAVTVMUBW
|
|
LVMUCUDUEVMWLBEUFTWNWGBWMWFVMWFWMEWEVJNROPTWHVGVHVKMZVMSZSZFKZBKWSBKFKZVS
|
|
WGWTBWGVMVGWQSZFKZSVMXBSZFKWTWFXCVMWFWQVGSFKXCVGWFFVKVJBUGUHVHVKWEUIUJWQV
|
|
GFUKUDOVMXBFQXDWSFXDXBVMSWSVMXBUCVGWQVMUBULPUMPWSBFUAXAVGWRBKZSZFKVSVGWRF
|
|
BUFXFVRFXEVQVGVQXEVOXEAVHFUGVIVHMZVNWRBXGVLWQVMVIVHVKUNUOUPUQROPULTTURDEV
|
|
CUSDFVPUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A y z $.
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` A ( x ) ` . (Contributed by NM, 14-Oct-2010.) $)
|
|
eusv1 $p |- ( E! y A. x y = A <-> E. y A. x y = A ) $=
|
|
( vz cv wceq wal weu wex wa wi sp eqtr3 syl2an gen2 eqeq1 albidv mpbiran2
|
|
eu4 ) BEZCFZAGZBHUBBIUBDEZCFZAGZJTUCFZKZDGBGUGBDUBUAUDUFUEUAALUDALTUCCMNO
|
|
UBUEBDUFUAUDATUCCPQSR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d A y z w $.
|
|
$( Even if ` x ` is free in ` A ` , it is effectively bound when
|
|
` A ( x ) ` is single-valued. (Contributed by NM, 14-Oct-2010.)
|
|
(Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
eusvnf $p |- ( E! y A. x y = A -> F/_ x A ) $=
|
|
( vz vw cv wceq wal csb cvv wcel wi vex nfcv nfcsb1v nfeq2 csbeq1a eqeq2d
|
|
spcgf ax-mp weu wex wnfc euex eqtr3d alrimivv sbnfc2 sylibr exlimiv syl )
|
|
BFZCGZAHZBUAUMBUBACUCZUMBUDUMUNBUMADFZCIZAEFZCIZGZEHDHUNUMUSDEUMUKUPURUOJ
|
|
KUMUKUPGZLDMULUTAUOJAUONAUKUPAUOCOPAFZUOGCUPUKAUOCQRSTUQJKUMUKURGZLEMULVB
|
|
AUQJAUQNAUKURAUQCOPVAUQGCURUKAUQCQRSTUEUFADECUGUHUIUJ $.
|
|
|
|
$( Two ways to say that ` A ( x ) ` is a set expression that does not
|
|
depend on ` x ` . (Contributed by Mario Carneiro, 18-Nov-2016.) $)
|
|
eusvnfb $p |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) ) $=
|
|
( cv wceq wal weu wnfc cvv wcel eusvnf wex euex vex syl6eqelr sps exlimiv
|
|
wa id syl jca isset nfcvd nfeqd nfrd eximdv syl5bi eusv1 sylibr impbii
|
|
imp ) BDZCEZAFZBGZACHZCIJZRZUOUPUQABCKUOUNBLZUQUNBMUNUQBUMUQAUMCULIUMSBNO
|
|
PQTUAURUSUOUPUQUSUQUMBLUPUSBCUBUPUMUNBUPUMAUPAULCUPAULUCUPSUDUEUFUGUKABCU
|
|
HUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $.
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` A ( x ) ` . (Contributed by NM, 14-Oct-2010.) (Revised by Mario
|
|
Carneiro, 18-Nov-2016.) $)
|
|
eusv2i $p |- ( E! y A. x y = A -> E! y E. x y = A ) $=
|
|
( cv wceq wal weu wex nfeu1 wnf nfcvd eusvnf nfeqd nf2 sylib 19.2 impbid1
|
|
wi eubid ibir ) BDZCEZAFZBGZUBAHZBGUDUEUCBUCBIUDUEUCUDUBAJUEUCRUDAUACUDAU
|
|
AKABCLMUBANOUBAAPQST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $.
|
|
eusv2.1 $e |- A e. _V $.
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` A ( x ) ` . (Contributed by Mario Carneiro, 18-Nov-2016.) $)
|
|
eusv2nf $p |- ( E! y E. x y = A <-> F/_ x A ) $=
|
|
( cv wceq wex weu wnfc wnf wal nfeu1 wi nfe1 nfeu wa isseti alrimi sylibr
|
|
cvv 19.8a ancri eximii eupick mpan2 nf3 wcel wb dfnfc2 mpg eusvnfb eusv2i
|
|
mpbiran2 sylbir impbii ) BECFZAGZBHZACIZURUPAJZBKZUSURUTBUQBLURUQUPMZAKUT
|
|
URVBAUQABUPANOURUQUPPZBGVBUPVCBBCDQUPUQUPAUAUBUCUQUPBUDUERUPAUFSRCTUGZUSV
|
|
AUHAABCTUIDUJSUSUPAKBHZURVEUSVDDABCUKUMABCULUNUO $.
|
|
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` A ( x ) ` . (Contributed by NM, 15-Oct-2010.) (Proof shortened by
|
|
Mario Carneiro, 18-Nov-2016.) $)
|
|
eusv2 $p |- ( E! y E. x y = A <-> E! y A. x y = A ) $=
|
|
( cv wceq wex weu wnfc wal eusv2nf cvv wcel eusvnfb mpbiran2 bitr4i ) BEC
|
|
FZAGBHACIZQAJBHZABCDKSRCLMDABCNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ph $. $d x y $.
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` C ( y ) ` . (Contributed by NM, 16-Dec-2012.) (Proof shortened by
|
|
Mario Carneiro, 18-Nov-2016.) $)
|
|
reusv1 $p |- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C )
|
|
<-> E. x e. A A. y e. B ( ph -> x = C ) ) ) $=
|
|
( wrex cv wceq wi wral wmo wrmo wreu wb nfra1 nfmo wcel wa wal impd com12
|
|
rsp alrimiv moeq moim mpisyl ex rexlimi mormo reu5 rbaib 3syl ) ACEGABHFI
|
|
ZJZCEKZBLZUPBDMZUPBDNZUPBDGZOAUQCEUPCBUOCEPQCHERZAUQVAASZUPUNJZBTUNBLUQVB
|
|
VCBUPVBUNUPVAAUNUOCEUCUAUBUDBFUEUPUNBUFUGUHUIUPBDUJUSUTURUPBDUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z B $. $d x z C $. $d x y D $. $d x z ph $. $d x y ps $.
|
|
reusv3.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
reusv3.2 $e |- ( y = z -> C = D ) $.
|
|
$( Two ways of expressing existential uniqueness via an indirect equality.
|
|
(Contributed by NM, 23-Dec-2012.) $)
|
|
reusv3i $p |- ( E. x e. A A. y e. B ( ph -> x = C )
|
|
-> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) ) $=
|
|
( cv wceq wi wral wa eqeq2d imbi12d cbvralv ralimi biimpi prth eqtr2 syl6
|
|
raaanv sylbir mpdan rexlimivw ) ACLZHMZNZDGOZABPZHIMZNZEGOZDGOZCFULBUIIMZ
|
|
NZEGOZUQULUTUKUSDEGDLELMZABUJURJVAHIUIKQRSUAULUTPUKUSPZEGOZDGOUQUKUSDEGUE
|
|
VCUPDGVBUOEGVBUMUJURPUNAUJBURUBUIHIUCUDTTUFUGUH $.
|
|
|
|
$d x y z A $.
|
|
$( Two ways to express single-valuedness of a class expression
|
|
` C ( y ) ` . See ~ reusv1 for the connection to uniqueness.
|
|
(Contributed by NM, 27-Dec-2012.) $)
|
|
reusv3 $p |- ( E. y e. B ( ph /\ C e. A )
|
|
-> ( A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D )
|
|
<-> E. x e. A A. y e. B ( ph -> x = C ) ) ) $=
|
|
( wcel wa wrex wceq wi wral cv bitri ralbii eleq1d anbi12d cbvrexv risset
|
|
nfra2xy nfv nfim ralcom impexp bi2.04 r19.21v rsp sylbi com3l imp31 eqeq1
|
|
eqcom syl6bb imbi2d ralbidv syl5ibrcom reximdv ex expimpd rexlimi reusv3i
|
|
com23 syl5bi impbid1 ) AHFLZMZDGNZABMHIOZPZEGQDGQZACRZHOZPZDGQZCFNZVLBIFL
|
|
ZMZEGNVOVTPZVKWBDEGDRERZOZABVJWAJWEHIFKUAUBUCWBWCEGVOVTEVNDEGGUEVTEUFUGWD
|
|
GLZBWAWCWAVPIOZCFNZWFBMZWCCIFUDWIVOWHVTWIVOWHVTPWIVOMZWGVSCFWJVSWGAVMPZDG
|
|
QZWFBVOWLVOWFBWLVOBWLPZEGQZWFWMPVOVNDGQZEGQWNVNDEGGUHWOWMEGWOBWKPZDGQWMVN
|
|
WPDGVNABVMPPWPABVMUIABVMUJSTBWKDGUKSTSWMEGULUMUNUOWGVRWKDGWGVQVMAWGVQIHOV
|
|
MVPIHUPIHUQURUSUTVAVBVCVGVHVDVEUMABCDEFGHIJKVFVI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y ph $. $d x ps $. $d x y $.
|
|
alxfr.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
18-Feb-2007.) $)
|
|
alxfr $p |- ( ( A. y A e. B /\ A. x E. y x = A ) ->
|
|
( A. x ph <-> A. y ps ) ) $=
|
|
( wcel wal cv wceq wex wa wi spcgv com12 alimdv adantr nfa1 nfv sp exlimd
|
|
syl5ibrcom adantl impbid ) EFHZDIZCJEKZDLZCIZMACIZBDIZUGUKULNUJUKUGULUKUF
|
|
BDUFUKBABCEFGOPQPRUJULUKNUGULUJUKULUIACULUHADBDSADTULAUHBBDUAGUCUBQPUDUE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x C $. $d x ch $. $d x y ph $. $d y ps $.
|
|
ralxfrd.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
|
|
ralxfrd.2 $e |- ( ( ph /\ x e. B ) -> E. y e. C x = A ) $.
|
|
ralxfrd.3 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) $)
|
|
ralxfrd $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
|
|
( wral cv wcel wa wceq wb adantlr ralrimdva wrex rspcdv r19.29 wi biimprd
|
|
expimpd ancomsd ad2antrr rexlimdva syl5 mpan2d impbid ) ABDGLZCEHLZAULCEH
|
|
AEMHNZOBCDFGIADMZFPZBCQUNKRUASAUMBDGAUOGNZOZUMUPEHTZBJUMUSOCUPOZEHTURBCUP
|
|
EHUBURUTBEHAUTBUCUQUNAUPCBAUPCBAUPOBCKUDUEUFUGUHUIUJSUK $.
|
|
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by FL,
|
|
10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.) $)
|
|
rexxfrd $p |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) ) $=
|
|
( wrex wal cv wcel wa nfv wex df-rex rexlimdva 19.3 wceq 19.29 an12 exbii
|
|
wi sylib sylibr sylan2b biimpd expimpd ancomsd reximdv syl5 adantr mpan2d
|
|
syl5bir wb adantlr rspcedv impbid ) ABDGLZCEHLZABVCDGBBEMZADNZGOZPZVCBEBE
|
|
QUAVGVDVEFUBZEHLZVCJAVDVIPZVCUFVFVJBVHPZEHLZAVCVIVDENHOZVHPZERZVLVHEHSVDV
|
|
OPZVMVKPZERZVLVPBVNPZERVRBVNEUCVSVQEBVMVHUDUEUGVKEHSUHUIAVKCEHAVHBCAVHBCA
|
|
VHPBCKUJUKULUMUNUOUPUQTACVBEHAVMPBCDFGIAVHBCURVMKUSUTTVA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x C $. $d x ch $. $d x y ph $. $d y ps $.
|
|
ralxfr2d.1 $e |- ( ( ph /\ y e. C ) -> A e. V ) $.
|
|
ralxfr2d.2 $e |- ( ph -> ( x e. B <-> E. y e. C x = A ) ) $.
|
|
ralxfr2d.3 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by Mario
|
|
Carneiro, 20-Aug-2014.) $)
|
|
ralxfr2d $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
|
|
( cv wcel wa wceq wex elisset syl wi wrex biimprd r19.23v sylibr r19.21bi
|
|
wral eleq1 mpbidi exlimdv mpd biimpa ralxfrd ) ABCDEFGHAEMHNOZDMZFPZDQZFG
|
|
NZUMFINUPJDFIRSUMUOUQDUOUNGNZUQUMAUOURTZEHAUOEHUAZURTUSEHUFAURUTKUBUOUREH
|
|
UCUDUEUNFGUGUHUIUJAURUTKUKLUL $.
|
|
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by Mario
|
|
Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro,
|
|
19-Nov-2016.) $)
|
|
rexxfr2d $p |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) ) $=
|
|
( cv wcel wa wceq wex elisset syl wi wrex biimprd r19.23v sylibr r19.21bi
|
|
wral eleq1 mpbidi exlimdv mpd biimpa rexxfrd ) ABCDEFGHAEMHNOZDMZFPZDQZFG
|
|
NZUMFINUPJDFIRSUMUOUQDUOUNGNZUQUMAUOURTZEHAUOEHUAZURTUSEHUFAURUTKUBUOUREH
|
|
UCUDUEUNFGUGUHUIUJAURUTKUKLUL $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x A $. $d x y B $. $d x C $.
|
|
ralxfr.1 $e |- ( y e. C -> A e. B ) $.
|
|
ralxfr.2 $e |- ( x e. B -> E. y e. C x = A ) $.
|
|
ralxfr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) $)
|
|
ralxfr $p |- ( A. x e. B ph <-> A. y e. C ps ) $=
|
|
( wral wb wtru cv wcel adantl wceq wrex ralxfrd trud ) ACFKBDGKLMABCDEFGD
|
|
NGOEFOMHPCNZFOUAEQZDGRMIPUBABLMJPST $.
|
|
|
|
$( Transfer universal quantification from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . This proof does not use
|
|
~ ralxfrd . (Contributed by NM, 10-Jun-2005.) (Revised by Mario
|
|
Carneiro, 15-Aug-2014.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ralxfrALT $p |- ( A. x e. B ph <-> A. y e. C ps ) $=
|
|
( wral cv wcel wi rspcv syl com12 ralrimiv wceq wrex nfv biimprcd rexlimd
|
|
nfra1 rsp syl6 syl5 impbii ) ACFKZBDGKZUIBDGDLGMZUIBUKEFMUIBNHABCEFJOPQRU
|
|
JACFCLZFMULESZDGTUJAIUJUMADGBDGUDADUAUJUKBUMANBDGUEUMABJUBUFUCUGRUH $.
|
|
|
|
$( Transfer existence from a variable ` x ` to another variable ` y `
|
|
contained in expression ` A ` . (Contributed by NM, 10-Jun-2005.)
|
|
(Revised by Mario Carneiro, 15-Aug-2014.) $)
|
|
rexxfr $p |- ( E. x e. B ph <-> E. y e. C ps ) $=
|
|
( wrex wb wtru cv wcel adantl wceq rexxfrd trud ) ACFKBDGKLMABCDEFGDNGOEF
|
|
OMHPCNZFOTEQZDGKMIPUAABLMJPRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y D $. $d y ph $. $d y ps $. $d x ch $.
|
|
rabxfrd.1 $e |- F/_ y B $.
|
|
rabxfrd.2 $e |- F/_ y C $.
|
|
rabxfrd.3 $e |- ( ( ph /\ y e. D ) -> A e. D ) $.
|
|
rabxfrd.4 $e |- ( x = A -> ( ps <-> ch ) ) $.
|
|
rabxfrd.5 $e |- ( y = B -> A = C ) $.
|
|
$( Class builder membership after substituting an expression ` A `
|
|
(containing ` y ` ) for ` x ` in the class expression ` ch ` .
|
|
(Contributed by NM, 16-Jan-2012.) $)
|
|
rabxfrd $p |- ( ( ph /\ B e. D ) ->
|
|
( C e. { x e. D | ps } <-> B e. { y e. D | ch } ) ) $=
|
|
( wcel crab wb wa wi imp cv ibibr sylib anbi1d elrab rabid 3bitr4g eleq2d
|
|
ex rabbidva nfcv nfel1 wceq eleq1d elrabf nfrab1 nfel eleq1 pm5.32 sylibr
|
|
3bitr3g ) AGIOZHBDIPZOZGCEIPZOZQZAVBVDRZVBVFRZQVBVGSAGFVCOZEIPZOGEUAZVEOZ
|
|
EIPZOVHVIAVKVNGAVJVMEIAVLIOZRZFIOZCRVOCRVJVMVPVQVOCAVOVQVOQZAVOVQSVOVRSAV
|
|
OVQLUIVOVQUBUCTUDBCDFIMUECEIUFUGUJUHVJVDEGIJEIUKZEHVCKULVLGUMFHVCNUNUOVMV
|
|
FEGIJVSEGVEJCEIUPUQVLGVEURUOVAVBVDVFUSUTT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y D $. $d y ph $. $d x ps $.
|
|
rabxfr.1 $e |- F/_ y B $.
|
|
rabxfr.2 $e |- F/_ y C $.
|
|
rabxfr.3 $e |- ( y e. D -> A e. D ) $.
|
|
rabxfr.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
rabxfr.5 $e |- ( y = B -> A = C ) $.
|
|
$( Class builder membership after substituting an expression ` A `
|
|
(containing ` y ` ) for ` x ` in the class expression ` ph ` .
|
|
(Contributed by NM, 10-Jun-2005.) $)
|
|
rabxfr $p |- ( B e. D ->
|
|
( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) ) $=
|
|
( wtru wcel crab wb tru cv adantl rabxfrd mpan ) NFHOGACHPOFBDHPOQRNABCDE
|
|
FGHIJDSHOEHONKTLMUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d y B $. $d y C $. $d x y $.
|
|
reuhypd.1 $e |- ( ( ph /\ x e. C ) -> B e. C ) $.
|
|
reuhypd.2 $e |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) ) $.
|
|
$( A theorem useful for eliminating restricted existential uniqueness
|
|
hypotheses. (Contributed by NM, 16-Jan-2012.) $)
|
|
reuhypd $p |- ( ( ph /\ x e. C ) -> E! y e. C x = A ) $=
|
|
( cv wcel wa wceq weu wreu cvv elex syl eueq sylib eleq1 syl5ibrcom 3expa
|
|
pm4.71rd wb pm5.32da bitr4d eubidv mpbid df-reu sylibr ) ABIZFJZKZCIZFJZU
|
|
KDLZKZCMZUPCFNUMUNELZCMZURUMEOJZUTUMEFJZVAGEFPQCERSUMUSUQCUMUSUOUSKUQUMUS
|
|
UOUMUOUSVBGUNEFTUAUCUMUOUPUSAULUOUPUSUDHUBUEUFUGUHUPCFUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d y B $. $d y C $. $d x y $.
|
|
reuhyp.1 $e |- ( x e. C -> B e. C ) $.
|
|
reuhyp.2 $e |- ( ( x e. C /\ y e. C ) -> ( x = A <-> y = B ) ) $.
|
|
$( A theorem useful for eliminating restricted existential uniqueness
|
|
hypotheses. (Contributed by NM, 15-Nov-2004.) $)
|
|
reuhyp $p |- ( x e. C -> E! y e. C x = A ) $=
|
|
( wtru cv wcel wceq wreu tru adantl wb 3adant1 reuhypd mpan ) HAIZEJZSCKZ
|
|
BELMHABCDETDEJHFNTBIZEJUAUBDKOHGPQR $.
|
|
$}
|
|
|
|
$( The Axiom of Union and its converse. A class is a set iff its union is a
|
|
set. (Contributed by NM, 11-Nov-2003.) $)
|
|
uniexb $p |- ( A e. _V <-> U. A e. _V ) $=
|
|
( cvv wcel cuni uniexg cpw wss pwuni pwexg ssexg sylancr impbii ) ABCZADZBC
|
|
ZABEOANFZGPBCMAHNBIAPBJKL $.
|
|
|
|
$( The Axiom of Power Sets and its converse. A class is a set iff its power
|
|
class is a set. (Contributed by NM, 11-Nov-2003.) $)
|
|
pwexb $p |- ( A e. _V <-> ~P A e. _V ) $=
|
|
( cpw cvv wcel cuni uniexb unipw eleq1i bitr2i ) ABZCDJEZCDACDJFKACAGHI $.
|
|
|
|
$( The union of the universe is the universe. Exercise 4.12(c) of
|
|
[Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) $)
|
|
univ $p |- U. _V = _V $=
|
|
( cvv cpw cuni pwv unieqi unipw eqtr3i ) ABZCACAHADEAFG $.
|
|
|
|
${
|
|
eldifpw.1 $e |- C e. _V $.
|
|
$( Membership in a power class difference. (Contributed by NM,
|
|
25-Mar-2007.) $)
|
|
eldifpw $p |- ( ( A e. ~P B /\ -. C C_ B ) ->
|
|
( A u. C ) e. ( ~P ( B u. C ) \ ~P B ) ) $=
|
|
( cpw wcel wss wn wa cun elpwi unss1 cvv wb unexg mpan2 elpwg syl syl5ibr
|
|
cdif mpd unssbd con3i anim12i eldif sylibr ) ABEZFZCBGZHZIACJZBCJZEZFZUKU
|
|
GFZHZIUKUMUGTFUHUNUJUPUHABGZUNABKUQUNUHUKULGZABCLUHUKMFZUNURNUHCMFUSDACUG
|
|
MOPUKULMQRSUAUOUIUOACBUKBKUBUCUDUKUMUGUEUF $.
|
|
$}
|
|
|
|
${
|
|
op1stb.1 $e |- A e. _V $.
|
|
op1stb.2 $e |- B e. _V $.
|
|
$( Extract the first member of an ordered pair. Theorem 73 of [Suppes]
|
|
p. 42. (Contributed by NM, 25-Nov-2003.) $)
|
|
op1stb $p |- |^| |^| <. A , B >. = A $=
|
|
( cop cint csn cpr dfop inteqi cin snex wcel prexg mp2an intpr wceq eqtri
|
|
cvv wss snsspr1 df-ss mpbi intsn ) ABEZFZFAGZFAUFUGUFUGABHZHZFZUGUEUIABCD
|
|
IJUJUGUHKZUGUGUHACLASMBSMUHSMCDABSSNOPUGUHTUKUGQABUAUGUHUBUCRRJACUDR $.
|
|
$}
|
|
|
|
$( Extract the first member of an ordered pair. Theorem 73 of [Suppes]
|
|
p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) $)
|
|
op1stbg $p |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A ) $=
|
|
( wcel wa cop cint csn cpr dfopg inteqd cin cvv snexg prexg intprg syl2an2r
|
|
wceq eqtrd wss snsspr1 df-ss mpbi syl6eq intsng adantr ) ACEZBDEZFZABGZHZHA
|
|
IZHZAUJULUMUJULUMABJZJZHZUMUJUKUPABCDKLUJUQUMUOMZUMUHUMNEUIUONEUQURSACOABCD
|
|
PUMUONNQRUMUOUAURUMSABUBUMUOUCUDUETLUHUNASUIACUFUGT $.
|
|
|
|
${
|
|
$d x y A $.
|
|
iunpw.1 $e |- A e. _V $.
|
|
$( An indexed union of a power class in terms of the power class of the
|
|
union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
|
|
(Contributed by NM, 29-Nov-2003.) $)
|
|
iunpw $p |- ( E. x e. A x = U. A <-> ~P U. A = U_ x e. A ~P x ) $=
|
|
( vy cuni wceq wrex cpw ciun wcel sseq2 biimprcd reximdv com12 ssiun elpw
|
|
cv wss eliun wa uniiun syl6sseqr impbid1 df-pw abeq2i bitri 3bitr4g eqrdv
|
|
vex rexbii ssid uniex eleq2 syl5bbr mpbii sylib elssuni elpwi eqss sylibr
|
|
anim12i ex reximia syl impbii ) AQZBEZFZABGZVGHZABVFHZIZFZVIDVJVLVIDQZVGR
|
|
ZVNVFRZABGZVNVJJVNVLJZVIVOVQVOVIVQVOVHVPABVHVPVOVFVGVNKLMNVQVNABVFIVGABVF
|
|
VNOABUAUBUCVNVGDUIPVRVNVKJZABGVQAVNBVKSVSVPABVPDVKDVFUDUEUJUFUGUHVMVGVKJZ
|
|
ABGZVIVMVGVLJZWAVMVGVGRZWBVGUKWCVGVJJVMWBVGVGBCULPVJVLVGUMUNUOAVGBVKSUPVT
|
|
VHABVFBJZVTVHWDVTTVFVGRZVGVFRZTVHWDWEVTWFVFBUQVGVFURVAVFVGUSUTVBVCVDVE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordinals (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The class of all ordinal numbers is ordinal. Proposition 7.12 of
|
|
[TakeutiZaring] p. 38, but without using the Axiom of Regularity.
|
|
(Contributed by NM, 17-May-1994.) $)
|
|
ordon $p |- Ord On $=
|
|
( vx con0 word wtr cv wral tron wcel df-on abeq2i ordtr sylbi rgen mpbir2an
|
|
dford3 ) BCBDAEZDZABFGQABPBHPCZQRABAIJPKLMABON $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The union of a class of ordinal numbers is ordinal. Proposition 7.19 of
|
|
[TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof
|
|
shortened by Andrew Salmon, 12-Aug-2011.) $)
|
|
ssorduni $p |- ( A C_ On -> Ord U. A ) $=
|
|
( vx vy con0 wss cuni wtr word cv wral wcel wrex eluni2 wa wi ssel onelss
|
|
syl6 rexlimdv syl5bi anc2r syl ssuni syl8 ralrimiv dftr3 sylibr onelon ex
|
|
ssrdv ordon trssord 3exp mpii sylc ) ADEZAFZGZUQDEZUQHZUPBIZUQEZBUQJURUPV
|
|
BBUQVAUQKZVACIZKZCALZUPVBCVAAMZUPVEVBCAUPVDAKZVEVAVDEZVHNZVBUPVHVEVIOZOVH
|
|
VEVJOOUPVHVDDKZVKADVDPZVDVAQRVHVEVIUAUBVAVDAUCUDSTUEBUQUFUGUPBUQDVCVFUPVA
|
|
DKZVGUPVEVNCAUPVHVLVEVNOVMVLVEVNVDVAUHUIRSTUJURUSDHZUTUKURUSVOUTUQDULUMUN
|
|
UO $.
|
|
$}
|
|
|
|
$( The union of a set of ordinal numbers is an ordinal number. Theorem 9 of
|
|
[Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) $)
|
|
ssonuni $p |- ( A e. V -> ( A C_ On -> U. A e. On ) ) $=
|
|
( con0 wss cuni wcel word ssorduni cvv wb uniexg elong syl syl5ibr ) ACDAEZ
|
|
CFZABFZOGZAHQOIFPRJABKOILMN $.
|
|
|
|
${
|
|
ssonuni.1 $e |- A e. _V $.
|
|
$( The union of a set of ordinal numbers is an ordinal number. Corollary
|
|
7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.) $)
|
|
ssonunii $p |- ( A C_ On -> U. A e. On ) $=
|
|
( cvv wcel con0 wss cuni wi ssonuni ax-mp ) ACDAEFAGEDHBACIJ $.
|
|
$}
|
|
|
|
$( The union of two ordinal numbers is an ordinal number. (Contributed by
|
|
Jim Kingdon, 25-Jul-2019.) $)
|
|
onun2 $p |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On ) $=
|
|
( con0 wcel wa cpr wss cun prssi cuni wi prexg ssonuni uniprg eleq1d sylibd
|
|
cvv syl mpd ) ACDBCDEZABFZCGZABHZCDZABCITUBUAJZCDZUDTUAQDUBUFKABCCLUAQMRTUE
|
|
UCCABCCNOPS $.
|
|
|
|
${
|
|
onun2i.1 $e |- A e. On $.
|
|
onun2i.2 $e |- B e. On $.
|
|
$( The union of two ordinal numbers is an ordinal number. (Contributed by
|
|
NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.) $)
|
|
onun2i $p |- ( A u. B ) e. On $=
|
|
( con0 wcel cun onun2 mp2an ) AEFBEFABGEFCDABHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Any ordinal class is a subclass of the class of ordinal numbers.
|
|
Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM,
|
|
18-May-1994.) $)
|
|
ordsson $p |- ( Ord A -> A C_ On ) $=
|
|
( vx word con0 cv wcel ordelon ex ssrdv ) ACZBADJBEZAFKDFAKGHI $.
|
|
$}
|
|
|
|
$( An ordinal number is a subset of the class of ordinal numbers.
|
|
(Contributed by NM, 5-Jun-1994.) $)
|
|
onss $p |- ( A e. On -> A C_ On ) $=
|
|
( con0 wcel word wss eloni ordsson syl ) ABCADABEAFAGH $.
|
|
|
|
$( The union of an ordinal number is an ordinal number. (Contributed by NM,
|
|
29-Sep-2006.) $)
|
|
onuni $p |- ( A e. On -> U. A e. On ) $=
|
|
( con0 wcel wss cuni onss ssonuni mpd ) ABCABDAEBCAFABGH $.
|
|
|
|
$( The union of an ordinal class is ordinal. (Contributed by NM,
|
|
12-Sep-2003.) $)
|
|
orduni $p |- ( Ord A -> Ord U. A ) $=
|
|
( word con0 wss cuni ordsson ssorduni syl ) ABACDAEBAFAGH $.
|
|
|
|
${
|
|
$d x y A $.
|
|
bm2.5ii.1 $e |- A e. _V $.
|
|
$( Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM,
|
|
20-Sep-2003.) $)
|
|
bm2.5ii $p |- ( A C_ On -> U. A = |^| { x e. On | A. y e. A y C_ x } ) $=
|
|
( con0 wss cuni wcel cv wral crab cint wceq ssonunii wb unissb a1i inteqi
|
|
rabbiia intmin syl5reqr syl ) CEFCGZEHZUCBIAIZFBCJZAEKZLZMCDNUDUHUCUEFZAE
|
|
KZLUCUJUGUIUFAEUIUFOUEEHBCUEPQSRAUCETUAUB $.
|
|
$}
|
|
|
|
$( A successor exists iff its class argument exists. (Contributed by NM,
|
|
22-Jun-1998.) $)
|
|
sucexb $p |- ( A e. _V <-> suc A e. _V ) $=
|
|
( cvv wcel csn wa cun csuc unexb snexg pm4.71i df-suc eleq1i 3bitr4i ) ABCZ
|
|
ADZBCZEAOFZBCNAGZBCAOHNPABIJRQBAKLM $.
|
|
|
|
$( The successor of a set is a set (generalization). (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
sucexg $p |- ( A e. V -> suc A e. _V ) $=
|
|
( wcel cvv csuc elex sucexb sylib ) ABCADCAEDCABFAGH $.
|
|
|
|
${
|
|
sucex.1 $e |- A e. _V $.
|
|
$( The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) $)
|
|
sucex $p |- suc A e. _V $=
|
|
( cvv wcel csuc sucexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The successor of an ordinal class is ordinal. (Contributed by Jim
|
|
Kingdon, 8-Nov-2018.) $)
|
|
ordsucim $p |- ( Ord A -> Ord suc A ) $=
|
|
( vx word csuc wtr cv wral ordtr suctr syl wcel wceq wo csn df-suc eleq2i
|
|
cun elun velsn dford3 orbi2i 3bitri wal simprbi df-ral sylib 19.21bi treq
|
|
wi syl5ibrcom jaod syl5bi ralrimiv sylanbrc ) ACZADZEZBFZEZBUPGUPCUOAEZUQ
|
|
AHZAIJUOUSBUPURUPKZURAKZURALZMZUOUSVBURAANZQZKVCURVFKZMVEUPVGURAOPURAVFRV
|
|
HVDVCBASUAUBUOVCUSVDUOVCUSUIZBUOUSBAGZVIBUCUOUTVJBATUDUSBAUEUFUGUOUSVDUTV
|
|
AURAUHUJUKULUMBUPTUN $.
|
|
$}
|
|
|
|
$( The successor of an ordinal number is an ordinal number. Proposition 7.24
|
|
of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) $)
|
|
suceloni $p |- ( A e. On -> suc A e. On ) $=
|
|
( con0 wcel csuc word eloni ordsucim syl cvv wb sucexg elong mpbird ) ABCZA
|
|
DZBCZOEZNAEQAFAGHNOICPQJABKOILHM $.
|
|
|
|
$( The successor of an ordinal class is ordinal. (Contributed by Jim
|
|
Kingdon, 20-Nov-2018.) $)
|
|
ordsucg $p |- ( A e. _V -> ( Ord A <-> Ord suc A ) ) $=
|
|
( cvv wcel word csuc ordsucim sucidg ordelord ex syl5com impbid2 ) ABCZADZA
|
|
EZDZAFLANCZOMABGOPMNAHIJK $.
|
|
|
|
$( The successor of an ordinal number is an ordinal number. (Contributed by
|
|
NM, 9-Sep-2003.) $)
|
|
sucelon $p |- ( A e. On <-> suc A e. On ) $=
|
|
( con0 wcel csuc suceloni word eloni cvv wb elex sucexb elong ordsucg bitrd
|
|
sylibr syl mpbird impbii ) ABCZADZBCZAEUASTFZTGUAAHCZSUBIUATHCUCTBJAKOUCSAF
|
|
UBAHLAMNPQR $.
|
|
|
|
$( The successor of an element of an ordinal class is a subset of it.
|
|
(Contributed by NM, 21-Jun-1998.) $)
|
|
ordsucss $p |- ( Ord B -> ( A e. B -> suc A C_ B ) ) $=
|
|
( word wtr wcel csuc wss wi ordtr csn cun trss snssi a1i jcad syl6ib df-suc
|
|
wa unss sseq1i syl6ibr syl ) BCBDZABEZAFZBGZHBIUCUDAAJZKZBGZUFUCUDABGZUGBGZ
|
|
RUIUCUDUJUKBALUDUKHUCABMNOAUGBSPUEUHBAQTUAUB $.
|
|
|
|
$( A set belongs to an ordinal iff its successor is a subset of the ordinal.
|
|
Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM,
|
|
29-Nov-2003.) $)
|
|
ordelsuc $p |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) ) $=
|
|
( wcel word wa csuc wss wi ordsucss adantl sucssel adantr impbid ) ACDZBEZF
|
|
ABDZAGBHZPQRIOABJKORQIPABCLMN $.
|
|
|
|
${
|
|
onsucssi.1 $e |- A e. On $.
|
|
onsucssi.2 $e |- B e. On $.
|
|
$( A set belongs to an ordinal number iff its successor is a subset of the
|
|
ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse.
|
|
(Contributed by NM, 16-Sep-1995.) $)
|
|
onsucssi $p |- ( A e. B <-> suc A C_ B ) $=
|
|
( con0 wcel word csuc wss wb onordi ordelsuc mp2an ) AEFBGABFAHBIJCBDKABE
|
|
LM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The successor of an ordinal number is the smallest larger ordinal
|
|
number. (Contributed by NM, 28-Nov-2003.) $)
|
|
onsucmin $p |- ( A e. On -> suc A = |^| { x e. On | A e. x } ) $=
|
|
( con0 wcel cv crab cint csuc wss word wb ordelsuc sylan2 rabbidva inteqd
|
|
eloni wceq sucelon intmin sylbi eqtr2d ) BCDZBAEZDZACFZGBHZUCIZACFZGZUFUB
|
|
UEUHUBUDUGACUCCDUBUCJUDUGKUCPBUCCLMNOUBUFCDUIUFQBRAUFCSTUA $.
|
|
$}
|
|
|
|
$( Membership is inherited by predecessors. The converse, for all ordinals,
|
|
implies excluded middle, as shown at ~ onsucelsucexmid . However, the
|
|
converse does hold where ` B ` is a natural number, as seen at
|
|
~ nnsucelsuc . (Contributed by Jim Kingdon, 17-Jul-2019.) $)
|
|
onsucelsucr $p |- ( B e. On -> ( suc A e. suc B -> A e. B ) ) $=
|
|
( csuc wcel con0 cvv wi elex sucexb sylibr wa wceq wo wss onelss eqimss a1i
|
|
jaod adantl wb elsucg sylbi adantr ordelsuc sylan2 3imtr4d impancom mpancom
|
|
word eloni com12 ) ACZBCZDZBEDZABDZAFDZUNUOUPGUNULFDZUQULUMHAIZJUQUOUNUPUQU
|
|
OKULBDZULBLZMZULBNZUNUPUOVBVCGUQUOUTVCVABULOVAVCGUOULBPQRSUQUNVBTZUOUQURVDU
|
|
SULBFUAUBUCUOUQBUIUPVCTBUJABFUDUEUFUGUHUK $.
|
|
|
|
$( The subclass relationship between two ordinals is inherited by their
|
|
predecessors. The converse implies excluded middle, as shown at
|
|
~ onsucsssucexmid . (Contributed by Mario Carneiro and Jim Kingdon,
|
|
29-Jul-2019.) $)
|
|
onsucsssucr $p |- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) ) $=
|
|
( con0 wcel word wa csuc wss wb ordsucim ordelsuc sylan2 wi wtr trsucss syl
|
|
ordtr adantl sylbird ) ACDZBEZFAGBGZHZAUBDZABHZUATUBEUDUCIBJAUBCKLUAUDUEMZT
|
|
UABNUFBQBAOPRS $.
|
|
|
|
$( Successor and union. The converse (where ` B ` is an ordinal) implies
|
|
excluded middle, as seen at ~ ordsucunielexmid . (Contributed by Jim
|
|
Kingdon, 2-Aug-2019.) $)
|
|
sucunielr $p |- ( suc A e. B -> A e. U. B ) $=
|
|
( csuc wcel cuni cvv elex sucexb sylibr sucidg syl elunii mpancom ) AACZDZN
|
|
BDZABEDPAFDZOPNFDQNBGAHIAFJKANBLM $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The class of all ordinal numbers is its own union. Exercise 11 of
|
|
[TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) $)
|
|
unon $p |- U. On = On $=
|
|
( vx vy con0 cuni cv wcel wrex eluni2 onelon rexlimiva sylbi vex suceloni
|
|
csuc sucid elunii sylancr impbii eqriv ) ACDZCAEZTFZUACFZUBUABEZFZBCGUCBU
|
|
ACHUEUCBCUDUAIJKUCUAUANZFUFCFUBUAALOUAMUAUFCPQRS $.
|
|
|
|
$( The union of the ordinal subsets of an ordinal number is that number.
|
|
(Contributed by Jim Kingdon, 2-Aug-2019.) $)
|
|
onuniss2 $p |- ( A e. On -> U. { x e. On | x C_ A } = A ) $=
|
|
( con0 unimax ) ABCD $.
|
|
$}
|
|
|
|
$( The class of ordinal numbers is a limit ordinal. (Contributed by NM,
|
|
24-Mar-1995.) $)
|
|
limon $p |- Lim On $=
|
|
( con0 wlim word c0 wcel cuni wceq ordon 0elon unon eqcomi dflim2 mpbir3an
|
|
) ABACDAEAAFZGHINAJKALM $.
|
|
|
|
${
|
|
$d x A $.
|
|
|
|
$( An ordinal which contains the successor of each of its members is equal
|
|
to its union. (Contributed by Jim Kingdon, 14-Nov-2018.) $)
|
|
ordunisuc2r $p |- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) ) $=
|
|
( word cv csuc wcel wral cuni wss wa wceq wi wal sucid elunii mpan imim2i
|
|
vex alimi df-ral dfss2 3imtr4i a1i orduniss jctird eqss syl6ibr ) BCZADZE
|
|
ZBFZABGZBBHZIZUMBIZJBUMKUHULUNUOULUNLUHUIBFZUKLZAMUPUIUMFZLZAMULUNUQUSAUK
|
|
URUPUIUJFUKURUIARNUIUJBOPQSUKABTABUMUAUBUCBUDUEBUMUFUG $.
|
|
$}
|
|
|
|
${
|
|
onssi.1 $e |- A e. On $.
|
|
$( An ordinal number is a subset of ` On ` . (Contributed by NM,
|
|
11-Aug-1994.) $)
|
|
onssi $p |- A C_ On $=
|
|
( con0 wcel wss onss ax-mp ) ACDACEBAFG $.
|
|
|
|
$( The successor of an ordinal number is an ordinal number. Corollary
|
|
7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) $)
|
|
onsuci $p |- suc A e. On $=
|
|
( con0 wcel csuc suceloni ax-mp ) ACDAECDBAFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The intersection of an inhabited collection of ordinal numbers is an
|
|
ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.) $)
|
|
onintonm $p |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On ) $=
|
|
( con0 wss cv wcel wex wa cint word wtr wral ssel eloni syl syl6 ralrimiv
|
|
ordtr trint cvv nfv nfe1 nfan cuni intssuni2m unon syl6sseq sseld ralrimi
|
|
adantr dford3 sylanbrc wb inteximm adantl elong mpbird ) BCDZAEZBFZAGZHZB
|
|
IZCFZVCJZVBVCKZUSKZAVCLVEURVFVAURVGABLVFURVGABURUTUSCFZVGBCUSMVHUSJZVGUSN
|
|
ZUSRZOPQABSOUJVBVGAVCURVAAURAUAUTAUBUCVBUSVCFZVIVGVBVLVHVIVBVCCUSVBVCCUDC
|
|
ABCUEUFUGUHVJPVKPUIAVCUKULVBVCTFZVDVEUMVAVMURABUNUOVCTUPOUQ $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d x y $.
|
|
$( An existence condition which implies an intersection is an ordinal
|
|
number. (Contributed by Jim Kingdon, 30-Aug-2021.) $)
|
|
onintrab2im $p |- ( E. x e. On ph -> |^| { x e. On | ph } e. On ) $=
|
|
( vy con0 wrex crab wss cv wcel cint ssrab2 nfrab1 nfcri nfex rabid elex2
|
|
wex wa sylbir ex rexlimi onintonm sylancr ) ABDEABDFZDGCHUDIZCQZUDJDIABDK
|
|
AUFBDUEBCBCUDABDLMNBHZDIZAUFUHARUGUDIUFABDOCUGUDPSTUACUDUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph y z $.
|
|
$( Lemma for decidability and ordinals. The set ` { x e. { (/) } | ph } `
|
|
is a way of connecting statements about ordinals (such as trichotomy in
|
|
~ ordtriexmid or weak linearity in ~ ordsoexmid ) with a proposition
|
|
` ph ` . Our lemma states that it is an ordinal number. (Contributed
|
|
by Jim Kingdon, 28-Jan-2019.) $)
|
|
ordtriexmidlem $p |- { x e. { (/) } | ph } e. On $=
|
|
( vy vz c0 csn crab con0 wcel word wtr wss cv wa wi wal simpl mpbir ax-mp
|
|
wceq elrabi velsn sylib noel eleq2 mtbiri syl adantl pm2.21dd gen2 ssrab2
|
|
wn dftr2 csuc ord0 ordsucim suc0 ordeq mpbi trssord mp3an p0ex rabex elon
|
|
wb ) ABEFZGZHIVGJZVGKZVGVFLVFJZVHVICMZDMZIZVLVGIZNZVKVGIZOZDPCPVQCDVOVMVP
|
|
VMVNQVNVMULZVMVNVLETZVRVNVLVFIVSABVLVFUADEUBUCVSVMVKEIVKUDVLEVKUEUFUGUHUI
|
|
UJCDVGUMRABVFUKEUNZJZVJEJWAUOEUPSVTVFTWAVJVEUQVTVFURSUSVGVFUTVAVGABVFVBVC
|
|
VDR $.
|
|
$}
|
|
|
|
${
|
|
$d ph x $.
|
|
$( Lemma for decidability and ordinals. The set ` { x e. { (/) } | ph } `
|
|
is a way of connecting statements about ordinals (such as trichotomy in
|
|
~ ordtriexmid or weak linearity in ~ ordsoexmid ) with a proposition
|
|
` ph ` . Our lemma helps connect that set to excluded middle.
|
|
(Contributed by Jim Kingdon, 28-Jan-2019.) $)
|
|
ordtriexmidlem2 $p |- ( { x e. { (/) } | ph } = (/) -> -. ph ) $=
|
|
( c0 csn crab wceq wcel noel eleq2 mtbiri wb 0ex snid biidd elrab3 sylnib
|
|
cv ax-mp ) ABCDZEZCFZCTGZAUAUBCCGCHTCCIJCSGUBAKCLMAABCSBQCFANORP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d x ph z $.
|
|
ordtriexmid.1 $e |- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) $.
|
|
$( Ordinal trichotomy implies the law of the excluded middle (that is,
|
|
decidability of an arbitrary proposition).
|
|
|
|
This theorem is stated in "Constructive ordinals", [Crosilla], p.
|
|
"Set-theoretic principles incompatible with intuitionistic logic".
|
|
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) $)
|
|
ordtriexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz wn wo c0 wceq wcel w3o con0 cv eleq1 eleq2 3orbi123d wa wi 0ex ax-mp
|
|
csn crab noel ordtriexmidlem eqeq1 0elon anbi2d eqeq2 imbi12d vtocl mpan2
|
|
rspec2 vtoclga 3orass mpbi mtpor ordtriexmidlem2 snid biidd elrab3 biimpi
|
|
wb orim12i orcom mpbir ) AAFZGVFAGZAEHUAZUBZHIZHVIJZGZVGVIHJZVLVIUCVMVJVK
|
|
KZVMVLGVILJVNAEUDBMZHJZVOHIZHVOJZKZVNBVILVOVIIVPVMVQVJVRVKVOVIHNVOVIHUEVO
|
|
VIHOPVOLJZHLJZVSUFVTCMZLJZQZVOWBJZVOWBIZWBVOJZKZRVTWAQZVSRCHSWBHIZWDWIWHV
|
|
SWJWCWAVTWBHLNUGWJWEVPWFVQWGVRWBHVOOWBHVOUHWBHVONPUIWHBCLLDULUJUKUMTVMVJV
|
|
KUNUOUPVJVFVKAAEUQVKAHVHJVKAVBHSURAAEHVHEMHIAUSUTTVAVCTAVFVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
ordtri2orexmid.1 $e |- A. x e. On A. y e. On ( x e. y \/ y C_ x ) $.
|
|
$( Ordinal trichotomy implies excluded middle. (Contributed by Jim
|
|
Kingdon, 31-Jul-2019.) $)
|
|
ordtri2orexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz wn wo c0 csn crab wcel wss cv con0 wral 0elon wceq orbi12d ax-mp wb
|
|
ordtriexmidlem csuc suc0 eqeltrri eleq1 sseq2 eleq2 sseq1 rspc2va mpanl12
|
|
onsuci elsni ordtriexmidlem2 syl snssg biidd elrab3 biimpi sylbir orim12i
|
|
0ex snid orcom mpbi ) AFZAGZAVEGAEHIZJZVGKZVGVHLZGZVFBMZCMZKZVMVLLZGZCNOB
|
|
NOZVKDVHNKVGNKVQVKAEUAHUBVGNUCHPUKUDVPVKVHVMKZVMVHLZGBCVHVGNNVLVHQVNVRVOV
|
|
SVLVHVMUEVLVHVMUFRVMVGQVRVIVSVJVMVGVHUGVMVGVHUHRUIUJSVIVEVJAVIVHHQVEVHHUL
|
|
AEUMUNVJHVHKZAHNKVTVJTPHVHNUOSVTAHVGKVTATHVAVBAAEHVGEMHQAUPUQSURUSUTSVEAV
|
|
CVD $.
|
|
$}
|
|
|
|
$( Version of ~ 2on with the definition of ` 2o ` expanded and expressed in
|
|
terms of ` Ord ` . (Contributed by Jim Kingdon, 29-Aug-2021.) $)
|
|
2ordpr $p |- Ord { (/) , { (/) } } $=
|
|
( c0 csuc word csn cpr ord0 ordsucim mp2b wceq cun df-suc suceq ax-mp df-pr
|
|
wb suc0 3eqtr4i ordeq mpbi ) ABZBZCZAADZEZCZACTCUBFAGTGHUAUDIUBUEOUCBZUCUCD
|
|
JUAUDUCKTUCIUAUFIPTUCLMAUCNQUAUDRMS $.
|
|
|
|
${
|
|
$d ph w x y z $.
|
|
ontr2exmid.1 $e |- A. x e. On A. y A. z e. On
|
|
( ( x C_ y /\ y e. z ) -> x e. z ) $.
|
|
$( An ordinal transitivity law which implies excluded middle. (Contributed
|
|
by Jim Kingdon, 17-Sep-2021.) $)
|
|
ontr2exmid $p |- ( ph \/ -. ph ) $=
|
|
( vw wo c0 wceq wcel wss p0ex con0 cv wa wi wral wal eleq1 ax-mp csn crab
|
|
cpr ssrab2 prid2 word 2ordpr pp0ex elon mpbir ordtriexmidlem sseq1 anbi1d
|
|
wn imbi12d ralbidv albidv rspcv mp2 sseq2 anbi12d imbi1d spcv eleq2 mp2an
|
|
anbi2d elpri ordtriexmidlem2 0ex biidd rabsnt orim12i orcom mpbi ) AUNZAG
|
|
ZAVOGAFHUAZUBZHIZVRVQIZGZVPVRHVQUCZJZWAVRVQKZVQWBJZWCAFVQUDHVQLUEWBMJZWDV
|
|
QDNZJZOZVRWGJZPZDMQZWDWEOZWCPZWFWBUFUGWBUHUIUJVRCNZKZWOWGJZOZWJPZDMQZCRZW
|
|
LVRMJBNZWOKZWQOZXBWGJZPZDMQZCRZBMQXAAFUKEXHXABVRMXBVRIZXGWTCXIXFWSDMXIXDW
|
|
RXEWJXIXCWPWQXBVRWOULUMXBVRWGSUOUPUQURUSWTWLCVQLWOVQIZWSWKDMXJWRWIWJXJWPW
|
|
DWQWHWOVQVRUTWOVQWGSVAVBUPVCTWKWNDWBMWGWBIZWIWMWJWCXKWHWEWDWGWBVQVDVFWGWB
|
|
VRVDUOURUSVEVRHVQVGTVSVOVTAAFVHAAFVQHVIFNHIAVJVKVLTVOAVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
$( A set which is ` 2o ` if ` ph ` or ` (/) ` if ` -. ph ` is an ordinal.
|
|
(Contributed by Jim Kingdon, 29-Aug-2021.) $)
|
|
ordtri2or2exmidlem $p |- { x e. { (/) , { (/) } } | ph } e. On $=
|
|
( vy vz c0 wcel word cv wa wal wceq eleq2 adantl sylib wo vex biidd elrab
|
|
elpr mpbir csn cpr crab con0 wtr wss wi simpll wn mtbiri pm2.21dd biimpac
|
|
noel velsn orc sylibr syl adantlr simprbi ad2antlr sylanbrc mpjaodan gen2
|
|
elrabi dftr2 ssrab2 2ordpr trssord mp3an pp0ex rabex elon ) ABEEUAZUBZUCZ
|
|
UDFVOGZVOUEZVOVNUFVNGVPVQCHZDHZFZVSVOFZIZVRVOFZUGZDJCJWDCDWBVSEKZWCVSVMKZ
|
|
WBWEIVTWCVTWAWEUHWEVTUIWBWEVTVREFVRUMVSEVRLUJMUKWBWFIVRVNFZAWCVTWFWGWAVTW
|
|
FIZVREKZWGWHVRVMFZWIWFVTWJVSVMVRLULCEUNNWIWIVRVMKZOWGWIWKUOVREVMCPSUPUQUR
|
|
WAAVTWFWAVSVNFZAAABVSVNBHZVSKAQRUSUTAABVRVNWMVRKAQRVAWAWEWFOZVTWAWLWNABVS
|
|
VNVDVSEVMDPSNMVBVCCDVOVETABVNVFVGVOVNVHVIVOABVNVJVKVLT $.
|
|
$}
|
|
|
|
${
|
|
$d ph x z $. $d x y $.
|
|
onsucsssucexmid.1 $e |- A. x e. On A. y e. On
|
|
( x C_ y -> suc x C_ suc y ) $.
|
|
$( The converse of ~ onsucsssucr implies excluded middle. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 29-Jul-2019.) $)
|
|
onsucsssucexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz wo c0 wcel wceq csuc wss con0 wi cv suceq imbi12d onsuci wa ax-mp wb
|
|
wn crab ssrab2 ordtriexmidlem sseq1 sseq1d suc0 0elon eqeltrri p0ex eleq1
|
|
anbi2d sseq2 sseq2d rspec2 vtocl mpan2 vtoclga word onordi ordelsuc mp2an
|
|
csn mpbir elsucg elsni ordtriexmidlem2 syl 0ex biidd rabsnt orim12i orcom
|
|
mpbi ) AUAZAFZAVOFAEGVCZUBZVQHZVRVQIZFZVPVRVQJZHZWAWCVRJZWBKZVRVQKZWEAEVQ
|
|
UCVRLHZWFWEMZAEUDZBNZVQKZWJJZWBKZMZWHBVRLWJVRIZWKWFWMWEWJVRVQUEWOWLWDWBWJ
|
|
VROUFPWJLHZVQLHZWNGJVQLUGGUHQUIZWPCNZLHZRZWJWSKZWLWSJZKZMZMWPWQRZWNMCVQUJ
|
|
WSVQIZXAXFXEWNXGWTWQWPWSVQLUKULXGXBWKXDWMWSVQWJUMXGXCWBWLWSVQOUNPPXEBCLLD
|
|
UOUPUQURSSWGWBUSWCWETWIWBVQWRQUTVRWBLVAVBVDWGWCWATWIVRVQLVESVNVSVOVTAVSVR
|
|
GIVOVRGVFAEVGVHAAEVQGVIENGIAVJVKVLSVOAVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
$( Lemma for ~ onsucelsucexmid . (Contributed by Jim Kingdon,
|
|
2-Aug-2019.) $)
|
|
onsucelsucexmidlem1 $p |- (/) e.
|
|
{ x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } $=
|
|
( c0 cv wceq csn cpr crab wcel 0ex prid1 eqid eqeq1 orbi1d elrab mpbir2an
|
|
wo orci ) CBDZCEZAQZBCCFZGZHICUCICCEZAQZCUBJKUDACLRUAUEBCUCTTUDASCCMNOP
|
|
$.
|
|
|
|
$( Lemma for ~ onsucelsucexmid . The set
|
|
` { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ` appears as ` A ` in
|
|
the proof of Theorem 1.3 in [Bauer] p. 483 (see ~ acexmidlema ), and
|
|
similar sets also appear in other proofs that various propositions imply
|
|
excluded middle, for example in ~ ordtriexmidlem . (Contributed by Jim
|
|
Kingdon, 2-Aug-2019.) $)
|
|
onsucelsucexmidlem $p |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
|
|
e. On $=
|
|
( vy vz cv c0 wceq wo csn cpr wcel word wa wi eleq2 adantl ex sylib mpbir
|
|
wal crab con0 wtr wss simpll wn noel pm2.21dd biimpac onsucelsucexmidlem1
|
|
mtbiri velsn syl6eqel adantr elrabi elpr mpjaod gen2 dftr2 ssrab2 trssord
|
|
vex 2ordpr mp3an pp0ex rabex elon ) BEFGAHZBFFIZJZUAZUBKVKLZVKUCZVKVJUDVJ
|
|
LVLVMCEZDEZKZVOVKKZMZVNVKKZNZDTCTVTCDVRVOFGZVSVOVIGZVRWAVSVRWAMVPVSVPVQWA
|
|
UEWAVPUFVRWAVPVNFKVNUGVOFVNOUKPUHQVPWBVSNVQVPWBVSVPWBMZVNFVKWCVNVIKZVNFGW
|
|
BVPWDVOVIVNOUICFULRABUJUMQUNVQWAWBHZVPVQVOVJKWEVHBVOVJUOVOFVIDVBUPRPUQURC
|
|
DVKUSSVHBVJUTVCVKVJVAVDVKVHBVJVEVFVGS $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
onsucelsucexmid.1 $e |- A. x e. On A. y e. On
|
|
( x e. y -> suc x e. suc y ) $.
|
|
$( The converse of ~ onsucelsucr implies excluded middle. On the other
|
|
hand, if ` y ` is constrained to be a natural number, instead of an
|
|
arbitrary ordinal, then the converse of ~ onsucelsucr does hold, as seen
|
|
at ~ nnsucelsuc . (Contributed by Jim Kingdon, 2-Aug-2019.) $)
|
|
onsucelsucexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz c0 csuc cv wceq wo wcel wn con0 wral suceq imbi12d eleq2 ax-mp suc0
|
|
wi csn cpr crab onsucelsucexmidlem1 0elon onsucelsucexmidlem pm3.2i eleq1
|
|
wa eleq1d eleq2d rspc2va mp2an elsuci wb p0ex prid2 eqeltri orbi1d elrab3
|
|
eqeq1 wne cvv 0ex nsuceq0g df-ne mpbi pm2.53 sylbi eqeq1i mtbi elsn mtbir
|
|
mpi mtbii olc sylibr nsyl orim12i ) FGZEHZFIZAJZEFFUAZUBZUCZKZVTWFIZJZAAL
|
|
ZJVTWFGZKZWIFWFKZWLAEUDFMKZWFMKZUIBHZCHZKZWPGZWQGZKZTZCMNBMNWMWLTZWNWOUEA
|
|
EUFUGDXBXCFWQKZVTWTKZTBCFWFMMWPFIZWRXDXAXEWPFWQUHXFWSVTWTWPFOUJPWQWFIZXDW
|
|
MXEWLWQWFFQXGWTWKVTWQWFOUKPULUMRVTWFUNRWGAWHWJWGVTFIZAJZAVTWEKWGXIUOVTWDW
|
|
ESFWDUPUQZURWCXIEVTWEWAVTIWBXHAWAVTFVAUSUTRXIXHLZAVTFVBZXKFVCKXLVDFVCVERV
|
|
TFVFVGZXHAVHVNVIWHWDWFKZAWHWDWFIZXNLVTWDWFSVJXOWDWDKZXNXPWDFIZXHXQXMVTWDF
|
|
SVJVKWDFUPVLVMWDWFWDQVOVIAXQAJZXNAXQVPWDWEKXNXRUOXJWCXREWDWEWAWDIWBXQAWAW
|
|
DFVAUSUTRVQVRVSR $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y a b $.
|
|
ordsucunielexmid.1 $e |- A. x e. On A. y e. On
|
|
( x e. U. y -> suc x e. y ) $.
|
|
$( The converse of ~ sucunielr (where ` B ` is an ordinal) implies excluded
|
|
middle. (Contributed by Jim Kingdon, 2-Aug-2019.) $)
|
|
ordsucunielexmid $p |- ( ph \/ -. ph ) $=
|
|
( va vb cv wcel csuc wi con0 wa cuni wb wtr wceq word eleq2d wral imbi12d
|
|
eloni ordtr syl vex unisuc sylib adantl suceloni eleq1 suceq eleq1d unieq
|
|
eleq2 rspc2va mpan2 sylan2 sylbird rgen2a onsucelsucexmid ) AEFEGZFGZHZUT
|
|
IZVAIZHZJEFKUTKHZVAKHZLVBUTVDMZHZVEVGVIVBNVFVGVHVAUTVGVAOZVHVAPVGVAQVJVAU
|
|
AVAUBUCVAFUDUEUFRUGVGVFVDKHZVIVEJZVAUHVFVKLBGZCGZMZHZVMIZVNHZJZCKSBKSVLDV
|
|
SVLUTVOHZVCVNHZJBCUTVDKKVMUTPZVPVTVRWAVMUTVOUIWBVQVCVNVMUTUJUKTVNVDPZVTVI
|
|
WAVEWCVOVHUTVNVDULRVNVDVCUMTUNUOUPUQURUS $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
IZF Set Theory - add the Axiom of Set Induction
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The ZF Axiom of Foundation would imply Excluded Middle
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A y z $. $d ph x y $.
|
|
regexmidlemm.a $e |-
|
|
A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } $.
|
|
$( Lemma for ~ regexmid . ` A ` is inhabited. (Contributed by Jim Kingdon,
|
|
3-Sep-2019.) $)
|
|
regexmidlemm $p |- E. y y e. A $=
|
|
( c0 csn wcel cv wex cpr wceq wa wo p0ex prid2 eqid orci eqeq1 anbi1d
|
|
orbi12d elrab2 mpbir2an elex2 ax-mp ) FGZDHZCIDHCJUGUFFUFKZHUFUFLZUFFLZAM
|
|
ZNZFUFOPUIUKUFQRBIZUFLZUMFLZAMZNULBUFUHDUNUNUIUPUKUMUFUFSUNUOUJAUMUFFSTUA
|
|
EUBUCCUFDUDUE $.
|
|
|
|
$( Lemma for ~ regexmid . If ` A ` has a minimal element, excluded middle
|
|
follows. (Contributed by Jim Kingdon, 3-Sep-2019.) $)
|
|
regexmidlem1 $p |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) ->
|
|
( ph \/ -. ph ) ) $=
|
|
( cv wcel wn wi wa wo c0 wceq eqeq1 anbi1d orbi12d elrab2 0ex eleq1 eleq2
|
|
wal csn cpr simprbi snid mpbiri notbid imbi12d spcv syl5com prid1 mpbiran
|
|
pm2.46 sylnbi eqid biantrur sylnibr olcd syl6 orc adantl a1d jaoi syl imp
|
|
exlimiv ) CGZEHZDGZVHHZVJEHZIZJZDUBZKAAIZLZCVIVOVQVIVHMUCZNZVHMNZAKZLZVOV
|
|
QJZVIVHMVRUDZHWBBGZVRNZWEMNZAKZLZWBBVHWDEWEVHNZWFVSWHWAWEVHVROWJWGVTAWEVH
|
|
MOPQFRUEVSWCWAVSVOMEHZIZVQVSMVHHZVOWLVSWMMVRHMSUFVHVRMUAUGVNWMWLJDMSVJMNZ
|
|
VKWMVMWLVJMVHTWNVLWKVJMETUHUIUJUKWLVPAWLMMNZAKZAWKMVRNZWPLZWPIWKMWDHWRMVR
|
|
SULWIWRBMWDEWGWFWQWHWPWEMVROWGWGWOAWEMMOPQFRUMWQWPUNUOWOAMUPUQURUSUTWAVQV
|
|
OAVQVTAVPVAVBVCVDVEVFVG $.
|
|
|
|
${
|
|
$d A v $. $d ph u x $. $d u v $.
|
|
$( Lemma for ~ reg2exmid . If ` A ` has a minimal element (expressed by
|
|
` C_ ` ), excluded middle follows. (Contributed by Jim Kingdon,
|
|
2-Oct-2021.) $)
|
|
reg2exmidlema $p |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) ) $=
|
|
( cv wss wral wo wcel wa c0 wceq 0ex sylibr olcd eqeq1 anbi1d orbi12d
|
|
wn csn simplr sseq1 ralbidv adantl mpbid snss ralbii noel cpr eqid jctl
|
|
wb wi prid1 jctil elrab2 eleq2 rspcv syl com12 mtoi simprr orcd simprbi
|
|
adantr mpjaodan rexlimiva ) DGZCGZHZCEIZAAUAZJZDEVJEKZVMLZVJMUBZNZVOVJM
|
|
NZALZVQVSLZVNAWBMVKKZCEIZVNWBVRVKHZCEIZWDWBVMWFVPVMVSUCVSVMWFUNVQVSVLWE
|
|
CEVJVRVKUDUEUFUGWCWECEMVKOUHUIPWDAMMKZMUJAWDWGAMEKZWDWGUOAMMVRUKZKZMVRN
|
|
ZMMNZALZJZLWHAWNWJAWMWKAWLMULUMQMVROUPUQBGZVRNZWOMNZALZJZWNBMWIEWQWPWKW
|
|
RWMWOMVRRWQWQWLAWOMMRSTFURPWCWGCMEVKMMUSUTVAVBVCVAQVQWALAVNVQVTAVDVEVPV
|
|
SWAJZVMVPVJWIKWTWSWTBVJWIEWOVJNZWPVSWRWAWOVJVRRXAWQVTAWOVJMRSTFURVFVGVH
|
|
VI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d ph x y z w $.
|
|
regexmid.1 $e |- ( E. y y e. x ->
|
|
E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) ) $.
|
|
$( The axiom of foundation implies excluded middle.
|
|
|
|
By foundation (or regularity), we mean the principle that every
|
|
inhabited set has an element which is minimal (when arranged by
|
|
` e. ` ). The statement of foundation here is taken from Metamath Proof
|
|
Explorer's ax-reg, and is identical (modulo one unnecessary quantifier)
|
|
to the statement of foundation in Theorem "Foundation implies instances
|
|
of EM" of [Crosilla], p. "Set-theoretic principles incompatible with
|
|
intuitionistic logic".
|
|
|
|
For this reason, IZF does not adopt foundation as an axiom and instead
|
|
replaces it with ~ ax-setind . (Contributed by Jim Kingdon,
|
|
3-Sep-2019.) $)
|
|
regexmid $p |- ( ph \/ -. ph ) $=
|
|
( vw cv c0 csn wceq wa wo wcel wex wel wn wi wal eleq2 exbidv pp0ex rabex
|
|
crab regexmidlemm notbid imbi2d albidv anbi12d imbi12d vtocl regexmidlem1
|
|
cpr eqid mp2b ) CGZFGZHIZJUPHJAKLZFHUQULZUCZMZCNZVADCOZDGZUTMZPZQZDRZKZCN
|
|
ZAAPLAFCUTUTUMZUDCBOZCNZVLVCDBOZPZQZDRZKZCNZQVBVJQBUTURFUSUAUBBGZUTJZVMVB
|
|
VSVJWAVLVACVTUTUOSZTWAVRVICWAVLVAVQVHWBWAVPVGDWAVOVFVCWAVNVEVTUTVDSUEUFUG
|
|
UHTUIEUJAFCDUTVKUKUN $.
|
|
$}
|
|
|
|
${
|
|
$d ph u w z $. $d ph u x z $. $d ph u y z $. $d x y $.
|
|
reg2exmid.1 $e |- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y ) $.
|
|
$( If any inhabited set has a minimal element (when expressed by ` C_ ` ),
|
|
excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) $)
|
|
reg2exmid $p |- ( ph \/ -. ph ) $=
|
|
( vu cv wss c0 csn wceq wa wo wral wrex wcel wex wi ax-mp wn regexmidlemm
|
|
cpr crab eqid wal pp0ex rabex eleq2 exbidv raleq rexeqbi1dv reg2exmidlema
|
|
imbi12d spcv ) BHCHIZCGHZJKZLUQJLAMNZGJURUCZUDZOZBVAPZAAUANEHZVAQZERZVCAG
|
|
EVAVAUEZUBVDDHZQZERZUPCVHOZBVHPZSZDUFVFVCSZFVMVNDVAUSGUTUGUHVHVALZVJVFVLV
|
|
CVOVIVEEVHVAVDUIUJVKVBBVHVAUPCVHVAUKULUNUOTTAGCBVAVGUMT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Set Induction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d y a $. $d ph y $.
|
|
$( Axiom of ` e. `-Induction (also known as set induction). An axiom of
|
|
Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p.
|
|
"Axioms of CZF and IZF". This replaces the Axiom of Foundation (also
|
|
called Regularity) from Zermelo-Fraenkel set theory.
|
|
|
|
For more on axioms which might be adopted which are incompatible with
|
|
this axiom (that is, Non-wellfounded Set Theory but in the absence of
|
|
excluded middle), see Chapter 20 of [AczelRathjen], p. 183.
|
|
(Contributed by Jim Kingdon, 19-Oct-2018.) $)
|
|
ax-setind $a |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y S $.
|
|
$( ` e. `-Induction in terms of membership in a class. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 22-Oct-2018.) $)
|
|
setindel $p |- ( A. x ( A. y ( y e. x -> y e. S ) -> x e. S ) ->
|
|
S = _V ) $=
|
|
( cv wcel wi wal cvv wceq wsb wral clelsb3 ralbii df-ral imbi1i ax-setind
|
|
bitri albii sylbir eqv sylibr ) BDZADZEUBCEZFBGZUCCEZFZAGZUFAGZCHIUHUFABJ
|
|
ZBUCKZUFFZAGUIULUGAUKUEUFUKUDBUCKUEUJUDBUCBACLMUDBUCNQORUFBAPSACTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21.
|
|
(Contributed by NM, 17-Sep-2003.) $)
|
|
setind $p |- ( A. x ( x C_ A -> x e. A ) -> A = _V ) $=
|
|
( vy cv wss wcel wi wal cvv wceq dfss2 imbi1i albii setindel sylbi ) ADZB
|
|
EZPBFZGZAHCDZPFTBFGCHZRGZAHBIJSUBAQUARCPBKLMACBNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Set (epsilon) induction, stated compactly. Given as a homework problem
|
|
in 1992 by George Boolos (1940-1996). (Contributed by NM,
|
|
17-Sep-2003.) $)
|
|
setind2 $p |- ( ~P A C_ A -> A = _V ) $=
|
|
( vx cpw wss cv wcel wi wal cvv wceq pwss setind sylbi ) ACADBEZADNAFGBHA
|
|
IJBAAKBALM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.
|
|
|
|
The reason that this theorem is marked as discouraged is a bit subtle.
|
|
If we wanted to reduce usage of ~ ax-setind , we could redefine
|
|
` Ord A ` ( ~ df-iord ) to also require ` _E Fr A ` ( ~ df-frind ) and
|
|
in that case any theorem related to irreflexivity of ordinals could use
|
|
~ ordirr (which under that definition would presumably not need
|
|
~ ax-setind to prove it). But since ordinals have not yet been defined
|
|
that way, we cannot rely on the "don't add additional axiom use" feature
|
|
of the minimizer to get theorems to use ~ ordirr . To encourage
|
|
~ ordirr when possible, we mark this theorem as discouraged.
|
|
|
|
(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and
|
|
Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) $)
|
|
elirr $p |- -. A e. A $=
|
|
( vx vy wcel cvv csn cdif cv wal wsb wral wi wn neldifsnd eleq1 spcgv mpd
|
|
wceq sylibr albii wa w3a simp1 imbi12d pm2.43b 3ad2ant2 wb eleq2 3ad2ant3
|
|
imbi1d mpbid 3expia mtod vex eldif mpbiran xchbinx alrimiv df-ral clelsb3
|
|
velsn ex imbi2i bitri imbi1i ax-setind syl pm2.65i ) AADZAEAFZGZDZVIBHZVK
|
|
DZBIZVLVIVNBCJZCVMKZVNLZBIZVOVICHZVMDZVTVKDZLZCIZVNLZBIVSVIWEBVIWDVNVIWDU
|
|
AZVMARZMVNWFWGVLWFAENVIWDWGVLVIWDWGUBZVIVLVIWDWGUCWHAVMDZVLLZVIVLLZWDVIWJ
|
|
WGWDWIVLWCWJCAVMVTARWAWIWBVLVTAVMOVTAVKOUDPUEUFWGVIWJWKUGWDWGWIVIVLVMAAUH
|
|
UJUIUKQULUMVNVMVJDZWGVNVMEDWLMBUNVMEVJUOUPBAVAUQSVBURVRWEBVQWDVNVQWAVPLZC
|
|
IWDVPCVMUSWMWCCVPWBWACBVKUTVCTVDVETSVNCBVFVGVNVLBAAVMAVKOPQVIAENVH $.
|
|
$}
|
|
|
|
$( Epsilon irreflexivity of ordinals: no ordinal class is a member of itself.
|
|
Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The
|
|
present proof requires ~ ax-setind . If in the definition of ordinals
|
|
~ df-iord , we also required that membership be well-founded on any
|
|
ordinal (see ~ df-frind ), then we could prove ~ ordirr without
|
|
~ ax-setind . (Contributed by NM, 2-Jan-1994.) $)
|
|
ordirr $p |- ( Ord A -> -. A e. A ) $=
|
|
( wcel wn word elirr a1i ) AABCADAEF $.
|
|
|
|
${
|
|
onirri.1 $e |- A e. On $.
|
|
$( An ordinal number is not a member of itself. Theorem 7M(c) of
|
|
[Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) $)
|
|
onirri $p |- -. A e. A $=
|
|
( word wcel wn onordi ordirr ax-mp ) ACAADEABFAGH $.
|
|
$}
|
|
|
|
$( A member of an ordinal class is not equal to it. (Contributed by NM,
|
|
25-May-1998.) $)
|
|
nordeq $p |- ( ( Ord A /\ B e. A ) -> A =/= B ) $=
|
|
( word wcel wne wn wceq ordirr eleq1 notbid syl5ibcom necon2ad imp ) ACZBAD
|
|
ZABENOABNAADZFABGZOFAHQPOABAIJKLM $.
|
|
|
|
$( An ordinal class cannot be an element of one of its members. Variant of
|
|
first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by
|
|
NM, 3-Apr-1994.) $)
|
|
ordn2lp $p |- ( Ord A -> -. ( A e. B /\ B e. A ) ) $=
|
|
( word wcel wa ordirr wtr wi ordtr trel syl mtod ) ACZABDBADEZAADZAFMAGNOHA
|
|
IAABJKL $.
|
|
|
|
$( An ordinal class and its singleton are disjoint. (Contributed by NM,
|
|
19-May-1998.) $)
|
|
orddisj $p |- ( Ord A -> ( A i^i { A } ) = (/) ) $=
|
|
( word wcel wn csn cin c0 wceq ordirr disjsn sylibr ) ABAACDAAEFGHAIAAJK $.
|
|
|
|
$( Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) $)
|
|
orddif $p |- ( Ord A -> A = ( suc A \ { A } ) ) $=
|
|
( word csn cin wceq csuc cdif orddisj disj3 cun df-suc difeq1i difun2 eqtri
|
|
c0 eqeq2i bitr4i sylib ) ABAACZDOEZAAFZSGZEZAHTAASGZEUCASIUBUDAUBASJZSGUDUA
|
|
UESAKLASMNPQR $.
|
|
|
|
$( The membership relation is irreflexive: no set is a member of itself.
|
|
Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) $)
|
|
elirrv $p |- -. x e. x $=
|
|
( cv elirr ) ABC $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A class is equal to its successor iff it is a proper class (assuming the
|
|
Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.) $)
|
|
sucprcreg $p |- ( -. A e. _V <-> suc A = A ) $=
|
|
( vx cvv wcel wn csuc sucprc cv wi wal elirr nfv eleq1 ceqsalg mtbiri csn
|
|
wceq velsn wss syl5bir wo olc cun elun ssid wb df-suc eqeq1i sseq1 mpbiri
|
|
sylbi sseld syl5 alrimiv nsyl3 impbii ) ACDZEAFZAQZAGUQBHZAQZUTADZIZBJZUS
|
|
UQVDAADZAKVBVEBACVEBLUTAAMNOUSVCBVAUTAPZDZUSVBBARVGVBVGUAZUSVBVGVBUBVHUTA
|
|
VFUCZDUSVBUTAVFUDUSVIAUTUSVIASZAASZAUEUSVIAQVJVKUFURVIAAUGUHVIAAUIUKUJULT
|
|
UMTUNUOUP $.
|
|
$}
|
|
|
|
$( The Russell class is equal to the universe ` _V ` . Exercise 5 of
|
|
[TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) $)
|
|
ruv $p |- { x | x e/ x } = _V $=
|
|
( cvv weq cab cv wnel df-v equid elirrv nelir 2th abbii eqtr2i ) BAACZADAEZ
|
|
OFZADAGNPANPAHOOAIJKLM $.
|
|
|
|
$( Alternate proof of Russell's Paradox ~ ru , simplified using (indirectly)
|
|
the Axiom of Set Induction ~ ax-setind . (Contributed by Alan Sare,
|
|
4-Oct-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ruALT $p |- { x | x e/ x } e/ _V $=
|
|
( cv wnel cab cvv wcel wn vprc df-nel mpbir wceq wb ruv neleq1 ax-mp ) ABZP
|
|
CADZECZEECZSEEFGHEEIJQEKRSLAMQEENOJ $.
|
|
|
|
$( No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring]
|
|
p. 38. This is also known as the Burali-Forti paradox (remark in
|
|
[Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the
|
|
"set" of all ordinal numbers is an ordinal class ( ~ ordon ), it must be
|
|
both an element of the set of all ordinal numbers yet greater than every
|
|
such element. ZF set theory resolves this paradox by not allowing the
|
|
class of all ordinal numbers to be a set (so instead it is a proper
|
|
class). Here we prove the denial of its existence. (Contributed by NM,
|
|
18-May-1994.) $)
|
|
onprc $p |- -. On e. _V $=
|
|
( con0 cvv wcel word wn ordon ordirr ax-mp elong mpbiri mto ) ABCZAACZADZME
|
|
FAGHLMNFABIJK $.
|
|
|
|
$( The class of all ordinal numbers is its own successor. (Contributed by
|
|
NM, 12-Sep-2003.) $)
|
|
sucon $p |- suc On = On $=
|
|
( con0 cvv wcel wn csuc wceq onprc sucprc ax-mp ) ABCDAEAFGAHI $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton]
|
|
p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario
|
|
Carneiro and Jim Kingdon, 27-Nov-2018.) $)
|
|
en2lp $p |- -. ( A e. B /\ B e. A ) $=
|
|
( vx vy wcel wa cvv cv wal wi wceq wn elex eldif pm3.4 eleq1 spcgv adantr
|
|
syl mpd cpr cdif wsb wral wo prid2g sylbi com12 mt2d ad2antlr w3a imbi12d
|
|
simp1r pm2.43b 3ad2ant2 wb eleq2 imbi1d 3ad2ant3 mpbid 3expia mtod prid1g
|
|
simp1l sylanbrc vex mpbiran elpr xchbinx sylibr ex alrimiv df-ral clelsb3
|
|
ioran imbi2i albii bitri imbi1i ax-setind pm2.65i ) ABEZBAEZFZAGABUAZUBZE
|
|
ZWDCHZWFEZCIZWGWDWICDUCZDWHUDZWIJZCIZWJWDDHZWHEZWOWFEZJZDIZWIJZCIWNWDWTCW
|
|
DWSWIWDWSFZWHAKZWHBKZUEZLZWIXAXBLXCLXEXAXBBWFEZWCXFLZWBWSWCBGEZXGBAMXHXFB
|
|
WEEZABGUFXFXHXILZXFXHXJFXHXJJBGWENXHXJOUGUHUISUJWDWSXBXFWDWSXBUKZWCXFWBWC
|
|
WSXBUMXKBWHEZXFJZWCXFJZWSWDXMXBWSXLXFWRXMDBWHWOBKWPXLWQXFWOBWHPWOBWFPULQU
|
|
NUOXBWDXMXNUPWSXBXLWCXFWHABUQURUSUTTVAVBXAXCWGWDWGLZWSWBXOWCWBAGEZXOABMXP
|
|
WGAWEEZABGVCWGXPXQLZWGXPXRFXPXRJAGWENXPXROUGUHUISRZRWDWSXCWGWDWSXCUKZWBWG
|
|
WBWCWSXCVDXTAWHEZWGJZWBWGJZWSWDYBXCWSYAWGWRYBDAWHWOAKWPYAWQWGWOAWHPWOAWFP
|
|
ULQUNUOXCWDYBYCUPWSXCYAWBWGWHBAUQURUSUTTVAVBXBXCVOVEWIWHWEEZXDWIWHGEYDLCV
|
|
FZWHGWENVGWHABYEVHVIVJVKVLWMWTCWLWSWIWLWPWKJZDIWSWKDWHVMYFWRDWKWQWPDCWFVN
|
|
VPVQVRVSVQVJWIDCVTSWBWJWGJWCWIWGCABWHAWFPQRTXSWA $.
|
|
$}
|
|
|
|
${
|
|
preleq.1 $e |- A e. _V $.
|
|
preleq.2 $e |- B e. _V $.
|
|
preleq.3 $e |- C e. _V $.
|
|
preleq.4 $e |- D e. _V $.
|
|
$( Equality of two unordered pairs when one member of each pair contains
|
|
the other member. (Contributed by NM, 16-Oct-1996.) $)
|
|
preleq $p |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) ->
|
|
( A = C /\ B = D ) ) $=
|
|
( wcel wa cpr wceq wn en2lp eleq12 anbi1d mtbiri con2i adantr wo preq12b
|
|
biimpi adantl ecased ) ABIZCDIZJZABKCDKLZJACLBDLJZADLBCLJZUGUJMUHUJUGUJUG
|
|
DCIZUFJDCNUJUEUKUFADBCOPQRSUHUIUJTZUGUHULABCDEFGHUAUBUCUD $.
|
|
|
|
$( Theorem for alternate representation of ordered pairs, requiring the
|
|
Axiom of Set Induction ~ ax-setind (via the ~ preleq step). See ~ df-op
|
|
for a description of other ordered pair representations. Exercise 34 of
|
|
[Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) $)
|
|
opthreg $p |- ( { A , { A , B } } = { C , { C , D } } <->
|
|
( A = C /\ B = D ) ) $=
|
|
( cpr wceq wa wcel prid1 cvv prexg mp2an preleq mpanl12 preq1 eqeq1d syl
|
|
preqr2 syl6bi imdistani adantr preq12 preq2d eqtrd impbii ) AABIZIZCCDIZI
|
|
ZJZACJZBDJZKZUNUOUJULJZKZUQAUJLCULLUNUSABEMCDGMAUJCULEANLBNLUJNLEFABNNOPG
|
|
CNLDNLULNLGHCDNNOPQRUOURUPUOURCBIZULJUPUOUJUTULACBSTBDCFHUBUCUDUAUQUKCUJI
|
|
ZUMUOUKVAJUPACUJSUEUQUJULCABCDUFUGUHUI $.
|
|
$}
|
|
|
|
$( The successor operation behaves like a one-to-one function (assuming the
|
|
Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and
|
|
its converse. (Contributed by NM, 25-Oct-2003.) $)
|
|
suc11g $p |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) ) $=
|
|
( wcel wa csuc wceq w3a wn en2lp wo sucidg syl5ibrcom elsucg sylibd 3adant1
|
|
eleq2 imp syl5ibcom 3adant2 jca eqcom orbi2i anbi1i sylib sylibr ord 3expia
|
|
ordir mpi suceq impbid1 ) ACEZBDEZFAGZBGZHZABHZUNUOURUSUNUOURIZBAEZABEZFZJU
|
|
SBAKUTVCUSUTVAUSLZVBUSLZFZVCUSLUTVABAHZLZVEFVFUTVHVEUOURVHUNUOURVHUOURBUPEZ
|
|
VHUOVIURBUQEBDMUPUQBRNBADOPSQUNURVEUOUNURVEUNURAUQEZVEUNAUPEURVJACMUPUQARTA
|
|
BCOPSUAUBVHVDVEVGUSVABAUCUDUEUFVAVBUSUJUGUHUKUIABULUM $.
|
|
|
|
$( The successor operation behaves like a one-to-one function. Compare
|
|
Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) $)
|
|
suc11 $p |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) ) $=
|
|
( con0 suc11g ) ABCCD $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( At least two sets exist (or in terms of first-order logic, the universe
|
|
of discourse has two or more objects). Although ~ dtruarb can also be
|
|
summarized as "at least two sets exist", the difference is that
|
|
~ dtruarb shows the existence of two sets which are not equal to each
|
|
other, but this theorem says that given a specific ` y ` , we can
|
|
construct a set ` x ` which does not equal it. (Contributed by Jim
|
|
Kingdon, 29-Dec-2018.) $)
|
|
dtruex $p |- E. x -. x = y $=
|
|
( cv wceq wn wex csn vex snex isseti wa wcel elirrv vsnid mpbiri mto eqtr
|
|
eleq2 ancom mtbi imnani eximii equcom notbii exbii mpbi ) BCZACZDZEZAFUHU
|
|
GDZEZAFUHUGGZDZUJAAUMUGBHIJUNUIUIUNKZUNUIKUOUGUMDZUPUGUGLZBMUPUQUGUMLBNUG
|
|
UMUGROPUGUHUMQPUIUNSTUAUBUJULAUIUKBAUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( At least two sets exist (or in terms of first-order logic, the universe
|
|
of discourse has two or more objects). If we assumed the law of the
|
|
excluded middle this would be equivalent to ~ dtruex . (Contributed by
|
|
Jim Kingdon, 29-Dec-2018.) $)
|
|
dtru $p |- -. A. x x = y $=
|
|
( weq wn wex wal dtruex exnalim ax-mp ) ABCZDAEJAFDABGJAHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Existential uniqueness implies there is a value for which the wff
|
|
argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) $)
|
|
eunex $p |- ( E! x ph -> E. x -. ph ) $=
|
|
( vy weu wex cv wceq wi wal wa wn nfv eu3 dtruex nfa1 con3d eximd exlimiv
|
|
sp mpi adantl sylbi ) ABDABEZABFCFGZHZBIZCEZJAKZBEZABCACLMUGUIUCUFUICUFUD
|
|
KZBEUIBCNUFUJUHBUEBOUFAUDUEBSPQTRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w ph $.
|
|
ordsoexmid.1 $e |- _E Or On $.
|
|
$( Weak linearity of ordinals implies the law of the excluded middle (that
|
|
is, decidability of an arbitrary proposition). (Contributed by Mario
|
|
Carneiro and Jim Kingdon, 29-Jan-2019.) $)
|
|
ordsoexmid $p |- ( ph \/ -. ph ) $=
|
|
( vw vx vy vz wo c0 wcel con0 wi cv w3a wceq eleq1 imbi12d eleq2 cep wral
|
|
wbr wn csn crab csuc ordtriexmidlem elexi sucid onsuci suc0 0elon 3anbi1d
|
|
eqeltrri orbi1d 3anbi2d orbi2d p0ex 3anbi3d orbi12d imbi2d wpo wor df-iso
|
|
wa mpbi simpri epel orbi12i imbi12i 2ralbii ralbii rspec3 ordtriexmidlem2
|
|
vtocl mp3an sylbi elirr elrabi mto elsuci ord mpi 0ex biidd rabsnt eqcoms
|
|
elsn syl orim12i mp2b orcom ) AUAZAGZAWKGACHUBZUCZWNUDZIZWNWMIZWMWOIZGZWL
|
|
WNWNJACUEZUFZUGWNJIZWOJIZWMJIZWPWSKZWTWNWTUHZHUDWMJUIHUJUHULDLZJIZXCXDMZX
|
|
GWOIZXGWMIZWRGZKZKZXBXCXDMZXEKDWNXAXGWNNZXIXOXMXEXPXHXBXCXDXGWNJOUKXPXJWP
|
|
XLWSXGWNWOOXPXKWQWRXGWNWMOUMPPXHELZJIZXDMZXGXQIZXKWMXQIZGZKZKZXNEWOWOJXFU
|
|
FXQWONZXSXIYCXMYEXRXCXHXDXQWOJOUNYEXTXJYBXLXQWOXGQYEYAWRXKXQWOWMQUOPPXHXR
|
|
FLZJIZMZXTXGYFIZYFXQIZGZKZKYDFWMUPYFWMNZYHXSYLYCYMYGXDXHXRYFWMJOUQYMYKYBX
|
|
TYMYIXKYJYAYFWMXGQYFWMXQOURUSPYLDEFJJJXGXQRTZXGYFRTZYFXQRTZGZKZFJSEJSZDJS
|
|
ZYLFJSEJSZDJSJRUTZYTJRVAUUBYTVCBDEFJRVBVDVEYSUUADJYRYLEFJJYNXTYQYKDEVFYOY
|
|
IYPYJDFVFFEVFVGVHVIVJVDVKVMVMVMVNWQWKWRAWQWNHNWKWNHXAWFACVLVOWRWMWNNZAWRW
|
|
MWNIZUAUUCUUDWMWMIWMVPACWMWMVQVRWRUUDUUCWMWNVSVTWAAWNWMAACWMHWBCLHNAWCWDW
|
|
EWGWHWIWKAWJVD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The successor of an ordinal class is ordinal. (Contributed by NM,
|
|
3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon,
|
|
20-Jul-2019.) $)
|
|
ordsuc $p |- ( Ord A <-> Ord suc A ) $=
|
|
( vy vx word csuc ordsucim wtr cv wcel wa wi wal wceq en2lp eleq1 biimpac
|
|
wn anim2i expr wss mtoi adantl elelsuc adantr ordelss sylan2 pm2.43d impr
|
|
wo sseld elsuci syl ecased ancom2s alrimivv sylibr sssucid trssord mp3an2
|
|
ex dftr2 mpancom impbii ) ADZAEZDZAFAGZVFVDVFBHZCHZIZVIAIZJZVHAIZKZCLBLVG
|
|
VFVNBCVFVLVMVFVKVJVMVFVKVJJZJZVMVHAMZVOVQQVFVOVQVKAVIIZJZVIANVKVJVQVSVJVQ
|
|
JVRVKVQVJVRVHAVIOPRSUAUBVPVHVEIZVMVQUIVFVKVJVTVFVKJVJVTVFVKVJVJVTKVPVIVEV
|
|
HVOVFVIVEIZVIVETVKWAVJVIAUCUDVEVIUEUFUJSUGUHVHAUKULUMUNUTUOBCAVAUPVGAVETV
|
|
FVDAUQAVEURUSVBVC $.
|
|
$}
|
|
|
|
$( A successor ordinal is the successor of its union. (Contributed by NM,
|
|
10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
onsucuni2 $p |- ( ( A e. On /\ A = suc B ) -> suc U. A = A ) $=
|
|
( con0 wcel csuc wceq wa cuni eleq1 biimpac sucelon wtr eloni ordtr unisucg
|
|
word syl mpbid suceq sylbir eqtr4d unieqd eqeq12d syl5ibr anabsi7 adantr
|
|
unieq eqtrd ) ACDZABEZFZGZAHZEZAEZHZAUIUKUNUPFZULUQUKUJHZEZUJEZHZFZULUJCDZV
|
|
BUKUIVCAUJCIJVCUSUJVAVCBCDZUSUJFZBKVDURBFZVEVDBLZVFVDBPVGBMBNQBCORURBSQTVCU
|
|
JLZVAUJFVCUJPVHUJMUJNQUJCORUAQUKUNUSUPVAUKUMURFUNUSFAUJUGUMURSQUKUOUTAUJSUB
|
|
UCUDUEUIUPAFZUKUIALZVIUIAPVJAMANQACORUFUH $.
|
|
|
|
${
|
|
$d ph x y $.
|
|
0elsucexmid.1 $e |- A. x e. On (/) e. suc x $.
|
|
$( If the successor of any ordinal class contains the empty set, excluded
|
|
middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) $)
|
|
0elsucexmid $p |- ( ph \/ -. ph ) $=
|
|
( vy c0 csn crab wcel wceq wo wn csuc con0 cv ordtriexmidlem suceq eleq2d
|
|
wral 0ex ax-mp rspcv mp2 elsuc mpbi wb snid elrab3 biimpi ordtriexmidlem2
|
|
biidd eqcoms orim12i ) EADEFZGZHZEUNIZJZAAKZJEUNLZHZUQUNMHEBNZLZHZBMRUTAD
|
|
OCVCUTBUNMVAUNIVBUSEVAUNPQUAUBEUNSUCUDUOAUPURUOAEUMHUOAUEESUFAADEUMDNEIAU
|
|
JUGTUHURUNEADUIUKULT $.
|
|
$}
|
|
|
|
$( A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
nlimsucg $p |- ( A e. V -> -. Lim suc A ) $=
|
|
( wcel csuc wlim wn word cuni wceq wa limord ordsuc sylibr limuni jca ordtr
|
|
wtr unisucg biimpa sylan2 eqeq2d ordirr eleq2 notbid syl5ibrcom sucidg syl6
|
|
wi con3i adantl sylbid expimpd syl5 con2d pm2.43i ) ABCZADZEZFUPURUPURAGZUQ
|
|
UQHZIZJUPUPFZURUSVAURUQGUSUQKALMUQNOUPUSVAVBUPUSJZVAUQAIZVBVCUTAUQUSUPAQZUT
|
|
AIZAPUPVEVFABRSTUAUSVDVBUHUPUSVDAUQCZFZVBUSVHVDAACZFAUBVDVGVIUQAAUCUDUEUPVG
|
|
ABUFUIUGUJUKULUMUNUO $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The collection of ordinals in the power class of an ordinal is a
|
|
superset of its successor.
|
|
|
|
We can think of ` ( ~P A i^i On ) ` as another possible definition of
|
|
successor, which would be equivalent to ~ df-suc given excluded middle.
|
|
It is an ordinal, and has some successor-like properties. For example,
|
|
if ` A e. On ` then both ` U. suc A = A ` ( ~ onunisuci ) and
|
|
` U. { x e. On | x C_ A } = A ` ( ~ onuniss2 ).
|
|
|
|
Constructively ` ( ~P A i^i On ) ` and ` suc A ` cannot be shown to be
|
|
equivalent (as proved at ~ ordpwsucexmid ). (Contributed by Jim
|
|
Kingdon, 21-Jul-2019.) $)
|
|
ordpwsucss $p |- ( Ord A -> suc A C_ ( ~P A i^i On ) ) $=
|
|
( vx word csuc cpw con0 cin cv wcel wss wa wi ordsuc ordelon ex sylbi wtr
|
|
ordtr trsucss syl jcad elin selpw anbi2ci bitri syl6ibr ssrdv ) ACZBADZAE
|
|
ZFGZUHBHZUIIZULFIZULAJZKZULUKIZUHUMUNUOUHUICZUMUNLAMURUMUNUIULNOPUHAQUMUO
|
|
LARAULSTUAUQULUJIZUNKUPULUJFUBUSUOUNBAUCUDUEUFUG $.
|
|
$}
|
|
|
|
$( No member of a set of ordinal numbers belongs to its minimum.
|
|
(Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro
|
|
and Jim Kingdon, 21-Jul-2019.) $)
|
|
onnmin $p |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A ) $=
|
|
( wcel cint wn con0 wss intss1 elirr ssel mtoi syl adantl ) BACZBADZCZEZAFG
|
|
NOBGZQBAHRPBBCBIOBBJKLM $.
|
|
|
|
$( Relationship between subset and elementhood. In the context of ordinals
|
|
this can be seen as an ordering law. (Contributed by Jim Kingdon,
|
|
22-Jul-2019.) $)
|
|
ssnel $p |- ( A C_ B -> -. B e. A ) $=
|
|
( wss wcel elirr ssel mtoi ) ABCBADBBDBEABBFG $.
|
|
|
|
${
|
|
$d ph x z $.
|
|
ordpwsucexmid.1 $e |- A. x e. On suc x = ( ~P x i^i On ) $.
|
|
$( The subset in ~ ordpwsucss cannot be equality. That is, strengthening
|
|
it to equality implies excluded middle. (Contributed by Jim Kingdon,
|
|
30-Jul-2019.) $)
|
|
ordpwsucexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz c0 csn crab wcel wceq wo wn csuc cpw con0 cin 0elpw 0elon elin ax-mp
|
|
cv mpbir2an ordtriexmidlem pweq ineq1d eqeq12d vtoclri eleqtrri elsuci wb
|
|
suceq 0ex snid biidd elrab3 biimpi ordtriexmidlem2 eqcoms orim12i ) EADEF
|
|
ZGZHZEUTIZJZAAKZJEUTLZHVCEUTMZNOZVEEVGHEVFHENHUTPQEVFNRUAUTNHVEVGIZADUBBT
|
|
ZLZVIMZNOZIVHBUTNVIUTIZVJVEVLVGVIUTUJVMVKVFNVIUTUCUDUECUFSUGEUTUHSVAAVBVD
|
|
VAAEUSHVAAUIEUKULAADEUSDTEIAUMUNSUOVDUTEADUPUQURS $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
ordtri2or2exmid.1 $e |- A. x e. On A. y e. On ( x C_ y \/ y C_ x ) $.
|
|
$( Ordinal trichotomy implies excluded middle. (Contributed by Jim
|
|
Kingdon, 29-Aug-2021.) $)
|
|
ordtri2or2exmid $p |- ( ph \/ -. ph ) $=
|
|
( vz wo c0 wss cv con0 wral wcel 0elon sseq1 sseq2 orbi12d ax-mp wb biidd
|
|
wceq wn csn cpr crab ordtri2or2exmidlem csuc suc0 onsuci eqeltrri rspc2va
|
|
mpanl12 elirr wa simpl simpr p0ex prid2 elrab3 sylibr sseldd ex snssg 0ex
|
|
mtoi prid1 biimpi sylbir orim12i orcom mpbi ) AUAZAFZAVKFAEGGUBZUCZUDZVMH
|
|
ZVMVOHZFZVLBIZCIZHZVTVSHZFZCJKBJKZVRDVOJLVMJLWDVRAEUEGUFVMJUGGMUHUIWCVRVO
|
|
VTHZVTVOHZFBCVOVMJJVSVOTWAWEWBWFVSVOVTNVSVOVTOPVTVMTWEVPWFVQVTVMVOOVTVMVO
|
|
NPUJUKQVPVKVQAVPAVMVMLZVMULVPAWGVPAUMZVOVMVMVPAUNWHAVMVOLZVPAUOVMVNLWIARG
|
|
VMUPUQAAEVMVNEIZVMTASURQUSUTVAVDVQGVOLZAGJLWKVQRMGVOJVBQWKAGVNLWKARGVMVCV
|
|
EAAEGVNWJGTASURQVFVGVHQVKAVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d ph u v $. $d x y u v $.
|
|
onintexmid.onint $e |- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y ) $.
|
|
$( If the intersection (infimum) of an inhabited class of ordinal numbers
|
|
belongs to the class, excluded middle follows. The hypothesis would be
|
|
provable given excluded middle. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 29-Aug-2021.) $)
|
|
onintexmid $p |- ( ph \/ -. ph ) $=
|
|
( vu vv cv wss wo con0 wcel wa cint wceq wex wi cin eqeq1i dfss1 vex prmg
|
|
cpr prssi adantr zfpair2 sseq1 eleq2 exbidv anbi12d inteq eleq12d imbi12d
|
|
vtocl syl2anc elpri syl incom intpr 3bitr4ri bitr4i sylib ordtri2or2exmid
|
|
id orbi12i rgen2a ) AEFEGZFGZHZVGVFHZIZEFJVFJKZVGJKZLZVFVGUBZMZVFNZVOVGNZ
|
|
IZVJVMVOVNKZVRVMVNJHZBGZVNKZBOZVSVFVGJUCVKWCVLBVFVGJUAUDCGZJHZWAWDKZBOZLZ
|
|
WDMZWDKZPVTWCLZVSPCVNEFUEWDVNNZWHWKWJVSWLWEVTWGWCWDVNJUFWLWFWBBWDVNWAUGUH
|
|
UIWLWIVOWDVNWDVNUJWLVCUKULDUMUNVOVFVGUOUPVPVHVQVIVGVFQZVFNVFVGQZVFNVHVPWM
|
|
WNVFVGVFUQRVFVGSVOWNVFVFVGETFTURZRUSVQWNVGNVIVOWNVGWORVGVFSUTVDVAVEVB $.
|
|
$}
|
|
|
|
${
|
|
$d A s x y $.
|
|
$( The epsilon relation is well-founded on any class. (Contributed by NM,
|
|
26-Nov-1995.) $)
|
|
zfregfr $p |- _E Fr A $=
|
|
( vs vy vx cep wfr cv wfrfor df-frind wel wi wral wcel wsb bitr4i clelsb3
|
|
wal bitri ralbii imbi1i wbr bi2.04 albii df-ral sbim imbi12i ralcom3 epel
|
|
wss ax-setind dfss2 sylibr sylbir df-frfor mpbir mpgbir ) AEFAEBGZHZBAEBI
|
|
URCGZDGZEUAZCBJZKZCALZDBJZKZDALZAUQUIZKVGUTAMZVEKZDCNZCUTLZVJKZDQZVHVNVLV
|
|
EKZDALZVGVNVIVOKZDQVPVMVQDVLVIVEUBUCVODAUDOVOVFDAVLVDVEVLCDJZVBKZCALZVDVL
|
|
USAMZVBKZCUTLVTVKWBCUTVKVIDCNZVEDCNZKWBVIVEDCUEWCWAWDVBCDAPCDUQPUFRSVBCUT
|
|
AUGRVCVSCAVAVRVBCDUHTSOTSRVNVJDQVHVJCDUJDAUQUKULUMDCAEUQUNUOUP $.
|
|
$}
|
|
|
|
$( Epsilon is well-founded on an ordinal class. (Contributed by NM,
|
|
22-Apr-1994.) $)
|
|
ordfr $p |- ( Ord A -> _E Fr A ) $=
|
|
( cep wfr word zfregfr a1i ) ABCADAEF $.
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring]
|
|
p. 36. (Contributed by NM, 3-Apr-1994.) $)
|
|
ordwe $p |- ( Ord A -> _E We A ) $=
|
|
( vx vy vz word cep wfr cv wbr wa wi wral wwe wcel w3a ordelord 3ad2antr3
|
|
ordfr wel epel ordtr1 anbi12i 3imtr4g syl ralrimivvva df-wetr sylanbrc )
|
|
AEZAFGBHZCHZFIZUJDHZFIZJZUIULFIZKZDALCALBALAFMARUHUPBCDAAAUHUIANZUJANZULA
|
|
NZOJULEZUPUHUQUSUTURAULPQUTBCSZCDSZJBDSUNUOUIUJULUAUKVAUMVBBCTCDTUBBDTUCU
|
|
DUEBCDAFUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d A z $. $d B a b $. $d B z $. $d C b $. $d C z $.
|
|
$d R a b $. $d R z $.
|
|
wetriext.we $e |- ( ph -> R We A ) $.
|
|
wetriext.a $e |- ( ph -> A e. V ) $.
|
|
wetriext.tri $e |- ( ph ->
|
|
A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) ) $.
|
|
wetriext.b $e |- ( ph -> B e. A ) $.
|
|
wetriext.c $e |- ( ph -> C e. A ) $.
|
|
wetriext.ext $e |- ( ph -> A. z e. A ( z R B <-> z R C ) ) $.
|
|
$( A trichotomous well-order is extensional. (Contributed by Jim Kingdon,
|
|
26-Sep-2021.) $)
|
|
wetriext $p |- ( ph -> B = C ) $=
|
|
( wbr wceq wcel cv breq1 wa wb wral bibi12d rspcv sylc biimpar wn wfr wwe
|
|
syl frirrg syl3anc adantr pm2.21dd simpr biimpa w3o eqeq1 breq2 3orbi123d
|
|
wefr wi eqeq2 rspc2v syl2anc mpd mpjao3dan ) ADEFPZDEQZVJEDFPZAVIUADDFPZV
|
|
JAVLVIADCRZBSZDFPZVNEFPZUBZBCUCZVLVIUBZMOVQVSBDCVNDQVOVLVPVIVNDDFTVNDEFTU
|
|
DUEUFUGAVLUHZVIACFUIZCGRZVMVTACFUJWAJCFVBUKZKMCDFGULUMUNUOAVJUPAVKUAEEFPZ
|
|
VJAVKWDAECRZVRVKWDUBZNOVQWFBECVNEQVOVKVPWDVNEDFTVNEEFTUDUEUFUQAWDUHZVKAWA
|
|
WBWEWGWCKNCEFGULUMUNUOAHSZISZFPZWHWIQZWIWHFPZURZICUCHCUCZVIVJVKURZLAVMWEW
|
|
NWOVCMNWMWODWIFPZDWIQZWIDFPZURHIDECCWHDQWJWPWKWQWLWRWHDWIFTWHDWIUSWHDWIFU
|
|
TVAWIEQWPVIWQVJWRVKWIEDFUTWIEDVDWIEDFTVAVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( A subset of a set well-ordered by set membership is well-ordered by set
|
|
membership. (Contributed by Jim Kingdon, 30-Sep-2021.) $)
|
|
wessep $p |- ( ( _E We A /\ B C_ A ) -> _E We B ) $=
|
|
( vx vy vz cep wwe wss wa cv wbr wral wcel w3a ssel 3anim123d adantl epel
|
|
wi wel wetrep adantlr anbi12i 3imtr4g syl ralrimivvva wfr zfregfr df-wetr
|
|
imdistani mpbiran sylibr ) AFGZBAHZIZCJZDJZFKZUQEJZFKZIZUPUSFKZSZEBLDBLCB
|
|
LZBFGZUOVCCDEBBBUOUPBMZUQBMZUSBMZNZIUOUPAMZUQAMZUSAMZNZIZVCUOVIVMUNVIVMSU
|
|
MUNVFVJVGVKVHVLBAUPOBAUQOBAUSOPQUJVNCDTZDETZIZCETZVAVBUMVMVQVRSUNCDEAUAUB
|
|
URVOUTVPCDRDERUCCERUDUEUFVEBFUGVDBUHCDEBFUIUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c $. $d b c x $. $d ph x $.
|
|
reg3exmidlemwe.a $e |-
|
|
A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) } $.
|
|
$( Lemma for ~ reg3exmid . Our counterexample ` A ` satisfies ` We ` .
|
|
(Contributed by Jim Kingdon, 3-Oct-2021.) $)
|
|
reg3exmidlemwe $p |- _E We A $=
|
|
( va vb vc cep cv wbr wa wral wcel epel c0 wceq noel eleq2 wo adantr crab
|
|
wwe wfr wi zfregfr w3a anbi12i simpr csn elirr simprr mtbiri simprl nsyl3
|
|
wn cpr elrabi eleq2s elpri syl orcomd 3ad2ant2 ecased elpr sylib 3ad2ant3
|
|
vex 3eltr3d ex mtoi pm2.21dd sylan2b rgen3 df-wetr mpbir2an ) CHUBCHUCEIZ
|
|
FIZHJZVQGIZHJZKZVPVSHJZUDZGCLFCLECLCUEWCEFGCCCVPCMZVQCMZVSCMZUFZWAWBWAWGV
|
|
PVQMZVQVSMZKZWBVRWHVTWIEFNFGNUGWGWJKZWJWBWGWJUHWGWJUOWJWGWJOUIZWLMZWLUJWG
|
|
WJWMWKVQVSWLWLWGWHWIUKZWKVQWLPZVQOPZWPWHWKWPWHVPOMVPQVQOVPRULWGWHWIUMUNWG
|
|
WOWPSZWJWEWDWQWFWEWPWOWEVQOWLUPZMZWPWOSWSVQBIZWLPWTOPAKSZBWRUAZCXABVQWRUQ
|
|
DURVQOWLUSUTVAVBTVCWKVSWLPZVSOPZXDWIWKXDWIVQOMVQQVSOVQRULWNUNWGXCXDSZWJWF
|
|
WDXEWEWFXDXCWFVSWRMZXDXCSXFVSXBCXABVSWRUQDURVSOWLGVGVDVEVAVFTVCVHVIVJTVKV
|
|
LVIVMEFGCHVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d ph u w z $. $d ph u x y z $.
|
|
reg3exmid.1 $e |- ( ( _E We z /\ E. w w e. z ) ->
|
|
E. x e. z A. y e. z x C_ y ) $.
|
|
$( If any inhabited set satisfying ~ df-wetr for ` _E ` has a minimal
|
|
element, excluded middle follows. (Contributed by Jim Kingdon,
|
|
3-Oct-2021.) $)
|
|
reg3exmid $p |- ( ph \/ -. ph ) $=
|
|
( vu cv c0 csn wceq wa wo wcel wex wral wrex cep wwe wi cpr crab wss eqid
|
|
wn regexmidlemm reg3exmidlemwe pp0ex rabex weeq2 eleq2 anbi12d rexeqbi1dv
|
|
exbidv raleq imbi12d vtocl mpan reg2exmidlema mp2b ) EHZGHZIJZKVBIKALMZGI
|
|
VCUAZUBZNZEOZBHCHUCZCVFPZBVFQZAAUEMAGEVFVFUDZUFVFRSZVHVKAGVFVLUGDHZRSZVAV
|
|
NNZEOZLZVICVNPZBVNQZTVMVHLZVKTDVFVDGVEUHUIVNVFKZVRWAVTVKWBVOVMVQVHVNVFRUJ
|
|
WBVPVGEVNVFVAUKUNULVSVJBVNVFVICVNVFUOUMUPFUQURAGCBVFVLUSUT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Transfinite induction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
|
|
p. 39. This principle states that if ` A ` is a class of ordinal
|
|
numbers with the property that every ordinal number included in ` A `
|
|
also belongs to ` A ` , then every ordinal number is in ` A ` .
|
|
|
|
(Contributed by NM, 18-Feb-2004.) $)
|
|
tfi $p |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On ) $=
|
|
( vy con0 wss cv wcel wi wral wa wal df-ral albii bitri imbi1i spi imim1i
|
|
dfss2 impexp wsb wceq imdi imbi2i 19.21v bitr4i simpl wtr tron dftr2 mpbi
|
|
bi2.04 3imtr3i alimi sylbi adantl clelsb3 imbi12i ralbii ax-setind sylbir
|
|
jca sbim sylibr syl eqss biimpri syldan ) BDEZAFZBEZVIBGZHZADIZDBEZBDUAZV
|
|
HVMJCFZVIGZVPDGZVPBGZHZHZCKZVIDGZVKHZHZAKZVNVMWFVHVMWCVQVSHZHZCKZWDHZAKZW
|
|
FVMWCVJHZWDHZAKZWKVMWCVLHZAKWNVLADLWOWMAWCVJVKUBMNWMWJAWLWIWDWLWCWGCKZHWI
|
|
VJWPWCCVIBRUCWCWGCUDUEOMNWJWEAWBWIWDWAWHCVQVRJZVSHVQWCJZVSHZWAWHWRWQVSWRV
|
|
QVRVQWCUFWRVRHZAWTAKZCDUGXACKUHCADUIUJPPVAQVQVRVSSWSVQWCVSHHWHVQWCVSSVQWC
|
|
VSUKNULUMQUMUNUOWFWDAKZVNWFWDACTZCVIIZWDHZAKXBXEWEAXDWBWDXDVTCVIIWBXCVTCV
|
|
IXCWCACTZVKACTZHVTWCVKACVBXFVRXGVSCADUPCABUPUQNURVTCVILNOMWDCAUSUTADBRVCV
|
|
DVOVHVNJBDVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d w y z ph $. $d w x y z $.
|
|
tfis.1 $e |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) ) $.
|
|
$( Transfinite Induction Schema. If all ordinal numbers less than a given
|
|
number ` x ` have a property (induction hypothesis), then all ordinal
|
|
numbers have the property (conclusion). Exercise 25 of [Enderton]
|
|
p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro,
|
|
20-Nov-2016.) $)
|
|
tfis $p |- ( x e. On -> ph ) $=
|
|
( vz vw cv con0 wcel crab wss wi wral wceq ssrab2 wa nfcv syl5bbr simprbi
|
|
wsb nfrab1 nfss nfcri dfss3 sseq1 rabid eleq1 imbi12d sbequ nfs1v sbequ12
|
|
nfim nfv cbvrab elrab2 ralimi syl5 anc2li vtoclgaf rgen tfi mp2an rabeq2i
|
|
eqcomi ) BGZHIZVFAABHHABHJZHVGHKEGZVGKZVHVGIZLZEHMVGHNABHOVKEHCGZVGIZCVEM
|
|
ZVFAPZLVKBVHHBVHQZVIVJBBVHVGVPABHUAZUBBEVGVQUCULVEVHNZVNVIVOVJVNVEVGKVRVI
|
|
CVEVGUDVEVHVGUERVOVEVGIVRVJABHUFVEVHVGUGRUHVFVNAVNABCTZCVEMVFAVMVSCVEVMVL
|
|
HIVSABFTZVSFVLHVGAFCBUIAVTBFHBHQFHQAFUMABFUJABFUKUNUOSUPDUQURUSUTEVGVAVBV
|
|
DVCS $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x y $.
|
|
tfis2f.1 $e |- F/ x ps $.
|
|
tfis2f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
tfis2f.3 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
|
|
$( Transfinite Induction Schema, using implicit substitution. (Contributed
|
|
by NM, 18-Aug-1994.) $)
|
|
tfis2f $p |- ( x e. On -> ph ) $=
|
|
( wsb cv wral con0 wcel sbie ralbii syl5bi tfis ) ACDACDHZDCIZJBDRJRKLAQB
|
|
DRABCDEFMNGOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x y $.
|
|
tfis2.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
tfis2.2 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
|
|
$( Transfinite Induction Schema, using implicit substitution. (Contributed
|
|
by NM, 18-Aug-1994.) $)
|
|
tfis2 $p |- ( x e. On -> ph ) $=
|
|
( nfv tfis2f ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x ch $. $d x A $. $d x y $.
|
|
tfis3.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
tfis3.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
tfis3.3 $e |- ( x e. On -> ( A. y e. x ps -> ph ) ) $.
|
|
$( Transfinite Induction Schema, using implicit substitution. (Contributed
|
|
by NM, 4-Nov-2003.) $)
|
|
tfis3 $p |- ( A e. On -> ch ) $=
|
|
( con0 tfis2 vtoclga ) ACDFJHABDEGIKL $.
|
|
$}
|
|
|
|
${
|
|
$d x v w y z T $. $d v w y z R $. $d x v w z S $. $d x v w z ch $.
|
|
$d x v w y z ph $. $d w y z ps $. $d x A $. $d x th $.
|
|
tfisi.a $e |- ( ph -> A e. V ) $.
|
|
tfisi.b $e |- ( ph -> T e. On ) $.
|
|
tfisi.c $e |- ( ( ph /\ ( R e. On /\ R C_ T ) /\
|
|
A. y ( S e. R -> ch ) ) -> ps ) $.
|
|
tfisi.d $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
tfisi.e $e |- ( x = A -> ( ps <-> th ) ) $.
|
|
tfisi.f $e |- ( x = y -> R = S ) $.
|
|
tfisi.g $e |- ( x = A -> R = T ) $.
|
|
$( A transfinite induction scheme in "implicit" form where the induction is
|
|
done on an object derived from the object of interest. (Contributed by
|
|
Stefan O'Rear, 24-Aug-2015.) $)
|
|
tfisi $p |- ( ph -> th ) $=
|
|
( vv wi vz vw wss ssid wceq wa eqid wcel wal con0 weq eqeq2 anbi2d imbi1d
|
|
cv sseq1 imbi12d albidv eqeq1d imbi2d cbvalv syl6bb wral w3a simp3l simp2
|
|
simp1l eqeltrd simp3r eqsstrd csb wsb simpl3l simpl1l simpr simpl2 onelss
|
|
eleqtrd simpl3r sstrd simpl1r rspcva syl2anc eqidd csbhypf eqcomd equcoms
|
|
sylc nfcv wb nfv sbhypf bicomd spv mp2and ex alrimiv sylib syl121anc 3exp
|
|
eleq1d tfis3 syl spcgv mpi expd pm2.43i ) AJJUCZDJUDAXHDTAAXHDAJJUEZAXHUF
|
|
ZDTZJUGAGKUHHJUEZXJBTZTZEUIZXIXKTZLAJUJUHXOMHUAUOZUEZAXQJUCZUFZBTZTZEUIZI
|
|
UBUOZUEZAYDJUCZUFZCTZTZFUIZXOUAUBJUAUBUKZYCHYDUEZYGBTZTZEUIYJYKYBYNEYKXRY
|
|
LYAYMXQYDHULYKXTYGBYKXSYFAXQYDJUPUMUNUQURYNYIEFEFUKZYLYEYMYHYOHIYDQUSYOBC
|
|
YGOUTUQVAVBXQJUEZYBXNEYPXRXLYAXMXQJHULYPXTXJBYPXSXHAXQJJUPUMUNUQURXQUJUHZ
|
|
YJUBXQVCZYCYQYRUFZYBEYSXRXTBYSXRXTVDZAHUJUHHJUCIHUHZCTZFUIZBYSXRAXSVEYTHX
|
|
QUJYSXRXTVFZYQYRXRXTVGVHYTHXQJUUDYSXRAXSVIVJYTESUOHVKZHUHZBESVLZTZSUIUUCY
|
|
TUUHSYTUUFUUGYTUUFUFZAUUEJUCZUUGAXSYSXRUUFVMUUIUUEXQJUUIYQUUEXQUHZUUEXQUC
|
|
YQYRXRXTUUFVNUUIUUEHXQYTUUFVOYSXRXTUUFVPVRZXQUUEVQWHAXSYSXRUUFVSVTUUIIUUE
|
|
UEZAUUJUFZCTZTZFUIZUUEUUEUEZUUNUUGTZUUIUUKYRUUQUULYQYRXRXTUUFWAYJUUQUBUUE
|
|
XQYDUUEUEZYIUUPFUUTYEUUMYHUUOYDUUEIULUUTYGUUNCUUTYFUUJAYDUUEJUPUMUNUQURWB
|
|
WCUUIUUEWDUUPUURUUSTFSFSUKZUUMUURUUOUUSUVAIUUEUUEUUMSFSFUKZUUEIESFUOZHIEU
|
|
VCWIEIWIQWEZWFWGUSUVACUUGUUNCUUGWJSFUVBUUGCBCESUVCCEWKOWLZWMWGUTUQWNWHWOW
|
|
PWQUUHUUBSFUVBUUFUUAUUGCUVBUUEIHUVDXAUVEUQVAWRNWSWTWQWPXBXCXNXPEGKEUOGUEZ
|
|
XLXIXMXKUVFHJJRUSUVFBDXJPUTUQXDWHXEXFXGXE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
IZF Set Theory - add the Axiom of Infinity
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Infinity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF".
|
|
(Contributed by Jim Kingdon, 16-Nov-2018.) $)
|
|
ax-iinf $a |- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) ) $.
|
|
|
|
$( A standard version of the Axiom of Infinity, using definitions to
|
|
abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
zfinf2 $p |- E. x ( (/) e. x /\ A. y e. x suc y e. x ) $=
|
|
( c0 cv wcel csuc wral wa wex wi wal ax-iinf df-ral anbi2i exbii mpbir )
|
|
CADZEZBDZFQEZBQGZHZAIRSQETJBKZHZAIABLUBUDAUAUCRTBQMNOP $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The natural numbers (i.e. finite ordinals)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbol. $)
|
|
$c _om $. $( Greek letter omega $)
|
|
|
|
$( Extend class notation to include the class of natural numbers. $)
|
|
com $a class _om $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the class of natural numbers as the smallest inductive set, which
|
|
is valid provided we assume the Axiom of Infinity. Definition 6.3 of
|
|
[Eisenberg] p. 82.
|
|
|
|
_Note_: the natural numbers ` _om ` are a subset of the ordinal numbers
|
|
~ df-on . Later, when we define complex numbers, we will be able to
|
|
also define a subset of the complex numbers with analogous properties
|
|
and operations, but they will be different sets. (Contributed by NM,
|
|
6-Aug-1994.) Use its alias ~ dfom3 instead for naming consistency with
|
|
set.mm. (New usage is discouraged.) $)
|
|
df-iom $a |- _om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Alias for ~ df-iom . Use it instead of ~ df-iom for naming consistency
|
|
with set.mm. (Contributed by NM, 6-Aug-1994.) $)
|
|
dfom3 $p |- _om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } $=
|
|
( df-iom ) ABC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The existence of omega (the class of natural numbers). Axiom 7 of
|
|
[TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) $)
|
|
omex $p |- _om e. _V $=
|
|
( vy vx com cvv wcel c0 cv csuc wral wa cint zfinf2 intexabim ax-mp dfom3
|
|
cab wex eleq1i mpbir ) CDEFAGZEBGHTEBTIJZAPKZDEZUAAQUCABLUAAMNCUBDABORS
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Peano's postulates
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Zero is a natural number. One of Peano's five postulates for
|
|
arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed
|
|
by NM, 15-May-1994.) $)
|
|
peano1 $p |- (/) e. _om $=
|
|
( vy vx vz c0 cv wcel csuc wral wa cab com wi 0ex elint wsb df-clab simpl
|
|
cint sbimi clelsb4 sylib sylbi mpgbir dfom3 eleqtrri ) DDAEZFZBEGUFFBUFHZ
|
|
IZAJZRZKDUKFCEZUJFZDULFZLCCDUJMNUMUIACOZUNUICAPUOUGACOUNUIUGACUGUHQSCADTU
|
|
AUBUCABUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The successor of any natural number is a natural number. One of Peano's
|
|
five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring]
|
|
p. 42. (Contributed by NM, 3-Sep-2003.) $)
|
|
peano2 $p |- ( A e. _om -> suc A e. _om ) $=
|
|
( vy vx vz cvv wcel com csuc cv wa wi wb imbi12d adantl wsb wal sylib nfv
|
|
wral nfan elex c0 cint simpl wceq eleq1 suceq eleq1d df-clab simpr df-ral
|
|
cab sbimi sbim elsb4 clelsb4 imbi12i bitri sbalv sylbi 19.21bi nfra1 nfvd
|
|
nfsab nfcvd vtocldf ralrimiva ralim elintg sucexg syl syl5ibr mpd 3imtr4g
|
|
dfom3 eleq2i mpcom ) AEFZAGFZAHZGFZAGUAVRAUBBIZFZCIZHZWBFZCWBSZJZBULZUCZF
|
|
ZVTWJFZVSWAVRADIZFZVTWMFZKZDWISZWKWLKZVRWPDWIVRWMWIFZJZWDWMFZWEWMFZKZWPCA
|
|
EVRWSUDWDAUEZXCWPLWTXDXAWNXBWOWDAWMUFXDWEVTWMWDAUGUHMNWSXCVRWSXCCWSWHBDOZ
|
|
XCCPZWHDBUIXEWDWBFZWFKZCPZBDOXFWHXIBDWHWGXIWCWGUJWFCWBUKQUMXHXCBDCXHBDOXG
|
|
BDOZWFBDOZKXCXGWFBDUNXJXAXKXBDBCUODBWEUPUQURUSQUTVANVRWSCVRCRWHCBDWCWGCWC
|
|
CRWFCWBVBTVDTWTCAVEWTWPCVCVFVGWQWRVRWNDWISZWODWISZKWNWODWIVHVRWKXLWLXMDAW
|
|
IEVIVRVTEFWLXMLAEVJDVTWIEVIVKMVLVMGWJABCVOZVPGWJVTXNVPVNVQ $.
|
|
$}
|
|
|
|
$( The successor of any natural number is not zero. One of Peano's five
|
|
postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42.
|
|
(Contributed by NM, 3-Sep-2003.) $)
|
|
peano3 $p |- ( A e. _om -> suc A =/= (/) ) $=
|
|
( com nsuceq0g ) ABC $.
|
|
|
|
$( Two natural numbers are equal iff their successors are equal, i.e. the
|
|
successor function is one-to-one. One of Peano's five postulates for
|
|
arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed
|
|
by NM, 3-Sep-2003.) $)
|
|
peano4 $p |- ( ( A e. _om /\ B e. _om ) -> ( suc A = suc B <-> A = B ) ) $=
|
|
( com suc11g ) ABCCD $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The induction postulate: any class containing zero and closed under the
|
|
successor operation contains all natural numbers. One of Peano's five
|
|
postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring]
|
|
p. 43. The more traditional statement of mathematical induction as a
|
|
theorem schema, with a basis and an induction step, is derived from this
|
|
theorem as theorem ~ findes . (Contributed by NM, 18-Feb-2004.) $)
|
|
peano5 $p |- ( ( (/) e. A /\
|
|
A. x e. _om ( x e. A -> suc x e. A ) ) -> _om C_ A ) $=
|
|
( vy c0 wcel cv csuc wi com wral wa cin wss cint elin df-ral sylibr eleq2
|
|
cab wal dfom3 peano1 mpbiran biimpri peano2 adantr a1i jcad alimi imbi12i
|
|
pm3.31 albii 3imtr4i anim12i omex wceq raleqbi1dv anbi12d elab intss1 syl
|
|
inex1 syl5eqss ssid biantrur ssin bitri ) DBEZAFZBEZVIGZBEZHZAIJZKZIIBLZM
|
|
ZIBMZVOIDCFZEZVKVSEZAVSJZKZCSZNZVPCAUAVOVPWDEZWEVPMVODVPEZVKVPEZAVPJZKZWF
|
|
VHWGVNWIWGVHWGDIEVHUBDIBOUCUDVNVIVPEZWHHZATZWIVIIEZVMHZATWNVJKZVKIEZVLKZH
|
|
ZATVNWMWOWSAWOWPWQVLWPWQHWOWNWQVJVIUEUFUGWNVJVLUKUHUIVMAIPWLWSAWKWPWHWRVI
|
|
IBOVKIBOUJULUMWHAVPPQUNWCWJCVPIBUOVBVSVPUPVTWGWBWIVSVPDRWAWHAVSVPVSVPVKRU
|
|
QURUSQVPWDUTVAVCVRIIMZVRKVQWTVRIVDVEIIBVFVGQ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite induction (for finite ordinals)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x A $.
|
|
find.1 $e |- ( A C_ _om /\ (/) e. A /\ A. x e. A suc x e. A ) $.
|
|
$( The Principle of Finite Induction (mathematical induction). Corollary
|
|
7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was
|
|
suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states
|
|
that ` A ` is a set of natural numbers, zero belongs to ` A ` , and
|
|
given any member of ` A ` the member's successor also belongs to ` A ` .
|
|
The conclusion is that every natural number is in ` A ` . (Contributed
|
|
by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
find $p |- A = _om $=
|
|
( com wss c0 wcel cv csuc wral simp1i wi wa w3a 3simpc ax-mp df-ral alral
|
|
wal sylbi anim2i peano5 eqssi ) BDBDEZFBGZAHZIBGZABJZCKUEUFBGUGLZADJZMZDB
|
|
EUEUHMZUKUDUEUHNULCUDUEUHOPUHUJUEUHUIASUJUGABQUIADRTUAPABUBPUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ps $. $d x ch $. $d x th $. $d x ta $.
|
|
$d y ph $.
|
|
$( Substitutions. $)
|
|
finds.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
finds.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
finds.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
|
|
finds.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
finds.5 $e |- ps $.
|
|
$( Induction step. $)
|
|
finds.6 $e |- ( y e. _om -> ( ch -> th ) ) $.
|
|
$( Principle of Finite Induction (inference schema), using implicit
|
|
substitutions. The first four hypotheses establish the substitutions we
|
|
need. The last two are the basis and the induction step. Theorem
|
|
Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74.
|
|
(Contributed by NM, 14-Apr-1995.) $)
|
|
finds $p |- ( A e. _om -> ta ) $=
|
|
( com wcel cab c0 cv elab csuc wi wral wss 0ex mpbir sucex 3imtr4g peano5
|
|
vex rgen mp2an sseli elabg mpbid ) HOPHAFQZPEOUPHRUPPZGSZUPPZURUAZUPPZUBZ
|
|
GOUCOUPUDUQBMABFRUEITUFVBGOUROPCDUSVANACFURGUJZJTADFUTURVCUGKTUHUKGUPUIUL
|
|
UMAEFHOLUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y ta $. $d x ps $. $d x ch $. $d x th $. $d y ph $.
|
|
$( Substitutions. $)
|
|
finds2.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
finds2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
finds2.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
|
|
$( Basis. $)
|
|
finds2.4 $e |- ( ta -> ps ) $.
|
|
$( Induction step. $)
|
|
finds2.5 $e |- ( y e. _om -> ( ta -> ( ch -> th ) ) ) $.
|
|
$( Principle of Finite Induction (inference schema), using implicit
|
|
substitutions. The first three hypotheses establish the substitutions
|
|
we need. The last two are the basis and the induction step. Theorem
|
|
Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.) $)
|
|
finds2 $p |- ( x e. _om -> ( ta -> ph ) ) $=
|
|
( cv com wcel wi c0 wceq imbi2d elab cab csuc wss 0ex mpbir a2d vex sucex
|
|
wral 3imtr4g rgen peano5 mp2an sseli abid sylib ) FMZNOUQEAPZFUAZOURNUSUQ
|
|
QUSOZGMZUSOZVAUBZUSOZPZGNUINUSUCUTEBPZKURVFFQUDUQQRABEHSTUEVEGNVANOZECPZE
|
|
DPZVBVDVGECDLUFURVHFVAGUGZUQVARACEISTURVIFVCVAVJUHUQVCRADEJSTUJUKGUSULUMU
|
|
NURFUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ps $. $d x ch $. $d x th $. $d y ph $.
|
|
$( Substitutions. $)
|
|
finds1.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
finds1.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
finds1.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
|
|
$( Basis. $)
|
|
finds1.4 $e |- ps $.
|
|
$( Induction step. $)
|
|
finds1.5 $e |- ( y e. _om -> ( ch -> th ) ) $.
|
|
$( Principle of Finite Induction (inference schema), using implicit
|
|
substitutions. The first three hypotheses establish the substitutions
|
|
we need. The last two are the basis and the induction step. Theorem
|
|
Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.) $)
|
|
finds1 $p |- ( x e. _om -> ph ) $=
|
|
( cv com wcel c0 wceq eqid a1i wi a1d finds2 mpi ) ELMNOOPZAOQABCDUCEFGHI
|
|
BUCJRFLMNCDSUCKTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z ph $.
|
|
findes.1 $e |- [. (/) / x ]. ph $.
|
|
findes.2 $e |- ( x e. _om -> ( ph -> [. suc x / x ]. ph ) ) $.
|
|
$( Finite induction with explicit substitution. The first hypothesis is
|
|
the basis and the second is the induction step. Theorem Schema 22 of
|
|
[Suppes] p. 136. This is an alternative for Metamath 100 proof #74.
|
|
(Contributed by Raph Levien, 9-Jul-2003.) $)
|
|
findes $p |- ( x e. _om -> ph ) $=
|
|
( vz vy wsb c0 wsbc csuc dfsbcq2 sbequ sbequ12r com wcel nfv nfim imbi12d
|
|
cv wi nfs1v nfsbc1v weq eleq1 sbequ12 wceq suceq dfsbcq syl chvar finds
|
|
wb ) ABEGABHIABFGZABFSZJZIZAEFBSZABEHKAEFBLABEUOKAEBMCUQNOZAABUQJZIZTZTUN
|
|
NOZUMUPTZTBFVBVCBVBBPUMUPBABFUAABUOUBQQBFUCZURVBVAVCUQUNNUDVDAUMUTUPABFUE
|
|
VDUSUOUFUTUPULUQUNUGABUSUOUHUIRRDUJUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The Natural Numbers (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( A natural number is either 0 or a successor. Similar theorems for
|
|
arbitrary sets or real numbers will not be provable (without the law of
|
|
the excluded middle), but equality of natural numbers is decidable.
|
|
(Contributed by NM, 27-May-1998.) $)
|
|
nn0suc $p |- ( A e. _om -> ( A = (/) \/ E. x e. _om A = suc x ) ) $=
|
|
( vy vz cv c0 wceq csuc com wrex wo eqeq1 rexbidv orbi12d eqid orci suceq
|
|
wcel eqeq2d rspcev mpan2 olcd a1d finds ) CEZFGZUEAEZHZGZAIJZKFFGZFUHGZAI
|
|
JZKDEZFGZUNUHGZAIJZKZUNHZFGZUSUHGZAIJZKZBFGZBUHGZAIJZKCDBUFUFUKUJUMUEFFLU
|
|
FUIULAIUEFUHLMNUEUNGZUFUOUJUQUEUNFLVGUIUPAIUEUNUHLMNUEUSGZUFUTUJVBUEUSFLV
|
|
HUIVAAIUEUSUHLMNUEBGZUFVDUJVFUEBFLVIUIVEAIUEBUHLMNUKUMFOPUNIRZVCURVJVBUTV
|
|
JUSUSGZVBUSOVAVKAUNIUGUNGUHUSUSUGUNQSTUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y B $.
|
|
$( A member of a natural number is a natural number. (Contributed by NM,
|
|
21-Jun-1998.) $)
|
|
elnn $p |- ( ( A e. B /\ B e. _om ) -> A e. _om ) $=
|
|
( vy vx com wcel wss cv c0 csuc sseq1 0ss wa csn cun unss vex snss anbi2i
|
|
df-suc sseq1i 3bitr4i biimpi expcom finds ssel2 ancoms sylan2 ) BEFABFZBE
|
|
GZAEFZCHZEGIEGDHZEGZUMJZEGZUJCDBULIEKULUMEKULUOEKULBEKELUNUMEFZUPUNUQMZUP
|
|
UNUMNZEGZMUMUSOZEGURUPUMUSEPUQUTUNUMEDQRSUOVAEUMTUAUBUCUDUEUJUIUKBEAUFUGU
|
|
H $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed
|
|
by NM, 18-Oct-1995.) $)
|
|
ordom $p |- Ord _om $=
|
|
( vx vy com word wtr cv wral wcel wa wi wal elnn gen2 dftr2 mpbir c0 csuc
|
|
treq tr0 suctr a1i finds rgen dford3 mpbir2an ) CDCEZAFZEZACGUFUGBFZHUICH
|
|
IUGCHZJZBKAKUKABUGUILMABCNOUHACUIEPEUHUGQZEZUHBAUGUIPRUIUGRZUIULRUNSUHUMJ
|
|
UJUGTUAUBUCACUDUE $.
|
|
$}
|
|
|
|
$( Omega is an ordinal number. (Contributed by Mario Carneiro,
|
|
30-Jan-2013.) $)
|
|
omelon2 $p |- ( _om e. _V -> _om e. On ) $=
|
|
( com cvv wcel con0 word ordom elong mpbiri ) ABCADCAEFABGH $.
|
|
|
|
$( Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised
|
|
by Mario Carneiro, 30-Jan-2013.) $)
|
|
omelon $p |- _om e. On $=
|
|
( com cvv wcel con0 omex omelon2 ax-mp ) ABCADCEFG $.
|
|
|
|
$( A natural number is an ordinal number. (Contributed by NM,
|
|
27-Jun-1994.) $)
|
|
nnon $p |- ( A e. _om -> A e. On ) $=
|
|
( com omelon oneli ) BACD $.
|
|
|
|
${
|
|
nnoni.1 $e |- A e. _om $.
|
|
$( A natural number is an ordinal number. (Contributed by NM,
|
|
27-Jun-1994.) $)
|
|
nnoni $p |- A e. On $=
|
|
( com wcel con0 nnon ax-mp ) ACDAEDBAFG $.
|
|
$}
|
|
|
|
$( A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) $)
|
|
nnord $p |- ( A e. _om -> Ord A ) $=
|
|
( com wcel con0 word nnon eloni syl ) ABCADCAEAFAGH $.
|
|
|
|
$( Omega is a subset of ` On ` . (Contributed by NM, 13-Jun-1994.) $)
|
|
omsson $p |- _om C_ On $=
|
|
( vx com con0 cv nnon ssriv ) ABCADEF $.
|
|
|
|
${
|
|
$d x z $.
|
|
$( Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473.
|
|
(Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon,
|
|
5-Jan-2019.) $)
|
|
limom $p |- Lim _om $=
|
|
( vx vz com wlim word c0 wcel cuni wceq ordom peano1 cv wa wex csuc sucex
|
|
wi vex isseti peano2 sucid jctil eleq2 eleq1 anbi12d syl5ibr eximii eluni
|
|
19.37aiv sylibr ssriv wss orduniss ax-mp eqssi dflim2 mpbir3an ) CDCEZFCG
|
|
CCHZIJKCUSACUSALZCGZUTBLZGZVBCGZMZBNUTUSGVAVEBVBUTOZIZVAVEQBBVFUTARZPSVAV
|
|
EVGUTVFGZVFCGZMVAVJVIUTTUTVHUAUBVGVCVIVDVJVBVFUTUCVBVFCUDUEUFUGUIBUTCUHUJ
|
|
UKURUSCULJCUMUNUOCUPUQ $.
|
|
$}
|
|
|
|
$( A class belongs to omega iff its successor does. (Contributed by NM,
|
|
3-Dec-1995.) $)
|
|
peano2b $p |- ( A e. _om <-> suc A e. _om ) $=
|
|
( com wcel csuc peano2 cvv elex sucexb sylibr sucidg elnn mpancom impbii
|
|
syl ) ABCZADZBCZAEAPCZQOQAFCZRQPFCSPBGAHIAFJNAPKLM $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( A nonzero natural number is a successor. (Contributed by NM,
|
|
18-Feb-2004.) $)
|
|
nnsuc $p |- ( ( A e. _om /\ A =/= (/) ) -> E. x e. _om A = suc x ) $=
|
|
( c0 wne com wcel wceq wn cv csuc wrex df-ne nn0suc ord imp sylan2b ) BCD
|
|
BEFZBCGZHZBAIJGAEKZBCLQSTQRTABMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A natural number is either zero or nonzero. Decidable equality for
|
|
natural numbers is a special case of the law of the excluded middle
|
|
which holds in most constructive set theories including ours.
|
|
(Contributed by Jim Kingdon, 5-Jan-2019.) $)
|
|
nndceq0 $p |- ( A e. _om -> DECID A = (/) ) $=
|
|
( vx vy com wcel c0 wceq wn wo wdc cv csuc eqeq1 notbid orbi12d eqid orci
|
|
peano3 neneqd olcd a1d finds df-dc sylibr ) ADEAFGZUEHZIZUEJBKZFGZUIHZIFF
|
|
GZUKHZICKZFGZUNHZIZUMLZFGZURHZIZUGBCAUIUIUKUJULUHFFMZUIUIUKVANOUHUMGZUIUN
|
|
UJUOUHUMFMZVBUIUNVCNOUHUQGZUIURUJUSUHUQFMZVDUIURVENOUHAGZUIUEUJUFUHAFMZVF
|
|
UIUEVGNOUKULFPQUMDEZUTUPVHUSURVHUQFUMRSTUAUBUEUCUD $.
|
|
|
|
$( A natural number is either the empty set or has the empty set as an
|
|
element. (Contributed by Jim Kingdon, 23-Aug-2019.) $)
|
|
0elnn $p |- ( A e. _om -> ( A = (/) \/ (/) e. A ) ) $=
|
|
( vx vy cv c0 wceq wcel wo csuc eqeq1 eleq2 orbi12d eqid com wi 0ex sucid
|
|
orci suceq a1i syl5eleqr wss sssucid sseld jaod olc syl6 finds ) BDZEFZEU
|
|
IGZHEEFZEEGZHCDZEFZEUNGZHZUNIZEFZEURGZHZAEFZEAGZHBCAUJUJULUKUMUIEEJUIEEKL
|
|
UIUNFUJUOUKUPUIUNEJUIUNEKLUIURFUJUSUKUTUIUREJUIUREKLUIAFUJVBUKVCUIAEJUIAE
|
|
KLULUMEMRUNNGZUQUTVAVDUOUTUPUOUTOVDUOEEIUREPQUNESUATVDUNUREUNURUBVDUNUCTU
|
|
DUEUTUSUFUGUH $.
|
|
$}
|
|
|
|
$( A natural number is nonempty iff it contains the empty set. Although in
|
|
constructive mathematics it is generally more natural to work with
|
|
inhabited sets and ignore the whole concept of nonempty sets, in the
|
|
specific case of natural numbers this theorem may be helpful in converting
|
|
proofs which were written assuming excluded middle. (Contributed by Jim
|
|
Kingdon, 28-Aug-2019.) $)
|
|
nn0eln0 $p |- ( A e. _om -> ( (/) e. A <-> A =/= (/) ) ) $=
|
|
( com wcel c0 wceq wo wne wb 0elnn noel eleq2 mtbiri nner 2falsed ne0i 2thd
|
|
id jaoi syl ) ABCADEZDACZFUAADGZHZAITUCUATUAUBTUADDCDJADDKLADMNUAUAUBUAQADO
|
|
PRS $.
|
|
|
|
${
|
|
$d ph w x y z $.
|
|
nnregexmid.1 $e |- ( ( x C_ _om /\ E. y y e. x ) ->
|
|
E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) ) $.
|
|
$( If inhabited sets of natural numbers always have minimal elements,
|
|
excluded middle follows. The argument is essentially the same as
|
|
~ regexmid and the larger lesson is that although natural numbers may
|
|
behave "non-constructively" even in a constructive set theory (for
|
|
example see ~ nndceq or ~ nntri3or ), sets of natural numbers are a
|
|
different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) $)
|
|
nnregexmid $p |- ( ph \/ -. ph ) $=
|
|
( vw cv c0 wceq wa wo wcel wel wn wi wal wex com wss peano1 csn crab csuc
|
|
cpr ssrab2 suc0 peano2 ax-mp eqeltrri prssi mp2an sstri eqid regexmidlemm
|
|
pp0ex rabex sseq1 eleq2 exbidv anbi12d notbid imbi2d albidv imbi12d vtocl
|
|
regexmidlem1 ) CGZFGZHUAZIVHHIAJKZFHVIUDZUBZLZDCMZDGZVLLZNZOZDPZJZCQZAANK
|
|
VLRSZVMCQZWAVLVKRVJFVKUEHRLZVIRLVKRSTHUCZVIRUFWDWERLTHUGUHUIHVIRUJUKULAFC
|
|
VLVLUMZUNBGZRSZCBMZCQZJZWIVNDBMZNZOZDPZJZCQZOWBWCJZWAOBVLVJFVKUOUPWGVLIZW
|
|
KWRWQWAWSWHWBWJWCWGVLRUQWSWIVMCWGVLVGURZUSUTWSWPVTCWSWIVMWOVSWTWSWNVRDWSW
|
|
MVQVNWSWLVPWGVLVOURVAVBVCUTUSVDEVEUKAFCDVLWFVFUH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c X. $. $( Times symbol (cross product symbol) (read: 'cross') $)
|
|
$c `' $. $( Small elevated smiley (converse operation) $)
|
|
$c dom $. $( Domain $)
|
|
$c ran $. $( Range $)
|
|
$c |` $. $( Right hook (restriction symbol) $)
|
|
$c " $. $( Left quote (image symbol) $)
|
|
$c o. $. $( Small circle (composition symbol) $)
|
|
$c Rel $. $( Relation predicate $)
|
|
|
|
$( Extend the definition of a class to include the cross product. $)
|
|
cxp $a class ( A X. B ) $.
|
|
|
|
$( Extend the definition of a class to include the converse of a class. $)
|
|
ccnv $a class `' A $.
|
|
|
|
$( Extend the definition of a class to include the domain of a class. $)
|
|
cdm $a class dom A $.
|
|
|
|
$( Extend the definition of a class to include the range of a class. $)
|
|
crn $a class ran A $.
|
|
|
|
$( Extend the definition of a class to include the restriction of a class.
|
|
(Read: The restriction of ` A ` to ` B ` .) $)
|
|
cres $a class ( A |` B ) $.
|
|
|
|
$( Extend the definition of a class to include the image of a class. (Read:
|
|
The image of ` B ` under ` A ` .) $)
|
|
cima $a class ( A " B ) $.
|
|
|
|
$( Extend the definition of a class to include the composition of two
|
|
classes. (Read: The composition of ` A ` and ` B ` .) $)
|
|
ccom $a class ( A o. B ) $.
|
|
|
|
$( Extend the definition of a wff to include the relation predicate. (Read:
|
|
` A ` is a relation.) $)
|
|
wrel $a wff Rel A $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Define the cross product of two classes. Definition 9.11 of [Quine]
|
|
p. 64. For example, ( { 1 , 5 } ` X. ` { 2 , 7 } ) = ( { ` <. ` 1 , 2
|
|
` >. ` , ` <. ` 1 , 7 ` >. ` } ` u. ` { ` <. ` 5 , 2 ` >. ` , ` <. ` 5 ,
|
|
7 ` >. ` } ) . Another example is that the set of rational numbers are
|
|
defined in using the cross-product ( Z ` X. ` N ) ; the left- and
|
|
right-hand sides of the cross-product represent the top (integer) and
|
|
bottom (natural) numbers of a fraction. (Contributed by NM,
|
|
4-Jul-1994.) $)
|
|
df-xp $a |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) } $.
|
|
|
|
$( Define the relation predicate. Definition 6.4(1) of [TakeutiZaring]
|
|
p. 23. For alternate definitions, see ~ dfrel2 and ~ dfrel3 .
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
df-rel $a |- ( Rel A <-> A C_ ( _V X. _V ) ) $.
|
|
|
|
$( Define the converse of a class. Definition 9.12 of [Quine] p. 64. The
|
|
converse of a binary relation swaps its arguments, i.e., if ` A e. _V `
|
|
and ` B e. _V ` then ` ( A ``' R B <-> B R A ) ` , as proven in ~ brcnv
|
|
(see ~ df-br and ~ df-rel for more on relations). For example, ` ``' `
|
|
{ ` <. ` 2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` } = { ` <. ` 6 , 2 ` >. ` ,
|
|
` <. ` 9 , 3 ` >. ` } . We use Quine's breve accent (smile) notation.
|
|
Like Quine, we use it as a prefix, which eliminates the need for
|
|
parentheses. Many authors use the postfix superscript "to the minus
|
|
one." "Converse" is Quine's terminology; some authors call it
|
|
"inverse," especially when the argument is a function. (Contributed by
|
|
NM, 4-Jul-1994.) $)
|
|
df-cnv $a |- `' A = { <. x , y >. | y A x } $.
|
|
|
|
$( Define the composition of two classes. Definition 6.6(3) of
|
|
[TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63
|
|
reverses ` A ` and ` B ` , uses a slash instead of ` o. ` , and calls
|
|
the operation "relative product." (Contributed by NM, 4-Jul-1994.) $)
|
|
df-co $a |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) } $.
|
|
|
|
$( Define the domain of a class. Definition 3 of [Suppes] p. 59. For
|
|
example, F = { ` <. ` 2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` } ` -> ` dom F
|
|
= { 2 , 3 } . Contrast with range (defined in ~ df-rn ). For alternate
|
|
definitions see ~ dfdm2 , ~ dfdm3 , and ~ dfdm4 . The
|
|
notation " ` dom ` " is used by Enderton; other authors sometimes use
|
|
script D. (Contributed by NM, 1-Aug-1994.) $)
|
|
df-dm $a |- dom A = { x | E. y x A y } $.
|
|
|
|
$( Define the range of a class. For example, F = { ` <. ` 2 , 6 ` >. ` ,
|
|
` <. ` 3 , 9 ` >. ` } -> ran F = { 6 , 9 } . Contrast with domain
|
|
(defined in ~ df-dm ). For alternate definitions, see ~ dfrn2 ,
|
|
~ dfrn3 , and ~ dfrn4 . The notation " ` ran ` " is used by Enderton;
|
|
other authors sometimes use script R or script W. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
df-rn $a |- ran A = dom `' A $.
|
|
|
|
$( Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring]
|
|
p. 24. For example ( F = { ` <. ` 2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` }
|
|
` /\ ` B = { 1 , 2 } ) -> ( F ` |`` ` B ) = { ` <. ` 2 , 6 ` >. ` } .
|
|
(Contributed by NM, 2-Aug-1994.) $)
|
|
df-res $a |- ( A |` B ) = ( A i^i ( B X. _V ) ) $.
|
|
|
|
$( Define the image of a class (as restricted by another class).
|
|
Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = { ` <. `
|
|
2 , 6 ` >. ` , ` <. ` 3 , 9 ` >. ` } /\ B = { 1 , 2 } ) -> ( F ` " ` B )
|
|
= { 6 } . Contrast with restriction ( ~ df-res ) and range ( ~ df-rn ).
|
|
For an alternate definition, see ~ dfima2 . (Contributed by NM,
|
|
2-Aug-1994.) $)
|
|
df-ima $a |- ( A " B ) = ran ( A |` B ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) $)
|
|
xpeq1 $p |- ( A = B -> ( A X. C ) = ( B X. C ) ) $=
|
|
( vx vy wceq cv wcel wa copab cxp eleq2 anbi1d opabbidv df-xp 3eqtr4g ) A
|
|
BFZDGZAHZEGCHZIZDEJRBHZTIZDEJACKBCKQUAUCDEQSUBTABRLMNDEACODEBCOP $.
|
|
|
|
$( Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) $)
|
|
xpeq2 $p |- ( A = B -> ( C X. A ) = ( C X. B ) ) $=
|
|
( vx vy wceq cv wcel wa copab cxp eleq2 anbi2d opabbidv df-xp 3eqtr4g ) A
|
|
BFZDGCHZEGZAHZIZDEJRSBHZIZDEJCAKCBKQUAUCDEQTUBRABSLMNDECAODECBOP $.
|
|
|
|
$( Membership in a cross product. Uses fewer axioms than ~ elxp .
|
|
(Contributed by NM, 4-Jul-1994.) $)
|
|
elxpi $p |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\
|
|
( x e. B /\ y e. C ) ) ) $=
|
|
( vz cv cop wceq wcel wa wex cab cxp eqeq1 anbi1d 2exbidv elabg ibi copab
|
|
df-xp df-opab eqtri eleq2s ) CAGZBGZHZIZUEDJUFEJKZKZBLALZCFGZUGIZUIKZBLAL
|
|
ZFMZDENZCUPJUKUOUKFCUPULCIZUNUJABURUMUHUIULCUGOPQRSUQUIABTUPABDEUAUIABFUB
|
|
UCUD $.
|
|
|
|
$( Membership in a cross product. (Contributed by NM, 4-Jul-1994.) $)
|
|
elxp $p |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\
|
|
( x e. B /\ y e. C ) ) ) $=
|
|
( cxp wcel cv wa copab cop wceq wex df-xp eleq2i elopab bitri ) CDEFZGCAH
|
|
ZDGBHZEGIZABJZGCSTKLUAIBMAMRUBCABDENOUAABCPQ $.
|
|
|
|
$( Membership in a cross product. (Contributed by NM, 23-Feb-2004.) $)
|
|
elxp2 $p |- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. ) $=
|
|
( cv wcel cop wceq wrex wa wex cxp df-rex r19.42v an13 exbii 3bitr3i elxp
|
|
3bitr4ri ) AFZDGZCUABFZHIZBEJZKZALUDUBUCEGZKKZBLZALUEADJCDEMGUFUIAUBUDKZB
|
|
EJUGUJKZBLUFUIUJBENUBUDBEOUKUHBUGUBUDPQRQUEADNABCDEST $.
|
|
$}
|
|
|
|
$( Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) $)
|
|
xpeq12 $p |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) ) $=
|
|
( wceq cxp xpeq1 xpeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
xpeq1i.1 $e |- A = B $.
|
|
$( Equality inference for cross product. (Contributed by NM,
|
|
21-Dec-2008.) $)
|
|
xpeq1i $p |- ( A X. C ) = ( B X. C ) $=
|
|
( wceq cxp xpeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for cross product. (Contributed by NM,
|
|
21-Dec-2008.) $)
|
|
xpeq2i $p |- ( C X. A ) = ( C X. B ) $=
|
|
( wceq cxp xpeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
xpeq12i.1 $e |- A = B $.
|
|
xpeq12i.2 $e |- C = D $.
|
|
$( Equality inference for cross product. (Contributed by FL,
|
|
31-Aug-2009.) $)
|
|
xpeq12i $p |- ( A X. C ) = ( B X. D ) $=
|
|
( wceq cxp xpeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
xpeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for cross product. (Contributed by Jeff Madsen,
|
|
17-Jun-2010.) $)
|
|
xpeq1d $p |- ( ph -> ( A X. C ) = ( B X. C ) ) $=
|
|
( wceq cxp xpeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for cross product. (Contributed by Jeff Madsen,
|
|
17-Jun-2010.) $)
|
|
xpeq2d $p |- ( ph -> ( C X. A ) = ( C X. B ) ) $=
|
|
( wceq cxp xpeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
xpeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for cross product. (Contributed by NM,
|
|
8-Dec-2013.) $)
|
|
xpeq12d $p |- ( ph -> ( A X. C ) = ( B X. D ) ) $=
|
|
( wceq cxp xpeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
nfxp.1 $e |- F/_ x A $.
|
|
nfxp.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for cross product. (Contributed by
|
|
NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfxp $p |- F/_ x ( A X. B ) $=
|
|
( vy vz cxp cv wcel wa copab df-xp nfcri nfan nfopab nfcxfr ) ABCHFIBJZGI
|
|
CJZKZFGLFGBCMTFGARSAAFBDNAGCENOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( The empty set is not a member of a cross product. (Contributed by NM,
|
|
2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
0nelxp $p |- -. (/) e. ( A X. B ) $=
|
|
( vx vy c0 cxp wcel cv cop wceq wa wex wne wn opnzi simpl eqcomd necon3ai
|
|
vex nex ax-mp elxp mtbir ) EABFGECHZDHZIZJZUDAGUEBGKZKZDLZCLUJCUIDUFEMUIN
|
|
UDUECSDSOUIUFEUIEUFUGUHPQRUATTCDEABUBUC $.
|
|
|
|
$( A member of a cross product (ordered pair) doesn't contain the empty
|
|
set. (Contributed by NM, 15-Dec-2008.) $)
|
|
0nelelxp $p |- ( C e. ( A X. B ) -> -. (/) e. C ) $=
|
|
( vx vy cxp wcel cv cop wceq wa wex c0 wn elxp 0nelop simpl eleq2d mtbiri
|
|
exlimivv sylbi ) CABFGCDHZEHZIZJZUBAGUCBGKZKZELDLMCGZNZDECABOUGUIDEUGUHMU
|
|
DGUBUCPUGCUDMUEUFQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( Ordered pair membership in a cross product. (Contributed by NM,
|
|
15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opelxp $p |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) ) $=
|
|
( vx vy cop cxp wcel cv wceq wrex wa elxp2 wb opth2 eleq1 bi2anan9 eqeq2d
|
|
vex sylbi biimprcd rexlimivv eqid opeq1 opeq2 rspc2ev mp3an3 impbii bitri
|
|
) ABGZCDHIUKEJZFJZGZKZFDLECLZACIZBDIZMZEFUKCDNUPUSUOUSEFCDUOUSULCIZUMDIZM
|
|
ZUOAULKZBUMKZMUSVBOABULUMETFTPVCUQUTVDURVAAULCQBUMDQRUAUBUCUQURUKUKKZUPUK
|
|
UDUOVEUKAUMGZKEFABCDULAKUNVFUKULAUMUESUMBKVFUKUKUMBAUFSUGUHUIUJ $.
|
|
|
|
$( Binary relation on a cross product. (Contributed by NM,
|
|
22-Apr-2004.) $)
|
|
brxp $p |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) ) $=
|
|
( cxp wbr cop wcel wa df-br opelxp bitri ) ABCDEZFABGMHACHBDHIABMJABCDKL
|
|
$.
|
|
$}
|
|
|
|
$( Ordered pair membership in a cross product (implication). (Contributed by
|
|
NM, 28-May-1995.) $)
|
|
opelxpi $p |- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) ) $=
|
|
( cop cxp wcel wa opelxp biimpri ) ABECDFGACGBDGHABCDIJ $.
|
|
|
|
$( The first member of an ordered pair of classes in a cross product belongs
|
|
to first cross product argument. (Contributed by NM, 28-May-2008.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opelxp1 $p |- ( <. A , B >. e. ( C X. D ) -> A e. C ) $=
|
|
( cop cxp wcel opelxp simplbi ) ABECDFGACGBDGABCDHI $.
|
|
|
|
$( The second member of an ordered pair of classes in a cross product belongs
|
|
to second cross product argument. (Contributed by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
opelxp2 $p |- ( <. A , B >. e. ( C X. D ) -> B e. D ) $=
|
|
( cop cxp wcel opelxp simprbi ) ABECDFGACGBDGABCDHI $.
|
|
|
|
$( The first member of an ordered triple of classes in a cross product
|
|
belongs to first cross product argument. (Contributed by NM,
|
|
28-May-2008.) $)
|
|
otelxp1 $p |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T )
|
|
-> A e. R ) $=
|
|
( cop cxp wcel opelxp1 syl ) ABGZCGDEHZFHILMIADILCMFJABDEJK $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d y z ph $. $d x ps $.
|
|
rabxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class builder restricted to a cross product.
|
|
(Contributed by NM, 20-Feb-2014.) $)
|
|
rabxp $p |- { x e. ( A X. B ) | ph }
|
|
= { <. y , z >. | ( y e. A /\ z e. B /\ ps ) } $=
|
|
( cv cxp wcel wa cab cop wceq w3a wex crab copab elxp anbi1i anass anbi2d
|
|
19.41vv df-3an syl6bbr pm5.32i bitri 2exbii 3bitr2i abbii df-opab 3eqtr4i
|
|
df-rab ) CIZFGJZKZALZCMUODIZEIZNOZUSFKZUTGKZBPZLZEQDQZCMACUPRVDDESURVFCUR
|
|
VAVBVCLZLZEQDQZALVHALZEQDQVFUQVIADEUOFGTUAVHADEUDVJVEDEVJVAVGALZLVEVAVGAU
|
|
BVAVKVDVAVKVGBLVDVAABVGHUCVBVCBUEUFUGUHUIUJUKACUPUNVDDECULUM $.
|
|
$}
|
|
|
|
$( A true binary relation on a relation implies the arguments are sets.
|
|
(This is a property of our ordered pair definition.) (Contributed by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
brrelex12 $p |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) ) $=
|
|
( wrel wbr wa cvv cxp wcel wss df-rel biimpi ssbrd imp brxp sylib ) CDZABCE
|
|
ZFABGGHZEZAGIBGIFQRTQCSABQCSJCKLMNABGGOP $.
|
|
|
|
$( A true binary relation on a relation implies the first argument is a set.
|
|
(This is a property of our ordered pair definition.) (Contributed by NM,
|
|
18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
brrelex $p |- ( ( Rel R /\ A R B ) -> A e. _V ) $=
|
|
( wrel wbr wa cvv wcel brrelex12 simpld ) CDABCEFAGHBGHABCIJ $.
|
|
|
|
$( A true binary relation on a relation implies the second argument is a set.
|
|
(This is a property of our ordered pair definition.) (Contributed by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
brrelex2 $p |- ( ( Rel R /\ A R B ) -> B e. _V ) $=
|
|
( wrel wbr wa cvv wcel brrelex12 simprd ) CDABCEFAGHBGHABCIJ $.
|
|
|
|
${
|
|
brrelexi.1 $e |- Rel R $.
|
|
$( The first argument of a binary relation exists. (An artifact of our
|
|
ordered pair definition.) (Contributed by NM, 4-Jun-1998.) $)
|
|
brrelexi $p |- ( A R B -> A e. _V ) $=
|
|
( wrel wbr cvv wcel brrelex mpan ) CEABCFAGHDABCIJ $.
|
|
|
|
$( The second argument of a binary relation exists. (An artifact of our
|
|
ordered pair definition.) (Contributed by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
brrelex2i $p |- ( A R B -> B e. _V ) $=
|
|
( wrel wbr cvv wcel brrelex2 mpan ) CEABCFBGHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
nprrel.1 $e |- Rel R $.
|
|
nprrel.2 $e |- -. A e. _V $.
|
|
$( No proper class is related to anything via any relation. (Contributed
|
|
by Roy F. Longton, 30-Jul-2005.) $)
|
|
nprrel $p |- -. A R B $=
|
|
( wbr cvv wcel brrelexi mto ) ABCFAGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Representation of a constant function using the mapping operation.
|
|
(Note that ` x ` cannot appear free in ` B ` .) (Contributed by NM,
|
|
12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.) $)
|
|
fconstmpt $p |- ( A X. { B } ) = ( x e. A |-> B ) $=
|
|
( vy cv wcel csn wa copab wceq cxp cmpt velsn anbi2i opabbii df-xp df-mpt
|
|
3eqtr4i ) AEBFZDEZCGZFZHZADISTCJZHZADIBUAKABCLUCUEADUBUDSDCMNOADBUAPADBCQ
|
|
R $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y z C $. $d x y z R $.
|
|
vtoclr.1 $e |- Rel R $.
|
|
vtoclr.2 $e |- ( ( x R y /\ y R z ) -> x R z ) $.
|
|
$( Variable to class conversion of transitive relation. (Contributed by
|
|
NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
vtoclr $p |- ( ( A R B /\ B R C ) -> A R C ) $=
|
|
( wbr wa wi cvv wcel brrelex2i cv wceq breq1 imbi12d breq2 anbi1d anbi12d
|
|
brrelexi jca imbi2d imbi1d anbi2d vtoclg vtocl2g syl2im imp pm2.43i ) DEG
|
|
JZEFGJZKZDFGJZUMUNUOUPLZUMDMNZEMNZKUNFMNZUQUMURUSDEGHUCDEGHOUDEFGHOUTAPZB
|
|
PZGJZVBFGJZKZVAFGJZLZLUTDVBGJZVDKZUPLZLUTUQLABDEMMVADQZVGVJUTVKVEVIVFUPVK
|
|
VCVHVDVADVBGRUAVADFGRSUEVBEQZVJUQUTVLVIUOUPVLVHUMVDUNVBEDGTVBEFGRUBUFUEVC
|
|
VBCPZGJZKZVAVMGJZLVGCFMVMFQZVOVEVPVFVQVNVDVCVMFVBGTUGVMFVAGTSIUHUIUJUKUL
|
|
$.
|
|
$}
|
|
|
|
$( Ordered pair membership in the universal class of ordered pairs.
|
|
(Contributed by Mario Carneiro, 3-May-2015.) $)
|
|
opelvvg $p |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) ) $=
|
|
( wcel cvv cop cxp elex opelxpi syl2an ) ACEAFEBFEABGFFHEBDEACIBDIABFFJK $.
|
|
|
|
${
|
|
opelvv.1 $e |- A e. _V $.
|
|
opelvv.2 $e |- B e. _V $.
|
|
$( Ordered pair membership in the universal class of ordered pairs.
|
|
(Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
opelvv $p |- <. A , B >. e. ( _V X. _V ) $=
|
|
( cvv wcel cop cxp opelxpi mp2an ) AEFBEFABGEEHFCDABEEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
$( Justification theorem for an ordered pair definition that works for any
|
|
classes, including proper classes. This is a possible definition
|
|
implied by the footnote in [Jech] p. 78, which says, "The sophisticated
|
|
reader will not object to our use of a pair of classes." (Contributed
|
|
by NM, 28-Sep-2003.) $)
|
|
opthprc $p |- ( ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ) =
|
|
( ( C X. { (/) } ) u. ( D X. { { (/) } } ) )
|
|
<-> ( A = C /\ B = D ) ) $=
|
|
( vx c0 csn cxp cun wceq wa wcel wo opelxp mpbiran2 bianfi bitr4i orbi12i
|
|
elun 3bitr4ri cv cop eleq2 0ex snid 0nep0 elsn nemtbir 3bitr4g eqrdv p0ex
|
|
biorfi eqcom bitri wn wb biorf ax-mp jca xpeq1 uneq12 syl2an impbii ) AFG
|
|
ZHZBVDGZHZIZCVDHZDVFHZIZJZACJZBDJZKVLVMVNVLEACVLEUAZFUBZVHLZVPVKLZVOALZVO
|
|
CLZVHVKVPUCVPVELZVPVGLZMVSFVFLZMVQVSWAVSWBWCWAVSFVDLZFUDUEZVOFAVDNOWBVOBL
|
|
ZWCKWCVOFBVFNWCWFWCFVDUFFVDUDUGUHZPQRVPVEVGSWCVSWGULTVPVILZVPVJLZMVTWCMVR
|
|
VTWHVTWIWCWHVTWDWEVOFCVDNOWIVODLZWCKWCVOFDVFNWCWJWGPQRVPVIVJSWCVTWGULTUIU
|
|
JVLEBDVLVOVDUBZVHLZWKVKLZWFWJVHVKWKUCWKVELZWKVGLZMVDVDLZWFMZWLWFWNWPWOWFW
|
|
NVSWPKWPVOVDAVDNWPVSWPFVDUFWPVDFJFVDJVDFUKUGVDFUMUNUHZPQWOWFVDVFLZVDUKUEZ
|
|
VOVDBVFNORWKVEVGSWPUOZWFWQUPWRWPWFUQURTWKVILZWKVJLZMWPWJMZWMWJXBWPXCWJXBV
|
|
TWPKWPVOVDCVDNWPVTWRPQXCWJWSWTVOVDDVFNORWKVIVJSXAWJXDUPWRWPWJUQURTUIUJUSV
|
|
MVEVIJVGVJJVLVNACVDUTBDVFUTVEVIVGVJVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
brel.1 $e |- R C_ ( C X. D ) $.
|
|
$( Two things in a binary relation belong to the relation's domain.
|
|
(Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
brel $p |- ( A R B -> ( A e. C /\ B e. D ) ) $=
|
|
( wbr cxp wcel wa ssbri brxp sylib ) ABEGABCDHZGACIBDIJENABFKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y ps $.
|
|
brab2a.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
brab2a.2 $e |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } $.
|
|
$( Ordered pair membership in an ordered pair class abstraction.
|
|
(Contributed by Mario Carneiro, 9-Nov-2015.) $)
|
|
brab2a $p |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) ) $=
|
|
( wbr wcel wa cv copab cxp simpl ssopab2i df-xp 3sstr4i brel df-br eleq2i
|
|
cop bitri opelopab2a syl5bb biadan2 ) EFILZEGMFHMNZBEFGHICOGMDOHMNZANZCDP
|
|
ZULCDPIGHQUMULCDULARSKCDGHTUAUBUJEFUEZUNMZUKBUJUOIMUPEFIUCIUNUOKUDUFABCDE
|
|
FGHJUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in a cross product. (Contributed by NM, 5-Mar-1995.) $)
|
|
elxp3 $p |- ( A e. ( B X. C ) <->
|
|
E. x E. y ( <. x , y >. = A /\ <. x , y >. e. ( B X. C ) ) ) $=
|
|
( cxp wcel cv cop wceq wa wex elxp eqcom opelxp anbi12i 2exbii bitr4i ) C
|
|
DEFZGCAHZBHZIZJZTDGUAEGKZKZBLALUBCJZUBSGZKZBLALABCDEMUHUEABUFUCUGUDUBCNTU
|
|
ADEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d y z C $. $d x y z $.
|
|
$( Membership in a union of cross products. (Contributed by Mario
|
|
Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) $)
|
|
opeliunxp $p |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <->
|
|
( x e. A /\ C e. B ) ) $=
|
|
( vz vy cv cop csn cxp ciun wcel cvv elex wex wceq eleq2d anbi12d bitri
|
|
wa opexg wsb csb wrex nfv nfs1v nfcv nfcsb1v nfxp nfcri nfan sbequ12 sneq
|
|
df-rex csbeq1a cbvex eleq1 anbi2d exbidv syl5bb df-iun elab2g opelxp an12
|
|
xpeq12d anbi2i velsn equcom anbi1i 3bitri sbequ12r equcoms eqcomd ceqsexv
|
|
exbii vex syl6bb pm5.21nii ) AGZDHZABVSIZCJZKZLZVTMLZVSBLZDCLZTZVTWCNVSDB
|
|
CUAWEWDWFAEUBZVTEGZIZAWJCUCZJZLZTZEOZWHFGZWBLZABUDZWPFVTWCMWSWIWQWMLZTZEO
|
|
ZWQVTPZWPWSWFWRTZAOXBWRABUNXDXAAEXDEUEWIWTAWFAEUFAFWMAWKWLAWKUGAWJCUHUIUJ
|
|
UKVSWJPZWFWIWRWTWFAEULXEWBWMWQXEWAWKCWLVSWJUMAWJCUOZVEQRUPSXCXAWOEXCWTWNW
|
|
IWQVTWMUQURUSUTAFBWBVAVBWPWJVSPZWIDWLLZTZTZEOWHWOXJEWOWIVSWKLZXHTZTXKXITX
|
|
JWNXLWIVSDWKWLVCVFWIXKXHVDXKXGXIXKXEXGAWJVGAEVHSVIVJVOXIWHEVSAVPXGWIWFXHW
|
|
GWFEAVKXGWLCDXGCWLCWLPAEXFVLVMQRVNSVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Distributive law for cross product over union. Theorem 103 of [Suppes]
|
|
p. 52. (Contributed by NM, 12-Aug-2004.) $)
|
|
xpundi $p |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) ) $=
|
|
( vx vy cun cxp cv wcel wa copab df-xp uneq12i wo elun andi bitri opabbii
|
|
anbi2i eqtr4i unopab ) ABCFZGDHAIZEHZUBIZJZDEKZABGZACGZFZDEAUBLUJUCUDBIZJ
|
|
ZDEKZUCUDCIZJZDEKZFZUGUHUMUIUPDEABLDEACLMUGULUONZDEKUQUFURDEUFUCUKUNNZJUR
|
|
UEUSUCUDBCOSUCUKUNPQRULUODEUATTT $.
|
|
|
|
$( Distributive law for cross product over union. Similar to Theorem 103
|
|
of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) $)
|
|
xpundir $p |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) ) $=
|
|
( vx vy cun cxp cv wcel wa copab df-xp uneq12i wo elun anbi1i andir bitri
|
|
opabbii eqtr4i unopab ) ABFZCGDHZUBIZEHCIZJZDEKZACGZBCGZFZDEUBCLUJUCAIZUE
|
|
JZDEKZUCBIZUEJZDEKZFZUGUHUMUIUPDEACLDEBCLMUGULUONZDEKUQUFURDEUFUKUNNZUEJU
|
|
RUDUSUEUCABOPUKUNUEQRSULUODEUATTT $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w y z B $. $d w x y z C $.
|
|
$( Distributive law for cross product over indexed union. (Contributed by
|
|
Mario Carneiro, 27-Apr-2014.) $)
|
|
xpiundi $p |- ( C X. U_ x e. A B ) = U_ x e. A ( C X. B ) $=
|
|
( vz vw vy ciun cxp cv wrex wcel wa wex eliun exbii df-rex rexbii 3bitr4i
|
|
elxp2 cop wceq rexcom anbi1i rexcom4 r19.41v 3bitri eqriv ) EDABCHZIZABDC
|
|
IZHZEJZFJGJZUAUBZGUIKZFDKZUMUKLZABKZUMUJLUMULLUOGCKZABKZFDKUTFDKZABKUQUSU
|
|
TFADBUCUPVAFDUNUILZUOMZGNUNCLZABKZUOMZGNZUPVAVDVGGVCVFUOAUNBCOUDPUOGUIQVA
|
|
VEUOMZGNZABKVIABKZGNVHUTVJABUOGCQRVIAGBUEVKVGGVEUOABUFPUGSRURVBABFGUMDCTR
|
|
SFGUMDUITAUMBUKOSUH $.
|
|
|
|
$( Distributive law for cross product over indexed union. (Contributed by
|
|
Mario Carneiro, 27-Apr-2014.) $)
|
|
xpiundir $p |- ( U_ x e. A B X. C ) = U_ x e. A ( B X. C ) $=
|
|
( vz vy vw ciun cxp cv cop wrex wcel wa df-rex rexbii eliun elxp2 3bitr4i
|
|
wex wceq rexcom4 anbi1i r19.41v bitr4i exbii 3bitr4ri eqriv ) EABCHZDIZAB
|
|
CDIZHZEJZFJZGJKUAGDLZFUILZUMUKMZABLZUMUJMUMULMUNUIMZUONZFTZUOFCLZABLZUPUR
|
|
UNCMZUONZFTZABLVEABLZFTVCVAVEAFBUBVBVFABUOFCOPUTVGFUTVDABLZUONVGUSVHUOAUN
|
|
BCQUCVDUOABUDUEUFUGUOFUIOUQVBABFGUMCDRPSFGUMUIDRAUMBUKQSUH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Membership in a union of cross products when the second factor is
|
|
constant. (Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
iunxpconst $p |- U_ x e. A ( { x } X. B ) = ( A X. B ) $=
|
|
( cv csn ciun cxp xpiundir iunid xpeq1i eqtr3i ) ABADEZFZCGABLCGFBCGABLCH
|
|
MBCABIJK $.
|
|
$}
|
|
|
|
$( The cross product of two unions. (Contributed by NM, 12-Aug-2004.) $)
|
|
xpun $p |- ( ( A u. B ) X. ( C u. D ) ) =
|
|
( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) ) $=
|
|
( cun cxp xpundi xpundir uneq12i un4 3eqtri ) ABEZCDEFLCFZLDFZEACFZBCFZEZAD
|
|
FZBDFZEZEOREPSEELCDGMQNTABCHABDHIOPRSJK $.
|
|
|
|
${
|
|
$d w x y z A $.
|
|
$( Membership in universal class of ordered pairs. (Contributed by NM,
|
|
4-Jul-1994.) $)
|
|
elvv $p |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. ) $=
|
|
( cvv cxp wcel cv cop wceq wa wex elxp vex pm3.2i biantru 2exbii bitr4i )
|
|
CDDEFCAGZBGZHIZRDFZSDFZJZJZBKAKTBKAKABCDDLTUDABUCTUAUBAMBMNOPQ $.
|
|
|
|
$( Membership in universal class of ordered triples. (Contributed by NM,
|
|
17-Dec-2008.) $)
|
|
elvvv $p |- ( A e. ( ( _V X. _V ) X. _V )
|
|
<-> E. x E. y E. z A = <. <. x , y >. , z >. ) $=
|
|
( vw cvv cxp wcel cv cop wceq wa wex elxp anass 19.42vv 2exbii vex bitr3i
|
|
bitri ancom biantru elvv anbi2i 3bitr4ri exrot4 excom opex eqeq2d ceqsexv
|
|
opeq1 exbii ) DFFGZFGHDEIZCIZJZKZUNUMHZUOFHZLLZCMEMZDAIZBIZJZUOJZKZCMZBMA
|
|
MZECDUMFNVAUNVDKZUQLZBMAMZCMEMZVHUTVKECUTUQURLZUSLZVKUQURUSOUQVILZBMAMUQV
|
|
IBMAMZLZVKVNUQVIABPVJVOABVIUQUAQVNVMVQUSVMCRUBURVPUQABUNUCUDSUESQVLVJCMEM
|
|
ZBMAMVHVJABECUFVRVGABVRVJEMZCMVGVJECUGVSVFCUQVFEVDVBVCARBRUHVIUPVEDUNVDUO
|
|
UKUIUJULTQSTT $.
|
|
|
|
$( An ordered pair contains its union. (Contributed by NM,
|
|
16-Sep-2006.) $)
|
|
elvvuni $p |- ( A e. ( _V X. _V ) -> U. A e. A ) $=
|
|
( vx vy cvv cxp wcel cv cop wceq wex cuni elvv cpr vex uniop opi2 eqeltri
|
|
unieq id eleq12d mpbiri exlimivv sylbi ) ADDEFABGZCGZHZIZCJBJAKZAFZBCALUG
|
|
UIBCUGUIUFKZUFFUJUDUEMUFUDUEBNZCNZOUDUEUKULPQUGUHUJAUFAUFRUGSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( "At most one" remains true inside ordered pair quantification.
|
|
(Contributed by NM, 28-Aug-2007.) $)
|
|
mosubopt $p |- ( A. y A. z E* x ph ->
|
|
E* x E. y E. z ( A = <. y , z >. /\ ph ) ) $=
|
|
( wmo wal cv cop wceq wex wa wi nfa1 nfe1 nfmo nfex copsexg sps exlimd
|
|
mobidv biimpcd moanimv simpl 2eximi ancri moimi sylbir syl ) ABFZDGZCGZEC
|
|
HDHIJZDKZCKZUMALZDKZCKZBFZMZUSULUNUSCUKCNURCBUQCOPUKUNUSMCUKUMUSDUJDNURDB
|
|
UQDCUPDOQPUJUMUSMDUMUJUSUMAURBACDERUAUBSTSTUTUOURLZBFUSUOURBUCURVABURUOUP
|
|
UMCDUMAUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
mosubop.1 $e |- E* x ph $.
|
|
$( "At most one" remains true inside ordered pair quantification.
|
|
(Contributed by NM, 28-May-1995.) $)
|
|
mosubop $p |- E* x E. y E. z ( A = <. y , z >. /\ ph ) $=
|
|
( wmo wal cv cop wceq wa wex gen2 mosubopt ax-mp ) ABGZDHCHECIDIJKALDMCMB
|
|
GQCDFNABCDEOP $.
|
|
$}
|
|
|
|
$( Intersection of binary relation with cross product. (Contributed by NM,
|
|
3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
brinxp2 $p |- ( A ( R i^i ( C X. D ) ) B <->
|
|
( A e. C /\ B e. D /\ A R B ) ) $=
|
|
( cxp cin wbr wa wcel w3a brin ancom brxp anbi1i df-3an bitr4i 3bitri ) ABE
|
|
CDFZGHABEHZABSHZIUATIZACJZBDJZTKZABESLTUAMUBUCUDIZTIUEUAUFTABCDNOUCUDTPQR
|
|
$.
|
|
|
|
$( Intersection of binary relation with cross product. (Contributed by NM,
|
|
9-Mar-1997.) $)
|
|
brinxp $p |- ( ( A e. C /\ B e. D ) ->
|
|
( A R B <-> A ( R i^i ( C X. D ) ) B ) ) $=
|
|
( cxp cin wbr wcel wa w3a brinxp2 df-3an bitri baibr ) ABECDFGHZACIZBDIZJZA
|
|
BEHZPQRTKSTJABCDELQRTMNO $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z R $.
|
|
$( Intersection of partial order with cross product of its field.
|
|
(Contributed by Mario Carneiro, 10-Jul-2014.) $)
|
|
poinxp $p |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A ) $=
|
|
( vx vy vz cv wbr wn wa wi wral cxp cin wpo wcel wb brinxp ralbidva df-po
|
|
anbi12d simpll syl2anc notbid adantll adantlr imbi12d ralbiia 3bitr4i
|
|
adantr ) CFZUJBGZHZUJDFZBGZUMEFZBGZIZUJUOBGZJZIZEAKZDAKZCAKUJUJBAALMZGZHZ
|
|
UJUMVCGZUMUOVCGZIZUJUOVCGZJZIZEAKZDAKZCAKABNAVCNVBVMCAUJAOZVAVLDAVNUMAOZI
|
|
ZUTVKEAVPUOAOZIZULVEUSVJVRUKVDVRVNVNUKVDPVNVOVQUAZVSUJUJAABQUBUCVRUQVHURV
|
|
IVRUNVFUPVGVPUNVFPVQUJUMAABQUIVOVQUPVGPVNUMUOAABQUDTVNVQURVIPVOUJUOAABQUE
|
|
UFTRRUGCDEABSCDEAVCSUH $.
|
|
|
|
$( Intersection of linear order with cross product of its field.
|
|
(Contributed by Mario Carneiro, 10-Jul-2014.) $)
|
|
soinxp $p |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A ) $=
|
|
( vx vy vz wpo cv wbr wo wi wral wa cxp cin wor poinxp wcel brinxp df-iso
|
|
wb 3adant3 3adant2 ancoms 3adant1 orbi12d imbi12d 3expb 2ralbidva ralbiia
|
|
w3a anbi12i 3bitr4i ) ABFZCGZDGZBHZUNEGZBHZUQUOBHZIZJZEAKDAKZCAKZLABAAMNZ
|
|
FZUNUOVDHZUNUQVDHZUQUOVDHZIZJZEAKDAKZCAKZLABOAVDOUMVEVCVLABPVBVKCAUNAQZVA
|
|
VJDEAAVMUOAQZUQAQZVAVJTVMVNVOUJZUPVFUTVIVMVNUPVFTVOUNUOAABRUAVPURVGUSVHVM
|
|
VOURVGTVNUNUQAABRUBVNVOUSVHTZVMVOVNVQUQUOAABRUCUDUEUFUGUHUIUKCDEABSCDEAVD
|
|
SUL $.
|
|
|
|
$( Intersection of set-like relation with cross product of its field.
|
|
(Contributed by Mario Carneiro, 22-Jun-2015.) $)
|
|
seinxp $p |- ( R Se A <-> ( R i^i ( A X. A ) ) Se A ) $=
|
|
( vy vx cv wbr crab cvv wcel cxp cin wse wb brinxp ancoms rabbidva eleq1d
|
|
wral ralbiia df-se 3bitr4i ) CEZDEZBFZCAGZHIZDARUBUCBAAJKZFZCAGZHIZDARABL
|
|
AUGLUFUJDAUCAIZUEUIHUKUDUHCAUBAIUKUDUHMUBUCAABNOPQSDCABTDCAUGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z R $.
|
|
$( Partial ordering of a singleton. (Contributed by Jim Kingdon,
|
|
5-Dec-2018.) $)
|
|
posng $p |- ( ( Rel R /\ A e. _V ) -> ( R Po { A } <-> -. A R A ) ) $=
|
|
( vz vy vx csn cv wbr wn wa wi wral cvv wb wceq breq2 anbi2d ralsng bitrd
|
|
ralbidv wpo wrel df-po imbi12d simpl syl5ib biantrud bicomd breq12 anidms
|
|
wcel notbid adantl syl5bb ) AFZBUACGZUPBHZIZUPDGZBHZUSEGZBHZJZUPVABHZKZJZ
|
|
EUOLZDUOLZCUOLZBUBZAMUKZJAABHZIZCDEUOBUCVKVIVMNVJVKVIURCUOLVMVKVHURCUOVKV
|
|
HURUTUSABHZJZUPABHZKZJZDUOLURVKVGVRDUOVFVREAMVAAOZVEVQURVSVCVOVDVPVSVBVNU
|
|
TVAAUSBPQVAAUPBPUDQRTVRURDAMUSAOZURVRVTVQURVOUTVTVPUTVNUEUSAUPBPUFUGUHRST
|
|
URVMCAMUPAOZUQVLWAUQVLNUPAUPABUIUJULRSUMUN $.
|
|
|
|
$( Strict linear ordering on a singleton. (Contributed by Jim Kingdon,
|
|
5-Dec-2018.) $)
|
|
sosng $p |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } <-> -. A R A ) ) $=
|
|
( vx vy vz wrel cvv wcel wa csn wor wbr wn wpo sopo cv wral elsni adantl
|
|
wi posng syl5ib wo ax-in2 adantr wb breqan12d imbi1d ralrimivw ralrimivva
|
|
biimpar mpbird df-iso sylanbrc ex impbid ) BFAGHIZAJZBKZAABLZMZUSURBNZUQV
|
|
AURBOABUAZUBUQVAUSUQVAIVBCPZDPZBLZVDEPZBLVGVEBLUCZTZEURQZDURQCURQZUSUQVBV
|
|
AVCUKVAVKUQVAVJCDURURVAVDURHZVEURHZIZIZVIEURVOVIUTVHTZVAVPVNUTVHUDUEVNVIV
|
|
PUFVAVNVFUTVHVLVMVDAVEABVDARVEARUGUHSULUIUJSCDEURBUMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( An abstraction relation is a subset of a related cross product.
|
|
(Contributed by NM, 16-Jul-1995.) $)
|
|
opabssxp $p |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) }
|
|
C_ ( A X. B ) $=
|
|
( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtr4i ) BFDGCFEGHZAHZBCIOB
|
|
CIDEJPOBCOAKLBCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y ps $.
|
|
brab2ga.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
brab2ga.2 $e |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } $.
|
|
$( The law of concretion for a binary relation. See ~ brab2a for alternate
|
|
proof. TODO: should one of them be deleted? (Contributed by Mario
|
|
Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) $)
|
|
brab2ga $p |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) ) $=
|
|
( wbr wcel wa cv copab cxp opabssxp eqsstri brel eleq2i opelopab2a syl5bb
|
|
cop df-br bitri biadan2 ) EFILZEGMFHMNZBEFGHIICOGMDOHMNANCDPZGHQKACDGHRST
|
|
UHEFUDZUJMZUIBUHUKIMULEFIUEIUJUKKUAUFABCDEFGHJUBUCUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ps $.
|
|
optocl.1 $e |- D = ( B X. C ) $.
|
|
optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
optocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
|
|
$( Implicit substitution of class for ordered pair. (Contributed by NM,
|
|
5-Mar-1995.) $)
|
|
optocl $p |- ( A e. D -> ps ) $=
|
|
( cxp wcel cv cop wceq wa wex elxp3 sylbi opelxp syl5ib exlimivv eleq2s
|
|
imp ) BEFGLZHEUFMCNZDNZOZEPZUIUFMZQZDRCRBCDEFGSULBCDUJUKBUKAUJBUKUGFMUHGM
|
|
QAUGUHFGUAKTJUBUEUCTIUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y ps $. $d z w ch $. $d z w R $.
|
|
2optocl.1 $e |- R = ( C X. D ) $.
|
|
2optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
2optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
|
|
2optocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) ->
|
|
ph ) $.
|
|
$( Implicit substitution of classes for ordered pairs. (Contributed by NM,
|
|
12-Mar-1995.) $)
|
|
2optocl $p |- ( ( A e. R /\ B e. R ) -> ch ) $=
|
|
( wcel wi cv cop wceq imbi2d wa ex optocl com12 impcom ) ILQHLQZCUHBRUHCR
|
|
FGIJKLMFSZGSZTIUABCUHOUBUHUIJQUJKQUCZBUKARUKBRDEHJKLMDSZESZTHUAABUKNUBULJ
|
|
QUMKQUCUKAPUDUEUFUEUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d v u C $. $d x y z w v u D $.
|
|
$d x y z w v u F $. $d z w v u R $. $d x y ps $. $d z w ch $.
|
|
$d v u th $.
|
|
3optocl.1 $e |- R = ( D X. F ) $.
|
|
3optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
3optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
|
|
3optocl.4 $e |- ( <. v , u >. = C -> ( ch <-> th ) ) $.
|
|
3optocl.5 $e |- ( ( ( x e. D /\ y e. F ) /\ ( z e. D /\ w e. F )
|
|
/\ ( v e. D /\ u e. F ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for ordered pairs. (Contributed by NM,
|
|
12-Mar-1995.) $)
|
|
3optocl $p |- ( ( A e. R /\ B e. R /\ C e. R ) -> th ) $=
|
|
( wcel wa wi cv cop wceq imbi2d 3expia 2optocl com12 optocl impcom 3impa
|
|
) KOUBZLOUBZMOUBZDUQUOUPUCZDURCUDURDUDIJMNPOQIUEZJUEZUFMUGCDURTUHURUSNUBU
|
|
TPUBUCZCVAAUDVABUDVACUDEFGHKLNPOQEUEZFUEZUFKUGABVARUHGUEZHUEZUFLUGBCVASUH
|
|
VBNUBVCPUBUCVDNUBVEPUBUCVAAUAUIUJUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $. $d x y z w v u S $. $d x y ph $. $d z w v u ps $.
|
|
opbrop.1 $e |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) ->
|
|
( ph <-> ps ) ) $.
|
|
opbrop.2 $e |- R = { <. x , y >. | ( ( x e. ( S X. S ) /\
|
|
y e. ( S X. S ) ) /\
|
|
E. z E. w E. v E. u ( ( x = <. z , w >. /\
|
|
y = <. v , u >. ) /\ ph ) ) } $.
|
|
$( Ordered pair membership in a relation. Special case. (Contributed by
|
|
NM, 5-Aug-1995.) $)
|
|
opbrop $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
|
|
( <. A , B >. R <. C , D >. <-> ps ) ) $=
|
|
( wcel wa cv wex cop cxp wbr copsex4g anbi2d cvv opexg eleq1 anbi1d eqeq1
|
|
wceq wb 4exbidv anbi12d brabg syl2an opelxpi anim12i biantrurd 3bitr4d )
|
|
INQJNQRZKNQLNQRZRZIJUAZNNUBZQZKLUAZVEQZRZVDESFSUAZUKZVGGSHSUAZUKZRZARZHTG
|
|
TFTETZRZVIBRVDVGMUCZBVCVPBVIABEFGHIJKLNNOUDUEVAVDUFQVGUFQVRVQULVBIJNNUGKL
|
|
NNUGCSZVEQZDSZVEQZRZVSVJUKZWAVLUKZRZARZHTGTFTETZRVFWBRZVKWERZARZHTGTFTETZ
|
|
RVQCDVDVGUFUFMVSVDUKZWCWIWHWLWMVTVFWBVSVDVEUHUIWMWGWKEFGHWMWFWJAWMWDVKWEV
|
|
SVDVJUJUIUIUMUNWAVGUKZWIVIWLVPWNWBVHVFWAVGVEUHUEWNWKVOEFGHWNWJVNAWNWEVMVK
|
|
WAVGVLUJUEUIUMUNPUOUPVCVIBVAVFVBVHIJNNUQKLNNUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The cross product with the empty set is empty. Part of Theorem 3.13(ii)
|
|
of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.) $)
|
|
0xp $p |- ( (/) X. A ) = (/) $=
|
|
( vz vx vy c0 cxp cv wcel cop wceq wa wex elxp noel simprl mto nex 2false
|
|
bitri eqriv ) BEAFZEBGZUAHUBCGZDGZIJZUCEHZUDAHZKKZDLZCLZUBEHZCDUBEAMUJUKU
|
|
ICUHDUHUFUCNUEUFUGOPQQUBNRST $.
|
|
$}
|
|
|
|
${
|
|
$d A w y z $. $d B w y z $. $d C w y z $. $d D w y z $. $d w x y z $.
|
|
$( Distribute proper substitution through the cross product of two classes.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbxpg $p |- ( A e. D -> [_ A / x ]_ ( B X. C ) =
|
|
( [_ A / x ]_ B X. [_ A / x ]_ C ) ) $=
|
|
( vz vw vy wcel cv wa wex cab csb cxp wsbc sbcexg sbcang sbcel2g bitrd
|
|
wceq csbabg anbi12d exbidv abbidv eqtrd copab df-xp df-opab eqtri csbeq2i
|
|
cop sbcg 3eqtr4g ) BEIZABFJGJZHJZULUAZUPCIZUQDIZKZKZHLZGLZFMZNZURUPABCNZI
|
|
ZUQABDNZIZKZKZHLZGLZFMZABCDOZNVGVIOZUOVFVDABPZFMVOVDAFBEUBUOVRVNFUOVRVCAB
|
|
PZGLVNVCGABEQUOVSVMGUOVSVBABPZHLVMVBHABEQUOVTVLHUOVTURABPZVAABPZKVLURVAAB
|
|
ERUOWAURWBVKURABEUMUOWBUSABPZUTABPZKVKUSUTABERUOWCVHWDVJABUPCESABUQDESUCT
|
|
UCTUDTUDTUEUFABVPVEVPVAGHUGVEGHCDUHVAGHFUIUJUKVQVKGHUGVOGHVGVIUHVKGHFUIUJ
|
|
UN $.
|
|
$}
|
|
|
|
$( Equality theorem for the relation predicate. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
releq $p |- ( A = B -> ( Rel A <-> Rel B ) ) $=
|
|
( wceq cvv cxp wss wrel sseq1 df-rel 3bitr4g ) ABCADDEZFBKFAGBGABKHAIBIJ $.
|
|
|
|
${
|
|
releqi.1 $e |- A = B $.
|
|
$( Equality inference for the relation predicate. (Contributed by NM,
|
|
8-Dec-2006.) $)
|
|
releqi $p |- ( Rel A <-> Rel B ) $=
|
|
( wceq wrel wb releq ax-mp ) ABDAEBEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
releqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for the relation predicate. (Contributed by NM,
|
|
8-Mar-2014.) $)
|
|
releqd $p |- ( ph -> ( Rel A <-> Rel B ) ) $=
|
|
( wceq wrel wb releq syl ) ABCEBFCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
nfrel.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for a relation. (Contributed by NM,
|
|
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfrel $p |- F/ x Rel A $=
|
|
( wrel cvv cxp wss df-rel nfcv nfss nfxfr ) BDBEEFZGABHABLCALIJK $.
|
|
$}
|
|
|
|
$( Distribute proper substitution through a relation predicate. (Contributed
|
|
by Alexander van der Vekens, 23-Jul-2017.) $)
|
|
sbcrel $p |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) ) $=
|
|
( wcel cvv cxp wss wsbc sbcssg csbconstg sseq2d bitrd df-rel sbcbii 3bitr4g
|
|
csb wrel ) BDEZCFFGZHZABIZABCQZTHZCRZABIUCRSUBUCABTQZHUDABCTDJSUFTUCABTDKLM
|
|
UEUAABCNOUCNP $.
|
|
|
|
$( Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
|
|
(Contributed by NM, 15-Aug-1994.) $)
|
|
relss $p |- ( A C_ B -> ( Rel B -> Rel A ) ) $=
|
|
( wss cvv cxp wrel sstr2 df-rel 3imtr4g ) ABCBDDEZCAJCBFAFABJGBHAHI $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( A subclass relationship depends only on a relation's ordered pairs.
|
|
Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
ssrel $p |- ( Rel A -> ( A C_ B <->
|
|
A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) ) $=
|
|
( vz wrel wss cv cop wcel wi wal ssel alrimivv wceq wex eleq1 imbi12d cvv
|
|
dfss2 biimprcd 2alimi 19.23vv sylib com23 alimdv df-rel elvv imbi2i albii
|
|
a2d cxp 3bitri 3imtr4g com12 impbid2 ) CFZCDGZAHBHIZCJZUSDJZKZBLALZURVBAB
|
|
CDUSMNVCUQURVCEHZCJZVDUSOZBPAPZKZELZVEVDDJZKZELUQURVCVHVKEVCVEVGVJVCVGVEV
|
|
JVCVFVKKZBLALVGVKKVBVLABVFVKVBVFVEUTVJVAVDUSCQVDUSDQRUAUBVFVKABUCUDUEUKUF
|
|
UQCSSULZGVEVDVMJZKZELVICUGECVMTVOVHEVNVGVEABVDUHUIUJUMECDTUNUOUP $.
|
|
|
|
$( Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
|
|
p. 33. (Contributed by NM, 2-Aug-1994.) $)
|
|
eqrel $p |- ( ( Rel A /\ Rel B ) -> ( A = B <->
|
|
A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) ) $=
|
|
( wrel wa wss cv cop wcel wi wal wceq ssrel bi2anan9 eqss 2albiim 3bitr4g
|
|
wb ) CEZDEZFCDGZDCGZFAHBHIZCJZUDDJZKBLALZUFUEKBLALZFCDMUEUFSBLALTUBUGUAUC
|
|
UHABCDNABDCNOCDPUEUFABQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z R $. $d x y z S $.
|
|
$( A subclass relationship depends only on a relation's ordered pairs.
|
|
This version of ~ ssrel is restricted to the relation's domain.
|
|
(Contributed by Thierry Arnoux, 25-Jan-2018.) $)
|
|
ssrel2 $p |- ( R C_ ( A X. B ) -> ( R C_ S <-> A. x e. A A. y e. B
|
|
( <. x , y >. e. R -> <. x , y >. e. S ) ) ) $=
|
|
( vz cxp wss cv wcel wi wral wrex wal eleq1 ralimi r19.23v bitri dfss2 wa
|
|
cop ssel a1d ralrimivv wceq imbi12d biimprcd ralbii sylib com23 a2d elxp2
|
|
alimdv imbi2i albii 3imtr4g com12 impbid2 ) ECDHZIZEFIZAJZBJZUBZEKZVEFKZL
|
|
ZBDMZACMZVBVHABCDVBVHVCCKVDDKUAEFVEUCUDUEVJVAVBVJGJZEKZVKVEUFZBDNZACNZLZG
|
|
OZVLVKFKZLZGOVAVBVJVPVSGVJVLVOVRVJVOVLVRVJVMVSLZBDMZACMZVOVSLZVIWAACVHVTB
|
|
DVMVSVHVMVLVFVRVGVKVEEPVKVEFPUGUHQQWBVNVSLZACMWCWAWDACVMVSBDRUIVNVSACRSUJ
|
|
UKULUNVAVLVKUTKZLZGOVQGEUTTWFVPGWEVOVLABVKCDUMUOUPSGEFTUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
relssi.1 $e |- Rel A $.
|
|
relssi.2 $e |- ( <. x , y >. e. A -> <. x , y >. e. B ) $.
|
|
$( Inference from subclass principle for relations. (Contributed by NM,
|
|
31-Mar-1998.) $)
|
|
relssi $p |- A C_ B $=
|
|
( wss cv cop wcel wi wal wrel wb ssrel ax-mp ax-gen mpgbir ) CDGZAHBHIZCJ
|
|
TDJKZBLZACMSUBALNEABCDOPUABFQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ph $.
|
|
relssdv.1 $e |- ( ph -> Rel A ) $.
|
|
relssdv.2 $e |- ( ph -> ( <. x , y >. e. A -> <. x , y >. e. B ) ) $.
|
|
$( Deduction from subclass principle for relations. (Contributed by NM,
|
|
11-Sep-2004.) $)
|
|
relssdv $p |- ( ph -> A C_ B ) $=
|
|
( wss cv cop wcel wi wal alrimivv wrel wb ssrel syl mpbird ) ADEHZBICIJZD
|
|
KUAEKLZCMBMZAUBBCGNADOTUCPFBCDEQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
eqrelriv.1 $e |- ( <. x , y >. e. A <-> <. x , y >. e. B ) $.
|
|
$( Inference from extensionality principle for relations. (Contributed by
|
|
FL, 15-Oct-2012.) $)
|
|
eqrelriv $p |- ( ( Rel A /\ Rel B ) -> A = B ) $=
|
|
( wrel wa wceq cv cop wcel wb wal gen2 eqrel mpbiri ) CFDFGCDHAIBIJZCKQDK
|
|
LZBMAMRABENABCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
eqreliiv.1 $e |- Rel A $.
|
|
eqreliiv.2 $e |- Rel B $.
|
|
eqreliiv.3 $e |- ( <. x , y >. e. A <-> <. x , y >. e. B ) $.
|
|
$( Inference from extensionality principle for relations. (Contributed by
|
|
NM, 17-Mar-1995.) $)
|
|
eqrelriiv $p |- A = B $=
|
|
( wrel wceq eqrelriv mp2an ) CHDHCDIEFABCDGJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
eqbrriv.1 $e |- Rel A $.
|
|
eqbrriv.2 $e |- Rel B $.
|
|
eqbrriv.3 $e |- ( x A y <-> x B y ) $.
|
|
$( Inference from extensionality principle for relations. (Contributed by
|
|
NM, 12-Dec-2006.) $)
|
|
eqbrriv $p |- A = B $=
|
|
( cv wbr cop wcel df-br 3bitr3i eqrelriiv ) ABCDEFAHZBHZCIOPDIOPJZCKQDKGO
|
|
PCLOPDLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqrelrdv.1 $e |- Rel A $.
|
|
eqrelrdv.2 $e |- Rel B $.
|
|
eqrelrdv.3 $e |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $.
|
|
$( Deduce equality of relations from equivalence of membership.
|
|
(Contributed by Rodolfo Medina, 10-Oct-2010.) $)
|
|
eqrelrdv $p |- ( ph -> A = B ) $=
|
|
( cv cop wcel wb wal wceq alrimivv wrel eqrel mp2an sylibr ) ABICIJZDKTEK
|
|
LZCMBMZDENZAUABCHODPEPUCUBLFGBCDEQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqbrrdv.1 $e |- ( ph -> Rel A ) $.
|
|
eqbrrdv.2 $e |- ( ph -> Rel B ) $.
|
|
eqbrrdv.3 $e |- ( ph -> ( x A y <-> x B y ) ) $.
|
|
$( Deduction from extensionality principle for relations. (Contributed by
|
|
Mario Carneiro, 3-Jan-2017.) $)
|
|
eqbrrdv $p |- ( ph -> A = B ) $=
|
|
( wceq cv cop wcel wb wal wbr df-br 3bitr3g alrimivv wrel eqrel syl2anc
|
|
mpbird ) ADEIZBJZCJZKZDLZUFELZMZCNBNZAUIBCAUDUEDOUDUEEOUGUHHUDUEDPUDUEEPQ
|
|
RADSESUCUJMFGBCDETUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqbrrdiv.1 $e |- Rel A $.
|
|
eqbrrdiv.2 $e |- Rel B $.
|
|
eqbrrdiv.3 $e |- ( ph -> ( x A y <-> x B y ) ) $.
|
|
$( Deduction from extensionality principle for relations. (Contributed by
|
|
Rodolfo Medina, 10-Oct-2010.) $)
|
|
eqbrrdiv $p |- ( ph -> A = B ) $=
|
|
( cv wbr cop wcel df-br 3bitr3g eqrelrdv ) ABCDEFGABIZCIZDJPQEJPQKZDLRELH
|
|
PQDMPQEMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqrelrdv2.1 $e |- ( ph
|
|
-> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $.
|
|
$( A version of ~ eqrelrdv . (Contributed by Rodolfo Medina,
|
|
10-Oct-2010.) $)
|
|
eqrelrdv2 $p |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B ) $=
|
|
( wrel wa wceq cv cop wcel wb wal alrimivv adantl eqrel adantr mpbird ) D
|
|
GEGHZAHDEIZBJCJKZDLUBELMZCNBNZAUDTAUCBCFOPTUAUDMABCDEQRS $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $.
|
|
$( A subclass relationship determined by ordered triples. Use ~ relrelss
|
|
to express the antecedent in terms of the relation predicate.
|
|
(Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
ssrelrel $p |- ( A C_ ( ( _V X. _V ) X. _V ) -> ( A C_ B <->
|
|
A. x A. y A. z ( <. <. x , y >. , z >. e. A
|
|
-> <. <. x , y >. , z >. e. B ) ) ) $=
|
|
( vw cvv cxp wss cv cop wcel wi wal ssel alrimiv wex eleq1 sylib dfss2
|
|
alrimivv wceq elvvv imbi12d biimprcd alimi 19.23v 2alimi syl5bi com23 a2d
|
|
19.23vv alimdv 3imtr4g com12 impbid2 ) DGGHGHZIZDEIZAJBJKCJKZDLZUTELZMZCN
|
|
ZBNANZUSVDABUSVCCDEUTOPUAVEURUSVEFJZDLZVFUQLZMZFNVGVFELZMZFNURUSVEVIVKFVE
|
|
VGVHVJVEVHVGVJVHVFUTUBZCQZBQAQZVEVKABCVFUCVEVMVKMZBNANVNVKMVDVOABVDVLVKMZ
|
|
CNVOVCVPCVLVKVCVLVGVAVJVBVFUTDRVFUTERUDUEUFVLVKCUGSUHVMVKABULSUIUJUKUMFDU
|
|
QTFDETUNUOUP $.
|
|
|
|
$( Extensionality principle for ordered triples, analogous to ~ eqrel .
|
|
Use ~ relrelss to express the antecedent in terms of the relation
|
|
predicate. (Contributed by NM, 17-Dec-2008.) $)
|
|
eqrelrel $p |- ( ( A u. B ) C_ ( ( _V X. _V ) X. _V ) -> ( A = B <->
|
|
A. x A. y A. z ( <. <. x , y >. , z >. e. A
|
|
<-> <. <. x , y >. , z >. e. B ) ) ) $=
|
|
( cun cvv cxp wss wa wceq cv cop wcel wb wal unss wi ssrelrel bi2anan9
|
|
eqss 2albiim albii 19.26 bitri 3bitr4g sylbir ) DEFGGHGHZIDUHIZEUHIZJZDEK
|
|
ZALBLMCLMZDNZUMENZOCPBPZAPZODEUHQUKDEIZEDIZJUNUORCPBPZAPZUOUNRCPBPZAPZJZU
|
|
LUQUIURVAUJUSVCABCDESABCEDSTDEUAUQUTVBJZAPVDUPVEAUNUOBCUBUCUTVBAUDUEUFUG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A member of a relation is an ordered pair. (Contributed by NM,
|
|
17-Sep-2006.) $)
|
|
elrel $p |- ( ( Rel R /\ A e. R ) -> E. x E. y A = <. x , y >. ) $=
|
|
( wrel wcel wa cvv cxp cop wceq wex wss df-rel biimpi sselda elvv sylib
|
|
cv ) DEZCDFGCHHIZFCASBSJKBLALTDUACTDUAMDNOPABCQR $.
|
|
$}
|
|
|
|
$( A singleton is a relation iff it is an ordered pair. (Contributed by NM,
|
|
24-Sep-2013.) (Revised by BJ, 12-Feb-2022.) $)
|
|
relsng $p |- ( A e. V -> ( Rel { A } <-> A e. ( _V X. _V ) ) ) $=
|
|
( wcel cvv cxp csn wss wrel snssg df-rel syl6rbbr ) ABCADDEZCAFZLGMHALBIMJK
|
|
$.
|
|
|
|
$( A singleton of an ordered pair is a relation. (Contributed by NM,
|
|
17-May-1998.) (Revised by BJ, 12-Feb-2022.) $)
|
|
relsnopg $p |- ( ( A e. V /\ B e. W ) -> Rel { <. A , B >. } ) $=
|
|
( wcel wa cop csn wrel cvv cxp opelvvg wb opexg relsng syl mpbird ) ACEBDEF
|
|
ZABGZHIZSJJKEZABCDLRSJETUAMABCDNSJOPQ $.
|
|
|
|
${
|
|
relsn.1 $e |- A e. _V $.
|
|
$( A singleton is a relation iff it is an ordered pair. (Contributed by
|
|
NM, 24-Sep-2013.) $)
|
|
relsn $p |- ( Rel { A } <-> A e. ( _V X. _V ) ) $=
|
|
( csn wrel cvv cxp wss wcel df-rel snss bitr4i ) ACZDLEEFZGAMHLIAMBJK $.
|
|
|
|
relsnop.2 $e |- B e. _V $.
|
|
$( A singleton of an ordered pair is a relation. (Contributed by NM,
|
|
17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
relsnop $p |- Rel { <. A , B >. } $=
|
|
( cop csn wrel cvv cxp wcel opelvv opex relsn mpbir ) ABEZFGOHHIJABCDKOAB
|
|
CDLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( Subset theorem for cross product. Generalization of Theorem 101 of
|
|
[Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) $)
|
|
xpss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) ) $=
|
|
( vx vy wss wa cv wcel copab cxp ssel im2anan9 ssopab2dv df-xp 3sstr4g )
|
|
ABGZCDGZHZEIZAJZFIZCJZHZEFKUABJZUCDJZHZEFKACLBDLTUEUHEFRUBUFSUDUGABUAMCDU
|
|
CMNOEFACPEFBDPQ $.
|
|
$}
|
|
|
|
$( A cross product is included in the ordered pair universe. Exercise 3 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) $)
|
|
xpss $p |- ( A X. B ) C_ ( _V X. _V ) $=
|
|
( cvv wss cxp ssv xpss12 mp2an ) ACDBCDABECCEDAFBFACBCGH $.
|
|
|
|
$( A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
|
|
(Contributed by NM, 2-Aug-1994.) $)
|
|
relxp $p |- Rel ( A X. B ) $=
|
|
( cxp wrel cvv wss xpss df-rel mpbir ) ABCZDJEECFABGJHI $.
|
|
|
|
$( Subset relation for cross product. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
xpss1 $p |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) ) $=
|
|
( wss cxp ssid xpss12 mpan2 ) ABDCCDACEBCEDCFABCCGH $.
|
|
|
|
$( Subset relation for cross product. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
xpss2 $p |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) ) $=
|
|
( wss cxp ssid xpss12 mpan ) CCDABDCAECBEDCFCCABGH $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( A cross product is included in the power of the power of the union of
|
|
its arguments. (Contributed by NM, 13-Sep-2006.) $)
|
|
xpsspw $p |- ( A X. B ) C_ ~P ~P ( A u. B ) $=
|
|
( vz vx vy cun cpw cv wcel wss wceq wa wex csn cpr snssi sseq1 syl5ibrcom
|
|
vex syl cxp cop elxpi dfop wo ssun3 adantr df-pr ssun4 anim12i unss sylib
|
|
syl5eqss jaod elpr 3imtr4g ssrdv biimpar sylan2 exlimivv sylibr ssriv
|
|
elpw ) CABUAZABFZGZGZCHZVDIZVHVFJZVHVGIVIVHDHZEHZUBZKZVKAIZVLBIZLZLZEMDMV
|
|
JDEVHABUCVRVJDEVQVNVMVFJZVJVQVMVKNZVKVLOZOZVFVKVLDSESUDVQCWBVFVQVHVTKZVHW
|
|
AKZUEVHVEJZVHWBIVHVFIVQWCWEWDVQWEWCVTVEJZVOWFVPVOVTAJWFVKAPVTABUFTZUGVHVT
|
|
VEQRVQWEWDWAVEJVQWAVTVLNZFZVEVKVLUHVQWFWHVEJZLWIVEJVOWFVPWJWGVPWHBJWJVLBP
|
|
WHBAUITUJVTWHVEUKULUMVHWAVEQRUNVHVTWACSZUOVHVEWKVCUPUQUMVNVJVSVHVMVFQURUS
|
|
UTTVHVFWKVCVAVB $.
|
|
$}
|
|
|
|
$( The double class union of a cross product is included in the union of its
|
|
arguments. (Contributed by NM, 16-Sep-2006.) $)
|
|
unixpss $p |- U. U. ( A X. B ) C_ ( A u. B ) $=
|
|
( cxp cuni cun cpw xpsspw unissi unipw sseqtri ) ABCZDZDABEZFZDMLNLNFZDNKOA
|
|
BGHNIJHMIJ $.
|
|
|
|
$( The cross product of two sets is a set. Proposition 6.2 of
|
|
[TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) $)
|
|
xpexg $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V ) $=
|
|
( wcel wa cxp cun cpw wss cvv xpsspw unexg pwexg 3syl ssexg sylancr ) ACEBD
|
|
EFZABGZABHZIZIZJUBKEZSKEABLRTKEUAKEUCABCDMTKNUAKNOSUBKPQ $.
|
|
|
|
${
|
|
xpex.1 $e |- A e. _V $.
|
|
xpex.2 $e |- B e. _V $.
|
|
$( The cross product of two sets is a set. Proposition 6.2 of
|
|
[TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) $)
|
|
xpex $p |- ( A X. B ) e. _V $=
|
|
( cvv wcel cxp xpexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( The union of two relations is a relation. Compare Exercise 5 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.) $)
|
|
relun $p |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) ) $=
|
|
( cvv cxp wss wa cun wrel unss df-rel anbi12i 3bitr4ri ) ACCDZEZBMEZFABGZME
|
|
AHZBHZFPHABMIQNROAJBJKPJL $.
|
|
|
|
$( The intersection with a relation is a relation. (Contributed by NM,
|
|
16-Aug-1994.) $)
|
|
relin1 $p |- ( Rel A -> Rel ( A i^i B ) ) $=
|
|
( cin wss wrel wi inss1 relss ax-mp ) ABCZADAEJEFABGJAHI $.
|
|
|
|
$( The intersection with a relation is a relation. (Contributed by NM,
|
|
17-Jan-2006.) $)
|
|
relin2 $p |- ( Rel B -> Rel ( A i^i B ) ) $=
|
|
( cin wss wrel wi inss2 relss ax-mp ) ABCZBDBEJEFABGJBHI $.
|
|
|
|
$( A difference cutting down a relation is a relation. (Contributed by NM,
|
|
31-Mar-1998.) $)
|
|
reldif $p |- ( Rel A -> Rel ( A \ B ) ) $=
|
|
( cdif wss wrel wi difss relss ax-mp ) ABCZADAEJEFABGJAHI $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( An indexed union is a relation iff each member of its indexed family is
|
|
a relation. (Contributed by NM, 19-Dec-2008.) $)
|
|
reliun $p |- ( Rel U_ x e. A B <-> A. x e. A Rel B ) $=
|
|
( vy ciun wrel cv wcel wrex cab cvv cxp wss wral df-iun releqi df-rel wal
|
|
wi 3bitri abss dfss2 bitri ralbii ralcom4 r19.23v albii bitr4i ) ABCEZFDG
|
|
ZCHZABIZDJZFUMKKLZMZCFZABNZUIUMADBCOPUMQUOULUJUNHZSZDRZUQULDUNUAUQUKURSZD
|
|
RZABNVAABNZDRUTUPVBABUPCUNMVBCQDCUNUBUCUDVAADBUEVCUSDUKURABUFUGTUHT $.
|
|
$}
|
|
|
|
$( An indexed intersection is a relation if at least one of the member of the
|
|
indexed family is a relation. (Contributed by NM, 8-Mar-2014.) $)
|
|
reliin $p |- ( E. x e. A Rel B -> Rel |^|_ x e. A B ) $=
|
|
( cvv cxp wss wrex ciin wrel iinss df-rel rexbii 3imtr4i ) CDDEZFZABGABCHZN
|
|
FCIZABGPIABCNJQOABCKLPKM $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The union of a class is a relation iff any member is a relation.
|
|
Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by
|
|
NM, 13-Aug-2004.) $)
|
|
reluni $p |- ( Rel U. A <-> A. x e. A Rel x ) $=
|
|
( cuni wrel cv ciun wral uniiun releqi reliun bitri ) BCZDABAEZFZDMDABGLN
|
|
ABHIABMJK $.
|
|
|
|
$( The intersection of a class is a relation if at least one member is a
|
|
relation. (Contributed by NM, 8-Mar-2014.) $)
|
|
relint $p |- ( E. x e. A Rel x -> Rel |^| A ) $=
|
|
( cv wrel wrex ciin cint reliin intiin releqi sylibr ) ACZDABEABLFZDBGZDA
|
|
BLHNMABIJK $.
|
|
$}
|
|
|
|
$( The empty set is a relation. (Contributed by NM, 26-Apr-1998.) $)
|
|
rel0 $p |- Rel (/) $=
|
|
( c0 wrel cvv cxp wss 0ss df-rel mpbir ) ABACCDZEIFAGH $.
|
|
|
|
${
|
|
$d ph z $. $d x z $. $d y z $.
|
|
relopabi.1 $e |- A = { <. x , y >. | ph } $.
|
|
$( A class of ordered pairs is a relation. (Contributed by Mario Carneiro,
|
|
21-Dec-2013.) $)
|
|
relopabi $p |- Rel A $=
|
|
( vz wrel cvv cxp wss cv cop wceq wa wex cab copab df-opab wcel vex eqtri
|
|
opelvv eleq1 mpbiri adantr exlimivv abssi eqsstri df-rel mpbir ) DGDHHIZJ
|
|
DFKZBKZCKZLZMZANZCOBOZFPZUKDABCQUSEABCFRUAURFUKUQULUKSZBCUPUTAUPUTUOUKSUM
|
|
UNBTCTUBULUOUKUCUDUEUFUGUHDUIUJ $.
|
|
$}
|
|
|
|
$( A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.)
|
|
(Unnecessary distinct variable restrictions were removed by Alan Sare,
|
|
9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) $)
|
|
relopab $p |- Rel { <. x , y >. | ph } $=
|
|
( copab eqid relopabi ) ABCABCDZGEF $.
|
|
|
|
${
|
|
$d w x y z A $. $d x y B $. $d x y C $. $d x y D $. $d ph z w $.
|
|
$d ps z w $.
|
|
$( The identity relation is a relation. Part of Exercise 4.12(p) of
|
|
[Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by
|
|
Mario Carneiro, 21-Dec-2013.) $)
|
|
reli $p |- Rel _I $=
|
|
( vx vy cv wceq cid df-id relopabi ) ACBCDABEABFG $.
|
|
|
|
$( The membership relation is a relation. (Contributed by NM,
|
|
26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.) $)
|
|
rele $p |- Rel _E $=
|
|
( vx vy wel cep df-eprel relopabi ) ABCABDABEF $.
|
|
|
|
$( A relation expressed as an ordered pair abstraction. (Contributed by
|
|
NM, 11-Dec-2006.) $)
|
|
opabid2 $p |- ( Rel A -> { <. x , y >. | <. x , y >. e. A } = A ) $=
|
|
( vz vw wrel cv cop wcel copab wceq wb wal vex opeq1 eleq1d opelopab gen2
|
|
opeq2 relopab eqrel mpan mpbiri ) CFZAGZBGZHZCIZABJZCKZDGZEGZHZUIIUMCIZLZ
|
|
EMDMZUODEUHUKUFHZCIUNABUKULDNENUEUKKUGUQCUEUKUFOPUFULKUQUMCUFULUKSPQRUIFU
|
|
DUJUPLUHABTDEUICUAUBUC $.
|
|
|
|
$( Intersection of two ordered pair class abstractions. (Contributed by
|
|
NM, 30-Sep-2002.) $)
|
|
inopab $p |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) =
|
|
{ <. x , y >. | ( ph /\ ps ) } $=
|
|
( vz vw copab cin wa wrel relopab relin1 ax-mp cv cop wcel wsb sban sbbii
|
|
opelopabsbALT anbi12i 3bitr4ri elin 3bitr4i eqrelriiv ) EFACDGZBCDGZHZABI
|
|
ZCDGZUFJUHJACDKUFUGLMUICDKENFNOZUFPZUKUGPZIZUICEQZDFQZUKUHPUKUJPACEQZBCEQ
|
|
ZIZDFQUQDFQZURDFQZIUPUNUQURDFRUOUSDFABCERSULUTUMVAACDEFTBCDEFTUAUBUKUFUGU
|
|
CUICDEFTUDUE $.
|
|
|
|
$( The difference of two ordered-pair abstractions. (Contributed by Stefan
|
|
O'Rear, 17-Jan-2015.) $)
|
|
difopab $p |- ( { <. x , y >. | ph } \ { <. x , y >. | ps } ) =
|
|
{ <. x , y >. | ( ph /\ -. ps ) } $=
|
|
( vz vw copab wn wa wrel relopab ax-mp cv wcel wsbc sbcbii opelopabsb cvv
|
|
sbcan wb cdif reldif cop vex sbcng notbii anbi12i eldif 3bitr4i eqrelriiv
|
|
3bitr4ri ) EFACDGZBCDGZUAZABHZIZCDGZULJUNJACDKULUMUBLUPCDKEMZFMZUCZULNZUT
|
|
UMNZHZIZUPDUSOZCUROZUTUNNUTUQNADUSOZUODUSOZIZCUROVGCUROZVHCUROZIVFVDVGVHC
|
|
URSVEVICURAUODUSSPVAVJVCVKACDURUSQBDUSOZHZCUROZVLCUROZHZVKVCURRNVNVPTEUDV
|
|
LCURRUELVHVMCURUSRNVHVMTFUDBDUSRUELPVBVOBCDURUSQUFUKUGUKUTULUMUHUPCDURUSQ
|
|
UIUJ $.
|
|
|
|
$( The intersection of two cross products. Exercise 9 of [TakeutiZaring]
|
|
p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
inxp $p |- ( ( A X. B ) i^i ( C X. D ) ) =
|
|
( ( A i^i C ) X. ( B i^i D ) ) $=
|
|
( vx vy cv wcel wa copab cin cxp inopab elin anbi12i bitr4i opabbii eqtri
|
|
an4 df-xp ineq12i 3eqtr4i ) EGZAHZFGZBHZIZEFJZUCCHZUEDHZIZEFJZKZUCACKZHZU
|
|
EBDKZHZIZEFJZABLZCDLZKUNUPLUMUGUKIZEFJUSUGUKEFMVBUREFVBUDUIIZUFUJIZIURUDU
|
|
FUIUJSUOVCUQVDUCACNUEBDNOPQRUTUHVAULEFABTEFCDTUAEFUNUPTUB $.
|
|
|
|
$( Distributive law for cross product over intersection. Theorem 102 of
|
|
[Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) $)
|
|
xpindi $p |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) ) $=
|
|
( cxp cin inxp inidm xpeq1i eqtr2i ) ABDACDEAAEZBCEZDAKDABACFJAKAGHI $.
|
|
|
|
$( Distributive law for cross product over intersection. Similar to
|
|
Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) $)
|
|
xpindir $p |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) ) $=
|
|
( cxp cin inxp inidm xpeq2i eqtr2i ) ACDBCDEABEZCCEZDJCDACBCFKCJCGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w C $. $d z w B $.
|
|
$( Distributive law for cross product over indexed intersection.
|
|
(Contributed by Jim Kingdon, 7-Dec-2018.) $)
|
|
xpiindim $p |- ( E. y y e. A ->
|
|
( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) ) $=
|
|
( vw vz ciin cxp wrel wa cv wcel wex wral relxp wb cvv vex eliin wceq syl
|
|
wrex rgenw eleq1 cbvexv r19.2m sylanbr mpan2 reliin jctil r19.28mv sylbir
|
|
cop bicomd ax-mp anbi2i opelxp ralbii 3bitr4g opex eqrelrdv2 mpancom ) EA
|
|
CDHZIZJZACEDIZHZJZKBLZCMZBNZVEVHUAVLVIVFVLVGJZACUCZVIVLVMACOZVNVMACEDPUDV
|
|
LALZCMZANZVOVNVQVKABVPVJCUEUFZVMACUGUHUIACVGUJUBEVDPUKVLFGVEVHVLFLZEMZGLZ
|
|
VDMZKZVTWBUNZVGMZACOZWEVEMWEVHMZVLWAWBDMZACOZKZWAWIKZACOZWDWGVLWMWKVLVRWM
|
|
WKQVSWAWIACULUMUOWCWJWAWBRMWCWJQGSZAWBCDRTUPUQWFWLACVTWBEDURUSUTVTWBEVDUR
|
|
WERMWHWGQVTWBFSWNVAAWECVGRTUPUTVBVC $.
|
|
|
|
$( Distributive law for cross product over relativized indexed
|
|
intersection. (Contributed by Jim Kingdon, 7-Dec-2018.) $)
|
|
xpriindim $p |- ( E. y y e. A ->
|
|
( C X. ( D i^i |^|_ x e. A B ) ) =
|
|
( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) ) $=
|
|
( cv wcel wex ciin cin cxp xpindi xpiindim ineq2d syl5eq ) BGCHBIZEFACDJZ
|
|
KLEFLZERLZKSACEDLJZKEFRMQTUASABCDENOP $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y C $. $d x y D $. $d x E $.
|
|
$( Membership in a union of cross products. Analogue of ~ elxp for
|
|
nonconstant ` B ( x ) ` . (Contributed by Mario Carneiro,
|
|
29-Dec-2014.) $)
|
|
eliunxp $p |- ( C e. U_ x e. A ( { x } X. B ) <->
|
|
E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) ) $=
|
|
( cv csn cxp ciun wcel cop wceq wex wa wrel wral relxp rgenw reliun exbii
|
|
mpbir elrel mpan pm4.71ri nfiu1 nfel2 19.41 19.41v eleq1 opeliunxp syl6bb
|
|
pm5.32i bitr3i 3bitr2i ) EACAFZGZDHZIZJZEUOBFZKZLZBMZAMZUSNVCUSNZAMVBUOCJ
|
|
UTDJNZNZBMZAMUSVDUROZUSVDVIUQOZACPVJACUPDQRACUQSUAABEURUBUCUDVCUSAAEURACU
|
|
QUEUFUGVEVHAVEVBUSNZBMVHVBUSBUHVKVGBVBUSVFVBUSVAURJVFEVAURUIACDUTUJUKULTU
|
|
MTUN $.
|
|
|
|
$d x A $.
|
|
opeliunxp2.1 $e |- ( x = C -> B = E ) $.
|
|
$( Membership in a union of cross products. (Contributed by Mario
|
|
Carneiro, 14-Feb-2015.) $)
|
|
opeliunxp2 $p |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <->
|
|
( C e. A /\ D e. E ) ) $=
|
|
( cop cv csn cxp ciun wcel cvv wa wbr df-br wrel wral wb relxp rgenw elex
|
|
reliun mpbir brrelexi sylbir adantr nfcv nfiu1 nfel2 nfv nfbi wceq eleq1d
|
|
opeq1 eleq1 eleq2d anbi12d bibi12d opeliunxp vtoclgf pm5.21nii ) DEHZABAI
|
|
ZJZCKZLZMZDNMZDBMZEFMZOZVIDEVHPVJDEVHQDEVHVHRVGRZABSVNABVFCUAUBABVGUDUEUF
|
|
UGVKVJVLDBUCUHVEEHZVHMZVEBMZECMZOZTVIVMTADNADUIVIVMAAVDVHABVGUJUKVMAULUMV
|
|
EDUNZVPVIVSVMVTVOVDVHVEDEUPUOVTVQVKVRVLVEDBUQVTCFEGURUSUTABCEVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x z B $. $d y z ph $. $d x ps $.
|
|
ralxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Write a double restricted quantification as one universal quantifier.
|
|
In this version of ~ ralxp , ` B ( y ) ` is not assumed to be constant.
|
|
(Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
raliunxp $p |- ( A. x e. U_ y e. A ( { y } X. B ) ph <->
|
|
A. y e. A A. z e. B ps ) $=
|
|
( cv csn cxp wcel wi wal wa wral wex albii vex bitri ciun cop wceq imbi1i
|
|
eliunxp 19.23vv bitr4i alrot3 impexp imbi2d ceqsalv 2albii df-ral 3bitr4i
|
|
opex r2al ) CIZDFDIZJGKUAZLZAMZCNZURFLEIZGLOZBMZENDNZACUSPBEGPDFPVBUQURVC
|
|
UBZUCZVDOZAMZENDNZCNZVFVAVKCVAVIEQDQZAMVKUTVMADEFGUQUEUDVIADEUFUGRVLVJCNZ
|
|
ENDNVFVJCDEUHVNVEDEVNVHVDAMZMZCNVEVJVPCVHVDAUIRVOVECVGURVCDSESUOVHABVDHUJ
|
|
UKTULTTACUSUMBDEFGUPUN $.
|
|
|
|
$( Write a double restricted quantification as one universal quantifier.
|
|
In this version of ~ rexxp , ` B ( y ) ` is not assumed to be constant.
|
|
(Contributed by Mario Carneiro, 14-Feb-2015.) $)
|
|
rexiunxp $p |- ( E. x e. U_ y e. A ( { y } X. B ) ph <->
|
|
E. y e. A E. z e. B ps ) $=
|
|
( cv csn cxp ciun wcel wa wex wrex cop exbii vex bitri wceq anbi1i bitr4i
|
|
eliunxp 19.41vv exrot3 anass opex anbi2d ceqsexv 2exbii df-rex 3bitr4i
|
|
r2ex ) CIZDFDIZJGKLZMZANZCOZUPFMEIZGMNZBNZEODOZACUQPBEGPDFPUTUOUPVAQZUAZV
|
|
BNZANZEODOZCOZVDUSVICUSVGEODOZANVIURVKADEFGUOUDUBVGADEUEUCRVJVHCOZEODOVDV
|
|
HCDEUFVLVCDEVLVFVBANZNZCOVCVHVNCVFVBAUGRVMVCCVEUPVADSESUHVFABVBHUIUJTUKTT
|
|
ACUQULBDEFGUNUM $.
|
|
|
|
$d y B $.
|
|
$( Universal quantification restricted to a cross product is equivalent to
|
|
a double restricted quantification. The hypothesis specifies an
|
|
implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
ralxp $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
|
|
( cxp wral cv csn ciun iunxpconst raleqi raliunxp bitr3i ) ACFGIZJACDFDKL
|
|
GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.
|
|
|
|
$( Existential quantification restricted to a cross product is equivalent
|
|
to a double restricted quantification. (Contributed by NM,
|
|
11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.) $)
|
|
rexxp $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
|
|
( cxp wrex cv csn ciun iunxpconst rexeqi rexiunxp bitr3i ) ACFGIZJACDFDKL
|
|
GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Disjoint union is a subset of a cross product. (Contributed by Stefan
|
|
O'Rear, 21-Nov-2014.) $)
|
|
djussxp $p |- U_ x e. A ( { x } X. B ) C_ ( A X. _V ) $=
|
|
( cv csn cxp ciun cvv wss iunss wcel snssi ssv xpss12 sylancl mprgbir ) A
|
|
BADZEZCFZGBHFZISTIZABABSTJQBKRBICHIUAQBLCMRBCHNOP $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x y A $. $d u v w x y z B $. $d u v w ph $. $d u v w ps $.
|
|
ralxpf.1 $e |- F/ y ph $.
|
|
ralxpf.2 $e |- F/ z ph $.
|
|
ralxpf.3 $e |- F/ x ps $.
|
|
ralxpf.4 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ ralxp with bound-variable hypotheses. (Contributed by NM,
|
|
18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
ralxpf $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
|
|
( vv vw vu wral wsb cv wceq vex nfsb cxp cbvralsv ralbii nfv nfcv nfralxy
|
|
nfs1v sbequ12 ralbidv cbvral cop wa wex eqvinop nfbi sbhypf opth sylan9bb
|
|
wb sylbi exlimi ralxp 3bitr4ri bitri ) ACFGUAZOACLPZLVEOZBEGOZDFOZACLVEUB
|
|
BDMPZEGOZMFOVJENPZNGOZMFOVIVGVKVMMFVJENGUBUCVHVKDMFVHMUDVJDEGDGUEBDMUGZUF
|
|
DQZMQZRZBVJEGBDMUHZUIUJVFVLLMNFGLQZVPNQZUKZRVSVOEQZUKZRZWCWARZULZEUMZDUMV
|
|
FVLUSZDEVSVPVTMSNSUNWGWHDVFVLDACLDHTVJENDVNTUOWFWHEVFVLEACLEITVJENUGUOWDV
|
|
FBWEVLABCLWCJKUPWEVQWBVTRZULBVLUSVOWBVPVTDSESUQVQBVJWIVLVRVJENUHURUTURVAV
|
|
AUTVBVCVD $.
|
|
|
|
$( Version of ~ rexxp with bound-variable hypotheses. (Contributed by NM,
|
|
19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
rexxpf $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
|
|
( vv vw vu wrex wsb cv wceq vex nfsb cxp cbvrexsv rexbii nfv nfcv nfrexxy
|
|
nfs1v sbequ12 rexbidv cbvrex cop wa wex eqvinop nfbi sbhypf opth sylan9bb
|
|
wb sylbi exlimi rexxp 3bitr4ri bitri ) ACFGUAZOACLPZLVEOZBEGOZDFOZACLVEUB
|
|
BDMPZEGOZMFOVJENPZNGOZMFOVIVGVKVMMFVJENGUBUCVHVKDMFVHMUDVJDEGDGUEBDMUGZUF
|
|
DQZMQZRZBVJEGBDMUHZUIUJVFVLLMNFGLQZVPNQZUKZRVSVOEQZUKZRZWCWARZULZEUMZDUMV
|
|
FVLUSZDEVSVPVTMSNSUNWGWHDVFVLDACLDHTVJENDVNTUOWFWHEVFVLEACLEITVJENUGUOWDV
|
|
FBWEVLABCLWCJKUPWEVQWBVTRZULBVLUSVOWBVPVTDSESUQVQBVJWIVLVRVJENUHURUTURVAV
|
|
AUTVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d w x y A $. $d w x y z B $. $d w C $. $d w D $.
|
|
iunxpf.1 $e |- F/_ y C $.
|
|
iunxpf.2 $e |- F/_ z C $.
|
|
iunxpf.3 $e |- F/_ x D $.
|
|
iunxpf.4 $e |- ( x = <. y , z >. -> C = D ) $.
|
|
$( Indexed union on a cross product is equals a double indexed union. The
|
|
hypothesis specifies an implicit substitution. (Contributed by NM,
|
|
19-Dec-2008.) $)
|
|
iunxpf $p |- U_ x e. ( A X. B ) C = U_ y e. A U_ z e. B D $=
|
|
( vw cxp ciun cv wcel wrex nfcri cop eliun wceq eleq2d rexxpf bitri eqriv
|
|
rexbii 3bitr4i ) LADEMZFNZBDCEGNZNZLOZFPZAUHQULGPZCEQZBDQZULUIPULUKPZUMUN
|
|
ABCDEBLFHRCLFIRALGJRAOBOCOSUAFGULKUBUCAULUHFTUQULUJPZBDQUPBULDUJTURUOBDCU
|
|
LEGTUFUDUGUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y ph $.
|
|
opabbi2dv.1 $e |- Rel A $.
|
|
opabbi2dv.3 $e |- ( ph -> ( <. x , y >. e. A <-> ps ) ) $.
|
|
$( Deduce equality of a relation and an ordered-pair class builder.
|
|
Compare ~ abbi2dv . (Contributed by NM, 24-Feb-2014.) $)
|
|
opabbi2dv $p |- ( ph -> A = { <. x , y >. | ps } ) $=
|
|
( cv cop wcel copab wrel wceq opabid2 ax-mp opabbidv syl5eqr ) AECHDHIEJZ
|
|
CDKZBCDKELSEMFCDENOARBCDGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z A $. $d v w x y z B $.
|
|
relop.1 $e |- A e. _V $.
|
|
relop.2 $e |- B e. _V $.
|
|
$( A necessary and sufficient condition for a Kuratowski ordered pair to be
|
|
a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this
|
|
detail.) $)
|
|
relop $p |- ( Rel <. A , B >.
|
|
<-> E. x E. y ( A = { x } /\ B = { x , y } ) ) $=
|
|
( vz vw vv cop cvv cv wceq wa wex wi wcel bitri weq eqeq2d cxp wss df-rel
|
|
wrel csn cpr wal dfss2 vex elop elvv imbi12i jaob albii 19.26 eqeq1 eqcom
|
|
wo snex opeqsn syl6bb 2exbidv imbi12d spcv sneq cbvexv a9ev equcom 19.41v
|
|
exbii mpbi mpbiran eqid a1bi 3bitr2ri sylib prexg mp2an mpi opeqpr preqsn
|
|
idd eqtr2 simplbi syl dfsn2 preq2 syl5req syl5eq biimpd expd com12 adantr
|
|
anbi12d mpd expcom impd jaod syl5bi 2eximdv exlimiv imp sylbi simpr equid
|
|
syl2an jctl sylibr eqtr4d opeq12 spc2ev adantlr preq12 biimpa dfop jaodan
|
|
syl6eqr ex 3imtr4g ssrdv exlimivv impbii ) CDJZUDYCKKUAZUBZCALZUEZMZDYFBL
|
|
ZUFZMZNZBOAOZYCUCYEYMYEGLZCUEZMZYNYFYIJZMZBOAOZPZGUGZYNCDUFZMZYSPZGUGZNZY
|
|
MYEYNYCQZYNYDQZPZGUGZUUFGYCYDUHUUJYTUUDNZGUGUUFUUIUUKGUUIYPUUCURZYSPUUKUU
|
|
GUULUUHYSYNCDGUIEFUJZABYNUKULYPYSUUCUMRUNYTUUDGUORRUUACHLZUEZMZHOZUUBYQMZ
|
|
BOAOZYMUUEUUAYOYOMZABSZYHNZBOZAOZPZUUQYTUVEGYOCEUSYPYPUUTYSUVDYNYOYOUPYPY
|
|
RUVBABYPYRYOYQMZUVBYNYOYQUPUVFYQYOMUVBYOYQUQYFYICAUIZBUIZEUTRVAVBVCVDUUQY
|
|
HAOUVDUVEUUPYHHAHASZUUOYGCUUNYFVETVFUVCYHAUVCUVABOZYHBASZBOUVJBAVGUVKUVAB
|
|
BAVHVJVKUVAYHBVIVLVJUUTUVDYOVMVNVOVPUUEUUBUUBMZUUSUUBVMUUDUVLUUSPGUUBCKQD
|
|
KQUUBKQEFCDKKVQVRUUCUUCUVLYSUUSYNUUBUUBUPUUCYRUURABYNUUBYQUPVBVCVDVSUUQUU
|
|
SYMUUPUUSYMPHUUPUURYLABUURYLCYJMZDYGMZNZURZUUPYLUURYQUUBMUVPUUBYQUQYFYICD
|
|
UVGUVHEFVTRUUPYLYLUVOUUPYLWBUUPUVMUVNYLUVMUUPUVNYLPZUVMUUPNZUVAUVQUVRYJUU
|
|
OMZUVACYJUUOWCUVSUVABHSYFYIUUNUVGUVHHUIWAWDWEUVMUVAUVQPUUPUVAUVMUVQUVAUVM
|
|
UVNYLUVAUVOYLUVAUVMYHUVNYKUVAYJYGCUVAYGYFYFUFZYJYFWFZYFYIYFWGZWHTUVAYGYJD
|
|
UVAYGUVTYJUWAUWBWITWNWJWKWLWMWOWPWQWRWSWTXAXBXFXCYLYEABYLGYCYDYLUULYNUUNI
|
|
LZJZMZIOHOZUUGUUHYLUULUWFYLYPUWFUUCYHYPUWFYKYHYPNZYNYFYFJZMZUWFUWGYNYOUWH
|
|
YHYPXDYHUWHYOMZYPYHAASZYHNUWJYHUWKAXEXGYFYFCUVGUVGEUTXHWMXIUWEUWIHIYFYFUV
|
|
GUVGUVIIASNUWDUWHYNUUNUWCYFYFXJTXKWEXLYLUUCNZYRUWFUWLYNYGYJUFZYQYLUUCYNUW
|
|
MMYLUUBUWMYNCDYGYJXMTXNYFYIUVGUVHXOXQUWEYRHIYFYIUVGUVHUVIIBSNUWDYQYNUUNUW
|
|
CYFYIXJTXKWEXPXRUUMHIYNUKXSXTYAYBR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( For sets, the identity relation is the same as equality. (Contributed
|
|
by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
ideqg $p |- ( B e. V -> ( A _I B <-> A = B ) ) $=
|
|
( vx vy wcel cid wbr wceq cvv wa brrelexi adantl simpl jca eleq1 biimparc
|
|
reli elex cv syl eqeq1 eqeq2 df-id brabg pm5.21nd ) BCFZABGHZABIZAJFZUGKU
|
|
GUHKUJUGUHUJUGABGRLMUGUHNOUGUIKZUJUGUKACFZUJUIULUGABCPQACSUAUGUINODTZETZI
|
|
AUNIUIDEABJCGUMAUNUBUNBAUCDEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
ideq.1 $e |- B e. _V $.
|
|
$( For sets, the identity relation is the same as equality. (Contributed
|
|
by NM, 13-Aug-1995.) $)
|
|
ideq $p |- ( A _I B <-> A = B ) $=
|
|
( cvv wcel cid wbr wceq wb ideqg ax-mp ) BDEABFGABHICABDJK $.
|
|
$}
|
|
|
|
$( A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof
|
|
shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
ididg $p |- ( A e. V -> A _I A ) $=
|
|
( wcel cid wbr wceq eqid ideqg mpbiri ) ABCAADEAAFAGAABHI $.
|
|
|
|
$( Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
issetid $p |- ( A e. _V <-> A _I A ) $=
|
|
( cvv wcel cid wbr ididg reli brrelexi impbii ) ABCAADEABFAADGHI $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.) $)
|
|
coss1 $p |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) ) $=
|
|
( vx vy vz wss cv wbr wa wex copab id ssbrd anim2d eximdv ssopab2dv df-co
|
|
ccom 3sstr4g ) ABGZDHEHZCIZUBFHZAIZJZEKZDFLUCUBUDBIZJZEKZDFLACSBCSUAUGUJD
|
|
FUAUFUIEUAUEUHUCUAABUBUDUAMNOPQDFEACRDFEBCRT $.
|
|
|
|
$( Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.) $)
|
|
coss2 $p |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) ) $=
|
|
( vx vy vz wss cv wbr wa wex copab id ssbrd anim1d eximdv ssopab2dv df-co
|
|
ccom 3sstr4g ) ABGZDHZEHZAIZUCFHCIZJZEKZDFLUBUCBIZUEJZEKZDFLCASCBSUAUGUJD
|
|
FUAUFUIEUAUDUHUEUAABUBUCUAMNOPQDFECARDFECBRT $.
|
|
$}
|
|
|
|
$( Equality theorem for composition of two classes. (Contributed by NM,
|
|
3-Jan-1997.) $)
|
|
coeq1 $p |- ( A = B -> ( A o. C ) = ( B o. C ) ) $=
|
|
( wss wa ccom wceq coss1 anim12i eqss 3imtr4i ) ABDZBADZEACFZBCFZDZONDZEABG
|
|
NOGLPMQABCHBACHIABJNOJK $.
|
|
|
|
$( Equality theorem for composition of two classes. (Contributed by NM,
|
|
3-Jan-1997.) $)
|
|
coeq2 $p |- ( A = B -> ( C o. A ) = ( C o. B ) ) $=
|
|
( wss wa ccom wceq coss2 anim12i eqss 3imtr4i ) ABDZBADZECAFZCBFZDZONDZEABG
|
|
NOGLPMQABCHBACHIABJNOJK $.
|
|
|
|
${
|
|
coeq1i.1 $e |- A = B $.
|
|
$( Equality inference for composition of two classes. (Contributed by NM,
|
|
16-Nov-2000.) $)
|
|
coeq1i $p |- ( A o. C ) = ( B o. C ) $=
|
|
( wceq ccom coeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for composition of two classes. (Contributed by NM,
|
|
16-Nov-2000.) $)
|
|
coeq2i $p |- ( C o. A ) = ( C o. B ) $=
|
|
( wceq ccom coeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
coeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for composition of two classes. (Contributed by NM,
|
|
16-Nov-2000.) $)
|
|
coeq1d $p |- ( ph -> ( A o. C ) = ( B o. C ) ) $=
|
|
( wceq ccom coeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for composition of two classes. (Contributed by NM,
|
|
16-Nov-2000.) $)
|
|
coeq2d $p |- ( ph -> ( C o. A ) = ( C o. B ) ) $=
|
|
( wceq ccom coeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
coeq12i.1 $e |- A = B $.
|
|
coeq12i.2 $e |- C = D $.
|
|
$( Equality inference for composition of two classes. (Contributed by FL,
|
|
7-Jun-2012.) $)
|
|
coeq12i $p |- ( A o. C ) = ( B o. D ) $=
|
|
( ccom coeq1i coeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
coeq12d.1 $e |- ( ph -> A = B ) $.
|
|
coeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for composition of two classes. (Contributed by FL,
|
|
7-Jun-2012.) $)
|
|
coeq12d $p |- ( ph -> ( A o. C ) = ( B o. D ) ) $=
|
|
( ccom coeq1d coeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d y z w A $. $d y z w B $.
|
|
nfco.1 $e |- F/_ x A $.
|
|
nfco.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for function value. (Contributed by
|
|
NM, 1-Sep-1999.) $)
|
|
nfco $p |- F/_ x ( A o. B ) $=
|
|
( vy vw vz ccom cv wbr wa wex copab df-co nfcv nfbr nfan nfex nfopab
|
|
nfcxfr ) ABCIFJZGJZCKZUCHJZBKZLZGMZFHNFHGBCOUHFHAUGAGUDUFAAUBUCCAUBPEAUCP
|
|
ZQAUCUEBUIDAUEPQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $.
|
|
$( Ordered pair membership in a composition. (Contributed by NM,
|
|
24-Feb-2015.) $)
|
|
brcog $p |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <->
|
|
E. x ( A D x /\ x C B ) ) ) $=
|
|
( vy vz cv wbr wa wex ccom wceq breq1 breq2 bi2anan9 exbidv df-co brabga
|
|
) HJZAJZEKZUCIJZDKZLZAMBUCEKZUCCDKZLZAMHIBCDENFGUBBOZUECOZLUGUJAUKUDUHULU
|
|
FUIUBBUCEPUECUCDQRSHIADETUA $.
|
|
|
|
$( Ordered pair membership in a composition. (Contributed by NM,
|
|
27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.) $)
|
|
opelco2g $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( C o. D ) <->
|
|
E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) ) ) $=
|
|
( wcel wa ccom wbr cv wex cop brcog df-br anbi12i exbii 3bitr3g ) BFHCGHI
|
|
BCDEJZKBALZEKZUACDKZIZAMBCNTHBUANEHZUACNDHZIZAMABCDEFGOBCTPUDUGAUBUEUCUFB
|
|
UAEPUACDPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $. $d x X $.
|
|
$( Ordered pair membership in a composition. (Contributed by Thierry
|
|
Arnoux, 14-Jan-2018.) $)
|
|
brcogw $p |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) )
|
|
-> A ( C o. D ) B ) $=
|
|
( vx wcel w3a wbr wa cv wex ccom simpl1 simpl2 wceq breq2 breq1 3ad2antl3
|
|
anbi12d spcegv imp brcog biimpar syl21anc ) AEJZBFJZGHJZKAGDLZGBCLZMZMUIU
|
|
JAINZDLZUOBCLZMZIOZABCDPLZUIUJUKUNQUIUJUKUNRUKUIUNUSUJUKUNUSURUNIGHUOGSUP
|
|
ULUQUMUOGADTUOGBCUAUCUDUEUBUIUJMUTUSIABCDEFUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqbrrdva.1 $e |- ( ph -> A C_ ( C X. D ) ) $.
|
|
eqbrrdva.2 $e |- ( ph -> B C_ ( C X. D ) ) $.
|
|
eqbrrdva.3 $e |- ( ( ph /\ x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) $.
|
|
$( Deduction from extensionality principle for relations, given an
|
|
equivalence only on the relation's domain and range. (Contributed by
|
|
Thierry Arnoux, 28-Nov-2017.) $)
|
|
eqbrrdva $p |- ( ph -> A = B ) $=
|
|
( cvv cxp wss wrel syl6ss df-rel sylibr cv wcel wbr xpss wa ssbrd brxp wb
|
|
syl6ib 3expib pm5.21ndd eqbrrdv ) ABCDEADKKLZMDNADFGLZUJHFGUAZODPQAEUJMEN
|
|
AEUKUJIULOEPQABRZFSZCRZGSZUBZUMUODTZUMUOETZAURUMUOUKTZUQADUKUMUOHUCUMUOFG
|
|
UDZUFAUSUTUQAEUKUMUOIUCVAUFAUNUPURUSUEJUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
opelco.1 $e |- A e. _V $.
|
|
opelco.2 $e |- B e. _V $.
|
|
$( Binary relation on a composition. (Contributed by NM, 21-Sep-2004.)
|
|
(Revised by Mario Carneiro, 24-Feb-2015.) $)
|
|
brco $p |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) $=
|
|
( cvv wcel ccom wbr cv wa wex wb brcog mp2an ) BHICHIBCDEJKBALZEKRCDKMANO
|
|
FGABCDEHHPQ $.
|
|
|
|
$( Ordered pair membership in a composition. (Contributed by NM,
|
|
27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.) $)
|
|
opelco $p |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) ) $=
|
|
( cop ccom wcel wbr cv wa wex df-br brco bitr3i ) BCHDEIZJBCRKBALZEKSCDKM
|
|
ANBCROABCDEFGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) $)
|
|
cnvss $p |- ( A C_ B -> `' A C_ `' B ) $=
|
|
( vy vx wss wbr copab ccnv cop wcel ssel 3imtr4g ssopab2dv df-cnv 3sstr4g
|
|
cv df-br ) ABEZCPZDPZAFZDCGSTBFZDCGAHBHRUAUBDCRSTIZAJUCBJUAUBABUCKSTAQSTB
|
|
QLMDCANDCBNO $.
|
|
$}
|
|
|
|
$( Equality theorem for converse. (Contributed by NM, 13-Aug-1995.) $)
|
|
cnveq $p |- ( A = B -> `' A = `' B ) $=
|
|
( wss wa ccnv wceq cnvss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMN
|
|
FKOLPABGBAGHABIMNIJ $.
|
|
|
|
${
|
|
cnveqi.1 $e |- A = B $.
|
|
$( Equality inference for converse. (Contributed by NM, 23-Dec-2008.) $)
|
|
cnveqi $p |- `' A = `' B $=
|
|
( wceq ccnv cnveq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
cnveqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for converse. (Contributed by NM, 6-Dec-2013.) $)
|
|
cnveqd $p |- ( ph -> `' A = `' B ) $=
|
|
( wceq ccnv cnveq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
|
|
by NM, 24-Mar-1998.) $)
|
|
elcnv $p |- ( A e. `' R <-> E. x E. y ( A = <. x , y >. /\ y R x ) ) $=
|
|
( ccnv wcel cv wbr copab cop wceq wa wex df-cnv eleq2i elopab bitri ) CDE
|
|
ZFCBGZAGZDHZABIZFCTSJKUALBMAMRUBCABDNOUAABCPQ $.
|
|
|
|
$( Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
|
|
by NM, 11-Aug-2004.) $)
|
|
elcnv2 $p |- ( A e. `' R <->
|
|
E. x E. y ( A = <. x , y >. /\ <. y , x >. e. R ) ) $=
|
|
( ccnv wcel cv cop wceq wbr wa wex elcnv df-br anbi2i 2exbii bitri ) CDEF
|
|
CAGZBGZHIZSRDJZKZBLALTSRHDFZKZBLALABCDMUBUDABUAUCTSRDNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfcnv.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for converse. (Contributed by NM,
|
|
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfcnv $p |- F/_ x `' A $=
|
|
( vz vy ccnv cv wbr copab df-cnv nfcv nfbr nfopab nfcxfr ) ABFDGZEGZBHZED
|
|
IEDBJQEDAAOPBAOKCAPKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $.
|
|
$( Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
opelcnvg $p |- ( ( A e. C /\ B e. D ) ->
|
|
( <. A , B >. e. `' R <-> <. B , A >. e. R ) ) $=
|
|
( vy vx wcel wa ccnv wbr cop cv breq2 breq1 df-cnv brabg df-br 3bitr3g )
|
|
ACHBDHIABEJZKBAEKZABLTHBALEHFMZGMZEKUBAEKUAGFABCDTUCAUBENUBBAEOGFEPQABTRB
|
|
AERS $.
|
|
$}
|
|
|
|
$( The converse of a binary relation swaps arguments. Theorem 11 of [Suppes]
|
|
p. 61. (Contributed by NM, 10-Oct-2005.) $)
|
|
brcnvg $p |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) ) $=
|
|
( wcel wa cop ccnv wbr opelcnvg df-br 3bitr4g ) ACFBDFGABHEIZFBAHEFABNJBAEJ
|
|
ABCDEKABNLBAELM $.
|
|
|
|
${
|
|
opelcnv.1 $e |- A e. _V $.
|
|
opelcnv.2 $e |- B e. _V $.
|
|
$( Ordered-pair membership in converse. (Contributed by NM,
|
|
13-Aug-1995.) $)
|
|
opelcnv $p |- ( <. A , B >. e. `' R <-> <. B , A >. e. R ) $=
|
|
( cvv wcel cop ccnv wb opelcnvg mp2an ) AFGBFGABHCIGBAHCGJDEABFFCKL $.
|
|
|
|
$( The converse of a binary relation swaps arguments. Theorem 11 of
|
|
[Suppes] p. 61. (Contributed by NM, 13-Aug-1995.) $)
|
|
brcnv $p |- ( A `' R B <-> B R A ) $=
|
|
( cvv wcel ccnv wbr wb brcnvg mp2an ) AFGBFGABCHIBACIJDEABFFCKL $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z F $. $d y z V $. $d x y z $.
|
|
$( Move class substitution in and out of the converse of a function.
|
|
(Contributed by Thierry Arnoux, 8-Feb-2017.) $)
|
|
csbcnvg $p |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F ) $=
|
|
( vz vy wcel csb ccnv cv wbr copab wsbc sbcbrg csbconstg breq12d opabbidv
|
|
bitrd csbopabg df-cnv wceq a1i 3eqtr4rd csbeq2i syl6eqr ) BDGZABCHZIZABEJ
|
|
ZFJZCKZFELZHZABCIZHUFUKABMZFELUIUJUGKZFELZUMUHUFUOUPFEUFUOABUIHZABUJHZUGK
|
|
UPABUIUJDCNUFURUIUSUJUGABUIDOABUJDOPRQUKAFEBDSUHUQUAUFFEUGTUBUCABUNULFECT
|
|
UDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Distributive law of converse over class composition. Theorem 26 of
|
|
[Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) $)
|
|
cnvco $p |- `' ( A o. B ) = ( `' B o. `' A ) $=
|
|
( vx vy vz cv ccom wbr copab ccnv wa wex exancom brco brcnv anbi12i exbii
|
|
vex 3bitr4i opabbii df-cnv df-co 3eqtr4i ) CFZDFZABGZHZDCIUEEFZAJZHZUHUDB
|
|
JZHZKZELZDCIUFJUKUIGUGUNDCUDUHBHZUHUEAHZKELUPUOKZELUGUNUOUPEMEUDUEABCRZDR
|
|
ZNUMUQEUJUPULUOUEUHAUSERZOUHUDBUTUROPQSTDCUFUADCEUKUIUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $.
|
|
$( The converse of a class union is the (indexed) union of the converses of
|
|
its members. (Contributed by NM, 11-Aug-2004.) $)
|
|
cnvuni $p |- `' U. A = U_ x e. A `' x $=
|
|
( vy vz vw cuni ccnv cv ciun wcel wrex cop wa elcnv2 eluni2 anbi2i bitr4i
|
|
wceq wex rexcom4 r19.42v 2exbii rexbii exbii 3bitrri 3bitri eliun eqriv )
|
|
CBFZGZABAHZGZIZCHZUJJZUNULJZABKZUNUMJUOUNDHZEHZLRZUSURLZUIJZMZESDSUTVAUKJ
|
|
ZMZABKZESZDSZUQDEUNUINVCVFDEVCUTVDABKZMVFVBVIUTAVABOPUTVDABUAQUBUQVEESZDS
|
|
ZABKVJABKZDSVHUPVKABDEUNUKNUCVJADBTVLVGDVEAEBTUDUEUFAUNBULUGQUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring]
|
|
p. 24. (Contributed by NM, 28-Dec-1996.) $)
|
|
dfdm3 $p |- dom A = { x | E. y <. x , y >. e. A } $=
|
|
( cdm cv wbr wex cab cop wcel df-dm df-br exbii abbii eqtri ) CDAEZBEZCFZ
|
|
BGZAHPQICJZBGZAHABCKSUAARTBPQCLMNO $.
|
|
|
|
$( Alternate definition of range. Definition 4 of [Suppes] p. 60.
|
|
(Contributed by NM, 27-Dec-1996.) $)
|
|
dfrn2 $p |- ran A = { y | E. x x A y } $=
|
|
( crn ccnv cdm cv wbr wex cab df-rn df-dm vex brcnv exbii abbii 3eqtri )
|
|
CDCEZFBGZAGZRHZAIZBJTSCHZAIZBJCKBARLUBUDBUAUCASTCBMAMNOPQ $.
|
|
|
|
$( Alternate definition of range. Definition 6.5(2) of [TakeutiZaring]
|
|
p. 24. (Contributed by NM, 28-Dec-1996.) $)
|
|
dfrn3 $p |- ran A = { y | E. x <. x , y >. e. A } $=
|
|
( crn cv wbr wex cab cop wcel dfrn2 df-br exbii abbii eqtri ) CDAEZBEZCFZ
|
|
AGZBHPQICJZAGZBHABCKSUABRTAPQCLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) $)
|
|
elrn2g $p |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) ) $=
|
|
( vy cv cop wcel wex crn wceq opeq2 eleq1d exbidv dfrn3 elab2g ) AFZEFZGZ
|
|
CHZAIQBGZCHZAIEBCJDRBKZTUBAUCSUACRBQLMNAECOP $.
|
|
|
|
$( Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) $)
|
|
elrng $p |- ( A e. V -> ( A e. ran B <-> E. x x B A ) ) $=
|
|
( wcel crn cv cop wex wbr elrn2g df-br exbii syl6bbr ) BDEBCFEAGZBHCEZAIO
|
|
BCJZAIABCDKQPAOBCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of domain. (Contributed by NM, 28-Dec-1996.) $)
|
|
dfdm4 $p |- dom A = ran `' A $=
|
|
( vy vx cv wbr wex cab crn cdm vex brcnv exbii abbii dfrn2 df-dm 3eqtr4ri
|
|
ccnv ) BDZCDZAQZEZBFZCGSRAEZBFZCGTHAIUBUDCUAUCBRSABJCJKLMBCTNCBAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v $. $d w v A $.
|
|
dfdmf.1 $e |- F/_ x A $.
|
|
dfdmf.2 $e |- F/_ y A $.
|
|
$( Definition of domain, using bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 8-Mar-1995.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
dfdmf $p |- dom A = { x | E. y x A y } $=
|
|
( vw vv cdm cv wbr wex cab df-dm nfcv nfbr nfv breq2 cbvex abbii nfex weq
|
|
breq1 exbidv cbvab 3eqtri ) CHFIZGIZCJZGKZFLUFBIZCJZBKZFLAIZUJCJZBKZALFGC
|
|
MUIULFUHUKGBBUFUGCBUFNEBUGNOUKGPUGUJUFCQRSULUOFAUKABAUFUJCAUFNDAUJNOTUOFP
|
|
FAUAUKUNBUFUMUJCUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A w y $. $d B w y $. $d V w y $. $d x w y $.
|
|
$( Distribute proper substitution through the domain of a class.
|
|
(Contributed by Jim Kingdon, 8-Dec-2018.) $)
|
|
csbdmg $p |- ( A e. V -> [_ A / x ]_ dom B = dom [_ A / x ]_ B ) $=
|
|
( vy vw wcel cop wex cab csb cdm wsbc csbabg sbcex2 sbcel2g exbidv syl5bb
|
|
cv dfdm3 abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABESFSHZCGZFIZEJZKZUFABCKZGZF
|
|
IZEJZABCLZKUKLUEUJUHABMZEJUNUHAEBDNUEUPUMEUPUGABMZFIUEUMUGFABOUEUQULFABUF
|
|
CDPQRUAUBABUOUIEFCTUCEFUKTUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
eldmg $p |- ( A e. V -> ( A e. dom B <-> E. y A B y ) ) $=
|
|
( vx cv wbr wex cdm wceq breq1 exbidv df-dm elab2g ) EFZAFZCGZAHBPCGZAHEB
|
|
CIDOBJQRAOBPCKLEACMN $.
|
|
|
|
$( Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM,
|
|
27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
eldm2g $p |- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) ) $=
|
|
( wcel cdm cv wbr wex cop eldmg df-br exbii syl6bb ) BDEBCFEBAGZCHZAIBOJC
|
|
EZAIABCDKPQABOCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $.
|
|
eldm.1 $e |- A e. _V $.
|
|
$( Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by
|
|
NM, 2-Apr-2004.) $)
|
|
eldm $p |- ( A e. dom B <-> E. y A B y ) $=
|
|
( cvv wcel cdm cv wbr wex wb eldmg ax-mp ) BEFBCGFBAHCIAJKDABCELM $.
|
|
|
|
$( Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by
|
|
NM, 1-Aug-1994.) $)
|
|
eldm2 $p |- ( A e. dom B <-> E. y <. A , y >. e. B ) $=
|
|
( cvv wcel cdm cv cop wex wb eldm2g ax-mp ) BEFBCGFBAHICFAJKDABCELM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Subset theorem for domain. (Contributed by NM, 11-Aug-1994.) $)
|
|
dmss $p |- ( A C_ B -> dom A C_ dom B ) $=
|
|
( vx vy wss cdm cv cop wcel wex ssel eximdv vex eldm2 3imtr4g ssrdv ) ABE
|
|
ZCAFZBFZQCGZDGHZAIZDJUABIZDJTRITSIQUBUCDABUAKLDTACMZNDTBUDNOP $.
|
|
$}
|
|
|
|
$( Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) $)
|
|
dmeq $p |- ( A = B -> dom A = dom B ) $=
|
|
( wss wa cdm wceq dmss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMNFK
|
|
OLPABGBAGHABIMNIJ $.
|
|
|
|
${
|
|
dmeqi.1 $e |- A = B $.
|
|
$( Equality inference for domain. (Contributed by NM, 4-Mar-2004.) $)
|
|
dmeqi $p |- dom A = dom B $=
|
|
( wceq cdm dmeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
dmeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for domain. (Contributed by NM, 4-Mar-2004.) $)
|
|
dmeqd $p |- ( ph -> dom A = dom B ) $=
|
|
( wceq cdm dmeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $.
|
|
opeldm.1 $e |- A e. _V $.
|
|
opeldm.2 $e |- B e. _V $.
|
|
$( Membership of first of an ordered pair in a domain. (Contributed by NM,
|
|
30-Jul-1995.) $)
|
|
opeldm $p |- ( <. A , B >. e. C -> A e. dom C ) $=
|
|
( vy cop wcel cv wex cdm wceq opeq2 eleq1d spcev eldm2 sylibr ) ABGZCHZAF
|
|
IZGZCHZFJACKHUBSFBETBLUARCTBAMNOFACDPQ $.
|
|
|
|
$( Membership of first of a binary relation in a domain. (Contributed by
|
|
NM, 30-Jul-1995.) $)
|
|
breldm $p |- ( A R B -> A e. dom R ) $=
|
|
( wbr cop wcel cdm df-br opeldm sylbi ) ABCFABGCHACIHABCJABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $.
|
|
$( Membership of first of an ordered pair in a domain. (Contributed by Jim
|
|
Kingdon, 9-Jul-2019.) $)
|
|
opeldmg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. e. C -> A e. dom C ) ) $=
|
|
( vy wcel wa cop cv wex cdm wi wceq opeq2 eleq1d spcegv adantl wb eldm2g
|
|
adantr sylibrd ) ADGZBEGZHABIZCGZAFJZIZCGZFKZACLGZUDUFUJMUCUIUFFBEUGBNUHU
|
|
ECUGBAOPQRUCUKUJSUDFACDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x R $.
|
|
$( Membership of first of a binary relation in a domain. (Contributed by
|
|
NM, 21-Mar-2007.) $)
|
|
breldmg $p |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R ) $=
|
|
( vx wbr w3a cdm cv wex breq2 spcegv imp 3adant1 wb eldmg 3ad2ant1 mpbird
|
|
wcel ) ACTZBDTZABEGZHAEITZAFJZEGZFKZUBUCUGUAUBUCUGUFUCFBDUEBAELMNOUAUBUDU
|
|
GPUCFAECQRS $.
|
|
|
|
$( The domain of a union is the union of domains. Exercise 56(a) of
|
|
[Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened
|
|
by Andrew Salmon, 27-Aug-2011.) $)
|
|
dmun $p |- dom ( A u. B ) = ( dom A u. dom B ) $=
|
|
( vy vx cv wbr wex cab cun cdm wo unab brun exbii 19.43 abbii eqtri df-dm
|
|
bitr2i uneq12i 3eqtr4ri ) CEZDEZAFZDGZCHZUBUCBFZDGZCHZIZUBUCABIZFZDGZCHZA
|
|
JZBJZIUKJUJUEUHKZCHUNUEUHCLUQUMCUMUDUGKZDGUQULURDUBUCABMNUDUGDOSPQUOUFUPU
|
|
ICDARCDBRTCDUKRUA $.
|
|
|
|
$( The domain of an intersection belong to the intersection of domains.
|
|
Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) $)
|
|
dmin $p |- dom ( A i^i B ) C_ ( dom A i^i dom B ) $=
|
|
( vx vy cin cdm cv cop wcel wa wex 19.40 eldm2 elin exbii anbi12i 3imtr4i
|
|
vex bitri ssriv ) CABEZFZAFZBFZEZCGZDGHZAIZUGBIZJZDKZUHDKZUIDKZJZUFUBIZUF
|
|
UEIZUHUIDLUOUGUAIZDKUKDUFUACRZMUQUJDUGABNOSUPUFUCIZUFUDIZJUNUFUCUDNUSULUT
|
|
UMDUFAURMDUFBURMPSQT $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d y z B $.
|
|
$( The domain of an indexed union. (Contributed by Mario Carneiro,
|
|
26-Apr-2016.) $)
|
|
dmiun $p |- dom U_ x e. A B = U_ x e. A dom B $=
|
|
( vy vz ciun cdm cv cop wcel wex wrex rexcom4 eldm2 rexbii eliun 3bitr4ri
|
|
vex exbii 3bitr4i eqriv ) DABCFZGZABCGZFZDHZEHIZUBJZEKZUFUDJZABLZUFUCJUFU
|
|
EJUGCJZEKZABLULABLZEKUKUIULAEBMUJUMABEUFCDRZNOUHUNEAUGBCPSQEUFUBUONAUFBUD
|
|
PTUA $.
|
|
|
|
$d x A $.
|
|
$( The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|
|
(Contributed by NM, 3-Feb-2004.) $)
|
|
dmuni $p |- dom U. A = U_ x e. A dom x $=
|
|
( vy vz cuni cdm cv ciun cop wcel wex wrex excom ancom 19.41v vex 3bitr4i
|
|
wa eldm2 exbii anbi2i bitri eluni df-rex eliun eqriv ) CBEZFZABAGZFZHZCGZ
|
|
DGIZUGJZDKZULUJJZABLZULUHJULUKJUMUIJZUIBJZRZAKZDKZUSUPRZAKZUOUQVBUTDKZAKV
|
|
DUTDAMVEVCAURDKZUSRUSVFRVEVCVFUSNURUSDOUPVFUSDULUICPZSUAQTUBUNVADAUMBUCTU
|
|
PABUDQDULUGVGSAULBUJUEQUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The domain of a class of ordered pairs. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) $)
|
|
dmopab $p |- dom { <. x , y >. | ph } = { x | E. y ph } $=
|
|
( copab cdm wbr wex cab nfopab1 nfopab2 dfdmf cop wcel df-br opabid bitri
|
|
cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZCGZBHACGZBHBCUAABCIABCJKUEUFBUDACU
|
|
DUBUCLUAMAUBUCUANABCOPRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Upper bound for the domain of a restricted class of ordered pairs.
|
|
(Contributed by NM, 31-Jan-2004.) $)
|
|
dmopabss $p |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A $=
|
|
( cv wcel wa copab cdm wex cab dmopab 19.42v abbii ssab2 eqsstri ) BEDFZA
|
|
GZBCHIRCJZBKZDRBCLTQACJZGZBKDSUBBQACMNUABDOPP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The domain of a restricted class of ordered pairs. (Contributed by NM,
|
|
31-Jan-2004.) $)
|
|
dmopab3 $p |- ( A. x e. A E. y ph <->
|
|
dom { <. x , y >. | ( x e. A /\ ph ) } = A ) $=
|
|
( wex wral cv wcel wi wal wa wb copab cdm wceq df-ral pm4.71 albii dmopab
|
|
cab 19.42v abbii eqtri eqeq1i eqcom abeq2 3bitr2ri 3bitri ) ACEZBDFBGDHZU
|
|
IIZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCE
|
|
ZBTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
|
|
p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
dm0 $p |- dom (/) = (/) $=
|
|
( vx vy c0 cdm wceq cv wcel eq0 cop wex noel nex vex eldm2 mtbir mpgbir
|
|
wn ) CDZCEAFZRGZQAARHTSBFIZCGZBJUBBUAKLBSCAMNOP $.
|
|
|
|
$( The domain of the identity relation is the universe. (Contributed by
|
|
NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
dmi $p |- dom _I = _V $=
|
|
( vx vy cid cdm cvv wceq cv wcel eqv wbr wex a9ev ideq equcom bitri exbii
|
|
vex mpbir eldm mpgbir ) CDZEFAGZUAHZAAUAIUCUBBGZCJZBKZUFUDUBFZBKBALUEUGBU
|
|
EUBUDFUGUBUDBQMABNOPRBUBCAQSRT $.
|
|
|
|
$( The domain of the universe is the universe. (Contributed by NM,
|
|
8-Aug-2003.) $)
|
|
dmv $p |- dom _V = _V $=
|
|
( cvv cdm ssv cid dmi wss dmss ax-mp eqsstr3i eqssi ) ABZAKCADBZKEDAFLKFD
|
|
CDAGHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( An empty domain implies an empty range. (Contributed by NM,
|
|
21-May-1998.) $)
|
|
dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
|
|
( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii abeq1 3bitr4i
|
|
wn nbn eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i df-dm dfrn2 ) BDZCDZAE
|
|
ZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULRZBLZUORZC
|
|
LZVBVEVGUOCFZRVIVGULBFVJULBMUKBCUCUDUOCMUEVFVABUTULUINSOVHVDCVCUOUJNSOUFU
|
|
LBHPUOCHPQURUMHBCAUGTUSUPHBCAUHTQ $.
|
|
|
|
$( A relation is empty iff its domain is empty. (Contributed by NM,
|
|
15-Sep-2004.) $)
|
|
reldm0 $p |- ( Rel A -> ( A = (/) <-> dom A = (/) ) ) $=
|
|
( vx vy wrel c0 wceq cv cop wcel wb wal cdm rel0 eqrel mpan2 wn eq0 alnex
|
|
wex albii vex eldm2 xchbinxr noel nbn bitr3i bitr2i syl6bb ) ADZAEFZBGZCG
|
|
HZAIZULEIZJZCKZBKZALZEFZUIEDUJUQJMBCAENOUSUKURIZPZBKUQBURQVAUPBVAUMPZCKZU
|
|
PVCUMCSUTUMCRCUKABUAUBUCVBUOCUNUMULUDUETUFTUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z A $.
|
|
$( A domain is inhabited if and only if the range is inhabited.
|
|
(Contributed by Jim Kingdon, 15-Dec-2018.) $)
|
|
dmmrnm $p |- ( E. x x e. dom A <-> E. y y e. ran A ) $=
|
|
( vz cv cdm wcel wex wbr crn cab df-dm exbii abid bitri dfrn2 excom eleq1
|
|
eleq2i cbvexv 3bitr2i ) AEZCFZGZAHZUBDEZCIZDHZAHZUFCJZGZDHZBEZUJGZBHUEUBU
|
|
HAKZGZAHUIUDUPAUCUOUBADCLSMUPUHAUHANMOULUFUGAHZDKZGZDHZUIUKUSDUJURUFADCPS
|
|
MUTUQDHUIUSUQDUQDNMUGDAQOOUKUNDBUFUMUJRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z B $.
|
|
$( The domain of a cross product. Part of Theorem 3.13(x) of [Monk1]
|
|
p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
dmxpm $p |- ( E. x x e. B -> dom ( A X. B ) = A ) $=
|
|
( vz vy cv wcel wex cxp cdm wceq eleq1 cbvexv copab df-xp dmeqi ralrimivw
|
|
wa wral id dmopab3 sylib syl5eq sylbi ) AFZCGZAHDFZCGZDHZBCIZJZBKUFUHADUE
|
|
UGCLMUIUKEFBGUHREDNZJZBUJULEDBCOPUIUIEBSUMBKUIUIEBUITQUHEDBUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The domain of the intersection of two square cross products. Unlike
|
|
~ dmin , equality holds. (Contributed by NM, 29-Jan-2008.) $)
|
|
dmxpinm $p |- ( E. x x e. ( A i^i B ) ->
|
|
dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B ) ) $=
|
|
( cv cin wcel wex cxp cdm wceq inxp dmeqi a1i dmxpm eqtrd ) ADBCEZFAGZBBH
|
|
CCHEZIZPPHZIZPSUAJQRTBBCCKLMAPPNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The cross product of a class with itself is one-to-one. (Contributed by
|
|
Jim Kingdon, 8-Dec-2018.) $)
|
|
xpid11m $p |- ( ( E. x x e. A /\ E. x x e. B ) ->
|
|
( ( A X. A ) = ( B X. B ) <-> A = B ) ) $=
|
|
( cv wcel wex wa cxp wceq cdm dmxpm adantr dmeq sylan9req ad2antlr xpeq12
|
|
eqtrd ex anidms impbid1 ) ADZBEAFZUACEAFZGZBBHZCCHZIZBCIZUDUGUHUDUGGBUFJZ
|
|
CUDUGBUEJZUIUBUJBIUCABBKLUEUFMNUCUICIUBUGACCKOQRUHUGBCBCPST $.
|
|
$}
|
|
|
|
$( The domain of the double converse of a class (which doesn't have to be a
|
|
relation as in ~ dfrel2 ). (Contributed by NM, 8-Apr-2007.) $)
|
|
dmcnvcnv $p |- dom `' `' A = dom A $=
|
|
( cdm ccnv crn dfdm4 df-rn eqtr2i ) ABACZDHCBAEHFG $.
|
|
|
|
$( The range of the double converse of a class. (Contributed by NM,
|
|
8-Apr-2007.) $)
|
|
rncnvcnv $p |- ran `' `' A = ran A $=
|
|
( crn ccnv cdm df-rn dfdm4 eqtr2i ) ABACZDHCBAEHFG $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The first member of an ordered pair in a relation belongs to the domain
|
|
of the relation. (Contributed by NM, 28-Jul-2004.) $)
|
|
elreldm $p |- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A ) $=
|
|
( vx vy wrel wcel cint cdm cv cop wceq wex cvv cxp wss wi df-rel ssel vex
|
|
sylbi elvv syl6ib eleq1 opeldm syl6bi inteqd op1stb syl6eq eleq1d sylibrd
|
|
inteq exlimivv syli imp ) AEZBAFZBGZGZAHZFZUPUOBCIZDIZJZKZDLCLZUTUOUPBMMN
|
|
ZFZVEUOAVFOUPVGPAQAVFBRTCDBUAUBVDUPUTPCDVDUPVAUSFZUTVDUPVCAFVHBVCAUCVAVBA
|
|
CSZDSZUDUEVDURVAUSVDURVCGZGVAVDUQVKBVCUKUFVAVBVIVJUGUHUIUJULUMUN $.
|
|
$}
|
|
|
|
$( Equality theorem for range. (Contributed by NM, 29-Dec-1996.) $)
|
|
rneq $p |- ( A = B -> ran A = ran B ) $=
|
|
( wceq ccnv cdm crn cnveq dmeqd df-rn 3eqtr4g ) ABCZADZEBDZEAFBFKLMABGHAIBI
|
|
J $.
|
|
|
|
${
|
|
rneqi.1 $e |- A = B $.
|
|
$( Equality inference for range. (Contributed by NM, 4-Mar-2004.) $)
|
|
rneqi $p |- ran A = ran B $=
|
|
( wceq crn rneq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
rneqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for range. (Contributed by NM, 4-Mar-2004.) $)
|
|
rneqd $p |- ( ph -> ran A = ran B ) $=
|
|
( wceq crn rneq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Subset theorem for range. (Contributed by NM, 22-Mar-1998.) $)
|
|
rnss $p |- ( A C_ B -> ran A C_ ran B ) $=
|
|
( wss ccnv cdm crn cnvss dmss syl df-rn 3sstr4g ) ABCZADZEZBDZEZAFBFLMOCNPC
|
|
ABGMOHIAJBJK $.
|
|
|
|
$( The second argument of a binary relation belongs to its range.
|
|
(Contributed by NM, 29-Jun-2008.) $)
|
|
brelrng $p |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C ) $=
|
|
( wcel wbr w3a ccnv cdm crn wb brcnvg ancoms biimp3ar 3com12 syld3an3 df-rn
|
|
breldmg syl6eleqr ) ADFZBEFZABCGZHBCIZJZCKUAUBUCBAUDGZBUEFZUAUBUFUCUBUAUFUC
|
|
LBAEDCMNOUBUAUFUGBAEDUDSPQCRT $.
|
|
|
|
$( Membership of second member of an ordered pair in a range. (Contributed
|
|
by Jim Kingdon, 26-Jan-2019.) $)
|
|
opelrng $p |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C ) $=
|
|
( cop wcel wbr crn df-br brelrng syl3an3br ) ABFCGADGBEGABCHBCIGABCJABCDEKL
|
|
$.
|
|
|
|
${
|
|
brelrn.1 $e |- A e. _V $.
|
|
brelrn.2 $e |- B e. _V $.
|
|
$( The second argument of a binary relation belongs to its range.
|
|
(Contributed by NM, 13-Aug-2004.) $)
|
|
brelrn $p |- ( A C B -> B e. ran C ) $=
|
|
( cvv wcel wbr crn brelrng mp3an12 ) AFGBFGABCHBCIGDEABCFFJK $.
|
|
|
|
$( Membership of second member of an ordered pair in a range. (Contributed
|
|
by NM, 23-Feb-1997.) $)
|
|
opelrn $p |- ( <. A , B >. e. C -> B e. ran C ) $=
|
|
( cop wcel wbr crn df-br brelrn sylbir ) ABFCGABCHBCIGABCJABCDEKL $.
|
|
$}
|
|
|
|
$( The first argument of a binary relation belongs to its domain.
|
|
(Contributed by NM, 2-Jul-2008.) $)
|
|
releldm $p |- ( ( Rel R /\ A R B ) -> A e. dom R ) $=
|
|
( wrel wbr wa cvv wcel cdm brrelex brrelex2 simpr breldmg syl3anc ) CDZABCE
|
|
ZFAGHBGHPACIHABCJABCKOPLABGGCMN $.
|
|
|
|
$( The second argument of a binary relation belongs to its range.
|
|
(Contributed by NM, 2-Jul-2008.) $)
|
|
relelrn $p |- ( ( Rel R /\ A R B ) -> B e. ran R ) $=
|
|
( wrel wbr wa cvv wcel crn brrelex brrelex2 simpr brelrng syl3anc ) CDZABCE
|
|
ZFAGHBGHPBCIHABCJABCKOPLABCGGMN $.
|
|
|
|
${
|
|
$d x A $. $d x R $.
|
|
$( Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.) $)
|
|
releldmb $p |- ( Rel R -> ( A e. dom R <-> E. x A R x ) ) $=
|
|
( wrel cdm wcel cv wbr wex eldmg ibi releldm ex exlimdv impbid2 ) CDZBCEZ
|
|
FZBAGZCHZAIZRUAABCQJKPTRAPTRBSCLMNO $.
|
|
|
|
$( Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.) $)
|
|
relelrnb $p |- ( Rel R -> ( A e. ran R <-> E. x x R A ) ) $=
|
|
( wrel crn wcel cv wbr wex elrng ibi relelrn ex exlimdv impbid2 ) CDZBCEZ
|
|
FZAGZBCHZAIZRUAABCQJKPTRAPTRSBCLMNO $.
|
|
$}
|
|
|
|
${
|
|
releldm.1 $e |- Rel R $.
|
|
$( The first argument of a binary relation belongs to its domain.
|
|
(Contributed by NM, 28-Apr-2015.) $)
|
|
releldmi $p |- ( A R B -> A e. dom R ) $=
|
|
( wrel wbr cdm wcel releldm mpan ) CEABCFACGHDABCIJ $.
|
|
|
|
$( The second argument of a binary relation belongs to its range.
|
|
(Contributed by NM, 28-Apr-2015.) $)
|
|
relelrni $p |- ( A R B -> B e. ran R ) $=
|
|
( wrel wbr crn wcel relelrn mpan ) CEABCFBCGHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v $. $d w v A $.
|
|
dfrnf.1 $e |- F/_ x A $.
|
|
dfrnf.2 $e |- F/_ y A $.
|
|
$( Definition of range, using bound-variable hypotheses instead of distinct
|
|
variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
dfrnf $p |- ran A = { y | E. x x A y } $=
|
|
( vv vw crn cv wbr wex cab dfrn2 nfcv nfbr nfv breq1 cbvex abbii nfex weq
|
|
breq2 exbidv cbvab 3eqtri ) CHFIZGIZCJZFKZGLAIZUGCJZAKZGLUJBIZCJZAKZBLFGC
|
|
MUIULGUHUKFAAUFUGCAUFNDAUGNOUKFPUFUJUGCQRSULUOGBUKBABUJUGCBUJNEBUGNOTUOGP
|
|
GBUAUKUNAUGUMUJCUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
elrn.1 $e |- A e. _V $.
|
|
$( Membership in a range. (Contributed by NM, 10-Jul-1994.) $)
|
|
elrn2 $p |- ( A e. ran B <-> E. x <. x , A >. e. B ) $=
|
|
( vy cv cop wcel wex crn wceq opeq2 eleq1d exbidv dfrn3 elab2 ) AFZEFZGZC
|
|
HZAIQBGZCHZAIEBCJDRBKZTUBAUCSUACRBQLMNAECOP $.
|
|
|
|
$( Membership in a range. (Contributed by NM, 2-Apr-2004.) $)
|
|
elrn $p |- ( A e. ran B <-> E. x x B A ) $=
|
|
( crn wcel cv cop wex wbr elrn2 df-br exbii bitr4i ) BCEFAGZBHCFZAIOBCJZA
|
|
IABCDKQPAOBCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $.
|
|
nfrn.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for domain. (Contributed by NM,
|
|
30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfdm $p |- F/_ x dom A $=
|
|
( vy vz cdm cv wbr wex cab df-dm nfcv nfbr nfex nfab nfcxfr ) ABFDGZEGZBH
|
|
ZEIZDJDEBKTADSAEAQRBAQLCARLMNOP $.
|
|
|
|
$( Bound-variable hypothesis builder for range. (Contributed by NM,
|
|
1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfrn $p |- F/_ x ran A $=
|
|
( crn ccnv cdm df-rn nfcnv nfdm nfcxfr ) ABDBEZFBGAKABCHIJ $.
|
|
$}
|
|
|
|
$( Domain of an intersection. (Contributed by FL, 15-Oct-2012.) $)
|
|
dmiin $p |- dom |^|_ x e. A B C_ |^|_ x e. A dom B $=
|
|
( ciin cdm wss nfii1 nfdm ssiinf cv wcel iinss2 dmss syl mprgbir ) ABCDZEZA
|
|
BCEZDFQRFZABABRQAPABCGHIAJBKPCFSABCLPCMNO $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( The range of a class of ordered pairs. (Contributed by NM,
|
|
14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) $)
|
|
rnopab $p |- ran { <. x , y >. | ph } = { y | E. x ph } $=
|
|
( copab crn wbr wex cab nfopab1 nfopab2 dfrnf cop wcel df-br opabid bitri
|
|
cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZBGZCHABGZCHBCUAABCIABCJKUEUFCUDABU
|
|
DUBUCLUAMAUBUCUANABCOPRST $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z C $.
|
|
rnmpt.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The range of a function in maps-to notation. (Contributed by Scott
|
|
Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
rnmpt $p |- ran F = { y | E. x e. A y = B } $=
|
|
( cv wcel wceq wa copab crn wex cab wrex rnopab cmpt df-mpt eqtri rneqi
|
|
df-rex abbii 3eqtr4i ) AGCHBGDIZJZABKZLUEAMZBNELUDACOZBNUEABPEUFEACDQUFFA
|
|
BCDRSTUHUGBUDACUAUBUC $.
|
|
|
|
$( The range of a function in maps-to notation. (Contributed by Mario
|
|
Carneiro, 20-Feb-2015.) $)
|
|
elrnmpt $p |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) ) $=
|
|
( vy cv wceq wrex crn eqeq1 rexbidv rnmpt elab2g ) HIZCJZABKDCJZABKHDELFQ
|
|
DJRSABQDCMNAHBCEGOP $.
|
|
|
|
${
|
|
$d x A $. $d x D $.
|
|
elrnmpt1s.1 $e |- ( x = D -> B = C ) $.
|
|
$( Elementhood in an image set. (Contributed by Mario Carneiro,
|
|
12-Sep-2015.) $)
|
|
elrnmpt1s $p |- ( ( D e. A /\ C e. V ) -> C e. ran F ) $=
|
|
( wcel wceq wrex crn eqid cv eqeq2d rspcev mpan2 elrnmpt biimparc sylan
|
|
) EBJZDCKZABLZDGJZDFMJZUBDDKZUDDNUCUGAEBAOEKCDDIPQRUEUFUDABCDFGHSTUA $.
|
|
$}
|
|
|
|
$( Elementhood in an image set. (Contributed by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
elrnmpt1 $p |- ( ( x e. A /\ B e. V ) -> B e. ran F ) $=
|
|
( vz vy wcel cv crn csb wceq wa wex vex wb id csbeq1a nfcsb1v bitr2d wrex
|
|
eleq12d biantrud equcoms spcev df-rex nfv nfcri nfeq2 nfan eqeq2d anbi12d
|
|
cbvex bitri eqeq1 anbi2d exbidv syl5bb rnmpt elab2g syl5ibr impcom ) CEIZ
|
|
AJZBIZCDKZIZVFVHVDGJZAVIBLZIZCAVICLZMZNZGOZVNVFGVEAPVNVFQAGVEVIMZVFVKVNVP
|
|
VEVIBVJVPRAVIBSUCZVPVMVKAVICSZUDUAUEUFHJZCMZABUBZVOHCVGEWAVKVSVLMZNZGOZVT
|
|
VOWAVFVTNZAOWDVTABUGWEWCAGWEGUHVKWBAAGVJAVIBTUIAVSVLAVICTUJUKVPVFVKVTWBVQ
|
|
VPCVLVSVRULUMUNUOVTWCVNGVTWBVMVKVSCVLUPUQURUSAHBCDFUTVAVBVC $.
|
|
|
|
$( Membership in the range of a function. (Contributed by NM,
|
|
27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
elrnmptg $p |- ( A. x e. A B e. V ->
|
|
( C e. ran F <-> E. x e. A C = B ) ) $=
|
|
( vy crn wcel cv wceq wrex cab wral rnmpt eleq2i cvv wa syl wi eleq1 elex
|
|
wb r19.29 biimparc rexlimivw ex eqeq1 rexbidv elab3g syl5bb ) DEIZJDHKZCL
|
|
ZABMZHNZJZCFJZABOZDCLZABMZUMUQDAHBCEGPQUTVBDRJZUAURVBUDUTVBVCUTVBSUSVASZA
|
|
BMVCUSVAABUEVDVCABVDDFJZVCVAVEUSDCFUBUFDFUCTUGTUHUPVBHDRUNDLUOVAABUNDCUIU
|
|
JUKTUL $.
|
|
|
|
elrnmpti.2 $e |- B e. _V $.
|
|
$( Membership in the range of a function. (Contributed by NM,
|
|
30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
elrnmpti $p |- ( C e. ran F <-> E. x e. A C = B ) $=
|
|
( cvv wcel wral crn wceq wrex wb rgenw elrnmptg ax-mp ) CHIZABJDEKIDCLABM
|
|
NRABGOABCDEHFPQ $.
|
|
$}
|
|
|
|
$( The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
|
|
p. 36. (Contributed by NM, 4-Jul-1994.) $)
|
|
rn0 $p |- ran (/) = (/) $=
|
|
( c0 cdm wceq crn dm0 dm0rn0 mpbi ) ABACADACEAFG $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. (Contributed
|
|
by Mario Carneiro, 31-Aug-2015.) $)
|
|
dfiun3g $p |- ( A. x e. A B e. C ->
|
|
U_ x e. A B = U. ran ( x e. A |-> B ) ) $=
|
|
( vy wcel wral ciun wceq wrex cab cuni cmpt crn dfiun2g eqid rnmpt unieqi
|
|
cv syl6eqr ) CDFABGABCHESCIABJEKZLABCMZNZLAEBCDOUCUAAEBCUBUBPQRT $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
(Contributed by Mario Carneiro, 31-Aug-2015.) $)
|
|
dfiin3g $p |- ( A. x e. A B e. C
|
|
-> |^|_ x e. A B = |^| ran ( x e. A |-> B ) ) $=
|
|
( vy wcel wral ciin wceq wrex cab cint cmpt crn dfiin2g eqid rnmpt inteqi
|
|
cv syl6eqr ) CDFABGABCHESCIABJEKZLABCMZNZLAEBCDOUCUAAEBCUBUBPQRT $.
|
|
$}
|
|
|
|
${
|
|
dfiun3.1 $e |- B e. _V $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. (Contributed
|
|
by Mario Carneiro, 31-Aug-2015.) $)
|
|
dfiun3 $p |- U_ x e. A B = U. ran ( x e. A |-> B ) $=
|
|
( cvv wcel ciun cmpt crn cuni wceq dfiun3g cv a1i mprg ) CEFZABCGABCHIJKA
|
|
BABCELPAMBFDNO $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
(Contributed by Mario Carneiro, 31-Aug-2015.) $)
|
|
dfiin3 $p |- |^|_ x e. A B = |^| ran ( x e. A |-> B ) $=
|
|
( cvv wcel ciin cmpt crn cint wceq dfiin3g cv a1i mprg ) CEFZABCGABCHIJKA
|
|
BABCELPAMBFDNO $.
|
|
$}
|
|
|
|
${
|
|
$d V k $. $d X k $.
|
|
$( Express a relative indexed intersection as an intersection.
|
|
(Contributed by Stefan O'Rear, 22-Feb-2015.) $)
|
|
riinint $p |- ( ( X e. V /\ A. k e. I S C_ X ) ->
|
|
( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) ) $=
|
|
( wcel wss wral wa ciin cin cmpt crn cint csn cun cvv wceq ssexg expcom
|
|
ralimdv imp dfiin3g syl ineq2d intun intsng adantr ineq1d syl5eq eqtr4d )
|
|
EDFZAEGZBCHZIZEBCAJZKEBCALMZNZKZEOZUQPNZUOUPUREUOAQFZBCHZUPURRULUNVCULUMV
|
|
BBCUMULVBAEDSTUAUBBCAQUCUDUEUOVAUTNZURKUSUTUQUFUOVDEURULVDERUNEDUGUHUIUJU
|
|
K $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A relation is empty iff its range is empty. (Contributed by NM,
|
|
15-Sep-2004.) $)
|
|
relrn0 $p |- ( Rel A -> ( A = (/) <-> ran A = (/) ) ) $=
|
|
( wrel c0 wceq cdm crn reldm0 dm0rn0 syl6bb ) ABACDAECDAFCDAGAHI $.
|
|
|
|
$( The domain and range of a class are included in its double union.
|
|
(Contributed by NM, 13-May-2008.) $)
|
|
dmrnssfld $p |- ( dom A u. ran A ) C_ U. U. A $=
|
|
( vx vy cdm crn cuni cv cop wex vex eldm2 cpr prid1 wss sseld mpi exlimiv
|
|
wcel sylbi ssriv uniop uniopel syl5eqelr elssuni syl elrn2 prid2 unssi )
|
|
ADZAEZAFZFZBUIULBGZUIRUMCGZHZARZCIUMULRZCUMABJZKUPUQCUPUMUMUNLZRUQUMUNURM
|
|
UPUSULUMUPUSUKRUSULNUPUSUOFUKUMUNURCJZUAUMUNAURUTUBUCUSUKUDUEZOPQSTCUJULU
|
|
NUJRUPBIUNULRZBUNAUTUFUPVBBUPUNUSRVBUMUNUTUGUPUSULUNVAOPQSTUH $.
|
|
$}
|
|
|
|
$( The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
|
|
(Contributed by NM, 7-Apr-1995.) $)
|
|
dmexg $p |- ( A e. V -> dom A e. _V ) $=
|
|
( wcel cuni cvv cdm uniexg wss crn ssun1 dmrnssfld sstri ssexg mpan 3syl
|
|
cun ) ABCADZECQDZECZAFZECZABGQEGTRHSUATTAIZPRTUBJAKLTREMNO $.
|
|
|
|
$( The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26.
|
|
Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM,
|
|
31-Mar-1995.) $)
|
|
rnexg $p |- ( A e. V -> ran A e. _V ) $=
|
|
( wcel cuni cvv crn uniexg wss cdm ssun2 dmrnssfld sstri ssexg mpan 3syl
|
|
cun ) ABCADZECQDZECZAFZECZABGQEGTRHSUATAIZTPRTUBJAKLTREMNO $.
|
|
|
|
${
|
|
dmex.1 $e |- A e. _V $.
|
|
$( The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring]
|
|
p. 26. (Contributed by NM, 7-Jul-2008.) $)
|
|
dmex $p |- dom A e. _V $=
|
|
( cvv wcel cdm dmexg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26.
|
|
Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM,
|
|
7-Jul-2008.) $)
|
|
rnex $p |- ran A e. _V $=
|
|
( cvv wcel crn rnexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( The identity function is a proper class. This means, for example, that we
|
|
cannot use it as a member of the class of continuous functions unless it
|
|
is restricted to a set. (Contributed by NM, 1-Jan-2007.) $)
|
|
iprc $p |- -. _I e. _V $=
|
|
( cid cvv wcel cdm vprc dmi eleq1i mtbir dmexg mto ) ABCADZBCZLBBCEKBBFGHAB
|
|
IJ $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by
|
|
NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
dmcoss $p |- dom ( A o. B ) C_ dom B $=
|
|
( vx vy vz ccom cdm cv cop wcel wex wbr nfe1 wa exsimpl vex opelco cbvexv
|
|
breq2 3imtr4i exlimi eldm2 eldm ssriv ) CABFZGZBGZCHZDHZIUEJZDKUHUIBLZDKZ
|
|
UHUFJUHUGJUJULDUKDMUHEHZBLZUMUIALZNEKUNEKUJULUNUOEOEUHUIABCPZDPQUKUNDEUIU
|
|
MUHBSRTUADUHUEUPUBDUHBUPUCTUD $.
|
|
$}
|
|
|
|
$( Range of a composition. (Contributed by NM, 19-Mar-1998.) $)
|
|
rncoss $p |- ran ( A o. B ) C_ ran A $=
|
|
( ccnv ccom cdm crn dmcoss df-rn cnvco dmeqi eqtri 3sstr4i ) BCZACZDZEZNEAB
|
|
DZFZAFMNGRQCZEPQHSOABIJKAHL $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof
|
|
shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
dmcosseq $p |- ( ran B C_ dom A -> dom ( A o. B ) = dom B ) $=
|
|
( vx vy vz crn cdm wss ccom dmcoss a1i cv wbr wex wcel wi vex eldm eximdv
|
|
syl6ibr wa ssel elrn imbi12i 19.8a imim1i pm3.2 sylcom sylbi excom opelco
|
|
cop syl exbii eldm2 3imtr4g ssrdv eqssd ) BFZAGZHZABIZGZBGZVCVDHVAABJKVAC
|
|
VDVCVACLZDLZBMZDNZVEELZULVBOZENZVEVDOVEVCOVAVHVGVFVIAMZUAZDNZENZVKVAVHVME
|
|
NZDNVOVAVGVPDVAVFUSOZVFUTOZPZVGVPPZUSUTVFUBVSVGCNZVLENZPZVTVQWAVRWBCVFBDQ
|
|
ZUCEVFAWDRUDWCVGWBVPVGWAWBVGCUEUFVGVLVMEVGVLUGSUHUIUMSVMEDUJTVJVNEDVEVIAB
|
|
CQZEQUKUNTDVEBWEREVEVBWEUOUPUQUR $.
|
|
|
|
$( Domain of a composition. (Contributed by NM, 19-Mar-1998.) $)
|
|
dmcoeq $p |- ( dom A = ran B -> dom ( A o. B ) = dom B ) $=
|
|
( cdm crn wceq wss ccom eqimss2 dmcosseq syl ) ACZBDZELKFABGCBCELKHABIJ
|
|
$.
|
|
$}
|
|
|
|
$( Range of a composition. (Contributed by NM, 19-Mar-1998.) $)
|
|
rncoeq $p |- ( dom A = ran B -> ran ( A o. B ) = ran A ) $=
|
|
( ccnv cdm crn wceq ccom dmcoeq eqcom df-rn dfdm4 eqeq12i bitri cnvco dmeqi
|
|
eqtri 3imtr4i ) BCZDZACZEZFZRTGZDZTDZFADZBEZFZABGZEZAEZFRTHUHUGUFFUBUFUGIUG
|
|
SUFUABJAKLMUJUDUKUEUJUICZDUDUIJULUCABNOPAJLQ $.
|
|
|
|
$( Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.) $)
|
|
reseq1 $p |- ( A = B -> ( A |` C ) = ( B |` C ) ) $=
|
|
( wceq cvv cxp cin cres ineq1 df-res 3eqtr4g ) ABDACEFZGBLGACHBCHABLIACJBCJ
|
|
K $.
|
|
|
|
$( Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.) $)
|
|
reseq2 $p |- ( A = B -> ( C |` A ) = ( C |` B ) ) $=
|
|
( wceq cvv cxp cin cres xpeq1 ineq2d df-res 3eqtr4g ) ABDZCAEFZGCBEFZGCAHCB
|
|
HMNOCABEIJCAKCBKL $.
|
|
|
|
${
|
|
reseqi.1 $e |- A = B $.
|
|
$( Equality inference for restrictions. (Contributed by NM,
|
|
21-Oct-2014.) $)
|
|
reseq1i $p |- ( A |` C ) = ( B |` C ) $=
|
|
( wceq cres reseq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for restrictions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
reseq2i $p |- ( C |` A ) = ( C |` B ) $=
|
|
( wceq cres reseq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
reseqi.2 $e |- C = D $.
|
|
$( Equality inference for restrictions. (Contributed by NM,
|
|
21-Oct-2014.) $)
|
|
reseq12i $p |- ( A |` C ) = ( B |` D ) $=
|
|
( cres reseq1i reseq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
reseqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for restrictions. (Contributed by NM,
|
|
21-Oct-2014.) $)
|
|
reseq1d $p |- ( ph -> ( A |` C ) = ( B |` C ) ) $=
|
|
( wceq cres reseq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for restrictions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
reseq2d $p |- ( ph -> ( C |` A ) = ( C |` B ) ) $=
|
|
( wceq cres reseq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
reseqd.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for restrictions. (Contributed by NM,
|
|
21-Oct-2014.) $)
|
|
reseq12d $p |- ( ph -> ( A |` C ) = ( B |` D ) ) $=
|
|
( cres reseq1d reseq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
nfres.1 $e |- F/_ x A $.
|
|
nfres.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for restriction. (Contributed by NM,
|
|
15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.) $)
|
|
nfres $p |- F/_ x ( A |` B ) $=
|
|
( cres cvv cxp cin df-res nfcv nfxp nfin nfcxfr ) ABCFBCGHZIBCJABODACGEAG
|
|
KLMN $.
|
|
$}
|
|
|
|
$( Distribute proper substitution through the restriction of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbresg $p |- ( A e. V -> [_ A / x ]_ ( B |` C ) =
|
|
( [_ A / x ]_ B |` [_ A / x ]_ C ) ) $=
|
|
( wcel cvv cxp cin cres csbing csbxpg csbconstg xpeq2d eqtrd ineq2d csbeq2i
|
|
csb df-res 3eqtr4g ) BEFZABCDGHZIZRZABCRZABDRZGHZIZABCDJZRUEUFJUAUDUEABUBRZ
|
|
IUHABECUBKUAUJUGUEUAUJUFABGRZHUGABDGELUAUKGUFABGEMNOPOABUIUCCDSQUEUFST $.
|
|
|
|
$( A restriction to the empty set is empty. (Contributed by NM,
|
|
12-Nov-1994.) $)
|
|
res0 $p |- ( A |` (/) ) = (/) $=
|
|
( c0 cres cvv cxp cin df-res 0xp ineq2i in0 3eqtri ) ABCABDEZFABFBABGLBADHI
|
|
AJK $.
|
|
|
|
${
|
|
opelres.1 $e |- B e. _V $.
|
|
$( Ordered pair membership in a restriction. Exercise 13 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) $)
|
|
opelres $p |- ( <. A , B >. e. ( C |` D ) <->
|
|
( <. A , B >. e. C /\ A e. D ) ) $=
|
|
( cop cres wcel cvv cxp cin wa df-res eleq2i elin opelxp mpbiran2 anbi2i
|
|
3bitri ) ABFZCDGZHTCDIJZKZHTCHZTUBHZLUDADHZLUAUCTCDMNTCUBOUEUFUDUEUFBIHEA
|
|
BDIPQRS $.
|
|
|
|
$( Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.) $)
|
|
brres $p |- ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) $=
|
|
( cop cres wcel wa wbr opelres df-br anbi1i 3bitr4i ) ABFZCDGZHOCHZADHZIA
|
|
BPJABCJZRIABCDEKABPLSQRABCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y D $.
|
|
$( Ordered pair membership in a restriction. Exercise 13 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.) $)
|
|
opelresg $p |- ( B e. V -> ( <. A , B >. e. ( C |` D ) <->
|
|
( <. A , B >. e. C /\ A e. D ) ) ) $=
|
|
( vy cv cop cres wcel wa wceq opeq2 eleq1d anbi1d vex opelres vtoclbg ) A
|
|
FGZHZCDIZJTCJZADJZKABHZUAJUDCJZUCKFBESBLZTUDUASBAMZNUFUBUEUCUFTUDCUGNOASC
|
|
DFPQR $.
|
|
|
|
$( Binary relation on a restriction. (Contributed by Mario Carneiro,
|
|
4-Nov-2015.) $)
|
|
brresg $p |- ( B e. V -> ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) ) $=
|
|
( wcel cop cres wa wbr opelresg df-br anbi1i 3bitr4g ) BEFABGZCDHZFOCFZAD
|
|
FZIABPJABCJZRIABCDEKABPLSQRABCLMN $.
|
|
$}
|
|
|
|
${
|
|
opres.1 $e |- B e. _V $.
|
|
$( Ordered pair membership in a restriction when the first member belongs
|
|
to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof
|
|
shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
opres $p |- ( A e. D ->
|
|
( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) ) $=
|
|
( cop cres wcel opelres rbaib ) ABFZCDGHKCHADHABCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( A restricted identity relation is equivalent to equality in its domain.
|
|
(Contributed by NM, 30-Apr-2004.) $)
|
|
resieq $p |- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) ) $=
|
|
( vx wcel cid cres wbr wceq wb cv wi breq2 eqeq2 bibi12d imbi2d cop opres
|
|
vex df-br ideq bitr3i 3bitr4g vtoclg impcom ) CAEBAEZBCFAGZHZBCIZJZUFBDKZ
|
|
UGHZBUKIZJZLUFUJLDCAUKCIZUNUJUFUOULUHUMUIUKCBUGMUKCBNOPUFBUKQZUGEUPFEZULU
|
|
MBUKFADSZRBUKUGTUMBUKFHUQBUKURUABUKFTUBUCUDUE $.
|
|
$}
|
|
|
|
$( ` <. A , A >. ` belongs to a restriction of the identity class iff ` A `
|
|
belongs to the restricting class. (Contributed by FL, 27-Oct-2008.)
|
|
(Revised by NM, 30-Mar-2016.) $)
|
|
opelresi $p |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) ) $=
|
|
( wcel cop cid cres wa opelresg wbr ididg df-br sylib biantrurd bitr4d ) AC
|
|
DZAAEZFBGDQFDZABDZHSAAFBCIPRSPAAFJRACKAAFLMNO $.
|
|
|
|
$( The restriction of a restriction. (Contributed by NM, 27-Mar-2008.) $)
|
|
resres $p |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) ) $=
|
|
( cres cvv cxp cin df-res ineq1i xpindir ineq2i inass 3eqtr4ri 3eqtri ) ABD
|
|
ZCDOCEFZGABEFZGZPGZABCGZDZOCHORPABHIATEFZGAQPGZGUASUBUCABCEJKATHAQPLMN $.
|
|
|
|
$( Distributive law for restriction over union. Theorem 31 of [Suppes]
|
|
p. 65. (Contributed by NM, 30-Sep-2002.) $)
|
|
resundi $p |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) ) $=
|
|
( cun cvv cxp cin cres xpundir ineq2i indi eqtri df-res uneq12i 3eqtr4i ) A
|
|
BCDZEFZGZABEFZGZACEFZGZDZAPHABHZACHZDRASUADZGUCQUFABCEIJASUAKLAPMUDTUEUBABM
|
|
ACMNO $.
|
|
|
|
$( Distributive law for restriction over union. (Contributed by NM,
|
|
23-Sep-2004.) $)
|
|
resundir $p |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) ) $=
|
|
( cun cvv cxp cin cres indir df-res uneq12i 3eqtr4i ) ABDZCEFZGANGZBNGZDMCH
|
|
ACHZBCHZDABNIMCJQORPACJBCJKL $.
|
|
|
|
$( Class restriction distributes over intersection. (Contributed by FL,
|
|
6-Oct-2008.) $)
|
|
resindi $p |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) ) $=
|
|
( cin cvv cxp cres xpindir ineq2i inindi eqtri df-res ineq12i 3eqtr4i ) ABC
|
|
DZEFZDZABEFZDZACEFZDZDZAOGABGZACGZDQARTDZDUBPUEABCEHIARTJKAOLUCSUDUAABLACLM
|
|
N $.
|
|
|
|
$( Class restriction distributes over intersection. (Contributed by NM,
|
|
18-Dec-2008.) $)
|
|
resindir $p |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) ) $=
|
|
( cin cvv cxp cres inindir df-res ineq12i 3eqtr4i ) ABDZCEFZDAMDZBMDZDLCGAC
|
|
GZBCGZDABMHLCIPNQOACIBCIJK $.
|
|
|
|
$( Move intersection into class restriction. (Contributed by NM,
|
|
18-Dec-2008.) $)
|
|
inres $p |- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C ) $=
|
|
( cin cvv cxp cres inass df-res ineq2i 3eqtr4ri ) ABDZCEFZDABMDZDLCGABCGZDA
|
|
BMHLCIONABCIJK $.
|
|
|
|
${
|
|
$d x C $.
|
|
$( Distribution of restriction over indexed union. (Contributed by Mario
|
|
Carneiro, 29-May-2015.) $)
|
|
resiun1 $p |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C ) $=
|
|
( cvv cxp cin ciun cres iunin2 wceq cv wcel df-res incom iuneq2i 3eqtr4ri
|
|
eqtri a1i ) ABDEFZCGZHTABCHZGZABCDIZHUBDIZABTCJABUDUAUDUAKALBMUDCTGUACDNC
|
|
TORSPUEUBTGUCUBDNUBTORQ $.
|
|
|
|
$( Distribution of restriction over indexed union. (Contributed by Mario
|
|
Carneiro, 29-May-2015.) $)
|
|
resiun2 $p |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B ) $=
|
|
( ciun cres cvv cxp cin df-res wceq cv a1i iuneq2i xpiundir ineq2i iunin2
|
|
wcel eqtr4i ) DABCEZFDTGHZIZABDCFZEZDTJUDABDCGHZIZEZUBABUCUFUCUFKALBRDCJM
|
|
NUBDABUEEZIUGUAUHDABCGOPABDUEQSSS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
dmres $p |- dom ( A |` B ) = ( B i^i dom A ) $=
|
|
( vx vy cdm cin cres cv wcel cop wex wa eldm2 19.41v opelres exbii anbi1i
|
|
vex 3bitr4i bitr2i ineqri incom eqtr3i ) AEZBFABGZEZBUDFCUDBUFCHZUFIUGDHZ
|
|
JZUEIZDKZUGUDIZUGBIZLZDUGUECRZMUIAIZUMLZDKUPDKZUMLUKUNUPUMDNUJUQDUGUHABDR
|
|
OPULURUMDUGAUOMQSTUAUDBUBUC $.
|
|
$}
|
|
|
|
$( A domain restricted to a subclass equals the subclass. (Contributed by
|
|
NM, 2-Mar-1997.) $)
|
|
ssdmres $p |- ( A C_ dom B <-> dom ( B |` A ) = A ) $=
|
|
( cdm wss cin wceq cres df-ss dmres eqeq1i bitr4i ) ABCZDALEZAFBAGCZAFALHNM
|
|
ABAIJK $.
|
|
|
|
$( The domain of a restriction to a set exists. (Contributed by NM,
|
|
7-Apr-1995.) $)
|
|
dmresexg $p |- ( B e. V -> dom ( A |` B ) e. _V ) $=
|
|
( wcel cres cdm cin cvv dmres inex1g syl5eqel ) BCDABEFBAFZGHABIBLCJK $.
|
|
|
|
$( A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 2-Aug-1994.) $)
|
|
resss $p |- ( A |` B ) C_ A $=
|
|
( cres cvv cxp cin df-res inss1 eqsstri ) ABCABDEZFAABGAJHI $.
|
|
|
|
$( Commutative law for restriction. (Contributed by NM, 27-Mar-1998.) $)
|
|
rescom $p |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B ) $=
|
|
( cin cres incom reseq2i resres 3eqtr4i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
|
|
ACBHI $.
|
|
|
|
$( Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.) $)
|
|
ssres $p |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) ) $=
|
|
( wss cvv cxp cin cres ssrin df-res 3sstr4g ) ABDACEFZGBLGACHBCHABLIACJBCJK
|
|
$.
|
|
|
|
$( Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
ssres2 $p |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) ) $=
|
|
( wss cvv cxp cin cres xpss1 sslin syl df-res 3sstr4g ) ABDZCAEFZGZCBEFZGZC
|
|
AHCBHNOQDPRDABEIOQCJKCALCBLM $.
|
|
|
|
$( A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
relres $p |- Rel ( A |` B ) $=
|
|
( cres cvv cxp wss wrel cin df-res inss2 eqsstri relxp relss mp2 ) ABCZBDEZ
|
|
FPGOGOAPHPABIAPJKBDLOPMN $.
|
|
|
|
$( Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 9-Aug-1994.) $)
|
|
resabs1 $p |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) ) $=
|
|
( wss cres cin resres wceq sseqin2 reseq2 sylbi syl5eq ) BCDZACEBEACBFZEZAB
|
|
EZACBGMNBHOPHBCINBAJKL $.
|
|
|
|
$( Absorption law for restriction. (Contributed by NM, 27-Mar-1998.) $)
|
|
resabs2 $p |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) ) $=
|
|
( wss cres rescom resabs1 syl5eq ) BCDABEZCEACEBEIABCFABCGH $.
|
|
|
|
$( Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.) $)
|
|
residm $p |- ( ( A |` B ) |` B ) = ( A |` B ) $=
|
|
( wss cres wceq ssid resabs2 ax-mp ) BBCABDZBDIEBFABBGH $.
|
|
|
|
$( A restriction to an image. (Contributed by NM, 29-Sep-2004.) $)
|
|
resima $p |- ( ( A |` B ) " B ) = ( A " B ) $=
|
|
( cres crn cima residm rneqi df-ima 3eqtr4i ) ABCZBCZDJDJBEABEKJABFGJBHABHI
|
|
$.
|
|
|
|
$( Image under a restricted class. (Contributed by FL, 31-Aug-2009.) $)
|
|
resima2 $p |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) ) $=
|
|
( wss cres cima crn df-ima cin resres rneqi wceq df-ss incom reseq2d reseq2
|
|
a1i rneqd syl6eqr syl5eq eqtrd sylbi ) BCDZACEZBFUDBEZGZABFZUDBHUCUFACBIZEZ
|
|
GZUGUEUIACBJKUCBCIZBLZUJUGLBCMULUJAUKEZGZUGULUIUMULUHUKAUHUKLULCBNQORULUNAB
|
|
EZGUGULUMUOUKBAPRABHSUAUBTT $.
|
|
|
|
$( Restriction of a constant function (or other cross product). (Contributed
|
|
by Stefan O'Rear, 24-Jan-2015.) $)
|
|
xpssres $p |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) ) $=
|
|
( wss cxp cres cin df-res inxp incom inv1 xpeq12i 3eqtri wceq biimpi xpeq1d
|
|
cvv df-ss syl5eq ) CADZABEZCFZCAGZBEZCBEUBUACQEGACGZBQGZEUDUACHABCQIUEUCUFB
|
|
ACJBKLMTUCCBTUCCNCAROPS $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in a restriction. (Contributed by Scott Fenton,
|
|
17-Mar-2011.) $)
|
|
elres $p |- ( A e. ( B |` C )
|
|
<-> E. x e. C E. y ( A = <. x , y >. /\ <. x , y >. e. B ) ) $=
|
|
( cres wcel cv cop wceq wex wrex wrel relres elrel mpan eleq1 biimpd vex
|
|
wa opelres biimpi ancomd syl6com ancld an12 syl6ib 2eximdv rexcom4 df-rex
|
|
mpd exbii excom 3bitri sylibr simplbi2com biimprd exlimdv rexlimiv impbii
|
|
syl9 impd ) CDEFZGZCAHZBHZIZJZVGDGZTZBKZAELZVDVEEGZVJTZBKAKZVLVDVHBKAKZVO
|
|
VCMVDVPDENABCVCOPVDVHVNABVDVHVHVMVITZTVNVDVHVQVHVDVGVCGZVQVHVDVRCVGVCQZRV
|
|
RVIVMVRVIVMTVEVFDEBSUAZUBUCUDUEVHVMVIUFUGUHUKVLVJAELZBKVNAKZBKVOVJABEUIWA
|
|
WBBVJAEUJULVNBAUMUNUOVKVDAEVMVJVDBVMVHVIVDVMVIVRVHVDVRVIVMVTUPVHVDVRVSUQV
|
|
AVBURUSUT $.
|
|
|
|
${
|
|
elsnres.1 $e |- C e. _V $.
|
|
$( Memebership in restriction to a singleton. (Contributed by Scott
|
|
Fenton, 17-Mar-2011.) $)
|
|
elsnres $p |- ( A e. ( B |` { C } )
|
|
<-> E. y ( A = <. C , y >. /\ <. C , y >. e. B ) ) $=
|
|
( vx csn cres wcel cv cop wceq wa wex elres rexcom4 opeq1 eqeq2d eleq1d
|
|
wrex anbi12d rexsn exbii 3bitri ) BCDGZHIBFJZAJZKZLZUHCIZMZANFUETUKFUET
|
|
ZANBDUGKZLZUMCIZMZANFABCUEOUKFAUEPULUPAUKUPFDEUFDLZUIUNUJUOUQUHUMBUFDUG
|
|
QZRUQUHUMCURSUAUBUCUD $.
|
|
$}
|
|
|
|
$( Simplification law for restriction. (Contributed by NM,
|
|
16-Aug-1994.) $)
|
|
relssres $p |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A ) $=
|
|
( vx vy wrel cdm wss wa cres wceq simpl cv cop wcel vex opeldm ssel ancld
|
|
wi syl5 opelres syl6ibr adantl relssdv resss jctil eqss sylibr ) AEZAFZBG
|
|
ZHZABIZAGZAUMGZHUMAJULUOUNULCDAUMUIUKKUKCLZDLZMZANZURUMNZSUIUKUSUSUPBNZHU
|
|
TUKUSVAUSUPUJNUKVAUPUQACODOZPUJBUPQTRUPUQABVBUAUBUCUDABUEUFUMAUGUH $.
|
|
$}
|
|
|
|
$( A relation restricted to its domain equals itself. (Contributed by NM,
|
|
12-Dec-2006.) $)
|
|
resdm $p |- ( Rel A -> ( A |` dom A ) = A ) $=
|
|
( wrel cdm wss cres wceq ssid relssres mpan2 ) ABACZJDAJEAFJGAJHI $.
|
|
|
|
$( The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
resexg $p |- ( A e. V -> ( A |` B ) e. _V ) $=
|
|
( cres wss wcel cvv resss ssexg mpan ) ABDZAEACFKGFABHKACIJ $.
|
|
|
|
${
|
|
resex.1 $e |- A e. _V $.
|
|
$( The restriction of a set is a set. (Contributed by Jeff Madsen,
|
|
19-Jun-2011.) $)
|
|
resex $p |- ( A |` B ) e. _V $=
|
|
( cvv wcel cres resexg ax-mp ) ADEABFDECABDGH $.
|
|
$}
|
|
|
|
$( When restricting a relation, intersecting with the domain of the relation
|
|
has no effect. (Contributed by FL, 6-Oct-2008.) $)
|
|
resindm $p |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) ) $=
|
|
( wrel cres cdm cin resdm resindi incom inres inidm reseq1i 3eqtrri 3eqtr4g
|
|
ineq2d ) ACZABDZAAEZDZFQAFZABRFDQPSAQAGOABRHTAQFAAFZBDQQAIAABJUAABAKLMN $.
|
|
|
|
$( Restricting a relation to its domain without a set is the same as
|
|
restricting the relation to the universe without this set. (Contributed
|
|
by AV, 2-Dec-2018.) $)
|
|
resdmdfsn $p |- ( Rel R -> ( R |` ( _V \ { X } ) )
|
|
= ( R |` ( dom R \ { X } ) ) ) $=
|
|
( wrel cdm csn cdif cres cvv cin indif1 incom eqtri difeq1i reseq2i resindm
|
|
inv1 syl5reqr ) ACAADZBEZFZGAHSFZRIZGAUAGUBTAUBHRIZSFTHRSJUCRSUCRHIRHRKRPLM
|
|
LNAUAOQ $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Restriction of a class abstraction of ordered pairs. (Contributed by
|
|
NM, 5-Nov-2002.) $)
|
|
resopab $p |- ( { <. x , y >. | ph } |` A ) =
|
|
{ <. x , y >. | ( x e. A /\ ph ) } $=
|
|
( copab cres cvv cxp cin cv wa df-res df-xp biantru opabbii eqtr4i ineq2i
|
|
wcel vex eqtri incom inopab ) ABCEZDFUCDGHZIZBJDRZAKBCEZUCDLUEUFBCEZUCIZU
|
|
GUEUCUHIUIUDUHUCUDUFCJGRZKZBCEUHBCDGMUFUKBCUJUFCSNOPQUCUHUATUFABCUBTT $.
|
|
|
|
$( The existence of a restricted identity function, proved without using
|
|
the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) $)
|
|
resiexg $p |- ( A e. V -> ( _I |` A ) e. _V ) $=
|
|
( vx vy wcel cid cres cxp wss cvv relres weq cv wa cop simpr eleq1 biimpa
|
|
jca vex opelres wbr df-br bitr3i anbi1i bitri opelxp 3imtr4i relssi xpexg
|
|
ideq anidms ssexg sylancr ) ABEZFAGZAAHZIUQJEZUPJECDUPUQFAKCDLZCMZAEZNZVA
|
|
DMZAEZNUTVCOZUPEZVEUQEVBVAVDUSVAPUSVAVDUTVCAQRSVFVEFEZVANVBUTVCFADTZUAVGU
|
|
SVAVGUTVCFUBUSUTVCFUCUTVCVHUKUDUEUFUTVCAAUGUHUIUOURAABBUJULUPUQJUMUN $.
|
|
|
|
$( A subclass of the identity function is the identity function restricted
|
|
to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) $)
|
|
iss $p |- ( A C_ _I <-> A = ( _I |` dom A ) ) $=
|
|
( vx vy cid wss cdm cres wceq cv cop wcel wb wal wa ssel wi opeldm syl5bi
|
|
vex wrel a1i jcad wbr df-br bitr3i wex eldm2 opeq2 eleq1d biimprcd sylcom
|
|
ideq exlimdv imbi2d syl5ibcom impd impbid opelres alrimivv reli relss mpi
|
|
syl6bbr relres eqrel sylancl mpbird resss sseq1 mpbiri impbii ) ADEZADAFZ
|
|
GZHZVLVOBIZCIZJZAKZVRVNKZLZCMBMZVLWABCVLVSVRDKZVPVMKZNZVTVLVSWEVLVSWCWDAD
|
|
VROZVSWDPVLVPVQABSZCSZQUAUBVLWCWDVSWCVPVQHZVLWDVSPZWCVPVQDUCWIVPVQDUDVPVQ
|
|
WHULUEZVLWDVPVPJZAKZPWIWJWDVSCUFVLWMCVPAWGUGVLVSWMCVLVSWCWMWFWCWIVSWMWKWI
|
|
WMVSWIWLVRAVPVQVPUHUIZUJRUKUMRWIWMVSWDWNUNUORUPUQVPVQDVMWHURVCUSVLATZVNTV
|
|
OWBLVLDTWOUTADVAVBDVMVDBCAVNVEVFVGVOVLVNDEDVMVHAVNDVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( Restriction of a class abstraction of ordered pairs. (Contributed by
|
|
NM, 24-Aug-2007.) $)
|
|
resopab2 $p |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) =
|
|
{ <. x , y >. | ( x e. A /\ ph ) } ) $=
|
|
( wss cv wcel wa copab cres resopab pm4.71d anbi1d anass syl6rbb opabbidv
|
|
ssel syl5eq ) DEFZBGZEHZAIZBCJDKUADHZUCIZBCJUDAIZBCJUCBCDLTUEUFBCTUFUDUBI
|
|
ZAIUETUDUGATUDUBDEUARMNUDUBAOPQS $.
|
|
|
|
$( Restriction of the mapping operation. (Contributed by Mario Carneiro,
|
|
15-Jul-2013.) $)
|
|
resmpt $p |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) ) $=
|
|
( vy wss cv wcel wceq wa copab cres cmpt resopab2 df-mpt reseq1i 3eqtr4g
|
|
) CBFAGZBHEGDIZJAEKZCLRCHSJAEKABDMZCLACDMSAECBNUATCAEBDOPAECDOQ $.
|
|
|
|
$( Unconditional restriction of the mapping operation. (Contributed by
|
|
Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro,
|
|
22-Mar-2015.) $)
|
|
resmpt3 $p |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C ) $=
|
|
( cmpt cres cin resres wss wceq ssid resmpt ax-mp reseq1i inss1 3eqtr3i )
|
|
ABDEZBFZCFQBCGZFZQCFASDEZQBCHRQCBBIRQJBKABBDLMNSBITUAJBCOABSDLMP $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z R $.
|
|
$( Alternate definition of the restriction operation. (Contributed by
|
|
Mario Carneiro, 5-Nov-2013.) $)
|
|
dfres2 $p |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) } $=
|
|
( vz vw cres cv wbr wa copab relres relopab cop vex brres df-br ancom weq
|
|
wcel 3bitr3i eleq1 breq1 anbi12d breq2 anbi2d opelopab bitr4i eqrelriiv )
|
|
EFDCGZAHZCTZUKBHZDIZJZABKZDCLUOABMEHZFHZNZUJTZUQCTZUQURDIZJZUSUPTUQURUJIV
|
|
BVAJUTVCUQURDCFOZPUQURUJQVBVARUAUOVAUQUMDIZJVCABUQUREOVDAESULVAUNVEUKUQCU
|
|
BUKUQUMDUCUDBFSVEVBVAUMURUQDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The restricted identity expressed with the class builder. (Contributed
|
|
by FL, 25-Apr-2012.) $)
|
|
opabresid $p |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A ) $=
|
|
( weq copab cres cv wcel cid resopab equcom opabbii eqtr4i reseq1i eqtr3i
|
|
wa df-id ) BADZABEZCFAGCHRPABEICFRABCJSICSABDZABEIRTABBAKLABQMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The restricted identity expressed with the "maps to" notation.
|
|
(Contributed by FL, 25-Apr-2012.) $)
|
|
mptresid $p |- ( x e. A |-> x ) = ( _I |` A ) $=
|
|
( vy cv cmpt wcel weq wa copab cid cres df-mpt opabresid eqtri ) ABADZEOB
|
|
FCAGHACIJBKACBOLACBMN $.
|
|
$}
|
|
|
|
$( The domain of a restricted identity function. (Contributed by NM,
|
|
27-Aug-2004.) $)
|
|
dmresi $p |- dom ( _I |` A ) = A $=
|
|
( cid cdm wss cres wceq cvv ssv dmi sseqtr4i ssdmres mpbi ) ABCZDBAECAFAGMA
|
|
HIJABKL $.
|
|
|
|
$( TODO - delete this and replace w/ dfres3 (in FL's mathbox) $)
|
|
$( Any relation restricted to the universe is itself. (Contributed by NM,
|
|
16-Mar-2004.) $)
|
|
resid $p |- ( Rel A -> ( A |` _V ) = A ) $=
|
|
( wrel cdm cvv wss cres wceq ssv relssres mpan2 ) ABACZDEADFAGKHADIJ $.
|
|
|
|
$( Equality theorem for image. (Contributed by NM, 14-Aug-1994.) $)
|
|
imaeq1 $p |- ( A = B -> ( A " C ) = ( B " C ) ) $=
|
|
( wceq cres crn cima reseq1 rneqd df-ima 3eqtr4g ) ABDZACEZFBCEZFACGBCGLMNA
|
|
BCHIACJBCJK $.
|
|
|
|
$( Equality theorem for image. (Contributed by NM, 14-Aug-1994.) $)
|
|
imaeq2 $p |- ( A = B -> ( C " A ) = ( C " B ) ) $=
|
|
( wceq cres crn cima reseq2 rneqd df-ima 3eqtr4g ) ABDZCAEZFCBEZFCAGCBGLMNA
|
|
BCHICAJCBJK $.
|
|
|
|
${
|
|
imaeq1i.1 $e |- A = B $.
|
|
$( Equality theorem for image. (Contributed by NM, 21-Dec-2008.) $)
|
|
imaeq1i $p |- ( A " C ) = ( B " C ) $=
|
|
( wceq cima imaeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality theorem for image. (Contributed by NM, 21-Dec-2008.) $)
|
|
imaeq2i $p |- ( C " A ) = ( C " B ) $=
|
|
( wceq cima imaeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
imaeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for image. (Contributed by FL, 15-Dec-2006.) $)
|
|
imaeq1d $p |- ( ph -> ( A " C ) = ( B " C ) ) $=
|
|
( wceq cima imaeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality theorem for image. (Contributed by FL, 15-Dec-2006.) $)
|
|
imaeq2d $p |- ( ph -> ( C " A ) = ( C " B ) ) $=
|
|
( wceq cima imaeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
imaeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality theorem for image. (Contributed by Mario Carneiro,
|
|
4-Dec-2016.) $)
|
|
imaeq12d $p |- ( ph -> ( A " C ) = ( B " D ) ) $=
|
|
( cima imaeq1d imaeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Alternate definition of image. Compare definition (d) of [Enderton]
|
|
p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
dfima2 $p |- ( A " B ) = { y | E. x e. B x A y } $=
|
|
( cima cres crn cv wbr wex cab wrex df-ima dfrn2 wa vex brres ancom bitri
|
|
wcel exbii df-rex bitr4i abbii 3eqtri ) CDECDFZGAHZBHZUFIZAJZBKUGUHCIZADL
|
|
ZBKCDMABUFNUJULBUJUGDTZUKOZAJULUIUNAUIUKUMOUNUGUHCDBPQUKUMRSUAUKADUBUCUDU
|
|
E $.
|
|
|
|
$( Alternate definition of image. Compare definition (d) of [Enderton]
|
|
p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
dfima3 $p |- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) } $=
|
|
( cima cv wbr wrex cab wcel cop wa dfima2 df-br rexbii df-rex bitri abbii
|
|
wex eqtri ) CDEAFZBFZCGZADHZBIUADJUAUBKCJZLASZBIABCDMUDUFBUDUEADHUFUCUEAD
|
|
UAUBCNOUEADPQRT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
NM, 20-Jan-2007.) $)
|
|
elimag $p |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) ) $=
|
|
( vy cv wbr wrex cima wceq breq2 rexbidv dfima2 elab2g ) AGZFGZCHZADIPBCH
|
|
ZADIFBCDJEQBKRSADQBPCLMAFCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
elima.1 $e |- A e. _V $.
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
NM, 19-Apr-2004.) $)
|
|
elima $p |- ( A e. ( B " C ) <-> E. x e. C x B A ) $=
|
|
( cvv wcel cima cv wbr wrex wb elimag ax-mp ) BFGBCDHGAIBCJADKLEABCDFMN
|
|
$.
|
|
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
NM, 11-Aug-2004.) $)
|
|
elima2 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ x B A ) ) $=
|
|
( cima wcel cv wbr wrex wa wex elima df-rex bitri ) BCDFGAHZBCIZADJPDGQKA
|
|
LABCDEMQADNO $.
|
|
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
NM, 14-Aug-1994.) $)
|
|
elima3 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) ) $=
|
|
( cima wcel cv wbr wa wex cop elima2 df-br anbi2i exbii bitri ) BCDFGAHZD
|
|
GZRBCIZJZAKSRBLCGZJZAKABCDEMUAUCATUBSRBCNOPQ $.
|
|
$}
|
|
|
|
${
|
|
nfima.1 $e |- F/_ x A $.
|
|
nfima.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for image. (Contributed by NM,
|
|
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
nfima $p |- F/_ x ( A " B ) $=
|
|
( cima cres crn df-ima nfres nfrn nfcxfr ) ABCFBCGZHBCIAMABCDEJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d B z $. $d A z $.
|
|
nfimad.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfimad.3 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfima .
|
|
(Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfimad $p |- ( ph -> F/_ x ( A " B ) ) $=
|
|
( vz cv wcel wal cab cima wnfc nfaba1 nfima wb wa nfnfc1 nfan abidnf
|
|
imaeq1d imaeq2d sylan9eq nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUGDIZBJGKZL
|
|
ZMZBCDLZMZBUIUKUHBGNUJBGNOABCMZBDMZUMUOPEFUPUQQBULUNUPUQBBCRBDRSUPUQULCUK
|
|
LUNUPUICUKBGCTUAUQUKDCBGDTUBUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The image of the domain of a class is the range of the class.
|
|
(Contributed by NM, 14-Aug-1994.) $)
|
|
imadmrn $p |- ( A " dom A ) = ran A $=
|
|
( vx vy cv cdm wcel cop wa wex cab cima crn vex opeldm ancom bitr2i exbii
|
|
pm4.71i abbii dfima3 dfrn3 3eqtr4i ) BDZAEZFZUCCDZGAFZHZBIZCJUGBIZCJAUDKA
|
|
LUIUJCUHUGBUGUGUEHUHUGUEUCUFABMCMNRUGUEOPQSBCAUDTBCAUAUB $.
|
|
|
|
$( The image of a class is a subset of its range. Theorem 3.16(xi) of
|
|
[Monk1] p. 39. (Contributed by NM, 31-Mar-1995.) $)
|
|
imassrn $p |- ( A " B ) C_ ran A $=
|
|
( vx vy cv wcel cop wex cab cima crn exsimpr ss2abi dfima3 dfrn3 3sstr4i
|
|
wa ) CEZBFZRDEGAFZQCHZDITCHZDIABJAKUAUBDSTCLMCDABNCDAOP $.
|
|
$}
|
|
|
|
$( The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed
|
|
by NM, 24-Jul-1995.) $)
|
|
imaexg $p |- ( A e. V -> ( A " B ) e. _V ) $=
|
|
( wcel cima crn wss cvv imassrn rnexg ssexg sylancr ) ACDABEZAFZGNHDMHDABIA
|
|
CJMNHKL $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
|
|
(Contributed by NM, 30-Apr-1998.) $)
|
|
imai $p |- ( _I " A ) = A $=
|
|
( vx vy cid cima cv wcel cop wex cab dfima3 weq wbr df-br vex ideq bitr3i
|
|
wa anbi2i bitri ancom exbii eleq1 ceqsexv abbii abid2 3eqtri ) DAEBFZAGZU
|
|
HCFZHDGZRZBIZCJUJAGZCJABCDAKUMUNCUMBCLZUIRZBIUNULUPBULUIUORUPUKUOUIUKUHUJ
|
|
DMUOUHUJDNUHUJCOZPQSUIUOUATUBUIUNBUJUQUHUJAUCUDTUECAUFUG $.
|
|
$}
|
|
|
|
$( The range of the restricted identity function. (Contributed by NM,
|
|
27-Aug-2004.) $)
|
|
rnresi $p |- ran ( _I |` A ) = A $=
|
|
( cid cima cres crn df-ima imai eqtr3i ) BACBADEABAFAGH $.
|
|
|
|
$( The image of a restriction of the identity function. (Contributed by FL,
|
|
31-Dec-2006.) $)
|
|
resiima $p |- ( B C_ A -> ( ( _I |` A ) " B ) = B ) $=
|
|
( wss cid cres cima crn wceq df-ima a1i resabs1 rneqd rnresi 3eqtrd ) BACZD
|
|
AEZBFZPBEZGZDBEZGZBQSHOPBIJORTDBAKLUABHOBMJN $.
|
|
|
|
$( Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed
|
|
by NM, 20-May-1998.) $)
|
|
ima0 $p |- ( A " (/) ) = (/) $=
|
|
( c0 cima cres crn df-ima res0 rneqi rn0 3eqtri ) ABCABDZEBEBABFKBAGHIJ $.
|
|
|
|
$( Image under the empty relation. (Contributed by FL, 11-Jan-2007.) $)
|
|
0ima $p |- ( (/) " A ) = (/) $=
|
|
( c0 cima crn imassrn rn0 sseqtri 0ss eqssi ) BACZBJBDBBAEFGJHI $.
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $. $d x y $. $d F y $.
|
|
$( Move class substitution in and out of the image of a function.
|
|
(Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro,
|
|
4-Dec-2016.) $)
|
|
csbima12g $p |- ( A e. C -> [_ A / x ]_ ( F " B ) =
|
|
( [_ A / x ]_ F " [_ A / x ]_ B ) ) $=
|
|
( vy cv cima csb csbeq1 imaeq12d eqeq12d vex nfcsb1v nfima csbeq1a csbief
|
|
wceq vtoclg ) AFGZECHZIZATEIZATCIZHZRABUAIZABEIZABCIZHZRFBDTBRZUBUFUEUIAT
|
|
BUAJUJUCUGUDUHATBEJATBCJKLATUAUEFMAUCUDATENATCNOAGTREUCCUDATEPATCPKQS $.
|
|
$}
|
|
|
|
$( A class whose image under another is empty is disjoint with the other's
|
|
domain. (Contributed by FL, 24-Jan-2007.) $)
|
|
imadisj $p |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) ) $=
|
|
( cima wceq cres crn cdm cin df-ima eqeq1i dm0rn0 dmres incom eqtri 3bitr2i
|
|
c0 ) ABCZPDABEZFZPDRGZPDAGZBHZPDQSPABIJRKTUBPTBUAHUBABLBUAMNJO $.
|
|
|
|
$( A preimage under any class is included in the domain of the class.
|
|
(Contributed by FL, 29-Jan-2007.) $)
|
|
cnvimass $p |- ( `' A " B ) C_ dom A $=
|
|
( ccnv cima crn cdm imassrn dfdm4 sseqtr4i ) ACZBDJEAFJBGAHI $.
|
|
|
|
$( The preimage of the range of a class is the domain of the class.
|
|
(Contributed by Jeff Hankins, 15-Jul-2009.) $)
|
|
cnvimarndm $p |- ( `' A " ran A ) = dom A $=
|
|
( ccnv cdm cima crn imadmrn df-rn imaeq2i dfdm4 3eqtr4i ) ABZKCZDKEKAEZDACK
|
|
FMLKAGHAIJ $.
|
|
|
|
${
|
|
$d x y A $. $d x B $. $d x y R $.
|
|
$( The image of a singleton. (Contributed by NM, 8-May-2005.) $)
|
|
imasng $p |- ( A e. B -> ( R " { A } ) = { y | A R y } ) $=
|
|
( vx wcel cvv csn cima wbr cab wceq elex wrex dfima2 rexsng abbidv syl5eq
|
|
cv breq1 syl ) BCFBGFZDBHZIZBASZDJZAKZLBCMUBUDESZUEDJZEUCNZAKUGEADUCOUBUJ
|
|
UFAUIUFEBGUHBUEDTPQRUA $.
|
|
|
|
$( Elementhood in the image of a singleton. (Contributed by Jim Kingdon,
|
|
10-Dec-2018.) $)
|
|
elreimasng $p |- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <->
|
|
A R B ) ) $=
|
|
( vx wcel csn cima cv wbr cab wrel imasng eleq2d cvv wi wb brrelex2 breq2
|
|
ex elab3g syl sylan9bbr ) ADFZBCAGHZFBAEIZCJZEKZFZCLZABCJZUDUEUHBEADCMNUJ
|
|
UKBOFZPUIUKQUJUKULABCRTUGUKEBOUFBACSUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
elimasn.1 $e |- B e. _V $.
|
|
elimasn.2 $e |- C e. _V $.
|
|
$( Membership in an image of a singleton. (Contributed by NM,
|
|
15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
elimasn $p |- ( C e. ( A " { B } ) <-> <. B , C >. e. A ) $=
|
|
( vx csn cima wcel wbr cop cv breq2 cvv cab wceq imasng ax-mp elab2 df-br
|
|
bitri ) CABGHZIBCAJZBCKAIBFLZAJZUCFCUBEUDCBAMBNIUBUEFOPDFBNAQRSBCATUA $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B y z $. $d C y z $.
|
|
$( Membership in an image of a singleton. (Contributed by Raph Levien,
|
|
21-Oct-2006.) $)
|
|
elimasng $p |- ( ( B e. V /\ C e. W ) ->
|
|
( C e. ( A " { B } ) <-> <. B , C >. e. A ) ) $=
|
|
( vz vy cv csn cima wcel cop wceq sneq imaeq2d eleq2d eleq1d bibi12d vex
|
|
wb opeq1 eleq1 opeq2 elimasn vtocl2g ) FHZAGHZIZJZKZUGUFLZAKZTUFABIZJZKZB
|
|
UFLZAKZTCUNKZBCLZAKZTGFBCDEUGBMZUJUOULUQVAUIUNUFVAUHUMAUGBNOPVAUKUPAUGBUF
|
|
UAQRUFCMZUOURUQUTUFCUNUBVBUPUSAUFCBUCQRAUGUFGSFSUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d x y $.
|
|
$( Two ways to express the class of unique-valued arguments of ` F ` ,
|
|
which is the same as the domain of ` F ` whenever ` F ` is a function.
|
|
The left-hand side of the equality is from Definition 10.2 of [Quine]
|
|
p. 65. Quine uses the notation "arg ` F ` " for this class (for which
|
|
we have no separate notation). (Contributed by NM, 8-May-2005.) $)
|
|
args $p |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y } $=
|
|
( cv csn cima wceq wex wbr weu cab cvv wcel vex imasng ax-mp eqeq1i exbii
|
|
euabsn bitr4i abbii ) CADZEFZBDZEZGZBHZUBUDCIZBJZAUGUHBKZUEGZBHUIUFUKBUCU
|
|
JUEUBLMUCUJGANBUBLCOPQRUHBSTUA $.
|
|
$}
|
|
|
|
${
|
|
eliniseg.1 $e |- C e. _V $.
|
|
$( Membership in an initial segment. The idiom ` ( ``' A " { B } ) ` ,
|
|
meaning ` { x | x A B } ` , is used to specify an initial segment in
|
|
(for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
|
|
NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
eliniseg $p |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) ) $=
|
|
( wcel cvv ccnv csn cima wbr wb wa cop elimasng df-br syl6bbr bitrd mpan2
|
|
brcnvg ) BDFZCGFZCAHZBIJFZCBAKZLEUAUBMZUDBCUCKZUEUFUDBCNUCFUGUCBCDGOBCUCP
|
|
QBCDGATRS $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
epini.1 $e |- A e. _V $.
|
|
$( Any set is equal to its preimage under the converse epsilon relation.
|
|
(Contributed by Mario Carneiro, 9-Mar-2013.) $)
|
|
epini $p |- ( `' _E " { A } ) = A $=
|
|
( vx cep ccnv csn cima cv wbr cvv wb vex eliniseg ax-mp epelc bitri eqriv
|
|
wcel ) CDEAFGZACHZSRZTADIZTARAJRUAUBKBDATJCLMNTABOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An idiom that signifies an initial segment of an ordering, used, for
|
|
example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
|
|
NM, 28-Apr-2004.) $)
|
|
iniseg $p |- ( B e. V -> ( `' A " { B } ) = { x | x A B } ) $=
|
|
( wcel cvv ccnv csn cima cv wbr cab wceq elex vex eliniseg abbi2dv syl )
|
|
CDECFEZBGCHIZAJZCBKZALMCDNSUBATBCUAFAOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $. $d x V $.
|
|
$( Alternate definition of set-like relation. (Contributed by Mario
|
|
Carneiro, 23-Jun-2015.) $)
|
|
dfse2 $p |- ( R Se A <-> A. x e. A ( A i^i ( `' R " { x } ) ) e. _V ) $=
|
|
( vy wse cv wbr crab cvv wcel wral ccnv csn cima cin df-se cab dfrab3 vex
|
|
wceq iniseg ax-mp ineq2i eqtr4i eleq1i ralbii bitri ) BCEDFAFZCGZDBHZIJZA
|
|
BKBCLUHMNZOZIJZABKADBCPUKUNABUJUMIUJBUIDQZOUMUIDBRULUOBUHIJULUOTASDCUHIUA
|
|
UBUCUDUEUFUG $.
|
|
|
|
$( Any set relation is set-like. (Contributed by Mario Carneiro,
|
|
22-Jun-2015.) $)
|
|
exse2 $p |- ( R e. V -> R Se A ) $=
|
|
( vy vx wcel cv wbr crab cvv wral wse cdm wss wa cab df-rab breldm adantl
|
|
vex abssi eqsstri dmexg ssexg sylancr ralrimivw df-se sylibr ) BCFZDGZEGZ
|
|
BHZDAIZJFZEAKABLUIUNEAUIUMBMZNUOJFUNUMUJAFZULOZDPUOULDAQUQDUOULUJUOFUPUJU
|
|
KBDTETRSUAUBBCUCUMUOJUDUEUFEDABUGUH $.
|
|
$}
|
|
|
|
$( Subset theorem for image. (Contributed by NM, 16-Mar-2004.) $)
|
|
imass1 $p |- ( A C_ B -> ( A " C ) C_ ( B " C ) ) $=
|
|
( wss cres crn cima ssres rnss syl df-ima 3sstr4g ) ABDZACEZFZBCEZFZACGBCGM
|
|
NPDOQDABCHNPIJACKBCKL $.
|
|
|
|
$( Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
|
|
(Contributed by NM, 22-Mar-1998.) $)
|
|
imass2 $p |- ( A C_ B -> ( C " A ) C_ ( C " B ) ) $=
|
|
( wss cres crn cima ssres2 rnss syl df-ima 3sstr4g ) ABDZCAEZFZCBEZFZCAGCBG
|
|
MNPDOQDABCHNPIJCAKCBKL $.
|
|
|
|
$( The image of a singleton outside the domain is empty. (Contributed by NM,
|
|
22-May-1998.) $)
|
|
ndmima $p |- ( -. A e. dom B -> ( B " { A } ) = (/) ) $=
|
|
( cdm wcel wn csn cima cres crn c0 df-ima wceq cin dmres incom eqtri disjsn
|
|
biimpri syl5eq dm0rn0 sylib ) ABCZDEZBAFZGBUDHZIZJBUDKUCUECZJLUFJLUCUGUBUDM
|
|
ZJUGUDUBMUHBUDNUDUBOPUHJLUCUBAQRSUETUAS $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed
|
|
by NM, 29-Oct-1996.) $)
|
|
relcnv $p |- Rel `' A $=
|
|
( vy vx cv wbr ccnv df-cnv relopabi ) BDCDAECBAFCBAGH $.
|
|
$}
|
|
|
|
$( When ` R ` is a relation, the sethood assumptions on ~ brcnv can be
|
|
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) $)
|
|
relbrcnvg $p |- ( Rel R -> ( A `' R B <-> B R A ) ) $=
|
|
( wrel cvv wcel wa ccnv wbr wi relcnv brrelex12 mpan ancomd ex wb pm5.21ndd
|
|
a1i brcnvg ) CDZAEFZBEFZGZABCHZIZBACIZUEUCJTUDDUEUCCKABUDLMRTUFUCTUFGUBUABA
|
|
CLNOUCUEUFPJTABEECSRQ $.
|
|
|
|
${
|
|
relbrcnv.1 $e |- Rel R $.
|
|
$( When ` R ` is a relation, the sethood assumptions on ~ brcnv can be
|
|
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) $)
|
|
relbrcnv $p |- ( A `' R B <-> B R A ) $=
|
|
( wrel ccnv wbr wb relbrcnvg ax-mp ) CEABCFGBACGHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $.
|
|
$( Two ways of saying a relation is transitive. Definition of transitivity
|
|
in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof
|
|
shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
cotr $p |- ( ( R o. R ) C_ R <->
|
|
A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) $=
|
|
( ccom wss cv cop wcel wi wal wbr wa wrel wb wex df-co vex albii bitri
|
|
relopabi ssrel ax-mp opelco df-br bicomi imbi12i 19.23v bitr4i alcom ) DD
|
|
EZDFZAGZCGZHZUKIZUODIZJZCKZAKZUMBGZDLVAUNDLMZUMUNDLZJZCKBKZAKUKNULUTOVBBP
|
|
ZACUKACBDDQUAACUKDUBUCUSVEAUSVDBKZCKVEURVGCURVFVCJVGUPVFUQVCBUMUNDDARCRUD
|
|
VCUQUMUNDUEUFUGVBVCBUHUISVDCBUJTST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z R $. $d x y z S $. $d z V $.
|
|
$d z W $.
|
|
$( Two ways to state a relation is reflexive. Adapted from Tarski.
|
|
(Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.) $)
|
|
issref $p |- ( ( _I |` A ) C_ R <-> A. x e. A x R x ) $=
|
|
( vy vz cv wbr wral wcel wi wal cop cid wss df-ral cvv vex opelresi ax-mp
|
|
sylbi cres wb df-br bicomi imbi12i albii ralidm bitri cxp pm2.27 opelresg
|
|
ralv wa weq ideq opeq2 eleq1d biimpcd syl6bir pm2.43i com3r sylbir syl6bi
|
|
syl6 imp ralrimiv sps ralimi wceq eleq1 imbi12d sylibr wrel relres df-rel
|
|
ralxp mpbi sseli ancri pm3.31 syl5 alimi syl ssel alrimiv impbii 3bitr2ri
|
|
dfss2 ) AFZWICGZABHWIBIZWJJZAKWIWILZMBUAZIZWMCIZJZAKZWNCNZWJABOWQWLAWOWKW
|
|
PWJWIPIZWOWKUBAQZWIBPRSWJWPWIWICUCUDUEUFWRWSWRDFZWNIZXBCIZJZDKZWSWRWQAPHZ
|
|
APHZXFXHXGWRWQAPUGWQAULUHXHXEDPPUIZHZXFXHWIEFZLZWNIZXLCIZJZEPHZAPHXJXGXPA
|
|
PXGWTWQJZAKXPWQAPOXQXPAWTXQXPJXAWTXQWQXPWTWQUJWQXOEPXKPIZXMWQXNXRXMXLMIZW
|
|
KUMWQXNJZWIXKMBPUKXSWKXTXSWIXKMGZWKXTJZWIXKMUCYAAEUNZYBWIXKEQUOWKWQYCXNWK
|
|
WQYCXNJZJZWKWKWOYEWIBBRWOWQWPYDWOWPUJYCWPXNYCWMXLCWIXKWIUPUQURVDUSUTVATVB
|
|
VEVCVAVFVDSVGTVHXEXODAEPPXBXLVIXCXMXDXNXBXLWNVJXBXLCVJVKVPVLXJXBXIIZXEJZD
|
|
KXFXEDXIOYGXEDXCYFXCUMYGXDXCYFWNXIXBWNVMWNXINMBVNWNVOVQVRVSYFXCXDVTWAWBTW
|
|
CVBDWNCWHVLWSWQAWNCWMWDWEWFWG $.
|
|
|
|
$( Two ways of saying a relation is symmetric. Similar to definition of
|
|
symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
cnvsym $p |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) $=
|
|
( cv cop ccnv wcel wi wal wss wbr alcom wrel relcnv ssrel ax-mp vex brcnv
|
|
wb df-br bitr3i imbi12i 2albii 3bitr4i ) BDZADZEZCFZGZUGCGZHZAIBIZUKBIAIU
|
|
HCJZUFUECKZUEUFCKZHZBIAIUKBALUHMUMULSCNBAUHCOPUPUKABUNUIUOUJUNUEUFUHKUIUE
|
|
UFCBQAQRUEUFUHTUAUEUFCTUBUCUD $.
|
|
|
|
$( Two ways of saying a relation is antisymmetric. Definition of
|
|
antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
intasym $p |- ( ( R i^i `' R ) C_ _I <->
|
|
A. x A. y ( ( x R y /\ y R x ) -> x = y ) ) $=
|
|
( ccnv cin cid wss cv cop wcel wi wal wbr wa weq wrel wb df-br vex bitr3i
|
|
relcnv relin2 ssrel mp2b elin brcnv anbi12i bitr4i imbi12i 2albii bitri
|
|
ideq ) CCDZEZFGZAHZBHZIZUNJZURFJZKZBLALZUPUQCMZUQUPCMZNZABOZKZBLALUMPUNPU
|
|
OVBQCUACUMUBABUNFUCUDVAVGABUSVEUTVFUSURCJZURUMJZNVEURCUMUEVCVHVDVIUPUQCRV
|
|
DUPUQUMMVIUPUQCASBSZUFUPUQUMRTUGUHUTUPUQFMVFUPUQFRUPUQVJULTUIUJUK $.
|
|
|
|
$( Two ways of saying a relation is antisymmetric and reflexive.
|
|
` U. U. R ` is the field of a relation by ~ relfld . (Contributed by
|
|
NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
asymref $p |- ( ( R i^i `' R ) = ( _I |` U. U. R ) <->
|
|
A. x e. U. U. R A. y ( ( x R y /\ y R x ) <-> x = y ) ) $=
|
|
( cv wcel cid cuni wb wal wbr wa wceq wi df-br bitr3i bitri 3bitr4i albii
|
|
vex wrel cop ccnv cin cres wral opeluu simpld adantr pm4.71ri bibi1i elin
|
|
sylbi brcnv anbi12i bitr4i opelres ideq pm5.32 19.21v relcnv relin2 ax-mp
|
|
anbi2ci bibi12i relres eqrel mp2an df-ral ) ADZBDZUAZCCUBZUCZEZVKFCGGZUDZ
|
|
EZHZBIZAIZVIVOEZVIVJCJZVJVICJZKZVIVJLZHZBIZMZAIVMVPLZWGAVOUEVSWHAVSWAWFMZ
|
|
BIWHVRWJBWDWAWEKZHWAWDKZWKHVRWJWDWLWKWDWAWBWAWCWBWAVJVOEZWBVKCEZWAWMKVIVJ
|
|
CNZVIVJCASZBSZUFULUGUHUIUJVNWDVQWKVNWNVKVLEZKWDVKCVLUKWBWNWCWRWOWCVIVJVLJ
|
|
WRVIVJCWPWQUMVIVJVLNOUNUOVQVKFEZWAKWKVIVJFVOWQUPWSWEWAWSVIVJFJWEVIVJFNVIV
|
|
JWQUQOVCPVDWAWDWEURQRWAWFBUSPRVMTZVPTWIVTHVLTWTCUTCVLVAVBFVOVEABVMVPVFVGW
|
|
GAVOVHQ $.
|
|
|
|
$( Two ways of saying a relation is irreflexive. Definition of
|
|
irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
intirr $p |- ( ( R i^i _I ) = (/) <-> A. x -. x R x ) $=
|
|
( vy cid cin c0 wceq cv cop wcel cvv cdif wi wal wbr wn wss df-br 3bitr2i
|
|
vex incom eqeq1i disj2 wrel wb reli ssrel ax-mp 3bitri equcom wa biantrur
|
|
ideq opex eldif bitr4i xchnxbir imbi12i 2albii breq2 notbid equsal albii
|
|
nfv ) BDEZFGZAHZCHZIZDJZVIKBLZJZMZCNANZVHVGGZVGVHBOZPZMZCNZANVGVGBOZPZANV
|
|
FDBEZFGDVKQZVNVEWBFBDUAUBDBUCDUDWCVNUEUFACDVKUGUHUIVRVMACVOVJVQVLVOVGVHGV
|
|
GVHDOVJCAUJVGVHCTZUMVGVHDRSVIBJZVLVPWEPZVIKJZWFUKVLWGWFVGVHATWDUNULVIKBUO
|
|
UPVGVHBRUQURUSVSWAAVQWACAWACVDVOVPVTVHVGVGBUTVAVBVCS $.
|
|
|
|
$( Two ways of saying that two elements have an upper bound. (Contributed
|
|
by Mario Carneiro, 3-Nov-2015.) $)
|
|
brcodir $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A ( `' R o. R ) B <-> E. z ( A R z /\ B R z ) ) ) $=
|
|
( wcel wa ccnv ccom wbr cv wex brcog wb cvv vex brcnvg mpan anbi2d adantl
|
|
exbidv bitrd ) BEGZCFGZHZBCDIZDJKBALZDKZUHCUGKZHZAMUICUHDKZHZAMABCUGDEFNU
|
|
FUKUMAUEUKUMOUDUEUJULUIUHPGUEUJULOAQUHCPFDRSTUAUBUC $.
|
|
|
|
$( Two ways of saying a relation is directed. (Contributed by Mario
|
|
Carneiro, 22-Nov-2013.) $)
|
|
codir $p |- ( ( A X. B ) C_ ( `' R o. R ) <-> A. x e. A A. y e. B
|
|
E. z ( x R z /\ y R z ) ) $=
|
|
( cv cop cxp wcel ccnv ccom wi wal wa wbr wral cvv wb vex wex wss brcodir
|
|
opelxp df-br mp2an bitr3i imbi12i 2albii wrel relxp ssrel ax-mp 3bitr4i
|
|
r2al ) AGZBGZHZDEIZJZURFKFLZJZMZBNANZUPDJUQEJOZUPCGZFPUQVFFPOCUAZMZBNANUS
|
|
VAUBZVGBEQADQVCVHABUTVEVBVGUPUQDEUDVBUPUQVAPZVGUPUQVAUEUPRJUQRJVJVGSATBTC
|
|
UPUQFRRUCUFUGUHUIUSUJVIVDSDEUKABUSVAULUMVGABDEUOUN $.
|
|
|
|
$( A quantifier-free way of expressing the total order predicate.
|
|
(Contributed by Mario Carneiro, 22-Nov-2013.) $)
|
|
qfto $p |- ( ( A X. B ) C_ ( R u. `' R ) <->
|
|
A. x e. A A. y e. B ( x R y \/ y R x ) ) $=
|
|
( cv cop cxp wcel ccnv cun wi wal wa wbr wo wss wral opelxp vex brun wrel
|
|
df-br brcnv orbi2i 3bitr3i imbi12i 2albii relxp ssrel ax-mp r2al 3bitr4i
|
|
wb ) AFZBFZGZCDHZIZUQEEJZKZIZLZBMAMZUOCIUPDINZUOUPEOZUPUOEOZPZLZBMAMURVAQ
|
|
ZVHBDRACRVCVIABUSVEVBVHUOUPCDSUOUPVAOVFUOUPUTOZPVBVHUOUPEUTUAUOUPVAUCVKVG
|
|
VFUOUPEATBTUDUEUFUGUHURUBVJVDUNCDUIABURVAUJUKVHABCDULUM $.
|
|
|
|
$( A square cross product ` ( A X. A ) ` is a transitive relation.
|
|
(Contributed by FL, 31-Jul-2009.) $)
|
|
xpidtr $p |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) $=
|
|
( vx vy vz cxp ccom wss cv wbr wa wal wcel simplbi2com adantl sylbi com12
|
|
wi brxp adantr imp ax-gen gen2 cotr mpbir ) AAEZUEFUEGBHZCHZUEIZUGDHZUEIZ
|
|
JUFUIUEIZQZDKZCKBKUMBCULDUHUJUKUHUFALZUGALZJUJUKQZUFUGAARUNUPUOUJUNUKUJUO
|
|
UIALZJUNUKQZUGUIAARUQURUOUKUNUQUFUIAARMNOPSOTUAUBBCDUEUCUD $.
|
|
|
|
$( The intersection of two transitive classes is transitive. (Contributed
|
|
by FL, 31-Jul-2009.) $)
|
|
trin2 $p |- ( ( ( R o. R ) C_ R /\ ( S o. S ) C_ S )
|
|
-> ( ( R i^i S ) o. ( R i^i S ) ) C_ ( R i^i S ) ) $=
|
|
( vx vy vz ccom wss wa cv cin wbr wal cotr brin simpr simpl com12 alanimi
|
|
wi sylbi anim12d an4s syl2anb syl6ibr ex imp sylibr ) AAFAGZBBFBGZHCIZDIZ
|
|
ABJZKZUKEIZULKZHZUJUNULKZSZELZDLZCLZULULFULGUHUIVAUHUJUKAKZUKUNAKZHZUJUNA
|
|
KZSZELZDLZCLZUIVASCDEAMUIVIVAUIUJUKBKZUKUNBKZHZUJUNBKZSZELZDLZCLZVIVASCDE
|
|
BMVQVIVAVPVHUTCVOVGUSDVNVFUREVNVFHZUPVEVMHZUQUPVRVSUMVBVJHVCVKHVRVSSZUOUJ
|
|
UKABNUKUNABNVBVCVJVKVTVRVDVLHVSVRVDVEVLVMVNVFOVNVFPUAQUBUCQUJUNABNUDRRRUE
|
|
TQTUFCDEULMUG $.
|
|
|
|
$( A partial order relation is irreflexive. (Contributed by Mario
|
|
Carneiro, 2-Nov-2015.) $)
|
|
poirr2 $p |- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) ) $=
|
|
( vx vy wpo cid cres cin c0 wss wceq wrel relres relin2 cv wcel wbr wa wn
|
|
syl5bi mp1i cop df-br brin bitr3i wi vex brres poirr wb ideq breq2 notbid
|
|
sylbi syl5ibcom expimpd ancomsd con2d imnan sylib pm2.21d relssdv ss0 syl
|
|
) ABEZBFAGZHZIJVGIKVECDVGIVFLVGLVEFAMBVFNUACOZDOZUBZVGPZVHVIBQZVHVIVFQZRZ
|
|
VEVJIPZVKVHVIVGQVNVHVIVGUCVHVIBVFUDUEVEVNVOVEVLVMSUFVNSVEVMVLVMVHVIFQZVHA
|
|
PZRVEVLSZVHVIFADUGZUHVEVQVPVRVEVQVPVRVEVQRVHVHBQZSVPVRAVHBUIVPVTVLVPVHVIK
|
|
VTVLUJVHVIVSUKVHVIVHBULUNUMUOUPUQTURVLVMUSUTVATVBVGVCVD $.
|
|
$}
|
|
|
|
$( The relation induced by a transitive relation on a part of its field is
|
|
transitive. (Taking the intersection of a relation with a square cross
|
|
product is a way to restrict it to a subset of its field.) (Contributed
|
|
by FL, 31-Jul-2009.) $)
|
|
trinxp $p |- ( ( R o. R ) C_ R ->
|
|
( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ ( R i^i ( A X. A ) ) ) $=
|
|
( ccom wss cxp cin xpidtr trin2 mpan2 ) BBCBDAAEZJCJDBJFZKCKDAGBJHI $.
|
|
|
|
${
|
|
soi.1 $e |- R Or S $.
|
|
soi.2 $e |- R C_ ( S X. S ) $.
|
|
$( A strict order relation is irreflexive. (Contributed by NM,
|
|
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) $)
|
|
soirri $p |- -. A R A $=
|
|
( wbr id wor wcel wn brel simpld sonr sylancr pm2.65i ) AABFZPPGPCBHACIZP
|
|
JDPQQAACCBEKLCABMNO $.
|
|
|
|
$( A strict order relation is a transitive relation. (Contributed by NM,
|
|
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) $)
|
|
sotri $p |- ( ( A R B /\ B R C ) -> A R C ) $=
|
|
( wcel wa wbr brel simpld anim12i wi wor w3a sotr mpan 3expb mpcom ) AEHZ
|
|
BEHZCEHZIZIABDJZBCDJZIZACDJZUEUAUFUDUEUAUBABEEDGKLBCEEDGKMUAUBUCUGUHNZEDO
|
|
UAUBUCPUIFEABCDQRST $.
|
|
|
|
$( A strict order relation has no 2-cycle loops. (Contributed by NM,
|
|
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) $)
|
|
son2lpi $p |- -. ( A R B /\ B R A ) $=
|
|
( wbr wa soirri sotri mto ) ABCGBACGHAACGACDEFIABACDEFJK $.
|
|
|
|
$( A transitivity relation. (Read ` -. ` B < A and B < C implies A < C .)
|
|
(Contributed by Mario Carneiro, 10-May-2013.) $)
|
|
sotri2 $p |- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C ) $=
|
|
( wcel wbr wn w3a simp2 wo wa brel 3ad2ant3 simp1 df-3an sylanbrc simp3
|
|
wor wi sowlin mpan sylc ord mpd ) AEHZBADIZJZBCDIZKZUJACDIZUHUJUKLULUIUMU
|
|
LBEHZCEHZUHKZUKUIUMMZULUNUONZUHUPUKUHURUJBCEEDGOPUHUJUKQUNUOUHRSUHUJUKTED
|
|
UAUPUKUQUBFEBCADUCUDUEUFUG $.
|
|
|
|
$( A transitivity relation. (Read A < B and ` -. ` C < B implies A < C .)
|
|
(Contributed by Mario Carneiro, 10-May-2013.) $)
|
|
sotri3 $p |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C ) $=
|
|
( wcel wbr wn w3a simp3 wo wa brel 3ad2ant2 simp1 df-3an sylanbrc simp2
|
|
wor wi sowlin mpan sylc ecased ) CEHZABDIZCBDIZJZKZACDIZUIUGUHUJLUKAEHZBE
|
|
HZUGKZUHULUIMZUKUMUNNZUGUOUHUGUQUJABEEDGOPUGUHUJQUMUNUGRSUGUHUJTEDUAUOUHU
|
|
PUBFEABCDUCUDUEUF $.
|
|
$}
|
|
|
|
$( Express "less than or equals" for general strict orders. (Contributed by
|
|
Stefan O'Rear, 17-Jan-2015.) $)
|
|
poleloe $p |- ( B e. V -> ( A ( R u. _I ) B <-> ( A R B \/ A = B ) ) ) $=
|
|
( cid cun wbr wo wcel wceq brun ideqg orbi2d syl5bb ) ABCEFGABCGZABEGZHBDIZ
|
|
OABJZHABCEKQPROABDLMN $.
|
|
|
|
$( Transitive law for general strict orders. (Contributed by Stefan O'Rear,
|
|
17-Jan-2015.) $)
|
|
poltletr $p |- ( ( R Po X /\ ( A e. X /\ B e. X /\ C e. X ) ) ->
|
|
( ( A R B /\ B ( R u. _I ) C ) -> A R C ) ) $=
|
|
( wpo wcel w3a wa wbr cid cun wceq wo poleloe 3ad2ant3 adantl anbi2d com12
|
|
wb wi potr breq2 biimpac a1d jaodan sylbid ) EDFZAEGZBEGZCEGZHZIZABDJZBCDKL
|
|
JZIUNBCDJZBCMZNZIZACDJZUMUOURUNULUOURTZUHUKUIVAUJBCDEOPQRUSUMUTUNUPUMUTUAUQ
|
|
UMUNUPIUTEABCDUBSUNUQIUTUMUQUNUTBCADUCUDUEUFSUG $.
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( The converse of a class abstraction of ordered pairs. (Contributed by
|
|
NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
cnvopab $p |- `' { <. x , y >. | ph } = { <. y , x >. | ph } $=
|
|
( vz vw copab ccnv relcnv relopab cop wcel wsb opelopabsbALT sbcom2 bitri
|
|
cv vex opelcnv 3bitr4i eqrelriiv ) DEABCFZGZACBFZUAHACBIEPZDPZJUAKZACDLBE
|
|
LZUEUDJZUBKUHUCKUFABELCDLUGABCEDMABECDNOUEUDUADQEQRACBDEMST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The converse of the empty set. (Contributed by NM, 6-Apr-1998.) $)
|
|
cnv0 $p |- `' (/) = (/) $=
|
|
( vx vy c0 ccnv relcnv rel0 cv cop wcel vex opelcnv noel 2false eqrelriiv
|
|
bitr4i ) ABCDZCCEFAGZBGZHZPIRQHZCIZSCIZQRCAJBJKUBUASLTLMON $.
|
|
|
|
$( The converse of the identity relation. Theorem 3.7(ii) of [Monk1]
|
|
p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew
|
|
Salmon, 27-Aug-2011.) $)
|
|
cnvi $p |- `' _I = _I $=
|
|
( vy vx cid wbr copab weq ccnv vex ideq equcom bitri opabbii df-cnv df-id
|
|
cv 3eqtr4i ) AOZBOZCDZBAEBAFZBAECGCSTBASABFTQRBHIABJKLBACMBANP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The converse of a union is the union of converses. Theorem 16 of
|
|
[Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) $)
|
|
cnvun $p |- `' ( A u. B ) = ( `' A u. `' B ) $=
|
|
( vy vx cun ccnv cv wbr copab df-cnv unopab brun opabbii eqtr4i uneq12i
|
|
wo ) ABEZFZCGZDGZAHZDCIZSTBHZDCIZEZAFZBFZERSTQHZDCIZUEDCQJUEUAUCPZDCIUIUA
|
|
UCDCKUHUJDCSTABLMNNUFUBUGUDDCAJDCBJON $.
|
|
|
|
$( Distributive law for converse over set difference. (Contributed by
|
|
Mario Carneiro, 26-Jun-2014.) $)
|
|
cnvdif $p |- `' ( A \ B ) = ( `' A \ `' B ) $=
|
|
( vx vy cdif ccnv relcnv wss wrel difss relss mp2 cv cop wcel wn wa eldif
|
|
vex opelcnv notbii anbi12i bitri 3bitr4i eqrelriiv ) CDABEZFZAFZBFZEZUFGU
|
|
JUHHUHIUJIUHUIJAGUJUHKLDMZCMZNZUFOUMAOZUMBOZPZQZULUKNZUGOURUJOZUMABRULUKU
|
|
FCSZDSZTUSURUHOZURUIOZPZQUQURUHUIRVBUNVDUPULUKAUTVATVCUOULUKBUTVATUAUBUCU
|
|
DUE $.
|
|
|
|
$( Distributive law for converse over intersection. Theorem 15 of [Suppes]
|
|
p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro,
|
|
26-Jun-2014.) $)
|
|
cnvin $p |- `' ( A i^i B ) = ( `' A i^i `' B ) $=
|
|
( vy vx cin ccnv cv wbr copab df-cnv inopab brin opabbii eqtr4i ineq12i
|
|
wa ) ABEZFZCGZDGZAHZDCIZSTBHZDCIZEZAFZBFZERSTQHZDCIZUEDCQJUEUAUCPZDCIUIUA
|
|
UCDCKUHUJDCSTABLMNNUFUBUGUDDCAJDCBJON $.
|
|
$}
|
|
|
|
$( Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
|
|
(Contributed by NM, 24-Mar-1998.) $)
|
|
rnun $p |- ran ( A u. B ) = ( ran A u. ran B ) $=
|
|
( cun ccnv cdm crn cnvun dmeqi dmun eqtri df-rn uneq12i 3eqtr4i ) ABCZDZEZA
|
|
DZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.
|
|
|
|
$( The range of an intersection belongs the intersection of ranges. Theorem
|
|
9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) $)
|
|
rnin $p |- ran ( A i^i B ) C_ ( ran A i^i ran B ) $=
|
|
( cin ccnv cdm crn cnvin dmeqi dmin eqsstri df-rn ineq12i 3sstr4i ) ABCZDZE
|
|
ZADZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d y z B $.
|
|
$( The range of an indexed union. (Contributed by Mario Carneiro,
|
|
29-May-2015.) $)
|
|
rniun $p |- ran U_ x e. A B = U_ x e. A ran B $=
|
|
( vz vy ciun crn cv cop wcel wex wrex rexcom4 elrn2 rexbii eliun 3bitr4ri
|
|
vex exbii 3bitr4i eqriv ) DABCFZGZABCGZFZEHDHZIZUBJZEKZUFUDJZABLZUFUCJUFU
|
|
EJUGCJZEKZABLULABLZEKUKUIULAEBMUJUMABEUFCDRZNOUHUNEAUGBCPSQEUFUBUONAUFBUD
|
|
PTUA $.
|
|
|
|
$d x A $.
|
|
$( The range of a union. Part of Exercise 8 of [Enderton] p. 41.
|
|
(Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro,
|
|
29-May-2015.) $)
|
|
rnuni $p |- ran U. A = U_ x e. A ran x $=
|
|
( cuni crn cv ciun uniiun rneqi rniun eqtri ) BCZDABAEZFZDABLDFKMABGHABLI
|
|
J $.
|
|
$}
|
|
|
|
$( Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
|
|
(Contributed by NM, 30-Sep-2002.) $)
|
|
imaundi $p |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) ) $=
|
|
( cun cres crn cima resundi rneqi rnun eqtri df-ima uneq12i 3eqtr4i ) ABCDZ
|
|
EZFZABEZFZACEZFZDZAOGABGZACGZDQRTDZFUBPUEABCHIRTJKAOLUCSUDUAABLACLMN $.
|
|
|
|
$( The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) $)
|
|
imaundir $p |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) ) $=
|
|
( cun cima cres crn df-ima resundir rneqi rnun 3eqtri uneq12i eqtr4i ) ABDZ
|
|
CEZACFZGZBCFZGZDZACEZBCEZDPOCFZGQSDZGUAOCHUDUEABCIJQSKLUBRUCTACHBCHMN $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $.
|
|
$( An upper bound for intersection with a domain. Theorem 40 of [Suppes]
|
|
p. 66, who calls it "somewhat surprising." (Contributed by NM,
|
|
11-Aug-2004.) $)
|
|
dminss $p |- ( dom R i^i A ) C_ ( `' R " ( R " A ) ) $=
|
|
( vx vy cdm cin ccnv cima cv wbr wcel wa 19.8a ancoms elima2 sylibr simpl
|
|
wex vex brcnv jca eximi eldm anbi1i elin 19.41v 3bitr4i 3imtr4i ssriv ) C
|
|
BEZAFZBGZBAHZHZCIZDIZBJZUOAKZLZDRZUPUMKZUPUOULJZLZDRUOUKKZUOUNKUSVCDUSVAV
|
|
BUSURUQLZCRZVAURUQVFVECMNCUPBADSZOPUSUQVBUQURQUPUOBVGCSZTPUAUBUOUJKZURLUQ
|
|
DRZURLVDUTVIVJURDUOBVHUCUDUOUJAUEUQURDUFUGDUOULUMVHOUHUI $.
|
|
|
|
$( An upper bound for intersection with an image. Theorem 41 of [Suppes]
|
|
p. 66. (Contributed by NM, 11-Aug-2004.) $)
|
|
imainss $p |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) ) $=
|
|
( vy vx cima cin ccnv cv wcel wbr wa wex brcnv 19.8a sylan2br elin elima2
|
|
vex anbi1i ancoms anim2i simprl anassrs anbi2i bitri sylibr eximi 3bitr4i
|
|
jca 19.41v 3imtr4i ssriv ) DCAFZBGZCACHZBFZGZFZEIZAJZUTDIZCKZLZVBBJZLZEMZ
|
|
UTURJZVCLZEMVBUOJZVBUSJVFVIEVFVAVEVBUTUPKZLZDMZLZVCLZVIVAVCVEVOVAVCVELZLV
|
|
NVCVPVMVAVEVCVMVCVEVKVMVBUTCDSZESZNVLDOPUAUBVAVCVEUCUJUDVHVNVCVHVAUTUQJZL
|
|
VNUTAUQQVSVMVADUTUPBVRRUEUFTUGUHVBUNJZVELVDEMZVELVJVGVTWAVEEVBCAVQRTVBUNB
|
|
QVDVEEUKUIEVBCURVQRULUM $.
|
|
$}
|
|
|
|
$( The image of an intersection (Contributed by Thierry Arnoux,
|
|
16-Dec-2017.) $)
|
|
inimass $p |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) ) $=
|
|
( cres cin crn cima rnin df-ima resindir rneqi eqtri ineq12i 3sstr4i ) ACDZ
|
|
BCDZEZFZOFZPFZEABEZCGZACGZBCGZEOPHUBUACDZFRUACIUEQABCJKLUCSUDTACIBCIMN $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x V $.
|
|
$( The intersection of the image of singleton (Contributed by Thierry
|
|
Arnoux, 16-Dec-2017.) $)
|
|
inimasn $p |- ( C e. V
|
|
-> ( ( A i^i B ) " { C } ) = ( ( A " { C } ) i^i ( B " { C } ) ) ) $=
|
|
( vx wcel cin csn cima cv wa elin cop a1i cvv vex elimasng mpan2 anbi12d
|
|
wb 3bitr4rd syl5rbb eqrdv ) CDFZEABGZCHZIZAUFIZBUFIZGZEJZUJFUKUHFZUKUIFZK
|
|
ZUDUKUGFZUKUHUILUDCUKMZUEFZUPAFZUPBFZKZUOUNUQUTTUDUPABLNUDUKOFZUOUQTEPZUE
|
|
CUKDOQRUDULURUMUSUDVAULURTVBACUKDOQRUDVAUMUSTVBBCUKDOQRSUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The converse of a cross product. Exercise 11 of [Suppes] p. 67.
|
|
(Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
cnvxp $p |- `' ( A X. B ) = ( B X. A ) $=
|
|
( vy vx cv wcel copab ccnv cxp cnvopab ancom opabbii eqtri cnveqi 3eqtr4i
|
|
wa df-xp ) CEAFZDEBFZPZCDGZHZSRPZDCGZABIZHBAIUBTDCGUDTCDJTUCDCRSKLMUEUACD
|
|
ABQNDCBAQO $.
|
|
$}
|
|
|
|
$( The cross product with the empty set is empty. Part of Theorem 3.13(ii)
|
|
of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) $)
|
|
xp0 $p |- ( A X. (/) ) = (/) $=
|
|
( c0 cxp ccnv 0xp cnveqi cnvxp cnv0 3eqtr3i ) BACZDBDABCBJBAEFBAGHI $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( The cross product of inhabited classes is inhabited. (Contributed by
|
|
Jim Kingdon, 11-Dec-2018.) $)
|
|
xpmlem $p |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) ) $=
|
|
( cv wcel wex wa cxp eeanv cop vex opex wceq eleq1 opelxp syl6bb exlimivv
|
|
spcev sylbir elxp simpr 2eximi sylbi exlimiv sylib impbii ) AFZDGZAHBFZEG
|
|
ZBHIZCFZDEJZGZCHZUMUJULIZBHAHZUQUJULABKZURUQABUPURCUIUKLZUIUKAMBMNUNVAOZU
|
|
PVAUOGURUNVAUOPUIUKDEQRTSUAUQUSUMUPUSCUPVBURIZBHAHUSABUNDEUBVCURABVBURUCU
|
|
DUEUFUTUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y B $. $d z A $. $d z B $. $d a b w A $. $d a b w B $.
|
|
$d a x $. $d b y $. $d z w $.
|
|
$( The cross product of inhabited classes is inhabited. (Contributed by
|
|
Jim Kingdon, 13-Dec-2018.) $)
|
|
xpm $p |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) ) $=
|
|
( va vb vw cv wcel wex wa cxp xpmlem eleq1 cbvexv anbi12i 3bitr3i ) FIZDJ
|
|
ZFKZGIZEJZGKZLHIZDEMZJZHKAIZDJZAKZBIZEJZBKZLCIZUFJZCKFGHDENUAUJUDUMTUIFAS
|
|
UHDOPUCULGBUBUKEOPQUGUOHCUEUNUFOPR $.
|
|
$}
|
|
|
|
$( A cross product is empty if at least one member is empty. (Contributed by
|
|
Jim Kingdon, 12-Dec-2018.) $)
|
|
xpeq0r $p |- ( ( A = (/) \/ B = (/) ) -> ( A X. B ) = (/) ) $=
|
|
( c0 wceq cxp xpeq1 0xp syl6eq xpeq2 xp0 jaoi ) ACDZABEZCDBCDZLMCBECACBFBGH
|
|
NMACECBCAIAJHK $.
|
|
|
|
$( Cross products with disjoint sets are disjoint. (Contributed by NM,
|
|
13-Sep-2004.) $)
|
|
xpdisj1 $p |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) ) $=
|
|
( cin c0 wceq cxp inxp xpeq1 0xp syl6eq syl5eq ) ABEZFGZACHBDHENCDEZHZFACBD
|
|
IOQFPHFNFPJPKLM $.
|
|
|
|
$( Cross products with disjoint sets are disjoint. (Contributed by NM,
|
|
13-Sep-2004.) $)
|
|
xpdisj2 $p |- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) ) $=
|
|
( cin c0 wceq cxp inxp xpeq2 xp0 syl6eq syl5eq ) ABEZFGZCAHDBHECDEZNHZFCADB
|
|
IOQPFHFNFPJPKLM $.
|
|
|
|
$( Cross products with two different singletons are disjoint. (Contributed
|
|
by NM, 28-Jul-2004.) $)
|
|
xpsndisj $p |- ( B =/= D -> ( ( A X. { B } ) i^i ( C X. { D } ) ) = (/) ) $=
|
|
( wne csn cin c0 wceq cxp disjsn2 xpdisj2 syl ) BDEBFZDFZGHIANJCOJGHIBDKNOA
|
|
CLM $.
|
|
|
|
${
|
|
$d x A $. $d y B $.
|
|
$( Disjoint unions with disjoint index sets are disjoint. (Contributed by
|
|
Stefan O'Rear, 21-Nov-2014.) $)
|
|
djudisj $p |- ( ( A i^i B ) = (/) -> ( U_ x e. A ( { x } X. C ) i^i
|
|
U_ y e. B ( { y } X. D ) ) = (/) ) $=
|
|
( cin c0 wceq cv csn cxp ciun cvv wss djussxp incom syl5eq ssdisj sylancr
|
|
xpdisj1 ) CDGHIZACAJKELMZCNLZOUDBDBJKFLMZGZHIUCUEGHIACEPUBUFUEUDGZHUDUEQU
|
|
BUEDNLZOUHUDGZHIUGHIBDFPUBUIUDUHGHUHUDQCDNNUARUEUHUDSTRUCUDUEST $.
|
|
$}
|
|
|
|
$( A double restriction to disjoint classes is the empty set. (Contributed
|
|
by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
resdisj $p |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) ) $=
|
|
( cin c0 wceq cres resres reseq2 res0 syl6eq syl5eq ) ABDZEFZCAGBGCMGZECABH
|
|
NOCEGEMECICJKL $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37,
|
|
with non-empty changed to inhabited. (Contributed by Jim Kingdon,
|
|
12-Dec-2018.) $)
|
|
rnxpm $p |- ( E. x x e. A -> ran ( A X. B ) = B ) $=
|
|
( cv wcel wex cxp crn cdm ccnv df-rn cnvxp dmeqi eqtri dmxpm syl5eq ) ADB
|
|
EAFBCGZHZCBGZIZCRQJZITQKUASBCLMNACBOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The domain of a cross product is a subclass of the first factor.
|
|
(Contributed by NM, 19-Mar-2007.) $)
|
|
dmxpss $p |- dom ( A X. B ) C_ A $=
|
|
( vx vy cxp cdm cv wcel cop wex vex eldm2 opelxp1 exlimiv sylbi ssriv ) C
|
|
ABEZFZACGZRHSDGZIQHZDJSAHZDSQCKLUAUBDSTABMNOP $.
|
|
$}
|
|
|
|
$( The range of a cross product is a subclass of the second factor.
|
|
(Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
rnxpss $p |- ran ( A X. B ) C_ B $=
|
|
( cxp crn ccnv cdm df-rn cnvxp dmeqi dmxpss eqsstri ) ABCZDLEZFZBLGNBACZFBM
|
|
OABHIBAJKK $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The range of a square cross product. (Contributed by FL,
|
|
17-May-2010.) $)
|
|
rnxpid $p |- ran ( A X. A ) = A $=
|
|
( vx vy cxp crn rnxpss cv wcel cop wex wa opelxp anidm bitri weq wb opeq1
|
|
eleq1d equcoms biimpd spimev sylbir vex elrn2 sylibr ssriv eqssi ) AADZEZ
|
|
AAAFBAUIBGZAHZCGZUJIZUHHZCJZUJUIHUKUJUJIZUHHZUOUQUKUKKUKUJUJAALUKMNUQUNCB
|
|
CBOUQUNUQUNPBCBCOUPUMUHUJULUJQRSTUAUBCUJUHBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x a b A $. $d x a b B $.
|
|
$( A cross-product subclass relationship is equivalent to the relationship
|
|
for its components. (Contributed by Jim Kingdon, 12-Dec-2018.) $)
|
|
ssxpbm $p |- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <->
|
|
( A C_ C /\ B C_ D ) ) ) $=
|
|
( va vb cv cxp wcel wex wss cdm wceq adantl sylbir adantr eqsstr3d syl6ss
|
|
wa xpm dmxpm dmss dmxpss crn rnxpm rnss rnxpss jca ex xpss12 impbid1 ) AH
|
|
BCIZJAKZUMDEIZLZBDLZCELZTZUNUPUSUNUPTZUQURUTBUOMZDUTBUMMZVAUNVBBNZUPUNFHB
|
|
JFKZGHCJGKZTZVCFGABCUAZVEVCVDGBCUBOPQUPVBVALUNUMUOUCORDEUDSUTCUOUEZEUTCUM
|
|
UEZVHUNVICNZUPUNVFVJVGVDVJVEFBCUFQPQUPVIVHLUNUMUOUGORDEUHSUIUJBDCEUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Cross product subset cancellation. (Contributed by Jim Kingdon,
|
|
14-Dec-2018.) $)
|
|
ssxp1 $p |- ( E. x x e. C ->
|
|
( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) ) $=
|
|
( cv wcel wex cxp wss cdm wceq dmxpm adantr adantl eqsstr3d dmxpss syl6ss
|
|
wa dmss ex xpss1 impbid1 ) AEDFAGZBDHZCDHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
|
|
UIUCUJBKUFABDLMUFUJUIIUCUDUESNOCDPQTBCDUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Cross product subset cancellation. (Contributed by Jim Kingdon,
|
|
14-Dec-2018.) $)
|
|
ssxp2 $p |- ( E. x x e. C ->
|
|
( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) ) $=
|
|
( cv wcel wex cxp wss crn wceq rnxpm adantr adantl eqsstr3d rnxpss syl6ss
|
|
wa rnss ex xpss2 impbid1 ) AEDFAGZDBHZDCHZIZBCIZUCUFUGUCUFRZBUEJZCUHBUDJZ
|
|
UIUCUJBKUFADBLMUFUJUIIUCUDUESNODCPQTBCDUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y B $. $d A z $. $d B z $. $d C z $. $d D z $.
|
|
$( The cross product of inhabited classes is one-to-one. (Contributed by
|
|
Jim Kingdon, 13-Dec-2018.) $)
|
|
xp11m $p |- ( ( E. x x e. A /\ E. y y e. B )
|
|
-> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) ) $=
|
|
( vz cv wcel wex wa cxp wceq wi xpm anidm wss ssxpbm syl5ibcom eqss eleq2
|
|
exbidv anbi2d syl5bbr eqimss eqimss2 anim12d anbi12i bitr4i syl6ib sylbid
|
|
an4 com12 sylbi xpeq12 impbid1 ) AHCIAJBHDIBJKZCDLZEFLZMZCEMZDFMZKZUQGHZU
|
|
RIZGJZUTVCNABGCDOUTVFVCUTVFVFVDUSIZGJZKZVCVFVFVFKUTVIVFPUTVFVHVFUTVEVGGUR
|
|
USVDUAUBUCUDUTVICEQZDFQZKZECQZFDQZKZKZVCUTVFVLVHVOUTURUSQVFVLURUSUEGCDEFR
|
|
SUTUSURQVHVOUSURUFGEFCDRSUGVPVJVMKZVKVNKZKVCVJVKVMVNULVAVQVBVRCETDFTUHUIU
|
|
JUKUMUNCEDFUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Cancellation law for cross-product. (Contributed by Jim Kingdon,
|
|
14-Dec-2018.) $)
|
|
xpcanm $p |- ( E. x x e. C -> ( ( C X. A ) = ( C X. B ) <-> A = B ) ) $=
|
|
( cv wcel wex cxp wss wa wceq ssxp2 anbi12d eqss 3bitr4g ) AEDFAGZDBHZDCH
|
|
ZIZRQIZJBCIZCBIZJQRKBCKPSUATUBABCDLACBDLMQRNBCNO $.
|
|
|
|
$( Cancellation law for cross-product. (Contributed by Jim Kingdon,
|
|
14-Dec-2018.) $)
|
|
xpcan2m $p |- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) ) $=
|
|
( cv wcel wex cxp wss wa wceq ssxp1 anbi12d eqss 3bitr4g ) AEDFAGZBDHZCDH
|
|
ZIZRQIZJBCIZCBIZJQRKBCKPSUATUBABCDLACBDLMQRNBCNO $.
|
|
$}
|
|
|
|
${
|
|
$d x a A $. $d x b B $.
|
|
$( If a nonempty cross product is a set, so are both of its components.
|
|
(Contributed by Jim Kingdon, 14-Dec-2018.) $)
|
|
xpexr2m $p |- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) ->
|
|
( A e. _V /\ B e. _V ) ) $=
|
|
( va vb cv cxp wcel wex wa cvv xpm cdm dmxpm adantl dmexg adantr eqeltrrd
|
|
wceq crn rnxpm rnexg anim12dan ancom2s sylan2br ) AGBCHZIAJUGDIZEGBIEJZFG
|
|
CIFJZKBLIZCLIZKZEFABCMUHUJUIUMUHUJUKUIULUHUJKUGNZBLUJUNBTUHFBCOPUHUNLIUJU
|
|
GDQRSUHUIKUGUAZCLUIUOCTUHEBCUBPUHUOLIUIUGDUCRSUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Subset of the range of a restriction. (Contributed by NM,
|
|
16-Jan-2006.) $)
|
|
ssrnres $p |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B ) $=
|
|
( vy vx cxp cin crn wceq wss rnss ax-mp cvv cv wex wa elrn2 bitr2i 3bitri
|
|
wcel cres inss2 rnxpss sstri eqss mpbiran ssv xpss12 mp2an sslin sseqtr4i
|
|
ssid df-res sstr mpan2 cop ssel syl6ib ancrd opelxp anbi2i opelres anbi1i
|
|
vex elin anass exbii 19.41v syl6ibr ssrdv impbii ) CABFZGZHZBIZBVNJZBCAUA
|
|
ZHZJZVOVNBJVPVNVLHZBVMVLJVNVTJCVLUBVMVLKLABUCUDVNBUEUFVPVSVPVNVRJZVSVMVQJ
|
|
WAVMCAMFZGZVQVLWBJZVMWCJAAJBMJWDAULBUGAABMUHUIVLWBCUJLCAUMUKVMVQKLBVNVRUN
|
|
UOVSDBVNVSDNZBTZENZWEUPZVQTZEOZWFPZWEVNTZVSWFWJVSWFWEVRTWJBVRWEUQEWEVQDVD
|
|
ZQURUSWLWHVMTZEOWIWFPZEOWKEWEVMWMQWNWOEWNWHCTZWHVLTZPWPWGATZWFPZPZWOWHCVL
|
|
VEWQWSWPWGWEABUTVAWOWPWRPZWFPWTWIXAWFWGWECAWMVBVCWPWRWFVFRSVGWIWFEVHSVIVJ
|
|
VKR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $.
|
|
$( Range of the intersection with a cross product. (Contributed by NM,
|
|
17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
rninxp $p |- ( ran ( C i^i ( A X. B ) ) = B <->
|
|
A. y e. B E. x e. A x C y ) $=
|
|
( cres crn wss wcel wral cxp cin wceq wbr wrex dfss3 ssrnres cima df-ima
|
|
cv eleq2i vex elima bitr3i ralbii 3bitr3i ) DECFGZHBTZUGIZBDJECDKLGDMATUH
|
|
ENACOZBDJBDUGPCDEQUIUJBDUIUHECRZIUJUKUGUHECSUAAUHECBUBUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y C $.
|
|
$( Domain of the intersection with a cross product. (Contributed by NM,
|
|
17-Jan-2006.) $)
|
|
dminxp $p |- ( dom ( C i^i ( A X. B ) ) = A <->
|
|
A. x e. A E. y e. B x C y ) $=
|
|
( cxp cin cdm wceq ccnv crn cv wbr wrex wral dfdm4 cnvin cnvxp eqtri vex
|
|
ineq2i rneqi eqeq1i rninxp brcnv rexbii ralbii 3bitri ) ECDFZGZHZCIEJZDCF
|
|
ZGZKZCIBLZALZULMZBDNZACOUQUPEMZBDNZACOUKUOCUKUJJZKUOUJPVBUNVBULUIJZGUNEUI
|
|
QVCUMULCDRUASUBSUCBADCULUDUSVAACURUTBDUPUQEBTATUEUFUGUH $.
|
|
$}
|
|
|
|
$( Image of a relation restricted to a rectangular region. (Contributed by
|
|
Stefan O'Rear, 19-Feb-2015.) $)
|
|
imainrect $p |- ( ( G i^i ( A X. B ) ) " Y ) =
|
|
( ( G " ( Y i^i A ) ) i^i B ) $=
|
|
( cxp cin cres crn cima df-res rneqi df-ima eqtri ineq1i ccnv ineq2i eqtr4i
|
|
cvv cdm 3eqtr4ri cnvin inxp inv1 incom xpeq12i eqtr2i xpindir inass 3eqtr4i
|
|
in32 cnveqi cnvxp dmeqi dmres df-rn ) CABEZFZDGZHUQDREZFZHZUQDICDAFZIZBFZUR
|
|
UTUQDJKUQDLVDCVBREZFZHZBFZVAVCVGBVCCVBGZHVGCVBLVIVFCVBJKMNVFOZBGZSZUTOZSVHV
|
|
AVKVMVFRBEZFZOVJVNOZFZVMVKVFVNUAUTVOCUSFZUPFVRAREZVNFZFZUTVOUPVTVRVTARFZRBF
|
|
ZEUPARRBUBWBAWCBAUCWCBRFBRBUDBUCMUEUFPCUPUSUJVOVRVSFZVNFWAVFWDVNVFCUSVSFZFW
|
|
DVEWECDARUGPCUSVSUHQNVRVSVNUHMUIUKVKVJBREZFVQVJBJVPWFVJRBULPQTUMBVJSZFWGBFV
|
|
LVHBWGUDVJBUNVGWGBVFUONTUTUOTQUI $.
|
|
|
|
$( The image by a cross product. (Contributed by Thierry Arnoux,
|
|
16-Dec-2017.) $)
|
|
xpima1 $p |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) ) $=
|
|
( cin c0 wceq cxp cima cvv crn cres df-ima df-res rneqi 3eqtri xpeq1 syl6eq
|
|
inxp 0xp rneq rn0 syl syl5eq ) ACDZEFZABGZCHZUDBIDZGZJZEUGUFCKZJUFCIGDZJUJU
|
|
FCLUKULUFCMNULUIABCIRNOUEUIEFZUJEFUEUIEUHGEUDEUHPUHSQUMUJEJEUIETUAQUBUC $.
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
$( The image by a cross product. (Contributed by Thierry Arnoux,
|
|
16-Dec-2017.) $)
|
|
xpima2m $p |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B ) $=
|
|
( cv cin wcel wex cxp cima cvv cres df-ima df-res rneqi inxp 3eqtri rnxpm
|
|
crn syl5eq inv1 syl6eq ) AEBDFZGAHZBCIZDJZCKFZCUDUFUCUGIZSZUGUFUEDLZSUEDK
|
|
IFZSUIUEDMUJUKUEDNOUKUHBCDKPOQAUCUGRTCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d X x $.
|
|
$( The image of a singleton by a cross product. (Contributed by Thierry
|
|
Arnoux, 14-Jan-2018.) $)
|
|
xpimasn $p |- ( X e. A -> ( ( A X. B ) " { X } ) = B ) $=
|
|
( vx wcel cv csn cin wex cxp cima wceq snmg wss snssi dfss1 eleq2d exbidv
|
|
sylib mpbird xpima2m syl ) CAEZDFZACGZHZEZDIZABJUEKBLUCUHUDUEEZDIDCAMUCUG
|
|
UIDUCUFUEUDUCUEANUFUELCAOUEAPSQRTDABUEUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y R $.
|
|
$( The set of all ordered pairs in a class is the same as the double
|
|
converse. (Contributed by Mario Carneiro, 16-Aug-2015.) $)
|
|
cnvcnv3 $p |- `' `' R = { <. x , y >. | x R y } $=
|
|
( ccnv cv wbr copab df-cnv vex brcnv opabbii eqtri ) CDZDBEZAEZMFZABGONCF
|
|
ZABGABMHPQABNOCBIAIJKL $.
|
|
|
|
$( Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 29-Dec-1996.) $)
|
|
dfrel2 $p |- ( Rel R <-> `' `' R = R ) $=
|
|
( vx vy wrel ccnv wceq relcnv cv cop wcel vex opelcnv bitri eqrelriv mpan
|
|
releq mpbii impbii ) ADZAEZEZAFZUADZSUBTGZBCUAABHZCHZIZUAJUFUEITJUGAJUEUF
|
|
TBKZCKZLUFUEAUIUHLMNOUBUCSUDUAAPQR $.
|
|
|
|
$( A relation can be expressed as the set of ordered pairs in it.
|
|
(Contributed by Mario Carneiro, 16-Aug-2015.) $)
|
|
dfrel4v $p |- ( Rel R <-> R = { <. x , y >. | x R y } ) $=
|
|
( wrel ccnv wceq cv wbr copab dfrel2 eqcom cnvcnv3 eqeq2i 3bitri ) CDCEEZ
|
|
CFCOFCAGBGCHABIZFCJOCKOPCABCLMN $.
|
|
$}
|
|
|
|
$( The double converse of a class strips out all elements that are not
|
|
ordered pairs. (Contributed by NM, 8-Dec-2003.) $)
|
|
cnvcnv $p |- `' `' A = ( A i^i ( _V X. _V ) ) $=
|
|
( ccnv cvv cxp cin wceq wrel relcnv df-rel mpbi relxp dfrel2 sseqtr4i cnvin
|
|
wss dfss cnveqi inss2 mpbir eqtr3i 3eqtr2i ) ABZBZUCCCDZBZBZEZUBUEEZBZAUDEZ
|
|
UCUFOUCUGFUCUDUFUCGUCUDOUBHUCIJUDGUFUDFCCKUDLJMUCUFPJUBUENUJBZBZUIUJUKUHAUD
|
|
NQUJGZULUJFUMUJUDOAUDRUJISUJLJTUA $.
|
|
|
|
$( The double converse of a class equals its restriction to the universe.
|
|
(Contributed by NM, 8-Oct-2007.) $)
|
|
cnvcnv2 $p |- `' `' A = ( A |` _V ) $=
|
|
( ccnv cvv cxp cin cres cnvcnv df-res eqtr4i ) ABBACCDEACFAGACHI $.
|
|
|
|
$( The double converse of a class is a subclass. Exercise 2 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) $)
|
|
cnvcnvss $p |- `' `' A C_ A $=
|
|
( ccnv cvv cxp cin cnvcnv inss1 eqsstri ) ABBACCDZEAAFAIGH $.
|
|
|
|
$( Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) $)
|
|
cnveqb $p |- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) ) $=
|
|
( wrel wa wceq ccnv cnveq wi dfrel2 eqeq2 syl5ibr eqcoms sylbi eqeq1 imbi2d
|
|
imp impbid2 ) ACZBCZDABEZAFZBFZEZABGRSUCTHZRUAFZAESUDHZAIUFAUESUDAUEEZUCUEB
|
|
EZHZSUBFZBEUIBIUIBUJUCUHBUJEUEUJEUAUBGBUJUEJKLMUGTUHUCAUEBNOKLMPQ $.
|
|
|
|
$( A relation empty iff its converse is empty. (Contributed by FL,
|
|
19-Sep-2011.) $)
|
|
cnveq0 $p |- ( Rel A -> ( A = (/) <-> `' A = (/) ) ) $=
|
|
( c0 ccnv wceq wrel wb wi cnv0 rel0 cnveqb mpan2 eqeq2 bibi2d syl5ibr ax-mp
|
|
eqcoms ) BCZBDAEZABDZACZBDZFZGZHUCBQRUBBQDZSTQDZFZRBEUFIABJKUDUAUESBQTLMNPO
|
|
$.
|
|
|
|
$( Alternate definition of relation. (Contributed by NM, 14-May-2008.) $)
|
|
dfrel3 $p |- ( Rel R <-> ( R |` _V ) = R ) $=
|
|
( wrel ccnv wceq cvv cres dfrel2 cnvcnv2 eqeq1i bitri ) ABACCZADAEFZADAGKLA
|
|
AHIJ $.
|
|
|
|
$( The domain of a universal restriction. (Contributed by NM,
|
|
14-May-2008.) $)
|
|
dmresv $p |- dom ( A |` _V ) = dom A $=
|
|
( cvv cres cdm cin dmres incom inv1 3eqtri ) ABCDBADZEJBEJABFBJGJHI $.
|
|
|
|
$( The range of a universal restriction. (Contributed by NM,
|
|
14-May-2008.) $)
|
|
rnresv $p |- ran ( A |` _V ) = ran A $=
|
|
( ccnv crn cvv cres cnvcnv2 rneqi rncnvcnv eqtr3i ) ABBZCADEZCACJKAFGAHI $.
|
|
|
|
$( Range defined in terms of image. (Contributed by NM, 14-May-2008.) $)
|
|
dfrn4 $p |- ran A = ( A " _V ) $=
|
|
( cvv cima cres crn df-ima rnresv eqtr2i ) ABCABDEAEABFAGH $.
|
|
|
|
${
|
|
$d A w y $. $d B w y $. $d V w y $. $d x w y $.
|
|
$( Distribute proper substitution through the range of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbrng $p |- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B ) $=
|
|
( vw vy wcel cv cop wex cab csb crn wsbc csbabg sbcexg exbidv bitrd dfrn3
|
|
sbcel2g abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHIZCGZEJZFKZLZUFABCLZGZEJ
|
|
ZFKZABCMZLUKMUEUJUHABNZFKUNUHAFBDOUEUPUMFUEUPUGABNZEJUMUGEABDPUEUQULEABUF
|
|
CDTQRUAUBABUOUIEFCSUCEFUKSUD $.
|
|
$}
|
|
|
|
$( The restriction of the double converse of a class. (Contributed by NM,
|
|
8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
rescnvcnv $p |- ( `' `' A |` B ) = ( A |` B ) $=
|
|
( ccnv cres cvv cin cnvcnv2 reseq1i resres wss wceq ssv sseqin2 mpbi 3eqtri
|
|
reseq2i ) ACCZBDAEDZBDAEBFZDABDQRBAGHAEBISBABEJSBKBLBEMNPO $.
|
|
|
|
$( The double converse of the restriction of a class. (Contributed by NM,
|
|
3-Jun-2007.) $)
|
|
cnvcnvres $p |- `' `' ( A |` B ) = ( `' `' A |` B ) $=
|
|
( cres ccnv wrel wceq relres dfrel2 mpbi rescnvcnv eqtr4i ) ABCZDDZLADDBCLE
|
|
MLFABGLHIABJK $.
|
|
|
|
$( The image of the double converse of a class. (Contributed by NM,
|
|
8-Apr-2007.) $)
|
|
imacnvcnv $p |- ( `' `' A " B ) = ( A " B ) $=
|
|
( ccnv cres crn cima rescnvcnv rneqi df-ima 3eqtr4i ) ACCZBDZEABDZEKBFABFLM
|
|
ABGHKBIABIJ $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The domain of a singleton is inhabited iff the singleton argument is an
|
|
ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) $)
|
|
dmsnm $p |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } ) $=
|
|
( vy cvv cxp wcel cop wceq wex csn cdm elvv wbr vex eldm df-br opex exbii
|
|
cv elsn eqcom 3bitri bitr2i bitri ) BDDEFBASZCSZGZHZCIZAIUEBJZKFZAIACBLUI
|
|
UKAUKUEUFUJMZCIUICUEUJANZOULUHCULUGUJFUGBHUHUEUFUJPUGBUEUFUMCNQTUGBUAUBRU
|
|
CRUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The range of a singleton is inhabited iff the singleton argument is an
|
|
ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) $)
|
|
rnsnm $p |- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } ) $=
|
|
( cvv cxp wcel cv csn cdm wex crn dmsnm dmmrnm bitri ) BCCDEAFZBGZHEAINOJ
|
|
EAIABKAAOLM $.
|
|
$}
|
|
|
|
$( The domain of the singleton of the empty set is empty. (Contributed by
|
|
NM, 30-Jan-2004.) $)
|
|
dmsn0 $p |- dom { (/) } = (/) $=
|
|
( vx c0 csn cdm wceq cv wcel wn wal wex cvv cxp 0nelxp dmsnm mtbi alnex eq0
|
|
mpbir ) BCDZBEAFSGZHAIZUATAJZHBKKLGUBKKMABNOTAPRASQR $.
|
|
|
|
$( The converse of the singleton of the empty set is empty. (Contributed by
|
|
Mario Carneiro, 30-Aug-2015.) $)
|
|
cnvsn0 $p |- `' { (/) } = (/) $=
|
|
( c0 csn ccnv wceq crn dfdm4 dmsn0 eqtr3i wrel wb relcnv relrn0 ax-mp mpbir
|
|
cdm ) ABZCZADZQEZADZPOSAPFGHQIRTJPKQLMN $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The domain of a singleton is empty if the singleton's argument contains
|
|
the empty set. (Contributed by NM, 15-Dec-2008.) $)
|
|
dmsn0el $p |- ( (/) e. A -> dom { A } = (/) ) $=
|
|
( vx c0 wcel cv csn cdm wn wal wceq wex 0nelelxp con2i dmsnm sylnib alnex
|
|
cvv cxp sylibr eq0 ) CADZBEAFGZDZHBIZUBCJUAUCBKZHUDUAAQQRDZUEUFUAQQALMBAN
|
|
OUCBPSBUBTS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
relsn2m.1 $e |- A e. _V $.
|
|
$( A singleton is a relation iff it has an inhabited domain. (Contributed
|
|
by Jim Kingdon, 16-Dec-2018.) $)
|
|
relsn2m $p |- ( Rel { A } <-> E. x x e. dom { A } ) $=
|
|
( csn wrel cvv cxp wcel cv cdm wex relsn dmsnm bitri ) BDZEBFFGHAIOJHAKBC
|
|
LABMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x V $.
|
|
$( The domain of a singleton of an ordered pair is the singleton of the
|
|
first member. (Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
dmsnopg $p |- ( B e. V -> dom { <. A , B >. } = { A } ) $=
|
|
( vx vy wcel cop csn cdm cv wceq wex vex opth1 exlimiv opeq1 opeq2 eqeq1d
|
|
spcegv syl5 impbid2 eldm2 opex elsn exbii bitri velsn 3bitr4g eqrdv ) BCF
|
|
ZDABGZHZIZAHZUJDJZEJZGZUKKZELZUOAKZUOUMFZUOUNFUJUSUTURUTEUOUPABDMZEMZNOUT
|
|
UOBGZUKKZUJUSUOABPURVEEBCUPBKUQVDUKUPBUOQRSTUAVAUQULFZELUSEUOULVBUBVFUREU
|
|
QUKUOUPVBVCUCUDUEUFDAUGUHUI $.
|
|
|
|
$( The domain of an unordered pair of ordered pairs. (Contributed by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
dmpropg $p |- ( ( B e. V /\ D e. W ) ->
|
|
dom { <. A , B >. , <. C , D >. } = { A , C } ) $=
|
|
( wcel wa cop csn cdm cun cpr wceq dmsnopg uneq12 syl2an df-pr dmeqi dmun
|
|
eqtri 3eqtr4g ) BEGZDFGZHABIZJZKZCDIZJZKZLZAJZCJZLZUEUHMZKZACMUCUGULNUJUM
|
|
NUKUNNUDABEOCDFOUGULUJUMPQUPUFUILZKUKUOUQUEUHRSUFUITUAACRUB $.
|
|
$}
|
|
|
|
${
|
|
dmsnop.1 $e |- B e. _V $.
|
|
$( The domain of a singleton of an ordered pair is the singleton of the
|
|
first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
dmsnop $p |- dom { <. A , B >. } = { A } $=
|
|
( cvv wcel cop csn cdm wceq dmsnopg ax-mp ) BDEABFGHAGICABDJK $.
|
|
|
|
dmprop.1 $e |- D e. _V $.
|
|
$( The domain of an unordered pair of ordered pairs. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
dmprop $p |- dom { <. A , B >. , <. C , D >. } = { A , C } $=
|
|
( cvv wcel cop cpr cdm wceq dmpropg mp2an ) BGHDGHABICDIJKACJLEFABCDGGMN
|
|
$.
|
|
|
|
dmtpop.1 $e |- F e. _V $.
|
|
$( The domain of an unordered triple of ordered pairs. (Contributed by NM,
|
|
14-Sep-2011.) $)
|
|
dmtpop $p |- dom { <. A , B >. , <. C , D >. , <. E , F >. }
|
|
= { A , C , E } $=
|
|
( cop ctp cdm cpr csn cun df-tp dmeqi dmun dmprop dmsnop uneq12i 3eqtri
|
|
eqtr4i ) ABJZCDJZEFJZKZLZACMZENZOZACEKUHUDUEMZUFNZOZLULLZUMLZOUKUGUNUDUEU
|
|
FPQULUMRUOUIUPUJABCDGHSEFITUAUBACEPUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Double converse of a singleton of an ordered pair. (Unlike ~ cnvsn ,
|
|
this does not need any sethood assumptions on ` A ` and ` B ` .)
|
|
(Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
cnvcnvsn $p |- `' `' { <. A , B >. } = `' { <. B , A >. } $=
|
|
( vy vx cop csn ccnv relcnv cv wcel vex opelcnv wceq wa opth 3bitr4i opex
|
|
ancom elsn bitri eqrelriiv ) CDABEZFZGZGZBAEZFZGZUDHUGHCIZDIZEZUEJUJUIEZU
|
|
DJZUKUHJZUIUJUDCKZDKZLUKUCJZULUGJZUMUNUKUBMZULUFMZUQURUIAMZUJBMZNVBVANUSU
|
|
TVAVBRUIUJABUOUPOUJUIBAUPUOOPUKUBUIUJUOUPQSULUFUJUIUPUOQSPUJUIUCUPUOLUIUJ
|
|
UGUOUPLPTUA $.
|
|
|
|
$( The domain of the singleton of the singleton of a singleton.
|
|
(Contributed by Jim Kingdon, 16-Dec-2018.) $)
|
|
dmsnsnsng $p |- ( A e. _V -> dom { { { A } } } = { A } ) $=
|
|
( vx cop csn cdm wceq cvv vex opid sneq sneqd syl5eq dmeqd eqeq12d dmsnop
|
|
cv vtoclg ) BPZRCZDZEZRDZFADZDZDZEZUCFBAGRAFZUAUFUBUCUGTUEUGSUDUGSUBDUDRB
|
|
HZIUGUBUCRAJZKLKMUINRRUHOQ $.
|
|
$}
|
|
|
|
$( The range of a singleton of an ordered pair is the singleton of the second
|
|
member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro,
|
|
30-Apr-2015.) $)
|
|
rnsnopg $p |- ( A e. V -> ran { <. A , B >. } = { B } ) $=
|
|
( wcel cop csn crn cdm ccnv df-rn dfdm4 cnvcnvsn dmeqi 3eqtri eqtr4i syl5eq
|
|
dmsnopg ) ACDABEFZGZBAEFZHZBFSRIZHZUARJUATIZGUDIZHUCTKUDJUEUBBALMNOBACQP $.
|
|
|
|
$( The range of a pair of ordered pairs is the pair of second members.
|
|
(Contributed by Thierry Arnoux, 3-Jan-2017.) $)
|
|
rnpropg $p |- ( ( A e. V /\ B e. W )
|
|
-> ran { <. A , C >. , <. B , D >. } = { C , D } ) $=
|
|
( wcel cop cpr crn csn cun df-pr rneqi wceq rnsnopg adantr adantl uneq12d
|
|
wa rnun 3eqtr4g syl5eq ) AEGZBFGZTZACHZBDHZIZJUGKZUHKZLZJZCDIZUIULUGUHMNUFU
|
|
JJZUKJZLCKZDKZLUMUNUFUOUQUPURUDUOUQOUEACEPQUEUPUROUDBDFPRSUJUKUACDMUBUC $.
|
|
|
|
${
|
|
cnvsn.1 $e |- A e. _V $.
|
|
$( The range of a singleton of an ordered pair is the singleton of the
|
|
second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
rnsnop $p |- ran { <. A , B >. } = { B } $=
|
|
( cvv wcel cop csn crn wceq rnsnopg ax-mp ) ADEABFGHBGICABDJK $.
|
|
|
|
cnvsn.2 $e |- B e. _V $.
|
|
$( Extract the first member of an ordered pair. (See ~ op2nda to extract
|
|
the second member and ~ op1stb for an alternate version.) (Contributed
|
|
by Raph Levien, 4-Dec-2003.) $)
|
|
op1sta $p |- U. dom { <. A , B >. } = A $=
|
|
( cop csn cdm cuni dmsnop unieqi unisn eqtri ) ABEFGZHAFZHAMNABDIJACKL $.
|
|
|
|
$( Converse of a singleton of an ordered pair. (Contributed by NM,
|
|
11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
cnvsn $p |- `' { <. A , B >. } = { <. B , A >. } $=
|
|
( cop csn ccnv cnvcnvsn wrel wceq relsnop dfrel2 mpbi eqtr3i ) BAEFZGGZAB
|
|
EFGOBAHOIPOJBADCKOLMN $.
|
|
|
|
$( Extract the second member of an ordered pair. Theorem 5.12(ii) of
|
|
[Monk1] p. 52. (See ~ op1stb to extract the first member and ~ op2nda
|
|
for an alternate version.) (Contributed by NM, 25-Nov-2003.) $)
|
|
op2ndb $p |- |^| |^| |^| `' { <. A , B >. } = B $=
|
|
( cop csn ccnv cint cnvsn inteqi opex intsn eqtri op1stb ) ABEFGZHZHZHBAE
|
|
ZHZHBQSPRPRFZHROTABCDIJRBADCKLMJJBADCNM $.
|
|
|
|
$( Extract the second member of an ordered pair. (See ~ op1sta to extract
|
|
the first member and ~ op2ndb for an alternate version.) (Contributed
|
|
by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon,
|
|
27-Aug-2011.) $)
|
|
op2nda $p |- U. ran { <. A , B >. } = B $=
|
|
( cop csn crn cuni rnsnop unieqi unisn eqtri ) ABEFGZHBFZHBMNABCIJBDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Converse of a singleton of an ordered pair. (Contributed by NM,
|
|
23-Jan-2015.) $)
|
|
cnvsng $p |- ( ( A e. V /\ B e. W ) ->
|
|
`' { <. A , B >. } = { <. B , A >. } ) $=
|
|
( vx vy cv cop csn ccnv wceq opeq1 sneqd cnveqd opeq2 eqeq12d vex vtocl2g
|
|
cnvsn ) EGZFGZHZIZJZUATHZIZKAUAHZIZJZUAAHZIZKABHZIZJZBAHZIZKEFABCDTAKZUDU
|
|
IUFUKUQUCUHUQUBUGTAUALMNUQUEUJTAUAOMPUABKZUIUNUKUPURUHUMURUGULUABAOMNURUJ
|
|
UOUABALMPTUAEQFQSR $.
|
|
|
|
$( Swap the members of an ordered pair. (Contributed by Jim Kingdon,
|
|
16-Dec-2018.) $)
|
|
opswapg $p |- ( ( A e. V /\ B e. W ) ->
|
|
U. `' { <. A , B >. } = <. B , A >. ) $=
|
|
( wcel cop csn ccnv cuni cnvsng unieqd cvv wceq elex opexg syl2anr unisng
|
|
wa syl eqtrd ) ACEZBDEZRZABFGHZIBAFZGZIZUEUCUDUFABCDJKUCUELEZUGUEMUBBLEAL
|
|
EUHUABDNACNBALLOPUELQST $.
|
|
|
|
$( Membership in a cross product. This version requires no quantifiers or
|
|
dummy variables. See also ~ elxp5 . (Contributed by NM,
|
|
17-Feb-2004.) $)
|
|
elxp4 $p |- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >.
|
|
/\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) ) $=
|
|
( vx vy wcel cvv csn cdm cuni crn cop wceq wa elex eleq1 wex unieqd syl
|
|
wb cxp anim12i opexg adantl adantr mpbird sylan2 cv elxp a1i rneqd op2nda
|
|
sneq vex syl6req anbi1i anass bitri exbii snexg rnexg uniexg opeq2 eqeq2d
|
|
pm4.71ri anbi2d anbi12d ceqsexgv syl5bb dmeqd dmsnopg unisn syl6eq eqtr2d
|
|
ex pm4.71rd anbi1d 3bitrd exbidv dmexg opeq1 pm5.21nii ) ABCUAZFZAGFZAAHZ
|
|
IZJZWFKZJZLZMZWHBFZWJCFZNZNZAWCOWOWLWHGFZWJGFZNZWEWMWQWNWRWHBOWJCOUBWLWSN
|
|
WEWKGFZWSWTWLWHWJGGUCUDWLWEWTTWSAWKGPUEUFUGWEWDADUHZEUHZLZMZXABFZXBCFZNZN
|
|
ZEQZDQZXAWHMZAXAWJLZMZXEWNNZNZNZDQZWPWDXJTWEDEABCUIUJWEXIXPDWEXIXOXKXMNZX
|
|
NNZXPXIXBWJMZXHNZEQZWEXOXHYAEXHXTXDNZXGNYAXDYCXGXDXTXDWJXCHZKZJXBXDWIYEXD
|
|
WFYDAXCUMUKRXAXBDUNZEUNULUOVEUPXTXDXGUQURUSWEWRYBXOTWEWIGFZWRWEWFGFZYGAGU
|
|
TZWFGVASWIGVBSZXHXOEWJGXTXDXMXGXNXTXCXLAXBWJXAVCVDXTXFWNXEXBWJCPVFVGVHSVI
|
|
WEXMXRXNWEXMXKWEXMXKWEXMNWHXLHZIZJZXAXMWHYMMWEXMWGYLXMWFYKAXLUMVJRUDWEYMX
|
|
AMXMWEYMXAHZJXAWEYLYNWEWRYLYNMYJXAWJGVKSRXAYFVLVMUEVNVOVPVQXSXPTWEXKXMXNU
|
|
QUJVRVSWEWQXQWPTWEWGGFZWQWEYHYOYIWFGVTSWGGVBSXOWPDWHGXKXMWLXNWOXKXLWKAXAW
|
|
HWJWAVDXKXEWMWNXAWHBPVQVGVHSVRWB $.
|
|
|
|
$( Membership in a cross product requiring no quantifiers or dummy
|
|
variables. Provides a slightly shorter version of ~ elxp4 when the
|
|
double intersection does not create class existence problems (caused by
|
|
~ int0 ). (Contributed by NM, 1-Aug-2004.) $)
|
|
elxp5 $p |- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >.
|
|
/\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) $=
|
|
( vx vy wcel cvv cint csn crn cuni cop wceq wa elex eleq1 adantr wex syl
|
|
wb cxp anim12i opexg adantl mpbird sylan2 cv elxp rneqd unieqd vex op2nda
|
|
syl6req pm4.71ri anbi1i anass bitri exbii snexg rnexg uniexg opeq2 eqeq2d
|
|
sneq anbi2d anbi12d ceqsexgv syl5bb inteq inteqd op1stbg sylancr pm4.71rd
|
|
eqtr2d anbi1d a1i 3bitrd exbidv eqvisset exlimiv ad2antrl opeq1 pm5.21nii
|
|
ex syl6bb ) ABCUAZFZAGFZAAHZHZAIZJZKZLZMZWJBFZWMCFZNZNZAWFOWRWOWJGFZWMGFZ
|
|
NZWHWPWTWQXAWJBOZWMCOUBWOXBNWHWNGFZXBXDWOWJWMGGUCUDWOWHXDTXBAWNGPQUEUFWHW
|
|
GDUGZWJMZAXEWMLZMZXEBFZWQNZNZNZDRZWSWGAXEEUGZLZMZXIXNCFZNZNZERZDRWHXMDEAB
|
|
CUHWHXTXLDWHXTXKXFXHNZXJNZXLXTXNWMMZXSNZERZWHXKXSYDEXSYCXPNZXRNYDXPYFXRXP
|
|
YCXPWMXOIZJZKXNXPWLYHXPWKYGAXOVDUIUJXEXNDUKZEUKULUMUNUOYCXPXRUPUQURWHXAYE
|
|
XKTWHWLGFZXAWHWKGFYJAGUSWKGUTSWLGVASZXSXKEWMGYCXPXHXRXJYCXOXGAXNWMXEVBVCY
|
|
CXQWQXIXNWMCPVEVFVGSVHWHXHYAXJWHXHXFWHXHXFWHXHNWJXGHZHZXEXHWJYMMWHXHWIYLA
|
|
XGVIVJUDWHYMXEMZXHWHXEGFXAYNYIYKXEWMGGVKVLQVNWDVMVOYBXLTWHXFXHXJUPVPVQVRV
|
|
HXMWTWSXLWTDXFWTXKDWJVSQVTWPWTWOWQXCWAXKWSDWJGXFXHWOXJWRXFXGWNAXEWJWMWBVC
|
|
XFXIWPWQXEWJBPVOVFVGWCWEWC $.
|
|
$}
|
|
|
|
${
|
|
$d s t A $. $d s t B $. $d s t F $.
|
|
$( An image under the converse of a restriction. (Contributed by Jeff
|
|
Hankins, 12-Jul-2009.) $)
|
|
cnvresima $p |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A ) $=
|
|
( vt vs cres ccnv cima cin cv cop wa wex vex elima3 anbi1i opelcnv bitr4i
|
|
wcel bitri elin opelres anbi2i anass exbii 19.41v 3bitr4ri eqriv ) DCAFZG
|
|
ZBHZCGZBHZAIZDJZUKSEJZBSZUPUOKZUJSZLZEMZUOUNSZEUOUJBDNZOUOUMSZUOASZLUQURU
|
|
LSZLZEMZVELZVBVAVDVHVEEUOULBVCOPUOUMAUAVAVGVELZEMVIUTVJEUTUQVFVELZLVJUSVK
|
|
UQUSUOUPKZUISZVKUPUOUIENZVCQVMVLCSZVELVKUOUPCAVNUBVFVOVEUPUOCVNVCQPRTUCUQ
|
|
VFVEUDRUEVGVEEUFTUGTUH $.
|
|
$}
|
|
|
|
$( A class restricted to its domain equals its double converse. (Contributed
|
|
by NM, 8-Apr-2007.) $)
|
|
resdm2 $p |- ( A |` dom A ) = `' `' A $=
|
|
( ccnv cdm cres rescnvcnv wrel relcnv resdm ax-mp dmcnvcnv reseq2i 3eqtr3ri
|
|
wceq ) ABZBZOCZDZAPDOAACZDAPEOFQOMNGOHIPRAAJKL $.
|
|
|
|
$( Restriction to the domain of a restriction. (Contributed by NM,
|
|
8-Apr-2007.) $)
|
|
resdmres $p |- ( A |` dom ( A |` B ) ) = ( A |` B ) $=
|
|
( cres cdm ccnv cvv cxp in12 df-res resdm2 eqtr3i ineq2i incom 3eqtri dmres
|
|
cin xpeq1i xpindir eqtri 3eqtr4i rescnvcnv ) AABCZDZCZAEEZBCZUBABFGZADZFGZP
|
|
ZPZUEUGPZUDUFUKUGAUIPZPUGUEPULAUGUIHUMUEUGAUHCUMUEAUHIAJKLUGUEMNUDAUCFGZPUK
|
|
AUCIUNUJAUNBUHPZFGUJUCUOFABOQBUHFRSLSUEBITABUAS $.
|
|
|
|
$( The image of the domain of a restriction. (Contributed by NM,
|
|
8-Apr-2007.) $)
|
|
imadmres $p |- ( A " dom ( A |` B ) ) = ( A " B ) $=
|
|
( cres cdm crn cima resdmres rneqi df-ima 3eqtr4i ) AABCZDZCZEKEALFABFMKABG
|
|
HALIABIJ $.
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $. $d y F $. $d x V $.
|
|
dmmpt2.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The preimage of a function in maps-to notation. (Contributed by Stefan
|
|
O'Rear, 25-Jan-2015.) $)
|
|
mptpreima $p |- ( `' F " C ) = { x e. A | B e. C } $=
|
|
( vy ccnv cima cv wcel wceq wa copab crab eqtri crn wex cab bitri cnvopab
|
|
cmpt df-mpt cnveqi imaeq1i df-ima resopab rneqi ancom anass exbii df-clel
|
|
cres 19.42v bicomi anbi2i abbii rnopab df-rab 3eqtr4i ) EHZDIAJBKZGJZCLZM
|
|
ZGANZDIZCDKZABOZVAVFDVAVEAGNZHVFEVJEABCUBVJFAGBCUCPUDVEAGUAPUEVGVFDUMZQZV
|
|
IVFDUFVLVCDKZVEMZGANZQZVIVKVOVEGADUGUHVNGRZASVBVHMZASVPVIVQVRAVQVBVDVMMZM
|
|
ZGRZVRVNVTGVNVEVMMVTVMVEUIVBVDVMUJTUKWAVBVSGRZMVRVBVSGUNWBVHVBVHWBGCDULUO
|
|
UPTTUQVNGAURVHABUSUTPPP $.
|
|
|
|
$( Converse singleton image of a function defined by maps-to. (Contributed
|
|
by Stefan O'Rear, 25-Jan-2015.) $)
|
|
mptiniseg $p |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } ) $=
|
|
( wcel ccnv csn cima crab wceq mptpreima elsn2g rabbidv syl5eq ) DFHZEIDJ
|
|
ZKCSHZABLCDMZABLABCSEGNRTUAABCDFOPQ $.
|
|
|
|
$( The domain of the mapping operation in general. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) $)
|
|
dmmpt $p |- dom F = { x e. A | B e. _V } $=
|
|
( cdm ccnv crn cvv cima wcel crab dfdm4 dfrn4 mptpreima 3eqtri ) DFDGZHQI
|
|
JCIKABLDMQNABCIDEOP $.
|
|
|
|
$d x A $.
|
|
$( The domain of a mapping is a subset of its base class. (Contributed by
|
|
Scott Fenton, 17-Jun-2013.) $)
|
|
dmmptss $p |- dom F C_ A $=
|
|
( cdm cvv wcel crab dmmpt ssrab2 eqsstri ) DFCGHZABIBABCDEJMABKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The domain of the mapping operation is the stated domain, if the
|
|
function value is always a set. (Contributed by Mario Carneiro,
|
|
9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.) $)
|
|
dmmptg $p |- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A ) $=
|
|
( wcel wral cvv crab cmpt cdm wceq elex ralimi rabid2 eqid dmmpt syl6reqr
|
|
sylibr ) CDEZABFZBCGEZABHZABCIZJTUAABFBUBKSUAABCDLMUAABNRABCUCUCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $.
|
|
$( A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25.
|
|
(Contributed by NM, 26-Jan-1997.) $)
|
|
relco $p |- Rel ( A o. B ) $=
|
|
( vx vz vy cv wbr wa wex ccom df-co relopabi ) CFDFZBGMEFAGHDICEABJCEDABK
|
|
L $.
|
|
|
|
$( Alternate definition of a class composition, using only one bound
|
|
variable. (Contributed by NM, 19-Dec-2008.) $)
|
|
dfco2 $p |- ( A o. B )
|
|
= U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) $=
|
|
( vy vz ccom cvv ccnv cv csn cima cxp wrel wcel cop wex vex elimasn bitri
|
|
wa ciun relco reliun relxp a1i mprgbir wb opelco2g mp2an wrex rexv opelxp
|
|
eliun opelcnv anbi12i exbii 3bitrri eqrelriiv ) DEBCFZAGCHZAIZJZKZBVBKZLZ
|
|
UAZBCUBVFMVEMZAGAGVEUCVGVAGNVCVDUDUEUFDIZEIZOZUSNZVHVAOCNZVAVIOBNZTZAPZVJ
|
|
VFNZVHGNVIGNVKVOUGDQZEQZAVHVIBCGGUHUIVPVJVENZAGUJVSAPVOAVJGVEUMVSAUKVSVNA
|
|
VSVHVCNZVIVDNZTVNVHVIVCVDULVTVLWAVMVTVAVHOUTNVLUTVAVHAQZVQRVAVHCWBVQUNSBV
|
|
AVIWBVRRUOSUPUQSUR $.
|
|
|
|
$( Generalization of ~ dfco2 , where ` C ` can have any value between
|
|
` dom A i^i ran B ` and ` _V ` . (Contributed by NM, 21-Dec-2008.)
|
|
(Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
dfco2a $p |- ( ( dom A i^i ran B ) C_ C -> ( A o. B )
|
|
= U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) ) $=
|
|
( vy vz vw cvv cv cima ciun wcel wrex wex wa cop vex sylbi 3bitr4g eliun
|
|
cdm crn cin wss ccom ccnv csn cxp dfco2 wceq wbr wb eliniseg ax-mp brelrn
|
|
elimasn opeldm anim12ci adantl exlimivv elxp elin 3imtr4i pm4.71rd exbidv
|
|
ssel syl5 rexv df-rex eqrdv syl5eq ) BUAZCUBZUCZDUDZBCUEAHCUFAIZUGZJZBVQJ
|
|
ZUHZKZADVTKZABCUIVOEWAWBVOEIZVTLZAHMZWDADMZWCWALWCWBLVOWDANVPDLZWDOZANWEW
|
|
FVOWDWHAVOWDWGWDVPVNLZVOWGWCFIZGIZPUJZWJVRLZWKVSLZOZOZGNFNVPVLLZVPVMLZOZW
|
|
DWIWPWSFGWOWSWLWMWRWNWQWMWJVPCUKZWRVPHLWMWTULAQZCVPWJHFQZUMUNWJVPCXBXAUOR
|
|
WNVPWKPBLWQBVPWKXAGQZUPVPWKBXAXCUQRURUSUTFGWCVRVSVAVPVLVMVBVCVNDVPVFVGVDV
|
|
EWDAVHWDADVISAWCHVTTAWCDVTTSVJVK $.
|
|
|
|
$( Class composition distributes over union. (Contributed by NM,
|
|
21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
coundi $p |- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) ) $=
|
|
( vx vz vy cv wbr wa wex copab cun ccom wo unopab brun anbi1i andir bitri
|
|
df-co exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGZEGZBHZUIFGAH
|
|
ZIZEJZDFKZUHUICHZUKIZEJZDFKZLZUHUIBCLZHZUKIZEJZDFKZABMZACMZLAUTMUSUMUQNZD
|
|
FKVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUJUONZUKIVHVAVIUKUHUIBCPQUJUOUKRSUAUL
|
|
UPEUBUCUDUEVEUNVFURDFEABTDFEACTUFDFEAUTTUG $.
|
|
|
|
$( Class composition distributes over union. (Contributed by NM,
|
|
21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
coundir $p |- ( ( A u. B ) o. C ) = ( ( A o. C ) u. ( B o. C ) ) $=
|
|
( vx vy vz cv wbr wa wex copab cun ccom wo unopab brun anbi2i bitri df-co
|
|
andi exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGEGZCHZUHFGZAHZ
|
|
IZEJZDFKZUIUHUJBHZIZEJZDFKZLZUIUHUJABLZHZIZEJZDFKZACMZBCMZLUTCMUSUMUQNZDF
|
|
KVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUIUKUONZIVHVAVIUIUHUJABPQUIUKUOTRUAULU
|
|
PEUBUCUDUEVEUNVFURDFEACSDFEBCSUFDFEUTCSUG $.
|
|
|
|
$( Restricted first member of a class composition. (Contributed by NM,
|
|
12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
cores $p |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) ) $=
|
|
( vz vy vx crn wss cv wbr cres wa wex copab ccom wcel wb vex brelrn df-co
|
|
ssel brres rbaib syl56 pm5.32d exbidv opabbidv 3eqtr4g ) BGZCHZDIZEIZBJZU
|
|
LFIZACKZJZLZEMZDFNUMULUNAJZLZEMZDFNUOBOABOUJURVADFUJUQUTEUJUMUPUSUMULUIPU
|
|
JULCPZUPUSQUKULBDRERSUICULUAUPUSVBULUNACFRUBUCUDUEUFUGDFEUOBTDFEABTUH $.
|
|
|
|
$( Associative law for the restriction of a composition. (Contributed by
|
|
NM, 12-Dec-2006.) $)
|
|
resco $p |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) ) $=
|
|
( vx vy vz ccom cres relres relco cv wbr wcel wex vex anbi1i 19.41v brres
|
|
wa brco an32 bitr4i exbii 3bitr2i 3bitr4i eqbrriv ) DEABGZCHZABCHZGZUGCIA
|
|
UIJDKZEKZUGLZUKCMZSZUKFKZUILZUPULALZSZFNZUKULUHLUKULUJLUOUKUPBLZURSZFNZUN
|
|
SVBUNSZFNUTUMVCUNFUKULABDOZEOZTPVBUNFQVDUSFVDVAUNSZURSUSVAURUNUAUQVGURUKU
|
|
PBCFORPUBUCUDUKULUGCVFRFUKULAUIVEVFTUEUF $.
|
|
|
|
$( Image of the composition of two classes. (Contributed by Jason
|
|
Orendorff, 12-Dec-2006.) $)
|
|
imaco $p |- ( ( A o. B ) " C ) = ( A " ( B " C ) ) $=
|
|
( vx vy vz ccom cima cv wbr wrex wcel wa wex df-rex vex elima exbii bitri
|
|
rexcom4 r19.41v brco rexbii anbi1i 3bitr4i 3bitr4ri eqriv ) DABGZCHZABCHZ
|
|
HZEIZDIZAJZEUJKULUJLZUNMZENZUMUKLUMUILZUNEUJOEUMAUJDPZQFIZULBJZUNMZENZFCK
|
|
ZVAFCKZUNMZENZURUQVDVBFCKZENVGVBFECTVHVFEVAUNFCUARSURUTUMUHJZFCKVDFUMUHCU
|
|
SQVIVCFCEUTUMABFPUSUBUCSUPVFEUOVEUNFULBCEPQUDRUEUFUG $.
|
|
|
|
$( The range of the composition of two classes. (Contributed by NM,
|
|
12-Dec-2006.) $)
|
|
rnco $p |- ran ( A o. B ) = ran ( A |` ran B ) $=
|
|
( vy vx vz ccom crn cres cv wbr wex wcel wa vex exbii excom ancom 3bitr4i
|
|
brco elrn 19.41v anbi2i brres bitr4i 3bitri eqriv ) CABFZGZABGZHZGZDIZCIZ
|
|
UGJZDKZEIZUMUJJZEKZUMUHLUMUKLUOULUPBJZUPUMAJZMZEKZDKVADKZEKURUNVBDEULUMAB
|
|
DNCNZSOVADEPVCUQEVCUTUPUILZMZUQUSDKZUTMUTVGMVCVFVGUTQUSUTDUAVEVGUTDUPBENT
|
|
UBRUPUMAUIVDUCUDOUEDUMUGVDTEUMUJVDTRUF $.
|
|
$}
|
|
|
|
$( The range of the composition of two classes. (Contributed by NM,
|
|
27-Mar-2008.) $)
|
|
rnco2 $p |- ran ( A o. B ) = ( A " ran B ) $=
|
|
( ccom crn cres cima rnco df-ima eqtr4i ) ABCDABDZEDAJFABGAJHI $.
|
|
|
|
$( The domain of a composition. Exercise 27 of [Enderton] p. 53.
|
|
(Contributed by NM, 4-Feb-2004.) $)
|
|
dmco $p |- dom ( A o. B ) = ( `' B " dom A ) $=
|
|
( ccom cdm ccnv crn cima dfdm4 cnvco rneqi rnco2 imaeq2i eqtr4i 3eqtri ) AB
|
|
CZDOEZFBEZAEZCZFZQADZGZOHPSABIJTQRFZGUBQRKUAUCQAHLMN $.
|
|
|
|
${
|
|
$d w x y z A $. $d w y z B $. $d w y z C $.
|
|
$( Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) $)
|
|
coiun $p |- ( A o. U_ x e. C B ) = U_ x e. C ( A o. B ) $=
|
|
( vy vz vw ciun ccom relco wrel cv wcel wbr wa wex wrex cop eliun df-br
|
|
reliun a1i mprgbir rexbii 3bitr4i anbi1i r19.41v bitr4i exbii rexcom4 vex
|
|
opelco bitri eqrelriiv ) EFBADCHZIZADBCIZHZBUOJURKUQKZADADUQUAUSALDMBCJUB
|
|
UCELZGLZUONZVAFLZBNZOZGPZUTVACNZVDOZGPZADQZUTVCRZUPMVKURMZVFVHADQZGPVJVEV
|
|
MGVEVGADQZVDOVMVBVNVDUTVARZUOMVOCMZADQVBVNAVODCSUTVAUOTVGVPADUTVACTUDUEUF
|
|
VGVDADUGUHUIVHAGDUJUHGUTVCBUOEUKZFUKZULVLVKUQMZADQVJAVKDUQSVSVIADGUTVCBCV
|
|
QVRULUDUMUEUN $.
|
|
$}
|
|
|
|
$( A composition is not affected by a double converse of its first argument.
|
|
(Contributed by NM, 8-Oct-2007.) $)
|
|
cocnvcnv1 $p |- ( `' `' A o. B ) = ( A o. B ) $=
|
|
( ccnv ccom cvv cres cnvcnv2 coeq1i crn wss wceq ssv cores ax-mp eqtri ) AC
|
|
CZBDAEFZBDZABDZPQBAGHBIZEJRSKTLABEMNO $.
|
|
|
|
$( A composition is not affected by a double converse of its second argument.
|
|
(Contributed by NM, 8-Oct-2007.) $)
|
|
cocnvcnv2 $p |- ( A o. `' `' B ) = ( A o. B ) $=
|
|
( ccnv ccom cres cnvcnv2 coeq2i resco wrel wceq relco dfrel3 mpbi 3eqtr2i
|
|
cvv ) ABCCZDABOEZDABDZOEZRPQABFGABOHRISRJABKRLMN $.
|
|
|
|
$( Absorption of a reverse (preimage) restriction of the second member of a
|
|
class composition. (Contributed by NM, 11-Dec-2006.) $)
|
|
cores2 $p |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) ) $=
|
|
( cdm wss ccnv cres ccom wceq dfdm4 sseq1i cores sylbi cnvco cocnvcnv1 wrel
|
|
crn relco dfrel2 mpbi eqtri 3eqtr4g cnveqd 3eqtr3g ) ADZCEZABFZCGZFZHZFZFZA
|
|
BHZFZFZUJUMUFUKUNUFUHAFZHZUGUPHZUKUNUFUPQZCEUQURIUEUSCAJKUGUPCLMUKUIFUPHUQA
|
|
UINUHUPOUAABNUBUCUJPULUJIAUIRUJSTUMPUOUMIABRUMSTUD $.
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Composition with the empty set. Theorem 20 of [Suppes] p. 63.
|
|
(Contributed by NM, 24-Apr-2004.) $)
|
|
co02 $p |- ( A o. (/) ) = (/) $=
|
|
( vx vy vz c0 ccom relco rel0 cv cop wcel wbr wa noel df-br mtbir intnanr
|
|
wex nex vex opelco 2false eqrelriiv ) BCAEFZEAEGHBIZCIZJZUDKZUGEKUHUEDIZE
|
|
LZUIUFALZMZDRULDUJUKUJUEUIJZEKUMNUEUIEOPQSDUEUFAEBTCTUAPUGNUBUC $.
|
|
|
|
$( Composition with the empty set. (Contributed by NM, 24-Apr-2004.) $)
|
|
co01 $p |- ( (/) o. A ) = (/) $=
|
|
( c0 ccnv ccom cnv0 cnvco coeq2i co02 3eqtri eqtr4i cnveqi wrel wceq rel0
|
|
dfrel2 mpbi relco 3eqtr3ri ) BCZCZBADZCZCZBUASUBSBUBEUBACZSDUDBDBBAFSBUDE
|
|
GUDHIJKBLTBMNBOPUALUCUAMBAQUAOPR $.
|
|
|
|
$( Composition with the identity relation. Part of Theorem 3.7(i) of
|
|
[Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) $)
|
|
coi1 $p |- ( Rel A -> ( A o. _I ) = A ) $=
|
|
( vx vy vz cid ccom wrel wceq relco cv cop wcel wbr wex vex opelco equcom
|
|
wa ideq bitri anbi1i exbii breq1 ceqsexv df-br eqrelriv mpan ) AEFZGAGUHA
|
|
HAEIBCUHABJZCJZKZUHLZUIUJAMZUKALULUIDJZEMZUNUJAMZRZDNZUMDUIUJAEBOZCOPURUN
|
|
UIHZUPRZDNUMUQVADUOUTUPUOUIUNHUTUIUNDOSBDQTUAUBUPUMDUIUSUNUIUJAUCUDTTUIUJ
|
|
AUETUFUG $.
|
|
|
|
$( Composition with the identity relation. Part of Theorem 3.7(i) of
|
|
[Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) $)
|
|
coi2 $p |- ( Rel A -> ( _I o. A ) = A ) $=
|
|
( wrel ccnv ccom cnvco wceq relcnv ax-mp cnveqi eqtr3i dfrel2 coeq2 coeq1
|
|
cid coi1 cnvi sylan9eq mpan2 sylbi biimpi 3eqtr3a ) ABZNCZACZCZDZUENADZAU
|
|
DNDZCUFUEUDNEUHUDUDBUHUDFAGUDOHIJUBUEAFZUFUGFZAKZUIUCNFZUJPUIULUFUCADUGUE
|
|
AUCLUCNAMQRSUBUIUKTUA $.
|
|
$}
|
|
|
|
$( Composition with a restricted identity relation. (Contributed by FL,
|
|
19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) $)
|
|
coires1 $p |- ( A o. ( _I |` B ) ) = ( A |` B ) $=
|
|
( ccnv cres ccom cocnvcnv1 wrel wceq relcnv coi1 ax-mp eqtr3i reseq1i resco
|
|
cid rescnvcnv ) ACZCZBDZAOBDEZABDAOEZBDSTUARBROEZUARAOFRGUBRHQIRJKLMAOBNLAB
|
|
PL $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $.
|
|
$( Associative law for class composition. Theorem 27 of [Suppes] p. 64.
|
|
Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for
|
|
any classes whatsoever, not just functions or even relations.
|
|
(Contributed by NM, 27-Jan-1997.) $)
|
|
coass $p |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) ) $=
|
|
( vx vy vz vw ccom relco cv wbr wa wex cop wcel brco exbii opelco 3bitr4i
|
|
vex excom anass 2exbii bitr4i anbi2i exdistr anbi1i 19.41v eqrelriiv ) DE
|
|
ABHZCHZABCHZHZUJCIAULIDJZFJZCKZUOGJZBKZUQEJZAKZLZLZGMFMZUPURLZUTLZFMZGMZU
|
|
NUSNZUKOZVHUMOZVCVBFMGMVGVBFGUAVEVBGFUPURUTUBUCUDUPUOUSUJKZLZFMUPVAGMZLZF
|
|
MVIVCVLVNFVKVMUPGUOUSABFTETZPUEQFUNUSUJCDTZVORUPVAFGUFSUNUQULKZUTLZGMVDFM
|
|
ZUTLZGMVJVGVRVTGVQVSUTFUNUQBCVPGTPUGQGUNUSAULVPVORVFVTGVDUTFUHQSSUI $.
|
|
$}
|
|
|
|
$( A relation is transitive iff its converse is transitive. (Contributed by
|
|
FL, 19-Sep-2011.) $)
|
|
relcnvtr $p |- ( Rel R ->
|
|
( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) ) $=
|
|
( wrel ccom wss ccnv cnvco cnvss syl5eqssr wceq wi sseq1 dfrel2 coeq1 coeq2
|
|
eqtrd id sseq12d biimpd sylbi com12 syl6bi mpsyl impbid2 ) ABZAACZADZAEZUGC
|
|
ZUGDZUFUHUEEUGAAFUEAGHUIUDUFUHEZUGEZUKCZIZUIUJUKDZUDUFJZUGUGFUHUGGUMUNULUKD
|
|
ZUOUJULUKKUDUPUFUDUKAIZUPUFJALUQUPUFUQULUEUKAUQULAUKCUEUKAUKMUKAANOUQPQRSTU
|
|
AUBTUC $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A relation is included in the cross product of its domain and range.
|
|
Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
relssdmrn $p |- ( Rel A -> A C_ ( dom A X. ran A ) ) $=
|
|
( vx vy wrel cdm crn cxp id cv cop wi wex 19.8a wa opelxp vex eldm2 elrn2
|
|
wcel anbi12i bitri sylanbrc a1i relssdv ) ADZBCAAEZAFZGZUEHBIZCIZJZASZUKU
|
|
HSZKUEULULCLZULBLZUMULCMULBMUMUIUFSZUJUGSZNUNUONUIUJUFUGOUPUNUQUOCUIABPQB
|
|
UJACPRTUAUBUCUD $.
|
|
$}
|
|
|
|
$( The converse is a subset of the cartesian product of range and domain.
|
|
(Contributed by Mario Carneiro, 2-Jan-2017.) $)
|
|
cnvssrndm $p |- `' A C_ ( ran A X. dom A ) $=
|
|
( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtr4i
|
|
wss ) ABZOCZODZEZADZACZEOFORNAGOHISPTQAJAKLM $.
|
|
|
|
$( Composition as a subset of the cross product of factors. (Contributed by
|
|
Mario Carneiro, 12-Jan-2017.) $)
|
|
cossxp $p |- ( A o. B ) C_ ( dom B X. ran A ) $=
|
|
( ccom cdm crn cxp wrel wss relco relssdmrn ax-mp dmcoss rncoss mp2an sstri
|
|
xpss12 ) ABCZQDZQEZFZBDZAEZFZQGQTHABIQJKRUAHSUBHTUCHABLABMRUASUBPNO $.
|
|
|
|
$( Two ways to describe the structure of a two-place operation. (Contributed
|
|
by NM, 17-Dec-2008.) $)
|
|
relrelss $p |- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) ) $=
|
|
( vx wrel cdm wa cvv cxp wss df-rel anbi2i crn relssdmrn ssv mpan2 sylan9ss
|
|
xpss12 xpss sstr sylibr dmss cv wcel wex wceq vn0m dmxpm ax-mp syl6sseq jca
|
|
impbii bitri ) ACZADZCZEULUMFFGZHZEZAUOFGZHZUNUPULUMIJUQUSULUPAUMAKZGZURALU
|
|
PUTFHVAURHUTMUMUOUTFPNOUSULUPUSAUOHZULUSURUOHVBUOFQAURUORNAISUSUMURDZUOAURT
|
|
BUAFUBBUCVCUOUDBUEBUOFUFUGUHUIUJUK $.
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( The membership relation for a relation is inherited by class union.
|
|
(Contributed by NM, 17-Sep-2006.) $)
|
|
unielrel $p |- ( ( Rel R /\ A e. R ) -> U. A e. U. R ) $=
|
|
( vx vy wrel wcel wa cv cop wceq wex elrel simpr wi vex uniopel a1i eleq1
|
|
cuni unieq eleq1d 3imtr4d exlimivv sylc ) BEZABFZGACHZDHZIZJZDKCKUFASZBSZ
|
|
FZCDABLUEUFMUJUFUMNCDUJUIBFZUISZULFZUFUMUNUPNUJUGUHBCODOPQAUIBRUJUKUOULAU
|
|
ITUAUBUCUD $.
|
|
$}
|
|
|
|
$( The double union of a relation is its field. (Contributed by NM,
|
|
17-Sep-2006.) $)
|
|
relfld $p |- ( Rel R -> U. U. R = ( dom R u. ran R ) ) $=
|
|
( wrel cuni cdm crn cun cxp wss relssdmrn 3syl unixpss syl6ss dmrnssfld a1i
|
|
uniss eqssd ) ABZACZCZADZAEZFZQSTUAGZCZCZUBQAUCHRUDHSUEHAIAUCORUDOJTUAKLUBS
|
|
HQAMNP $.
|
|
|
|
$( Restriction of a relation to its field. (Contributed by FL,
|
|
15-Apr-2012.) $)
|
|
relresfld $p |- ( Rel R -> ( R |` U. U. R ) = R ) $=
|
|
( wrel cuni cres wceq cdm crn wi relfld reseq2d resundi wa eqtr resss resdm
|
|
cun wss ssequn2 uneq1 eqeq2d ex syl6bi com3r sylbi syl5com sylancl pm2.43i
|
|
mpsyl ) ABZAACCZDZAEZUIUKAAFZAGZPZDZEZUPAUMDZAUNDZPZEZUIULHUIUJUOAAIJAUMUNK
|
|
UQVALUKUTEZUIULUKUPUTMUSAQZUIURAEZVBULHZAUNNAOVCAUSPZAEZVDVEHUSARVDVBVGULVD
|
|
VBUKVFEZVGULHVDUTVFUKURAUSSTVHVGULUKVFAMUAUBUCUDUHUEUFUG $.
|
|
|
|
$( Composition with the identity relation restricted to a relation's field.
|
|
(Contributed by FL, 2-May-2011.) $)
|
|
relcoi2 $p |- ( Rel R -> ( ( _I |` U. U. R ) o. R ) = R ) $=
|
|
( wrel cid cuni cres ccom crn wss wceq cdm cun dmrnssfld simpr sylbir ax-mp
|
|
wa unss cores mp1i coi2 eqtrd ) ABZCADDZEAFZCAFZAAGZUCHZUDUEIUBAJZUFKUCHZUG
|
|
ALUIUHUCHZUGPUGUHUFUCQUJUGMNOCAUCRSATUA $.
|
|
|
|
$( Composition with the identity relation restricted to a relation's field.
|
|
(Contributed by FL, 8-May-2011.) $)
|
|
relcoi1 $p |- ( Rel R -> ( R o. ( _I |` U. U. R ) ) = R ) $=
|
|
( wrel cid cuni cres ccom cun wceq wi resco reseq1 wa eqtr eqeq1 syl5ibr ex
|
|
syl com3l eqcoms mpcom cdm crn relfld resundi coeq2 coundi coi1 resdm uneq1
|
|
uneq2d wss resss ssequn2 mpbi syl6reqr mpsyl syl5com mpi syl5eq mp2b reseq2
|
|
coeq2d eqeq1d eqtrd ) ABZACADDZEZFZACFZAVFAUAZAUBZGZHZVEVHVIHZAUCVEVNVMACVL
|
|
EZFZVIHZVOCVJEZCVKEZGZHVPAVTFZHZVEVQICVJVKUDVOVTAUEVEVQWBWAVIHVEWAAVRFZAVSF
|
|
ZGZVIAVRVSUFVEVIVJEZWCHZWEVIHZACVJJVIAHZVEWGWHIZAUGZWIWFAVJEZHZVEWJVIAVJKWL
|
|
AHZVEWMWJIAUHWMWNVEWJWMWNVEWJIZWMWNLWFAHZWOWFWLAMWGWPVEWHWPVEWHIZIWCWFWCWFH
|
|
ZWPWQWRWPLWCAHZWQWCWFAMVIVKEZWDHZWSWEAWDGZHZWQACVKJWCAWDUIVEXAXCWHWIVEXAXCW
|
|
HIZIZWKWIWTAVKEZHZVEXEIVIAVKKXAXGVEXDXGVEXDIZIWDWTWDWTHZXGXHXIXGLZXBAXFGZHZ
|
|
XHXJWDXFAWDWTXFMUJXCXLVEWHXCXLWQXCXLLWEXKHZWQWEXBXKMVEWHXMXKVIHVEVIAXKWKXFA
|
|
UKXKAHAVKULXFAUMUNUOWEXKVINOQPRQPSRQTRUPQPSRQPRTUQTURUSVPWAVINOUTVMVHVPVIVM
|
|
VGVOAVFVLCVAVBVCOTWKVD $.
|
|
|
|
$( The double union of the converse of a class is its field. (Contributed by
|
|
NM, 4-Jun-2008.) $)
|
|
unidmrn $p |- U. U. `' A = ( dom A u. ran A ) $=
|
|
( ccnv cuni crn cdm cun wrel wceq relcnv ax-mp equncomi dfdm4 df-rn uneq12i
|
|
relfld eqtr4i ) ABZCCZQDZQEZFAEZADZFRTSQGRTSFHAIQOJKUASUBTALAMNP $.
|
|
|
|
$( if ` R ` is a relation, its double union equals the double union of its
|
|
converse. (Contributed by FL, 5-Jan-2009.) $)
|
|
relcnvfld $p |- ( Rel R -> U. U. R = U. U. `' R ) $=
|
|
( wrel cuni cdm crn cun ccnv relfld unidmrn syl6eqr ) ABACCADAEFAGCCAHAIJ
|
|
$.
|
|
|
|
$( Alternate definition of domain ~ df-dm that doesn't require dummy
|
|
variables. (Contributed by NM, 2-Aug-2010.) $)
|
|
dfdm2 $p |- dom A = U. U. ( `' A o. A ) $=
|
|
( ccnv ccom cuni cdm crn cun cnvco cocnvcnv2 eqtri unieqi eqtr3i wceq df-rn
|
|
unidmrn eqcomi dmcoeq ax-mp rncoeq dfdm4 eqtr4i uneq12i unidm 3eqtrri ) ABZ
|
|
ACZDZDZUFEZUFFZGZAEZULGULUFBZDZDUHUKUNUGUMUFUMUEUEBCUFUEAHUEAIJKKUFOLUIULUJ
|
|
ULUEEZAFZMZUIULMUPUOANPZUEAQRUJUEFZULUQUJUSMURUEASRATUAUBULUCUD $.
|
|
|
|
${
|
|
$d x a A $. $d x b B $.
|
|
$( The double class union of an inhabited cross product is the union of its
|
|
members. (Contributed by Jim Kingdon, 18-Dec-2018.) $)
|
|
unixpm $p |- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) ) $=
|
|
( vb va cv cxp wcel wex cuni cdm crn cun wrel relxp relfld ax-mp wa ancom
|
|
wceq xpm bitri dmxpm rnxpm uneq12 syl2an sylbir syl5eq ) AFBCGZHAIZUIJJZU
|
|
IKZUILZMZBCMZUINUKUNTBCOUIPQUJDFCHDIZEFBHEIZRZUNUOTZURUQUPRUJUPUQSEDABCUA
|
|
UBUPULBTUMCTUSUQDBCUCEBCUDULBUMCUEUFUGUH $.
|
|
$}
|
|
|
|
$( The union of an empty cross product is empty. (Contributed by Jim
|
|
Kingdon, 18-Dec-2018.) $)
|
|
unixp0im $p |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) ) $=
|
|
( cxp c0 wceq cuni unieq uni0 syl6eq ) ABCZDEJFDFDJDGHI $.
|
|
|
|
$( The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring]
|
|
p. 26. (Contributed by NM, 17-Mar-1998.) $)
|
|
cnvexg $p |- ( A e. V -> `' A e. _V ) $=
|
|
( wcel ccnv cdm crn cxp wss cvv wrel relcnv relssdmrn ax-mp df-rn syl5eqelr
|
|
rnexg dfdm4 dmexg xpexg syl2anc ssexg sylancr ) ABCZADZUDEZUDFZGZHZUGICZUDI
|
|
CUDJUHAKUDLMUCUEICUFICUIUCUEAFIANABPOUCUFAEIAQABROUEUFIISTUDUGIUAUB $.
|
|
|
|
${
|
|
cnvex.1 $e |- A e. _V $.
|
|
$( The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring]
|
|
p. 26. (Contributed by NM, 19-Dec-2003.) $)
|
|
cnvex $p |- `' A e. _V $=
|
|
( cvv wcel ccnv cnvexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( A relation is a set iff its converse is a set. (Contributed by FL,
|
|
3-Mar-2007.) $)
|
|
relcnvexb $p |- ( Rel R -> ( R e. _V <-> `' R e. _V ) ) $=
|
|
( wrel cvv wcel ccnv cnvexg wceq wi dfrel2 eleq1 syl5ib sylbi impbid2 ) ABZ
|
|
ACDZAEZCDZACFNPEZAGZQOHAIQRCDSOPCFRACJKLM $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Restriction of a class to a singleton. (Contributed by Mario Carneiro,
|
|
28-Dec-2014.) $)
|
|
ressn $p |- ( A |` { B } ) = ( { B } X. ( A " { B } ) ) $=
|
|
( vx vy csn cres cima cxp relres relxp cv cop wcel wa ancom elimasn elsni
|
|
vex sneqd imaeq2d eleq2d syl5bbr pm5.32i opelres opelxp 3bitr4i eqrelriiv
|
|
bitri ) CDABEZFZUIAUIGZHZAUIIUIUKJCKZDKZLZAMZUMUIMZNZUQUNUKMZNZUOUJMUOULM
|
|
URUQUPNUTUPUQOUQUPUSUPUNAUMEZGZMUQUSAUMUNCRDRZPUQVBUKUNUQVAUIAUQUMBUMBQST
|
|
UAUBUCUHUMUNAUIVCUDUMUNUIUKUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A a b x $. $d B a b $. $d A a b y $.
|
|
$( The converse of an intersection is the intersection of the converse.
|
|
(Contributed by Jim Kingdon, 18-Dec-2018.) $)
|
|
cnviinm $p |- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B ) $=
|
|
( va vb cv wcel wex ciin ccnv eleq1 cbvexv wrel relcnv cvv wss df-rel cop
|
|
wral wceq cxp wrex wi r19.2m expcom mpbi a1i mprg iinss syl sylibr wb vex
|
|
eliin ax-mp opelcnv ralbii bitri 3bitr4i eqrelriv sylancr sylbir sylbi
|
|
opex ) BGZCHZBIEGZCHZEIZACDJZKZACDKZJZUAZVGVIBEVFVHCLMVJAGZCHZAIZVOVQVIAE
|
|
VPVHCLMVRVLNVNNZVOVKOVRVNPPUBZQZVSVRVMVTQZACUCZWAWBVRWCUDACVRWBACTWCWBACU
|
|
EUFWBVQVMNWBDOVMRUGUHUIACVMVTUJUKVNRULEFVLVNFGZVHSZVKHZWEDHZACTZVHWDSZVLH
|
|
WIVNHZWEPHWFWHUMWDVHFUNZEUNZVEAWECDPUOUPVHWDVKWLWKUQWJWIVMHZACTZWHWIPHWJW
|
|
NUMVHWDWLWKVEAWICVMPUOUPWMWGACVHWDDWLWKUQURUSUTVAVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w R $.
|
|
$( The converse of a partial order relation is a partial order relation.
|
|
(Contributed by NM, 15-Jun-2005.) $)
|
|
cnvpom $p |- ( E. x x e. A -> ( R Po A <-> `' R Po A ) ) $=
|
|
( vw vy vz cv wbr wn wa wral wpo r19.26 ralbidv ralbii 3bitr4g vex ralcom
|
|
wi brcnv wcel wex ralidm r19.3rmv syl5rbb anbi1d syl5bb id breq12d notbid
|
|
ccnv weq cbvralv anbi12ci imbi12i anbi12i bitr2i bitri syl6bb df-po ) AGB
|
|
UAAUBZDGZVBCHZIZVBEGZCHZVEFGZCHZJZVBVGCHZSZJFBKZEBKDBKZVGVGCUKZHZIZVGVEVN
|
|
HZVEVBVNHZJZVGVBVNHZSZJZDBKZEBKFBKZBCLBVNLVAVLDBKZEBKWCFBKZEBKVMWDVAWEWFE
|
|
BVAWEVDDBKZVKFBKZJZDBKZWFVAVDFBKZWHJZDBKZWGDBKZWHDBKZJZWEWJWMWKDBKZWOJVAW
|
|
PWKWHDBMVAWQWNWOWNWGVAWQVDDBUCVAVDWKDBVDFABUDNUEUFUGVLWLDBVDVKFBMOWGWHDBM
|
|
PWJWBFBKZDBKWFWIWRDBWRVPFBKZWAFBKZJWIVPWAFBMWSWGWTWHVPVDFDBFDULZVOVCVOVGV
|
|
GCHXAVCVGVGCFQZXBTXAVGVBVGVBCXAUHZXCUIUGUJUMWAVKFBVSVIVTVJVQVHVRVFVGVECXB
|
|
EQZTVEVBCXDDQZTUNVGVBCXBXETUOOUPUQOWBDFBBRURUSNVLDEBBRWCFEBBRPDEFBCUTFEDB
|
|
VNUTP $.
|
|
|
|
$( The converse of a strict order relation is a strict order relation.
|
|
(Contributed by Jim Kingdon, 19-Dec-2018.) $)
|
|
cnvsom $p |- ( E. x x e. A -> ( R Or A <-> `' R Or A ) ) $=
|
|
( vy vz cv wcel wex wpo wbr wo wi wral ccnv wor cnvpom vex brcnv df-iso
|
|
wa orbi12i orcom imbi12i ralbii 2ralbii ralcom bitr3i a1i anbi12d 3bitr4g
|
|
wb bitri ) AFZBGAHZBCIZUMDFZCJZUMEFZCJZURUPCJZKZLZEBMZDBMABMZTBCNZIZUPUMV
|
|
EJZUPURVEJZURUMVEJZKZLZEBMZABMDBMZTBCOBVEOUNUOVFVDVMABCPVDVMUKUNVDVLDBMAB
|
|
MVMVLVCADBBVKVBEBVGUQVJVAUPUMCDQZAQZRVJUTUSKVAVHUTVIUSUPURCVNEQZRURUMCVPV
|
|
ORUAUTUSUBULUCUDUEVLADBBUFUGUHUIADEBCSDAEBVESUJ $.
|
|
$}
|
|
|
|
$( The composition of two sets is a set. (Contributed by NM,
|
|
19-Mar-1998.) $)
|
|
coexg $p |- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V ) $=
|
|
( wcel wa ccom cdm crn cxp wss cvv cossxp dmexg rnexg xpexg syl2anr sylancr
|
|
ssexg ) ACEZBDEZFABGZBHZAIZJZKUELEZUBLEABMUAUCLEUDLEUFTBDNACOUCUDLLPQUBUELS
|
|
R $.
|
|
|
|
${
|
|
coex.1 $e |- A e. _V $.
|
|
coex.2 $e |- B e. _V $.
|
|
$( The composition of two sets is a set. (Contributed by NM,
|
|
15-Dec-2003.) $)
|
|
coex $p |- ( A o. B ) e. _V $=
|
|
( cvv wcel ccom coexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x a c A $. $d x a c B $. $d x a c C $.
|
|
$( Composition of two cross products. (Contributed by Jim Kingdon,
|
|
20-Dec-2018.) $)
|
|
xpcom $p |- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) ) $=
|
|
( va vc cv wcel wex cxp wbr wa copab ccom ibar anbi1i brxp anbi12i anandi
|
|
ancom 3bitr4i exbii 19.41v bitr2i syl6rbb opabbidv df-co df-xp 3eqtr4g )
|
|
AGZCHZAIZEGZUJBCJZKZUJFGZCDJZKZLZAIZEFMUMBHZUPDHZLZEFMUQUNNBDJULUTVCEFULV
|
|
CULVCLZUTULVCOUTUKVCLZAIVDUSVEAVAUKLZUKVBLZLUKVALZVGLUSVEVFVHVGVAUKTPUOVF
|
|
URVGUMUJBCQUJUPCDQRUKVAVBSUAUBUKVCAUCUDUEUFEFAUQUNUGEFBDUHUI $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Definite description binder (inverted iota)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c iota $.
|
|
$( Extend class notation with Russell's definition description binder
|
|
(inverted iota). $)
|
|
cio $a class ( iota x ph ) $.
|
|
|
|
${
|
|
$d w x z $. $d ph w z $. $d ph w y $. $d x y $.
|
|
$( Soundness justification theorem for ~ df-iota . (Contributed by Andrew
|
|
Salmon, 29-Jun-2011.) $)
|
|
iotajust $p |- U. { y | { x | ph } = { y } } =
|
|
U. { z | { x | ph } = { z } } $=
|
|
( vw cab cv csn wceq sneq eqeq2d cbvabv eqtri unieqi ) ABFZCGZHZIZCFZODGZ
|
|
HZIZDFZSOEGZHZIZEFUCRUFCEPUDIQUEOPUDJKLUFUBEDUDTIUEUAOUDTJKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
$( Define Russell's definition description binder, which can be read as
|
|
"the unique ` x ` such that ` ph ` ," where ` ph ` ordinarily contains
|
|
` x ` as a free variable. Our definition is meaningful only when there
|
|
is exactly one ` x ` such that ` ph ` is true (see ~ iotaval );
|
|
otherwise, it evaluates to the empty set (see ~ iotanul ). Russell used
|
|
the inverted iota symbol ` iota ` to represent the binder.
|
|
|
|
Sometimes proofs need to expand an iota-based definition. That is,
|
|
given "X = the x for which ... x ... x ..." holds, the proof needs to
|
|
get to "... X ... X ...". A general strategy to do this is to use
|
|
~ iotacl (for unbounded iota). This can be easier than applying a
|
|
version that applies an explicit substitution, because substituting an
|
|
iota into its own property always has a bound variable clash which must
|
|
be first renamed or else guarded with NF.
|
|
|
|
(Contributed by Andrew Salmon, 30-Jun-2011.) $)
|
|
df-iota $a |- ( iota x ph ) = U. { y | { x | ph } = { y } } $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
$( Alternate definition for descriptions. Definition 8.18 in [Quine]
|
|
p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) $)
|
|
dfiota2 $p |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) } $=
|
|
( cio cab cv csn wceq cuni wb wal df-iota df-sn eqeq2i abbi bitr4i unieqi
|
|
abbii eqtri ) ABDABEZCFZGZHZCEZIABFUAHZJBKZCEZIABCLUDUGUCUFCUCTUEBEZHUFUB
|
|
UHTBUAMNAUEBOPRQS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for the ` iota ` class. (Contributed
|
|
by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfiota1 $p |- F/_ x ( iota x ph ) $=
|
|
( vy cio weq wb wal cab cuni dfiota2 nfaba1 nfuni nfcxfr ) BABDABCEFZBGCH
|
|
ZIABCJBONBCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d z ph $. $d x y z $.
|
|
nfiotadxy.1 $e |- F/ y ph $.
|
|
nfiotadxy.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfiotaxy . (Contributed by Jim Kingdon,
|
|
21-Dec-2018.) $)
|
|
nfiotadxy $p |- ( ph -> F/_ x ( iota y ps ) ) $=
|
|
( vz cio cv wceq wb wal cab cuni dfiota2 nfv wnf nfcv nfeq a1i nfcxfrd
|
|
nfbid nfald nfabd nfunid ) ACBDHBDIZGIZJZKZDLZGMZNBDGOACUKAUJCGAGPAUICDEA
|
|
BUHCFUHCQACUFUGCUFRCUGRSTUBUCUDUEUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nfiota.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for the ` iota ` class. (Contributed
|
|
by NM, 23-Aug-2011.) $)
|
|
nfiotaxy $p |- F/_ x ( iota y ph ) $=
|
|
( cio wnfc wtru nftru wnf a1i nfiotadxy trud ) BACEFGABCCHABIGDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d z w x $. $d z w y $. $d z w ph $. $d z w ps $.
|
|
cbviota.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
cbviota.2 $e |- F/ y ph $.
|
|
cbviota.3 $e |- F/ x ps $.
|
|
$( Change bound variables in a description binder. (Contributed by Andrew
|
|
Salmon, 1-Aug-2011.) $)
|
|
cbviota $p |- ( iota x ph ) = ( iota y ps ) $=
|
|
( vw vz weq wb wal cab cuni cio wsb nfv nfbi equequ1 bibi12d sbequ12 nfsb
|
|
nfs1v cbval sbequ sbie syl6bb bitri abbii unieqi dfiota2 3eqtr4i ) ACHJZK
|
|
ZCLZHMZNBDHJZKZDLZHMZNACOBDOUPUTUOUSHUOACIPZIHJZKZILUSUNVCCIUNIQVAVBCACIU
|
|
CVBCQRCIJAVAUMVBACIUACIHSTUDVCURIDVAVBDACIDFUBVBDQRURIQIDJZVABVBUQVDVAACD
|
|
PBAIDCUEABCDGEUFUGIDHSTUDUHUIUJACHUKBDHUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps x $.
|
|
cbviotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variables in a description binder. (Contributed by Andrew
|
|
Salmon, 1-Aug-2011.) $)
|
|
cbviotav $p |- ( iota x ph ) = ( iota y ps ) $=
|
|
( nfv cbviota ) ABCDEADFBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d w z ph $. $d w z x $. $d w z y $.
|
|
sb8iota.1 $e |- F/ y ph $.
|
|
$( Variable substitution in description binder. Compare ~ sb8eu .
|
|
(Contributed by NM, 18-Mar-2013.) $)
|
|
sb8iota $p |- ( iota x ph ) = ( iota y [ y / x ] ph ) $=
|
|
( vz vw weq wal cab cuni wsb cio nfv sb8 sbbi nfsb equsb3 nfxfr dfiota2
|
|
wb nfbi sbequ cbval sblbis albii 3bitri abbii unieqi 3eqtr4i ) ABEGZTZBHZ
|
|
EIZJABCKZCEGZTZCHZEIZJABLUNCLUMURULUQEULUKBFKZFHUKBCKZCHUQUKBFUKFMNUSUTFC
|
|
USABFKZUJBFKZTCAUJBFOVAVBCABFCDPVBFEGZCFBEQVCCMRUARUTFMUKFCBUBUCUTUPCUJUO
|
|
ABCCBEQUDUEUFUGUHABESUNCESUI $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x z $. $d ph z $.
|
|
$( Equality theorem for descriptions. (Contributed by Andrew Salmon,
|
|
30-Jun-2011.) $)
|
|
iotaeq $p |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) ) $=
|
|
( vz cv wceq wal cab csn cuni cio wcel drsb1 df-clab 3bitr4g eqrdv eqeq1d
|
|
wsb abbidv df-iota unieqd 3eqtr4g ) BECEFBGZABHZDEZIZFZDHZJACHZUFFZDHZJAB
|
|
KACKUCUHUKUCUGUJDUCUDUIUFUCDUDUIUCABDRACDRUEUDLUEUILABCDMADBNADCNOPQSUAAB
|
|
DTACDTUB $.
|
|
$}
|
|
|
|
${
|
|
$d ph z $. $d ps z $. $d x y z $.
|
|
$( Equivalence theorem for descriptions. (Contributed by Andrew Salmon,
|
|
30-Jun-2011.) $)
|
|
iotabi $p |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) ) $=
|
|
( vz wb wal cab cv csn wceq cuni abbi biimpi eqeq1d abbidv unieqd df-iota
|
|
cio 3eqtr4g ) ABECFZACGZDHIZJZDGZKBCGZUBJZDGZKACRBCRTUDUGTUCUFDTUAUEUBTUA
|
|
UEJABCLMNOPACDQBCDQS $.
|
|
|
|
$( Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma
|
|
for the fundamental property of iota. (Contributed by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
uniabio $p |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y ) $=
|
|
( cv wceq wb wal cab cuni csn abbi biimpi df-sn syl6eqr unieqd vex syl6eq
|
|
unisn ) ABDCDZEZFBGZABHZISJZISUAUBUCUAUBTBHZUCUAUBUDEATBKLBSMNOSCPRQ $.
|
|
|
|
$( Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property
|
|
of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) $)
|
|
iotaval $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $=
|
|
( vz weq wb wal cio cab cuni cv dfiota2 wceq cvv wcel vex sbeqalb equcomi
|
|
wa wi ax-mp ex equequ2 equcoms bibi2d biimpd alimdv com12 alrimiv uniabio
|
|
syl6 impbid syl syl5eq ) ABCEZFZBGZABHABDEZFZBGZDIJZCKZABDLUQUTDCEZFZDGVA
|
|
VBMUQVDDUQUTVCUQUTVCVBNOZUQUTSZVCTCPVEVFCDEVCABVBDKNQCDRUKUAUBVCUQUTVCUPU
|
|
SBVCUPUSVCUOURAUOURFCDCDBUCUDUEUFUGUHULUIUTDCUJUMUN $.
|
|
|
|
$( Equivalence between two different forms of ` iota ` . (Contributed by
|
|
Andrew Salmon, 12-Jul-2011.) $)
|
|
iotauni $p |- ( E! x ph -> ( iota x ph ) = U. { x | ph } ) $=
|
|
( vz weu cv wceq wb wal wex cio cuni df-eu iotaval uniabio eqtr4d exlimiv
|
|
cab sylbi ) ABDABECEZFGBHZCIABJZABQKZFZABCLTUCCTUASUBABCMABCNOPR $.
|
|
|
|
$( Equivalence between two different forms of ` iota ` . (Contributed by
|
|
Mario Carneiro, 24-Dec-2016.) $)
|
|
iotaint $p |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } ) $=
|
|
( weu cio cab cuni cint iotauni uniintabim eqtrd ) ABCABDABEZFKGABHABIJ
|
|
$.
|
|
|
|
$( Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
iota1 $p |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) ) $=
|
|
( vz weu cv wceq wb wal wex cio df-eu iotaval eqeq2d bitr4d eqcom exlimiv
|
|
sp syl6bb sylbi ) ABDABEZCEZFZGZBHZCIAABJZTFZGZABCKUDUGCUDATUEFZUFUDAUBUH
|
|
UCBQUDUEUATABCLMNTUEORPS $.
|
|
|
|
$( Theorem 8.22 in [Quine] p. 57. This theorem is the result if there
|
|
isn't exactly one ` x ` that satisfies ` ph ` . (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) $)
|
|
iotanul $p |- ( -. E! x ph -> ( iota x ph ) = (/) ) $=
|
|
( vz weu wn cio c0 wss wceq weq wb wal wex df-eu cab dfiota2 alnex ax-in2
|
|
cuni wi alimi ss2ab sylibr dfnul2 syl6sseqr sylbir uni0 syl6sseq syl5eqss
|
|
unissd sylnbi ss0 syl ) ABDZEABFZGHZUOGIUNABCJKBLZCMZUPABCNUREZUOUQCOZSZG
|
|
ABCPUSVAGSGUSUTGUSUQEZCLZUTGHUQCQVCUTCCJEZCOZGVCUQVDTZCLUTVEHVBVFCUQVDRUA
|
|
UQVDCUBUCCUDUEUFUJUGUHUIUKUOULUM $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d x y $.
|
|
$( Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition
|
|
added. This theorem proves the existence of the ` iota ` class under
|
|
our definition. (Contributed by Jim Kingdon, 21-Dec-2018.) $)
|
|
euiotaex $p |- ( E! x ph -> ( iota x ph ) e. _V ) $=
|
|
( vy cv wceq wb wal wex cio weu cvv wcel iotaval eqcomd eximi df-eu isset
|
|
3imtr4i ) ABDCDZEFBGZCHSABIZEZCHABJUAKLTUBCTUASABCMNOABCPCUAQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Value of iota based on a proposition which holds only for values which
|
|
are subsets of a given class. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 21-Dec-2018.) $)
|
|
iotass $p |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A ) $=
|
|
( vy cv wss wal cio cab csn wceq cuni df-iota cpw unieq vex df-pw sspwuni
|
|
wi sylib unisn syl6eq sseq2i ss2ab bitri biimpri sseq1 syl2anr ex ss2abdv
|
|
biimpa syl6sseqr syl5eqss ) ABECFZSBGZABHABIZDEZJZKZDIZLZCABDMUOUTCNZFVAC
|
|
FUOUTUQCFZDIVBUOUSVCDUOUSVCUSUPLZUQKZVDCFZVCUOUSVDURLUQUPUROUQDPUAUBUOUPV
|
|
BFZVFVGUOVGUPUNBIZFUOVBVHUPBCQUCAUNBUDUEUFUPCRTVEVFVCVDUQCUGUKUHUIUJDCQUL
|
|
UTCRTUM $.
|
|
$}
|
|
|
|
${
|
|
$d ph z $. $d x z $.
|
|
$( Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew
|
|
Salmon, 12-Jul-2011.) $)
|
|
iota4 $p |- ( E! x ph -> [. ( iota x ph ) / x ]. ph ) $=
|
|
( vz weu weq wb wal wex cio wsbc df-eu wsb bi2 alimi sb2 syl wceq iotaval
|
|
wi cv eqcomd dfsbcq2 mpbid exlimiv sylbi ) ABDABCEZFZBGZCHABABIZJZABCKUHU
|
|
JCUHABCLZUJUHUFASZBGUKUGULBAUFMNABCOPUHCTZUIQUKUJFUHUIUMABCRUAABCUIUBPUCU
|
|
DUE $.
|
|
$}
|
|
|
|
$( Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew
|
|
Salmon, 12-Jul-2011.) $)
|
|
iota4an $p |- ( E! x ( ph /\ ps )
|
|
-> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) $=
|
|
( wa weu cio wsbc iota4 wi cvv euiotaex simpl sbcth syl wb sbcimg mpbid mpd
|
|
wcel ) ABDZCEZTCTCFZGZACUBGZTCHUATAIZCUBGZUCUDIZUAUBJSZUFTCKZUECUBJABLMNUAU
|
|
HUFUGOUITACUBJPNQR $.
|
|
|
|
${
|
|
$d x y A $. $d x V $. $d x ph $. $d y ps $.
|
|
iota5.1 $e |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) ) $.
|
|
$( A method for computing iota. (Contributed by NM, 17-Sep-2013.) $)
|
|
iota5 $p |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A ) $=
|
|
( vy wcel wa cv wceq wb wal cio alrimiv wi eqeq2 bibi2d albidv imbi12d
|
|
iotaval vtoclg adantl mpd ) ADEHZIZBCJZDKZLZCMZBCNZDKZUFUICFOUEUJULPZABUG
|
|
GJZKZLZCMZUKUNKZPUMGDEUNDKZUQUJURULUSUPUICUSUOUHBUNDUGQRSUNDUKQTBCGUAUBUC
|
|
UD $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
iotabidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for iota. (Contributed by NM,
|
|
20-Aug-2011.) $)
|
|
iotabidv $p |- ( ph -> ( iota x ps ) = ( iota x ch ) ) $=
|
|
( wb wal cio wceq alrimiv iotabi syl ) ABCFZDGBDHCDHIAMDEJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
iotabii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building deduction rule for iota. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
iotabii $p |- ( iota x ph ) = ( iota x ps ) $=
|
|
( wb cio wceq iotabi mpg ) ABEACFBCFGCABCHDI $.
|
|
$}
|
|
|
|
$( Membership law for descriptions.
|
|
|
|
This can useful for expanding an unbounded iota-based definition (see
|
|
~ df-iota ).
|
|
|
|
(Contributed by Andrew Salmon, 1-Aug-2011.) $)
|
|
iotacl $p |- ( E! x ph -> ( iota x ph ) e. { x | ph } ) $=
|
|
( weu cio wsbc cab wcel iota4 df-sbc sylib ) ABCABABDZEKABFGABHABKIJ $.
|
|
|
|
${
|
|
iota2df.1 $e |- ( ph -> B e. V ) $.
|
|
iota2df.2 $e |- ( ph -> E! x ps ) $.
|
|
iota2df.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
|
|
${
|
|
iota2df.4 $e |- F/ x ph $.
|
|
iota2df.5 $e |- ( ph -> F/ x ch ) $.
|
|
iota2df.6 $e |- ( ph -> F/_ x B ) $.
|
|
$( A condition that allows us to represent "the unique element such that
|
|
` ph ` " with a class expression ` A ` . (Contributed by NM,
|
|
30-Dec-2014.) $)
|
|
iota2df $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
|
|
( cio cv wceq wb wa simpr eqeq2d bibi12d weu iota1 syl wnfc nfiota1 a1i
|
|
nfeqd nfbid vtocldf ) ABBDMZDNZOZPZCUJEOZPDEFGAUKEOZQZBCULUNIUPUKEUJAUO
|
|
RSTABDUAUMHBDUBUCJLACUNDKADUJEDUJUDABDUEUFLUGUHUI $.
|
|
$}
|
|
|
|
$d x B $. $d x ph $. $d x ch $.
|
|
$( A condition that allows us to represent "the unique element such that
|
|
` ph ` " with a class expression ` A ` . (Contributed by NM,
|
|
30-Dec-2014.) $)
|
|
iota2d $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
|
|
( nfv nfvd nfcvd iota2df ) ABCDEFGHIADJACDKADELM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
iota2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( The unique element such that ` ph ` . (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) $)
|
|
iota2 $p |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) ) $=
|
|
( wcel cvv weu cio wceq wb elex wa simpl simpr cv adantl nfv nfeu1 nfcvd
|
|
nfan nfvd iota2df sylan ) DEGDHGZACIZBACJDKLDEMUFUGNZABCDHUFUGOUFUGPCQDKA
|
|
BLUHFRUFUGCUFCSACTUBUHBCUCUHCDUAUDUE $.
|
|
$}
|
|
|
|
$( A class abstraction with a unique member can be expressed as a singleton.
|
|
(Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
sniota $p |- ( E! x ph -> { x | ph } = { ( iota x ph ) } ) $=
|
|
( weu cv cab wcel cio csn wb wal wceq nfeu1 iota1 eqcom syl6bb abid 3bitr4g
|
|
vex elsn alrimi nfab1 nfiota1 nfsn cleqf sylibr ) ABCZBDZABEZFZUGABGZHZFZIZ
|
|
BJUHUKKUFUMBABLUFAUGUJKZUIULUFAUJUGKUNABMUJUGNOABPUGUJBRSQTBUHUKABUABUJABUB
|
|
UCUDUE $.
|
|
|
|
${
|
|
$d A y z $. $d x y z $. $d ph z $.
|
|
$( Class substitution within a description binder. (Contributed by Scott
|
|
Fenton, 6-Oct-2017.) $)
|
|
csbiotag $p |- ( A e. V ->
|
|
[_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) ) $=
|
|
( vz cio csb wsb wceq wsbc csbeq1 dfsbcq2 iotabidv eqeq12d nfs1v nfiotaxy
|
|
cv vex weq sbequ12 csbief vtoclg ) BFRZACGZHZABFIZCGZJBDUEHZABDKZCGZJFDEU
|
|
DDJZUFUIUHUKBUDDUELULUGUJCABFDMNOBUDUEUHFSUGBCABFPQBFTAUGCABFUANUBUC $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Functions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c : $. $( Colon $)
|
|
$c Fun $. $( Function predicate $)
|
|
$c Fn $. $( Function connective $)
|
|
$c --> $. $( Domain-codomain connective $)
|
|
$c -1-1-> $. $( 'One-to-one' domain-codomain connective $)
|
|
$c -onto-> $. $( 'Onto' domain-codomain connective $)
|
|
$c -1-1-onto-> $. $( 'One-to-one' and 'onto' domain-codomain connective $)
|
|
$c ` $. $( Left apostrophe (function value symbol) $)
|
|
$c Isom $. $( Isomorphism $)
|
|
|
|
$( Extend the definition of a wff to include the function predicate. (Read:
|
|
` A ` is a function.) $)
|
|
wfun $a wff Fun A $.
|
|
|
|
$( Extend the definition of a wff to include the function predicate with a
|
|
domain. (Read: ` A ` is a function on ` B ` .) $)
|
|
wfn $a wff A Fn B $.
|
|
|
|
$( Extend the definition of a wff to include the function predicate with
|
|
domain and codomain. (Read: ` F ` maps ` A ` into ` B ` .) $)
|
|
wf $a wff F : A --> B $.
|
|
|
|
$( Extend the definition of a wff to include one-to-one functions. (Read:
|
|
` F ` maps ` A ` one-to-one into ` B ` .) The notation ("1-1" above the
|
|
arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. $)
|
|
wf1 $a wff F : A -1-1-> B $.
|
|
|
|
$( Extend the definition of a wff to include onto functions. (Read: ` F `
|
|
maps ` A ` onto ` B ` .) The notation ("onto" below the arrow) is from
|
|
Definition 6.15(4) of [TakeutiZaring] p. 27. $)
|
|
wfo $a wff F : A -onto-> B $.
|
|
|
|
$( Extend the definition of a wff to include one-to-one onto functions.
|
|
(Read: ` F ` maps ` A ` one-to-one onto ` B ` .) The notation ("1-1"
|
|
above the arrow and "onto" below the arrow) is from Definition 6.15(6) of
|
|
[TakeutiZaring] p. 27. $)
|
|
wf1o $a wff F : A -1-1-onto-> B $.
|
|
|
|
$( Extend the definition of a class to include the value of a function.
|
|
(Read: The value of ` F ` at ` A ` , or " ` F ` of ` A ` .") $)
|
|
cfv $a class ( F ` A ) $.
|
|
|
|
$( Extend the definition of a wff to include the isomorphism property.
|
|
(Read: ` H ` is an ` R ` , ` S ` isomorphism of ` A ` onto ` B ` .) $)
|
|
wiso $a wff H Isom R , S ( A , B ) $.
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( Define predicate that determines if some class ` A ` is a function.
|
|
Definition 10.1 of [Quine] p. 65. For example, the expression
|
|
` Fun _I ` is true ( ~ funi ). This is not the same as defining a
|
|
specific function's mapping, which is typically done using the format of
|
|
~ cmpt with the maps-to notation (see ~ df-mpt ). Contrast this
|
|
predicate with the predicates to determine if some class is a function
|
|
with a given domain ( ~ df-fn ), a function with a given domain and
|
|
codomain ( ~ df-f ), a one-to-one function ( ~ df-f1 ), an onto function
|
|
( ~ df-fo ), or a one-to-one onto function ( ~ df-f1o ). For alternate
|
|
definitions, see ~ dffun2 , ~ dffun4 , ~ dffun6 , ~ dffun7 , ~ dffun8 ,
|
|
and ~ dffun9 . (Contributed by NM, 1-Aug-1994.) $)
|
|
df-fun $a |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) ) $.
|
|
|
|
$( Define a function with domain. Definition 6.15(1) of [TakeutiZaring]
|
|
p. 27. (Contributed by NM, 1-Aug-1994.) $)
|
|
df-fn $a |- ( A Fn B <-> ( Fun A /\ dom A = B ) ) $.
|
|
|
|
$( Define a function (mapping) with domain and codomain. Definition
|
|
6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) $)
|
|
df-f $a |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) ) $.
|
|
|
|
$( Define a one-to-one function. Compare Definition 6.15(5) of
|
|
[TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
df-f1 $a |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) ) $.
|
|
|
|
$( Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27.
|
|
We use their notation ("onto" under the arrow). (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
df-fo $a |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) ) $.
|
|
|
|
$( Define a one-to-one onto function. Compare Definition 6.15(6) of
|
|
[TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and
|
|
"onto" below the arrow). (Contributed by NM, 1-Aug-1994.) $)
|
|
df-f1o $a |- ( F : A -1-1-onto-> B <->
|
|
( F : A -1-1-> B /\ F : A -onto-> B ) ) $.
|
|
|
|
$( Define the value of a function, ` ( F `` A ) ` , also known as function
|
|
application. For example, ` ( _I `` (/) ) = (/) ` . Typically,
|
|
function ` F ` is defined using maps-to notation (see ~ df-mpt ), but
|
|
this is not required. For example, F = { ` <. ` 2 , 6 ` >. ` , ` <. `
|
|
3 , 9 ` >. ` } -> ( F ` `` ` 3 ) = 9 . We will later define
|
|
two-argument functions using ordered pairs as
|
|
` ( A F B ) = ( F `` <. A , B >. ) ` . This particular definition is
|
|
quite convenient: it can be applied to any class and evaluates to the
|
|
empty set when it is not meaningful. The left apostrophe notation
|
|
originated with Peano and was adopted in Definition *30.01 of
|
|
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
|
|
Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as
|
|
the more familiar ` F ( A ) ` notation for a function's value at ` A ` ,
|
|
i.e. " ` F ` of ` A ` ," but without context-dependent notational
|
|
ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use ` iota ` .
|
|
(Revised by Scott Fenton, 6-Oct-2017.) $)
|
|
df-fv $a |- ( F ` A ) = ( iota x A F x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$( Define the isomorphism predicate. We read this as " ` H ` is an ` R ` ,
|
|
` S ` isomorphism of ` A ` onto ` B ` ." Normally, ` R ` and ` S ` are
|
|
ordering relations on ` A ` and ` B ` respectively. Definition 6.28 of
|
|
[TakeutiZaring] p. 32, whose notation is the same as ours except that
|
|
` R ` and ` S ` are subscripts. (Contributed by NM, 4-Mar-1997.) $)
|
|
df-isom $a |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\
|
|
A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Alternate definition of a function. (Contributed by NM,
|
|
29-Dec-1996.) $)
|
|
dffun2 $p |- ( Fun A <-> ( Rel A /\
|
|
A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) ) $=
|
|
( wfun wrel cid wss wa cv wbr wal wex copab 3bitri vex albii alcom bitri
|
|
wi ccnv ccom df-fun df-id sseq2i df-co sseq1i ssopab2b brcnv anbi1i exbii
|
|
weq imbi1i 19.23v bitr4i anbi2i ) DEDFZDDUAZUBZGHZIUQAJZBJZDKZVACJDKZIZBC
|
|
ULZTZCLZBLALZIDUCUTVIUQUTVBVAURKZVDIZAMZVFTZCLZBLZVHALZBLVIUTUSVFBCNZHVLB
|
|
CNZVQHVOGVQUSBCUDUEUSVRVQBCADURUFUGVLVFBCUHOVNVPBVNVGALZCLVPVMVSCVMVEAMZV
|
|
FTVSVLVTVFVKVEAVJVCVDVBVADBPAPUIUJUKUMVEVFAUNUOQVGCARSQVHBAROUPS $.
|
|
|
|
$( Alternate definition of a function. Definition 6.4(4) of
|
|
[TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) $)
|
|
dffun4 $p |- ( Fun A <-> ( Rel A /\
|
|
A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A )
|
|
-> y = z ) ) ) $=
|
|
( wfun wrel cv wbr wa weq wal cop wcel dffun2 df-br anbi12i imbi1i 2albii
|
|
wi albii anbi2i bitri ) DEDFZAGZBGZDHZUDCGZDHZIZBCJZSZCKZBKAKZIUCUDUELDMZ
|
|
UDUGLDMZIZUJSZCKZBKAKZIABCDNUMUSUCULURABUKUQCUIUPUJUFUNUHUOUDUEDOUDUGDOPQ
|
|
TRUAUB $.
|
|
|
|
$( A way of proving a relation is a function, analogous to ~ mo2r .
|
|
(Contributed by Jim Kingdon, 27-May-2020.) $)
|
|
dffun5r $p |- ( ( Rel A /\
|
|
A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) -> Fun A ) $=
|
|
( wrel cv cop wcel weq wi wal wex wa wfun wmo nfv mo2r opeq2 eleq1d mo4
|
|
sylib alimi anim2i dffun4 sylibr ) DEZAFZBFZGZDHZBCIZJBKCLZAKZMUFUJUGCFZG
|
|
ZDHZMUKJCKBKZAKZMDNUMURUFULUQAULUJBOUQUJBCUJCPQUJUPBCUKUIUODUHUNUGRSTUAUB
|
|
UCABCDUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v u $. $d A w v u $.
|
|
dffun6f.1 $e |- F/_ x A $.
|
|
dffun6f.2 $e |- F/_ y A $.
|
|
$( Definition of function, using bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 9-Mar-1995.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
dffun6f $p |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) ) $=
|
|
( vw vv vu wfun cv wbr wa weq wal wmo nfcv nfbr nfv breq2 albii wi dffun2
|
|
wrel cbvmo mo4 nfmo breq1 mobidv cbval 3bitr3ri anbi2i bitr4i ) CICUCZFJZ
|
|
GJZCKZUNHJZCKZLGHMUAHNGNZFNZLUMAJZBJZCKZBOZANZLFGHCUBVEUTUMUPGOZFNUNVBCKZ
|
|
BOZFNUTVEVFVHFUPVGGBBUNUOCBUNPEBUOPQVGGRUOVBUNCSUDTVFUSFUPURGHUOUQUNCSUET
|
|
VHVDFAVGABAUNVBCAUNPDAVBPQUFVDFRFAMVGVCBUNVAVBCUGUHUIUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Alternate definition of a function using "at most one" notation.
|
|
(Contributed by NM, 9-Mar-1995.) $)
|
|
dffun6 $p |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) ) $=
|
|
( nfcv dffun6f ) ABCACDBCDE $.
|
|
|
|
$( A function has at most one value for each argument. (Contributed by NM,
|
|
24-May-1998.) $)
|
|
funmo $p |- ( Fun F -> E* y A F y ) $=
|
|
( vx wfun cv wbr cvv wcel wa wi wal wmo wrel dffun6 simplbi brrelex ancrd
|
|
ex syl alrimiv wceq breq1 mobidv imbi2d simprbi 19.21bi vtoclg com12 moim
|
|
moanimv sylibr sylc ) CEZBAFZCGZBHIZUPJZKZALURAMZUPAMZUNUSAUNUPUQUNCNZUPU
|
|
QKUNVBDFZUOCGZAMZDLZDACOZPVBUPUQBUOCQSTRUAUNUQVAKUTUQUNVAUNVEKUNVAKDBHVCB
|
|
UBZVEVAUNVHVDUPAVCBUOCUCUDUEUNVEDUNVBVFVGUFUGUHUIUQUPAUKULUPURAUJUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d A w $.
|
|
dffun4f.1 $e |- F/_ x A $.
|
|
dffun4f.2 $e |- F/_ y A $.
|
|
dffun4f.3 $e |- F/_ z A $.
|
|
$( Definition of function like ~ dffun4 but using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by Jim Kingdon,
|
|
17-Mar-2019.) $)
|
|
dffun4f $p |- ( Fun A <-> ( Rel A /\
|
|
A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A )
|
|
-> y = z ) ) ) $=
|
|
( vw cv wbr wal wa weq wi cop wcel nfcv nfbr breq2 albii wfun wmo dffun6f
|
|
wrel mo4f nfv nfan nfim anbi2d equequ2 imbi12d cbval bitr4i df-br anbi12i
|
|
anbi2i imbi1i 2albii 3bitri ) DUADUDZAIZBIZDJZBUBZAKZLUTVCVACIZDJZLZBCMZN
|
|
ZCKZBKZAKZLUTVAVBODPZVAVFODPZLZVINZCKBKZAKZLABDEFUCVEVMUTVDVLAVDVCVAHIZDJ
|
|
ZLZBHMZNZHKZBKVLVCWABHBVAVTDBVAQFBVTQRVBVTVADSUEVKWEBVJWDCHVJHUFWBWCCVCWA
|
|
CCVAVBDCVAQZGCVBQRCVAVTDWFGCVTQRUGWCCUFUHCHMZVHWBVIWCWGVGWAVCVFVTVADSUICH
|
|
BUJUKULTUMTUPVMVSUTVLVRAVJVQBCVHVPVIVCVNVGVOVAVBDUNVAVFDUNUOUQURTUPUS $.
|
|
$}
|
|
|
|
$( A function is a relation. (Contributed by NM, 1-Aug-1994.) $)
|
|
funrel $p |- ( Fun A -> Rel A ) $=
|
|
( wfun wrel ccnv ccom cid wss df-fun simplbi ) ABACAADEFGAHI $.
|
|
|
|
$( Subclass theorem for function predicate. (Contributed by NM,
|
|
16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) $)
|
|
funss $p |- ( A C_ B -> ( Fun B -> Fun A ) ) $=
|
|
( wss wrel ccnv ccom cid wa wfun relss wi coss1 cnvss coss2 syl sstrd sstr2
|
|
anim12d df-fun 3imtr4g ) ABCZBDZBBEZFZGCZHADZAAEZFZGCZHBIAIUAUBUFUEUIABJUAU
|
|
HUDCUEUIKUAUHBUGFZUDABUGLUAUGUCCUJUDCABMUGUCBNOPUHUDGQORBSAST $.
|
|
|
|
$( Equality theorem for function predicate. (Contributed by NM,
|
|
16-Aug-1994.) $)
|
|
funeq $p |- ( A = B -> ( Fun A <-> Fun B ) ) $=
|
|
( wceq wfun wss wi eqimss2 funss syl eqimss impbid ) ABCZADZBDZLBAEMNFBAGBA
|
|
HILABENMFABJABHIK $.
|
|
|
|
${
|
|
funeqi.1 $e |- A = B $.
|
|
$( Equality inference for the function predicate. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
funeqi $p |- ( Fun A <-> Fun B ) $=
|
|
( wceq wfun wb funeq ax-mp ) ABDAEBEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
funeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for the function predicate. (Contributed by NM,
|
|
23-Feb-2013.) $)
|
|
funeqd $p |- ( ph -> ( Fun A <-> Fun B ) ) $=
|
|
( wceq wfun wb funeq syl ) ABCEBFCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
nffun.1 $e |- F/_ x F $.
|
|
$( Bound-variable hypothesis builder for a function. (Contributed by NM,
|
|
30-Jan-2004.) $)
|
|
nffun $p |- F/ x Fun F $=
|
|
( wfun wrel ccnv ccom cid wa df-fun nfrel nfcnv nfco nfcv nfss nfan nfxfr
|
|
wss ) BDBEZBBFZGZHRZIABJSUBAABCKAUAHABTCABCLMAHNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A w y z $. $d F w y z $. $d V w y z $. $d x w y z $.
|
|
$( Distribute proper substitution through the function predicate.
|
|
(Contributed by Alexander van der Vekens, 23-Jul-2017.) $)
|
|
sbcfung $p |- ( A e. V -> ( [. A / x ]. Fun F <-> Fun [_ A / x ]_ F ) ) $=
|
|
( vw vy vz wrel cv wbr wa wi wal wsbc sbcal csbconstg bitrd syl5bb albidv
|
|
csb wcel weq wfun sbcan sbcrel sbcimg sbcbrg breq12d anbi12d sbcg imbi12d
|
|
dffun2 sbcbii 3bitr4g ) BDUAZCHZEIZFIZCJZUQGIZCJZKZFGUBZLZGMZFMZEMZKZABNZ
|
|
ABCTZHZUQURVJJZUQUTVJJZKZVCLZGMZFMZEMZKZCUCZABNVJUCVIUPABNZVGABNZKUOVSUPV
|
|
GABUDUOWAVKWBVRABCDUEWBVFABNZEMUOVRVFEABOUOWCVQEWCVEABNZFMUOVQVEFABOUOWDV
|
|
PFWDVDABNZGMUOVPVDGABOUOWEVOGUOWEVBABNZVCABNZLVOVBVCABDUFUOWFVNWGVCWFUSAB
|
|
NZVAABNZKUOVNUSVAABUDUOWHVLWIVMUOWHABUQTZABURTZVJJVLABUQURDCUGUOWJUQWKURV
|
|
JABUQDPZABURDPUHQUOWIWJABUTTZVJJVMABUQUTDCUGUOWJUQWMUTVJWLABUTDPUHQUIRVCA
|
|
BDUJUKQSRSRSRUIRVTVHABEFGCULUMEFGVJULUN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y F $.
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
funeu $p |- ( ( Fun F /\ A F B ) -> E! y A F y ) $=
|
|
( wfun wbr wa cv wex weu cdm wcel wrel funrel releldm sylan eldmg ibi syl
|
|
wmo wi funmo adantr df-mo sylib mpd ) DEZBCDFZGZBAHDFZAIZUJAJZUIBDKZLZUKU
|
|
GDMUHUNDNBCDOPUNUKABDUMQRSUIUJATZUKULUAUGUOUHABDUBUCUJAUDUEUF $.
|
|
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
funeu2 $p |- ( ( Fun F /\ <. A , B >. e. F ) -> E! y <. A , y >. e. F ) $=
|
|
( cop wcel wfun wbr cv weu df-br wa funeu eubii sylib sylan2br ) BCEDFDGZ
|
|
BCDHZBAIZEDFZAJZBCDKQRLBSDHZAJUAABCDMUBTABSDKNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of a function. One possibility for the definition
|
|
of a function in [Enderton] p. 42. (Enderton's definition is ambiguous
|
|
because "there is only one" could mean either "there is at most one" or
|
|
"there is exactly one." However, ~ dffun8 shows that it doesn't matter
|
|
which meaning we pick.) (Contributed by NM, 4-Nov-2002.) $)
|
|
dffun7 $p |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) ) $=
|
|
( wfun wrel cv wbr wmo wal wa cdm wral dffun6 wcel wi wex vex eldm bitr4i
|
|
moabs imbi1i albii df-ral anbi2i bitri ) CDCEZAFZBFCGZBHZAIZJUFUIACKZLZJA
|
|
BCMUJULUFUJUGUKNZUIOZAIULUIUNAUIUHBPZUIOUNUHBTUMUOUIBUGCAQRUASUBUIAUKUCSU
|
|
DUE $.
|
|
|
|
$( Alternate definition of a function. One possibility for the definition
|
|
of a function in [Enderton] p. 42. Compare ~ dffun7 . (Contributed by
|
|
NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
dffun8 $p |- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) ) $=
|
|
( wfun wrel cv wbr wmo cdm wral wa weu dffun7 wex wi wcel df-mo vex eldm
|
|
wb pm5.5 sylbi syl5bb ralbiia anbi2i bitri ) CDCEZAFZBFCGZBHZACIZJZKUGUIB
|
|
LZAUKJZKABCMULUNUGUJUMAUKUJUIBNZUMOZUHUKPZUMUIBQUQUOUPUMTBUHCARSUOUMUAUBU
|
|
CUDUEUF $.
|
|
|
|
$( Alternate definition of a function. (Contributed by NM, 28-Mar-2007.)
|
|
(Revised by NM, 16-Jun-2017.) $)
|
|
dffun9 $p |- ( Fun A <->
|
|
( Rel A /\ A. x e. dom A E* y e. ran A x A y ) ) $=
|
|
( wfun wrel cv wbr wmo cdm wral wa wrmo dffun7 wcel brelrn pm4.71ri mobii
|
|
crn vex df-rmo bitr4i ralbii anbi2i bitri ) CDCEZAFZBFZCGZBHZACIZJZKUEUHB
|
|
CRZLZAUJJZKABCMUKUNUEUIUMAUJUIUGULNZUHKZBHUMUHUPBUHUOUFUGCASBSOPQUHBULTUA
|
|
UBUCUD $.
|
|
$}
|
|
|
|
$( An equivalence for the function predicate. (Contributed by NM,
|
|
13-Aug-2004.) $)
|
|
funfn $p |- ( Fun A <-> A Fn dom A ) $=
|
|
( wfun cdm wceq wa wfn eqid biantru df-fn bitr4i ) ABZKACZLDZEALFMKLGHALIJ
|
|
$.
|
|
|
|
$( The identity relation is a function. Part of Theorem 10.4 of [Quine]
|
|
p. 65. (Contributed by NM, 30-Apr-1998.) $)
|
|
funi $p |- Fun _I $=
|
|
( cid wfun wrel ccnv ccom wss reli wceq relcnv coi2 ax-mp cnvi eqtri df-fun
|
|
eqimssi mpbir2an ) ABACAADZEZAFGRARQAQCRQHAIQJKLMOANP $.
|
|
|
|
$( The universe is not a function. (Contributed by Raph Levien,
|
|
27-Jan-2004.) $)
|
|
nfunv $p |- -. Fun _V $=
|
|
( cvv wfun wrel cxp wcel 0nelxp 0ex wss df-rel biimpi sseld mpi mto funrel
|
|
c0 ) ABACZPOAADZEZAAFPOAERGPAQOPAQHAIJKLMANM $.
|
|
|
|
${
|
|
$d t u v w x y z A $. $d t u v w x y z B $.
|
|
$( A Kuratowski ordered pair is a function only if its components are
|
|
equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
funopg $p |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B ) $=
|
|
( vu vt vx vy vz vw vv wcel cop wceq cv weq wi cpr vex cvv funeqd imbi12d
|
|
wfun opeq1 eqeq1 opeq2 eqeq2 csn wa wex wrel funrel relop sylib opth opid
|
|
preq1i dfop preq2i snex zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi
|
|
prid1 eleq2 mpbiri prid2 jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb
|
|
opex eqeq12 3adant1 spc3gv mp3an syl2im syl5bi dfsn2 preq2 syl5req eqeq2d
|
|
eqtr3 expcom syl6bi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) ACLBDL
|
|
ABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXFXIXD
|
|
AXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHOZRZN
|
|
ZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNMZXNX
|
|
QMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYLYKXN
|
|
GSZUPUQYGYKYFXNXQYMHSZURUSXOXRXNYMUTGHVAURVBVCVDXGIOZJOZMZXFLZYOKOZMZXFLZ
|
|
UIZJKPZQZKVEJVEIVEZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVFVGYIUUFUUGYIUUFYFYHLY
|
|
FYGXNXNYMYMVTVHXFYHYFVIVJYIUUGYGYHLYFYGXNXQYMYNVTVKXFYHYGVIVJVLXNTLZUUIXQ
|
|
TLUUEUUHYEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRUUFUU
|
|
AUUGUUNYQYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSXNXQV
|
|
NVQVPVRUULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWFXPXSYEXHQYEXSXPXHYEXSXEXONZX
|
|
PXHQYEXRXOXEYEXOXNXNRXRXNWGXNXQXNWHWIWJXPUUOXHXDXEXOWKWLWMWNWOWPWQWRWSWT
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A class of ordered pairs is a function when there is at most one second
|
|
member for each pair. (Contributed by NM, 16-May-1995.) $)
|
|
funopab $p |- ( Fun { <. x , y >. | ph } <-> A. x E* y ph ) $=
|
|
( copab wfun wbr wmo wal wrel relopab nfopab1 nfopab2 dffun6f mpbiran cop
|
|
cv wcel df-br opabid bitri mobii albii ) ABCDZEZBPZCPZUCFZCGZBHZACGZBHUDU
|
|
CIUIABCJBCUCABCKABCLMNUHUJBUGACUGUEUFOUCQAUEUFUCRABCSTUAUBT $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( A class of ordered pairs of values is a function. (Contributed by NM,
|
|
14-Nov-1995.) $)
|
|
funopabeq $p |- Fun { <. x , y >. | y = A } $=
|
|
( cv wceq copab wfun wmo funopab moeq mpgbir ) BDCEZABFGLBHALABIBCJK $.
|
|
|
|
$( A class of ordered pairs of values in the form used by ~ df-mpt is a
|
|
function. (Contributed by NM, 17-Feb-2013.) $)
|
|
funopab4 $p |- Fun { <. x , y >. | ( ph /\ y = A ) } $=
|
|
( cv wceq wa copab wss wfun simpr ssopab2i funopabeq funss mp2 ) ACEDFZGZ
|
|
BCHZPBCHZISJRJQPBCAPKLBCDMRSNO $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
$( A function in maps-to notation is a function. (Contributed by Mario
|
|
Carneiro, 13-Jan-2013.) $)
|
|
funmpt $p |- Fun ( x e. A |-> B ) $=
|
|
( vy cmpt wfun cv wcel wceq wa copab funopab4 df-mpt funeqi mpbir ) ABCEZ
|
|
FAGBHZDGCIJADKZFQADCLPRADBCMNO $.
|
|
$}
|
|
|
|
${
|
|
funmpt2.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( Functionality of a class given by a "maps to" notation. (Contributed by
|
|
FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) $)
|
|
funmpt2 $p |- Fun F $=
|
|
( wfun cmpt funmpt funeqi mpbir ) DFABCGZFABCHDKEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z F $. $d x y z G $.
|
|
$( The composition of two functions is a function. Exercise 29 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof
|
|
shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
funco $p |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) ) $=
|
|
( vx vy vz wfun ccom wrel wbr wmo cdm wral wex wss dmcoss weu sylibr wcel
|
|
wa cv wal funmo alrimiv ralrimivw dffun8 simprbi r19.26 nfv euexex ralimi
|
|
anim12ci ssralv mpsyl copab df-br df-co eleq2i opabid 3bitri mobii ralbii
|
|
syl cop relco jctil dffun7 ) AFZBFZSZABGZHZCTZDTZVJIZDJZCVJKZLZSVJFVIVQVK
|
|
VIVLETZBIZVRVMAIZSEMZDJZCVPLZVQVPBKZNVIWBCWDLZWCABOVIVSEPZVTDJZEUAZSZCWDL
|
|
ZWEVIWFCWDLZWHCWDLZSWJVGWLVHWKVGWHCWDVGWGEDVRAUBUCUDVHBHWKCEBUEUFUKWFWHCW
|
|
DUGQWIWBCWDVSVTEDVSDUHUIUJVBWBCVPWDULUMVOWBCVPVNWADVNVLVMVCZVJRWMWACDUNZR
|
|
WAVLVMVJUOVJWNWMCDEABUPUQWACDURUSUTVAQABVDVECDVJVFQ $.
|
|
$}
|
|
|
|
$( A restriction of a function is a function. Compare Exercise 18 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) $)
|
|
funres $p |- ( Fun F -> Fun ( F |` A ) ) $=
|
|
( cres wss wfun wi resss funss ax-mp ) BACZBDBEJEFBAGJBHI $.
|
|
|
|
${
|
|
$d x y F $. $d x y G $.
|
|
$( The restriction of a function to the domain of a subclass equals the
|
|
subclass. (Contributed by NM, 15-Aug-1994.) $)
|
|
funssres $p |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G ) $=
|
|
( vx vy wfun wss wa cdm cres wceq cv cop wcel wb wal wi vex wex imp wrel
|
|
ssel opeldm a1i adantl weu funeu2 eldm2 ancrd eximdv syl5bi eupick syl2an
|
|
jcad exp43 com23 com34 pm2.43d impd impbid opelres syl6rbbr relres funrel
|
|
alrimivv relss mpan9 eqrel sylancr mpbird ) AEZBAFZGZABHZIZBJZCKZDKZLZVNM
|
|
ZVRBMZNZDOCOZVLWACDVLVTVRAMZVPVMMZGZVSVLVTWEVKVTWEPVJVKVTWCWDBAVRUAZVTWDP
|
|
VKVPVQBCQZDQZUBUCUMUDVLWCWDVTVLWCWDVTPVLWCWDWCVTVJVKWCWDWCVTPZPZPVJWCVKWJ
|
|
VJWCVKWDWIVJWCGWCDUEWCVTGZDRZWIVKWDGDVPVQAUFVKWDWLWDVTDRVKWLDVPBWGUGVKVTW
|
|
KDVKVTWCWFUHUIUJSWCVTDUKULUNUOSUPUQURUSVPVQAVMWHUTVAVDVLVNTBTZVOWBNAVMVBV
|
|
JATVKWMAVCBAVEVFCDVNBVGVHVI $.
|
|
$}
|
|
|
|
$( Equality of restrictions of a function and a subclass. (Contributed by
|
|
NM, 16-Aug-1994.) $)
|
|
fun2ssres $p |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) ->
|
|
( F |` A ) = ( G |` A ) ) $=
|
|
( wfun wss cdm cres wceq wa resabs1 eqcomd funssres reseq1d sylan9eqr 3impa
|
|
) BDZCBEZACFZEZBAGZCAGZHSPQIZTBRGZAGZUASUDTBARJKUBUCCABCLMNO $.
|
|
|
|
${
|
|
$d x y z F $. $d x y z G $.
|
|
$( The union of functions with disjoint domains is a function. Theorem 4.6
|
|
of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) $)
|
|
funun $p |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) ->
|
|
Fun ( F u. G ) ) $=
|
|
( vx vy vz wfun wa cdm wrel cv wi wal anim12i wo wn 19.21bi opeldm dffun4
|
|
wcel vex cin c0 wceq cun cop funrel relun sylibr elun anbi12i anddi bitri
|
|
adantr disj1 biimpi imnan sylib nsyl orel2 syl con2d orel1 orim12d adantl
|
|
syl5bi simprbi 19.21bbi jaao syld alrimiv alrimivv sylanbrc ) AFZBFZGZAHZ
|
|
BHZUAUBUCZGZABUDZIZCJZDJZUEZVTSZWBEJZUEZVTSZGZWCWFUCZKZELZDLCLVTFVOWAVRVO
|
|
AIZBIZGWAVMWMVNWNAUFBUFMABUGUHUMVSWLCDVSWKEVSWIWDASZWGASZGZWDBSZWGBSZGZNZ
|
|
WJWIWQWOWSGZNZWRWPGZWTNZNZVSXAWIWOWRNZWPWSNZGXFWEXGWHXHWDABUIWGABUIUJWOWR
|
|
WPWSUKULVRXFXAKVOVRXCWQXEWTVRXBOXCWQKVRWBVPSZWBVQSZGZXBVRXIXJOKZXKOVRXLCV
|
|
RXLCLCVPVQUNUOPZXIXJUPUQWOXIWSXJWBWCACTZDTZQWBWFBXNETZQMURXBWQUSUTVRXDOXE
|
|
WTKVRXJXIGZXDVRXJXIOKXQOVRXIXJXMVAXJXIUPUQWRXJWPXIWBWCBXNXOQWBWFAXNXPQMUR
|
|
XDWTVBUTVCVDVEVOXAWJKVRVMWQWJVNWTVMWQWJKZDEVMXRELDLZCVMWMXSCLCDEARVFPVGVN
|
|
WTWJKZDEVNXTELDLZCVNWNYACLCDEBRVFPVGVHUMVIVJVKCDEVTRVL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The converse singleton of an ordered pair is a function. This is
|
|
equivalent to ~ funsn via ~ cnvsn , but stating it this way allows us to
|
|
skip the sethood assumptions on ` A ` and ` B ` . (Contributed by NM,
|
|
30-Apr-2015.) $)
|
|
funcnvsn $p |- Fun `' { <. A , B >. } $=
|
|
( vx vy cop csn ccnv wfun wrel cv wbr wmo wal relcnv wceq moeq wcel brcnv
|
|
vex df-br bitri elsni opth1 syl sylbi moimi ax-mp ax-gen dffun6 mpbir2an
|
|
) ABEZFZGZHUMICJZDJZUMKZDLZCMULNUQCUOAOZDLUQDAPUPURDUPUOUNEZULQZURUPUOUNU
|
|
LKUTUNUOULCSZDSZRUOUNULTUAUTUSUKOURUSUKUBUOUNABVBVAUCUDUEUFUGUHCDUMUIUJ
|
|
$.
|
|
|
|
$( A singleton of an ordered pair is a function. Theorem 10.5 of [Quine]
|
|
p. 65. (Contributed by NM, 28-Jun-2011.) $)
|
|
funsng $p |- ( ( A e. V /\ B e. W ) -> Fun { <. A , B >. } ) $=
|
|
( wcel wa cop csn ccnv wfun funcnvsn wceq cnvsng ancoms funeqd mpbii ) AC
|
|
EZBDEZFZBAGHIZJABGHZJBAKSTUARQTUALBADCMNOP $.
|
|
|
|
$( Functionality and domain of the singleton of an ordered pair.
|
|
(Contributed by Mario Carneiro, 30-Apr-2015.) $)
|
|
fnsng $p |- ( ( A e. V /\ B e. W ) -> { <. A , B >. } Fn { A } ) $=
|
|
( wcel wa cop csn wfun cdm wceq wfn funsng dmsnopg adantl df-fn sylanbrc
|
|
) ACEZBDEZFABGHZITJAHZKZTUALABCDMSUBRABDNOTUAPQ $.
|
|
|
|
funsn.1 $e |- A e. _V $.
|
|
funsn.2 $e |- B e. _V $.
|
|
$( A singleton of an ordered pair is a function. Theorem 10.5 of [Quine]
|
|
p. 65. (Contributed by NM, 12-Aug-1994.) $)
|
|
funsn $p |- Fun { <. A , B >. } $=
|
|
( cvv wcel cop csn wfun funsng mp2an ) AEFBEFABGHICDABEEJK $.
|
|
$}
|
|
|
|
$( A set of two pairs is a function if their first members are different.
|
|
(Contributed by FL, 26-Jun-2011.) $)
|
|
funprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
|
|
-> Fun { <. A , C >. , <. B , D >. } ) $=
|
|
( wcel wa cop csn wfun cdm cin c0 wceq funsng syl2anc dmsnopg simp1l simp2l
|
|
wne w3a cun cpr simp1r simp2r ineq12d disjsn2 3ad2ant3 eqtrd funun syl21anc
|
|
syl df-pr funeqi sylibr ) AEIZBFIZJZCGIZDHIZJZABUCZUDZACKZLZBDKZLZUEZMZVGVI
|
|
UFZMVFVHMZVJMZVHNZVJNZOZPQVLVFUSVBVNUSUTVDVEUAVAVBVCVEUBZACEGRSVFUTVCVOUSUT
|
|
VDVEUGVAVBVCVEUHZBDFHRSVFVRALZBLZOZPVFVPWAVQWBVFVBVPWAQVSACGTUOVFVCVQWBQVTB
|
|
DHTUOUIVEVAWCPQVDABUJUKULVHVJUMUNVMVKVGVIUPUQUR $.
|
|
|
|
$( A set of three pairs is a function if their first members are different.
|
|
(Contributed by Alexander van der Vekens, 5-Dec-2017.) $)
|
|
funtpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
|
|
/\ ( A e. F /\ B e. G /\ C e. H )
|
|
/\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
|
|
-> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } ) $=
|
|
( wcel w3a wne cop wfun wceq syl wn cpr csn cun ctp cdm cin c0 3simpa simp1
|
|
funprg syl3an simp13 simp23 funsng syl2anc 3ad2ant2 dmpropg dmsnopg ineq12d
|
|
wa wo elpri w3o nner eqcoms 3mix2 3mix3 jaoi 3ianorr disjsn sylibr 3ad2ant3
|
|
con2i eqtrd funun syl21anc df-tp funeqi ) JDMZKHMZLIMZNZAEMZBFMZCGMZNZJKOZJ
|
|
LOZKLOZNZNZJAPZKBPZUAZLCPZUBZUCZQZWLWMWOUDZQWKWNQZWPQZWNUEZWPUEZUFZUGRWRWBV
|
|
SVTUTWFWCWDUTZWJWGWTVSVTWAUHWCWDWEUHZWGWHWIUIJKABDHEFUJUKWKWAWEXAVSVTWAWFWJ
|
|
ULWBWCWDWEWJUMZLCIGUNUOWKXDJKUAZLUBZUFZUGWKXBXHXCXIWKXEXBXHRWFWBXEWJXFUPJAK
|
|
BEFUQSWKWEXCXIRXGLCGURSUSWJWBXJUGRZWFWJLXHMZTXKXLWJXLLJRZLKRZVAZWJTZLJKVBXO
|
|
WGTZWHTZWITZVCZXPXMXTXNXMXRXTXRJLJLVDVEXRXQXSVFSXNXSXTXSKLKLVDVEXSXQXRVGSVH
|
|
WGWHWIVISSVMXHLVJVKVLVNWNWPVOVPWSWQWLWMWOVQVRVK $.
|
|
|
|
${
|
|
funpr.1 $e |- A e. _V $.
|
|
funpr.2 $e |- B e. _V $.
|
|
funpr.3 $e |- C e. _V $.
|
|
funpr.4 $e |- D e. _V $.
|
|
$( A function with a domain of two elements. (Contributed by Jeff Madsen,
|
|
20-Jun-2010.) $)
|
|
funpr $p |- ( A =/= B -> Fun { <. A , C >. , <. B , D >. } ) $=
|
|
( cvv wcel wa wne cop cpr wfun pm3.2i funprg mp3an12 ) AIJZBIJZKCIJZDIJZK
|
|
ABLACMBDMNOSTEFPUAUBGHPABCDIIIIQR $.
|
|
$}
|
|
|
|
${
|
|
funtp.1 $e |- A e. _V $.
|
|
funtp.2 $e |- B e. _V $.
|
|
funtp.3 $e |- C e. _V $.
|
|
funtp.4 $e |- D e. _V $.
|
|
funtp.5 $e |- E e. _V $.
|
|
funtp.6 $e |- F e. _V $.
|
|
$( A function with a domain of three elements. (Contributed by NM,
|
|
14-Sep-2011.) $)
|
|
funtp $p |- ( ( A =/= B /\ A =/= C /\ B =/= C )
|
|
-> Fun { <. A , D >. , <. B , E >. , <. C , F >. } ) $=
|
|
( wne cop csn wfun wa cin c0 wceq w3a cpr cun ctp funpr funsn jctir df-pr
|
|
cdm dmprop eqtri dmsnop ineq12i disjsn2 anim12i sylib syl5eq funun syl2an
|
|
undisj1 3impb df-tp funeqi sylibr ) ABMZACMZBCMZUAADNZBENZUBZCFNZOZUCZPZV
|
|
HVIVKUDZPVEVFVGVNVEVJPZVLPZQVJUIZVLUIZRZSTVNVFVGQZVEVPVQABDEGHJKUECFILUFU
|
|
GWAVTAOZBOZUCZCOZRZSVRWDVSWEVRABUBWDADBEJKUJABUHUKCFLULUMWAWBWERSTZWCWERS
|
|
TZQWFSTVFWGVGWHACUNBCUNUOWBWCWEUTUPUQVJVLURUSVAVOVMVHVIVKVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
fnsn.1 $e |- A e. _V $.
|
|
fnsn.2 $e |- B e. _V $.
|
|
$( Functionality and domain of the singleton of an ordered pair.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
fnsn $p |- { <. A , B >. } Fn { A } $=
|
|
( cvv wcel cop csn wfn fnsng mp2an ) AEFBEFABGHAHICDABEEJK $.
|
|
$}
|
|
|
|
$( Function with a domain of two different values. (Contributed by FL,
|
|
26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
fnprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
|
|
-> { <. A , C >. , <. B , D >. } Fn { A , B } ) $=
|
|
( wcel wa wne w3a cop cpr wfun cdm wceq wfn funprg dmpropg 3ad2ant2 df-fn
|
|
sylanbrc ) AEIBFIJZCGIDHIJZABKZLACMBDMNZOUGPABNZQZUGUHRABCDEFGHSUEUDUIUFACB
|
|
DGHTUAUGUHUBUC $.
|
|
|
|
$( Function with a domain of three different values. (Contributed by
|
|
Alexander van der Vekens, 5-Dec-2017.) $)
|
|
fntpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
|
|
/\ ( A e. F /\ B e. G /\ C e. H )
|
|
/\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
|
|
-> { <. X , A >. , <. Y , B >. , <. Z , C >. } Fn { X , Y , Z } ) $=
|
|
( wcel w3a wne cop cdm wceq csn cun ctp wfun wfn funtpg wa dmsnopg 3ad2ant1
|
|
cpr 3ad2ant2 jca uneq12 syl df-pr syl6eqr dmeqi eqeq1i dmun sylibr 3ad2ant3
|
|
bitri uneq12d df-tp eqtri 3eqtr4g df-fn sylanbrc ) JDMKHMLIMNZAEMZBFMZCGMZN
|
|
ZJKOJLOKLONZNZJAPZKBPZLCPZUAZUBVQQZJKLUAZRVQVSUCABCDEFGHIJKLUDVMVNVOUHZQZVP
|
|
SZQZTZJKUHZLSZTVRVSVMWAWEWCWFVMVNSZQZVOSZQZTZWERZWAWERZVMWKJSZKSZTZWEVMWHWN
|
|
RZWJWORZUEZWKWPRVKVGWSVLVKWQWRVHVIWQVJJAEUFUGVIVHWRVJKBFUFUIUJUIWHWNWJWOUKU
|
|
LJKUMUNWMWGWITZQZWERWLWAXAWEVTWTVNVOUMUOUPXAWKWEWGWIUQUPUTURVKVGWCWFRZVLVJV
|
|
HXBVILCGUFUSUIVAVRVTWBTZQWDVQXCVNVOVPVBUOVTWBUQVCJKLVBVDVQVSVEVF $.
|
|
|
|
${
|
|
fntp.1 $e |- A e. _V $.
|
|
fntp.2 $e |- B e. _V $.
|
|
fntp.3 $e |- C e. _V $.
|
|
fntp.4 $e |- D e. _V $.
|
|
fntp.5 $e |- E e. _V $.
|
|
fntp.6 $e |- F e. _V $.
|
|
$( A function with a domain of three elements. (Contributed by NM,
|
|
14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
fntp $p |- ( ( A =/= B /\ A =/= C /\ B =/= C )
|
|
-> { <. A , D >. , <. B , E >. , <. C , F >. } Fn { A , B , C } ) $=
|
|
( wne w3a cop ctp wfun cdm wceq wfn funtp dmtpop a1i df-fn sylanbrc ) ABM
|
|
ACMBCMNZADOBEOCFOPZQUGRABCPZSZUGUHTABCDEFGHIJKLUAUIUFADBECFJKLUBUCUGUHUDU
|
|
E $.
|
|
$}
|
|
|
|
$( The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed
|
|
by NM, 7-Apr-1998.) $)
|
|
fun0 $p |- Fun (/) $=
|
|
( c0 cop csn wss wfun 0ss 0ex funsn funss mp2 ) AAABCZDKEAEKFAAGGHAKIJ $.
|
|
|
|
$( The double converse of a function is a function. (Contributed by NM,
|
|
21-Sep-2004.) $)
|
|
funcnvcnv $p |- ( Fun A -> Fun `' `' A ) $=
|
|
( ccnv wss wfun wi cnvcnvss funss ax-mp ) ABBZACADIDEAFIAGH $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A simpler equivalence for single-rooted (see ~ funcnv ). (Contributed
|
|
by NM, 9-Aug-2004.) $)
|
|
funcnv2 $p |- ( Fun `' A <-> A. y E* x x A y ) $=
|
|
( ccnv wfun cv wbr wmo wrel relcnv dffun6 mpbiran brcnv mobii albii bitri
|
|
wal vex ) CDZEZBFZAFZSGZAHZBQZUBUACGZAHZBQTSIUECJBASKLUDUGBUCUFAUAUBCBRAR
|
|
MNOP $.
|
|
|
|
$( The converse of a class is a function iff the class is single-rooted,
|
|
which means that for any ` y ` in the range of ` A ` there is at most
|
|
one ` x ` such that ` x A y ` . Definition of single-rooted in
|
|
[Enderton] p. 43. See ~ funcnv2 for a simpler version. (Contributed by
|
|
NM, 13-Aug-2004.) $)
|
|
funcnv $p |- ( Fun `' A <-> A. y e. ran A E* x x A y ) $=
|
|
( cv wbr wmo wal crn wcel wi ccnv wfun wral wa vex pm4.71ri mobii moanimv
|
|
brelrn bitri albii funcnv2 df-ral 3bitr4i ) ADZBDZCEZAFZBGUFCHZIZUHJZBGCK
|
|
LUHBUIMUHUKBUHUJUGNZAFUKUGULAUGUJUEUFCAOBOSPQUJUGARTUAABCUBUHBUIUCUD $.
|
|
|
|
$( A condition showing a class is single-rooted. (See ~ funcnv ).
|
|
(Contributed by NM, 26-May-2006.) $)
|
|
funcnv3 $p |- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y ) $=
|
|
( cv wbr wmo crn wral wex ccnv wfun cdm wreu wcel dfrn2 abeq2i biimpi weu
|
|
wa vex biantrurd ralbiia funcnv df-reu breldm pm4.71ri eu5 3bitr2i ralbii
|
|
eubii 3bitr4i ) ADZBDZCEZAFZBCGZHUNAIZUOSZBUPHCJKUNACLZMZBUPHUOURBUPUMUPN
|
|
ZUQUOVAUQUQBUPABCOPQUAUBABCUCUTURBUPUTULUSNZUNSZARUNARURUNAUSUDUNVCAUNVBU
|
|
LUMCATBTUEUFUJUNAUGUHUIUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Another way of expressing that a class is single-rooted. Counterpart to
|
|
~ dffun2 . (Contributed by Jim Kingdon, 24-Dec-2018.) $)
|
|
funcnveq $p |- ( Fun `' A <->
|
|
A. x A. y A. z ( ( x A y /\ z A y ) -> x = z ) ) $=
|
|
( ccnv wfun cv wbr wa weq wi wal wrel relcnv dffun2 mpbiran alcom anbi12i
|
|
vex brcnv imbi1i 2albii albii 3bitri ) DEZFZBGZAGZUEHZUGCGZUEHZIZACJZKZCL
|
|
ZALBLZUOBLZALUHUGDHZUJUGDHZIZUMKZCLBLZALUFUEMUPDNBACUEOPUOBAQUQVBAUNVABCU
|
|
LUTUMUIURUKUSUGUHDBSZASTUGUJDVCCSTRUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d f g x y z w v A $. $d x y B $. $d x y R $.
|
|
$( The double converse of a class is a function iff the class is
|
|
single-valued. Each side is equivalent to Definition 6.4(2) of
|
|
[TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued.
|
|
Note that ` A ` is not necessarily a function. (Contributed by NM,
|
|
13-Aug-2004.) $)
|
|
fun2cnv $p |- ( Fun `' `' A <-> A. x E* y x A y ) $=
|
|
( ccnv wfun cv wbr wmo wal funcnv2 vex brcnv mobii albii bitri ) CDZDEBFZ
|
|
AFZPGZBHZAIRQCGZBHZAIBAPJTUBASUABQRCBKAKLMNO $.
|
|
|
|
$( A single-valued relation is a function. (See ~ fun2cnv for
|
|
"single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.
|
|
(Contributed by NM, 17-Jan-2006.) $)
|
|
svrelfun $p |- ( Fun A <-> ( Rel A /\ Fun `' `' A ) ) $=
|
|
( vx vy wfun wrel cv wbr wmo wal wa ccnv dffun6 fun2cnv anbi2i bitr4i ) A
|
|
DAEZBFCFAGCHBIZJPAKKDZJBCALRQPBCAMNO $.
|
|
|
|
$( Single-rootedness (see ~ funcnv ) of a class cut down by a cross
|
|
product. (Contributed by NM, 5-Mar-2007.) $)
|
|
fncnv $p |- ( `' ( R i^i ( A X. B ) ) Fn B <->
|
|
A. y e. B E! x e. A x R y ) $=
|
|
( cxp cin ccnv wfn wfun cdm wceq wa cv wbr wral wmo wcel wi 3bitr4i df-fn
|
|
crn wreu df-rn eqeq1i anbi2i wrex wrmo rninxp anbi1i funcnv raleq moanimv
|
|
biimt w3a brinxp2 3anan12 bitri mobii df-rmo imbi2i ralbiia syl6bb syl5bb
|
|
syl6rbbr pm5.32i r19.26 ancom reu5 ralbii 3bitr2i ) ECDFGZHZDIVMJZVMKZDLZ
|
|
MVNVLUBZDLZMZANZBNZEOZACUCZBDPZVMDUAVRVPVNVQVODVLUDUEUFVRVNMZWBACUGZWBACU
|
|
HZMZBDPZVSWDVRWGBDPZMWFBDPZWJMWEWIVRWKWJABCDEUIUJVRVNWJVNVTWAVLOZAQZBVQPZ
|
|
VRWJABVLUKVRWNWMBDPWJWMBVQDULWMWGBDWADRZWGWOWGSZWMWOWGUNWOVTCRZWBMZMZAQWO
|
|
WRAQZSWMWPWOWRAUMWLWSAWLWQWOWBUOWSVTWACDEUPWQWOWBUQURUSWGWTWOWBACUTVATVEV
|
|
BVCVDVFWFWGBDVGTVNVRVHWCWHBDWBACVIVJTVK $.
|
|
|
|
$( Two ways of stating that ` A ` is one-to-one (but not necessarily a
|
|
function). Each side is equivalent to Definition 6.4(3) of
|
|
[TakeutiZaring] p. 24, who use the notation "Un_2 (A)" for one-to-one
|
|
(but not necessarily a function). (Contributed by NM, 17-Jan-2006.) $)
|
|
fun11 $p |- ( ( Fun `' `' A /\ Fun `' A ) <->
|
|
A. x A. y A. z A. w ( ( x A y /\ z A w ) -> ( x = z <-> y = w ) ) ) $=
|
|
( cv wbr wa weq wi wal ccnv wfun bi2.04 anbi12i 2albii alcom albii 3bitri
|
|
19.26-2 wb dfbi2 imbi2i pm4.76 nfv breq1 anbi1d imbi1d equsal bitri breq2
|
|
3bitr2i bitr2i wmo fun2cnv mo4 funcnv2 alrot4 3bitr4i ) CFZBFZEGZUTDFZEGZ
|
|
HZBDIZJZBKZDKZCKZAFZVCEGZVDHZACIZJZAKZDKCKZHZVKVAEGZVDHZVNVFUAZJZBKAKZDKC
|
|
KZELZLMZWEMZHWBDKCKBKAKWDVHVPHZDKCKVRWCWHCDWCVNVTVFJZJZVFVTVNJZJZHZBKAKWJ
|
|
BKAKZWLBKZAKZHWHWBWMABWBVTVNVFJZVFVNJZHZJVTWQJZVTWRJZHWMWAWSVTVNVFUBUCVTW
|
|
QWRUDWTWJXAWLVTVNVFNVTVFVNNOULPWJWLABTWNVHWPVPWNWJAKZBKVHWJABQXBVGBWIVGAC
|
|
VGAUEVNVTVEVFVNVSVBVDVKUTVAEUFUGUHUIRUJWOVOAWKVOBDVOBUEVFVTVMVNVFVSVLVDVA
|
|
VCVKEUKUGUHUIROSPVHVPCDTUMWFVJWGVQWFVBBUNZCKVGDKBKZCKVJCBEUOXCXDCVBVDBDVA
|
|
VCUTEUKUPRXDVICVGBDQRSWGVLAUNZDKVOCKAKZDKZVQADEUQXEXFDVLVDACVKUTVCEUFUPRX
|
|
GVPCKZDKVQXFXHDVOACQRVPDCQUJSOWBABCDURUS $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of functions is a
|
|
function. (Contributed by NM, 10-Aug-2004.) $)
|
|
fununi $p |- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) ->
|
|
Fun U. A ) $=
|
|
( vx vy vz vw vv cv wfun wss wo wral wa wrel wcel weq wi wal wex cuni cop
|
|
funrel adantr ralimi reluni sylibr r19.28av anim1d dffun4 simprbi 19.21bi
|
|
ssel 19.21bbi syl9r adantl anim2d jaod funeq sseq1 orbi12d anbi12d anbi2d
|
|
sseq2 cbvral2v ralcom orcom syl5bb bitri anbi12i anandir 2ralbii r19.26-2
|
|
imp anidm bitr2i 3bitr3i eluni eeanv an4 ancom 2exbii 3bitr2i imbi1i r2al
|
|
19.23v impexp 2albii albii 3bitr2ri 3imtr4i alrimiv alrimivv syl sylanbrc
|
|
) BIZJZWPCIZKZWRWPKZLZCAMZNZBAMZAUAZOZDIZEIUBZXEPZXGFIUBZXEPZNZEFQZRZFSZE
|
|
SDSZXEJXDWPOZBAMXFXCXQBAWQXQXBWPUCUDUEBAUFUGXDWQXANZCAMZBAMZXPXCXSBAWQXAC
|
|
AUHUEXTXODEXTXNFGIZJZHIZJZNZYAYCKZYCYAKZLZNZHAMZGAMZXHYAPZXJYCPZNZXMRZHAM
|
|
ZGAMZXTXNYJYPGAYIYOHAYEYHYOYEYFYOYGYDYFYORYBYFYNXHYCPZYMNZYDXMYFYLYRYMYAY
|
|
CXHUMUIYDYSXMRZFYDYTFSZDEYDYCOUUAESDSDEFYCUJUKUNULUOUPYBYGYORYDYGYNYLXJYA
|
|
PZNZYBXMYGYMUUBYLYCYAXJUMUQYBUUCXMRZFYBUUDFSZDEYBYAOUUEESDSDEFYAUJUKUNULU
|
|
OUDURVNUEUEXTXTNYBYHNZHAMGAMZYDYHNZHAMGAMZNZXTYKXTUUGXTUUIXRUUFYBYAWRKZWR
|
|
YAKZLZNBCGHAABGQZWQYBXAUUMWPYAUSUUNWSUUKWTUULWPYAWRUTWPYAWRVDVAVBCHQZUUMY
|
|
HYBUUOUUKYFUULYGWRYCYAVDWRYCYAUTVAVCVEXTXRBAMCAMUUIXRBCAAVFXRUUHWQYAWPKZW
|
|
PYAKZLZNCBGHAACGQZXAUURWQXAWTWSLUUSUURWSWTVGUUSWTUUPWSUUQWRYAWPUTWRYAWPVD
|
|
VAVHVCBHQZWQYDUURYHWPYCUSUUTUUPYFUUQYGWPYCYAVDWPYCYAUTVAVBVEVIVJXTVOYKUUF
|
|
UUHNZHAMGAMUUJYIUVAGHAAYBYDYHVKVLUUFUUHGHAAVMVPVQXNYAAPZYCAPZNZYNNZHTZGTZ
|
|
XMRUVFXMRZGSZYQXLUVGXMXLYLUVBNZGTZYMUVCNZHTZNUVJUVLNZHTGTUVGXIUVKXKUVMGXH
|
|
AVRHXJAVRVJUVJUVLGHVSUVNUVEGHUVNYNUVDNUVEYLUVBYMUVCVTYNUVDWAVIWBWCWDUVFXM
|
|
GWFYQUVDYORZHSGSUVEXMRZHSZGSUVIYOGHAAWEUVPUVOGHUVDYNXMWGWHUVQUVHGUVEXMHWF
|
|
WIWJWCWKWLWMWNDEFXEUJWO $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of single-rooted sets
|
|
is single-rooted. (See ~ funcnv for "single-rooted" definition.)
|
|
(Contributed by NM, 11-Aug-2004.) $)
|
|
funcnvuni $p |- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) )
|
|
-> Fun `' U. A ) $=
|
|
( vy vx vz vw vv cv ccnv wfun wss wo wral wa wceq wrex wi wal weq cbvrexv
|
|
cab cuni cnveq eqeq2d wcel funeqd sseq1 sseq2 orbi12d ralbidv rspcv funeq
|
|
anbi12d biimprcd cnvss orim12i wb sseq12 syl5ibrcom expd syl6com rexlimdv
|
|
ancoms com23 alrimdv anim12ii syl5bi alrimiv vex eqeq1 rexbidv elab ralab
|
|
df-ral anbi2i imbi12i albii bitr2i sylib fununi cnvuni cnvex dfiun2 eqtri
|
|
syl ciun funeqi sylibr ) BIZJZKZWJCIZLZWMWJLZMZCANZOZBANZDIZEIZJZPZEAQZDU
|
|
BZUCZKZAUCJZKWSFIZKZXIGIZLZXKXILZMZGXENZOZFXENZXGWSXIXBPZEAQZXJXKXBPZEAQZ
|
|
XNRZGSZOZRZFSZXQWSYEFXSXIHIZJZPZHAQWSYDXRYIEHAEHTXBYHXIXAYGUDUEUAWSYIYDHA
|
|
YGAUFWSYHKZYGWMLZWMYGLZMZCANZOZYIYDRWRYOBYGABHTZWLYJWQYNYPWKYHWJYGUDUGYPW
|
|
PYMCAYPWNYKWOYLWJYGWMUHWJYGWMUIUJUKUNULYJYIXJYNYCYIXJYJXIYHUMUOYNYIYBGYNY
|
|
AYIXNYNXTYIXNRZEAXAAUFYNYGXALZXAYGLZMZXTYQRYMYTCXAACETYKYRYLYSWMXAYGUIWMX
|
|
AYGUHUJULYTXTYIXNYTXNXTYIOZYHXBLZXBYHLZMYRUUBYSUUCYGXAUPXAYGUPUQUUAXLUUBX
|
|
MUUCYIXTXLUUBURXIYHXKXBUSVDXKXBXIYHUSUJUTVAVBVCVEVFVGVBVCVHVIXQXIXEUFZXPR
|
|
ZFSYFXPFXEVOUUEYEFUUDXSXPYDXDXSDXIFVJDFTXCXREAWTXIXBVKVLVMXOYCXJXDYAXNGDD
|
|
GTXCXTEAWTXKXBVKVLVNVPVQVRVSVTXEFGWAWFXHXFXHEAXBWGXFEAWBEDAXBXAEVJWCWDWEW
|
|
HWI $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of one-to-one functions
|
|
is a one-to-one function. (Contributed by NM, 11-Aug-2004.) $)
|
|
fun11uni $p |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\
|
|
A. g e. A ( f C_ g \/ g C_ f ) ) ->
|
|
( Fun U. A /\ Fun `' U. A ) ) $=
|
|
( cv wfun ccnv wa wo wral cuni simpl anim1i ralimi fununi simpr funcnvuni
|
|
wss syl jca ) BDZEZTFEZGZTCDZQUDTQHCAIZGZBAIZAJZEZUHFEZUGUAUEGZBAIUIUFUKB
|
|
AUCUAUEUAUBKLMABCNRUGUBUEGZBAIUJUFULBAUCUBUEUAUBOLMABCPRS $.
|
|
$}
|
|
|
|
$( The intersection with a function is a function. Exercise 14(a) of
|
|
[Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by
|
|
Andrew Salmon, 17-Sep-2011.) $)
|
|
funin $p |- ( Fun F -> Fun ( F i^i G ) ) $=
|
|
( cin wss wfun wi inss1 funss ax-mp ) ABCZADAEJEFABGJAHI $.
|
|
|
|
$( The restriction of a one-to-one function is one-to-one. (Contributed by
|
|
NM, 25-Mar-1998.) $)
|
|
funres11 $p |- ( Fun `' F -> Fun `' ( F |` A ) ) $=
|
|
( cres wss ccnv wfun wi resss cnvss funss mp2b ) BACZBDLEZBEZDNFMFGBAHLBIMN
|
|
JK $.
|
|
|
|
$( The converse of a restricted function. (Contributed by NM,
|
|
27-Mar-1998.) $)
|
|
funcnvres $p |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) ) $=
|
|
( ccnv wfun cima cres cdm df-ima df-rn eqtri reseq2i wceq resss cnvss ax-mp
|
|
crn wss funssres mpan2 syl5req ) BCZDZUABAEZFUABAFZCZGZFZUEUCUFUAUCUDPUFBAH
|
|
UDIJKUBUEUAQZUGUELUDBQUHBAMUDBNOUAUERST $.
|
|
|
|
$( Converse of a restricted identity function. (Contributed by FL,
|
|
4-Mar-2007.) $)
|
|
cnvresid $p |- `' ( _I |` A ) = ( _I |` A ) $=
|
|
( cid ccnv wceq wfun cres cnvi eqcomi funi funeq cima funcnvres imai syl6eq
|
|
mpbii reseq12i mp2b ) BBCZDZREZBAFZCZUADRBGHSBETIBRJOTUBRBAKZFUAABLRBUCAGAM
|
|
PNQ $.
|
|
|
|
$( The converse of a restriction of the converse of a function equals the
|
|
function restricted to the image of its converse. (Contributed by NM,
|
|
4-May-2005.) $)
|
|
funcnvres2 $p |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) ) $=
|
|
( wfun ccnv cres cima wceq funcnvcnv funcnvres syl wrel funrel dfrel2 sylib
|
|
reseq1d eqtrd ) BCZBDZAEDZRDZRAFZEZBUAEQTCSUBGBHARIJQTBUAQBKTBGBLBMNOP $.
|
|
|
|
$( The image of the preimage of a function. (Contributed by NM,
|
|
25-May-2004.) $)
|
|
funimacnv $p |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) ) $=
|
|
( wfun ccnv cima cres crn cin funcnvres2 rneqd df-ima syl6reqr df-rn ineq2i
|
|
cdm dmres dfdm4 3eqtr2ri syl6eq ) BCZBBDZAEZEZUAAFZDZGZABGZHZTUFBUBFZGUCTUE
|
|
UIABIJBUBKLUHAUAOZHUDOUFUGUJABMNUAAPUDQRS $.
|
|
|
|
$( A kind of contraposition law that infers a subclass of an image from a
|
|
preimage subclass. (Contributed by NM, 25-May-2004.) $)
|
|
funimass1 $p |- ( ( Fun F /\ A C_ ran F ) ->
|
|
( ( `' F " A ) C_ B -> A C_ ( F " B ) ) ) $=
|
|
( ccnv cima wss wfun crn wa imass2 funimacnv wceq dfss biimpi eqcomd sseq1d
|
|
cin sylan9eq syl5ib ) CDAEZBFCTEZCBEZFCGZACHZFZIZAUBFTBCJUFUAAUBUCUEUAAUDQZ
|
|
AACKUEAUGUEAUGLAUDMNORPS $.
|
|
|
|
$( A kind of contraposition law that infers an image subclass from a subclass
|
|
of a preimage. (Contributed by NM, 25-May-2004.) $)
|
|
funimass2 $p |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B ) $=
|
|
( ccnv cima wss wfun imass2 crn cin funimacnv sseq2d inss1 sstr2 mpi syl6bi
|
|
imp sylan2 ) ACDBEZFCGZCAEZCSEZFZUABFZASCHTUCUDTUCUABCIZJZFZUDTUBUFUABCKLUG
|
|
UFBFUDBUEMUAUFBNOPQR $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( One direction of ~ imadif . This direction does not require
|
|
` Fun ``' F ` . (Contributed by Jim Kingdon, 25-Dec-2018.) $)
|
|
imadiflem $p |- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) ) $=
|
|
( vx vy cv wbr wrex wn wa cab cdif cima wcel wex df-rex notbii wal sylibr
|
|
dfima2 alnex 19.29r sylan2br simpl simplr wi simpr ancom imnan bitr4i mpd
|
|
sylib jca32 eldif anbi1i anandir bitri eximi syl2anb difeq12i difab eqtri
|
|
syl ss2abi 3sstr4i ) DFZEFCGZDAHZVGDBHZIZJZEKZVGDABLZHZEKCAMZCBMZLZCVMMVK
|
|
VNEVHVFANZVGJZDOZVFBNZVGJZDOZIZVNVJVGDAPVIWCVGDBPQVTWDJZVFVMNZVGJZDOZVNWE
|
|
VSWBIZJZDOZWHWDVTWIDRWKWBDUAVSWIDUBUCWJWGDWJVSWAIZVGJJZWGWJVSWLVGVSWIUDWJ
|
|
VGWLVRVGWIUEZWJWIVGWLUFZVSWIUGWIVGWAJZIWOWBWPWAVGUHQVGWAUIUJULUKWNUMWGVRW
|
|
LJZVGJWMWFWQVGVFABUNUOVRWLVGUPUQSURVCVGDVMPSUSVDVQVHEKZVIEKZLVLVOWRVPWSDE
|
|
CATDECBTUTVHVIEVAVBDECVMTVE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( The image of a difference is the difference of images. (Contributed by
|
|
NM, 24-May-1998.) $)
|
|
imadif $p |- ( Fun `' F ->
|
|
( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) ) $=
|
|
( vx vy cdif cima cv wbr wrex cab wn wa wcel wex exbii wmo df-rex dfima2
|
|
wi ccnv wfun anandir 19.40 sylbi wal nfv nfe1 funmo vex brcnv mobii sylib
|
|
nfan mopick sylan con2d imnan alrimi ex exancom alnex 3imtr3g anim2d syl5
|
|
eldif anbi1i bitri notbii anbi12i 3imtr4g ss2abdv difab eqtri 3sstr4g wss
|
|
difeq12i imadiflem a1i eqssd ) CUAZUBZCABFZGZCAGZCBGZFZWBDHZEHZCIZDWCJZEK
|
|
WJDAJZWJDBJZLZMZEKZWDWGWBWKWOEWBWHANZWHBNZLZMZWJMZDOZWQWJMZDOZWRWJMZDOZLZ
|
|
MZWKWOXBXDWSWJMZDOZMZWBXHXBXCXIMZDOXKXAXLDWQWSWJUCPXCXIDUDUEWBXJXGXDWBWJW
|
|
SMZDOZXELZDUFZXJXGWBXNXPWBXNMZXODWBXNDWBDUGXMDUHUNXQWRWJLTXOXQWJWRWBWJDQZ
|
|
XNWJWSTWBWIWHWAIZDQXRDWIWAUIXSWJDWIWHCEUJDUJUKULUMWJWSDUOUPUQWRWJURUMUSUT
|
|
WJWSDVAXEDVBVCVDVEWKWHWCNZWJMZDOXBWJDWCRYAXADXTWTWJWHABVFVGPVHWLXDWNXGWJD
|
|
ARWMXFWJDBRVIVJVKVLDECWCSWGWLEKZWMEKZFWPWEYBWFYCDECASDECBSVQWLWMEVMVNVOWG
|
|
WDVPWBABCVRVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( One direction of ~ imain . This direction does not require
|
|
` Fun ``' F ` . (Contributed by Jim Kingdon, 25-Dec-2018.) $)
|
|
imainlem $p |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) ) $=
|
|
( vx vy cv wbr cin wrex cab wa cima wcel df-rex elin anbi1i anandir sylbi
|
|
wex dfima2 bitri exbii 19.40 anbi12i sylibr ss2abi ineq12i eqtri 3sstr4i
|
|
inab ) DFZEFCGZDABHZIZEJULDAIZULDBIZKZEJZCUMLCALZCBLZHZUNUQEUNUKAMZULKZDS
|
|
ZUKBMZULKZDSZKZUQUNUKUMMZULKZDSZVHULDUMNVKVCVFKZDSVHVJVLDVJVBVEKZULKVLVIV
|
|
MULUKABOPVBVEULQUAUBVCVFDUCRRUOVDUPVGULDANULDBNUDUEUFDECUMTVAUOEJZUPEJZHU
|
|
RUSVNUTVODECATDECBTUGUOUPEUJUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z F $.
|
|
$( The image of an intersection is the intersection of images.
|
|
(Contributed by Paul Chapman, 11-Apr-2009.) $)
|
|
imain $p |- ( Fun `' F ->
|
|
( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) ) $=
|
|
( vx vy vz ccnv cin cima cv wbr wrex cab wcel wex simpr wal df-rex dfima2
|
|
wa wfun wss imainlem a1i eeanv simprll anim12i wi funcnveq biimpi 19.21bi
|
|
weq 19.21bbi imp sylan2 simprrl eqeltrd elin sylanbrc simprlr jca exlimdv
|
|
eximdv syl5bir anbi12i 3imtr4g ss2abdv ineq12i inab eqtri 3sstr4g eqssd
|
|
ex ) CGUAZCABHZIZCAIZCBIZHZVPVSUBVNABCUCUDVNDJZEJZCKZDALZFJZWACKZFBLZTZEM
|
|
ZWBDVOLZEMVSVPVNWGWIEVNVTANZWBTZDOZWDBNZWETZFOZTZVTVONZWBTZDOZWGWIWPWKWNT
|
|
ZFOZDOVNWSWKWNDFUEVNXAWRDVNWTWRFVNWTWRVNWTTZWQWBXBWJVTBNWQVNWJWBWNUFXBVTW
|
|
DBWTVNWBWETZDFULZWKWBWNWEWJWBPWMWEPUGVNXCXDVNXCXDUHZEFVNXEFQEQZDVNXFDQDEF
|
|
CUIUJUKUMUNUOVNWKWMWEUPUQVTABURUSVNWJWBWNUTVAVMVBVCVDWCWLWFWOWBDARWEFBRVE
|
|
WBDVORVFVGVSWCEMZWFEMZHWHVQXGVRXHDECASFECBSVHWCWFEVIVJDECVOSVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A b x y z $. $d B b x y z $. $d C z $.
|
|
$( Lemma for ~ funimaexg . It constitutes the interesting part of
|
|
~ funimaexg , in which ` B C_ dom A ` . (Contributed by Jim Kingdon,
|
|
27-Dec-2018.) $)
|
|
funimaexglem $p |- ( ( Fun A /\ B e. C /\ B C_ dom A ) ->
|
|
( A " B ) e. _V ) $=
|
|
( vz vx vy vb wcel wss w3a cv wex wrex wel wi wal wral wa simprbi exbii
|
|
wfun cdm cima cvv wbr cab wmo dffun7 3ad2ant1 ssralv 3ad2ant3 mpd alrimiv
|
|
wrel wceq weu sseq1 biimpar 3adant1 jca dffun8 adantl ssel adantr sylsyld
|
|
simp1 rsp ralrimiv zfrep6 3syl raleq exbidv 3ad2ant2 mpbid 3com12 vtocleg
|
|
wb 3expib 3impib df-rex exancom bitri sylib 19.29 nfcv nfmo1 nfralxy nfe1
|
|
ralbii r19.26 mopick ralimi sylbir alrimi eximi syl syl2anc r19.23v albii
|
|
nfan abss sylibr dfima2 sseq1i vex ssex exlimiv ) AUAZBCHZBAUBZIZJZABUCZD
|
|
KZIZDLZXMUDHZXLEKZFKAUEZEBMZFUFZXNIZDLZXPXLXTFDNZOZFPZDLZYCXLXSYDOZEBQZFP
|
|
ZDLZYGXLXSFUGZEBQZDPZXSYDRZFLZEBQZDLZYKXLYMDXLYLEXJQZYMXHXIYSXKXHAUNZYSEF
|
|
AUHSUIXKXHYSYMOXIYLEBXJUJUKULUMXLXSFXNMZEBQZDLZYRXIXHXKUUCXIXHXKUUCXHXKRU
|
|
UCOGBCGKZBUOZXHXKUUCXHUUEXKUUCXHUUEXKJZUUAEUUDQZDLZUUCUUFUUDXJIZXHRZXSFUP
|
|
ZEUUDQUUHUUFUUIXHUUEXKUUIXHUUEUUIXKUUDBXJUQURUSXHUUEXKVFUTUUJUUKEUUDUUJUU
|
|
KEXJQZEGNZXRXJHZUUKXHUULUUIXHYTUULEFAVASVBUUIUUMUUNOXHUUDXJXRVCVDUUKEXJVG
|
|
VEVHXSEFGDVIVJUUEXHUUHUUCVQXKUUEUUGUUBDUUAEUUDBVKVLVMVNVOVRVPVSVOUUBYQDUU
|
|
AYPEBUUAYDXSRFLYPXSFXNVTYDXSFWAWBWITWCYNYRRYMYQRZDLYKYMYQDWDUUOYJDUUOYIFY
|
|
MYQFYLFEBFBWEZXSFWFWGYPFEBUUPYOFWHWGWTUUOYLYPRZEBQYIYLYPEBWJUUQYHEBXSYDFW
|
|
KWLWMWNWOWPWQYJYFDYIYEFXSYDEBWRWSTWCYBYFDXTFXNXATXBXOYBDXMYAXNEFABXCXDTXB
|
|
XOXQDXMXNDXEXFXGWP $.
|
|
$}
|
|
|
|
$( Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284.
|
|
Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM,
|
|
10-Sep-2006.) $)
|
|
funimaexg $p |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V ) $=
|
|
( wfun wcel cdm cin cima cvv wrel wceq simpl funrel cres crn resres reseq2i
|
|
wa incom df-ima eqtr4i resdm reseq1d syl5eqr rneqd 3eqtr4g inex1g wss inss2
|
|
3syl funimaexglem mp3an3 sylan2 eqeltrrd ) ADZBCEZRZABAFZGZHZABHZIUQUOAJZUT
|
|
VAKUOUPLAMVBAUSNZOABNZOUTVAVBVCVDVBVCAURNZBNZVDVFAURBGZNVCAURBPUSVGABURSQUA
|
|
VBVEABAUBUCUDUEAUSTABTUFUJUPUOUSIEZUTIEZBURCUGUOVHUSURUHVIBURUIAUSIUKULUMUN
|
|
$.
|
|
|
|
${
|
|
zfrep5.1 $e |- B e. _V $.
|
|
$( The image of a set under any function is also a set. Equivalent of
|
|
Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise
|
|
9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.) $)
|
|
funimaex $p |- ( Fun A -> ( A " B ) e. _V ) $=
|
|
( wfun cvv wcel cima funimaexg mpan2 ) ADBEFABGEFCABEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d b x y z $. $d z ph $.
|
|
$( Part of a study of the Axiom of Replacement used by the Isabelle prover.
|
|
The object PrimReplace is apparently the image of the function encoded
|
|
by ` ph ( x , y ) ` i.e. the class ` ( { <. x , y >. | ph } " A ) ` .
|
|
If so, we can prove Isabelle's "Axiom of Replacement" conclusion without
|
|
using the Axiom of Replacement, for which I (N. Megill) currently have
|
|
no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by
|
|
Mario Carneiro, 4-Dec-2016.) $)
|
|
isarep1 $p |- ( b e. ( { <. x , y >. | ph } " A ) <->
|
|
E. x e. A [ b / y ] ph ) $=
|
|
( vz copab cima wcel wbr wrex wsb vex elima cop wsbc df-br sbsbc 3bitri
|
|
cv opelopabsb sbbii bitr2i rexbii nfs1v nfv sbequ12r cbvrex ) ETZABCGZDHI
|
|
FTZUIUJJZFDKACELZBFLZFDKUMBDKFUIUJDEMNULUNFDULUKUIOUJIACUIPZBUKPZUNUKUIUJ
|
|
QABCUKUIUAUNUOBFLUPUMUOBFACERUBUOBFRUCSUDUNUMFBDUMBFUEUMFUFUMFBUGUHS $.
|
|
$}
|
|
|
|
${
|
|
$d w x y A $. $d y z $. $d w ph $. $d z ph $.
|
|
isarep2.1 $e |- A e. _V $.
|
|
isarep2.2 $e |- A. x e. A A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) $.
|
|
$( Part of a study of the Axiom of Replacement used by the Isabelle prover.
|
|
In Isabelle, the sethood of PrimReplace is apparently postulated
|
|
implicitly by its type signature " ` [ ` i, ` [ ` i, i ` ] ` => o ` ] `
|
|
=> i", which automatically asserts that it is a set without using any
|
|
axioms. To prove that it is a set in Metamath, we need the hypotheses
|
|
of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement
|
|
in the form ~ funimaex . (Contributed by NM, 26-Oct-2006.) $)
|
|
isarep2 $p |- E. w w = ( { <. x , y >. | ph } " A ) $=
|
|
( copab cima cv wcel wa cvv cres resima resopab wmo wi wal imaeq1i eqtr3i
|
|
wfun funopab wsb weq rspec nfv sylibr moanimv mpbir mpgbir funimaex ax-mp
|
|
mo3 eqeltri isseti ) EABCIZFJZUSBKFLZAMZBCIZFJZNURFOZFJUSVCURFPVDVBFABCFQ
|
|
UAUBVBUCZVCNLVEVACRZBVABCUDVFUTACRZSUTAACDUEMCDUFSDTCTZVGVHBFHUGACDADUHUO
|
|
UIUTACUJUKULVBFGUMUNUPUQ $.
|
|
$}
|
|
|
|
$( Equality theorem for function predicate with domain. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
fneq1 $p |- ( F = G -> ( F Fn A <-> G Fn A ) ) $=
|
|
( wceq wfun cdm wa wfn funeq dmeq eqeq1d anbi12d df-fn 3bitr4g ) BCDZBEZBFZ
|
|
ADZGCEZCFZADZGBAHCAHOPSRUABCIOQTABCJKLBAMCAMN $.
|
|
|
|
$( Equality theorem for function predicate with domain. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
fneq2 $p |- ( A = B -> ( F Fn A <-> F Fn B ) ) $=
|
|
( wceq wfun cdm wa wfn eqeq2 anbi2d df-fn 3bitr4g ) ABDZCEZCFZADZGNOBDZGCAH
|
|
CBHMPQNABOIJCAKCBKL $.
|
|
|
|
${
|
|
fneq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq1d $p |- ( ph -> ( F Fn A <-> G Fn A ) ) $=
|
|
( wceq wfn wb fneq1 syl ) ACDFCBGDBGHEBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
fneq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq2d $p |- ( ph -> ( F Fn A <-> F Fn B ) ) $=
|
|
( wceq wfn wb fneq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
fneq12d.1 $e |- ( ph -> F = G ) $.
|
|
fneq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
NM, 26-Jun-2011.) $)
|
|
fneq12d $p |- ( ph -> ( F Fn A <-> G Fn B ) ) $=
|
|
( wfn fneq1d fneq2d bitrd ) ADBHEBHECHABDEFIABCEGJK $.
|
|
$}
|
|
|
|
$( Equality theorem for function predicate with domain. (Contributed by
|
|
Thierry Arnoux, 31-Jan-2017.) $)
|
|
fneq12 $p |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) ) $=
|
|
( wceq wa simpl simpr fneq12d ) CDEZABEZFABCDJKGJKHI $.
|
|
|
|
${
|
|
fneq1i.1 $e |- F = G $.
|
|
$( Equality inference for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq1i $p |- ( F Fn A <-> G Fn A ) $=
|
|
( wceq wfn wb fneq1 ax-mp ) BCEBAFCAFGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
fneq2i.1 $e |- A = B $.
|
|
$( Equality inference for function predicate with domain. (Contributed by
|
|
NM, 4-Sep-2011.) $)
|
|
fneq2i $p |- ( F Fn A <-> F Fn B ) $=
|
|
( wceq wfn wb fneq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
nffn.1 $e |- F/_ x F $.
|
|
nffn.2 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for a function with domain.
|
|
(Contributed by NM, 30-Jan-2004.) $)
|
|
nffn $p |- F/ x F Fn A $=
|
|
( wfn wfun cdm wceq wa df-fn nffun nfdm nfeq nfan nfxfr ) CBFCGZCHZBIZJAC
|
|
BKQSAACDLARBACDMENOP $.
|
|
$}
|
|
|
|
$( A function with domain is a function. (Contributed by NM, 1-Aug-1994.) $)
|
|
fnfun $p |- ( F Fn A -> Fun F ) $=
|
|
( wfn wfun cdm wceq df-fn simplbi ) BACBDBEAFBAGH $.
|
|
|
|
$( A function with domain is a relation. (Contributed by NM, 1-Aug-1994.) $)
|
|
fnrel $p |- ( F Fn A -> Rel F ) $=
|
|
( wfn wfun wrel fnfun funrel syl ) BACBDBEABFBGH $.
|
|
|
|
$( The domain of a function. (Contributed by NM, 2-Aug-1994.) $)
|
|
fndm $p |- ( F Fn A -> dom F = A ) $=
|
|
( wfn wfun cdm wceq df-fn simprbi ) BACBDBEAFBAGH $.
|
|
|
|
${
|
|
funfni.1 $e |- ( ( Fun F /\ B e. dom F ) -> ph ) $.
|
|
$( Inference to convert a function and domain antecedent. (Contributed by
|
|
NM, 22-Apr-2004.) $)
|
|
funfni $p |- ( ( F Fn A /\ B e. A ) -> ph ) $=
|
|
( wfn wcel wa wfun cdm fnfun adantr fndm eleq2d biimpar syl2anc ) DBFZCBG
|
|
ZHDIZCDJZGZAQSRBDKLQUARQTBCBDMNOEP $.
|
|
$}
|
|
|
|
$( A function has a unique domain. (Contributed by NM, 11-Aug-1994.) $)
|
|
fndmu $p |- ( ( F Fn A /\ F Fn B ) -> A = B ) $=
|
|
( wfn cdm fndm sylan9req ) CADCBDACEBACFBCFG $.
|
|
|
|
$( The first argument of binary relation on a function belongs to the
|
|
function's domain. (Contributed by NM, 7-May-2004.) $)
|
|
fnbr $p |- ( ( F Fn A /\ B F C ) -> B e. A ) $=
|
|
( wfn wbr cdm wcel wrel fnrel releldm sylan fndm eleq2d biimpa syldan ) DAE
|
|
ZBCDFZBDGZHZBAHZQDIRTADJBCDKLQTUAQSABADMNOP $.
|
|
|
|
$( The first argument of an ordered pair in a function belongs to the
|
|
function's domain. (Contributed by NM, 8-Aug-1994.) $)
|
|
fnop $p |- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A ) $=
|
|
( cop wcel wfn wbr df-br fnbr sylan2br ) BCEDFDAGBCDHBAFBCDIABCDJK $.
|
|
|
|
${
|
|
$d y F $. $d y B $.
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fneu $p |- ( ( F Fn A /\ B e. A ) -> E! y B F y ) $=
|
|
( cv wbr weu wfun cdm wa wmo funmo adantr wex wb eldmg ibi adantl exmoeu2
|
|
wcel syl mpbid funfni ) CAEDFZAGZBCDDHZCDIZTZJZUDAKZUEUFUJUHACDLMUIUDANZU
|
|
JUEOUHUKUFUHUKACDUGPQRUDASUAUBUC $.
|
|
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
7-Nov-1995.) $)
|
|
fneu2 $p |- ( ( F Fn A /\ B e. A ) -> E! y <. B , y >. e. F ) $=
|
|
( wfn wcel wa cv wbr weu cop fneu df-br eubii sylib ) DBECBFGCAHZDIZAJCPK
|
|
DFZAJABCDLQRACPDMNO $.
|
|
$}
|
|
|
|
$( The union of two functions with disjoint domains. (Contributed by NM,
|
|
22-Sep-2004.) $)
|
|
fnun $p |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) ->
|
|
( F u. G ) Fn ( A u. B ) ) $=
|
|
( wfn wa cin c0 wceq cun wfun cdm wi df-fn ineq12 eqeq1d funun syl6bir dmun
|
|
anbi2d uneq12 syl5eq jctird syl6ibr expd impcom an4s syl2anb imp ) CAEZDBEZ
|
|
FABGZHIZCDJZABJZEZUJCKZCLZAIZFDKZDLZBIZFUMUPMZUKCANDBNUQUTUSVBVCUSVBFZUQUTF
|
|
ZVCVDVEUMUPVDVEUMFZUNKZUNLZUOIZFUPVDVFVGVIVDVFVEURVAGZHIZFVGVDVKUMVEVDVJULH
|
|
URAVABOPTCDQRVDVHURVAJUOCDSURAVABUAUBUCUNUONUDUEUFUGUHUI $.
|
|
|
|
${
|
|
fnunop.x $e |- ( ph -> X e. _V ) $.
|
|
fnunop.y $e |- ( ph -> Y e. _V ) $.
|
|
fnunop.f $e |- ( ph -> F Fn D ) $.
|
|
fnunop.g $e |- G = ( F u. { <. X , Y >. } ) $.
|
|
fnunop.e $e |- E = ( D u. { X } ) $.
|
|
fnunop.d $e |- ( ph -> -. X e. D ) $.
|
|
$( Extension of a function with a new ordered pair. (Contributed by NM,
|
|
28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) $)
|
|
fnunsn $p |- ( ph -> G Fn E ) $=
|
|
( cop csn cun wfn cvv wcel sylibr cin c0 fnsng syl2anc wn disjsn syl21anc
|
|
wceq fnun fneq1i fneq2i bitri ) ADFGNOZPZBFOZPZQZECQZADBQUMUOQZBUOUAUBUHZ
|
|
UQJAFRSGRSUSHIFGRRUCUDAFBSUEUTMBFUFTBUODUMUIUGURUNCQUQCEUNKUJCUPUNLUKULT
|
|
$.
|
|
$}
|
|
|
|
$( Composition of two functions. (Contributed by NM, 22-May-2006.) $)
|
|
fnco $p |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B ) $=
|
|
( wfn crn wss w3a ccom wfun cdm wceq fnfun funco syl2an 3adant3 fndm sseq2d
|
|
wa biimpar dmcosseq syl 3adant2 3ad2ant2 eqtrd df-fn sylanbrc ) CAEZDBEZDFZ
|
|
AGZHZCDIZJZUMKZBLUMBEUHUIUNUKUHCJDJUNUIACMBDMCDNOPULUODKZBUHUKUOUPLZUIUHUKS
|
|
UJCKZGZUQUHUSUKUHURAUJACQRTCDUAUBUCUIUHUPBLUKBDQUDUEUMBUFUG $.
|
|
|
|
$( A function does not change when restricted to its domain. (Contributed by
|
|
NM, 5-Sep-2004.) $)
|
|
fnresdm $p |- ( F Fn A -> ( F |` A ) = F ) $=
|
|
( wfn wrel cdm wss cres wceq fnrel fndm eqimss syl relssres syl2anc ) BACZB
|
|
DBEZAFZBAGBHABIOPAHQABJPAKLBAMN $.
|
|
|
|
$( A function restricted to a class disjoint with its domain is empty.
|
|
(Contributed by NM, 23-Sep-2004.) $)
|
|
fnresdisj $p |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) ) $=
|
|
( cres c0 wceq cdm wfn wrel wb relres reldm0 ax-mp dmres incom eqtri ineq1d
|
|
cin fndm syl5eq eqeq1d syl5rbb ) CBDZEFZUCGZEFZCAHZABRZEFUCIUDUFJCBKUCLMUGU
|
|
EUHEUGUECGZBRZUHUEBUIRUJCBNBUIOPUGUIABACSQTUAUB $.
|
|
|
|
$( Membership in two functions restricted by each other's domain.
|
|
(Contributed by NM, 8-Aug-1994.) $)
|
|
2elresin $p |- ( ( F Fn A /\ G Fn B ) ->
|
|
( ( <. x , y >. e. F /\ <. x , z >. e. G ) <->
|
|
( <. x , y >. e. ( F |` ( A i^i B ) ) /\
|
|
<. x , z >. e. ( G |` ( A i^i B ) ) ) ) ) $=
|
|
( wfn wa cv cop wcel cin cres fnop anim12i vex opres resss sseli wi anbi12d
|
|
an4s elin sylibr biimprd syl ex pm2.43d impbid1 ) FDHZGEHZIZAJZBJZKZFLZUNCJ
|
|
ZKZGLZIZUPFDEMZNZLZUSGVBNZLZIZUMVAVGUMVAVAVGUAZUMVAIZUNVBLZVHVIUNDLZUNELZIZ
|
|
VJUKUQULUTVMUKUQIVKULUTIVLDUNUOFOEUNURGOPUCUNDEUDUEVJVGVAVJVDUQVFUTUNUOFVBB
|
|
QRUNURGVBCQRUBUFUGUHUIVDUQVFUTVCFUPFVBSTVEGUSGVBSTPUJ $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
NM, 10-Oct-2007.) $)
|
|
fnssresb $p |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) ) $=
|
|
( cres wfn wfun cdm wceq wa wss df-fn fnfun funres biantrurd ssdmres sseq2d
|
|
syl fndm syl5bbr bitr3d syl5bb ) CBDZBEUBFZUBGBHZIZCAEZBAJZUBBKUFUDUEUGUFUC
|
|
UDUFCFUCACLBCMQNUDBCGZJUFUGBCOUFUHABACRPSTUA $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
NM, 2-Aug-1994.) $)
|
|
fnssres $p |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B ) $=
|
|
( wfn cres wss fnssresb biimpar ) CADCBEBDBAFABCGH $.
|
|
|
|
$( Restriction of a function's domain with an intersection. (Contributed by
|
|
NM, 9-Aug-1994.) $)
|
|
fnresin1 $p |- ( F Fn A -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) ) $=
|
|
( wfn cin wss cres inss1 fnssres mpan2 ) CADABEZAFCKGKDABHAKCIJ $.
|
|
|
|
$( Restriction of a function's domain with an intersection. (Contributed by
|
|
NM, 9-Aug-1994.) $)
|
|
fnresin2 $p |- ( F Fn A -> ( F |` ( B i^i A ) ) Fn ( B i^i A ) ) $=
|
|
( wfn cin wss cres inss2 fnssres mpan2 ) CADBAEZAFCKGKDBAHAKCIJ $.
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( An equivalence for functionality of a restriction. Compare ~ dffun8 .
|
|
(Contributed by Mario Carneiro, 20-May-2015.) $)
|
|
fnres $p |- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y ) $=
|
|
( cres wfun cdm wa cv wbr wmo wral ancom wal wcel vex mpbiran 3bitr4i wss
|
|
bitri wceq wex wfn wi brres mobii moanimv albii wrel relres dffun6 df-ral
|
|
weu dmres inss1 eqsstri eqss dfss3 elin2 baib eldm syl6bb ralbiia anbi12i
|
|
cin r19.26 df-fn eu5 ralbii ) DCEZFZVJGZCUAZHZAIZBIZDJZBUBZVQBKZHZACLZVJC
|
|
UCVQBUMZACLVSACLZVRACLZHWDWCHVNWAWCWDMVKWCVMWDVOVPVJJZBKZANZVOCOZVSUDZANV
|
|
KWCWFWIAWFWHVQHZBKWIWEWJBWEVQWHHWJVOVPDCBPUEVQWHMTUFWHVQBUGTUHVKVJUIWGDCU
|
|
JABVJUKQVSACULRVMCVLSZWDVMVLCSWKVLCDGZVECDCUNZCWLUOUPVLCUQQWKVOVLOZACLWDA
|
|
CVLURWNVRACWHWNVOWLOZVRWNWHWOVOCWLVLWMUSUTBVODAPVAVBVCTTVDVRVSACVFRVJCVGW
|
|
BVTACVQBVHVIR $.
|
|
$}
|
|
|
|
$( Functionality and domain of restricted identity. (Contributed by NM,
|
|
27-Aug-2004.) $)
|
|
fnresi $p |- ( _I |` A ) Fn A $=
|
|
( cid cres wfn wfun cdm wceq funi funres ax-mp dmresi df-fn mpbir2an ) BACZ
|
|
ADNEZNFAGBEOHABIJAKNALM $.
|
|
|
|
$( The image of a function's domain is its range. (Contributed by NM,
|
|
4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fnima $p |- ( F Fn A -> ( F " A ) = ran F ) $=
|
|
( wfn cima cres crn df-ima fnresdm rneqd syl5eq ) BACZBADBAEZFBFBAGKLBABHIJ
|
|
$.
|
|
|
|
$( A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.)
|
|
(Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fn0 $p |- ( F Fn (/) <-> F = (/) ) $=
|
|
( c0 wfn wceq wrel cdm fnrel fndm reldm0 biimpar syl2anc wfun fun0 mpbir2an
|
|
dm0 df-fn fneq1 mpbiri impbii ) ABCZABDZTAEZAFBDZUABAGBAHUBUAUCAIJKUATBBCZU
|
|
DBLBFBDMOBBPNBABQRS $.
|
|
|
|
$( A class that is disjoint with the domain of a function has an empty image
|
|
under the function. (Contributed by FL, 24-Jan-2007.) $)
|
|
fnimadisj $p |- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) ) $=
|
|
( wfn cin c0 wceq wa cdm cima fndm ineq1d eqeq1d biimpar imadisj sylibr ) C
|
|
ADZABEZFGZHCIZBEZFGZCBJFGQUBSQUARFQTABACKLMNCBOP $.
|
|
|
|
$( Images under a function never map nonempty sets to empty sets.
|
|
(Contributed by Stefan O'Rear, 21-Jan-2015.) $)
|
|
fnimaeq0 $p |- ( ( F Fn A /\ B C_ A ) ->
|
|
( ( F " B ) = (/) <-> B = (/) ) ) $=
|
|
( cima c0 wceq cdm cin wfn wa imadisj incom fndm sseq2d biimpar df-ss sylib
|
|
wss syl5eq eqeq1d syl5bb ) CBDEFCGZBHZEFCAIZBARZJZBEFCBKUFUCBEUFUCBUBHZBUBB
|
|
LUFBUBRZUGBFUDUHUEUDUBABACMNOBUBPQSTUA $.
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
$( Alternate definition for the "maps to" notation ~ df-mpt . (Contributed
|
|
by Mario Carneiro, 30-Dec-2016.) $)
|
|
dfmpt3 $p |- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } ) $=
|
|
( vy vz cmpt cv wcel wceq copab csn cxp ciun df-mpt cop wex anbi2i 2exbii
|
|
wa velsn eliunxp elopab 3bitr4i eqriv eqtr4i ) ABCFAGZBHZDGZCIZSZADJZABUF
|
|
KCKZLMZADBCNEUMUKEGZUFUHOIZUGUHULHZSZSZDPAPUOUJSZDPAPUNUMHUNUKHURUSADUQUJ
|
|
UOUPUIUGDCTQQRADBULUNUAUJADUNUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
fnopabg.1 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro,
|
|
4-Dec-2016.) $)
|
|
fnopabg $p |- ( A. x e. A E! y ph <-> F Fn A ) $=
|
|
( wmo wex wa wral cv wcel copab wfn weu wfun cdm wceq wal 3bitr4i moanimv
|
|
wi albii funopab df-ral 3bitr4ri dmopab3 anbi12i r19.26 df-fn ancom bitri
|
|
eu5 ralbii fneq1i ) ACGZACHZIZBDJZBKDLZAIZBCMZDNZACOZBDJEDNUPBDJZUQBDJZIV
|
|
BPZVBQDRZIUSVCVEVGVFVHVACGZBSUTUPUBZBSVGVEVIVJBUTACUAUCVABCUDUPBDUEUFABCD
|
|
UGUHUPUQBDUIVBDUJTVDURBDVDUQUPIURACUMUQUPUKULUNDEVBFUOT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
fnopab.1 $e |- ( x e. A -> E! y ph ) $.
|
|
fnopab.2 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by NM, 5-Mar-1996.) $)
|
|
fnopab $p |- F Fn A $=
|
|
( weu wral wfn rgen fnopabg mpbi ) ACHZBDIEDJNBDFKABCDEGLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
mptfng.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The maps-to notation defines a function with domain. (Contributed by
|
|
Scott Fenton, 21-Mar-2011.) $)
|
|
mptfng $p |- ( A. x e. A B e. _V <-> F Fn A ) $=
|
|
( vy cvv wcel wral cv wceq weu wfn eueq ralbii cmpt wa copab df-mpt eqtri
|
|
fnopabg bitri ) CGHZABIFJCKZFLZABIDBMUCUEABFCNOUDAFBDDABCPAJBHUDQAFREAFBC
|
|
STUAUB $.
|
|
|
|
$( The maps-to notation defines a function with domain. (Contributed by
|
|
NM, 9-Apr-2013.) $)
|
|
fnmpt $p |- ( A. x e. A B e. V -> F Fn A ) $=
|
|
( wcel wral cvv wfn elex ralimi mptfng sylib ) CEGZABHCIGZABHDBJOPABCEKLA
|
|
BCDFMN $.
|
|
$}
|
|
|
|
$( A mapping operation with empty domain. (Contributed by Mario Carneiro,
|
|
28-Dec-2014.) $)
|
|
mpt0 $p |- ( x e. (/) |-> A ) = (/) $=
|
|
( c0 cmpt wfn wceq cvv wcel wral ral0 eqid fnmpt ax-mp fn0 mpbi ) ACBDZCEZP
|
|
CFBGHZACIQRAJACBPGPKLMPNO $.
|
|
|
|
${
|
|
$d x A $.
|
|
fnmpti.1 $e |- B e. _V $.
|
|
fnmpti.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
fnmpti $p |- F Fn A $=
|
|
( cvv wcel wral wfn rgenw mptfng mpbi ) CGHZABIDBJNABEKABCDFLM $.
|
|
|
|
$( Domain of an ordered-pair class abstraction that specifies a function.
|
|
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
dmmpti $p |- dom F = A $=
|
|
( wfn cdm wceq fnmpti fndm ax-mp ) DBGDHBIABCDEFJBDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Union of mappings which are mutually compatible. (Contributed by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
mptun $p |- ( x e. ( A u. B ) |-> C ) =
|
|
( ( x e. A |-> C ) u. ( x e. B |-> C ) ) $=
|
|
( vy cmpt cv wcel wceq wa copab df-mpt uneq12i wo elun anbi1i andir bitri
|
|
cun eqtr4i opabbii unopab ) ABCSZDFAGZUCHZEGDIZJZAEKZABDFZACDFZSZAEUCDLUK
|
|
UDBHZUFJZAEKZUDCHZUFJZAEKZSZUHUIUNUJUQAEBDLAECDLMUHUMUPNZAEKURUGUSAEUGULU
|
|
ONZUFJUSUEUTUFUDBCOPULUOUFQRUAUMUPAEUBTTT $.
|
|
$}
|
|
|
|
$( Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
feq1 $p |- ( F = G -> ( F : A --> B <-> G : A --> B ) ) $=
|
|
( wceq wfn crn wss wa wf fneq1 rneq sseq1d anbi12d df-f 3bitr4g ) CDEZCAFZC
|
|
GZBHZIDAFZDGZBHZIABCJABDJQRUATUCACDKQSUBBCDLMNABCOABDOP $.
|
|
|
|
$( Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
feq2 $p |- ( A = B -> ( F : A --> C <-> F : B --> C ) ) $=
|
|
( wceq wfn crn wss wa wf fneq2 anbi1d df-f 3bitr4g ) ABEZDAFZDGCHZIDBFZQIAC
|
|
DJBCDJOPRQABDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
feq3 $p |- ( A = B -> ( F : C --> A <-> F : C --> B ) ) $=
|
|
( wceq wfn crn wss wa wf sseq2 anbi2d df-f 3bitr4g ) ABEZDCFZDGZAHZIPQBHZIC
|
|
ADJCBDJORSPABQKLCADMCBDMN $.
|
|
|
|
$( Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof
|
|
shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
feq23 $p |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) ) $=
|
|
( wceq wf feq2 feq3 sylan9bb ) ACFABEGCBEGBDFCDEGACBEHBDCEIJ $.
|
|
|
|
${
|
|
feq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for functions. (Contributed by NM, 19-Feb-2008.) $)
|
|
feq1d $p |- ( ph -> ( F : A --> B <-> G : A --> B ) ) $=
|
|
( wceq wf wb feq1 syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
feq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq2d $p |- ( ph -> ( F : A --> C <-> F : B --> C ) ) $=
|
|
( wceq wf wb feq2 syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
feq12d.1 $e |- ( ph -> F = G ) $.
|
|
feq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq12d $p |- ( ph -> ( F : A --> C <-> G : B --> C ) ) $=
|
|
( wf feq1d feq2d bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
|
|
feq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq123d $p |- ( ph -> ( F : A --> C <-> G : B --> D ) ) $=
|
|
( wf feq12d wceq wb feq3 syl bitrd ) ABDFKCDGKZCEGKZABCDFGHILADEMRSNJDECG
|
|
OPQ $.
|
|
$}
|
|
|
|
$( Equality theorem for functions. (Contributed by FL, 16-Nov-2008.) $)
|
|
feq123 $p |- ( ( F = G /\ A = C /\ B = D ) ->
|
|
( F : A --> B <-> G : C --> D ) ) $=
|
|
( wceq w3a simp1 simp2 simp3 feq123d ) EFGZACGZBDGZHACBDEFMNOIMNOJMNOKL $.
|
|
|
|
${
|
|
feq1i.1 $e |- F = G $.
|
|
$( Equality inference for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq1i $p |- ( F : A --> B <-> G : A --> B ) $=
|
|
( wceq wf wb feq1 ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
feq2i.1 $e |- A = B $.
|
|
$( Equality inference for functions. (Contributed by NM, 5-Sep-2011.) $)
|
|
feq2i $p |- ( F : A --> C <-> F : B --> C ) $=
|
|
( wceq wf wb feq2 ax-mp ) ABFACDGBCDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
feq23i.1 $e |- A = C $.
|
|
feq23i.2 $e |- B = D $.
|
|
$( Equality inference for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq23i $p |- ( F : A --> B <-> F : C --> D ) $=
|
|
( wceq wf wb feq23 mp2an ) ACHBDHABEICDEIJFGABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
feq23d.1 $e |- ( ph -> A = C ) $.
|
|
feq23d.2 $e |- ( ph -> B = D ) $.
|
|
$( Equality deduction for functions. (Contributed by NM, 8-Jun-2013.) $)
|
|
feq23d $p |- ( ph -> ( F : A --> B <-> F : C --> D ) ) $=
|
|
( eqidd feq123d ) ABDCEFFAFIGHJ $.
|
|
$}
|
|
|
|
${
|
|
nff.1 $e |- F/_ x F $.
|
|
nff.2 $e |- F/_ x A $.
|
|
nff.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a mapping. (Contributed by NM,
|
|
29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nff $p |- F/ x F : A --> B $=
|
|
( wf wfn crn wss wa df-f nffn nfrn nfss nfan nfxfr ) BCDHDBIZDJZCKZLABCDM
|
|
SUAAABDEFNATCADEOGPQR $.
|
|
$}
|
|
|
|
${
|
|
$d V x $. $d X x $.
|
|
$( Distribute proper substitution through the function predicate with a
|
|
domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.) $)
|
|
sbcfng $p |- ( X e. V -> ( [. X / x ]. F Fn A
|
|
<-> [_ X / x ]_ F Fn [_ X / x ]_ A ) ) $=
|
|
( wcel wfn wsbc wfun cdm wa csb wb df-fn a1i sbcbidv sbcfung sbceqg bitrd
|
|
wceq csbdmg eqeq1d anbi12d sbcan 3bitr4g ) EDFZCBGZAEHCIZCJZBTZKZAEHZAECL
|
|
ZAEBLZGZUFUGUKAEUGUKMUFCBNOPUFUHAEHZUJAEHZKUMIZUMJZUNTZKULUOUFUPURUQUTAEC
|
|
DQUFUQAEUILZUNTUTAEUIBDRUFVAUSUNAECDUAUBSUCUHUJAEUDUMUNNUES $.
|
|
|
|
$( Distribute proper substitution through the function predicate with
|
|
domain and codomain. (Contributed by Alexander van der Vekens,
|
|
15-Jul-2018.) $)
|
|
sbcfg $p |- ( X e. V -> ( [. X / x ]. F : A --> B
|
|
<-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) ) $=
|
|
( wcel wf wsbc wfn crn wss wa csb wb df-f a1i sbcbidv sbcfng bitrd sbcssg
|
|
csbrng sseq1d anbi12d sbcan 3bitr4g ) FEGZBCDHZAFIDBJZDKZCLZMZAFIZAFBNZAF
|
|
CNZAFDNZHZUGUHULAFUHULOUGBCDPQRUGUIAFIZUKAFIZMUPUNJZUPKZUOLZMUMUQUGURUTUS
|
|
VBABDEFSUGUSAFUJNZUOLVBAFUJCEUAUGVCVAUOAFDEUBUCTUDUIUKAFUEUNUOUPPUFT $.
|
|
$}
|
|
|
|
$( A mapping is a function. (Contributed by NM, 2-Aug-1994.) $)
|
|
ffn $p |- ( F : A --> B -> F Fn A ) $=
|
|
( wf wfn crn wss df-f simplbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( Any function is a mapping into ` _V ` . (Contributed by NM, 31-Oct-1995.)
|
|
(Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
dffn2 $p |- ( F Fn A <-> F : A --> _V ) $=
|
|
( wfn crn cvv wss wa wf ssv biantru df-f bitr4i ) BACZMBDZEFZGAEBHOMNIJAEBK
|
|
L $.
|
|
|
|
$( A mapping is a function. (Contributed by NM, 3-Aug-1994.) $)
|
|
ffun $p |- ( F : A --> B -> Fun F ) $=
|
|
( wf wfn wfun ffn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( A mapping is a relation. (Contributed by NM, 3-Aug-1994.) $)
|
|
frel $p |- ( F : A --> B -> Rel F ) $=
|
|
( wf wfn wrel ffn fnrel syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( The domain of a mapping. (Contributed by NM, 2-Aug-1994.) $)
|
|
fdm $p |- ( F : A --> B -> dom F = A ) $=
|
|
( wf wfn cdm wceq ffn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
${
|
|
fdmi.1 $e |- F : A --> B $.
|
|
$( The domain of a mapping. (Contributed by NM, 28-Jul-2008.) $)
|
|
fdmi $p |- dom F = A $=
|
|
( wf cdm wceq fdm ax-mp ) ABCECFAGDABCHI $.
|
|
$}
|
|
|
|
$( The range of a mapping. (Contributed by NM, 3-Aug-1994.) $)
|
|
frn $p |- ( F : A --> B -> ran F C_ B ) $=
|
|
( wf wfn crn wss df-f simprbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( A function maps to its range. (Contributed by NM, 1-Sep-1999.) $)
|
|
dffn3 $p |- ( F Fn A <-> F : A --> ran F ) $=
|
|
( wfn crn wss wa wf ssid biantru df-f bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJK
|
|
$.
|
|
|
|
$( Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.)
|
|
(Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fss $p |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C ) $=
|
|
( wss wf wfn crn wa sstr2 com12 anim2d df-f 3imtr4g impcom ) BCEZABDFZACDFZ
|
|
PDAGZDHZBEZISTCEZIQRPUAUBSUAPUBTBCJKLABDMACDMNO $.
|
|
|
|
${
|
|
fssd.f $e |- ( ph -> F : A --> B ) $.
|
|
fssd.b $e |- ( ph -> B C_ C ) $.
|
|
$( Expanding the codomain of a mapping, deduction form. (Contributed by
|
|
Glauco Siliprandi, 11-Dec-2019.) $)
|
|
fssd $p |- ( ph -> F : A --> C ) $=
|
|
( wf wss fss syl2anc ) ABCEHCDIBDEHFGBCDEJK $.
|
|
$}
|
|
|
|
$( Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
|
|
shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fco $p |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C ) $=
|
|
( wf wa ccom wfn crn wss df-f wi fnco 3expib adantr rncoss sstr mpan adantl
|
|
jctird imp syl2anb sylibr ) BCDFZABEFZGDEHZAIZUGJZCKZGZACUGFUEDBIZDJZCKZGZE
|
|
AIZEJBKZGZUKUFBCDLABELUOURUKUOURUHUJULURUHMUNULUPUQUHBADENOPUNUJULUIUMKUNUJ
|
|
DEQUIUMCRSTUAUBUCACUGLUD $.
|
|
|
|
$( Functionality of a composition with weakened out of domain condition on
|
|
the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.) $)
|
|
fco2 $p |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) ->
|
|
( F o. G ) : A --> C ) $=
|
|
( cres wf wa ccom fco wceq crn wss frn cores syl adantl feq1d mpbid ) BCDBF
|
|
ZGZABEGZHZACTEIZGACDEIZGABCTEJUCACUDUEUBUDUEKZUAUBELBMUFABENDEBOPQRS $.
|
|
|
|
$( A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.)
|
|
(Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fssxp $p |- ( F : A --> B -> F C_ ( A X. B ) ) $=
|
|
( wf cdm crn cxp wrel wss frel relssdmrn syl wceq fdm eqimss xpss12 syl2anc
|
|
frn sstrd ) ABCDZCCEZCFZGZABGZTCHCUCIABCJCKLTUAAIZUBBIUCUDITUAAMUEABCNUAAOL
|
|
ABCRUAAUBBPQS $.
|
|
|
|
$( A function with bounded domain and range is a set. This version is proven
|
|
without the Axiom of Replacement. (Contributed by Mario Carneiro,
|
|
24-Jun-2015.) $)
|
|
fex2 $p |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V ) $=
|
|
( wf wcel w3a cxp cvv xpexg 3adant1 wss fssxp 3ad2ant1 ssexd ) ABCFZADGZBEG
|
|
ZHCABIZJRSTJGQABDEKLQRCTMSABCNOP $.
|
|
|
|
$( Two ways of specifying a partial function from ` A ` to ` B ` .
|
|
(Contributed by NM, 13-Nov-2007.) $)
|
|
funssxp $p |- ( ( Fun F /\ F C_ ( A X. B ) ) <->
|
|
( F : dom F --> B /\ dom F C_ A ) ) $=
|
|
( wfun cxp wss wa cdm wf wfn funfn biimpi rnss rnxpss syl6ss anim12i sylibr
|
|
crn df-f jca dmss dmxpss adantl ffun adantr fssxp xpss1 sylan9ss impbii ) C
|
|
DZCABEZFZGZCHZBCIZUNAFZGZUMUOUPUMCUNJZCRZBFZGUOUJURULUTUJURCKLULUSUKRBCUKMA
|
|
BNOPUNBCSQULUPUJULUNUKHACUKUAABUBOUCTUQUJULUOUJUPUNBCUDUEUOUPCUNBEUKUNBCUFU
|
|
NABUGUHTUI $.
|
|
|
|
$( A mapping is a partial function. (Contributed by NM, 25-Nov-2007.) $)
|
|
ffdm $p |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) ) $=
|
|
( wf cdm wss fdm feq2d ibir wceq eqimss syl jca ) ABCDZCEZBCDZOAFZNPNOABCAB
|
|
CGZHINOAJQROAKLM $.
|
|
|
|
$( The members of an ordered pair element of a mapping belong to the
|
|
mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.)
|
|
(Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
opelf $p |- ( ( F : A --> B /\ <. C , D >. e. F ) ->
|
|
( C e. A /\ D e. B ) ) $=
|
|
( wf cop wcel wa cxp fssxp sseld opelxp syl6ib imp ) ABEFZCDGZEHZCAHDBHIZPR
|
|
QABJZHSPETQABEKLCDABMNO $.
|
|
|
|
$( The union of two functions with disjoint domains. (Contributed by NM,
|
|
22-Sep-2004.) $)
|
|
fun $p |- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) ->
|
|
( F u. G ) : ( A u. B ) --> ( C u. D ) ) $=
|
|
( cin c0 wceq wf wa cun wfn crn wss fnun expcom wi rnun df-f unss12 anim12d
|
|
syl5eqss a1i anbi12i an4 bitri 3imtr4g impcom ) ABGHIZACEJZBDFJZKZABLZCDLZE
|
|
FLZJZUJEAMZFBMZKZENZCOZFNZDOZKZKZUPUNMZUPNZUOOZKUMUQUJUTVGVEVIUTUJVGABEFPQV
|
|
EVIRUJVEVHVAVCLUOEFSVACVCDUAUCUDUBUMURVBKZUSVDKZKVFUKVJULVKACETBDFTUEURVBUS
|
|
VDUFUGUNUOUPTUHUI $.
|
|
|
|
$( The union of two functions with disjoint domains. (Contributed by Mario
|
|
Carneiro, 12-Mar-2015.) $)
|
|
fun2 $p |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) ->
|
|
( F u. G ) : ( A u. B ) --> C ) $=
|
|
( wf wa cin c0 wceq cun fun wb unidm feq3 ax-mp sylib ) ACDFBCEFGABHIJGABKZ
|
|
CCKZDEKZFZRCTFZABCCDELSCJUAUBMCNSCRTOPQ $.
|
|
|
|
$( Composition of two functions. (Contributed by NM, 22-May-2006.) $)
|
|
fnfco $p |- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B ) $=
|
|
( wf wfn crn wss wa ccom df-f fnco 3expb sylan2b ) BADECAFZDBFZDGAHZICDJBFZ
|
|
BADKOPQRABCDLMN $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
NM, 23-Sep-2004.) $)
|
|
fssres $p |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B ) $=
|
|
( wf wss cres wfn crn df-f fnssres resss rnss ax-mp sstr mpan anim12i an32s
|
|
wa sylanb sylibr ) ABDEZCAFZSDCGZCHZUDIZBFZSZCBUDEUBDAHZDIZBFZSUCUHABDJUIUC
|
|
UKUHUIUCSUEUKUGACDKUFUJFZUKUGUDDFULDCLUDDMNUFUJBOPQRTCBUDJUA $.
|
|
|
|
$( Restriction of a restricted function with a subclass of its domain.
|
|
(Contributed by NM, 21-Jul-2005.) $)
|
|
fssres2 $p |- ( ( ( F |` A ) : A --> B /\ C C_ A ) ->
|
|
( F |` C ) : C --> B ) $=
|
|
( cres wf wss wa fssres wb resabs1 feq1d adantl mpbid ) ABDAEZFZCAGZHCBOCEZ
|
|
FZCBDCEZFZABCOIQSUAJPQCBRTDCAKLMN $.
|
|
|
|
$( An identity for the mapping relationship under restriction. (Contributed
|
|
by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro,
|
|
26-May-2016.) $)
|
|
fresin $p |- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B ) $=
|
|
( wf cin cres wss inss1 fssres mpan2 resres wfn ffn fnresdm reseq1d syl5eqr
|
|
wceq syl feq1d mpbid ) ABCEZADFZBCUCGZEZUCBCDGZEUBUCAHUEADIABUCCJKUBUCBUDUF
|
|
UBUDCAGZDGUFCADLUBUGCDUBCAMUGCRABCNACOSPQTUA $.
|
|
|
|
$( If two functions agree on their common domain, their union contains a
|
|
union of three functions with pairwise disjoint domains. If we assumed
|
|
the law of the excluded middle, this would be equality rather than subset.
|
|
(Contributed by Jim Kingdon, 28-Dec-2018.) $)
|
|
resasplitss $p |- ( ( F Fn A /\ G Fn B /\
|
|
( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
|
|
( ( F |` ( A i^i B ) ) u.
|
|
( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) )
|
|
C_ ( F u. G )
|
|
) $=
|
|
( wfn cin cres wceq w3a cdif cun uneq1i wss inundifss ssres2 ax-mp eqsstr3i
|
|
resundi syl5eqssr fnresdm unidm un4 simp3 uneq1d uneq2d incom eqsstri mp2an
|
|
unss12 syl6eqss uneq12 syl2an 3adant3 sseqtrd ) CAEZDBEZCABFZGZDUQGZHZIZURC
|
|
ABJZGZDBAJZGZKZKZCAGZDBGZKZCDKZVAVGURURKZVFKZVJVLURVFURUALVAVMURVCKZURVEKZK
|
|
ZVJURVCURVEUBVAVPVNUSVEKZKZVJVAVOVQVNVAURUSVEUOUPUTUCUDUEVNVHMVQVIMVRVJMVNC
|
|
UQVBKZGZVHCUQVBRVSAMVTVHMABNVSACOPQVQDUQVDKZGZVIDUQVDRWABMWBVIMWABAFZVDKBUQ
|
|
WCVDABUFLBANUGWABDOPQVNVHVQVIUIUHUJSSUOUPVJVKHZUTUOVHCHVIDHWDUPACTBDTVHCVID
|
|
UKULUMUN $.
|
|
|
|
$( Composition of a mapping and restricted identity. (Contributed by NM,
|
|
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fcoi1 $p |- ( F : A --> B -> ( F o. ( _I |` A ) ) = F ) $=
|
|
( wf wfn cid cres ccom wceq ffn wfun cdm df-fn wss eqimss ccnv cnvi reseq1i
|
|
wa syl cnveqi eqtr2i coeq2i cores2 syl5eq wrel funrel coi1 sylan9eqr sylbi
|
|
cnvresid ) ABCDCAEZCFAGZHZCIZABCJULCKZCLZAIZSUOCAMURUPUNCFHZCURUQANZUNUSIUQ
|
|
AOUTUNCFPZAGZPZHUSUMVCCVCUMPUMVBUMVAFAQRUAAUKUBUCCFAUDUETUPCUFUSCICUGCUHTUI
|
|
UJT $.
|
|
|
|
$( Composition of restricted identity and a mapping. (Contributed by NM,
|
|
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fcoi2 $p |- ( F : A --> B -> ( ( _I |` B ) o. F ) = F ) $=
|
|
( wf wfn crn wss wa cid cres ccom wceq df-f cores wrel fnrel coi2 sylan9eqr
|
|
syl sylbi ) ABCDCAEZCFBGZHIBJCKZCLABCMUBUAUCICKZCICBNUACOUDCLACPCQSRT $.
|
|
|
|
${
|
|
$d y F $. $d y A $. $d y B $. $d y C $.
|
|
$( There is exactly one value of a function in its codomain. (Contributed
|
|
by NM, 10-Dec-2003.) $)
|
|
feu $p |- ( ( F : A --> B /\ C e. A ) -> E! y e. B <. C , y >. e. F ) $=
|
|
( wf wcel wa cv cop weu wreu wfn ffn fneu2 sylan wb opelf simprd ex mpbid
|
|
pm4.71rd eubidv adantr df-reu sylibr ) BCEFZDBGZHZAIZCGZDUJJEGZHZAKZULACL
|
|
UIULAKZUNUGEBMUHUOBCENABDEOPUGUOUNQUHUGULUMAUGULUKUGULUKUGULHUHUKBCDUJERS
|
|
TUBUCUDUAULACUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y F $. $d x y A $. $d x y B $.
|
|
$( The converse of a restriction of a function. (Contributed by NM,
|
|
26-Mar-1998.) $)
|
|
fcnvres $p |- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) ) $=
|
|
( vy vx wf cres ccnv relcnv cv cop wcel wa ex pm4.71d vex opelcnv opelres
|
|
bitri syl6bbr relres opelf simpld simprd anbi1i bitr3d eqrelrdv ) ABCFZDE
|
|
CAGZHZCHZBGZUIIUKBUAUHEJZDJZKZCLZUNUMKZUJLZUQULLZUHUPUPUMALZMZURUHUPUTUHU
|
|
PUTUHUPMZUTUNBLZABUMUNCUBZUCNOURUOUILVAUNUMUIDPZEPZQUMUNCAVERSTUHUPUPVCMZ
|
|
USUHUPVCUHUPVCVBUTVCVDUDNOUSUQUKLZVCMVGUNUMUKBVFRVHUPVCUNUMCVEVFQUESTUFUG
|
|
$.
|
|
$}
|
|
|
|
$( The preimage of a class disjoint with a mapping's codomain is empty.
|
|
(Contributed by FL, 24-Jan-2007.) $)
|
|
fimacnvdisj $p |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) ->
|
|
( `' F " C ) = (/) ) $=
|
|
( wf cin c0 wceq wa ccnv cdm cima wss crn df-rn frn adantr syl5eqssr ssdisj
|
|
sylancom imadisj sylibr ) ABDEZBCFGHZIZDJZKZCFGHZUFCLGHUCUDUGBMUHUEUGDNZBDO
|
|
UCUIBMUDABDPQRUGBCSTUFCUAUB $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
fintm.1 $e |- E. x x e. B $.
|
|
$( Function into an intersection. (Contributed by Jim Kingdon,
|
|
28-Dec-2018.) $)
|
|
fintm $p |- ( F : A --> |^| B <-> A. x e. B F : A --> x ) $=
|
|
( wfn crn cint wss wa cv wral wf ssint anbi2i wcel wex wb r19.28mv df-f
|
|
ax-mp bitr4i ralbii 3bitr4i ) DBFZDGZCHZIZJZUEUFAKZIZJZACLZBUGDMBUJDMZACL
|
|
UIUEUKACLZJZUMUHUOUEAUFCNOUJCPAQUMUPREUEUKACSUAUBBUGDTUNULACBUJDTUCUD $.
|
|
$}
|
|
|
|
$( Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
|
|
shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fin $p |- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) ) $=
|
|
( wfn crn cin wss wa wf ssin anbi2i anandi bitr3i df-f anbi12i 3bitr4i ) DA
|
|
EZDFZBCGZHZIZRSBHZIZRSCHZIZIZATDJABDJZACDJZIUBRUCUEIZIUGUJUARSBCKLRUCUEMNAT
|
|
DOUHUDUIUFABDOACDOPQ $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
fabexg.1 $e |- F = { x | ( x : A --> B /\ ph ) } $.
|
|
$( Existence of a set of functions. (Contributed by Paul Chapman,
|
|
25-Feb-2008.) $)
|
|
fabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
|
|
( wcel wa cxp cvv cpw xpexg pwexg wss cv cab wf fssxp selpw sylibr anim1i
|
|
ss2abi eqsstri ssab2 sstri ssexg mpan 3syl ) CEIDFIJCDKZLIUKMZLIZGLIZCDEF
|
|
NUKLOGULPUMUNGBQZULIZAJZBRZULGCDUOSZAJZBRURHUTUQBUSUPAUSUOUKPUPCDUOTBUKUA
|
|
UBUCUDUEABULUFUGGULLUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
fabex.1 $e |- A e. _V $.
|
|
fabex.2 $e |- B e. _V $.
|
|
fabex.3 $e |- F = { x | ( x : A --> B /\ ph ) } $.
|
|
$( Existence of a set of functions. (Contributed by NM, 3-Dec-2007.) $)
|
|
fabex $p |- F e. _V $=
|
|
( cvv wcel fabexg mp2an ) CIJDIJEIJFGABCDIIEHKL $.
|
|
$}
|
|
|
|
$( If a mapping is a set, its domain is a set. (Contributed by NM,
|
|
27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
dmfex $p |- ( ( F e. C /\ F : A --> B ) -> A e. _V ) $=
|
|
( wf wcel cvv cdm wceq wi fdm dmexg eleq1 syl5ib syl impcom ) ABDEZDCFZAGFZ
|
|
QDHZAIZRSJABDKRTGFUASDCLTAGMNOP $.
|
|
|
|
$( The empty function. (Contributed by NM, 14-Aug-1999.) $)
|
|
f0 $p |- (/) : (/) --> A $=
|
|
( c0 wf wfn crn wss wceq eqid fn0 mpbir rn0 0ss eqsstri df-f mpbir2an ) BAB
|
|
CBBDZBEZAFPBBGBHBIJQBAKALMBABNO $.
|
|
|
|
$( A class is a function with empty codomain iff it and its domain are empty.
|
|
(Contributed by NM, 10-Dec-2003.) $)
|
|
f00 $p |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wf wceq wa wfn wfun cdm ffun crn wss frn ss0 syl dm0rn0 df-fn sylanbrc
|
|
sylibr fn0 sylib fdm eqtr3d jca f0 feq1 feq2 sylan9bb mpbiri impbii ) ACBDZ
|
|
BCEZACEZFZUKULUMUKBCGZULUKBHBIZCEZUOACBJUKBKZCEZUQUKURCLUSACBMURNOBPSZBCQRB
|
|
TUAUKUPACACBUBUTUCUDUNUKCCCDZCUEULUKACCDUMVAACBCUFACCCUGUHUIUJ $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
fconst.1 $e |- B e. _V $.
|
|
$( A cross product with a singleton is a constant function. (Contributed
|
|
by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
|
|
17-Sep-2011.) $)
|
|
fconst $p |- ( A X. { B } ) : A --> { B } $=
|
|
( vx csn cxp wf wfn crn wss fconstmpt fnmpti rnxpss df-f mpbir2an ) ABEZA
|
|
PFZGQAHQIPJDABQCDABKLAPMAPQNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A cross product with a singleton is a constant function. (Contributed
|
|
by NM, 19-Oct-2004.) $)
|
|
fconstg $p |- ( B e. V -> ( A X. { B } ) : A --> { B } ) $=
|
|
( vx cv csn cxp wf wceq sneq xpeq2d feq1 feq3 sylan9bb syl2anc vex fconst
|
|
wb vtoclg ) ADEZFZAUAGZHZABFZAUDGZHZDBCTBIZUBUEIZUAUDIZUCUFRUGUAUDATBJZKU
|
|
JUHUCAUAUEHUIUFAUAUBUELUAUDAUEMNOATDPQS $.
|
|
$}
|
|
|
|
$( A cross product with a singleton is a constant function. (Contributed by
|
|
NM, 24-Jul-2014.) $)
|
|
fnconstg $p |- ( B e. V -> ( A X. { B } ) Fn A ) $=
|
|
( wcel csn cxp wf wfn fconstg ffn syl ) BCDABEZALFZGMAHABCIALMJK $.
|
|
|
|
$( Constant function with loose range. (Contributed by Stefan O'Rear,
|
|
1-Feb-2015.) $)
|
|
fconst6g $p |- ( B e. C -> ( A X. { B } ) : A --> C ) $=
|
|
( wcel csn cxp wf wss fconstg snssi fss syl2anc ) BCDABEZAMFZGMCHACNGABCIBC
|
|
JAMCNKL $.
|
|
|
|
${
|
|
fconst6.1 $e |- B e. C $.
|
|
$( A constant function as a mapping. (Contributed by Jeff Madsen,
|
|
30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.) $)
|
|
fconst6 $p |- ( A X. { B } ) : A --> C $=
|
|
( wcel csn cxp wf fconst6g ax-mp ) BCEACABFGHDABCIJ $.
|
|
$}
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1eq1 $p |- ( F = G -> ( F : A -1-1-> B <-> G : A -1-1-> B ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq1 cnveq funeqd anbi12d df-f1 3bitr4g ) CDEZAB
|
|
CFZCGZHZIABDFZDGZHZIABCJABDJQRUATUCABCDKQSUBCDLMNABCOABDOP $.
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1eq2 $p |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq2 anbi1d df-f1 3bitr4g ) ABEZACDFZDGHZIBCDFZQ
|
|
IACDJBCDJOPRQABCDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1eq3 $p |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq3 anbi1d df-f1 3bitr4g ) ABEZCADFZDGHZICBDFZQ
|
|
ICADJCBDJOPRQABCDKLCADMCBDMN $.
|
|
|
|
${
|
|
nff1.1 $e |- F/_ x F $.
|
|
nff1.2 $e |- F/_ x A $.
|
|
nff1.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a one-to-one function.
|
|
(Contributed by NM, 16-May-2004.) $)
|
|
nff1 $p |- F/ x F : A -1-1-> B $=
|
|
( wf1 wf ccnv wfun wa df-f1 nff nfcnv nffun nfan nfxfr ) BCDHBCDIZDJZKZLA
|
|
BCDMSUAAABCDEFGNATADEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y F $.
|
|
$( Alternate definition of a one-to-one function. (Contributed by NM,
|
|
31-Dec-1996.) $)
|
|
dff12 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) ) $=
|
|
( wf1 wf ccnv wfun wa cv wbr wmo wal df-f1 funcnv2 anbi2i bitri ) CDEFCDE
|
|
GZEHIZJSAKBKELAMBNZJCDEOTUASABEPQR $.
|
|
$}
|
|
|
|
$( A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.) $)
|
|
f1f $p |- ( F : A -1-1-> B -> F : A --> B ) $=
|
|
( wf1 wf ccnv wfun df-f1 simplbi ) ABCDABCECFGABCHI $.
|
|
|
|
$( A one-to-one mapping is a function on its domain. (Contributed by NM,
|
|
8-Mar-2014.) $)
|
|
f1fn $p |- ( F : A -1-1-> B -> F Fn A ) $=
|
|
( wf1 wf wfn f1f ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) $)
|
|
f1fun $p |- ( F : A -1-1-> B -> Fun F ) $=
|
|
( wf1 wfn wfun f1fn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( A one-to-one onto mapping is a relation. (Contributed by NM,
|
|
8-Mar-2014.) $)
|
|
f1rel $p |- ( F : A -1-1-> B -> Rel F ) $=
|
|
( wf1 wfn wrel f1fn fnrel syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) $)
|
|
f1dm $p |- ( F : A -1-1-> B -> dom F = A ) $=
|
|
( wf1 wfn cdm wceq f1fn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
$( A function that is one-to-one is also one-to-one on some superset of its
|
|
range. (Contributed by Mario Carneiro, 12-Jan-2013.) $)
|
|
f1ss $p |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C ) $=
|
|
( wf1 wss wa wf ccnv wfun f1f fss sylan df-f1 simprbi adantr sylanbrc ) ABD
|
|
EZBCFZGACDHZDIJZACDERABDHZSTABDKABCDLMRUASRUBUAABDNOPACDNQ $.
|
|
|
|
$( Combine a one-to-one function with a restriction on the domain.
|
|
(Contributed by Stefan O'Rear, 20-Feb-2015.) $)
|
|
f1ssr $p |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C ) $=
|
|
( wf1 crn wss wa wf ccnv wfun f1fn adantr simpr df-f sylanbrc df-f1 simprbi
|
|
wfn ) ABDEZDFCGZHZACDIZDJKZACDEUBDASZUAUCTUEUAABDLMTUANACDOPTUDUATABDIUDABD
|
|
QRMACDQP $.
|
|
|
|
$( A function that is one-to-one is also one-to-one on some aubset of its
|
|
domain. (Contributed by Mario Carneiro, 17-Jan-2015.) $)
|
|
f1ssres $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B ) $=
|
|
( wf1 wss wa cres wf ccnv wfun f1f fssres sylan simprbi funres11 syl adantr
|
|
df-f1 sylanbrc ) ABDEZCAFZGCBDCHZIZUCJKZCBUCEUAABDIZUBUDABDLABCDMNUAUEUBUAD
|
|
JKZUEUAUFUGABDSOCDPQRCBUCST $.
|
|
|
|
$( Two ways to express that a set ` A ` (not necessarily a function) is
|
|
one-to-one. Each side is equivalent to Definition 6.4(3) of
|
|
[TakeutiZaring] p. 24, who use the notation "Un_2 (A)" for one-to-one. We
|
|
do not introduce a separate notation since we rarely use it. (Contributed
|
|
by NM, 13-Aug-2004.) $)
|
|
f1cnvcnv $p |- ( `' `' A : dom A -1-1-> _V
|
|
<-> ( Fun `' A /\ Fun `' `' A ) ) $=
|
|
( cdm cvv ccnv wf1 wf wfun wa df-f1 wfn dffn2 wceq dmcnvcnv mpbiran2 bitr3i
|
|
df-fn wrel relcnv dfrel2 mpbi funeqi anbi12ci bitri ) ABZCADZDZEUDCUFFZUFDZ
|
|
GZHUEGZUFGZHUDCUFIUGUKUIUJUGUFUDJZUKUDUFKULUKUFBUDLAMUFUDPNOUHUEUEQUHUELARU
|
|
ESTUAUBUC $.
|
|
|
|
$( Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
|
|
p. 25. (Contributed by NM, 28-May-1998.) $)
|
|
f1co $p |- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) ->
|
|
( F o. G ) : A -1-1-> C ) $=
|
|
( wf1 wa ccom wf ccnv wfun df-f1 fco funco cnvco funeqi sylibr anim12i an4s
|
|
ancoms syl2anb ) BCDFZABEFZGACDEHZIZUDJZKZGZACUDFUBBCDIZDJZKZGABEIZEJZKZGUH
|
|
UCBCDLABELUIULUKUNUHUIULGUEUKUNGUGABCDEMUNUKUGUNUKGUMUJHZKUGUMUJNUFUODEOPQT
|
|
RSUAACUDLQ $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
foeq1 $p |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) ) $=
|
|
( wceq wfn crn wa wfo fneq1 rneq eqeq1d anbi12d df-fo 3bitr4g ) CDEZCAFZCGZ
|
|
BEZHDAFZDGZBEZHABCIABDIPQTSUBACDJPRUABCDKLMABCNABDNO $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
foeq2 $p |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) ) $=
|
|
( wceq wfn crn wa wfo fneq2 anbi1d df-fo 3bitr4g ) ABEZDAFZDGCEZHDBFZPHACDI
|
|
BCDINOQPABDJKACDLBCDLM $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) $)
|
|
foeq3 $p |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) ) $=
|
|
( wceq wfn crn wa wfo eqeq2 anbi2d df-fo 3bitr4g ) ABEZDCFZDGZAEZHOPBEZHCAD
|
|
ICBDINQROABPJKCADLCBDLM $.
|
|
|
|
${
|
|
nffo.1 $e |- F/_ x F $.
|
|
nffo.2 $e |- F/_ x A $.
|
|
nffo.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for an onto function. (Contributed by
|
|
NM, 16-May-2004.) $)
|
|
nffo $p |- F/ x F : A -onto-> B $=
|
|
( wfo wfn crn wceq wa df-fo nffn nfrn nfeq nfan nfxfr ) BCDHDBIZDJZCKZLAB
|
|
CDMSUAAABDEFNATCADEOGPQR $.
|
|
$}
|
|
|
|
$( An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) $)
|
|
fof $p |- ( F : A -onto-> B -> F : A --> B ) $=
|
|
( wfn crn wceq wa wss wfo wf eqimss anim2i df-fo df-f 3imtr4i ) CADZCEZBFZG
|
|
PQBHZGABCIABCJRSPQBKLABCMABCNO $.
|
|
|
|
$( An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) $)
|
|
fofun $p |- ( F : A -onto-> B -> Fun F ) $=
|
|
( wfo wf wfun fof ffun syl ) ABCDABCECFABCGABCHI $.
|
|
|
|
$( An onto mapping is a function on its domain. (Contributed by NM,
|
|
16-Dec-2008.) $)
|
|
fofn $p |- ( F : A -onto-> B -> F Fn A ) $=
|
|
( wfo wf wfn fof ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( The codomain of an onto function is its range. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
forn $p |- ( F : A -onto-> B -> ran F = B ) $=
|
|
( wfo wfn crn wceq df-fo simprbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( Alternate definition of an onto function. (Contributed by NM,
|
|
22-Mar-2006.) $)
|
|
dffo2 $p |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) ) $=
|
|
( wfo wf crn wceq wa fof forn jca wfn ffn df-fo biimpri sylan impbii ) ABCD
|
|
ZABCEZCFBGZHRSTABCIABCJKSCALZTRABCMRUATHABCNOPQ $.
|
|
|
|
$( The image of the domain of an onto function. (Contributed by NM,
|
|
29-Nov-2002.) $)
|
|
foima $p |- ( F : A -onto-> B -> ( F " A ) = B ) $=
|
|
( wfo cdm cima crn imadmrn wf wceq fof fdm syl imaeq2d forn 3eqtr3a ) ABCDZ
|
|
CCEZFCGCAFBCHQRACQABCIRAJABCKABCLMNABCOP $.
|
|
|
|
$( A function maps onto its range. (Contributed by NM, 10-May-1998.) $)
|
|
dffn4 $p |- ( F Fn A <-> F : A -onto-> ran F ) $=
|
|
( wfn crn wceq wa wfo eqid biantru df-fo bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJ
|
|
K $.
|
|
|
|
$( A function maps its domain onto its range. (Contributed by NM,
|
|
23-Jul-2004.) $)
|
|
funforn $p |- ( Fun A <-> A : dom A -onto-> ran A ) $=
|
|
( wfun cdm wfn crn wfo funfn dffn4 bitri ) ABAACZDJAEAFAGJAHI $.
|
|
|
|
$( An onto function has unique domain and range. (Contributed by NM,
|
|
5-Nov-2006.) $)
|
|
fodmrnu $p |- ( ( F : A -onto-> B /\ F : C -onto-> D ) ->
|
|
( A = C /\ B = D ) ) $=
|
|
( wfo wa wceq wfn fofn fndmu syl2an crn forn sylan9req jca ) ABEFZCDEFZGACH
|
|
ZBDHQEAIECISRABEJCDEJACEKLQRBEMDABENCDENOP $.
|
|
|
|
$( Restriction of a function. (Contributed by NM, 4-Mar-1997.) $)
|
|
fores $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F |` A ) : A -onto-> ( F " A ) ) $=
|
|
( wfun cdm wss cres cima wfo funres anim1i wfn wceq df-fn crn df-ima eqcomi
|
|
wa df-fo mpbiran2 ssdmres anbi2i 3bitr4i sylibr ) BCZABDEZQBAFZCZUEQZABAGZU
|
|
FHZUDUGUEABIJUFAKZUGUFDALZQUJUHUFAMUJUKUFNZUILUIUMBAOPAUIUFRSUEULUGABTUAUBU
|
|
C $.
|
|
|
|
$( Composition of onto functions. (Contributed by NM, 22-Mar-2006.) $)
|
|
foco $p |- ( ( F : B -onto-> C /\ G : A -onto-> B ) ->
|
|
( F o. G ) : A -onto-> C ) $=
|
|
( wfo wa ccom crn wceq dffo2 fco ad2ant2r cdm fdm eqtr3 sylan rncoeq eqeq1d
|
|
wf biimpar an32s adantrl jca syl2anb sylibr ) BCDFZABEFZGACDEHZTZUIIZCJZGZA
|
|
CUIFUGBCDTZDIZCJZGZABETZEIZBJZGZUMUHBCDKABEKUQVAGUJULUNURUJUPUTABCDELMUQUTU
|
|
LURUNUTUPULUNUTGDNZUSJZUPULUNVBBJUTVCBCDOVBUSBPQVCULUPVCUKUOCDERSUAQUBUCUDU
|
|
EACUIKUF $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1oeq1 $p |- ( F = G -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq1 foeq1 anbi12d df-f1o 3bitr4g ) CDEZABCFZABCGZH
|
|
ABDFZABDGZHABCIABDIOPRQSABCDJABCDKLABCMABDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1oeq2 $p |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq2 foeq2 anbi12d df-f1o 3bitr4g ) ABEZACDFZACDGZH
|
|
BCDFZBCDGZHACDIBCDIOPRQSABCDJABCDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
f1oeq3 $p |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq3 foeq3 anbi12d df-f1o 3bitr4g ) ABEZCADFZCADGZH
|
|
CBDFZCBDGZHCADICBDIOPRQSABCDJABCDKLCADMCBDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by FL,
|
|
14-Jul-2012.) $)
|
|
f1oeq23 $p |- ( ( A = B /\ C = D ) ->
|
|
( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) ) $=
|
|
( wceq wf1o f1oeq2 f1oeq3 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
f1eq123d.1 $e |- ( ph -> F = G ) $.
|
|
f1eq123d.2 $e |- ( ph -> A = B ) $.
|
|
f1eq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for one-to-one functions. (Contributed by Mario
|
|
Carneiro, 27-Jan-2017.) $)
|
|
f1eq123d $p |- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) ) $=
|
|
( wf1 wceq wb f1eq1 syl f1eq2 f1eq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFGLSTM
|
|
HBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.
|
|
|
|
$( Equality deduction for onto functions. (Contributed by Mario Carneiro,
|
|
27-Jan-2017.) $)
|
|
foeq123d $p |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) ) $=
|
|
( wfo wceq wb foeq1 syl foeq2 foeq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFGLSTM
|
|
HBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.
|
|
|
|
$( Equality deduction for one-to-one onto functions. (Contributed by Mario
|
|
Carneiro, 27-Jan-2017.) $)
|
|
f1oeq123d $p |- ( ph ->
|
|
( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) ) $=
|
|
( wf1o wceq wb f1oeq1 syl f1oeq2 f1oeq3 3bitrd ) ABDFKZBDGKZCDGKZCEGKZAFG
|
|
LSTMHBDFGNOABCLTUAMIBCDGPOADELUAUBMJDECGQOR $.
|
|
$}
|
|
|
|
${
|
|
nff1o.1 $e |- F/_ x F $.
|
|
nff1o.2 $e |- F/_ x A $.
|
|
nff1o.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a one-to-one onto function.
|
|
(Contributed by NM, 16-May-2004.) $)
|
|
nff1o $p |- F/ x F : A -1-1-onto-> B $=
|
|
( wf1o wf1 wfo wa df-f1o nff1 nffo nfan nfxfr ) BCDHBCDIZBCDJZKABCDLQRAAB
|
|
CDEFGMABCDEFGNOP $.
|
|
$}
|
|
|
|
$( A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
|
|
12-Dec-2003.) $)
|
|
f1of1 $p |- ( F : A -1-1-onto-> B -> F : A -1-1-> B ) $=
|
|
( wf1o wf1 wfo df-f1o simplbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( A one-to-one onto mapping is a mapping. (Contributed by NM,
|
|
12-Dec-2003.) $)
|
|
f1of $p |- ( F : A -1-1-onto-> B -> F : A --> B ) $=
|
|
( wf1o wf1 wf f1of1 f1f syl ) ABCDABCEABCFABCGABCHI $.
|
|
|
|
$( A one-to-one onto mapping is function on its domain. (Contributed by NM,
|
|
12-Dec-2003.) $)
|
|
f1ofn $p |- ( F : A -1-1-onto-> B -> F Fn A ) $=
|
|
( wf1o wf wfn f1of ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( A one-to-one onto mapping is a function. (Contributed by NM,
|
|
12-Dec-2003.) $)
|
|
f1ofun $p |- ( F : A -1-1-onto-> B -> Fun F ) $=
|
|
( wf1o wfn wfun f1ofn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( A one-to-one onto mapping is a relation. (Contributed by NM,
|
|
13-Dec-2003.) $)
|
|
f1orel $p |- ( F : A -1-1-onto-> B -> Rel F ) $=
|
|
( wf1o wfun wrel f1ofun funrel syl ) ABCDCECFABCGCHI $.
|
|
|
|
$( The domain of a one-to-one onto mapping. (Contributed by NM,
|
|
8-Mar-2014.) $)
|
|
f1odm $p |- ( F : A -1-1-onto-> B -> dom F = A ) $=
|
|
( wf1o wfn cdm wceq f1ofn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (Contributed by NM,
|
|
10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
dff1o2 $p |- ( F : A -1-1-onto-> B
|
|
<-> ( F Fn A /\ Fun `' F /\ ran F = B ) ) $=
|
|
( wf1o wf1 wfo wa wfn ccnv wfun crn w3a df-f1o wf df-f1 df-fo anbi12i anass
|
|
wceq bitri 3anan12 anbi1i eqimss df-f biimpri sylan2 3adant2 ancom 3bitr4ri
|
|
wss pm4.71i ) ABCDABCEZABCFZGZCAHZCIJZCKZBSZLZABCMUNABCNZUPGZUOURGZGZUSULVA
|
|
UMVBABCOABCPQVCUTUPVBGZGZUSUTUPVBRUSUTGVDUTGUSVEUSVDUTUOUPURUAUBUSUTUOURUTU
|
|
PURUOUQBUJZUTUQBUCUTUOVFGABCUDUEUFUGUKUTVDUHUITTT $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (Contributed by NM,
|
|
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
dff1o3 $p |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) ) $=
|
|
( wfn ccnv wfun crn wceq w3a wf1o wfo 3anan32 dff1o2 df-fo anbi1i 3bitr4i
|
|
wa ) CADZCEFZCGBHZIRTQZSQABCJABCKZSQRSTLABCMUBUASABCNOP $.
|
|
|
|
$( A one-to-one onto function is an onto function. (Contributed by NM,
|
|
28-Apr-2004.) $)
|
|
f1ofo $p |- ( F : A -1-1-onto-> B -> F : A -onto-> B ) $=
|
|
( wf1o wfo ccnv wfun dff1o3 simplbi ) ABCDABCECFGABCHI $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (Contributed by NM,
|
|
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
dff1o4 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) ) $=
|
|
( wf1o wfn ccnv wfun crn w3a wa dff1o2 3anass cdm df-rn eqeq1i anbi2i df-fn
|
|
wceq bitr4i 3bitri ) ABCDCAEZCFZGZCHZBRZIUAUCUEJZJUAUBBEZJABCKUAUCUELUFUGUA
|
|
UFUCUBMZBRZJUGUEUIUCUDUHBCNOPUBBQSPT $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (Contributed by NM,
|
|
10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
dff1o5 $p |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) ) $=
|
|
( wf1o wf1 wfo wa crn wceq df-f1o wf biantrurd dffo2 syl6rbbr pm5.32i bitri
|
|
f1f ) ABCDABCEZABCFZGRCHBIZGABCJRSTRTABCKZTGSRUATABCQLABCMNOP $.
|
|
|
|
$( A one-to-one function maps onto its range. (Contributed by NM,
|
|
13-Aug-2004.) $)
|
|
f1orn $p |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) ) $=
|
|
( crn wf1o wfn ccnv wfun wceq w3a wa dff1o2 eqid df-3an mpbiran2 bitri ) AB
|
|
CZBDBAEZBFGZPPHZIZQRJZAPBKTUASPLQRSMNO $.
|
|
|
|
$( A one-to-one function maps one-to-one onto its range. (Contributed by NM,
|
|
4-Sep-2004.) $)
|
|
f1f1orn $p |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F ) $=
|
|
( wf1 wfn ccnv wfun crn wf1o f1fn wf df-f1 simprbi f1orn sylanbrc ) ABCDZCA
|
|
ECFGZACHCIABCJPABCKQABCLMACNO $.
|
|
|
|
${
|
|
$d A f $. $d B f $.
|
|
f1oabexg.1 $e |- F = { f | ( f : A -1-1-onto-> B /\ ph ) } $.
|
|
$( The class of all 1-1-onto functions mapping one set to another is a set.
|
|
(Contributed by Paul Chapman, 25-Feb-2008.) $)
|
|
f1oabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
|
|
( wcel wa cv wf1o cab cvv wf wss f1of anim1i ss2abi eqid sylancr syl5eqel
|
|
fabexg ssexg ) BDICEIJZGBCFKZLZAJZFMZNHUEUIBCUFOZAJZFMZPULNIUINIUHUKFUGUJ
|
|
ABCUFQRSAFBCDEULULTUCUIULNUDUAUB $.
|
|
$}
|
|
|
|
$( The converse of a one-to-one onto function is also one-to-one onto.
|
|
(Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
f1ocnv $p |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A ) $=
|
|
( wfn ccnv wa wf1o wrel fnrel wceq dfrel2 fneq1 biimprd sylbi anim2i ancoms
|
|
wi mpcom dff1o4 3imtr4i ) CADZCEZBDZFUCUBEZADZFZABCGBAUBGUCUAUFUAUEUCCHZUAU
|
|
EACIUGUDCJZUAUEQCKUHUEUAAUDCLMNROPABCSBAUBST $.
|
|
|
|
$( A relation is a one-to-one onto function iff its converse is a one-to-one
|
|
onto function with domain and range interchanged. (Contributed by NM,
|
|
8-Dec-2003.) $)
|
|
f1ocnvb $p |- ( Rel F ->
|
|
( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) ) $=
|
|
( wrel wf1o ccnv f1ocnv wceq wb dfrel2 f1oeq1 sylbi syl5ib impbid2 ) CDZABC
|
|
EZBACFZEZABCGRABQFZEZOPBAQGOSCHTPICJABSCKLMN $.
|
|
|
|
$( The restriction of a one-to-one function maps one-to-one onto the image.
|
|
(Contributed by NM, 25-Mar-1998.) $)
|
|
f1ores $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto->
|
|
( F " C ) ) $=
|
|
( wf1 wss wa cres crn wf1o cima f1ssres f1f1orn wceq wb df-ima f1oeq3 ax-mp
|
|
syl sylibr ) ABDECAFGZCDCHZIZUBJZCDCKZUBJZUACBUBEUDABCDLCBUBMSUEUCNUFUDODCP
|
|
UEUCCUBQRT $.
|
|
|
|
$( The converse of a one-to-one-onto restricted function. (Contributed by
|
|
Paul Chapman, 21-Apr-2008.) $)
|
|
f1orescnv $p |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) ->
|
|
( `' F |` P ) : P -1-1-onto-> R ) $=
|
|
( ccnv wfun cres wf1o wa f1ocnv adantl wceq wb cima funcnvres df-ima dff1o5
|
|
crn wf1 simprbi syl5eq reseq2d sylan9eq f1oeq1 syl mpbid ) CDZEZBACBFZGZHZA
|
|
BUHDZGZABUFAFZGZUIULUGBAUHIJUJUKUMKULUNLUGUIUKUFCBMZFUMBCNUIUOAUFUIUOUHQZAC
|
|
BOUIBAUHRUPAKBAUHPSTUAUBABUKUMUCUDUE $.
|
|
|
|
$( Preimage of an image. (Contributed by NM, 30-Sep-2004.) $)
|
|
f1imacnv $p |- ( ( F : A -1-1-> B /\ C C_ A )
|
|
-> ( `' F " ( F " C ) ) = C ) $=
|
|
( wf1 wss wa ccnv cima cres resima wfun wceq df-f1 simprbi adantr funcnvres
|
|
wf syl wf1o imaeq1d f1ores f1ocnv cdm crn imadmrn f1odm imaeq2d wfo 3eqtr3a
|
|
f1ofo forn eqtr3d syl5eqr ) ABDEZCAFZGZDHZDCIZIURUSJZUSIZCURUSKUQDCJZHZUSIZ
|
|
VACUQVCUTUSUQURLZVCUTMUOVEUPUOABDRVEABDNOPCDQSUAUQUSCVCTZVDCMUQCUSVBTVFABCD
|
|
UBCUSVBUCSVFVCVCUDZIVCUEZVDCVCUFVFVGUSVCUSCVCUGUHVFUSCVCUIVHCMUSCVCUKUSCVCU
|
|
LSUJSUMUN $.
|
|
|
|
$( A reverse version of ~ f1imacnv . (Contributed by Jeff Hankins,
|
|
16-Jul-2009.) $)
|
|
foimacnv $p |- ( ( F : A -onto-> B /\ C C_ B )
|
|
-> ( F " ( `' F " C ) ) = C ) $=
|
|
( wfo wss wa ccnv cima cres resima wfun wceq fofun adantr syl crn cdm df-rn
|
|
syl5eqr funcnvres2 imaeq1d wfn resss cnvss ax-mp cnvcnvss sstri funss mpsyl
|
|
df-ima eqtr2i jctir df-fn sylibr dfdm4 forn sseq2d biimpar syl6sseq ssdmres
|
|
sylib df-fo sylanbrc foima eqtr3d ) ABDEZCBFZGZDDHZCIZIDVKJZVKIZCDVKKVIVJCJ
|
|
ZHZVKIZVMCVIVOVLVKVIDLZVOVLMVGVQVHABDNZOCDUAPUBVIVKCVOEZVPCMVIVOVKUCZVOQZCM
|
|
VSVIVOLZVORZVKMZGVTVIWBWDVGWBVHVODFVGVQWBVOVJHZDVNVJFVOWEFVJCUDVNVJUEUFDUGU
|
|
HVRVODUIUJOVKVNQWCVJCUKVNSULUMVOVKUNUOVIWAVNRZCVNUPVICVJRZFWFCMVICDQZWGVGCW
|
|
HFVHVGWHBCABDUQURUSDSUTCVJVAVBTVKCVOVCVDVKCVOVEPVFT $.
|
|
|
|
$( The union of two onto functions with disjoint domains is an onto function.
|
|
(Contributed by Mario Carneiro, 22-Jun-2016.) $)
|
|
foun $p |- ( ( ( F : A -onto-> B /\ G : C -onto-> D )
|
|
/\ ( A i^i C ) = (/) )
|
|
-> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) ) $=
|
|
( wfo wa cin c0 wceq cun wfn crn fofn anim12i fnun sylan rnun forn ad2antrr
|
|
ad2antlr uneq12d syl5eq df-fo sylanbrc ) ABEGZCDFGZHZACIJKZHZEFLZACLZMZULNZ
|
|
BDLZKUMUPULGUIEAMZFCMZHUJUNUGUQUHURABEOCDFOPACEFQRUKUOENZFNZLUPEFSUKUSBUTDU
|
|
GUSBKUHUJABETUAUHUTDKUGUJCDFTUBUCUDUMUPULUEUF $.
|
|
|
|
$( The union of two one-to-one onto functions with disjoint domains and
|
|
ranges. (Contributed by NM, 26-Mar-1998.) $)
|
|
f1oun $p |- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D )
|
|
/\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) )
|
|
-> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) ) $=
|
|
( wf1o wa cin c0 wceq cun wfn ccnv wi dff1o4 fnun ex cnvun fneq1i im2anan9
|
|
sylibr an4s syl2anb syl6ibr imp ) ABEGZCDFGZHZACIJKZBDIJKZHZACLZBDLZEFLZGZU
|
|
IULUOUMMZUONZUNMZHZUPUGEAMZENZBMZHFCMZFNZDMZHULUTOZUHABEPCDFPVAVDVCVFVGVAVD
|
|
HZUJUQVCVFHZUKUSVHUJUQACEFQRVIUKUSVIUKHVBVELZUNMUSBDVBVEQUNURVJEFSTUBRUAUCU
|
|
DUMUNUOPUEUF $.
|
|
|
|
${
|
|
$d A u v x z $. $d A u v y $. $d B u v y $. $d B u v z $. $d C u v x $.
|
|
$d D u v $. $d S u v x $.
|
|
fun11iun.1 $e |- ( x = y -> B = C ) $.
|
|
fun11iun.2 $e |- B e. _V $.
|
|
$( The union of a chain (with respect to inclusion) of one-to-one functions
|
|
is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.)
|
|
(Revised by Mario Carneiro, 24-Jun-2015.) $)
|
|
fun11iun $p |- ( A. x e. A ( B : D -1-1-> S /\
|
|
A. y e. A ( B C_ C \/ C C_ B ) ) ->
|
|
U_ x e. A B : U_ x e. A D -1-1-> S ) $=
|
|
( vz vu vv wss wral wa wfun wceq cv wrex wcel wf1 wo ciun wf ccnv wfn crn
|
|
cdm cab cuni vex eqeq1 rexbidv elab r19.29 nfv nfre1 nfab nfralxy nfan wi
|
|
f1eq1 biimparc df-f1 ffun anim1i syl adantlr eqeq2d cbvrexv sseq12 ancoms
|
|
sylbi orbi12d biimprcd expdimp rexlimivw imp sylan an32s adantlll sylan2b
|
|
wb ralrimiva jca a1i rexlimi fun11uni simpld dfiun2 funeqi sylibr cop wex
|
|
nfra1 eldm2 f1dm eleq2d syl5bbr adantr syl6 rexbida eliun rexcom4 3bitr4i
|
|
rsp exbii 3bitr4g eqrdv df-fn sylanbrc rniun f1f frn ralimi syl5eqss df-f
|
|
iunss simprd cnveqi ) FGDUAZDEMZEDMZUBZBCNZOZACNZACFUCZGACDUCZUDZYIUEZPZY
|
|
HGYIUAYGYIYHUFZYIUGZGMYJYGYIPZYIUHZYHQYMYGJRZDQZACSZJUIZUJZPZYOYGUUBUUAUE
|
|
ZPZYGKRZPZUUEUEPZOZUUELRZMZUUIUUEMZUBZLYTNZOZKYTNUUBUUDOYGUUNKYTUUEYTTYGU
|
|
UEDQZACSZUUNYSUUPJUUEKUKZYQUUEQYRUUOACYQUUEDULUMUNYGUUPOYFUUOOZACSUUNYFUU
|
|
OACUOUURUUNACUUHUUMAUUHAUPUULALYTYSAJYRACUQURUULAUPUSUTUURUUNVAARZCTZUURU
|
|
UHUUMYAUUOUUHYEYAUUOOFGUUEUAZUUHUUOUVAYAFGUUEDVBVCUVAFGUUEUDZUUGOUUHFGUUE
|
|
VDUVBUUFUUGFGUUEVEVFVMVGVHUURUULLYTUUIYTTUURUUIDQZACSZUULYSUVDJUUILUKYQUU
|
|
IQYRUVCACYQUUIDULUMUNUVDUURUUIEQZBCSZUULUVCUVEABCUUSBRQDEUUIHVIVJYEUUOUVF
|
|
UULYAYEUVFUUOUULYEUVFOYDUVEOZBCSZUUOUULYDUVEBCUOUVHUUOUULUVGUUOUULVABCYDU
|
|
VEUUOUULUVEUUOOZUULYDUVIUUJYBUUKYCUUOUVEUUJYBWCUUEDUUIEVKVLUUIEUUEDVKVNVO
|
|
VPVQVRVSVTWAWBWBWDWEWFWGVGWBWDYTKLWHVGZWIYIUUAAJCDIWJZWKWLYGKYPYHYGUUEUUI
|
|
WMZDTZLWNZACSZUUEFTZACSUUEYPTZUUEYHTYGUVNUVPACYFACWOYGUUTUVNUVPWCZYGUUTYF
|
|
UVRYFACXFYAUVRYEUVNUUEDUHZTYAUVPLUUEDUUQWPYAUVSFUUEFGDWQWRWSWTXAVRXBUVLYI
|
|
TZLWNUVMACSZLWNUVQUVOUVTUWALAUVLCDXCXGLUUEYIUUQWPUVMALCXDXEAUUECFXCXHXIYI
|
|
YHXJXKYGYNACDUGZUCZGACDXLYGUWBGMZACNUWCGMYFUWDACYAUWDYEYAFGDUDUWDFGDXMFGD
|
|
XNVGWTXOACUWBGXRWLXPYHGYIXQXKYGUUDYLYGUUBUUDUVJXSYKUUCYIUUAUVKXTWKWLYHGYI
|
|
VDXK $.
|
|
$}
|
|
|
|
$( The restriction of a one-to-one onto function to a difference maps onto
|
|
the difference of the images. (Contributed by Paul Chapman,
|
|
11-Apr-2009.) $)
|
|
resdif $p |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\
|
|
( F |` B ) : B -onto-> D ) ->
|
|
( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
|
|
( ccnv wfun cres wfo cdif cima wf1o wceq syl wb ax-mp crn df-ima wa forn wf
|
|
w3a cdm wss fofun difss fof fdm syl5sseqr fores syl2anc indif reseq2i eqtri
|
|
cin resres foeq1 rneqi 3eqtr4i foeq3 bitri funres11 biimpri syl2anr 3adant3
|
|
sylib dff1o3 syl5eq anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3
|
|
mpbid ) EFGZACEAHZIZBDEBHZIZUBZABJZEWCKZEWCHZLZWCCDJZWELZVQVSWFWAVSWCWDWEIZ
|
|
WEFGZWFVQVSWCVRWCKZVRWCHZIZWIVSVRGWCVRUCZUDWMACVRUEVSAWCWNABUFVSACVRUAWNAMA
|
|
CVRUGACVRUHNUIWCVRUJUKWMWCWKWEIZWIWLWEMWMWOOWLEAWCUOZHWEEAWCUPWPWCEABULUMUN
|
|
ZWCWKWLWEUQPWKWDMWOWIOWLQWEQWKWDWLWEWQURVRWCREWCRUSWKWDWCWEUTPVAVFWCEVBWFWI
|
|
WJSWCWDWEVGVCVDVEWBWDWGMZWFWHOVQVSWAWRVSWASVQEAKZCMZEBKZDMZSZWRVSWTWAXBVSWS
|
|
VRQCEARACVRTVHWAXAVTQDEBRBDVTTVHVIVQXCWDWSXAJWGABEVJWSCXADVKVLVMVNWDWGWCWEV
|
|
ONVP $.
|
|
|
|
$( Composition of one-to-one onto functions. (Contributed by NM,
|
|
19-Mar-1998.) $)
|
|
f1oco $p |- ( ( F : B -1-1-onto-> C /\ G : A -1-1-onto-> B ) ->
|
|
( F o. G ) : A -1-1-onto-> C ) $=
|
|
( wf1o wa ccom wf1 wfo df-f1o f1co foco anim12i an4s syl2anb sylibr ) BCDFZ
|
|
ABEFZGACDEHZIZACTJZGZACTFRBCDIZBCDJZGABEIZABEJZGUCSBCDKABEKUDUFUEUGUCUDUFGU
|
|
AUEUGGUBABCDELABCDEMNOPACTKQ $.
|
|
|
|
$( The converse of an injective function is bijective. (Contributed by FL,
|
|
11-Nov-2011.) $)
|
|
f1cnv $p |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A ) $=
|
|
( wf1 crn wf1o ccnv f1f1orn f1ocnv syl ) ABCDACEZCFKACGFABCHAKCIJ $.
|
|
|
|
$( Composition with the converse. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
funcocnv2 $p |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) ) $=
|
|
( wfun ccnv ccom cid wss crn cres wceq wrel df-fun simprbi cdm dfdm4 dmcoeq
|
|
iss ax-mp df-rn eqtr4i a1i reseq2d eqeq2d syl5bb mpbid ) ABZAACZDZEFZUGEAGZ
|
|
HZIZUEAJUHAKLUHUGEUGMZHZIUEUKUGPUEUMUJUGUEULUIEULUIIUEULUFMZUIAMUFGIULUNIAN
|
|
AUFOQARSTUAUBUCUD $.
|
|
|
|
$( The composition of an onto function and its converse. (Contributed by
|
|
Stefan O'Rear, 12-Feb-2015.) $)
|
|
fococnv2 $p |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) ) $=
|
|
( wfo ccnv ccom cid crn cres wfun wceq fofun funcocnv2 forn reseq2d eqtrd
|
|
syl ) ABCDZCCEFZGCHZIZGBIRCJSUAKABCLCMQRTBGABCNOP $.
|
|
|
|
$( The composition of a one-to-one onto function and its converse equals the
|
|
identity relation restricted to the function's range. (Contributed by NM,
|
|
13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.) $)
|
|
f1ococnv2 $p |- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) ) $=
|
|
( wf1o wfo ccnv ccom cid cres wceq f1ofo fococnv2 syl ) ABCDABCECCFGHBIJABC
|
|
KABCLM $.
|
|
|
|
$( Composition of an injective function with its converse. (Contributed by
|
|
FL, 11-Nov-2011.) $)
|
|
f1cocnv2 $p |- ( F : A -1-1-> B -> ( F o. `' F ) = ( _I |` ran F ) ) $=
|
|
( wf1 wfun ccnv ccom cid crn cres wceq f1fun funcocnv2 syl ) ABCDCECCFGHCIJ
|
|
KABCLCMN $.
|
|
|
|
$( The composition of a one-to-one onto function's converse and itself equals
|
|
the identity relation restricted to the function's domain. (Contributed
|
|
by NM, 13-Dec-2003.) $)
|
|
f1ococnv1 $p |- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
|
|
( wf1o ccnv ccom cres wrel wceq f1orel dfrel2 sylib coeq2d f1ocnv f1ococnv2
|
|
cid syl eqtr3d ) ABCDZCEZTEZFZTCFPAGZSUACTSCHUACIABCJCKLMSBATDUBUCIABCNBATO
|
|
QR $.
|
|
|
|
$( Composition of an injective function with its converse. (Contributed by
|
|
FL, 11-Nov-2011.) $)
|
|
f1cocnv1 $p |- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
|
|
( wf1 crn wf1o ccnv ccom cid cres wceq f1f1orn f1ococnv1 syl ) ABCDACEZCFCG
|
|
CHIAJKABCLAOCMN $.
|
|
|
|
$( Re-express a constraint on a composition as a constraint on the composand.
|
|
(Contributed by Stefan O'Rear, 7-Mar-2015.) $)
|
|
funcoeqres $p |- ( ( Fun G /\ ( F o. G ) = H ) ->
|
|
( F |` ran G ) = ( H o. `' G ) ) $=
|
|
( wfun ccom wceq crn cres cid funcocnv2 coeq2d coass eqcomi coires1 3eqtr3g
|
|
ccnv coeq1 sylan9req ) BDZABEZCFABGZHZTBPZEZCUCESABUCEZEZAIUAHZEUDUBSUEUGAB
|
|
JKUDUFABUCLMAUANOTCUCQR $.
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
f11o.1 $e |- F e. _V $.
|
|
$( Relationship between a mapping and an onto mapping. Figure 38 of
|
|
[Enderton] p. 145. (Contributed by NM, 10-May-1998.) $)
|
|
ffoss $p |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) ) $=
|
|
( wf cv wfo wss wa wex crn wfn df-f dffn4 anbi1i bitri rnex wceq foeq3
|
|
sseq1 anbi12d spcev sylbi fof fss sylan exlimiv impbii ) BCDFZBAGZDHZUKCI
|
|
ZJZAKZUJBDLZDHZUPCIZJZUOUJDBMZURJUSBCDNUTUQURBDOPQUNUSAUPDERUKUPSULUQUMUR
|
|
UKUPBDTUKUPCUAUBUCUDUNUJAULBUKDFUMUJBUKDUEBUKCDUFUGUHUI $.
|
|
|
|
$( Relationship between one-to-one and one-to-one onto function.
|
|
(Contributed by NM, 4-Apr-1998.) $)
|
|
f11o $p |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) ) $=
|
|
( wf ccnv wfun wa cv wfo wss wex wf1 wf1o ffoss anbi1i df-f1 dff1o3 bitri
|
|
an32 exbii 19.41v 3bitr4i ) BCDFZDGHZIBAJZDKZUGCLZIZAMZUFIZBCDNBUGDOZUIIZ
|
|
AMZUEUKUFABCDEPQBCDRUOUJUFIZAMULUNUPAUNUHUFIZUIIUPUMUQUIBUGDSQUHUFUIUATUB
|
|
UJUFAUCTUD $.
|
|
$}
|
|
|
|
$( The empty set maps one-to-one into any class. (Contributed by NM,
|
|
7-Apr-1998.) $)
|
|
f10 $p |- (/) : (/) -1-1-> A $=
|
|
( c0 wf1 wf ccnv wfun f0 fun0 cnv0 funeqi mpbir df-f1 mpbir2an ) BABCBABDBE
|
|
ZFZAGOBFHNBIJKBABLM $.
|
|
|
|
$( One-to-one onto mapping of the empty set. (Contributed by NM,
|
|
15-Apr-1998.) $)
|
|
f1o00 $p |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wf1o wfn ccnv wceq dff1o4 fn0 biimpi adantr cdm dm0 cnveq syl6eq sylbi
|
|
wa cnv0 fneq1d jca biimpa fndm syl syl5reqr biimpri eqid mpbir fneq2 mpbiri
|
|
sylan9bb impbii bitri ) CABDBCEZBFZAEZQZBCGZACGZQZCABHUPUSUPUQURUMUQUOUMUQB
|
|
IZJKUPCCLZAMUPCAEZVAAGUMUOVBUMAUNCUMUQUNCGUTUQUNCFCBCNROZPSUAACUBUCUDTUSUMU
|
|
OUQUMURUMUQUTUEKUSUOCCEZVDCCGCUFCIUGUQUOVBURVDUQAUNCVCSACCUHUJUITUKUL $.
|
|
|
|
$( Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.) $)
|
|
fo00 $p |- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wfo wf1o wceq wf1 wfn fofn fn0 f10 f1eq1 mpbiri sylbi syl ancri df-f1o
|
|
wa sylibr f1ofo impbii f1o00 bitri ) CABDZCABEZBCFZACFRUDUEUDCABGZUDRUEUDUG
|
|
UDBCHZUGCABIUHUFUGBJUFUGCACGAKCABCLMNOPCABQSCABTUAABUBUC $.
|
|
|
|
$( One-to-one onto mapping of the empty set. (Contributed by NM,
|
|
10-Sep-2004.) $)
|
|
f1o0 $p |- (/) : (/) -1-1-onto-> (/) $=
|
|
( c0 wf1o wceq eqid f1o00 mpbir2an ) AAABAACZGADZHAAEF $.
|
|
|
|
$( A restriction of the identity relation is a one-to-one onto function.
|
|
(Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
f1oi $p |- ( _I |` A ) : A -1-1-onto-> A $=
|
|
( cid cres wf1o wfn ccnv fnresi cnvresid fneq1i mpbir dff1o4 mpbir2an ) AAB
|
|
ACZDMAEZMFZAEZAGZPNQAOMAHIJAAMKL $.
|
|
|
|
$( The identity relation is a one-to-one onto function on the universe.
|
|
(Contributed by NM, 16-May-2004.) $)
|
|
f1ovi $p |- _I : _V -1-1-onto-> _V $=
|
|
( cvv cid cres wf1o f1oi wceq wb wrel reli dfrel3 mpbi f1oeq1 ax-mp ) AABAC
|
|
ZDZAABDZAENBFZOPGBHQIBJKAANBLMK $.
|
|
|
|
${
|
|
f1osn.1 $e |- A e. _V $.
|
|
f1osn.2 $e |- B e. _V $.
|
|
$( A singleton of an ordered pair is one-to-one onto function.
|
|
(Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
f1osn $p |- { <. A , B >. } : { A } -1-1-onto-> { B } $=
|
|
( csn cop wf1o wfn ccnv fnsn cnvsn fneq1i mpbir dff1o4 mpbir2an ) AEZBEZA
|
|
BFEZGRPHRIZQHZABCDJTBAFEZQHBADCJQSUAABCDKLMPQRNO $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $.
|
|
$( A singleton of an ordered pair is one-to-one onto function.
|
|
(Contributed by Mario Carneiro, 12-Jan-2013.) $)
|
|
f1osng $p |- ( ( A e. V /\ B e. W ) ->
|
|
{ <. A , B >. } : { A } -1-1-onto-> { B } ) $=
|
|
( va vb cv csn cop wf1o wceq sneq f1oeq2 syl opeq1 sneqd f1oeq1 bitrd vex
|
|
wb f1oeq3 opeq2 f1osn vtocl2g ) EGZHZFGZHZUEUGIZHZJZAHZUHAUGIZHZJZULBHZAB
|
|
IZHZJZEFABCDUEAKZUKULUHUJJZUOUTUFULKUKVATUEALUFULUHUJMNUTUJUNKVAUOTUTUIUM
|
|
UEAUGOPULUHUJUNQNRUGBKZUOULUPUNJZUSVBUHUPKUOVCTUGBLUHUPULUNUANVBUNURKVCUS
|
|
TVBUMUQUGBAUBPULUPUNURQNRUEUGESFSUCUD $.
|
|
$}
|
|
|
|
$( An unordered pair of ordered pairs with different elements is a one-to-one
|
|
onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.) $)
|
|
f1oprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) )
|
|
-> ( ( A =/= C /\ B =/= D )
|
|
-> { <. A , B >. , <. C , D >. } : { A , C } -1-1-onto-> { B , D } ) ) $=
|
|
( wcel wa wne cpr cop wf1o csn cun wceq df-pr eqcomi a1i c0 f1osng ad2antrr
|
|
cin ad2antlr disjsn2 ad2antrl ad2antll f1oun syl22anc f1oeq123d mpbid ex )
|
|
AEIBFIJZCGIDHIJZJZACKZBDKZJZACLZBDLZABMZCDMZLZNZUPUSJZAOZCOZPZBOZDOZPZVBOZV
|
|
COZPZNZVEVFVGVJVMNZVHVKVNNZVGVHUDUAQZVJVKUDUAQZVPUNVQUOUSABEFUBUCUOVRUNUSCD
|
|
GHUBUEUQVSUPURACUFUGURVTUPUQBDUFUHVGVJVHVKVMVNUIUJVFVIUTVLVAVOVDVOVDQVFVDVO
|
|
VBVCRSTVIUTQVFUTVIACRSTVLVAQVFVAVLBDRSTUKULUM $.
|
|
|
|
${
|
|
$d x F $. $d x A $.
|
|
$( Function value when ` F ` is not a function. Theorem 6.12(2) of
|
|
[TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof
|
|
shortened by Mario Carneiro, 31-Aug-2015.) $)
|
|
tz6.12-2 $p |- ( -. E! x A F x -> ( F ` A ) = (/) ) $=
|
|
( cv wbr weu wn cfv cio c0 df-fv iotanul syl5eq ) BADCEZAFGBCHNAIJABCKNAL
|
|
M $.
|
|
|
|
$( The value of a function at a unique point. (Contributed by Scott
|
|
Fenton, 6-Oct-2017.) $)
|
|
fveu $p |- ( E! x A F x -> ( F ` A ) = U. { x | A F x } ) $=
|
|
( cv wbr weu cfv cio cab cuni df-fv iotauni syl5eq ) BADCEZAFBCGNAHNAIJAB
|
|
CKNALM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( If ` A ` is a proper class and ` F ` is any class, then there is no
|
|
unique set which is related to ` A ` through the binary relation ` F ` .
|
|
(Contributed by Scott Fenton, 7-Oct-2017.) $)
|
|
brprcneu $p |- ( -. A e. _V -> -. E! x A F x ) $=
|
|
( vy wcel wn cv wbr wex wi wa weq cop c0 notbii exbii sylibr opprc1 df-br
|
|
eleq1d cvv wmo weu dtruex equcom mpbir 19.42v anbi12d anidm syl6bb anbi1d
|
|
exbidv imbi12d mpbiri anbi12i anbi1i 3imtr4g eximdv wal exanaliim exnalim
|
|
jctr eximi syl breq2 mo4 syl6 eu5 imnan bitr4i ) BUAEFZBAGZCHZAIZVMAUBZFZ
|
|
JZVMAUCZFZVKVNVMBDGZCHZKZADLZFZKZDIZAIZVPVKVMWFAVKBVLMZCEZWIBVTMZCEZKZWDK
|
|
ZDIZVMWFVKWIWNJNCEZWOWDKZDIZJWOWOWDDIZKWQWOWRWRDALZFZDIDAUDWDWTDWCWSADUEO
|
|
PUFVBWOWDDUGQVKWIWOWNWQVKWHNCBVLRTZVKWMWPDVKWLWOWDVKWLWOWOKWOVKWIWOWKWOXA
|
|
VKWJNCBVTRTUHWOUIUJUKULUMUNBVLCSZWEWMDWBWLWDVMWIWAWKXBBVTCSUOUPPUQURWGWBW
|
|
CJDUSZAUSZFZVPWGXCFZAIXEWFXFAWBWCDUTVCXCAVAVDVOXDVMWAADVLVTBCVEVFOQVGVSVN
|
|
VOKZFVQVRXGVMAVHOVNVOVIVJQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( A function's value at a proper class is the empty set. (Contributed by
|
|
NM, 20-May-1998.) $)
|
|
fvprc $p |- ( -. A e. _V -> ( F ` A ) = (/) ) $=
|
|
( vx cvv wcel wn cv wbr weu cfv c0 wceq brprcneu tz6.12-2 syl ) ADEFACGBH
|
|
CIFABJKLCABMCABNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Alternate definition of function value. Definition 10.11 of [Quine]
|
|
p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew
|
|
Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fv2 $p |- ( F ` A ) = U. { x | A. y ( A F y <-> y = x ) } $=
|
|
( cfv cv wbr cio weq wb wal cab cuni df-fv dfiota2 eqtri ) CDECBFDGZBHQBA
|
|
IJBKALMBCDNQBAOP $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d A x $. $d V x $.
|
|
$( A definition of function value in terms of iota. (Contributed by Jim
|
|
Kingdon, 29-Dec-2018.) $)
|
|
dffv3g $p |- ( A e. V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) ) $=
|
|
( wcel cv csn cima cio wbr cfv cvv wb vex wa elimasng df-br syl6bbr mpan2
|
|
cop iotabidv df-fv syl6reqr ) BDEZAFZCBGHEZAIBUECJZAIBCKUDUFUGAUDUELEZUFU
|
|
GMANUDUHOUFBUETCEUGCBUEDLPBUECQRSUAABCUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $. $d x y V $.
|
|
$( The previous definition of function value, from before the ` iota `
|
|
operator was introduced. Although based on the idea embodied by
|
|
Definition 10.2 of [Quine] p. 65 (see ~ args ), this definition
|
|
apparently does not appear in the literature. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
dffv4g $p |- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } ) $=
|
|
( vy wcel cfv cv csn cima cio wceq cuni dffv3g df-iota abid2 eqeq1i abbii
|
|
cab unieqi eqtri syl6eq ) BDFBCGEHCBIJZFZEKZUCAHIZLZASZMZEBCDNUEUDESZUFLZ
|
|
ASZMUIUDEAOULUHUKUGAUJUCUFEUCPQRTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y F $.
|
|
$( Membership in a function value. (Contributed by NM, 30-Apr-2004.) $)
|
|
elfv $p |- ( A e. ( F ` B ) <->
|
|
E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) ) $=
|
|
( cfv wcel cv wbr weq wb wal cab cuni wa wex fv2 eleq2i eluniab bitri ) C
|
|
DEFZGCDBHEIBAJKBLZAMNZGCAHGUBOAPUAUCCABDEQRUBACST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $. $d x G $.
|
|
$( Equality theorem for function value. (Contributed by NM,
|
|
29-Dec-1996.) $)
|
|
fveq1 $p |- ( F = G -> ( F ` A ) = ( G ` A ) ) $=
|
|
( vx wceq cv wbr cio cfv breq iotabidv df-fv 3eqtr4g ) BCEZADFZBGZDHAOCGZ
|
|
DHABIACINPQDAOBCJKDABLDACLM $.
|
|
|
|
$( Equality theorem for function value. (Contributed by NM,
|
|
29-Dec-1996.) $)
|
|
fveq2 $p |- ( A = B -> ( F ` A ) = ( F ` B ) ) $=
|
|
( vx wceq cv wbr cio cfv breq1 iotabidv df-fv 3eqtr4g ) ABEZADFZCGZDHBOCG
|
|
ZDHACIBCINPQDABOCJKDACLDBCLM $.
|
|
$}
|
|
|
|
${
|
|
fveq1i.1 $e |- F = G $.
|
|
$( Equality inference for function value. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
fveq1i $p |- ( F ` A ) = ( G ` A ) $=
|
|
( wceq cfv fveq1 ax-mp ) BCEABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
fveq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for function value. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
fveq1d $p |- ( ph -> ( F ` A ) = ( G ` A ) ) $=
|
|
( wceq cfv fveq1 syl ) ACDFBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
fveq2i.1 $e |- A = B $.
|
|
$( Equality inference for function value. (Contributed by NM,
|
|
28-Jul-1999.) $)
|
|
fveq2i $p |- ( F ` A ) = ( F ` B ) $=
|
|
( wceq cfv fveq2 ax-mp ) ABEACFBCFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
fveq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function value. (Contributed by NM,
|
|
29-May-1999.) $)
|
|
fveq2d $p |- ( ph -> ( F ` A ) = ( F ` B ) ) $=
|
|
( wceq cfv fveq2 syl ) ABCFBDGCDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
fveq12i.1 $e |- F = G $.
|
|
fveq12i.2 $e |- A = B $.
|
|
$( Equality deduction for function value. (Contributed by FL,
|
|
27-Jun-2014.) $)
|
|
fveq12i $p |- ( F ` A ) = ( G ` B ) $=
|
|
( cfv fveq1i fveq2i eqtri ) ACGADGBDGACDEHABDFIJ $.
|
|
$}
|
|
|
|
${
|
|
fveq12d.1 $e |- ( ph -> F = G ) $.
|
|
fveq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function value. (Contributed by FL,
|
|
22-Dec-2008.) $)
|
|
fveq12d $p |- ( ph -> ( F ` A ) = ( G ` B ) ) $=
|
|
( cfv fveq1d fveq2d eqtrd ) ABDHBEHCEHABDEFIABCEGJK $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d y A $. $d x y $.
|
|
nffv.1 $e |- F/_ x F $.
|
|
nffv.2 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for function value. (Contributed by
|
|
NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nffv $p |- F/_ x ( F ` A ) $=
|
|
( vy cfv cv wbr cio df-fv nfcv nfbr nfiotaxy nfcxfr ) ABCGBFHZCIZFJFBCKQA
|
|
FABPCEDAPLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Bound-variable hypothesis builder for mapping, special case.
|
|
(Contributed by Mario Carneiro, 25-Dec-2016.) $)
|
|
nffvmpt1 $p |- F/_ x ( ( x e. A |-> B ) ` C ) $=
|
|
( cmpt nfmpt1 nfcv nffv ) ADABCEABCFADGH $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z F $. $d x z $.
|
|
nffvd.2 $e |- ( ph -> F/_ x F ) $.
|
|
nffvd.3 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nffv .
|
|
(Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nffvd $p |- ( ph -> F/_ x ( F ` A ) ) $=
|
|
( vz cv wcel wal cab cfv wnfc nfaba1 nffv wb wa nfnfc1 wceq abidnf adantr
|
|
nfan adantl fveq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
|
|
DLZMZBUJULUKBGNUIBGNOABDMZBCMZUNUPPEFUQURQZBUMUOUQURBBDRBCRUBUSUJCULDUQUL
|
|
DSURBGDTUAURUJCSUQBGCTUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( A function has one value given an argument in its domain. (Contributed
|
|
by Jim Kingdon, 29-Dec-2018.) $)
|
|
funfveu $p |- ( ( Fun F /\ A e. dom F ) -> E! y A F y ) $=
|
|
( vx wfun cdm wcel cv wbr weu wa wi wceq eleq1 anbi2d eubidv imbi12d wrel
|
|
breq1 wral dffun8 simprbi r19.21bi vtoclg anabsi7 ) CEZBCFZGZBAHZCIZAJZUF
|
|
DHZUGGZKZULUICIZAJZLUFUHKZUKLDBUGULBMZUNUQUPUKURUMUHUFULBUGNOURUOUJAULBUI
|
|
CSPQUFUPDUGUFCRUPDUGTDACUAUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( The value of a function is a subset of ` B ` if every element that could
|
|
be a candidate for the value is a subset of ` B ` . (Contributed by
|
|
Mario Carneiro, 24-May-2019.) $)
|
|
fvss $p |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B ) $=
|
|
( cv wbr wss wi wal cfv cio df-fv iotass syl5eqss ) BAEZDFZOCGHAIBDJPAKCA
|
|
BDLPACMN $.
|
|
|
|
$( The result of a function value is always a subset of the union of the
|
|
range, if the input is a set. (Contributed by Stefan O'Rear,
|
|
2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.) $)
|
|
fvssunirng $p |- ( A e. _V -> ( F ` A ) C_ U. ran F ) $=
|
|
( vx cvv wcel cv wbr crn cuni wss wi wal cfv vex brelrng 3exp mpi elssuni
|
|
syl6 alrimiv fvss syl ) ADEZACFZBGZUDBHZIZJZKZCLABMUGJUCUICUCUEUDUFEZUHUC
|
|
UDDEZUEUJKCNUCUKUEUJAUDBDDOPQUDUFRSTCAUGBUAUB $.
|
|
|
|
$( The result of a function value is always a subset of the union of the
|
|
range, even if it is invalid and thus empty. (Contributed by Stefan
|
|
O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.) $)
|
|
relfvssunirn $p |- ( Rel F -> ( F ` A ) C_ U. ran F ) $=
|
|
( vx wrel cv wbr crn cuni wss wi wal wcel relelrn ex elssuni syl6 alrimiv
|
|
cfv fvss syl ) BDZACEZBFZUBBGZHZIZJZCKABRUEIUAUGCUAUCUBUDLZUFUAUCUHAUBBMN
|
|
UBUDOPQCAUEBST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( The value of a function exists. A special case of Corollary 6.13 of
|
|
[TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.) $)
|
|
funfvex $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V ) $=
|
|
( vy wfun cdm wcel wa cfv wbr cio cvv df-fv weu funfveu euiotaex syl5eqel
|
|
cv syl ) BDABEFGZABHACQBIZCJZKCABLSTCMUAKFCABNTCORP $.
|
|
|
|
$( If a function has a set range, then the function value exists
|
|
unconditional on the domain. (Contributed by Mario Carneiro,
|
|
24-May-2019.) $)
|
|
relrnfvex $p |- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V ) $=
|
|
( wrel cfv crn cuni wss cvv wcel relfvssunirn uniexg ssexg syl2an ) BCABD
|
|
ZBEZFZGPHINHIOHIABJOHKNPHLM $.
|
|
|
|
$( Evaluating a set function at a set exists. (Contributed by Mario
|
|
Carneiro and Jim Kingdon, 28-May-2019.) $)
|
|
fvexg $p |- ( ( F e. V /\ A e. W ) -> ( F ` A ) e. _V ) $=
|
|
( wcel cfv crn cuni wss cvv elex fvssunirng rnexg uniexg ssexg syl2anr
|
|
syl ) ADEZABFZBGZHZIZUAJEZSJEBCEZRAJEUBADKABLQUDTJEUCBCMTJNQSUAJOP $.
|
|
|
|
${
|
|
fvex.1 $e |- F e. V $.
|
|
fvex.2 $e |- A e. W $.
|
|
$( Evaluating a set function at a set exists. (Contributed by Mario
|
|
Carneiro and Jim Kingdon, 28-May-2019.) $)
|
|
fvex $p |- ( F ` A ) e. _V $=
|
|
( wcel cfv cvv fvexg mp2an ) BCGADGABHIGEFABCDJK $.
|
|
$}
|
|
|
|
$( If a function is set-like, then the function value exists if the input
|
|
does. (Contributed by Mario Carneiro, 24-May-2019.) $)
|
|
sefvex $p |- ( ( `' F Se _V /\ A e. _V ) -> ( F ` A ) e. _V ) $=
|
|
( vy vx cvv ccnv wse wcel wa cfv cv wbr crab cuni wss wi wal w3a vex syl
|
|
a1i simp3 simp2 brcnvg sylancr mpbird breq1 elrab sylanbrc elssuni 3expia
|
|
wb alrimiv fvss seex uniexg ssexg syl2anc ) EBFZGZAEHZIZABJZCKZAUSLZCEMZN
|
|
ZOZVGEHZVCEHVBADKZBLZVJVGOZPZDQVHVBVMDUTVAVKVLUTVAVKRZVJVFHZVLVNVJEHZVJAU
|
|
SLZVOVPVNDSZUAVNVQVKUTVAVKUBVNVPVAVQVKULVRUTVAVKUCVJAEEBUDUEUFVEVQCVJEVDV
|
|
JAUSUGUHUIVJVFUJTUKUMDAVGBUNTVBVFEHVICEAUSUOVFEUPTVCVGEUQUR $.
|
|
$}
|
|
|
|
$( Move a conditional outside of a function. (Contributed by Jim Kingdon,
|
|
1-Jan-2022.) $)
|
|
fvifdc $p |- ( DECID ph
|
|
-> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) ) $=
|
|
( cif cfv fveq2 ifsbdc ) ABCABCEZDFBDFCDFIBDGICDGH $.
|
|
|
|
${
|
|
$d x y z F $. $d x y z A $.
|
|
$( Alternate definition of the value of a function. Definition 6.11 of
|
|
[TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
fv3 $p |- ( F ` A ) = { x | ( E. y ( x e. y /\ A F y ) /\
|
|
E! y A F y ) } $=
|
|
( vz cv wcel wbr wa wex weu cfv weq wb wal elfv wi bi2 breq2 sylib anim2i
|
|
alimi vex ceqsalv eximi elequ2 anbi12d cbvexv exsimpr df-eu jca nfeu1 nfv
|
|
sylibr nfa1 nfan nfex nfim bi1 ax-14 syl6 com23 impd anc2ri eximdv syl5bi
|
|
sps com12 exlimi imp impbii bitri abbi2i ) AFZBFZGZCVODHZIZBJZVQBKZIZACDL
|
|
ZVNWBGVNEFZGZVQBEMZNZBOZIZEJZWAEBVNCDPWIWAWIVSVTWIWDCWCDHZIZEJVSWHWKEWGWJ
|
|
WDWGWEVQQZBOWJWFWLBVQWERUBVQWJBWCEUCVOWCCDSUDTUAUEWKVREBEBMWDVPWJVQEBAUFW
|
|
CVOCDSUGUHTWIWGEJZVTWDWGEUIVQBEUJZUNUKVSVTWIVRVTWIQBVTWIBVQBULWHBEWDWGBWD
|
|
BUMWFBUOUPUQURVTWMVRWIWNVRWGWHEWGVRWHWGVRWDWFVRWDQBWFVPVQWDWFVQVPWDWFVQWE
|
|
VPWDQVQWEUSBEAUTVAVBVCVGVDVHVEVFVIVJVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
$( The value of a restricted function. (Contributed by NM, 2-Aug-1994.) $)
|
|
fvres $p |- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) ) $=
|
|
( vx wcel cv cres wbr cio cfv vex brres rbaib iotabidv df-fv 3eqtr4g ) AB
|
|
EZADFZCBGZHZDIARCHZDIASJACJQTUADTUAQARCBDKLMNDASODACOP $.
|
|
$}
|
|
|
|
$( The value of a member of the domain of a subclass of a function.
|
|
(Contributed by NM, 15-Aug-1994.) $)
|
|
funssfv $p |- ( ( Fun F /\ G C_ F /\ A e. dom G ) ->
|
|
( F ` A ) = ( G ` A ) ) $=
|
|
( wfun wss cdm wcel cfv wceq wa cres fvres eqcomd funssres fveq1d sylan9eqr
|
|
3impa ) BDZCBEZACFZGZABHZACHZIUARSJZUBABTKZHZUCUAUFUBATBLMUDAUECBCNOPQ $.
|
|
|
|
${
|
|
$d y F $. $d y A $.
|
|
$( Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
|
|
by NM, 30-Apr-2004.) $)
|
|
tz6.12-1 $p |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y ) $=
|
|
( cv wbr weu wa cfv cio df-fv wceq iota1 biimpac syl5eq ) BADZCEZPAFZGBCH
|
|
PAIZOABCJQPROKPALMN $.
|
|
|
|
$( Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
|
|
by NM, 10-Jul-1994.) $)
|
|
tz6.12 $p |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) ->
|
|
( F ` A ) = y ) $=
|
|
( cv cop wcel wbr weu cfv wceq df-br eubii tz6.12-1 syl2anbr ) BADZECFZBO
|
|
CGZQAHBCIOJPAHBOCKZQPARLABCMN $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d z F $.
|
|
tz6.12f.1 $e |- F/_ y F $.
|
|
$( Function value, using bound-variable hypotheses instead of distinct
|
|
variable conditions. (Contributed by NM, 30-Aug-1999.) $)
|
|
tz6.12f $p |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) ->
|
|
( F ` A ) = y ) $=
|
|
( vz cv cop wcel weu wa cfv wceq wi opeq2 eleq1d wb nfel2 nfv cbveu a1i
|
|
anbi12d eqeq2 imbi12d tz6.12 chvarv ) BEFZGZCHZUHEIZJZBCKZUFLZMBAFZGZCHZU
|
|
OAIZJZUKUMLZMEAUFUMLZUJUQULURUSUHUOUIUPUSUGUNCUFUMBNOZUIUPPUSUHUOEAAUGCDQ
|
|
UOERUTSTUAUFUMUKUBUCEBCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d y A $.
|
|
$( Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by
|
|
NM, 30-Apr-2004.) $)
|
|
tz6.12c $p |- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) ) $=
|
|
( cv wbr weu cfv wceq wex euex wi nfeu1 nfv tz6.12-1 expcom breq2 biimprd
|
|
nfim syli com12 exlimi mpcom syl5ibcom impbid ) BADZCEZAFZBCGZUEHZUFUGBUH
|
|
CEZUIUFUFAIUGUJUFAJUFUGUJKAUGUJAUFALUJAMRUGUFUJUFUGUIUJUFUGUIABCNOZUIUJUF
|
|
UHUEBCPZQSTUAUBULUCUKUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( The value of a class outside its domain is the empty set. (Contributed
|
|
by Jim Kingdon, 15-Jan-2019.) $)
|
|
ndmfvg $p |- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) ) $=
|
|
( vx cvv wcel cdm wn cfv c0 wceq cv wbr weu wex euex eldmg con3d tz6.12-2
|
|
syl5ibr syl6 imp ) ADEZABFEZGZABHIJZUBUDACKBLZCMZGUEUBUGUCUGUCUBUFCNUFCOC
|
|
ABDPSQCABRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( If a function value has a member, the argument belongs to the domain.
|
|
(Contributed by Jim Kingdon, 22-Jan-2019.) $)
|
|
relelfvdm $p |- ( ( Rel F /\ A e. ( F ` B ) ) -> B e. dom F ) $=
|
|
( vx vy wrel cfv wcel wa cv wbr wex cdm weq wb wal elfv exsimpr wsb syl
|
|
sylbi equsb1 spsbbi mpbiri breq2 sylib eximi anim2i 19.42v sylibr releldm
|
|
nfv sbie exlimiv ) CFZABCGHZIZUOBDJZCKZIZDLZBCMHZUQUOUSDLZIVAUPVCUOUPBEJZ
|
|
CKZEDNZOEPZDLZVCUPAURHZVGIDLVHDEABCQVIVGDRUAVGUSDVGVEEDSZUSVGVJVFEDSEDUBV
|
|
EVFEDUCUDVEUSEDUSEULVDURBCUEUMUFUGTUHUOUSDUIUJUTVBDBURCUKUNT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w F $.
|
|
$( The value of a non-member of a restriction is the empty set.
|
|
(Contributed by NM, 13-Nov-1995.) $)
|
|
nfvres $p |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) ) $=
|
|
( vz vx vw vy wcel wn cres cfv cv wex cab csn wceq eleq2i bitri sylbi cvv
|
|
wbr weu cuni cio df-fv df-iota eqtri eluni exsimpr wsb df-clab nfv eqeq2d
|
|
wa sneq sbie exbii sylib euabsn2 sylibr euex cop cxp df-br df-res simprbi
|
|
cin elin opelxp1 syl exlimiv 3syl con3i eq0rdv ) ABHZIDACBJZKZDLZVQHZVOVS
|
|
AELZVPUAZEUBZWAEMVOVSWAENZFLZOZPZFMZWBVSWDWCGLZOZPZGNZHZFMZWGVSVRWDHZWLUN
|
|
FMZWMVSVRWKUCZHWOVQWPVRVQWAEUDWPEAVPUEWAEGUFUGQFVRWKUHRWNWLFUISWLWFFWLWJG
|
|
FUJWFWJFGUKWJWFGFWFGULWHWDPWIWEWCWHWDUOUMUPRUQURWAEFUSUTWAEVAWAVOEWAAVTVB
|
|
ZBTVCZHZVOWAWQCWRVGZHZWSWAWQVPHXAAVTVPVDVPWTWQCBVEQRXAWQCHWSWQCWRVHVFSAVT
|
|
BTVIVJVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( If the restriction of a class to a singleton is not a function, its
|
|
value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof
|
|
shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nfunsn $p |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) ) $=
|
|
( vy vx csn cres wfun wn cv wbr weu cfv c0 wceq wrel wmo wal wa sylbi syl
|
|
eumo wcel vex brres velsn breq1 biimpac moimi alrimiv relres jctil dffun6
|
|
wb sylibr con3i tz6.12-2 ) BAEZFZGZHACIZBJZCKZHABLMNVBUSVBUROZDIZUTURJZCP
|
|
ZDQZRUSVBVGVCVBVFDVBVACPVFVACUAVEVACVEVDUTBJZVDUQUBZRVAVDUTBUQCUCUDVIVHVA
|
|
VIVDANVHVAUMDAUEVDAUTBUFSUGSUHTUIBUQUJUKDCURULUNUOCABUPT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Function value of the empty set. (Contributed by Stefan O'Rear,
|
|
26-Nov-2014.) $)
|
|
0fv $p |- ( (/) ` A ) = (/) $=
|
|
( vx c0 cfv cv wbr cio df-fv weu wn wceq wex cop wcel noel df-br nex euex
|
|
mtbir mto iotanul ax-mp eqtri ) ACDABEZCFZBGZCBACHUEBIZJUFCKUGUEBLUEBUEAU
|
|
DMZCNUHOAUDCPSQUEBRTUEBUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y F $. $d x y $.
|
|
$( Move class substitution in and out of a function value. (Contributed by
|
|
NM, 11-Nov-2005.) $)
|
|
csbfv12g $p |- ( A e. C ->
|
|
[_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) ) $=
|
|
( vy wcel wbr cio csb cfv csbiotag sbcbrg csbconstg breq2d bitrd iotabidv
|
|
cv wsbc df-fv eqtrd csbeq2i 3eqtr4g ) BDGZABCFRZEHZFIZJZABCJZUEABEJZHZFIZ
|
|
ABCEKZJUIUJKUDUHUFABSZFIULUFAFBDLUDUNUKFUDUNUIABUEJZUJHUKABCUEDEMUDUOUEUI
|
|
UJABUEDNOPQUAABUMUGFCETUBFUIUJTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x F $.
|
|
$( Move class substitution in and out of a function value. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbfv2g $p |- ( A e. C -> [_ A / x ]_ ( F ` B ) =
|
|
( F ` [_ A / x ]_ B ) ) $=
|
|
( wcel cfv csb csbfv12g csbconstg fveq1d eqtrd ) BDFZABCEGHABCHZABEHZGNEG
|
|
ABCDEIMNOEABEDJKL $.
|
|
|
|
$( Substitution for a function value. (Contributed by NM, 1-Jan-2006.) $)
|
|
csbfvg $p |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) ) $=
|
|
( wcel cv cfv csb csbfv2g csbvarg fveq2d eqtrd ) BCEZABAFZDGHABNHZDGBDGAB
|
|
NCDIMOBDABCJKL $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y F $. $d y B $.
|
|
$( The second argument of a binary relation on a function is the function's
|
|
value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
funbrfv $p |- ( Fun F -> ( A F B -> ( F ` A ) = B ) ) $=
|
|
( vy wfun wbr cfv wceq cvv wcel wa wrel funrel brrelex2 sylan cv wi breq2
|
|
anbi2d eqeq2 imbi12d weu funeu tz6.12-1 sylan2 anabss7 vtoclg mpcom ex )
|
|
CEZABCFZACGZBHZBIJZUJUKKZUMUJCLUKUNCMABCNOUJADPZCFZKZULUPHZQUOUMQDBIUPBHZ
|
|
URUOUSUMUTUQUKUJUPBACRSUPBULTUAUJUQUSURUQUQDUBUSDAUPCUCDACUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
$( The second element in an ordered pair member of a function is the
|
|
function's value. (Contributed by NM, 19-Jul-1996.) $)
|
|
funopfv $p |- ( Fun F -> ( <. A , B >. e. F -> ( F ` A ) = B ) ) $=
|
|
( cop wcel wbr wfun cfv wceq df-br funbrfv syl5bir ) ABDCEABCFCGACHBIABCJAB
|
|
CKL $.
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $. $d x C $.
|
|
$( Equivalence of function value and binary relation. (Contributed by NM,
|
|
19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
fnbrfvb $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F ` B ) = C <-> B F C ) ) $=
|
|
( vx wfn wcel wa cfv wceq wbr eqid cvv wb funfvex funfni cv wi breq2 syl
|
|
eqeq2 bibi12d imbi2d fneu tz6.12c vtoclg mpcom mpbii syl5ibcom wfun fnfun
|
|
weu funbrfv adantr impbid ) DAFZBAGZHZBDIZCJZBCDKZURBUSDKZUTVAURUSUSJZVBU
|
|
SLUSMGZURVCVBNZVDABDBDOPURUSEQZJZBVFDKZNZRURVEREUSMVFUSJZVIVEURVJVGVCVHVB
|
|
VFUSUSUAVFUSBDSUBUCURVHEULVIEABDUDEBDUETUFUGUHUSCBDSUIUPVAUTRZUQUPDUJVKAD
|
|
UKBCDUMTUNUO $.
|
|
$}
|
|
|
|
$( Equivalence of function value and ordered pair membership. (Contributed
|
|
by NM, 7-Nov-1995.) $)
|
|
fnopfvb $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F ` B ) = C <-> <. B , C >. e. F ) ) $=
|
|
( wfn wcel wa cfv wceq wbr cop fnbrfvb df-br syl6bb ) DAEBAFGBDHCIBCDJBCKDF
|
|
ABCDLBCDMN $.
|
|
|
|
$( Equivalence of function value and binary relation. (Contributed by NM,
|
|
26-Mar-2006.) $)
|
|
funbrfvb $p |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) ) $=
|
|
( wfun cdm wfn wcel cfv wceq wbr wb funfn fnbrfvb sylanb ) CDCCEZFAOGACHBIA
|
|
BCJKCLOABCMN $.
|
|
|
|
$( Equivalence of function value and ordered pair membership. Theorem
|
|
4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.) $)
|
|
funopfvb $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( ( F ` A ) = B <-> <. A , B >. e. F ) ) $=
|
|
( wfun cdm wfn wcel cfv wceq cop wb funfn fnopfvb sylanb ) CDCCEZFAOGACHBIA
|
|
BJCGKCLOABCMN $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( Function value in terms of a binary relation. (Contributed by Mario
|
|
Carneiro, 19-Mar-2014.) $)
|
|
funbrfv2b $p |- ( Fun F ->
|
|
( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) ) $=
|
|
( wfun wbr cdm wcel wa cfv wceq wrel funrel releldm syl pm4.71rd funbrfvb
|
|
wi ex pm5.32da bitr4d ) CDZABCEZACFGZUBHUCACIBJZHUAUBUCUACKZUBUCQCLUEUBUC
|
|
ABCMRNOUAUCUDUBABCPST $.
|
|
|
|
$( Representation of a function in terms of its values. The converse holds
|
|
given the law of the excluded middle; as it is we have most of the
|
|
converse via ~ funmpt and ~ dmmptss . (Contributed by Jim Kingdon,
|
|
31-Dec-2018.) $)
|
|
dffn5im $p |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) ) $=
|
|
( vy wfn cv wcel cfv wceq wa copab cmpt wbr wrel fnrel dfrel4v sylib fnbr
|
|
ex pm4.71rd eqcom fnbrfvb syl5bb pm5.32da bitr4d opabbidv df-mpt syl6eqr
|
|
eqtrd ) CBEZCAFZBGZDFZUKCHZIZJZADKZABUNLUJCUKUMCMZADKZUQUJCNCUSIBCOADCPQU
|
|
JURUPADUJURULURJUPUJURULUJURULBUKUMCRSTUJULUOURUOUNUMIUJULJURUMUNUABUKUMC
|
|
UBUCUDUEUFUIADBUNUGUH $.
|
|
|
|
$( The range of a function expressed as a collection of the function's
|
|
values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
fnrnfv $p |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } ) $=
|
|
( wfn crn cv cfv cmpt wceq wrex cab dffn5im rneqd eqid rnmpt syl6eq ) DCE
|
|
ZDFACAGDHZIZFBGSJACKBLRDTACDMNABCSTTOPQ $.
|
|
|
|
$( A member of a function's range is a value of the function. (Contributed
|
|
by NM, 31-Oct-1995.) $)
|
|
fvelrnb $p |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) ) $=
|
|
( vy cv cfv wceq wrex cvv wcel wi wfn crn wb wa wex df-rex 19.41v simpl
|
|
anim1i ancomd funfvex funfni syl simpr eleq1d adantr mpbid exlimiv sylbir
|
|
sylanb expcom cab fnrnfv eleq2d eqeq1 eqcom syl6bb rexbidv elab3g mpancom
|
|
sylan9bbr ) AFZDGZCHZABIZCJKZLZDBMZCDNZKZVGOVGVJVHVGVDBKZVFPZAQZVJVHVFABR
|
|
VOVJPVNVJPZAQVHVNVJASVPVHAVPVEJKZVHVPVJVMPVQVPVMVJVNVMVJVMVFTUAUBVQBVDDVD
|
|
DUCUDUEVNVQVHOVJVNVECJVMVFUFUGUHUIUJUKULUMVJVLCEFZVEHZABIZEUNZKVIVGVJVKWA
|
|
CAEBDUOUPVTVGECJVRCHZVSVFABWBVSCVEHVFVRCVEUQCVEURUSUTVAVCVB $.
|
|
|
|
$( Alternate definition of the image of a function. (Contributed by Raph
|
|
Levien, 20-Nov-2006.) $)
|
|
dfimafn $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F " A ) = { y | E. x e. A ( F ` x ) = y } ) $=
|
|
( wfun cdm wss wa cv cfv wceq wrex cab wbr cima wcel wb ssel funbrfvb ex
|
|
syl9r imp31 rexbidva abbidv dfima2 syl6reqr ) DEZCDFZGZHZAIZDJBIZKZACLZBM
|
|
UKULDNZACLZBMDCOUJUNUPBUJUMUOACUGUIUKCPZUMUOQZUIUQUKUHPZUGURCUHUKRUGUSURU
|
|
KULDSTUAUBUCUDABDCUEUF $.
|
|
|
|
$( Alternate definition of the image of a function as an indexed union of
|
|
singletons of function values. (Contributed by Raph Levien,
|
|
20-Nov-2006.) $)
|
|
dfimafn2 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F " A ) = U_ x e. A { ( F ` x ) } ) $=
|
|
( vy wfun cdm wss wa cima cv cfv wceq cab ciun wrex dfimafn iunab syl6eqr
|
|
csn wcel df-sn eqcom abbii eqtri a1i iuneq2i ) CEBCFGHZCBIZABAJZCKZDJZLZD
|
|
MZNZABUJSZNUGUHULABODMUNADBCPULADBQRABUOUMUOUMLUIBTUOUKUJLZDMUMDUJUAUPULD
|
|
UKUJUBUCUDUEUFR $.
|
|
|
|
$( Membership relation for the values of a function whose image is a
|
|
subclass. (Contributed by Raph Levien, 20-Nov-2006.) $)
|
|
funimass4 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vy cima wss cv wcel wi wal wfun cdm wa wral wceq wb wrex syl5bb cvv cfv
|
|
dfss2 wbr eqcom ssel funbrfvb ex imp31 rexbidva vex elima syl6rbbr imbi1d
|
|
syl9 r19.23v syl6bbr albidv ancoms ralcom4 w3a ssel2 anim2i 3impb funfvex
|
|
nfv eleq1 ceqsalg 3syl 3expa ralbidva syl5bbr bitrd ) DBFZCGEHZVMIZVNCIZJ
|
|
ZEKZDLZBDMZGZNZAHZDUAZCIZABOZEVMCUBWBVRVNWDPZVPJZABOZEKZWFWAVSVRWJQWAVSNZ
|
|
VQWIEWKVQWGABRZVPJWIWKVOWLVPWKWLWCVNDUCZABRVOWKWGWMABWGWDVNPZWKWCBIZNWMVN
|
|
WDUDWAVSWOWNWMQZWAWOWCVTIZVSWPBVTWCUEVSWQWPWCVNDUFUGUNUHSUIAVNDBEUJUKULUM
|
|
WGVPABUOUPUQURWJWHEKZABOWBWFWHAEBUSWBWRWEABVSWAWOWRWEQZVSWAWOUTVSWQNZWDTI
|
|
WSVSWAWOWTWAWONWQVSBVTWCVAVBVCWCDVDVPWEEWDTWEEVEVNWDCVFVGVHVIVJVKVLS $.
|
|
|
|
$( Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
|
|
(Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
fvelima $p |- ( ( Fun F /\ A e. ( F " B ) ) ->
|
|
E. x e. B ( F ` x ) = A ) $=
|
|
( wfun cima wcel cv cfv wceq wrex wbr elimag ibi funbrfv reximdv syl5 imp
|
|
) DEZBDCFZGZAHZDIBJZACKZUAUBBDLZACKZSUDUAUFABDCTMNSUEUCACUBBDOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x F $.
|
|
feqmptd.1 $e |- ( ph -> F : A --> B ) $.
|
|
$( Deduction form of ~ dffn5im . (Contributed by Mario Carneiro,
|
|
8-Jan-2015.) $)
|
|
feqmptd $p |- ( ph -> F = ( x e. A |-> ( F ` x ) ) ) $=
|
|
( wfn cv cfv cmpt wceq wf ffn syl dffn5im ) AECGZEBCBHEIJKACDELPFCDEMNBCE
|
|
ON $.
|
|
|
|
feqresmpt.2 $e |- ( ph -> C C_ A ) $.
|
|
$( Express a restricted function as a mapping. (Contributed by Mario
|
|
Carneiro, 18-May-2016.) $)
|
|
feqresmpt $p |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) ) $=
|
|
( cres cv cfv cmpt wf wss fssres syl2anc feqmptd fvres mpteq2ia syl6eq )
|
|
AFEIZBEBJZUAKZLBEUBFKZLABEDUAACDFMECNEDUAMGHCDEFOPQBEUCUDUBEFRST $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d x z A $. $d z F $.
|
|
dffn5imf.1 $e |- F/_ x F $.
|
|
$( Representation of a function in terms of its values. (Contributed by
|
|
Jim Kingdon, 31-Dec-2018.) $)
|
|
dffn5imf $p |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) ) $=
|
|
( vz wfn cv cfv cmpt dffn5im nfcv nffv fveq2 cbvmpt syl6eq ) CBFCEBEGZCHZ
|
|
IABAGZCHZIEBCJEABQSAPCDAPKLESKPRCMNO $.
|
|
$}
|
|
|
|
${
|
|
$d u v A $. $d x u v B $. $d x u v C $. $d x u v F $.
|
|
$( Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof
|
|
shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy,
|
|
17-Dec-2011.) $)
|
|
fvelimab $p |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <->
|
|
E. x e. B ( F ` x ) = C ) ) $=
|
|
( vu vv wfn wss wa cima wcel cv cfv wceq wrex cvv eleq1 wb wi elex anim2i
|
|
ssel2 funfvex funfni sylan2 anassrs syl5ibcom rexlimdva imdistani rexbidv
|
|
eqeq2 bibi12d imbi2d wfun cdm fnfun adantr sseq2d biimpar dfimafn syl2anc
|
|
cab fndm abeq2d vtoclg impcom pm5.21nd fveq2 eqeq1d cbvrexv syl6bb ) EBHZ
|
|
CBIZJZDECKZLZFMZENZDOZFCPZAMZENZDOZACPVOVQWAVODQLZJVQWEVODVPUAUBVOWAWEVOV
|
|
TWEFCVOVRCLZJVSQLZVTWEVMVNWFWGVNWFJVMVRBLWGCBVRUCWGBVREVREUDUEUFUGVSDQRUH
|
|
UIUJWEVOVQWASZVOGMZVPLZVSWIOZFCPZSZTVOWHTGDQWIDOZWMWHVOWNWJVQWLWAWIDVPRWN
|
|
WKVTFCWIDVSULUKUMUNVOWLGVPVOEUOZCEUPZIZVPWLGVCOVMWOVNBEUQURVMWQVNVMWPBCBE
|
|
VDUSUTFGCEVAVBVEVFVGVHVTWDFACVRWBOVSWCDVRWBEVIVJVKVL $.
|
|
$}
|
|
|
|
$( The value of the identity function. (Contributed by NM, 1-May-2004.)
|
|
(Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
fvi $p |- ( A e. V -> ( _I ` A ) = A ) $=
|
|
( cid wfun wcel wbr cfv wceq funi ididg funbrfv mpsyl ) CDABEAACFACGAHIABJA
|
|
ACKL $.
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y F $.
|
|
$( The indexed intersection of a function's values is the intersection of
|
|
its range. (Contributed by NM, 20-Oct-2005.) $)
|
|
fniinfv $p |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F ) $=
|
|
( vy wfn cv cfv ciin wceq wrex cab cint crn wcel funfvex funfni ralrimiva
|
|
cvv wral dfiin2g syl fnrnfv inteqd eqtr4d ) CBEZABAFZCGZHZDFUGIABJDKZLZCM
|
|
ZLUEUGRNZABSUHUJIUEULABULBUFCUFCOPQADBUGRTUAUEUKUIADBCUBUCUD $.
|
|
|
|
$( Singleton of function value. (Contributed by NM, 22-May-1998.) $)
|
|
fnsnfv $p |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) $=
|
|
( vy wfn wcel wa cv cfv wceq cab wbr csn cima eqcom fnbrfvb syl5bb abbidv
|
|
df-sn a1i imasng adantl 3eqtr4d ) CAEZBAFZGZDHZBCIZJZDKZBUGCLZDKZUHMZCBMN
|
|
ZUFUIUKDUIUHUGJUFUKUGUHOABUGCPQRUMUJJUFDUHSTUEUNULJUDDBACUAUBUC $.
|
|
$}
|
|
|
|
$( The image of a pair under a function. (Contributed by Jeff Madsen,
|
|
6-Jan-2011.) $)
|
|
fnimapr $p |- ( ( F Fn A /\ B e. A /\ C e. A ) ->
|
|
( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } ) $=
|
|
( wfn wcel w3a csn cima cun cfv fnsnfv 3adant3 3adant2 uneq12d eqcomd df-pr
|
|
cpr wceq imaeq2i imaundi eqtri 3eqtr4g ) DAEZBAFZCAFZGZDBHZIZDCHZIZJZBDKZHZ
|
|
CDKZHZJZDBCRZIZUMUORUGUQULUGUNUIUPUKUDUEUNUISUFABDLMUDUFUPUKSUEACDLNOPUSDUH
|
|
UJJZIULURUTDBCQTDUHUJUAUBUMUOQUC $.
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z F $.
|
|
ssimaex.1 $e |- A e. _V $.
|
|
$( The existence of a subimage. (Contributed by NM, 8-Apr-2007.) $)
|
|
ssimaex $p |- ( ( Fun F /\ B C_ ( F " A ) ) ->
|
|
E. x ( x C_ A /\ B = ( F " x ) ) ) $=
|
|
( vy vz vw cima wss cdm cv wceq wa wex cfv wcel wi ex adantr wfun imaeq2i
|
|
cin cres dmres imadmres eqtr3i sseq2i ssrab2 ssel2 adantll fvelima eleq1a
|
|
crab wrex anim2d fveq2 eleq1d elrab syl6ibr simpr jcad reximdv2 adantl wb
|
|
a1i wfn funfn inss2 sstri fvelimab mpan2 sylbi sylibrd syld adantlr eleq1
|
|
mpd biimpcd rexlimiv impbid eqrdv inex1 rabex sseq1 imaeq2 eqeq2d anbi12d
|
|
syl6 spcev sylancr inss1 sstr anim1i eximi syl sylan2br ) CDBIZJDUAZCDBDK
|
|
ZUCZIZJZALZBJZCDXDIZMZNZAOZXBWRCDDBUDKZIXBWRXJXADDBUEUBDBUFUGUHWSXCNZXDXA
|
|
JZXGNZAOZXIXKFLZDPZCQZFXAUNZXAJZCDXRIZMZXNXQFXAUIZXKGCXTXKGLZCQZYCXTQZXKY
|
|
DYEXKYDNYCXBQZYEXCYDYFWSCXBYCUJUKWSYDYFYERXCWSYDNZYFHLZDPZYCMZHXAUOZYEWSY
|
|
FYKRYDWSYFYKHYCXADULSTYGYKYJHXRUOZYEYDYKYLRWSYDYJYJHXAXRYDYHXAQZYJNZYHXRQ
|
|
ZYJYDYNYMYICQZNZYOYDYJYPYMYCCYIUMUPXQYPFYHXAXOYHMXPYICXOYHDUQURUSZUTYNYJR
|
|
YDYMYJVAVFVBVCVDWSYEYLVEZYDWSDWTVGZYSDVHYTXRWTJYSXRXAWTYBBWTVIVJHWTXRYCDV
|
|
KVLVMTVNVOVPVRSWSYEYDRXCWSYEYLYDWSYEYLHYCXRDULSYJYDHXRYOYQYJYDRZYRYPUUAYM
|
|
YJYPYDYIYCCVQVSVDVMVTWITWAWBXMXSYANAXRXQFXABWTEWCWDXDXRMZXLXSXGYAXDXRXAWE
|
|
UUBXFXTCXDXRDWFWGWHWJWKXMXHAXLXEXGXLXABJXEBWTWLXDXABWMVLWNWOWPWQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $.
|
|
$( The existence of a subimage. (Contributed by FL, 15-Apr-2007.) $)
|
|
ssimaexg $p |- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) ->
|
|
E. x ( x C_ A /\ B = ( F " x ) ) ) $=
|
|
( vy wcel wfun cima wss cv wceq wa wex imaeq2 sseq2d anbi2d sseq2 anbi1d
|
|
wi exbidv imbi12d vex ssimaex vtoclg 3impib ) BDGEHZCEBIZJZAKZBJZCEUJILZM
|
|
ZANZUGCEFKZIZJZMZUJUOJZULMZANZTUGUIMZUNTFBDUOBLZURVBVAUNVCUQUIUGVCUPUHCUO
|
|
BEOPQVCUTUMAVCUSUKULUOBUJRSUAUBAUOCEFUCUDUEUF $.
|
|
$}
|
|
|
|
$( A simplified expression for the value of a function when we know it's a
|
|
function. (Contributed by Jim Kingdon, 1-Jan-2019.) $)
|
|
funfvdm $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) ) $=
|
|
( wfun cdm wcel wa cfv csn cuni cima cvv wceq funfvex unisng syl eqid df-fn
|
|
wfn mpbiran2 fnsnfv sylanbr unieqd eqtr3d ) BCZABDZEZFZABGZHZIZUHBAHJZIUGUH
|
|
KEUJUHLABMUHKNOUGUIUKUDBUERZUFUIUKLULUDUEUELUEPBUEQSUEABTUAUBUC $.
|
|
|
|
${
|
|
$d y A $. $d y F $.
|
|
$( The value of a function. Definition of function value in [Enderton]
|
|
p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.) $)
|
|
funfvdm2 $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( F ` A ) = U. { y | A F y } ) $=
|
|
( wfun cdm wcel wa cfv csn cima cuni wbr cab funfvdm imasng adantl unieqd
|
|
cv wceq eqtrd ) CDZBCEZFZGZBCHCBIJZKBARCLAMZKBCNUDUEUFUCUEUFSUAABUBCOPQT
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d w A $. $d w F $. $d w y $.
|
|
funfvdm2f.1 $e |- F/_ y A $.
|
|
funfvdm2f.2 $e |- F/_ y F $.
|
|
$( The value of a function. Version of ~ funfvdm2 using a bound-variable
|
|
hypotheses instead of distinct variable conditions. (Contributed by Jim
|
|
Kingdon, 1-Jan-2019.) $)
|
|
funfvdm2f $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( F ` A ) = U. { y | A F y } ) $=
|
|
( vw wfun cdm wcel wa cfv cv wbr cab cuni funfvdm2 nfcv nfbr nfv breq2
|
|
cbvab unieqi syl6eq ) CGBCHIJBCKBFLZCMZFNZOBALZCMZANZOFBCPUFUIUEUHFAABUDC
|
|
DEAUDQRUHFSUDUGBCTUAUBUC $.
|
|
$}
|
|
|
|
$( The value of a union when the argument is in the first domain.
|
|
(Contributed by Scott Fenton, 29-Jun-2013.) $)
|
|
fvun1 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) ->
|
|
( ( F u. G ) ` X ) = ( F ` X ) ) $=
|
|
( wfn cin c0 wceq wcel wa w3a wfun cdm cun cfv fnfun 3ad2ant1 cima cuni csn
|
|
3ad2ant2 ineqan12d eqeq1d biimprd adantrd 3impia simp3r eleq2d mpbird funun
|
|
fndm wb wss ssun1 ax-mp sseli anim12i anasss 3impa funfvdm syl imaundir a1i
|
|
dmss unieqd disjel ndmima 3ad2ant3 uneq2d un0 syl6eq eqcomd adantrl 3adant2
|
|
wn eqtrd 3eqtrd syl112anc ) CAFZDBFZABGZHIZEAJZKZLZCMZDMZCNZDNZGZHIZEWIJZEC
|
|
DOZPZECPZIVTWAWGWEACQRWAVTWHWEBDQUBVTWAWEWLVTWAKZWCWLWDWQWLWCWQWKWBHVTWAWIA
|
|
WJBACULZBDULUCUDUEUFUGWFWMWDVTWAWCWDUHVTWAWMWDUMWEVTWIAEWRUIRUJWGWHWLWMKZLZ
|
|
WOWNEUAZSZTZCXASZTZWPWTWNMZEWNNZJZKZWOXCIWGWHWSXIWGWHKZWLWMXIXJWLKXFWMXHCDU
|
|
KWIXGECWNUNWIXGUNCDUOCWNVEUPUQURUSUTEWNVAVBWTXCXDDXASZOZTXEWTXBXLXBXLIWTCDX
|
|
AVCVDVFWTXLXDWTXLXDHOXDWTXKHXDWSWGXKHIZWHWSEWJJVPXMWIWJEVGEDVHVBVIVJXDVKVLV
|
|
FVQWGWSXEWPIZWHWGWMXNWLWGWMKWPXEECVAVMVNVOVRVS $.
|
|
|
|
$( The value of a union when the argument is in the second domain.
|
|
(Contributed by Scott Fenton, 29-Jun-2013.) $)
|
|
fvun2 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) ->
|
|
( ( F u. G ) ` X ) = ( G ` X ) ) $=
|
|
( wfn cin c0 wceq wcel w3a cun cfv uncom fveq1i incom eqeq1i anbi1i fvun1
|
|
wa syl3an3b 3com12 syl5eq ) CAFZDBFZABGZHIZEBJZTZKECDLZMEDCLZMZEDMZEUJUKCDN
|
|
OUEUDUIULUMIZUIUEUDBAGZHIZUHTUNUGUPUHUFUOHABPQRBADCESUAUBUC $.
|
|
|
|
${
|
|
$d x y A $. $d x y F $. $d x y G $. $d x X $.
|
|
$( Domains of a function composition. (Contributed by NM, 27-Jan-1997.) $)
|
|
dmfco $p |- ( ( Fun G /\ A e. dom G ) ->
|
|
( A e. dom ( F o. G ) <-> ( G ` A ) e. dom F ) ) $=
|
|
( vy vx wfun cdm wcel wa cfv cv cop wex ccom wceq cvv syl exbidv eldm2g
|
|
wb opeq1 eleq1d ceqsexgv eqcom funopfvb syl5bb anbi1d bitr3d vex opelco2g
|
|
funfvex mpan2 bitrd adantl 3bitr4rd ) CFZACGZHZIZACJZDKZLZBHZDMZAEKZLCHZV
|
|
EVALZBHZIZEMZDMZUTBGHZABCNZGHZUSVCVJDUSVEUTOZVHIZEMZVCVJUSUTPHZVQVCTACUKZ
|
|
VHVCEUTPVOVGVBBVEUTVAUAUBUCQUSVPVIEUSVOVFVHVOUTVEOUSVFVEUTUDAVECUEUFUGRUH
|
|
RUSVRVLVDTVSDUTBPSQURVNVKTUPURVNAVALVMHZDMVKDAVMUQSURVTVJDURVAPHVTVJTDUIE
|
|
AVABCUQPUJULRUMUNUO $.
|
|
|
|
$( Value of a function composition. Similar to second part of Theorem 3H
|
|
of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened
|
|
by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear,
|
|
16-Oct-2014.) $)
|
|
fvco2 $p |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) =
|
|
( F ` ( G ` X ) ) ) $=
|
|
( vx wfn wcel wa cv ccom csn cima cio cfv fnsnfv imaeq2d imaco dffv3g cvv
|
|
wceq syl6reqr eleq2d iotabidv adantl funfvex funfni syl 3eqtr4d ) CAFZDAG
|
|
ZHZEIZBCJZDKZLZGZEMZULBDCNZKZLZGZEMZDUMNZURBNZUKUPVAEUKUOUTULUKUTBCUNLZLU
|
|
OUKUSVEBADCOPBCUNQUAUBUCUJVCUQTUIEDUMARUDUKURSGZVDVBTVFADCDCUEUFEURBSRUGU
|
|
H $.
|
|
$}
|
|
|
|
$( Value of a function composition. Similar to Exercise 5 of [TakeutiZaring]
|
|
p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario
|
|
Carneiro, 26-Dec-2014.) $)
|
|
fvco $p |- ( ( Fun G /\ A e. dom G ) ->
|
|
( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) ) $=
|
|
( wfun cdm wfn wcel ccom cfv wceq funfn fvco2 sylanb ) CDCCEZFANGABCHIACIBI
|
|
JCKNBCALM $.
|
|
|
|
$( Value of a function composition. (Contributed by NM, 3-Jan-2004.)
|
|
(Revised by Mario Carneiro, 26-Dec-2014.) $)
|
|
fvco3 $p |- ( ( G : A --> B /\ C e. A ) ->
|
|
( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) ) $=
|
|
( wf wfn wcel ccom cfv wceq ffn fvco2 sylan ) ABEFEAGCAHCDEIJCEJDJKABELADEC
|
|
MN $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ch $.
|
|
fvopab3g.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
fvopab3g.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
fvopab3g.4 $e |- ( x e. C -> E! y ph ) $.
|
|
fvopab3g.5 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fvopab3g $p |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) ) $=
|
|
( wcel wa cv wceq wb adantr cop copab cfv anbi12d anbi2d opelopabg fnopab
|
|
eleq1 wfn fnopfvb mpan eleq2i syl6bb ibar 3bitr4d ) FHOZGIOZPFGUAZDQZHOZA
|
|
PZDEUBZOZUPCPZFJUCGRZCVAUPBPVDDEFGHIUSFRUTUPABUSFHUHKUDEQGRBCUPLUEUFUPVEV
|
|
CSUQUPVEURJOZVCJHUIUPVEVFSADEHJMNUGHFGJUJUKJVBURNULUMTUPCVDSUQUPCUNTUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ch $.
|
|
fvopab3ig.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
fvopab3ig.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
fvopab3ig.3 $e |- ( x e. C -> E* y ph ) $.
|
|
fvopab3ig.4 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by NM, 23-Oct-1999.) $)
|
|
fvopab3ig $p |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) ) $=
|
|
( wcel wa cv cfv wceq wi cop copab eleq1 anbi12d anbi2d opelopabg biimpar
|
|
exp43 pm2.43a imp fveq1i wfun funopab moanimv mpbir mpgbir funopfv syl5eq
|
|
wmo ax-mp syl6 ) FHOZGIOZPZCFGUADQZHOZAPZDEUBZOZFJRZGSVBVCCVITZVCVBVKVBVC
|
|
VBCVIVDVIVBCPZVGVBBPVLDEFGHIVEFSVFVBABVEFHUCKUDEQGSBCVBLUEUFUGUHUIUJVIVJF
|
|
VHRZGFJVHNUKVHULZVIVMGSTVNVGEUSZDVGDEUMVOVFAEUSTMVFAEUNUOUPFGVHUQUTURVA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $. $d y F $.
|
|
fvmptss2.1 $e |- ( x = D -> B = C ) $.
|
|
fvmptss2.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( A mapping always evaluates to a subset of the substituted expression in
|
|
the mapping, even if this is a proper class, or we are out of the
|
|
domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by
|
|
Mario Carneiro, 3-Jul-2019.) $)
|
|
fvmptss2 $p |- ( F ` D ) C_ C $=
|
|
( vy cv wbr wss wi cfv fvss cvv wcel nfcv wceq sylbi funmpt2 funrel ax-mp
|
|
wfun wrel brrelexi cmpt nfmpt1 nfcxfr nfbr nfv breq1 sseq2d imbi12d df-br
|
|
nfim cop copab opabid eqimss adantl df-mpt eqtri eleq2s vtoclgf mpcom mpg
|
|
wa ) EIJZFKZVIDLZMZEFNDLIIEDFOEPQVJVKEVIFFUDFUEABCFHUAFUBUCUFAJZVIFKZVICL
|
|
ZMVLAEPAERZVJVKAAEVIFVPAFABCUGZHABCUHUIAVIRUJVKAUKUPVMESZVNVJVOVKVMEVIFUL
|
|
VRCDVIGUMUNVNVMVIUQZFQVOVMVIFUOVOVSVMBQZVICSZVHZAIURZFVSWCQWBVOWBAIUSWAVO
|
|
VTVICUTVATFVQWCHAIBCVBVCVDTVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C y $. $d x D y $.
|
|
fvmptg.1 $e |- ( x = A -> B = C ) $.
|
|
fvmptg.2 $e |- F = ( x e. D |-> B ) $.
|
|
$( Value of a function given in maps-to notation. (Contributed by NM,
|
|
2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fvmptg $p |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C ) $=
|
|
( vy wcel wa wceq cfv eqid cv eqeq2d eqeq1 wmo moeq a1i cmpt copab df-mpt
|
|
eqtri fvopab3ig mpi ) BEKDFKLDDMZBGNDMDOJPZCMZUIDMUHAJBDEFGAPZBMCDUIHQUID
|
|
DRUJJSUKEKZJCTUAGAECUBULUJLAJUCIAJECUDUEUFUG $.
|
|
|
|
${
|
|
fvmpt.3 $e |- C e. _V $.
|
|
$( Value of a function given in maps-to notation. (Contributed by NM,
|
|
17-Aug-2011.) $)
|
|
fvmpt $p |- ( A e. D -> ( F ` A ) = C ) $=
|
|
( wcel cvv cfv wceq fvmptg mpan2 ) BEJDKJBFLDMIABCDEKFGHNO $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y C $.
|
|
fvmpts.1 $e |- F = ( x e. C |-> B ) $.
|
|
$( Value of a function given in maps-to notation, using explicit class
|
|
substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
fvmpts $p |- ( ( A e. C /\ [_ A / x ]_ B e. V ) ->
|
|
( F ` A ) = [_ A / x ]_ B ) $=
|
|
( vy cv csb csbeq1 cmpt nfcv nfcsb1v csbeq1a cbvmpt eqtri fvmptg ) HBAHIZ
|
|
CJZABCJDFEASBCKEADCLHDTLGAHDCTHCMASCNASCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $. $d x V $.
|
|
fvmpt3.a $e |- ( x = A -> B = C ) $.
|
|
fvmpt3.b $e |- F = ( x e. D |-> B ) $.
|
|
${
|
|
fvmpt3.c $e |- ( x e. D -> B e. V ) $.
|
|
$( Value of a function given in maps-to notation, with a slightly
|
|
different sethood condition. (Contributed by Stefan O'Rear,
|
|
30-Jan-2015.) $)
|
|
fvmpt3 $p |- ( A e. D -> ( F ` A ) = C ) $=
|
|
( wcel cfv wceq cv eleq1d vtoclga fvmptg mpdan ) BEKDGKZBFLDMCGKSABEANB
|
|
MCDGHOJPABCDEGFHIQR $.
|
|
$}
|
|
|
|
fvmpt3i.c $e |- B e. _V $.
|
|
$( Value of a function given in maps-to notation, with a slightly different
|
|
sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.) $)
|
|
fvmpt3i $p |- ( A e. D -> ( F ` A ) = C ) $=
|
|
( cvv wcel cv a1i fvmpt3 ) ABCDEFJGHCJKALEKIMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $. $d x ph $.
|
|
fvmptd.1 $e |- ( ph -> F = ( x e. D |-> B ) ) $.
|
|
fvmptd.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
fvmptd.3 $e |- ( ph -> A e. D ) $.
|
|
fvmptd.4 $e |- ( ph -> C e. V ) $.
|
|
$( Deduction version of ~ fvmpt . (Contributed by Scott Fenton,
|
|
18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fvmptd $p |- ( ph -> ( F ` A ) = C ) $=
|
|
( cfv cmpt csb fveq1d wcel wceq csbied eqeltrd eqid fvmpts syl2anc 3eqtrd
|
|
) ACGMCBFDNZMZBCDOZEACGUEIPACFQUGHQUFUGRKAUGEHABCDEFKJSZLTBCDFUEHUEUAUBUC
|
|
UHUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y D $. $d y F $. $d y V $.
|
|
fvmpt2.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( Value of a function given by the "maps to" notation. (Contributed by
|
|
FL, 21-Jun-2010.) $)
|
|
fvmpt2 $p |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B ) $=
|
|
( vy cv csb weq csbeq1 csbid syl6eq cmpt nfcv nfcsb1v cbvmpt eqtri fvmptg
|
|
csbeq1a ) GAHZAGHZCIZCBDEGAJUCAUACICAUBUACKACLMEABCNGBUCNFAGBCUCGCOAUBCPA
|
|
UBCTQRS $.
|
|
|
|
$d x y C $.
|
|
$( If all the values of the mapping are subsets of a class ` C ` , then so
|
|
is any evaluation of the mapping at a value in the domain of the
|
|
mapping. (Contributed by Jim Kingdon, 3-Jan-2018.) $)
|
|
fvmptssdm $p |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C ) $=
|
|
( vy wcel wss cfv wi cvv cv wceq fveq2 sseq1d imbi2d nfcv nfim cdm nfrab1
|
|
wral crab nfcri nfra1 cmpt nfmpt1 nfcxfr nffv nfss eleq1 imbi12d dmmpt wa
|
|
eleq2i rabeq2i fvmpt2 eqimss syl sylbi adantr dmmptss sseli rsp ex sylbir
|
|
mpan9 sstrd chvar vtoclga eleq2s imp ) EFUAZICDJZABUCZEFKZDJZVPVRLZECMIZA
|
|
BUDZVNVPHNZFKZDJZLZVSHEWAWBEOZWDVRVPWFWCVQDWBEFPQRANZWAIZVPWGFKZDJZLZLWBW
|
|
AIZWELAHWLWEAAHWAVTABUBUEVPWDAVOABUFAWCDAWBFAFABCUGGABCUHUIAWBSUJADSUKTTW
|
|
GWBOZWHWLWKWEWGWBWAULWMWJWDVPWMWIWCDWGWBFPQRUMWHWGVNIZWKVNWAWGABCFGUNZUPW
|
|
NVPWJWNVPUOWICDWNWICJZVPWNWGBIZVTUOZWPVTAVNBWOUQWRWICOWPABCMFGURWICUSUTVA
|
|
VBWNWQVPVOVNBWGABCFGVCVDVOABVEVHVIVFVGVJVKWOVLVM $.
|
|
|
|
$( Sufficient condition for a maps-to notation to be set-like.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
mptfvex $p |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V ) $=
|
|
( vy wcel wal wa cfv csb cvv cv cmpt nfel1 wceq eleq1d wss csbeq1 nfcsb1v
|
|
nfcv cbvmpt eqtri fvmptss2 elex alimi cbval sylib spcgv syl5 impcom ssexg
|
|
csbeq1a sylancr ) CFJZAKZDGJZLDEMZADCNZUAVBOJZVAOJIBAIPZCNZVBDEAVDDCUBZEA
|
|
BCQIBVEQHAIBCVEICUDZAVDCUCZAVDCUPZUEUFUGUTUSVCUSVEOJZIKZUTVCUSCOJZAKVKURV
|
|
LACFUHUIVLVJAIICOVGRAVEOVHRAPVDSCVEOVITUJUKVJVCIDGVDDSVEVBOVFTULUMUNVAVBO
|
|
UOUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
fvmpt2d.1 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
|
|
fvmpt2d.4 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
|
|
$( Deduction version of ~ fvmpt2 . (Contributed by Thierry Arnoux,
|
|
8-Dec-2016.) $)
|
|
fvmpt2d $p |- ( ( ph /\ x e. A ) -> ( F ` x ) = B ) $=
|
|
( cv wcel wa cfv cmpt wceq fveq1d adantr simpr eqid fvmpt2 syl2anc eqtrd
|
|
) ABIZCJZKZUBELZUBBCDMZLZDAUEUGNUCAUBEUFGOPUDUCDFJUGDNAUCQHBCDFUFUFRSTUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x D $. $d x ph $.
|
|
fvmptdf.1 $e |- ( ph -> A e. D ) $.
|
|
fvmptdf.2 $e |- ( ( ph /\ x = A ) -> B e. V ) $.
|
|
fvmptdf.3 $e |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) ) $.
|
|
${
|
|
fvmptdf.4 $e |- F/_ x F $.
|
|
fvmptdf.5 $e |- F/ x ps $.
|
|
$( Alternate deduction version of ~ fvmpt , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
fvmptdf $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
|
|
( cv wceq cmpt wi nfv wcel cfv nfmpt1 nfeq nfim cvv wex syl isset sylib
|
|
elex fveq1 simpr fveq2d adantr eqeltrd eqid fvmpt2 eqtr3d eqeq2d sylbid
|
|
wa syl2anc syl5 exlimdd ) ACNZDOZGCFEPZOZBQCACRVGBCCGVFLCFEUAUBMUCADUDS
|
|
ZVECUEADFSZVHIDFUIUFCDUGUHVGDGTZDVFTZOZAVEUTZBDGVFUJVMVLVJEOBVMVKEVJVMV
|
|
DVFTZVKEVMVDDVFAVEUKZULVMVDFSEHSVNEOVMVDDFVOAVIVEIUMUNJCFEHVFVFUOUPVAUQ
|
|
URKUSVBVC $.
|
|
$}
|
|
|
|
$d x F $. $d x ps $.
|
|
$( Alternate deduction version of ~ fvmpt , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
fvmptdv $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
|
|
( nfcv nfv fvmptdf ) ABCDEFGHIJKCGLBCMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $. $d x ph $.
|
|
fvmptdv2.1 $e |- ( ph -> A e. D ) $.
|
|
fvmptdv2.2 $e |- ( ( ph /\ x = A ) -> B e. V ) $.
|
|
fvmptdv2.3 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
$( Alternate deduction version of ~ fvmpt , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
fvmptdv2 $p |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) ) $=
|
|
( cfv wceq cmpt cvv eqidd cv wcel elex syl wex isset wa eqeltrrd exlimddv
|
|
sylib fvmptd fveq1 eqeq1d syl5ibrcom ) ACGLZEMGBFDNZMZCULLZEMABCDEFULOAUL
|
|
PKIABQCMZEORBACORZUOBUAACFRUPICFSTBCUBUFAUOUCZDEOKUQDHRDORJDHSTUDUEUGUMUK
|
|
UNECGULUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bidirectional equality theorem for a mapping abstraction. Equivalent to
|
|
~ eqfnfv . (Contributed by Mario Carneiro, 14-Nov-2014.) $)
|
|
mpteqb $p |- ( A. x e. A B e. V ->
|
|
( ( x e. A |-> B ) = ( x e. A |-> C ) <->
|
|
A. x e. A B = C ) ) $=
|
|
( wcel wral cvv cmpt wceq wb wfn eqid mptfng wa wi nfmpt1 cfv fvmpt2 syl
|
|
elex ralimi fneq1 3bitr4g biimpd r19.26 nfeq cv simpll ad2ant2lr ad2ant2l
|
|
fveq1d 3eqtr3d exp31 ralrimi ralim syl5bir expd mpdd mpteq12 mpan impbid1
|
|
com12 ) CEFZABGCHFZABGZABCIZABDIZJZCDJZABGZKVDVEABCEUAUBVFVIVKVIVFVKVIVFD
|
|
HFZABGZVKVIVFVMVIVGBLVHBLVFVMBVGVHUCABCVGVGMZNABDVHVHMZNUDUEVIVFVMVKVFVMO
|
|
VEVLOZABGZVIVKVEVLABUFVIVPVJPZABGVQVKPVIVRABAVGVHABCQABDQUGVIAUHZBFZVPVJV
|
|
IVTOVPOZVSVGRZVSVHRZCDWAVSVGVHVIVTVPUIULVTVEWBCJVIVLABCHVGVNSUJVTVLWCDJVI
|
|
VEABDHVHVOSUKUMUNUOVPVJABUPTUQURUSVCBBJVKVIBMABCBDUTVAVBT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $.
|
|
$( Closed theorem form of ~ fvmpt . (Contributed by Scott Fenton,
|
|
21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.) $)
|
|
fvmptt $p |- ( ( A. x ( x = A -> B = C ) /\
|
|
F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C ) $=
|
|
( cv wceq wi wal cmpt wcel wa w3a cfv simp2 fveq1d wrex cvv elex nfa1 nfv
|
|
risset nfeq1 nfim simprl simplr simprr eqeltrd eqid fvmpt2 syl2anc simpll
|
|
nffvmpt1 fveq2d 3eqtr3d exp43 a2i com23 rexlimd syl7 syl5bi imp32 3adant2
|
|
sps eqtrd ) AHZBIZCDIZJZAKZFAECLZIZBEMZDGMZNZOZBFPBVMPZDVRBFVMVLVNVQQRVLV
|
|
QVSDIZVNVLVOVPVTVOVIAESZVLVPVTJABEUDVPDTMZVLWAVTDGUAVLVIWBVTJZAEVKAUBWBVT
|
|
AWBAUCAVSDAECBUOUEUFVKVHEMZVIWCJJAVKVIWDWCVIVJWDWCJVIVJWDWBVTVIVJNZWDWBNZ
|
|
NZVHVMPZCVSDWGWDCTMWHCIWEWDWBUGWGCDTVIVJWFUHZWEWDWBUIUJAECTVMVMUKULUMWGVH
|
|
BVMVIVJWFUNUPWIUQURUSUTVFVAVBVCVDVEVG $.
|
|
$}
|
|
|
|
${
|
|
$d x D $.
|
|
fvmptf.1 $e |- F/_ x A $.
|
|
fvmptf.2 $e |- F/_ x C $.
|
|
fvmptf.3 $e |- ( x = A -> B = C ) $.
|
|
fvmptf.4 $e |- F = ( x e. D |-> B ) $.
|
|
$( Value of a function given by an ordered-pair class abstraction. This
|
|
version of ~ fvmptg uses bound-variable hypotheses instead of distinct
|
|
variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
fvmptf $p |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C ) $=
|
|
( wcel cfv wceq cvv elex cv wi nfel1 cmpt nfmpt1 nfcxfr nffv eleq1d fveq2
|
|
nfeq nfim eqeq12d imbi12d fvmpt2 ex vtoclgaf syl5 imp ) BELZDGLZBFMZDNZUP
|
|
DOLZUOURDGPCOLZAQZFMZCNZRUSURRABEHUSURAADOISAUQDABFAFAECTKAECUAUBHUCIUFUG
|
|
VABNZUTUSVCURVDCDOJUDVDVBUQCDVABFUEJUHUIVAELUTVCAECOFKUJUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d ps x y $. $d B y $. $d C x y $.
|
|
fvopab6.1 $e |- F = { <. x , y >. | ( ph /\ y = B ) } $.
|
|
fvopab6.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
fvopab6.3 $e |- ( x = A -> B = C ) $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro,
|
|
11-Sep-2015.) $)
|
|
fvopab6 $p |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C ) $=
|
|
( wcel cfv wceq cvv cv wa copab elex eqeq2d anbi12d iba bicomd moeq moani
|
|
wi wmo a1i vex biantrur opabbii eqtri fvopab3ig sylan 3impia ) EHNZGINZBE
|
|
JOGPZUREQNUSBUTUHEHUAADRZFPZSZBVAGPZSZBCDEGQIJCRZEPZABVBVDLVGFGVAMUBUCVDB
|
|
VEVDBUDUEVCDUIVFQNZVBADDFUFUGUJJVCCDTVHVCSZCDTKVCVICDVHVCCUKULUMUNUOUPUQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $. $d x G $. $d x ph $.
|
|
$( Equality of functions is determined by their values. Special case of
|
|
Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|
|
(Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) $)
|
|
eqfnfv $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
|
|
A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
|
|
( wfn wa wceq cv cfv cmpt wral dffn5im eqeqan12d wb cvv funfvex ralrimiva
|
|
wcel funfni mpteqb syl adantr bitrd ) CBEZDBEZFCDGABAHZCIZJZABUFDIZJZGZUG
|
|
UIGABKZUDUECUHDUJABCLABDLMUDUKULNZUEUDUGORZABKUMUDUNABUNBUFCUFCPSQABUGUIO
|
|
TUAUBUC $.
|
|
|
|
$( Equality of functions is determined by their values. Exercise 4 of
|
|
[TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
eqfnfv2 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <->
|
|
( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) ) $=
|
|
( wfn wa wceq cv cfv wral dmeq fndm eqeqan12d syl5ib pm4.71rd wb biimparc
|
|
cdm fneq2 eqfnfv sylan2 anassrs pm5.32da bitrd ) DBFZECFZGZDEHZBCHZUIGUJA
|
|
IZDJUKEJHABKZGUHUIUJUIDSZESZHUHUJDELUFUGUMBUNCBDMCEMNOPUHUJUIULUFUGUJUIUL
|
|
QZUGUJGUFEBFZUOUJUPUGBCETRABDEUAUBUCUDUE $.
|
|
|
|
$d x B $.
|
|
$( Derive equality of functions from equality of their values.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
eqfnfv3 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A
|
|
( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) ) $=
|
|
( wfn wa wceq cv cfv wral wss wcel eqfnfv2 ancom bitri anbi1i anass dfss3
|
|
eqss r19.26 bitr4i anbi2i syl6bb ) DBFECFGDEHBCHZAIZDJUFEJHZABKZGZCBLZUFC
|
|
MZUGGABKZGZABCDENUIUJBCLZGZUHGZUMUEUOUHUEUNUJGUOBCTUNUJOPQUPUJUNUHGZGUMUJ
|
|
UNUHRUQULUJUQUKABKZUHGULUNURUHABCSQUKUGABUAUBUCPPUD $.
|
|
|
|
eqfnfvd.1 $e |- ( ph -> F Fn A ) $.
|
|
eqfnfvd.2 $e |- ( ph -> G Fn A ) $.
|
|
eqfnfvd.3 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) ) $.
|
|
$( Deduction for equality of functions. (Contributed by Mario Carneiro,
|
|
24-Jul-2014.) $)
|
|
eqfnfvd $p |- ( ph -> F = G ) $=
|
|
( wceq cv cfv wral ralrimiva wfn wb eqfnfv syl2anc mpbird ) ADEIZBJZDKTEK
|
|
IZBCLZAUABCHMADCNECNSUBOFGBCDEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d z F $. $d z G $.
|
|
eqfnfv2f.1 $e |- F/_ x F $.
|
|
eqfnfv2f.2 $e |- F/_ x G $.
|
|
$( Equality of functions is determined by their values. Special case of
|
|
Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|
|
This version of ~ eqfnfv uses bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 29-Jan-2004.) $)
|
|
eqfnfv2f $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
|
|
A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
|
|
( vz wfn wa wceq cv cfv wral eqfnfv nfcv nffv nfeq nfv fveq2 eqeq12d
|
|
cbvral syl6bb ) CBHDBHICDJGKZCLZUCDLZJZGBMAKZCLZUGDLZJZABMGBCDNUFUJGABAUD
|
|
UEAUCCEAUCOZPAUCDFUKPQUJGRUCUGJUDUHUEUIUCUGCSUCUGDSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d G x $.
|
|
$( Equality of functions is determined by their values. (Contributed by
|
|
Scott Fenton, 19-Jun-2011.) $)
|
|
eqfunfv $p |- ( ( Fun F /\ Fun G ) -> ( F = G <->
|
|
( dom F = dom G /\
|
|
A. x e. dom F ( F ` x ) = ( G ` x ) ) ) ) $=
|
|
( wfun cdm wfn wceq cv cfv wral wa wb funfn eqfnfv2 syl2anb ) BDBBEZFCCEZ
|
|
FBCGPQGAHZBIRCIGAPJKLCDBMCMAPQBCNO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x F $. $d x G $.
|
|
$( Equality of restricted functions is determined by their values.
|
|
(Contributed by NM, 3-Aug-1994.) $)
|
|
fvreseq $p |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) ->
|
|
( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) $=
|
|
( wfn wa wss cres wceq cv cfv wral wb fnssres anim12i anandirs wcel fvres
|
|
eqfnfv eqeq12d ralbiia syl6bb syl ) DBFZEBFZGCBHZGDCIZCFZECIZCFZGZUHUJJZA
|
|
KZDLZUNELZJZACMZNUEUFUGULUEUGGUIUFUGGUKBCDOBCEOPQULUMUNUHLZUNUJLZJZACMURA
|
|
CUHUJTVAUQACUNCRUSUOUTUPUNCDSUNCESUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d G x y $. $d A x y $.
|
|
$( Two ways to express the locus of differences between two functions.
|
|
(Contributed by Stefan O'Rear, 17-Jan-2015.) $)
|
|
fndmdif $p |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) =
|
|
{ x e. A | ( F ` x ) =/= ( G ` x ) } ) $=
|
|
( vy wfn wa cdm cv cfv wss wceq wcel wbr wex wn wb eqcom fnbrfvb syl5bb
|
|
cdif cin wne crab difss dmss ax-mp fndm adantr syl5sseq dfss1 vex adantll
|
|
sylib eldm necon3abid funfvex funfni adantlr breq2 notbid ceqsexgv anbi1d
|
|
cvv syl brdif syl6bbr exbidv 3bitr2rd rabbi2dva eqtr3d ) CBFZDBFZGZBCDUAZ
|
|
HZUBZVPAIZCJZVRDJZUCZABUDVNVPBKVQVPLVNCHZVPBVOCKVPWBKCDUEVOCUFUGVLWBBLVMB
|
|
CUHUIUJVPBUKUNVNWAABVPVRVPMVREIZVONZEOZVNVRBMZGZWAEVRVOAULUOWGWAVRVSDNZPZ
|
|
WCVSLZVRWCDNZPZGZEOZWEWGWHVSVTVMWFVSVTLZWHQVLWOVTVSLVMWFGWHVSVTRBVRVSDSTU
|
|
MUPWGVSVDMZWNWIQVLWFWPVMWPBVRCVRCUQURUSWLWIEVSVDWJWKWHWCVSVRDUTVAVBVEWGWM
|
|
WDEWGWMVRWCCNZWLGWDWGWJWQWLVLWFWJWQQVMWJVSWCLVLWFGWQWCVSRBVRWCCSTUSVCVRWC
|
|
CDVFVGVHVITVJVK $.
|
|
|
|
$( The difference set between two functions is commutative. (Contributed
|
|
by Stefan O'Rear, 17-Jan-2015.) $)
|
|
fndmdifcom $p |- ( ( F Fn A /\ G Fn A ) ->
|
|
dom ( F \ G ) = dom ( G \ F ) ) $=
|
|
( vx wfn wa cv cfv wne crab cdif cdm wb wcel necom rabbiia fndmdif ancoms
|
|
a1i wceq 3eqtr4a ) BAEZCAEZFDGZBHZUDCHZIZDAJUFUEIZDAJZBCKLCBKLZUGUHDAUGUH
|
|
MUDANUEUFOSPDABCQUCUBUJUITDACBQRUA $.
|
|
|
|
$( Two ways to express the locus of equality between two functions.
|
|
(Contributed by Stefan O'Rear, 17-Jan-2015.) $)
|
|
fndmin $p |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) =
|
|
{ x e. A | ( F ` x ) = ( G ` x ) } ) $=
|
|
( vy wfn wa cin cdm cv wcel cfv wceq copab cmpt dffn5im df-mpt syl6eq wex
|
|
cab crab ineqan12d inopab dmeqd anandi exbii 19.42v bitr3i wb cvv funfvex
|
|
wfun eqeq1 ceqsexgv syl funfni syl5bb abbidv dmopab df-rab 3eqtr4g adantr
|
|
pm5.32da eqtrd ) CBFZDBFZGZCDHZIAJZBKZEJZVICLZMZGZVJVKVIDLZMZGZGZAENZIZVL
|
|
VOMZABUAZVGVHVSVGVHVNAENZVQAENZHVSVEVFCWCDWDVECABVLOWCABCPAEBVLQRVFDABVOO
|
|
WDABDPAEBVOQRUBVNVQAEUCRUDVEVTWBMVFVEVRESZATVJWAGZATVTWBVEWEWFAWEVJVMVPGZ
|
|
ESZGZVEWFWEVJWGGZESWIWJVREVJVMVPUEUFVJWGEUGUHVEVJWHWAWHWAUIZBVICCULVICIKG
|
|
VLUJKWKVICUKVPWAEVLUJVKVLVOUMUNUOUPVCUQURVRAEUSWAABUTVAVBVD $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d G x $. $d A x $.
|
|
$( Two functions are equal iff their equalizer is the whole domain.
|
|
(Contributed by Stefan O'Rear, 7-Mar-2015.) $)
|
|
fneqeql $p |- ( ( F Fn A /\ G Fn A ) ->
|
|
( F = G <-> dom ( F i^i G ) = A ) ) $=
|
|
( vx wfn wa wceq cfv crab cin cdm wral eqfnfv eqcom rabid2 syl6bbr fndmin
|
|
cv bitri eqeq1d bitr4d ) BAECAEFZBCGZDRZBHUDCHGZDAIZAGZBCJKZAGUBUCUEDALZU
|
|
GDABCMUGAUFGUIUFANUEDAOSPUBUHUFADABCQTUA $.
|
|
|
|
$( Two functions are equal iff their equalizer contains the whole domain.
|
|
(Contributed by Stefan O'Rear, 9-Mar-2015.) $)
|
|
fneqeql2 $p |- ( ( F Fn A /\ G Fn A ) ->
|
|
( F = G <-> A C_ dom ( F i^i G ) ) ) $=
|
|
( wfn wceq cin cdm wss fneqeql inss1 dmss ax-mp adantr syl5sseq biantrurd
|
|
wa fndm eqss syl6rbbr bitrd ) BADZCADZPZBCEBCFZGZAEZAUEHZABCIUCUGUEAHZUGP
|
|
UFUCUHUGUCBGZUEAUDBHUEUIHBCJUDBKLUAUIAEUBABQMNOUEARST $.
|
|
|
|
$( Two functions are equal on a subset iff their equalizer contains that
|
|
subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) $)
|
|
fnreseql $p |- ( ( F Fn A /\ G Fn A /\ X C_ A ) ->
|
|
( ( F |` X ) = ( G |` X ) <-> X C_ dom ( F i^i G ) ) ) $=
|
|
( wfn wss w3a cin cdm wb fnssres 3adant2 3adant1 fneqeql syl2anc resindir
|
|
cres wceq dmeqi dmres eqtr3i eqeq1i df-ss bitr4i syl6bb ) BAEZCAEZDAFZGZB
|
|
DQZCDQZRZUJUKHZIZDRZDBCHZIZFZUIUJDEZUKDEZULUOJUFUHUSUGADBKLUGUHUTUFADCKMD
|
|
UJUKNOUODUQHZDRURUNVADUPDQZIUNVAVBUMBCDPSUPDTUAUBDUQUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( The range of a choice function (a function that chooses an element from
|
|
each member of its domain) is included in the union of its domain.
|
|
(Contributed by NM, 31-Aug-1999.) $)
|
|
chfnrn $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A ) $=
|
|
( vy wfn cv cfv wcel wral wa crn cuni wrex wceq fvelrnb biimpd wi biimpcd
|
|
eleq1 ralimi rexim syl sylan9 eluni2 syl6ibr ssrdv ) CBEZAFZCGZUHHZABIZJZ
|
|
DCKZBLZULDFZUMHZUOUHHZABMZUOUNHUGUPUIUONZABMZUKURUGUPUTABUOCOPUKUSUQQZABI
|
|
UTURQUJVAABUSUJUQUIUOUHSRTUSUQABUAUBUCAUOBUDUEUF $.
|
|
$}
|
|
|
|
$( Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1]
|
|
p. 41. (Contributed by NM, 14-Oct-1996.) $)
|
|
funfvop $p |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F ) $=
|
|
( wfun cdm wcel wa cfv wceq cop eqid funopfvb mpbii ) BCABDEFABGZMHAMIBEMJA
|
|
MBKL $.
|
|
|
|
$( Two ways to say that ` A ` is in the domain of ` F ` . (Contributed by
|
|
Mario Carneiro, 1-May-2014.) $)
|
|
funfvbrb $p |- ( Fun F -> ( A e. dom F <-> A F ( F ` A ) ) ) $=
|
|
( wfun cdm wcel cfv wbr wa funfvop df-br sylibr wrel funrel releldm impbida
|
|
cop sylan ) BCZABDEZAABFZBGZRSHATPBEUAABIATBJKRBLUASBMATBNQO $.
|
|
|
|
$( A member of a preimage is a function value argument. (Contributed by NM,
|
|
4-May-2007.) $)
|
|
fvimacnvi $p |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B ) $=
|
|
( wfun ccnv cima wcel wa cfv csn wss snssi funimass2 sylan2 cdm wb cnvimass
|
|
sseli cvv funfvex snssg syl wceq funfn fnsnfv sylanb sseq1d bitrd mpbird
|
|
wfn ) CDZACEBFZGZHZACIZBGZCAJZFZBKZUMUKUQULKUSAULLUQBCMNUNUPUOJZBKZUSUMUKAC
|
|
OZGZUPVAPZULVBACBQRZUKVCHUOSGVDACTUOBSUAUBNUNUTURBUMUKVCUTURUCZVEUKCVBUJVCV
|
|
FCUDVBACUEUFNUGUHUI $.
|
|
|
|
$( The argument of a function value belongs to the preimage of any class
|
|
containing the function value. Raph Levien remarks: "This proof is
|
|
unsatisfying, because it seems to me that ~ funimass2 could probably be
|
|
strengthened to a biconditional." (Contributed by Raph Levien,
|
|
20-Nov-2006.) $)
|
|
fvimacnv $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( ( F ` A ) e. B <-> A e. ( `' F " B ) ) ) $=
|
|
( wfun cdm wcel wa cfv ccnv cima csn cop funfvop cvv wb sylancom mpbird wss
|
|
wi ex funfvex opelcnvg elimasng snssg syl imass2 syl6bi imp sseld fvimacnvi
|
|
mpid adantr impbid ) CDZACEZFZGZACHZBFZACIZBJZFZUQUSAUTURKZJZFZVBUQVEURALUT
|
|
FZUQVFAURLCFZACMUNUPURNFZVFVGOACUAZURANUOCUBPQUNUPVHVEVFOVIUTURANUOUCPQUQUS
|
|
VEVBSUQUSGVDVAAUQUSVDVARZUQUSVCBRZVJUQVHUSVKOVIURBNUDUEVCBUTUFUGUHUITUKUNVB
|
|
USSUPUNVBUSABCUJTULUM $.
|
|
|
|
${
|
|
$d F x $. $d A x $. $d B x $.
|
|
$( A kind of contraposition law that infers an image subclass from a
|
|
subclass of a preimage. Raph Levien remarks: "Likely this could be
|
|
proved directly, and ~ fvimacnv would be the special case of ` A ` being
|
|
a singleton, but it works this way round too." (Contributed by Raph
|
|
Levien, 20-Nov-2006.) $)
|
|
funimass3 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) ) $=
|
|
( vx wfun cdm wss wa cima cv ccnv wcel wral funimass4 wb ssel fvimacnv ex
|
|
cfv syl9r imp31 ralbidva bitrd dfss3 syl6bbr ) CEZACFZGZHZCAIBGZDJZCKBIZL
|
|
ZDAMZAULGUIUJUKCSBLZDAMUNDABCNUIUOUMDAUFUHUKALZUOUMOZUHUPUKUGLZUFUQAUGUKP
|
|
UFURUQUKBCQRTUAUBUCDAULUDUE $.
|
|
|
|
$( A subclass of a preimage in terms of function values. (Contributed by
|
|
NM, 15-May-2007.) $)
|
|
funimass5 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( A C_ ( `' F " B ) <-> A. x e. A ( F ` x ) e. B ) ) $=
|
|
( wfun cdm wss wa cima ccnv cv cfv wcel wral funimass3 funimass4 bitr3d )
|
|
DEBDFGHDBICGBDJCIGAKDLCMABNBCDOABCDPQ $.
|
|
|
|
$( Two ways of specifying that a function is constant on a subdomain.
|
|
(Contributed by NM, 8-Mar-2007.) $)
|
|
funconstss $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) ) $=
|
|
( wfun cdm wss wa cima csn cv cfv wcel wral ccnv wceq funimass4 funimass3
|
|
cvv wb ssel2 anim2i anassrs funfvex elsng 3syl ralbidva 3bitr3rd ) DEZBDF
|
|
ZGZHZDBICJZGAKZDLZUMMZABNBDOUMIGUOCPZABNABUMDQBUMDRULUPUQABULUNBMZHUIUNUJ
|
|
MZHZUOSMUPUQTUIUKURUTUKURHUSUIBUJUNUAUBUCUNDUDUOCSUEUFUGUH $.
|
|
$}
|
|
|
|
$( Membership in the preimage of a set under a function. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
elpreima $p |- ( F Fn A -> ( B e. ( `' F " C )
|
|
<-> ( B e. A /\ ( F ` B ) e. C ) ) ) $=
|
|
( wfn ccnv cima wcel cfv wa cdm cnvimass sseli fndm eleq2d syl5ib fvimacnvi
|
|
wfun fnfun sylan ex jcad wb fvimacnv funfni biimpd expimpd impbid ) DAEZBDF
|
|
CGZHZBAHZBDICHZJUIUKULUMUKBDKZHUIULUJUNBDCLMUIUNABADNOPUIUKUMUIDRUKUMADSBCD
|
|
QTUAUBUIULUMUKUIULJUMUKUMUKUCABDBCDUDUEUFUGUH $.
|
|
|
|
$( Membership in the preimage of a singleton, under a function. (Contributed
|
|
by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fniniseg $p |- ( F Fn A -> ( C e. ( `' F " { B } ) <->
|
|
( C e. A /\ ( F ` C ) = B ) ) ) $=
|
|
( wfn ccnv csn cima wcel cfv wa wceq elpreima wb wfun cdm cvv funfvex elsng
|
|
syl funfni pm5.32da bitrd ) DAEZCDFBGZHICAIZCDJZUEIZKUFUGBLZKACUEDMUDUFUHUI
|
|
UHUINZACDDOCDPIKUGQIUJCDRUGBQSTUAUBUC $.
|
|
|
|
${
|
|
$d x A $. $d x F $. $d x B $.
|
|
$( Inverse images under functions expressed as abstractions. (Contributed
|
|
by Stefan O'Rear, 1-Feb-2015.) $)
|
|
fncnvima2 $p |- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } ) $=
|
|
( wfn ccnv cima cv wcel cfv wa cab crab elpreima abbi2dv df-rab syl6eqr )
|
|
DBEZDFCGZAHZBITDJCIZKZALUAABMRUBASBTCDNOUAABPQ $.
|
|
|
|
$( Inverse point images under functions expressed as abstractions.
|
|
(Contributed by Stefan O'Rear, 1-Feb-2015.) $)
|
|
fniniseg2 $p |- ( F Fn A -> ( `' F " { B } ) =
|
|
{ x e. A | ( F ` x ) = B } ) $=
|
|
( wfn ccnv csn cima cv cfv wcel crab wceq fncnvima2 wb cdm wa cvv funfvex
|
|
wfun elsng syl funfni rabbidva eqtrd ) DBEZDFCGZHAIZDJZUGKZABLUICMZABLABU
|
|
GDNUFUJUKABUJUKOZBUHDDTUHDPKQUIRKULUHDSUICRUAUBUCUDUE $.
|
|
|
|
$( Support sets of functions expressed as abstractions. (Contributed by
|
|
Stefan O'Rear, 1-Feb-2015.) $)
|
|
fnniniseg2 $p |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) =
|
|
{ x e. A | ( F ` x ) =/= B } ) $=
|
|
( wfn ccnv cvv csn cdif cima cv cfv wcel crab fncnvima2 wa funfvex funfni
|
|
wne biantrurd eldifsn syl6rbbr rabbidva eqtrd ) DBEZDFGCHIZJAKZDLZUFMZABN
|
|
UHCSZABNABUFDOUEUIUJABUEUGBMPZUJUHGMZUJPUIUKULUJULBUGDUGDQRTUHGCUAUBUCUD
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d A x $.
|
|
$( Existential quantification restricted to a support. (Contributed by
|
|
Stefan O'Rear, 23-Mar-2015.) $)
|
|
rexsupp $p |- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <->
|
|
E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) ) $=
|
|
( wfn cv cfv wne wa ccnv cvv csn cdif cima wcel elpreima funfni biantrurd
|
|
funfvex eldifsn syl6rbbr pm5.32da bitrd anbi1d anass syl6bb rexbidv2 ) DC
|
|
FZABGZDHZEIZAJZBDKLEMNZOZCUIUJUOPZAJUJCPZULJZAJUQUMJUIUPURAUIUPUQUKUNPZJU
|
|
RCUJUNDQUIUQUSULUIUQJZULUKLPZULJUSUTVAULVACUJDUJDTRSUKLEUAUBUCUDUEUQULAUF
|
|
UGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
$( Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
unpreima $p |- ( Fun F -> ( `' F " ( A u. B ) )
|
|
= ( ( `' F " A ) u. ( `' F " B ) ) ) $=
|
|
( vx wfun cdm wfn ccnv cun cima wceq funfn cv wcel wa elpreima wo orbi12d
|
|
cfv elun syl5bb anbi2i andi bitri syl6rbbr bitrd eqrdv sylbi ) CECCFZGZCH
|
|
ZABIZJZUKAJZUKBJZIZKCLUJDUMUPUJDMZUMNUQUINZUQCSZULNZOZUQUPNZUIUQULCPUJVBU
|
|
RUSANZOZURUSBNZOZQZVAVBUQUNNZUQUONZQUJVGUQUNUOTUJVHVDVIVFUIUQACPUIUQBCPRU
|
|
AVAURVCVEQZOVGUTVJURUSABTUBURVCVEUCUDUEUFUGUH $.
|
|
|
|
$( Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.)
|
|
(Proof shortened by Mario Carneiro, 14-Jun-2016.) $)
|
|
inpreima $p |- ( Fun F -> ( `' F " ( A i^i B ) )
|
|
= ( ( `' F " A ) i^i ( `' F " B ) ) ) $=
|
|
( wfun ccnv cin cima wceq funcnvcnv imain syl ) CDCEZEDLABFGLAGLBGFHCIABL
|
|
JK $.
|
|
|
|
$( Preimage of a difference. (Contributed by Mario Carneiro,
|
|
14-Jun-2016.) $)
|
|
difpreima $p |- ( Fun F -> ( `' F " ( A \ B ) )
|
|
= ( ( `' F " A ) \ ( `' F " B ) ) ) $=
|
|
( wfun ccnv cdif cima wceq funcnvcnv imadif syl ) CDCEZEDLABFGLAGLBGFHCIA
|
|
BLJK $.
|
|
|
|
$( The preimage of a restricted function. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
respreima $p |- ( Fun F -> ( `' ( F |` B ) " A )
|
|
= ( ( `' F " A ) i^i B ) ) $=
|
|
( vx wfun cres ccnv cima cin cdm wfn cv wcel wb cfv wa bitri syl elpreima
|
|
elin funfn ancom anbi1i fvres eleq1d adantl pm5.32i a1i an32 syl6bb fnfun
|
|
wceq funres dmres jctir df-fn sylibr anbi1d syl5bb 3bitr4d sylbi eqrdv )
|
|
CEZDCBFZGAHZCGAHZBIZVCCCJZKZDLZVEMZVJVGMZNCUAVIVJBVHIZMZVJVDOZAMZPZVJVHMZ
|
|
VJCOZAMZPZVJBMZPZVKVLVIVQVRWBPZVTPZWCVQWENVIVQWDVPPWEVNWDVPVNWBVRPWDVJBVH
|
|
TWBVRUBQUCWDVPVTWBVPVTNVRWBVOVSAVJBCUDUEUFUGQUHVRWBVTUIUJVIVDVMKZVKVQNVIV
|
|
DEZVDJVMULZPWFVIWGWHVIVCWGVHCUKBCUMRCBUNUOVDVMUPUQVMVJAVDSRVLVJVFMZWBPVIW
|
|
CVJVFBTVIWIWAWBVHVJACSURUSUTVAVB $.
|
|
$}
|
|
|
|
$( The preimage of the codomain of a mapping is the mapping's domain.
|
|
(Contributed by FL, 25-Jan-2007.) $)
|
|
fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
|
|
( wf ccnv cima crn imassrn cdm dfdm4 fdm ssid syl6eqss syl5eqssr syl5ss wss
|
|
frn wfun wb ffun syl5sseqr funimass3 syl2anc mpbid eqssd ) ABCDZCEZBFZAUFUH
|
|
UGGZAUGBHUFUICIZACJUFUJAAABCKZALZMNOUFCAFZBPZAUHPZUFUMCGBCAHABCQOUFCRAUJPUN
|
|
UOSABCTUFAAUJULUKUAABCUBUCUDUE $.
|
|
|
|
$( Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1]
|
|
p. 41. (Contributed by NM, 30-Sep-2004.) $)
|
|
fnopfv $p |- ( ( F Fn A /\ B e. A ) -> <. B , ( F ` B ) >. e. F ) $=
|
|
( cfv cop wcel funfvop funfni ) BBCDECFABCBCGH $.
|
|
|
|
${
|
|
$d x y F $. $d x A $.
|
|
$( A function's value belongs to its range. (Contributed by NM,
|
|
14-Oct-1996.) $)
|
|
fvelrn $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) $=
|
|
( vx vy wfun cdm wcel cfv crn cv wa wi wceq eleq1 anbi2d fveq2 eleq1d cop
|
|
syl cvv imbi12d wex funfvop vex opeq1 spcev funfvex elrn2g mpbird anabsi7
|
|
wb vtoclg ) BEZABFZGZABHZBIZGZUMCJZUNGZKZUSBHZUQGZLUMUOKZURLCAUNUSAMZVAVD
|
|
VCURVEUTUOUMUSAUNNOVEVBUPUQUSABPQUAVAVCDJZVBRZBGZDUBZVAUSVBRZBGZVIUSBUCVH
|
|
VKDUSCUDVFUSMVGVJBVFUSVBUEQUFSVAVBTGVCVIUKUSBUGDVBBTUHSUIULUJ $.
|
|
$}
|
|
|
|
$( A function's value belongs to its range. (Contributed by NM,
|
|
15-Oct-1996.) $)
|
|
fnfvelrn $p |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F ) $=
|
|
( cfv crn wcel fvelrn funfni ) BCDCEFABCBCGH $.
|
|
|
|
$( A function's value belongs to its codomain. (Contributed by NM,
|
|
12-Aug-1999.) $)
|
|
ffvelrn $p |- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B ) $=
|
|
( wf wcel wa cfv crn wfn ffn fnfvelrn sylan wi frn sseld adantr mpd ) ABDEZ
|
|
CAFZGCDHZDIZFZUABFZSDAJTUCABDKACDLMSUCUDNTSUBBUAABDOPQR $.
|
|
|
|
${
|
|
ffvrni.1 $e |- F : A --> B $.
|
|
$( A function's value belongs to its codomain. (Contributed by NM,
|
|
6-Apr-2005.) $)
|
|
ffvelrni $p |- ( C e. A -> ( F ` C ) e. B ) $=
|
|
( wf wcel cfv ffvelrn mpan ) ABDFCAGCDHBGEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
ffvelrnd.1 $e |- ( ph -> F : A --> B ) $.
|
|
$( A function's value belongs to its codomain. (Contributed by Mario
|
|
Carneiro, 29-Dec-2016.) $)
|
|
ffvelrnda $p |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) $=
|
|
( wf wcel cfv ffvelrn sylan ) ABCEGDBHDEICHFBCDEJK $.
|
|
|
|
ffvelrnd.2 $e |- ( ph -> C e. A ) $.
|
|
$( A function's value belongs to its codomain. (Contributed by Mario
|
|
Carneiro, 29-Dec-2016.) $)
|
|
ffvelrnd $p |- ( ph -> ( F ` C ) e. B ) $=
|
|
( wcel cfv ffvelrnda mpdan ) ADBHDEICHGABCDEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $. $d x ps $. $d y ph $.
|
|
rexrn.1 $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
|
|
$( Restricted existential quantification over the range of a function.
|
|
(Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario
|
|
Carneiro, 20-Aug-2014.) $)
|
|
rexrn $p |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) ) $=
|
|
( wfn cv cfv crn cvv wcel funfvex funfni wceq wrex fvelrnb eqcom rexbii
|
|
syl6bb wb adantl rexxfr2d ) FEHZABCDDIZFJZFKZELUGLMEUFFUFFNOUECIZUHMUGUIP
|
|
ZDEQUIUGPZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.
|
|
|
|
$( Restricted universal quantification over the range of a function.
|
|
(Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario
|
|
Carneiro, 20-Aug-2014.) $)
|
|
ralrn $p |- ( F Fn A -> ( A. x e. ran F ph <-> A. y e. A ps ) ) $=
|
|
( wfn cv cfv crn cvv wcel funfvex funfni wceq wrex fvelrnb eqcom rexbii
|
|
syl6bb wb adantl ralxfr2d ) FEHZABCDDIZFJZFKZELUGLMEUFFUFFNOUECIZUHMUGUIP
|
|
ZDEQUIUGPZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d Y x y $.
|
|
$( For any element in the range of a function there is an element in the
|
|
domain of the function for which the function value is the element of
|
|
the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.) $)
|
|
elrnrexdm $p |- ( Fun F -> ( Y e. ran F
|
|
-> E. x e. dom F Y = ( F ` x ) ) ) $=
|
|
( vy wfun crn wcel cv wceq cfv cdm wa eqidd ancli adantl eqeq2 rspcev syl
|
|
wrex ex wfn wb funfn rexrn sylbi sylibd ) BEZCBFZGZCDHZIZDUHSZCAHBJZIZABK
|
|
ZSZUGUIULUGUILUICCIZLZULUIURUGUIUQUICMNOUKUQDCUHUJCCPQRTUGBUOUAULUPUBBUCU
|
|
KUNDAUOBUJUMCPUDUEUF $.
|
|
|
|
$( For any element in the range of a function there is an element in the
|
|
domain of the function for which the function value is the element of
|
|
the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.) $)
|
|
elrnrexdmb $p |- ( Fun F -> ( Y e. ran F
|
|
<-> E. x e. dom F Y = ( F ` x ) ) ) $=
|
|
( wfun crn wcel cv cfv wceq cdm wrex wfn funfn fvelrnb sylbi eqcom rexbii
|
|
wb syl6bbr ) BDZCBEFZAGBHZCIZABJZKZCUBIZAUDKTBUDLUAUERBMAUDCBNOUFUCAUDCUB
|
|
PQS $.
|
|
|
|
$( For any element in the domain of a function there is an element in the
|
|
range of the function which is the function value for the element of the
|
|
domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.) $)
|
|
eldmrexrn $p |- ( Fun F -> ( Y e. dom F
|
|
-> E. x e. ran F x = ( F ` Y ) ) ) $=
|
|
( wfun cdm wcel cv cfv wceq crn wrex wa fvelrn eqid eqeq1 rspcev sylancl
|
|
ex ) BDZCBEFZAGZCBHZIZABJZKZSTLUBUDFUBUBIZUECBMUBNUCUFAUBUDUAUBUBOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d w x z A $. $d y B $. $d y ch $. $d w y z F $. $d w x z ps $.
|
|
ralrnmpt.1 $e |- F = ( x e. A |-> B ) $.
|
|
ralrnmpt.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( A restricted quantifier over an image set. (Contributed by Mario
|
|
Carneiro, 20-Aug-2015.) $)
|
|
ralrnmpt $p |- ( A. x e. A B e. V ->
|
|
( A. y e. ran F ps <-> A. x e. A ch ) ) $=
|
|
( vw vz wcel wral cv wsbc wb dfsbcq syl nfv crn cfv fnmpt nfsbc1v sbceq1a
|
|
wfn ralrn cbvral bicomi cmpt nfmpt1 nfcxfr nfcv nffv nfsbc weq wceq fveq2
|
|
3bitr3g wa fvmpt2 sbcieg adantl bitrd ralimiaa ralbi ) FHMZCENZADGUAZNZAD
|
|
COZGUBZPZCENZBCENZVHADKOZPZKVINZADLOZGUBZPZLENZVJVNVHGEUFVRWBQCEFGHIUCVQW
|
|
AKLEGADVPVTRUGSVJVRAVQDKVIAKTADVPUDADVPUEUHUIWAVMLCEACDVTCVSGCGCEFUJICEFU
|
|
KULCVSUMUNACTUOVMLTLCUPVTVLUQWAVMQVSVKGURADVTVLRSUHUSVHVMBQZCENVNVOQVGWCC
|
|
EVKEMZVGUTZVMADFPZBWEVLFUQVMWFQCEFHGIVAADVLFRSVGWFBQWDABDFHJVBVCVDVEVMBCE
|
|
VFSVD $.
|
|
|
|
$( A restricted quantifier over an image set. (Contributed by Mario
|
|
Carneiro, 20-Aug-2015.) $)
|
|
rexrnmpt $p |- ( A. x e. A B e. V ->
|
|
( E. y e. ran F ps <-> E. x e. A ch ) ) $=
|
|
( vw vz wcel wral wrex cv wsbc wb nfv wa crn cfv wfn fnmpt dfsbcq nfsbc1v
|
|
rexrn syl sbceq1a cbvrex bicomi cmpt nfmpt1 nfcxfr nfcv nfsbc weq sbceq1d
|
|
fveq2 3bitr3g fvmpt2 sbcieg adantl bitrd ralimiaa wi wal wex pm5.32 albii
|
|
nffv exbi sylbi df-ral df-rex bibi12i 3imtr4i ) FHMZCENZADGUAZOZADCPZGUBZ
|
|
QZCEOZBCEOZVSADKPZQZKVTOZADLPZGUBZQZLEOZWAWEVSGEUCWIWMRCEFGHIUDWHWLKLEGAD
|
|
WGWKUEUGUHWAWIAWHDKVTAKSADWGUFADWGUIUJUKWLWDLCEACDWKCWJGCGCEFULICEFUMUNCW
|
|
JUOVKACSUPWDLSLCUQADWKWCWJWBGUSURUJUTVSWDBRZCENZWEWFRZVRWNCEWBEMZVRTZWDAD
|
|
FQZBWRADWCFCEFHGIVAURVRWSBRWQABDFHJVBVCVDVEWQWNVFZCVGZWQWDTZCVHZWQBTZCVHZ
|
|
RZWOWPXAXBXDRZCVGXFWTXGCWQWDBVIVJXBXDCVLVMWNCEVNWEXCWFXEWDCEVOBCEVOVPVQUH
|
|
VD $.
|
|
$}
|
|
|
|
$( Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.) $)
|
|
dff2 $p |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) ) $=
|
|
( wf wfn cxp wss wa ffn fssxp jca crn rnss rnxpss syl6ss anim2i df-f sylibr
|
|
impbii ) ABCDZCAEZCABFZGZHZTUAUCABCIABCJKUDUACLZBGZHTUCUFUAUCUEUBLBCUBMABNO
|
|
PABCQRS $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) $)
|
|
dff3im $p |- ( F : A --> B ->
|
|
( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) ) $=
|
|
( wf cxp wss cv wbr weu wral fssxp wcel wa wex adantr syl2anc cvv syl wmo
|
|
cfv cop wfun cdm fdm eleq2d biimpar funfvop df-br sylibr wi funfvex breq2
|
|
ffun spcegv mpd funmo eu5 sylanbrc ralrimiva jca ) CDEFZECDGHAIZBIZEJZBKZ
|
|
ACLCDEMVCVGACVCVDCNZOZVFBPZVFBUAZVGVIVDVDEUBZEJZVJVIVDVLUCENZVMVIEUDZVDEU
|
|
EZNZVNVCVOVHCDEUOZQZVCVQVHVCVPCVDCDEUFUGUHZVDEUIRVDVLEUJUKVIVOVQVMVJULZVS
|
|
VTVOVQOVLSNWAVDEUMVFVMBVLSVEVLVDEUNUPTRUQVCVKVHVCVOVKVRBVDEURTQVFBUSUTVAV
|
|
B $.
|
|
|
|
$( Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) $)
|
|
dff4im $p |- ( F : A --> B ->
|
|
( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) ) $=
|
|
( wf cxp wss cv wbr weu wral wa wreu dff3im wcel cop df-br ssel opelxp2
|
|
syl6 syl5bi pm4.71rd eubidv df-reu syl6bbr ralbidv pm5.32i sylib ) CDEFEC
|
|
DGZHZAIZBIZEJZBKZACLZMUKUNBDNZACLZMABCDEOUKUPURUKUOUQACUKUOUMDPZUNMZBKUQU
|
|
KUNUTBUKUNUSUNULUMQZEPZUKUSULUMERUKVBVAUJPUSEUJVASULUMCDTUAUBUCUDUNBDUEUF
|
|
UGUHUI $.
|
|
|
|
$( An onto mapping expressed in terms of function values. (Contributed by
|
|
NM, 29-Oct-2006.) $)
|
|
dffo3 $p |- ( F : A -onto-> B <-> ( F : A --> B /\
|
|
A. y e. B E. x e. A y = ( F ` x ) ) ) $=
|
|
( wfo wf crn wceq wa cv cfv wrex wral dffo2 cab wb wcel wal wi wfn fnrnfv
|
|
ffn eqeq1d simpr ffvelrn adantr eqeltrd exp31 rexlimdv biantrurd syl6rbbr
|
|
syl dfbi2 albidv abeq1 df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZD
|
|
IZJVBBKZAKZELZIZACMZBDNZJCDEOVBVDVJVBVDVIBPZDIZVJVBECUAZVDVLQCDEUCVMVCVKD
|
|
ABCEUBUDUMVBVIVEDRZQZBSVNVITZBSVLVJVBVOVPBVBVPVIVNTZVPJVOVBVQVPVBVHVNACVB
|
|
VFCRZVHVNVBVRJZVHJVEVGDVSVHUEVSVGDRVHCDVFEUFUGUHUIUJUKVIVNUNULUOVIBDUPVIB
|
|
DUQURUSUTVA $.
|
|
|
|
$( Alternate definition of an onto mapping. (Contributed by NM,
|
|
20-Mar-2007.) $)
|
|
dffo4 $p |- ( F : A -onto-> B <->
|
|
( F : A --> B /\ A. y e. B E. x e. A x F y ) ) $=
|
|
( wfo wf cv wbr wrex wral wa crn wceq dffo2 simpl wcel wex vex wi syl5bbr
|
|
elrn eleq2 biimpar adantll wfn ffn fnbr syl ancrd eximdv syl6ibr ad2antrr
|
|
ex df-rex mpd ralrimiva jca sylbi cfv fnbrfvb eqcom syl6ib sylan reximdva
|
|
biimprd ralimdv imdistani dffo3 sylibr impbii ) CDEFZCDEGZAHZBHZEIZACJZBD
|
|
KZLZVLVMEMZDNZLZVSCDEOWBVMVRVMWAPWBVQBDWBVODQZLVPARZVQWAWCWDVMWAWDWCWDVOV
|
|
TQWAWCAVOEBSUBVTDVOUCUAUDUEVMWDVQTWAWCVMWDVNCQZVPLZARVQVMVPWFAVMVPWEVMECU
|
|
FZVPWETCDEUGZWGVPWECVNVOEUHUNUIUJUKVPACUOULUMUPUQURUSVSVMVOVNEUTZNZACJZBD
|
|
KZLVLVMVRWLVMVQWKBDVMVPWJACVMWGWEVPWJTWHWGWELZVPWIVONZWJWMWNVPCVNVOEVAVFW
|
|
IVOVBVCVDVEVGVHABCDEVIVJVK $.
|
|
|
|
$( Alternate definition of an onto mapping. (Contributed by NM,
|
|
20-Mar-2007.) $)
|
|
dffo5 $p |- ( F : A -onto-> B <->
|
|
( F : A --> B /\ A. y e. B E. x x F y ) ) $=
|
|
( wfo wf cv wbr wrex wral wa wex dffo4 rexex ralimi anim2i wcel wfn wi ex
|
|
ffn fnbr syl ancrd eximdv df-rex syl6ibr ralimdv imdistani impbii bitri )
|
|
CDEFCDEGZAHZBHZEIZACJZBDKZLZUMUPAMZBDKZLZABCDENUSVBURVAUMUQUTBDUPACOPQUMV
|
|
AURUMUTUQBDUMUTUNCRZUPLZAMUQUMUPVDAUMUPVCUMECSZUPVCTCDEUBVEUPVCCUNUOEUCUA
|
|
UDUEUFUPACUGUHUIUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d A x y $. $d B x y $. $d C x y $.
|
|
$( Property of a surjective function. (Contributed by Jeff Madsen,
|
|
4-Jan-2011.) $)
|
|
foelrn $p |- ( ( F : A -onto-> B /\ C e. B )
|
|
-> E. x e. A C = ( F ` x ) ) $=
|
|
( vy wfo cv wceq wrex wral wcel dffo3 simprbi eqeq1 rexbidv rspccva sylan
|
|
cfv wf ) BCEGZFHZAHESZIZABJZFCKZDCLDUCIZABJZUABCETUFAFBCEMNUEUHFDCUBDIUDU
|
|
GABUBDUCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d F x y z $. $d G x y z $. $d A y z $. $d B x y z $. $d C x y z $.
|
|
$( If a composition of two functions is surjective, then the function on
|
|
the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) $)
|
|
foco2 $p |- ( ( F : B --> C /\ G : A --> B /\
|
|
( F o. G ) : A -onto-> C ) -> F : B -onto-> C ) $=
|
|
( vy vx vz wf ccom wfo w3a cv cfv wceq wrex wral wa wcel adantll sylanbrc
|
|
simp1 foelrn ffvelrn fvco3 fveq2 eqeq2d rspcev syl2anc rexbidv syl5ibrcom
|
|
eqeq1 rexlimdva syl5 impl ralrimiva 3impa dffo3 ) BCDIZABEIZACDEJZKZLUSFM
|
|
ZGMZDNZOZGBPZFCQZBCDKUSUTVBUBUSUTVBVHUSUTRZVBRVGFCVIVBVCCSZVGVBVJRVCHMZVA
|
|
NZOZHAPVIVGHACVCVAUCVIVMVGHAVIVKASZRZVGVMVLVEOZGBPZVOVKENZBSZVLVRDNZOZVQU
|
|
TVNVSUSABVKEUDTUTVNWAUSABVKDEUETVPWAGVRBVDVROVEVTVLVDVRDUFUGUHUIVMVFVPGBV
|
|
CVLVEULUJUKUMUNUOUPUQGFBCDURUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d y z C $. $d y z F $.
|
|
fmpt.1 $e |- F = ( x e. A |-> C ) $.
|
|
$( Functionality of the mapping operation. (Contributed by Mario Carneiro,
|
|
26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fmpt $p |- ( A. x e. A C e. B <-> F : A --> B ) $=
|
|
( vy wcel wral wf wfn crn wss fnmpt cv wceq wrex cab rnmpt wa biimparc ex
|
|
r19.29 rexlimivw syl abssdv syl5eqss df-f sylanbrc crab ccnv cima fimacnv
|
|
eleq1 mptpreima syl5reqr rabid2 sylib impbii ) DCHZABIZBCEJZVAEBKELZCMVBA
|
|
BDECFNVAVCGOZDPZABQZGRCAGBDEFSVAVFGCVAVFVDCHZVAVFTUTVETZABQVGUTVEABUCVHVG
|
|
ABVEVGUTVDDCUNUAUDUEUBUFUGBCEUHUIVBBUTABUJZPVAVBVIEUKCULBABDCEFUOBCEUMUPU
|
|
TABUQURUS $.
|
|
|
|
$( Express bijection for a mapping operation. (Contributed by Mario
|
|
Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.) $)
|
|
f1ompt $p |- ( F : A -1-1-onto-> B <->
|
|
( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) ) $=
|
|
( vz wa cv wceq wral wcel wfn wbr weu nfcv bitri vex 3bitr4i wf wf1o wreu
|
|
ccnv wb ffn dff1o4 baib syl cres fnres cmpt nfmpt1 nfcxfr nfv breq1 copab
|
|
nfbr df-mpt eqtri breqi cop df-br opabid syl6bb cbveu brcnv df-reu ralbii
|
|
eubii wrel cdm wss relcnv crn df-rn frn syl5eqssr relssres sylancr fneq1d
|
|
syl5bbr bitr4d pm5.32i f1of pm4.71ri fmpt anbi1i ) CDFUAZCDFUBZIWIBJZEKZA
|
|
CUCZBDLZIWJEDMACLZWNIWIWJWNWIWJFUDZDNZWNWIFCNZWJWQUECDFUFWJWRWQCDFUGUHUIW
|
|
NWPDUJZDNZWIWQWTWKHJZWPOZHPZBDLWNBHDWPUKXCWMBDXAWKFOZHPAJZCMWLIZAPXCWMXDX
|
|
FHAAXAWKFAXAQAFACEULZGACEUMUNAWKQURXFHUOXAXEKXDXEWKFOZXFXAXEWKFUPXHXEWKXF
|
|
ABUQZOZXFXEWKFXIFXGXIGABCEUSUTVAXJXEWKVBXIMXFXEWKXIVCXFABVDRRVEVFXBXDHWKX
|
|
AFBSHSVGVJWLACVHTVIRWIDWSWPWIWPVKWPVLZDVMWSWPKFVNWIXKFVODFVPCDFVQVRWPDVSV
|
|
TWAWBWCWDWJWICDFWEWFWOWIWNACDEFGWGWHT $.
|
|
|
|
fmpti.2 $e |- ( x e. A -> C e. B ) $.
|
|
$( Functionality of the mapping operation. (Contributed by NM,
|
|
19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
fmpti $p |- F : A --> B $=
|
|
( wcel wral wf rgen fmpt mpbi ) DCHZABIBCEJNABGKABCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x ph $.
|
|
fmptd.1 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
|
|
fmptd.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( Domain and codomain of the mapping operation; deduction form.
|
|
(Contributed by Mario Carneiro, 13-Jan-2013.) $)
|
|
fmptd $p |- ( ph -> F : A --> C ) $=
|
|
( wcel wral wf ralrimiva fmpt sylib ) ADEIZBCJCEFKAOBCGLBCEDFHMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( A function maps to a class to which all values belong. (Contributed by
|
|
NM, 3-Dec-2003.) $)
|
|
ffnfv $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vy wf wfn cv cfv wcel wral ffn ffvelrn ralrimiva jca crn wss simpl wceq
|
|
wa wrex fvelrnb biimpd nfra1 nfv wi rsp eleq1 biimpcd syl6 rexlimd sylan9
|
|
ssrdv df-f sylanbrc impbii ) BCDFZDBGZAHZDIZCJZABKZTZUQURVBBCDLUQVAABBCUS
|
|
DMNOVCURDPZCQUQURVBRVCEVDCUREHZVDJZUTVESZABUAZVBVECJZURVFVHABVEDUBUCVBVGV
|
|
IABVAABUDVIAUEVBUSBJVAVGVIUFVAABUGVGVAVIUTVECUHUIUJUKULUMBCDUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z F $. $d x z $.
|
|
ffnfvf.1 $e |- F/_ x A $.
|
|
ffnfvf.2 $e |- F/_ x B $.
|
|
ffnfvf.3 $e |- F/_ x F $.
|
|
$( A function maps to a class to which all values belong. This version of
|
|
~ ffnfv uses bound-variable hypotheses instead of distinct variable
|
|
conditions. (Contributed by NM, 28-Sep-2006.) $)
|
|
ffnfvf $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vz wf wfn cv cfv wcel wral wa ffnfv nfcv nffv nfel nfv weq fveq2 eleq1d
|
|
cbvralf anbi2i bitri ) BCDIDBJZHKZDLZCMZHBNZOUGAKZDLZCMZABNZOHBCDPUKUOUGU
|
|
JUNHABHBQEAUICAUHDGAUHQRFSUNHTHAUAUIUMCUHULDUBUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( An upper bound for range determined by function values. (Contributed by
|
|
NM, 8-Oct-2004.) $)
|
|
fnfvrnss $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B ) $=
|
|
( wfn cv cfv wcel wral wa wf crn wss ffnfv frn sylbir ) DBEAFDGCHABIJBCDK
|
|
DLCMABCDNBCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
rnmptss.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The range of an operation given by the "maps to" notation as a subset.
|
|
(Contributed by Thierry Arnoux, 24-Sep-2017.) $)
|
|
rnmptss $p |- ( A. x e. A B e. C -> ran F C_ C ) $=
|
|
( wcel wral wf crn wss fmpt frn sylbi ) CDGABHBDEIEJDKABDCEFLBDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y C $. $d y F $. $d x ph $. $d y ph $.
|
|
fmpt2d.2 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
|
|
fmpt2d.1 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
|
|
fmpt2d.3 $e |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C ) $.
|
|
$( Domain and codomain of the mapping operation; deduction form.
|
|
(Contributed by NM, 27-Dec-2014.) $)
|
|
fmpt2d $p |- ( ph -> F : A --> C ) $=
|
|
( wfn cv cfv wcel wral wf cmpt ralrimiva eqid fnmpt fneq1d ffnfv sylanbrc
|
|
syl mpbird ) AGDLZCMGNFOZCDPDFGQAUGBDERZDLZAEHOZBDPUJAUKBDISBDEUIHUITUAUE
|
|
ADGUIJUBUFAUHCDKSCDFGUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( A necessary and sufficient condition for a restricted function.
|
|
(Contributed by Mario Carneiro, 14-Nov-2013.) $)
|
|
ffvresb $p |- ( Fun F -> ( ( F |` A ) : A --> B <->
|
|
A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) ) $=
|
|
( wfun cres wf cv cdm wcel cfv wa wral fdm cin dmres inss2 adantl wfn wss
|
|
eqsstri syl6eqssr sselda wceq fvres ffvelrn eqeltrrd jca ralrimiva ralimi
|
|
crn simpl dfss3 sylibr funfn fnssres sylanb sylan2 eleq1d syl5ibr ralimia
|
|
simpr fnfvrnss syl2anc df-f sylanbrc ex impbid2 ) DEZBCDBFZGZAHZDIZJZVLDK
|
|
ZCJZLZABMZVKVQABVKVLBJZLZVNVPVKBVMVLVKBVJIZVMBCVJNWABVMOVMDBPBVMQUAUBUCVT
|
|
VLVJKZVOCVSWBVOUDVKVLBDUEZRBCVLVJUFUGUHUIVIVRVKVIVRLZVJBSZVJUKCTZVKVRVIBV
|
|
MTZWEVRVNABMWGVQVNABVNVPULUJABVMUMUNVIDVMSWGWEDUOVMBDUPUQURZWDWEWBCJZABMZ
|
|
WFWHVRWJVIVQWIABVQWIVSVPVNVPVBVSWBVOCWCUSUTVARABCVJVCVDBCVJVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x y ph $. $d y ps $. $d x ch $.
|
|
f1oresrab.1 $e |- F = ( x e. A |-> C ) $.
|
|
f1oresrab.2 $e |- ( ph -> F : A -1-1-onto-> B ) $.
|
|
f1oresrab.3 $e |- ( ( ph /\ x e. A /\ y = C ) -> ( ch <-> ps ) ) $.
|
|
$( Build a bijection between restricted abstract builders, given a
|
|
bijection between the base classes, deduction version. (Contributed by
|
|
Thierry Arnoux, 17-Aug-2018.) $)
|
|
f1oresrab $p |- ( ph -> ( F |` { x e. A | ps } ) :
|
|
{ x e. A | ps } -1-1-onto-> { y e. B | ch } ) $=
|
|
( crab ccnv cres wf1o syl wceq wb wcel wfun f1ofun funcnvcnv 3syl wf1 wss
|
|
cima f1ocnv f1of1 ssrab2 f1ores sylancl mptpreima cv wa wi 3expia alrimiv
|
|
wal wral f1of fmpt sylibr r19.21bi elrab3t syl2anc rabbidva syl5eq f1oeq3
|
|
wf mpbid f1orescnv rescnvcnv f1oeq1 ax-mp sylib ) ABDFMZCEGMZINZNZVQOZPZV
|
|
QVRIVQOZPZAVTUAZVRVQVSVROZPZWBAFGIPZIUAWEKFGIUBIUCUDAVRVSVRUGZWFPZWGAGFVS
|
|
UEZVRGUFWJAGFVSPZWKAWHWLKFGIUHQGFVSUIQCEGUJGFVRVSUKULAWIVQRWJWGSAWIHVRTZD
|
|
FMVQDFHVRIJUMAWMBDFADUNFTZUOZEUNHRZCBSZUPZEUSHGTZWMBSWOWREAWNWPWQLUQURAWS
|
|
DFAFGIVJZWSDFUTAWHWTKFGIVAQDFGHIJVBVCVDCBEHGVEVFVGVHWIVQVRWFVIQVKVQVRVSVL
|
|
VFWAWCRWBWDSIVQVMVQVRWAWCVNVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x z A $. $d u x y B $. $d u w z F $. $d u w z G $. $d u y R $.
|
|
$d u w x z ph $. $d u x S $. $d u v w y z T $.
|
|
fmptco.1 $e |- ( ( ph /\ x e. A ) -> R e. B ) $.
|
|
fmptco.2 $e |- ( ph -> F = ( x e. A |-> R ) ) $.
|
|
fmptco.3 $e |- ( ph -> G = ( y e. B |-> S ) ) $.
|
|
fmptco.4 $e |- ( y = R -> S = T ) $.
|
|
$( Composition of two functions expressed as ordered-pair class
|
|
abstractions. If ` F ` has the equation ( x + 2 ) and ` G ` the
|
|
equation ( 3 * z ) then ` ( G o. F ) ` has the equation ( 3 * ( x +
|
|
2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro,
|
|
24-Jul-2014.) $)
|
|
fmptco $p |- ( ph -> ( G o. F ) = ( x e. A |-> T ) ) $=
|
|
( vu cv wbr wa wcel wceq vz vw vv ccom cmpt relco wfun wrel funmpt funrel
|
|
ax-mp wex csb cop cfv eqid fmptd feq1d mpbird ffun syl funbrfv imp eqcomd
|
|
wf sylan a1d expimpd pm4.71rd exbidv cvv wi exsimpl sylibr a1i cdm adantr
|
|
isset eleq2d biimpar funfvex syl2anc adantrr ex wb breq2 anbi12d ceqsexgv
|
|
fdm breq1 funfvbrb bitr3d fveq1d eqidd breq123d nfcv nfv nffvmpt1 nfcsb1v
|
|
nfbr nfeq2 nfbi nfim fveq2 breq1d csbeq1a eqeq2d bibi12d imbi2d vex simpl
|
|
eleq1d simpr eqeq12d df-mpt brabga sylancl fvmpt2 3bitr4d expcom vtoclgaf
|
|
biantrurd impcom bitrd sylan9bbr pm5.21ndd opelco copab eleq2i nfan eleq1
|
|
pm5.32da eqeq1 anbi2d opelopabf bitri 3bitr4g eqrelrdv ) AUAUBJIUDZBDHUEZ
|
|
JIUFYTUGYTUHBDHUIYTUJUKAUAPZOPZIQZUUBUBPZJQZRZOULZUUADSZUUDBUUAHUMZTZRZUU
|
|
AUUDUNZYSSUULYTSZAUUGUUBUUAIUOZTZUUFRZOULZUUKAUUFUUPOAUUFUUOAUUCUUEUUOAUU
|
|
CRZUUOUUEUURUUNUUBAIUGZUUCUUNUUBTZADEIVEZUUSAUVADEBDFUEZVEABDFEUVBKUVBUPZ
|
|
UQADEIUVBLURUSZDEIUTVAZUUSUUCUUTUUAUUBIVBVCVFVDVGVHVIVJAUUNVKSZUUQUUKUUQU
|
|
VFVLAUUQUUOOULUVFUUOUUFOVMOUUNVRVNVOAUUKUVFAUUHUVFUUJAUUHRUUSUUAIVPZSZUVF
|
|
AUUSUUHUVEVQAUVHUUHAUVGDUUAAUVAUVGDTUVDDEIWIVAVSZVTUUAIWAWBWCWDAUVFUUQUUK
|
|
WEUVFUUQUUAUUNIQZUUNUUDJQZRZAUUKUUFUVLOUUNVKUUOUUCUVJUUEUVKUUBUUNUUAIWFUU
|
|
BUUNUUDJWJWGWHAUVLUUHUUAUVBUOZUUDCEGUEZQZRUUKAUVJUUHUVKUVOAUVHUVJUUHAUUSU
|
|
VHUVJWEUVEUUAIWKVAUVIWLAUUNUVMUUDUUDJUVNAUUAIUVBLWMMAUUDWNWOWGAUUHUVOUUJU
|
|
UHAUVOUUJWEZABPZUVBUOZUUDUVNQZUUDHTZWEZVLAUVPVLBUUADBUUAWPAUVPBABWQUVOUUJ
|
|
BBUVMUUDUVNBDFUUAWRBUVNWPBUUDWPWTBUUDUUIBUUAHWSZXAXBXCUVQUUATZUWAUVPAUWCU
|
|
VSUVOUVTUUJUWCUVRUVMUUDUVNUVQUUAUVBXDXEUWCHUUIUUDBUUAHXFZXGXHXIAUVQDSZUWA
|
|
AUWERZFUUDUVNQZFESZUVTRZUVSUVTUWFUWHUUDVKSUWGUWIWEKUBXJZCPZESZUUBGTZRUWIC
|
|
OFUUDUVNEVKUWKFTZUUBUUDTZRZUWLUWHUWMUVTUWPUWKFEUWNUWOXKXLUWPUUBUUDGHUWNUW
|
|
OXMUWNGHTUWONVQXNWGCOEGXOXPXQUWFUVRFUUDUVNUWFUWEUWHUVRFTAUWEXMKBDFEUVBUVC
|
|
XRWBXEUWFUWHUVTKYBXSXTYAYCYLYDYEWDYFYDOUUAUUDJIUAXJZUWJYGUUMUULUWEUCPZHTZ
|
|
RZBUCYHZSUUKYTUXAUULBUCDHXOYIUWTUUHUWRUUITZRUUKBUCUUAUUDUUHUXBBUUHBWQBUWR
|
|
UUIUWBXAYJUUKUCWQUWQUWJUWCUWEUUHUWSUXBUVQUUADYKUWCHUUIUWRUWDXGWGUWRUUDTUX
|
|
BUUJUUHUWRUUDUUIYMYNYOYPYQYR $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z B $. $d w y z R $. $d w x z S $. $d x z A $. $d y z T $.
|
|
$d z ph $.
|
|
fmptcof.1 $e |- ( ph -> A. x e. A R e. B ) $.
|
|
fmptcof.2 $e |- ( ph -> F = ( x e. A |-> R ) ) $.
|
|
fmptcof.3 $e |- ( ph -> G = ( y e. B |-> S ) ) $.
|
|
${
|
|
fmptcof.4 $e |- ( y = R -> S = T ) $.
|
|
$( Version of ~ fmptco where ` ph ` needn't be distinct from ` x ` .
|
|
(Contributed by NM, 27-Dec-2014.) $)
|
|
fmptcof $p |- ( ph -> ( G o. F ) = ( x e. A |-> T ) ) $=
|
|
( vz vw csb cmpt wceq nfcv ccom cv wcel wral nfcsb1v nfel1 csbeq1a rspc
|
|
eleq1d mpan9 cbvmpt syl6eq csbeq1 fmptco csbeq1d syl6eqr csbiegf ralimi
|
|
nfcsb eqid nfcvd mpteq12 sylancr syl eqtrd ) AJIUAZBDCFGQZRZBDHRZAVFODC
|
|
BOUBZFQZGQZRVHAOPDEVKCPUBZGQZVLIJAFEUCZBDUDZVJDUCVKEUCZKVOVQBVJDBVKEBVJ
|
|
FUEZUFBUBVJSZFVKEBVJFUGZUIUHUJAIBDFRODVKRLBODFVKOFTVRVTUKULAJCEGRPEVNRM
|
|
CPEGVNPGTCVMGUECVMGUGUKULCVMVKGUMUNBODVGVLOVGTBCVKGVRBGTUSVSCFVKGVTUOUK
|
|
UPAVPVHVISZKVPDDSVGHSZBDUDWADUTVOWBBDCFGHEVOCHVANUQURBDVGDHVBVCVDVE $.
|
|
$}
|
|
|
|
$( Composition of two functions expressed as mapping abstractions.
|
|
(Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
fmptcos $p |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) ) $=
|
|
( vz cv csb cmpt nfcv nfcsb1v csbeq1a cbvmpt syl6eq csbeq1 fmptcof ) ABMD
|
|
EFCMNZGOZCFGOHIJKAICEGPMEUEPLCMEGUEMGQCUDGRCUDGSTUACUDFGUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x C $. $d x y D $. $d x E $.
|
|
$( Express composition of two functions as a maps-to applying both in
|
|
sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened
|
|
by Mario Carneiro, 27-Dec-2014.) $)
|
|
fcompt $p |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |->
|
|
( A ` ( B ` x ) ) ) ) $=
|
|
( vy wf wa cv cfv wcel ffvelrn adantll wfn cmpt wceq ffn dffn5im syl
|
|
adantl adantr fveq2 fmptco ) EFBHZDECHZIZAGDEAJZCKZGJZBKZUIBKCBUFUHDLUIEL
|
|
UEDEUHCMNUGCDOZCADUIPQUFULUEDECRUAADCSTUGBEOZBGEUKPQUEUMUFEFBRUBGEBSTUJUI
|
|
BUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d I x $. $d X x y $. $d Y x y $.
|
|
$( Composition with a constant function. (Contributed by Stefan O'Rear,
|
|
11-Mar-2015.) $)
|
|
fcoconst $p |- ( ( F Fn X /\ Y e. X ) ->
|
|
( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) ) $=
|
|
( vx vy wfn wcel wa csn cxp ccom cfv cmpt simplr wceq fconstmpt a1i cvv
|
|
cv wf simpl dffn2 sylib feqmptd fveq2 fmptco syl6eqr ) ACGZDCHZIZABDJKZLE
|
|
BDAMZNBUMJKUKEFBCDFTZAMUMULAUIUJETBHOULEBDNPUKEBDQRUKFCSAUKUICSAUAUIUJUBC
|
|
AUCUDUEUNDAUFUGEBUMQUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
fsn.1 $e |- A e. _V $.
|
|
fsn.2 $e |- B e. _V $.
|
|
$( A function maps a singleton to a singleton iff it is the singleton of an
|
|
ordered pair. (Contributed by NM, 10-Dec-2003.) $)
|
|
fsn $p |- ( F : { A } --> { B } <-> F = { <. A , B >. } ) $=
|
|
( vx vy csn cop wceq cv wcel wb wal wa velsn weu eleq1d bitr4i bitr2i feu
|
|
wf opelf anbi12i sylib ex wreu snid mpan2 opeq2 pm5.32i ancom eubii eueq1
|
|
anbi1i biantru euanv df-reu sylibr opeq12 syl5ibrcom impbid vex opex elsn
|
|
3bitr4i opth2 syl6bb alrimivv wrel frel relsnop eqrel sylancl mpbird wf1o
|
|
f1osn f1oeq1 mpbiri f1of syl impbii ) AHZBHZCUBZCABIZHZJZWEWHFKZGKZIZCLZW
|
|
KWGLZMZGNFNZWEWNFGWEWLWIAJZWJBJZOZWMWEWLWRWEWLWRWEWLOWIWCLZWJWDLZOWRWCWDW
|
|
IWJCUCWSWPWTWQFAPGBPZUDUEUFWEWLWRWFCLZWEAWJIZCLZGWDUGZXBWEAWCLXEADUHGWCWD
|
|
ACUAUIXBWQOZGQZWTXDOZGQXBXEXFXHGXHWQXDOZXFWTWQXDXAUOXIWQXBOXFWQXDXBWQXCWF
|
|
CWJBAUJRUKXBWQULSTUMXBXBWQGQZOXGXJXBGBEUNUPXBWQGUQSXDGWDURVFUSWRWKWFCWIWJ
|
|
ABUTRVAVBWMWKWFJWRWKWFWIWJFVCGVCVDVEWIWJABDEVGTVHVIWECVJWGVJWHWOMWCWDCVKA
|
|
BDEVLFGCWGVMVNVOWHWCWDCVPZWEWHXKWCWDWGVPABDEVQWCWDCWGVRVSWCWDCVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $. $d F a b $.
|
|
$( A function maps a singleton to a singleton iff it is the singleton of an
|
|
ordered pair. (Contributed by NM, 26-Oct-2012.) $)
|
|
fsng $p |- ( ( A e. C /\ B e. D ) ->
|
|
( F : { A } --> { B } <-> F = { <. A , B >. } ) ) $=
|
|
( va vb cv csn wf cop wceq wb sneq feq2d opeq1 sneqd eqeq2d bibi12d vex
|
|
feq3 syl opeq2 fsn vtocl2g ) FHZIZGHZIZEJZEUFUHKZIZLZMAIZUIEJZEAUHKZIZLZM
|
|
UNBIZEJZEABKZIZLZMFGABCDUFALZUJUOUMURVDUGUNUIEUFANOVDULUQEVDUKUPUFAUHPQRS
|
|
UHBLZUOUTURVCVEUIUSLUOUTMUHBNUIUSUNEUAUBVEUQVBEVEUPVAUHBAUCQRSUFUHEFTGTUD
|
|
UE $.
|
|
$}
|
|
|
|
${
|
|
fsn2.1 $e |- A e. _V $.
|
|
$( A function that maps a singleton to a class is the singleton of an
|
|
ordered pair. (Contributed by NM, 19-May-2004.) $)
|
|
fsn2 $p |- ( F : { A } --> B <->
|
|
( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) $=
|
|
( csn wf cfv cvv wcel cop wceq wa wfn ffn snid funfvex mpan2 syl wb cima
|
|
funfni adantr ffvelrn crn dffn3 biimpi cdm imadmrn imaeq2d syl5eqr fnsnfv
|
|
elex fndm eqtr4d feq3 mpbid jca wss snssi ancoms sylan impbii fsng anbi2d
|
|
fss mpan syl5bb pm5.21nii ) AEZBCFZACGZHIZVKBIZCAVKJEKZLZVJCVIMZVLVIBCNZV
|
|
PAVIIZVLADOZVLVIACACPUAQRVMVLVNVKBULUBVJVMVIVKEZCFZLZVLVOVJWBVJVMWAVJVRVM
|
|
VSVIBACUCQVJVPWAVQVPVICUDZCFZWAVPWDVICUEUFVPWCVTKWDWASVPWCCVITZVTVPWCCCUG
|
|
ZTWECUHVPWFVICVICUMUIUJVPVRVTWEKVSVIACUKQUNWCVTVICUORUPRUQVMVTBURZWAVJVKB
|
|
USWAWGVJVIVTBCVEUTVAVBVLWAVNVMAHIVLWAVNSDAVKHHCVCVFVDVGVH $.
|
|
$}
|
|
|
|
$( The cross product of two singletons. (Contributed by Mario Carneiro,
|
|
30-Apr-2015.) $)
|
|
xpsng $p |- ( ( A e. V /\ B e. W ) ->
|
|
( { A } X. { B } ) = { <. A , B >. } ) $=
|
|
( wcel wa csn cxp wf cop wceq fconstg adantl fsng mpbid ) ACEZBDEZFAGZBGZRS
|
|
HZIZTABJGKQUAPRBDLMABCDTNO $.
|
|
|
|
${
|
|
xpsn.1 $e |- A e. _V $.
|
|
xpsn.2 $e |- B e. _V $.
|
|
$( The cross product of two singletons. (Contributed by NM,
|
|
4-Nov-2006.) $)
|
|
xpsn $p |- ( { A } X. { B } ) = { <. A , B >. } $=
|
|
( cvv wcel csn cxp cop wceq xpsng mp2an ) AEFBEFAGBGHABIGJCDABEEKL $.
|
|
$}
|
|
|
|
${
|
|
dfmpt.1 $e |- B e. _V $.
|
|
$( Alternate definition for the "maps to" notation ~ df-mpt (although it
|
|
requires that ` B ` be a set). (Contributed by NM, 24-Aug-2010.)
|
|
(Revised by Mario Carneiro, 30-Dec-2016.) $)
|
|
dfmpt $p |- ( x e. A |-> B ) = U_ x e. A { <. x , B >. } $=
|
|
( cmpt cv csn cxp ciun cop dfmpt3 wceq wcel vex xpsn a1i iuneq2i eqtri )
|
|
ABCEABAFZGCGHZIABSCJGZIABCKABTUATUALSBMSCANDOPQR $.
|
|
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( A function expressed as the range of another function. (Contributed by
|
|
Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
fnasrn $p |- ( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. ) $=
|
|
( vy cmpt cv cop csn ciun crn dfmpt wcel wrex cab wceq rnmpt velsn eqtr4i
|
|
eqid rexbii abbii df-iun ) ABCFABAGCHZIZJZABUDFZKZABCDLUHEGZUEMZABNZEOZUF
|
|
UHUIUDPZABNZEOULAEBUDUGUGTQUKUNEUJUMABEUDRUAUBSAEBUEUCSS $.
|
|
$}
|
|
|
|
$( Alternate definition for the "maps to" notation ~ df-mpt (which requires
|
|
that ` B ` be a set). (Contributed by Jim Kingdon, 9-Jan-2019.) $)
|
|
dfmptg $p |- ( A. x e. A B e. V ->
|
|
( x e. A |-> B ) = U_ x e. A { <. x , B >. } ) $=
|
|
( wcel wral cmpt cv csn cxp ciun cop dfmpt3 wceq cvv vex mpan ralimi iuneq2
|
|
xpsng syl syl5eq ) CDEZABFZABCGABAHZICIJZKZABUECLIZKZABCMUDUFUHNZABFUGUINUC
|
|
UJABUEOEUCUJAPUECODTQRABUFUHSUAUB $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( A function expressed as the range of another function. (Contributed by
|
|
Jim Kingdon, 9-Jan-2019.) $)
|
|
fnasrng $p |- ( A. x e. A B e. V ->
|
|
( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. ) ) $=
|
|
( vy wcel wral cmpt cv cop csn ciun crn dfmptg wrex cab wceq rnmpt eqtr4i
|
|
eqid velsn rexbii abbii df-iun syl6eqr ) CDFABGABCHABAICJZKZLZABUFHZMZABC
|
|
DNUJEIZUGFZABOZEPZUHUJUKUFQZABOZEPUNAEBUFUIUITRUMUPEULUOABEUFUAUBUCSAEBUG
|
|
UDSUE $.
|
|
$}
|
|
|
|
$( If ` A ` is not in ` C ` , then the restriction of a singleton of
|
|
` <. A , B >. ` to ` C ` is null. (Contributed by Scott Fenton,
|
|
15-Apr-2011.) $)
|
|
ressnop0 $p |- ( -. A e. C -> ( { <. A , B >. } |` C ) = (/) ) $=
|
|
( wcel wn cop cvv cxp csn cres c0 wceq opelxp1 con3i cin df-res incom eqtri
|
|
disjsn biimpri syl5eq syl ) ACDZEABFZCGHZDZEZUDIZCJZKLUFUCABCGMNUGUIUEUHOZK
|
|
UIUHUEOUJUHCPUHUEQRUJKLUGUEUDSTUAUB $.
|
|
|
|
${
|
|
fpr.1 $e |- A e. _V $.
|
|
fpr.2 $e |- B e. _V $.
|
|
fpr.3 $e |- C e. _V $.
|
|
fpr.4 $e |- D e. _V $.
|
|
$( A function with a domain of two elements. (Contributed by Jeff Madsen,
|
|
20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
fpr $p |- ( A =/= B
|
|
-> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } ) $=
|
|
( wne cop cpr wfn crn wa jctir sylibr csn cun df-pr rnsnop wss wfun funpr
|
|
wf cdm wceq dmprop df-fn rneqi rnun uneq12i eqtr4i 3eqtri eqimssi df-f )
|
|
ABIZACJZBDJZKZABKZLZUSMZCDKZUAZNUTVCUSUDUPVAVDUPUSUBZUSUEUTUFZNVAUPVEVFAB
|
|
CDEFGHUCACBDGHUGOUSUTUHPVBVCVBUQQZURQZRZMVGMZVHMZRZVCUSVIUQURSUIVGVHUJVLC
|
|
QZDQZRVCVJVMVKVNACETBDFTUKCDSULUMUNOUTVCUSUOP $.
|
|
$}
|
|
|
|
$( A function with a domain of two elements. (Contributed by FL,
|
|
2-Feb-2014.) $)
|
|
fprg $p |- ( ( ( A e. E /\ B e. F ) /\ ( C e. G /\ D e. H ) /\ A =/= B )
|
|
-> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } ) $=
|
|
( wcel wa wne cop cpr crn wceq csn cun rnsnopg 3ad2ant1 df-pr w3a wfn fnprg
|
|
wss adantr adantl uneq12d rneqi rnun eqtri 3eqtr4g eqimss syl df-f sylanbrc
|
|
wf ) AEIZBFIZJZCGIDHIJZABKZUAZACLZBDLZMZABMZUBVENZCDMZUDZVFVHVEUPABCDEFGHUC
|
|
VBVGVHOVIVBVCPZNZVDPZNZQZCPZDPZQVGVHVBVKVOVMVPUSUTVKVOOZVAUQVQURACERUESUSUT
|
|
VMVPOZVAURVRUQBDFRUFSUGVGVJVLQZNVNVEVSVCVDTUHVJVLUIUJCDTUKVGVHULUMVFVHVEUNU
|
|
O $.
|
|
|
|
$( A function with a domain of three elements. (Contributed by Alexander van
|
|
der Vekens, 4-Dec-2017.) $)
|
|
ftpg $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
|
|
/\ ( A e. F /\ B e. G /\ C e. H )
|
|
/\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
|
|
-> { <. X , A >. , <. Y , B >. , <. Z , C >. } :
|
|
{ X , Y , Z } --> { A , B , C } ) $=
|
|
( wcel w3a wne cpr wf wceq wa wn csn cun cop ctp cin c0 3simpa simp1 syl3an
|
|
fprg eqidd wb simp3 anim12i 3adant3 syl mpbird df-ne wo elpri eqcom orbi12i
|
|
fsng sylib oranim con2i syl2anb 3adant1 3ad2ant3 disjsn sylibr fun syl21anc
|
|
df-tp feq1i feq23i bitri ) JDMZKHMZLIMZNZAEMZBFMZCGMZNZJKOZJLOZKLOZNZNZJKPZ
|
|
LUAZUBZABPZCUAZUBZJAUCZKBUCZPZLCUCZUAZUBZQZJKLUDZABCUDZWQWRWTUDZQZWJWKWNWSQ
|
|
ZWLWOXAQZWKWLUEUFRZXCWAVRVSSWEWBWCSWIWFXHVRVSVTUGWBWCWDUGWFWGWHUHJKABDHEFUJ
|
|
UIWJXIXAXARZWJXAUKWJVTWDSZXIXKULWAWEXLWIWAVTWEWDVRVSVTUMWBWCWDUMUNUOLCIGXAV
|
|
CUPUQWJLWKMZTZXJWIWAXNWEWGWHXNWFWGJLRZTZKLRZTZXNWHJLURKLURXMXPXRSZXMXOXQUSZ
|
|
XSTXMLJRZLKRZUSXTLJKUTYAXOYBXQLJVALKVAVBVDXOXQVEUPVFVGVHVIWKLVJVKWKWLWNWOWS
|
|
XAVLVMXGXDXEXBQXCXDXEXFXBWQWRWTVNVOXDXEWMWPXBJKLVNABCVNVPVQVK $.
|
|
|
|
${
|
|
ftp.a $e |- A e. _V $.
|
|
ftp.b $e |- B e. _V $.
|
|
ftp.c $e |- C e. _V $.
|
|
ftp.d $e |- X e. _V $.
|
|
ftp.e $e |- Y e. _V $.
|
|
ftp.f $e |- Z e. _V $.
|
|
ftp.g $e |- A =/= B $.
|
|
ftp.h $e |- A =/= C $.
|
|
ftp.i $e |- B =/= C $.
|
|
$( A function with a domain of three elements. (Contributed by Stefan
|
|
O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens,
|
|
23-Jan-2018.) $)
|
|
ftp $p |- { <. A , X >. , <. B , Y >. , <. C , Z >. } : { A , B , C } -->
|
|
{ X , Y , Z } $=
|
|
( cvv wcel w3a wne ctp cop wf 3pm3.2i ftpg mp3an ) APQZBPQZCPQZRDPQZEPQZF
|
|
PQZRABSZACSZBCSZRABCTDEFTADUABEUACFUATUBUFUGUHGHIUCUIUJUKJKLUCULUMUNMNOUC
|
|
DEFPPPPPPABCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( A function restricted to a singleton. (Contributed by NM,
|
|
9-Oct-2004.) $)
|
|
fnressn $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( F |` { B } ) = { <. B , ( F ` B ) >. } ) $=
|
|
( vx wcel wfn csn cres cfv cop wceq cv wi sneq reseq2d fveq2 opeq12 mpdan
|
|
wa cvv sneqd eqeq12d imbi2d wss vex snss fnssres sylan2b dffn2 fsn2 bitri
|
|
vsnid fvres ax-mp opeq2i sneqi eqeq2i snssi sylan2 funfvex funfni sylancl
|
|
wf biantrurd syl5rbbr syl5bb mpbid expcom vtoclga impcom ) BAECAFZCBGZHZB
|
|
BCIZJZGZKZVKCDLZGZHZVRVRCIZJZGZKZMVKVQMDBAVRBKZWDVQVKWEVTVMWCVPWEVSVLCVRB
|
|
NOWEWBVOWEWAVNKWBVOKVRBCPVRWABVNQRUAUBUCVKVRAEZWDVKWFSZVTVSFZWDWFVKVSAUDZ
|
|
WHVRADUEZUFAVSCUGZUHWHVRVTIZTEZVTVRWLJZGZKZSZWGWDWHVSTVTVCWQVSVTUIVRTVTWJ
|
|
UJUKWDWPWGWQWOWCVTWNWBWLWAVRVRVSEZWLWAKDULZVRVSCUMUNUOUPUQWGWMWPWGWHWRWMW
|
|
FVKWIWHVRAURWKUSWSWMVSVRVTVRVTUTVAVBVDVEVFVGVHVIVJ $.
|
|
|
|
$( The value of a function restricted to a singleton. (Contributed by NM,
|
|
9-Oct-2004.) $)
|
|
fressnfv $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) $=
|
|
( vx wcel wfn csn cres wf cfv wb cv wi wceq sneq reseq2 syl wa cop eleq1d
|
|
feq1d feq2 bitrd fveq2 bibi12d imbi2d fnressn vsnid fvres ax-mp sneqi vex
|
|
opeq2i eqeq2i fsn2 eleq1i syl5rbbr syl5bb sylbir expcom vtoclga impcom
|
|
iba ) BAFDAGZBHZCDVFIZJZBDKZCFZLZVEEMZHZCDVMIZJZVLDKZCFZLZNVEVKNEBAVLBOZV
|
|
RVKVEVSVOVHVQVJVSVMVFOZVOVHLVLBPVTVOVMCVGJVHVTVMCVNVGVMVFDQUBVMVFCVGUCUDR
|
|
VSVPVICVLBDUEUAUFUGVEVLAFZVRVEWASVNVLVPTZHZOZVRAVLDUHWDVNVLVLVNKZTZHZOZVR
|
|
WGWCVNWFWBWEVPVLVLVMFWEVPOEUIVLVMDUJUKZUNULUOVOWECFZWHSZWHVQVLCVNEUMUPVQW
|
|
JWHWKWEVPCWIUQWHWJVDURUSUTRVAVBVC $.
|
|
$}
|
|
|
|
$( The value of a constant function. (Contributed by NM, 30-May-1999.) $)
|
|
fvconst $p |- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B ) $=
|
|
( csn wf wcel wa cfv wceq ffvelrn elsni syl ) ABEZDFCAGHCDIZNGOBJANCDKOBLM
|
|
$.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Express a singleton function in maps-to notation. (Contributed by NM,
|
|
6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised
|
|
by Stefan O'Rear, 28-Feb-2015.) $)
|
|
fmptsn $p |- ( ( A e. V /\ B e. W ) ->
|
|
{ <. A , B >. } = ( x e. { A } |-> B ) ) $=
|
|
( wcel wa csn cmpt cxp cop fconstmpt xpsng syl5reqr ) BDFCEFGABHZCIOCHJBC
|
|
KHAOCLBCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x S $.
|
|
fmptap.0a $e |- A e. _V $.
|
|
fmptap.0b $e |- B e. _V $.
|
|
fmptap.1 $e |- ( R u. { A } ) = S $.
|
|
fmptap.2 $e |- ( x = A -> C = B ) $.
|
|
$( Append an additional value to a function. (Contributed by NM,
|
|
6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fmptap $p |- ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) $=
|
|
( cmpt cop csn cun cvv wcel wceq fmptsn mp2an cv elsni syl mpteq2ia mptun
|
|
eqtr4i uneq2i mpteq1 ax-mp 3eqtr2i ) AEDKZBCLMZNUJABMZDKZNAEULNZDKZAFDKZU
|
|
KUMUJUKAULCKZUMBOPCOPUKUQQGHABCOORSAULDCATZULPURBQDCQURBUAJUBUCUEUFAEULDU
|
|
DUNFQUOUPQIAUNFDUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x S $. $d x ph $.
|
|
fmptapd.0a $e |- ( ph -> A e. _V ) $.
|
|
fmptapd.0b $e |- ( ph -> B e. _V ) $.
|
|
fmptapd.1 $e |- ( ph -> ( R u. { A } ) = S ) $.
|
|
fmptapd.2 $e |- ( ( ph /\ x = A ) -> C = B ) $.
|
|
$( Append an additional value to a function. (Contributed by Thierry
|
|
Arnoux, 3-Jan-2017.) $)
|
|
fmptapd $p |- ( ph ->
|
|
( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) ) $=
|
|
( cmpt cop csn cun cvv wcel wceq fmptsn syl2anc cv elsni sylan2 mpteq2dva
|
|
eqtr4d uneq2d mptun a1i mpteq1d 3eqtr2d ) ABFELZCDMNZOUKBCNZELZOZBFUMOZEL
|
|
ZBGELAULUNUKAULBUMDLZUNACPQDPQULURRHIBCDPPSTABUMEDBUAZUMQAUSCREDRUSCUBKUC
|
|
UDUEUFUQUORABFUMEUGUHABUPGEJUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $. $d x ph $.
|
|
fmptpr.1 $e |- ( ph -> A e. V ) $.
|
|
fmptpr.2 $e |- ( ph -> B e. W ) $.
|
|
fmptpr.3 $e |- ( ph -> C e. X ) $.
|
|
fmptpr.4 $e |- ( ph -> D e. Y ) $.
|
|
fmptpr.5 $e |- ( ( ph /\ x = A ) -> E = C ) $.
|
|
fmptpr.6 $e |- ( ( ph /\ x = B ) -> E = D ) $.
|
|
$( Express a pair function in maps-to notation. (Contributed by Thierry
|
|
Arnoux, 3-Jan-2017.) $)
|
|
fmptpr $p |- ( ph ->
|
|
{ <. A , C >. , <. B , D >. } = ( x e. { A , B } |-> E ) ) $=
|
|
( cun c0 wcel cop cpr csn cmpt wceq df-pr a1i uneq1i uncom un0 3eqtri cvv
|
|
mpt0 elex syl eqtr3i fmptapd syl5eqr uneq1d eqcomi 3eqtrd ) ACEUAZDFUAZUB
|
|
ZVBUCZVCUCZRZBCUCZGUDZVFRBCDUBZGUDVDVGUEAVBVCUFUGAVEVIVFAVEBSGUDZVERZVIVL
|
|
SVERVESRVEVKSVEBGUMUHSVEUIVEUJUKABCEGSVHACHTCULTLCHUNUOAEJTEULTNEJUNUOSVH
|
|
RZVHUEAVHSRVMVHVHSUIVHUJUPUGPUQURUSABDFGVHVJADITDULTMDIUNUOAFKTFULTOFKUNU
|
|
OVHDUCRZVJUEAVJVNCDUFUTUGQUQVA $.
|
|
$}
|
|
|
|
$( The value of a restricted identity function. (Contributed by NM,
|
|
19-May-2004.) $)
|
|
fvresi $p |- ( B e. A -> ( ( _I |` A ) ` B ) = B ) $=
|
|
( wcel cid cres cfv fvres fvi eqtrd ) BACBDAEFBDFBBADGBAHI $.
|
|
|
|
$( Remove an ordered pair not participating in a function value.
|
|
(Contributed by Jim Kingdon, 7-Jan-2019.) $)
|
|
fvunsng $p |- ( ( D e. V /\ B =/= D )
|
|
-> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) ) $=
|
|
( wcel wne wa cop csn cun cfv cres wceq snidg fvres syl resundir c0 wn un0
|
|
elsni necon3ai ressnop0 uneq2d syl6eq syl5eq fveq1d sylan9req adantr eqtrd
|
|
) DEFZBDGZHDABCIJZKZLZDADJZMZLZDALZULUMUPDUOUQMZLZUSULDUQFZVBUPNDEOZDUQUOPQ
|
|
UMDVAURUMVAURUNUQMZKZURAUNUQRUMVFURSKURUMVESURUMBUQFZTVESNVGBDBDUBUCBCUQUDQ
|
|
UEURUAUFUGUHUIULUSUTNZUMULVCVHVDDUQAPQUJUK $.
|
|
|
|
${
|
|
fvsn.1 $e |- A e. _V $.
|
|
fvsn.2 $e |- B e. _V $.
|
|
$( The value of a singleton of an ordered pair is the second member.
|
|
(Contributed by NM, 12-Aug-1994.) $)
|
|
fvsn $p |- ( { <. A , B >. } ` A ) = B $=
|
|
( cop csn wfun wcel cfv wceq funsn opex snid funopfv mp2 ) ABEZFZGPQHAQIB
|
|
JABCDKPABCDLMABQNO $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $.
|
|
$( The value of a singleton of an ordered pair is the second member.
|
|
(Contributed by NM, 26-Oct-2012.) $)
|
|
fvsng $p |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B ) $=
|
|
( va vb cv cop csn cfv opeq1 sneqd id fveq12d eqeq1d opeq2 fveq1d eqeq12d
|
|
wceq vex fvsn vtocl2g ) EGZUCFGZHZIZJZUDSAAUDHZIZJZUDSAABHZIZJZBSEFABCDUC
|
|
ASZUGUJUDUNUCAUFUIUNUEUHUCAUDKLUNMNOUDBSZUJUMUDBUOAUIULUOUHUKUDBAPLQUOMRU
|
|
CUDETFTUAUB $.
|
|
$}
|
|
|
|
${
|
|
fvsnun.1 $e |- A e. _V $.
|
|
fvsnun.2 $e |- B e. _V $.
|
|
fvsnun.3 $e |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) $.
|
|
$( The value of a function with one of its ordered pairs replaced, at the
|
|
replaced ordered pair. See also ~ fvsnun2 . (Contributed by NM,
|
|
23-Sep-2007.) $)
|
|
fvsnun1 $p |- ( G ` A ) = B $=
|
|
( csn cres cfv cop cdif cun c0 cin wceq eqtri ax-mp fvres reseq1i disjdif
|
|
resundir incom resdisj uneq2i un0 fveq1i wcel snid fvsn 3eqtr3i ) AEAIZJZ
|
|
KZAABLIZUMJZKZAEKZBAUNUQUNUPDCUMMZJZNZUMJZUQEVBUMHUAVCUQVAUMJZNZUQUPVAUMU
|
|
CVEUQONUQVDOUQUTUMPZOQVDOQVFUMUTPOUTUMUDUMCUBRUTUMDUESUFUQUGRRRUHAUMUIZUO
|
|
USQAFUJZAUMETSURAUPKZBVGURVIQVHAUMUPTSABFGUKRUL $.
|
|
|
|
$( The value of a function with one of its ordered pairs replaced, at
|
|
arguments other than the replaced one. See also ~ fvsnun1 .
|
|
(Contributed by NM, 23-Sep-2007.) $)
|
|
fvsnun2 $p |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) ) $=
|
|
( csn cdif wcel cres cfv cop cun c0 wceq 3eqtri fvres reseq1i cin disjdif
|
|
resundir wfn wb fnsn fnresdisj ax-mp mpbi residm uneq12i uncom un0 fveq1i
|
|
3eqtr3a ) DCAJZKZLDFURMZNDEURMZNDFNDENDUSUTUSABOJZUTPZURMVAURMZUTURMZPZUT
|
|
FVBURIUAVAUTURUDVEQUTPUTQPUTVCQVDUTUQURUBQRZVCQRZUQCUCVAUQUEVFVGUFABGHUGU
|
|
QURVAUHUIUJEURUKULQUTUMUTUNSSUODURFTDURETUP $.
|
|
$}
|
|
|
|
$( Adjoining a point to a function gives a function. (Contributed by Stefan
|
|
O'Rear, 28-Feb-2015.) $)
|
|
fsnunf $p |- ( ( F : S --> T /\ ( X e. V /\ -. X e. S ) /\ Y e. T ) ->
|
|
( F u. { <. X , Y >. } ) : ( S u. { X } ) --> T ) $=
|
|
( wf wcel wn wa w3a csn cun cop cin c0 wceq simp1 wf1o syl simp2l simp3 fun
|
|
f1osng syl2anc f1of simp2r disjsn sylibr syl21anc wb snssi 3ad2ant3 ssequn2
|
|
wss sylib feq3 mpbid ) ABCGZEDHZEAHIZJZFBHZKZAELZMZBFLZMZCEFNLZMZGZVFBVJGZV
|
|
DUSVEVGVIGZAVEOPQZVKUSVBVCRVDVEVGVISZVMVDUTVCVOUSUTVAVCUAUSVBVCUBEFDBUDUEVE
|
|
VGVIUFTVDVAVNUSUTVAVCUGAEUHUIAVEBVGCVIUCUJVDVHBQZVKVLUKVDVGBUOZVPVCUSVQVBFB
|
|
ULUMVGBUNUPVHBVFVJUQTUR $.
|
|
|
|
$( Recover the added point from a point-added function. (Contributed by
|
|
Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.) $)
|
|
fsnunfv $p |- ( ( X e. V /\ Y e. W /\ -. X e. dom F ) ->
|
|
( ( F u. { <. X , Y >. } ) ` X ) = Y ) $=
|
|
( wcel cdm wn w3a cop csn cun cres cfv c0 wceq cin dmres 3adant3 syl disjsn
|
|
incom eqtri biimpri syl5eq 3ad2ant3 wb relres reldm0 ax-mp sylibr wfn fnsng
|
|
wrel fnresdm uneq12d resundir uncom un0 eqtr2i 3eqtr4g snidg 3ad2ant1 fvres
|
|
fveq1d fvsng 3eqtr3d ) DBFZECFZDAGZFHZIZDADEJKZLZDKZMZNZDVMNZDVNNZEVLDVPVMV
|
|
LAVOMZVMVOMZLOVMLZVPVMVLVTOWAVMVLVTGZOPZVTOPZVKVHWDVIVKWCVJVOQZOWCVOVJQWFAV
|
|
ORVOVJUBUCWFOPVKVJDUAUDUEUFVTUNWEWDUGAVOUHVTUIUJUKVLVMVOULZWAVMPVHVIWGVKDEB
|
|
CUMSVOVMUOTUPAVMVOUQWBVMOLVMOVMURVMUSUTVAVEVLDVOFZVQVSPVHVIWHVKDBVBVCDVOVNV
|
|
DTVHVIVREPVKDEBCVFSVG $.
|
|
|
|
$( Recover the original function from a point-added function. (Contributed
|
|
by Stefan O'Rear, 28-Feb-2015.) $)
|
|
fsnunres $p |- ( ( F Fn S /\ -. X e. S ) ->
|
|
( ( F u. { <. X , Y >. } ) |` S ) = F ) $=
|
|
( wfn wcel wn wa cres cop csn cun c0 fnresdm adantr ressnop0 adantl uneq12d
|
|
wceq resundir un0 eqcomi 3eqtr4g ) BAEZCAFGZHZBAIZCDJKZAIZLBMLZBUHLAIBUFUGB
|
|
UIMUDUGBSUEABNOUEUIMSUDCDAPQRBUHATUJBBUAUBUC $.
|
|
|
|
${
|
|
fvpr1.1 $e |- A e. _V $.
|
|
fvpr1.2 $e |- C e. _V $.
|
|
$( The value of a function with a domain of two elements. (Contributed by
|
|
Jeff Madsen, 20-Jun-2010.) $)
|
|
fvpr1 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
|
|
( wne cop cpr cfv csn cun df-pr fveq1i wceq necom cvv wcel fvunsng mpan
|
|
sylbi syl5eq fvsn syl6eq ) ABGZAACHZBDHZIZJZAUFKZJZCUEUIAUJUGKLZJZUKAUHUL
|
|
UFUGMNUEBAGZUMUKOZABPAQRUNUOEUJBDAQSTUAUBACEFUCUD $.
|
|
$}
|
|
|
|
${
|
|
fvpr2.1 $e |- B e. _V $.
|
|
fvpr2.2 $e |- D e. _V $.
|
|
$( The value of a function with a domain of two elements. (Contributed by
|
|
Jeff Madsen, 20-Jun-2010.) $)
|
|
fvpr2 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
|
|
( wne cop cpr cfv prcom fveq1i wceq necom fvpr1 sylbi syl5eq ) ABGZBACHZB
|
|
DHZIZJBTSIZJZDBUAUBSTKLRBAGUCDMABNBADCEFOPQ $.
|
|
$}
|
|
|
|
$( The value of a function with a domain of (at most) two elements.
|
|
(Contributed by Alexander van der Vekens, 3-Dec-2017.) $)
|
|
fvpr1g $p |- ( ( A e. V /\ C e. W /\ A =/= B )
|
|
-> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
|
|
( wcel wne w3a cop cpr cfv csn wceq wa cun df-pr fveq1i necom fvunsng fvsng
|
|
sylan2b syl5eq 3adant2 3adant3 eqtrd ) AEGZCFGZABHZIAACJZBDJZKZLZAUJMZLZCUG
|
|
UIUMUONUHUGUIOUMAUNUKMPZLZUOAULUPUJUKQRUIUGBAHUQUONABSUNBDAETUBUCUDUGUHUOCN
|
|
UIACEFUAUEUF $.
|
|
|
|
$( The value of a function with a domain of (at most) two elements.
|
|
(Contributed by Alexander van der Vekens, 3-Dec-2017.) $)
|
|
fvpr2g $p |- ( ( B e. V /\ D e. W /\ A =/= B )
|
|
-> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
|
|
( wcel wne w3a cop cpr cfv csn wceq wa cun prcom df-pr eqtri fveq1i fvunsng
|
|
syl5eq 3adant2 fvsng 3adant3 eqtrd ) BEGZDFGZABHZIBACJZBDJZKZLZBUKMZLZDUGUI
|
|
UMUONUHUGUIOUMBUNUJMPZLUOBULUPULUKUJKUPUJUKQUKUJRSTUNACBEUAUBUCUGUHUODNUIBD
|
|
EFUDUEUF $.
|
|
|
|
$( The value of a function with a domain of (at most) three elements.
|
|
(Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
|
|
fvtp1g $p |- ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D ) $=
|
|
( wcel wa wne cop ctp cfv cpr csn cun df-tp fveq1i wceq necom fvunsng 3expa
|
|
sylan2b ad2ant2rl fvpr1g adantrr eqtrd syl5eq ) AGIZDHIZJZABKZACKZJJZAADLZB
|
|
ELZCFLZMZNAUPUQOZURPQZNZDAUSVAUPUQURRSUOVBAUTNZDUJUNVBVCTZUKUMUNUJCAKVDACUA
|
|
UTCFAGUBUDUEULUMVCDTZUNUJUKUMVEABDEGHUFUCUGUHUI $.
|
|
|
|
$( The value of a function with a domain of (at most) three elements.
|
|
(Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
|
|
fvtp2g $p |- ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E ) $=
|
|
( wcel wa wne cop ctp cfv tprot fveq1i wceq wi necom fvtp1g expcom ancoms
|
|
sylanb impcom syl5eq ) BGIEHIJZABKZBCKZJZJBADLZBELZCFLZMZNBUKULUJMZNZEBUMUN
|
|
UJUKULOPUIUFUOEQZUGBAKZUHUFUPRZABSUHUQURUFUHUQJUPBCAEFDGHTUAUBUCUDUE $.
|
|
|
|
$( The value of a function with a domain of (at most) three elements.
|
|
(Contributed by Alexander van der Vekens, 4-Dec-2017.) $)
|
|
fvtp3g $p |- ( ( ( C e. V /\ F e. W ) /\ ( A =/= C /\ B =/= C ) )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F ) $=
|
|
( wcel wa wne cop ctp cfv tprot fveq1i wceq wi necom fvtp2g expcom sylan2b
|
|
ancoms impcom syl5eq ) CGIFHIJZACKZBCKZJZJCADLZBELZCFLZMZNCUKULUJMZNZFCUMUN
|
|
UJUKULOPUIUFUOFQZUHUGUFUPRZUGUHCAKZUQACSUFUHURJUPBCAEFDGHTUAUBUCUDUE $.
|
|
|
|
${
|
|
fvtp1.1 $e |- A e. _V $.
|
|
fvtp1.4 $e |- D e. _V $.
|
|
$( The first value of a function with a domain of three elements.
|
|
(Contributed by NM, 14-Sep-2011.) $)
|
|
fvtp1 $p |- ( ( A =/= B /\ A =/= C )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D ) $=
|
|
( cvv wcel wne wa cop ctp cfv wceq fvtp1g mpanl12 ) AIJDIJABKACKLAADMBEMC
|
|
FMNODPGHABCDEFIIQR $.
|
|
$}
|
|
|
|
${
|
|
fvtp2.1 $e |- B e. _V $.
|
|
fvtp2.4 $e |- E e. _V $.
|
|
$( The second value of a function with a domain of three elements.
|
|
(Contributed by NM, 14-Sep-2011.) $)
|
|
fvtp2 $p |- ( ( A =/= B /\ B =/= C )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E ) $=
|
|
( wne wa cop ctp cfv tprot fveq1i wceq necom fvtp1 ancoms sylanb syl5eq )
|
|
ABIZBCIZJBADKZBEKZCFKZLZMBUEUFUDLZMZEBUGUHUDUEUFNOUBBAIZUCUIEPZABQUCUJUKB
|
|
CAEFDGHRSTUA $.
|
|
$}
|
|
|
|
${
|
|
fvtp3.1 $e |- C e. _V $.
|
|
fvtp3.4 $e |- F e. _V $.
|
|
$( The third value of a function with a domain of three elements.
|
|
(Contributed by NM, 14-Sep-2011.) $)
|
|
fvtp3 $p |- ( ( A =/= C /\ B =/= C )
|
|
-> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F ) $=
|
|
( wne wa cop ctp cfv tprot fveq1i wceq necom fvtp2 sylan2b ancoms syl5eq
|
|
) ACIZBCIZJCADKZBEKZCFKZLZMCUEUFUDLZMZFCUGUHUDUEUFNOUCUBUIFPZUBUCCAIUJACQ
|
|
BCAEFDGHRSTUA $.
|
|
$}
|
|
|
|
$( The value of a constant function. (Contributed by NM, 20-Aug-2005.) $)
|
|
fvconst2g $p |- ( ( B e. D /\ C e. A ) -> ( ( A X. { B } ) ` C ) = B ) $=
|
|
( wcel csn cxp wf cfv wceq fconstg fvconst sylan ) BDEABFZANGZHCAECOIBJABDK
|
|
ABCOLM $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( A constant function expressed as a cross product. (Contributed by NM,
|
|
27-Nov-2007.) $)
|
|
fconst2g $p |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) ) $=
|
|
( vx wcel csn wf cxp wceq wa cv cfv wral fvconst adantlr fvconst2g eqtr4d
|
|
adantll wfn ralrimiva wb ffn fnconstg eqfnfv syl2an mpbird expcom fconstg
|
|
feq1 syl5ibrcom impbid ) BCFZABGZDHZDAUNIZJZUOUMUQUOUMKZUQELZDMZUSUPMZJZE
|
|
ANZURVBEAURUSAFZKUTBVAUOVDUTBJUMABUSDOPUMVDVABJUOABUSCQSRUAUODATUPATUQVCU
|
|
BUMAUNDUCABCUDEADUPUEUFUGUHUMUOUQAUNUPHABCUIAUNDUPUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
fvconst2.1 $e |- B e. _V $.
|
|
$( The value of a constant function. (Contributed by NM, 16-Apr-2005.) $)
|
|
fvconst2 $p |- ( C e. A -> ( ( A X. { B } ) ` C ) = B ) $=
|
|
( cvv wcel csn cxp cfv wceq fvconst2g mpan ) BEFCAFCABGHIBJDABCEKL $.
|
|
|
|
$( A constant function expressed as a cross product. (Contributed by NM,
|
|
20-Aug-1999.) $)
|
|
fconst2 $p |- ( F : A --> { B } <-> F = ( A X. { B } ) ) $=
|
|
( cvv wcel csn wf cxp wceq wb fconst2g ax-mp ) BEFABGZCHCANIJKDABECLM $.
|
|
$}
|
|
|
|
${
|
|
$d x w z A $. $d x w z B $. $d x w z F $. $d y w A $.
|
|
$( A constant function expressed in terms of its functionality, domain, and
|
|
value. See also ~ fconst2 . (Contributed by Jim Kingdon,
|
|
8-Jan-2019.) $)
|
|
fconstfvm $p |- ( E. y y e. A -> (
|
|
F : A --> { B } <->
|
|
( F Fn A /\ A. x e. A ( F ` x ) = B ) ) ) $=
|
|
( vw vz cv wcel wex csn wf wfn cfv wceq wral wa wrex eqeq1d sylan9bbr ffn
|
|
fvconst ralrimiva jca fvelrnb fveq2 rspccva rexbidva r19.9rmv velsn eqcom
|
|
crn bicomd bitr2i syl6bb eqrdv an32s exp31 imdistand wfo df-fo fof sylbir
|
|
syl6 impbid2 ) BHCIBJZCDKZELZECMZAHZENZDOZACPZQZVHVIVMCVGEUAVHVLACCDVJEUB
|
|
UCUDVFVNVIEULZVGOZQZVHVFVIVMVPVFVIVMVPVFVMVIVPVFVMQZVIQZFVOVGVSFHZVOIZDVT
|
|
OZVTVGIZVIWAGHZENZVTOZGCRZVRWBGCVTEUEVMWGWBGCRZVFWBVMWFWBGCVMWDCIQWEDVTVL
|
|
WEDOAWDCVJWDOVKWEDVJWDEUFSUGSUHVFWBWHWBGBCUIUMTTWCVTDOWBFDUJVTDUKUNUOUPUQ
|
|
URUSVQCVGEUTVHCVGEVACVGEVBVCVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y B $. $d y F $. $d y A $. $d y x $.
|
|
$( Two ways to express a constant function. (Contributed by Jim Kingdon,
|
|
8-Jan-2019.) $)
|
|
fconst3m $p |- ( E. x x e. A -> (
|
|
F : A --> { B } <->
|
|
( F Fn A /\ A C_ ( `' F " { B } ) ) ) ) $=
|
|
( vy cv wcel wex csn wf wfn cfv wceq wral wa ccnv cima wss fconstfvm wfun
|
|
cdm wb fnfun fndm eqimss2 syl funconstss syl2anc pm5.32i syl6bb ) AFBGAHB
|
|
CIZDJDBKZEFDLCMEBNZOULBDPUKQRZOEABCDSULUMUNULDTBDUAZRZUMUNUBBDUCULUOBMUPB
|
|
DUDBUOUEUFEBCDUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to express a constant function. (Contributed by NM,
|
|
8-Mar-2007.) $)
|
|
fconst4m $p |- ( E. x x e. A -> (
|
|
F : A --> { B } <->
|
|
( F Fn A /\ ( `' F " { B } ) = A ) ) ) $=
|
|
( cv wcel wex csn wf wfn ccnv cima wss wa wceq fconst3m cdm cnvimass fndm
|
|
syl5sseq biantrurd eqss syl6bbr pm5.32i syl6bb ) AEBFAGBCHZDIDBJZBDKUFLZM
|
|
ZNUGUHBOZNABCDPUGUIUJUGUIUHBMZUINUJUGUKUIUGDQUHBDUFRBDSTUAUHBUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The restriction of a function to a set exists. Compare Proposition 6.17
|
|
of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by
|
|
Mario Carneiro, 22-Jun-2013.) $)
|
|
resfunexg $p |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V ) $=
|
|
( vx wfun wcel cres cdm cfv cmpt cima cvv wceq wral ralrimiva 3syl adantr
|
|
wa cv sylancr cop crn funfvex fnasrng wfn funfn sylib dffn5im syl imadmrn
|
|
funres vex opexg dmmptg imaeq2d syl5reqr funmpt dmresexg adantl funimaexg
|
|
3eqtr4d eqeltrd ) AEZBCFZRZABGZDVFHZDSZVHVFIZUAZJZVGKZLVEDVGVIJZVKUBZVFVL
|
|
VCVMVNMZVDVCVFEZVILFZDVGNVOBAUKZVPVQDVGVHVFUCZODVGVILUDPQVEVFVGUEZVFVMMVE
|
|
VPVTVCVPVDVRQVFUFUGDVGVFUHUIVCVLVNMVDVCVNVKVKHZKVLVKUJVCWAVGVKVCVPVJLFZDV
|
|
GNWAVGMVRVPWBDVGVPVHVGFRVHLFVQWBDULVSVHVILLUMTODVGVJLUNPUOUPQVAVEVKEVGLFZ
|
|
VLLFDVGVJUQVDWCVCABCURUSVKVGLUTTVB $.
|
|
$}
|
|
|
|
$( If the domain of a function is a set, the function is a set. Theorem
|
|
6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom
|
|
of Replacement in the form of ~ resfunexg . (Contributed by NM,
|
|
14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
fnex $p |- ( ( F Fn A /\ A e. B ) -> F e. _V ) $=
|
|
( wfn wcel wrel cdm cres cvv fnrel adantr wfun wceq eleq1a impcom resfunexg
|
|
wa df-fn sylan2 anassrs sylanb resdm eleq1d biimpa syl2anc ) CADZABEZQCFZCC
|
|
GZHZIEZCIEZUFUHUGACJKUFCLZUIAMZQUGUKCARUMUNUGUKUNUGQUMUIBEZUKUGUNUOABUINOCU
|
|
IBPSTUAUHUKULUHUJCICUBUCUDUE $.
|
|
|
|
$( If the domain of a function exists, so does the function. Part of Theorem
|
|
4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of
|
|
Replacement in the form of ~ fnex . (Note: Any resemblance between
|
|
F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by
|
|
Swedish chefs.) (Contributed by NM, 11-Nov-1995.) $)
|
|
funex $p |- ( ( Fun F /\ dom F e. B ) -> F e. _V ) $=
|
|
( wfun cdm wfn wcel cvv funfn fnex sylanb ) BCBBDZEKAFBGFBHKABIJ $.
|
|
|
|
${
|
|
$d x y A $.
|
|
opabex.1 $e |- A e. _V $.
|
|
opabex.2 $e |- ( x e. A -> E* y ph ) $.
|
|
$( Existence of a function expressed as class of ordered pairs.
|
|
(Contributed by NM, 21-Jul-1996.) $)
|
|
opabex $p |- { <. x , y >. | ( x e. A /\ ph ) } e. _V $=
|
|
( cv wcel wa copab wfun cdm cvv wmo funopab moanimv mpbir mpgbir dmopabss
|
|
wi ssexi funex mp2an ) BGDHZAIZBCJZKZUFLZMHUFMHUGUECNZBUEBCOUIUDACNTFUDAC
|
|
PQRUHDEABCDSUAMUFUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( If the domain of a function given by maps-to notation is a set, the
|
|
function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
mptexg $p |- ( A e. V -> ( x e. A |-> B ) e. _V ) $=
|
|
( wcel cmpt wfun cdm cvv funmpt wss eqid dmmptss ssexg mpan funex sylancr
|
|
) BDEZABCFZGSHZIEZSIEABCJTBKRUAABCSSLMTBDNOISPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
mptex.1 $e |- A e. _V $.
|
|
$( If the domain of a function given by maps-to notation is a set, the
|
|
function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario
|
|
Carneiro, 20-Dec-2013.) $)
|
|
mptex $p |- ( x e. A |-> B ) e. _V $=
|
|
( cvv wcel cmpt mptexg ax-mp ) BEFABCGEFDABCEHI $.
|
|
$}
|
|
|
|
$( If the domain of a mapping is a set, the function is a set. (Contributed
|
|
by NM, 3-Oct-1999.) $)
|
|
fex $p |- ( ( F : A --> B /\ A e. C ) -> F e. _V ) $=
|
|
( wf wfn wcel cvv ffn fnex sylan ) ABDEDAFACGDHGABDIACDJK $.
|
|
|
|
${
|
|
$d f x y A $. $d f y B $.
|
|
eufnfv.1 $e |- A e. _V $.
|
|
eufnfv.2 $e |- B e. _V $.
|
|
$( A function is uniquely determined by its values. (Contributed by NM,
|
|
31-Aug-2011.) $)
|
|
eufnfv $p |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B ) $=
|
|
( vy cv wfn cfv wceq wral wa weu wb wal wex cmpt mptex cvv eqeq2 pm4.71ri
|
|
bibi2d albidv spcev eqid fnmpti mpbiri dffn5im eqeq1d wcel funfvex funfni
|
|
fneq1 ralrimiva mpteqb syl bitrd pm5.32i bitr2i mpg df-eu mpbir ) DHZBIZA
|
|
HZVDJZCKABLZMZDNVIVDGHZKZOZDPZGQZVIVDABCRZKZOZVNDVMVQDPGVOABCESVJVOKZVLVQ
|
|
DVRVKVPVIVJVOVDUAUCUDUEVPVEVPMVIVPVEVPVEVOBIABCVOFVOUFUGBVDVOUNUHUBVEVPVH
|
|
VEVPABVGRZVOKZVHVEVDVSVOABVDUIUJVEVGTUKZABLVTVHOVEWAABWABVFVDVFVDULUMUOAB
|
|
VGCTUPUQURUSUTVAVIDGVBVC $.
|
|
$}
|
|
|
|
$( A function's value in a preimage belongs to the image. (Contributed by
|
|
NM, 23-Sep-2003.) $)
|
|
funfvima $p |- ( ( Fun F /\ B e. dom F ) -> ( B e. A ->
|
|
( F ` B ) e. ( F " A ) ) ) $=
|
|
( wfun cdm wcel wa cfv cima cres dmres elin2 crn funres fvelrn sylan eleq1d
|
|
wi fvres df-ima eleq2i syl6rbbr syl5ibrcom syl5bir expd com12 impd pm2.43b
|
|
ex ) CDZBCEZFZGBAFZBCHZCAIZFZUMUJULUMUPRZUJUMULUQRUJUMULUQUMULGBCAJZEZFZUJU
|
|
QBAUKUSCAKLUJUTUQUJUTGUPUMBURHZURMZFZUJURDUTVCACNBUROPUMVCUNVBFUPUMVAUNVBBA
|
|
CSQUOVBUNCATUAUBUCUIUDUEUFUGUH $.
|
|
|
|
$( A function's value in an included preimage belongs to the image.
|
|
(Contributed by NM, 3-Feb-1997.) $)
|
|
funfvima2 $p |- ( ( Fun F /\ A C_ dom F ) -> ( B e. A ->
|
|
( F ` B ) e. ( F " A ) ) ) $=
|
|
( wfun cdm wss wcel cfv cima wi ssel funfvima ex com23 a2d syl5 imp ) CDZAC
|
|
EZFZBAGZBCHCAIGZJZTUABSGZJRUCASBKRUAUDUBRUDUAUBRUDUCABCLMNOPQ $.
|
|
|
|
${
|
|
$d x A $. $d x F $. $d x G $.
|
|
$( A class including a function contains the function's value in the image
|
|
of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) $)
|
|
funfvima3 $p |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F ->
|
|
( F ` A ) e. ( G " { A } ) ) ) $=
|
|
( vx wss wfun cdm wcel cfv csn cima wi wa cop funfvop ssel syl5 wb adantl
|
|
cvv imp cv simpr wceq sneq imaeq2d eleq2d eleq1d bibi12d funfvex elimasng
|
|
opeq1 vex sylancr vtocld mpbird exp32 impcom ) BCEZBFZABGZHZABIZCAJZKZHZL
|
|
USUTVBVFUSUTVBMZMVFAVCNZCHZUSVGVIVGVHBHUSVIABOBCVHPQUAVGVFVIRZUSVGVCCDUBZ
|
|
JZKZHZVKVCNZCHZRZVJDAVAUTVBUCVKAUDZVQVJRVGVRVNVFVPVIVRVMVEVCVRVLVDCVKAUEU
|
|
FUGVRVOVHCVKAVCULUHUISVGVKTHVCTHVQDUMABUJCVKVCTTUKUNUOSUPUQUR $.
|
|
$}
|
|
|
|
$( The function value of an operand in a set is contained in the image of
|
|
that set, using the ` Fn ` abbreviation. (Contributed by Stefan O'Rear,
|
|
10-Mar-2015.) $)
|
|
fnfvima $p |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) ) $=
|
|
( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtr4d
|
|
cfv jca simp3 funfvima2 sylc ) CAEZBAFZDBGZHZCIZBCJZFZKUFDCSCBLGUGUHUJUDUEU
|
|
HUFACMNUGBAUIUDUEUFOUDUEUIAPUFACQNRTUDUEUFUABDCUBUC $.
|
|
|
|
${
|
|
$d ph y $. $d ps x $. $d F x y $. $d B x y $. $d A x y $.
|
|
rexima.x $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
|
|
$( Existential quantification under an image in terms of the base set.
|
|
(Contributed by Stefan O'Rear, 21-Jan-2015.) $)
|
|
rexima $p |- ( ( F Fn A /\ B C_ A ) ->
|
|
( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $=
|
|
( wfn wss wa cv cfv cima cvv wcel ssel2 funfvex wceq wrex funfni fvelimab
|
|
sylan2 anassrs eqcom rexbii syl6bb wb adantl rexxfr2d ) GEIZFEJZKZABCDDLZ
|
|
GMZGFNZFOUKULUNFPZUOOPZULUQKUKUNEPURFEUNQUREUNGUNGRUAUCUDUMCLZUPPUOUSSZDF
|
|
TUSUOSZDFTDEFUSGUBUTVADFUOUSUEUFUGVAABUHUMHUIUJ $.
|
|
|
|
$( Universal quantification under an image in terms of the base set.
|
|
(Contributed by Stefan O'Rear, 21-Jan-2015.) $)
|
|
ralima $p |- ( ( F Fn A /\ B C_ A ) ->
|
|
( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $=
|
|
( wfn wss wa cv cfv cima cvv wcel ssel2 funfvex wceq wrex funfni fvelimab
|
|
sylan2 anassrs eqcom rexbii syl6bb wb adantl ralxfr2d ) GEIZFEJZKZABCDDLZ
|
|
GMZGFNZFOUKULUNFPZUOOPZULUQKUKUNEPURFEUNQUREUNGUNGRUAUCUDUMCLZUPPUOUSSZDF
|
|
TUSUOSZDFTDEFUSGUBUTVADFUOUSUEUFUGVAABUHUMHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d R x $.
|
|
$( TODO: This is the same as ~ issref (which has a much longer proof).
|
|
Should we replace ~ issref with this one? - NM 9-May-2016.
|
|
|
|
Two ways to state a relation is reflexive. (Adapted from Tarski.)
|
|
(Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro,
|
|
3-Nov-2015.) (Proof modification is discouraged.) $)
|
|
idref $p |- ( ( _I |` A ) C_ R <-> A. x e. A x R x ) $=
|
|
( cv cop wcel wral cmpt crn wss wbr cid cres wf eqid fmpt wfn opex fnmpti
|
|
vex mpbiran bitri df-br ralbii mptresid fnasrn eqtr3i sseq1i 3bitr4ri
|
|
df-f ) ADZUKEZCFZABGZABULHZIZCJZUKUKCKZABGLBMZCJUNBCUONZUQABCULUOUOOZPUTU
|
|
OBQUQABULUOUKUKATZVBRVASBCUOUJUAUBURUMABUKUKCUCUDUSUPCABUKHUSUPABUEABUKVB
|
|
UFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d y z B $. $d x y z A $.
|
|
elabrex.1 $e |- B e. _V $.
|
|
$( Elementhood in an image set. (Contributed by Mario Carneiro,
|
|
14-Jan-2014.) $)
|
|
elabrex $p |- ( x e. A -> B e. { y | E. x e. A y = B } ) $=
|
|
( vz cv wcel csb wceq wrex cab wtru tru csbeq1a equcoms a1tru 2thd rspcev
|
|
mpan2 eqeq1 rexbidv elab sylibr nfv nfcsb1v nfeq2 eqeq2d cbvrex syl6eleqr
|
|
abbii ) AGZCHZDBGZAFGZDIZJZFCKZBLZUNDJZACKZBLUMDUPJZFCKZDUSHUMMVCNVBMFULC
|
|
UOULJZVBMVBAFAUODOZPVDQRSTURVCBDEUTUQVBFCUNDUPUAUBUCUDVAURBUTUQAFCUTFUEAU
|
|
NUPAUODUFUGULUOJDUPUNVEUHUIUKUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B y z $. $d C w $. $d D y $. $d w x y $. $d w z y $.
|
|
abrexco.1 $e |- B e. _V $.
|
|
abrexco.2 $e |- ( y = B -> C = D ) $.
|
|
$( Composition of two image maps ` C ( y ) ` and ` B ( w ) ` .
|
|
(Contributed by NM, 27-May-2013.) $)
|
|
abrexco $p |- { x | E. y e. { z | E. w e. A z = B } x = C } =
|
|
{ x | E. w e. A x = D } $=
|
|
( cv wceq wrex cab wa wex wcel df-rex bitr4i bitri vex eqeq1 rexbidv elab
|
|
anbi1i r19.41v exbii rexcom4 eqeq2d ceqsexv rexbii abbii ) AKZGLZBCKZFLZD
|
|
EMZCNZMZUMHLZDEMZAUSBKZFLZUNOZBPZDEMZVAUSVDDEMZBPZVFUSVBURQZUNOZBPVHUNBUR
|
|
RVJVGBVJVCDEMZUNOVGVIVKUNUQVKCVBBUAUOVBLUPVCDEUOVBFUBUCUDUEVCUNDEUFSUGTVD
|
|
DBEUHSVEUTDEUNUTBFIVCGHUMJUIUJUKTUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d y z C $.
|
|
$( The image of an indexed union is the indexed union of the images.
|
|
(Contributed by Mario Carneiro, 18-Jun-2014.) $)
|
|
imaiun $p |- ( A " U_ x e. B C ) = U_ x e. B ( A " C ) $=
|
|
( vy vz ciun cima cv wcel cop wa wex wrex rexcom4 vex elima3 rexbii eliun
|
|
anbi1i r19.41v bitr4i exbii 3bitr4ri 3bitr4i eqriv ) EBACDGZHZACBDHZGZFIZ
|
|
UGJZUKEIZKBJZLZFMZUMUIJZACNZUMUHJUMUJJUKDJZUNLZFMZACNUTACNZFMURUPUTAFCOUQ
|
|
VAACFUMBDEPZQRUOVBFUOUSACNZUNLVBULVDUNAUKCDSTUSUNACUAUBUCUDFUMBUGVCQAUMCU
|
|
ISUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The image of a union is the indexed union of the images. Theorem 3K(a)
|
|
of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened
|
|
by Mario Carneiro, 18-Jun-2014.) $)
|
|
imauni $p |- ( A " U. B ) = U_ x e. B ( A " x ) $=
|
|
( cuni cima cv ciun uniiun imaeq2i imaiun eqtri ) BCDZEBACAFZGZEACBMEGLNB
|
|
ACHIABCMJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( The indexed union of a function's values is the union of its range.
|
|
Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM,
|
|
27-Sep-2004.) $)
|
|
fniunfv $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F ) $=
|
|
( vy wfn cv cfv ciun wceq wrex cab cuni crn wcel funfvex funfni ralrimiva
|
|
cvv wral dfiun2g syl fnrnfv unieqd eqtr4d ) CBEZABAFZCGZHZDFUGIABJDKZLZCM
|
|
ZLUEUGRNZABSUHUJIUEULABULBUFCUFCOPQADBUGRTUAUEUKUIADBCUBUCUD $.
|
|
|
|
$( The indexed union of a function's values is the union of its image under
|
|
the index class. This theorem is a slight variation of ~ fniunfv .
|
|
(Contributed by Jim Kingdon, 10-Jan-2019.) $)
|
|
funiunfvdm $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
|
|
( wfn cv cfv ciun crn cuni cima fniunfv cdm imadmrn imaeq2d syl5eqr eqtrd
|
|
fndm unieqd ) CBDZABAECFGCHZICBJZIABCKSTUASTCCLZJUACMSUBBCBCQNORP $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d z F $.
|
|
funiunfvf.1 $e |- F/_ x F $.
|
|
$( The indexed union of a function's values is the union of its image under
|
|
the index class. This version of ~ funiunfvdm uses a bound-variable
|
|
hypothesis in place of a distinct variable condition. (Contributed by
|
|
Jim Kingdon, 10-Jan-2019.) $)
|
|
funiunfvdmf $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
|
|
( vz wfn cv cfv ciun cima cuni nfcv nffv fveq2 cbviun funiunfvdm syl5eqr
|
|
) CBFABAGZCHZIEBEGZCHZICBJKEABUASATCDATLMESLTRCNOEBCPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( Membership in the union of an image of a function. (Contributed by Jim
|
|
Kingdon, 10-Jan-2019.) $)
|
|
eluniimadm $p |- ( F Fn A ->
|
|
( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) ) $=
|
|
( cv cfv wcel wrex ciun wfn cima cuni eliun funiunfvdm eleq2d syl5rbbr )
|
|
CAEDFZGABHCABQIZGDBJZCDBKLZGACBQMSRTCABDNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Membership in the union of the range of a function. (Contributed by NM,
|
|
24-Sep-2006.) $)
|
|
elunirn $p |- ( Fun F -> ( A e. U. ran F <->
|
|
E. x e. dom F A e. ( F ` x ) ) ) $=
|
|
( vy crn cuni wcel cv wa wex wfun cfv cdm wrex eluni wceq wfn eleq2 cvv
|
|
wb funfn fvelrnb anbi2d r19.42v syl6bbr biimparc reximi syl6bi exlimdv wi
|
|
sylbi fvelrn funfvex eleq1 anbi12d spcegv mpan2d rexlimdva impbid syl5bb
|
|
syl ) BCEZFGBDHZGZVCVBGZIZDJZCKZBAHZCLZGZACMZNZDBVBOVHVGVMVHVFVMDVHVFVDVJ
|
|
VCPZIZAVLNZVMVHVFVDVNAVLNZIVPVHVEVQVDVHCVLQVEVQTCUAAVLVCCUBUKUCVDVNAVLUDU
|
|
EVOVKAVLVNVKVDVJVCBRUFUGUHUIVHVKVGAVLVHVIVLGIZVKVJVBGZVGVICULVRVJSGVKVSIZ
|
|
VGUJVICUMVFVTDVJSVCVJPVDVKVEVSVCVJBRVCVJVBUNUOUPVAUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x I $. $d x F $.
|
|
$( Membership in a union of some function-defined family of sets.
|
|
(Contributed by Stefan O'Rear, 30-Jan-2015.) $)
|
|
fnunirn $p |- ( F Fn I -> ( A e. U. ran F <->
|
|
E. x e. I A e. ( F ` x ) ) ) $=
|
|
( wfn crn cuni wcel cv cfv cdm wrex wfun fnfun elunirn fndm rexeqdv bitrd
|
|
wb syl ) CDEZBCFGHZBAICJHZACKZLZUCADLUACMUBUESDCNABCOTUAUCAUDDDCPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d z B $. $d x y z F $.
|
|
$( A one-to-one function in terms of function values. Compare Theorem
|
|
4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.) $)
|
|
dff13 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( vz cv wbr wal wa cfv wceq wi wral wb wcel vex breldm eleq2d syl5bb fndm
|
|
wf1 wf wmo dff12 wfn ffn syl5ib anim12d pm4.71rd fnbrfvb bi2anan9 anandis
|
|
cdm pm5.32da bitr4d imbi1d impexp syl6bb albidv 19.21v wex funfvex funfni
|
|
eqcom cvv eqvincg syl syl6rbbr adantrr pm5.74da bitrd 2albidv breq1 albii
|
|
19.23v mo4 alrot3 bitri r2al 3bitr4g pm5.32i ) CDEUBCDEUCZAGZFGZEHZAUDZFI
|
|
ZJWCWDEKZBGZEKZLZWDWJLZMZBCNACNZJAFCDEUEWCWHWOWCECUFZWHWOOCDEUGWPWFWJWEEH
|
|
ZJZWMMZFIZBIAIZWDCPZWJCPZJZWNMZBIAIWHWOWPWTXEABWPWTXDWEWILZWEWKLZJZWMMZMZ
|
|
FIZXEWPWSXJFWPWSXDXHJZWMMXJWPWRXLWMWPWRXDWRJXLWPWRXDWPWFXBWQXCWFWDEUNZPWP
|
|
XBWDWEEAQFQZRWPXMCWDCEUAZSUHWQWJXMPWPXCWJWEEBQXNRWPXMCWJXOSUHUIUJWPXDXHWR
|
|
WPXBXCXHWROWPXBJZXFWFWPXCJZXGWQXFWIWELXPWFWEWIVECWDWEEUKTXGWKWELXQWQWEWKV
|
|
ECWJWEEUKTULUMUOUPUQXDXHWMURUSUTXKXDXIFIZMWPXEXDXIFVAWPXDXRWNWPXBXRWNOXCX
|
|
PWNXHFVBZWMMXRXPWLXSWMXPWIVFPZWLXSOXTCWDEWDEVCVDFWIWKVFVGVHUQXHWMFVPVIVJV
|
|
KTVLVMWHWSBIAIZFIXAWGYAFWFWQABWDWJWEEVNVQVOWSFABVRVSWNABCCVTWAVHWBVS $.
|
|
$}
|
|
|
|
${
|
|
$d A c d $. $d C c d $. $d D d $. $d F c d $.
|
|
$( If the values of a one-to-one function for two arguments are equal, the
|
|
arguments themselves must be equal. (Contributed by Alexander van der
|
|
Vekens, 12-Nov-2017.) $)
|
|
f1veqaeq $p |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) ->
|
|
( ( F ` C ) = ( F ` D ) -> C = D ) ) $=
|
|
( vc vd wf1 wcel wa cfv wceq wi wf cv weq wral dff13 fveq2 imbi12d eqeq1d
|
|
eqeq1 eqeq2d eqeq2 rspc2v com12 adantl sylbi imp ) ABEHZCAIDAIJZCEKZDEKZL
|
|
ZCDLZMZUJABENZFOZEKZGOZEKZLZFGPZMZGAQFAQZJUKUPMZFGABERVEVFUQUKVEUPVDUPULV
|
|
ALZCUTLZMFGCDAAURCLZVBVGVCVHVIUSULVAURCESUAURCUTUBTUTDLZVGUNVHUOVJVAUMULU
|
|
TDESUCUTDCUDTUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v A $. $d w v B $. $d w v F $.
|
|
dff13f.1 $e |- F/_ x F $.
|
|
dff13f.2 $e |- F/_ y F $.
|
|
$( A one-to-one function in terms of function values. Compare Theorem
|
|
4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) $)
|
|
dff13f $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( vw vv cv cfv wceq weq wi wral wa nfcv nffv nfeq nfv wf dff13 nfim fveq2
|
|
eqeq2d equequ2 imbi12d cbvral ralbii nfralxy eqeq1d equequ1 ralbidv bitri
|
|
wf1 anbi2i ) CDEUOCDEUAZHJZEKZIJZEKZLZHIMZNZICOZHCOZPUQAJZEKZBJZEKZLZABMZ
|
|
NZBCOZACOZPHICDEUBVFVOUQVFUSVJLZHBMZNZBCOZHCOVOVEVSHCVDVRIBCVBVCBBUSVABUR
|
|
EGBURQRBUTEGBUTQRSVCBTUCVRITIBMZVBVPVCVQVTVAVJUSUTVIEUDUEIBHUFUGUHUIVSVNH
|
|
ACVRABCACQVPVQAAUSVJAUREFAURQRAVIEFAVIQRSVQATUCUJVNHTHAMZVRVMBCWAVPVKVQVL
|
|
WAUSVHVJURVGEUDUKHABULUGUMUHUNUPUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $. $d y F $.
|
|
f1mpt.1 $e |- F = ( x e. A |-> C ) $.
|
|
f1mpt.2 $e |- ( x = y -> C = D ) $.
|
|
$( Express injection for a mapping operation. (Contributed by Mario
|
|
Carneiro, 2-Jan-2017.) $)
|
|
f1mpt $p |- ( F : A -1-1-> B <->
|
|
( A. x e. A C e. B /\ A. x e. A A. y e. A ( C = D -> x = y ) ) ) $=
|
|
( wf1 wf cv cfv wceq wi wral wa wcel wb ralbi nfmpt1 nfcxfr dff13f anbi1i
|
|
cmpt nfcv fmpt eleq1d raaanv fvmpt2 fvmptg eqeqan12d an4s imbi1d ralimdva
|
|
cbvralv ex syl6 ralimia syl sylbir sylan2b anidms pm5.32i 3bitr2i ) CDGJC
|
|
DGKZALZGMZBLZGMZNZVGVINZOZBCPZACPZQEDRZACPZVOQVQEFNZVLOZBCPZACPZQABCDGAGA
|
|
CEUEHACEUAUBBGUFUCVQVFVOACDEGHUGUDVQVOWAVQVOWASZVQVQFDRZBCPZWBVPWCABCVLEF
|
|
DIUHUPVQWDQVPWCQZBCPZACPZWBVPWCABCUIWGVNVTSZACPWBWFWHACVGCRZWFVMVSSZBCPWH
|
|
WIWEWJBCWIVICRZQZWEWJWLWEQVKVRVLWIVPWKWCVKVRSWIVPQWKWCQVHEVJFACEDGHUJAVIE
|
|
FCDGIHUKULUMUNUQUOVMVSBCTURUSVNVTACTUTVAVBVCVDVE $.
|
|
$}
|
|
|
|
$( Equality of function values for a one-to-one function. (Contributed by
|
|
NM, 11-Feb-1997.) $)
|
|
f1fveq $p |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) ->
|
|
( ( F ` C ) = ( F ` D ) <-> C = D ) ) $=
|
|
( wf1 wcel wa cfv wceq f1veqaeq fveq2 impbid1 ) ABEFCAGDAGHHCEIDEIJCDJABCDE
|
|
KCDELM $.
|
|
|
|
${
|
|
$d F z $. $d A z $. $d Y z $. $d X z $. $d B z $.
|
|
$( Membership in the image of a 1-1 map. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
f1elima $p |- ( ( F : A -1-1-> B /\ X e. A /\ Y C_ A )
|
|
-> ( ( F ` X ) e. ( F " Y ) <-> X e. Y ) ) $=
|
|
( vz wf1 wcel wss w3a cfv cima cv wceq wrex wb wfn wi wa anassrs fvelimab
|
|
f1fn sylan 3adant2 ssel impac f1fveq ancom2s biimpd biimpcd sylan9 anasss
|
|
eleq1 sylan2 rexlimdva 3impa eqid fveq2 eqeq1d rspcev mpan2 impbid1 bitrd
|
|
) ABCGZDAHZEAIZJZDCKZCELHZFMZCKZVHNZFEOZDEHZVDVFVIVMPZVEVDCAQVFVOABCUBFAE
|
|
VHCUAUCUDVGVMVNVDVEVFVMVNRVDVESZVFSVLVNFEVPVFVJEHZVLVNRZVFVQSVPVJAHZVQSVR
|
|
VFVQVSEAVJUEUFVPVSVQVRVPVSSVLVJDNZVQVNVDVEVSVLVTRVDVEVSSSVLVTVDVSVEVLVTPA
|
|
BVJDCUGUHUITVTVQVNVJDEUMUJUKULUNTUOUPVNVHVHNZVMVHUQVLWAFDEVTVKVHVHVJDCURU
|
|
SUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d F a $. $d A a $. $d B a $. $d C a $. $d D a $.
|
|
$( Taking images under a one-to-one function preserves subsets.
|
|
(Contributed by Stefan O'Rear, 30-Oct-2014.) $)
|
|
f1imass $p |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) ->
|
|
( ( F " C ) C_ ( F " D ) <-> C C_ D ) ) $=
|
|
( va wf1 wss wa cima cv wcel wi simplrl sseld wb 3expa f1elima syl3anc ex
|
|
simplr simplll simpr simp1rl simp1rr 3imtr3d pm2.43d ssrdv imass2 impbid1
|
|
cfv syld ) ABEGZCAHZDAHZIZIZECJZEDJZHZCDHZUQUTVAUQUTIZFCDVBFKZCLZVCDLZVBV
|
|
DVCALZVDVEMZVBCAVCUMUNUOUTNOVBVFVGVBVFIZVCEUKZURLZVIUSLZVDVEVHURUSVIUQUTV
|
|
FUAOVHUMVFUNVJVDPUMUPUTVFUBZVBVFUCZUQUTVFUNUNUOUMUTVFUDQABEVCCRSVHUMVFUOV
|
|
KVEPVLVMUQUTVFUOUNUOUMUTVFUEQABEVCDRSUFTULUGUHTCDEUIUJ $.
|
|
|
|
$( Taking images under a one-to-one function preserves equality.
|
|
(Contributed by Stefan O'Rear, 30-Oct-2014.) $)
|
|
f1imaeq $p |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) ->
|
|
( ( F " C ) = ( F " D ) <-> C = D ) ) $=
|
|
( wf1 wss wa cima wceq f1imass wb ancom2s anbi12d eqss 3bitr4g ) ABEFZCAG
|
|
ZDAGZHHZECIZEDIZGZUBUAGZHCDGZDCGZHUAUBJCDJTUCUEUDUFABCDEKQSRUDUFLABDCEKMN
|
|
UAUBOCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( A one-to-one onto function in terms of function values. (Contributed by
|
|
NM, 29-Mar-2008.) $)
|
|
dff1o6 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( wf1o wf1 wfo wa wf cv cfv wceq wi wral wfn crn w3a df-f1o dff13 anbi12i
|
|
df-fo df-3an wss eqimss anim2i df-f sylibr pm4.71ri anbi1i 3bitrri 3bitri
|
|
an32 ) CDEFCDEGZCDEHZICDEJZAKZELBKZELMUQURMNBCOACOZIZECPZEQZDMZIZIZVAVCUS
|
|
RZCDESUNUTUOVDABCDETCDEUBUAVFVDUSIUPVDIZUSIVEVAVCUSUCVDVGUSVDUPVDVAVBDUDZ
|
|
IUPVCVHVAVBDUEUFCDEUGUHUIUJUPVDUSUMUKUL $.
|
|
$}
|
|
|
|
$( The converse value of the value of a one-to-one onto function.
|
|
(Contributed by NM, 20-May-2004.) $)
|
|
f1ocnvfv1 $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
|
|
( `' F ` ( F ` C ) ) = C ) $=
|
|
( wf1o wcel wa ccnv ccom cfv cid cres wceq f1ococnv1 fveq1d adantr wf fvco3
|
|
f1of sylan fvresi adantl 3eqtr3d ) ABDEZCAFZGCDHZDIZJZCKALZJZCDJUFJZCUDUHUJ
|
|
MUEUDCUGUIABDNOPUDABDQUEUHUKMABDSABCUFDRTUEUJCMUDACUAUBUC $.
|
|
|
|
$( The value of the converse value of a one-to-one onto function.
|
|
(Contributed by NM, 20-May-2004.) $)
|
|
f1ocnvfv2 $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
( F ` ( `' F ` C ) ) = C ) $=
|
|
( wf1o wcel ccnv ccom cfv cid cres wceq f1ococnv2 fveq1d adantr f1ocnv f1of
|
|
wa wf syl fvco3 sylan fvresi adantl 3eqtr3d ) ABDEZCBFZRCDDGZHZIZCJBKZIZCUH
|
|
IDIZCUFUJULLUGUFCUIUKABDMNOUFBAUHSZUGUJUMLUFBAUHEUNABDPBAUHQTBACDUHUAUBUGUL
|
|
CLUFBCUCUDUE $.
|
|
|
|
$( Relationship between the value of a one-to-one onto function and the value
|
|
of its converse. (Contributed by Raph Levien, 10-Apr-2004.) $)
|
|
f1ocnvfv $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
|
|
( ( F ` C ) = D -> ( `' F ` D ) = C ) ) $=
|
|
( cfv wceq ccnv wf1o wcel wa fveq2 eqcoms f1ocnvfv1 eqeq2d syl5ib ) CEFZDGD
|
|
EHZFZQRFZGZABEICAJKZSCGUADQDQRLMUBTCSABCENOP $.
|
|
|
|
$( Relationship between the value of a one-to-one onto function and the value
|
|
of its converse. (Contributed by NM, 20-May-2004.) $)
|
|
f1ocnvfvb $p |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) ->
|
|
( ( F ` C ) = D <-> ( `' F ` D ) = C ) ) $=
|
|
( wf1o wcel w3a cfv wceq ccnv wi f1ocnvfv 3adant3 wa fveq2 eqcoms f1ocnvfv2
|
|
eqeq2d syl5ib 3adant2 impbid ) ABEFZCAGZDBGZHCEIZDJZDEKIZCJZUCUDUGUILUEABCD
|
|
EMNUCUEUIUGLUDUIUFUHEIZJZUCUEOZUGUKCUHCUHEPQULUJDUFABDERSTUAUB $.
|
|
|
|
$( The value of the converse of a one-to-one onto function belongs to its
|
|
domain. (Contributed by NM, 26-May-2006.) $)
|
|
f1ocnvdm $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
( `' F ` C ) e. A ) $=
|
|
( wf1o ccnv wf f1ocnv f1of syl ffvelrnda ) ABDEZBACDFZLBAMEBAMGABDHBAMIJK
|
|
$.
|
|
|
|
$( If the values of a one-to-one function for two arguments from the range of
|
|
the function are equal, the arguments themselves must be equal.
|
|
(Contributed by Alexander van der Vekens, 12-Nov-2017.) $)
|
|
f1ocnvfvrneq $p |- ( ( F : A -1-1-> B /\ ( C e. ran F /\ D e. ran F ) ) ->
|
|
( ( `' F ` C ) = ( `' F ` D ) -> C = D ) ) $=
|
|
( wf1 crn wcel wa ccnv cfv wceq wi wf1o f1f1orn f1ocnv f1of1 f1veqaeq 4syl
|
|
ex imp ) ABEFZCEGZHDUCHIZCEJZKDUEKLCDLMZUBAUCENUCAUENUCAUEFZUDUFMABEOAUCEPU
|
|
CAUEQUGUDUFUCACDUERTSUA $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $. $d R x y $.
|
|
$( An application is injective if a retraction exists. Proposition 8 of
|
|
[BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised
|
|
by Mario Carneiro, 27-Dec-2014.) $)
|
|
fcof1 $p |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) )
|
|
-> F : A -1-1-> B ) $=
|
|
( vx vy wf ccom wceq wa cfv wral wcel fvco3 syl2anc fveq1d 3eqtr3d fvresi
|
|
cv syl cid cres wi wf1 simpl simprr fveq2d simpll simprll simprlr 3eqtr4d
|
|
simplr expr ralrimivva dff13 sylanbrc ) ABDGZCDHZUAAUBZIZJZUQESZDKZFSZDKZ
|
|
IZVBVDIZUCZFALEALABDUDUQUTUEVAVHEFAAVAVBAMZVDAMZJZVFVGVAVKVFJZJZVBUSKZVDU
|
|
SKZVBVDVMVBURKZVDURKZVNVOVMVCCKZVECKZVPVQVMVCVECVAVKVFUFUGVMUQVIVPVRIUQUT
|
|
VLUHZVAVIVJVFUIZABVBCDNOVMUQVJVQVSIVTVAVIVJVFUJZABVDCDNOUKVMVBURUSUQUTVLU
|
|
LZPVMVDURUSWCPQVMVIVNVBIWAAVBRTVMVJVOVDIWBAVDRTQUMUNEFABDUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $. $d S x y $.
|
|
$( An application is surjective if a section exists. Proposition 8 of
|
|
[BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof
|
|
shortened by Mario Carneiro, 27-Dec-2014.) $)
|
|
fcofo $p |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) )
|
|
-> F : A -onto-> B ) $=
|
|
( vy vx wf ccom cid cres wceq w3a cfv wrex wral wfo simp1 wcel 3ad2antl2
|
|
cv ffvelrn simpl3 fveq1d fvco3 fvresi adantl 3eqtr3rd fveq2 eqeq2d rspcev
|
|
wa syl2anc ralrimiva dffo3 sylanbrc ) ABDGZBACGZDCHZIBJZKZLZUPETZFTZDMZKZ
|
|
FANZEBOABDPUPUQUTQVAVFEBVAVBBRZUKZVBCMZARZVBVIDMZKZVFUQUPVGVJUTBAVBCUASVH
|
|
VBURMZVBUSMZVKVBVHVBURUSUPUQUTVGUBUCUQUPVGVMVKKUTBAVBDCUDSVGVNVBKVABVBUEU
|
|
FUGVEVLFVIAVCVIKVDVKVBVCVIDUHUIUJULUMFEABDUNUO $.
|
|
$}
|
|
|
|
${
|
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$d x y A $. $d y B $. $d x y F $. $d y ph $. $d x ps $.
|
|
cbvfo.1 $e |- ( ( F ` x ) = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variable between domain and range of function.
|
|
(Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro,
|
|
21-Mar-2015.) $)
|
|
cbvfo $p |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) ) $=
|
|
( wfo crn wral wfn wb fofn cv cfv wceq bicomd eqcoms ralrn raleqdv bitr3d
|
|
syl forn ) EFGIZBDGJZKZACEKZBDFKUEGELUGUHMEFGNBADCEGBAMCOGPZDOZUIUJQABHRS
|
|
TUCUEBDUFFEFGUDUAUB $.
|
|
|
|
$( Change bound variable between domain and range of function.
|
|
(Contributed by NM, 23-Feb-1997.) $)
|
|
cbvexfo $p |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) ) $=
|
|
( wfo crn wrex wfn wb fofn cv cfv wceq bicomd eqcoms rexrn rexeqdv bitr3d
|
|
syl forn ) EFGIZBDGJZKZACEKZBDFKUEGELUGUHMEFGNBADCEGBAMCOGPZDOZUIUJQABHRS
|
|
TUCUEBDUFFEFGUDUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d F x $. $d H x $. $d K x $.
|
|
$( An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.)
|
|
(Revised by Mario Carneiro, 21-Mar-2015.) $)
|
|
cocan1 $p |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) ->
|
|
( ( F o. H ) = ( F o. K ) <-> H = K ) ) $=
|
|
( vx wf ccom cfv wceq wral wcel fvco3 3ad2antl2 wb wfn ffn syl syl2anc cv
|
|
wf1 wa 3ad2antl3 eqeq12d simpl1 ffvelrn f1fveq syl12anc ralbidva 3ad2ant1
|
|
w3a bitrd f1f simp2 fnfco simp3 eqfnfv 3bitr4d ) BCDUBZABEHZABFHZULZGUAZD
|
|
EIZJZVDDFIZJZKZGALZVDEJZVDFJZKZGALZVEVGKZEFKZVCVIVMGAVCVDAMZUCZVIVKDJZVLD
|
|
JZKZVMVRVFVSVHVTVAUTVQVFVSKVBABVDDENOVBUTVQVHVTKVAABVDDFNUDUEVRUTVKBMZVLB
|
|
MZWAVMPUTVAVBVQUFVAUTVQWBVBABVDEUGOVBUTVQWCVAABVDFUGUDBCVKVLDUHUIUMUJVCVE
|
|
AQZVGAQZVOVJPVCDBQZVAWDVCBCDHZWFUTVAWGVBBCDUNUKBCDRSZUTVAVBUOZBADEUPTVCWF
|
|
VBWEWHUTVAVBUQZBADFUPTGAVEVGURTVCEAQZFAQZVPVNPVCVAWKWIABERSVCVBWLWJABFRSG
|
|
AEFURTUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $. $d x y H $. $d x y K $.
|
|
$( A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.)
|
|
(Proof shortened by Mario Carneiro, 21-Mar-2015.) $)
|
|
cocan2 $p |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) ->
|
|
( ( H o. F ) = ( K o. F ) <-> H = K ) ) $=
|
|
( vy vx wfn cv ccom cfv wceq wral 3ad2ant1 fvco3 sylan eqeq12d wb syl2anc
|
|
fveq2 wfo w3a wcel wa fof ralbidva cbvfo bitrd simp2 fnfco eqfnfv 3bitr4d
|
|
wf simp3 ) ABCUAZDBHZEBHZUBZFIZDCJZKZUSECJZKZLZFAMZGIZDKZVFEKZLZGBMZUTVBL
|
|
ZDELZURVEUSCKZDKZVMEKZLZFAMZVJURVDVPFAURUSAUCZUDVAVNVCVOURABCUMZVRVAVNLUO
|
|
UPVSUQABCUENZABUSDCOPURVSVRVCVOLVTABUSECOPQUFUOUPVQVJRUQVPVIFGABCVMVFLVNV
|
|
GVOVHVMVFDTVMVFETQUGNUHURUTAHZVBAHZVKVERURUPVSWAUOUPUQUIZVTBADCUJSURUQVSW
|
|
BUOUPUQUNZVTBAECUJSFAUTVBUKSURUPUQVLVJRWCWDGBDEUKSUL $.
|
|
$}
|
|
|
|
$( Show that two functions are inverse to each other by computing their
|
|
compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) $)
|
|
fcof1o $p |- ( ( ( F : A --> B /\ G : B --> A ) /\
|
|
( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) ->
|
|
( F : A -1-1-onto-> B /\ `' F = G ) ) $=
|
|
( wf wa ccom cid cres wceq wf1o wf1 wfo fcof1 ad2ant2rl fcofo 3expa adantrr
|
|
ccnv df-f1o sylanbrc simprl coeq2d coass f1ococnv1 syl fcoi2 ad2antlr eqtrd
|
|
coeq1d syl5eqr f1ocnv f1of fcoi1 4syl 3eqtr3rd jca ) ABCEZBADEZFZCDGZHBIZJZ
|
|
DCGHAIZJZFZFZABCKZCSZDJVGABCLZABCMZVHURVEVJUSVCABDCNOUTVCVKVEURUSVCVKABDCPQ
|
|
RABCTUAZVGVIVAGZVIVBGZDVIVGVAVBVIUTVCVEUBUCVGVMVICGZDGZDVICDUDVGVPVDDGZDVGV
|
|
OVDDVGVHVOVDJVLABCUEUFUJUSVQDJURVFBADUGUHUIUKVGVHBAVIKBAVIEVNVIJVLABCULBAVI
|
|
UMBAVIUNUOUPUQ $.
|
|
|
|
${
|
|
$d F x y $. $d G x y $. $d A x y $. $d B x y $.
|
|
$( Condition for function equality in terms of vanishing of the composition
|
|
with the converse. _EDITORIAL_: Is there a relation-algebraic proof of
|
|
this? (Contributed by Stefan O'Rear, 12-Feb-2015.) $)
|
|
foeqcnvco $p |- ( ( F : A -onto-> B /\ G : A -onto-> B ) ->
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|
( F = G <-> ( F o. `' G ) = ( _I |` B ) ) ) $=
|
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( vx vy wfo wa wceq ccnv ccom adantr wfn wcel wbr cvv syl syl2anc adantlr
|
|
wb cid cres fococnv2 cnveq coeq2d eqeq1d syl5ibcom fofn ad2antrr ad2antlr
|
|
wi cv cfv wex cop adantl fnopfv anim1i adantll funfvex funfni vex sylancl
|
|
sylan brcnvg df-br syl6bb mpbird sylibr breq2 breq1 anbi12d spcev breq wf
|
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brcog mpbid fof ffvelrnda resieq eqcomd eqfnfvd ex impbid ) ABCGZABDGZHZC
|
|
DIZCDJZKZUABUBZIZWEWHWLUKWFWECCJZKZWKIWHWLABCUCWHWNWJWKWHWMWICCDUDUEUFUGL
|
|
WGWLWHWGWLHZEACDWECAMZWFWLABCUHZUIWFDAMZWEWLABDUHZUJWOEULZANZHZWTDUMZWTCU
|
|
MZXBXCXDWKOZXCXDIZXBXCXDWJOZXEWGXAXGWLWGXAHZXGXCFULZWIOZXIXDCOZHZFUNZXHXC
|
|
WTWIOZWTXDCOZXMXHXNWTXCUODNZWGWRXAXPWFWRWEWSUPAWTDUQVDXHWRXAHZXNXPTWFXAXQ
|
|
WEWFWRXAWSURUSZXQXNWTXCDOZXPXQXCPNZWTPNXNXSTXTAWTDWTDUTVAZEVBZXCWTPPDVEVC
|
|
WTXCDVFVGQVHXHWTXDUOCNZXOWGWPXAYCWEWPWFWQLAWTCUQVDWTXDCVFVIXLXNXOHFWTYBXI
|
|
WTIXJXNXKXOXIWTXCWIVJXIWTXDCVKVLVMRXHXTXDPNZXGXMTXHXQXTXRYAQXHWPXAHZYDWEX
|
|
AYEWFWEWPXAWQURSYDAWTCWTCUTVAQFXCXDCWIPPVPRVHSWLXGXETWGXAXCXDWJWKVNUJVQWG
|
|
XAXEXFTZWLXHXCBNXDBNYFWGABWTDWFABDVOWEABDVRUPVSWGABWTCWEABCVOWFABCVRLVSBX
|
|
CXDVTRSVQWAWBWCWD $.
|
|
|
|
$( Condition for function equality in terms of vanishing of the composition
|
|
with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) $)
|
|
f1eqcocnv $p |- ( ( F : A -1-1-> B /\ G : A -1-1-> B ) ->
|
|
( F = G <-> ( `' F o. G ) = ( _I |` A ) ) ) $=
|
|
( vx vy wf1 wa wceq ccom wi adantr wfn f1fn adantl cv wcel wbr wb syl cid
|
|
ccnv cres f1cocnv1 coeq2 eqeq1d syl5ibcom equid resieq mpbiri anidms breq
|
|
cfv ad2antlr mpbird wex vex brco cop wfun cdm fnfun fndm biimpar funopfvb
|
|
eleq2d syl2anc bicomd df-br 3bitr4g biimpd syl6rbbr anim12d eximdv syl5bi
|
|
eqcom brcnv cvv anim1i adantll funfvex funfni eqvincg sylibrd adantlr mpd
|
|
3syl eqfnfvd eqcomd ex impbid ) ABCGZABDGZHZCDIZCUBZDJZUAAUCZIZWLWOWSKWMW
|
|
LWPCJZWRIWOWSABCUDWOWTWQWRCDWPUEUFUGLWNWSWOWNWSHZDCXAEADCWNDAMZWSWMXBWLAB
|
|
DNZOZLWNCAMZWSWLXEWMABCNLZLXAEPZAQZHZXGXGWQRZXGDUMZXGCUMZIZXIXJXGXGWRRZXH
|
|
XNXAXHXNXHXHHXNXGXGIEUHAXGXGUIUJUKOWSXJXNSWNXHXGXGWQWRULUNUOWNXHXJXMKWSWN
|
|
XHHZXJFPZXKIZXPXLIZHZFUPZXMXJXGXPDRZXPXGWPRZHZFUPXOXTFXGXGWPDEUQZYDURXOYC
|
|
XSFXOYAXQYBXRXOYAXQXOXGXPUSZDQZXKXPIZYAXQXOYGYFXODUTZXGDVAZQZYGYFSWNYHXHW
|
|
NXBYHXDADVBTLWNYJXHWNYIAXGWNXBYIAIXDADVCTVFVDXGXPDVEVGVHXGXPDVIXPXKVPVJVK
|
|
XOYBXRXOXGXPCRZXLXPIZYBXRXOYLYECQZYKXOCUTZXGCVAZQZYLYMSWNYNXHWNXEYNXFACVB
|
|
TLWNYPXHWNYOAXGWNXEYOAIXFACVCTVFVDXGXPCVEVGXGXPCVIVLXPXGCFUQYDVQXPXLVPVJV
|
|
KVMVNVOXOXBXHHZXKVRQZXMXTSWMXHYQWLWMXBXHXCVSVTYRAXGDXGDWAWBFXKXLVRWCWGWDW
|
|
EWFWHWIWJWK $.
|
|
$}
|
|
|
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${
|
|
$d u v y z A $. $d u v y z B $. $d u v x z C $. $d x y z R $. $d x Y $.
|
|
$d u v x z D $. $d u v y z F $. $d u v x y z ph $. $d u v x y z X $.
|
|
$d x y z S $.
|
|
flift.1 $e |- F = ran ( x e. X |-> <. A , B >. ) $.
|
|
flift.2 $e |- ( ( ph /\ x e. X ) -> A e. R ) $.
|
|
flift.3 $e |- ( ( ph /\ x e. X ) -> B e. S ) $.
|
|
$( ` F ` , a function lift, is a subset of ` R X. S ` . (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
fliftrel $p |- ( ph -> F C_ ( R X. S ) ) $=
|
|
( cop cmpt crn cxp wf wss cv wcel wa opelxpi syl2anc eqid fmptd syl5eqss
|
|
frn syl ) AGBHCDLZMZNZEFOZIAHUKUIPUJUKQABHUHUKUIABRHSTCESDFSUHUKSJKCDEFUA
|
|
UBUIUCUDHUKUIUFUGUE $.
|
|
|
|
$( Elementhood in the relation ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
fliftel $p |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) ) $=
|
|
( cop wceq wrex wa wcel cvv wb wbr cmpt crn df-br eleq2i bitri wral opexg
|
|
cv syl2anc ralrimiva eqid elrnmptg syl syl5bb opthg2 rexbidva bitrd ) AEF
|
|
IUAZEFNZCDNZOZBJPZECOFDOQZBJPUSUTBJVAUBZUCZRZAVCUSUTIRVGEFIUDIVFUTKUEUFAV
|
|
ASRZBJUGVGVCTAVHBJABUIJRQZCGRZDHRZVHLMCDGHUHUJUKBJVAUTVESVEULUMUNUOAVBVDB
|
|
JVIVJVKVBVDTLMEFCDGHUPUJUQUR $.
|
|
|
|
$( Elementhood in the relation ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
fliftel1 $p |- ( ( ph /\ x e. X ) -> A F B ) $=
|
|
( cv wcel wa cop wbr cmpt crn cvv opexg syl2anc eqid elrnmpt1 mpdan df-br
|
|
adantll syl6eleqr sylibr ) ABLHMZNZCDOZGMCDGPUJUKBHUKQZRZGUJUKSMZUKUMMZUJ
|
|
CEMDFMUNJKCDEFTUAUIUNUOABHUKULSULUBUCUFUDIUGCDGUEUH $.
|
|
|
|
$( Converse of the relation ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
fliftcnv $p |- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) ) $=
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|
( vy vz wrel cop wa wceq cv wbr wcel ccnv cmpt crn cxp wss fliftrel relxp
|
|
eqid relss mpisyl relcnv jctil wrex fliftel vex brcnv ancom rexbii bitr4d
|
|
3bitr4g df-br 3bitr3g eqrelrdv2 mpancom ) GUAZNZBHDCOUBUCZNZPAVEVGQAVHVFA
|
|
VGFEUDZUEVINVHABDCFEVGHVGUHZKJUFFEUGVGVIUIUJGUKULALMVEVGALRZMRZVESZVKVLVG
|
|
SZVKVLOZVETVOVGTAVMVKDQZVLCQZPZBHUMZVNAVLVKGSVQVPPZBHUMVMVSABCDVLVKEFGHIJ
|
|
KUNVKVLGLUOMUOUPVRVTBHVPVQUQURUTABDCVKVLFEVGHVJKJUNUSVKVLVEVAVKVLVGVAVBVC
|
|
VD $.
|
|
|
|
${
|
|
fliftfun.4 $e |- ( x = y -> A = C ) $.
|
|
fliftfun.5 $e |- ( x = y -> B = D ) $.
|
|
$( The function ` F ` is the unique function defined by ` F `` A = B ` ,
|
|
provided that the well-definedness condition holds. (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
fliftfun $p |- ( ph -> ( Fun F <->
|
|
A. x e. X A. y e. X ( A = C -> B = D ) ) ) $=
|
|
( wceq cv wa wrex vz vu vv wfun wi wral nfv cop cmpt nfmpt1 nfrn nfcxfr
|
|
crn nffun wcel cfv fveq2 wbr simplr fliftel1 ad2ant2r sylc simprr eqidd
|
|
funbrfv eqeq2d anbi12d rspcev syl12anc wb fliftel mpbird eqeq12d syl5ib
|
|
anassrs ralrimiva exp31 ralrimd wal cbvrexv syl6bb biimpd reeanv r19.29
|
|
ad2antrr eqtr2 imim1i imp simprlr simprrr 3eqtr4d rexlimivw syl syl5bir
|
|
ex syl9 alrimdv wrel cxp wss fliftrel relxp relss mpisyl dffun2 sylibrd
|
|
baib impbid ) AJUDZDFQZEGQZUEZCKUFZBKUFZAXIXMBKABUGBJBJBKDEUHZUIZUMLBXP
|
|
BKXOUJUKULUNAXIBRZKUOZXMAXISZXRSXLCKXSXRCRZKUOZXLXJDJUPZFJUPZQXSXRYASZS
|
|
ZXKDFJUQYEYBEYCGYEXIDEJURZYBEQAXIYDUSZAXRYFXIYAABDEHIJKLMNUTVADEJVEVBYE
|
|
XIFGJURZYCGQYGYEYHFDQZGEQZSZBKTZYEYAFFQZGGQZYLXSXRYAVCYEFVDYEGVDYKYMYNS
|
|
BXTKXQXTQZYIYMYJYNYODFFOVFYOEGGPVFVGVHVIAYHYLVJXIYDABDEFGHIJKLMNVKWEVLF
|
|
GJVEVBVMVNVOVPVQVRAXNUARZUBRZJURZYPUCRZJURZSZYQYSQZUEZUCVSZUBVSZUAVSZXI
|
|
AXNUUEUAAXNUUDUBAXNUUCUCAUUAYPDQZYQEQZSZBKTZYPFQZYSGQZSZCKTZSZXNUUBAUUA
|
|
UUOAYRUUJYTUUNABDEYPYQHIJKLMNVKAYTUUGYSEQZSZBKTUUNABDEYPYSHIJKLMNVKUUQU
|
|
UMBCKYOUUGUUKUUPUULYODFYPOVFYOEGYSPVFVGVTWAVGWBUUOUUIUUMSZCKTZBKTZXNUUB
|
|
UUIUUMBCKKWCXNUUTUUBXNUUTSXMUUSSZBKTUUBXMUUSBKWDUVAUUBBKUVAXLUURSZCKTUU
|
|
BXLUURCKWDUVBUUBCKUVBEGYQYSXLUURXKUURXJXKUUGUUKXJUUHUULYPDFWFVAWGWHXLUU
|
|
GUUHUUMWIXLUUIUUKUULWJWKWLWMWLWMWOWNWPWQWQWQAJWRZXIUUFVJAJHIWSZWTUVDWRU
|
|
VCABDEHIJKLMNXAHIXBJUVDXCXDXIUVCUUFUAUBUCJXEXGWMXFXH $.
|
|
|
|
fliftfund.6 $e |- ( ( ph /\ ( x e. X /\ y e. X /\ A = C ) ) -> B = D ) $.
|
|
$( The function ` F ` is the unique function defined by ` F `` A = B ` ,
|
|
provided that the well-definedness condition holds. (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
fliftfund $p |- ( ph -> Fun F ) $=
|
|
( wceq wral cv wfun wi wcel 3exp2 imp32 ralrimivva fliftfun mpbird ) AJ
|
|
UADFRZEGRZUBZCKSBKSAUKBCKKABTKUCZCTKUCZUKAULUMUIUJQUDUEUFABCDEFGHIJKLMN
|
|
OPUGUH $.
|
|
$}
|
|
|
|
$( The function ` F ` is the unique function defined by ` F `` A = B ` ,
|
|
provided that the well-definedness condition holds. (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
fliftfuns $p |- ( ph -> ( Fun F <-> A. y e. X A. z e. X
|
|
( [_ y / x ]_ A = [_ z / x ]_ A -> [_ y / x ]_ B = [_ z / x ]_ B ) ) ) $=
|
|
( cv csb cop cmpt crn nfcsb1v wcel nfcv nfop csbeq1a opeq12d cbvmpt rneqi
|
|
weq eqtri wral ralrimiva nfel1 eleq1d rspc mpan9 csbeq1 fliftfun ) ACDBCN
|
|
ZEOZBUQFOZBDNZEOBUTFOGHIJIBJEFPZQZRCJURUSPZQZRKVBVDBCJVAVCCVAUABURUSBUQES
|
|
ZBUQFSZUBBCUGZEURFUSBUQEUCZBUQFUCZUDUEUFUHAEGTZBJUIUQJTZURGTZAVJBJLUJVJVL
|
|
BUQJBURGVEUKVGEURGVHULUMUNAFHTZBJUIVKUSHTZAVMBJMUJVMVNBUQJBUSHVFUKVGFUSHV
|
|
IULUMUNBUQUTEUOBUQUTFUOUP $.
|
|
|
|
$( The domain and range of the function ` F ` . (Contributed by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
fliftf $p |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) ) $=
|
|
( vy vz crn wa wss wceq cv wex wrex wfun cmpt wf wfn cdm simpr wbr cab wb
|
|
fliftel exbidv adantr rexcom4 wcel elisset syl biantrud syl6rbbr rexbidva
|
|
19.42v syl5bbr bitrd df-dm eqid rnmpt 3eqtr4g df-fn sylanbrc cxp fliftrel
|
|
abbidv rnss rnxpss syl6ss df-f ex ffun impbid1 ) AGUAZBHCUBZNZFGUCZAVSWBA
|
|
VSOZGWAUDZGNZFPWBWCVSGUEZWAQWDAVSUFWCLRZMRZGUGZMSZLUHWGCQZBHTZLUHWFWAWCWJ
|
|
WLLWCWJWKWHDQZOZBHTZMSZWLAWJWPUIVSAWIWOMABCDWGWHEFGHIJKUJUKULWPWNMSZBHTZW
|
|
CWLWNBMHUMAWRWLUIVSAWQWKBHABRHUNOZWKWKWMMSZOWQWSWTWKWSDFUNWTKMDFUOUPUQWKW
|
|
MMUTURUSULVAVBVKLMGVCBLHCVTVTVDVEVFGWAVGVHWCWEEFVIZNZFWCGXAPZWEXBPAXCVSAB
|
|
CDEFGHIJKVJULGXAVLUPEFVMVNWAFGVOVHVPWAFGVQVR $.
|
|
|
|
fliftval.4 $e |- ( x = Y -> A = C ) $.
|
|
fliftval.5 $e |- ( x = Y -> B = D ) $.
|
|
fliftval.6 $e |- ( ph -> Fun F ) $.
|
|
$( The value of the function ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
fliftval $p |- ( ( ph /\ Y e. X ) -> ( F ` C ) = D ) $=
|
|
( wa wceq adantr wcel wfun wbr cfv simpr eqidd anim12ci cv eqeq2d anbi12d
|
|
wrex rspcev syl2anc wb fliftel mpbird funbrfv sylc ) AKJUAZRZIUBZEFIUCZEI
|
|
UDFSAVAUSQTUTVBECSZFDSZRZBJUKZUTUSEESZFFSZRZVFAUSUEAVHUSVGAFUFUSEUFUGVEVI
|
|
BKJBUHKSZVCVGVDVHVJCEEOUIVJDFFPUIUJULUMAVBVFUNUSABCDEFGHIJLMNUOTUPEFIUQUR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y H $. $d x y G $.
|
|
$d x y R $. $d x y S $. $d x y T $.
|
|
$( Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) $)
|
|
isoeq1 $p |- ( H = G ->
|
|
( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq1 fveq1 df-isom breq12d
|
|
bibi2d 2ralbidv anbi12d 3bitr4g ) FEIZABFJZGKZHKZCLZUHFMZUIFMZDLZNZHAOGAO
|
|
ZPABEJZUJUHEMZUIEMZDLZNZHAOGAOZPABCDFQABCDEQUFUGUPUOVAABFERUFUNUTGHAAUFUM
|
|
USUJUFUKUQULURDUHFESUIFESUAUBUCUDGHABCDFTGHABCDETUE $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) $)
|
|
isoeq2 $p |- ( R = T ->
|
|
( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi1d df-isom 2ralbidv
|
|
anbi2d 3bitr4g ) CEIZABFJZGKZHKZCLZUFFMUGFMDLZNZHAOGAOZPUEUFUGELZUINZHAOG
|
|
AOZPABCDFQABEDFQUDUKUNUEUDUJUMGHAAUDUHULUIUFUGCERSUAUBGHABCDFTGHABEDFTUC
|
|
$.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) $)
|
|
isoeq3 $p |- ( S = T ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi2d df-isom 2ralbidv
|
|
anbi2d 3bitr4g ) DEIZABFJZGKZHKZCLZUFFMZUGFMZDLZNZHAOGAOZPUEUHUIUJELZNZHA
|
|
OGAOZPABCDFQABCEFQUDUMUPUEUDULUOGHAAUDUKUNUHUIUJDERSUAUBGHABCDFTGHABCEFTU
|
|
C $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) $)
|
|
isoeq4 $p |- ( A = C ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq2 raleq df-isom anbi12d
|
|
raleqbi1dv 3bitr4g ) ACIZABFJZGKZHKZDLUFFMUGFMELNZHAOZGAOZPCBFJZUHHCOZGCO
|
|
ZPABDEFQCBDEFQUDUEUKUJUMACBFRUIULGACUHHACSUBUAGHABDEFTGHCBDEFTUC $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) $)
|
|
isoeq5 $p |- ( B = C ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq3 anbi1d df-isom
|
|
3bitr4g ) BCIZABFJZGKZHKZDLUDFMUEFMELNHAOGAOZPACFJZUFPABDEFQACDEFQUBUCUGU
|
|
FBCAFRSGHABDEFTGHACDEFTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y z H $. $d y z R $. $d y z S $. $d y z A $. $d y z B $.
|
|
$d x y z $.
|
|
nfiso.1 $e |- F/_ x H $.
|
|
nfiso.2 $e |- F/_ x R $.
|
|
nfiso.3 $e |- F/_ x S $.
|
|
nfiso.4 $e |- F/_ x A $.
|
|
nfiso.5 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for an isomorphism. (Contributed by
|
|
NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nfiso $p |- F/ x H Isom R , S ( A , B ) $=
|
|
( vy vz cv wbr cfv wral nfcv nfbr nffv wiso wf1o wb wa df-isom nff1o nfbi
|
|
nfralxy nfan nfxfr ) BCDEFUABCFUBZLNZMNZDOZULFPZUMFPZEOZUCZMBQZLBQZUDALMB
|
|
CDEFUEUKUTAABCFGJKUFUSALBJURAMBJUNUQAAULUMDAULRZHAUMRZSAUOUPEAULFGVATIAUM
|
|
FGVBTSUGUHUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$( An isomorphism is a one-to-one onto function. (Contributed by NM,
|
|
27-Apr-2004.) $)
|
|
isof1o $p |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B ) $=
|
|
( vx vy wiso wf1o cv wbr cfv wb wral df-isom simplbi ) ABCDEHABEIFJZGJZCK
|
|
QELRELDKMGANFANFGABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$d x y C $. $d x y D $.
|
|
$( An isomorphism connects binary relations via its function values.
|
|
(Contributed by NM, 27-Apr-2004.) $)
|
|
isorel $p |- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) ->
|
|
( C R D <-> ( H ` C ) S ( H ` D ) ) ) $=
|
|
( vx vy wiso cv wbr cfv wb wral wcel wa wceq fveq2 bibi12d df-isom breq1d
|
|
wf1o simprbi breq1 breq2 breq2d rspc2v mpan9 ) ABEFGJZHKZIKZELZUKGMZULGMZ
|
|
FLZNZIAOHAOZCAPDAPQCDELZCGMZDGMZFLZNZUJABGUCURHIABEFGUAUDUQVCCULELZUTUOFL
|
|
ZNHICDAAUKCRZUMVDUPVEUKCULEUEVFUNUTUOFUKCGSUBTULDRZVDUSVEVBULDCEUFVGUOVAU
|
|
TFULDGSUGTUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $. $d x y R $. $d x y S $.
|
|
$( A consequence of isomorphism on two relations for a function's
|
|
restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) $)
|
|
isoresbr $p |- ( ( F |` A ) Isom R , S ( A , ( F " A ) ) ->
|
|
A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) ) $=
|
|
( cima cres wiso cv wbr cfv wi wcel wa isorel wb fvres breqan12d adantl
|
|
bitrd biimpd ralrimivva ) CFCGZDEFCHZIZAJZBJZDKZUGFLZUHFLZEKZMABCCUFUGCNZ
|
|
UHCNZOZOZUIULUPUIUGUELZUHUELZEKZULCUDUGUHDEUEPUOUSULQUFUMUNUQUJURUKEUGCFR
|
|
UHCFRSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring]
|
|
p. 33. (Contributed by NM, 27-Apr-2004.) $)
|
|
isoid $p |- ( _I |` A ) Isom R , R ( A , A ) $=
|
|
( vx vy cid cres wiso wf1o wbr cfv wral f1oi wcel fvresi breqan12d bicomd
|
|
cv wb wa rgen2a df-isom mpbir2an ) AABBEAFZGAAUCHCQZDQZBIZUDUCJZUEUCJZBIZ
|
|
RZDAKCAKALUJCDAUDAMZUEAMZSUIUFUKULUGUDUHUEBAUDNAUENOPTCDAABBUCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z H $. $d w x y z R $.
|
|
$d w x y z S $.
|
|
$( Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring]
|
|
p. 33. (Contributed by NM, 27-Apr-2004.) $)
|
|
isocnv $p |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) ) $=
|
|
( vx vy vz vw wf1o cv wbr cfv wb wral wa wiso wcel wceq f1ocnvfv2 adantrr
|
|
ccnv f1ocnv adantr adantrl breq12d adantlr wf syl ffvelrn anim12dan breq1
|
|
f1of fveq2 breq1d bibi12d bicom syl6bb breq2d breq2 rspc2va sylan sylanl1
|
|
an32s bitr3d ralrimivva jca df-isom 3imtr4i ) ABEJZFKZGKZCLZVKEMZVLEMZDLZ
|
|
NZGAOFAOZPZBAEUBZJZHKZIKZDLZWBVTMZWCVTMZCLZNZIBOHBOZPABCDEQBADCVTQVSWAWIV
|
|
JWAVRABEUCZUDVSWHHIBBVSWBBRZWCBRZPZPWEEMZWFEMZDLZWDWGVJWMWPWDNVRVJWMPWNWB
|
|
WOWCDVJWKWNWBSWLABWBETUAVJWLWOWCSWKABWCETUEUFUGVJBAVTUHZVRWMWPWGNZVJWAWQW
|
|
JBAVTUMUIWQWMVRWRWQWMPWEARZWFARZPVRWRWQWKWSWLWTBAWBVTUJBAWCVTUJUKVQWRWNVO
|
|
DLZWEVLCLZNZFGWEWFAAVKWESZVQXBXANXCXDVMXBVPXAVKWEVLCULXDVNWNVODVKWEEUNUOU
|
|
PXBXAUQURVLWFSZXAWPXBWGXEVOWOWNDVLWFEUNUSVLWFWECUTUPVAVBVDVCVEVFVGFGABCDE
|
|
VHHIBADCVTVHVI $.
|
|
|
|
$( Converse law for isomorphism. (Contributed by Mario Carneiro,
|
|
30-Jan-2014.) $)
|
|
isocnv2 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom `' R , `' S ( A , B ) ) $=
|
|
( vy vx wiso ccnv isof1o syl cv wbr cfv wb wral wa wcel vex cvv wfn f1ofn
|
|
wf1o funfvex funfni adantr adantlr brcnvg syl2anc bibi12d ralbidva ralcom
|
|
brcnv a1i syl6rbbr anbi2d df-isom 3bitr4g pm5.21nii ) ABCDEHZEAUAZABCIZDI
|
|
ZEHZUTABEUCZVAABCDEJABEUBZKVDVEVAABVBVCEJVFKVAVEFLZGLZCMZVGENZVHENZDMZOZG
|
|
APFAPZQVEVHVGVBMZVKVJVCMZOZFAPZGAPZQUTVDVAVNVSVEVAVSVMFAPZGAPVNVAVRVTGAVA
|
|
VHARZQZVQVMFAWBVGARZQZVOVIVPVLVOVIOWDVHVGCGSFSUMUNWDVKTRZVJTRZVPVLOWBWEWC
|
|
WEAVHEVHEUDUEUFVAWCWFWAWFAVGEVGEUDUEUGVKVJTTDUHUIUJUKUKVMFGAAULUOUPFGABCD
|
|
EUQGFABVBVCEUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d H x y $. $d R x y $. $d S x y $.
|
|
$( An isomorphism from one well-order to another can be restricted on
|
|
either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) $)
|
|
isores2 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom R , ( S i^i ( B X. B ) ) ( A , B ) ) $=
|
|
( vx vy wf1o cv wbr cfv wb wral wa cxp wiso wcel ffvelrn ralbidva df-isom
|
|
cin wf f1of adantrr adantrl brinxp syl2anc anassrs bibi2d pm5.32i 3bitr4i
|
|
sylan ) ABEHZFIZGIZCJZUNEKZUOEKZDJZLZGAMZFAMZNUMUPUQURDBBOUAZJZLZGAMZFAMZ
|
|
NABCDEPABCVCEPUMVBVGUMVAVFFAUMUNAQZNZUTVEGAVIUOAQZNUSVDUPUMVHVJUSVDLZUMAB
|
|
EUBZVHVJNZVKABEUCVLVMNUQBQZURBQZVKVLVHVNVJABUNERUDVLVJVOVHABUOERUEUQURBBD
|
|
UFUGULUHUISSUJFGABCDETFGABCVCETUK $.
|
|
$}
|
|
|
|
$( An isomorphism from one well-order to another can be restricted on either
|
|
well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) $)
|
|
isores1 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $=
|
|
( wiso cxp cin ccnv isocnv isores2 sylib syl wf1o isof1o isoeq1 sylbi mpbid
|
|
wb 3syl wrel f1orel wceq dfrel2 sylibr impbii ) ABCDEFZABCAAGHZDEFZUGABUHDE
|
|
IZIZFZUIUGBADUHUJFZULUGBADCUJFZUMABCDEJBADCUJKZLBADUHUJJMUGABENZEUAZULUISZA
|
|
BCDEOABEUBZUQUKEUCZUREUDZABUHDEUKPQTRUIABCDUKFZUGUIUNVBUIUMUNABUHDEJUOUEBAD
|
|
CUJJMUIUPUQVBUGSZABUHDEOUSUQUTVCVAABCDEUKPQTRUF $.
|
|
|
|
${
|
|
$d H a b $. $d R a b $. $d S a b $. $d K a b $. $d A a b $.
|
|
$d B a b $. $d X a b $.
|
|
$( Induced isomorphism on a subset. (Contributed by Stefan O'Rear,
|
|
5-Nov-2014.) $)
|
|
isores3 $p |- ( ( H Isom R , S ( A , B ) /\ K C_ A /\ X = ( H " K ) ) ->
|
|
( H |` K ) Isom R , S ( K , X ) ) $=
|
|
( va vb wiso wa wf1o cv wbr cfv wb wral ssralv wcel fvres cima wceq f1of1
|
|
wss cres wf1 f1ores expcom syl5 adantr breqan12d adantll biimprd ralimdva
|
|
wi bibi2d syld anim12d df-isom 3imtr4g impcom isoeq5 syl5ibrcom 3impia )
|
|
ABCDEJZFAUDZGEFUAZUBZFGCDEFUEZJZVEVFKVJVHFVGCDVIJZVFVEVKVFABELZHMZIMZCNZV
|
|
MEOZVNEOZDNZPZIAQZHAQZKFVGVILZVOVMVIOZVNVIOZDNZPZIFQZHFQZKVEVKVFVLWBWAWHV
|
|
LABEUFZVFWBABEUCWIVFWBABFEUGUHUIVFWAVTHFQWHVTHFARVFVTWGHFVFVMFSZKZVTVSIFQ
|
|
ZWGVFVTWLUOWJVSIFARUJWKVSWFIFWKVNFSZKZWFVSWNWEVRVOWJWMWEVRPVFWJWMWCVPWDVQ
|
|
DVMFETVNFETUKULUPUMUNUQUNUQURHIABCDEUSHIFVGCDVIUSUTVAFGVGCDVIVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w R $.
|
|
$d x y z w S $. $d x y z w T $. $d x y z w G $. $d x y z w H $.
|
|
$( Composition (transitive) law for isomorphism. Proposition 6.30(3) of
|
|
[TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof
|
|
shortened by Mario Carneiro, 5-Dec-2016.) $)
|
|
isotr $p |- ( ( H Isom R , S ( A , B ) /\ G Isom S , T ( B , C ) ) ->
|
|
( G o. H ) Isom R , T ( A , C ) ) $=
|
|
( vx vy vz vw cv wbr cfv wb wral wa wcel wceq wf1o simpl f1oco syl2anr wf
|
|
ccom wiso f1of ad2antrr simprl ffvelrnd simprr simplrr breq1 fveq2 breq1d
|
|
bibi12d breq2 breq2d rspc2va syl21anc fvco3 syl2anc breq12d bitr4d bibi2d
|
|
2ralbidva biimpd impancom imp jca df-isom anbi12i 3imtr4i ) ABHUAZIMZJMZD
|
|
NZVPHOZVQHOZENZPZJAQIAQZRZBCGUAZKMZLMZENZWFGOZWGGOZFNZPZLBQKBQZRZRZACGHUF
|
|
ZUAZVRVPWPOZVQWPOZFNZPZJAQIAQZRABDEHUGZBCEFGUGZRACDFWPUGWOWQXBWNWEVOWQWDW
|
|
EWMUBVOWCUBABCGHUCUDWDWNXBVOWNWCXBVOWNRZWCXBXEWBXAIJAAXEVPASZVQASZRZRZWAW
|
|
TVRXIWAVSGOZVTGOZFNZWTXIVSBSVTBSWMWAXLPZXIABVPHVOABHUEZWNXHABHUHUIZXEXFXG
|
|
UJZUKXIABVQHXOXEXFXGULZUKVOWEWMXHUMWLXMVSWGENZXJWJFNZPKLVSVTBBWFVSTZWHXRW
|
|
KXSWFVSWGEUNXTWIXJWJFWFVSGUOUPUQWGVTTZXRWAXSXLWGVTVSEURYAWJXKXJFWGVTGUOUS
|
|
UQUTVAXIWRXJWSXKFXIXNXFWRXJTXOXPABVPGHVBVCXIXNXGWSXKTXOXQABVQGHVBVCVDVEVF
|
|
VGVHVIVJVKXCWDXDWNIJABDEHVLKLBCEFGVLVMIJACDFWPVLVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$d x y D $.
|
|
$( Isomorphisms preserve initial segments. Proposition 6.31(2) of
|
|
[TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) $)
|
|
isoini $p |- ( ( H Isom R , S ( A , B ) /\ D e. A ) ->
|
|
( H " ( A i^i ( `' R " { D } ) ) ) =
|
|
( B i^i ( `' S " { ( H ` D ) } ) ) ) $=
|
|
( vx vy wcel wa ccnv cfv csn cima cin wbr wrex wb 3syl syl6bb wiso cv cab
|
|
elin wceq crn wf1o wfo isof1o f1ofo forn eleq2d wfn fvelrnb bitr3d adantr
|
|
f1ofn cvv syl anim1i funfvex funfni eliniseg anbi12d anbi2d syl5bb anbi1d
|
|
vex anass adantl wi isorel fnbrfvb bicomd sylan adantrr ancom breq1 bitri
|
|
pm5.32i exp32 com23 pm5.32d bitrd rexbidv2 r19.41v bitr4d dfima2 syl6reqr
|
|
imp abbi2dv ) ABDEFUAZCAIZJZBEKCFLZMNZOZGUBZHUBZFPZGADKCMNZOZQZHUCFXBNWNX
|
|
CHWQWSWQIWSBIZWSWPIZJZWNXCWSBWPUDWNXFWRFLZWSUEZGAQZWSWOEPZJZXCWNXDXIXEXJW
|
|
LXDXIRWMWLWSFUFZIZXDXIWLABFUGZABFUHZXMXDRABDEFUIZABFUJXOXLBWSABFUKULSWLXN
|
|
FAUMZXMXIRXPABFUQZGAWSFUNSUOUPWNXQWMJWOURIZXEXJRWLXQWMWLXNXQXPXRUSZUTXSAC
|
|
FCFVAVBEWOWSURHVHVCSVDWNXCXHXJJZGAQXKWNWTYAGXBAWNWRXBIZWTJZWRAIZWRCDPZWTJ
|
|
ZJZYDYAJWMYCYGRWLWMYCYDYEJZWTJYGWMYBYHWTYBYDWRXAIZJWMYHWRAXAUDWMYIYEYDDCW
|
|
RAGVHVCVEVFVGYDYEWTVITVJWNYDYFYAWLWMYDYFYARZVKWLYDWMYJWLYDWMYJWLYDWMJJZYF
|
|
XGWOEPZXHJZYAYKYEYLWTXHABWRCDEFVLWLYDWTXHRZWMWLXQYDYNXTXQYDJXHWTAWRWSFVMV
|
|
NVOVPVDYMXHYLJYAYLXHVQXHYLXJXGWSWOEVRVTVSTWAWBWJWCWDWEXHXJGAWFTWGVFWKGHFX
|
|
BWHWI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y H $.
|
|
$d x y R $. $d x y S $.
|
|
isoini2.1 $e |- C = ( A i^i ( `' R " { X } ) ) $.
|
|
isoini2.2 $e |- D = ( B i^i ( `' S " { ( H ` X ) } ) ) $.
|
|
$( Isomorphisms are isomorphisms on their initial segments. (Contributed
|
|
by Mario Carneiro, 29-Mar-2014.) $)
|
|
isoini2 $p |- ( ( H Isom R , S ( A , B ) /\ X e. A ) ->
|
|
( H |` C ) Isom R , S ( C , D ) ) $=
|
|
( vx vy wiso wcel wf1o wbr cfv wb wral cima wa cres cv wf1 wss isof1o syl
|
|
f1of1 adantr ccnv csn inss1 eqsstri f1ores sylancl isoini imaeq2i 3eqtr4g
|
|
cin wceq f1oeq3 mpbid df-isom simprbi ssralv mpsyl fvres breqan12d bibi2d
|
|
ralimdv ralbidva ralbiia sylibr sylanbrc ) ABEFGMZHANZUAZCDGCUBZOZKUCZLUC
|
|
ZEPZVTVRQZWAVRQZFPZRZLCSZKCSZCDEFVRMVQCGCTZVROZVSVQABGUDZCAUEZWJVOWKVPVOA
|
|
BGOZWKABEFGUFABGUHUGUICAEUJHUKTZUSZAIAWNULUMZABCGUNUOVQWIDUTWJVSRVQGWOTBF
|
|
UJHGQUKTUSWIDABHEFGUPCWOGIUQJURWIDCVRVAUGVBVQWBVTGQZWAGQZFPZRZLCSZKCSZWHW
|
|
LVQXAKASZXBWPWLVQWTLASZKASZXCWPVOXEVPVOWMXEKLABEFGVCVDUIWLXDXAKAWTLCAVEVJ
|
|
VFXAKCAVEVFWGXAKCVTCNZWFWTLCXFWACNZUAWEWSWBXFXGWCWQWDWRFVTCGVGWACGVGVHVIV
|
|
KVLVMKLCDEFVRVCVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z H $. $d x y z ph $. $d x y z R $.
|
|
$d x y z S $.
|
|
isofrlem.1 $e |- ( ph -> H Isom R , S ( A , B ) ) $.
|
|
isofrlem.2 $e |- ( ph -> ( H " x ) e. _V ) $.
|
|
$( Lemma for ~ isose . (Contributed by Mario Carneiro, 23-Jun-2015.) $)
|
|
isoselem $p |- ( ph -> ( R Se A -> S Se B ) ) $=
|
|
( vy vz cv csn cima cin cvv wcel wral wi wceq wse ccnv cfv dfse2 r19.21bi
|
|
biimpi expcom adantl imaeq2 eleq1d imbi2d vtoclg com12 adantr wiso isoini
|
|
wa sylan sylibd syld ralrimdva crn wf1o wfn wb isof1o sneq imaeq2d ineq2d
|
|
f1ofn ralrn 4syl wfo f1ofo forn raleqdv bitr3d syl6ibr ) ACEUAZDFUBZJLZMZ
|
|
NZOZPQZJDRZDFUAAVSDVTKLZGUCZMZNZOZPQZKCRZWFAVSWLKCAWGCQZUQZVSCEUBWGMNOZPQ
|
|
ZWLWNVSWQSAVSWNWQVSWQKCVSWQKCRKCEUDUFUEUGUHWOWQGWPNZPQZWLAWQWSSWNWQAWSAGB
|
|
LZNZPQZSAWSSBWPPWTWPTZXBWSAXCXAWRPWTWPGUIUJUKIULUMUNWOWRWKPACDEFGUOZWNWRW
|
|
KTHCDWGEFGUPURUJUSUTVAAWEJGVBZRZWMWFAXDCDGVCZGCVDXFWMVEHCDEFGVFZCDGVJWEWL
|
|
JKCGWAWHTZWDWKPXIWCWJDXIWBWIVTWAWHVGVHVIUJVKVLAWEJXEDAXDXGCDGVMXEDTHXHCDG
|
|
VNCDGVOVLVPVQUSJDFUDVR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x H $. $d x R $. $d x S $.
|
|
$( An isomorphism preserves set-like relations. (Contributed by Mario
|
|
Carneiro, 23-Jun-2015.) $)
|
|
isose $p |- ( H Isom R , S ( A , B ) -> ( R Se A <-> S Se B ) ) $=
|
|
( vx wiso wse wf1o wfun cima cvv wcel isof1o f1ofun vex funimaex isoselem
|
|
id cv 3syl ccnv isocnv 4syl impbid ) ABCDEGZACHBDHUFFABCDEUFSUFABEIEJEFTZ
|
|
KLMABCDENABEOEUGFPZQUARUFFBADCEUBZABCDEUCZUFBADCUIGBAUIIUIJUIUGKLMUJBADCU
|
|
INBAUIOUIUGUHQUDRUE $.
|
|
$}
|
|
|
|
${
|
|
$d H a b c d e f $. $d R a b c d e f $. $d S a b c d e f $.
|
|
$d A a b c d e f $. $d B a b c d e f $.
|
|
$( Lemma for ~ isopo . (Contributed by Stefan O'Rear, 16-Nov-2014.) $)
|
|
isopolem $p |- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) ) $=
|
|
( va vb vc vd ve vf cv wbr wa wi wral wcel ex wb anbi12d wiso wpo w3a cfv
|
|
wn wf1o isof1o f1of ffvelrn 3anim123d 3syl imp breq12 anidms notbid breq1
|
|
wf wceq anbi1d imbi12d breq2 imbi1d anbi2d rspc3v syl simpl simpr1 isorel
|
|
syl12anc simpr2 simpr3 sylibrd com23 imp31 ralrimivvva df-po 3imtr4g ) AB
|
|
CDEUAZFLZVSDMZUEZVSGLZDMZWBHLZDMZNZVSWDDMZOZNZHBPGBPFBPZILZWKCMZUEZWKJLZC
|
|
MZWNKLZCMZNZWKWPCMZOZNZKAPJAPIAPZBDUBACUBVRWJXBVRWJNXAIJKAAAVRWJWKAQZWNAQ
|
|
ZWPAQZUCZXAVRXFWJXAVRXFWJXAOVRXFNZWJWKEUDZXHDMZUEZXHWNEUDZDMZXKWPEUDZDMZN
|
|
ZXHXMDMZOZNZXAXGXHBQZXKBQZXMBQZUCZWJXROVRXFYBVRABEUFABEUQZXFYBOABCDEUGABE
|
|
UHYCXCXSXDXTXEYAYCXCXSABWKEUIRYCXDXTABWNEUIRYCXEYAABWPEUIRUJUKULWIXRXJXHW
|
|
BDMZWENZXHWDDMZOZNXJXLXKWDDMZNZYFOZNFGHXHXKXMBBBVSXHURZWAXJWHYGYKVTXIYKVT
|
|
XISVSXHVSXHDUMUNUOYKWFYEWGYFYKWCYDWEVSXHWBDUPUSVSXHWDDUPUTTWBXKURZYGYJXJY
|
|
LYEYIYFYLYDXLWEYHWBXKXHDVAWBXKWDDUPTVBVCWDXMURZYJXQXJYMYIXOYFXPYMYHXNXLWD
|
|
XMXKDVAVCWDXMXHDVAUTVCVDVEXGWMXJWTXQXGWLXIXGVRXCXCWLXISVRXFVFZVRXCXDXEVGZ
|
|
YOABWKWKCDEVHVIUOXGWRXOWSXPXGWOXLWQXNXGVRXCXDWOXLSYNYOVRXCXDXEVJZABWKWNCD
|
|
EVHVIXGVRXDXEWQXNSYNYPVRXCXDXEVKZABWNWPCDEVHVITXGVRXCXEWSXPSYNYOYQABWKWPC
|
|
DEVHVIUTTVLRVMVNVORFGHBDVPIJKACVPVQ $.
|
|
|
|
$( An isomorphism preserves partial ordering. (Contributed by Stefan
|
|
O'Rear, 16-Nov-2014.) $)
|
|
isopo $p |- ( H Isom R , S ( A , B ) -> ( R Po A <-> S Po B ) ) $=
|
|
( wiso wpo ccnv wi isocnv isopolem syl impbid ) ABCDEFZACGZBDGZNBADCEHZFO
|
|
PIABCDEJBADCQKLABCDEKM $.
|
|
$}
|
|
|
|
${
|
|
$d H a b c x y z $. $d R a b c x y z $. $d S a b c x y z $.
|
|
$d A a b c x y z $. $d B a b c x y z $.
|
|
$( Lemma for ~ isoso . (Contributed by Stefan O'Rear, 16-Nov-2014.) $)
|
|
isosolem $p |- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) ) $=
|
|
( vx vy vz va vb vc cv wbr wo wi wral wa wcel cfv ffvelrn wiso wpo df-3an
|
|
wor isopolem w3a wf1o wf isof1o f1of 3anim123d 3syl imp wceq breq1 orbi1d
|
|
ex imbi12d breq2 orbi2d orbi12d imbi2d rspc3v wb isorel 3adantr3 3adantr2
|
|
syl ancom2s 3adantr1 sylibrd sylan2br anassrs ralrimdva ralrimdvva df-iso
|
|
anim12d 3imtr4g ) ABCDEUAZBDUBZFLZGLZDMZWAHLZDMZWDWBDMZNZOZHBPGBPFBPZQACU
|
|
BZILZJLZCMZWKKLZCMZWNWLCMZNZOZKAPZJAPIAPZQBDUDACUDVSVTWJWIWTABCDEUEVSWIWS
|
|
IJAAVSWKARZWLARZQZQWIWRKAVSXCWNARZWIWROZXCXDQVSXAXBXDUFZXEXAXBXDUCVSXFQZW
|
|
IWKESZWLESZDMZXHWNESZDMZXKXIDMZNZOZWRXGXHBRZXIBRZXKBRZUFZWIXOOVSXFXSVSABE
|
|
UGABEUHZXFXSOABCDEUIABEUJXTXAXPXBXQXDXRXTXAXPABWKETUQXTXBXQABWLETUQXTXDXR
|
|
ABWNETUQUKULUMWHXOXHWBDMZXHWDDMZWFNZOXJYBWDXIDMZNZOFGHXHXIXKBBBWAXHUNZWCY
|
|
AWGYCWAXHWBDUOYFWEYBWFWAXHWDDUOUPURWBXIUNZYAXJYCYEWBXIXHDUSYGWFYDYBWBXIWD
|
|
DUSUTURWDXKUNZYEXNXJYHYBXLYDXMWDXKXHDUSWDXKXIDUOVAVBVCVHXGWMXJWQXNVSXAXBW
|
|
MXJVDXDABWKWLCDEVEVFXGWOXLWPXMVSXAXDWOXLVDXBABWKWNCDEVEVGVSXBXDWPXMVDZXAV
|
|
SXDXBYIABWNWLCDEVEVIVJVAURVKVLVMVNVOVQFGHBDVPIJKACVPVR $.
|
|
$}
|
|
|
|
$( An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear,
|
|
16-Nov-2014.) $)
|
|
isoso $p |- ( H Isom R , S ( A , B ) -> ( R Or A <-> S Or B ) ) $=
|
|
( wiso wor ccnv wi isocnv isosolem syl impbid ) ABCDEFZACGZBDGZNBADCEHZFOPI
|
|
ABCDEJBADCQKLABCDEKM $.
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y v u B $. $d x y z w v u H $.
|
|
$d x y z w v u R $. $d v u S $.
|
|
$( Any one-to-one onto function determines an isomorphism with an induced
|
|
relation ` S ` . Proposition 6.33 of [TakeutiZaring] p. 34.
|
|
(Contributed by NM, 30-Apr-2004.) $)
|
|
f1oiso $p |- ( ( H : A -1-1-onto-> B /\ S = { <. z , w >. |
|
|
E. x e. A E. y e. A ( ( z = ( H ` x ) /\ w = ( H ` y ) ) /\ x R y ) } ) ->
|
|
H Isom R , S ( A , B ) ) $=
|
|
( vv vu cv cfv wceq wa wbr wrex wb wcel anbi1d wf1o copab wral wiso simpl
|
|
wf1 f1of1 cop df-br eleq2 wfn cvv funfvex funfni anim12dan eqeq1 2rexbidv
|
|
f1fn anbi2d opelopabg syl sylan anass f1fveq equcom syl6bb anassrs syl5bb
|
|
rexbidv r19.42v rexbidva breq1 ceqsrexv adantl sylan9bb anandis sylan9bbr
|
|
bitrd breq2 an32s syl5rbb ralrimivva df-isom sylanbrc ) EFIUAZHCLZALZIMZN
|
|
ZDLZBLZIMZNZOZWGWKGPZOZBEQAEQZCDUBZNZOWEJLZKLZGPZWTIMZXAIMZHPZRZKEUCJEUCZ
|
|
EFGHIUDWEWSUEWEEFIUFZWSXGEFIUGXHWSOZXFJKEEXEXCXDUHZHSZXIWTESZXAESZOZOXBXC
|
|
XDHUIXHXNWSXKXBRWSXKXJWRSZXHXNOZXBHWRXJUJXPXOXCWHNZXDWLNZOZWOOZBEQZAEQZXB
|
|
XHIEUKZXNXOYBRZEFIURYCXNOXCULSZXDULSZOYDYCXLYEXMYFYEEWTIWTIUMUNYFEXAIXAIU
|
|
MUNUOWQXQWMOZWOOZBEQAEQYBCDXCXDULULWFXCNZWPYHABEEYIWNYGWOYIWIXQWMWFXCWHUP
|
|
TTUQWJXDNZYHXTABEEYJYGXSWOYJWMXRXQWJXDWLUPUSTUQUTVAVBXHXLXMYBXBRXHXLOZYBX
|
|
RWTWKGPZOZBEQZXHXMOZXBYKYBWGWTNZXRWOOZBEQZOZAEQZYNYKYAYSAEYKWGESZOZYAYPYQ
|
|
OZBEQYSUUBXTUUCBEXTXQYQOUUBUUCXQXRWOVCUUBXQYPYQXHXLUUAXQYPRXHXLUUAOOXQWTW
|
|
GNYPEFWTWGIVDJAVEVFVGTVHVIYPYQBEVJVFVKXLYTYNRXHYRYNAWTEYPYQYMBEYPWOYLXRWG
|
|
WTWKGVLUSVIVMVNVRYOYNWKXANZYLOZBEQZXBYOYMUUEBEYOWKESZOXRUUDYLXHXMUUGXRUUD
|
|
RXHXMUUGOOXRXAWKNUUDEFXAWKIVDKBVEVFVGTVKXMUUFXBRXHYLXBBXAEWKXAWTGVSVMVNVR
|
|
VOVPVRVQVTWAWBVBJKEFGHIWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $. $d H w x y z $. $d R w x y z $.
|
|
f1oiso2.1 $e |- S = { <. x , y >. |
|
|
( ( x e. B /\ y e. B ) /\ ( `' H ` x ) R ( `' H ` y ) ) } $.
|
|
$( Any one-to-one onto function determines an isomorphism with an induced
|
|
relation ` S ` . (Contributed by Mario Carneiro, 9-Mar-2013.) $)
|
|
f1oiso2 $p |- ( H : A -1-1-onto-> B -> H Isom R , S ( A , B ) ) $=
|
|
( vz vw cv cfv wceq wa wbr wrex wcel 3adant3 eqcomd syl2anc wf1o wiso w3a
|
|
copab ccnv f1ocnvdm adantrr f1ocnvfv2 anim12dan simp3 fveq2 eqeq2d anbi2d
|
|
adantrl breq2 anbi12d rspcev syl12anc anbi1d breq1 rexbidv 3expib simp3ll
|
|
simp1 simp2l f1of ffvelrnda eqeltrd simp3lr simp2r simp3r wi f1ocnvfv mpd
|
|
3brtr4d jca31 3exp rexlimdvv impbid opabbidv syl5eq f1oiso mpdan ) CDGUAZ
|
|
FAKZIKZGLZMZBKZJKZGLZMZNZWFWJEOZNZJCPZICPZABUDZMCDEFGUBWDFWEDQZWIDQZNZWEG
|
|
UEZLZWIXBLZEOZNZABUDWRHWDXFWQABWDXFWQWDXAXEWQWDXAXEUCZXCCQZWEXCGLZMZWLNZX
|
|
CWJEOZNZJCPZWQWDXAXHXEWDWSXHWTCDWEGUFUGRXGXDCQZXJWIXDGLZMZNZXEXNWDXAXOXEW
|
|
DWTXOWSCDWIGUFUNRWDXAXRXEWDWSXJWTXQWDWSNXIWECDWEGUHSWDWTNXPWICDWIGUHSUIRW
|
|
DXAXEUJXMXRXENJXDCWJXDMZXKXRXLXEXSWLXQXJXSWKXPWIWJXDGUKULUMWJXDXCEUOUPUQU
|
|
RWPXNIXCCWFXCMZWOXMJCXTWMXKWNXLXTWHXJWLXTWGXIWEWFXCGUKULUSWFXCWJEUTUPVAUQ
|
|
TVBWDWOXFIJCCWDWFCQZWJCQZNZWOXFWDYCWOUCZWSWTXEYDWEWGDWHWLWNWDYCVCZYDWDYAW
|
|
GDQWDYCWOVDZWDYAYBWOVEZWDCDWFGCDGVFZVGTVHYDWIWKDWHWLWNWDYCVIZYDWDYBWKDQYF
|
|
WDYAYBWOVJZWDCDWJGYHVGTVHYDWFWJXCXDEWDYCWMWNVKYDWGWEMZXCWFMZYDWEWGYESYDWD
|
|
YAYKYLVLYFYGCDWFWEGVMTVNYDWKWIMZXDWJMZYDWIWKYISYDWDYBYMYNVLYFYJCDWJWIGVMT
|
|
VNVOVPVQVRVSVTWAIJABCDEFGWBWC $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Restricted iota (description binder)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c iota_ $.
|
|
|
|
$( Extend class notation with restricted description binder. $)
|
|
crio $a class ( iota_ x e. A ph ) $.
|
|
|
|
$( Define restricted description binder. In case there is no unique ` x `
|
|
such that ` ( x e. A /\ ph ) ` holds, it evaluates to the empty set. See
|
|
also comments for ~ df-iota . (Contributed by NM, 15-Sep-2011.) (Revised
|
|
by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.) $)
|
|
df-riota $a |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) ) $.
|
|
|
|
${
|
|
$d x ph $.
|
|
riotaeqdv.1 $e |- ( ph -> A = B ) $.
|
|
$( Formula-building deduction rule for iota. (Contributed by NM,
|
|
15-Sep-2011.) $)
|
|
riotaeqdv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) ) $=
|
|
( cv wcel wa cio crio eleq2d anbi1d iotabidv df-riota 3eqtr4g ) ACGZDHZBI
|
|
ZCJQEHZBIZCJBCDKBCEKASUACARTBADEQFLMNBCDOBCEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
riotabidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for restricted iota. (Contributed by
|
|
NM, 15-Sep-2011.) $)
|
|
riotabidv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) ) $=
|
|
( cv wcel wa cio crio biidd anbi12d iotabidv df-riota 3eqtr4g ) ADGEHZBIZ
|
|
DJQCIZDJBDEKCDEKARSDAQQBCAQLFMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
riotaeqbidv.1 $e |- ( ph -> A = B ) $.
|
|
riotaeqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 15-Sep-2011.) $)
|
|
riotaeqbidv $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) ) $=
|
|
( crio riotabidv riotaeqdv eqtrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) $)
|
|
riotaexg $p |- ( A e. V -> ( iota_ x e. A ps ) e. _V ) $=
|
|
( wcel crio cv cio cvv df-riota cuni uniexg wss iotass elssuni adantr mpg
|
|
wa wi a1i ssexd syl5eqel ) CDEZABCFBGZCEZARZBHZIABCJUCUGCKZICDLUGUHMZUCUF
|
|
UDUHMZSUIBUFBUHNUEUJAUDCOPQTUAUB $.
|
|
$}
|
|
|
|
$( An iota restricted to the universe is unrestricted. (Contributed by NM,
|
|
18-Sep-2011.) $)
|
|
riotav $p |- ( iota_ x e. _V ph ) = ( iota x ph ) $=
|
|
( cvv crio cv wcel wa cio df-riota vex biantrur iotabii eqtr4i ) ABCDBECFZA
|
|
GZBHABHABCIAOBNABJKLM $.
|
|
|
|
$( Restricted iota in terms of class union. (Contributed by NM,
|
|
11-Oct-2011.) $)
|
|
riotauni $p |- ( E! x e. A ph
|
|
-> ( iota_ x e. A ph ) = U. { x e. A | ph } ) $=
|
|
( wreu cv wcel wa cio cab cuni crio crab wceq df-reu iotauni sylbi df-riota
|
|
weu df-rab unieqi 3eqtr4g ) ABCDZBECFAGZBHZUCBIZJZABCKABCLZJUBUCBRUDUFMABCN
|
|
UCBOPABCQUGUEABCSTUA $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The abstraction variable in a restricted iota descriptor isn't free.
|
|
(Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfriota1 $p |- F/_ x ( iota_ x e. A ph ) $=
|
|
( crio cv wcel wa cio df-riota nfiota1 nfcxfr ) BABCDBECFAGZBHABCILBJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nfriotadxy.1 $e |- F/ y ph $.
|
|
nfriotadxy.2 $e |- ( ph -> F/ x ps ) $.
|
|
nfriotadxy.3 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfriota . (Contributed by Jim Kingdon,
|
|
12-Jan-2019.) $)
|
|
nfriotadxy $p |- ( ph -> F/_ x ( iota_ y e. A ps ) ) $=
|
|
( crio cv wcel wa cio df-riota wnfc nfcv a1i nfeld nfand nfiotadxy
|
|
nfcxfrd ) ACBDEIDJZEKZBLZDMBDENAUDCDFAUCBCACUBECUBOACUBPQHRGSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nfriota.1 $e |- F/ x ph $.
|
|
nfriota.2 $e |- F/_ x A $.
|
|
$( A variable not free in a wff remains so in a restricted iota descriptor.
|
|
(Contributed by NM, 12-Oct-2011.) $)
|
|
nfriota $p |- F/_ x ( iota_ y e. A ph ) $=
|
|
( crio wnfc wtru nftru wnf a1i nfriotadxy trud ) BACDGHIABCDCJABKIELBDHIF
|
|
LMN $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $. $d z ps $.
|
|
cbvriota.1 $e |- F/ y ph $.
|
|
cbvriota.2 $e |- F/ x ps $.
|
|
cbvriota.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variable in a restricted description binder. (Contributed
|
|
by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvriota $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
|
|
( vz cv wcel wa cio crio wsb weq eleq1 anbi12d nfv nfan nfs1v sbie syl6bb
|
|
sbequ12 cbviota sbequ nfsb eqtri df-riota 3eqtr4i ) CJZEKZALZCMZDJZEKZBLZ
|
|
DMZACENBDENUNIJZEKZACIOZLZIMURUMVBCICIPULUTAVAUKUSEQACIUDRUMISUTVACUTCSAC
|
|
IUATUEVBUQIDIDPZUTUPVABUSUOEQVCVAACDOBAIDCUFABCDGHUBUCRUTVADUTDSACIDFUGTU
|
|
QISUEUHACEUIBDEUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y ph $. $d x ps $.
|
|
cbvriotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variable in a restricted description binder. (Contributed
|
|
by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvriotav $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
|
|
( nfv cbvriota ) ABCDEADGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d z ph $. $d x y $.
|
|
$( Interchange class substitution and restricted description binder.
|
|
(Contributed by NM, 24-Feb-2013.) $)
|
|
csbriotag $p |- ( A e. V
|
|
-> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) ) $=
|
|
( vz cv crio csb wsb wceq wsbc csbeq1 dfsbcq2 riotabidv eqeq12d vex nfs1v
|
|
nfcv nfriota weq sbequ12 csbief vtoclg ) BGHZACEIZJZABGKZCEIZLBDUGJZABDMZ
|
|
CEIZLGDFUFDLZUHUKUJUMBUFDUGNUNUIULCEABGDOPQBUFUGUJGRUIBCEABGSBETUABGUBAUI
|
|
CEABGUCPUDUE $.
|
|
$}
|
|
|
|
$( Membership law for "the unique element in ` A ` such that ` ph ` ."
|
|
|
|
(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
riotacl2 $p |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } ) $=
|
|
( wreu cv wcel wa cio cab crio crab weu df-reu iotacl sylbi df-riota df-rab
|
|
3eltr4g ) ABCDZBECFAGZBHZTBIZABCJABCKSTBLUAUBFABCMTBNOABCPABCQR $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) $)
|
|
riotacl $p |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A ) $=
|
|
( wreu crab crio ssrab2 riotacl2 sseldi ) ABCDABCECABCFABCGABCHI $.
|
|
$}
|
|
|
|
$( Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.)
|
|
(Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
|
|
riotasbc $p |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph ) $=
|
|
( wreu crio cab wcel wsbc crab rabssab riotacl2 sseldi df-sbc sylibr ) ABCD
|
|
ZABCEZABFZGABPHOABCIQPABCJABCKLABPMN $.
|
|
|
|
${
|
|
$d x ph $.
|
|
riotabidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (deduction
|
|
rule). ( ~ rabbidva analog.) (Contributed by NM, 17-Jan-2012.) $)
|
|
riotabidva $p |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) ) $=
|
|
( cv wcel wa cio crio pm5.32da iotabidv df-riota 3eqtr4g ) ADGEHZBIZDJPCI
|
|
ZDJBDEKCDEKAQRDAPBCFLMBDENCDENO $.
|
|
$}
|
|
|
|
${
|
|
riotabiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Equivalent wff's yield equal restricted iotas (inference rule).
|
|
( ~ rabbiia analog.) (Contributed by NM, 16-Jan-2012.) $)
|
|
riotabiia $p |- ( iota_ x e. A ph ) = ( iota_ x e. A ps ) $=
|
|
( cvv wceq crio eqid cv wcel wb adantl riotabidva ax-mp ) FFGZACDHBCDHGFI
|
|
PABCDCJDKABLPEMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Property of restricted iota. Compare ~ iota1 . (Contributed by Mario
|
|
Carneiro, 15-Oct-2016.) $)
|
|
riota1 $p |- ( E! x e. A ph ->
|
|
( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) ) $=
|
|
( wreu cv wcel wa cio wceq crio weu wb df-reu iota1 sylbi df-riota eqeq1i
|
|
syl6bbr ) ABCDZBEZCFAGZUABHZTIZABCJZTISUABKUAUCLABCMUABNOUDUBTABCPQR $.
|
|
$}
|
|
|
|
$( Property of iota. (Contributed by NM, 23-Aug-2011.) $)
|
|
riota1a $p |- ( ( x e. A /\ E! x e. A ph ) ->
|
|
( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) ) $=
|
|
( cv wcel wa wreu cio wceq ibar weu wb df-reu iota1 sylbi sylan9bb ) BDZCEZ
|
|
ARAFZABCGZSBHQIZRAJTSBKSUALABCMSBNOP $.
|
|
|
|
${
|
|
$d x A $.
|
|
riota2df.1 $e |- F/ x ph $.
|
|
riota2df.2 $e |- ( ph -> F/_ x B ) $.
|
|
riota2df.3 $e |- ( ph -> F/ x ch ) $.
|
|
riota2df.4 $e |- ( ph -> B e. A ) $.
|
|
riota2df.5 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
|
|
$( A deduction version of ~ riota2f . (Contributed by NM, 17-Feb-2013.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
riota2df $p |- ( ( ph /\ E! x e. A ps ) ->
|
|
( ch <-> ( iota_ x e. A ps ) = B ) ) $=
|
|
( wreu wa cv wcel cio wceq crio adantr simpr weu df-reu eqeltrd biantrurd
|
|
sylib adantlr bitr3d nfreu1 nfan wnf wnfc iota2df df-riota eqeq1i syl6bbr
|
|
wb ) ABDELZMZCDNZEOZBMZDPZFQBDERZFQURVACDFEAFEOZUQJSZURUQVADUAAUQTBDEUBUE
|
|
URUSFQZMZBVACVGUTBVGUSFEURVFTURVDVFVESUCUDAVFBCUPUQKUFUGAUQDGBDEUHUIACDUJ
|
|
UQISADFUKUQHSULVCVBFBDEUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
riota2f.1 $e |- F/_ x B $.
|
|
riota2f.2 $e |- F/ x ps $.
|
|
riota2f.3 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( This theorem shows a condition that allows us to represent a descriptor
|
|
with a class expression ` B ` . (Contributed by NM, 23-Aug-2011.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
riota2f $p |- ( ( B e. A /\ E! x e. A ph ) ->
|
|
( ps <-> ( iota_ x e. A ph ) = B ) ) $=
|
|
( wcel nfel1 wnfc a1i wnf id cv wceq wb adantl riota2df ) EDIZABCDECEDFJC
|
|
EKTFLBCMTGLTNCOEPABQTHRS $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $. $d x B $.
|
|
riota2.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( This theorem shows a condition that allows us to represent a descriptor
|
|
with a class expression ` B ` . (Contributed by NM, 23-Aug-2011.)
|
|
(Revised by Mario Carneiro, 10-Dec-2016.) $)
|
|
riota2 $p |- ( ( B e. A /\ E! x e. A ph ) ->
|
|
( ps <-> ( iota_ x e. A ph ) = B ) ) $=
|
|
( nfcv nfv riota2f ) ABCDECEGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
riotaprop.0 $e |- F/ x ps $.
|
|
riotaprop.1 $e |- B = ( iota_ x e. A ph ) $.
|
|
riotaprop.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Properties of a restricted definite description operator. Todo
|
|
( ~ df-riota update): can some uses of ~ riota2f be shortened with this?
|
|
(Contributed by NM, 23-Nov-2013.) $)
|
|
riotaprop $p |- ( E! x e. A ph -> ( B e. A /\ ps ) ) $=
|
|
( wreu wcel crio riotacl syl5eqel wa wceq eqcomi nfriota1 nfcxfr riota2f
|
|
mpbiri mpancom jca ) ACDIZEDJZBUCEACDKZDGACDLMZUDUCBUFUDUCNBUEEOEUEGPABCD
|
|
ECEUEGACDQRFHSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ph $. $d y ps $.
|
|
riota5f.1 $e |- ( ph -> F/_ x B ) $.
|
|
riota5f.2 $e |- ( ph -> B e. A ) $.
|
|
riota5f.3 $e |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) ) $.
|
|
$( A method for computing restricted iota. (Contributed by NM,
|
|
16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
riota5f $p |- ( ph -> ( iota_ x e. A ps ) = B ) $=
|
|
( vy cv wceq wb wral ralrimiva wi wcel wa wtru a1tru nfv crio wsbc adantl
|
|
wreu reu6i nfra1 nfan nfcvd nfvd simprl simpr simplrr simplrl eqeltrd rsp
|
|
sylc mpbird 2thd mpdan mpbid expr rspsbc nfeqd nfan1 eqeq2d bibi2d ralbid
|
|
riota2df imbi12d sbcied mpd ) ABCJZEKZLZCDMZBCDUAZEKZAVNCDHNABVLIJZKZLZCD
|
|
MZVPVRKZOZIEUBZVOVQOZAEDPWCIDMWDGAWCIDAVRDPZWAWBAWFWAQZQZRWBWHSWHBCDUDZRW
|
|
BLWGWIABCDVRUEUCWHBRCDVRAWGCACTZWFWACWFCTVTCDUFUGUGWHCVRUHWHRCUIAWFWAUJWH
|
|
VSQZBRWKBVSWHVSUKZWKWAVLDPVTAWFWAVSULWKVLVRDWLAWFWAVSUMUNVTCDUOUPUQWKSURV
|
|
HUSUTVANWCIEDVBUPAWCWEIEDGAVREKZQZWAVOWBVQWNVTVNCDAWMCWJACVREACVRUHFVCVDW
|
|
NVSVMBWNVREVLAWMUKZVEVFVGWNVREVPWOVEVIVJUTVK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
riota5.1 $e |- ( ph -> B e. A ) $.
|
|
riota5.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) ) $.
|
|
$( A method for computing restricted iota. (Contributed by NM,
|
|
20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) $)
|
|
riota5 $p |- ( ph -> ( iota_ x e. A ps ) = B ) $=
|
|
( nfcvd riota5f ) ABCDEACEHFGI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Restriction of a unique element to a smaller class. (Contributed by NM,
|
|
21-Aug-2011.) (Revised by NM, 22-Mar-2013.) $)
|
|
riotass2 $p |- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) )
|
|
/\ ( E. x e. A ph /\ E! x e. B ps ) )
|
|
-> ( iota_ x e. A ph ) = ( iota_ x e. B ps ) ) $=
|
|
( wi wral wa wrex wreu crio wsbc wceq reuss2 simplr riotasbc wcel riotacl
|
|
wss syl rspsbc sbcimg sylibd mpid sylc wb ssel ad2antrr mpd simprr nfsbc1
|
|
nfriota1 sbceq1a riota2f syl2anc mpbid eqcomd ) DESZABFZCDGZHZACDIZBCEJZH
|
|
ZHZBCEKZACDKZVEBCVGLZVFVGMZVEACDJZUTVHABCDENZURUTVDOVJUTACVGLZVHACDPVJVGD
|
|
QZUTVLVHFZFACDRZVMUTUSCVGLVNUSCVGDUAABCVGDUBUCTUDUEVEVGEQZVCVHVIUFVEVMVPV
|
|
EVJVMVKVOTURVMVPFUTVDDEVGUGUHUIVAVBVCUJBVHCEVGACDULZBCVGVQUKBCVGUMUNUOUPU
|
|
Q $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Restriction of a unique element to a smaller class. (Contributed by NM,
|
|
19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) $)
|
|
riotass $p |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) ->
|
|
( iota_ x e. A ph ) = ( iota_ x e. B ph ) ) $=
|
|
( wss wrex wreu w3a crio wsbc wceq reuss riotasbc syl wcel riotacl sseldd
|
|
wb simp1 simp3 nfriota1 nfsbc1 sbceq1a riota2f syl2anc mpbid eqcomd ) CDE
|
|
ZABCFZABDGZHZABDIZABCIZUKABUMJZULUMKZUKABCGZUNABCDLZABCMNUKUMDOUJUNUORUKC
|
|
DUMUHUIUJSUKUPUMCOUQABCPNQUHUIUJTAUNBDUMABCUAZABUMURUBABUMUCUDUEUFUG $.
|
|
|
|
$( Restriction of a unique element to a smaller class. (Contributed by NM,
|
|
19-Feb-2006.) (Revised by NM, 16-Jun-2017.) $)
|
|
moriotass $p |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) ->
|
|
( iota_ x e. A ph ) = ( iota_ x e. B ph ) ) $=
|
|
( wss wrex wrmo wreu crio wceq w3a ssrexv imp 3adant3 simp3 reu5 sylanbrc
|
|
riotass syld3an3 ) CDEZABCFZABDGZABDHZABCIABDIJTUAUBKABDFZUBUCTUAUDUBTUAU
|
|
DABCDLMNTUAUBOABDPQABCDRS $.
|
|
$}
|
|
|
|
$( A restricted class abstraction with a unique member can be expressed as a
|
|
singleton. (Contributed by NM, 30-May-2006.) $)
|
|
snriota $p |- ( E! x e. A ph ->
|
|
{ x e. A | ph } = { ( iota_ x e. A ph ) } ) $=
|
|
( wreu cv wcel wa cab cio csn crab crio weu wceq df-reu sniota sylbi df-rab
|
|
df-riota sneqi 3eqtr4g ) ABCDZBECFAGZBHZUCBIZJZABCKABCLZJUBUCBMUDUFNABCOUCB
|
|
PQABCRUGUEABCSTUA $.
|
|
|
|
${
|
|
$d x y z A $. $d x z B $.
|
|
eusvobj1.1 $e |- B e. _V $.
|
|
$( Specify the same property in two ways when class ` B ( y ) ` is
|
|
single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by
|
|
Mario Carneiro, 24-Dec-2016.) $)
|
|
eusvobj2 $p |- ( E! x E. y e. A x = B
|
|
-> ( E. y e. A x = B <-> A. y e. A x = B ) ) $=
|
|
( vz cv wceq wrex weu wral cab csn wex wi euabsn2 wcel eleq2 abid exlimiv
|
|
velsn 3bitr3g nfre1 nfeq1 elabrex eqcom bitri syl6bb syl5ib ralrimi eqeq1
|
|
nfab elsn ralbidv syl5ibrcom sylbid sylbi euex rexm r19.2m ex 3syl impbid
|
|
) AGZDHZBCIZAJZVFVEBCKZVGVFALZFGZMZHZFNVFVHOZVFAFPVLVMFVLVFVDVJHZVHVLVDVI
|
|
QVDVKQVFVNVIVKVDRVFASAVJUAUBVLVHVNVJDHZBCKVLVOBCBVIVKVFBAVEBCUCULUDBGCQZD
|
|
VIQZVLVOBACDEUEVLVQDVKQZVOVIVKDRVRDVJHVODVJEUMDVJUFUGUHUIUJVNVEVOBCVDVJDU
|
|
KUNUOUPTUQVGVFANVPBNZVHVFOVFAURVFVSAVEBCUSTVSVHVFVEBCUTVAVBVC $.
|
|
|
|
$( Specify the same object in two ways when class ` B ( y ) ` is
|
|
single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by
|
|
Mario Carneiro, 19-Nov-2016.) $)
|
|
eusvobj1 $p |- ( E! x E. y e. A x = B
|
|
-> ( iota x E. y e. A x = B ) = ( iota x A. y e. A x = B ) ) $=
|
|
( cv wceq wrex weu wral wb wal cio nfeu1 eusvobj2 alrimi iotabi syl ) AFD
|
|
GZBCHZAIZTSBCJZKZALTAMUBAMGUAUCATANABCDEOPTUBAQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( There is one domain element for each value of a one-to-one onto
|
|
function. (Contributed by NM, 26-May-2006.) $)
|
|
f1ofveu $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
E! x e. A ( F ` x ) = C ) $=
|
|
( wf1o wcel wa cv cfv wceq wreu cop ccnv wf f1ocnv f1of syl wb wfn 3com23
|
|
feu sylan f1ocnvfvb dff1o4 simprbi fnopfvb 3adant3 syl3an1 bitrd reubidva
|
|
w3a 3expa mpbird ) BCEFZDCGZHZAIZEJDKZABLDURMENZGZABLZUOCBUTOZUPVBUOCBUTF
|
|
VCBCEPCBUTQRACBDUTUBUCUQUSVAABUOUPURBGZUSVASUOUPVDULUSDUTJURKZVAUOVDUPUSV
|
|
ESBCURDEUDUAUOUTCTZUPVDVEVASZUOEBTVFBCEUEUFVFUPVGVDCDURUTUGUHUIUJUMUKUN
|
|
$.
|
|
|
|
$( Value of the converse of a one-to-one onto function. (Contributed by
|
|
NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
|
|
f1ocnvfv3 $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) ) $=
|
|
( wf1o wcel wa cv cfv wceq crio ccnv f1ocnvdm f1ocnvfvb 3expa an32s eqcom
|
|
wb syl6bbr riota5 eqcomd ) BCEFZDCGZHZAIZEJDKZABLDEMJZUEUGABUHBCDENUEUFBG
|
|
ZHUGUHUFKZUFUHKUCUIUDUGUJSZUCUIUDUKBCUFDEOPQUFUHRTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Restricted iota equals the empty set when not meaningful. (Contributed
|
|
by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
|
|
(Revised by NM, 13-Sep-2018.) $)
|
|
riotaund $p |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) ) $=
|
|
( wreu wn crio cv wcel cio df-riota weu wceq df-reu iotanul sylnbi syl5eq
|
|
wa c0 ) ABCDZEABCFBGCHAQZBIZRABCJSTBKUARLABCMTBNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u ph $.
|
|
acexmidlem.a $e |- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } $.
|
|
acexmidlem.b $e |- B = { x e. { (/) , { (/) } }
|
|
| ( x = { (/) } \/ ph ) } $.
|
|
acexmidlem.c $e |- C = { A , B } $.
|
|
$( Lemma for ~ acexmid . (Contributed by Jim Kingdon, 6-Aug-2019.) $)
|
|
acexmidlema $p |- ( { (/) } e. A -> ph ) $=
|
|
( c0 csn wcel wceq wo cv cpr crab eleq2i wb p0ex ax-mp prid2 eqeq1 orbi1d
|
|
elrab3 bitri wn wi noel 0ex snid eleq2 mpbii mto orel1 sylbi ) IJZCKZUPIL
|
|
ZAMZAUQUPBNZILZAMZBIUPOZPZKZUSCVDUPFQUPVCKVEUSRIUPSUAVBUSBUPVCUTUPLVAURAU
|
|
TUPIUBUCUDTUEURUFUSAUGURIIKZIUHURIUPKVFIUIUJUPIIUKULUMURAUNTUO $.
|
|
|
|
$( Lemma for ~ acexmid . (Contributed by Jim Kingdon, 6-Aug-2019.) $)
|
|
acexmidlemb $p |- ( (/) e. B -> ph ) $=
|
|
( c0 wcel csn wceq wo cv cpr crab eleq2i wb 0ex ax-mp prid1 orbi1d elrab3
|
|
eqeq1 bitri wn wi noel snid eleq2 mpbiri mto orel1 sylbi ) IDJZIIKZLZAMZA
|
|
UOIBNZUPLZAMZBIUPOZPZJZURDVCIGQIVBJVDURRIUPSUAVAURBIVBUSILUTUQAUSIUPUDUBU
|
|
CTUEUQUFURAUGUQIIJZIUHUQVEIUPJISUIIUPIUJUKULUQAUMTUN $.
|
|
|
|
$( Lemma for ~ acexmid . (Contributed by Jim Kingdon, 6-Aug-2019.) $)
|
|
acexmidlemph $p |- ( ph -> A = B ) $=
|
|
( c0 csn wceq wo wral olc ralrimivw crab eqeq2i rabid2 bitri sylibr cpr
|
|
cv eqtr3d ) AIIJZUAZCDABUBZIKZALZBUEMZUECKZAUHBUEAUGNOUJUEUHBUEPZKUICUKUE
|
|
FQUHBUERSTAUFUDKZALZBUEMZUEDKZAUMBUEAULNOUOUEUMBUEPZKUNDUPUEGQUMBUERSTUC
|
|
$.
|
|
|
|
$( Lemma for ~ acexmid . (Contributed by Jim Kingdon, 6-Aug-2019.) $)
|
|
acexmidlemab $p |- (
|
|
( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\
|
|
( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) ->
|
|
-. ph ) $=
|
|
( cv wcel wa wrex crio c0 wceq csn noel 0ex eleq2 mpbiri mto acexmidlemph
|
|
snid id eleq1 anbi1d rexbidv riotaeqbidv syl eqeq1d biimpa adantrr simprr
|
|
eqtr3d ex mtoi con2i ) AFELZMZDLVAMZNZECLZOZDFPZQRZGVAMZVCNZEVEOZDGPZQSZR
|
|
ZNZAVOQVMRZVPQQMZQTVPVQQVMMQUAUFQVMQUBUCUDAVOVPAVONVLQVMAVHVLQRZVNAVHVRAV
|
|
GVLQAFGRZVGVLRABFGHIJKUEVSVFVKDFGVSUGVSVDVJEVEVSVBVIVCFGVAUHUIUJUKULUMUNU
|
|
OAVHVNUPUQURUSUT $.
|
|
|
|
$( Lemma for ~ acexmid . Here we divide the proof into cases (based on the
|
|
disjunction implicit in an unordered pair, not the sort of case
|
|
elimination which relies on excluded middle).
|
|
|
|
The cases are (1) the choice function evaluated at ` A ` equals
|
|
` { (/) } ` , (2) the choice function evaluated at ` B ` equals
|
|
` (/) ` , and (3) the choice function evaluated at ` A ` equals ` (/) `
|
|
and the choice function evaluated at ` B ` equals ` { (/) } ` .
|
|
|
|
Because of the way we represent the choice function ` y ` , the choice
|
|
function evaluated at ` A ` is
|
|
` ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) ` and the choice
|
|
function evaluated at ` B ` is
|
|
` ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) ` . Other than the
|
|
difference in notation these work just as ` ( y `` A ) ` and
|
|
` ( y `` B ) ` would if ` y ` were a function as defined by ~ df-fun .
|
|
|
|
Although it isn't exactly about the division into cases, it is also
|
|
convenient for this lemma to also include the step that if the choice
|
|
function evaluated at ` A ` equals ` { (/) } ` , then ` { (/) } e. A `
|
|
and likewise for ` B ` .
|
|
|
|
(Contributed by Jim Kingdon, 7-Aug-2019.) $)
|
|
acexmidlemcase $p |- ( A. z e. C E! v e. z E. u e. y
|
|
( z e. u /\ v e. u ) ->
|
|
( { (/) } e. A \/
|
|
(/) e. B \/
|
|
( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\
|
|
( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) ) ) $=
|
|
( cv wcel wa c0 wceq wo eleq1 syl wrex wreu wral csn crio w3o cpr wi con0
|
|
crab onsucelsucexmidlem eqeltri prid1g ax-mp eleqtrri anbi1d reueqd rspcv
|
|
rexbidv riotacl elrabi eleq2s elpri 3syl syl5ibcom orim2d mpd pp0ex rabex
|
|
cvv prid2 orim1d anddi sylib simpl jaoi orim2i orcomd simpr orim1i 3orass
|
|
jca sylibr ) DMZFMZNZEMWENZOZFCMZUAZEWDUBZDIUCZPUDZGNZPHNZGWENZWGOZFWIUAZ
|
|
EGUEZPQZHWENZWGOZFWIUAZEHUEZWMQZOZRZRZWNWOXFUFWLWNWTWOOZXFRZRXHWLXJWNWLXJ
|
|
WNWOOZWNXEOZRZRZXJWNRWLWTWNRZWOXERZOXNWLXOXPWLWTWSWMQZRZXOWLWSGNZWSPWMUGZ
|
|
NZXRWLWREGUBZXSGINWLYBUHGGHUGZIGUINGYCNGBMZPQARZBXTUJZUIJABUKULGHUIUMUNLU
|
|
OWKYBDGIWJWREWDGWDGQZWHWQFWIYGWFWPWGWDGWESUPUSUQURUNWREGUTTZYAWSYFGYEBWSX
|
|
TVAJVBWSPWMVCVDWLXQWNWTWLXSXQWNYHWSWMGSVEVFVGWLXDPQZXERZXPWLXDHNZXDXTNZYJ
|
|
WLXCEHUBZYKHINWLYMUHHYCIGHHYDWMQARZBXTUJZVJKYNBXTVHVIULVKLUOWKYMDHIWJXCEW
|
|
DHWDHQZWHXBFWIYPWFXAWGWDHWESUPUSUQURUNXCEHUTTZYLXDYOHYNBXDXTVAKVBXDPWMVCV
|
|
DWLYIWOXEWLYKYIWOYQXDPHSVEVLVGWBWTWNWOXEVMVNXMWNXJXKWNXLWNWOVOWNXEVOVPVQT
|
|
VRXJXGWNXIWOXFWTWOVSVTVQTWNWOXFWAWC $.
|
|
|
|
$( Lemma for ~ acexmid . List the cases identified in ~ acexmidlemcase and
|
|
hook them up to the lemmas which handle each case. (Contributed by Jim
|
|
Kingdon, 7-Aug-2019.) $)
|
|
acexmidlem1 $p |- ( A. z e. C E! v e. z E. u e. y
|
|
( z e. u /\ v e. u ) -> ( ph \/ -. ph ) ) $=
|
|
( cv wcel wa wrex c0 crio wceq orcd csn w3o wn acexmidlemcase acexmidlema
|
|
wreu wral wo acexmidlemb acexmidlemab olcd 3jaoi syl ) DMZFMZNEMUONZOFCMZ
|
|
PEUNUFDIUGQUAZGNZQHNZGUONUPOFUQPEGRQSHUONUPOFUQPEHRURSOZUBAAUCZUHZABCDEFG
|
|
HIJKLUDUSVCUTVAUSAVBABGHIJKLUETUTAVBABGHIJKLUITVAVBAABCEFGHIJKLUJUKULUM
|
|
$.
|
|
|
|
$( Lemma for ~ acexmid . This builds on ~ acexmidlem1 by noting that every
|
|
element of ` C ` is inhabited.
|
|
|
|
(Note that ` y ` is not quite a function in the ~ df-fun sense because
|
|
it uses ordered pairs as described in ~ opthreg rather than ~ df-op ).
|
|
|
|
The set ` A ` is also found in ~ onsucelsucexmidlem .
|
|
|
|
(Contributed by Jim Kingdon, 5-Aug-2019.) $)
|
|
acexmidlem2 $p |- ( A. z e. C A. w e. z E! v e. z E. u e. y
|
|
( z e. u /\ v e. u ) -> ( ph \/ -. ph ) ) $=
|
|
( wel cv wral wo wcel wceq c0 wa wrex wreu wn wex wi df-ral 19.23v bitr2i
|
|
wal cpr eleq2i vex elpr bitri csn crab onsucelsucexmidlem1 eleqtrri elex2
|
|
ax-mp eleq2 exbidv mpbiri p0ex prid2 eqid orci eqeq1 orbi1d mpbir2an jaoi
|
|
elrab sylbi pm2.27 syl imp sylan2br ralimiaa acexmidlem1 ) DGNFGNUAGCOUBF
|
|
DOZUCZEWAPZDJPWBDJPAAUDQWCWBDJWCWAJRZEDNZEUEZWBUFZWBWCWEWBUFEUJWGWBEWAUGW
|
|
EWBEUHUIWDWGWBWDWFWGWBUFWDWAHSZWAISZQZWFWDWAHIUKZRWJJWKWAMULWAHIDUMUNUOWH
|
|
WFWIWHWFEOZHRZEUEZTHRWNTBOZTSAQBTTUPZUKZUQHABURKUSETHUTVAWHWEWMEWAHWLVBVC
|
|
VDWIWFWLIRZEUEZWPIRWSWPWOWPSZAQZBWQUQZIWPXBRWPWQRWPWPSZAQZTWPVEVFXCAWPVGV
|
|
HXAXDBWPWQWTWTXCAWOWPWPVIVJVMVKLUSEWPIUTVAWIWEWREWAIWLVBVCVDVLVNWFWBVOVPV
|
|
QVRVSABCDFGHIJKLMVTVP $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z w v u s t $.
|
|
acexmidlemv.choice $e |- E. y A. z e. x A. w e. z E! v e. z E. u e. y
|
|
( z e. u /\ v e. u ) $.
|
|
$( Lemma for ~ acexmid .
|
|
|
|
This is ~ acexmid with additional distinct variable constraints, most
|
|
notably between ` ph ` and ` x ` .
|
|
|
|
(Contributed by Jim Kingdon, 6-Aug-2019.) $)
|
|
acexmidlemv $p |- ( ph \/ -. ph ) $=
|
|
( vs vt cv wcel wral c0 wceq wo cpr crab wex cvv wa wrex wreu csn wn con0
|
|
onsucelsucexmidlem pp0ex rabex prexg mp2an raleq exbidv vtocl orbi1d eqid
|
|
eqeq1 cbvrabv acexmidlem2 exlimiv ax-mp ) DKZGKZLFKVCLUAGCKUBFVBUCEVBMZDI
|
|
KZNOZAPZINNUDZQZRZVEVHOZAPZIVIRZQZMZCSZAAUEPZVDDBKZMZCSVPBVNVJUFLVMTLVNTL
|
|
AIUGVLIVIUHUIVJVMUFTUJUKVRVNOVSVOCVDDVRVNULUMHUNVOVQCAJCDEFGVJVMVNVGJKZNO
|
|
ZAPIJVIVEVTOZVFWAAVEVTNUQUOURVLVTVHOZAPIJVIWBVKWCAVEVTVHUQUOURVNUPUSUTVA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $. $d a b c d e f ph $. $d a b c d e f u v w x y z $.
|
|
acexmid.choice $e |- E. y A. z e. x A. w e. z E! v e. z E. u e. y
|
|
( z e. u /\ v e. u ) $.
|
|
$( The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer]
|
|
p. 483.
|
|
|
|
The statement of the axiom of choice given here is ac2 in the Metamath
|
|
Proof Explorer (version of 3-Aug-2019). In particular, note that the
|
|
choice function ` y ` provides a value when ` z ` is inhabited (as
|
|
opposed to non-empty as in some statements of the axiom of choice).
|
|
|
|
Essentially the same proof can also be found at "The axiom of choice
|
|
implies instances of EM", [Crosilla], p. "Set-theoretic principles
|
|
incompatible with intuitionistic logic".
|
|
|
|
Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill
|
|
theorem, after Radu Diaconescu who discovered it in 1975 in the
|
|
framework of topos theory and N. D. Goodman and John Myhill in 1978 in
|
|
the framework of set theory (although it already appeared as an exercise
|
|
in Errett Bishop's book _Foundations of Constructive Analysis_ from
|
|
1967).
|
|
|
|
(Contributed by Jim Kingdon, 4-Aug-2019.) $)
|
|
acexmid $p |- ( ph \/ -. ph ) $=
|
|
( va vb vc vd vf ve wel wa cv wral weq wb wrex wreu wex wal weu wsb sb8eu
|
|
nfv w3a eleq12 ancoms 3adant3 3ad2antl1 3ad2antl2 simpl3 cbvrexdva2 3expa
|
|
anbi12d 3com23 sbiedv eubidv syl5bb df-reu 3bitr4g adantr simpll cbvexdva
|
|
cbvraldva2 adantll cbvalv mpgbir spi acexmidlemv ) AIJKLMNKNOZMNOZPZNJQZU
|
|
AZMKQZUBZLVSRZKIQZRZJUCZIWDIUDDGOZFGOZPZGCQZUAZFDQZUBZEWJRZDBQZRZCUCZBWDW
|
|
OIBIBSZWCWNJCWPJCSZPWAWLKDWBWMWQKDSZWAWLTZWPWRWQWSWRWQPZVTWKLEVSWJWTVTWKT
|
|
LESZWTMKOZVRPZMUEZFDOZWIPZFUEZVTWKXDXCMFUFZFUEWTXGXCMFXCFUHUGWTXHXFFWTXCX
|
|
FMFWRWQMFSZXCXFTZWRXIWQXJWRXIWQUIZXBXEVRWIWRXIXBXETZWQXIWRXLMQZFQZVSWJUJU
|
|
KULXKVPWGNGVQWHXKNGSZPVNWEVOWFWRXIXOVNWETWQVSWJNQZGQZUJUMXIWRXOVOWFTWQXMX
|
|
NXPXQUJUNURWRXIWQXOUOUPURUSUQUTVAVBVRMVSVCWIFWJVCVDVEWRWQXAVFVHUKVIWPWQWR
|
|
VFVHVGVJHVKVLVM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Operations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend class notation to include the value of an operation ` F ` (such as
|
|
+ ) for two arguments ` A ` and ` B ` . Note that the syntax is simply
|
|
three class symbols in a row surrounded by parentheses. Since operation
|
|
values are the only possible class expressions consisting of three class
|
|
expressions in a row surrounded by parentheses, the syntax is
|
|
unambiguous. $)
|
|
co $a class ( A F B ) $.
|
|
|
|
$( Extend class notation to include class abstraction (class builder) of
|
|
nested ordered pairs. $)
|
|
coprab $a class { <. <. x , y >. , z >. | ph } $.
|
|
|
|
$( Extend the definition of a class to include maps-to notation for defining
|
|
an operation via a rule. $)
|
|
cmpt2 $a class ( x e. A , y e. B |-> C ) $.
|
|
|
|
$( Define the value of an operation. Definition of operation value in
|
|
[Enderton] p. 79. Note that the syntax is simply three class expressions
|
|
in a row bracketed by parentheses. There are no restrictions of any kind
|
|
on what those class expressions may be, although only certain kinds of
|
|
class expressions - a binary operation ` F ` and its arguments ` A ` and
|
|
` B ` - will be useful for proving meaningful theorems. For example, if
|
|
class ` F ` is the operation + and arguments ` A ` and ` B ` are 3 and 2 ,
|
|
the expression ( 3 + 2 ) can be proved to equal 5 . This definition is
|
|
well-defined, although not very meaningful, when classes ` A ` and/or
|
|
` B ` are proper classes (i.e. are not sets); see ~ ovprc1 and ~ ovprc2 .
|
|
On the other hand, we often find uses for this definition when ` F ` is a
|
|
proper class. ` F ` is normally equal to a class of nested ordered pairs
|
|
of the form defined by ~ df-oprab . (Contributed by NM, 28-Feb-1995.) $)
|
|
df-ov $a |- ( A F B ) = ( F ` <. A , B >. ) $.
|
|
|
|
${
|
|
$d x w $. $d y w $. $d z w $. $d w ph $.
|
|
$( Define the class abstraction (class builder) of a collection of nested
|
|
ordered pairs (for use in defining operations). This is a special case
|
|
of Definition 4.16 of [TakeutiZaring] p. 14. Normally ` x ` , ` y ` ,
|
|
and ` z ` are distinct, although the definition doesn't strictly require
|
|
it. See ~ df-ov for the value of an operation. The brace notation is
|
|
called "class abstraction" by Quine; it is also called a "class builder"
|
|
in the literature. The value of the most common operation class builder
|
|
is given by ~ ovmpt2 . (Contributed by NM, 12-Mar-1995.) $)
|
|
df-oprab $a |- { <. <. x , y >. , z >. | ph } =
|
|
{ w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z C $.
|
|
$( Define maps-to notation for defining an operation via a rule. Read as
|
|
"the operation defined by the map from ` x , y ` (in ` A X. B ` ) to
|
|
` B ( x , y ) ` ." An extension of ~ df-mpt for two arguments.
|
|
(Contributed by NM, 17-Feb-2008.) $)
|
|
df-mpt2 $a |- ( x e. A , y e. B |-> C ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } $.
|
|
$}
|
|
|
|
$( Equality theorem for operation value. (Contributed by NM,
|
|
28-Feb-1995.) $)
|
|
oveq $p |- ( F = G -> ( A F B ) = ( A G B ) ) $=
|
|
( wceq cop cfv co fveq1 df-ov 3eqtr4g ) CDEABFZCGLDGABCHABDHLCDIABCJABDJK
|
|
$.
|
|
|
|
$( Equality theorem for operation value. (Contributed by NM,
|
|
28-Feb-1995.) $)
|
|
oveq1 $p |- ( A = B -> ( A F C ) = ( B F C ) ) $=
|
|
( wceq cop cfv co opeq1 fveq2d df-ov 3eqtr4g ) ABEZACFZDGBCFZDGACDHBCDHMNOD
|
|
ABCIJACDKBCDKL $.
|
|
|
|
$( Equality theorem for operation value. (Contributed by NM,
|
|
28-Feb-1995.) $)
|
|
oveq2 $p |- ( A = B -> ( C F A ) = ( C F B ) ) $=
|
|
( wceq cop cfv co opeq2 fveq2d df-ov 3eqtr4g ) ABEZCAFZDGCBFZDGCADHCBDHMNOD
|
|
ABCIJCADKCBDKL $.
|
|
|
|
$( Equality theorem for operation value. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
oveq12 $p |- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq1 oveq2 sylan9eq ) ABFCDFACEGBCEGBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
oveq1i.1 $e |- A = B $.
|
|
$( Equality inference for operation value. (Contributed by NM,
|
|
28-Feb-1995.) $)
|
|
oveq1i $p |- ( A F C ) = ( B F C ) $=
|
|
( wceq co oveq1 ax-mp ) ABFACDGBCDGFEABCDHI $.
|
|
|
|
$( Equality inference for operation value. (Contributed by NM,
|
|
28-Feb-1995.) $)
|
|
oveq2i $p |- ( C F A ) = ( C F B ) $=
|
|
( wceq co oveq2 ax-mp ) ABFCADGCBDGFEABCDHI $.
|
|
|
|
${
|
|
oveq12i.2 $e |- C = D $.
|
|
$( Equality inference for operation value. (Contributed by NM,
|
|
28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
oveq12i $p |- ( A F C ) = ( B F D ) $=
|
|
( wceq co oveq12 mp2an ) ABHCDHACEIBDEIHFGABCDEJK $.
|
|
$}
|
|
|
|
$( Equality inference for operation value. (Contributed by NM,
|
|
24-Nov-2007.) $)
|
|
oveqi $p |- ( C A D ) = ( C B D ) $=
|
|
( wceq co oveq ax-mp ) ABFCDAGCDBGFECDABHI $.
|
|
$}
|
|
|
|
${
|
|
oveq123i.1 $e |- A = C $.
|
|
oveq123i.2 $e |- B = D $.
|
|
oveq123i.3 $e |- F = G $.
|
|
$( Equality inference for operation value. (Contributed by FL,
|
|
11-Jul-2010.) $)
|
|
oveq123i $p |- ( A F B ) = ( C G D ) $=
|
|
( co oveq12i oveqi eqtri ) ABEJCDEJCDFJACBDEGHKEFCDILM $.
|
|
$}
|
|
|
|
${
|
|
oveq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
13-Mar-1995.) $)
|
|
oveq1d $p |- ( ph -> ( A F C ) = ( B F C ) ) $=
|
|
( wceq co oveq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
13-Mar-1995.) $)
|
|
oveq2d $p |- ( ph -> ( C F A ) = ( C F B ) ) $=
|
|
( wceq co oveq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
9-Sep-2006.) $)
|
|
oveqd $p |- ( ph -> ( C A D ) = ( C B D ) ) $=
|
|
( wceq co oveq syl ) ABCGDEBHDECHGFDEBCIJ $.
|
|
|
|
${
|
|
oveq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
oveq12d $p |- ( ph -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq12 syl2anc ) ABCIDEIBDFJCEFJIGHBCDEFKL $.
|
|
$}
|
|
|
|
${
|
|
opreqan12i.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
oveqan12d $p |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq12 syl2an ) ACDJEFJCEGKDFGKJBHICDEFGLM $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
oveqan12rd $p |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) ) $=
|
|
( co wceq oveqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
oveq123d.1 $e |- ( ph -> F = G ) $.
|
|
oveq123d.2 $e |- ( ph -> A = B ) $.
|
|
oveq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for operation value. (Contributed by FL,
|
|
22-Dec-2008.) $)
|
|
oveq123d $p |- ( ph -> ( A F C ) = ( B G D ) ) $=
|
|
( co oveqd oveq12d eqtrd ) ABDFKBDGKCEGKAFGBDHLABCDEGIJMN $.
|
|
$}
|
|
|
|
${
|
|
nfovd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfovd.3 $e |- ( ph -> F/_ x F ) $.
|
|
nfovd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfov .
|
|
(Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
nfovd $p |- ( ph -> F/_ x ( A F B ) ) $=
|
|
( co cop cfv df-ov nfopd nffvd nfcxfrd ) ABCDEICDJZEKCDELABPEGABCDFHMNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfov.1 $e |- F/_ x A $.
|
|
nfov.2 $e |- F/_ x F $.
|
|
nfov.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for operation value. (Contributed by
|
|
NM, 4-May-2004.) $)
|
|
nfov $p |- F/_ x ( A F B ) $=
|
|
( co wnfc wtru a1i nfovd trud ) ABCDHIJABCDABIJEKADIJFKACIJGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Slight elaboration of ~ exdistrfor . A lemma for ~ oprabid .
|
|
(Contributed by Jim Kingdon, 15-Jan-2019.) $)
|
|
oprabidlem $p |-
|
|
( E. x E. y ( x = z /\ ps ) -> E. x ( x = z /\ E. y ps ) ) $=
|
|
( weq wal wi wo wnf ax-bndl ax-10 wn dtru pm2.53 mpi df-nf sylibr orim12i
|
|
albii ax-mp exdistrfor ) BDEZABCCBECFZCDECFZUBUBCFGCFZBFZHZHBCEBFZUBCIZBF
|
|
ZHBDCJUCUHUGUJCBKUGUFUJUGUDLUFCDMUDUFNOUIUEBUBCPSQRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d a ph r s t w $. $d a r s t w x $. $d a r s t w y $. $d a r s t w z $.
|
|
$( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.
|
|
Although this theorem would be useful with a distinct variable
|
|
constraint between ` x ` , ` y ` , and ` z ` , we use ~ ax-bndl to
|
|
eliminate that constraint. (Contributed by Mario Carneiro,
|
|
20-Mar-2013.) $)
|
|
oprabid $p |- ( <. <. x , y >. , z >. e.
|
|
{ <. <. x , y >. , z >. | ph } <-> ph ) $=
|
|
( vw va vt vr vs cv cop wceq wa wex cvv wcel vex wi oprabidlem weu coprab
|
|
opex opexg mp2an eqvinop biimpi eqeq1 opth1 syl6bi opeq1 eqeq2d w3a otth2
|
|
df-3an bitri anbi1i anass 3bitri 3exbii eximi excom 3imtr4i anim2i euequ1
|
|
3syl sylbi eupick mpan syl6 3impd syl5bi com12 syl5 syl6bb anbi1d 3exbidv
|
|
eqcom imbi1d imbi12d mpbiri adantr exlimivv com3l mpdd mpcom 19.8a impbid
|
|
ex df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWPABCDUAWNOPWOOPWPOPW
|
|
LWMBQZCQZUBZDQZWNWOOOUCUDWQXAAWKFJZGJZKZLZXHWPLZMZGNFNZWQXAARZWQXLFGWKWNW
|
|
OXDXEUEUFXKWQXMRZFGXIXNXJXIWQXFWNLZXMXIWQXJXOWKXHWPUGXFXGWNWOFQGQUHUIXOXI
|
|
WQXMXOXFHJZIJZKZLZXRWNLZMZINHNXIXNRZHIXFWLWMXBXCUEYAYBHIXSYBXTXSXIWKXRXGK
|
|
ZLZXNXSXHYCWKXFXRXGUJUKYDXNWPYCLZYEAMZDNCNBNZARZRYGWLXPLZWMXQLZWOXGLZAMZD
|
|
NZMCNZMZBNZYEAYGYIYJYLMZMZDNZCNBNZYPYFYRBCDYFYIYJMZYKMZAMUUAYLMYRYEUUBAYE
|
|
YIYJYKULZUUBWLWMXPXQWOXGXBXCXEUMZYIYJYKUNUOUPUUAYKAUQYIYJYLUQURUSYTYIYQDN
|
|
ZMZCNBNZYIUUECNZMZBNYPYSBNZCNUUFBNZCNYTUUGUUJUUKCYQBDHSUTYSBCVAUUFBCVAVBU
|
|
UEBCHSUUIYOBUUHYNYIYLCDISVCUTVEVFYPYEAYEUUCYPAUUDYPYIYJYKAYPYIYNYJYKARZRY
|
|
IBTYPYIYNRBHVDYIYNBVGVHYNYJYMUULYJCTYNYJYMRCIVDYJYMCVGVHYKDTYMUULDGVDYKAD
|
|
VGVHVIVIVJVKVLVMYDWQYEXMYHYDWQYCWPLYEWKYCWPUGYCWPVQVNZYDXAYGAYDWRYFBCDYDW
|
|
QYEAUUMVOVPVRVSVTUIWAWBVFWCWDWAWBWEWQAXAWRWSWTXAWRDWFWSCWFWTBWFVEWHWGABCD
|
|
EWIWJ $.
|
|
$}
|
|
|
|
$( The result of an operation is a set. (Contributed by Jim Kingdon,
|
|
15-Jan-2019.) $)
|
|
fnovex $p |- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) ->
|
|
( A F B ) e. _V ) $=
|
|
( cxp wfn wcel w3a co cop cfv df-ov wa opelxp funfvex funfni sylan2br 3impb
|
|
cvv syl5eqel ) ECDFZGZACHZBDHZIABEJABKZELZTABEMUCUDUEUGTHZUDUENUCUFUBHUHABC
|
|
DOUHUBUFEUFEPQRSUA $.
|
|
|
|
$( Evaluating a set operation at two sets gives a set. (Contributed by Jim
|
|
Kingdon, 19-Aug-2021.) $)
|
|
ovexg $p |- ( ( A e. V /\ F e. W /\ B e. X ) -> ( A F B ) e. _V ) $=
|
|
( wcel w3a co cop cfv cvv df-ov simp2 opexg 3adant2 fvexg syl2anc syl5eqel
|
|
) ADGZCEGZBFGZHZABCIABJZCKZLABCMUCUAUDLGZUELGTUAUBNTUBUFUAABDFOPUDCELQRS $.
|
|
|
|
${
|
|
ovprc1.1 $e |- Rel dom F $.
|
|
$( The value of an operation when the one of the arguments is a proper
|
|
class. Note: this theorem is dependent on our particular definitions of
|
|
operation value, function value, and ordered pair. (Contributed by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
ovprc $p |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) ) $=
|
|
( cvv wcel wa wn co cop cfv df-ov cdm wceq opprc 0ex syl6eqel wbr df-br
|
|
c0 wrel brrelex12 mpan sylbir con3i ndmfvg syl2anc syl5eq ) AEFBEFGZHZABC
|
|
IABJZCKZTABCLUJUKEFUKCMZFZHULTNUJUKTEABOPQUNUIUNABUMRZUIABUMSUMUAUOUIDABU
|
|
MUBUCUDUEUKCUFUGUH $.
|
|
|
|
$( The value of an operation when the first argument is a proper class.
|
|
(Contributed by NM, 16-Jun-2004.) $)
|
|
ovprc1 $p |- ( -. A e. _V -> ( A F B ) = (/) ) $=
|
|
( cvv wcel wn wa co c0 wceq simpl con3i ovprc syl ) AEFZGPBEFZHZGABCIJKRP
|
|
PQLMABCDNO $.
|
|
|
|
$( The value of an operation when the second argument is a proper class.
|
|
(Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
ovprc2 $p |- ( -. B e. _V -> ( A F B ) = (/) ) $=
|
|
( cvv wcel wn wa co c0 wceq simpr con3i ovprc syl ) BEFZGAEFZPHZGABCIJKRP
|
|
QPLMABCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y D $. $d y F $. $d x y $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) $)
|
|
csbov123g $p |- ( A e. D -> [_ A / x ]_ ( B F C ) =
|
|
( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) ) $=
|
|
( vy cv co csb wceq csbeq1 oveq123d eqeq12d nfcsb1v csbeq1a csbief vtoclg
|
|
vex nfov ) AGHZCDFIZJZAUACJZAUADJZAUAFJZIZKABUBJZABCJZABDJZABFJZIZKGBEUAB
|
|
KZUCUHUGULAUABUBLUMUDUIUEUJUFUKAUABFLAUABCLAUABDLMNAUAUBUGGSAUDUEUFAUACOA
|
|
UAFOAUADOTAHUAKCUDDUEFUFAUAFPAUACPAUADPMQR $.
|
|
$}
|
|
|
|
${
|
|
$d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov12g $p |- ( A e. V ->
|
|
[_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) ) $=
|
|
( wcel co csb csbov123g csbconstg oveqd eqtrd ) BFGZABCDEHIABCIZABDIZABEI
|
|
ZHOPEHABCDFEJNQEOPABEFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x C $. $d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov1g $p |- ( A e. V ->
|
|
[_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) ) $=
|
|
( wcel co csb csbov12g csbconstg oveq2d eqtrd ) BFGZABCDEHIABCIZABDIZEHOD
|
|
EHABCDEFJNPDOEABDFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov2g $p |- ( A e. V ->
|
|
[_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) ) $=
|
|
( wcel co csb csbov12g csbconstg oveq1d eqtrd ) BFGZABCDEHIABCIZABDIZEHCP
|
|
EHABCDEFJNOCPEABCFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y C $. $d y D $. $d x y F $. $d x y S $.
|
|
$( A frequently used special case of ~ rspc2ev for operation values.
|
|
(Contributed by NM, 21-Mar-2007.) $)
|
|
rspceov $p |- ( ( C e. A /\ D e. B /\ S = ( C F D ) ) ->
|
|
E. x e. A E. y e. B S = ( x F y ) ) $=
|
|
( cv co wceq oveq1 eqeq2d oveq2 rspc2ev ) GAIZBIZHJZKGEFHJZKGEQHJZKABEFCD
|
|
PEKRTGPEQHLMQFKTSGQFEHNMO $.
|
|
$}
|
|
|
|
$( Equivalence of operation value and ordered triple membership, analogous to
|
|
~ fnopfvb . (Contributed by NM, 17-Dec-2008.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
fnotovb $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
|
|
( ( C F D ) = R <-> <. C , D , R >. e. F ) ) $=
|
|
( cxp wfn wcel w3a cop cfv wceq co cotp wb wa opelxpi fnopfvb sylan2 eqeq1i
|
|
3impb df-ov df-ot eleq1i 3bitr4g ) FABGZHZCAIZDBIZJCDKZFLZEMZUKEKZFIZCDFNZE
|
|
MCDEOZFIUHUIUJUMUOPZUIUJQUHUKUGIURCDABRUGUKEFSTUBUPULECDFUCUAUQUNFCDEUDUEUF
|
|
$.
|
|
|
|
${
|
|
$d E f p z $. $d V f p z $. $d W z $. $d th z $. $d ps z $.
|
|
opabbrex.1 $e |- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) ) $.
|
|
opabbrex.2 $e |- ( ( V e. _V /\ E e. _V )
|
|
-> { <. f , p >. | th } e. _V ) $.
|
|
$( A collection of ordered pairs with an extension of a binary relation is
|
|
a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) $)
|
|
opabbrex $p |- ( ( V e. _V /\ E e. _V )
|
|
-> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V ) $=
|
|
( vz cvv wcel wa cv co wbr copab wex cab df-opab syl5eqelr adantrd anim2d
|
|
cop wceq 2eximdv ss2abdv syl5eqss ssexd ) EKLDKLMZCNZGNZEDFOPZAMZCGQZJNUK
|
|
ULUDUEZBMZGRCRZJSZKUJUSBCGQKBCGJTIUAUJUOUPUNMZGRCRZJSUSUNCGJTUJVAURJUJUTU
|
|
QCGUJUNBUPUJUMBAHUBUCUFUGUHUI $.
|
|
$}
|
|
|
|
$( The empty set is never an element in an ordered-pair class abstraction.
|
|
(Contributed by Alexander van der Vekens, 5-Nov-2017.) $)
|
|
0neqopab $p |- -. (/) e. { <. x , y >. | ph } $=
|
|
( c0 copab wcel id cv cop wceq wa wex wn elopab nfopab1 nfel2 nfn vex sylbi
|
|
exlimi nfopab2 wne wi opnzi nesym pm2.21 ax-mp adantr pm2.65i ) DABCEZFZUKU
|
|
KGUKDBHZCHZIZJZAKZCLZBLUKMZABCDNUQURBUKBBDUJABCOPQUPURCUKCCDUJABCUAPQUOURAU
|
|
NDUBZUOURUCZULUMBRCRUDUSUOMUTUNDUEUOURUFSUGUHTTSUI $.
|
|
|
|
${
|
|
$d x y X $. $d x y Y $.
|
|
$( If two classes are in a relationship given by an ordered-pair class
|
|
abstraction, the classes are sets. (Contributed by Jim Kingdon,
|
|
16-Jan-2019.) $)
|
|
brabvv $p |- ( X { <. x , y >. | ph } Y -> ( X e. _V /\ Y e. _V ) ) $=
|
|
( copab wbr cv wceq wex cvv wcel cop df-br elopab bitri exsimpl vex isset
|
|
wa eximi sylbi opth biimpi eqcoms 2eximi syl eeanv sylib anbi12i sylibr )
|
|
DEABCFZGZBHZDIZBJZCHZEIZCJZTZDKLZEKLZTUMUOURTZCJBJZUTUMDEMZUNUQMZIZCJZBJZ
|
|
VDUMVGATCJZBJZVIUMVEULLVKDEULNABCVEOPVJVHBVGACQUAUBVGVCBCVCVFVEVFVEIVCUNU
|
|
QDEBRCRUCUDUEUFUGUOURBCUHUIVAUPVBUSBDSCESUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x z w v $. $d y z w v $. $d w ph v $.
|
|
$( Class abstraction for operations in terms of class abstraction of
|
|
ordered pairs. (Contributed by NM, 12-Mar-1995.) $)
|
|
dfoprab2 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } $=
|
|
( vv cv cop wceq wa wex cab coprab copab excom exrot4 opeq1 3bitr3i bitri
|
|
vex eqeq2d pm5.32ri anbi1i anass an32 exbii isseti 19.42v mpbiran2 3exbii
|
|
opex 19.42vv 2exbii abbii df-oprab df-opab 3eqtr4i ) FGZBGZCGZHZDGZHZIZAJ
|
|
ZDKCKBKZFLUREGZVBHZIZVGVAIZAJZCKBKZJZDKEKZFLABCDMVLEDNVFVNFVIVKJZCKBKZEKD
|
|
KZVPDKEKVFVNVPDEOVQVOEKZDKCKBKVFVODEBCPVRVEBCDVRVEVJJZEKZVEVOVSEVIVJJZAJV
|
|
DVJJZAJVOVSWAWBAVJVIVDVJVHVCURVGVAVBQUAUBUCVIVJAUDVDVJAUERUFVTVEVJEKEVAUS
|
|
UTBTCTUKUGVEVJEUHUISUJSVPVMEDVIVKBCULUMRUNABCDFUOVLEDFUPUQ $.
|
|
|
|
$( An operation class abstraction is a relation. (Contributed by NM,
|
|
16-Jun-2004.) $)
|
|
reloprab $p |- Rel { <. <. x , y >. , z >. | ph } $=
|
|
( vw cv cop wceq wa wex coprab dfoprab2 relopabi ) EFBFCFGHAICJBJEDABCDKA
|
|
BCDELM $.
|
|
$}
|
|
|
|
$(
|
|
@{
|
|
@d x y z w v @. @d ph v @.
|
|
dfoprab2f.1 @e |- ( ph -> A. w ph ) @.
|
|
@( Class abstraction for operations in terms of class abstraction of
|
|
ordered pairs. This is a version of ~ dfoprab2 with bound-variable
|
|
hypothesis instead of distinct variable requirement. @)
|
|
dfoprab2f @p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } @=
|
|
( vv coprab cv cop wceq wa wex copab dfoprab2 ax-5 hban hbex weq
|
|
eqeq1 anbi1d 2exbidv cbvopab1
|
|
eqtri ) ABCDHGIZBICIJZKZALZCMZBMZGDNEIZUFKZALZCMBM
|
|
ZEDNABCDGOUJUNGDEUIEBUHECUGAEUGEPFQRRUNGPGESZUHUMBCUOUGULAUEUKUFTUAUBUCUD
|
|
@.
|
|
@}
|
|
$)
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w ph $.
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy,
|
|
19-Jun-2012.) $)
|
|
nfoprab1 $p |- F/_ x { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfab nfcxfr ) BABCDFEGBG
|
|
CGHDGHIAJDKCKZBKZELABCDEMRBEQBNOP $.
|
|
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy,
|
|
30-Jul-2012.) $)
|
|
nfoprab2 $p |- F/_ y { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) CABCD
|
|
FEGBGCGHDGHIAJDKZCKZBKZELABCDEMTCESCBRCNOPQ $.
|
|
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 22-Aug-2013.) $)
|
|
nfoprab3 $p |- F/_ z { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) DABCD
|
|
FEGBGCGHDGHIAJZDKZCKZBKZELABCDEMUADETDBSDCRDNOOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d v w x $. $d v w y $. $d v w z $. $d v ph $.
|
|
nfoprab.1 $e |- F/ w ph $.
|
|
$( Bound-variable hypothesis builder for an operation class abstraction.
|
|
(Contributed by NM, 22-Aug-2013.) $)
|
|
nfoprab $p |- F/_ w { <. <. x , y >. , z >. | ph } $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab nfv nfan nfex nfab nfcxfr ) E
|
|
ABCDHGIBICIJDIJKZALZDMZCMZBMZGNABCDGOUEEGUDEBUCECUBEDUAAEUAEPFQRRRST $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d w ps $. $d w ch $.
|
|
oprabbid.1 $e |- F/ x ph $.
|
|
oprabbid.2 $e |- F/ y ph $.
|
|
oprabbid.3 $e |- F/ z ph $.
|
|
oprabbid.4 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro,
|
|
24-Jun-2014.) $)
|
|
oprabbid $p |- ( ph ->
|
|
{ <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
|
|
( vw cv cop wceq wa wex cab coprab exbid df-oprab anbi2d abbidv 3eqtr4g )
|
|
AKLDLELMFLMNZBOZFPZEPZDPZKQUDCOZFPZEPZDPZKQBDEFRCDEFRAUHULKAUGUKDGAUFUJEH
|
|
AUEUIFIABCUDJUASSSUBBDEFKTCDEFKTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x z ph $. $d y z ph $.
|
|
oprabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) $)
|
|
oprabbidv $p |- ( ph ->
|
|
{ <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
|
|
( nfv oprabbid ) ABCDEFADHAEHAFHGI $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d w ps $.
|
|
oprabbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions. (Contributed
|
|
by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) $)
|
|
oprabbii $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. x , y >. , z >. | ps } $=
|
|
( vw cv wceq coprab eqid wb a1i oprabbidv ax-mp ) GHZPIZACDEJBCDEJIPKQABC
|
|
DEABLQFMNO $.
|
|
$}
|
|
|
|
${
|
|
$d ph w $. $d ps w $. $d x w $. $d y w $. $d z w $.
|
|
$( Equivalence of ordered pair abstraction subclass and implication.
|
|
Compare ~ ssopab2 . (Contributed by FL, 6-Nov-2013.) (Proof shortened
|
|
by Mario Carneiro, 11-Dec-2016.) $)
|
|
ssoprab2 $p |- ( A. x A. y A. z ( ph -> ps ) ->
|
|
{ <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } ) $=
|
|
( vw wi wal cv cop wceq wa wex cab coprab id alimi exim syl df-oprab
|
|
anim2d ss2abdv 3sstr4g ) ABGZEHZDHZCHZFICIDIJEIJKZALZEMZDMZCMZFNUHBLZEMZD
|
|
MZCMZFNACDEOBCDEOUGULUPFUGUKUOGZCHULUPGUFUQCUFUJUNGZDHUQUEURDUEUIUMGZEHUR
|
|
UDUSEUDABUHUDPUAQUIUMERSQUJUNDRSQUKUOCRSUBACDEFTBCDEFTUC $.
|
|
$}
|
|
|
|
$( Equivalence of ordered pair abstraction subclass and implication. Compare
|
|
~ ssopab2b . (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario
|
|
Carneiro, 11-Dec-2016.) $)
|
|
ssoprab2b $p |- ( { <. <. x , y >. , z >. | ph } C_
|
|
{ <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph -> ps ) ) $=
|
|
( coprab wss wi wal nfoprab1 nfss nfoprab2 nfoprab3 cv wcel oprabid 3imtr3g
|
|
cop ssel alrimi ssoprab2 impbii ) ACDEFZBCDEFZGZABHZEIZDIZCIUEUHCCUCUDACDEJ
|
|
BCDEJKUEUGDDUCUDACDELBCDELKUEUFEEUCUDACDEMBCDEMKUECNDNRENRZUCOUIUDOABUCUDUI
|
|
SACDEPBCDEPQTTTABCDEUAUB $.
|
|
|
|
$( Equivalence of ordered pair abstraction subclass and biconditional.
|
|
Compare ~ eqopab2b . (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
eqoprab2b $p |- ( { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) ) $=
|
|
( coprab wss wa wi wceq wb ssoprab2b anbi12i eqss 2albiim albii 19.26 bitri
|
|
wal 3bitr4i ) ACDEFZBCDEFZGZUBUAGZHABIESDSZCSZBAIESDSZCSZHZUAUBJABKESDSZCSZ
|
|
UCUFUDUHABCDELBACDELMUAUBNUKUEUGHZCSUIUJULCABDEOPUEUGCQRT $.
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d x y z D $. $d y z E $. $d z C $.
|
|
$d z F $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) $)
|
|
mpt2eq123 $p |- ( ( A = D /\ A. x e. A ( B = E /\ A. y e. B C = F ) ) ->
|
|
( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
|
|
( vz wceq wral wa cv wcel coprab cmpt2 nfv nfra1 nfan wb nfcv nfralxy rsp
|
|
eqeq2 syl6 pm5.32d eleq2 sylan9bbr anass 3bitr4g oprabbid df-mpt2 3eqtr4g
|
|
anbi1d ) CFJZDGJZEHJZBDKZLZACKZLZAMZCNZBMZDNZLIMZEJZLZABIOVBFNZVDGNZLVFHJ
|
|
ZLZABIOABCDEPABFGHPVAVHVLABIUOUTAUOAQUSACRSUOUTBUOBQUSBACBCUAUPURBUPBQUQB
|
|
DRSUBSVAIQVAVCVEVGLZLZVIVJVKLZLZVHVLUTVNVCVOLUOVPUTVCVMVOUTVCUSVMVOTUSACU
|
|
CURVMVEVKLUPVOURVEVGVKURVEUQVGVKTUQBDUCEHVFUDUEUFUPVEVJVKDGVDUGUNUHUEUFUO
|
|
VCVIVOCFVBUGUNUHVCVEVGUIVIVJVKUIUJUKABICDEULABIFGHULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpt2eq12 $p |- ( ( A = C /\ B = D ) ->
|
|
( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) ) $=
|
|
( wceq wral wa cmpt2 eqid rgenw jctr ralrimivw mpt2eq123 sylan2 ) DFHZCEH
|
|
RGGHZBDIZJZACIABCDGKABEFGKHRUAACRTSBDGLMNOABCDGEFGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z C $. $d z D $. $d z E $. $d x z ph $.
|
|
$d z F $. $d y z ph $.
|
|
mpt2eq123dv.1 $e |- ( ph -> A = D ) $.
|
|
${
|
|
mpt2eq123dva.2 $e |- ( ( ph /\ x e. A ) -> B = E ) $.
|
|
mpt2eq123dva.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F ) $.
|
|
$( An equality deduction for the maps to notation. (Contributed by Mario
|
|
Carneiro, 26-Jan-2017.) $)
|
|
mpt2eq123dva $p |- ( ph
|
|
-> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
|
|
( vz cv wcel wa wceq coprab cmpt2 pm5.32da eqeq2d eleq2d anbi1d df-mpt2
|
|
bitrd oprabbidv 3eqtr4g ) ABNZDOZCNZEOZPZMNZFQZPZBCMRUHGOZUJHOZPZUMIQZP
|
|
ZBCMRBCDEFSBCGHISAUOUTBCMAUOULUSPUTAULUNUSAULPFIUMLUATAULURUSAULUIUQPUR
|
|
AUIUKUQAUIPEHUJKUBTAUIUPUQADGUHJUBUCUEUCUEUFBCMDEFUDBCMGHIUDUG $.
|
|
$}
|
|
|
|
mpt2eq123dv.2 $e |- ( ph -> B = E ) $.
|
|
mpt2eq123dv.3 $e |- ( ph -> C = F ) $.
|
|
$( An equality deduction for the maps to notation. (Contributed by NM,
|
|
12-Sep-2011.) $)
|
|
mpt2eq123dv $p |- ( ph
|
|
-> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
|
|
( wceq cv wcel adantr wa mpt2eq123dva ) ABCDEFGHIJAEHMBNDOZKPAFIMSCNEOQLP
|
|
R $.
|
|
$}
|
|
|
|
${
|
|
mpt2eq123i.1 $e |- A = D $.
|
|
mpt2eq123i.2 $e |- B = E $.
|
|
mpt2eq123i.3 $e |- C = F $.
|
|
$( An equality inference for the maps to notation. (Contributed by NM,
|
|
15-Jul-2013.) $)
|
|
mpt2eq123i $p |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) $=
|
|
( cmpt2 wceq wtru a1i mpt2eq123dv trud ) ABCDELABFGHLMNABCDEFGHCFMNIODGMN
|
|
JOEHMNKOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x z ph $. $d y z ph $. $d z A $. $d z B $. $d z C $. $d z D $.
|
|
mpt2eq3dva.1 $e |- ( ( ph /\ x e. A /\ y e. B ) -> C = D ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by NM, 17-Oct-2013.) $)
|
|
mpt2eq3dva $p |- ( ph -> ( x e. A , y e. B |-> C )
|
|
= ( x e. A , y e. B |-> D ) ) $=
|
|
( vz cv wcel wa wceq coprab cmpt2 3expb eqeq2d pm5.32da oprabbidv df-mpt2
|
|
3eqtr4g ) ABJDKZCJEKZLZIJZFMZLZBCINUDUEGMZLZBCINBCDEFOBCDEGOAUGUIBCIAUDUF
|
|
UHAUDLFGUEAUBUCFGMHPQRSBCIDEFTBCIDEGTUA $.
|
|
$}
|
|
|
|
${
|
|
mpt2eq3ia.1 $e |- ( ( x e. A /\ y e. B ) -> C = D ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpt2eq3ia $p |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) $=
|
|
( cmpt2 wceq wtru cv wcel 3adant1 mpt2eq3dva trud ) ABCDEHABCDFHIJABCDEFA
|
|
KCLBKDLEFIJGMNO $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z C $. $d z x $. $d z y $.
|
|
$( Bound-variable hypothesis builder for an operation in maps-to notation.
|
|
(Contributed by NM, 27-Aug-2013.) $)
|
|
nfmpt21 $p |- F/_ x ( x e. A , y e. B |-> C ) $=
|
|
( vz cmpt2 cv wcel wa wceq coprab df-mpt2 nfoprab1 nfcxfr ) AABCDEGAHCIBH
|
|
DIJFHEKJZABFLABFCDEMPABFNO $.
|
|
|
|
$( Bound-variable hypothesis builder for an operation in maps-to notation.
|
|
(Contributed by NM, 27-Aug-2013.) $)
|
|
nfmpt22 $p |- F/_ y ( x e. A , y e. B |-> C ) $=
|
|
( vz cmpt2 cv wcel wa wceq coprab df-mpt2 nfoprab2 nfcxfr ) BABCDEGAHCIBH
|
|
DIJFHEKJZABFLABFCDEMPABFNO $.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d w y z $. $d w A $. $d w B $. $d w C $.
|
|
nfmpt2.1 $e |- F/_ z A $.
|
|
nfmpt2.2 $e |- F/_ z B $.
|
|
nfmpt2.3 $e |- F/_ z C $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by NM, 20-Feb-2013.) $)
|
|
nfmpt2 $p |- F/_ z ( x e. A , y e. B |-> C ) $=
|
|
( vw cmpt2 cv wcel wa wceq coprab df-mpt2 nfcri nfan nfeq2 nfoprab nfcxfr
|
|
) CABDEFKALDMZBLEMZNZJLZFOZNZABJPABJDEFQUHABJCUEUGCUCUDCCADGRCBEHRSCUFFIT
|
|
SUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d w y z $. $d w z B $. $d w z C $.
|
|
$( A mapping operation with empty domain. (Contributed by Stefan O'Rear,
|
|
29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) $)
|
|
mpt20 $p |- ( x e. (/) , y e. B |-> C ) = (/) $=
|
|
( vz vw c0 cmpt2 cv wcel wa wceq coprab cop wex cab df-mpt2 df-oprab noel
|
|
nex simprll mto abf 3eqtri ) ABGCDHAIZGJZBIZCJZKEIZDLZKZABEMFIUEUGNUINLZU
|
|
KKZEOZBOZAOZFPGABEGCDQUKABEFRUPFUOAUNBUMEUMUFUESULUFUHUJUAUBTTTUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Two ways to state the domain of an operation. (Contributed by FL,
|
|
24-Jan-2010.) $)
|
|
oprab4 $p |-
|
|
{ <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
|
|
( cv cop cxp wcel wa opelxp anbi1i oprabbii ) BGZCGZHEFIJZAKOEJPFJKZAKBCD
|
|
QRAOPEFLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d v ph $. $d v ps $.
|
|
cbvoprab1.1 $e |- F/ w ph $.
|
|
cbvoprab1.2 $e |- F/ x ps $.
|
|
cbvoprab1.3 $e |- ( x = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first bound variable in an operation abstraction,
|
|
using implicit substitution. (Contributed by NM, 20-Dec-2008.)
|
|
(Revised by Mario Carneiro, 5-Dec-2016.) $)
|
|
cbvoprab1 $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , y >. , z >. | ps } $=
|
|
( vv cv cop wceq wa wex copab coprab nfv nfan nfex eqeq2d anbi12d opabbii
|
|
opeq1 exbidv cbvex dfoprab2 3eqtr4i ) JKZCKZDKZLZMZANZDOZCOZJEPUIFKZUKLZM
|
|
ZBNZDOZFOZJEPACDEQBFDEQUPVBJEUOVACFUNFDUMAFUMFRGSTUTCDUSBCUSCRHSTUJUQMZUN
|
|
UTDVCUMUSABVCULURUIUJUQUKUDUAIUBUEUFUCACDEJUGBFDEJUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z $. $d ph v $. $d ps v $.
|
|
cbvoprab2.1 $e |- F/ w ph $.
|
|
cbvoprab2.2 $e |- F/ y ps $.
|
|
cbvoprab2.3 $e |- ( y = w -> ( ph <-> ps ) ) $.
|
|
$( Change the second bound variable in an operation abstraction.
|
|
(Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
cbvoprab2 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , w >. , z >. | ps } $=
|
|
( vv cv cop wceq wa wex cab coprab nfv nfan nfex opeq2 opeq1d cbvex exbii
|
|
eqeq2d anbi12d exbidv abbii df-oprab 3eqtr4i ) JKZCKZDKZLZEKZLZMZANZEOZDO
|
|
ZCOZJPUKULFKZLZUOLZMZBNZEOZFOZCOZJPACDEQBCFEQVAVIJUTVHCUSVGDFURFEUQAFUQFR
|
|
GSTVFDEVEBDVEDRHSTUMVBMZURVFEVJUQVEABVJUPVDUKVJUNVCUOUMVBULUAUBUEIUFUGUCU
|
|
DUHACDEJUIBCFEJUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $. $d u ph $. $d u ps $.
|
|
cbvoprab12.1 $e |- F/ w ph $.
|
|
cbvoprab12.2 $e |- F/ v ph $.
|
|
cbvoprab12.3 $e |- F/ x ps $.
|
|
cbvoprab12.4 $e |- F/ y ps $.
|
|
cbvoprab12.5 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first two bound variables in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
cbvoprab12 $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , v >. , z >. | ps } $=
|
|
( vu cv cop wceq wa wex nfv nfan coprab anbi12d opabbii dfoprab2 3eqtr4i
|
|
copab opeq12 eqeq2d cbvex2 ) MNZCNZDNZOZPZAQZDRCRZMEUFUJFNZGNZOZPZBQZGRFR
|
|
ZMEUFACDEUABFGEUAUPVBMEUOVACDFGUNAFUNFSHTUNAGUNGSITUTBCUTCSJTUTBDUTDSKTUK
|
|
UQPULURPQZUNUTABVCUMUSUJUKULUQURUGUHLUBUIUCACDEMUDBFGEMUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d w v ph $. $d x y ps $.
|
|
cbvoprab12v.1 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first two bound variables in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
8-Oct-2004.) $)
|
|
cbvoprab12v $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , v >. , z >. | ps } $=
|
|
( nfv cbvoprab12 ) ABCDEFGAFIAGIBCIBDIHJ $.
|
|
$}
|
|
|
|
${
|
|
$d x z w v $. $d y z w v $. $d v ph $. $d v ps $.
|
|
cbvoprab3.1 $e |- F/ w ph $.
|
|
cbvoprab3.2 $e |- F/ z ps $.
|
|
cbvoprab3.3 $e |- ( z = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the third bound variable in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
22-Aug-2013.) $)
|
|
cbvoprab3 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , y >. , w >. | ps } $=
|
|
( vv cv wceq wa wex copab coprab nfv nfan nfex dfoprab2 2exbidv cbvopab2
|
|
cop anbi2d 3eqtr4i ) JKCKDKUCLZAMZDNZCNZJEOUFBMZDNZCNZJFOACDEPBCDFPUIULJE
|
|
FUHFCUGFDUFAFUFFQGRSSUKECUJEDUFBEUFEQHRSSEKFKLZUGUJCDUMABUFIUDUAUBACDEJTB
|
|
CDFJTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d z ps $.
|
|
cbvoprab3v.1 $e |- ( z = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the third bound variable in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.) $)
|
|
cbvoprab3v $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , y >. , w >. | ps } $=
|
|
( nfv cbvoprab3 ) ABCDEFAFHBEHGI $.
|
|
$}
|
|
|
|
${
|
|
$d u w x y z $. $d u w x y z A $. $d u w B $. $d u C $. $d u y D $.
|
|
$d u E $.
|
|
cbvmpt2x.1 $e |- F/_ z B $.
|
|
cbvmpt2x.2 $e |- F/_ x D $.
|
|
cbvmpt2x.3 $e |- F/_ z C $.
|
|
cbvmpt2x.4 $e |- F/_ w C $.
|
|
cbvmpt2x.5 $e |- F/_ x E $.
|
|
cbvmpt2x.6 $e |- F/_ y E $.
|
|
cbvmpt2x.7 $e |- ( x = z -> B = D ) $.
|
|
cbvmpt2x.8 $e |- ( ( x = z /\ y = w ) -> C = E ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. This version of ~ cbvmpt2 allows ` B ` to be a function
|
|
of ` x ` . (Contributed by NM, 29-Dec-2014.) $)
|
|
cbvmpt2x $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E ) $=
|
|
( vu cv nfan wcel wa wceq coprab cmpt2 nfv nfcri nfeq2 nfcv weq wb adantr
|
|
eleq1 eleq2d sylan9bb anbi12d eqeq2d cbvoprab12 df-mpt2 3eqtr4i ) ASZEUAZ
|
|
BSZFUAZUBZRSZGUCZUBZABRUDCSZEUAZDSZHUAZUBZVFIUCZUBZCDRUDABEFGUECDEHIUEVHV
|
|
OABRCDVEVGCVBVDCVBCUFCBFJUGTCVFGLUHTVEVGDVBVDDVBDUFDBFDFUIUGTDVFGMUHTVMVN
|
|
AVJVLAVJAUFADHKUGTAVFINUHTVMVNBVMBUFBVFIOUHTACUJZBDUJZUBZVEVMVGVNVRVBVJVD
|
|
VLVPVBVJUKVQVAVIEUMULVPVDVCHUAVQVLVPFHVCPUNVCVKHUMUOUPVRGIVFQUQUPURABREFG
|
|
USCDREHIUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $.
|
|
cbvmpt2.1 $e |- F/_ z C $.
|
|
cbvmpt2.2 $e |- F/_ w C $.
|
|
cbvmpt2.3 $e |- F/_ x D $.
|
|
cbvmpt2.4 $e |- F/_ y D $.
|
|
cbvmpt2.5 $e |- ( ( x = z /\ y = w ) -> C = D ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. (Contributed by NM, 17-Dec-2013.) $)
|
|
cbvmpt2 $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
|
|
( nfcv weq eqidd cbvmpt2x ) ABCDEFGFHCFNAFNIJKLACOFPMQ $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w z C $. $d x y D $.
|
|
cbvmpt2v.1 $e |- ( x = z -> C = E ) $.
|
|
cbvmpt2v.2 $e |- ( y = w -> E = D ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. With a longer proof analogous to ~ cbvmpt , some distinct
|
|
variable requirements could be eliminated. (Contributed by NM,
|
|
11-Jun-2013.) $)
|
|
cbvmpt2v $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
|
|
( nfcv weq sylan9eq cbvmpt2 ) ABCDEFGHCGLDGLAHLBHLACMBDMGIHJKNO $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $.
|
|
$( The domain of an operation class abstraction. (Contributed by NM,
|
|
17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.) $)
|
|
dmoprab $p |- dom { <. <. x , y >. , z >. | ph } =
|
|
{ <. x , y >. | E. z ph } $=
|
|
( vw coprab cdm cv cop wceq wa wex copab cab dfoprab2 dmeqi dmopab exrot3
|
|
19.42v 2exbii bitri abbii df-opab eqtr4i 3eqtri ) ABCDFZGEHBHCHIJZAKZCLBL
|
|
ZEDMZGUIDLZENZADLZBCMZUFUJABCDEOPUIEDQULUGUMKZCLBLZENUNUKUPEUKUHDLZCLBLUP
|
|
UHDBCRUQUOBCUGADSTUAUBUMBCEUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( The domain of an operation class abstraction. (Contributed by NM,
|
|
24-Aug-1995.) $)
|
|
dmoprabss $p |- dom { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B ) $=
|
|
( cv wcel wa coprab cdm wex copab dmoprab 19.42v opabbii opabssxp eqsstri
|
|
cxp ) BGEHCGFHIZAIZBCDJKUADLZBCMZEFSZUABCDNUCTADLZIZBCMUDUBUFBCTADOPUEBCE
|
|
FQRR $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $.
|
|
$( The range of an operation class abstraction. (Contributed by NM,
|
|
30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.) $)
|
|
rnoprab $p |- ran { <. <. x , y >. , z >. | ph } =
|
|
{ z | E. x E. y ph } $=
|
|
( vw coprab crn cv cop wceq wa wex copab cab dfoprab2 rneqi rnopab exrot3
|
|
vex opex isseti 19.41v mpbiran 2exbii bitri abbii 3eqtri ) ABCDFZGEHBHZCH
|
|
ZIZJZAKZCLBLZEDMZGUNELZDNACLBLZDNUHUOABCDEOPUNEDQUPUQDUPUMELZCLBLUQUMEBCR
|
|
URABCURULELAEUKUIUJBSCSTUAULAEUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d x y z $.
|
|
$( The range of a restricted operation class abstraction. (Contributed by
|
|
Scott Fenton, 21-Mar-2012.) $)
|
|
rnoprab2 $p |- ran { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } =
|
|
{ z | E. x e. A E. y e. B ph } $=
|
|
( cv wcel wa coprab crn wex cab wrex rnoprab r2ex abbii eqtr4i ) BGEHCGFH
|
|
IAIZBCDJKSCLBLZDMACFNBENZDMSBCDOUATDABCEFPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The domain of an operation class abstraction is a relation.
|
|
(Contributed by NM, 17-Mar-1995.) $)
|
|
reldmoprab $p |- Rel dom { <. <. x , y >. , z >. | ph } $=
|
|
( wex coprab cdm dmoprab relopabi ) ADEBCABCDFGABCDHI $.
|
|
|
|
$( Structure of an operation class abstraction. (Contributed by NM,
|
|
28-Nov-2006.) $)
|
|
oprabss $p |- { <. <. x , y >. , z >. | ph } C_ ( ( _V X. _V ) X. _V ) $=
|
|
( coprab cdm crn cxp wrel reloprab relssdmrn ax-mp reldmoprab df-rel mpbi
|
|
cvv wss ssv xpss12 mp2an sstri ) ABCDEZUBFZUBGZHZPPHZPHZUBIUBUEQABCDJUBKL
|
|
UCUFQZUDPQUEUGQUCIUHABCDMUCNOUDRUCUFUDPSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d w ph $.
|
|
$d x y z w ps $.
|
|
eloprabga.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion for operation class abstraction. Compare
|
|
~ elopab . (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|
|
(Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
eloprabga $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) $=
|
|
( vw wcel cvv cop wa cv wceq wex coprab wb w3a opexg sylan 3impa wi simpr
|
|
elex eqeq1d eqcom vex otth2 syl6bb anbi1d pm5.32i 3exbidv df-oprab eleq2i
|
|
bitri cab abid bitr2i eleq1 syl5bb adantl elisset 3anim123i eeeanv sylibr
|
|
biantrurd 19.41vvv syl6rbbr adantr 3bitr3d expcom vtocleg mpcom syl3an )
|
|
FINFONZGJNGONZHKNHONZFGPZHPZACDEUAZNZBUBZFIUIGJUIHKUIWDONZVTWAWBUCZWGVTWA
|
|
WBWHVTWAQWCONWBWHFGOOUDWCHOOUDUEUFWIWGUGMWDOWIMRZWDSZWGWIWKQZWJCRZDRZPERZ
|
|
PZSZAQZETDTCTZWMFSZWNGSZWOHSZUCZBQZETDTCTZWFBWLWRXDCDEWLWRXCAQXDWLWQXCAWL
|
|
WQWDWPSZXCWLWJWDWPWIWKUHUJXFWPWDSXCWDWPUKWMWNFGWOHCULDULEULUMUTUNUOXCABLU
|
|
PUNUQWKWSWFUBWIWSWJWENZWKWFXGWJWSMVAZNWSWEXHWJACDEMURUSWSMVBVCWJWDWEVDVEV
|
|
FWIXEBUBWKWIBXCETDTCTZBQXEWIXIBWIWTCTZXADTZXBETZUCXIVTXJWAXKWBXLCFOVGDGOV
|
|
GEHOVGVHWTXAXBCDEVIVJVKXCBCDEVLVMVNVOVPVQVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z th $.
|
|
eloprabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
eloprabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
eloprabg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
$( The law of concretion for operation class abstraction. Compare
|
|
~ elopab . (Contributed by NM, 14-Sep-1999.) (Revised by David
|
|
Abernethy, 19-Jun-2012.) $)
|
|
eloprabg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) ) $=
|
|
( cv wceq syl3an9b eloprabga ) ADEFGHIJKLMEQHRABFQIRCGQJRDNOPST $.
|
|
$}
|
|
|
|
${
|
|
$d ph w $. $d ps w $. $d x z w $. $d y z w $.
|
|
ssoprab2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of operation class abstraction subclass from implication.
|
|
(Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy,
|
|
19-Jun-2012.) $)
|
|
ssoprab2i $p |- { <. <. x , y >. , z >. | ph } C_
|
|
{ <. <. x , y >. , z >. | ps } $=
|
|
( vw cv cop wceq wex copab coprab anim2i 2eximi ssopab2i dfoprab2 3sstr4i
|
|
wa ) GHCHDHIJZASZDKCKZGELTBSZDKCKZGELACDEMBCDEMUBUDGEUAUCCDABTFNOPACDEGQB
|
|
CDEGQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z C $.
|
|
$( Operation with universal domain in maps-to notation. (Contributed by
|
|
NM, 16-Aug-2013.) $)
|
|
mpt2v $p |- ( x e. _V , y e. _V |-> C )
|
|
= { <. <. x , y >. , z >. | z = C } $=
|
|
( cvv cmpt2 cv wcel wa coprab df-mpt2 vex pm3.2i biantrur oprabbii eqtr4i
|
|
wceq ) ABEEDFAGEHZBGEHZIZCGDQZIZABCJUAABCJABCEEDKUAUBABCTUARSALBLMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w y z B $. $d w x y C $. $d w z D $.
|
|
mpt2mpt.1 $e |- ( z = <. x , y >. -> C = D ) $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. In this version ` B ( x ) ` is not assumed to be constant
|
|
w.r.t ` x ` . (Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
mpt2mptx $p |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) =
|
|
( x e. A , y e. B |-> D ) $=
|
|
( vw cv csn cxp ciun cmpt wcel wceq wa copab wex eqtr4i df-mpt coprab cop
|
|
cmpt2 df-mpt2 eliunxp anbi1i 19.41vv eqeq2d anbi2d pm5.32i 2exbii 3bitr2i
|
|
anass bitri opabbii dfoprab2 ) CADAJZKELMZFNCJZUSOZIJZFPZQZCIRZABDEGUDZCI
|
|
USFUAVFURDOBJZEOQZVBGPZQZABIUBZVEABIDEGUEVEUTURVGUCPZVJQZBSASZCIRVKVDVNCI
|
|
VDVLVHQZBSASZVCQVOVCQZBSASVNVAVPVCABDEUTUFUGVOVCABUHVQVMABVQVLVHVCQZQVMVL
|
|
VHVCUNVLVRVJVLVCVIVHVLFGVBHUIUJUKUOULUMUPVJABICUQTTT $.
|
|
|
|
$d x B $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
mpt2mpt $p |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D ) $=
|
|
( cv csn cxp ciun cmpt cmpt2 wceq iunxpconst mpteq1 ax-mp mpt2mptx eqtr3i
|
|
) CADAIJEKLZFMZCDEKZFMZABDEGNUAUCOUBUDOADEPCUAUCFQRABCDEFGHST $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w ph $.
|
|
$( Restriction of an operation class abstraction. (Contributed by NM,
|
|
10-Feb-2007.) $)
|
|
resoprab $p |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
|
|
( vw cv cop wceq wex copab cxp cres wcel coprab resopab 19.42vv dfoprab2
|
|
wa eleq1 opelxp syl6bb anbi1d pm5.32i bitri 2exbii bitr3i opabbii reseq1i
|
|
an12 eqtri 3eqtr4i ) GHZBHZCHZIZJZATZCKBKZGDLZEFMZNZURUOEOUPFOTZATZTZCKBK
|
|
ZGDLZABCDPZVBNVEBCDPVCUNVBOZUTTZGDLVHUTGDVBQVKVGGDVKVJUSTZCKBKVGVJUSBCRVL
|
|
VFBCVLURVJATZTVFVJURAUKURVMVEURVJVDAURVJUQVBOVDUNUQVBUAUOUPEFUBUCUDUEUFUG
|
|
UHUIULVIVAVBABCDGSUJVEBCDGSUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. $d E z $.
|
|
$( Restriction of an operator abstraction. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
resoprab2 $p |- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } ) $=
|
|
( wss wa cv wcel coprab cxp cres resoprab anass ssel pm4.71d bicomd an4
|
|
bi2anan9 syl5bb anbi1d syl5bbr oprabbidv syl5eq ) GEIZHFIZJZBKZELZCKZFLZJ
|
|
ZAJZBCDMGHNOUKGLZUMHLZJZUPJZBCDMUSAJZBCDMUPBCDGHPUJUTVABCDUTUSUOJZAJUJVAU
|
|
SUOAQUJVBUSAVBUQULJZURUNJZJUJUSUQURULUNUAUHVCUQUIVDURUHUQVCUHUQULGEUKRSTU
|
|
IURVDUIURUNHFUMRSTUBUCUDUEUFUG $.
|
|
|
|
$( Restriction of the mapping operation. (Contributed by Mario Carneiro,
|
|
17-Dec-2013.) $)
|
|
resmpt2 $p |- ( ( C C_ A /\ D C_ B ) ->
|
|
( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) =
|
|
( x e. C , y e. D |-> E ) ) $=
|
|
( vz wss wa cv wcel wceq coprab cxp cres cmpt2 resoprab2 df-mpt2 reseq1i
|
|
3eqtr4g ) ECIFDIJAKZCLBKZDLJHKGMZJABHNZEFOZPUBELUCFLJUDJABHNABCDGQZUFPABE
|
|
FGQUDABHCDEFRUGUEUFABHCDGSTABHEFGSUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d w ph $.
|
|
$( "At most one" is a sufficient condition for an operation class
|
|
abstraction to be a function. (Contributed by NM, 28-Aug-2007.) $)
|
|
funoprabg $p |- ( A. x A. y E* z ph ->
|
|
Fun { <. <. x , y >. , z >. | ph } ) $=
|
|
( vw wmo wal cv cop wceq wa wex coprab wfun mosubopt alrimiv copab funeqi
|
|
dfoprab2 funopab bitr2i sylib ) ADFCGBGZEHZBHCHIJAKCLBLZDFZEGZABCDMZNZUCU
|
|
FEADBCUDOPUIUEEDQZNUGUHUJABCDESRUEEDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
funoprab.1 $e |- E* z ph $.
|
|
$( "At most one" is a sufficient condition for an operation class
|
|
abstraction to be a function. (Contributed by NM, 17-Mar-1995.) $)
|
|
funoprab $p |- Fun { <. <. x , y >. , z >. | ph } $=
|
|
( wmo wal coprab wfun gen2 funoprabg ax-mp ) ADFZCGBGABCDHIMBCEJABCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
$( Functionality and domain of an operation class abstraction.
|
|
(Contributed by NM, 28-Aug-2007.) $)
|
|
fnoprabg $p |- ( A. x A. y ( ph -> E! z ps ) ->
|
|
{ <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } ) $=
|
|
( weu wi wal wa coprab wfun cdm copab wceq wfn wmo eumo imim2i wex sps wb
|
|
moanimv sylibr 2alimi funoprabg syl dmoprab nfa1 nfa2 simpl exlimiv ancld
|
|
euex 19.42v syl6ibr impbid2 opabbid syl5eq df-fn sylanbrc ) ABEFZGZDHZCHZ
|
|
ABIZCDEJZKZVFLZACDMZNVFVIOVDVEEPZDHCHVGVBVJCDVBABEPZGVJVAVKABEQRABEUBUCUD
|
|
VECDEUEUFVDVHVEESZCDMVIVECDEUGVDVLACDVCCUHVBDCUIVCVLAUAZCVBVMDVBVLAVEAEAB
|
|
UJUKVBAABESZIVLVBAVNVAVNABEUMRULABEUNUOUPTTUQURVFVIUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A w z $. $d B w z $. $d C w z $. $d x y w z $.
|
|
mpt2fun.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( The maps-to notation for an operation is always a function.
|
|
(Contributed by Scott Fenton, 21-Mar-2012.) $)
|
|
mpt2fun $p |- Fun F $=
|
|
( vz vw wfun cv wcel wa wceq coprab wmo weq wi wal mpbir eqtr3 gen2 eqeq1
|
|
ad2ant2l anbi2d mo4 funoprab cmpt2 df-mpt2 eqtri funeqi ) FJAKCLBKDLMZHKZ
|
|
ENZMZABHOZJUOABHUOHPUOULIKZENZMZMHIQZRZISHSVAHIUNURUTULULUMUQEUAUDUBUOUSH
|
|
IUTUNURULUMUQEUCUEUFTUGFUPFABCDEUHUPGABHCDEUIUJUKT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
fnoprab.1 $e |- ( ph -> E! z ps ) $.
|
|
$( Functionality and domain of an operation class abstraction.
|
|
(Contributed by NM, 15-May-1995.) $)
|
|
fnoprab $p |- { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn
|
|
{ <. x , y >. | ph } $=
|
|
( weu wi wal wa coprab copab wfn gen2 fnoprabg ax-mp ) ABEGHZDICIABJCDEKA
|
|
CDLMQCDFNABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y w A $. $d x y w B $. $d x y w C $. $d x y w F $.
|
|
$( An operation maps to a class to which all values belong. (Contributed
|
|
by NM, 7-Feb-2004.) $)
|
|
ffnov $p |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\
|
|
A. x e. A A. y e. B ( x F y ) e. C ) ) $=
|
|
( vw cxp wf wfn cv cfv wcel wral wa co ffnfv cop wceq fveq2 df-ov syl6eqr
|
|
eleq1d ralxp anbi2i bitri ) CDHZEFIFUGJZGKZFLZEMZGUGNZOUHAKZBKZFPZEMZBDNA
|
|
CNZOGUGEFQULUQUHUKUPGABCDUIUMUNRZSZUJUOEUSUJURFLUOUIURFTUMUNFUAUBUCUDUEUF
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y F $. $d x y R $. $d x y S $.
|
|
fovcl.1 $e |- F : ( R X. S ) --> C $.
|
|
$( Closure law for an operation. (Contributed by NM, 19-Apr-2007.) $)
|
|
fovcl $p |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C ) $=
|
|
( vx vy wcel wa cv co wral cxp wf wfn ffnov wceq eleq1d ax-mp oveq1 oveq2
|
|
simprbi rspc2v mpi ) ADJBEJKHLZILZFMZCJZIENHDNZABFMZCJZDEOZCFPZUKGUOFUNQU
|
|
KHIDECFRUDUAUJUMAUHFMZCJHIABDEUGASUIUPCUGAUHFUBTUHBSUPULCUHBAFUCTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d z C $. $d z D $. $d x y z F $.
|
|
$d x y z G $.
|
|
$( Equality of two operations is determined by their values. (Contributed
|
|
by NM, 1-Sep-2005.) $)
|
|
eqfnov $p |- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <->
|
|
( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) ) $=
|
|
( vz cxp wfn wa wceq cv cfv wral co eqfnfv2 fveq2 df-ov cop eqeq12d ralxp
|
|
eqeq12i syl6bbr anbi2i syl6bb ) GCDJZKHEFJZKLGHMUHUIMZINZGOZUKHOZMZIUHPZL
|
|
UJANZBNZGQZUPUQHQZMZBDPACPZLIUHUIGHRUOVAUJUNUTIABCDUKUPUQUAZMZUNVBGOZVBHO
|
|
ZMUTVCULVDUMVEUKVBGSUKVBHSUBURVDUSVEUPUQGTUPUQHTUDUEUCUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $. $d G x y $.
|
|
$( Two operators with the same domain are equal iff their values at each
|
|
point in the domain are equal. (Contributed by Jeff Madsen,
|
|
7-Jun-2010.) $)
|
|
eqfnov2 $p |- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) ->
|
|
( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) $=
|
|
( cxp wfn wa wceq cv co wral eqfnov simpr eqidd ancri impbii syl6bb ) ECD
|
|
GZHFTHIEFJTTJZAKZBKZELUBUCFLJBDMACMZIZUDABCDCDEFNUEUDUAUDOUDUAUDTPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z F $.
|
|
$( Representation of a function in terms of its values. (Contributed by
|
|
Jim Kingdon, 16-Jan-2019.) $)
|
|
fnovim $p |- ( F Fn ( A X. B ) ->
|
|
F = ( x e. A , y e. B |-> ( x F y ) ) ) $=
|
|
( vz cxp wfn cv cfv cmpt co cmpt2 dffn5im cop fveq2 df-ov syl6eqr mpt2mpt
|
|
wceq eqeq2i sylib ) ECDGZHEFUCFIZEJZKZTEABCDAIZBIZELZMZTFUCENUFUJEABFCDUE
|
|
UIUDUGUHOZTUEUKEJUIUDUKEPUGUHEQRSUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $. $d z D $.
|
|
$( Bidirectional equality theorem for a mapping abstraction. Equivalent to
|
|
~ eqfnov2 . (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
mpt22eqb $p |- ( A. x e. A A. y e. B C e. V ->
|
|
( ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) <->
|
|
A. x e. A A. y e. B C = D ) ) $=
|
|
( vz wcel wral wceq cv wb wal cmpt2 ralimi ralbi syl wa coprab df-mpt2 wi
|
|
pm13.183 eqeq12i eqoprab2b pm5.32 19.21v bitr3i 2albii r2al bitr4i 3bitri
|
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albii syl6rbbr ) EGIZBDJZACJZEFKZBDJZACJZHLZEKZVAFKZMZHNZBDJZACJZABCDEOZA
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BCDFOZKZUQUSVFMZACJUTVGMUPVKACUPURVEMZBDJVKUOVLBDHEFGUCPURVEBDQRPUSVFACQR
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VJALCIBLDISZVBSZABHTZVMVCSZABHTZKVNVPMZHNZBNANZVGVHVOVIVQABHCDEUAABHCDFUA
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UDVNVPABHUEVTVMVEUBZBNANVGVSWAABVSVMVDUBZHNWAWBVRHVMVBVCUFUMVMVDHUGUHUIVE
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ABCDUJUKULUN $.
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$}
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${
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$d w x $. $d w y z A $. $d w z B $. $d w z C $. $d w z F $. $d z ps $.
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$d x y z D $. $d x y ph $.
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rngop.1 $e |- F = ( x e. A , y e. B |-> C ) $.
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$( The range of an operation given by the "maps to" notation. (Contributed
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by FL, 20-Jun-2011.) $)
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rnmpt2 $p |- ran F = { z | E. x e. A E. y e. B z = C } $=
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( crn cv wcel wa wceq coprab wrex cab cmpt2 df-mpt2 eqtri rneqi rnoprab2
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) GIAJDKBJEKLCJFMZLABCNZIUBBEOADOCPGUCGABDEFQUCHABCDEFRSTUBABCDEUAS $.
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$( The domain of an operation defined by maps-to notation is a relation.
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(Contributed by Stefan O'Rear, 27-Nov-2014.) $)
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reldmmpt2 $p |- Rel dom F $=
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( vz cdm wrel cv wcel wa wceq coprab reldmoprab cmpt2 df-mpt2 eqtri dmeqi
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releqi mpbir ) FIZJAKCLBKDLMHKENMZABHOZIZJUDABHPUCUFFUEFABCDEQUEGABHCDERS
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TUAUB $.
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$( Membership in the range of an operation class abstraction. (Contributed
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by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
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elrnmpt2g $p |- ( D e. V ->
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( D e. ran F <-> E. x e. A E. y e. B D = C ) ) $=
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( vz cv wceq wrex crn eqeq1 2rexbidv rnmpt2 elab2g ) JKZELZBDMACMFELZBDMA
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CMJFGNHSFLTUAABCDSFEOPABJCDEGIQR $.
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${
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elrnmpt2.1 $e |- C e. _V $.
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$( Membership in the range of an operation class abstraction.
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(Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro,
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31-Aug-2015.) $)
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elrnmpt2 $p |- ( D e. ran F <-> E. x e. A E. y e. B D = C ) $=
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( vz crn wcel cv wceq wrex cab rnmpt2 eleq2i cvv rexlimivw eleq1 mpbiri
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eqeq1 2rexbidv elab3 bitri ) FGKZLFJMZENZBDOACOZJPZLFENZBDOZACOZUGUKFAB
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JCDEGHQRUJUNJFUMFSLZACULUOBDULUOESLIFESUAUBTTUHFNUIULABCDUHFEUCUDUEUF
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$.
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$}
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ralrnmpt2.2 $e |- ( z = C -> ( ph <-> ps ) ) $.
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$( A restricted quantifier over an image set. (Contributed by Mario
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Carneiro, 1-Sep-2015.) $)
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ralrnmpt2 $p |- ( A. x e. A A. y e. B C e. V ->
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( A. z e. ran F ph <-> A. x e. A A. y e. B ps ) ) $=
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( vw wral cv wceq wrex wi wal wb crn wcel cab rnmpt2 eqeq1 2rexbidv ralab
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raleqi ralcom4 r19.23v albii bitr2i 3bitri bitri nfv ceqsalg ralimi ralbi
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syl syl5bbr syl5bb ) AEIUAZNZEOZHPZDGQZARZESZCFNZHJUBZDGNZCFNZBDGNZCFNZVC
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AEMOZHPZDGQCFQZMUCZNVFCFQZARZESZVIAEVBVRCDMFGHIKUDUHVQVSAEMVOVDPVPVECDFGV
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OVDHUEUFUGVIVGCFNZESWAVGCEFUIWBVTEVFACFUJUKULUMVLVHVMTZCFNVIVNTVKWCCFVHVE
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ARZESZDGNZVKVMWFWDDGNZESVHWDDEGUIWGVGEVEADGUJUKUNVKWEBTZDGNWFVMTVJWHDGABE
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HJBEUOLUPUQWEBDGURUSUTUQVHVMCFURUSVA $.
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$( A restricted quantifier over an image set. (Contributed by Mario
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Carneiro, 1-Sep-2015.) $)
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rexrnmpt2 $p |- ( A. x e. A A. y e. B C e. V ->
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( E. z e. ran F ph <-> E. x e. A E. y e. B ps ) ) $=
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( vw wrex cv wceq wa wex wral wb crn wcel cab rnmpt2 eqeq1 2rexbidv rexab
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rexeqi rexcom4 r19.41v exbii bitr2i 3bitri bitri ralimi rexbi syl syl5bbr
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ceqsexgv syl5bb ) AEIUAZNZEOZHPZDGNZAQZERZCFNZHJUBZDGSZCFSZBDGNZCFNZVBAEM
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OZHPZDGNCFNZMUCZNVECFNZAQZERZVHAEVAVQCDMFGHIKUDUHVPVRAEMVNVCPVOVDCDFGVNVC
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HUEUFUGVHVFCFNZERVTVFCEFUIWAVSEVEACFUJUKULUMVKVGVLTZCFSVHVMTVJWBCFVGVDAQZ
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ERZDGNZVJVLWEWCDGNZERVGWCDEGUIWFVFEVDADGUJUKUNVJWDBTZDGSWEVLTVIWGDGABEHJL
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USUOWDBDGUPUQURUOVGVLCFUPUQUT $.
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$}
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${
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$d x y z $. $d z R $. $d z S $.
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ovid.1 $e |- ( ( x e. R /\ y e. S ) -> E! z ph ) $.
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ovid.2 $e |- F =
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{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
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$( The value of an operation class abstraction. (Contributed by NM,
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16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) $)
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ovid $p |- ( ( x e. R /\ y e. S ) -> ( ( x F y ) = z <-> ph ) ) $=
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( cv co wceq cop cfv wcel wa df-ov eqeq1i copab wfn coprab fnoprab fneq1i
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wb mpbir opabid biimpri fnopfvb sylancr eleq2i oprabid bitri bitrd syl5bb
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baib ) BJZCJZGKZDJZLUPUQMZGNZUSLZUPEOUQFOPZAURVAUSUPUQGQRVCVBUTUSMZGOZAVC
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GVCBCSZTZUTVFOZVBVEUDVGVCAPZBCDUAZVFTVCABCDHUBVFGVJIUCUEVHVCVCBCUFUGVFUTU
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SGUHUIVEVCAVEVDVJOVIGVJVDIUJVIBCDUKULUOUMUN $.
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$}
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${
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$d x y z $.
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ovidig.1 $e |- E* z ph $.
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ovidig.2 $e |- F = { <. <. x , y >. , z >. | ph } $.
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$( The value of an operation class abstraction. Compare ~ ovidi . The
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condition ` ( x e. R /\ y e. S ) ` is been removed. (Contributed by
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Mario Carneiro, 29-Dec-2014.) $)
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ovidig $p |- ( ph -> ( x F y ) = z ) $=
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( cv co cop cfv df-ov wfun wcel wceq coprab funoprab funeqi mpbir oprabid
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biimpri syl6eleqr funopfv mpsyl syl5eq ) ABHZCHZEIUFUGJZEKZDHZUFUGELEMZAU
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HUJJZENUIUJOUKABCDPZMABCDFQEUMGRSAULUMEULUMNAABCDTUAGUBUHUJEUCUDUE $.
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$}
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${
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$d x y z $. $d z R $. $d z S $.
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ovidi.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
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ovidi.3 $e |- F =
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{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
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$( The value of an operation class abstraction (weak version).
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(Contributed by Mario Carneiro, 29-Dec-2014.) $)
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ovidi $p |- ( ( x e. R /\ y e. S ) -> ( ph -> ( x F y ) = z ) ) $=
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( cv wcel wa co wceq wmo wi moanimv mpbir ovidig ex ) BJZEKCJZFKLZAUAUBGM
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DJNUCALZBCDGUDDOUCADOPHUCADQRIST $.
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$}
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${
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$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z S $.
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$d x y z th $.
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ov.1 $e |- C e. _V $.
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ov.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
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ov.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
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ov.4 $e |- ( z = C -> ( ch <-> th ) ) $.
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ov.5 $e |- ( ( x e. R /\ y e. S ) -> E! z ph ) $.
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ov.6 $e |- F =
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{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
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$( The value of an operation class abstraction. (Contributed by NM,
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16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) $)
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ov $p |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) ) $=
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( wcel wa co wceq cop cv coprab cfv df-ov fveq1i eqtri eqeq1i wfn fnoprab
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copab eleq1 anbi1d anbi2d opelopabg ibir fnopfvb sylancr anbi12d eloprabg
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wb cvv mp3an3 bitrd syl5bb bianabs ) HKTZILTZUAZHIMUBZJUCZDVNHIUDZEUEZKTZ
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FUEZLTZUAZAUAZEFGUFZUGZJUCZVLVLDUAZVMWCJVMVOMUGWCHIMUHVOMWBSUIUJUKVLWDVOJ
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UDWBTZWEVLWBVTEFUNZULVOWGTZWDWFVDVTAEFGRUMVLWHVTVJVSUAZVLEFHIKLVPHUCZVQVJ
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VSVPHKUOUPZVRIUCZVSVKVJVRILUOUQZURUSWGVOJWBUTVAVJVKJVETWFWEVDNWAWIBUAVLCU
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AWEEFGHIJKLVEWJVTWIABWKOVBWLWIVLBCWMPVBGUEJUCCDVLQUQVCVFVGVHVI $.
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$}
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${
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$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
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ovigg.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
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ovigg.4 $e |- E* z ph $.
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ovigg.5 $e |- F = { <. <. x , y >. , z >. | ph } $.
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$( The value of an operation class abstraction. Compare ~ ovig . The
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condition ` ( x e. R /\ y e. S ) ` is been removed. (Contributed by FL,
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24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) $)
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ovigg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
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( ps -> ( A F B ) = C ) ) $=
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( wcel w3a cop wceq cfv coprab eloprabga df-ov fveq1i eqtri wfun funoprab
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co wi funopfv ax-mp syl5eq syl6bir ) FJPGKPHLPQBFGRZHRACDEUAZPZFGIUHZHSAB
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CDEFGHJKLMUBUPUQUNUOTZHUQUNITURFGIUCUNIUOOUDUEUOUFUPURHSUIACDENUGUNHUOUJU
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KULUM $.
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$}
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${
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$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z S $.
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$d x y z ps $.
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ovig.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
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ovig.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
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ovig.3 $e |- F =
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{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
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$( The value of an operation class abstraction (weak version).
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(Unnecessary distinct variable restrictions were removed by David
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Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by
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Mario Carneiro, 19-Dec-2013.) $)
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ovig $p |- ( ( A e. R /\ B e. S /\ C e. D ) ->
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( ps -> ( A F B ) = C ) ) $=
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( wcel w3a wa wceq cv co 3simpa wb eleq1 bi2anan9 3adant3 anbi12d moanimv
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wmo wi mpbir ovigg mpand ) FJPZGKPZHIPZQUNUORZBFGLUAHSUNUOUPUBCTZJPZDTZKP
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ZRZARZUQBRCDEFGHLJKIURFSZUTGSZETHSZQVBUQABVDVEVBUQUCVFVDUSUNVEVAUOURFJUDU
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TGKUDUEUFMUGVCEUIVBAEUIUJNVBAEUHUKOULUM $.
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$}
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${
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$d x y z $. $d z A $. $d z B $. $d z C $. $d z F $.
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ovmpt4g.3 $e |- F = ( x e. A , y e. B |-> C ) $.
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$( Value of a function given by the "maps to" notation. (This is the
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operation analog of ~ fvmpt2 .) (Contributed by NM, 21-Feb-2004.)
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(Revised by Mario Carneiro, 1-Sep-2015.) $)
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ovmpt4g $p |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C ) $=
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( vz cv wcel co wceq wex wa elisset wmo moeq a1i cmpt2 coprab eqtri ovidi
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df-mpt2 eqeq2 mpbidi exlimdv syl5 3impia ) AJZCKZBJZDKZEGKZUJULFLZEMZUNIJ
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ZEMZINUKUMOZUPIEGPUSURUPIURUOUQMUPUSURABICDFURIQUSIERSFABCDETUSUROABIUAHA
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BICDEUDUBUCUQEUOUEUFUGUHUI $.
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$}
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${
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$d x y A $. $d x y B $. $d x y C $. $d x y D $.
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ovmpt2s.3 $e |- F = ( x e. C , y e. D |-> R ) $.
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$( Value of a function given by the "maps to" notation, expressed using
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explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.) $)
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ovmpt2s $p |- ( ( A e. C /\ B e. D /\ [_ A / x ]_ [_ B / y ]_ R e. V ) ->
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( A F B ) = [_ A / x ]_ [_ B / y ]_ R ) $=
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( wcel csb co wceq cvv cv wi nfcv nfcsb1v eleq1d elex nfel1 cmpt2 nfmpt21
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nfcxfr nfov nfeq nfim nfmpt22 csbeq1a oveq1 eqeq12d imbi12d oveq2 ovmpt4g
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wa 3expia vtocl2gaf csbcomg eqeq2d 3imtr4d syl5 3impia ) CEKZDFKZACBDGLLZ
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IKZCDHMZVFNZVGVFOKZVDVEUPZVIVFIUAVKBDACGLZLZOKZVHVMNZVJVIGOKZAPZBPZHMZGNZ
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QVLOKZCVRHMZVLNZQVNVOQABCDEFACRZBCRZBDRZWAWCAAVLOACGSZUBAWBVLACVRHWDAHABE
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FGUCZJABEFGUDUEAVRRUFWGUGUHVNVOBBVMOBDVLSZUBBVHVMBCDHWEBHWHJABEFGUIUEWFUF
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WIUGUHVQCNZVPWAVTWCWJGVLOACGUJZTWJVSWBGVLVQCVRHUKWKULUMVRDNZWAVNWCVOWLVLV
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MOBDVLUJZTWLWBVHVLVMVRDCHUNWMULUMVQEKVRFKVPVTABEFGHOJUOUQURVKVFVMOABCDGEF
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USZTVKVFVMVHWNUTVAVBVC $.
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$}
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${
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$d x y C $. $d x y D $.
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ov2gf.a $e |- F/_ x A $.
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ov2gf.c $e |- F/_ y A $.
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ov2gf.d $e |- F/_ y B $.
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ov2gf.1 $e |- F/_ x G $.
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ov2gf.2 $e |- F/_ y S $.
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ov2gf.3 $e |- ( x = A -> R = G ) $.
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ov2gf.4 $e |- ( y = B -> G = S ) $.
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ov2gf.5 $e |- F = ( x e. C , y e. D |-> R ) $.
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$( The value of an operation class abstraction. A version of ~ ovmpt2g
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using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.)
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(Revised by Mario Carneiro, 19-Dec-2013.) $)
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ov2gf $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
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( wcel co wceq cvv wa elex cv wi nfel1 nfmpt21 nfcxfr nfcv nfov nfeq nfim
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cmpt2 nfmpt22 eleq1d oveq1 eqeq12d imbi12d oveq2 ovmpt4g 3expia vtocl2gaf
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syl5 3impia ) CETZDFTZHKTZCDIUAZHUBZVIHUCTZVGVHUDVKHKUEGUCTZAUFZBUFZIUAZG
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UBZUGJUCTZCVOIUAZJUBZUGVLVKUGABCDEFLMNVRVTAAJUCOUHAVSJACVOILAIABEFGUOZSAB
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EFGUIUJAVOUKULOUMUNVLVKBBHUCPUHBVJHBCDIMBIWASABEFGUPUJNULPUMUNVNCUBZVMVRV
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QVTWBGJUCQUQWBVPVSGJVNCVOIURQUSUTVODUBZVRVLVTVKWCJHUCRUQWCVSVJJHVODCIVARU
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SUTVNETVOFTVMVQABEFGIUCSVBVCVDVEVF $.
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$}
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${
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$d x y $. $d x A $. $d y B $.
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ovmpt2dx.1 $e |- ( ph -> F = ( x e. C , y e. D |-> R ) ) $.
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ovmpt2dx.2 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
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ovmpt2dx.3 $e |- ( ( ph /\ x = A ) -> D = L ) $.
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ovmpt2dx.4 $e |- ( ph -> A e. C ) $.
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ovmpt2dx.5 $e |- ( ph -> B e. L ) $.
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ovmpt2dx.6 $e |- ( ph -> S e. X ) $.
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${
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ovmpt2dxf.px $e |- F/ x ph $.
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ovmpt2dxf.py $e |- F/ y ph $.
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ovmpt2dxf.ay $e |- F/_ y A $.
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ovmpt2dxf.bx $e |- F/_ x B $.
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ovmpt2dxf.sx $e |- F/_ x S $.
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ovmpt2dxf.sy $e |- F/_ y S $.
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$( Value of an operation given by a maps-to rule, deduction form.
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(Contributed by Mario Carneiro, 29-Dec-2014.) $)
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ovmpt2dxf $p |- ( ph -> ( A F B ) = S ) $=
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( co cmpt2 oveqd cv wcel cvv w3a wceq wi wsbc ovmpt4g a1i alrimi spsbcd
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eqid wa adantr simplr ad2antrr eqeltrd simpr 3eltr4d anassrs elex biimt
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wb syl syl3anc oveq12d eqeq12d bitr3d nfeq2 nfan nfmpt22 nfcv nfov nfeq
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wnf sbciedf nfmpt21 mpbid eqtrd ) ADEJUEDEBCFGHUFZUEZIAJWGDEMUGABUHZFUI
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ZCUHZGUIZHUJUIZUKZWIWKWGUEZHULZUMZCEUNZBDUNWHIULZAWRBDFPAWRBSAWQCEKQAWQ
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|
CTWQABCFGHWGUJWGUSUOUPUQURUQURAWRWSBDFPAWIDULZUTZWQWSCEKAEKUIZWTQVAXAWK
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EULZUTZWPWQWSXDWJWLWMWPWQVJXDWIDFAWTXCVBZADFUIWTXCPVCVDXDEKWKGAXBWTXCQV
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|
CXAXCVEZXAGKULXCOVAVFXDHIUJAWTXCHIULNVGZAIUJUIZWTXCAILUIXHRILVHVKVCVDWN
|
|
WPVIVLXDWOWHHIXDWIDWKEWGXEXFVMXGVNVOAWTCTCWIDUAVPVQWSCWBXACWHICDEWGUABC
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FGHVRCEVSVTUDWAUPWCSWSBWBABWHIBDEWGBDVSBCFGHWDUBVTUCWAUPWCWEWF $.
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$}
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$d y A $. $d x B $. $d x y S $. $d x y ph $.
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$( Value of an operation given by a maps-to rule, deduction form.
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(Contributed by Mario Carneiro, 29-Dec-2014.) $)
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ovmpt2dx $p |- ( ph -> ( A F B ) = S ) $=
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( nfv nfcv ovmpt2dxf ) ABCDEFGHIJKLMNOPQRABSACSCDTBETBITCITUA $.
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$}
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${
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$d x y A $. $d x y B $. $d x y S $. $d x y ph $.
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ovmpt2d.1 $e |- ( ph -> F = ( x e. C , y e. D |-> R ) ) $.
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ovmpt2d.2 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
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ovmpt2d.3 $e |- ( ph -> A e. C ) $.
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ovmpt2d.4 $e |- ( ph -> B e. D ) $.
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ovmpt2d.5 $e |- ( ph -> S e. X ) $.
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$( Value of an operation given by a maps-to rule, deduction form.
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|
(Contributed by Mario Carneiro, 7-Dec-2014.) $)
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ovmpt2d $p |- ( ph -> ( A F B ) = S ) $=
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|
( cv wceq wa eqidd ovmpt2dx ) ABCDEFGHIJGKLMABQDRSGTNOPUA $.
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$}
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${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y L $. $d x y S $.
|
|
ovmpt2x.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
|
|
ovmpt2x.2 $e |- ( x = A -> D = L ) $.
|
|
ovmpt2x.3 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( The value of an operation class abstraction. Variant of ~ ovmpt2ga
|
|
which does not require ` D ` and ` x ` to be distinct. (Contributed by
|
|
Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.) $)
|
|
ovmpt2x $p |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq cv adantl elex w3a cmpt2 wa simp1 simp2 simp3 ovmpt2dx
|
|
a1i syl3an3 ) HJOCEOZDKOZHPOZCDIQHRHJUAUKULUMUBZABCDEFGHIKPIABEFGUCRUNNUI
|
|
ASCRZBSDRUDGHRUNLTUOFKRUNMTUKULUMUEUKULUMUFUKULUMUGUHUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y S $.
|
|
ovmpt2ga.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
|
|
ovmpt2ga.2 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( Value of an operation given by a maps-to rule. (Contributed by Mario
|
|
Carneiro, 19-Dec-2013.) $)
|
|
ovmpt2ga $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq elex w3a cmpt2 cv a1i adantl simp1 simp2 simp3 ovmpt2d
|
|
wa syl3an3 ) HJMCEMZDFMZHNMZCDIOHPHJQUIUJUKRZABCDEFGHINIABEFGSPULLUAATCPB
|
|
TDPUGGHPULKUBUIUJUKUCUIUJUKUDUIUJUKUEUFUH $.
|
|
|
|
ovmpt2a.4 $e |- S e. _V $.
|
|
$( Value of an operation given by a maps-to rule. (Contributed by NM,
|
|
19-Dec-2013.) $)
|
|
ovmpt2a $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq ovmpt2ga mp3an3 ) CEMDFMHNMCDIOHPLABCDEFGHINJKQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ph $.
|
|
ovmpt2df.1 $e |- ( ph -> A e. C ) $.
|
|
ovmpt2df.2 $e |- ( ( ph /\ x = A ) -> B e. D ) $.
|
|
ovmpt2df.3 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V ) $.
|
|
ovmpt2df.4 $e |- ( ( ph /\ ( x = A /\ y = B ) ) ->
|
|
( ( A F B ) = R -> ps ) ) $.
|
|
${
|
|
ovmpt2df.5 $e |- F/_ x F $.
|
|
ovmpt2df.6 $e |- F/ x ps $.
|
|
ovmpt2df.7 $e |- F/_ y F $.
|
|
ovmpt2df.8 $e |- F/ y ps $.
|
|
$( Alternate deduction version of ~ ovmpt2 , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
ovmpt2df $p |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) $=
|
|
( wcel cv wceq cmpt2 nfv nfmpt21 nfeq nfim cvv wex elex syl isset sylib
|
|
wi wa nfmpt22 co oveq simprl simprr oveq12d adantr eqeltrd adantrr eqid
|
|
ovmpt4g syl3anc eqtr3d eqeq2d sylbid syl5 expr exlimd mpd exlimdd ) ACU
|
|
AZEUBZJCDGHIUCZUBZBUNZCACUDVSBCCJVRPCDGHIUEUFQUGAEUHTZVQCUIAEGTZWALEGUJ
|
|
UKCEULUMAVQUOZDUAZFUBZDUIZVTWCFUHTZWFWCFHTZWGMFHUJUKDFULUMWCWEVTDWCDUDV
|
|
SBDDJVRRCDGHIUPUFSUGAVQWEVTVSEFJUQZEFVRUQZUBZAVQWEUOZUOZBEFJVRURWMWKWII
|
|
UBBWMWJIWIWMVPWDVRUQZWJIWMVPEWDFVRAVQWEUSZAVQWEUTZVAWMVPGTWDHTIKTWNIUBW
|
|
MVPEGWOAWBWLLVBVCWMWDFHWPAVQWHWEMVDVCNCDGHIVRKVRVEVFVGVHVIOVJVKVLVMVNVO
|
|
$.
|
|
$}
|
|
|
|
$d x y F $. $d x y ps $.
|
|
$( Alternate deduction version of ~ ovmpt2 , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
ovmpt2dv $p |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) ) $=
|
|
( nfcv nfv ovmpt2df ) ABCDEFGHIJKLMNOCJPBCQDJPBDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ph $. $d x y S $.
|
|
ovmpt2dv2.1 $e |- ( ph -> A e. C ) $.
|
|
ovmpt2dv2.2 $e |- ( ( ph /\ x = A ) -> B e. D ) $.
|
|
ovmpt2dv2.3 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V ) $.
|
|
ovmpt2dv2.4 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S ) $.
|
|
$( Alternate deduction version of ~ ovmpt2 , suitable for iteration.
|
|
(Contributed by Mario Carneiro, 7-Jan-2017.) $)
|
|
ovmpt2dv2 $p |- ( ph ->
|
|
( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) ) $=
|
|
( co wceq cv wa nfcv cmpt2 eqidd eqeq2d biimpd nfmpt21 nfov nfeq1 nfmpt22
|
|
ovmpt2df mpd oveq eqeq1d syl5ibrcom ) ADEJPZIQJBCFGHUAZQZDEUOPZIQZAUOUOQU
|
|
RAUOUBAURBCDEFGHUOKLMNABRDQCREQSSZUQHQURUSHIUQOUCUDBCFGHUEZBUQIBDEUOBDTUT
|
|
BETUFUGBCFGHUHZCUQICDEUOCDTVACETUFUGUIUJUPUNUQIDEJUOUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y S $.
|
|
ovmpt2g.1 $e |- ( x = A -> R = G ) $.
|
|
ovmpt2g.2 $e |- ( y = B -> G = S ) $.
|
|
ovmpt2g.3 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( Value of an operation given by a maps-to rule. Special case.
|
|
(Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy,
|
|
19-Jun-2012.) $)
|
|
ovmpt2g $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( cv wceq sylan9eq ovmpt2ga ) ABCDEFGHIKAOCPBODPGJHLMQNR $.
|
|
|
|
ovmpt2.4 $e |- S e. _V $.
|
|
$( Value of an operation given by a maps-to rule. Special case.
|
|
(Contributed by NM, 16-May-1995.) (Revised by David Abernethy,
|
|
19-Jun-2012.) $)
|
|
ovmpt2 $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq ovmpt2g mp3an3 ) CEODFOHPOCDIQHRNABCDEFGHIJPKLMST $.
|
|
$}
|
|
|
|
${
|
|
$d f u v w x y z A $. $d f u v w x y z B $. $d x y z R $.
|
|
$d f u v w y z C $. $d f u v w y z D $. $d f u v w x y z H $.
|
|
$d f u v w z S $.
|
|
ovi3.1 $e |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) ->
|
|
S e. ( H X. H ) ) $.
|
|
ovi3.2 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) ->
|
|
R = S ) $.
|
|
ovi3.3 $e |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\
|
|
y e. ( H X. H ) ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = R ) ) } $.
|
|
$( The value of an operation class abstraction. Special case.
|
|
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
|
|
29-Dec-2014.) $)
|
|
ovi3 $p |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) ->
|
|
( <. A , B >. F <. C , D >. ) = S ) $=
|
|
( wa wex wcel cv wceq cop co cvv cxp elex syl isset sylib nfv nfcv coprab
|
|
nfoprab3 nfcxfr nfov nfeq1 eqeq2d copsex4g opelxpi nfoprab1 nfim nfoprab2
|
|
eqeq1 anbi1d 4exbidv oveq1 eqeq1d imbi12d anbi2d oveq2 moeq mosubop anass
|
|
wi wmo 2exbii 19.42vv bitri mobii mpbir a1i ovidi vtocl2gaf sylbird eqeq2
|
|
syl2an mpbidi exlimd mpd ) GOUAHOUASZIOUAJOUASZSZCUBZLUCZCTZGHUDZIJUDZNUE
|
|
ZLUCZWNLUFUAZWQWNLOOUGZUAXBPLXCUHUICLUJUKWNWPXACWNCULCWTLCWRWSNCWRUMCNAUB
|
|
ZXCUABUBZXCUASZXDDUBZEUBZUDZUCZXEFUBZMUBZUDZUCZSZWOKUCZSZMTFTZETDTZSZABCU
|
|
NZRXTABCUOUPCWSUMUQURWPWTWOUCZXAWNWNWPWRXIUCZWSXMUCZSZXPSZMTFTETDTZYBXPWP
|
|
DEFMGHIJOOXGGUCXHHUCSXKIUCXLJUCSSKLWOQUSUTWLWRXCUAWSXCUAYGYBVPZWMGHOOVAIJ
|
|
OOVAXSXDXENUEZWOUCZVPYCXNSZXPSZMTFTETDTZWRXENUEZWOUCZVPYHABWRWSXCXCAWRUMZ
|
|
BWRUMZBWSUMZYMYOAYMAULAYNWOAWRXENYPANYARXTABCVBUPAXEUMUQURVCYGYBBYGBULBWT
|
|
WOBWRWSNYQBNYARXTABCVDUPYRUQURVCXDWRUCZXSYMYJYOYSXQYLDEFMYSXOYKXPYSXJYCXN
|
|
XDWRXIVEVFVFVGYSYIYNWOXDWRXENVHVIVJXEWSUCZYMYGYOYBYTYLYFDEFMYTYKYEXPYTXNY
|
|
DYCXEWSXMVEVKVFVGYTYNWTWOXEWSWRNVLVIVJXSABCXCXCNXSCVQZXFUUAXJXNXPSZMTFTZS
|
|
ZETDTZCVQUUCCDEXDXPCFMXECKVMVNVNXSUUECXRUUDDEXRXJUUBSZMTFTUUDXQUUFFMXJXNX
|
|
PVOVRXJUUBFMVSVTVRWAWBWCRWDWEWHWFWOLWTWGWIWJWK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $. $d w z R $.
|
|
$d w x y z S $.
|
|
ov6g.1 $e |- ( <. x , y >. = <. A , B >. -> R = S ) $.
|
|
ov6g.2 $e |- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C
|
|
/\ z = R ) } $.
|
|
$( The value of an operation class abstraction. Special case.
|
|
(Contributed by NM, 13-Nov-2006.) $)
|
|
ov6g $p |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J )
|
|
-> ( A F B ) = S ) $=
|
|
( vw wcel wa cv wceq wex cop w3a co cfv df-ov eqid biidd copsex2g 3adant3
|
|
mpbiri adantr wi eqeq1 anbi1d eqeq2d eqcoms pm5.32i syl6bb 2exbidv anbi2d
|
|
wb wmo moeq mosubop a1i coprab copab dfoprab2 eleq1 bitr3i 2exbii 19.42vv
|
|
an12 bitri opabbii 3eqtri fvopab3ig 3ad2antl3 mpd syl5eq ) DJPZEKPZDEUAZF
|
|
PZUBZHLPZQZDEIUCWCIUDZHDEIUEWGWCARZBRZUAZSZHHSZQZBTATZWHHSZWEWOWFWAWBWOWD
|
|
WAWBQWOWMHUFWMWMABDEJKWIDSWJESQWMUGUHUJUIUKWDWAWFWOWPULWBORZWKSZCRZGSZQZB
|
|
TATZWLWSHSZQZBTATWOOCWCHFLIWQWCSZXAXDABXEXAWLWTQXDXEWRWLWTWQWCWKUMUNWLWTX
|
|
CWTXCVAWKWCWKWCSGHWSMUOUPUQURUSXCXDWNABXCXCWMWLWSHHUMUTUSXBCVBWQFPZWTCABW
|
|
QCGVCVDVEIWKFPZWTQZABCVFWRXHQZBTATZOCVGXFXBQZOCVGNXHABCOVHXJXKOCXJXFXAQZB
|
|
TATXKXIXLABXIWRXFWTQZQXLWRXMXHWRXFXGWTWQWKFVIUNUQWRXFWTVMVJVKXFXAABVLVNVO
|
|
VPVQVRVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d ph c $. $d ps x $. $d ch x y $. $d th x y z $. $d ta x y c $.
|
|
$d R x y z c $. $d S x y z c $. $d A x y z c $. $d B x y z c $.
|
|
$d C x y z c $.
|
|
ovg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ovg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ovg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ovg.4 $e |- ( ( ta /\ ( x e. R /\ y e. S ) ) -> E! z ph ) $.
|
|
ovg.5 $e |- F = { <. <. x , y >. , z >. |
|
|
( ( x e. R /\ y e. S ) /\ ph ) } $.
|
|
$( The value of an operation class abstraction. (Contributed by Jeff
|
|
Madsen, 10-Jun-2010.) $)
|
|
ovg $p |- ( ( ta /\ ( A e. R /\ B e. S /\ C e. D ) )
|
|
-> ( ( A F B ) = C <-> th ) ) $=
|
|
( vc wcel w3a wa co wceq cop cv coprab df-ov fveq1i eqtri eqeq1i wb eqeq2
|
|
cfv wi opeq2 eleq1d bibi12d imbi2d copab wfn weu wal ex alrimivv fnoprabg
|
|
syl eleq1 anbi1d anbi2d opelopabg fnopfvb syl2an vtoclg com12 exp32 3imp2
|
|
ibir anbi12d eloprabg adantl bitrd syl5bb biidd bianabs 3adant3 ) EIMUBZJ
|
|
NUBZKLUBZUCZUDZIJOUEZKUFZWIWJUDZDUDZDWOIJUGZFUHZMUBZGUHZNUBZUDZAUDZFGHUIZ
|
|
UPZKUFZWMWQWNXFKWNWROUPXFIJOUJWROXETUKULUMWMXGWRKUGZXEUBZWQEWIWJWKXGXIUNZ
|
|
EWIWJWKXJUQWKEWPUDZXJXKXFUAUHZUFZWRXLUGZXEUBZUNZUQXKXJUQUAKLXLKUFZXPXJXKX
|
|
QXMXGXOXIXLKXFUOXQXNXHXEXLKWRURUSUTVAEXEXCFGVBZVCZWRXRUBZXPWPEXCAHVDZUQZG
|
|
VEFVEXSEYBFGEXCYASVFVGXCAFGHVHVIWPXTXCWIXBUDZWPFGIJMNWSIUFZWTWIXBWSIMVJVK
|
|
ZXAJUFZXBWJWIXAJNVJVLZVMVTXRWRXLXEVNVOVPVQVRVSWLXIWQUNEXDYCBUDWPCUDWQFGHI
|
|
JKMNLYDXCYCABYEPWAYFYCWPBCYGQWAHUHKUFCDWPRVLWBWCWDWEWLWQDUNZEWIWJYHWKWPWQ
|
|
DWPWQWFWGWHWCWD $.
|
|
$}
|
|
|
|
$( The value of a restricted operation. (Contributed by FL, 10-Nov-2006.) $)
|
|
ovres $p |- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B )
|
|
= ( A F B ) ) $=
|
|
( wcel wa cop cxp cres cfv co wceq opelxpi fvres syl df-ov 3eqtr4g ) ACFBDF
|
|
GZABHZECDIZJZKZTEKZABUBLABELSTUAFUCUDMABCDNTUAEOPABUBQABEQR $.
|
|
|
|
${
|
|
ovresd.1 $e |- ( ph -> A e. X ) $.
|
|
ovresd.2 $e |- ( ph -> B e. X ) $.
|
|
$( Lemma for converting metric theorems to metric space theorems.
|
|
(Contributed by Mario Carneiro, 2-Oct-2015.) $)
|
|
ovresd $p |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) ) $=
|
|
( wcel cxp cres co wceq ovres syl2anc ) ABEHCEHBCDEEIJKBCDKLFGBCEEDMN $.
|
|
$}
|
|
|
|
$( The value of a member of the domain of a subclass of an operation.
|
|
(Contributed by NM, 23-Aug-2007.) $)
|
|
oprssov $p |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\
|
|
( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) ) $=
|
|
( wfun cxp wfn wss w3a wcel wa cres co wceq ovres adantl cdm eqtr3d reseq2d
|
|
fndm 3ad2ant2 funssres 3adant2 oveqd adantr ) EGZFCDHZIZFEJZKZACLBDLMZMABEU
|
|
INZOZABEOZABFOZUMUOUPPULABCDEQRULUOUQPUMULUNFABULEFSZNZUNFUJUHUSUNPUKUJURUI
|
|
EUIFUBUAUCUHUKUSFPUJEFUDUETUFUGT $.
|
|
|
|
$( An operation's value belongs to its codomain. (Contributed by NM,
|
|
27-Aug-2006.) $)
|
|
fovrn $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) ->
|
|
( A F B ) e. C ) $=
|
|
( cxp wf wcel co wa cop opelxpi cfv df-ov ffvelrn syl5eqel sylan2 3impb ) D
|
|
EGZCFHZADIZBEIZABFJZCIZUBUCKUAABLZTIZUEABDEMUAUGKUDUFFNCABFOTCUFFPQRS $.
|
|
|
|
${
|
|
fovrnd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $.
|
|
$( An operation's value belongs to its codomain. (Contributed by Mario
|
|
Carneiro, 29-Dec-2016.) $)
|
|
fovrnda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $=
|
|
( wcel co cxp wf fovrn syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFGM
|
|
NO $.
|
|
|
|
fovrnd.2 $e |- ( ph -> A e. R ) $.
|
|
fovrnd.3 $e |- ( ph -> B e. S ) $.
|
|
$( An operation's value belongs to its codomain. (Contributed by Mario
|
|
Carneiro, 29-Dec-2016.) $)
|
|
fovrnd $p |- ( ph -> ( A F B ) e. C ) $=
|
|
( cxp wf wcel co fovrn syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w z C $. $d w x y z F $.
|
|
$( The range of an operation expressed as a collection of the operation's
|
|
values. (Contributed by NM, 29-Oct-2006.) $)
|
|
fnrnov $p |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B
|
|
z = ( x F y ) } ) $=
|
|
( vw cxp wfn crn cv cfv wceq wrex cab co fnrnfv cop fveq2 df-ov syl6eqr
|
|
eqeq2d rexxp abbii syl6eq ) FDEHZIFJCKZGKZFLZMZGUFNZCOUGAKZBKZFPZMZBENADN
|
|
ZCOGCUFFQUKUPCUJUOGABDEUHULUMRZMZUIUNUGURUIUQFLUNUHUQFSULUMFTUAUBUCUDUE
|
|
$.
|
|
|
|
$( An onto mapping of an operation expressed in terms of operation values.
|
|
(Contributed by NM, 29-Oct-2006.) $)
|
|
foov $p |- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\
|
|
A. z e. C E. x e. A E. y e. B z = ( x F y ) ) ) $=
|
|
( vw cxp wfo wf cv cfv wceq wrex wral wa co dffo3 cop fveq2 df-ov syl6eqr
|
|
eqeq2d rexxp ralbii anbi2i bitri ) DEIZFGJUIFGKZCLZHLZGMZNZHUIOZCFPZQUJUK
|
|
ALZBLZGRZNZBEOADOZCFPZQHCUIFGSUPVBUJUOVACFUNUTHABDEULUQURTZNZUMUSUKVDUMVC
|
|
GMUSULVCGUAUQURGUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
$( An operation's value belongs to its range. (Contributed by NM,
|
|
10-Feb-2007.) $)
|
|
fnovrn $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
|
|
( C F D ) e. ran F ) $=
|
|
( cxp wfn wcel co crn cop opelxpi cfv df-ov fnfvelrn syl5eqel sylan2 3impb
|
|
wa ) EABFZGZCAHZDBHZCDEIZEJZHZUBUCSUACDKZTHZUFCDABLUAUHSUDUGEMUECDENTUGEOPQ
|
|
R $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z F $.
|
|
$( A member of an operation's range is a value of the operation.
|
|
(Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro,
|
|
30-Jan-2014.) $)
|
|
ovelrn $p |- ( F Fn ( A X. B ) -> ( C e. ran F <->
|
|
E. x e. A E. y e. B C = ( x F y ) ) ) $=
|
|
( vz cxp wfn crn wcel cv co wceq wrex cab fnrnov cvv wi a1i eleq2d fnovex
|
|
elex w3a eleq1 syl5ibrcom 3expb rexlimdvva eqeq1 2rexbidv elabg pm5.21ndd
|
|
wb bitrd ) FCDHIZEFJZKEGLZALZBLZFMZNZBDOACOZGPZKZEUTNZBDOACOZUOUPVCEABGCD
|
|
FQUAUOERKZVDVFVDVGSUOEVCUCTUOVEVGABCDUOURCKZUSDKZVEVGSUOVHVIUDVGVEUTRKURU
|
|
SCDFUBEUTRUEUFUGUHVGVDVFUMSUOVBVFGERUQENVAVEABCDUQEUTUIUJUKTULUN $.
|
|
|
|
$( Membership relation for the values of a function whose image is a
|
|
subclass. (Contributed by Mario Carneiro, 23-Dec-2013.) $)
|
|
funimassov $p |- ( ( Fun F /\ ( A X. B ) C_ dom F ) ->
|
|
( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) ) $=
|
|
( vz wfun cxp cdm wss wa cima cv cfv wcel wral co funimass4 cop syl6eqr
|
|
wceq fveq2 df-ov eleq1d ralxp syl6bb ) FHCDIZFJKLFUHMEKGNZFOZEPZGUHQANZBN
|
|
ZFRZEPZBDQACQGUHEFSUKUOGABCDUIULUMTZUBZUJUNEUQUJUPFOUNUIUPFUCULUMFUDUAUEU
|
|
FUG $.
|
|
|
|
$( Operation value in an image. (Contributed by Mario Carneiro,
|
|
23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.) $)
|
|
ovelimab $p |- ( ( F Fn A /\ ( B X. C ) C_ A ) ->
|
|
( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) ) $=
|
|
( vz wfn cxp wss wa cima wcel cv cfv wceq wrex co syl6bb fvelimab syl6eqr
|
|
cop fveq2 df-ov eqeq1d eqcom rexxp ) GCIDEJZCKLFGUIMNHOZGPZFQZHUIRFAOZBOZ
|
|
GSZQZBERADRHCUIFGUAULUPHABDEUJUMUNUCZQZULUOFQUPURUKUOFURUKUQGPUOUJUQGUDUM
|
|
UNGUEUBUFUOFUGTUHT $.
|
|
$}
|
|
|
|
${
|
|
oprvalconst2.1 $e |- C e. _V $.
|
|
$( The value of a constant operation. (Contributed by NM, 5-Nov-2006.) $)
|
|
ovconst2 $p |- ( ( R e. A /\ S e. B ) ->
|
|
( R ( ( A X. B ) X. { C } ) S ) = C ) $=
|
|
( wcel wa cxp csn co cop cfv df-ov wceq opelxpi fvconst2 syl syl5eq ) DAG
|
|
EBGHZDEABIZCJIZKDELZUBMZCDEUBNTUCUAGUDCODEABPUACUCFQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $. $d x y E $. $d x y ph $.
|
|
$d x y F $.
|
|
caovclg.1 $e |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E ) $.
|
|
$( Convert an operation closure law to class notation. (Contributed by
|
|
Mario Carneiro, 26-May-2014.) $)
|
|
caovclg $p |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E ) $=
|
|
( cv co wcel wral wa ralrimivva wceq oveq1 eleq1d oveq2 rspc2v mpan9 ) AB
|
|
KZCKZILZHMZCGNBFNDFMEGMODEILZHMZAUFBCFGJPUFUHDUDILZHMBCDEFGUCDQUEUIHUCDUD
|
|
IRSUDEQUIUGHUDEDITSUAUB $.
|
|
|
|
caovcld.2 $e |- ( ph -> A e. C ) $.
|
|
caovcld.3 $e |- ( ph -> B e. D ) $.
|
|
$( Convert an operation closure law to class notation. (Contributed by
|
|
Mario Carneiro, 30-Dec-2014.) $)
|
|
caovcld $p |- ( ph -> ( A F B ) e. E ) $=
|
|
( wcel co id caovclg syl12anc ) AADFMEGMDEINHMAOKLABCDEFGHIJPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y F $. $d x y S $.
|
|
caovcl.1 $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S ) $.
|
|
$( Convert an operation closure law to class notation. (Contributed by NM,
|
|
4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) $)
|
|
caovcl $p |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S ) $=
|
|
( wtru wcel wa co tru cv adantl caovclg mpan ) HCEIDEIJCDFKEILHABCDEEEFAM
|
|
ZEIBMZEIJQRFKEIHGNOP $.
|
|
$}
|
|
|
|
${
|
|
$( General laws for commutative, associative, distributive operations. $)
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z ph $.
|
|
$d x y z F $. $d x y z G $. $d x y z H $. $d x y z K $. $d x y z R $.
|
|
$d x y z S $. $d x y z T $.
|
|
${
|
|
caovcomg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
$( Convert an operation commutative law to class notation. (Contributed
|
|
by Mario Carneiro, 1-Jun-2013.) $)
|
|
caovcomg $p |- ( ( ph /\ ( A e. S /\ B e. S ) ) ->
|
|
( A F B ) = ( B F A ) ) $=
|
|
( cv co wceq wral wcel wa ralrimivva oveq1 oveq2 eqeq12d rspc2v mpan9 )
|
|
ABIZCIZGJZUBUAGJZKZCFLBFLDFMEFMNDEGJZEDGJZKZAUEBCFFHOUEUHDUBGJZUBDGJZKB
|
|
CDEFFUADKUCUIUDUJUADUBGPUADUBGQRUBEKUIUFUJUGUBEDGQUBEDGPRST $.
|
|
|
|
caovcomd.2 $e |- ( ph -> A e. S ) $.
|
|
caovcomd.3 $e |- ( ph -> B e. S ) $.
|
|
$( Convert an operation commutative law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovcomd $p |- ( ph -> ( A F B ) = ( B F A ) ) $=
|
|
( wcel co wceq id caovcomg syl12anc ) AADFKEFKDEGLEDGLMANIJABCDEFGHOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
caovcom.1 $e |- A e. _V $.
|
|
caovcom.2 $e |- B e. _V $.
|
|
caovcom.3 $e |- ( x F y ) = ( y F x ) $.
|
|
$( Convert an operation commutative law to class notation. (Contributed
|
|
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.) $)
|
|
caovcom $p |- ( A F B ) = ( B F A ) $=
|
|
( cvv wcel wa co wceq pm3.2i cv a1i caovcomg mp2an ) CIJZSDIJZKCDELDCEL
|
|
MFSTFGNSABCDIEAOZBOZELUBUAELMSUAIJUBIJKKHPQR $.
|
|
$}
|
|
|
|
${
|
|
caovassg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
|
|
$( Convert an operation associative law to class notation. (Contributed
|
|
by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro,
|
|
26-May-2014.) $)
|
|
caovassg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 oveq1d eqeq12d oveq2 oveq2d rspc3v mpan9
|
|
w3a ralrimivvva ) ABKZCKZILZDKZILZUEUFUHILZILZMZDHNCHNBHNEHOFHOGHOUCEFI
|
|
LZGILZEFGILZILZMZAULBCDHHHJUDULUQEUFILZUHILZEUJILZMUMUHILZEFUHILZILZMBC
|
|
DEFGHHHUEEMZUIUSUKUTVDUGURUHIUEEUFIPQUEEUJIPRUFFMZUSVAUTVCVEURUMUHIUFFE
|
|
ISQVEUJVBEIUFFUHIPTRUHGMZVAUNVCUPUHGUMISVFVBUOEIUHGFISTRUAUB $.
|
|
|
|
caovassd.2 $e |- ( ph -> A e. S ) $.
|
|
caovassd.3 $e |- ( ph -> B e. S ) $.
|
|
caovassd.4 $e |- ( ph -> C e. S ) $.
|
|
$( Convert an operation associative law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovassd $p |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
|
|
( wcel co wceq id caovassg syl13anc ) AAEHNFHNGHNEFIOGIOEFGIOIOPAQKLMAB
|
|
CDEFGHIJRS $.
|
|
$}
|
|
|
|
${
|
|
caovass.1 $e |- A e. _V $.
|
|
caovass.2 $e |- B e. _V $.
|
|
caovass.3 $e |- C e. _V $.
|
|
caovass.4 $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
$( Convert an operation associative law to class notation. (Contributed
|
|
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) $)
|
|
caovass $p |- ( ( A F B ) F C ) = ( A F ( B F C ) ) $=
|
|
( cvv wcel co wceq wtru w3a tru cv wa a1i caovassg mpan mp3an ) DLMZELM
|
|
ZFLMZDEGNFGNDEFGNGNOZHIJPUEUFUGQUHRPABCDEFLGASZBSZGNCSZGNUIUJUKGNGNOPUI
|
|
LMUJLMUKLMQTKUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
caovcang.1 $e |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) = ( x F z ) <-> y = z ) ) $.
|
|
$( Convert an operation cancellation law to class notation. (Contributed
|
|
by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovcang $p |- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) ->
|
|
( ( A F B ) = ( A F C ) <-> B = C ) ) $=
|
|
( cv co wceq wb wral wcel oveq1 oveq2 bibi12d ralrimivvva bibi1d eqeq1d
|
|
w3a eqeq12d eqeq1 eqeq2d eqeq2 rspc3v mpan9 ) ABLZCLZJMZUKDLZJMZNZULUNN
|
|
ZOZDHPCHPBIPEIQFHQGHQUDEFJMZEGJMZNZFGNZOZAURBCDIHHKUAURVCEULJMZEUNJMZNZ
|
|
UQOUSVENZFUNNZOBCDEFGIHHUKENZUPVFUQVIUMVDUOVEUKEULJRUKEUNJRUEUBULFNZVFV
|
|
GUQVHVJVDUSVEULFEJSUCULFUNUFTUNGNZVGVAVHVBVKVEUTUSUNGEJSUGUNGFUHTUIUJ
|
|
$.
|
|
|
|
caovcand.2 $e |- ( ph -> A e. T ) $.
|
|
caovcand.3 $e |- ( ph -> B e. S ) $.
|
|
caovcand.4 $e |- ( ph -> C e. S ) $.
|
|
$( Convert an operation cancellation law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovcand $p |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) ) $=
|
|
( wcel co wceq wb id caovcang syl13anc ) AAEIOFHOGHOEFJPEGJPQFGQRASLMNA
|
|
BCDEFGHIJKTUA $.
|
|
|
|
caovcanrd.5 $e |- ( ph -> A e. S ) $.
|
|
caovcanrd.6 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
$( Commute the arguments of an operation cancellation law. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovcanrd $p |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) ) $=
|
|
( co wceq caovcomd eqeq12d caovcand bitr3d ) AEFJQZEGJQZRFEJQZGEJQZRFGR
|
|
AUCUEUDUFABCEFHJPOMSABCEGHJPONSTABCDEFGHIJKLMNUAUB $.
|
|
$}
|
|
|
|
${
|
|
caovcan.1 $e |- C e. _V $.
|
|
caovcan.2 $e |- ( ( x e. S /\ y e. S ) ->
|
|
( ( x F y ) = ( x F z ) -> y = z ) ) $.
|
|
$( Convert an operation cancellation law to class notation. (Contributed
|
|
by NM, 20-Aug-1995.) $)
|
|
caovcan $p |- ( ( A e. S /\ B e. S ) ->
|
|
( ( A F B ) = ( A F C ) -> B = C ) ) $=
|
|
( cv co wceq wi oveq1 eqeq12d imbi1d oveq2 imbi12d wcel eqeq1d eqeq1 wa
|
|
eqeq2d eqeq2 imbi2d vtocl vtocl2ga ) AKZBKZHLZUIFHLZMZUJFMZNZDUJHLZDFHL
|
|
ZMZUNNDEHLZUQMZEFMZNABDEGGUIDMZUMURUNVBUKUPULUQUIDUJHOUIDFHOPQUJEMZURUT
|
|
UNVAVCUPUSUQUJEDHRUAUJEFUBSUIGTUJGTUCZUKUICKZHLZMZUJVEMZNZNVDUONCFIVEFM
|
|
ZVIUOVDVJVGUMVHUNVJVFULUKVEFUIHRUDVEFUJUESUFJUGUH $.
|
|
$}
|
|
|
|
${
|
|
caovordig.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( x R y -> ( z F x ) R ( z F y ) ) ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
Mario Carneiro, 31-Dec-2014.) $)
|
|
caovordig $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( A R B -> ( C F A ) R ( C F B ) ) ) $=
|
|
( cv wbr co wi wral wcel wceq oveq2 imbi12d w3a ralrimivvva breq1 breq2
|
|
breq1d breq2d oveq1 breq12d imbi2d rspc3v mpan9 ) ABLZCLZHMZDLZULJNZUOU
|
|
MJNZHMZOZDIPCIPBIPEIQFIQGIQUAEFHMZGEJNZGFJNZHMZOZAUSBCDIIIKUBUSVDEUMHMZ
|
|
UOEJNZUQHMZOUTVFUOFJNZHMZOBCDEFGIIIULERZUNVEURVGULEUMHUCVJUPVFUQHULEUOJ
|
|
SUETUMFRZVEUTVGVIUMFEHUDVKUQVHVFHUMFUOJSUFTUOGRZVIVCUTVLVFVAVHVBHUOGEJU
|
|
GUOGFJUGUHUIUJUK $.
|
|
|
|
caovordid.2 $e |- ( ph -> A e. S ) $.
|
|
caovordid.3 $e |- ( ph -> B e. S ) $.
|
|
caovordid.4 $e |- ( ph -> C e. S ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
Mario Carneiro, 31-Dec-2014.) $)
|
|
caovordid $p |- ( ph -> ( A R B -> ( C F A ) R ( C F B ) ) ) $=
|
|
( wcel wbr co wi id caovordig syl13anc ) AAEIOFIOGIOEFHPGEJQGFJQHPRASLM
|
|
NABCDEFGHIJKTUA $.
|
|
$}
|
|
|
|
${
|
|
caovordg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( x R y <-> ( z F x ) R ( z F y ) ) ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovordg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( A R B <-> ( C F A ) R ( C F B ) ) ) $=
|
|
( cv wbr co wb wral wcel wceq oveq2 bibi12d w3a ralrimivvva breq1 breq2
|
|
breq1d breq2d oveq1 breq12d bibi2d rspc3v mpan9 ) ABLZCLZHMZDLZULJNZUOU
|
|
MJNZHMZOZDIPCIPBIPEIQFIQGIQUAEFHMZGEJNZGFJNZHMZOZAUSBCDIIIKUBUSVDEUMHMZ
|
|
UOEJNZUQHMZOUTVFUOFJNZHMZOBCDEFGIIIULERZUNVEURVGULEUMHUCVJUPVFUQHULEUOJ
|
|
SUETUMFRZVEUTVGVIUMFEHUDVKUQVHVFHUMFUOJSUFTUOGRZVIVCUTVLVFVAVHVBHUOGEJU
|
|
GUOGFJUGUHUIUJUK $.
|
|
|
|
caovordd.2 $e |- ( ph -> A e. S ) $.
|
|
caovordd.3 $e |- ( ph -> B e. S ) $.
|
|
caovordd.4 $e |- ( ph -> C e. S ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
Mario Carneiro, 30-Dec-2014.) $)
|
|
caovordd $p |- ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
|
|
( wcel wbr co wb id caovordg syl13anc ) AAEIOFIOGIOEFHPGEJQGFJQHPRASLMN
|
|
ABCDEFGHIJKTUA $.
|
|
|
|
caovord2d.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
$( Operation ordering law with commuted arguments. (Contributed by Mario
|
|
Carneiro, 30-Dec-2014.) $)
|
|
caovord2d $p |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) ) $=
|
|
( wbr co caovordd caovcomd breq12d bitrd ) AEFHPGEJQZGFJQZHPEGJQZFGJQZH
|
|
PABCDEFGHIJKLMNRAUBUDUCUEHABCGEIJONLSABCGFIJONMSTUA $.
|
|
|
|
caovord3d.5 $e |- ( ph -> D e. S ) $.
|
|
$( Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovord3d $p |- ( ph ->
|
|
( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) ) $=
|
|
( co wbr wb wceq breq1 caovord2d caovordd bibi12d syl5ibr ) EFKRZGHKRZU
|
|
AEGISZHFISZTAUGGFKRZISZUHUKISZTUGUHUKIUBAUIULUJUMABCDEGFIJKLMONPUCABCDH
|
|
FGIJKLQNOUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
caovord.1 $e |- A e. _V $.
|
|
caovord.2 $e |- B e. _V $.
|
|
caovord.3 $e |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
NM, 19-Feb-1996.) $)
|
|
caovord $p |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
|
|
( wbr cv co wb wceq oveq1 wi oveq2 breq12d bibi2d wcel wa breq1 bibi12d
|
|
breq1d breq2 breq2d sylan9bb imbi2d vtocl2 vtoclga ) DEGMZCNZDIOZUOEIOZ
|
|
GMZPZUNFDIOZFEIOZGMZPCFHUOFQZURVBUNVCUPUTUQVAGUOFDIRUOFEIRUAUBUOHUCZANZ
|
|
BNZGMZUOVEIOZUOVFIOZGMZPZSVDUSSABDEJKVEDQZVFEQZUDVKUSVDVLVKDVFGMZUPVIGM
|
|
ZPVMUSVLVGVNVJVOVEDVFGUEVLVHUPVIGVEDUOITUGUFVMVNUNVOURVFEDGUHVMVIUQUPGV
|
|
FEUOITUIUFUJUKLULUM $.
|
|
|
|
$( (We don't bother to eliminate this redundant hypothesis.) $)
|
|
caovord2.3 $e |- C e. _V $.
|
|
caovord2.com $e |- ( x F y ) = ( y F x ) $.
|
|
$( Operation ordering law with commuted arguments. (Contributed by NM,
|
|
27-Feb-1996.) $)
|
|
caovord2 $p |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) ) $=
|
|
( wcel wbr co caovord caovcom breq12i syl6bb ) FHODEGPFDIQZFEIQZGPDFIQZ
|
|
EFIQZGPABCDEFGHIJKLRUBUDUCUEGABFDIMJNSABFEIMKNSTUA $.
|
|
|
|
$( (We don't bother to eliminate redundant hypotheses.) $)
|
|
caovord3.4 $e |- D e. _V $.
|
|
$( Ordering law. (Contributed by NM, 29-Feb-1996.) $)
|
|
caovord3 $p |- ( ( ( B e. S /\ C e. S ) /\
|
|
( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) ) $=
|
|
( wcel wa co wbr wceq wb caovord2 adantr breq1 sylan9bb ad2antlr bitr4d
|
|
caovord ) EIQZFIQZRZDEJSZFGJSZUAZRDFHTZUNFEJSZHTZGEHTZULUPUMUQHTZUOURUJ
|
|
UPUTUBUKABCDFEHIJKNMLOUCUDUMUNUQHUEUFUKUSURUBUJUOABCGEFHIJPLMUIUGUH $.
|
|
$}
|
|
|
|
${
|
|
caovdig.1 $e |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) ->
|
|
( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.) $)
|
|
caovdig $p |- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) ->
|
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( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 eqeq12d oveq12d oveq2d oveq2 oveq1d rspc3v
|
|
w3a ralrimivvva mpan9 ) ABNZCNZDNZIOZJOZUIUJJOZUIUKJOZKOZPZDHQCHQBLQELR
|
|
FHRGHRUFEFGIOZJOZEFJOZEGJOZKOZPZAUQBCDLHHMUGUQVCEULJOZEUJJOZEUKJOZKOZPE
|
|
FUKIOZJOZUTVFKOZPBCDEFGLHHUIEPZUMVDUPVGUIEULJSVKUNVEUOVFKUIEUJJSUIEUKJS
|
|
UATUJFPZVDVIVGVJVLULVHEJUJFUKISUBVLVEUTVFKUJFEJUCUDTUKGPZVIUSVJVBVMVHUR
|
|
EJUKGFIUCUBVMVFVAUTKUKGEJUCUBTUEUH $.
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|
|
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caovdid.2 $e |- ( ph -> A e. K ) $.
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caovdid.3 $e |- ( ph -> B e. S ) $.
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caovdid.4 $e |- ( ph -> C e. S ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovdid $p |- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) $=
|
|
( wcel co wceq id caovdig syl13anc ) AAELQFHQGHQEFGIRJREFJREGJRKRSATNOP
|
|
ABCDEFGHIJKLMUAUB $.
|
|
$}
|
|
|
|
${
|
|
caovdir2d.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) ) $.
|
|
caovdir2d.2 $e |- ( ph -> A e. S ) $.
|
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caovdir2d.3 $e |- ( ph -> B e. S ) $.
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caovdir2d.4 $e |- ( ph -> C e. S ) $.
|
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caovdir2d.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) e. S ) $.
|
|
caovdir2d.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x G y ) = ( y G x ) ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovdir2d $p |- ( ph
|
|
-> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) ) $=
|
|
( co caovdid caovcld caovcomd oveq12d 3eqtr4d ) AGEFIQZJQGEJQZGFJQZIQUC
|
|
GJQEGJQZFGJQZIQABCDGEFHIJIHKNLMRABCUCGHJPABCEFHHHIOLMSNTAUFUDUGUEIABCEG
|
|
HJPLNTABCFGHJPMNTUAUB $.
|
|
$}
|
|
|
|
${
|
|
caovdirg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) ->
|
|
( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) ) $.
|
|
$( Convert an operation reverse distributive law to class notation.
|
|
(Contributed by Mario Carneiro, 19-Oct-2014.) $)
|
|
caovdirg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) ->
|
|
( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 oveq2 w3a ralrimivvva oveq1d oveq2d rspc3v
|
|
eqeq12d oveq12d mpan9 ) ABNZCNZIOZDNZJOZUIULJOZUJULJOZKOZPZDLQCHQBHQEHR
|
|
FHRGLRUAEFIOZGJOZEGJOZFGJOZKOZPZAUQBCDHHLMUBUQVCEUJIOZULJOZEULJOZUOKOZP
|
|
URULJOZVFFULJOZKOZPBCDEFGHHLUIEPZUMVEUPVGVKUKVDULJUIEUJISUCVKUNVFUOKUIE
|
|
ULJSUCUFUJFPZVEVHVGVJVLVDURULJUJFEITUCVLUOVIVFKUJFULJSUDUFULGPZVHUSVJVB
|
|
ULGURJTVMVFUTVIVAKULGEJTULGFJTUGUFUEUH $.
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caovdird.2 $e |- ( ph -> A e. S ) $.
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caovdird.3 $e |- ( ph -> B e. S ) $.
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caovdird.4 $e |- ( ph -> C e. K ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by Mario Carneiro, 30-Dec-2014.) $)
|
|
caovdird $p |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) $=
|
|
( wcel co wceq id caovdirg syl13anc ) AAEHQFHQGLQEFIRGJREGJRFGJRKRSATNO
|
|
PABCDEFGHIJKLMUAUB $.
|
|
$}
|
|
|
|
${
|
|
caovdi.1 $e |- A e. _V $.
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caovdi.2 $e |- B e. _V $.
|
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caovdi.3 $e |- C e. _V $.
|
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caovdi.4 $e |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.) $)
|
|
caovdi $p |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) $=
|
|
( cvv wcel co wceq wtru w3a tru cv wa a1i caovdig mpan mp3an ) DMNZEMNZ
|
|
FMNZDEFGOHODEHODFHOGOPZIJKQUFUGUHRUISQABCDEFMGHGMATZBTZCTZGOHOUJUKHOUJU
|
|
LHOGOPQUJMNUKMNULMNRUALUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
caovd.1 $e |- ( ph -> A e. S ) $.
|
|
caovd.2 $e |- ( ph -> B e. S ) $.
|
|
caovd.3 $e |- ( ph -> C e. S ) $.
|
|
caovd.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
caovd.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov32d $p |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) ) $=
|
|
( co caovcomd oveq2d caovassd 3eqtr4d ) AEFGIOZIOEGFIOZIOEFIOGIOEGIOFIO
|
|
ATUAEIABCFGHIMKLPQABCDEFGHINJKLRABCDEGFHINJLKRS $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov12d $p |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) ) $=
|
|
( co caovcomd oveq1d caovassd 3eqtr3d ) AEFIOZGIOFEIOZGIOEFGIOIOFEGIOIO
|
|
ATUAGIABCEFHIMJKPQABCDEFGHINJKLRABCDFEGHINKJLRS $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov31d $p |- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) ) $=
|
|
( co caovcomd oveq1d caov32d 3eqtr4d ) AEGIOZFIOGEIOZFIOEFIOGIOGFIOEIOA
|
|
TUAFIABCEGHIMJLPQABCDEFGHIJKLMNRABCDGFEHILKJMNRS $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov13d $p |- ( ph -> ( A F ( B F C ) ) = ( C F ( B F A ) ) ) $=
|
|
( co caov31d caovassd 3eqtr3d ) AEFIOGIOGFIOEIOEFGIOIOGFEIOIOABCDEFGHIJ
|
|
KLMNPABCDEFGHINJKLQABCDGFEHINLKJQR $.
|
|
|
|
${
|
|
caovd.4 $e |- ( ph -> D e. S ) $.
|
|
caovd.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S ) $.
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov4d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
|
|
( ( A F C ) F ( B F D ) ) ) $=
|
|
( co caovcld caovassd caov12d oveq2d 3eqtr4d ) AEFGHJRZJRZJREGFHJRZJR
|
|
ZJREFJRUDJREGJRUFJRAUEUGEJABCDFGHIJLMPNOUAUBABCDEFUDIJOKLABCGHIIIJQMP
|
|
STABCDEGUFIJOKMABCFHIIIJQLPSTUC $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov411d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
|
|
( ( C F B ) F ( A F D ) ) ) $=
|
|
( co caovcomd oveq1d caov4d 3eqtr3d ) AFEJRZGHJRZJRFGJRZEHJRZJREFJRZU
|
|
DJRGFJRZUFJRABCDFEGHIJLKMNOPQUAAUCUGUDJABCFEIJNLKSTAUEUHUFJABCFGIJNLM
|
|
STUB $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
30-Dec-2014.) $)
|
|
caov42d $p |- ( ph -> ( ( A F B ) F ( C F D ) ) =
|
|
( ( A F C ) F ( D F B ) ) ) $=
|
|
( co caov4d caovcomd oveq2d eqtrd ) AEFJRGHJRJREGJRZFHJRZJRUCHFJRZJRA
|
|
BCDEFGHIJKLMNOPQSAUDUEUCJABCFHIJNLPTUAUB $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
caov.1 $e |- A e. _V $.
|
|
caov.2 $e |- B e. _V $.
|
|
caov.3 $e |- C e. _V $.
|
|
caov.com $e |- ( x F y ) = ( y F x ) $.
|
|
caov.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) $)
|
|
caov32 $p |- ( ( A F B ) F C ) = ( ( A F C ) F B ) $=
|
|
( co caovcom oveq2i caovass 3eqtr4i ) DEFGMZGMDFEGMZGMDEGMFGMDFGMEGMRSD
|
|
GABEFGIJKNOABCDEFGHIJLPABCDFEGHJILPQ $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) $)
|
|
caov12 $p |- ( A F ( B F C ) ) = ( B F ( A F C ) ) $=
|
|
( co caovcom oveq1i caovass 3eqtr3i ) DEGMZFGMEDGMZFGMDEFGMGMEDFGMGMRSF
|
|
GABDEGHIKNOABCDEFGHIJLPABCEDFGIHJLPQ $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) $)
|
|
caov31 $p |- ( ( A F B ) F C ) = ( ( C F B ) F A ) $=
|
|
( co caovass caov12 eqtri caov32 eqtr3i 3eqtr4i ) DFGMEGMZFDEGMZGMZUAFG
|
|
MFEGMZDGMZTDUCGMUBABCDFEGHJILNABCDFEGHJIKLOPABCDEFGHIJKLQFDGMEGMUDUBABC
|
|
FDEGJHIKLQABCFDEGJHILNRS $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by NM, 26-Aug-1995.) $)
|
|
caov13 $p |- ( A F ( B F C ) ) = ( C F ( B F A ) ) $=
|
|
( co caov31 caovass 3eqtr3i ) DEGMFGMFEGMDGMDEFGMGMFEDGMGMABCDEFGHIJKLN
|
|
ABCDEFGHIJLOABCFEDGJIHLOP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $.
|
|
$d r s t x y z ph $. $d r s t x y z F $. $d r s t x y z G $.
|
|
$d x y z H $. $d x y z R $. $d r s t x y z S $.
|
|
caovdilemd.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x G y ) = ( y G x ) ) $.
|
|
caovdilemd.distr $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) ) $.
|
|
caovdilemd.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x G y ) G z ) = ( x G ( y G z ) ) ) $.
|
|
caovdilemd.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x G y ) e. S ) $.
|
|
caovdilemd.a $e |- ( ph -> A e. S ) $.
|
|
caovdilemd.b $e |- ( ph -> B e. S ) $.
|
|
caovdilemd.c $e |- ( ph -> C e. S ) $.
|
|
caovdilemd.d $e |- ( ph -> D e. S ) $.
|
|
caovdilemd.h $e |- ( ph -> H e. S ) $.
|
|
$( Lemma used by real number construction. (Contributed by Jim Kingdon,
|
|
16-Sep-2019.) $)
|
|
caovdilemd $p |- ( ph -> ( ( ( A G C ) F ( B G D ) ) G H ) =
|
|
( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) ) $=
|
|
( co caovcld caovdird caovassd oveq12d eqtrd ) AEGKUBZFHKUBZJUBLKUBUHLKUB
|
|
ZUILKUBZJUBEGLKUBKUBZFHLKUBKUBZJUBABCDUHUILIJKJINABCEGIIIKPQSUCABCFHIIIKP
|
|
RTUCUAUDAUJULUKUMJABCDEGLIKOQSUAUEABCDFHLIKORTUAUEUFUG $.
|
|
|
|
caovdl2.6 $e |- ( ph -> R e. S ) $.
|
|
caovdl2.com $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
caovdl2.ass $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
|
|
caovdl2.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) e. S ) $.
|
|
$( Rearrangement of expression involving multiplication ( ` G ` ) and
|
|
addition ( ` F ` ). (Contributed by Jim Kingdon, 3-Jan-2020.) $)
|
|
caovlem2d $p |- ( ph -> ( ( ( ( A G C ) F ( B G D ) ) G H ) F
|
|
( ( ( A G D ) F ( B G C ) ) G R ) ) =
|
|
( ( A G ( ( C G H ) F ( D G R ) ) ) F
|
|
( B G ( ( C G R ) F ( D G H ) ) ) ) ) $=
|
|
( vr vs vt co caovcld caov42d caovdilemd oveq12d cv wcel w3a wceq caovclg
|
|
wa simpr1 3adantr1 caovcomg syldan 3anrot caovdirg sylan2b eqcomd ancom2s
|
|
jca 3adantr3 3adantr2 3eqtrd caovdid 3eqtr4d ) AEGMLUJZLUJZFHMLUJZLUJZKUJ
|
|
ZEHILUJZLUJZFGILUJZLUJZKUJZKUJVQWBKUJZWDVSKUJZKUJEGLUJFHLUJKUJMLUJZEHLUJF
|
|
GLUJKUJILUJZKUJEVPWAKUJLUJZFWCVRKUJLUJZKUJABCDVQVSWBWDJKABCEVPJJJLQRABCGM
|
|
JJJLQTUBUKZUKABCFVRJJJLQSABCHMJJJLQUAUBUKZUKABCEWAJJJLQRABCHIJJJLQUAUCUKZ
|
|
UKUDUEABCFWCJJJLQSABCGIJJJLQTUCUKZUKUFULAWHVTWIWEKABCDEFGHJKLMNOPQRSTUAUB
|
|
UMABCDEFHGJKLINOPQRSUATUCUMUNAWJWFWKWGKABCDEVPWAJKLKJABUOZJUPZCUOZJUPZDUO
|
|
ZJUPZUQZUTZWPWRWTKUJZLUJZXDWPLUJZWRWPLUJZWTWPLUJZKUJZWPWRLUJZWPWTLUJZKUJA
|
|
XBWQXDJUPZUTXEXFURXCWQXLAWQWSXAVAAWSXAXLWQAUGUHWRWTJJJKABCUGUOZUHUOZJJJKU
|
|
FUSUSVBVJAUGUHWPXDJLABCXMXNJLNVCZVCVDXBAWSXAWQUQXFXIURWQWSXAVEAUGUHUIWRWT
|
|
WPJKLKJABCDXMXNUIUOJKLKJOVFVFVGXCXGXJXHXKKAWQWSXGXJURXAAWQWSUTUTXJXGNVHVK
|
|
AWQXAXHXKURZWSAXAWQXPAUGUHWTWPJLXOVCVIVLUNVMZRWLWNVNABCDFWCVRJKLKJXQSWOWM
|
|
VNUNVO $.
|
|
$}
|
|
|
|
${
|
|
$d A v w x y z $. $d B v w x y $. $d F v w x y z $. $d S v w x y z $.
|
|
caovimo.idel $e |- B e. S $.
|
|
caovimo.com $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) ) $.
|
|
caovimo.ass $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
|
|
( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
|
|
caovimo.id $e |- ( x e. S -> ( x F B ) = x ) $.
|
|
$( Uniqueness of inverse element in commutative, associative operation with
|
|
identity. The identity element is ` B ` . (Contributed by Jim Kingdon,
|
|
18-Sep-2019.) $)
|
|
caovimo $p |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) ) $=
|
|
( vv wcel cv co wceq wa w3a adantl wi wal wmo oveq1 3ad2ant2 df-3an simp1
|
|
simp2 simp3 caovassd caov12d eqtrd adantr oveq2 eqeq12d vtoclga sylan9eqr
|
|
id 3ad2antl2 sylanbr anasss 3adant2r a1i caovcomd 3ad2ant3 3eqtr3d 3expib
|
|
3impa alrimivv eleq1 eqeq1d anbi12d mo4 sylibr ) EGNZDOZGNZEVPHPZFQZRZMOZ
|
|
GNZEWAHPZFQZRZRVPWAQZUAZMUBDUBVTDUCVOWGDMVOVTWEWFVOVTWESVRWAHPZFWAHPZVPWA
|
|
VTVOWHWIQZWEVSWJVQVRFWAHUDTUEVOVQWEWHVPQZVSVOVQWEWKVOVQRZWBWDWKWLWBRVOVQW
|
|
BSZWDWKVOVQWBUFWMWDRWHVPWCHPZVPWMWHWNQWDWMWHEVPWAHPHPWNWMABCEVPWAGHAOZGNZ
|
|
BOZGNZCOZGNSWOWQHPZWSHPWOWQWSHPHPQWMKTZVOVQWBUGZVOVQWBUHZVOVQWBUIZUJWMABC
|
|
EVPWAGHXBXCXDWPWRRZWTWQWOHPQZWMJTXAUKULUMVQVOWDWNVPQWBWDVQWNVPFHPZVPWCFVP
|
|
HUNWOFHPZWOQZXGVPQAVPGWOVPQZXHXGWOVPWOVPFHUDXJURUOLUPUQUSULUTVAVHVBWEVOWI
|
|
WAQZVTWBXKWDWBWIWAFHPZWAWBABFWAGHXEXFWBJTFGNWBIVCWBURVDXIXLWAQAWAGWOWAQZX
|
|
HXLWOWAWOWAFHUDXMURUOLUPULUMVEVFVGVIVTWEDMWFVQWBVSWDVPWAGVJWFVRWCFVPWAEHU
|
|
NVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d n u v w x y z B $. $d n u v w x y z O $. $d n u v w x y z ph $.
|
|
$d u v w y z N $. $d n u v w x y z .+ $. $d u v w y z X $.
|
|
$d u v w y ps $.
|
|
grprinvlem.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B ) $.
|
|
grprinvlem.o $e |- ( ph -> O e. B ) $.
|
|
grprinvlem.i $e |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x ) $.
|
|
grprinvlem.a $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) )
|
|
-> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) $.
|
|
grprinvlem.n $e |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O ) $.
|
|
${
|
|
grprinvlem.x $e |- ( ( ph /\ ps ) -> X e. B ) $.
|
|
grprinvlem.e $e |- ( ( ph /\ ps ) -> ( X .+ X ) = X ) $.
|
|
$( Lemma for ~ grprinvd . (Contributed by NM, 9-Aug-2013.) $)
|
|
grprinvlem $p |- ( ( ph /\ ps ) -> X = O ) $=
|
|
( cv co wceq wcel vu vv vw wa wrex wral ralrimiva oveq2 rexbidv cbvralv
|
|
eqeq1d sylib rspccva syldan oveq2d adantr simprr oveq1d simpll caovassg
|
|
sylan w3a simprl caovassd id eqeq12d rspcv sylc 3eqtr3d rexlimddv ) ABU
|
|
DZDQZIGRZHSZIHSDFABIFTZVNDFUEZOAVLEQZGRZHSZDFUEZEFUFZVOVPAVLCQZGRZHSZDF
|
|
UEZCFUFWAAWECFNUGWEVTCEFWBVQSZWDVSDFWFWCVRHWBVQVLGUHUKUIUJULVTVPEIFVQIS
|
|
ZVSVNDFWGVRVMHVQIVLGUHUKUIUMVAUNVKVLFTZVNUDZUDZVLIIGRZGRZVMIHVKWLVMSWIV
|
|
KWKIVLGPUOUPWJVMIGRHIGRZWLIWJVMHIGVKWHVNUQZURWJUAUBUCVLIIFGWJAUAQZFTUBQ
|
|
ZFTUCQZFTVBWOWPGRWQGRWOWPWQGRGRSABWIUSACDEWOWPWQFGMUTVAVKWHVNVCVKVOWIOU
|
|
PZWRVDVKWMISZWIVKVOHVLGRZVLSZDFUFZWSOAXBBAHWBGRZWBSZCFUFXBAXDCFLUGXDXAC
|
|
DFWBVLSZXCWTWBVLWBVLHGUHXEVEVFUJULUPXAWSDIFVLISZWTWMVLIVLIHGUHXFVEVFVGV
|
|
HUPVIWNVIVJ $.
|
|
$}
|
|
|
|
${
|
|
grprinvd.x $e |- ( ( ph /\ ps ) -> X e. B ) $.
|
|
grprinvd.n $e |- ( ( ph /\ ps ) -> N e. B ) $.
|
|
grprinvd.e $e |- ( ( ph /\ ps ) -> ( N .+ X ) = O ) $.
|
|
$( Deduce right inverse from left inverse and left identity in an
|
|
associative structure (such as a group). (Contributed by NM,
|
|
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) $)
|
|
grprinvd $p |- ( ( ph /\ ps ) -> ( X .+ N ) = O ) $=
|
|
( co wcel vu vv vw wa cv caovclg adantlr caovcld wceq caovassg caovassd
|
|
3expb oveq1d wral ralrimiva oveq2 id eqeq12d cbvralv sylib adantr rspcv
|
|
w3a sylc 3eqtr3d oveq2d eqtrd grprinvlem ) ABCDEFGIJHGSZKLMNOABUDZUAUBJ
|
|
HFFFGAUAUEZFTZUBUEZFTZUDVKVMGSZFTBACDVKVMFFFGACUEZFTDUEZFTVPVQGSFTKULUF
|
|
UGPQUHZVJVIVIGSJHVIGSZGSVIVJUAUBUCJHVIFGAVLVNUCUEZFTVCVOVTGSVKVMVTGSGSU
|
|
IBACDEVKVMVTFGNUJUGZPQVRUKVJVSHJGVJHJGSZHGSIHGSZVSHVJWBIHGRUMVJUAUBUCHJ
|
|
HFGWAQPQUKVJHFTIVQGSZVQUIZDFUNZWCHUIZQAWFBAIVPGSZVPUIZCFUNWFAWICFMUOWIW
|
|
ECDFVPVQUIZWHWDVPVQVPVQIGUPWJUQURUSUTVAWEWGDHFVQHUIZWDWCVQHVQHIGUPWKUQU
|
|
RVBVDVEVFVGVH $.
|
|
$}
|
|
|
|
$( Deduce right identity from left inverse and left identity in an
|
|
associative structure (such as a group). (Contributed by NM,
|
|
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) $)
|
|
grpridd $p |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x ) $=
|
|
( vn vu vv cv wcel wa co wceq vw wrex oveq1 eqeq1d cbvrexv sylib caovassg
|
|
w3a adantlr simprl simprrl caovassd simprrr oveq1d oveq2d 3eqtr3d anassrs
|
|
grprinvd rexlimddv eqtr3d ) ABPZEQZRZGVAFSZVAGFSZVAVCMPZVAFSZGTZVDVETZMEV
|
|
CCPZVAFSZGTZCEUBVHMEUBLVLVHCMEVJVFTVKVGGVJVFVAFUCUDUEUFAVBVFEQZVHRZVIAVBV
|
|
NRZRZVAVFFSZVAFSVAVGFSVDVEVPNOUAVAVFVAEFANPZEQOPZEQUAPZEQUHVRVSFSVTFSVRVS
|
|
VTFSFSTVOABCDVRVSVTEFKUGUIAVBVNUJZAVBVMVHUKZWAULVPVQGVAFAVOBCDEFVFGVAHIJK
|
|
LWAWBAVBVMVHUMZURUNVPVGGVAFWCUOUPUQUSJUT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
"Maps to" notation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C z $.
|
|
elmpt2cl.f $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( If a two-parameter class is not empty, constrain the implicit pair.
|
|
(Contributed by Stefan O'Rear, 7-Mar-2015.) $)
|
|
elmpt2cl $p |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) ) $=
|
|
( vz co wcel cop cxp wa cdm cv wceq coprab cmpt2 df-mpt2 eqtri dmeqi wrel
|
|
dmoprabss eqsstri wfun mpt2fun funrel ax-mp relelfvdm df-ov eleq2s sseldi
|
|
cfv mpan opelxp sylib ) IFGHLZMZFGNZCDOZMFCMGDMPVAHQZVCVBVDARCMBRDMPKRESZ
|
|
PABKTZQVCHVFHABCDEUAVFJABKCDEUBUCUDVEABKCDUFUGVBVDMZIVBHUPZUTHUEZIVHMVGHU
|
|
HVIABCDEHJUIHUJUKIVBHULUQFGHUMUNUOFGCDURUS $.
|
|
|
|
$( If a two-parameter class is not empty, the first argument is in its
|
|
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
|
|
O'Rear, 7-Mar-2015.) $)
|
|
elmpt2cl1 $p |- ( X e. ( S F T ) -> S e. A ) $=
|
|
( co wcel elmpt2cl simpld ) IFGHKLFCLGDLABCDEFGHIJMN $.
|
|
|
|
$( If a two-parameter class is not empty, the second argument is in its
|
|
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
|
|
O'Rear, 7-Mar-2015.) $)
|
|
elmpt2cl2 $p |- ( X e. ( S F T ) -> T e. B ) $=
|
|
( co wcel elmpt2cl simprd ) IFGHKLFCLGDLABCDEFGHIJMN $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B a b $. $d E a b $. $d F a b $. $d X a b $.
|
|
$d Y a b $.
|
|
elovmpt2.d $e |- D = ( a e. A , b e. B |-> C ) $.
|
|
elovmpt2.c $e |- C e. _V $.
|
|
elovmpt2.e $e |- ( ( a = X /\ b = Y ) -> C = E ) $.
|
|
$( Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear,
|
|
21-Jan-2015.) $)
|
|
elovmpt2 $p |- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) ) $=
|
|
( co wcel wa cvv wceq wal cv w3a elmpt2cl gen2 eleq1d spc2gv mpi ovmpt2ga
|
|
mpd3an3 eleq2d biadan2 df-3an bitr4i ) FGHDNZOZGAOZHBOZPZFEOZPUOUPURUAUNU
|
|
QURIJABCGHDFKUBUQUMEFUOUPEQOZUMERUQCQOZJSISUSUTIJLUCUTUSIJGHABITGRJTHRPCE
|
|
QMUDUEUFIJGHABCEDQMKUGUHUIUJUOUPURUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $. $d x y ph $.
|
|
f1od.1 $e |- F = ( x e. A |-> C ) $.
|
|
${
|
|
f1od.2 $e |- ( ( ph /\ x e. A ) -> C e. W ) $.
|
|
f1od.3 $e |- ( ( ph /\ y e. B ) -> D e. X ) $.
|
|
f1od.4 $e |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) $.
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 30-Apr-2015.) $)
|
|
f1ocnvd $p |- ( ph ->
|
|
( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) $=
|
|
( ccnv cmpt wceq wfn wcel copab wf1o wral ralrimiva fnmpt eqid opabbidv
|
|
cv wa df-mpt eqtri cnveqi cnvopab 3eqtr4g fneq1d mpbird dff1o4 sylanbrc
|
|
syl jca ) ADEHUAZHOZCEGPZQAHDRZVAERZUTAFISZBDUBVCAVEBDLUCBDFHIKUDURAVDV
|
|
BERZAGJSZCEUBVFAVGCEMUCCEGVBJVBUEUDURAEVAVBABUGZDSCUGZFQUHZCBTZVIESVHGQ
|
|
UHZCBTVAVBAVJVLCBNUFVAVJBCTZOVKHVMHBDFPVMKBCDFUIUJUKVJBCULUJCBEGUIUMZUN
|
|
UODEHUPUQVNUS $.
|
|
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 12-May-2014.) $)
|
|
f1od $p |- ( ph -> F : A -1-1-onto-> B ) $=
|
|
( wf1o ccnv cmpt wceq f1ocnvd simpld ) ADEHOHPCEGQRABCDEFGHIJKLMNST $.
|
|
$}
|
|
|
|
f1o2d.2 $e |- ( ( ph /\ x e. A ) -> C e. B ) $.
|
|
f1o2d.3 $e |- ( ( ph /\ y e. B ) -> D e. A ) $.
|
|
f1o2d.4 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) ->
|
|
( x = D <-> y = C ) ) $.
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 30-Apr-2015.) $)
|
|
f1ocnv2d $p |- ( ph ->
|
|
( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) $=
|
|
( cv wcel wceq wa wi eleq1a syl impr biimpar exp42 com34 imp32 jcai com23
|
|
biimpa impbida f1ocnvd ) ABCDEFGHEDIJKABMZDNZCMZFOZPZULENZUJGOZPZAUNPUOUP
|
|
AUKUMUOAUKPFENUMUOQJFEULRSTAUKUMUOUPQAUKUOUMUPAUKUOUMUPAUKUOPPZUPUMLUAUBU
|
|
CUDUEAUQPUKUMAUOUPUKAUOPGDNUPUKQKGDUJRSTAUOUPUKUMQAUOUKUPUMAUKUOUPUMQAUKU
|
|
OUPUMURUPUMLUGUBUFUCUDUEUHUI $.
|
|
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 12-May-2014.) $)
|
|
f1o2d $p |- ( ph -> F : A -1-1-onto-> B ) $=
|
|
( wf1o ccnv cmpt wceq f1ocnv2d simpld ) ADEHMHNCEGOPABCDEFGHIJKLQR $.
|
|
$}
|
|
|
|
${
|
|
$d a b A $. $d a b B $. $d a b F $. $d a b ph $.
|
|
f1opw2.1 $e |- ( ph -> F : A -1-1-onto-> B ) $.
|
|
f1opw2.2 $e |- ( ph -> ( `' F " a ) e. _V ) $.
|
|
f1opw2.3 $e |- ( ph -> ( F " b ) e. _V ) $.
|
|
$( A one-to-one mapping induces a one-to-one mapping on power sets. This
|
|
version of ~ f1opw avoids the Axiom of Replacement. (Contributed by
|
|
Mario Carneiro, 26-Jun-2015.) $)
|
|
f1opw2 $p |- ( ph ->
|
|
( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B ) $=
|
|
( cpw cv cima wcel wss crn imassrn wceq syl cvv adantr ccnv cmpt eqid wfo
|
|
wf1o f1ofo forn syl5sseq wb elpwg mpbird cdm dfdm4 f1odm syl5eqr wa elpwi
|
|
adantl foimacnv syl2an eqcomd imaeq2 eqeq2d syl5ibrcom wf1 f1of1 f1imacnv
|
|
impbid f1o2d ) AFEBJZCJZDFKZLZDUAZEKZLZFVJVMUBZVQUCAVMVKMZVLVJMZAVRVMCNZA
|
|
DOZVMCDVLPABCDUDZWACQABCDUEZWBGBCDUFRZBCDUGRUHAVMSMVRVTUIIVMCSUJRUKTAVPVJ
|
|
MZVOVKMZAWEVPBNZAVNOZVPBVNVOPAWHDULZBDUMAWCWIBQGBCDUNRUOUHAVPSMWEWGUIHVPB
|
|
SUJRUKTAVSWFUPZUPZVLVPQZVOVMQZWKWMWLVODVPLZQWKWNVOAWBVOCNZWNVOQWJWDWFWOVS
|
|
VOCUQURBCVODUSUTVAWLVMWNVOVLVPDVBVCVDWKWLWMVLVNVMLZQWKWPVLABCDVEZVLBNZWPV
|
|
LQWJAWCWQGBCDVFRVSWRWFVLBUQTBCVLDVGUTVAWMVPWPVLVOVMVNVBVCVDVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d a b A $. $d a b B $. $d a b F $.
|
|
$( A one-to-one mapping induces a one-to-one mapping on power sets.
|
|
(Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario
|
|
Carneiro, 26-Jun-2015.) $)
|
|
f1opw $p |- ( F : A -1-1-onto-> B -> ( b e. ~P A |-> ( F " b ) ) :
|
|
~P A -1-1-onto-> ~P B ) $=
|
|
( va wf1o id ccnv wfun cv cima cvv wfo dff1o3 simprbi vex funimaex f1ofun
|
|
wcel syl f1opw2 ) ABCFZABCEDUBGUBCHZIZUCEJZKLSUBABCMUDABCNOUCUEEPQTUBCICD
|
|
JZKLSABCRCUFDPQTUA $.
|
|
$}
|
|
|
|
${
|
|
$d ph x $. $d Y x $. $d Z x $.
|
|
suppssfv.a $e |- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L ) $.
|
|
suppssfv.f $e |- ( ph -> ( F ` Y ) = Z ) $.
|
|
suppssfv.v $e |- ( ( ph /\ x e. D ) -> A e. V ) $.
|
|
$( Formula building theorem for support restriction, on a function which
|
|
preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
|
|
suppssfv $p |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) "
|
|
( _V \ { Z } ) ) C_ L ) $=
|
|
( cfv cmpt ccnv cvv csn cdif wcel wceq cima crab wne cv eldifsni elex syl
|
|
wa adantr wi fveq2 eqeq1d syl5ibrcom necon3d imp eldifsn sylanbrc ex syl5
|
|
ss2rabdv eqid mptpreima 3sstr4g sstrd ) ABDCEMZNZOPIQRZUAZBDCNZOPHQRZUAZF
|
|
AVEVGSZBDUBCVJSZBDUBVHVKAVLVMBDVLVEIUCZABUDDSZUHZVMVEPIUEVPVNVMVPVNUHCPSZ
|
|
CHUCZVMVPVQVNVPCGSVQLCGUFUGUIVPVNVRAVNVRUJVOACHVEIAVEITCHTZHEMZITKVSVEVTI
|
|
CHEUKULUMUNUIUOCPHUPUQURUSUTBDVEVGVFVFVAVBBDCVJVIVIVAVBVCJVD $.
|
|
$}
|
|
|
|
${
|
|
$d ph v $. $d ph x $. $d B v $. $d O v $. $d R v $. $d Y v $.
|
|
$d Y x $. $d Z v $. $d Z x $.
|
|
suppssov1.s $e |- ( ph ->
|
|
( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L ) $.
|
|
suppssov1.o $e |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z ) $.
|
|
suppssov1.a $e |- ( ( ph /\ x e. D ) -> A e. V ) $.
|
|
suppssov1.b $e |- ( ( ph /\ x e. D ) -> B e. R ) $.
|
|
$( Formula building theorem for support restrictions: operator with left
|
|
annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
|
|
suppssov1 $p |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) "
|
|
( _V \ { Z } ) ) C_ L ) $=
|
|
( co cvv wcel wceq cmpt ccnv csn cdif cima crab cv wa wne elex syl adantr
|
|
eldifsni wral ralrimiva oveq2 eqeq1d rspcva syl2anc oveq1 syl5ibrcom syl5
|
|
necon3d imp eldifsn sylanbrc ex ss2rabdv eqid mptpreima 3sstr4g sstrd ) A
|
|
BFDEIQZUAZUBRLUCUDZUEZBFDUAZUBRKUCUDZUEZHAVMVOSZBFUFDVRSZBFUFVPVSAVTWABFA
|
|
BUGFSZUHZVTWAWCVTUHDRSZDKUIZWAWCWDVTWCDJSWDODJUJUKULWCVTWEVTVMLUIWCWEVMRL
|
|
UMWCDKVMLWCVMLTDKTZKEIQZLTZWCEGSKCUGZIQZLTZCGUNZWHPAWLWBAWKCGNUOULWKWHCEG
|
|
WIETWJWGLWIEKIUPUQURUSWFVMWGLDKEIUTUQVAVCVBVDDRKVEVFVGVHBFVMVOVNVNVIVJBFD
|
|
VRVQVQVIVJVKMVL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Function operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c oF $.
|
|
$c oR $.
|
|
|
|
$( Extend class notation to include mapping of an operation to a function
|
|
operation. $)
|
|
cof $a class oF R $.
|
|
|
|
$( Extend class notation to include mapping of a binary relation to a
|
|
function relation. $)
|
|
cofr $a class oR R $.
|
|
|
|
${
|
|
$d f g x R $.
|
|
$( Define the function operation map. The definition is designed so that
|
|
if ` R ` is a binary operation, then ` oF R ` is the analogous operation
|
|
on functions which corresponds to applying ` R ` pointwise to the values
|
|
of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.) $)
|
|
df-of $a |- oF R = ( f e. _V , g e. _V |->
|
|
( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) $.
|
|
|
|
$( Define the function relation map. The definition is designed so that if
|
|
` R ` is a binary relation, then ` oF R ` is the analogous relation on
|
|
functions which is true when each element of the left function relates
|
|
to the corresponding element of the right function. (Contributed by
|
|
Mario Carneiro, 28-Jul-2014.) $)
|
|
df-ofr $a |- oR R = { <. f , g >. |
|
|
A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) } $.
|
|
$}
|
|
|
|
${
|
|
$d f g x R $. $d f g x S $.
|
|
$( Equality theorem for function operation. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
ofeq $p |- ( R = S -> oF R = oF S ) $=
|
|
( vf vg vx wceq cvv cv cdm cin cfv co cmpt cmpt2 cof wcel w3a simp1 oveqd
|
|
df-of mpteq2dv mpt2eq3dva 3eqtr4g ) ABFZCDGGECHZIDHZIJZEHZUEKZUHUFKZALZMZ
|
|
NCDGGEUGUIUJBLZMZNAOBOUDCDGGULUNUDUEGPZUFGPZQZEUGUKUMUQABUIUJUDUOUPRSUAUB
|
|
EACDTEBCDTUC $.
|
|
|
|
$( Equality theorem for function relation. (Contributed by Mario Carneiro,
|
|
28-Jul-2014.) $)
|
|
ofreq $p |- ( R = S -> oR R = oR S ) $=
|
|
( vx vf vg wceq cv cfv wbr cdm cin wral cofr breq ralbidv opabbidv df-ofr
|
|
copab 3eqtr4g ) ABFZCGZDGZHZUAEGZHZAIZCUBJUDJKZLZDERUCUEBIZCUGLZDERAMBMTU
|
|
HUJDETUFUICUGUCUEABNOPCADEQCBDEQS $.
|
|
|
|
$( A function operation restricted to a set is a set. (Contributed by NM,
|
|
28-Jul-2014.) $)
|
|
ofexg $p |- ( A e. V -> ( oF R |` A ) e. _V ) $=
|
|
( vf vg vx cof wfun wcel cres cvv cv cdm cin cmpt df-of mpt2fun resfunexg
|
|
cfv co mpan ) BGZHACIUBAJKIDEKKFDLZMELZMNFLZUCSUEUDSBTOUBFBDEPQUBACRUA $.
|
|
|
|
nfof.1 $e |- F/_ x R $.
|
|
$( Hypothesis builder for function operation. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
nfof $p |- F/_ x oF R $=
|
|
( cof nfcv ) ABDE $.
|
|
|
|
$( Hypothesis builder for function relation. (Contributed by Mario
|
|
Carneiro, 28-Jul-2014.) $)
|
|
nfofr $p |- F/_ x oR R $=
|
|
( cofr nfcv ) ABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d f g x F $. $d f g x G $. $d x ph $. $d x S $. $d x X $.
|
|
$d f g x R $.
|
|
offval.1 $e |- ( ph -> F Fn A ) $.
|
|
offval.2 $e |- ( ph -> G Fn B ) $.
|
|
offval.3 $e |- ( ph -> A e. V ) $.
|
|
offval.4 $e |- ( ph -> B e. W ) $.
|
|
offval.5 $e |- ( A i^i B ) = S $.
|
|
${
|
|
offval.6 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = C ) $.
|
|
offval.7 $e |- ( ( ph /\ x e. B ) -> ( G ` x ) = D ) $.
|
|
$( Value of an operation applied to two functions. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
offval $p |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) ) $=
|
|
( cvv vf vg cof co cdm cin cfv cmpt wcel wceq wfn fnex syl2anc fndm syl
|
|
cv ineq12d syl6eq mpteq1d inex1g syl5eqelr mptexg 3syl eqeltrd wa fveq1
|
|
dmeq ineqan12d oveqan12d mpteq12dv df-of ovmpt2ga syl3anc eleq2i bitr3i
|
|
elin adantrr adantrl oveq12d sylan2b mpteq2dva 3eqtrd ) AIJGUCZUDZBIUEZ
|
|
JUEZUFZBUPZIUGZWHJUGZGUDZUHZBHWKUHZBHEFGUDZUHAITUIZJTUIZWLTUIWDWLUJAICU
|
|
KZCKUIZWOMOCKIULUMAJDUKZDLUIWPNPDLJULUMAWLWMTABWGHWKAWGCDUFZHAWECWFDAWQ
|
|
WECUJMCIUNUOAWSWFDUJNDJUNUOUQQURUSZAWRHTUIWMTUIOWRHWTTQCDKUTVABHWKTVBVC
|
|
VDUAUBIJTTBUAUPZUEZUBUPZUEZUFZWHXBUGZWHXDUGZGUDZUHWLWCTXBIUJZXDJUJZVEBX
|
|
FXIWGWKXJXKXCWEXEWFXBIVGXDJVGVHXJXKXGWIXHWJGWHXBIVFWHXDJVFVIVJBGUAUBVKV
|
|
LVMXAABHWKWNWHHUIZAWHCUIZWHDUIZVEZWKWNUJXLWHWTUIXOWTHWHQVNWHCDVPVOAXOVE
|
|
WIEWJFGAXMWIEUJXNRVQAXNWJFUJXMSVRVSVTWAWB $.
|
|
|
|
$( Value of a relation applied to two functions. (Contributed by Mario
|
|
Carneiro, 28-Jul-2014.) $)
|
|
ofrfval $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
|
|
( wcel vf vg cofr wbr cv cfv cdm cin wral cvv wb fnex syl2anc wceq dmeq
|
|
wfn wa ineqan12d fveq1 breqan12d raleqbidv df-ofr brabga ineq12d syl6eq
|
|
fndm raleqdv inss1 eqsstr3i sseli sylan2 inss2 breq12d ralbidva 3bitrd
|
|
syl ) AIJGUCZUDZBUEZIUFZVSJUFZGUDZBIUGZJUGZUHZUIZWBBHUIEFGUDZBHUIAIUJTZ
|
|
JUJTZVRWFUKAICUPZCKTWHMOCKIULUMAJDUPZDLTWINPDLJULUMVSUAUEZUFZVSUBUEZUFZ
|
|
GUDZBWLUGZWNUGZUHZUIWFUAUBIJVQUJUJWLIUNZWNJUNZUQWPWBBWSWEWTXAWQWCWRWDWL
|
|
IUOWNJUOURWTXAWMVTWOWAGVSWLIUSVSWNJUSUTVABGUAUBVBVCUMAWBBWEHAWECDUHZHAW
|
|
CCWDDAWJWCCUNMCIVFVPAWKWDDUNNDJVFVPVDQVEVGAWBWGBHAVSHTZUQVTEWAFGXCAVSCT
|
|
VTEUNHCVSHXBCQCDVHVIVJRVKXCAVSDTWAFUNHDVSHXBDQCDVLVIVJSVKVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
ofval.6 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
|
|
ofval.7 $e |- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) $.
|
|
ofval.8 $e |- ( ph -> R Fn ( U X. V ) ) $.
|
|
ofval.9 $e |- ( ph -> C e. U ) $.
|
|
ofval.10 $e |- ( ph -> D e. V ) $.
|
|
$( Evaluate a function operation at a point. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
fnofval $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
|
|
( vx wcel wa cof co cfv cv cmpt wceq eqidd offval fveq1d adantr cvv cxp
|
|
simpr wfn inss1 eqsstr3i sseli sylan2 eleq1d mpbird inss2 syl3anc fveq2
|
|
cin fnovex oveq12d eqid fvmptg syl2anc 3eqtrd ) AMGUEZUFZMIJFUGUHZUIZMU
|
|
DGUDUJZIUIZWAJUIZFUHZUKZUIZMIUIZMJUIZFUHZDEFUHAVTWFULVQAMVSWEAUDBCWBWCF
|
|
GIJKLNOPQRAWABUEUFWBUMAWACUEUFWCUMUNUOUPVRVQWIUQUEZWFWIULAVQUSVRFHKURUT
|
|
ZWGHUEZWHKUEZWJAWKVQUAUPVRWLDHUEZAWNVQUBUPVRWGDHVQAMBUEWGDULGBMGBCVJZBR
|
|
BCVAVBVCSVDZVEVFVRWMEKUEZAWQVQUCUPVRWHEKVQAMCUEWHEULGCMGWOCRBCVGVBVCTVD
|
|
ZVEVFWGWHHKFVKVHUDMWDWIGUQWEWAMULWBWGWCWHFWAMIVIWAMJVIVLWEVMVNVOVRWGDWH
|
|
EFWPWRVLVP $.
|
|
$}
|
|
|
|
${
|
|
ofrval.6 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
|
|
ofrval.7 $e |- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) $.
|
|
$( Exhibit a function relation at a point. (Contributed by Mario
|
|
Carneiro, 28-Jul-2014.) $)
|
|
ofrval $p |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) $=
|
|
( wcel vx cofr wbr w3a cfv wa cv wral eqidd ofrfval biimpa wceq breq12d
|
|
wi fveq2 rspccv syl 3impia simp1 cin inss1 eqsstr3i simp3 syl2anc inss2
|
|
sseldi 3brtr3d ) AHIFUBUCZLGTZUDZLHUEZLIUEZDEFAVHVIVKVLFUCZAVHUFUAUGZHU
|
|
EZVNIUEZFUCZUAGUHZVIVMUNAVHVRAUABCVOVPFGHIJKMNOPQAVNBTUFVOUIAVNCTUFVPUI
|
|
UJUKVQVMUALGVNLULVOVKVPVLFVNLHUOVNLIUOUMUPUQURVJALBTVKDULAVHVIUSZVJGBLG
|
|
BCUTZBQBCVAVBAVHVIVCZVFRVDVJALCTVLEULVSVJGCLGVTCQBCVEVBWAVFSVDVG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ofmresval.f $e |- ( ph -> F e. A ) $.
|
|
ofmresval.g $e |- ( ph -> G e. B ) $.
|
|
$( Value of a restriction of the function operation map. (Contributed by
|
|
NM, 20-Oct-2014.) $)
|
|
ofmresval $p |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) ) $=
|
|
( wcel cof cxp cres co wceq ovres syl2anc ) AEBIFCIEFDJZBCKLMEFQMNGHEFBCQ
|
|
OP $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z C $. $d y z G $. $d x y z ph $. $d x y S $. $d x y T $.
|
|
$d x y z F $. $d x y z R $. $d x y z U $.
|
|
off.1 $e |- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U ) $.
|
|
off.2 $e |- ( ph -> F : A --> S ) $.
|
|
off.3 $e |- ( ph -> G : B --> T ) $.
|
|
off.4 $e |- ( ph -> A e. V ) $.
|
|
off.5 $e |- ( ph -> B e. W ) $.
|
|
off.6 $e |- ( A i^i B ) = C $.
|
|
$( The function operation produces a function. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
off $p |- ( ph -> ( F oF R G ) : C --> U ) $=
|
|
( vz cof co wf cv cfv cmpt wcel wa wral cin inss1 eqsstr3i ffvelrn syl2an
|
|
sseli inss2 ralrimivva adantr wceq oveq1 eleq1d oveq2 syl21anc eqid fmptd
|
|
rspc2va wfn ffn syl eqidd offval feq1d mpbird ) AFJKLGUBUCZUDFJUAFUAUEZKU
|
|
FZVPLUFZGUCZUGZUDAUAFVSJVTAVPFUHZUIVQHUHZVRIUHZBUEZCUEZGUCZJUHZCIUJBHUJZV
|
|
SJUHZADHKUDZVPDUHZWBWAPFDVPFDEUKZDTDEULUMUPDHVPKUNUOAEILUDZVPEUHZWCWAQFEV
|
|
PFWLETDEUQUMUPEIVPLUNUOAWHWAAWGBCHIOURUSWGWIVQWEGUCZJUHBCVQVRHIWDVQUTWFWO
|
|
JWDVQWEGVAVBWEVRUTWOVSJWEVRVQGVCVBVGVDVTVEVFAFJVOVTAUADEVQVRGFKLMNAWJKDVH
|
|
PDHKVIVJAWMLEVHQEILVIVJRSTAWKUIVQVKAWNUIVRVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x F $. $d x G $. $d x ph $. $d x R $.
|
|
ofres.1 $e |- ( ph -> F Fn A ) $.
|
|
ofres.2 $e |- ( ph -> G Fn B ) $.
|
|
ofres.3 $e |- ( ph -> A e. V ) $.
|
|
ofres.4 $e |- ( ph -> B e. W ) $.
|
|
ofres.5 $e |- ( A i^i B ) = C $.
|
|
$( Restrict the operands of a function operation to the same domain as that
|
|
of the operation itself. (Contributed by Mario Carneiro,
|
|
15-Sep-2014.) $)
|
|
ofres $p |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) ) $=
|
|
( vx co cfv wcel cvv wfn cof cv cmpt cres eqidd offval wss inss1 eqsstr3i
|
|
wa cin fnssres sylancl inss2 ssexg sylancr inidm wceq fvres adantl eqtr4d
|
|
) AFGEUAZPODOUBZFQZVCGQZEPUCFDUDZGDUDZVBPAOBCVDVEEDFGHIJKLMNAVCBRUJVDUEAV
|
|
CCRUJVEUEUFAODDVDVEEDVFVGSSAFBTDBUGZVFDTJDBCUKZBNBCUHUIZBDFULUMAGCTDCUGVG
|
|
DTKDVICNBCUNUICDGULUMAVHBHRDSRVJLDBHUOUPZVKDUQVCDRZVCVFQVDURAVCDFUSUTVLVC
|
|
VGQVEURAVCDGUSUTUFVA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y C $. $d y F $. $d y G $. $d x y ph $.
|
|
$d x y R $.
|
|
offval2.1 $e |- ( ph -> A e. V ) $.
|
|
offval2.2 $e |- ( ( ph /\ x e. A ) -> B e. W ) $.
|
|
offval2.3 $e |- ( ( ph /\ x e. A ) -> C e. X ) $.
|
|
offval2.4 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
|
|
offval2.5 $e |- ( ph -> G = ( x e. A |-> C ) ) $.
|
|
$( The function operation expressed as a mapping. (Contributed by Mario
|
|
Carneiro, 20-Jul-2014.) $)
|
|
offval2 $p |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) ) $=
|
|
( vy co cmpt wceq cof cv cfv wcel wral ralrimiva eqid fnmpt fneq1d mpbird
|
|
wfn inidm wa adantr fveq1d offval nffvmpt1 nfcv nfov fveq2 oveq12d cbvmpt
|
|
syl simpr fvmpt2 syl2anc mpteq2dva syl5eq eqtrd ) AGHFUARQCQUBZBCDSZUCZVJ
|
|
BCESZUCZFRZSZBCDEFRZSZAQCCVLVNFCGHIIAGCUKVKCUKZADJUDZBCUEVSAVTBCMUFBCDVKJ
|
|
VKUGZUHVCACGVKOUIUJAHCUKVMCUKZAEKUDZBCUEWBAWCBCNUFBCEVMKVMUGZUHVCACHVMPUI
|
|
UJLLCULAVJCUDZUMZVJGVKAGVKTWEOUNUOWFVJHVMAHVMTWEPUNUOUPAVPBCBUBZVKUCZWGVM
|
|
UCZFRZSVRQBCVOWJBVLVNFBCDVJUQBFURBCEVJUQUSQWJURVJWGTVLWHVNWIFVJWGVKUTVJWG
|
|
VMUTVAVBABCWJVQAWGCUDZUMZWHDWIEFWLWKVTWHDTAWKVDZMBCDJVKWAVEVFWLWKWCWIETWM
|
|
NBCEKVMWDVEVFVAVGVHVI $.
|
|
|
|
$( The function relation acting on maps. (Contributed by Mario Carneiro,
|
|
20-Jul-2014.) $)
|
|
ofrfval2 $p |- ( ph -> ( F oR R G <-> A. x e. A B R C ) ) $=
|
|
( vy wbr wral wceq cofr cmpt cfv wfn wcel ralrimiva eqid fnmpt syl fneq1d
|
|
cv mpbird inidm wa adantr fveq1d ofrfval nffvmpt1 nfcv nfbr fveq2 breq12d
|
|
nfv cbvral simpr fvmpt2 syl2anc ralbidva syl5bb bitrd ) AGHFUARQUKZBCDUBZ
|
|
UCZVKBCEUBZUCZFRZQCSZDEFRZBCSZAQCCVMVOFCGHIIAGCUDVLCUDZADJUEZBCSVTAWABCMU
|
|
FBCDVLJVLUGZUHUIACGVLOUJULAHCUDVNCUDZAEKUEZBCSWCAWDBCNUFBCEVNKVNUGZUHUIAC
|
|
HVNPUJULLLCUMAVKCUEZUNZVKGVLAGVLTWFOUOUPWGVKHVNAHVNTWFPUOUPUQVQBUKZVLUCZW
|
|
HVNUCZFRZBCSAVSVPWKQBCBVMVOFBCDVKURBFUSBCEVKURUTWKQVCVKWHTVMWIVOWJFVKWHVL
|
|
VAVKWHVNVAVBVDAWKVRBCAWHCUEZUNZWIDWJEFWMWLWAWIDTAWLVEZMBCDJVLWBVFVGWMWLWD
|
|
WJETWNNBCEKVNWEVFVGVBVHVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d ph v x $. $d A x $. $d B v x $. $d D x $. $d O v x $. $d R v $.
|
|
$d Y v x $. $d Z v x $.
|
|
suppssof1.s $e |- ( ph -> ( `' A " ( _V \ { Y } ) ) C_ L ) $.
|
|
suppssof1.o $e |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z ) $.
|
|
suppssof1.a $e |- ( ph -> A : D --> V ) $.
|
|
suppssof1.b $e |- ( ph -> B : D --> R ) $.
|
|
suppssof1.d $e |- ( ph -> D e. W ) $.
|
|
$( Formula building theorem for support restrictions: vector operation with
|
|
left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) $)
|
|
suppssof1 $p |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L ) $=
|
|
( vx ccnv cvv cof co csn cdif cima cv cfv cmpt wf wfn ffn syl inidm eqidd
|
|
wcel wa offval cnveqd imaeq1d feqmptd eqsstr3d funfni ffvelrnda suppssov1
|
|
funfvex sylan eqsstrd ) ACDHUAUBZSZTLUCUDZUERERUFZCUGZVKDUGZHUBUHZSZVJUEG
|
|
AVIVOVJAVHVNAREEVLVMHECDJJAEICUICEUJZOEICUKULZAEFDUIDEUJPEFDUKULQQEUMAVKE
|
|
UOZUPZVLUNVSVMUNUQURUSARBVLVMEFGHTKLAREVLUHZSZTKUCUDZUECSZWBUEGAWCWAWBACV
|
|
TAREICOUTURUSMVANAVPVRVLTUOZVQWDEVKCVKCVEVBVFAEFVKDPVCVDVG $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y C $. $d x y F $. $d x y G $. $d x y H $. $d x y ph $.
|
|
$d x D $. $d x y R $.
|
|
ofco.1 $e |- ( ph -> F Fn A ) $.
|
|
ofco.2 $e |- ( ph -> G Fn B ) $.
|
|
ofco.3 $e |- ( ph -> H : D --> C ) $.
|
|
ofco.4 $e |- ( ph -> A e. V ) $.
|
|
ofco.5 $e |- ( ph -> B e. W ) $.
|
|
ofco.6 $e |- ( ph -> D e. X ) $.
|
|
ofco.7 $e |- ( A i^i B ) = C $.
|
|
$( The composition of a function operation with another function.
|
|
(Contributed by Mario Carneiro, 19-Dec-2014.) $)
|
|
ofco $p |- ( ph ->
|
|
( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) $=
|
|
( cfv vx vy cof co ccom cv cmpt ffvelrnda feqmptd wcel eqidd offval fveq2
|
|
wa oveq12d fmptco wfn wf wss cin inss1 eqsstr3i fss sylancl fnfco syl2anc
|
|
wceq inss2 inidm ffn syl fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGTZ
|
|
VRHTZFUDZUGGIUEZHIUEZVOUDAUAUBEDVRUBUFZGTZWDHTZFUDWAIVPAEDVQIOUHAUAEDIOUI
|
|
AUBBCWEWFFDGHJKMNPQSAWDBUJUNWEUKAWDCUJUNWFUKULWDVRVGWEVSWFVTFWDVRGUMWDVRH
|
|
UMUOUPAUAEEVSVTFEWBWCLLAGBUQEBIURZWBEUQMAEDIURZDBUSWGODBCUTZBSBCVAVBEDBIV
|
|
CVDBEGIVEVFAHCUQECIURZWCEUQNAWHDCUSWJODWICSBCVHVBEDCIVCVDCEHIVEVFRREVIAIE
|
|
UQZVQEUJZVQWBTVSVGAWHWKOEDIVJVKZEGIVQVLVMAWKWLVQWCTVTVGWMEHIVQVLVMULVN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $. $d x G $. $d x H $. $d x ph $. $d x R $.
|
|
offveq.1 $e |- ( ph -> A e. V ) $.
|
|
offveq.2 $e |- ( ph -> F Fn A ) $.
|
|
offveq.3 $e |- ( ph -> G Fn A ) $.
|
|
offveq.4 $e |- ( ph -> H Fn A ) $.
|
|
offveq.5 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = B ) $.
|
|
offveq.6 $e |- ( ( ph /\ x e. A ) -> ( G ` x ) = C ) $.
|
|
|
|
$d x F $.
|
|
$( Equivalent expressions for equality with a function operation.
|
|
(Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro,
|
|
5-Dec-2016.) $)
|
|
offveqb $p |- ( ph
|
|
-> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) ) $=
|
|
( co wceq cmpt wral cof cfv wfn dffn5im syl inidm offval eqeq12d cvv wcel
|
|
cv wb funfvex funfni sylan ralrimiva mpteqb bitrd ) AIGHFUAQZRBCBUKZIUBZS
|
|
ZBCDEFQZSZRZVAVCRBCTZAIVBUSVDAICUCZIVBRNBCIUDUEABCCDEFCGHJJLMKKCUFOPUGUHA
|
|
VAUIUJZBCTVEVFULAVHBCAVGUTCUJVHNVHCUTIUTIUMUNUOUPBCVAVCUIUQUEUR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ph $. $d x R $. $d x W $.
|
|
$d x X $.
|
|
ofc12.1 $e |- ( ph -> A e. V ) $.
|
|
ofc12.2 $e |- ( ph -> B e. W ) $.
|
|
ofc12.3 $e |- ( ph -> C e. X ) $.
|
|
$( Function operation on two constant functions. (Contributed by Mario
|
|
Carneiro, 28-Jul-2014.) $)
|
|
ofc12 $p |- ( ph ->
|
|
( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) ) $=
|
|
( vx csn cxp co cmpt wcel adantr wceq fconstmpt cof a1i offval2 syl6eqr
|
|
cv ) ABCMNZBDMNZEUAOLBCDEOZPBUHMNALBCDEUFUGFGHIACGQLUEBQZJRADHQUIKRUFLBCP
|
|
SALBCTUBUGLBDPSALBDTUBUCLBUHTUD $.
|
|
$}
|
|
|
|
${
|
|
$d w x B $. $d w x y z F $. $d w x y z G $. $d w x y z H $.
|
|
$d w x y z ph $. $d w x y z R $. $d w A $. $d w x y z S $.
|
|
$d w x y z T $. $d w x y z U $.
|
|
caofref.1 $e |- ( ph -> A e. V ) $.
|
|
caofref.2 $e |- ( ph -> F : A --> S ) $.
|
|
${
|
|
caofref.3 $e |- ( ( ph /\ x e. S ) -> x R x ) $.
|
|
$( Transfer a reflexive law to the function relation. (Contributed by
|
|
Mario Carneiro, 28-Jul-2014.) $)
|
|
caofref $p |- ( ph -> F oR R F ) $=
|
|
( vw cofr wbr cv cfv wral wcel wa ffvelrnda ralrimiva adantr id breq12d
|
|
wceq rspcv sylc wf wfn ffn syl inidm eqidd ofrfval mpbird ) AFFDLMKNZFO
|
|
ZUPDMZKCPAUQKCAUOCQZRZUPEQBNZUTDMZBEPZUQACEUOFISAVBURAVABEJTUAVAUQBUPEU
|
|
TUPUDZUTUPUTUPDVCUBZVDUCUEUFTAKCCUPUPDCFFGGACEFUGFCUHICEFUIUJZVEHHCUKUS
|
|
UPULZVFUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d v A $. $d v F $. $d x v N $. $d v S $. $d v ph $. $d v w $.
|
|
caofinv.3 $e |- ( ph -> B e. W ) $.
|
|
caofinv.4 $e |- ( ph -> N : S --> S ) $.
|
|
caofinv.5 $e |- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) ) $.
|
|
${
|
|
caofinvl.6 $e |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B ) $.
|
|
$( Transfer a left inverse law to the function operation. (Contributed
|
|
by NM, 22-Oct-2014.) $)
|
|
caofinvl $p |- ( ph -> ( G oF R F ) = ( A X. { B } ) ) $=
|
|
( vw cfv cof co cmpt csn cv wf wcel wa adantr ffvelrnda ffvelrnd eqid
|
|
cxp fmptd feq1d mpbird wfn wceq wral ralrimiva fneq1d dffn5im feqmptd
|
|
fnmpt offval2 fveq1d simpr fveq2 fveq2d fvmptg syl2anc oveq1d oveq12d
|
|
syl eqtrd id eqeq1d rspcva mpteq2dva fconstmpt syl6eqr ) AIHFUAUBZSDE
|
|
UCZDEUDUMAWBSDSUEZITZWDHTZFUBZUCWCASDWEWFFIHKGGMADGWDIADGIUFDGCDCUEZH
|
|
TZJTZUCZUFACDWJGWKAWHDUGZUHGGWIJAGGJUFZWLPUIADGWHHNUJUKZWKULZUNADGIWK
|
|
QUOUPUJADGWDHNUJZAIDUQZISDWEUCURAWQWKDUQZAWJGUGZCDUSWRAWSCDWNUTCDWJWK
|
|
GWOVDVNADIWKQVAUPSDIVBVNASDGHNVCVEASDWGEAWDDUGZUHZWGWFJTZWFFUBZEXAWEX
|
|
BWFFXAWEWDWKTZXBAWEXDURWTAWDIWKQVFUIXAWTXBGUGXDXBURAWTVGXAGGWFJAWMWTP
|
|
UIWPUKCWDWJXBDGWKWHWDURWIWFJWHWDHVHVIWOVJVKVOVLXAWFGUGBUEZJTZXEFUBZEU
|
|
RZBGUSZXCEURZWPAXIWTAXHBGRUTUIXHXJBWFGXEWFURZXGXCEXKXFXBXEWFFXEWFJVHX
|
|
KVPVMVQVRVKVOVSVOSDEVTWA $.
|
|
$}
|
|
$}
|
|
|
|
caofcom.3 $e |- ( ph -> G : A --> S ) $.
|
|
${
|
|
caofcom.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x R y ) = ( y R x ) ) $.
|
|
$( Transfer a commutative law to the function operation. (Contributed by
|
|
Mario Carneiro, 26-Jul-2014.) $)
|
|
caofcom $p |- ( ph -> ( F oF R G ) = ( G oF R F ) ) $=
|
|
( vw cfv co cmpt wcel wa ffvelrnda cv cof jca caovcomg syldan mpteq2dva
|
|
wceq feqmptd offval2 3eqtr4d ) ANDNUAZGOZUKHOZEPZQNDUMULEPZQGHEUBZPHGUP
|
|
PANDUNUOAUKDRZULFRZUMFRZSUNUOUGAUQSURUSADFUKGKTZADFUKHLTZUCABCULUMFEMUD
|
|
UEUFANDULUMEGHIFFJUTVAANDFGKUHZANDFHLUHZUIANDUMULEHGIFFJVAUTVCVBUIUJ $.
|
|
$}
|
|
|
|
${
|
|
caofrss.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x R y -> x T y ) ) $.
|
|
$( Transfer a relation subset law to the function relation. (Contributed
|
|
by Mario Carneiro, 28-Jul-2014.) $)
|
|
caofrss $p |- ( ph -> ( F oR R G -> F oR T G ) ) $=
|
|
( vw cv wbr wral wcel wi cofr wa ffvelrnda ralrimivva adantr wceq breq1
|
|
cfv imbi12d breq2 rspc2va syl21anc ralimdva wfn ffn inidm eqidd ofrfval
|
|
wf syl 3imtr4d ) AOPZHUHZVBIUHZEQZODRVCVDGQZODRHIEUAQHIGUAQAVEVFODAVBDS
|
|
ZUBZVCFSVDFSBPZCPZEQZVIVJGQZTZCFRBFRZVEVFTZADFVBHLUCADFVBIMUCAVNVGAVMBC
|
|
FFNUDUEVMVOVCVJEQZVCVJGQZTBCVCVDFFVIVCUFVKVPVLVQVIVCVJEUGVIVCVJGUGUIVJV
|
|
DUFVPVEVQVFVJVDVCEUJVJVDVCGUJUIUKULUMAODDVCVDEDHIJJADFHUSHDUNLDFHUOUTZA
|
|
DFIUSIDUNMDFIUOUTZKKDUPZVHVCUQZVHVDUQZURAODDVCVDGDHIJJVRVSKKVTWAWBURVA
|
|
$.
|
|
$}
|
|
|
|
caofass.4 $e |- ( ph -> H : A --> S ) $.
|
|
|
|
${
|
|
caoftrn.5 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x R y /\ y T z ) -> x U z ) ) $.
|
|
$( Transfer a transitivity law to the function relation. (Contributed by
|
|
Mario Carneiro, 28-Jul-2014.) $)
|
|
caoftrn $p |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) ) $=
|
|
( vw wbr cv cfv wa wral cofr wcel wi ralrimivvva adantr ffvelrnda breq1
|
|
wceq anbi1d imbi12d breq2 anbi12d imbi1d anbi2d rspc3v syl3anc ralimdva
|
|
mpd wf wfn ffn syl inidm eqidd ofrfval r19.26 syl6bbr 3imtr4d ) ASUAZJU
|
|
BZVMKUBZFTZVOVMLUBZHTZUCZSEUDZVNVQITZSEUDJKFUETZKLHUETZUCZJLIUETAVSWASE
|
|
AVMEUFZUCZBUAZCUAZFTZWHDUAZHTZUCZWGWJITZUGZDGUDCGUDBGUDZVSWAUGZAWOWEAWN
|
|
BCDGGGRUHUIWFVNGUFVOGUFVQGUFWOWPUGAEGVMJOUJAEGVMKPUJAEGVMLQUJWNWPVNWHFT
|
|
ZWKUCZVNWJITZUGVPVOWJHTZUCZWSUGBCDVNVOVQGGGWGVNULZWLWRWMWSXBWIWQWKWGVNW
|
|
HFUKUMWGVNWJIUKUNWHVOULZWRXAWSXCWQVPWKWTWHVOVNFUOWHVOWJHUKUPUQWJVQULZXA
|
|
VSWSWAXDWTVRVPWJVQVOHUOURWJVQVNIUOUNUSUTVBVAAWDVPSEUDZVRSEUDZUCVTAWBXEW
|
|
CXFASEEVNVOFEJKMMAEGJVCJEVDOEGJVEVFZAEGKVCKEVDPEGKVEVFZNNEVGZWFVNVHZWFV
|
|
OVHZVIASEEVOVQHEKLMMXHAEGLVCLEVDQEGLVEVFZNNXIXKWFVQVHZVIUPVPVRSEVJVKASE
|
|
EVNVQIEJLMMXGXLNNXIXJXMVIVL $.
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Functions (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The restriction of a function to a set exists. Compare Proposition 6.17
|
|
of [TakeutiZaring] p. 28. This version has a shorter proof than
|
|
~ resfunexg but requires ~ ax-pow and ~ ax-un . (Contributed by NM,
|
|
7-Apr-1995.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
resfunexgALT $p |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V ) $=
|
|
( wfun wcel cres cdm cvv crn cxp dmresexg adantl df-ima funimaexg syl5eqelr
|
|
wa cima jca xpexg wss wrel relres relssdmrn ax-mp ssexg mpan 3syl ) ADZBCEZ
|
|
PZABFZGZHEZUKIZHEZPULUNJZHEZUKHEZUJUMUOUIUMUHABCKLUJUNABQHABMABCNORULUNHHSU
|
|
KUPTZUQURUKUAUSABUBUKUCUDUKUPHUEUFUG $.
|
|
|
|
$( Existence of a composition when the first member is a function.
|
|
(Contributed by NM, 8-Oct-2007.) $)
|
|
cofunexg $p |- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V ) $=
|
|
( wfun wcel wa ccom cdm crn cxp wss wrel relco relssdmrn ax-mp dmcoss ssexg
|
|
cvv sylancr rnexg dmexg adantl cres rnco resfunexg sylan2 syl xpexg syl2anc
|
|
syl5eqel ) ADZBCEZFZABGZUNHZUNIZJZKZUQREZUNREUNLURABMUNNOUMUOREZUPREUSULUTU
|
|
KULUOBHZKVAREUTABPBCUAUOVARQSUBUMUPABIZUCZIZRABUDUMVCREZVDREULUKVBREVEBCTAV
|
|
BRUEUFVCRTUGUJUOUPRRUHUIUNUQRQS $.
|
|
|
|
$( Existence of a composition when the second member is one-to-one.
|
|
(Contributed by NM, 8-Oct-2007.) $)
|
|
cofunex2g $p |- ( ( A e. V /\ Fun `' B ) -> ( A o. B ) e. _V ) $=
|
|
( ccnv wfun wcel ccom cvv wa cnvexg cofunexg sylan2 cnvco cocnvcnv2 3eqtrri
|
|
cocnvcnv1 syl5eqel syl ancoms ) BDZEZACFZABGZHFZUAUBITADZGZHFZUDUBUAUEHFUGA
|
|
CJTUEHKLUGUCUFDZHUHUEDZTDGUIBGUCTUEMUIBNABPOUFHJQRS $.
|
|
|
|
$( If the domain of a function is a set, the function is a set. Theorem
|
|
6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom
|
|
of Replacement in the form of ~ funimaexg . This version of ~ fnex uses
|
|
~ ax-pow and ~ ax-un , whereas ~ fnex does not. (Contributed by NM,
|
|
14-Aug-1994.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
fnexALT $p |- ( ( F Fn A /\ A e. B ) -> F e. _V ) $=
|
|
( wfn wcel wa cdm crn cxp wss cvv fnrel relssdmrn syl adantr eleq1d biimpar
|
|
wrel cima syl2anc fndm fnfun funimaexg sylan imadmrn imaeq2d syl5eqr syldan
|
|
wfun xpexg ssexg ) CADZABEZFZCCGZCHZIZJZUQKEZCKEULURUMULCRURACLCMNOUNUOBEZU
|
|
PKEZUSULUTUMULUOABACUAZPQULUMCASZKEZVAULCUIUMVDACUBCABUCUDULVAVDULUPVCKULUP
|
|
CUOSVCCUEULUOACVBUFUGPQUHUOUPBKUJTCUQKUKT $.
|
|
|
|
$( If the domain of a function exists, so does its range. Part of Theorem
|
|
4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of
|
|
Replacement in the form of ~ funex . (Contributed by NM, 11-Nov-1995.) $)
|
|
funrnex $p |- ( dom F e. B -> ( Fun F -> ran F e. _V ) ) $=
|
|
( wfun cdm wcel cvv crn funex ex rnexg syl6com ) BCZBDAEZBFEZBGFELMNABHIBFJ
|
|
K $.
|
|
|
|
$( If the domain of an onto function exists, so does its codomain.
|
|
(Contributed by NM, 23-Jul-2004.) $)
|
|
fornex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $=
|
|
( wfo wcel cvv cdm crn wfun funrnex syl5com wf wceq fof fdm syl eleq1d forn
|
|
fofun 3imtr3d com12 ) ABDEZACFZBGFZUCDHZCFZDIZGFZUDUEUCDJUGUIABDTCDKLUCUFAC
|
|
UCABDMUFANABDOABDPQRUCUHBGABDSRUAUB $.
|
|
|
|
$( If the codomain of a one-to-one function exists, so does its domain. This
|
|
can be thought of as a form of the Axiom of Replacement. (Contributed by
|
|
NM, 4-Sep-2004.) $)
|
|
f1dmex $p |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V ) $=
|
|
( wf1 wcel cvv crn wss wf f1f frn syl ssexg sylan ex ccnv wfo wf1o f1cnv
|
|
f1ofo fornex syl5com syld imp ) ABDEZBCFZAGFZUFUGDHZGFZUHUFUGUJUFUIBIZUGUJU
|
|
FABDJUKABDKABDLMUIBCNOPUFUIADQZRZUJUHUFUIAULSUMABDTUIAULUAMUIAGULUBUCUDUE
|
|
$.
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
abrexex.1 $e |- A e. _V $.
|
|
$( Existence of a class abstraction of existentially restricted sets. ` x `
|
|
is normally a free-variable parameter in the class expression
|
|
substituted for ` B ` , which can be thought of as ` B ( x ) ` . This
|
|
simple-looking theorem is actually quite powerful and appears to involve
|
|
the Axiom of Replacement in an intrinsic way, as can be seen by tracing
|
|
back through the path ~ mptexg , ~ funex , ~ fnex , ~ resfunexg , and
|
|
~ funimaexg . See also ~ abrexex2 . (Contributed by NM, 16-Oct-2003.)
|
|
(Proof shortened by Mario Carneiro, 31-Aug-2015.) $)
|
|
abrexex $p |- { y | E. x e. A y = B } e. _V $=
|
|
( cmpt crn cv wceq wrex cab cvv eqid rnmpt mptex rnex eqeltrri ) ACDFZGBH
|
|
DIACJBKLABCDRRMNRACDEOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( Existence of a class abstraction of existentially restricted sets. ` x `
|
|
is normally a free-variable parameter in ` B ` . The antecedent assures
|
|
us that ` A ` is a set. (Contributed by NM, 3-Nov-2003.) $)
|
|
abrexexg $p |- ( A e. V -> { y | E. x e. A y = B } e. _V ) $=
|
|
( wcel wceq wrex cab cmpt crn cvv eqid rnmpt mptexg rnexg syl syl5eqelr
|
|
cv ) CEFZBSDGACHBIACDJZKZLABCDUAUAMNTUALFUBLFACDEOUALPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The existence of an indexed union. ` x ` is normally a free-variable
|
|
parameter in ` B ` . (Contributed by NM, 23-Mar-2006.) $)
|
|
iunexg $p |- ( ( A e. V /\ A. x e. A B e. W ) -> U_ x e. A B e. _V ) $=
|
|
( vy wcel wral ciun wceq wrex cab cuni cvv dfiun2g adantl abrexexg uniexg
|
|
wa cv syl adantr eqeltrd ) BDGZCEGABHZSABCIZFTCJABKFLZMZNUEUFUHJUDAFBCEOP
|
|
UDUHNGZUEUDUGNGUIAFBCDQUGNRUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d V x y z $. $d W x y z $. $d ph z $.
|
|
$( Existence of an existentially restricted class abstraction.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
abrexex2g $p |- ( ( A e. V /\ A. x e. A { y | ph } e. W )
|
|
-> { y | E. x e. A ph } e. _V ) $=
|
|
( vz wcel cab wral wa wrex cv cvv wsb nfv nfcv nfs1v nfrexxy wceq sbequ12
|
|
rexbidv cbvab df-clab rexbii eqtr4i ciun df-iun iunexg syl5eqelr syl5eqel
|
|
abbii ) DEHACIZFHBDJKZABDLZCIZGMZUMHZBDLZGIZNUPACGOZBDLZGIUTUOVBCGUOGPVAC
|
|
BDCDQACGRSCMUQTAVABDACGUAUBUCUSVBGURVABDAGCUDUEULUFUNUTBDUMUGNBGDUMUHBDUM
|
|
EFUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y v w z $. $d ps v w z $. $d ph x $.
|
|
opabex3d.1 $e |- ( ph -> A e. _V ) $.
|
|
opabex3d.2 $e |- ( ( ph /\ x e. A ) -> { y | ps } e. _V ) $.
|
|
$( Existence of an ordered pair abstraction, deduction version.
|
|
(Contributed by Alexander van der Vekens, 19-Oct-2017.) $)
|
|
opabex3d $p |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V ) $=
|
|
( vz vv vw cv wcel wa cvv wex cop wceq an12 exbii bitri copab csn cab cxp
|
|
ciun 19.42v excom velsn anbi1i vex opeq1 eqeq2d anbi1d ceqsexv nfv nfsab1
|
|
elxp nfan opeq2 wsb sbequ12 equcoms df-clab syl6rbbr anbi12d cbvex 3bitri
|
|
wb anbi2i 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral snexg ax-mp xpexg
|
|
elopab sylancr ralrimiva iunexg syl2anc syl5eqelr ) ACKZELZBMZCDUAZCEWFUB
|
|
ZBDUCZUDZUEZNHWMWIWGHKZWLLZMZCOZWNWFDKZPZQZWHMZDOZCOWNWMLZWNWILWPXBCWGWTB
|
|
MZMZDOWGXDDOZMXBWPWGXDDUFXAXEDWTWGBRSWOXFWGWOWNIKZJKZPZQZXGWJLZXHWKLZMMZJ
|
|
OIOZWNWFXHPZQZXLMZJOZXFIJWNWJWKUQXNXMIOZJOXRXMIJUGXSXQJXSXGWFQZXJXLMZMZIO
|
|
XQXMYBIXMXKYAMYBXJXKXLRXKXTYAIWFUHUITSYAXQIWFCUJZXTXJXPXLXTXIXOWNXGWFXHUK
|
|
ULUMUNTSTXQXDJDXPXLDXPDUOBDJUPURXDJUOXHWRQZXPWTXLBYDXOWSWNXHWRWFUSULYDBBD
|
|
JUTZXLBYEVHDJBDJVAVBBJDVCVDVEVFVGVIVJSXCWOCEVKWQCWNEWLVLWOCEVMTWHCDWNVTVN
|
|
VOAENLWLNLZCEVPWMNLFAYFCEAWGMWJNLZWKNLYFWFNLYGYCWFNVQVRGWJWKNNVSWAWBCEWLN
|
|
NWCWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d A x y v w z $. $d ph v w z $.
|
|
opabex3.1 $e |- A e. _V $.
|
|
opabex3.2 $e |- ( x e. A -> { y | ph } e. _V ) $.
|
|
$( Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
opabex3 $p |- { <. x , y >. | ( x e. A /\ ph ) } e. _V $=
|
|
( vz vv vw cv wcel wa cvv wex cop wceq an12 exbii bitri eqeq2d ciun copab
|
|
csn cab cxp 19.42v excom velsn anbi1i vex opeq1 anbi1d ceqsexv nfv nfsab1
|
|
elxp nfan opeq2 wsb sbequ12 equcoms df-clab syl6rbbr anbi12d cbvex 3bitri
|
|
wb anbi2i 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral snexg ax-mp xpexg
|
|
elopab sylancr rgen iunexg mp2an eqeltrri ) BDBJZUCZACUDZUEZUAZWEDKZALZBC
|
|
UBZMGWIWLWJGJZWHKZLZBNZWMWECJZOZPZWKLZCNZBNWMWIKZWMWLKWOXABWJWSALZLZCNWJX
|
|
CCNZLXAWOWJXCCUFWTXDCWSWJAQRWNXEWJWNWMHJZIJZOZPZXFWFKZXGWGKZLLZINHNZWMWEX
|
|
GOZPZXKLZINZXEHIWMWFWGUPXMXLHNZINXQXLHIUGXRXPIXRXFWEPZXIXKLZLZHNXPXLYAHXL
|
|
XJXTLYAXIXJXKQXJXSXTHWEUHUISRXTXPHWEBUJZXSXIXOXKXSXHXNWMXFWEXGUKTULUMSRSX
|
|
PXCICXOXKCXOCUNACIUOUQXCIUNXGWQPZXOWSXKAYCXNWRWMXGWQWEURTYCAACIUSZXKAYDVG
|
|
CIACIUTVAAICVBVCVDVEVFVHVIRXBWNBDVJWPBWMDWHVKWNBDVLSWKBCWMVSVMVNDMKWHMKZB
|
|
DVOWIMKEYEBDWJWFMKZWGMKYEWEMKYFYBWEMVPVQFWFWGMMVRVTWABDWHMMWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
iunex.1 $e |- A e. _V $.
|
|
iunex.2 $e |- B e. _V $.
|
|
$( The existence of an indexed union. ` x ` is normally a free-variable
|
|
parameter in the class expression substituted for ` B ` , which can be
|
|
read informally as ` B ( x ) ` . (Contributed by NM, 13-Oct-2003.) $)
|
|
iunex $p |- U_ x e. A B e. _V $=
|
|
( cvv wcel wral ciun rgenw iunexg mp2an ) BFGCFGZABHABCIFGDMABEJABCFFKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d z ph $.
|
|
abrexex2.1 $e |- A e. _V $.
|
|
abrexex2.2 $e |- { y | ph } e. _V $.
|
|
$( Existence of an existentially restricted class abstraction. ` ph ` is
|
|
normally has free-variable parameters ` x ` and ` y ` . See also
|
|
~ abrexex . (Contributed by NM, 12-Sep-2004.) $)
|
|
abrexex2 $p |- { y | E. x e. A ph } e. _V $=
|
|
( vz wrex cab cv wcel cvv wsb nfv nfcv nfs1v nfrexxy weq sbequ12 rexbidv
|
|
cbvab df-clab rexbii abbii eqtr4i ciun df-iun iunex eqeltrri eqeltri ) AB
|
|
DHZCIZGJACIZKZBDHZGIZLULACGMZBDHZGIUPUKURCGUKGNUQCBDCDOACGPQCGRAUQBDACGST
|
|
UAUOURGUNUQBDAGCUBUCUDUEBDUMUFUPLBGDUMUGBDUMEFUHUIUJ $.
|
|
|
|
$( Existence of a class abstraction with an existentially quantified
|
|
expression. Both ` x ` and ` y ` can be free in ` ph ` . (Contributed
|
|
by NM, 29-Jul-2006.) $)
|
|
abexssex $p |- { y | E. x ( x C_ A /\ ph ) } e. _V $=
|
|
( cpw wrex cab cv wss wa wex cvv wcel df-rex selpw anbi1i exbii bitri
|
|
abbii pwex abrexex2 eqeltrri ) ABDGZHZCIBJZDKZALZBMZCINUFUJCUFUGUEOZALZBM
|
|
UJABUEPULUIBUKUHABDQRSTUAABCUEDEUBFUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
abexex.1 $e |- A e. _V $.
|
|
abexex.2 $e |- ( ph -> x e. A ) $.
|
|
abexex.3 $e |- { y | ph } e. _V $.
|
|
$( A condition where a class builder continues to exist after its wff is
|
|
existentially quantified. (Contributed by NM, 4-Mar-2007.) $)
|
|
abexex $p |- { y | E. x ph } e. _V $=
|
|
( wrex cab wex cvv cv wcel wa df-rex pm4.71ri exbii bitr4i abbii abrexex2
|
|
eqeltrri ) ABDHZCIABJZCIKUBUCCUBBLDMZANZBJUCABDOAUEBAUDFPQRSABCDEGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d ph x y z $.
|
|
oprabexd.1 $e |- ( ph -> A e. _V ) $.
|
|
oprabexd.2 $e |- ( ph -> B e. _V ) $.
|
|
oprabexd.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps ) $.
|
|
oprabexd.4 $e |- ( ph -> F = { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ps ) } ) $.
|
|
$( Existence of an operator abstraction. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
oprabexd $p |- ( ph -> F e. _V ) $=
|
|
( cv wcel wa coprab cvv wmo wal syl2anc cdm wi ex moanimv sylibr alrimivv
|
|
wfun funoprabg syl cxp wss dmoprabss xpexg ssexg sylancr funex eqeltrd )
|
|
AHCMFNDMGNOZBOZCDEPZQLAUTUGZUTUAZQNZUTQNAUSERZDSCSVAAVDCDAURBERZUBVDAURVE
|
|
KUCURBEUDUEUFUSCDEUHUIAVBFGUJZUKVFQNZVCBCDEFGULAFQNGQNVGIJFGQQUMTVBVFQUNU
|
|
OQUTUPTUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
oprabex.1 $e |- A e. _V $.
|
|
oprabex.2 $e |- B e. _V $.
|
|
oprabex.3 $e |- ( ( x e. A /\ y e. B ) -> E* z ph ) $.
|
|
oprabex.4 $e |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B )
|
|
/\ ph ) } $.
|
|
$( Existence of an operation class abstraction. (Contributed by NM,
|
|
19-Oct-2004.) $)
|
|
oprabex $p |- F e. _V $=
|
|
( cv wcel wa coprab cvv wfun cdm wmo wi mpbir funoprab cxp xpex dmoprabss
|
|
moanimv ssexi funex mp2an eqeltri ) GBLEMCLFMNZANZBCDOZPKUMQUMRZPMUMPMULB
|
|
CDULDSUKADSTJUKADUFUAUBUNEFUCEFHIUDABCDEFUEUGPUMUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f H $. $d x y z R $.
|
|
oprabex3.1 $e |- H e. _V $.
|
|
oprabex3.2 $e |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\
|
|
y e. ( H X. H ) ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = R ) ) } $.
|
|
$( Existence of an operation class abstraction (special case).
|
|
(Contributed by NM, 19-Oct-2004.) $)
|
|
oprabex3 $p |- F e. _V $=
|
|
( cv cop wceq wa wex wmo wcel mosubop cxp xpex anass 2exbii 19.42vv bitri
|
|
moeq mobii mpbir a1i oprabex ) AMZDMEMNOZBMZFMHMNOZPCMGOZPZHQFQZEQDQZABCJ
|
|
JUAZUTIJJKKUBZVAUSCRZULUTSUNUTSPVBUMUOUPPZHQFQZPZEQDQZCRVDCDEULUPCFHUNCGU
|
|
GTTUSVFCURVEDEURUMVCPZHQFQVEUQVGFHUMUOUPUCUDUMVCFHUEUFUDUHUIUJLUK $.
|
|
$}
|
|
|
|
${
|
|
$d A v x y z w $. $d ph v $.
|
|
oprabrexex2.1 $e |- A e. _V $.
|
|
oprabrexex2.2 $e |- { <. <. x , y >. , z >. | ph } e. _V $.
|
|
$( Existence of an existentially restricted operation abstraction.
|
|
(Contributed by Jeff Madsen, 11-Jun-2010.) $)
|
|
oprabrexex2 $p |- { <. <. x , y >. , z >. | E. w e. A ph } e. _V $=
|
|
( vv wrex coprab cv cop wa wex cab cvv df-oprab rexcom4 exbii wceq bitr2i
|
|
r19.42v bitri abbii eqtri eqeltrri abrexex2 eqeltri ) AEFJZBCDKZILBLCLMDL
|
|
MUAZANZDOZCOZBOZEFJZIPZQUKULUJNZDOZCOZBOZIPURUJBCDIRVBUQIUQUOEFJZBOVBUOEB
|
|
FSVCVABVCUNEFJZCOVAUNECFSVDUTCVDUMEFJZDOUTUMEDFSVEUSDULAEFUCTUDTUDTUBUEUF
|
|
UPEIFGABCDKUPIPQABCDIRHUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z B $. $d z C $.
|
|
ab2rexex.1 $e |- A e. _V $.
|
|
ab2rexex.2 $e |- B e. _V $.
|
|
$( Existence of a class abstraction of existentially restricted sets.
|
|
Variables ` x ` and ` y ` are normally free-variable parameters in the
|
|
class expression substituted for ` C ` , which can be thought of as
|
|
` C ( x , y ) ` . See comments for ~ abrexex . (Contributed by NM,
|
|
20-Sep-2011.) $)
|
|
ab2rexex $p |- { z | E. x e. A E. y e. B z = C } e. _V $=
|
|
( cv wceq wrex abrexex abrexex2 ) CIFJBEKACDGBCEFHLM $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z B $.
|
|
ab2rexex2.1 $e |- A e. _V $.
|
|
ab2rexex2.2 $e |- B e. _V $.
|
|
ab2rexex2.3 $e |- { z | ph } e. _V $.
|
|
$( Existence of an existentially restricted class abstraction. ` ph `
|
|
normally has free-variable parameters ` x ` , ` y ` , and ` z ` .
|
|
Compare ~ abrexex2 . (Contributed by NM, 20-Sep-2011.) $)
|
|
ab2rexex2 $p |- { z | E. x e. A E. y e. B ph } e. _V $=
|
|
( wrex abrexex2 ) ACFJBDEGACDFHIKK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d V y $.
|
|
$( The cross product of two sets is a set. Proposition 6.2 of
|
|
[TakeutiZaring] p. 23. This version is proven using Replacement; see
|
|
~ xpexg for a version that uses the Power Set axiom instead.
|
|
(Contributed by Mario Carneiro, 20-May-2013.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
xpexgALT $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V ) $=
|
|
( vy vx wcel wa cxp cv csn ciun iunid xpeq2i xpiundi eqtr3i wral syl5eqel
|
|
cvv id cmpt fconstmpt mptexg ralrimivw iunexg syl2anr ) ACGZBDGZHABIZEBAE
|
|
JZKZIZLZSAEBUKLZIUIUMUNBAEBMNEBUKAOPUHUHULSGZEBQUMSGUGUHTUGUOEBUGULFAUJUA
|
|
SFAUJUBFAUJCUCRUDEBULDSUEUFR $.
|
|
$}
|
|
|
|
${
|
|
$d F x a b $. $d G x a b $. $d V x $. $d W x $. $d R x a b $.
|
|
$d D x $.
|
|
$( General value of ` ( F oF R G ) ` with no assumptions on functionality
|
|
of ` F ` and ` G ` . (Contributed by Stefan O'Rear, 24-Jan-2015.) $)
|
|
offval3 $p |- ( ( F e. V /\ G e. W ) -> ( F oF R G ) =
|
|
( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) ) $=
|
|
( va vb wcel wa cvv cdm cin cv cfv co cmpt wceq elex adantr adantl inex1g
|
|
dmexg mptexg 3syl dmeq ineqan12d fveq1 oveqan12d mpteq12dv df-of ovmpt2ga
|
|
cof syl3anc ) CEIZDFIZJCKIZDKIZACLZDLZMZANZCOZVBDOZBPZQZKIZCDBUMZPVFRUOUQ
|
|
UPCESTUPURUODFSUAUOVGUPUOUSKIVAKIVGCEUCUSUTKUBAVAVEKUDUETGHCDKKAGNZLZHNZL
|
|
ZMZVBVIOZVBVKOZBPZQVFVHKVICRZVKDRZJAVMVPVAVEVQVRVJUSVLUTVICUFVKDUFUGVQVRV
|
|
NVCVOVDBVBVICUHVBVKDUHUIUJABGHUKULUN $.
|
|
|
|
$( Pointwise combination commutes with restriction. (Contributed by Stefan
|
|
O'Rear, 24-Jan-2015.) $)
|
|
offres $p |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) =
|
|
( ( F |` D ) oF R ( G |` D ) ) ) $=
|
|
( vx wcel cdm cin cfv co cmpt cres wceq fvres dmres 3eqtr4ri offval3 cvv
|
|
wa cv cof inss2 sseli oveq12d mpteq2ia inindi incom ineq12i eqid mpteq12i
|
|
syl resmpt3 reseq1d resexg syl2an 3eqtr4a ) CEHZDFHZUAZGCIZDIZJZGUBZCKZVE
|
|
DKZBLZMZANZGCANZIZDANZIZJZVEVKKZVEVMKZBLZMZCDBUCZLZANVKVMVTLZGVDAJZVRMGWC
|
|
VHMVSVJGWCVRVHVEWCHVEAHZVRVHOWCAVEVDAUDUEWDVPVFVQVGBVEACPVEADPUFUMUGGVOVR
|
|
WCVRAVDJAVBJZAVCJZJWCVOAVBVCUHVDAUIVLWEVNWFCAQDAQUJRVRUKULGVDAVHUNRVAWAVI
|
|
AGBCDEFSUOUSVKTHVMTHWBVSOUTCAEUPDAFUPGBVKVMTTSUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d f g A $. $d f g B $. $d f g x R $.
|
|
$( Equivalent expressions for a restriction of the function operation map.
|
|
Unlike ` oF R ` which is a proper class, ` ( oF R | `` ( A X. B ) ) `
|
|
can be a set by ~ ofmresex , allowing it to be used as a function or
|
|
structure argument. By ~ ofmresval , the restricted operation map
|
|
values are the same as the original values, allowing theorems for
|
|
` oF R ` to be reused. (Contributed by NM, 20-Oct-2014.) $)
|
|
ofmres $p |- ( oF R |` ( A X. B ) ) =
|
|
( f e. A , g e. B |-> ( f oF R g ) ) $=
|
|
( vx cvv cv cdm cin cfv co cmpt2 cres wss wceq ssv eqid wcel vex cmpt cxp
|
|
cof resmpt2 mp2an df-of reseq1i dmex inex1 mptex ovmpt4g mp3an mpt2eq123i
|
|
3eqtr4i ) DEGGFDHZIZEHZIZJZFHZUOKUTUQKCLZUAZMZABUBZNZDEABVBMZCUCZVDNDEABU
|
|
OUQVGLZMAGOBGOVEVFPAQBQDEGGABVBUDUEVGVCVDFCDEUFZUGDEABVHABVBARBRUOGSUQGSV
|
|
BGSVHVBPDTZETFUSVAUPURUOVJUHUIUJDEGGVBVGGVIUKULUMUN $.
|
|
|
|
ofmresex.a $e |- ( ph -> A e. V ) $.
|
|
ofmresex.b $e |- ( ph -> B e. W ) $.
|
|
$( Existence of a restriction of the function operation map. (Contributed
|
|
by NM, 20-Oct-2014.) $)
|
|
ofmresex $p |- ( ph -> ( oF R |` ( A X. B ) ) e. _V ) $=
|
|
( cxp cvv wcel cof cres xpexg syl2anc ofexg syl ) ABCIZJKZDLRMJKABEKCFKSG
|
|
HBCEFNORDJPQ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
First and second members of an ordered pair
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c 1st $. $( First member of an ordered pair $)
|
|
$c 2nd $. $( Second member of an ordered pair $)
|
|
|
|
$( Extend the definition of a class to include the first member an ordered
|
|
pair function. $)
|
|
c1st $a class 1st $.
|
|
|
|
$( Extend the definition of a class to include the second member an ordered
|
|
pair function. $)
|
|
c2nd $a class 2nd $.
|
|
|
|
$( Define a function that extracts the first member, or abscissa, of an
|
|
ordered pair. Theorem ~ op1st proves that it does this. For example,
|
|
( ` 1st `` <. ` 3 , 4 ` >. ` ) = 3 . Equivalent to Definition 5.13 (i) of
|
|
[Monk1] p. 52 (compare ~ op1sta and ~ op1stb ). The notation is the same
|
|
as Monk's. (Contributed by NM, 9-Oct-2004.) $)
|
|
df-1st $a |- 1st = ( x e. _V |-> U. dom { x } ) $.
|
|
|
|
$( Define a function that extracts the second member, or ordinate, of an
|
|
ordered pair. Theorem ~ op2nd proves that it does this. For example,
|
|
` ( 2nd `` <. ` 3 , 4 ` >. ` ) = 4 . Equivalent to Definition 5.13 (ii)
|
|
of [Monk1] p. 52 (compare ~ op2nda and ~ op2ndb ). The notation is the
|
|
same as Monk's. (Contributed by NM, 9-Oct-2004.) $)
|
|
df-2nd $a |- 2nd = ( x e. _V |-> U. ran { x } ) $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The value of the function that extracts the first member of an ordered
|
|
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
1stvalg $p |- ( A e. _V -> ( 1st ` A ) = U. dom { A } ) $=
|
|
( vx cvv wcel csn cdm cuni c1st cfv wceq snexg dmexg uniexg 3syl cv dmeqd
|
|
sneq unieqd df-1st fvmptg mpdan ) ACDZAEZFZGZCDZAHIUEJUBUCCDUDCDUFACKUCCL
|
|
UDCMNBABOZEZFZGUECCHUGAJZUIUDUJUHUCUGAQPRBSTUA $.
|
|
|
|
$( The value of the function that extracts the second member of an ordered
|
|
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
2ndvalg $p |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } ) $=
|
|
( vx cvv wcel csn crn cuni c2nd cfv wceq snexg rnexg uniexg 3syl cv rneqd
|
|
sneq unieqd df-2nd fvmptg mpdan ) ACDZAEZFZGZCDZAHIUEJUBUCCDUDCDUFACKUCCL
|
|
UDCMNBABOZEZFZGUECCHUGAJZUIUDUJUHUCUGAQPRBSTUA $.
|
|
$}
|
|
|
|
$( The value of the first-member function at the empty set. (Contributed by
|
|
NM, 23-Apr-2007.) $)
|
|
1st0 $p |- ( 1st ` (/) ) = (/) $=
|
|
( c1st cfv csn cdm cuni cvv wcel wceq 0ex 1stvalg ax-mp dmsn0 unieqi 3eqtri
|
|
c0 uni0 ) OABZOCDZEZOEOOFGQSHIOJKROLMPN $.
|
|
|
|
$( The value of the second-member function at the empty set. (Contributed by
|
|
NM, 23-Apr-2007.) $)
|
|
2nd0 $p |- ( 2nd ` (/) ) = (/) $=
|
|
( c0 c2nd cfv csn crn cuni cvv wcel wceq 0ex 2ndvalg ax-mp cdm dmsn0 dm0rn0
|
|
mpbi unieqi uni0 3eqtri ) ABCZADZEZFZAFAAGHTUCIJAKLUBAUAMAIUBAINUAOPQRS $.
|
|
|
|
${
|
|
op1st.1 $e |- A e. _V $.
|
|
op1st.2 $e |- B e. _V $.
|
|
$( Extract the first member of an ordered pair. (Contributed by NM,
|
|
5-Oct-2004.) $)
|
|
op1st $p |- ( 1st ` <. A , B >. ) = A $=
|
|
( cop c1st cfv csn cdm cuni cvv wcel wceq opexg mp2an 1stvalg ax-mp eqtri
|
|
op1sta ) ABEZFGZTHIJZATKLZUAUBMAKLBKLUCCDABKKNOTPQABCDSR $.
|
|
|
|
$( Extract the second member of an ordered pair. (Contributed by NM,
|
|
5-Oct-2004.) $)
|
|
op2nd $p |- ( 2nd ` <. A , B >. ) = B $=
|
|
( cop c2nd cfv csn crn cuni cvv wcel wceq opexg mp2an 2ndvalg ax-mp eqtri
|
|
op2nda ) ABEZFGZTHIJZBTKLZUAUBMAKLBKLUCCDABKKNOTPQABCDSR $.
|
|
|
|
$( Extract the first member of an ordered pair. (Contributed by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
op1std $p |- ( C = <. A , B >. -> ( 1st ` C ) = A ) $=
|
|
( cop wceq c1st cfv fveq2 op1st syl6eq ) CABFZGCHIMHIACMHJABDEKL $.
|
|
|
|
$( Extract the second member of an ordered pair. (Contributed by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
op2ndd $p |- ( C = <. A , B >. -> ( 2nd ` C ) = B ) $=
|
|
( cop wceq c2nd cfv fveq2 op2nd syl6eq ) CABFZGCHIMHIBCMHJABDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Extract the first member of an ordered pair. (Contributed by NM,
|
|
19-Jul-2005.) $)
|
|
op1stg $p |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A ) $=
|
|
( vx vy cv cop c1st cfv wceq opeq1 fveq2d id eqeq12d opeq2 eqeq1d vtocl2g
|
|
vex op1st ) EGZFGZHZIJZUAKAUBHZIJZAKABHZIJZAKEFABCDUAAKZUDUFUAAUIUCUEIUAA
|
|
UBLMUINOUBBKZUFUHAUJUEUGIUBBAPMQUAUBESFSTR $.
|
|
|
|
$( Extract the second member of an ordered pair. (Contributed by NM,
|
|
19-Jul-2005.) $)
|
|
op2ndg $p |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B ) $=
|
|
( vx vy cv cop c2nd cfv wceq opeq1 fveq2d eqeq1d opeq2 id eqeq12d vtocl2g
|
|
vex op2nd ) EGZFGZHZIJZUBKAUBHZIJZUBKABHZIJZBKEFABCDUAAKZUDUFUBUIUCUEIUAA
|
|
UBLMNUBBKZUFUHUBBUJUEUGIUBBAOMUJPQUAUBESFSTR $.
|
|
|
|
$( Extract the first member of an ordered triple. (Due to infrequent
|
|
usage, it isn't worthwhile at this point to define special extractors
|
|
for triples, so we reuse the ordered pair extractors for ~ ot1stg ,
|
|
~ ot2ndg , ~ ot3rdgg .) (Contributed by NM, 3-Apr-2015.) (Revised by
|
|
Mario Carneiro, 2-May-2015.) $)
|
|
ot1stg $p |- ( ( A e. V /\ B e. W /\ C e. X )
|
|
-> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A ) $=
|
|
( wcel w3a cotp c1st cfv cop df-ot fveq2i wceq wa cvv opexg op1stg sylan
|
|
3impa syl5eq fveq2d 3adant3 eqtrd ) ADGZBEGZCFGZHZABCIZJKZJKABLZJKZAUIUKU
|
|
LJUIUKULCLZJKZULUJUNJABCMNUFUGUHUOULOZUFUGPULQGUHUPABDERULCQFSTUAUBUCUFUG
|
|
UMAOUHABDESUDUE $.
|
|
|
|
$( Extract the second member of an ordered triple. (See ~ ot1stg comment.)
|
|
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
|
|
2-May-2015.) $)
|
|
ot2ndg $p |- ( ( A e. V /\ B e. W /\ C e. X )
|
|
-> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B ) $=
|
|
( wcel w3a cotp c1st cfv c2nd cop df-ot fveq2i wceq wa cvv opexg op1stg
|
|
sylan 3impa syl5eq fveq2d op2ndg 3adant3 eqtrd ) ADGZBEGZCFGZHZABCIZJKZLK
|
|
ABMZLKZBUKUMUNLUKUMUNCMZJKZUNULUPJABCNOUHUIUJUQUNPZUHUIQUNRGUJURABDESUNCR
|
|
FTUAUBUCUDUHUIUOBPUJABDEUEUFUG $.
|
|
|
|
$( Extract the third member of an ordered triple. (See ~ ot1stg comment.)
|
|
(Contributed by NM, 3-Apr-2015.) $)
|
|
ot3rdgg $p |- ( ( A e. V /\ B e. W /\ C e. X )
|
|
-> ( 2nd ` <. A , B , C >. ) = C ) $=
|
|
( wcel w3a cotp c2nd cfv cop df-ot fveq2i wceq wa cvv opexg op2ndg sylan
|
|
3impa syl5eq ) ADGZBEGZCFGZHABCIZJKABLZCLZJKZCUFUHJABCMNUCUDUEUICOZUCUDPU
|
|
GQGUEUJABDERUGCQFSTUAUB $.
|
|
|
|
$( Alternate value of the function that extracts the first member of an
|
|
ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by
|
|
NM, 18-Aug-2006.) $)
|
|
1stval2 $p |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A ) $=
|
|
( vx vy cvv cxp wcel cv cop wceq wex c1st cfv cint elvv vex op1stb eqtr4i
|
|
op1st fveq2 inteq inteqd 3eqtr4a exlimivv sylbi ) ADDEFABGZCGZHZIZCJBJAKL
|
|
ZAMZMZIZBCANUHULBCUHUGKLZUGMZMZUIUKUMUEUOUEUFBOZCOZRUEUFUPUQPQAUGKSUHUJUN
|
|
AUGTUAUBUCUD $.
|
|
|
|
$( Alternate value of the function that extracts the second member of an
|
|
ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by
|
|
NM, 18-Aug-2006.) $)
|
|
2ndval2 $p |- ( A e. ( _V X. _V )
|
|
-> ( 2nd ` A ) = |^| |^| |^| `' { A } ) $=
|
|
( vx vy cvv cxp wcel cv cop wceq wex c2nd cfv ccnv cint elvv op2nd op2ndb
|
|
csn vex inteqd eqtr4i fveq2 sneq cnveqd 3eqtr4a exlimivv sylbi ) ADDEFABG
|
|
ZCGZHZIZCJBJAKLZARZMZNZNZNZIZBCAOUKURBCUKUJKLZUJRZMZNZNZNZULUQUSUIVDUHUIB
|
|
SZCSZPUHUIVEVFQUAAUJKUBUKUPVCUKUOVBUKUNVAUKUMUTAUJUCUDTTTUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( The ` 1st ` function maps the universe onto the universe. (Contributed
|
|
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
fo1st $p |- 1st : _V -onto-> _V $=
|
|
( vx vy cvv c1st wfo wfn crn wceq csn cdm cuni vex snex dmex uniex df-1st
|
|
cv fnmpti wrex wcel cab rnmpt opex op1sta eqcomi sneq dmeqd unieqd eqeq2d
|
|
cop rspcev mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAQZIZJZ
|
|
KZDVAUTUSALMNOAPZRURBQZVBHZACSZBUACABCVBDVCUBVFBCVDCTVFBLZVDVDUJZCTVDVHIZ
|
|
JZKZHZVFVDVDVGVGUCVKVDVDVDVGVGUDUEVEVLAVHCUSVHHZVBVKVDVMVAVJVMUTVIUSVHUFU
|
|
GUHUIUKULUMUNUOCCDUPUQ $.
|
|
|
|
$( The ` 2nd ` function maps the universe onto the universe. (Contributed
|
|
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
fo2nd $p |- 2nd : _V -onto-> _V $=
|
|
( vx vy cvv c2nd wfo wfn crn wceq cv csn cuni vex snex rnex df-2nd fnmpti
|
|
uniex wrex cab wcel rnmpt cop opex op2nda eqcomi sneq rneqd unieqd eqeq2d
|
|
rspcev mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZGZKZDU
|
|
TUSURALMNQAOZPUQBIZVAHZACRZBSCABCVADVBUAVEBCVCCTVEBLZVCVCUBZCTVCVGJZGZKZH
|
|
ZVEVCVCVFVFUCVJVCVCVCVFVFUDUEVDVKAVGCURVGHZVAVJVCVLUTVIVLUSVHURVGUFUGUHUI
|
|
UJUKULUMUNCCDUOUP $.
|
|
|
|
$( Mapping of a restriction of the ` 1st ` (first member of an ordered
|
|
pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario
|
|
Carneiro, 8-Sep-2013.) $)
|
|
f1stres $p |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A $=
|
|
( vx vy vz cv csn cdm cuni wcel cxp wral c1st cres cop vex wceq cvv cmpt
|
|
wf op1sta eleq1i adantr rgen2 sneq dmeqd unieqd eleq1d ralxp mpbir df-1st
|
|
biimpri reseq1i wss ssv resmpt ax-mp eqtri fmpt mpbi ) CFZGZHZIZAJZCABKZL
|
|
ZVFAMVFNZTVGDFZEFZOZGZHZIZAJZEBLDALVODEABVIAJZVOVJBJVOVPVNVIAVIVJDPEPUAUB
|
|
ULUCUDVEVOCDEABVAVKQZVDVNAVQVCVMVQVBVLVAVKUEUFUGUHUIUJCVFAVDVHVHCRVDSZVFN
|
|
ZCVFVDSZMVRVFCUKUMVFRUNVSVTQVFUOCRVFVDUPUQURUSUT $.
|
|
|
|
$( Mapping of a restriction of the ` 2nd ` (second member of an ordered
|
|
pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario
|
|
Carneiro, 8-Sep-2013.) $)
|
|
f2ndres $p |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B $=
|
|
( vx vy vz cv csn crn cuni wcel cxp wral c2nd cres cop vex wceq cvv cmpt
|
|
wf op2nda eleq1i adantl rgen2 sneq rneqd unieqd eleq1d ralxp mpbir df-2nd
|
|
biimpri reseq1i wss ssv resmpt ax-mp eqtri fmpt mpbi ) CFZGZHZIZBJZCABKZL
|
|
ZVFBMVFNZTVGDFZEFZOZGZHZIZBJZEBLDALVODEABVJBJZVOVIAJVOVPVNVJBVIVJDPEPUAUB
|
|
ULUCUDVEVOCDEABVAVKQZVDVNBVQVCVMVQVBVLVAVKUEUFUGUHUIUJCVFBVDVHVHCRVDSZVFN
|
|
ZCVFVDSZMVRVFCUKUMVFRUNVSVTQVFUOCRVFVDUPUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d y B $. $d u v y $. $d u v A $. $d u v B $.
|
|
$( Onto mapping of a restriction of the ` 1st ` (first member of an ordered
|
|
pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) $)
|
|
fo1stresm $p |- ( E. y y e. B ->
|
|
( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A ) $=
|
|
( vv vu cv wcel wex cxp c1st wfo wa wss cfv vex ax-mp sylbir jctil sylibr
|
|
cres eleq1 cbvexv wf crn wceq wi cop opelxp fvres syl6req wfn f1stres ffn
|
|
op1st fnfvelrn mpan eqeltrd expcom exlimiv ssrdv frn eqss dffo2 ) AFZCGZA
|
|
HDFZCGZDHZBCIZBJVITZKZVGVEDAVFVDCUAUBVHVIBVJUCZVJUDZBUEZLVKVHVNVLVHVMBMZB
|
|
VMMZLVNVHVPVOVHEBVMVGEFZBGZVQVMGZUFDVRVGVSVRVGLVQVFUGZVIGZVSVQVFBCUHWAVQV
|
|
TVJNZVMWAWBVTJNVQVTVIJUIVQVFEODOUNUJVJVIUKZWAWBVMGVLWCBCULZVIBVJUMPVIVTVJ
|
|
UOUPUQQURUSUTVLVOWDVIBVJVAPRVMBVBSWDRVIBVJVCSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d u v x $. $d u v A $. $d u v B $.
|
|
$( Onto mapping of a restriction of the ` 2nd ` (second member of an
|
|
ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) $)
|
|
fo2ndresm $p |- ( E. x x e. A ->
|
|
( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B ) $=
|
|
( vu vv cv wcel wex cxp c2nd wfo wa wss cfv vex ax-mp sylbir jctil sylibr
|
|
cres eleq1 cbvexv wf crn wceq wi cop opelxp fvres syl6req wfn f2ndres ffn
|
|
op2nd fnfvelrn mpan eqeltrd ex exlimiv ssrdv frn eqss dffo2 ) AFZBGZAHDFZ
|
|
BGZDHZBCIZCJVITZKZVGVEDAVFVDBUAUBVHVICVJUCZVJUDZCUEZLVKVHVNVLVHVMCMZCVMMZ
|
|
LVNVHVPVOVHECVMVGEFZCGZVQVMGZUFDVGVRVSVGVRLVFVQUGZVIGZVSVFVQBCUHWAVQVTVJN
|
|
ZVMWAWBVTJNVQVTVIJUIVFVQDOEOUNUJVJVIUKZWAWBVMGVLWCBCULZVICVJUMPVIVTVJUOUP
|
|
UQQURUSUTVLVOWDVICVJVAPRVMCVBSWDRVICVJVCSQ $.
|
|
$}
|
|
|
|
$( Composition of the first member function with another function.
|
|
(Contributed by NM, 12-Oct-2007.) $)
|
|
1stcof $p |- ( F : A --> ( B X. C ) -> ( 1st o. F ) : A --> B ) $=
|
|
( cxp wf c1st ccom wfn crn wss cvv wfo fo1st fofn ax-mp ffn dffn2 cres frn
|
|
sylib fnfco sylancr rnco ssres2 rnss 3syl f1stres syl6ss syl5eqss sylanbrc
|
|
df-f ) ABCEZDFZGDHZAIZUOJZBKABUOFUNGLIZALDFZUPLLGMURNLLGOPUNDAIUSAUMDQADRUA
|
|
LAGDUBUCUNUQGDJZSZJZBGDUDUNVBGUMSZJZBUNUTUMKVAVCKVBVDKAUMDTUTUMGUEVAVCUFUGU
|
|
MBVCFVDBKBCUHUMBVCTPUIUJABUOULUK $.
|
|
|
|
$( Composition of the second member function with another function.
|
|
(Contributed by FL, 15-Oct-2012.) $)
|
|
2ndcof $p |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C ) $=
|
|
( cxp wf c2nd ccom wfn crn wss cvv wfo fo2nd fofn ax-mp ffn dffn2 cres frn
|
|
sylib fnfco sylancr rnco ssres2 rnss 3syl f2ndres syl6ss syl5eqss sylanbrc
|
|
df-f ) ABCEZDFZGDHZAIZUOJZCKACUOFUNGLIZALDFZUPLLGMURNLLGOPUNDAIUSAUMDQADRUA
|
|
LAGDUBUCUNUQGDJZSZJZCGDUDUNVBGUMSZJZCUNUTUMKVAVCKVBVDKAUMDTUTUMGUEVAVCUFUGU
|
|
MCVCFVDCKBCUHUMCVCTPUIUJACUOULUK $.
|
|
|
|
${
|
|
$d A b c $. $d B b c $. $d C b c $.
|
|
$( Location of the first element of a Cartesian product. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
xp1st $p |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B ) $=
|
|
( vb vc cxp wcel cv cop wceq wex c1st cfv elxp vex op1std biimpar adantrr
|
|
wa eleq1d exlimivv sylbi ) ABCFGADHZEHZIJZUCBGZUDCGZSSZEKDKALMZBGZDEABCNU
|
|
HUJDEUEUFUJUGUEUJUFUEUIUCBUCUDADOEOPTQRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A b c $. $d B b c $. $d C b c $.
|
|
$( Location of the second element of a Cartesian product. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
xp2nd $p |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C ) $=
|
|
( vb vc cxp wcel cv cop wceq wex c2nd cfv elxp vex op2ndd biimpar adantrl
|
|
wa eleq1d exlimivv sylbi ) ABCFGADHZEHZIJZUCBGZUDCGZSSZEKDKALMZCGZDEABCNU
|
|
HUJDEUEUGUJUFUEUJUGUEUIUDCUCUDADOEOPTQRUAUB $.
|
|
$}
|
|
|
|
$( Existence of the first member of a set. (Contributed by Jim Kingdon,
|
|
26-Jan-2019.) $)
|
|
1stexg $p |- ( A e. V -> ( 1st ` A ) e. _V ) $=
|
|
( wcel cvv c1st cfv elex wfn wfo fo1st fofn ax-mp funfvex funfni mpan syl )
|
|
ABCADCZAEFDCZABGEDHZQRDDEISJDDEKLRDAEAEMNOP $.
|
|
|
|
$( Existence of the first member of a set. (Contributed by Jim Kingdon,
|
|
26-Jan-2019.) $)
|
|
2ndexg $p |- ( A e. V -> ( 2nd ` A ) e. _V ) $=
|
|
( wcel cvv c2nd cfv elex wfn wfo fo2nd fofn ax-mp funfvex funfni mpan syl )
|
|
ABCADCZAEFDCZABGEDHZQRDDEISJDDEKLRDAEAEMNOP $.
|
|
|
|
$( Membership in a cross product. This version requires no quantifiers or
|
|
dummy variables. See also ~ elxp4 . (Contributed by NM, 9-Oct-2004.) $)
|
|
elxp6 $p |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >.
|
|
/\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) $=
|
|
( cxp wcel cvv c1st cfv c2nd wceq wa elex opexg adantl wb eleq1 cuni eleq1d
|
|
cop anbi12d adantr mpbird csn cdm crn 1stvalg 2ndvalg eqeq2d elxp4 syl6rbbr
|
|
opeq12d pm5.21nii ) ABCDZEZAFEZAAGHZAIHZSZJZUPBEZUQCEZKZKZAUMLVCUOURFEZVBVD
|
|
USUPUQBCMNUSUOVDOVBAURFPUAUBUOVCAAUCZUDQZVEUEQZSZJZVFBEZVGCEZKZKUNUOUSVIVBV
|
|
LUOURVHAUOUPVFUQVGAUFZAUGZUKUHUOUTVJVAVKUOUPVFBVMRUOUQVGCVNRTTABCUIUJUL $.
|
|
|
|
$( Membership in a cross product. This version requires no quantifiers or
|
|
dummy variables. See also ~ elxp4 . (Contributed by NM, 19-Aug-2006.) $)
|
|
elxp7 $p |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V )
|
|
/\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) $=
|
|
( cxp wcel cvv c1st cfv c2nd wa elex adantr cop wceq 1stexg 2ndexg biantrud
|
|
jca elxp6 syl6rbbr anbi1d pm5.21nii ) ABCDZEZAFEZAFFDZEZAGHZBEAIHZCEJZJZAUC
|
|
KUGUEUJAUFKLUEUKAUHUIMNZUJJUDUEUGULUJUEULULUHFEZUIFEZJZJUGUEUOULUEUMUNAFOAF
|
|
PRQAFFSTUAABCSTUB $.
|
|
|
|
$( Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.)
|
|
(Revised by Mario Carneiro, 23-Feb-2014.) $)
|
|
eqopi $p |- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) )
|
|
-> A = <. B , C >. ) $=
|
|
( cxp wcel cvv c1st cfv wceq c2nd wa cop xpss sseli simplbi opeq12 sylan9eq
|
|
elxp6 sylan ) ADEFZGAHHFZGZAIJZBKALJZCKMZABCNZKUBUCADEOPUDUGAUEUFNZUHUDAUIK
|
|
UEHGUFHGMAHHTQUEUFBCRSUA $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Representation of cross product based on ordered pair component
|
|
functions. (Contributed by NM, 16-Sep-2006.) $)
|
|
xp2 $p |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\
|
|
( 2nd ` x ) e. B ) } $=
|
|
( cxp cv cvv wcel c1st cfv c2nd wa cab crab elxp7 abbi2i df-rab eqtr4i )
|
|
BCDZAEZFFDZGSHIBGSJICGKZKZALUAATMUBARSBCNOUAATPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( The membership relation for a cross product is inherited by union.
|
|
(Contributed by NM, 16-Sep-2006.) $)
|
|
unielxp $p |- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) ) $=
|
|
( vx cxp wcel cvv c1st cfv c2nd wa cuni elxp7 elvvuni adantr cv cab fveq2
|
|
eleq1d anbi12d wex simprl wceq eleq2 eleq1 spcegv eluniab sylibr crab xp2
|
|
mpcom df-rab eqtri unieqi syl6eleqr mpancom sylbi ) ABCEZFAGGEZFZAHIZBFZA
|
|
JIZCFZKZKZALZURLZFZABCMVGAFZVFVIUTVJVEANOVJVFKZVGDPZUSFZVLHIZBFZVLJIZCFZK
|
|
ZKZDQZLZVHVKVGVLFZVSKZDUAZVGWAFUTVKWDVJUTVEUBWCVKDAUSVLAUCZWBVJVSVFVLAVGU
|
|
DWEVMUTVRVEVLAUSUEWEVOVBVQVDWEVNVABVLAHRSWEVPVCCVLAJRSTTTUFUKVSDVGUGUHURV
|
|
TURVRDUSUIVTDBCUJVRDUSULUMUNUOUPUQ $.
|
|
$}
|
|
|
|
$( Reconstruction of a member of a cross product in terms of its ordered pair
|
|
components. (Contributed by NM, 20-Oct-2013.) $)
|
|
1st2nd2 $p |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
|
|
( cxp wcel c1st cfv c2nd cop wceq wa elxp6 simplbi ) ABCDEAAFGZAHGZIJNBEOCE
|
|
KABCLM $.
|
|
|
|
$( An ordered pair theorem for members of cross products. (Contributed by
|
|
NM, 20-Jun-2007.) $)
|
|
xpopth $p |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) ->
|
|
( ( ( 1st ` A ) = ( 1st ` B ) /\
|
|
( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) ) $=
|
|
( cxp wcel wa wceq c1st cfv c2nd cop 1st2nd2 eqeqan12d wb cvv 1stexg 2ndexg
|
|
opthg syl2anc adantr bitr2d ) ACDGZHZBEFGHZIABJAKLZAMLZNZBKLZBMLZNZJZUHUKJU
|
|
IULJIZUFUGAUJBUMACDOBEFOPUFUNUOQZUGUFUHRHUIRHUPAUESAUETUHUIUKULRRUAUBUCUD
|
|
$.
|
|
|
|
$( Two ways to express equality with an ordered pair. (Contributed by NM,
|
|
3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) $)
|
|
eqop $p |- ( A e. ( V X. W ) -> ( A = <. B , C >.
|
|
<-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) ) $=
|
|
( cxp wcel cop wceq c1st cfv c2nd wa 1st2nd2 eqeq1d cvv 1stexg 2ndexg opthg
|
|
wb syl2anc bitrd ) ADEFZGZABCHZIAJKZALKZHZUEIZUFBIUGCIMZUDAUHUEADENOUDUFPGU
|
|
GPGUIUJTAUCQAUCRUFUGBCPPSUAUB $.
|
|
|
|
${
|
|
eqop2.1 $e |- B e. _V $.
|
|
eqop2.2 $e |- C e. _V $.
|
|
$( Two ways to express equality with an ordered pair. (Contributed by NM,
|
|
25-Feb-2014.) $)
|
|
eqop2 $p |- ( A = <. B , C >.
|
|
<-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) ) $=
|
|
( cop wceq cvv cxp wcel c1st cfv c2nd wa opelvv eleq1 mpbiri eqop biadan2
|
|
) ABCFZGZAHHIZJZAKLBGAMLCGNUAUCTUBJBCDEOATUBPQABCHHRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Two ways of expressing that an element is the first member of an ordered
|
|
pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro,
|
|
23-Feb-2014.) $)
|
|
op1steq $p |- ( A e. ( V X. W )
|
|
-> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) ) $=
|
|
( cxp wcel cvv c1st cfv wceq cv cop wex wb xpss sseli wa c2nd syl eqid wi
|
|
eqopi mpanr2 2ndexg opeq2 eqeq2d spcegv adantr mpd ex eqop syl6bi exlimdv
|
|
simpl impbid ) BDEFZGBHHFZGZBIJCKZBCALZMZKZANZOUQURBDEPQUSUTVDUSUTVDUSUTR
|
|
BCBSJZMZKZVDUSUTVEVEKVGVEUABCVEHHUCUDUSVGVDUBZUTUSVEHGVHBURUEVCVGAVEHVAVE
|
|
KVBVFBVAVECUFUGUHTUIUJUKUSVCUTAUSVCUTVEVAKZRUTBCVAHHULUTVIUOUMUNUPT $.
|
|
$}
|
|
|
|
$( Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) $)
|
|
2nd1st $p |- ( A e. ( B X. C ) ->
|
|
U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. ) $=
|
|
( cxp wcel csn ccnv cuni c1st cfv c2nd cop 1st2nd2 sneqd cnveqd unieqd wceq
|
|
cvv 1stexg 2ndexg opswapg syl2anc eqtrd ) ABCDZEZAFZGZHAIJZAKJZLZFZGZHZUIUH
|
|
LZUEUGULUEUFUKUEAUJABCMNOPUEUHREUIREUMUNQAUDSAUDTUHUIRRUAUBUC $.
|
|
|
|
$( Reconstruction of a member of a relation in terms of its ordered pair
|
|
components. (Contributed by NM, 29-Aug-2006.) $)
|
|
1st2nd $p |- ( ( Rel B /\ A e. B ) ->
|
|
A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
|
|
( wrel wcel wa cvv cxp c1st cfv c2nd cop wceq wss df-rel sylanb 1st2nd2 syl
|
|
ssel2 ) BCZABDZEAFFGZDZAAHIAJIKLSBUAMTUBBNBUAAROAFFPQ $.
|
|
|
|
$( The first ordered pair component of a member of a relation belongs to the
|
|
domain of the relation. (Contributed by NM, 17-Sep-2006.) $)
|
|
1stdm $p |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R ) $=
|
|
( wrel wcel c1st cfv cint cdm cvv cxp wceq wss df-rel biimpi sselda 1stval2
|
|
wa syl elreldm eqeltrd ) BCZABDQZAEFZAGGZBHUBAIIJZDUCUDKUABUEAUABUELBMNOAPR
|
|
BAST $.
|
|
|
|
$( The second ordered pair component of a member of a relation belongs to the
|
|
range of the relation. (Contributed by NM, 17-Sep-2006.) $)
|
|
2ndrn $p |- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R ) $=
|
|
( wrel wcel wa c1st cfv cop crn simpr 1st2nd eqeltrrd cvv 1stexg 2ndexg jca
|
|
c2nd opelrng 3expa sylan syl2anc ) BCZABDZEZUCAFGZAQGZHZBDZUFBIDZUBUCJZUDAU
|
|
GBABKUJLUCUEMDZUFMDZEUHUIUCUKULABNABOPUKULUHUIUEUFBMMRSTUA $.
|
|
|
|
$( Express an element of a relation as a relationship between first and
|
|
second components. (Contributed by Mario Carneiro, 22-Jun-2016.) $)
|
|
1st2ndbr $p |- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) ) $=
|
|
( wrel wcel wa c1st cfv c2nd cop wbr 1st2nd simpr eqeltrrd df-br sylibr ) B
|
|
CZABDZEZAFGZAHGZIZBDSTBJRAUABABKPQLMSTBNO $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Two ways of expressing membership in the domain of a relation.
|
|
(Contributed by NM, 22-Sep-2013.) $)
|
|
releldm2 $p |- ( Rel A
|
|
-> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) ) $=
|
|
( vy wrel cdm wcel cv c1st cfv wceq wrex cvv wa elex anim2i id vex wex wb
|
|
1stexg ax-mp syl6eqelr rexlimivw cop eldm2g adantl cxp wss wi df-rel ssel
|
|
sylbi imp op1steq syl rexbidva adantr rexcom4 risset bitr4i syl6bb bitr4d
|
|
exbii pm5.21nd ) BEZCBFZGZAHZIJZCKZABLZVFCMGZNZVHVMVFCVGOPVLVMVFVKVMABVKC
|
|
VJMVKQVIMGVJMGARVIMUAUBUCUDPVNVHCDHUEZBGZDSZVLVMVHVQTVFDCBMUFUGVNVLVIVOKZ
|
|
DSZABLZVQVFVLVTTVMVFVKVSABVFVIBGZNVIMMUHZGZVKVSTVFWAWCVFBWBUIWAWCUJBUKBWB
|
|
VIULUMUNDVICMMUOUPUQURVTVRABLZDSVQVRADBUSVPWDDAVOBUTVDVAVBVCVE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( An expression for the domain of a relation. (Contributed by NM,
|
|
22-Sep-2013.) $)
|
|
reldm $p |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) ) $=
|
|
( vy vz wrel cdm c1st cfv cmpt crn wcel wceq wrex releldm2 cvv vex 1stexg
|
|
cv wb ax-mp wfn eqid fnmpti fvelrnb fveq2 fvmpt rexbiia a1i syl5rbb bitrd
|
|
eqeq1d eqrdv ) BEZCBFZABARZGHZIZJZUMCRZUNKDRZGHZUSLZDBMZUSURKZDBUSNVDUTUQ
|
|
HZUSLZDBMZUMVCUQBUAVDVGSABUPUQUOOKUPOKAPUOOQTUQUBZUCDBUSUQUDTVGVCSUMVFVBD
|
|
BUTBKVEVAUSAUTUPVABUQUOUTGUEVHUTOKVAOKDPUTOQTUFUKUGUHUIUJUL $.
|
|
$}
|
|
|
|
$( Equality theorem for substitution of a class for an ordered pair (analog
|
|
of ~ sbceq1a that avoids the existential quantifiers of ~ copsexg ).
|
|
(Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
sbcopeq1a $p |- ( A = <. x , y >. ->
|
|
( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) ) $=
|
|
( cv cop wceq c2nd cfv wsbc c1st wb vex op2ndd eqcomd sbceq1a op1std bitr2d
|
|
syl ) DBEZCEZFGZAACDHIZJZUDBDKIZJZUBUAUCGAUDLUBUCUATUADBMZCMZNOACUCPSUBTUEG
|
|
UDUFLUBUETTUADUGUHQOUDBUEPSR $.
|
|
|
|
$( Equality theorem for substitution of a class ` A ` for an ordered pair
|
|
` <. x , y >. ` in ` B ` (analog of ~ csbeq1a ). (Contributed by NM,
|
|
19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
csbopeq1a $p |- ( A = <. x , y >. ->
|
|
[_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B ) $=
|
|
( cv cop wceq c2nd cfv csb c1st vex op2ndd eqcomd csbeq1a syl op1std eqtr2d
|
|
) CAEZBEZFGZDBCHIZDJZACKIZUCJZUATUBGDUCGUAUBTSTCALZBLZMNBUBDOPUASUDGUCUEGUA
|
|
UDSSTCUFUGQNAUDUCOPR $.
|
|
|
|
${
|
|
$d z ph $. $d x y z $.
|
|
$( A way to define an ordered-pair class abstraction without using
|
|
existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
dfopab2 $p |- { <. x , y >. | ph } = { z e. ( _V X. _V ) |
|
|
[. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph } $=
|
|
( cv cop wceq wex cab cvv cxp wcel c2nd cfv wsbc c1st nfsbc1v 19.41 exbii
|
|
wa copab crab sbcopeq1a pm5.32i nfsbc bitr3i anbi1i 3bitr4i abbii df-opab
|
|
nfcv elvv df-rab 3eqtr4i ) DEZBECEFGZATZCHZBHZDIUOJJKZLZACUOMNZOZBUOPNZOZ
|
|
TZDIABCUAVEDUTUBUSVFDUPCHZVETZBHVGBHZVETUSVFVGVEBVCBVDQRURVHBURUPVETZCHVH
|
|
VJUQCUPVEAABCUOUCUDSUPVECVCCBVDCVDUKACVBQUERUFSVAVIVEBCUOULUGUHUIABCDUJVE
|
|
DUTUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d w ph $. $d x y z w $.
|
|
$( A way to define an operation class abstraction without using existential
|
|
quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
dfoprab3s $p |- { <. <. x , y >. , z >. | ph } = { <. w , z >. |
|
|
( w e. ( _V X. _V )
|
|
/\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) } $=
|
|
( coprab cv cop wceq wex copab cvv cxp wcel cfv wsbc nfsbc1v 19.41 exbii
|
|
wa c2nd c1st dfoprab2 sbcopeq1a pm5.32i nfcv nfsbc bitr3i 3bitr4i opabbii
|
|
elvv anbi1i eqtri ) ABCDFEGZBGCGHIZATZCJZBJZEDKUNLLMNZACUNUAOZPZBUNUBOZPZ
|
|
TZEDKABCDEUCURVDEDUOCJZVCTZBJVEBJZVCTURVDVEVCBVABVBQRUQVFBUQUOVCTZCJVFVHU
|
|
PCUOVCAABCUNUDUESUOVCCVACBVBCVBUFACUTQUGRUHSUSVGVCBCUNUKULUIUJUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d w ps $. $d x y z w $.
|
|
dfoprab3.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
|
|
$( Operation class abstraction expressed without existential quantifiers.
|
|
(Contributed by NM, 16-Dec-2008.) $)
|
|
dfoprab3 $p |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } =
|
|
{ <. <. x , y >. , z >. | ps } $=
|
|
( coprab cv cvv cxp wcel cfv wsbc wa copab ax-mp wceq wb eqcom vex 1stexg
|
|
c2nd c1st dfoprab3s 2ndexg cop anbi12i eqopi sylan2b syl sbc2iedv pm5.32i
|
|
bicomd ex opabbii eqtr2i ) BCDEHFIZJJKLZBDURUCMZNCURUDMZNZOZFEPUSAOZFEPBC
|
|
DEFUEVCVDFEUSVBAUSBACDVAUTURJLZVAJLFUAZURJUBQVEUTJLVFURJUFQUSCIZVARZDIZUT
|
|
RZOZBASUSVKOZABVLURVGVIUGRZABSVKUSVAVGRZUTVIRZOVMVHVNVJVOVGVATVIUTTUHURVG
|
|
VIJJUIUJGUKUNUOULUMUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d w x y A $. $d w x y B $. $d x y ph $. $d w ps $. $d w x y z $.
|
|
dfoprab4.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
|
|
$( Operation class abstraction expressed without existential quantifiers.
|
|
(Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
dfoprab4 $p |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } $=
|
|
( cv cxp wcel wa copab cvv coprab xpss sseli adantr pm4.71ri opabbii wceq
|
|
cop eleq1 opelxp syl6bb anbi12d dfoprab3 eqtri ) FJZGHKZLZAMZFENUJOOKZLZU
|
|
MMZFENCJZGLDJZHLMZBMZCDEPUMUPFEUMUOULUOAUKUNUJGHQRSTUAUMUTCDEFUJUQURUCZUB
|
|
ZULUSABVBULVAUKLUSUJVAUKUDUQURGHUEUFIUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d t u w x y z $. $d t u w x y A $. $d t u w x y B $. $d t u w ps $.
|
|
$d t u ph $.
|
|
dfoprab4f.x $e |- F/ x ph $.
|
|
dfoprab4f.y $e |- F/ y ph $.
|
|
dfoprab4f.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $.
|
|
$( Operation class abstraction expressed without existential quantifiers.
|
|
(Unnecessary distinct variable restrictions were removed by David
|
|
Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
dfoprab4f $p |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } $=
|
|
( vt vu cv wcel wa cop wceq wb nfv cxp copab coprab nfs1v nfbi nfim opeq1
|
|
wsb wi eqeq2d sbequ12 bibi2d imbi12d opeq2 chvar dfoprab4 nfan nfsb eleq1
|
|
bi2anan9 sylan9bbr anbi12d cbvoprab12 eqtr4i ) FNZGHUAOAPFEUBLNZGOZMNZHOZ
|
|
PZBDMUHZCLUHZPZLMEUCCNZGOZDNZHOZPZBPZCDEUCAVLLMEFGHVEVNVHQZRZAVKSZUIZVEVF
|
|
VHQZRZAVLSZUICLWEWFCWECTAVLCIVKCLUDZUEUFVNVFRZWAWEWBWFWHVTWDVEVNVFVHUGUJW
|
|
HVKVLAVKCLUKZULUMVEVNVPQZRZABSZUIWCDMWAWBDWADTAVKDJBDMUDZUEUFVPVHRZWKWAWL
|
|
WBWNWJVTVEVPVHVNUNUJWNBVKABDMUKZULUMKUOUOUPVSVMCDELMVSLTVSMTVJVLCVJCTWGUQ
|
|
VJVLDVJDTVKCLDWMURUQWHWNPVRVJBVLWHVOVGWNVQVIVNVFGUSVPVHHUSUTWNBVKWHVLWOWI
|
|
VAVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z u A $. $d x y z u B $. $d x y z u C $.
|
|
$( Define the cross product of three classes. Compare ~ df-xp .
|
|
(Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro,
|
|
3-Nov-2015.) $)
|
|
dfxp3 $p |- ( ( A X. B ) X. C ) =
|
|
{ <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) } $=
|
|
( vu cv cxp wcel wa copab coprab w3a cop wceq biidd dfoprab4 df-xp df-3an
|
|
oprabbii 3eqtr4i ) GHZDEIZJCHFJZKGCLAHZDJZBHZEJZKUEKZABCMUDFIUGUIUENZABCM
|
|
UEUEABCGDEUCUFUHOPUEQRGCUDFSUKUJABCUGUIUETUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y ch $.
|
|
elopabi.1 $e |- ( x = ( 1st ` A ) -> ( ph <-> ps ) ) $.
|
|
elopabi.2 $e |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) ) $.
|
|
$( A consequence of membership in an ordered-pair class abstraction, using
|
|
ordered pair extractors. (Contributed by NM, 29-Aug-2006.) $)
|
|
elopabi $p |- ( A e. { <. x , y >. | ph } -> ch ) $=
|
|
( copab wcel c1st cfv c2nd cop wrel wceq relopab 1st2nd mpan cvv eqeltrrd
|
|
id wb 1stexg 2ndexg opelopabg syl2anc mpbid ) FADEIZJZFKLZFMLZNZUIJZCUJFU
|
|
MUIUIOUJFUMPADEQFUIRSUJUBUAUJUKTJULTJUNCUCFUIUDFUIUEABCDEUKULTTGHUFUGUH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w ph $. $d x y z th $.
|
|
eloprabi.1 $e |- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) ) $.
|
|
eloprabi.2 $e |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) ) $.
|
|
eloprabi.3 $e |- ( z = ( 2nd ` A ) -> ( ch <-> th ) ) $.
|
|
$( A consequence of membership in an operation class abstraction, using
|
|
ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by
|
|
David Abernethy, 19-Jun-2012.) $)
|
|
eloprabi $p |- ( A e. { <. <. x , y >. , z >. | ph } -> th ) $=
|
|
( vw cv wceq wex c1st cfv wb syl c2nd coprab wcel wa eqeq1 anbi1d 3exbidv
|
|
cop df-oprab elab2g ibi vex opex op1std fveq2d op1st syl6req op2nd op2ndd
|
|
eqcomd 3bitrd biimpa exlimiv ) HAEFGUAZUBZHEMZFMZUGZGMZUGZNZAUCZGOZFOZEOZ
|
|
DVDVNLMZVINZAUCZGOFOEOVNLHVCVCVOHNZVQVKEFGVRVPVJAVOHVIUDUEUFAEFGLUHUIUJVM
|
|
DEVLDFVKDGVJADVJABCDVJVEHPQZPQZNABRVJVTVGPQVEVJVSVGPVGVHHVEVFEUKZFUKZULZG
|
|
UKZUMZUNVEVFWAWBUOUPISVJVFVSTQZNBCRVJWFVGTQVFVJVSVGTWEUNVEVFWAWBUQUPJSVJV
|
|
HHTQZNCDRVJWGVHVGVHHWCWDURUSKSUTVAVBVBVBS $.
|
|
$}
|
|
|
|
${
|
|
$d u v x y z A $. $d u v y z B $. $d u v z C $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) $)
|
|
mpt2mptsx $p |- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B )
|
|
|-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) $=
|
|
( vu vv cv csn csb cxp ciun cfv cmpt cmpt2 wceq nfcv nfcsb1v csbeq1a c1st
|
|
c2nd cop vex op1std csbeq1d op2ndd csbeq2dv eqtrd mpt2mptx xpeq12d cbviun
|
|
nfxp sneq mpteq1 ax-mp nfcsb sylan9eqr cbvmpt2x 3eqtr4ri ) CGDGIZJZAVAEKZ
|
|
LZMZACIZUANZBVFUBNZFKZKZOZGHDVCAVABHIZFKZKZPCADAIZJZELZMZVJOZABDEFPGHCDVC
|
|
VJVNVFVAVLUCQZVJAVAVIKVNVTAVGVAVIVAVLVFGUDZHUDZUEUFVTAVAVIVMVTBVHVLFVAVLV
|
|
FWAWBUGUFUHUIUJVRVEQVSVKQAGDVQVDGVQRAVBVCAVBRAVAESZUMVOVAQZVPVBEVCVOVAUNA
|
|
VAETZUKULCVRVEVJUOUPABGHDEFVCVNGERWCGFRHFRAVAVMSBAVAVMBVARBVLFSUQWEBIVLQW
|
|
DFVMVNBVLFTAVAVMTURUSUT $.
|
|
|
|
$d x B $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) $)
|
|
mpt2mpts $p |- ( x e. A , y e. B |-> C ) =
|
|
( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) $=
|
|
( cv csn cxp ciun c1st cfv c2nd csb cmpt mpt2mptsx wceq iunxpconst mpteq1
|
|
cmpt2 ax-mp eqtri ) ABDEFTCADAGHEIJZACGZKLBUDMLFNNZOZCDEIZUEOZABCDEFPUCUG
|
|
QUFUHQADERCUCUGUESUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d t u v w x y z A $. $d t u v w y z B $. $d t u v w z C $.
|
|
$d v w x y z D $.
|
|
fmpt2x.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( The domain of a mapping is a subset of its base class. (Contributed by
|
|
Mario Carneiro, 9-Feb-2015.) $)
|
|
dmmpt2ssx $p |- dom F C_ U_ x e. A ( { x } X. B ) $=
|
|
( vu vt vv cv csn csb cxp ciun cfv nfcv nfcsb1v csbeq1a wceq cdm cbvmpt2x
|
|
c1st c2nd cmpt2 cmpt nfcsb sylan9eqr cop vex op1std op2ndd csbeq2dv eqtrd
|
|
csbeq1d mpt2mptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtr4i ) FUAH
|
|
CHKZLZAVDDMZNZOZACAKZLZDNZOIVHAIKZUCPZBVLUDPZEMZMZFABCDEUEHJCVFAVDBJKZEMZ
|
|
MZUEFIVHVPUFABHJCDEVFVSHDQAVDDRZHEQJEQAVDVRRBAVDVRBVDQBVQERUGAVDDSZBKVQTV
|
|
IVDTZEVRVSBVQESAVDVRSUHUBGHJICVFVPVSVLVDVQUITZVPAVDVOMVSWCAVMVDVOVDVQVLHU
|
|
JZJUJZUKUOWCAVDVOVRWCBVNVQEVDVQVLWDWEULUOUMUNUPUQURAHCVKVGHVKQAVEVFAVEQVT
|
|
USWBVJVEDVFVIVDUTWAVAVBVC $.
|
|
|
|
$( Functionality, domain and codomain of a class given by the "maps to"
|
|
notation, where ` B ( x ) ` is not constant but depends on ` x ` .
|
|
(Contributed by NM, 29-Dec-2014.) $)
|
|
fmpt2x $p |- ( A. x e. A A. y e. B C e. D <->
|
|
F : U_ x e. A ( { x } X. B ) --> D ) $=
|
|
( vz vw vv cv csb wcel wral wceq eleq1d wa nfv nfcsb1v csn ciun c1st c2nd
|
|
cxp wf cfv cop op1std csbeq1d op2ndd csbeq2dv eqtrd raliunxp cmpt2 coprab
|
|
cmpt nfcri nfan nfeq2 nfcv nfcsb wb eleq1 adantr csbeq1a eleq2d sylan9bbr
|
|
anbi12d sylan9eqr eqeq2d cbvoprab12 df-mpt2 3eqtr4i mpt2mptx bitr3i nfel1
|
|
vex fmpt nfralxy cbvral raleqbidv syl5bb nfxp sneq xpeq12d cbviun 3bitr4i
|
|
feq2i ) AILZBJLZEMZMZFNZJAWJDMZOZICOZICWJUAZWOUEZUBZFGUFZEFNZBDOZACOACALZ
|
|
UAZDUEZUBZFGUFWQAKLZUCUGZBXHUDUGZEMZMZFNZKWTOXAXMWNKIJCWOXHWJWKUHPZXLWMFX
|
|
NXLAWJXKMWMXNAXIWJXKWJWKXHIVRZJVRZUIUJXNAWJXKWLXNBXJWKEWJWKXHXOXPUKUJULUM
|
|
ZQUNKWTFXLGABCDEUOZIJCWOWMUOZGKWTXLUQXDCNZBLZDNZRZXHEPZRZABKUPWJCNZWKWONZ
|
|
RZXHWMPZRZIJKUPXRXSYEYJABKIJYEISYEJSYHYIAYFYGAYFASAJWOAWJDTZURUSAXHWMAWJW
|
|
LTZUTUSYHYIBYHBSBXHWMBAWJWLBWJVABWKETZVBUTUSXDWJPZYAWKPZRZYCYHYDYIYPXTYFY
|
|
BYGYNXTYFVCYOXDWJCVDVEYOYBWKDNYNYGYAWKDVDYNDWOWKAWJDVFZVGVHVIYPEWMXHYOYNE
|
|
WLWMBWKEVFZAWJWLVFZVJVKVIVLABKCDEVMIJKCWOWMVMVNHIJKCWOXLWMXQVOVNVSVPXCWPA
|
|
ICXCISWNAJWOYKAWMFYLVQVTXCWLFNZJDOYNWPXBYTBJDXBJSBWLFYMVQYOEWLFYRQWAYNYTW
|
|
NJDWOYQYNWLWMFYSQWBWCWAXGWTFGAICXFWSIXFVAAWRWOAWRVAYKWDYNXEWRDWOXDWJWEYQW
|
|
FWGWIWH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C z $. $d D x y z $.
|
|
fmpt2.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Functionality, domain and range of a class given by the "maps to"
|
|
notation. (Contributed by FL, 17-May-2010.) $)
|
|
fmpt2 $p |- ( A. x e. A A. y e. B C e. D <-> F : ( A X. B ) --> D ) $=
|
|
( wcel wral cv csn cxp ciun wf fmpt2x iunxpconst feq2i bitri ) EFIBDJACJA
|
|
CAKLDMNZFGOCDMZFGOABCDEFGHPTUAFGACDQRS $.
|
|
|
|
$( Functionality and domain of a class given by the "maps to" notation.
|
|
(Contributed by FL, 17-May-2010.) $)
|
|
fnmpt2 $p |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) ) $=
|
|
( wcel wral cvv cxp wfn elex ralimi wf fmpt2 dffn2 bitr4i sylib ) EGIZBDJ
|
|
ZACJEKIZBDJZACJZFCDLZMZUBUDACUAUCBDEGNOOUEUFKFPUGABCDEKFHQUFFRST $.
|
|
|
|
$d R z $. $d S z $. $d V z $.
|
|
$( Sufficient condition for an operation maps-to notation to be set-like.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
mpt2fvex $p |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) ->
|
|
( R F S ) e. _V ) $=
|
|
( vz wcel wal cfv cvv cv csb alimi w3a co cop df-ov c1st c2nd elex wi vex
|
|
2ndexg nfcv nfcsb1v nfel1 wceq csbeq1a eleq1d spcgf mp2b alrimiv 3ad2ant1
|
|
syl opexg 3adant1 csn cxp ciun cmpt2 cmpt mpt2mptsx eqtri mptfvex syl2anc
|
|
1stexg syl5eqel ) EINZBOZAOZFJNZGKNZUAZFGHUBFGUCZHPZQFGHUDVTAMRZUEPZBWCUF
|
|
PZESZSZQNZMOZWAQNZWBQNVQVRWIVSVQWHMVQWFQNZAOZWHVPWKAVPEQNZBOZWKVOWMBEIUGT
|
|
WCQNZWEQNWNWKUHMUIZWCQUJWMWKBWEQBWEUKBWFQBWEEULUMBRWEUNEWFQBWEEUOUPUQURVA
|
|
TWOWDQNWLWHUHWPWCQVMWKWHAWDQAWDUKAWGQAWDWFULUMARZWDUNWFWGQAWDWFUOUPUQURVA
|
|
USUTVRVSWJVQFGJKVBVCMACWQVDDVEVFZWGWAHQQHABCDEVGMWRWGVHLABMCDEVIVJVKVLVN
|
|
$.
|
|
|
|
fnmpt2i.2 $e |- C e. _V $.
|
|
$( Functionality and domain of a class given by the "maps to" notation.
|
|
(Contributed by FL, 17-May-2010.) $)
|
|
fnmpt2i $p |- F Fn ( A X. B ) $=
|
|
( cvv wcel wral cxp wfn rgen2w fnmpt2 ax-mp ) EIJZBDKACKFCDLMQABCDHNABCDE
|
|
FIGOP $.
|
|
|
|
$( Domain of a class given by the "maps to" notation. (Contributed by FL,
|
|
17-May-2010.) $)
|
|
dmmpt2 $p |- dom F = ( A X. B ) $=
|
|
( cxp wfn cdm wceq fnmpt2i fndm ax-mp ) FCDIZJFKPLABCDEFGHMPFNO $.
|
|
|
|
mpt2fvexi.3 $e |- R e. _V $.
|
|
mpt2fvexi.4 $e |- S e. _V $.
|
|
$( Sufficient condition for an operation maps-to notation to be set-like.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
mpt2fvexi $p |- ( R F S ) e. _V $=
|
|
( cvv wcel wal co gen2 mpt2fvex mp3an ) EMNZBOAOFMNGMNFGHPMNTABJQKLABCDEF
|
|
GHMMMIRS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B y $.
|
|
mpt2exg.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Existence of an operation class abstraction (version for dependent
|
|
domains). (Contributed by Mario Carneiro, 30-Dec-2016.) $)
|
|
mpt2exxg $p |- ( ( A e. R /\ A. x e. A B e. S ) -> F e. _V ) $=
|
|
( wcel wral wa wfun cdm cvv mpt2fun cv csn cxp sylancr ciun wss dmmpt2ssx
|
|
vex snexg ax-mp xpexg mpan ralimi iunexg sylan2 ssexg funex ) CFJZDGJZACK
|
|
ZLZHMHNZOJZHOJABCDEHIPUQURACAQZRZDSZUAZUBVCOJZUSABCDEHIUCUPUNVBOJZACKVDUO
|
|
VEACVAOJZUOVEUTOJVFAUDUTOUEUFVADOGUGUHUIACVBFOUJUKURVCOULTOHUMT $.
|
|
|
|
$d x B $.
|
|
$( Existence of an operation class abstraction (special case).
|
|
(Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
mpt2exg $p |- ( ( A e. R /\ B e. S ) -> F e. _V ) $=
|
|
( wcel cvv wral elex ralrimivw syl mpt2exxg sylan2 ) DGJZCFJDKJZACLZHKJRS
|
|
TDGMSSACDKMNOABCDEFKHIPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( If the domain of a function given by maps-to notation is a set, the
|
|
function is a set. (Contributed by NM, 12-Sep-2011.) $)
|
|
mpt2exga $p |- ( ( A e. V /\ B e. W )
|
|
-> ( x e. A , y e. B |-> C ) e. _V ) $=
|
|
( cmpt2 eqid mpt2exg ) ABCDEFGABCDEHZKIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
mpt2ex.1 $e |- A e. _V $.
|
|
mpt2ex.2 $e |- B e. _V $.
|
|
$( If the domain of a function given by maps-to notation is a set, the
|
|
function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) $)
|
|
mpt2ex $p |- ( x e. A , y e. B |-> C ) e. _V $=
|
|
( cvv wcel wral cmpt2 rgenw eqid mpt2exxg mp2an ) CHIDHIZACJABCDEKZHIFPAC
|
|
GLABCDEHHQQMNO $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x y B $. $d u w x y z C $. $d x y ph $. $d u v w x y S $.
|
|
$d u v w x y A $. $d u v w z R $. $d z T $.
|
|
fmpt2co.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C ) $.
|
|
fmpt2co.2 $e |- ( ph -> F = ( x e. A , y e. B |-> R ) ) $.
|
|
fmpt2co.3 $e |- ( ph -> G = ( z e. C |-> S ) ) $.
|
|
fmpt2co.4 $e |- ( z = R -> S = T ) $.
|
|
$( Composition of two functions. Variation of ~ fmptco when the second
|
|
function has two arguments. (Contributed by Mario Carneiro,
|
|
8-Feb-2015.) $)
|
|
fmpt2co $p |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) ) $=
|
|
( vw vu vv csb ccom cxp cv c2nd c1st cmpt cmpt2 wcel wral ralrimivva eqid
|
|
cfv wf fmpt2 sylib nfcv nfcsb1v weq csbeq1a sylan9eq cbvmpt2 cop wceq vex
|
|
nfcsb op2ndd csbeq1d op1std csbeq2dv mpt2mpt eqtr4i sylibr syl6eq fmptcos
|
|
eqtrd fmpt wa w3a 3impb nfcvd csbiegf syl mpt2eq3dva syl5eq ) ALKUAQEFUBZ
|
|
DCQUCZUDULZBWFUEULZHTZTZITZUFZBCEFJUGZAQDWEGWJIKLAWEGBCEFHUGZUMZWJGUHQWEU
|
|
IAHGUHZCFUIBEUIWOAWPBCEFMUJBCEFHGWNWNUKUNUOQWEGWJWNWNRSEFCSUCZBRUCZHTZTZU
|
|
GQWEWJUFZBCRSEFHWTRHUPSHUPBCWQWSBWQUPBWRHUQVEZCWQWSUQZBRURZCSURZHWSWTBWRH
|
|
USCWQWSUSUTZVARSQEFWJWTWFWRWQVBVCZWJCWQWITWTXGCWGWQWIWRWQWFRVDZSVDZVFVGXG
|
|
CWQWIWSXGBWHWRHWRWQWFXHXIVHVGVIVOZVJVKZVPVLAKWNXANXKVMOVNAWLBCEFDHITZUGZW
|
|
MWLRSEFDWTITZUGXMRSQEFWKXNXGDWJWTIXJVGVJBCRSEFXLXNRXLUPSXLUPBDWTIXBBIUPVE
|
|
CDWTIXCCIUPVEXDXEVQDHWTIXFVGVAVKABCEFXLJABUCEUHZCUCFUHZVRWPXLJVCAXOXPWPMV
|
|
SDHIJGWPDJVTPWAWBWCWDVO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z D $. $d x y z H $. $d z C $.
|
|
oprabco.1 $e |- ( ( x e. A /\ y e. B ) -> C e. D ) $.
|
|
oprabco.2 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
oprabco.3 $e |- G = ( x e. A , y e. B |-> ( H ` C ) ) $.
|
|
$( Composition of a function with an operator abstraction. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro,
|
|
26-Sep-2015.) $)
|
|
oprabco $p |- ( H Fn D -> G = ( H o. F ) ) $=
|
|
( vz wfn ccom cfv cmpt2 cv wcel wa adantl wceq a1i dffn5im fveq2 syl6reqr
|
|
fmpt2co ) IFNZIGOABCDEIPZQHUHABMCDFEMRZIPUIGIARCSBRDSTEFSUHJUAGABCDEQUBUH
|
|
KUCMFIUDUJEIUEUGLUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y M $. $d x y R $. $d x y S $.
|
|
oprab2co.1 $e |- ( ( x e. A /\ y e. B ) -> C e. R ) $.
|
|
oprab2co.2 $e |- ( ( x e. A /\ y e. B ) -> D e. S ) $.
|
|
oprab2co.3 $e |- F = ( x e. A , y e. B |-> <. C , D >. ) $.
|
|
oprab2co.4 $e |- G = ( x e. A , y e. B |-> ( C M D ) ) $.
|
|
$( Composition of operator abstractions. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) $)
|
|
oprab2co $p |- ( M Fn ( R X. S ) -> G = ( M o. F ) ) $=
|
|
( cop cxp cv wcel cmpt2 wa opelxpi syl2anc cfv wceq df-ov mpt2eq3ia eqtri
|
|
co a1i oprabco ) ABCDEFPZGHQZIJKARCSBRDSUAZEGSFHSULUMSLMEFGHUBUCNJABCDEFK
|
|
UIZTABCDULKUDZTOABCDUOUPUOUPUEUNEFKUFUJUGUHUK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $.
|
|
$( An alternate possible definition of the ` 1st ` function. (Contributed
|
|
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
df1st2 $p |- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) ) $=
|
|
( vw c1st cvv cxp cres cv cfv wceq copab wcel wa coprab wfo wfn fo1st vex
|
|
cmpt fofn dffn5im mp2b mptv eqtri reseq1i resopab op1std dfoprab3 3eqtrri
|
|
cop eqeq2d ) EFFGZHCIZDIZEJZKZDCLZUMHUOUMMUQNDCLUNAIZKZABCOEURUMEDFUPTZUR
|
|
FFEPEFQEVAKRFFEUADFEUBUCDCUPUDUEUFUQDCUMUGUQUTABCDUOUSBIZUKKUPUSUNUSVBUOA
|
|
SBSUHULUIUJ $.
|
|
|
|
$( An alternate possible definition of the ` 2nd ` function. (Contributed
|
|
by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
df2nd2 $p |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) ) $=
|
|
( vw c2nd cvv cxp cres cv cfv wceq copab wcel wa coprab wfo wfn fo2nd vex
|
|
cmpt fofn dffn5im mp2b mptv eqtri reseq1i resopab op2ndd dfoprab3 3eqtrri
|
|
cop eqeq2d ) EFFGZHCIZDIZEJZKZDCLZUMHUOUMMUQNDCLUNBIZKZABCOEURUMEDFUPTZUR
|
|
FFEPEFQEVAKRFFEUADFEUBUCDCUPUDUEUFUQDCUMUGUQUTABCDUOAIZUSUKKUPUSUNVBUSUOA
|
|
SBSUHULUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y V $.
|
|
$( The mapping of a restriction of the ` 1st ` function to a constant
|
|
function. (Contributed by NM, 14-Dec-2008.) $)
|
|
1stconst $p |- ( B e. V ->
|
|
( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A ) $=
|
|
( vx vy wcel csn cxp c1st wfo cv wbr wmo wceq wa vex cfv cvv jca syl2anc
|
|
cres ccnv wfun wf1o wex snmg fo1stresm syl wal cop moeq moani brres fo1st
|
|
wfn wb fofn ax-mp fnbrfvb mp2an anbi1i elxp7 eleq1 biimpa adantrr adantrl
|
|
c2nd elsni eqopi an12s sylanr2 adantrrl adantl simprr fveq2d simprl simpl
|
|
sylan2b op1stg eqtrd adantr opelxpi eqeltrd impbida syl5bbr syl5bb mobidv
|
|
snidg mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) BCFZABGZHZAIWQUAZJZ
|
|
WRUBUCZWQAWRUDWODKZWPFDUEWSDBCUFDAWPUGUHWOXAEKZWRLZDMZEUIWTWOXDEWOXDXBAFZ
|
|
XAXBBUJZNZOZDMXGXEDDXFUKULWOXCXHDXCXAXBILZXAWQFZOZWOXHXAXBIWQEPUMXKXAIQZX
|
|
BNZXJOZWOXHXMXIXJIRUOZXARFXMXIUPRRIJXOUNRRIUQURDPRXAXBIUSUTVAWOXNXHXNXHWO
|
|
XJXMXARRHFZXLAFZXAVGQZWPFZOZOZXHXAAWPVBXMYAOXEXGXMXTXEXPXMXQXEXSXMXQXEXLX
|
|
BAVCVDVEVFXMXPXSXGXQXSXMXPXRBNZXGXRBVHXPXMYBXGXAXBBRRVIVJVKVLSVRVMWOXHOZX
|
|
MXJYCXLXFIQZXBYCXAXFIWOXEXGVNZVOYCXEWOYDXBNWOXEXGVPZWOXHVQXBBACVSTVTYCXAX
|
|
FWQYEYCXEBWPFZXFWQFYFWOYGXHBCWHWAXBBAWPWBTWCSWDWEWFWGWIWJDEWRWKWLWQAWRWMW
|
|
N $.
|
|
|
|
$( The mapping of a restriction of the ` 2nd ` function to a converse
|
|
constant function. (Contributed by NM, 27-Mar-2008.) $)
|
|
2ndconst $p |- ( A e. V ->
|
|
( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B ) $=
|
|
( vx vy wcel csn cxp c2nd wfo cv wbr wmo wceq wa vex cfv cvv adantrl jca
|
|
cres ccnv wfun wf1o wex snmg fo2ndresm syl wal cop moeq moani brres fo2nd
|
|
wfn fofn ax-mp fnbrfvb mp2an anbi1i c1st elxp7 eleq1 biimpa elsni ancom2s
|
|
eqopi an12s sylanr2 adantrrr sylan2b adantl fveq2 op2ndg sylan9eqr simprr
|
|
mpan2 adantr simprl opelxpi syl2anc eqeltrd impbida syl5bbr syl5bb mobidv
|
|
wb snidg mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) ACFZAGZBHZBIWQUA
|
|
ZJZWRUBUCZWQBWRUDWODKZWPFDUEWSDACUFDWPBUGUHWOXAEKZWRLZDMZEUIWTWOXDEWOXDXB
|
|
BFZXAAXBUJZNZOZDMXGXEDDXFUKULWOXCXHDXCXAXBILZXAWQFZOZWOXHXAXBIWQEPZUMXKXA
|
|
IQZXBNZXJOZWOXHXNXIXJIRUOZXARFXNXIWGRRIJXPUNRRIUPUQDPRXAXBIURUSUTWOXOXHXO
|
|
XHWOXJXNXARRHFZXAVAQZWPFZXMBFZOZOZXHXAWPBVBXNYBOXEXGXNYAXEXQXNXTXEXSXNXTX
|
|
EXMXBBVCVDSSXNXQXSXGXTXSXNXQXRANZXGXRAVEXQXNYCXGXQYCXNXGXAAXBRRVGVFVHVIVJ
|
|
TVKVLWOXHOZXNXJWOXGXNXEXGWOXMXFIQZXBXAXFIVMWOXBRFYEXBNXLAXBCRVNVQVOSYDXAX
|
|
FWQWOXEXGVPYDAWPFZXEXFWQFWOYFXHACWHVRWOXEXGVSAXBWPBVTWAWBTWCWDWEWFWIWJDEW
|
|
RWKWLWQBWRWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d w x y A $. $d w x y B $. $d w C $.
|
|
dfmpt2.1 $e |- C e. _V $.
|
|
$( Alternate definition for the "maps to" notation ~ df-mpt2 (although it
|
|
requires that ` C ` be a set). (Contributed by NM, 19-Dec-2008.)
|
|
(Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
dfmpt2 $p |- ( x e. A , y e. B |-> C )
|
|
= U_ x e. A U_ y e. B { <. <. x , y >. , C >. } $=
|
|
( vw cv cfv csb cop csn ciun cvv wcel ax-mp csbexa nfcv nfcsb1v nfop c1st
|
|
cmpt2 cxp c2nd cmpt mpt2mpts vex 1stexg 2ndexg dfmpt nfsn nfcsb csbopeq1a
|
|
wceq id opeq12d sneqd iunxpf 3eqtri ) ABCDEUBGCDUCZAGHZUAIZBVAUDIZEJZJZUE
|
|
GUTVAVEKZLZMACBDAHBHKZEKZLZMMABGCDEUFGUTVEAVBVDVANOZVBNOGUGZVANUHPBVCEVKV
|
|
CNOVLVANUIPFQQUJGABCDVGVJAVFAVAVEAVARAVBVDSTUKBVFBVAVEBVARBAVBVDBVBRBVCES
|
|
ULTUKGVJRVAVHUNZVFVIVMVAVHVEEVMUOABVAEUMUPUQURUS $.
|
|
$}
|
|
|
|
$( Lemma for ~ cnvf1o . (Contributed by Mario Carneiro, 27-Apr-2014.) $)
|
|
cnvf1olem $p |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) ->
|
|
( C e. `' A /\ B = U. `' { C } ) ) $=
|
|
( wrel wcel csn ccnv cuni wceq c2nd cfv cop sneqd cnveqd unieqd cvv opswapg
|
|
wa syl2anc ad2antrl c1st simprr 1st2nd 1stexg 2ndexg 3eqtrd simprl eqeltrrd
|
|
adantrr wb opelcnvg mpbird eqeltrd eqcomd 3eqtr4d jca ) ADZBAEZCBFZGZHZIZRR
|
|
ZCAGZEBCFZGZHZIVCCBJKZBUAKZLZVDVCCVAVIVHLZFZGZHZVJUQURVBUBVCUTVMVCUSVLVCBVK
|
|
UQURBVKIVBBAUCUIZMNOURVNVJIZUQVBURVIPEZVHPEZVPBAUDZBAUEZVIVHPPQSTUFZVCVJVDE
|
|
ZVKAEZVCBVKAVOUQURVBUGUHURWBWCUJZUQVBURVRVQWDVTVSVHVIPPAUKSTULUMVCVKVJFZGZH
|
|
ZBVGURVKWGIUQVBURWGVKURVRVQWGVKIVTVSVHVIPPQSUNTVOVCVFWFVCVEWEVCCVJWAMNOUOUP
|
|
$.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Describe a function that maps the elements of a set to its converse
|
|
bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) $)
|
|
cnvf1o $p |- ( Rel A ->
|
|
( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A ) $=
|
|
( vy wrel ccnv cv csn cuni cmpt wcel snexg cnvexg uniexg 3syl adantl wceq
|
|
cvv wa cnvf1olem wb relcnv simpr sylancr dfrel2 eleq2 sylbi anbi1d adantr
|
|
eqid mpbid impbida f1od ) BDZACBBEZAFZGZEZHZCFZGZEZHZABURIZQQVCUIUOBJZURQ
|
|
JZUMVDUPQJUQQJVEUOBKUPQLUQQMNOUSUNJZVBQJZUMVFUTQJVAQJVGUSUNKUTQLVAQMNOUMV
|
|
DUSURPZRZVFUOVBPRZBUOUSSUMVJRZUOUNEZJZVHRZVIVKUNDVJVNBUAUMVJUBUNUSUOSUCUM
|
|
VNVITVJUMVMVDVHUMVLBPVMVDTBUDVLBUOUEUFUGUHUJUKUL $.
|
|
$}
|
|
|
|
$( The ` 2nd ` (second member of an ordered pair) function restricted to a
|
|
function ` F ` is a function of ` F ` into the codomain of ` F ` .
|
|
(Contributed by Alexander van der Vekens, 4-Feb-2018.) $)
|
|
f2ndf $p |- ( F : A --> B -> ( 2nd |` F ) : F --> B ) $=
|
|
( wf c2nd cres cxp wss f2ndres fssxp fssres sylancr wceq resabs1 syl eqcomd
|
|
feq1d mpbird ) ABCDZCBECFZDCBEABGZFZCFZDZSUABUBDCUAHZUDABIABCJZUABCUBKLSCBT
|
|
UCSUCTSUEUCTMUFECUANOPQR $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $.
|
|
$( The ` 2nd ` (second member of an ordered pair) function restricted to a
|
|
function ` F ` is a function of ` F ` onto the range of ` F ` .
|
|
(Contributed by Alexander van der Vekens, 4-Feb-2018.) $)
|
|
fo2ndf $p |- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F ) $=
|
|
( vy vx wf crn c2nd cres wceq wfo wfn ffn dffn3 f2ndf syl cv wcel cfv vex
|
|
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one-to-one function ` F ` is a one-to-one function of ` F ` onto the
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range of ` F ` . (Contributed by Alexander van der Vekens,
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4-Feb-2018.) $)
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f1o2ndf1 $p |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F ) $=
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$}
|
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${
|
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algrflem.1 $e |- B e. _V $.
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algrflem.2 $e |- C e. _V $.
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$( Lemma for algrf and related theorems. (Contributed by Mario Carneiro,
|
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28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) $)
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algrflem $p |- ( B ( F o. 1st ) C ) = ( F ` B ) $=
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( c1st ccom co cop cfv df-ov cvv wf wcel wceq wfo fo1st fof ax-mp mp2an
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opexg fvco3 op1st fveq2i 3eqtri ) ABCFGZHABIZUFJZUGFJZCJZACJABUFKLLFMZUGL
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$}
|
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22-Jul-2021.) $)
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algrflemg $p |- ( ( B e. V /\ C e. W ) ->
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( wcel wa c1st ccom co cop cfv df-ov cvv wf wceq wfo fo1st fof ax-mp op1stg
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$.
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|
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${
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$d b x y $. $d c x y $. $d d x y $.
|
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xporderlem.1 $e |- T = { <. x , y >. | ( ( x e. ( A X. B )
|
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/\ y e. ( A X. B ) )
|
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/\ ( ( 1st ` x ) R ( 1st ` y )
|
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\/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } $.
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$( Lemma for lexicographical ordering theorems. (Contributed by Scott
|
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Fenton, 16-Mar-2011.) $)
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xporderlem $p |- ( <. a , b >. T <. c , d >. <->
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( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\
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( a R c \/ ( a = c /\ b S d ) ) ) ) $=
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bitri opex eleq1 opelxp syl6bb anbi1d op1std breq1d eqeq1d op2ndd orbi12d
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cxp anbi2d breq2d eqeq2d opelopab an4 anbi1i 3bitri ) HMZIMZUAZJMZKMZUAZG
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|
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$}
|
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${
|
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$d A a b c d e f t u v x y $. $d B a b c d e f t u v x y $.
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$d R a b c d e f t u v x y $. $d S a b c d e f t u v x y $.
|
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$d T a b c d e f t u v $.
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poxp.1 $e |- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) )
|
|
/\ ( ( 1st ` x ) R ( 1st ` y ) \/
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( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } $.
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$( A lexicographical ordering of two posets. (Contributed by Scott Fenton,
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16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) $)
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poxp $p |- ( ( R Po A /\ S Po B ) -> T Po ( A X. B ) ) $=
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( va vb vc vd ve vf wa cv wbr wn wi wcel vt vu wpo cxp wral cop wceq elxp
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anidms notbid 3ad2ant1 3adant3 3adant1 anbi12d imbi12d xporderlem anbi12i
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notbii imbi12i syl6bb expcomd sylbi com3l exlimivv syl3anb com3r ralrimiv
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ZGQZOZYDYIGQZSZOZUICDUDZUEZUBYOUEUAYOUEYOGUCYCYPUAUBYOYOYCYDYOTZYGYOTZOZO
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YNUIYOYCYSYIYOTZYNSYSYTYCYNYQYRYTYCYNSZYQYDIPZJPZUFZUGZUUBCTZUUCDTZOZOZJU
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JIUJZYRYGKPZLPZUFZUGZUUKCTZUULDTZOZOZLUJKUJZYTYIMPZNPZUFZUGZUUTCTZUVADTZO
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$}
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${
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$d x y A $. $d x y B $. $d x y ph $.
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spc2ed.x $e |- F/ x ch $.
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spc2ed.y $e |- F/ y ch $.
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spc2ed.1 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) ) $.
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$( Existential specialization with 2 quantifiers, using implicit
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spc2ed $p |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )
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$=
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$}
|
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|
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${
|
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$d a w x y z $. $d a w ph $. $d w ps $.
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cnvoprab.x $e |- F/ x ps $.
|
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cnvoprab.y $e |- F/ y ps $.
|
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cnvoprab.1 $e |- ( a = <. x , y >. -> ( ps <-> ph ) ) $.
|
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cnvoprab.2 $e |- ( ps -> a e. ( _V X. _V ) ) $.
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$( TODO - simplify with ~ dfoprab3 $)
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|
$( The converse of a class abstraction of nested ordered pairs.
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(Contributed by Thierry Arnoux, 17-Aug-2017.) $)
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cnvoprab $p |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps } $=
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( vw copab cv cop wceq wa wex vex cvv wcel ccnv coprab cab excom nfv nfan
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CXAXBXACWSCFWRBCWRCUEGUFZUGWNXADWSDFWRBDWRDUEHUFZUGWSWNFWJWHWICRDRUHWPWJO
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$}
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${
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$d a x y z A $. $d a x y z B $. $d a z C $. $d a x y z D $.
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$d a x y I $. $d a x y J $. $d a x y z ph $.
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f1od2.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
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f1od2.2 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W ) $.
|
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f1od2.3 $e |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) ) $.
|
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f1od2.4 $e |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <->
|
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( z e. D /\ ( x = I /\ y = J ) ) ) ) $.
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|
$( Describe an implicit one-to-one onto function of two variables.
|
|
(Contributed by Thierry Arnoux, 17-Aug-2017.) $)
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f1od2 $p |- ( ph -> F : ( A X. B ) -1-1-onto-> D ) $=
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( wa wsbc va cxp wfn ccnv wf1o wcel wral ralrimivva fnmpt2 syl cv opelxpi
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cop cmpt ralrimiva eqid fnmpt c1st cfv c2nd wceq copab elxp7 anbi1i anass
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csb cvv sbcbidv sbcan vex 2ndexg ax-mp sbcg sbcel1v anbi12i bitri sbceq2g
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wb sbcbii 1stexg 3bitri sbceq1g csbvarg eqeq1i 3bitr3g anbi2d syl5bb xpss
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simprr adantrr eqeltrd sseldi ex pm4.71rd pm5.32i syl6rbb opabbidv coprab
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eqop bitrd cmpt2 df-mpt2 eqtri cnveqi nfcsb1v nfeq2 nfan nfcv nfcsb eleq1
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nfv opelxp syl6bb csbopeq1a eqeq2d anbi12d adantr cnvoprab df-mpt 3eqtr4g
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sseli fneq1d mpbird dff1o4 sylanbrc ) AIEFUBZUCZIUDZHUCZYFHIUEAGLUFZCFUGB
|
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|
|
FZSJMUFKNUFSYOQJKMNULUJZUODHYKYLYNYLUPUQUJAHYHYLAUAUKZYFUFZYPBYSURUSZCYSU
|
|
TUSZGVFZVFZVAZSZDUAVBZYQYSYKVAZSZDUAVBYHYLAUUFUUIDUAUUFYSVGVGUBZUFZUUAEUF
|
|
ZUUBFUFZSZSZUUESZAUUIYTUUOUUEYSEFVCVDAUUPUUKYQUUAJVAZUUBKVAZSZSZSZUUIUUPU
|
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|
|
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|
|
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|
|
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|
|
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|
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|
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|
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|
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UXCUUBKUWGUXCUUBVAUWJCUUBVGWCVLWDVPVOVPVOVPVSYQUWOBUUAVIUWQYQUWRUUSUWMUWQ
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|
YQVRUWNYQBUUAVGVMVLUWRUVLBUUATZUURBUUATZSUUSUVLUURBUUAVIUXEUUQUXFUURUXEBU
|
|
UAUVCVFZJVAZUUQUWMUXEUXHVRUWNBUUAUVCJVGWBVLUXGUUAJUWMUXGUUAVAUWNBUUAVGWCV
|
|
LWDVPUWMUXFUURVRUWNUURBUUAVGVMVLVOVPVOWAWEWFWGAUUIUUKUUISUVAAUUIUUKAUUIUU
|
|
KAUUISZYNUUJYSMNWHUXIYSYKYNAYQUUHWIAYQYOUUHYRWJWKWLWMWNUUKUUIUUTUUKUUHUUS
|
|
YQYSJKVGVGWSWFWOWPWTWGWQYHUVIBCDWRZUDUUGIUXJIBCEFGXAUXJOBCDEFGXBXCXDUVIUU
|
|
FBCDUAYTUUEBYTBXKBYPUUDBUUAUUCXEXFXGYTUUECYTCXKCYPUUDCBUUAUUCCUUAXHCUUBGX
|
|
EXIXFXGYSUVCUVEUMZVAZYTUVGUUEUVHUXLYTUXKYFUFUVGYSUXKYFXJUVCUVEEFXLXMUXLUU
|
|
DGYPBCYSGXNXOXPYTUUKUUEYFUUJYSEFWHYAXQXRXCDUAHYKXSXTYBYCYFHIYDYE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Special "Maps to" operations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The following theorems are about maps-to operations (see ~ df-mpt2 ) where the
|
|
first argument is a pair and the base set of the second argument is the first
|
|
component of the first argument, in short "x-maps-to operations". For labels,
|
|
the abbreviations "mpt2x" are used (since "x" usually denotes the first
|
|
argument). This is in line with the currently used conventions for such cases
|
|
(see ~ cbvmpt2x , ~ ovmpt2x and ~ fmpt2x ). However, there is a proposal by
|
|
Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2"
|
|
(see beginning of set.mm). If this proposal will be realized, the labels in the
|
|
following should also be adapted. If the first argument is an ordered pair,
|
|
as in the following, the abbreviation is extended to "mpt2xop", and the maps-to
|
|
operations are called "x-op maps-to operations" for short.
|
|
$)
|
|
|
|
${
|
|
$d x y $. $d K x $. $d V x $. $d W x $.
|
|
mpt2xopn0yelv.f $e |- F = ( x e. _V , y e. ( 1st ` x ) |-> C ) $.
|
|
$( If there is an element of the value of an operation given by a maps-to
|
|
rule, where the first argument is a pair and the base set of the second
|
|
argument is the first component of the first argument, then the second
|
|
argument is an element of the first component of the first argument.
|
|
(Contributed by Alexander van der Vekens, 10-Oct-2017.) $)
|
|
mpt2xopn0yelv $p |- ( ( V e. X /\ W e. Y )
|
|
-> ( N e. ( <. V , W >. F K ) -> K e. V ) ) $=
|
|
( cop co wcel c1st cfv wa cvv cv csn cxp ciun dmmpt2ssx wrel wfun mpt2fun
|
|
cdm funrel ax-mp relelfvdm mpan df-ov eleq2s fveq2 opeliunxp2 simprbi syl
|
|
sseldi op1stg eleq2d syl5ib ) FGHLZEDMZNZEVBOPZNZGINHJNQZEGNVDVBELZARASZT
|
|
VIOPZUAUBZNZVFVDDUGZVKVHABRVJCDKUCVHVMNZFVHDPZVCDUDZFVONVNDUEVPABRVJCDKUF
|
|
DUHUIFVHDUJUKVBEDULUMURVLVBRNVFARVJVBEVEVIVBOUNUOUPUQVGVEGEGHIJUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d K n x y $. $d V n x y $. $d W n x y $. $d X n x y $. $d Y n x y $.
|
|
mpt2xopoveq.f $e |- F = ( x e. _V , y e. ( 1st ` x )
|
|
|-> { n e. ( 1st ` x ) | ph } ) $.
|
|
$( Value of an operation given by a maps-to rule, where the first argument
|
|
is a pair and the base set of the second argument is the first component
|
|
of the first argument. (Contributed by Alexander van der Vekens,
|
|
11-Oct-2017.) $)
|
|
mpt2xopoveq $p |- ( ( ( V e. X /\ W e. Y ) /\ K e. V )
|
|
-> ( <. V , W >. F K )
|
|
= { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) $=
|
|
( wcel wa cvv cv c1st wceq adantr adantl nfcv cop cfv crab wsbc cmpt2 a1i
|
|
vz fveq2 op1stg sylan9eqr adantrr wb sbceq1a bitrd rabeqbidv opexg rabexg
|
|
simpr ad2antrr wnf equid nfvd ax-mp nfsbc1v nfrabxy nfsbc ovmpt2dxf ) GIL
|
|
ZHJLZMZFGLZMZBCGHUAZFNBOZPUBZADVOUCZACFUDZBVMUDZDGUCZEGNEBCNVOVPUEQVLKUFV
|
|
LVNVMQZCOFQZMZMZAVRDVOGVLVTVOGQWAVTVLVOVMPUBZGVNVMPUHVJWDGQVKGHIJUIRUJZUK
|
|
WCAVQVRWBAVQULZVLWAWFVTACFUMSSWBVQVRULZVLVTWGWAVQBVMUMRSUNUOWEVJVMNLVKGHI
|
|
JUPRVJVKURVHVSNLVIVKVRDGIUQUSUGOZWHQZVLBUTUGVAZWIVLBVBVCWIVLCUTWJWIVLCVBV
|
|
CCVMTZBFTVRBDGVQBVMVDBGTVEVRCDGVQCBVMWKACFVDVFCGTVEVG $.
|
|
|
|
$d N x y $.
|
|
$( Element of the value of an operation given by a maps-to rule, where the
|
|
first argument is a pair and the base set of the second argument is the
|
|
first component of the first argument. (Contributed by Alexander van
|
|
der Vekens and Mario Carneiro, 10-Oct-2017.) $)
|
|
mpt2xopovel $p |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K )
|
|
<-> ( K e. V /\ N e. V /\ [. <. V , W >. / x ].
|
|
[. K / y ]. [. N / n ]. ph ) ) ) $=
|
|
( wcel wa cop co wsbc crab sbccom bitri w3a cv cfv mpt2xopn0yelv pm4.71rd
|
|
c1st mpt2xopoveq eleq2d nfcv elrabsf sbcbii anbi2i syl6bb pm5.32da 3anass
|
|
syl6bbr bitrd ) HJMIKMNZGHIOZFEPZMZFHMZVANZVBGHMZADGQCFQZBUSQZUAZURVAVBBC
|
|
ADBUBUFUCREFGHIJKLUDUEURVCVBVDVFNZNVGURVBVAVHURVBNZVAGACFQZBUSQZDHRZMZVHV
|
|
IUTVLGABCDEFHIJKLUGUHVMVDVKDGQZNVHVKDGHDHUIUJVNVFVDVNVJDGQZBUSQVFVJDBGUSS
|
|
VOVEBUSADCGFSUKTULTUMUNVBVDVFUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d E e f p v $. $d V e f p v $. $d W e v $. $d ps e v $.
|
|
sprmpt2.1 $e |- M = ( v e. _V , e e. _V
|
|
|-> { <. f , p >. | ( f ( v W e ) p /\ ch ) } ) $.
|
|
sprmpt2.2 $e |- ( ( v = V /\ e = E ) -> ( ch <-> ps ) ) $.
|
|
sprmpt2.3 $e |- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) ) $.
|
|
sprmpt2.4 $e |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V ) $.
|
|
$( The extension of a binary relation which is the value of an operation
|
|
given in maps-to notation. (Contributed by Alexander van der Vekens,
|
|
30-Oct-2017.) $)
|
|
sprmpt2 $p |- ( ( V e. _V /\ E e. _V ) -> ( V M E )
|
|
= { <. f , p >. | ( f ( V W E ) p /\ ps ) } ) $=
|
|
( cvv wcel wa cv wceq co wbr copab cmpt2 a1i oveq12 adantl breqd opabbidv
|
|
wb anbi12d simpl simpr opabbrex ovmpt2d ) IPQZGPQZRZDEIGPPFSZKSZDSZESZJUA
|
|
ZUBZBRZFKUCZUSUTIGJUAZUBZARZFKUCHPHDEPPVFUDTURLUEURVAITVBGTRZRZVEVIFKVKVD
|
|
VHBAVKVCVGUSUTVJVCVGTURVAIVBGJUFUGUHVJBAUJURMUGUKUIUPUQULUPUQUMACFGIJKNOU
|
|
NUO $.
|
|
$}
|
|
|
|
${
|
|
$d F f p $. $d P f p $. $d W f p $. $d ch f p $.
|
|
isprmpt2.1 $e |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } ) $.
|
|
isprmpt2.2 $e |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) ) $.
|
|
$( Properties of a pair in an extended binary relation. (Contributed by
|
|
Alexander van der Vekens, 30-Oct-2017.) $)
|
|
isprmpt2 $p |- ( ph -> ( ( F e. X /\ P e. Y )
|
|
-> ( F M P <-> ( F W P /\ ch ) ) ) ) $=
|
|
( wcel wa wbr wb cop cv wceq copab adantr eleq2d syl5bb breq12 opelopabga
|
|
df-br anbi12d adantl bitrd ex ) AFINDJNOZFDGPZFDHPZCOZQAULOZUMFDRZESZKSZH
|
|
PZBOZEKUAZNZUOUMUQGNUPVCFDGUGUPGVBUQAGVBTULLUBUCUDULVCUOQAVAUOEKFDIJURFTU
|
|
SDTOUTUNBCURFUSDHUEMUHUFUIUJUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Function transposition
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c tpos $. $( Function transposition $)
|
|
|
|
$( The transposition of a function. $)
|
|
ctpos $a class tpos F $.
|
|
|
|
${
|
|
$d F x $.
|
|
$( Define the transposition of a function, which is a function
|
|
` G = tpos F ` satisfying ` G ( x , y ) = F ( y , x ) ` . (Contributed
|
|
by Mario Carneiro, 10-Sep-2015.) $)
|
|
df-tpos $a |- tpos F = ( F o.
|
|
( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d w x y z F $. $d x G $.
|
|
$( Subset theorem for transposition. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
tposss $p |- ( F C_ G -> tpos F C_ tpos G ) $=
|
|
( vx wss cdm ccnv c0 csn cun cv cuni cmpt ccom ctpos coss1 cres wceq dmss
|
|
cnvss df-tpos unss1 resmpt 4syl resss syl6eqssr coss2 syl sstrd 3sstr4g )
|
|
ABDZACAEZFZGHZIZCJHFKZLZMZBCBEZFZUMIZUOLZMZANBNUJUQBUPMZVBABUPOUJUPVADVCV
|
|
BDUJUPVAUNPZVAUJUKURDULUSDUNUTDVDUPQABRUKURSULUSUMUACUTUNUOUBUCVAUNUDUEUP
|
|
VABUFUGUHCATCBTUI $.
|
|
|
|
$( Equality theorem for transposition. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
tposeq $p |- ( F = G -> tpos F = tpos G ) $=
|
|
( wceq ctpos wss eqimss tposss syl eqimss2 eqssd ) ABCZADZBDZKABELMEABFAB
|
|
GHKBAEMLEBAIBAGHJ $.
|
|
|
|
${
|
|
tposeqd.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality theorem for transposition. (Contributed by Mario Carneiro,
|
|
7-Jan-2017.) $)
|
|
tposeqd $p |- ( ph -> tpos F = tpos G ) $=
|
|
( wceq ctpos tposeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( The transposition is a subset of a cross product. (Contributed by Mario
|
|
Carneiro, 12-Jan-2017.) $)
|
|
tposssxp $p |- tpos F C_ ( ( `' dom F u. { (/) } ) X. ran F ) $=
|
|
( vx ctpos cdm ccnv csn cun cuni cmpt crn cxp ccom df-tpos cossxp eqsstri
|
|
c0 cv wss eqid dmmptss xpss1 ax-mp sstri ) ACZBADEPFGZBQFEHZIZDZAJZKZUEUI
|
|
KZUDAUGLUJBAMAUGNOUHUERUJUKRBUEUFUGUGSTUHUEUIUAUBUC $.
|
|
|
|
$( The transposition is a relation. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
reltpos $p |- Rel tpos F $=
|
|
( ctpos cdm ccnv c0 csn cun crn cxp wss wrel tposssxp relxp relss mp2 ) A
|
|
BZACDEFGZAHZIZJSKPKALQRMPSNO $.
|
|
|
|
$( Value of the transposition at a pair ` <. A , B >. ` . (Contributed by
|
|
Mario Carneiro, 10-Sep-2015.) $)
|
|
brtpos2 $p |- ( B e. V -> ( A tpos F B <->
|
|
( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) ) $=
|
|
( vy vx wcel cvv wbr cdm ccnv csn cuni wa wi wb wex wceq syl bitri c0 cun
|
|
ctpos reltpos brrelexi a1i elex adantr cv cmpt df-tpos breqi brcog syl5bb
|
|
ccom cfv wfun funmpt funbrfv2b ax-mp snexg cnvex uniex eqid dmmpti eleq2i
|
|
vex eqcom anbi12i cnvexg uniexg cnveqd unieqd fvmptg mpdan eqeq2d pm5.32i
|
|
ancom anbi1i anass exbii exsimpr exsimpl 19.9v sylib simpl breq1 ceqsexgv
|
|
sneq anbi2d pm5.21nii syl6bb expcom pm5.21ndd ) BDGZAHGZABCUCZIZACJKUALUB
|
|
ZGZALZKZMZBCIZNZWRWPOWOABWQCUDUEUFXEWPOWOWTWPXDAWSUGUHUFWPWOWRXEPWPWONZWR
|
|
AEUIZFWSFUIZLZKZMZUJZIZXGBCIZNZEQZXEWRABCXLUOZIXFXPABWQXQFCUKULEABCXLHDUM
|
|
UNXPXGXCRZWTXNNZNZEQZXEXOXTEXOXRWTNZXNNXTXMYBXNXMWTXRNZYBXMAXLJZGZAXLUPZX
|
|
GRZNZYCXLUQXMYHPFWSXKURAXGXLUSUTYHWTXGYFRZNYCYEWTYGYIYDWSAFWSXKXLXJXIXHHG
|
|
XIHGFVGXHHVAUTVBVCXLVDZVEVFYFXGVHVIWTYIXRWTYFXCXGWTXCHGZYFXCRWTXBHGZYKWTX
|
|
AHGYLAWSVAXAHVJSXBHVKSZFAXKXCWSHXLXHARZXJXBYNXIXAXHAWIVLVMYJVNVOVPVQTTWTX
|
|
RVRTVSXRWTXNVTTWAYAWTXEYAXSEQZWTXRXSEWBYOWTEQWTWTXNEWCWTEWDWESWTXDWFWTYKY
|
|
AXEPYMXSXEEXCHXRXNXDWTXGXCBCWGWJWHSWKTWLWMWN $.
|
|
|
|
$( The behavior of ` tpos ` when the left argument is the empty set (which
|
|
is not an ordered pair but is the "default" value of an ordered pair
|
|
when the arguments are proper classes). (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
brtpos0 $p |- ( A e. V -> ( (/) tpos F A <-> (/) F A ) ) $=
|
|
( wcel c0 ctpos wbr cdm ccnv csn cun cuni brtpos2 ssun2 0ex snid biantrur
|
|
wa sselii cnvsn0 unieqi uni0 eqtri breq1i bitr3i syl6bb ) ACDEABFGEBHIZEJ
|
|
ZKZDZUHIZLZABGZRZEABGZEABCMUNUMUOUJUMUHUIEUHUGNEOPSQULEABULELEUKETUAUBUCU
|
|
DUEUF $.
|
|
|
|
$( Necessary and sufficient condition for ` dom tpos F ` to be a relation.
|
|
(Contributed by Mario Carneiro, 10-Sep-2015.) $)
|
|
reldmtpos $p |- ( Rel dom tpos F <-> -. (/) e. dom F ) $=
|
|
( vy vx cdm wrel c0 wcel wn cv wbr wex 0ex eldm cvv wb vex ax-mp wi sylbi
|
|
wss ctpos brtpos0 cxp 0nelxp df-rel ssel mtoi breldm nsyl3 sylbir exlimiv
|
|
con2i wa ccnv csn relcnv mpbi sseli a1i elsni breq1d pm2.24d syl6bi com3l
|
|
impcom wo cuni brtpos2 simplbi elun sylib adantl mpjaod ex exlimdv syl5bi
|
|
cun ssrdv sylibr impbii ) AUAZDZEZFADZGZHZWEWCWEFBIZAJZBKWCHZBFALMWHWIBWH
|
|
FWGWAJZWIWGNGZWJWHOBPZWGANUBQZWCFWBGZWJWCWNFNNUCZGZNNUDWCWBWOTZWNWPRWBUEZ
|
|
WBWOFUFSUGFWGWALWLUHUIUJUKSULWFWQWCWFCWBWOCIZWBGWSWGWAJZBKWFWSWOGZBWSWACP
|
|
MWFWTXABWFWTXAWFWTUMZWSWDUNZGZXAWSFUOZGZXDXARXBXCWOWSXCEXCWOTWDUPXCUEUQUR
|
|
USWTWFXFXARXFWTWFXAXFWTWJWFXARZXFWSFWGWAWSFUTVAWJWHXGWMWHWEXAFWGALWLUHVBS
|
|
VCVDVEWTXDXFVFZWFWTWSXCXEVQGZXHWTXIWSUOUNVGWGAJZWKWTXIXJUMOWLWSWGANVHQVIW
|
|
SXCXEVJVKVLVMVNVOVPVRWRVSVT $.
|
|
|
|
$( The transposition swaps arguments of a three-parameter relation.
|
|
(Contributed by Jim Kingdon, 31-Jan-2019.) $)
|
|
brtposg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. A , B >. tpos F C <-> <. B , A >. F C ) ) $=
|
|
( wcel w3a cop cdm ccnv c0 csn cun wbr wa wb 3adant3 cvv opswapg 3ad2ant3
|
|
cuni ctpos breq1d anbi2d brtpos2 opexg ancoms anim1i 3impa breldmg 3expia
|
|
wi syl opelcnvg sylibrd elun1 syl6 pm4.71rd 3bitr4d ) AEHZBFHZCGHZIZABJZD
|
|
KZLZMNZOHZVFNLUCZCDPZQZVJBAJZCDPZQVFCDUDPZVOVEVLVOVJVBVCVLVORVDVBVCQZVKVN
|
|
CDABEFUAUESUFVDVBVPVMRVCVFCDGUGUBVEVOVJVEVOVFVHHZVJVEVOVNVGHZVRVEVNTHZVDQ
|
|
ZVOVSUNVBVCVDWAVQVTVDVCVBVTBAFEUHUIUJUKVTVDVOVSVNCTGDULUMUOVBVCVRVSRVDABE
|
|
FVGUPSUQVFVHVIURUSUTVA $.
|
|
|
|
$( The transposition swaps the first two elements in a collection of
|
|
ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) $)
|
|
ottposg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. A , B , C >. e. tpos F <-> <. B , A , C >. e. F ) ) $=
|
|
( wcel w3a cop ctpos cotp wbr brtposg df-br 3bitr3g df-ot eleq1i 3bitr4g
|
|
) AEHBFHCGHIZABJZCJZDKZHZBAJZCJZDHZABCLZUCHBACLZDHTUACUCMUECDMUDUGABCDEFG
|
|
NUACUCOUECDOPUHUBUCABCQRUIUFDBACQRS $.
|
|
|
|
$( The domain of ` tpos F ` when ` dom F ` is a relation. (Contributed by
|
|
Mario Carneiro, 10-Sep-2015.) $)
|
|
dmtpos $p |- ( Rel dom F -> dom tpos F = `' dom F ) $=
|
|
( vx vy vz ctpos cdm wrel ccnv wa wceq cvv wcel cop wbr wex vex opex eldm
|
|
c0 cv cxp wss wn 0nelxp ssel mtoi df-rel reldmtpos 3imtr4i relcnv brtposg
|
|
jctir wb mp3an a1i exbidv opelcnv bitri 3bitr4g eqrelrdv2 mpancom ) AEZFZ
|
|
GZAFZHZGZIVEGZVCVFJVHVDVGVEKKUAZUBZSVELZUCVHVDVJVKSVILKKUDVEVISUEUFVEUGAU
|
|
HUIVEUJULVHBCVCVFVHBTZCTZMZDTZVBNZDOVMVLMZVOANZDOZVNVCLVNVFLZVHVPVRDVPVRU
|
|
MZVHVLKLVMKLVOKLWABPZCPZDPVLVMVOAKKKUKUNUOUPDVNVBVLVMWBWCQRVTVQVELVSVLVMV
|
|
EWBWCUQDVQAVMVLWCWBQRURUSUTVA $.
|
|
|
|
$( The range of ` tpos F ` when ` dom F ` is a relation. (Contributed by
|
|
Mario Carneiro, 10-Sep-2015.) $)
|
|
rntpos $p |- ( Rel dom F -> ran tpos F = ran F ) $=
|
|
( vx vy vw vz cdm wrel crn cv wcel wbr wex vex elrn cop wceq breldm elrel
|
|
wi cvv ctpos ccnv dmtpos eleq2d syl5ib relcnv mpan breq1 wb brtposg mp3an
|
|
syl6 syl6bb opex brelrn syl6bi syli exlimdv syl5bi ex syl5 syl6bbr impbid
|
|
exlimivv eqrdv ) AFZGZBAUAZHZAHZVGBIZVIJZVKVJJZVLCIZVKVHKZCLVGVMCVKVHBMZN
|
|
VGVOVMCVOVGVNDIZEIZOZPZELDLZVMVGVOVNVFUBZJZWAVOVNVHFZJVGWCVNVKVHCMZVPQVGW
|
|
DWBVNAUCUDUEWBGWCWAVFUFDEVNWBRUGULVTVOVMSDEVTVOVRVQOZVKAKZVMVTVOVSVKVHKZW
|
|
GVNVSVKVHUHVQTJVRTJVKTJWHWGUIDMZEMZVPVQVRVKATTTUJUKZUMWFVKAVRVQWJWIUNVPUO
|
|
UPVDUQURUSVMVNVKAKZCLVGVLCVKAVPNVGWLVLCWLVGVNWFPZDLELZVLWLVNVFJZVGWNVNVKA
|
|
WEVPQVGWOWNEDVNVFRUTVAWMWLVLSEDWMWLWHVLWMWLWGWHVNWFVKAUHWKVBVSVKVHVQVRWIW
|
|
JUNVPUOUPVDUQURUSVCVE $.
|
|
|
|
$( The transposition of a set is a set. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
tposexg $p |- ( F e. V -> tpos F e. _V ) $=
|
|
( wcel ctpos cdm ccnv c0 csn cun crn cxp wss cvv tposssxp cnvexg syl p0ex
|
|
dmexg unexg sylancl rnexg xpexg syl2anc ssexg sylancr ) ABCZADZAEZFZGHZIZ
|
|
AJZKZLUMMCZUGMCANUFUKMCZULMCUNUFUIMCZUJMCUOUFUHMCUPABRUHMOPQUIUJMMSTABUAU
|
|
KULMMUBUCUGUMMUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y F $. $d y V $. $d y W $.
|
|
$( The transposition swaps the arguments in a two-argument function. When
|
|
` F ` is a matrix, which is to say a function from ( 1 ... m ) ` X. `
|
|
( 1 ... n ) to the reals or some ring, ` tpos F ` is the transposition
|
|
of ` F ` , which is where the name comes from. (Contributed by Mario
|
|
Carneiro, 10-Sep-2015.) $)
|
|
ovtposg $p |- ( ( A e. V /\ B e. W ) -> ( A tpos F B ) = ( B F A ) ) $=
|
|
( vy wcel wa cop ctpos cfv co cv wbr cio cvv wb df-fv 3eqtr4g df-ov vex
|
|
brtposg mp3an3 iotabidv ) ADGZBEGZHZABIZCJZKZBAIZCKZABUILBACLUGUHFMZUINZF
|
|
OUKUMCNZFOUJULUGUNUOFUEUFUMPGUNUOQFUAABUMCDEPUBUCUDFUHUIRFUKCRSABUITBACTS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d w x y z F $. $d x G $.
|
|
$( The transposition of a function is a function. (Contributed by Mario
|
|
Carneiro, 10-Sep-2015.) $)
|
|
tposfun $p |- ( Fun F -> Fun tpos F ) $=
|
|
( vx wfun cdm ccnv c0 csn cun cv cuni cmpt ccom ctpos funco mpan2 df-tpos
|
|
funmpt funeqi sylibr ) ACZABADEFGHZBIGEJZKZLZCZAMZCTUCCUEBUAUBQAUCNOUFUDB
|
|
APRS $.
|
|
|
|
$( Alternate definition of ` tpos ` when ` F ` has relational domain.
|
|
(Contributed by Mario Carneiro, 10-Sep-2015.) $)
|
|
dftpos2 $p |- ( Rel dom F -> tpos F =
|
|
( F o. ( x e. `' dom F |-> U. `' { x } ) ) ) $=
|
|
( cdm wrel ctpos cres ccnv csn cuni cmpt ccom dmtpos reseq2d wceq reltpos
|
|
cv resdm ax-mp c0 cun df-tpos reseq1i resco resmpt coeq2i 3eqtri 3eqtr3g
|
|
wss ssun1 ) BCZDZBEZULCZFZULUJGZFZULBAUOAPHGIZJZKZUKUMUOULBLMULDUNULNBOUL
|
|
QRUPBAUOSHZTZUQJZKZUOFBVBUOFZKUSULVCUOABUAUBBVBUOUCVDURBUOVAUHVDURNUOUTUI
|
|
AVAUOUQUDRUEUFUG $.
|
|
|
|
$( Alternate definition of ` tpos ` when ` F ` has relational domain.
|
|
Compare ~ df-cnv . (Contributed by Mario Carneiro, 10-Sep-2015.) $)
|
|
dftpos3 $p |- ( Rel dom F -> tpos F =
|
|
{ <. <. x , y >. , z >. | <. y , x >. F z } ) $=
|
|
( vw cdm wrel ctpos cv cop wceq wbr wa wex wcel cvv cxp vex bitr3i syl6bb
|
|
cab coprab wss ccnv relcnv dmtpos releqd mpbiri reltpos jctil sylib sseld
|
|
relrelss elvvv syl6ib pm4.71rd 19.41vvv eleq1 df-br brtposg mp3an pm5.32i
|
|
wb 3exbii abbi2dv df-oprab syl6eqr ) DFZGZDHZEIZAIZBIZJZCIZJZKZVMVLJVODLZ
|
|
MZCNBNANZEUAVRABCUBVIVTEVJVIVKVJOZVQCNBNANZWAMZVTVIWAWBVIWAVKPPQPQZOWBVIV
|
|
JWDVKVIVJGZVJFZGZMVJWDUCVIWGWEVIWGVHUDZGVHUEVIWFWHDUFUGUHDUIUJVJUMUKULABC
|
|
VKUNUOUPWCVQWAMZCNBNANVTVQWAABCUQWIVSABCVQWAVRVQWAVPVJOZVRVKVPVJURWJVNVOV
|
|
JLZVRVNVOVJUSVLPOVMPOVOPOWKVRVCARBRCRVLVMVODPPPUTVASTVBVDSTVEVRABCEVFVG
|
|
$.
|
|
|
|
$( Alternate definition of ` tpos ` . (Contributed by Mario Carneiro,
|
|
4-Oct-2015.) $)
|
|
dftpos4 $p |- tpos F =
|
|
( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) ) $=
|
|
( vy vz vw cvv csn cun cv ccnv cuni wss wceq cop wcel wbr wa wex vex wb
|
|
ctpos cxp cmpt ccom cdm df-tpos cres wrel relcnv df-rel mpbi unss1 resmpt
|
|
c0 mp2b resss eqsstr3i coss2 ax-mp eqsstri relco opelco eleq1 sneq cnveqd
|
|
unieqd eqeq2d anbi12d anbi2d df-mpt brab wi simplr breldm adantl eqeltrrd
|
|
eqeq1 opswapg mp2an eleq1i opelcnv bitr4i eleq1d bibi12d exlimivv biimpcd
|
|
elvv mpbiri sylbi elun1 syl6 syl elun2 a1i simpll elun sylib mpjaod simpr
|
|
wo eqbrtrrd jca sylanb brtpos2 sylibr df-br exlimiv relssi eqssi ) BUAZBA
|
|
FFUBZUNGZHZAIZGZJZKZUCZUDZXJBABUEZJZXLHZXQUCZUDZXSABUFYCXRLYDXSLYCXRYBUGZ
|
|
XRYAXKLZYBXMLYEYCMYAUHYFXTUIYAUJUKYAXKXLULAXMYBXQUMUOXRYBUPUQYCXRBURUSUTC
|
|
DXSXJBXRVACIZDIZNZXSOYGEIZXRPZYJYHBPZQZERYIXJOZEYGYHBXRCSZDSZVBYMYNEYMYGY
|
|
HXJPZYNYMYGYBOZYGGZJZKZYHBPZQZYQYKYGXMOZYJUUAMZQZYLUUCXNXMOZYHXQMZQUUDYHU
|
|
UAMZQUUFADYGYJXRYOESZXNYGMZUUGUUDUUHUUIXNYGXMVCUUKXQUUAYHUUKXPYTUUKXOYSXN
|
|
YGVDVEVFVGVHYHYJMUUIUUEUUDYHYJUUAVQVIADXMXQVJVKUUFYLQZYRUUBUULYGXKOZYRYGX
|
|
LOZUULUUAXTOZUUMYRVLUULYJUUAXTUUDUUEYLVMZYLYJXTOUUFYJYHBUUJYPVNVOVPUUOUUM
|
|
YGYAOZYRUUMUUOUUQUUMYGYHYJNZMZERDRUUOUUQTZDEYGWGUUSUUTDEUUSUUTUURGZJZKZXT
|
|
OZUURYAOZTUVDYJYHNZXTOUVEUVCUVFXTYHFOZYJFOUVCUVFMYPUUJYHYJFFVRVSVTYHYJXTY
|
|
PUUJWAWBUUSUUOUVDUUQUVEUUSUUAUVCXTUUSYTUVBUUSYSUVAYGUURVDVEVFWCYGUURYAVCW
|
|
DWHWEWIWFYGYAXLWJWKWLUUNYRVLUULYGXLYAWMWNUULUUDUUMUUNWTUUDUUEYLWOYGXKXLWP
|
|
WQWRUULYJUUAYHBUUPUUFYLWSXAXBXCUVGYQUUCTYPYGYHBFXDUSXEYGYHXJXFWQXGWIXHXI
|
|
$.
|
|
|
|
$( Value of the double transposition for a general class ` F ` .
|
|
(Contributed by Mario Carneiro, 16-Sep-2015.) $)
|
|
tpostpos $p |- tpos tpos F =
|
|
( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) $=
|
|
( vz vx vy cvv c0 csn wrel cv ccnv wcel wbr wa wo wb syl6bb syl6eq breq1d
|
|
cuni vex vw ctpos cxp cun cin reltpos wss inss2 relxp relss relcnv df-rel
|
|
mp2 cdm mpbi simpl sseldi simpr cop wceq elvv eleq1 opelcnv cnveqd unieqd
|
|
wex sneq opswapg mp2an anbi12d breldm pm4.71ri brtposg mp3an bitr3i breq1
|
|
opex bitr4d exlimivv sylbi iba bitrd pm5.21nii elsni sneqd cnvsn0 brtpos0
|
|
uni0 ax-mp pm5.32i ancom bitri orbi12i andir andi 3bitr4i anbi1i mpbiran2
|
|
elun brxp anbi2i brtpos2 brin eqbrriv ) UABAUBZUBZAEEUCZFGZUDZEUCZUEZXEUF
|
|
XKXJUGXJHXKHAXJUHXIEUIXKXJUJUMUAIZXEUNZJZXHUDKZXLGZJZSZBIZXELZMZXLXSALZXL
|
|
XSXJLZMZXLXSXFLZXLXSXKLXLXNKZXLXHKZNZXTMZYBXLXGKZYGNZMZYAYDYFXTMZYGXTMZNY
|
|
BYJMZYBYGMZNYIYLYMYOYNYPYMYJYOYMXNXGXLXNHXNXGUGXMUKXNULUOYFXTUPUQYBYJURYJ
|
|
YMYBYOYJXLCIZDIZUSZUTZDVFCVFYMYBOZCDXLVAYTUUACDYTYMYSXSALZYBYTYMYRYQUSZXM
|
|
KZUUCXSXELZMZUUBYTYFUUDXTUUEYTYFYSXNKUUDXLYSXNVBYQYRXMCTZDTZVCPYTXRUUCXSX
|
|
EYTXRYSGZJZSZUUCYTXQUUJYTXPUUIXLYSVGVDVEYQEKZYREKZUUKUUCUTUUGUUHYQYREEVHV
|
|
IQRVJUUFUUEUUBUUEUUDUUCXSXEYRYQUUHUUGVQBTZVKVLUUMUULXSEKZUUEUUBOUUHUUGUUN
|
|
YRYQXSAEEEVMVNVOPXLYSXSAVPVRVSVTYJYBWAWBWCYNYGYBMYPYGXTYBYGXTFXSALZYBYGXT
|
|
FXSXELZUUPYGXRFXSXEYGXRFSFYGXQFYGXQXHJFYGXPXHYGXLFXLFWDZWEVDWFQVEWHQRUUOU
|
|
UQUUPOUUNXSAEWGWIPYGXLFXSAUURRVRWJYGYBWKWLWMYFYGXTWNYBYJYGWOWPXOYHXTXLXNX
|
|
HWSWQYCYKYBYCXLXIKZYKYCUUSUUOUUNXLXSXIEWTWRXLXGXHWSWLXAWPUUOYEYAOUUNXLXSX
|
|
EEXBWIXLXSAXJXCWPXD $.
|
|
|
|
$( Value of the double transposition for a relation on triples.
|
|
(Contributed by Mario Carneiro, 16-Sep-2015.) $)
|
|
tpostpos2 $p |- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F ) $=
|
|
( wrel cdm wa ctpos cvv cxp csn cun cin tpostpos wss relrelss ssun1 xpss1
|
|
c0 wceq ax-mp sstr mpan2 sylbi df-ss sylib syl5eq ) ABACBDZAEEAFFGZPHZIZF
|
|
GZJZAAKUEAUILZUJAQUEAUFFGZLZUKAMUMULUILZUKUFUHLUNUFUGNUFUHFORAULUISTUAAUI
|
|
UBUCUD $.
|
|
|
|
$( The domain of a transposition. (Contributed by NM, 10-Sep-2015.) $)
|
|
tposfn2 $p |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) ) $=
|
|
( wrel wfun cdm wceq wa ctpos ccnv wfn wi tposfun a1i dmtpos releq eqeq2d
|
|
cnveq 3imtr3d com12 df-fn anim12d 3imtr4g ) ACZBDZBEZAFZGBHZDZUGEZAIZFZGB
|
|
AJUGUJJUCUDUHUFUKUDUHKUCBLMUFUCUKUFUECZUIUEIZFZUCUKULUNKUFBNMUEAOUFUMUJUI
|
|
UEAQPRSUABATUGUJTUB $.
|
|
|
|
$( Condition for a surjective transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposfo2 $p |- ( Rel A ->
|
|
( F : A -onto-> B -> tpos F : `' A -onto-> B ) ) $=
|
|
( wrel wfn crn wceq wa ctpos ccnv wfo tposfn2 adantrd cdm releqd biimparc
|
|
fndm rntpos syl df-fo eqeq1d biimprd expimpd jcad 3imtr4g ) ADZCAEZCFZBGZ
|
|
HZCIZAJZEZUKFZBGZHABCKULBUKKUFUJUMUOUFUGUMUIACLMUFUGUIUOUFUGHZUOUIUPUNUHB
|
|
UPCNZDZUNUHGUGURUFUGUQAACQOPCRSUAUBUCUDABCTULBUKTUE $.
|
|
|
|
$( The domain and range of a transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposf2 $p |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) ) $=
|
|
( wrel wf ccnv ctpos crn wss wfo wfn ffn dffn4 sylib tposfo2 syl5 imp fof
|
|
wa syl frn adantl fss syl2anc ex ) ADZABCEZAFZBCGZEZUFUGSZUHCHZUIEZULBIZU
|
|
JUKUHULUIJZUMUFUGUOUGAULCJZUFUOUGCAKUPABCLACMNAULCOPQUHULUIRTUGUNUFABCUAU
|
|
BUHULBUIUCUDUE $.
|
|
|
|
$( Condition for an injective transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposf12 $p |- ( Rel A -> ( F : A -1-1-> B -> tpos F : `' A -1-1-> B ) ) $=
|
|
( vx wrel wf1 ccnv ctpos wa cdm cv csn cuni cmpt wceq wb syl f1eq1 mpbird
|
|
3syl ccom simpr relcnv cnvf1o f1of1 simpl dfrel2 sylib f1eq3 mpbii cnveqd
|
|
wf1o mp2b f1dm mpteq1 f1co syl2anc releqd biimparc dftpos2 ex ) AEZABCFZA
|
|
GZBCHZFZVBVCIZVFVDBCDCJZGZDKLGMZNZUAZFZVGVCVDAVKFZVMVBVCUBZVGVNVDADVDVJNZ
|
|
FZVGVDVDGZVPFZVQVDEVDVRVPULVSAUCDVDUDVDVRVPUEUMVGVRAOZVSVQPVGVBVTVBVCUFAU
|
|
GUHVRAVDVPUIQUJVGVIVDOVKVPOVNVQPVGVHAVGVCVHAOVOABCUNZQUKDVIVDVJUOVDAVKVPR
|
|
TSVDABCVKUPUQVGVHEZVEVLOVFVMPVCWBVBVCVHAWAURUSDCUTVDBVEVLRTSVA $.
|
|
|
|
$( Condition of a bijective transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposf1o2 $p |- ( Rel A ->
|
|
( F : A -1-1-onto-> B -> tpos F : `' A -1-1-onto-> B ) ) $=
|
|
( wrel wf1 wfo wa ccnv ctpos wf1o tposf12 tposfo2 anim12d df-f1o 3imtr4g
|
|
) ADZABCEZABCFZGAHZBCIZEZSBTFZGABCJSBTJPQUARUBABCKABCLMABCNSBTNO $.
|
|
|
|
$( The domain and range of a transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposfo $p |- ( F : ( A X. B ) -onto-> C ->
|
|
tpos F : ( B X. A ) -onto-> C ) $=
|
|
( cxp wfo ccnv ctpos wrel wi relxp tposfo2 ax-mp wceq cnvxp foeq2 sylib
|
|
wb ) ABEZCDFZSGZCDHZFZBAEZCUBFZSITUCJABKSCDLMUAUDNUCUERABOUAUDCUBPMQ $.
|
|
|
|
$( The domain and range of a transposition. (Contributed by NM,
|
|
10-Sep-2015.) $)
|
|
tposf $p |- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C ) $=
|
|
( cxp wf ccnv ctpos wrel wi relxp tposf2 ax-mp cnvxp feq2i sylib ) ABEZCD
|
|
FZQGZCDHZFZBAEZCTFQIRUAJABKQCDLMSUBCTABNOP $.
|
|
|
|
$( Functionality of a transposition. (Contributed by Mario Carneiro,
|
|
4-Oct-2015.) $)
|
|
tposfn $p |- ( F Fn ( A X. B ) -> tpos F Fn ( B X. A ) ) $=
|
|
( cxp cvv wf ctpos wfn tposf dffn2 3imtr4i ) ABDZECFBADZECGZFCLHNMHABECIL
|
|
CJMNJK $.
|
|
|
|
$( Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) $)
|
|
tpos0 $p |- tpos (/) = (/) $=
|
|
( c0 ctpos wfn wceq ccnv wrel rel0 eqid fn0 mpbir tposfn2 mp2 cnv0 fneq2i
|
|
mpbi ) ABZACZPADPAEZCZQAFAACZSGTAADAHAIJAAKLRAPMNOPIO $.
|
|
|
|
$( Transposition of a composition. (Contributed by Mario Carneiro,
|
|
4-Oct-2015.) $)
|
|
tposco $p |- tpos ( F o. G ) = ( F o. tpos G ) $=
|
|
( vx ccom cvv cxp c0 csn cun cv ccnv cuni cmpt ctpos coass dftpos4 coeq2i
|
|
3eqtr4i ) ABDZCEEFGHICJHKLMZDABTDZDSNABNZDABTOCSPUBUAACBPQR $.
|
|
|
|
$( Two ways to say a function is symmetric. (Contributed by Mario
|
|
Carneiro, 4-Oct-2015.) $)
|
|
tpossym $p |- ( F Fn ( A X. A ) ->
|
|
( tpos F = F <-> A. x e. A A. y e. A ( x F y ) = ( y F x ) ) ) $=
|
|
( cxp wfn ctpos wceq cv co wral tposfn eqfnov2 mpancom eqcom cvv wcel vex
|
|
wb ovtposg mp2an eqeq2i bitri 2ralbii syl6bb ) DCCEZFZDGZDHZAIZBIZUHJZUJU
|
|
KDJZHZBCKACKZUMUKUJDJZHZBCKACKUHUFFUGUIUOSCCDLABCCUHDMNUNUQABCCUNUMULHUQU
|
|
LUMOULUPUMUJPQUKPQULUPHARBRUJUKDPPTUAUBUCUDUE $.
|
|
|
|
$}
|
|
|
|
${
|
|
tposeqi.1 $e |- F = G $.
|
|
$( Equality theorem for transposition. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
tposeqi $p |- tpos F = tpos G $=
|
|
( wceq ctpos tposeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
tposex.1 $e |- F e. _V $.
|
|
$( A transposition is a set. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
tposex $p |- tpos F e. _V $=
|
|
( cvv wcel ctpos tposexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y F $.
|
|
nftpos.1 $e |- F/_ x F $.
|
|
$( Hypothesis builder for transposition. (Contributed by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
nftpos $p |- F/_ x tpos F $=
|
|
( vy ctpos cvv cxp c0 csn cv ccnv cuni cmpt ccom dftpos4 nfcv nfco nfcxfr
|
|
cun ) ABEBDFFGHISDJIKLMZNDBOABTCATPQR $.
|
|
$}
|
|
|
|
${
|
|
$d a b c x y z $. $d a b c ph $.
|
|
tposoprab.1 $e |- F = { <. <. x , y >. , z >. | ph } $.
|
|
$( Transposition of a class of ordered triples. (Contributed by Mario
|
|
Carneiro, 10-Sep-2015.) $)
|
|
tposoprab $p |- tpos F = { <. <. y , x >. , z >. | ph } $=
|
|
( vb va vc ctpos coprab cv cop wbr tposeqi wceq nfcv nfbr nfv weq dftpos3
|
|
cdm reldmoprab ax-mp nfoprab2 nfoprab1 wa opeq12 ancoms breq1d cbvoprab12
|
|
wrel nfoprab3 breq2 df-br oprabid bitri syl6bb cbvoprab3 eqtri 3eqtri
|
|
wcel ) EJABCDKZJZGLZHLZMZILZVCNZHGIKZACBDKZEVCFOVCUBULVDVJPABCDUCHGIVCUAU
|
|
DVJBLZCLZMZVHVCNZCBIKVKVIVOHGICBCVGVHVCCVGQABCDUECVHQRBVGVHVCBVGQABCDUFBV
|
|
HQRVOHSVOGSHCTZGBTZUGVGVNVHVCVQVPVGVNPVEVFVLVMUHUIUJUKVOACBIDDVNVHVCDVNQA
|
|
BCDUMDVHQRAISIDTVOVNDLZVCNZAVHVRVNVCUNVSVNVRMVCVBAVNVRVCUOABCDUPUQURUSUTV
|
|
A $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z A $. $d z B $. $d z C $.
|
|
tposmpt2.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Transposition of a two-argument mapping. (Contributed by Mario
|
|
Carneiro, 10-Sep-2015.) $)
|
|
tposmpt2 $p |- tpos F = ( y e. B , x e. A |-> C ) $=
|
|
( vz ctpos cv wcel wceq coprab cmpt2 df-mpt2 ancom anbi1i oprabbii 3eqtri
|
|
wa tposoprab eqtr4i ) FIBJDKZAJCKZTZHJELZTZBAHMBADCENUGABHFFABCDENUDUCTZU
|
|
FTZABHMUGABHMGABHCDEOUIUGABHUHUEUFUDUCPQRSUABAHDCEOUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Undefined values
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The power set of the union of a set does not belong to the set. This
|
|
theorem provides a way of constructing a new set that doesn't belong to a
|
|
given set. (Contributed by Stefan O'Rear, 22-Feb-2015.) $)
|
|
pwuninel2 $p |- ( U. A e. V -> -. ~P U. A e. A ) $=
|
|
( cuni wcel cpw wss pwnss elssuni nsyl ) ACZBDJEZJFKADJBGKAHI $.
|
|
|
|
$( The power set of the power set of the union of a set does not belong to
|
|
the set. This theorem provides a way of constructing a new set that
|
|
doesn't belong to a given set. (Contributed by Jim Kingdon,
|
|
14-Jan-2020.) $)
|
|
2pwuninelg $p |- ( A e. V -> -. ~P ~P U. A e. A ) $=
|
|
( wcel cuni cpw wa en2lp wi wss pwuni elpwg mpbiri ax-ia3 syl mtoi ) ABCZAD
|
|
EZEZACZARCZSFZARGPTSUAHPTAQIAJAQBKLTSMNO $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Functions on ordinals; strictly monotone ordinal functions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x A $.
|
|
$( The indexed union of a set of ordinal numbers ` B ( x ) ` is an ordinal
|
|
number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro,
|
|
5-Dec-2016.) $)
|
|
iunon $p |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On ) $=
|
|
( wcel con0 wral wa ciun cmpt crn cuni wceq dfiun3g adantl cvv wss mptexg
|
|
rnexg syl wf eqid fmpt frn sylbi ssonuni imp syl2an eqeltrd ) BDEZCFEABGZ
|
|
HABCIZABCJZKZLZFUKULUOMUJABCFNOUJUNPEZUNFQZUOFEZUKUJUMPEUPABCDRUMPSTUKBFU
|
|
MUAUQABFCUMUMUBUCBFUMUDUEUPUQURUNPUFUGUHUI $.
|
|
$}
|
|
|
|
$c Smo $.
|
|
|
|
$( Introduce the strictly monotone ordinal function. A strictly monotone
|
|
function is one that is constantly increasing across the ordinals. $)
|
|
wsmo $a wff Smo A $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Definition of a strictly monotone ordinal function. Definition 7.46 in
|
|
[TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) $)
|
|
df-smo $a |- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A
|
|
A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $.
|
|
$( Alternate definition of a strictly monotone ordinal function.
|
|
(Contributed by Mario Carneiro, 4-Mar-2013.) $)
|
|
dfsmo2 $p |- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F
|
|
A. y e. x ( F ` y ) e. ( F ` x ) ) ) $=
|
|
( wsmo cdm con0 wf word wel cfv wcel wral w3a df-smo ralcom impexp 3anass
|
|
cv wi wa simpr ordtr1 3impib 3com23 simp3 3expia impbid2 syl5bbr ralbidv2
|
|
jca imbi1d ralbidva syl5bb pm5.32i anbi2i 3bitr4i bitri ) CDCEZFCGZURHZBA
|
|
IZBRZCJARZCJKZSZAURLBURLZMZUSUTVDBVCLZAURLZMZBACNUSUTVFTZTUSUTVITZTVGVJVK
|
|
VLUSUTVFVIVFVEBURLZAURLUTVIVEBAURUROUTVMVHAURUTVCURKZTZVEVDBURVCVBURKZVES
|
|
VPVATZVDSVOVEVPVAVDPVOVQVAVDVOVQVAVPVAUAUTVNVAVQUTVNVAMVPVAUTVAVNVPUTVAVN
|
|
VPVBVCURUBUCUDUTVNVAUEUJUFUGUKUHUIULUMUNUOUSUTVFQUSUTVIQUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
issmo.1 $e |- A : B --> On $.
|
|
issmo.2 $e |- Ord B $.
|
|
issmo.3 $e |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y
|
|
) ) ) $.
|
|
issmo.4 $e |- dom A = B $.
|
|
$( Conditions for which ` A ` is a strictly monotone ordinal function.
|
|
(Contributed by Andrew Salmon, 15-Nov-2011.) $)
|
|
issmo $p |- Smo A $=
|
|
( wsmo cdm con0 wf word cv wcel cfv wi wral mpbir eleq2i feq2i wceq ordeq
|
|
wb ax-mp syl2anb rgen2a df-smo mpbir3an ) CICJZKCLZUJMZANZBNZOUMCPUNCPOQZ
|
|
BUJRAUJRUKDKCLEUJDKCHUASULDMZFUJDUBULUPUDHUJDUCUESUOABUJUMUJOUMDOUNDOUOUN
|
|
UJOUJDUMHTUJDUNHTGUFUGABCUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d F x y $.
|
|
$( Alternate definition of a strictly monotone ordinal function.
|
|
(Contributed by Mario Carneiro, 12-Mar-2013.) $)
|
|
issmo2 $p |- ( F : A --> B -> ( ( B C_ On /\ Ord A /\
|
|
A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) ) $=
|
|
( wf con0 wss word cv cfv wcel wral w3a cdm wsmo fss ex fdm biimprd feq2d
|
|
sylibrd wceq wb ordeq syl raleqdv 3anim123d dfsmo2 syl6ibr ) CDEFZDGHZCIZ
|
|
BJEKAJZEKLBUNMZACMZNEOZGEFZUQIZUOAUQMZNEPUKULURUMUSUPUTUKULCGEFZURUKULVAC
|
|
DGEQRUKUQCGECDESZUAUBUKUSUMUKUQCUCUSUMUDVBUQCUEUFTUKUTUPUKUOAUQCVBUGTUHAB
|
|
EUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Equality theorem for strictly monotone functions. (Contributed by
|
|
Andrew Salmon, 16-Nov-2011.) $)
|
|
smoeq $p |- ( A = B -> ( Smo A <-> Smo B ) ) $=
|
|
( vx vy wceq cdm con0 wf word cv wcel cfv wi wral w3a wsmo raleqdv df-smo
|
|
id fveq1 dmeq feq12d wb ordeq syl eleq12d imbi2d ralbidv 3bitrd 3anbi123d
|
|
2ralbidv 3bitr4g ) ABEZAFZGAHZUNIZCJZDJZKZUQALZURALZKZMZDUNNCUNNZOBFZGBHZ
|
|
VEIZUSUQBLZURBLZKZMZDVENZCVENZOAPBPUMUOVFUPVGVDVMUMUNVEGABUMSABUAZUBUMUNV
|
|
EEUPVGUCVNUNVEUDUEUMVDVKDUNNZCUNNVLCUNNVMUMVCVKCDUNUNUMVBVJUSUMUTVHVAVIUQ
|
|
ABTURABTUFUGUKUMVOVLCUNUMVKDUNVEVNQUHUMVLCUNVEVNQUIUJCDARCDBRUL $.
|
|
|
|
$( The domain of a strictly monotone function is an ordinal. (Contributed
|
|
by Andrew Salmon, 16-Nov-2011.) $)
|
|
smodm $p |- ( Smo A -> Ord dom A ) $=
|
|
( vx vy wsmo cdm con0 wf word cv wcel cfv wi wral df-smo simp2bi ) ADAEZF
|
|
AGPHBIZCIZJQAKRAKJLCPMBPMBCANO $.
|
|
|
|
$( A strictly monotone function restricted to an ordinal remains strictly
|
|
monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof
|
|
shortened by Mario Carneiro, 5-Dec-2016.) $)
|
|
smores $p |- ( ( Smo A /\ B e. dom A ) -> Smo ( A |` B ) ) $=
|
|
( vx vy cdm wcel wsmo con0 wf word cv cfv wi wral w3a wss ax-mp wceq syl
|
|
wa cres wfn crn wfun funres funfn 3imtr3i resss rnss sstr anim12i 3imtr4i
|
|
mpan df-f a1i ordelord expcom ordin ex syli wb dmres ordeq syl6ibr ssralv
|
|
cin ralimi inss1 eqsstri simpl sseldi fvres simpr eleq12d imbi2d ralbidva
|
|
dmss ralbiia sylibr 3anim123d df-smo 3imtr4g impcom ) BAEZFZAGZABUAZGZWEW
|
|
DHAIZWDJZCKZDKZFZWKALZWLALZFZMZDWDNZCWDNZOWGEZHWGIZWTJZWMWKWGLZWLWGLZFZMZ
|
|
DWTNZCWTNZOWFWHWEWIXAWJXBWSXHWIXAMWEAWDUBZAUCZHPZTWGWTUBZWGUCZHPZTWIXAXIX
|
|
LXKXNAUDWGUDXIXLBAUEAUFWGUFUGXMXJPZXKXNWGAPZXOABUHZWGAUIQXMXJHUJUMUKWDHAU
|
|
NWTHWGUNULUOWEWJBWDVFZJZXBWJWEBJZXSWJWEXTWDBUPUQXTWJXSBWDURUSUTWTXRRXBXSV
|
|
AABVBZWTXRVCQVDWSXHMWEWSWQDWTNZCWTNZXHWSWRCWTNZYCWTWDPZWSYDMXPYEXQWGAVQQZ
|
|
WRCWTWDVEQWRYBCWTYEWRYBMYFWQDWTWDVEQVGSXGYBCWTWKWTFZXFWQDWTYGWLWTFZTZXEWP
|
|
WMYIXCWNXDWOYIWKBFXCWNRYIWTBWKWTXRBYABWDVHVIZYGYHVJVKWKBAVLSYIWLBFXDWORYI
|
|
WTBWLYJYGYHVMVKWLBAVLSVNVOVPVRVSUOVTCDAWACDWGWAWBWC $.
|
|
|
|
$( A strictly monotone function restricted to an ordinal remains strictly
|
|
monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) $)
|
|
smores3 $p |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo
|
|
( A |` C ) ) $=
|
|
( cres wsmo cdm cin wcel word w3a dmres incom eqtri eleq2i smores 3adant3
|
|
sylan2br wss wceq wb inss2 sseli ordelss sylan 3adant1 resabs1 smoeq 3syl
|
|
ancoms mpbid ) ABDZEZCAFZBGZHZBIZJZUKCDZEZACDZEZULUOUSUPUOULCUKFZHUSVBUNC
|
|
VBBUMGUNABKBUMLMNUKCOQPUQCBRZURUTSUSVATUOUPVCULUOCBHZUPVCUNBCUMBUAUBUPVDV
|
|
CBCUCUIUDUEACBUFURUTUGUHUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d F x y $.
|
|
$( A strictly monotone ordinal function restricted to an ordinal is still
|
|
monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) $)
|
|
smores2 $p |- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) ) $=
|
|
( vy vx wsmo word wa cdm con0 wf cv cfv wcel wral crn wss wfun dfsmo2 syl
|
|
wceq cres wfn simp1bi ffun funres funfn sylib df-ima imassrn eqsstr3i frn
|
|
cima syl5ss df-f sylanbrc adantr smodm cin ordin dmres ordeq ax-mp sylibr
|
|
wb ancoms sylan resss dmss simp3bi ssralv mpsyl wel wi ordtr1 inss1 sseli
|
|
eqsstri expcomd imp31 fvres ad2antlr eleq12d ralbidva mpbird syl3anbrc
|
|
syl6 ) BEZAFZGZBAUAZHZIWJJZWKFZCKZWJLZDKZWJLZMZCWPNZDWKNZWJEWGWLWHWGWJWKU
|
|
BZWJOZIPWLWGBQZXAWGBHZIBJZXCWGXEXDFZWNBLZWPBLZMZCWPNZDXDNZDCBRZUCZXDIBUDS
|
|
XCWJQXAABUEWJUFUGSWGXBBOZIXBBAULXNBAUHBAUIUJWGXEXNIPXMXDIBUKSUMWKIWJUNUOU
|
|
PWGXFWHWMBUQWHXFWMWHXFGAXDURZFZWMAXDUSWKXOTWMXPVDBAUTZWKXOVAVBVCVEVFZWIWT
|
|
XJDWKNZWGXSWHWKXDPZWGXKXSWJBPXTBAVGWJBVHVBWGXEXFXKXLVIXJDWKXDVJVKUPWIWSXJ
|
|
DWKWIWPWKMZGZWRXICWPYBCDVLZGZWOXGWQXHYDWNAMZWOXGTWIYAYCYEWIYCYAYEWIYCYAGZ
|
|
WNWKMZYEWIWMYFYGVMXRWNWPWKVNSWKAWNWKXOAXQAXDVOVQZVPWFVRVSWNABVTSYAWQXHTZW
|
|
IYCYAWPAMYIWKAWPYHVPWPABVTSWAWBWCWCWDDCWJRWE $.
|
|
$}
|
|
|
|
$( The domain of a strictly monotone ordinal function is an ordinal.
|
|
(Contributed by Mario Carneiro, 12-Mar-2013.) $)
|
|
smodm2 $p |- ( ( F Fn A /\ Smo F ) -> Ord A ) $=
|
|
( wsmo wfn cdm word smodm wceq wb fndm ordeq syl biimpa sylan2 ) BCBADZBEZF
|
|
ZAFZBGOQROPAHQRIABJPAKLMN $.
|
|
|
|
${
|
|
$d F x y $.
|
|
$( The function values of a strictly monotone ordinal function are
|
|
ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) $)
|
|
smofvon2dm $p |- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On ) $=
|
|
( vy vx wsmo cdm con0 wf word cv cfv wcel wral dfsmo2 simp1bi ffvelrnda )
|
|
BEZBFZGABQRGBHRICJBKDJZBKLCSMDRMDCBNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
iordsmo.1 $e |- Ord A $.
|
|
$( The identity relation restricted to the ordinals is a strictly monotone
|
|
function. (Contributed by Andrew Salmon, 16-Nov-2011.) $)
|
|
iordsmo $p |- Smo ( _I |` A ) $=
|
|
( vx vy cid cres con0 wf wfn crn wss fnresi rnresi word ordsson wcel wceq
|
|
cv cfv fvresi ax-mp eqsstri df-f mpbir2an wa adantr adantl eleq12d dmresi
|
|
biimprd issmo ) CDEAFZAAGULHULAIULJZGKALUMAGAMANAGKBAOUAUBAGULUCUDBCRZAPZ
|
|
DRZAPZUEZUNULSZUPULSZPUNUPPURUSUNUTUPUOUSUNQUQAUNTUFUQUTUPQUOAUPTUGUHUJAU
|
|
IUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y F $.
|
|
$( The null set is a strictly monotone ordinal function. (Contributed by
|
|
Andrew Salmon, 20-Nov-2011.) $)
|
|
smo0 $p |- Smo (/) $=
|
|
( cid c0 cres wsmo ord0 iordsmo wceq wb res0 smoeq ax-mp mpbi ) ABCZDZBDZ
|
|
BEFMBGNOHAIMBJKL $.
|
|
|
|
$( If ` B ` is a strictly monotone ordinal function, and ` A ` is in the
|
|
domain of ` B ` , then the value of the function at ` A ` is an ordinal.
|
|
(Contributed by Andrew Salmon, 20-Nov-2011.) $)
|
|
smofvon $p |- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On ) $=
|
|
( vx vy wsmo cdm con0 wf word cv wcel cfv wral df-smo simp1bi ffvelrnda
|
|
wi ) BEZBFZGABRSGBHSICJZDJZKTBLUABLKQDSMCSMCDBNOP $.
|
|
|
|
$( If ` x ` is less than ` y ` then a strictly monotone function's value
|
|
will be strictly less at ` x ` than at ` y ` . (Contributed by Andrew
|
|
Salmon, 22-Nov-2011.) $)
|
|
smoel $p |- ( ( Smo B /\ A e. dom B /\ C e. A ) -> ( B ` C ) e. ( B ` A )
|
|
) $=
|
|
( vx vy wsmo cdm wcel cfv wa wi word smodm ordtr1 expdimp wral wceq fveq2
|
|
cv imbi12d ancomsd sylan con0 w3a df-smo eleq1 eleq1d eleq2 eleq2d rspc2v
|
|
wf ancoms com12 3ad2ant3 sylbi syld pm2.43d 3impia ) BFZABGZHZCAHZCBIZABI
|
|
ZHZUSVAJZVBVEVFVBCUTHZVBVEKZUSUTLZVAVBVGKBMVIVAVBVGVIVBVAVGCAUTNUAOUBUSVA
|
|
VGVHUSUTUCBUKZVIDSZESZHZVKBIZVLBIZHZKZEUTPDUTPZUDVAVGJZVHKZDEBUEVRVJVTVIV
|
|
SVRVHVGVAVRVHKVQVHCVLHZVCVOHZKDECAUTUTVKCQZVMWAVPWBVKCVLUFWCVNVCVOVKCBRUG
|
|
TVLAQZWAVBWBVEVLACUHWDVOVDVCVLABRUITUJULUMUNUOOUPUQUR $.
|
|
|
|
$( The value of a strictly monotone ordinal function contains its indexed
|
|
union. (Contributed by Andrew Salmon, 22-Nov-2011.) $)
|
|
smoiun $p |- ( ( Smo B /\ A e. dom B ) -> U_ x e. A ( B ` x )
|
|
C_ ( B ` A ) ) $=
|
|
( vy wsmo cdm wcel wa cv cfv ciun wrex eliun con0 wi smofvon smoel 3expia
|
|
ontr1 expcomd sylsyld rexlimdv syl5bi ssrdv ) CEZBCFGZHZDABAIZCJZKZBCJZDI
|
|
ZUJGULUIGZABLUGULUKGZAULBUIMUGUMUNABUGUKNGZUHBGZUIUKGZUMUNOBCPUEUFUPUQBCU
|
|
HQRUOUMUQUNULUIUKSTUAUBUCUD $.
|
|
|
|
$( If ` F ` is an isomorphism from an ordinal ` A ` onto ` B ` , which is a
|
|
subset of the ordinals, then ` F ` is a strictly monotonic function.
|
|
Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon,
|
|
24-Nov-2011.) $)
|
|
smoiso $p |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On )
|
|
-> Smo F ) $=
|
|
( vx vy cep word con0 wss wf cv wcel cfv wral syl wb wa eleq2d wbr cvv wi
|
|
wiso w3a cdm wsmo wf1o isof1o f1of ffdm simpld sylan 3adant2 syl3an1 wceq
|
|
fss fdm eqcomd ordeq 4syl biimpa 3adant3 anbi12d 3syl epel isorel syl5bbr
|
|
wfn ffn adantr simprr funfvex funfni epelg syl2anc bitrd biimpd ex sylbid
|
|
ralrimivv 3ad2ant1 df-smo syl3anbrc ) ABFFCUBZAGZBHIZUCCUDZHCJZWFGZDKZEKZ
|
|
LZWICMZWJCMZLZUAZEWFNDWFNZCUEWCABCJZWDWEWGWCABCUFZWQABFFCUGZABCUHZOZWQWEW
|
|
GWDWQWFBCJZWEWGWQXBWFAIABCUIUJWFBHCUOUKULUMWCWDWHWEWCWDWHWCWRWQAWFUNWDWHP
|
|
WSWTWQWFAABCUPZUQAWFURUSUTVAWCWDWPWEWCWODEWFWFWCWIWFLZWJWFLZQZWIALZWJALZQ
|
|
ZWOWCWRWQXFXIPWSWTWQXDXGXEXHWQWFAWIXCRWQWFAWJXCRVBVCWCXIWOWCXIQZWKWNXJWKW
|
|
LWMFSZWNWKWIWJFSXJXKDEVDABWIWJFFCVEVFXJCAVGZXHXKWNPZWCXLXIWCWQXLXAABCVHOV
|
|
IWCXGXHVJXLXHQWMTLZXMXNAWJCWJCVKVLWLWMTVMOVNVOVPVQVRVSVTDECWAWB $.
|
|
$}
|
|
|
|
$( A strictly monotone ordinal function preserves the epsilon relation.
|
|
(Contributed by Mario Carneiro, 12-Mar-2013.) $)
|
|
smoel2 $p |- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) )
|
|
-> ( F ` C ) e. ( F ` B ) ) $=
|
|
( wfn wsmo wcel cfv cdm fndm eleq2d anbi1d biimprd smoel 3expib sylan9 imp
|
|
wa ) DAEZDFZRBAGZCBGZRZCDHBDHGZSUCBDIZGZUBRZTUDSUGUCSUFUAUBSUEABADJKLMTUFUB
|
|
UDBDCNOPQ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
"Strong" transfinite recursion
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c recs $.
|
|
|
|
$( Notation for a function defined by strong transfinite recursion. $)
|
|
crecs $a class recs ( F ) $.
|
|
|
|
${
|
|
$d F f x y $.
|
|
$( Define a function ` recs ( F ) ` on ` On ` , the class of ordinal
|
|
numbers, by transfinite recursion given a rule ` F ` which sets the next
|
|
value given all values so far. See ~ df-irdg for more details on why
|
|
this definition is desirable. Unlike ~ df-irdg which restricts the
|
|
update rule to use only the previous value, this version allows the
|
|
update rule to use _all_ previous values, which is why it is described
|
|
as "strong", although it is actually more primitive. See ~ tfri1d and
|
|
~ tfri2d for the primary contract of this definition.
|
|
|
|
(Contributed by Stefan O'Rear, 18-Jan-2015.) $)
|
|
df-recs $a |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d F a b c $. $d G a b c $. $d x a b c $.
|
|
$( Equality theorem for ` recs ` . (Contributed by Stefan O'Rear,
|
|
18-Jan-2015.) $)
|
|
recseq $p |- ( F = G -> recs ( F ) = recs ( G ) ) $=
|
|
( va vb vc wceq cv wfn cfv cres wral wa con0 wrex cuni crecs fveq1 eqeq2d
|
|
cab df-recs ralbidv anbi2d rexbidv abbidv unieqd 3eqtr4g ) ABFZCGZDGZHZEG
|
|
ZUHIZUHUKJZAIZFZEUIKZLZDMNZCSZOUJULUMBIZFZEUIKZLZDMNZCSZOAPBPUGUSVEUGURVD
|
|
CUGUQVCDMUGUPVBUJUGUOVAEUIUGUNUTULUMABQRUAUBUCUDUEDECATDECBTUF $.
|
|
|
|
nfrecs.f $e |- F/_ x F $.
|
|
$( Bound-variable hypothesis builder for ` recs ` . (Contributed by Stefan
|
|
O'Rear, 18-Jan-2015.) $)
|
|
nfrecs $p |- F/_ x recs ( F ) $=
|
|
( va vb vc crecs cv wfn cfv cres wceq wral wa con0 wrex cuni df-recs nfcv
|
|
cab nfv nffv nfeq2 nfralxy nfan nfrexxy nfab nfuni nfcxfr ) ABGDHZEHZIZFH
|
|
ZUJJZUJUMKZBJZLZFUKMZNZEOPZDTZQEFDBRAVAUTADUSAEOAOSULURAULAUAUQAFUKAUKSAU
|
|
NUPAUOBCAUOSUBUCUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A u w x y z $. $d B x $. $d F u w x y z $. $d G u w x y z $.
|
|
$d ph u w y z $.
|
|
tfrlem1.1 $e |- ( ph -> A e. On ) $.
|
|
tfrlem1.2 $e |- ( ph -> ( Fun F /\ A C_ dom F ) ) $.
|
|
tfrlem1.3 $e |- ( ph -> ( Fun G /\ A C_ dom G ) ) $.
|
|
tfrlem1.4 $e |- ( ph -> A. x e. A ( F ` x ) = ( B ` ( F |` x ) ) ) $.
|
|
tfrlem1.5 $e |- ( ph -> A. x e. A ( G ` x ) = ( B ` ( G |` x ) ) ) $.
|
|
$( A technical lemma for transfinite recursion. Compare Lemma 1 of
|
|
[TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by
|
|
Mario Carneiro, 24-May-2019.) $)
|
|
tfrlem1 $p |- ( ph -> A. x e. A ( F ` x ) = ( G ` x ) ) $=
|
|
( vz wss cv cfv wceq wral wcel wi wa vy vw ssid con0 sseq1 imbi12d imbi2d
|
|
vu raleq r19.21v cres cdm wfn wfun simplll adantr simpld funfn sylib word
|
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syl simpllr eloni ordelss sylan simplr sstrd simprd fnssres syl2anc simpr
|
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rspcv syl3c fveq2 eqeq12d sylc fvres adantl 3eqtr4d eqfnfvd fveq2d sselda
|
|
ad2antrr reseq2 rspcva ralrimiva cbvralv sylibr exp31 expcom syl5bi tfis2
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|
a2d vtoclga mpcom mpi ) ACCMZBNZEOZWRFOZPZBCQZCUCCUDRAWQXBSZGAUANZCMZXABX
|
|
DQZSZSZAXCSUACUDXDCPZXGXCAXIXEWQXFXBXDCCUEXABXDCUIUFUGXHALNZCMZXABXJQZSZS
|
|
ZUALXDXJPZXGXMAXOXEXKXFXLXDXJCUEXABXDXJUIUFUGXNLXDQAXMLXDQZSXDUDRZXHAXMLX
|
|
DUJXQAXPXGAXQXPXGSAXQTZXPXEXFXRXPTZXETZUBNZEOZYAFOZPZUBXDQXFXTYDUBXDXTYAX
|
|
DRZTZEYAUKZDOZFYAUKZDOZYBYCYFYGYIDYFUHYAYGYIYFEEULZUMZYAYKMYGYAUMYFEUNZYL
|
|
YFYMCYKMZYFAYMYNTXTAYEAXQXPXEUOUPZHVAZUQEURUSYFYACYKYFYAXDCXTXDUTZYEYAXDM
|
|
XTXQYQAXQXPXEVBXDVCVAXDYAVDVEXSXEYEVFVGZYFYMYNYPVHVGYKYAEVIVJYFFFULZUMZYA
|
|
YSMYIYAUMYFFUNZYTYFUUACYSMZYFAUUAUUBTYOIVAZUQFURUSYFYACYSYRYFUUAUUBUUCVHV
|
|
GYSYAFVIVJYFUHNZYARZTZUUDEOZUUDFOZUUDYGOZUUDYIOZUUFUUEXABYAQZUUGUUHPZYFUU
|
|
EVKUUFYEXPYACMZUUKXTYEUUEVFXTXPYEUUEXRXPXEVFWCYFUUMUUEYRUPXMUUMUUKSLYAXDX
|
|
JYAPXKUUMXLUUKXJYACUEXABXJYAUIUFVLVMXAUULBUUDYAWRUUDPWSUUGWTUUHWRUUDEVNWR
|
|
UUDFVNVOVLVPUUEUUIUUGPYFUUDYAEVQVRUUEUUJUUHPYFUUDYAFVQVRVSVTWAYFYACRZWSEW
|
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RUKZDOZPZBCQZYBYHPZXTXDCYAXSXEVKWBZYFAUURYOJVAUUQUUSBYACWRYAPZWSYBUUPYHWR
|
|
YAEVNZUVAUUOYGDWRYAEWDWAVOWEVJYFUUNWTFWRUKZDOZPZBCQZYCYJPZUUTYFAUVFYOKVAU
|
|
VEUVGBYACUVAWTYCUVDYJWRYAFVNZUVAUVCYIDWRYAFWDWAVOWEVJVSWFXAYDBUBXDUVAWSYB
|
|
WTYCUVBUVHVOWGWHWIWJWMWKWLWNWOWP $.
|
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$}
|
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|
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${
|
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$d g A $. $d f g w x y z F $. $d f w x y z G $.
|
|
tfrlem3.1 $e |- A = { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
$( Lemma for transfinite recursion. This lemma just changes some bound
|
|
variables in ` A ` for later use. (Contributed by Jim Kingdon,
|
|
5-Jul-2019.) $)
|
|
tfrlem3ag $p |- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\
|
|
A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) ) $=
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|
( cv wfn cfv cres wceq wral wa con0 wrex cvv fneq12 simpll simpr reseq12d
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fveq12d fveq2d eqeq12d simplr cbvraldva2 anbi12d cbvrexdva elab2g ) FJZAJ
|
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ZKZBJZULLZULUOMZGLZNZBUMOZPZAQRHCJZKZDJZHLZHVDMZGLZNZDVBOZPZCQRFHESULHNZV
|
|
AVJACQVKUMVBNZPZUNVCUTVIUMVBULHTVMUSVHBDUMVBVMUOVDNZPZUPVEURVGVOUOVDULHVK
|
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VLVNUAZVMVNUBZUDVOUQVFGVOULHUOVDVPVQUCUEUFVKVLVNUGUHUIUJIUK $.
|
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|
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${
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tfrlem3.2 $e |- G e. _V $.
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$( Lemma for transfinite recursion. Let ` A ` be the class of
|
|
"acceptable" functions. The final thing we're interested in is the
|
|
union of all these acceptable functions. This lemma just changes some
|
|
bound variables in ` A ` for later use. (Contributed by NM,
|
|
9-Apr-1995.) $)
|
|
tfrlem3a $p |- ( G e. A <-> E. z e. On ( G Fn z /\
|
|
A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) $=
|
|
( cv wfn cfv cres wceq wral wa con0 wrex simpr fveq12d reseq12d eqeq12d
|
|
fneq12 simpll fveq2d adantr cbvraldva2 anbi12d cbvrexdva elab2 ) FKZAKZ
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LZBKZULMZULUONZGMZOZBUMPZQZARSHCKZLZDKZHMZHVDNZGMZOZDVBPZQZCRSFHEJULHOZ
|
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VAVJACRVKUMVBOZQZUNVCUTVIUMVBULHUDVMUSVHBDUMVBVMUOVDOZQZUPVEURVGVOUOVDU
|
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LHVKVLVNUEZVMVNTZUAVOUQVFGVOULHUOVDVPVQUBUFUCVMVLVNVKVLTUGUHUIUJIUK $.
|
|
$}
|
|
|
|
$( Lemma for transfinite recursion. Let ` A ` be the class of "acceptable"
|
|
functions. The final thing we're interested in is the union of all
|
|
these acceptable functions. This lemma just changes some bound
|
|
variables in ` A ` for later use. (Contributed by NM, 9-Apr-1995.) $)
|
|
tfrlem3 $p |- A = { g | E. z e. On ( g Fn z /\
|
|
A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) } $=
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|
( cv wfn cfv cres wceq wral wa con0 wrex vex tfrlem3a abbi2i ) GJZCJZKDJZ
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UBLUBUDMHLNDUCOPCQRGEABCDEFHUBIGSTUA $.
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$}
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${
|
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$d g x F $.
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tfrlem3-2d.1 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
|
|
$( Lemma for transfinite recursion which changes a bound variable
|
|
(Contributed by Jim Kingdon, 2-Jul-2019.) $)
|
|
tfrlem3-2d $p |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) ) $=
|
|
( wfun cv cfv cvv wcel wal weq fveq2 eleq1d anbi2d cbvalv sylib 19.21bi
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wa ) ADFZCGZDHZIJZSZCATBGZDHZIJZSZBKUDCKEUHUDBCBCLZUGUCTUIUFUBIUEUADMNOPQ
|
|
R $.
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|
$}
|
|
|
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${
|
|
$d f g x y z B $. $d a f g h u v w x y z F $. $d g h z A $.
|
|
tfrlem.1 $e |- A = { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
$( Lemma for transfinite recursion. ` A ` is the class of all "acceptable"
|
|
functions, and ` F ` is their union. First we show that an acceptable
|
|
function is in fact a function. (Contributed by NM, 9-Apr-1995.) $)
|
|
tfrlem4 $p |- ( g e. A -> Fun g ) $=
|
|
( vz vw cv wcel wfn cfv cres wceq wral wa con0 wrex wfun abeq2i rexlimivw
|
|
tfrlem3 fnfun adantr sylbi ) EJZCKUGHJZLZIJZUGMUGUJNFMOIUHPZQZHRSZUGTZUME
|
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CABHICDEFGUCUAULUNHRUIUNUKUHUGUDUEUBUF $.
|
|
|
|
$( Lemma for transfinite recursion. The values of two acceptable functions
|
|
are the same within their domains. (Contributed by NM, 9-Apr-1995.)
|
|
(Revised by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlem5 $p |- ( ( g e. A /\ h e. A ) ->
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|
( ( x g u /\ x h v ) -> u = v ) ) $=
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( vz va vw cv wcel cfv wceq wral wa con0 wfn cres wrex wi tfrlem3a reeanv
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wbr vex w3a simp2ll simp3l fnbr syl2anc simp2rl simp3r elin sylanbrc onin
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cin 3ad2ant1 wfun cdm wss fnfun syl inss1 fndm syl5sseqr jca inss2 ssralv
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simp2lr mpsyl simp2rr tfrlem1 eqeq12d sylc funbrfv 3eqtr3d 3exp rexlimdva
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fveq2 rspcv rexlimiv sylbir syl2anb ) GNZEOWGKNZUAZLNZWGPZWGWJUBIPQZLWHRZ
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SZKTUCZHNZMNZUAZWJWPPZWPWJUBIPQZLWQRZSZMTUCZANZDNZWGUGZXDCNZWPUGZSZXEXGQZ
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UDZWPEOABKLEFIWGJGUHUEABMLEFIWPJHUHUEWOXCSWNXBSZMTUCZKTUCXKWNXBKMTTUFXMXK
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KTWHTOZXLXKMTXNWQTOSZXLXIXJXOXLXIUIZXDWGPZXDWPPZXEXGXPXDWHWQUSZOZWKWSQZLX
|
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SRXQXRQZXPXDWHOZXDWQOZXTXPWIXFYCWIWMXBXOXIUJZXOXLXFXHUKZWHXDXEWGULUMXPWRX
|
|
HYDWRXAWNXOXIUNZXOXLXFXHUOZWQXDXGWPULUMXDWHWQUPUQXPLXSIWGWPXOXLXSTOXIWHWQ
|
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URUTXPWGVAZXSWGVBZVCXPWIYIYEWHWGVDVEZXPWHXSYJWHWQVFZXPWIYJWHQYEWHWGVGVEVH
|
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VIXPWPVAZXSWPVBZVCXPWRYMYGWQWPVDVEZXPWQXSYNWHWQVJZXPWRYNWQQYGWQWPVGVEVHVI
|
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XSWHVCXPWMWLLXSRYLWIWMXBXOXIVLWLLXSWHVKVMXSWQVCXPXAWTLXSRYPWRXAWNXOXIVNWT
|
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LXSWQVKVMVOYAYBLXDXSWJXDQWKXQWSXRWJXDWGWBWJXDWPWBVPWCVQXPYIXFXQXEQYKYFXDX
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EWGVRVQXPYMXHXRXGQYOYHXDXGWPVRVQVSVTWAWDWEWF $.
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$( Lemma for transfinite recursion. The definition ` recs ` is the union
|
|
of all acceptable functions. (Contributed by Mario Carneiro,
|
|
9-May-2015.) $)
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|
recsfval $p |- recs ( F ) = U. A $=
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( crecs cv wfn cfv cres wceq wral wa con0 wrex cab cuni df-recs unieqi
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eqtr4i ) EGDHZAHZIBHZUBJUBUDKEJLBUCMNAOPDQZRCRABDESCUEFTUA $.
|
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|
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$( Lemma for transfinite recursion. The union of all acceptable functions
|
|
is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario
|
|
Carneiro, 9-May-2015.) $)
|
|
tfrlem6 $p |- Rel recs ( F ) $=
|
|
( vg crecs wrel cuni reluni wcel wfun tfrlem4 funrel syl mprgbir recsfval
|
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cv releqi mpbir ) EHZICJZIZUDGSZIZGCGCKUECLUEMUFABCDGEFNUEOPQUBUCABCDEFRT
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UA $.
|
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$( Lemma for transfinite recursion. The union of all acceptable functions
|
|
is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario
|
|
Carneiro, 24-May-2019.) $)
|
|
tfrlem7 $p |- Fun recs ( F ) $=
|
|
( vu vv vg vh cv cop wcel wa wal wex eleq2i eluni bitri anbi12i wfun wrel
|
|
crecs wceq wi tfrlem6 cuni recsfval eeanv bitr4i wbr df-br tfrlem5 impcom
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sylanbr an4s exlimivv sylbi ax-gen gen2 dffun4 mpbir2an ) EUCZUAVCUBAKZGK
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ZLZVCMZVDHKZLZVCMZNZVEVHUDZUEZHOZGOAOABCDEFUFVNAGVMHVKVFIKZMZVOCMZNZVIJKZ
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MZVSCMZNZNZJPIPZVLVKVRIPZWBJPZNWDVGWEVJWFVGVFCUGZMWEVCWGVFABCDEFUHZQIVFCR
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SVJVIWGMWFVCWGVIWHQJVICRSTVRWBIJUIUJWCVLIJVPVTVQWAVLVPVTNVDVEVOUKZVDVHVSU
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KZNZVQWANZVLWIVPWJVTVDVEVOULVDVHVSULTWLWKVLABHGCDIJEFUMUNUOUPUQURUSUTAGHV
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|
CVAVB $.
|
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|
|
$( Lemma for transfinite recursion. The domain of ` recs ` is ordinal.
|
|
(Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare,
|
|
11-Mar-2008.) $)
|
|
tfrlem8 $p |- Ord dom recs ( F ) $=
|
|
( vz vg vw cdm word cv wceq wrex cuni con0 wcel cfv rexlimiv ax-mp cab wi
|
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crecs wss wfn cres wral tfrlem3 abeq2i fndm adantr eleq1d biimprcd eleq1a
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wa sylbi syl abssi ssorduni wb ciun recsfval dmeqi vex dmex dfiun2 3eqtri
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dmuni ordeq mpbir ) EUCZJZKZGLZHLZJZMZHCNZGUAZOZKZVSPUDWAVRGPVQVNPQZHCVOC
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QZVPPQZVQWBUBWCVOVNUEZILZVORVOWFUFERMIVNUGZUOZGPNZWDWIHCABGICDHEFUHUIWHWD
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GPWHWDWBWHVPVNPWEVPVNMWGVNVOUJUKULUMSUPVPPVNUNUQSURVSUSTVLVTMVMWAUTVLCOZJ
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HCVPVAVTVKWJABCDEFVBVCHCVHHGCVPVOHVDVEVFVGVLVTVITVJ $.
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|
$( Lemma for transfinite recursion. Here we compute the value of ` recs `
|
|
(the union of all acceptable functions). (Contributed by NM,
|
|
17-Aug-1994.) $)
|
|
tfrlem9 $p |- ( B e. dom recs ( F ) ->
|
|
( recs ( F ) ` B ) = ( F ` ( recs ( F ) |` B ) ) ) $=
|
|
( vz cdm wcel cv wex cfv cres wceq wa con0 wi syl com3l crecs cop ibi wfn
|
|
eldm2g wral wrex cab cuni df-recs eleq2i eluniab bitri wss abeq2i elssuni
|
|
fnop rspe recsfval syl6sseqr sylbir fveq2 reseq2 fveq2d rspcv fndm eleq2d
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eqeq12d wfun tfrlem7 funssfv mp3an1 adantrl eleq1d onelss imp31 fun2ssres
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syl6bir w3a sylan2 exbiri exp31 com34 com24 sylbird syld imp4d mpdi exp4d
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ex com4r pm2.43i imp4a rexlimdv imp exlimiv sylbi ) DFUAZIZJZDHKZUBZWRJZH
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LZDWRMZWRDNZFMZOZWTXDHDWRWSUEUCXCXHHXCXBEKZJZXIAKZUDZBKZXIMZXIXMNZFMZOZBX
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KUFZPZAQUGZPZELZXHXCXBXTEUHUIZJYBWRYCXBABEFUJUKXTEXBULUMYAXHEXJXTXHXJXSXH
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AQXJXKQJZXLXRXHXLXJYDXRXHRZXLXJYDYERRXLXJYDXLYEXLXJYDXLYERRXLXJPZYDXLXRXH
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YFDXKJZYDXSPZXHRXKDXAXIUQYGYHXIWRUNZXHYHXTYIXSAQURXTXICJZYIXTECGUOYJXICUI
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WRXICUPABCEFGUSUTVASYGYDXLXRYIXHRZYGXRXLYDYKYGXRDXIMZXIDNZFMZOZXLYDYKRZRX
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QYOBDXKXMDOZXNYLXPYNXMDXIVBYQXOYMFXMDXIVCVDVHVEXLYGYOYPXLYGDXIIZJZYOYPRXL
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YRXKDXKXIVFZVGXLYDYOYSYKXLYDYSYOYKXLYDYSYOYKRYIXLYDPZYSPZYOXHYIUUBXHYOYIU
|
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UBPXEYLXGYNYIYSXEYLOZUUAWRVIZYIYSUUCABCEFGVJZDWRXIVKVLVMUUBYIDYRUNZXGYNOZ
|
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XLYDYSUUFXLYDYRQJYSUUFRXLYRXKQYTVNYRDVOVRVPUUDYIUUFUUGUUEUUDYIUUFVSXFYMFD
|
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WRXIVQVDVLVTVHWATWBWCWDWETWFWDWGWHSWIWJWKWLTWMWNWOWPWQWPS $.
|
|
$}
|
|
|
|
${
|
|
$d F f x y $.
|
|
$( Transfinite recursion produces a function. (Contributed by Jim Kingdon,
|
|
20-Aug-2021.) $)
|
|
tfrfun $p |- Fun recs ( F ) $=
|
|
( vx vy vf cv wfn cfv cres wceq wral wa con0 wrex cab eqid tfrlem7 ) BCDE
|
|
ZBEZFCEZQGQSHAGICRJKBLMDNZDATOP $.
|
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$}
|
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${
|
|
$d f x y A $. $d x y F $. $d f x y G $.
|
|
tfr.1 $e |- F = recs ( G ) $.
|
|
$( A weak version of transfinite recursion. (Contributed by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
tfr2a $p |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
|
|
( vx vy vf cdm wcel crecs cfv cres wceq cv wfn wral wa con0 wrex cab eqid
|
|
tfrlem9 dmeqi eleq2s fveq1i reseq1i fveq2i 3eqtr4g ) ABHZIACJZKZUJALZCKZA
|
|
BKBALZCKUKUMMAUJHUIEFGNZENZOFNZUOKUOUQLCKMFUPPQERSGTZAGCURUAUBBUJDUCUDABU
|
|
JDUEUNULCBUJADUFUGUH $.
|
|
|
|
$( Transfinite recursion is defined at the empty set. (Contributed by Jim
|
|
Kingdon, 8-Mar-2022.) $)
|
|
tfr0dm $p |- ( ( G ` (/) ) e. V -> (/) e. dom F ) $=
|
|
( vx vy vf c0 cfv wcel cv wfn cres wceq wral wa con0 cvv 0ex mpan cop cdm
|
|
csn wrex opexg snidg fnsng fvsng res0 fveq2i syl6eqr fveq2 reseq2 eqeq12d
|
|
syl fveq2d ralsn sylibr csuc suc0 0elon onsuci fneq2 raleq anbi12d rspcev
|
|
eqeltrri syl2anc wi snexg eleq2 fneq1 fveq1 reseq1 ralbidv rexbidv spcegv
|
|
wex 3syl crecs cab cuni eleq2i df-recs eluniab 3bitri syl6ibr opeldmg mpd
|
|
mp2and ) HBIZCJZHWKUAZAJZHAUBJZWLWMWMUCZJZWPEKZLZFKZWPIZWPWTMZBIZNZFWROZP
|
|
ZEQUDZWNWLWMRJZWQHRJZWLXHSHWKRCUETZWMRUFUOWLWPHUCZLZXDFXKOZXGXIWLXLSHWKRC
|
|
UGTWLHWPIZWPHMZBIZNZXMWLXNWKXPXIWLXNWKNSHWKRCUHTXOHBWPUIUJUKXDXQFHSWTHNZX
|
|
AXNXCXPWTHWPULXRXBXOBWTHWPUMUPUNUQURXKQJXLXMPZXGHUSXKQUTHVAVBVGXFXSEXKQWR
|
|
XKNWSXLXEXMWRXKWPVCXDFWRXKVDVEVFTVHWLWQXGPZWMGKZJZYAWRLZWTYAIZYAWTMZBIZNZ
|
|
FWROZPZEQUDZPZGVRZWNWLXHWPRJXTYLVIXJWMRVJYKXTGWPRYAWPNZYBWQYJXGYAWPWMVKYM
|
|
YIXFEQYMYCWSYHXEWRYAWPVLYMYGXDFWRYMYDXAYFXCWTYAWPVMYMYEXBBYAWPWTVNUPUNVOV
|
|
EVPVEVQVSWNWMBVTZJWMYJGWAWBZJYLAYNWMDWCYNYOWMEFGBWDWCYJGWMWEWFWGWJXIWLWNW
|
|
OVISHWKARCWHTWI $.
|
|
|
|
$( Transfinite recursion at the empty set. (Contributed by Jim Kingdon,
|
|
8-May-2020.) $)
|
|
tfr0 $p |- ( ( G ` (/) ) e. V -> ( F ` (/) ) = ( G ` (/) ) ) $=
|
|
( c0 cfv wcel cres cdm wceq tfr0dm tfr2a syl res0 fveq2i syl6eq ) EBFZCGZ
|
|
EAFZAEHZBFZQREAIGSUAJABCDKEABDLMTEBANOP $.
|
|
$}
|
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|
|
${
|
|
$d e f g h k t u v w x y z A $. $d e f g h k t u v w x y z F $.
|
|
$d ph v w y $.
|
|
tfrlemisucfn.1 $e |- A = { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
tfrlemisucfn.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
|
|
${
|
|
$d u w y ph $.
|
|
tfrlemisucfn.3 $e |- ( ph -> z e. On ) $.
|
|
tfrlemisucfn.4 $e |- ( ph -> g Fn z ) $.
|
|
tfrlemisucfn.5 $e |- ( ph -> g e. A ) $.
|
|
$( We can extend an acceptable function by one element to produce a
|
|
function. Lemma for ~ tfrlemi1 . (Contributed by Jim Kingdon,
|
|
2-Jul-2019.) $)
|
|
tfrlemisucfn $p |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) Fn suc z ) $=
|
|
( cv csuc cfv cop cvv wcel a1i csn cun wfun tfrlem3-2d simprd df-suc wn
|
|
vex eqid elirrv fnunsn ) ADNZULOGNZUMULUMHPZQUAUBZULUNULRSADUHTAHUCUNRS
|
|
ABGHJUDUELUOUIULUFULULSUGADUJTUK $.
|
|
|
|
$( We can extend an acceptable function by one element to produce an
|
|
acceptable function. Lemma for ~ tfrlemi1 . (Contributed by Jim
|
|
Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro,
|
|
24-May-2019.) $)
|
|
tfrlemisucaccv $p |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) e. A ) $=
|
|
( vu cfv wcel wceq wa con0 cvv vw vv cv cop csn cun cres wral wrex csuc
|
|
wfn suceloni syl tfrlemisucfn vex elsuc tfrlem3a simprrr simprrl adantr
|
|
wo sylib fndmu syl2anc raleqdv mpbid rexlimddv wne elirrv elequ2 mtbiri
|
|
r19.21bi necon2ai adantl fvunsng sylancr wss word eloni ordelss resabs1
|
|
sylan wn fsnunres sylancl reseq1d eqtr3d fveq2d 3eqtr4d wfun tfrlem3-2d
|
|
cdm simprd eleq2d fsnunfv syl3anc simpr reseq2 sylan9eqr jaodan sylan2b
|
|
fndm ralrimiva fneq2 raleq anbi12d rspcev syl12anc wb opexg snexg unexg
|
|
tfrlem3ag mpbird ) AGUCZDUCZXOHOZUDZUEZUFZEPZXTUAUCZUKZNUCZXTOZXTYDUGZH
|
|
OZQZNYBUHZRZUASUIZAXPUJZSPZXTYLUKZYHNYLUHZYKAXPSPZYMKXPULUMABCDEFGHIJKL
|
|
MUNAYHNYLYDYLPAYDXPPZYDXPQZVAYHYDXPNUOZUPAYQYHYRAYQRZYDXOOZXOYDUGZHOZYE
|
|
YGAUUAUUCQZNXPAXOUBUCZUKZUUDNUUEUHZRZUUDNXPUHZUBSAXOEPUUHUBSUIMBCUBNEFH
|
|
XOIGUOZUQVBAUUESPZUUHRZRZUUGUUIAUUKUUFUUGURUUMUUDNUUEXPUUMUUFXOXPUKZUUE
|
|
XPQAUUKUUFUUGUSAUUNUULLUTUUEXPXOVCVDVEVFVGVLYTYDTPXPYDVHZYEUUAQYSYQUUOA
|
|
YQXPYDXPYDQYQYDYDPNVIDNNVJVKVMVNXOXPXQYDTVOVPYTYFUUBHYTXTXPUGZYDUGZYFUU
|
|
BYTYDXPVQZUUQYFQAXPVRZYQUURAYPUUSKXPVSUMXPYDVTWBXTYDXPWAUMAUUQUUBQYQAUU
|
|
PXOYDAUUNXPXPPZWCUUPXOQLDVIZXPXOXPXQWDWEZWFUTWGWHWIAYRRZXPXTOZXQYEYGAUV
|
|
DXQQZYRAYPXQTPZXPXOWLZPZWCUVEKAHWJUVFABGHJWKWMZAUVHUUTUVAAUVGXPXPAUUNUV
|
|
GXPQLXPXOXBUMWNVKXOSTXPXQWOWPUTUVCYDXPXTAYRWQWHUVCYFXOHYRAYFUUPXOYDXPXT
|
|
WRUVBWSWHWIWTXAXCYJYNYORUAYLSYBYLQYCYNYIYOYBYLXTXDYHNYBYLXEXFXGXHAXTTPZ
|
|
YAYKXIAXOTPXSTPZUVJUUJAXRTPZUVKAXPTPUVFUVLDUOUVIXPXQTTXJVPXRTXKUMXOXSTT
|
|
XLVPBCUANEFHXTIXMUMXN $.
|
|
$}
|
|
|
|
${
|
|
$d u w f g h z B $. $d g h y z w ph $.
|
|
tfrlemi1.3 $e |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\
|
|
h = ( g u. { <. z , ( F ` g ) >. } ) ) } $.
|
|
tfrlemi1.4 $e |- ( ph -> x e. On ) $.
|
|
tfrlemi1.5 $e |- ( ph -> A. z e. x E. g ( g Fn z /\
|
|
A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) ) $.
|
|
$( Each element of ` B ` is an acceptable function. Lemma for
|
|
~ tfrlemi1 . (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof
|
|
shortened by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlemibacc $p |- ( ph -> B C_ A ) $=
|
|
( cv wcel cfv wa wfn cop csn cun wceq w3a wex wrex simpr3 wfun ad2antrr
|
|
cab cvv con0 simplr onelon syl2anc simpr1 simpr2 tfrlemisucaccv eqeltrd
|
|
wal ex exlimdv rexlimdva abssdv syl5eqss ) AGIQZDQZUAZVHFRZJQZVHVIVHKSU
|
|
BUCUDZUEZUFZIUGZDBQZUHZJULFNAVRJFAVPVLFRZDVQAVIVQRZTZVOVSIWAVOVSWAVOTZV
|
|
LVMFWAVJVKVNUIWBBCDFHIKLAKUJVQKSUMRTBVBVTVOMUKWBVQUNRZVTVIUNRAWCVTVOOUK
|
|
AVTVOUOVQVIUPUQWAVJVKVNURWAVJVKVNUSUTVAVCVDVEVFVG $.
|
|
|
|
$( The union of ` B ` is defined on all ordinals. Lemma for ~ tfrlemi1 .
|
|
(Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario
|
|
Carneiro, 24-May-2019.) $)
|
|
tfrlemibxssdm $p |- ( ph -> x C_ dom U. B ) $=
|
|
( cv wcel wa cvv cuni cdm wral wss wfn cfv cres wceq con0 wi tfrlem3-2d
|
|
wex w3a cop wfun simprd 3ad2ant1 csn cun opexg sylancr snidg elun2 3syl
|
|
simp2r simp3l wrex onelon rspe sylan tfrlem3a sylibr 3adant1 3jca snexg
|
|
vex unexg isset sylib simpr3 19.8a abeq2i sylan2 eqeltrrd 3exp2 exlimdv
|
|
mpan 3imp sylc elunii syl2anc opeq2 eleq1d spcegv eldm2 syl6ibr anassrs
|
|
3expia ralimdva mpdan mpd dfss3 ) ADQZGUAZUBZRZDBQZUCZXGXEUDAIQZXCUEZEQ
|
|
ZXIUFXIXKUGKUFUHEXCUCZSZIULZDXGUCZXHPAXGUIRZXOXHUJOAXPSXNXFDXGAXPXCXGRZ
|
|
XNXFUJAXPXQSZSXMXFIAXRXMXFAXRXMUMZXIKUFZTRZXCXTUNZXDRZXFAXRYAXMAKUOYAAB
|
|
IKMUKUPZUQXSYBXIYBURZUSZRZYFGRZYCAXRYGXMAYBTRZYBYERYGAXCTRYAYIDVPZYDXCX
|
|
TTTUTVAZYBTVBYBYEXIVCVDUQXSXQXJXIFRZUMZJQZYFUHZJULZYHXSXQXJYLAXPXQXMVEA
|
|
XRXJXLVFXRXMYLAXRXMSXMDUIVGZYLXRXCUIRXMYQXGXCVHXMDUIVIVJBCDEFHKXILIVPZV
|
|
KVLVMVNAXRYPXMAYFTRZYPAYIYETRZYSYKYBTVOXITRYTYSYRXIYETTVQWGVDJYFVRVSUQY
|
|
MYOYHJXQXJYLYOYHUJXQXJYLYOYHXQXJYLYOUMZSYNYFGXQXJYLYOVTUUAXQUUAIULZYNGR
|
|
ZUUAIWAXQUUBSUUBDXGVGZUUCUUBDXGVIUUDJGNWBVLWCWDWEWHWFWIYBYFGWJWKYAYCXCX
|
|
KUNZXDRZEULXFUUFYCEXTTXKXTUHUUEYBXDXKXTXCWLWMWNEXCXDYJWOWPWIWRWFWQWSWTX
|
|
ADXGXEXBVL $.
|
|
|
|
$( The union of ` B ` is a function defined on ` x ` . Lemma for
|
|
~ tfrlemi1 . (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof
|
|
shortened by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlemibfn $p |- ( ph -> U. B Fn x ) $=
|
|
( wss cvv wcel syl cuni wfun cdm cv wceq wfn crecs tfrlemibacc recsfval
|
|
unissd syl6sseqr tfrlem7 funss cxp cpw cfv cop csn cun w3a wex wrex cab
|
|
mpisyl wa simpr3 csuc wf wal ad2antrr con0 simplr onelon syl2anc simpr1
|
|
simpr2 tfrlemisucfn dffn2 sylib fssxp word eloni ordsucss sylc xpss1 wb
|
|
sstrd vex tfrlem3-2d simprd opexg sylancr snexg unexg mpbird eqeltrd ex
|
|
elpwg exlimdv rexlimdva abssdv sspwuni dmss dmxpss syl6ss tfrlemibxssdm
|
|
syl5eqss eqssd df-fn sylanbrc ) AGUAZUBZXKUCZBUDZUEXKXNUFAXKKUGZQXOUBXL
|
|
AXKFUAXOAGFABCDEFGHIJKLMNOPUHUJBCFHKLUIUKBCFHKLULXKXOUMVDAXMXNAXMXNRUNZ
|
|
UCZXNAXKXPQZXMXQQAGXPUOZQXRAGIUDZDUDZUFZXTFSZJUDZXTYAXTKUPZUQZURZUSZUEZ
|
|
UTZIVAZDXNVBZJVCXSNAYLJXSAYKYDXSSZDXNAYAXNSZVEZYJYMIYOYJYMYOYJVEZYDYHXS
|
|
YOYBYCYIVFYPYHXSSZYHXPQZYPYHYAVGZRUNZXPYPYSRYHVHZYHYTQYPYHYSUFUUAYPBCDF
|
|
HIKLAKUBZXNKUPRSVEBVIYNYJMVJYPXNVKSZYNYAVKSAUUCYNYJOVJZAYNYJVLZXNYAVMVN
|
|
YOYBYCYIVOYOYBYCYIVPVQYSYHVRVSYSRYHVTTYPYSXNQZYTXPQYPXNWAZYNUUFYPUUCUUG
|
|
UUDXNWBTUUEYAXNWCWDYSXNRWETWGAYQYRWFZYNYJAYHRSZUUHAXTRSYGRSZUUIIWHAYFRS
|
|
ZUUJAYARSYERSZUUKDWHAUUBUULABIKMWIWJYAYERRWKWLYFRWMTXTYGRRWNWLYHXPRWRTV
|
|
JWOWPWQWSWTXAXGGXPXBVSXKXPXCTXNRXDXEABCDEFGHIJKLMNOPXFXHXKXNXIXJ $.
|
|
|
|
$( The set ` B ` exists. Lemma for ~ tfrlemi1 . (Contributed by Jim
|
|
Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro,
|
|
24-May-2019.) $)
|
|
tfrlemibex $p |- ( ph -> B e. _V ) $=
|
|
( cuni cvv wcel cv wfn tfrlemibfn vex fnex sylancl uniexb sylibr ) AGQZ
|
|
RSZGRSAUHBTZUAUJRSUIABCDEFGHIJKLMNOPUBBUCUJRUHUDUEGUFUG $.
|
|
|
|
$( The union of ` B ` satisfies the recursion rule (lemma for
|
|
~ tfrlemi1 ). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof
|
|
shortened by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlemiubacc $p |- ( ph ->
|
|
A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) $=
|
|
( cfv wceq wcel cv cuni cres wral crecs cdm wfn tfrlemibfn fndm syl wss
|
|
tfrlemibacc unissd recsfval syl6sseqr dmss eqsstr3d sselda tfrlem9 wfun
|
|
wa tfrlem7 a1i adantr eleq2d biimpar funssfv syl3anc word eloni ordelss
|
|
sylan sseqtr4d fun2ssres fveq2d 3eqtr3d ralrimiva fveq2 eqeq12d cbvralv
|
|
con0 reseq2 sylibr ) AEUAZHUBZRZWEWDUCZLRZSZEBUAZUDFUAZWERZWEWKUCZLRZSZ
|
|
FWJUDAWIEWJAWDWJTZVAZWDLUEZRZWRWDUCZLRZWFWHWQWDWRUFZTWSXASAWJXBWDAWJWEU
|
|
FZXBAWEWJUGXCWJSZABCDEGHIJKLMNOPQUHWJWEUIUJZAWEWRUKZXCXBUKAWEGUBWRAHGAB
|
|
CDEGHIJKLMNOPQULUMBCGILMUNUOZWEWRUPUJUQURBCGWDILMUSUJWQWRUTZXFWDXCTZWSW
|
|
FSXHWQBCGILMVBVCZAXFWPXGVDZAXIWPAXCWJWDXEVEVFWDWRWEVGVHWQWTWGLWQXHXFWDX
|
|
CUKWTWGSXJXKWQWDWJXCAWJVIZWPWDWJUKAWJWATXLPWJVJUJWJWDVKVLAXDWPXEVDVMWDW
|
|
RWEVNVHVOVPVQWOWIFEWJWKWDSZWLWFWNWHWKWDWEVRXMWMWGLWKWDWEWBVOVSVTWC $.
|
|
|
|
$( Lemma for ~ tfrlemi1 . (Contributed by Jim Kingdon, 18-Mar-2019.)
|
|
(Proof shortened by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlemiex $p |- ( ph ->
|
|
E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) ) $=
|
|
( cvv cv cfv cuni wcel wfn cres wceq wral wex tfrlemibex syl tfrlemibfn
|
|
wa uniexg tfrlemiubacc fneq1 fveq1 reseq1 fveq2d eqeq12d ralbidv spcegv
|
|
jca anbi12d sylc ) AHUAZRUBZVDBSZUCZFSZVDTZVDVHUDZLTZUEZFVFUFZUKZISZVFU
|
|
CZVHVOTZVOVHUDZLTZUEZFVFUFZUKZIUGAHRUBVEABCDEGHIJKLMNOPQUHHRULUIAVGVMAB
|
|
CDEGHIJKLMNOPQUJABCDEFGHIJKLMNOPQUMVAWBVNIVDRVOVDUEZVPVGWAVMVFVOVDUNWCV
|
|
TVLFVFWCVQVIVSVKVHVOVDUOWCVRVJLVOVDVHUPUQURUSVBUTVC $.
|
|
$}
|
|
|
|
$d g u z C $. $d ph e f h z $.
|
|
$( We can define an acceptable function on any ordinal.
|
|
|
|
As with many of the transfinite recursion theorems, we have a hypothesis
|
|
that states that ` F ` is a function and that it is defined for all
|
|
ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened
|
|
by Mario Carneiro, 24-May-2019.) $)
|
|
tfrlemi1 $p |- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\
|
|
A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) $=
|
|
( vz vk vw vv cv cfv wral wa weq ve vt vh con0 wcel wfn cres wex wi simpr
|
|
wceq simpl fneq12d fveq1d reseq1d fveq2d eqeq12d raleqbidv anbi12d imbi2d
|
|
cbvexdva fneq2 raleq exbidv r19.21v cop csn cun w3a wrex cab tfrlem3 wfun
|
|
cvv fveq2 eleq1d anbi2d cbvalv sylib adantr simplr simpll opeq12d uneq12d
|
|
wal 3anbi123d cbvrexdva cbvabv adantl fveq12d reseq12d cbvraldva2 cbvralv
|
|
sneqd tfrlemiex expr expcom a2d syl5bi tfis3 impcom ) FUDUEAHPZFUFZDPZXBQ
|
|
ZXBXDUGZIQZUKZDFRZSZHUHZAXBLPZUFZXHDXLRZSZHUHZUIZAMPZNPZUFZXDXRQZXRXDUGZI
|
|
QZUKZDXSRZSZMUHZUIZAXKUILNFLNTZXPYGAYIXOYFHMYIHMTZSZXMXTXNYEYKXLXSXBXRYIY
|
|
JUJZYIYJULZUMYKXHYDDXLXSYMYKXEYAXGYCYKXDXBXRYLUNYKXFYBIYKXBXRXDYLUOUPUQUR
|
|
USVAUTXLFUKZXPXKAYNXOXJHYNXMXCXNXIXLFXBVBXHDXLFVCUSVDUTYHNXLRAYGNXLRZUIXL
|
|
UDUEZXQAYGNXLVEYPAYOXPAYPYOXPUIAYPYOXPAYPYOSZSLUAOCDEXTXREUEZUBPZXRXSXRIQ
|
|
ZVFZVGZVHZUKZVIZMUHZNXLVJZUBVKHGUCIBCLUAEGHIJVLAIVMZXLIQZVNUEZSZLWEZYQAUU
|
|
HBPZIQZVNUEZSZBWEUULKUUPUUKBLBLTZUUOUUJUUHUUQUUNUUIVNUUMXLIVOVPVQVRVSVTUU
|
|
GGPZOPZUFZUUREUEZUCPZUURUUSUURIQZVFZVGZVHZUKZVIZGUHZOXLVJUBUCUBUCTZUUFUVI
|
|
NOXLUVJNOTZSZUUEUVHMGUVLMGTZSZXTUUTYRUVAUUDUVGUVNXSUUSXRUURUVLUVMUJZUVJUV
|
|
KUVMWAZUMUVNXRUUREUVOVPUVNYSUVBUUCUVFUVJUVKUVMWBUVNXRUURUUBUVEUVOUVNUUAUV
|
|
DUVNXSUUSYTUVCUVPUVNXRUURIUVOUPWCWNWDUQWFVAWGWHYQYPAYPYOULWIYQUUTCPZUURQZ
|
|
UURUVQUGZIQZUKZCUUSRZSZGUHZOXLRZAYQYOUWEYPYOUJYGUWDNOXLUVKYFUWCMGUVKUVMSZ
|
|
XTUUTYEUWBUWFXSUUSXRUURUVKUVMUJUVKUVMULUMUWFYDUWADCXSUUSUWFDCTZSZYAUVRYCU
|
|
VTUWHXDUVQXRUURUVKUVMUWGWAZUWFUWGUJZWJUWHYBUVSIUWHXRUURXDUVQUWIUWJWKUPUQU
|
|
VKUVMUWGWBWLUSVAWMVSWIWOWPWQWRWSWTXA $.
|
|
$}
|
|
|
|
${
|
|
$d f g h u w x y z A $. $d f g h u w x y z F $. $d f g ph w y z $.
|
|
tfrlemi14d.1 $e |- A = { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
tfrlemi14d.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
|
|
$( The domain of ` recs ` is all ordinals (lemma for transfinite
|
|
recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) $)
|
|
tfrlemi14d $p |- ( ph -> dom recs ( F ) = On ) $=
|
|
( vh vz vg vu vw cdm con0 cv wcel wa cfv cvv crecs word tfrlem8 mp1i ciun
|
|
wss ordsson wrex wfn cres wceq wral tfrlemi1 cop csn wfun ad2antrr simplr
|
|
cun wal simprl fneq2 raleq anbi12d rspcev adantll tfrlem3a tfrlemisucaccv
|
|
vex sylibr tfrlem3-2d simprd opexg sylancr snidg 3syl wi opeldmg mpd dmeq
|
|
elun2 eleq2d syl2anc exlimddv eliun ssrdv cuni recsfval dmeqi dmuni eqtri
|
|
ex syl6sseqr eqssd ) AFUAZNZOWPUBWPOUFABCDEFGUCWPUGUDAOIDIPZNZUEZWPAJOWSA
|
|
JPZOQZWTWSQZAXARZWTWRQZIDUHZXBXCKPZWTUIZLPZXFSXFXHUJFSUKZLWTULZRZXEKABCLD
|
|
WTEKFGHUMXCXKRZXFWTXFFSZUNZUOZUSZDQWTXPNZQZXEXLBCJDEKFGAFUPZBPFSTQRBUTXAX
|
|
KHUQAXAXKURXCXGXJVAXLXFMPZUIZXILXTULZRZMOUHZXFDQXAXKYDAYCXKMWTOXTWTUKYAXG
|
|
YBXJXTWTXFVBXILXTWTVCVDVEVFBCMLDEFXFGKVIVGVJVHXLXNXPQZXRAYEXAXKAXNTQZXNXO
|
|
QYEAWTTQZXMTQZYFJVIZAXSYHABKFHVKVLZWTXMTTVMVNXNTVOXNXOXFWAVPUQAYEXRVQZXAX
|
|
KAYGYHYKYIYJWTXMXPTTVRVNUQVSXDXRIXPDWQXPUKWRXQWTWQXPVTWBVEWCWDIWTDWRWEVJW
|
|
LWFWPDWGZNWSWOYLBCDEFGWHWIIDWJWKWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d e f g h u v t w x y z A $. $d e f g h u v t w x y z F $. $d z C $.
|
|
$d u v z ph $.
|
|
tfrexlem.1 $e |- A = { f | E. x e. On ( f Fn x /\
|
|
A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $.
|
|
tfrexlem.2 $e |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) ) $.
|
|
$( The transfinite recursion function is set-like if the input is.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
tfrexlem $p |- ( ( ph /\ C e. V ) -> ( recs ( F ) ` C ) e. _V ) $=
|
|
( vz vv wcel cfv cvv cv wceq con0 wa syl vg vw vu vh vt ve crecs wi fveq2
|
|
eleq1d imbi2d csuc cin cuni wfn cres wral wex word wss inss2 ssorduni vex
|
|
ax-mp sucex inex1 uniex elon2 mpbir2an tfrlem3 wal anbi2d cbvalv tfrlemi1
|
|
wfun sylib mpan2 crn wbr cop recsfval breqi df-br eluni 3bitri sucid wrex
|
|
simpr tfrlem3a simprl simprrl simpll fnop syl2anc onelon rexlimddv adantl
|
|
cdm suceloni onss df-ss unieqd wtr eloni ordtr 3syl unisuc syl5eleqr fndm
|
|
eqtrd ad2antrr eleqtrrd fneq2 raleq anbi12d rspcev sylibr simplrr simplrl
|
|
eldm mpan tfrlem5 imp syl22anc breqtrd exlimddv brelrn elssuni ex exlimdv
|
|
syl5bi alrimiv fvss rnex ssex exlimiv vtoclg impcom ) EHMAEGUGZNZOMZAKPZY
|
|
SNZOMZUHAUUAUHKEHUUBEQZUUDUUAAUUEUUCYTOUUBEYSUIUJUKAUAPZUUBULZULZRUMZUNZU
|
|
OZUBPZUUFNUUFUULUPGNQZUBUUJUQZSZUAURZUUDAUUJRMZUUPUUQUUJUSZUUJOMUUIRUTUUR
|
|
UUHRVAUUIVBVDUUIUUHRUUGUUBKVCZVEZVEVFVGUUJVHVIZAKUCUBDUUJLUAGBCKUCDFLGIVJ
|
|
ZAGVOZBPZGNZOMZSZBVKUVCUUBGNZOMZSZKVKJUVGUVJBKUVDUUBQZUVFUVIUVCUVKUVEUVHO
|
|
UVDUUBGUIUJVLVMVPVNVQUUOUUDUAUUOUUCUUFVRZUNZUTZUUDUUOUUBCPZYSVSZUVOUVMUTZ
|
|
UHZCVKUVNUUOUVRCUVPUUBUVOVTZUDPZMZUVTDMZSZUDURZUUOUVQUVPUUBUVODUNZVSUVSUW
|
|
EMUWDUUBUVOYSUWEKUCDLGUVBWAWBUUBUVOUWEWCUDUVSDWDWEUUOUWCUVQUDUUOUWCUVQUUO
|
|
UWCSZUVOUVLMZUVQUWFUUBUVOUUFVSZUWGUWFUUBUVDUUFVSZUWHBUWFUUBUUFWRZMUWIBURU
|
|
WFUUBUUJUWJUWFUUBUUGUUJUUBUUSWFUWFUUJUUHUNZUUGUWFUUIUUHUWFUUHRUTZUUIUUHQU
|
|
WFUUHRMZUWLUWFUUGRMZUWMUWFUUBRMZUWNUWCUWOUUOUWCUVTUEPZUOZUFPZUVTNUVTUWRUP
|
|
GNQUFUWPUQZSZUWOUERUWCUWBUWTUERWGUWAUWBWHBCUEUFDFGUVTIUDVCWIVPUWCUWPRMZUW
|
|
TSZSZUXAUUBUWPMZUWOUWCUXAUWTWJUXCUWQUWAUXDUWCUXAUWQUWSWKUWAUWBUXBWLUWPUUB
|
|
UVOUVTWMWNUWPUUBWOWNWPWQUUBWSTZUUGWSTUUHWTTUUHRXAVPXBUWFUUGXCZUWKUUGQUWFU
|
|
WNUUGUSUXFUXEUUGXDUUGXEXFUUGUUTXGVPXJXHUUKUWJUUJQUUNUWCUUJUUFXIXKXLBUUBUU
|
|
FUUSXTVPUWFUWISZUUBUVDUVOUUFUWFUWIWHZUXGUUFDMZUWBUWIUUBUVOUVTVSZUVDUVOQZU
|
|
UOUXIUWCUWIUUOUUFLPZUOZUUMUBUXLUQZSZLRWGZUXIUUQUUOUXPUVAUXOUUOLUUJRUXLUUJ
|
|
QUXMUUKUXNUUNUXLUUJUUFXMUUMUBUXLUUJXNXOXPYABCLUBDFGUUFIUAVCZWIXQXKUUOUWAU
|
|
WBUWIXRUXHUXGUWAUXJUUOUWAUWBUWIXSUUBUVOUVTWCXQUXIUWBSUWIUXJSUXKKUCCBDLUAU
|
|
DGUVBYBYCYDYEYFUUBUVOUUFUUSCVCYGTUVOUVLYHTYIYJYKYLCUUBUVMYSYMTUUCUVMUVLUU
|
|
FUXQYNVGYOTYPTYQYR $.
|
|
$}
|
|
|
|
${
|
|
$d G f g u w x y z $. $d ph f y $.
|
|
tfri1d.1 $e |- F = recs ( G ) $.
|
|
tfri1d.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
|
|
$( Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
|
|
[TakeutiZaring] p. 47, with an additional condition.
|
|
|
|
The condition is that ` G ` is defined "everywhere", which is stated
|
|
here as ` ( G `` x ) e. _V ` . Alternately,
|
|
` A. x e. On A. f ( f Fn x -> f e. dom G ) ` would suffice.
|
|
|
|
Given a function ` G ` satisfying that condition, we define a class
|
|
` A ` of all "acceptable" functions. The final function we're
|
|
interested in is the union ` F = recs ( G ) ` of them. ` F ` is then
|
|
said to be defined by transfinite recursion. The purpose of the 3 parts
|
|
of this theorem is to demonstrate properties of ` F ` . In this first
|
|
part we show that ` F ` is a function whose domain is all ordinal
|
|
numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario
|
|
Carneiro, 24-May-2019.) $)
|
|
tfri1d $p |- ( ph -> F Fn On ) $=
|
|
( vy vg vz vu vf vw con0 wfn wceq wa cv cfv cres wral crecs wfun cdm wrex
|
|
cab eqid tfrlem3 tfrlemi14d tfrlem7 jctil df-fn sylibr fneq1i ) ADUAZMNZC
|
|
MNAUNUBZUNUCMOZPUOAUQUPABGHQZIQZNJQZURRURUTSDROJUSTPIMUDHUEZKDIJBGVAHKDVA
|
|
UFUGFUHGILQZGQZNUSVBRVBUSSDROIVCTPGMUDLUEZLDVDUFUIUJUNMUKULMCUNEUMUL $.
|
|
|
|
$( Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of
|
|
[TakeutiZaring] p. 47, with an additional condition on the recursion
|
|
rule ` G ` ( as described at ~ tfri1 ). Here we show that the function
|
|
` F ` has the property that for any function ` G ` satisfying that
|
|
condition, the "next" value of ` F ` is ` G ` recursively applied to all
|
|
"previous" values of ` F ` . (Contributed by Jim Kingdon,
|
|
4-May-2019.) $)
|
|
tfri2d $p |- ( ( ph /\ A e. On ) -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
|
|
( con0 wcel wa cdm cfv cres wceq wfn tfri1d fndm syl eleq2d biimpar tfr2a
|
|
) ACHIZJCDKZIZCDLDCMELNAUDUBAUCHCADHOUCHNABDEFGPHDQRSTCDEFUAR $.
|
|
$}
|
|
|
|
${
|
|
$d G f w x y z $. $d H f w x y z $. $d X f x z $.
|
|
tfr1onlem3ag.1 $e |- A = { f | E. x e. X ( f Fn x
|
|
/\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
|
|
$( Lemma for transfinite recursion. This lemma changes some bound
|
|
variables in ` A ` (version of ~ tfrlem3ag but for ~ tfr1on related
|
|
lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) $)
|
|
tfr1onlem3ag $p |- ( H e. V -> ( H e. A <-> E. z e. X ( H Fn z
|
|
/\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) ) $=
|
|
( cv wfn cfv cres wceq wral wa wrex fneq12 simpll fveq12d reseq12d fveq2d
|
|
simpr eqeq12d simplr cbvraldva2 anbi12d cbvrexdva elab2g ) FLZALZMZBLZULN
|
|
ZULUOOZGNZPZBUMQZRZAJSHCLZMZDLZHNZHVDOZGNZPZDVBQZRZCJSFHEIULHPZVAVJACJVKU
|
|
MVBPZRZUNVCUTVIUMVBULHTVMUSVHBDUMVBVMUOVDPZRZUPVEURVGVOUOVDULHVKVLVNUAZVM
|
|
VNUEZUBVOUQVFGVOULHUOVDVPVQUCUDUFVKVLVNUGUHUIUJKUK $.
|
|
|
|
$d A g $. $d f g w x y z $.
|
|
$( Lemma for transfinite recursion. This lemma changes some bound
|
|
variables in ` A ` (version of ~ tfrlem3 but for ~ tfr1on related
|
|
lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.) $)
|
|
tfr1onlem3 $p |- A = { g | E. z e. X ( g Fn z
|
|
/\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) } $=
|
|
( cv wfn cfv cres wceq wral wa wrex cvv wcel wb tfr1onlem3ag ax-mp abbi2i
|
|
vex ) GKZCKZLDKZUFMUFUHNHMODUGPQCIRZGEUFSTUFETUIUAGUEABCDEFHUFSIJUBUCUD
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d G f x y $. $d X x $. $d f ph $.
|
|
tfr1onlemssrecs.1 $e |- A = { f | E. x e. X ( f Fn x
|
|
/\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
|
|
tfr1onlemssrecs.x $e |- ( ph -> Ord X ) $.
|
|
$( Lemma for ~ tfr1on . The union of functions acceptable for ~ tfr1on is
|
|
a subset of ` recs ` . (Contributed by Jim Kingdon, 15-Mar-2022.) $)
|
|
tfr1onlemssrecs $p |- ( ph -> U. A C_ recs ( G ) ) $=
|
|
( cuni cv wfn cfv cres wceq wral wa con0 wrex cab word wss ordsson ssrexv
|
|
crecs wi 3syl ss2abdv syl5eqss unissd df-recs syl6sseqr ) ADJEKZBKZLCKZUM
|
|
MUMUONFMOCUNPQZBRSZETZJFUEADURADUPBGSZETURHAUSUQEAGUAGRUBUSUQUFIGUCUPBGRU
|
|
DUGUHUIUJBCEFUKUL $.
|
|
$}
|
|
|
|
${
|
|
tfr1on.f $e |- F = recs ( G ) $.
|
|
tfr1on.g $e |- ( ph -> Fun G ) $.
|
|
tfr1on.x $e |- ( ph -> Ord X ) $.
|
|
tfr1on.ex $e |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V ) $.
|
|
${
|
|
tfr1onlemsucfn.1 $e |- A = { f | E. x e. X ( f Fn x
|
|
/\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
|
|
|
|
${
|
|
$d G f x $. $d X f x $. $d f g $. $d f ph x $. $d f x z $.
|
|
tfr1onlemsucfn.3 $e |- ( ph -> z e. X ) $.
|
|
tfr1onlemsucfn.4 $e |- ( ph -> g Fn z ) $.
|
|
tfr1onlemsucfn.5 $e |- ( ph -> g e. A ) $.
|
|
$( We can extend an acceptable function by one element to produce a
|
|
function. Lemma for ~ tfr1on . (Contributed by Jim Kingdon,
|
|
12-Mar-2022.) $)
|
|
tfr1onlemsucfn $p |- ( ph
|
|
-> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z ) $=
|
|
( cv wcel csuc cfv cop csn cun elexd wfn cvv wi wal weq imbi1d albidv
|
|
fneq2 3expia alrimiv ralrimiva rspcdva fneq1 fveq2 eleq1d imbi12d spv
|
|
wa sylc eqid df-suc con0 word wn ordelon syl2anc eloni ordirr fnunsn
|
|
3syl ) ADSZVQUAGSZVRVQVRIUBZUCUDUEZVQVSAVQJPUFAFSZVQUGZWAIUBZUHTZUIZF
|
|
UJZVRVQUGZVSUHTZAWABSZUGZWDUIZFUJZWFBJVQBDUKZWKWEFWMWJWBWDWIVQWAUNULU
|
|
MAWLBJAWIJTZVDWKFAWNWJWDNUOUPUQPURQWEWGWHUIFGFGUKZWBWGWDWHVQWAVRUSWOW
|
|
CVSUHWAVRIUTVAVBVCVEQVTVFVQVGAVQVHTZVQVIVQVQTVJAJVIVQJTWPMPJVQVKVLVQV
|
|
MVQVNVPVO $.
|
|
$}
|
|
|
|
${
|
|
$d G f u v x y $. $d G f u w x y $. $d X f v x $. $d X f w x $.
|
|
$d f g u v x y $. $d f ph u v x $. $d f u v x y z $. $d g u w x y $.
|
|
$d w x y z $.
|
|
tfr1onlemsucaccv.yx $e |- ( ph -> Y e. X ) $.
|
|
tfr1onlemsucaccv.zy $e |- ( ph -> z e. Y ) $.
|
|
tfr1onlemsucaccv.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
tfr1onlemsucaccv.gfn $e |- ( ph -> g Fn z ) $.
|
|
tfr1onlemsucaccv.gacc $e |- ( ph -> g e. A ) $.
|
|
$( Lemma for ~ tfr1on . We can extend an acceptable function by one
|
|
element to produce an acceptable function. (Contributed by Jim
|
|
Kingdon, 12-Mar-2022.) $)
|
|
tfr1onlemsucaccv $p |- ( ph
|
|
-> ( g u. { <. z , ( G ` g ) >. } ) e. A ) $=
|
|
( vw vu vv cv cfv cop csn cun wcel wfn cres wceq wral wrex csuc suceq
|
|
wa cuni eleq1d ralrimiva elunii syl2anc rspcdva ordtr1 tfr1onlemsucfn
|
|
word jca wo vex elsuc cvv wb tfr1onlem3ag ax-mp sylib simprrr simprrl
|
|
sylc adantr fndmu raleqdv mpbid rexlimddv r19.21bi wne con0 wn onelon
|
|
ordelon sylan eloni ordirr 3syl wi elequ2 biimpcd adantl mtod fvunsng
|
|
neqned sylancr wss syl ordelss resabs1 fsnunres reseq1d eqtr3d fveq2d
|
|
3eqtr4d cdm wal fneq2 imbi1d albidv 3expia alrimiv fneq1 imbi12d fndm
|
|
fveq2 neleqtrrd fsnunfv syl3anc simpr reseq2 sylan9eqr jaodan sylan2b
|
|
spv raleq anbi12d rspcev syl12anc opexg snexg unexg mpbird ) AGUEZDUE
|
|
ZYTIUFZUGZUHZUIZEUJZUUEUBUEZUKZUCUEZUUEUFZUUEUUIULZIUFZUMZUCUUGUNZURZ
|
|
UBJUOZAUUAUPZJUJZUUEUUQUKZUUMUCUUQUNZUUPABUEZUPZJUJZUURBJUSZUUAUVAUUA
|
|
UMZUVBUUQJUVAUUAUQUTAUVCBUVDSVAAUUAKUJZKJUJZUUAUVDUJRQUUAKJVBVCVDABCD
|
|
EFGHIJLMNOPAJVGZUVFUVGURUUAJUJZNAUVFUVGRQVHUUAKJVEVSZTUAVFAUUMUCUUQUU
|
|
IUUQUJAUUIUUAUJZUUIUUAUMZVIUUMUUIUUAUCVJZVKAUVKUUMUVLAUVKURZUUIYTUFZY
|
|
TUUIULZIUFZUUJUULAUVOUVQUMZUCUUAAYTUDUEZUKZUVRUCUVSUNZURZUVRUCUUAUNZU
|
|
DJAYTEUJZUWBUDJUOZUAYTVLUJZUWDUWEVMGVJZBCUDUCEFIYTVLJPVNVOVPAUVSJUJZU
|
|
WBURZURZUWAUWCAUWHUVTUWAVQUWJUVRUCUVSUUAUWJUVTYTUUAUKZUVSUUAUMAUWHUVT
|
|
UWAVRAUWKUWITVTUVSUUAYTWAVCWBWCWDWEUVNUUIVLUJUUAUUIWFUUJUVOUMUVMUVNUU
|
|
AUUIUVNUUAUUIUMZUUIUUIUJZUVNUUIWGUJZUUIVGUWMWHAUUAWGUJZUVKUWNAUVHUVIU
|
|
WONUVJJUUAWJVCZUUAUUIWIWKUUIWLUUIWMWNUVKUWLUWMWOAUWLUVKUWMDUCUCWPWQWR
|
|
WSXAYTUUAUUBUUIVLWTXBUVNUUKUVPIUVNUUEUUAULZUUIULZUUKUVPUVNUUIUUAXCZUW
|
|
RUUKUMAUUAVGZUVKUWSAUWOUWTUWPUUAWLXDZUUAUUIXEWKUUEUUIUUAXFXDAUWRUVPUM
|
|
UVKAUWQYTUUIAUWKUUAUUAUJWHZUWQYTUMTAUWTUXBUXAUUAWMXDZUUAYTUUAUUBXGVCZ
|
|
XHVTXIXJXKAUVLURZUUAUUEUFZUUBUUJUULAUXFUUBUMZUVLAUVFUUBVLUJZUUAYTXLZU
|
|
JWHUXGRAFUEZUUAUKZUXJIUFZVLUJZWOZFXMZUWKUXHAUXJUVAUKZUXMWOZFXMZUXOBJU
|
|
UAUVEUXQUXNFUVEUXPUXKUXMUVAUUAUXJXNXOXPAUXRBJAUVAJUJZURUXQFAUXSUXPUXM
|
|
OXQXRVAUVJVDTUXNUWKUXHWOFGUXJYTUMZUXKUWKUXMUXHUUAUXJYTXSUXTUXLUUBVLUX
|
|
JYTIYBUTXTYKVSZAUXIUUAUUAUXCAUWKUXIUUAUMTUUAYTYAXDYCYTKVLUUAUUBYDYEVT
|
|
UXEUUIUUAUUEAUVLYFXJUXEUUKYTIUVLAUUKUWQYTUUIUUAUUEYGUXDYHXJXKYIYJVAUU
|
|
OUUSUUTURUBUUQJUUGUUQUMUUHUUSUUNUUTUUGUUQUUEXNUUMUCUUGUUQYLYMYNYOAUUE
|
|
VLUJZUUFUUPVMAUWFUUDVLUJZUYBUWGAUUCVLUJZUYCAUUAVLUJUXHUYDDVJUYAUUAUUB
|
|
VLVLYPXBUUCVLYQXDYTUUDVLVLYRXBBCUBUCEFIUUEVLJPVNXDYS $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h x z $. $d D f g x $. $d G f x y $. $d X f x $.
|
|
$d f g h ph x z $. $d f g x y z $.
|
|
tfr1onlembacc.3 $e |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\
|
|
h = ( g u. { <. z , ( G ` g ) >. } ) ) } $.
|
|
tfr1onlembacc.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
tfr1onlembacc.4 $e |- ( ph -> D e. X ) $.
|
|
tfr1onlembacc.5 $e |- ( ph -> A. z e. D E. g ( g Fn z /\
|
|
A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) $.
|
|
$( Lemma for ~ tfr1on . Each element of ` B ` is an acceptable
|
|
function. (Contributed by Jim Kingdon, 14-Mar-2022.) $)
|
|
tfr1onlembacc $p |- ( ph -> B C_ A ) $=
|
|
( wfn wcel cfv cop csn cun wceq w3a wex wrex cab simpr3 wfun ad2antrr
|
|
cv word cvv 3adant1r simplr cuni csuc adantlr simpr1 tfr1onlemsucaccv
|
|
wa simpr2 eqeltrd ex exlimdv rexlimdva abssdv syl5eqss ) AGJURZDURZUD
|
|
ZVPFUEZKURZVPVQVPMUFUGUHUIZUJZUKZJULZDHUMZKUNFTAWEKFAWDVTFUEZDHAVQHUE
|
|
ZVHZWCWFJWHWCWFWHWCVHZVTWAFWHVRVSWBUOWIBCDFIJLMNHOAMUPWGWCPUQANUSWGWC
|
|
QUQWHBURZNUEZIURZWJUDZWLMUFUTUEZWCAWKWMWNWGRVAVASAHNUEWGWCUBUQAWGWCVB
|
|
WHWJNVCUEZWJVDNUEZWCAWOWPWGUAVEVEWHVRVSWBVFWHVRVSWBVIVGVJVKVLVMVNVO
|
|
$.
|
|
|
|
${
|
|
$d A f g h x z $. $d B g h z $. $d B g w z $. $d D f g h x z $.
|
|
$d G f h x z $. $d G f w x y z $. $d X f g x z $.
|
|
$d f g h ph x z $. $d f g x y z $.
|
|
$( Lemma for ~ tfr1on . The union of ` B ` is defined on all
|
|
elements of ` X ` . (Contributed by Jim Kingdon, 14-Mar-2022.) $)
|
|
tfr1onlembxssdm $p |- ( ph -> D C_ dom U. B ) $=
|
|
( cv cuni cdm wcel wral wss wfn cfv cres wceq wex w3a cvv cop simp1
|
|
wa simp2 syl word wi ordtr1 imp syl12anc simp3l fneq2 imbi1d albidv
|
|
wal 3expia alrimiv ralrimiva adantr simpr rspcdva fneq1 imbi12d spv
|
|
fveq2 eleq1d syl21anc csn cun opexg sylancr snidg elun2 3syl simp3r
|
|
vex wrex rspe tfr1onlem3ag ax-mp sylibr 3jca snexg unexg mpan isset
|
|
sylib simpr3 19.8a abeq2i sylan2 eqeltrrd 3exp2 3imp exlimdv elunii
|
|
wb sylc syl2anc opeq2 spcegv eldm2 syl6ibr ralimdva mpd dfss3 ) ADU
|
|
DZGUEZUFZUGZDHUHZHYEUIAJUDZYCUJZEUDZYHUKYHYJULMUKUMEYCUHZUSZJUNZDHU
|
|
HYGUCAYMYFDHAYCHUGZUSYLYFJAYNYLYFAYNYLUOZYHMUKZUPUGZYCYPUQZYDUGZYFY
|
|
OAYCNUGZYIYQAYNYLURZYOAYNHNUGZYTUUAAYNYLUTZYOAUUBUUAUBVAAYNUUBUSZYT
|
|
ANVBUUDYTVCQYCHNVDVAVEVFZAYNYIYKVGZAYTUSZYIYQUUGIUDZYCUJZUUHMUKZUPU
|
|
GZVCZIVKZYIYQVCZUUGUUHBUDZUJZUUKVCZIVKZUUMBNYCUUOYCUMZUUQUULIUUSUUP
|
|
UUIUUKUUOYCUUHVHVIVJAUURBNUHYTAUURBNAUUONUGZUSUUQIAUUTUUPUUKRVLVMVN
|
|
VOAYTVPVQUULUUNIJUUHYHUMZUUIYIUUKYQYCUUHYHVRUVAUUJYPUPUUHYHMWAWBVSV
|
|
TVAVEWCZYOYRYHYRWDZWEZUGZUVDGUGZYSYOYRUPUGZYRUVCUGUVEYOYCUPUGYQUVGD
|
|
WLZUVBYCYPUPUPWFWGZYRUPWHYRUVCYHWIWJYOYNYIYHFUGZUOZKUDZUVDUMZKUNZUV
|
|
FYOYNYIUVJUUCUUFYOYLDNWMZUVJYOYTYIYKUVOUUEUUFAYNYIYKWKYLDNWNVFYHUPU
|
|
GZUVJUVOXMJWLZBCDEFIMYHUPNSWOWPWQWRYOUVDUPUGZUVNYOUVGUVCUPUGZUVRUVI
|
|
YRUPWSUVPUVSUVRUVQYHUVCUPUPWTXAWJKUVDXBXCUVKUVMUVFKYNYIUVJUVMUVFVCY
|
|
NYIUVJUVMUVFYNYIUVJUVMUOZUSUVLUVDGYNYIUVJUVMXDUVTYNUVTJUNZUVLGUGZUV
|
|
TJXEYNUWAUSUWADHWMZUWBUWADHWNUWCKGTXFWQXGXHXIXJXKXNYRUVDGXLXOYQYSYC
|
|
YJUQZYDUGZEUNYFUWEYSEYPUPYJYPUMUWDYRYDYJYPYCXPWBXQEYCYDUVHXRXSXNVLX
|
|
KXTYADHYEYBWQ $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h x z $. $d B g h z w $. $d D f g h x z $.
|
|
$d G f h x y z w $. $d X f g x z $. $d f g h ph x z $.
|
|
$d f g x y z w $.
|
|
$( Lemma for ~ tfr1on . The union of ` B ` is a function defined on
|
|
` x ` . (Contributed by Jim Kingdon, 15-Mar-2022.) $)
|
|
tfr1onlembfn $p |- ( ph -> U. B Fn D ) $=
|
|
( cuni wfun cdm wceq wfn crecs tfr1onlembacc unissd tfr1onlemssrecs
|
|
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funss wrex cab wa simpr3 csuc wf simpl word adantr simpr jca ordtr1
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dmss ) AGUDZUEZYTUFZHUGYTHUHAYTMUIZUMUUCUEUUAAYTFUDUUCAGFABCDEFGHIJ
|
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BUVLBUTZNVAZIUTZUVQUHZUVSMVBZUQVAZUURAUVRUVTUWBUVKRWJWJSAUVKUURWCUV
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NXEUUJAUWOBNAUVRVKUWNIAUVRUVTUWBRXFXGXHXIAUVKUUJXJXLAUVKUUJXMUWLUUJ
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$}
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${
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$d G f h w x y z $. $d X f g x z $. $d f g h ph x z $.
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|
$d f g x y z $.
|
|
$( Lemma for ~ tfr1on . The set ` B ` exists. (Contributed by Jim
|
|
Kingdon, 14-Mar-2022.) $)
|
|
tfr1onlembex $p |- ( ph -> B e. _V ) $=
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( cuni cvv wcel wfn tfr1onlembfn fnex syl2anc uniexb sylibr ) AGUDZ
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UEUFZGUEUFAUMHUGHNUFUNABCDEFGHIJKLMNOPQRSTUAUBUCUHUBHNUMUIUJGUKUL
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$.
|
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$}
|
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|
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${
|
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$d A f g h x z $. $d B g h w z $. $d B u w $. $d D f g h w x z $.
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$d f g h ph x z $. $d f g x y z $. $d ph w x z $. $d w x y z $.
|
|
$( Lemma for ~ tfr1on . The union of ` B ` satisfies the recursion
|
|
rule. (Contributed by Jim Kingdon, 15-Mar-2022.) $)
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tfr1onlemubacc $p |- ( ph ->
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A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
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( cv cuni cfv cres wceq wral wcel wa crecs cdm wfn tfr1onlembfn syl
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fndm wss tfr1onlembacc unissd tfr1onlemssrecs sstrd eqsstr3d sselda
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dmss con0 wrex eqid tfrlem9 tfrfun eleq2d biimpar funssfv mp3an2ani
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cab wfun word ordelon syl2anc eloni sylan adantr sseqtr4d fun2ssres
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ordelss fveq2d 3eqtr3d ralrimiva reseq2 eqeq12d cbvralv sylibr
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CJUEZBUEZUOCUEZXRUGXRXTUHNUGUICXSUJULBVGVHJVPZWOJNYAVIVJUQXHVQZAXPX
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FWOXMUKZXIWQUINVKZXQAYCXFAXMIWOXOVLVMWOXHWPVNVOXGXJWRNYBAXPXFWOXMUS
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$}
|
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$d X f g x z $. $d f g h ph x z $. $d f g x y z $. $d ph w x z $.
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$d w x y z $.
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|
$( Lemma for ~ tfr1on . (Contributed by Jim Kingdon,
|
|
16-Mar-2022.) $)
|
|
tfr1onlemex $p |- ( ph ->
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E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) ) $=
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( cuni cvv wcel wfn cv cfv cres wceq wa wex tfr1onlembex uniexg syl
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|
wral tfr1onlembfn tfr1onlemubacc fneq1 fveq1 reseq1 eqeq12d ralbidv
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jca fveq2d anbi12d spcegv sylc ) AHUEZUFUGZVKIUHZFUIZVKUJZVKVNUKZNU
|
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HIJKLMNOPQRSTUAUBUCUDUOHUFUPUQAVMVSABCDEGHIJKLMNOPQRSTUAUBUCUDUSABC
|
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|
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$}
|
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|
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$d A a b c d h u x $. $d A a b c d h r s t u $. $d C g u z $.
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$d G a b c d g h u w x z $. $d G f x y z $. $d G a b c d h r s t u $.
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$d X a b c d h w x z $. $d X f h x z $. $d a b c d h ph w x z $.
|
|
$d a b c d g h x y z $. $d f h ph x z $. $d s t u z $.
|
|
tfr1onlemaccex.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
$( We can define an acceptable function on any element of ` X ` .
|
|
|
|
As with many of the transfinite recursion theorems, we have
|
|
hypotheses that state that ` F ` is a function and that it is
|
|
defined up to ` X ` . (Contributed by Jim Kingdon, 16-Mar-2022.) $)
|
|
tfr1onlemaccex $p |- ( ( ph /\ C e. X ) -> E. g ( g Fn C /\
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A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) ) $=
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( vw wa cfv vz vh vb vc vr vt vs va vd con0 wcel cv wfn cres wceq wex
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wral word ordelon sylan weq eleq1 anbi2d fneq2 anbi12d exbidv imbi12d
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wi raleq cop csn cun w3a wrex cab wfun ad3antrrr cvv wal 3expia fneq1
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alrimiv fveq2 eleq1d cbvalv sylib 3impia 3adant1r fveq1 reseq1 fveq2d
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sneqd 3anbi123d cbvexv rexbii opeq1 uneq2d 3anbi13d bitri abbii eqeq1
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syl mp2and nfv nfra1 nfan rspcda reseq2 anbi2i exbii syl6ib ralrimiva
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cbvralv mpd tfr1onlemex exp31 expcom a2d impexp ralbii r19.21v tfis3
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3imtr4g mpcom ) FUJUKZAFKUKZSZHULZFUMZDULZUUHTZUUHUUJUNZJTZUOZDFUQZSZ
|
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HUPZAKURZUUFUUENKFUSUTAUAULZKUKZSZUUHUUSUMZUUNDUUSUQZSZHUPZVHZARULZKU
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|
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|
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VDUUNDUUSFVIVEVFVGUUSUJUKZAUVHUVMVHZRUUSUQZVHZAUUTUVEVHZVHUVNRUUSUQZU
|
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VFUVQAUVSUWAAUVQUVSUWAVHAUVQSZUVSUUTUVEUWCUVSSZUUTSZUBULZUUSUMZUUJUWF
|
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TZUWFUUJUNZJTZUOZDUUSUQZSZUBUPUVEUWEBCUCUDDEUEULZUFULZUMZUWNEUKZUGULZ
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UWNUWOUWNJTZVJZVKZVLZUOZVMZUEUPZUFUUSVNZUGVOZUUSUBUHUIIJKLAJVPUVQUVSU
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UTMVQAUURUVQUVSUUTNVQUWDBULZKUKZUWFUXHUMZUWFJTZVRUKZUUTUWCUXIUXJUXLUV
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SAUXIUXJUXLUVQAUXIUXJUXLAUXISZUXJUXLVHZUBUXMGULZUXHUMZUXOJTZVRUKZVHZG
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XBVUQUWRVVAUWNUYQUXAVUPVVAWRVVAUWTVUOVVAUWSVUAUWOUWNUYQJWCWSXBWTXAXCX
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UGVULUCUUSVVCVUFVUKUHVVCVUEVUJUYSUYTUWRVUIVUDXKXMVFWOWPWQUWDUXHKXNUKZ
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$}
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${
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$d A g h z $. $d A g u x z $. $d G f g u w x y $. $d G g h $.
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$d X f w x $. $d Y f g x z $. $d f g ph x z $. $d w x y z $.
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tfr1onlemres.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
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tfr1onlemres.yx $e |- ( ph -> Y e. X ) $.
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$( Lemma for ~ tfr1on . Recursion is defined on an ordinal if the
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characteristic function is defined up to a suitable point.
|
|
(Contributed by Jim Kingdon, 18-Mar-2022.) $)
|
|
tfr1onlemres $p |- ( ph -> Y C_ dom F ) $=
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( vg cv wcel cvv vh vz vu vw crecs cdm ciun wa wrex wfn cfv cres wceq
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wral wex word adantr simpr jca ordtr1 sylc tfr1onlemaccex cop csn cun
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syldan wfun ad2antrr 3adant1r cuni adantlr simprl fneq2 raleq anbi12d
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rspcev sylan wb vex tfr1onlem3ag ax-mp sylibr tfr1onlemsucaccv imbi1d
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csuc wi wal 3expia alrimiv fneq1 fveq2 eleq1d imbi12d spv syl rspcdva
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ralrimiva imp opexg sylancr snidg elun2 3syl opeldmg mpd dmeq syl2anc
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eleq2d exlimddv eliun ssrdv dmuni wss tfr1onlemssrecs syl5eqssr sstrd
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ex dmss dmeqi syl6sseqr ) AIGUEZUFZFUFAIUADUARZUFZUGZYBAUBIYEAUBRZISZ
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AYGYFHSZYQQUOYIHUPZYGIHSZUHYRAYSYGLUQYIYGYTAYGURZAYTYGPUQUSYFIHUTVAZA
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DEQFGHIJAGVGYGYQKVHAYSYGYQLVHYIBRZHSZERZUUJUJZUULGUKZTSZYQAUUKUUMUUOY
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GMVIVINAYTYGYQPVHYIYGYQUUAUQYIUUJHVJSZUUJWEHSZYQAUUPUUQYGOVKVKYIYMYPV
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$}
|
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$}
|
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|
|
$d G a b c f x y $. $d X a b f x $. $d Y f x $. $d f ph x $.
|
|
tfr1on.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
tfr1on.yx $e |- ( ph -> Y e. X ) $.
|
|
$( Recursion is defined on an ordinal if the characteristic function is
|
|
defined up to a suitable point. (Contributed by Jim Kingdon,
|
|
12-Mar-2022.) $)
|
|
tfr1on $p |- ( ph -> Y C_ dom F ) $=
|
|
( vy va vb vc cv wfn cfv cres wceq wral wrex eqid tfr1onlem3 tfr1onlemres
|
|
wa cab ) ABNORZPRZSQRZUJTUJULUAETUBQUKUCUHPFUDOUIZCDEFGHIJKPQBNUMOCEFUMUE
|
|
UFLMUG $.
|
|
$}
|
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|
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${
|
|
$d F z $. $d G a b c $. $d G f x y z $. $d f ph y z $. $d G f x y z $.
|
|
$d ph f y $.
|
|
tfri1dALT.1 $e |- F = recs ( G ) $.
|
|
tfri1dALT.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
|
|
$( Alternate proof of ~ tfri1d in terms of ~ tfr1on .
|
|
|
|
Although this does show that the ~ tfr1on proof is general enough to
|
|
also prove ~ tfri1d , the ~ tfri1d proof is simpler in places because it
|
|
does not need to deal with ` X ` being any ordinal. For that reason, we
|
|
have both proofs. (Proof modification is discouraged.)
|
|
(New usage is discouraged.) (Contributed by Jim Kingdon,
|
|
20-Mar-2022.) $)
|
|
tfri1dALT $p |- ( ph -> F Fn On ) $=
|
|
( vb vc va vz vf wfun con0 wceq wfn word cv cfv wa wcel cdm tfrfun funeqi
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|
vy crecs mpbir a1i wss cres wral wrex eqid tfrlem8 wb dmeqi ordeq ordsson
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syl fveq2 spv 3syl 3ad2ant1 cuni suceloni unon eleq2s adantl tfr1on sucid
|
|
vex ssel2 sylancl ex ssrdv eqssd df-fn sylanbrc ) ACLZCUAZMNCMOWKAWKDUEZL
|
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|
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|
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CWBXEWLXAWDWEWFWGWHCMWIWJ $.
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$}
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${
|
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$d G f x y $. $d X x $. $d f ph $.
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tfrcllemssrecs.1 $e |- A = { f | E. x e. X ( f : x --> S
|
|
/\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } $.
|
|
tfrcllemssrecs.x $e |- ( ph -> Ord X ) $.
|
|
$( Lemma for ~ tfrcl . The union of functions acceptable for ~ tfrcl is a
|
|
subset of ` recs ` . (Contributed by Jim Kingdon, 25-Mar-2022.) $)
|
|
tfrcllemssrecs $p |- ( ph -> U. A C_ recs ( G ) ) $=
|
|
( cuni cv wf cfv cres wceq wa con0 wrex cab wral crecs word wss wi ssrexv
|
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ordsson 3syl ss2abdv syl5eqss unissd wfn ffn anim1i reximi ss2abi df-recs
|
|
unissi sseqtr4i syl6ss ) ADKBLZEFLZMZCLZVBNVBVDOGNPCVAUAZQZBRSZFTZKZGUBZA
|
|
DVHADVFBHSZFTVHIAVKVGFAHUCHRUDVKVGUEJHUGVFBHRUFUHUIUJUKVIVBVAULZVEQZBRSZF
|
|
TZKVJVHVOVGVNFVFVMBRVCVLVEVAEVBUMUNUOUPURBCFGUQUSUT $.
|
|
$}
|
|
|
|
${
|
|
tfrcl.f $e |- F = recs ( G ) $.
|
|
tfrcl.g $e |- ( ph -> Fun G ) $.
|
|
tfrcl.x $e |- ( ph -> Ord X ) $.
|
|
tfrcl.ex $e |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S ) $.
|
|
|
|
${
|
|
tfrcllemsucfn.1 $e |- A = { f | E. x e. X ( f : x --> S
|
|
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|
|
|
|
${
|
|
$d G f x $. $d S f x $. $d X f x $. $d f g $. $d f ph x $.
|
|
$d f x z $.
|
|
tfrcllemsucfn.3 $e |- ( ph -> z e. X ) $.
|
|
tfrcllemsucfn.4 $e |- ( ph -> g : z --> S ) $.
|
|
tfrcllemsucfn.5 $e |- ( ph -> g e. A ) $.
|
|
$( We can extend an acceptable function by one element to produce a
|
|
function. Lemma for ~ tfrcl . (Contributed by Jim Kingdon,
|
|
24-Mar-2022.) $)
|
|
tfrcllemsucfn $p |- ( ph
|
|
-> ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S ) $=
|
|
( wcel cv csn cun cfv cop wf csuc cvv elexd con0 word ordelon syl2anc
|
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eloni ordirr 3syl wal weq feq2 imbi1d albidv 3expia alrimiv ralrimiva
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|
wn wi wa rspcdva feq1 eleq1d imbi12d spv sylc fsnunf syl121anc df-suc
|
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fveq2 feq2i sylibr ) ADUAZVTUBUCZFHUAZVTWBJUDZUEUBUCZUFZVTUGZFWDUFAVT
|
|
FWBUFZVTUHTVTVTTVEZWCFTZWERAVTKQUIAVTUJTZVTUKWHAKUKVTKTWJNQKVTULUMVTU
|
|
NVTUOUPAVTFGUAZUFZWKJUDZFTZVFZGUQZWGWIABUAZFWKUFZWNVFZGUQZWPBKVTBDURZ
|
|
WSWOGXAWRWLWNWQVTFWKUSUTVAAWTBKAWQKTZVGWSGAXBWRWNOVBVCVDQVHRWOWGWIVFG
|
|
HGHURZWLWGWNWIVTFWKWBVIXCWMWCFWKWBJVQVJVKVLVMVTFWBUHVTWCVNVOWFWAFWDVT
|
|
VPVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d G f x y $. $d G w x y $. $d S f x $. $d S w x $. $d X f x $.
|
|
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|
|
$d g w x y $. $d w x y z $.
|
|
tfrcllemsucaccv.yx $e |- ( ph -> Y e. X ) $.
|
|
tfrcllemsucaccv.zy $e |- ( ph -> z e. Y ) $.
|
|
tfrcllemsucaccv.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
tfrcllemsucaccv.gfn $e |- ( ph -> g : z --> S ) $.
|
|
tfrcllemsucaccv.gacc $e |- ( ph -> g e. A ) $.
|
|
$( Lemma for ~ tfrcl . We can extend an acceptable function by one
|
|
element to produce an acceptable function. (Contributed by Jim
|
|
Kingdon, 24-Mar-2022.) $)
|
|
tfrcllemsucaccv $p |- ( ph
|
|
-> ( g u. { <. z , ( G ` g ) >. } ) e. A ) $=
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( vw cv cfv cop csn cun wcel wf cres wceq wral wa wrex csuc weq suceq
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cuni eleq1d ralrimiva elunii syl2anc rspcdva jca ordtr1 tfrcllemsucfn
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word sylc wel wo vex elsuc feq1 reseq1 fveq2d eqeq12d ralbidv anbi12d
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fveq1 rexbidv elab2 sylib simprrr wfn simprrl ffn fndmu raleqdv mpbid
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syl adantr rexlimddv r19.21bi cvv wne con0 ordelon onelon sylan eloni
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ordirr 3syl elequ2 biimpcd adantl mtod neqned fvunsng sylancr ordelss
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albidv alrimiv fveq2 imbi12d spv fndm neleqtrrd fsnunfv syl3anc simpr
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3expia reseq2 sylan9eqr jaodan sylan2b raleq rspcev syl12anc wb opexg
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cbvrexv snexg unexg elab2g mpbird ) AHUDZDUDZUUIJUEZUFZUGZUHZEUIZBUDZ
|
|
FUUNUJZCUDZUUNUEZUUNUURUKZJUEZULZCUUPUMZUNZBKUOZAUUJUPZKUIZUVFFUUNUJZ
|
|
UVBCUVFUMZUVEAUUPUPZKUIZUVGBKUSZUUJBDUQZUVJUVFKUUPUUJURUTAUVKBUVLTVAA
|
|
UUJLUIZLKUIZUUJUVLUISRUUJLKVBVCVDABCDEFGHIJKMNOPQAKVHZUVNUVOUNUUJKUIZ
|
|
OAUVNUVOSRVEUUJLKVFVIZUAUBVGAUVBCUVFUURUVFUIACDVJZCDUQZVKUVBUURUUJCVL
|
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ZVMAUVSUVBUVTAUVSUNZUURUUIUEZUUIUURUKZJUEZUUSUVAAUWCUWEULZCUUJAUUPFUU
|
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IUJZUWFCUUPUMZUNZUWFCUUJUMZBKAUUIEUIUWIBKUOZUBUUPFGUDZUJZUURUWLUEZUWL
|
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UURUKZJUEZULZCUUPUMZUNZBKUOZUWKGUUIEHVLZGHUQZUWSUWIBKUXBUWMUWGUWRUWHU
|
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UPFUWLUUIVNUXBUWQUWFCUUPUXBUWNUWCUWPUWEUURUWLUUIVTUXBUWOUWDJUWLUUIUUR
|
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VOVPVQVRVSWAQWBWCAUUPKUIZUWIUNZUNZUWHUWJAUXCUWGUWHWDUXEUWFCUUPUUJUXEU
|
|
UIUUPWEZUUIUUJWEZUVMUXEUWGUXFAUXCUWGUWHWFUUPFUUIWGWKUXEUUJFUUIUJZUXGA
|
|
UXHUXDUAWLUUJFUUIWGZWKUUPUUJUUIWHVCWIWJWMWNUWBUURWOUIUUJUURWPUUSUWCUL
|
|
UWAUWBUUJUURUWBDCUQZCCVJZUWBUURWQUIZUURVHUXKXLAUUJWQUIZUVSUXLAUVPUVQU
|
|
XMOUVRKUUJWRVCZUUJUURWSWTUURXAUURXBXCUVSUXJUXKXMAUXJUVSUXKDCCXDXEXFXG
|
|
XHUUIUUJUUKUURWOXIXJUWBUUTUWDJUWBUUNUUJUKZUURUKZUUTUWDUWBUURUUJXNZUXP
|
|
UUTULAUUJVHZUVSUXQAUXMUXRUXNUUJXAWKZUUJUURXKWTUUNUURUUJXOWKAUXPUWDULU
|
|
VSAUXOUUIUURAUXGDDVJXLZUXOUUIULAUXHUXGUAUXIWKZAUXRUXTUXSUUJXBWKZUUJUU
|
|
IUUJUUKXPVCZXQWLXRVPXSAUVTUNZUUJUUNUEZUUKUUSUVAAUYEUUKULZUVTAUVNUUKFU
|
|
IZUUJUUIXTZUIXLUYFSAUUJFUWLUJZUWLJUEZFUIZXMZGYAZUXHUYGAUWMUYKXMZGYAZU
|
|
YMBKUUJUVMUYNUYLGUVMUWMUYIUYKUUPUUJFUWLYBYCYDAUYOBKAUXCUNUYNGAUXCUWMU
|
|
YKPYNYEVAUVRVDUAUYLUXHUYGXMGHUXBUYIUXHUYKUYGUUJFUWLUUIVNUXBUYJUUKFUWL
|
|
UUIJYFUTYGYHVIZAUYHUUJUUJUYBAUXGUYHUUJULUYAUUJUUIYIWKYJUUILFUUJUUKYKY
|
|
LWLUYDUURUUJUUNAUVTYMVPUYDUUTUUIJUVTAUUTUXOUUIUURUUJUUNYOUYCYPVPXSYQY
|
|
RVAUVGUVHUVIUNZUNUCUDZFUUNUJZUVBCUYRUMZUNZUCKUOUVEVUAUYQUCUVFKUYRUVFU
|
|
LUYSUVHUYTUVIUYRUVFFUUNYBUVBCUYRUVFYSVSYTVUAUVDUCBKUCBUQUYSUUQUYTUVCU
|
|
YRUUPFUUNYBUVBCUYRUUPYSVSUUDWCUUAAUUNWOUIZUUOUVEUUBAUUIWOUIUUMWOUIZVU
|
|
BUXAAUULWOUIZVUCAUUJWOUIUYGVUDDVLUYPUUJUUKWOFUUCXJUULWOUUEWKUUIUUMWOW
|
|
OUUFXJUWTUVEGUUNEWOUWLUUNULZUWSUVDBKVUEUWMUUQUWRUVCUUPFUWLUUNVNVUEUWQ
|
|
UVBCUUPVUEUWNUUSUWPUVAUURUWLUUNVTVUEUWOUUTJUWLUUNUURVOVPVQVRVSWAQUUGW
|
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KUUH $.
|
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$}
|
|
|
|
${
|
|
$d A f g h x y z $. $d D f g x y $. $d G f x y $. $d S f x y $.
|
|
$d X f x $. $d f g h ph x y z $.
|
|
tfrcllembacc.3 $e |- B = { h | E. z e. D E. g ( g : z --> S
|
|
/\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) } $.
|
|
tfrcllembacc.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
|
tfrcllembacc.4 $e |- ( ph -> D e. X ) $.
|
|
tfrcllembacc.5 $e |- ( ph -> A. z e. D E. g ( g : z --> S
|
|
/\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) $.
|
|
${
|
|
$( Lemma for ~ tfrcl . Each element of ` B ` is an acceptable
|
|
function. (Contributed by Jim Kingdon, 25-Mar-2022.) $)
|
|
tfrcllembacc $p |- ( ph -> B C_ A ) $=
|
|
( cv wf wcel cfv cop csn cun wceq w3a wex wrex cab wa wfun ad2antrr
|
|
simpr3 word simp1ll syld3an1 simplr cuni csuc adantlr simpr1 simpr2
|
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tfrcllemsucaccv eqeltrd ex exlimdv rexlimdva abssdv syl5eqss ) AGDU
|
|
EZIKUEZUFZVRFUGZLUEZVRVQVRNUHUIUJUKZULZUMZKUNZDHUOZLUPFUAAWFLFAWEWA
|
|
FUGZDHAVQHUGZUQZWDWGKWIWDWGWIWDUQZWAWBFWIVSVTWCUTWJBCDFIJKMNOHPANUR
|
|
WHWDQUSAOVAWHWDRUSABUEZOUGZWJWKIJUEZUFZWMNUHIUGAWHWDWLWNVBSVCTAHOUG
|
|
WHWDUCUSAWHWDVDWIWKOVEUGZWKVFOUGZWDAWOWPWHUBVGVGWIVSVTWCVHWIVSVTWCV
|
|
IVJVKVLVMVNVOVP $.
|
|
$}
|
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|
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${
|
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$d A f g h x y z $. $d B g h z $. $d B g w z $.
|
|
$d D f g h x y z $. $d G f h x y z $. $d G w y z $.
|
|
$d S f h x y z $. $d X f x z $. $d f g h ph x y z $.
|
|
$( Lemma for ~ tfrcl . The union of ` B ` is defined on all elements
|
|
of ` X ` . (Contributed by Jim Kingdon, 25-Mar-2022.) $)
|
|
tfrcllembxssdm $p |- ( ph -> D C_ dom U. B ) $=
|
|
( cv cuni cdm wcel wral wss wf cfv cres wceq wa fveq2 reseq2 fveq2d
|
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wex weq eqeq12d cbvralv anbi2i exbii ralbii sylib w3a cop simp1 syl
|
|
simp2 word ordtr1 imp syl12anc simp3l wal feq2 imbi1d albidv 3expia
|
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alrimiv ralrimiva adantr simpr rspcdva feq1 eleq1d imbi12d syl21anc
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spv csn cun cvv vex opexg sylancr snidg elun2 3syl wrex simp3r rspe
|
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raleq anbi12d cbvrexv fveq1 reseq1 ralbidv rexbidv elab2 3jca snexg
|
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wi sylibr unexg mpan isset simpr3 19.8a abeq2i sylan2 eqeltrrd 3imp
|
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3exp2 exlimdv sylc elunii syl2anc opeq2 spcegv syl6ibr ralimdva mpd
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eldm2 dfss3 ) ADUEZGUFZUGZUHZDHUIZHYSUJAYQIKUEZUKZCUEZUUBULZUUBUUDU
|
|
MZNULZUNZCYQUIZUOZKUSZDHUIZUUAAUUCEUEZUUBULZUUBUUMUMZNULZUNZEYQUIZU
|
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OZKUSZDHUIUULUDUUTUUKDHUUSUUJKUURUUIUUCUUQUUHECYQECUTZUUNUUEUUPUUGU
|
|
UMUUDUUBUPUVAUUOUUFNUUMUUDUUBUQURVAVBVCVDVEVFAUUKYTDHAYQHUHZUOUUJYT
|
|
KAUVBUUJYTAUVBUUJVGZUUBNULZIUHZYQUVDVHZYRUHZYTUVCAYQOUHZUUCUVEAUVBU
|
|
UJVIZUVCAUVBHOUHZUVHUVIAUVBUUJVKZUVCAUVJUVIUCVJAUVBUVJUOZUVHAOVLUVL
|
|
UVHXNRYQHOVMVJVNVOZAUVBUUCUUIVPZAUVHUOZUUCUVEUVOYQIJUEZUKZUVPNULZIU
|
|
HZXNZJVQZUUCUVEXNZUVOBUEZIUVPUKZUVSXNZJVQZUWABOYQBDUTZUWEUVTJUWGUWD
|
|
UVQUVSUWCYQIUVPVRVSVTAUWFBOUIUVHAUWFBOAUWCOUHZUOUWEJAUWHUWDUVSSWAWB
|
|
WCWDAUVHWEWFUVTUWBJKJKUTZUVQUUCUVSUVEYQIUVPUUBWGUWIUVRUVDIUVPUUBNUP
|
|
WHWIWKVJVNWJZUVCUVFUUBUVFWLZWMZUHZUWLGUHZUVGUVCUVFWNUHZUVFUWKUHUWMU
|
|
VCYQWNUHUVEUWODWOZUWJYQUVDWNIWPWQZUVFWNWRUVFUWKUUBWSWTUVCUVBUUCUUBF
|
|
UHZVGZLUEZUWLUNZLUSZUWNUVCUVBUUCUWRUVKUVNUVCUWCIUUBUKZUUHCUWCUIZUOZ
|
|
BOXAZUWRUVCUUJDOXAZUXFUVCUVHUUCUUIUXGUVMUVNAUVBUUCUUIXBUUJDOXCVOUUJ
|
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UXEDBODBUTUUCUXCUUIUXDYQUWCIUUBVRUUHCYQUWCXDXEXFVFUWDUUDUVPULZUVPUU
|
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DUMZNULZUNZCUWCUIZUOZBOXAUXFJUUBFKWOZUWIUXMUXEBOUWIUWDUXCUXLUXDUWCI
|
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UVPUUBWGUWIUXKUUHCUWCUWIUXHUUEUXJUUGUUDUVPUUBXGUWIUXIUUFNUVPUUBUUDX
|
|
HURVAXIXEXJTXKXOXLUVCUWLWNUHZUXBUVCUWOUWKWNUHZUXOUWQUVFWNXMUUBWNUHU
|
|
XPUXOUXNUUBUWKWNWNXPXQWTLUWLXRVFUWSUXAUWNLUVBUUCUWRUXAUWNXNUVBUUCUW
|
|
RUXAUWNUVBUUCUWRUXAVGZUOUWTUWLGUVBUUCUWRUXAXSUXQUVBUXQKUSZUWTGUHZUX
|
|
QKXTUVBUXRUOUXRDHXAZUXSUXRDHXCUXTLGUAYAXOYBYCYEYDYFYGUVFUWLGYHYIUVE
|
|
UVGYQUUMVHZYRUHZEUSYTUYBUVGEUVDIUUMUVDUNUYAUVFYRUUMUVDYQYJWHYKEYQYR
|
|
UWPYOYLYGWAYFYMYNDHYSYPXO $.
|
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$}
|
|
|
|
${
|
|
$d A f g h x y z $. $d B g h z $. $d B g w z $.
|
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$d D f g h x y z $. $d G f h x y z $. $d G w y z $.
|
|
$d S f g h x y z $. $d X f x z $. $d f g h ph x y z $.
|
|
$( Lemma for ~ tfrcl . The union of ` B ` is a function defined on
|
|
` x ` . (Contributed by Jim Kingdon, 25-Mar-2022.) $)
|
|
tfrcllembfn $p |- ( ph -> U. B : D --> S ) $=
|
|
( cuni wfn crn wss wfun cdm wceq tfrcllembacc unissd tfrcllemssrecs
|
|
wf crecs sstrd tfrfun funss mpisyl cxp cpw wcel cfv cop csn cun w3a
|
|
cv wex wrex cab simpr3 csuc simpl word adantr simpr jca ordtr1 sylc
|
|
wa ad2antrr 3adant1r simplr simpr1 simpr2 tfrcllemsucfn sylan fssxp
|
|
syl con0 ordelon syl2anc eloni ordsucss xpss1 cvv wb vex wi wal weq
|
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feq2 imbi1d albidv wral 3expia alrimiv 3ad2ant1 simp2 rspcdva simp3
|
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ralrimiva feq1 fveq2 eleq1d imbi12d spv syl3anc opexg sylancr snexg
|
|
unexg elpwg mpbird eqeltrd exlimdv rexlimdva syl5eqss sspwuni sylib
|
|
ex abssdv dmss dmxpss syl6ss tfrcllembxssdm eqssd sylanbrc rnxpss
|
|
df-fn rnss df-f ) AGUEZHUFZUUEUGZIUHHIUUEUOAUUEUIZUUEUJZHUKUUFAUUEN
|
|
UPZUHUUJUIUUHAUUEFUEUUJAGFABCDEFGHIJKLMNOPQRSTUAUBUCUDULUMABCFIJNOT
|
|
RUNUQNURUUEUUJUSUTAUUIHAUUIHIVAZUJZHAUUEUUKUHZUUIUULUHAGUUKVBZUHUUM
|
|
AGDVIZIKVIZUOZUUPFVCZLVIZUUPUUOUUPNVDZVEZVFZVGZUKZVHZKVJZDHVKZLVLUU
|
|
NUAAUVGLUUNAUVFUUSUUNVCZDHAUUOHVCZWBZUVEUVHKUVJUVEUVHUVJUVEWBZUUSUV
|
|
CUUNUVJUUQUURUVDVMUVKUVCUUNVCZUVCUUKUHZUVKUVCUUOVNZIVAZUUKUVKUVNIUV
|
|
CUOZUVCUVOUHUVJAUUOOVCZWBZUVEUVPUVJAUVQAUVIVOZUVJOVPZUVIHOVCZWBUVQA
|
|
UVTUVIRVQUVJUVIUWAAUVIVRAUWAUVIUCVQVSUUOHOVTWAZVSUVRUVEWBBCDFIJKMNO
|
|
PANUIUVQUVEQWCAUVTUVQUVERWCUVRBVIZOVCZUWCIJVIZUOZUWENVDZIVCZUVEAUWD
|
|
UWFUWHUVQSWDWDTAUVQUVEWEUVRUUQUURUVDWFUVRUUQUURUVDWGWHWIUVNIUVCWJWK
|
|
UVKUVNHUHZUVOUUKUHUVKHVPZUVIUWIAUWJUVIUVEAHWLVCZUWJAUVTUWAUWKRUCOHW
|
|
MWNHWOWKWCAUVIUVEWEUUOHWPWAUVNHIWQWKUQUVKUVCWRVCZUVLUVMWSUVKUUPWRVC
|
|
UVBWRVCZUWLKWTUVKUVAWRVCZUWMUVKUUOWRVCUUTIVCZUWNDWTUVKAUVQUUQUWOUVJ
|
|
AUVEUVSVQUVJUVQUVEUWBVQUVJUUQUURUVDWFAUVQUUQVHZUUOIUWEUOZUWHXAZJXBZ
|
|
UUQUWOUWPUWFUWHXAZJXBZUWSBOUUOBDXCZUWTUWRJUXBUWFUWQUWHUWCUUOIUWEXDX
|
|
EXFAUVQUXABOXGUUQAUXABOAUWDWBUWTJAUWDUWFUWHSXHXIXNXJAUVQUUQXKXLAUVQ
|
|
UUQXMUWRUUQUWOXAJKJKXCZUWQUUQUWHUWOUUOIUWEUUPXOUXCUWGUUTIUWEUUPNXPX
|
|
QXRXSWAXTUUOUUTWRIYAYBUVAWRYCWKUUPUVBWRWRYDYBUVCUUKWRYEWKYFYGYMYHYI
|
|
YNYJGUUKYKYLZUUEUUKYOWKHIYPYQABCDEFGHIJKLMNOPQRSTUAUBUCUDYRYSUUEHUU
|
|
BYTAUUGUUKUGZIAUUMUUGUXEUHUXDUUEUUKUUCWKHIUUAYQHIUUEUUDYT $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h x y z $. $d B g h z $. $d B g w z $.
|
|
$d D f g h x y z $. $d G f h x y z $. $d G w y z $.
|
|
$d S f g h x y z $. $d X f x z $. $d f g h ph x y z $.
|
|
$( Lemma for ~ tfrcl . The set ` B ` exists. (Contributed by Jim
|
|
Kingdon, 25-Mar-2022.) $)
|
|
tfrcllembex $p |- ( ph -> B e. _V ) $=
|
|
( cuni cvv wcel wf tfrcllembfn fex syl2anc uniexb sylibr ) AGUEZUFU
|
|
GZGUFUGAHIUNUHHOUGUOABCDEFGHIJKLMNOPQRSTUAUBUCUDUIUCHIOUNUJUKGULUM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A f g h x y z $. $d B g h z $. $d B u w $. $d D f g h x y z $.
|
|
$d D u w $. $d G e t v w $. $d G f h x y z $. $d G u w $.
|
|
$d S f g h x y z $. $d X f x z $. $d f g h ph x y z $.
|
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$d g ph w y z $.
|
|
$( Lemma for ~ tfrcl . The union of ` B ` satisfies the recursion
|
|
rule. (Contributed by Jim Kingdon, 25-Mar-2022.) $)
|
|
tfrcllemubacc $p |- ( ph ->
|
|
A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) $=
|
|
( vv vt ve cv cuni cfv cres wceq wral wcel wa crecs cdm tfrcllembfn
|
|
wf fdm syl wss tfrcllembacc unissd tfrcllemssrecs sstrd dmss sselda
|
|
eqsstr3d wfn con0 wrex cab eqid tfrlem9 wfun tfrfun biimpar funssfv
|
|
eleq2d mp3an2ani word ordelon syl2anc eloni ordelss adantr sseqtr4d
|
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|
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|
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|
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|
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|
|
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|
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|
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|
|
tfrcllemex $p |- ( ph ->
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|
E. f ( f : D --> S
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|
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|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
|
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|
|
$( We can define an acceptable function on any element of ` X ` .
|
|
|
|
As with many of the transfinite recursion theorems, we have
|
|
hypotheses that state that ` F ` is a function and that it is
|
|
defined up to ` X ` . (Contributed by Jim Kingdon, 26-Mar-2022.) $)
|
|
tfrcllemaccex $p |- ( ( ph /\ C e. X ) -> E. g ( g : C --> S
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/\ A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) ) $=
|
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|
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|
|
HURZUCUUSVOZUIVPUXFVUHUGUXFUWNGUYRUMZUYTUWRUYRUWNVUAVKZVLZVMZUPZVNZUH
|
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UYQSUCVCZUVHVVPUVMVWBUVGUYQLVDVWCUVLVWAIVWCUVJVVSUVKVVTUVGUYQGUUHVFUU
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NDUVGUYQVGVHVJVIUWCUVSUUTVVNYBVVQUWEVVNSUWDUUTSUWCUVSSUWCSYDUVRSUUSYE
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|
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DVCZVWDVVGVWFVVIUUJVVFUYRWBVWKVWEVVHKUUJVVFUYRYGWKWLYHYIYJXIYKYLYMYOU
|
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WMUVDUBIUBIVCZUWGUVBUWLUVCUUSGUWFUUHWAVWLUWKUUNDUUSVWLUWHUUKUWJUUMUUJ
|
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UWFUUHWIVWLUWIUULKUWFUUHUUJWJWKWLWNVHXCWFYPYQYRUWBAUVRVBZSUUSUQUVTUVN
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VWMSUUSAUVHUVMYSYTAUVRSUUSUUAXIAUUTUVEYSUUBUUCUUD $.
|
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$}
|
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|
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${
|
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|
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$d S f g u x y $. $d X f x y $. $d Y f g x y z $. $d f g ph x y z $.
|
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$d x y z $.
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tfrcllemres.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
|
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tfrcllemres.yx $e |- ( ph -> Y e. X ) $.
|
|
$( Lemma for ~ tfr1on . Recursion is defined on an ordinal if the
|
|
characteristic function is defined up to a suitable point.
|
|
(Contributed by Jim Kingdon, 18-Mar-2022.) $)
|
|
tfrcllemres $p |- ( ph -> Y C_ dom F ) $=
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( wcel wa cfv vh vz vg vu crecs cdm ciun wrex cres wceq wral wex word
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cv adantr simpr jca ordtr1 sylc tfrcllemaccex syldan cop csn cun wfun
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|
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|
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ULYQIUMZYOJIRZSUUIAUUJYOMUOYQYOUUKAYOUPZAUUKYOQUOUQYNJIURUSZABCUDDYNE
|
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FUCGHIKLMNOPUTVAYQUUHSZYTYNYTHTZVBZVCZVDZDRYNUURUFZRZYSUUNBCUBDEFUCGH
|
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IJKAHVEYOUUHLVGAUUJYOUUHMVGYQBUNZIRZUVAEFUNZVFZUVCHTZERZUUHAUVBUVDUVF
|
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UUAUUGVLZUUNUVAEYTVFZCUNZYTTZYTUVKUIZHTZUJZCUVAUKZSZBIUHZYTDRYQUUIUUH
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|
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UWCUVLUWEUVNUVKUVCYTWHUWIUWDUVMHUVCYTUVKWFVRVSWIVQWJOWKWLWMUUNUUPUURR
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YQUVJUWNXGZUUAUWNXGBIYNUVSUVJUUAUWNUWAWOAUWPBIUKYOAUWPBIAUVBSZUVDUVFX
|
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GZFWPUWPUWQUWRFAUVBUVDUVFNWQWRUWRUWPFUCUWIUVDUVJUVFUWNUWJUWIUVEUUOEUV
|
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CYTHVOWSWTXAXBXCUOUUMXDXHVAZYNUUOWNEXEXFUUPWNXIUUPUUQYTXJXKUUNUWMUWNU
|
|
WKUUTXGUWOUWSYNUUOUURWNEXLXFXSYRUUTUAUURDYKUURUJYLUUSYNYKUURXMXNWCXOX
|
|
PUAYNDYLXQWLXRXTAYMDVIZUFZYJUADYAAUWTYIYHUXAYJYHABCDEFHIOMYBUWTYIYCXB
|
|
YDYEGYIKYFYG $.
|
|
$}
|
|
$}
|
|
|
|
tfrcl.u $e |- ( ( ph /\ x e. U. X ) -> suc x e. X ) $.
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|
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|
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${
|
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$d F z $. $d G a b f x y $. $d S a f x y $. $d X a f x y $.
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$d X f x y z $. $d Y f x y z $. $d f ph x y z $.
|
|
$( Recursion is defined on an ordinal if the characteristic function
|
|
satisfies a closure hypothesis up to a suitable point. (Contributed
|
|
by Jim Kingdon, 26-Mar-2022.) $)
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|
tfrcldm $p |- ( ph -> Y e. dom F ) $=
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|
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tfrcllemres simprl sseldd exlimddv ) AHOQZRZVRGRZSZHEUCZROAHGUDZRWAOUEN
|
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OHGUFUGAWASZVRWBHWDBUAUBQZCDQZUHZPQZWFTZWFWHUIZFTZUJZPWEUKZSZUBGULZDVCC
|
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DEFGVRIAFUMWAJUNAGUOWAKUNABQZGRWPCWFUHZWFFTCRWALUPWOWQUAQZWFTZWFWRUIZFT
|
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ZUJZUAWPUKZSZBGULZDWOWQWLPWPUKZSZBGULXEWNXGUBBGWEWPUJWGWQWMXFWEWPCWFUQW
|
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|
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|
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$}
|
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|
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${
|
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$d F f k w x $. $d G f x $. $d S f k w x $. $d X f k w x $. $d Y w $.
|
|
$d f k ph w x $.
|
|
$( Closure for transfinite recursion. As with ~ tfr1on , the
|
|
characteristic function must be defined up to a suitable point, not
|
|
necessarily on all ordinals. (Contributed by Jim Kingdon,
|
|
25-Mar-2022.) $)
|
|
tfrcl $p |- ( ph -> ( F ` Y ) e. S ) $=
|
|
( vk wcel wa cfv wi ad2antrl vw con0 cuni word orduni syl ordelon ancli
|
|
syl2anc cv wceq eleq1 anbi2d fveq2 eleq1d imbi12d wral cres cdm wfun wf
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|
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|
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ralimdva imp an32s crecs tfrfun funeqi mpbir a1i ffvresb mpbird vex fex
|
|
sylancl feq2 imbi1d albidv 3expia alrimiv ralrimiva sucid suceq rspcdva
|
|
wb feq1 spcgv syl3c eqeltrd exp31 tfis3 ) AHUBPZAHGUCZPZQZHERZCPZAXNUDZ
|
|
XOXMAGUDZXSKGUEZUFNXNHUGUIAXONUHAUAUJZXNPZQZYBERZCPZSAOUJZXNPZQZYGERZCP
|
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ZSZXPXRSUAOHYBYGUKZYDYIYFYKYMYCYHAYBYGXNULUMYMYEYJCYBYGEUNUOUPYBHUKZYDX
|
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PYFXRYNYCXOAYBHXNULUMYNYEXQCYBHEUNUOUPYBUBPZYLOYBUQZYDYFYOYPQZYDQZYEEYB
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URZFRZCYRYBEUSZPYEYTUKYRBCDEFGYBIAFUTZYQYCJTAXTYQYCKTZYDBUJZGPZUUDCDUJZ
|
|
VAZUUFFRZCPZYQAUUEUUGUUIYCLVBVCYDUUDXNPZUUDVDZGPZYQAUUJUULYCMVEVFYQAYCV
|
|
GZVHYBEFIVIUFYRYSVJPZYBCUUFVAZUUISZDVLZYBCYSVAZYTCPZYRUURYBVJPUUNYRUURY
|
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GUUAPZYKQZOYBUQZYOYDYPUVBYOYDQZYPUVBUVCYLUVAOYBUVCYGYBPZQZYLUVAUVEYLQZU
|
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UTYKUVFBCDEFGYGIUVEUUBYLUVCUUBUVDAUUBYOYCJTVMVMUVEXTYLUVCXTUVDAXTYOYCKT
|
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|
|
HMVOVEUVEYHYLUVEXSUVDYCQYHUVEXTXSUVGYAUFUVEUVDYCUVCUVDVPYOAYCUVDVQVRYGY
|
|
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|
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YRUVJFWGZUTFWHEUVKIWIWJWKOYBCEWLUFWMZUAWNZYBCVJYSWOWPYRUUGUUISZDVLZUUQB
|
|
GYBUUDYBUKZUVNUUPDUVPUUGUUOUUIUUDYBCUUFWQWRWSAUVOBGUQYQYCAUVOBGAUUEQUVN
|
|
DAUUEUUGUUILWTXAXBTYRXTYBYBVDZPZUVQGPZQYBGPUUCYRUVRUVSUVRYRYBUVMXCWKYRU
|
|
ULUVSBXNYBUVPUUKUVQGUUDYBXDUOAUULBXNUQYQYCAUULBXNMXBTUUMXEVRYBUVQGVSVTX
|
|
EUVLUUPUURUUSSDYSVJUUFYSUKZUUOUURUUIUUSYBCUUFYSXGUVTUUHYTCUUFYSFUNUOUPX
|
|
HXIXJXKXLVT $.
|
|
$}
|
|
$}
|
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|
|
${
|
|
$d G x $.
|
|
tfri1.1 $e |- F = recs ( G ) $.
|
|
tfri1.2 $e |- ( Fun G /\ ( G ` x ) e. _V ) $.
|
|
$( Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
|
|
[TakeutiZaring] p. 47, with an additional condition.
|
|
|
|
The condition is that ` G ` is defined "everywhere", which is stated
|
|
here as ` ( G `` x ) e. _V ` . Alternately,
|
|
` A. x e. On A. f ( f Fn x -> f e. dom G ) ` would suffice.
|
|
|
|
Given a function ` G ` satisfying that condition, we define a class
|
|
` A ` of all "acceptable" functions. The final function we're
|
|
interested in is the union ` F = recs ( G ) ` of them. ` F ` is then
|
|
said to be defined by transfinite recursion. The purpose of the 3 parts
|
|
of this theorem is to demonstrate properties of ` F ` . In this first
|
|
part we show that ` F ` is a function whose domain is all ordinal
|
|
numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario
|
|
Carneiro, 24-May-2019.) $)
|
|
tfri1 $p |- F Fn On $=
|
|
( con0 wfn wtru wfun cv cfv cvv wcel wa wal ax-gen a1i tfri1d trud ) BFGH
|
|
ABCDCIAJCKLMNZAOHTAEPQRS $.
|
|
|
|
$( Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of
|
|
[TakeutiZaring] p. 47, with an additional condition on the recursion
|
|
rule ` G ` ( as described at ~ tfri1 ). Here we show that the function
|
|
` F ` has the property that for any function ` G ` satisfying that
|
|
condition, the "next" value of ` F ` is ` G ` recursively applied to all
|
|
"previous" values of ` F ` . (Contributed by Jim Kingdon,
|
|
4-May-2019.) $)
|
|
tfri2 $p |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
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|
( con0 wcel cdm cfv cres wceq wfn tfri1 fndm ax-mp eleq2i tfr2a sylbir )
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BGHBCIZHBCJCBKDJLTGBCGMTGLACDEFNGCOPQBCDERS $.
|
|
$}
|
|
|
|
${
|
|
$d B x y $. $d F x y $. $d G x y $.
|
|
tfri3.1 $e |- F = recs ( G ) $.
|
|
tfri3.2 $e |- ( Fun G /\ ( G ` x ) e. _V ) $.
|
|
$( Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of
|
|
[TakeutiZaring] p. 47, with an additional condition on the recursion
|
|
rule ` G ` ( as described at ~ tfri1 ). Finally, we show that ` F ` is
|
|
unique. We do this by showing that any class ` B ` with the same
|
|
properties of ` F ` that we showed in parts 1 and 2 is identical to
|
|
` F ` . (Contributed by Jim Kingdon, 4-May-2019.) $)
|
|
tfri3 $p |- ( ( B Fn On /\
|
|
A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F ) $=
|
|
( vy con0 wfn cv cfv cres wceq wral wa nfv wi fveq2 wb imp nfan wcel nfim
|
|
nfra1 eqeq12d imbi2d r19.21v rsp onss tfri1 fvreseq mpanl2 syl6bir sylan2
|
|
wss ancoms adantr tfri2 jctr jcab sylibr eqeq12 adantl mpbird exp43 com4t
|
|
syl6 exp4a pm2.43d syl com3l a2d syl5bi tfis2f com12 ralrimi eqfnfv mpan2
|
|
impd biimpar syldan ) BHIZAJZBKZBWCLZDKZMZAHNZWDWCCKZMZAHNZBCMZWBWHOZWJAH
|
|
WBWHAWBAPWGAHUDUAZWCHUBZWMWJWMWJQZWMGJZBKZWQCKZMZQZAGWMWTAWNWTAPUCWCWQMZW
|
|
JWTWMXBWDWRWIWSWCWQBRWCWQCRUEUFXAGWCNWMWTGWCNZQWOWPWMWTGWCUGWOWMXCWJWOWBW
|
|
HXCWJQZWHWOWBXDWHWOWGQZWOWBXDQZQWGAHUHXEWOXFXEWOWOWBXDWOWBOZXCXEWOWJXGXCX
|
|
EWOWJXGXCOZXEWOOZOWJWFCWCLZDKZMZXHXLXIXGXCXLWBWOXCXLQZWOWBWCHUOZXMWCUIWBX
|
|
NOXCWEXJMZXLWBCHIZXNXOXCSACDEFUJZGHWCBCUKULWEXJDRUMUNUPTUQXIWJXLSZXHXEWOX
|
|
RXEWOWGWIXKMZOZXRXEXEWOXSQZOWOXTQXEYAAWCCDEFURUSWOWGXSUTVAWDWFWIXKVBVGTVC
|
|
VDVEVFVHVIVJVKVSVLVMVNVOVPWBWLWKWBXPWLWKSXQAHBCVQVRVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d f g u x y z G $.
|
|
tfrex.1 $e |- F = recs ( G ) $.
|
|
tfrex.2 $e |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) ) $.
|
|
$( The transfinite recursion function is set-like if the input is.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
tfrex $p |- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V ) $=
|
|
( vy vg vz vu vf wcel wa cfv crecs cvv fveq1i cv cres wceq wral con0 wrex
|
|
wfn cab eqid tfrlem3 tfrexlem syl5eqel ) ACFNOCDPCEQZPRCDULGSABIJTZKTZUFL
|
|
TZUMPUMUOUAEPUBLUNUCOKUDUEJUGZCMEFKLBIUPJMEUPUHUIHUJUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Recursive definition generator
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Define a constant for the recursive definition generator. $)
|
|
$c rec $.
|
|
|
|
$( Extend class notation with the recursive definition generator, with
|
|
characteristic function ` F ` and initial value ` I ` . $)
|
|
crdg $a class rec ( F , I ) $.
|
|
|
|
${
|
|
$d x g F $. $d x g I $.
|
|
$( Define a recursive definition generator on ` On ` (the class of ordinal
|
|
numbers) with characteristic function ` F ` and initial value ` I ` .
|
|
This rather amazing operation allows us to define, with compact direct
|
|
definitions, functions that are usually defined in textbooks only with
|
|
indirect self-referencing recursive definitions. A recursive definition
|
|
requires advanced metalogic to justify - in particular, eliminating a
|
|
recursive definition is very difficult and often not even shown in
|
|
textbooks. On the other hand, the elimination of a direct definition is
|
|
a matter of simple mechanical substitution. The price paid is the
|
|
daunting complexity of our ` rec ` operation (especially when ~ df-recs
|
|
that it is built on is also eliminated). But once we get past this
|
|
hurdle, definitions that would otherwise be recursive become relatively
|
|
simple. In classical logic it would be easier to divide this definition
|
|
into cases based on whether the domain of ` g ` is zero, a successor, or
|
|
a limit ordinal. Cases do not (in general) work that way in
|
|
intuitionistic logic, so instead we choose a definition which takes the
|
|
union of all the results of the characteristic function for ordinals in
|
|
the domain of ` g ` . This means that this definition has the expected
|
|
properties for increasing and continuous ordinal functions, which
|
|
include ordinal addition and multiplication.
|
|
|
|
For finite recursion we also define ~ df-frec and for suitable
|
|
characteristic functions ~ df-frec yields the same result as ` rec `
|
|
restricted to ` _om ` , as seen at ~ frecrdg .
|
|
|
|
_Note: We introduce_ ` rec ` _with the philosophical goal of being_
|
|
_able to eliminate all definitions with direct mechanical substitution_
|
|
_and to verify easily the soundness of definitions. Metamath itself_
|
|
_has no built-in technical limitation that prevents multiple-part_
|
|
_recursive definitions in the traditional textbook style_. (Contributed
|
|
by Jim Kingdon, 19-May-2019.) $)
|
|
df-irdg $a |- rec ( F , I ) = recs ( ( g e. _V |-> ( I u.
|
|
U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x g F $. $d x g G $. $d x g A $. $d x g B $.
|
|
$( Equality theorem for the recursive definition generator. (Contributed
|
|
by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) $)
|
|
rdgeq1 $p |- ( F = G -> rec ( F , A ) = rec ( G , A ) ) $=
|
|
( vg vx wceq cvv cv cdm cfv ciun cun cmpt crecs crdg fveq1 iuneq2d uneq2d
|
|
mpteq2dv df-irdg recseq syl 3eqtr4g ) BCFZDGAEDHZIZEHUEJZBJZKZLZMZNZDGAEU
|
|
FUGCJZKZLZMZNZBAOCAOUDUKUPFULUQFUDDGUJUOUDUIUNAUDEUFUHUMUGBCPQRSUKUPUAUBE
|
|
DBATEDCATUC $.
|
|
|
|
$( Equality theorem for the recursive definition generator. (Contributed
|
|
by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) $)
|
|
rdgeq2 $p |- ( A = B -> rec ( F , A ) = rec ( F , B ) ) $=
|
|
( vg vx wceq cvv cv cdm cfv ciun cun cmpt crecs uneq1 mpteq2dv recseq syl
|
|
crdg df-irdg 3eqtr4g ) ABFZDGAEDHZIEHUCJCJKZLZMZNZDGBUDLZMZNZCASCBSUBUFUI
|
|
FUGUJFUBDGUEUHABUDOPUFUIQREDCATEDCBTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z f g F $. $d x y z f g A $.
|
|
$( The recursive definition generator is a function. (Contributed by Mario
|
|
Carneiro, 16-Nov-2014.) $)
|
|
rdgfun $p |- Fun rec ( F , A ) $=
|
|
( vg vx vy vz vf crdg wfun cvv cdm cfv ciun cun cmpt crecs wfn cres wceq
|
|
cv wral wa con0 wrex cab eqid tfrlem7 df-irdg funeqi mpbir ) BAHZICJADCTZ
|
|
KDTULLBLMNOZPZIEFGTZETZQFTZUOLUOUQRUMLSFUPUAUBEUCUDGUEZGUMURUFUGUKUNDCBAU
|
|
HUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A g $. $d g x z F $.
|
|
$( The recursion rule for the recursive definition generator is defined
|
|
everywhere. (Contributed by Jim Kingdon, 14-May-2020.) $)
|
|
rdgtfr $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> (
|
|
Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
|
|
/\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
|
|
e. _V ) ) $=
|
|
( wcel cv cfv cvv wal cdm wa vex wral wceq ralrimivw sylancr sylan2 fveq2
|
|
ciun cun cmpt wfun elex funmpt dmex fvex eleq1d spcv iunexg ancoms dmmptg
|
|
unexg syl syl5eleqr funfvex jctil ) CGHBIZFJZKHZBLZCKHZEKCAEIZMZAIZVEJZFJ
|
|
ZUBZUCZUDZUEZDIZVLJKHZNCGUFVCVDNZVOVMVPVMVNVLMZHVOEKVKUGZVPVNKVQDOVPVKKHZ
|
|
EKPVQKQVPVSEKVDVCVSVCVDVJKHZVSVCVFKHVIKHZAVFPVTVEEOZUHVCWAAVFVBWABVHVGVEK
|
|
KWBAOUIUTVHQVAVIKUTVHFUAUJUKRAVFVIKKULSCVJKKUOTUMREKVKKUNUPUQVNVLURSVRUST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A g $. $d g x F $.
|
|
$( The recursion rule for the recursive definition generator is defined
|
|
everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) $)
|
|
rdgruledefgg $p |- ( ( F Fn _V /\ A e. V ) -> (
|
|
Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
|
|
/\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
|
|
e. _V ) ) $=
|
|
( wcel cvv wfn cv cdm cfv ciun wa vex wral funfvex ralrimivw syl sylan2
|
|
cun cmpt wfun elex funmpt wceq fvex funfni mpan2 dmex iunexg unexg ancoms
|
|
mpan dmmptg syl5eleqr sylancr jctil ) BFGEHIZBHGZDHBADJZKZAJZVALZELZMZUAZ
|
|
UBZUCZCJZVHLHGZNBFUDUSUTNZVKVIVLVIVJVHKZGVKDHVGUEZVLVJHVMCOVLVGHGZDHPVMHU
|
|
FVLVODHUTUSVOUSUTVFHGZVOUSVEHGZAVBPZVPUSVQAVBUSVDHGVQVCVAHHDOZAOUGVQHVDEV
|
|
DEQUHUIRVBHGVRVPVAVSUJAVBVEHHUKUNSBVFHHULTUMRDHVGHUOSUPVJVHQUQVNURT $.
|
|
$}
|
|
|
|
${
|
|
$d A g $. $d V g $. $d g x F $.
|
|
rdgruledefg.1 $e |- F Fn _V $.
|
|
$( The recursion rule for the recursive definition generator is defined
|
|
everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) $)
|
|
rdgruledefg $p |- ( A e. V -> (
|
|
Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
|
|
/\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f )
|
|
e. _V ) ) $=
|
|
( cvv wfn wcel cv cdm cfv ciun cun cmpt wfun wa rdgruledefgg mpan ) EHIBF
|
|
JDHBADKZLAKUAMEMNOPZQCKUBMHJRGABCDEFST $.
|
|
$}
|
|
|
|
${
|
|
$d A g x y $. $d F g x y $. $d V g y $.
|
|
$( The recursive definition generator produces a set on a set input.
|
|
(Contributed by Jim Kingdon, 4-Jul-2019.) $)
|
|
rdgexggg $p |- ( ( F Fn _V /\ A e. V /\ B e. W ) ->
|
|
( rec ( F , A ) ` B ) e. _V ) $=
|
|
( vy vg vx cvv wfn wcel crdg cfv wa cv cdm ciun cun cmpt df-irdg alrimiv
|
|
wfun rdgruledefgg tfrex 3impa ) CIJZADKZBEKBCALZMIKUFUGNZFBUHGIAHGOZPHOUJ
|
|
MCMQRSZEHGCATUIUKUBFOUKMIKNFHAFGCDUCUAUDUE $.
|
|
$}
|
|
|
|
${
|
|
rdgexgg.1 $e |- F Fn _V $.
|
|
$( The recursive definition generator produces a set on a set input.
|
|
(Contributed by Jim Kingdon, 4-Jul-2019.) $)
|
|
rdgexgg $p |- ( ( A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V ) $=
|
|
( cvv wfn wcel crdg cfv rdgexggg mp3an1 ) CGHADIBEIBCAJKGIFABCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d A f g x $. $d F f g x $. $d V f $.
|
|
$( The recursive definition generator is a function on ordinal numbers.
|
|
The ` F Fn _V ` condition states that the characteristic function is
|
|
defined for all sets (being defined for all ordinals might be enough if
|
|
being used in a manner similar to ~ rdgon ; in cases like ~ df-oadd
|
|
either presumably could work). (Contributed by Jim Kingdon,
|
|
13-Jul-2019.) $)
|
|
rdgifnon $p |- ( ( F Fn _V /\ A e. V ) ->
|
|
rec ( F , A ) Fn On ) $=
|
|
( vf vg vx cvv wfn wcel wa crdg cv cdm cfv ciun cmpt df-irdg rdgruledefgg
|
|
cun wfun alrimiv tfri1d ) BGHACIJZDBAKEGAFELZMFLUDNBNOSPZFEBAQUCUETDLUENG
|
|
IJDFADEBCRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A f g x $. $d F f g x z $. $d V f $.
|
|
$( The recursive definition generator is a function on ordinal numbers.
|
|
(Contributed by Jim Kingdon, 14-May-2020.) $)
|
|
rdgifnon2 $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
|
|
rec ( F , A ) Fn On ) $=
|
|
( vf vg vx cv cfv cvv wcel wal wa crdg cdm ciun cun cmpt df-irdg wfun
|
|
rdgtfr alrimiv tfri1d ) AHCIJKALBDKMZECBNFJBGFHZOGHUEICIPQRZGFCBSUDUFTEHU
|
|
FIJKMEGABEFCDUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A g x y $. $d B g x $. $d F g x y $. $d V g x y $.
|
|
$( Value of the recursive definition generator. Lemma for ~ rdgival which
|
|
simplifies the value further. (Contributed by Jim Kingdon,
|
|
13-Jul-2019.) (New usage is discouraged.) $)
|
|
rdgivallem $p |- ( ( F Fn _V /\ A e. V /\ B e. On ) ->
|
|
( rec ( F , A ) ` B ) =
|
|
( A u. U_ x e. B ( F ` ( ( rec ( F , A ) |` B ) ` x ) ) ) ) $=
|
|
( vg vy cvv wfn wcel con0 cfv cv cdm ciun cun wceq wa wfun wss w3a tfri2d
|
|
crdg cres cmpt df-irdg rdgruledefgg alrimiv 3impa eqidd 3ad2ant3 rdgifnon
|
|
dmeq onss fndm syl 3adant3 sseqtr4d ssdmres sylib sylan9eqr fveq2d adantl
|
|
fveq1 iuneq12d uneq2d rdgfun resfunexg mpan simpr fvexg sylancl ralrimivw
|
|
wral vex wi funfvex funfni ex ralimdv adantr iunexg syl2anc 3adant2 unexg
|
|
mpd 3ad2ant2 fvmptd eqtrd ) DHIZBEJZCKJZUAZCDBUCZLZWNCUDZFHBAFMZNZAMZWQLZ
|
|
DLZOZPZUEZLZBACWSWPLZDLZOZPZWJWKWLWOXEQWJWKRZGCWNXDAFDBUFXJXDSGMXDLHJRGAB
|
|
GFDEUGUHUBUIWMFWPXCXIHXDHWMXDUJWMWQWPQZRZXBXHBXLAWRCXAXGXKWMWRWPNZCWQWPUM
|
|
WMCWNNZTXMCQWMCKXNWLWJCKTWKCUNUKWJWKXNKQZWLXJWNKIXOBDEULKWNUOUPUQURCWNUSU
|
|
TVAXKXAXGQWMXKWTXFDWSWQWPVDVBVCVEVFWLWJWPHJZWKWNSWLXPBDVGWNCKVHVIZUKWMXHH
|
|
JZXIHJZWJWLXRWKWJWLRZWLXGHJZACVNZXRWJWLVJXTXFHJZACVNZYBWLYDWJWLYCACWLXPWS
|
|
HJYCXQAVOWSWPHHVKVLVMVCWJYDYBVPWLWJYCYAACWJYCYAYAHXFDXFDVQVRVSVTWAWFACXGK
|
|
HWBWCWDWKWJXRXSVPWLWKXRXSBXHEHWEVSWGWFWHWI $.
|
|
|
|
$( Value of the recursive definition generator. (Contributed by Jim
|
|
Kingdon, 26-Jul-2019.) $)
|
|
rdgival $p |- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) =
|
|
( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) ) $=
|
|
( cvv wfn wcel con0 w3a crdg cfv cv cres ciun cun rdgivallem fvres fveq2d
|
|
iuneq2i uneq2i syl6eq ) DFGBEHCIHJCDBKZLBACAMZUCCNLZDLZOZPBACUDUCLZDLZOZP
|
|
ABCDEQUGUJBACUFUIUDCHUEUHDUDCUCRSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $. $d I x y $. $d V x y $. $d ph x $.
|
|
rdgss.1 $e |- ( ph -> F Fn _V ) $.
|
|
rdgss.2 $e |- ( ph -> I e. V ) $.
|
|
rdgss.3 $e |- ( ph -> A e. On ) $.
|
|
rdgss.4 $e |- ( ph -> B e. On ) $.
|
|
rdgss.5 $e |- ( ph -> A C_ B ) $.
|
|
$( Subset and recursive definition generator. (Contributed by Jim Kingdon,
|
|
15-Jul-2019.) $)
|
|
rdgss $p |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) ) $=
|
|
( vx vy cv cfv ciun cun wss wcel wceq crdg wrex wral fveq2d sseq2d rspcev
|
|
ssel ssid fveq2 mpan2 syl6 ralrimiv syl iunss2 unss2 3syl cvv wfn rdgival
|
|
con0 syl3anc 3sstr4d ) AELBLNZDEUAZOZDOZPZQZEMCMNZVDOZDOZPZQZBVDOZCVDOZAV
|
|
FVKRZMCUBZLBUCZVGVLRVHVMRABCRZVRKVSVQLBVSVCBSVCCSZVQBCVCUGVTVFVFRZVQVFUHV
|
|
PWAMVCCVIVCTZVKVFVFWBVJVEDVIVCVDUIUDUEUFUJUKULUMLMBCVFVKUNVGVLEUOUPADUQUR
|
|
ZEFSZBUTSVNVHTGHILEBDFUSVAAWCWDCUTSVOVMTGHJMECDFUSVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d A x $. $d B x $. $d V x $.
|
|
rdgisuc1.1 $e |- ( ph -> F Fn _V ) $.
|
|
rdgisuc1.2 $e |- ( ph -> A e. V ) $.
|
|
rdgisuc1.3 $e |- ( ph -> B e. On ) $.
|
|
$( One way of describing the value of the recursive definition generator at
|
|
a successor. There is no condition on the characteristic function ` F `
|
|
other than ` F Fn _V ` . Given that, the resulting expression
|
|
encompasses both the expected successor term
|
|
` ( F `` ( rec ( F , A ) `` B ) ) ` but also terms that correspond to
|
|
the initial value ` A ` and to limit ordinals
|
|
` U_ x e. B ( F `` ( rec ( F , A ) `` x ) ) ` .
|
|
|
|
If we add conditions on the characteristic function, we can show tighter
|
|
results such as ~ rdgisucinc . (Contributed by Jim Kingdon,
|
|
9-Jun-2019.) $)
|
|
rdgisuc1 $p |- ( ph -> ( rec ( F , A ) ` suc B ) =
|
|
( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) )
|
|
u. ( F ` ( rec ( F , A ) ` B ) ) ) ) ) $=
|
|
( csuc crdg cfv cv ciun cun wcel con0 wceq syl uneq2d cvv rdgival syl3anc
|
|
wfn suceloni df-suc iuneq1 ax-mp iunxun eqtri fveq2 fveq2d iunxsng syl5eq
|
|
csn eqtrd ) ADJZECKZLZCBUQBMZURLZELZNZOZCBDVBNZDURLZELZOZOZAEUAUDCFPUQQPZ
|
|
USVDRGHADQPZVJIDUESBCUQEFUBUCAVKVDVIRIVKVCVHCVKVCVEBDUOZVBNZOZVHVCBDVLOZV
|
|
BNZVNUQVORVCVPRDUFBUQVOVBUGUHBDVLVBUIUJVKVMVGVEBDVBVGQUTDRVAVFEUTDURUKULU
|
|
MTUNTSUP $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d F x $.
|
|
rdgisucinc.inc $e |- ( ph -> A. x x C_ ( F ` x ) ) $.
|
|
$( Value of the recursive definition generator at a successor.
|
|
|
|
This can be thought of as a generalization of ~ oasuc and ~ omsuc .
|
|
(Contributed by Jim Kingdon, 29-Aug-2019.) $)
|
|
rdgisucinc $p |- ( ph -> ( rec ( F , A ) ` suc B ) =
|
|
( F ` ( rec ( F , A ) ` B ) ) ) $=
|
|
( csuc crdg cfv cun cvv wcel con0 wceq syl3anc wss cv ciun rdgisuc1 wfn
|
|
unass syl6eqr rdgival uneq1d wal rdgexggg id fveq2 sseq12d sylc ssequn1
|
|
spcgv sylib 3eqtr2d ) ADKECLZMZCBDBUAZUSMEMUBZNZDUSMZEMZNZVDVENZVEAUTCV
|
|
BVENNVFABCDEFGHIUCCVBVEUEUFAVDVCVEAEOUDZCFPZDQPZVDVCRGHIBCDEFUGSUHAVDVE
|
|
TZVGVERAVDOPZVAVAEMZTZBUIVKAVHVIVJVLGHICDEFQUJSJVNVKBVDOVAVDRZVAVDVMVEV
|
|
OUKVAVDEULUMUPUNVDVEUOUQUR $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A f g x y $. $d B f x y z $. $d F f g x y $. $d F f x y z $.
|
|
$d f ph x y z $.
|
|
rdgon.2 $e |- ( ph -> A e. On ) $.
|
|
rdgon.3 $e |- ( ph -> A. x e. On ( F ` x ) e. On ) $.
|
|
$( Evaluating the recursive definition generator produces an ordinal.
|
|
There is a hypothesis that the characteristic function produces ordinals
|
|
on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) $)
|
|
rdgon $p |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On ) $=
|
|
( vy vf vg vz con0 wcel wa cvv cv cfv wceq adantr wral crdg cdm ciun cmpt
|
|
cun df-irdg wfun funmpt a1i word ordon w3a vex 3ad2ant1 dmex fveq2 eleq1d
|
|
wf simpl3 simpr wb fdm eleq2d syl mpbid ffvelrnd rspcdva ralrimiva fveq2d
|
|
cbvralv sylibr iunon sylancr onun2 dmeq fveq1 iuneq12d uneq2d eqid fvmptg
|
|
syl2anc eqeltrd cuni csuc eleq2i biimpi adantl suceloni biimpri tfrcl
|
|
unon ) ADLMZNZHLIECUAJOCBJPZUBZBPZWNQZEQZUCZUEZUDZLDBJECUFXAUGWMJOWTUHUIL
|
|
UJWMUKUIWMHPZLMZXBLIPZURZULZXDXAQZCBXDUBZWPXDQZEQZUCZUEZLXFXDOMXLLMZXGXLR
|
|
IUMZXFCLMZXKLMZXMWMXCXOXEAXOWLFSUNXFXHOMXJLMZBXHTZXPXDXNUOXFKPZXDQZEQZLMZ
|
|
KXHTXRXFYBKXHXFXSXHMZNZWPEQZLMZYBBLXTWPXTRYEYALWPXTEUPUQXFYFBLTZYCWMXCYGX
|
|
EAYGWLGSUNSYDXBLXSXDWMXCXEYCUSZYDYCXSXBMZXFYCUTYDXEYCYIVAYHXEXHXBXSXBLXDV
|
|
BVCVDVEVFVGVHXQYBBKXHWPXSRZXJYALYJXIXTEWPXSXDUPVIUQVJVKBXHXJOVLVMCXKVNWAZ
|
|
JXDWTXLOLXAWNXDRZWSXKCYLBWOXHWRXJWNXDVOYLWQXIEWPWNXDVPVIVQVRXAVSVTVMYKWBW
|
|
MXBLWCZMZNXCXBWDLMYNXCWMYNXCYMLXBWKWEWFWGXBWHVDWLDYMMZAYOWLYMLDWKWEWIWGWJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x g F $. $d x g A $.
|
|
${
|
|
rdg.1 $e |- A e. _V $.
|
|
$( The initial value of the recursive definition generator. (Contributed
|
|
by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) $)
|
|
rdg0 $p |- ( rec ( F , A ) ` (/) ) = A $=
|
|
( vg vx c0 crdg cfv cvv cdm ciun cun cmpt wcel wceq ax-mp eqtri eqeltri
|
|
cv 0ex dmeq fveq1 fveq2d iuneq12d uneq2d eqid iuneq1 0iun fvmpt df-irdg
|
|
dm0 unex tfr0 uneq2i un0 ) FBAGZHZFDIAEDSZJZESZURHZBHZKZLZMZHZAVFINUQVF
|
|
OVFAEFJZUTFHZBHZKZLZIFINVFVKOTDFVDVKIVEURFOZVCVJAVLEUSVGVBVIURFUAVLVAVH
|
|
BUTURFUBUCUDUEVEUFAVJCVJFIVJEFVIKZFVGFOVJVMOUKEVGFVIUGPEVIUHQZTRULZUIPZ
|
|
VORUPVEIEDBAUJUMPVFAFLZAVFVKVQVPVJFAVNUNQAUOQQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( The initial value of the recursive definition generator. (Contributed
|
|
by NM, 25-Apr-1995.) $)
|
|
rdg0g $p |- ( A e. C -> ( rec ( F , A ) ` (/) ) = A ) $=
|
|
( vx c0 cv crdg cfv wceq rdgeq2 fveq1d id eqeq12d vex rdg0 vtoclg ) ECDFZ
|
|
GZHZQIECAGZHZAIDABQAIZSUAQAUBERTQACJKUBLMQCDNOP $.
|
|
$}
|
|
|
|
${
|
|
rdg0.1 $e |- A e. _V $.
|
|
rdg0.2 $e |- F Fn _V $.
|
|
$( The recursive definition generator produces a set on a set input.
|
|
(Contributed by Mario Carneiro, 3-Jul-2019.) $)
|
|
rdgexg $p |- ( B e. V -> ( rec ( F , A ) ` B ) e. _V ) $=
|
|
( cvv wcel crdg cfv rdgexgg mpan ) AGHBDHBCAIJGHEABCGDFKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite recursion
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Define a constant for the finite recursive definition generator. $)
|
|
$c frec $.
|
|
|
|
$( Extend class notation with the fnite recursive definition generator, with
|
|
characteristic function ` F ` and initial value ` I ` . $)
|
|
cfrec $a class frec ( F , I ) $.
|
|
|
|
${
|
|
$d x g m F $. $d x g m I $.
|
|
$( Define a recursive definition generator on ` _om ` (the class of finite
|
|
ordinals) with characteristic function ` F ` and initial value ` I ` .
|
|
This rather amazing operation allows us to define, with compact direct
|
|
definitions, functions that are usually defined in textbooks only with
|
|
indirect self-referencing recursive definitions. A recursive definition
|
|
requires advanced metalogic to justify - in particular, eliminating a
|
|
recursive definition is very difficult and often not even shown in
|
|
textbooks. On the other hand, the elimination of a direct definition is
|
|
a matter of simple mechanical substitution. The price paid is the
|
|
daunting complexity of our ` frec ` operation (especially when ~ df-recs
|
|
that it is built on is also eliminated). But once we get past this
|
|
hurdle, definitions that would otherwise be recursive become relatively
|
|
simple; see ~ frec0g and ~ frecsuc .
|
|
|
|
Unlike with transfinite recursion, finite recurson can readily divide
|
|
definitions and proofs into zero and successor cases, because even
|
|
without excluded middle we have theorems such as ~ nn0suc . The
|
|
analogous situation with transfinite recursion - being able to say that
|
|
an ordinal is zero, successor, or limit - is enabled by excluded middle
|
|
and thus is not available to us. For the characteristic functions which
|
|
satisfy the conditions given at ~ frecrdg , this definition and
|
|
~ df-irdg restricted to ` _om ` produce the same result.
|
|
|
|
_Note: We introduce_ ` frec ` _with the philosophical goal of being_
|
|
_able to eliminate all definitions with direct mechanical substitution_
|
|
_and to verify easily the soundness of definitions. Metamath itself_
|
|
_has no built-in technical limitation that prevents multiple-part_
|
|
_recursive definitions in the traditional textbook style_. (Contributed
|
|
by Mario Carneiro and Jim Kingdon, 10-Aug-2019.) $)
|
|
df-frec $a |- frec ( F , I ) = ( recs ( ( g e. _V |-> { x |
|
|
( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/
|
|
( dom g = (/) /\ x e. I ) ) } ) ) |` _om ) $.
|
|
$}
|
|
|
|
${
|
|
$d x g m F $. $d x g m G $. $d x g m A $. $d x g m B $.
|
|
$( Equality theorem for the finite recursive definition generator.
|
|
(Contributed by Jim Kingdon, 30-May-2020.) $)
|
|
freceq1 $p |- ( F = G -> frec ( F , A ) = frec ( G , A ) ) $=
|
|
( vg vm vx wceq cvv cv cfv wcel wa com wrex wo cab cmpt crecs cres cfrec
|
|
csuc c0 simpl fveq1d eleq2d anbi2d rexbidv orbi1d abbidv mpteq2dva recseq
|
|
cdm syl reseq1d df-frec 3eqtr4g ) BCGZDHDIZULZEIZUAGZFIZUTURJZBJZKZLZEMNZ
|
|
USUBGVBAKLZOZFPZQZRZMSDHVAVBVCCJZKZLZEMNZVHOZFPZQZRZMSBATCATUQVLVTMUQVKVS
|
|
GVLVTGUQDHVJVRUQURHKZLZVIVQFWBVGVPVHWBVFVOEMWBVEVNVAWBVDVMVBWBVCBCUQWAUCU
|
|
DUEUFUGUHUIUJVKVSUKUMUNFDEBAUOFDECAUOUP $.
|
|
|
|
$( Equality theorem for the finite recursive definition generator.
|
|
(Contributed by Jim Kingdon, 30-May-2020.) $)
|
|
freceq2 $p |- ( A = B -> frec ( F , A ) = frec ( F , B ) ) $=
|
|
( vg vm vx wceq cvv cv cfv wcel wa com cab cmpt crecs cres cfrec df-frec
|
|
wo cdm csuc c0 simpl eleq2d anbi2d orbi2d abbidv mpteq2dva recseq reseq1d
|
|
wrex syl 3eqtr4g ) ABGZDHDIZUAZEIZUBGFIZURUPJCJKLEMULZUQUCGZUSAKZLZTZFNZO
|
|
ZPZMQDHUTVAUSBKZLZTZFNZOZPZMQCARCBRUOVGVMMUOVFVLGVGVMGUODHVEVKUOUPHKZLZVD
|
|
VJFVOVCVIUTVOVBVHVAVOABUSUOVNUDUEUFUGUHUIVFVLUJUMUKFDECASFDECBSUN $.
|
|
$}
|
|
|
|
${
|
|
$d A g m x $. $d F g m x $.
|
|
$( Finite recursion produces a set. (Contributed by Jim Kingdon,
|
|
20-Aug-2021.) $)
|
|
frecex $p |- frec ( F , A ) e. _V $=
|
|
( vg vm vx cfrec cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt
|
|
crecs cres df-frec wfun tfrfun omex resfunexg mp2an eqeltri ) BAFCGCHZIZD
|
|
HZJKEHZULUJLBLMNDOPUKQKUMAMNRESTZUAZOUBZGECDBAUCUOUDOGMUPGMUNUEUFUOOGUGUH
|
|
UI $.
|
|
$}
|
|
|
|
${
|
|
$d A g m x $. $d F g m x $.
|
|
$( Finite recursion produces a function. See also ~ frecfnom which also
|
|
states that the domain of that function is ` _om ` but which puts
|
|
conditions on ` A ` and ` F ` . (Contributed by Jim Kingdon,
|
|
13-Feb-2022.) $)
|
|
frecfun $p |- Fun frec ( F , A ) $=
|
|
( vg vm vx cfrec wfun cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab
|
|
cmpt crecs cres tfrfun funres ax-mp df-frec funeqi mpbir ) BAFZGCHCIZJZDI
|
|
ZKLEIZUMUKMBMNODPQULRLUNANOSETUAZUBZPUCZGZUPGURUOUDPUPUEUFUJUQECDBAUGUHUI
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d g m y F $. $d g m y A $. $d x g m y $.
|
|
nffrec.1 $e |- F/_ x F $.
|
|
nffrec.2 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for the finite recursive definition
|
|
generator. (Contributed by Jim Kingdon, 30-May-2020.) $)
|
|
nffrec $p |- F/_ x frec ( F , A ) $=
|
|
( vg vm vy cfrec cvv cv wceq cfv wcel wa com nfcv nfv nfcri nfan cdm csuc
|
|
wrex c0 wo cab cmpt crecs cres df-frec nffv nfrexya nfor nfab nfmpt nfres
|
|
nfrecs nfcxfr ) ACBIFJFKZUAZGKZUBLZHKZVAUSMZCMZNZOZGPUCZUTUDLZVCBNZOZUEZH
|
|
UFZUGZUHZPUIHFGCBUJAVOPAVNAFJVMAJQVLAHVHVKAVGAGPAPQZVBVFAVBARAHVEAVDCDAVD
|
|
QUKSTULVIVJAVIARAHBESTUMUNUOUQVPUPUR $.
|
|
$}
|
|
|
|
${
|
|
$d A g m x $. $d F g m x $.
|
|
$( The initial value resulting from finite recursive definition generation.
|
|
(Contributed by Jim Kingdon, 7-May-2020.) $)
|
|
frec0g $p |- ( A e. V -> ( frec ( F , A ) ` (/) ) = A ) $=
|
|
( vg vm vx wcel c0 cfv cvv cv cdm wceq wa com wrex wo cab dm0 ax-mp cfrec
|
|
csuc cmpt biantrur wn wne vex nsuceq0g nesymi eqeq1i mtbir intnanr biorfi
|
|
a1i nrex orcom 3bitri abbii abid2 eqtr3i elex syl5eqel dmeq eqeq1d fveq2d
|
|
0ex fveq1 eleq2d anbi12d rexbidv anbi1d orbi12d abbidv eqid fvmptg syl6eq
|
|
mpan syl eqeltrd crecs cres df-frec fveq1i peano1 fvres eqtri tfr0 syl5eq
|
|
eqtrd ) ACGZHBAUAZIZHDJDKZLZEKZUBZMZFKZWOWMIZBIZGZNZEOPZWNHMZWRAGZNZQZFRZ
|
|
UCZIZAWJXJJGZWLXJMWJXJAJWJHLZWPMZWRWOHIZBIZGZNZEOPZXLHMZXENZQZFRZJGZXJAMW
|
|
JYBAJXEFRYBAXEYAFXEXTXTXRQYAXSXESUDXRXTXQEOXQUEWOOGXMXPXMHWPMWPHWOJGWPHUF
|
|
EUGWOJUHTUIXLHWPSUJUKULUNUOUMXTXRUPUQURFAUSUTZACVAZVBYCXJYBAHJGYCXJYBMVFD
|
|
HXHYBJJXIWMHMZXGYAFYFXCXRXFXTYFXBXQEOYFWQXMXAXPYFWNXLWPWMHVCZVDYFWTXOWRYF
|
|
WSXNBWOWMHVGVEVHVIVJYFXDXSXEYFWNXLHYGVDVKVLVMXIVNVOVQYDVPVRZYEVSXKWLHXIVT
|
|
ZIZXJWLHYIOWAZIZYJHWKYKFDEBAWBWCHOGYLYJMWDHOYIWETWFYIXIJYIVNWGWHVRYHWI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d F x $. $d S x y $. $d m ph $. $d m x y $. $d F y $.
|
|
frecabex.sex $e |- ( ph -> S e. V ) $.
|
|
frecabex.fvex $e |- ( ph -> A. y ( F ` y ) e. _V ) $.
|
|
frecabex.aex $e |- ( ph -> A e. W ) $.
|
|
$( The class abstraction from ~ df-frec exists. This is a lemma for other
|
|
finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) $)
|
|
frecabex $p |- ( ph -> { x |
|
|
( E. m e. _om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/
|
|
( dom S = (/) /\ x e. A ) ) } e. _V ) $=
|
|
( cv wceq cfv wcel wa com cab cvv cdm csuc wrex c0 wo wral omex wss simpr
|
|
abssi wal fvexg sylancl fveq2 eleq1d spcgv sylc ssexg ralrimivw abrexex2g
|
|
vex sylancr jca cun unexb unab eleq1i bitri sylib ) AEUAZFMZUBNZBMZVKEOZG
|
|
OZPZQZFRUCZBSZTPZVJUDNZVMDPZQZBSZTPZQZVRWCUEBSZTPZAVTWEARTPVQBSZTPZFRUFVT
|
|
UGAWJFRAWIVOUHVOTPZWJVQBVOVLVPUIUJAVNTPZCMZGOZTPZCUKWKAEHPVKTPWLJFVAVKEHT
|
|
ULUMKWOWKCVNTWMVNNWNVOTWMVNGUNUOUPUQWIVOTURVBUSVQFBRTTUTVBAWDDUHDIPWEWCBD
|
|
WAWBUIUJLWDDIURVBVCWFVSWDVDZTPWHVSWDVEWPWGTVRWCBVFVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d A k m x z $. $d F k m x y $. $d F k m x z $. $d G k m x y $.
|
|
$d G k m x z $. $d N k m x y $. $d S k x y $. $d S k x z $.
|
|
$d k m ph x y $.
|
|
frecabcl.n $e |- ( ph -> N e. _om ) $.
|
|
frecabcl.g $e |- ( ph -> G : N --> S ) $.
|
|
frecabcl.fs $e |- ( ph -> A. y e. S ( F ` y ) e. S ) $.
|
|
frecabcl.as $e |- ( ph -> A e. S ) $.
|
|
$( The class abstraction from ~ df-frec exists. Unlike ~ frecabex the
|
|
function ` F ` only needs to be defined on ` S ` , not all sets. This
|
|
is a lemma for other finite recursion proofs. (Contributed by Jim
|
|
Kingdon, 21-Mar-2022.) $)
|
|
frecabcl $p |- ( ph -> { x |
|
|
( E. m e. _om ( dom G = suc m /\ x e. ( F ` ( G ` m ) ) ) \/
|
|
( dom G = (/) /\ x e. A ) ) } e. S ) $=
|
|
( cv wcel wceq wa com c0 wb vz vk va cdm csuc cfv wrex wal wex cab adantr
|
|
wo peano1 elex2 r19.9rmv mp2b wf fdm syl eqeq2 adantl mpbid biantrurd wne
|
|
peano3 neeqtrrd necomd neneqd intnanrd biorf bitrd rexbidva syl5bb syl6bb
|
|
r19.44mv alrimiv jca eleq1 eleq2 bibi1d albidv anbi12d spcegv sylc eleq1d
|
|
wn fveq2 wral ad2antrr sucid mpbiri ffvelrnd rspcdva simpllr simplr eqtrd
|
|
simpr suceq eqeq2d fveq2d eleq2d rspcev syl2anc ex simprl 3eqtr3rd peano4
|
|
mpbird rexlimdva impbid neeq1 eqnetrd orcom bibi2d nn0suc mpjaodan clabel
|
|
vex imp sylibr ) AUANZEOZBNZYAOZHUDZFNZUEZPZYCYFHUFZGUFZOZQZFRUGZYESPZYCD
|
|
OZQZULZTZBUHZQZUAUIZYQBUJEOAISPZUUAIUBNZUEZPZUBRUGZAUUBQZDEOZUUHYOYQTZBUH
|
|
ZQZUUAAUUHUUBMUKZUUGUUHUUJUULUUGUUIBUUGYOYLYPULZFRUGZYQYOYOFRUGZUUGUUNSRO
|
|
ZUCNROUCUIZYOUUOTUMUCSRUNZYOFUCRUOUPUUGYOUUMFRUUGYFROZQZYOYPUUMUUTYNYOUUG
|
|
YNUUSUUGYEIPZYNAUVAUUBAIEHUQZUVAKIEHURZUSUKUUBUVAYNTAISYEUTVAVBUKZVCUUTYL
|
|
WFYPUUMTUUTYHYKUUTYEYGUUTYGYEUUTYGSYEUUSYGSVDUUGYFVEVAUVDVFVGVHVIYLYPVJUS
|
|
VKVLVMUUPUUQUUNYQTUMUURYLYPFUCRVOUPVNVPVQYTUUKUADEYADPZYBUUHYSUUJYADEVRUV
|
|
EYRUUIBUVEYDYOYQYADYCVSVTWAWBWCWDAUUFUUAAUUEUUAUBRAUUCROZQZUUEUUAUVGUUEQZ
|
|
UUCHUFZGUFZEOZUVKYCUVJOZYQTZBUHZQZUUAUVHCNZGUFZEOZUVKCEUVIUVPUVIPUVQUVJEU
|
|
VPUVIGWGWEAUVRCEWHUVFUUELWIUVHIEUUCHAUVBUVFUUEKWIZUUEUUCIOZUVGUUEUVTUUCUU
|
|
DOUUCUBXRWJIUUDUUCVSWKVAWLWMZUVHUVKUVNUWAUVHUVLYMTZBUHUVNUVHUWBBUVHUVLYMU
|
|
VHUVLYMUVHUVLQZUVFYEUUDPZUVLQZYMAUVFUUEUVLWNUWCUWDUVLUWCYEIUUDUVHUVAUVLUV
|
|
HUVBUVAUVSUVCUSZUKUVGUUEUVLWOWPUVHUVLWQVQYLUWEFUUCRYFUUCPZYHUWDYKUVLUWGYG
|
|
UUDYEYFUUCWRWSUWGYJUVJYCUWGYIUVIGYFUUCHWGWTXAWBXBXCXDUVHYLUVLFRUVHUUSQZYL
|
|
UVLUWHYLQZUVLYKYLYKUWHYHYKWQVAUWIUVJYJYCUWIUVIYIGUWIUUCYFHUWIUUDYGPZUUCYF
|
|
PZUWIYEIYGUUDUVHUVAUUSYLUWFWIUWHYHYKXEUVGUUEUUSYLWNXFUWIUVFUUSUWJUWKTUVHU
|
|
VFUUSYLAUVFUUEWOZWIUVHUUSYLWOUUCYFXGXCVBWTWTXAXHXDXIXJVPUVHUWBUVMBUVHYMYQ
|
|
UVLUVHYMYPYMULZYQUVHYPWFYMUWMTUVHYNYOUVHYESUVHYEISUWFUVHISVDZUUDSVDZUVHUV
|
|
FUWOUWLUUCVEUSUUEUWNUWOTUVGIUUDSXKVAXHXLVHVIYPYMVJUSYPYMXMVNXNWAVBVQYTUVO
|
|
UAUVJEYAUVJPZYBUVKYSUVNYAUVJEVRUWPYRUVMBUWPYDUVLYQYAUVJYCVSVTWAWBWCWDXDXI
|
|
XSAIROUUBUUFULJUBIXOUSXPYQBUAEXQXT $.
|
|
$}
|
|
|
|
${
|
|
$d g m x y A $. $d g m x y z F $. $d g m y V $.
|
|
frectfr.1 $e |- G = ( g e. _V |-> { x | ( E. m e. _om ( dom g = suc m /\
|
|
x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) $.
|
|
$( Lemma to connect transfinite recursion theorems with finite recursion.
|
|
That is, given the conditions ` F Fn _V ` and ` A e. V ` on
|
|
` frec ( F , A ) ` , we want to be able to apply ~ tfri1d or ~ tfri2d ,
|
|
and this lemma lets us satisfy hypotheses of those theorems.
|
|
|
|
(Contributed by Jim Kingdon, 15-Aug-2019.) $)
|
|
frectfr $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
|
|
A. y ( Fun G /\ ( G ` y ) e. _V ) ) $=
|
|
( cv cfv cvv wcel wal wa wfun wfn wceq vex cdm csuc com wrex cab wral a1i
|
|
c0 wo simpl simpr frecabex ralrimivw fnmpt funfvex funfni sylancl funmpt2
|
|
syl jctil alrimiv ) CKGLMNCOZDINZPZHQZBKZHLMNZPBVDVGVEVDHMRZVFMNVGVDEKZUA
|
|
ZFKZUBSAKZVKVILGLNPFUCUDVJUHSVLDNPUIAUEZMNZEMUFVHVDVNEMVDACDVIFGMIVIMNVDE
|
|
TUGVBVCUJVBVCUKULUMEMVMHMJUNUSBTVGMVFHVFHUOUPUQEMVMHJURUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A g m x y z $. $d B g m x z $. $d F g m x y z $. $d G g m x y z $.
|
|
$d V g m x y $.
|
|
|
|
$( The function generated by finite recursive definition generation is a
|
|
function on omega. (Contributed by Jim Kingdon, 13-May-2020.) $)
|
|
frecfnom $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) ->
|
|
frec ( F , A ) Fn _om ) $=
|
|
( vg vm vx vy cv cfv cvv wcel wal wa wceq com con0 cres wfn eqid cdm csuc
|
|
wrex c0 wo cab cmpt crecs cin cfrec frectfr tfri1d fnresin1 syl wb omsson
|
|
wss sseqin2 mpbi reseq2i df-frec eqtr4i fneq12 mp2an sylib ) AICJKLAMBDLN
|
|
ZEKEIZUAZFIZUBOGIZVIVGJCJLNFPUCVHUDOVJBLNUEGUFUGZUHZQPUIZRZVMSZCBUJZPSZVF
|
|
VLQSVOVFHVLVKVLTGHABEFCVKDVKTUKULQPVLUMUNVNVPOVMPOZVOVQUOVNVLPRVPVMPVLPQU
|
|
QVRUPPQURUSZUTGEFCBVAVBVSVMPVNVPVCVDVE $.
|
|
|
|
${
|
|
frecsuclem1.h $e |- G = ( g e. _V |-> { x | (
|
|
E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
|
|
\/ ( dom g = (/) /\ x e. A ) ) } ) $.
|
|
$( Lemma for ~ frecsuc . (Contributed by Jim Kingdon, 13-Aug-2019.) $)
|
|
frecsuclem1 $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. _om ) ->
|
|
( frec ( F , A ) ` suc B ) = ( G ` ( recs ( G ) |` suc B ) ) ) $=
|
|
( vy cv cfv cvv wcel com csuc cres wceq wa wal w3a cfrec crecs cdm wrex
|
|
c0 cab cmpt df-frec recseq ax-mp reseq1i eqtr4i fveq1i peano2 fvres syl
|
|
wo syl5eq 3ad2ant3 con0 nnon eqid frectfr tfri2d sylan2 3impa eqtrd ) B
|
|
LGMNOBUAZCIOZDPOZUBDQZGCUCZMZVMHUDZMZVPVMRHMZVLVJVOVQSVKVLVOVMVPPRZMZVQ
|
|
VMVNVSVNENELZUEZFLZQSALZWCWAMGMOTFPUFWBUGSWDCOTUSAUHUIZUDZPRVSAEFGCUJVP
|
|
WFPHWESVPWFSJHWEUKULUMUNUOVLVMPOZVTVQSDUPZVMPVPUQURUTVAVJVKVLVQVRSZVLVJ
|
|
VKTZVMVBOZWIVLWGWKWHVMVCURWJKVMVPHVPVDAKBCEFGHIJVEVFVGVHVI $.
|
|
|
|
$( Lemma for ~ frecsuc . (Contributed by Jim Kingdon, 15-Aug-2019.) $)
|
|
frecsuclemdm $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. _om ) ->
|
|
dom ( recs ( G ) |` suc B ) = suc B ) $=
|
|
( vy cv cfv cvv wcel wal cdm wss wceq con0 com w3a csuc crecs cres nnon
|
|
suceloni onss 3syl 3ad2ant3 wb wfn eqid frectfr tfri1d fndm syl 3adant3
|
|
wa sseq2d mpbird ssdmres sylib ) BLGMNOBPZCIOZDUAOZUBZDUCZHUDZQZRZVIVHU
|
|
EQVHSVGVKVHTRZVFVDVLVEVFDTOVHTOVLDUFDUGVHUHUIUJVDVEVKVLUKVFVDVEUSZVJTVH
|
|
VMVITULVJTSVMKVIHVIUMAKBCEFGHIJUNUOTVIUPUQUTURVAVHVIVBVC $.
|
|
|
|
$( Lemma for ~ frecsuc . (Contributed by Jim Kingdon, 15-Aug-2019.) $)
|
|
frecsuclem2 $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. _om ) ->
|
|
( ( recs ( G ) |` suc B ) ` B ) = ( frec ( F , A ) ` B ) ) $=
|
|
( com wcel cv cfv cvv crecs csuc cres wceq fvres wal cfrec syl cdm wrex
|
|
sucidg wa c0 cab cmpt df-frec recseq ax-mp reseq1i eqtr4i fveq1i syl5eq
|
|
wo eqtr4d 3ad2ant3 ) DKLZBMGNOLBUADHPZDQZRNZDGCUBZNZSCILVAVDDVBNZVFVADV
|
|
CLVDVGSDKUFDVCVBTUCVAVFDVBKRZNVGDVEVHVEEOEMZUDZFMZQSAMZVKVINGNLUGFKUEVJ
|
|
UHSVLCLUGURAUIUJZPZKRVHAEFGCUKVBVNKHVMSVBVNSJHVMULUMUNUOUPDKVBTUQUSUT
|
|
$.
|
|
|
|
$( Lemma for ~ frecsuc . (Contributed by Jim Kingdon, 15-Aug-2019.) $)
|
|
frecsuclem3 $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. _om ) ->
|
|
( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
|
|
( cv cfv cvv wcel com wceq wa wrex c0 eqeq1d vy wal w3a csuc cfrec cres
|
|
crecs cdm wo cab wfun con0 wfn eqid frectfr tfri1d fnfun 3adant3 peano2
|
|
syl 3ad2ant3 resfunexg syl2anc simp1 simp2 frecabex fveq1 fveq2d eleq2d
|
|
wi dmeq anbi12d rexbidv anbi1d orbi12d abbidv fvmptg ex frecsuclem1 mpd
|
|
sylibrd abeq2d wn frecsuclemdm wne peano3 eqnetrd neneqd intnanrd biorf
|
|
orcom syl6bb vex suc11g mpan2 bitrd eqcom 3bitr2d fveq2 ceqsrexbv eqrdv
|
|
wb 3anibar frecsuclem2 eqtrd ) BKGLMNBUBZCINZDONZUCZDUDZGCUEZLZDHUGZXJU
|
|
FZLZGLZDXKLZGLXIAXLXPXFXGXHAKZXLNZXRXPNZXIXSFKZDPZXRYAXNLZGLZNZQZFORZXH
|
|
XTQXIXSXNUHZYAUDZPZYEQZFORZYHSPZXRCNZQZUIZYLYGXIYPAXLXIYPAUJZMNZXLYQPZX
|
|
IABCXNFGMIXIXMUKZXJONZXNMNZXFXGYTXHXFXGQZXMULUMYTUUCUAXMHXMUNAUABCEFGHI
|
|
JUOUPULXMUQUTURXHXFUUAXGDUSVAXMXJOVBVCZXFXGXHVDXFXGXHVEVFXIYRXNHLZYQPZY
|
|
SXIUUBYRUUFVJUUDUUBYRUUFEXNEKZUHZYIPZXRYAUUGLZGLZNZQZFORZUUHSPZYNQZUIZA
|
|
UJYQMMHUUGXNPZUUQYPAUURUUNYLUUPYOUURUUMYKFOUURUUIYJUULYEUURUUHYHYIUUGXN
|
|
VKZTUURUUKYDXRUURUUJYCGYAUUGXNVGVHVIVLVMUURUUOYMYNUURUUHYHSUUSTVNVOVPJV
|
|
QVRUTXIXLUUEYQABCDEFGHIJVSTWAVTWBXIYLYOYLUIZYPXIYOWCYLUUTXBXIYMYNXIYHSX
|
|
IYHXJSABCDEFGHIJWDZXHXFXJSWEXGDWFVAWGWHWIYOYLWJUTYOYLWKWLXIYKYFFOXIYJYB
|
|
YEXIYJDYAPZYBXIYJXJYIPZUVBXIYHXJYIUVATXHXFUVCUVBXBZXGXHYAMNUVDFWMDYAOMW
|
|
NWOVAWPDYAWQWLVNVMWRYEXTFDOYBYDXPXRYBYCXOGYADXNWSVHVIWTWLXCXAXIXOXQGABC
|
|
DEFGHIJXDVHXE $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A f g m x y $. $d A g m y z $. $d B g m y z $. $d F f g m n x y $.
|
|
$d F g m y z $. $d V g m y $. $d f x n z $.
|
|
$( The successor value resulting from finite recursive definition
|
|
generation. (Proof modification is discouraged.)
|
|
(New usage is discouraged.) (Contributed by Jim Kingdon,
|
|
15-Aug-2019.) $)
|
|
frecsucOLD $p |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. _om ) ->
|
|
( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
|
|
( vy vg vm vf vx cvv cv wceq cfv wcel wa com wrex wo cab vn cdm csuc cmpt
|
|
c0 suceq eqeq2d fveq2 fveq2d eleq2d anbi12d cbvrexv orbi1i anbi2d rexbidv
|
|
abbii eleq1 orbi12d cbvabv eqtri mpteq2i dmeq eqeq1d fveq1 anbi1d cbvmptv
|
|
abbidv frecsuclem3 ) FABCGHDIKILZUBZUALZUCZMZJLZVKVINZDNZOZPZUAQRZVJUEMZV
|
|
NBOZPZSZJTZUDZEWEIKVJHLZUCZMZFLZWFVINZDNZOZPZHQRZVTWIBOZPZSZFTZUDGKGLZUBZ
|
|
WGMZWIWFWSNZDNZOZPZHQRZWTUEMZWOPZSZFTZUDIKWDWRWDWHVNWKOZPZHQRZWBSZJTWRWCX
|
|
NJVSXMWBVRXLUAHQVKWFMZVMWHVQXKXOVLWGVJVKWFUFUGXOVPWKVNXOVOWJDVKWFVIUHUIUJ
|
|
UKULUMUPXNWQJFVNWIMZXMWNWBWPXPXLWMHQXPXKWLWHVNWIWKUQUNUOXPWAWOVTVNWIBUQUN
|
|
URUSUTVAIGKWRXJVIWSMZWQXIFXQWNXFWPXHXQWMXEHQXQWHXAWLXDXQVJWTWGVIWSVBZVCXQ
|
|
WKXCWIXQWJXBDWFVIWSVDUIUJUKUOXQVTXGWOXQVJWTUEXRVCVEURVGVFUTVH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x $. $d F x y z $. $d S x y z $. $d ph x y z $.
|
|
frecclOLD.ex $e |- ( ph -> A. z ( F ` z ) e. _V ) $.
|
|
frecclOLD.a $e |- ( ph -> A e. S ) $.
|
|
frecclOLD.cl $e |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S ) $.
|
|
frecclOLD.b $e |- ( ph -> B e. _om ) $.
|
|
$( Closure for finite recursion. New proofs should use ~ freccl which is
|
|
the same but without the ` A. z ( F `` z ) e. _V ` hypothesis.
|
|
(Proof modification is discouraged.) (New usage is discouraged.)
|
|
(Contributed by Jim Kingdon, 25-May-2020.) $)
|
|
frecclOLD $p |- ( ph -> ( frec ( F , A ) ` B ) e. S ) $=
|
|
( vx com wcel cfv cv wi wceq fveq2 eleq1d adantr cfrec imbi2d csuc frec0g
|
|
vy c0 syl eqeltrd wa wex cvv wfn wal frecfnom syl2anc funfvex sylan isset
|
|
funfni sylib ex eleq1 adantl 3imtr3d exlimdv mpd simpr frecsucOLD syl3anc
|
|
wb sylibrd expcom finds2 vtoclga mpcom ) DLMADFCUAZNZEMZJAKOZVPNZEMZPAVRP
|
|
KDLVSDQZWAVRAWBVTVQEVSDVPRSUBWAUFVPNZEMUEOZVPNZEMZWDUCZVPNZEMZAKUEVSUFQVT
|
|
WCEVSUFVPRSVSWDQVTWEEVSWDVPRSVSWGQVTWHEVSWGVPRSAWCCEACEMZWCCQHCFEUDUGHUHA
|
|
WDLMZWFWIPAWKUIZWFWEFNZEMZWIWLBOZWEQZBUJZWFWNPZWLWEUKMZWQAVPLULZWKWSAWOFN
|
|
ZUKMBUMZWJWTGHBCFEUNUOWSLWDVPWDVPUPUSUQBWEURUTAWQWRPWKAWPWRBAWPWRAWPUIWOE
|
|
MZXAEMZWFWNAXCXDPWPAXCXDIVATWPXCWFVJAWOWEEVBVCWPXDWNVJAWPXAWMEWOWEFRSVCVD
|
|
VAVETVFWLWHWMEWLXBWJWKWHWMQAXBWKGTAWJWKHTAWKVGBCWDFEVHVISVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
freccl.a $e |- ( ph -> A e. S ) $.
|
|
freccl.cl $e |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S ) $.
|
|
freccl.b $e |- ( ph -> B e. _om ) $.
|
|
|
|
${
|
|
$d A f g m x y $. $d A f m x y z $. $d B x $. $d F f g m x y $.
|
|
$d F f m x y z $. $d G f y $. $d S f m x y z $. $d f m ph x y z $.
|
|
freccllem.g $e |- G = recs ( ( g e. _V |-> { x |
|
|
( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
|
|
\/ ( dom g = (/) /\ x e. A ) ) } ) ) $.
|
|
$( Lemma for ~ freccl . Just giving a name to a common expression to
|
|
simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.) $)
|
|
freccllem $p |- ( ph -> ( frec ( F , A ) ` B ) e. S ) $=
|
|
( cfv com cvv cv wceq wcel vy vf cfrec cres cdm csuc wa wrex c0 wo cmpt
|
|
cab crecs df-frec reseq1i eqtr4i fveq1i fvres syl wfun funmpt a1i ordom
|
|
word wf w3a vex simp2 simp3 wral ralrimiva 3ad2ant1 frecabcl dmeq fveq1
|
|
eqeq1d fveq2d eleq2d anbi12d rexbidv anbi1d orbi12d abbidv eqid sylancr
|
|
fvmptg eqeltrd cuni limom limuni eleq2i peano2 adantl sylan2br syl6eleq
|
|
wlim ax-mp tfrcl syl5eqel ) AEIDUCZOEJPUDZOZFEWTXAWTGQGRZUEZHRZUFZSZBRZ
|
|
XEXCOZIOZTZUGZHPUHZXDUISZXHDTZUGZUJZBULZUKZUMZPUDXABGHIDUNJXTPNUOUPUQAX
|
|
BEJOZFAEPTXBYASMEPJURUSAUAFUBJXSPENXSUTAGQXRVAVBPVDAVCVBAUARZPTZYBFUBRZ
|
|
VEZVFZYDXSOZYDUEZXFSZXHXEYDOZIOZTZUGZHPUHZYHUISZXOUGZUJZBULZFYFYDQTYRFT
|
|
YGYRSUBVGYFBCDFHIYDYBAYCYEVHAYCYEVIAYCCRIOFTZCFVJYEAYSCFLVKVLAYCDFTYEKV
|
|
LVMZGYDXRYRQFXSXCYDSZXQYQBUUAXMYNXPYPUUAXLYMHPUUAXGYIXKYLUUAXDYHXFXCYDV
|
|
NZVPUUAXJYKXHUUAXIYJIXEXCYDVOVQVRVSVTUUAXNYOXOUUAXDYHUIUUBVPWAWBWCXSWDW
|
|
FWEYTWGYBPWHZTAYCYBUFPTZPUUCYBPWPPUUCSWIPWJWQZWKYCUUDAYBWLWMWNAEPUUCMUU
|
|
EWOWRWGWS $.
|
|
$}
|
|
|
|
$d ph x z m $. $d S x z m $. $d F x z g m $. $d A x z g m $. $d B x $.
|
|
$( Closure for finite recursion. (Contributed by Jim Kingdon,
|
|
27-Mar-2022.) $)
|
|
freccl $p |- ( ph -> ( frec ( F , A ) ` B ) e. S ) $=
|
|
( vx vg vm cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt crecs
|
|
eqid freccllem ) AJBCDEKLFKMKNZOZLNZPQJNZULUJRFRSTLUAUBUKUCQUMCSTUDJUEUFU
|
|
GZGHIUNUHUI $.
|
|
$}
|
|
|
|
${
|
|
${
|
|
$d A f g m x y $. $d A f k m w x y $. $d F f g m x y $.
|
|
$d F f k m w x y z $. $d G f k y $. $d S f k m w x y z $. $d m x z $.
|
|
frecfcllem.g $e |- G = recs ( ( g e. _V |-> { x |
|
|
( E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
|
|
\/ ( dom g = (/) /\ x e. A ) ) } ) ) $.
|
|
$( Lemma for ~ frecfcl . Just giving a name to a common expression to
|
|
simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.) $)
|
|
frecfcllem $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S )
|
|
-> frec ( F , A ) : _om --> S ) $=
|
|
( vk vy vf vw cv cfv wcel wa com cvv wceq wral cres cfrec cdm csuc wrex
|
|
wf c0 cab cmpt wfun funmpt a1i word ordom w3a simp2 simp3 simp1ll fveq2
|
|
wo vex eleq1d cbvralv sylib simp1lr frecabcl eqeq1d fveq1 fveq2d eleq2d
|
|
dmeq anbi12d rexbidv anbi1d orbi12d abbidv fvmptg sylancr eqeltrd limom
|
|
eqid cuni limuni ax-mp eleq2i peano2 sylbir adantl biimpi tfrcldm tfrcl
|
|
wlim jca ralrimiva wb crecs tfrfun funeqi mpbir ffvresb df-frec reseq1i
|
|
sylibr eqtr4i feq1i ) BNZGOZDPZBDUAZCDPZQZRDHRUBZUGZRDGCUCZUGXLJNZHUDPZ
|
|
XPHODPZQZJRUAZXNXLXSJRXLXPRPZQZXQXRYBKDLHESENZUDZFNZUEZTZANZYEYCOZGOZPZ
|
|
QZFRUFZYDUHTZYHCPZQZVAZAUIZUJZRXPIYSUKYBESYRULUMZRUNYBUOUMZYBKNZRPZUUBD
|
|
LNZUGZUPZUUDYSOZUUDUDZYFTZYHYEUUDOZGOZPZQZFRUFZUUHUHTZYOQZVAZAUIZDUUFUU
|
|
DSPUURDPUUGUURTLVBUUFAMCDFGUUDUUBYBUUCUUEUQYBUUCUUEURUUFXJMNZGOZDPZMDUA
|
|
XJXKYAUUCUUEUSXIUVABMDXGUUSTXHUUTDXGUUSGUTVCVDVEXJXKYAUUCUUEVFVGZEUUDYR
|
|
UURSDYSYCUUDTZYQUUQAUVCYMUUNYPUUPUVCYLUUMFRUVCYGUUIYKUULUVCYDUUHYFYCUUD
|
|
VLZVHUVCYJUUKYHUVCYIUUJGYEYCUUDVIVJVKVMVNUVCYNUUOYOUVCYDUUHUHUVDVHVOVPV
|
|
QYSWBVRVSUVBVTZUUBRWCZPZUUBUERPZYBUVGUUCUVHRUVFUUBRWMRUVFTWARWDWEZWFUUB
|
|
WGWHWIZYAXPUVFPZXLYAUVKRUVFXPUVIWFWJWIZWKYBKDLHYSRXPIYTUUAUVEUVJUVLWLWN
|
|
WOHUKZXNXTWPUVMYSWQZUKYSWRHUVNIWSWTJRDHXAWEXDRDXOXMXOUVNRUBXMAEFGCXBHUV
|
|
NRIXCXEXFXD $.
|
|
$}
|
|
|
|
$d A g m x $. $d F g m x $. $d F m x z $. $d S m x z $.
|
|
$( Finite recursion yields a function on the natural numbers. (Contributed
|
|
by Jim Kingdon, 30-Mar-2022.) $)
|
|
frecfcl $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S )
|
|
-> frec ( F , A ) : _om --> S ) $=
|
|
( vx vg vm cvv cv cdm csuc wceq cfv wcel wa com wrex c0 wo cab cmpt crecs
|
|
eqid frecfcllem ) EABCFGDFHFIZJZGIZKLEIZUGUEMDMNOGPQUFRLUHBNOSETUAUBZUIUC
|
|
UD $.
|
|
$}
|
|
|
|
${
|
|
$d A f g m x $. $d A f m w x y $. $d B f g m x $. $d B f m w x y $.
|
|
$d F f g m x $. $d F f m w x y z $. $d G f g m x $. $d G f m w x y $.
|
|
$d S f m w x y z $.
|
|
frecsuclem.g $e |- G = ( g e. _V |-> { x | (
|
|
E. m e. _om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
|
|
\/ ( dom g = (/) /\ x e. A ) ) } ) $.
|
|
$( Lemma for ~ frecsuc . Just giving a name to a common expression to
|
|
simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.) $)
|
|
frecsuclem $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. _om ) ->
|
|
( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
|
|
( vw cv cfv wcel com wceq wa c0 cvv 3ad2ant3 vy wral w3a csuc cfrec crecs
|
|
vf cres wrex cdm wo cab df-frec recseq ax-mp reseq1i eqtr4i fveq1i peano2
|
|
cmpt fvres syl syl5eq eqid wfun funmpt2 a1i word ordom wf vex simp2 simp3
|
|
simp11 fveq2 eleq1d sylib simp12 frecabcl dmeq eqeq1d fveq1 fveq2d eleq2d
|
|
cbvralv anbi12d rexbidv anbi1d orbi12d abbidv fvmptg syl2anc eqeltrd cuni
|
|
wlim limom limuni eleq2i sylbir adantl tfrcldm tfr2a tfrfun resfunexg wss
|
|
eqtrd frecfcl feq1i 3adant3 ordelsuc mpan2 mpbid fssres2 simp1 abeq2d fdm
|
|
wb wn wne peano3 eqnetrd neneqd intnanrd biorf orcom syl6bb bitrd 3bitr2d
|
|
suc11g eqcom ceqsrexbv 3anibar eqrdv sucidg eqtr4d ) BLZHMZENZBEUBZCENZDO
|
|
NZUCZDUDZHCUEZMZDIUFZUUCUHZMZHMZDUUDMZHMUUBAUUEUUIYSYTUUAALZUUENZUUKUUINZ
|
|
UUBUULGLZDPZUUKUUNUUGMZHMZNZQZGOUIZUUAUUMQUUBUULUUGUJZUUNUDZPZUURQZGOUIZU
|
|
VARPZUUKCNZQZUKZUVEUUTUUBUVIAUUEUUBUUEUUGIMZUVIAULZUUBUUEUUCUUFMZUVJUUAYS
|
|
UUEUVLPYTUUAUUEUUCUUFOUHZMZUVLUUCUUDUVMUUDFSFLZUJZUVBPZUUKUUNUVOMZHMZNZQZ
|
|
GOUIZUVPRPZUVGQZUKZAULZUTZUFZOUHUVMAFGHCUMUUFUWHOIUWGPUUFUWHPJIUWGUNUOUPU
|
|
QZURUUAUUCONZUVNUVLPDUSZUUCOUUFVAVBVCTUUBUUCUUFUJNUVLUVJPUUBUAEUGUUFIOUUC
|
|
UUFVDZIVEUUBFSUWFIJVFVGOVHZUUBVIVGUUBUALZONZUWNEUGLZVJZUCZUWPIMZUWPUJZUVB
|
|
PZUUKUUNUWPMZHMZNZQZGOUIZUWTRPZUVGQZUKZAULZEUWRUWPSNZUXJENUWSUXJPUXKUWRUG
|
|
VKVGUWRAKCEGHUWPUWNUUBUWOUWQVLUUBUWOUWQVMUWRYSKLZHMZENZKEUBZYSYTUUAUWOUWQ
|
|
VNYRUXNBKEYPUXLPYQUXMEYPUXLHVOVPWEZVQYSYTUUAUWOUWQVRVSZFUWPUWFUXJSEIUVOUW
|
|
PPZUWEUXIAUXRUWBUXFUWDUXHUXRUWAUXEGOUXRUVQUXAUVTUXDUXRUVPUWTUVBUVOUWPVTZW
|
|
AUXRUVSUXCUUKUXRUVRUXBHUUNUVOUWPWBWCWDWFWGUXRUWCUXGUVGUXRUVPUWTRUXSWAWHWI
|
|
WJJWKWLUXQWMUWNOWNZNZUWNUDONZUUBUYAUWOUYBOUXTUWNOWOOUXTPWPOWQUOZWRUWNUSWS
|
|
WTUUAYSUUCUXTNZYTUUAUWJUYDUWKOUXTUUCUYCWRVQTXAUUCUUFIUWLXBVBXFUUBUUGSNZUV
|
|
KENUVJUVKPUUBUUFVEZUWJUYEUYFUUBIXCVGUUAYSUWJYTUWKTZUUFUUCOXDWLUUBAKCEGHUU
|
|
GUUCUYGUUBOEUVMVJZUUCOXEZUUCEUUGVJZYSYTUYHUUAYSYTQOEUUDVJUYHBCEHXGOEUUDUV
|
|
MUWIXHVQXIUUBUUAUYIYSYTUUAVMUUAYSUUAUYIXQZYTUUAUWMUYKVIDOOXJXKTXLOEUUCUUF
|
|
XMWLZUUBYSUXOYSYTUUAXNUXPVQYSYTUUAVLVSFUUGUWFUVKSEIUVOUUGPZUWEUVIAUYMUWBU
|
|
VEUWDUVHUYMUWAUVDGOUYMUVQUVCUVTUURUYMUVPUVAUVBUVOUUGVTZWAUYMUVSUUQUUKUYMU
|
|
VRUUPHUUNUVOUUGWBWCWDWFWGUYMUWCUVFUVGUYMUVPUVARUYNWAWHWIWJJWKWLXFXOUUBUVE
|
|
UVHUVEUKZUVIUUBUVHXRUVEUYOXQUUBUVFUVGUUBUVARUUBUVAUUCRUUBUYJUVAUUCPUYLUUC
|
|
EUUGXPVBZUUAYSUUCRXSYTDXTTYAYBYCUVHUVEYDVBUVHUVEYEYFUUBUVDUUSGOUUBUVCUUOU
|
|
URUUBUVCDUUNPZUUOUUBUVCUUCUVBPZUYQUUBUVAUUCUVBUYPWAUUAYSUYRUYQXQZYTUUAUUN
|
|
SNUYSGVKDUUNOSYIXKTYGDUUNYJYFWHWGYHUURUUMGDOUUOUUQUUIUUKUUOUUPUUHHUUNDUUG
|
|
VOWCWDYKYFYLYMUUBUUHUUJHUUAYSUUHUUJPYTUUAUUHDUUFMZUUJUUADUUCNUUHUYTPDOYND
|
|
UUCUUFVAVBUUAUUJDUVMMUYTDUUDUVMUWIURDOUUFVAVCYOTWCXF $.
|
|
$}
|
|
|
|
${
|
|
$d A f g m x y $. $d B g m x $. $d F f g m n x y $. $d F m x z $.
|
|
$d S m x z $.
|
|
$( The successor value resulting from finite recursive definition
|
|
generation. (Contributed by Jim Kingdon, 31-Mar-2022.) $)
|
|
frecsuc $p |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. _om ) ->
|
|
( frec ( F , A ) ` suc B ) = ( F ` ( frec ( F , A ) ` B ) ) ) $=
|
|
( vx vg vm vn vy cvv cv wceq cfv wcel wa com wrex wo cab vf cdm csuc cmpt
|
|
c0 dmeq eqeq1d fveq1 fveq2d eleq2d anbi12d rexbidv anbi1d orbi12d cbvmptv
|
|
abbidv eleq1 anbi2d cbvabv mpteq2i suceq eqeq2d fveq2 orbi1i abbii 3eqtri
|
|
cbvrexv frecsuclem ) FABCDGHEUAKUALZUBZILZUCZMZJLZVKVINZENZOZPZIQRZVJUEMZ
|
|
VNBOZPZSZJTZUDZWEGKGLZUBZVLMZVNVKWFNZENZOZPZIQRZWGUEMZWAPZSZJTZUDGKWHFLZW
|
|
JOZPZIQRZWNWRBOZPZSZFTZUDGKWGHLZUCZMZWRXFWFNZENZOZPZHQRZXCSZFTZUDUAGKWDWQ
|
|
VIWFMZWCWPJXPVSWMWBWOXPVRWLIQXPVMWHVQWKXPVJWGVLVIWFUFZUGXPVPWJVNXPVOWIEVK
|
|
VIWFUHUIUJUKULXPVTWNWAXPVJWGUEXQUGUMUNUPUOGKWQXEWPXDJFVNWRMZWMXAWOXCXRWLW
|
|
TIQXRWKWSWHVNWRWJUQURULXRWAXBWNVNWRBUQURUNUSUTGKXEXOXDXNFXAXMXCWTXLIHQVKX
|
|
FMZWHXHWSXKXSVLXGWGVKXFVAVBXSWJXJWRXSWIXIEVKXFWFVCUIUJUKVGVDVEUTVFVH $.
|
|
$}
|
|
|
|
${
|
|
frecrdg.1 $e |- ( ph -> F Fn _V ) $.
|
|
frecrdg.2 $e |- ( ph -> A e. V ) $.
|
|
frecrdg.inc $e |- ( ph -> A. x x C_ ( F ` x ) ) $.
|
|
$d A x y z $. $d F x y z $. $d V x $. $d ph x y $.
|
|
$( Transfinite recursion restricted to omega.
|
|
|
|
Given a suitable characteristic function, ~ df-frec produces the same
|
|
results as ~ df-irdg restricted to ` _om ` .
|
|
|
|
Presumably the theorem would also hold if ` F Fn _V ` were changed to
|
|
` A. z ( F `` z ) e. _V ` . (Contributed by Jim Kingdon,
|
|
29-Aug-2019.) $)
|
|
frecrdg $p |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` _om ) ) $=
|
|
( vz com cfv cvv wcel wfn syl con0 wceq c0 fveq2 adantr vy cfrec crdg wal
|
|
cv vex funfvex funfni mpan2 alrimiv frecfnom syl2anc wss rdgifnon2 omsson
|
|
cres fnssres sylancl csuc eqeq12d frec0g peano1 fvres ax-mp syl5eq eqtr4d
|
|
rdg0g wi simpr ad2antlr eqtrd fveq2d jca w3a wral simp1 ralv sylibr simp2
|
|
wa elexd simp3 frecsuc syl3anc 3expa sylan nnon rdgisucinc 3eqtr4d peano2
|
|
ex expcom finds2 impcom eqfnfvd ) ABJDCUBZDCUCZJUPZAIUEZDKLMZIUDZCEMZWPJN
|
|
ADLNZXAFXCWTIXCWSLMWTIUFWTLWSDWSDUGUHUIUJOZGICDEUKULAWQPNZJPUMWRJNAXAXBXE
|
|
XDGICDEUNULUOPJWQUQURBUEZJMAXFWPKZXFWRKZQZXIRWPKZRWRKZQUAUEZWPKZXLWRKZQZX
|
|
LUSZWPKZXPWRKZQZABUAXFRQXGXJXHXKXFRWPSXFRWRSUTXFXLQXGXMXHXNXFXLWPSXFXLWRS
|
|
UTXFXPQXGXQXHXRXFXPWPSXFXPWRSUTAXJCXKAXBXJCQGCDEVAOAXKRWQKZCRJMXKXTQVBRJW
|
|
QVCVDAXBXTCQGCEDVGOVEVFAXLJMZXOXSVHAYAVTZXOXSYBXOVTZXQXPWQKZXRYCXMDKZXLWQ
|
|
KZDKZXQYDYCXMYFDYCXMXNYFYBXOVIYAXNYFQAXOXLJWQVCVJVKVLYBXQYEQZXOAXAXBVTYAY
|
|
HAXAXBXDGVMXAXBYAYHXAXBYAVNZWTILVOZCLMYAYHYIXAYJXAXBYAVPWTIVQVRYICEXAXBYA
|
|
VSWAXAXBYAWBICXLLDWCWDWEWFTYBYDYGQXOYBBCXLDEAXCYAFTAXBYAGTYBYAXLPMAYAVIXL
|
|
WGOAXFXFDKUMBUDYAHTWHTWIYAXRYDQZAXOYAXPJMYKXLWJXPJWQVCOVJVFWKWLWMWNWO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordinal arithmetic
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c 1o $. $( The ordinal number 1 $)
|
|
$c 2o $. $( The ordinal number 2 $)
|
|
$c 3o $. $( The ordinal number 3 $)
|
|
$c 4o $. $( The ordinal number 4 $)
|
|
$c +o $. $( Ordinal addition operation (plus subscript o) $)
|
|
$c .o $. $( Ordinal multiplication operation (center dot subscript o) $)
|
|
$c ^oi $. $( Ordinal exponentiation operation (caret subscript oi) $)
|
|
|
|
$( Extend the definition of a class to include the ordinal number 1. $)
|
|
c1o $a class 1o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal number 2. $)
|
|
c2o $a class 2o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal number 3. $)
|
|
c3o $a class 3o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal number 4. $)
|
|
c4o $a class 4o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal addition
|
|
operation. $)
|
|
coa $a class +o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal multiplication
|
|
operation. $)
|
|
comu $a class .o $.
|
|
|
|
$( Extend the definition of a class to include the ordinal exponentiation
|
|
operation. $)
|
|
coei $a class ^oi $.
|
|
|
|
$( Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) $)
|
|
df-1o $a |- 1o = suc (/) $.
|
|
|
|
$( Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) $)
|
|
df-2o $a |- 2o = suc 1o $.
|
|
|
|
$( Define the ordinal number 3. (Contributed by Mario Carneiro,
|
|
14-Jul-2013.) $)
|
|
df-3o $a |- 3o = suc 2o $.
|
|
|
|
$( Define the ordinal number 4. (Contributed by Mario Carneiro,
|
|
14-Jul-2013.) $)
|
|
df-4o $a |- 4o = suc 3o $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define the ordinal addition operation. (Contributed by NM,
|
|
3-May-1995.) $)
|
|
df-oadd $a |- +o = ( x e. On , y e. On |->
|
|
( rec ( ( z e. _V |-> suc z ) , x ) ` y ) ) $.
|
|
|
|
$( Define the ordinal multiplication operation. (Contributed by NM,
|
|
26-Aug-1995.) $)
|
|
df-omul $a |- .o = ( x e. On , y e. On |->
|
|
( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) ) $.
|
|
|
|
$( Define the ordinal exponentiation operation.
|
|
|
|
This definition is similar to a conventional definition of
|
|
exponentiation except that it defines ` (/) ^oi A ` to be ` 1o ` for all
|
|
` A e. On ` , in order to avoid having different cases for whether the
|
|
base is ` (/) ` or not. (Contributed by Mario Carneiro, 4-Jul-2019.) $)
|
|
df-oexpi $a |- ^oi = ( x e. On , y e. On |->
|
|
( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) $.
|
|
$}
|
|
|
|
$( Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) $)
|
|
1on $p |- 1o e. On $=
|
|
( c1o c0 csuc con0 df-1o 0elon onsuci eqeltri ) ABCDEBFGH $.
|
|
|
|
$( Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof
|
|
shortened by Andrew Salmon, 12-Aug-2011.) $)
|
|
2on $p |- 2o e. On $=
|
|
( c2o c1o csuc con0 df-2o 1on onsuci eqeltri ) ABCDEBFGH $.
|
|
|
|
$( Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) $)
|
|
2on0 $p |- 2o =/= (/) $=
|
|
( c2o c1o csuc c0 df-2o con0 wcel wne 1on nsuceq0g ax-mp eqnetri ) ABCZDEBF
|
|
GMDHIBFJKL $.
|
|
|
|
$( Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
3on $p |- 3o e. On $=
|
|
( c3o c2o csuc con0 df-3o 2on onsuci eqeltri ) ABCDEBFGH $.
|
|
|
|
$( Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
4on $p |- 4o e. On $=
|
|
( c4o c3o csuc con0 df-4o 3on onsuci eqeltri ) ABCDEBFGH $.
|
|
|
|
$( Expanded value of the ordinal number 1. (Contributed by NM,
|
|
4-Nov-2002.) $)
|
|
df1o2 $p |- 1o = { (/) } $=
|
|
( c1o c0 csuc csn df-1o suc0 eqtri ) ABCBDEFG $.
|
|
|
|
$( Expanded value of the ordinal number 2. (Contributed by Mario Carneiro,
|
|
14-Aug-2015.) $)
|
|
df2o3 $p |- 2o = { (/) , 1o } $=
|
|
( c2o c1o csuc csn cun c0 cpr df-2o df-suc df1o2 uneq1i df-pr eqtr4i 3eqtri
|
|
) ABCBBDZEZFBGZHBIPFDZOEQBROJKFBLMN $.
|
|
|
|
$( Expanded value of the ordinal number 2. (Contributed by NM,
|
|
29-Jan-2004.) $)
|
|
df2o2 $p |- 2o = { (/) , { (/) } } $=
|
|
( c2o c0 c1o cpr csn df2o3 df1o2 preq2i eqtri ) ABCDBBEZDFCJBGHI $.
|
|
|
|
$( Ordinal one is not equal to ordinal zero. (Contributed by NM,
|
|
26-Dec-2004.) $)
|
|
1n0 $p |- 1o =/= (/) $=
|
|
( c1o c0 csn df1o2 0ex snnz eqnetri ) ABCBDBEFG $.
|
|
|
|
$( Cartesian products with the singletons of ordinals 0 and 1 are disjoint.
|
|
(Contributed by NM, 2-Jun-2007.) $)
|
|
xp01disj $p |- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/) $=
|
|
( c0 c1o wne csn cxp cin wceq 1n0 necomi xpsndisj ax-mp ) CDEACFGBDFGHCIDCJ
|
|
KACBDLM $.
|
|
|
|
$( Two ways to express that an ordinal class is positive. (Contributed by
|
|
NM, 21-Dec-2004.) $)
|
|
ordgt0ge1 $p |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) ) $=
|
|
( word c0 wcel csuc wss c1o con0 0elon ordelsuc mpan df-1o sseq1i syl6bbr
|
|
wb ) ABZCADZCEZAFZGAFCHDPQSOICAHJKGRALMN $.
|
|
|
|
$( An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim
|
|
Kingdon, 26-Jun-2019.) $)
|
|
ordge1n0im $p |- ( Ord A -> ( 1o C_ A -> A =/= (/) ) ) $=
|
|
( word c1o wss c0 wcel wne ordgt0ge1 ne0i syl6bir ) ABCADEAFAEGAHAEIJ $.
|
|
|
|
$( Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) $)
|
|
el1o $p |- ( A e. 1o <-> A = (/) ) $=
|
|
( c1o wcel c0 csn wceq df1o2 eleq2i 0ex elsn2 bitri ) ABCADEZCADFBLAGHADIJK
|
|
$.
|
|
|
|
$( Two ways to say that ` A ` is a nonzero number of the set ` B ` .
|
|
(Contributed by Mario Carneiro, 21-May-2015.) $)
|
|
dif1o $p |- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) ) $=
|
|
( c1o cdif wcel c0 csn wne wa df1o2 difeq2i eleq2i eldifsn bitri ) ABCDZEAB
|
|
FGZDZEABEAFHIOQACPBJKLABFMN $.
|
|
|
|
$( Closure of the pair swapping function on ` 2o ` . (Contributed by Mario
|
|
Carneiro, 27-Sep-2015.) $)
|
|
2oconcl $p |- ( A e. 2o -> ( 1o \ A ) e. 2o ) $=
|
|
( c1o cdif c2o wcel c0 cpr wceq wo elpri difeq2 syl6eq difid orim12i orcomd
|
|
dif0 syl con0 cvv df2o3 1on difexg ax-mp elpr sylibr syl6eleqr eleq2s ) BAC
|
|
ZDEAFBGZDAUIEZUHUIDUJUHFHZUHBHZIZUHUIEUJAFHZABHZIZUMAFBJUPULUKUNULUOUKUNUHB
|
|
FCBAFBKBPLUOUHBBCFABBKBMLNOQUHFBBREUHSEUABARUBUCUDUETUFTUG $.
|
|
|
|
$( Ordinal zero is less than ordinal one. (Contributed by NM,
|
|
5-Jan-2005.) $)
|
|
0lt1o $p |- (/) e. 1o $=
|
|
( c0 c1o wcel wceq eqid el1o mpbir ) ABCAADAEAFG $.
|
|
|
|
$( The characteristic function for ordinal addition is defined everywhere.
|
|
(Contributed by Jim Kingdon, 27-Jul-2019.) $)
|
|
oafnex $p |- ( x e. _V |-> suc x ) Fn _V $=
|
|
( cvv cv csuc cmpt vex sucex eqid fnmpti ) ABACZDZABKEZJAFGLHI $.
|
|
|
|
${
|
|
$d x z $.
|
|
sucinc.1 $e |- F = ( z e. _V |-> suc z ) $.
|
|
$( Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) $)
|
|
sucinc $p |- A. x x C_ ( F ` x ) $=
|
|
( cv cfv wss csuc sssucid cvv wcel wceq sucex suceq fvmptg mp2an sseqtr4i
|
|
vex ax-gen ) AEZTCFZGATTHZUATITJKUBJKUAUBLARZTUCMBTBEZHUBJJCUDTNDOPQS $.
|
|
|
|
$d A z $. $d B z $.
|
|
$( Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) $)
|
|
sucinc2 $p |- ( ( B e. On /\ A e. B ) ->
|
|
( F ` A ) C_ ( F ` B ) ) $=
|
|
( con0 wcel wa csuc cfv wss word syl wceq cvv sucexg suceq fvmptg syl2anc
|
|
elex wi eloni ordsucss imp sssucid syl6ss onelon cv adantr 3sstr4d ) CFGZ
|
|
BCGZHZBIZCIZBDJZCDJZUMUNCUOUKULUNCKZUKCLULURUACUBBCUCMUDCUEUFUMBFGZUPUNNZ
|
|
CBUGUSBOGUNOGUTBFTBFPABAUHZIZUNOODVABQERSMUKUQUONZULUKCOGUOOGVCCFTCFPACVB
|
|
UOOODVACQERSUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Functionality and domain of ordinal addition. (Contributed by NM,
|
|
26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.) $)
|
|
fnoa $p |- +o Fn ( On X. On ) $=
|
|
( vx vy vz con0 cvv csuc cmpt crdg cfv coa df-oadd wcel vex oafnex rdgexg
|
|
cv ax-mp fnmpt2i ) ABDDBPZCECPFGZAPZHIZJABCKSELUBELBMUASTEAMCNOQR $.
|
|
|
|
$( Ordinal addition is a set. (Contributed by Mario Carneiro,
|
|
3-Jul-2019.) $)
|
|
oaexg $p |- ( ( A e. V /\ B e. W ) -> ( A +o B ) e. _V ) $=
|
|
( vy vz vx cv cvv csuc cmpt crdg cfv wcel wal coa co vex oafnex con0 gen2
|
|
rdgexg ax-mp df-oadd mpt2fvex mp3an1 ) EHZFIFHJKZGHZLMZINZEOGOACNBDNABPQI
|
|
NUKGEUGINUKERUIUGUHIGRFSUBUCUAGETTUJABPICDGEFUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x V $.
|
|
$( The characteristic function for ordinal multiplication is defined
|
|
everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.) $)
|
|
omfnex $p |- ( A e. V -> ( x e. _V |-> ( x +o A ) ) Fn _V ) $=
|
|
( wcel cv coa cvv wral cmpt wfn vex oaexg mpan ralrimivw eqid fnmpt syl
|
|
co ) BCDZAEZBFRZGDZAGHAGUAIZGJSUBAGTGDSUBAKTBGCLMNAGUAUCGUCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Functionality and domain of ordinal multiplication. (Contributed by NM,
|
|
26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.) $)
|
|
fnom $p |- .o Fn ( On X. On ) $=
|
|
( vx vy vz con0 cv cvv coa co cmpt crdg cfv comu df-omul wcel vex 0ex wfn
|
|
c0 omfnex ax-mp rdgexg fnmpt2i ) ABDDBEZCFCEAEZGHIZRJKZLABCMUCFNUFFNBORUC
|
|
UEFPUDFNUEFQAOCUDFSTUATUB $.
|
|
|
|
$( Ordinal multiplication is a set. (Contributed by Mario Carneiro,
|
|
3-Jul-2019.) $)
|
|
omexg $p |- ( ( A e. V /\ B e. W ) -> ( A .o B ) e. _V ) $=
|
|
( vy vz vx cv cvv coa co cmpt c0 crdg wcel wal comu vex ax-mp con0 omfnex
|
|
cfv 0ex wfn rdgexg gen2 df-omul mpt2fvex mp3an1 ) EHZFIFHGHZJKLZMNUBZIOZE
|
|
PGPACOBDOABQKIOUNGEUJIOUNERMUJULIUCUKIOULIUDGRFUKIUASUESUFGETTUMABQICDGEF
|
|
UGUHUI $.
|
|
|
|
$( Functionality and domain of ordinal exponentiation. (Contributed by
|
|
Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro,
|
|
3-Jul-2019.) $)
|
|
fnoei $p |- ^oi Fn ( On X. On ) $=
|
|
( vx vy vz con0 cv cvv comu co cmpt c1o crdg cfv coei df-oexpi wcel elexi
|
|
vex 1on omexg mp2an eqid fnmpti rdgexg ax-mp fnmpt2i ) ABDDBEZCFCEZAEZGHZ
|
|
IZJKLZMABCNUFFOUKFOBQJUFUJFJDRPCFUIUJUGFOUHFOUIFOCQAQUGUHFFSTUJUAUBUCUDUE
|
|
$.
|
|
|
|
$( Ordinal exponentiation is a set. (Contributed by Mario Carneiro,
|
|
3-Jul-2019.) $)
|
|
oeiexg $p |- ( ( A e. V /\ B e. W ) -> ( A ^oi B ) e. _V ) $=
|
|
( vy vz vx cv cvv comu co cmpt c1o crdg cfv wcel wal coei vex con0 fnmpti
|
|
1on elexi omexg mp2an eqid rdgexg ax-mp gen2 df-oexpi mpt2fvex mp3an1 ) E
|
|
HZFIFHZGHZJKZLZMNOZIPZEQGQACPBDPABRKIPUSGEUMIPUSESMUMUQIMTUBUCFIUPUQUNIPU
|
|
OIPUPIPFSGSUNUOIIUDUEUQUFUAUGUHUIGETTURABRICDGEFUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $.
|
|
$( Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised
|
|
by Mario Carneiro, 8-Sep-2013.) $)
|
|
oav $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A +o B ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` B ) ) $=
|
|
( vy vz con0 wcel cvv cv csuc cmpt crdg cfv co wceq oafnex rdgexgg rdgeq2
|
|
coa fveq1d fveq2 df-oadd ovmpt2g mpd3an3 ) BFGCFGCAHAIJKZBLZMZHGBCSNUGOBC
|
|
UEFFAPQDEBCFFEIZUEDIZLZMUGSUHUFMHUIBOUHUJUFUIBUERTUHCUFUADEAUBUCUD $.
|
|
|
|
$( Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.)
|
|
(Revised by Mario Carneiro, 23-Aug-2014.) $)
|
|
omv $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) $=
|
|
( vy vz con0 wcel cvv cv coa co cmpt crdg cfv comu wceq 0elon wfn omfnex
|
|
c0 rdgexggg syl3an1 mp3an2 oveq2 mpteq2dv rdgeq1 syl fveq1d fveq2 df-omul
|
|
ovmpt2g mpd3an3 ) BFGZCFGZCAHAIZBJKZLZTMZNZHGZBCOKUSPUMTFGZUNUTQUMUQHRVAU
|
|
NUTABFSTCUQFFUAUBUCDEBCFFEIZAHUODIZJKZLZTMZNUSOVBURNHVCBPZVBVFURVGVEUQPVF
|
|
URPVGAHVDUPVCBUOJUDUETVEUQUFUGUHVBCURUIDEAUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $.
|
|
$( Value of ordinal exponentiation. (Contributed by Jim Kingdon,
|
|
9-Jul-2019.) $)
|
|
oeiv $p |- ( ( A e. On /\ B e. On ) -> ( A ^oi B ) =
|
|
( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) $=
|
|
( vy vz con0 wcel cvv cv comu co cmpt c1o crdg cfv coei wceq 1on wfn syl
|
|
wral vex omexg mpan ralrimivw eqid fnmpt rdgexggg syl3an1 mp3an2 mpteq2dv
|
|
oveq2 rdgeq1 fveq1d fveq2 df-oexpi ovmpt2g mpd3an3 ) BFGZCFGZCAHAIZBJKZLZ
|
|
MNZOZHGZBCPKVEQUSMFGZUTVFRUSVCHSZVGUTVFUSVBHGZAHUAVHUSVIAHVAHGUSVIAUBVABH
|
|
FUCUDUEAHVBVCHVCUFUGTMCVCFFUHUIUJDEBCFFEIZAHVADIZJKZLZMNZOVEPVJVDOHVKBQZV
|
|
JVNVDVOVMVCQVNVDQVOAHVLVBVKBVAJULUKMVMVCUMTUNVJCVDUODEAUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.
|
|
(Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
oa0 $p |- ( A e. On -> ( A +o (/) ) = A ) $=
|
|
( vx con0 wcel c0 coa co cvv cv csuc cmpt crdg cfv wceq 0elon mpan2 rdg0g
|
|
oav eqtrd ) ACDZAEFGZEBHBIJKZALMZATECDUAUCNOBAERPACUBQS $.
|
|
|
|
$( Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring]
|
|
p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
om0 $p |- ( A e. On -> ( A .o (/) ) = (/) ) $=
|
|
( vx con0 wcel c0 comu co cvv coa cmpt crdg cfv wceq 0elon omv mpan2 rdg0
|
|
cv 0ex syl6eq ) ACDZAEFGZEBHBRAIGJZEKLZEUAECDUBUDMNBAEOPEUCSQT $.
|
|
|
|
$( Ordinal exponentiation with zero exponent. Definition 8.30 of
|
|
[TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by
|
|
Mario Carneiro, 8-Sep-2013.) $)
|
|
oei0 $p |- ( A e. On -> ( A ^oi (/) ) = 1o ) $=
|
|
( vx con0 wcel c0 coei co cvv cv comu cmpt c1o crdg wceq 0elon oeiv mpan2
|
|
cfv 1on rdg0g ax-mp syl6eq ) ACDZAEFGZEBHBIAJGKZLMRZLUCECDUDUFNOBAEPQLCDU
|
|
FLNSLCUETUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y w B $.
|
|
$( Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring]
|
|
p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim
|
|
Kingdon, 26-Jul-2019.) $)
|
|
oacl $p |- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On ) $=
|
|
( vz vw con0 wcel wa coa co cvv cv csuc cmpt crdg cfv oav wral wceq vex
|
|
id suceq eqid sucex fvmpt ax-mp eleq1i suceloni mprgbir a1i rdgon eqeltrd
|
|
ralbii ) AEFZBEFGABHIBCJCKZLZMZANOECABPUMDABUPUMTDKZUPOZEFZDEQZUMUTUQLZEF
|
|
ZDEUSVBDEURVAEUQJFURVARDSZCUQUOVAJUPUNUQUAUPUBUQVCUCUDUEUFULUQUGUHUIUJUK
|
|
$.
|
|
|
|
$( Closure law for ordinal multiplication. Proposition 8.16 of
|
|
[TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive
|
|
proof by Jim Kingdon, 26-Jul-2019.) $)
|
|
omcl $p |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On ) $=
|
|
( vx vy con0 wcel wa comu co cvv cv coa cmpt c0 cfv omv 0elon a1i eqeltrd
|
|
crdg wceq vex oacl oveq1 eqid fvmptg sylancr ancoms ralrimiva rdgon ) AEF
|
|
ZBEFGABHIBCJCKZALIZMZNTOECABPUKDNBUNNEFUKQRUKDKZUNOZEFZDEUOEFZUKUQURUKGZU
|
|
PUOALIZEUSUOJFUTEFUPUTUADUBUOAUCZCUOUMUTJEUNULUOALUDUNUEUFUGVASUHUIUJS $.
|
|
|
|
$( Closure law for ordinal exponentiation. (Contributed by Jim Kingdon,
|
|
26-Jul-2019.) $)
|
|
oeicl $p |- ( ( A e. On /\ B e. On ) -> ( A ^oi B ) e. On ) $=
|
|
( vx vy con0 wcel wa coei cvv comu cmpt c1o crdg cfv oeiv 1on a1i eqeltrd
|
|
co cv wceq vex omcl oveq1 eqid fvmptg sylancr ancoms ralrimiva rdgon ) AE
|
|
FZBEFGABHSBCICTZAJSZKZLMNECABOUKDLBUNLEFUKPQUKDTZUNNZEFZDEUOEFZUKUQURUKGZ
|
|
UPUOAJSZEUSUOIFUTEFUPUTUADUBUOAUCZCUOUMUTIEUNULUOAJUDUNUEUFUGVARUHUIUJR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Value of ordinal addition. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 12-Aug-2019.) $)
|
|
oav2 $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A +o B ) = ( A u. U_ x e. B suc ( A +o x ) ) ) $=
|
|
( vy con0 wcel wa cvv cv csuc cmpt crdg cfv ciun cun coa co wfn wceq oav
|
|
oafnex rdgival mp3an1 onelon vex oaexg mpan2 sucexg syl suceq eqid fvmptg
|
|
syl2anc adantr fveq2d eqtr3d sylan2 anassrs iuneq2dv uneq2d 3eqtr4d ) BEF
|
|
ZCEFZGZCDHDIZJZKZBLZMZBACAIZVHMZVGMZNZOZBCPQBACBVJPQZJZNZOVGHRVBVCVIVNSDU
|
|
AABCVGEUBUCDBCTVDVQVMBVDACVPVLVBVCVJCFZVPVLSZVCVRGVBVJEFZVSCVJUDVBVTGZVOV
|
|
GMZVPVLVBWBVPSZVTVBVOHFZVPHFZWCVBVJHFWDAUEBVJEHUFUGZVBWDWEWFVOHUHUIDVOVFV
|
|
PHHVGVEVOUJVGUKULUMUNWAVOVKVGDBVJTUOUPUQURUSUTVA $.
|
|
|
|
$( Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56.
|
|
(Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
oasuc $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A +o suc B ) = suc ( A +o B ) ) $=
|
|
( vx con0 wcel wa csuc coa co cun cv ciun wceq suceloni sylan2 csn df-suc
|
|
oav2 iuneq1 uneq2d ax-mp iunxun eqtri oveq2 suceq syl syl5eq adantl eqtrd
|
|
iunxsng unass syl6eqr uneq1d eqtr4d wss sssucid ssequn1 mpbi syl6eq ) ADE
|
|
ZBDEZFZABGZHIZABHIZVEGZJZVFVBVDACBACKZHIZGZLZJZVFJZVGVBVDAVKVFJZJZVMVBVDA
|
|
CVCVJLZJZVOVAUTVCDEVDVQMBNCAVCROVAVQVOMUTVAVPVNAVAVPVKCBPZVJLZJZVNVPCBVRJ
|
|
ZVJLZVTVCWAMVPWBMBQCVCWAVJSUACBVRVJUBUCVAVSVFVKCBVJVFDVHBMVIVEMVJVFMVHBAH
|
|
UDVIVEUEUFUJTUGTUHUIAVKVFUKULVBVEVLVFCABRUMUNVEVFUOVGVFMVEUPVEVFUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Value of ordinal multiplication. (Contributed by Jim Kingdon,
|
|
23-Aug-2019.) $)
|
|
omv2 $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) ) $=
|
|
( vy con0 wcel wa comu co c0 coa ciun cun cvv cmpt cfv wceq omv syl2anc
|
|
cv crdg wfn omfnex 0elon rdgival mp3an2 sylan omexg omcl simpl oacl oveq1
|
|
onelon eqid fvmptg fveq2d eqtr3d sylan2 anassrs iuneq2dv uneq2d uncom un0
|
|
3eqtr4d eqtri syl6eq ) BEFZCEFZGZBCHIZJACBATZHIZBKIZLZMZVNVICDNDTZBKIZOZJ
|
|
UAZPZJACVKVSPZVRPZLZMZVJVOVGVRNUBZVHVTWDQZDBEUCWEJEFVHWFUDAJCVREUEUFUGDBC
|
|
RVIVNWCJVIACVMWBVGVHVKCFZVMWBQZVHWGGVGVKEFZWHCVKUMVGWIGZVLVRPZVMWBWJVLNFV
|
|
MEFZWKVMQBVKEEUHWJVLEFVGWLBVKUIVGWIUJVLBUKSDVLVQVMNEVRVPVLBKULVRUNUOSWJVL
|
|
WAVRDBVKRUPUQURUSUTVAVDVOVNJMVNJVNVBVNVCVEVF $.
|
|
$}
|
|
|
|
$( Addition with successor. Theorem 4I(A2) of [Enderton] p. 79.
|
|
(Contributed by Mario Carneiro, 16-Nov-2014.) $)
|
|
onasuc $p |- ( ( A e. On /\ B e. _om ) ->
|
|
( A +o suc B ) = suc ( A +o B ) ) $=
|
|
( com wcel con0 csuc coa co wceq nnon oasuc sylan2 ) BCDAEDBEDABFGHABGHFIBJ
|
|
ABKL $.
|
|
|
|
$( Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson]
|
|
p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro,
|
|
16-Nov-2014.) $)
|
|
oa1suc $p |- ( A e. On -> ( A +o 1o ) = suc A ) $=
|
|
( con0 wcel c1o coa co c0 csuc df-1o oveq2i wceq peano1 onasuc mpan2 syl5eq
|
|
com oa0 suceq syl eqtrd ) ABCZADEFZAGEFZHZAHZUAUBAGHZEFZUDDUFAEIJUAGPCUGUDK
|
|
LAGMNOUAUCAKUDUEKAQUCARST $.
|
|
|
|
$( 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) $)
|
|
o1p1e2 $p |- ( 1o +o 1o ) = 2o $=
|
|
( c1o coa co csuc c2o con0 wcel wceq 1on oa1suc ax-mp df-2o eqtr4i ) AABCZA
|
|
DZEAFGNOHIAJKLM $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Weak ordering property of ordinal addition. (Contributed by Jim
|
|
Kingdon, 27-Jul-2019.) $)
|
|
oawordi $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
|
|
( A C_ B -> ( C +o A ) C_ ( C +o B ) ) ) $=
|
|
( vx con0 wcel w3a wss coa co wa cvv cv csuc cmpt cfv wceq jca oav syl ex
|
|
crdg wfn oafnex a1i simpl3 simpl1 simpl2 simpr rdgss 3sstr4d ) AEFZBEFZCE
|
|
FZGZABHZCAIJZCBIJZHUOUPKZADLDMNOZCUBZPZBVAPZUQURUSABUTCEUTLUCUSDUDUEULUMU
|
|
NUPUFZULUMUNUPUGZULUMUNUPUHZUOUPUIUJUSUNULKUQVBQUSUNULVDVERDCASTUSUNUMKUR
|
|
VCQUSUNUMVDVFRDCBSTUKUA $.
|
|
$}
|
|
|
|
$( An ordinal is less than or equal to its sum with another. Part of
|
|
Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.) $)
|
|
oaword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) ) $=
|
|
( con0 wcel wa c0 coa co wceq oa0 adantr wss wi 0elon oawordi 3com13 mp3an3
|
|
0ss mpi eqsstr3d ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBRUAUBFCDZUF
|
|
UGMZNUHUBUAUIFBAOPQST $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Multiplication with successor. Definition 8.15 of [TakeutiZaring]
|
|
p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
omsuc $p |- ( ( A e. On /\ B e. On ) ->
|
|
( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
|
|
( vx con0 wcel wa csuc comu co coa cun ciun wceq csn df-suc iuneq1 iunxun
|
|
cv ax-mp omv2 oveq1d iunxsng uneq2d syl5eq adantl suceloni sylan2 3eqtr4d
|
|
eqtri oveq2 uneq1d omcl simpl wss oaword1 ssequn1 sylib syl2anc eqtrd ) A
|
|
DEZBDEZFZABGZHIZABHIZVEAJIZKZVFVBCVCACRZHIZAJIZLZCBVJLZVFKZVDVGVAVKVMMUTV
|
|
AVKVLCBNZVJLZKZVMVKCBVNKZVJLZVPVCVQMVKVRMBOCVCVQVJPSCBVNVJQUIVAVOVFVLCBVJ
|
|
VFDVHBMVIVEAJVHBAHUJUAUBUCUDUEVAUTVCDEVDVKMBUFCAVCTUGVBVEVLVFCABTUKUHVBVE
|
|
DEZUTVGVFMZABULUTVAUMVSUTFVEVFUNVTVEAUOVEVFUPUQURUS $.
|
|
$}
|
|
|
|
$( Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80.
|
|
(Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
14-Nov-2014.) $)
|
|
onmsuc $p |- ( ( A e. On /\ B e. _om ) ->
|
|
( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
|
|
( com wcel con0 csuc comu co coa wceq nnon omsuc sylan2 ) BCDAEDBEDABFGHABG
|
|
HAIHJBKABLM $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Natural number arithmetic
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by
|
|
NM, 20-Sep-1995.) $)
|
|
nna0 $p |- ( A e. _om -> ( A +o (/) ) = A ) $=
|
|
( com wcel con0 c0 coa co wceq nnon oa0 syl ) ABCADCAEFGAHAIAJK $.
|
|
|
|
$( Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80.
|
|
(Contributed by NM, 20-Sep-1995.) $)
|
|
nnm0 $p |- ( A e. _om -> ( A .o (/) ) = (/) ) $=
|
|
( com wcel con0 c0 comu co wceq nnon om0 syl ) ABCADCAEFGEHAIAJK $.
|
|
|
|
$( Addition with successor. Theorem 4I(A2) of [Enderton] p. 79.
|
|
(Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
14-Nov-2014.) $)
|
|
nnasuc $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A +o suc B ) = suc ( A +o B ) ) $=
|
|
( com wcel con0 csuc coa co wceq nnon onasuc sylan ) ACDAEDBCDABFGHABGHFIAJ
|
|
ABKL $.
|
|
|
|
$( Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80.
|
|
(Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
14-Nov-2014.) $)
|
|
nnmsuc $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A .o suc B ) = ( ( A .o B ) +o A ) ) $=
|
|
( com wcel con0 csuc comu co coa wceq nnon onmsuc sylan ) ACDAEDBCDABFGHABG
|
|
HAIHJAKABLM $.
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
$( Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81.
|
|
(Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
14-Nov-2014.) $)
|
|
nna0r $p |- ( A e. _om -> ( (/) +o A ) = A ) $=
|
|
( vx vy c0 cv coa co wceq csuc oveq2 id eqeq12d con0 wcel 0elon oa0 ax-mp
|
|
com peano1 nnasuc mpan suceq eqeq2d syl5ibcom finds ) DBEZFGZUFHDDFGZDHZD
|
|
CEZFGZUJHZDUJIZFGZUMHZDAFGZAHBCAUFDHZUGUHUFDUFDDFJUQKLUFUJHZUGUKUFUJUFUJD
|
|
FJURKLUFUMHZUGUNUFUMUFUMDFJUSKLUFAHZUGUPUFAUFADFJUTKLDMNUIODPQUJRNZUNUKIZ
|
|
HZULUODRNVAVCSDUJTUAULVBUMUNUKUJUBUCUDUE $.
|
|
|
|
$( Multiplication with zero. Exercise 16 of [Enderton] p. 82.
|
|
(Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
nnm0r $p |- ( A e. _om -> ( (/) .o A ) = (/) ) $=
|
|
( vx vy c0 cv comu co wceq csuc oveq2 eqeq1d con0 0elon om0 ax-mp com coa
|
|
wcel oveq1 oa0 syl6eq peano1 nnmsuc mpan syl5ibr finds ) DBEZFGZDHDDFGZDH
|
|
ZDCEZFGZDHZDUKIZFGZDHZDAFGZDHBCAUGDHUHUIDUGDDFJKUGUKHUHULDUGUKDFJKUGUNHUH
|
|
UODUGUNDFJKUGAHUHUQDUGADFJKDLRZUJMDNOUMUPUKPRZULDQGZDHUMUTDDQGZDULDDQSURV
|
|
ADHMDTOUAUSUOUTDDPRUSUOUTHUBDUKUCUDKUEUF $.
|
|
|
|
$( Closure of addition of natural numbers. Proposition 8.9 of
|
|
[TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof
|
|
shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nnacl $p |- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) e. _om ) $=
|
|
( vx vy com wcel coa co cv wceq oveq2 eleq1d imbi2d csuc nna0 ibir peano2
|
|
wi c0 wa nnasuc syl5ibr expcom finds2 vtoclga impcom ) BEFAEFZABGHZEFZUGA
|
|
CIZGHZEFZRUGUIRCBEUJBJZULUIUGUMUKUHEUJBAGKLMULASGHZEFZADIZGHZEFZAUPNZGHZE
|
|
FZUGCDUJSJUKUNEUJSAGKLUJUPJUKUQEUJUPAGKLUJUSJUKUTEUJUSAGKLUGUOUGUNAEAOLPU
|
|
GUPEFZURVARURVAUGVBTZUQNZEFUQQVCUTVDEAUPUALUBUCUDUEUF $.
|
|
|
|
$( Closure of multiplication of natural numbers. Proposition 8.17 of
|
|
[TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof
|
|
shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nnmcl $p |- ( ( A e. _om /\ B e. _om ) -> ( A .o B ) e. _om ) $=
|
|
( vx vy wcel comu co cv wi wceq oveq2 eleq1d imbi2d c0 csuc nnm0 syl6eqel
|
|
com peano1 expcom coa nnacl adantr nnmsuc sylibrd finds2 vtoclga impcom
|
|
wa ) BREAREZABFGZREZUJACHZFGZREZIUJULICBRUMBJZUOULUJUPUNUKRUMBAFKLMUOANFG
|
|
ZREADHZFGZREZAUROZFGZREZUJCDUMNJUNUQRUMNAFKLUMURJUNUSRUMURAFKLUMVAJUNVBRU
|
|
MVAAFKLUJUQNRAPSQUJURREZUTVCIUJVDUIZUTUSAUAGZREZVCUJUTVGIVDUTUJVGUSAUBTUC
|
|
VEVBVFRAURUDLUETUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
nncli.1 $e |- A e. _om $.
|
|
nncli.2 $e |- B e. _om $.
|
|
$( ` _om ` is closed under addition. Inference form of ~ nnacl .
|
|
(Contributed by Scott Fenton, 20-Apr-2012.) $)
|
|
nnacli $p |- ( A +o B ) e. _om $=
|
|
( com wcel coa co nnacl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( ` _om ` is closed under multiplication. Inference form of ~ nnmcl .
|
|
(Contributed by Scott Fenton, 20-Apr-2012.) $)
|
|
nnmcli $p |- ( A .o B ) e. _om $=
|
|
( com wcel comu co nnmcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y z B $.
|
|
$( Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton]
|
|
p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
nnacom $p |- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) = ( B +o A ) ) $=
|
|
( vx vy com wcel coa co wceq cv wi oveq1 oveq2 eqeq12d c0 csuc nna0 suceq
|
|
syl nnasuc vz imbi2d nna0r eqtr4d wa peano2 syl5ibr expcom finds2 vtoclga
|
|
sylan imp ) AEFBEFZABGHZBAGHZIZUMCJZBGHZBUQGHZIZKUMUPKCAEUQAIZUTUPUMVAURU
|
|
NUSUOUQABGLUQABGMNUBUTOBGHZBOGHZIDJZBGHZBVDGHZIZVDPZBGHZBVHGHZIZUMCDUQOIZ
|
|
URVBUSVCUQOBGLUQOBGMNUQVDIURVEUSVFUQVDBGLUQVDBGMNUQVHIURVIUSVJUQVHBGLUQVH
|
|
BGMNUMVBBVCBUCBQUDUMVDEFZVGVKKVGVKUMVMUEZVEPZVFPZIVEVFRVNVIVOVJVPUMVMVIVO
|
|
IZVMVHUQGHZVDUQGHZPZIZKVMVQKCBEUQBIZWAVQVMWBVRVIVTVOUQBVHGMWBVSVEIVTVOIUQ
|
|
BVDGMVSVERSNUBWAVHOGHZVDOGHZPZIVHUAJZGHZVDWFGHZPZIZVHWFPZGHZVDWKGHZPZIZVM
|
|
CUAVLVRWCVTWEUQOVHGMVLVSWDIVTWEIUQOVDGMVSWDRSNUQWFIZVRWGVTWIUQWFVHGMWPVSW
|
|
HIVTWIIUQWFVDGMVSWHRSNUQWKIZVRWLVTWNUQWKVHGMWQVSWMIVTWNIUQWKVDGMVSWMRSNVM
|
|
WCVHWEVMVHEFZWCVHIVDUFZVHQSVMWDVDIWEVHIVDQWDVDRSUDVMWFEFZWJWOKWJWOVMWTUEZ
|
|
WGPZWIPZIWGWIRXAWLXBWNXCVMWRWTWLXBIWSVHWFTUKXAWMWIIWNXCIVDWFTWMWIRSNUGUHU
|
|
IUJULBVDTNUGUHUIUJUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x C $.
|
|
$( Addition of natural numbers is associative. Theorem 4K(1) of [Enderton]
|
|
p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
nnaass $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) $=
|
|
( vx vy com wcel coa co wceq wa cv wi oveq2 oveq2d eqeq12d c0 csuc nnasuc
|
|
nnacl imbi2d nna0 adantl eqtr4d suceq sylan sylan2 anassrs syl5ibr expcom
|
|
syl eqtrd finds2 vtoclga com12 3impia ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJZU
|
|
SUQURKZVDVEUTDLZHIZABVFHIZHIZJZMVEVDMDCFVFCJZVJVDVEVKVGVAVIVCVFCUTHNVKVHV
|
|
BAHVFCBHNOPUAVJUTQHIZABQHIZHIZJUTELZHIZABVOHIZHIZJZUTVORZHIZABVTHIZHIZJZV
|
|
EDEVFQJZVGVLVIVNVFQUTHNWEVHVMAHVFQBHNOPVFVOJZVGVPVIVRVFVOUTHNWFVHVQAHVFVO
|
|
BHNOPVFVTJZVGWAVIWCVFVTUTHNWGVHWBAHVFVTBHNOPVEVLUTVNVEUTFGZVLUTJABTZUTUBU
|
|
KURVNUTJUQURVMBAHBUBOUCUDVEVOFGZVSWDMVSWDVEWJKZVPRZVRRZJVPVRUEWKWAWLWCWMV
|
|
EWHWJWAWLJWIUTVOSUFUQURWJWCWMJUQURWJKZKWCAVQRZHIZWMWNWCWPJUQWNWBWOAHBVOSO
|
|
UCWNUQVQFGWPWMJBVOTAVQSUGULUHPUIUJUMUNUOUP $.
|
|
|
|
$( Distributive law for natural numbers (left-distributivity). Theorem
|
|
4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised
|
|
by Mario Carneiro, 15-Nov-2014.) $)
|
|
nndi $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( A .o ( B +o C ) ) = ( ( A .o B ) +o ( A .o C ) ) ) $=
|
|
( vx vy com wcel coa co comu wceq wa cv wi oveq2 oveq2d eqeq12d c0 eqtr4d
|
|
csuc imbi2d nna0 adantl nnmcl syl nnm0 adantr oveq1 nnasuc 3adant1 nnmsuc
|
|
w3a nnacl sylan2 3impb eqtrd 3adant2 nnaass syl3an1 syl3an2 exp4b pm2.43a
|
|
3exp com4r pm2.43i 3imp syl5ibr com3r impd finds2 vtoclga expdcom ) AFGZB
|
|
FGZCFGZABCHIZJIZABJIZACJIZHIZKZVOVMVNWAVMVNLZABDMZHIZJIZVRAWCJIZHIZKZNWBW
|
|
ANDCFWCCKZWHWAWBWIWEVQWGVTWIWDVPAJWCCBHOPWIWFVSVRHWCCAJOPQUAWHABRHIZJIZVR
|
|
ARJIZHIZKABEMZHIZJIZVRAWNJIZHIZKZABWNTZHIZJIZVRAWTJIZHIZKZWBDEWCRKZWEWKWG
|
|
WMXFWDWJAJWCRBHOPXFWFWLVRHWCRAJOPQWCWNKZWEWPWGWRXGWDWOAJWCWNBHOPXGWFWQVRH
|
|
WCWNAJOPQWCWTKZWEXBWGXDXHWDXAAJWCWTBHOPXHWFXCVRHWCWTAJOPQWBWKVRRHIZWMWBWK
|
|
VRXIWBWJBAJVNWJBKVMBUBUCPWBVRFGZXIVRKABUDZVRUBUESWBWLRVRHVMWLRKVNAUFUGPSW
|
|
NFGZVMVNWSXENZVMVNXLXMVMVNXLXMWSXEVMVNXLULZWPAHIZWRAHIZKWPWRAHUHXNXBXOXDX
|
|
PXNXBAWOTZJIZXOXNXAXQAJVNXLXAXQKVMBWNUIUJPVMVNXLXRXOKZVNXLLVMWOFGXSBWNUMA
|
|
WOUKUNUOUPXNXDVRWQAHIZHIZXPXNXCXTVRHVMXLXCXTKVNAWNUKUQPVMVNXLXPYAKZVMVNXL
|
|
YBNNVMVNXLVMYBVNVMXLVMYBNZNVMVNVMXLYCWBVMXLLZVMYBYDWBWQFGZVMYBAWNUDWBXJYE
|
|
VMYBXKVRWQAURUSUTVCVAVBVDVEVFSQVGVCVHVIVJVKVLVF $.
|
|
|
|
$( Multiplication of natural numbers is associative. Theorem 4K(4) of
|
|
[Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario
|
|
Carneiro, 15-Nov-2014.) $)
|
|
nnmass $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( ( A .o B ) .o C ) = ( A .o ( B .o C ) ) ) $=
|
|
( vx vy com wcel comu co wceq wa cv wi oveq2 oveq2d eqeq12d c0 nnmcl nnm0
|
|
coa imbi2d csuc syl sylan9eqr eqtr4d w3a oveq1 nnmsuc sylan 3impa 3adant1
|
|
nndi syl3an2 3exp expd com34 pm2.43d 3imp eqtrd syl5ibr com3r impd finds2
|
|
vtoclga expdcom ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJZVHVFVGVMVFVGKZVIDLZHIZA
|
|
BVOHIZHIZJZMVNVMMDCFVOCJZVSVMVNVTVPVJVRVLVOCVIHNVTVQVKAHVOCBHNOPUAVSVIQHI
|
|
ZABQHIZHIZJVIELZHIZABWDHIZHIZJZVIWDUBZHIZABWIHIZHIZJZVNDEVOQJZVPWAVRWCVOQ
|
|
VIHNWNVQWBAHVOQBHNOPVOWDJZVPWEVRWGVOWDVIHNWOVQWFAHVOWDBHNOPVOWIJZVPWJVRWL
|
|
VOWIVIHNWPVQWKAHVOWIBHNOPVNWAQWCVNVIFGZWAQJABRZVISUCVGVFWCAQHIQVGWBQAHBSO
|
|
ASUDUEWDFGZVFVGWHWMMZVFVGWSWTVFVGWSWTWHWMVFVGWSUFZWEVITIZWGVITIZJWEWGVITU
|
|
GXAWJXBWLXCVFVGWSWJXBJZVNWQWSXDWRVIWDUHUIUJXAWLAWFBTIZHIZXCXAWKXEAHVGWSWK
|
|
XEJVFBWDUHUKOVFVGWSXFXCJZVFVGWSXGMVFVGWSVGXGVFVGWSVGXGMVFVGWSKZVGXGXHVFWF
|
|
FGVGXGBWDRAWFBULUMUNUOUPUQURUSPUTUNVAVBVCVDVEUR $.
|
|
|
|
$( Multiplication with successor. Exercise 16 of [Enderton] p. 82.
|
|
(Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
nnmsucr $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( suc A .o B ) = ( ( A .o B ) +o B ) ) $=
|
|
( vx vy com wcel csuc comu co coa wceq cv wi oveq2 id oveq12d eqeq12d syl
|
|
c0 eqtr4d imbi2d peano2 nnm0 peano1 nnmcl mpan2 nna0 oveq1 peano2b nnmsuc
|
|
wa sylanb nnaass syl3an3b syl3an1 3expb anidms oveq1d an42s nnacom nnasuc
|
|
suceq ancoms 3eqtr4d oveq2d syl5ibr expcom finds2 vtoclga impcom ) BEFAEF
|
|
ZAGZBHIZABHIZBJIZKZVKVLCLZHIZAVQHIZVQJIZKZMVKVPMCBEVQBKZWAVPVKWBVRVMVTVOV
|
|
QBVLHNWBVSVNVQBJVQBAHNWBOPQUAWAVLSHIZASHIZSJIZKVLDLZHIZAWFHIZWFJIZKZVLWFG
|
|
ZHIZAWKHIZWKJIZKZVKCDVQSKZVRWCVTWEVQSVLHNWPVSWDVQSJVQSAHNWPOPQVQWFKZVRWGV
|
|
TWIVQWFVLHNWQVSWHVQWFJVQWFAHNWQOPQVQWKKZVRWLVTWNVQWKVLHNWRVSWMVQWKJVQWKAH
|
|
NWROPQVKWCWDWEVKWCSWDVKVLEFZWCSKAUBVLUCRAUCTVKWDEFZWEWDKVKSEFWTUDASUEUFWD
|
|
UGRTVKWFEFZWJWOMWJWOVKXAUKZWGVLJIZWIVLJIZKWGWIVLJUHXBWLXCWNXDVKWSXAWLXCKA
|
|
UIZVLWFUJULXBWHAJIZWKJIZWHAWKJIZJIZWNXDXBXGXIKZXBVKXAXJXBWHEFZVKXAXJAWFUE
|
|
ZXAXKVKWKEFXJWFUIWHAWKUMUNUOUPUQXBWMXFWKJAWFUJURXBXDWHWFVLJIZJIZXIXBXDXNK
|
|
ZVKXAXAVKXOXBXAVKXOXBXKXAVKXOXLVKXKXAWSXOXEWHWFVLUMUNUOUPUSUQXBXHXMWHJXBA
|
|
WFJIZGZWFAJIZGZXHXMXBXPXRKXQXSKAWFUTXPXRVBRAWFVAXAVKXMXSKWFAVAVCVDVETVDQV
|
|
FVGVHVIVJ $.
|
|
|
|
$( Multiplication of natural numbers is commutative. Theorem 4K(5) of
|
|
[Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened
|
|
by Andrew Salmon, 22-Oct-2011.) $)
|
|
nnmcom $p |- ( ( A e. _om /\ B e. _om ) -> ( A .o B ) = ( B .o A ) ) $=
|
|
( vx vy com wcel comu co wceq cv wi oveq1 oveq2 eqeq12d imbi2d csuc nnm0r
|
|
c0 nnm0 coa eqtr4d wa nnmsucr nnmsuc ancoms syl5ibr ex finds2 vtoclga imp
|
|
) AEFBEFZABGHZBAGHZIZUKCJZBGHZBUOGHZIZKUKUNKCAEUOAIZURUNUKUSUPULUQUMUOABG
|
|
LUOABGMNOURRBGHZBRGHZIDJZBGHZBVBGHZIZVBPZBGHZBVFGHZIZUKCDUORIUPUTUQVAUORB
|
|
GLUORBGMNUOVBIUPVCUQVDUOVBBGLUOVBBGMNUOVFIUPVGUQVHUOVFBGLUOVFBGMNUKUTRVAB
|
|
QBSUAVBEFZUKVEVIKVEVIVJUKUBZVCBTHZVDBTHZIVCVDBTLVKVGVLVHVMVBBUCUKVJVHVMIB
|
|
VBUDUENUFUGUHUIUJ $.
|
|
$}
|
|
|
|
$( Distributive law for natural numbers (right-distributivity). (Contributed
|
|
by Jim Kingdon, 3-Dec-2019.) $)
|
|
nndir $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( ( A +o B ) .o C ) = ( ( A .o C ) +o ( B .o C ) ) ) $=
|
|
( com wcel w3a coa co comu wceq nndi 3coml nnacl nnmcom sylan2 ancoms 3impa
|
|
wa 3adant2 3adant1 oveq12d 3eqtr3d ) ADEZBDEZCDEZFZCABGHZIHZCAIHZCBIHZGHZUG
|
|
CIHZACIHZBCIHZGHUEUCUDUHUKJCABKLUCUDUEUHULJZUEUCUDRZUOUPUEUGDEUOABMCUGNOPQU
|
|
FUIUMUJUNGUCUEUIUMJZUDUEUCUQCANPSUDUEUJUNJZUCUEUDURCBNPTUAUB $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Membership is inherited by successors. The reverse direction holds for
|
|
all ordinals, as seen at ~ onsucelsucr , but the forward direction, for
|
|
all ordinals, implies excluded middle as seen as ~ onsucelsucexmid .
|
|
(Contributed by Jim Kingdon, 25-Aug-2019.) $)
|
|
nnsucelsuc $p |- ( B e. _om -> ( A e. B <-> suc A e. suc B ) ) $=
|
|
( vx vy com wcel csuc cv wi c0 wceq eleq2 suceq eleq2d imbi12d pm2.21i wa
|
|
noel wo a1i elsuci adantl simpl orim12d mpd vex sucex elsuc2 sylibr finds
|
|
ex con0 nnon onsucelsucr syl impbid ) BEFZABFZAGZBGZFZACHZFZUSVBGZFZIAJFZ
|
|
USJGZFZIADHZFZUSVIGZFZIZAVKFZUSVKGZFZIZURVAICDBVBJKZVCVFVEVHVBJALVRVDVGUS
|
|
VBJMNOVBVIKZVCVJVEVLVBVIALVSVDVKUSVBVIMNOVBVKKZVCVNVEVPVBVKALVTVDVOUSVBVK
|
|
MNOVBBKZVCURVEVAVBBALWAVDUTUSVBBMNOVFVHARPVMVQIVIEFVMVNVPVMVNQZVLUSVKKZSZ
|
|
VPWBVJAVIKZSZWDVNWFVMAVIUAUBWBVJVLWEWCVMVNUCWEWCIWBAVIMTUDUEVKUSVIDUFUGUH
|
|
UIUKTUJUQBULFVAURIBUMABUNUOUP $.
|
|
|
|
$( Membership is inherited by successors. The reverse direction holds for
|
|
all ordinals, as seen at ~ onsucsssucr , but the forward direction, for
|
|
all ordinals, implies excluded middle as seen as ~ onsucsssucexmid .
|
|
(Contributed by Jim Kingdon, 25-Aug-2019.) $)
|
|
nnsucsssuc $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A C_ B <-> suc A C_ suc B ) ) $=
|
|
( vx vy com wcel wa wss csuc wi wceq sseq1 suceq sseq1d imbi12d c0 syl wb
|
|
ordelsuc mpbid cv imbi2d wn peano3 neneqd peano2 0elnn ord mpd word nnord
|
|
ordsucim cvv 0ex mpan 3syl a1d w3a simp3 simp1l simp1r syl2anc nnsucelsuc
|
|
wo mpbird 3expia exp31 finds2 vtoclga con0 nnon onsucsssucr syl2an impbid
|
|
imp ) AEFZBEFZGABHZAIZBIZHZVPVQVRWAJZVQCUAZBHZWCIZVTHZJZJVQWBJCAEWCAKZWGW
|
|
BVQWHWDVRWFWAWCABLWHWEVSVTWCAMNOUBWGPBHZPIZVTHZJDUAZBHZWLIZVTHZJZWNBHZWNI
|
|
ZVTHZJZVQCDWCPKZWDWIWFWKWCPBLXAWEWJVTWCPMNOWCWLKZWDWMWFWOWCWLBLXBWEWNVTWC
|
|
WLMNOWCWNKZWDWQWFWSWCWNBLXCWEWRVTWCWNMNOVQWKWIVQPVTFZWKVQVTPKZUCXDVQVTPBU
|
|
DUEVQXEXDVQVTEFXEXDVDBUFVTUGQUHUIVQBUJZVTUJZXDWKRZBUKZBULZPUMFXGXHUNPVTUM
|
|
SUOUPTUQWLEFZVQWPWTXKVQGZWPWQWSXLWPWQURZWNVTFZWSXMWLBFZXNXMXOWQXLWPWQUSXM
|
|
XKXFXOWQRXKVQWPWQUTZXMVQXFXKVQWPWQVAZXIQZWLBESVBVEXMVQXOXNRXQWLBVCQTXMWNE
|
|
FZXGXNWSRXMXKXSXPWLUFQXMXFXGXRXJQWNVTESVBTVFVGVHVIVOVPAVJFXFWAVRJVQAVKXIA
|
|
BVLVMVN $.
|
|
|
|
$( Trichotomy for natural numbers. (Contributed by Jim Kingdon,
|
|
25-Aug-2019.) $)
|
|
nntri3or $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A e. B \/ A = B \/ B e. A ) ) $=
|
|
( vx vy com wcel wceq cv wi eleq2 eqeq2 eleq1 3orbi123d c0 csuc wo 3orass
|
|
w3o sylibr syl imbi2d 0elnn olc wa df-3or elex elsuc2g syl6bir nnsucelsuc
|
|
cvv 3mix1 elsuci syl6bi orbi2i biimpi orcomd olcd syl6 jaao syl5bi finds2
|
|
eqcom ex vtoclga impcom ) BEFAEFZABFZABGZBAFZRZVFACHZFZAVKGZVKAFZRZIVFVJI
|
|
CBEVKBGZVOVJVFVPVLVGVMVHVNVIVKBAJVKBAKVKBALMUAVOANFZANGZNAFZRZADHZFZAWAGZ
|
|
WAAFZRZAWAOZFZAWFGZWFAFZRZVFCDVKNGVLVQVMVRVNVSVKNAJVKNAKVKNALMVKWAGVLWBVM
|
|
WCVNWDVKWAAJVKWAAKVKWAALMVKWFGVLWGVMWHVNWIVKWFAJVKWFAKVKWFALMVFVRVSPZVTAU
|
|
BWKVQWKPVTWKVQUCVQVRVSQSTWAEFZVFWEWJIWEWBWCPZWDPWLVFUDWJWBWCWDUEWLWMWJVFW
|
|
DWLWAUJFZWMWJIWAEUFWNWMWGWJAWAUJUGWGWHWIUKUHTVFWDWIWFAGZPZWJVFWDWFAOFWPWA
|
|
AUIWFAULUMWPWGWHWIPZPWJWPWQWGWPWIWHWPWIWHPWOWHWIWFAVBUNUOUPUQWGWHWIQSURUS
|
|
UTVCVAVDVE $.
|
|
|
|
$( A trichotomy law for natural numbers. (Contributed by Jim Kingdon,
|
|
28-Aug-2019.) $)
|
|
nntri2 $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A e. B <-> -. ( A = B \/ B e. A ) ) ) $=
|
|
( com wcel wa wceq wo elirr eleq2 mtbii con2i en2lp imnani ioran sylanbrc
|
|
wn w3o nntri3or 3orass sylib orcomd ord impbid2 ) ACDBCDEZABDZABFZBADZGZP
|
|
ZUEUFPUGPUIUFUEUFAADUEAHABAIJKUEUGABLMUFUGNOUDUHUEUDUEUHUDUEUFUGQUEUHGABR
|
|
UEUFUGSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A n $. $d B n $.
|
|
$( Given an element ` A ` of the union of a natural number ` B ` ,
|
|
` suc A ` is an element of ` B ` itself. The reverse direction holds
|
|
for all ordinals ( ~ sucunielr ). The forward direction for all
|
|
ordinals implies excluded middle ( ~ ordsucunielexmid ). (Contributed
|
|
by Jim Kingdon, 13-Mar-2022.) $)
|
|
nnsucuniel $p |- ( B e. _om -> ( A e. U. B <-> suc A e. B ) ) $=
|
|
( vn wcel cuni csuc c0 wceq wi cv wrex noel uni0 unieq eleq2d wa wb mpbid
|
|
com syl eleq2i mtbir mtbiri pm2.21d adantl ad2antll biimpa wtr word nnord
|
|
simplrl ordtr vex unisuc sylib nnsucelsuc simplrr eleqtrrd rexlimdvaa imp
|
|
ex nn0suc mpjaodan sucunielr impbid1 ) BSDZABEZDZAFZBDZVFBGHZVHVJIZBCJZFZ
|
|
HZCSKZVKVLVFVKVHVJVKVHAGEZDZVRAGDALVQGAMUAUBVKVGVQABGNOUCUDUEVFVPVLVFVOVL
|
|
CSVFVMSDZVOPPZVHVJVTVHPZVIVNBWAAVMDZVIVNDZWAAVNEZDZWBVTVHWEVOVHWEQVFVSVOV
|
|
GWDABVNNOUFUGWAWDVMAWAVSWDVMHZVFVSVOVHUKZVSVMUHZWFVSVMUIWHVMUJVMULTVMCUMU
|
|
NUOTORWAVSWBWCQWGAVMUPTRVFVSVOVHUQURVAUSUTCBVBVCABVDVE $.
|
|
$}
|
|
|
|
$( A trichotomy law for natural numbers. (Contributed by Jim Kingdon,
|
|
28-Aug-2019.) $)
|
|
nntri1 $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B <-> -. B e. A ) ) $=
|
|
( com wcel wa wss wn ssnel wceq wo w3o wi nntri3or df-3or biimpi orcomd ord
|
|
syl word nnord ordelss sylan ex adantl eqimss a1i jaod syld impbid2 ) ACDZB
|
|
CDZEZABFZBADZGZABHULUOABDZABIZJZUMULUPUQUNKZUOURLABMUSUNURUSURUNUSURUNJUPUQ
|
|
UNNOPQRULUPUMUQUKUPUMLUJUKUPUMUKBSUPUMBTBAUAUBUCUDUQUMLULABUEUFUGUHUI $.
|
|
|
|
$( A trichotomy law for natural numbers. (Contributed by Jim Kingdon,
|
|
15-May-2020.) $)
|
|
nntri3 $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A = B <-> ( -. A e. B /\ -. B e. A ) ) ) $=
|
|
( com wcel wa wceq wn wb elirr eleq2 mtbii con2i adantl simpl 2falsed simpr
|
|
eleq1 jca 2thd nntri3or mpjao3dan ) ACDBCDEZABDZABFZUCGZBADZGZEZHUDUFUBUCEU
|
|
DUHUCUDGZUBUDUCUDAADZUCAIZABAJKZLMUCUHGZUBUHUCUEUGNLMOUBUDEUDUHUBUDPUDUHUBU
|
|
DUEUGULUDUJUFUKABAQKZRMSUBUFEUDUHUFUIUBUDUFUNLMUFUMUBUHUFUEUGPLMOABTUA $.
|
|
|
|
$( A trichotomy law for natural numbers. (Contributed by Jim Kingdon,
|
|
15-Sep-2021.) $)
|
|
nntri2or2 $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B \/ B C_ A ) ) $=
|
|
( com wcel wa wss wo wceq con0 wi nnon adantl onelss syl orcd eqimss adantr
|
|
imp olcd nntri3or mpjao3dan ) ACDZBCDZEZABDZABFZBAFZGABHZBADZUDUEEUFUGUDUEU
|
|
FUDBIDZUEUFJUCUJUBBKLBAMNROUDUHEUFUGUHUFUDABPLOUDUIEUGUFUDUIUGUDAIDZUIUGJUB
|
|
UKUCAKQABMNRSABTUA $.
|
|
|
|
$( Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p.
|
|
(varies). For the specific case where ` B ` is zero, see ~ nndceq0 .
|
|
(Contributed by Jim Kingdon, 31-Aug-2019.) $)
|
|
nndceq $p |- ( ( A e. _om /\ B e. _om ) -> DECID A = B ) $=
|
|
( com wcel wa wceq wn wo wdc w3o nntri3or elirr eleq2 mtbii olcd orc mtbiri
|
|
con2i 3jaoi syl df-dc sylibr ) ACDBCDEZABFZUDGZHZUDIUCABDZUDBADZJUFABKUGUFU
|
|
DUHUGUEUDUDUGUDAADUGALABAMNROUDUEPUHUEUDUDUHUDUHBBDBLABBMQROSTUDUAUB $.
|
|
|
|
$( Set membership between two natural numbers is decidable. (Contributed by
|
|
Jim Kingdon, 6-Sep-2019.) $)
|
|
nndcel $p |- ( ( A e. _om /\ B e. _om ) -> DECID A e. B ) $=
|
|
( com wcel wa wn wo wdc wceq w3o nntri3or orc elirr eleq1 mtbiri olcd en2lp
|
|
imnani 3jaoi syl df-dc sylibr ) ACDBCDEZABDZUDFZGZUDHUCUDABIZBADZJUFABKUDUF
|
|
UGUHUDUELUGUEUDUGUDBBDBMABBNOPUHUEUDUHUDBAQRPSTUDUAUB $.
|
|
|
|
$( For natural numbers, inclusion is equivalent to membership or equality.
|
|
(Contributed by Jim Kingdon, 16-Sep-2021.) $)
|
|
nnsseleq $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( A C_ B <-> ( A e. B \/ A = B ) ) ) $=
|
|
( com wcel wa wss wceq wo wn nntri1 w3o nntri3or df-3or sylib orcomd sylbid
|
|
ord word wi nnord adantl ordelss ex syl eqimss a1i jaod impbid ) ACDZBCDZEZ
|
|
ABFZABDZABGZHZUKULBADZIUOABJUKUPUOUKUOUPUKUMUNUPKUOUPHABLUMUNUPMNOQPUKUMULU
|
|
NUKBRZUMULSUJUQUIBTUAUQUMULBAUBUCUDUNULSUKABUEUFUGUH $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( If we remove a single element from a natural number then put it back in,
|
|
we end up with the original natural number. This strengthens ~ difsnss
|
|
from subset to equality but the proof relies on equality being
|
|
decidable. (Contributed by Jim Kingdon, 31-Aug-2021.) $)
|
|
nndifsnid $p |- ( ( A e. _om /\ B e. A ) ->
|
|
( ( A \ { B } ) u. { B } ) = A ) $=
|
|
( vx com wcel wa csn cdif cun wss difsnss adantl cv wceq simpr syl simplr
|
|
wn elnn syl2anc velsn sylibr elun2 sylnibr eldifd elun1 wdc simpll nndceq
|
|
wo df-dc sylib mpjaodan ex ssrdv eqssd ) ADEZBAEZFZABGZHZUTIZAURVBAJUQABK
|
|
LUSCAVBUSCMZAEZVCVBEZUSVDFZVCBNZVEVGRZVFVGFZVCUTEZVEVIVGVJVFVGOCBUAZUBVCU
|
|
TVAUCPVFVHFZVCVAEVEVLVCAUTUSVDVHQVLVGVJVFVHOVKUDUEVCVAUTUFPVFVGUGZVGVHUJV
|
|
FVCDEZBDEZVMVFVDUQVNUSVDOUQURVDUHZVCASTVFURUQVOUQURVDQVPBASTVCBUITVGUKULU
|
|
MUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x $.
|
|
|
|
$( Ordering property of addition. Proposition 8.4 of [TakeutiZaring]
|
|
p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnaordi $p |- ( ( B e. _om /\ C e. _om ) ->
|
|
( A e. B -> ( C +o A ) e. ( C +o B ) ) ) $=
|
|
( vx vy com wcel coa co wa wi wceq oveq2 eleq12d c0 csuc syl syl2anc con0
|
|
nnon cv imbi2d simpr elnn ancoms nna0 adantr 3eltr4d wb simprl nnsucelsuc
|
|
simpl nnacl adantl oasuc ad2antrl bitr4d biimpd finds2 vtoclga imp nnacom
|
|
ex adantrr 3eltr3d 3impb 3com12 3expia ) BFGZCFGZABGZCAHIZCBHIZGZVJVIVKVN
|
|
VJVIVKVNVJVIVKJZJZACHIZBCHIZVLVMVJVOVQVRGZVOADUAZHIZBVTHIZGZKVOVSKDCFVTCL
|
|
ZWCVSVOWDWAVQWBVRVTCAHMVTCBHMNUBWCAOHIZBOHIZGAEUAZHIZBWGHIZGZAWGPZHIZBWKH
|
|
IZGZVODEVTOLWAWEWBWFVTOAHMVTOBHMNVTWGLWAWHWBWIVTWGAHMVTWGBHMNVTWKLWAWLWBW
|
|
MVTWKAHMVTWKBHMNVOABWEWFVIVKUCVOAFGZWEALVKVIWOABUDUEZAUFQVIWFBLVKBUFUGUHW
|
|
GFGZVOWJWNKWQVOJZWJWNWRWJWHPZWIPZGZWNWRWIFGZWJXAUIWRVIWQXBWQVIVKUJWQVOULB
|
|
WGUMRWHWIUKQWRWLWSWMWTWRASGZWGSGZWLWSLWRWOXCVOWOWQWPUNATQWQXDVOWGTUGZAWGU
|
|
ORWRBSGZXDWMWTLVIXFWQVKBTUPXEBWGUORNUQURVCUSUTVAVPWOVJVQVLLVOWOVJWPUNVJVO
|
|
ULACVBRVJVIVRVMLZVKVIVJXGBCVBUEVDVEVFVGVH $.
|
|
$}
|
|
|
|
$( Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58,
|
|
limited to natural numbers, and its converse. (Contributed by NM,
|
|
7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnaord $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( A e. B <-> ( C +o A ) e. ( C +o B ) ) ) $=
|
|
( com wcel w3a coa co wi nnaordi 3adant1 wo wn oveq2 a1i wa wb nnacl nntri2
|
|
wceq 3adant2 orim12d con3d df-3an ancom anandi 3bitri anim12i sylbi 3adant3
|
|
syl 3imtr4d impbid ) ADEZBDEZCDEZFZABEZCAGHZCBGHZEZUOUPURVAIUNABCJKUQUSUTTZ
|
|
UTUSEZLZMZABTZBAEZLZMZVAURUQVHVDUQVFVBVGVCVFVBIUQABCGNOUNUPVGVCIUOBACJUAUBU
|
|
CUQUSDEZUTDEZPZVAVEQUQUPUNPZUPUOPZPZVLUQUNUOPZUPPUPVPPVOUNUOUPUDVPUPUEUPUNU
|
|
OUFUGVMVJVNVKCARCBRUHUIUSUTSUKUNUOURVIQUPABSUJULUM $.
|
|
|
|
$( Ordering property of addition of natural numbers. (Contributed by NM,
|
|
9-Nov-2002.) $)
|
|
nnaordr $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( A e. B <-> ( A +o C ) e. ( B +o C ) ) ) $=
|
|
( com wcel w3a coa nnaord wceq nnacom ancoms 3adant2 3adant1 eleq12d bitrd
|
|
co ) ADEZBDEZCDEZFZABECAGPZCBGPZEACGPZBCGPZEABCHTUAUCUBUDQSUAUCIZRSQUECAJKL
|
|
RSUBUDIZQSRUFCBJKMNO $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x $.
|
|
|
|
$( Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnaword $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( A C_ B <-> ( C +o A ) C_ ( C +o B ) ) ) $=
|
|
( vx vy com wcel wss co wb wa wi wceq oveq1 sseq12d bibi2d c0 csuc nnacom
|
|
coa cv imbi2d nna0r eqcomd adantr adantl nnacl 3adant3 3adant2 nnsucsssuc
|
|
w3a syl2anc nnasuc peano2 sylan2 syl 3eqtr3rd ancoms biimpd 3expib finds2
|
|
suceq bitrd vtoclga impcom 3impa ) AFGZBFGZCFGZABHZCATIZCBTIZHZJZVIVGVHKZ
|
|
VNVOVJDUAZATIZVPBTIZHZJZLVOVNLDCFVPCMZVTVNVOWAVSVMVJWAVQVKVRVLVPCATNVPCBT
|
|
NOPUBVTVJQATIZQBTIZHZJVJEUAZATIZWEBTIZHZJZVJWERZATIZWJBTIZHZJZVODEVPQMZVS
|
|
WDVJWOVQWBVRWCVPQATNVPQBTNOPVPWEMZVSWHVJWPVQWFVRWGVPWEATNVPWEBTNOPVPWJMZV
|
|
SWMVJWQVQWKVRWLVPWJATNVPWJBTNOPVOAWBBWCVGAWBMVHVGWBAAUCUDUEVHBWCMVGVHWCBB
|
|
UCUDUFOWEFGZVGVHWIWNLWRVGVHUKZWIWNWSWHWMVJWSWHWFRZWGRZHZWMWSWFFGZWGFGZWHX
|
|
BJWRVGXCVHWEAUGUHWRVHXDVGWEBUGUIWFWGUJULWSWTWKXAWLWRVGWTWKMZVHVGWRXEVGWRK
|
|
ZAWJTIZAWETIZRZWKWTAWEUMWRVGWJFGZXGWKMWEUNZAWJSUOXFXHWFMXIWTMAWESXHWFVBUP
|
|
UQURUHWRVHXAWLMZVGVHWRXLVHWRKZBWJTIZBWETIZRZWLXABWEUMWRVHXJXNWLMXKBWJSUOX
|
|
MXOWGMXPXAMBWESXOWGVBUPUQURUIOVCPUSUTVAVDVEVF $.
|
|
$}
|
|
|
|
$( Cancellation law for addition of natural numbers. (Contributed by NM,
|
|
27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnacan $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( ( A +o B ) = ( A +o C ) <-> B = C ) ) $=
|
|
( com wcel w3a coa co wss wa wb nnaword 3comr 3com13 anbi12d bicomd 3bitr4g
|
|
wceq eqss ) ADEZBDEZCDEZFZABGHZACGHZIZUEUDIZJZBCIZCBIZJZUDUERBCRUCUKUHUCUIU
|
|
FUJUGUAUBTUIUFKBCALMUBUATUJUGKCBALNOPUDUESBCSQ $.
|
|
|
|
$( Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnaword1 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( A +o B ) ) $=
|
|
( com wcel con0 coa co wss nnon oaword1 syl2an ) ACDAEDBEDAABFGHBCDAIBIABJK
|
|
$.
|
|
|
|
$( Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) $)
|
|
nnaword2 $p |- ( ( A e. _om /\ B e. _om ) -> A C_ ( B +o A ) ) $=
|
|
( com wcel wa coa co nnaword1 nnacom sseqtrd ) ACDBCDEAABFGBAFGABHABIJ $.
|
|
|
|
$( Adding to both sides of an inequality in ` _om ` (Contributed by Scott
|
|
Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.) $)
|
|
nnawordi $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( A C_ B -> ( A +o C ) C_ ( B +o C ) ) ) $=
|
|
( com wcel w3a wss co nnaword biimpd nnacom 3adant2 3adant1 sseq12d sylibrd
|
|
coa wceq ) ADEZBDEZCDEZFZABGZCAPHZCBPHZGZACPHZBCPHZGUAUBUEABCIJUAUFUCUGUDRT
|
|
UFUCQSACKLSTUGUDQRBCKMNO $.
|
|
|
|
${
|
|
$d x y A $. $d x B $. $d x y C $.
|
|
$( Ordering property of multiplication. Half of Proposition 8.19 of
|
|
[TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM,
|
|
18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnmordi $p |- ( ( ( B e. _om /\ C e. _om ) /\ (/) e. C ) ->
|
|
( A e. B -> ( C .o A ) e. ( C .o B ) ) ) $=
|
|
( vx vy com wcel c0 comu co wi wa wceq eleq2 oveq2 eleq2d imbi12d coa imp
|
|
cv elnn expcom imbi2d csuc pm2.21i a1i wo elsuci nnmcl simpl jca nnaword1
|
|
noel sseld imim2d adantrl nna0 ad2antrr ancoms eqeltrrd eleq1d syl5ibrcom
|
|
nnaordi adantrr jaod sylan syl5 nnmsuc adantr sylibrd exp43 com12 adantld
|
|
wb impd finds2 vtoclga com23 exp4a mpdd com34 com24 imp31 ) BFGZCFGZHCGZA
|
|
BGZCAIJZCBIJZGZKZWDWGWFWEWJWDWGWEWFWJWDWGAFGZWEWFWJKZKWGWDWLABUAUBWDWGWLW
|
|
EWMWDWGWLWELZWFWJWDWNWFLZWGWJWOADTZGZWHCWPIJZGZKZKWOWKKDBFWPBMZWTWKWOXAWQ
|
|
WGWSWJWPBANXAWRWIWHWPBCIOPQUCWTAHGZWHCHIJZGZKZAETZGZWHCXFIJZGZKZAXFUDZGZW
|
|
HCXKIJZGZKZWODEWPHMZWQXBWSXDWPHANXPWRXCWHWPHCIOPQWPXFMZWQXGWSXIWPXFANXQWR
|
|
XHWHWPXFCIOPQWPXKMZWQXLWSXNWPXKANXRWRXMWHWPXKCIOPQXEWOXBXDAUMUEUFXFFGZWNW
|
|
FXJXOKZXSWEWFXTKZWLWEXSYAWEXSWFXJXOWEXSLZWFXJLZLZXLWHXHCRJZGZXNXLXGAXFMZU
|
|
GZYDYFAXFUHYBXHFGZWELZYCYHYFKYBYIWECXFUIWEXSUJUKYJYCLXGYFYGYJXJXGYFKZWFYJ
|
|
XJYKYJXIYFXGYJXHYEWHXHCULUNUOSUPYJWFYGYFKXJYJWFLZYFYGXHYEGYLXHHRJZXHYEYIY
|
|
MXHMWEWFXHUQURYJWFYMYEGZWEYIWFYNKHCXHVCUSSUTYGWHXHYEAXFCIOVAVBVDVEVFVGYBX
|
|
NYFVNYCYBXMYEWHCXFVHPVIVJVKVLVMVOVPVQVRVSVSVTWAWBWC $.
|
|
$}
|
|
|
|
$( Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring]
|
|
p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnmord $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
|
|
( ( A e. B /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) ) $=
|
|
( com wcel c0 wa comu co wi nnmordi ex com23 3adant1 wne wceq wb wo syl2anc
|
|
wn w3a impd ne0i nnm0r eqeq1d syl5ibrcom necon3d syl5 adantr nn0eln0 adantl
|
|
oveq1 sylibrd oveq2 a1i 3adantl2 orim12d con3d simpl3 simpl1 simpl2 3imtr4d
|
|
nnmcl nntri2 mpdd jcad impbid ) ADEZBDEZCDEZUAZABEZFCEZGZCAHIZCBHIZEZVIVJVN
|
|
VQJVHVIVJGZVLVMVQVRVMVLVQVRVMVLVQJABCKLMUBNVKVQVLVMVKVQVMVLVIVJVQVMJVHVRVQC
|
|
FOZVMVIVQVSJVJVQVPFOVIVSVPVOUCVICFVPFVIVPFPCFPZFBHIZFPBUDVTVPWAFCFBHULUEUFU
|
|
GUHUIVJVMVSQVICUJUKUMNZVKVMVQVLVKVMVQVLJVKVMGZVOVPPZVPVOEZRZTZABPZBAEZRZTZV
|
|
QVLWCWJWFWCWHWDWIWEWHWDJWCABCHUNUOVHVJVMWIWEJVIBACKUPUQURWCVODEZVPDEZVQWGQW
|
|
CVJVHWLVHVIVJVMUSZVHVIVJVMUTZCAVCSWCVJVIWMWNVHVIVJVMVAZCBVCSVOVPVDSWCVHVIVL
|
|
WKQWOWPABVDSVBLMVEWBVFVG $.
|
|
|
|
$( Weak ordering property of ordinal multiplication. (Contributed by Mario
|
|
Carneiro, 17-Nov-2014.) $)
|
|
nnmword $p |- ( ( ( A e. _om /\ B e. _om /\ C e. _om ) /\ (/) e. C ) ->
|
|
( A C_ B <-> ( C .o A ) C_ ( C .o B ) ) ) $=
|
|
( com wcel w3a c0 wa wn comu wss iba nnmord 3com12 sylan9bbr nntri1 syl2anc
|
|
co wb nnmcl notbid simpl1 simpl2 simpl3 3bitr4d ) ADEZBDEZCDEZFZGCEZHZBAEZI
|
|
ZCBJRZCAJRZEZIZABKZUOUNKZUKULUPUJULULUJHZUIUPUJULLUGUFUHUTUPSBACMNOUAUKUFUG
|
|
URUMSUFUGUHUJUBZUFUGUHUJUCZABPQUKUODEZUNDEZUSUQSUKUHUFVCUFUGUHUJUDZVACATQUK
|
|
UHUGVDVEVBCBTQUOUNPQUE $.
|
|
|
|
$( Cancellation law for multiplication of natural numbers. (Contributed by
|
|
NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnmcan $p |- ( ( ( A e. _om /\ B e. _om /\ C e. _om ) /\ (/) e. A ) ->
|
|
( ( A .o B ) = ( A .o C ) <-> B = C ) ) $=
|
|
( com wcel w3a c0 comu wss wceq 3anrot nnmword sylanb 3anrev anbi12d bicomd
|
|
wa co wb eqss 3bitr4g ) ADEZBDEZCDEZFZGAEZQZABHRZACHRZIZUIUHIZQZBCIZCBIZQZU
|
|
HUIJBCJUGUOULUGUMUJUNUKUEUCUDUBFUFUMUJSUBUCUDKBCALMUEUDUCUBFUFUNUKSUBUCUDNC
|
|
BALMOPUHUITBCTUA $.
|
|
|
|
$( One is a natural number. (Contributed by NM, 29-Oct-1995.) $)
|
|
1onn $p |- 1o e. _om $=
|
|
( c1o c0 csuc com df-1o wcel peano1 peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHI
|
|
J $.
|
|
|
|
$( The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) $)
|
|
2onn $p |- 2o e. _om $=
|
|
( c2o c1o csuc com df-2o wcel 1onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
|
|
$.
|
|
|
|
$( The ordinal 3 is a natural number. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
3onn $p |- 3o e. _om $=
|
|
( c3o c2o csuc com df-3o wcel 2onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
|
|
$.
|
|
|
|
$( The ordinal 4 is a natural number. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
4onn $p |- 4o e. _om $=
|
|
( c4o c3o csuc com df-4o wcel 3onn peano2 ax-mp eqeltri ) ABCZDEBDFKDFGBHIJ
|
|
$.
|
|
|
|
$( Multiply an element of ` _om ` by ` 1o ` . (Contributed by Mario
|
|
Carneiro, 17-Nov-2014.) $)
|
|
nnm1 $p |- ( A e. _om -> ( A .o 1o ) = A ) $=
|
|
( com wcel c1o comu co c0 csuc df-1o oveq2i wceq peano1 nnmsuc mpan2 oveq1d
|
|
coa nnm0 nna0r 3eqtrd syl5eq ) ABCZADEFAGHZEFZADUBAEIJUAUCAGEFZAPFZGAPFAUAG
|
|
BCUCUEKLAGMNUAUDGAPAQOARST $.
|
|
|
|
$( Multiply an element of ` _om ` by ` 2o ` (Contributed by Scott Fenton,
|
|
18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
nnm2 $p |- ( A e. _om -> ( A .o 2o ) = ( A +o A ) ) $=
|
|
( com wcel c2o comu c1o csuc coa df-2o oveq2i wceq 1onn nnmsuc mpan2 oveq1d
|
|
co nnm1 eqtrd syl5eq ) ABCZADEPAFGZEPZAAHPZDUAAEIJTUBAFEPZAHPZUCTFBCUBUEKLA
|
|
FMNTUDAAHAQORS $.
|
|
|
|
$( Multiply an element of ` _om ` by ` 2o ` (Contributed by Scott Fenton,
|
|
16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
nn2m $p |- ( A e. _om -> ( 2o .o A ) = ( A +o A ) ) $=
|
|
( com wcel c2o comu co coa wceq 2onn nnmcom mpan nnm2 eqtrd ) ABCZDAEFZADEF
|
|
ZAAGFDBCNOPHIDAJKALM $.
|
|
|
|
${
|
|
$d A b x y $. $d A x y z $. $d B b x y $.
|
|
$( Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88.
|
|
(Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
nnaordex $p |- ( ( A e. _om /\ B e. _om ) -> ( A e. B <->
|
|
E. x e. _om ( (/) e. x /\ ( A +o x ) = B ) ) ) $=
|
|
( vz com wcel wa c0 cv co wceq wrex wi eleq2 eqeq2 anbi2d rexbidv imbi12d
|
|
coa c1o vb vy imbi2d csuc noel pm2.21i a1i wo elsuci simpr peano2 elelsuc
|
|
ad2antlr nnasuc suceq sylan9eq ex anim12d imp oveq2 eqeq1d anbi12d rspcev
|
|
syl2anc rexlimdva cbvrexv syl6ib syld con0 nnon oa1suc syl 1onn mpan jaod
|
|
0lt1o syl5 exp31 finds2 vtoclga impcom peano1 nnaord mp3an1 ancoms adantr
|
|
wb nna0 eleq1d bitrd anbi1d biimpac syl6bi impbid ) BEFZCEFZGBCFZHAIZFZBW
|
|
RSJZCKZGZAELZWPWOWQXCMZWOBUAIZFZWSWTXEKZGZAELZMZMWOXDMUACEXECKZXJXDWOXKXF
|
|
WQXIXCXECBNXKXHXBAEXKXGXAWSXECWTOPQRUCXJBHFZWSWTHKZGZAELZMZBUBIZFZWSWTXQK
|
|
ZGZAELZMZBXQUDZFZWSWTYCKZGZAELZMZWOUAUBXEHKZXFXLXIXOXEHBNYIXHXNAEYIXGXMWS
|
|
XEHWTOPQRXEXQKZXFXRXIYAXEXQBNYJXHXTAEYJXGXSWSXEXQWTOPQRXEYCKZXFYDXIYGXEYC
|
|
BNYKXHYFAEYKXGYEWSXEYCWTOPQRXPWOXLXOBUEUFUGXQEFZWOYBYHYDXRBXQKZUHYLWOGZYB
|
|
GZYGBXQUIYOXRYGYMYOXRYAYGYNYBUJWOYAYGMYLYBWOYAHDIZFZBYPSJZYCKZGZDELZYGWOX
|
|
TUUAAEWOWREFZGZXTUUAUUCXTGWRUDZEFZHUUDFZBUUDSJZYCKZGZUUAUUBUUEWOXTWRUKUMU
|
|
UCXTUUIUUCWSUUFXSUUHWSUUFMUUCHWRULUGUUCXSUUHUUCXSUUGWTUDYCBWRUNWTXQUOUPUQ
|
|
URUSYTUUIDUUDEYPUUDKZYQUUFYSUUHYPUUDHNUUJYRUUGYCYPUUDBSUTVAVBVCVDUQVEYTYF
|
|
DAEYPWRKZYQWSYSYEYPWRHNUUKYRWTYCYPWRBSUTVAVBVFVGUMVHWOYMYGMYLYBWOYMYGWOYM
|
|
GZHTFZBTSJZYCKZYGUUMUULVPUGWOYMUUNBUDZYCWOBVIFUUNUUPKBVJBVKVLBXQUOUPTEFUU
|
|
MUUOGZYGVMYFUUQATEWRTKZWSUUMYEUUOWRTHNUURWTUUNYCWRTBSUTVAVBVCVNVDUQUMVOVQ
|
|
VRVSVTWAWOXCWQMWPWOXBWQAEUUCXBBWTFZXAGWQUUCWSUUSXAUUCWSBHSJZWTFZUUSUUBWOW
|
|
SUVAWGZHEFUUBWOUVBWBHWRBWCWDWEUUCUUTBWTWOUUTBKUUBBWHWFWIWJWKXAUUSWQWTCBNW
|
|
LWMVEWFWN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equivalence for weak ordering of natural numbers. (Contributed by NM,
|
|
8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
nnawordex $p |- ( ( A e. _om /\ B e. _om ) -> ( A C_ B <->
|
|
E. x e. _om ( A +o x ) = B ) ) $=
|
|
( com wcel wa wss cv coa co wceq wrex w3a nntri3or 3adant3 wi c0 nnaordex
|
|
w3o syl5ibcom simpr reximi syl6bi nna0 3ad2ant1 eqeq2 peano1 oveq2 eqeq1d
|
|
rspcev mpan wn nntri1 biimp3a pm2.21d 3jaod mpd 3expia nnaword1 rexlimdva
|
|
syl6 sseq2 adantr impbid ) BDEZCDEZFZBCGZBAHZIJZCKZADLZVEVFVHVLVEVFVHMZBC
|
|
EZBCKZCBEZSZVLVEVFVQVHBCNOVMVNVLVOVPVEVFVNVLPVHVGVNQVIEZVKFZADLVLABCRVSVK
|
|
ADVRVKUAUBUCOVMVOBQIJZCKZVLVMVTBKZVOWAVEVFWBVHBUDUEBCVTUFTQDEWAVLUGVKWAAQ
|
|
DVIQKVJVTCVIQBIUHUIUJUKVAVMVPVLVEVFVHVPULBCUMUNUOUPUQURVEVLVHPVFVEVKVHADV
|
|
EVIDEFBVJGVKVHBVIUSVJCBVBTUTVCVD $.
|
|
$}
|
|
|
|
$( The product of two natural numbers is zero iff at least one of them is
|
|
zero. (Contributed by Jim Kingdon, 11-Nov-2004.) $)
|
|
nnm00 $p |- ( ( A e. _om /\ B e. _om ) -> ( ( A .o B ) = (/) <->
|
|
( A = (/) \/ B = (/) ) ) ) $=
|
|
( com wcel wa comu co c0 wceq wo wi simpl jaoi orcd a1i adantr ex sylan9eqr
|
|
jaod 0elnn olcd simplr wn nnmordi expimpd ancoms nnm0 eleq1d sylibd imp n0i
|
|
simpr pm2.21dd anim12i anddi sylib mpjaod oveq1 nnm0r adantl oveq2 impbid
|
|
syl ) ACDZBCDZEZABFGZHIZAHIZBHIZJZVFVHVKVFVHEZVIVJEZVIHBDZEZJZVKHADZVJEZVQV
|
|
NEZJZVPVKKVLVPVIVJVMVIVOVIVJLVIVNLMNOVLVRVKVSVRVKKVLVRVJVIVQVJULUAOVLVSVKVL
|
|
VSEZVHVKVFVHVSUBWAHVGDZVHUCVLVSWBVFVSWBKVHVFVSAHFGZVGDZWBVEVDVSWDKVEVDEVQVN
|
|
WDHBAUDUEUFVFWCHVGVDWCHIVEAUGZPUHUIPUJVGHUKVCUMQSVFVPVTJZVHVFVIVQJZVJVNJZEW
|
|
FVDWGVEWHATBTUNVIVQVJVNUOUPPUQQVFVIVHVJVEVIVHKVDVEVIVHVIVEVGHBFGHAHBFURBUSR
|
|
QUTVDVJVHKVEVDVJVHVJVDVGWCHBHAFVAWERQPSVB $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equivalence relations and classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c Er $. $( Equivalence predicate $)
|
|
$c /. $. $( Long slash for quotient set $)
|
|
|
|
$( Extend the definition of a wff to include the equivalence predicate. $)
|
|
wer $a wff R Er A $.
|
|
|
|
$( Extend the definition of a class to include equivalence class. $)
|
|
cec $a class [ A ] R $.
|
|
|
|
$( Extend the definition of a class to include quotient set. $)
|
|
cqs $a class ( A /. R ) $.
|
|
|
|
$( Define the equivalence relation predicate. Our notation is not standard.
|
|
A formal notation doesn't seem to exist in the literature; instead only
|
|
informal English tends to be used. The present definition, although
|
|
somewhat cryptic, nicely avoids dummy variables. In ~ dfer2 we derive a
|
|
more typical definition. We show that an equivalence relation is
|
|
reflexive, symmetric, and transitive in ~ erref , ~ ersymb , and ~ ertr .
|
|
(Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro,
|
|
2-Nov-2015.) $)
|
|
df-er $a |- ( R Er A <->
|
|
( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) ) $.
|
|
|
|
${
|
|
$d x y z R $.
|
|
$( Alternate definition of equivalence predicate. (Contributed by NM,
|
|
3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
dfer2 $p |- ( R Er A <->
|
|
( Rel R /\ dom R = A /\ A. x A. y A. z
|
|
( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) ) ) $=
|
|
( wer wrel cdm wceq ccnv wss w3a cv wbr wi wa wal albii 19.26 bitri df-er
|
|
ccom cun cnvsym cotr anbi12i unss 19.28v bitr2i 3bitr3i 3anbi3i ) DEFEGZE
|
|
HDIZEJZEEUBZUCEKZLULUMAMZBMZENZURUQENOZUSURCMZENPUQVAENOZPCQZBQZAQZLDEUAU
|
|
PVEULUMUNEKZUOEKZPUTBQZAQZVBCQZBQZAQZPZUPVEVFVIVGVLABEUDABCEUEUFUNUOEUGVE
|
|
VHVKPZAQVMVDVNAVDUTVJPZBQVNVCVOBUTVBCUHRUTVJBSTRVHVKASUIUJUKT $.
|
|
$}
|
|
|
|
$( Define the ` R ` -coset of ` A ` . Exercise 35 of [Enderton] p. 61. This
|
|
is called the equivalence class of ` A ` modulo ` R ` when ` R ` is an
|
|
equivalence relation (i.e. when ` Er R ` ; see ~ dfer2 ). In this case,
|
|
` A ` is a representative (member) of the equivalence class ` [ A ] R ` ,
|
|
which contains all sets that are equivalent to ` A ` . Definition of
|
|
[Enderton] p. 57 uses the notation ` [ A ] ` (subscript) ` R ` , although
|
|
we simply follow the brackets by ` R ` since we don't have subscripted
|
|
expressions. For an alternate definition, see ~ dfec2 . (Contributed by
|
|
NM, 23-Jul-1995.) $)
|
|
df-ec $a |- [ A ] R = ( R " { A } ) $.
|
|
|
|
${
|
|
$d y A $. $d y R $.
|
|
$( Alternate definition of ` R ` -coset of ` A ` . Definition 34 of
|
|
[Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by
|
|
Mario Carneiro, 9-Jul-2014.) $)
|
|
dfec2 $p |- ( A e. V -> [ A ] R = { y | A R y } ) $=
|
|
( wcel cec csn cima cv wbr cab df-ec imasng syl5eq ) BDEBCFCBGHBAICJAKBCL
|
|
ABDCMN $.
|
|
$}
|
|
|
|
$( An equivalence class modulo a set is a set. (Contributed by NM,
|
|
24-Jul-1995.) $)
|
|
ecexg $p |- ( R e. B -> [ A ] R e. _V ) $=
|
|
( wcel cec csn cima cvv df-ec imaexg syl5eqel ) CBDACECAFZGHACICLBJK $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d R x $.
|
|
$( An inhabited equivalence class implies the representative is a set.
|
|
(Contributed by Mario Carneiro, 9-Jul-2014.) $)
|
|
ecexr $p |- ( A e. [ B ] R -> B e. _V ) $=
|
|
( vx cec wcel cv wceq wex cvv wbr csn wrex elimag ibi df-ec eleq2s df-rex
|
|
cima wa simpl velsn sylib eximi sylbi syl isset sylibr ) ABCEZFZDGZBHZDIZ
|
|
BJFUJUKACKZDBLZMZUMUPACUOSZUIAUQFUPDACUOUQNOBCPQUPUKUOFZUNTZDIUMUNDUORUSU
|
|
LDUSURULURUNUADBUBUCUDUEUFDBUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Define quotient set. ` R ` is usually an equivalence relation.
|
|
Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.) $)
|
|
df-qs $a |- ( A /. R ) = { y | E. x e. A y = [ x ] R } $.
|
|
$}
|
|
|
|
$( Equality theorem for equivalence predicate. (Contributed by NM,
|
|
4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
ereq1 $p |- ( R = S -> ( R Er A <-> S Er A ) ) $=
|
|
( wceq wrel cdm ccnv ccom cun wss releq dmeq eqeq1d cnveq coeq1 coeq2 eqtrd
|
|
w3a wer df-er uneq12d sseq1d sseq2 bitrd 3anbi123d 3bitr4g ) BCDZBEZBFZADZB
|
|
GZBBHZIZBJZRCEZCFZADZCGZCCHZIZCJZRABSACSUGUHUOUJUQUNVABCKUGUIUPABCLMUGUNUTB
|
|
JVAUGUMUTBUGUKURULUSBCNUGULCBHUSBCBOBCCPQUAUBBCUTUCUDUEABTACTUF $.
|
|
|
|
$( Equality theorem for equivalence predicate. (Contributed by Mario
|
|
Carneiro, 12-Aug-2015.) $)
|
|
ereq2 $p |- ( A = B -> ( R Er A <-> R Er B ) ) $=
|
|
( wceq wrel cdm ccnv ccom cun wss w3a wer eqeq2 3anbi2d df-er 3bitr4g ) ABD
|
|
ZCEZCFZADZCGCCHICJZKRSBDZUAKACLBCLQTUBRUAABSMNACOBCOP $.
|
|
|
|
$( An equivalence relation is a relation. (Contributed by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
errel $p |- ( R Er A -> Rel R ) $=
|
|
( wer wrel cdm wceq ccnv ccom cun wss df-er simp1bi ) ABCBDBEAFBGBBHIBJABKL
|
|
$.
|
|
|
|
$( The domain of an equivalence relation. (Contributed by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
erdm $p |- ( R Er A -> dom R = A ) $=
|
|
( wer wrel cdm wceq ccnv ccom cun wss df-er simp2bi ) ABCBDBEAFBGBBHIBJABKL
|
|
$.
|
|
|
|
${
|
|
ersym.1 $e |- ( ph -> R Er X ) $.
|
|
ersym.2 $e |- ( ph -> A R B ) $.
|
|
$( Elementhood in the field of an equivalence relation. (Contributed by
|
|
Mario Carneiro, 12-Aug-2015.) $)
|
|
ercl $p |- ( ph -> A e. X ) $=
|
|
( cdm wrel wbr wcel wer errel syl releldm syl2anc wceq erdm eleqtrd ) ABD
|
|
HZEADIZBCDJBTKAEDLZUAFEDMNGBCDOPAUBTEQFEDRNS $.
|
|
|
|
$( An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.)
|
|
(Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
ersym $p |- ( ph -> B R A ) $=
|
|
( ccnv wbr cvv wcel wa wb wrel wer errel syl brrelex12 syl2anc brcnvg cun
|
|
ancoms mpbird ccom wss cdm wceq df-er simp3bi unssad ssbrd mpd ) ACBDHZIZ
|
|
CBDIAUNBCDIZGABJKZCJKZLZUNUOMZADNZUOURAEDOZUTFEDPQGBCDRSUQUPUSCBJJDTUBQUC
|
|
AUMDCBAUMDDUDZDAVAUMVBUADUEZFVAUTDUFEUGVCEDUHUIQUJUKUL $.
|
|
|
|
$( Elementhood in the field of an equivalence relation. (Contributed by
|
|
Mario Carneiro, 12-Aug-2015.) $)
|
|
ercl2 $p |- ( ph -> B e. X ) $=
|
|
( ersym ercl ) ACBDEFABCDEFGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ph $. $d x R $.
|
|
ersymb.1 $e |- ( ph -> R Er X ) $.
|
|
$( An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.)
|
|
(Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
ersymb $p |- ( ph -> ( A R B <-> B R A ) ) $=
|
|
( wbr wa wer adantr simpr ersym impbida ) ABCDGZCBDGZANHBCDEAEDIZNFJANKLA
|
|
OHCBDEAPOFJAOKLM $.
|
|
|
|
$( An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.)
|
|
(Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
ertr $p |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) ) $=
|
|
( vx wbr wa ccom cv wex cvv wcel syl simpr brrelex syl2an wceq wrel errel
|
|
breq2 breq1 anbi12d spcegv sylc wb simpl brrelex2 brcog syl2anc mpbird ex
|
|
wer ccnv cun wss cdm df-er simp3bi unssbd ssbrd syld ) ABCEIZCDEIZJZBDEEK
|
|
ZIZBDEIAVGVIAVGJZVIBHLZEIZVKDEIZJZHMZVJCNOZVGVOAEUAZVFVPVGAFEUOZVQGFEUBPZ
|
|
VEVFQZCDERSAVGQVNVGHCNVKCTVLVEVMVFVKCBEUCVKCDEUDUEUFUGVJBNOZDNOZVIVOUHAVQ
|
|
VEWAVGVSVEVFUIBCERSAVQVFWBVGVSVTCDEUJSHBDEENNUKULUMUNAVHEBDAEUPZVHEAVRWCV
|
|
HUQEURZGVRVQEUSFTWDFEUTVAPVBVCVD $.
|
|
|
|
${
|
|
ertrd.5 $e |- ( ph -> A R B ) $.
|
|
ertrd.6 $e |- ( ph -> B R C ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ertrd $p |- ( ph -> A R C ) $=
|
|
( wbr ertr mp2and ) ABCEJCDEJBDEJHIABCDEFGKL $.
|
|
|
|
$( A transitivity relation for equivalences. (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ertr2d $p |- ( ph -> C R A ) $=
|
|
( ertrd ersym ) ABDEFGABCDEFGHIJK $.
|
|
$}
|
|
|
|
${
|
|
ertr3d.5 $e |- ( ph -> B R A ) $.
|
|
ertr3d.6 $e |- ( ph -> B R C ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ertr3d $p |- ( ph -> A R C ) $=
|
|
( ersym ertrd ) ABCDEFGACBEFGHJIK $.
|
|
$}
|
|
|
|
${
|
|
ertr4d.5 $e |- ( ph -> A R B ) $.
|
|
ertr4d.6 $e |- ( ph -> C R B ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ertr4d $p |- ( ph -> A R C ) $=
|
|
( ersym ertrd ) ABCDEFGHADCEFGIJK $.
|
|
$}
|
|
|
|
erref.2 $e |- ( ph -> A e. X ) $.
|
|
$( An equivalence relation is reflexive on its field. Compare Theorem 3M
|
|
of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.)
|
|
(Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
erref $p |- ( ph -> A R A ) $=
|
|
( vx cv wbr cdm wcel wex wer wceq erdm syl eleqtrrd wb eldmg mpbid adantr
|
|
wa simpr ertr4d exlimddv ) ABGHZCIZBBCIGABCJZKZUGGLZABDUHFADCMZUHDNEDCOPQ
|
|
ABDKUIUJRFGBCDSPTAUGUBBUFBCDAUKUGEUAAUGUCZULUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( The converse of an equivalence relation is itself. (Contributed by
|
|
Mario Carneiro, 12-Aug-2015.) $)
|
|
ercnv $p |- ( R Er A -> `' R = R ) $=
|
|
( vx vy wrel wer ccnv wceq errel relcnv cv wbr wcel id ersymb brcnv df-br
|
|
cop vex bitr3i 3bitr3g eqrelrdv2 mpanl1 mpancom ) BEZABFZBGZBHZABIUGEUEUF
|
|
UHBJUFCDUGBUFDKZCKZBLZUJUIBLUJUIRZUGMZULBMUFUIUJBAUFNOUKUJUIUGLUMUJUIBCSD
|
|
SPUJUIUGQTUJUIBQUAUBUCUD $.
|
|
$}
|
|
|
|
$( The range and domain of an equivalence relation are equal. (Contributed
|
|
by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
errn $p |- ( R Er A -> ran R = A ) $=
|
|
( wer crn ccnv cdm df-rn ercnv dmeqd erdm eqtrd syl5eq ) ABCZBDBEZFZABGMOBF
|
|
AMNBABHIABJKL $.
|
|
|
|
$( An equivalence relation is a subset of the cartesian product of the field.
|
|
(Contributed by Mario Carneiro, 12-Aug-2015.) $)
|
|
erssxp $p |- ( R Er A -> R C_ ( A X. A ) ) $=
|
|
( wer cdm crn cxp wrel wss errel relssdmrn syl erdm errn xpeq12d sseqtrd )
|
|
ABCZBBDZBEZFZAAFPBGBSHABIBJKPQARAABLABMNO $.
|
|
|
|
$( An equivalence relation is a set if its domain is a set. (Contributed by
|
|
Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
erex $p |- ( R Er A -> ( A e. V -> R e. _V ) ) $=
|
|
( wer wcel cvv cxp wss erssxp xpexg anidms ssexg syl2an ex ) ABDZACEZBFEZOB
|
|
AAGZHRFEZQPABIPSAACCJKBRFLMN $.
|
|
|
|
$( An equivalence relation is a set if and only if its domain is a set.
|
|
(Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
erexb $p |- ( R Er A -> ( R e. _V <-> A e. _V ) ) $=
|
|
( wer cvv wcel cdm dmexg erdm eleq1d syl5ib erex impbid ) ABCZBDEZADEZNBFZD
|
|
EMOBDGMPADABHIJABDKL $.
|
|
|
|
${
|
|
$d x y z R $. $d x A $. $d x y z ph $.
|
|
iserd.1 $e |- ( ph -> Rel R ) $.
|
|
iserd.2 $e |- ( ( ph /\ x R y ) -> y R x ) $.
|
|
iserd.3 $e |- ( ( ph /\ ( x R y /\ y R z ) ) -> x R z ) $.
|
|
iserd.4 $e |- ( ph -> ( x e. A <-> x R x ) ) $.
|
|
$( A reflexive, symmetric, transitive relation is an equivalence relation
|
|
on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised
|
|
by Mario Carneiro, 12-Aug-2015.) $)
|
|
iserd $p |- ( ph -> R Er A ) $=
|
|
( wer wceq cv wbr wi wa wal ex alrimiv wcel cdm eqidd jca dfer2 syl3anbrc
|
|
wrel adantr simpr erref vex breldm impbid1 bitr4d eqrdv ereq2 syl mpbid
|
|
wb ) AFUAZFKZEFKZAFUFUSUSLBMZCMZFNZVCVBFNZOZVDVCDMZFNPZVBVGFNZOZPZDQZCQZB
|
|
QUTGAUSUBAVMBAVLCAVKDAVFVJAVDVEHRAVHVIIRUCSSSBCDUSFUDUEZAUSELUTVAURABUSEA
|
|
VBUSTZVBVBFNZVBETAVOVPAVOVPAVOPVBFUSAUTVOVNUGAVOUHUIRVBVBFBUJZVQUKULJUMUN
|
|
USEFUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
swoer.1 $e |- R = ( ( X X. X ) \ ( .< u. `' .< ) ) $.
|
|
$( Evaluate the incomparability relation. (Contributed by Mario Carneiro,
|
|
9-Jul-2014.) $)
|
|
brdifun $p |- ( ( A e. X /\ B e. X ) ->
|
|
( A R B <-> -. ( A .< B \/ B .< A ) ) ) $=
|
|
( wcel wa wbr ccnv cun wn wo cxp wb cop opelxpi df-br sylibr cdif brcnvg
|
|
breqi brdif bitri baib syl brun orbi2d syl5bb notbid bitrd ) AEGBEGHZABCI
|
|
ZABDDJZKZIZLZABDIZBADIZMZLULABEENZIZUMUQOULABPVAGVBABEEQABVARSUMVBUQUMABV
|
|
AUOTZIVBUQHABCVCFUBABVAUOUCUDUEUFULUPUTUPURABUNIZMULUTABDUNUGULVDUSURABEE
|
|
DUAUHUIUJUK $.
|
|
|
|
$d u v w R $. $d x y z .< $. $d x y z A $. $d x y z B $. $d x y z C $.
|
|
$d u v w x y z ph $. $d u x y z X $.
|
|
swoer.2 $e |- ( ( ph /\ ( y e. X /\ z e. X ) ) ->
|
|
( y .< z -> -. z .< y ) ) $.
|
|
swoer.3 $e |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) ->
|
|
( x .< y -> ( x .< z \/ z .< y ) ) ) $.
|
|
$( Incomparability under a strict weak partial order is an equivalence
|
|
relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by
|
|
Mario Carneiro, 12-Aug-2015.) $)
|
|
swoer $p |- ( ph -> R Er X ) $=
|
|
( cv wbr wa wo wn wb wcel brdifun syl syl2anc vu vv wrel cxp wss ccnv cun
|
|
vw cdif difss eqsstri relxp relss mp2 a1i simpr orcom notbid ssbri adantl
|
|
brxp sylib simprd simpld 3bitr4d mpbid simprl simplbi simprbi simprr brel
|
|
ad2antrl wi simpl swopolem syl13anc syl6ibr orim12d or4 syl6ib mpbird wpo
|
|
mtord swopo poirr sylan pm1.2 nsyl impbida iserd ) AUAUBUHGEEUCZAEGGUDZUE
|
|
WLUCWKEWLFFUFUGZUIWLHWLWMUJUKZGGULEWLUMUNUOAUAKZUBKZELZMZWQWPWOELZAWQUPWR
|
|
WOWPFLZWPWOFLZNZOZXAWTNZOZWQWSWRXBXDXBXDPWRWTXAUQUOURWRWOGQZWPGQZMZWQXCPZ
|
|
WRWOWPWLLZXHWQXJAEWLWOWPWNUSZUTWOWPGGVAZVBZWOWPEFGHRZSWRXGXFWSXEPWRXFXGXM
|
|
VCWRXFXGXMVDWPWOEFGHRTVEVFAWQWPUHKZELZMZMZWOXOELZWOXOFLZXOWOFLZNZOZXRYBXB
|
|
WPXOFLZXOWPFLZNZXRWQXCAWQXPVGXRXFXGXIXRXJXFWQXJAXPXKVLZXJXFXGXLVHSZXRXJXG
|
|
YGXJXFXGXLVISZXNTVFXRXPYFOZAWQXPVJZXRXGXOGQZXPYJPYIXRXPYLYKXPXGYLWPXOGGEW
|
|
NVKVCSZWPXOEFGHRTVFXRYBWTYDNZXAYENZNXBYFNXRXTYNYAYOXRAXFYLXGXTYNVMAXQVNZY
|
|
HYMYIABCDGFWOXOWPJVOVPXRYAYEXANZYOXRAYLXFXGYAYQVMYPYMYHYIABCDGFXOWOWPJVOV
|
|
PXAYEUQVQVRWTYDXAYEVSVTWCXRXFYLXSYCPYHYMWOXOEFGHRTWAAXFWOWOELZAXFMZYRWOWO
|
|
FLZYTNZOZYSYTUUAAGFWBXFYTOABCDGFIJWDGWOFWEWFYTWGWHYSXFXFYRUUBPAXFUPZUUCWO
|
|
WOEFGHRTWAYRXFAYRWOWOWLLZXFEWLWOWOWNUSUUDXFXFWOWOGGVAVHSUTWIWJ $.
|
|
|
|
${
|
|
swoord.4 $e |- ( ph -> B e. X ) $.
|
|
swoord.5 $e |- ( ph -> C e. X ) $.
|
|
swoord.6 $e |- ( ph -> A R B ) $.
|
|
$( The incomparability equivalence relation is compatible with the
|
|
original order. (Contributed by Mario Carneiro, 31-Dec-2014.) $)
|
|
swoord1 $p |- ( ph -> ( A .< C <-> B .< C ) ) $=
|
|
( wbr wo wcel wn wi cxp ccnv cun cdif difss eqsstri ssbri df-br opelxp1
|
|
id cop sylbi 3syl swopolem syl13anc wb brdifun syl2anc mpbid nsyl biorf
|
|
orc syl sylibrd olc impbid ) AEGIQZFGIQZAVHEFIQZVIRZVIAAEJSZGJSZFJSZVHV
|
|
KUAAUKZAEFHQZEFJJUBZQZVLPHVQEFHVQIIUCUDZUEVQKVQVSUFUGUHVREFULVQSVLEFVQU
|
|
IEFJJUJUMUNZONABCDJIEGFMUOUPAVJTVIVKUQAVJFEIQZRZVJAVPWBTZPAVLVNVPWCUQVT
|
|
NEFHIJKURUSUTZVJWAVCVAVJVIVBVDVEAVIWAVHRZVHAAVNVMVLVIWEUAVONOVTABCDJIFG
|
|
EMUOUPAWATVHWEUQAWBWAWDWAVJVFVAWAVHVBVDVEVG $.
|
|
|
|
$( The incomparability equivalence relation is compatible with the
|
|
original order. (Contributed by Mario Carneiro, 31-Dec-2014.) $)
|
|
swoord2 $p |- ( ph -> ( C .< A <-> C .< B ) ) $=
|
|
( wbr wo wcel wi id cxp ccnv cun cdif difss eqsstri ssbri df-br opelxp1
|
|
cop sylbi swopolem syl13anc idd wn wb brdifun syl2anc mpbid olc pm2.21d
|
|
3syl nsyl jaod syld orc impbid ) AGEIQZGFIQZAVIVJFEIQZRZVJAAGJSZEJSZFJS
|
|
ZVIVLTAUAZOAEFHQZEFJJUBZQZVNPHVREFHVRIIUCUDZUEVRKVRVTUFUGUHVSEFUKVRSVNE
|
|
FVRUIEFJJUJULVCZNABCDJIGEFMUMUNAVJVJVKAVJUOAVKVJAEFIQZVKRZVKAVQWCUPZPAV
|
|
NVOVQWDUQWANEFHIJKURUSUTZVKWBVAVDVBVEVFAVJVIWBRZVIAAVMVOVNVJWFTVPONWAAB
|
|
CDJIGFEMUMUNAVIVIWBAVIUOAWBVIAWCWBWEWBVKVGVDVBVEVFVH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d w x y $. $d x y z $. $d y A $. $d v x B $.
|
|
eqer.1 $e |- ( x = y -> A = B ) $.
|
|
eqer.2 $e |- R = { <. x , y >. | A = B } $.
|
|
$( Lemma for ~ eqer . (Contributed by NM, 17-Mar-2008.) (Proof shortened
|
|
by Mario Carneiro, 6-Dec-2016.) $)
|
|
eqerlem $p |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A ) $=
|
|
( cv wceq wsbc csb cvv wcel wb vex nfcsb1v weq sbciegf brabsb nfeq csbief
|
|
wbr nfv nfcv csbeq1 syl5eqr eqeq2d ax-mp csbeq1a eqeq1d syl5bb bitri ) CJ
|
|
ZDJZGUDEFKZBUPLZAUOLZAUOEMZAUPEMZKZUQABUOUPGIUAUONOUSVBPCQURVBAUONAUTVAAU
|
|
OERAUPERUBUREVAKZACSZVBUPNOURVCPDQUQVCBUPNVCBUEBDSZFVAEVEFABJZEMVAAVFEFBQ
|
|
AFUFHUCAVFUPEUGUHUITUJVDEUTVAAUOEUKULUMTUJUN $.
|
|
|
|
$d v w x y z $. $d v w z R $.
|
|
$( Equivalence relation involving equality of dependent classes ` A ( x ) `
|
|
and ` B ( y ) ` . (Contributed by NM, 17-Mar-2008.) (Revised by Mario
|
|
Carneiro, 12-Aug-2015.) $)
|
|
eqer $p |- R Er _V $=
|
|
( vz vw vv cvv wtru wceq a1i cv wbr csb eqerlem 3imtr4i adantl wer eqcomd
|
|
wrel relopabi id wa eqtr anbi12i wcel wb vex eqid mpbir 2th iserd trud )
|
|
KEUALHIJKEEUCLCDMABEGUDNHOZIOZEPZURUQEPZLAUQCQZAURCQZMZVBVAMUSUTVCVAVBVCU
|
|
EUBABHICDEFGRZABIHCDEFGRSTUSURJOZEPZUFZUQVEEPZLVCVBAVECQZMZUFVAVIMVGVHVAV
|
|
BVIUGUSVCVFVJVDABIJCDEFGRUHABHJCDEFGRSTUQKUIZUQUQEPZUJLVKVLHUKVLVAVAMVAUL
|
|
ABHHCDEFGRUMUNNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The identity relation is an equivalence relation. (Contributed by NM,
|
|
10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof
|
|
shortened by Mario Carneiro, 9-Jul-2014.) $)
|
|
ider $p |- _I Er _V $=
|
|
( vx vy cv cid wceq id df-id eqer ) ABACZBCZDIJEFABGH $.
|
|
|
|
$( The empty set is an equivalence relation on the empty set. (Contributed
|
|
by Mario Carneiro, 5-Sep-2015.) $)
|
|
0er $p |- (/) Er (/) $=
|
|
( vx vy vz c0 wer wtru wrel rel0 a1i cv wbr wcel df-br noel pm2.21i sylbi
|
|
cop adantl ad2antrl wb 2false bitr4i iserd trud ) DDEFABCDDDGFHIAJZBJZDKZ
|
|
UFUEDKZFUGUEUFQZDLZUHUEUFDMZUJUHUINZOPRUGUECJZDKZFUFUMDKUGUJUNUKUJUNULOPS
|
|
UEDLZUEUEDKZTFUOUEUEQZDLZUPUOURUENUQNUAUEUEDMUBIUCUD $.
|
|
$}
|
|
|
|
$( Equality theorem for equivalence class. (Contributed by NM,
|
|
23-Jul-1995.) $)
|
|
eceq1 $p |- ( A = B -> [ A ] C = [ B ] C ) $=
|
|
( wceq csn cima cec sneq imaeq2d df-ec 3eqtr4g ) ABDZCAEZFCBEZFACGBCGLMNCAB
|
|
HIACJBCJK $.
|
|
|
|
${
|
|
eceq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for equivalence class (deduction form). (Contributed
|
|
by Jim Kingdon, 31-Dec-2019.) $)
|
|
eceq1d $p |- ( ph -> [ A ] C = [ B ] C ) $=
|
|
( wceq cec eceq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
$}
|
|
|
|
$( Equality theorem for equivalence class. (Contributed by NM,
|
|
23-Jul-1995.) $)
|
|
eceq2 $p |- ( A = B -> [ C ] A = [ C ] B ) $=
|
|
( wceq csn cima cec imaeq1 df-ec 3eqtr4g ) ABDACEZFBKFCAGCBGABKHCAICBIJ $.
|
|
|
|
$( Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|
|
(Contributed by Mario Carneiro, 9-Jul-2014.) $)
|
|
elecg $p |- ( ( A e. V /\ B e. W ) -> ( A e. [ B ] R <-> B R A ) ) $=
|
|
( wcel wa csn cima cop cec wbr elimasng ancoms df-ec eleq2i df-br 3bitr4g
|
|
wb ) ADFZBEFZGACBHIZFZBAJCFZABCKZFBACLUATUCUDSCBAEDMNUEUBABCOPBACQR $.
|
|
|
|
${
|
|
elec.1 $e |- A e. _V $.
|
|
elec.2 $e |- B e. _V $.
|
|
$( Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|
|
(Contributed by NM, 23-Jul-1995.) $)
|
|
elec $p |- ( A e. [ B ] R <-> B R A ) $=
|
|
( cvv wcel cec wbr wb elecg mp2an ) AFGBFGABCHGBACIJDEABCFFKL $.
|
|
$}
|
|
|
|
$( Membership in an equivalence class when ` R ` is a relation. (Contributed
|
|
by Mario Carneiro, 11-Sep-2015.) $)
|
|
relelec $p |- ( Rel R -> ( A e. [ B ] R <-> B R A ) ) $=
|
|
( wrel cec wcel wbr cvv wa ecexr jca adantl brrelex12 ancomd elecg pm5.21nd
|
|
elex ) CDZABCEZFZBACGZAHFZBHFZIZTUDRTUBUCASQABCJKLRUAIUCUBBACMNABCHHOP $.
|
|
|
|
${
|
|
ecss.1 $e |- ( ph -> R Er X ) $.
|
|
$( An equivalence class is a subset of the domain. (Contributed by NM,
|
|
6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
ecss $p |- ( ph -> [ A ] R C_ X ) $=
|
|
( crn cec csn cima df-ec imassrn eqsstri wer wceq errn syl syl5sseq ) ACF
|
|
ZBCGZDSCBHZIRBCJCTKLADCMRDNEDCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x R $. $d x A $.
|
|
$( A representative of an inhabited equivalence class belongs to the domain
|
|
of the equivalence relation. (Contributed by Jim Kingdon,
|
|
21-Aug-2019.) $)
|
|
ecdmn0m $p |- ( A e. dom R <-> E. x x e. [ A ] R ) $=
|
|
( cdm wcel cvv cv cec wex elex ecexr exlimiv wbr eldmg wb vex mpan exbidv
|
|
elecg bitr4d pm5.21nii ) BCDZEZBFEZAGZBCHEZAIZBUBJUFUDAUEBCKLUDUCBUECMZAI
|
|
UGABCFNUDUFUHAUEFEUDUFUHOAPUEBCFFSQRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d R x $. $d ph x $.
|
|
ereldm.1 $e |- ( ph -> R Er X ) $.
|
|
ereldm.2 $e |- ( ph -> [ A ] R = [ B ] R ) $.
|
|
$( Equality of equivalence classes implies equivalence of domain
|
|
membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario
|
|
Carneiro, 12-Aug-2015.) $)
|
|
ereldm $p |- ( ph -> ( A e. X <-> B e. X ) ) $=
|
|
( vx cdm wcel cv cec wex eleq2d exbidv ecdmn0m 3bitr4g wer wceq erdm syl
|
|
3bitr3d ) ABDIZJZCUCJZBEJCEJAHKZBDLZJZHMUFCDLZJZHMUDUEAUHUJHAUGUIUFGNOHBD
|
|
PHCDPQAUCEBAEDRUCESFEDTUAZNAUCECUKNUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x ph $.
|
|
erth.1 $e |- ( ph -> R Er X ) $.
|
|
erth.2 $e |- ( ph -> A e. X ) $.
|
|
$( Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82.
|
|
(Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro,
|
|
6-Jul-2015.) $)
|
|
erth $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
|
|
( vx wbr cec wa wcel ertr impl sylan impbida cvv wb adantr elecg cv simpl
|
|
wceq ersymb biimpa jca vex sylancr wrel wer errel syl 3bitr4d eqrdv erref
|
|
brrelex2 syl2anc mpbird simpr eleqtrd ereldm mpbid ersym ) ABCDIZBDJZCDJZ
|
|
UCZAVDKZHVEVFVHBHUAZDIZCVIDIZVIVELZVIVFLZVHVJVKVHACBDIZKVJVKVHAVNAVDUBAVD
|
|
VNABCDEFUDUEUFAVNVJVKACBVIDEFMNOAVDVKVJABCVIDEFMNPVHVIQLZBELZVLVJRHUGZAVP
|
|
VDGSVIBDQETUHVHVOCQLZVMVKRVQADUIZVDVRAEDUJZVSFEDUKULBCDUPOVICDQQTUHUMUNAV
|
|
GKZCBDEAVTVGFSZWABVFLZVNWABVEVFWABVELZBBDIZAWEVGABDEFGUOSWAVPVPWDWERAVPVG
|
|
GSZWFBBDEETUQURAVGUSZUTWAVPCELZWCVNRWFWAVPWHWFWABCDEWBWGVAVBBCDEETUQVBVCP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
erth2.1 $e |- ( ph -> R Er X ) $.
|
|
erth2.2 $e |- ( ph -> B e. X ) $.
|
|
$( Basic property of equivalence relations. Compare Theorem 73 of [Suppes]
|
|
p. 82. Assumes membership of the second argument in the domain.
|
|
(Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro,
|
|
6-Jul-2015.) $)
|
|
erth2 $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
|
|
( wbr cec wceq ersymb erth eqcom syl6bb bitrd ) ABCDHCBDHZBDIZCDIZJZABCDE
|
|
FKAPRQJSACBDEFGLRQMNO $.
|
|
$}
|
|
|
|
${
|
|
erthi.1 $e |- ( ph -> R Er X ) $.
|
|
erthi.2 $e |- ( ph -> A R B ) $.
|
|
$( Basic property of equivalence relations. Part of Lemma 3N of [Enderton]
|
|
p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro,
|
|
9-Jul-2014.) $)
|
|
erthi $p |- ( ph -> [ A ] R = [ B ] R ) $=
|
|
( wbr cec wceq ercl erth mpbid ) ABCDHBDICDIJGABCDEFABCDEFGKLM $.
|
|
$}
|
|
|
|
$( An equivalence class modulo the identity relation is a singleton.
|
|
(Contributed by NM, 24-Oct-2004.) $)
|
|
ecidsn $p |- [ A ] _I = { A } $=
|
|
( cid cec csn cima df-ec imai eqtri ) ABCBADZEIABFIGH $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) $)
|
|
qseq1 $p |- ( A = B -> ( A /. C ) = ( B /. C ) ) $=
|
|
( vy vx wceq cv cec wrex cab cqs rexeq abbidv df-qs 3eqtr4g ) ABFZDGEGCHF
|
|
ZEAIZDJQEBIZDJACKBCKPRSDQEABLMEDACNEDBCNO $.
|
|
|
|
$( Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) $)
|
|
qseq2 $p |- ( A = B -> ( C /. A ) = ( C /. B ) ) $=
|
|
( vy vx wceq cec wrex cab cqs eceq2 eqeq2d rexbidv abbidv df-qs 3eqtr4g
|
|
cv ) ABFZDQZEQZAGZFZECHZDISTBGZFZECHZDICAJCBJRUCUFDRUBUEECRUAUDSABTKLMNED
|
|
CAOEDCBOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $.
|
|
$( Closed form of ~ elqs . (Contributed by Rodolfo Medina,
|
|
12-Oct-2010.) $)
|
|
elqsg $p |- ( B e. V ->
|
|
( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) ) $=
|
|
( vy cv cec wceq wrex cqs eqeq1 rexbidv df-qs elab2g ) FGZAGDHZIZABJCQIZA
|
|
BJFCBDKEPCIRSABPCQLMAFBDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $.
|
|
elqs.1 $e |- B e. _V $.
|
|
$( Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) $)
|
|
elqs $p |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) $=
|
|
( cvv wcel cqs cv cec wceq wrex wb elqsg ax-mp ) CFGCBDHGCAIDJKABLMEABCDF
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $.
|
|
$( Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) $)
|
|
elqsi $p |- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R ) $=
|
|
( cqs wcel cv cec wceq wrex elqsg ibi ) CBDEZFCAGDHIABJABCDMKL $.
|
|
$}
|
|
|
|
${
|
|
$d R x $. $d B x $. $d A x $.
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
ecelqsg $p |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) ) $=
|
|
( vx wcel cec cv wceq wrex cqs eqid eceq1 eqeq2d rspcev mpan2 ecexg elqsg
|
|
cvv wb syl biimpar sylan2 ) BAFZCDFZBCGZEHZCGZIZEAJZUFACKFZUDUFUFIZUJUFLU
|
|
IULEBAUGBIUHUFUFUGBCMNOPUEUKUJUEUFSFUKUJTBDCQEAUFCSRUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
ecelqsi.1 $e |- R e. _V $.
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
ecelqsi $p |- ( B e. A -> [ B ] R e. ( A /. R ) ) $=
|
|
( cvv wcel cec cqs ecelqsg mpan ) CEFBAFBCGACHFDABCEIJ $.
|
|
$}
|
|
|
|
${
|
|
ecopqsi.1 $e |- R e. _V $.
|
|
ecopqsi.2 $e |- S = ( ( A X. A ) /. R ) $.
|
|
$( "Closure" law for equivalence class of ordered pairs. (Contributed by
|
|
NM, 25-Mar-1996.) $)
|
|
ecopqsi $p |- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S ) $=
|
|
( wcel wa cop cxp cec opelxpi cqs ecelqsi syl6eleqr syl ) BAHCAHIBCJZAAKZ
|
|
HZRDLZEHBCAAMTUASDNESRDFOGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by
|
|
Mario Carneiro, 9-Jul-2014.) $)
|
|
qsexg $p |- ( A e. V -> ( A /. R ) e. _V ) $=
|
|
( vy vx wcel cqs cv cec wceq wrex cab cvv df-qs abrexexg syl5eqel ) ACFAB
|
|
GDHEHBIZJEAKDLMEDABNEDAQCOP $.
|
|
$}
|
|
|
|
${
|
|
qsex.1 $e |- A e. _V $.
|
|
$( A quotient set exists. (Contributed by NM, 14-Aug-1995.) $)
|
|
qsex $p |- ( A /. R ) e. _V $=
|
|
( cvv wcel cqs qsexg ax-mp ) ADEABFDECABDGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $. $d x V $.
|
|
$( The union of a quotient set. (Contributed by NM, 9-Dec-2008.) $)
|
|
uniqs $p |- ( R e. V -> U. ( A /. R ) = ( R " A ) ) $=
|
|
( vy vx wcel cv cec wceq wrex cab cuni ciun cqs cima wral ecexg ralrimivw
|
|
cvv dfiun2g syl eqcomd df-qs unieqi csn df-ec a1i iuneq2i imaiun 3eqtr2ri
|
|
iunid imaeq2i 3eqtr4g ) BCFZDGEGZBHZIEAJDKZLZEAUPMZABNZLBAOZUNUSURUNUPSFZ
|
|
EAPUSURIUNVBEAUOCBQREDAUPSTUAUBUTUQEDABUCUDUSEABUOUEZOZMBEAVCMZOVAEAUPVDU
|
|
PVDIUOAFUOBUFUGUHEBAVCUIVEABEAUKULUJUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y ph $. $d x y R $.
|
|
qsss.1 $e |- ( ph -> R Er A ) $.
|
|
$( A quotient set is a set of subsets of the base set. (Contributed by
|
|
Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
qsss $p |- ( ph -> ( A /. R ) C_ ~P A ) $=
|
|
( vx vy cqs cpw cv wcel cec wceq wrex vex elqs wss sseq1 syl5ibrcom selpw
|
|
ecss syl6ibr rexlimdvw syl5bi ssrdv ) AEBCGZBHZEIZUEJUGFIZCKZLZFBMAUGUFJZ
|
|
FBUGCENOAUJUKFBAUJUGBPZUKAULUJUIBPAUHCBDTUGUIBQREBSUAUBUCUD $.
|
|
|
|
qsss.2 $e |- ( ph -> R e. V ) $.
|
|
$( The union of a quotient set. (Contributed by Mario Carneiro,
|
|
11-Jul-2014.) $)
|
|
uniqs2 $p |- ( ph -> U. ( A /. R ) = A ) $=
|
|
( cqs cuni crn cdm cima wcel wceq uniqs syl erdm imaeq2d eqtr4d imadmrn
|
|
wer syl6eq errn eqtrd ) ABCGHZCIZBAUDCCJZKZUEAUDCBKZUGACDLUDUHMFBCDNOAUFB
|
|
CABCTZUFBMEBCPOQRCSUAAUIUEBMEBCUBOUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
snec.1 $e |- A e. _V $.
|
|
$( The singleton of an equivalence class. (Contributed by NM,
|
|
29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
snec $p |- { [ A ] R } = ( { A } /. R ) $=
|
|
( vy vx cv cec wceq csn wrex cab cqs eceq1 eqeq2d rexsn abbii df-qs df-sn
|
|
3eqtr4ri ) DFZEFZBGZHZEAIZJZDKTABGZHZDKUDBLUFIUEUGDUCUGEACUAAHUBUFTUAABMN
|
|
OPEDUDBQDUFRS $.
|
|
$}
|
|
|
|
${
|
|
ecqs.1 $e |- R e. _V $.
|
|
$( Equivalence class in terms of quotient set. (Contributed by NM,
|
|
29-Jan-1999.) $)
|
|
ecqs $p |- [ A ] R = U. ( { A } /. R ) $=
|
|
( cec csn cima cqs cuni df-ec cvv wcel wceq uniqs ax-mp eqtr4i ) ABDBAEZF
|
|
ZPBGHZABIBJKRQLCPBJMNO $.
|
|
$}
|
|
|
|
${
|
|
$d y A $.
|
|
ecid.1 $e |- A e. _V $.
|
|
$( A set is equal to its converse epsilon coset. (Note: converse epsilon
|
|
is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.)
|
|
(Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
ecid $p |- [ A ] `' _E = A $=
|
|
( vy cep ccnv cec cv wcel wbr vex elec brcnv epelc 3bitri eqriv ) CADEZFZ
|
|
ACGZQHARPIRADIRAHRAPCJZBKARDBSLRABMNO $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y V $.
|
|
$( A set is equal to its converse epsilon coset. (Note: converse epsilon
|
|
is not an equivalence relation.) (Contributed by Jim Kingdon,
|
|
8-Jan-2020.) $)
|
|
ecidg $p |- ( A e. V -> [ A ] `' _E = A ) $=
|
|
( vy wcel cep ccnv cec cv wbr cvv wb elecg mpan brcnvg mpan2 epelg 3bitrd
|
|
vex eqrdv ) ABDZCAEFZGZATCHZUBDZAUCUAIZUCAEIZUCADUCJDZTUDUEKCRZUCAUAJBLMT
|
|
UGUEUFKUHAUCBJENOUCABPQS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A set is equal to its quotient set mod converse epsilon. (Note:
|
|
converse epsilon is not an equivalence relation.) (Contributed by NM,
|
|
13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
qsid $p |- ( A /. `' _E ) = A $=
|
|
( vy vx cep ccnv cqs cv cec wceq wrex wcel vex eqeq2i equcom bitri rexbii
|
|
ecid elqs risset 3bitr4i eqriv ) BADEZFZABGZCGZUBHZIZCAJUEUDIZCAJUDUCKUDA
|
|
KUGUHCAUGUDUEIUHUFUEUDUECLQMBCNOPCAUDUBBLRCUDASTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x ps $. $d x ch $.
|
|
ectocl.1 $e |- S = ( B /. R ) $.
|
|
ectocl.2 $e |- ( [ x ] R = A -> ( ph <-> ps ) ) $.
|
|
${
|
|
ectocld.3 $e |- ( ( ch /\ x e. B ) -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class. (Contributed by
|
|
Mario Carneiro, 9-Jul-2014.) $)
|
|
ectocld $p |- ( ( ch /\ A e. S ) -> ps ) $=
|
|
( wcel cv cec wceq wrex cqs elqsi eleq2s wa wb syl5ibcom rexlimdva syl5
|
|
eqcoms imp ) CEHLZBUGEDMZGNZOZDFPZCBUKEFGQHDFEGRISCUJBDFCUHFLTAUJBKABUA
|
|
UIEJUEUBUCUDUF $.
|
|
$}
|
|
|
|
ectocl.3 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class. (Contributed by
|
|
NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
ectocl $p |- ( A e. S -> ps ) $=
|
|
( wtru wcel tru cv adantl ectocld mpan ) KDGLBMABKCDEFGHICNELAKJOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y R $. $d x y A $. $d x y B $.
|
|
$( An element of a quotient set is inhabited. (Contributed by Jim Kingdon,
|
|
21-Aug-2019.) $)
|
|
elqsn0m $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B ) $=
|
|
( vy cv cec wcel wex cdm wceq cqs eqid eleq2 exbidv biimpar ecdmn0m sylib
|
|
wa ectocld ) AFZEFZDGZHZAIZUACHZAIDJZBKZECBDBDLZUIMUCCKUDUFAUCCUANOUHUBBH
|
|
ZSUBUGHZUEUHUKUJUGBUBNPAUBDQRT $.
|
|
|
|
$( A quotient set doesn't contain the empty set. (Contributed by NM,
|
|
24-Aug-1995.) $)
|
|
elqsn0 $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) ) $=
|
|
( vx cdm wceq cqs wcel wa cv wex c0 wne elqsn0m n0r syl ) CEAFBACGHIDJBHD
|
|
KBLMDABCNDBOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d R x $.
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
NM, 30-Jul-1995.) $)
|
|
ecelqsdm $p |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A ) $=
|
|
( vx cdm wceq cec cqs wcel wa cv wex elqsn0m ecdmn0m sylibr simpl eleqtrd
|
|
) CEZAFZBCGZACHIZJZBRAUBDKTIDLBRIDATCMDBCNOSUAPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A square Cartesian product is an equivalence relation (in general it's
|
|
not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.) $)
|
|
xpiderm $p |- ( E. x x e. A -> ( A X. A ) Er A ) $=
|
|
( wcel wex cxp wrel cdm wceq ccnv ccom cun wss wer relxp a1i dmxpm xpidtr
|
|
cv cnvxp uneq1 unss2 wi unidm wa eqtr sseq2 biimpd syl mpan2 syl2im df-er
|
|
mp2 syl3anbrc ) ARBCADZBBEZFZUOGBHUOIZUOUOJZKZUOLZBUOMUPUNBBNOABBPUTUNUQU
|
|
OHZURUOLZUTBBSBQVAUQUOKZUOUOKZHZVBUSVCLZUTUQUOUOTURUOUQUAVEVDUOHZVFUTUBZU
|
|
OUCVEVGUDVCUOHZVHVCVDUOUEVIVFUTVCUOUSUFUGUHUIUJULOBUOUKUM $.
|
|
$}
|
|
|
|
${
|
|
$d a u v w x A $. $d u v w x B $. $d u v w R $. $d a y A $.
|
|
$( The intersection of a nonempty family of equivalence relations is an
|
|
equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) $)
|
|
iinerm $p |- ( ( E. y y e. A /\ A. x e. A R Er B ) ->
|
|
|^|_ x e. A R Er B ) $=
|
|
( va vu cv wcel wral bitri wa cvv wbr cop df-br wb opex eliin ax-mp vv vw
|
|
wex wer ciin eleq1 cbvexv cxp wrel r19.2m errel df-rel sylib reximi iinss
|
|
wss wrex 3syl sylibr ersymb biimpd 3imtr3g ral2imi adantl vex 3imtr4g imp
|
|
wi r19.26 ertr anbi12i syl5bir simpl simpr erref expcom ralimdv com12 cdm
|
|
id opeldm erdm eleq2d biimpa sylan2 rexlimivw syl ex expdimp impbid iserd
|
|
syl6bbr sylanbr ) BHZCIZBUCZAHZCIZAUCZDEUDZACJZDACEUEZUDWSFHZCIZFUCWPWRXD
|
|
AFWQXCCUFUGXDWOFBXCWNCUFUGKWSXALZGUAUBDXBXEXBMMUHZUPZXBUIXEWTACUQEXFUPZAC
|
|
UQXGWTACUJWTXHACWTEUIXHDEUKEULUMUNACEXFUOURXBULUSXEGHZUAHZXBNZXJXIXBNZXEX
|
|
IXJOZEIZACJZXJXIOZEIZACJZXKXLXAXOXRVHWSWTXNXQACWTXIXJENZXJXIENZXNXQWTXSXT
|
|
WTXIXJEDWTVTZUTVAXIXJEPZXJXIEPVBVCVDXKXMXBIZXOXIXJXBPXMMIYCXOQXIXJGVEZUAV
|
|
EZRAXMCEMSTKZXLXPXBIZXRXJXIXBPXPMIYGXRQXJXIYEYDRAXPCEMSTKVFVGXEXKXJUBHZXB
|
|
NZLZXIYHXBNZXEXOXJYHOZEIZACJZLZXIYHOZEIZACJZYJYKYOXNYMLZACJZXEYRXNYMACVIX
|
|
AYTYRVHWSWTYSYQACWTXSXJYHENZLXIYHENYSYQWTXIXJYHEDYAVJXSXNUUAYMYBXJYHEPVKX
|
|
IYHEPVBVCVDVLXKXOYIYNYFYIYLXBIZYNXJYHXBPYLMIUUBYNQXJYHYEUBVEZRAYLCEMSTKVK
|
|
YKYPXBIZYRXIYHXBPYPMIUUDYRQXIYHYDUUCRAYPCEMSTKVFVGXEXIDIZXIXIOZEIZACJZXIX
|
|
IXBNZXEUUEUUHXAUUEUUHVHWSUUEXAUUHUUEWTUUGACWTUUEUUGWTUUELZXIXIENUUGUUJXIE
|
|
DWTUUEVMWTUUEVNVOXIXIEPUMVPVQVRVDWSXAUUHUUEXAUUHLWTUUGLZACJZWSUUEWTUUGACV
|
|
IWSUULUUEWSUULLUUKACUQUUEUUKACUJUUKUUEACUUGWTXIEVSZIZUUEXIXIEYDYDWAWTUUNU
|
|
UEWTUUMDXIDEWBWCWDWEWFWGWHVLWIWJUUIUUFXBIZUUHXIXIXBPUUFMIUUOUUHQXIXIYDYDR
|
|
AUUFCEMSTKWLWKWM $.
|
|
|
|
$( The relative intersection of a family of equivalence relations is an
|
|
equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) $)
|
|
riinerm $p |- ( ( E. y y e. A /\ A. x e. A R Er B ) ->
|
|
( ( B X. B ) i^i |^|_ x e. A R ) Er B ) $=
|
|
( va cv wcel wex wer wral wa cxp ciin cin iinerm wb eleq1 cbvexv bitri
|
|
wceq wss erssxp ralimi riinm sylan ereq1 syl sylan2br ancoms mpbird ) BGZ
|
|
CHZBIZDEJZACKZLDDDMZACENZOZJZDURJZABCDEPUPUNUTVAQZUNUPAGZCHZAIZVBVEFGZCHZ
|
|
FIUNVDVGAFVCVFCRSVGUMFBVFULCRSTUPVELUSURUAZVBUPEUQUBZACKVEVHUOVIACDEUCUDA
|
|
UQECUEUFDUSURUGUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z B $. $d x y z ph $. $d x y z R $.
|
|
erinxp.r $e |- ( ph -> R Er A ) $.
|
|
erinxp.a $e |- ( ph -> B C_ A ) $.
|
|
$( A restricted equivalence relation is an equivalence relation.
|
|
(Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario
|
|
Carneiro, 12-Aug-2015.) $)
|
|
erinxp $p |- ( ph -> ( R i^i ( B X. B ) ) Er B ) $=
|
|
( vx vy vz wrel cv wbr wa wcel w3a brinxp2 sylib simp2d adantr simp3d cxp
|
|
cin wss inss2 relxp relss mp2 a1i simpr simp1d wer ersym syl3anbrc simprr
|
|
adantrr ertrd sselda erref ex pm4.71rd brin brxp anidm bitri anbi2i iserd
|
|
syl6bbr ) AGHICDCCUAZUBZVIJZAVIVHUCVHJVJDVHUDCCUEVIVHUFUGUHAGKZHKZVILZMZV
|
|
LCNZVKCNZVLVKDLVLVKVILVNVPVOVKVLDLZVNVMVPVOVQOAVMUIVKVLCCDPQZRVNVPVOVQVRU
|
|
JZVNVKVLDBABDUKZVMESVNVPVOVQVRTZULVLVKCCDPUMAVMVLIKZVILZMZMZVPWBCNZVKWBDL
|
|
VKWBVILAVMVPWCVSUOWEVOWFVLWBDLZWEWCVOWFWGOAVMWCUNVLWBCCDPQZRWEVKVLWBDBAVT
|
|
WDESAVMVQWCWAUOWEVOWFWGWHTUPVKWBCCDPUMAVPVKVKDLZVPMZVKVKVILZAVPWIAVPWIAVP
|
|
MVKDBAVTVPESACBVKFUQURUSUTWKWIVKVKVHLZMWJVKVKDVHVAWLVPWIWLVPVPMVPVKVKCCVB
|
|
VPVCVDVEVDVGVF $.
|
|
$}
|
|
|
|
$( Restrict the relation in an equivalence class to a base set. (Contributed
|
|
by Mario Carneiro, 10-Jul-2015.) $)
|
|
ecinxp $p |- ( ( ( R " A ) C_ A /\ B e. A ) ->
|
|
[ B ] R = [ B ] ( R i^i ( A X. A ) ) ) $=
|
|
( cima wss wcel csn cxp cin cec wceq simpr snssd df-ss sylib imaeq2d ineq1d
|
|
wa imass2 df-ec syl simpl sstrd eqtr2d imainrect syl6eqr 3eqtr4g ) CADZAEZB
|
|
AFZRZCBGZDZCAAHIZULDZBCJBUNJUKUMCULAIZDZAIZUOUKURUMAIZUMUKUQUMAUKUPULCUKULA
|
|
EZUPULKUKBAUIUJLMZULANOPQUKUMAEUSUMKUKUMUHAUKUTUMUHEVAULACSUAUIUJUBUCUMANOU
|
|
DAACULUEUFBCTBUNTUG $.
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Restrict the equivalence relation in a quotient set to the base set.
|
|
(Contributed by Mario Carneiro, 23-Feb-2015.) $)
|
|
qsinxp $p |- ( ( R " A ) C_ A ->
|
|
( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) ) $=
|
|
( vy vx cima wss cv cec wceq wrex cab cxp cin cqs wcel wa ecinxp rexbidva
|
|
eqeq2d df-qs abbidv 3eqtr4g ) BAEAFZCGZDGZBHZIZDAJZCKUDUEBAALMZHZIZDAJZCK
|
|
ABNAUINUCUHULCUCUGUKDAUCUEAOPUFUJUDAUEBQSRUADCABTDCAUITUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x R $. $d x X $.
|
|
$( If an element of a quotient set contains a given element, it is equal to
|
|
the equivalence class of the element. (Contributed by Mario Carneiro,
|
|
12-Aug-2015.) $)
|
|
qsel $p |- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R ) $=
|
|
( vx wer cqs wcel cec wceq cv wi eqid eleq2 eqeq1 imbi12d wbr wa cvv syl5
|
|
wb vex elecg mpan2 ibi simpll simpr erthi ex ectocld 3impia ) EDGZBADHZIC
|
|
BIZBCDJZKZCFLZDJZIZUSUPKZMUOUQMUMFBADUNUNNUSBKUTUOVAUQUSBCOUSBUPPQUTURCDR
|
|
ZUMURAIZSZVAUTVBUTURTIUTVBUBFUCCURDUSTUDUEUFVDVBVAVDVBSURCDEUMVCVBUGVDVBU
|
|
HUIUJUAUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x B $. $d x C $. $d x D $. $d x y z ph $. $d x y z R $.
|
|
$d y z F $. $d x y z X $. $d x y z Y $.
|
|
qlift.1 $e |- F = ran ( x e. X |-> <. [ x ] R , A >. ) $.
|
|
qlift.2 $e |- ( ( ph /\ x e. X ) -> A e. Y ) $.
|
|
qlift.3 $e |- ( ph -> R Er X ) $.
|
|
qlift.4 $e |- ( ph -> X e. _V ) $.
|
|
$( ` F ` , a function lift, is a subset of ` R X. S ` . (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
qliftlem $p |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) ) $=
|
|
( cvv wcel cv cec cqs wer erex sylc ecelqsg sylan ) ADLMZBNZFMUCDOFDPMAFD
|
|
QFLMUBJKFDLRSFUCDLTUA $.
|
|
|
|
$( ` F ` , a function lift, is a subset of ` R X. S ` . (Contributed by
|
|
Mario Carneiro, 23-Dec-2016.) $)
|
|
qliftrel $p |- ( ph -> F C_ ( ( X /. R ) X. Y ) ) $=
|
|
( cv cec cqs qliftlem fliftrel ) ABBLDMCFDNGEFHABCDEFGHIJKOIP $.
|
|
|
|
$( Elementhood in the relation ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
qliftel $p |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) ) $=
|
|
( cec wbr cv wceq wa wrex cqs qliftlem wcel wer adantr simpr erth2 anbi1d
|
|
fliftel rexbidva bitr4d ) ADFNZEGOUKBPZFNZQZECQZRZBHSDULFOZUORZBHSABUMCUK
|
|
EHFTIGHJABCFGHIJKLMUAKUHAURUPBHAULHUBZRZUQUNUOUTDULFHAHFUCUSLUDAUSUEUFUGU
|
|
IUJ $.
|
|
|
|
$( Elementhood in the relation ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
qliftel1 $p |- ( ( ph /\ x e. X ) -> [ x ] R F A ) $=
|
|
( cv cec cqs qliftlem fliftel1 ) ABBLDMCFDNGEFHABCDEFGHIJKOIP $.
|
|
|
|
${
|
|
qliftfun.4 $e |- ( x = y -> A = B ) $.
|
|
$( The function ` F ` is the unique function defined by
|
|
` F `` [ x ] = A ` , provided that the well-definedness condition
|
|
holds. (Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
qliftfun $p |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) ) $=
|
|
( cv cec wceq wi wal wa wfun wral wbr cqs qliftlem fliftfun wcel adantr
|
|
eceq1 wer simpr ercl ercl2 jca ex pm4.71rd simprl pm5.32da bitrd imbi1d
|
|
erth impexp syl6bb 2albidv r2al syl6bbr bitr4d ) AGUABOZFPZCOZFPZQZDEQZ
|
|
RZCHUBBHUBZVHVJFUCZVMRZCSBSZABCVIDVKEHFUDIGHJABDFGHIJKLMUEKVHVJFUINUFAV
|
|
RVHHUGZVJHUGZTZVNRZCSBSVOAVQWBBCAVQWAVLTZVMRWBAVPWCVMAVPWAVPTWCAVPWAAVP
|
|
WAAVPTZVSVTWDVHVJFHAHFUJZVPLUHZAVPUKZULWDVHVJFHWFWGUMUNUOUPAWAVPVLAWATV
|
|
HVJFHAWEWALUHAVSVTUQVAURUSUTWAVLVMVBVCVDVNBCHHVEVFVG $.
|
|
|
|
qliftfund.6 $e |- ( ( ph /\ x R y ) -> A = B ) $.
|
|
$( The function ` F ` is the unique function defined by
|
|
` F `` [ x ] = A ` , provided that the well-definedness condition
|
|
holds. (Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
qliftfund $p |- ( ph -> Fun F ) $=
|
|
( wfun cv wbr wceq wal wi ex alrimivv qliftfun mpbird ) AGPBQCQFRZDESZU
|
|
AZCTBTAUHBCAUFUGOUBUCABCDEFGHIJKLMNUDUE $.
|
|
$}
|
|
|
|
$( The function ` F ` is the unique function defined by
|
|
` F `` [ x ] = A ` , provided that the well-definedness condition holds.
|
|
(Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
qliftfuns $p |- ( ph -> ( Fun F <->
|
|
A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) ) $=
|
|
( cv csb cec cop cmpt crn wcel nfcv nfcsb1v nfop weq eceq1 csbeq1a cbvmpt
|
|
opeq12d rneqi eqtri wral ralrimiva nfel1 eleq1d mpan9 csbeq1 qliftfun
|
|
rspc ) ACDBCNZEOZBDNZEOFGHIGBHBNZFPZEQZRZSCHUSFPZUTQZRZSJVEVHBCHVDVGCVDUA
|
|
BVFUTBVFUABUSEUBZUCBCUDZVCVFEUTVBUSFUEBUSEUFZUHUGUIUJAEITZBHUKUSHTUTITZAV
|
|
LBHKULVLVMBUSHBUTIVIUMVJEUTIVKUNURUOLMBUSVAEUPUQ $.
|
|
|
|
$( The domain and range of the function ` F ` . (Contributed by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
qliftf $p |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) ) $=
|
|
( vy wfun cv cec cmpt crn wf cqs wceq qliftlem fliftf wrex cab df-qs eqid
|
|
rnmpt eqtr4i a1i feq2d bitr4d ) AEMBFBNDOZPZQZGERFDSZGERABULCUOGEFHABCDEF
|
|
GHIJKUAIUBAUOUNGEUOUNTAUOLNULTBFUCLUDUNBLFDUEBLFULUMUMUFUGUHUIUJUK $.
|
|
|
|
qliftval.4 $e |- ( x = C -> A = B ) $.
|
|
qliftval.6 $e |- ( ph -> Fun F ) $.
|
|
$( The value of the function ` F ` . (Contributed by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
qliftval $p |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B ) $=
|
|
( cv cec cqs qliftlem eceq1 fliftval ) ABBPZFQCEFQDHFRIGHEJABCFGHIJKLMSKU
|
|
BEFTNOUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z ps $.
|
|
ecoptocl.1 $e |- S = ( ( B X. C ) /. R ) $.
|
|
ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
ecoptocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class of ordered pair.
|
|
(Contributed by NM, 23-Jul-1995.) $)
|
|
ecoptocl $p |- ( A e. S -> ps ) $=
|
|
( vz cxp cqs wcel cv cec wceq wi wrex elqsi cop eceq1 eqeq2d imbi1d wa wb
|
|
eqid eqcoms syl5ibcom optocl rexlimiv syl eleq2s ) BEFGNZHOZIEUQPEMQZHRZS
|
|
ZMUPUABMUPEHUBUTBMUPECQZDQZUCZHRZSZBTUTBTCDURFGUPUPUIVCURSZVEUTBVFVDUSEVC
|
|
URHUDUEUFVAFPVBGPUGAVEBLABUHVDEKUJUKULUMUNJUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w C $. $d x y z w D $. $d z w S $.
|
|
$d x y z w R $. $d x y ps $. $d z w ch $.
|
|
2ecoptocl.1 $e |- S = ( ( C X. D ) /. R ) $.
|
|
2ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
2ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
|
|
2ecoptocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\
|
|
( z e. C /\ w e. D ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for equivalence classes of ordered
|
|
pairs. (Contributed by NM, 23-Jul-1995.) $)
|
|
2ecoptocl $p |- ( ( A e. S /\ B e. S ) -> ch ) $=
|
|
( wcel wi cv cop cec wceq imbi2d wa ex ecoptocl com12 impcom ) IMRHMRZCUJ
|
|
BSUJCSFGIJKLMNFTZGTZUALUBIUCBCUJPUDUJUKJRULKRUEZBUMASUMBSDEHJKLMNDTZETZUA
|
|
LUBHUCABUMOUDUNJRUOKRUEUMAQUFUGUHUGUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d v u C $. $d x y z w v u D $.
|
|
$d z w v u S $. $d x y z w v u R $. $d x y ps $. $d z w ch $.
|
|
$d v u th $.
|
|
3ecoptocl.1 $e |- S = ( ( D X. D ) /. R ) $.
|
|
3ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
3ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
|
|
3ecoptocl.4 $e |- ( [ <. v , u >. ] R = C -> ( ch <-> th ) ) $.
|
|
3ecoptocl.5 $e |- ( ( ( x e. D /\ y e. D ) /\
|
|
( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for equivalence classes of ordered
|
|
pairs. (Contributed by NM, 9-Aug-1995.) $)
|
|
3ecoptocl $p |- ( ( A e. S /\ B e. S /\ C e. S ) -> th ) $=
|
|
( wcel wa wi cop cec wceq imbi2d 3expib ecoptocl com12 2ecoptocl 3impib
|
|
cv ) KPUBZLPUBZMPUBZDUPUQUCUODUOBUDUOCUDUODUDGHIJLMNNOPQGUNZHUNZUEOUFLUGB
|
|
CUOSUHIUNZJUNZUEOUFMUGCDUOTUHUOURNUBUSNUBUCZUTNUBVANUBUCZUCZBVDAUDVDBUDEF
|
|
KNNOPQEUNZFUNZUEOUFKUGABVDRUHVENUBVFNUBUCVBVCAUAUIUJUKULUKUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $. $d x y z w v u .~ $. $d x y H $. $d z w v u G $.
|
|
$d x y ph $. $d z w v u ps $.
|
|
brecop.1 $e |- .~ e. _V $.
|
|
brecop.2 $e |- .~ Er ( G X. G ) $.
|
|
brecop.4 $e |- H = ( ( G X. G ) /. .~ ) $.
|
|
brecop.5 $e |- .<_ = { <. x , y >. | ( ( x e. H /\ y e. H ) /\
|
|
E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\
|
|
y = [ <. v , u >. ] .~ ) /\ ph ) ) } $.
|
|
brecop.6 $e |- ( ( ( ( z e. G /\ w e. G ) /\ ( A e. G /\ B e. G ) ) /\
|
|
( ( v e. G /\ u e. G ) /\ ( C e. G /\ D e. G ) ) ) ->
|
|
( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\
|
|
[ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) ->
|
|
( ph <-> ps ) ) ) $.
|
|
$( Binary relation on a quotient set. Lemma for real number construction.
|
|
(Contributed by NM, 29-Jan-1996.) $)
|
|
brecop $p |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) ->
|
|
( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> ps ) ) $=
|
|
( wcel wa cop cec wbr cv wceq wex ecopqsi copab df-br eleq2i bitri anbi1d
|
|
wb eqeq1 4exbidv anbi2d opelopab2 syl5bb syl2an opeq12 eceq1d anim12i cxp
|
|
wi opelxpi opelxp wer a1i id ereldm syl5bbr syl5ibr im2anan9 an4s ex mpdd
|
|
com13 pm5.74d cgsex4g eqcom anbi12i biimt anbi12d 3bitr4d bitrd ) INUBJNU
|
|
BUCZKNUBLNUBUCZUCZIJUDZMUEZKLUDZMUEZPUFZWMEUGZFUGZUDZMUEZUHZWOGUGZHUGZUDZ
|
|
MUEZUHZUCZAUCZHUIGUIFUIEUIZBWIWMOUBZWOOUBZWPXIUPWJNIJMOQSUJNKLMOQSUJWPWMW
|
|
OUDZCUGZOUBDUGZOUBUCXMWTUHZXNXEUHZUCZAUCZHUIGUIFUIEUIZUCCDUKZUBZXJXKUCXIW
|
|
PXLPUBYAWMWOPULPXTXLTUMUNXSXAXPUCZAUCZHUIGUIFUIEUIXICDWMWOOOXMWMUHZXRYCEF
|
|
GHYDXQYBAYDXOXAXPXMWMWTUQUOUOURXNWOUHZYCXHEFGHYEYBXGAYEXPXFXAXNWOXEUQUSUO
|
|
URUTVAVBWKWTWMUHZXEWOUHZUCZWKAVGZUCZHUIGUIFUIEUIWKBVGZXIBYIYKYHEFGHIJKLNN
|
|
WQIUHWRJUHUCZYFXBKUHXCLUHUCZYGYLWSWLMWQWRIJVCVDYMXDWNMXBXCKLVCVDVEYHWKABY
|
|
HWKWQNUBWRNUBUCZXBNUBXCNUBUCZUCZABUPZYFWIYNYGWJYOWIYNYFWLNNVFZUBZIJNNVHYN
|
|
WSYRUBYFYSWQWRNNVIYFWSWLMYRYRMVJZYFRVKYFVLVMVNVOWJYOYGWNYRUBZKLNNVHYOXDYR
|
|
UBYGUUAXBXCNNVIYGXDWNMYRYTYGRVKYGVLVMVNVOVPYPWKYHYQYPWKYHYQVGZYNWIYOWJUUB
|
|
UAVQVRVTVSWAWBWKXHYJEFGHWKXGYHAYIXGYHUPWKXAYFXFYGWMWTWCWOXEWCWDVKWKAWEWFU
|
|
RWKBWEWGWH $.
|
|
$}
|
|
|
|
${
|
|
$d p q r s t u w x y z A $. $d p q r s t u w x y z B $. $d p q x y z L $.
|
|
$d p q w x y z J $. $d p q r s t u w x y z R $. $d p q w x y z K $.
|
|
$d p q r s t u w x y z S $. $d p q r s t u w x y z .+ $.
|
|
$d p q r s t u w x y z ph $. $d p q r s t u w x y z T $.
|
|
$d p q r s t u w z X $. $d p q r s t u w z Y $.
|
|
eropr.1 $e |- J = ( A /. R ) $.
|
|
eropr.2 $e |- K = ( B /. S ) $.
|
|
eropr.3 $e |- ( ph -> T e. Z ) $.
|
|
eropr.4 $e |- ( ph -> R Er U ) $.
|
|
eropr.5 $e |- ( ph -> S Er V ) $.
|
|
eropr.6 $e |- ( ph -> T Er W ) $.
|
|
eropr.7 $e |- ( ph -> A C_ U ) $.
|
|
eropr.8 $e |- ( ph -> B C_ V ) $.
|
|
eropr.9 $e |- ( ph -> C C_ W ) $.
|
|
eropr.10 $e |- ( ph -> .+ : ( A X. B ) --> C ) $.
|
|
eropr.11 $e |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) )
|
|
-> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) ) $.
|
|
$( Lemma for ~ eroprf . (Contributed by Jeff Madsen, 10-Jun-2010.)
|
|
(Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
eroveu $p |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E! z E. p e. A E. q e. B
|
|
( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) $=
|
|
( vw wcel wa cv cec wceq co wrex wex weq wal weu cqs elqsi eleq2s anim12i
|
|
wi adantl reeanv sylibr adantr ecexg elisset 3syl biantrud 2rexbidv mpbid
|
|
19.42v bicomi rexbii rexcom4 bitri sylib eceq1 eqeq2d anbi1d oveq1 eceq1d
|
|
cvv anbi12d anbi2d oveq2 cbvrex2v anbi12i bitr4i wbr wer wss simprll erth
|
|
sseldd simprrl cxp wf fovrnd 3imtr3d eqeq2 biimprcd syl6 impd wb bi2anan9
|
|
eqeq1 imbi12d syl5ibrcom anassrs rexlimdvva syl5bir alrimivv eu4 sylanbrc
|
|
) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQZUBF
|
|
VBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYIYPYT
|
|
BVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUSAYFU
|
|
UQYGUURUUQQEIVGMUCEQIVHUDVIUURRFJVGNUBFRJVHUEVIVJVLYLYOUCUBEFVMVNYIYPUUMU
|
|
CUBEFYIUULYPYIKSUPZYSWMUPUULAUUTYHUFVOYRSKVPBYSWMVQVRVSVTWAUUOUUBBVCZUCEV
|
|
BUUDUUNUVAUCEUUNUUABVCZUBFVBUVAUUMUVBUBFUVBUUMYPYTBWBWCWDUUAUBBFWEWFWDUUB
|
|
UCBEWEWFWGYIUUKBUOAUUKYHUUIQUAURZIUSZUTZRDURZJUSZUTZUQZYQUVCUVFHVAZKUSZUT
|
|
ZUQZDFVBZQTURZIUSZUTZRCURZJUSZUTZUQZUUEUVOUVRHVAZKUSZUTZUQZCFVBZUQZTEVBUA
|
|
EVBZAUUJUWHUVNUAEVBZUWFTEVBZUQUUIUVNUWFUATEEVMUUCUWIUUHUWJUUAUVMUVEYOUQZY
|
|
QUVCYMHVAZKUSZUTZUQUCUBUADEFUCUAVDZYPUWKYTUWNUWOYLUVEYOUWOYKUVDQYJUVCIWHW
|
|
IWJUWOYSUWMYQUWOYRUWLKYJUVCYMHWKWLWIWNUBDVDZUWKUVIUWNUVLUWPYOUVHUVEUWPYNU
|
|
VGRYMUVFJWHWIWOUWPUWMUVKYQUWPUWLUVJKYMUVFUVCHWPWLWIWNWQUUGUWEUVQYOUQZUUEU
|
|
VOYMHVAZKUSZUTZUQUCUBTCEFUCTVDZYPUWQUUFUWTUXAYLUVQYOUXAYKUVPQYJUVOIWHWIWJ
|
|
UXAYSUWSUUEUXAYRUWRKYJUVOYMHWKWLWIWNUBCVDZUWQUWAUWTUWDUXBYOUVTUVQUXBYNUVS
|
|
RYMUVRJWHWIWOUXBUWSUWCUUEUXBUWRUWBKYMUVRUVOHWPWLWIWNWQWRWSAUWGUUJUATEEUWG
|
|
UVMUWEUQZCFVBDFVBAUVCEUPZUVOEUPZUQZUQZUUJUVMUWEDCFFVMUXGUXCUUJDCFFAUXFUVF
|
|
FUPZUVRFUPZUQZUXCUUJVKAUXFUXJUQZUQZUVMUWEUUJUXLUWEUUJVKUVMUVDUVPUTZUVGUVS
|
|
UTZUQZUWDUQZUVKUUEUTZVKUXLUXOUWDUXQUXLUXOUVKUWCUTZUWDUXQVKUXLUVCUVOIWTZUV
|
|
FUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLELUVC
|
|
AELXBUXKUJVOAUXDUXEUXJXCZXEXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKUKVOA
|
|
UXFUXHUXIXFZXEXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUVFGEF
|
|
HAEFXGGHXHUXKUMVOUYAUYBXIXEXDXJUWDUXQUXRUUEUWCUVKXKXLXMXNUVMUWEUXPUUJUXQU
|
|
VIUWEUXPXOUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXQRUVGUVSXQXPWJVOUVLUUJ
|
|
UXQXOUVIYQUVKUUEXQVLXRXSXNXTYAYBYAYBVOYCUUCUUHBUOUUJUUAUUGUCUBEFUUJYTUUFY
|
|
PYQUUEYSXQWOVTYDYE $.
|
|
|
|
eropr.12 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B
|
|
( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) } $.
|
|
$( Lemma for ~ eroprf . (Contributed by Jeff Madsen, 10-Jun-2010.)
|
|
(Revised by Mario Carneiro, 30-Dec-2014.) $)
|
|
erovlem $p |- ( ph -> .+^ = ( x e. J , y e. K |->
|
|
( iota z E. p e. A E. q e. B
|
|
( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) ) $=
|
|
( vw cv cec wceq wa co wrex coprab wcel cio cmpt2 simpl reximi cqs eleq2i
|
|
vex elqs bitri anbi12i reeanv bitr4i sylibr pm4.71ri weu eroveu iota1 syl
|
|
wb eqcom syl6bb pm5.32da syl5bb oprabbidv df-mpt2 nfv nfiota1 nfeq2 eqeq1
|
|
nfan anbi2d cbvoprab3 eqtr4i 3eqtr4g ) ABURZUDURZLUSUTZCURZUCURZMUSUTZVAZ
|
|
DURZXAXDJVBNUSUTZVAZUCHVCZUDGVCZBCDVDWTPVEZXCQVEZVAZXGXKDVFZUTZVAZBCDVDZK
|
|
BCPQXOVGZAXKXQBCDXKXNXKVAAXQXKXNXKXFUCHVCZUDGVCZXNXJXTUDGXIXFUCHXFXHVHVIV
|
|
IXNXBUDGVCZXEUCHVCZVAYAXLYBXMYCXLWTGLVJZVEYBPYDWTUEVKUDGWTLBVLVMVNXMXCHMV
|
|
JZVEYCQYEXCUFVKUCHXCMCVLVMVNVOXBXEUDUCGHVPVQVRVSAXNXKXPAXNVAZXKXOXGUTZXPY
|
|
FXKDVTXKYGWDADEFGHIJLMNOPQRSWTXCTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOWAXKDWBWCX
|
|
OXGWEWFWGWHWIUPXSXNUQURZXOUTZVAZBCUQVDXRBCUQPQXOWJXQYJBCDUQXQUQWKXNYIDXND
|
|
WKDYHXOXKDWLWMWOXGYHUTXPYIXNXGYHXOWNWPWQWRWS $.
|
|
|
|
eropr.13 $e |- ( ph -> R e. X ) $.
|
|
eropr.14 $e |- ( ph -> S e. Y ) $.
|
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eropr.15 $e |- L = ( C /. T ) $.
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$( Functionality of an operation defined on equivalence classes.
|
|
(Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro,
|
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30-Dec-2014.) $)
|
|
eroprf $p |- ( ph -> .+^ : ( J X. K ) --> L ) $=
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|
( cxp wf cv cec wceq wa co wrex cio cmpt2 wcel cab wi cqs ad2antrr adantr
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fovrnda ecelqsg syl2anc syl6eleqr eleq1a adantld rexlimdvva abssdv eroveu
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wral syl iotacl sseldd ralrimivva eqid fmpt2 sylib erovlem feq1d mpbird
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weu ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZ
|
|
VHZUFHVJUGGVJZDVKZVLZVDZAXLRVMZCQWHBPWHXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVNZRX
|
|
LXQXKDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVOYAXHINVPZRYAN
|
|
UCVMZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRVSIXGNUCVTWAVBWBXHRXF
|
|
WCWIWDWEWFXQXKDWSXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQ
|
|
URWGXKDWJWIWKWLBCPQXLRXMXMWMWNWOAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIU
|
|
JUKULUMUNUOUPUQURUSWPWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d p q r s t u x y z A $. $d p q r s t u z X $.
|
|
$d p q r s t u x y z .+ $. $d p q r s t u x y z .~ $. $d p q x y z J $.
|
|
$d p q r s t u x y z ph $.
|
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eropr2.1 $e |- J = ( A /. .~ ) $.
|
|
eropr2.2 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. A
|
|
( ( x = [ p ] .~ /\ y = [ q ] .~ ) /\ z = [ ( p .+ q ) ] .~ ) } $.
|
|
eropr2.3 $e |- ( ph -> .~ e. X ) $.
|
|
eropr2.4 $e |- ( ph -> .~ Er U ) $.
|
|
eropr2.5 $e |- ( ph -> A C_ U ) $.
|
|
eropr2.6 $e |- ( ph -> .+ : ( A X. A ) --> A ) $.
|
|
eropr2.7 $e |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. A /\ u e. A ) ) )
|
|
-> ( ( r .~ s /\ t .~ u ) -> ( r .+ t ) .~ ( s .+ u ) ) ) $.
|
|
$( Functionality of an operation defined on equivalence classes.
|
|
(Contributed by Jeff Madsen, 10-Jun-2010.) $)
|
|
eroprf2 $p |- ( ph -> .+^ : ( J X. J ) --> J ) $=
|
|
( eroprf ) ABCDEFGGGHIJJJKLLLKKMMMNOPQRRTUAUAUAUBUBUBUCUDSTTRUE $.
|
|
$}
|
|
|
|
${
|
|
$d f g h t s r A $. $d f g h t s r B $. $d f g h t s r C $.
|
|
$d f g h t s r D $. $d x y z w v u f g h t s r .+ $. $d f g h t s r .~ $.
|
|
$d x y z w v u f g h t s r S $.
|
|
ecopopr.1 $e |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S )
|
|
) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\
|
|
( z .+ u ) = ( w .+ v ) ) ) } $.
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $.
|
|
$( This is the first of several theorems about equivalence relations of
|
|
the kind used in construction of fractions and signed reals, involving
|
|
operations on equivalent classes of ordered pairs. This theorem
|
|
expresses the relation ` .~ ` (specified by the hypothesis) in terms
|
|
of its operation ` F ` . (Contributed by NM, 16-Aug-1995.) $)
|
|
ecopoveq $p |- ( ( ( A e. S /\ B e. S ) /\
|
|
( C e. S /\ D e. S ) ) ->
|
|
( <. A , B >. .~ <. C , D >. <-> ( A .+ D ) = ( B .+ C ) ) ) $=
|
|
( cv co wceq wb wa oveq12 eqeqan12d an42s opbrop ) COZFOZKPZDOZEOZKPZQZ
|
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GJKPZHIKPZQZABCDEFGHIJLMUDGQZUEJQZUGHQZUHIQZUJUMRUNUOSUPUQSUFUKUIULUDGU
|
|
EJKTUGHUHIKTUAUBNUC $.
|
|
$}
|
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|
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${
|
|
ecopopr.com $e |- ( x .+ y ) = ( y .+ x ) $.
|
|
$( Assuming the operation ` F ` is commutative, show that the relation
|
|
` .~ ` , specified by the first hypothesis, is symmetric.
|
|
(Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
ecopovsym $p |- ( A .~ B -> B .~ A ) $=
|
|
( wbr wcel wa wb cv wceq co vf vg vh cxp cop wex copab opabssxp eqsstri
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brel eqid breq1 breq2 bibi12d ecopoveq vex caovcom eqeq12i eqcom syl6bb
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|
vt bitri ancoms bitr4d 2optocl syl ibi ) GHJNZHGJNZVHGKKUDZOHVJOPVHVIQZ
|
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GHVJVJJJARZVJOBRZVJOPVLCRZDRZUESVMERZFRZUESPVNVQITVOVPITSPFUFEUFDUFCUFZ
|
|
PABUGVJVJUDLVRABVJVJUHUIUJUARZUBRZUEZUCRZVARZUEZJNZWDWAJNZQGWDJNZWDGJNZ
|
|
QVKUAUBUCVAGHKKVJVJUKWAGSWEWGWFWHWAGWDJULWAGWDJUMUNWDHSWGVHWHVIWDHGJUMW
|
|
DHGJULUNVSKOVTKOPZWBKOWCKOPZPZWEWBVTITZWCVSITZSZWFWKWEVSWCITZVTWBITZSZW
|
|
NABCDEFVSVTWBWCIJKLUOWQWMWLSWNWOWMWPWLABVSWCIUAUPVAUPMUQABVTWBIUBUPUCUP
|
|
MUQURWMWLUSVBUTWJWIWFWNQABCDEFWBWCVSVTIJKLUOVCVDVEVFVG $.
|
|
|
|
${
|
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$( Assume the operation is closed. $)
|
|
ecopopr.cl $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) $.
|
|
$( Assume the operation is associative. $)
|
|
ecopopr.ass $e |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) $.
|
|
$( Assume the operation has the cancellation property. $)
|
|
ecopopr.can $e |- ( ( x e. S /\ y e. S ) ->
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( ( x .+ y ) = ( x .+ z ) -> y = z ) ) $.
|
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$( Assuming that operation ` F ` is commutative (second hypothesis),
|
|
closed (third hypothesis), associative (fourth hypothesis), and has
|
|
the cancellation property (fifth hypothesis), show that the relation
|
|
` .~ ` , specified by the first hypothesis, is transitive.
|
|
(Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
ecopovtrn $p |- ( ( A .~ B /\ B .~ C ) -> A .~ C ) $=
|
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( wcel wa co vf vg vh vt vs vr cxp w3a wbr cv cop wceq copab opabssxp
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wex eqsstri brel simpld anim12i 3anass sylibr wi breq1 anbi1d imbi12d
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breq2 anbi12d imbi1d anbi2d wb ecopoveq 3adant3 3adant1 oveq12 simp2l
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eqid simp2r simp1l simp3r adantl caov411d simp1r simp3l caov4d eqtr3d
|
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eqeq12d syl5ibr sylbid impbid1 caovcl syl2anc caovcld caovcand sylibd
|
|
a1i oveq2 3adant2 sylibrd 3optocl mpcom ) GLLUGZRZHXARZIXARZUHZGHKUIZ
|
|
HIKUIZSZGIKUIZXHXBXCXDSZSXEXFXBXGXJXFXBXCGHXAXAKKAUJZXARBUJZXARSXKCUJ
|
|
ZDUJZUKULXLEUJZFUJZUKULSXMXPJTXNXOJTULSFUOEUODUOCUOZSABUMXAXAUGMXQABX
|
|
AXAUNUPZUQURHIXAXAKXRUQUSXBXCXDUTVAUAUJZUBUJZUKZUCUJZUDUJZUKZKUIZYDUE
|
|
UJZUFUJZUKZKUIZSZYAYHKUIZVBGYDKUIZYISZGYHKUIZVBXFHYHKUIZSZYNVBXHXIVBU
|
|
AUBUCUDUEUFGHILXALXAVPYAGULZYJYMYKYNYQYEYLYIYAGYDKVCVDYAGYHKVCVEYDHUL
|
|
ZYMYPYNYRYLXFYIYOYDHGKVFYDHYHKVCVGVHYHIULZYPXHYNXIYSYOXGXFYHIHKVFVIYH
|
|
IGKVFVEXSLRZXTLRZSZYBLRZYCLRZSZYFLRZYGLRZSZUHZYJXSYGJTZXTYFJTZULZYKUU
|
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IYJYBYCJTZUUJJTZUUMUUKJTZULZUULUUIYJXSYCJTZXTYBJTZULZYBYGJTZYCYFJTZUL
|
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ZSZUUPUUIYEUUSYIUVBUUBUUEYEUUSVJUUHABCDEFXSXTYBYCJKLMVKVLUUEUUHYIUVBV
|
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JUUBABCDEFYBYCYFYGJKLMVKVMVGUVCUUPUUIUUQUUTJTZUURUVAJTZULUUQUURUUTUVA
|
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JVNUUIUUNUVDUUOUVEUUIABCYBYCXSYGLJUUBUUCUUDUUHVOZUUBUUCUUDUUHVQZYTUUA
|
|
UUEUUHVRZXKXLJTZXLXKJTULUUIXKLRZXLLRZSZSNWOZUVIXMJTXKXLXMJTJTULUUIUVJ
|
|
UVKXMLRZUHZSPWOZUUBUUEUUFUUGVSZUVLUVILRUUIOVTZWAUUIXTYCJTYBYFJTJTUUOU
|
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VEUUIABCXTYCYBYFLJYTUUAUUEUUHWBZUVGUVFUVMUVPUUBUUEUUFUUGWCZUVRWAUUIAB
|
|
CXTYCYBYFLJUVSUVGUVFUVMUVPUVTUVRWDWEWFWGWHUUIABCUUMUUJUUKLLJUVOUVIXKX
|
|
MJTULZXLXMULZVJUUIUVOUWAUWBUVJUVKUWAUWBVBUVNQVLXLXMXKJWPWIVTUUIUUCUUD
|
|
UUMLRUVFUVGABYBYCLJOWJWKUUIYTUUGUUJLRUVHUVQABXSYGLJOWJWKUUIABXTYFLLLJ
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UVRUVSUVTWLWMWNUUBUUHYKUULVJUUEABCDEFXSXTYFYGJKLMVKWQWRWSWT $.
|
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$( Assuming that operation ` F ` is commutative (second hypothesis),
|
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closed (third hypothesis), associative (fourth hypothesis), and has
|
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the cancellation property (fifth hypothesis), show that the relation
|
|
` .~ ` , specified by the first hypothesis, is an equivalence
|
|
relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario
|
|
Carneiro, 12-Aug-2015.) $)
|
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ecopover $p |- .~ Er ( S X. S ) $=
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( vg vh wtru cv wa wbr vf cxp wer wrel wcel cop wceq wex relopabi a1i
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SVSCRZDRZUFUGVTERZFRZUFUGSWAWDGUKWBWCGUKUGSFUHEUHDUHCUHZSZABHJUIUJUAR
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ZORZHTZWHWGHTQABCDEFWGWHGHIJKULUMWIWHPRZHTSWGWJHTQABCDEFWGWHWJGHIJKLM
|
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NUNUMQWGVRUEZWGWGHTZWKWLVAQWLUAVRWLUAVRUOWHWJUFZWMHTZPIUOOIUOWNOPIWHI
|
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UEWJIUESZWNWOWOSWNWHWJGUKWJWHGUKUGABWHWJGOUPPUPKUQABCDEFWHWJWHWJGHIJU
|
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RUSUTVBWLWNUAOPIIWGWMUGWLWNVLWGWMWGWMHVMUTVCVDVEUJWLWGWGVRVRUBZTZWKHW
|
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PWGWGHWFABVFWPJWEABVRVRVGVHVIWQWKWKWGWGVRVRVJVKVNVOVPVQ $.
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$}
|
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|
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|
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|
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${
|
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ecopoprg.com $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) ) $.
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$( Assuming the operation ` F ` is commutative, show that the relation
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` .~ ` , specified by the first hypothesis, is symmetric.
|
|
(Contributed by Jim Kingdon, 1-Sep-2019.) $)
|
|
ecopovsymg $p |- ( A .~ B -> B .~ A ) $=
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( wbr wcel wa wb cv wceq co vf vg vh cxp cop wex copab opabssxp eqsstri
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vt brel breq1 breq2 bibi12d adantl simpll simprr caovcomd simplr simprl
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|
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NZHGJNZVKGKKUDZOHVMOPVKVLQZGHVMVMJJARZVMOBRZVMOPVOCRZDRZUESVPERZFRZUESP
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VQVTITVRVSITSPFUFEUFDUFCUFZPABUGVMVMUDLWAABVMVMUHUIUKUARZUBRZUEZUCRZUJR
|
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ZUEZJNZWGWDJNZQGWGJNZWGGJNZQVNUAUBUCUJGHKKVMVMVAWDGSWHWJWIWKWDGWGJULWDG
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WGJUMUNWGHSWJVKWKVLWGHGJUMWGHGJULUNWBKOZWCKOZPZWEKOZWFKOZPZPZWBWFITZWCW
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EITZSZWEWCITZWFWBITZSZWHWIWRXAXCXBSXDWRWSXCWTXBWRABWBWFKIVOKOVPKOPVOVPI
|
|
TVPVOITSWRMUOZWLWMWQUPWNWOWPUQURWRABWCWEKIXEWLWMWQUSWNWOWPUTURVBXCXBVCV
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DABCDEFWBWCWEWFIJKLVEWQWNWIXDQABCDEFWEWFWBWCIJKLVEVFVGVHVIVJ $.
|
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|
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${
|
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$( Assume the operation is closed. $)
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ecopoprg.cl $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) $.
|
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$( Assume the operation is associative. $)
|
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ecopoprg.ass $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
|
|
( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) $.
|
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$( Assume the operation has the cancellation property. $)
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|
ecopoprg.can $e |- ( ( x e. S /\ y e. S /\ z e. S ) ->
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( ( x .+ y ) = ( x .+ z ) -> y = z ) ) $.
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$( Assuming that operation ` F ` is commutative (second hypothesis),
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closed (third hypothesis), associative (fourth hypothesis), and has
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the cancellation property (fifth hypothesis), show that the relation
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` .~ ` , specified by the first hypothesis, is transitive.
|
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(Contributed by Jim Kingdon, 1-Sep-2019.) $)
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ecopovtrng $p |- ( ( A .~ B /\ B .~ C ) -> A .~ C ) $=
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( wcel wa co vf vg vh vt vs vr cxp w3a wbr cv cop wceq copab opabssxp
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wex eqsstri brel simpld anim12i 3anass sylibr wi breq1 anbi1d imbi12d
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breq2 anbi12d imbi1d anbi2d wb ecopoveq 3adant3 3adant1 oveq12 simp2l
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eqid simp2r simp1l adantl simp3r caov411d simp1r simp3l caov4d eqtr3d
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oveq2 3adant2 sylibrd 3optocl mpcom ) GLLUGZRZHWTRZIWTRZUHZGHKUIZHIKU
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IZSZGIKUIZXGXAXBXCSZSXDXEXAXFXIXEXAXBGHWTWTKKAUJZWTRBUJZWTRSXJCUJZDUJ
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ZUKULXKEUJZFUJZUKULSXLXOJTXMXNJTULSFUOEUODUOCUOZSABUMWTWTUGMXPABWTWTU
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NUPZUQURHIWTWTKXQUQUSXAXBXCUTVAUAUJZUBUJZUKZUCUJZUDUJZUKZKUIZYCUEUJZU
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FUJZUKZKUIZSZXTYGKUIZVBGYCKUIZYHSZGYGKUIZVBXEHYGKUIZSZYMVBXGXHVBUAUBU
|
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CUDUEUFGHILWTLWTVPXTGULZYIYLYJYMYPYDYKYHXTGYCKVCVDXTGYGKVCVEYCHULZYLY
|
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OYMYQYKXEYHYNYCHGKVFYCHYGKVCVGVHYGIULZYOXGYMXHYRYNXFXEYGIHKVFVIYGIGKV
|
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FVEXRLRZXSLRZSZYALRZYBLRZSZYELRZYFLRZSZUHZYIXRYFJTZXSYEJTZULZYJUUHYIY
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AYBJTZUUIJTZUULUUJJTZULZUUKUUHYIXRYBJTZXSYAJTZULZYAYFJTZYBYEJTZULZSZU
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UOUUHYDUURYHUVAUUAUUDYDUURVJUUGABCDEFXRXSYAYBJKLMVKVLUUDUUGYHUVAVJUUA
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ABCDEFYAYBYEYFJKLMVKVMVGUVBUUOUUHUUPUUSJTZUUQUUTJTZULUUPUUQUUSUUTJVNU
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UHUUMUVCUUNUVDUUHABCYAYBXRYFLJUUAUUBUUCUUGVOZUUAUUBUUCUUGVQZYSYTUUDUU
|
|
GVRZXJLRZXKLRZSZXJXKJTZXKXJJTULUUHNVSZUVHUVIXLLRUHZUVKXLJTXJXKXLJTJTU
|
|
LUUHPVSZUUAUUDUUEUUFVTZUVJUVKLRUUHOVSZWAUUHXSYBJTYAYEJTJTUUNUVDUUHABC
|
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XSYBYAYELJYSYTUUDUUGWBZUVFUVEUVLUVNUUAUUDUUEUUFWCZUVPWAUUHABCXSYBYAYE
|
|
LJUVQUVFUVEUVLUVNUVRUVPWDWEWFWGWHUUHABCUULUUIUUJLLJUVMUVKXJXLJTULZXKX
|
|
LULZVJUUHUVMUVSUVTQXKXLXJJWOWIVSUUHUUBUUCUULLRUVEUVFABYAYBLJOWJWKUUHY
|
|
SUUFUUILRUVGUVOABXRYFLJOWJWKUUHABXSYELLLJUVPUVQUVRWLWMWNUUAUUGYJUUKVJ
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$( Assuming that operation ` F ` is commutative (second hypothesis),
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closed (third hypothesis), associative (fourth hypothesis), and has
|
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the cancellation property (fifth hypothesis), show that the relation
|
|
` .~ ` , specified by the first hypothesis, is an equivalence
|
|
relation. (Contributed by Jim Kingdon, 1-Sep-2019.) $)
|
|
ecopoverg $p |- .~ Er ( S X. S ) $=
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( wtru cv wcel wa wceq wbr vf vg vh cxp wer wrel cop wex relopabi a1i
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co ecopovsymg adantl ecopovtrng wi wral simpll simplr caovcomd mpbird
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|
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BHJUIUJUAPZUBPZHTZWIWHHTOABCDEFWHWIGHIJKULUMWJWIUCPZHTRWHWKHTOABCDEFW
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HWIWKGHIJKLMNUNUMOWHVSQZWHWHHTZWLWMUOOWMUAVSWMUAVSUPWIWKUGZWNHTZUCIUP
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UBIUPWOUBUCIWIIQZWKIQZRZWOWRWRRZWOWIWKGUKWKWIGUKSWSABWIWKIGVTIQWAIQRV
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TWAGUKWAVTGUKSWSKUMWPWQWRUQWPWQWRURUSABCDEFWIWKWIWKGHIJVAUTVBVCWMWOUA
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$}
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${
|
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$d x y z w v u .+ $. $d x y z w v u .~ $. $d x y z w v u S $.
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$d x y z w v u A $. $d x y z w v u B $.
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th3qlem1.1 $e |- .~ Er S $.
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$( Compatibility assumption. $)
|
|
th3qlem1.3 $e |- ( ( ( y e. S /\ w e. S ) /\ ( z e. S /\ v e. S ) ) ->
|
|
( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) ) $.
|
|
$( Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The
|
|
third hypothesis is the compatibility assumption. (Contributed by NM,
|
|
3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) $)
|
|
th3qlem1 $p |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> E* x
|
|
E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\
|
|
x = [ ( y .+ z ) ] .~ ) ) $=
|
|
( vu wcel wa cv cec wceq wex ecelqsdm cqs co wi wal ee4anv eleq1 bi2anan9
|
|
wmo an4 wb adantr biimpac anim12i an4s adantl wer a1i wbr simprl cdm erdm
|
|
eqtr2 ax-mp simpll sylancr mpbird simprr simplr eqeltrrd syl22anc syl2anc
|
|
erth mp2and erthi eqeq12 syl5ibrcom expimpd syl5bi exlimdvv syl5bir eqeq1
|
|
alrimivv anbi2d 2exbidv eceq1 eqeq2d oveq12 eceq1d anbi12d cbvex2v syl6bb
|
|
mo4 sylibr ) FJIUAZNZGWNNZOZFBPZIQZRZGCPZIQZRZOZAPZWRXAHUBZIQZRZOZCSBSZFD
|
|
PZIQZRZGEPZIQZRZOZMPZXKXNHUBZIQZRZOZESDSZOZXEXRRZUCZMUDAUDXJAUHWQYFAMYDXI
|
|
YBOZESDSZCSBSWQYEXIYBBCDEUEWQYHYEBCWQYGYEDEYGXDXQOZXHYAOZOWQYEXDXHXQYAUIW
|
|
QYIYJYEWQYIOZYEYJXGXTRZYKWSWNNZXBWNNZOZWSXLRZXBXORZOZYLYIWQYOXDWQYOUJXQWT
|
|
WOYMXCWPYNFWSWNUFGXBWNUFUGUKULYIYRWQWTXMXCXPYRWTXMOYPXCXPOYQFWSXLVBGXBXOV
|
|
BUMUNUOYOYROZXFXSIJJIUPZYSKUQZYSWRXKIURZXAXNIURZXFXSIURZYSUUBYPYOYPYQUSZY
|
|
SWRXKIJUUAYSIUTJRZYMWRJNZYTUUFKJIVAVCZYMYNYRVDZJWRITVEZVLVFYSUUCYQYOYPYQV
|
|
GZYSXAXNIJUUAYSUUFYNXAJNZUUHYMYNYRVHZJXAITVEZVLVFYSUUGXKJNZUULXNJNZUUBUUC
|
|
OUUDUCUUJYSUUFXLWNNUUOUUHYSWSXLWNUUEUUIVIJXKITVEUUNYSUUFXOWNNUUPUUHYSXBXO
|
|
WNUUKUUMVIJXNITVELVJVMVNVKXEXGXRXTVOVPVQVRVSVSVTWBXJYCAMYEXJXDXRXGRZOZCSB
|
|
SYCYEXIUURBCYEXHUUQXDXEXRXGWAWCWDUURYBBCDEWRXKRZXAXNRZOZXDXQUUQYAUUSWTXMU
|
|
UTXCXPUUSWSXLFWRXKIWEWFUUTXBXOGXAXNIWEWFUGUVAXGXTXRUVAXFXSIWRXKXAXNHWGWHW
|
|
FWIWJWKWLWM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u t s f g h .~ $. $d x y z w v u t s f g h S $.
|
|
$d x y z w v u t s f A $. $d x y z w v u t s f B $. $d x y z w v u t C $.
|
|
$d x y z w v u t D $. $d x y z w v u t s f g h .+ $.
|
|
th3q.1 $e |- .~ e. _V $.
|
|
th3q.2 $e |- .~ Er ( S X. S ) $.
|
|
$( Compatibility assumption. $)
|
|
th3q.4 $e |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) )
|
|
/\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) ->
|
|
( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) ->
|
|
( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) $.
|
|
$( Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60,
|
|
extended to operations on ordered pairs. The fourth hypothesis is the
|
|
compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by
|
|
Mario Carneiro, 12-Aug-2015.) $)
|
|
th3qlem2 $p |- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) )
|
|
-> E* z E. w E. v E. u E. t (
|
|
( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\
|
|
z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) $=
|
|
( wcel wa wceq vx vy cxp cqs cv cec co wex wmo cop wbr breq1 anbi1d oveq1
|
|
eqid breq1d imbi12d imbi2d breq2 breq2d anbi2d oveq2 expcom 2optocl com12
|
|
wi imp th3qlem1 opex eceq1 eqeq2d bi2anan9 oveq12 eceq1d anbi12d exlimivv
|
|
vex spc2ev moimi syl ) FJJUCZIUDZRGWBRSFNUEZIUFZTZGUAUEZIUFZTZSZAUEZWCWFH
|
|
UGZIUFZTZSZUAUHNUHZAUIFBUEZCUEZUJZIUFZTZGDUEZEUEZUJZIUFZTZSZWJWRXCHUGZIUF
|
|
ZTZSZEUHDUHZCUHBUHZAUIANUAKUBFGHIWAPWCWARKUEZWARSWFWARUBUEZWARSZWCXMIUKZW
|
|
FXNIUKZSZWKXMXNHUGZIUKZVFZXOWRXCIUKZXQSZWRWFHUGZXCXNHUGZIUKZVFZVFXOWCXCIU
|
|
KZXQSZWKYEIUKZVFZVFXOYAVFBCDEWCXMJJWAWAUOZWRWCTZYGYKXOYMYCYIYFYJYMYBYHXQW
|
|
RWCXCIULUMYMYDWKYEIWRWCWFHUNUPUQURXCXMTZYKYAXOYNYIXRYJXTYNYHXPXQXCXMWCIUS
|
|
UMYNYEXSWKIXCXMXNHUNUTUQURXOWPJRWQJRSXAJRXBJRSSZYGYOYBWCXMUJZLUEZMUEZUJZI
|
|
UKZSZWRYPHUGZXCYSHUGZIUKZVFZVFYOYBWFYSIUKZSZYDUUCIUKZVFZVFYOYGVFNKLMWFXNJ
|
|
JWAYLYPWFTZUUEUUIYOUUJUUAUUGUUDUUHUUJYTUUFYBYPWFYSIULVAUUJUUBYDUUCIYPWFWR
|
|
HVBUPUQURYSXNTZUUIYGYOUUKUUGYCUUHYFUUKUUFXQYBYSXNWFIUSVAUUKUUCYEYDIYSXNXC
|
|
HVBUTUQURYOWCJRXMJRSYQJRYRJRSSUUEQVCVDVEVDVGVHXLWOAXKWOBCXJWODEWNXJNUAWRX
|
|
CWPWQBVQCVQVIXAXBDVQEVQVIWCWRTZWFXCTZSZWIXFWMXIUULWEWTUUMWHXEUULWDWSFWCWR
|
|
IVJVKUUMWGXDGWFXCIVJVKVLUUNWLXHWJUUNWKXGIWCWRWFXCHVMVNVKVOVRVPVPVSVT $.
|
|
|
|
${
|
|
th3q.5 $e |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ )
|
|
/\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( (
|
|
x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\
|
|
z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) } $.
|
|
$( Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM,
|
|
12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.) $)
|
|
th3qcor $p |- Fun G $=
|
|
( cv wfun cxp cqs wcel wa cop cec co wex coprab wmo wi th3qlem2 moanimv
|
|
wceq mpbir funoprab funeqi ) NUAATZJJUBIUCZUDBTZUTUDUEZUSDTETUFZIUGUOVA
|
|
FTGTUFZIUGUOUECTVCVDHUHIUGUOUEGUIFUIEUIDUIZUEZABCUJZUAVFABCVFCUKVBVECUK
|
|
ULCDEFGUSVAHIJKLMOPQRUMVBVECUNUPUQNVGSURUP $.
|
|
|
|
$( Theorem 3Q of [Enderton] p. 60, extended to operations on ordered
|
|
pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro,
|
|
19-Dec-2013.) $)
|
|
th3q $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
|
|
( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) =
|
|
[ ( <. A , B >. .+ <. C , D >. ) ] .~ ) $=
|
|
( wcel wa cop cec cxp cqs cv wceq co wex opelxpi ecelqsi anim12i pm3.2i
|
|
syl eqid opeq12 eceq1 eqeq2d anbi1d oveq1 eceq1d anbi12d spc2egv anbi2d
|
|
wb oveq2 2eximdv sylan9 mp2ani cvv ecexg ax-mp w3a eqeq1 bi2anan9 3impa
|
|
wi 4exbidv th3qlem2 ovig mp3an3 sylc ) HNUDINUDUEZJNUDKNUDUEZUEZHIUFZMU
|
|
GZNNUHZMUIZUDZJKUFZMUGZWMUDZUEWKDUJZEUJZUFZMUGZUKZWPFUJZGUJZUFZMUGZUKZU
|
|
EZWJWOLULZMUGZWTXELULZMUGZUKZUEZGUMFUMZEUMDUMZWKWPRULXJUKZWGWNWHWQWGWJW
|
|
LUDWNHINNUNWLWJMTUOURWHWOWLUDWQJKNNUNWLWOMTUOURUPWIWKWKUKZWPWPUKZUEZXJX
|
|
JUKZXPXRXSWKUSWPUSUQXJUSWGXTYAUEZXBXSUEZXJWTWOLULZMUGZUKZUEZEUMDUMWHXPY
|
|
GYBDEHINNWRHUKWSIUKUEWTWJUKZYGYBVIWRWSHIUTYHYCXTYFYAYHXBXRXSYHXAWKWKWTW
|
|
JMVAVBVCYHYEXJXJYHYDXIMWTWJWOLVDVEVBVFURVGWHYGXODEXNYGFGJKNNXCJUKXDKUKU
|
|
EXEWOUKZXNYGVIXCXDJKUTYIXHYCXMYFYIXGXSXBYIXFWPWPXEWOMVAVBVHYIXLYEXJYIXK
|
|
YDMXEWOWTLVJVEVBVFURVGVKVLVMWNWQXJVNUDZXPXQWAMVNUDYJTXIVNMVOVPAUJZXAUKZ
|
|
BUJZXFUKZUEZCUJZXLUKZUEZGUMFUMEUMDUMXPABCWKWPXJVNWMWMRYKWKUKZYMWPUKZYPX
|
|
JUKZVQYRXNDEFGYSYTUUAYRXNVIYSYTUEYOXHUUAYQXMYSYLXBYTYNXGYKWKXAVRYMWPXFV
|
|
RVSYPXJXLVRVSVTWBCDEFGYKYMLMNOPQSTUAUBWCUCWDWEWF $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d a b c d f u v w x y z C $. $d a b c d f u v w x y z D $. $d x y z J $.
|
|
$d a b c d f g h u v w x y z A $. $d u v w z ch $. $d f u v w x y z H $.
|
|
$d a b c d f g h u v w x y z B $. $d f u v w x y z K $. $d u v w z ps $.
|
|
$d f u v w x y z L $. $d x y ph $. $d a b c d f g h s t u v w x y z S $.
|
|
$d a b c d g h s t x y z .+ $. $d a b c d g h s t x y z .~ $.
|
|
oviec.1 $e |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
|
|
H e. ( S X. S ) ) $.
|
|
oviec.2 $e |- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) ->
|
|
K e. ( S X. S ) ) $.
|
|
oviec.3 $e |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) ->
|
|
L e. ( S X. S ) ) $.
|
|
oviec.4 $e |- .~ e. _V $.
|
|
oviec.5 $e |- .~ Er ( S X. S ) $.
|
|
$( Equivalence relation. $)
|
|
oviec.7 $e |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\
|
|
y e. ( S X. S ) ) /\
|
|
E. z E. w E. v E. u ( ( x = <. z , w >. /\
|
|
y = <. v , u >. ) /\ ph ) ) } $.
|
|
$( Substitutions for equivalence relation. $)
|
|
oviec.8 $e |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) ->
|
|
( ph <-> ps ) ) $.
|
|
oviec.9 $e |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) ->
|
|
( ph <-> ch ) ) $.
|
|
$( Pre-operation. $)
|
|
oviec.10 $e |- .+ = { <. <. x , y >. , z >. | ( ( x e. ( S X. S ) /\
|
|
y e. ( S X. S ) ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = J ) ) } $.
|
|
$( Substitutions for pre-operation. $)
|
|
oviec.11 $e |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> J = K ) $.
|
|
oviec.12 $e |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> J = L ) $.
|
|
oviec.13 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> J = H ) $.
|
|
$( Main operation. $)
|
|
oviec.14 $e |- .+^ = { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\
|
|
E. a E. b E. c E. d ( (
|
|
x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\
|
|
z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) } $.
|
|
$( Quotient class for main operation. $)
|
|
oviec.15 $e |- Q = ( ( S X. S ) /. .~ ) $.
|
|
$( Lemma to establish compatibility of pre-operation. $)
|
|
oviec.16 $e |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\
|
|
( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) ->
|
|
( ( ps /\ ch ) -> K .~ L ) ) $.
|
|
$( Express an operation on equivalence classes of ordered pairs in terms of
|
|
equivalence class of operations on ordered pairs. See iset.mm for
|
|
additional comments describing the hypotheses. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 4-Jun-2013.)
|
|
(Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro,
|
|
4-Jun-2013.) $)
|
|
oviec $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
|
|
( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) =
|
|
[ H ] .~ ) $=
|
|
( wcel wa cop cec co wbr opbrop bi2anan9 ovi3 breqan12d an4s 3imtr4d wceq
|
|
cv wb wex coprab cxp cqs eleq2i anbi12i anbi1i oprabbii eqtri th3q eceq1d
|
|
eqtrd ) KSVGLSVGVHMSVGNSVGVHVHZKLVIZRVJMNVIZRVJPVKWOWPOVKZRVJUCRVJDEFUHUI
|
|
UJUKKLMNORSUBJUGPUAUOUPUHVTZSVGUIVTZSVGVHZUJVTZSVGUKVTZSVGVHZVHZUAVTZSVGU
|
|
BVTZSVGVHZJVTZSVGUGVTZSVGVHZVHZVHBCVHUEUFRVLZWRWSVIZXAXBVIZRVLZXEXFVIZXHX
|
|
IVIZRVLZVHXMXPOVKZXNXQOVKZRVLZVFXDXOBXKXRCABDEFGHIWRWSXAXBRSURUQVMACDEFGH
|
|
IXEXFXHXIRSUSUQVMVNWTXGXCXJYAXLWAWTXGVHXCXJVHXSUEXTUFRDEFGHIWRWSXEXFUDUET
|
|
OSUMVAUTVODEFGHIXAXBXHXIUDUFTOSUNVBUTVOVPVQVRPDVTZQVGZEVTZQVGZVHZYBXMRVJV
|
|
SYDXNRVJVSVHFVTXMXNOVKRVJVSVHUKWBUJWBUIWBUHWBZVHZDEFWCYBSSWDRWEZVGZYDYIVG
|
|
ZVHZYGVHZDEFWCVDYHYMDEFYFYLYGYCYJYEYKQYIYBVEWFQYIYDVEWFWGWHWIWJWKWNWQUCRD
|
|
EFGHIKLMNUDUCTOSULVCUTVOWLWM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w .+ $. $d x y z w .~ $.
|
|
$d x y z w S $. $d z w C $.
|
|
ecovcom.1 $e |- C = ( ( S X. S ) /. .~ ) $.
|
|
$( D,G for AFB $)
|
|
ecovcom.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ ) $.
|
|
$( H,J for BFA $)
|
|
ecovcom.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ ) $.
|
|
ecovcom.4 $e |- D = H $.
|
|
ecovcom.5 $e |- G = J $.
|
|
$( Lemma used to transfer a commutative law via an equivalence relation.
|
|
Most uses will want ~ ecovicom instead. (Contributed by NM,
|
|
29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) $)
|
|
ecovcom $p |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) ) $=
|
|
( wceq cv cop cec co oveq1 oveq2 eqeq12d wcel opeq12 eceq1d mp2an 3eqtr4a
|
|
wa ancoms 2ecoptocl ) AUAZBUAZUBJUCZCUAZDUAZUBJUCZIUDZVAURIUDZTEVAIUDZVAE
|
|
IUDZTEFIUDZFEIUDZTABCDEFKKJGOURETVBVDVCVEUREVAIUEUREVAIUFUGVAFTVDVFVEVGVA
|
|
FEIUFVAFEIUEUGUPKUHUQKUHUMZUSKUHUTKUHUMZUMHLUBZJUCZMNUBZJUCZVBVCHMTZLNTZV
|
|
KVMTRSVNVOUMVJVLJHLMNUIUJUKPVIVHVCVMTQUNULUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w .+ $. $d x y z w .~ $.
|
|
$d x y z w S $. $d z w C $.
|
|
ecovicom.1 $e |- C = ( ( S X. S ) /. .~ ) $.
|
|
$( D,G for AFB $)
|
|
ecovicom.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ ) $.
|
|
$( H,J for BFA $)
|
|
ecovicom.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ ) $.
|
|
ecovicom.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
D = H ) $.
|
|
ecovicom.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
G = J ) $.
|
|
$( Lemma used to transfer a commutative law via an equivalence relation.
|
|
(Contributed by Jim Kingdon, 15-Sep-2019.) $)
|
|
ecovicom $p |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) ) $=
|
|
( wceq cv cop co oveq1 oveq2 eqeq12d wcel wa opeq12 eceq1d syl2anc ancoms
|
|
cec 3eqtr4d 2ecoptocl ) AUAZBUAZUBJUMZCUAZDUAZUBJUMZIUCZVAURIUCZTEVAIUCZV
|
|
AEIUCZTEFIUCZFEIUCZTABCDEFKKJGOURETVBVDVCVEUREVAIUDUREVAIUEUFVAFTVDVFVEVG
|
|
VAFEIUEVAFEIUDUFUPKUGUQKUGUHZUSKUGUTKUGUHZUHZHLUBZJUMZMNUBZJUMZVBVCVJHMTZ
|
|
LNTZVLVNTRSVOVPUHVKVMJHLMNUIUJUKPVIVHVCVNTQULUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u .+ $. $d x y z w v u .~ $. $d x y z w v u S $.
|
|
$d z w v u D $.
|
|
ecovass.1 $e |- D = ( ( S X. S ) /. .~ ) $.
|
|
ecovass.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ ) $.
|
|
ecovass.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ ) $.
|
|
ecovass.4 $e |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ ) $.
|
|
ecovass.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ ) $.
|
|
ecovass.6 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( G e. S /\ H e. S ) ) $.
|
|
ecovass.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( N e. S /\ Q e. S ) ) $.
|
|
ecovass.8 $e |- J = L $.
|
|
ecovass.9 $e |- K = M $.
|
|
$( Lemma used to transfer an associative law via an equivalence relation.
|
|
In most cases ~ ecoviass will be more useful. (Contributed by NM,
|
|
31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) $)
|
|
ecovass $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
|
|
( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) $=
|
|
( cv cop cec co wceq oveq1 oveq1d eqeq12d oveq2 oveq2d wcel wa w3a opeq12
|
|
eceq1d mp2an adantr sylan eqtrd 3impa adantl sylan2 3eqtr4a 3ecoptocl
|
|
3impb ) AUKZBUKZULMUMZCUKZDUKZULMUMZKUNZEUKZFUKZULMUMZKUNZVRWAWEKUNZKUNZU
|
|
OGWAKUNZWEKUNZGWGKUNZUOGHKUNZWEKUNZGHWEKUNZKUNZUOWLIKUNZGHIKUNZKUNZUOABCD
|
|
EFGHINMJUBVRGUOZWFWJWHWKWSWBWIWEKVRGWAKUPUQVRGWGKUPURWAHUOZWJWMWKWOWTWIWL
|
|
WEKWAHGKUSUQWTWGWNGKWAHWEKUPUTURWEIUOZWMWPWOWRWEIWLKUSXAWNWQGKWEIHKUSUTUR
|
|
VPNVAVQNVAVBZVSNVAVTNVAVBZWCNVAWDNVAVBZVCQRULZMUMZSTULZMUMZWFWHQSUOZRTUOZ
|
|
XFXHUOUIUJXIXJVBXEXGMQRSTVDVEVFXBXCXDWFXFUOXBXCVBZXDVBWFOPULMUMZWEKUNZXFX
|
|
KWFXMUOXDXKWBXLWEKUCUQVGXKONVAPNVAVBXDXMXFUOUGUEVHVIVJXBXCXDWHXHUOXBXCXDV
|
|
BZVBWHVRUALULMUMZKUNZXHXNWHXPUOXBXNWGXOVRKUDUTVKXNXBUANVALNVAVBXPXHUOUHUF
|
|
VLVIVOVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u .+ $. $d x y z w v u .~ $. $d x y z w v u S $.
|
|
$d z w v u D $.
|
|
ecoviass.1 $e |- D = ( ( S X. S ) /. .~ ) $.
|
|
ecoviass.2 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ ) $.
|
|
ecoviass.3 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ ) $.
|
|
ecoviass.4 $e |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ ) $.
|
|
ecoviass.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ ) $.
|
|
ecoviass.6 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( G e. S /\ H e. S ) ) $.
|
|
ecoviass.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( N e. S /\ Q e. S ) ) $.
|
|
ecoviass.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
|
|
/\ ( v e. S /\ u e. S ) ) -> J = L ) $.
|
|
ecoviass.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
|
|
/\ ( v e. S /\ u e. S ) ) -> K = M ) $.
|
|
$( Lemma used to transfer an associative law via an equivalence relation.
|
|
(Contributed by Jim Kingdon, 16-Sep-2019.) $)
|
|
ecoviass $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
|
|
( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) $=
|
|
( cv cop cec co wceq oveq1 oveq1d eqeq12d oveq2 oveq2d wcel wa w3a opeq12
|
|
eceq1d syl2anc adantr sylan eqtrd 3impa adantl sylan2 3eqtr4d 3ecoptocl
|
|
3impb ) AUKZBUKZULMUMZCUKZDUKZULMUMZKUNZEUKZFUKZULMUMZKUNZVRWAWEKUNZKUNZU
|
|
OGWAKUNZWEKUNZGWGKUNZUOGHKUNZWEKUNZGHWEKUNZKUNZUOWLIKUNZGHIKUNZKUNZUOABCD
|
|
EFGHINMJUBVRGUOZWFWJWHWKWSWBWIWEKVRGWAKUPUQVRGWGKUPURWAHUOZWJWMWKWOWTWIWL
|
|
WEKWAHGKUSUQWTWGWNGKWAHWEKUPUTURWEIUOZWMWPWOWRWEIWLKUSXAWNWQGKWEIHKUSUTUR
|
|
VPNVAVQNVAVBZVSNVAVTNVAVBZWCNVAWDNVAVBZVCZQRULZMUMZSTULZMUMZWFWHXEQSUOZRT
|
|
UOZXGXIUOUIUJXJXKVBXFXHMQRSTVDVEVFXBXCXDWFXGUOXBXCVBZXDVBWFOPULMUMZWEKUNZ
|
|
XGXLWFXNUOXDXLWBXMWEKUCUQVGXLONVAPNVAVBXDXNXGUOUGUEVHVIVJXBXCXDWHXIUOXBXC
|
|
XDVBZVBWHVRUALULMUMZKUNZXIXOWHXQUOXBXOWGXPVRKUDUTVKXOXBUANVALNVAVBXQXIUOU
|
|
HUFVLVIVOVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d w v u C $. $d x y z w v u .+ $.
|
|
$d x y z w v u .~ $. $d x y z w v u S $. $d x y z w v u .x. $.
|
|
$d z w v u D $.
|
|
ecovdi.1 $e |- D = ( ( S X. S ) /. .~ ) $.
|
|
$( M,N for BFC $)
|
|
ecovdi.2 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ ) $.
|
|
$( H,J for AG(BFC) $)
|
|
ecovdi.3 $e |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ ) $.
|
|
$( W,X for AGB $)
|
|
ecovdi.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ ) $.
|
|
$( Y,Z for AGC $)
|
|
ecovdi.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ ) $.
|
|
$( K,L for (AGB)F(AGC) $)
|
|
ecovdi.6 $e |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) ->
|
|
( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ ) $.
|
|
ecovdi.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( M e. S /\ N e. S ) ) $.
|
|
ecovdi.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( W e. S /\ X e. S ) ) $.
|
|
ecovdi.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( Y e. S /\ Z e. S ) ) $.
|
|
ecovdi.10 $e |- H = K $.
|
|
ecovdi.11 $e |- J = L $.
|
|
$( Lemma used to transfer a distributive law via an equivalence relation.
|
|
Most likely ~ ecovidi will be more helpful. (Contributed by NM,
|
|
2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.) $)
|
|
ecovdi $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
|
|
( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) ) $=
|
|
( cv cop co wceq oveq1 oveq12d eqeq12d oveq2d oveq2 oveq1d wcel wa opeq12
|
|
cec w3a eceq1d mp2an adantl sylan2 eqtrd 3impb oveqan12d syl2an 3ecoptocl
|
|
3impdi 3eqtr4a ) AUPZBUPZUQLVIZCUPZDUPZUQLVIZEUPZFUPZUQLVIZKURZNURZWDWGNU
|
|
RZWDWJNURZKURZUSGWKNURZGWGNURZGWJNURZKURZUSGHWJKURZNURZGHNURZWRKURZUSGHIK
|
|
URZNURZXBGINURZKURZUSABCDEFGHIMLJUEWDGUSZWLWPWOWSWDGWKNUTXHWMWQWNWRKWDGWG
|
|
NUTWDGWJNUTVAVBWGHUSZWPXAWSXCXIWKWTGNWGHWJKUTVCXIWQXBWRKWGHGNVDVEVBWJIUSZ
|
|
XAXEXCXGXJWTXDGNWJIHKVDVCXJWRXFXBKWJIGNVDVCVBWBMVFWCMVFVGZWEMVFWFMVFVGZWH
|
|
MVFWIMVFVGZVJOPUQZLVIZQRUQZLVIZWLWOOQUSZPRUSZXOXQUSUNUOXRXSVGXNXPLOPQRVHV
|
|
KVLXKXLXMWLXOUSXKXLXMVGZVGWLWDSTUQLVIZNURZXOXTWLYBUSXKXTWKYAWDNUFVCVMXTXK
|
|
SMVFTMVFVGYBXOUSUKUGVNVOVPXKXLXMWOXQUSXKXLVGZXKXMVGZVGWOUAUBUQLVIZUCUDUQL
|
|
VIZKURZXQYCYDWMYEWNYFKUHUIVQYCUAMVFUBMVFVGUCMVFUDMVFVGYGXQUSYDULUMUJVRVOV
|
|
TWAVS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d w v u C $. $d x y z w v u .+ $.
|
|
$d x y z w v u .~ $. $d x y z w v u S $. $d x y z w v u .x. $.
|
|
$d z w v u D $.
|
|
ecovidi.1 $e |- D = ( ( S X. S ) /. .~ ) $.
|
|
$( M,N for BFC $)
|
|
ecovidi.2 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ ) $.
|
|
$( H,J for AG(BFC) $)
|
|
ecovidi.3 $e |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ ) $.
|
|
$( W,X for AGB $)
|
|
ecovidi.4 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ ) $.
|
|
$( Y,Z for AGC $)
|
|
ecovidi.5 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ ) $.
|
|
$( K,L for (AGB)F(AGC) $)
|
|
ecovidi.6 $e |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) ->
|
|
( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ ) $.
|
|
ecovidi.7 $e |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( M e. S /\ N e. S ) ) $.
|
|
ecovidi.8 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ->
|
|
( W e. S /\ X e. S ) ) $.
|
|
ecovidi.9 $e |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ->
|
|
( Y e. S /\ Z e. S ) ) $.
|
|
ecovidi.10 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
|
|
/\ ( v e. S /\ u e. S ) ) -> H = K ) $.
|
|
ecovidi.11 $e |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S )
|
|
/\ ( v e. S /\ u e. S ) ) -> J = L ) $.
|
|
$( Lemma used to transfer a distributive law via an equivalence relation.
|
|
(Contributed by Jim Kingdon, 17-Sep-2019.) $)
|
|
ecovidi $p |- ( ( A e. D /\ B e. D /\ C e. D ) ->
|
|
( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) ) $=
|
|
( cv cop co wceq oveq1 oveq12d eqeq12d oveq2d oveq2 oveq1d wcel wa opeq12
|
|
cec w3a eceq1d syl2anc adantl sylan2 eqtrd 3impb oveqan12d syl2an 3eqtr4d
|
|
3impdi 3ecoptocl ) AUPZBUPZUQLVIZCUPZDUPZUQLVIZEUPZFUPZUQLVIZKURZNURZWDWG
|
|
NURZWDWJNURZKURZUSGWKNURZGWGNURZGWJNURZKURZUSGHWJKURZNURZGHNURZWRKURZUSGH
|
|
IKURZNURZXBGINURZKURZUSABCDEFGHIMLJUEWDGUSZWLWPWOWSWDGWKNUTXHWMWQWNWRKWDG
|
|
WGNUTWDGWJNUTVAVBWGHUSZWPXAWSXCXIWKWTGNWGHWJKUTVCXIWQXBWRKWGHGNVDVEVBWJIU
|
|
SZXAXEXCXGXJWTXDGNWJIHKVDVCXJWRXFXBKWJIGNVDVCVBWBMVFWCMVFVGZWEMVFWFMVFVGZ
|
|
WHMVFWIMVFVGZVJZOPUQZLVIZQRUQZLVIZWLWOXNOQUSZPRUSZXPXRUSUNUOXSXTVGXOXQLOP
|
|
QRVHVKVLXKXLXMWLXPUSXKXLXMVGZVGWLWDSTUQLVIZNURZXPYAWLYCUSXKYAWKYBWDNUFVCV
|
|
MYAXKSMVFTMVFVGYCXPUSUKUGVNVOVPXKXLXMWOXRUSXKXLVGZXKXMVGZVGWOUAUBUQLVIZUC
|
|
UDUQLVIZKURZXRYDYEWMYFWNYGKUHUIVQYDUAMVFUBMVFVGUCMVFUDMVFVGYHXRUSYEULUMUJ
|
|
VRVOVTVSWA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equinumerosity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c ~~ $. $( Equinumerosity relation ("approximately equals" symbol) $)
|
|
$c ~<_ $. $( Dominance relation (curly less-than-or-equal) $)
|
|
$c Fin $. $( Class of all finite sets. $)
|
|
|
|
$( Extend class definition to include the equinumerosity relation
|
|
("approximately equals" symbol) $)
|
|
cen $a class ~~ $.
|
|
|
|
$( Extend class definition to include the dominance relation (curly
|
|
less-than-or-equal) $)
|
|
cdom $a class ~<_ $.
|
|
|
|
$( Extend class definition to include the class of all finite sets. $)
|
|
cfn $a class Fin $.
|
|
|
|
${
|
|
$d x y f $.
|
|
$( Define the equinumerosity relation. Definition of [Enderton] p. 129.
|
|
We define ` ~~ ` to be a binary relation rather than a connective, so
|
|
its arguments must be sets to be meaningful. This is acceptable because
|
|
we do not consider equinumerosity for proper classes. We derive the
|
|
usual definition as ~ bren . (Contributed by NM, 28-Mar-1998.) $)
|
|
df-en $a |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y } $.
|
|
|
|
$( Define the dominance relation. Compare Definition of [Enderton] p. 145.
|
|
Typical textbook definitions are derived as ~ brdom and ~ domen .
|
|
(Contributed by NM, 28-Mar-1998.) $)
|
|
df-dom $a |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y } $.
|
|
|
|
$( Define the (proper) class of all finite sets. Similar to Definition
|
|
10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to
|
|
our " ` a e. Fin ` ". This definition is meaningful whether or not we
|
|
accept the Axiom of Infinity ~ ax-inf2 . (Contributed by NM,
|
|
22-Aug-2008.) $)
|
|
df-fin $a |- Fin = { x | E. y e. _om x ~~ y } $.
|
|
$}
|
|
|
|
${
|
|
$d x y f $. $d x y f $.
|
|
$( Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.) $)
|
|
relen $p |- Rel ~~ $=
|
|
( vx vy vf cv wf1o wex cen df-en relopabi ) ADBDCDECFABGABCHI $.
|
|
|
|
$( Dominance is a relation. (Contributed by NM, 28-Mar-1998.) $)
|
|
reldom $p |- Rel ~<_ $=
|
|
( vx vy vf cv wf1 wex cdom df-dom relopabi ) ADBDCDECFABGABCHI $.
|
|
$}
|
|
|
|
$( If two classes are equinumerous, both classes are sets. (Contributed by
|
|
AV, 21-Mar-2019.) $)
|
|
encv $p |- ( A ~~ B -> ( A e. _V /\ B e. _V ) ) $=
|
|
( cen wrel wbr cvv wcel wa relen brrelex12 mpan ) CDABCEAFGBFGHIABCJK $.
|
|
|
|
${
|
|
$d f x y A $. $d f x y B $. $d y C $.
|
|
$( Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) $)
|
|
bren $p |- ( A ~~ B <-> E. f f : A -1-1-onto-> B ) $=
|
|
( vx vy cen wbr cvv wcel wa cv wf1o wex encv wfn f1ofn syl6eqelr syl wceq
|
|
exbidv cdm fndm vex dmex crn wfo f1ofo forn rnex jca exlimiv f1oeq2 df-en
|
|
f1oeq3 brabg pm5.21nii ) ABFGAHIZBHIZJZABCKZLZCMZABNVAUSCVAUQURVAUTAOZUQA
|
|
BUTPVCAUTUAHAUTUBUTCUCZUDQRVABUTUEZHVAABUTUFVEBSABUTUGABUTUHRUTVDUIQUJUKD
|
|
KZEKZUTLZCMAVGUTLZCMVBDEABHHFVFASVHVICVFAVGUTULTVGBSVIVACVGBAUTUNTDECUMUO
|
|
UP $.
|
|
|
|
$( Dominance relation. (Contributed by NM, 15-Jun-1998.) $)
|
|
brdomg $p |- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $=
|
|
( vx vy wcel cvv cdom wbr cv wf1 wex wi reldom brrelexi a1i wceq exbidv
|
|
wf f1f cdm fdm vex dmex syl6eqelr syl exlimiv wb f1eq2 f1eq3 df-dom brabg
|
|
expcom pm5.21ndd ) BCGZAHGZABIJZABDKZLZDMZURUQNUPABIOPQVAUQNUPUTUQDUTABUS
|
|
TZUQABUSUAVBAUSUBHABUSUCUSDUDUEUFUGUHQUQUPURVAUIEKZFKZUSLZDMAVDUSLZDMVAEF
|
|
ABHCIVCARVEVFDVCAVDUSUJSVDBRVFUTDVDBAUSUKSEFDULUMUNUO $.
|
|
|
|
$( Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
brdomi $p |- ( A ~<_ B -> E. f f : A -1-1-> B ) $=
|
|
( cdom wbr cv wf1 wex cvv wcel wb reldom brrelex2i brdomg syl ibi ) ABDEZ
|
|
ABCFGCHZQBIJQRKABDLMABICNOP $.
|
|
$}
|
|
|
|
${
|
|
$d f x A $. $d f x B $.
|
|
bren.1 $e |- B e. _V $.
|
|
$( Dominance relation. (Contributed by NM, 15-Jun-1998.) $)
|
|
brdom $p |- ( A ~<_ B <-> E. f f : A -1-1-> B ) $=
|
|
( cvv wcel cdom wbr cv wf1 wex wb brdomg ax-mp ) BEFABGHABCIJCKLDABECMN
|
|
$.
|
|
|
|
$( Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
|
|
(Contributed by NM, 15-Jun-1998.) $)
|
|
domen $p |- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) $=
|
|
( vf cdom wbr cv wf1 wex cen wss brdom wf1o vex f11o exbii bitri bitr4i
|
|
wa excom bren anbi1i 19.41v ) BCFGBCEHZIZEJZBAHZKGZUHCLZTZAJZBCEDMUGBUHUE
|
|
NZUJTZEJZAJZULUGUNAJZEJUPUFUQEABCUEEOPQUNEAUARUKUOAUKUMEJZUJTUOUIURUJBUHE
|
|
UBUCUMUJEUDSQSR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Dominance in terms of equinumerosity, with the sethood requirement
|
|
expressed as an antecedent. Example 1 of [Enderton] p. 146.
|
|
(Contributed by NM, 24-Apr-2004.) $)
|
|
domeng $p |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) ) $=
|
|
( vy cv cdom wbr cen wss wex breq2 wceq sseq2 anbi2d exbidv domen vtoclbg
|
|
wa vex ) BEFZGHBAFZIHZUBUAJZSZAKBCGHUCUBCJZSZAKECDUACBGLUACMZUEUGAUHUDUFU
|
|
CUACUBNOPABUAETQR $.
|
|
$}
|
|
|
|
${
|
|
$d f A $. $d f B $. $d f F $.
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
This variation of ~ f1oeng does not require the Axiom of Replacement.
|
|
(Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro,
|
|
10-Sep-2015.) $)
|
|
f1oen3g $p |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
|
|
( vf wcel wf1o wa cv wex cen wbr f1oeq1 spcegv imp bren sylibr ) CDFZABCG
|
|
ZHABEIZGZEJZABKLRSUBUASECDABTCMNOABEPQ $.
|
|
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
This variation of ~ f1oeng does not require the Axiom of Replacement.
|
|
(Contributed by Mario Carneiro, 10-Sep-2015.) $)
|
|
f1oen2g $p |- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
|
|
( wcel wf1o w3a cvv cen wbr f1of fex2 syl3an1 3coml simp3 f1oen3g syl2anc
|
|
wf ) ADFZBEFZABCGZHCIFZUBABJKUBTUAUCUBABCSTUAUCABCLABCDEMNOTUAUBPABCIQR
|
|
$.
|
|
|
|
$( The domain of a one-to-one function is dominated by its codomain. This
|
|
variation of ~ f1domg does not require the Axiom of Replacement.
|
|
(Contributed by Mario Carneiro, 24-Jun-2015.) $)
|
|
f1dom2g $p |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B ) $=
|
|
( vf wcel wf1 w3a cdom wbr cv wex cvv wf f1f fex2 syl3an1 3coml simp3 wb
|
|
f1eq1 spcegv sylc brdomg 3ad2ant2 mpbird ) ADGZBEGZABCHZIZABJKZABFLZHZFMZ
|
|
UKCNGZUJUOUJUHUIUPUJABCOUHUIUPABCPABCDEQRSUHUIUJTUNUJFCNABUMCUBUCUDUIUHUL
|
|
UOUAUJABEFUEUFUG $.
|
|
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
(Contributed by NM, 19-Jun-1998.) $)
|
|
f1oeng $p |- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
|
|
( wcel wf1o cvv cen wbr wfo f1ofo fornex syl5 imp f1oen2g 3com23 mpd3an3
|
|
) ACEZABDFZBGEZABHIZRSTSABDJRTABDKABCDLMNRTSUAABDCGOPQ $.
|
|
|
|
$( The domain of a one-to-one function is dominated by its codomain.
|
|
(Contributed by NM, 4-Sep-2004.) $)
|
|
f1domg $p |- ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) ) $=
|
|
( vf wcel wf1 cv wex cdom wbr cvv f1dmex wf f1f sylan syldan expcom f1eq1
|
|
fex spcegv syli brdomg sylibrd ) BCFZABDGZABEHZGZEIZABJKUFUEDLFZUIUFUEUJU
|
|
FUEALFZUJABCDMUFABDNUKUJABDOABLDTPQRUHUFEDLABUGDSUAUBABCEUCUD $.
|
|
$}
|
|
|
|
${
|
|
f1oen.1 $e |- A e. _V $.
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
(Contributed by NM, 19-Jun-1998.) $)
|
|
f1oen $p |- ( F : A -1-1-onto-> B -> A ~~ B ) $=
|
|
( cvv wcel wf1o cen wbr f1oeng mpan ) AEFABCGABHIDABECJK $.
|
|
$}
|
|
|
|
${
|
|
f1dom.1 $e |- B e. _V $.
|
|
$( The domain of a one-to-one function is dominated by its codomain.
|
|
(Contributed by NM, 19-Jun-1998.) $)
|
|
f1dom $p |- ( F : A -1-1-> B -> A ~<_ B ) $=
|
|
( cvv wcel wf1 cdom wbr wi f1domg ax-mp ) BEFABCGABHIJDABECKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y f A $.
|
|
$( Express " ` A ` is finite." Definition 10.29 of [TakeutiZaring] p. 91
|
|
(whose " ` Fin ` " is a predicate instead of a class). (Contributed by
|
|
NM, 22-Aug-2008.) $)
|
|
isfi $p |- ( A e. Fin <-> E. x e. _om A ~~ x ) $=
|
|
( vy cfn wcel cen wbr com wrex cab df-fin eleq2i relen brrelexi rexlimivw
|
|
cv cvv wceq breq1 rexbidv elab3 bitri ) BDEBCPZAPZFGZAHIZCJZEBUDFGZAHIZDU
|
|
GBCAKLUFUICBUHBQEAHBUDFMNOUCBRUEUHAHUCBUDFSTUAUB $.
|
|
|
|
$( Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) $)
|
|
enssdom $p |- ~~ C_ ~<_ $=
|
|
( vx vy vf cen cdom relen cop wf1o wex copab wcel wf1 f1of1 eximi 3imtr4i
|
|
cv opabid df-en eleq2i df-dom relssi ) ABDEFAPZBPZGZUBUCCPZHZCIZABJZKZUDU
|
|
BUCUELZCIZABJZKZUDDKUDEKUGUKUIUMUFUJCUBUCUEMNUGABQUKABQODUHUDABCRSEULUDAB
|
|
CTSOUA $.
|
|
$}
|
|
|
|
$( Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94.
|
|
(Contributed by NM, 28-May-1998.) $)
|
|
endom $p |- ( A ~~ B -> A ~<_ B ) $=
|
|
( cen cdom enssdom ssbri ) CDABEF $.
|
|
|
|
$( Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
|
|
by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
enrefg $p |- ( A e. V -> A ~~ A ) $=
|
|
( wcel cen wbr cid cres wf1o f1oi f1oen2g mp3an3 anidms ) ABCZAADEZMMAAFAGZ
|
|
HNAIAAOBBJKL $.
|
|
|
|
${
|
|
enref.1 $e |- A e. _V $.
|
|
$( Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
|
|
by NM, 25-Sep-2004.) $)
|
|
enref $p |- A ~~ A $=
|
|
( cvv wcel cen wbr enrefg ax-mp ) ACDAAEFBACGH $.
|
|
$}
|
|
|
|
$( Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) $)
|
|
eqeng $p |- ( A e. V -> ( A = B -> A ~~ B ) ) $=
|
|
( wcel cen wbr wceq enrefg breq2 syl5ibcom ) ACDAAEFABGABEFACHABAEIJ $.
|
|
|
|
$( Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) $)
|
|
domrefg $p |- ( A e. V -> A ~<_ A ) $=
|
|
( wcel cen wbr cdom enrefg endom syl ) ABCAADEAAFEABGAAHI $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $. $d x y ph $.
|
|
en2d.1 $e |- ( ph -> A e. _V ) $.
|
|
en2d.2 $e |- ( ph -> B e. _V ) $.
|
|
en2d.3 $e |- ( ph -> ( x e. A -> C e. _V ) ) $.
|
|
en2d.4 $e |- ( ph -> ( y e. B -> D e. _V ) ) $.
|
|
en2d.5 $e |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) $.
|
|
$( Equinumerosity inference from an implicit one-to-one onto function.
|
|
(Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro,
|
|
12-May-2014.) $)
|
|
en2d $p |- ( ph -> A ~~ B ) $=
|
|
( cvv wcel cmpt wf1o cen wbr cv imp eqid f1od f1oen2g syl3anc ) ADMNEMNDE
|
|
BDFOZPDEQRHIABCDEFGUEMMUEUAABSDNFMNJTACSENGMNKTLUBDEUEMMUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $. $d x y ph $.
|
|
en3d.1 $e |- ( ph -> A e. _V ) $.
|
|
en3d.2 $e |- ( ph -> B e. _V ) $.
|
|
en3d.3 $e |- ( ph -> ( x e. A -> C e. B ) ) $.
|
|
en3d.4 $e |- ( ph -> ( y e. B -> D e. A ) ) $.
|
|
en3d.5 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) ) $.
|
|
$( Equinumerosity inference from an implicit one-to-one onto function.
|
|
(Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro,
|
|
12-May-2014.) $)
|
|
en3d $p |- ( ph -> A ~~ B ) $=
|
|
( cvv wcel cmpt wf1o cen cv imp wceq wbr eqid wa wb f1o2d f1oen2g syl3anc
|
|
) ADMNEMNDEBDFOZPDEQUAHIABCDEFGUHUHUBABRZDNZFENJSACRZENZGDNKSAUJULUCUIGTU
|
|
KFTUDLSUEDEUHMMUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $.
|
|
en2i.1 $e |- A e. _V $.
|
|
en2i.2 $e |- B e. _V $.
|
|
en2i.3 $e |- ( x e. A -> C e. _V ) $.
|
|
en2i.4 $e |- ( y e. B -> D e. _V ) $.
|
|
en2i.5 $e |- ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) $.
|
|
$( Equinumerosity inference from an implicit one-to-one onto function.
|
|
(Contributed by NM, 4-Jan-2004.) $)
|
|
en2i $p |- A ~~ B $=
|
|
( cen wtru cvv wcel a1i cv wi wceq wa wbr wb en2d trud ) CDLUAMABCDEFCNOM
|
|
GPDNOMHPAQZCOZENORMIPBQZDOZFNORMJPUFUGESTUHUEFSTUBMKPUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $.
|
|
en3i.1 $e |- A e. _V $.
|
|
en3i.2 $e |- B e. _V $.
|
|
en3i.3 $e |- ( x e. A -> C e. B ) $.
|
|
en3i.4 $e |- ( y e. B -> D e. A ) $.
|
|
en3i.5 $e |- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) $.
|
|
$( Equinumerosity inference from an implicit one-to-one onto function.
|
|
(Contributed by NM, 19-Jul-2004.) $)
|
|
en3i $p |- A ~~ B $=
|
|
( cen wbr wtru cvv wcel a1i cv wi wceq wa wb en3d trud ) CDLMNABCDEFCOPNG
|
|
QDOPNHQARZCPZEDPSNIQBRZDPZFCPSNJQUFUHUAUEFTUGETUBSNKQUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d y z C $. $d x z D $. $d x y ph $.
|
|
dom2d.1 $e |- ( ph -> ( x e. A -> C e. B ) ) $.
|
|
dom2d.2 $e |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) ) $.
|
|
$( A mapping (first hypothesis) that is one-to-one (second hypothesis)
|
|
implies its domain is dominated by its codomain. (Contributed by NM,
|
|
24-Jul-2004.) $)
|
|
dom2lem $p |- ( ph -> ( x e. A |-> C ) : A -1-1-> B ) $=
|
|
( cv cfv wceq wi wral wcel wa imp anbi2d wb eqeq12d cmpt wf ralrimiv eqid
|
|
wf1 fmpt sylib fvmpt2 adantll mpdan adantrr nfv nffvmpt1 nfeq1 nfim eleq1
|
|
imbi1d anbi1d anidm syl6bb fveq2 adantr biimparc ex sylbird pm5.74d bitrd
|
|
chvar adantrl biimpd sylbid ralrimivva nfmpt1 nfcv dff13f sylanbrc ) ADEB
|
|
DFUAZUBZBJZVQKZCJZVQKZLZVSWALZMZCDNBDNDEVQUEAFEOZBDNVRAWFBDHUCBDEFVQVQUDZ
|
|
UFUGAWEBCDDAVSDOZWADOZPZPZWCFGLZWDWKVTFWBGAWHVTFLZWIAWHPZWFWMAWHWFHQWHWFW
|
|
MABDFEVQWGUHUIUJZUKAWIWBGLZWHWNWMMZAWIPZWPMZBCWRWPBWRBULBWBGBDFWAUMUNUOWD
|
|
WQWRWMMWSWDWNWRWMWDWHWIAVSWADUPZRUQWDWRWMWPWDWRWKWMWPSZWDWJWIAWDWJWIWIPWI
|
|
WDWHWIWIWTURWIUSUTRWDWKXAWDWKPVTWBFGWDWCWKVSWAVQVAVBWKWLWDAWJWLWDSIQZVCTV
|
|
DVEVFVGWOVHVITWKWLWDXBVJVKVLBCDEVQBDFVMCVQVNVOVP $.
|
|
|
|
$( A mapping (first hypothesis) that is one-to-one (second hypothesis)
|
|
implies its domain is dominated by its codomain. (Contributed by NM,
|
|
24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.) $)
|
|
dom2d $p |- ( ph -> ( B e. R -> A ~<_ B ) ) $=
|
|
( cmpt wf1 wcel cdom wbr dom2lem f1domg syl5com ) ADEBDFKZLEHMDENOABCDEFG
|
|
IJPDEHSQR $.
|
|
|
|
dom3d.3 $e |- ( ph -> A e. V ) $.
|
|
dom3d.4 $e |- ( ph -> B e. W ) $.
|
|
$( A mapping (first hypothesis) that is one-to-one (second hypothesis)
|
|
implies its domain is dominated by its codomain. (Contributed by Mario
|
|
Carneiro, 20-May-2013.) $)
|
|
dom3d $p |- ( ph -> A ~<_ B ) $=
|
|
( vz cdom wbr wf1 cvv wcel syl cv wex cmpt dom2lem f1f fex2 syl3anc f1eq1
|
|
wf spcegv sylc wb brdomg mpbird ) ADEOPZDENUAZQZNUBZABDFUCZRSZDEUSQZURADE
|
|
USUIZDHSEISZUTAVAVBABCDEFGJKUDZDEUSUETLMDEUSHIUFUGVDUQVANUSRDEUPUSUHUJUKA
|
|
VCUOURULMDEINUMTUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $.
|
|
dom2.1 $e |- ( x e. A -> C e. B ) $.
|
|
dom2.2 $e |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) $.
|
|
$( A mapping (first hypothesis) that is one-to-one (second hypothesis)
|
|
implies its domain is dominated by its codomain. ` C ` and ` D ` can be
|
|
read ` C ( x ) ` and ` D ( y ) ` , as can be inferred from their
|
|
distinct variable conditions. (Contributed by NM, 26-Oct-2003.) $)
|
|
dom2 $p |- ( B e. V -> A ~<_ B ) $=
|
|
( wceq wcel cdom wbr wi eqid cv a1i wa wb dom2d ax-mp ) CCJZDGKCDLMNCOUBA
|
|
BCDEFGAPZCKZEDKNUBHQUDBPZCKREFJUCUEJSNUBIQTUA $.
|
|
|
|
$d x y V $. $d x y W $.
|
|
$( A mapping (first hypothesis) that is one-to-one (second hypothesis)
|
|
implies its domain is dominated by its codomain. ` C ` and ` D ` can be
|
|
read ` C ( x ) ` and ` D ( y ) ` , as can be inferred from their
|
|
distinct variable conditions. (Contributed by Mario Carneiro,
|
|
20-May-2013.) $)
|
|
dom3 $p |- ( ( A e. V /\ B e. W ) -> A ~<_ B ) $=
|
|
( wcel wa cv wi a1i wceq wb simpl simpr dom3d ) CGKZDHKZLZABCDEFGHAMZCKZE
|
|
DKNUCIOUEBMZCKLEFPUDUFPQNUCJOUAUBRUAUBST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
idssen $p |- _I C_ ~~ $=
|
|
( vx vy cid cen reli cv wbr cop wcel weq vex ideq eqeng ax-mp sylbi df-br
|
|
cvv wi 3imtr3i relssi ) ABCDEAFZBFZCGZUAUBDGZUAUBHZCIUEDIUCABJZUDUAUBBKLU
|
|
AQIUFUDRAKUAUBQMNOUAUBCPUAUBDPST $.
|
|
$}
|
|
|
|
$( A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed
|
|
by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) $)
|
|
ssdomg $p |- ( B e. V -> ( A C_ B -> A ~<_ B ) ) $=
|
|
( wss wcel cdom wbr wa cvv cid cres wf1 ssexg simpr wf ccnv wfun wf1o ax-mp
|
|
wfo f1oi dff1o3 mpbi simpli fof fss mpan funi cnvi funeqi mpbir jctir df-f1
|
|
funres11 sylibr adantr f1dom2g syl3anc expcom ) ABDZBCEZABFGZUTVAHAIEVAABJA
|
|
KZLZVBABCMUTVANUTVDVAUTABVCOZVCPQZHVDUTVEVFAAVCOZUTVEAAVCTZVGVHVFAAVCRVHVFH
|
|
AUAAAVCUBUCUDAAVCUESAABVCUFUGJPZQZVFVJJQUHVIJUIUJUKAJUNSULABVCUMUOUPABVCICU
|
|
QURUS $.
|
|
|
|
${
|
|
$d f g x y z $.
|
|
$( Equinumerosity is an equivalence relation. (Contributed by NM,
|
|
19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
ener $p |- ~~ Er _V $=
|
|
( vx vy vz vf cvv cen wtru a1i wbr wf1o wex bren wcel vex f1oen2g mp3an12
|
|
vg cv syl wa wrel relen ccnv f1ocnv exlimiv sylbi adantl eeanv ccom f1oco
|
|
wer ancoms exlimivv sylbir syl2anb wb enref 2th iserd trud ) EFUKGABCEFFU
|
|
AGUBHARZBRZFIZVBVAFIZGVCVAVBDRZJZDKVDVAVBDLVFVDDVFVBVAVEUCZJZVDVAVBVEUDVB
|
|
EMVAEMZVHVDBNANZVBVAVGEEOPSUEUFUGVCVBCRZFIZTVAVKFIZGVCVAVBQRZJZQKZVBVKVEJ
|
|
ZDKZVMVLVAVBQLVBVKDLVPVRTVOVQTZDKQKVMVOVQQDUHVSVMQDVSVAVKVEVNUIZJZVMVQVOW
|
|
AVAVBVKVEVNUJULVIVKEMWAVMVJCNVAVKVTEEOPSUMUNUOUGVIVAVAFIZUPGVIWBVJVAVJUQU
|
|
RHUSUT $.
|
|
$}
|
|
|
|
$( Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
|
|
Mario Carneiro, 26-Apr-2015.) $)
|
|
ensymb $p |- ( A ~~ B <-> B ~~ A ) $=
|
|
( cen wbr wb wtru cvv wer ener a1i ersymb trud ) ABCDBACDEFABCGGCHFIJKL $.
|
|
|
|
$( Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
|
|
NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) $)
|
|
ensym $p |- ( A ~~ B -> B ~~ A ) $=
|
|
( cen wbr ensymb biimpi ) ABCDBACDABEF $.
|
|
|
|
${
|
|
ensymi.2 $e |- A ~~ B $.
|
|
$( Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed
|
|
by NM, 25-Sep-2004.) $)
|
|
ensymi $p |- B ~~ A $=
|
|
( cen wbr ensym ax-mp ) ABDEBADECABFG $.
|
|
$}
|
|
|
|
${
|
|
ensymd.1 $e |- ( ph -> A ~~ B ) $.
|
|
$( Symmetry of equinumerosity. Deduction form of ~ ensym . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
ensymd $p |- ( ph -> B ~~ A ) $=
|
|
( cen wbr ensym syl ) ABCEFCBEFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z f g h A $. $d x y z f g B $. $d x y z f g h C $.
|
|
$( Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92.
|
|
(Contributed by NM, 9-Jun-1998.) $)
|
|
entr $p |- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
|
|
( cen wbr wa wi wtru cvv wer ener a1i ertr trud ) ABDEBCDEFACDEGHABCDIIDJ
|
|
HKLMN $.
|
|
|
|
$( Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94.
|
|
(Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
domtr $p |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C ) $=
|
|
( vx vy vz vg vf vh cdom reldom cv wbr wf1 wex vex brdom wa eeanv ccom
|
|
f1co ancoms coex f1eq1 spcev syl sylibr exlimivv sylbir syl2anb vtoclr )
|
|
DEFABCJKDLZELZJMULUMGLZNZGOZUMFLZHLZNZHOZULUQJMZUMUQJMULUMGEPQUMUQHFPZQUP
|
|
UTRUOUSRZHOGOVAUOUSGHSVCVAGHVCULUQILZNZIOZVAVCULUQURUNTZNZVFUSUOVHULUMUQU
|
|
RUNUAUBVEVHIVGURUNHPGPUCULUQVDVGUDUEUFULUQIVBQUGUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
entri.1 $e |- A ~~ B $.
|
|
entri.2 $e |- B ~~ C $.
|
|
$( A chained equinumerosity inference. (Contributed by NM,
|
|
25-Sep-2004.) $)
|
|
entri $p |- A ~~ C $=
|
|
( cen wbr entr mp2an ) ABFGBCFGACFGDEABCHI $.
|
|
$}
|
|
|
|
${
|
|
entr2i.1 $e |- A ~~ B $.
|
|
entr2i.2 $e |- B ~~ C $.
|
|
$( A chained equinumerosity inference. (Contributed by NM,
|
|
25-Sep-2004.) $)
|
|
entr2i $p |- C ~~ A $=
|
|
( entri ensymi ) ACABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
entr3i.1 $e |- A ~~ B $.
|
|
entr3i.2 $e |- A ~~ C $.
|
|
$( A chained equinumerosity inference. (Contributed by NM,
|
|
25-Sep-2004.) $)
|
|
entr3i $p |- B ~~ C $=
|
|
( ensymi entri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
entr4i.1 $e |- A ~~ B $.
|
|
entr4i.2 $e |- C ~~ B $.
|
|
$( A chained equinumerosity inference. (Contributed by NM,
|
|
25-Sep-2004.) $)
|
|
entr4i $p |- A ~~ C $=
|
|
( ensymi entri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( Transitivity of equinumerosity and dominance. (Contributed by NM,
|
|
7-Jun-1998.) $)
|
|
endomtr $p |- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C ) $=
|
|
( cen wbr cdom endom domtr sylan ) ABDEABFEBCFEACFEABGABCHI $.
|
|
|
|
$( Transitivity of dominance and equinumerosity. (Contributed by NM,
|
|
7-Jun-1998.) $)
|
|
domentr $p |- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C ) $=
|
|
( cen wbr cdom endom domtr sylan2 ) BCDEABFEBCFEACFEBCGABCHI $.
|
|
|
|
$( A one-to-one function's image under a subset of its domain is equinumerous
|
|
to the subset. (Contributed by Mario Carneiro, 15-May-2015.) $)
|
|
f1imaeng $p |- ( ( F : A -1-1-> B /\ C C_ A /\ C e. V ) ->
|
|
( F " C ) ~~ C ) $=
|
|
( wf1 wss wcel w3a cima cres wf1o cen f1ores f1oeng ancoms stoic3 ensymd
|
|
wbr ) ABDFZCAGZCEHZICDCJZTUACUCDCKZLZUBCUCMSZABCDNUBUEUFCUCEUDOPQR $.
|
|
|
|
$( A one-to-one function's image under a subset of its domain is equinumerous
|
|
to the subset. (This version of ~ f1imaen does not need ~ ax-setind .)
|
|
(Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro,
|
|
25-Jun-2015.) $)
|
|
f1imaen2g $p |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) ->
|
|
( F " C ) ~~ C ) $=
|
|
( wf1 wcel wa wss cima cvv cres wf1o cen wbr simprr simplr wf f1f crn ssexd
|
|
imassrn frn syl5ss syl ad2antrr f1ores ad2ant2r f1oen2g syl3anc ensymd ) AB
|
|
DFZBEGZHZCAIZCEGZHZHZCDCJZURUPUSKGCUSDCLZMZCUSNOUNUOUPPURUSBEULUMUQQULUSBIZ
|
|
UMUQULABDRZVBABDSVCUSDTBDCUBABDUCUDUEUFUAULUOVAUMUPABCDUGUHCUSUTEKUIUJUK $.
|
|
|
|
${
|
|
f1imaen.1 $e |- C e. _V $.
|
|
$( A one-to-one function's image under a subset of its domain is
|
|
equinumerous to the subset. (Contributed by NM, 30-Sep-2004.) $)
|
|
f1imaen $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C ) $=
|
|
( wf1 wss cvv wcel cima cen wbr f1imaeng mp3an3 ) ABDFCAGCHIDCJCKLEABCDHM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d f A $.
|
|
$( The empty set is equinumerous only to itself. Exercise 1 of
|
|
[TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) $)
|
|
en0 $p |- ( A ~~ (/) <-> A = (/) ) $=
|
|
( vf c0 cen wbr wceq cv wf1o wex bren ccnv f1ocnv f1o00 simprbi syl sylbi
|
|
exlimiv 0ex enref breq1 mpbiri impbii ) ACDEZACFZUCACBGZHZBIUDACBJUFUDBUF
|
|
CAUEKZHZUDACUELUHUGCFUDAUGMNOQPUDUCCCDECRSACCDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d f A $.
|
|
ensn1.1 $e |- A e. _V $.
|
|
$( A singleton is equinumerous to ordinal one. (Contributed by NM,
|
|
4-Nov-2002.) $)
|
|
ensn1 $p |- { A } ~~ 1o $=
|
|
( vf csn c0 c1o cen wbr cv wf1o wex cop 0ex f1osn opex f1oeq1 spcev ax-mp
|
|
snex bren mpbir df1o2 breqtrri ) ADZEDZFGUDUEGHUDUECIZJZCKZUDUEAELZDZJZUH
|
|
AEBMNUGUKCUJUIAEBMOSUDUEUFUJPQRUDUECTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A singleton is equinumerous to ordinal one. (Contributed by NM,
|
|
23-Apr-2004.) $)
|
|
ensn1g $p |- ( A e. V -> { A } ~~ 1o ) $=
|
|
( vx cv csn c1o cen wbr wceq sneq breq1d vex ensn1 vtoclg ) CDZEZFGHAEZFG
|
|
HCABOAIPQFGOAJKOCLMN $.
|
|
$}
|
|
|
|
$( ` { A , A } ` has only one element. (Contributed by FL, 15-Feb-2010.) $)
|
|
enpr1g $p |- ( A e. V -> { A , A } ~~ 1o ) $=
|
|
( wcel cpr csn c1o cen dfsn2 ensn1g syl5eqbrr ) ABCAADAEFGAHABIJ $.
|
|
|
|
${
|
|
$d x f y A $. $d y ph $.
|
|
$( A set is equinumerous to ordinal one iff it is a singleton.
|
|
(Contributed by NM, 25-Jul-2004.) $)
|
|
en1 $p |- ( A ~~ 1o <-> E. x A = { x } ) $=
|
|
( vf c1o cen wbr cv csn wceq wex c0 wf1o df1o2 breq2i crn syl 0ex exlimiv
|
|
wcel cvv bren bitri ccnv cfv f1ocnv wfo f1ofo forn cop wf f1of fsn2 rneqd
|
|
simprbi rnsnop syl6eq eqtr3d wfn f1ofn snid funfvex funfni sylancl eqeq2d
|
|
wi sneq spcegv sylc sylbi vex ensn1 breq1 mpbiri impbii ) BDEFZBAGZHZIZAJ
|
|
ZVOBKHZCGZLZCJZVSVOBVTEFWCDVTBEMNBVTCUAUBWBVSCWBVTBWAUCZLZBKWDUDZHZIZVSBV
|
|
TWAUEZWBWEWHWIWEWDOZBWGWEVTBWDUFWJBIVTBWDUGVTBWDUHPWEWJKWFUIHZOWGWEWDWKWE
|
|
VTBWDUJZWDWKIZVTBWDUKWLWFBSWMKBWDQULUNPUMKWFQUOUPUQPWEWFTSZWHVSVEWEWDVTUR
|
|
KVTSWNVTBWDUSKQUTWNVTKWDKWDVAVBVCVRWHAWFTVPWFIVQWGBVPWFVFVDVGPVHRVIVRVOAV
|
|
RVOVQDEFVPAVJVKBVQDEVLVMRVN $.
|
|
|
|
$( A set is equinumerous to ordinal one iff it is a singleton.
|
|
(Contributed by Jim Kingdon, 13-Apr-2020.) $)
|
|
en1bg $p |- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) ) $=
|
|
( vx wcel c1o cen wbr cuni csn cv wex en1 id unieq vex unisn syl6eq sneqd
|
|
wceq cvv eqtr4d exlimiv sylbi uniexg ensn1g syl breq1 syl5ibrcom impbid2
|
|
) ABDZAEFGZAAHZIZSZUKACJZIZSZCKUNCALUQUNCUQAUPUMUQMUQULUOUQULUPHUOAUPNUOC
|
|
OPQRUAUBUCUJUKUNUMEFGZUJULTDURABUDULTUEUFAUMEFUGUHUI $.
|
|
|
|
$( Two ways to express "exactly one". (Contributed by Stefan O'Rear,
|
|
28-Oct-2014.) $)
|
|
reuen1 $p |- ( E! x e. A ph <-> { x e. A | ph } ~~ 1o ) $=
|
|
( vy wreu crab cv csn wceq wex c1o cen wbr reusn en1 bitr4i ) ABCEABCFZDG
|
|
HIDJQKLMABDCNDQOP $.
|
|
|
|
$( Two ways to express "exactly one". (Contributed by Stefan O'Rear,
|
|
28-Oct-2014.) $)
|
|
euen1 $p |- ( E! x ph <-> { x | ph } ~~ 1o ) $=
|
|
( cvv wreu crab c1o cen wbr weu cab reuen1 reuv rabab breq1i 3bitr3i ) AB
|
|
CDABCEZFGHABIABJZFGHABCKABLPQFGABMNO $.
|
|
|
|
$( Two ways to express " ` A ` has a unique element". (Contributed by
|
|
Mario Carneiro, 9-Apr-2015.) $)
|
|
euen1b $p |- ( A ~~ 1o <-> E! x x e. A ) $=
|
|
( cv wcel weu cab c1o cen wbr euen1 abid2 breq1i bitr2i ) ACBDZAENAFZGHIB
|
|
GHINAJOBGHABKLM $.
|
|
$}
|
|
|
|
$( A singleton contains its sole element. (Contributed by Stefan O'Rear,
|
|
16-Aug-2015.) $)
|
|
en1uniel $p |- ( S ~~ 1o -> U. S e. S ) $=
|
|
( c1o cen wbr cuni csn wcel relen brrelexi uniexg snidg 3syl wceq wb simpld
|
|
cvv encv en1bg syl ibi eleqtrrd ) ABCDZAEZUCFZAUBAPGZUCPGUCUDGABCHIAPJUCPKL
|
|
UBAUDMZUBUEUBUFNUBUEBPGABQOAPRSTUA $.
|
|
|
|
${
|
|
$d x y f A $.
|
|
$( A set that dominates ordinal 2 has at least 2 different members.
|
|
(Contributed by NM, 25-Jul-2004.) $)
|
|
2dom $p |- ( 2o ~<_ A -> E. x e. A E. y e. A -. x = y ) $=
|
|
( vf c2o cdom wbr c0 csn cpr cv wf1 wceq wn wrex cfv wcel ffvelrn sylancl
|
|
notbid wex df2o2 breq1i brdomi sylbi wf f1f 0ex prid1 p0ex prid2 0nep0 wb
|
|
neii f1fveq mpanr12 mtbiri eqeq1 eqeq2 rspc2ev syl3anc exlimiv syl ) ECFG
|
|
ZHHIZJZCDKZLZDUAZAKZBKZMZNZBCOACOZVDVFCFGVIEVFCFUBUCVFCDUDUEVHVNDVHHVGPZC
|
|
QZVEVGPZCQZVOVQMZNZVNVHVFCVGUFZHVFQZVPVFCVGUGZHVEUHUIZVFCHVGRSVHWAVEVFQZV
|
|
RWCHVEUJUKZVFCVEVGRSVHVSHVEMZHVEULUNVHWBWEVSWGUMWDWFVFCHVEVGUOUPUQVMVTVOV
|
|
KMZNABVOVQCCVJVOMVLWHVJVOVKURTVKVQMWHVSVKVQVOUSTUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w F $.
|
|
fundmen.1 $e |- F e. _V $.
|
|
$( A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
|
|
(Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro,
|
|
15-Nov-2014.) $)
|
|
fundmen $p |- ( Fun F -> dom F ~~ F ) $=
|
|
( vx vy vz vw cv cop cint cvv wcel a1i ex wi wa wceq wb wex vex adantl
|
|
wfun cdm cfv dmex funfvop wrel funrel elreldm syl cxp df-rel sylib sselda
|
|
wss elvv inteq inteqd op1stb syl6eq eqeq1 syl5ibr opeq1 syl6 imp biimprcd
|
|
eqeq2 ancoms eleq1d funopfv adantr sylbid exp32 com24 imp43 opeq2d eqtr4d
|
|
mpd exlimdvv adantrl funfvex op1stbg sylancr eqtr2d adantrr impbid en3d )
|
|
AUAZCDAUBZACGZWIAUCZHZDGZIZIZWHJKWGABUDLAJKWGBLWGWIWHKZWKAKWIAUEMWGAUFZWL
|
|
AKZWNWHKZNAUGZWPWQWRAWLUHMUIWGWOWQOZWIWNPZWLWKPZQWGWTOXAXBWGWQXAXBNZWOWGW
|
|
QOZWLEGZFGZHZPZFRERZXCXDWLJJUJZKXIWGAXJWLWGWPAXJUNWSAUKULUMEFWLUOULXDXHXC
|
|
EFXDXHXAXBXDXHXAOZOZWLWIXFHZWKXKWLXMPZXDXAXHXNXAXHOZXMXGPZXNXAXHXPXAXHWIX
|
|
EPZXPXHXQXAWNXEPXHWNXGIZIXEXHWMXRWLXGUPUQXEXFESFSURUSWIWNXEUTVAWIXEXFVBVC
|
|
VDXHXPXNNXAXPXNXHXMXGWLVFVETVQZVGTXLWJXFWIWGWQXHXAWJXFPZWGXAXHWQXTWGXAXHW
|
|
QXTNWGXOOWQXMAKZXTXOWQYAQWGXOWLXMAXSVHTWGYAXTNXOWIXFAVIVJVKVLVMVNVOVPVLVR
|
|
VQVSWGWOXBXANWQWGWOOZXBXAYBXBOWNWKIZIZWIXBWNYDPYBXBWMYCWLWKUPUQTYBYDWIPZX
|
|
BYBWIJKWJJKYECSWIAVTWIWJJJWAWBVJWCMWDWEMWF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
|
|
(Contributed by NM, 17-Sep-2013.) $)
|
|
fundmeng $p |- ( ( F e. V /\ Fun F ) -> dom F ~~ F ) $=
|
|
( vx wcel wfun cdm cen wbr cv wceq funeq dmeq breq12d imbi12d vex fundmen
|
|
wi id vtoclg imp ) ABDAEZAFZAGHZCIZEZUDFZUDGHZQUAUCQCABUDAJZUEUAUGUCUDAKU
|
|
HUFUBUDAGUDALUHRMNUDCOPST $.
|
|
|
|
$( A relational set is equinumerous to its converse. (Contributed by Mario
|
|
Carneiro, 28-Dec-2014.) $)
|
|
cnven $p |- ( ( Rel A /\ A e. V ) -> A ~~ `' A ) $=
|
|
( vx wrel wcel wa ccnv cvv csn cuni cmpt wf1o cen wbr simpr cnvexg adantl
|
|
cv cnvf1o adantr f1oen2g syl3anc ) ADZABEZFUDAGZHEZAUECACRIGJKZLZAUEMNUCU
|
|
DOUDUFUCABPQUCUHUDCASTAUEUGBHUAUB $.
|
|
$}
|
|
|
|
$( A function is equinumerate to its domain. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
fndmeng $p |- ( ( F Fn A /\ A e. C ) -> A ~~ F ) $=
|
|
( wfn wcel wa cdm cen wbr wfun fnex fnfun adantr fundmeng syl2anc wb breq1d
|
|
cvv fndm mpbid ) CADZABEZFZCGZCHIZACHIZUCCRECJZUEABCKUAUGUBACLMCRNOUAUEUFPU
|
|
BUAUDACHACSQMT $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y V $.
|
|
$( Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) $)
|
|
en2sn $p |- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } ) $=
|
|
( wcel csn c1o cen wbr ensn1g ensymd entr syl2an ) ACEAFZGHIGBFZHINOHIBDE
|
|
ZACJPOGBDJKNGOLM $.
|
|
|
|
$( A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.) $)
|
|
snfig $p |- ( A e. V -> { A } e. Fin ) $=
|
|
( vx wcel csn cv cen wbr com wrex cfn c1o 1onn ensn1g rspcev sylancr isfi
|
|
breq2 sylibr ) ABDZAEZCFZGHZCIJZUAKDTLIDUALGHZUDMABNUCUECLIUBLUAGROPCUAQS
|
|
$.
|
|
|
|
$( The class of finite sets is a proper class. (Contributed by Jeff
|
|
Hankins, 3-Oct-2008.) $)
|
|
fiprc $p |- Fin e/ _V $=
|
|
( vx vy cv csn wceq wex cab cvv wnel cfn snnex wcel wss snfig ax-mp eleq1
|
|
wn vex mpbiri df-nel exlimiv abssi ssexg mpan con3i 3imtr4i ) ACZBCZDZEZB
|
|
FZAGZHIZJHIZABKULHLZQJHLZQUMUNUPUOULJMUPUOUKAJUJUGJLZBUJUQUIJLZUHHLURBRUH
|
|
HNOUGUIJPSUAUBULJHUCUDUEULHTJHTUFO $.
|
|
|
|
$( Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
|
|
(Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
unen $p |- ( ( ( A ~~ B /\ C ~~ D ) /\
|
|
( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) ) $=
|
|
( vx vy cen wbr wa cin c0 wceq cun cv wf1o wex wi bren cvv vex eeanv wcel
|
|
unex f1oun f1oen3g sylancr ex exlimivv sylbir syl2anb imp ) ABGHZCDGHZIAC
|
|
JKLBDJKLIZACMZBDMZGHZULABENZOZEPZCDFNZOZFPZUNUQQZUMABERCDFRUTVCIUSVBIZFPE
|
|
PVDUSVBEFUAVEVDEFVEUNUQVEUNIURVAMZSUBUOUPVFOUQURVAETFTUCABCDURVAUDUOUPVFS
|
|
UEUFUGUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( A singleton (whether of a set or a proper class) is dominated by one.
|
|
(Contributed by Jim Kingdon, 1-Mar-2022.) $)
|
|
1domsn $p |- { A } ~<_ 1o $=
|
|
( vx vy csn c1o c0 cmpt wf1 cdom wbr wcel wral cv wi 0lt1o rgenw wa elsni
|
|
wceq adantr adantl eqtr4d a1d rgen2a eqid eqidd f1mpt mpbir2an 1onn elexi
|
|
com f1dom ax-mp ) ADZEBUNFGZHZUNEIJUPFEKZBUNLFFSZBMZCMZSZNZCUNLBUNLUQBUNO
|
|
PVBBCUNUSUNKZUTUNKZQZVAURVEUSAUTVCUSASVDUSARTVDUTASVCUTARUAUBUCUDBCUNEFFU
|
|
OUOUEVAFUFUGUHUNEUOEUKUIUJULUM $.
|
|
$}
|
|
|
|
${
|
|
$d f x y A $. $d f x y B $.
|
|
$( A set equinumerous to an inhabited set is inhabited. (Contributed by
|
|
Jim Kingdon, 19-May-2020.) $)
|
|
enm $p |- ( ( A ~~ B /\ E. x x e. A ) -> E. y y e. B ) $=
|
|
( vf cv wcel wex cen wbr wi wf1o bren wf f1of wa cfv ffvelrn syl exlimiv
|
|
elex2 ex sylbi com12 impcom ) AFZCGZAHCDIJZBFDGBHZUGUHUIKAUHUGUIUHCDEFZLZ
|
|
EHUGUIKZCDEMUKULEUKCDUJNZULCDUJOUMUGUIUMUGPUFUJQZDGUICDUFUJRBUNDUASUBSTUC
|
|
UDTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
xpsnen.1 $e |- A e. _V $.
|
|
xpsnen.2 $e |- B e. _V $.
|
|
$( A set is equinumerous to its Cartesian product with a singleton.
|
|
Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM,
|
|
4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
xpsnen $p |- ( A X. { B } ) ~~ A $=
|
|
( vy vx vz cv cint cop wcel wceq wex cvv inteq inteqd vex op1stb adantr
|
|
wa csn cxp snex xpex elxp syl6eq syl6eqel exlimivv sylbi opex wb eqvisset
|
|
a1i ancom anass velsn anbi1i 3bitr3i exbii eqeq2d anbi1d ceqsexv pm4.71ri
|
|
opeq2 syl6req bitri 3bitri opeq1 anbi12d ceqsexgv syl5bb pm5.32ri pm4.71i
|
|
eleq1 syl bitr2i en2i ) EFABUAZUBZAEHZIZIZFHZBJZAVRCBDUCUDCVTVSKZVTWCGHZJ
|
|
ZLZWCAKZWFVRKZTZTZGMZFMZWBNKZFGVTAVRUEZWLWOFGWHWOWKWHWBWCNWHWBWGIZIWCWHWA
|
|
WQVTWGOPWCWFFQZGQRUFWRUGSUHUIWDNKWIWCBWRDUJUMWEWCWBLZTVTWBBJZLZWBAKZTZWST
|
|
ZVTWDLZWITZWIXETWSWEXCWSWOWEXCUKFWBULWEWSXFTZFMZWOXCWEWNXHWPWMXGFWMWFBLZW
|
|
HWITZTZGMXFXGWLXKGXJWJTWJXJTWLXKXJWJUNWHWIWJUOWJXIXJGBUPUQURUSXJXFGBDXIWH
|
|
XEWIXIWGWDVTWFBWCVDUTVAVBXFWSXETZWITXGXEXLWIXEWSXEWBWDIZIWCXEWAXMVTWDOPWC
|
|
BWRDRVEZVCUQWSXEWIUOVFVGUSVFXFXCFWBNWSXEXAWIXBWSWDWTVTWCWBBVHUTWCWBAVNVIZ
|
|
VJVKVOVLXFXFWSTXDXFWSXEWSWIXNSVMWSXFXCXOVLVPXEWIUNVGVQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( A set is equinumerous to its Cartesian product with a singleton.
|
|
Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM,
|
|
22-Oct-2004.) $)
|
|
xpsneng $p |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A ) $=
|
|
( vx vy cv csn cxp cen wbr wceq xpeq1 id breq12d xpeq2d breq1d vex xpsnen
|
|
sneq vtocl2g ) EGZFGZHZIZUBJKAUDIZAJKABHZIZAJKEFABCDUBALZUEUFUBAJUBAUDMUI
|
|
NOUCBLZUFUHAJUJUDUGAUCBTPQUBUCERFRSUA $.
|
|
$}
|
|
|
|
$( One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised
|
|
by Mario Carneiro, 29-Apr-2015.) $)
|
|
xp1en $p |- ( A e. V -> ( A X. 1o ) ~~ A ) $=
|
|
( wcel c1o cxp c0 csn cen df1o2 xpeq2i cvv wbr 0ex xpsneng mpan2 syl5eqbr )
|
|
ABCZADEAFGZEZAHDRAIJQFKCSAHLMAFBKNOP $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
endisj.1 $e |- A e. _V $.
|
|
endisj.2 $e |- B e. _V $.
|
|
$( Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
|
|
[Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) $)
|
|
endisj $p |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) ) $=
|
|
( c0 csn cxp cen wbr c1o wa cin wceq cv wex xpsnen xpex breq1 con0 pm3.2i
|
|
0ex elexi xp01disj p0ex snex bi2anan9 ineq12 eqeq1d anbi12d spc2ev mp2an
|
|
1on ) CGHZIZCJKZDLHZIZDJKZMZUPUSNZGOZAPZCJKZBPZDJKZMZVDVFNZGOZMZBQAQUQUTC
|
|
GEUCRDLFLUAUNUDZRUBCDUEVKVAVCMABUPUSCUOEUFSDURFLVLUGSVDUPOZVFUSOZMZVHVAVJ
|
|
VCVMVEUQVNVGUTVDUPCJTVFUSDJTUHVOVIVBGVDUPVFUSUIUJUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x y z A $. $d u v w x y z B $. $d u v C $. $d u v w y z F $.
|
|
$d u v w G $.
|
|
xpcomf1o.1 $e |- F = ( x e. ( A X. B ) |-> U. `' { x } ) $.
|
|
$( The canonical bijection from ` ( A X. B ) ` to ` ( B X. A ) ` .
|
|
(Contributed by Mario Carneiro, 23-Apr-2014.) $)
|
|
xpcomf1o $p |- F : ( A X. B ) -1-1-onto-> ( B X. A ) $=
|
|
( cxp ccnv wf1o cv cuni cmpt wrel relxp cnvf1o ax-mp wceq wb f1oeq1 mpbir
|
|
csn cnvxp f1oeq3 mpbi ) BCFZUDGZDHZUDCBFZDHZUFUDUEAUDAITGJKZHZUDLUJBCMAUD
|
|
NODUIPUFUJQEUDUEDUIROSUEUGPUFUHQBCUAUEUGUDDUBOUC $.
|
|
|
|
xpcomco.1 $e |- G = ( y e. B , z e. A |-> C ) $.
|
|
$( Composition with the bijection of ~ xpcomf1o swaps the arguments to a
|
|
mapping. (Contributed by Mario Carneiro, 30-May-2015.) $)
|
|
xpcomco $p |- ( G o. F ) = ( z e. A , y e. B |-> C ) $=
|
|
( vu vw vv cv wbr wa wex wceq wcel cvv copab cop ccom cmpt2 cfv wf1o wfun
|
|
cxp cdm wb xpcomf1o f1ofun funbrfv2b mp2b ancom eqcom f1odm ax-mp anbi12i
|
|
eleq2i 3bitri anbi1i anass bitri exbii csn ccnv cuni wal vex mptfvex snex
|
|
mpan2 cnvex uniex mpg breq1 anbi2d ceqsexv elxp nfcv nfmpt22 nfcxfr 19.41
|
|
nfbr nfmpt21 fveq2 opelxpi sneq cnveqd unieqd mp2an syl6eq opex fvmpt syl
|
|
opswapg sylan9eq breq1d coprab df-br df-mpt2 oprabid baib ancoms pm5.32da
|
|
eqtri adantl bitrd pm5.32i bitr3i 3bitr2i opabbii df-co dfoprab2 3eqtr4i
|
|
) KNZLNZGOZXRMNZHOZPZLQZKMUAXQCNZBNZUBZRZYDDSZYEESZPZXTFRZPZPZBQZCQZKMUAZ
|
|
HGUCCBDEFUDZYCYOKMYCXRXQGUEZRZXQDEUHZSZYAPZPZLQUUAYRXTHOZPZYOYBUUCLYBYSUU
|
|
APZYAPUUCXSUUFYAXSXQGUIZSZYRXRRZPZUUIUUHPUUFYTEDUHZGUFZGUGXSUUJUJADEGIUKZ
|
|
YTUUKGULXQXRGUMUNUUHUUIUOUUIYSUUHUUAYRXRUPUUGYTXQUULUUGYTRUUMYTUUKGUQURUT
|
|
USVAVBYSUUAYAVCVDVEUUBUUELYRANZVFZVGZVHZTSZYRTSZAUURAVIXQTSUUSKVJAYTUUQXQ
|
|
GTTIVKVMUUPUUOUUNAVJVLVNVOVPYSYAUUDUUAXRYRXTHVQVRVSUUEYGYJPZBQZCQZUUDPUVA
|
|
UUDPZCQYOUUAUVBUUDCBXQDEVTVBUVAUUDCCYRXTHCYRWACHBCEDFUDZJBCEDFWBWCCXTWAWE
|
|
WDUVCYNCUVCUUTUUDPZBQYNUUTUUDBBYRXTHBYRWABHUVDJBCEDFWFWCBXTWAWEWDUVEYMBUV
|
|
EYGYJUUDPZPYMYGYJUUDVCYGUVFYLYGYJUUDYKUUTUUDYEYDUBZXTHOZYKUUTYRUVGXTHYGYJ
|
|
YRYFGUEZUVGXQYFGWGYJYFYTSUVIUVGRYDYEDEWHAYFUUQUVGYTGUUNYFRZUUQYFVFZVGZVHZ
|
|
UVGUVJUUPUVLUVJUUOUVKUUNYFWIWJWKYDTSYETSUVMUVGRCVJZBVJZYDYETTWQWLWMIYEYDU
|
|
VOUVNWNWOWPWRWSYJUVHYKUJZYGYIYHUVPUVHYIYHPZYKUVHUVGXTUBZHSUVRUVQYKPZBCMWT
|
|
ZSUVSUVGXTHXAHUVTUVRHUVDUVTJBCMEDFXBXGUTUVSBCMXCVAXDXEXHXIXFXJVDVEXKVEXLV
|
|
AXMKMLHGXNYQYLCBMWTYPCBMDEFXBYLCBMKXOXGXP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
xpcomen.1 $e |- A e. _V $.
|
|
xpcomen.2 $e |- B e. _V $.
|
|
$( Commutative law for equinumerosity of Cartesian product. Proposition
|
|
4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
xpcomen $p |- ( A X. B ) ~~ ( B X. A ) $=
|
|
( cxp cvv wcel csn ccnv cuni cmpt wf1o cen wbr xpex eqid xpcomf1o f1oen2g
|
|
vx cv mp3an ) ABEZFGBAEZFGUBUCSUBSTHIJKZLUBUCMNABCDOBADCOSABUDUDPQUBUCUDF
|
|
FRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( Commutative law for equinumerosity of Cartesian product. Proposition
|
|
4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.) $)
|
|
xpcomeng $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) ) $=
|
|
( vx vy cv cxp cen wbr wceq xpeq1 xpeq2 breq12d vex xpcomen vtocl2g ) EGZ
|
|
FGZHZSRHZIJASHZSAHZIJABHZBAHZIJEFABCDRAKTUBUAUCIRASLRASMNSBKUBUDUCUEISBAM
|
|
SBALNRSEOFOPQ $.
|
|
$}
|
|
|
|
$( A set is equinumerous to its Cartesian product with a singleton on the
|
|
left. (Contributed by Stefan O'Rear, 21-Nov-2014.) $)
|
|
xpsnen2g $p |- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B ) $=
|
|
( wcel csn cxp cen wbr cvv snexg xpcomeng sylan xpsneng ancoms entr syl2anc
|
|
wa ) ACEZBDEZRAFZBGZBUAGZHIZUCBHIZUBBHISUAJETUDACKUABJDLMTSUEBADCNOUBUCBPQ
|
|
$.
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
xpassen.1 $e |- A e. _V $.
|
|
xpassen.2 $e |- B e. _V $.
|
|
xpassen.3 $e |- C e. _V $.
|
|
$( Associative law for equinumerosity of Cartesian product. Proposition
|
|
4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.)
|
|
(Revised by Mario Carneiro, 15-Nov-2014.) $)
|
|
xpassen $p |- ( ( A X. B ) X. C ) ~~ ( A X. ( B X. C ) ) $=
|
|
( vz vw vv vu csn cdm cuni crn cop wcel uniex wceq wa wex vx cxp xpex cvv
|
|
vy vex snex dmex rnex opex a1i sneq dmeqd unieqd sneqd op1sta sneqi dmeqi
|
|
unieqi eqtri syl6req rneqd rneqi op2nda opeq12d eq2tri anass anbi12i an32
|
|
3bitr4i exbii 19.41v 3bitr3i 2exbii 19.41vv elxp excom anbi1i an12 exrot4
|
|
cv opeq1 eqeq2d anbi1d anbi2d ceqsexv 3bitri 3exbii anbi2i 19.42vv bitr3i
|
|
bitri exrot3 opeq2 en2i ) UAUEABUBZCUBZABCUBZUBZUAWAZKZLZMZKZLZMZXDNZMZXA
|
|
NZMZOZOZUEWAZKZLZMZXNNZMZKZLZMZOZXSNZMZOZWPCABDEUCFUCAWRDBCEFUCUCXLUDPWTW
|
|
QPZXFXKXEXDXCXBXAWTUAUFUGZUHQUGZUHQXHXJXGXDYHUIQXIXAYGUIQUJUJUKYEUDPXMWSP
|
|
ZYBYDXPYAXOXNXMUEUFUGZUHQXTXSXRXQXNYJUIQUGZUHQUJYCXSYKUIQUJUKWTGWAZHWAZOZ
|
|
IWAZOZRZYLAPZYMBPZSZYOCPZSZSZITZHTGTZXMXLRZSZXMYLYMYOOZOZRZYRYSUUASZSZSZI
|
|
TZHTZGTZWTYERZSZYFUUFSYIUUQSUUDUUFSZHTGTUUNUUQSZHTGTUUGUURUUSUUTGHUUCUUFS
|
|
ZITUUMUUQSZITUUSUUTUVAUVBIYQUUFSZUUBSUUJUUQSZUULSUVAUVBUVCUVDUUBUULWTXMYP
|
|
UUIXLYEYQYLXFUUHXKYQXFYPKZLZMZKZLZMZYLYQXEUVIYQXDUVHYQXCUVGYQXBUVFYQXAUVE
|
|
WTYPULZUMUNUOZUMUNUVJYNKZLZMYLUVIUVNUVHUVMUVGYNYNYOYLYMGUFZHUFZUJZIUFZUPU
|
|
QZURUSYLYMUVOUVPUPUTVAYQYMXHYOXJYQXHUVHNZMZYMYQXGUVTYQXDUVHUVLVBUNUWAUVMN
|
|
ZMYMUVTUWBUVHUVMUVSVCUSYLYMUVOUVPVDUTVAYQXJUVENZMYOYQXIUWCYQXAUVEUVKVBUNY
|
|
NYOUVQUVRVDVAVEVEUUJYNYBYOYDUUJYLXPYMYAUUJXPUUIKZLZMYLUUJXOUWEUUJXNUWDXMU
|
|
UIULZUMUNYLUUHUVOYMYOUVPUVRUJZUPVAUUJYAUWDNZMZKZLZMZYMUUJXTUWKUUJXSUWJUUJ
|
|
XRUWIUUJXQUWHUUJXNUWDUWFVBUNUOZUMUNUWLUUHKZLZMYMUWKUWOUWJUWNUWIUUHYLUUHUV
|
|
OUWGVDUQZURUSYMYOUVPUVRUPUTVAVEUUJYDUWJNZMZYOUUJYCUWQUUJXSUWJUWMVBUNUWRUW
|
|
NNZMYOUWQUWSUWJUWNUWPVCUSYMYOUVPUVRVDUTVAVEVFYRYSUUAVGVHYQUUBUUFVIUUJUULU
|
|
UQVIVJVKUUCUUFIVLUUMUUQIVLVMVNUUDUUFGHVOUUNUUQGHVOVMYFUUEUUFYFWTJWAZYOOZR
|
|
ZUWTWPPZUUASSZITJTUXDJTITZUUEJIWTWPCVPUXDJIVQUXEUWTYNRZYTSZUXBUUASZSZHTGT
|
|
ZJTITUXIJTZITHTGTUUEUXDUXJIJUXCUXHSUXGHTGTZUXHSUXDUXJUXCUXLUXHGHUWTABVPVR
|
|
UXBUXCUUAVSUXGUXHGHVOVJVNUXIIJGHVTUXKUUCGHIUXKUXFYTUXHSZSZJTYTYQUUASZSZUU
|
|
CUXIUXNJUXFYTUXHVGVKUXMUXPJYNUVQUXFUXHUXOYTUXFUXBYQUUAUXFUXAYPWTUWTYNYOWB
|
|
WCWDWEWFYTYQUUAVSWGWHWGWGVRYIUUPUUQYIXMYLUWTOZRZYRUWTWRPZSSZJTZGTUUPGJXMA
|
|
WRVPUYAUUOGUYAUWTUUHRZUXRUULSZSZITHTZJTUYDJTZITHTUUOUXTUYEJUXRYRSZUXSSUYG
|
|
UYBUUKSZITHTZSZUXTUYEUXSUYIUYGHIUWTBCVPWIUXRYRUXSVGUYJUYGUYHSZITHTUYEUYGU
|
|
YHHIWJUYKUYDHIUYKUYBUYGUUKSZSUYDUYGUYBUUKVSUYLUYCUYBUXRYRUUKVGWIWLVNWKVMV
|
|
KUYDJHIWMUYFUUMHIUYCUUMJUUHUWGUYBUXRUUJUULUYBUXQUUIXMUWTUUHYLWNWCWDWFVNWG
|
|
VKWLVRVJWO $.
|
|
$}
|
|
|
|
${
|
|
$d f u v w x y z A $. $d f u v w x y z B $. $d f u v w x y z C $.
|
|
xpdom.2 $e |- C e. _V $.
|
|
$( Dominance law for Cartesian product. Proposition 10.33(2) of
|
|
[TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by
|
|
Mario Carneiro, 15-Nov-2014.) $)
|
|
xpdom2 $p |- ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) $=
|
|
( vf vz vw vv vu cdom cv wa cuni cop cvv wcel wi wceq wb vex vx vy brdomi
|
|
wbr wf1 cxp csn cdm crn cfv wf f1f ffvelrn ex anim2d adantld elxp4 opelxp
|
|
syl 3imtr4g adantl wrex elxp2 fvex opth f1fveq ancoms anbi2d ad2ant2l imp
|
|
syl5bb adantlr dmeqd unieqd op1sta syl6eq op2nda fveq2d opeq12d eqeqan12d
|
|
sneq rneqd ad2antlr eqeq12 syl6bb 3bitr4d exp53 com23 rexlimivv rexlimdvv
|
|
syl2anb com12 reldom brrelexi xpexg sylancr adantr brrelex2i exlimddv
|
|
dom3d ) ABJUDZABEKZUEZCAUFZCBUFZJUDEABEUCXAXCLUAUBXDXEUAKZUGZUHZMZXGUIZMZ
|
|
XBUJZNZUBKZUGZUHZMZXOUIZMZXBUJZNZOOXCXFXDPZXMXEPZQXAXCXFXIXKNRZXICPZXKAPZ
|
|
LZLYEXLBPZLZYBYCXCYGYIYDXCYFYHYEXCABXBUKZYFYHQABXBULYJYFYHABXKXBUMUNUSUOU
|
|
PXFCAUQXIXLCBURUTVAXCYBXNXDPZLZXMYARZXFXNRZSZQXAYLXCYOYBXFFKZGKZNZRZGAVBF
|
|
CVBZXNHKZIKZNZRZIAVBHCVBZXCYOQZYKFGXFCAVCHIXNCAVCYTUUEUUFYTUUDUUFHICAYSUU
|
|
ACPZUUBAPZLZUUDUUFQZQFGCAYPCPZYQAPZLZUUIYSUUJUUMUUIYSUUDXCYOUUMUUILZYSUUD
|
|
LZLXCLYPYQXBUJZNZUUAUUBXBUJZNZRZYPUUARZYQUUBRZLZYMYNUUNXCUUTUVCSZUUOUUNXC
|
|
UVDUULUUHXCUVDQUUKUUGUULUUHLZXCUVDUUTUVAUUPUURRZLUVEXCLZUVCYPUUPUUAUURFTZ
|
|
YQXBOOETGTZVDVEUVGUVFUVBUVAXCUVEUVFUVBSABYQUUBXBVFVGVHVKUNVIVJVLUUOYMUUTS
|
|
UUNXCYSUUDXMUUQYAUUSYSXIYPXLUUPYSXIYRUGZUHZMYPYSXHUVKYSXGUVJXFYRWAZVMVNYP
|
|
YQUVHUVIVOVPYSXKYQXBYSXKUVJUIZMYQYSXJUVMYSXGUVJUVLWBVNYPYQUVHUVIVQVPVRVSU
|
|
UDXQUUAXTUURUUDXQUUCUGZUHZMUUAUUDXPUVOUUDXOUVNXNUUCWAZVMVNUUAUUBHTZITZVOV
|
|
PUUDXSUUBXBUUDXSUVNUIZMUUBUUDXRUVSUUDXOUVNUVPWBVNUUAUUBUVQUVRVQVPVRVSVTWC
|
|
UUOYNUVCSUUNXCUUOYNYRUUCRUVCXFYRXNUUCWDYPYQUUAUUBUVHUVIVEWEWCWFWGWHWIWJVJ
|
|
WKWLVAXAXDOPZXCXACOPZAOPUVTDABJWMWNCAOOWOWPWQXAXEOPZXCXAUWABOPUWBDABJWMWR
|
|
CBOOWOWPWQWTWS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x V $. $d x W $.
|
|
$( Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
|
|
p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
xpdom2g $p |- ( ( C e. V /\ A ~<_ B ) ->
|
|
( C X. A ) ~<_ ( C X. B ) ) $=
|
|
( vx wcel cdom wbr cxp cv wceq xpeq1 breq12d imbi2d vex xpdom2 vtoclg imp
|
|
wi ) CDFABGHZCAIZCBIZGHZTEJZAIZUDBIZGHZSTUCSECDUDCKZUGUCTUHUEUAUFUBGUDCAL
|
|
UDCBLMNABUDEOPQR $.
|
|
|
|
$( Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
|
|
p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
xpdom1g $p |- ( ( C e. V /\ A ~<_ B ) ->
|
|
( A X. C ) ~<_ ( B X. C ) ) $=
|
|
( wcel cdom wbr wa cxp cen reldom brrelexi xpcomeng ancoms sylan2 xpdom2g
|
|
cvv brrelex2i domentr syl2anc endomtr ) CDEZABFGZHZACIZCAIZJGZUFBCIZFGZUE
|
|
UHFGUCUBAQEZUGABFKLUJUBUGACQDMNOUDUFCBIZFGUKUHJGZUIABCDPUCUBBQEULABFKRCBD
|
|
QMOUFUKUHSTUEUFUHUAT $.
|
|
|
|
$( A set is dominated by its Cartesian product with an inhabited set.
|
|
Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon,
|
|
15-Apr-2020.) $)
|
|
xpdom3m $p |- ( ( A e. V /\ B e. W /\ E. x x e. B ) ->
|
|
A ~<_ ( A X. B ) ) $=
|
|
( wcel cv wex cxp cdom wbr wa w3a csn cen xpsneng 3adant2 ensymd cvv wss
|
|
xpexg 3adant3 simp3 snssd xpss2 syl ssdomg endomtr syl2anc 3expia exlimdv
|
|
sylc 3impia ) BDFZCEFZAGZCFZAHBBCIZJKZUNUOLUQUSAUNUOUQUSUNUOUQMZBBUPNZIZO
|
|
KVBURJKZUSUTVBBUNUQVBBOKUOBUPDCPQRUTURSFZVBURTZVCUNUOVDUQBCDEUAUBUTVACTVE
|
|
UTUPCUNUOUQUCUDVACBUEUFVBURSUGULBVBURUHUIUJUKUM $.
|
|
$}
|
|
|
|
${
|
|
xpdom1.2 $e |- C e. _V $.
|
|
$( Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
|
|
p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM,
|
|
29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.) $)
|
|
xpdom1 $p |- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) ) $=
|
|
( cvv wcel cdom wbr cxp xpdom1g mpan ) CEFABGHACIBCIGHDABCEJK $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B a b $. $d F a b $.
|
|
$( Covering implies injection on power sets. (Contributed by Stefan
|
|
O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) $)
|
|
fopwdom $p |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A ) $=
|
|
( va vb cvv wcel wa cpw cv cima wss crn syl adantl imaeq2 elpwid foimacnv
|
|
wceq wb wfo ccnv imassrn cdm dfdm4 fof fdm syl5eqr syl5sseq cnvexg adantr
|
|
wf imaexg 3syl mpbird a1d simpllr simplrl syl2anc simplrr 3eqtr3d impbid1
|
|
elpwg ex rnexg forn eleq1d syl5ibcom imp pwexg dmfex sylan2 dom3d ) CFGZA
|
|
BCUAZHZDEBIZAIZCUBZDJZKZVSEJZKZFFVPWAVRGZVTVQGZVPWDWAALZVOWFVNVOVSMZWAAVS
|
|
VTUCVOWGCUDZACUEVOABCULZWHASABCUFZABCUGNUHUIOVPVSFGZWAFGWDWFTVNWKVOCFUJUK
|
|
VSVTFUMWAAFVCUNUOUPVPWEWBVQGZHZWAWCSZVTWBSZTVPWMHZWNWOWPWNWOWPWNHZCWAKZCW
|
|
CKZVTWBWNWRWSSWPWAWCCPOWQVOVTBLWRVTSVNVOWMWNUQZWQVTBVPWEWLWNURQABVTCRUSWQ
|
|
VOWBBLWSWBSWTWQWBBVPWEWLWNUTQABWBCRUSVAVDVTWBVSPVBVDVPBFGZVQFGVNVOXAVNCMZ
|
|
FGVOXACFVEVOXBBFABCVFVGVHVIBFVJNVPAFGZVRFGVOVNWIXCWJABFCVKVLAFVJNVM $.
|
|
$}
|
|
|
|
$( Equality-like theorem for equinumerosity. (Contributed by NM,
|
|
18-Dec-2003.) $)
|
|
enen1 $p |- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) ) $=
|
|
( cen wbr ensym entr sylan impbida ) ABDEZACDEZBCDEZJBADEKLABFBACGHABCGI $.
|
|
|
|
$( Equality-like theorem for equinumerosity. (Contributed by NM,
|
|
18-Dec-2003.) $)
|
|
enen2 $p |- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) ) $=
|
|
( cen wbr entr ancoms ensym sylan impbida ) ABDEZCADEZCBDEZLKMCABFGKBADEZML
|
|
ABHMNLCBAFGIJ $.
|
|
|
|
$( Equality-like theorem for equinumerosity and dominance. (Contributed by
|
|
NM, 8-Nov-2003.) $)
|
|
domen1 $p |- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) ) $=
|
|
( cen wbr cdom ensym endomtr sylan impbida ) ABDEZACFEZBCFEZKBADELMABGBACHI
|
|
ABCHJ $.
|
|
|
|
$( Equality-like theorem for equinumerosity and dominance. (Contributed by
|
|
NM, 8-Nov-2003.) $)
|
|
domen2 $p |- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) ) $=
|
|
( cen wbr cdom domentr ancoms ensym sylan impbida ) ABDEZCAFEZCBFEZMLNCABGH
|
|
LBADEZNMABINOMCBAGHJK $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Pigeonhole Principle
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Lemma for Pigeonhole Principle. If we join a natural number to itself
|
|
minus an element, we end up with its successor minus the same element.
|
|
(Contributed by NM, 25-May-1998.) $)
|
|
phplem1 $p |- ( ( A e. _om /\ B e. A ) ->
|
|
( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) ) $=
|
|
( com wcel wa csn cin c0 wceq cdif cun csuc word nnord nordeq disjsn2 sylan
|
|
wne syl undif4 df-suc equncomi difeq1i syl6eqr ) ACDZBADZEAFZBFZGHIZUGAUHJK
|
|
ZALZUHJZIUEAMZUFUIANUMUFEABRUIABOABPSQUIUJUGAKZUHJULUGAUHTUKUNUHUKAUGAUAUBU
|
|
CUDS $.
|
|
|
|
${
|
|
$d f A $. $d f B $.
|
|
phplem2.1 $e |- A e. _V $.
|
|
phplem2.2 $e |- B e. _V $.
|
|
$( Lemma for Pigeonhole Principle. A natural number is equinumerous to its
|
|
successor minus one of its elements. (Contributed by NM, 11-Jun-1998.)
|
|
(Revised by Mario Carneiro, 16-Nov-2014.) $)
|
|
phplem2 $p |- ( ( A e. _om /\ B e. A ) -> A ~~ ( suc A \ { B } ) ) $=
|
|
( com wcel wa csn cdif cun csuc cen wbr cin c0 wceq cvv wss syl syl5eq
|
|
cop wf1o opex snex f1osn mp2an difss ssexi enref pm3.2i incom ssrin ax-mp
|
|
f1oen3g word nnord orddisj syl5sseq ss0 disjdif jctil unen sylancr adantr
|
|
uncom nndifsnid phplem1 3brtr3d ) AEFZBAFZGZBHZAVLIZJZAHZVMJZAAKVLILVIVNV
|
|
PLMZVJVIVLVOLMZVMVMLMZGVLVMNOPZVOVMNZOPZGVQVRVSBAUAZHZQFVLVOWDUBVRWCBADCU
|
|
CUDBADCUEVLVOWDQUNUFVMVMACAVLUGZUHUIUJVIWBVTVIWAVMVONZOVOVMUKVIWFORWFOPVI
|
|
AVONZWFOVMARWFWGRWEVMAVOULUMVIAUOWGOPAUPAUQSURWFUSSTVLAUTVAVLVOVMVMVBVCVD
|
|
VKVNVMVLJAVLVMVEABVFTABVGVH $.
|
|
|
|
$( Lemma for Pigeonhole Principle. A natural number is equinumerous to its
|
|
successor minus any element of the successor. For a version without the
|
|
redundant hypotheses, see ~ phplem3g . (Contributed by NM,
|
|
26-May-1998.) $)
|
|
phplem3 $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $=
|
|
( csuc wcel com wceq wo csn cdif cen wbr elsuci phplem2 enref word orddif
|
|
wa nnord syl sneq difeq2d eqcoms sylan9eq syl5breq jaodan sylan2 ) BAEZFA
|
|
GFZBAFZBAHZIAUIBJZKZLMZBANUJUKUOULABCDOUJULSAAUNLACPUJULAUIAJZKZUNUJAQAUQ
|
|
HATARUAUQUNHABABHUPUMUIABUBUCUDUEUFUGUH $.
|
|
|
|
$( Lemma for Pigeonhole Principle. Equinumerosity of successors implies
|
|
equinumerosity of the original natural numbers. (Contributed by NM,
|
|
28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) $)
|
|
phplem4 $p |- ( ( A e. _om /\ B e. _om )
|
|
-> ( suc A ~~ suc B -> A ~~ B ) ) $=
|
|
( csuc cen wbr com wcel csn cdif cima cvv sylancl ensymd wceq syl imaeq2d
|
|
vf wa cv wf1o wex bren cfv wf1 f1of1 adantl wss sssucid f1imaen2g mpanr12
|
|
sucex word nnord orddif wfn f1ofn sucid fnsnfv difeq2d crn imadmrn eqcomi
|
|
cdm wfo f1ofo forn f1odm 3eqtr3a difeq1d ccnv wfun dff1o3 imadif 3eqtr4rd
|
|
simprbi sylan9eq breqtrd fnfvelrn wb eleq2d mpbid vex fvex phplem3 sylan2
|
|
entr syl2an anandirs ex exlimdv syl5bi ) AEZBEZFGWNWOSUAZUBZSUCAHIZBHIZTZ
|
|
ABFGZWNWOSUDWTWQXASWTWQXAWRWSWQXAWRWQTZAWOAWPUEZJZKZFGXEBFGXAWSWQTZXBAWPA
|
|
LZXEFXBXGAXBWNWOWPUFZWOMIZXGAFGZWQXHWRWNWOWPUGUHBDUMXHXITAWNUIAMIXJAUJCWN
|
|
WOAWPMUKULNOWRWQXGWPWNAJZKZLZXEWRAXLWPWRAUNAXLPAUOAUPQRWQWPWNLZXDKXNWPXKL
|
|
ZKZXEXMWQXDXOXNWQWPWNUQZAWNIZXDXOPWNWOWPURZACUSZWNAWPUTNVAWQWOXNXDWQWPVBZ
|
|
WPWPVEZLZWOXNYCYAWPVCVDWQWNWOWPVFZYAWOPWNWOWPVGZWNWOWPVHZQWQYBWNWPWNWOWPV
|
|
IRVJVKWQWPVLVMZXMXPPWQYDYGWNWOWPVNVQWNXKWPVOQVPVRVSXFBXEWQWSXCWOIZBXEFGWQ
|
|
XCYAIZYHWQXQXRYIXSXTWNAWPVTNWQYDYIYHWAYEYDYAWOXCYFWBQWCBXCDAWPMMSWDCWEWFW
|
|
GOAXEBWHWIWJWKWLWM $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $.
|
|
$( A natural number is equinumerous to its successor minus any element of
|
|
the successor. Version of ~ phplem3 with unnecessary hypotheses
|
|
removed. (Contributed by Jim Kingdon, 1-Sep-2021.) $)
|
|
phplem3g $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $=
|
|
( vb va com wcel csuc csn cdif cen wbr cv wa wi wceq eleq1 anbi2d imbi12d
|
|
vex vtoclg difeq2d breq2d suceq eleq2d anbi12d id difeq1d breq12d phplem3
|
|
sneq anabsi5 anabsi7 ) AEFZBAGZFZAUNBHZIZJKZUMCLZUNFZMZAUNUSHZIZJKZNZUMUO
|
|
MZURNCBUNUSBOZVAVFVDURVGUTUOUMUSBUNPQVGVCUQAJVGVBUPUNUSBUJUAUBRUMUTVDDLZE
|
|
FZUSVHGZFZMZVHVJVBIZJKZNVEDAEVHAOZVLVAVNVDVOVIUMVKUTVHAEPVOVJUNUSVHAUCZUD
|
|
UEVOVHAVMVCJVOUFVOVJUNVBVPUGUHRVHUSDSCSUITUKTUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Two equinumerous natural numbers are equal. Proposition 10.20 of
|
|
[TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of
|
|
[Enderton] p. 136. (Contributed by NM, 28-May-1998.) $)
|
|
nneneq $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~~ B <-> A = B ) ) $=
|
|
( vz vy vw com wcel cen wceq cv wi wral breq1 eqeq1 imbi12d ralbidv breq2
|
|
wbr c0 eqeq2 vx wa csuc ensym en0 eqcom bitri sylib rgenw wrex wo bibi12d
|
|
nn0suc wb mpbiri biimpd a1i nfv nfra1 nfan rsp vex phplem4 imim1d ex syl5
|
|
a2d imp suceq syl8 biimprcd syl6 rexlimd jaod syl7 ralrimdv cbvralv finds
|
|
syl6ib rspcv mpan9 eqeng adantr impbid ) AFGZBFGZUBABHRZABIZWEACJZHRZAWII
|
|
ZKZCFLZWFWGWHKZUAJZWIHRZWOWIIZKZCFLSWIHRZSWIIZKZCFLDJZWIHRZXBWIIZKZCFLZXB
|
|
UCZWIHRZXGWIIZKZCFLZWMUADAWOSIZWRXACFXLWPWSWQWTWOSWIHMWOSWINOPWOXBIZWRXEC
|
|
FXMWPXCWQXDWOXBWIHMWOXBWINOPWOXGIZWRXJCFXNWPXHWQXIWOXGWIHMWOXGWINOPWOAIZW
|
|
RWLCFXOWPWJWQWKWOAWIHMWOAWINOPXACFWSWISHRZWTSWIUDXPWISIWTWIUEWISUFUGUHUIX
|
|
BFGZXFXGEJZHRZXGXRIZKZEFLXKXQXFYAEFXRFGXRSIZXRWIUCZIZCFUJZUKZXQXFYACXRUMX
|
|
QXFYFYAKXQXFUBZYBYAYEYBYAKYGYBXSXTYBXSXTUNXGSHRZXGSIZUNXGUEYBXSYHXTYIXRSX
|
|
GHQXRSXGTULUOUPUQYGYDYACFXQXFCXQCURXECFUSUTYACURYGWIFGZXGYCHRZXGYCIZKZYDY
|
|
AKYGYJYKXDYLXQXFYJYKXDKZKZXFYJXEKXQYOXECFVAXQYJXEYNXQYJXEYNKXQYJUBYKXCXDX
|
|
BWIDVBCVBVCVDVEVGVFVHXBWIVIVJYDYAYMYDXSYKXTYLXRYCXGHQXRYCXGTOVKVLVMVNVEVO
|
|
VPYAXJECFXRWIIXSXHXTXIXRWIXGHQXRWIXGTOVQVSVRWLWNCBFWIBIWJWGWKWHWIBAHQWIBA
|
|
TOVTWAWEWHWGKWFABFWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d k w $.
|
|
$( A natural number is not equinumerous to its successor. Corollary
|
|
10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.) $)
|
|
php5 $p |- ( A e. _om -> -. A ~~ suc A ) $=
|
|
( vw vk cv csuc cen wbr wn c0 wceq id suceq breq12d notbid com wne peano1
|
|
wcel peano3 ax-mp en0 nemtbir ensymb mtbi wi peano2 vex sucex mpdan con3d
|
|
phplem4 finds ) BDZUMEZFGZHIIEZFGZHCDZUREZFGZHUSUSEZFGZHAAEZFGZHBCAUMIJZU
|
|
OUQVEUMIUNUPFVEKUMILMNUMURJZUOUTVFUMURUNUSFVFKUMURLMNUMUSJZUOVBVGUMUSUNVA
|
|
FVGKUMUSLMNUMAJZUOVDVHUMAUNVCFVHKUMALMNUPIFGZUQVIUPIIORUPIPQISTUPUAUBUPIU
|
|
CUDURORZVBUTVJUSORVBUTUEURUFURUSCUGZURVKUHUKUIUJUL $.
|
|
$}
|
|
|
|
$( A singleton ` { A } ` is never equinumerous with the ordinal number 2. If
|
|
` A ` is a proper class, see ~ snnen2oprc . (Contributed by Jim Kingdon,
|
|
1-Sep-2021.) $)
|
|
snnen2og $p |- ( A e. V -> -. { A } ~~ 2o ) $=
|
|
( c1o csuc cen wbr wn wcel csn c2o com 1onn php5 ax-mp ensn1g eqcomi breq2i
|
|
wi df-2o ensymb entr ex sylbi con3rr3 sylnbi mpsyl ) CCDZEFZGZABHAIZCEFZUJJ
|
|
EFZGZCKHUILCMNABOUHCJEFZUKUMRUGJCEJUGSPQUKULUNUKCUJEFZULUNRUJCTUOULUNCUJJUA
|
|
UBUCUDUEUF $.
|
|
|
|
$( A singleton ` { A } ` is never equinumerous with the ordinal number 2. If
|
|
` A ` is a set, see ~ snnen2og . (Contributed by Jim Kingdon,
|
|
1-Sep-2021.) $)
|
|
snnen2oprc $p |- ( -. A e. _V -> -. { A } ~~ 2o ) $=
|
|
( cvv wcel wn csn c2o cen wbr c0 2on0 ensymb en0 bitri nemtbir snprc biimpi
|
|
wceq breq1d mtbiri ) ABCDZAEZFGHIFGHZUBFIJUBFIGHFIQIFKFLMNTUAIFGTUAIQAOPRS
|
|
$.
|
|
|
|
$( One and two are not equinumerous. (Contributed by Jim Kingdon,
|
|
25-Jan-2022.) $)
|
|
1nen2 $p |- -. 1o ~~ 2o $=
|
|
( c1o c2o cen wbr csuc com wcel wn 1onn php5 ax-mp df-2o breq2i mtbir ) ABC
|
|
DAAEZCDZAFGPHIAJKBOACLMN $.
|
|
|
|
${
|
|
$d A f $. $d B f $.
|
|
$( Dominance of successors implies dominance of the original natural
|
|
numbers. (Contributed by Jim Kingdon, 1-Sep-2021.) $)
|
|
phplem4dom $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( suc A ~<_ suc B -> A ~<_ B ) ) $=
|
|
( com wcel csuc cdom wbr adantl syl csn cdif cen cima wss ensymd cvv wceq
|
|
vf wa syl2anc cv wf1 wex wb peano2 brdomg biimpa cfv ad2antrr sssucid a1i
|
|
simpr simplll f1imaen2g syl22anc difexg word nnord orddif wfn f1fn sucidg
|
|
imaeq2d fnsnfv difeq2d ccnv wfun wf simprbi imadif eqtr4d f1f crn imassrn
|
|
df-f1 syl5ss ssdifd eqsstr3d eqsstrd ssdomg sylc endomtr simpllr ffvelrnd
|
|
frn phplem3g domentr exlimddv ex ) ACDZBCDZSZAEZBEZFGZABFGZWLWOSZWMWNRUAZ
|
|
UBZWPRWLWOWSRUCZWLWNCDZWOWTUDWKXAWJBUEHZWMWNCRUFIUGWQWSSZAWNAWRUHZJZKZFGZ
|
|
XFBLGWPXCAWRAMZLGXHXFFGZXGXCXHAXCWSXAAWMNZWJXHALGWQWSULWLXAWOWSXBUIZXJXCA
|
|
UJUKWJWKWOWSUMZWMWNAWRCUNUOOXCXFPDZXHXFNXIXCXAXMXKWNXECUPIXCXHWRWMAJZKZMZ
|
|
XFXCWJXHXPQXLWJAXOWRWJAUQAXOQAURAUSIVCIXCXPWRWMMZXEKZXFXCXRXQWRXNMZKZXPXC
|
|
XEXSXQXCWRWMUTZAWMDZXEXSQWSYAWQWMWNWRVAHXCWJYBXLACVBIZWMAWRVDTVEWSXPXTQZW
|
|
QWSWRVFVGZYDWSWMWNWRVHZYEWMWNWRVOVIWMXNWRVJIHVKXCXQWNXEXCYFXQWNNWSYFWQWMW
|
|
NWRVLHZYFXQWRVMWNWRWMVNWMWNWRWEVPIVQVRVSXHXFPVTWAAXHXFWBTXCBXFXCWKXDWNDBX
|
|
FLGWJWKWOWSWCXCWMWNAWRYGYCWDBXDWFTOAXFBWGTWHWI $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d k w $.
|
|
$( A natural number does not dominate its successor. (Contributed by Jim
|
|
Kingdon, 1-Sep-2021.) $)
|
|
php5dom $p |- ( A e. _om -> -. suc A ~<_ A ) $=
|
|
( vw vk vx cv csuc cdom wbr wn c0 wceq suceq id breq12d notbid cen peano1
|
|
com wcel php5 ax-mp wss wa wex 0ex domen ss0 en0 sylibr entr sylan2 sylbi
|
|
exlimiv ensymd mto wi peano2 phplem4dom mpancom con3d finds ) BEZFZVBGHZI
|
|
JFZJGHZICEZFZVGGHZIVHFZVHGHZIAFZAGHZIBCAVBJKZVDVFVNVCVEVBJGVBJLVNMNOVBVGK
|
|
ZVDVIVOVCVHVBVGGVBVGLVOMNOVBVHKZVDVKVPVCVJVBVHGVBVHLVPMNOVBAKZVDVMVQVCVLV
|
|
BAGVBALVQMNOVFJVEPHZJRSVRIQJTUAVFVEJVFVEDEZPHZVSJUBZUCZDUDVEJPHZDVEJUEUFW
|
|
BWCDWAVTVSJPHZWCWAVSJKWDVSUGVSUHUIVEVSJUJUKUMULUNUOVGRSZVKVIVHRSWEVKVIUPV
|
|
GUQVHVGURUSUTVA $.
|
|
$}
|
|
|
|
$( Cardinal ordering agrees with natural number ordering. Example 3 of
|
|
[Enderton] p. 146. (Contributed by NM, 17-Jun-1998.) $)
|
|
nndomo $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~<_ B <-> A C_ B ) ) $=
|
|
( com wcel wa cdom wbr wn csuc php5dom ad2antlr wi domtr expcom adantl mtod
|
|
wss ssdomg ad2antrr word nnord ordsucss syl wb nntri1 adantr mpbird impbid
|
|
ex ) ACDZBCDZEZABFGZABQZULUMUNULUMEZUNBADZHZUOUPBIZAQZUOUSURAFGZUOUTURBFGZU
|
|
KVAHUJUMBJKUMUTVALULUTUMVAURABMNOPUJUSUTLUKUMURACRSPUJUPUSLZUKUMUJATVBAUABA
|
|
UBUCSPULUNUQUDUMABUEUFUGUIUKUNUMLUJABCROUH $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Pigeonhole Principle. A natural number is not equinumerous to a proper
|
|
subset of itself. By "proper subset" here we mean that there is an
|
|
element which is in the natural number and not in the subset, or in
|
|
symbols ` E. x x e. ( A \ B ) ` (which is stronger than not being equal
|
|
in the absence of excluded middle). Theorem (Pigeonhole Principle) of
|
|
[Enderton] p. 134. The theorem is so-called because you can't put _n_ +
|
|
1 pigeons into _n_ holes (if each hole holds only one pigeon). The
|
|
proof consists of lemmas ~ phplem1 through ~ phplem4 , ~ nneneq , and
|
|
this final piece of the proof. (Contributed by NM, 29-May-1998.) $)
|
|
phpm $p |- ( ( A e. _om /\ B C_ A /\ E. x x e. ( A \ B ) ) ->
|
|
-. A ~~ B ) $=
|
|
( vy com wcel wss cv cdif cen wn wa c0 wceq simpr ad2antlr simplr syl2anc
|
|
wbr cdom wex csuc wrex wne eldifi syl neneqd pm2.21dd php5dom csn cvv vex
|
|
sucex difss ssexi eldifn ad3antlr simpllr adantr sseqtrd ssdif cin disjsn
|
|
ne0i wb disj3 bitr3i sseq1 sylbi syl5ibr sylc ssdomg mpsyl eleqtrd ensymd
|
|
phplem3g domentr endomtr eqbrtrrd mtand ex rexlimdva wo ad2antrr mpjaodan
|
|
imp nn0suc exlimdv 3impia ) BEFZCBGZAHZBCIFZAUABCJSZKZWJWKLZWMWOAWPWMWOWP
|
|
WMLZBMNZWOBDHZUBZNZDEUCZWQWRLWRWOWQWROWMWRKWPWRWMBMWMWLBFZBMUDWLBCUEZBWLV
|
|
DUFUGPUHWQXBWOWQXAWODEWQWSEFZLZXAWOXFXALZWNWTWSTSZXEXHKWQXAWSUIPXGWNLZBWT
|
|
WSTXFXAWNQXIWNCWSTSZBWSTSXGWNOXGXJWNXGCWTWLUJZIZTSZXLWSJSZXJXLUKFXGCXLGZX
|
|
MXLWTWSDULUMWTXKUNUOXGWLCFKZCWTGZXOWMXPWPXEXAWLBCUPUQXGCBWTXFWKXAWJWKWMXE
|
|
URUSXFXAOZUTXQXOXPCXKIZXLGZCWTXKVAXPCXSNZXOXTVEXPCXKVBMNYACWLVCCXKVFVGCXS
|
|
XLVHVIVJVKCXLUKVLVMXGXEWLWTFZXNWQXEXAQXGWLBWTWMXCWPXEXAXDUQXRVNXEYBLWSXLW
|
|
SWLVPVORCXLWSVQRUSBCWSVRRVSVTWAWBWFWJWRXBWCWKWMDBWGWDWEWAWHWI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
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$( Pigeonhole Principle. A natural number is not equinumerous to an
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|
element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.) $)
|
|
phpelm $p |- ( ( A e. _om /\ B e. A ) -> -. A ~~ B ) $=
|
|
( vx com wcel wa wss cv cdif wex cen wbr wn simpl con0 wi nnon onelss syl
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AGZCHZABIZEZCJZABKLMUJUKNUJUKUMUJAOEUKUMPAQABRSTULUKBUOEZUQUJUKUAZULBABUS
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|
BBEMULBUBUCUDUPURCBAUNBUOUEUFUGCABUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A f $. $d B f $.
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|
$( Equinumerosity of successors of an ordinal and a natural number implies
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|
equinumerosity of the originals. (Contributed by Jim Kingdon,
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5-Sep-2021.) $)
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phplem4on $p |- ( ( A e. On /\ B e. _om ) ->
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( suc A ~~ suc B -> A ~~ B ) ) $=
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( vf con0 wcel com wa csuc cen wbr cv adantl csn cdif cima ensymd imaeq2d
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syl wceq syl2anc wf1o wex bren biimpi cfv wf1 wss peano2 ad3antlr sssucid
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f1of1 nnon a1i simplll f1imaen2g syl22anc word eloni orddif ad3antrrr wfn
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f1ofn sucidg fnsnfv difeq2d crn cdm imadmrn eqcomi wfo f1ofo forn 3eqtr3a
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difeq1d ccnv wfun dff1o3 simprbi imadif 3eqtr4rd breqtrd simpllr fnfvelrn
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f1odm eqtrd wb eleq2d mpbid phplem3g entr exlimddv ex ) ADEZBFEZGZAHZBHZI
|
|
JZABIJZWOWRGZWPWQCKZUAZWSCWRXBCUBZWOWRXCWPWQCUCUDLWTXBGZAWQAXAUEZMZNZIJXG
|
|
BIJWSXDAXAAOZXGIXDXHAXDWPWQXAUFZWQDEZAWPUGZWMXHAIJXBXIWTWPWQXAUKLWNXJWMWR
|
|
XBWNWQFEXJBUHWQULRUIXKXDAUJUMWMWNWRXBUNZWPWQAXADUOUPPXDXHXAWPAMZNZOZXGWMX
|
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HXOSWNWRXBWMAXNXAWMAUQAXNSAURAUSRQUTXDXAWPOZXFNZXPXAXMOZNZXGXOXDXFXRXPXDX
|
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AWPVAZAWPEZXFXRSXBXTWTWPWQXAVBLZXDWMYAXLADVCRZWPAXAVDTVEXBXGXQSWTXBWQXPXF
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|
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$}
|
|
|
|
|
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$(
|
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
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Finite sets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A n $.
|
|
$( A finite set is countable. (Contributed by Thierry Arnoux,
|
|
27-Mar-2018.) $)
|
|
fict $p |- ( A e. Fin -> A ~<_ _om ) $=
|
|
( vn cfn wcel cv cen wbr com cdom wrex isfi biimpi wa simprr cvv wss omex
|
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word ordom ordelss mpan ssdomg mpsyl ad2antrl endomtr syl2anc rexlimddv )
|
|
ACDZABEZFGZAHIGZBHUHUJBHJBAKLUHUIHDZUJMMUJUIHIGZUKUHULUJNULUMUHUJHODULUIH
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PZUMQHRULUNSHUITUAUIHOUBUCUDAUIHUEUFUG $.
|
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$}
|
|
|
|
${
|
|
$d A f x $. $d B f x $. $d C f x $.
|
|
$( Equality of members of a finite set is decidable. This may be
|
|
counterintuitive: cannot any two sets be elements of a finite set?
|
|
Well, to show, for example, that ` { B , C } ` is finite would require
|
|
showing it is equinumerous to ` 1o ` or to ` 2o ` but to show that you'd
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|
need to know ` B = C ` or ` -. B = C ` , respectively. (Contributed by
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|
Jim Kingdon, 5-Sep-2021.) $)
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fidceq $p |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C ) $=
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( vx vf wcel cv wceq wdc com biimpi wa wn wo adantl ffvelrnd elnn syl2anc
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cfv wi cfn w3a cen wbr wrex isfi 3ad2ant1 wf1o bren ad2antll f1of simpll2
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wex wf simplrl simpll3 nndceq exmiddc f1of1 f1veqaeq syl12anc fveq2 con3i
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syl wf1 a1i orim12d mpd df-dc sylibr exlimddv rexlimddv ) AUAFZBAFZCAFZUB
|
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ZADGZUCUDZBCHZIZDJVMVNVRDJUEZVOVMWADAUFKUGVPVQJFZVRLZLZAVQEGZUHZVTEVRWFEU
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|
MZVPWBVRWGAVQEUIKUJWDWFLZVSVSMZNZVTWHBWESZCWESZHZWMMZNZWJWHWMIZWOWHWKJFZW
|
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|
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|
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$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d X x y $.
|
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$( All decrements of a finite set are equinumerous. (Contributed by Jim
|
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Kingdon, 9-Sep-2021.) $)
|
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fidifsnen $p |- ( ( X e. Fin /\ A e. X /\ B e. X ) ->
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( X \ { A } ) ~~ ( X \ { B } ) ) $=
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( vx vy wcel wceq wn wa adantr syl adantl eqeltrd sylanbrc ad2antlr simpr
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cvv enrefg sneq difeq2d breqtrd cv cif cmpt wf1o eqid wne simpll2 simpllr
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iftrue eqeq1d mtbird eldifsn iffalse eldifi wdc wo simpll1 simpll3 fidceq
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syl3anc exmiddc simpl3 necomd wb wi biimpar a1d mpbid eqtr3d simprr velsn
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eldifbd sylnib pm2.21dd ex ad2antll biimprcd sylbid impbid eqtrd ad3antlr
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simplr eqtr2d simprl impbida adantrr f1o2d f1oeng syl2anc ) CUAFZACFZBCFZ
|
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UBZABGZCAUCZUDZCBUCZUDZUEUFZXGHZXFXGIZXIXIXKUEXNXIUIFZXIXIUEUFXFXOXGXCXDX
|
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OXECXHUAUGUHZJXIUIUJKXGXIXKGXFXGXHXJCABUKULLUMXFXMIZXOXIXKDXIDUNZBGZAXRUO
|
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ZUPZUQXLXFXOXMXPJXQDEXIXKXTEUNZAGZBYBUOZYAYAURXQXRXIFZIZXSXTXKFZXSHZYFXSI
|
|
ZXTCFZXTBUSZYGYIXTACXSXTAGZYFXSAXRVBZLZYFXDXSXCXDXEXMYEUTJMYIXTBYIXTBGZXG
|
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XFXMYEXSVAYIXTABYNVCVDQXTCBVEZNYFYHIZYJYKYGYQXTXRCYHXTXRGZYFXSAXRVFZLZYEX
|
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RCFZXQYHXRCXHVGZOMYQXTBYQYOXSYFYHPYQXTXRBYTVCVDQYPNYFXSVHZXSYHVIZYFXCUUAX
|
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EUUCXCXDXEXMYEVJYEUUAXQUUBLXCXDXEXMYEVKCXRBVLVMXSVNKZRXQYBXKFZIZYCYDXIFYC
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HZUUGYCIYDBXIYCYDBGZUUGYCBYBVBZLXQBXIFZUUFYCXQXEBAUSUUKXCXDXEXMVOXQABXQAB
|
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XFXMPQVPBCAVENSMUUGUUHIZYDYBXIUUHYDYBGZUUGYCBYBVFZLUULYBCFZYBAUSYBXIFUUFU
|
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UOXQUUHYBCXJVGZOUULYBAUUGUUHPQYBCAVENMUUGYCVHZYCUUHVIZUUGXCUUOXDUUQXCXDXE
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XMUUFVJUUFUUOXQUUPLXCXDXEXMUUFUTCYBAVLZVMYCVNZKRXQYEUUFIZIZXSXRYDGZYBXTGZ
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VQYHUVBXSIZUVCUVDUVEYCUVCUVDVRUUHUVEYCIUVDUVCUVEUVDYCUVEXTAYBXSYLUVBYMLTZ
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VSVTUVEUUHIZUVCUVDUVGUVCIZYBBGZUVDUVHXRYBBUVHUVCXRYBGZUVGUVCPUUHUVCUVJVQU
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VEUVCUUHYDYBXRUUNTOWAUVBXSUUHUVCVAWBUVHYBXJFZUVIUVGUVKHUVCUVGYBCXJUVBUUFX
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SUUHXQYEUUFWCSWEJEBWDWFWGWHUVBUURXSUVBUUQUURUVBXCUUOXDUUQXCXDXEXMUVAVJUUF
|
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UUOXQYEUUPWIXCXDXEXMUVAUTUUSVMUUTKZJRUVEUVDYCUVCUVFXSYCUVCVRUVBYCUVCXSYCY
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DBXRUUJTWJLWKWLUVBYHIZUVCUVDUVMUVCIZYCUVDUUHUVNYCIZXSUVDUVOXRYDBUVMUVCYCW
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OYCUUIUVNUUJLWMUVBYHUVCYCVAWGUVNUUHIZXTXRYBYHYRUVBUVCUUHYSWNUVPXRYDYBUVMU
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VCUUHWOUUHUUMUVNUUNLWMWPUVBUURYHUVCUVLSRUVMUVDIZYDYBXRUVQUUHUUMUVQYCXRAGZ
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UVBUVRHYHUVDUVBXRXHFUVRUVBXRCXHXQYEUUFWQWEDAWDWFSUVQYBXRAUVQUVDYBXRGZUVMU
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VDPYHUVDUVSVQUVBUVDYHXTXRYBYSTOWAZVCVDUUNKUVTWPWRXQYEUUDUUFUUEWSRWTXIXKUI
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YAXAXBXFXGVHXGXMVICABVLXGVNKR $.
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$}
|
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|
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${
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$d A x $. $d B x $.
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$( If we remove a single element from a finite set then put it back in, we
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end up with the original finite set. This strengthens ~ difsnss from
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subset to equality when the set is finite. (Contributed by Jim Kingdon,
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9-Sep-2021.) $)
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fidifsnid $p |- ( ( A e. Fin /\ B e. A ) ->
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( ( A \ { B } ) u. { B } ) = A ) $=
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( vx cfn wcel wa csn cdif cun wss difsnss adantl cv wn simpr velsn sylibr
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wceq syl simplr elun2 sylnibr eldifd elun1 wo simpll fidceq syl3anc df-dc
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wdc sylib mpjaodan ex ssrdv eqssd ) ADEZBAEZFZABGZHZUSIZAUQVAAJUPABKLURCA
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VAURCMZAEZVBVAEZURVCFZVBBRZVDVFNZVEVFFZVBUSEZVDVHVFVIVEVFOCBPZQVBUSUTUASV
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EVGFZVBUTEVDVKVBAUSURVCVGTVKVFVIVEVGOVJUBUCVBUTUSUDSVEVFUJZVFVGUEVEUPVCUQ
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VLUPUQVCUFURVCOUPUQVCTAVBBUGUHVFUIUKULUMUNUO $.
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$}
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${
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$( Natural numbers are finite sets. (Contributed by Stefan O'Rear,
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21-Mar-2015.) $)
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nnfi $p |- ( A e. _om -> A e. Fin ) $=
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( vx com wcel cv cen wbr wrex cfn enrefg breq2 rspcev mpdan isfi sylibr )
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ACDZABEZFGZBCHZAIDPAAFGZSACJRTBACQAAFKLMBANO $.
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$}
|
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${
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$d x A $. $d x B $.
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$( Equinumerous sets have the same finiteness. (Contributed by NM,
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22-Aug-2008.) $)
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enfi $p |- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) ) $=
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EZCGHBPDEZCGHAIJBIJOQRCGABPKLCAMCBMN $.
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$( A set equinumerous to a finite set is finite. (Contributed by Mario
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Carneiro, 12-Mar-2015.) $)
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enfii $p |- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin ) $=
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$}
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${
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ssfilem.1 $e |- { z e. { (/) } | ph } e. Fin $.
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$( Lemma for ~ ssfiexmid . (Contributed by Jim Kingdon, 3-Feb-2022.) $)
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ssfilem $p |- ( ph \/ -. ph ) $=
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( vn vw vy vx c0 csn crab cv cen wbr com wo wcel wceq wa wex ax-mp wn cfn
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|
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CDVOUDUEVQVTDNVQVPNPZVTWAVQVPHQZHVPPZOVTVPUFVQWBVTWCVQWBRZVSAWDVOHQZVSWBV
|
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QWEWBVQVOHLMWEVPHVOLUGVOUHUIUJWEVSBVNUKZVSABVNULEKVNPESVSWFUMEHUNUPVSBEVN
|
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UOTUQURUSVQWCRFKZVOPZFSZVTVQVPVOLMGKVPPGSWIWCVOVPUTGHVPVAGFVPVOVHVBWHVTFW
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HAVSWHWGVNPAAABWGVNBKWGQAVCVDVEVFVGVIVJVKVLVMT $.
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$}
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${
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$d ph x y z $.
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ssfiexmid.1 $e |- A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) $.
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$( If any subset of a finite set is finite, excluded middle follows. One
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|
direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim
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Kingdon, 19-May-2020.) $)
|
|
ssfiexmid $p |- ( ph \/ -. ph ) $=
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( vz c0 csn cfn wcel wss cvv ax-mp cv wa wi wal p0ex wceq eleq1 spcv crab
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|
snfig ssrab2 sseq2 anbi12d imbi1d albidv rabex sseq1 anbi2d imbi12d mp2an
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0ex ssfilem ) AEFGZHIZAEUOUAZUOJZUQHIZFKIUPUMFKUBLAEUOUCUPCMZUOJZNZUTHIZO
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ZCPZUPURNZUSOZBMZHIZUTVHJZNZVCOZCPZBPVEDVMVEBUOQVHUORZVLVDCVNVKVBVCVNVIUP
|
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VJVAVHUOHSVHUOUTUDUEUFUGTLVDVGCUQAEUOQUHUTUQRZVBVFVCUSVOVAURUPUTUQUOUIUJU
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TUQHSUKTLULUN $.
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$}
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${
|
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$d ph x y $.
|
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infiexmid.1 $e |- ( x e. Fin -> ( x i^i y ) e. Fin ) $.
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$( If the intersection of any finite set and any other set is finite,
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excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.) $)
|
|
infiexmid $p |- ( ph \/ -. ph ) $=
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( cv cfn wcel wss wa wi cin wceq dfss1 biimpi adantl adantr eqeltrrd gen2
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ssfiexmid ) ABCBEZFGZCEZTHZIZUBFGJBCUDTUBKZUBFUCUEUBLZUAUCUFUBTMNOUAUEFGU
|
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CDPQRS $.
|
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$}
|
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|
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${
|
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$d ph y z $. $d x y $.
|
|
domfiexmid.1 $e |- ( ( x e. Fin /\ y ~<_ x ) -> y e. Fin ) $.
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|
$( If any set dominated by a finite set is finite, excluded middle follows.
|
|
(Contributed by Jim Kingdon, 3-Feb-2022.) $)
|
|
domfiexmid $p |- ( ph \/ -. ph ) $=
|
|
( vz c0 cfn wcel cdom wbr cvv ax-mp cv wa wi wal p0ex wceq eleq1 spcv csn
|
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crab 0ex snfig wss ssrab2 ssdomg mp2 gen2 breq2 imbi1d albidv rabex breq1
|
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anbi12d anbi2d imbi12d mp2an ssfilem ) AEFUAZGHZAEUTUBZUTIJZVBGHZFKHVAUCF
|
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KUDLZVAVBUTUEVCVEAEUTUFVBUTGUGUHVACMZUTIJZNZVFGHZOZCPZVAVCNZVDOZBMZGHZVFV
|
|
NIJZNZVIOZCPZBPVKVRBCDUIVSVKBUTQVNUTRZVRVJCVTVQVHVIVTVOVAVPVGVNUTGSVNUTVF
|
|
IUJUOUKULTLVJVMCVBAEUTQUMVFVBRZVHVLVIVDWAVGVCVAVFVBUTIUNUPVFVBGSUQTLURUS
|
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$.
|
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$}
|
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|
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${
|
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$d A f $. $d X f $. $d M f $.
|
|
$( If a set ` A ` is equinumerous to the successor of a natural number
|
|
` M ` , then ` A ` with an element removed is equinumerous to ` M ` .
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear,
|
|
16-Aug-2015.) $)
|
|
dif1en $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) ->
|
|
( A \ { X } ) ~~ M ) $=
|
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( vf com wcel cen wbr wf1o csn cdif ensymd 3ad2ant1 syl2anc adantr adantl
|
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syl wceq cres wfo csuc w3a cv wex simp2 bren sylib wa cfv cfn peano2 nnfi
|
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enfii simpl3 wf f1of sucidg ffvelrnd fidifsnen syl3anc word orddif eleq1d
|
|
nnord ibi ccnv wfun wfn crn dff1o2 simp2bi f1ofo wb cdm wrel f1orel resdm
|
|
f1odm reseq2d eqtr3d foeq1 mpbid simpl1 f1osng f1ofn mpbird resdif f1oeng
|
|
cop fnressn eqbrtrd entr exlimddv ) BEFZABUAZGHZCAFZUBZWOADUCZIZACJKZBGHZ
|
|
DWRWOAGHWTDUDWRAWOWNWPWQUEZLWOADUFUGWRWTUHZXAABWSUIZJZKZGHZXGBGHXBXDAUJFZ
|
|
WQXEAFZXHWRXIWTWRWOUJFZWPXIWNWPXKWQWNWOEFXKBUKWOULQMXCAWOUMNOWNWPWQWTUNXD
|
|
WOABWSWTWOAWSUOWRWOAWSUPPWRBWOFZWTWNWPXLWQBEUQMOZURZCXEAUSUTXDBXGXDBWOBJZ
|
|
KZXGGWRBXPRZWTWNWPXQWQWNBVAXQBVDBVBQZMOXDXPEFZXPXGWSXPSZIZXPXGGHWRXSWTWNW
|
|
PXSWQWNXSWNBXPEXRVCVEMOXDWSVFVGZWOAWSWOSZTZXOXFWSXOSZTZYAWTYBWRWTWSWOVHZY
|
|
BWSVIARWOAWSVJVKPXDWOAWSTZYDWTYHWRWOAWSVLPXDWSYCRYHYDVMXDWSWSVNZSZWSYCXDW
|
|
SVOZYJWSRWTYKWRWOAWSVPPWSVQQWTYJYCRWRWTYIWOWSWOAWSVRVSPVTWOAWSYCWAQWBXDYF
|
|
XOXFBXEWIJZTZXDXOXFYLIZYMXDWNXJYNWNWPWQWTWCXNBXEEAWDNXOXFYLVLQXDYEYLRZYFY
|
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MVMXDYGXLYOWTYGWRWOAWSWEPXMWOBWSWJNXOXFYEYLWAQWFWOXOAXFWSWGUTXPXGEXTWHNWK
|
|
LXAXGBWLNWM $.
|
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$}
|
|
|
|
$( Adding one element to a finite set which is equinumerous to a natural
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number. (Contributed by Jim Kingdon, 13-Sep-2021.) $)
|
|
fiunsnnn $p |- ( ( ( A e. Fin /\ B e. ( _V \ A ) )
|
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/\ ( N e. _om /\ A ~~ N ) ) ->
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( cfn wcel cvv cdif wa com cen wbr csn cun csuc cin c0 wceq simprr disjsn
|
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wn en2sn ad2ant2lr simplr eldifbd sylibr elirr mpbir a1i syl22anc syl6breqr
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unen df-suc ) ADEZBFAGZEZHZCIEZACJKZHZHZABLZMZCCLZMZCNJUTURVAVCJKZAVAOPQZCV
|
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COPQZVBVDJKUPUQURRUOUQVEUMURBCUNIUAUBUTBAETVFUTBFAUMUOUSUCUDABSUEVGUTVGCCET
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CUFCCSUGUHACVAVCUKUICULUJ $.
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|
|
${
|
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$d A n $. $d B n $.
|
|
$( A finite set is not equinumerous to a set which adds one element.
|
|
(Contributed by Jim Kingdon, 13-Sep-2021.) $)
|
|
php5fin $p |- ( ( A e. Fin /\ B e. ( _V \ A ) ) ->
|
|
-. A ~~ ( A u. { B } ) ) $=
|
|
( vn cfn wcel cvv cdif wa cv cen wbr csn cun wn com wrex biimpi adantr wb
|
|
isfi csuc php5 ad2antrl enen1 ad2antll fiunsnnn enen2 syl bitrd rexlimddv
|
|
mtbird ) ADEZBFAGEZHZACIZJKZAABLMZJKZNCOULUPCOPZUMULUSCATQRUNUOOEZUPHHZUR
|
|
UOUOUAZJKZUTVCNUNUPUOUBUCVAURUOUQJKZVCUPURVDSUNUTAUOUQUDUEVAUQVBJKVDVCSAB
|
|
UOUFUQVBUOUGUHUIUKUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A m n $. $d B m n $.
|
|
$( Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim
|
|
Kingdon, 12-Sep-2021.) $)
|
|
fisbth $p |- ( ( ( A e. Fin /\ B e. Fin ) /\
|
|
( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B ) $=
|
|
( vn vm cfn wcel wa cdom wbr cen com wrex isfi biimpi ad2antrr wss ensymd
|
|
cv simprl syl2anc ad3antlr simplrr endomtr simprr domentr wb nndomo mpbid
|
|
simplrl eqssd breqtrd entr rexlimddv ) AEFZBEFZGZABHIZBAHIZGZGZACRZJIZABJ
|
|
IZCKUNVBCKLZUOUSUNVDCAMNOUTVAKFZVBGZGZBDRZJIZVCDKUOVIDKLZUNUSVFUOVJDBMNUA
|
|
VGVHKFZVIGZGZAVHJIVHBJIZVCVMAVAVHJUTVEVBVLUBZVMVAVHVMVAVHHIZVAVHPZVMVABHI
|
|
ZVIVPVMVAAJIUQVRVMAVAVOQUTUQVFVLUPUQURSOVAABUCTVGVKVIUDZVABVHUETVMVEVKVPV
|
|
QUFUTVEVBVLUIZVGVKVISZVAVHUGTUHVMVHVAHIZVHVAPZVMVHAHIZVBWBVMVNURWDVMBVHVS
|
|
QZUTURVFVLUPUQURUDOVHBAUCTVOVHAVAUETVMVKVEWBWCUFWAVTVHVAUGTUHUJUKWEAVHBUL
|
|
TUMUM $.
|
|
$}
|
|
|
|
$( The empty set is finite. (Contributed by FL, 14-Jul-2008.) $)
|
|
0fin $p |- (/) e. Fin $=
|
|
( c0 com wcel cfn peano1 nnfi ax-mp ) ABCADCEAFG $.
|
|
|
|
${
|
|
$d A f m n x $.
|
|
$( A nonempty finite set has at least one element. (Contributed by Jim
|
|
Kingdon, 10-Sep-2021.) $)
|
|
fin0 $p |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) ) $=
|
|
( vn vm vf cfn wcel cv cen wbr c0 wex com wrex wa wceq simplrr sylib syl
|
|
wn wne wb isfi biimpi csuc simpr breqtrd en0 nner necon2bi 2falsed adantr
|
|
n0r wf1o ensymd bren cfv wf f1of adantl sucidg ad3antlr eleqtrrd ffvelrnd
|
|
simplr elex2 exlimddv ex rexlimdva imp nn0suc ad2antrl mpjaodan rexlimddv
|
|
2thd wo ) BFGZBCHZIJZBKUAZAHBGALZUBZCMVQVSCMNCBUCUDVQVRMGZVSOOZVRKPZWBVRD
|
|
HZUEZPZDMNZWDWEOZVTWAWJBKPZVTTWJBKIJWKWJBVRKIVQWCVSWEQWDWEUFUGBUHRZBKUISW
|
|
JWKWATWLWABKABUMZUJSUKWDWIWBWDWHWBDMWDWFMGZOZWHWBWOWHOZVRBEHZUNZWBEWPVRBI
|
|
JWRELWPBVRWOVSWHVQWCVSWNQULUOVRBEUPRWPWROZVTWAWSWAVTWSWFWQUQZBGWAWSVRBWFW
|
|
QWRVRBWQURWPVRBWQUSUTWSWFWGVRWNWFWGGWDWHWRWFMVAVBWOWHWRVEVCVDAWTBVFSZWMSX
|
|
AVOVGVHVIVJWCWEWIVPVQVSDVRVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d A f m n x $.
|
|
$( A finite set is either empty or inhabited. (Contributed by Jim Kingdon,
|
|
30-Sep-2021.) $)
|
|
fin0or $p |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) ) $=
|
|
( vn vm vf cfn wcel cv cen wbr c0 wceq wex wo com wrex wa simplrr sylib
|
|
ex isfi biimpi csuc nn0suc ad2antrl simpr breqtrd wf1o adantr ensymd bren
|
|
en0 cfv wf f1of adantl sucidg ad3antlr simplr eleqtrrd ffvelrnd elex2 syl
|
|
exlimddv rexlimdva orim12d mpd rexlimddv ) BFGZBCHZIJZBKLZAHBGAMZNZCOVIVK
|
|
COPCBUAUBVIVJOGZVKQQZVJKLZVJDHZUCZLZDOPZNZVNVOWBVIVKDVJUDUEVPVQVLWAVMVPVQ
|
|
VLVPVQQZBKIJVLWCBVJKIVIVOVKVQRVPVQUFUGBULSTVPVTVMDOVPVROGZQZVTVMWEVTQZVJB
|
|
EHZUHZVMEWFVJBIJWHEMWFBVJWEVKVTVIVOVKWDRUIUJVJBEUKSWFWHQZVRWGUMZBGVMWIVJB
|
|
VRWGWHVJBWGUNWFVJBWGUOUPWIVRVSVJWDVRVSGVPVTWHVROUQURWEVTWHUSUTVAAWJBVBVCV
|
|
DTVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d b ph x $. $d n ph w x y $.
|
|
diffitest.1 $e |- A. a e. Fin A. b ( a \ b ) e. Fin $.
|
|
$( If subtracting any set from a finite set gives a finite set, any
|
|
proposition of the form ` -. ph ` is decidable. This is not a proof of
|
|
full excluded middle, but it is close enough to show we won't be able to
|
|
prove ` A e. Fin -> ( A \ B ) e. Fin ` . (Contributed by Jim Kingdon,
|
|
8-Sep-2021.) $)
|
|
diffitest $p |- ( -. ph \/ -. -. ph ) $=
|
|
( vx vn vw vy c0 cdif cv cen wbr com cfn wcel cvv ax-mp wceq wex csn crab
|
|
wrex wn wo wal wral 0ex snfig difeq1 eleq1d albidv mp2 rabexg difeq2 spcv
|
|
rspcv isfi mpbi 0elnn wa breq2 en0 syl6bb biimpac rabeq0 notrab eqeq1i wb
|
|
snm r19.3rmv 3bitr4i sylib olcd ensym elex2 enm syl2an biidd simprbi orcd
|
|
elrab eleq2s exlimiv syl jaodan sylan2 ancoms rexlimiva ) IUAZAEWJUBZJZFK
|
|
ZLMZFNUCZAUDZWPUDZUEZWLOPZWOWJCKZJZOPZCUFZWSWJOPZBKZWTJZOPZCUFZBOUGXCIQPX
|
|
DUHIQUIRZDXHXCBWJOXEWJSZXGXBCXJXFXAOXEWJWTUJUKULUQUMXBWSCWKXDWKQPXIAEWJOU
|
|
NRWTWKSXAWLOWTWKWJUOUKUPRFWLURUSWNWRFNWNWMNPZWRXKWNWMISZIWMPZUEWRWMUTWNXL
|
|
WRXMWNXLVAZWQWPXNWLISZWQXLWNXOXLWNWLILMXOWMIWLLVBWLVCVDVEWPEWJUBZISWQEWJU
|
|
GZXOWQWPEWJVFWLXPIAEWJVGZVHGKZWJPGTWQXQVIGIUHVJWQEGWJVKRVLVMVNWNXMVAHKZWL
|
|
PZHTZWRWNWMWLLMXSWMPGTYBXMWLWMVOGIWMVPGHWMWLVQVRYAWRHWRXTXPWLXTXPPZWPWQYC
|
|
XTWJPWPWPWPEXTWJEKXTSWPVSWBVTWAXRWCWDWEWFWGWHWIR $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z $. $d A x y z $. $d ps x $. $d ch x $. $d th x $.
|
|
$d ta x $. $d ph v w y z $.
|
|
findcard.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
findcard.2 $e |- ( x = ( y \ { z } ) -> ( ph <-> ch ) ) $.
|
|
findcard.3 $e |- ( x = y -> ( ph <-> th ) ) $.
|
|
findcard.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
findcard.5 $e |- ps $.
|
|
findcard.6 $e |- ( y e. Fin -> ( A. z e. y ch -> th ) ) $.
|
|
$( Schema for induction on the cardinality of a finite set. The inductive
|
|
hypothesis is that the result is true on the given set with any one
|
|
element removed. The result is then proven to be true for all finite
|
|
sets. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
findcard $p |- ( A e. Fin -> ta ) $=
|
|
( vw wcel cen wbr com vv cfn cv wrex isfi wi wal csuc breq2 imbi1d albidv
|
|
c0 wceq en0 mpbiri sylbi ax-gen w3a wa peano2 rspcev sylan sylibr 3adant2
|
|
wral csn cdif dif1en 3expa cvv vex difexg ax-mp imbi12d syl5com ralrimdva
|
|
breq1 spcv imp an32s 3impa alrimdv cbvalv syl6ibr finds1 19.21bi rexlimiv
|
|
sylc 3exp vtoclga ) AEFIUBMFUCZUBQWKPUCZRSZPTUDAPWKUEWMAPTWLTQWMAUFZFWNFU
|
|
GWKULRSZAUFZFUGWKUAUCZRSZAUFZFUGZWKWQUHZRSZAUFZFUGZPUAWLULUMZWNWPFXEWMWOA
|
|
WLULWKRUIUJUKWLWQUMZWNWSFXFWMWRAWLWQWKRUIUJUKWLXAUMZWNXCFXGWMXBAWLXAWKRUI
|
|
UJUKWPFWOWKULUMZAWKUNXHABNJUOUPUQWQTQZWTGUCZXARSZDUFZGUGXDXIWTXLGXIWTXKDX
|
|
IWTXKURXJUBQZCHXJVEZDXIXKXMWTXIXKUSZXJWLRSZPTUDZXMXIXATQXKXQWQUTXPXKPXATW
|
|
LXAXJRUIVAVBPXJUEVCVDXIWTXKXNXIXKWTXNXOWTXNXOWTCHXJXOHUCZXJQZUSXJXRVFZVGZ
|
|
WQRSZWTCXIXKXSYBXJWQXRVHVIWSYBCUFFYAXJVJQYAVJQGVKXJXTVJVLVMWKYAUMWRYBACWK
|
|
YAWQRVQKVNVRVOVPVSVTWAOWHWIWBXCXLFGWKXJUMXBXKADWKXJXARVQLVNWCWDWEWFWGUPWJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z $. $d A w x y z $. $d ps x $. $d ch x $. $d th x $.
|
|
$d ta x $. $d ph v w y z $.
|
|
findcard2.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
findcard2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
findcard2.3 $e |- ( x = ( y u. { z } ) -> ( ph <-> th ) ) $.
|
|
findcard2.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
findcard2.5 $e |- ps $.
|
|
findcard2.6 $e |- ( y e. Fin -> ( ch -> th ) ) $.
|
|
$( Schema for induction on the cardinality of a finite set. The inductive
|
|
step shows that the result is true if one more element is added to the
|
|
set. The result is then proven to be true for all finite sets.
|
|
(Contributed by Jeff Madsen, 8-Jul-2010.) $)
|
|
findcard2 $p |- ( A e. Fin -> ta ) $=
|
|
( vw wcel cen wi c0 vv cfn wbr com wrex isfi wal csuc breq2 imbi1d albidv
|
|
cv wceq en0 mpbiri sylbi ax-gen wsbc wne wa peano3 adantr breq1 anbi2d wb
|
|
peano1 peano2 nneneq sylancr biimpa eqcomd syl6bi necon3d mpd ex wex nnfi
|
|
com12 syl enfi adantl mpbird fin0 csn simpll dif1en 3expa fidifsnid sylan
|
|
cdif cun cvv difexg ax-mp uneq1 sbceq1d imbi2d imbi12d rspe sylibr pm2.27
|
|
vex spv sylsyld syl5 snex unex sbcie syl6ibr dfsbcq syl5ib mp2and exlimdv
|
|
vtocl sylbid mpdd com23 alrimdv nfsbc1v nfim sbceq1a cbval finds1 19.21bi
|
|
nfv rexlimiv vtoclga ) AEFIUBMFULZUBQYHPULZRUCZPUDUEAPYHUFYJAPUDYIUDQYJAS
|
|
ZFYKFUGYHTRUCZASZFUGYHUAULZRUCZASZFUGZYHYNUHZRUCZASZFUGZPUAYITUMZYKYMFUUB
|
|
YJYLAYITYHRUIUJUKYIYNUMZYKYPFUUCYJYOAYIYNYHRUIUJUKYIYRUMZYKYTFUUDYJYSAYIY
|
|
RYHRUIUJUKYMFYLYHTUMZAYHUNUUEABNJUOUPUQYNUDQZYQYIYRRUCZAFYIURZSZPUGUUAUUF
|
|
YQUUIPUUFUUGYQUUHUUFUUGYITUSZYQUUHSZUUFUUGUUJUUFUUGUTZYRTUSZUUJUUFUUMUUGY
|
|
NVAVBUULYITYRTUUBUULYRTUMZUUBUULUUFTYRRUCZUTZUUNUUBUUGUUOUUFYITYRRVCVDUUP
|
|
TYRUUFUUOTYRUMZUUFTUDQYRUDQZUUOUUQVEVFYNVGZTYRVHVIVJVKVLVRVMVNVOUUFUUGUUJ
|
|
UUKSUULUUJHULZYIQZHVPZUUKUULYIUBQZUUJUVBVEUULUVCYRUBQZUUFUVDUUGUUFUURUVDU
|
|
USYRVQVSVBUUGUVCUVDVEUUFYIYRVTWAWBZHYIWCVSUULUVAUUKHUULUVAUUKUULUVAUTZUUF
|
|
YIUUTWDZWJZYNRUCZUUKUUFUUGUVAWEUUFUUGUVAUVIYIYNUUTWFWGUVFUVHUVGWKZYIUMZUU
|
|
FUVIUTZUUKSUULUVCUVAUVKUVEYIUUTWHWIUVLYQAFUVJURZSZUVKUUKUUFGULZYNRUCZUTZY
|
|
QAFUVOUVGWKZURZSZSUVLUVNSGUVHYIWLQUVHWLQPXBYIUVGWLWMWNUVOUVHUMZUVQUVLUVTU
|
|
VNUWAUVPUVIUUFUVOUVHYNRVCVDUWAUVSUVMYQUWAAFUVRUVJUVOUVHUVGWOWPWQWRUVQYQDU
|
|
VSYQUVPCSZUVQDYPUWBFGYHUVOUMYOUVPACYHUVOYNRVCKWRXCUVQUVOUBQZUWBCDUVQUVPUA
|
|
UDUEUWCUVPUAUDWSUAUVOUFWTUVPUWBCSUUFUVPCXAWAOXDXEADFUVRUVOUVGGXBUUTHXBXFX
|
|
GLXHXIXNUVKUVMUUHYQAFUVJYIXJWQXKVSXLVOXMXOVOXPXQXRYTUUIFPYTPYEUUGUUHFUUGF
|
|
YEAFYIXSXTYHYIUMYSUUGAUUHYHYIYRRVCAFYIYAWRYBXIYCYDYFUPYG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d ch x $. $d ph v w y z $. $d ps x $. $d ta x $.
|
|
$d th x $. $d v w x y z $.
|
|
findcard2s.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
|
|
findcard2s.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
findcard2s.3 $e |- ( x = ( y u. { z } ) -> ( ph <-> th ) ) $.
|
|
findcard2s.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
findcard2s.5 $e |- ps $.
|
|
findcard2s.6 $e |- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) ) $.
|
|
$( Variation of ~ findcard2 requiring that the element added in the
|
|
induction step not be a member of the original set. (Contributed by
|
|
Paul Chapman, 30-Nov-2012.) $)
|
|
findcard2s $p |- ( A e. Fin -> ta ) $=
|
|
( vw wcel cen wi c0 vv cfn wbr com wrex isfi wal csuc breq2 imbi1d albidv
|
|
cv wceq en0 mpbiri sylbi ax-gen wsbc wne wa peano3 adantr breq1 anbi2d wb
|
|
peano1 peano2 nneneq sylancr biimpa eqcomd syl6bi necon3d mpd ex wex nnfi
|
|
com12 syl enfi adantl mpbird fin0 csn simpll dif1en 3expa fidifsnid sylan
|
|
cdif cun neldifsn cvv vex difexg ax-mp eleq2 notbid anbi12d uneq1 sbceq1d
|
|
wn imbi2d imbi12d spv pm2.27 rspe syld syl5 snex unex sbcie syl6ibr vtocl
|
|
sylibr mpan2 dfsbcq syl5ib mp2and exlimdv sylbid mpdd alrimdv nfv nfsbc1v
|
|
com23 nfim sbceq1a cbval finds1 19.21bi rexlimiv vtoclga ) AEFIUBMFULZUBQ
|
|
YNPULZRUCZPUDUEAPYNUFYPAPUDYOUDQYPASZFYQFUGYNTRUCZASZFUGYNUAULZRUCZASZFUG
|
|
ZYNYTUHZRUCZASZFUGZPUAYOTUMZYQYSFUUHYPYRAYOTYNRUIUJUKYOYTUMZYQUUBFUUIYPUU
|
|
AAYOYTYNRUIUJUKYOUUDUMZYQUUFFUUJYPUUEAYOUUDYNRUIUJUKYSFYRYNTUMZAYNUNUUKAB
|
|
NJUOUPUQYTUDQZUUCYOUUDRUCZAFYOURZSZPUGUUGUULUUCUUOPUULUUMUUCUUNUULUUMYOTU
|
|
SZUUCUUNSZUULUUMUUPUULUUMUTZUUDTUSZUUPUULUUSUUMYTVAVBUURYOTUUDTUUHUURUUDT
|
|
UMZUUHUURUULTUUDRUCZUTZUUTUUHUUMUVAUULYOTUUDRVCVDUVBTUUDUULUVATUUDUMZUULT
|
|
UDQUUDUDQZUVAUVCVEVFYTVGZTUUDVHVIVJVKVLVRVMVNVOUULUUMUUPUUQSUURUUPHULZYOQ
|
|
ZHVPZUUQUURYOUBQZUUPUVHVEUURUVIUUDUBQZUULUVJUUMUULUVDUVJUVEUUDVQVSVBUUMUV
|
|
IUVJVEUULYOUUDVTWAWBZHYOWCVSUURUVGUUQHUURUVGUUQUURUVGUTZUULYOUVFWDZWJZYTR
|
|
UCZUUQUULUUMUVGWEUULUUMUVGUVOYOYTUVFWFWGUVLUVNUVMWKZYOUMZUULUVOUTZUUQSUUR
|
|
UVIUVGUVQUVKYOUVFWHWIUVRUUCAFUVPURZSZUVQUUQUVRUVFUVNQZXBZUVTUVFYOWLUULGUL
|
|
ZYTRUCZUTZUVFUWCQZXBZUTZUUCAFUWCUVMWKZURZSZSUVRUWBUTZUVTSGUVNYOWMQUVNWMQP
|
|
WNYOUVMWMWOWPUWCUVNUMZUWHUWLUWKUVTUWMUWEUVRUWGUWBUWMUWDUVOUULUWCUVNYTRVCV
|
|
DUWMUWFUWAUWCUVNUVFWQWRWSUWMUWJUVSUUCUWMAFUWIUVPUWCUVNUVMWTXAXCXDUWHUUCDU
|
|
WJUUCUWDCSZUWHDUUBUWNFGYNUWCUMUUAUWDACYNUWCYTRVCKXDXEUWHUWNCDUWEUWNCSZUWG
|
|
UWDUWOUULUWDCXFWAVBUWEUWCUBQZUWGCDSUWEUWDUAUDUEUWPUWDUAUDXGUAUWCUFXOOWIXH
|
|
XIADFUWIUWCUVMGWNUVFHWNXJXKLXLXMXNXPUVQUVSUUNUUCAFUVPYOXQXCXRVSXSVOXTYAVO
|
|
YBYFYCUUFUUOFPUUFPYDUUMUUNFUUMFYDAFYOYEYGYNYOUMUUEUUMAUUNYNYOUUDRVCAFYOYH
|
|
XDYIXMYJYKYLUPYM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d ph x y z $. $d ps y z $. $d ch x $. $d th x $.
|
|
$d ta x $. $d et x $.
|
|
findcard2d.ch $e |- ( x = (/) -> ( ps <-> ch ) ) $.
|
|
findcard2d.th $e |- ( x = y -> ( ps <-> th ) ) $.
|
|
findcard2d.ta $e |- ( x = ( y u. { z } ) -> ( ps <-> ta ) ) $.
|
|
findcard2d.et $e |- ( x = A -> ( ps <-> et ) ) $.
|
|
findcard2d.z $e |- ( ph -> ch ) $.
|
|
findcard2d.i $e |- ( ( ph /\ ( y C_ A /\ z e. ( A \ y ) ) ) ->
|
|
( th -> ta ) ) $.
|
|
findcard2d.a $e |- ( ph -> A e. Fin ) $.
|
|
$( Deduction version of ~ findcard2 . If you also need ` y e. Fin ` (which
|
|
doesn't come for free due to ~ ssfiexmid ), use ~ findcard2sd instead.
|
|
(Contributed by SO, 16-Jul-2018.) $)
|
|
findcard2d $p |- ( ph -> et ) $=
|
|
( wcel wa wi wss ssid cfn adantr cv csn cun wceq sseq1 anbi2d imbi12d weq
|
|
c0 wel wn simprl simprr unssad jca cdif id vsnid elun2 mp1i sseldd simplr
|
|
ad2antll eldifd syl12anc embantd ex com23 findcard2s mpcom mpan2 ) AJJUAZ
|
|
FJUBJUCRZAVPSZFAVQVPQUDAGUEZJUAZSZBTAUMJUAZSZCTAHUEZJUAZSZDTZAWDIUEZUFZUG
|
|
ZJUAZSZETVRFTGHIJVSUMUHZWAWCBCWMVTWBAVSUMJUIUJKUKGHULZWAWFBDWNVTWEAVSWDJU
|
|
IUJLUKVSWJUHZWAWLBEWOVTWKAVSWJJUIUJMUKVSJUHZWAVRBFWPVTVPAVSJJUIUJNUKACWBO
|
|
UDWDUCRZIHUNUOZSZWLWGEWSWLWGETWSWLSZWFDEWTAWEWSAWKUPZWTWDWIJWSAWKUQURZUSW
|
|
TAWEWHJWDUTRDETXAXBWTWHJWDWKWHJRWSAWKWJJWHWKVAWHWIRWHWJRWKIVBWHWIWDVCVDVE
|
|
VGWQWRWLVFVHPVIVJVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d ph x y z $. $d ps y z $. $d ch x $. $d th x $.
|
|
$d ta x $. $d et x $.
|
|
findcard2sd.ch $e |- ( x = (/) -> ( ps <-> ch ) ) $.
|
|
findcard2sd.th $e |- ( x = y -> ( ps <-> th ) ) $.
|
|
findcard2sd.ta $e |- ( x = ( y u. { z } ) -> ( ps <-> ta ) ) $.
|
|
findcard2sd.et $e |- ( x = A -> ( ps <-> et ) ) $.
|
|
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|
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findcard2sd.i $e |- ( ( ( ph /\ y e. Fin ) /\
|
|
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|
|
findcard2sd.a $e |- ( ph -> A e. Fin ) $.
|
|
$( Deduction form of finite set induction . (Contributed by Jim Kingdon,
|
|
14-Sep-2021.) $)
|
|
findcard2sd $p |- ( ph -> et ) $=
|
|
( wcel wa wi wss ssid cfn adantr cv c0 csn cun wceq anbi2d imbi12d simprl
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|
|
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|
|
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|
|
UMUJLUKVRWIUIZVTWKBEWNVSWJAVRWIJUMUJMUKVRJUIZVTVQBFWOVSVOAVRJJUMUJNUKACWA
|
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QWSAWPWDWGJWCURRDETWTWPWQWKUSXAWSWGJWCWJWGJRWRAWJWIJWGWJVEWGWHRWGWIRWJIUT
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|
|
$}
|
|
|
|
${
|
|
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|
|
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|
|
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|
|
diffisn $p |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin ) $=
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( vn vm vx cfn wcel wa cv cen wbr com wrex adantr wceq adantl simplrr en0
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|
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|
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$}
|
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|
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|
|
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|
|
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|
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diffifi $p |- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) ->
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|
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( vw vy vz cfn wcel wss cdif wa cv wi c0 wceq sseq1 anbi2d difeq2 imbi12d
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difun1 simprl unssad simplr mp2and vsnid unssbd sseld mpi adantllr eldifd
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simpllr diffisn syl2anc syl5eqel exp31 findcard2s imp syl12anc ) AFGZBFGZ
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|
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PJZWQWIWMIZFAWFWMULXGWJWLWIGXHFGXGVKWGWJXFVKWOUMXGWFWMAXFVKWOTUNXEWKWPUOU
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|
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$}
|
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|
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|
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$( An infinite set is not finite. (Contributed by Jim Kingdon,
|
|
20-Feb-2022.) $)
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|
infnfi $p |- ( _om ~<_ A -> -. A e. Fin ) $=
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( vn com cdom wbr cfn wcel wa cv cen wrex isfi biimpi adantl csuc cvv wss
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|
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|
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|
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|
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|
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because we can, the more natural way to express " ` _om ` is infinite" is
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` _om ~<_ _om ` which is an instance of ~ domrefg . (Contributed by NM,
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2-Jun-1998.) $)
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ominf $p |- -. _om e. Fin $=
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ac6sfi.1 $e |- ( y = ( f ` x ) -> ( ph <-> ps ) ) $.
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ac6sfi $p |- ( ( A e. Fin /\ A. x e. A E. y e. B ph ) ->
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( wcel wral cv wf wa wex wi c0 wceq raleq anbi12d cvv vu vw vz vg cfn csn
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|
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18-Feb-2022.) $)
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infm $p |- ( _om ~<_ A -> E. x x e. A ) $=
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( vf com cdom wbr cv wf1 wcel wex brdomi wa c0 cfv wf f1f adantl ffvelrnd
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$( An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) $)
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infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $=
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$( Building a set with two elements. (Contributed by FL, 11-Aug-2008.)
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(Revised by Mario Carneiro, 10-Sep-2015.) $)
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en2eqpr $p |- ( ( C ~~ 2o /\ A e. C /\ B e. C )
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-> ( A =/= B -> C = { A , B } ) ) $=
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( vf c2o wcel wa adantr c0 c1o simplr simpr eqtr4d ad2antrr mpbid eqeltrd
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15-Sep-2021.) $)
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fientri3 $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) ) $=
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Kingdon, 25-Sep-2021.) $)
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nnwetri $p |- ( A e. _om -> ( _E We A /\
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|
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20-Oct-2021.) $)
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onunsnss $p |- ( ( B e. V /\ ( A u. { B } ) e. On ) ->
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Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro
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and Jim Kingdon, 5-Mar-2022.) $)
|
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unfiexmid $p |- ( ph \/ -. ph ) $=
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$}
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Jim Kingdon, 3-Feb-2022.) $)
|
|
unsnfi $p |- ( ( A e. Fin /\ B e. V /\ -. B e. A )
|
|
-> ( A u. { B } ) e. Fin ) $=
|
|
( vn vm cfn wcel wn cv cen wbr csn cun com wrex isfi wa ad2antrl cin c0
|
|
w3a biimpi 3ad2ant1 csuc peano2 simprr simpl2 simprl en2sn syl2anc disjsn
|
|
wceq biimpri 3ad2ant3 adantr word nnord ordirr syl sylibr syl22anc df-suc
|
|
unen syl6breqr breq2 rspcev rexlimddv ) AFGZBCGZBAGHZUAZADIZJKZABLZMZFGZD
|
|
NVHVIVMDNOZVJVHVQDAPUBUCVKVLNGZVMQZQZVOEIZJKZENOZVPVTVLUDZNGZVOWDJKZWCVRW
|
|
EVKVMVLUERVTVOVLVLLZMZWDJVTVMVNWGJKZAVNSTULZVLWGSTULZVOWHJKVKVRVMUFVTVIVR
|
|
WIVHVIVJVSUGVKVRVMUHBVLCNUIUJVKWJVSVJVHWJVIWJVJABUKUMUNUOVRWKVKVMVRVLVLGH
|
|
ZWKVRVLUPWLVLUQVLURUSVLVLUKUTRAVLVNWGVCVAVLVBVDWBWFEWDNWAWDVOJVEVFUJEVOPU
|
|
TVG $.
|
|
$}
|
|
|
|
${
|
|
$d A m n $. $d B m n $.
|
|
$( The ` B e. V ` condition in ~ unsnfi . This is intended to show that
|
|
~ unsnfi without that condition would not be provable but it probably
|
|
would need to be strengthened (for example, to imply included middle) to
|
|
fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.) $)
|
|
unsnfidcex $p |- ( ( A e. Fin /\ -. B e. A /\ ( A u. { B } ) e. Fin )
|
|
-> DECID -. B e. _V ) $=
|
|
( vn vm cfn wcel wn cun cvv wo wdc cen wbr com biimpi wa ad2antrr syl2anc
|
|
wceq c0 csn cv wrex isfi 3ad2ant1 3ad2ant3 adantr simprr ad3antrrr simplr
|
|
w3a breqtrrd ensymd entr cdif simp1 ad4antr simpr eldifd php5fin pm2.65da
|
|
simp2 orcd snprc uneq2d un0 syl6eq adantl eqbrtrrd wb simprl nneneq mpbid
|
|
simplrl eqcomd olcd nndceq exmiddc syl mpjaodan rexlimddv df-dc sylibr )
|
|
AEFZBAFGZABUAZHZEFZUKZBIFZGZWKGZJZWKKWIACUBZLMZWMCNWDWEWOCNUCZWHWDWPCAUDO
|
|
UEWIWNNFZWOPZPZWGDUBZLMZWMDNWIXADNUCZWRWHWDXBWEWHXBDWGUDOUFUGWSWTNFZXAPZP
|
|
ZWTWNSZWMXFGZXEXFPZWKWLXHWJAWGLMZXHWJPZAWTLMZWTWGLMXIXJAWNWTLWSWOXDXFWJWI
|
|
WQWOUHZUIXEXFWJUJULXJWGWTXEXAXFWJWSXCXAUHZQUMAWTWGUNRXJWDBIAUOFXIGWIWDWRX
|
|
DXFWJWDWEWHUPUQXJBIAXHWJURWIWEWRXDXFWJWDWEWHVBUQUSABUTRVAVCXEXGPZWLWKXNWK
|
|
XFXNWKPZWNWTXOWNWTLMZWNWTSZXOWNALMXKXPXOAWNWSWOXDXGWKXLUIUMXOWGAWTLWKWGAS
|
|
XNWKWGATHAWKWFTAWKWFTSBVDOVEAVFVGVHXEXAXGWKXMQVIWNAWTUNRXOWQXCXPXQVJXEWQX
|
|
GWKWIWQWOXDVNZQXEXCXGWKWSXCXAVKZQWNWTVLRVMVOXEXGWKUJVAVPXEXFKZXFXGJXEXCWQ
|
|
XTXSXRWTWNVQRXFVRVSVTWAWAWKWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d A m n $. $d B m n $. $d V m n $.
|
|
$( The ` -. B e. A ` condition in ~ unsnfi . This is intended to show that
|
|
~ unsnfi without that condition would not be provable but it probably
|
|
would need to be strengthened (for example, to imply included middle) to
|
|
fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.) $)
|
|
unsnfidcel $p |- ( ( A e. Fin /\ B e. V /\ ( A u. { B } ) e. Fin )
|
|
-> DECID -. B e. A ) $=
|
|
( vn vm cfn wcel wn wo wdc cv cen wbr com wrex ad2antrr ad3antrrr syl2anc
|
|
wa ensymd csn cun isfi biimpi 3ad2ant1 3ad2ant3 adantr wceq simprr simplr
|
|
w3a breqtrrd entr cdif simp1 ad4antr simpl2 elexd eldifd php5fin pm2.65da
|
|
cvv simpr olcd wb wss snssi ssequn2 sylib breq1d adantl mpbid simprl orcd
|
|
nneneq nndceq exmiddc syl mpjaodan rexlimddv df-dc sylibr ) AFGZBCGZABUAZ
|
|
UBZFGZUKZBAGZHZWJHZIZWJJWHADKZLMZWLDNWCWDWNDNOZWGWCWODAUCUDUEWHWMNGZWNSZS
|
|
ZWFEKZLMZWLENWHWTENOZWQWGWCXAWDWGXAEWFUCUDUFUGWRWSNGZWTSZSZWSWMUHZWLXEHZX
|
|
DXESZWKWJXGWJAWFLMZXGWJSZWFAXIWTWSALMZWFALMXDWTXEWJWRXBWTUIZPXIAWSXIAWMWS
|
|
LWRWNXCXEWJWHWPWNUIZQXDXEWJUJULTWFWSAUMRTXIWCBVBAUNGXHHWHWCWQXCXEWJWCWDWG
|
|
UOUPXIBVBAXIBCWRWDXCXEWJWCWDWGWQUQQURXGWJVCUSABUTRVAVDXDXFSZWJWKXMWIXEXMW
|
|
ISZWSWMLMZXEXNXJWNXOXNAWSXNWTAWSLMZXDWTXFWIXKPWIWTXPVEXMWIWFAWSLWIWEAVFWF
|
|
AUHBAVGWEAVHVIVJVKVLTWRWNXCXFWIXLQWSAWMUMRXNXBWPXOXEVEXDXBXFWIWRXBWTVMZPW
|
|
RWPXCXFWIWHWPWNVMZQWSWMVORVLXDXFWIUJVAVNXDXEJZXEXFIXDXBWPXSXQWRWPXCXRUGWS
|
|
WMVPRXEVQVRVSVTVTWJWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d A w y z $. $d B w y z $.
|
|
$( The union of two disjoint finite sets is finite. (Contributed by Jim
|
|
Kingdon, 25-Feb-2022.) $)
|
|
unfidisj $p |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) )
|
|
-> ( A u. B ) e. Fin ) $=
|
|
( vw vy vz cfn wcel cin c0 wceq w3a cv cun csn uneq2 eleq1d un0 wa cvv wn
|
|
simp1 syl5eqel wss unass simpr vex a1i wo simplrr eldifad simp3 ad3antrrr
|
|
cdif minel syl2anc eldifbd sylanbrc elun sylnibr unsnfi syl3anc syl5eqelr
|
|
ioran ex simp2 findcard2sd ) AFGZBFGZABHIJZKZACLZMZFGAIMZFGADLZMZFGZAVNEL
|
|
ZNZMZMZFGZABMZFGCDEBVKIJVLVMFVKIAOPVKVNJVLVOFVKVNAOPVKVSJVLVTFVKVSAOPVKBJ
|
|
VLWBFVKBAOPVJVMAFAQVGVHVIUAUBVJVNFGZRZVNBUCZVQBVNUMGZRZRZVPWAWHVPRZVTVOVR
|
|
MZFAVNVRUDWIVPVQSGZVQVOGZTWJFGWHVPUEWKWIEUFUGWIVQAGZVQVNGZUHZWLWIWMTZWNTW
|
|
OTWIVQBGVIWPWIVQBVNWDWEWFVPUIZUJVJVIWCWGVPVGVHVIUKULVQBAUNUOWIVQBVNWQUPWM
|
|
WNVCUQVQAVNURUSVOVQSUTVAVBVDVGVHVIVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d A w y z $. $d B w y z $.
|
|
|
|
$( Union of complementary parts into whole. This is a case where we can
|
|
strengthen ~ undifss from subset to equality. (Contributed by Jim
|
|
Kingdon, 2-Mar-2022.) $)
|
|
undiffi $p |- ( ( A e. Fin /\ B e. Fin /\ B C_ A )
|
|
-> A = ( B u. ( A \ B ) ) ) $=
|
|
( vw vy cfn wcel wss cv cdif cun c0 id difeq2 uneq12d eqeq2d uncom wa cin
|
|
wceq syl5eq w3a csn un0 dif0 3eqtr3ri a1i difundi uneq2i undi eqtri simp3
|
|
ad3antrrr simplrr eldifad sseldd snssd ssequn1 sylib uneq2d eqtr3d uneq1i
|
|
vz simpr unass syl6reqr simp1 fidifsnid syl2anc simplrl sstrd eqtrd inidm
|
|
ineq12d syl6req ex simp2 findcard2sd ) AEFZBEFZBAGZUAZACHZAWBIZJZSAKAKIZJ
|
|
ZSZADHZAWHIZJZSZAWHVBHZUBZJZAWNIZJZSZABABIZJZSCDVBBWBKSZWDWFAWTWBKWCWEWTL
|
|
WBKAMNOWBWHSZWDWJAXAWBWHWCWIXALWBWHAMNOWBWNSZWDWPAXBWBWNWCWOXBLWBWNAMNOWB
|
|
BSZWDWSAXCWBBWCWRXCLWBBAMNOWGWAWEKJWEWFAWEUCWEKPAUDUEUFWAWHEFZQZWHBGZWLBW
|
|
HIFZQZQZWKWQXIWKQZWPAARZAXJWPWNWIJZWNAWMIZJZRZXKWPWNWIXMRZJXOWOXPWNAWHWMU
|
|
GUHWNWIXMUIUJXJXLAXNAXJAWMWJJZXLXJWMAJZAXQXJWMAGXRASXJWLAXJBAWLWAVTXDXHWK
|
|
VRVSVTUKULZXJWLBWHXEXFXGWKUMUNUOZUPWMAUQURXJAWJWMXIWKVCUSUTXLWMWHJZWIJXQW
|
|
NYAWIWHWMPVAWMWHWIVDUJVEXJXNWHAJZAXJXNWHWMXMJZJYBWHWMXMVDXJYCAWHXJYCXMWMJ
|
|
ZAWMXMPXJVRWLAFYDASWAVRXDXHWKVRVSVTVFULXTAWLVGVHTUSTXJWHAGYBASXJWHBAXEXFX
|
|
GWKVIXSVJWHAUQURVKVMTAVLVNVOVRVSVTVPVQ $.
|
|
$}
|
|
|
|
$( The union of two finite sets is finite if their intersection is.
|
|
(Contributed by Jim Kingdon, 2-Mar-2022.) $)
|
|
unfiin $p |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) e. Fin )
|
|
-> ( A u. B ) e. Fin ) $=
|
|
( cfn wcel cin cun wa cdif wss simpll a1i undiffi syl3anc c0 diffifi eqtr3i
|
|
wceq ineq2i disjdif unfidisj simpr inss1 simplr inss2 uneq12d syl6eqr incom
|
|
unundi difeq2i difin difss ssdisj mp2an eqtri difundir eqeltrd 3impa ) ACDZ
|
|
BCDZABEZCDZABFZCDURUSGZVAGZVBUTAUTHZBUTHZFZFZCVDVBUTVEFZUTVFFZFVHVDAVIBVJVD
|
|
URVAUTAIZAVIQURUSVAJZVCVAUAZVKVDABUBKZAUTLMVDUSVAUTBIZBVJQURUSVAUCZVMVOVDAB
|
|
UDKZBUTLMUEUTVEVFUHUFVDVAVGCDZUTVGEZNQZVHCDVMVDVECDZVFCDZVEVFEZNQZVRVDURVAV
|
|
KWAVLVMVNAUTOMVDUSVAVOWBVPVMVQBUTOMWDVDWCVEBAHZEZNVFWEVEBBAEZHVFWEWGUTBBAUG
|
|
UIBAUJPRVEAIAWEENQWFNQAUTUKABSVEAWEULUMUNKVEVFTMVTVDUTVBUTHZEVSNWHVGUTABUTU
|
|
ORUTVBSPKUTVGTMUPUQ $.
|
|
|
|
${
|
|
$d A x $. $d V x $.
|
|
$( An ordinal which is a singleton is ` { (/) } ` . (Contributed by Jim
|
|
Kingdon, 19-Oct-2021.) $)
|
|
snon0 $p |- ( ( A e. V /\ { A } e. On ) -> A = (/) ) $=
|
|
( vx wcel con0 wa cv elirr wceq snidg adantr wi ontr1 adantl mpan2d elsni
|
|
csn syl6 eleq1 biimpcd sylcom mtoi eq0rdv ) ABDZAQZEDZFZCAUGCGZADZAADZAHU
|
|
GUIUHAIZUJUGUIUHUEDZUKUGUIAUEDZULUDUMUFABJKUFUIUMFULLUDUHAUEMNOUHAPRUKUIU
|
|
JUHAASTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y z w A $. $d y z w F $.
|
|
$( A version of ~ fnex for finite sets that does not require Replacement.
|
|
(Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario
|
|
Carneiro, 24-Jun-2015.) $)
|
|
fnfi $p |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin ) $=
|
|
( vw vy vz cfn wcel wa cres cv c0 csn cun reseq2 eleq1d syl2anc cvv simpr
|
|
wceq cdm wfn fnresdm adantr res0 0fin eqeltri a1i wss resundi cfv simp-4l
|
|
cdif simplrr eldifad fnressn uneq2d wn elexd funfvex funfni opexg eldifbd
|
|
cop cin opeldmg dmres eleq2i syl6ib elin simplbi syl6 mtod unsnfi syl3anc
|
|
wi eqeltrd syl5eqel ex findcard2sd eqeltrrd ) BAUAZAFGZHZBAIZBFWAWDBSWBAB
|
|
UBUCWCBCJZIZFGBKIZFGZBDJZIZFGZBWIEJZLZMZIZFGZWDFGCDEAWEKSWFWGFWEKBNOWEWIS
|
|
WFWJFWEWIBNOWEWNSWFWOFWEWNBNOWEASWFWDFWEABNOWHWCWGKFBUDUEUFUGWCWIFGZHZWIA
|
|
UHZWLAWIULZGZHZHZWKWPXCWKHZWOWJBWMIZMZFBWIWMUIXDXFWJWLWLBUJZVCZLZMZFXDXEX
|
|
IWJXDWAWLAGZXEXISWAWBWQXBWKUKZXDWLAWIWRWSXAWKUMZUNZAWLBUOPUPXDWKXHQGZXHWJ
|
|
GZUQXJFGXCWKRXDWLQGXGQGZXOXDWLWTXMURXDWAXKXQXLXNXQAWLBWLBUSUTPZWLXGQQVAPX
|
|
DXPWLWIGZXDWLAWIXMVBXDXPWLWIBTZVDZGZXSXDXPWLWJTZGZYBXDXKXQXPYDVOXNXRWLXGW
|
|
JAQVEPYCYAWLBWIVFVGVHYBXSWLXTGWLWIXTVIVJVKVLWJXHQVMVNVPVQVRWAWBRVSVT $.
|
|
$}
|
|
|
|
$( The domain of a finite function is finite. (Contributed by Jim Kingdon,
|
|
5-Feb-2022.) $)
|
|
fundmfi $p |- ( ( A e. Fin /\ Fun A ) -> dom A e. Fin ) $=
|
|
( cfn wcel wfun cdm cen wbr fundmeng enfii syldan ) ABCADAEZAFGKBCABHKAIJ
|
|
$.
|
|
|
|
$( A function is finite if and only if its domain is finite. (Contributed by
|
|
AV, 10-Jan-2020.) $)
|
|
fundmfibi $p |- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) ) $=
|
|
( wfun cfn wcel cdm fundmfi ancoms wfn funfn fnfi sylanb impbida ) ABZACDZA
|
|
EZCDZNMPAFGMAOHPNAIOAJKL $.
|
|
|
|
$( If a relation is finite, its converse is as well. (Contributed by Jim
|
|
Kingdon, 5-Feb-2022.) $)
|
|
relcnvfi $p |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin ) $=
|
|
( wrel cfn wcel wa ccnv cen wbr wceq dfrel2 biimpi adantr simpr eqeltrd cvv
|
|
relcnv cnvexg cnven sylancr adantl enfii syl2anc ) ABZACDZEZAFZFZCDUFUGGHZU
|
|
FCDUEUGACUCUGAIZUDUCUIAJKLUCUDMNUDUHUCUDUFBUFODUHAPACQUFORSTUFUGUAUB $.
|
|
|
|
$( The range of a finite relation is finite if its converse is a function.
|
|
(Contributed by Jim Kingdon, 5-Feb-2022.) $)
|
|
funrnfi $p |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin ) $=
|
|
( wrel ccnv wfun cfn wcel w3a crn cdm df-rn relcnvfi 3adant2 simp2 syl5eqel
|
|
fundmfi syl2anc ) ABZACZDZAEFZGZAHRIZEAJUAREFZSUBEFQTUCSAKLQSTMROPN $.
|
|
|
|
$( If a 1-1 and onto function has a finite domain, its range is finite.
|
|
(Contributed by Jim Kingdon, 21-Feb-2022.) $)
|
|
f1ofi $p |- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> B e. Fin ) $=
|
|
( cfn wcel wf1o cen wbr wa f1oeng ensymd enfii syldan ) ADEZABCFZBAGHBDENOI
|
|
ABABDCJKBALM $.
|
|
|
|
$( A one-to-one function whose domain is a set is finite if and only if its
|
|
range is finite. See also ~ f1vrnfibi . (Contributed by AV,
|
|
10-Jan-2020.) $)
|
|
f1dmvrnfibi $p |- ( ( A e. V /\ F : A -1-1-> B )
|
|
-> ( F e. Fin <-> ran F e. Fin ) ) $=
|
|
( wcel wf1 wa cfn crn wrel ccnv wfun f1rel ad2antlr wf1o f1cnv syl simpr wb
|
|
wceq f1ofun funrnfi syl3anc cdm cen wbr wi f1dm f1f1orn eleq1 f1oeq2 eqcoms
|
|
anbi12d biimpd expcomd sylc impcom adantr f1oeng enfii syl2anc f1fun mpbird
|
|
fundmfibi impbida ) ADEZABCFZGZCHEZCIZHEZVHVIGCJZCKZLZVIVKVGVLVFVIABCMNVGVN
|
|
VFVIVGVJAVMOVNABCPVJAVMUAQNVHVIRCUBUCVHVKGZVICUDZHEZVOVKVPVJUEUFZVQVHVKRVOV
|
|
PDEZVPVJCOZGZVRVHWAVKVGVFWAVGVPATZAVJCOZVFWAUGABCUHABCUIWBVFWCWAWBVFWCGZWAW
|
|
DWASAVPAVPTVFVSWCVTAVPDUJAVPVJCUKUMULUNUOUPUQURVPVJDCUSQVPVJUTVAVOCLZVIVQSV
|
|
GWEVFVKABCVBNCVDQVCVE $.
|
|
|
|
$( A one-to-one function which is a set is finite if and only if its range is
|
|
finite. See also ~ f1dmvrnfibi . (Contributed by AV, 10-Jan-2020.) $)
|
|
f1vrnfibi $p |- ( ( F e. V /\ F : A -1-1-> B )
|
|
-> ( F e. Fin <-> ran F e. Fin ) ) $=
|
|
( wcel wf1 cvv cfn crn wb cdm wceq wi dmexg eleq1 eqcoms syl5ibr syl impcom
|
|
f1dm f1dmvrnfibi sylancom ) CDEZABCFZAGEZCHECIHEJUDUCUEUDCKZALZUCUEMABCTUCU
|
|
EUGUFGEZCDNUEUHJAUFAUFGOPQRSABCGUAUB $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Supremum and infimum
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Add new constant symbols. $)
|
|
$c sup $. $( Supremum. $)
|
|
$c inf $. $( Infimum. $)
|
|
|
|
$( Extend class notation to include supremum of class ` A ` . Here ` R ` is
|
|
ordinarily a relation that strictly orders class ` B ` . For example,
|
|
` R ` could be 'less than' and ` B ` could be the set of real numbers. $)
|
|
csup $a class sup ( A , B , R ) $.
|
|
|
|
$( Extend class notation to include infimum of class ` A ` . Here ` R ` is
|
|
ordinarily a relation that strictly orders class ` B ` . For example,
|
|
` R ` could be 'less than' and ` B ` could be the set of real numbers. $)
|
|
cinf $a class inf ( A , B , R ) $.
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z B $.
|
|
$( Define the supremum of class ` A ` . It is meaningful when ` R ` is a
|
|
relation that strictly orders ` B ` and when the supremum exists.
|
|
(Contributed by NM, 22-May-1999.) $)
|
|
df-sup $a |- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\
|
|
A. y e. B ( y R x -> E. z e. A y R z ) ) } $.
|
|
$}
|
|
|
|
$( Define the infimum of class ` A ` . It is meaningful when ` R ` is a
|
|
relation that strictly orders ` B ` and when the infimum exists. For
|
|
example, ` R ` could be 'less than', ` B ` could be the set of real
|
|
numbers, and ` A ` could be the set of all positive reals; in this case
|
|
the infimum is 0. The infimum is defined as the supremum using the
|
|
converse ordering relation. In the given example, 0 is the supremum of
|
|
all reals (greatest real number) for which all positive reals are greater.
|
|
(Contributed by AV, 2-Sep-2020.) $)
|
|
df-inf $a |- inf ( A , B , R ) = sup ( A , B , `' R ) $.
|
|
|
|
${
|
|
$d x y z R $. $d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Equality theorem for supremum. (Contributed by NM, 22-May-1999.) $)
|
|
supeq1 $p |- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) ) $=
|
|
( vx vy vz wceq cv wbr wn wral wrex wi wa crab cuni csup raleq df-sup
|
|
rexeq imbi2d ralbidv anbi12d rabbidv unieqd 3eqtr4g ) BCHZEIZFIZDJKZFBLZU
|
|
JUIDJZUJGIDJZGBMZNZFALZOZEAPZQUKFCLZUMUNGCMZNZFALZOZEAPZQBADRCADRUHUSVEUH
|
|
URVDEAUHULUTUQVCUKFBCSUHUPVBFAUHUOVAUMUNGBCUAUBUCUDUEUFEFGBADTEFGCADTUG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
supeq1d.1 $e |- ( ph -> B = C ) $.
|
|
$( Equality deduction for supremum. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
supeq1d $p |- ( ph -> sup ( B , A , R ) = sup ( C , A , R ) ) $=
|
|
( wceq csup supeq1 syl ) ACDGCBEHDBEHGFBCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
supeq1i.1 $e |- B = C $.
|
|
$( Equality inference for supremum. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
supeq1i $p |- sup ( B , A , R ) = sup ( C , A , R ) $=
|
|
( wceq csup supeq1 ax-mp ) BCFBADGCADGFEABCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d R x y z $.
|
|
$( Equality theorem for supremum. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
supeq2 $p |- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) ) $=
|
|
( vx vy vz wceq cv wbr wn wral wrex wi wa crab cuni csup rabeq df-sup
|
|
raleq anbi2d rabbidv eqtrd unieqd 3eqtr4g ) BCHZEIZFIZDJKFALZUIUHDJUIGIDJ
|
|
GAMNZFBLZOZEBPZQUJUKFCLZOZECPZQABDRACDRUGUNUQUGUNUMECPUQUMEBCSUGUMUPECUGU
|
|
LUOUJUKFBCUAUBUCUDUEEFGABDTEFGACDTUF $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d R x y z $. $d S x y z $.
|
|
$( Equality theorem for supremum. (Contributed by Scott Fenton,
|
|
13-Jun-2018.) $)
|
|
supeq3 $p |- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) ) $=
|
|
( vx vy vz cv wbr wn wral wrex wi wa crab cuni csup breq ralbidv df-sup
|
|
wceq notbid rexbidv imbi12d anbi12d rabbidv unieqd 3eqtr4g ) CDUAZEHZFHZC
|
|
IZJZFAKZUKUJCIZUKGHZCIZGALZMZFBKZNZEBOZPUJUKDIZJZFAKZUKUJDIZUKUPDIZGALZMZ
|
|
FBKZNZEBOZPABCQABDQUIVBVLUIVAVKEBUIUNVEUTVJUIUMVDFAUIULVCUJUKCDRUBSUIUSVI
|
|
FBUIUOVFURVHUKUJCDRUIUQVGGAUKUPCDRUCUDSUEUFUGEFGABCTEFGABDTUH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. $d E x y z $.
|
|
$d F x y z $. $d ph x y z $.
|
|
supeq123d.a $e |- ( ph -> A = D ) $.
|
|
supeq123d.b $e |- ( ph -> B = E ) $.
|
|
supeq123d.c $e |- ( ph -> C = F ) $.
|
|
$( Equality deduction for supremum. (Contributed by Stefan O'Rear,
|
|
20-Jan-2015.) $)
|
|
supeq123d $p |- ( ph -> sup ( A , B , C ) = sup ( D , E , F ) ) $=
|
|
( vx vy vz cv wbr wn wral wrex wi breqd wa crab cuni csup imbi12d anbi12d
|
|
notbid raleqbidv rexeqbidv rabeqbidv unieqd df-sup 3eqtr4g ) AKNZLNZDOZPZ
|
|
LBQZUOUNDOZUOMNZDOZMBRZSZLCQZUAZKCUBZUCUNUOGOZPZLEQZUOUNGOZUOUTGOZMERZSZL
|
|
FQZUAZKFUBZUCBCDUDEFGUDAVFVPAVEVOKCFIAURVIVDVNAUQVHLBEHAUPVGADGUNUOJTUGUH
|
|
AVCVMLCFIAUSVJVBVLADGUOUNJTAVAVKMBEHADGUOUTJTUIUEUHUFUJUKKLMBCDULKLMEFGUL
|
|
UM $.
|
|
$}
|
|
|
|
${
|
|
$d A u v w $. $d B u v w $. $d R u v w $. $d u v w x $.
|
|
nfsup.1 $e |- F/_ x A $.
|
|
nfsup.2 $e |- F/_ x B $.
|
|
nfsup.3 $e |- F/_ x R $.
|
|
$( Hypothesis builder for supremum. (Contributed by Mario Carneiro,
|
|
20-Mar-2014.) $)
|
|
nfsup $p |- F/_ x sup ( A , B , R ) $=
|
|
( vu vv vw csup cv wbr wn wral wrex wi nfcv nfbr nfralya crab cuni df-sup
|
|
wa nfn nfrexya nfim nfan nfrabxy nfuni nfcxfr ) ABCDKHLZILZDMZNZIBOZUMULD
|
|
MZUMJLZDMZJBPZQZICOZUDZHCUAZUBHIJBCDUCAVDVCAHCUPVBAUOAIBEUNAAULUMDAULRZGA
|
|
UMRZSUETVAAICFUQUTAAUMULDVFGVESUSAJBEAUMURDVFGAURRSUFUGTUHFUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x $. $d A v w x $. $d A w x y $. $d A z $. $d B w x y z $.
|
|
$d R u v x $. $d R v w x $. $d R w x y z $. $d ph u v x $.
|
|
$d ph v w x $.
|
|
supmoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
$( Any class ` B ` has at most one supremum in ` A ` (where ` R ` is
|
|
interpreted as 'less than'). The hypothesis is satisfied by real
|
|
numbers (see ~ lttri3 ) or other orders which correspond to tight
|
|
apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.) $)
|
|
supmoti $p |- ( ph -> E* x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $=
|
|
( vw cv wbr wn wral wrex wi wa weq breq1 wrmo wcel anbi2ci 3bitr4i ralnex
|
|
ancom rexbidv imbi12d rspcva breq2 cbvrexv syl6ibr con3d expimpd ad2antrl
|
|
an42 syl5bi ad2antll anim12d wb ralrimivva equequ1 notbid anbi12d bibi12d
|
|
equequ2 rspc2v mpan9 sylibrd ralbidv imbi1d rmo4 sylibr ) ABLZCLZIMZNZCHO
|
|
ZVOVNIMZVODLZIMZDHPZQZCGOZRZKLZVOIMZNZCHOZVOWFIMZWBQZCGOZRZRZBKSZQZKGOBGO
|
|
WEBGUAAWPBKGGAVNGUBZWFGUBZRZRZWNVNWFIMZNZWFVNIMZNZRZWOWNWLVRRZWDWIRZRZWTX
|
|
EVRWIRZWLWDRZRXJWIVRRZRWNXHXIXKXJVRWIUFUCVRWDWIWLUPWLVRWDWIUPUDWTXFXBXGXD
|
|
WQXFXBQAWRWQWLVRXBVRVPCHPZNWQWLRZXBVPCHUEXMXAXLXMXAVNVTIMZDHPZXLWKXAXOQCV
|
|
NGCBSZWJXAWBXOVOVNWFITXPWAXNDHVOVNVTITUGUHUIVPXNCDHVOVTVNIUJUKULUMUQUNUOW
|
|
RXGXDQAWQWRWDWIXDWIWGCHPZNWRWDRZXDWGCHUEXRXCXQXRXCWFVTIMZDHPZXQWCXCXTQCWF
|
|
GCKSZVSXCWBXTVOWFVNITYAWAXSDHVOWFVTITUGUHUIWGXSCDHVOVTWFIUJUKULUMUQUNURUS
|
|
UQAFESZFLZELZIMZNZYDYCIMZNZRZUTZEGOFGOWSWOXEUTZAYJFEGGJVAYJYKBESZVNYDIMZN
|
|
ZYDVNIMZNZRZUTFEVNWFGGFBSZYBYLYIYQFBEVBYRYFYNYHYPYRYEYMYCVNYDITVCYRYGYOYC
|
|
VNYDIUJVCVDVEEKSZYLWOYQXEEKBVFYSYNXBYPXDYSYMXAYDWFVNIUJVCYSYOXCYDWFVNITVC
|
|
VDVEVGVHVIVAWEWMBKGWOVRWIWDWLWOVQWHCHWOVPWGVNWFVOITVCVJWOWCWKCGWOVSWJWBVN
|
|
WFVOIUJVKVJVDVLVM $.
|
|
|
|
${
|
|
supeuti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
|
|
$( A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but
|
|
for suprema in general). (Contributed by Jim Kingdon,
|
|
23-Nov-2021.) $)
|
|
supeuti $p |- ( ph ->
|
|
E! x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $=
|
|
( cv wbr wn wral wrex wi wa wrmo wreu supmoti reu5 sylanbrc ) ABLZCLZIM
|
|
NCHOUEUDIMUEDLIMDHPQCGORZBGPUFBGSUFBGTKABCDEFGHIJUAUFBGUBUC $.
|
|
$}
|
|
|
|
${
|
|
supval2ti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
|
|
$( Alternate expression for the supremum. (Contributed by Jim Kingdon,
|
|
23-Nov-2021.) $)
|
|
supval2ti $p |- ( ph -> sup ( B , A , R ) =
|
|
( iota_ x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) ) $=
|
|
( cv wbr wn wral wrex wi wa crio crab cuni csup supeuti riotauni df-sup
|
|
wreu wceq syl syl6reqr ) ABLZCLZIMNCHOUKUJIMUKDLIMDHPQCGORZBGSZULBGTUAZ
|
|
HGIUBAULBGUFUMUNUGABCDEFGHIJKUCULBGUDUHBCDHGIUEUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y u v C $. $d B u v $. $d u v z $.
|
|
$( Sufficient condition for an element to be equal to the supremum.
|
|
(Contributed by Jim Kingdon, 23-Nov-2021.) $)
|
|
eqsupti $p |- ( ph -> ( ( C e. A /\ A. y e. B -. C R y /\
|
|
A. y e. A ( y R C -> E. z e. B y R z ) ) -> sup ( B , A , R ) = C ) ) $=
|
|
( vx wcel cv wbr wn wral wrex wi wceq wa w3a csup crio wb adantlr breq1
|
|
notbid ralbidv breq2 imbi1d anbi12d rspcev adantl supval2ti 3simpc wreu
|
|
3impb simpr1 supeuti riota2 syl2anc mpbid eqtrd ex ) AHFLZHBMZINZOZBGPZ
|
|
VFHINZVFCMINCGQZRZBFPZUAZGFIUBZHSAVNTZVOKMZVFINZOZBGPZVFVQINZVKRZBFPZTZ
|
|
KFUCZHVPKBCDEFGIAEMZFLDMZFLTWFWGSWFWGINOWGWFINOTUDVNJUEZVNWDKFQZAVEVIVM
|
|
WIWDVIVMTZKHFVQHSZVTVIWCVMWKVSVHBGWKVRVGVQHVFIUFUGUHWKWBVLBFWKWAVJVKVQH
|
|
VFIUIUJUHUKZULUQUMZUNVPWJWEHSZVNWJAVEVIVMUOUMVPVEWDKFUPWJWNUDAVEVIVMURV
|
|
PKBCDEFGIWHWMUSWDWJKFHWLUTVAVBVCVD $.
|
|
|
|
$d y ph $.
|
|
eqsuptid.2 $e |- ( ph -> C e. A ) $.
|
|
eqsuptid.3 $e |- ( ( ph /\ y e. B ) -> -. C R y ) $.
|
|
eqsuptid.4 $e |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. z e. B y R z ) $.
|
|
$( Sufficient condition for an element to be equal to the supremum.
|
|
(Contributed by Jim Kingdon, 24-Nov-2021.) $)
|
|
eqsuptid $p |- ( ph -> sup ( B , A , R ) = C ) $=
|
|
( wcel cv wbr wn wral wrex ralrimiva wi csup wceq expr eqsupti mp3and )
|
|
AHFNHBOZIPQZBGRUGHIPZUGCOIPCGSZUAZBFRGFIUBHUCKAUHBGLTAUKBFAUGFNUIUJMUDT
|
|
ABCDEFGHIJUEUF $.
|
|
$}
|
|
|
|
supclti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
|
|
$( A supremum belongs to its base class (closure law). See also ~ supubti
|
|
and ~ suplubti . (Contributed by Jim Kingdon, 24-Nov-2021.) $)
|
|
supclti $p |- ( ph -> sup ( B , A , R ) e. A ) $=
|
|
( csup cv wbr wn wral wrex wi wa crio supval2ti wreu wcel supeuti riotacl
|
|
syl eqeltrd ) AHGILBMZCMZINOCHPUIUHINUIDMINDHQRCGPSZBGTZGABCDEFGHIJKUAAUJ
|
|
BGUBUKGUCABCDEFGHIJKUDUJBGUEUFUG $.
|
|
|
|
${
|
|
$d C w $.
|
|
$( A supremum is an upper bound. See also ~ supclti and ~ suplubti .
|
|
|
|
This proof demonstrates how to expand an iota-based definition
|
|
( ~ df-iota ) using ~ riotacl2 .
|
|
|
|
(Contributed by Jim Kingdon, 24-Nov-2021.) $)
|
|
supubti $p |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) ) $=
|
|
( vw cv wbr wn wral wcel wi wceq csup crab wrex simpl ss2rabi supval2ti
|
|
wa a1i crio wreu supeuti riotacl2 syl eqeltrd sseldi breq2 notbid breq1
|
|
cbvralv ralbidv syl5bb elrab simprbi rspccv 3syl ) AHGJUAZBNZCNZJOZPZCH
|
|
QZBGUBZRZVFMNZJOZPZMHQZIHRVFIJOZPZSAVKVHVGJOVHDNJODHUCSCGQZUGZBGUBZVLVF
|
|
WAVKBGWAVKSVGGRVKVTUDUHUEAVFWABGUIZWBABCDEFGHJKLUFAWABGUJWCWBRABCDEFGHJ
|
|
KLUKWABGULUMUNUOVMVFGRVQVKVQBVFGVKVGVNJOZPZMHQVGVFTZVQVJWECMHVHVNTVIWDV
|
|
HVNVGJUPUQUSWFWEVPMHWFWDVOVGVFVNJURUQUTVAVBVCVPVSMIHVNITVOVRVNIVFJUPUQV
|
|
DVE $.
|
|
$}
|
|
|
|
${
|
|
$d C w z $.
|
|
$( A supremum is the least upper bound. See also ~ supclti and
|
|
~ supubti . (Contributed by Jim Kingdon, 24-Nov-2021.) $)
|
|
suplubti $p |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) ->
|
|
E. z e. B C R z ) ) $=
|
|
( vw cv wbr wrex wi wral wcel breq1 wa wn simpr rexbidv imbi12d cbvralv
|
|
csup crab wceq a1i ss2rabi crio supval2ti wreu supeuti riotacl2 eqeltrd
|
|
sylib syl sseldi breq2 imbi1d ralbidv elrab simprbi rspccv impd 3syl )
|
|
AHGJUGZMNZBNZJOZVJDNZJOZDHPZQZMGRZBGUHZSZVJVIJOZVOQZMGRZIGSZIVIJOZUAIVM
|
|
JOZDHPZQAVKCNZJOUBCHRZWGVKJOZWGVMJOZDHPZQZCGRZUAZBGUHZVRVIWNVQBGWNVQQVK
|
|
GSWNWMVQWHWMUCWLVPCMGWGVJUIZWIVLWKVOWGVJVKJTWPWJVNDHWGVJVMJTUDUEUFURUJU
|
|
KAVIWNBGULZWOABCDEFGHJKLUMAWNBGUNWQWOSABCDEFGHJKLUOWNBGUPUSUQUTVSVIGSWB
|
|
VQWBBVIGVKVIUIZVPWAMGWRVLVTVOVKVIVJJVAVBVCVDVEWBWCWDWFWAWDWFQMIGVJIUIZV
|
|
TWDVOWFVJIVIJTWSVNWEDHVJIVMJTUDUEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d A v w x $. $d A w x y z $. $d B w x y z $. $d C w z $.
|
|
$d R v w x $. $d R w x y z $. $d ph v w x $.
|
|
suplub2ti.or $e |- ( ph -> R Or A ) $.
|
|
suplub2ti.3 $e |- ( ph -> B C_ A ) $.
|
|
$( Bidirectional form of ~ suplubti . (Contributed by Jim Kingdon,
|
|
17-Jan-2022.) $)
|
|
suplub2ti $p |- ( ( ph /\ C e. A ) ->
|
|
( C R sup ( B , A , R ) <-> E. z e. B C R z ) ) $=
|
|
( vw wcel wa wbr cv ad3antrrr suplubti expdimp breq2 cbvrexv wn simplll
|
|
csup wrex simplr supubti sylc wo simpr wor wi simpllr wss sseldd sowlin
|
|
supclti syl13anc mpd ecased ex rexlimdva syl5bi impbid ) AIGPZQZIHGJUGZ
|
|
JRZIDSZJRZDHUHZAVHVKVNABCDEFGHIJKLUAUBVNIOSZJRZOHUHVIVKVMVPDOHVLVOIJUCU
|
|
DVIVPVKOHVIVOHPZQZVPVKVRVPQZVKVJVOJRZVSAVQVTUEAVHVQVPUFVIVQVPUIZABCDEFG
|
|
HVOJKLUJUKVSVPVKVTULZVRVPUMVSGJUNZVHVOGPVJGPZVPWBUOAWCVHVQVPMTAVHVQVPUP
|
|
VSHGVOAHGUQVHVQVPNTWAURAWDVHVQVPABCDEFGHJKLUTTGIVOVJJUSVAVBVCVDVEVFVG
|
|
$.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A u v x $. $d A x y z $. $d B x y z $. $d C x $. $d R u v x $.
|
|
$d R x y z $. $d ph u v x $.
|
|
supelti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
supelti.ex $e |- ( ph -> E. x e. C ( A. y e. B -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. B y R z ) ) ) $.
|
|
supelti.ss $e |- ( ph -> C C_ A ) $.
|
|
$( Supremum membership in a set. (Contributed by Jim Kingdon,
|
|
16-Jan-2022.) $)
|
|
supelti $p |- ( ph -> sup ( B , A , R ) e. C ) $=
|
|
( cv wceq wcel wex wbr wrex wa csup wss wn wral wi ssrexv supclti elisset
|
|
sylc eqcom exbii sylib simpr crio supval2ti eqeq1d biimpa wb wreu supeuti
|
|
syl riota1 adantr mpbird simpld simprd reupick syl2an2r eqeltrd exlimddv
|
|
jca32 ) AHGJUAZBNZOZVLIPBAVMVLOZBQZVNBQAVLGPVPABCDEFGHJKAIGUBZVMCNZJRUCCH
|
|
UDVRVMJRVRDNJRDHSUECGUDTZBISZVSBGSMLVSBIGUFUIZUGBVLGUHVAVOVNBVMVLUJUKULAV
|
|
NTZVLVMIAVNUMWBVMIPZVMGPZWBWDVSWBWDVSTZVSBGUNZVMOZAVNWGAVLWFVMABCDEFGHJKW
|
|
AUOUPUQAWEWGURZVNAVSBGUSZWHABCDEFGHJKWAUTZVSBGVBVAVCVDZVEAVQVTWITTVNVSWCW
|
|
DURAVQVTWIMLWJVKWBWDVSWKVFVSBIGVGVHVDVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d B x y z $. $d R x y z $.
|
|
$( The supremum under an empty base set is always the empty set.
|
|
(Contributed by AV, 4-Sep-2020.) $)
|
|
sup00 $p |- sup ( B , (/) , R ) = (/) $=
|
|
( vx vy vz c0 csup cv wbr wn wral wrex wi wa crab cuni df-sup rab0 unieqi
|
|
uni0 3eqtri ) AFBGCHZDHZBIJDAKUCUBBIUCEHBIEALMDFKNZCFOZPFPFCDEAFBQUEFUDCR
|
|
STUA $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x y $. $d B u v x y $. $d C u v x y $. $d R u v x y $.
|
|
$d ph u v y $.
|
|
supmaxti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
supmaxti.2 $e |- ( ph -> C e. A ) $.
|
|
supmaxti.3 $e |- ( ph -> C e. B ) $.
|
|
supmaxti.4 $e |- ( ( ph /\ y e. B ) -> -. C R y ) $.
|
|
$( The greatest element of a set is its supremum. Note that the converse
|
|
is not true; the supremum might not be an element of the set considered.
|
|
(Contributed by Jim Kingdon, 24-Nov-2021.) $)
|
|
supmaxti $p |- ( ph -> sup ( B , A , R ) = C ) $=
|
|
( vx wcel cv wbr wa wrex simprr breq2 rspcev syl2an2r eqsuptid ) ABMCDEFG
|
|
HIJLAGFNBOZENZUDGHPZQUFUDMOZHPZMFRKAUEUFSUHUFMGFUGGUDHTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x $. $d B u v x $. $d R u v x $. $d ph u v x $.
|
|
supsnti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
supsnti.b $e |- ( ph -> B e. A ) $.
|
|
$( The supremum of a singleton. (Contributed by Jim Kingdon,
|
|
26-Nov-2021.) $)
|
|
supsnti $p |- ( ph -> sup ( { B } , A , R ) = B ) $=
|
|
( vx wcel cv wa wbr wn wceq wb wral breq1 notbid breq2 csn snidg syl eqid
|
|
ralrimivva wi eqeq1 anbi12d bibi12d eqeq2 rspc2v syl2anc mpd mpbii simpld
|
|
adantr elsni breq2d adantl mpbird supmaxti ) AIBCDEUAZEFGHAEDJZEVBJHEDUBU
|
|
CAIKZVBJZLEVDFMZNZEEFMZNZAVIVEAVIVIAEEOZVIVILZEUDACKZBKZOZVLVMFMZNZVMVLFM
|
|
ZNZLZPZBDQCDQZVJVKPZAVTCBDDGUEAVCVCWAWBUFHHVTWBEVMOZEVMFMZNZVMEFMZNZLZPCB
|
|
EEDDVLEOZVNWCVSWHVLEVMUGWIVPWEVRWGWIVOWDVLEVMFRSWIVQWFVLEVMFTSUHUIVMEOZWC
|
|
VJWHVKVMEEUJWJWEVIWGVIWJWDVHVMEEFTSWJWFVHVMEEFRSUHUIUKULUMUNUOUPVEVGVIPAV
|
|
EVFVHVEVDEEFVDEUQURSUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A u v $. $d B u v x y $. $d F u v x y $. $d R u v $. $d S u v x y $.
|
|
$( Lemma for ~ isoti . (Contributed by Jim Kingdon, 26-Nov-2021.) $)
|
|
isotilem $p |- ( F Isom R , S ( A , B ) -> (
|
|
A. x e. B A. y e. B ( x = y <-> ( -. x S y /\ -. y S x ) ) ->
|
|
A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) ) ) $=
|
|
( cv wceq wbr wn wa wb wral wcel notbid anbi12d bibi12d wiso wi wf isof1o
|
|
cfv wf1o f1of ffvelrn anim12d 3syl imp eqeq1 breq1 breq2 eqeq2 rspc2v syl
|
|
ex wf1 f1of1 f1fveq sylan bicomd isorel ancom2s sylibrd ralrimdvva ) EFGH
|
|
IUAZAJZBJZKZVIVJHLZMZVJVIHLZMZNZOZBFPAFPZDJZCJZKZVSVTGLZMZVTVSGLZMZNZOZDC
|
|
EEVHVSEQZVTEQZNZNZVRVSIUEZVTIUEZKZWLWMHLZMZWMWLHLZMZNZOZWGWKWLFQZWMFQZNZV
|
|
RWTUBVHWJXCVHEFIUFZEFIUCZWJXCUBEFGHIUDZEFIUGXEWHXAWIXBXEWHXAEFVSIUHURXEWI
|
|
XBEFVTIUHURUIUJUKVQWTWLVJKZWLVJHLZMZVJWLHLZMZNZOABWLWMFFVIWLKZVKXGVPXLVIW
|
|
LVJULXMVMXIVOXKXMVLXHVIWLVJHUMRXMVNXJVIWLVJHUNRSTVJWMKZXGWNXLWSVJWMWLUOXN
|
|
XIWPXKWRXNXHWOVJWMWLHUNRXNXJWQVJWMWLHUMRSTUPUQWKWAWNWFWSWKWNWAVHEFIUSZWJW
|
|
NWAOVHXDXOXFEFIUTUQEFVSVTIVAVBVCWKWCWPWEWRWKWBWOEFVSVTGHIVDRVHWIWHWEWROVH
|
|
WIWHNNWDWQEFVTVSGHIVDRVESTVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x y $. $d B u v x y $. $d F u v x y $. $d R u v x y $.
|
|
$d S u v x y $.
|
|
$( An isomorphism preserves tightness. (Contributed by Jim Kingdon,
|
|
26-Nov-2021.) $)
|
|
isoti $p |- ( F Isom R , S ( A , B ) -> (
|
|
A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) <->
|
|
A. u e. B A. v e. B ( u = v <-> ( -. u S v /\ -. v S u ) ) ) ) $=
|
|
( vx vy wiso weq cv wbr wn wa wb wral isotilem breq1 notbid wi isocnv syl
|
|
ccnv impbid equequ1 breq2 anbi12d bibi12d equequ2 cbvral2v syl6bb ) CDEFG
|
|
JZBAKZBLZALZEMNUPUOEMNOPACQBCQZHIKZHLZILZFMZNZUTUSFMZNZOZPZIDQHDQZUNUOUPF
|
|
MZNZUPUOFMZNZOZPZADQBDQUMUQVGUMDCFEGUDZJUQVGUACDEFGUBBAIHDCFEVNRUCHIABCDE
|
|
FGRUEVFVMBIKZUOUTFMZNZUTUOFMZNZOZPHIBADDHBKZURVOVEVTHBIUFWAVBVQVDVSWAVAVP
|
|
USUOUTFSTWAVCVRUSUOUTFUGTUHUIIAKZVOUNVTVLIABUJWBVQVIVSVKWBVPVHUTUPUOFUGTW
|
|
BVRVJUTUPUOFSTUHUIUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x y z A $. $d u v w x y z C $. $d w y z D $. $d u w ph $.
|
|
$d u v w x y z F $. $d u w x y z R $. $d u v w x y z S $.
|
|
$d u v w x y z B $.
|
|
supiso.1 $e |- ( ph -> F Isom R , S ( A , B ) ) $.
|
|
supiso.2 $e |- ( ph -> C C_ A ) $.
|
|
$( Lemma for ~ supisoti . (Contributed by Mario Carneiro, 24-Dec-2016.) $)
|
|
supisolem $p |- ( ( ph /\ D e. A ) ->
|
|
( ( A. y e. C -. D R y /\ A. y e. A ( y R D -> E. z e. C y R z ) ) <->
|
|
( A. w e. ( F " C ) -. ( F ` D ) S w /\
|
|
A. w e. B ( w S ( F ` D ) -> E. v e. ( F " C ) w S v ) ) ) ) $=
|
|
( wa wcel wbr wral wb adantr wiso wss cv wn wrex wi cfv jca simpll simplr
|
|
cima sselda isorel syl12anc notbid ralbidva wfn wf1o syl f1ofn wceq breq2
|
|
isof1o ralima syl2anc bitr4d simpr rexima imbi12d wfo f1ofo breq1 rexbidv
|
|
rexbidva cbvfo 3syl bitrd anbi12d sylan ) AFGJKLUAZHFUBZOZIFPZIBUCZJQZUDZ
|
|
BHRZWDIJQZWDCUCZJQZCHUEZUFZBFRZOILUGZDUCZKQZUDZDLHUKZRZWOWNKQZWOEUCZKQZEW
|
|
RUEZUFZDGRZOSAVTWAMNUHWBWCOZWGWSWMXEXFWGWNWDLUGZKQZUDZBHRZWSXFWFXIBHXFWDH
|
|
PZOZWEXHXLVTWCWDFPZWEXHSXFVTXKVTWAWCUIZTWBWCXKUJXFHFWDVTWAWCUJZULFGIWDJKL
|
|
UMUNUOUPXFLFUQZWAWSXJSXFFGLURZXPXFVTXQXNFGJKLVCUSZFGLUTUSZXOWQXIDBFHLWOXG
|
|
VAWPXHWOXGWNKVBUOVDVEVFXFWMXGWNKQZXGXAKQZEWRUEZUFZBFRZXEXFWLYCBFXFXMOZWHX
|
|
TWKYBYEVTXMWCWHXTSXFVTXMXNTZXFXMVGWBWCXMUJFGWDIJKLUMUNYEWKXGWILUGZKQZCHUE
|
|
ZYBYEWJYHCHYEWIHPZOVTXMWIFPWJYHSYEVTYJYFTXFXMYJUJYEHFWIXFWAXMXOTZULFGWDWI
|
|
JKLUMUNVNYEXPWAYBYISXFXPXMXSTYKYAYHECFHLXAYGXGKVBVHVEVFVIUPXFXQFGLVJYDXES
|
|
XRFGLVKYCXDBDFGLXGWOVAZXTWTYBXCXGWOWNKVLYLYAXBEWRXGWOXAKVLVMVIVOVPVQVRVS
|
|
$.
|
|
|
|
supisoex.3 $e |- ( ph -> E. x e. A ( A. y e. C -. x R y /\
|
|
A. y e. A ( y R x -> E. z e. C y R z ) ) ) $.
|
|
$( Lemma for ~ supisoti . (Contributed by Mario Carneiro, 24-Dec-2016.) $)
|
|
supisoex $p |- ( ph -> E. u e. B ( A. w e. ( F " C ) -. u S w /\
|
|
A. w e. B ( w S u -> E. v e. ( F " C ) w S v ) ) ) $=
|
|
( cv wbr wral wi wn wrex wa cima wiso wss wcel simpl simpr supisolem wf1o
|
|
cfv wf isof1o f1of 3syl ffvelrnda wceq breq1 notbid ralbidv breq2 anbi12d
|
|
imbi1d rspcev ex syl sylbid rexlimdva syl2anc mpd ) ABQZCQZKRUACJSVMVLKRV
|
|
MDQKRDJUBTCHSUCZBHUBZGQZEQZLRZUAZEMJUDZSZVQVPLRZVQFQLRFVTUBZTZEISZUCZGIUB
|
|
ZPAHIKLMUEZJHUFZVOWGTNOWHWIUCZVNWGBHWJVLHUGUCZVNVLMULZVQLRZUAZEVTSZVQWLLR
|
|
ZWCTZEISZUCZWGWJCDEFHIJVLKLMWHWIUHZWHWIUIUJWKWLIUGZWSWGTWJHIVLMWJWHHIMUKH
|
|
IMUMWTHIKLMUNHIMUOUPUQXAWSWGWFWSGWLIVPWLURZWAWOWEWRXBVSWNEVTXBVRWMVPWLVQL
|
|
USUTVAXBWDWQEIXBWBWPWCVPWLVQLVBVDVAVCVEVFVGVHVIVJVK $.
|
|
|
|
$d A j k w z $. $d B j k w z $. $d C j k w z $. $d F j k w z $.
|
|
$d R j w z v $. $d S j k w z $. $d ph j k w v x $. $d j k u v $.
|
|
supisoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
$( Image of a supremum under an isomorphism. (Contributed by Jim Kingdon,
|
|
26-Nov-2021.) $)
|
|
supisoti $p |- ( ph ->
|
|
sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) ) $=
|
|
( vw vj wbr wral vk cima csup cfv cv wcel weq wn wa ralrimivva wiso isoti
|
|
wb syl mpbid r19.21bi anasss wf1o wf isof1o f1of 3syl supclti ffvelrnd wi
|
|
wrex supubti ralrimiv suplubti expd supisolem mpbi2and simpld simprd impr
|
|
mpdan eqsuptid ) AQUAEFHLIUBZIGJUCZLUDZKAFUEZHUFZEUEZHUFFEUGZWAWCKSUHWCWA
|
|
KSUHUIUMZAWBUIWEEHAWEEHTZFHAWDWAWCJSUHWCWAJSUHUIUMZEGTFGTZWFFHTZAWGFEGGPU
|
|
JAGHJKLUKZWHWIUMMEFGHJKLULUNUOUPUPUQAGHVSLAWJGHLURGHLUSMGHJKLUTGHLVAVBABC
|
|
DEFGIJPOVCZVDAVTQUEZKSUHZQVRAWMQVRTZWLVTKSZWLUAUEKSUAVRVFZVEZQHTZAVSRUEZJ
|
|
SUHZRITZWSVSJSZWSDUEJSDIVFZVEZRGTZWNWRUIZAWTRIABCDEFGIWSJPOVGVHAXDRGAWSGU
|
|
FXBXCABCDEFGIWSJPOVIVJVHAVSGUFXAXEUIXFUMWKARDQUAGHIVSJKLMNVKVPVLZVMUPAWLH
|
|
UFWOWPAWQQHAWNWRXGVNUPVOVQ $.
|
|
$}
|
|
|
|
$( Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq1 $p |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) ) $=
|
|
( wceq ccnv csup cinf supeq1 df-inf 3eqtr4g ) BCEBADFZGCALGBADHCADHABCLIBAD
|
|
JCADJK $.
|
|
|
|
${
|
|
infeq1d.1 $e |- ( ph -> B = C ) $.
|
|
$( Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq1d $p |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) ) $=
|
|
( wceq cinf infeq1 syl ) ACDGCBEHDBEHGFBCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
infeq1i.1 $e |- B = C $.
|
|
$( Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq1i $p |- inf ( B , A , R ) = inf ( C , A , R ) $=
|
|
( wceq cinf infeq1 ax-mp ) BCFBADGCADGFEABCDHI $.
|
|
$}
|
|
|
|
$( Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq2 $p |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) ) $=
|
|
( wceq ccnv csup cinf supeq2 df-inf 3eqtr4g ) BCEABDFZGACLGABDHACDHABCLIABD
|
|
JACDJK $.
|
|
|
|
$( Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq3 $p |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) ) $=
|
|
( wceq ccnv csup cinf cnveq supeq3 syl df-inf 3eqtr4g ) CDEZABCFZGZABDFZGZA
|
|
BCHABDHNOQEPRECDIABOQJKABCLABDLM $.
|
|
|
|
${
|
|
infeq123d.a $e |- ( ph -> A = D ) $.
|
|
infeq123d.b $e |- ( ph -> B = E ) $.
|
|
infeq123d.c $e |- ( ph -> C = F ) $.
|
|
$( Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
infeq123d $p |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) ) $=
|
|
( ccnv csup cinf cnveqd supeq123d df-inf 3eqtr4g ) ABCDKZLEFGKZLBCDMEFGMA
|
|
BCREFSHIADGJNOBCDPEFGPQ $.
|
|
$}
|
|
|
|
${
|
|
nfinf.1 $e |- F/_ x A $.
|
|
nfinf.2 $e |- F/_ x B $.
|
|
nfinf.3 $e |- F/_ x R $.
|
|
$( Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) $)
|
|
nfinf $p |- F/_ x inf ( A , B , R ) $=
|
|
( cinf ccnv csup df-inf nfcnv nfsup nfcxfr ) ABCDHBCDIZJBCDKABCOEFADGLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d ph x $. $d ph y $. $d ph z $.
|
|
cnvinfex.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
|
|
$( Two ways of expressing existence of an infimum (one in terms of
|
|
converse). (Contributed by Jim Kingdon, 17-Dec-2021.) $)
|
|
cnvinfex $p |- ( ph -> E. x e. A ( A. y e. B -. x `' R y
|
|
/\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) ) $=
|
|
( cv wbr wn wral wrex wi wa wb vex brcnv a1i ralbidv ccnv rexbidv imbi12d
|
|
notbid anbi12d mpbird ) ABIZCIZGUAZJZKZCFLZUHUGUIJZUHDIZUIJZDFMZNZCELZOZB
|
|
EMUHUGGJZKZCFLZUGUHGJZUNUHGJZDFMZNZCELZOZBEMHAUSVHBEAULVBURVGAUKVACFAUJUT
|
|
UJUTPAUGUHGBQZCQZRSUDTAUQVFCEAUMVCUPVEUMVCPAUHUGGVJVIRSAUOVDDFUOVDPAUHUNG
|
|
VJDQRSUBUCTUEUBUF $.
|
|
$}
|
|
|
|
${
|
|
eqinfti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
$d A u v y z $. $d ph u v $. $d R u v y z $.
|
|
|
|
$( If a relation satisfies a condition corresponding to tightness of an
|
|
apartness generated by an order, so does its converse. (Contributed by
|
|
Jim Kingdon, 17-Dec-2021.) $)
|
|
cnvti $p |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) ) $=
|
|
( cv wcel wa weq wbr wn ccnv ancom syl6bb wb brcnvg notbid ancoms anbi12d
|
|
adantl bitr4d ) ACGZDHZBGZDHZIZIZCBJZUEUCEKZLZUCUEEKZLZIZUCUEEMZKZLZUEUCU
|
|
OKZLZIZUHUIUMUKIUNFUMUKNOUGUTUNPAUGUQUKUSUMUGUPUJUCUEDDEQRUGURULUFUDURULP
|
|
UEUCDDEQSRTUAUB $.
|
|
|
|
$d B u v y z $.
|
|
${
|
|
$d C u v y z $.
|
|
$( Sufficient condition for an element to be equal to the infimum.
|
|
(Contributed by Jim Kingdon, 16-Dec-2021.) $)
|
|
eqinfti $p |- ( ph -> ( ( C e. A /\ A. y e. B -. y R C /\
|
|
A. y e. A ( C R y -> E. z e. B z R y ) ) -> inf ( B , A , R ) = C ) ) $=
|
|
( wcel cv wbr wn wral wrex wa cvv wb bicomd wi w3a cinf wceq ccnv cnvti
|
|
csup df-inf eqsupti vex brcnvg mpan2 ralbidv mpan brcnv rexbidv imbi12d
|
|
notbid a1i anbi12d pm5.32i 3anass 3bitr4i biimpi impel syl5eq ex ) AHFK
|
|
ZBLZHIMZNZBGOZHVIIMZCLZVIIMZCGPZUAZBFOZUBZGFIUCZHUDAVSQVTGFIUEZUGZHGFIU
|
|
HAVHHVIWAMZNZBGOZVIHWAMZVIVNWAMZCGPZUAZBFOZUBZWBHUDVSABCDEFGHWAADEFIJUF
|
|
UIVSWKVHVLVRQZQVHWEWJQZQVSWKVHWLWMVHVLWEVRWJVHVKWDBGVHVJWCVHVIRKZVJWCSB
|
|
UJZVHWNQWCVJHVIFRIUKTULURUMVHVQWIBFVHVMWFVPWHVHWFVMWNVHWFVMSWOVIHRFIUKU
|
|
NTVHVOWGCGVHWGVOWGVOSVHVIVNIWOCUJUOUSTUPUQUMUTVAVHVLVRVBVHWEWJVBVCVDVEV
|
|
FVG $.
|
|
|
|
$d ph y $.
|
|
eqinftid.2 $e |- ( ph -> C e. A ) $.
|
|
eqinftid.3 $e |- ( ( ph /\ y e. B ) -> -. y R C ) $.
|
|
eqinftid.4 $e |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y ) $.
|
|
$( Sufficient condition for an element to be equal to the infimum.
|
|
(Contributed by Jim Kingdon, 16-Dec-2021.) $)
|
|
eqinftid $p |- ( ph -> inf ( B , A , R ) = C ) $=
|
|
( wcel cv wbr wn wral wrex ralrimiva wi cinf wceq expr eqinfti mp3and )
|
|
AHFNBOZHIPQZBGRHUGIPZCOUGIPCGSZUAZBFRGFIUBHUCKAUHBGLTAUKBFAUGFNUIUJMUDT
|
|
ABCDEFGHIJUEUF $.
|
|
$}
|
|
|
|
infvalti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
|
|
$d A x $. $d B x $. $d R x $. $d ph u v x y z $.
|
|
$( Alternate expression for the infimum. (Contributed by Jim Kingdon,
|
|
17-Dec-2021.) $)
|
|
infvalti $p |- ( ph -> inf ( B , A , R ) =
|
|
( iota_ x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) ) $=
|
|
( cv wbr wn wral wrex wb vex brcnv a1i cinf ccnv csup wi wa crio cnvinfex
|
|
df-inf supval2ti notbid ralbidv rexbidv imbi12d anbi12d riotabidv syl5eq
|
|
cnvti eqtrd ) AHGIUAHGIUBZUCZCLZBLZIMZNZCHOZVBVAIMZDLZVAIMZDHPZUDZCGOZUEZ
|
|
BGUFZHGIUHAUTVBVAUSMZNZCHOZVAVBUSMZVAVGUSMZDHPZUDZCGOZUEZBGUFVMABCDEFGHUS
|
|
AEFGIJUQABCDGHIKUGUIAWBVLBGAVPVEWAVKAVOVDCHAVNVCVNVCQAVBVAIBRZCRZSTUJUKAV
|
|
TVJCGAVQVFVSVIVQVFQAVAVBIWDWCSTAVRVHDHVRVHQAVAVGIWDDRSTULUMUKUNUOURUP $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x y z $. $d B u v x y z $. $d R u v x y z $. $d ph u v x y z $.
|
|
infclti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
infclti.ex $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
|
|
$( An infimum belongs to its base class (closure law). See also ~ inflbti
|
|
and ~ infglbti . (Contributed by Jim Kingdon, 17-Dec-2021.) $)
|
|
infclti $p |- ( ph -> inf ( B , A , R ) e. A ) $=
|
|
( cinf ccnv csup df-inf cnvti cnvinfex supclti syl5eqel ) AHGILHGIMZNGHGI
|
|
OABCDEFGHTAEFGIJPABCDGHIKQRS $.
|
|
|
|
$( An infimum is a lower bound. See also ~ infclti and ~ infglbti .
|
|
(Contributed by Jim Kingdon, 18-Dec-2021.) $)
|
|
inflbti $p |- ( ph -> ( C e. B -> -. C R inf ( B , A , R ) ) ) $=
|
|
( wcel cinf wbr wn wa ccnv csup cnvti cnvinfex supubti wceq df-inf breq2d
|
|
imp a1i wb supclti brcnvg bicomd sylan bitrd mtbird ex ) AIHMZIHGJNZJOZPA
|
|
UPQZURHGJRZSZIUTOZAUPVBPABCDEFGHIUTAEFGJKTZABCDGHJLUAZUBUFUSURIVAJOZVBUSU
|
|
QVAIJUQVAUCUSHGJUDUGUEAVAGMZUPVEVBUHABCDEFGHUTVCVDUIVFUPQVBVEVAIGHJUJUKUL
|
|
UMUNUO $.
|
|
|
|
$d C z $. $d ph z $.
|
|
$( An infimum is the greatest lower bound. See also ~ infclti and
|
|
~ inflbti . (Contributed by Jim Kingdon, 18-Dec-2021.) $)
|
|
infglbti $p |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) ->
|
|
E. z e. B z R C ) ) $=
|
|
( wcel cinf wbr wrex wa wb brcnvg cvv ccnv csup df-inf breq1i simpr cnvti
|
|
cv cnvinfex supclti adantr bicomd syl2anc syl5bb suplubti expdimp sylancl
|
|
vex rexbidv sylibd sylbid expimpd ) AIGMZHGJNZIJOZDUGZIJOZDHPZAVBQZVDIHGJ
|
|
UAZUBZVIOZVGVDVJIJOZVHVKVCVJIJHGJUCUDVHVBVJGMZVLVKRAVBUEZAVMVBABCDEFGHVIA
|
|
EFGJKUFZABCDGHJLUHZUIUJVBVMQVKVLIVJGGJSUKULUMVHVKIVEVIOZDHPZVGAVBVKVRABCD
|
|
EFGHIVIVOVPUNUOVHVQVFDHVHVBVETMVQVFRVNDUQIVEGTJSUPURUSUTVA $.
|
|
|
|
$( A lower bound is not greater than the infimum. (Contributed by Jim
|
|
Kingdon, 18-Dec-2021.) $)
|
|
infnlbti $p |- ( ph -> ( ( C e. A /\ A. z e. B -. z R C ) ->
|
|
-. inf ( B , A , R ) R C ) ) $=
|
|
( wcel cv wbr wn wral cinf wa wrex infglbti expdimp rexalim con2d expimpd
|
|
syl6 ) AIGMZDNIJOZPDHQZHGJRIJOZPAUGSZUJUIUKUJUHDHTZUIPAUGUJULABCDEFGHIJKL
|
|
UAUBUHDHUCUFUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A u v y z $. $d B u v y z $. $d C u v y z $. $d R u v y z $.
|
|
$d u v y z ph $.
|
|
infminti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
infminti.2 $e |- ( ph -> C e. A ) $.
|
|
infminti.3 $e |- ( ph -> C e. B ) $.
|
|
infminti.4 $e |- ( ( ph /\ y e. B ) -> -. y R C ) $.
|
|
$( The smallest element of a set is its infimum. Note that the converse is
|
|
not true; the infimum might not be an element of the set considered.
|
|
(Contributed by Jim Kingdon, 18-Dec-2021.) $)
|
|
infminti $p |- ( ph -> inf ( B , A , R ) = C ) $=
|
|
( vz wcel cv wbr wa wrex simprr breq1 rspcev syl2an2r eqinftid ) ABMCDEFG
|
|
HIJLAGFNBOZENZGUDHPZQUFMOZUDHPZMFRKAUEUFSUHUFMGFUGGUDHTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d u v x y z A $. $d u v x y z R $. $d u v x y z B $. $d u v x ph $.
|
|
infmoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
$( Any class ` B ` has at most one infimum in ` A ` (where ` R ` is
|
|
interpreted as 'less than'). (Contributed by Jim Kingdon,
|
|
18-Dec-2021.) $)
|
|
infmoti $p |- ( ph -> E* x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $=
|
|
( cv wbr wn wral wrex wi wa wrmo vex brcnv supmoti notbii imbi12i anbi12i
|
|
ccnv cnvti ralbii rexbii rmobii sylib ) ABKZCKZIUEZLZMZCHNZULUKUMLZULDKZU
|
|
MLZDHOZPZCGNZQZBGRULUKILZMZCHNZUKULILZURULILZDHOZPZCGNZQZBGRABCDEFGHUMAEF
|
|
GIJUFUAVCVLBGUPVFVBVKUOVECHUNVDUKULIBSZCSZTUBUGVAVJCGUQVGUTVIULUKIVNVMTUS
|
|
VHDHULURIVNDSTUHUCUGUDUIUJ $.
|
|
|
|
infeuti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $.
|
|
$( An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.) $)
|
|
infeuti $p |- ( ph ->
|
|
E! x e. A ( A. y e. B -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. B z R y ) ) ) $=
|
|
( cv wbr wn wral wrex wi wa wrmo wreu infmoti reu5 sylanbrc ) ACLZBLZIMNC
|
|
HOUEUDIMDLUDIMDHPQCGORZBGPUFBGSUFBGTKABCDEFGHIJUAUFBGUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A u v $. $d B u v $. $d R u v $. $d ph u v $.
|
|
infsnti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
infsnti.b $e |- ( ph -> B e. A ) $.
|
|
$( The infimum of a singleton. (Contributed by Jim Kingdon,
|
|
19-Dec-2021.) $)
|
|
infsnti $p |- ( ph -> inf ( { B } , A , R ) = B ) $=
|
|
( csn cinf ccnv csup df-inf cnvti supsnti syl5eq ) AEIZDFJQDFKZLEQDFMABCD
|
|
ERABCDFGNHOP $.
|
|
$}
|
|
|
|
$( The infimum regarding an empty base set is always the empty set.
|
|
(Contributed by AV, 4-Sep-2020.) $)
|
|
inf00 $p |- inf ( B , (/) , R ) = (/) $=
|
|
( c0 cinf ccnv csup df-inf sup00 eqtri ) ACBDACBEZFCACBGAJHI $.
|
|
|
|
${
|
|
$d A u v x y z $. $d B u v x y z $. $d C u v x y z $. $d F u v x y z $.
|
|
$d R u v x y z $. $d S u v x y z $. $d ph u v x y z $.
|
|
infisoti.1 $e |- ( ph -> F Isom R , S ( A , B ) ) $.
|
|
infisoti.2 $e |- ( ph -> C C_ A ) $.
|
|
infisoti.3 $e |- ( ph -> E. x e. A ( A. y e. C -. y R x /\
|
|
A. y e. A ( x R y -> E. z e. C z R y ) ) ) $.
|
|
infisoti.ti $e |- ( ( ph /\ ( u e. A /\ v e. A ) ) ->
|
|
( u = v <-> ( -. u R v /\ -. v R u ) ) ) $.
|
|
$( Image of an infimum under an isomorphism. (Contributed by Jim Kingdon,
|
|
19-Dec-2021.) $)
|
|
infisoti $p |- ( ph ->
|
|
inf ( ( F " C ) , B , S ) = ( F ` inf ( C , A , R ) ) ) $=
|
|
( ccnv csup cfv cinf cima wiso isocnv2 sylib cnvinfex cnvti df-inf fveq2i
|
|
supisoti 3eqtr4g ) ALIUAZHKQZRIGJQZRZLSUKHKTIGJTZLSABCDEFGHIUMULLAGHJKLUB
|
|
GHUMULLUBMGHJKLUCUDNABCDGIJOUEAEFGJPUFUIUKHKUGUOUNLIGJUGUHUJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordinal isomorphism
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d f w x y z A $. $d f w x y z B $. $d w x y z F $.
|
|
$( Generalize ~ ordiso to proper classes. (Contributed by Mario Carneiro,
|
|
24-Jun-2015.) $)
|
|
ordiso2 $p |- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) ->
|
|
A = B ) $=
|
|
( vx vy vz vw cep cv cfv wceq wcel wi weq fveq2 id eqeq12d wa wb syl wiso
|
|
word w3a wral con0 wss ordsson sseld eleq1 imbi12d imbi2d r19.21v ordelss
|
|
3ad2ant2 3ad2antl2 sselda pm5.5 ralbidva isof1o 3ad2ant1 ad2antrr simpll3
|
|
ccnv wf1o simpr wf f1of simplrl ffvelrnd ordtr1 f1ocnvfv2 syl2anc eqeltrd
|
|
jca sylc wbr simpll1 f1ocnv isorel syl12anc vex epelc a1i cvv wfn funfvex
|
|
3syl f1ofn funfni sylan epelg 3bitr3d mpbird simplrr rspcv eqtr3d rspccva
|
|
simprr epel biimpri adantl simpl2 simprl mpbid eqeltrrd eqrdv expr sylbid
|
|
impbida ex com23 a2i syl5bi tfis2 com3l mpdd ralrimiv ffvelrnda 3ad2antl1
|
|
adantll adantlr wrex crn simpl1 wfo f1ofo forn eleq2d adantr bitr3d simpl
|
|
fvelrnb simplr exp43 syldd imp rexlimdv impbid mpdan ) ABHHCUAZAUBZBUBZUC
|
|
ZDIZCJZUUDKZDAUDZABKUUCUUFDAUUCUUDALZUUDUELZUUFUUCAUEUUDUUAYTAUEUFUUBAUGU
|
|
NUHUUIUUCUUHUUFUUCUUHUUFMZMZUUCEIZALZUULCJZUULKZMZMZDEDENZUUJUUPUUCUURUUH
|
|
UUMUUFUUOUUDUULAUIUURUUEUUNUUDUULUUDUULCOUURPQUJUKUUQEUUDUDUUCUUPEUUDUDZM
|
|
ZUUIUUKUUCUUPEUUDULUUTUUKMUUIUUCUUSUUJUUCUUHUUSUUFUUCUUHUUSUUFMUUCUUHRZUU
|
|
SUUOEUUDUDZUUFUVAUUPUUOEUUDUVAUULUUDLRUUMUUPUUOSUVAUUDAUULUUAYTUUHUUDAUFZ
|
|
UUBAUUDUMZUOUPUUMUUOUQTURUUCUUHUVBUUFUUCUUHUVBRZRZFUUEUUDUVFFIZUUELZUVGUU
|
|
DLZUVFUVHRZUVGUVGCVCZJZUUDUVJUVLCJZUVGUVLUVJABCVDZUVGBLZUVMUVGKUUCUVNUVEU
|
|
VHYTUUAUVNUUBABHHCUSZUTVAZUVJUUBUVHUUEBLZRUVOYTUUAUUBUVEUVHVBUVJUVHUVRUVF
|
|
UVHVEZUVJABUUDCUUCABCVFZUVEUVHYTUUAUVTUUBYTUVNUVTUVPABCVGTZUTVAUUCUUHUVBU
|
|
VHVHZVIVNUVGUUEBVJVOZABUVGCVKVLZUVJUVLUUDLZUVBUVMUVLKZUVJUWEUVMUUELZUVJUV
|
|
MUVGUUEUWDUVSVMUVJUVLUUDHVPZUVMUUEHVPZUWEUWGUVJYTUVLALUUHUWHUWISYTUUAUUBU
|
|
VEUVHVQZUVJBAUVGUVKUVJUVNBAUVKVDBAUVKVFUVQABCVRBAUVKVGWGUWCVIUWBABUVLUUDH
|
|
HCVSVTUWHUWESUVJUVLUUDDWAWBWCUVJUUEWDLZUWIUWGSUVJYTUUHUWKUWJUWBYTCAWEZUUH
|
|
UWKYTUVNUWLUVPABCWHTZUWKAUUDCUUDCWFWIWJZVLUVMUUEWDWKTWLWMZUUCUUHUVBUVHWNU
|
|
UOUWFEUVLUUDUULUVLKZUUNUVMUULUVLUULUVLCOUWPPQWOVOWPUWOVMUVFUVIRZUVGCJZUVG
|
|
UUEUVFUVBUVIUWRUVGKZUUCUUHUVBWRUUOUWSEUVGUUDEFNZUUNUWRUULUVGUULUVGCOUWTPQ
|
|
WQWJUWQUWRUUEHVPZUWRUUELZUWQUVGUUDHVPZUXAUVIUXCUVFUXCUVIFDWSWTXAUWQYTUVGA
|
|
LZUUHUXCUXASYTUUAUUBUVEUVIVQZUVFUUDAUVGUVFUUAUUHUVCYTUUAUUBUVEXBUUCUUHUVB
|
|
XCUVDVLUPUUCUUHUVBUVIVHZABUVGUUDHHCVSVTXDUWQUWKUXAUXBSUWQYTUUHUWKUXEUXFUW
|
|
NVLUWRUUEWDWKTXDXEXIXFXGXHXJXKXLWCXMXNXOXPXQUUCUUGRZFABUXGUXDUVOUXGUXDUVO
|
|
UXGUXDRUWRUVGBUUGUXDUWSUUCUUFUWSDUVGADFNZUUEUWRUUDUVGUUDUVGCOUXHPQWQXTUUC
|
|
UXDUWRBLZUUGYTUUAUXDUXIUUBYTABUVGCUWAXRXSYAXEXJUXGUVOGIZCJZUVGKZGAYBZUXDU
|
|
XGUVGCYCZLZUVOUXMUXGUXNBUVGUXGYTUXNBKZYTUUAUUBUUGYDYTUVNABCYEUXPUVPABCYFA
|
|
BCYGWGTYHUXGUWLUXOUXMSUUCUWLUUGYTUUAUWLUUBUWMUTYIGAUVGCYLTYJUXGUXLUXDGAUU
|
|
CUUGUXJALZUXLUXDMZMUUCUXQUUGUXRUUCUXQUUGUXKUXJKZUXRUXQUUGUXSMMUUCUUFUXSDU
|
|
XJADGNZUUEUXKUUDUXJUUDUXJCOUXTPQWOWCUUCUXQUXSUXLUXDUUCUXQRZUXSUXLRZRUVGUX
|
|
JAUYBFGNUYAUYBUXKUVGUXJUXSUXLVEUXSUXLYKWPXAUUCUXQUYBYMVMYNYOXKYPYQXHYRXFY
|
|
S $.
|
|
|
|
$( Order-isomorphic ordinal numbers are equal. (Contributed by Jeff
|
|
Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro,
|
|
24-Jun-2015.) $)
|
|
ordiso $p |- ( ( A e. On /\ B e. On ) -> ( A = B
|
|
<-> E. f f Isom _E , _E ( A , B ) ) ) $=
|
|
( con0 wcel wa wceq cep wiso wex cid cres cvv resiexg isoid isoeq1 spcegv
|
|
cv word eloni mpisyl adantr isoeq5 exbidv syl5ibcom ordiso2 3expia syl2an
|
|
wi 3coml exlimdv impbid ) ADEZBDEZFZABGZABHHCRZIZCJZUOAAHHUQIZCJZUPUSUMVA
|
|
UNUMKALZMEAAHHVBIZVAADNAHOUTVCCVBMAAHHVBUQPQUAUBUPUTURCAABHHUQUCUDUEUOURU
|
|
PCUMASZBSZURUPUIUNATBTVDVEURUPURVDVEUPABUQUFUJUGUHUKUL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Cardinal numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c card $. $( Cardinal size function $)
|
|
|
|
$( Extend class definition to include the cardinal size function. $)
|
|
ccrd $a class card $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the cardinal number function. The cardinal number of a set is
|
|
the least ordinal number equinumerous to it. In other words, it is the
|
|
"size" of the set. Definition of [Enderton] p. 197. Our notation is
|
|
from Enderton. Other textbooks often use a double bar over the set to
|
|
express this function. (Contributed by NM, 21-Oct-2003.) $)
|
|
df-card $a |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The cardinality of a well-orderable set is an ordinal. (Contributed by
|
|
Jim Kingdon, 30-Aug-2021.) $)
|
|
cardcl $p |- ( E. y e. On y ~~ A -> ( card ` A ) e. On ) $=
|
|
( vx cen wbr con0 wrex ccrd cfv crab cint cvv cmpt wceq df-card a1i breq2
|
|
cv rabbidv wcel inteqd adantl simprd rexlimivw fvmptd onintrab2im eqeltrd
|
|
encv intexrabim ) ARZBDEZAFGZBHIUKAFJZKZFULCBUJCRZDEZAFJZKZUNLHLHCLURMNUL
|
|
CAOPUOBNZURUNNULUSUQUMUSUPUKAFUOBUJDQSUAUBUKBLTZAFUKUJLTUTUJBUHUCUDUKAFUI
|
|
UEUKAUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( A set equinumerous to an ordinal is numerable. (Contributed by Mario
|
|
Carneiro, 29-Apr-2015.) $)
|
|
isnumi $p |- ( ( A e. On /\ A ~~ B ) -> B e. dom card ) $=
|
|
( vy vx con0 wcel cen wbr wa crab cint cvv ccrd cdm wrex breq1 intexrabim
|
|
cv rspcev syl encv simprd wceq rabbidv inteqd eleq1d elrab3 adantl mpbird
|
|
wb breq2 df-card dmmpt syl6eleqr ) AEFZABGHZIZBCRZDRZGHZCEJZKZLFZDLJZMNUQ
|
|
BVDFZURBGHZCEJZKZLFZUQVFCEOVIVFUPCAEURABGPSVFCEQTUPVEVIUJZUOUPBLFZVJUPALF
|
|
VKABUAUBVCVIDBLUSBUCZVBVHLVLVAVGVLUTVFCEUSBURGUKUDUEUFUGTUHUIDLVBMDCULUMU
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Every finite set is numerable. (Contributed by Mario Carneiro,
|
|
4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) $)
|
|
finnum $p |- ( A e. Fin -> A e. dom card ) $=
|
|
( vx cfn wcel cv cen wbr com wrex ccrd isfi con0 nnon ensym isnumi syl2an
|
|
cdm rexlimiva sylbi ) ACDABEZFGZBHIAJQDZBAKUAUBBHTHDTLDTAFGUBUATMATNTAOPR
|
|
S $.
|
|
|
|
$( Every ordinal number is numerable. (Contributed by Mario Carneiro,
|
|
29-Apr-2015.) $)
|
|
onenon $p |- ( A e. On -> A e. dom card ) $=
|
|
( con0 wcel cen wbr ccrd cdm enrefg isnumi mpdan ) ABCAADEAFGCABHAAIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d A y z $.
|
|
$( The value of ` ( card `` A ) ` . (Contributed by Jim Kingdon,
|
|
30-Aug-2021.) $)
|
|
cardval3ex $p |- ( E. x e. On x ~~ A ->
|
|
( card ` A ) = |^| { y e. On | y ~~ A } ) $=
|
|
( vz cv cen wbr con0 wrex cvv wcel crab cint ccrd cfv wceq encv rexlimivw
|
|
simprd breq1 cbvrexv sylbir breq2 rabbidv inteqd df-card fvmptg syl2anc
|
|
intexrabim ) AEZCFGZAHIZCJKZBEZCFGZBHLZMZJKZCNOUQPUKUMAHUKUJJKUMUJCQSRULU
|
|
OBHIURUOUKBAHUNUJCFTUAUOBHUIUBDCUNDEZFGZBHLZMUQJJNUSCPZVAUPVBUTUOBHUSCUNF
|
|
UCUDUEDBUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The value of the cardinal number function with an ordinal number as its
|
|
argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario
|
|
Carneiro, 13-Sep-2013.) $)
|
|
oncardval $p |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } ) $=
|
|
( vy con0 wcel cv cen wbr wrex ccrd cfv crab cint wceq enrefg breq1 mpdan
|
|
rspcev cardval3ex syl ) BDEZCFZBGHZCDIZBJKAFBGHADLMNUABBGHZUDBDOUCUECBDUB
|
|
BBGPRQCABST $.
|
|
|
|
$( The cardinal of an ordinal number is less than or equal to the ordinal
|
|
number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by
|
|
NM, 22-Oct-2003.) $)
|
|
cardonle $p |- ( A e. On -> ( card ` A ) C_ A ) $=
|
|
( vx con0 wcel ccrd cfv cv cen wbr crab cint oncardval wss breq1 intminss
|
|
enrefg mpdan eqsstrd ) ACDZAEFBGZAHIZBCJKZABALSAAHIZUBAMACPUAUCBACTAAHNOQ
|
|
R $.
|
|
$}
|
|
|
|
$( The cardinality of the empty set is the empty set. (Contributed by NM,
|
|
25-Oct-2003.) $)
|
|
card0 $p |- ( card ` (/) ) = (/) $=
|
|
( c0 ccrd cfv wss wceq con0 wcel 0elon cardonle ax-mp ss0b mpbi ) ABCZADZMA
|
|
EAFGNHAIJMKL $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( If two numerable sets are equinumerous, then they have equal
|
|
cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) $)
|
|
carden2bex $p |- ( ( A ~~ B /\ E. x e. On x ~~ A ) ->
|
|
( card ` A ) = ( card ` B ) ) $=
|
|
( vy cen wbr cv con0 wrex wa crab cint ccrd cfv wceq enen2 rabbidv inteqd
|
|
adantr cardval3ex adantl entr expcom reximdv imp syl 3eqtr4d ) BCEFZAGZBE
|
|
FZAHIZJZDGZBEFZDHKZLZUMCEFZDHKZLZBMNZCMNZUHUPUSOUKUHUOURUHUNUQDHBCUMPQRSU
|
|
KUTUPOUHADBTUAULUICEFZAHIZVAUSOUHUKVCUHUJVBAHUJUHVBUIBCUBUCUDUEADCTUFUG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM,
|
|
4-Apr-2007.) $)
|
|
pm54.43 $p |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <->
|
|
( A u. B ) ~~ 2o ) ) $=
|
|
( vx vy c1o cen wbr wa cin c0 wceq cun c2o csn wi 1on ensn1 entr wn ex cv
|
|
con0 elexi ensymi mpan2 wcel onirri disjsn mpbir unen mpanr2 sylan2 df-2o
|
|
csuc df-suc eqtri breq2i syl6ibr wex en1 1nen2 unidm sneq uneq2d syl5reqr
|
|
wne a1i syl6eqbr ensymd sylan mtand necon2ai disjsn2 uneq12 breq1d ineq12
|
|
vex syl eqeq1d 3imtr4d exlimdv exlimiv imp syl2anb impbid ) AEFGZBEFGZHZA
|
|
BIZJKZABLZMFGZWHWJWKEENZLZFGZWLWGWFBWMFGZWJWOOWGEWMFGWPWMEEEUBPUCQUDBEWMR
|
|
UEWFWPHZWJWOWQWJEWMIJKZWOWREEUFSEPUGEEUHUIAEBWMUJUKTULMWNWKFMEUNWNUMEUOUP
|
|
UQURWFACUAZNZKZCUSZBDUAZNZKZDUSZWLWJOZWGCAUTDBUTXBXFXGXAXFXGOCXAXEXGDXAXE
|
|
XGXAXEHZWTXDLZMFGZWTXDIZJKZWLWJXJXLOXHXJWSXCVFXLXJWSXCWSXCKZXJEMFGZXNSXMV
|
|
AVGXMEXIFGXJXNXMXIEXMXIWTEFXMWTWTWTLXIWTVBXMWTXDWTWSXCVCVDVEWSCVQQVHVIEXI
|
|
MRVJVKVLWSXCVMVRVGXHWKXIMFAWTBXDVNVOXHWIXKJAWTBXDVPVSVTTWAWBWCWDWE $.
|
|
$}
|
|
|
|
$( Lemma for ~ pr2ne . (Contributed by FL, 17-Aug-2008.) $)
|
|
pr2nelem $p |- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o ) $=
|
|
( wcel wne cpr c2o cen wbr csn cin c0 wceq disjsn2 wi c1o ensn1g wa cun ex
|
|
pm54.43 df-pr breq1i syl6bbr biimpd syl2an syl7 3imp ) ACEZBDEZABFZABGZHIJZ
|
|
ULAKZBKZLMNZUJUKUNABOUJUKUQUNPZUJUOQIJZUPQIJZURUKACRBDRUSUTSZUQUNVAUQUOUPTZ
|
|
HIJUNUOUPUBUMVBHIABUCUDUEUFUGUAUHUI $.
|
|
|
|
$( If an unordered pair has two elements they are different. (Contributed by
|
|
FL, 14-Feb-2010.) $)
|
|
pr2ne $p |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) ) $=
|
|
( wcel wa cpr c2o cen wbr wne wceq wn preq2 c1o wi cvv syl entr ex mtoi a1d
|
|
eqcoms enpr1g adantr prexg eqeng 1nen2 ensym syld mpid syl5 necon2ad 3expia
|
|
com3r pr2nelem impbid ) ACEZBDEZFZABGZHIJZABKZUTVBABABLVAAAGZLZUTVBMZVEBABA
|
|
ANUCUTVEVDOIJZVFURVGUSACUDUEUTVEVAVDIJZVGVFPUTVAQEVEVHPABCDUFVAVDQUGRVHVGUT
|
|
VFVHVGUTVFPZVHVGFVAOIJZVIVAVDOSVJVFUTVJVBOHIJZUHVJOVAIJZVBVKPVAOUIVLVBVKOVA
|
|
HSTRUAUBRTUOUJUKULUMURUSVCVBABCDUPUNUQ $.
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
REAL AND COMPLEX NUMBERS
|
|
###############################################################################
|
|
|
|
This section derives the basics of real and complex numbers.
|
|
|
|
To construct the real numbers constructively, we follow two main sources.
|
|
The first is Metamath Proof Explorer, which has the advantage of being
|
|
already formalized in metamath. Its disadvantage, for our purposes, is that
|
|
it assumes the law of the excluded middle throughout. Since we have already
|
|
developed natural numbers ( for example, ~ nna0 and similar theorems ), going
|
|
from there to positive integers ( ~ df-ni ) and then positive rational numbers
|
|
( ~ df-nqqs ) does not involve a major change in approach compared with
|
|
the Metamath Proof Explorer.
|
|
|
|
It is when we proceed to Dedekind cuts that we bring in more material from
|
|
Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which
|
|
are different without excluded middle. With excluded middle, it is natural
|
|
to define the cut as the lower set only (as Metamath Proof Explorer does),
|
|
but we define the cut as a pair of both the lower and upper sets, as [HoTT]
|
|
does. There are also differences in how we handle order and replacing
|
|
"not equal to zero" with "apart from zero".
|
|
$)
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Construction and axiomatization of real and complex numbers
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Dedekind-cut construction of real and complex numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
|
|
$c N. $. $( Set of positive integers $)
|
|
|
|
$c +N $. $( Positive integer addition $)
|
|
$c .N $. $( Positive integer multiplication $)
|
|
$c <N $. $( Positive integer ordering relation $)
|
|
|
|
$c +pQ $. $( Positive pre-fraction addition $)
|
|
$c .pQ $. $( Positive pre-fraction multiplication $)
|
|
$c <pQ $. $( Positive pre-fraction ordering relation $)
|
|
|
|
$c ~Q $. $( Equivalence relation to construct positive fractions $)
|
|
$c Q. $. $( Set of positive fractions $)
|
|
$c 1Q $. $( Positive fraction constant 1 $)
|
|
$c +Q $. $( Positive fraction addition $)
|
|
$c .Q $. $( Positive fraction multiplication $)
|
|
$c *Q $. $( Positive fraction reciprocal $)
|
|
$c <Q $. $( Positive fraction ordering relation $)
|
|
|
|
$c ~Q0 $. $( Equivalence relation to construct non-negative fractions $)
|
|
$c Q0. $. $( Set of non-negative fractions $)
|
|
$c 0Q0 $. $( Non-negative fraction constant 0 $)
|
|
$c +Q0 $. $( Non-negative fraction addition $)
|
|
$c .Q0 $. $( Non-negative fraction multiplication $)
|
|
|
|
$c P. $. $( Set of positive reals $)
|
|
$c 1P $. $( Positive real constant 1 $)
|
|
$c +P. $. $( Positive real addition $)
|
|
$c .P. $. $( Positive real multiplication $)
|
|
$c <P $. $( Positive real ordering relation $)
|
|
|
|
$c ~R $. $( Equivalence relation to construct signed reals $)
|
|
$c R. $. $( Set of signed reals $)
|
|
$c 0R $. $( Signed real constant 0 $)
|
|
$c 1R $. $( Signed real constant 1 $)
|
|
$c -1R $. $( Signed real constant minus 1 $)
|
|
$c +R $. $( Signed real addition $)
|
|
$c .R $. $( Signed real multiplication $)
|
|
$c <R $. $( Signed real ordering relation $)
|
|
$c <RR $. $( 'Less than' relation (restricted to non-extended reals) $)
|
|
|
|
$c CC $. $( The set of complex numbers (blackboard bold C) $)
|
|
$c RR $. $( The set of real numbers (blackboard bold R) $)
|
|
$c 0 $. $( The real number 0. $)
|
|
$c 1 $. $( The real number 1. $)
|
|
$c _i $. $( Letter i (the imaginary unit = square root of -1) $)
|
|
$c + $. $( Plus (addition) symbol $)
|
|
$c x. $. $( Multiplication symbol (center dot) $)
|
|
|
|
$( Extend class notation with new sets (constants) used for positive
|
|
integers: $)
|
|
|
|
$( The set of positive integers, which is the set of natural numbers ` _om `
|
|
with 0 removed.
|
|
|
|
_Note: This is the start of the Dedekind-cut construction of real and_
|
|
_complex numbers. $)
|
|
cnpi $a class N. $.
|
|
$( Positive integer addition. $)
|
|
cpli $a class +N $.
|
|
$( Positive integer multiplication. $)
|
|
cmi $a class .N $.
|
|
$( Positive integer ordering relation. $)
|
|
clti $a class <N $.
|
|
|
|
$( Extend class notation with new sets (constants) used in construction of
|
|
positive fractions: $)
|
|
|
|
$( Positive pre-fraction addition. $)
|
|
cplpq $a class +pQ $.
|
|
$( Positive pre-fraction multiplication. $)
|
|
cmpq $a class .pQ $.
|
|
$( Positive pre-fraction ordering relation. $)
|
|
cltpq $a class <pQ $.
|
|
$( Equivalence class used to construct positive fractions. $)
|
|
ceq $a class ~Q $.
|
|
$( Set of positive fractions. $)
|
|
cnq $a class Q. $.
|
|
$( The positive fraction constant 1. $)
|
|
c1q $a class 1Q $.
|
|
$( Positive fraction addition. $)
|
|
cplq $a class +Q $.
|
|
$( Positive fraction multiplication. $)
|
|
cmq $a class .Q $.
|
|
$( Positive fraction reciprocal operation. $)
|
|
crq $a class *Q $.
|
|
$( Positive fraction ordering relation. $)
|
|
cltq $a class <Q $.
|
|
|
|
$( Equivalence class used to construct non-negative fractions. $)
|
|
ceq0 $a class ~Q0 $.
|
|
$( Set of non-negative fractions. $)
|
|
cnq0 $a class Q0. $.
|
|
$( The non-negative fraction constant 0. $)
|
|
c0q0 $a class 0Q0 $.
|
|
$( Non-negative fraction addition. $)
|
|
cplq0 $a class +Q0 $.
|
|
$( Non-negative fraction multiplication. $)
|
|
cmq0 $a class .Q0 $.
|
|
|
|
$( Extend class notation with new sets (constants) used in construction of
|
|
positive real numbers: $)
|
|
|
|
$( Set of positive reals. $)
|
|
cnp $a class P. $.
|
|
$( Positive real constant 1. $)
|
|
c1p $a class 1P $.
|
|
$( Positive real addition. $)
|
|
cpp $a class +P. $.
|
|
$( Positive real multiplication. $)
|
|
cmp $a class .P. $.
|
|
$( Positive real ordering relation. $)
|
|
cltp $a class <P $.
|
|
|
|
$( Extend class notation with new sets (constants) used in construction of
|
|
signed real numbers: $)
|
|
|
|
$( Equivalence class used to construct signed reals. $)
|
|
cer $a class ~R $.
|
|
$( Set of signed reals. $)
|
|
cnr $a class R. $.
|
|
$( The signed real constant 0. $)
|
|
c0r $a class 0R $.
|
|
$( The signed real constant 1. $)
|
|
c1r $a class 1R $.
|
|
$( The signed real constant -1. $)
|
|
cm1r $a class -1R $.
|
|
$( Signed real addition. $)
|
|
cplr $a class +R $.
|
|
$( Signed real multiplication. $)
|
|
cmr $a class .R $.
|
|
$( Signed real ordering relation. $)
|
|
cltr $a class <R $.
|
|
|
|
$( Define the class of positive integers. This is a "temporary" set used in
|
|
the construction of complex numbers, and is intended to be used only by
|
|
the construction. (Contributed by NM, 15-Aug-1995.) $)
|
|
df-ni $a |- N. = ( _om \ { (/) } ) $.
|
|
|
|
$( Define addition on positive integers. This is a "temporary" set used in
|
|
the construction of complex numbers, and is intended to be used only by
|
|
the construction. (Contributed by NM, 26-Aug-1995.) $)
|
|
df-pli $a |- +N = ( +o |` ( N. X. N. ) ) $.
|
|
|
|
$( Define multiplication on positive integers. This is a "temporary" set
|
|
used in the construction of complex numbers and is intended to be used
|
|
only by the construction. (Contributed by NM, 26-Aug-1995.) $)
|
|
df-mi $a |- .N = ( .o |` ( N. X. N. ) ) $.
|
|
|
|
$( Define 'less than' on positive integers. This is a "temporary" set used
|
|
in the construction of complex numbers, and is intended to be used only by
|
|
the construction. (Contributed by NM, 6-Feb-1996.) $)
|
|
df-lti $a |- <N = ( _E i^i ( N. X. N. ) ) $.
|
|
|
|
$( Membership in the class of positive integers. (Contributed by NM,
|
|
15-Aug-1995.) $)
|
|
elni $p |- ( A e. N. <-> ( A e. _om /\ A =/= (/) ) ) $=
|
|
( cnpi wcel com c0 csn cdif wne wa df-ni eleq2i eldifsn bitri ) ABCADEFGZCA
|
|
DCAEHIBNAJKADELM $.
|
|
|
|
$( A positive integer is a natural number. (Contributed by NM,
|
|
15-Aug-1995.) $)
|
|
pinn $p |- ( A e. N. -> A e. _om ) $=
|
|
( cnpi com c0 csn cdif df-ni difss eqsstri sseli ) BCABCDEZFCGCKHIJ $.
|
|
|
|
$( A positive integer is an ordinal number. (Contributed by NM,
|
|
23-Mar-1996.) $)
|
|
pion $p |- ( A e. N. -> A e. On ) $=
|
|
( cnpi wcel com con0 pinn nnon syl ) ABCADCAECAFAGH $.
|
|
|
|
$( A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) $)
|
|
piord $p |- ( A e. N. -> Ord A ) $=
|
|
( cnpi wcel com word pinn nnord syl ) ABCADCAEAFAGH $.
|
|
|
|
$( The class of positive integers is a set. (Contributed by NM,
|
|
15-Aug-1995.) $)
|
|
niex $p |- N. e. _V $=
|
|
( cnpi com omex c0 csn cdif df-ni difss eqsstri ssexi ) ABCABDEZFBGBKHIJ $.
|
|
|
|
$( The empty set is not a positive integer. (Contributed by NM,
|
|
26-Aug-1995.) $)
|
|
0npi $p |- -. (/) e. N. $=
|
|
( c0 wceq cnpi wcel wn eqid com wne elni simprbi necon2bi ax-mp ) AABACDZEA
|
|
FMAAMAGDAAHAIJKL $.
|
|
|
|
$( Membership in the class of positive integers. (Contributed by NM,
|
|
27-Nov-1995.) $)
|
|
elni2 $p |- ( A e. N. <-> ( A e. _om /\ (/) e. A ) ) $=
|
|
( cnpi wcel com c0 wa pinn wceq wn 0npi eleq1 mtbiri con2i wo 0elnn syl ord
|
|
mpd sylib wi jca nndceq0 df-dc anim1i ancom andi bitr3i noel pm2.21d impcom
|
|
wdc eleq2 a1i wne df-ne elni simplbi2 syl5bir adantld jaod adantr impbii )
|
|
ABCZADCZEACZFZVCVDVEAGZVCAEHZIZVEVHVCVHVCEBCJAEBKLMVCVHVEVCVDVHVENVGAOPQRUA
|
|
VFVEVHFZVEVIFZNZVCVFVHVINZVEFZVLVDVMVEVDVHUKVMAUBVHUCSUDVNVEVMFVLVEVMUEVEVH
|
|
VIUFUGSVDVLVCTVEVDVJVCVKVJVCTVDVHVEVCVHVEVCVHVEEECEUHAEEULLUIUJUMVDVIVCVEVI
|
|
AEUNZVDVCAEUOVCVDVOAUPUQURUSUTVARVB $.
|
|
|
|
$( Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) $)
|
|
1pi $p |- 1o e. N. $=
|
|
( c1o cnpi wcel com c0 wne 1onn 1n0 elni mpbir2an ) ABCADCAEFGHAIJ $.
|
|
|
|
$( Positive integer addition in terms of ordinal addition. (Contributed by
|
|
NM, 27-Aug-1995.) $)
|
|
addpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) ) $=
|
|
( cnpi wcel wa cop cxp cpli co coa wceq opelxpi cres cfv fvres df-ov df-pli
|
|
fveq1i eqtri 3eqtr4g syl ) ACDBCDEABFZCCGZDZABHIZABJIZKABCCLUDUBJUCMZNZUBJN
|
|
UEUFUBUCJOUEUBHNUHABHPUBHUGQRSABJPTUA $.
|
|
|
|
$( Positive integer multiplication in terms of ordinal multiplication.
|
|
(Contributed by NM, 27-Aug-1995.) $)
|
|
mulpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( A .o B ) ) $=
|
|
( cnpi wcel wa cop cxp cmi co comu wceq opelxpi cres cfv fvres df-ov fveq1i
|
|
df-mi eqtri 3eqtr4g syl ) ACDBCDEABFZCCGZDZABHIZABJIZKABCCLUDUBJUCMZNZUBJNU
|
|
EUFUBUCJOUEUBHNUHABHPUBHUGRQSABJPTUA $.
|
|
|
|
$( 1 is an identity element for multiplication on positive integers.
|
|
(Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro,
|
|
17-Nov-2014.) $)
|
|
mulidpi $p |- ( A e. N. -> ( A .N 1o ) = A ) $=
|
|
( cnpi wcel c1o cmi co comu wceq 1pi mulpiord mpan2 com pinn nnm1 syl eqtrd
|
|
) ABCZADEFZADGFZAQDBCRSHIADJKQALCSAHAMANOP $.
|
|
|
|
$( Positive integer 'less than' in terms of ordinal membership. (Contributed
|
|
by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
ltpiord $p |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) ) $=
|
|
( clti wbr cep cnpi cxp cin wcel wa df-lti breqi brinxp epelg adantl bitr3d
|
|
wb syl5bb ) ABCDABEFFGHZDZAFIZBFIZJZABIZABCSKLUCABEDZTUDABFFEMUBUEUDQUAABFN
|
|
OPR $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Positive integer 'less than' is a strict ordering. (Contributed by NM,
|
|
8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) $)
|
|
ltsopi $p |- <N Or N. $=
|
|
( vx vy vz cnpi clti wor wtru cv wcel wbr wn wel wb ltpiord adantl wa w3o
|
|
wi com pinn elirrv anidms mtbiri w3a con0 pion ontr1 syl 3ad2ant3 3adant3
|
|
3adant1 anbi12d 3adant2 3imtr4d ispod weq nntri3or syl2an biidd 3orbi123d
|
|
ancoms mpbird issod trud ) DEFGABDEGABCDEAHZDIZVEVEEJZKGVFVGAALZAUAVFVGVH
|
|
MVEVENUBUCOVFBHZDIZCHZDIZUDZVEVIEJZVIVKEJZPZVEVKEJZRGVMABLZBCLZPZACLZVPVQ
|
|
VLVFVTWARZVJVLVKUEIWBVKUFVEVIVKUGUHUIVMVNVRVOVSVFVJVNVRMVLVEVINZUJVJVLVOV
|
|
SMVFVIVKNUKULVFVLVQWAMVJVEVKNUMUNOUOVFVJPZVNABUPZVIVEEJZQZGWDWGVRWEBALZQZ
|
|
VFVESIVISIWIVJVETVITVEVIUQURWDVNVRWEWEWFWHWCWDWEUSVJVFWFWHMVIVENVAUTVBOVC
|
|
VD $.
|
|
$}
|
|
|
|
$( Trichotomy for positive integers. (Contributed by Jim Kingdon,
|
|
21-Sep-2019.) $)
|
|
pitric $p |- ( ( A e. N. /\ B e. N. ) ->
|
|
( A <N B <-> -. ( A = B \/ B <N A ) ) ) $=
|
|
( cnpi wcel wa wceq wo wn clti wbr com wb pinn nntri2 syl2an ltpiord ancoms
|
|
orbi2d notbid 3bitr4d ) ACDZBCDZEZABDZABFZBADZGZHZABIJUEBAIJZGZHUAAKDBKDUDU
|
|
HLUBAMBMABNOABPUCUJUGUCUIUFUEUBUAUIUFLBAPQRST $.
|
|
|
|
$( Trichotomy for positive integers. (Contributed by Jim Kingdon,
|
|
21-Sep-2019.) $)
|
|
pitri3or $p |- ( ( A e. N. /\ B e. N. ) ->
|
|
( A <N B \/ A = B \/ B <N A ) ) $=
|
|
( cnpi wcel clti wbr wceq w3o com pinn nntri3or syl2an ltpiord biidd ancoms
|
|
wa wb 3orbi123d mpbird ) ACDZBCDZPZABEFZABGZBAEFZHABDZUDBADZHZTAIDBIDUHUAAJ
|
|
BJABKLUBUCUFUDUDUEUGABMUBUDNUATUEUGQBAMORS $.
|
|
|
|
$( Less-than for positive integers is decidable. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
ltdcpi $p |- ( ( A e. N. /\ B e. N. ) -> DECID A <N B ) $=
|
|
( cnpi wcel wa clti wbr wdc com pinn nndcel syl2an ltpiord dcbid mpbird ) A
|
|
CDZBCDZEZABFGZHABDZHZPAIDBIDUAQAJBJABKLRSTABMNO $.
|
|
|
|
$( Positive integer 'less than' is a relation on positive integers.
|
|
(Contributed by NM, 8-Feb-1996.) $)
|
|
ltrelpi $p |- <N C_ ( N. X. N. ) $=
|
|
( clti cep cnpi cxp cin df-lti inss2 eqsstri ) ABCCDZEIFBIGH $.
|
|
|
|
$( Domain of addition on positive integers. (Contributed by NM,
|
|
26-Aug-1995.) $)
|
|
dmaddpi $p |- dom +N = ( N. X. N. ) $=
|
|
( coa cnpi cxp cres cdm con0 cin cpli dmres wfn wceq fnoa fndm ax-mp ineq2i
|
|
eqtri df-pli dmeqi wss com wa c0 csn df-ni difss eqsstri omsson sstri anidm
|
|
cdif mpbir xpss12 dfss mpbi 3eqtr4i ) ABBCZDZEZUPFFCZGZHEUPURUPAEZGUTAUPIVA
|
|
USUPAUSJVAUSKLUSAMNOPHUQQRUPUSSZUPUTKBFSZVCUAZVBVDVCBTFBTUBUCZUJTUDTVEUEUFU
|
|
GUHVCUIUKBFBFULNUPUSUMUNUO $.
|
|
|
|
$( Domain of multiplication on positive integers. (Contributed by NM,
|
|
26-Aug-1995.) $)
|
|
dmmulpi $p |- dom .N = ( N. X. N. ) $=
|
|
( comu cnpi cxp cres cdm con0 cin cmi dmres wfn wceq fnom fndm ax-mp ineq2i
|
|
eqtri df-mi dmeqi wss com wa c0 cdif df-ni difss eqsstri omsson sstri anidm
|
|
csn mpbir xpss12 dfss mpbi 3eqtr4i ) ABBCZDZEZUPFFCZGZHEUPURUPAEZGUTAUPIVAU
|
|
SUPAUSJVAUSKLUSAMNOPHUQQRUPUSSZUPUTKBFSZVCUAZVBVDVCBTFBTUBUJZUCTUDTVEUEUFUG
|
|
UHVCUIUKBFBFULNUPUSUMUNUO $.
|
|
|
|
$( Closure of addition of positive integers. (Contributed by NM,
|
|
18-Oct-1995.) $)
|
|
addclpi $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. ) $=
|
|
( cnpi wcel wa cpli co coa addpiord com pinn wne nnacl sylan2 elni2 nnaordi
|
|
c0 wi ne0i syl6 expcom imp32 sylan2b elni sylanbrc sylan eqeltrd ) ACDZBCDZ
|
|
EABFGABHGZCABIUHAJDZUIUJCDZAKUKUIEUJJDZUJQLZULUIUKBJDZUMBKABMNUIUKUOQBDZEUN
|
|
BOUKUOUPUNUOUKUPUNRUOUKEUPAQHGZUJDUNQBAPUJUQSTUAUBUCUJUDUEUFUG $.
|
|
|
|
$( Closure of multiplication of positive integers. (Contributed by NM,
|
|
18-Oct-1995.) $)
|
|
mulclpi $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) e. N. ) $=
|
|
( cnpi wcel wa cmi co comu mulpiord com wne pinn nnmcl syl2an elni2 simprbi
|
|
c0 adantl wi adantr nnmordi syl21anc mpd ne0i syl elni sylanbrc eqeltrd ) A
|
|
CDZBCDZEZABFGABHGZCABIUKULJDZULQKZULCDUIAJDZBJDZUMUJALZBLZABMNUKAQHGZULDZUN
|
|
UKQBDZUTUJVAUIUJUPVABOPRUKUPUOQADZVAUTSUJUPUIURRUIUOUJUQTUIVBUJUIUOVBAOPTQB
|
|
AUAUBUCULUSUDUEULUFUGUH $.
|
|
|
|
$( Addition of positive integers is commutative. (Contributed by Jim
|
|
Kingdon, 26-Aug-2019.) $)
|
|
addcompig $p |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) ) $=
|
|
( cnpi wcel wa coa cpli com wceq pinn nnacom syl2an addpiord ancoms 3eqtr4d
|
|
co ) ACDZBCDZEABFPZBAFPZABGPBAGPZQAHDBHDSTIRAJBJABKLABMRQUATIBAMNO $.
|
|
|
|
$( Addition of positive integers is associative. (Contributed by Jim
|
|
Kingdon, 26-Aug-2019.) $)
|
|
addasspig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
( ( A +N B ) +N C ) = ( A +N ( B +N C ) ) ) $=
|
|
( cnpi wcel w3a coa co cpli com wceq pinn nnaass syl3an wa addclpi addpiord
|
|
sylan oveq1d eqtrd adantr 3impa sylan2 oveq2d adantl 3impb 3eqtr4d ) ADEZBD
|
|
EZCDEZFABGHZCGHZABCGHZGHZABIHZCIHZABCIHZIHZUHAJEUIBJEUJCJEULUNKALBLCLABCMNU
|
|
HUIUJUPULKUHUIOZUJOUPUOCGHZULUSUODEUJUPUTKABPUOCQRUSUTULKUJUSUOUKCGABQSUATU
|
|
BUHUIUJURUNKUHUIUJOZOURAUQGHZUNVAUHUQDEURVBKBCPAUQQUCVAVBUNKUHVAUQUMAGBCQUD
|
|
UETUFUG $.
|
|
|
|
$( Multiplication of positive integers is commutative. (Contributed by Jim
|
|
Kingdon, 26-Aug-2019.) $)
|
|
mulcompig $p |- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( B .N A ) ) $=
|
|
( cnpi wcel wa comu cmi com wceq pinn nnmcom syl2an mulpiord ancoms 3eqtr4d
|
|
co ) ACDZBCDZEABFPZBAFPZABGPBAGPZQAHDBHDSTIRAJBJABKLABMRQUATIBAMNO $.
|
|
|
|
$( Multiplication of positive integers is associative. (Contributed by Jim
|
|
Kingdon, 26-Aug-2019.) $)
|
|
mulasspig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. )
|
|
-> ( ( A .N B ) .N C ) = ( A .N ( B .N C ) ) ) $=
|
|
( cnpi wcel w3a comu co cmi com wceq pinn nnmass syl3an wa mulclpi mulpiord
|
|
sylan oveq1d eqtrd adantr 3impa sylan2 oveq2d adantl 3impb 3eqtr4d ) ADEZBD
|
|
EZCDEZFABGHZCGHZABCGHZGHZABIHZCIHZABCIHZIHZUHAJEUIBJEUJCJEULUNKALBLCLABCMNU
|
|
HUIUJUPULKUHUIOZUJOUPUOCGHZULUSUODEUJUPUTKABPUOCQRUSUTULKUJUSUOUKCGABQSUATU
|
|
BUHUIUJURUNKUHUIUJOZOURAUQGHZUNVAUHUQDEURVBKBCPAUQQUCVAVBUNKUHVAUQUMAGBCQUD
|
|
UETUFUG $.
|
|
|
|
$( Multiplication of positive integers is distributive. (Contributed by Jim
|
|
Kingdon, 26-Aug-2019.) $)
|
|
distrpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. )
|
|
-> ( A .N ( B +N C ) ) = ( ( A .N B ) +N ( A .N C ) ) ) $=
|
|
( cnpi wcel w3a coa co comu cpli cmi wceq pinn nndi mulpiord addpiord eqtrd
|
|
com wa mulclpi syl3an addclpi sylan2 oveq2d adantl syl2an oveqan12d 3eqtr4d
|
|
3impb 3impdi ) ADEZBDEZCDEZFABCGHZIHZABIHZACIHZGHZABCJHZKHZABKHZACKHZJHZUKA
|
|
REULBREUMCREUOURLAMBMCMABCNUAUKULUMUTUOLUKULUMSZSUTAUSIHZUOVDUKUSDEUTVELBCU
|
|
BAUSOUCVDVEUOLUKVDUSUNAIBCPUDUEQUIUKULUMVCURLUKULSZUKUMSZSVCVAVBGHZURVFVADE
|
|
VBDEVCVHLVGABTACTVAVBPUFVFVGVAUPVBUQGABOACOUGQUJUH $.
|
|
|
|
$( Addition cancellation law for positive integers. (Contributed by Jim
|
|
Kingdon, 27-Aug-2019.) $)
|
|
addcanpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
( ( A +N B ) = ( A +N C ) <-> B = C ) ) $=
|
|
( cnpi wcel w3a cpli co wceq coa addpiord 3adant3 3adant2 eqeq12d wi nnacan
|
|
com pinn biimpd syl3an sylbid oveq2 impbid1 ) ADEZBDEZCDEZFZABGHZACGHZIZBCI
|
|
ZUGUJABJHZACJHZIZUKUGUHULUIUMUDUEUHULIUFABKLUDUFUIUMIUEACKMNUDAQEZUEBQEZUFC
|
|
QEZUNUKOARBRCRUOUPUQFUNUKABCPSTUABCAGUBUC $.
|
|
|
|
$( Multiplication cancellation law for positive integers. (Contributed by
|
|
Jim Kingdon, 29-Aug-2019.) $)
|
|
mulcanpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
( ( A .N B ) = ( A .N C ) <-> B = C ) ) $=
|
|
( cnpi wcel w3a cmi co wceq wi wa comu mulpiord adantr adantlr eqeq12d pinn
|
|
com c0 elni2 simprbi nnmcan biimpd sylan2 ex syl3an 3exp com4r imp31 sylbid
|
|
pm2.43i 3impa oveq2 impbid1 ) ADEZBDEZCDEZFABGHZACGHZIZBCIZUOUPUQUTVAJUOUPK
|
|
ZUQKZUTABLHZACLHZIZVAVCURVDUSVEVBURVDIUQABMNUOUQUSVEIUPACMOPUOUPUQVFVAJZUOU
|
|
PUQVGJJUOUPUQUOVGUOUPUQUOVGJZUOAREZUPBREZUQCREZVHAQBQCQVIVJVKFZUOVGUOVLSAEZ
|
|
VGUOVIVMATUAVLVMKVFVAABCUBUCUDUEUFUGUHUKUIUJULBCAGUMUN $.
|
|
|
|
$( There is no identity element for addition on positive integers.
|
|
(Contributed by NM, 28-Nov-1995.) $)
|
|
addnidpig $p |- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A ) $=
|
|
( cnpi wcel wa cpli co wceq coa com wn pinn c0 elni2 wi nnaordi nna0 eleq1d
|
|
word nnord ordirr eleq2 notbid syl5ibrcom sylbid adantl syld expcom sylan2b
|
|
syl con2d imp32 sylan addpiord eqeq1d mtbird ) ACDZBCDZEZABFGZAHABIGZAHZUQA
|
|
JDZURVBKZALURVCBJDZMBDZEVDBNVCVEVFVDVEVCVFVDOVEVCEVFAMIGZVADZVDMBAPVCVHVDOV
|
|
EVCVHAVADZVDVCVGAVAAQRVCVBVIVCVIKVBAADZKZVCASVKATAUAUJVBVIVJVAAAUBUCUDUKUEU
|
|
FUGUHULUIUMUSUTVAAABUNUOUP $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Ordering on positive integers in terms of existence of sum.
|
|
(Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro,
|
|
14-Jun-2013.) $)
|
|
ltexpi $p |- ( ( A e. N. /\ B e. N. ) ->
|
|
( A <N B <-> E. x e. N. ( A +N x ) = B ) ) $=
|
|
( cnpi wcel wa c0 cv coa wceq com wrex clti wbr cpli pinn nnaordex syl2an
|
|
co wb ltpiord addpiord eqeq1d pm5.32da elni2 anbi1i anass syl6bb rexbidv2
|
|
bitri adantr 3bitr4d ) BDEZCDEZFBCEZGAHZEZBUPISZCJZFZAKLZBCMNBUPOSZCJZADL
|
|
ZUMBKECKEUOVATUNBPCPABCQRBCUAUMVDVATUNUMVCUTADKUMUPDEZVCFVEUSFZUPKEZUTFZU
|
|
MVEVCUSUMVEFVBURCBUPUBUCUDVFVGUQFZUSFVHVEVIUSUPUEUFVGUQUSUGUJUHUIUKUL $.
|
|
$}
|
|
|
|
$( Ordering property of addition for positive integers. (Contributed by Jim
|
|
Kingdon, 31-Aug-2019.) $)
|
|
ltapig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
( A <N B <-> ( C +N A ) <N ( C +N B ) ) ) $=
|
|
( cnpi wcel clti wbr cpli co wb wa coa com pinn nnaord ltpiord addclpi wceq
|
|
adantr addpiord syl3an 3expa syl2an adantl eleq12d anandis ancoms 3bitr4d
|
|
bitrd 3impa ) ADEZBDEZCDEZABFGZCAHIZCBHIZFGZJUKULKZUMKABEZCALIZCBLIZEZUNUQU
|
|
KULUMUSVBJZUKAMEULBMEUMCMEVCANBNCNABCOUAUBURUNUSJUMABPSUMURUQVBJZUMUKULVDUM
|
|
UKKZUMULKZKZUQUOUPEZVBVEUODEUPDEUQVHJVFCAQCBQUOUPPUCVGUOUTUPVAVEUOUTRVFCATS
|
|
VFUPVARVECBTUDUEUIUFUGUHUJ $.
|
|
|
|
$( Ordering property of multiplication for positive integers. (Contributed
|
|
by Jim Kingdon, 31-Aug-2019.) $)
|
|
ltmpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
( A <N B <-> ( C .N A ) <N ( C .N B ) ) ) $=
|
|
( cnpi wcel clti wbr cmi co wb wa comu com pinn syl2an ltpiord mulclpi wceq
|
|
adantr mulpiord wi c0 elni2 w3a iba nnmord sylan9bbr 3exp1 imp4b syl5bi imp
|
|
adantl eleq12d bitrd anandis ancoms 3bitr4d 3impa ) ADEZBDEZCDEZABFGZCAHIZC
|
|
BHIZFGZJUSUTKZVAKABEZCALIZCBLIZEZVBVEVFVAVGVJJZUSAMEZBMEZVAVKUAUTANBNVACMEZ
|
|
UBCEZKVLVMKVKCUCVLVMVNVOVKVLVMVNVOVKVOVGVGVOKVLVMVNUDVJVOVGUEABCUFUGUHUIUJO
|
|
UKVFVBVGJVAABPSVAVFVEVJJZVAUSUTVPVAUSKZVAUTKZKZVEVCVDEZVJVQVCDEVDDEVEVTJVRC
|
|
AQCBQVCVDPOVSVCVHVDVIVQVCVHRVRCATSVRVDVIRVQCBTULUMUNUOUPUQUR $.
|
|
|
|
$( One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) $)
|
|
1lt2pi $p |- 1o <N ( 1o +N 1o ) $=
|
|
( c1o cpli co clti wbr wcel coa c0 com wceq 1onn nna0 ax-mp 0lt1o wb peano1
|
|
nnaord cnpi 1pi mp2an mp3an mpbi eqeltrri addpiord eleqtrri addclpi ltpiord
|
|
mpbir ) AAABCZDEZAUIFZAAAGCZUIAHGCZAULAIFZUMAJKALMHAFZUMULFZNHIFUNUNUOUPOPK
|
|
KHAAQUAUBUCARFZUQUIULJSSAAUDTUEUQUIRFZUJUKOSUQUQURSSAAUFTAUIUGTUH $.
|
|
|
|
$( No positive integer is less than one. (Contributed by Jim Kingdon,
|
|
31-Aug-2019.) $)
|
|
nlt1pig $p |- ( A e. N. -> -. A <N 1o ) $=
|
|
( cnpi wcel c0 wne c1o clti wbr wn com elni simprbi wceq wa noel wo ltpiord
|
|
wb 1pi mpan2 csuc df-1o eleq2i elsucg syl5bb biimpa ord mpi ex necon3ad mpd
|
|
bitrd ) ABCZADEZAFGHZIUMAJCUNAKLUMUOADUMUOADMZUMUONZADCZIUPAOUQURUPUMUOURUP
|
|
PZUMUOAFCZUSUMFBCUOUTRSAFQTUTADUAZCUMUSFVAAUBUCADBUDUEULUFUGUHUIUJUK $.
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ps $. $d x ch $. $d x th $. $d x ta $.
|
|
$d y ph $.
|
|
$( Substitutions. $)
|
|
indpi.1 $e |- ( x = 1o -> ( ph <-> ps ) ) $.
|
|
indpi.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
indpi.3 $e |- ( x = ( y +N 1o ) -> ( ph <-> th ) ) $.
|
|
indpi.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
indpi.5 $e |- ps $.
|
|
$( Induction step. $)
|
|
indpi.6 $e |- ( y e. N. -> ( ch -> th ) ) $.
|
|
$( Principle of Finite Induction on positive integers. (Contributed by NM,
|
|
23-Mar-1996.) $)
|
|
indpi $p |- ( A e. N. -> ta ) $=
|
|
( wcel com c0 wceq wo c1o cnpi cv wne wa elni cab csuc wral wss wsbc eqid
|
|
orci nfv nfsbc1v nfor 0ex eqeq1 sbceq1a orbi12d elabf mpbir suceq syl6eqr
|
|
wi df-1o olci 1onn elexi elab syl6eqel a1d a1i wn wb simprbi neneqd biorf
|
|
syl vex syl6bbr cpli co 1pi addclpi mpan2 sylib simprd elabg coa addpiord
|
|
con0 pion oa1suc eqtrd eleq1d 3bitr2d 3imtr3d wdc nndceq0 necon3bd anc2li
|
|
df-dc idd syl6ibr orim2d mpjaod rgen peano5 mp2an sseli abid adantr df-ne
|
|
mpd sylbi adantl mpbird vtoclga ) AEFHUALFUBZUAOXSPOZXSQUCZUDZAXSUEYBAXSQ
|
|
RZASZXTYDYAXTXSYDFUFZOYDPYEXSQYEOZGUBZYEOZYGUGZYEOZVDZGPUHPYEUIYFQQRZAFQU
|
|
JZSZYLYMQUKULYDYNFQYLYMFYLFUMAFQUNUOUPYCYCYLAYMXSQQUQAFQURUSUTVAYKGPYGPOZ
|
|
YGQRZYKYGUAOZYPYKVDYOYPYJYHYPYITYEYPYIQUGTYGQVBVEVCTYEOTQRZBSZBYRMVFYDYSF
|
|
TTPVGVHXSTRYCYRABXSTQUQIUSVIVAVJVKVLYQYKVDYOYQCDYHYJNYQCYPCSZYHYQYPVMZCYT
|
|
VNYQYGQYQYOYGQUCZYGUEZVOVPYPCVQVRYDYTFYGGVSXSYGRYCYPACXSYGQUQJUSVIVTYQDYG
|
|
TWAWBZQRZDSZUUDYEOZYJYQUUEVMDUUFVNYQUUDQYQUUDPOZUUDQUCZYQUUDUAOZUUHUUIUDY
|
|
QTUAOZUUJWCYGTWDWEZUUDUEWFWGVPUUEDVQVRYQUUJUUGUUFVNUULYDUUFFUUDUAXSUUDRYC
|
|
UUEADXSUUDQUQKUSWHVRYQUUDYIYEYQUUDYGTWIWBZYIYQUUKUUDUUMRWCYGTWJWEYQYGWKOU
|
|
UMYIRYGWLYGWMVRWNWOWPWQVLYOYPUUASZYPYQSYOYPWRUUNYGWSYPXBWFYOUUAYQYPYOUUAY
|
|
OUUBUDYQYOUUAUUBYOYPYGQYOYPXCWTXAUUCXDXEXNXFXGGYEXHXIXJYDFXKWFXLYAAYDVNZX
|
|
TYAYCVMUUOXSQXMYCAVQXOXPXQXOXR $.
|
|
$}
|
|
|
|
$( A natural number plus a positive integer is a positive integer.
|
|
(Contributed by Jim Kingdon, 10-Nov-2019.) $)
|
|
nnppipi $p |- ( ( A e. _om /\ B e. N. ) -> ( A +o B ) e. N. ) $=
|
|
( com wcel cnpi wa coa co pinn nnacl sylan2 wss nnaword2 sylan ancoms elni2
|
|
c0 simprbi adantl sseldd sylanbrc ) ACDZBEDZFZABGHZCDZQUEDUEEDUCUBBCDZUFBIZ
|
|
ABJKUDBUEQUCUBBUELZUCUGUBUIUHBAMNOUCQBDZUBUCUGUJBPRSTUEPUA $.
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Define pre-addition on positive fractions. This is a "temporary" set
|
|
used in the construction of complex numbers, and is intended to be used
|
|
only by the construction. This "pre-addition" operation works directly
|
|
with ordered pairs of integers. The actual positive fraction addition
|
|
` +Q ` ( ~ df-plqqs ) works with the equivalence classes of these
|
|
ordered pairs determined by the equivalence relation ` ~Q `
|
|
( ~ df-enq ). (Analogous remarks apply to the other "pre-" operations
|
|
in the complex number construction that follows.) From Proposition
|
|
9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) $)
|
|
df-plpq $a |- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |->
|
|
<. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N
|
|
( ( 1st ` y ) .N ( 2nd ` x ) ) ) ,
|
|
( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) $.
|
|
|
|
$( Define pre-multiplication on positive fractions. This is a "temporary"
|
|
set used in the construction of complex numbers, and is intended to be
|
|
used only by the construction. From Proposition 9-2.4 of [Gleason]
|
|
p. 119. (Contributed by NM, 28-Aug-1995.) $)
|
|
df-mpq $a |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |->
|
|
<. ( ( 1st ` x ) .N ( 1st ` y ) ) ,
|
|
( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) $.
|
|
|
|
$( Define pre-ordering relation on positive fractions. This is a
|
|
"temporary" set used in the construction of complex numbers, and is
|
|
intended to be used only by the construction. Similar to Definition 5
|
|
of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) $)
|
|
df-ltpq $a |- <pQ = { <. x , y >. |
|
|
( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\
|
|
( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) } $.
|
|
|
|
$( Define equivalence relation for positive fractions. This is a
|
|
"temporary" set used in the construction of complex numbers, and is
|
|
intended to be used only by the construction. From Proposition 9-2.1 of
|
|
[Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) $)
|
|
df-enq $a |- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\
|
|
y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u
|
|
( ( x = <. z , w >. /\ y = <. v , u >. ) /\
|
|
( z .N u ) = ( w .N v ) ) ) } $.
|
|
|
|
$( Define class of positive fractions. This is a "temporary" set used in
|
|
the construction of complex numbers, and is intended to be used only by
|
|
the construction. From Proposition 9-2.2 of [Gleason] p. 117.
|
|
(Contributed by NM, 16-Aug-1995.) $)
|
|
df-nqqs $a |- Q. = ( ( N. X. N. ) /. ~Q ) $.
|
|
|
|
$( Define addition on positive fractions. This is a "temporary" set used
|
|
in the construction of complex numbers, and is intended to be used only
|
|
by the construction. From Proposition 9-2.3 of [Gleason] p. 117.
|
|
(Contributed by NM, 24-Aug-1995.) $)
|
|
df-plqqs $a |- +Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\
|
|
z = [ ( <. w , v >. +pQ <. u , f >. ) ] ~Q ) ) } $.
|
|
|
|
$( Define multiplication on positive fractions. This is a "temporary" set
|
|
used in the construction of complex numbers, and is intended to be used
|
|
only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
|
|
(Contributed by NM, 24-Aug-1995.) $)
|
|
df-mqqs $a |- .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\
|
|
z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) } $.
|
|
|
|
$( Define positive fraction constant 1. This is a "temporary" set used in
|
|
the construction of complex numbers, and is intended to be used only by
|
|
the construction. From Proposition 9-2.2 of [Gleason] p. 117.
|
|
(Contributed by NM, 29-Oct-1995.) $)
|
|
df-1nqqs $a |- 1Q = [ <. 1o , 1o >. ] ~Q $.
|
|
|
|
$( Define reciprocal on positive fractions. It means the same thing as one
|
|
divided by the argument (although we don't define full division since we
|
|
will never need it). This is a "temporary" set used in the construction
|
|
of complex numbers, and is intended to be used only by the construction.
|
|
From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to
|
|
denote this unary operation. (Contributed by Jim Kingdon,
|
|
20-Sep-2019.) $)
|
|
df-rq $a |- *Q = { <. x , y >. |
|
|
( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) } $.
|
|
|
|
$( Define ordering relation on positive fractions. This is a "temporary"
|
|
set used in the construction of complex numbers, and is intended to be
|
|
used only by the construction. Similar to Definition 5 of [Suppes]
|
|
p. 162. (Contributed by NM, 13-Feb-1996.) $)
|
|
df-ltnqqs $a |- <Q = { <. x , y >. | ( ( x e. Q. /\ y e. Q. ) /\
|
|
E. z E. w E. v E. u
|
|
( ( x = [ <. z , w >. ] ~Q /\ y = [ <. v , u >. ] ~Q ) /\
|
|
( z .N u ) <N ( w .N v ) ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Alternate definition of pre-addition on positive fractions.
|
|
(Contributed by Jim Kingdon, 12-Sep-2019.) $)
|
|
dfplpq2 $p |- +pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\
|
|
y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) ,
|
|
( v .N f ) >. ) ) } $=
|
|
( cnpi cv cfv cmi co cpli cop wcel wa wceq wex vex oveq12d cxp c1st cmpt2
|
|
c2nd coprab cplpq df-mpt2 df-plpq 1st2nd2 eqeq1d bi2anan9 anbi1d ad2antlr
|
|
xp1st biimpa simprd opth2 simplbi eleq1d syl mpbid xp2nd ad2antrr simprbi
|
|
simpld mulcompig syl2anc oveq2d opeq1d eqeq2d pm5.32da bitr3d 4exbidv jca
|
|
wb simpll simprr simprl simplr opeq12d copsex4g pm5.32i oprabbii 3eqtr4i
|
|
syl2an ) ABHHUAZWFAIZUBJZBIZUDJZKLZWIUBJZWGUDJZKLZMLZWMWJKLZNZUCWGWFOZWIW
|
|
FOZPZCIZWQQZPZABCUEUFWTWGDIZEIZNZQZWIFIZGIZNZQZPZXAXDXIKLZXEXHKLZMLZXEXIK
|
|
LZNZQZPZGRFRERDRZPZABCUEABCWFWFWQUGABUHYAXCABCWTXTXBWTWHWMNZXFQZWLWJNZXJQ
|
|
ZPZXAXMXHXEKLZMLZXPNZQZPZGRFRERDRZXTXBWTYKXSDEFGWTXLYJPYKXSWTXLYFYJWRXGYC
|
|
WSXKYEWRWGYBXFWGHHUIUJWSWIYDXJWIHHUIUJUKZULWTXLYJXRWTXLPZYIXQXAYNYHXOXPYN
|
|
YGXNXMMYNXHHOZXEHOZYGXNQYNWLHOZYOWSYQWRXLWIHHUNZUMYNYEYQYOVOYNYCYEWTXLYFY
|
|
MUOZUPYEWLXHHYEWLXHQWJXIQWLWJXHXIFSGSUQURUSUTVAYNWMHOZYPWRYTWSXLWGHHVBZVC
|
|
YNYCYTYPVOYNYCYEYSVEYCWMXEHYCWHXDQWMXEQWHWMXDXEDSESUQVDUSUTVAXHXEVFVGVHVI
|
|
VJVKVLVMWRWHHOZYTPYQWJHOZPYLXBVOWSWRUUBYTWGHHUNUUAVNWSYQUUCYRWIHHVBVNYJXB
|
|
DEFGWHWMWLWJHHXDWHQZXEWMQZPZXHWLQZXIWJQZPZPZYIWQXAUUJYHWOXPWPUUJXMWKYGWNM
|
|
UUJXDWHXIWJKUUDUUEUUIVPUUFUUGUUHVQZTUUJXHWLXEWMKUUFUUGUUHVRUUDUUEUUIVSZTT
|
|
UUJXEWMXIWJKUULUUKTVTVJWAWEVLWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Alternate definition of pre-multiplication on positive fractions.
|
|
(Contributed by Jim Kingdon, 13-Sep-2019.) $)
|
|
dfmpq2 $p |- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y
|
|
e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) } $=
|
|
( cnpi cv c1st cfv cmi co c2nd cop wcel wa wceq coprab wex cxp cmpt2 cmpq
|
|
df-mpt2 df-mpq 1st2nd2 eqeq1d bi2anan9 anbi1d bicomd 4exbidv wb xp1st jca
|
|
simpll simprl oveq12d simplr simprr opeq12d eqeq2d copsex4g syl2an bitr3d
|
|
xp2nd pm5.32i oprabbii 3eqtr4i ) ABHHUAZVIAIZJKZBIZJKZLMZVJNKZVLNKZLMZOZU
|
|
BVJVIPZVLVIPZQZCIZVRRZQZABCSUCWAVJDIZEIZOZRZVLFIZGIZOZRZQZWBWEWILMZWFWJLM
|
|
ZOZRZQZGTFTETDTZQZABCSABCVIVIVRUDABUEWTWDABCWAWSWCWAVKVOOZWGRZVMVPOZWKRZQ
|
|
ZWQQZGTFTETDTZWSWCWAXFWRDEFGWAWRXFWAWMXEWQVSWHXBVTWLXDVSVJXAWGVJHHUFUGVTV
|
|
LXCWKVLHHUFUGUHUIUJUKVSVKHPZVOHPZQVMHPZVPHPZQXGWCULVTVSXHXIVJHHUMVJHHVEUN
|
|
VTXJXKVLHHUMVLHHVEUNWQWCDEFGVKVOVMVPHHWEVKRZWFVORZQZWIVMRZWJVPRZQZQZWPVRW
|
|
BXRWNVNWOVQXRWEVKWIVMLXLXMXQUOXNXOXPUPUQXRWFVOWJVPLXLXMXQURXNXOXPUSUQUTVA
|
|
VBVCVDVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $.
|
|
$( Equivalence relation for positive fractions in terms of positive
|
|
integers. (Contributed by NM, 27-Aug-1995.) $)
|
|
enqbreq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
|
|
( <. A , B >. ~Q <. C , D >. <-> ( A .N D ) = ( B .N C ) ) ) $=
|
|
( vx vy vz vw vv vu cmi ceq cnpi df-enq ecopoveq ) EFGHIJABCDKLMEFGHIJNO
|
|
$.
|
|
$}
|
|
|
|
$( Equivalence relation for positive fractions in terms of positive integers.
|
|
(Contributed by Mario Carneiro, 8-May-2013.) $)
|
|
enqbreq2 $p |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) ->
|
|
( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) =
|
|
( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) $=
|
|
( cnpi cxp wcel wa ceq wbr c1st cfv c2nd cop cmi co 1st2nd2 xp1st xp2nd jca
|
|
wceq syl2an breqan12d wb enqbreq mulcompig eqeq2d 3bitrd ) ACCDZEZBUGEZFZAB
|
|
GHAIJZAKJZLZBIJZBKJZLZGHZUKUOMNZULUNMNZSZURUNULMNZSUHUIAUMBUPGACCOBCCOUAUHU
|
|
KCEZULCEZFUNCEZUOCEZFUQUTUBUIUHVBVCACCPACCQZRUIVDVEBCCPZBCCQRUKULUNUOUCTUJU
|
|
SVAURUHVCVDUSVASUIVFVGULUNUDTUEUF $.
|
|
|
|
${
|
|
$d x y z w v u $.
|
|
$( The equivalence relation for positive fractions is an equivalence
|
|
relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM,
|
|
27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) $)
|
|
enqer $p |- ~Q Er ( N. X. N. ) $=
|
|
( vx vy vz vw vv cmi ceq cnpi df-enq mulcompig mulclpi mulasspig wcel w3a
|
|
vu cv co wceq mulcanpig biimpd ecopoverg ) ABCDEOFGHABCDEOIAPZBPZJUBUCKUB
|
|
UCCPZLUBHMUCHMUDHMNUBUCFQUBUDFQRUCUDRUBUCUDSTUA $.
|
|
$}
|
|
|
|
$( Equivalence class equality of positive fractions in terms of positive
|
|
integers. (Contributed by NM, 29-Nov-1995.) $)
|
|
enqeceq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
|
|
( [ <. A , B >. ] ~Q = [ <. C , D >. ] ~Q <->
|
|
( A .N D ) = ( B .N C ) ) ) $=
|
|
( cnpi wcel wa cop ceq wbr cec wceq cmi co cxp wer enqer a1i opelxpi adantr
|
|
erth enqbreq bitr3d ) AEFBEFGZCEFDEFGZGZABHZCDHZIJUGIKUHIKLADMNBCMNLUFUGUHI
|
|
EEOZUIIPUFQRUDUGUIFUEABEESTUAABCDUBUC $.
|
|
|
|
${
|
|
$d x y z w v u $.
|
|
$( The equivalence relation for positive fractions exists. (Contributed by
|
|
NM, 3-Sep-1995.) $)
|
|
enqex $p |- ~Q e. _V $=
|
|
( vx vy vz vw vv vu ceq cnpi cxp niex xpex cv wcel wa cop wceq cmi co wex
|
|
copab df-enq opabssxp eqsstri ssexi ) GHHIZUEIZUEUEHHJJKZUGKGALZUEMBLZUEM
|
|
NUHCLZDLZOPUIELZFLZOPNUJUMQRUKULQRPNFSESDSCSZNABTUFABCDEFUAUNABUEUEUBUCUD
|
|
$.
|
|
$}
|
|
|
|
$( The equivalence relation for positive fractions is decidable.
|
|
(Contributed by Jim Kingdon, 7-Sep-2019.) $)
|
|
enqdc $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
|
|
DECID <. A , B >. ~Q <. C , D >. ) $=
|
|
( cnpi wcel wa cop ceq wbr wdc cmi co wceq mulclpi pinn nndceq syl2an an42s
|
|
com enqbreq dcbid mpbird ) AEFZBEFZGCEFZDEFZGGZABHCDHIJZKADLMZBCLMZNZKZUDUG
|
|
UEUFUMUDUGGUJEFZUKEFZUMUEUFGADOBCOUNUJTFUKTFUMUOUJPUKPUJUKQRRSUHUIULABCDUAU
|
|
BUC $.
|
|
|
|
$( The equivalence relation for positive fractions is decidable.
|
|
(Contributed by Jim Kingdon, 7-Sep-2019.) $)
|
|
enqdc1 $p |- ( ( ( A e. N. /\ B e. N. ) /\ C e. ( N. X. N. ) ) ->
|
|
DECID <. A , B >. ~Q C ) $=
|
|
( cnpi wcel wa cxp cop ceq wbr wdc c1st cfv xp1st xp2nd jca enqdc sylan2 wb
|
|
c2nd 1st2nd2 breq2d dcbid adantl mpbird ) ADEBDEFZCDDGEZFABHZCIJZKZUHCLMZCT
|
|
MZHZIJZKZUGUFUKDEZULDEZFUOUGUPUQCDDNCDDOPABUKULQRUGUJUOSUFUGUIUNUGCUMUHICDD
|
|
UAUBUCUDUE $.
|
|
|
|
${
|
|
$d x y z w u v $.
|
|
$( The class of positive fractions exists. (Contributed by NM,
|
|
16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) $)
|
|
nqex $p |- Q. e. _V $=
|
|
( cnq cnpi cxp ceq cqs cvv df-nqqs niex xpex qsex eqeltri ) ABBCZDEFGLDBB
|
|
HHIJK $.
|
|
|
|
$( The empty set is not a positive fraction. (Contributed by NM,
|
|
24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) $)
|
|
0nnq $p |- -. (/) e. Q. $=
|
|
( c0 cnq wcel cnpi cxp ceq cqs wne neirr cdm wceq enqer erdm ax-mp elqsn0
|
|
wer mpan mto df-nqqs eleq2i mtbir ) ABCADDEZFGZCZUDAAHZAIFJUBKZUDUEUBFPUF
|
|
LUBFMNUBAFOQRBUCASTUA $.
|
|
|
|
$( Positive fraction 'less than' is a relation on positive fractions.
|
|
(Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro,
|
|
27-Apr-2013.) $)
|
|
ltrelnq $p |- <Q C_ ( Q. X. Q. ) $=
|
|
( vx vy vz vw vv vu cltq cv cnq wcel wa cop ceq cec wceq cmi clti wbr wex
|
|
co copab cxp df-ltnqqs opabssxp eqsstri ) GAHZIJBHZIJKUFCHZDHZLMNOUGEHZFH
|
|
ZLMNOKUHUKPTUIUJPTQRKFSESDSCSZKABUAIIUBABCDEFUCULABIIUDUE $.
|
|
$}
|
|
|
|
$( The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) $)
|
|
1nq $p |- 1Q e. Q. $=
|
|
( c1o cop ceq cec cnpi cxp cqs c1q cnq wcel 1pi opelxpi mp2an enqex ecelqsi
|
|
ax-mp df-1nqqs df-nqqs 3eltr4i ) AABZCDZEEFZCGZHITUBJZUAUCJAEJZUEUDKKAAEELM
|
|
UBTCNOPQRS $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z F $.
|
|
$d x y z G $. $d x y z R $. $d x y z S $.
|
|
|
|
$( Lemma showing compatibility of addition. (Contributed by NM,
|
|
27-Aug-1995.) $)
|
|
addcmpblnq $p |- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. )
|
|
) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) ->
|
|
( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) ->
|
|
<. ( ( A .N G ) +N ( B .N F ) ) , ( B .N G ) >. ~Q
|
|
<. ( ( C .N S ) +N ( D .N R ) ) , ( D .N S ) >. ) ) $=
|
|
( vx vy vz cnpi wcel wa cmi co wceq cpli adantl mulclpi distrpig ad2ant2l
|
|
cop ceq wbr w3a simplll simprlr syl2anc simpllr simprll addclpi mulcompig
|
|
cv caovdir2d simplrr mulasspig simprrr caov4d eqtrd oveq1 oveq2 oveqan12d
|
|
oveq12d sylan9eq simplrl simprrl syl3anc adantr eqtr4d syl2an jca anim12i
|
|
wb an42s an4s enqbreq syl mpbird ex ) ALMZBLMZNZCLMZDLMZNZNZGLMZHLMZNZELM
|
|
ZFLMZNZNZNZADOPZBCOPZQZGFOPZHEOPZQZNZAHOPZBGOPZRPZBHOPZUCCFOPZDEOPZRPZDFO
|
|
PZUCUDUEZWOXBNZXKXEXJOPZXFXIOPZQZXLXMWQHFOPZOPZBDOPZWTOPZRPZXNWOXBXMWPXPO
|
|
PZXRWSOPZRPZXTWOXMXCXJOPZXDXJOPZRPYCWOIJKXCXDXJLROIUNZLMZJUNZLMZKUNZLMUFZ
|
|
YFYHYJRPOPYFYHOPZYFYJOPRPQWOYFYHYJUASWOWAWIXCLMZWAWBWFWNUGZWGWHWIWMUHZAHT
|
|
ZUIWOWBWHXDLMZWAWBWFWNUJZWGWHWIWMUKZBGTZUIWFWMXJLMZWCWJWEWLUUAWDWKDFTUBZU
|
|
BYGYINZYFYHRPLMWOYFYHULSUUCYLYHYFOPQWOYFYHUMSZUOWOYDYAYEYBRWOIJKAHDFLOYNY
|
|
OWCWDWEWNUPZUUDYKYLYJOPYFYHYJOPOPQWOYFYHYJUQSZWGWJWKWLURZUUCYLLMWOYFYHTSZ
|
|
USWOIJKBGDFLOYRYSUUEUUDUUFUUGUUHUSVDUTWRXAYAXQYBXSRWPWQXPOVAWSWTXROVBVCVE
|
|
WOXNXTQXBWOXNXFXGOPZXFXHOPZRPZXTWOXFLMZXGLMZXHLMZXNUUKQWOWBWIUULYRYOBHTZU
|
|
IWOWDWLUUMWCWDWEWNVFZUUGCFTZUIWOWEWKUUNUUEWGWJWKWLVGZDETZUIXFXGXHUAVHWOUU
|
|
IXQUUJXSRWOIJKBHCFLOYRYOUUPUUDUUFUUGUUHUSWOIJKBHDELOYRYOUUEUUDUUFUURUUHUS
|
|
VDUTVIVJWOXKXOVNZXBWOXELMZUULNZXILMZUUANZNZUUTWCWJWFWMUVEWCWJNZUVBWFWMNZU
|
|
VDUVFUVAUULWAWIWBWHUVAWAWINYMYQUVAWBWHNYPYTXCXDULVKVOWBWIUULWAWHUUOUBVLUV
|
|
GUVCUUAWDWLWEWKUVCWDWLNUUMUUNUVCWEWKNUUQUUSXGXHULVKVOUUBVLVMVPXEXFXIXJVQV
|
|
RVIVSVT $.
|
|
|
|
$( Lemma showing compatibility of multiplication. (Contributed by NM,
|
|
27-Aug-1995.) $)
|
|
mulcmpblnq $p |- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. )
|
|
) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) ->
|
|
( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) ->
|
|
<. ( A .N F ) , ( B .N G ) >. ~Q <. ( C .N R ) , ( D .N S ) >. ) ) $=
|
|
( vx vy vz cmi co wceq wa cnpi wcel mulclpi anim12i an4s cop oveq12 wb cv
|
|
ceq wbr enqbreq simplll simprll simplrr mulcompig adantl mulasspig caov4d
|
|
syl w3a simprrr simpllr simprlr simplrl simprrl eqeq12d bitrd syl5ibr ) A
|
|
DLMZBCLMZNGFLMZHELMZNOAGLMZBHLMZUACELMZDFLMZUAUEUFZAPQZBPQZOZCPQZDPQZOZOZ
|
|
GPQZHPQZOZEPQZFPQZOZOZOZVEVGLMZVFVHLMZNZVEVFVGVHLUBWHVMVIVLLMZVJVKLMZNZWK
|
|
WHVIPQZVJPQZOZVKPQZVLPQZOZOZVMWNUCVPWCVSWFXAVPWCOWQVSWFOWTVNWAVOWBWQVNWAO
|
|
WOVOWBOWPAGRBHRSTVQWDVRWEWTVQWDOWRVRWEOWSCERDFRSTSTVIVJVKVLUGUOWHWLWIWMWJ
|
|
WHIJKAGDFPLVNVOVSWGUHVTWAWBWFUIVPVQVRWGUJIUDZPQZJUDZPQZOZXBXDLMZXDXBLMNWH
|
|
XBXDUKULZXCXEKUDZPQUPXGXILMXBXDXILMLMNWHXBXDXIUMULZVTWCWDWEUQXFXGPQWHXBXD
|
|
RULZUNWHIJKBHCEPLVNVOVSWGURVTWAWBWFUSVPVQVRWGUTXHXJVTWCWDWEVAXKUNVBVCVD
|
|
$.
|
|
$}
|
|
|
|
$( Lemma for ~ addpipqqs . (Contributed by Jim Kingdon, 11-Sep-2019.) $)
|
|
addpipqqslem $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
|
|
<. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. e. ( N. X. N. ) ) $=
|
|
( cnpi wcel wa cmi co cop cxp mulclpi addclpi syl2an an42s ad2ant2l opelxpi
|
|
cpli syl2anc ) AEFZBEFZGCEFZDEFZGGADHIZBCHIZRIZEFZBDHIZEFZUFUHJEEKFTUCUAUBU
|
|
GTUCGUDEFUEEFUGUAUBGADLBCLUDUEMNOUAUCUITUBBDLPUFUHEEQS $.
|
|
|
|
${
|
|
$d x y z w v u t s f g h a b c d A $.
|
|
$d x y z w v u t s f g h a b c d B $.
|
|
$d x y z w v u t s f g h a b c d C $.
|
|
$d x y z w v u t s f g h a b c d D $.
|
|
$( Addition of positive fractions in terms of positive integers.
|
|
(Contributed by NM, 28-Aug-1995.) $)
|
|
addpipqqs $p |- ( ( ( A e. N. /\ B e. N. ) /\
|
|
( C e. N. /\ D e. N. ) ) ->
|
|
( [ <. A , B >. ] ~Q +Q [ <. C , D >. ] ~Q ) =
|
|
[ <. ( ( A .N D ) +N ( B .N C ) ) ,
|
|
( B .N D ) >. ] ~Q ) $=
|
|
( vz vu vw vv va vd vb vc cv cmi co wceq cpli weq wa oveq12 vg vs vh cplq
|
|
vt vx vy cplpq cnq ceq cnpi cop addpipqqslem enqex enqer df-enq eqeqan12d
|
|
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|
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(Contributed by Mario Carneiro, 8-May-2013.) $)
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mulpipq2 $p |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) ->
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( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) ,
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( vx vy cnpi wcel c1st cfv cmi co c2nd cop cmpq wceq xp1st mulclpi syl2an
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ABULULCUJZGHZDUJZGHZIJZVEKHZVGKHZIJZLVAMUOVHIJZURVKIJZLRVEANZVIVMVLVNVOVF
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|
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(Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro,
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8-May-2013.) $)
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mulpipq $p |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) ->
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( <. A , B >. .pQ <. C , D >. ) = <. ( A .N C ) , ( B .N D ) >. ) $=
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$d x y z w v u t s f g h a b c d D $.
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$( Multiplication of positive fractions in terms of positive integers.
|
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(Contributed by NM, 28-Aug-1995.) $)
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mulpipqqs $p |- ( ( ( A e. N. /\ B e. N. ) /\
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( C e. N. /\ D e. N. ) ) ->
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KOPWLOPXHXEXFQACRBDRWKWLOOUOUPUQVSOPZWEOPZWBOPZWHOPZWSXGPZXIXJQWQOPWROPXM
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(Contributed by Jim Kingdon, 14-Sep-2019.) $)
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vu vw vv cltq enqex enqer df-nqqs df-ltnqqs enqeceq eqcom syl6bb bi2anan9
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|
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29-Aug-1995.) $)
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29-Aug-1995.) $)
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20-Sep-2019.) $)
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24-Aug-1995.) $)
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24-Aug-1995.) $)
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Kingdon, 16-Sep-2019.) $)
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Jim Kingdon, 17-Sep-2019.) $)
|
|
mulcomnqg $p |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( B .Q A ) ) $=
|
|
( vx vy vz vw cnq cv cmi co cmq ceq cnpi df-nqqs mulpipqqs wcel mulcompig
|
|
wceq ad2ant2r ad2ant2l ecovicom ) CDEFABGCHZEHZIJZKLMDHZFHZIJZUCUBIJZUFUE
|
|
IJZNUBUEUCUFOUCUFUBUEOUBMPZUCMPZUDUHRUEMPZUFMPZUBUCQSULUMUGUIRUJUKUEUFQTU
|
|
A $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$( Multiplication of positive fractions is associative. (Contributed by
|
|
Jim Kingdon, 17-Sep-2019.) $)
|
|
mulassnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
|
|
( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) ) ) $=
|
|
( vx vy vz vw vv vu cv cmi co cnpi mulpipqqs wa mulclpi ad2ant2r ad2ant2l
|
|
wcel jca cnq cmq ceq df-nqqs 3adant1r 3adant2r 3adant3r 3adant1l 3adant2l
|
|
wceq mulasspig 3adant3l ecoviass ) DEFGHIABCUAUBGJZIJZKLZUCMDJZFJZKLZEJZU
|
|
NKLZUSHJZKLZVAUOKLZUQURVBKLZKLZUTUPKLZVEUDUQUTURUNNURUNVBUONUSVAVBUONUQUT
|
|
VEUPNUQMSZUTMSZOZURMSZUNMSZOZOUSMSZVAMSZVHVKVNVIVLUQURPQVIVLVOVHVKUTUNPRT
|
|
VMVBMSZUOMSZOOVEMSZUPMSZVKVPVRVLVQURVBPQVLVQVSVKVPUNUOPRTVJVMVPVCVFUJZVQV
|
|
JVKVPVTVLVHVKVPVTVIUQURVBUKUEUFUGVJVMVQVDVGUJZVPVJVLVQWAVKVIVLVQWAVHUTUNU
|
|
OUKUHUIULUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Lemma for distributive law: cancellation of common factor. (Contributed
|
|
by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) $)
|
|
mulcanenq $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
<. ( A .N B ) , ( A .N C ) >. ~Q <. B , C >. ) $=
|
|
( vx vy vz cnpi wcel w3a cmi co cop ceq wbr wceq simp1 cv adantl mulclpi
|
|
wa simp2 simp3 mulcompig mulasspig caov32d anim12i simpr an4s jca enqbreq
|
|
wb 3impdi syl mpbird ) AGHZBGHZCGHZIZABJKZACJKZLBCLMNZUSCJKUTBJKOZURDEFAB
|
|
CGJUOUPUQPUOUPUQUAUOUPUQUBDQZGHZEQZGHZTVCVEJKZVEVCJKOURVCVEUCRVDVFFQZGHIV
|
|
GVHJKVCVEVHJKJKOURVCVEVHUDRUEURUSGHZUTGHZTZUPUQTZTZVAVBUKUOUPUQVMUOUPTZUO
|
|
UQTZTVKVLVNVIVOVJABSACSUFUOUOUPUQVLUOUOTVLUGUHUIULUSUTBCUJUMUN $.
|
|
$}
|
|
|
|
$( Lemma for distributive law: cancellation of common factor. (Contributed
|
|
by Jim Kingdon, 17-Sep-2019.) $)
|
|
mulcanenqec $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) ->
|
|
[ <. ( A .N B ) , ( A .N C ) >. ] ~Q = [ <. B , C >. ] ~Q ) $=
|
|
( cnpi wcel w3a cmi co cop ceq cxp wer enqer a1i mulcanenq erthi ) ADEBDECD
|
|
EFZABGHACGHIBCIJDDKZRJLQMNABCOP $.
|
|
|
|
${
|
|
$d A u v w x y z $. $d B u v w x y z $. $d C u v w x y z $.
|
|
$d f g h u v w x y z $.
|
|
$( Multiplication of positive fractions is distributive. (Contributed by
|
|
Jim Kingdon, 17-Sep-2019.) $)
|
|
distrnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
|
|
( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) ) $=
|
|
( vf vg vh ceq cnpi cv cmi co cpli wcel cop cec w3a wceq mulclpi adantl
|
|
wa vx vy vz vw vv vu cnq cplq cmq df-nqqs addpipqqs mulpipqqs jca anim12i
|
|
simpl an12 3anass bitr4i sylib an4s mulcanenqec syl eqtr4d addclpi syl2an
|
|
an42s ad2ant2l an42 anbi2i 3bitr4i ancoms distrpig syl3an simp1r 3ad2ant2
|
|
simp1l 3ad2ant3 syl2anc syl3anc oveq1d adantr simpll simplr simprl simprr
|
|
mulasspig mulcompig caov4d eqtr3d 3adant3 3adant2 oveq12d 3eqtr3d ecovidi
|
|
sylbir ) UAUBUCUDUEUFABCUGUHGHUIUBIZUAIZUCIZUFIZJKZUDIZUEIZJKZLKZJKZJKZWP
|
|
WPXAWSJKZJKZJKZWQWRJKZWPWSJKZJKZWPXAJKZWQXBJKZJKZLKZXMXKJKZXDXGXJXMXNXKUJ
|
|
WRXAXBWSUKWQHMZWPHMZTZXDHMZXGHMZTTZWQWPNGOXDXGNGOUIKXEXHNGOZXFXINGOZWQWPX
|
|
DXGULYCXSXEHMZXHHMZPZYEYDQXRYAXSYBYHXRYATZXSYBTZTYFXSYGTZTZYHYIYFYJYKWQXD
|
|
RYJXSYGXSYBUOWPXGRUMUNYLXSYFYGTTYHYFXSYGUPXSYFYGUQURUSUTWPXEXHVAVBVCWQWPW
|
|
RXAULWQWPXBWSULXJXMXNXKUKWRHMZXAHMZTZXBHMZWSHMZTZTZYAYBYMYQYNYPYAYMYQTZWT
|
|
HMZXCHMZYAYNYPTZWRWSRZXAXBRZWTXCVDZVEVFYNYQYBYMYPXAWSRZVGUMXRYMXSYNXJHMZX
|
|
MHMZTXRYMTUUHXSYNTUUIWQWRRWPXARUNUTXRYPXSYQXNHMZXKHMZTXRYPTUUJXSYQTUUKWQX
|
|
BRWPWSRUNUTXTYOYRPZXTYTUUCPZXFXPQXTYTUUCTZTXTYSTUUMUULUUNYSXTYMYQYNYPVHVI
|
|
XTYTUUCUQXTYOYRUQVJZUUMWPWQJKZXDJKZUUPWTJKZUUPXCJKZLKZXFXPXTUUPHMZYTUUAUU
|
|
CUUBUUQUUTQXSXRUVAWPWQRVKUUDUUEUUPWTXCVLVMUUMXSXRYAUUQXFQXRXSYTUUCVNZXRXS
|
|
YTUUCVPUUMUUAUUBYAYTXTUUAUUCUUDVOUUCXTUUBYTUUEVQUUFVRWPWQXDWFVSUUMUURXLUU
|
|
SXOLXTYTUURXLQUUCXTYTTZWQWPJKZWTJKZUURXLXTUVEUURQYTXTUVDUUPWTJWQWPWGVTWAU
|
|
VCDEFWQWPWRWSHJXRXSYTWBXRXSYTWCXTYMYQWDDIZHMZEIZHMZTZUVFUVHJKZUVHUVFJKQZU
|
|
VCUVFUVHWGZSUVGUVIFIZHMPZUVKUVNJKUVFUVHUVNJKJKQZUVCUVFUVHUVNWFZSXTYMYQWEZ
|
|
UVJUVKHMZUVCUVFUVHRZSWHWIWJXTUUCUUSXOQYTXTUUCTZDEFWPWQXAXBHJXRXSUUCWCXRXS
|
|
UUCWBXTYNYPWDZUVJUVLUWAUVMSUVOUVPUWAUVQSXTYNYPWEUVJUVSUWAUVTSWHWKWLWMWOUU
|
|
LUUMXIXQQUUOUUMWPWPJKXGJKZXIXQUUMXSXSYBUWCXIQUVBUVBUUMYNYQYBXTUUCYNYTUWBW
|
|
KZXTYTYQUUCUVRWJZUUGVRWPWPXGWFVSUUMDEFWPWPXAWSHJUVBUVBUWDUVJUVLUUMUVMSUVO
|
|
UVPUUMUVQSUWEUVJUVSUUMUVTSWHWIWOWN $.
|
|
$}
|
|
|
|
$( The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) $)
|
|
1qec $p |- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q ) $=
|
|
( cnpi wcel c1q c1o cop ceq cec df-1nqqs cmi co wceq mulcanenqec mp3an23 wa
|
|
1pi mulidpi jca opeq12 eceq1 3syl eqtr3d syl5eq ) ABCZDEEFGHZAAFZGHZIUDAEJK
|
|
ZUHFZGHZUEUGUDEBCZUKUJUELPPAEEMNUDUHALZULOUIUFLUJUGLUDULULAQZUMRUHUHAASUIUF
|
|
GTUAUBUC $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Multiplication identity element for positive fractions. (Contributed by
|
|
NM, 3-Mar-1996.) $)
|
|
mulidnq $p |- ( A e. Q. -> ( A .Q 1Q ) = A ) $=
|
|
( vx vy cv cop ceq cec c1q cmq co wceq cnpi cnq df-nqqs c1o cmi mulcompig
|
|
wcel 1pi mpan oveq1 id eqeq12d wa df-1nqqs oveq2i mulpipqqs syl5eq adantr
|
|
mpanr12 adantl opeq12d eceq1d mulcanenqec mp3an1 3eqtr2d ecoptocl ) BDZCD
|
|
ZEFGZHIJZUTKAHIJZAKBCALLFMNUTAKZVAVBUTAUTAHIUAVCUBUCURLRZUSLRZUDZVAUROPJZ
|
|
USOPJZEZFGZOURPJZOUSPJZEZFGZUTVFVAUTOOEFGZIJZVJHVOUTIUEUFVFOLRZVQVPVJKSSU
|
|
RUSOOUGUJUHVFVMVIFVFVKVGVLVHVDVKVGKZVEVQVDVRSOURQTUIVEVLVHKZVDVQVEVSSOUSQ
|
|
TUKULUMVQVDVEVNUTKSOURUSUNUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Existence of positive fraction reciprocal. (Contributed by Jim Kingdon,
|
|
20-Sep-2019.) $)
|
|
recexnq $p |- ( A e. Q. -> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) ) $=
|
|
( vx vz cv cnq wcel cop ceq cec cmq co c1q wceq wa wex df-nqqs eqeq1d syl
|
|
cnpi oveq1 anbi2d exbidv cxp cqs opelxpi ancoms enqex syl6eleqr mulcompig
|
|
ecelqsi cmi opeq2d eceq1d mulclpi 1qec mulpipqqs an42s 3eqtr4rd jca eleq1
|
|
anidms oveq2 anbi12d spcegv sylc ecoptocl ) AEZFGZCEZDEZHIJZVHKLZMNZOZAPZ
|
|
VIBVHKLZMNZOZAPCDBTTIFQVLBNZVOVSAVTVNVRVIVTVMVQMVLBVHKUARUBUCVJTGZVKTGZOZ
|
|
VKVJHZIJZFGZWFVLWEKLZMNZOZVPWCWETTUDZIUEZFWCWDWJGZWEWKGWBWAWLVKVJTTUFUGWJ
|
|
WDIUHUKSQUIZWCWFWHWMWCVJVKULLZWNHZIJZWNVKVJULLZHZIJZMWGWCWOWRIWCWNWQWNVJV
|
|
KUJUMUNWCWNTGMWPNVJVKUOWNUPSWCWGWSNZWAWBWBWAWTVJVKVKVJUQURVBUSUTVOWIAWEFV
|
|
HWENZVIWFVNWHVHWEFVAXAVMWGMVHWEVLKVCRVDVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y z w v $.
|
|
$( Relationship between reciprocal and multiplication on positive
|
|
fractions. (Contributed by Jim Kingdon, 19-Sep-2019.) $)
|
|
recmulnqg $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) ) $=
|
|
( vy vx vz vw vv cnq wcel wa crq cfv wceq cmq co c1q oveq1 eqeq1d copab
|
|
cv anbi2d eleq1 oveq2 anbi12d wex wmo weu recexnq 1nq mulcomnqg mulassnqg
|
|
mulidnq caovimo eu5 sylanbrc w3a df-rq 3anass opabbii eqtri fvopab3g ibar
|
|
wb adantl bitr4d ) AHIZBHIZJAKLBMVGABNOZPMZJZVICTZHIZDTZVKNOZPMZJZVLAVKNO
|
|
ZPMZJVJDCABHHKVMAMZVOVRVLVSVNVQPVMAVKNQRUAVKBMZVLVGVRVIVKBHUBVTVQVHPVKBAN
|
|
UCRUDVMHIZVPCUEVPCUFVPCUGCVMUHEFGCVMPHNUIETZFTZUJWBWCGTUKWBULUMVPCUNUOKWA
|
|
VLVOUPZDCSWAVPJZDCSDCUQWDWEDCWAVLVOURUSUTVAVGVIVJVCVFVGVIVBVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d A y $.
|
|
$( Closure law for positive fraction reciprocal. (Contributed by NM,
|
|
6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) $)
|
|
recclnq $p |- ( A e. Q. -> ( *Q ` A ) e. Q. ) $=
|
|
( vy cnq wcel cv cmq co c1q wceq wa wex crq cfv recexnq recmulnqg biimpar
|
|
wi eleq1a ad2antlr mpd expl exlimdv ) ACDZBEZCDZAUDFGHIZJZBKALMZCDZBANUCU
|
|
GUIBUCUEUFUIUCUEJZUFJUHUDIZUIUJUKUFAUDOPUEUKUIQUCUFUDCUHRSTUAUBT $.
|
|
$}
|
|
|
|
$( A positive fraction times its reciprocal is 1. (Contributed by NM,
|
|
6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) $)
|
|
recidnq $p |- ( A e. Q. -> ( A .Q ( *Q ` A ) ) = 1Q ) $=
|
|
( cnq wcel crq cfv cmq co c1q wceq recclnq wa eqid recmulnqg mpbii mpdan )
|
|
ABCZADEZBCZAQFGHIZAJPRKQQISQLAQMNO $.
|
|
|
|
$( Reciprocal of reciprocal of positive fraction. (Contributed by NM,
|
|
26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) $)
|
|
recrecnq $p |- ( A e. Q. -> ( *Q ` ( *Q ` A ) ) = A ) $=
|
|
( cnq wcel crq cfv wceq cmq c1q recclnq mulcomnqg mpancom recidnq recmulnqg
|
|
co eqtrd wb mpbird ) ABCZADEZDEAFZSAGNZHFZRUAASGNZHSBCZRUAUCFAIZSAJKALOUDRT
|
|
UBPUESAMKQ $.
|
|
|
|
$( Reciprocal of positive fraction one. (Contributed by Jim Kingdon,
|
|
29-Dec-2019.) $)
|
|
rec1nq $p |- ( *Q ` 1Q ) = 1Q $=
|
|
( c1q crq cfv cmq co cnq wcel wceq 1nq recclnq ax-mp mulcomnqg mulidnq mp2b
|
|
mp2an recidnq 3eqtr3i ) ABCZADEZARDEZRARFGZAFGZSTHUBUAIAJZKIRALOUBUASRHIUCR
|
|
MNUBTAHIAPKQ $.
|
|
|
|
${
|
|
$d A u v w z $. $d B u v $.
|
|
$( Trichotomy for positive fractions. (Contributed by Jim Kingdon,
|
|
21-Sep-2019.) $)
|
|
nqtri3or $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( A <Q B \/ A = B \/ B <Q A ) ) $=
|
|
( vz vw vu vv cv cop ceq cltq wbr wceq w3o cnpi 3orbi123d wcel wa co clti
|
|
cmi cec cnq df-nqqs breq1 eqeq1 breq2 mulclpi ad2ant2rl ad2ant2lr syl2anc
|
|
eqeq2 pitri3or ordpipqqs enqeceq ancoms mulcompig bitr4d mpbird 2ecoptocl
|
|
wb breq12d ) CGZDGZHIUAZEGZFGZHIUAZJKZVDVGLZVGVDJKZMZAVGJKZAVGLZVGAJKZMAB
|
|
JKZABLZBAJKZMCDEFABNNIUBUCVDALVHVLVIVMVJVNVDAVGJUDVDAVGUEVDAVGJUFOVGBLVLV
|
|
OVMVPVNVQVGBAJUFVGBAUKVGBAJUDOVBNPZVCNPZQZVENPZVFNPZQZQZVKVBVFTRZVCVETRZS
|
|
KZWEWFLZWFWESKZMZWDWENPZWFNPZWJVRWBWKVSWAVBVFUGUHVSWAWLVRWBVCVEUGUIWEWFUL
|
|
UJWDVHWGVIWHVJWIVBVCVEVFUMVBVCVEVFUNWDVJVEVCTRZVFVBTRZSKZWIWCVTVJWOUTVEVF
|
|
VBVCUMUOWDWFWMWEWNSVSWAWFWMLVRWBVCVEUPUIVRWBWEWNLVSWAVBVFUPUHVAUQOURUS $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $.
|
|
$( Less-than for positive fractions is decidable. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
ltdcnq $p |- ( ( A e. Q. /\ B e. Q. ) -> DECID A <Q B ) $=
|
|
( vx vy vz vw cnq wcel wa cv cnpi cop ceq cec wceq wex cltq wbr wdc nqpi
|
|
anim12i ee4anv sylibr co clti mulclpi ltdcpi syl2an an42s ordpipqqs dcbid
|
|
cmi mpbird ad2ant2r wb breq12 ad2ant2l exlimivv syl ) AGHZBGHZIZCJZKHZDJZ
|
|
KHZIZAVCVELMNZOZIZEJZKHZFJZKHZIZBVKVMLMNZOZIZIZFPEPZDPCPZABQRZSZVBVJDPCPZ
|
|
VRFPEPZIWAUTWDVAWECDATEFBTUAVJVRCDEFUBUCVTWCCDVSWCEFVSWCVHVPQRZSZVGVOWGVI
|
|
VQVGVOIZWGVCVMULUDZVEVKULUDZUERZSZVDVNVFVLWLVDVNIWIKHWJKHWLVFVLIVCVMUFVEV
|
|
KUFWIWJUGUHUIWHWFWKVCVEVKVMUJUKUMUNVSWBWFVIVQWBWFUOVGVOAVHBVPQUPUQUKUMURU
|
|
RUS $.
|
|
$}
|
|
|
|
${
|
|
$d a b c d e f x y z w $.
|
|
$( 'Less than' is a strict ordering on positive fractions. (Contributed by
|
|
NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) $)
|
|
ltsonq $p |- <Q Or Q. $=
|
|
( vx vy vz cnq cltq wtru cv wcel wbr ceq cnpi wceq wa cmi co clti mulclpi
|
|
wb adantl syl2anc vw va vb vc vd ve vf wor cop cec df-nqqs breq12d notbid
|
|
wn ltsopi ltrelpi soirri ordpipqqs anidms mulcompig breq1d bitrd ecoptocl
|
|
id mtbiri w3a wi breq1 anbi1d imbi12d breq2 anbi12d imbi1d anbi2d 3adant3
|
|
simp1l simp2r simp1r simp2l simp3r ltmpig syl3anc adantrr breqtrd 3adant1
|
|
biimpa adantr simp3l adantrl sotri mulasspig caov411d eqtrd caov4d oveq1d
|
|
caovcomd mpbird 3adant2 ex 3ecoptocl ispod w3o nqtri3or issod trud ) DEUH
|
|
FABDEFABCDEAGZDHZXFXFEIZUNZFCGZUAGZUIJUJZXLEIZUNXICUAXFKKJDUKXLXFLZXMXHXN
|
|
XLXFXLXFEXNVDZXOULUMXJKHZXKKHMZXMXKXJNOZXRPIZXRPKUOUPUQXQXMXJXKNOZXRPIZXS
|
|
XQXMYARXJXKXJXKURUSXQXTXRXRPXJXKUTVAVBVEVCSXGBGZDHZXJDHVFXFYBEIZYBXJEIZMZ
|
|
XFXJEIZVGZFUBGZUCGZUIJUJZUDGZUEGZUIJUJZEIZYNUFGZUGGZUIJUJZEIZMZYKYREIZVGX
|
|
FYNEIZYSMZXFYREIZVGYDYBYREIZMZUUDVGYHUBUCUDUEUFUGXFYBXJKJDUKYKXFLZYTUUCUU
|
|
AUUDUUGYOUUBYSYKXFYNEVHVIYKXFYREVHVJYNYBLZUUCUUFUUDUUHUUBYDYSUUEYNYBXFEVK
|
|
YNYBYREVHVLVMYRXJLZUUFYFUUDYGUUIUUEYEYDYRXJYBEVKVNYRXJXFEVKVJYIKHZYJKHZMZ
|
|
YLKHZYMKHZMZYPKHZYQKHZMZVFZYTUUAUUSYTMZUUAYIYQNOZYJYPNOZPIZUUTUVCYLYMNOZU
|
|
VANOZUVDUVBNOZPIZUUTUVGYLYQNOZYIYMNOZNOZYJYLNOZYMYPNOZNOZPIZUUTUVJUVKUVHN
|
|
OZPIUVOUVMPIZUVNUUTUVJUVHUVKNOZUVOPUUSYOUVJUVQPIZYSUUSYOUVRUUSYOUVIUVKPIZ
|
|
UVRUULUUOYOUVSRUURYIYJYLYMURVOUUSUVIKHZUVKKHZUVHKHZUVSUVRRUUSUUJUUNUVTUUJ
|
|
UUKUUOUURVPZUULUUMUUNUURVQZYIYMQTZUUSUUKUUMUWAUUJUUKUUOUURVRZUULUUMUUNUUR
|
|
VSZYJYLQTZUUSUUMUUQUWBUWGUULUUOUUPUUQVTZYLYQQTZUVIUVKUVHWAWBVBWFWCUUSUVQU
|
|
VOLZYTUUSUWBUWAUWKUWJUWHUVHUVKUTTWGWDUUSYSUVPYOUUSYSUVPUUSYSUVHUVLPIZUVPU
|
|
UOUURYSUWLRUULYLYMYPYQURWEUUSUWBUVLKHZUWAUWLUVPRUWJUUSUUNUUPUWMUWDUULUUOU
|
|
UPUUQWHZYMYPQTUWHUVHUVLUVKWAWBVBWFWIUVJUVOUVMPKUOUPWJTUUSUVGUVNRYTUUSUVEU
|
|
VJUVFUVMPUUSUVEUVIUVHNOUVJUUSABCYLYMYIYQKNUWGUWDUWCXFKHZYBKHZMZXFYBNOZYBX
|
|
FNOLUUSXFYBUTSZUWOUWPXPVFUWRXJNOXFYBXJNONOLUUSXFYBXJWKSZUWIUWQUWRKHUUSXFY
|
|
BQSZWLUUSABUVIUVHKNUWSUWEUWJWPWMUUSUVFYLYJNOZUVLNOUVMUUSABCYLYMYJYPKNUWGU
|
|
WDUWFUWSUWTUWNUXAWNUUSUXBUVKUVLNUUSABYLYJKNUWSUWGUWFWPWOWMULWGWQUUSUVCUVG
|
|
RZYTUUSUVAKHZUVBKHZUVDKHZUXCUUSUUJUUQUXDUWCUWIYIYQQTUUSUUKUUPUXEUWFUWNYJY
|
|
PQTUUSUUMUUNUXFUWGUWDYLYMQTUVAUVBUVDWAWBWGWQUUSUUAUVCRZYTUULUURUXGUUOYIYJ
|
|
YPYQURWRWGWQWSWTSXAXGYCMYDXFYBLYBXFEIXBFXFYBXCSXDXE $.
|
|
$}
|
|
|
|
$( Trichotomy for positive fractions. (Contributed by Jim Kingdon,
|
|
21-Sep-2019.) $)
|
|
nqtric $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( A <Q B <-> -. ( A = B \/ B <Q A ) ) ) $=
|
|
( cnq cltq ltsonq nqtri3or sotritric ) CABDEABFG $.
|
|
|
|
${
|
|
$d A x y z w v u f g h $. $d B x y z w v u f g h $.
|
|
$d C x y z w v u f g h $.
|
|
$( Ordering property of addition for positive fractions. Proposition
|
|
9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon,
|
|
22-Sep-2019.) $)
|
|
ltanqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
|
|
( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) ) $=
|
|
( vf vg vh cv cltq wbr cplq co wb cnpi wceq wcel cmi cpli mulclpi syl2anc
|
|
clti vx vy vz vw vv vu cop ceq cec cnq df-nqqs breq1 oveq2 breq1d bibi12d
|
|
breq2 breq2d oveq1 breq12d bibi2d w3a addclpi adantl simp3l simp1r simp3r
|
|
wa simp1l caovcld simp2r ordpipqqs syl22anc simp3 simp1 addpipqqs 3adant3
|
|
simp2l simp2 ltmpig syl3anc ltapig mulcompig mulasspig distrpig caovdir2d
|
|
caov4d caovdid caov411d oveq1d eqtrd 3bitr4d 3bitrd 3bitr4rd 3ecoptocl )
|
|
UAGZUBGZUGUHUIZUCGZUDGZUGUHUIZHIZUEGZUFGZUGUHUIZWQJKZXDWTJKZHIZLAWTHIZXDA
|
|
JKZXFHIZLABHIZXIXDBJKZHIZLXKCAJKZCBJKZHIZLUAUBUCUDUEUFABCMUHUJUKWQANZXAXH
|
|
XGXJWQAWTHULXQXEXIXFHWQAXDJUMUNUOWTBNZXHXKXJXMWTBAHUPXRXFXLXIHWTBXDJUMUQU
|
|
OXDCNZXMXPXKXSXIXNXLXOHXDCAJURXDCBJURUSUTWOMOZWPMOZVGZWRMOZWSMOZVGZXBMOZX
|
|
CMOZVGZVAZXBWPPKZXCWOPKZQKZXCWPPKZUGUHUIZXBWSPKZXCWRPKZQKZXCWSPKZUGUHUIZH
|
|
IZYLYRPKZYMYQPKZTIZXGXAYIYLMOYMMOZYQMOYRMOZYTUUCLYIDEYJYKMMMQDGZMOZEGZMOZ
|
|
VGZUUFUUHQKMOYIUUFUUHVBVCZYIYFYAYJMOZYBYEYFYGVDZXTYAYEYHVEZXBWPRSZYIYGXTY
|
|
KMOZYBYEYFYGVFZXTYAYEYHVHZXCWORSZVIYIYGYAUUDUUQUUNXCWPRSZYIDEYOYPMMMQUUKY
|
|
IDEXBWSMMMPUUJUUFUUHPKZMOYIUUFUUHRVCZUUMYBYCYDYHVJZVIZYIYGYCYPMOZUUQYBYCY
|
|
DYHVQZXCWRRSZVIYIYGYDUUEUUQUVCXCWSRSZYLYMYQYRVKVLYIXEYNXFYSHYIYHYBXEYNNYB
|
|
YEYHVMZYBYEYHVNXBXCWOWPVOSYIYHYEXFYSNUVIYBYEYHVRXBXCWRWSVOSUSYIXAWOWSPKZW
|
|
PWRPKZTIZXCXCPKZUVJPKZUVMUVKPKZTIZUUCYBYEXAUVLLYHWOWPWRWSVKVPYIUVJMOZUVKM
|
|
OZUVMMOZUVLUVPLYIXTYDUVQUURUVCWOWSRSYIYAYCUVRUUNUVFWPWRRSYIYGYGUVSUUQUUQX
|
|
CXCRSUVJUVKUVMVSVTYIYKYRPKZYMYPPKZTIZYJYRPKZUVTQKZUWCUWAQKZTIZUVPUUCYIUVT
|
|
MOZUWAMOZUWCMOZUWBUWFLYIUUPUUEUWGUUSUVHYKYRRSYIUUDUVEUWHUUTUVGYMYPRSYIUUL
|
|
UUEUWIUUOUVHYJYRRSUVTUWAUWCWAVTYIUVNUVTUVOUWATYIDEFXCXCWOWSMPUUQUUQUURUUJ
|
|
UVAUUHUUFPKNYIUUFUUHWBVCZUUGUUIFGZMOVAZUVAUWKPKUUFUUHUWKPKPKNYIUUFUUHUWKW
|
|
CVCZUVCUVBWFYIDEFXCXCWPWRMPUUQUUQUUNUWJUWMUVFUVBWFUSYIUUAUWDUUBUWETYIDEFY
|
|
JYKYRMQPUWLUUFUUHUWKQKPKUVAUUFUWKPKQKNYIUUFUUHUWKWDVCZUUOUUSUVHUUKUWJWEYI
|
|
UUBYMYOPKZUWAQKUWEYIDEFYMYOYPMQPQMUWNUUTUVDUVGWGYIUWOUWCUWAQYIDEFXCWPXBWS
|
|
MPUUQUUNUUMUWJUWMUVCUVBWHWIWJUSWKWLWMWN $.
|
|
|
|
$( Ordering property of multiplication for positive fractions. Proposition
|
|
9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon,
|
|
22-Sep-2019.) $)
|
|
ltmnqg $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
|
|
( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) ) $=
|
|
( vf vg cv cop ceq cec cltq wbr cmq co wb cnpi wceq wcel wa cmi caovcld
|
|
vx vy vz vw vv vu vh cnq df-nqqs breq1 oveq2 breq1d bibi12d breq2 breq12d
|
|
breq2d oveq1 bibi2d w3a clti mulclpi adantl simp1l simp2r simp1r 3ad2ant3
|
|
simp2l ltmpig syl3anc simp3l simp3r mulcompig caov4d oveq1d ancoms bitr4d
|
|
mulasspig ordpipqqs 3adant3 syl22anc 3bitr4d mulpipqqs 3adant2 3ecoptocl
|
|
eqtrd 3adant1 ) UAFZUBFZGHIZUCFZUDFZGHIZJKZUEFZUFFZGHIZWILMZWPWLLMZJKZNAW
|
|
LJKZWPALMZWRJKZNABJKZXAWPBLMZJKZNXCCALMZCBLMZJKZNUAUBUCUDUEUFABCOHUHUIWIA
|
|
PZWMWTWSXBWIAWLJUJXIWQXAWRJWIAWPLUKULUMWLBPZWTXCXBXEWLBAJUNXJWRXDXAJWLBWP
|
|
LUKUPUMWPCPZXEXHXCXKXAXFXDXGJWPCALUQWPCBLUQUOURWGOQZWHOQZRZWJOQZWKOQZRZWN
|
|
OQZWOOQZRZUSZWMWNWGSMZWOWHSMZGHIZWNWJSMZWOWKSMZGHIZJKZWSYAWGWKSMZWHWJSMZU
|
|
TKZYBYFSMZYCYESMZUTKZWMYHYAYKWNWOSMZYISMZYOYJSMZUTKZYNYAYIOQYJOQYOOQZYKYR
|
|
NYADEWGWKOOOSDFZOQZEFZOQZRZYTUUBSMZOQYAYTUUBVAVBZXLXMXQXTVCZXNXOXPXTVDZTY
|
|
ADEWHWJOOOSUUFXLXMXQXTVEZXNXOXPXTVGZTXTXNYSXQWNWOVAVFYIYJYOVHVIYAYLYPYMYQ
|
|
UTYADEUGWNWGWOWKOSXNXQXRXSVJZUUGXNXQXRXSVKZUUDUUEUUBYTSMPYAYTUUBVLVBZUUAU
|
|
UCUGFZOQUSUUEUUNSMYTUUBUUNSMSMPYAYTUUBUUNVQVBZUUHUUFVMYAYMWOWNSMZYJSMZYQY
|
|
ADEUGWOWHWNWJOSUULUUIUUKUUMUUOUUJUUFVMXTXNUUQYQPZXQXSXRUURXSXRRUUPYOYJSWO
|
|
WNVLVNVOVFWEUOVPXNXQWMYKNXTWGWHWJWKVRVSYAYBOQYCOQYEOQYFOQYHYNNYADEWNWGOOO
|
|
SUUFUUKUUGTYADEWOWHOOOSUUFUULUUITYADEWNWJOOOSUUFUUKUUJTYADEWOWKOOOSUUFUUL
|
|
UUHTYBYCYEYFVRVTWAYAWQYDWRYGJXNXTWQYDPZXQXTXNUUSWNWOWGWHWBVOWCXQXTWRYGPZX
|
|
NXTXQUUTWNWOWJWKWBVOWFUOVPWD $.
|
|
$}
|
|
|
|
$( Ordering property of addition for positive fractions. One direction of
|
|
~ ltanqg . (Contributed by Jim Kingdon, 9-Dec-2019.) $)
|
|
ltanqi $p |- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) ) $=
|
|
( cltq wbr cnq wcel wa cplq co simpl ltrelnq brel ltanqg 3expa sylan mpbid
|
|
wb ) ABDEZCFGZHSCAIJCBIJDEZSTKSAFGZBFGZHTSUARZABFFDLMUBUCTUDABCNOPQ $.
|
|
|
|
$( Ordering property of multiplication for positive fractions. One direction
|
|
of ~ ltmnqg . (Contributed by Jim Kingdon, 9-Dec-2019.) $)
|
|
ltmnqi $p |- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) ) $=
|
|
( cltq wbr cnq wcel wa cmq co simpl ltrelnq brel ltmnqg 3expa sylan mpbid
|
|
wb ) ABDEZCFGZHSCAIJCBIJDEZSTKSAFGZBFGZHTSUARZABFFDLMUBUCTUDABCNOPQ $.
|
|
|
|
$( Ordering property of addition for positive fractions. (Contributed by Jim
|
|
Kingdon, 7-Dec-2019.) $)
|
|
lt2addnq $p |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) ->
|
|
( ( A <Q B /\ C <Q D ) -> ( A +Q C ) <Q ( B +Q D ) ) ) $=
|
|
( cnq wcel wa cltq wbr cplq co ltanqg 3expa adantrr wceq addcomnqg ad2ant2r
|
|
wb ancoms ad2ant2lr breq12d bitrd adantll anbi12d ltsonq ltrelnq syl6bi
|
|
sotri ) AEFZBEFZGZCEFZDEFZGZGZABHIZCDHIZGACJKZBCJKZHIZUSBDJKZHIZGURVAHIUOUP
|
|
UTUQVBUOUPCAJKZCBJKZHIZUTUKULUPVERZUMUIUJULVFABCLMNUOVCURVDUSHUIULVCUROZUJU
|
|
MULUIVGCAPSQUJULVDUSOZUIUMULUJVHCBPSTUAUBUJUNUQVBRZUIUNUJVIULUMUJVICDBLMSUC
|
|
UDURUSVAHEUEUFUHUG $.
|
|
|
|
$( Ordering property of multiplication for positive fractions. (Contributed
|
|
by Jim Kingdon, 18-Jul-2021.) $)
|
|
lt2mulnq $p |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) ->
|
|
( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) ) $=
|
|
( cnq wcel wa cltq wbr cmq co wb ltmnqg 3expa adantrr wceq mulcomnqg ancoms
|
|
ad2ant2r ad2ant2lr breq12d bitrd adantll anbi12d ltsonq ltrelnq syl6bi
|
|
sotri ) AEFZBEFZGZCEFZDEFZGZGZABHIZCDHIZGACJKZBCJKZHIZUSBDJKZHIZGURVAHIUOUP
|
|
UTUQVBUOUPCAJKZCBJKZHIZUTUKULUPVELZUMUIUJULVFABCMNOUOVCURVDUSHUIULVCURPZUJU
|
|
MULUIVGCAQRSUJULVDUSPZUIUMULUJVHCBQRTUAUBUJUNUQVBLZUIUNUJVIULUMUJVICDBMNRUC
|
|
UDURUSVAHEUEUFUHUG $.
|
|
|
|
$( One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
|
|
(Revised by Mario Carneiro, 10-May-2013.) $)
|
|
1lt2nq $p |- 1Q <Q ( 1Q +Q 1Q ) $=
|
|
( c1o cop ceq cec cmi co cpli c1q cplq cltq wbr clti cnpi wcel wceq oveq12i
|
|
1pi 3brtr4i wb df-1nqqs 1lt2pi mulidpi ax-mp mulclpi mp2an ltmpig ordpipqqs
|
|
addclpi mp3an mpbi mp4an mpbir addpipqqs eqtri ) AABCDZAAEFZUPGFZUPBCDZHHHI
|
|
FZJUOURJKZAUPEFAUQEFLKZUPUQLKZVAAAAGFUPUQLUAAMNZUPAOQAUBUCZUPAUPAGVDVDPRUPM
|
|
NZUQMNZVCVBVASVCVCVEQQAAUDUEZVEVEVFVGVGUPUPUHUEZQUPUQAUFUIUJVCVCVFVEUTVASQQ
|
|
VHVGAAUQUPUGUKULTUSUOUOIFZURHUOHUOITTPVCVCVCVCVIUROQQQQAAAAUMUKUNR $.
|
|
|
|
${
|
|
$d A r s t $. $d B r s t $.
|
|
$( The sum of two fractions is greater than one of them. (Contributed by
|
|
NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) $)
|
|
ltaddnq $p |- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) ) $=
|
|
( vr vs vt cnq wcel wa cplq co cltq wbr c1q cmq wb addclnq wceq adantl cv
|
|
1nq 1lt2nq mp2an ltmnqg mp3an12 mpbii mulidnq mp3an23 oveq12d eqtrd simpr
|
|
3brtr3d anidms simpl ltanqg syl3anc mpbid addcomnqg w3a addassnqg caov12d
|
|
distrnqg mpbird ) AFGZBFGZHZAABIJZKLZBAIJZBVFIJZKLZVEVFABBIJZIJZVHVIKVEBV
|
|
KKLZVFVLKLZVDVMVCVDBMNJZBMMIJZNJZBVKKVDMVPKLZVOVQKLZUAMFGZVPFGZVDVRVSOTVT
|
|
VTWATTMMPUBMVPBUCUDUEBUFZVDVQVOVOIJZVKVDVTVTVQWCQTTBMMVAUGVDVOBVOBIWBWBUH
|
|
UIUKRVEVDVKFGZVCVMVNOVCVDUJZVDWDVCVDWDBBPULRVCVDUMZBVKAUNUOUPABUQVECDEABB
|
|
FIWFWEWECSZFGZDSZFGZHWGWIIJZWIWGIJQVEWGWIUQRWHWJESZFGURWKWLIJWGWIWLIJIJQV
|
|
EWGWIWLUSRUTUKVEVCVFFGVDVGVJOWFABPWEAVFBUNUOVB $.
|
|
$}
|
|
|
|
${
|
|
$d f g h x y z w v u A $. $d x y z w v u B $.
|
|
$( Ordering on positive fractions in terms of existence of sum. Definition
|
|
in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon,
|
|
23-Sep-2019.) $)
|
|
ltexnqq $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( A <Q B <-> E. x e. Q. ( A +Q x ) = B ) ) $=
|
|
( vu cnq wcel wa cltq wbr cv cplq co wceq wrex cop ceq cec cnpi cmi syl
|
|
vy vz vw vv vf vg vh df-nqqs breq1 oveq1 rexbidv imbi12d breq2 eqeq2 clti
|
|
wi eqeq1d ordpipqqs cpli wex wb mulclpi an42s ltexpi df-rex syl6bb simpll
|
|
anim12i adantr adantrr addpipqqs simplll simpllr simplrr mulcompig adantl
|
|
simpr w3a mulasspig caov12d oveq1d syl2anc simprl distrpig syl3anc eqtr4d
|
|
jca32 opeq1d eceq1d ad2ant2rl addclpi sylan 3jca mulcanenqec eqtrd 3anass
|
|
biimpri adantll anim1i adantrl opeq1 sylan9eqr 3eqtrd cxp opelxpi ecelqsi
|
|
cqs enqex syl6eleqr oveq2 rspcedv mpd ex exlimdv sylbid 2ecoptocl ltaddnq
|
|
syl5ibcom rexlimdva impbid ) BEFZCEFZGBCHIZBAJZKLZCMZAENZUAJZUBJZOPQZUCJZ
|
|
UDJZOPQZHIZYJYDKLZYMMZAENZUPBYMHIZYEYMMZAENZUPYCYGUPUAUBUCUDBCRRPEUHYJBMZ
|
|
YNYRYQYTYJBYMHUIUUAYPYSAEUUAYOYEYMYJBYDKUJUQUKULYMCMZYRYCYTYGYMCBHUMUUBYS
|
|
YFAEYMCYEUNUKULYHRFZYIRFZGZYKRFZYLRFZGZGZYNYHYLSLZYIYKSLZUOIZYQYHYIYKYLUR
|
|
UUIUULDJZRFZUUJUUMUSLZUUKMZGZDUTZYQUUIUULUUPDRNZUURUUIUUJRFZUUKRFZGZUULUU
|
|
SVAUUCUUGUUDUUFUVBUUCUUGGUUTUUDUUFGUVAYHYLVBZYIYKVBVHVCDUUJUUKVDTUUPDRVEV
|
|
FUUIUUQYQDUUIUUQYQUUIUUQGZYJUUMYIYLSLZOZPQZKLZYMMZYQUVDUVHYHUVESLZYIUUMSL
|
|
ZUSLZYIUVESLZOZPQZUUOUVEOZPQZYMUVDUUEUUNUVERFZGZGZUVHUVOMUUIUUNUVTUUPUUIU
|
|
UNGZUUEUUNUVRUUEUUHUUNVGUUIUUNVQUWAUUDUUGGZUVRUUIUWBUUNUUEUUDUUHUUGUUCUUD
|
|
VQUUFUUGVQVHVIYIYLVBZTZWGVJYHYIUUMUVEVKTUVDUVOYIUUOSLZUVMOZPQZUVQUVDUVNUW
|
|
FPUVDUVLUWEUVMUVDUVLYIUUJSLZUVKUSLZUWEUVDUVJUWHUVKUSUVDUEUFUGYHYIYLRSUUCU
|
|
UDUUHUUQVLZUUCUUDUUHUUQVMZUUEUUFUUGUUQVNZUEJZRFZUFJZRFZGUWMUWOSLZUWOUWMSL
|
|
MUVDUWMUWOVOVPUWNUWPUGJZRFVRUWQUWRSLUWMUWOUWRSLSLMUVDUWMUWOUWRVSVPVTWAUVD
|
|
UUDUUTUUNUWEUWIMUWKUVDUUCUUGUUTUWJUWLUVCWBUUIUUNUUPWCZYIUUJUUMWDWEWFWHWIU
|
|
VDUUDUUORFZUVRVRZUWGUVQMUUIUUNUXAUUPUWAUUDUWTUVRUUCUUDUUHUUNVMUUIUUTUUNUW
|
|
TUUCUUGUUTUUDUUFUVCWJUUJUUMWKWLUWDWMVJYIUUOUVEWNTWOUVDUUDUUFUUGVRZUUPGZUV
|
|
QYMMUUIUUPUXCUUNUUIUXBUUPUUDUUHUXBUUCUXBUUDUUHGUUDUUFUUGWPWQWRWSWTUUPUXBU
|
|
VQUUKUVEOZPQYMUUPUVPUXDPUUOUUKUVEXAWIYIYKYLWNXBTXCUVDYPUVIAUVGEUVDUVGRRXD
|
|
ZPXGZEUVDUUNUVRUVGUXFFZUWSUVDUUDUUGUVRUWKUWLUWCWBUVSUVFUXEFUXGUUMUVERRXEU
|
|
XEUVFPXHXFTWBUHXIYDUVGMZYPUVIVAUVDUXHYOUVHYMYDUVGYJKXJUQVPXKXLXMXNXOXOXPY
|
|
AYGYCUPYBYAYFYCAEYAYDEFGBYEHIYFYCBYDXQYECBHUMXRXSVIXT $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Ordering on positive fractions in terms of existence of sum.
|
|
(Contributed by Jim Kingdon, 30-Apr-2020.) $)
|
|
ltexnqi $p |- ( A <Q B -> E. x e. Q. ( A +Q x ) = B ) $=
|
|
( cnq wcel wa cltq cv cplq co wceq wrex ltrelnq brel ltexnqq biimpd mpcom
|
|
wbr ) BDECDEFZBCGRZBAHIJCKADLZBCDDGMNSTUAABCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( One-half of any positive fraction is a fraction. (Contributed by Jim
|
|
Kingdon, 23-Sep-2019.) $)
|
|
halfnqq $p |- ( A e. Q. -> E. x e. Q. ( x +Q x ) = A ) $=
|
|
( cnq wcel c1q cplq co crq cmq wceq 1nq mp2an ax-mp distrnqg mp3an oveq1i
|
|
addclnq oveq2i mulidnq 3eqtr3i cfv wrex recclnq recidnq oveq12i mulassnqg
|
|
eqtri mulcomnqg eqtr3i syl2anc mp2b mp3an23 3eqtr3a mulclnq mpan2 oveq12d
|
|
cv wb id eqeq1d adantl rspcedv mpd ) BCDZBEEFGZHUAZIGZVGFGZBJZAUQZVJFGZBJ
|
|
ZACUBVDBVFVFFGZIGZBEIGVHBVMEBIVEVMIGZVFIGZVEVFIGZVMEVOVEVFIVOVQVQFGZVEVEC
|
|
DZVFCDZVTVOVRJECDZWAVSKKEEQLZVSVTWBVEUCZMZWDVEVFVFNOVQEVQEFVSVQEJWBVEUDMZ
|
|
WEUEUGPVMVQIGZVMEIGZVPVMVQEVMIWERVMVEIGZVFIGZWFVPVMCDZVSVTWIWFJVTVTWJWDWD
|
|
VFVFQZLZWBWDVMVEVFUFOWHVOVFIWJVSWHVOJWLWBVMVEUHLPUIVSWJWGVMJWBVSVTVTWJWCW
|
|
CWKUJVMSUKTWETRVDVTVTVNVHJWDWDBVFVFNULBSUMVDVLVIAVGCVDVTVGCDWDBVFUNUOVJVG
|
|
JZVLVIURVDWMVKVHBWMVJVGVJVGFWMUSZWNUPUTVAVBVC $.
|
|
|
|
$( One-half of any positive fraction exists. Lemma for Proposition
|
|
9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.)
|
|
(Revised by Mario Carneiro, 10-May-2013.) $)
|
|
halfnq $p |- ( A e. Q. -> E. x ( x +Q x ) = A ) $=
|
|
( cnq wcel cv cplq co wceq wrex wex halfnqq rexex syl ) BCDAEZNFGBHZACIOA
|
|
JABKOACLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( There is no smallest positive fraction. (Contributed by Jim Kingdon,
|
|
24-Sep-2019.) $)
|
|
nsmallnqq $p |- ( A e. Q. -> E. x e. Q. x <Q A ) $=
|
|
( cnq wcel cv cplq co wceq wrex cltq wbr halfnqq ltaddnq anidms syl5ibcom
|
|
breq2 reximia syl ) BCDAEZSFGZBHZACISBJKZACIABLUAUBACSCDZSTJKZUAUBUCUDSSM
|
|
NTBSJPOQR $.
|
|
|
|
$( There is no smallest positive fraction. (Contributed by NM,
|
|
26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) $)
|
|
nsmallnq $p |- ( A e. Q. -> E. x x <Q A ) $=
|
|
( cnq wcel cv cltq wbr wrex wex nsmallnqq rexex syl ) BCDAEBFGZACHMAIABJM
|
|
ACKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( There is a number which is less than half of any positive fraction. The
|
|
case where ` A ` is one is Lemma 11.4 of [BauerTaylor], p. 50, and they
|
|
use the word "approximate half" for such a number (since there may be
|
|
constructions, for some structures other than the rationals themselves,
|
|
which rely on such an approximate half but do not require division by
|
|
two as seen at ~ halfnqq ). (Contributed by Jim Kingdon,
|
|
25-Nov-2019.) $)
|
|
subhalfnqq $p |- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A ) $=
|
|
( vy cnq wcel cv cplq co cltq wbr wa wex wrex halfnqq df-rex eximi sylibr
|
|
wceq syl wb adantr ancli sylbi anbi2i exbii exdistr simprl simpll ltaddnq
|
|
sylib syl2anc breq2 ad2antlr mpbid breq1 ad2antll mpbird jca exlimiv ) BD
|
|
EZAFZDEZVAVAGHZBIJZKZALZVDADMUTCFZDEZVGVGGHZBRZKZVBVCVGRZKZKZALZCLZVFUTVK
|
|
VMALZKZCLZVPUTVKVLADMZKZCLZVSUTVJCDMZWBCBNWCVKCLWBVJCDOVKWACVKVTVHVTVJAVG
|
|
NUAUBPUCSWAVRCVTVQVKVLADOUDUEUJVKVMCAUFQVOVFCVNVEAVNVBVDVKVBVLUGVNVDVGBIJ
|
|
ZVNVGVIIJZWDVNVHVHWEVHVJVMUHZWFVGVGUIUKVJWEWDTVHVMVIBVGIULUMUNVLVDWDTVKVB
|
|
VCVGBIUOUPUQURPUSSVDADOQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( There exists a number between any two positive fractions. Proposition
|
|
9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon,
|
|
24-Sep-2019.) $)
|
|
ltbtwnnqq $p |- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) ) $=
|
|
( vy vz cltq wbr cv wa cnq wrex wcel cplq wceq ltrelnq brel simpld sylan2
|
|
co adantr ltexnqi wex nsmallnq ltaddnq ancoms ltanqi breq2 adantl addclnq
|
|
wb mpbid breq1 anbi12d rspcedv mp2and 3impa 3coml 3expia exlimdv impancom
|
|
syl5 rexlimdva sylc ltsonq sotri rexlimivw impbii ) BCFGZBAHZFGZVICFGZIZA
|
|
JKZVHBJLZBDHZMSZCNZDJKVMVHVNCJLBCJJFOPQDBCUAVNVQVMDJVNVQVOJLZVMVREHZVOFGZ
|
|
EUBVNVQIZVMEVOUCWAVTVMEVNVQVTVMVTVNVQVMVTVNVQVMVTVNIZVQIZBBVSMSZFGZWDCFGZ
|
|
VMWBWEVQVNVTWEVTVNVSJLZWEVTWGVRVSVOJJFOPQZBVSUDRUETWCWDVPFGZWFWBWIVQVSVOB
|
|
UFTVQWIWFUJWBVPCWDFUGUHUKWCVLWEWFIZAWDJWBWDJLZVQVNVTWKVTVNWGWKWHBVSUIRUET
|
|
VIWDNZVLWJUJWCWLVJWEVKWFVIWDBFUGVIWDCFULUMUHUNUOUPUQURUSVAUTVBVCVLVHAJBVI
|
|
CFJVDOVEVFVG $.
|
|
|
|
$( There exists a number between any two positive fractions. Proposition
|
|
9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.)
|
|
(Revised by Mario Carneiro, 10-May-2013.) $)
|
|
ltbtwnnq $p |- ( A <Q B <-> E. x ( A <Q x /\ x <Q B ) ) $=
|
|
( cv cltq wbr wa cnq wrex wcel wex df-rex ltbtwnnqq ltrelnq simprd adantr
|
|
brel pm4.71ri exbii 3bitr4i ) BADZEFZUACEFZGZAHIUAHJZUDGZAKBCEFUDAKUDAHLA
|
|
BCMUDUFAUDUEUBUEUCUBBHJUEBUAHHENQOPRST $.
|
|
$}
|
|
|
|
${
|
|
$d A w x z $.
|
|
$( For any fraction, there is an integer that is greater than it. This is
|
|
also known as the "archimedean property". (Contributed by Jim Kingdon,
|
|
1-Dec-2019.) $)
|
|
archnqq $p |- ( A e. Q. -> E. x e. N. A <Q [ <. x , 1o >. ] ~Q ) $=
|
|
( vz vw wcel cnpi wa ceq wceq c1o cltq wbr co adantr clti comu 1onn sylan
|
|
com c0 cnq cv cop cec wex wrex nqpi cpli 1pi addclpi mpan2 cmi pinn nnacl
|
|
coa sylancl nnm1 syl wss elni2 word nnord ordgt0ge1 biimpa sylbi nnaword1
|
|
adantl wb simprbi sseldd nnmword mp3anl1 syl21anc mpbid eqsstr3d 0lt1o wi
|
|
nna0 nnaordi mpi eqeltrrd mulclpi ltpiord syldan mulpiord addpiord oveq1d
|
|
eqtrd eleq2d bitrd mpbird mulcompig breq2d jctir ordpipqqs mulidpi breq1d
|
|
mpan mpdan breq1 opeq1 eceq1d rspcev syl2anc exlimivv ) BUAECUBZFEZDUBZFE
|
|
ZGZBXFXHUCHUDZIZGZDUECUEBAUBZJUCZHUDZKLZAFUFZCDBUGXMXRCDXMXFJUHMZFEZBXSJU
|
|
CZHUDZKLZXRXJXTXLXGXTXIXGJFEZXTUIXFJUJUKZNZNXMYCXKYBKLZXJYGXLXJYGXFXHXSUL
|
|
MZOLZXJXFXSXHULMZOLZYIXJYKXFXFJUOMZXHPMZEZXJYLYMXFXJYLYLJPMZYMXJYLSEZYOYL
|
|
IXGYPXIXGXFSEZJSEZYPXFUMZQXFJUNUPNZYLUQURXJJXHUSZYOYMUSZXIUUAXGXIXHSEZTXH
|
|
EZGUUAXHUTUUCXHVAZUUDUUAXHVBUUEUUDUUAXHVCVDRVEVGXJUUCYPTYLEZUUAUUBVHZXIUU
|
|
CXGXHUMVGYTXGUUFXIXGXFYLTXGYQYRXFYLUSYSQXFJVFUPXGYQTXFEXFUTVIVJNYRUUCYPUU
|
|
FUUGQJXHYLVKVLVMVNVOXGXFYLEZXIXGYQUUHYSYQXFTUOMZXFYLXFVRYQTJEZUUIYLEZVPYR
|
|
YQUUJUUKVQQTJXFVSWRVTWAURNVJXJYKXFYJEZYNXGXIYJFEZYKUULVHXGXTXIUUMYEXSXHWB
|
|
RXFYJWCWDXJYJYMXFXJYJXSXHPMZYMXGXTXIYJUUNIYEXSXHWERXJXSYLXHPXGXSYLIZXIXGY
|
|
DUUOUIXFJWFUKNWGWHWIWJWKXJYJYHXFOXGXTXIYJYHIYEXSXHWLRWMVNXJYGXFJULMZYHOLZ
|
|
YIXJXTYDGYGUUQVHXJXTYDYFUIWNXFXHXSJWOWSXJUUPXFYHOXGUUPXFIXIXFWPNWQWJWKNXL
|
|
YCYGVHXJBXKYBKWTVGWKXQYCAXSFXNXSIZXPYBBKUURXOYAHXNXSJXAXBWMXCXDXEUR $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( A version of the Archimedean property. This variation is "stronger"
|
|
than ~ archnqq in the sense that we provide an integer which is larger
|
|
than a given rational ` A ` even after being multiplied by a second
|
|
rational ` B ` . (Contributed by Jim Kingdon, 30-Nov-2019.) $)
|
|
prarloclemarch $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) ) $=
|
|
( cnq wcel wa cmq c1o ceq cltq wbr cnpi wrex mulclnq syl2anc syl3anc wceq
|
|
co syl mulcomnqg crq cfv cv cop cec recclnq sylan2 archnqq simpll cxp cqs
|
|
wb 1pi opelxpi mpan2 enqex ecelqsi df-nqqs syl6eleqr adantl simplr ltmnqg
|
|
c1q mulassnqg recidnq oveq2d mulidnq eqtrd breq12d bitrd biimprd reximdva
|
|
3eqtrd mpd ) BDEZCDEZFZBCUAUBZGRZAUCZHUDZIUEZJKZALMZBWBCGRZJKZALMVQVSDEZW
|
|
DVPVOVRDEZWGCUFZBVRNUGAVSUHSVQWCWFALVQVTLEZFZWFWCWKWFVRBGRZVRWEGRZJKZWCWK
|
|
VOWEDEZWHWFWNULVOVPWJUIZWKWBDEZVPWOWJWQVQWJWBLLUJZIUKZDWJWAWREZWBWSEWJHLE
|
|
WTUMVTHLLUNUOWRWAIUPUQSURUSUTZVOVPWJVAZWBCNOZWKVPWHXBWISZBWEVRVBPWKWLVSWM
|
|
WBJWKWHVOWLVSQXDWPVRBTOWKWMWEVRGRZWBWKWHWOWMXEQXDXCVRWETOWKXEWBCVRGRZGRZW
|
|
BVCGRZWBWKWQVPWHXEXGQXAXBXDWBCVRVDPWKVPXGXHQXBVPXFVCWBGCVEVFSWKWQXHWBQXAW
|
|
BVGSVMVHVIVJVKVLVN $.
|
|
$}
|
|
|
|
${
|
|
$d A x z $. $d B x z $. $d C x z $.
|
|
$( Like ~ prarloclemarch but the integer must be at least two, and there is
|
|
also ` B ` added to the right hand side. These details follow
|
|
straightforwardly but are chosen to be helpful in the proof of
|
|
~ prarloc . (Contributed by Jim Kingdon, 25-Nov-2019.) $)
|
|
prarloclemarch2 $p |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) ->
|
|
E. x e. N. ( 1o <N x /\
|
|
A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) ) $=
|
|
( cnq wcel c1o ceq cmq co cltq wbr clti wa cnpi c2o 1pi syl wceq c0 vz cv
|
|
w3a cop cec cplq wrex prarloclemarch 3adant2 cpli coa com pinn csuc elexi
|
|
sucid df-2o eleqtrri wss 2onn nnaword2 mpan sseld o1p1e2 addpiord addclpi
|
|
mpi mp2an eqeltrri mpan2 eleqtrrd ltpiord mpbird adantl adantrr cmi 0lt1o
|
|
wb nna0 con0 wi 1on onsuci ontr1 ax-mp nnaordi eqeltrrd mulidpi mulcompig
|
|
mpdan eqtr3d 3brtr4d ordpipqqs mpanl2 mpanr2 cxp cqs opelxpi ecelqsi 3syl
|
|
df-nqqs syl6eleqr ltmnqg syl3an1 syl3an2 3anidm12 ancoms mulcomnqg sylan2
|
|
enqex mpbid 3brtr3d 3ad2antl3 ltsonq ltrelnq sotri ex mpd mulclnq ltaddnq
|
|
sylan simpl2 syl2anc breq2d simpr breq2 opeq1 eceq1d oveq1d oveq2d rspcev
|
|
addcomnqg anbi12d mp2and rexlimddv ) BEFZCEFZDEFZUCZBUAUBZGUDZHUEZDIJZKLZ
|
|
GAUBZMLZBCUUEGUDZHUEZDIJZUFJZKLZNZAOUGZUAOYPYRUUDUAOUGYQUABDUHUIYSYTOFZUU
|
|
DNZNZGYTPUJJZMLZBCUUQGUDZHUEZDIJZUFJZKLZUUMYSUUNUURUUDUUNUURYSUUNUURGUUQF
|
|
ZUUNGYTPUKJZUUQUUNYTULFZGUVEFZYTUMZUVFGPFUVGGGUNZPGGOQUOUPZUQURUVFPUVEGPU
|
|
LFZUVFPUVEUSUTPYTVAVBVCVGRUUNPOFZUUQUVESGGUKJZPOVDGGUJJZUVMOGOFZUVOUVNUVM
|
|
SQQGGVEVHUVOUVOUVNOFQQGGVFVHVIVIZYTPVEVJZVKUUNUUQOFZUURUVDVRZUUNUVLUVRUVP
|
|
YTPVFVJZUVOUVRUVSQGUUQVLVBRVMVNVOUUPBUVACUFJZKLZUVCUUPBUVAKLZUVAUWAKLZUWB
|
|
UUPUUCUVAKLZUWCYSUUNUWEUUDYRYPUUNUWEYQYRUUNNZDUUBIJZDUUTIJZUUCUVAKUWFUUBU
|
|
UTKLZUWGUWHKLZUUNUWIYRUUNUWIYTGVPJZGUUQVPJZMLZUUNYTUUQUWKUWLMUUNYTUUQMLZY
|
|
TUUQFZUUNYTUVEUUQUUNUVFYTUVEFUVHUVFYTTUKJZYTUVEYTVSUVFTPFZUWPUVEFZTUVIPTG
|
|
FZGUVIFZTUVIFZVQUVJUVIVTFUWSUWTNUXAWAGWBWCTGUVIWDWEVHUQURUVKUVFUWQUWRWAUT
|
|
TPYTWFVBVGWGRUVQVKUUNUVRUWNUWOVRUVTYTUUQVLWJVMYTWHUUNUUQGVPJZUWLUUQUUNUVR
|
|
UXBUWLSZUVTUVRUVOUXCQUUQGWIVJRUUNUVRUXBUUQSUVTUUQWHRWKWLUUNUVRUWIUWMVRZUV
|
|
TUUNUVRUVOUXDQUUNUVOUVRUVONUXDQYTGUUQGWMWNWOWJVMVNUUNYRUWIUWJVRZUUNYRUXEU
|
|
UNUUNUUTEFZYRUXEUUNUUTOOWPZHWQZEUUNUVRUUSUXGFZUUTUXHFUVTUVRUVOUXIQUUQGOOW
|
|
RVJUXGUUSHXJWSWTXAXBZUUNUUBEFZUXFYRUXEUUNUUBUXHEUUNUUAUXGFZUUBUXHFUUNUVOU
|
|
XLQYTGOOWRVJUXGUUAHXJWSRXAXBZUUBUUTDXCXDXEXFXGXKUUNYRUXKUWGUUCSUXMDUUBXHX
|
|
IUUNYRUXFUWHUVASUXJDUUTXHXIXLXMVOUUOUWEUWCWAZYSUUDUXNUUNUUDUWEUWCBUUCUVAK
|
|
EXNXOXPXQVNVNXRYSUUNUWDUUDYSUUNNZUVAEFZYQUWDYRYPUUNUXPYQUUNYRUXPUUNUXFYRU
|
|
XPUXJUUTDXSYAXGXMZYPYQYRUUNYBZUVACXTYCVOBUVAUWAKEXNXOXPYCYSUUNUWBUVCVRUUD
|
|
UXOUWAUVBBKUXOUXPYQUWAUVBSUXQUXRUVACYLYCYDVOXKYSUUNUURUVCNZUUMWAZUUDUXOUU
|
|
NUVRUXTYSUUNYEUVTUVRUXSUUMUULUXSAUUQOUUEUUQSZUUFUURUUKUVCUUEUUQGMYFUYAUUJ
|
|
UVBBKUYAUUIUVACUFUYAUUHUUTDIUYAUUGUUSHUUEUUQGYGYHYIYJYDYMYKXQWTVOYNYO $.
|
|
$}
|
|
|
|
$( Ordering property of reciprocal for positive fractions. For a simplified
|
|
version of the forward implication, see ~ ltrnqi . (Contributed by Jim
|
|
Kingdon, 29-Dec-2019.) $)
|
|
ltrnqg $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) ) $=
|
|
( cnq wcel cltq wbr crq cfv cmq co recclnq mulclnq wceq mulcomnqg mulassnqg
|
|
syl2anc syl3anc c1q recidnq oveq2d wa wb syl2an ltmnqg mpd3an3 simpl adantr
|
|
adantl 3eqtr2d mulidnq sylan9eq eqtrd simpr sylan9eqr 3eqtrd breq12d bitrd
|
|
syl ) ACDZBCDZUAZABEFZAGHZBGHZIJZAIJZVEBIJZEFZVDVCEFUSUTVECDZVBVHUBUSVCCDZV
|
|
DCDZVIUTAKZBKZVCVDLUCZABVEUDUEVAVFVDVGVCEVAVFVDAVCIJZIJZVDVAVFAVEIJZVOVDIJZ
|
|
VPVAVIUSVFVQMVNUSUTUFZVEANPVAUSVJVKVRVQMVSUSVJUTVLUGZUTVKUSVMUHZAVCVDOQVAVO
|
|
CDZVKVRVPMVAUSVJWBVSVTAVCLPWAVOVDNPUIUSUTVPVDRIJZVDUSVORVDIASTUTVKWCVDMVMVD
|
|
UJURUKULVAVGVCVDBIJZIJZVCBVDIJZIJZVCVAVJVKUTVGWEMVTWAUSUTUMZVCVDBOQVAWDWFVC
|
|
IVAVKUTWDWFMWAWHVDBNPTUTUSWGVCRIJZVCUTWFRVCIBSTUSVJWIVCMVLVCUJURUNUOUPUQ $.
|
|
|
|
$( Ordering property of reciprocal for positive fractions. For the converse,
|
|
see ~ ltrnqg . (Contributed by Jim Kingdon, 24-Sep-2019.) $)
|
|
ltrnqi $p |- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) ) $=
|
|
( cltq wbr crq cfv cnq wcel wa wb ltrelnq brel ltrnqg syl ibi ) ABCDZBEFAEF
|
|
CDZPAGHBGHIPQJABGGCKLABMNO $.
|
|
|
|
$( The canonical embedding of positive integers into positive fractions.
|
|
(Contributed by Jim Kingdon, 26-Apr-2020.) $)
|
|
nnnq $p |- ( A e. N. -> [ <. A , 1o >. ] ~Q e. Q. ) $=
|
|
( cnpi wcel c1o cop ceq cec cxp cqs cnq 1pi opelxpi mpan2 enqex ecelqsi syl
|
|
df-nqqs syl6eleqr ) ABCZADEZFGZBBHZFIZJSTUBCZUAUCCSDBCUDKADBBLMUBTFNOPQR $.
|
|
|
|
$( Ordering of positive integers via ` <N ` or ` <Q ` is equivalent.
|
|
(Contributed by Jim Kingdon, 3-Oct-2020.) $)
|
|
ltnnnq $p |- ( ( A e. N. /\ B e. N. ) -> ( A <N B <->
|
|
[ <. A , 1o >. ] ~Q <Q [ <. B , 1o >. ] ~Q ) ) $=
|
|
( cnpi wcel wa c1o cop ceq cec cltq wbr cmi co clti wb 1pi wceq mulidpi syl
|
|
simpl a1i simpr ordpipqqs syl22anc mulcompig sylancr eqtrd breq12d bitr2d )
|
|
ACDZBCDZEZAFGHIBFGHIJKZAFLMZFBLMZNKZABNKULUJFCDZUKUQUMUPOUJUKTZUQULPUAZUJUK
|
|
UBZUSAFBFUCUDULUNAUOBNULUJUNAQURARSULUOBFLMZBULUQUKUOVAQPUTFBUEUFULUKVABQUT
|
|
BRSUGUHUI $.
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Define equivalence relation for non-negative fractions. This is a
|
|
"temporary" set used in the construction of complex numbers, and is
|
|
intended to be used only by the construction. (Contributed by Jim
|
|
Kingdon, 2-Nov-2019.) $)
|
|
df-enq0 $a |- ~Q0 = { <. x , y >. | ( ( x e. ( _om X. N. ) /\
|
|
y e. ( _om X. N. ) ) /\ E. z E. w E. v E. u
|
|
( ( x = <. z , w >. /\ y = <. v , u >. ) /\
|
|
( z .o u ) = ( w .o v ) ) ) } $.
|
|
|
|
$( Define class of non-negative fractions. This is a "temporary" set used
|
|
in the construction of complex numbers, and is intended to be used only
|
|
by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) $)
|
|
df-nq0 $a |- Q0. = ( ( _om X. N. ) /. ~Q0 ) $.
|
|
|
|
$( Define non-negative fraction constant 0. This is a "temporary" set used
|
|
in the construction of complex numbers, and is intended to be used only
|
|
by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.) $)
|
|
df-0nq0 $a |- 0Q0 = [ <. (/) , 1o >. ] ~Q0 $.
|
|
|
|
$( Define addition on non-negative fractions. This is a "temporary" set
|
|
used in the construction of complex numbers, and is intended to be used
|
|
only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) $)
|
|
df-plq0 $a |- +Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0. /\ y e. Q0. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
|
|
z = [ <. ( ( w .o f ) +o ( v .o u ) ) ,
|
|
( v .o f ) >. ] ~Q0 ) ) } $.
|
|
|
|
$( Define multiplication on non-negative fractions. This is a "temporary"
|
|
set used in the construction of complex numbers, and is intended to be
|
|
used only by the construction. (Contributed by Jim Kingdon,
|
|
2-Nov-2019.) $)
|
|
df-mq0 $a |- .Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0. /\ y e. Q0. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
|
|
z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Multiplication on non-negative fractions. This definition is similar to
|
|
~ df-mq0 but expands ` Q0. ` (Contributed by Jim Kingdon,
|
|
22-Nov-2019.) $)
|
|
dfmq0qs $p |- .Q0 = { <. <. x , y >. , z >. |
|
|
( ( x e. ( ( _om X. N. ) /. ~Q0 ) /\ y e. ( ( _om X. N. ) /. ~Q0 ) ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
|
|
z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) } $=
|
|
( cv cnq0 wcel wa cop ceq0 cec wceq comu co wex coprab df-nq0 com cxp cqs
|
|
cmq0 cnpi df-mq0 eleq2i anbi12i anbi1i oprabbii eqtri ) UDAHZIJZBHZIJZKZU
|
|
LDHZEHZLMNOUNFHZGHZLMNOKCHUQUSPQURUTPQLMNOKGRFRERDRZKZABCSULUAUEUBMUCZJZU
|
|
NVCJZKZVAKZABCSABCDEFGUFVBVGABCUPVFVAUMVDUOVEIVCULTUGIVCUNTUGUHUIUJUK $.
|
|
|
|
$( Addition on non-negative fractions. This definition is similar to
|
|
~ df-plq0 but expands ` Q0. ` (Contributed by Jim Kingdon,
|
|
24-Nov-2019.) $)
|
|
dfplq0qs $p |- +Q0 = { <. <. x , y >. , z >. |
|
|
( ( x e. ( ( _om X. N. ) /. ~Q0 ) /\ y e. ( ( _om X. N. ) /. ~Q0 ) ) /\
|
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E. w E. v E. u E. f (
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|
( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\
|
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z = [ <. ( ( w .o f ) +o ( v .o u ) ) ,
|
|
( v .o f ) >. ] ~Q0 ) ) } $=
|
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( cv cnq0 wcel wa cop ceq0 cec wceq comu co wex coprab df-nq0 coa com cxp
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cplq0 cnpi cqs df-plq0 eleq2i anbi12i anbi1i oprabbii eqtri ) UDAHZIJZBHZ
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IJZKZUMDHZEHZLMNOUOFHZGHZLMNOKCHURVAPQUSUTPQUAQUSVAPQLMNOKGRFRERDRZKZABCS
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TUHUIUJUKUL $.
|
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$}
|
|
|
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${
|
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$d u v w x y z $.
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$( Equivalence on positive fractions in terms of equivalence on
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non-negative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.) $)
|
|
enq0enq $p |- ~Q = ( ~Q0 i^i ( ( N. X. N. ) X. ( N. X. N. ) ) ) $=
|
|
( vx vy vz vw vv vu cnpi cv com wcel wa wceq co copab bitr4i anbi1i anass
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wex 3bitr3i 19.42vv ceq0 cxp cin cop ceq df-enq0 df-xp ineq12i inopab cmi
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comu an32 an4 wss pinn ssriv xpss1 ax-mp sseli anbi12i bitri eleq1 opelxp
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pm4.71ri syl6bb bi2anan9 pm5.32i wb mulpiord eqeqan12d an42s anbi2i ancom
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$}
|
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|
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${
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$d f a b c d u v w x y z $. $d g a b c d u v w x y z $.
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$( The equivalence relation for non-negative fractions is symmetric. Lemma
|
|
for ~ enq0er . (Contributed by Jim Kingdon, 14-Nov-2019.) $)
|
|
enq0sym $p |- ( f ~Q0 g -> g ~Q0 f ) $=
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( vc vd va vb vz vw vv vu cv com wcel wa wceq comu co wex anbi1d anbi12d
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vx vy ceq0 wbr cnpi cxp cop vex eleq1 eqeq1 4exbidv anbi2d df-enq0 biimpi
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$}
|
|
|
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${
|
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$d f u v w x y z $.
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|
$( The equivalence relation for non-negative fractions is reflexive. Lemma
|
|
for ~ enq0er . (Contributed by Jim Kingdon, 14-Nov-2019.) $)
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|
enq0ref $p |- ( f e. ( _om X. N. ) <-> f ~Q0 f ) $=
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( vz vw vv vu vx vy cv com cnpi wcel ceq0 wceq comu wex vex anbi2d anbi1d
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wa co cxp wbr elxpi ee4anv sylanbrc eqtr2 opth sylib oveq1 oveq2 sylan9eq
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cop equcoms syl ancli ad2ant2r pinn nnmcom sylan2 eqeq2d ancoms ad2ant2lr
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VR $.
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$}
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${
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$d a b c d e f s t u v w x y z $. $d a b c d e g s t u v w x y z $.
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$d a b c d e h s t u v w x y z $.
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|
$( The equivalence relation for non-negative fractions is transitive.
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|
Lemma for ~ enq0er . (Contributed by Jim Kingdon, 14-Nov-2019.) $)
|
|
enq0tr $p |- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h ) $=
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( vz vw vs vt vv vu va vb cv wa com wcel wceq comu co wex c0 vx vy vc wbr
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vd ve ceq0 cnpi cxp cop w3a vex eleq1 anbi1d eqeq1 4exbidv anbi12d anbi2d
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bitr4i sylibr ee8anv 19.42vvvv 2exbii bitri 3simpb adantr simplll simprlr
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$}
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${
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$d f g h x y z w v u $.
|
|
$( The equivalence relation for non-negative fractions is an equivalence
|
|
relation. (Contributed by Jim Kingdon, 12-Nov-2019.) $)
|
|
enq0er $p |- ~Q0 Er ( _om X. N. ) $=
|
|
( vf vg vh vx vy vz vw vv vu ceq0 wtru cv wcel wa cop wceq comu wex wbr
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co com cnpi cxp wer df-enq0 relopabi a1i enq0sym adantl enq0tr wb enq0ref
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wrel iserd trud ) UAUBUCZJUDKABCUPJJUMKDLZUPMELZUPMNUQFLZGLZOPURHLZILZOPN
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|
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$}
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${
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$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
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$d x y z w v u D $.
|
|
$( Equivalence relation for non-negative fractions in terms of natural
|
|
numbers. (Contributed by NM, 27-Aug-1995.) $)
|
|
enq0breq $p |- ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. ) ) ->
|
|
( <. A , B >. ~Q0 <. C , D >. <-> ( A .o D ) = ( B .o C ) ) ) $=
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|
( vz vw vv vu com wcel cnpi wa cop cv wceq comu co wex cvv anbi1d vx ceq0
|
|
vy cxp wbr wb oveq12 eqeqan12d an42s copsex4g opexg eleq1 4exbidv anbi12d
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|
anbi2d eqeq1 df-enq0 brabg syl2an opelxpi anim12i biantrurd 3bitr4d ) AIJ
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|
|
PQZOZLZHRGRFRERZLZVLADPQZBCPQZOZLVGVJUBUEZWJVFWFWJVLWDWJEFGHABCDIKVMAOZVR
|
|
DOZVNBOZVQCOZWDWJUFWLWMLWNWOLWBWHWCWIVMAVRDPUGVNBVQCPUGUHUIUJUOVDVGSJVJSJ
|
|
WKWGUFVEABIKUKCDIKUKUANZVHJZUCNZVHJZLZWPVOOZWRVSOZLZWDLZHRGRFRERZLVIWSLZV
|
|
PXBLZWDLZHRGRFRERZLWGUAUCVGVJSSUBWPVGOZWTXFXEXIXJWQVIWSWPVGVHULTXJXDXHEFG
|
|
HXJXCXGWDXJXAVPXBWPVGVOUPTTUMUNWRVJOZXFVLXIWFXKWSVKVIWRVJVHULUOXKXHWEEFGH
|
|
XKXGWAWDXKXBVTVPWRVJVSUPUOTUMUNUAUCEFGHUQURUSVFVLWJVDVIVEVKABIKUTCDIKUTVA
|
|
VBVC $.
|
|
$}
|
|
|
|
$( Equivalence class equality of non-negative fractions in terms of natural
|
|
numbers. (Contributed by Jim Kingdon, 24-Nov-2019.) $)
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|
enq0eceq $p |- ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. ) ) ->
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|
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adantr erth enq0breq bitr3d ) AEFBGFHZCEFDGFHZHZABIZCDIZJKUHJLUIJLMADNOBCNO
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MUGUHUIJEGPZUJJQUGRSUEUHUJFUFABEGTUAUBABCDUCUD $.
|
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|
${
|
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$d u v w x y z $.
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$( A non-negative fraction is a positive fraction if its numerator and
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denominator are positive integers. (Contributed by Jim Kingdon,
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10-Nov-2019.) $)
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nqnq0pi $p |- ( ( A e. N. /\ B e. N. ) ->
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[ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q ) $=
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( vy vx vz vw vu vv cnpi wcel wa cop ceq0 wceq cv wex com wb c0 wn opelxp
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|
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SNXIUEXRXOXJXLXM $.
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|
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Jim Kingdon, 18-Nov-2019.) $)
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enq0ex $p |- ~Q0 e. _V $=
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wa co niex copab df-enq0 opabssxp eqsstri ssexi ) GHIJZUGJZUGUGHIKUALZUIL
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UPABUGUGUDUEUF $.
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$}
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18-Nov-2019.) $)
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nq0ex $p |- Q0. e. _V $=
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Kingdon, 18-Nov-2019.) $)
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nqnq0 $p |- Q. C_ Q0. $=
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$}
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$( Lemma showing compatibility of addition on non-negative fractions.
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(Contributed by Jim Kingdon, 23-Nov-2019.) $)
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eqtr4d wb cmi ad2ant2r enq0breq syl22anc mpbird ex ) AKLZBMLZNZCKLZDMLZNZ
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WMWHWOUVHWGWNWHWONDFVRPXMMDFVLDFVMVNVOVOXHXIXLXMVTWAVKWBWC $.
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$( Lemma showing compatibility of multiplication on non-negative fractions.
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(Contributed by Jim Kingdon, 20-Nov-2019.) $)
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mulcmpblnq0 $p |- ( ( ( ( A e. _om /\ B e. N. ) /\ ( C e. _om /\ D e. N. )
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by Jim Kingdon, 29-Nov-2019.) $)
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mulcanenq0ec $p |- ( ( A e. N. /\ B e. _om /\ C e. N. ) ->
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~ addnnnq0 and ~ mulnnnq0 . (Contributed by Jim Kingdon,
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23-Nov-2019.) $)
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nnnq0lem1 $p |- ( (
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|
|
(Contributed by Jim Kingdon, 23-Nov-2019.) $)
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impd simpr 2albii sylibr eqeq1 4exbidv mo4 ) FUCUTUDZKUEZLGWILMZFBNZCNZOZ
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co c0 cnq0 wex nq0nn df-0nq0 oveq12 peano1 1pi addnnnq0 mpanr12 sylan9eqr
|
|
mpan2 pinn nnm0 oveq2d nnm1 oveq1d nna0 eqtrd adantl eceq1d eqeq2d eqtr4d
|
|
opeq12d biimpar exlimivv ) AUADBEZFDZCEZGDZHZAVFVHIZJKZLZHZCUBBUBAMNSZALZ
|
|
BCAUCVNVPBCVNVOVFOPSZVHTPSZQSZVHOPSZIZJKZAVMVJVOVLTOIJKZNSZWBVMMWCLVOWDLU
|
|
DAVLMWCNUEUKVJTFDOGDWDWBLUFUGVFVHTOUHUIUJVJAWBLVMVJWBVLAVJWAVKJVJVSVFVTVH
|
|
VIVGVSVQTQSZVFVIVHFDZVSWELVHULZWFVRTVQQVHUMUNRVGWEVFTQSVFVGVQVFTQVFUOUPVF
|
|
UQURUJVIVTVHLZVGVIWFWHWGVHUORUSVCUTVAVDVBVER $.
|
|
$}
|
|
|
|
$( Addition of non-negative fractions with a common denominator. You can add
|
|
two fractions with the same denominator by adding their numerators and
|
|
keeping the same denominator. (Contributed by Jim Kingdon,
|
|
1-Dec-2019.) $)
|
|
nnanq0 $p |- ( ( N e. _om /\ M e. _om /\ A e. N. ) ->
|
|
[ <. ( N +o M ) , A >. ] ~Q0 =
|
|
( [ <. N , A >. ] ~Q0 +Q0 [ <. M , A >. ] ~Q0 ) ) $=
|
|
( com wcel cnpi w3a cop ceq0 cec cplq0 co comu wceq addnnnq0 3impdir nnmcom
|
|
coa pinn sylan2 3adant2 oveq1d nndi 3coml syl3an3 eqtr4d opeq1d simp3 nnacl
|
|
eceq1d 3adant3 mulcanenq0ec syl3anc 3eqtrrd ) CDEZBDEZAFEZGZCAHIJBAHIJKLZCA
|
|
MLZABMLZRLZAAMLZHZIJZACBRLZMLZVCHZIJZVFAHIJZUOUQUPUSVENCABAOPURVDVHIURVBVGV
|
|
CURVBACMLZVARLZVGURUTVKVARUOUQUTVKNZUPUQUOADEZVMASZCAQTUAUBUQUOUPVNVGVLNZVO
|
|
VNUOUPVPACBUCUDUEUFUGUJURUQVFDEZUQVIVJNUOUPUQUHZUOUPVQUQCBUIUKVRAVFAULUMUN
|
|
$.
|
|
|
|
${
|
|
$d A u v w x y z $. $d B u v w x y z $. $d C u v w x y z $.
|
|
$d f g h u v w x y z $.
|
|
$( Multiplication of non-negative fractions is distributive. (Contributed
|
|
by Jim Kingdon, 27-Nov-2019.) $)
|
|
distrnq0 $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
|
|
( A .Q0 ( B +Q0 C ) ) = ( ( A .Q0 B ) +Q0 ( A .Q0 C ) ) ) $=
|
|
( cnq0 wcel cplq0 co cmq0 wceq wa cop ceq0 cec com cnpi comu nnmcl adantl
|
|
cv syl vz vw vv vu vx vy vf vg vh df-nq0 oveq1 oveq2d oveq2 oveq1d imbi2d
|
|
wi eqeq12d oveq12d w3a coa an42 anbi2i 3anass 3bitr4i sylan ancoms sylan2
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|
pinn syl3an simp1r simp1l 3ad2ant2 3ad2ant3 nnacl syl2anc syl3an1 syl3anc
|
|
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|
|
nndi eqtr3d 3eqtr3d sylbir cmi mulpiord ad2ant2lr 3adant1 eqeltrrd opeq12
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|
mulclpi eceq1d addnnnq0 syl2an an42s ad2ant2l eleq1d mpbid mulnnnq0 simpl
|
|
wb jca anim12i an12 bitr4i sylib an4s mulcanenq0ec eqtr4d eqtrd oveqan12d
|
|
3impb 3impdi 3eqtr4d 3expib ecoptocl com12 2ecoptocl 3impib ) ADEZBDEZCDE
|
|
ZABCFGZHGZABHGZACHGZFGZIZYHYIJYGYOYGAUASZUBSZKLMZUCSZUDSZKLMZFGZHGZAYRHGZ
|
|
AUUAHGZFGZIZUPYGABUUAFGZHGZYLUUEFGZIZUPYGYOUPUAUBUCUDBCNOLDUJYRBIZUUGUUKY
|
|
GUULUUCUUIUUFUUJUULUUBUUHAHYRBUUAFUKULUULUUDYLUUEFYRBAHUMUNUQUOUUACIZUUKY
|
|
OYGUUMUUIYKUUJYNUUMUUHYJAHUUACBFUMULUUMUUEYMYLFUUACAHUMULUQUOYGYPNEZYQOEZ
|
|
JZYSNEZYTOEZJZJZUUGUUTUESZUFSZKLMZUUBHGZUVCYRHGZUVCUUAHGZFGZIZUPUUTUUGUPU
|
|
EUFANOLDUJUVCAIZUVHUUGUUTUVIUVDUUCUVGUUFUVCAUUBHUKUVIUVEUUDUVFUUEFUVCAYRH
|
|
UKUVCAUUAHUKURUQUOUVANEZUVBOEZJZUUPUUSUVHUVLUUPUUSUSZUVBUVAYPYTPGZYQYSPGZ
|
|
UTGZPGZPGZUVBUVBYQYTPGZPGZPGZKZLMZUVAYPPGZUVBYTPGZPGZUVBYQPGZUVAYSPGZPGZU
|
|
TGZUWGUWEPGZKZLMZUVDUVGUVMUVRUWJIZUWAUWKIZUWCUWMIUVMUVLUUNUURJZUUOUUQJZUS
|
|
ZUWNUVLUWPUWQJZJUVLUUTJZUWRUVMUWSUUTUVLUUNUURUUOUUQVAVBUVLUWPUWQVCUVLUUPU
|
|
USVCVDZUWRUVBUVAPGZUVPPGZUXBUVNPGZUXBUVOPGZUTGZUVRUWJUVLUXBNEZUWPUVNNEZUW
|
|
QUVONEZUXCUXFIUVKUVJUXGUVKUVBNEZUVJUXGUVBVHZUVBUVAQVEVFUURUUNYTNEZUXHYTVH
|
|
ZYPYTQVGZUUOYQNEZUUQUXIYQVHZYQYSQVEZUXBUVNUVOWHVIUWRUVKUVJUVPNEZUXCUVRIZU
|
|
VJUVKUWPUWQVJZUVJUVKUWPUWQVKUWRUXHUXIUXRUWPUVLUXHUWQUXNVLUWQUVLUXIUWPUXQV
|
|
MUVNUVOVNZVOUVKUXJUVJUXRUXSUXKUVBUVAUVPVRVPVQUWRUXDUWFUXEUWIUTUVLUWPUXDUW
|
|
FIUWQUVLUWPJZUVAUVBPGZUVNPGZUXDUWFUVLUYDUXDIUWPUVLUYCUXBUVNPUVKUVJUXJUYCU
|
|
XBIUXKUVAUVBVSVGUNVTUYBUGUHUIUVAUVBYPYTNPUVJUVKUWPWAUVKUXJUVJUWPUXKWBUVLU
|
|
UNUURWCUGSZNEZUHSZNEZJZUYEUYGPGZUYGUYEPGIZUYBUYEUYGVSZRUYFUYHUISZNEUSZUYJ
|
|
UYMPGUYEUYGUYMPGPGIZUYBUYEUYGUYMVRZRUYBUURUXLUVLUUNUURWDZUXMTUYIUYJNEZUYB
|
|
UYEUYGQZRWEWIWFUVLUWQUXEUWIIUWPUVLUWQJZUGUHUIUVBUVAYQYSNPUVKUXJUVJUWQUXKW
|
|
BUVJUVKUWQWAUYTUUOUXOUVLUUOUUQWCZUXPTUYIUYKUYTUYLRUYNUYOUYTUYPRUVLUUOUUQW
|
|
DUYIUYRUYTUYSRWEWGURWJWKUVMUWRUWOUXAUWRUVBUVBPGUVSPGZUWAUWKUWRUXJUXJUVSNE
|
|
VUBUWAIUWRUVKUXJUXTUXKTZVUCUWRYQYTWLGZUVSNUWPUWQVUDUVSIZUVLUURUUOVUEUUNUU
|
|
QUUOUURVUEYQYTWMZVFWNWOUWRVUDOEZVUDNEUWRUUOUURVUGUVLUWQUUOUWPVUAWGZUVLUWP
|
|
UURUWQUYQWFZYQYTWRZVOVUDVHTWPUVBUVBUVSVRVQUWRUGUHUIUVBUVBYQYTNPVUCVUCUWRU
|
|
UOUXOVUHUXPTUYIUYKUWRUYLRUYNUYOUWRUYPRUWRUURUXLVUIUXMTUYIUYRUWRUYSRWEWIWK
|
|
UWNUWOJUWBUWLLUVRUWAUWJUWKWQWSVOUVLUUPUUSUVDUWCIUWTUVDUVCUVPUVSKLMZHGZUWC
|
|
UUTUVDVULIUVLUUTUUBVUKUVCHYPYQYSYTWTULRUUTUVLUXRUVSOEZJZVULUWCIUUTUXRVUMU
|
|
UNUURUUOUUQUXRUWPUXHUXIUXRUWQUXNUXQUYAXAXBUUTVUGVUMUUOUURVUGUUNUUQVUJXCUU
|
|
OUURVUGVUMXHUUNUUQUUOUURJVUDUVSOVUFXDXCXEXIUVLVUNJZVULUVQUVTKLMZUWCUVAUVB
|
|
UVPUVSXFVUOUVKUVQNEZUVTOEZUSZUWCVUPIUVJUXRUVKVUMVUSUVJUXRJZUVKVUMJZJVUQUV
|
|
KVURJZJZVUSVUTVUQVVAVVBUVAUVPQVVAUVKVURUVKVUMXGVVAUVBUVSWLGUVTOUVBUVSWMUV
|
|
BUVSWRWPXIXJVVCUVKVUQVURJJVUSVUQUVKVURXKUVKVUQVURVCXLXMXNUVBUVQUVTXOTXPVG
|
|
XQXSUVLUUPUUSUVGUWMIUVLUUPJZUVLUUSJZJUVGUWDUWGKLMZUWHUWEKLMZFGZUWMVVDVVEU
|
|
VEVVFUVFVVGFUVAUVBYPYQXFUVAUVBYSYTXFXRVVDUWDNEZUWGOEZJZUWHNEZUWEOEZJZVVHU
|
|
WMIVVEUVJUUNUVKUUOVVKUVJUUNJVVIUVKUUOJZVVJUVAYPQVVOUVBYQWLGUWGOUVBYQWMUVB
|
|
YQWRWPXJXNUVJUUQUVKUURVVNUVJUUQJVVLUVKUURJZVVMUVAYSQVVPUVBYTWLGUWEOUVBYTW
|
|
MUVBYTWRWPXJXNUWDUWGUWHUWEWTXAXQXTYAYBYCYDYEYDYF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Multiplication of non-negative fractions is commutative. (Contributed
|
|
by Jim Kingdon, 27-Nov-2019.) $)
|
|
mulcomnq0 $p |- ( ( A e. Q0. /\ B e. Q0. ) ->
|
|
( A .Q0 B ) = ( B .Q0 A ) ) $=
|
|
( vx vy vz vw cv cop ceq0 cec cmq0 co wceq com cnpi oveq1 oveq2 wcel comu
|
|
wa cnq0 df-nq0 eqeq12d nnmcom ad2ant2r pinn syl2an ad2ant2l opeq12 eceq1d
|
|
syl2anc mulnnnq0 ancoms 3eqtr4d 2ecoptocl ) CGZDGZHIJZEGZFGZHIJZKLZVAURKL
|
|
ZMAVAKLZVAAKLZMABKLZBAKLZMCDEFABNOIUAUBURAMVBVDVCVEURAVAKPURAVAKQUCVABMVD
|
|
VFVEVGVABAKQVABAKPUCUPNRZUQORZTZUSNRZUTORZTZTZUPUSSLZUQUTSLZHZIJZUSUPSLZU
|
|
TUQSLZHZIJZVBVCVNVOVSMZVPVTMZVRWBMVHVKWCVIVLUPUSUDUEVIVLWDVHVKVIUQNRUTNRW
|
|
DVLUQUFUTUFUQUTUDUGUHWCWDTVQWAIVOVPVSVTUIUJUKUPUQUSUTULVMVJVCWBMUSUTUPUQU
|
|
LUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u f g h $.
|
|
|
|
$( A natural number closure law. Lemma for ~ addassnq0 . (Contributed by
|
|
Jim Kingdon, 3-Dec-2019.) $)
|
|
addassnq0lemcl $p |- (
|
|
( ( I e. _om /\ J e. N. ) /\ ( K e. _om /\ L e. N. ) ) ->
|
|
( ( ( I .o L ) +o ( J .o K ) ) e. _om /\ ( J .o L ) e. N. ) ) $=
|
|
( com wcel cnpi wa comu co coa pinn nnmcl ad2ant2rl sylan ad2ant2lr nnacl
|
|
sylan2 syl2anc cmi mulpiord mulclpi eqeltrrd ad2ant2l jca ) AEFZBGFZHCEFZ
|
|
DGFZHHZADIJZBCIJZKJEFZBDIJZGFZUJUKEFZULEFZUMUFUIUPUGUHUIUFDEFUPDLADMRNUGU
|
|
HUQUFUIUGBEFUHUQBLBCMOPUKULQSUGUIUOUFUHUGUIHBDTJUNGBDUABDUBUCUDUE $.
|
|
|
|
$( Addition of non-negaative fractions is associative. (Contributed by Jim
|
|
Kingdon, 29-Nov-2019.) $)
|
|
addassnq0 $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
|
|
( ( A +Q0 B ) +Q0 C ) = ( A +Q0 ( B +Q0 C ) ) ) $=
|
|
( cnq0 wcel cplq0 co wceq wa cv cop ceq0 cec wi com cnpi comu coa syl2anc
|
|
nnmcl vz vw vv vu vx vy vf vg vh df-nq0 oveq2 oveq1d oveq1 oveq2d eqeq12d
|
|
imbi2d w3a simp1l simp2r simp3r simp1r simp2l simp3l nnaass nnmcom adantl
|
|
pinn syl3anc nndir nnmass caovdilemd oveq12d nndi 3eqtr4d opeq12 addnnnq0
|
|
syl eceq1d adantr addassnq0lemcl sylan eqtrd 3impa sylan2 3expib ecoptocl
|
|
3impb com12 2ecoptocl 3impib ) ADEZBDEZCDEZABFGZCFGZABCFGZFGZHZWLWMIWKWRW
|
|
KAUAJZUBJZKLMZFGZUCJZUDJZKLMZFGZAXAXEFGZFGZHZNWKWNXEFGZABXEFGZFGZHZNWKWRN
|
|
UAUBUCUDBCOPLDUJXABHZXIXMWKXNXFXJXHXLXNXBWNXEFXABAFUKULXNXGXKAFXABXEFUMUN
|
|
UOUPXECHZXMWRWKXOXJWOXLWQXECWNFUKXOXKWPAFXECBFUKUNUOUPWKWSOEZWTPEZIZXCOEZ
|
|
XDPEZIZIZXIYBUEJZUFJZKLMZXAFGZXEFGZYEXGFGZHZNYBXINUEUFAOPLDUJYEAHZYIXIYBY
|
|
JYGXFYHXHYJYFXBXEFYEAXAFUMULYEAXGFUMUOUPYCOEZYDPEZIZXRYAYIYMXRYAUQZYCWTQG
|
|
YDWSQGRGZXDQGZYDWTQGZXCQGZRGZYQXDQGZKZLMZYCWTXDQGZQGZYDWSXDQGZWTXCQGZRGZQ
|
|
GZRGZYDUUCQGZKZLMZYGYHYNYSUUIHZYTUUJHZUUBUULHYNUUDYDUUEQGZRGZYDUUFQGZRGZU
|
|
UDUUOUUQRGZRGZYSUUIYNUUDOEZUUOOEZUUQOEZUURUUTHYNYKUUCOEZUVAYKYLXRYAURZYNW
|
|
TOEZXDOEZUVDYNXQUVFYMXPXQYAUSWTVGVQZYNXTUVGYMXRXSXTUTXDVGVQZWTXDTSYCUUCTS
|
|
YNYDOEZUUEOEZUVBYNYLUVJYKYLXRYAVAYDVGVQZYNXPUVGUVKYMXPXQYAVBZUVIWSXDTSZYD
|
|
UUETSYNUVJUUFOEZUVCUVLYNUVFXSUVOUVHYMXRXSXTVCZWTXCTSZYDUUFTSUUDUUOUUQVDVH
|
|
YNYPUUPYRUUQRYNUGUHUIYCYDWTWSORQXDUGJZOEZUHJZOEZIZUVRUVTQGZUVTUVRQGHYNUVR
|
|
UVTVEVFUVSUWAUIJZOEUQZUVRUVTRGUWDQGUVRUWDQGUVTUWDQGZRGHYNUVRUVTUWDVIVFUWE
|
|
UWCUWDQGUVRUWFQGHYNUVRUVTUWDVJVFUWBUWCOEYNUVRUVTTVFUVEUVLUVHUVMUVIVKYNUVJ
|
|
UVFXSYRUUQHUVLUVHUVPYDWTXCVJVHVLYNUUHUUSUUDRYNUVJUVKUVOUUHUUSHUVLUVNUVQYD
|
|
UUEUUFVMVHUNVNYNUVJUVFUVGUUNUVLUVHUVIYDWTXDVJVHUUMUUNIUUAUUKLYSYTUUIUUJVO
|
|
VRSYMXRYAYGUUBHYMXRIZYAIYGYOYQKLMZXEFGZUUBUWGYGUWIHYAUWGYFUWHXEFYCYDWSWTV
|
|
PULVSUWGYOOEYQPEIYAUWIUUBHYCYDWSWTVTYOYQXCXDVPWAWBWCYMXRYAYHUULHYMYBIYHYE
|
|
UUGUUCKLMZFGZUULYBYHUWKHYMYBXGUWJYEFWSWTXCXDVPUNVFYBYMUUGOEUUCPEIUWKUULHW
|
|
SWTXCXDVTYCYDUUGUUCVPWDWBWGVNWEWFWHWIWHWJ $.
|
|
$}
|
|
|
|
$( Multiplication of non-negative fractions is distributive. Version of
|
|
~ distrnq0 with the multiplications commuted. (Contributed by Jim
|
|
Kingdon, 29-Nov-2019.) $)
|
|
distnq0r $p |- ( ( A e. Q0. /\ B e. Q0. /\ C e. Q0. ) ->
|
|
( ( B +Q0 C ) .Q0 A ) = ( ( B .Q0 A ) +Q0 ( C .Q0 A ) ) ) $=
|
|
( cnq0 wcel w3a cplq0 co cmq0 distrnq0 wceq addclnq0 mulcomnq0 sylan2 3impb
|
|
wa 3adant3 3adant2 oveq12d 3eqtr3d ) ADEZBDEZCDEZFZABCGHZIHZABIHZACIHZGHUEA
|
|
IHZBAIHZCAIHZGHABCJUAUBUCUFUIKZUBUCPUAUEDEULBCLAUEMNOUDUGUJUHUKGUAUBUGUJKUC
|
|
ABMQUAUCUHUKKUBACMRST $.
|
|
|
|
$( Addition of one to the numerator of a fraction whose denominator is one.
|
|
(Contributed by Jim Kingdon, 26-Apr-2020.) $)
|
|
addpinq1 $p |- ( A e. N. ->
|
|
[ <. ( A +N 1o ) , 1o >. ] ~Q = ( [ <. A , 1o >. ] ~Q +Q 1Q ) ) $=
|
|
( cnpi wcel c1o cop ceq cec c1q cplq co cpli df-1nqqs oveq2i wceq addpipqqs
|
|
cmi 1pi wa mulidpi ax-mp mpanr12 syl5eq opeq12i syl6eq oveq1d opeq1d eceq1d
|
|
mpan2 eceq1 eqtr2d ) ABCZADEFGZHIJZADPJZDKJZDEZFGZADKJZDEZFGUKUMUNDDPJZKJZU
|
|
TEZFGZUQUKUMULDDEFGZIJZVCHVDULILMUKDBCZVEVCNZQUKVFRVFVFVGQQADDDOUAUHUBVBUPN
|
|
VCUQNVAUOUTDUTDUNKVFUTDNQDSTZMVHUCVBUPFUITUDUKUPUSFUKUOURDUKUNADKASUEUFUGUJ
|
|
$.
|
|
|
|
${
|
|
$d A w z $.
|
|
$( Multiply a non-negative fraction by two. (Contributed by Jim Kingdon,
|
|
29-Nov-2019.) $)
|
|
nq02m $p |- ( A e. Q0. ->
|
|
( [ <. 2o , 1o >. ] ~Q0 .Q0 A ) = ( A +Q0 A ) ) $=
|
|
( vz vw wcel cv com cnpi wa cop ceq0 cec wceq wex c2o c1o cmq0 cplq0 comu
|
|
co adantr cnq0 nq0nn coa 2onn mulnnnq0 mpanl12 nn2m pinn 1onn nnmcom mpan
|
|
1pi nnm1 eqtrd adantl opeq12d eceq1d nnanq0 3anidm12 3eqtrd oveq2 oveq12d
|
|
syl wb id eqeq12d mpbird exlimivv ) AUADBEZFDZCEZGDZHZAVIVKIJKZLZHZCMBMNO
|
|
IJKZAPSZAAQSZLZBCAUBVPVTBCVPVTVQVNPSZVNVNQSZLZVMWCVOVMWANVIRSZOVKRSZIZJKZ
|
|
VIVIUCSZVKIZJKZWBNFDOGDVMWAWGLUDULNOVIVKUEUFVMWFWIJVMWDWHWEVKVJWDWHLVLVIU
|
|
GTVLWEVKLZVJVLVKFDZWKVKUHWLWEVKORSZVKOFDWLWEWMLUIOVKUJUKVKUMUNVCUOUPUQVJV
|
|
LWJWBLVKVIVIURUSUTTVOVTWCVDVMVOVRWAVSWBAVNVQPVAVOAVNAVNQVOVEZWNVBVFUOVGVH
|
|
VC $.
|
|
$}
|
|
|
|
${
|
|
$d l u q r $.
|
|
$( Define the set of positive reals. A "Dedekind cut" is a partition of
|
|
the positive rational numbers into two classes such that all the numbers
|
|
of one class are less than all the numbers of the other.
|
|
|
|
Here we follow the definition of a Dedekind cut from Definition 11.2.1
|
|
of [HoTT], p. (varies) with the one exception that we define it over
|
|
positive rational numbers rather than all rational numbers.
|
|
|
|
A Dedekind cut is an ordered pair of a lower set ` l ` and an upper set
|
|
` u ` which is inhabited ( ` E. q e. Q. q e. l /\ E. r e. Q. r e. u ` ),
|
|
rounded ( ` A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) `
|
|
and likewise for ` u ` ), disjoint
|
|
( ` A. q e. Q. -. ( q e. l /\ q e. u ) ` ) and located
|
|
( ` A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) ) ` ). See
|
|
HoTT for more discussion of those terms and different ways of defining
|
|
Dedekind cuts.
|
|
|
|
(Note: This is a "temporary" definition used in the construction of
|
|
complex numbers, and is intended to be used only by the construction.)
|
|
(Contributed by Jim Kingdon, 25-Sep-2019.) $)
|
|
df-inp $a |- P. = { <. l , u >. | ( ( ( l C_ Q. /\ u C_ Q. ) /\
|
|
( E. q e. Q. q e. l /\ E. r e. Q. r e. u ) ) /\ (
|
|
( A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) /\
|
|
A. r e. Q. ( r e. u <-> E. q e. Q. ( q <Q r /\ q e. u ) ) ) /\
|
|
A. q e. Q. -. ( q e. l /\ q e. u ) /\
|
|
A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) )
|
|
) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d l u $.
|
|
$( Define the positive real constant 1. This is a "temporary" set used in
|
|
the construction of complex numbers and is intended to be used only by
|
|
the construction. (Contributed by Jim Kingdon, 25-Sep-2019.) $)
|
|
df-i1p $a |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >. $.
|
|
$}
|
|
|
|
${
|
|
$d x y q r s $.
|
|
$( Define addition on positive reals. From Section 11.2.1 of [HoTT], p.
|
|
(varies). We write this definition to closely resemble the definition
|
|
in HoTT although some of the conditions are redundant (for example,
|
|
` r e. ( 1st `` x ) ` implies ` r e. Q. ` ) and can be simplified as
|
|
shown at ~ genpdf .
|
|
|
|
This is a "temporary" set used in the construction of complex numbers,
|
|
and is intended to be used only by the construction. (Contributed by
|
|
Jim Kingdon, 26-Sep-2019.) $)
|
|
df-iplp $a |- +P. = ( x e. P. , y e. P. |->
|
|
<. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\
|
|
s e. ( 1st ` y ) /\ q = ( r +Q s ) ) } ,
|
|
{ q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\
|
|
s e. ( 2nd ` y ) /\ q = ( r +Q s ) ) } >. ) $.
|
|
|
|
$( Define multiplication on positive reals. Here we use a simple
|
|
definition which is similar to ~ df-iplp or the definition of
|
|
multiplication on positive reals in Metamath Proof Explorer. This is as
|
|
opposed to the more complicated definition of multiplication given in
|
|
Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by
|
|
handling negative numbers or handling modified Dedekind cuts in which
|
|
locatedness is omitted.
|
|
|
|
This is a "temporary" set used in the construction of complex numbers,
|
|
and is intended to be used only by the construction. (Contributed by
|
|
Jim Kingdon, 29-Sep-2019.) $)
|
|
df-imp $a |- .P. = ( x e. P. , y e. P. |->
|
|
<. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\
|
|
s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } ,
|
|
{ q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\
|
|
s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y q $.
|
|
$( Define ordering on positive reals. We define ` x <P y ` if there is a
|
|
positive fraction ` q ` which is an element of the upper cut of ` x `
|
|
and the lower cut of ` y ` . From the definition of < in Section 11.2.1
|
|
of [HoTT], p. (varies).
|
|
|
|
This is a "temporary" set used in the construction of complex numbers,
|
|
and is intended to be used only by the construction. (Contributed by
|
|
Jim Kingdon, 29-Sep-2019.) $)
|
|
df-iltp $a |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\
|
|
E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d l u q r $.
|
|
$( Lemma for proving existence of reals. (Contributed by Jim Kingdon,
|
|
27-Sep-2019.) $)
|
|
npsspw $p |- P. C_ ( ~P Q. X. ~P Q. ) $=
|
|
( vl vu vq vr cv cnq wss wa wcel wrex cltq wbr wb wral wn wo wi w3a copab
|
|
selpw cpw cnp cxp simpll anbi12i sylibr ssopab2i df-inp df-xp 3sstr4i ) A
|
|
EZFGZBEZFGZHZCEZUKIZCFJDEZUMIZDFJHZHUQUPURKLZURUKIHDFJMCFNUSVAUPUMIZHCFJM
|
|
DFNHUQVBHOCFNVAUQUSPQDFNCFNRZHZABSUKFUAZIZUMVEIZHZABSUBVEVEUCVDVHABVDUOVH
|
|
UOUTVCUDVFULVGUNAFTBFTUEUFUGBDCAUHABVEVEUIUJ $.
|
|
$}
|
|
|
|
$( Two reals are equal if and only if their lower and upper cuts are.
|
|
(Contributed by Jim Kingdon, 11-Dec-2019.) $)
|
|
preqlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A = B <->
|
|
( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) ) $=
|
|
( cnp wcel wa wceq c1st cfv c2nd cop cnq cpw npsspw sseli 1st2nd2 eqeqan12d
|
|
cxp syl wb xp1st xp2nd opthg syl2anc adantr bitrd ) ACDZBCDZEABFAGHZAIHZJZB
|
|
GHZBIHZJZFZUHUKFUIULFEZUFUGAUJBUMUFAKLZUPQZDZAUJFCUQAMNZAUPUPORUGBUQDBUMFCU
|
|
QBMNBUPUPORPUFUNUOSZUGUFUHUPDZUIUPDZUTUFURVAUSAUPUPTRUFURVBUSAUPUPUARUHUIUK
|
|
ULUPUPUBUCUDUE $.
|
|
|
|
$( The class of positive reals is a set. (Contributed by NM,
|
|
31-Oct-1995.) $)
|
|
npex $p |- P. e. _V $=
|
|
( cnp cnq cpw cxp nqex pwex xpex npsspw ssexi ) ABCZJDJJBEFZKGHI $.
|
|
|
|
${
|
|
$d l u q r L $. $d l u q r U $.
|
|
$( Membership in positive reals. (Contributed by Jim Kingdon,
|
|
27-Sep-2019.) $)
|
|
elinp $p |- ( <. L , U >. e. P. <-> ( ( ( L C_ Q. /\ U C_ Q. ) /\
|
|
( E. q e. Q. q e. L /\ E. r e. Q. r e. U ) ) /\ (
|
|
( A. q e. Q. ( q e. L <-> E. r e. Q. ( q <Q r /\ r e. L ) ) /\
|
|
A. r e. Q. ( r e. U <-> E. q e. Q. ( q <Q r /\ q e. U ) ) ) /\
|
|
A. q e. Q. -. ( q e. L /\ q e. U ) /\
|
|
A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. L \/ r e. U ) )
|
|
) ) ) $=
|
|
( vl vu wcel cvv wa cnq wss cv wrex wb wral anbi1d rexbidv anbi12d anbi2d
|
|
eleq2 cop cnp cltq wbr wn wo wi w3a cpw npsspw sseli opelxp sylib anim12i
|
|
cxp elex syl nqex ssex ad2antrr copab df-inp eleq2i sseq1 bibi12d ralbidv
|
|
notbid orbi1d imbi2d 2ralbidv 3anbi123d orbi2d opelopabg syl5bb pm5.21nii
|
|
wceq ) BAUAZUBGZBHGZAHGZIZBJKZAJKZIZDLZBGZDJMZCLZAGZCJMZIZIZWFWEWHUCUDZWH
|
|
BGZIZCJMZNZDJOZWIWMWEAGZIZDJMZNZCJOZIZWFWSIZUEZDJOZWMWFWIUFZUGZCJODJOZUHZ
|
|
IZVRBJUIZGZAXMGZIZWAVRVQXMXMUOZGXPUBXQVQUJUKBAXMXMULUMXNVSXOVTBXMUPAXMUPU
|
|
NUQWDWAWKXKWBVSWCVTBJURUSAJURUSUNUTVRVQELZJKZFLZJKZIZWEXRGZDJMZWHXTGZCJMZ
|
|
IZIZYCWMWHXRGZIZCJMZNZDJOZYEWMWEXTGZIZDJMZNZCJOZIZYCYNIZUEZDJOZWMYCYEUFZU
|
|
GZCJODJOZUHZIZEFVAZGWAXLUBUUHVQFCDEVBVCUUGWBYAIZWGYFIZIZWRYRIZWFYNIZUEZDJ
|
|
OZWMWFYEUFZUGZCJODJOZUHZIXLEFBAHHXRBVPZYHUUKUUFUUSUUTYBUUIYGUUJUUTXSWBYAX
|
|
RBJVDPUUTYDWGYFUUTYCWFDJXRBWETZQPRUUTYSUULUUBUUOUUEUURUUTYMWRYRUUTYLWQDJU
|
|
UTYCWFYKWPUVAUUTYJWOCJUUTYIWNWMXRBWHTSQVEVFPUUTUUAUUNDJUUTYTUUMUUTYCWFYNU
|
|
VAPVGVFUUTUUDUUQDCJJUUTUUCUUPWMUUTYCWFYEUVAVHVIVJVKRXTAVPZUUKWLUUSXKUVBUU
|
|
IWDUUJWKUVBYAWCWBXTAJVDSUVBYFWJWGUVBYEWICJXTAWHTZQSRUVBUULXDUUOXGUURXJUVB
|
|
YRXCWRUVBYQXBCJUVBYEWIYPXAUVCUVBYOWTDJUVBYNWSWMXTAWETZSQVEVFSUVBUUNXFDJUV
|
|
BUUMXEUVBYNWSWFUVDSVGVFUVBUUQXIDCJJUVBUUPXHWMUVBYEWIWFUVCVLVIVJVKRVMVNVO
|
|
$.
|
|
$}
|
|
|
|
$( A positive real is an ordered pair of a lower cut and an upper cut.
|
|
(Contributed by Jim Kingdon, 27-Sep-2019.) $)
|
|
prop $p |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. ) $=
|
|
( cnp wcel c1st cfv c2nd cop cnq cpw cxp npsspw sseli 1st2nd2 eleq1 biimpcd
|
|
wceq syl mpd ) ABCZAADEAFEGZPZTBCZSAHIZUCJZCUABUDAKLAUCUCMQUASUBATBNOR $.
|
|
|
|
${
|
|
$d q r A $.
|
|
$( Membership in positive reals, using ` 1st ` and ` 2nd ` to refer to the
|
|
lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.) $)
|
|
elnp1st2nd $p |- ( A e. P. <-> ( (
|
|
A e. ( ~P Q. X. ~P Q. ) /\
|
|
( E. q e. Q. q e. ( 1st ` A ) /\ E. r e. Q. r e. ( 2nd ` A ) ) ) /\ (
|
|
( A. q e. Q. ( q e. ( 1st ` A ) <->
|
|
E. r e. Q. ( q <Q r /\ r e. ( 1st ` A ) ) ) /\
|
|
A. r e. Q. ( r e. ( 2nd ` A ) <->
|
|
E. q e. Q. ( q <Q r /\ q e. ( 2nd ` A ) ) ) ) /\
|
|
A. q e. Q. -. ( q e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) /\
|
|
A. q e. Q. A. r e. Q. ( q <Q r ->
|
|
( q e. ( 1st ` A ) \/ r e. ( 2nd ` A ) ) )
|
|
) ) ) $=
|
|
( cnp wcel cnq cpw cxp cv c1st cfv wrex wral wss simprd jca elpwid anim1i
|
|
wa wb c2nd cltq wbr wn wo wi w3a npsspw sseli cop prop elinp sylib simpld
|
|
wceq 1st2nd2 ad2antrr xp1st xp2nd sylibr eqeltrd impbii ) ADEZAFGZVDHZEZC
|
|
IZAJKZEZCFLBIZAUAKZEZBFLSZSZVIVGVJUBUCZVJVHESBFLTCFMVLVOVGVKEZSCFLTBFMSVI
|
|
VPSUDCFMVOVIVLUEUFBFMCFMUGZSZVCVNVQVCVFVMDVEAUHUIVCVHFNZVKFNZSZVMVCWAVMSZ
|
|
VQVCVHVKUJZDEZWBVQSZAUKVKVHBCULZUMZUNOPVCWBVQWGOPVRAWCDVFAWCUOVMVQAVDVDUP
|
|
UQVRWEWDVNWBVQVFWAVMVFVSVTVFVHFAVDVDURQVFVKFAVDVDUSQPRRWFUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x y L $. $d U x y $.
|
|
$( A positive real's lower cut is inhabited. (Contributed by Jim Kingdon,
|
|
27-Sep-2019.) $)
|
|
prml $p |- ( <. L , U >. e. P. -> E. x e. Q. x e. L ) $=
|
|
( vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a elinp
|
|
simplrl sylbi ) CBEFGCHIBHIJZAKZCGZAHLZDKZBGZDHLZJJUFUEUHMNZUHCGJDHLOAHPU
|
|
IUKUEBGZJAHLODHPJUFULJQAHPUKUFUIRSDHPAHPTZJUGBCDAUAUDUGUJUMUBUC $.
|
|
|
|
$( A positive real's upper cut is inhabited. (Contributed by Jim Kingdon,
|
|
27-Sep-2019.) $)
|
|
prmu $p |- ( <. L , U >. e. P. -> E. x e. Q. x e. U ) $=
|
|
( vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a elinp
|
|
simplrr sylbi ) CBEFGCHIBHIJZDKZCGZDHLZAKZBGZAHLZJJUFUEUHMNZUHCGJAHLODHPU
|
|
IUKUEBGZJDHLOAHPJUFULJQDHPUKUFUIRSAHPDHPTZJUJBCADUAUDUGUJUMUBUC $.
|
|
|
|
$( The lower cut of a positive real is a subset of the positive fractions.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
prssnql $p |- ( <. L , U >. e. P. -> L C_ Q. ) $=
|
|
( vx vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a
|
|
elinp simplll sylbi ) BAEFGBHIZAHIZJCKZBGZCHLDKZAGZDHLJZJUGUFUHMNZUHBGJDH
|
|
LOCHPUIUKUFAGZJCHLODHPJUGULJQCHPUKUGUIRSDHPCHPTZJUDABDCUAUDUEUJUMUBUC $.
|
|
|
|
$( The upper cut of a positive real is a subset of the positive fractions.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
prssnqu $p |- ( <. L , U >. e. P. -> U C_ Q. ) $=
|
|
( vx vy cop cnp wcel cnq wss wa cv wrex cltq wbr wb wral wn wo wi w3a
|
|
elinp simpllr sylbi ) BAEFGBHIZAHIZJCKZBGZCHLDKZAGZDHLJZJUGUFUHMNZUHBGJDH
|
|
LOCHPUIUKUFAGZJCHLODHPJUGULJQCHPUKUGUIRSDHPCHPTZJUEABDCUAUDUEUJUMUBUC $.
|
|
$}
|
|
|
|
$( An element of a positive real's lower cut is a positive fraction.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
elprnql $p |- ( ( <. L , U >. e. P. /\ B e. L ) -> B e. Q. ) $=
|
|
( cop cnp wcel cnq prssnql sselda ) CBDEFCGABCHI $.
|
|
|
|
$( An element of a positive real's upper cut is a positive fraction.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
elprnqu $p |- ( ( <. L , U >. e. P. /\ B e. U ) -> B e. Q. ) $=
|
|
( cop cnp wcel cnq prssnqu sselda ) CBDEFBGABCHI $.
|
|
|
|
$( The empty set is not a positive real. (Contributed by NM,
|
|
15-Nov-1995.) $)
|
|
0npr $p |- -. (/) e. P. $=
|
|
( vx c0 cnp wcel c1st cfv c2nd cop cv cnq wrex noel 1st0 eleq2i mtbir rexex
|
|
wex nex mto prml prop ) BCDBEFZBGFZHCDZUDAIZUBDZAJKZUGUFAQUFAUFUEBDUELUBBUE
|
|
MNORUFAJPSAUCUBTSBUAS $.
|
|
|
|
${
|
|
$d b c L $. $d b c U $. $d b c B $. $d b c C $.
|
|
$( A lower cut is closed downwards under the positive fractions.
|
|
(Contributed by Jim Kingdon, 28-Sep-2019.) $)
|
|
prcdnql $p |- ( ( <. L , U >. e. P. /\ B e. L ) ->
|
|
( C <Q B -> C e. L ) ) $=
|
|
( vc vb wcel wa cltq wbr cnq ltrelnq brel cv wi wceq eleq1 wrex wb wral
|
|
cop cnp simpld adantl breq1 imbi12d imbi2d ancomd an42 anbi12d rspcev wss
|
|
breq2 wn wo w3a elinp simpr1l sylbi r19.21bi syl5ibrcom 3impb 3com12 impd
|
|
3expib syl5bi mpand com12 ancoms vtoclg mpcom ex ) DCUAUBGZADGZHZBAIJZBDG
|
|
ZBKGZVOVPHVQVPVRVOVPVRAKGZBAKKILMUCUDVRVOVPVQVOENZAIJZVTDGZOZOVOVPVQOZOEB
|
|
KVTBPZWCWDVOWEWAVPWBVQVTBAIUEVTBDQUFUGVNVMWCWAVNVMHZWBWAVSVTKGZHZWFWBWAWG
|
|
VSVTAKKILMUHWHWFHVSVNHZVMWGHZHWAWBVSWGVNVMUIWAWIWJWBWAVSVNWJWBOZVSWAVNWKV
|
|
SWAVNWKVSWAVNHZHWBWJVTFNZIJZWMDGZHZFKRZWPWLFAKWMAPWNWAWOVNWMAVTIUMWMADQUJ
|
|
UKVMWBWQSZEKVMDKULCKULHWBEKRWMCGZFKRHHZWREKTZWSWNVTCGZHEKRSFKTZHWBXBHUNEK
|
|
TZWNWBWSUOOFKTEKTZUPHXACDFEUQXAXCXDXEWTURUSUTVAVBVCVEVDVFVGVHVIVJVDVKVL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d b c L $. $d b c U $. $d b c B $. $d b c C $.
|
|
$( An upper cut is closed upwards under the positive fractions.
|
|
(Contributed by Jim Kingdon, 25-Nov-2019.) $)
|
|
prcunqu $p |- ( ( <. L , U >. e. P. /\ C e. U ) ->
|
|
( C <Q B -> B e. U ) ) $=
|
|
( vb vc wcel wa cltq wbr cnq ltrelnq brel cv wi wceq eleq1 wrex wb wral
|
|
cop cnp simprd adantl breq2 imbi12d imbi2d breq1 anbi12d rspcev wss wn wo
|
|
an42 w3a elinp simpr1r sylbi r19.21bi syl5ibrcom 3impb 3com12 3expib impd
|
|
syl5bi mpand com12 ancoms vtoclg mpcom ex ) DCUAUBGZBCGZHZBAIJZACGZAKGZVN
|
|
VOHVPVOVQVNVOBKGZVQBAKKILMUCUDVQVNVOVPVNBENZIJZVSCGZOZOVNVOVPOZOEAKVSAPZW
|
|
BWCVNWDVTVOWAVPVSABIUEVSACQUFUGVMVLWBVTVMVLHZWAVTVRVSKGZHZWEWABVSKKILMWGW
|
|
EHVRVMHZVLWFHZHVTWAVRWFVMVLUNVTWHWIWAVTVRVMWIWAOZVRVTVMWJVRVTVMWJVRVTVMHZ
|
|
HWAWIFNZVSIJZWLCGZHZFKRZWOWKFBKWLBPWMVTWNVMWLBVSIUHWLBCQUIUJVLWAWPSZEKVLD
|
|
KUKCKUKHWLDGZFKRWAEKRHHZWRWMVSDGHEKRSFKTZWQEKTZHWRWNHULFKTZWMWRWAUMOEKTFK
|
|
TZUOHXACDEFUPWTXAXBXCWSUQURUSUTVAVBVCVDVEVFVGVHVIVDVJVK $.
|
|
$}
|
|
|
|
$( A positive fraction not in a lower cut is an upper bound. (Contributed by
|
|
Jim Kingdon, 29-Sep-2019.) $)
|
|
prubl $p |- ( ( ( <. L , U >. e. P. /\ B e. L ) /\ C e. Q. ) ->
|
|
( -. C e. L -> B <Q C ) ) $=
|
|
( cop cnp wcel wa cnq wn wceq cltq wbr wo eleq1 biimpcd adantl prcdnql jaod
|
|
wi con3d adantr wb elprnql nqtric sylan sylibrd ) DCEFGZADGZHZBIGZHBDGZJZAB
|
|
KZBALMZNZJZABLMZUJUMUQTUKUJUPULUJUNULUOUIUNULTUHUNUIULABDOPQABCDRSUAUBUJAIG
|
|
UKURUQUCACDUDABUEUFUG $.
|
|
|
|
${
|
|
$d C q $. $d L q r $. $d U q r $.
|
|
$( An element of a lower cut is less than an element of the corresponding
|
|
upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.) $)
|
|
prltlu $p |- ( ( <. L , U >. e. P. /\ B e. L /\ C e. U ) -> B <Q C ) $=
|
|
( vq vr wcel w3a wn cltq wbr wa wi cnq cv wral wss wrex wb eleq1 simp3 wo
|
|
cop elprnqu 3adant2 elinp simpr2 sylbi 3ad2ant1 wceq anbi12d notbid rspcv
|
|
cnp sylc ancom notbii imnan bitr4i sylib mpd 3simpa prubl syl2anc ) DCUCU
|
|
NGZADGZBCGZHZBDGZIZABJKZVHVGVJVEVFVGUAVHVIVGLZIZVGVJMZVHBNGZEOZDGZVPCGZLZ
|
|
IZENPZVMVEVGVOVFBCDUDUEZVEVFWAVGVEDNQCNQLVQENRFOZCGZFNRLLZVQVPWCJKZWCDGLF
|
|
NRSENPWDWFVRLENRSFNPLZWAWFVQWDUBMFNPENPZHLWACDFEUFWEWGWAWHUGUHUIVTVMEBNVP
|
|
BUJZVSVLWIVQVIVRVGVPBDTVPBCTUKULUMUOVMVGVILZIVNVLWJVIVGUPUQVGVIURUSUTVAVH
|
|
VEVFLVOVJVKMVEVFVGVBWBABCDVCVDVA $.
|
|
$}
|
|
|
|
${
|
|
$d x y B $. $d x y L $. $d x y U $.
|
|
$( A lower cut has no largest member. (Contributed by Jim Kingdon,
|
|
29-Sep-2019.) $)
|
|
prnmaxl $p |- ( ( <. L , U >. e. P. /\ B e. L ) -> E. x e. L B <Q x ) $=
|
|
( vy cop wcel wa cv cltq wbr wex wrex cnq wb wral wi wss df-rex sylibr wn
|
|
cnp elprnql wo w3a elinp simpr1l sylbi eleq1 breq1 anbi1d rexbidv bibi12d
|
|
wceq rspcv bi1 syl56 impd mpcom sylib ltrelnq brel simprd pm4.71ri anbi1i
|
|
ancom anass 3bitr3i exbii ) DCFUBGZBDGZHZAIZDGZBVMJKZHZALZVOADMVLVMNGZVOV
|
|
NHZHZALZVQVLVSANMZWABNGZVLWBBCDUCWCVJVKWBVJEIZDGZWDVMJKZVNHZANMZOZENPZWCV
|
|
KWBOZVKWBQVJDNRCNRHWEENMVMCGZANMHHZWJWLWFWDCGZHENMOANPZHWEWNHUAENPZWFWEWL
|
|
UDQANPENPZUEHWJCDAEUFWJWOWPWQWMUGUHWIWKEBNWDBUNZWEVKWHWBWDBDUIWRWGVSANWRW
|
|
FVOVNWDBVMJUJUKULUMUOVKWBUPUQURUSVSANSUTVPVTAVSVRVOHZVNHVPVTVOWSVNVOVRVOW
|
|
CVRBVMNNJVAVBVCVDVEVOVNVFVRVOVNVGVHVITVOADST $.
|
|
|
|
$( An upper cut has no smallest member. (Contributed by Jim Kingdon,
|
|
7-Nov-2019.) $)
|
|
prnminu $p |- ( ( <. L , U >. e. P. /\ B e. U ) -> E. x e. U x <Q B ) $=
|
|
( vy cop wcel wa cv cltq wbr wex wrex cnq wb wral wi wss df-rex sylibr wn
|
|
cnp elprnqu wo w3a elinp simpr1r sylbi eleq1 breq2 anbi1d rexbidv bibi12d
|
|
wceq rspcv bi1 syl56 impd mpcom sylib ltrelnq brel simpld pm4.71ri anbi1i
|
|
ancom anass 3bitr3i exbii ) DCFUBGZBCGZHZAIZCGZVMBJKZHZALZVOACMVLVMNGZVOV
|
|
NHZHZALZVQVLVSANMZWABNGZVLWBBCDUCWCVJVKWBVJEIZCGZVMWDJKZVNHZANMZOZENPZWCV
|
|
KWBOZVKWBQVJDNRCNRHVMDGZANMWEENMHHZWLWFWDDGHENMOANPZWJHWLVNHUAANPZWFWLWEU
|
|
DQENPANPZUEHWJCDEAUFWNWJWOWPWMUGUHWIWKEBNWDBUNZWEVKWHWBWDBCUIWQWGVSANWQWF
|
|
VOVNWDBVMJUJUKULUMUOVKWBUPUQURUSVSANSUTVPVTAVSVRVOHZVNHVPVTVOWRVNVOVRVOVR
|
|
WCVMBNNJVAVBVCVDVEVOVNVFVRVOVNVGVHVITVOACST $.
|
|
$}
|
|
|
|
${
|
|
$d x y B $. $d x y L $. $d x y U $.
|
|
$( A lower cut has no largest member. Addition version. (Contributed by
|
|
Jim Kingdon, 29-Sep-2019.) $)
|
|
prnmaddl $p |- ( ( <. L , U >. e. P. /\ B e. L ) ->
|
|
E. x e. Q. ( B +Q x ) e. L ) $=
|
|
( vy cop cnp wcel wa cv cltq wbr wrex cplq co prnmaxl wceq ltexnqi eleq1a
|
|
cnq reximdv syl5 rexlimiv syl ) DCFGHBDHIBEJZKLZEDMBAJNOZDHZATMZEBCDPUFUI
|
|
EDUFUGUEQZATMUEDHZUIABUERUKUJUHATUEDUGSUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d A q r $. $d B q r $. $d L q r $. $d U q r $.
|
|
$( A Dedekind cut is located. (Contributed by Jim Kingdon,
|
|
23-Oct-2019.) $)
|
|
prloc $p |- ( ( <. L , U >. e. P. /\ A <Q B ) -> ( A e. L \/ B e. U ) ) $=
|
|
( vq vr wcel cltq wbr wa cv wo wi cnq wral wss wrex wb simpr adantl elinp
|
|
cop cnp wn w3a simpr3 sylbi adantr ltrelnq brel simpld wceq breq1d eleq1d
|
|
orbi1d imbi12d ralbidv rspcdv simprd breq2d orbi2d syld mp2d ) DCUBUCGZAB
|
|
HIZJZEKZFKZHIZVGDGZVHCGZLZMZFNOZENOZVEADGZBCGZLZVDVOVEVDDNPCNPJVJENQVKFNQ
|
|
JJZVJVIVHDGJFNQRENOVKVIVGCGZJENQRFNOJZVJVTJUDENOZVOUEJVOCDFEUAVSWAWBVOUFU
|
|
GUHVDVESVFVOAVHHIZVPVKLZMZFNOZVEVRMZVFVNWFEANVEANGZVDVEWHBNGZABNNHUIUJZUK
|
|
TVFVGAULZJZVMWEFNWLVIWCVLWDWLVGAVHHVFWKSZUMWLVJVPVKWLVGADWMUNUOUPUQURVFWE
|
|
WGFBNVEWIVDVEWHWIWJUSTVFVHBULZJZWCVEWDVRWOVHBAHVFWNSZUTWOVKVQVPWOVHBCWPUN
|
|
VAUPURVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d A q r $. $d L q r $. $d U q r $.
|
|
$( A Dedekind cut is disjoint. (Contributed by Jim Kingdon,
|
|
15-Dec-2019.) $)
|
|
prdisj $p |- ( ( <. L , U >. e. P. /\ A e. Q. ) ->
|
|
-. ( A e. L /\ A e. U ) ) $=
|
|
( vq vr cop cnp wcel cnq wa wn cv wi wceq eleq1 anbi2d wss wrex wb wral
|
|
anbi12d notbid imbi12d cltq wbr wo w3a elinp simpr2 sylbi r19.21bi vtoclg
|
|
anabsi7 ) CBFGHZAIHZACHZABHZJZKZUNDLZIHZJZUTCHZUTBHZJZKZMUNUOJZUSMDAIUTAN
|
|
ZVBVGVFUSVHVAUOUNUTAIOPVHVEURVHVCUPVDUQUTACOUTABOUAUBUCUNVFDIUNCIQBIQJVCD
|
|
IRELZBHZEIRJJZVCUTVIUDUEZVICHJEIRSDITVJVLVDJDIRSEITJZVFDITZVLVCVJUFMEITDI
|
|
TZUGJVNBCEDUHVKVMVNVOUIUJUKULUM $.
|
|
$}
|
|
|
|
$( Two possible ways of contracting an interval which straddles a Dedekind
|
|
cut. Lemma for ~ prarloc . (Contributed by Jim Kingdon, 10-Nov-2019.) $)
|
|
prarloclemlt $p |- ( ( ( X e. _om
|
|
/\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
|
|
/\ y e. _om ) ->
|
|
( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) <Q
|
|
( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) ) $=
|
|
( com wcel cnq wa c1o coa co ceq c2o wbr cnpi 1pi wceq syl2anc cop cnp cltq
|
|
w3a cv cec cmq cplq cmi clti wss 2onn nnacl mpan2 nnaword1 sylan csuc elexi
|
|
1onn sucid eleqtrri wi nnaordi mpan mpi adantr sseldd ancoms nnppipi adantl
|
|
df-2o o1p1e2 mp2an eqeltrri pinn syl nnacom sylan2 eqeltrrd ltpiord mulidpi
|
|
mpbird mulcompig sylancl eqtr3d breq12d ordpipqqs mpanl2 mpanr2 adantlr cxp
|
|
wb simpr cqs opelxpi enqex ecelqsi df-nqqs syl6eleqr simplr3 ltmnqg syl3anc
|
|
mpbid mulcomnqg mulclnq simplr1 simplr2 elprnql ltanqg ) FGHZEDUAUBHZBEHZCI
|
|
HZUDZJAUEZGHZJZXOKLMZKUAZNUFZCUGMZXOOLMZFLMZKUAZNUFZCUGMZUCPZBYAUHMBYFUHMUC
|
|
PZXQCXTUGMZCYEUGMZUCPZYGXQXTYEUCPZYKXJXPYLXNXJXPJZYLXRKUIMZKYCUIMZUJPZYMYPX
|
|
RYCUJPZYMYQXRYCHZXPXJYRXPXJJYBYCXRXPYBGHZXJYBYCUKXPOGHZYSULXOOUMUNYBFUOUPXP
|
|
XRYBHZXJXPKOHZUUAKKUQOKKGUSURUTVKVAYTXPUUBUUAVBULKOXOVCVDVEVFVGVHYMXRQHZYCQ
|
|
HZYQYRWLXPUUCXJXPKQHZUUCRXOKVIUNZVJZYMFYBLMZYCQXPXJYSUUHYCSXPYBQHZYSXPOQHUU
|
|
IKKLMZOQVLKGHUUEUUJQHUSRKKVIVMVNXOOVIUNZYBVOVPFYBVQVRXPXJUUIUUHQHUUKFYBVIVR
|
|
VSZXRYCVTTWBYMYNXRYOYCUJYMUUCYNXRSUUGXRWAVPYMYCKUIMZYOYCYMUUDUUEUUMYOSUULRY
|
|
CKWCWDYMUUDUUMYCSUULYCWAVPWEWFWBYMXPUUDYLYPWLZXJXPWMUULXPUUCUUDUUNUUFUUCUUD
|
|
UUEUUNRUUCUUEUUDUUEJUUNRXRKYCKWGWHWIUPTWBWJXQXTIHZYEIHZXMYLYKWLXJXPUUOXNYMX
|
|
TQQWKZNWNZIYMXSUUQHZXTUURHYMUUCUUEUUSUUGRXRKQQWOWDUUQXSNWPWQVPWRWSWJZXJXPUU
|
|
PXNYMYEUURIYMYDUUQHZYEUURHYMUUDUUEUVAUULRYCKQQWOWDUUQYDNWPWQVPWRWSWJZXKXLXM
|
|
XJXPWTZXTYECXAXBXCXQYIYAYJYFUCXQXMUUOYIYASUVCUUTCXTXDTXQXMUUPYJYFSUVCUVBCYE
|
|
XDTWFXCXQYAIHZYFIHZBIHZYGYHWLXQUUOXMUVDUUTUVCXTCXETXQUUPXMUVEUVBUVCYECXETXQ
|
|
XKXLUVFXKXLXMXJXPXFXKXLXMXJXPXGBDEXHTYAYFBXIXBXC $.
|
|
|
|
${
|
|
$d A f g h y z $. $d L f g h y z $. $d P f g h y z $. $d U f g h y z $.
|
|
$d X f g h y z $.
|
|
$( Contracting the lower side of an interval which straddles a Dedekind
|
|
cut. Lemma for ~ prarloc . (Contributed by Jim Kingdon,
|
|
10-Nov-2019.) $)
|
|
prarloclemlo $p |- ( ( ( X e. _om
|
|
/\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
|
|
/\ y e. _om ) -> (
|
|
( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) e. L -> (
|
|
( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
|
|
E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) ) $=
|
|
( com wcel cop wa c1o coa co ceq cec cmq cplq c2o wceq oveq1d vf vg vh vz
|
|
cnp cnq w3a cv ceq0 cmq0 cplq0 csuc wrex nnaass adantl simpr 1onn sylancl
|
|
wi nnacl 2onn a1i simpll caovassd nnacom caov4d caovcomd con0 nnon oa1suc
|
|
eqtrd oveq2d 3eqtr2rd opeq1d eceq1d eleq1d biimpd simplr1 simplr2 elprnql
|
|
3syl syl2anc cnpi cxp 1pi nnppipi opelxpi mpan2 cqs enqex ecelqsi df-nqqs
|
|
syl6eleqr simplr3 mulclnq nqnq0a nqnq0m nqnq0pi eqtr4d anbi1d opeq1 oveq1
|
|
anbi12d rspcev ex syl sylbid cbvrexv syl6ib sylan2d expdimp adantld ) FGH
|
|
ZEDIUEHZBEHZCUFHZUGZJZAUHZGHZJZBXSKLMZKIZNOZCPMZQMZEHZBXSKIZUIOZCUJMZUKMZ
|
|
EHZBXSRLMZFULZLMZKIZNOZCPMZQMZDHZJYLBYMFLMZKIZNOZCPMZQMZDHZJZAGUMZUSYAYGJ
|
|
YTUUHYLYAYGYTUUHYAYTBYBRLMZFLMZKIZNOZCPMZQMZDHZYGUUHYAYTUUOYAYSUUNDYAYRUU
|
|
MBQYAYQUULCPYAYPUUKNYAYOUUJKYAUUJYBRFLMLMYMKFLMZLMYOYAUAUBUCYBRFGLUAUHZGH
|
|
ZUBUHZGHZUCUHZGHUGUUQUUSLMZUVALMUUQUUSUVALMLMSYAUUQUUSUVAUNUOZYAXTKGHZYBG
|
|
HZXRXTUPZUQXSKUTURZRGHYAVAVBZXMXQXTVCZVDYAUAUBUCXSRKFGLUVFUVHUVDYAUQVBZUU
|
|
RUUTJZUVBUUSUUQLMSYAUUQUUSVEUOZUVCUVIUVKUVBGHYAUUQUUSUTUOVFYAUUPYNYMLYAUU
|
|
PFKLMZYNYAUAUBKFGLUVLUVJUVIVGYAXMFVHHUVMYNSUVIFVIFVJWAVKVLVMVNVOTVLVPVQYA
|
|
YGUUOJZBUDUHZKIZUIOZCUJMZUKMZEHZBUVORLMZFLMZKIZNOZCPMZQMZDHZJZUDGUMZUUHYA
|
|
UVNBYCUIOZCUJMZUKMZEHZUUOJZUWIYAYGUWMUUOYAYFUWLEYAYFBYEUKMZUWLYABUFHZYEUF
|
|
HZYFUWOSYAXNXOUWPXNXOXPXMXTVRXNXOXPXMXTVSBDEVTWBYAYDUFHZXPUWQYAYBWCHZYCWC
|
|
WCWDZHZUWRYAXTKWCHZUWSUVFWEXSKWFURZUWSUXBUXAWEYBKWCWCWGWHUXAYDUWTNWIUFUWT
|
|
YCNWJWKWLWMWAZXNXOXPXMXTWNZYDCWOWBBYEWPWBYAYEUWKBUKYAYEYDCUJMZUWKYAUWRXPY
|
|
EUXFSUXDUXEYDCWQWBYAUWJYDCUJYAUWSUXBUWJYDSUXCWEYBKWRURTWSVLVKVPWTYAUVEUWN
|
|
UWIUSUVGUVEUWNUWIUWHUWNUDYBGUVOYBSZUVTUWMUWGUUOUXGUVSUWLEUXGUVRUWKBUKUXGU
|
|
VQUWJCUJUXGUVPYCUIUVOYBKXAVOTVLVPUXGUWFUUNDUXGUWEUUMBQUXGUWDUULCPUXGUWCUU
|
|
KNUXGUWBUUJKUXGUWAUUIFLUVOYBRLXBTVNVOTVLVPXCXDXEXFXGUWHUUGUDAGUVOXSSZUVTY
|
|
LUWGUUFUXHUVSYKEUXHUVRYJBUKUXHUVQYICUJUXHUVPYHUIUVOXSKXAVOTVLVPUXHUWFUUED
|
|
UXHUWEUUDBQUXHUWDUUCCPUXHUWCUUBNUXHUWBUUAKUXHUWAYMFLUVOXSRLXBTVNVOTVLVPXC
|
|
XHXIXJXKXLXE $.
|
|
$}
|
|
|
|
$( Contracting the upper side of an interval which straddles a Dedekind cut.
|
|
Lemma for ~ prarloc . (Contributed by Jim Kingdon, 10-Nov-2019.) $)
|
|
prarloclemup $p |- ( ( ( X e. _om
|
|
/\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
|
|
/\ y e. _om ) -> (
|
|
( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U -> (
|
|
( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
|
|
E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) ) $=
|
|
( com wcel cop cnp cnq w3a wa coa co c1o ceq cec cmq cplq cv c2o ceq0 cplq0
|
|
cmq0 csuc wrex simpllr simprl simplr rspe syl12anc exp31 ) FGHEDIJHBEHCKHLM
|
|
ZAUAZGHZMZBUOUBNOZFNOPIQRCSOTODHZBUOPIUCRCUEOUDOEHZBURFUFNOPIQRCSOTODHZMZUT
|
|
USMZAGUGZUQUSMZVBMUPUTUSVDUNUPUSVBUHVEUTVAUIUQUSVBUJVCAGUKULUM $.
|
|
|
|
${
|
|
$d A y $. $d L y $. $d P y $. $d U y $. $d X y $.
|
|
$( Induction step for ~ prarloclem3 . (Contributed by Jim Kingdon,
|
|
9-Nov-2019.) $)
|
|
prarloclem3step $p |- (
|
|
( ( X e. _om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) )
|
|
/\ E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ->
|
|
E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
|
|
( com wcel cop wa c1o cec co coa ceq cmq cplq wrex wi ex cnp cnq w3a ceq0
|
|
cmq0 cplq0 c2o csuc nfv nfre1 prarloclemlo prarloclemup cltq prarloclemlt
|
|
cv wbr wo prloc 3ad2ant1 ad2antlr mpd mpjaod rexlimd imp ) FGHZEDIUAHZBEH
|
|
ZCUBHZUCZJZBAUOZKIUDLCUEMUFMEHZBVKUGNMZFUHNMKIOLCPMQMDHJZAGRVLBVMFNMKIOLC
|
|
PMQMZDHZJZAGRZVJVNVRAGVJAUIVQAGUJVJVKGHZVNVRSZVJVSJZBVKKNMKIOLCPMQMZEHZVT
|
|
VPABCDEFUKABCDEFULWAWBVOUMUPZWCVPUQZABCDEFUNVIWDWESZVEVSVFVGWFVHVFWDWEWBV
|
|
ODEURTUSUTVAVBTVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d A j x y z $. $d L j x y z $. $d P j x y z $. $d U j x y z $.
|
|
$d X x y $.
|
|
$( Contracting an interval which straddles a Dedekind cut. Lemma for
|
|
~ prarloc . (Contributed by Jim Kingdon, 27-Oct-2019.) $)
|
|
prarloclem3 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
|
|
( X e. _om /\ P e. Q. ) /\
|
|
E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ->
|
|
E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
|
|
( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
|
|
( cop wcel wa com c1o cec co coa ceq cmq cplq wrex wi vx vz cnp ceq0 cmq0
|
|
cnq cv cplq0 c2o simprl simpll simplr simprr w3a wceq oveq2 opeq1d eceq1d
|
|
oveq1d oveq2d eleq1d anbi2d rexbidv imbi1d imbi2d c0 csuc 2onn nnacl nna0
|
|
mpan2 rexbiia opeq1 oveq1 anbi12d cbvrexv bitri biimpi prarloclem3step ex
|
|
syl a1i imim1d finds2 vtoclga imp syl13anc 3impia ) FDHUCIZBFIZJZGKIZCUFI
|
|
ZJZBAUGZLHZUDMZCUENZUHNZFIZBWOUIONZGONZLHZPMZCQNZRNZDIZJZAKSZBEUGZLHZUDMZ
|
|
CUENZUHNZFIZBXJUIONZLHZPMZCQNZRNZDIZJZEKSZWKWNJWLWIWJWMXIYCTZWKWLWMUJWIWJ
|
|
WNUKWIWJWNULWKWLWMUMWLWIWJWMUNZYDYEWTBXAUAUGZONZLHZPMZCQNZRNZDIZJZAKSZYCT
|
|
ZTYEYDTUAGKYFGUOZYOYDYEYPYNXIYCYPYMXHAKYPYLXGWTYPYKXFDYPYJXEBRYPYIXDCQYPY
|
|
HXCPYPYGXBLYFGXAOUPUQURUSUTVAVBVCVDVEYOWTBXAVFONZLHZPMZCQNZRNZDIZJZAKSZYC
|
|
TZWTBXAUBUGZONZLHZPMZCQNZRNZDIZJZAKSZYCTZWTBXAUUFVGZONZLHZPMZCQNZRNZDIZJZ
|
|
AKSZYCTZYEUAUBYFVFUOZYNUUDYCUVFYMUUCAKUVFYLUUBWTUVFYKUUADUVFYJYTBRUVFYIYS
|
|
CQUVFYHYRPUVFYGYQLYFVFXAOUPUQURUSUTVAVBVCVDYFUUFUOZYNUUNYCUVGYMUUMAKUVGYL
|
|
UULWTUVGYKUUKDUVGYJUUJBRUVGYIUUICQUVGYHUUHPUVGYGUUGLYFUUFXAOUPUQURUSUTVAV
|
|
BVCVDYFUUPUOZYNUVDYCUVHYMUVCAKUVHYLUVBWTUVHYKUVADUVHYJUUTBRUVHYIUUSCQUVHY
|
|
HUURPUVHYGUUQLYFUUPXAOUPUQURUSUTVAVBVCVDUUEYEUUDYCUUDWTBXALHZPMZCQNZRNZDI
|
|
ZJZAKSYCUUCUVNAKWOKIZUUBUVMWTUVOUUAUVLDUVOYTUVKBRUVOYSUVJCQUVOYRUVIPUVOYQ
|
|
XALUVOUIKIZYQXAUOZVHUVOUVPJXAKIUVQWOUIVIXAVJWAVKUQURUSUTVAVBVLUVNYBAEKWOX
|
|
JUOZWTXOUVMYAUVRWSXNFUVRWRXMBUHUVRWQXLCUEUVRWPXKUDWOXJLVMURUSUTVAUVRUVLXT
|
|
DUVRUVKXSBRUVRUVJXRCQUVRUVIXQPUVRXAXPLWOXJUIOVNUQURUSUTVAVOVPVQVRWBUUFKIZ
|
|
YEUUOUVETUVSYEJZUVDUUNYCUVTUVDUUNABCDFUUFVSVTWCVTWDWEWFWGWH $.
|
|
$}
|
|
|
|
${
|
|
$d A j x y $. $d L j x y $. $d P j x y $. $d U j x y $.
|
|
$( A slight rearrangement of ~ prarloclem3 . Lemma for ~ prarloc .
|
|
(Contributed by Jim Kingdon, 4-Nov-2019.) $)
|
|
prarloclem4 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ P e. Q. ) -> (
|
|
E. x e. _om E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ->
|
|
E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
|
|
( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U )
|
|
) ) $=
|
|
( cop wcel wa cv c1o ceq0 cec cmq0 co cplq0 c2o coa com cnp cnq cplq wrex
|
|
ceq cmq wi prarloclem3 3expia ancom2s anassrs rexlimdva ) GEHUAICGIJZDUBI
|
|
ZJCBKZLHMNDOPQPGICUORSPAKZSPLHUENDUFPUCPEIJBTUDZCFKZLHMNDOPQPGICURRSPLHUE
|
|
NDUFPUCPEIJFTUDZATUMUNUPTIZUQUSUGZUMUTUNVAUMUTUNJUQUSBCDEFGUPUHUIUJUKUL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d N x $.
|
|
$( Subtracting two from a positive integer. Lemma for ~ prarloc .
|
|
(Contributed by Jim Kingdon, 5-Nov-2019.) $)
|
|
prarloclemn $p |- ( ( N e. N. /\ 1o <N N ) ->
|
|
E. x e. _om ( 2o +o x ) = N ) $=
|
|
( cnpi wcel c1o clti wbr wa csuc wss c2o cv coa co wceq com wrex mpan syl
|
|
wb simpl 1pi ltpiord biimpa word wi piord ordsucss sylc df-2o sseq1i pinn
|
|
2onn nnawordex adantr syl5bbr mpbid ) BCDZEBFGZHZEIZBJZKALMNBOAPQZUTUREBD
|
|
ZVBURUSUAURUSVDECDURUSVDTUBEBUCRUDURBUEVDVBUFBUGEBUHSUIVBKBJZUTVCKVABUJUK
|
|
URVEVCTZUSURBPDZVFBULKPDVGVFUMAKBUNRSUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d L x y $. $d N x $. $d P x y $. $d U x y $.
|
|
$( A substitution of zero for ` y ` and ` N ` minus two for ` x ` . Lemma
|
|
for ~ prarloc . (Contributed by Jim Kingdon, 4-Nov-2019.) $)
|
|
prarloclem5 $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
|
|
( N e. N. /\ P e. Q. /\ 1o <N N ) /\
|
|
( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) ->
|
|
E. x e. _om E. y e. _om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 .Q0 P ) ) e. L
|
|
/\ ( A +Q
|
|
( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
|
|
( cop wcel c1o ceq cec cmq co c2o coa wceq com c0 c0q0 cnp wa cnq wbr w3a
|
|
cnpi clti cplq wrex ceq0 cmq0 cplq0 prarloclemn 3adant2 3ad2ant2 3ad2ant1
|
|
cv elprnql simp22 cnq0 nqnq0 sseli nq0a0 syl df-0nq0 oveq1i nq0m0r oveq2d
|
|
syl5reqr sylan9req syl2anc simp1r eqeltrrd 2onn nna0r ax-mp eqeq1i opeq1d
|
|
biimpri eceq1d oveq1d eleq1d biimprcd 3ad2ant3 peano1 opeq1 oveq1 anbi12d
|
|
wi rspcev mpan syl6an reximdv mpd ) FEHUAIZCFIZUBZGUFIZDUCIZJGUGUDZUEZCGJ
|
|
HZKLZDMNZUHNZEIZUEZOAUQZPNZGQZARUIZCBUQZJHZUJLZDUKNZULNZFIZCXLOPNZXHPNZJH
|
|
ZKLZDMNZUHNZEIZUBZBRUIZARUIXAWQXKXFWRWTXKWSAGUMUNUOXGXJYFARXGCSJHZUJLZDUK
|
|
NZULNZFIZXJCSOPNZXHPNZJHZKLZDMNZUHNZEIZYFXGCYJFXGCUCIZWSCYJQWQXAYSXFCEFUR
|
|
UPWQWRWSWTXFUSYSWSCCTULNZYJYSCUTIYTCQUCUTCVAVBCVCVDWSTYICULWSYITDUKNZTTYH
|
|
DUKVEVFWSDUTIUUATQUCUTDVAVBDVGVDVIVHVJVKWOWPXAXFVLVMXFWQXJYRWIXAXJYRXFXJY
|
|
QXEEXJYPXDCUHXJYOXCDMXJYNXBKXJYMGJYMGQXJYMXIGYLOXHPORIYLOQVNOVOVPVFVQVSVR
|
|
VTWAVHWBWCWDSRIYKYRUBZYFWEYEUUBBSRXLSQZXQYKYDYRUUCXPYJFUUCXOYICULUUCXNYHD
|
|
UKUUCXMYGUJXLSJWFVTWAVHWBUUCYCYQEUUCYBYPCUHUUCYAYODMUUCXTYNKUUCXSYMJUUCXR
|
|
YLXHPXLSOPWGWAVRVTWAVHWBWHWJWKWLWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d A j x y $. $d L j x y $. $d N j x $. $d P j x y $. $d U j x y $.
|
|
$( A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly
|
|
spaced rational numbers from ` A ` to ` A +Q ( N .Q P ) ` (which are in
|
|
the lower and upper cuts, respectively, of a real number), there are a
|
|
pair of numbers, two positions apart in the even spacing, which straddle
|
|
the cut. (Contributed by Jim Kingdon, 22-Oct-2019.) $)
|
|
prarloclem $p |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\
|
|
( N e. N. /\ P e. Q. /\ 1o <N N ) /\
|
|
( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) ->
|
|
E. j e. _om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 .Q0 P ) ) e. L /\
|
|
( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) $=
|
|
( vy vx cop wcel wa c1o ceq cec cmq co cplq cv coa com cnp cnpi clti ceq0
|
|
cnq wbr w3a cmq0 cplq0 c2o wrex prarloclem5 prarloclem4 3ad2antr2 3adant3
|
|
wi mpd ) ECIUAJAEJKZFUBJZBUEJZLFUCUFZUGZAFLIMNBOPQPCJZUGAGRZLIUDNBUHPUIPE
|
|
JAVDUJSPHRSPLIMNBOPQPCJKGTUKHTUKZADRZLIUDNBUHPUIPEJAVFUJSPLIMNBOPQPCJKDTU
|
|
KZHGABCEFULURVBVEVGUPZVCURUSUTVHVAHGABCDEUMUNUOUQ $.
|
|
$}
|
|
|
|
$( Some calculations for ~ prarloc . (Contributed by Jim Kingdon,
|
|
26-Oct-2019.) $)
|
|
prarloclemcalc $p |- ( ( (
|
|
A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 .Q0 Q ) )
|
|
/\
|
|
B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
|
|
/\ (
|
|
( Q e. Q. /\ ( Q +Q Q ) <Q P )
|
|
/\ ( X e. Q. /\ M e. _om )
|
|
) ) -> B <Q ( A +Q P ) ) $=
|
|
( c1o ceq0 cmq0 co cplq0 wceq c2o wa cnq wcel syl2anc cnq0 cnpi syl cop cec
|
|
coa ceq cmq cplq cltq wbr com simprll nqnq0a oveq2d simprrl simprrr cxp cqs
|
|
simpll 1pi opelxpi enq0ex ecelqsi df-nq0 syl6eleqr sseldi mulclnq0 nqpnq0nq
|
|
mpan2 eqeltrd addclnq simplr c0 wne 2onn 2on0 elni mpbir2an nnppipi sylancl
|
|
nqnq0 enqex df-nqqs mulclnq nqnq0m oveq1d eqtr4d 3eqtrd nnanq0 mp3an3 mp2an
|
|
nqnq0pi ax-mp eleqtrri distnq0r eqtrd nq02m addclnq0 addassnq0 oveq1 eqeq2d
|
|
syl3anc wb mpbird 3eqtr4rd simprlr ltrelnq brel ltanqg 3expa mpbid eqbrtrd
|
|
) AFEGUAZHUBZDIJZKJZLZBFEMUCJZGUAZUDUBZDUEJZUFJZLZNZDOPZDDUFJZCUGUHZNZFOPZE
|
|
UIPZNZNZNZBAYDUFJZACUFJZUGYKAYDKJZADDKJZKJZYLBYKYDYOAKYKYCYCYDYOLYBYCYEYIUJ
|
|
ZYQDDUKQULYKAOPZYDOPZYLYNLYKAXNOXOYAYJUQZYKYGXMRPZXNOPYBYFYGYHUMZYKXLRPZDRP
|
|
ZUUAYKYHUUCYBYFYGYHUNZYHXLUISUOZHUPZRYHXKUUFPZXLUUGPYHGSPZUUHUREGUISUSVGUUF
|
|
XKHUTVATVBVCTZYKORDVSYQVDZXLDVEQZFXMVFQVHZYKYCYCYSYQYQDDVIQAYDUKQYKBYPLZBXN
|
|
YOKJZLZYKBFXQHUBZDIJZKJZFXMMGUAZHUBZDIJZKJZKJZUUOYKBXTFXSKJZUUSXOYAYJVJYKYG
|
|
XSOPZXTUVELUUBYKXROPZYCUVFYKYHUVGUUEYHXRSSUOZUDUPZOYHXQUVHPZXRUVIPYHXPSPZUU
|
|
IUVJYHMSPZUVKUVLMUIPZMVKVLVMVNMVOVPEMVQVGZURXPGSSUSVRUVHXQUDVTVATWAVCTZYQXR
|
|
DWBQFXSUKQYKXSUURFKYKXSXRDIJZUURYKUVGYCXSUVPLUVOYQXRDWCQYKUUQXRDIYKYHUUQXRL
|
|
ZUUEYHUVKUUIUVQUVNURXPGWJVRTWDWEULWFYKUURUVCFKYKUURXLUVAKJZDIJZUVCYKUUQUVRD
|
|
IYKYHUVMUUQUVRLZUUEVMYHUVMUUIUVTURGMEWGWHVRWDYKUUDUUCUVSUVCLZUUKUUJUUDUUCUV
|
|
ARPUWAUVAUUGRUUTUUFPZUVAUUGPUVMUUIUWBVMURMGUISUSWIUUFUUTHUTVAWKVBWLDXLUVAWM
|
|
WHQWNULYKUVDFXMYOKJZKJZUUOYKUUDUVDUWDLUUKUUDUVCUWCFKUUDUVBYOXMKDWOULULTYKFR
|
|
PUUAYORPZUUOUWDLYKORFVSUUBVDUULYKUUDUUDUWEUUKUUKDDWPQFXMYOWQWTWEWFYKXOUUNUU
|
|
PXAYTXOYPUUOBAXNYOKWRWSTXBXCYKYEYLYMUGUHZYBYCYEYIXDZYKYSCOPZNZYRYEUWFXAZYKY
|
|
EUWIUWGYDCOOUGXEXFTUUMYSUWHYRUWJYDCAXGXHQXIXJ $.
|
|
|
|
${
|
|
$d L a b m n q x y $. $d P a b m n q x y $. $d U a b m n q x y $.
|
|
$( A Dedekind cut is arithmetically located. Part of Proposition 11.15 of
|
|
[BauerTaylor], p. 52, slightly modified. It states that given a
|
|
tolerance ` P ` , there are elements of the lower and upper cut which
|
|
are within that tolerance of each other.
|
|
|
|
Usually, proofs will be shorter if they use ~ prarloc2 instead.
|
|
(Contributed by Jim Kingdon, 22-Oct-2019.) $)
|
|
prarloc $p |- ( ( <. L , U >. e. P. /\ P e. Q. ) ->
|
|
E. a e. L E. b e. U b <Q ( a +Q P ) ) $=
|
|
( vx vy vq vn vm wcel cnq wa cv co wex wrex df-rex sylib anbi2d cplq cltq
|
|
cop cnp wbr w3a cnpi c1o clti ceq cec prml adantr subhalfnqq adantl ancli
|
|
cmq prmu 19.42v sylibr eeeanv syl3anbrc prarloclemarch2 3adant1r 3adant2r
|
|
3com12 3adant3r 3adant3l 2eximi eximi com ceq0 cmq0 cplq0 c2o coa simpl1l
|
|
simp3rl simp3rr simp3ll simpl1r simprl simprrl simprrr wi simpl2r prcunqu
|
|
3jca syl2anc mpd prarloclem syl231anc wceq eleq1 anbi1d ceqsexgv biimprcd
|
|
exbidv sylc exbii simprl2 simprl3 simprl1 prarloclemcalc syl12anc ancom1s
|
|
jca simpl anasss syl exlimiv exlimivv 3syl excom anbi2i bitr4i ) CBUCUDKZ
|
|
ALKZMZDNZCKZENZBKZYBXTAUAOUBUEZMZMZEPZDPZYDEBQZDCQZXSYFDPEPZYHXSFNZLKZYLC
|
|
KZMZGNZLKZYPBKZMZXSHNZLKZYTYTUAOAUBUEZMZMZUFZHPGPZFPZUUEINZUGKZUHUUHUIUEZ
|
|
YPYLUUHUHUCUJUKYTUQOUAOZUBUEZMZMZMZIPZHPZGPZFPYKXSYOFPZYSGPZUUDHPZUUGXQUU
|
|
SXRXQYNFLQUUSFBCULYNFLRSUMXQUUTXRXQYRGLQUUTGBCURYRGLRSUMXSXSUUCHPZMUVAXSU
|
|
VBXSUUBHLQZUVBXRUVCXQHAUNUOUUBHLRSUPXSUUCHUSUTYOYSUUDFGHVAVBUUFUURFUUEUUP
|
|
GHUUEUUEUUNIPZMUUPUUEUVDYOYSUUCUVDXSYOYSUUAUVDUUBYOYQUUAUVDYRYMYQUUAUVDYN
|
|
YQYMUUAUVDYQYMUUAUFUUMIUGQUVDIYPYLYTVCUUMIUGRSVFVDVEVGVHUPUUEUUNIUSUTVIVJ
|
|
UUQYKFGUUOYKHIUUOYMUUAUUBUFZJNZVKKZYLUVFUHUCVLUKYTVMOVNOZCKZYLUVFVOVPOUHU
|
|
CUJUKYTUQOUAOZBKZMZMZMZJPZYKUUOUVEUVMJPZMUVOUUOUVEUVPUUOYMUUAUUBYMYNYSUUD
|
|
UUNVQUUEUUAUUNUUAUUBXSYOYSVRUMZUUEUUBUUNUUAUUBXSYOYSVSUMWHUUOUVLJVKQZUVPU
|
|
UOXQYNUUIUUAUUJUUKBKZUVRUUEXQUUNXQXRUUCYOYSVTUMZYMYNYSUUDUUNWAUUEUUIUUMWB
|
|
UVQUUEUUIUUJUULWCUUOUULUVSUUEUUIUUJUULWDUUOXQYRUULUVSWEUVTYQYRYOUUDUUNWFU
|
|
UKYPBCWGWIWJYLYTBJCUUHWKWLUVLJVKRSXGUVEUVMJUSUTUVNYKJUVNYBUVJWMZXTUVHWMZU
|
|
VEUVGYAYCMZMZMZMZMZDPZEPZYKUVNUWAUWFDPZMZEPZUWIUVNUWBUVEUVGYAUVKMZMZMZMZD
|
|
PZUVKUWLUVNUVIUWQUVEUVGUVIUVKWCUVIUWQUVNUWOUVNDUVHCUWBUWNUVMUVEUWBUWMUVLU
|
|
VGUWBYAUVIUVKXTUVHCWNWOTTWPWQWJUVEUVGUVIUVKWDUVKUWLUWQUWJUWQEUVJBUWAUWFUW
|
|
PDUWAUWEUWOUWBUWAUWDUWNUVEUWAUWCUWMUVGUWAYCUVKYAYBUVJBWNTTTTWRWPWQWSUWHUW
|
|
KEUWAUWFDUSWTUTUWGYFEDUWAUWBUWEYFUWBUWAUWEYFUWBUWAMZUWEMZYAYEUWEYAUWRUVEU
|
|
VGYAYCWCUOUWSYCYDUWEYCUWRUVEUVGYAYCWDUOUWSUWRUUCYMUVGMYDUWRUWEXHUWSUUAUUB
|
|
YMUUAUUBUWDUWRXAYMUUAUUBUWDUWRXBXGUWSYMUVGYMUUAUUBUWDUWRXCUWRUVEUVGUWCWCX
|
|
GXTYBAYTUVFYLXDXEXGXGXFXIVIXJXKXJXLXLXMYFEDXNSYHYAYIMZDPYJYGUWTDYGYAYEEPZ
|
|
MUWTYAYEEUSYIUXAYAYDEBRXOXPWTYIDCRXPS $.
|
|
$}
|
|
|
|
${
|
|
$d L a b $. $d P a b $. $d U a b $.
|
|
$( A Dedekind cut is arithmetically located. This is a variation of
|
|
~ prarloc which only constructs one (named) point and is therefore often
|
|
easier to work with. It states that given a tolerance ` P ` , there are
|
|
elements of the lower and upper cut which are exactly that tolerance
|
|
from each other. (Contributed by Jim Kingdon, 26-Dec-2019.) $)
|
|
prarloc2 $p |- ( ( <. L , U >. e. P. /\ P e. Q. ) ->
|
|
E. a e. L ( a +Q P ) e. U ) $=
|
|
( vb cop cnp wcel cnq wa cv cplq co cltq wbr prarloc wi prcunqu rexlimdva
|
|
wrex reximdv adantr mpd ) CBFGHZAIHZJEKZDKALMZNOZEBTZDCTZUGBHZDCTZABCDEPU
|
|
DUJULQUEUDUIUKDCUDUHUKEBUGUFBCRSUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d q x y $.
|
|
$( Positive real 'less than' is a relation on positive reals. (Contributed
|
|
by NM, 14-Feb-1996.) $)
|
|
ltrelpr $p |- <P C_ ( P. X. P. ) $=
|
|
( vx vy vq cltp cv cnp wcel c2nd cfv c1st cnq wrex copab df-iltp opabssxp
|
|
wa cxp eqsstri ) DAEZFGBEZFGPCEZSHIGUATJIGPCKLZPABMFFQABCNUBABFFOR $.
|
|
$}
|
|
|
|
${
|
|
$d A q x y $. $d B q x y $.
|
|
$( More convenient form of ~ df-iltp . (Contributed by Jim Kingdon,
|
|
15-Dec-2019.) $)
|
|
ltdfpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <->
|
|
E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) $=
|
|
( vx vy cltp wbr cop cv cnp wcel wa c2nd cfv c1st wrex wceq fveq2d eleq2d
|
|
cnq copab df-br df-iltp eleq2i bitri simpl simpr anbi12d rexbidv syl5bb
|
|
opelopab2a ) ABFGZABHZDIZJKEIZJKLCIZUNMNZKZUPUOONZKZLZCTPZLDEUAZKZAJKBJKL
|
|
UPAMNZKZUPBONZKZLZCTPZULUMFKVDABFUBFVCUMDECUCUDUEVBVJDEABJJUNAQZUOBQZLZVA
|
|
VICTVMURVFUTVHVMUQVEUPVMUNAMVKVLUFRSVMUSVGUPVMUOBOVKVLUGRSUHUIUKUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A s $. $d ph q r s $.
|
|
genpdflem.r $e |- ( ( ph /\ r e. A ) -> r e. Q. ) $.
|
|
genpdflem.s $e |- ( ( ph /\ s e. B ) -> s e. Q. ) $.
|
|
$( Simplification of upper or lower cut expression. Lemma for ~ genpdf .
|
|
(Contributed by Jim Kingdon, 30-Sep-2019.) $)
|
|
genpdflem $p |- ( ph -> { q e. Q. | E. r e. Q. E. s e. Q.
|
|
( r e. A /\ s e. B /\ q = ( r G s ) ) } =
|
|
{ q e. Q. | E. r e. A E. s e. B q = ( r G s ) } ) $=
|
|
( cv wcel cnq wrex wa wex ex pm4.71rd anbi1d exbidv df-rex co wceq 3anass
|
|
w3a rexbii r19.42v bitri anass exbii bitr4i syl6bbr rexbidv bitrd rabbidv
|
|
syl6rbbr ) AFJZBKZEJZCKZGJUPURDUAUBZUDZELMZFLMZUTECMZFBMZGLAVCUSUTNZELMZF
|
|
BMZVEAVCUQVGNZFOZVHAVJUPLKZUQNZVGNZFOZVCAVIVMFAUQVLVGAUQVKAUQVKHPQRSVCVKV
|
|
INZFOZVNVCVIFLMVPVBVIFLVBUQVFNZELMVIVAVQELUQUSUTUCUEUQVFELUFUGUEVIFLTUGVM
|
|
VOFVKUQVGUHUIUJUOVGFBTUKAVGVDFBAVGVFEOZVDAVRURLKZUSNZUTNZEOZVGAVFWAEAUSVT
|
|
UTAUSVSAUSVSIPQRSVGVSVFNZEOWBVFELTWAWCEVSUSUTUHUIUJUOUTECTUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d q r s v w $.
|
|
genpdf.1 $e |- F = ( w e. P. , v e. P. |->
|
|
<. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\
|
|
s e. ( 1st ` v ) /\ q = ( r G s ) ) } ,
|
|
{ q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\
|
|
s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. ) $.
|
|
$( Simplified definition of addition or multiplication on positive reals.
|
|
(Contributed by Jim Kingdon, 30-Sep-2019.) $)
|
|
genpdf $p |- F = ( w e. P. , v e. P. |-> <.
|
|
{ q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } ,
|
|
{ q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) }
|
|
>. ) $=
|
|
( cnp cv c1st cfv wcel w3a cnq wrex crab c2nd cop sylan wceq prop elprnql
|
|
co cmpt2 wa adantlr adantll genpdflem elprnqu opeq12d mpt2eq3ia eqtri ) C
|
|
ABIIFJZAJZKLZMZEJZBJZKLZMZGJUNURDUDUAZNEOPFOPGOQZUNUORLZMZURUSRLZMZVBNEOP
|
|
FOPGOQZSZUEABIIVBEUTPFUPPGOQZVBEVFPFVDPGOQZSZUEHABIIVIVLUOIMZUSIMZUFZVCVJ
|
|
VHVKVOUPUTDEFGVMUQUNOMZVNVMUPVDSIMZUQVPUOUBZUNVDUPUCTUGVNVAUROMZVMVNUTVFS
|
|
IMZVAVSUSUBZURVFUTUCTUHUIVOVDVFDEFGVMVEVPVNVMVQVEVPVRUNVDUPUJTUGVNVGVSVMV
|
|
NVTVGVSWAURVFUTUJTUHUIUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z f g q r s A $. $d x y z f g q r s B $.
|
|
$d x y z f g w v q r s G $. $d f g F $.
|
|
genp.1 $e |- F = ( w e. P. , v e. P. |->
|
|
<. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\
|
|
z e. ( 1st ` v ) /\ x = ( y G z ) ) } ,
|
|
{ x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\
|
|
z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. ) $.
|
|
genp.2 $e |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. ) $.
|
|
$( Value of general operation (addition or multiplication) on positive
|
|
reals. (Contributed by Jim Kingon, 3-Oct-2019.) $)
|
|
genipv $p |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <.
|
|
{ q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } ,
|
|
{ q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) }
|
|
>. ) $=
|
|
( wcel wceq cfv wrex cnq crab vf vg cnp wa co c1st c2nd cop oveq1 rexeqdv
|
|
cv fveq2 rabbidv opeq12d eqeq12d oveq2 rexbidv cvv cxp a1i cab rabssab wi
|
|
nqex elprnql sylan eleq1 syl5ibrcom syl2an rexlimdvva abssdv syl5ss ssexd
|
|
prop an4s elprnqu opelxp sylanbrc ovmpt2g mpd3an3 vtocl2ga eqeq1 2rexbidv
|
|
genpdf eqeq2d cbvrex2v syl6bb cbvrabv opeq12i syl6eq ) FUCOGUCOUDFGHUEZAU
|
|
KZBUKZCUKZIUEZPZCGUFQZRZBFUFQZRZASTZWPCGUGQZRZBFUGQZRZASTZUHZLUKZKUKZJUKZ
|
|
IUEZPZJWQRKWSRZLSTZXLJXBRKXDRZLSTZUHUAUKZUBUKZHUEZWPCXRUFQZRZBXQUFQZRZAST
|
|
ZWPCXRUGQZRZBXQUGQZRZASTZUHZPZFXRHUEZYABWSRZASTZYFBXDRZASTZUHZPWKXGPUAUBF
|
|
GUCUCXQFPZXSYLYJYQXQFXRHUIYRYDYNYIYPYRYCYMASYRYABYBWSXQFUFULUJUMYRYHYOASY
|
|
RYFBYGXDXQFUGULUJUMUNUOXRGPZYLWKYQXGXRGFHUPYSYNXAYPXFYSYMWTASYSYAWRBWSYSW
|
|
PCXTWQXRGUFULUJUQUMYSYOXEASYSYFXCBXDYSWPCYEXBXRGUGULUJUQUMUNUOXQUCOZXRUCO
|
|
ZYJURURUSZOZYKYTUUAUDZYDUROYIUROUUCUUDYDSURSUROUUDVDUTZUUDYDYCAVASYCASVBU
|
|
UDYCASUUDWPWLSOZBCYBXTYTWMYBOZUUAWNXTOZWPUUFVCZYTUUGUDWMSOZWNSOZUUIUUAUUH
|
|
UDYTYBYGUHUCOZUUGUUJXQVNZWMYGYBVEVFUUAXTYEUHUCOZUUHUUKXRVNZWNYEXTVEVFUUJU
|
|
UKUDUUFWPWOSONWLWOSVGVHZVIVOVJVKVLVMUUDYISURUUEUUDYIYHAVASYHASVBUUDYHASUU
|
|
DWPUUFBCYGYEYTWMYGOZUUAWNYEOZUUIYTUUQUDUUJUUKUUIUUAUURUDYTUULUUQUUJUUMWMY
|
|
GYBVPVFUUAUUNUURUUKUUOWNYEXTVPVFUUPVIVOVJVKVLVMYDYIURURVQVRDEXQXRUCUCWPCE
|
|
UKZUFQZRZBDUKZUFQZRZASTZWPCUUSUGQZRZBUVBUGQZRZASTZUHYJHUVABYBRZASTZUVGBYG
|
|
RZASTZUHUUBUVBXQPZUVEUVLUVJUVNUVOUVDUVKASUVOUVABUVCYBUVBXQUFULUJUMUVOUVIU
|
|
VMASUVOUVGBUVHYGUVBXQUGULUJUMUNUUSXRPZUVLYDUVNYIUVPUVKYCASUVPUVAYABYBUVPW
|
|
PCUUTXTUUSXRUFULUJUQUMUVPUVMYHASUVPUVGYFBYGUVPWPCUVFYEUUSXRUGULUJUQUMUNDE
|
|
HICBAMWDVSVTWAXAXNXFXPWTXMALSWLXHPZWTXHWOPZCWQRBWSRXMUVQWPUVRBCWSWQWLXHWO
|
|
WBZWCUVRXLXHXIWNIUEZPZBCKJWSWQWMXIPWOUVTXHWMXIWNIUIWEZWNXJPUVTXKXHWNXJXII
|
|
UPWEZWFWGWHXEXOALSUVQXEUVRCXBRBXDRXOUVQWPUVRBCXDXBUVSWCUVRXLUWABCKJXDXBUW
|
|
BUWCWFWGWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. $d G x y z $.
|
|
genplt2i.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
|
|
( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
|
|
genplt2i.com $e |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) ) $.
|
|
$( Operating on both sides of two inequalities, when the operation is
|
|
consistent with ` <Q ` . (Contributed by Jim Kingdon, 6-Oct-2019.) $)
|
|
genplt2i $p |- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) ) $=
|
|
( cltq wbr wa co cnq cv wcel adantl ltrelnq syl2an simpl wb simpll simplr
|
|
w3a brel simprl wceq caovord2d mpbid simpr simprr caovordd ltsonq syl2anc
|
|
sotri ) DEKLZFGKLZMZDFHNZEFHNZKLZVAEGHNZKLZUTVCKLUSUQVBUQURUAUSABCDEFKOHA
|
|
PZOQZBPZOQZCPZOQUEVEVGKLVIVEHNVIVGHNKLUBUSIRZUQDOQZEOQZMZFOQZGOQZMZVKURDE
|
|
OOKSUFZFGOOKSUFZVKVLVPUCTUQVMVPVLURVQVRVKVLVPUDTZUQVMVPVNURVQVRVMVNVOUGTZ
|
|
VFVHMVEVGHNVGVEHNUHUSJRUIUJUSURVDUQURUKUSABCFGEKOHVJVTUQVMVPVOURVQVRVMVNV
|
|
OULTVSUMUJUTVAVCKOUNSUPUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z f g h w v q A $. $d x y z f g h w v q B $.
|
|
$d x y z f g h w v q G $. $d f g q F $. $d f g h C $. $d f g h D $.
|
|
genpelvl.1 $e |- F = ( w e. P. , v e. P. |->
|
|
<. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\
|
|
z e. ( 1st ` v ) /\ x = ( y G z ) ) } ,
|
|
{ x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\
|
|
z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. ) $.
|
|
$( Set containing the result of adding or multiplying positive reals.
|
|
(Contributed by Jim Kingdon, 5-Dec-2019.) $)
|
|
genpelxp $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
( A F B ) e. ( ~P Q. X. ~P Q. ) ) $=
|
|
( cnp wcel cv c1st cfv w3a cnq wrex crab c2nd wceq cop cpw cxp wss ssrab2
|
|
wa co nqex elpw2 mpbir opelxpi mp2an fveq2 eleq2d 3anbi1d rabbidv opeq12d
|
|
2rexbidv 3anbi2d ovmpt2g mp3an3 syl6eqel ) FKLZGKLZUGFGHUHZBMZFNOZLZCMZGN
|
|
OZLZAMVGVJIUHUAZPZCQRBQRZAQSZVGFTOZLZVJGTOZLZVMPZCQRBQRZAQSZUBZQUCZWEUDZV
|
|
DVEWDWFLZVFWDUAVPWELZWCWELZWGWHVPQUEVOAQUFVPQUIUJUKWIWCQUEWBAQUFWCQUIUJUK
|
|
VPWCWEWEULUMZDEFGKKVGDMZNOZLZVJEMZNOZLZVMPZCQRBQRZAQSZVGWKTOZLZVJWNTOZLZV
|
|
MPZCQRBQRZAQSZUBWDHVIWPVMPZCQRBQRZAQSZVRXCVMPZCQRBQRZAQSZUBWFWKFUAZWSXIXF
|
|
XLXMWRXHAQXMWQXGBCQQXMWMVIWPVMXMWLVHVGWKFNUNUOUPUSUQXMXEXKAQXMXDXJBCQQXMX
|
|
AVRXCVMXMWTVQVGWKFTUNUOUPUSUQURWNGUAZXIVPXLWCXNXHVOAQXNXGVNBCQQXNWPVLVIVM
|
|
XNWOVKVJWNGNUNUOUTUSUQXNXKWBAQXNXJWABCQQXNXCVTVRVMXNXBVSVJWNGTUNUOUTUSUQU
|
|
RJVAVBWJVC $.
|
|
|
|
genpelvl.2 $e |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. ) $.
|
|
$( Membership in lower cut of general operation (addition or
|
|
multiplication) on positive reals. (Contributed by Jim Kingdon,
|
|
2-Oct-2019.) $)
|
|
genpelvl $p |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) <->
|
|
E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) ) $=
|
|
( vf wcel wa cnq cfv wrex cnp co c1st cv wceq crab c2nd cop genipv fveq2d
|
|
rabex op1st syl6eq eleq2d elrabi syl6bi elprnql sylan caovcl syl2an eleq1
|
|
nqex prop an4s syl5ibrcom rexlimdvva wb eqeq1 2rexbidv elrab3 sylan9bb ex
|
|
pm5.21ndd ) FUAPZGUAPZQZHRPZHFGKUBZUCSZPZHIUDZJUDZLUBZUEZJGUCSZTIFUCSZTZV
|
|
PVTHOUDZWCUEZJWETIWFTZORUFZPZVQVPVSWKHVPVSWKWIJGUGSZTIFUGSZTZORUFZUHZUCSW
|
|
KVPVRWQUCABCDEFGKLJIOMNUIUJWKWPWJORVBUKWOORVBUKULUMUNZWJOHRUOUPVPWDVQIJWF
|
|
WEVPWAWFPZWBWEPZQQVQWDWCRPZVNWSVOWTXAVNWSQWARPZWBRPZXAVOWTQVNWFWNUHUAPWSX
|
|
BFVCWAWNWFUQURVOWEWMUHUAPWTXCGVCWBWMWEUQURBCWAWBRLNUSUTVDHWCRVAVEVFVPVQVT
|
|
WGVGVPVTWLVQWGWRWJWGOHRWHHUEWIWDIJWFWEWHHWCVHVIVJVKVLVM $.
|
|
|
|
$( Membership in upper cut of general operation (addition or
|
|
multiplication) on positive reals. (Contributed by Jim Kingdon,
|
|
15-Oct-2019.) $)
|
|
genpelvu $p |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <->
|
|
E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) ) $=
|
|
( vf wcel wa cnq cfv wrex cnp co c2nd cv wceq crab c1st cop genipv fveq2d
|
|
rabex op2nd syl6eq eleq2d elrabi syl6bi elprnqu sylan caovcl syl2an eleq1
|
|
nqex prop an4s syl5ibrcom rexlimdvva wb eqeq1 2rexbidv elrab3 sylan9bb ex
|
|
pm5.21ndd ) FUAPZGUAPZQZHRPZHFGKUBZUCSZPZHIUDZJUDZLUBZUEZJGUCSZTIFUCSZTZV
|
|
PVTHOUDZWCUEZJWETIWFTZORUFZPZVQVPVSWKHVPVSWIJGUGSZTIFUGSZTZORUFZWKUHZUCSW
|
|
KVPVRWQUCABCDEFGKLJIOMNUIUJWPWKWOORVBUKWJORVBUKULUMUNZWJOHRUOUPVPWDVQIJWF
|
|
WEVPWAWFPZWBWEPZQQVQWDWCRPZVNWSVOWTXAVNWSQWARPZWBRPZXAVOWTQVNWNWFUHUAPWSX
|
|
BFVCWAWFWNUQURVOWMWEUHUAPWTXCGVCWBWEWMUQURBCWAWBRLNUSUTVDHWCRVAVEVFVPVQVT
|
|
WGVGVPVTWLVQWGWRWJWGOHRWHHUEWIWDIJWFWEWHHWCVHVIVJVKVLVM $.
|
|
|
|
$( Pre-closure law for general operation on lower cuts. (Contributed by
|
|
Jim Kingdon, 2-Oct-2019.) $)
|
|
genpprecll $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
( ( C e. ( 1st ` A ) /\ D e. ( 1st ` B ) ) ->
|
|
( C G D ) e. ( 1st ` ( A F B ) ) ) ) $=
|
|
( vg vh c1st cfv wcel wa co cnp cv wceq wrex eqid rspceov mp3an3 genpelvl
|
|
syl5ibr ) HFPQZRZIGPQZRZSHIKTZFGJTPQRFUARGUARSUNNUBOUBKTUCOULUDNUJUDZUKUM
|
|
UNUNUCUOUNUENOUJULHIUNKUFUGABCDEFGUNNOJKLMUHUI $.
|
|
|
|
$( Pre-closure law for general operation on upper cuts. (Contributed by
|
|
Jim Kingdon, 7-Nov-2019.) $)
|
|
genppreclu $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
( ( C e. ( 2nd ` A ) /\ D e. ( 2nd ` B ) ) ->
|
|
( C G D ) e. ( 2nd ` ( A F B ) ) ) ) $=
|
|
( vg vh c2nd cfv wcel wa co cnp cv wceq wrex eqid rspceov mp3an3 genpelvu
|
|
syl5ibr ) HFPQZRZIGPQZRZSHIKTZFGJTPQRFUARGUARSUNNUBOUBKTUCOULUDNUJUDZUKUM
|
|
UNUNUCUOUNUENOUJULHIUNKUFUGABCDEFGUNNOJKLMUHUI $.
|
|
|
|
$( Domain of general operation on positive reals. (Contributed by Jim
|
|
Kingdon, 2-Oct-2019.) $)
|
|
genipdm $p |- dom F = ( P. X. P. ) $=
|
|
( cnp cv c1st cfv wcel w3a cnq wrex crab c2nd nqex co wceq cop rabex opex
|
|
dmmpt2 ) DEJJBKZDKZLMNCKZEKZLMNAKUGUIGUAUBZOCPQBPQZAPRZUGUHSMNUIUJSMNUKOC
|
|
PQBPQZAPRZUCFHUMUOULAPTUDUNAPTUDUEUF $.
|
|
|
|
$( The lower cut produced by addition or multiplication on positive reals
|
|
is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) $)
|
|
genpml $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
E. q e. Q. q e. ( 1st ` ( A F B ) ) ) $=
|
|
( vf vg cnp wcel wa cv cfv cnq c1st co wrex wex c2nd prop prml rexex 3syl
|
|
cop adantr ad2antlr genpprecll imp elprnql sylan anim12i an4s caovcl wceq
|
|
syl simpr eleq1d rspcedv mpd anassrs exlimddv ) FOPZGOPZQZMRZFUASZPZJRZFG
|
|
HUBUASZPZJTUCZMVHVMMUDZVIVHVLFUESZUJOPZVMMTUCVRFUFZMVSVLUGVMMTUHUIUKVJVMQ
|
|
NRZGUASZPZVQNVIWDNUDZVHVMVIWCGUESZUJOPZWDNTUCWEGUFZNWFWCUGWDNTUHUIULVJVMW
|
|
DVQVJVMWDQZQZVKWBIUBZVOPZVQVJWIWLABCDEFGVKWBHIKLUMUNWJVPWLJWKTWJVKTPZWBTP
|
|
ZQZWKTPVHVMVIWDWOVHVMQWMVIWDQWNVHVTVMWMWAVKVSVLUOUPVIWGWDWNWHWBWFWCUOUPUQ
|
|
URBCVKWBTILUSVAWJVNWKUTZQVNWKVOWJWPVBVCVDVEVFVGVG $.
|
|
|
|
$( The upper cut produced by addition or multiplication on positive reals
|
|
is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) $)
|
|
genpmu $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
E. q e. Q. q e. ( 2nd ` ( A F B ) ) ) $=
|
|
( vf vg cnp wcel wa cv cfv cnq c2nd co wrex wex c1st prop prmu rexex 3syl
|
|
cop adantr ad2antlr genppreclu imp elprnqu sylan anim12i an4s caovcl wceq
|
|
syl simpr eleq1d rspcedv mpd anassrs exlimddv ) FOPZGOPZQZMRZFUASZPZJRZFG
|
|
HUBUASZPZJTUCZMVHVMMUDZVIVHFUESZVLUJOPZVMMTUCVRFUFZMVLVSUGVMMTUHUIUKVJVMQ
|
|
NRZGUASZPZVQNVIWDNUDZVHVMVIGUESZWCUJOPZWDNTUCWEGUFZNWCWFUGWDNTUHUIULVJVMW
|
|
DVQVJVMWDQZQZVKWBIUBZVOPZVQVJWIWLABCDEFGVKWBHIKLUMUNWJVPWLJWKTWJVKTPZWBTP
|
|
ZQZWKTPVHVMVIWDWOVHVMQWMVIWDQWNVHVTVMWMWAVKVLVSUOUPVIWGWDWNWHWBWCWFUOUPUQ
|
|
URBCVKWBTILUSVAWJVNWKUTZQVNWKVOWJWPVBVCVDVEVFVGVG $.
|
|
|
|
${
|
|
$d F h $.
|
|
genpcdl.2 $e |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\
|
|
( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) ->
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|
( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) ) $.
|
|
$( Downward closure of an operation on positive reals. (Contributed by
|
|
Jim Kingdon, 14-Oct-2019.) $)
|
|
genpcdl $p |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) ->
|
|
( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) ) $=
|
|
( wcel wa cv cltq wi cnp wbr c1st cfv cnq ltrelnq brel simpld wceq wrex
|
|
co genpelvl adantr breq2 biimpd sylan9r exp31 impancom rexlimdvv sylbid
|
|
wb an4s ex syl5 com34 pm2.43d com23 ) FUAPZGUAPZQZARZHRZSUBZVLFGKUKUCUD
|
|
ZPZVKVNPZVJVMVOVPTVJVMVOVMVPVMVKUEPZVJVOVMVPTZTZVMVQVLUEPVKVLUEUESUFUGU
|
|
HVJVQVSVJVQQZVOVLIRZJRZLUKZUIZJGUCUDZUJIFUCUDZUJZVRVJVOWGVAVQABCDEFGVLI
|
|
JKLMNULUMVTWDVRIJWFWEVJWAWFPZWBWEPZQVQWDVRTZVHWHVIWIVQWJTVHWHQVIWIQQZVQ
|
|
WDVRWDVMVKWCSUBZWKVQQVPWDVMWLVLWCVKSUNUOOUPUQVBURUSUTVCVDVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d F h $.
|
|
genpcuu.2 $e |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\
|
|
( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) ->
|
|
( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) $.
|
|
$( Upward closure of an operation on positive reals. (Contributed by Jim
|
|
Kingdon, 8-Nov-2019.) $)
|
|
genpcuu $p |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) ->
|
|
( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) $=
|
|
( wcel wa cv cltq wi cnp wbr c2nd cfv cnq ltrelnq brel simprd wceq wrex
|
|
co genpelvu adantr breq1 biimpd sylan9r exp31 impancom rexlimdvv sylbid
|
|
wb an4s ex syl5 com34 pm2.43d com23 ) FUAPZGUAPZQZHRZARZSUBZVKFGKUKUCUD
|
|
ZPZVLVNPZVJVMVOVPTVJVMVOVMVPVMVLUEPZVJVOVMVPTZTZVMVKUEPVQVKVLUEUESUFUGU
|
|
HVJVQVSVJVQQZVOVKIRZJRZLUKZUIZJGUCUDZUJIFUCUDZUJZVRVJVOWGVAVQABCDEFGVKI
|
|
JKLMNULUMVTWDVRIJWFWEVJWAWFPZWBWEPZQVQWDVRTZVHWHVIWIVQWJTVHWHQVIWIQQZVQ
|
|
WDVRWDVMWCVLSUBZWKVQQVPWDVMWLVKWCVLSUNUOOUPUQVBURUSUTVCVDVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c d q r v w x y z $. $d B a b c d g h q r v w x y z $.
|
|
$d F a b c d g h q r v w x y z $. $d G a b c d g h q r v w x y z $.
|
|
genprndl.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
|
|
( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
|
|
genprndl.com $e |- ( ( x e. Q. /\ y e. Q. ) ->
|
|
( x G y ) = ( y G x ) ) $.
|
|
genprndl.lower $e |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\
|
|
( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) ->
|
|
( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) ) $.
|
|
$( The lower cut produced by addition or multiplication on positive reals
|
|
is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) $)
|
|
genprndl $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
A. q e. Q. ( q e. ( 1st ` ( A F B ) ) <->
|
|
E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) ) $=
|
|
( wcel wa vc vd va vb cnp cv co c1st cfv cltq wbr wrex wb wceq genpelvl
|
|
cnq wex r2ex syl6bb biimpa adantrl c2nd prop prnmaxl sylan anim12i an4s
|
|
wi cop reeanv sylibr genplt2i reximi syl adantrr breq1 biimprd ad2antll
|
|
reximdv mpd exlimdvv adantr genpprecll imp elprnql caovcl breq2 anbi12d
|
|
eleq1 adantl rspcedv mpan2d rexlimdvva expr wal genpcdl alrimdv imbi12d
|
|
ex cbvalv syl6ib syl6 impd ancomsd ad2antrr rexlimdva impbid ralrimiva
|
|
sp ) FUESZGUESZTZMUFZFGJUGUHUIZSZXMLUFZUJUKZXPXNSZTZLUPULZUMMUPXLXMUPSZ
|
|
TZXOXTXLYAXOXTXLYAXOTZTZXMUAUFZUBUFZKUGZUJUKZUBGUHUIZULZUAFUHUIZULZXTYD
|
|
UCUFZYKSZUDUFZYISZTZXMYMYOKUGZUNZTZUDUQUCUQZYLXLXOUUAYAXLXOUUAXLXOYSUDY
|
|
IULUCYKULUUAABCDEFGXMUCUDJKNOUOYSUCUDYKYIURUSUTVAXLUUAYLVHYCXLYTYLUCUDX
|
|
LYTYLXLYTTYRYGUJUKZUBYIULZUAYKULZYLXLYQUUDYSXLYQTZYMYEUJUKZYOYFUJUKZTZU
|
|
BYIULZUAYKULZUUDUUEUUFUAYKULZUUGUBYIULZTZUUJXJYNXKYPUUMXJYNTUUKXKYPTUUL
|
|
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$}
|
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( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
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genprndu.com $e |- ( ( x e. Q. /\ y e. Q. ) ->
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( x G y ) = ( y G x ) ) $.
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genprndu.upper $e |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\
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genprndu $p |- ( ( A e. P. /\ B e. P. ) ->
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BYIULZUAYKULZUUDUUEUUFUAYKULZUUGUBYIULZTZUUJXJYNXKYPUUMXJYNTUUKXKYPTUUL
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XJFVBUIZYKVIUESZYNUUKFVCZUAYMYKUUNVDVEXKGVBUIZYIVIUESZYPUULGVCZUBYOYIUU
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$}
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$d F a b c d q $. $d G a b c d q v w x y z $.
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genpdisj.ord $e |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) ->
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( x <Q y <-> ( z G x ) <Q ( z G y ) ) ) $.
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genpdisj.com $e |-
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( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) ) $.
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$( The lower and upper cuts produced by addition or multiplication on
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positive reals are disjoint. (Contributed by Jim Kingdon,
|
|
15-Oct-2019.) $)
|
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genpdisj $p |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q.
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cltq prop prltlu 3expib syl im2anan9 genplt2i syl6 syl5bir imp adantrlr
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wi adantlr adantrrr eqtr2 ad2ant2l adantl ltsonq soirri breq2 pm2.21fal
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ltrelnq mtbii ex exlimdvv mpd inegd ralrimivw ) FUASZGUASZTZJUBZFGHUCZU
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DUESZWOWPUFUESZTZUGJUHWNWSWNWSTZOUBZFUDUEZSZPUBZGUDUEZSZTZWOXAXDIUCZUIZ
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HYGYHXCXMTZXFXPTZTZWNYGXCXMXFXPVBWNYKXAXKVDVAZXDXNVDVAZTYGWLYIYLWMYJYMW
|
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LXBXLVCUASZYIYLVOFVEYNXCXMYLXAXKXLXBVFVGVHWMXEXOVCUASZYJYMVOGVEYOXFXPYM
|
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XDXNXOXEVFVGVHVIABCXAXKXDXNIMNVJVKVLVMVPVNVQYFXHXRUIZYGUGYAYPWTXIXSYPXG
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XQWOXHXRVRVSVTYPXHXHVDVAYGXHVDUHWAWEWBXHXRXHVDWCWFVHWDWGWHWHWIWJWK $.
|
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$}
|
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${
|
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$d A f g h t v w x y z $. $d B f g h t v w x y z $.
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$d C f g h t v w x y z $. $d F f g h t v w x y z $.
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$d G f g h t v w x y z $.
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genpassg.4 $e |- dom F = ( P. X. P. ) $.
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genpassg.5 $e |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. ) $.
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genpassg.6 $e |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) ->
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Jim Kingdon, 11-Dec-2019.) $)
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genpassl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
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( vt wa cnp wcel w3a co c1st cfv cv wceq wrex cnq c2nd cop prop elprnql
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rexcom4 3bitr4rd 3impb 3adant3 3impa oveq1 rexbidv pm5.32i exbii rexbii
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|
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UFZXBAUGZIUGZSUGZMUDZUHZSXEUEUFZUIZIFUEUFZUIZXGXHJUGZMUDZKUGZMUDZUHZKHU
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EUFZUIZJGUEUFZUIZIXNUIZXGXFUBZXGXDUBZWSWTXAXOYEUOZWTXATZWSYHYIWSTXMYDIX
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|
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|
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STZUUQXRUJUBZUUTXAYAHUKUFZULUAUBUUSUVBHUMXRUVCYAUNUPZYMUUMUVBUUTYMUUMUV
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BUCZYRXKXTYRUVEXKXTUOYRUVETZXJXSXGUVFXJXHYQMUDZXSYRXJUVGUHUVEXIYQXHMURU
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SUVEXSUVGUHYRRUTVAVBVCVDVEVGVHVIVJVKVGVHVIUUCXKJYCUQVLVPVMYIYDUUBUOYMYI
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VSVTVGVHVIYSKSYAWFVLWAVGVHVIWAYTJSYCWFVLUSYIXMUUFUOYMXMXIXLUBZXKTZSVFYI
|
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UUFXKSXLWBYIUVQUUESYIUVPUUDXKABCDEGHXIJKLMNOWCWDVMWEUSWGVGVHVIWAWHWSWTX
|
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AYFXOUOZYIWSXEUAUBUVRIJGHUALQVNABCDEFXEXGISLMNOWCVGWHXBXGXIXRMUDZUHZKYA
|
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UIZSXCUEUFZUIZXIXQUHZJYCUIZIXNUIZUWATZSVFZYGYEUWCXIUWBUBZUWATZSVFXBUWHU
|
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WASUWBWBXBUWJUWGSXBUWIUWFUWAWSWTUWIUWFUOXAABCDEFGXIIJLMNOWCWIWDVMWEWSWT
|
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XAYGUWCUOZWSWTTXCUAUBXAUWKIJFGUALQVNABCDEXCHXGSKLMNOWCUPWJWSWTYEUWHUOZX
|
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AWTWSUWLWTWSTZYEUWDUWATZJYCUIZIXNUIZSVFZUWHUWMYEUWOSVFZIXNUIUWQUWMYDUWR
|
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UWPUWEUWATZIXNUIUWGUWOUXFIXNUWDUWAJYCUQWOUWEUWAIXNUQWPWNVLWAWIWQWGWR $.
|
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|
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$( Associativity of upper cuts. Lemma for ~ genpassg . (Contributed by
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Jim Kingdon, 11-Dec-2019.) $)
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genpassu $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
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( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) ) $=
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( vt wa cnp wcel w3a co c2nd cfv cv wceq wrex cnq c1st cop prop elprnqu
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wb sylan r19.41v oveq2 adantr adantl eqtr4d eqeq2d expcom pm5.32d 3expa
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wex sylan2 anassrs rexbidva syl5rbbr an32s syl6bb exbidv caovcl elisset
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an31s syl biantrurd 19.41v syl6bbr ancoms df-rex genpelvu anbi1d syl5bb
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rexcom4 3bitr4rd 3impb 3adant3 3impa oveq1 rexbidv pm5.32i exbii rexbii
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bitri 3bitr4d eqrdv ) FUAUBZGUAUBZHUAUBZUCZAFGLUDZHLUDUEUFZFGHLUDZLUDUE
|
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UFZXBAUGZIUGZSUGZMUDZUHZSXEUEUFZUIZIFUEUFZUIZXGXHJUGZMUDZKUGZMUDZUHZKHU
|
|
EUFZUIZJGUEUFZUIZIXNUIZXGXFUBZXGXDUBZWSWTXAXOYEUOZWTXATZWSYHYIWSTXMYDIX
|
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|
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ESYMXAWTUUAUUEUOYMXATZWTTZUUAUUCXKTZJYCUIUUEUUHYTUUIJYCUUGWTXPYCUBZYTUU
|
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|
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UUIYRXKTZKYAUIYMUUMTZXATZYTYRXKKYAUQUURUUPYSKYAUUQXAXRYAUBZUUPYSUOZXAUU
|
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STZUUQXRUJUBZUUTXAHUKUFZYAULUAUBUUSUVBHUMXRYAUVCUNUPZYMUUMUVBUUTYMUUMUV
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UUFXKSXLWBYIUVQUUESYIUVPUUDXKABCDEGHXIJKLMNOWCWDVMWEUSWGVGVHVIWAWHWSWTX
|
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AYFXOUOZYIWSXEUAUBUVRIJGHUALQVNABCDEFXEXGISLMNOWCVGWHXBXGXIXRMUDZUHZKYA
|
|
UIZSXCUEUFZUIZXIXQUHZJYCUIZIXNUIZUWATZSVFZYGYEUWCXIUWBUBZUWATZSVFXBUWHU
|
|
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|
|
AWTWSUWLWTWSTZYEUWDUWATZJYCUIZIXNUIZSVFZUWHUWMYEUWOSVFZIXNUIUWQUWMYDUWR
|
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IXNWTWSYJYDUWRUOZYLWTYMUWSYOYMWTUWSYMWTTZYDUWNSVFZJYCUIUWRUWTYBUXAJYCYM
|
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WTUUJYBUXAUOZUULYMUUMUXBUUOUUQYBUWDSVFZYBTZUXAUUQUXCYBUUQXQUJUBUXCBCXHX
|
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PUJMOVNSXQUJVOVQVRUXAUWDYBTZSVFUXDUWNUXESUWDUWAYBUWDUVTXTKYAUWDUVSXSXGX
|
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IXQXRMWKVBWLWMWNUWDYBSVSWPVTVGVHVIUWNJSYCWFVLWAVGVHVIUWOISXNWFVLUWPUWGS
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UWPUWEUWATZIXNUIUWGUWOUXFIXNUWDUWAJYCUQWOUWEUWAIXNUQWPWNVLWAWIWQWGWR $.
|
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|
|
$( Associativity of an operation on reals. (Contributed by Jim Kingdon,
|
|
11-Dec-2019.) $)
|
|
genpassg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
|
|
( cnp wcel w3a co wceq c1st cfv genpassl genpassu wa caovcl sylan 3impa
|
|
c2nd wb sylan2 3impb preqlu syl2anc mpbir2and ) FSTZGSTZHSTZUAZFGLUBZHL
|
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UBZFGHLUBZLUBZUCZVDUDUEVFUDUEUCZVDULUEVFULUEUCZABCDEFGHIJKLMNOPQRUFABCD
|
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EFGHIJKLMNOPQRUGVBVDSTZVFSTZVGVHVIUHUMUSUTVAVJUSUTUHVCSTVAVJIJFGSLQUIIJ
|
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VCHSLQUIUJUKUSUTVAVKUTVAUHUSVESTVKIJGHSLQUIIJFVESLQUIUNUOVDVFUPUQUR $.
|
|
$}
|
|
$}
|
|
|
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${
|
|
$d G w y z $. $d L w y z $. $d S w y z $. $d U w y z $. $d X w y z $.
|
|
$( Lemma to prove downward closure in positive real addition. (Contributed
|
|
by Jim Kingdon, 7-Dec-2019.) $)
|
|
addnqprllem $p |- ( ( ( <. L , U >. e. P. /\ G e. L ) /\ X e. Q. ) ->
|
|
( X <Q S -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. L ) ) $=
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|
( vy vz vw wcel wa cnq cltq wbr cmq co wb adantl cv syl2anc c1q cop simpr
|
|
cnp crq cfv ltrnqi ltrelnq brel simprd recclnq syl simplr syl3anc mulclnq
|
|
ltmnqg w3a elprnql ad2antrr wceq mulcomnqg caovord2d bitrd syl5ib recidnq
|
|
mpd oveq1d 1nq mpan mulidnq eqtrd sylan9eqr breq2d mpbid wi prcdnql ex )
|
|
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|
|
MZWFWEVTWJVSVTUBVTWAWGLMZWEWJEAUFWEWKWBWHLMZWJWEWAKIZWGKIZVRWKWLPWEAKIZWM
|
|
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|
|
HCLKNFRZKIZGRZKIZHRZKIUPWSXALMXCWSNOXCXANOLMPWEWSXAXCUOQWEVRWMWBKIWQWPEWA
|
|
UNSWEVRWNWHKIWQWREWGUNSVQCKIZVRVTCBDUQURZWTXBJWSXANOXAWSNOUSWEWSXAUTQVAVB
|
|
VCVEWEXDVRWJWFPXEWQXDVRJWICWCLVRXDWITCNOZCVRWHTCNEVDVFXDXFCTNOZCTKIXDXFXG
|
|
USVGTCUTVHCVIVJVKVLSVMVQWFWDVNVRVTCWCBDVOURVEVP $.
|
|
|
|
$( Lemma to prove upward closure in positive real addition. (Contributed
|
|
by Jim Kingdon, 7-Dec-2019.) $)
|
|
addnqprulem $p |- ( ( ( <. L , U >. e. P. /\ G e. U ) /\ X e. Q. ) ->
|
|
( S <Q X -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. U ) ) $=
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( vy vz vw wcel wa cnq cltq wbr cmq co wb adantl cv syl2anc c1q cop simpr
|
|
cnp crq cfv ltrnqi simplr recclnq syl ltrelnq brel simpld syl3anc mulclnq
|
|
ltmnqg w3a elprnqu ad2antrr wceq mulcomnqg caovord2d bitrd syl5ib recidnq
|
|
mpd oveq1d 1nq mpan mulidnq eqtrd sylan9eqr breq1d mpbid wi prcunqu ex )
|
|
DBUAUCICBIJZEKIZJZAELMZEAUDUEZNOZCNOZBIZVSVTJZCWCLMZWDWEEEUDUEZNOZCNOZWCL
|
|
MZWFWEVTWJVSVTUBVTWGWALMZWEWJAEUFWEWKWHWBLMZWJWEWGKIZWAKIZVRWKWLPWEVRWMVQ
|
|
VRVTUGZEUHUIZWEAKIZWNWEWQVRVTWQVRJVSAEKKLUJUKQULAUHUIZWOWGWAEUOUMWEFGHWHW
|
|
BCLKNFRZKIZGRZKIZHRZKIUPWSXALMXCWSNOXCXANOLMPWEWSXAXCUOQWEVRWMWHKIWOWPEWG
|
|
UNSWEVRWNWBKIWOWREWAUNSVQCKIZVRVTCBDUQURZWTXBJWSXANOXAWSNOUSWEWSXAUTQVAVB
|
|
VCVEWEXDVRWJWFPXEWOXDVRJWICWCLVRXDWITCNOZCVRWHTCNEVDVFXDXFCTNOZCTKIXDXFXG
|
|
USVGTCUTVHCVIVJVKVLSVMVQWFWDVNVRVTWCCBDVOURVEVP $.
|
|
$}
|
|
|
|
${
|
|
$d q r s x y A $. $d q r s x y B $.
|
|
$( Lemma to prove downward closure in positive real addition. (Contributed
|
|
by Jim Kingdon, 5-Dec-2019.) $)
|
|
addnqprl $p |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\
|
|
( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) ->
|
|
( X <Q ( G +Q H ) -> X e. ( 1st ` ( A +P. B ) ) ) ) $=
|
|
( vr vs cnp wcel c1st cfv wa cnq cplq co cmq wi simpl wceq c1q vq vx cltq
|
|
vy wbr crq cpp c2nd prop addnqprllem sylanl1 adantlr adantll jcad anim12i
|
|
cop df-iplp cv addclnq genpprecll 3syl syld simpr sylan ad2antrr ad2antlr
|
|
elprnql syl2anc recclnq mulassnqg syl3anc mulclnq mulcomnqg recidnq eqtrd
|
|
syl distrnqg oveq2d mulidnq adantl 3eqtr3d eleq1d sylibd ) AHIZCAJKZIZLZB
|
|
HIZDBJKZIZLZLZEMIZLZECDNOZUCUEZEWOUFKZPOZCPOZWRDPOZNOZABUGOJKZIZEXBIWNWPW
|
|
SWEIZWTWIIZLZXCWNWPXDXEWGWMWPXDQZWKWDWEAUHKZUPHIZWFWMXGAUIZWOXHCWEEUJUKUL
|
|
WKWMWPXEQZWGWHWIBUHKZUPHIZWJWMXKBUIZWOXLDWIEUJUKUMUNWNWLWDWHLXFXCQWLWMRWG
|
|
WDWKWHWDWFRWHWJRUOUAFGUBUDABWSWTUGNUBUDGFUAUQFURGURUSUTVAVBWNXAEXBWNWRWOP
|
|
OZEWQWOPOZPOZXAEWNWMWQMIZWOMIZXOXQSWLWMVCZWNXSXRWNCMIZDMIZXSWGYAWKWMWDXIW
|
|
FYAXJCXHWEVGVDVEZWKYBWGWMWHXMWJYBXNDXLWIVGVDVFZCDUSVHZWOVIVPZYEEWQWOVJVKW
|
|
NWRMIZYAYBXOXASWNWMXRYGXTYFEWQVLVHYCYDWRCDVQVKWNXQETPOZEWNXPTEPWNXPWOWQPO
|
|
ZTWNXRXSXPYISYFYEWQWOVMVHWNXSYITSYEWOVNVPVOVRWMYHESWLEVSVTVOWAWBWC $.
|
|
|
|
$( Lemma to prove upward closure in positive real addition. (Contributed
|
|
by Jim Kingdon, 5-Dec-2019.) $)
|
|
addnqpru $p |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\
|
|
( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) ->
|
|
( ( G +Q H ) <Q X -> X e. ( 2nd ` ( A +P. B ) ) ) ) $=
|
|
( vr vs cnp wcel c2nd cfv wa cnq cplq co cmq wi simpl wceq c1q vq vx cltq
|
|
vy wbr crq cpp c1st prop addnqprulem sylanl1 adantlr adantll jcad anim12i
|
|
cop df-iplp cv addclnq genppreclu 3syl syld simpr sylan ad2antrr ad2antlr
|
|
elprnqu syl2anc recclnq mulassnqg syl3anc mulclnq mulcomnqg recidnq eqtrd
|
|
syl distrnqg oveq2d mulidnq adantl 3eqtr3d eleq1d sylibd ) AHIZCAJKZIZLZB
|
|
HIZDBJKZIZLZLZEMIZLZCDNOZEUCUEZEWOUFKZPOZCPOZWRDPOZNOZABUGOJKZIZEXBIWNWPW
|
|
SWEIZWTWIIZLZXCWNWPXDXEWGWMWPXDQZWKWDAUHKZWEUPHIZWFWMXGAUIZWOWECXHEUJUKUL
|
|
WKWMWPXEQZWGWHBUHKZWIUPHIZWJWMXKBUIZWOWIDXLEUJUKUMUNWNWLWDWHLXFXCQWLWMRWG
|
|
WDWKWHWDWFRWHWJRUOUAFGUBUDABWSWTUGNUBUDGFUAUQFURGURUSUTVAVBWNXAEXBWNWRWOP
|
|
OZEWQWOPOZPOZXAEWNWMWQMIZWOMIZXOXQSWLWMVCZWNXSXRWNCMIZDMIZXSWGYAWKWMWDXIW
|
|
FYAXJCWEXHVGVDVEZWKYBWGWMWHXMWJYBXNDWIXLVGVDVFZCDUSVHZWOVIVPZYEEWQWOVJVKW
|
|
NWRMIZYAYBXOXASWNWMXRYGXTYFEWQVLVHYCYDWRCDVQVKWNXQETPOZEWNXPTEPWNXPWOWQPO
|
|
ZTWNXRXSXPYISYFYEWQWOVMVHWNXSYITSYEWOVNVPVOVRWMYHESWLEVSVTVOWAWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d D f g h $. $d E f g h $. $d P f g h $. $d f g h ph $.
|
|
addlocprlem.a $e |- ( ph -> A e. P. ) $.
|
|
addlocprlem.b $e |- ( ph -> B e. P. ) $.
|
|
addlocprlem.qr $e |- ( ph -> Q <Q R ) $.
|
|
addlocprlem.p $e |- ( ph -> P e. Q. ) $.
|
|
addlocprlem.qppr $e |- ( ph -> ( Q +Q ( P +Q P ) ) = R ) $.
|
|
addlocprlem.dlo $e |- ( ph -> D e. ( 1st ` A ) ) $.
|
|
addlocprlem.uup $e |- ( ph -> U e. ( 2nd ` A ) ) $.
|
|
addlocprlem.du $e |- ( ph -> U <Q ( D +Q P ) ) $.
|
|
addlocprlem.elo $e |- ( ph -> E e. ( 1st ` B ) ) $.
|
|
addlocprlem.tup $e |- ( ph -> T e. ( 2nd ` B ) ) $.
|
|
addlocprlem.et $e |- ( ph -> T <Q ( E +Q P ) ) $.
|
|
$( Lemma for ~ addlocpr . The ` Q <Q ( D +Q E ) ` case. (Contributed by
|
|
Jim Kingdon, 6-Dec-2019.) $)
|
|
addlocprlemlt $p |- ( ph ->
|
|
( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) ) $=
|
|
( cnp wcel c1st cfv wa cnq cplq co cltq wbr cpp wi jca ltrelnq simpld syl
|
|
brel addnqprl syl21anc ) ABUBUCZDBUDUEUCZUFCUBUCZJCUDUEUCZUFFUGUCZFDJUHUI
|
|
UJUKFBCULUIUDUEUCUMAVAVBKPUNAVCVDLSUNAFGUJUKZVEMVFVEGUGUCFGUGUGUJUOURUPUQ
|
|
BCDJFUSUT $.
|
|
|
|
$( Lemma for ~ addlocpr . This is a step used in both the
|
|
` Q = ( D +Q E ) ` and ` ( D +Q E ) <Q Q ` cases. (Contributed by Jim
|
|
Kingdon, 7-Dec-2019.) $)
|
|
addlocprlemeqgt $p |- ( ph ->
|
|
( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) ) $=
|
|
( vf vg vh cplq co cltq wbr cnq wcel wa wi c1st cfv c2nd cop cnp prop syl
|
|
elprnqu syl2anc elprnql addclnq lt2addnq syl22anc mp2and addcomnqg adantl
|
|
cv wceq w3a addassnqg caov4d breqtrd ) AIHUEUFZDEUEUFZJEUEUFZUEUFZDJUEUFE
|
|
EUEUFUEUFUGAIVPUGUHZHVQUGUHZVOVRUGUHZRUAAIUIUJZVPUIUJZHUIUJZVQUIUJZVSVTUK
|
|
WAULABUMUNZBUOUNZUPUQUJZIWGUJWBABUQUJWHKBURUSZQIWGWFUTVAADUIUJZEUIUJZWCAW
|
|
HDWFUJWJWIPDWGWFVBVAZNDEVCVAACUMUNZCUOUNZUPUQUJZHWNUJWDACUQUJWOLCURUSZTHW
|
|
NWMUTVAAJUIUJZWKWEAWOJWMUJWQWPSJWNWMVBVAZNJEVCVAIVPHVQVDVEVFAUBUCUDDEJEUI
|
|
UEWLNWRUBVIZUIUJZUCVIZUIUJZUKZWSXAUEUFZXAWSUEUFVJAWSXAVGVHWTXBUDVIZUIUJVK
|
|
XDXEUEUFWSXAXEUEUFUEUFVJAWSXAXEVLVHNXCXDUIUJAWSXAVCVHVMVN $.
|
|
|
|
$( Lemma for ~ addlocpr . The ` Q = ( D +Q E ) ` case. (Contributed by
|
|
Jim Kingdon, 6-Dec-2019.) $)
|
|
addlocprlemeq $p |- ( ph ->
|
|
( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) ) $=
|
|
( cplq co wceq cpp c2nd cfv wcel wa cltq wbr addlocprlemeqgt adantr oveq1
|
|
sylan9req breqtrrd wi cnp cnq jca ltrelnq simprd syl addnqpru syl21anc ex
|
|
brel mpd ) AFDJUBUCZUDZGBCUEUCUFUGUHZAVJUIZIHUBUCZGUJUKZVKVLVMVIEEUBUCZUB
|
|
UCZGUJAVMVPUJUKVJABCDEFGHIJKLMNOPQRSTUAULUMAVJGFVOUBUCVPOFVIVOUBUNUOUPAVN
|
|
VKUQZVJABURUHZIBUFUGUHZUICURUHZHCUFUGUHZUIGUSUHZVQAVRVSKQUTAVTWALTUTAFGUJ
|
|
UKZWBMWCFUSUHWBFGUSUSUJVAVGVBVCBCIHGVDVEUMVHVF $.
|
|
|
|
$( Lemma for ~ addlocpr . The ` ( D +Q E ) <Q Q ` case. (Contributed by
|
|
Jim Kingdon, 6-Dec-2019.) $)
|
|
addlocprlemgt $p |- ( ph ->
|
|
( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) ) $=
|
|
( cplq co cltq wbr cpp c2nd cfv wcel wa addlocprlemeqgt adantr cnq wb cop
|
|
c1st cnp prop syl elprnql syl2anc addclnq ltrelnq brel simpld ltanqg wceq
|
|
syl3anc addcomnqg breq12d bitrd biimpa breq2d mpbid ltsonq sotri addnqpru
|
|
jca wi simprd syl21anc mpd ex ) ADJUBUCZFUDUEZGBCUFUCUGUHUIZAWEUJZIHUBUCZ
|
|
GUDUEZWFWGWHWDEEUBUCZUBUCZUDUEZWKGUDUEZUJWIWGWLWMAWLWEABCDEFGHIJKLMNOPQRS
|
|
TUAUKULWGWKFWJUBUCZUDUEZWMAWEWOAWEWJWDUBUCZWJFUBUCZUDUEZWOAWDUMUIZFUMUIZW
|
|
JUMUIZWEWRUNADUMUIZJUMUIZWSABUPUHZBUGUHZUOUQUIZDXDUIXBABUQUIZXFKBURUSPDXE
|
|
XDUTVAACUPUHZCUGUHZUOUQUIZJXHUIXCACUQUIZXJLCURUSSJXIXHUTVADJVBVAZAWTGUMUI
|
|
ZAFGUDUEWTXMUJMFGUMUMUDVCVDUSZVEZAEUMUIZXPXANNEEVBVAZWDFWJVFVHAWPWKWQWNUD
|
|
AXAWSWPWKVGXQXLWJWDVIVAAXAWTWQWNVGXQXOWJFVIVAVJVKVLAWOWMUNWEAWNGWKUDOVMUL
|
|
VNVRWHWKGUDUMVOVCVPUSAWIWFVSZWEAXGIXEUIZUJXKHXIUIZUJXMXRAXGXSKQVRAXKXTLTV
|
|
RAWTXMXNVTBCIHGVQWAULWBWC $.
|
|
|
|
$( Lemma for ~ addlocpr . The result, in deduction form. (Contributed by
|
|
Jim Kingdon, 6-Dec-2019.) $)
|
|
addlocprlem $p |- ( ph ->
|
|
( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) ) $=
|
|
( cplq co cltq wbr wceq w3o cpp c1st cfv wcel c2nd wo ltrelnq brel simpld
|
|
cnq syl cop cnp prop elprnql syl2anc nqtri3or addlocprlemlt addlocprlemeq
|
|
addclnq orc syl6 olc addlocprlemgt 3jaod mpd ) AFDJUBUCZUDUEZFVNUFZVNFUDU
|
|
EZUGZFBCUHUCZUIUJUKZGVSULUJUKZUMZAFUQUKZVNUQUKZVRAFGUDUEZWCMWEWCGUQUKFGUQ
|
|
UQUDUNUOUPURADUQUKZJUQUKZWDABUIUJZBULUJZUSUTUKZDWHUKWFABUTUKWJKBVAURPDWIW
|
|
HVBVCACUIUJZCULUJZUSUTUKZJWKUKWGACUTUKWMLCVAURSJWLWKVBVCDJVGVCFVNVDVCAVOW
|
|
BVPVQAVOVTWBABCDEFGHIJKLMNOPQRSTUAVEVTWAVHVIAVPWAWBABCDEFGHIJKLMNOPQRSTUA
|
|
VFWAVTVJZVIAVQWAWBABCDEFGHIJKLMNOPQRSTUAVKWNVIVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d A d e h p q r t u $. $d B d e h p q r t u $.
|
|
$( Locatedness of addition on positive reals. Lemma 11.16 in
|
|
[BauerTaylor], p. 53. The proof in BauerTaylor relies on signed
|
|
rationals, so we replace it with another proof which applies ~ prarloc
|
|
to both ` A ` and ` B ` , and uses ~ nqtri3or rather than ~ prloc to
|
|
decide whether ` q ` is too big to be in the lower cut of ` A +P. B `
|
|
(and deduce that if it is, then ` r ` must be in the upper cut). What
|
|
the two proofs have in common is that they take the difference between
|
|
` q ` and ` r ` to determine how tight a range they need around the real
|
|
numbers. (Contributed by Jim Kingdon, 5-Dec-2019.) $)
|
|
addlocpr $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
A. q e. Q. A. r e. Q. ( q <Q r ->
|
|
( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) ) $=
|
|
( vp vh vu vd vt ve cnp wcel wa cv co cfv cnq cplq wrex adantr wbr cpp wo
|
|
cltq c1st c2nd wi w3a ltexnqq biimpa 3adant1 halfnqq ad2antrl cop prarloc
|
|
wceq prop sylan adantlr 3ad2antl1 ad2ant2r adantll ad2antrr simpld simprd
|
|
simpll1 simpll3 simplrl simplrr wb eqeq1d ad2antll mpbird simprll simprlr
|
|
oveq2 simprr addlocprlem expr rexlimdvva mpd rexlimddv 3expia ralrimivva
|
|
) AKLZBKLZMZDNZCNZUDUAZWHABUBOZUEPLWIWKUFPLUCZUGDCQQWGWHQLWIQLMZWJWLWGWMW
|
|
JUHZWHENZROZWIUPZWLEQWMWJWQEQSZWGWMWJWREWHWIUIUJUKWNWOQLZWQMZMZFNZXBROZWO
|
|
UPZWLFQWSXDFQSWNWQFWOULUMXAXBQLZXDMZMZGNZHNZXBROUDUAZGAUFPZSHAUEPZSZWLWNX
|
|
EXMWTXDWGWMXEXMWJWEXEXMWFWEXLXKUNKLXEXMAUQXBXKXLHGUOURUSUTVAXGXJWLHGXLXKX
|
|
GXIXLLZXHXKLZMZXJWLXGXPXJMZMZINZJNZXBROUDUAZIBUFPZSJBUEPZSZWLXGYDXQWNXEYD
|
|
WTXDWGWMXEYDWJWFXEYDWEWFYCYBUNKLXEYDBUQXBYBYCJIUOURVBUTVATXRYAWLJIYCYBXRX
|
|
TYCLZXSYBLZMZYAWLXRYGYAMZMZABXIXBWHWIXSXHXTYIWEWFXGWGXQYHWGWMWJWTXFVFVCZV
|
|
DYIWEWFYJVEXGWJXQYHWGWMWJWTXFVGVCXRXEYHXAXEXDXQVHTXGWHXCROZWIUPZXQYHXGYLW
|
|
QWNWSWQXFVIXDYLWQVJXAXEXDYKWPWIXCWOWHRVPVKVLVMVCXRXNYHXGXNXOXJVNTXRXOYHXG
|
|
XNXOXJVOTXGXPXJYHVIXRYEYFYAVNXRYEYFYAVOXRYGYAVQVRVSVTWAVSVTWAWBWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v g h q r A $. $d x y z w v g h q r B $.
|
|
$( Closure of addition on positive reals. First statement of Proposition
|
|
9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16
|
|
in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.) $)
|
|
addclpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. ) $=
|
|
( vq vr vx vy vz vw vv vg vh cnp wcel wa cpp cnq cv wrex wral cplq co cpw
|
|
cxp c1st cfv c2nd cltq wbr wb wn wo wi w3a df-iplp genpelxp genpml genpmu
|
|
addclnq ltanqg addcomnqg addnqprl genprndl addnqpru genprndu jca genpdisj
|
|
jca32 addlocpr 3jca elnp1st2nd sylanbrc ) ALMBLMNZABOUAZPUBZVNUCMZCQZVMUD
|
|
UEZMZCPRZDQZVMUFUEZMZDPRZNNVRVPVTUGUHZVTVQMNDPRUICPSZWBWDVPWAMZNCPRUIDPSZ
|
|
NZVRWFNUJCPSZWDVRWBUKULDPSCPSZUMVMLMVLVOVSWCEFGHIABOTHIGFEUNZUOEFGHIABOTC
|
|
WKFQZGQZURZUPEFGHIABOTDWKWNUQVGVLWHWIWJVLWEWGEFGHIABJKOTDCWKWNEQZWLWMUSZW
|
|
OWLUTZABJQZKQZWOVAVBEFGHIABJKOTDCWKWNWPWQABWRWSWOVCVDVEEFGHIABOTCWKWNWPWQ
|
|
VFABDCVHVIVMDCVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z f g h A $. $d x y z f g h B $. $d x y z w v f g h $.
|
|
$( Value of addition on positive reals. (Contributed by Jim Kingdon,
|
|
8-Dec-2019.) $)
|
|
plpvlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = <.
|
|
{ x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y +Q z ) } ,
|
|
{ x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y +Q z ) }
|
|
>. ) $=
|
|
( vf vg vh vw vv cpp cplq df-iplp cv addclnq genipv ) FGHIJDEKLCBAIJHGFMG
|
|
NHNOP $.
|
|
|
|
$( Value of multiplication on positive reals. (Contributed by Jim Kingdon,
|
|
8-Dec-2019.) $)
|
|
mpvlu $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = <.
|
|
{ x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } ,
|
|
{ x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) }
|
|
>. ) $=
|
|
( vf vg vh vw vv cmp cmq df-imp cv mulclnq genipv ) FGHIJDEKLCBAIJHGFMGNH
|
|
NOP $.
|
|
|
|
$( Domain of addition on positive reals. (Contributed by NM,
|
|
18-Nov-1995.) $)
|
|
dmplp $p |- dom +P. = ( P. X. P. ) $=
|
|
( vv vw vz vx vy cpp cplq df-iplp cv addclnq genipdm ) ABCDEFGDECBAHBICIJ
|
|
K $.
|
|
|
|
$( Domain of multiplication on positive reals. (Contributed by NM,
|
|
18-Nov-1995.) $)
|
|
dmmp $p |- dom .P. = ( P. X. P. ) $=
|
|
( vv vw vz vx vy cmp cmq df-imp cv mulclnq genipdm ) ABCDEFGDECBAHBICIJK
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x n r q $.
|
|
$( A cut produced from a rational is inhabited. Lemma for ~ nqprlu .
|
|
(Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
nqprm $p |- ( A e. Q. -> ( E. q e. Q. q e. { x | x <Q A }
|
|
/\ E. r e. Q. r e. { x | A <Q x } ) ) $=
|
|
( vn cnq wcel cv cltq wbr cab wrex vex elab rexbii sylibr cnpi c1o ceq wa
|
|
nsmallnqq breq1 cop cec wex archnqq sylib cxp cqs 1pi opelxpi ecelqsi syl
|
|
df-rex enqex mpan2 df-nqqs syl6eleqr breq2 rspcev sylan exlimiv jca ) BFG
|
|
ZDHZAHZBIJZAKGZDFLZCHZBVFIJZAKGZCFLZVDVEBIJZDFLVIDBUAVHVNDFVGVNAVEDMVFVEB
|
|
IUBNOPVDBVJIJZCFLZVMVDEHZQGZBVQRUCZSUDZIJZTZEUEZVPVDWAEQLWCEBUFWAEQUNUGWB
|
|
VPEVRVTFGWAVPVRVTQQUHZSUIZFVRRQGZVTWEGZUJVRWFTVSWDGWGVQRQQUKWDVSSUOULUMUP
|
|
UQURVOWACVTFVJVTBIUSUTVAVBUMVLVOCFVKVOAVJCMVFVJBIUSNOPVC $.
|
|
|
|
$( A cut produced from a rational is rounded. Lemma for ~ nqprlu .
|
|
(Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
nqprrnd $p |- ( A e. Q. -> ( A. q e. Q. ( q e. { x | x <Q A }
|
|
<-> E. r e. Q. ( q <Q r /\ r e. { x | x <Q A } ) )
|
|
/\ A. r e. Q. ( r e. { x | A <Q x }
|
|
<-> E. q e. Q. ( q <Q r /\ q e. { x | A <Q x } ) ) ) ) $=
|
|
( cnq wcel cv cltq wbr cab wa wrex wral ltbtwnnqq rexbii vex breq2 anbi2i
|
|
wb elab ancom bitri 3bitr4i rgenw a1i breq1 jctil ) BEFZCGZBAGZHIZAJZFZDG
|
|
ZUIHIZUNULFZKZDELZSZCEMZUNUJBHIZAJZFZUOUIVBFZKZCELZSZDEMUTUHUSCEBUIHIZUOB
|
|
UNHIZKZDELZUMURVHVIUOKZDELVKDBUINVLVJDEVIUOUAOUBUKVHAUICPZUJUIBHQTUQVJDEU
|
|
PVIUOUKVIAUNDPZUJUNBHQTROUCUDUEVGDEUNBHIZUOUIBHIZKZCELVCVFCUNBNVAVOAUNVNU
|
|
JUNBHUFTVEVQCEVDVPUOVAVPAUIVMUJUIBHUFTROUCUDUG $.
|
|
|
|
$( A cut produced from a rational is disjoint. Lemma for ~ nqprlu .
|
|
(Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
nqprdisj $p |- ( A e. Q. -> A. q e. Q.
|
|
-. ( q e. { x | x <Q A } /\ q e. { x | A <Q x } ) ) $=
|
|
( cv cltq wbr cab wcel wa cnq wral ltsonq ltrelnq son2lpi vex breq1 breq2
|
|
wn elab anbi12i mtbir rgenw a1i ) CDZADZBEFZAGHZUDBUEEFZAGHZIZRZCJKBJHUKC
|
|
JUJUDBEFZBUDEFZIUDBEJLMNUGULUIUMUFULAUDCOZUEUDBEPSUHUMAUDUNUEUDBEQSTUAUBU
|
|
C $.
|
|
|
|
$( A cut produced from a rational is located. Lemma for ~ nqprlu .
|
|
(Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
nqprloc $p |- ( A e. Q. -> A. q e. Q. A. r e. Q. ( q <Q r
|
|
-> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) ) $=
|
|
( cnq wcel cv cltq wbr cab wo wi wral wceq w3o vex breq1 elab ralrimiva
|
|
wa nqtri3or ancoms ad2antrr biimpri orcd a1i simpr syl5ibcom breq2 sylbir
|
|
olc syl6 ltsonq ltrelnq sotri syl expcom adantl 3jaod mpd ex ) BEFZDGZCGZ
|
|
HIZVCAGZBHIZAJFZVDBVFHIZAJFZKZLZCEMDEVBVCEFZTZVLCEVNVDEFZTZVEVKVPVETZVCBH
|
|
IZVCBNZBVCHIZOZVKVNWAVOVEVMVBWAVCBUAUBUCVQVRVKVSVTVRVKLVQVRVHVJVHVRVGVRAV
|
|
CDPVFVCBHQRUDUEUFVQVSBVDHIZVKVQVEVSWBVPVEUGVCBVDHQUHWBVJVKVIWBAVDCPVFVDBH
|
|
UIRVJVHUKUJZULVEVTVKLVPVTVEVKVTVETWBVKBVCVDHEUMUNUOWCUPUQURUSUTVASS $.
|
|
|
|
$( The canonical embedding of the rationals into the reals, expressed with
|
|
the same variable for the lower and upper cuts. (Contributed by Jim
|
|
Kingdon, 8-Dec-2019.) $)
|
|
nqprxx $p |- ( A e. Q. -> <. { x | x <Q A } , { x | A <Q x } >. e. P. ) $=
|
|
( vq vr cnq wcel cv cltq wbr cab wa wrex wb wral wn wo ltrelnq brel abssi
|
|
wss w3a cop cnp nqprm simpld simprd pm3.2i jctil nqprrnd nqprdisj nqprloc
|
|
wi 3jca elinp sylanbrc ) BEFZAGZBHIZAJZETZBUQHIZAJZETZKZCGZUSFZCELDGZVBFZ
|
|
DELKZKVFVEVGHIZVGUSFKDELMCENVHVJVEVBFZKCELMDENKZVFVKKOCENZVJVFVHPULDENCEN
|
|
ZUAUSVBUBUCFUPVIVDABDCUDUTVCURAEURUQEFZUPUQBEEHQRUESVAAEVAUPVOBUQEEHQRUFS
|
|
UGUHUPVLVMVNABDCUIABCUJABDCUKUMVBUSDCUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d A l a $. $d A u a $.
|
|
$( The canonical embedding of the rationals into the reals. (Contributed
|
|
by Jim Kingdon, 24-Jun-2020.) $)
|
|
nqprlu $p |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. ) $=
|
|
( va cnq wcel cv cltq wbr cab breq2 cbvabv eqtr4i opeq2i nqprxx syl5eqelr
|
|
cop cnp ) BEFCGZBHICJZBAGZHIZAJZQTBSHIZCJZQRUEUCTUEBDGZHIZDJUCUDUGCDSUFBH
|
|
KLUBUGADUAUFBHKLMNCBOP $.
|
|
$}
|
|
|
|
${
|
|
$d A l u $.
|
|
$( The reciprocal of a positive integer, as a positive real. (Contributed
|
|
by Jim Kingdon, 27-Feb-2021.) $)
|
|
recnnpr $p |- ( A e. N. ->
|
|
<. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. e. P. ) $=
|
|
( cnpi wcel c1o cop ceq cec cnq crq cfv cv cltq wbr cab cnp nnnq recclnq
|
|
nqprlu 3syl ) BDEBFGHIZJEUBKLZJECMUCNOCPUCAMNOAPGQEBRUBSAUCCTUA $.
|
|
$}
|
|
|
|
$( The class of rationals less than a given rational is a set. (Contributed
|
|
by Jim Kingdon, 13-Dec-2019.) $)
|
|
ltnqex $p |- { x | x <Q A } e. _V $=
|
|
( cv cltq wbr cab cnq nqex wcel ltrelnq brel simpld abssi ssexi ) ACZBDEZAF
|
|
GHPAGPOGIBGIOBGGDJKLMN $.
|
|
|
|
$( The class of rationals greater than a given rational is a set.
|
|
(Contributed by Jim Kingdon, 13-Dec-2019.) $)
|
|
gtnqex $p |- { x | A <Q x } e. _V $=
|
|
( cv cltq wbr cab cnq nqex wcel ltrelnq brel simprd abssi ssexi ) BACZDEZAF
|
|
GHPAGPBGIOGIBOGGDJKLMN $.
|
|
|
|
${
|
|
$d A l u x $. $d B x $.
|
|
$( Comparing a fraction to a real can be done by whether it is an element
|
|
of the lower cut, or by ` <P ` . (Contributed by Jim Kingdon,
|
|
8-Jul-2020.) $)
|
|
nqprl $p |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <->
|
|
<. { l | l <Q A } , { u | A <Q u } >. <P B ) ) $=
|
|
( vx cnq wcel cnp wa cfv cv cltq wbr cab cop wi c2nd wrex sylan elprnql
|
|
c1st cltp prop prnmaxl wex ad2ant2r vex breq2 biimpri ltnqex gtnqex op2nd
|
|
elab eleq2i sylibr ad2antll simprl 19.8a syl12anc df-rex wb nqprlu ltdfpr
|
|
simpl syl2anc adantr mpbird rexlimddv adantl biimpa bitri biimpi ad2antrl
|
|
ex simpllr simprrr prcdnql mpd impbid ) BFGZCHGZIZBCUAJZGZDKBLMDNZBAKZLMZ
|
|
ANZOZCUBMZWAWDWJPVTWAWDWJWAWDIZBEKZLMZWJEWCWAWCCQJZOHGZWDWMEWCRCUCZEBWNWC
|
|
UDSWKWLWCGZWMIZIZWJWLWIQJZGZWQIZEFRZWSWLFGZXBIZEUEZXCWSXDXAWQXFWAWQXDWDWM
|
|
WAWOWQXDWPWLWNWCTSUFWMXAWKWQWMWLWHGZXAXGWMWGWMAWLEUGWFWLBLUHUMZUIWTWHWLWE
|
|
WHDBUJABUKULUNZUOUPWKWQWMUQXEEURUSXBEFUTUOWKWJXCVAZWRWKVTWAXJWAWOWDVTWPBW
|
|
NWCTSWAWDVDVTWIHGWAXJABDVBWICEVCSZVEVFVGVHVNVIWBWJWDWBWJIZXBWDEFWBWJXCXKV
|
|
JXLXEIZWMWDXEWMXLXAWMXDWQXAWMXAXGWMXIXHVKVLVMVIXMWAWQWMWDPZVTWAWJXEVOXLXD
|
|
XAWQVPWAWOWQXNWPWLBWNWCVQSVEVRVHVNVS $.
|
|
$}
|
|
|
|
${
|
|
$d A l u x $. $d B x $.
|
|
$( Comparing a fraction to a real can be done by whether it is an element
|
|
of the upper cut, or by ` <P ` . (Contributed by Jim Kingdon,
|
|
29-Nov-2020.) $)
|
|
nqpru $p |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) <->
|
|
B <P <. { l | l <Q A } , { u | A <Q u } >. ) ) $=
|
|
( vx cnq wcel cnp wa cfv cv cltq wbr cab cop wi c1st wrex sylan elprnqu
|
|
c2nd cltp prop prnminu wex ad2ant2r simprl vex elab biimpri ltnqex gtnqex
|
|
breq1 op1st eleq2i sylibr ad2antll 19.8a syl12anc df-rex wb nqprlu ltdfpr
|
|
sylan2 syldan adantr mpbird rexlimddv adantl ancoms biimpa biimpi simpllr
|
|
ex bitri simprrl prcunqu syl2anc mpd impbid ) BFGZCHGZIZBCUAJZGZCDKZBLMZD
|
|
NZBAKLMANZOZUBMZWBWEWKPWAWBWEWKWBWEIZEKZBLMZWKEWDWBCQJZWDOHGZWEWNEWDRCUCZ
|
|
EBWDWOUDSWLWMWDGZWNIZIZWKWRWMWJQJZGZIZEFRZWTWMFGZXCIZEUEZXDWTXEWRXBXGWBWR
|
|
XEWEWNWBWPWRXEWQWMWDWOTSUFWLWRWNUGWNXBWLWRWNWMWHGZXBXHWNWGWNDWMEUHWFWMBLU
|
|
MUIZUJXAWHWMWHWIDBUKABULUNUOZUPUQXFEURUSXCEFUTUPWLWKXDVAZWSWBWEWAXKWBWPWE
|
|
WAWQBWDWOTSWAWBWJHGXKABDVBCWJEVCVDZVEVFVGVHVNVIWCWKWEWCWKIZXCWEEFWCWKXDWB
|
|
WAXKXLVJVKXMXFIZWNWEXFWNXMXBWNXEWRXBWNXBXHWNXJXIVOVLUQVIXNWBWRWNWEPZWAWBW
|
|
KXFVMXMXEWRXBVPWBWPWRXOWQBWMWDWOVQSVRVSVHVNVT $.
|
|
$}
|
|
|
|
${
|
|
$d A l u $.
|
|
$( The canonical embedding of positive integers into the positive reals.
|
|
(Contributed by Jim Kingdon, 23-Apr-2020.) $)
|
|
nnprlu $p |- ( A e. N. -> <. { l | l <Q [ <. A , 1o >. ] ~Q } ,
|
|
{ u | [ <. A , 1o >. ] ~Q <Q u } >. e. P. ) $=
|
|
( cnpi wcel c1o cop ceq cec cnq cv cltq wbr cab cnp nnnq nqprlu syl ) BDE
|
|
BFGHIZJECKSLMCNSAKLMANGOEBPASCQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The positive real number 'one'. (Contributed by NM, 13-Mar-1996.)
|
|
(Revised by Mario Carneiro, 12-Jun-2013.) $)
|
|
1pr $p |- 1P e. P. $=
|
|
( vx vy c1p cv c1q cltq wbr cab cop cnp df-i1p cnq wcel 1nq ax-mp eqeltri
|
|
nqprlu ) CADEFGAHEBDFGBHIZJBAKELMRJMNBEAQOP $.
|
|
|
|
$( The lower cut of the positive real number 'one'. (Contributed by Jim
|
|
Kingdon, 28-Dec-2019.) $)
|
|
1prl $p |- ( 1st ` 1P ) = { x | x <Q 1Q } $=
|
|
( vy c1p c1st cfv cv c1q cltq wbr df-i1p fveq2i ltnqex gtnqex op1st eqtri
|
|
cab cop ) CDEAFGHIAPZGBFHIBPZQZDERCTDBAJKRSAGLBGMNO $.
|
|
|
|
$( The upper cut of the positive real number 'one'. (Contributed by Jim
|
|
Kingdon, 28-Dec-2019.) $)
|
|
1pru $p |- ( 2nd ` 1P ) = { x | 1Q <Q x } $=
|
|
( vy c1p c2nd cfv cv c1q cltq wbr df-i1p fveq2i ltnqex gtnqex op2nd eqtri
|
|
cab cop ) CDEBFGHIBPZGAFHIAPZQZDESCTDABJKRSBGLAGMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h l r s t u x y $. $d B f g h l r s t u x y $.
|
|
$( Lemma for ~ addnqpr . The reverse subset relationship for the lower
|
|
cut. (Contributed by Jim Kingdon, 19-Aug-2020.) $)
|
|
addnqprlemrl $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) C_
|
|
( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
|
|
) $=
|
|
( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c1st cfv breq1 vf vx cpp
|
|
vy co cplq wceq wrex cnp wb nqprlu df-iplp addclnq genpelvl syl2an biimpa
|
|
ltnqex gtnqex op1st elab2 biimpi ad2antrl adantr ad2antll wi ltrelnq brel
|
|
vex syl lt2addnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
|
|
ex ) BJKZCJKZLZEDMZBNOZDPZBAMZNOAPZQZWCCNOZDPZCWFNOAPZQZUCUERSZWCBCUFUEZN
|
|
OZDPZWNWFNOAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUHFWHRSZUHZX
|
|
AWBWTXIVTWHUIKWLUIKWTXIUJWAABDUKACDUKUAHIUBUDWHWLWSFGUCUFUBUDIHUAULHMIMUM
|
|
UNUOUPXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWSWNNOZXAXMXNXEWNNOZXMXCBNOZ
|
|
XDCNOZXOXLXPXFXJXPXBXKXJXPWDXPDXCXHFVHWCXCBNTWEWGDBUQABURUSUTVAVBVCZXLXQX
|
|
FXKXQXBXJXKXQWIXQDXDXGGVHWCXDCNTWJWKDCUQACURUSUTVAVDVCZXMXCJKVTLZXDJKWALZ
|
|
XPXQLXOVEXMXPXTXRXCBJJNVFVGVIXMXQYAXSXDCJJNVFVGVIXCBXDCVJVKVLXFXNXOUJXLWS
|
|
XEWNNTVMVNWOXNDWSWREVHWCWSWNNTWPWQDWNUQAWNURUSUTVOVSVPVQVSVR $.
|
|
|
|
$( Lemma for ~ addnqpr . The reverse subset relationship for the upper
|
|
cut. (Contributed by Jim Kingdon, 19-Aug-2020.) $)
|
|
addnqprlemru $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) C_
|
|
( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
|
|
) $=
|
|
( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c2nd cfv breq2 vf vx cpp
|
|
vy co cplq wceq wrex cnp wb nqprlu df-iplp addclnq genpelvu syl2an biimpa
|
|
ltnqex gtnqex op2nd elab2 biimpi ad2antrl adantr ad2antll wi ltrelnq brel
|
|
vex syl lt2addnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
|
|
ex ) BJKZCJKZLZEDMZBNODPZBAMZNOZAPZQZWCCNODPZCWENOZAPZQZUCUERSZWCBCUFUEZN
|
|
ODPZWNWENOZAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUHFWHRSZUHZX
|
|
AWBWTXIVTWHUIKWLUIKWTXIUJWAABDUKACDUKUAHIUBUDWHWLWSFGUCUFUBUDIHUAULHMIMUM
|
|
UNUOUPXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWNWSNOZXAXMXNWNXENOZXMBXCNOZ
|
|
CXDNOZXOXLXPXFXJXPXBXKXJXPWFXPAXCXHFVHWEXCBNTWDWGDBUQABURUSUTVAVBVCZXLXQX
|
|
FXKXQXBXJXKXQWJXQAXDXGGVHWEXDCNTWIWKDCUQACURUSUTVAVDVCZXMVTXCJKLZWAXDJKLZ
|
|
XPXQLXOVEXMXPXTXRBXCJJNVFVGVIXMXQYAXSCXDJJNVFVGVIBXCCXDVJVKVLXFXNXOUJXLWS
|
|
XEWNNTVMVNWPXNAWSWREVHWEWSWNNTWOWQDWNUQAWNURUSUTVOVSVPVQVSVR $.
|
|
|
|
$( Lemma for ~ addnqpr . The forward subset relationship for the lower
|
|
cut. (Contributed by Jim Kingdon, 19-Aug-2020.) $)
|
|
addnqprlemfl $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
|
|
C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
|
|
( vr cnq wcel wa cv co cltq wbr cab cop c1st cfv c2nd wn syl cnp cplq cpp
|
|
addnqprlemru wor ltsonq addclnq sonr sylancr cvv ltrelnq brel simpld elex
|
|
breq2 elab3 sylnibr ltnqex gtnqex eleq2i ssneldd adantr wo nqprlu addclpr
|
|
op2nd syl2an prop vex breq1 op1st elab2 biimpi prloc ecased ex ssrdv ) BF
|
|
GZCFGZHZEDIZBCUAJZKLZDMZWAAIZKLZAMZNZOPZVTBKLDMBWDKLAMNZVTCKLDMCWDKLAMNZU
|
|
BJZOPZVSEIZWHGZWMWLGZVSWNHWOWAWKQPZGZVSWQRWNVSWPWGQPZWAABCDUCVSWAWFGZWAWR
|
|
GVSWAWAKLZWSVSFKUDWAFGZWTRUEBCUFFWAKUGUHWEWTAWAWTXAWAUIGWTXAXAWAWAFFKUJUK
|
|
ULWAFUMSWDWAWAKUNUOUPWRWFWAWCWFDWAUQZAWAURZVEUSUPUTVAVSWLWPNTGZWMWAKLZWOW
|
|
QVBWNVSWKTGZXDVQWITGWJTGXFVRABDVCACDVCWIWJVDVFWKVGSWNXEWBXEDWMWHEVHVTWMWA
|
|
KVIWCWFXBXCVJVKVLWMWAWPWLVMVFVNVOVP $.
|
|
|
|
$( Lemma for ~ addnqpr . The forward subset relationship for the upper
|
|
cut. (Contributed by Jim Kingdon, 19-Aug-2020.) $)
|
|
addnqprlemfu $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. )
|
|
C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
|
|
( vr cnq wcel wa cv co cltq wbr cab cop c2nd cfv c1st wn syl cnp cplq cpp
|
|
addnqprlemrl wor ltsonq addclnq sonr sylancr cvv ltrelnq brel simpld elex
|
|
breq1 elab3 sylnibr ltnqex gtnqex eleq2i ssneldd adantr wo nqprlu addclpr
|
|
op1st syl2an prop vex breq2 op2nd elab2 biimpi prloc orcomd ecased ssrdv
|
|
ex ) BFGZCFGZHZEDIZBCUAJZKLZDMZWBAIZKLZAMZNZOPZWABKLDMBWEKLAMNZWACKLDMCWE
|
|
KLAMNZUBJZOPZVTEIZWIGZWNWMGZVTWOHZWPWBWLQPZGZVTWSRWOVTWRWHQPZWBABCDUCVTWB
|
|
WDGZWBWTGVTWBWBKLZXAVTFKUDWBFGZXBRUEBCUFFWBKUGUHWCXBDWBXBXCWBUIGXBXCXCWBW
|
|
BFFKUJUKULWBFUMSWAWBWBKUNUOUPWTWDWBWDWGDWBUQZAWBURZVEUSUPUTVAWQWSWPVTWRWM
|
|
NTGZWBWNKLZWSWPVBWOVTWLTGZXFVRWJTGWKTGXHVSABDVCACDVCWJWKVDVFWLVGSWOXGWFXG
|
|
AWNWIEVHWEWNWBKVIWDWGXDXEVJVKVLWBWNWMWRVMVFVNVOVQVP $.
|
|
|
|
$( Addition of fractions embedded into positive reals. One can either add
|
|
the fractions as fractions, or embed them into positive reals and add
|
|
them as positive reals, and get the same result. (Contributed by Jim
|
|
Kingdon, 19-Aug-2020.) $)
|
|
addnqpr $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
<. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. =
|
|
( <. { l | l <Q A } , { u | A <Q u } >. +P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) $=
|
|
( cnq wcel wa cv co cltq wbr cab cop wceq c1st cfv c2nd eqssd cnp nqprlu
|
|
cpp addnqprlemfl addnqprlemrl addnqprlemfu addnqprlemru wb addclnq syl2an
|
|
cplq syl addclpr preqlu syl2anc mpbir2and ) BEFZCEFZGZDHZBCUIIZJKDLUSAHZJ
|
|
KALMZURBJKDLBUTJKALMZURCJKDLCUTJKALMZUAIZNZVAOPZVDOPZNZVAQPZVDQPZNZUQVFVG
|
|
ABCDUBABCDUCRUQVIVJABCDUDABCDUERUQVASFZVDSFZVEVHVKGUFUQUSEFVLBCUGAUSDTUJU
|
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OVBSFVCSFVMUPABDTACDTVBVCUKUHVAVDULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A l u $.
|
|
$( Addition of one to a fraction embedded into a positive real. One can
|
|
either add the fraction one to the fraction, or the positive real one to
|
|
the positive real, and get the same result. Special case of ~ addnqpr .
|
|
(Contributed by Jim Kingdon, 26-Apr-2020.) $)
|
|
addnqpr1 $p |- ( A e. Q. ->
|
|
<. { l | l <Q ( A +Q 1Q ) } , { u | ( A +Q 1Q ) <Q u } >. =
|
|
( <. { l | l <Q A } , { u | A <Q u } >. +P. 1P ) ) $=
|
|
( cnq wcel cv c1q cplq co cltq wbr cab cop cpp c1p wceq 1nq addnqpr mpan2
|
|
df-i1p oveq2i syl6eqr ) BDEZCFZBGHIZJKCLUEAFZJKALMZUDBJKCLBUFJKALMZUDGJKC
|
|
LGUFJKALMZNIZUHONIUCGDEUGUJPQABGCRSOUIUHNACTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d B m $. $d C m $.
|
|
$( Approximate division for positive rationals. Proposition 12.7 of
|
|
[BauerTaylor], p. 55 (a special case where ` A ` and ` B ` are positive,
|
|
as well as ` C ` ). Our proof is simpler than the one in BauerTaylor
|
|
because we have reciprocals. (Contributed by Jim Kingdon,
|
|
8-Dec-2019.) $)
|
|
appdivnq $p |- ( ( A <Q B /\ C e. Q. ) ->
|
|
E. m e. Q. ( A <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) ) $=
|
|
( cltq wbr cnq wcel wa cmq co adantr ltmnqg syl3anc syl2anc c1q mulcomnqg
|
|
wb wceq bitrd crq cv wrex simpl ltrelnq brel simpld simprd recclnq adantl
|
|
cfv mpbid ltbtwnnqq sylib mulclnq simpr simplr recidnq ad2antlr mulassnqg
|
|
oveq1d 1nq mpan mulidnq syl 3eqtr3d breq1d breq2d anbi12d biimpd reximdva
|
|
eqtrd mpd ) ABEFZCGHZIZCUAUKZAJKZDUBZEFZVSVQBJKZEFZIZDGUCZAVSCJKZEFZWEBEF
|
|
ZIZDGUCVPVRWAEFZWDVPVNWIVNVOUDVPAGHZBGHZVQGHZVNWIRVPWJWKVNWJWKIVOABGGEUEU
|
|
FLZUGZVPWJWKWMUHZVOWLVNCUIUJZABVQMNULDVRWAUMUNVPWCWHDGVPVSGHZIZWCWHWRWCAC
|
|
VSJKZEFZWSBEFZIWHWRVTWTWBXAWRVTCVRJKZWSEFZWTWRVRGHZWQVOVTXCRWRWLWJXDVPWLW
|
|
QWPLZVPWJWQWNLZVQAUOOVPWQUPZVNVOWQUQZVRVSCMNWRXBAWSEWRCVQJKZAJKZPAJKZXBAV
|
|
OXJXKSVNWQVOXIPAJCURZVAUSWRVOWLWJXJXBSXHXEXFCVQAUTNWRWJXKASXFWJXKAPJKZAPG
|
|
HZWJXKXMSVBPAQVCAVDVLVEVFVGTWRWBWSCWAJKZEFZXAWRWQWAGHZVOWBXPRXGWRWLWKXQXE
|
|
VPWKWQWOLZVQBUOOXHVSWACMNWRXOBWSEWRXIBJKZPBJKZXOBVOXSXTSVNWQVOXIPBJXLVAUS
|
|
WRVOWLWKXSXOSXHXEXRCVQBUTNWRWKXTBSXRWKXTBPJKZBXNWKXTYASVBPBQVCBVDVLVEVFVH
|
|
TVIWRWTWFXAWGWRWSWEAEWRVOWQWSWESXHXGCVSQOZVHWRWSWEBEYBVGVITVJVKVM $.
|
|
$}
|
|
|
|
${
|
|
$d B m x $. $d C m x $.
|
|
$( Approximate division for positive rationals. This can be thought of as
|
|
a variation of ~ appdivnq in which ` A ` is zero, although it can be
|
|
stated and proved in terms of positive rationals alone, without zero as
|
|
such. (Contributed by Jim Kingdon, 9-Dec-2019.) $)
|
|
appdiv0nq $p |- ( ( B e. Q. /\ C e. Q. ) ->
|
|
E. m e. Q. ( m .Q C ) <Q B ) $=
|
|
( vx cnq wcel wa cv cltq wbr cmq co wrex nsmallnqq adantr appdivnq reximi
|
|
simpr syl ancoms ad2ant2l rexlimddv ) AEFZBEFZGDHZAIJZCHBKLZAIJZCEMZDEUCU
|
|
FDEMUDDANOUDUFUIUCUEEFUFUDUIUFUDGUEUGIJZUHGZCEMUIUEABCPUKUHCEUJUHRQSTUAUB
|
|
$.
|
|
$}
|
|
|
|
${
|
|
prmuloclemcalc.ru $e |- ( ph -> R <Q U ) $.
|
|
prmuloclemcalc.udp $e |- ( ph -> U <Q ( D +Q P ) ) $.
|
|
prmuloclemcalc.axb $e |- ( ph -> ( A +Q X ) = B ) $.
|
|
prmuloclemcalc.pbrx $e |- ( ph -> ( P .Q B ) <Q ( R .Q X ) ) $.
|
|
prmuloclemcalc.a $e |- ( ph -> A e. Q. ) $.
|
|
prmuloclemcalc.b $e |- ( ph -> B e. Q. ) $.
|
|
prmuloclemcalc.d $e |- ( ph -> D e. Q. ) $.
|
|
prmuloclemcalc.p $e |- ( ph -> P e. Q. ) $.
|
|
prmuloclemcalc.x $e |- ( ph -> X e. Q. ) $.
|
|
$( Calculations for ~ prmuloc . (Contributed by Jim Kingdon,
|
|
9-Dec-2019.) $)
|
|
prmuloclemcalc $p |- ( ph -> ( U .Q A ) <Q ( D .Q B ) ) $=
|
|
( co cltq syl2anc cmq wbr cplq oveq2d cnq wcel wceq wa ltrelnq syl simprd
|
|
brel distrnqg syl3anc eqtr3d ltmnqi breqtrd eqbrtrrd mulclnq ltanqi sotri
|
|
mulcomnqg ltsonq simpld 3brtr3d addcomnqg wb ltanqg mpbird ) AGBUARZCDUAR
|
|
ZDCUARZSAVJVKSUBZGHUARZVJUCRZVNVKUCRZSUBZAVJVNUCRZVKVNUCRZVOVPSAGCUARZVRV
|
|
SSAGBHUCRZUARZVTVRAWACGUAKUDAGUEUFZBUEUFZHUEUFZWBVRUGAFUEUFZWCAFGSUBZWFWC
|
|
UHIFGUEUESUIULUJZUKZMQGBHUMUNUOACGUARZVTVSSACUEUFZWCWJVTUGNWICGVBTAWJVKFH
|
|
UARZUCRZSUBZWMVSSUBZWJVSSUBAWJVKCEUARZUCRZSUBWQWMSUBZWNAWJCDEUCRZUARZWQSA
|
|
GWSSUBWKWJWTSUBJNGWSCUPTAWKDUEUFZEUEUFZWTWQUGNOPCDEUMUNUQAWPWLSUBVKUEUFZW
|
|
RAECUARZWPWLSAXBWKXDWPUGPNECVBTLURAWKXAXCNOCDUSTZWPWLVKUTTWJWQWMSUEVCUIVA
|
|
TAWLVNSUBXCWOAHFUARZHGUARZWLVNSAWGWEXFXGSUBIQFGHUPTAWEWFXFWLUGQAWFWCWHVDH
|
|
FVBTAWEWCXGVNUGQWIHGVBTVEXEWLVNVKUTTWJWMVSSUEVCUIVATURURAVJUEUFZVNUEUFZVR
|
|
VOUGAWCWDXHWIMGBUSTZAWCWEXIWIQGHUSTZVJVNVFTAXCXIVSVPUGXEXKVKVNVFTVEAXHXCX
|
|
IVMVQVGXJXEXKVJVKVNVHUNVIAWKXAVKVLUGNOCDVBTUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A d p r u x $. $d B d p r u x $. $d L d p r u x $. $d U d p r u x $.
|
|
$( Positive reals are multiplicatively located. Lemma 12.8 of
|
|
[BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
prmuloc $p |- ( ( <. L , U >. e. P. /\ A <Q B ) -> E. d e. Q. E. u e. Q.
|
|
( d e. L /\ u e. U /\ ( u .Q A ) <Q ( d .Q B ) ) ) $=
|
|
( vx vr vp wcel cltq wbr wa cv co cmq cnq wrex ad2antrr adantlr cplq wceq
|
|
cop cnp ltexnqi adantl prml simprl simplrl mulclnq syl2anc ltrelnq simprd
|
|
w3a brel ad3antlr appdiv0nq prarloc ad2ant2r r2ex elprnql adantrr simplll
|
|
wex sylib elprnqu prltlu 3adant1r 3adant2l 3adant3l simprr simplrr simpld
|
|
3expa prmuloclemcalc df-3an sylanbrc jca31 2eximdv mpd sylibr rexlimddv
|
|
ex ) EDUCUDJZBCKLZMZBGNZUAOCUBZFNZEJZANZDJZWKBPOWICPOKLZUNZAQRFQRZGQWEWHG
|
|
QRWDGBCUEUFWFWGQJZWHMZMZHNZEJZWOHQWDWTHQRWEWQHDEUGSWRWSQJZWTMZMZINZCPOWSW
|
|
GPOZKLZWOIQXCXEQJZCQJZXFIQRXCXAWPXGWRXAWTUHWFWPWHXBUIZWSWGUJUKWEXHWDWQXBW
|
|
EBQJZXHBCQQKULUOZUMUPZXECIUQUKXCXDQJZXFMZMZWIQJZWKQJZMWNMZAVDFVDZWOXOWJWL
|
|
MZWKWIXDUAOKLZMZAVDFVDZXSXOYAADRFERZYCWRXMYDXBXFWFXMYDWQWDXMYDWEXDDEFAURT
|
|
TUSYAFAEDUTVEXOYBXRFAXOYBXRXOYBMZXPXQWNXOXTXPYAXCWJXPXNWLWRWJXPXBWFWJXPWQ
|
|
WDWJXPWEWIDEVATTTUSVBZYEWDWLXQXCWDXNYBWDWEWQXBVCSYEWJWLXOXTYAUHZUMWKDEVFU
|
|
KYEXTWMWNYGYEBCWIXDWSWKWGXCXTWSWKKLZXNYAWRXBXTYHWFXBXTYHWQWFXBWLYHWJWFWTW
|
|
LYHXAWDWTWLYHWEWSWKDEVGVHVIVJVHVNUSXOXTYAVKXCWHXNYBWFWPWHXBVLSXCXMXFYBVLX
|
|
CXJXNYBWEXJWDWQXBWEXJXHXKVMUPSXCXHXNYBXLSYFXCXMXFYBUIXCWPXNYBXISVOWJWLWMV
|
|
PVQVRWCVSVTWNFAQQUTWAWBWBWB $.
|
|
$}
|
|
|
|
${
|
|
$d B x y $. $d L x y $. $d U x y $.
|
|
$( Positive reals are multiplicatively located. This is a variation of
|
|
~ prmuloc which only constructs one (named) point and is therefore often
|
|
easier to work with. It states that given a ratio ` B ` , there are
|
|
elements of the lower and upper cut which have exactly that ratio
|
|
between them. (Contributed by Jim Kingdon, 28-Dec-2019.) $)
|
|
prmuloc2 $p |- ( ( <. L , U >. e. P. /\ 1Q <Q B ) ->
|
|
E. x e. L ( x .Q B ) e. U ) $=
|
|
( vy cop wcel c1q cltq wbr wa cv cmq co cnq wrex wi syl2anc mpd ex simpr1
|
|
cnp w3a prmuloc nfv nfre1 simpr3 wceq wb simplrr mulidnq breq1 3syl mpbid
|
|
simplll simpr2 prcunqu rspe anassrs rexlimdva rexlimd ) DCFUBGZHBIJZKZALZ
|
|
DGZELZCGZVGHMNZVEBMNZIJZUCZEOPZAOPVJCGZADPZEHBCDAUDVDVMVOAOVDAUEVNADUFVDV
|
|
EOGZVMVOQVDVPKVLVOEOVDVPVGOGZVLVOQVDVPVQKZKZVLVOVSVLKZVFVNVOVSVFVHVKUAVTV
|
|
GVJIJZVNVTVKWAVSVFVHVKUGVTVQVIVGUHVKWAUIVDVPVQVLUJVGUKVIVGVJIULUMUNVTVBVH
|
|
WAVNQVBVCVRVLUOVSVFVHVKUPVJVGCDUQRSVNADURRTUSUTTVAS $.
|
|
$}
|
|
|
|
${
|
|
$d A v w x y z $. $d B v w x y z $. $d G w y z $. $d H w y z $.
|
|
$d X w y z $.
|
|
$( Lemma to prove downward closure in positive real multiplication.
|
|
(Contributed by Jim Kingdon, 10-Dec-2019.) $)
|
|
mulnqprl $p |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\
|
|
( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) ->
|
|
( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) ) $=
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|
( vy vz vw cnp wcel cfv wa cnq cmq co cltq wbr syl wceq c1q vx vv crq cmp
|
|
c1st cv w3a ltmnqg adantl simpr c2nd prop elprnql sylan ad2antrr ad2antlr
|
|
cop mulclnq syl2anc recclnq mulcomnqg caovord2d mulassnqg syl3anc recidnq
|
|
wb oveq2d mulidnq 3eqtrd breq2d bitrd wi prcdnql sylbid df-imp genpprecll
|
|
exp4b com34 imp32 adantlr adantr syld eqtrd eleq1d sylibd ) AIJZCAUEKZJZL
|
|
ZBIJZDBUEKZJZLZLZEMJZLZECDNOZPQZEDUCKZNOZDNOZABUDOUEKZJZEXBJWPWRWTWGJZXCW
|
|
PWRWTCPQZXDWPWRWTWQWSNOZPQXEWPFGHEWQWSPMNFUFZMJZGUFZMJZHUFZMJUGXGXIPQXKXG
|
|
NOXKXINOPQVFWPXGXIXKUHUIWNWOUJZWPCMJZDMJZWQMJWIXMWMWOWFWGAUKKZUQIJZWHXMAU
|
|
LZCXOWGUMUNUOZWMXNWIWOWJWKBUKKZUQIJWLXNBULDXSWKUMUNUPZCDURUSWPXNWSMJZXTDU
|
|
TRZXHXJLXGXINOXIXGNOSWPXGXIVAUIVBWPXFCWTPWPXFCDWSNOZNOZCTNOZCWPXMXNYAXFYD
|
|
SXRXTYBCDWSVCVDWPXNYDYESXTXNYCTCNDVEZVGRWPXMYECSXRCVHRVIVJVKWIXEXDVLZWMWO
|
|
WFXPWHYGXQCWTXOWGVMUNUOVNWNXDXCVLZWOWFWMYHWHWFWJWLYHWFWJXDWLXCWFWJXDWLXCU
|
|
AFGHUBABWTDUDNHUBGFUAVOXGXIURVPVQVRVSVTWAWBWPXAEXBWPXAEWSDNOZNOZETNOZEWPW
|
|
OYAXNXAYJSXLYBXTEWSDVCVDWPYITENWPYIYCTWPYAXNYIYCSYBXTWSDVAUSWPXNYCTSXTYFR
|
|
WCVGWOYKESWNEVHUIVIWDWE $.
|
|
|
|
$( Lemma to prove upward closure in positive real multiplication.
|
|
(Contributed by Jim Kingdon, 10-Dec-2019.) $)
|
|
mulnqpru $p |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\
|
|
( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) ->
|
|
( ( G .Q H ) <Q X -> X e. ( 2nd ` ( A .P. B ) ) ) ) $=
|
|
( vy vz vw cnp wcel cfv wa cnq cmq co cltq wbr syl wceq c1q vx vv crq cmp
|
|
c2nd cv w3a wb ltmnqg adantl c1st cop prop elprnqu sylan ad2antrr mulclnq
|
|
ad2antlr syl2anc simpr recclnq mulcomnqg caovord2d syl3anc recidnq oveq2d
|
|
mulassnqg mulidnq 3eqtrd breq1d bitrd wi prcunqu sylbid df-imp genppreclu
|
|
exp4b com34 imp32 adantlr adantr syld eqtrd eleq1d sylibd ) AIJZCAUEKZJZL
|
|
ZBIJZDBUEKZJZLZLZEMJZLZCDNOZEPQZEDUCKZNOZDNOZABUDOUEKZJZEXBJWPWRWTWGJZXCW
|
|
PWRCWTPQZXDWPWRWQWSNOZWTPQXEWPFGHWQEWSPMNFUFZMJZGUFZMJZHUFZMJUGXGXIPQXKXG
|
|
NOXKXINOPQUHWPXGXIXKUIUJWPCMJZDMJZWQMJWIXLWMWOWFAUKKZWGULIJZWHXLAUMZCWGXN
|
|
UNUOUPZWMXMWIWOWJBUKKZWKULIJWLXMBUMDWKXRUNUOURZCDUQUSWNWOUTZWPXMWSMJZXSDV
|
|
ARZXHXJLXGXINOXIXGNOSWPXGXIVBUJVCWPXFCWTPWPXFCDWSNOZNOZCTNOZCWPXLXMYAXFYD
|
|
SXQXSYBCDWSVGVDWPXMYDYESXSXMYCTCNDVEZVFRWPXLYECSXQCVHRVIVJVKWIXEXDVLZWMWO
|
|
WFXOWHYGXPWTCWGXNVMUOUPVNWNXDXCVLZWOWFWMYHWHWFWJWLYHWFWJXDWLXCWFWJXDWLXCU
|
|
AFGHUBABWTDUDNHUBGFUAVOXGXIUQVPVQVRVSVTWAWBWPXAEXBWPXAEWSDNOZNOZETNOZEWPW
|
|
OYAXMXAYJSXTYBXSEWSDVGVDWPYITENWPYIYCTWPYAXMYIYCSYBXSWSDVBUSWPXMYCTSXSYFR
|
|
WCVFWOYKESWNEVHUJVIWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d D x y z $. $d E x y z $. $d T x y z $. $d U x y z $. $d ph x y z $.
|
|
mullocprlem.ab $e |- ( ph -> ( A e. P. /\ B e. P. ) ) $.
|
|
mullocprlem.uqedu $e |- ( ph -> ( U .Q Q ) <Q ( E .Q ( D .Q U ) ) ) $.
|
|
mullocprlem.edutdu $e |- ( ph ->
|
|
( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) ) $.
|
|
mullocprlem.tdudr $e |- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) ) $.
|
|
mullocprlem.qr $e |- ( ph -> ( Q e. Q. /\ R e. Q. ) ) $.
|
|
mullocprlem.duq $e |- ( ph -> ( D e. Q. /\ U e. Q. ) ) $.
|
|
mullocprlem.du $e |- ( ph -> ( D e. ( 1st ` A ) /\ U e. ( 2nd ` A ) ) ) $.
|
|
mullocprlem.et $e |- ( ph -> ( E e. Q. /\ T e. Q. ) ) $.
|
|
$( Calculations for ~ mullocpr . (Contributed by Jim Kingdon,
|
|
10-Dec-2019.) $)
|
|
mullocprlem $p |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/
|
|
R e. ( 2nd ` ( A .P. B ) ) ) ) $=
|
|
( wcel co cmq vx vy vz c1st cfv cmp c2nd wo wa cltq wbr cnq simpld simprd
|
|
cv wceq mulcomnqg adantl w3a mulassnqg caov13d breqtrd wb mulclnq syl2anc
|
|
ltmnqg syl3anc mpbird adantr cnp wi jca anim1i mulnqprl syl21anc mpd orcd
|
|
caov12d breq1d bitr4d eqbrtrrd mulnqpru olcd syl breq12d bitrd prop prloc
|
|
cop sylan mpjaodan ) AICUDUEZRZEBCUFSZUDUERZFWNUGUERZUHGCUGUEZRZAWMUIZWOW
|
|
PWSEDITSZUJUKZWOAXAWMAXAHETSZHWTTSZUJUKZAXBIDHTSZTSZXCUJKAUAUBUCIDHULTAIU
|
|
LRZGULRZQUMZADULRZHULRZOUMZAXJXKOUNZUAUOZULRZUBUOZULRZUIXNXPTSZXPXNTSUPAX
|
|
NXPUQURZXOXQUCUOZULRUSXRXTTSXNXPXTTSTSUPAXNXPXTUTURZVAVBAEULRZWTULRZXKXAX
|
|
DVCAYBFULRZNUMZAXJXGYCXLXIDIVDVEXMEWTHVFVGVHVIWSBVJRZDBUDUERZUIZCVJRZWMUI
|
|
YBXAWOVKAYHWMAYFYGAYFYIJUMZAYGHBUGUERZPUMVLVIAYIWMAYFYIJUNZVMAYBWMYEVIBCD
|
|
IEVNVOVPVQAWRUIZWPWOYMHGTSZFUJUKZWPAYOWRAGHTSZYNFUJAXHXKYPYNUPAXGXHQUNZXM
|
|
GHUQVEAYPFUJUKZGXETSZDFTSZUJUKZMAYRDYPTSZYTUJUKZUUAAYPULRZYDXJYRUUCVCAXHX
|
|
KUUDYQXMGHVDVEAYBYDNUNZXLYPFDVFVGAYSUUBYTUJAUAUBUCGDHULTYQXLXMXSYAVRVSVTV
|
|
HWAVIYMYFYKUIZYIWRUIYDYOWPVKAUUFWRAYFYKYJAYGYKPUNVLVIAYIWRYLVMAYDWRUUEVIB
|
|
CHGFWBVOVPWCAYIIGUJUKZWMWRUHZYLAUUGXFYSUJUKZLAUUGXEITSZXEGTSZUJUKZUUIAXGX
|
|
HXEULRZUUGUULVCXIYQAXJXKUIUUMODHVDWDZIGXEVFVGAUUJXFUUKYSUJAUUMXGUUJXFUPUU
|
|
NXIXEIUQVEAUUMXHUUKYSUPUUNYQXEGUQVEWEWFVHYIWLWQWIVJRUUGUUHCWGIGWQWLWHWJVE
|
|
WK $.
|
|
$}
|
|
|
|
${
|
|
$d A d e q r t u $. $d B d e q r t u $.
|
|
$( Locatedness of multiplication on positive reals. Lemma 12.9 in
|
|
[BauerTaylor], p. 56 (but where both ` A ` and ` B ` are positive, not
|
|
just ` A ` ). (Contributed by Jim Kingdon, 8-Dec-2019.) $)
|
|
mullocpr $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
A. q e. Q. A. r e. Q. ( q <Q r ->
|
|
( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) ) $=
|
|
( vd vu ve vt cnp wcel wa cv cltq wbr co cfv cnq cmq wrex ad2antrr cmp wo
|
|
c1st c2nd wi w3a wex prop prmuloc sylan r2ex sylib adantlr simprr3 simprl
|
|
cop mulclnq syl appdivnq syl2anc simprrr simplll ad2antlr simprrl simpllr
|
|
adantr ad2antll simplrl jca mullocprlem rexlimddv exlimdvv mpd ralrimivva
|
|
3simpa ex ) AIJZBIJZKZDLZCLZMNZVTABUAOZUCPJWAWCUDPJUBZUEDCQQVSVTQJWAQJKZK
|
|
ZWBWDWFWBKZELZQJFLZQJKZWHAUCPZJZWIAUDPZJZWIVTROZWHWAROZMNZUFZKZFUGEUGZWDV
|
|
SWBWTWEVQWBWTVRVQWBKWRFQSEQSZWTVQWKWMUPIJWBXAAUHFVTWAWMWKEUIUJWREFQQUKULU
|
|
MUMWGWSWDEFWGWSWDWGWSKZWOGLZWHWIROZROZMNZXEWPMNZKZWDGQXBWQXDQJZXHGQSWLWNW
|
|
QWJWGUNXBWJXIWGWJWRUOZWHWIUQURZWOWPXDGUSUTXBXCQJZXHKZKZXEHLZXDROZMNZXPWPM
|
|
NZKZWDHQXNXGXIXSHQSXBXLXFXGVAXBXIXMXKVFXEWPXDHUSUTXNXOQJZXSKZKZABWHVTWAXO
|
|
WIXCXBVSXMYAVSWEWBWSVBTXMXFXBYAXLXFXGUOVCXNXTXQXRVDXNXTXQXRVAXBWEXMYAVSWE
|
|
WBWSVETXBWJXMYAXJTXBWLWNKZXMYAWRYCWGWJWLWNWQVOVGTYBXLXTXBXLXHYAVHXNXTXSUO
|
|
VIVJVKVKVPVLVMVPVN $.
|
|
$}
|
|
|
|
${
|
|
$d A q r t u v w x y z $. $d B q r t u v w x y z $.
|
|
$( Closure of multiplication on positive reals. First statement of
|
|
Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM,
|
|
13-Mar-1996.) $)
|
|
mulclpr $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. ) $=
|
|
( vq vr vx vy vz vw vv vu vt cnp wcel wa cmp cnq cv wrex wral cmq cpw cxp
|
|
co c1st cfv c2nd cltq wbr wb wn w3a df-imp genpelxp mulclnq genpml genpmu
|
|
wo wi jca32 ltmnqg mulcomnqg mulnqprl genprndl mulnqpru genprndu genpdisj
|
|
jca mullocpr 3jca elnp1st2nd sylanbrc ) ALMBLMNZABOUCZPUAZVNUBMZCQZVMUDUE
|
|
ZMZCPRZDQZVMUFUEZMZDPRZNNVRVPVTUGUHZVTVQMNDPRUICPSZWBWDVPWAMZNCPRUIDPSZNZ
|
|
VRWFNUJCPSZWDVRWBUQURDPSCPSZUKVMLMVLVOVSWCEFGHIABOTHIGFEULZUMEFGHIABOTCWK
|
|
FQZGQZUNZUOEFGHIABOTDWKWNUPUSVLWHWIWJVLWEWGEFGHIABJKOTDCWKWNEQZWLWMUTZWOW
|
|
LVAZABJQZKQZWOVBVCEFGHIABJKOTDCWKWNWPWQABWRWSWOVDVEVGEFGHIABOTCWKWNWPWQVF
|
|
ABDCVHVIVMDCVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h l r s t u x y $. $d B f g h l r s t u x y $.
|
|
$( Lemma for ~ mulnqpr . The reverse subset relationship for the lower
|
|
cut. (Contributed by Jim Kingdon, 18-Jul-2021.) $)
|
|
mulnqprlemrl $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) C_
|
|
( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
|
|
) $=
|
|
( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c1st cfv breq1 vf vx cmp
|
|
vy co cmq wceq cnp wb nqprlu df-imp mulclnq genpelvl syl2an biimpa ltnqex
|
|
wrex vex gtnqex op1st biimpi ad2antrl adantr ad2antll wi ltrelnq brel syl
|
|
elab2 lt2mulnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
|
|
ex ) BJKZCJKZLZEDMZBNOZDPZBAMZNOAPZQZWCCNOZDPZCWFNOAPZQZUCUERSZWCBCUFUEZN
|
|
OZDPZWNWFNOAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUQFWHRSZUQZX
|
|
AWBWTXIVTWHUHKWLUHKWTXIUIWAABDUJACDUJUAHIUBUDWHWLWSFGUCUFUBUDIHUAUKHMIMUL
|
|
UMUNUOXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWSWNNOZXAXMXNXEWNNOZXMXCBNOZ
|
|
XDCNOZXOXLXPXFXJXPXBXKXJXPWDXPDXCXHFURWCXCBNTWEWGDBUPABUSUTVIVAVBVCZXLXQX
|
|
FXKXQXBXJXKXQWIXQDXDXGGURWCXDCNTWJWKDCUPACUSUTVIVAVDVCZXMXCJKVTLZXDJKWALZ
|
|
XPXQLXOVEXMXPXTXRXCBJJNVFVGVHXMXQYAXSXDCJJNVFVGVHXCBXDCVJVKVLXFXNXOUIXLWS
|
|
XEWNNTVMVNWOXNDWSWREURWCWSWNNTWPWQDWNUPAWNUSUTVIVOVSVPVQVSVR $.
|
|
|
|
$( Lemma for ~ mulnqpr . The reverse subset relationship for the upper
|
|
cut. (Contributed by Jim Kingdon, 18-Jul-2021.) $)
|
|
mulnqprlemru $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) C_
|
|
( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
|
|
) $=
|
|
( vr vs vt vg vh cnq wcel wa cv cltq wbr cab cop c2nd cfv breq2 vf vx cmp
|
|
vy co cmq wceq cnp wb nqprlu df-imp mulclnq genpelvu syl2an biimpa ltnqex
|
|
wrex vex gtnqex op2nd biimpi ad2antrl adantr ad2antll wi ltrelnq brel syl
|
|
elab2 lt2mulnq syl2anc mp2and adantl mpbird sylibr rexlimdvva mpd ssrdv
|
|
ex ) BJKZCJKZLZEDMZBNODPZBAMZNOZAPZQZWCCNODPZCWENOZAPZQZUCUERSZWCBCUFUEZN
|
|
ODPZWNWENOZAPZQRSZWBEMZWMKZWSWRKZWBWTLZWSFMZGMZUFUEZUGZGWLRSZUQFWHRSZUQZX
|
|
AWBWTXIVTWHUHKWLUHKWTXIUIWAABDUJACDUJUAHIUBUDWHWLWSFGUCUFUBUDIHUAUKHMIMUL
|
|
UMUNUOXBXFXAFGXHXGXBXCXHKZXDXGKZLLZXFXAXLXFLZWNWSNOZXAXMXNWNXENOZXMBXCNOZ
|
|
CXDNOZXOXLXPXFXJXPXBXKXJXPWFXPAXCXHFURWEXCBNTWDWGDBUPABUSUTVIVAVBVCZXLXQX
|
|
FXKXQXBXJXKXQWJXQAXDXGGURWEXDCNTWIWKDCUPACUSUTVIVAVDVCZXMVTXCJKLZWAXDJKLZ
|
|
XPXQLXOVEXMXPXTXRBXCJJNVFVGVHXMXQYAXSCXDJJNVFVGVHBXCCXDVJVKVLXFXNXOUIXLWS
|
|
XEWNNTVMVNWPXNAWSWREURWEWSWNNTWOWQDWNUPAWNUSUTVIVOVSVPVQVSVR $.
|
|
|
|
$( Lemma for ~ mulnqpr . The forward subset relationship for the lower
|
|
cut. (Contributed by Jim Kingdon, 18-Jul-2021.) $)
|
|
mulnqprlemfl $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
|
|
C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
|
|
( vr cnq wcel wa cv co cltq wbr cab cop c1st cfv c2nd wn syl cnp cmq sonr
|
|
cmp mulnqprlemru wor ltsonq mulclnq sylancr cvv ltrelnq brel simpld breq2
|
|
elab3 sylnibr ltnqex gtnqex op2nd eleq2i ssneldd adantr wo nqprlu mulclpr
|
|
elex syl2an prop vex breq1 op1st elab2 biimpi prloc ecased ex ssrdv ) BFG
|
|
ZCFGZHZEDIZBCUAJZKLZDMZWAAIZKLZAMZNZOPZVTBKLDMBWDKLAMNZVTCKLDMCWDKLAMNZUC
|
|
JZOPZVSEIZWHGZWMWLGZVSWNHWOWAWKQPZGZVSWQRWNVSWPWGQPZWAABCDUDVSWAWFGZWAWRG
|
|
VSWAWAKLZWSVSFKUEWAFGZWTRUFBCUGFWAKUBUHWEWTAWAWTXAWAUIGWTXAXAWAWAFFKUJUKU
|
|
LWAFVESWDWAWAKUMUNUOWRWFWAWCWFDWAUPZAWAUQZURUSUOUTVAVSWLWPNTGZWMWAKLZWOWQ
|
|
VBWNVSWKTGZXDVQWITGWJTGXFVRABDVCACDVCWIWJVDVFWKVGSWNXEWBXEDWMWHEVHVTWMWAK
|
|
VIWCWFXBXCVJVKVLWMWAWPWLVMVFVNVOVP $.
|
|
|
|
$( Lemma for ~ mulnqpr . The forward subset relationship for the upper
|
|
cut. (Contributed by Jim Kingdon, 18-Jul-2021.) $)
|
|
mulnqprlemfu $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. )
|
|
C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) ) $=
|
|
( vr cnq wcel wa cv co cltq wbr cab cop c2nd cfv c1st wn syl cnp cmq sonr
|
|
cmp mulnqprlemrl wor ltsonq mulclnq sylancr cvv ltrelnq brel simpld breq1
|
|
elab3 sylnibr ltnqex gtnqex op1st eleq2i ssneldd adantr wo nqprlu mulclpr
|
|
elex syl2an prop vex breq2 op2nd elab2 biimpi prloc orcomd ecased ssrdv
|
|
ex ) BFGZCFGZHZEDIZBCUAJZKLZDMZWBAIZKLZAMZNZOPZWABKLDMBWEKLAMNZWACKLDMCWE
|
|
KLAMNZUCJZOPZVTEIZWIGZWNWMGZVTWOHZWPWBWLQPZGZVTWSRWOVTWRWHQPZWBABCDUDVTWB
|
|
WDGZWBWTGVTWBWBKLZXAVTFKUEWBFGZXBRUFBCUGFWBKUBUHWCXBDWBXBXCWBUIGXBXCXCWBW
|
|
BFFKUJUKULWBFVESWAWBWBKUMUNUOWTWDWBWDWGDWBUPZAWBUQZURUSUOUTVAWQWSWPVTWRWM
|
|
NTGZWBWNKLZWSWPVBWOVTWLTGZXFVRWJTGWKTGXHVSABDVCACDVCWJWKVDVFWLVGSWOXGWFXG
|
|
AWNWIEVHWEWNWBKVIWDWGXDXEVJVKVLWBWNWMWRVMVFVNVOVQVP $.
|
|
|
|
$( Multiplication of fractions embedded into positive reals. One can
|
|
either multiply the fractions as fractions, or embed them into positive
|
|
reals and multiply them as positive reals, and get the same result.
|
|
(Contributed by Jim Kingdon, 18-Jul-2021.) $)
|
|
mulnqpr $p |- ( ( A e. Q. /\ B e. Q. ) ->
|
|
<. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. =
|
|
( <. { l | l <Q A } , { u | A <Q u } >. .P.
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) $=
|
|
( cnq wcel wa cv co cltq wbr cab cop wceq c1st cfv c2nd eqssd cnp nqprlu
|
|
cmq mulnqprlemfl mulnqprlemrl mulnqprlemfu mulnqprlemru wb mulclnq syl2an
|
|
cmp syl mulclpr preqlu syl2anc mpbir2and ) BEFZCEFZGZDHZBCUAIZJKDLUSAHZJK
|
|
ALMZURBJKDLBUTJKALMZURCJKDLCUTJKALMZUIIZNZVAOPZVDOPZNZVAQPZVDQPZNZUQVFVGA
|
|
BCDUBABCDUCRUQVIVJABCDUDABCDUERUQVASFZVDSFZVEVHVKGUFUQUSEFVLBCUGAUSDTUJUO
|
|
VBSFVCSFVMUPABDTACDTVBVCUKUHVAVDULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d f g h v w x y z A $. $d f g h v w x y z B $. $d f g h v w x y z C $.
|
|
$( Addition of positive reals is commutative. Proposition 9-3.5(ii) of
|
|
[Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) $)
|
|
addcomprg $p |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) ) $=
|
|
( vx vy vz cnp wcel wa cv wceq cfv wrex cnq crab cop sylan sylan2 anassrs
|
|
co rexbidva cplq c1st c2nd wb prop elprnql addcomnqg eqeq2d ancoms rexcom
|
|
cpp syl6bb rabbidv elprnqu opeq12d plpvlu 3eqtr4rd ) AFGZBFGZHZCIZDIZEIZU
|
|
ASZJZEAUBKZLZDBUBKZLZCMNZVEEAUCKZLZDBUCKZLZCMNZOZVAVCVBUASZJZDVHLEVFLZCMN
|
|
ZVRDVMLEVKLZCMNZOBAUKSZABUKSUTVJVTVOWBUTVIVSCMUTVIVREVFLZDVHLVSUTVGWDDVHU
|
|
RUSVBVHGZVGWDUDZUSWEHURVBMGZWFUSVHVMOFGZWEWGBUEZVBVMVHUFPWGURWFWGURHZVEVR
|
|
EVFWGURVCVFGZVEVRUDZURWKHWGVCMGZWLURVFVKOFGZWKWMAUEZVCVKVFUFPWGWMHVDVQVAV
|
|
BVCUGUHZQRTUIQRTVRDEVHVFUJULUMUTVNWACMUTVNVREVKLZDVMLWAUTVLWQDVMURUSVBVMG
|
|
ZVLWQUDZUSWRHURWGWSUSWHWRWGWIVBVMVHUNPWGURWSWJVEVREVKWGURVCVKGZWLURWTHWGW
|
|
MWLURWNWTWMWOVCVKVFUNPWPQRTUIQRTVRDEVMVKUJULUMUOUSURWCVPJCDEBAUPUICEDABUP
|
|
UQ $.
|
|
|
|
$( Addition of positive reals is associative. Proposition 9-3.5(i) of
|
|
[Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) $)
|
|
addassprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( ( A +P. B ) +P. C ) = ( A +P. ( B +P. C ) ) ) $=
|
|
( vx vy vz vw vv vf vg vh cplq df-iplp cv addclnq dmplp addclpr addassnqg
|
|
cpp genpassg ) DEFGHABCIJKSLGHFEDMENFNOPINZJNZQUAUBKNRT $.
|
|
|
|
$( Multiplication of positive reals is commutative. Proposition 9-3.7(ii)
|
|
of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) $)
|
|
mulcomprg $p |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) ) $=
|
|
( vx vz vy cnp wcel wa cv wceq cfv wrex cnq crab cop sylan sylan2 anassrs
|
|
co rexbidva cmq c1st c2nd cmp prop elprnql mulcomnqg eqeq2d ancoms rexcom
|
|
wb syl6bb rabbidv elprnqu opeq12d mpvlu 3eqtr4rd ) AFGZBFGZHZCIZDIZEIZUAS
|
|
ZJZEAUBKZLZDBUBKZLZCMNZVEEAUCKZLZDBUCKZLZCMNZOZVAVCVBUASZJZDVHLEVFLZCMNZV
|
|
RDVMLEVKLZCMNZOBAUDSZABUDSUTVJVTVOWBUTVIVSCMUTVIVREVFLZDVHLVSUTVGWDDVHURU
|
|
SVBVHGZVGWDUKZUSWEHURVBMGZWFUSVHVMOFGZWEWGBUEZVBVMVHUFPWGURWFWGURHZVEVREV
|
|
FWGURVCVFGZVEVRUKZURWKHWGVCMGZWLURVFVKOFGZWKWMAUEZVCVKVFUFPWGWMHVDVQVAVBV
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|
CUGUHZQRTUIQRTVRDEVHVFUJULUMUTVNWACMUTVNVREVKLZDVMLWAUTVLWQDVMURUSVBVMGZV
|
|
LWQUKZUSWRHURWGWSUSWHWRWGWIVBVMVHUNPWGURWSWJVEVREVKWGURVCVKGZWLURWTHWGWMW
|
|
LURWNWTWMWOVCVKVFUNPWPQRTUIQRTVRDEVMVKUJULUMUOUSURWCVPJCDEBAUPUICEDABUPUQ
|
|
$.
|
|
|
|
$( Multiplication of positive reals is associative. Proposition 9-3.7(i)
|
|
of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) $)
|
|
mulassprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( ( A .P. B ) .P. C ) = ( A .P. ( B .P. C ) ) ) $=
|
|
( vx vy vz vw vv vf vg vh cmp cmq df-imp cv mulclnq dmmp mulclpr genpassg
|
|
mulassnqg ) DEFGHABCIJKLMGHFEDNEOFOPQIOZJOZRUAUBKOTS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f g h A $. $d x y z w v u f g h B $.
|
|
$d x y z w v u f g h C $.
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem1prl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( 1st ` ( A .P. ( B +P. C ) ) ) C_
|
|
( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) $=
|
|
( vw vx vv vf vg vh vy vz cnp wcel co cmp c1st cfv cv cmq wa w3a cpp wceq
|
|
wrex wb addclpr df-imp mulclnq genpelvl sylan2 3impb cplq df-iplp addclnq
|
|
3adant1 adantr cnq c2nd prop elprnql sylan 3ad2antl1 adantrr anim12i an4s
|
|
cop 3adantl1 ad2ant2r 3anass sylanbrc simprr simpr oveq2 biimpac distrnqg
|
|
wi eqeq2d syl5ib sylc mulclpr 3adant3 ad2antrr simpll genpprecll adantlrr
|
|
3adant2 impl simplr imp syl22anc eqeltrd exp32 rexlimdvv com34 impd ssrdv
|
|
sylbid ) ALMZBLMZCLMZUAZDABCUBNZONPQZABONZACONZUBNPQZXADRZXCMZXGERZFRZSNZ
|
|
UCZFXBPQZUDEAPQZUDZXGXFMZWRWSWTXHXOUEZWSWTTWRXBLMXQBCUFGHIJKAXBXGEFOSJKIH
|
|
GUGZHRZIRZUHZUIUJUKXAXLXPEFXNXMXAXIXNMZXJXMMZXLXPVPXAYBXLYCXPXAYBXLYCXPVP
|
|
XAYBXLTZTZYCXJJRZKRZULNZUCZKCPQZUDJBPQZUDZXPXAYCYLUEZYDWSWTYMWRGHIDEBCXJJ
|
|
KUBULDEIHGUMZXSXTUNZUIUOUPYEYIXPJKYKYJYEYFYKMZYGYJMZTZYIXPYEYRYITZTZXGXIY
|
|
FSNZXIYGSNZULNZXFYTXIUQMZYFUQMZYGUQMZUAZXLYITZXGUUCUCZYTUUDUUEUUFTZUUGYEU
|
|
UDYSXAYBUUDXLWRWSYBUUDWTWRXNAURQZVFLMYBUUDAUSXIUUKXNUTVAVBVCUPXAYRUUJYDYI
|
|
WSWTYRUUJWRWSYPWTYQUUJWSYPTUUEWTYQTUUFWSYKBURQZVFLMYPUUEBUSYFUULYKUTVAWTY
|
|
JCURQZVFLMYQUUFCUSYGUUMYJUTVAVDVEVGVHUUDUUEUUFVIVJYEXLYSYIXAYBXLVKYRYIVLV
|
|
DUUHXGXIYHSNZUCZUUGUUIYIXLUUOYIXKUUNXGXJYHXISVMVQVNUUGUUNUUCXGXIYFYGVOVQV
|
|
RVSYTXDLMZXELMZUUAXDPQMZUUBXEPQMZUUCXFMZXAUUPYDYSWRWSUUPWTABVTWAWBXAUUQYD
|
|
YSWRWTUUQWSACVTWFWBYSYEYPUURYPYQYIWCXAYBYPUURXLXAYBYPUURWRWSYBYPTUURVPWTG
|
|
HIJKABXIYFOSXRYAWDWAWGWEUJYSYEYQUUSYPYQYIWHXAYBYQUUSXLXAYBYQUUSWRWTYBYQTU
|
|
USVPWSGHIJKACXIYGOSXRYAWDWFWGWEUJUUPUUQTUURUUSTUUTGHIDEXDXEUUAUUBUBULYNYO
|
|
WDWIWJWKWLWMWQWLWNWOWMWQWP $.
|
|
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem1pru $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( 2nd ` ( A .P. ( B +P. C ) ) ) C_
|
|
( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) $=
|
|
( vw vx vv vf vg vh vy vz cnp wcel co cmp c2nd cfv cv cmq wa w3a cpp wceq
|
|
wrex wb addclpr df-imp mulclnq genpelvu sylan2 3impb cplq df-iplp addclnq
|
|
3adant1 adantr cnq c1st prop elprnqu sylan 3ad2antl1 adantrr anim12i an4s
|
|
cop 3adantl1 ad2ant2r 3anass sylanbrc simprr simpr oveq2 biimpac distrnqg
|
|
wi eqeq2d syl5ib sylc mulclpr 3adant3 ad2antrr simpll genppreclu adantlrr
|
|
3adant2 impl simplr imp syl22anc eqeltrd exp32 rexlimdvv com34 impd ssrdv
|
|
sylbid ) ALMZBLMZCLMZUAZDABCUBNZONPQZABONZACONZUBNPQZXADRZXCMZXGERZFRZSNZ
|
|
UCZFXBPQZUDEAPQZUDZXGXFMZWRWSWTXHXOUEZWSWTTWRXBLMXQBCUFGHIJKAXBXGEFOSJKIH
|
|
GUGZHRZIRZUHZUIUJUKXAXLXPEFXNXMXAXIXNMZXJXMMZXLXPVPXAYBXLYCXPXAYBXLYCXPVP
|
|
XAYBXLTZTZYCXJJRZKRZULNZUCZKCPQZUDJBPQZUDZXPXAYCYLUEZYDWSWTYMWRGHIDEBCXJJ
|
|
KUBULDEIHGUMZXSXTUNZUIUOUPYEYIXPJKYKYJYEYFYKMZYGYJMZTZYIXPYEYRYITZTZXGXIY
|
|
FSNZXIYGSNZULNZXFYTXIUQMZYFUQMZYGUQMZUAZXLYITZXGUUCUCZYTUUDUUEUUFTZUUGYEU
|
|
UDYSXAYBUUDXLWRWSYBUUDWTWRAURQZXNVFLMYBUUDAUSXIXNUUKUTVAVBVCUPXAYRUUJYDYI
|
|
WSWTYRUUJWRWSYPWTYQUUJWSYPTUUEWTYQTUUFWSBURQZYKVFLMYPUUEBUSYFYKUULUTVAWTC
|
|
URQZYJVFLMYQUUFCUSYGYJUUMUTVAVDVEVGVHUUDUUEUUFVIVJYEXLYSYIXAYBXLVKYRYIVLV
|
|
DUUHXGXIYHSNZUCZUUGUUIYIXLUUOYIXKUUNXGXJYHXISVMVQVNUUGUUNUUCXGXIYFYGVOVQV
|
|
RVSYTXDLMZXELMZUUAXDPQMZUUBXEPQMZUUCXFMZXAUUPYDYSWRWSUUPWTABVTWAWBXAUUQYD
|
|
YSWRWTUUQWSACVTWFWBYSYEYPUURYPYQYIWCXAYBYPUURXLXAYBYPUURWRWSYBYPTUURVPWTG
|
|
HIJKABXIYFOSXRYAWDWAWGWEUJYSYEYQUUSYPYQYIWHXAYBYQUUSXLXAYBYQUUSWRWTYBYQTU
|
|
USVPWSGHIJKACXIYGOSXRYAWDWFWGWEUJUUPUUQTUURUUSTUUTGHIDEXDXEUUAUUBUBULYNYO
|
|
WDWIWJWKWLWMWQWLWNWOWMWQWP $.
|
|
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem4prl $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
|
|
( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\
|
|
( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) ->
|
|
( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) $=
|
|
( vw vv vu cnp wcel cv wa cltq cmq co cplq cnq syl2anc vg vh w3a c1st cfv
|
|
wbr wo cpp cmp wn ltmnqg adantl simp1 simpll c2nd cop prop elprnql syl2an
|
|
wb sylan simprl simpl2 simprlr mulcomnqg caovord2d ltanqg mulclnq simprrr
|
|
wceq simpl3 addcomnqg bitrd simpl1 addclpr 3adant1 adantr mulclpr syl3anc
|
|
distrnqg simprrl df-iplp addclnq genpprecll imp syl22anc eqeltrrd prcdnql
|
|
wi df-imp sylbid simprll ltsonq nqtri3or sotritrieq oveq1 oveq2d sylan9eq
|
|
jaod ex sylbird wdc ltdcnq ancoms dcor sylc df-dc sylib mpjaod ) DKLZEKLZ
|
|
FKLZUCZAMZDUDUEZLZBMZEUDUEZLZNZGMZXOLZCMZFUDUEZLZNZNZNZXNYAOUFZYAXNOUFZUG
|
|
ZXNXQPQZYAYCPQZRQZDEFUHQZUIQZUDUEZLZYKUJZYHYIYRYJYHYIYNYAXQPQZYMRQZOUFZYR
|
|
YHYIYLYTOUFUUBYHHIJXNYAXQOSPHMZSLZIMZSLZJMZSLUCZUUCUUEOUFZUUGUUCPQUUGUUEP
|
|
QOUFUTYHUUCUUEUUGUKULZXMXJXPXNSLZYGXJXKXLUMZXPXSYFUNXJXODUOUEZUPKLZXPUUKD
|
|
UQZXNUUMXOURVAUSZXMXJYBYASLZYGUULXTYBYEVBXJUUNYBUUQUUOYAUUMXOURVAUSZYHXKX
|
|
SXQSLZXJXKXLYGVCZXMXPXSYFVDZXKXREUOUEZUPKLXSUUSEUQXQUVBXRURVATZUUDUUFNZUU
|
|
CUUEPQUUEUUCPQVJYHUUCUUEVEULZVFYHHIJYLYTYMOSRUUHUUIUUGUUCRQUUGUUERQOUFUTY
|
|
HUUCUUEUUGVGULYHUUKUUSYLSLZUUPUVCXNXQVHTZYHUUQUUSYTSLUURUVCYAXQVHTYHUUQYC
|
|
SLZYMSLZUURYHXLYEUVHXJXKXLYGVKZXMXTYBYEVIZXLYDFUOUEZUPKLYEUVHFUQYCUVLYDUR
|
|
VATZYAYCVHTZUVDUUCUUERQUUEUUCRQVJYHUUCUUEVLULVFVMYHYPKLZUUAYQLZUUBYRWIZYH
|
|
XJYOKLZUVOXJXKXLYGVNZXMUVRYGXKXLUVRXJEFVOVPVQZDYOVRTZYHYAXQYCRQZPQZUUAYQY
|
|
HUUQUUSUVHUWCUUAVJUURUVCUVMYAXQYCVTVSYHXJUVRYBUWBYOUDUELZUWCYQLZUVSUVTXMX
|
|
TYBYEWAYHXKXLXSYEUWDUUTUVJUVAUVKXKXLNXSYENUWDHUAUBJIEFXQYCUHRJIUBUAHWBUAM
|
|
ZUBMZWCWDWEWFZXJUVRNZYBUWDNUWEHUAUBJIDYOYAUWBUIPJIUBUAHWJZUWFUWGVHZWDWEWF
|
|
WGUVOYQYPUOUEZUPKLZUVPUVQYPUQZUUAYNUWLYQWHVATWKYHYJYNYLXNYCPQZRQZOUFZYRYH
|
|
YJYMUWOOUFZUWQYHHIJYAXNYCOSPUUJUURUUPUVMUVEVFYHUVIUWOSLZUVFUWRUWQUTUVNYHU
|
|
UKUVHUWSUUPUVMXNYCVHTUVGYMUWOYLVGVSVMYHUVOUWPYQLZUWQYRWIZUWAYHXNUWBPQZUWP
|
|
YQYHUUKUUSUVHUXBUWPVJUUPUVCUVMXNXQYCVTVSZYHXJUVRXPUWDUXBYQLZUVSUVTXMXPXSY
|
|
FWLUWHUWIXPUWDNUXDHUAUBJIDYOXNUWBUIPUWJUWKWDWEWFZWGUVOUWMUWTUXAUWNUWPYNUW
|
|
LYQWHVATWKWSYHYSXNYAVJZYRYHUUKUUQUXFYSUTUUPUURSXNYAOWMXNYAWNWOTYHUXFYRYHU
|
|
XFNUXBYNYQYHUXFUXBUWPYNUXCUXFUWOYMYLRXNYAYCPWPWQWRYHUXDUXFUXEVQWGWTXAYHYK
|
|
XBZYKYSUGYHUUKUUQUXGUUPUURUUKUUQNYIXBYJXBZUXGXNYAXCUUQUUKUXHYAXNXCXDYIYJX
|
|
EXFTYKXGXHXI $.
|
|
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem4pru $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
|
|
( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\
|
|
( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) ->
|
|
( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) $=
|
|
( vw vv vu cnp wcel cv wa cltq cmq co cplq cnq syl2anc vg vh w3a c2nd cfv
|
|
wbr wo cpp cmp wn ltmnqg adantl simp1 simpll c1st cop prop elprnqu syl2an
|
|
wb sylan simprl simpl3 simprrr mulcomnqg caovord2d mulclnq simpl2 simprlr
|
|
wceq ltanqg syl3anc bitrd simpl1 addclpr 3adant1 mulclpr distrnqg simprll
|
|
wi adantr df-iplp addclnq genppreclu imp syl22anc df-imp eqeltrrd prcunqu
|
|
sylbid addcomnqg simprrl ltsonq nqtri3or sotritrieq oveq1 oveq2d sylan9eq
|
|
jaod ex sylbird wdc ltdcnq ancoms dcor sylc df-dc sylib mpjaod ) DKLZEKLZ
|
|
FKLZUCZAMZDUDUEZLZBMZEUDUEZLZNZGMZXOLZCMZFUDUEZLZNZNZNZXNYAOUFZYAXNOUFZUG
|
|
ZXNXQPQZYAYCPQZRQZDEFUHQZUIQZUDUEZLZYKUJZYHYIYRYJYHYIYLXNYCPQZRQZYNOUFZYR
|
|
YHYIYTYMOUFZUUBYHHIJXNYAYCOSPHMZSLZIMZSLZJMZSLUCZUUDUUFOUFZUUHUUDPQUUHUUF
|
|
PQOUFUTYHUUDUUFUUHUKULZXMXJXPXNSLZYGXJXKXLUMZXPXSYFUNXJDUOUEZXOUPKLZXPUUL
|
|
DUQZXNXOUUNURVAUSZXMXJYBYASLZYGUUMXTYBYEVBXJUUOYBUURUUPYAXOUUNURVAUSZYHXL
|
|
YEYCSLZXJXKXLYGVCZXMXTYBYEVDZXLFUOUEZYDUPKLYEUUTFUQYCYDUVCURVATZUUEUUGNZU
|
|
UDUUFPQUUFUUDPQVJYHUUDUUFVEULZVFYHYTSLZYMSLZYLSLZUUCUUBUTYHUULUUTUVGUUQUV
|
|
DXNYCVGTYHUURUUTUVHUUSUVDYAYCVGTZYHUULXQSLZUVIUUQYHXKXSUVKXJXKXLYGVHZXMXP
|
|
XSYFVIZXKEUOUEZXRUPKLXSUVKEUQXQXRUVNURVATZXNXQVGTZYTYMYLVKVLVMYHYPKLZUUAY
|
|
QLZUUBYRVTZYHXJYOKLZUVQXJXKXLYGVNZXMUVTYGXKXLUVTXJEFVOVPWAZDYOVQTZYHXNXQY
|
|
CRQZPQZUUAYQYHUULUVKUUTUWEUUAVJUUQUVOUVDXNXQYCVRVLZYHXJUVTXPUWDYOUDUELZUW
|
|
EYQLZUWAUWBXMXPXSYFVSYHXKXLXSYEUWGUVLUVAUVMUVBXKXLNXSYENUWGHUAUBJIEFXQYCU
|
|
HRJIUBUAHWBUAMZUBMZWCWDWEWFZXJUVTNZXPUWGNUWHHUAUBJIDYOXNUWDUIPJIUBUAHWGZU
|
|
WIUWJVGZWDWEWFZWHUVQYPUOUEZYQUPKLZUVRUVSYPUQZYNUUAYQUWPWIVATWJYHYJYAXQPQZ
|
|
YMRQZYNOUFZYRYHYJUWSYLOUFUXAYHHIJYAXNXQOSPUUKUUSUUQUVOUVFVFYHHIJUWSYLYMOS
|
|
RUUIUUJUUHUUDRQUUHUUFRQOUFUTYHUUDUUFUUHVKULYHUURUVKUWSSLUUSUVOYAXQVGTUVPU
|
|
VJUVEUUDUUFRQUUFUUDRQVJYHUUDUUFWKULVFVMYHUVQUWTYQLZUXAYRVTZUWCYHYAUWDPQZU
|
|
WTYQYHUURUVKUUTUXDUWTVJUUSUVOUVDYAXQYCVRVLYHXJUVTYBUWGUXDYQLZUWAUWBXMXTYB
|
|
YEWLUWKUWLYBUWGNUXEHUAUBJIDYOYAUWDUIPUWMUWNWDWEWFWHUVQUWQUXBUXCUWRYNUWTYQ
|
|
UWPWIVATWJWSYHYSXNYAVJZYRYHUULUURUXFYSUTUUQUUSSXNYAOWMXNYAWNWOTYHUXFYRYHU
|
|
XFNUWEYNYQYHUXFUWEUUAYNUWFUXFYTYMYLRXNYAYCPWPWQWRYHUWHUXFUWOWAWHWTXAYHYKX
|
|
BZYKYSUGYHUULUURUXGUUQUUSUULUURNYIXBYJXBZUXGXNYAXCUURUULUXHYAXNXCXDYIYJXE
|
|
XFTYKXGXHXI $.
|
|
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem5prl $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_
|
|
( 1st ` ( A .P. ( B +P. C ) ) ) ) $=
|
|
( vw vv vf vg vh vx vy vz cnp wcel cmp co c1st cfv cv wrex wa vu w3a cplq
|
|
cpp wb mulclpr 3adant3 3adant2 df-iplp addclnq genpelvl syl2anc wi df-imp
|
|
cmq mulclnq anbi2d distrlem4prl oveq12 eqeq2d eleq1 syl6bi imp syl5ibrcom
|
|
wceq exp4b com3l com23 rexlimivv rexlimdvv com3r sylbid impd ssrdv ) ALMZ
|
|
BLMZCLMZUBZDABNOZACNOZUDOPQZABCUDONOPQZVRDRZWAMZWCERZUARZUCOZVEZUAVTPQZSE
|
|
VSPQZSZWCWBMZVRVSLMZVTLMZWDWKUEVOVPWMVQABUFUGVOVQWNVPACUFUHFGHIJVSVTWCEUA
|
|
UDUCIJHGFUIGRZHRZUJUKULVRWHWLEUAWJWIVRWEWJMZWFWIMZTWQWFFRZKRZUOOZVEZKCPQZ
|
|
SFAPQZSZTWHWLUMZVRWRXEWQVOVQWRXEUEVPIGHDEACWFFKNUODEHGIUNWOWPUPZUKUHUQVRW
|
|
QXEXFVRWQWEIRZJRZUOOZVEZJBPQZSIXDSZXEXFUMVOVPWQXMUEVQFGHDEABWEIJNUODEHGFU
|
|
NXGUKUGXMXEVRXFXMXBVRXFUMZFKXDXCXKWSXDMWTXCMTZXBXNUMZUMIJXDXLXHXDMXIXLMTZ
|
|
XOXKXPXQXOXKXBXNVRXQXOTZXKXBTZXFVRXRXSWHWLVRXRTWLXSWHTXJXAUCOZWBMZIJKABCF
|
|
URXSWHWLYAUEZXSWHWCXTVEYBXSWGXTWCWEXJWFXAUCUSUTWCXTWBVAVBVCVDVFVGVFVHVIVJ
|
|
VKVLVMVLVJVLVN $.
|
|
|
|
$( Lemma for distributive law for positive reals. (Contributed by Jim
|
|
Kingdon, 12-Dec-2019.) $)
|
|
distrlem5pru $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_
|
|
( 2nd ` ( A .P. ( B +P. C ) ) ) ) $=
|
|
( vw vv vf vg vh vx vy vz cnp wcel cmp co c2nd cfv cv wrex wa vu w3a cplq
|
|
cpp wb mulclpr 3adant3 3adant2 df-iplp addclnq genpelvu syl2anc wi df-imp
|
|
cmq mulclnq anbi2d distrlem4pru oveq12 eqeq2d eleq1 syl6bi imp syl5ibrcom
|
|
wceq exp4b com3l com23 rexlimivv rexlimdvv com3r sylbid impd ssrdv ) ALMZ
|
|
BLMZCLMZUBZDABNOZACNOZUDOPQZABCUDONOPQZVRDRZWAMZWCERZUARZUCOZVEZUAVTPQZSE
|
|
VSPQZSZWCWBMZVRVSLMZVTLMZWDWKUEVOVPWMVQABUFUGVOVQWNVPACUFUHFGHIJVSVTWCEUA
|
|
UDUCIJHGFUIGRZHRZUJUKULVRWHWLEUAWJWIVRWEWJMZWFWIMZTWQWFFRZKRZUOOZVEZKCPQZ
|
|
SFAPQZSZTWHWLUMZVRWRXEWQVOVQWRXEUEVPIGHDEACWFFKNUODEHGIUNWOWPUPZUKUHUQVRW
|
|
QXEXFVRWQWEIRZJRZUOOZVEZJBPQZSIXDSZXEXFUMVOVPWQXMUEVQFGHDEABWEIJNUODEHGFU
|
|
NXGUKUGXMXEVRXFXMXBVRXFUMZFKXDXCXKWSXDMWTXCMTZXBXNUMZUMIJXDXLXHXDMXIXLMTZ
|
|
XOXKXPXQXOXKXBXNVRXQXOTZXKXBTZXFVRXRXSWHWLVRXRTWLXSWHTXJXAUCOZWBMZIJKABCF
|
|
URXSWHWLYAUEZXSWHWCXTVEYBXSWGXTWCWEXJWFXAUCUSUTWCXTWBVAVBVCVDVFVGVFVHVIVJ
|
|
VKVLVMVLVJVLVN $.
|
|
$}
|
|
|
|
$( Multiplication of positive reals is distributive. Proposition 9-3.7(iii)
|
|
of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) $)
|
|
distrprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( A .P. ( B +P. C ) ) = ( ( A .P. B ) +P. ( A .P. C ) ) ) $=
|
|
( cnp wcel w3a cpp cmp wceq c1st cfv distrlem1prl distrlem5prl distrlem1pru
|
|
co c2nd eqssd addclpr syl2anc mulclpr distrlem5pru wa wb simp1 simp2 preqlu
|
|
simp3 mpbir2and ) ADEZBDEZCDEZFZABCGOZHOZABHOZACHOZGOZIZUNJKZUQJKZIZUNPKZUQ
|
|
PKZIZULUSUTABCLABCMQULVBVCABCNABCUAQULUNDEZUQDEZURVAVDUBUCULUIUMDEZVEUIUJUK
|
|
UDZULUJUKVGUIUJUKUEZUIUJUKUGZBCRSAUMTSULUODEZUPDEZVFULUIUJVKVHVIABTSULUIUKV
|
|
LVHVJACTSUOUPRSUNUQUFSUH $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( If a positive real is less than a second positive real, its lower cut is
|
|
a subset of the second's lower cut. (Contributed by Jim Kingdon,
|
|
23-Dec-2019.) $)
|
|
ltprordil $p |- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) ) $=
|
|
( vx vy cltp wbr cv c2nd cfv wcel c1st wa wss cnq cnp wrex adantr cop syl
|
|
prop ltrelpr brel ltdfpr biimpd mpcom simpll simpr simprrl simpld syl3an1
|
|
cltq prltlu syl3anc wi simprrr simprd prcdnql sylan syl2anc mpd rexlimddv
|
|
ex ssrdv ) ABEFZCGZAHIZJZVEBKIZJZLZAKIZVHMCNAOJZBOJZLZVDVJCNPZABOOEUAUBZV
|
|
NVDVOABCUCUDUEVDVENJZVJLZLZDVKVHVSDGZVKJZVTVHJZVSWALZVTVEUKFZWBWCVDWAVGWD
|
|
VDVRWAUFZVSWAUGVSVGWAVDVQVGVIUHQVDVKVFROJZWAVGWDVDVLWFVDVLVMVPUIATSVTVEVF
|
|
VKULUJUMWCVDVIWDWBUNZWEVSVIWAVDVQVGVIUOQVDVHBHIZROJZVIWGVDVMWIVDVLVMVPUPB
|
|
TSVEVTWHVHUQURUSUTVBVCVA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f g h A $.
|
|
$( Lemma for ~ 1idpr . (Contributed by Jim Kingdon, 13-Dec-2019.) $)
|
|
1idprl $p |- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) ) $=
|
|
( vf vg vw vy wcel c1p co cfv cv cmq wceq wrex cltq wbr wa cnq c1q adantl
|
|
wb vx vu vv vz cnp cmp c1st wss ssid rexss ax-mp r19.42v 1pr c2nd elprnql
|
|
cop prop sylan mpan w3a 3ad2ant3 1prl abeq2i ltmnqg mp3an2 ancoms mulidnq
|
|
breq1 breq2d adantr bitrd syl5rbb 3adant3 syl3an1 syl3an2 3expia pm5.32rd
|
|
1nq rexbidva syl5rbbr syl5bb df-imp mulclnq genpelvl mpan2 prnmaxl wi crq
|
|
ltrelnq brel simpl simpr recclnq mulcomnqg caovord2d recidnq biimpd mpcom
|
|
sylan2 elab2g mpbird mulassnqg caov12d oveq2d 3eqtrrd oveq2 eqeq2d rspcev
|
|
syl syl2anc a1i ancld reximia ex prcdnql adantrd rexlimdva impbid 3bitr4d
|
|
eqrdv ) AUEFZUAAGUFHUGIZAUGIZYAUAJZBJZCJZKHZLZCGUGIZMZBYCMZYDYENOZYJPZBYC
|
|
MZYDYBFZYDYCFZYAYJYMBYCYJYFYIFZYHPZCYIMZYAYEYCFZPZYMYIYIUHYJYSTYIUIYHCYIY
|
|
IUJUKYMYLYHPZCYIMUUAYSYLYHCYIULUUAUUBYRCYIUUAYQPYHYLYQUUAYQYHYLYQTZYQUUAY
|
|
FQFZYHUUCGUEFZYQUUDUMUUEYIGUNIZUPUEFYQUUDGUQYFUUFYIUOURUSUUAYEQFZUUDYHUUC
|
|
YAYCAUNIZUPUEFZYTUUGAUQZYEUUHYCUOURUUGUUDYHUTYLYGYENOZYQYHUUGYLUUKTUUDYDY
|
|
GYENVHVAUUGUUDUUKYQTYHYQYFRNOZUUGUUDPZUUKUULCYICVBZVCUUMUULYGYERKHZNOZUUK
|
|
UUDUUGUULUUPTZUUDRQFUUGUUQVRYFRYEVDVEVFUUGUUPUUKTUUDUUGUUOYEYGNYEVGVIVJVK
|
|
VLVMVKVNVOVPVQVSVTWAVSYAUUEYOYKTUMDUBUCEUDAGYDBCUFKEUDUCUBDWBUBJUCJWCWDWE
|
|
YAYPYNYAYPYNYAYPPYLBYCMZYNYAUUIYPUURUUJBYDUUHYCWFURYLYMBYCYTYLYJYLYJWGYTY
|
|
LYDYEWHIZKHZYIFZYDYEUUTKHZLZYJYLUVAUUTRNOZYDQFZUUGPZYLUVDYDYEQQNWIWJZUVFY
|
|
LUVDUVFYLUUTYEUUSKHZNOZUVDUVFEUDDYDYEUUSNQKEJZQFZUDJZQFZDJZQFUTZUVJUVLNOU
|
|
VNUVJKHUVNUVLKHNOTUVFUVJUVLUVNVDSUVEUUGWKZUVEUUGWLZUUGUUSQFZUVEYEWMZSZUVK
|
|
UVMPUVJUVLKHZUVLUVJKHLUVFUVJUVLWNSZWOUUGUVIUVDTUVEUUGUVHRUUTNYEWPZVISVKWQ
|
|
WRYLUUTQFZUVAUVDTYLUVFUWDUVGUUGUVEUVRUWDUVSYDUUSWCWSXIUULUVDCUUTYIQYFUUTR
|
|
NVHUUNWTXIXAYLUVFUVCUVGUVFUVBYDUVHKHZYDRKHZYDUVFEUDDYEYDUUSQKUVQUVPUVTUWB
|
|
UVOUWAUVNKHUVJUVLUVNKHKHLUVFUVJUVLUVNXBSXCUUGUWEUWFLUVEUUGUVHRYDKUWCXDSUV
|
|
EUWFYDLUUGYDVGVJXEXIYHUVCCUUTYIYFUUTLYGUVBYDYFUUTYEKXFXGXHXJXKXLXMXIXNYAY
|
|
MYPBYCUUAYLYPYJYAUUIYTYLYPWGUUJYEYDUUHYCXOURXPXQXRXSXT $.
|
|
|
|
$( Lemma for ~ 1idpr . (Contributed by Jim Kingdon, 13-Dec-2019.) $)
|
|
1idpru $p |- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) ) $=
|
|
( vf vh vw vy wcel c1p co cfv cv cmq wceq wrex cltq wbr wa cnq c1q adantl
|
|
wb vx vu vv vz cnp cmp c2nd wss ssid rexss ax-mp r19.42v 1pr c1st elprnqu
|
|
cop prop sylan mpan w3a 3ad2ant3 1pru abeq2i ltmnqg mp3an1 ancoms mulidnq
|
|
breq2 breq1d adantr bitrd syl5rbb 3adant3 syl3an1 syl3an2 3expia pm5.32rd
|
|
1nq rexbidva syl5rbbr syl5bb df-imp mulclnq genpelvu mpan2 prnminu wi crq
|
|
ltrelnq brel ancomd simpr simpl recclnq mulcomnqg caovord2d recidnq mpcom
|
|
biimpd sylan2 elab2g mpbird mulassnqg caov12d oveq2d 3eqtrrd oveq2 eqeq2d
|
|
3syl syl rspcev syl2anc ancld reximia ex prcunqu adantrd rexlimdva impbid
|
|
a1i 3bitr4d eqrdv ) AUEFZUAAGUFHUGIZAUGIZYCUAJZBJZCJZKHZLZCGUGIZMZBYEMZYG
|
|
YFNOZYLPZBYEMZYFYDFZYFYEFZYCYLYOBYEYLYHYKFZYJPZCYKMZYCYGYEFZPZYOYKYKUHYLU
|
|
UATYKUIYJCYKYKUJUKYOYNYJPZCYKMUUCUUAYNYJCYKULUUCUUDYTCYKUUCYSPYJYNYSUUCYS
|
|
YJYNYSTZYSUUCYHQFZYJUUEGUEFZYSUUFUMUUGGUNIZYKUPUEFYSUUFGUQYHYKUUHUOURUSUU
|
|
CYGQFZUUFYJUUEYCAUNIZYEUPUEFZUUBUUIAUQZYGYEUUJUOURUUIUUFYJUTYNYGYINOZYSYJ
|
|
UUIYNUUMTUUFYFYIYGNVHVAUUIUUFUUMYSTYJYSRYHNOZUUIUUFPZUUMUUNCYKCVBZVCUUOUU
|
|
NYGRKHZYINOZUUMUUFUUIUUNUURTZRQFUUFUUIUUSVRRYHYGVDVEVFUUIUURUUMTUUFUUIUUQ
|
|
YGYINYGVGVIVJVKVLVMVKVNVOVPVQVSVTWAVSYCUUGYQYMTUMDUBUCEUDAGYFBCUFKEUDUCUB
|
|
DWBUBJUCJWCWDWEYCYRYPYCYRYPYCYRPYNBYEMZYPYCUUKYRUUTUULBYFYEUUJWFURYNYOBYE
|
|
UUBYNYLYNYLWGUUBYNYFYGWHIZKHZYKFZYFYGUVBKHZLZYLYNUVCRUVBNOZYFQFZUUIPZYNUV
|
|
FYNUUIUVGYGYFQQNWIWJWKZUVHYNUVFUVHYNYGUVAKHZUVBNOZUVFUVHEUDDYGYFUVANQKEJZ
|
|
QFZUDJZQFZDJZQFUTZUVLUVNNOUVPUVLKHUVPUVNKHNOTUVHUVLUVNUVPVDSUVGUUIWLZUVGU
|
|
UIWMZUUIUVAQFZUVGYGWNZSZUVMUVOPUVLUVNKHZUVNUVLKHLUVHUVLUVNWOSZWPUUIUVKUVF
|
|
TUVGUUIUVJRUVBNYGWQZVISVKWSWRYNUVHUVBQFZUVCUVFTUVIUUIUVGUVTUWFUWAYFUVAWCW
|
|
TUUNUVFCUVBYKQYHUVBRNVHUUPXAXIXBYNUVHUVEUVIUVHUVDYFUVJKHZYFRKHZYFUVHEUDDY
|
|
GYFUVAQKUVRUVSUWBUWDUVQUWCUVPKHUVLUVNUVPKHKHLUVHUVLUVNUVPXCSXDUUIUWGUWHLU
|
|
VGUUIUVJRYFKUWEXESUVGUWHYFLUUIYFVGVJXFXJYJUVECUVBYKYHUVBLYIUVDYFYHUVBYGKX
|
|
GXHXKXLXTXMXNXJXOYCYOYRBYEUUCYNYRYLYCUUKUUBYNYRWGUULYFYGYEUUJXPURXQXRXSYA
|
|
YB $.
|
|
|
|
$( 1 is an identity element for positive real multiplication. Theorem
|
|
9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) $)
|
|
1idpr $p |- ( A e. P. -> ( A .P. 1P ) = A ) $=
|
|
( cnp wcel c1p cmp co wceq c1st cfv 1idprl 1idpru wa wb 1pr mulclpr mpan2
|
|
c2nd preqlu mpancom mpbir2and ) ABCZADEFZAGZUBHIAHIGZUBQIAQIGZAJAKUBBCZUA
|
|
UCUDUELMUADBCUFNADOPUBARST $.
|
|
$}
|
|
|
|
${
|
|
$d A l x $. $d A u x $. $d B l x $. $d B u x $.
|
|
$( We can order fractions via ` <Q ` or ` <P ` . (Contributed by Jim
|
|
Kingdon, 19-Jun-2021.) $)
|
|
ltnqpr $p |- ( ( A e. Q. /\ B e. Q. ) -> (
|
|
A <Q B <-> <. { l | l <Q A } , { u | A <Q u } >. <P
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) ) $=
|
|
( vx cnq wcel wa cltq wbr cab cop wrex cfv cnp nqprlu ltnqex gtnqex elab2
|
|
cv cltp c2nd c1st wb ltdfpr syl2an breq2 op2nd breq1 op1st anbi12i rexbii
|
|
vex syl6bb ltbtwnnqq syl6rbbr ) BFGZCFGZHZDTZBIJDKZBATZIJZAKZLZUTCIJZDKZC
|
|
VBIJAKZLZUAJZBETZIJZVKCIJZHZEFMZBCIJUSVJVKVEUBNZGZVKVIUCNZGZHZEFMZVOUQVEO
|
|
GVIOGVJWAUDURABDPACDPVEVIEUEUFVTVNEFVQVLVSVMVCVLAVKVPEUMZVBVKBIUGVAVDDBQA
|
|
BRUHSVFVMDVKVRWBUTVKCIUIVGVHDCQACRUJSUKULUNEBCUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A l x $. $d A u x $. $d B l x $. $d B u x $.
|
|
$( We can order fractions via ` <Q ` or ` <P ` . (Contributed by Jim
|
|
Kingdon, 8-Jan-2021.) $)
|
|
ltnqpri $p |- ( A <Q B -> <. { l | l <Q A } , { u | A <Q u } >. <P
|
|
<. { l | l <Q B } , { u | B <Q u } >. ) $=
|
|
( vx cltq wbr cv cab cop cnq wrex cfv wcel cnp nqprlu syl ltnqex gtnqex
|
|
wa cltp c2nd c1st wb ltrelnq simpld simprd ltdfpr syl2anc vex breq2 op2nd
|
|
brel elab2 breq1 op1st anbi12i rexbii syl6bb ltbtwnnqq syl6bbr ibir ) BCF
|
|
GZDHZBFGDIZBAHZFGZAIZJZVDCFGZDIZCVFFGAIZJZUAGZVCVNBEHZFGZVOCFGZTZEKLZVCVC
|
|
VNVOVIUBMZNZVOVMUCMZNZTZEKLZVSVCVIONZVMONZVNWEUDVCBKNZWFVCWHCKNZBCKKFUEUM
|
|
ZUFABDPQVCWIWGVCWHWIWJUGACDPQVIVMEUHUIWDVREKWAVPWCVQVGVPAVOVTEUJZVFVOBFUK
|
|
VEVHDBRABSULUNVJVQDVOWBWKVDVOCFUOVKVLDCRACSUPUNUQURUSEBCUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d q r s t u $.
|
|
$( Positive real 'less than' is a partial ordering. Remark ("< is
|
|
transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p.
|
|
(varies). Lemma for ~ ltsopr . (Contributed by Jim Kingdon,
|
|
15-Dec-2019.) $)
|
|
ltpopr $p |- <P Po P. $=
|
|
( vq vr cnp cltp wtru cv wcel wbr c2nd cfv c1st wa cnq wrex cop wb ltdfpr
|
|
prop wi mpd vs vt vu wpo wn prdisj sylnib nrexdv anidms mtbird adantl w3a
|
|
sylan ancom 3adant3 3adant1 anbi12d reeanv syl6bbr biimpa simprll syl3an1
|
|
cltq prltlu 3adant3r 3adant2l 3ad2antl2 adantlr prcdnql adantrl 3ad2antl3
|
|
3expb jca ex rexlimdvw reximdv 3adant2 biimprd adantr ispod trud ) CDUDEU
|
|
AUBUCCDUAFZCGZWBWBDHZUEEWCWDAFZWBIJZGZWEWBKJZGZLZAMNZWCWJAMWCWEMGZLWIWGLZ
|
|
WJWCWHWFOCGWLWMUEWBRWEWFWHUFUMWIWGUNUGUHWCWDWKPWBWBAQUIUJUKWCUBFZCGZUCFZC
|
|
GZULZWBWNDHZWNWPDHZLZWBWPDHZSEWRXAXBWRXALZWGWEWPKJZGZLZAMNZXBXCWGWEWNKJZG
|
|
ZLZBFZWNIJZGZXKXDGZLZLZBMNZAMNZXGWRXAXRWRXAXJAMNZXOBMNZLXRWRWSXSWTXTWCWOW
|
|
SXSPWQWBWNAQUOWOWQWTXTPWCWNWPBQUPUQXJXOABMMURUSUTXCXQXFAMXCXPXFBMXCXPXFXC
|
|
XPLZWGXEXCWGXIXOVAYAWEXKVCHZXEWRXPYBXAWOWCXPYBWQWOXJXOYBWOXIXOYBWGWOXIXMY
|
|
BXNWOXHXLOCGXIXMYBWNRWEXKXLXHVDVBVEVFVLVGVHWRXPYBXESZXAWQWCXPYCWOWQXOYCXJ
|
|
WQXNYCXMWQXDWPIJZOCGXNYCWPRXKWEYDXDVIUMVJVJVKVHTVMVNVOVPTWRXGXBSXAWRXBXGW
|
|
CWQXBXGPWOWBWPAQVQVRVSTVNUKVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d q r s x y z $.
|
|
$( Positive real 'less than' is a weak linear order (in the sense of
|
|
~ df-iso ). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed
|
|
by Jim Kingdon, 16-Dec-2019.) $)
|
|
ltsopr $p |- <P Or P. $=
|
|
( vx vy vz vq vr vs cnp cltp cv wbr wi wcel cfv wa cnq wrex cltq sylan ex
|
|
mpd wor wpo wo wral ltpopr w3a c2nd wb ltdfpr 3adant3 cop prnminu prnmaxl
|
|
c1st prop anim12i an4s reeanv sylibr 3adantl3 ltsonq ltrelnq sotri adantl
|
|
prloc 3ad2antl3 adantr ad2antrr wex elprnqu ax-ia3 19.8a syl6an 3ad2antl1
|
|
imp df-rex biimprd 3adant2 adantrr elprnql pm3.21 3adant1 adantrl orim12d
|
|
3ad2antl2 ancoms adantlr rexlimdvva rexlimdvw sylbid df-iso mpbir2an
|
|
rgen3 ) GHUAGHUBAIZBIZHJZWNCIZHJZWQWOHJZUCZKZCGUDBGUDAGUDUEXAABCGGGWNGLZW
|
|
OGLZWQGLZUFZWPDIZWNUGMZLZXFWOUNMZLZNZDOPZWTXBXCWPXLUHXDWNWODUIUJXEXKWTDOX
|
|
EXKWTXEXKNZEIZXFQJZXFFIZQJZNZFXIPEXGPZWTXBXCXKXSXDXBXCNXKNXOEXGPZXQFXIPZN
|
|
ZXSXBXHXCXJYBXBXHNXTXCXJNYAXBWNUNMZXGUKGLZXHXTWNUOZEXFXGYCULRXCXIWOUGMZUK
|
|
GLZXJYAWOUOZFXFYFXIUMRUPUQXOXQEFXGXIURUSUTXMXRWTEFXGXIXMXNXGLZXPXILZNZNZX
|
|
RWTYLXRNZXNWQUNMZLZXPWQUGMZLZUCZWTYMXNXPQJZYRXRYSYLXNXFXPQOVAVBVCVDXMYSYR
|
|
KZYKXRXEYTXKXEYSYRXDXBYSYRXCXDYNYPUKGLYSYRWQUOXNXPYPYNVERVFSVGVHTYLYRWTKZ
|
|
XRXEYKUUAXKXEYKNYOWRYQWSXEYIYOWRKYJXEYINZYOWRUUBYONZYIYONZEOPZWRUUCXNOLZU
|
|
UDNZEVIZUUEUUBYOUUHXBXCYIYOUUHKXDXBYINUUFYOUUDUUHXBYDYIUUFYEXNXGYCVJRYIYO
|
|
UUDKXBYIYOVKVDUUGEVLVMVNVOUUDEOVPUSXEUUEWRKZYIYOXBXDUUIXCXBXDNWRUUEWNWQEU
|
|
IVQVRVHTSVSXEYJYQWSKYIXEYJNZYQWSUUJYQNZYQYJNZFOPZWSUUKXPOLZUULNZFVIZUUMUU
|
|
JYQUUPXCXBYJYQUUPKXDXCYJNUUNYQUULUUPXCYGYJUUNYHXPYFXIVTRYJYQUULKXCYJYQWAV
|
|
DUUOFVLVMWEVOUULFOVPUSXEUUMWSKZYJYQXCXDUUQXBXDXCUUQXDXCNWSUUMWQWOFUIVQWFW
|
|
BVHTSWCWDWGVGTSWHTSWIWJWMABCGHWKWL $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h x y p q r $. $d B f g h x y p q r $.
|
|
$( The sum of two positive reals is greater than one of them. Proposition
|
|
9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.)
|
|
(Revised by Mario Carneiro, 12-Jun-2013.) $)
|
|
ltaddpr $p |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) ) $=
|
|
( vp vq vr vf vg vh cnp wcel wa cv c1st cfv cnq wrex c2nd cop prop sylan
|
|
vx vy cpp co cltp wbr prml syl cplq cltq prarloc ad2ant2r elprnqu adantlr
|
|
adantl ad2ant2rl adantr simplrr wi simprl simplr jca df-iplp addclnq syl5
|
|
genpprecll imdistani addclpr prcdnql anassrs imp rspe wb ltdfpr ad3antrrr
|
|
syl12anc syldan mpbird ex rexlimdvva mpd rexlimddv ) AIJZBIJZKZCLZBMNZJZA
|
|
ABUCUDZUEUFZCOWDWHCOPZWCWDWGBQNZRIJWKBSCWLWGUGUHUOWEWFOJZWHKZKZDLZELZWFUI
|
|
UDZUJUFZDAQNZPEAMNZPZWJWCWMXBWDWHWCXAWTRIJZWMXBASZWFWTXAEDUKTULWOWSWJEDXA
|
|
WTWOWQXAJZWPWTJZKZKZWSWJXHWSKZWJXFWPWIMNZJZKZDOPZXIWPOJZXFXKXMXHXNWSWEXFX
|
|
NWNXEWCXFXNWDWCXCXFXNXDWPWTXAUMTUNUPUQWOXEXFWSURXHWSXKWEWNXGWSXKUSZWEWNXG
|
|
KZKWEWRXJJZKXOWEXPXQXPXEWHKWEXQXPXEWHWNXEXFUTWMWHXGVAVBFGHUAUBABWQWFUCUIU
|
|
AUBHGFVCGLHLVDVFVEVGWEWIIJZXQXOABVHZXRXJWIQNZRIJXQXOWISWRWPXTXJVITTUHVJVK
|
|
XLDOVLVPWEWJXMVMZWNXGWSWCWDXRYAXSAWIDVNVQVOVRVSVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w u v f g h q r s t A $. $d x y z w u v f g h q r s t B $.
|
|
$d x y z w u v f g h q r s t C $.
|
|
ltexprlem.1 $e |- C = <. { x e. Q. | E. y (
|
|
y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y (
|
|
y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. $.
|
|
$( Element in lower cut of the constructed difference. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemell $p |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\
|
|
E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) $=
|
|
( cv c2nd cfv wcel cplq co c1st wa wex cnq crab nqex rabex wceq oveq2 cop
|
|
eleq1d anbi2d exbidv fveq2i op1st eqtri elrab2 ) BHZCIJKZUKAHZLMZDNJZKZOZ
|
|
BPZULUKFHZLMZUOKZOZBPAUSQENJZUMUSUAZUQVBBVDUPVAULVDUNUTUOUMUSUKLUBUDUEUFV
|
|
CURAQRZUKCNJKUNDIJKOBPZAQRZUCZNJVEEVHNGUGVEVGURAQSTVFAQSTUHUIUJ $.
|
|
|
|
$( Element in upper cut of the constructed difference. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemelu $p |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\
|
|
E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) $=
|
|
( cv c1st cfv wcel cplq co c2nd wa wex cnq crab nqex rabex wceq oveq2 cop
|
|
eleq1d anbi2d exbidv fveq2i op2nd eqtri elrab2 ) BHZCIJKZUKAHZLMZDNJZKZOZ
|
|
BPZULUKFHZLMZUOKZOZBPAUSQENJZUMUSUAZUQVBBVDUPVAULVDUNUTUOUMUSUKLUBUDUEUFV
|
|
CUKCNJKUNDIJKOBPZAQRZURAQRZUCZNJVGEVHNGUGVFVGVEAQSTURAQSTUHUIUJ $.
|
|
|
|
$( Our constructed difference is inhabited. Lemma for ~ ltexpri .
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemm $p |- ( A <P B -> ( E. q e. Q. q e. ( 1st ` C ) /\
|
|
E. r e. Q. r e. ( 2nd ` C ) ) ) $=
|
|
( vw cv cfv wcel cnq wrex wa wex cnp sylan syl sylibr cltp c1st c2nd cplq
|
|
wbr co ltrelpr brel ltdfpr biimpd mpcom simprrl simprd wceq cltq cop prop
|
|
prnmaxl ltexnqi reximi sylib r19.42v exbii eleq1 biimparc exlimiv adantrl
|
|
df-rex jca expr reximdva mpd rexbii rexcom simpld elprnqu pm4.71rd anbi1d
|
|
ex anass syl6bb exbidv rexbidv syl6bbr mpbird ltexprlemell wss ssid rexss
|
|
wb ax-mp bitr4i nfv nfre1 prmu rexex 3syl prml adantr w3a 3adant3 elprnql
|
|
simp3 3adant2 addcomnqg syl2anc ltaddnq eqeltrrd 19.8a ltexprlemelu 3expa
|
|
wi prcunqu exlimddv exlimdd ) CDUAUEZGJZEUBKLZGMNZFJZEUCKLZFMNZXPBJZCUCKZ
|
|
LZYCXQUDUFZDUBKZLZOZBPZGMNZXSXPYKYIBMNZGMNZXPYIGMNZBMNZYMXPYEYHGMNZOZBMNZ
|
|
YOXPYEYCYGLZOZBMNZYRCQLZDQLZOZXPUUACDQQUAUGUHZUUDXPUUACDBUIUJUKXPYTYQBMXP
|
|
YCMLZYTYQXPUUFYTOOYEYPXPUUFYEYSULXPYTYPUUFXPYSYPYEXPUUCYSYPXPUUBUUCUUEUMZ
|
|
UUCYSOZIJZYGLZYFUUIUNZOZGMNZIPZYPUUHUUJUUKGMNZOZIPZUUNUUHUUOIYGNZUUQUUHYC
|
|
UUIUOUEZIYGNZUURUUCYGDUCKZUPQLZYSUUTDUQZIYCUVAYGURRUUSUUOIYGGYCUUIUSUTSUU
|
|
OIYGVHVAUUMUUPIUUJUUKGMVBVCTUUMYPIUULYHGMUUKYHUUJYFUUIYGVDVEUTVFSRVGVGVIV
|
|
JVKVLYNYQBMYEYHGMVBVMTYIBGMMVNVAXPYKUUFYIOZBPZGMNYMXPYJUVEGMXPYIUVDBXPYIU
|
|
UFYEOZYHOUVDXPYEUVFYHXPYEUUFXPYEUUFXPUUBYEUUFXPUUBUUCUUEVOZUUBCUBKZYDUPQL
|
|
ZYEUUFCUQZYCYDUVHVPRRVSVQVRUUFYEYHVTWAWBWCYLUVEGMYIBMVHVMWDWEXSXQMLYJOZGM
|
|
NZYKXRUVKGMABCDEGHWFVMMMWGYKUVLWJMWHYJGMMWIWKWLTXPXTUVALZYBFXPFWMYAFMWNXP
|
|
UUCUVBUVMFPZUUGUVCUVBUVMFMNUVNFUVAYGWOUVMFMWPSWQXPUVMOZXTMLZYAOZFPZYBUVOU
|
|
VPYAUVRXPUUCUVMUVPUUGUUCUVBUVMUVPUVCXTUVAYGVPRRZUVOYCUVHLZYABXPUVTBPZUVMX
|
|
PUUBUVTBMNZUWAUVGUUBUVIUWBUVJBYDUVHWRSUVTBMWPWQWSXPUVMUVTYAXPUVMUVTWTZUVP
|
|
UVTYCXTUDUFZUVALZOZBPZOYAUWCUVPUWGXPUVMUVPUVTUVSXAZUWCUVTUWEUWGXPUVMUVTXC
|
|
UWCXTYCUDUFZUWDUVAUWCUVPUUFUWIUWDUNUWHXPUVTUUFUVMXPUUBUVTUUFUVGUUBUVIUVTU
|
|
UFUVJYCYDUVHXBRRXDZXTYCXEXFUWCXTUWIUOUEZUWIUVALZUWCUVPUUFUWKUWHUWJXTYCXGX
|
|
FXPUVMUWKUWLXLZUVTXPUUCUVMUWMUUGUUCUVBUVMUWMUVCUWIXTUVAYGXMRRXAVLXHUWFBXI
|
|
XFVIABCDEFHXJTXKXNUVQFXIXFYAFMVHTXOVI $.
|
|
|
|
$( The lower cut of our constructed difference is open. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemopl $p |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) )
|
|
-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) $=
|
|
( vs wbr cv cnq wcel cfv cltq wa wrex wex cnp syl2anc cltp c1st c2nd cplq
|
|
ltexprlemell simprbi 19.42v anbi2i bitri ltrelpr brel simprd prop prnmaxl
|
|
co cop sylan adantrl wceq simpld ad2antrr simplrr elprnqu simplrl ltaddnq
|
|
simprr ltsonq ltrelnq sotri wb simprl elprnql ltexnqq mpbid adantr ltanqg
|
|
breqtrrd syl3anc mpbird eqeltrd jca jca32 reximdva rexlimddv eximi sylbir
|
|
expr mpd rexcom4 sylibr rexbii sylib sylanr2 3impb ) CDUAJZGKZLMZWPEUBNZM
|
|
ZWPFKZOJZWTWRMZPZFLQZWSWOWQBKZCUCNZMZXEWPUDUOZDUBNZMZPZBRZXDWSWQXLABCDEGH
|
|
UEUFWOWQXLPZPZXAWTLMZXGXEWTUDUOZXIMZPZBRPZPZFLQZXDXNXAXOXRPZPZBRZFLQZYAXN
|
|
YCFLQZBRZYEXNWOWQXKPZPZBRZYGYJWOYHBRZPXNWOYHBUGYKXMWOWQXKBUGUHUIYIYFBYIXH
|
|
IKZOJZYFIXIWOXKYMIXIQZWQWOXJYNXGWODSMZXJYNWOCSMZYOCDSSUAUJUKZULZYOXIDUCNZ
|
|
UPSMZXJYNDUMZIXHYSXIUNUQUQURURYIYLXIMZYMPZPZXPYLUSZFLQZYFUUDXEYLOJZUUFUUD
|
|
XEXHOJZYMUUGUUDXELMZWQUUHUUDYPXGUUIWOYPYHUUCWOYPYOYQUTVAUUDXGXJWOWQXKUUCV
|
|
BUTZYPCUBNZXFUPSMXGUUICUMXEXFUUKVCUQTZWOWQXKUUCVDZXEWPVETYIUUBYMVFXEXHYLO
|
|
LVGVHVITUUDUUIYLLMZUUGUUFVJUULUUDYOUUBUUNWOYOYHUUCYRVAYIUUBYMVKYOYTUUBUUN
|
|
UUAYLYSXIVLUQTFXEYLVMTVNUUDUUEYCFLUUDXOUUEYCUUDXOUUEPZPZXAXOXRUUPXAXHXPOJ
|
|
ZUUPXHYLXPOYIUUBYMUUOVBUUDXOUUEVFZVQUUPWQXOUUIXAUUQVJUUDWQUUOUUMVOUUDXOUU
|
|
EVKZUUDUUIUUOUULVOWPWTXEVPVRVSUUSUUPXGXQUUDXGUUOUUJVOUUPXPYLXIUURYIUUBYMU
|
|
UOVDVTWAWBWGWCWHWDWEWFYCFBLWIWJYDXTFLYDXAYBBRZPXTXAYBBUGUUTXSXAXOXRBUGUHU
|
|
IWKWLXCXTFLXBXSXAABCDEFHUEUHWKWJWMWN $.
|
|
|
|
$( The lower cut of our constructed difference is lower. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemlol $p |- ( ( A <P B /\ q e. Q. ) ->
|
|
( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) ) $=
|
|
( wbr cv cnq wcel wa c1st cfv wex cnp sylan 19.42v bitr4i cltp co simprrr
|
|
cltq c2nd cplq simplr simpld simprl ltrelpr brel cop prop elprnqu syl2anc
|
|
simpll ltanqi wi simprd syl prcdnql mpd jca32 ltexprlemell anbi2i 3imtr4i
|
|
eximi ex rexlimdvw ) CDUAIZGJZKLZMZVKFJZUDIZVNENOZLZMZVKVPLZFKVMVRVSVMVOV
|
|
NKLZBJZCUEOZLZWAVNUFUBZDNOZLZMZMZMZMZBPZVLWCWAVKUFUBZWELZMZMZBPZVMVRMZVSW
|
|
JWOBWJVLWCWMVJVLWIUGWJWCWFVMVOVTWGUCZUHZWJWLWDUDIZWMWJVOWAKLZWTVMVOWHUIWJ
|
|
VJWCXAVJVLWIUPZWSVJCQLZWCXAVJXCDQLZCDQQUAUJUKZUHXCCNOZWBULQLWCXACUMWAWBXF
|
|
UNRRUOVKVNWAUQUOWJXDWFWTWMURZWJVJXDXBVJXCXDXEUSUTWJWCWFWRUSXDWEDUEOZULQLW
|
|
FXGDUMWDWLXHWEVARUOVBVCVGWQVMWIBPZMWKVRXIVMVRVOWHBPZMXIVQXJVOVQVTWGBPMXJA
|
|
BCDEFHVDVTWGBSTVEVOWHBSTVEVMWIBSTVSVLWNBPMWPABCDEGHVDVLWNBSTVFVHVI $.
|
|
|
|
$( The upper cut of our constructed difference is open. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemopu $p |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) )
|
|
-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) $=
|
|
( vs wbr cv cnq wcel cfv cltq wa wrex wex cnp adantr vt cltp c2nd c1st co
|
|
cplq ltexprlemelu simprbi 19.42v anbi2i bitri cop ltrelpr brel simprd syl
|
|
prnminu sylan adantrl ltdfpr biimpd mpcom ad2antrr simpld simplrr simprrl
|
|
prop prltlu syl3an1 syl3anc simplll simprrr simplrl ltrelnq sotri syl2anc
|
|
ltsonq rexlimddv wb elprnql elprnqu ad2ant2r ltexnqq mpbid simprr eqbrtrd
|
|
wceq simprl ltanqg mpbird eqeltrd jca jca32 reximdva eximi sylbir rexcom4
|
|
expr mpd sylibr rexbii sylib sylanr2 3impb ) CDUBJZFKZLMZXFEUCNZMZGKZXFOJ
|
|
ZXJXHMZPZGLQZXIXEXGBKZCUDNZMZXOXFUFUEZDUCNZMZPZBRZXNXIXGYBABCDEFHUGUHXEXG
|
|
YBPZPZXKXJLMZXQXOXJUFUEZXSMZPZBRPZPZGLQZXNYDXKYEYHPZPZBRZGLQZYKYDYMGLQZBR
|
|
ZYOYDXEXGYAPZPZBRZYQYTXEYRBRZPYDXEYRBUIUUAYCXEXGYABUIUJUKYSYPBYSIKZXROJZY
|
|
PIXSXEYAUUCIXSQZXGXEXTUUDXQXEDUDNZXSULSMZXTUUDXEDSMZUUFXECSMZUUGCDSSUBUMU
|
|
NZUODVGUPZIXRXSUUEUQURUSUSYSUUBXSMZUUCPZPZYFUUBWGZGLQZYPUUMXOUUBOJZUUOUUM
|
|
UAKZCUCNZMZUUQUUEMZPZUUPUALXEUVAUALQZYRUULUUHUUGPZXEUVBUUIUVCXEUVBCDUAUTV
|
|
AVBVCUUMUUQLMZUVAPZPZXOUUQOJZUUQUUBOJZUUPUVFUUHXQUUSUVGUUMUUHUVEXEUUHYRUU
|
|
LXEUUHUUGUUIVDVCZTUUMXQUVEUUMXQXTXEXGYAUULVEVDZTUUMUVDUUSUUTVFUUHXPUURULS
|
|
MZXQUUSUVGCVGZXOUUQUURXPVHVIVJUVFXEUUTUUKUVHXEYRUULUVEVKUUMUVDUUSUUTVLYSU
|
|
UKUUCUVEVMXEUUFUUTUUKUVHUUJUUQUUBXSUUEVHVIVJXOUUQUUBOLVQVNVOVPVRUUMXOLMZU
|
|
UBLMZUUPUUOVSUUMUUHXQUVMUVIUVJUUHUVKXQUVMUVLXOUURXPVTURVPZXEUUKUVNYRUUCXE
|
|
UUFUUKUVNUUJUUBXSUUEWAURWBGXOUUBWCVPWDUUMUUNYMGLUUMYEUUNYMUUMYEUUNPZPZXKY
|
|
EYHUVQXKYFXROJZUVQYFUUBXROUUMYEUUNWEZYSUUKUUCUVPVEWFUVQYEXGUVMXKUVRVSUUMY
|
|
EUUNWHZUUMXGUVPXEXGYAUULVMTUUMUVMUVPUVOTXJXFXOWIVJWJUVTUVQXQYGUUMXQUVPUVJ
|
|
TUVQYFUUBXSUVSYSUUKUUCUVPVMWKWLWMWRWNWSVRWOWPYMGBLWQWTYNYJGLYNXKYLBRZPYJX
|
|
KYLBUIUWAYIXKYEYHBUIUJUKXAXBXMYJGLXLYIXKABCDEGHUGUJXAWTXCXD $.
|
|
|
|
$( The upper cut of our constructed difference is upper. Lemma for
|
|
~ ltexpri . (Contributed by Jim Kingdon, 21-Dec-2019.) $)
|
|
ltexprlemupu $p |- ( ( A <P B /\ r e. Q. ) -> (
|
|
E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ->
|
|
r e. ( 2nd ` C ) ) ) $=
|
|
( wbr cv cnq wcel wa c2nd cfv wex cnp syl2anc 19.42v bitr4i simplr simpld
|
|
cltp cltq c1st cplq simprrr simprl simpll simprrl adantl cop ltrelpr brel
|
|
co prop syl elprnql sylan ltanqi wi simprd prcunqu mpd jca32 ltexprlemelu
|
|
eximi anbi2i 3imtr4i ex rexlimdvw ) CDUCIZFJZKLZMZGJZVMUDIZVPENOZLZMZVMVR
|
|
LZGKVOVTWAVOVQVPKLZBJZCUEOZLZWCVPUFUOZDNOZLZMZMZMZMZBPZVNWEWCVMUFUOZWGLZM
|
|
ZMZBPZVOVTMZWAWLWQBWLVNWEWOVLVNWKUAWLWEWHVOVQWBWIUGZUBWLWFWNUDIZWOWLVQWCK
|
|
LZXAVOVQWJUHWLVLWEXBVLVNWKUIZWKWEVOVQWBWEWHUJUKVLWDCNOZULQLZWEXBVLCQLZXEV
|
|
LXFDQLZCDQQUCUMUNZUBCUPUQWCXDWDURUSRVPVMWCUTRWLXGWHXAWOVAZWLVLXGXCVLXFXGX
|
|
HVBUQWLWEWHWTVBXGDUEOZWGULQLWHXIDUPWNWFWGXJVCUSRVDVEVGWSVOWKBPZMWMVTXKVOV
|
|
TVQWJBPZMXKVSXLVQVSWBWIBPMXLABCDEGHVFWBWIBSTVHVQWJBSTVHVOWKBSTWAVNWPBPMWR
|
|
ABCDEFHVFVNWPBSTVIVJVK $.
|
|
|
|
$( Our constructed difference is rounded. Lemma for ~ ltexpri .
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemrnd $p |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C )
|
|
<-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
|
|
/\ A. r e. Q. ( r e. ( 2nd ` C )
|
|
<-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) ) $=
|
|
( wbr cv cfv wcel wa cnq wrex wb wral 3expia impbid ralrimiva cltp c1st
|
|
cltq c2nd ltexprlemopl ltexprlemlol ltexprlemopu ltexprlemupu jca ) CDUAI
|
|
ZGJZEUBKZLZUKFJZUCIZUNULLMFNOZPZGNQUNEUDKZLZUOUKURLMGNOZPZFNQUJUQGNUJUKNL
|
|
ZMUMUPUJVBUMUPABCDEFGHUERABCDEFGHUFSTUJVAFNUJUNNLZMUSUTUJVCUSUTABCDEFGHUG
|
|
RABCDEFGHUHSTUI $.
|
|
|
|
$( Our constructed difference is disjoint. Lemma for ~ ltexpri .
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemdisj $p |- ( A <P B -> A. q e. Q.
|
|
-. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) ) $=
|
|
( vz wbr cv cfv wcel wa wn cnq cplq co cltq cnp wex vf cltp c1st c2nd wal
|
|
vg ltsonq ltrelnq son2lpi cop ltrelpr brel simprd prop syl prltlu syl3an1
|
|
vh 3expb adantlr adantrll adantrrl w3a ltanqg adantl simpld elprnqu sylan
|
|
ad2ant2r adantrr elprnql adantrl simplr addcomnqg caovord2d mpbird 3com23
|
|
wb wceq adantrlr adantrrr mtoi alrimivv ltexprlemell ltexprlemelu anbi12i
|
|
jca ex anandi bitr4i baib eleq1 oveq1 eleq1d anbi12d cbvexv anbi2i syl6bb
|
|
eeanv syl6bbr notbid alnex albii bitri ralrimiva ) CDUBIZFJZEUCKLZXGEUDKL
|
|
ZMZNZFOXFXGOLZMZXKBJZCUDKZLZXNXGPQZDUCKZLZMZHJZCUCKZLZYAXGPQZDUDKZLZMZMZN
|
|
ZHUEZBUEZXMYIBHXMYHXNYARIZYAXNRIZMZXNYAROUGUHUIXMYHYNXMYHMZYLYMYOYLXQYDRI
|
|
ZXMXTYFYPYCXMXSYFYPXPXFXSYFMYPXLXFXSYFYPXFXRYEUJSLZXSYFYPXFDSLZYQXFCSLZYR
|
|
CDSSUBUKULZUMDUNUOXQYDYEXRUPUQUSUTVAVBYOUAUFURXNYAXGROPUAJZOLZUFJZOLZURJZ
|
|
OLVCUUAUUCRIUUEUUAPQUUEUUCPQRIVRYOUUAUUCUUEVDVEXMXTXNOLZYGXFXPUUFXLXSXFYB
|
|
XOUJSLZXPUUFXFYSUUGXFYSYRYTVFCUNUOZXNXOYBVGVHVIVJXMYGYAOLZXTXFYCUUIXLYFXF
|
|
UUGYCUUIUUHYAXOYBVKVHVIVLXFXLYHVMUUBUUDMUUAUUCPQUUCUUAPQVSYOUUAUUCVNVEVOV
|
|
PXMXTYCYMYFXMXPYCYMXSXFXPYCMYMXLXFXPYCYMXFYCXPYMXFUUGYCXPYMUUHYAXNXOYBUPU
|
|
QVQUSUTVTWAWGWHWBWCXLXKYKVRXFXLXKYHHTZBTZNZYKXLXJUUKXLXJXTBTZYGHTZMZUUKXL
|
|
XJUUMXNYBLZXQYELZMZBTZMZUUOXJXLUUTXJXLUUMMZXLUUSMZMXLUUTMXHUVAXIUVBABCDEF
|
|
GWDABCDEFGWEWFXLUUMUUSWIWJWKUUSUUNUUMUURYGBHXNYAVSZUUPYCUUQYFXNYAYBWLUVCX
|
|
QYDYEXNYAXGPWMWNWOWPWQWRXTYGBHWSWTXAYKUUJNZBUEUULYJUVDBYHHXBXCUUJBXBXDWTV
|
|
EVPXE $.
|
|
|
|
$( Our constructed difference is located. Lemma for ~ ltexpri .
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemloc $p |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r
|
|
-> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) ) $=
|
|
( vz wbr cv cltq wcel wo cnq wa cplq co wrex wb vw vf vg vh cltp c1st cfv
|
|
c2nd wi wral wceq ltexnqi adantl cnp ltrelpr brel simpld cop prop prarloc
|
|
sylan ad2ant2r simprd ad2antrr w3a ltanqg elprnqu adantlr elprnql simplrl
|
|
ad2ant2rl addclnq syl2anc ltrelnq addcomnqg syl3anc oveq2d simplrr 3eqtrd
|
|
caovord2d addassnqg breq2d bitrd biimpa prloc anassrs reximdva prml rexex
|
|
ex 3syl r19.45mv adantr prmu r19.9rmv orbi2d r19.43 syl6rbbr rexbidv ibar
|
|
wex orbi12d syl ltexprlemell ltexprlemelu eleq1 oveq1 eleq1d cbvexv bitri
|
|
anbi12d anbi2i orbi12i syl6bbr 3bitr4rd sylibrd mpd rexlimddv ralrimivw
|
|
df-rex ) CDUEJZGKZFKZLJZYBEUFUGMZYCEUHUGMZNZUIZFOUJGOYAYHFOYAYDYGYAYDPZYB
|
|
UAKZQRZYCUKZYGUAOYDYLUAOSYAUAYBYCULUMYIYJOMZYLPZPZBKZIKZYJQRZLJZBCUHUGZSZ
|
|
ICUFUGZSZYGYAYMUUCYDYLYACUNMZYMUUCYAUUDDUNMZCDUNUNUEUOUPZUQZUUDUUBYTURUNM
|
|
ZYMUUCCUSZYJYTUUBIBUTVAVAVBYOUUCYPYBQRZDUFUGZMZYQYCQRZDUHUGZMZNZBYTSZIUUB
|
|
SZYGYOUUAUUQIUUBYOYQUUBMZPYSUUPBYTYOUUSYPYTMZYSUUPUIYOUUSUUTPZPZYSUUPUVBY
|
|
SPUUEUUJUUMLJZUUPYOUUEUVAYSYAUUEYDYNYAUUDUUEUUFVCVDVDUVBYSUVCUVBYSUUJYRYB
|
|
QRZLJUVCUVBUBUCUDYPYRYBLOQUBKZOMZUCKZOMZUDKZOMVEUVEUVGLJUVIUVEQRUVIUVGQRL
|
|
JTUVBUVEUVGUVIVFUMYIUUTYPOMZYNUUSYAUUTUVJYDYAUUDUUTUVJUUGUUDUUHUUTUVJUUIY
|
|
PYTUUBVGVAVAVHVKUVBYQOMZYMYROMYIUUSUVKYNUUTYAUUSUVKYDYAUUDUUSUVKUUGUUDUUH
|
|
UUSUVKUUIYQYTUUBVIVAVAVHVBZYIYMYLUVAVJZYQYJVLVMYIYBOMZYNUVAYDUVNYAYDUVNYC
|
|
OMZYBYCOOLVNUPZUQUMVDZUVFUVHPUVEUVGQRUVGUVEQRUKUVBUVEUVGVOUMVTUVBUVDUUMUU
|
|
JLUVBUVDYQYJYBQRZQRZYQYKQRUUMUVBUVKYMUVNUVDUVSUKUVLUVMUVQYQYJYBWAVPUVBUVR
|
|
YKYQQUVBYMUVNUVRYKUKUVMUVQYJYBVOVMVQUVBYKYCYQQYIYMYLUVAVRVQVSWBWCWDUUEUUK
|
|
UUNURUNMUVCUUPDUSUUJUUMUUNUUKWEVAVMWJWFWGWGYIYGUURTYNYIUULBYTSZUUONZIUUBS
|
|
ZUVTUUOIUUBSZNZUURYGYAUWBUWDTZYDYAUUDUUSIXAZUWEUUGUUDUUHUUSIOSUWFUUIIYTUU
|
|
BWHUUSIOWIWKUVTUUOIUUBWLWKWMYAUURUWBTYDYAUUQUWAIUUBYAUUDAKYTMZAXAZUUQUWAT
|
|
UUGUUDUUHUWGAOSUWHUUIAYTUUBWNUWGAOWIWKUWHUWAUVTUUOBYTSZNUUQUWHUUOUWIUVTUU
|
|
OBAYTWOWPUULUUOBYTWQWRWKWSWMYDYGUWDTYAYDYGUUTUULPBXAZUUSUUOPZIXAZNZUWDYDU
|
|
WMUVNUWJPZUVOUWLPZNZYGYDUVNUVOPZUWMUWPTUVPUWQUWJUWNUWLUWOUVNUWJUWNTUVOUVN
|
|
UWJWTWMUVOUWLUWOTUVNUVOUWLWTUMXBXCYEUWNYFUWOABCDEGHXDYFUVOYPUUBMZYPYCQRZU
|
|
UNMZPZBXAZPUWOABCDEFHXEUXBUWLUVOUXAUWKBIYPYQUKZUWRUUSUWTUUOYPYQUUBXFUXCUW
|
|
SUUMUUNYPYQYCQXGXHXKXIXLXJXMWRUVTUWJUWCUWLUULBYTXTUUOIUUBXTXMXNUMXOWMXPXQ
|
|
XRWJXSXS $.
|
|
|
|
$( Our constructed difference is a positive real. Lemma for ~ ltexpri .
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlempr $p |- ( A <P B -> C e. P. ) $=
|
|
( vq vr wbr cnq wcel cv c1st cfv wrex c2nd wa wb wral wex cltp cpw cxp wn
|
|
cltq wo wi w3a cnp ltexprlemm cplq crab cop wss ssrab2 nqex elpw2 opelxpi
|
|
co mpbir mp2an eqeltri jctil ltexprlemrnd ltexprlemdisj ltexprlemloc 3jca
|
|
elnp1st2nd sylanbrc ) CDUAIZEJUBZVKUCZKZGLZEMNZKZGJOHLZEPNZKZHJOQZQVPVNVQ
|
|
UEIZVQVOKQHJORGJSVSWAVNVRKZQGJORHJSQZVPWBQUDGJSZWAVPVSUFUGHJSGJSZUHEUIKVJ
|
|
VTVMABCDEHGFUJEBLZCPNKWFALUKUSZDMNKQBTZAJULZWFCMNKWGDPNKQBTZAJULZUMZVLFWI
|
|
VKKZWKVKKZWLVLKWMWIJUNWHAJUOWIJUPUQUTWNWKJUNWJAJUOWKJUPUQUTWIWKVKVKURVAVB
|
|
VCVJWCWDWEABCDEHGFVDABCDEGFVEABCDEHGFVFVGEHGVHVI $.
|
|
|
|
$( Lemma for ~ ltexpri . One directon of our result for lower cuts.
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemfl $p |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) ) $=
|
|
( vz vw vu vf wbr co cfv cv wcel cplq cnp wa cnq cltq vg vh cltp cpp c1st
|
|
wceq wrex ltrelpr brel simpld ltexprlempr df-iplp addclnq genpelvl simprr
|
|
wb syl2anc c2nd wex ltexprlemell biimpi ad2antlr adantl simprd w3a prltlu
|
|
cop prop syl3an1 3adant2r 3adant3r ltanqg ltrelnq elprnql adantrl adantrr
|
|
syl sylan 3adant3 addcomnqg caovord2d wi prcdnql 3adant2 sylbid mpd 3expa
|
|
exlimddv eqeltrd expr rexlimdvva ssrdv ) CDUCKZGCEUDLUEMZDUEMZWMGNZWNOZWP
|
|
HNZINZPLZUFZIEUEMZUGHCUEMZUGZWPWOOZWMCQOZEQOZWQXDUPWMXFDQOZCDQQUCUHUIZUJZ
|
|
ABCDEFUKZJUAUBGBCEWPHIUDPGBUBUAJULUANZUBNZUMUNUQWMXAXEHIXCXBWMWRXCOZWSXBO
|
|
ZRZXAXEWMXPXARZRZWPWTWOWMXPXAUOXRBNZCURMZOZXSWSPLZWOOZRZWTWOOZBXRWSSOZYDB
|
|
USZXQYFYGRZWMXOYHXNXAXOYHABCDEIFUTVAVBVCVDWMXQYDYEWMXQYDVEZWRXSTKZYEWMXQY
|
|
AYJYCWMXPYAYJXAWMXNYAYJXOWMXCXTVGQOZXNYAYJWMXFYKXJCVHVQWRXSXTXCVFVIVJVJVK
|
|
ZYIYJWTYBTKZYEYIJUAUBWRXSWSTSPJNZSOZXLSOZXMSOVEYNXLTKXMYNPLXMXLPLTKUPYIYN
|
|
XLXMVLVCYIWRSOZXSSOZYIYJYQYRRYLWRXSSSTVMUIVQZUJYIYQYRYSVDWMXQYFYDWMXPYFXA
|
|
WMXOYFXNWMXBEURMZVGQOZXOYFWMXGUUAXKEVHVQWSYTXBVNVRVOVPVSYOYPRYNXLPLXLYNPL
|
|
UFYIYNXLVTVCWAWMYDYMYEWBZXQWMYCUUBYAWMWODURMZVGQOZYCUUBWMXHUUDWMXFXHXIVDD
|
|
VHVQYBWTUUCWOWCVRVOWDWEWFWGWHWIWJWKWEWL $.
|
|
|
|
$( Lemma for ~ ltexpri . Reverse directon of our result for lower cuts.
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemrl $p |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) ) $=
|
|
( vv vz cfv co cv wcel wa cplq cnq cnp syl sylan syl2anc ad2antrr vw cltp
|
|
vu vs vf vg vh wbr c1st cpp c2nd cop wrex ltrelpr brel prop prnmaddl cltq
|
|
simprd simpld prarloc ad2ant2r wceq w3o simplll simplrl elprnql ad3antrrr
|
|
adantr nqtri3or wb ltexnqq biimpa simprr wex simprl simpr simplrr prcunqu
|
|
wi mpd addcomnqg adantl w3a addassnqg caov32d oveq1 eleq1d mpbird eqeltrd
|
|
eleq1 anbi12d spcegv anabsi5 ltexprlemell sylanbrc ltexprlempr genpprecll
|
|
df-iplp addclnq mp2and eqeltrrd rexlimddv ex ltaddpr ltprordil sseld sylc
|
|
prcdnql sylsyld 3jaod rexlimdvva ssrdv ) CDUBUHZUADUIIZCEUJJZUIIZXNUAKZXO
|
|
LZXRXQLZXNXSMZXRGKZNJZXOLZXTGOXNXODUKIZULPLZXSYDGOUMXNDPLZYFXNCPLZYGCDPPU
|
|
BUNUOZUSDUPQZGXRYEXOUQRYAYBOLZYDMZMZUCKZHKZYBNJZURUHZUCCUKIZUMHCUIIZUMZXT
|
|
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|
|
|
|
$( Lemma for ~ ltexpri . One direction of our result for upper cuts.
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
|
ltexprlemfu $p |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) ) $=
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( vz vw vu vf wbr co cfv cv wcel cplq cnp wa cnq cltq vg vh cltp cpp c2nd
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wceq wrex ltrelpr brel simpld ltexprlempr df-iplp addclnq genpelvu simprr
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wb syl2anc c1st wex ltexprlemelu biimpi ad2antlr simprd adantl w3a prltlu
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cop prop syl syl3an1 3com23 3adant2r ltanqg elprnql sylan adantrr 3adant2
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wi mpd exlimddv eqeltrd expr rexlimdvva ssrdv ) CDUCKZGCEUDLUEMZDUEMZWNGN
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SSOZYEWNXQYSXBWNXOYSXPWNYLXOYSYMWSXDYAVSVOVPVPVTWNXRYHYEWNXQYHXBWNXPYHXOW
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|
|
|
|
$( Lemma for ~ ltexpri . One direction of our result for upper cuts.
|
|
(Contributed by Jim Kingdon, 17-Dec-2019.) $)
|
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ltexprlemru $p |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) ) $=
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( vv vz wbr cfv co cv wcel wa cltq cnp syl cplq cnq ad2antrr vw cltp c2nd
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cop wceq simprr wb elprnqu ad2ant2r adantr ltexnqq syl2anc simpld prarloc
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mpbid adantlr simplll ltdfpr biimpd mpcom simplrl simprrl syl3an1 syl3anc
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prltlu simprrr ltsonq ltrelnq sotri rexlimddv ltexnqi oveq1 eqeq1d mpbird
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simplrr elprnql simprl addcomnqg adantl w3a addassnqg caov32d prcunqu mpd
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|
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$}
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${
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$( Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM,
|
|
13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) $)
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|
ltexpri $p |- ( A <P B -> E. x e. P. ( A +P. x ) = B ) $=
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( vz vy vu vv cv c2nd cfv wcel cplq co wa wex cnq crab cnp wceq eleq1d wb
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cltp wbr c1st cop cpp wrex simpr oveq12d anbi12d cbvexdva cbvrabv opeq12i
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$}
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${
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$d A f g h q r s t u v w $. $d B f g h q r s t u v w $.
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$d C f g h q r s t u v w $.
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$( Lemma for ~ addcanprg . (Contributed by Jim Kingdon, 25-Dec-2019.) $)
|
|
addcanprleml $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
|
|
( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) ) $=
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( vf vg vh cnp wcel co wceq cfv cplq cnq c2nd syl adantr syl2anc ad2antrr
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wa cv vv vw vt vu vr vs w3a cpp c1st wrex cop prop prnmaddl sylan adantlr
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LOUUFUVEYLUWJUVFYHYLYKUUEVGXOYIWSQXOUUGUUHXNWTQXAXBXBXBXCXD $.
|
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|
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$( Lemma for ~ addcanprg . (Contributed by Jim Kingdon, 25-Dec-2019.) $)
|
|
addcanprlemu $p |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\
|
|
( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) ) $=
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( vr vw cnp wcel co wceq wa cfv cv cltq wbr c1st adantlr cplq cnq syl2anc
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syl vv vt vu vf vg vh vq vs w3a cpp c2nd wrex cop prnminu sylan 3ad2antl2
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|
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$}
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$( Addition cancellation law for positive reals. Proposition 9-3.5(vi) of
|
|
[Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.) $)
|
|
addcanprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( ( A +P. B ) = ( A +P. C ) -> B = C ) ) $=
|
|
( cnp wcel w3a cpp co wceq wa c1st cfv addcanprleml wss 3ancomb eqcom sylbi
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|
c2nd eqssd addcanprlemu anbi12i jca wb preqlu 3adant1 adantr mpbird ex ) AD
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TQSUBULUPVDUCZUOUJUKVIUIBCUDUEUFUGUH $.
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${
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$d A x y $. $d B x y $.
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|
$( The difference from ~ ltexpri is unique. (Contributed by Jim Kingdon,
|
|
7-Jul-2021.) $)
|
|
lteupri $p |- ( A <P B -> E! x e. P. ( A +P. x ) = B ) $=
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|
( vy cltp wbr cv cpp co wceq cnp wrex wrmo wreu ltexpri wcel ltrelpr wral
|
|
wa ralrimiva brel simpld w3a eqtr3 addcanprg syl5 3expa oveq2 eqeq1d rmo4
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|
wi sylibr syl reu5 sylanbrc ) BCEFZBAGZHIZCJZAKLUSAKMZUSAKNABCOUPBKPZUTUP
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SAKUNUO $.
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$}
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${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM,
|
|
8-Apr-1996.) $)
|
|
ltaprlem $p |- ( C e. P. -> ( A <P B ->
|
|
( C +P. A ) <P ( C +P. B ) ) ) $=
|
|
( vx cltp wbr cnp wcel cpp co wa cv wceq wrex ltexpri adantr ltrelpr brel
|
|
simplr syl2anc simpld addclpr simprl ltaddpr syl3anc oveq2 ad2antll eqtrd
|
|
addassprg breqtrd rexlimddv expcom ) ABEFZCGHZCAIJZCBIJZEFZUMUNKZADLZIJZB
|
|
MZUQDGUMVADGNUNDABOPURUSGHZVAKZKZUOUOUSIJZUPEVDUOGHZVBUOVEEFVDUNAGHZVFUMU
|
|
NVCSZURVGVCUMVGUNUMVGBGHABGGEQRUAPPZCAUBTURVBVAUCZUOUSUDTVDVECUTIJZUPVDUN
|
|
VGVBVEVKMVHVIVJCAUSUIUEVAVKUPMURVBUTBCIUFUGUHUJUKUL $.
|
|
|
|
$( Ordering property of addition. Proposition 9-3.5(v) of [Gleason]
|
|
p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.) $)
|
|
ltaprg $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) ) $=
|
|
( vx cnp wcel w3a cltp wbr cpp co wi ltaprlem 3ad2ant3 wa cv wceq ltexpri
|
|
wrex syl2anc adantl simpl1 simprl ltaddpr addassprg 3com12 adantrr simprr
|
|
3expa eqtr3d 3adantl2 simpl3 addclpr simpl2 addcanprg syl3anc mpd breqtrd
|
|
adantlr rexlimddv ex impbid ) AEFZBEFZCEFZGZABHIZCAJKZCBJKZHIZVEVCVGVJLVD
|
|
ABCMNVFVJVGVFVJOVHDPZJKZVIQZVGDEVJVMDESVFDVHVIRUAVFVKEFZVMOZVGVJVFVOOZAAV
|
|
KJKZBHVPVCVNAVQHIVCVDVEVOUBZVFVNVMUCZAVKUDTVPCVQJKZVIQZVQBQZVCVEVOWAVDVCV
|
|
EOZVOOVLVTVIWCVNVLVTQZVMVCVEVNWDVEVCVNWDCAVKUEUFUIUGWCVNVMUHUJUKVPVEVQEFZ
|
|
VDWAWBLVCVDVEVOULVPVCVNWEVRVSAVKUMTVCVDVEVOUNCVQBUOUPUQURUSUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h l u z $. $d A y z $. $d Q f g h l u z $. $d Q y z $.
|
|
$d X f g h z $. $d X y z $. $d f g h ph z $.
|
|
prplnqu.x $e |- ( ph -> X e. P. ) $.
|
|
prplnqu.q $e |- ( ph -> Q e. Q. ) $.
|
|
prplnqu.sum $e |- ( ph -> A e.
|
|
( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) $.
|
|
$( Membership in the upper cut of a sum of a positive real and a fraction.
|
|
(Contributed by Jim Kingdon, 16-Jun-2021.) $)
|
|
prplnqu $p |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A ) $=
|
|
( co wceq cltq wbr cab cltp cpp cnp wcel syl2anc vz vf vg vh cv cplq c2nd
|
|
cfv wrex cnq cop nqprlu ltaddpr addcomprg breqtrd wb addclpr c1st elprnqu
|
|
syl prop sylan nqpru mpbid ltsopr ltrelpr ltnqpr mpbird ltexnqi wa adantr
|
|
sotri simprl addcomnqg simprr eqtr3d abbidv opeq12d addnqpr ltaprg adantl
|
|
breq2 breq1 w3a caovord2d oveq1 eqeq1d rspcev rexlimddv ) AEUAUEZUFKZDLZB
|
|
UEZEUFKZDLZBFUGUHZUIZUAUJAEDMNZWLUAUJUIAWRGUEZEMNGOECUEZMNCOUKZWSDMNZGOZD
|
|
WTMNZCOZUKZPNZAXAFXAQKZPNXHXFPNZXGAXAXAFQKZXHPAXARSZFRSZXAXJPNAEUJSZXKICE
|
|
GULZUTZHXAFUMTAXKXLXJXHLXOHXAFUNTUOADXHUGUHZSZXIJADUJSZXHRSZXQXIUPAXSXQXR
|
|
AXLXKXSHXOFXAUQTZJXSXHURUHZXPUKRSXQXRXHVADXPYAUSVBTZXTCDXHGVCTVDZXAXHXFPR
|
|
VEVFVLTAXMXRWRXGUPIYBCEDGVGTVHUAEDVIUTAWJUJSZWLVJZVJZWJWPSZWJEUFKZDLZWQYF
|
|
YGFWSWJMNGOWJWTMNCOUKZPNZYFYKXHYJXAQKZPNYFXHXFYLPAXIYEYCVKYFWSYHMNZGOZYHW
|
|
TMNZCOZUKZXFYLYFYIYQXFLYFWKYHDYFXMYDWKYHLAXMYEIVKZAYDWLVMZEWJVNTAYDWLVOVP
|
|
ZYIYNXCYPXEYIYMXBGYHDWSMWBVQYIYOXDCYHDWTMWCVQVRUTYFYDXMYQYLLYSYRCWJEGVSTV
|
|
PUOYFUBUCUDFYJXAPRQUBUEZRSZUCUEZRSZUDUEZRSWDUUAUUCPNUUEUUAQKUUEUUCQKPNUPY
|
|
FUUAUUCUUEVTWAAXLYEHVKZYFYDYJRSYSCWJGULUTYFXMXKYRXNUTUUBUUDVJUUAUUCQKUUCU
|
|
UAQKLYFUUAUUCUNWAWEVHYFYDXLYGYKUPYSUUFCWJFGVCTVHYTWOYIBWJWPWMWJLWNYHDWMWJ
|
|
EUFWFWGWHTWI $.
|
|
$}
|
|
|
|
${
|
|
$d A f g $. $d B f g $. $d C f g $. $d D f g $.
|
|
$( Strong extensionality of addition (ordering version). This is similar
|
|
to ~ addext but for positive reals and based on less-than rather than
|
|
apartness. (Contributed by Jim Kingdon, 17-Feb-2020.) $)
|
|
addextpr $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
|
|
( ( A +P. B ) <P ( C +P. D ) -> ( A <P C \/ B <P D ) ) ) $=
|
|
( vf vg cnp wcel wa cpp co cltp wbr wo addclpr adantl syl3anc wb ltaprg
|
|
cv wi adantr simprl simplr syl2anc wor w3a ltsopr sowlin mpan simpll wceq
|
|
addcomprg caovcomd breq12d bitr4d simprr orbi12d sylibrd ) AGHZBGHZIZCGHZ
|
|
DGHZIZIZABJKZCDJKZLMZVGCBJKZLMZVJVHLMZNZACLMZBDLMZNVFVGGHZVHGHZVJGHZVIVMU
|
|
AZVBVPVEABOUBVEVQVBCDOPVFVCVAVRVBVCVDUCZUTVAVEUDZCBOUEGLUFVPVQVRUGVSUHGVG
|
|
VHVJLUIUJQVFVNVKVOVLVFVNBAJKZBCJKZLMZVKVFUTVCVAVNWDRUTVAVEUKZVTWAACBSQVFV
|
|
GWBVJWCLVFEFABGJETZGHFTZGHIWFWGJKWGWFJKULVFWFWGUMPZWEWAUNVFEFCBGJWHVTWAUN
|
|
UOUPVFVAVDVCVOVLRWAVBVCVDUQVTBDCSQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d q r x y z w v u f g h A $. $d q r x y z w v u f g B $. $d x y C $.
|
|
recexpr.1 $e |- B = <.
|
|
{ x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } ,
|
|
{ x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. $.
|
|
$( Membership in the lower cut of ` B ` . Lemma for ~ recexpr .
|
|
(Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlemell $p |- ( C e. ( 1st ` B ) <->
|
|
E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) $=
|
|
( c1st cfv wcel cvv cv cltq wbr wa wex cnq ltrelnq brel adantr exlimiv
|
|
crq c2nd elex simpld syl wceq breq1 anbi1d exbidv fveq2i nqex abssi ssexi
|
|
cab cop simprd op1st eqtri elab2g pm5.21nii ) EDGHZIEJIZEBKZLMZVCUAHZCUBH
|
|
IZNZBOZEVAUCVGVBBVDVBVFVDEPIZVBVDVIVCPIZEVCPPLQRUDEPUCUESTAKZVCLMZVFNZBOZ
|
|
VHAEVAJVKEUFZVMVGBVOVLVDVFVKEVCLUGUHUIVAVNAUNZVCVKLMZVECGHIZNZBOZAUNZUOZG
|
|
HVPDWBGFUJVPWAVPPUKVNAPVMVKPIZBVLWCVFVLWCVJVKVCPPLQRUDSTULUMWAPUKVTAPVSWC
|
|
BVQWCVRVQVJWCVCVKPPLQRUPSTULUMUQURUSUT $.
|
|
|
|
$( Membership in the upper cut of ` B ` . Lemma for ~ recexpr .
|
|
(Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlemelu $p |- ( C e. ( 2nd ` B ) <->
|
|
E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) $=
|
|
( c2nd cfv wcel cvv cv cltq wbr wa wex cnq ltrelnq brel adantr exlimiv
|
|
crq c1st elex simprd syl wceq breq2 anbi1d exbidv cab fveq2i simpld abssi
|
|
cop nqex ssexi op2nd eqtri elab2g pm5.21nii ) EDGHZIEJIZBKZELMZVCUAHZCUBH
|
|
IZNZBOZEVAUCVGVBBVDVBVFVDEPIZVBVDVCPIZVIVCEPPLQRUDEPUCUESTVCAKZLMZVFNZBOZ
|
|
VHAEVAJVKEUFZVMVGBVOVLVDVFVKEVCLUGUHUIVAVKVCLMZVECGHIZNZBOZAUJZVNAUJZUNZG
|
|
HWADWBGFUKVTWAVTPUOVSAPVRVKPIZBVPWCVQVPWCVJVKVCPPLQRULSTUMUPWAPUOVNAPVMWC
|
|
BVLWCVFVLVJWCVCVKPPLQRUDSTUMUPUQURUSUT $.
|
|
|
|
$( ` B ` is inhabited. Lemma for ~ recexpr . (Contributed by Jim Kingdon,
|
|
27-Dec-2019.) $)
|
|
recexprlemm $p |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\
|
|
E. r e. Q. r e. ( 2nd ` B ) ) ) $=
|
|
( wcel cv cfv cnq wrex wa crq cltq wbr syl wi eleq1d c1q cnp c1st recclnq
|
|
c2nd cop prop prmu nsmallnqq adantr wex recrecnq wceq breq2 fveq2 anbi12d
|
|
anbi2d spcegv sylbird recexprlemell syl6ibr expcomd imp reximdv rexlimiva
|
|
mpd 3syl prml co 1nq addclnq sylancl ltaddnq rspcev syl2anc recexprlemelu
|
|
cplq breq1 jca ) CUAHZFIZDUBJHZFKLZEIZDUDJHZEKLZVSCUBJZCUDJZUEUAHZAIZWGHZ
|
|
AKLWBCUFZAWGWFUGWJWBAKWIKHZWJMZVTWINJZOPZFKLZWBWLWPWJWLWNKHZWPWIUCZFWNUHQ
|
|
UIWMWOWAFKWLWJWOWARWLWOWJWAWLWOWJMZVTBIZOPZWTNJZWGHZMZBUJZWAWLWSWOWNNJZWG
|
|
HZMZXEWLXGWJWOWLXFWIWGWIUKZSUPWLWQXHXERWRXDXHBWNKWTWNULZXAWOXCXGWTWNVTOUM
|
|
XJXBXFWGWTWNNUNZSUOUQQURABCDVTGUSUTVAVBVCVEVDVFVSWHWIWFHZAKLWEWKAWGWFVGXL
|
|
WEAKWLXLMZWNWCOPZEKLZWEWLXOXLWLWNTVPVHZKHZWNXPOPZXOWLWQTKHZXQWRVIWNTVJVKW
|
|
LWQXSXRWRVIWNTVLVKXNXREXPKWCXPWNOUMVMVNUIXMXNWDEKWLXLXNWDRWLXNXLWDWLXNXLM
|
|
ZWTWCOPZXBWFHZMZBUJZWDWLXTXNXFWFHZMZYDWLYEXLXNWLXFWIWFXISUPWLWQYFYDRWRYCY
|
|
FBWNKXJYAXNYBYEWTWNWCOVQXJXBXFWFXKSUOUQQURABCDWCGVOUTVAVBVCVEVDVFVR $.
|
|
|
|
$( The lower cut of ` B ` is open. Lemma for ~ recexpr . (Contributed by
|
|
Jim Kingdon, 28-Dec-2019.) $)
|
|
recexprlemopl $p |- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) )
|
|
-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) $=
|
|
( cv c1st cfv wcel cnp cltq wbr wa cnq wrex crq wex recexprlemell adantll
|
|
c2nd ltbtwnnqq biimpi simpll 19.8a sylibr jca reximdv mpan9 exlimiv sylbi
|
|
expcom 3ad2ant3 ) FHZDIJZKZCLKUOEHZMNZURUPKZOZEPQZUOPKUQUOBHZMNZVCRJCUBJK
|
|
ZOZBSVBABCDUOGTVFVBBVDUSURVCMNZOZEPQZVEVBVDVIEUOVCUCUDVEVHVAEPVHVEVAVHVEO
|
|
USUTUSVGVEUEVGVEUTUSVGVEOZVJBSUTVJBUFABCDURGTUGUAUHUMUIUJUKULUN $.
|
|
|
|
$( The lower cut of ` B ` is lower. Lemma for ~ recexpr . (Contributed by
|
|
Jim Kingdon, 28-Dec-2019.) $)
|
|
recexprlemlol $p |- ( ( A e. P. /\ q e. Q. ) ->
|
|
( E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) -> q e. ( 1st ` B ) ) ) $=
|
|
( cv cltq wbr c1st cfv wcel wa cnq wrex wi cnp wex recexprlemell crq c2nd
|
|
ltsonq ltrelnq sotri ex anim1d eximdv 3imtr4g imp rexlimivw a1i ) FHZEHZI
|
|
JZUNDKLZMZNZEOPUMUPMZQCRMUMOMNURUSEOUOUQUSUOUNBHZIJZUTUALCUBLMZNZBSUMUTIJ
|
|
ZVBNZBSUQUSUOVCVEBUOVAVDVBUOVAVDUMUNUTIOUCUDUEUFUGUHABCDUNGTABCDUMGTUIUJU
|
|
KUL $.
|
|
|
|
$( The upper cut of ` B ` is open. Lemma for ~ recexpr . (Contributed by
|
|
Jim Kingdon, 28-Dec-2019.) $)
|
|
recexprlemopu $p |- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) )
|
|
-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) $=
|
|
( cv c2nd cfv wcel cnp cltq wbr wa cnq wrex crq wex recexprlemelu adantlr
|
|
c1st ltbtwnnqq biimpi simplr 19.8a sylibr jca reximdv mpan9 exlimiv sylbi
|
|
expcom 3ad2ant3 ) EHZDIJZKZCLKFHZUOMNZURUPKZOZFPQZUOPKUQBHZUOMNZVCRJCUBJK
|
|
ZOZBSVBABCDUOGTVFVBBVDVCURMNZUSOZFPQZVEVBVDVIFVCUOUCUDVEVHVAFPVHVEVAVHVEO
|
|
USUTVGUSVEUEVGVEUTUSVGVEOZVJBSUTVJBUFABCDURGTUGUAUHUMUIUJUKULUN $.
|
|
|
|
$( The upper cut of ` B ` is upper. Lemma for ~ recexpr . (Contributed by
|
|
Jim Kingdon, 28-Dec-2019.) $)
|
|
recexprlemupu $p |- ( ( A e. P. /\ r e. Q. ) -> (
|
|
E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) ) $=
|
|
( cv cltq wbr c2nd cfv wcel wa cnq wrex wi cnp wex recexprlemelu crq c1st
|
|
ltsonq ltrelnq sotri expcom anim1d eximdv 3imtr4g imp rexlimivw a1i ) FHZ
|
|
EHZIJZUMDKLZMZNZFOPUNUPMZQCRMUNOMNURUSFOUOUQUSUOBHZUMIJZUTUALCUBLMZNZBSUT
|
|
UNIJZVBNZBSUQUSUOVCVEBUOVAVDVBVAUOVDUTUMUNIOUCUDUEUFUGUHABCDUMGTABCDUNGTU
|
|
IUJUKUL $.
|
|
|
|
$( ` B ` is rounded. Lemma for ~ recexpr . (Contributed by Jim Kingdon,
|
|
27-Dec-2019.) $)
|
|
recexprlemrnd $p |- ( A e. P. -> ( A. q e. Q. ( q e. ( 1st ` B )
|
|
<-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
|
|
/\ A. r e. Q. ( r e. ( 2nd ` B )
|
|
<-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) ) $=
|
|
( cnp wcel cv c1st cfv wa cnq wrex wb wral 3expia impbid ralrimiva cltq
|
|
wbr c2nd recexprlemopl recexprlemlol recexprlemopu recexprlemupu jca ) CH
|
|
IZFJZDKLZIZUJEJZUAUBZUMUKIMENOZPZFNQUMDUCLZIZUNUJUQIMFNOZPZENQUIUPFNUIUJN
|
|
IZMULUOUIVAULUOABCDEFGUDRABCDEFGUESTUIUTENUIUMNIZMURUSUIVBURUSABCDEFGUFRA
|
|
BCDEFGUGSTUH $.
|
|
|
|
$( ` B ` is disjoint. Lemma for ~ recexpr . (Contributed by Jim Kingdon,
|
|
27-Dec-2019.) $)
|
|
recexprlemdisj $p |- ( A e. P. -> A. q e. Q.
|
|
-. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) ) $=
|
|
( vz cnp wcel cv c1st cfv wa wn cnq cltq wbr crq wal wex ltsonq wi simprr
|
|
c2nd ltrelnq son2lpi simplr jca prltlu syl3an1 3expb sylan2 simprl simpll
|
|
cop prop sotri syl2anc ltrnqi syl adantl ex adantr alrimivv recexprlemell
|
|
mtoi recexprlemelu anbi12i breq1 fveq2 eleq1d anbi12d cbvexv anbi2i bitri
|
|
wceq eeanv bitr4i notbii alnex albii sylibr ralrimiva ) CHIZEJZDKLIZWEDUD
|
|
LIZMZNZEOWDWEOIZMZWEBJZPQZWLRLZCUDLZIZMZGJZWEPQZWRRLZCKLZIZMZMZNZGSZBSZWI
|
|
WKXEBGWKXDWTWNPQZWNWTPQZMZWTWNPOUAUEUFWDXDXJUBWJWDXDXJWDXDMXHXIXDWDXBWPMX
|
|
HXDXBWPWQWSXBUCWMWPXCUGUHWDXBWPXHWDXAWOUOHIXBWPXHCUPWTWNWOXAUIUJUKULXDXIW
|
|
DXDWRWLPQZXIXDWSWMXKWQWSXBUMWMWPXCUNWRWEWLPOUAUEUQURWRWLUSUTVAUHVBVCVFVDW
|
|
IXDGTZBTZNZXGWHXMWHWQBTZXCGTZMZXMWHXOWLWEPQZWNXAIZMZBTZMXQWFXOWGYAABCDWEF
|
|
VEABCDWEFVGVHYAXPXOXTXCBGWLWRVPZXRWSXSXBWLWRWEPVIYBWNWTXAWLWRRVJVKVLVMVNV
|
|
OWQXCBGVQVRVSXGXLNZBSXNXFYCBXDGVTWAXLBVTVOVRWBWC $.
|
|
|
|
$( ` B ` is located. Lemma for ~ recexpr . (Contributed by Jim Kingdon,
|
|
27-Dec-2019.) $)
|
|
recexprlemloc $p |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r
|
|
-> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) ) $=
|
|
( vu vv wcel cv cltq wbr cfv cnq wa crq sylan adantlr syl c1st c2nd wo wi
|
|
cnp wral wrex prop prnmaxl simprr wceq elprnql ad2ant2r recrecnq breqtrrd
|
|
cop wb recclnq ltrelnq brel adantl ad2antrr simprd ltrnqg syl2anc eqeltrd
|
|
mpbird simprl wex breq1 fveq2 eleq1d anbi12d spcegv recexprlemelu syl6ibr
|
|
mp2and rexlimddv olcd prnminu elprnqu eqbrtrd simpld recexprlemell ltrnqi
|
|
breq2 orcd prloc syl2an mpjaodan ex ralrimivw ) CUEJZFKZEKZLMZWNDUANJZWOD
|
|
UBNJZUCZUDZEOUFFOWMWTEOWMWPWSWMWPPZWOQNZCUANZJZWSWNQNZCUBNZJZXAXDPZWRWQXH
|
|
XBHKZLMZWRHXCWMXDXJHXCUGZWPWMXCXFUPUEJZXDXKCUHZHXBXFXCUIRSXHXIXCJZXJPZPZX
|
|
IQNZWOLMZXQQNZXCJZWRXPXRXBXSLMZXPXBXIXSLXHXNXJUJXPXIOJZXSXIUKXAXOYBXDWMXN
|
|
YBWPXJWMXLXNYBXMXIXFXCULRUMSZXIUNTZUOXPXQOJZWOOJZXRYAUQXPYBYEYCXIURTZXPWN
|
|
OJZYFXAYHYFPZXDXOWPYIWMWNWOOOLUSUTVAZVBVCXQWOVDVEVGXPXSXIXCYDXHXNXJVHVFXP
|
|
YEXRXTPZWRUDYGYEYKBKZWOLMZYLQNZXCJZPZBVIWRYPYKBXQOYLXQUKZYMXRYOXTYLXQWOLV
|
|
JYQYNXSXCYLXQQVKVLVMVNABCDWOGVOVPTVQVRVSXAXGPZWQWRYRIKZXELMZWQIXFWMXGYTIX
|
|
FUGZWPWMXLXGUUAXMIXEXFXCVTRSYRYSXFJZYTPZPZWNYSQNZLMZUUEQNZXFJZWQUUDUUFUUG
|
|
XELMZUUDUUGYSXELUUDYSOJZUUGYSUKXAUUBUUJXGYTWMUUBUUJWPWMXLUUBUUJXMYSXFXCWA
|
|
RSUMZYSUNTZYRUUBYTUJWBUUDYHUUEOJZUUFUUIUQUUDYHYFXAYIXGUUCYJVBWCUUDUUJUUMU
|
|
UKYSURTZWNUUEVDVEVGUUDUUGYSXFUULYRUUBYTVHVFUUDUUMUUFUUHPZWQUDUUNUUMUUOWNY
|
|
LLMZYNXFJZPZBVIWQUURUUOBUUEOYLUUEUKZUUPUUFUUQUUHYLUUEWNLWFUUSYNUUGXFYLUUE
|
|
QVKVLVMVNABCDWNGWDVPTVQVRWGWMXLXBXELMXDXGUCWPXMWNWOWEXBXEXFXCWHWIWJWKWLWL
|
|
$.
|
|
|
|
$( ` B ` is a positive real. Lemma for ~ recexpr . (Contributed by Jim
|
|
Kingdon, 27-Dec-2019.) $)
|
|
recexprlempr $p |- ( A e. P. -> B e. P. ) $=
|
|
( vq vr cnp wcel cnq cv c1st cfv wrex c2nd wa cltq wbr wb wral cpw cxp wn
|
|
wo w3a recexprlemm crq wex cab cop wss ltrelnq brel simpld adantr exlimiv
|
|
abssi nqex elpw2 mpbir opelxpi mp2an eqeltri recexprlemrnd recexprlemdisj
|
|
wi simprd jctil recexprlemloc 3jca elnp1st2nd sylanbrc ) CHIZDJUAZVNUBZIZ
|
|
FKZDLMZIZFJNGKZDOMZIZGJNPZPVSVQVTQRZVTVRIPGJNSFJTWBWDVQWAIZPFJNSGJTPZVSWE
|
|
PUCFJTZWDVSWBUDVFGJTFJTZUEDHIVMWCVPABCDGFEUFDAKZBKZQRZWJUGMZCOMIZPZBUHZAU
|
|
IZWJWIQRZWLCLMIZPZBUHZAUIZUJZVOEWPVNIZXAVNIZXBVOIXCWPJUKWOAJWNWIJIZBWKXEW
|
|
MWKXEWJJIZWIWJJJQULUMUNUOUPUQWPJURUSUTXDXAJUKWTAJWSXEBWQXEWRWQXFXEWJWIJJQ
|
|
ULUMVGUOUPUQXAJURUSUTWPXAVNVNVAVBVCVHVMWFWGWHABCDGFEVDABCDFEVEABCDGFEVIVJ
|
|
DGFVKVL $.
|
|
|
|
$( The lower cut of one is a subset of the lower cut of ` A .P. B ` .
|
|
Lemma for ~ recexpr . (Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlem1ssl $p |- ( A e. P. ->
|
|
( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) ) $=
|
|
( vw vz vf wcel cfv co cv c1q cltq wbr wa crq cmq wceq cnq vv vg cnp c1st
|
|
vh vu c1p cmp 1prl abeq2i c2nd wrex rec1nq ltrnqi syl5eqbrr prop prmuloc2
|
|
cop sylan sylan2 prnmaxl ad2ant2r w3a elprnql 3adant3 simp1r ltrelnq brel
|
|
simpld syl simp3 simp2r simpr wb ltmnqg adantl simprl simprd recclnq 3syl
|
|
ad2antrr simplr mulcomnqg caovord2d syl5ib 1nq mulidnq ax-mp mpdan eqtr3d
|
|
recidnq oveqan12d adantr simpll mulassnqg mulclnq caov4d syl5reqr syl2anc
|
|
recmulnqg mpbird eleq1d biimprd wi wex breq2 anbi12d spcegv recexprlemell
|
|
fveq2 syl6ibr syl2and mpd syl22anc 3ad2ant3 mpan2 oveq1d syl3an2 3anidm12
|
|
3eqtr2d eqeq2d rspcev 3expia reximdv recexprlempr df-imp genpelvl sylibrd
|
|
oveq2 rexlimddv ex syl5bi ssrdv ) CUCIZFUGUDJZCDUHKUDJZFLZYOIYQMNOZYNYQYP
|
|
IZYRFYOFUIUJYNYRYSYNYRPZUALZYQQJZRKZCUKJZIZYSUACUDJZYRYNMUUBNOZUUEUAUUFUL
|
|
ZYRMMQJUUBNUMYQMUNUOYNUUFUUDURUCIZUUGUUHCUPZUAUUBUUDUUFUQUSUTYTUUAUUFIZUU
|
|
EPZPZUUAGLZNOZGUUFULZYSYNUUKUUPYRUUEYNUUIUUKUUPUUJGUUAUUDUUFVAUSVBUUMUUPY
|
|
QUUNALZRKZSZADUDJZULZGUUFULZYSUUMUUOUVAGUUFYTUULUUOUVAYTUULUUOVCZUUNQJZYQ
|
|
RKZUUTIZYQUUNUVERKZSZUVAUVCUUATIZYQTIZUUOUUEUVFYTUULUVIUUOYNUUKUVIYRUUEYN
|
|
UUIUUKUVIUUJUUAUUDUUFVDUSVBVEUVCYRUVJYNYRUULUUOVFYRUVJMTIZYQMTTNVGVHVIVJZ
|
|
YTUULUUOVKYTUUKUUEUUOVLUVIUVJPZUUOUUEPZPZUVNUVFUVMUVNVMUVOUUOUVEUUAQJZYQR
|
|
KZNOZUUEUVQQJZUUDIZUVFUUOUVDUVPNOUVOUVRUUAUUNUNUVOHUBUEUVDUVPYQNTRHLZTIZU
|
|
BLZTIZUELZTIVCZUWAUWCNOUWEUWARKUWEUWCRKNOVNUVOUWAUWCUWEVOVPUVOUUOUUNTIZUV
|
|
DTIZUVMUUOUUEVQUUOUVIUWGUUAUUNTTNVGVHVRZUUNVSZVTUVIUVPTIZUVJUVNUUAVSZWAZU
|
|
VIUVJUVNWBZUWBUWDPZUWAUWCRKZUWCUWARKSUVOUWAUWCWCVPZWDWEUVOUVTUUEUVOUVSUUC
|
|
UUDUVOUVSUUCSZUVQUUCRKZMSZUVOMMMRKZUWSUVKUXAMSWFMWGWHUVOUVPUUARKZYQUUBRKZ
|
|
RKZUXAUWSUVMUXDUXASUVNUVIUVJUXBMUXCMRUVIUUAUVPRKZUXBMUVIUWKUXEUXBSUWLUUAU
|
|
VPWCWIUUAWKWJYQWKWLWMUVOHUBUEUVPUUAYQUUBTRUWMUVIUVJUVNWNUWNUWQUWFUWPUWERK
|
|
UWAUWCUWERKRKSUVOUWAUWCUWEWOVPUVOUVJUUBTIZUWNYQVSZVJUWOUWPTIUVOUWAUWCWPZV
|
|
PWQWJWRUVMUWRUWTVNZUVNUVMUVQTIZUUCTIZUXIUVIUWKUVJUXJUWLUVPYQWPUSZUVJUVIUX
|
|
FUXKUXGUUAUUBWPUTUVQUUCWTWSWMXAXBXCUVMUVRUVTPZUVFXDUVNUVMUXMUVEBLZNOZUXNQ
|
|
JZUUDIZPZBXEZUVFUVMUXJUXMUXSXDUXLUXRUXMBUVQTUXNUVQSZUXOUVRUXQUVTUXNUVQUVE
|
|
NXFUXTUXPUVSUUDUXNUVQQXJXBXGXHVJABCDUVEEXIXKWMXLXMXNUVCUWGUVJUVHUUOYTUWGU
|
|
ULUWIXOUVLUWGUVJPYQMYQRKZUUNUVDRKZYQRKZUVGUVJYQUYASUWGUVJYQMRKZYQUYAYQWGU
|
|
VJUVKUYDUYASWFYQMWCXPWJVPUWGUYCUYASUVJUWGUYBMYQRUUNWKXQWMUWGUVJUYCUVGSZUW
|
|
GUWGUWHUVJUYEUWJUUNUVDYQWOXRXSXTWSUUSUVHAUVEUUTUUQUVESUURUVGYQUUQUVEUUNRY
|
|
IYAYBWSYCYDYNYSUVBVNZYRUULYNDUCIUYFABCDEYEUFHUBBFCDYQGAUHRBFUBHUFYFUXHYGW
|
|
IWAYHXMYJYKYLYM $.
|
|
|
|
$( The upper cut of one is a subset of the upper cut of ` A .P. B ` .
|
|
Lemma for ~ recexpr . (Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlem1ssu $p |- ( A e. P. ->
|
|
( 2nd ` 1P ) C_ ( 2nd ` ( A .P. B ) ) ) $=
|
|
( vw vf vg wcel cfv co cv c1q cltq wbr wa cmq wceq cnq syl vv vz cnp c2nd
|
|
vh vu c1p cmp 1pru abeq2i c1st wrex prop prmuloc2 sylan prnminu ad2ant2rl
|
|
cop w3a simp3 simp2l elprnql ad2ant2r 3adant3 simp1r ltrelnq brel recclnq
|
|
crq simprd mulassnqg syl3anc oveq2d 3eqtrd eleq1d mpbird ltrnqi wb ltmnqg
|
|
recidnq mulidnq adantl mulclnq simpld mulcomnqg caovord2d 1nq ax-mp mpdan
|
|
syl2anc syl5ib eqtr3d oveqan12d caov4d syl5reqr caovcld recmulnqg biimprd
|
|
wex breq1 fveq2 anbi12d spcegv recexprlemelu syl6ibr syl2and mp2and mpan2
|
|
oveq1d adantr syl3an2 3anidm12 3eqtr2d oveq2 eqeq2d rspcev 3expia reximdv
|
|
wi recexprlempr df-imp genpelvu ad2antrr sylibrd mpd rexlimddv ex syl5bi
|
|
ssrdv ) CUCIZFUGUDJZCDUHKUDJZFLZYKIMYMNOZYJYMYLIZYNFYKFUIUJYJYNYOYJYNPZUA
|
|
LZYMQKZCUDJZIZYOUACUKJZYJUUAYSURUCIZYNYTUAUUAULCUMZUAYMYSUUAUNUOYPYQUUAIZ
|
|
YTPZPZUBLZYRNOZUBYSULZYOYJYTUUIYNUUDYJUUBYTUUIUUCUBYRYSUUAUPUOUQUUFUUIYMU
|
|
UGALZQKZRZADUDJZULZUBYSULZYOUUFUUHUUNUBYSYPUUEUUHUUNYPUUEUUHUSZUUGVIJZYMQ
|
|
KZUUMIZYMUUGUURQKZRZUUNUUPUUHYRYMVIJZQKZUUAIZUUSYPUUEUUHUTZUUPUVDUUDYPUUD
|
|
YTUUHVAUUPUVCYQUUAUUPUVCYQYMUVBQKZQKZYQMQKZYQUUPYQSIZYMSIZUVBSIZUVCUVGRYP
|
|
UUEUVIUUHYJUUDUVIYNYTYJUUBUUDUVIUUCYQYSUUAVBUOVCVDZUUPYNUVJYJYNUUEUUHVEYN
|
|
MSIZUVJMYMSSNVFVGVJTZUUPUVJUVKUVNYMVHTZYQYMUVBVKVLUUPUVFMYQQUUPUVJUVFMRUV
|
|
NYMVTZTVMUUPUVIUVHYQRUVLYQWATVNVOVPUUPUUHYRVIJZYMQKZUURNOZUVDUVRVIJZUUAIZ
|
|
UUSUUHUVQUUQNOUUPUVSUUGYRVQUUPGHUEUVQUUQYMNSQGLZSIZHLZSIZUELZSIUSZUWBUWDN
|
|
OUWFUWBQKUWFUWDQKNOVRUUPUWBUWDUWFVSWBUUPYRSIZUVQSIZUUPUVIUVJUWHUVLUVNYQYM
|
|
WCWJZYRVHZTZUUPUUGSIZUUQSIZUUPUUHUWMUVEUUHUWMUWHUUGYRSSNVFVGWDTZUUGVHZTUV
|
|
NUWCUWEPZUWBUWDQKZUWDUWBQKRUUPUWBUWDWEWBZWFWKUUPUWAUVDUUPUVTUVCUUAUUPUVTU
|
|
VCRZUVRUVCQKZMRZUUPMMMQKZUXAUVMUXCMRWGMWAWHUUPUVQYRQKZUVFQKZUXCUXAUUPUWHU
|
|
VJUXEUXCRUWJUVNUWHUVJUXDMUVFMQUWHYRUVQQKZUXDMUWHUWIUXFUXDRUWKYRUVQWEWIYRV
|
|
TWLUVPWMWJUUPGHUEUVQYRYMUVBSQUWLUWJUVNUWSUWGUWRUWFQKUWBUWDUWFQKQKRUUPUWBU
|
|
WDUWFVKWBUVOUWQUWRSIUUPUWBUWDWCZWBZWNWLWOUUPUVRSIZUVCSIUWTUXBVRUUPGHUVQYM
|
|
SSSQUXHUWLUVNWPZUUPGHYRUVBSSSQUXHUWJUVOWPUVRUVCWQWJVPVOWRUUPUVSUWAPZBLZUU
|
|
RNOZUXLVIJZUUAIZPZBWSZUUSUUPUXIUXKUXQXSUXJUXPUXKBUVRSUXLUVRRZUXMUVSUXOUWA
|
|
UXLUVRUURNWTUXRUXNUVTUUAUXLUVRVIXAVOXBXCTABCDUUREXDXEXFXGUUPUWMUVJUVAUWOU
|
|
VNUWMUVJPYMMYMQKZUUGUUQQKZYMQKZUUTUVJYMUXSRUWMUVJYMMQKZYMUXSYMWAUVJUVMUYB
|
|
UXSRWGYMMWEXHWLWBUWMUYAUXSRUVJUWMUXTMYMQUUGVTXIXJUWMUVJUYAUUTRZUWMUWMUWNU
|
|
VJUYCUWPUUGUUQYMVKXKXLXMWJUULUVAAUURUUMUUJUURRUUKUUTYMUUJUURUUGQXNXOXPWJX
|
|
QXRYJYOUUOVRZYNUUEYJDUCIUYDABCDEXTUFGHBFCDYMUBAUHQBFHGUFYAUXGYBWIYCYDYEYF
|
|
YGYHYI $.
|
|
|
|
$( The lower cut of ` A .P. B ` is a subset of the lower cut of one. Lemma
|
|
for ~ recexpr . (Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlemss1l $p |- ( A e. P. ->
|
|
( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) ) $=
|
|
( vw vz vq wcel co c1st cfv cv c1q cltq wbr cmq wi wa cnq vu cnp cmp wceq
|
|
vf vg c1p wrex wb recexprlempr df-imp mulclnq genpelvl mpdan crq c2nd wex
|
|
recexprlemell ltrelnq brel simprd prop elprnql sylan ltmnqi expcom adantr
|
|
cop prltlu syl3an1 3expia mulcomnqg recidnq breq12d sylibd ancoms anim12d
|
|
syl syld ltsonq sotri syl6 exp4b syl5 pm2.43d impd exlimdv breq1 biimprcd
|
|
syl5bi expimpd rexlimdvv sylbid 1prl abeq2i syl6ibr ssrdv ) CUBIZFCDUCJKL
|
|
ZUGKLZWRFMZWSIZXANOPZXAWTIWRXBXAGMZHMZQJZUDZHDKLZUHGCKLZUHZXCWRDUBIXBXJUI
|
|
ABCDEUJUAUEUFBFCDXAGHUCQBFUFUEUAUKUEMUFMULUMUNWRXGXCGHXIXHWRXDXIIZXEXHIZX
|
|
GXCRZWRXKSZXLXFNOPZXMXLXEBMZOPZXPUOLZCUPLZIZSZBUQXNXOABCDXEEURXNYAXOBXNXQ
|
|
XTXOXNXQXTXORZXQXPTIZXNXQYBRXQXETIYCXEXPTTOUSUTVAXNYCXQXTXOXNYCSZYAXFXDXP
|
|
QJZOPZYENOPZSXOYDXQYFXTYGXNXQYFRZYCXNXDTIZYHWRXIXSVHUBIZXKYICVBZXDXSXIVCV
|
|
DZXQYIYFXEXPXDVEVFVRVGYDXTXDXROPZYGXNXTYMRYCWRXKXTYMWRYJXKXTYMYKXDXRXSXIV
|
|
IVJVKVGXNYIYCYMYGRZYLYCYIYNYCYISZYMXPXDQJZXPXRQJZOPZYGYCYMYRRYIYMYCYRXDXR
|
|
XPVEVFVGYOYPYEYQNOXPXDVLYCYQNUDYIXPVMVGVNVOVPVDVSVQXFYENOTVTUSWAWBWCWDWEW
|
|
FWGWJXGXCXOXAXFNOWHWIWBWKWLWMXCFWTFWNWOWPWQ $.
|
|
|
|
$( The upper cut of ` A .P. B ` is a subset of the upper cut of one. Lemma
|
|
for ~ recexpr . (Contributed by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlemss1u $p |- ( A e. P. ->
|
|
( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) ) $=
|
|
( vw vz vq wcel co c2nd cfv cv c1q cltq wbr cmq wi wa cnq vu cnp cmp wceq
|
|
vf vg c1p wrex wb recexprlempr df-imp mulclnq genpelvu mpdan crq c1st wex
|
|
recexprlemelu ltrelnq brel simpld prop elprnqu sylan ltmnqi expcom adantr
|
|
cop syl prltlu syl3an1 3com23 3expia recidnq mulcomnqg sylibd ancoms syld
|
|
breq12d anim12d ltsonq sotri syl6 exp4b syl5 pm2.43d exlimdv syl5bi breq2
|
|
impd biimprcd expimpd rexlimdvv sylbid 1pru abeq2i syl6ibr ssrdv ) CUBIZF
|
|
CDUCJKLZUGKLZWSFMZWTIZNXBOPZXBXAIWSXCXBGMZHMZQJZUDZHDKLZUHGCKLZUHZXDWSDUB
|
|
IXCXKUIABCDEUJUAUEUFBFCDXBGHUCQBFUFUEUAUKUEMUFMULUMUNWSXHXDGHXJXIWSXEXJIZ
|
|
XFXIIZXHXDRZWSXLSZXMNXGOPZXNXMBMZXFOPZXQUOLZCUPLZIZSZBUQXOXPABCDXFEURXOYB
|
|
XPBXOXRYAXPXOXRYAXPRZXRXQTIZXOXRYCRXRYDXFTIXQXFTTOUSUTVAXOYDXRYAXPXOYDSZY
|
|
BXEXQQJZXGOPZNYFOPZSXPYEXRYGYAYHXOXRYGRZYDXOXETIZYIWSXTXJVHUBIZXLYJCVBZXE
|
|
XJXTVCVDZXRYJYGXQXFXEVEVFVIVGYEYAXSXEOPZYHXOYAYNRYDWSXLYAYNWSYAXLYNWSYKYA
|
|
XLYNYLXSXEXJXTVJVKVLVMVGXOYJYDYNYHRZYMYDYJYOYDYJSZYNXQXSQJZXQXEQJZOPZYHYD
|
|
YNYSRYJYNYDYSXSXEXQVEVFVGYPYQNYRYFOYDYQNUDYJXQVNVGXQXEVOVSVPVQVDVRVTYHYGX
|
|
PNYFXGOTWAUSWBVQWCWDWEWFWJWGWHXHXDXPXBXGNOWIWKWCWLWMWNXDFXAFWOWPWQWR $.
|
|
|
|
$( ` B ` is the reciprocal of ` A ` . Lemma for ~ recexpr . (Contributed
|
|
by Jim Kingdon, 27-Dec-2019.) $)
|
|
recexprlemex $p |- ( A e. P. -> ( A .P. B ) = 1P ) $=
|
|
( cnp wcel cmp c1p wceq c1st cfv c2nd recexprlemss1l recexprlem1ssl eqssd
|
|
co recexprlemss1u recexprlem1ssu wa recexprlempr mulclpr mpdan 1pr preqlu
|
|
wb sylancl mpbir2and ) CFGZCDHQZIJZUJKLZIKLZJZUJMLZIMLZJZUIULUMABCDENABCD
|
|
EOPUIUOUPABCDERABCDESPUIUJFGZIFGUKUNUQTUFUIDFGURABCDEUACDUBUCUDUJIUEUGUH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x u v w z $.
|
|
$( The reciprocal of a positive real exists. Part of Proposition 9-3.7(v)
|
|
of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by
|
|
Mario Carneiro, 12-Jun-2013.) $)
|
|
recexpr $p |- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P ) $=
|
|
( vz vw vu vv cnp wcel cv cltq wbr crq cfv wa wex cab cmp co c1p wceq cop
|
|
c2nd c1st wrex breq12 simpr fveq2d eleq1d anbi12d cbvexdva cbvabv breq12d
|
|
simpl opeq12i recexprlempr recexprlemex oveq2 eqeq1d rspcev syl2anc ) BGH
|
|
CIZDIZJKZVBLMZBUBMZHZNZDOZCPZVBVAJKZVDBUCMZHZNZDOZCPZUAZGHBVPQRZSTZBAIZQR
|
|
ZSTZAGUDEFBVPVIEIZFIZJKZWCLMZVEHZNZFOZEPVOWCWBJKZWEVKHZNZFOZEPVHWHCEVAWBT
|
|
ZVGWGDFWMVBWCTZNZVCWDVFWFVAWBVBWCJUEWOVDWEVEWOVBWCLWMWNUFZUGZUHUIUJUKVNWL
|
|
CEWMVMWKDFWOVJWIVLWJWOVBWCVAWBJWPWMWNUMULWOVDWEVKWQUHUIUJUKUNZUOEFBVPWRUP
|
|
WAVRAVPGVSVPTVTVQSVSVPBQUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h s t u v x $. $d B f g h s t u v x $.
|
|
$( Lemma for ~ aptipr . (Contributed by Jim Kingdon, 28-Jan-2020.) $)
|
|
aptiprleml $p |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) ->
|
|
( 1st ` A ) C_ ( 1st ` B ) ) $=
|
|
( vs vt vu vv cnp wcel wbr cfv cv wa cltq wrex sylan cplq co wceq syl2anc
|
|
cnq vx vf vg vh cltp wn w3a c1st c2nd cop prop prnmaxl ad2ant2rl ad2antll
|
|
wi ltexnqi simplr ad2antrr simprl prarloc2 adantr simpll ad3antrrr simprr
|
|
w3o elprnql nqtri3or simplrl addclnq wb ltanqg adantl addcomnqg caovord2d
|
|
simplrr eqeltrd prcdnql sylbid jctild eleq1 rspcev ltdfpr mpbird pm2.21dd
|
|
imp anbi12d ex syl5ibcom 3jaod mpd rexlimddv expr 3impa ssrdv ) AGHZBGHZB
|
|
AUEIZUFZUGUAAUHJZBUHJZWOWPWRUAKZWSHZXAWTHZUOWOWPLZWRXBXCXDWRXBLZLZXACKZMI
|
|
ZXCCWSWOXBXHCWSNZWPWRWOWSAUIJZUJGHZXBXIAUKZCXAXJWSULOUMXFXGWSHZXHLZLZXADK
|
|
ZPQZXGRZXCDTXHXRDTNXFXMDXAXGUPUNXOXPTHZXRLZLZEKZXPPQZBUIJZHZXCEWTYAWPXSYE
|
|
EWTNZXFWPXNXTWOWPXEUQURZXOXSXRUSWPWTYDUJGHZXSYFBUKZXPYDWTEUTOSYAYBWTHZYEL
|
|
ZLZYBXAMIZYBXARZXAYBMIZVEZXCYLYBTHZXATHZYPYLWPYJYQYAWPYKYGVAZYAYJYEUSZWPY
|
|
HYJYQYIYBYDWTVFOSZYLWOXBYRXFWOXNXTYKWOWPXEVBVCZXFXBXNXTYKXDWRXBVDVCWOXKXB
|
|
YRXLXAXJWSVFOSZYBXAVGSYLYMXCYNYOYLYMXCYLYMLZWQXCUUDWQFKZYDHZUUEWSHZLZFTNZ
|
|
UUDYCTHZYEYCWSHZLZUUIUUDYQXSUUJYLYQYMUUAVAYLXSYMXOXSXRYKVHZVAYBXPVISYLYMU
|
|
ULYLYMUUKYEYLYMYCXQMIZUUKYLUBUCUDYBXAXPMTPUBKZTHZUCKZTHZUDKZTHUGUUOUUQMIU
|
|
USUUOPQUUSUUQPQMIVJYLUUOUUQUUSVKVLUUAUUCUUMUUPUURLUUOUUQPQUUQUUOPQRYLUUOU
|
|
UQVMVLVNYLWOXQWSHZUUNUUKUOZUUBYLXQXGWSXOXSXRYKVOXOXMXTYKXFXMXHUSURVPWOXKU
|
|
UTUVAXLXQYCXJWSVQOSVRYAYJYEVDVSWEUUHUULFYCTUUEYCRUUFYEUUGUUKUUEYCYDVTUUEY
|
|
CWSVTWFWASYLWQUUIVJZYMYLWPWOUVBYSUUBBAFWBSVAWCXOWRXTYKYMXDWRXBXNVHVCWDWGY
|
|
LYJYNXCYTYBXAWTVTWHYLWPYJYOXCUOZYSYTWPYHYJUVCYIYBXAYDWTVQOSWIWJWKWKWKWLWM
|
|
WN $.
|
|
|
|
$( Lemma for ~ aptipr . (Contributed by Jim Kingdon, 28-Jan-2020.) $)
|
|
aptiprlemu $p |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) ->
|
|
( 2nd ` B ) C_ ( 2nd ` A ) ) $=
|
|
( vs vt vu cnp wcel wbr cfv cv cltq wrex sylan cplq cnq syl2anc ad3antrrr
|
|
wa co adantr vx vf vg vh cltp wn w3a c2nd c1st cop prop prnminu 3ad2antl2
|
|
wceq simprr ltexnqi syl simpl1 ad2antrr simprl prarloc2 w3o simpr elprnqu
|
|
simpl2 elprnql addclnq nqtri3or wi simplrr breq1 biimprd wb ltanqg adantl
|
|
imp addcomnqg caovord2d mpbird prcdnql mpd rspe syl12anc simpll3 pm2.21dd
|
|
ltdfpr ex eleq1 syl5ibrcom prcunqu 3jaod rexlimddv ssrdv ) AFGZBFGZBAUEHZ
|
|
UFZUGZUABUHIZAUHIZWRUAJZWSGZXAWTGZWRXBRZCJZXAKHZXCCWSWOWNXBXFCWSLZWQWOBUI
|
|
IZWSUJFGZXBXGBUKZCXAWSXHULMUMXDXEWSGZXFRZRZXEDJZNSZXAUNZXCDOXMXFXPDOLXDXK
|
|
XFUODXEXAUPUQXMXNOGZXPRZRZEJZXNNSZWTGZXCEAUIIZXSWNXQYBEYCLZXDWNXLXRWNWOWQ
|
|
XBURZUSXMXQXPUTZWNYCWTUJFGZXQYDAUKZXNWTYCEVAMPXSXTYCGZYBRZRZXAYAKHZXAYAUN
|
|
ZYAXAKHZVBZXCYKXAOGZYAOGZYOYKWOXBYPXDWOXLXRYJWNWOWQXBVEZQZXDXBXLXRYJWRXBV
|
|
CQWOXIXBYPXJXAWSXHVDMPYKXTOGZXQYQYKWNYIYTXDWNXLXRYJYEQZXSYIYBUTZWNYGYIYTY
|
|
HXTWTYCVFMPZXSXQYJYFTZXTXNVGPXAYAVHPYKYLXCYMYNYKYLXCYKYLRZWPXCUUEWPXKXEYC
|
|
GZRZCOLZUUEXEOGZXKUUFUUHXMUUIXRYJYLXMWOXKUUIXDWOXLYRTXDXKXFUTZWOXIXKUUIXJ
|
|
XEWSXHVDMPQZXMXKXRYJYLUUJQUUEXEXTKHZUUFUUEUULXOYAKHZYKYLUUMYKXPYLUUMVIXMX
|
|
QXPYJVJXPUUMYLXOXAYAKVKVLUQVPUUEUBUCUDXEXTXNKONUBJZOGZUCJZOGZUDJZOGUGUUNU
|
|
UPKHUURUUNNSUURUUPNSKHVMUUEUUNUUPUURVNVOUUKYKYTYLUUCTYKXQYLUUDTUUOUUQRUUN
|
|
UUPNSUUPUUNNSUNUUEUUNUUPVQVOVRVSUUEWNYIUULUUFVIZYKWNYLUUATZYKYIYLUUBTWNYG
|
|
YIUUSYHXTXEWTYCVTMPWAUUGCOWBWCUUEWOWNWPUUHVMYKWOYLYSTUUTBACWFPVSXMWQXRYJY
|
|
LWNWOWQXBXLWDQWEWGYKXCYMYBXSYIYBUOZXAYAWTWHWIYKWNYBYNXCVIZUUAUVAWNYGYBUVB
|
|
YHXAYAWTYCWJMPWKWAWLWLWLWGWM $.
|
|
|
|
$( Apartness of positive reals is tight. (Contributed by Jim Kingdon,
|
|
28-Jan-2020.) $)
|
|
aptipr $p |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) )
|
|
-> A = B ) $=
|
|
( cnp wcel cltp wbr wo wn w3a wceq c1st cfv c2nd simp1 aptiprleml syl3anc
|
|
wss wa eqssd aptiprlemu simp2 ioran biimpi 3ad2ant3 simprd simpld 3adant3
|
|
wb preqlu mpbir2and ) ACDZBCDZABEFZBAEFZGHZIZABJZAKLZBKLZJZAMLZBMLZJZUPUR
|
|
USUPUKULUNHZURUSQUKULUONZUKULUOUAZUPUMHZVDUOUKVGVDRZULUOVHUMUNUBUCUDZUEZA
|
|
BOPUPULUKVGUSURQVFVEUPVGVDVIUFZBAOPSUPVAVBUPULUKVGVAVBQVFVEVKBATPUPUKULVD
|
|
VBVAQVEVFVJABTPSUKULUQUTVCRUHUOABUIUGUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Ordering property of multiplication. (Contributed by Jim Kingdon,
|
|
18-Feb-2020.) $)
|
|
ltmprr $p |- ( ( A e. P. /\ B e. P. /\ C e. P. ) ->
|
|
( ( C .P. A ) <P ( C .P. B ) -> A <P B ) ) $=
|
|
( vy vx cnp wcel cmp co cltp wbr wa cv c1p wceq syl2anc syl3anc mulcomprg
|
|
cpp oveq1d w3a wrex recexpr 3ad2ant3 adantr ltexpri simplll simp1d simprl
|
|
ad2antlr simplrl mulclpr ltaddpr simprr oveq2d distrprg mulassprg simplrr
|
|
simp3d eqtrd 1pr mpan2 1idpr eqtr3d syl 3eqtr2d 3eqtr3d breqtrd rexlimddv
|
|
simp2d ex ) AFGZBFGZCFGZUAZCAHIZCBHIZJKZABJKZVOVRLZCDMZHIZNOZVSDFVOWCDFUB
|
|
ZVRVNVLWDVMDCUCUDUEVTWAFGZWCLZLZVPEMZSIZVQOZVSEFVRWJEFUBVOWFEVPVQUFUJWGWH
|
|
FGZWJLZLZAAWAWHHIZSIZBJWMVLWNFGZAWOJKWMVLVMVNVOVRWFWLUGZUHZWMWEWKWPVTWEWC
|
|
WLUKZWGWKWJUIZWAWHULPAWNUMPWMWAWIHIZWAVQHIZWOBWMWIVQWAHWGWKWJUNUOWMXAWAVP
|
|
HIZWNSIZWACHIZAHIZWNSIWOWMWEVPFGZWKXAXDOWSWMVNVLXGWMVLVMVNWQUSZWRCAULPWTW
|
|
AVPWHUPQWMXFXCWNSWMWEVNVLXFXCOWSXHWRWACAUQQTWMXFAWNSWMXFNAHIZAWMXENAHWMXE
|
|
WBNWMWEVNXEWBOWSXHWACRPVTWEWCWLURUTZTWMVLXIAOWRVLANHIZXIAVLNFGZXKXIOVAANR
|
|
VBAVCVDVEUTTVFWMXEBHIZNBHIZXBBWMXENBHXJTWMWEVNVMXMXBOWSXHWMVLVMVNWQVJZWAC
|
|
BUQQWMVMXNBOXOVMBNHIZXNBVMXLXPXNOVABNRVBBVCVDVEVGVGVHVIVIVK $.
|
|
$}
|
|
|
|
${
|
|
$d A l u w x z $.
|
|
$( For any positive real, there is an integer that is greater than it.
|
|
This is also known as the "archimedean property". The integer ` x ` is
|
|
embedded into the reals as described at ~ nnprlu . (Contributed by Jim
|
|
Kingdon, 22-Apr-2020.) $)
|
|
archpr $p |- ( A e. P. -> E. x e. N. A <P
|
|
<. { l | l <Q [ <. x , 1o >. ] ~Q } ,
|
|
{ u | [ <. x , 1o >. ] ~Q <Q u } >. ) $=
|
|
( vz vw cnp wcel cv cfv cop cltq wbr cab cnpi wrex cnq c1st wa ad2antrr
|
|
c2nd c1o ceq cec cltp prop syl archnqq ad2antrl simprl simprr simpr breq1
|
|
vex ltnqex gtnqex op1st elab2 sylibr weq eleq1 anbi12d rspcev syl12anc wb
|
|
prmu simplll nnprlu ad2antlr ltdfpr syl2anc mpbird reximdva mpd rexlimddv
|
|
ex ) CGHZEIZCUAJZHZCDIZAIZUBKUCUDZLMZDNZWCBILMBNZKZUEMZAOPZEQVQCRJZVSKGHV
|
|
TEQPCUFEVSWJVFUGVQVRQHZVTSZSZVRWCLMZAOPZWIWKWOVQVTAVRUHUIWMWNWHAOWMWBOHZS
|
|
ZWNWHWQWNSZWHFIZVSHZWSWGRJZHZSZFQPZWRWKVTVRXAHZXDWMWKWPWNVQWKVTUJTWMVTWPW
|
|
NVQWKVTUKTWRWNXEWQWNULWDWNDVRXAEUNWAVRWCLUMWEWFDWCUOBWCUPUQURUSXCVTXESFVR
|
|
QFEUTWTVTXBXEWSVRVSVAWSVRXAVAVBVCVDWRVQWGGHZWHXDVEVQWLWPWNVGWPXFWMWNBWBDV
|
|
HVICWGFVJVKVLVPVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d L f g h $. $d Q f g h l u $. $d R f g h l u $. $d S f g h l u $.
|
|
$d f g h ph $.
|
|
caucvgprlemcanl.l $e |- ( ph -> L e. P. ) $.
|
|
caucvgprlemcanl.s $e |- ( ph -> S e. Q. ) $.
|
|
caucvgprlemcanl.r $e |- ( ph -> R e. Q. ) $.
|
|
caucvgprlemcanl.q $e |- ( ph -> Q e. Q. ) $.
|
|
$( Lemma for ~ cauappcvgprlemladdrl . Cancelling a term from both sides.
|
|
(Contributed by Jim Kingdon, 15-Aug-2020.) $)
|
|
caucvgprlemcanl $p |- ( ph -> (
|
|
( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } ,
|
|
{ u | ( S +Q Q ) <Q u } >. ) ) <->
|
|
R e. ( 1st ` ( L +P. <. { l | l <Q S } ,
|
|
{ u | S <Q u } >. ) ) ) ) $=
|
|
( cltq wbr cab cpp co cltp wcel cnp syl2anc vf vg vh cv cop c1st cfv cplq
|
|
w3a wb ltaprg adantl cnq nqprlu addclpr wa wceq addcomprg caovord2d nqprl
|
|
addnqpr oveq2d breq12d addclnq addassprg syl3anc breq2d 3bitr4d 3bitr4rd
|
|
syl ) AGUDZDLMGNDBUDZLMBNUEZFVKELMGNEVLLMBNUEZOPZQMZVMVKCLMGNCVLLMBNUEZOP
|
|
ZVOVQOPZQMZDVOUFUGRZDCUHPZFVKECUHPZLMGNWCVLLMBNUEZOPZUFUGRZAUAUBUCVMVOVQQ
|
|
SOUAUDZSRZUBUDZSRZUCUDZSRUIWGWIQMWKWGOPWKWIOPQMUJAWGWIWKUKULADUMRZVMSRJBD
|
|
GUNVJAFSRZVNSRZVOSRZHAEUMRZWNIBEGUNVJZFVNUOTZACUMRZVQSRZKBCGUNVJZWHWJUPWG
|
|
WIOPWIWGOPUQAWGWIURULUSAWLWOWAVPUJJWRBDVOGUTTAVKWBLMGNWBVLLMBNUEZWEQMZVRF
|
|
VNVQOPZOPZQMWFVTAXBVRWEXEQAWLWSXBVRUQJKBDCGVATAWDXDFOAWPWSWDXDUQIKBECGVAT
|
|
VBVCAWBUMRZWESRZWFXCUJAWLWSXFJKDCVDTAWMWDSRZXGHAWCUMRZXHAWPWSXIIKECVDTBWC
|
|
GUNVJFWDUOTBWBWEGUTTAVSXEVRQAWMWNWTVSXEUQHWQXAFVNVQVEVFVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d A p $.
|
|
cauappcvgpr.f $e |- ( ph -> F : Q. --> Q. ) $.
|
|
cauappcvgpr.app $e |- ( ph -> A. p e. Q. A. q e. Q. (
|
|
( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\
|
|
( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) ) $.
|
|
cauappcvgpr.bnd $e |- ( ph -> A. p e. Q. A <Q ( F ` p ) ) $.
|
|
${
|
|
$d L p q $. $d ph p q $.
|
|
cauappcvgpr.lim $e |- L = <.
|
|
{ l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } ,
|
|
{ u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
|
|
>. $.
|
|
|
|
${
|
|
$d L r s t f g h x y $. $d A p s x $. $d F l u p q r s t f g h x y $.
|
|
$d ph r s t f g h x y $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is inhabited.
|
|
(Contributed by Jim Kingdon, 18-Jul-2020.) $)
|
|
cauappcvgprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
|
|
/\ E. r e. Q. r e. ( 2nd ` L ) ) ) $=
|
|
( wcel cnq cplq c1q cltq wbr cv c1st cfv wrex c2nd co wceq wral fveq2
|
|
1nq breq2d rspcv mpsyl ltrelnq brel simpld halfnqq wa simplr ad2antrr
|
|
syl wi ad2antlr mpd wb breq1 adantl mpbird oveq2 breq12d rspcev oveq1
|
|
syl2anc breq1d crab cop fveq2i nqex rabex op1st eqtri elrab2 sylanbrc
|
|
rexbidv ex reximdva a1i ffvelrnd addclnq ltaddnq id breq2 op2nd eleq1
|
|
oveq12d jca ) AFUAZEUBUCZOZFPUDZGUAZEUEUCZOZGPUDZAWQWQQUFZCUGZFPUDZWT
|
|
ACPOZXGACRDUCZSTZXHRPOZACIUAZDUCZSTZIPUHZXJUJMXNXJIRPXLRUGXMXICSXLRDU
|
|
IUKULUMXJXHXIPOZCXIPPSUNUOUPVAFCUQVAAXFWSFPAWQPOZURZXFWSXRXFURZXQWQHU
|
|
AZQUFZXTDUCZSTZHPUDZWSAXQXFUSZXSXQXEWQDUCZSTZYDYEXSYGCYFSTZXSXOYHAXOX
|
|
QXFMUTXQXOYHVBAXFXNYHIWQPXLWQUGXMYFCSXLWQDUIUKULVCVDXFYGYHVEXRXECYFSV
|
|
FVGVHYCYGHWQPXTWQUGYAXEYBYFSXTWQWQQVIXTWQDUIVJVKVMJUAZXTQUFZYBSTZHPUD
|
|
ZYDJWQPWRYIWQUGZYKYCHPYMYJYAYBSYIWQXTQVLVNWDWRYLJPVOZYBXTQUFZBUAZSTZH
|
|
PUDZBPVOZVPZUBUCYNEYTUBNVQYNYSYLJPVRVSZYRBPVRVSZVTWAWBWCWEWFVDAXIRQUF
|
|
ZRQUFZPOZUUDXBOZXDAUUCPOZXKUUEAXPXKUUGAPPRDKXKAUJWGZWHUUHXIRWIVMZUUHU
|
|
UCRWIVMZAUUEYOUUDSTZHPUDZUUFUUJAXKUUCUUDSTZUULUUHAUUGXKUUMUUIUUHUUCRW
|
|
JVMUUKUUMHRPXTRUGZYOUUCUUDSUUNYBXIXTRQXTRDUIUUNWKWOVNVKVMYRUULBUUDPXB
|
|
YPUUDUGYQUUKHPYPUUDYOSWLWDXBYTUEUCYSEYTUENVQYNYSUUAUUBWMWAWBWCXCUUFGU
|
|
UDPXAUUDXBWNVKVMWP $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The lower cut of the putative limit is
|
|
open. (Contributed by Jim Kingdon, 4-Aug-2020.) $)
|
|
cauappcvgprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
|
|
E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) $=
|
|
( wa cplq cltq wbr cnq wrex vt cv c1st cfv wcel co wceq oveq1 rexbidv
|
|
breq1d crab cop fveq2i rabex op1st eqtri elrab2 simprbi adantl simprr
|
|
nqex ltbtwnnqq sylib simplrl simplbi ad3antlr ltaddnq syl2anc breqtrd
|
|
addcomnqg simprrl ltsonq ltrelnq sotri wb simprl mpbid ad2antrr simpr
|
|
ltexnqq breqtrrd simplr ltanqg syl3anc mpbird simprrr eqtr3d sylanbrc
|
|
rspe jca ex reximdva mpd rexlimddv ) AFUBZEUCUDZUEZOZWOHUBZPUFZWSDUDZ
|
|
QRZWOGUBZQRZXCWPUEZOZGSTZHSWQXBHSTZAWQWOSUEZXHJUBZWSPUFZXAQRZHSTZXHJW
|
|
OSWPXJWOUGZXLXBHSXNXKWTXAQXJWOWSPUHUJUIWPXMJSUKZXAWSPUFBUBQRHSTZBSUKZ
|
|
ULZUCUDXOEXRUCNUMXOXQXMJSVAUNXPBSVAUNUOUPZUQZURUSWRWSSUEZXBOZOZWTUAUB
|
|
ZQRZYDXAQRZOZXGUASYCXBYGUASTWRYAXBUTUAWTXAVBVCYCYDSUEZYGOZOZWSXCPUFZY
|
|
DUGZGSTZXGYJWSYDQRZYMYJWSWTQRYEYNYJWSWSWOPUFZWTQYJYAXIWSYOQRWRYAXBYIV
|
|
DZWQXIAYBYIWQXIXHXTVEVFZWSWOVGVHYJYAXIYOWTUGYPYQWSWOVJVHVIYCYHYEYFVKZ
|
|
WSWTYDQSVLVMVNVHYJYAYHYNYMVOYPYCYHYGVPGWSYDVTVHVQYJYLXFGSYJXCSUEZOZYL
|
|
XFYTYLOZXDXEUUAXDYOYKQRZUUAYOYDYKQUUAYEYOYDQRYJYEYSYLYRVRUUAWTYOYDQUU
|
|
AXIYAWTYOUGYJXIYSYLYQVRZYJYAYSYLYPVRZWOWSVJVHUJVQYTYLVSZWAUUAXIYSYAXD
|
|
UUBVOUUCYJYSYLWBZUUDWOXCWSWCWDWEUUAYSXCWSPUFZXAQRZHSTZXEUUFUUAYAUUHUU
|
|
IUUDUUAUUHYFYJYFYSYLYCYHYEYFWFVRUUAUUGYDXAQUUAYKUUGYDUUAYAYSYKUUGUGUU
|
|
DUUFWSXCVJVHUUEWGUJWEUUHHSWIVHXMUUIJXCSWPXJXCUGZXLUUHHSUUJXKUUGXAQXJX
|
|
CWSPUHUJUIXSUQWHWJWKWLWMWNWN $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The lower cut of the putative limit is
|
|
lower. (Contributed by Jim Kingdon, 4-Aug-2020.) $)
|
|
cauappcvgprlemlol $p |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) ->
|
|
s e. ( 1st ` L ) ) $=
|
|
( cv cltq wbr wcel cnq cplq vf vg vh c1st cfv w3a wrex ltrelnq simpld
|
|
co brel 3ad2ant2 weq oveq1 breq1d rexbidv crab cop fveq2i rabex op1st
|
|
nqex eqtri elrab2 simprbi 3ad2ant3 wa simpll2 wb ltanqg adantl simprd
|
|
ad2antrr simplr addcomnqg caovord2d mpbid ltsonq sylancom ex reximdva
|
|
wceq sotri mpd sylanbrc ) AFOZGOZPQZWGEUDUEZRZUFZWFSRZWFHOZTUJZWMDUEZ
|
|
PQZHSUGZWFWIRWHAWLWJWHWLWGSRZWFWGSSPUHUKZUIULZWKWGWMTUJZWOPQZHSUGZWQW
|
|
JAXCWHWJWRXCJOZWMTUJZWOPQZHSUGZXCJWGSWIJGUMZXFXBHSXHXEXAWOPXDWGWMTUNU
|
|
OUPWIXGJSUQZWOWMTUJBOPQHSUGZBSUQZURZUDUEXIEXLUDNUSXIXKXGJSVBUTXJBSVBU
|
|
TVAVCZVDVEVFWKXBWPHSWKWMSRZVGZXBWPXOXBWNXAPQZWPXOXBVGZWHXPAWHWJXNXBVH
|
|
XQUAUBUCWFWGWMPSTUAOZSRZUBOZSRZUCOZSRUFXRXTPQYBXRTUJYBXTTUJPQVIXQXRXT
|
|
YBVJVKWKWLXNXBWTVMWKWRXNXBWHAWRWJWHWLWRWSVLULVMWKXNXBVNXSYAVGXRXTTUJX
|
|
TXRTUJWBXQXRXTVOVKVPVQWNXAWOPSVRUHWCVSVTWAWDXGWQJWFSWIJFUMZXFWPHSYCXE
|
|
WNWOPXDWFWMTUNUOUPXMVDWE $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The upper cut of the putative limit is
|
|
open. (Contributed by Jim Kingdon, 4-Aug-2020.) $)
|
|
cauappcvgprlemopu $p |- ( ( ph /\ r e. ( 2nd ` L ) ) ->
|
|
E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) $=
|
|
( cv wcel wa cltq cnq wrex c2nd cfv cplq co wbr weq breq2 rexbidv cop
|
|
crab fveq2i nqex rabex op2nd eqtri elrab2 simprbi adantl simprr sylib
|
|
ltbtwnnqq simplr simplrl adantr simprl rspe syl2anc sylanbrc reximdva
|
|
jca ex mpd rexlimddv ) AGOZEUAUBZPZQZHOZDUBZVRUCUDZVNRUEZFOZVNRUEZWBV
|
|
OPZQZFSTZHSVPWAHSTZAVPVNSPWGVTBOZRUEZHSTZWGBVNSVOBGUFWIWAHSWHVNVTRUGU
|
|
HVOJOVRUCUDVSRUEHSTZJSUJZWJBSUJZUIZUAUBWMEWNUANUKWLWMWKJSULUMWJBSULUM
|
|
UNUOZUPUQURVQVRSPZWAQQZVTWBRUEZWCQZFSTZWFWQWAWTVQWPWAUSFVTVNVAUTWQWSW
|
|
EFSWQWBSPZQZWSWEXBWSQZWCWDXBWRWCUSXCXAWRHSTZWDWQXAWSVBXCWPWRXDXBWPWSV
|
|
QWPWAXAVCVDXBWRWCVEWRHSVFVGWJXDBWBSVOBFUFWIWRHSWHWBVTRUGUHWOUPVHVJVKV
|
|
IVLVM $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The upper cut of the putative limit is
|
|
upper. (Contributed by Jim Kingdon, 4-Aug-2020.) $)
|
|
cauappcvgprlemupu $p |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) ->
|
|
r e. ( 2nd ` L ) ) $=
|
|
( cv cltq wbr wcel cnq wrex c2nd cfv w3a cplq ltrelnq simprd 3ad2ant2
|
|
brel weq breq2 rexbidv crab cop fveq2i nqex rabex op2nd eqtri simprbi
|
|
co elrab2 3ad2ant3 wi ltsonq sotri expcom reximdv mpd sylanbrc ) AFOZ
|
|
GOZPQZVJEUAUBZRZUCZVKSRZHOZDUBZVQUDUTZVKPQZHSTZVKVMRVLAVPVNVLVJSRZVPV
|
|
JVKSSPUEUHUFUGVOVSVJPQZHSTZWAVNAWDVLVNWBWDVSBOZPQZHSTZWDBVJSVMBFUIWFW
|
|
CHSWEVJVSPUJUKVMJOVQUDUTVRPQHSTZJSULZWGBSULZUMZUAUBWJEWKUANUNWIWJWHJS
|
|
UOUPWGBSUOUPUQURZVAUSVBVOWCVTHSVLAWCVTVCVNWCVLVTVSVJVKPSVDUEVEVFUGVGV
|
|
HWGWABVKSVMBGUIWFVTHSWEVKVSPUJUKWLVAVI $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is rounded.
|
|
(Contributed by Jim Kingdon, 18-Jul-2020.) $)
|
|
cauappcvgprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
|
|
E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\
|
|
A. r e. Q. ( r e. ( 2nd ` L ) <->
|
|
E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) ) $=
|
|
( cv cfv wcel wa cnq wrex c1st cltq wb wral c2nd cauappcvgprlemopl ex
|
|
cauappcvgprlemlol 3expib rexlimdvw impbid ralrimivw cauappcvgprlemopu
|
|
wbr cauappcvgprlemupu jca ) AFOZEUAPZQZUQGOZUBUNZUTURQZRZGSTZUCZFSUDU
|
|
TEUEPZQZVAUQVFQZRZFSTZUCZGSUDAVEFSAUSVDAUSVDABCDEFGHIJKLMNUFUGAVCUSGS
|
|
AVAVBUSABCDEFGHIJKLMNUHUIUJUKULAVKGSAVGVJAVGVJABCDEFGHIJKLMNUMUGAVIVG
|
|
FSAVAVHVGABCDEFGHIJKLMNUOUIUJUKULUP $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is disjoint.
|
|
(Contributed by Jim Kingdon, 18-Jul-2020.) $)
|
|
cauappcvgprlemdisj $p |- ( ph -> A. s e. Q.
|
|
-. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
|
|
( wcel wa cnq cplq co cltq wbr vf vg vh cv c1st cfv c2nd wn wrex wfal
|
|
wral simpl ralimi syl adantr weq oveq1 breq1d rexbidv crab cop fveq2i
|
|
nqex rabex op1st eqtri elrab2 simprbi oveq2 fveq2 breq12d sylib breq2
|
|
cbvrexv anim12i reeanv sylibr adantl r19.29d2r simprl jca wi ad3antlr
|
|
op2nd simplbi simplr addclnq syl2anc ad3antrrr ffvelrnd simpr wor w3a
|
|
wf ltsonq sotr mpan syl3anc syl5 simprr jcad wceq addcomnqg addassnqg
|
|
a1i caov12d anbi1d sylibd ltanqg sylibrd so2nr pm2.21d syld rexlimdva
|
|
wb mpd inegd ralrimivw ) AFUDZEUEUFZNZXSEUGUFZNZOZUHFPAYDAYDOZHUDZDUF
|
|
ZGUDZDUFZYFYHQRZQRZSTZXSYFQRZYGSTZYIYHQRZXSSTZOZOZGPUIZHPUIUJYEYLYQHG
|
|
PPAYLGPUKZHPUKZYDAYLYIYGYJQRSTZOZGPUKZHPUKUUAKUUDYTHPUUCYLGPYLUUBULUM
|
|
UMUNUOYDYQGPUIHPUIZAYDYNHPUIZYPGPUIZOUUEYAUUFYCUUGYAXSYHQRZYISTZGPUIZ
|
|
UUFYAXSPNZUUJIUDZYHQRZYISTZGPUIZUUJIXSPXTIFUPZUUNUUIGPUUPUUMUUHYISUUL
|
|
XSYHQUQURUSXTUUOIPUTZYOBUDZSTZGPUIZBPUTZVAZUEUFUUQEUVBUEMVBUUQUVAUUOI
|
|
PVCVDZUUTBPVCVDZVEVFVGZVHUUIYNGHPGHUPUUHYMYIYGSYHYFXSQVIYHYFDVJVKVNVL
|
|
YCUUKUUGUUTUUGBXSPYBBFUPUUSYPGPUURXSYOSVMUSYBUVBUGUFUVAEUVBUGMVBUUQUV
|
|
AUVCUVDWDVFVGVHVOYNYPHGPPVPVQVRVSYEYSUJHPYEYFPNZOZYRUJGPUVGYHPNZOZYRX
|
|
SYOSTZYPOZUJUVIYRYFXSQRZYFYOQRZSTZYPOZUVKUVIYRYMYKSTZYPOUVOUVIYRUVPYP
|
|
YRYNYLOZUVIUVPYRYNYLYLYNYPVTYLYQULWAUVIYMPNZYGPNZYKPNZUVQUVPWBZUVIUUK
|
|
UVFUVRYDUUKAUVFUVHYAUUKYCYAUUKUUJUVEWEUOWCZYEUVFUVHWFZXSYFWGWHUVIPPYF
|
|
DAPPDWNYDUVFUVHJWIZUWCWJUVIYIPNZYJPNZUVTUVIPPYHDUWDUVGUVHWKZWJZUVIUVF
|
|
UVHUWFUWCUWGYFYHWGWHYIYJWGWHPSWLZUVRUVSUVTWMUWAWOPYMYGYKSWPWQWRWSYRYP
|
|
WBUVIYLYNYPWTXEXAUVIUVPUVNYPUVIYMUVLYKUVMSUVIUUKUVFYMUVLXBUWBUWCXSYFX
|
|
CWHUVIUAUBUCYIYFYHPQUWHUWCUWGUAUDZPNZUBUDZPNZOUWJUWLQRZUWLUWJQRXBUVIU
|
|
WJUWLXCVRUWKUWMUCUDZPNWMUWNUWOQRUWJUWLUWOQRQRXBUVIUWJUWLUWOXDVRXFVKXG
|
|
XHUVIUVJUVNYPUVIUUKYOPNZUVFUVJUVNXOUWBUVIUWEUVHUWPUWHUWGYIYHWGWHZUWCX
|
|
SYOYFXIWRXGXJUVIUVKUJUVIUUKUWPUVKUHZUWBUWQUWIUUKUWPOUWRWOPXSYOSXKWQWH
|
|
XLXMXNXNXPXQXR $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is located.
|
|
(Contributed by Jim Kingdon, 18-Jul-2020.) $)
|
|
cauappcvgprlemloc $p |- ( ph -> A. s e. Q. A. r e. Q.
|
|
( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) $=
|
|
( cltq wbr wcel cnq cplq co vy vx vf vg vh cv c1st c2nd wo wi wa wceq
|
|
wrex ltexnqi adantl subhalfnqq ad2antrl simprr simplrl adantr syl2anc
|
|
cfv ltanqi simplrr breqtrd simprl addclnq wf ad4antr ffvelrnd wor w3a
|
|
ltsonq sowlin mpan syl3anc simpr addassnqg breq1d mpbird wb addcomnqg
|
|
ltanqg caovord2d oveq2 fveq2 breq12d rspcev oveq1 rexbidv crab fveq2i
|
|
mpd cop nqex rabex op1st eqtri elrab2 sylanbrc ex oveq12d sylan breq2
|
|
id op2nd orim12d rexlimddv ralrimivva ) AFUFZGUFZOPZXJEUGVBZQZXKEUHVB
|
|
ZQZUIZUJFGRRAXJRQZXKRQZUKZUKZXLXQYAXLUKZXJUAUFZSTZXKULZXQUARXLYEUARUM
|
|
YAUAXJXKUNUOYBYCRQZYEUKZUKZUBUFZYISTZYCOPZXQUBRYFYKUBRUMYBYEUBYCUPUQY
|
|
HYIRQZYKUKZUKZXJYJSTZYIDVBZYISTZOPZYQXKOPZUIZXQYNYOXKOPZYTYNYOYDXKOYN
|
|
YKXRYOYDOPYHYLYKURYHXRYMYBXRYGAXRXSXLUSUTUTZYJYCXJVCVAYBYFYEYMVDVEYNY
|
|
ORQZXSYQRQZUUAYTUJZYNXRYJRQZUUCUUBYNYLYLUUFYHYLYKVFZUUGYIYIVGVAXJYJVG
|
|
VAYHXSYMYBXSYGAXRXSXLVDUTUTZYNYPRQZYLUUDYNRRYIDARRDVHXTXLYGYMKVIUUGVJ
|
|
ZUUGYPYIVGVAROVKUUCXSUUDVLUUEVMRYOXKYQOVNVOVPWMYNYRXNYSXPYNYRXNYNYRUK
|
|
ZXRXJHUFZSTZUULDVBZOPZHRUMZXNYNXRYRUUBUTZUUKYLXJYISTZYPOPZUUPYHYLYKYR
|
|
USZUUKUUSUURYISTZYQOPZUUKUVBYRYNYRVQUUKUVAYOYQOUUKXRYLYLUVAYOULUUQUUT
|
|
UUTXJYIYIVRVPVSVTUUKUCUDUEUURYPYIORSUCUFZRQZUDUFZRQZUEUFZRQVLUVCUVEOP
|
|
UVGUVCSTUVGUVESTOPWAUUKUVCUVEUVGWCUOUUKXRYLUURRQUUQUUTXJYIVGVAYNUUIYR
|
|
UUJUTUUTUVDUVFUKUVCUVESTUVEUVCSTULUUKUVCUVEWBUOWDVTUUOUUSHYIRUULYIULZ
|
|
UUMUURUUNYPOUULYIXJSWEUULYIDWFZWGWHVAJUFZUULSTZUUNOPZHRUMZUUPJXJRXMUV
|
|
JXJULZUVLUUOHRUVNUVKUUMUUNOUVJXJUULSWIVSWJXMUVMJRWKZUUNUULSTZBUFZOPZH
|
|
RUMZBRWKZWNZUGVBUVOEUWAUGNWLUVOUVTUVMJRWOWPZUVSBRWOWPZWQWRWSWTXAYNYSX
|
|
PYNYSUKXSUVPXKOPZHRUMZXPYNXSYSUUHUTYNYLYSUWEUUGUWDYSHYIRUVHUVPYQXKOUV
|
|
HUUNYPUULYISUVIUVHXEXBVSWHXCUVSUWEBXKRXOUVQXKULUVRUWDHRUVQXKUVPOXDWJX
|
|
OUWAUHVBUVTEUWAUHNWLUVOUVTUWBUWCXFWRWSWTXAXGWMXHXHXAXI $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is a positive real.
|
|
(Contributed by Jim Kingdon, 20-Jun-2020.) $)
|
|
cauappcvgprlemcl $p |- ( ph -> L e. P. ) $=
|
|
( vs vr cnq wcel cv wrex wa wral cpw cxp c1st cfv c2nd cltq wbr wb wn
|
|
wo wi w3a cnp cauappcvgprlemm cplq co crab cop wss ssrab2 elpw2 mpbir
|
|
nqex opelxpi mp2an eqeltri jctil cauappcvgprlemrnd cauappcvgprlemdisj
|
|
cauappcvgprlemloc 3jca elnp1st2nd sylanbrc ) AEOUAZVNUBZPZMQZEUCUDZPZ
|
|
MORNQZEUEUDZPZNORSZSVSVQVTUFUGZVTVRPSNORUHMOTWBWDVQWAPZSMORUHNOTSZVSW
|
|
ESUIMOTZWDVSWBUJUKNOTMOTZULEUMPAWCVPABCDEMNFGHIJKLUNEHQFQZUOUPWIDUDZU
|
|
FUGFORZHOUQZWJWIUOUPBQUFUGFORZBOUQZURZVOLWLVNPZWNVNPZWOVOPWPWLOUSWKHO
|
|
UTWLOVCVAVBWQWNOUSWMBOUTWNOVCVAVBWLWNVNVNVDVEVFVGAWFWGWHABCDEMNFGHIJK
|
|
LVHABCDEMFGHIJKLVIABCDEMNFGHIJKLVJVKENMVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d F f g h l u p q r s t v $. $d L f g h q r s t $.
|
|
$d L f g h s t v x y $. $d S f g h l q r s t u $.
|
|
$d S f g h l s t u v x y $. $d ph f g h q r s t v $.
|
|
cauappcvgprlemladd.s $e |- ( ph -> S e. Q. ) $.
|
|
$( Lemma for ~ cauappcvgprlemladd . The forward subset relationship
|
|
for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) $)
|
|
cauappcvgprlemladdfu $p |- ( ph ->
|
|
( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
|
|
( 2nd ` <.
|
|
{ l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
|
|
{ u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
|
|
>. ) ) $=
|
|
( cltq wbr co cplq cnq wcel vr vs vt vf vg vh vx cab cop cpp c2nd cfv
|
|
vy cv wrex crab wa cnp wb cauappcvgprlemcl nqprlu syl df-iplp addclnq
|
|
genpelvu syl2anc biimpa breq2 rexbidv fveq2i rabex op2nd eqtri elrab2
|
|
wceq nqex biimpi adantr adantl simpld vex ltnqex gtnqex elab2 ltrelnq
|
|
brel sylbi simprd ad2antll mpbird ad2antrr ad5antr wf simplr ffvelrnd
|
|
eleq1 ltanqg syl3anc mpbid simpr w3a addcomnqg caovord2d ltsonq sotri
|
|
simpllr breqtrrd ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AUAFIUNZ
|
|
DOPIUHZDBUNZOPZBUHZUIZUJQUKULZXNGUNZRQZYAEULZDRQOPGSUOZISUPZYCYARQZDR
|
|
QZXPOPZGSUOZBSUPZUIUKULZAUAUNZXTTZYLYKTZAYMUQZYLUBUNZUCUNZRQZVOZUCXSU
|
|
KULZUOUBFUKULZUOZYNAYMUUBAFURTXSURTZYMUUBUSABCEFGHIJKLMUTADSTZUUCNBDI
|
|
VAVBUDUEUFUGUMFXSYLUBUCUJRUGUMUFUEUDVCUEUNZUFUNZVDVEVFVGYOYSYNUBUCUUA
|
|
YTYOYPUUATZYQYTTZUQZUQZYSYNUUJYSUQZYLSTZYGYLOPZGSUOZYNUUKUULYRSTZUUKY
|
|
PSTZYQSTZUUOUUKUUPYFYPOPZGSUOZUUJUUPUUSUQZYSUUIUUTYOUUGUUTUUHUUGUUTYF
|
|
XPOPZGSUOZUUSBYPSUUAXPYPVOUVAUURGSXPYPYFOVHVIUUAYBYCOPGSUOZISUPZUVBBS
|
|
UPZUIZUKULUVEFUVFUKMVJUVDUVEUVCISVPVKUVBBSVPVKVLVMVNVQVRVSVRZVTZUUJUU
|
|
QYSUUHUUQYOUUGUUHUUDUUQUUHDYQOPZUUDUUQUQXQUVIBYQYTUCWAXPYQDOVHXOXRIDW
|
|
BBDWCVLWDZDYQSSOWEWFWGWHWIVRZYPYQVDVFYSUULUUOUSUUJYLYRSWPVSWJUUKUUSUU
|
|
NUUKUUPUUSUVGWHUUKUURUUMGSUUKYASTZUQZUURUUMUVMUURUQZYGYRYLOUVNYGYFYQR
|
|
QZOPZUVOYROPZYGYROPUVNUVIUVPUUKUVIUVLUURUUJUVIYSUUHUVIYOUUGUUHUVIUVJV
|
|
QWIVRWKUVNUUDUUQYFSTZUVIUVPUSAUUDYMUUIYSUVLUURNWLUUKUUQUVLUURUVKWKZUV
|
|
NYCSTUVLUVRUVNSSYAEASSEWMYMUUIYSUVLUURJWLUUKUVLUURWNZWOUVTYCYAVDVFZDY
|
|
QYFWQWRWSUVNUURUVQUVMUURWTUVNUDUEUFYFYPYQOSRUDUNZSTZUUESTZUUFSTXAUWBU
|
|
UEOPUUFUWBRQUUFUUERQOPUSUVNUWBUUEUUFWQVSUWAUUKUUPUVLUURUVHWKUVSUWCUWD
|
|
UQUWBUUERQUUEUWBRQVOUVNUWBUUEXBVSXCWSYGUVOYROSXDWEXEVFUUJYSUVLUURXFXG
|
|
XHXIXJYIUUNBYLSYKXPYLVOYHUUMGSXPYLYGOVHVIYEYJYDISVPVKYIBSVPVKVLVNXKXH
|
|
XLXJXHXM $.
|
|
|
|
$( Lemma for ~ cauappcvgprlemladd . The forward subset relationship
|
|
for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) $)
|
|
cauappcvgprlemladdfl $p |- ( ph ->
|
|
( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
|
|
( 1st ` <.
|
|
{ l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
|
|
{ u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
|
|
>. ) ) $=
|
|
( cltq co cplq cnq wcel wa vr vs vt vf vg vh vx vy cv wbr cab cop cpp
|
|
c1st cfv wrex crab cnp wb cauappcvgprlemcl nqprlu syl df-iplp addclnq
|
|
genpelvl syl2anc biimpa oveq1 breq1d rexbidv fveq2i rabex op1st eqtri
|
|
wceq nqex elrab2 biimpi ad2antrl adantr simpld vex breq1 ltnqex elab2
|
|
gtnqex ad2antll ltrelnq eleq1 adantl mpbird simprd ad2antrr addcomnqg
|
|
simplr w3a addassnqg caov32d simpr wi lt2addnq syl2anr mp2and adantlr
|
|
brel eqbrtrrd ad3antlr ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AU
|
|
AFIUIZDOUJZIUKZDBUIZOUJBUKZULZUMPUNUOZXNGUIZQPZYAEUOZDQPZOUJZGRUPZIRU
|
|
QZYCYAQPZDQPXQOUJGRUPZBRUQZULUNUOZAUAUIZXTSZYLYKSZAYMTZYLUBUIZUCUIZQP
|
|
ZVOZUCXSUNUOZUPUBFUNUOZUPZYNAYMUUBAFURSXSURSZYMUUBUSABCEFGHIJKLMUTADR
|
|
SZUUCNBDIVAVBUDUEUFUGUHFXSYLUBUCUMQUGUHUFUEUDVCUEUIZUFUIZVDVEVFVGYOYS
|
|
YNUBUCUUAYTYOYPUUASZYQYTSZTTZYSYNUUIYSTZYLRSZYLYAQPZYDOUJZGRUPZYNUUJU
|
|
UKYRRSZUUJYPRSZYQRSZUUOUUJUUPYPYAQPZYCOUJZGRUPZUUIUUPUUTTZYSUUGUVAYOU
|
|
UHUUGUVAYBYCOUJZGRUPZUUTIYPRUUAXNYPVOZUVBUUSGRUVDYBUURYCOXNYPYAQVHVIV
|
|
JUUAUVCIRUQZYHXQOUJGRUPZBRUQZULZUNUOUVEFUVHUNMVKUVEUVGUVCIRVPVLUVFBRV
|
|
PVLVMVNVQVRVSVTZWAZUUJUUQUUDUUJYQDOUJZUUQUUDTZUUIUVKYSUUHUVKYOUUGUUHU
|
|
VKXOUVKIYQYTUCWBXNYQDOWCXPXRIDWDBDWFVMWEVRWGZVTYQDRROWHXEVBZWAZYPYQVD
|
|
VFYSUUKUUOUSUUIYLYRRWIWJWKUUJUUTUUNUUJUUPUUTUVIWLUUJUUSUUMGRUUJYARSZT
|
|
ZUUSUUMUVQUUSTZUUMYRYAQPZYDOUJZUVRUURYQQPZUVSYDOUVRUDUEUFYPYAYQRQUUJU
|
|
UPUVPUUSUVJWMUUJUVPUUSWOUUJUUQUVPUUSUVOWMUDUIZRSZUUERSZTUWBUUEQPZUUEU
|
|
WBQPVOUVRUWBUUEWNWJUWCUWDUUFRSWPUWEUUFQPUWBUUEUUFQPQPVOUVRUWBUUEUUFWQ
|
|
WJWRUUJUUSUWAYDOUJZUVPUUJUUSTUUSUVKUWFUUJUUSWSUUIUVKYSUUSUVMWMUUSUURR
|
|
SYCRSTUVLUUSUVKTUWFWTUUJUURYCRROWHXEUVNUURYCYQDXAXBXCXDXFYSUUMUVTUSUU
|
|
IUVPUUSYSUULUVSYDOYLYRYAQVHVIXGWKXHXIXJYFUUNIYLRYKXNYLVOZYEUUMGRUWGYB
|
|
UULYDOXNYLYAQVHVIVJYGYJYFIRVPVLYIBRVPVLVMVQXKXHXLXJXHXM $.
|
|
|
|
$( Lemma for ~ cauappcvgprlemladd . The reverse subset relationship
|
|
for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) $)
|
|
cauappcvgprlemladdru $p |- ( ph ->
|
|
( 2nd ` <.
|
|
{ l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
|
|
{ u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
|
|
>. ) C_
|
|
( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) $=
|
|
( cplq co cltq wbr cnq wcel vr vv vf vg vh cfv wrex crab cop c2nd cab
|
|
cv cpp weq breq2 rexbidv nqex rabex op2nd elrab2 cauappcvgprlemladdfl
|
|
wa c1st wn wss oveq2 fveq2 oveq1d breq12d cbvrexv a1i rabbiia oveq12d
|
|
wb id breq1d opeq12i fveq2i syl6sseq adantr wceq wf ad2antrr ffvelrnd
|
|
simplr addassnqg syl3anc addclnq syl2anc sylancom eqeltrrd wor ltsonq
|
|
simpr so2nr mpan addcomnqg adantl w3a caov32d ltanqg caovord2d biimpd
|
|
3bitr2d wral oveq1 oveq2d breq2d anbi12d ralbidv rspcv mpd rsp simpld
|
|
wi sylc breqtrd jctird mtod nrexdv ffvelrnda elrab3 syl mtbird eleq2i
|
|
op1st sylnibr ssneldd adantlr wo cauappcvgprlemcl nqprlu addclpr prop
|
|
cnp prloc sylan orcomd ecased ex rexlimdva expimpd syl5bi ssrdv ) AUA
|
|
IULZGULZOPZUUFEUFZDOPZQRZGSUGZISUHZUUHUUFOPZDOPZBULZQRZGSUGZBSUHZUIZU
|
|
JUFZFUUEDQRIUKDUUOQRBUKUIZUMPZUJUFZUAULZUUTTUVDSTZUUNUVDQRZGSUGZVBAUV
|
|
DUVCTZUUQUVGBUVDSUUTBUAUNUUPUVFGSUUOUVDUUNQUOUPUULUURUUKISUQURUUQBSUQ
|
|
URUSUTAUVEUVGUVHAUVEVBZUVFUVHGSUVIUUFSTZVBZUVFUVHUVKUVFVBZUVHUUNUVBVC
|
|
UFZTZUVKUVNVDZUVFAUVJUVOUVEAUVJVBZUVMUUEUBULZOPZUVQEUFZDOPZQRZUBSUGZI
|
|
SUHZUVSUVQOPZDOPZUUOQRZUBSUGZBSUHZUIZVCUFZUUNAUVMUWJVEUVJAUVMUUSVCUFU
|
|
WJABCDEFGHIJKLMNVAUUSUWIVCUULUWCUURUWHUUKUWBISUUKUWBVNUUESTUUJUWAGUBS
|
|
GUBUNZUUGUVRUUIUVTQUUFUVQUUEOVFUWKUUHUVSDOUUFUVQEVGZVHVIVJVKVLUUQUWGB
|
|
SUUQUWGVNUUOSTUUPUWFGUBSUWKUUNUWEUUOQUWKUUMUWDDOUWKUUHUVSUUFUVQOUWLUW
|
|
KVOVMVHVPVJVKVLVQVRVSVTUVPUUNUWCTZUUNUWJTUVPUWMUUNUVQOPZUVTQRZUBSUGZU
|
|
VPUWOUBSUVPUVQSTZVBZUWOUUHUUFUVQOPZOPZUVSQRZUVSUWTQRZVBZUWRUWTSTZUVSS
|
|
TZUXCVDZUWRUUMUVQOPZUWTSUWRUUHSTZUVJUWQUXGUWTWAUWRSSUUFEASSEWBUVJUWQJ
|
|
WCZAUVJUWQWEZWDZUXJUVPUWQWNZUUHUUFUVQWFWGZUVPUWQUUMSTZUXGSTUWRUXHUVJU
|
|
XNUXKUXJUUHUUFWHZWIZUUMUVQWHWJZWKUWRSSUVQEUXIUXLWDZSQWLUXDUXEVBUXFWMS
|
|
UWTUVSQWOWPWIUWRUWOUXAUXBUWRUWOUXAUWRUWOUXGDOPZUVTQRUXGUVSQRUXAUWRUWN
|
|
UXSUVTQUWRUCUDUEUUMDUVQSOUXPADSTZUVJUWQNWCZUXLUCULZSTZUDULZSTZVBUYBUY
|
|
DOPZUYDUYBOPWAUWRUYBUYDWQWRZUYCUYEUEULZSTWSZUYFUYHOPUYBUYDUYHOPOPWAUW
|
|
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${
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|
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for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) $)
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|
cauappcvgprlemladdrl $p |- ( ph ->
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>. ) C_
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|
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|
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|
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YKUYBUYPUYFUVQUYHUXTQUYEUVPEWOZUYPUYGUXPUUFOUYEUVPUUDOUOZWHWPUYPUYJ
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KUVGUVNUUGUUCUVPOPZUVQUVHOPQRUBSUHZISUIZUVSUULQRZUBSUHZBSUIZUJVBUGZ
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NVUMVUFUYGUYFOVUOWHWPWSXMXNXOACUYFQRHSWNUURUVEUUTLVFFUUEUUFQRZGSUHZ
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UVPUUCOXLUUDUVPEWOZWPXPYBXQVUTVVFBSVUTVVFXCUULSTVUSVVEGUBSVVHUUKUVR
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UULQVVHUUFUVQUUDUVPOVVIVVHXRWQUPXPYBXQXSXTUVGUXGUVEUVHSTZUXIUWRDUUD
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YRYSYTUUAUUB $.
|
|
$}
|
|
|
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${
|
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$d S p $.
|
|
|
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$( Lemma for ~ cauappcvgpr . This takes ` L ` and offsets it by the
|
|
positive fraction ` S ` . (Contributed by Jim Kingdon,
|
|
23-Jun-2020.) $)
|
|
cauappcvgprlemladd $p |- ( ph ->
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|
( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <.
|
|
{ l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } ,
|
|
{ u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
|
|
>. ) $=
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( co cfv cnq wceq wcel syl cltq wbr cab cop cpp cplq wrex crab c1st
|
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c2nd cauappcvgprlemladdfl cauappcvgprlemladdrl cauappcvgprlemladdfu
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|
cv eqssd cauappcvgprlemladdru wa wb cauappcvgprlemcl nqprlu addclpr
|
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cnp syl2anc cpw cxp npsspw sseli 1st2nd2 ssrab2 elpw2 mpbir opelxpi
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wss nqex mp2an mp1i eqeq12d xp1st xp2nd opthg bitrd mpbir2and ) AFI
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|
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FODUFOWDUAUBGQUGZBQUHZUDZRZWFUIPZWMUIPZRZWFUJPZWMUJPZRZAWOWPABCDEFG
|
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WFVBSZWNWQWTUQZURAFVBSWEVBSZXAABCEFGHIJKLMUSADQSXCNBDIUTTFWEVAVCXAW
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NWOWRUDZWPWSUDZRZXBXAWFXDWMXEXAWFQVDZXGVEZSZWFXDRVBXHWFVFVGZWFXGXGV
|
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HTWMXHSZWMXERXAWJXGSZWLXGSZXKXLWJQVMWIIQVIWJQVNVJVKXMWLQVMWKBQVIWLQ
|
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VNVJVKWJWLXGXGVLVOWMXGXGVHVPVQXAWOXGSZWRXGSZXFXBURXAXIXNXJWFXGXGVRT
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XAXIXOXJWFXGXGVSTWOWRWPWSXGXGVTVCWATWB $.
|
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$}
|
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$}
|
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|
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${
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$d F f g h p q l u x $. $d L q x $. $d Q f g h p q l u x $.
|
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$d R f g h p q l u x $. $d ph f g h q x $.
|
|
cauappcvgprlem.q $e |- ( ph -> Q e. Q. ) $.
|
|
cauappcvgprlem.r $e |- ( ph -> R e. Q. ) $.
|
|
$( Lemma for ~ cauappcvgpr . Part of showing the putative limit to be
|
|
a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) $)
|
|
cauappcvgprlem1 $p |- ( ph ->
|
|
<. { l | l <Q ( F ` Q ) } , { u | ( F ` Q ) <Q u } >. <P
|
|
( L +P.
|
|
<. { l | l <Q ( Q +Q R ) } , { u | ( Q +Q R ) <Q u } >. ) ) $=
|
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|
|
addassnqg syl3anc breqtrrd wb ltanqg adantl addcomnqg caovord2d mpbid
|
|
w3a addclnq simprr 3eqtrd breqtrd oveq1d rexlimddv cauappcvgprlemladd
|
|
rspcev crab fveq2d nqex rabex op1st syl6eq eleq2d breq1d elrab3 bitrd
|
|
rexbidv mpbird cnp cauappcvgprlemcl nqprlu addclpr nqprl ) ADFUEZGJUF
|
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ZDEQRZSUGJUHXNBUFZSUGBUHUIZUNRZUJUEZUKZXMXLSUGJUHXLXOSUGBUHUIXQULUGZA
|
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XSXLHUFZQRZYAFUEZXNQRZSUGZHTUMZAUAUFZYGQRZEUOZYFUATAETUKZYIUATUMPUAEU
|
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PUQAYGTUKZYIURZURZYKXLYGQRZYGFUEZXNQRZSUGZYFAYKYIUSZYMYNYODQRZYGQRZYG
|
|
QRZYPSYMXLYTSUGYNUUASUGYMXLYODYGQRZQRZYTSYMXLUUCSUGZYOXLUUBQRZSUGZYMI
|
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UFZFUEZYCUUGYAQRZQRZSUGZYCUUHUUIQRZSUGZURZHTUTITUTZUUDUUFURZAUUOYLLVA
|
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YMDTUKZYKUUOUUPVFAUUQYLOVAZYRUUNUUPXLYCDYAQRZQRZSUGZYCXLUUSQRZSUGZURI
|
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HDYGTTUUGDUOZUUKUVAUUMUVCUVDUUHXLUUJUUTSUUGDFVBZUVDUUIUUSYCQUUGDYAQVC
|
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ZVDVEUVDUULUVBYCSUVDUUHXLUUIUUSQUVEUVFVGVHVIYAYGUOZUVAUUDUVCUUFUVGUUT
|
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UUCXLSUVGYCYOUUSUUBQYAYGFVBZYAYGDQVJZVGVHUVGYCYOUVBUUESUVHUVGUUSUUBXL
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WBWDWEYMUUAYSYHQRZYSEQRZYPYMUVSYKYKUUAUWAUOUVTYRYRYSYGYGVQVRYMYHEYSQA
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|
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RUWFXLAXRUWFYCYAQRXNQRXOSUGHTUMZBTWOZUIZUJUEUWFAXQUWJUJABCXNFGHIJKLMN
|
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AUUQYJXNTUKZOPDEWGVLZWMWPUWFUWIUWEJTWQWRUWHBTWQWRWSWTXAAUVQUWGYFVTUVR
|
|
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|
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|
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VLWE $.
|
|
|
|
$( Lemma for ~ cauappcvgpr . Part of showing the putative limit to be
|
|
a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) $)
|
|
cauappcvgprlem2 $p |- ( ph ->
|
|
L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } ,
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|
{ u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) $=
|
|
( cltq wbr cnq wcel vx cfv cplq co cv wa cab cop cltp ltaddnq syl2anc
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wrex ffvelrnd ltanqi ltbtwnnqq sylib c2nd simprl adantr simprrl fveq2
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c1st wceq oveq12d breq1d rspcev breq2 rexbidv crab fveq2i rabex op2nd
|
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id nqex eqtri elrab2 sylanbrc simprrr breq1 elab sylibr ltnqex gtnqex
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wb syl ltdfpr mpbird rexlimddv ) ADFUBZDUCUDZUAUEZQRZWTWRDEUCUDZUCUDZ
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|
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|
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|
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|
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|
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|
|
$}
|
|
|
|
${
|
|
$d F l p q r u x y $. $d L q r x y $. $d ph x y q $.
|
|
$( Lemma for ~ cauappcvgpr . The putative limit is a limit.
|
|
(Contributed by Jim Kingdon, 20-Jun-2020.) $)
|
|
cauappcvgprlemlim $p |- ( ph -> A. q e. Q. A. r e. Q. (
|
|
<. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P
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|
( L +P. <. { l | l <Q ( q +Q r ) } ,
|
|
{ u | ( q +Q r ) <Q u } >. ) /\
|
|
L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } ,
|
|
{ u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) $=
|
|
( cltq wbr cab cplq co cltp cnq vx vy cv cfv cop cpp wral wcel adantr
|
|
wa simprl simprr cauappcvgprlem1 cauappcvgprlem2 jca ralrimivva fveq2
|
|
weq breq2d abbidv breq1d opeq12d oveq1 oveq2d breq12d oveq12d anbi12d
|
|
wf oveq2 cbvral2v sylib ) AIUCZUAUCZDUDZNOZIPZVNBUCZNOZBPZUEZEVLVMUBU
|
|
CZQRZNOZIPZWBVQNOZBPZUEZUFRZSOZEVLVNWBQRZNOZIPZWJVQNOZBPZUEZSOZUJZUBT
|
|
UGUATUGVLGUCZDUDZNOZIPZWSVQNOZBPZUEZEVLWRFUCZQRZNOZIPZXFVQNOZBPZUEZUF
|
|
RZSOZEVLWSXFQRZNOZIPZXNVQNOZBPZUEZSOZUJZFTUGGTUGAWQUAUBTTAVMTUHZWATUH
|
|
ZUJZUJZWIWPYEBCVMWADEGHIATTDVHYDJUIZAHUCZDUDZWSYGWRQRZQRNOWSYHYIQRNOU
|
|
JGTUGHTUGYDKUIZACYHNOHTUGYDLUIZMAYBYCUKZAYBYCULZUMYEBCVMWADEGHIYFYJYK
|
|
MYLYMUNUOUPWQYAXDEVLWRWAQRZNOZIPZYNVQNOZBPZUEZUFRZSOZEVLWSYNQRZNOZIPZ
|
|
UUBVQNOZBPZUEZSOZUJUAUBGFTTUAGURZWIUUAWPUUHUUIVTXDWHYTSUUIVPXAVSXCUUI
|
|
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|
|
YPWFYRUUIWCYOIUUIWBYNVLNVMWRWAQVCZUSUTUUIWEYQBUUIWBYNVQNUUKVAUTVBVDVE
|
|
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|
|
UIWMUUEBUUIWJUUBVQNUULVAUTVBUSVGUBFURZUUAXMUUHXTUUMYTXLXDSUUMYSXKEUFU
|
|
UMYPXHYRXJUUMYOXGIUUMYNXFVLNWAXEWRQVIZUSUTUUMYQXIBUUMYNXFVQNUUNVAUTVB
|
|
VDUSUUMUUGXSESUUMUUDXPUUFXRUUMUUCXOIUUMUUBXNVLNUUMYNXFWSQUUNVDZUSUTUU
|
|
MUUEXQBUUMUUBXNVQNUUOVAUTVBUSVGVJVK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d F q y z $. $d F r y z $. $d F u $. $d F p $. $d F l $.
|
|
$d l q y z $. $d l r y z $. $d q u y z $. $d r u y z $. $d q r $.
|
|
$d p u $. $d p r $. $d p q $. $d l u $. $d l p $. $d ph q $.
|
|
$d ph p $. $d p z $. $d q z $.
|
|
$( A Cauchy approximation has a limit. A Cauchy approximation, here
|
|
` F ` , is similar to a Cauchy sequence but is indexed by the desired
|
|
tolerance (that is, how close together terms needs to be) rather than
|
|
by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p.
|
|
(varies) with a few differences such as that we are proving the
|
|
existence of a limit without anything about how fast it converges
|
|
(that is, mere existence instead of existence, in HoTT terms), and
|
|
that the codomain of ` F ` is ` Q. ` rather than ` P. ` . We also
|
|
specify that every term needs to be larger than a fraction ` A ` , to
|
|
avoid the case where we have positive terms which "converge" to zero
|
|
(which is not a positive real).
|
|
|
|
This proof (including its lemmas) is similar to the proofs of
|
|
~ caucvgpr and ~ caucvgprpr but is somewhat simpler, so reading this
|
|
one first may help understanding the other two.
|
|
|
|
(Contributed by Jim Kingdon, 19-Jun-2020.) $)
|
|
cauappcvgpr $p |- ( ph -> E. y e. P. A. q e. Q. A. r e. Q. (
|
|
<. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P
|
|
( y +P. <. { l | l <Q ( q +Q r ) } ,
|
|
{ u | ( q +Q r ) <Q u } >. ) /\
|
|
y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } ,
|
|
{ u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) $=
|
|
( cv cplq co cltq wbr cnq cab cltp vz cfv wrex crab cop cnp wcel cpp wa
|
|
wral wb wceq oveq2 fveq2 breq12d cbvrexv a1i rabbiia id oveq12d opeq12i
|
|
breq1d cauappcvgprlemcl cauappcvgprlemlim oveq1 breq2d anbi12d 2ralbidv
|
|
breq1 rspcev syl2anc ) AIMZUAMZNOZVMEUBZPQZUARUCZIRUDZVOVMNOZCMZPQZUARU
|
|
CZCRUDZUEZUFUGVLGMZEUBZPQISWFVTPQCSUEZWDVLWEFMNOZPQISWHVTPQCSUEZUHOZTQZ
|
|
WDVLWFWHNOZPQISWLVTPQCSUEZTQZUIZFRUJGRUJZWGBMZWIUHOZTQZWQWMTQZUIZFRUJGR
|
|
UJZBUFUCACDEWDGHIJKLVRVLWENOZWFPQZGRUCZIRUDWCWFWENOZVTPQZGRUCZCRUDVQXEI
|
|
RVQXEUKVLRUGVPXDUAGRVMWEULZVNXCVOWFPVMWEVLNUMVMWEEUNZUOUPUQURWBXHCRWBXH
|
|
UKVTRUGWAXGUAGRXIVSXFVTPXIVOWFVMWENXJXIUSUTVBUPUQURVAZVCACDEWDFGHIJKLXK
|
|
VDXBWPBWDUFWQWDULZXAWOGFRRXLWSWKWTWNXLWRWJWGTWQWDWIUHVEVFWQWDWMTVIVGVHV
|
|
JVK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A j $.
|
|
$( Archimedean principle for fractions (reciprocal version). (Contributed
|
|
by Jim Kingdon, 27-Sep-2020.) $)
|
|
archrecnq $p |- ( A e. Q. -> E. j e. N.
|
|
( *Q ` [ <. j , 1o >. ] ~Q ) <Q A ) $=
|
|
( cnq wcel crq cfv c1o cop ceq cec cltq wbr cnpi wrex recclnq archnqq syl
|
|
cv wa wb nnnq ltrnqg syl2an recrecnq breq2d adantr bitrd rexbidva mpbid )
|
|
ACDZAEFZBRZGHIJZKLZBMNZUMEFZAKLZBMNUJUKCDZUOAOZBUKPQUJUNUQBMUJULMDZSUNUPU
|
|
KEFZKLZUQUJURUMCDUNVBTUTUSULUAUKUMUBUCUJVBUQTUTUJVAAUPKAUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A j x $. $d j l u $.
|
|
$( Archimedean principle for positive reals (reciprocal version).
|
|
(Contributed by Jim Kingdon, 25-Nov-2020.) $)
|
|
archrecpr $p |- ( A e. P. -> E. j e. N.
|
|
<. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u }
|
|
>. <P A ) $=
|
|
( vx cnp wcel cv c1o cop cfv cnpi wrex cltq wbr cab cnq syl wa syl2anc wi
|
|
ceq cec crq c1st cltp c2nd prop prml archrecnq ad2antrl ad2antrr reximdva
|
|
simplrr prcdnql mpd rexlimddv wb nnnq recclnq adantl simpl nqprl rexbidva
|
|
mpbid ) BFGZCHZIJUBUCZUDKZBUEKZGZCLMZDHVINODPVIAHNOAPJBUFOZCLMVFEHZVJGZVL
|
|
EQVFVJBUGKZJFGZVOEQMBUHZEVPVJUIRVFVNQGZVOSZSZVIVNNOZCLMZVLVSWCVFVOVNCUJUK
|
|
WAWBVKCLWAVGLGZSVQVOWBVKUAVFVQVTWDVRULVFVSVOWDUNVNVIVPVJUOTUMUPUQVFVKVMCL
|
|
VFWDSVIQGZVFVKVMURWDWEVFWDVHQGWEVGUSVHUTRVAVFWDVBAVIBDVCTVDVE $.
|
|
$}
|
|
|
|
${
|
|
caucvgprlemk.jk $e |- ( ph -> J <N K ) $.
|
|
caucvgprlemk.jkq $e |- ( ph ->
|
|
( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q ) $.
|
|
$( Lemma for ~ caucvgpr . Reciprocals of positive integers decrease as the
|
|
positive integers increase. (Contributed by Jim Kingdon,
|
|
9-Oct-2020.) $)
|
|
caucvgprlemk $p |- ( ph ->
|
|
( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q ) $=
|
|
( c1o cop ceq cec crq cfv cltq wbr clti cnpi wcel wa wb syl ltrelpi mpbid
|
|
brel ltnnnq ltrnqi cnq ltsonq ltrelnq sotri syl2anc ) ADGHIJZKLZCGHIJZKLZ
|
|
MNZUNBMNULBMNAUMUKMNZUOACDONZUPEACPQDPQRZUQUPSAUQURECDPPOUAUCTCDUDTUBUMUK
|
|
UETFULUNBMUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A j $.
|
|
caucvgpr.f $e |- ( ph -> F : N. --> Q. ) $.
|
|
caucvgpr.cau $e |- ( ph ->
|
|
A. n e. N. A. k e. N. ( n <N k ->
|
|
( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\
|
|
( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) ) $.
|
|
${
|
|
$d F a b k $. $d F f g h $. $d F a k n $. $d J a b $. $d J f g h $.
|
|
$d K a b $. $d K f g h $. $d S f g h $. $d f g h ph $.
|
|
caucvgprlemnkj.k $e |- ( ph -> K e. N. ) $.
|
|
caucvgprlemnkj.j $e |- ( ph -> J e. N. ) $.
|
|
caucvgprlemnkj.s $e |- ( ph -> S e. Q. ) $.
|
|
$( Lemma for ~ caucvgpr . Part of disjointness. (Contributed by Jim
|
|
Kingdon, 23-Oct-2020.) $)
|
|
caucvgprlemnkj $p |- ( ph -> -.
|
|
( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\
|
|
( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) $=
|
|
( clti wbr cplq co cltq wa cnq cnpi va vb vf vg c1o cop ceq cec crq cfv
|
|
vh wn wceq ltsonq ltrelnq son2lpi simprl cv wi breq1 fveq2 opeq1 eceq1d
|
|
wral fveq2d oveq2d breq12d oveq12d breq2d anbi12d imbi12d oveq1d breq1d
|
|
breq2 cbvral2v sylib wcel rspc2v syl2anc mpd imp simpld adantr sotri wb
|
|
ltanqg adantl ad2antrr ffvelrnd nnnq recclnq addcomnqg caovord2d mpbird
|
|
w3a 3syl ltaddnq simprr jca ex mtoi anbi1d addclnq syl6bi jctird simprd
|
|
bitrd w3o pitri3or mpjao3dan ) AGFMNZBGUEUFZUGUHZUIUJZOPZGEUJZQNZFEUJZF
|
|
UEUFZUGUHZUIUJZOPZBQNZRZULGFUMZFGMNZAXKRZYDBXRQNZXRBQNZRZBXRQSUNUOUPYGY
|
|
DYJYGYDRZYHYIYKYHXOXRXNOPZQNZYKXQXPYLQNZYMYGXQYCUQYGYNYDYGYNXRXPXNOPZQN
|
|
ZAXKYNYPRZAUAURZUBURZMNZYREUJZYSEUJZYRUEUFZUGUHZUIUJZOPZQNZUUBUUAUUEOPZ
|
|
QNZRZUSZUBTVDUATVDZXKYQUSZADURZCURZMNZUUNEUJZUUOEUJZUUNUEUFZUGUHZUIUJZO
|
|
PZQNZUURUUQUVAOPZQNZRZUSZCTVDDTVDUULIUVGUUKYRUUOMNZUUAUURUUEOPZQNZUURUU
|
|
HQNZRZUSDCUAUBTTUUNYRUMZUUPUVHUVFUVLUUNYRUUOMUTUVMUVCUVJUVEUVKUVMUUQUUA
|
|
UVBUVIQUUNYREVAZUVMUVAUUEUUROUVMUUTUUDUIUVMUUSUUCUGUUNYRUEVBVCVEZVFVGUV
|
|
MUVDUUHUURQUVMUUQUUAUVAUUEOUVNUVOVHVIVJVKUUOYSUMZUVHYTUVLUUJUUOYSYRMVNU
|
|
VPUVJUUGUVKUUIUVPUVIUUFUUAQUVPUURUUBUUEOUUOYSEVAZVLVIUVPUURUUBUUHQUVQVM
|
|
VJVKVOVPZAGTVQZFTVQZUULUUMUSJKUUKUUMGYSMNZXPUUBXNOPZQNZUUBYOQNZRZUSUAUB
|
|
GFTTYRGUMZYTUWAUUJUWEYRGYSMUTUWFUUGUWCUUIUWDUWFUUAXPUUFUWBQYRGEVAZUWFUU
|
|
EXNUUBOUWFUUDXMUIUWFUUCXLUGYRGUEVBVCVEZVFVGUWFUUHYOUUBQUWFUUAXPUUEXNOUW
|
|
GUWHVHVIVJVKYSFUMZUWAXKUWEYQYSFGMVNUWIUWCYNUWDYPUWIUWBYLXPQUWIUUBXRXNOY
|
|
SFEVAZVLVIUWIUUBXRYOQUWJVMVJVKVRVSVTWAWBWCXOXPYLQSUNUOWDVSYKUCUDUKBXRXN
|
|
QSOUCURZSVQZUDURZSVQZUKURZSVQWOZUWKUWMQNUWOUWKOPUWOUWMOPQNWEZYKUWKUWMUW
|
|
OWFZWGABSVQZXKYDLWHAXRSVQZXKYDATSFEHKWIZWHZAXNSVQZXKYDAUVSXMSVQUXCJGWJX
|
|
MWKWPZWHUWLUWNRZUWKUWMOPUWMUWKOPUMZYKUWKUWMWLZWGWMWNYKXRYBQNZYCYIYKUWTY
|
|
ASVQZUXHUXBAUXIXKYDAUVTXTSVQUXIKFWJXTWKWPZWHXRYAWQVSYGXQYCWRXRYBBQSUNUO
|
|
WDVSWSWTXAAYERZYDBYAOPZYAOPZBQNZBUXMQNZRUXMBQSUNUOUPUXKYDUXNUXOUXKYDUXM
|
|
YBQNZYCRZUXNUXKYDUXLXRQNZYCRZUXQYEYDUXSWEAYEXQUXRYCYEXOUXLXPXRQYEXNYABO
|
|
YEXMXTUIYEXLXSUGGFUEVBVCVEVFGFEVAVGXBWGUXKUXRUXPYCAUXRUXPWEYEAUCUDUKUXL
|
|
XRYAQSOUWPUWQAUWRWGAUWSUXIUXLSVQZLUXJBYAXCVSZUXAUXJUXEUXFAUXGWGWMWCXBXG
|
|
UXMYBBQSUNUOWDXDAUXOYEABUXLQNZUXLUXMQNZUXOAUWSUXIUYBLUXJBYAWQVSAUXTUXIU
|
|
YCUYAUXJUXLYAWQVSBUXLUXMQSUNUOWDVSWCXEXAAYFRZYDBYBQNZYCRZBYBQSUNUOUPUYD
|
|
YDUYFUYDYDRZUYEYCUYGBXOQNZXOYBQNZUYEUYGUWSUXCUYHAUWSYFYDLWHAUXCYFYDUXDW
|
|
HBXNWQVSUYGXQXPYBQNZUYIUYDXQYCUQUYDUYJYDUYDXRXPYAOPZQNZUYJAYFUYLUYJRZAU
|
|
ULYFUYMUSZUVRAUVTUVSUULUYNUSKJUUKUYNFYSMNZXRUUBYAOPZQNZUUBYBQNZRZUSUAUB
|
|
FGTTYRFUMZYTUYOUUJUYSYRFYSMUTUYTUUGUYQUUIUYRUYTUUAXRUUFUYPQYRFEVAZUYTUU
|
|
EYAUUBOUYTUUDXTUIUYTUUCXSUGYRFUEVBVCVEZVFVGUYTUUHYBUUBQUYTUUAXRUUEYAOVU
|
|
AVUBVHVIVJVKYSGUMZUYOYFUYSUYMYSGFMVNVUCUYQUYLUYRUYJVUCUYPUYKXRQVUCUUBXP
|
|
YAOYSGEVAZVLVIVUCUUBXPYBQVUDVMVJVKVRVSVTWAXFWCXOXPYBQSUNUOWDVSBXOYBQSUN
|
|
UOWDVSUYDXQYCWRWSWTXAAUVSUVTXKYEYFXHJKGFXIVSXJ $.
|
|
$}
|
|
|
|
${
|
|
$d B f g h $. $d B k n $. $d F f g h $. $d F k n $. $d J f g h $.
|
|
$d J k n $. $d f g h ph $.
|
|
caucvgprlemnbj.b $e |- ( ph -> B e. N. ) $.
|
|
caucvgprlemnbj.j $e |- ( ph -> J e. N. ) $.
|
|
$( Lemma for ~ caucvgpr . Non-existence of two elements of the sequence
|
|
which are too far from each other. (Contributed by Jim Kingdon,
|
|
18-Oct-2020.) $)
|
|
caucvgprlemnbj $p |- ( ph ->
|
|
-. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) )
|
|
+Q ( *Q ` [ <. J , 1o >. ] ~Q ) )
|
|
<Q ( F ` J ) ) $=
|
|
( cplq co cltq wbr clti wa cnpi wcel syl2anc cnq vf cfv c1o cop ceq cec
|
|
vg vh wn wceq cv wi wral breq1 fveq2 opeq1 eceq1d fveq2d oveq2d breq12d
|
|
crq oveq12d breq2d anbi12d imbi12d breq2 oveq1d breq1d mpd imp ffvelrnd
|
|
rspc2v simprd nnnq recclnq 3syl addclnq ltaddnq adantr ltrelnq sotri wb
|
|
ltsonq adantl mpbid simpld w3a ltanqg addcomnqg caovord2d w3o mpjao3dan
|
|
pitri3or wor so2nr mpan imnan sylibr ) AFEUBZBEUBZBUCUDZUEUFZVAUBZKLZFU
|
|
CUDZUEUFZVAUBZKLZMNZXHWSMNZUIZABFONZXIBFUJZFBONZAXLPZWSXDMNZXDXHMNZXIXO
|
|
WTWSXCKLZMNZXPAXLXSXPPZADUKZCUKZONZYAEUBZYBEUBZYAUCUDZUEUFZVAUBZKLZMNZY
|
|
EYDYHKLZMNZPZULZCQUMDQUMZXLXTULZHABQRZFQRZYOYPULIJYNYPBYBONZWTYEXCKLZMN
|
|
ZYEXDMNZPZULDCBFQQYABUJZYCYSYMUUCYABYBOUNUUDYJUUAYLUUBUUDYDWTYIYTMYABEU
|
|
OZUUDYHXCYEKUUDYGXBVAUUDYFXAUEYABUCUPUQURZUSUTUUDYKXDYEMUUDYDWTYHXCKUUE
|
|
UUFVBVCVDVEYBFUJZYSXLUUCXTYBFBOVFUUGUUAXSUUBXPUUGYTXRWTMUUGYEWSXCKYBFEU
|
|
OZVGVCUUGYEWSXDMUUHVHVDVEVLSVIVJVMAXQXLAXDTRZXGTRZXQAWTTRZXCTRZUUIAQTBE
|
|
GIVKZAYQXBTRUULIBVNXBVOVPZWTXCVQSZAYRXFTRUUJJFVNXFVOVPZXDXGVRSZVSWSXDXH
|
|
MTWCVTWAZSAXMPZXPXQXIUUSWTXDMNZXPAUUTXMAUUKUULUUTUUMUUNWTXCVRSZVSXMUUTX
|
|
PWBAXMWTWSXDMBFEUOVHWDWEAXQXMUUQVSUURSAXNPZWSWTXGKLZMNZUVCXHMNZXIUVBUVD
|
|
WTWSXGKLZMNZAXNUVDUVGPZAYOXNUVHULZHAYRYQYOUVIULJIYNUVIFYBONZWSYEXGKLZMN
|
|
ZYEUVFMNZPZULDCFBQQYAFUJZYCUVJYMUVNYAFYBOUNUVOYJUVLYLUVMUVOYDWSYIUVKMYA
|
|
FEUOZUVOYHXGYEKUVOYGXFVAUVOYFXEUEYAFUCUPUQURZUSUTUVOYKUVFYEMUVOYDWSYHXG
|
|
KUVPUVQVBVCVDVEYBBUJZUVJXNUVNUVHYBBFOVFUVRUVLUVDUVMUVGUVRUVKUVCWSMUVRYE
|
|
WTXGKYBBEUOZVGVCUVRYEWTUVFMUVSVHVDVEVLSVIVJWFAUVEXNAUUTUVEUVAAUAUGUHWTX
|
|
DXGMTKUAUKZTRZUGUKZTRZUHUKZTRWGUVTUWBMNUWDUVTKLUWDUWBKLMNWBAUVTUWBUWDWH
|
|
WDUUMUUOUUPUWAUWCPUVTUWBKLUWBUVTKLUJAUVTUWBWIWDWJWEVSWSUVCXHMTWCVTWASAY
|
|
QYRXLXMXNWKIJBFWMSWLAXIXJPUIZXIXKULAWSTRZXHTRZUWEAQTFEGJVKAUUIUUJUWGUUO
|
|
UUPXDXGVQSTMWNUWFUWGPUWEWCTWSXHMWOWPSXIXJWQWRVI $.
|
|
$}
|
|
|
|
caucvgpr.bnd $e |- ( ph -> A. j e. N. A <Q ( F ` j ) ) $.
|
|
|
|
${
|
|
caucvgpr.lim $e |- L = <.
|
|
{ l e. Q. | E. j e. N.
|
|
( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } ,
|
|
{ u e. Q. | E. j e. N.
|
|
( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
|
|
>. $.
|
|
|
|
${
|
|
$d A s $. $d F j l $. $d F r $. $d F j u $. $d L r $. $d j ph s $.
|
|
$d l s $.
|
|
$( Lemma for ~ caucvgpr . The putative limit is inhabited.
|
|
(Contributed by Jim Kingdon, 27-Sep-2020.) $)
|
|
caucvgprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
|
|
/\ E. r e. Q. r e. ( 2nd ` L ) ) ) $=
|
|
( wcel cnq c1o cltq cnpi cv c1st cfv wrex c2nd cplq wceq wbr wral 1pi
|
|
fveq2 breq2d rspcv mpsyl ltrelnq brel simpld halfnqq 3syl cop ceq cec
|
|
co wa crq simplr archrecnq syl simpr wb nnnq recclnq ad2antrr syl3anc
|
|
ltanqg mpbid simpllr breqtrd wi rsp ad4antr mpd ltsonq sotri reximdva
|
|
syl2anc ex oveq1 breq1d rexbidv crab fveq2i nqex rabex op1st sylanbrc
|
|
elrab2 c1q a1i ffvelrnd addclnq sylancl df-1nqqs rec1nq eqtr3i oveq2i
|
|
eqtri ltaddnq syl5eqbr opeq1 eceq1d fveq2d oveq12d rspcev breq2 op2nd
|
|
1nq eleq1 jca ) AIUAZHUBUCZPZIQUDZJUAZHUEUCZPZJQUDZAXTXTUFVCZCUGZIQUD
|
|
ZYCACRGUCZSUHZCQPZYJRTPZACDUAZGUCZSUHZDTUIZYLUJNYQYLDRTYORUGZYPYKCSYO
|
|
RGUKZULUMUNYLYMYKQPZCYKQQSUOUPUQICURUSAYIYBIQAXTQPZVDZYIYBUUCYIVDZUUB
|
|
XTYORUTZVAVBZVEUCZUFVCZYPSUHZDTUDZYBAUUBYIVFZUUDUUGXTSUHZDTUDZUUJUUDU
|
|
UBUUMUUKXTDVGVHUUDUULUUIDTUUDYOTPZVDZUULUUIUUOUULVDZUUHCSUHYQUUIUUPUU
|
|
HYHCSUUPUULUUHYHSUHZUUOUULVIUUPUUGQPZUUBUUBUULUUQVJUUPUUNUUFQPUURUUDU
|
|
UNUULVFZYOVKUUFVLUSUUDUUBUUNUULUUKVMZUUTUUGXTXTVOVNVPUUCYIUUNUULVQVRU
|
|
UPUUNYQUUSAUUNYQVSZUUBYIUUNUULAYRUVANYQDTVTVHWAWBUUHCYPSQWCUOWDWFWGWE
|
|
WBKUAZUUGUFVCZYPSUHZDTUDZUUJKXTQYAUVBXTUGZUVDUUIDTUVFUVCUUHYPSUVBXTUU
|
|
GUFWHWIWJYAUVEKQWKZYPUUGUFVCZBUAZSUHZDTUDZBQWKZUTZUBUCUVGHUVMUBOWLUVG
|
|
UVLUVEKQWMWNZUVKBQWMWNZWOXGWQWPWGWEWBAYKWRUFVCZWRUFVCZQPZUVQYEPZYGAUV
|
|
PQPZWRQPZUVRAUUAUWAUVTATQRGLYNAUJWSZWTXQYKWRXAXBZXQUVPWRXAXBZAUVRUVHU
|
|
VQSUHZDTUDZUVSUWDAYNYKRRUTZVAVBZVEUCZUFVCZUVQSUHZUWFUWBAUWJUVPUVQSUWI
|
|
WRYKUFWRVEUCUWIWRWRUWHVEXCWLXDXEXFAUVTUWAUVPUVQSUHUWCXQUVPWRXHXBXIUWE
|
|
UWKDRTYSUVHUWJUVQSYSYPYKUUGUWIUFYTYSUUFUWHVEYSUUEUWGVAYORRXJXKXLXMWIX
|
|
NWFUVKUWFBUVQQYEUVIUVQUGUVJUWEDTUVIUVQUVHSXOWJYEUVMUEUCUVLHUVMUEOWLUV
|
|
GUVLUVNUVOXPXGWQWPYFUVSJUVQQYDUVQYEXRXNWFXS $.
|
|
$}
|
|
|
|
${
|
|
$d F f g h l r $. $d F r s t $. $d F u $. $d L f g h j r t $.
|
|
$d L s $. $d j l r s $. $d j ph f g h r t $. $d ph s $.
|
|
$d f g h s t $. $d j u r s $.
|
|
$( Lemma for ~ caucvgpr . The lower cut of the putative limit is open.
|
|
(Contributed by Jim Kingdon, 20-Oct-2020.) $)
|
|
caucvgprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
|
|
E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) $=
|
|
( wa cltq wbr cnq cnpi vt c1st cfv wcel c1o cop ceq cec crq cplq wrex
|
|
cv co wceq oveq1 breq1d rexbidv crab fveq2i rabex op1st eqtri simprbi
|
|
nqex elrab2 adantl simprr ltbtwnnqq simplrl nnnq recclnq 3syl simplbi
|
|
sylib ad3antlr ltaddnq syl2anc addcomnqg breqtrd simprrl ltsonq sotri
|
|
ltrelnq wb simprl ltexnqq mpbid ad2antrr eqbrtrrd simpr simplr ltanqg
|
|
breqtrrd syl3anc mpbird simprrr eqtr3d rspe sylanbrc jca reximdva mpd
|
|
ex rexlimddv ) AIULZHUBUCZUDZPZXEDULZUEUFUGUHZUIUCZUJUMZXIGUCZQRZXEJU
|
|
LZQRZXOXFUDZPZJSUKZDTXGXNDTUKZAXGXESUDZXTKULZXKUJUMZXMQRZDTUKZXTKXESX
|
|
FYBXEUNZYDXNDTYFYCXLXMQYBXEXKUJUOUPUQXFYEKSURZXMXKUJUMBULQRDTUKZBSURZ
|
|
UFZUBUCYGHYJUBOUSYGYIYEKSVDUTYHBSVDUTVAVBZVEZVCVFXHXITUDZXNPZPZXLUAUL
|
|
ZQRZYPXMQRZPZXSUASYOXNYSUASUKXHYMXNVGUAXLXMVHVNYOYPSUDZYSPZPZXKXOUJUM
|
|
ZYPUNZJSUKZXSUUBXKYPQRZUUEUUBXKXLQRYQUUFUUBXKXKXEUJUMZXLQUUBXKSUDZYAX
|
|
KUUGQRUUBYMXJSUDUUHXHYMXNUUAVIZXIVJXJVKVLZXGYAAYNUUAXGYAXTYLVMVOZXKXE
|
|
VPVQUUBUUHYAUUGXLUNUUJUUKXKXEVRVQVSYOYTYQYRVTZXKXLYPQSWAWCWBVQUUBUUHY
|
|
TUUFUUEWDUUJYOYTYSWEJXKYPWFVQWGUUBUUDXRJSUUBXOSUDZPZUUDXRUUNUUDPZXPXQ
|
|
UUOXPUUGUUCQRZUUOUUGYPUUCQUUOXLUUGYPQUUOYAUUHXLUUGUNUUBYAUUMUUDUUKWHZ
|
|
UUBUUHUUMUUDUUJWHZXEXKVRVQUUBYQUUMUUDUULWHWIUUNUUDWJZWMUUOYAUUMUUHXPU
|
|
UPWDUUQUUBUUMUUDWKZUURXEXOXKWLWNWOUUOUUMXOXKUJUMZXMQRZDTUKZXQUUTUUOYM
|
|
UVBUVCUUBYMUUMUUDUUIWHUUOUVBYRUUBYRUUMUUDYOYTYQYRWPWHUUOUVAYPXMQUUOUU
|
|
CUVAYPUUOUUHUUMUUCUVAUNUURUUTXKXOVRVQUUSWQUPWOUVBDTWRVQYEUVCKXOSXFYBX
|
|
OUNZYDUVBDTUVDYCUVAXMQYBXOXKUJUOUPUQYKVEWSWTXCXAXBXDXD $.
|
|
|
|
$( Lemma for ~ caucvgpr . The lower cut of the putative limit is
|
|
lower. (Contributed by Jim Kingdon, 20-Oct-2020.) $)
|
|
caucvgprlemlol $p |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) ->
|
|
s e. ( 1st ` L ) ) $=
|
|
( cv cltq wcel cnq cplq vf vg vh wbr c1st cfv w3a c1o cop ceq cec crq
|
|
co cnpi wrex ltrelnq brel simpld 3ad2ant2 oveq1 breq1d rexbidv fveq2i
|
|
weq crab nqex rabex op1st eqtri elrab2 simprbi 3ad2ant3 wa simpll2 wb
|
|
ltanqg adantl ad2antrr simprd simplr nnnq recclnq 3syl wceq addcomnqg
|
|
caovord2d mpbid ltsonq sotri sylancom ex reximdva mpd sylanbrc ) AIPZ
|
|
JPZQUDZWPHUEUFZRZUGZWOSRZWODPZUHUIUJUKZULUFZTUMZXBGUFZQUDZDUNUOZWOWRR
|
|
WQAXAWSWQXAWPSRZWOWPSSQUPUQZURUSZWTWPXDTUMZXFQUDZDUNUOZXHWSAXNWQWSXIX
|
|
NKPZXDTUMZXFQUDZDUNUOZXNKWPSWRKJVDZXQXMDUNXSXPXLXFQXOWPXDTUTVAVBWRXRK
|
|
SVEZXFXDTUMBPQUDDUNUOZBSVEZUIZUEUFXTHYCUEOVCXTYBXRKSVFVGYABSVFVGVHVIZ
|
|
VJVKVLWTXMXGDUNWTXBUNRZVMZXMXGYFXMXEXLQUDZXGYFXMVMZWQYGAWQWSYEXMVNYHU
|
|
AUBUCWOWPXDQSTUAPZSRZUBPZSRZUCPZSRUGYIYKQUDYMYITUMYMYKTUMQUDVOYHYIYKY
|
|
MVPVQWTXAYEXMXKVRWTXIYEXMWQAXIWSWQXAXIXJVSUSVRYHYEXCSRXDSRWTYEXMVTXBW
|
|
AXCWBWCYJYLVMYIYKTUMYKYITUMWDYHYIYKWEVQWFWGXEXLXFQSWHUPWIWJWKWLWMXRXH
|
|
KWOSWRKIVDZXQXGDUNYNXPXEXFQXOWOXDTUTVAVBYDVJWN $.
|
|
|
|
$( Lemma for ~ caucvgpr . The upper cut of the putative limit is open.
|
|
(Contributed by Jim Kingdon, 20-Oct-2020.) $)
|
|
caucvgprlemopu $p |- ( ( ph /\ r e. ( 2nd ` L ) ) ->
|
|
E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) $=
|
|
( wa cltq cnq wrex cnpi cv c2nd cfv wcel c1o cop ceq cec crq cplq wbr
|
|
co weq breq2 rexbidv crab fveq2i nqex rabex op2nd eqtri elrab2 adantl
|
|
simprbi simprr ltbtwnnqq sylib simplr simplrl adantr syl2anc sylanbrc
|
|
simprl rspe jca ex reximdva mpd rexlimddv ) AJUAZHUBUCZUDZPZDUAZGUCZW
|
|
DUEUFUGUHUIUCZUJULZVTQUKZIUAZVTQUKZWIWAUDZPZIRSZDTWBWHDTSZAWBVTRUDWNW
|
|
GBUAZQUKZDTSZWNBVTRWABJUMWPWHDTWOVTWGQUNUOWAKUAWFUJULWEQUKDTSZKRUPZWQ
|
|
BRUPZUFZUBUCWTHXAUBOUQWSWTWRKRURUSWQBRURUSUTVAZVBVDVCWCWDTUDZWHPPZWGW
|
|
IQUKZWJPZIRSZWMXDWHXGWCXCWHVEIWGVTVFVGXDXFWLIRXDWIRUDZPZXFWLXIXFPZWJW
|
|
KXIXEWJVEXJXHXEDTSZWKXDXHXFVHXJXCXEXKXIXCXFWCXCWHXHVIVJXIXEWJVMXEDTVN
|
|
VKWQXKBWIRWABIUMWPXEDTWOWIWGQUNUOXBVBVLVOVPVQVRVS $.
|
|
|
|
$( Lemma for ~ caucvgpr . The upper cut of the putative limit is
|
|
upper. (Contributed by Jim Kingdon, 20-Oct-2020.) $)
|
|
caucvgprlemupu $p |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) ->
|
|
r e. ( 2nd ` L ) ) $=
|
|
( cv cltq wbr cnq cnpi c2nd cfv wcel w3a c1o cop ceq cec cplq co wrex
|
|
crq ltrelnq brel simprd 3ad2ant2 breq2 rexbidv crab fveq2i nqex rabex
|
|
weq op2nd eqtri elrab2 simprbi 3ad2ant3 ltsonq sotri reximdv sylanbrc
|
|
wi expcom mpd ) AIPZJPZQRZVPHUAUBZUCZUDZVQSUCZDPZGUBZWCUEUFUGUHULUBZU
|
|
IUJZVQQRZDTUKZVQVSUCVRAWBVTVRVPSUCZWBVPVQSSQUMUNUOUPWAWFVPQRZDTUKZWHV
|
|
TAWKVRVTWIWKWFBPZQRZDTUKZWKBVPSVSBIVCWMWJDTWLVPWFQUQURVSKPWEUIUJWDQRD
|
|
TUKZKSUSZWNBSUSZUFZUAUBWQHWRUAOUTWPWQWOKSVAVBWNBSVAVBVDVEZVFVGVHWAWJW
|
|
GDTVRAWJWGVMVTWJVRWGWFVPVQQSVIUMVJVNUPVKVOWNWHBVQSVSBJVCWMWGDTWLVQWFQ
|
|
UQURWSVFVL $.
|
|
$}
|
|
|
|
${
|
|
$d L r s $. $d ph r s $. $d F l r $. $d F s u $. $d L j r $.
|
|
$d j l r s $. $d j ph r $. $d j r u $.
|
|
$( Lemma for ~ caucvgpr . The putative limit is rounded. (Contributed
|
|
by Jim Kingdon, 27-Sep-2020.) $)
|
|
caucvgprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
|
|
E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\
|
|
A. r e. Q. ( r e. ( 2nd ` L ) <->
|
|
E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) ) $=
|
|
( cv cfv wcel wa cnq c1st cltq wbr wrex caucvgprlemopl caucvgprlemlol
|
|
wb wral c2nd ex 3expib rexlimdvw impbid caucvgprlemopu caucvgprlemupu
|
|
ralrimivw jca ) AIPZHUAQZRZURJPZUBUCZVAUSRZSZJTUDZUGZITUHVAHUIQZRZVBU
|
|
RVGRZSZITUDZUGZJTUHAVFITAUTVEAUTVEABCDEFGHIJKLMNOUEUJAVDUTJTAVBVCUTAB
|
|
CDEFGHIJKLMNOUFUKULUMUPAVLJTAVHVKAVHVKABCDEFGHIJKLMNOUNUJAVJVHITAVBVI
|
|
VHABCDEFGHIJKLMNOUOUKULUMUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d F j k $. $d F j l $. $d F j u $. $d F n $. $d L j k $.
|
|
$d j k ph s $. $d l s $. $d s u $. $d k n $.
|
|
$( Lemma for ~ caucvgpr . The putative limit is disjoint.
|
|
(Contributed by Jim Kingdon, 27-Sep-2020.) $)
|
|
caucvgprlemdisj $p |- ( ph -> A. s e. Q.
|
|
-. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
|
|
( cfv cnq cplq cltq wbr cnpi cv c1st wcel c2nd wa c1o cop ceq cec crq
|
|
wn co wrex wfal weq oveq1 breq1d rexbidv crab fveq2i nqex rabex op1st
|
|
eqtri elrab2 simprbi opeq1 eceq1d fveq2d oveq2d fveq2 breq12d cbvrexv
|
|
sylib breq2 anim12i reeanv sylibr adantl wf ad2antrr clti wral simprl
|
|
op2nd wi simprr simplbi ad2antrl adantr caucvgprlemnkj rexlimdvva mpd
|
|
pm2.21d inegd ralrimivw ) AIUAZHUBOZUCZWQHUDOZUCZUEZUKIPAXBAXBUEZWQEU
|
|
AZUFUGZUHUIZUJOZQULZXDGOZRSZDUAZGOZXKUFUGZUHUIZUJOZQULZWQRSZUEZDTUMET
|
|
UMZUNXBXSAXBXJETUMZXQDTUMZUEXSWSXTXAYAWSWQXOQULZXLRSZDTUMZXTWSWQPUCZY
|
|
DJUAZXOQULZXLRSZDTUMZYDJWQPWRJIUOZYHYCDTYJYGYBXLRYFWQXOQUPUQURWRYIJPU
|
|
SZXPBUAZRSZDTUMZBPUSZUGZUBOYKHYPUBNUTYKYOYIJPVAVBZYNBPVAVBZVCVDVEZVFY
|
|
CXJDETDEUOZYBXHXLXIRYTXOXGWQQYTXNXFUJYTXMXEUHXKXDUFVGVHVIVJXKXDGVKVLV
|
|
MVNXAYEYAYNYABWQPWTBIUOYMXQDTYLWQXPRVOURWTYPUDOYOHYPUDNUTYKYOYQYRWEVD
|
|
VEVFVPXJXQEDTTVQVRVSXCXRUNEDTTXCXDTUCZXKTUCZUEZUEZXRUNUUDWQEFGXKXDATP
|
|
GVTXBUUCKWAAFUAZXDWBSUUEGOZXIUUEUFUGUHUIUJOZQULRSXIUUFUUGQULRSUEWFETW
|
|
CFTWCXBUUCLWAXCUUAUUBWDXCUUAUUBWGXCYEUUCWSYEAXAWSYEYDYSWHWIWJWKWNWLWM
|
|
WOWP $.
|
|
$}
|
|
|
|
${
|
|
$d F f g h $. $d F j l $. $d F u $. $d f g h m j ph r s x y $.
|
|
$d m j l s $. $d m j r u $. $d L m x y $.
|
|
|
|
$( Lemma for ~ caucvgpr . The putative limit is located. (Contributed
|
|
by Jim Kingdon, 27-Sep-2020.) $)
|
|
caucvgprlemloc $p |- ( ph -> A. s e. Q. A. r e. Q.
|
|
( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) $=
|
|
( cltq wbr wcel cnq cplq vy vx vm vf vg vh cv c1st c2nd wo wi wa wceq
|
|
cfv wrex ltexnqi adantl subhalfnqq ad2antrl c1o cop ceq cec archrecnq
|
|
co crq cnpi simprr nnnq recclnq syl simplrl lt2addnq syl22anc simplrr
|
|
mp2and ltsonq ltrelnq sotri syl2anc ad3antrrr ltanqi ad2antrr breqtrd
|
|
addclnq wf ad5antr simprl ffvelrnd wor w3a sowlin mpan syl3anc adantr
|
|
mpd simpr addassnqg breq1d mpbird wb ltanqg addcomnqg caovord2d opeq1
|
|
weq eceq1d fveq2d oveq2d fveq2 breq12d rspcev oveq1 rexbidv crab nqex
|
|
fveq2i rabex op1st eqtri elrab2 sylanbrc ex oveq12d sylan breq2 op2nd
|
|
orim12d rexlimddv ralrimivva ) AIUGZJUGZPQZYKHUHUNZRZYLHUIUNZRZUJZUKI
|
|
JSSAYKSRZYLSRZULZULZYMYRUUBYMULZYKUAUGZTVEZYLUMZYRUASYMUUFUASUOUUBUAY
|
|
KYLUPUQUUCUUDSRZUUFULZULZUBUGZUUJTVEZUUDPQZYRUBSUUGUULUBSUOUUCUUFUBUU
|
|
DURUSUUIUUJSRZUULULZULZUCUGZUTVAZVBVCZVFUNZUUJPQZYRUCVGUUMUUTUCVGUOUU
|
|
IUULUUJUCVDUSUUOUUPVGRZUUTULZULZYKUUSUUSTVEZTVEZUUPGUNZUUSTVEZPQZUVGY
|
|
LPQZUJZYRUVCUVEYLPQZUVJUVCUVEUUEYLPUVCUVDUUDPQZYSUVEUUEPQUVCUVDUUKPQZ
|
|
UULUVLUVCUUTUUTUVMUUOUVAUUTVHZUVNUVCUUSSRZUUMUVOUUMUUTUUTULUVMUKUVAUV
|
|
OUUOUUTUVAUURSRUVOUUPVIUURVJVKUSZUUIUUMUULUVBVLZUVPUVQUUSUUJUUSUUJVMV
|
|
NVPUUIUUMUULUVBVOUVDUUKUUDPSVQVRVSVTUUCYSUUHUUNUVBAYSYTYMVLWAZUVDUUDY
|
|
KWBVTUUIUUFUUNUVBUUCUUGUUFVHWCWDUVCUVESRZYTUVGSRZUVKUVJUKZUVCYSUVDSRZ
|
|
UVSUVRUVCUVOUVOUWBUVPUVPUUSUUSWEVTYKUVDWEVTUUCYTUUHUUNUVBAYSYTYMVOWAZ
|
|
UVCUVFSRZUVOUVTUVCVGSUUPGAVGSGWFUUAYMUUHUUNUVBLWGUUOUVAUUTWHZWIZUVPUV
|
|
FUUSWEVTSPWJUVSYTUVTWKUWAVQSUVEYLUVGPWLWMWNWPUVCUVHYOUVIYQUVCUVHYOUVC
|
|
UVHULZYSYKDUGZUTVAZVBVCZVFUNZTVEZUWHGUNZPQZDVGUOZYOUVCYSUVHUVRWOZUWGU
|
|
VAYKUUSTVEZUVFPQZUWOUUOUVAUUTUVHVLUWGUWRUWQUUSTVEZUVGPQZUWGUWTUVHUVCU
|
|
VHWQUWGUWSUVEUVGPUWGYSUVOUVOUWSUVEUMUWPUVCUVOUVHUVPWOZUXAYKUUSUUSWRWN
|
|
WSWTUWGUDUEUFUWQUVFUUSPSTUDUGZSRZUEUGZSRZUFUGZSRWKUXBUXDPQUXFUXBTVEUX
|
|
FUXDTVEPQXAUWGUXBUXDUXFXBUQUWGYSUVOUWQSRUWPUXAYKUUSWEVTUVCUWDUVHUWFWO
|
|
UXAUXCUXEULUXBUXDTVEUXDUXBTVEUMUWGUXBUXDXCUQXDWTUWNUWRDUUPVGDUCXFZUWL
|
|
UWQUWMUVFPUXGUWKUUSYKTUXGUWJUURVFUXGUWIUUQVBUWHUUPUTXEXGXHZXIUWHUUPGX
|
|
JZXKXLVTKUGZUWKTVEZUWMPQZDVGUOZUWOKYKSYNKIXFZUXLUWNDVGUXNUXKUWLUWMPUX
|
|
JYKUWKTXMWSXNYNUXMKSXOZUWMUWKTVEZBUGZPQZDVGUOZBSXOZVAZUHUNUXOHUYAUHOX
|
|
QUXOUXTUXMKSXPXRZUXSBSXPXRZXSXTYAYBYCUVCUVIYQUVCUVIULYTUXPYLPQZDVGUOZ
|
|
YQUVCYTUVIUWCWOUVCUVAUVIUYEUWEUYDUVIDUUPVGUXGUXPUVGYLPUXGUWMUVFUWKUUS
|
|
TUXIUXHYDWSXLYEUXSUYEBYLSYPBJXFUXRUYDDVGUXQYLUXPPYFXNYPUYAUIUNUXTHUYA
|
|
UIOXQUXOUXTUYBUYCYGXTYAYBYCYHWPYIYIYIYCYJ $.
|
|
$}
|
|
|
|
${
|
|
$d A s a c d $. $d F j l r a c d $. $d F j s r u a c d $. $d F n $.
|
|
$d F j k $. $d L j r s a c d $. $d L k $. $d ph r s a c d $.
|
|
$d l r s $. $d u r s $. $d j k r s a c d $. $d n k a c d $.
|
|
$( Lemma for ~ caucvgpr . The putative limit is a positive real.
|
|
(Contributed by Jim Kingdon, 26-Sep-2020.) $)
|
|
caucvgprlemcl $p |- ( ph -> L e. P. ) $=
|
|
( va cnq cfv cltq wbr cnpi cplq vs vr vd vc cpw cxp wcel cv c1st wrex
|
|
c2nd wa wb wral wn wo wi w3a cnp weq fveq2 breq2d cbvralv c1o cop ceq
|
|
sylib cec crq crab opeq1 eceq1d fveq2d oveq2d breq12d cbvrexv rabbiia
|
|
co a1i oveq12d breq1d opeq12i eqtri caucvgprlemm wss nqex elpw2 mpbir
|
|
opelxpi mp2an eqeltri jctil caucvgprlemrnd clti breq1 anbi12d imbi12d
|
|
ssrab2 oveq1d cbvral2v caucvgprlemdisj caucvgprlemloc 3jca elnp1st2nd
|
|
breq2 sylanbrc ) AHOUEZXGUFZUGZUAUHZHUIPZUGZUAOUJUBUHZHUKPZUGZUBOUJUL
|
|
ZULXLXJXMQRZXMXKUGULUBOUJUMUAOUNXOXQXJXNUGZULUAOUJUMUBOUNULZXLXRULUOU
|
|
AOUNZXQXLXOUPUQUBOUNUAOUNZURHUSUGAXPXIABCNEFGHUAUBIJKACDUHZGPZQRZDSUN
|
|
CNUHZGPZQRZNSUNLYDYGDNSDNUTZYCYFCQYBYEGVAZVBVCVGZHIUHZYBVDVEZVFVHZVIP
|
|
ZTVRZYCQRZDSUJZIOVJZYCYNTVRZBUHZQRZDSUJZBOVJZVEZYKYEVDVEZVFVHZVIPZTVR
|
|
ZYFQRZNSUJZIOVJZYFUUGTVRZYTQRZNSUJZBOVJZVEMYRUUKUUCUUOYQUUJIOYQUUJUMY
|
|
KOUGYPUUIDNSYHYOUUHYCYFQYHYNUUGYKTYHYMUUFVIYHYLUUEVFYBYEVDVKVLVMZVNYI
|
|
VOVPVSVQUUBUUNBOUUBUUNUMYTOUGUUAUUMDNSYHYSUULYTQYHYCYFYNUUGTYIUUPVTWA
|
|
VPVSVQWBWCZWDHUUDXHMYRXGUGZUUCXGUGZUUDXHUGUURYROWEYQIOWRYROWFWGWHUUSU
|
|
UCOWEUUBBOWRUUCOWFWGWHYRUUCXGXGWIWJWKWLAXSXTYAABCNEFGHUAUBIJKYJUUQWMA
|
|
BCNUCUDGHUAIJAFUHZEUHZWNRZUUTGPZUVAGPZUUTVDVEZVFVHZVIPZTVRZQRZUVDUVCU
|
|
VGTVRZQRZULZUQZESUNFSUNUDUHZUCUHZWNRZUVNGPZUVOGPZUVNVDVEZVFVHZVIPZTVR
|
|
ZQRZUVRUVQUWATVRZQRZULZUQZUCSUNUDSUNKUVMUWGUVNUVAWNRZUVQUVDUWATVRZQRZ
|
|
UVDUWDQRZULZUQFEUDUCSSFUDUTZUVBUWHUVLUWLUUTUVNUVAWNWOUWMUVIUWJUVKUWKU
|
|
WMUVCUVQUVHUWIQUUTUVNGVAZUWMUVGUWAUVDTUWMUVFUVTVIUWMUVEUVSVFUUTUVNVDV
|
|
KVLVMZVNVOUWMUVJUWDUVDQUWMUVCUVQUVGUWATUWNUWOVTVBWPWQEUCUTZUWHUVPUWLU
|
|
WFUVAUVOUVNWNXEUWPUWJUWCUWKUWEUWPUWIUWBUVQQUWPUVDUVRUWATUVAUVOGVAZWSV
|
|
BUWPUVDUVRUWDQUWQWAWPWQWTVGYJUUQXAABCNEFGHUAUBIJKYJUUQXBXCHUBUAXDXF
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d F m j r s t u v w z $. $d F l $. $d F n k $.
|
|
$d L f g h s t x y $. $d L k $. $d L m j r s t v w z $.
|
|
$d S f g h l s t u x y $. $d S m j l r s t u v w z $.
|
|
$d ph m r s t v w z $. $d j k $.
|
|
caucvgprlemladd.s $e |- ( ph -> S e. Q. ) $.
|
|
$( Lemma for ~ caucvgpr . Adding ` S ` after embedding in positive
|
|
reals, or adding it as a rational. (Contributed by Jim Kingdon,
|
|
9-Oct-2020.) $)
|
|
caucvgprlemladdfu $p |- ( ph ->
|
|
( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_
|
|
{ u e. Q. | E. j e. N.
|
|
( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S )
|
|
<Q u } ) $=
|
|
( cltq wbr cplq cnq wcel vr vs vt vf vg vh vx vy vm vz vw cab cop cpp
|
|
vv cv co c2nd cfv c1o ceq cec crq cnpi wrex crab wa cnp caucvgprlemcl
|
|
wceq nqprlu syl df-iplp addclnq genpelvu syl2anc biimpa breq2 rexbidv
|
|
wb fveq2i rabex op2nd eqtri elrab2 biimpi adantr adantl simpld ltnqex
|
|
nqex vex gtnqex elab2 ltrelnq brel sylbi simprd ad2antll eleq1 mpbird
|
|
fveq2 opeq1 eceq1d fveq2d oveq12d breq1d cbvrexv sylib ad5antr simplr
|
|
ad2antrr wf ffvelrnd nnnq recclnq 3syl ltanqg syl3anc mpbid simpr w3a
|
|
addcomnqg caovord2d ltsonq sotri simpllr breqtrrd reximdva mpd oveq1d
|
|
ex sylibr elrab sylanbrc rexlimdvva ssrdv ) AUAIJUPZDPQJULZDBUPZPQZBU
|
|
LZUMZUNUQURUSZEUPZHUSZUUEUTUMZVAVBZVCUSZRUQZDRUQZYTPQZEVDVEZBSVFZAUAU
|
|
PZUUDTZUUOUUNTZAUUPVGZUUOUBUPZUCUPZRUQZVJZUCUUCURUSZVEUBIURUSZVEZUUQA
|
|
UUPUVEAIVHTUUCVHTZUUPUVEVTABCEFGHIJKLMNVIADSTZUVFOBDJVKVLUDUEUFUGUHIU
|
|
UCUUOUBUCUNRUGUHUFUEUDVMUEUPUFUPVNVOVPVQUURUVBUUQUBUCUVDUVCUURUUSUVDT
|
|
ZUUTUVCTZVGZVGZUVBUUQUVKUVBVGZUUOSTZUUKUUOPQZEVDVEZUUQUVLUVMUVASTZUVL
|
|
UUSSTZUUTSTZUVPUVLUVQUUJUUSPQZEVDVEZUVKUVQUVTVGZUVBUVJUWAUURUVHUWAUVI
|
|
UVHUWAUUJYTPQZEVDVEZUVTBUUSSUVDYTUUSVJUWBUVSEVDYTUUSUUJPVRVSUVDYRUUIR
|
|
UQUUFPQEVDVEZJSVFZUWCBSVFZUMZURUSUWFIUWGURNWAUWEUWFUWDJSWKWBUWCBSWKWB
|
|
WCWDWEWFWGWHWGZWIZUVKUVRUVBUVIUVRUURUVHUVIUVGUVRUVIDUUTPQZUVGUVRVGUUA
|
|
UWJBUUTUVCUCWLYTUUTDPVRYSUUBJDWJBDWMWCWNZDUUTSSPWOWPWQWRWSWGZUUSUUTVN
|
|
VPUVBUVMUVPVTUVKUUOUVASWTWHXAUVLUIUPZHUSZUWMUTUMZVAVBZVCUSZRUQZDRUQZU
|
|
UOPQZUIVDVEZUVOUVLUWRUUSPQZUIVDVEZUXAUVLUVTUXCUVLUVQUVTUWHWRUVSUXBEUI
|
|
VDUUEUWMVJZUUJUWRUUSPUXDUUFUWNUUIUWQRUUEUWMHXBUXDUUHUWPVCUXDUUGUWOVAU
|
|
UEUWMUTXCXDXEXFZXGXHXIUVLUXBUWTUIVDUVLUWMVDTZVGZUXBUWTUXGUXBVGZUWSUVA
|
|
UUOPUXHUWSUWRUUTRUQZPQZUXIUVAPQZUWSUVAPQUXHUWJUXJUVLUWJUXFUXBUVKUWJUV
|
|
BUVIUWJUURUVHUVIUWJUWKWFWSWGXLUXHUVGUVRUWRSTZUWJUXJVTAUVGUUPUVJUVBUXF
|
|
UXBOXJUVLUVRUXFUXBUWLXLZUXHUWNSTUWQSTZUXLUXHVDSUWMHAVDSHXMUUPUVJUVBUX
|
|
FUXBKXJUVLUXFUXBXKZXNUXHUXFUWPSTUXNUXOUWMXOUWPXPXQUWNUWQVNVPZDUUTUWRX
|
|
RXSXTUXHUXBUXKUXGUXBYAUXHUJUKUOUWRUUSUUTPSRUJUPZSTZUKUPZSTZUOUPZSTYBU
|
|
XQUXSPQUYAUXQRUQUYAUXSRUQPQVTUXHUXQUXSUYAXRWHUXPUVLUVQUXFUXBUWIXLUXMU
|
|
XRUXTVGUXQUXSRUQUXSUXQRUQVJUXHUXQUXSYCWHYDXTUWSUXIUVAPSYEWOYFVPUVKUVB
|
|
UXFUXBYGYHYLYIYJUVNUWTEUIVDUXDUUKUWSUUOPUXDUUJUWRDRUXEYKXGXHYMUUMUVOB
|
|
UUOSYTUUOVJUULUVNEVDYTUUOUUKPVRVSYNYOYLYPYJYLYQ $.
|
|
|
|
$d a b f g h l u $. $d S a b $. $d S k $. $d a b j $.
|
|
$d F a b f g h k n $. $d L a b $. $d A b $. $d b a r $.
|
|
$d f g h j $. $d f g h r $. $d f g h a b ph $. $d F l $.
|
|
$d r f g h a b k $.
|
|
$( Lemma for ~ caucvgpr . Adding ` S ` after embedding in positive
|
|
reals, or adding it as a rational. (Contributed by Jim Kingdon,
|
|
8-Oct-2020.) $)
|
|
caucvgprlemladdrl $p |- ( ph ->
|
|
{ l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q
|
|
( ( F ` j ) +Q S ) } C_
|
|
( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) $=
|
|
( cplq cltq cnpi cnq wcel va vr vb vf vg vh cv c1o cop ceq cec crq co
|
|
cfv wbr wrex crab cab c1st wb opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d
|
|
cpp weq breq12d cbvrexv rabbiia wa oveq1 breq1d rexbidv elrab c2nd wf
|
|
ad4antr clti wi wral simpr simpllr caucvgprlemnbj ad3antrrr ffvelrnda
|
|
a1i wceq nnnq recclnq addclnq syl2anc addassnqg syl3anc ltanqg adantl
|
|
3syl w3a ffvelrnd addcomnqg caovord2d 3bitr4rd mtbird intnand cbvralv
|
|
nrexdv breq2d sylib oveq12d opeq12i eqtri syl caucvgprlemladdfu sseld
|
|
ad2antlr breq2 syl6ib mtod wo caucvgprlemcl nqprlu addclpr prop prloc
|
|
cnp sylan sylancom ecased caucvgprlemcanl ex rexlimdva expimpd syl5bi
|
|
mpbid ssrdv syl5eqss ) AJUGZEUGZUHUIZUJUKZULUNZPUMZYSHUNZDPUMZQUOZERU
|
|
PZJSUQYRUAUGZUHUIZUJUKZULUNZPUMZUUHHUNZDPUMZQUOZUARUPZJSUQZIYRDQUOJUR
|
|
DBUGZQUOBURUIVGUMUSUNZUUGUUPJSUUGUUPUTYRSTZUUFUUOEUAREUAVHZUUCUULUUEU
|
|
UNQUVAUUBUUKYRPUVAUUAUUJULUVAYTUUIUJYSUUHUHVAVBVCVDUVAUUDUUMDPYSUUHHV
|
|
EVFVIVJWHVKAUBUUQUUSUBUGZUUQTUVBSTZUVBUUKPUMZUUNQUOZUARUPZVLAUVBUUSTZ
|
|
UUPUVFJUVBSJUBVHZUUOUVEUARUVHUULUVDUUNQYRUVBUUKPVMVNVOVPAUVCUVFUVGAUV
|
|
CVLZUVEUVGUARUVIUUHRTZVLZUVEUVGUVKUVEVLZUVDIYRDUUKPUMZQUOJURUVMUURQUO
|
|
BURUIZVGUMZUSUNZTZUVGUVLUVQUUNUVOVQUNZTZUVLUVSUUNSTZUCUGZHUNZUWAUHUIZ
|
|
UJUKZULUNZPUMZUVMPUMZUUNQUOZUCRUPZVLZUVLUWIUVTUVLUWHUCRUVLUWARTZVLZUW
|
|
HUWFUUKPUMZUUMQUOZUWLUWAFGHUUHARSHVRZUVCUVJUVEUWKKVSZAGUGZFUGZVTUOUWQ
|
|
HUNZUWRHUNZUWQUHUIUJUKULUNZPUMQUOUWTUWSUXAPUMQUOVLWAFRWBGRWBZUVCUVJUV
|
|
EUWKLVSUVLUWKWCZUVIUVJUVEUWKWDZWEUWLUWMDPUMZUUNQUOUWFUUKDPUMZPUMZUUNQ
|
|
UOUWNUWHUWLUXEUXGUUNQUWLUWFSTZUUKSTZDSTZUXEUXGWIUWLUWBSTUWESTZUXHUVLR
|
|
SUWAHAUWOUVCUVJUVEKWFZWGUWLUWKUWDSTUXKUXCUWAWJUWDWKWRUWBUWEWLWMZUWLUV
|
|
JUUJSTZUXIUXDUUHWJZUUJWKZWRZAUXJUVCUVJUVEUWKOVSZUWFUUKDWNWOVNUWLUDUEU
|
|
FUWMUUMDQSPUDUGZSTZUEUGZSTZUFUGZSTWSUXSUYAQUOUYCUXSPUMUYCUYAPUMQUOUTU
|
|
WLUXSUYAUYCWPWQUWLUXHUXIUWMSTUXMUXQUWFUUKWLWMUWLRSUUHHUWPUXDWTUXRUXTU
|
|
YBVLUXSUYAPUMUYAUXSPUMWIUWLUXSUYAXAWQXBUWLUWGUXGUUNQUWLUVMUXFUWFPUWLU
|
|
XJUXIUVMUXFWIUXRUXQDUUKXAWMVDVNXCXDXGXEUVLUVSUUNUWGUURQUOZUCRUPZBSUQZ
|
|
TUWJUVLUVRUYFUUNUVLBCUVMUCFGHIJUXLAUXBUVCUVJUVELWFACUWBQUOZUCRWBZUVCU
|
|
VJUVEACUUDQUOZERWBUYHMUYIUYGEUCREUCVHZUUDUWBCQYSUWAHVEZXHXFXIWFIUUCUU
|
|
DQUOZERUPZJSUQZUUDUUBPUMZUURQUOZERUPZBSUQZUIYRUWEPUMZUWBQUOZUCRUPZJSU
|
|
QZUWFUURQUOZUCRUPZBSUQZUINUYNVUBUYRVUEUYMVUAJSUYMVUAUTUUTUYLUYTEUCRUY
|
|
JUUCUYSUUDUWBQUYJUUBUWEYRPUYJUUAUWDULUYJYTUWCUJYSUWAUHVAVBVCZVDUYKVIV
|
|
JWHVKUYQVUDBSUYQVUDUTUURSTUYPVUCEUCRUYJUYOUWFUURQUYJUUDUWBUUBUWEPUYKV
|
|
UFXJVNVJWHVKXKXLUVLUXJUXIUVMSTZAUXJUVCUVJUVEOWFZUVJUXIUVIUVEUVJUXNUXI
|
|
UXOUXPXMXPZDUUKWLWMZXNXOUYEUWIBUUNSUURUUNWIUYDUWHUCRUURUUNUWGQXQVOVPX
|
|
RXSUVKUVEUVOYFTZUVQUVSXTZUVLIYFTZUVNYFTZVUKAVUMUVCUVJUVEABCEFGHIJKLMN
|
|
YAWFZUVLVUGVUNVUJBUVMJYBXMIUVNYCWMVUKUVPUVRUIYFTUVEVULUVOYDUVDUUNUVRU
|
|
VPYEYGYHYIUVLBUUKUVBDIJVUOVUHAUVCUVJUVEWDVUIYJYOYKYLYMYNYPYQ $.
|
|
$}
|
|
|
|
${
|
|
caucvgprlemlim.q $e |- ( ph -> Q e. Q. ) $.
|
|
caucvgprlemlim.jk $e |- ( ph -> J <N K ) $.
|
|
caucvgprlemlim.jkq $e |- ( ph ->
|
|
( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q ) $.
|
|
${
|
|
$d F j l $. $d F l u $. $d K j l $. $d K l u $. $d Q j l $.
|
|
$d Q l u $. $d Q k $. $d L j $. $d L k $. $d j l $. $d j u $.
|
|
$d F k n $. $d j k $.
|
|
$( Lemma for ~ caucvgpr . Part of showing the putative limit to be a
|
|
limit. (Contributed by Jim Kingdon, 3-Oct-2020.) $)
|
|
caucvgprlem1 $p |- ( ph ->
|
|
<. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >.
|
|
<P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) $=
|
|
( cltq cfv wbr cab cop cpp c1st wcel cltp c1o ceq cec crq cplq cnpi
|
|
cv co wrex clti wa ltrelpi brel simprd caucvgprlemk ffvelrnd ltanqi
|
|
syl cnq syl2anc wceq opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d rspcev
|
|
breq12d crab wb oveq1 breq1d rexbidv elrab3 caucvgprlemladdrl sseld
|
|
sylbird mpd cnp caucvgprlemcl nqprlu addclpr nqprl mpbid ) AJHUAZKL
|
|
UOZDTUBLUCDBUOZTUBBUCUDZUEUPZUFUAZUGZWOWNTUBLUCWNWPTUBBUCUDWRUHUBZA
|
|
WNEUOZUIUDZUJUKZULUAZUMUPZXBHUAZDUMUPZTUBZEUNUQZWTAJUNUGZWNJUIUDZUJ
|
|
UKZULUAZUMUPZWNDUMUPZTUBZXJAIUNUGZXKAIJURUBXRXKUSRIJUNUNURUTVAVFVBZ
|
|
AXNDTUBWNVGUGZXQADIJRSVCAUNVGJHMXSVDZXNDWNVEVHXIXQEJUNXBJVIZXFXOXHX
|
|
PTYBXEXNWNUMYBXDXMULYBXCXLUJXBJUIVJVKVLVMYBXGWNDUMXBJHVNVOVQVPVHAXJ
|
|
WNWOXEUMUPZXHTUBZEUNUQZLVGVRZUGZWTAXTYGXJVSYAYEXJLWNVGWOWNVIZYDXIEU
|
|
NYHYCXFXHTWOWNXEUMVTWAWBWCVFAYFWSWNABCDEFGHKLMNOPQWDWEWFWGAXTWRWHUG
|
|
ZWTXAVSYAAKWHUGWQWHUGZYIABCEFGHKLMNOPWIADVGUGYJQBDLWJVFKWQWKVHBWNWR
|
|
LWLVHWM $.
|
|
$}
|
|
|
|
${
|
|
$d A j $. $d F j u x $. $d F l u x $. $d F n $. $d F k $.
|
|
$d K j u x $. $d K l u x $. $d L x $. $d L j $. $d L k $.
|
|
$d Q l u x $. $d ph x $. $d j l $. $d j k $. $d k n $.
|
|
$( Lemma for ~ caucvgpr . Part of showing the putative limit to be a
|
|
limit. (Contributed by Jim Kingdon, 3-Oct-2020.) $)
|
|
caucvgprlem2 $p |- ( ph ->
|
|
L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } ,
|
|
{ u | ( ( F ` K ) +Q Q ) <Q u } >. ) $=
|
|
( cnq vx cfv c1o cop ceq cec crq cplq co cv cltq wbr cltp wrex wcel
|
|
cab caucvgprlemk cnpi clti ltrelpi brel syl simprd ffvelrnd syl2anc
|
|
wa ltanqi ltbtwnnqq sylib c2nd c1st simprl simprrl wceq fveq2 opeq1
|
|
adantr eceq1d fveq2d oveq12d breq1d rspcev rexbidv crab fveq2i nqex
|
|
breq2 rabex op2nd eqtri elrab2 sylanbrc simprrr breq1 sylibr ltnqex
|
|
vex elab gtnqex op1st syl6eleqr rspe syl12anc caucvgprlemcl addclnq
|
|
cnp wb nqprlu ltdfpr mpbird rexlimddv ) AJHUBZJUCUDZUEUFZUGUBZUHUIZ
|
|
UAUJZUKULZXQXLDUHUIZUKULZVFZKLUJZXSUKULZLUPZXSBUJZUKULBUPZUDZUMULZU
|
|
ATAXPXSUKULZYAUATUNAXODUKULXLTUOZYIADIJRSUQAURTJHMAIURUOZJURUOZAIJU
|
|
SULYKYLVFRIJURURUSUTVAVBVCZVDZXODXLVGVEUAXPXSVHVIAXQTUOZYAVFZVFZYHX
|
|
QKVJUBZUOZXQYGVKUBZUOZVFZUATUNZYQYOYSUUAUUCAYOYAVLZYQYOEUJZHUBZUUEU
|
|
CUDZUEUFZUGUBZUHUIZXQUKULZEURUNZYSUUDYQYLXRUULAYLYPYMVQAYOXRXTVMUUK
|
|
XREJURUUEJVNZUUJXPXQUKUUMUUFXLUUIXOUHUUEJHVOUUMUUHXNUGUUMUUGXMUEUUE
|
|
JUCVPVRVSVTWAWBVEUUJYEUKULZEURUNZUULBXQTYRYEXQVNUUNUUKEURYEXQUUJUKW
|
|
GWCYRYBUUIUHUIUUFUKULEURUNZLTWDZUUOBTWDZUDZVJUBUURKUUSVJPWEUUQUURUU
|
|
PLTWFWHUUOBTWFWHWIWJWKWLYQXQYDYTYQXTXQYDUOAYOXRXTWMYCXTLXQUAWQYBXQX
|
|
SUKWNWRWOYDYFLXSWPBXSWSWTXAUUBUATXBXCYQKXFUOZYGXFUOZYHUUCXGAUUTYPAB
|
|
CEFGHKLMNOPXDVQAUVAYPAXSTUOZUVAAYJDTUOUVBYNQXLDXEVEBXSLXHVBVQKYGUAX
|
|
IVEXJXK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A j $. $d F j u $. $d F l u $. $d F k n $. $d j k ph x $.
|
|
$d k l u x $. $d j l $. $d L j $. $d L k $.
|
|
$( Lemma for ~ caucvgpr . The putative limit is a limit. (Contributed
|
|
by Jim Kingdon, 1-Oct-2020.) $)
|
|
caucvgprlemlim $p |- ( ph ->
|
|
A. x e. Q. E. j e. N. A. k e. N. ( j <N k ->
|
|
( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >.
|
|
<P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\
|
|
L <P <. { l | l <Q ( ( F ` k ) +Q x ) } ,
|
|
{ u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) $=
|
|
( cv wbr cltq cab wa cnpi clti cfv cop cpp co cltp cplq wral wrex cnq
|
|
wi wcel c1o ceq cec archrecnq adantl wf ad5antr simpr ad4antr simpllr
|
|
crq caucvgprlem1 caucvgprlem2 jca ex ralrimiva reximdva mpd ) AEOZFOZ
|
|
UAPZJOZVLHUBZQPJRVOCOZQPCRUCIVNBOZQPJRVQVPQPCRUCUDUEUFPZIVNVOVQUGUEZQ
|
|
PJRVSVPQPCRUCUFPZSZUKZFTUHZETUIZBUJAVQUJULZSZVKUMUCUNUOVCUBVQQPZETUIZ
|
|
WDWEWHAVQEUPUQWFWGWCETWFVKTULZSZWGWCWJWGSZWBFTWKVLTULZSZVMWAWMVMSZVRV
|
|
TWNCDVQEFGHVKVLIJATUJHURWEWIWGWLVMKUSZAGOZVLUAPWPHUBZVOWPUMUCUNUOVCUB
|
|
ZUGUEQPVOWQWRUGUEQPSUKFTUHGTUHWEWIWGWLVMLUSZADVKHUBQPETUHWEWIWGWLVMMU
|
|
SZNWFWEWIWGWLVMAWEUTVAZWMVMUTZWJWGWLVMVBZVDWNCDVQEFGHVKVLIJWOWSWTNXAX
|
|
BXCVEVFVGVHVGVIVJVH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A j $. $d F j k n l u x y z $. $d j k ph x $.
|
|
$( A Cauchy sequence of positive fractions with a modulus of convergence
|
|
converges to a positive real. This is basically Corollary 11.2.13 of
|
|
[HoTT], p. (varies) (one key difference being that this is for
|
|
positive reals rather than signed reals). Also, the HoTT book theorem
|
|
has a modulus of convergence (that is, a rate of convergence)
|
|
specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
|
|
convergence to say that all terms after the nth term must be within
|
|
` 1 / n ` of the nth term (it should later be able to prove versions
|
|
of this theorem with a different fixed rate or a modulus of
|
|
convergence supplied as a hypothesis). We also specify that every
|
|
term needs to be larger than a fraction ` A ` , to avoid the case
|
|
where we have positive terms which "converge" to zero (which is not a
|
|
positive real).
|
|
|
|
This proof (including its lemmas) is similar to the proofs of
|
|
~ cauappcvgpr and ~ caucvgprpr . Reading ~ cauappcvgpr first (the
|
|
simplest of the three) might help understanding the other two.
|
|
|
|
(Contributed by Jim Kingdon, 18-Jun-2020.) $)
|
|
caucvgpr $p |- ( ph ->
|
|
E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k ->
|
|
( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >.
|
|
<P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\
|
|
y <P <. { l | l <Q ( ( F ` k ) +Q x ) } ,
|
|
{ u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) $=
|
|
( cv cplq co cltq wbr cnpi cnq vz c1o cop ceq cec crq cfv wrex crab cnp
|
|
wcel clti cab cpp cltp wa wi wral wceq opeq1 eceq1d fveq2d oveq2d fveq2
|
|
breq12d cbvrexv a1i rabbiia oveq12d breq1d caucvgprlemcl caucvgprlemlim
|
|
wb opeq12i oveq1 breq2d breq1 anbi12d imbi2d rexralbidv ralbidv syl2anc
|
|
rspcev ) AJNZUANZUBUCZUDUEZUFUGZOPZWEIUGZQRZUASUHZJTUIZWJWHOPZDNZQRZUAS
|
|
UHZDTUIZUCZUJUKFNZGNZULRZWDXAIUGZQRJUMXCWOQRDUMUCZWSWDBNZQRJUMXEWOQRDUM
|
|
UCZUNPZUORZWSWDXCXEOPZQRJUMXIWOQRDUMUCZUORZUPZUQZGSURFSUHZBTURZXBXDCNZX
|
|
FUNPZUORZXPXJUORZUPZUQZGSURFSUHZBTURZCUJUHADEFGHIWSJKLMWMWDWTUBUCZUDUEZ
|
|
UFUGZOPZWTIUGZQRZFSUHZJTUIWRYHYFOPZWOQRZFSUHZDTUIWLYJJTWLYJVMWDTUKWKYIU
|
|
AFSWEWTUSZWIYGWJYHQYNWHYFWDOYNWGYEUFYNWFYDUDWEWTUBUTVAVBZVCWEWTIVDZVEVF
|
|
VGVHWQYMDTWQYMVMWOTUKWPYLUAFSYNWNYKWOQYNWJYHWHYFOYPYOVIVJVFVGVHVNZVKABD
|
|
EFGHIWSJKLMYQVLYCXOCWSUJXPWSUSZYBXNBTYRYAXMFGSSYRXTXLXBYRXRXHXSXKYRXQXG
|
|
XDUOXPWSXFUNVOVPXPWSXJUOVQVRVSVTWAWCWB $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d J l $. $d J u $. $d K l $. $d K u $.
|
|
caucvgprprlemk.jk $e |- ( ph -> J <N K ) $.
|
|
caucvgprprlemk.jkq $e |- ( ph ->
|
|
<. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >.
|
|
<P Q ) $.
|
|
$( Lemma for ~ caucvgprpr . Reciprocals of positive integers decrease as
|
|
the positive integers increase. (Contributed by Jim Kingdon,
|
|
28-Nov-2020.) $)
|
|
caucvgprprlemk $p |- ( ph ->
|
|
<. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >.
|
|
<P Q ) $=
|
|
( cv c1o cop ceq cec crq cfv cltq wbr cab cltp cnpi clti wcel brel ltnnnq
|
|
wa wb ltrelpi 3syl mpbid ltrnqi ltnqpri cnp ltsopr ltrelpr sotri syl2anc
|
|
) AFIZEJKLMZNOZPQFRUSBIZPQBRKZUQDJKLMZNOZPQFRVCUTPQBRKZSQZVDCSQVACSQAVBUR
|
|
PQZUSVCPQVEADEUAQZVFGAVGDTUBETUBUEVGVFUFGDETTUAUGUCDEUDUHUIVBURUJBUSVCFUK
|
|
UHHVAVDCSULUMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d M l u $. $d S l u $. $d T l u $.
|
|
caucvgprprlemloccalc.st $e |- ( ph -> S <Q T ) $.
|
|
caucvgprprlemloccalc.y $e |- ( ph -> Y e. Q. ) $.
|
|
caucvgprprlemloccalc.syt $e |- ( ph -> ( S +Q Y ) = T ) $.
|
|
caucvgprprlemloccalc.x $e |- ( ph -> X e. Q. ) $.
|
|
caucvgprprlemloccalc.xxy $e |- ( ph -> ( X +Q X ) <Q Y ) $.
|
|
caucvgprprlemloccalc.m $e |- ( ph -> M e. N. ) $.
|
|
caucvgprprlemloccalc.mx $e |- ( ph ->
|
|
( *Q ` [ <. M , 1o >. ] ~Q ) <Q X ) $.
|
|
$( Lemma for ~ caucvgprpr . Rearranging some expressions for
|
|
~ caucvgprprlemloc . (Contributed by Jim Kingdon, 8-Feb-2021.) $)
|
|
caucvgprprlemloccalc $p |- ( ph ->
|
|
( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } ,
|
|
{ u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >.
|
|
+P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. )
|
|
<P <. { l | l <Q T } , { u | T <Q u } >. ) $=
|
|
( co cltq wbr cab cnq c1o cop ceq cec crq cfv cplq cltp wcel wceq ltrelnq
|
|
cv cpp wa brel simpld cnpi nnnq recclnq addclnq syl2anc addnqpr addassnqg
|
|
syl 3syl syl3anc wi lt2addnq syl22anc mp2and ltsonq sotri breqtrd eqbrtrd
|
|
ltanqi ltnqpri eqbrtrrd ) AHULZCEUAUBUCUDZUEUFZUGPZVTUGPZQRHSWBBULZQRBSUB
|
|
ZVRWAQRHSWAWCQRBSUBVRVTQRHSVTWCQRBSUBUMPZVRDQRHSDWCQRBSUBZUHAWATUIZVTTUIZ
|
|
WDWEUJACTUIZWHWGAWIDTUIZACDQRWIWJUNICDTTQUKUOVDUPZAEUQUIVSTUIWHNEURVSUSVE
|
|
ZCVTUTVAWLBWAVTHVBVAAWBDQRWDWFUHRAWBCVTVTUGPZUGPZDQAWIWHWHWBWNUJWKWLWLCVT
|
|
VTVCVFAWNCGUGPZDQAWMGQRZWIWNWOQRAWMFFUGPZQRZWQGQRWPAVTFQRZWSWROOAWHFTUIZW
|
|
HWTWSWSUNWRVGWLLWLLVTFVTFVHVIVJMWMWQGQTVKUKVLVAWKWMGCVOVAKVMVNBWBDHVPVDVQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d F a b $. $d F l r $. $d F u r $. $d X a b p $. $d X l p r $.
|
|
$d X p r u $. $d X a b q $. $d a p r $. $d l q r $. $d q r u $.
|
|
caucvgprprlemell.lim $e |- L = <.
|
|
{ l e. Q. | E. r e. N.
|
|
<. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q }
|
|
>. <P ( F ` r ) } ,
|
|
{ u e. Q. | E. r e. N.
|
|
( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q u } , { q | u <Q q } >. }
|
|
>. $.
|
|
$( Lemma for ~ caucvgprpr . Membership in the lower cut of the putative
|
|
limit. (Contributed by Jim Kingdon, 21-Jan-2021.) $)
|
|
caucvgprprlemell $p |- ( X e. ( 1st ` L ) <-> ( X e. Q. /\
|
|
E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q }
|
|
>. <P ( F ` b ) ) ) $=
|
|
( cfv cnq cv cop cplq cltq wbr cab cltp cnpi va c1st wcel c1o ceq cec crq
|
|
co wrex wa wceq breq2d abbidv breq1d opeq12d rexbidv crab cpp fveq2i nqex
|
|
oveq1 rabex op1st eqtri elrab2 opeq1 eceq1d fveq2d oveq2d breq12d cbvrexv
|
|
fveq2 bitri anbi2i ) DCUBKZUCDLUCZGMZDEMZUDNZUEUFZUGKZOUHZPQZGRZWBFMZPQZF
|
|
RZNZVRBKZSQZETUIZUJVPVQDHMZUDNZUEUFZUGKZOUHZPQZGRZWPWEPQZFRZNZWLBKZSQZHTU
|
|
IZUJVQIMZWAOUHZPQZGRZXFWEPQZFRZNZWISQZETUIZWKIDLVOXEDUKZXLWJETXNXKWHWISXN
|
|
XHWDXJWGXNXGWCGXNXFWBVQPXEDWAOVAZULUMXNXIWFFXNXFWBWEPXOUNUMUOUNUPVOXMILUQ
|
|
ZWIVQWAPQGRWAWEPQFRNURUHVQAMZPQGRXQWEPQFRNSQETUIZALUQZNZUBKXPCXTUBJUSXPXS
|
|
XMILUTVBXRALUTVBVCVDVEWKXDVPWKVQDUAMZUDNZUEUFZUGKZOUHZPQZGRZYEWEPQZFRZNZY
|
|
ABKZSQZUATUIXDWJYLEUATVRYAUKZWHYJWIYKSYMWDYGWGYIYMWCYFGYMWBYEVQPYMWAYDDOY
|
|
MVTYCUGYMVSYBUEVRYAUDVFVGVHVIZULUMYMWFYHFYMWBYEWEPYNUNUMUOVRYABVLVJVKYLXC
|
|
UAHTYAWLUKZYJXAYKXBSYOYGWRYIWTYOYFWQGYOYEWPVQPYOYDWODOYOYCWNUGYOYBWMUEYAW
|
|
LUDVFVGVHVIZULUMYOYHWSFYOYEWPWEPYPUNUMUOYAWLBVLVJVKVMVNVM $.
|
|
|
|
$( Lemma for ~ caucvgprpr . Membership in the upper cut of the putative
|
|
limit. (Contributed by Jim Kingdon, 28-Jan-2021.) $)
|
|
caucvgprprlemelu $p |- ( X e. ( 2nd ` L ) <-> ( X e. Q. /\ E. b e. N.
|
|
( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q X } , { q | X <Q q } >. ) ) $=
|
|
( cfv cnq cv cop cltq wbr cab cltp cnpi abbidv va c2nd c1o ceq cec crq co
|
|
wcel cpp wrex wa wceq breq2 breq1 opeq12d breq2d rexbidv cplq crab fveq2i
|
|
rabex op2nd eqtri elrab2 fveq2 opeq1 eceq1d fveq2d breq1d oveq12d cbvrexv
|
|
nqex bitri anbi2i ) DCUBKZUHDLUHZEMZBKZGMZVQUCNZUDUEZUFKZOPZGQZWBFMZOPZFQ
|
|
ZNZUIUGZVSDOPZGQZDWEOPZFQZNZRPZESUJZUKVPHMZBKZVSWQUCNZUDUEZUFKZOPZGQZXAWE
|
|
OPZFQZNZUIUGZWNRPZHSUJZUKWIVSAMZOPZGQZXJWEOPZFQZNZRPZESUJZWPADLVOXJDULZXP
|
|
WOESXRXOWNWIRXRXLWKXNWMXRXKWJGXJDVSOUMTXRXMWLFXJDWEOUNTUOUPUQVOVSIMWBURUG
|
|
ZOPGQXSWEOPFQNVRRPESUJZILUSZXQALUSZNZUBKYBCYCUBJUTYAYBXTILVLVAXQALVLVAVBV
|
|
CVDWPXIVPWPUAMZBKZVSYDUCNZUDUEZUFKZOPZGQZYHWEOPZFQZNZUIUGZWNRPZUASUJXIWOY
|
|
OEUASVQYDULZWIYNWNRYPVRYEWHYMUIVQYDBVEYPWDYJWGYLYPWCYIGYPWBYHVSOYPWAYGUFY
|
|
PVTYFUDVQYDUCVFVGVHZUPTYPWFYKFYPWBYHWEOYQVITUOVJVIVKYOXHUAHSYDWQULZYNXGWN
|
|
RYRYEWRYMXFUIYDWQBVEYRYJXCYLXEYRYIXBGYRYHXAVSOYRYGWTUFYRYFWSUDYDWQUCVFVGV
|
|
HZUPTYRYKXDFYRYHXAWEOYSVITUOVJVIVKVMVNVM $.
|
|
$}
|
|
|
|
${
|
|
caucvgprpr.f $e |- ( ph -> F : N. --> P. ) $.
|
|
caucvgprpr.cau $e |- ( ph ->
|
|
A. n e. N. A. k e. N. ( n <N k ->
|
|
( ( F ` n ) <P ( ( F ` k ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
) /\
|
|
( F ` k ) <P ( ( F ` n ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
) ) ) ) $.
|
|
${
|
|
$d F a b k $. $d F a k n $. $d a b k l $. $d a b k u $. $d l n $.
|
|
$d n u $.
|
|
$( Lemma for ~ caucvgprpr . Change bound variables in Cauchy condition.
|
|
(Contributed by Jim Kingdon, 12-Feb-2021.) $)
|
|
caucvgprprlemcbv $p |- ( ph ->
|
|
A. a e. N. A. b e. N. ( a <N b ->
|
|
( ( F ` a ) <P ( ( F ` b ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >.
|
|
) /\
|
|
( F ` b ) <P ( ( F ` a ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >.
|
|
) ) ) ) $=
|
|
( cv clti wbr cfv cop cltq cpp co cltp cnpi c1o ceq cec crq cab wa wral
|
|
weq breq1 fveq2 opeq1 eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq2d
|
|
wi breq12d oveq12d anbi12d imbi12d breq2 oveq1d cbvral2v sylib ) ADKZCK
|
|
ZLMZVHENZVIENZHKZVHUAOZUBUCZUDNZPMZHUEZVPBKZPMZBUEZOZQRZSMZVLVKWBQRZSMZ
|
|
UFZUSZCTUGDTUGFKZGKZLMZWIENZWJENZVMWIUAOZUBUCZUDNZPMZHUEZWPVSPMZBUEZOZQ
|
|
RZSMZWMWLXAQRZSMZUFZUSZGTUGFTUGJWHXGWIVILMZWLVLXAQRZSMZVLXDSMZUFZUSDCFG
|
|
TTDFUHZVJXHWGXLVHWIVILUIXMWDXJWFXKXMVKWLWCXISVHWIEUJZXMWBXAVLQXMVRWRWAW
|
|
TXMVQWQHXMVPWPVMPXMVOWOUDXMVNWNUBVHWIUAUKULUMZUNUOXMVTWSBXMVPWPVSPXOUPU
|
|
OUQZURUTXMWEXDVLSXMVKWLWBXAQXNXPVAUNVBVCCGUHZXHWKXLXFVIWJWILVDXQXJXCXKX
|
|
EXQXIXBWLSXQVLWMXAQVIWJEUJZVEUNXQVLWMXDSXRUPVBVCVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d A a b l $. $d A a b u $. $d A l p $. $d A q u $. $d B b $.
|
|
$d F a b k $. $d F a k n $. $d k l n $. $d k n u $. $d a b k u $.
|
|
$( Lemma for ~ caucvgprpr . Cauchy condition expressed in terms of
|
|
classes. (Contributed by Jim Kingdon, 3-Mar-2021.) $)
|
|
caucvgprprlemval $p |- ( ( ph /\ A <N B ) ->
|
|
( ( F ` A ) <P ( ( F ` B ) +P.
|
|
<. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >.
|
|
) /\
|
|
( F ` B ) <P ( ( F ` A ) +P.
|
|
<. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >.
|
|
) ) ) $=
|
|
( clti wbr wa cltq cpp co cltp cnpi va cfv c1o cop ceq cec crq cab wcel
|
|
vb cv wral ltrelpi brel adantl caucvgprprlemcbv adantr simpr wceq breq1
|
|
fveq2 eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq2d breq12d oveq12d
|
|
wi opeq1 anbi12d imbi12d breq2 oveq1d rspc2v syl3c cbvabv oveq2i breq2i
|
|
opeq12i anbi12i sylib ) ACDMNZOZCGUBZDGUBZJUKZCUCUDZUEUFZUGUBZPNZJUHZWL
|
|
BUKZPNZBUHZUDZQRZSNZWHWGWRQRZSNZOZWGWHIUKZWLPNZIUHZWLHUKZPNZHUHZUDZQRZS
|
|
NZWHWGXJQRZSNZOWFCTUIDTUIOZUAUKZUJUKZMNZXPGUBZXQGUBZWIXPUCUDZUEUFZUGUBZ
|
|
PNZJUHZYCWOPNZBUHZUDZQRZSNZXTXSYHQRZSNZOZVKZUJTULUATULZWEXCWEXOACDTTMUM
|
|
UNUOAYOWEABEFGUAUJJKLUPUQAWEURYNWEXCVKCXQMNZWGXTWRQRZSNZXTXASNZOZVKUAUJ
|
|
CDTTXPCUSZXRYPYMYTXPCXQMUTUUAYJYRYLYSUUAXSWGYIYQSXPCGVAZUUAYHWRXTQUUAYE
|
|
WNYGWQUUAYDWMJUUAYCWLWIPUUAYBWKUGUUAYAWJUEXPCUCVLVBVCZVDVEUUAYFWPBUUAYC
|
|
WLWOPUUCVFVEVGZVHVIUUAYKXAXTSUUAXSWGYHWRQUUBUUDVJVDVMVNXQDUSZYPWEYTXCXQ
|
|
DCMVOUUEYRWTYSXBUUEYQWSWGSUUEXTWHWRQXQDGVAZVPVDUUEXTWHXASUUFVFVMVNVQVRW
|
|
TXLXBXNWSXKWGSWRXJWHQWNXFWQXIWMXEJIWIXDWLPUTVSWPXHBHWOXGWLPVOVSWBZVTWAX
|
|
AXMWHSWRXJWGQUUGVTWAWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d F f g h $. $d F k n $. $d J f g h $. $d K f g h $. $d S f g h $.
|
|
$d f g h ph $.
|
|
caucvgprprlemnkj.k $e |- ( ph -> K e. N. ) $.
|
|
caucvgprprlemnkj.j $e |- ( ph -> J e. N. ) $.
|
|
caucvgprprlemnkj.s $e |- ( ph -> S e. Q. ) $.
|
|
${
|
|
$d J f g h p q $. $d K f g h p q $. $d K l p $. $d K q u $.
|
|
$d S f g h p q $. $d k l n $. $d k n u $.
|
|
$( Lemma for ~ caucvgprpr . Part of disjointness. (Contributed by Jim
|
|
Kingdon, 12-Feb-2021.) $)
|
|
caucvgprprlemnkltj $p |- ( ( ph /\ K <N J ) -> -.
|
|
( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
|
|
<P ( F ` K ) /\
|
|
( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q S } , { q | S <Q q } >. ) ) $=
|
|
( wbr cltp cnp wcel vf vg vh clti wa cv c1o cop ceq cec crq cplq cltq
|
|
cfv co cab ltsopr ltrelpr simprl caucvgprprlemval simpld adantr sotri
|
|
cpp son2lpi syl2anc w3a wb ltaprg adantl cnq ad2antrr nqprlu syl cnpi
|
|
ffvelrnd recnnpr wceq addcomprg caovord2d mpbird simprr jca mtoi nnnq
|
|
ltaddpr ex recclnq 3syl addnqpr breq1d anbi1d mtbird ) AHGUDQZUEZJUFZ
|
|
CHUGUHUIUJZUKUNZULUOZUMQJUPWSIUFZUMQIUPUHZHFUNZRQZGFUNZWPGUGUHUIUJUKU
|
|
NZUMQJUPXEWTUMQIUPUHZVDUOZWPCUMQJUPCWTUMQIUPUHZRQZUEXHWPWRUMQJUPWRWTU
|
|
MQIUPUHZVDUOZXBRQZXIUEZWOXMXHXDRQZXDXHRQZUEZXHXDRSUQURVEWOXMXPWOXMUEZ
|
|
XNXOXQXNXKXDXJVDUOZRQZXQXLXBXRRQZXSWOXLXIUSWOXTXMWOXTXDXBXJVDUORQABHG
|
|
DEFIJKLMUTVAVBXKXBXRRSUQURVCVFXQUAUBUCXHXDXJRSVDUAUFZSTZUBUFZSTZUCUFZ
|
|
STVGYAYCRQYEYAVDUOYEYCVDUORQVHXQYAYCYEVIVJXQCVKTZXHSTAYFWNXMPVLICJVMV
|
|
NAXDSTZWNXMAVOSGFLOVPVLZAXJSTZWNXMAHVOTZYINIHJVQVNVLYBYDUEYAYCVDUOYCY
|
|
AVDUOVRXQYAYCVSVJVTWAXQXDXGRQZXIXOXQYGXFSTZYKYHAYLWNXMAGVOTYLOIGJVQVN
|
|
VLXDXFWFVFWOXLXIWBXDXGXHRSUQURVCVFWCWGWDWOXCXLXIWOXAXKXBRWOYFWRVKTZXA
|
|
XKVRAYFWNPVBAYMWNAYJWQVKTYMNHWEWQWHWIVBICWRJWJVFWKWLWM $.
|
|
$}
|
|
|
|
${
|
|
$d J p q $. $d K p q $. $d S p q $.
|
|
$( Lemma for ~ caucvgprpr . Part of disjointness. (Contributed by Jim
|
|
Kingdon, 12-Feb-2021.) $)
|
|
caucvgprprlemnkeqj $p |- ( ( ph /\ K = J ) -> -.
|
|
( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
|
|
<P ( F ` K ) /\
|
|
( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q S } , { q | S <Q q } >. ) ) $=
|
|
( cltq wbr cab cltp wceq wa cv c1o cop ceq cec crq cfv cplq co ltsopr
|
|
cpp cnp ltrelpr son2lpi wcel ffvelrnd ad2antrr cnq adantr syl recclnq
|
|
cnpi nnnq nqprlu ltaddpr syl2anc simprr sotri simprl wb addnqpr mpbid
|
|
breq1d ex mtoi opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2
|
|
jca breq12d anbi1d adantl mtbird ) AHGUAZUBZJUCZCHUDUEZUFUGZUHUIZUJUK
|
|
ZQRZJSZWQIUCZQRZISZUEZHFUIZTRZGFUIZWMGUDUEZUFUGZUHUIZQRJSXIWTQRISUEZU
|
|
MUKZWMCQRJSCWTQRISUEZTRZUBZWMCXIUJUKZQRZJSZXOWTQRZISZUEZXFTRZXMUBZWLY
|
|
BXFXLTRZXLXFTRZUBZXFXLTUNULUOUPWLYBYEWLYBUBZYCYDYFXFXKTRZXMYCYFXFUNUQ
|
|
ZXJUNUQZYGAYHWKYBAVDUNGFLOURUSWLYIYBWLXIUTUQZYIWLXHUTUQZYJWLGVDUQZYKA
|
|
YLWKOVAGVEVBXHVCVBZIXIJVFVBZVAXFXJVGVHWLYAXMVIXFXKXLTUNULUOVJVHYFXLXL
|
|
XJUMUKZTRZYOXFTRZYDWLYPYBWLXLUNUQZYIYPWLCUTUQZYRAYSWKPVAZICJVFVBYNXLX
|
|
JVGVHVAYFYAYQWLYAXMVKWLYAYQVLYBWLXTYOXFTWLYSYJXTYOUAYTYMICXIJVMVHVOVA
|
|
VNXLYOXFTUNULUOVJVHWFVPVQWKXNYBVLAWKXEYAXMWKXCXTXDXFTWKWSXQXBXSWKWRXP
|
|
JWKWQXOWMQWKWPXICUJWKWOXHUHWKWNXGUFHGUDVRVSVTWAZWBWCWKXAXRIWKWQXOWTQU
|
|
UAVOWCWDHGFWEWGWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d J l p $. $d J q u $. $d K p $. $d K q $. $d S p $. $d S q $.
|
|
$d k u $. $d k l n $. $d n u $.
|
|
$( Lemma for ~ caucvgprpr . Part of disjointness. (Contributed by Jim
|
|
Kingdon, 12-Feb-2021.) $)
|
|
caucvgprprlemnjltk $p |- ( ( ph /\ J <N K ) -> -.
|
|
( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
|
|
<P ( F ` K ) /\
|
|
( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q S } , { q | S <Q q } >. ) ) $=
|
|
( wbr cltq cab cltp clti wa cv c1o cop ceq cec crq cfv cplq co ltsopr
|
|
cpp cnp ltrelpr son2lpi wcel ad2antrr recclnq ltaddnq syl2anc ltnqpri
|
|
cnq cnpi nnnq 3syl simprl caucvgprprlemval simprd adantr sotri simprr
|
|
syl jca ex mtoi ) AGHUAQZUBZJUCZCHUDUEUFUGZUHUIZUJUKZRQJSWBIUCZRQISUE
|
|
ZHFUIZTQZGFUIZVSGUDUEUFUGUHUIZRQJSWHWCRQISUEZUMUKZVSCRQJSCWCRQISUEZTQ
|
|
ZUBZWKWJTQZWLUBZWKWJTUNULUOUPVRWMWOVRWMUBZWNWLWPWKWDTQZWDWJTQZWNWPCWB
|
|
RQZWQWPCVCUQZWAVCUQZWSAWTVQWMPURWPHVDUQZVTVCUQXAAXBVQWMNURHVEVTUSVFCW
|
|
AUTVAICWBJVBVMWPWFWEWJTQZWRVRWFWLVGVRXCWMVRWGWEWIUMUKTQXCABGHDEFIJKLM
|
|
VHVIVJWDWEWJTUNULUOVKVAWKWDWJTUNULUOVKVAVRWFWLVLVNVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d J p q $. $d J u $. $d J l $. $d K p q $. $d K l $. $d K u $.
|
|
$d S p q $. $d n u $. $d l n $. $d k l $. $d k u $. $d q u $.
|
|
$d p l $.
|
|
$( Lemma for ~ caucvgprpr . Part of disjointness. (Contributed by Jim
|
|
Kingdon, 20-Jan-2021.) $)
|
|
caucvgprprlemnkj $p |- ( ph -> -.
|
|
( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >.
|
|
<P ( F ` K ) /\
|
|
( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q S } , { q | S <Q q } >. ) ) $=
|
|
( wbr cop cltq cab clti cv c1o ceq cec crq cfv cplq co cltp cpp wa wn
|
|
caucvgprprlemnkltj caucvgprprlemnkeqj caucvgprprlemnjltk w3o pitri3or
|
|
wceq cnpi wcel syl2anc mpjao3dan ) AHGUAQZJUBZCHUCRUDUEUFUGUHUIZSQJTV
|
|
FIUBZSQITRHFUGUJQGFUGVEGUCRUDUEUFUGZSQJTVHVGSQITRUKUIVECSQJTCVGSQITRU
|
|
JQULUMHGUSZGHUAQZABCDEFGHIJKLMNOPUNABCDEFGHIJKLMNOPUOABCDEFGHIJKLMNOP
|
|
UPAHUTVAGUTVAVDVIVJUQNOHGURVBVC $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d B k l n $. $d B l p $. $d B p q x y z $. $d B k n u $. $d F k n $.
|
|
$d F x y z $. $d J k l n $. $d J l p $. $d J p q x y z $.
|
|
$d J k n u $. $d ph x y z $. $d q u $.
|
|
caucvgprprlemnbj.b $e |- ( ph -> B e. N. ) $.
|
|
caucvgprprlemnbj.j $e |- ( ph -> J e. N. ) $.
|
|
$( Lemma for ~ caucvgprpr . Non-existence of two elements of the
|
|
sequence which are too far from each other. (Contributed by Jim
|
|
Kingdon, 17-Jun-2021.) $)
|
|
caucvgprprlemnbj $p |- ( ph ->
|
|
-. ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. )
|
|
+P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. )
|
|
<P ( F ` J ) ) $=
|
|
( vp cltq wbr cpp co cltp cnp wcel vq vx vy cfv c1o cop ceq cec crq cab
|
|
vz cv wn clti wceq wa caucvgprprlemval simprd cnpi ffvelrnd recnnpr syl
|
|
addclpr syl2anc ltaddpr adantr ltsopr ltrelpr sotri fveq2 breq1d adantl
|
|
wb mpbid simpld w3a ltaprg addcomprg caovord2d pitri3or mpjao3dan so2nr
|
|
w3o wor mpan imnan sylibr mpd breq1 cbvabv breq2 opeq12i oveq2i oveq12i
|
|
wi breq1i sylnib ) ACFUDZMULZCUEUFUGUHUIUDZNOZMUJZWTUAULZNOZUAUJZUFZPQZ
|
|
WSGUEUFUGUHUIUDZNOZMUJZXHXCNOZUAUJZUFZPQZGFUDZROZWRHULZWTNOZHUJZWTBULZN
|
|
OZBUJZUFZPQZXQXHNOZHUJZXHXTNOZBUJZUFZPQZXOROAXOXNROZXPUMZACGUNOZYKCGUOZ
|
|
GCUNOZAYMUPZXOXGROZXGXNROZYKYPWRXOXFPQROYQABCGDEFUAMHIJUQURAYRYMAXGSTZX
|
|
MSTZYRAWRSTZXFSTZYSAUSSCFIKUTZACUSTZUUBKUACMVAVBZWRXFVCVDZAGUSTZYTLUAGM
|
|
VAVBZXGXMVEVDZVFXOXGXNRSVGVHVIZVDAYNUPZYQYRYKUUKWRXGROZYQAUULYNAUUAUUBU
|
|
ULUUCUUEWRXFVEVDZVFYNUULYQVMAYNWRXOXGRCGFVJVKVLVNAYRYNUUIVFUUJVDAYOUPZX
|
|
OWRXMPQZROZUUOXNROZYKUUNUUPWRXOXMPQROABGCDEFUAMHIJUQVOAUUQYOAUULUUQUUMA
|
|
UBUCUKWRXGXMRSPUBULZSTZUCULZSTZUKULZSTVPUURUUTROUVBUURPQUVBUUTPQROVMAUU
|
|
RUUTUVBVQVLUUCUUFUUHUUSUVAUPUURUUTPQUUTUURPQUOAUURUUTVRVLVSVNVFXOUUOXNR
|
|
SVGVHVIVDAUUDUUGYMYNYOWCKLCGVTVDWAAYKXPUPUMZYKYLWOAXOSTZXNSTZUVCAUSSGFI
|
|
LUTAYSYTUVEUUFUUHXGXMVCVDSRWDUVDUVEUPUVCVGSXOXNRWBWEVDYKXPWFWGWHXNYJXOR
|
|
XGYDXMYIPXFYCWRPXBXSXEYBXAXRMHWSXQWTNWIWJXDYAUABXCXTWTNWKWJWLWMXJYFXLYH
|
|
XIYEMHWSXQXHNWIWJXKYGUABXCXTXHNWKWJWLWNWPWQ $.
|
|
$}
|
|
|
|
$d A m $. $d F m $.
|
|
caucvgprpr.bnd $e |- ( ph -> A. m e. N. A <P ( F ` m ) ) $.
|
|
|
|
${
|
|
caucvgprpr.lim $e |- L = <.
|
|
{ l e. Q. | E. r e. N.
|
|
<. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } ,
|
|
{ q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q }
|
|
>. <P ( F ` r ) } ,
|
|
{ u e. Q. | E. r e. N.
|
|
( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P
|
|
<. { p | p <Q u } , { q | u <Q q } >. }
|
|
>. $.
|
|
|
|
${
|
|
$d A m r $. $d A r s x $. $d F l $. $d F m $. $d L x $.
|
|
$d l p q r s $. $d l u x $. $d ph r s x $.
|
|
$( Lemma for ~ caucvgprpr . The lower cut of the putative limit is
|
|
inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) $)
|
|
caucvgprprlemml $p |- ( ph -> E. s e. Q. s e. ( 1st ` L ) ) $=
|
|
( cnq wbr cltq vx c1st cfv wcel wrex cnp c1o cltp cnpi wral 1pi fveq2
|
|
cv wceq breq2d rspcv mpsyl ltrelpr brel simpld syl c2nd cop prop prml
|
|
wa cplq co subhalfnqq ad2antrl ceq cec crq cab simplr archrecnq simpr
|
|
nnnq recclnq 3syl ad2antrr ltanqg syl3anc mpbid simpllr ltrelnq sotri
|
|
wb ltsonq syl2anc wi ad5antr simprr ad4antr prcdnql sylan mpd addclnq
|
|
nqprl sylc ltsopr ex reximdva oveq1 abbidv breq1d opeq12d rexbidv cpp
|
|
crab fveq2i nqex rabex op1st eqtri elrab2 sylanbrc rexlimddv ) AUAUMZ
|
|
CUBUCZUDZIUMZHUBUCZUDZIRUEZUARACUFUDZYAUARUEZACUGGUCZUHSZYFUGUIUDACEU
|
|
MZGUCZUHSZEUIUJZYIUKPYLYIEUGUIYJUGUNYKYHCUHYJUGGULUOUPUQYIYFYHUFUDCYH
|
|
UFUFUHURUSUTVAZYFXTCVBUCZVCUFUDZYGCVDZUAYOXTVEVAVAAXSRUDZYAVFZVFZYBYB
|
|
VGVHZXSTSZIRUEZYEYRUUCAYAIXSVIVJYTUUBYDIRYTYBRUDZVFZUUBYDUUEUUBVFZUUD
|
|
LUMZYBJUMZUGVCVKVLZVMUCZVGVHZTSZLVNZUUKKUMZTSZKVNZVCZUUHGUCZUHSZJUIUE
|
|
ZYDYTUUDUUBVOZUUFUUJYBTSZJUIUEZUUTUUFUUDUVCUVAYBJVPVAUUFUVBUUSJUIUUFU
|
|
UHUIUDZVFZUVBUUSUVEUVBVFZUUQCUHSZCUURUHSZUUSUVFUUKXTUDZUVGUVFUUKXSTSZ
|
|
UVIUVFUUKUUATSZUUBUVJUVFUVBUVKUVEUVBVQUVFUUJRUDZUUDUUDUVBUVKWHUVFUVDU
|
|
UIRUDUVLUUFUVDUVBVOZUUHVRUUIVSVTZUUFUUDUVDUVBUVAWAZUVOUUJYBYBWBWCWDUU
|
|
EUUBUVDUVBWEUUKUUAXSTRWIWFWGWJUVFYFYAUVJUVIWKZAYFYSUUDUUBUVDUVBYNWLZY
|
|
TYAUUDUUBUVDUVBAYRYAWMWNYFYPYAUVPYQXSUUKYOXTWOWPWJWQUVFUUKRUDZYFUVIUV
|
|
GWHUVFUUDUVLUVRUVOUVNYBUUJWRWJUVQKUUKCLWSWJWDUVFUVDYMUVHUVMAYMYSUUDUU
|
|
BUVDUVBPWLYLUVHEUUHUIYJUUHUNYKUURCUHYJUUHGULUOUPWTUUQCUURUHUFXAURWGWJ
|
|
XBXCWQUUGMUMZUUJVGVHZTSZLVNZUVTUUNTSZKVNZVCZUURUHSZJUIUEZUUTMYBRYCUVS
|
|
YBUNZUWFUUSJUIUWHUWEUUQUURUHUWHUWBUUMUWDUUPUWHUWAUULLUWHUVTUUKUUGTUVS
|
|
YBUUJVGXDZUOXEUWHUWCUUOKUWHUVTUUKUUNTUWIXFXEXGXFXHYCUWGMRXJZUURUUGUUJ
|
|
TSLVNUUJUUNTSKVNVCXIVHUUGBUMZTSLVNUWKUUNTSKVNVCUHSJUIUEZBRXJZVCZUBUCU
|
|
WJHUWNUBQXKUWJUWMUWGMRXLXMUWLBRXLXMXNXOXPXQXBXCWQXR $.
|
|
$}
|
|
|
|
${
|
|
$d A m r $. $d F f g h x $. $d F m $. $d F r u x $. $d L t x $.
|
|
$d f g h p q x $. $d f g h ph x $. $d p q r u x $.
|
|
$( Lemma for ~ caucvgprpr . The upper cut of the putative limit is
|
|
inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) $)
|
|
caucvgprprlemmu $p |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) ) $=
|
|
( wcel cltq wbr vx vf vg vh cv c1o cfv c2nd cnq wrex cnp c1st cop 1pi
|
|
cnpi a1i ffvelrnd prop prmu 3syl wa c1q co simprl 1nq addclnq sylancl
|
|
cplq ceq cec crq cab cltp simprr wb adantr nqpru syl2anc mpbid ltaprg
|
|
cpp w3a adantl nqprlu syl mp1i wceq addcomprg caovord2d fveq2i rec1nq
|
|
eqtr3i breq2i abbii breq1i opeq12i oveq2i addnqpr 3brtr4d fveq2 opeq1
|
|
eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq12d rspcev breq2 breq1
|
|
df-1nqqs rexbidv crab nqex rabex op2nd eqtri sylanbrc eleq1 rexlimddv
|
|
elrab2 ) AUAUEZUFHUGZUHUGZRZCUEZIUHUGZRZCUIUJZUAUIAYDUKRZYDULUGZYEUMU
|
|
KRYFUAUIUJAUOUKUFHNUFUORZAUNUPUQZYDURUAYEYLUSUTAYCUIRZYFVAZVAZYCVBVHV
|
|
CZUIRZYRYHRZYJYQYOVBUIRZYSAYOYFVDZVEYCVBVFVGZYQYSJUEZHUGZLUEZUUDUFUMZ
|
|
VIVJZVKUGZSTZLVLZUUIKUEZSTZKVLZUMZWAVCZUUFYRSTZLVLZYRUULSTZKVLZUMZVMT
|
|
ZJUOUJZYTUUCYQYMYDUUFUFUFUMZVIVJZVKUGZSTZLVLZUVFUULSTZKVLZUMZWAVCZUVA
|
|
VMTZUVCYMYQUNUPYQYDUUFVBSTZLVLZVBUULSTZKVLZUMZWAVCZUUFYCSTLVLYCUULSTK
|
|
VLUMZUVRWAVCZUVLUVAVMYQYDUVTVMTZUVSUWAVMTYQYFUWBAYOYFVNYQYOYKYFUWBVOU
|
|
UBAYKYPYNVPZKYCYDLVQVRVSYQUBUCUDYDUVTUVRVMUKWAUBUEZUKRZUCUEZUKRZUDUEZ
|
|
UKRWBUWDUWFVMTUWHUWDWAVCUWHUWFWAVCVMTVOYQUWDUWFUWHVTWCUWCYQYOUVTUKRUU
|
|
BKYCLWDWEUUAUVRUKRYQVEKVBLWDWFUWEUWGVAUWDUWFWAVCUWFUWDWAVCWGYQUWDUWFW
|
|
HWCWIVSUVLUVSWGYQUVKUVRYDWAUVHUVOUVJUVQUVGUVNLUVFVBUUFSVBVKUGUVFVBVBU
|
|
VEVKXLWJWKWLZWMWNUVIUVPKUVFVBUULSUWIWOWNWPWQUPYQYOUUAUVAUWAWGUUBVEKYC
|
|
VBLWRVGWSUVBUVMJUFUOUUDUFWGZUUPUVLUVAVMUWJUUEYDUUOUVKWAUUDUFHWTUWJUUK
|
|
UVHUUNUVJUWJUUJUVGLUWJUUIUVFUUFSUWJUUHUVEVKUWJUUGUVDVIUUDUFUFXAXBXCZX
|
|
DXEUWJUUMUVIKUWJUUIUVFUULSUWKXFXEXGXHXFXIVRUUPUUFBUEZSTZLVLZUWLUULSTZ
|
|
KVLZUMZVMTZJUOUJZUVCBYRUIYHUWLYRWGZUWRUVBJUOUWTUWQUVAUUPVMUWTUWNUURUW
|
|
PUUTUWTUWMUUQLUWLYRUUFSXJXEUWTUWOUUSKUWLYRUULSXKXEXGXDXMYHUUFMUEUUIVH
|
|
VCZSTLVLUXAUULSTKVLUMUUEVMTJUOUJZMUIXNZUWSBUIXNZUMZUHUGUXDIUXEUHQWJUX
|
|
CUXDUXBMUIXOXPUWSBUIXOXPXQXRYBXSYIYTCYRUIYGYRYHXTXIVRYA $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A s $. $d F l $. $d F u r $. $d L t $. $d ph r s $.
|
|
$d l u p q r s $. $d m r $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is inhabited.
|
|
(Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemm $p |- ( ph -> ( E. s e. Q. s e. ( 1st ` L )
|
|
/\ E. t e. Q. t e. ( 2nd ` L ) ) ) $=
|
|
( cv cfv c1st wcel cnq wrex c2nd caucvgprprlemml caucvgprprlemmu jca
|
|
) AJSIUATUBJUCUDCSIUETUBCUCUDABDEFGHIJKLMNOPQRUFABCDEFGHIKLMNOPQRUGUH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d F a b l t $. $d F m $. $d F r $. $d F t u $.
|
|
$d L a b t $. $d b r $. $d a b l p q r s t $. $d p q r s t u $.
|
|
$d ph a b r t $.
|
|
$( Lemma for ~ caucvgprpr . The lower cut of the putative limit is
|
|
open. (Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemopl $p |- ( ( ph /\ s e. ( 1st ` L ) ) ->
|
|
E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) $=
|
|
( cltq wbr vb va cv c1st cfv wcel wa c1o cop ceq cec crq cplq co cltp
|
|
cab cnq wrex cnpi caucvgprprlemell simprbi adantl cnp ad2antrr simprl
|
|
c2nd wf ffvelrnd prop syl simprr wb simplbi ad2antlr recclnq ad2antrl
|
|
nnnq addclnq syl2anc nqprl mpbird prnmaxl wceq adantr ltaddnq breqtrd
|
|
addcomnqg ltrelnq sotri elprnql ltexnqq mpbid eqbrtrrd simpr breqtrrd
|
|
ltsonq simplr ltanqg syl3anc eqtr3d opeq1 eceq1d fveq2d oveq2d breq2d
|
|
ad3antrrr eqeltrd abbidv breq1d opeq12d fveq2 breq12d rspcev sylanbrc
|
|
jca ex reximdva mpd rexlimddv ) AJUCZIUDUEZUFZUGZMUCZXTUAUCZUHUIZUJUK
|
|
ZULUEZUMUNZSTMUPYILUCZSTLUPUIYEHUEZUOTZXTCUCZSTZYMYAUFZUGZCUQURZUAUSY
|
|
BYLUAUSURZAYBXTUQUFZYRBHIXTKLMUANRUTVAVBYCYEUSUFZYLUGZUGZYIUBUCZSTZYQ
|
|
UBYKUDUEZUUBUUEYKVFUEZUIVCUFZYIUUEUFZUUDUBUUEURUUBYKVCUFZUUGUUBUSVCYE
|
|
HAUSVCHVGYBUUAOVDYCYTYLVEZVHZYKVIVJZUUBUUHYLYCYTYLVKUUBYIUQUFZUUIUUHY
|
|
LVLUUBYSYHUQUFZUUMYBYSAUUAYBYSYDXTKUCZUHUIZUJUKZULUEZUMUNZSTMUPUUSYJS
|
|
TLUPUIUUOHUEZUOTKUSURBHIXTKLMKNRUTVMVNZYTUUNYCYLYTYGUQUFUUNYEVQYGVOVJ
|
|
VPZXTYHVRVSUUKLYIYKMVTVSWAUBYIUUFUUEWBVSUUBUUCUUEUFZUUDUGZUGZYHYMUMUN
|
|
ZUUCWCZCUQURZYQUVEYHUUCSTZUVHUVEYHYISTUUDUVIUVEYHYHXTUMUNZYISUVEUUNYS
|
|
YHUVJSTUUBUUNUVDUVBWDZUUBYSUVDUVAWDZYHXTWEVSUVEUUNYSUVJYIWCUVKUVLYHXT
|
|
WGVSWFUUBUVCUUDVKZYHYIUUCSUQWPWHWIVSUVEUUNUUCUQUFZUVIUVHVLUVKUVEUUGUV
|
|
CUVNUUBUUGUVDUULWDUUBUVCUUDVEZUUCUUFUUEWJVSCYHUUCWKVSWLUVEUVGYPCUQUVE
|
|
YMUQUFZUGZUVGYPUVQUVGUGZYNYOUVRYNUVJUVFSTZUVRUVJUUCUVFSUVRYIUVJUUCSUV
|
|
RYSUUNYIUVJWCUVEYSUVPUVGUVLVDZUVEUUNUVPUVGUVKVDZXTYHWGVSUVEUUDUVPUVGU
|
|
VMVDWMUVQUVGWNZWOUVRYSUVPUUNYNUVSVLUVTUVEUVPUVGWQZUWAXTYMYHWRWSWAUVRU
|
|
VPYDYMUURUMUNZSTZMUPZUWDYJSTZLUPZUIZUUTUOTZKUSURZYOUWCUVRYTYDYMYHUMUN
|
|
ZSTZMUPZUWLYJSTZLUPZUIZYKUOTZUWKUUBYTUVDUVPUVGUUJXFUVRUWLUUEUFZUWRUVR
|
|
UWLUUCUUEUVRUVFUWLUUCUVRUUNUVPUVFUWLWCUWAUWCYHYMWGVSUWBWTUVEUVCUVPUVG
|
|
UVOVDXGUVRUWLUQUFZUUIUWSUWRVLUVRUVPUUNUWTUWCUWAYMYHVRVSUUBUUIUVDUVPUV
|
|
GUUKXFLUWLYKMVTVSWLUWJUWRKYEUSUUOYEWCZUWIUWQUUTYKUOUXAUWFUWNUWHUWPUXA
|
|
UWEUWMMUXAUWDUWLYDSUXAUURYHYMUMUXAUUQYGULUXAUUPYFUJUUOYEUHXAXBXCXDZXE
|
|
XHUXAUWGUWOLUXAUWDUWLYJSUXBXIXHXJUUOYEHXKXLXMVSBHIYMKLMKNRUTXNXOXPXQX
|
|
RXSXS $.
|
|
$}
|
|
|
|
${
|
|
$d F l $. $d F u $. $d F b x y z $. $d F r $. $d L b x y z $.
|
|
$d l p s $. $d l q s $. $d l r s $. $d l p t $. $d u q s $.
|
|
$d u r s $. $d u p t $. $d b r p s x y z $. $d ph b r x y z $.
|
|
$d b r q t x y z $. $d b r t x y z $.
|
|
$( Lemma for ~ caucvgprpr . The lower cut of the putative limit is
|
|
lower. (Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemlol $p |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) ->
|
|
s e. ( 1st ` L ) ) $=
|
|
( cltq wbr vb vx vy vz c1st cfv wcel w3a cnq c1o cop ceq cec crq cplq
|
|
cv co cab cltp cnpi wrex ltrelnq brel simpld caucvgprprlemell simprbi
|
|
3ad2ant2 3ad2ant3 wa simpll2 wb ltanqg adantl ad2antrr simprd recclnq
|
|
simplr nnnq 3syl addcomnqg caovord2d mpbid ltnqpri syl ltsopr ltrelpr
|
|
wceq cnp sotri sylancom ex reximdva opeq1 eceq1d fveq2d oveq2d breq2d
|
|
mpd abbidv breq1d opeq12d fveq2 breq12d cbvrexv sylib sylanbrc ) AJUP
|
|
ZCUPZSTZXHIUEUFZUGZUHZXGUIUGZMUPZXGKUPZUJUKZULUMZUNUFZUOUQZSTZMURZXSL
|
|
UPZSTZLURZUKZXOHUFZUSTZKUTVAZXGXJUGXIAXMXKXIXMXHUIUGZXGXHUIUISVBVCZVD
|
|
VGZXLXNXGUAUPZUJUKZULUMZUNUFZUOUQZSTZMURZYPYBSTZLURZUKZYLHUFZUSTZUAUT
|
|
VAZYHXLXNXHYOUOUQZSTMURUUEYBSTLURUKZUUBUSTZUAUTVAZUUDXKAUUHXIXKYIUUHB
|
|
HIXHKLMUANRVEVFVHXLUUGUUCUAUTXLYLUTUGZVIZUUGUUCUUJUUGUUAUUFUSTZUUCUUJ
|
|
UUGVIZYPUUESTZUUKUULXIUUMAXIXKUUIUUGVJUULUBUCUDXGXHYOSUIUOUBUPZUIUGZU
|
|
CUPZUIUGZUDUPZUIUGUHUUNUUPSTUURUUNUOUQUURUUPUOUQSTVKUULUUNUUPUURVLVMX
|
|
LXMUUIUUGYKVNXLYIUUIUUGXIAYIXKXIXMYIYJVOVGVNUULUUIYNUIUGYOUIUGXLUUIUU
|
|
GVQYLVRYNVPVSUUOUUQVIUUNUUPUOUQUUPUUNUOUQWGUULUUNUUPVTVMWAWBLYPUUEMWC
|
|
WDUUAUUFUUBUSWHWEWFWIWJWKWLWRUUCYGUAKUTYLXOWGZUUAYEUUBYFUSUUSYRYAYTYD
|
|
UUSYQXTMUUSYPXSXNSUUSYOXRXGUOUUSYNXQUNUUSYMXPULYLXOUJWMWNWOWPZWQWSUUS
|
|
YSYCLUUSYPXSYBSUUTWTWSXAYLXOHXBXCXDXEBHIXGKLMKNRVEXF $.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d F l r s $. $d F m $. $d F b r s u $. $d L b s $.
|
|
$d b l p q r s t $. $d p q r s t u $. $d b ph r s $.
|
|
$( Lemma for ~ caucvgprpr . The upper cut of the putative limit is
|
|
open. (Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemopu $p |- ( ( ph /\ t e. ( 2nd ` L ) ) ->
|
|
E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) $=
|
|
( cltq wbr vb cv c2nd cfv wcel wa c1o cop ceq cec crq cab cpp co cltp
|
|
cnq wrex cnpi caucvgprprlemelu simprbi adantl simprr cnp wb ffvelrnda
|
|
recnnpr addclpr syl2anc ad2ant2r simplbi ad2antlr nqprlu ltdfpr mpbid
|
|
c1st syl simpr adantr nqpru vex breq1 ltnqex gtnqex op1st a1i anbi12d
|
|
elab2 biimpd reximdva simplr simplrl simprl fveq2 opeq1 eceq1d fveq2d
|
|
mpd breq2d abbidv breq1d opeq12d oveq12d rspcev sylanbrc ex rexlimddv
|
|
weq jca ) ACUBZIUCUDZUEZUFZUAUBZHUDZMUBZXMUGUHZUIUJZUKUDZSTZMULZXRLUB
|
|
ZSTZLULZUHZUMUNZXOXISTZMULZXIYASTLULZUHZUOTZJUBZXISTZYKXJUEZUFZJUPUQZ
|
|
UAURXKYJUAURUQZAXKXIUPUEZYPBHIXIKLMUANRUSZUTVAXLXMURUEZYJUFZUFZYEXOYK
|
|
STMULYKYASTLULUHZUOTZYLUFZJUPUQZYOUUAYKYEUCUDUEZYKYIVOUDZUEZUFZJUPUQZ
|
|
UUEUUAYJUUJXLYSYJVBUUAYEVCUEZYIVCUEZYJUUJVDAYSUUKXKYJAYSUFXNVCUEYDVCU
|
|
EZUUKAURVCXMHOVEYSUUMALXMMVFVAXNYDVGVHVIZUUAYQUULXKYQAYTXKYQYPYRVJVKL
|
|
XIMVLVPYEYIJVMVHVNUUAUUIUUDJUPUUAYKUPUEZUFZUUIUUDUUPUUFUUCUUHYLUUPUUO
|
|
UUKUUFUUCVDUUAUUOVQUUAUUKUUOUUNVRLYKYEMVSVHUUHYLVDUUPYFYLMYKUUGJVTXOY
|
|
KXISWAYGYHMXIWBLXIWCWDWGWEWFWHWIWQUUAUUDYNJUPUUPUUDYNUUPUUDUFZYLYMUUP
|
|
UUCYLVBUUQUUOKUBZHUDZXOUURUGUHZUIUJZUKUDZSTZMULZUVBYASTZLULZUHZUMUNZU
|
|
UBUOTZKURUQZYMUUAUUOUUDWJUUQYSUUCUVJUUPYSUUDXLYSYJUUOWKVRUUPUUCYLWLUV
|
|
IUUCKXMURKUAXGZUVHYEUUBUOUVKUUSXNUVGYDUMUURXMHWMUVKUVDXTUVFYCUVKUVCXS
|
|
MUVKUVBXRXOSUVKUVAXQUKUVKUUTXPUIUURXMUGWNWOWPZWRWSUVKUVEYBLUVKUVBXRYA
|
|
SUVLWTWSXAXBWTXCVHBHIYKKLMKNRUSXDXHXEWIWQXF $.
|
|
|
|
$( Lemma for ~ caucvgprpr . The upper cut of the putative limit is
|
|
upper. (Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemupu $p |- ( ( ph /\ s <Q t /\ s e. ( 2nd ` L ) ) ->
|
|
t e. ( 2nd ` L ) ) $=
|
|
( cltq wbr vb cv c2nd cfv wcel w3a cnq c1o cop ceq cec crq cab cpp co
|
|
cltp cnpi wrex ltrelnq brel simprd 3ad2ant2 caucvgprprlemelu 3ad2ant3
|
|
simprbi wi ltnqpri cnp ltsopr ltrelpr expcom syl reximdv mpd sylanbrc
|
|
sotri ) AJUBZCUBZSTZVQIUCUDZUEZUFZVRUGUEZUAUBZHUDMUBZWDUHUIUJUKULUDZS
|
|
TMUMWFLUBZSTLUMUIUNUOZWEVRSTMUMVRWGSTLUMUIZUPTZUAUQURZVRVTUEVSAWCWAVS
|
|
VQUGUEZWCVQVRUGUGSUSUTVAVBWBWHWEVQSTMUMVQWGSTLUMUIZUPTZUAUQURZWKWAAWO
|
|
VSWAWLWOBHIVQKLMUANRVCVEVDWBWNWJUAUQWBWMWIUPTZWNWJVFVSAWPWALVQVRMVGVB
|
|
WNWPWJWHWMWIUPVHVIVJVPVKVLVMVNBHIVRKLMUANRVCVO $.
|
|
$}
|
|
|
|
${
|
|
$d F l t $. $d F t u $. $d F r $. $d F s $. $d L s t $.
|
|
$d l p q r s t $. $d p q r s t u $. $d ph r s t $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is rounded.
|
|
(Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemrnd $p |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <->
|
|
E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) /\
|
|
A. t e. Q. ( t e. ( 2nd ` L ) <->
|
|
E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) ) ) $=
|
|
( wcel cnq cv c1st cfv cltq wbr wa wrex wb wral c2nd caucvgprprlemopl
|
|
caucvgprprlemlol 3expib impbid caucvgprprlemopu caucvgprprlemupu jca
|
|
ex rexlimdvw ralrimivw ) AJUAZIUBUCZSZVACUAZUDUEZVDVBSZUFZCTUGZUHZJTU
|
|
IVDIUJUCZSZVEVAVJSZUFZJTUGZUHZCTUIAVIJTAVCVHAVCVHABCDEFGHIJKLMNOPQRUK
|
|
URAVGVCCTAVEVFVCABCDEFGHIJKLMNOPQRULUMUSUNUTAVOCTAVKVNAVKVNABCDEFGHIJ
|
|
KLMNOPQRUOURAVMVKJTAVEVLVKABCDEFGHIJKLMNOPQRUPUMUSUNUTUQ $.
|
|
$}
|
|
|
|
${
|
|
$d F a b k n $. $d F l r $. $d F m $. $d F r u $. $d L a b k $.
|
|
$d a b l $. $d a u $. $d a b k p r s $. $d ph a b r s $.
|
|
$d a b k q r s $. $d l p r s $. $d l q r s $. $d p r s u $.
|
|
$d b q r s u $. $d p q $. $d n u $. $d l n $. $d k l $. $d k u $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is disjoint.
|
|
(Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemdisj $p |- ( ph -> A. s e. Q.
|
|
-. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) $=
|
|
( cfv wbr cnpi va vb cv c1st wcel c2nd wa wn cnq c1o cop ceq cec cplq
|
|
crq cltq cab cltp wrex wfal caucvgprprlemell simprbi caucvgprprlemelu
|
|
co cpp anim12i reeanv sylibr adantl wf ad2antrr clti wi simprl simprr
|
|
cnp simplbi ad2antrl adantr caucvgprprlemnkj pm2.21d rexlimdvva inegd
|
|
wral mpd ralrimivw ) AIUCZHUDRUEZWGHUFRUEZUGZUHIUIAWJAWJUGZLUCZWGUAUC
|
|
ZUJUKULUMUORUNVDZUPSLUQWNKUCZUPSKUQUKWMGRURSZUBUCZGRWLWQUJUKULUMUORZU
|
|
PSLUQWRWOUPSKUQUKVEVDWLWGUPSLUQWGWOUPSKUQUKURSZUGZUBTUSUATUSZUTWJXAAW
|
|
JWPUATUSZWSUBTUSZUGXAWHXBWIXCWHWGUIUEZXBBGHWGJKLUAMQVAVBWIXDXCBGHWGJK
|
|
LUBMQVCVBVFWPWSUAUBTTVGVHVIWKWTUTUAUBTTWKWMTUEZWQTUEZUGZUGZWTUTXHBWGD
|
|
FGWQWMKLMATVPGVJWJXGNVKAFUCZDUCZVLSXIGRZXJGRZMUCXIUJUKULUMUORZUPSMUQX
|
|
MBUCUPSBUQUKZVEVDURSXLXKXNVEVDURSUGVMDTWDFTWDWJXGOVKWKXEXFVNWKXEXFVOW
|
|
KXDXGWHXDAWIWHXDWLWGJUCZUJUKULUMUORUNVDZUPSLUQXPWOUPSKUQUKXOGRURSJTUS
|
|
BGHWGJKLJMQVAVQVRVSVTWAWBWEWCWF $.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d F a $. $d F b $. $d F f g h $. $d F l r $.
|
|
$d F u r $. $d F m $. $d L c x y $. $d a c p q s $. $d b c p q t $.
|
|
$d c f g h p q s t $. $d c f g h ph s t x y $. $d l p q r s t $.
|
|
$d u p q r s t $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is located.
|
|
(Contributed by Jim Kingdon, 21-Dec-2020.) $)
|
|
caucvgprprlemloc $p |- ( ph -> A. s e. Q. A. t e. Q.
|
|
( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) ) $=
|
|
( wbr wcel vy vx vc va vf vg vh vb cv cltq c1st cfv c2nd wo wi cnq wa
|
|
cplq wceq wrex ltexnqi adantl subhalfnqq ad2antrl c1o cop ceq cec crq
|
|
co cnpi archrecnq cab cpp simpllr adantr simplrl caucvgprprlemloccalc
|
|
cltp simplrr simprl simprr cnp ad3antrrr nnnq recclnq addclnq syl2anc
|
|
syl nqprlu addclpr wf ad5antr ffvelrnd ltrelnq simpld ad2antll ltsopr
|
|
brel wor w3a sowlin mpan syl3anc mpd simpr ltaprg addcomprg caovord2d
|
|
mpbird opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2 breq12d
|
|
wb breq1d caucvgprprlemell sylanbrc ex oveq12d sylan caucvgprprlemelu
|
|
rspcev orim12d rexlimddv ralrimivva ) AJUIZCUIZUJSZYLIUKULTZYMIUMULTZ
|
|
UNZUOJCUPUPAYLUPTZYMUPTZUQZUQZYNYQUUAYNUQZYLUAUIZURVJYMUSZYQUAUPYNUUD
|
|
UAUPUTUUAUAYLYMVAVBUUBUUCUPTZUUDUQZUQZUBUIZUUHURVJUUCUJSZYQUBUPUUEUUI
|
|
UBUPUTUUBUUDUBUUCVCVDUUGUUHUPTZUUIUQZUQZUCUIZVEVFZVGVHZVIULZUUHUJSZYQ
|
|
UCVKUUJUUQUCVKUTUUGUUIUUHUCVLVDUULUUMVKTZUUQUQZUQZMUIZYLUUPURVJZUJSZM
|
|
VMZUVBLUIZUJSZLVMZVFZUVAUUPUJSZMVMZUUPUVEUJSZLVMZVFZVNVJZUUMHULZUVMVN
|
|
VJZVSSZUVPUVAYMUJSMVMYMUVEUJSLVMVFZVSSZUNZYQUUTUVNUVRVSSZUVTUUTLYLYMU
|
|
UMUUHUUCMUULYNUUSUUAYNUUFUUKVOVPUULUUEUUSUUBUUEUUDUUKVQVPUULUUDUUSUUB
|
|
UUEUUDUUKVTVPUUGUUJUUIUUSVQUUGUUJUUIUUSVTUULUURUUQWAZUULUURUUQWBVRUUT
|
|
UVNWCTZUVRWCTZUVPWCTZUWAUVTUOZUUTUVHWCTZUVMWCTZUWCUUTUVBUPTZUWGUUTYRU
|
|
UPUPTZUWIUUBYRUUFUUKUUSAYRYSYNVQWDZUUTUUOUPTZUWJUURUWLUULUUQUUMWEVDUU
|
|
OWFWIZYLUUPWGZWHLUVBMWJZWIUUTUWJUWHUWMLUUPMWJZWIUVHUVMWKWHUUTYSUWDUUB
|
|
YSUUFUUKUUSAYRYSYNVTWDZLYMMWJWIUUTUVOWCTZUWHUWEUUTVKWCUUMHAVKWCHWLYTY
|
|
NUUFUUKUUSOWMUWBWNZUUTUWJUWHUUQUWJUULUURUUQUWJUUJUUPUUHUPUPUJWOWSWPWQ
|
|
ZUWPWIUVOUVMWKWHWCVSWTUWCUWDUWEXAUWFWRWCUVNUVRUVPVSXBXCXDXEUUTUVQYOUV
|
|
SYPUUTUVQYOUUTUVQUQZYRUVAYLUDUIZVEVFZVGVHZVIULZURVJZUJSZMVMZUXFUVEUJS
|
|
ZLVMZVFZUXBHULZVSSZUDVKUTZYOUUTYRUVQUWKVPZUXAUURUVHUVOVSSZUXNUULUURUU
|
|
QUVQVQUXAUXPUVQUUTUVQXFUXAUEUFUGUVHUVOUVMVSWCVNUEUIZWCTZUFUIZWCTZUGUI
|
|
ZWCTXAUXQUXSVSSUYAUXQVNVJUYAUXSVNVJVSSXTUXAUXQUXSUYAXGVBUXAUWIUWGUXAY
|
|
RUWJUWIUXOUUTUWJUVQUWTVPZUWNWHUWOWIUUTUWRUVQUWSVPUXAUWJUWHUYBUWPWIUXR
|
|
UXTUQUXQUXSVNVJUXSUXQVNVJUSUXAUXQUXSXHVBXIXJUXMUXPUDUUMVKUXBUUMUSZUXK
|
|
UVHUXLUVOVSUYCUXHUVDUXJUVGUYCUXGUVCMUYCUXFUVBUVAUJUYCUXEUUPYLURUYCUXD
|
|
UUOVIUYCUXCUUNVGUXBUUMVEXKXLXMXNZXOXPUYCUXIUVFLUYCUXFUVBUVEUJUYDYAXPX
|
|
QUXBUUMHXRXSYHWHBHIYLKLMUDNRYBYCYDUUTUVSYPUUTUVSUQYSUHUIZHULZUVAUYEVE
|
|
VFZVGVHZVIULZUJSZMVMZUYIUVEUJSZLVMZVFZVNVJZUVRVSSZUHVKUTZYPUUTYSUVSUW
|
|
QVPUUTUURUVSUYQUWBUYPUVSUHUUMVKUYEUUMUSZUYOUVPUVRVSUYRUYFUVOUYNUVMVNU
|
|
YEUUMHXRUYRUYKUVJUYMUVLUYRUYJUVIMUYRUYIUUPUVAUJUYRUYHUUOVIUYRUYGUUNVG
|
|
UYEUUMVEXKXLXMZXOXPUYRUYLUVKLUYRUYIUUPUVEUJUYSYAXPXQYEYAYHYFBHIYMKLMU
|
|
HNRYGYCYDYIXEYJYJYJYDYK $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A s $. $d F l t $. $d F u r $. $d F k n $. $d F s $.
|
|
$d L s t $. $d L k $. $d ph t r s $. $d l u p q r s t $. $d m r $.
|
|
$d k p q r s $. $d n u $. $d l n $. $d k l $. $d k u $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is a positive real.
|
|
(Contributed by Jim Kingdon, 21-Nov-2020.) $)
|
|
caucvgprprlemcl $p |- ( ph -> L e. P. ) $=
|
|
( vs vt cnq wcel cpw cxp cv c1st cfv wrex c2nd wa cltq wbr wb wral wn
|
|
wo w3a cnp caucvgprprlemm c1o cop ceq cec crq cplq cab cltp cnpi crab
|
|
wi co cpp wss nqex elpw2 mpbir opelxpi mp2an eqeltri caucvgprprlemrnd
|
|
ssrab2 jctil caucvgprprlemdisj caucvgprprlemloc elnp1st2nd sylanbrc
|
|
3jca ) AHSUAZWFUBZTZQUCZHUDUEZTZQSUFRUCZHUGUEZTZRSUFUHZUHWKWIWLUIUJZW
|
|
LWJTUHRSUFUKQSULWNWPWIWMTZUHQSUFUKRSULUHZWKWQUHUMQSULZWPWKWNUNVHRSULQ
|
|
SULZUOHUPTAWOWHABRCDEFGHQIJKLMNOPUQHKUCZLUCIUCZURUSUTVAVBUEZVCVIZUIUJ
|
|
KVDXDJUCZUIUJJVDUSXBGUEZVEUJIVFUFZLSVGZXFXAXCUIUJKVDXCXEUIUJJVDUSVJVI
|
|
XABUCZUIUJKVDXIXEUIUJJVDUSVEUJIVFUFZBSVGZUSZWGPXHWFTZXKWFTZXLWGTXMXHS
|
|
VKXGLSVSXHSVLVMVNXNXKSVKXJBSVSXKSVLVMVNXHXKWFWFVOVPVQVTAWRWSWTABRCDEF
|
|
GHQIJKLMNOPVRABCDEFGHQIJKLMNOPWAABRCDEFGHQIJKLMNOPWBWEHRQWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A s $. $d F l $. $d F u r $. $d F k n $. $d F s $.
|
|
$d L s $. $d L k $. $d ph s $. $d l u p q r s $. $d m r $.
|
|
$d m s $. $d k p q r s $. $d n u $. $d l n $. $d k l $. $d k u $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is a positive real.
|
|
Like ~ caucvgprprlemcl but without a distinct variable constraint
|
|
between ` ph ` and ` r ` . (Contributed by Jim Kingdon,
|
|
19-Jun-2021.) $)
|
|
caucvgprprlemclphr $p |- ( ph -> L e. P. ) $=
|
|
( cop cltq wbr cab vs cv c1o ceq cec crq cfv cplq cltp cnpi wrex crab
|
|
co cnq cpp wcel weq eceq1d fveq2d oveq2d breq2d abbidv breq1d opeq12d
|
|
opeq1 fveq2 breq12d cbvrexv a1i rabbiia oveq12d eqtri caucvgprprlemcl
|
|
wb opeq12i ) ABCDEFGHUAJKLMNOHKUBZLUBZIUBZUCQZUDUEZUFUGZUHUMZRSZKTZWB
|
|
JUBZRSZJTZQZVRGUGZUISZIUJUKZLUNULZWIVPWARSZKTZWAWERSZJTZQZUOUMZVPBUBZ
|
|
RSKTWSWERSJTQZUISZIUJUKZBUNULZQVPVQUAUBZUCQZUDUEZUFUGZUHUMZRSZKTZXHWE
|
|
RSZJTZQZXDGUGZUISZUAUJUKZLUNULZXNVPXGRSZKTZXGWERSZJTZQZUOUMZWTUISZUAU
|
|
JUKZBUNULZQPWLXQXCYFWKXPLUNWKXPVNVQUNUPWJXOIUAUJIUAUQZWHXMWIXNUIYGWDX
|
|
JWGXLYGWCXIKYGWBXHVPRYGWAXGVQUHYGVTXFUFYGVSXEUDVRXDUCVEURUSZUTZVAVBYG
|
|
WFXKJYGWBXHWERYIVCVBVDVRXDGVFZVGVHVIVJXBYEBUNXBYEVNWSUNUPXAYDIUAUJYGW
|
|
RYCWTUIYGWIXNWQYBUOYJYGWNXSWPYAYGWMXRKYGWAXGVPRYHVAVBYGWOXTJYGWAXGWER
|
|
YHVCVBVDVKVCVHVIVJVOVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d A m r $. $d F b f g h x y $. $d F k l n u $. $d F m r $.
|
|
$d L b f g h x y $. $d L k $. $d Q b f g h p q x y $.
|
|
$d T b f g h x y $. $d b f g h ph x y $. $d b p q r y $.
|
|
$d k l p q r u $. $d ph y $. $d u y $.
|
|
caucvgprprlemexbt.q $e |- ( ph -> Q e. Q. ) $.
|
|
caucvgprprlemexbt.t $e |- ( ph -> T e. P. ) $.
|
|
caucvgprprlemexbt.lt $e |- ( ph -> ( L +P.
|
|
<. { p | p <Q Q } , { q | Q <Q q } >. ) <P T ) $.
|
|
$( Lemma for ~ caucvgprpr . Part of showing the putative limit to be a
|
|
limit. (Contributed by Jim Kingdon, 16-Jun-2021.) $)
|
|
caucvgprprlemexbt $p |- ( ph -> E. b e. N. ( ( ( F ` b ) +P.
|
|
<. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q }
|
|
>. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >.
|
|
) <P T ) $=
|
|
( vx vy vf vg vh cv cfv c1o cop ceq cec crq cltq wbr cab co cltp cnpi
|
|
cpp wrex c2nd wcel c1st cnq cnp caucvgprprlemclphr nqprlu syl addclpr
|
|
wa syl2anc ltdfpr mpbid cplq wceq adantr simprrl prplnqu simprl breq2
|
|
weq abbidv breq1 opeq12d breq2d rexbidv crab fveq2i rabex op2nd eqtri
|
|
wb nqex elrab2 biimpi simprd fveq2 opeq1 eceq1d fveq2d breq1d oveq12d
|
|
cbvrexv sylib simpr w3a ltaprg adantl ad4antr simplr ffvelrnd recnnpr
|
|
wf ad2antrr simpld addcomprg caovord2d addnqpr breqtrrd simplrr nqpru
|
|
simplrl mpbird simprrr ad3antrrr jca ex reximdva mpd rexlimddv rexcom
|
|
expr ffvelrnda rexbidva ) ANUHZIUIZMUHZYQUJUKZULUMZUNUIZUOUPZMUQZUUBL
|
|
UHZUOUPZLUQZUKZVAURZYSDUOUPMUQDUUEUOUPLUQUKZVAURZEUSUPZNUTVBUCUHZUUKV
|
|
CUIVDZUUMEVEUIVDZVLZUCVFVBZNUTVBZAUUPNUTVBZUCVFVBZUURAUUMJUUJVAURZVCU
|
|
IVDZUUOVLZUCVFVBZUUTAUVAEUSUPZUVDUBAUVAVGVDZEVGVDZUVEUVDWNAJVGVDZUUJV
|
|
GVDZUVFABCFGHIJKLMOPQRSVHZADVFVDZUVITLDMVIVJZJUUJVKVMUAUVAEUCVNVMVOAU
|
|
VCUUSUCVFAUUMVFVDZUVCUUSAUVMUVCVLZVLZUDUHZDVPURZUUMVQZUUSUDJVCUIZUVOU
|
|
DLUUMDJMAUVHUVNUVJVRAUVKUVNTVRAUVMUVBUUOVSVTUVOUVPUVSVDZUVRVLZVLZUUIY
|
|
SUVPUOUPZMUQZUVPUUEUOUPZLUQZUKZUSUPZNUTVBZUUSUWBKUHZIUIZYSUWJUJUKZULU
|
|
MZUNUIZUOUPZMUQZUWNUUEUOUPZLUQZUKZVAURZUWGUSUPZKUTVBZUWIUWBUVTUXBUVOU
|
|
VTUVRWAZUVTUVPVFVDZUXBUVTUXDUXBVLUWTYSBUHZUOUPZMUQZUXEUUEUOUPZLUQZUKZ
|
|
USUPZKUTVBZUXBBUVPVFUVSBUDWCZUXKUXAKUTUXMUXJUWGUWTUSUXMUXGUWDUXIUWFUX
|
|
MUXFUWCMUXEUVPYSUOWBWDUXMUXHUWELUXEUVPUUEUOWEWDWFWGWHUVSYSOUHUWNVPURZ
|
|
UOUPMUQUXNUUEUOUPLUQUKUWKUSUPKUTVBZOVFWIZUXLBVFWIZUKZVCUIUXQJUXRVCSWJ
|
|
UXPUXQUXOOVFWOWKUXLBVFWOWKWLWMWPWQZWRVJUXAUWHKNUTKNWCZUWTUUIUWGUSUXTU
|
|
WKYRUWSUUHVAUWJYQIWSUXTUWPUUDUWRUUGUXTUWOUUCMUXTUWNUUBYSUOUXTUWMUUAUN
|
|
UXTUWLYTULUWJYQUJWTXAXBZWGWDUXTUWQUUFLUXTUWNUUBUUEUOUYAXCWDWFXDXCXEXF
|
|
UWBUWHUUPNUTUWBYQUTVDZVLZUWHUUPUYCUWHVLZUUNUUOUYDUUNUUKYSUUMUOUPZMUQZ
|
|
UUMUUEUOUPZLUQZUKZUSUPZUYDUUKYSUVQUOUPZMUQZUVQUUEUOUPZLUQZUKZUSUPZUYJ
|
|
UYDUUKUWGUUJVAURZUYOUSUYDUWHUUKUYQUSUPUYCUWHXGUYDUEUFUGUUIUWGUUJUSVGV
|
|
AUEUHZVGVDZUFUHZVGVDZUGUHZVGVDXHUYRUYTUSUPVUBUYRVAURVUBUYTVAURUSUPWNU
|
|
YDUYRUYTVUBXIXJUYDYRVGVDZUUHVGVDZUUIVGVDZUYDUTVGYQIAUTVGIXOUVNUWAUYBU
|
|
WHPXKUWBUYBUWHXLZXMUYDUYBVUDVUFLYQMXNZVJYRUUHVKZVMZUYDUXDUWGVGVDUYDUV
|
|
TUXDUWBUVTUYBUWHUXCXPUVTUXDUXBUXSXQVJZLUVPMVIVJAUVIUVNUWAUYBUWHUVLXKZ
|
|
UYSVUAVLUYRUYTVAURUYTUYRVAURVQUYDUYRUYTXRXJXSVOUYDUXDUVKUYOUYQVQVUJAU
|
|
VKUVNUWAUYBUWHTXKLUVPDMXTVMYAUYDUVRUYPUYJWNUYCUVRUWHUVOUVTUVRUYBYBVRU
|
|
VRUYOUYIUUKUSUVRUYLUYFUYNUYHUVRUYKUYEMUVQUUMYSUOWBWDUVRUYMUYGLUVQUUMU
|
|
UEUOWEWDWFWGVJVOUYDUVMUUKVGVDZUUNUYJWNUWBUVMUYBUWHAUVMUVCUWAYDXPUYDVU
|
|
EUVIVULVUIVUKUUIUUJVKZVMLUUMUUKMYCVMYEUVOUUOUWAUYBUWHAUVMUVBUUOYFYGYH
|
|
YIYJYKYLYNYJYKUUPUCNVFUTYMXFAUULUUQNUTAUYBVLZVULUVGUULUUQWNVUNVUEUVIV
|
|
ULVUNVUCVUDVUEAUTVGYQIPYOUYBVUDAVUGXJVUHVMAUVIUYBUVLVRVUMVMAUVGUYBUAV
|
|
RUUKEUCVNVMYPYE $.
|
|
$}
|
|
|
|
${
|
|
$d A m r $. $d F b f g h $. $d F k l n u $. $d F m r $.
|
|
$d L b f g h $. $d L k $. $d Q f g h $. $d R b f g h p q $.
|
|
$d R b p q $. $d b f g h ph $. $d k l p q r u $. $d b r $.
|
|
caucvgprprlemexb.q $e |- ( ph -> Q e. P. ) $.
|
|
caucvgprprlemexb.r $e |- ( ph -> R e. N. ) $.
|
|
$( Lemma for ~ caucvgprpr . Part of showing the putative limit to be a
|
|
limit. (Contributed by Jim Kingdon, 15-Jun-2021.) $)
|
|
caucvgprprlemexb $p |- ( ph -> (
|
|
( ( L +P. Q ) +P.
|
|
<. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q }
|
|
>. ) <P ( ( F ` R ) +P. Q )
|
|
-> E. b e. N. ( ( ( F ` b ) +P.
|
|
<. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q }
|
|
>. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q }
|
|
>. ) ) <P ( ( F ` R ) +P. Q ) ) ) $=
|
|
( vf vg vh cpp co c1o cop ceq cec crq cfv cltq wbr cab cltp cnpi wrex
|
|
cv cnp wcel wb caucvgprprlemclphr recnnpr syl addclpr ffvelrnd ltaprg
|
|
syl2anc syl3anc addassprg addcomprg oveq1d eqtr3d breq12d bitrd wa wf
|
|
wceq adantr clti wi wral cnq nnnq recclnq simpr caucvgprprlemexbt w3a
|
|
3syl adantl ffvelrnda ad2antrr caovord2d breq1d oveq2d rexbidva mpbid
|
|
3bitrd ex sylbird ) AJDUEUFZMUSZEUGUHUIUJZUKULZUMUNMUOXELUSZUMUNLUOUH
|
|
ZUEUFZEIULZDUEUFZUPUNZJXGUEUFZXIUPUNZNUSZIULZXCXNUGUHUIUJUKULZUMUNMUO
|
|
XPXFUMUNLUOUHZUEUFZDXGUEUFZUEUFZXJUPUNZNUQURZAXMDXLUEUFZDXIUEUFZUPUNZ
|
|
XKAXLUTVAZXIUTVAZDUTVAZXMYEVBAJUTVAZXGUTVAZYFABCFGHIJKLMOPQRSVCZAEUQV
|
|
AZYJUALEMVDZVEZJXGVFVIAUQUTEIPUAVGZTXLXIDVHVJAYCXHYDXJUPADJUEUFZXGUEU
|
|
FZYCXHAYHYIYJYQYCVSTYKYNDJXGVKVJAYPXBXGUEAYHYIYPXBVSTYKDJVLVIVMVNAYHY
|
|
GYDXJVSTYODXIVLVIVOVPAXMYBAXMVQZXRXGUEUFZXIUPUNZNUQURYBYRBCXEXIFGHIJK
|
|
LMNOAUQUTIVRXMPVTZAHUSZFUSZWAUNUUBIULZUUCIULZOUSUUBUGUHUIUJUKULZUMUNO
|
|
UOUUFBUSUMUNBUOUHZUEUFUPUNUUEUUDUUGUEUFUPUNVQWBFUQWCHUQWCXMQVTACGUSIU
|
|
LUPUNGUQWCXMRVTSAXEWDVAZXMAYLXDWDVAUUHUAEWEXDWFWJVTAYGXMYOVTAXMWGWHYR
|
|
YTYANUQYRXNUQVAZVQZYTYSDUEUFZXJUPUNXRXGDUEUFZUEUFZXJUPUNYAUUJUBUCUDYS
|
|
XIDUPUTUEUBUSZUTVAZUCUSZUTVAZUDUSZUTVAWIUUNUUPUPUNUURUUNUEUFUURUUPUEU
|
|
FUPUNVBUUJUUNUUPUURVHWKUUJXRUTVAZYJYSUTVAUUJXOUTVAXQUTVAZUUSYRUQUTXNI
|
|
UUAWLUUIUUTYRLXNMVDWKXOXQVFVIZUUJYLYJAYLXMUUIUAWMYMVEZXRXGVFVIAYGXMUU
|
|
IYOWMAYHXMUUITWMZUUOUUQVQUUNUUPUEUFUUPUUNUEUFVSUUJUUNUUPVLWKWNUUJUUKU
|
|
UMXJUPUUJUUSYJYHUUKUUMVSUVAUVBUVCXRXGDVKVJWOUUJUUMXTXJUPUUJUULXSXRUEU
|
|
UJYJYHUULXSVSUVBUVCXGDVLVIWPWOWSWQWRWTXA $.
|
|
$}
|
|
|
|
${
|
|
$d A m r $. $d F b f g h l r u $. $d F b k l n r u $. $d F m r $.
|
|
$d L f g h $. $d L k $. $d Q b f g h r $. $d X b f g h r $.
|
|
$d b f g h l p q r u $. $d b f g h ph r $. $d k l p q r u $.
|
|
$d L b $.
|
|
caucvgprprlemaddq.x $e |- ( ph -> X e. P. ) $.
|
|
caucvgprprlemaddq.q $e |- ( ph -> Q e. P. ) $.
|
|
caucvgprprlemaddq.ex $e |- ( ph -> E. r e. N.
|
|
( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } ,
|
|
{ q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q }
|
|
>. ) <P ( ( F ` r ) +P. Q ) ) $.
|
|
$( Lemma for ~ caucvgprpr . Part of showing the putative limit to be a
|
|
limit. (Contributed by Jim Kingdon, 5-Jun-2021.) $)
|
|
caucvgprprlemaddq $p |- ( ph -> X <P ( L +P. Q ) ) $=
|
|
( vb vf vg vh c1o cop ceq cec crq cfv cltq wbr cab cpp cltp cnpi wrex
|
|
cv co nfv nfcv cplq cnq crab nfre1 nfrabxy nfop nfcxfr nfov nfbr wcel
|
|
wa wf ad2antrr clti wral simpr simplrl caucvgprprlemnbj wceq ffvelrnd
|
|
cnp wi recnnpr adantl addclpr syl2anc syl addassprg syl3anc breq1d wb
|
|
ltaprg addcomprg caovord2d oveq2d 3bitr4rd mtbird nrexdv breq1 cbvabv
|
|
w3a breq2 opeq12i oveq2i breq1i rexbii adantr simprl caucvgprprlemexb
|
|
oveq12i sylnibr mtod wo simprr caucvgprprlemcl wor ltsopr sowlin mpan
|
|
mpd ecased mpbird exp32 rexlimd ) AJMUSZKUSZUFUGUHUIUJUKZULUMZMUNZYIL
|
|
USZULUMZLUNZUGZUOUTZYHHUKZDUOUTZUPUMZKUQURJIDUOUTZUPUMZUAAYSUUAKUQAKV
|
|
AKJYTUPKJVBKUPVBKIDUOKIYGNUSZYIVCUTZULUMMUNUUCYLULUMLUNUGYQUPUMZKUQUR
|
|
ZNVDVEZYQYOUOUTYGBUSZULUMMUNUUGYLULUMLUNUGUPUMZKUQURZBVDVEZUGRKUUFUUJ
|
|
UUEKNVDUUDKUQVFKVDVBZVGUUIKBVDUUHKUQVFUUKVGVHVIKUOVBKDVBVJVKAYHUQVLZY
|
|
SUUAAUULYSVMZVMZUUAYPYTYOUOUTZUPUMZUUNUUPUUOYRUPUMZUUNUUQUBUSZHUKZYGU
|
|
URUFUGUHUIUJUKZULUMZMUNZUUTYLULUMZLUNZUGZUOUTZDYOUOUTZUOUTZYRUPUMZUBU
|
|
QURZUUNUUSUUBUUTULUMZNUNZUUTUUGULUMZBUNZUGZUOUTZDUUBYIULUMZNUNZYIUUGU
|
|
LUMZBUNZUGZUOUTZUOUTZYRUPUMZUBUQURUVJUUNUWDUBUQUUNUURUQVLZVMZUWDUVPUW
|
|
AUOUTZYQUPUMZUWFBUUREGHYHNAUQWCHVNZUUMUWEOVOZAGUSZEUSZVPUMUWKHUKZUWLH
|
|
UKZUUBUWKUFUGUHUIUJUKZULUMNUNUWOUUGULUMBUNUGZUOUTUPUMUWNUWMUWPUOUTUPU
|
|
MVMWDEUQVQGUQVQZUUMUWEPVOUUNUWEVRZAUULYSUWEVSZVTUWFUWGDUOUTZYRUPUMUVP
|
|
UWADUOUTZUOUTZYRUPUMUWHUWDUWFUWTUXBYRUPUWFUVPWCVLZUWAWCVLZDWCVLZUWTUX
|
|
BWAUWFUUSWCVLUVOWCVLZUXCUWFUQWCUURHUWJUWRWBUWEUXFUUNBUURNWEWFUUSUVOWG
|
|
WHZUWFUULUXDUWSBYHNWEWIZAUXEUUMUWETVOZUVPUWADWJWKWLUWFUCUDUEUWGYQDUPW
|
|
CUOUCUSZWCVLZUDUSZWCVLZUEUSZWCVLXCZUXJUXLUPUMUXNUXJUOUTUXNUXLUOUTUPUM
|
|
WMZUWFUXJUXLUXNWNZWFUWFUXCUXDUWGWCVLUXGUXHUVPUWAWGWHUWFUQWCYHHUWJUWSW
|
|
BUXIUXKUXMVMZUXJUXLUOUTUXLUXJUOUTWAZUWFUXJUXLWOZWFWPUWFUWCUXBYRUPUWFU
|
|
WBUXAUVPUOUWFUXEUXDUWBUXAWAUXIUXHDUWAWOWHWQWLWRWSWTUVIUWDUBUQUVHUWCYR
|
|
UPUVFUVPUVGUWBUOUVEUVOUUSUOUVBUVLUVDUVNUVAUVKMNYGUUBUUTULXAXBUVCUVMLB
|
|
YLUUGUUTULXDXBXEXFYOUWADUOYKUVRYNUVTYJUVQMNYGUUBYIULXAXBYMUVSLBYLUUGY
|
|
IULXDXBXEXFXLXGXHXMUUNBCDYHEFGHIKLMUBNAUWIUUMOXIZAUWQUUMPXIACFUSHUKUP
|
|
UMFUQVQUUMQXIRAUXEUUMTXIZAUULYSXJZXKXNUUNYSUUPUUQXOZAUULYSXPUUNYPWCVL
|
|
ZYRWCVLZUUOWCVLZYSUYDWDZUUNJWCVLZYOWCVLZUYEAUYIUUMSXIZUUNUULUYJUYCLYH
|
|
MWEWIZJYOWGWHUUNYQWCVLUXEUYFUUNUQWCYHHUYAUYCWBUYBYQDWGWHUUNYTWCVLZUYJ
|
|
UYGUUNIWCVLZUXEUYMAUYNUUMABCEFGHIKLMNOPQRXQXIUYBIDWGWHZUYLYTYOWGWHWCU
|
|
PXRUYEUYFUYGXCUYHXSWCYPYRUUOUPXTYAWKYBYCUUNUCUDUEJYTYOUPWCUOUXOUXPUUN
|
|
UXQWFUYKUYOUYLUXRUXSUUNUXTWFWPYDYEYFYB $.
|
|
$}
|
|
|
|
${
|
|
caucvgprprlemlim.q $e |- ( ph -> Q e. P. ) $.
|
|
caucvgprprlemlim.jk $e |- ( ph -> J <N K ) $.
|
|
caucvgprprlemlim.jkq $e |- ( ph ->
|
|
<. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >.
|
|
<P Q ) $.
|
|
|
|
${
|
|
$d A m $. $d A r $. $d F m $. $d F r $. $d F l u $. $d F n k $.
|
|
$d J l $. $d J u $. $d K l r $. $d K r u $. $d Q r $. $d L k $.
|
|
$d ph r $. $d p q r l u $. $d m r $. $d k l u $. $d k r $.
|
|
$d k p q $. $d n l u $. $d n r $.
|
|
$( Lemma for ~ caucvgprpr . Part of showing the putative limit to be
|
|
a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) $)
|
|
caucvgprprlem1 $p |- ( ph -> ( F ` K ) <P ( L +P. Q ) ) $=
|
|
( cfv cnpi cnp wcel clti wbr wa ltrelpi brel syl simprd ffvelrnd cv
|
|
c1o cop ceq cec crq cltq cab cpp co cltp wrex caucvgprprlemk wb cnq
|
|
nnnq recclnq nqprlu ltaprg syl3anc mpbid opeq1 eceq1d fveq2d breq2d
|
|
3syl wceq abbidv breq1d opeq12d oveq2d fveq2 oveq1d breq12d syl2anc
|
|
rspcev breq1 cbvabv opeq12i oveq2i breq1i rexbii caucvgprprlemaddq
|
|
breq2 sylib ) ABCDEFGHKJHUCZLMNOPQRSAUDUEJHPAIUDUFZJUDUFZAIJUGUHXAX
|
|
BUIUAIJUDUDUGUJUKULUMZUNZTAWTOUOZLUOZUPUQZURUSZUTUCZVAUHZOVBZXIBUOZ
|
|
VAUHZBVBZUQZVCVDZXFHUCZDVCVDZVEUHZLUDVFZWTNUOZXIVAUHZNVBZXIMUOZVAUH
|
|
ZMVBZUQZVCVDZXRVEUHZLUDVFAXBWTXEJUPUQZURUSZUTUCZVAUHZOVBZYLXLVAUHZB
|
|
VBZUQZVCVDZWTDVCVDZVEUHZXTXCAYQDVEUHZYTABDIJOUAUBVGAYQUEUFZDUEUFWTU
|
|
EUFUUAYTVHAYKVIUFZYLVIUFUUBAXBUUCXCJVJULYKVKBYLOVLVTTXDYQDWTVMVNVOX
|
|
SYTLJUDXFJWAZXPYRXRYSVEUUDXOYQWTVCUUDXKYNXNYPUUDXJYMOUUDXIYLXEVAUUD
|
|
XHYKUTUUDXGYJURXFJUPVPVQVRZVSWBUUDXMYOBUUDXIYLXLVAUUEWCWBWDWEUUDXQW
|
|
TDVCXFJHWFWGWHWJWIXSYILUDXPYHXRVEXOYGWTVCXKYCXNYFXJYBONXEYAXIVAWKWL
|
|
XMYEBMXLYDXIVAWRWLWMWNWOWPWSWQ $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d F r u x $. $d F l $. $d F k $. $d F n $.
|
|
$d K l p u x $. $d K p q r u x $. $d J l $. $d J u $. $d L x $.
|
|
$d L k $. $d Q x $. $d ph x $. $d ph r $. $d k n $. $d k r $.
|
|
$d l u p q r $. $d m r $. $d k p q $. $d n u $. $d l n $.
|
|
$d k l $. $d k u $.
|
|
$( Lemma for ~ caucvgprpr . Part of showing the putative limit to be
|
|
a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) $)
|
|
caucvgprprlem2 $p |- ( ph -> L <P ( ( F ` K ) +P. Q ) ) $=
|
|
( vx cv cfv c1o cop ceq cec crq cltq wbr cab co c2nd wcel c1st cltp
|
|
cpp wa cnq wrex caucvgprprlemk cnp wb cnpi clti ltrelpi brel simprd
|
|
syl nnnq recclnq nqprlu ltaprg syl3anc mpbid addclpr syl2anc ltdfpr
|
|
ffvelrnd simprl adantr simprrl breq1 cbvabv opeq12i oveq2i syl6eleq
|
|
breq2 fveq2i nqpru fveq2 eceq1d fveq2d breq2d abbidv breq1d opeq12d
|
|
wceq opeq1 oveq12d rspcev rexbidv cplq crab nqex rabex op2nd elrab2
|
|
eqtri sylanbrc simprrr syl12anc caucvgprprlemcl mpbird rexlimddv
|
|
rspe ) AUCUDZJHUEZOUDZJUFUGZUHUIZUJUEZUKULZOUMZYDBUDZUKULZBUMZUGZUS
|
|
UNZUOUEZUPZXSXTDUSUNZUQUEUPZUTZKYNURULZUCVAAYKYNURULZYPUCVAVBZAYJDU
|
|
RULZYRABDIJOUAUBVCAYJVDUPZDVDUPZXTVDUPZYTYRVEAYDVAUPZUUAAJVFUPZUUDA
|
|
IVFUPZUUEAIJVGULUUFUUEUTUAIJVFVFVGVHVIVKVJZUUEYCVAUPUUDJVLYCVMVKVKZ
|
|
BYDOVNVKZTAVFVDJHPUUGWAZYJDXTVOVPVQAYKVDUPZYNVDUPZYRYSVEAUUCUUAUUKU
|
|
UJUUIXTYJVRVSAUUCUUBUULUUJTXTDVRVSZYKYNUCVTVSVQAXSVAUPZYPUTZUTZYQXS
|
|
KUOUEZUPZYOUTZUCVAVBZUUPUUNUURYOUUTAUUNYPWBZUUPUUNLUDZHUEZNUDZUVBUF
|
|
UGZUHUIZUJUEZUKULZNUMZUVGMUDZUKULZMUMZUGZUSUNZUVDXSUKULZNUMZXSUVJUK
|
|
ULZMUMZUGZURULZLVFVBZUURUVAUUPUUEXTUVDYDUKULZNUMZYDUVJUKULZMUMZUGZU
|
|
SUNZUVSURULZUWAAUUEUUOUUGWCUUPXSUWGUOUEZUPZUWHUUPXSYLUWIAUUNYMYOWDY
|
|
KUWGUOYJUWFXTUSYFUWCYIUWEYEUWBONYAUVDYDUKWEWFYHUWDBMYGUVJYDUKWJWFWG
|
|
WHWKWIUUPUUNUWGVDUPZUWJUWHVEUVAAUWKUUOAUUCUWFVDUPZUWKUUJAUUDUWLUUHM
|
|
YDNVNVKXTUWFVRVSWCMXSUWGNWLVSVQUVTUWHLJVFUVBJWTZUVNUWGUVSURUWMUVCXT
|
|
UVMUWFUSUVBJHWMUWMUVIUWCUVLUWEUWMUVHUWBNUWMUVGYDUVDUKUWMUVFYCUJUWMU
|
|
VEYBUHUVBJUFXAWNWOZWPWQUWMUVKUWDMUWMUVGYDUVJUKUWNWRWQWSXBWRXCVSUVNU
|
|
VDYGUKULZNUMZYGUVJUKULZMUMZUGZURULZLVFVBZUWABXSVAUUQYGXSWTZUWTUVTLV
|
|
FUXBUWSUVSUVNURUXBUWPUVPUWRUVRUXBUWOUVONYGXSUVDUKWJWQUXBUWQUVQMYGXS
|
|
UVJUKWEWQWSWPXDUUQUVDYAUVGXEUNZUKULNUMUXCUVJUKULMUMUGUVCURULLVFVBZO
|
|
VAXFZUXABVAXFZUGZUOUEUXFKUXGUOSWKUXEUXFUXDOVAXGXHUXABVAXGXHXIXKXJXL
|
|
AUUNYMYOXMUUSUCVAXRXNUUPKVDUPZUULYQUUTVEAUXHUUOABCEFGHKLMNOPQRSXOWC
|
|
AUULUUOUUMWCKYNUCVTVSXPXQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A j $. $d F u r $. $d F l $. $d F k $. $d F n $.
|
|
$d ph k r $. $d L k $. $d j k ph x $. $d k l u $. $d k l p $.
|
|
$d k q r $. $d j r $. $d r x $. $d l u p q r $. $d m r $.
|
|
$d k n $. $d n u $. $d l n $. $d j l u $. $d n r $.
|
|
$( Lemma for ~ caucvgprpr . The putative limit is a limit.
|
|
(Contributed by Jim Kingdon, 21-Nov-2020.) $)
|
|
caucvgprprlemlim $p |- ( ph ->
|
|
A. x e. P. E. j e. N. A. k e. N. ( j <N k ->
|
|
( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) ) $=
|
|
( wbr cnpi cv clti cfv cpp co cltp wral wrex cnp wcel c1o cop ceq cec
|
|
wa wi crq cltq cab archrecpr adantl wf ad5antr ad4antr caucvgprprlem1
|
|
simpr simpllr caucvgprprlem2 jca ex ralrimiva reximdva mpd ) AEUAZFUA
|
|
ZUBSZVOIUCZJBUAZUDUEUFSZJVQVRUDUEUFSZUOZUPZFTUGZETUHZBUIAVRUIUJZUOZNU
|
|
AZVNUKULUMUNUQUCZURSNUSWHCUAZURSCUSULVRUFSZETUHZWDWEWKACVRENUTVAWFWJW
|
|
CETWFVNTUJZUOZWJWCWMWJUOZWBFTWNVOTUJZUOZVPWAWPVPUOZVSVTWQCDVRFGHIVNVO
|
|
JKLMNATUIIVBWEWLWJWOVPOVCZAHUAZVOUBSWSIUCZVQWGWSUKULUMUNUQUCZURSNUSXA
|
|
WIURSCUSULZUDUEUFSVQWTXBUDUEUFSUOUPFTUGHTUGWEWLWJWOVPPVCZADGUAIUCUFSG
|
|
TUGWEWLWJWOVPQVCZRWFWEWLWJWOVPAWEVFVDZWPVPVFZWMWJWOVPVGZVEWQCDVRFGHIV
|
|
NVOJKLMNWRXCXDRXEXFXGVHVIVJVKVJVLVMVK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A j $. $d F j y $. $d F k y $. $d F x y $. $d F u r $.
|
|
$d F l $. $d F n $. $d j l y $. $d j p y $. $d j q y $. $d j r y $.
|
|
$d j u y $. $d k l y $. $d k p y $. $d k q y $. $d k r y $.
|
|
$d k u y $. $d k l u $. $d l x y $. $d l p $. $d p x y $.
|
|
$d q x y $. $d r x y $. $d u x y $. $d p q r u $. $d k q r $.
|
|
$d l u p q r $. $d ph x k j $. $d ph r $. $d m r $. $d k n $.
|
|
$d n u $. $d l n $. $d n r $.
|
|
$( A Cauchy sequence of positive reals with a modulus of convergence
|
|
converges to a positive real. This is basically Corollary 11.2.13 of
|
|
[HoTT], p. (varies) (one key difference being that this is for
|
|
positive reals rather than signed reals). Also, the HoTT book theorem
|
|
has a modulus of convergence (that is, a rate of convergence)
|
|
specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
|
|
convergence to say that all terms after the nth term must be within
|
|
` 1 / n ` of the nth term (it should later be able to prove versions
|
|
of this theorem with a different fixed rate or a modulus of
|
|
convergence supplied as a hypothesis). We also specify that every
|
|
term needs to be larger than a given value ` A ` , to avoid the case
|
|
where we have positive terms which "converge" to zero (which is not a
|
|
positive real).
|
|
|
|
This is similar to ~ caucvgpr except that values of the sequence are
|
|
positive reals rather than positive fractions. Reading that proof
|
|
first (or ~ cauappcvgpr ) might help in understanding this one, as
|
|
they are slightly simpler but similarly structured. (Contributed by
|
|
Jim Kingdon, 14-Nov-2020.) $)
|
|
caucvgprpr $p |- ( ph ->
|
|
E. y e. P. A. x e. P. E. j e. N. A. k e. N. ( j <N k ->
|
|
( ( F ` k ) <P ( y +P. x ) /\
|
|
y <P ( ( F ` k ) +P. x ) ) ) ) $=
|
|
( vp vq cv wbr cltp cnpi c1o cop ceq cec crq cfv cplq cltq cab wrex cnq
|
|
vr co crab cpp cnp wcel clti wral eqid caucvgprprlemcl caucvgprprlemlim
|
|
wa wi wceq oveq1 breq2d breq1 anbi12d imbi2d rexralbidv ralbidv syl2anc
|
|
rspcev ) AOQZKQULQZUAUBUCUDUEUFZUGUMZUHROUIVRPQZUHRPUIUBVPJUFZSRULTUJKU
|
|
KUNVTVOVQUHROUIVQVSUHRPUIUBUOUMVODQZUHROUIWAVSUHRPUIUBSRULTUJDUKUNUBZUP
|
|
UQFQGQZURRZWCJUFZWBBQZUOUMZSRZWBWEWFUOUMZSRZVCZVDZGTUSFTUJZBUPUSZWDWECQ
|
|
ZWFUOUMZSRZWOWISRZVCZVDZGTUSFTUJZBUPUSZCUPUJADEGHIJWBULPOKLMNWBUTZVAABD
|
|
EFGHIJWBULPOKLMNXCVBXBWNCWBUPWOWBVEZXAWMBUPXDWTWLFGTTXDWSWKWDXDWQWHWRWJ
|
|
XDWPWGWESWOWBWFUOVFVGWOWBWISVHVIVJVKVLVNVM $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Define equivalence relation for signed reals. This is a "temporary" set
|
|
used in the construction of complex numbers, and is intended to be used
|
|
only by the construction. From Proposition 9-4.1 of [Gleason] p. 126.
|
|
(Contributed by NM, 25-Jul-1995.) $)
|
|
df-enr $a |- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\
|
|
y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u
|
|
( ( x = <. z , w >. /\ y = <. v , u >. ) /\
|
|
( z +P. u ) = ( w +P. v ) ) ) } $.
|
|
|
|
$( Define class of signed reals. This is a "temporary" set used in the
|
|
construction of complex numbers, and is intended to be used only by the
|
|
construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed
|
|
by NM, 25-Jul-1995.) $)
|
|
df-nr $a |- R. = ( ( P. X. P. ) /. ~R ) $.
|
|
|
|
$( Define addition on signed reals. This is a "temporary" set used in the
|
|
construction of complex numbers, and is intended to be used only by the
|
|
construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed
|
|
by NM, 25-Aug-1995.) $)
|
|
df-plr $a |- +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\
|
|
z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) } $.
|
|
|
|
$( Define multiplication on signed reals. This is a "temporary" set used
|
|
in the construction of complex numbers, and is intended to be used only
|
|
by the construction. From Proposition 9-4.3 of [Gleason] p. 126.
|
|
(Contributed by NM, 25-Aug-1995.) $)
|
|
df-mr $a |- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\
|
|
E. w E. v E. u E. f (
|
|
( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\
|
|
z = [ <. ( ( w .P. u ) +P. ( v .P. f ) ) ,
|
|
( ( w .P. f ) +P. ( v .P. u ) ) >. ] ~R ) ) } $.
|
|
|
|
$( Define ordering relation on signed reals. This is a "temporary" set
|
|
used in the construction of complex numbers, and is intended to be used
|
|
only by the construction. From Proposition 9-4.4 of [Gleason] p. 127.
|
|
(Contributed by NM, 14-Feb-1996.) $)
|
|
df-ltr $a |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\
|
|
E. z E. w E. v E. u
|
|
( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\
|
|
( z +P. u ) <P ( w +P. v ) ) ) } $.
|
|
|
|
$( Define signed real constant 0. This is a "temporary" set used in the
|
|
construction of complex numbers, and is intended to be used only by the
|
|
construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed
|
|
by NM, 9-Aug-1995.) $)
|
|
df-0r $a |- 0R = [ <. 1P , 1P >. ] ~R $.
|
|
|
|
$( Define signed real constant 1. This is a "temporary" set used in the
|
|
construction of complex numbers, and is intended to be used only by the
|
|
construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed
|
|
by NM, 9-Aug-1995.) $)
|
|
df-1r $a |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R $.
|
|
|
|
$( Define signed real constant -1. This is a "temporary" set used in the
|
|
construction of complex numbers, and is intended to be used only by the
|
|
construction. (Contributed by NM, 9-Aug-1995.) $)
|
|
df-m1r $a |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $.
|
|
$( Equivalence relation for signed reals in terms of positive reals.
|
|
(Contributed by NM, 3-Sep-1995.) $)
|
|
enrbreq $p |- ( ( ( A e. P. /\ B e. P. ) /\
|
|
( C e. P. /\ D e. P. ) ) ->
|
|
( <. A , B >. ~R <. C , D >. <-> ( A +P. D ) = ( B +P. C ) ) ) $=
|
|
( vx vy vz vw vv vu cpp cer cnp df-enr ecopoveq ) EFGHIJABCDKLMEFGHIJNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $.
|
|
$( The equivalence relation for signed reals is an equivalence relation.
|
|
Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.)
|
|
(Revised by Mario Carneiro, 6-Jul-2015.) $)
|
|
enrer $p |- ~R Er ( P. X. P. ) $=
|
|
( vx vy vz vw vv cpp cer cnp df-enr addcomprg addclpr addassprg addcanprg
|
|
vu cv ecopoverg ) ABCDENFGHABCDENIAOZBOZJQRKQRCOZLQRSMP $.
|
|
$}
|
|
|
|
$( Equivalence class equality of positive fractions in terms of positive
|
|
integers. (Contributed by NM, 29-Nov-1995.) $)
|
|
enreceq $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
|
|
( [ <. A , B >. ] ~R = [ <. C , D >. ] ~R <->
|
|
( A +P. D ) = ( B +P. C ) ) ) $=
|
|
( cnp wcel wa cop cer wbr cec wceq cpp co cxp wer enrer a1i opelxpi adantr
|
|
erth enrbreq bitr3d ) AEFBEFGZCEFDEFGZGZABHZCDHZIJUGIKUHIKLADMNBCMNLUFUGUHI
|
|
EEOZUIIPUFQRUDUGUIFUEABEESTUAABCDUBUC $.
|
|
|
|
${
|
|
$d x y z w v u $.
|
|
$( The equivalence relation for signed reals exists. (Contributed by NM,
|
|
25-Jul-1995.) $)
|
|
enrex $p |- ~R e. _V $=
|
|
( vx vy vz vw vv vu cer cnp cxp npex xpex cv wcel wa cop cpp co wex copab
|
|
wceq df-enr opabssxp eqsstri ssexi ) GHHIZUEIZUEUEHHJJKZUGKGALZUEMBLZUEMN
|
|
UHCLZDLZOTUIELZFLZOTNUJUMPQUKULPQTNFRERDRCRZNABSUFABCDEFUAUNABUEUEUBUCUD
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $.
|
|
$( Signed real 'less than' is a relation on signed reals. (Contributed by
|
|
NM, 14-Feb-1996.) $)
|
|
ltrelsr $p |- <R C_ ( R. X. R. ) $=
|
|
( vx vy vz vw vv vu cltr cv cnr wcel wa cop cer cec wceq cpp cltp wbr wex
|
|
co copab cxp df-ltr opabssxp eqsstri ) GAHZIJBHZIJKUFCHZDHZLMNOUGEHZFHZLM
|
|
NOKUHUKPTUIUJPTQRKFSESDSCSZKABUAIIUBABCDEFUCULABIIUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z F $.
|
|
$d x y z G $. $d x y z R $. $d x y z S $.
|
|
$( Lemma showing compatibility of addition. (Contributed by NM,
|
|
3-Sep-1995.) $)
|
|
addcmpblnr $p |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
|
|
/\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
|
|
( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) ->
|
|
<. ( A +P. F ) , ( B +P. G ) >. ~R <. ( C +P. R ) , ( D +P. S ) >. ) ) $=
|
|
( cpp co wceq wa cnp wcel addclpr anim12i an4s syl2anc addassprg syl3anc
|
|
cop cer wbr oveq12 wb enrbreq syl simprll simplrr addcomprg oveq1d oveq2d
|
|
simprrr 3eqtr3d simplll 3eqtr4d simprlr simplrl simprrl simpllr eqeq12d
|
|
bitrd syl5ibr ) ADIJZBCIJZKGFIJZHEIJZKLAGIJZBHIJZUACEIJZDFIJZUAUBUCZAMNZB
|
|
MNZLZCMNZDMNZLZLZGMNZHMNZLZEMNZFMNZLZLZLZVDVFIJZVEVGIJZKZVDVEVFVGIUDWGVLV
|
|
HVKIJZVIVJIJZKZWJWGVHMNZVIMNZLZVJMNZVKMNZLZLZVLWMUEVOWBVRWEWTVOWBLWPVRWEL
|
|
WSVMVTVNWAWPVMVTLWNVNWALWOAGOBHOPQVPWCVQWDWSVPWCLWQVQWDLWRCEOZDFOZPQPQVHV
|
|
IVJVKUFUGWGWKWHWLWIWGAGVKIJZIJZADVFIJZIJZWKWHWGXCXEAIWGGDIJZFIJZDGIJZFIJZ
|
|
XCXEWGXGXIFIWGVTVQXGXIKVSVTWAWEUHZVOVPVQWFUIZGDUJRUKWGVTVQWDXHXCKXKXLVSWB
|
|
WCWDUMZGDFSTWGVQVTWDXJXEKXLXKXMDGFSTUNULWGVMVTWRWKXDKVMVNVRWFUOZXKWGVQWDW
|
|
RXLXMXBRAGVKSTWGVMVQVFMNZWHXFKXNXLWGVTWDXOXKXMGFORADVFSTUPWGBHVJIJZIJZBCV
|
|
GIJZIJZWLWIWGXPXRBIWGHCIJZEIJZCHIJZEIJZXPXRWGXTYBEIWGWAVPXTYBKVSVTWAWEUQZ
|
|
VOVPVQWFURZHCUJRUKWGWAVPWCYAXPKYDYEVSWBWCWDUSZHCESTWGVPWAWCYCXRKYEYDYFCHE
|
|
STUNULWGVNWAWQWLXQKVMVNVRWFUTZYDWGVPWCWQYEYFXARBHVJSTWGVNVPVGMNZWIXSKYGYE
|
|
WGWAWCYHYDYFHEORBCVGSTUPVAVBVC $.
|
|
|
|
$( Lemma used in lemma showing compatibility of multiplication.
|
|
(Contributed by Jim Kingdon, 1-Jan-2020.) $)
|
|
mulcmpblnrlemg $p |- (
|
|
( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
|
|
/\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
|
|
( ( ( A +P. D ) = ( B +P. C ) /\
|
|
( F +P. S ) = ( G +P. R ) ) ->
|
|
( ( D .P. F ) +P. ( ( ( A .P. F ) +P. ( B .P. G ) ) +P.
|
|
( ( C .P. S ) +P. ( D .P. R ) ) ) ) =
|
|
( ( D .P. F ) +P. ( ( ( A .P. G ) +P. ( B .P. F ) ) +P.
|
|
( ( C .P. R ) +P. ( D .P. S ) ) ) ) ) ) $=
|
|
( cnp wcel wa cpp co wceq cmp mulclpr syl2anc syl3anc adantr mulcomprg vx
|
|
vy simpllr simprlr simplrr simprrl addclpr simplrl simprll oveq2 ad2antll
|
|
addassprg simprrr distrprg 3eqtr3d simplll oveq1 ad2antrl oveq12d 3eqtr4d
|
|
vz oveq2d oveq1d 3eqtr2d addcomprg adantl caov12d caov32d adantrr adantrl
|
|
cv w3a eqtrd 3eqtr4rd caov13d caov4d caov42d ex ) AIJZBIJZKZCIJZDIJZKZKZG
|
|
IJZHIJZKZEIJZFIJZKZKZKZADLMZBCLMZNZGFLMZHELMZNZKZDGOMZAGOMZBHOMZLMCFOMZDE
|
|
OMZLMLMZLMZXAAHOMZBGOMZLMCEOMZDFOMZLMLMZLMZNWMWTKZXAXBXDLMZXCXELMZLMZLMZX
|
|
AXHXKLMZXIXJLMZLMZLMZXGXMXNXPXOXALMZLMZXAXSLMZXTLMZXRYBXNXPCHOMZLMZXTLMZX
|
|
PYGXTLMZLMZYFYDWMYIYKNZWTWMXPIJZYGIJZXTIJZYLWMXCIJZXEIJZYMWMVTWGYPVSVTWDW
|
|
LUCZWEWFWGWKUDZBHPQZWMWCWIYQWAWBWCWLUEZWEWHWIWJUFZDEPQZXCXEUGQZWMWBWGYNWA
|
|
WBWCWLUHZYSCHPQZWMXIIJZXJIJZYOWMVTWFUUGYRWEWFWGWKUIZBGPQZWMWBWIUUHUUEUUBC
|
|
EPQZXIXJUGQZXPYGXTULRSXNYEYHXTLXNXHXAXKLMZLMZXCYGLMZXELMZYEYHXNUUNXHDHOMZ
|
|
XELMZLMZXHUUQLMZXELMZUUPXNUUMUURXHLXNDWQOMZDWROMZUUMUURWSUVBUVCNWMWPWQWRD
|
|
OUJUKWMUVBUUMNZWTWMWCWFWJUVDUUAUUIWEWHWIWJUMZDGFUNRSWMUVCUURNZWTWMWCWGWIU
|
|
VFUUAYSUUBDHEUNRSUOVBWMUVAUUSNZWTWMXHIJZUUQIJZYQUVGWMVSWGUVHVSVTWDWLUPZYS
|
|
AHPQZWMWCWGUVIUUAYSDHPQUUCXHUUQXEULRSXNUUTUUOXELXNWNHOMZWOHOMZUUTUUOWPUVL
|
|
UVMNWMWSWNWOHOUQURWMUVLUUTNWTWMHWNOMZHAOMZHDOMZLMZUVLUUTWMWGVSWCUVNUVQNYS
|
|
UVJUUAHADUNRWMWNIJZWGUVLUVNNWMVSWCUVRUVJUUAADUGQZYSWNHTQWMXHUVOUUQUVPLWMV
|
|
SWGXHUVONUVJYSAHTQWMWCWGUUQUVPNUUAYSDHTQUSUTSWMUVMUUONWTWMHWOOMZHBOMZHCOM
|
|
ZLMZUVMUUOWMWGVTWBUVTUWCNYSYRUUEHBCUNRWMWOIJZWGUVMUVTNWMVTWBUWDYRUUEBCUGQ
|
|
ZYSWOHTQWMXCUWAYGUWBLWMVTWGXCUWANYRYSBHTQWMWBWGYGUWBNUUEYSCHTQUSUTSUOVCVD
|
|
WMUUNYENWTWMUAUBVAXHXAXKILUVKWMWCWFXAIJZUUAUUIDGPQZWMWCWJXKIJZUUAUVEDFPQZ
|
|
UAVKZIJZUBVKZIJZKZUWJUWLLMZUWLUWJLMNWMUWJUWLVEVFZUWKUWMVAVKZIJVLUWOUWQLMU
|
|
WJUWLUWQLMLMNWMUWJUWLUWQULVFZVGSWMUUPYHNWTWMUAUBVAXCYGXEILYTUUFUUCUWPUWRV
|
|
HSUOVCXNYCYJXPLXNXBXALMZXDLMZXIYGXJLMZLMZYCYJXNUWTXICGOMZLMZXDLMZUXBWMWPU
|
|
WTUXENWSWMWPKZUWSUXDXDLUXFWNGOMZWOGOMZUWSUXDWPUXGUXHNWMWNWOGOUQVFWMUXGUWS
|
|
NWPWMGWNOMZGAOMZGDOMZLMZUXGUWSWMWFVSWCUXIUXLNUUIUVJUUAGADUNRWMUVRWFUXGUXI
|
|
NUVSUUIWNGTQWMXBUXJXAUXKLWMVSWFXBUXJNUVJUUIAGTQWMWCWFXAUXKNUUAUUIDGTQUSUT
|
|
SWMUXHUXDNWPWMGWOOMZGBOMZGCOMZLMZUXHUXDWMWFVTWBUXMUXPNUUIYRUUEGBCUNRWMUWD
|
|
WFUXHUXMNUWEUUIWOGTQWMXIUXNUXCUXOLWMVTWFXIUXNNYRUUIBGTQWMWBWFUXCUXONUUEUU
|
|
ICGTQUSUTSUOVCVIWMWSUXEUXBNWPWMWSKZUXEXIUXCXDLMZLMZUXBWMUXEUXSNZWSWMUUGUX
|
|
CIJZXDIJZUXTUUJWMWBWFUYAUUEUUICGPQWMWBWJUYBUUEUVECFPQZXIUXCXDULRSUXQUXRUX
|
|
AXILUXQCWQOMZCWROMZUXRUXAWSUYDUYENWMWQWRCOUJVFWMUYDUXRNZWSWMWBWFWJUYFUUEU
|
|
UIUVECGFUNRSWMUYEUXANZWSWMWBWGWIUYGUUEYSUUBCHEUNRSUOVBVMVJVMWMUWTYCNWTWMU
|
|
AUBVAXBXAXDILWMVSWFXBIJZUVJUUIAGPQZUWGUYCUWPUWRVHSWMUXBYJNWTWMUAUBVAXIYGX
|
|
JILUUJUUFUUKUWPUWRVGSUOVBVNWMYDXRNWTWMUAUBVAXPXOXAILUUDWMUYHUYBXOIJUYIUYC
|
|
XBXDUGQUWGUWPUWRVOSWMYFYBNZWTWMUWFXSIJZYOUYJUWGWMUVHUWHUYKUVKUWIXHXKUGQUU
|
|
LXAXSXTULRSUOWMXRXGNWTWMXQXFXALWMUAUBVAXBXDXCXEILUYIUYCYTUWPUWRUUCUWNUWOI
|
|
JWMUWJUWLUGVFZVPVBSWMYBXMNWTWMYAXLXALWMUAUBVAXHXKXIXJILUVKUWIUUJUWPUWRUUK
|
|
UYLVQVBSUOVR $.
|
|
|
|
$( Lemma showing compatibility of multiplication. (Contributed by NM,
|
|
5-Sep-1995.) $)
|
|
mulcmpblnr $p |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) )
|
|
/\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) ->
|
|
( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) ->
|
|
<. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R
|
|
<. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) ) $=
|
|
( cnp wcel wa cpp co wceq cmp cop cer mulclpr syl2anc addclpr wbr simplrr
|
|
mulcmpblnrlemg wi simprll simplll simpllr simprlr simplrl simprrr simprrl
|
|
addcanprg syl3anc syld wb enrbreq syl22anc sylibrd ) AIJZBIJZKZCIJZDIJZKZ
|
|
KZGIJZHIJZKZEIJZFIJZKZKZKZADLMBCLMNGFLMHELMNKZAGOMZBHOMZLMZCFOMZDEOMZLMZL
|
|
MZAHOMZBGOMZLMZCEOMZDFOMZLMZLMZNZVQWDPWGVTPQUAZVMVNDGOMZWALMWKWHLMNZWIABC
|
|
DEFGHUCVMWKIJZWAIJZWHIJZWLWIUDVMVCVFWMVAVBVCVLUBZVEVFVGVKUEZDGRSVMVQIJZVT
|
|
IJZWNVMVOIJZVPIJZWRVMUSVFWTUSUTVDVLUFZWQAGRSVMUTVGXAUSUTVDVLUGZVEVFVGVKUH
|
|
ZBHRSVOVPTSZVMVRIJZVSIJZWSVMVBVJXFVAVBVCVLUIZVEVHVIVJUJZCFRSVMVCVIXGWPVEV
|
|
HVIVJUKZDERSVRVSTSZVQVTTSVMWDIJZWGIJZWOVMWBIJZWCIJZXLVMUSVGXNXBXDAHRSVMUT
|
|
VFXOXCWQBGRSWBWCTSZVMWEIJZWFIJZXMVMVBVIXQXHXJCERSVMVCVJXRWPXIDFRSWEWFTSZW
|
|
DWGTSWKWAWHULUMUNVMWRXLXMWSWJWIUOXEXPXSXKVQWDWGVTUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d a b c d f g h s t u v w x y $.
|
|
$( Decomposing signed reals into positive reals. Lemma for ~ addsrpr and
|
|
~ mulsrpr . (Contributed by Jim Kingdon, 30-Dec-2019.) $)
|
|
prsrlem1 $p |- ( (
|
|
( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) )
|
|
/\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
|
|
/\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> (
|
|
( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) )
|
|
/\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) )
|
|
/\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) ) $=
|
|
( cnp cer wcel wa cv cop cec wceq cpp co vx vy va vb vc cxp cqs cdm enrer
|
|
vd wer erdm ax-mp simprll simpll eqeltrrd ecelqsdm sylancr opelxp simprrl
|
|
sylib jca simprlr simplr simprrr wbr eqtr3d a1i mpbird wb df-enr ecopoveq
|
|
erth syl2anc mpbid jca31 ) EKKUFZLUGZMZFVRMZNZEAOZBOZPZLQZRZFCOZDOZPZLQZR
|
|
ZNZEJOZGOZPZLQZRZFHOZIOZPZLQZRZNZNZNZWBKMWCKMNZWMKMWNKMNZNWGKMWHKMNZWRKMW
|
|
SKMNZNWBWNSTWCWMSTRZWGWSSTWHWRSTRZNXEXFXGXEWDVQMZXFXELUHVQRZWEVRMXLVQLUKZ
|
|
XMUIVQLULUMZXEEWEVRWAWFWKXCUNZVSVTXDUOZUPVQWDLUQURZWBWCKKUSVAZXEWOVQMZXGX
|
|
EXMWPVRMXTXOXEEWPVRWAWLWQXBUTZXQUPVQWOLUQURWMWNKKUSVAZVBXEXHXIXEWIVQMZXHX
|
|
EXMWJVRMYCXOXEFWJVRWAWFWKXCVCZVSVTXDVDZUPVQWILUQURZWGWHKKUSVAZXEWTVQMZXIX
|
|
EXMXAVRMYHXOXEFXAVRWAWLWQXBVEZYEUPVQWTLUQURWRWSKKUSVAZVBXEXJXKXEWDWOLVFZX
|
|
JXEYKWEWPRXEEWEWPXPYAVGXEWDWOLVQXNXEUIVHZXRVMVIXEXFXGYKXJVJXSYBUAUBUCUDUE
|
|
UJWBWCWMWNSLKUAUBUCUDUEUJVKZVLVNVOXEWIWTLVFZXKXEYNWJXARXEFWJXAYDYIVGXEWIW
|
|
TLVQYLYFVMVIXEXHXIYNXKVJYGYJUAUBUCUDUEUJWGWHWRWSSLKYMVLVNVOVBVP $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h q s t u v w z $. $d B f g h q s t u v w z $.
|
|
$( There is at most one result from adding signed reals. (Contributed by
|
|
Jim Kingdon, 30-Dec-2019.) $)
|
|
addsrmo $p |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\
|
|
B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t
|
|
( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
|
|
/\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) $=
|
|
( vq vs cnp cer wcel wa cv cop cec wceq cpp co wex vf cxp cqs wal wmo wer
|
|
vg vh wi enrer a1i wbr prsrlem1 addcmpblnr imp syl erthi adantrlr simprlr
|
|
adantrrr simprrr 3eqtr4d expr exlimdvv impd alrimivv opeq12 eceq1d eqeq2d
|
|
ex anbi1d simpl oveq1d simpr opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i
|
|
imbi1i 2albii sylibr eqeq1 4exbidv mo4 ) FJJUBZKUCZLGWHLMZFBNZCNZOZKPZQZG
|
|
DNZENZOZKPZQZMZANZWJWORSZWKWPRSZOZKPZQZMZETDTZCTBTZWTHNZXEQZMZETDTCTBTZMZ
|
|
XAXJQZUIZHUDAUDZXIAUEWIXIFINZUANZOZKPZQZGUGNZUHNZOZKPZQZMZXJXRYCRSZXSYDRS
|
|
ZOZKPZQZMZUHTUGTZUATITZMZXOUIZHUDAUDXQWIYRAHWIXIYPXOWIXHYPXOUIZBCWIXGYSDE
|
|
WIXGYSWIXGMZYOXOIUAYTYNXOUGUHWIXGYNXOWIXGYNMMXEYLXAXJWIXGYHXEYLQZYMWIWTYH
|
|
UUAXFWIWTYHMMZXDYKKWGWGKUFUUBUJUKUUBWJJLWKJLMXRJLXSJLMMWOJLWPJLMYCJLYDJLM
|
|
MMZWJXSRSWKXRRSQWOYDRSWPYCRSQMZMXDYKKULZBCDEFGUAUGUHIUMUUCUUDUUEWJWKXRXSY
|
|
CYDWOWPUNUOUPUQURUTWIWTXFYNUSWIXGYHYMVAVBVCVDVDVJVDVDVEVFXPYRAHXNYQXOXMYP
|
|
XIXLYBWSMZXJXRWORSZXSWPRSZOZKPZQZMYNBCDEIUAUGUHWJXRQZWKXSQZMZWTUUFXKUUKUU
|
|
NWNYBWSUUNWMYAFUUNWLXTKWJWKXRXSVGVHVIVKUUNXEUUJXJUUNXDUUIKUUNXBUUGXCUUHUU
|
|
NWJXRWORUULUUMVLVMUUNWKXSWPRUULUUMVNVMVOVHVIVPWOYCQZWPYDQZMZUUFYHUUKYMUUQ
|
|
WSYGYBUUQWRYFGUUQWQYEKWOWPYCYDVGVHVIVQUUQUUJYLXJUUQUUIYKKUUQUUGYIUUHYJUUQ
|
|
WOYCXRRUUOUUPVLVRUUQWPYDXSRUUOUUPVNVRVOVHVIVPVSVTWAWBWCXIXMAHXOXGXLBCDEXO
|
|
XFXKWTXAXJXEWDVQWEWFWC $.
|
|
|
|
$( There is at most one result from multiplying signed reals. (Contributed
|
|
by Jim Kingdon, 30-Dec-2019.) $)
|
|
mulsrmo $p |- ( ( A e. ( ( P. X. P. ) /. ~R ) /\
|
|
B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t
|
|
( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R )
|
|
/\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) ,
|
|
( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) ) $=
|
|
( vq cnp cer wcel wa cv cop cec wceq cmp co cpp wex vs vf cxp cqs wal wmo
|
|
vg vh wi wer enrer a1i wbr prsrlem1 mulcmpblnr imp erthi adantrlr simprlr
|
|
syl adantrrr simprrr 3eqtr4d expr exlimdvv ex impd alrimivv opeq12 eceq1d
|
|
eqeq2d anbi1d oveq1d oveq12d opeq12d anbi12d anbi2d oveq2d cbvex4v anbi2i
|
|
simpl simpr imbi1i 2albii sylibr eqeq1 4exbidv mo4 ) FIIUCZJUDZKGWJKLZFBM
|
|
ZCMZNZJOZPZGDMZEMZNZJOZPZLZAMZWLWQQRZWMWRQRZSRZWLWRQRZWMWQQRZSRZNZJOZPZLZ
|
|
ETDTZCTBTZXBHMZXKPZLZETDTCTBTZLZXCXPPZUIZHUEAUEZXOAUFWKXOFUAMZUBMZNZJOZPZ
|
|
GUGMZUHMZNZJOZPZLZXPYDYIQRZYEYJQRZSRZYDYJQRZYEYIQRZSRZNZJOZPZLZUHTUGTZUBT
|
|
UATZLZYAUIZHUEAUEYCWKUUHAHWKXOUUFYAWKXNUUFYAUIZBCWKXMUUIDEWKXMUUIWKXMLZUU
|
|
EYAUAUBUUJUUDYAUGUHWKXMUUDYAWKXMUUDLLXKUUBXCXPWKXMYNXKUUBPZUUCWKXBYNUUKXL
|
|
WKXBYNLLZXJUUAJWIWIJUJUULUKULUULWLIKWMIKLYDIKYEIKLLWQIKWRIKLYIIKYJIKLLLZW
|
|
LYESRWMYDSRPWQYJSRWRYISRPLZLXJUUAJUMZBCDEFGUBUGUHUAUNUUMUUNUUOWLWMYDYEYIY
|
|
JWQWRUOUPUTUQURVAWKXBXLUUDUSWKXMYNUUCVBVCVDVEVEVFVEVEVGVHYBUUHAHXTUUGYAXS
|
|
UUFXOXRYHXALZXPYDWQQRZYEWRQRZSRZYDWRQRZYEWQQRZSRZNZJOZPZLUUDBCDEUAUBUGUHW
|
|
LYDPZWMYEPZLZXBUUPXQUVEUVHWPYHXAUVHWOYGFUVHWNYFJWLWMYDYEVIVJVKVLUVHXKUVDX
|
|
PUVHXJUVCJUVHXFUUSXIUVBUVHXDUUQXEUURSUVHWLYDWQQUVFUVGWAZVMUVHWMYEWRQUVFUV
|
|
GWBZVMVNUVHXGUUTXHUVASUVHWLYDWRQUVIVMUVHWMYEWQQUVJVMVNVOVJVKVPWQYIPZWRYJP
|
|
ZLZUUPYNUVEUUCUVMXAYMYHUVMWTYLGUVMWSYKJWQWRYIYJVIVJVKVQUVMUVDUUBXPUVMUVCU
|
|
UAJUVMUUSYQUVBYTUVMUUQYOUURYPSUVMWQYIYDQUVKUVLWAZVRUVMWRYJYEQUVKUVLWBZVRV
|
|
NUVMUUTYRUVAYSSUVMWRYJYDQUVOVRUVMWQYIYEQUVNVRVNVOVJVKVPVSVTWCWDWEXOXSAHYA
|
|
XMXRBCDEYAXLXQXBXCXPXKWFVQWGWHWE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u t A $. $d x y z w v u t B $. $d x y z w v u t C $.
|
|
$d x y z w v u t D $.
|
|
$( Addition of signed reals in terms of positive reals. (Contributed by
|
|
NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
addsrpr $p |- ( ( ( A e. P. /\ B e. P. ) /\
|
|
( C e. P. /\ D e. P. ) ) ->
|
|
( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) =
|
|
[ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) $=
|
|
( vw vv vu vt vz cnp wcel wa cop cer cec cv wceq cpp co wex vx vy cxp cqs
|
|
cplr opelxpi enrex ecelqsi syl anim12i pm3.2i opeq12 eceq1d eqeq2d anbi1d
|
|
eqid simpl oveq1d simpr opeq12d anbi12d spc2egv anbi2d oveq2d 2eximdv cvv
|
|
sylan9 mp2ani wi ecexg ax-mp w3a simp1 eqeq1d simp2 simp3 4exbidv addsrmo
|
|
coprab df-plr df-nr eleq2i anbi12i anbi1i oprabbii eqtri ovig mp3an3 sylc
|
|
cnr ) AJKBJKLZCJKDJKLZLZABMZNOZJJUCZNUDZKZCDMZNOZWQKZLWOEPZFPZMZNOZQZWTGP
|
|
ZHPZMZNOZQZLZACRSZBDRSZMZNOZXBXGRSZXCXHRSZMZNOZQZLZHTGTZFTETZWOWTUESXPQZW
|
|
KWRWLXAWKWNWPKWRABJJUFWPWNNUGUHUIWLWSWPKXACDJJUFWPWSNUGUHUIUJWMWOWOQZWTWT
|
|
QZLZXPXPQZYDYFYGWOUPWTUPUKXPUPWKYHYILZXFYGLZXPXBCRSZXCDRSZMZNOZQZLZFTETWL
|
|
YDYQYJEFABJJXBAQZXCBQZLZYKYHYPYIYTXFYFYGYTXEWOWOYTXDWNNXBXCABULUMUNUOYTYO
|
|
XPXPYTYNXONYTYLXMYMXNYTXBACRYRYSUQURYTXCBDRYRYSUSURUTUMUNVAVBWLYQYCEFYBYQ
|
|
GHCDJJXGCQZXHDQZLZXLYKYAYPUUCXKYGXFUUCXJWTWTUUCXIWSNXGXHCDULUMUNVCUUCXTYO
|
|
XPUUCXSYNNUUCXQYLXRYMUUCXGCXBRUUAUUBUQVDUUCXHDXCRUUAUUBUSVDUTUMUNVAVBVEVG
|
|
VHWRXAXPVFKZYDYEVINVFKUUDUGXOVFNVJVKUAPZXEQZUBPZXJQZLZIPZXTQZLZHTGTFTETZY
|
|
DUAUBIWOWTXPVFWQWQUEUUEWOQZUUGWTQZUUJXPQZVLZUULYBEFGHUUQUUIXLUUKYAUUQUUFX
|
|
FUUHXKUUQUUEWOXEUUNUUOUUPVMVNUUQUUGWTXJUUNUUOUUPVOVNVAUUQUUJXPXTUUNUUOUUP
|
|
VPVNVAVQIEFGHUUEUUGVRUEUUEWJKZUUGWJKZLZUUMLZUAUBIVSUUEWQKZUUGWQKZLZUUMLZU
|
|
AUBIVSUAUBIEFGHVTUVAUVEUAUBIUUTUVDUUMUURUVBUUSUVCWJWQUUEWAWBWJWQUUGWAWBWC
|
|
WDWEWFWGWHWI $.
|
|
|
|
$( Multiplication of signed reals in terms of positive reals. (Contributed
|
|
by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) $)
|
|
mulsrpr $p |- ( ( ( A e. P. /\ B e. P. ) /\
|
|
( C e. P. /\ D e. P. ) ) ->
|
|
( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) =
|
|
[ <. ( ( A .P. C ) +P. ( B .P. D ) ) ,
|
|
( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R ) $=
|
|
( vw vv vu vt vz cnp wcel wa cop cer cec wceq cmp co cpp wex vx vy cxp cv
|
|
cqs cmr opelxpi enrex ecelqsi syl eqid pm3.2i opeq12 eceq1d eqeq2d anbi1d
|
|
anim12i simpl oveq1d simpr oveq12d opeq12d anbi12d spc2egv anbi2d 2eximdv
|
|
oveq2d sylan9 mp2ani cvv ecexg ax-mp w3a simp1 eqeq1d simp2 simp3 4exbidv
|
|
wi mulsrmo coprab df-mr df-nr eleq2i anbi12i anbi1i oprabbii eqtri mp3an3
|
|
cnr ovig sylc ) AJKBJKLZCJKDJKLZLZABMZNOZJJUCZNUEZKZCDMZNOZWSKZLWQEUDZFUD
|
|
ZMZNOZPZXBGUDZHUDZMZNOZPZLZACQRZBDQRZSRZADQRZBCQRZSRZMZNOZXDXIQRZXEXJQRZS
|
|
RZXDXJQRZXEXIQRZSRZMZNOZPZLZHTGTZFTETZWQXBUFRYBPZWMWTWNXCWMWPWRKWTABJJUGW
|
|
RWPNUHUIUJWNXAWRKXCCDJJUGWRXANUHUIUJUQWOWQWQPZXBXBPZLZYBYBPZYNYPYQWQUKXBU
|
|
KULYBUKWMYRYSLZXHYQLZYBXDCQRZXEDQRZSRZXDDQRZXECQRZSRZMZNOZPZLZFTETWNYNUUK
|
|
YTEFABJJXDAPZXEBPZLZUUAYRUUJYSUUNXHYPYQUUNXGWQWQUUNXFWPNXDXEABUMUNUOUPUUN
|
|
UUIYBYBUUNUUHYANUUNUUDXQUUGXTUUNUUBXOUUCXPSUUNXDACQUULUUMURZUSUUNXEBDQUUL
|
|
UUMUTZUSVAUUNUUEXRUUFXSSUUNXDADQUUOUSUUNXEBCQUUPUSVAVBUNUOVCVDWNUUKYMEFYL
|
|
UUKGHCDJJXICPZXJDPZLZXNUUAYKUUJUUSXMYQXHUUSXLXBXBUUSXKXANXIXJCDUMUNUOVEUU
|
|
SYJUUIYBUUSYIUUHNUUSYEUUDYHUUGUUSYCUUBYDUUCSUUSXICXDQUUQUURURZVGUUSXJDXEQ
|
|
UUQUURUTZVGVAUUSYFUUEYGUUFSUUSXJDXDQUVAVGUUSXICXEQUUTVGVAVBUNUOVCVDVFVHVI
|
|
WTXCYBVJKZYNYOVSNVJKUVBUHYAVJNVKVLUAUDZXGPZUBUDZXLPZLZIUDZYJPZLZHTGTFTETZ
|
|
YNUAUBIWQXBYBVJWSWSUFUVCWQPZUVEXBPZUVHYBPZVMZUVJYLEFGHUVOUVGXNUVIYKUVOUVD
|
|
XHUVFXMUVOUVCWQXGUVLUVMUVNVNVOUVOUVEXBXLUVLUVMUVNVPVOVCUVOUVHYBYJUVLUVMUV
|
|
NVQVOVCVRIEFGHUVCUVEVTUFUVCWJKZUVEWJKZLZUVKLZUAUBIWAUVCWSKZUVEWSKZLZUVKLZ
|
|
UAUBIWAUAUBIEFGHWBUVSUWCUAUBIUVRUWBUVKUVPUVTUVQUWAWJWSUVCWCWDWJWSUVEWCWDW
|
|
EWFWGWHWKWIWL $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w v u f $. $d B x y z w v u f $. $d C x y z w v u f $.
|
|
$d D x y z w v u f $.
|
|
$( Ordering of signed reals in terms of positive reals. (Contributed by
|
|
Jim Kingdon, 2-Jan-2019.) $)
|
|
ltsrprg $p |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ->
|
|
( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <->
|
|
( A +P. D ) <P ( B +P. C ) ) ) $=
|
|
( vx vy cv cpp co cltp wbr cer cnp wcel wa wceq syl2anc addassprg syl3anc
|
|
addclpr vz vu vw vv vf cnr cltr enrex enrer df-nr df-ltr cop cec wb eqcom
|
|
enreceq syl6bb bi2anan9 simprlr simplrr addcomprg simprrl 3eqtr3d simplll
|
|
oveq12 adantl oveq1d oveq2d ad2ant2lr anim12ci an4s simpld 3eqtr4d adantr
|
|
simprll simplrl simprrr simpllr sylbid w3a ltaprg simprd caovord3d brecop
|
|
ex syld ) UAGZUBGZHIZUCGZUDGZHIZJKZADHIZBCHIZJKZEFUAUCUDUBABCDLMUFUGUHUIU
|
|
JEFUAUCUDUBUKWGMNZWJMNZOZAMNZBMNZOZOZWKMNZWHMNZOZCMNZDMNZOZOZOZWGWJULLUMA
|
|
BULLUMPZWKWHULLUMCDULLUMPZOZWIWOHIZWLWNHIZPZWMWPUNXKXNWGBHIZWJAHIZPZWHCHI
|
|
ZWKDHIZPZOZXQXCXLXTXJXMYCWGWJABUPXJXMYBYAPYCWKWHCDUPYBYAUOUQURXKYDXQXKYDO
|
|
XRYAHIZXSYBHIZXOXPYDYEYFPXKXRXSYAYBHVEVFXKXOYEPYDXKWGWHWOHIZHIZWGBYAHIZHI
|
|
ZXOYEXKYGYIWGHXKWHBHIZCHIZBWHHIZCHIZYGYIXKXEXAYLYNPXCXDXEXIUSZWSWTXAXJUTZ
|
|
XEXAOYKYMCHWHBVAVGQXKXEXAXGYLYGPYOYPXCXFXGXHVBZWHBCRSXKXAXEXGYNYIPYPYOYQB
|
|
WHCRSVCVHXKWQXEWOMNZXOYHPWQWRXBXJVDZYOXKYRWLMNZWSXFXBXIYRYTOWSXFOYTXBXIOY
|
|
RWRXDYTWQXEWJWKTVIXAXGYRWTXHBCTVIVJVKZVLZWGWHWORSXKWQXAYAMNZYEYJPYSYPXKXE
|
|
XGUUCYOYQWHCTQWGBYARSVMVNXKXPYFPYDXKWJWKWNHIZHIZWJAYBHIZHIZXPYFXKUUDUUFWJ
|
|
HXKWKAHIZDHIZAWKHIZDHIZUUDUUFXKUUHUUJDHXKXDWTUUHUUJPXCXDXEXIVOZWSWTXAXJVP
|
|
ZWKAVAQVGXKXDWTXHUUIUUDPUULUUMXCXFXGXHVQZWKADRSXKWTXDXHUUKUUFPUUMUULUUNAW
|
|
KDRSVCVHXKWRXDWNMNZXPUUEPWQWRXBXJVRZUULXKWTXHUUOUUMUUNADTQZWJWKWNRSXKWRWT
|
|
YBMNZYFUUGPUUPUUMXKXDXHUURUULUUNWKDTQWJAYBRSVMVNVMWEVSXKEFUEWIWOWLWNJMHEG
|
|
ZMNZFGZMNZUEGZMNVTUUSUVAJKUVCUUSHIUVCUVAHIJKUNXKUUSUVAUVCWAVFXKWQXEWIMNYS
|
|
YOWGWHTQUUBXKYRYTUUAWBUUTUVBOUUSUVAHIUVAUUSHIPXKUUSUVAVAVFUUQWCWFWD $.
|
|
$}
|
|
|
|
$( Greater than zero in terms of positive reals. (Contributed by NM,
|
|
13-May-1996.) $)
|
|
gt0srpr $p |- ( 0R <R [ <. A , B >. ] ~R <-> B <P A ) $=
|
|
( c0r cop cer cec cltr wbr cnp wcel wa cltp cxp cnr brel c1p cpp co wb 1pr
|
|
cdm wceq cqs wer enrer ax-mp ltrelsr simprd df-nr syl6eleq ecelqsdm sylancr
|
|
erdm opelxp sylib ltrelpr ancomd df-0r breq1i ltsrprg mpanl12 syl5bb ltaprg
|
|
mp3an3 ancoms bitr4d pm5.21nii ) CABDZEFZGHZAIJZBIJZKZBALHZVJVHIIMZJZVMVJEU
|
|
AVOUBZVIVOEUCZJVPVOEUDVQUEVOEUMUFVJVINVRVJCNJVINJCVINNGUGOUHUIUJVOVHEUKULAB
|
|
IIUNUOVNVLVKBAIILUPOUQVMVJPBQRPAQRLHZVNVJPPDEFZVIGHZVMVSCVTVIGURUSPIJZWBVMW
|
|
AVSSTTPPABUTVAVBVLVKVNVSSZVLVKWBWCTBAPVCVDVEVFVG $.
|
|
|
|
$( The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.)
|
|
(Revised by Mario Carneiro, 10-Jul-2014.) $)
|
|
0nsr $p |- -. (/) e. R. $=
|
|
( c0 wceq cnr wcel eqid wne cnp cxp cer cqs cdm wer enrer erdm ax-mp elqsn0
|
|
wn mpan df-nr eleq2s necon2bi ) AABACDZQAEUBAAAAFZAGGHZIJZCIKUDBZAUEDUCUDIL
|
|
UFMUDINOUDAIPRSTUAO $.
|
|
|
|
$( The constant ` 0R ` is a signed real. (Contributed by NM, 9-Aug-1995.) $)
|
|
0r $p |- 0R e. R. $=
|
|
( c1p cop cer cec cnp cxp cqs c0r cnr 1pr opelxpi mp2an enrex ecelqsi ax-mp
|
|
wcel df-0r df-nr 3eltr4i ) AABZCDZEEFZCGZHITUBPZUAUCPAEPZUEUDJJAAEEKLUBTCMN
|
|
OQRS $.
|
|
|
|
$( The constant ` 1R ` is a signed real. (Contributed by NM, 9-Aug-1995.) $)
|
|
1sr $p |- 1R e. R. $=
|
|
( c1p cpp co cop cer cec cnp cxp cqs c1r cnr wcel 1pr addclpr mp2an opelxpi
|
|
enrex ecelqsi ax-mp df-1r df-nr 3eltr4i ) AABCZADZEFZGGHZEIZJKUDUFLZUEUGLUC
|
|
GLZAGLZUHUJUJUIMMAANOMUCAGGPOUFUDEQRSTUAUB $.
|
|
|
|
$( The constant ` -1R ` is a signed real. (Contributed by NM,
|
|
9-Aug-1995.) $)
|
|
m1r $p |- -1R e. R. $=
|
|
( c1p cpp cop cer cec cnp cxp cqs cm1r cnr wcel addclpr mp2an opelxpi enrex
|
|
co 1pr ecelqsi ax-mp df-m1r df-nr 3eltr4i ) AAABPZCZDEZFFGZDHZIJUDUFKZUEUGK
|
|
AFKZUCFKZUHQUIUIUJQQAALMAUCFFNMUFUDDORSTUAUB $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Closure of addition on signed reals. (Contributed by NM,
|
|
25-Jul-1995.) $)
|
|
addclsr $p |- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) e. R. ) $=
|
|
( vx vy vz vw cnr wcel wa cplr co cnp cer cop cec df-nr wceq eleq1d cpp
|
|
cv cxp cqs oveq1 oveq2 addsrpr addclpr anim12i an4s opelxpi enrex ecelqsi
|
|
3syl eqeltrd 2ecoptocl syl6eleqr ) AGHBGHIABJKZLLUAZMUBZGCTZDTZNMOZETZFTZ
|
|
NMOZJKZURHAVDJKZURHUPURHCDEFABLLMGPVAAQVEVFURVAAVDJUCRVDBQVFUPURVDBAJUDRU
|
|
SLHZUTLHZIVBLHZVCLHZIIZVEUSVBSKZUTVCSKZNZMOZURUSUTVBVCUEVKVLLHZVMLHZIZVNU
|
|
QHVOURHVGVIVHVJVRVGVIIVPVHVJIVQUSVBUFUTVCUFUGUHVLVMLLUIUQVNMUJUKULUMUNPUO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Closure of multiplication on signed reals. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
mulclsr $p |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) e. R. ) $=
|
|
( vx vy vz vw cnr wcel wa cmr co cnp cer cop cec df-nr wceq cmp mulclpr
|
|
cv cxp cqs oveq1 eleq1d oveq2 cpp mulsrpr addclpr syl2an an4s jca opelxpi
|
|
an42s enrex ecelqsi 3syl eqeltrd 2ecoptocl syl6eleqr ) AGHBGHIABJKZLLUAZM
|
|
UBZGCTZDTZNMOZETZFTZNMOZJKZVBHAVHJKZVBHUTVBHCDEFABLLMGPVEAQVIVJVBVEAVHJUC
|
|
UDVHBQVJUTVBVHBAJUEUDVCLHZVDLHZIVFLHZVGLHZIIZVIVCVFRKZVDVGRKZUFKZVCVGRKZV
|
|
DVFRKZUFKZNZMOZVBVCVDVFVGUGVOVRLHZWALHZIWBVAHWCVBHVOWDWEVKVMVLVNWDVKVMIVP
|
|
LHVQLHWDVLVNIVCVFSVDVGSVPVQUHUIUJVKVNVLVMWEVKVNIVSLHVTLHWEVLVMIVCVGSVDVFS
|
|
VSVTUHUIUMUKVRWALLULVAWBMUNUOUPUQURPUS $.
|
|
$}
|
|
|
|
${
|
|
$d f g h r s t u v w x y z A $. $d u v w z B $. $d u v w x y z C $.
|
|
$( Addition of signed reals is commutative. (Contributed by Jim Kingdon,
|
|
3-Jan-2020.) $)
|
|
addcomsrg $p |- ( ( A e. R. /\ B e. R. ) -> ( A +R B ) = ( B +R A ) ) $=
|
|
( vx vy vz vw cnr cv cpp co cplr cer cnp df-nr addsrpr wcel wceq ad2ant2r
|
|
addcomprg ad2ant2l ecovicom ) CDEFABGCHZEHZIJZKLMDHZFHZIJZUCUBIJZUFUEIJZN
|
|
UBUEUCUFOUCUFUBUEOUBMPZUCMPZUDUHQUEMPZUFMPZUBUCSRULUMUGUIQUJUKUEUFSTUA $.
|
|
|
|
$( Addition of signed reals is associative. (Contributed by Jim Kingdon,
|
|
3-Jan-2020.) $)
|
|
addasssrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
|
|
( ( A +R B ) +R C ) = ( A +R ( B +R C ) ) ) $=
|
|
( vx vy vz vw vv vu cv cpp co cnp addsrpr wcel addclpr anim12i an4s wceq
|
|
wa cnr cer 3adant1r 3adant2r 3adant3r 3adant1l 3adant2l 3adant3l ecoviass
|
|
cplr df-nr addassprg ) DEFGHIABCUAUJGJZIJZKLZUBMDJZFJZKLZEJZUMKLZURHJZKLZ
|
|
UTUNKLZUPUQVAKLZKLZUSUOKLZVDUKUPUSUQUMNUQUMVAUNNURUTVAUNNUPUSVDUONUPMOZUQ
|
|
MOZUSMOZUMMOZURMOZUTMOZTVGVHTVKVIVJTVLUPUQPUSUMPQRVHVAMOZVJUNMOZVDMOZUOMO
|
|
ZTVHVMTVOVJVNTVPUQVAPUMUNPQRVGVITZVHVJTZVMVBVESZVNVQVHVMVSVJVGVHVMVSVIUPU
|
|
QVAULUCUDUEVQVRVNVCVFSZVMVQVJVNVTVHVIVJVNVTVGUSUMUNULUFUGUHUI $.
|
|
|
|
$( Multiplication of signed reals is commutative. (Contributed by Jim
|
|
Kingdon, 3-Jan-2020.) $)
|
|
mulcomsrg $p |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) ) $=
|
|
( vx vy vz vw cv cmp co cpp mulsrpr wcel wceq mulcomprg oveq12d ad2ant2rl
|
|
cnp wa ad2ant2lr mulclpr cnr cmr df-nr ad2ant2r ad2ant2l ancoms addcomprg
|
|
cer syl2anc eqtrd ecovicom ) CDEFABUACGZEGZHIZDGZFGZHIZJIUBUHQULUPHIZUOUM
|
|
HIZJIZUMULHIZUPUOHIZJIUMUOHIZUPULHIZJIZUCULUOUMUPKUMUPULUOKULQLZUOQLZRUMQ
|
|
LZUPQLZRRZUNVAUQVBJVFVHUNVAMVGVIULUMNUDVGVIUQVBMVFVHUOUPNUEOVJUTVDVCJIZVE
|
|
VJURVDUSVCJVFVIURVDMVGVHULUPNPVGVHUSVCMVFVIUOUMNSOVJVDQLZVCQLZVKVEMVFVIVL
|
|
VGVHVIVFVLUPULTUFPVGVHVMVFVIVHVGVMUMUOTUFSVDVCUGUIUJUK $.
|
|
|
|
$( Multiplication of signed reals is associative. (Contributed by Jim
|
|
Kingdon, 3-Jan-2020.) $)
|
|
mulasssrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
|
|
( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) ) $=
|
|
( vf vg vh cv cmp co cpp cnp mulsrpr wcel wa mulclpr addclpr syl2anc wceq
|
|
w3a adantl vx vy vz vw vv vu vr vs vt cnr cmr cer df-nr ad2ant2r ad2ant2l
|
|
ad2ant2rl ad2ant2lr mulcomprg distrprg simp1 simp2 simp3 caovdir2d simp1l
|
|
mulassprg simp1r simp2l simp2r simp3l simp3r addcomprg addassprg ecoviass
|
|
jca caovlem2d ) UAUBUCUDUEUFABCUJUKUCGZUFGZHIZUDGZUEGZHIZJIZULKUAGZVPHIZU
|
|
BGZVSHIZJIZWCVSHIZWEVPHIZJIZWGVTHIWJVQHIJIWGVQHIWJVTHIJIWCVPVTHIZVSVQHIZJ
|
|
IZHIWEWBHIJIWCWBHIWEWMHIJIWMUMWCWEVPVSLVPVSVTVQLWGWJVTVQLWCWEWMWBLWCKMZWE
|
|
KMZNZVPKMZVSKMZNZNZWGKMZWJKMZWTWDKMZWFKMZXAWNWQXCWOWRWCVPOUNWOWRXDWNWQWEV
|
|
SOUOWDWFPQWTWHKMZWIKMZXBWNWRXEWOWQWCVSOUPWOWQXFWNWRWEVPOUQWHWIPQVNWSVTKMZ
|
|
VQKMZNZNZWMKMZWBKMZXJWKKMZWLKMZXKWQXGXMWRXHVPVTOUNWRXHXNWQXGVSVQOUOWKWLPQ
|
|
XJVRKMZWAKMZXLWQXHXOWRXGVPVQOUPWRXGXPWQXHVSVTOUQVRWAPQVNWPWSXISZDEFWCWEVP
|
|
VSVQKJHVTDGZKMZEGZKMZNZXRXTHIZXTXRHIRXQXRXTURTZXSYAFGZKMZSZXRXTJIZYEHIXRY
|
|
EHIXTYEHIZJIRXQYGUGUHUIXRXTYEKJHUGGZKMZUHGZKMZUIGZKMSYJYLYNJIHIYJYLHIZYJY
|
|
NHIJIRYGYJYLYNUSTXSYAYFUTXSYAYFVAXSYAYFVBYKYMNZYJYLJIKMYGYJYLPTYPYOYLYJHI
|
|
RYGYJYLURTVCTZYGYCYEHIXRYIHIRXQXRXTYEVETZYBYCKMXQXRXTOTZWNWOWSXIVDZWNWOWS
|
|
XIVFZWPWQWRXIVGZWPWQWRXIVHZWPWSXGXHVIZWPWSXGXHVJZYBYHXTXRJIRXQXRXTVKTZYGY
|
|
HYEJIXRXTYEJIJIRXQXRXTYEVLTZYBYHKMXQXRXTPTZVOXQDEFWCWEVPVSVTKJHVQYDYQYRYS
|
|
YTUUAUUBUUCUUEUUDUUFUUGUUHVOVM $.
|
|
|
|
$( Multiplication of signed reals is distributive. (Contributed by Jim
|
|
Kingdon, 4-Jan-2020.) $)
|
|
distrsrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
|
|
( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) ) $=
|
|
( vf vg vh cnp cv cpp co cmp mulsrpr wcel wa addclpr mulclpr syl2anc wceq
|
|
distrprg syl3anc vx vy vz vw vv vu cnr cplr cer addsrpr ad2ant2r ad2ant2l
|
|
cmr jca ad2ant2rl ad2ant2lr w3a simp1l simp2l simp3l simp1r simp2r simp3r
|
|
df-nr oveq12d addcomprg adantl addassprg caov4d eqtrd ecovidi ) UAUBUCUDU
|
|
EUFABCUGUHUIGUMUAHZUCHZUEHZIJZKJZUBHZUDHZUFHZIJZKJZIJZVLVTKJZVQVOKJZIJZVL
|
|
VMKJZVQVRKJZIJZVLVNKJZVQVSKJZIJZIJZVLVRKJZVQVMKJZIJZVLVSKJZVQVNKJZIJZIJZV
|
|
OVTWHWOWKWRVDVMVRVNVSUJVLVQVOVTLVLVQVMVRLVLVQVNVSLWHWOWKWRUJVMGMZVRGMZNZV
|
|
NGMZVSGMZNZNVOGMZVTGMZWTXCXFXAXDVMVNOUKXAXDXGWTXCVRVSOULUNVLGMZVQGMZNZXBN
|
|
ZWHGMZWOGMZXKWFGMZWGGMZXLXHWTXNXIXAVLVMPZUKXIXAXOXHWTVQVRPZULWFWGOQXKWMGM
|
|
ZWNGMZXMXHXAXRXIWTVLVRPZUOXIWTXSXHXAVQVMPZUPWMWNOQUNXJXENZWKGMZWRGMZYBWIG
|
|
MZWJGMZYCXHXCYEXIXDVLVNPZUKXIXDYFXHXCVQVSPZULWIWJOQYBWPGMZWQGMZYDXHXDYIXI
|
|
XCVLVSPZUOXIXCYJXHXDVQVNPZUPWPWQOQUNXJXBXEUQZWBWFWIIJZWGWJIJZIJWLYMVPYNWA
|
|
YOIYMXHWTXCVPYNRXHXIXBXEURZXJWTXAXEUSZXJXBXCXDUTZVLVMVNSTYMXIXAXDWAYORXHX
|
|
IXBXEVAZXJWTXAXEVBZXJXBXCXDVCZVQVRVSSTVEYMDEFWFWIWGWJGIYMXHWTXNYPYQXPQYMX
|
|
HXCYEYPYRYGQYMXIXAXOYSYTXQQDHZGMZEHZGMZNZUUBUUDIJZUUDUUBIJRYMUUBUUDVFVGZU
|
|
UCUUEFHZGMUQUUGUUIIJUUBUUDUUIIJIJRYMUUBUUDUUIVHVGZYMXIXDYFYSUUAYHQUUFUUGG
|
|
MYMUUBUUDOVGZVIVJYMWEWMWPIJZWNWQIJZIJWSYMWCUULWDUUMIYMXHXAXDWCUULRYPYTUUA
|
|
VLVRVSSTYMXIWTXCWDUUMRYSYQYRVQVMVNSTVEYMDEFWMWPWNWQGIYMXHXAXRYPYTXTQYMXHX
|
|
DYIYPUUAYKQYMXIWTXSYSYQYAQUUHUUJYMXIXCYJYSYRYLQUUKVIVJVK $.
|
|
$}
|
|
|
|
$( Minus one plus one is zero for signed reals. (Contributed by NM,
|
|
5-May-1996.) $)
|
|
m1p1sr $p |- ( -1R +R 1R ) = 0R $=
|
|
( cm1r c1r cplr co c1p cpp cop cer cec c0r df-m1r cnp wcel wceq 1pr addclpr
|
|
df-1r mp2an mp4an eqtr4i oveq12i df-0r addsrpr addassprg mp3an oveq2i mpbir
|
|
wb enreceq ) ABCDEEEFDZGHIZUJEGHIZCDZJAUKBULCKQUAJEEGHIZUMUBUMEUJFDZUJEFDZG
|
|
HIZUNELMZUJLMZUSURUMUQNOURURUSOOEEPRZUTOEUJUJEUCSUNUQNZEUPFDEUOFDNZUPUOEFUR
|
|
URURUPUONOOOEEEUDUEUFURURUOLMZUPLMZVAVBUHOOURUSVCOUTEUJPRUSURVDUTOUJEPREEUO
|
|
UPUISUGTTT $.
|
|
|
|
$( Minus one times minus one is plus one for signed reals. (Contributed by
|
|
NM, 14-May-1996.) $)
|
|
m1m1sr $p |- ( -1R .R -1R ) = 1R $=
|
|
( cm1r cmr c1p cpp cop cer cec c1r df-m1r oveq12i cmp cnp wcel wceq addclpr
|
|
co 1pr mp2an mulclpr eqtr4i df-1r mulsrpr mp4an addassprg mp3an 1idpr ax-mp
|
|
distrprg mulcomprg oveq1i oveq2i wb enreceq mpbir ) AABPCCCDPZEFGZUPBPZHAUP
|
|
AUPBIIJHUOCEFGZUQUAUQCCKPZUOUOKPZDPZCUOKPZUOCKPZDPZEFGZURCLMZUOLMZVFVGUQVEN
|
|
QVFVFVGQQCCORZQVHCUOCUOUBUCURVENZUOVDDPZCVADPZNZVJCCVDDPZDPZVKVFVFVDLMZVJVN
|
|
NQQVBLMZVCLMZVOVFVGVPQVHCUOSRVGVFVQVHQUOCSRVBVCORZCCVDUDUEVAVMCDUSCUTVDDVFU
|
|
SCNQCUFUGUTVCVCDPZVDVGVFVFUTVSNVHQQUOCCUHUEVBVCVCDVFVGVBVCNQVHCUOUIRUJTJUKT
|
|
VGVFVALMZVOVIVLULVHQUSLMZUTLMZVTVFVFWAQQCCSRVGVGWBVHVHUOUOSRUSUTORVRUOCVAVD
|
|
UMUCUNTTT $.
|
|
|
|
${
|
|
$d f g h r s t x y z w v u $.
|
|
$( Signed real 'less than' is a transitive relation. (Contributed by Jim
|
|
Kingdon, 4-Jan-2019.) $)
|
|
lttrsr $p |- ( ( f e. R. /\ g e. R. /\ h e. R. ) ->
|
|
( ( f <R g /\ g <R h ) -> f <R h ) ) $=
|
|
( vr vs vt cv cer cltr wbr wa cnp wceq wcel cpp co cltp addclpr syl2anc
|
|
wb vx vy vz vw vv vu cop cec cnr df-nr breq1 anbi1d imbi12d breq2 anbi12d
|
|
wi imbi1d anbi2d ltsrprg ltaprg adantl simp1l simp2r simp1r simp2l simp3r
|
|
3adant3 addcomprg caovord2d addassprg syl3anc breq12d bitrd simp3l ltsopr
|
|
w3a 3adant1 ltrelpr biimprd caov12d 3adant2 3imtr4d syl5 sylbid 3ecoptocl
|
|
sotri ) UAGZUBGZUGHUHZUCGZUDGZUGHUHZIJZWLUEGZUFGZUGHUHZIJZKZWIWPIJZUPAGZW
|
|
LIJZWQKZWTWPIJZUPWTBGZIJZXDWPIJZKZXCUPXEXDCGZIJZKZWTXHIJZUPUAUBUCUDUEUFWT
|
|
XDXHLHUIUJWIWTMZWRXBWSXCXLWMXAWQWIWTWLIUKULWIWTWPIUKUMWLXDMZXBXGXCXMXAXEW
|
|
QXFWLXDWTIUNWLXDWPIUKUOUQWPXHMZXGXJXCXKXNXFXIXEWPXHXDIUNURWPXHWTIUNUMWGLN
|
|
ZWHLNZKZWJLNZWKLNZKZWNLNZWOLNZKZVPZWRWGWKWOOPOPZWHWJWOOPZOPZQJZYGWHWKWNOP
|
|
ZOPZQJZKZWSYDWMYHWQYKYDWMWGWKOPZWHWJOPZQJZYHXQXTWMYOTYCWGWHWJWKUSVGYDYOYM
|
|
WOOPZYNWOOPZQJYHYDDEFYMYNWOQLODGZLNZEGZLNZFGZLNVPZYRYTQJUUBYROPUUBYTOPQJT
|
|
YDYRYTUUBUTVAYDXOXSYMLNXOXPXTYCVBZXQXRXSYCVCZWGWKRSYDXPXRYNLNXOXPXTYCVDZX
|
|
QXRXSYCVEZWHWJRSXQXTYAYBVFZYSUUAKYRYTOPZYTYROPMYDYRYTVHVAZVIYDYPYEYQYGQYD
|
|
XOXSYBYPYEMUUDUUEUUHWGWKWOVJVKYDXPXRYBYQYGMUUFUUGUUHWHWJWOVJVKVLVMVMYDWQY
|
|
FYIQJZYKXTYCWQUUKTXQWJWKWNWOUSVQYDYFLNZYILNZXPUUKYKTYDXRYBUULUUGUUHWJWORS
|
|
YDXSYAUUMUUEXQXTYAYBVNZWKWNRSUUFYFYIWHUTVKVMUOYLYEYJQJZYDWSYEYGYJQLVOVRWF
|
|
YDWKWGWOOPZOPZWKWHWNOPZOPZQJZUUPUURQJZUUOWSYDUVAUUTYDUUPLNZUURLNZXSUVAUUT
|
|
TYDXOYBUVBUUDUUHWGWORSYDXPYAUVCUUFUUNWHWNRSUUEUUPUURWKUTVKVSYDYEUUQYJUUSQ
|
|
YDDEFWGWKWOLOUUDUUEUUHUUJUUCUUIUUBOPYRYTUUBOPOPMYDYRYTUUBVJVAZVTYDDEFWHWK
|
|
WNLOUUFUUEUUNUUJUVDVTVLXQYCWSUVATXTWGWHWNWOUSWAWBWCWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d x y f g h $.
|
|
$( Signed real 'less than' is a partial order. (Contributed by Jim
|
|
Kingdon, 4-Jan-2019.) $)
|
|
ltposr $p |- <R Po R. $=
|
|
( vf vg vh vx vy cnr cltr wpo wtru cv wcel wbr wn cer cnp cpp cltp adantl
|
|
wa co cop cec df-nr wceq id breq12d notbid ltsopr soirri addcomprg breq2d
|
|
ltrelpr mtbii wb ltsrprg anidms mtbird ecoptocl w3a wi lttrsr ispod trud
|
|
) FGHIABCFGAJZFKZVDVDGLZMZIDJZEJZUANUBZVJGLZMVGDEVDOONFUCVJVDUDZVKVFVLVJV
|
|
DVJVDGVLUEZVMUFUGVHOKVIOKSZVKVHVIPTZVIVHPTZQLZVNVOVOQLVQVOQOUHULUIVNVOVPV
|
|
OQVHVIUJUKUMVNVKVQUNVHVIVHVIUOUPUQURRVEBJZFKCJZFKUSVDVRGLVRVSGLSVDVSGLUTI
|
|
ABCVARVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d a b c d e f r s t x y z $.
|
|
$( Signed real 'less than' is a strict ordering. (Contributed by NM,
|
|
19-Feb-1996.) $)
|
|
ltsosr $p |- <R Or R. $=
|
|
( vx vy vr vs vt cnr cltr cv wbr wo wi cnp wceq wcel cltp addclpr syl2anc
|
|
cpp co wb vz va vb vc vd ve vf wor wpo wral ltposr cop df-nr breq1 orbi1d
|
|
cer cec imbi12d breq2 orbi2d imbi2d wa simp1l simp3r simp2r simp2l simp1r
|
|
orbi12d simp3l ltsopr sowlin mpan syl3anc ltaprg addcomprg adantl caov12d
|
|
addassprg caovassd eqtr4d caov32d eqtr3d breq12d bitrd caovord2d caovcomd
|
|
w3a eqtrd breq1d bitr4d 3imtr4d ltsrprg 3adant3 3adant2 3adant1 3ecoptocl
|
|
ancoms rgen3 df-iso mpbir2an ) FGUHFGUIAHZBHZGIZXAUAHZGIZXDXBGIZJZKZUAFUJ
|
|
BFUJAFUJUKXHABUAFFFUBHZUCHZULUPUQZUDHZUEHZULUPUQZGIZXKUFHZUGHZULUPUQZGIZX
|
|
RXNGIZJZKXAXNGIZXAXRGIZXTJZKXCYCXRXBGIZJZKXHUBUCUDUEUFUGXAXBXDLUPFUMXKXAM
|
|
ZXOYBYAYDXKXAXNGUNYGXSYCXTXKXAXRGUNUOURXNXBMZYBXCYDYFXNXBXAGUSYHXTYEYCXNX
|
|
BXRGUSUTURXRXDMZYFXGXCYIYCXEYEXFXRXDXAGUSXRXDXBGUNVHVAXILNZXJLNZVBZXLLNZX
|
|
MLNZVBZXPLNZXQLNZVBZWGZXIXMRSZXJXLRSZOIZXIXQRSZXJXPRSZOIZXPXMRSZXQXLRSZOI
|
|
ZJZXOYAYSUUCXMRSZUUGXJRSZOIZUUJUUDXMRSZOIZUUMUUKOIZJZUUBUUIYSUUJLNZUUKLNZ
|
|
UUMLNZUULUUPKZYSUUCLNZYNUUQYSYJYQUVAYJYKYOYRVCZYLYOYPYQVDZXIXQPQZYLYMYNYR
|
|
VEZUUCXMPQYSUUGLNZYKUURYSYQYMUVFUVCYLYMYNYRVFZXQXLPQZYJYKYOYRVGZUUGXJPQYS
|
|
UUDLNZYNUUSYSYKYPUVJUVIYLYOYPYQVIZXJXPPQZUVEUUDXMPQLOUHUUQUURUUSWGUUTVJLU
|
|
UJUUKUUMOVKVLVMYSUUBXQYTRSZXQUUARSZOIZUULYSYTLNZUUALNZYQUUBUVOTYSYJYNUVPU
|
|
VBUVEXIXMPQYSYKYMUVQUVIUVGXJXLPQUVCYTUUAXQVNVMYSUVMUUJUVNUUKOYSUVMXIXQXMR
|
|
SRSUUJYSCDEXQXIXMLRUVCUVBUVECHZLNZDHZLNZVBUVRUVTRSZUVTUVRRSMYSUVRUVTVOVPZ
|
|
UVSUWAEHZLNWGZUWBUWDRSUVRUVTUWDRSRSMYSUVRUVTUWDVRVPZVQYSCDEXIXQXMLRUWFUVB
|
|
UVCUVEVSVTYSXQXJRSXLRSUVNUUKYSCDEXQXJXLLRUWFUVCUVIUVGVSYSCDEXQXJXLLRUVCUV
|
|
IUVGUWCUWFWAWBWCWDYSUUEUUNUUHUUOYSCDEUUCUUDXMOLRUWEUVRUVTOIUWDUVRRSUWDUVT
|
|
RSOITYSUVRUVTUWDVNVPZUVDUVLUVEUWCWEYSUUHUUFXJRSZUUKOIUUOYSCDEUUFUUGXJOLRU
|
|
WGYSYPYNUUFLNUVKUVEXPXMPQZUVHUVIUWCWEYSUUMUWHUUKOYSUUMXJUUFRSUWHYSCDEXJXP
|
|
XMLRUWFUVIUVKUVEVSYSCDXJUUFLRUWCUVIUWIWFWHWIWJVHWKYLYOXOUUBTYRXIXJXLXMWLW
|
|
MYSXSUUEXTUUHYLYRXSUUETYOXIXJXPXQWLWNYOYRXTUUHTZYLYRYOUWJXPXQXLXMWLWQWOVH
|
|
WKWPWRABUAFGWSWT $.
|
|
$}
|
|
|
|
$( 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) $)
|
|
0lt1sr $p |- 0R <R 1R $=
|
|
( c1p cop cer cec cpp co c0r c1r cltr wbr cltp cnp wcel 1pr addclpr ltaddpr
|
|
mp2an wceq addcomprg breqtrri wb ltsrprg mp4an mpbir df-0r df-1r 3brtr4i )
|
|
AABCDZAAEFZABCDZGHIUHUJIJZUIAUIEFZKJZUIUIAEFZULKUILMZALMZUIUNKJUPUPUONNAAOQ
|
|
ZNUIAPQUPUOULUNRNUQAUISQTUPUPUOUPUKUMUANNUQNAAUIAUBUCUDUEUFUG $.
|
|
|
|
$( 1 and 0 are distinct for signed reals. (Contributed by NM,
|
|
26-Mar-1996.) $)
|
|
1ne0sr $p |- -. 1R = 0R $=
|
|
( c1r c0r wceq cltr wbr cnr wor wcel wn ltsosr 1sr sonr mp2an 0lt1sr mpbiri
|
|
breq1 mto ) ABCZAADEZFDGAFHSIJKFADLMRSBADENABADPOQ $.
|
|
|
|
${
|
|
$d x y z w v $. $d x y A $.
|
|
$( The signed real number 0 is an identity element for addition of signed
|
|
reals. (Contributed by NM, 10-Apr-1996.) $)
|
|
0idsr $p |- ( A e. R. -> ( A +R 0R ) = A ) $=
|
|
( vx vy vz vw vv cv cop cer cec c0r cplr co wceq cnp wcel wa c1p cpp 1pr
|
|
df-nr oveq1 id eqeq12d df-0r oveq2i addsrpr mpanr12 simpl simpr addcomprg
|
|
cnr a1i adantl w3a addassprg caov12d addclpr mpan2 anim12i enreceq mpbird
|
|
wb mpdan eqtr4d syl5eq ecoptocl ) BGZCGZHIJZKLMZVJNAKLMZANBCAOOIULUAVJANZ
|
|
VKVLVJAVJAKLUBVMUCUDVHOPZVIOPZQZVKVJRRHIJZLMZVJKVQVJLUEUFVPVRVHRSMZVIRSMZ
|
|
HIJZVJVPROPZWBVRWANTTVHVIRRUGUHVPVJWANZVHVTSMVIVSSMNZVPDEFVHVIROSVNVOUIVN
|
|
VOUJWBVPTUMDGZOPZEGZOPZQWEWGSMZWGWESMNVPWEWGUKUNWFWHFGZOPUOWIWJSMWEWGWJSM
|
|
SMNVPWEWGWJUPUNUQVPVSOPZVTOPZQWCWDVCVNWKVOWLVNWBWKTVHRURUSVOWBWLTVIRURUSU
|
|
TVHVIVSVTVAVDVBVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y z w v $.
|
|
$( 1 is an identity element for multiplication. (Contributed by Jim
|
|
Kingdon, 5-Jan-2020.) $)
|
|
1idsr $p |- ( A e. R. -> ( A .R 1R ) = A ) $=
|
|
( cv cop cer cec c1r cmr co wceq cnp wcel c1p cpp cmp 1pr addclpr mulclpr
|
|
wa sylancl mpan2 vx vy vz vw vv df-nr oveq1 id eqeq12d df-1r oveq2i mp2an
|
|
cnr mulsrpr mpanr12 distrprg mp3an23 1idpr oveq1d eqtr2d oveqan12d adantl
|
|
eqtrd simpl addassprg syl3anc simpr addcomprg w3a caov12d 3eqtr3d anim12i
|
|
wb syl2an enreceq syldan anidms mpbird eqtr4d syl5eq ecoptocl ) UABZUBBZC
|
|
DEZFGHZWDIAFGHZAIUAUBAJJDUMUFWDAIZWEWFWDAWDAFGUGWGUHUIWBJKZWCJKZRZWEWDLLM
|
|
HZLCDEZGHZWDFWLWDGUJUKWJWMWBWKNHZWCLNHZMHZWBLNHZWCWKNHZMHZCDEZWDWJWKJKZLJ
|
|
KZWMWTIXBXBXAOOLLPULZOWBWCWKLUNUOWJWDWTIZWBWSMHZWCWPMHZIZWJWBWQMHZWRMHZWN
|
|
WCWOMHZMHXEXFWHWIXHWNWRXJMWHWNWQWQMHZXHWHXBXBWNXKIOOWBLLUPUQWHWQWBWQMWBUR
|
|
USUTWIWRWOWOMHZXJWIXBXBWRXLIOOWCLLUPUQWIWOWCWOMWCURUSVCVAWJWHWQJKZWRJKZXI
|
|
XEIWHWIVDZWJWHXBXMXOOWBLQZSWIXNWHWIXAXNXCWCWKQTZVBWBWQWRVEVFWJUCUDUEWNWCW
|
|
OJMWJWHXAWNJKZXOXCWBWKQZSWHWIVGZWJWIXBWOJKZXTOWCLQZSUCBZJKZUDBZJKZRYCYEMH
|
|
ZYEYCMHIWJYCYEVHVBYDYFUEBZJKVIYGYHMHYCYEYHMHMHIWJYCYEYHVEVBVJVKWJXDXGVMZW
|
|
JWJWPJKZWSJKZRYIWJYJWJYKWHXRYAYJWIWHXAXRXCXSTWIXBYAOYBTWNWOPVNWHXMXNYKWIW
|
|
HXBXMOXPTXQWQWRPVNVLWBWCWPWSVOVPVQVRVSVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) $)
|
|
00sr $p |- ( A e. R. -> ( A .R 0R ) = 0R ) $=
|
|
( vx vy cv cop cer cec c0r cmr co wceq cnp wa c1p cmp cpp mpanr12 mulclpr
|
|
wcel 1pr cnr df-nr oveq1 eqeq1d mulsrpr mpan2 addclpr syl2an anim12i eqid
|
|
enreceq mpbiri sylan anidms eqtrd df-0r oveq2i 3eqtr4g ecoptocl ) BDZCDZE
|
|
FGZHIJZHKAHIJZHKBCALLFUAUBVBAKVCVDHVBAHIUCUDUTLSZVALSZMZVBNNEFGZIJZVHVCHV
|
|
GVIUTNOJZVANOJZPJZVLEFGZVHVGNLSZVNVIVMKTTUTVANNUEQVGVMVHKZVGVGMZVNVNVOTTV
|
|
PVLLSZVQMZVNVNMZVOVGVQVGVQVEVJLSZVKLSZVQVFVEVNVTTUTNRUFVFVNWATVANRUFVJVKU
|
|
GUHZWBUIVRVSMVOVLNPJZWCKWCUJVLVLNNUKULUMQUNUOHVHVBIUPUQUPURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d r s t x y z w v u $.
|
|
$( Ordering property of addition. (Contributed by NM, 10-May-1996.) $)
|
|
ltasrg $p |- ( ( A e. R. /\ B e. R. /\ C e. R. ) ->
|
|
( A <R B <-> ( C +R A ) <R ( C +R B ) ) ) $=
|
|
( vr vs wcel cltr wbr cplr co wb cop cer cec cnp wceq cpp addclpr syl2anc
|
|
cv vx vy vz vw vv vu vt cnr df-nr oveq1 breq12d bibi2d breq1 oveq2 breq1d
|
|
bibi12d breq2 breq2d w3a cltp simp2l simp3r simp2r simp3l 3ad2ant1 ltaprg
|
|
syl3anc ltsrprg 3adant1 simp1l simp1r syl22anc addcomprg adantl addassprg
|
|
wa caovcomd caov42d eqtrd bitrd 3bitr4d addsrpr 3adant3 3adant2 3ecoptocl
|
|
caov4d bitr4d 3coml ) CUHFAUHFBUHFABGHZCAIJZCBIJZGHZKZUATZUBTZLMNZUCTZUDT
|
|
ZLMNZGHZUETZUFTZLMNZWPIJZXCWSIJZGHZKWTCWPIJZCWSIJZGHZKAWSGHZWJXHGHZKWMUEU
|
|
FUAUBUCUDCABOMUHUIXCCPZXFXIWTXLXDXGXEXHGXCCWPIUJXCCWSIUJUKULWPAPZWTXJXIXK
|
|
WPAWSGUMXMXGWJXHGWPACIUNUOUPWSBPZXJWIXKWLWSBAGUQXNXHWKWJGWSBCIUNURUPXAOFZ
|
|
XBOFZVPZWNOFZWOOFZVPZWQOFZWROFZVPZUSZWTXAWNQJZXBWOQJZLMNZXAWQQJZXBWRQJZLM
|
|
NZGHZXFYDWNWRQJZWOWQQJZUTHZXAXBQJZYLQJZYOYMQJZUTHZWTYKYDYLOFZYMOFZYOOFZYN
|
|
YRKYDXRYBYSXQXRXSYCVAZXQXTYAYBVBZWNWRRSYDXSYAYTXQXRXSYCVCZXQXTYAYBVDZWOWQ
|
|
RSXQXTUUAYCXAXBRVEYLYMYOVFVGXTYCWTYNKXQWNWOWQWRVHVIYDYKYEYIQJZYFYHQJZUTHZ
|
|
YRYDYEOFZYFOFZYHOFZYIOFZYKUUHKYDXOXRUUIXOXPXTYCVJZUUBXAWNRSYDXPXSUUJXOXPX
|
|
TYCVKZUUDXBWORSZYDXOYAUUKUUMUUEXAWQRSZYDXPYBUULUUNUUCXBWRRSYEYFYHYIVHVLYD
|
|
UUFYPUUGYQUTYDDEUGXAWNXBWROQUUMUUBUUNDTZOFZETZOFZVPZUUQUUSQJZUUSUUQQJPYDU
|
|
UQUUSVMVNZUURUUTUGTZOFUSUVBUVDQJUUQUUSUVDQJQJPYDUUQUUSUVDVOVNZUUCUVAUVBOF
|
|
YDUUQUUSRVNZWFYDUUGYHYFQJYQYDDEYFYHOQUVCUUOUUPVQYDDEUGXAWQXBWOOQUUMUUEUUN
|
|
UVCUVEUUDUVFVRVSUKVTWAYDXDYGXEYJGXQXTXDYGPYCXAXBWNWOWBWCXQYCXEYJPXTXAXBWQ
|
|
WRWBWDUKWGWEWH $.
|
|
$}
|
|
|
|
$( A signed real plus its negative is zero. (Contributed by NM,
|
|
14-May-1996.) $)
|
|
pn0sr $p |- ( A e. R. -> ( A +R ( A .R -1R ) ) = 0R ) $=
|
|
( cnr wcel c0r cmr co c1r cm1r cplr wceq m1r distrsrg mp3an23 m1p1sr oveq2i
|
|
1sr a1i mulclsr mpan2 addcomsrg syl2anc 3eqtr3d 00sr 1idsr oveq1d 3eqtr3rd
|
|
) ABCZADEFZAGEFZAHEFZIFZDAUJIFUGAHGIFZEFZUJUIIFZUHUKUGHBCZGBCZUMUNJKPAHGLMU
|
|
MUHJUGULDAENOQUGUJBCZUIBCZUNUKJUGUOUQKAHRSUGUPURPAGRSUJUITUAUBAUCUGUIAUJIAU
|
|
DUEUF $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Existence of negative signed real. Part of Proposition 9-4.3 of
|
|
[Gleason] p. 126. (Contributed by NM, 2-May-1996.) $)
|
|
negexsr $p |- ( A e. R. -> E. x e. R. ( A +R x ) = 0R ) $=
|
|
( cnr wcel cm1r cmr co cplr c0r wceq cv wrex m1r mpan2 pn0sr oveq2 eqeq1d
|
|
mulclsr rspcev syl2anc ) BCDZBEFGZCDZBUBHGZIJZBAKZHGZIJZACLUAECDUCMBERNBO
|
|
UHUEAUBCUFUBJUGUDIUFUBBHPQST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z w v f g h $.
|
|
$( The reciprocal of a positive signed real exists and is positive.
|
|
(Contributed by Jim Kingdon, 6-Feb-2020.) $)
|
|
recexgt0sr $p |- ( 0R <R A ->
|
|
E. x e. R. ( 0R <R x /\ ( A .R x ) = 1R ) ) $=
|
|
( cnr wcel c0r cltr wbr cv co c1r wceq wa cer cnp cpp cmp c1p 1pr syl2anc
|
|
adantr vy vz vw vv vf vg vh cmr wrex ltrelsr brel simprd cop cec wi df-nr
|
|
breq2 oveq1 eqeq1d anbi2d rexbidv imbi12d gt0srpr ltexpri recexpr addclpr
|
|
cltp sylbi adantl mpan2 ecopqsi sylancl ad2antlr simprr ltaddpr addcomprg
|
|
enrex mpan breqtrd sylibr syl anim2i mulsrpr adantlrl eqcomd ad2antll w3a
|
|
jctir mulcomprg 3adant2 3adant1 oveq12d 3coml simp3 3adant3 simplr simprl
|
|
distrprg 3eqtr2rd caovdird oveq2 sylan9eq adantrr eqtrd oveq1d a1i simpll
|
|
mulclpr addassprg caov32d syl3anc oveq2d caov12d 3eqtr4d wb eqeltrd mp2an
|
|
enreceq syl22anc mpbird syl6eqr anbi12d rspcev syl12anc anassrs rexlimdva
|
|
df-1r exp32 mpd syl5 ecoptocl mpcom ) BCDZEBFGZEAHZFGZBYOUHIZJKZLZACUIZYN
|
|
ECDYMEBCCFUJUKULEUAHZUBHZUMMUNZFGZYPUUCYOUHIZJKZLZACUIZUOYNYTUOUAUBBNNMCU
|
|
PUUCBKZUUDYNUUHYTUUCBEFUQUUIUUGYSACUUIUUFYRYPUUIUUEYQJUUCBYOUHURUSUTVAVBU
|
|
UDUUBUCHZOIZUUAKZUCNUIZUUANDZUUBNDZLZUUHUUDUUBUUAVGGUUMUUAUUBVCUCUUBUUAVD
|
|
VHUUPUULUUHUCNUUPUUJNDZLZUUJUDHZPIZQKZUDNUIZUULUUHUOZUUQUVBUUPUDUUJVEVIUU
|
|
RUVAUVCUDNUUPUUQUUSNDZUVAUVCUOUUPUUQUVDLZLZUVAUULUUHUVFUVAUULLZLZUUSQOIZQ
|
|
UMMUNZCDZEUVJFGZUUCUVJUHIZJKZUUHUVEUVKUUPUVGUVDUVKUUQUVDUVINDZQNDZUVKUVDU
|
|
VPUVORUUSQVFZVJZRNUVIQMCVQUPVKVLVIVMUVHUVDUVLUVFUVDUVGUUPUUQUVDVNZTUVDQUV
|
|
IVGGUVLUVDQQUUSOIZUVIVGUVPUVDQUVTVGGRQUUSVOVRUVPUVDUVTUVIKRQUUSVPVRVSUVIQ
|
|
VCVTWAUVHUVMQQOIZQUMMUNZJUVHUVMUUAUVIPIZUUBQPIZOIZUUAQPIZUUBUVIPIZOIZUMMU
|
|
NZUWBUUPUVDUVGUVMUWIKZUUQUUPUVDLZUVGLUUPUVOUVPLZLZUWJUWKUWMUVGUVDUWLUUPUV
|
|
DUVOUVPUVRRWHWBTUUAUUBUVIQWCWAWDUVHUWIUWBKZUWEQOIZUWHUWAOIZKZUVHUUAUUSPIZ
|
|
UWFUWDOIZOIZQOIZUUBUUSPIZUWSOIZUWAOIZUWOUWPUVHUXAUXCQOIZQOIZUXDUVHUWTUXEQ
|
|
OUVHUWTUXBQOIZUWSOIZUXEUVHUWRUXGUWSOUVHUWRUUKUUSPIZUXGUULUWRUXIKUVFUVAUUL
|
|
UXIUWRUUKUUAUUSPURWEWFUVFUVAUXIUXGKUULUVFUVAUXIUXBUUTOIUXGUVFUEUFUGUUBUUJ
|
|
UUSNOPONUEHZNDZUFHZNDZUGHZNDZWGZUXJUXLOIZUXNPIZUXJUXNPIZUXLUXNPIZOIZKUVFU
|
|
XPUYAUXNUXJPIZUXNUXLPIZOIZUXNUXQPIZUXRUXPUXSUYBUXTUYCOUXKUXOUXSUYBKUXMUXJ
|
|
UXNWIWJUXMUXOUXTUYCKUXKUXLUXNWIWKWLUXOUXKUXMUYEUYDKUXNUXJUXLWRWMUXPUXOUXQ
|
|
NDZUYEUXRKUXKUXMUXOWNUXKUXMUYFUXOUXJUXLVFWOUXNUXQWISWSVIUUNUUOUVEWPZUUPUU
|
|
QUVDWQUVSWTUUTQUXBOXAXBXCXDXEUVFUXHUXEKUVGUVFUEUFUGUXBQUWSNOUVFUUOUVDUXBN
|
|
DZUYGUVSUUBUUSXHSZUVPUVFRXFZUVFUWFNDZUWDNDZUWSNDZUVFUUNUVPUYKUUNUUOUVEXGZ
|
|
RUUAQXHVLZUVFUUOUVPUYLUYGRUUBQXHVLZUWFUWDVFSZUXKUXMLUXQUXLUXJOIKUVFUXJUXL
|
|
VPVIZUXPUXQUXNOIUXJUXLUXNOIOIKUVFUXJUXLUXNXIVIZXJTXDXEUVHUXCNDZUVPUVPUXFU
|
|
XDKUVFUYTUVGUVFUYHUYMUYTUYIUYQUXBUWSVFSZTUVPUVHRXFZVUBUXCQQXIXKXDUVFUWOUX
|
|
AKUVGUVFUWEUWTQOUVFUWEUWRUWFOIZUWDOIZUWTUVFUWCVUCUWDOUVFUUNUVDUVPUWCVUCKU
|
|
YNUVSUYJUUAUUSQWRXKXEUVFUWRNDZUYKUYLVUDUWTKUVFUUNUVDVUEUYNUVSUUAUUSXHSUYO
|
|
UYPUWRUWFUWDXIXKXDXETUVFUWPUXDKUVGUVFUWHUXCUWAOUVFUWHUWFUXBUWDOIZOIUXCUVF
|
|
UWGVUFUWFOUVFUUOUVDUVPUWGVUFKUYGUVSUYJUUBUUSQWRXKXLUVFUEUFUGUWFUXBUWDNOUY
|
|
OUYIUYPUYRUYSXMXDZXETXNUVFUWNUWQXOZUVGUVFUWENDZUWHNDUWANDZUVPVUHUVFUWCNDZ
|
|
UYLVUIUVFUUNUVOVUKUYNUVFUVDUVPUVOUVSRUVQVLUUAUVIXHSUYPUWCUWDVFSUVFUWHUXCN
|
|
VUGVUAXPVUJUVFUVPUVPVUJRRQQVFXQXFUYJUWEUWHUWAQXRXSTXTXDYGYAUUGUVLUVNLAUVJ
|
|
CYOUVJKZYPUVLUUFUVNYOUVJEFUQVULUUEUVMJYOUVJUUCUHXAUSYBYCYDYHYEYFYIYFYJYKY
|
|
L $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The reciprocal of a positive signed real exists. Part of Proposition
|
|
9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) $)
|
|
recexsrlem $p |- ( 0R <R A -> E. x e. R. ( A .R x ) = 1R ) $=
|
|
( c0r cltr wbr cv cmr co c1r wceq wa cnr wrex recexgt0sr simpr reximi syl
|
|
) CBDECAFZDEZBRGHIJZKZALMTALMABNUATALSTOPQ $.
|
|
$}
|
|
|
|
$( The sum of two positive signed reals is positive. (Contributed by NM,
|
|
14-May-1996.) $)
|
|
addgt0sr $p |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A +R B ) ) $=
|
|
( c0r cltr wbr cplr co wa simpr cnr wcel wb ltrelsr simprd 0r ltasrg mp3an1
|
|
brel syl2anr mpbid adantr 0idsr breq1d syl ltsosr sotri syldan ) CADEZCBDEZ
|
|
AABFGZDEZCUJDEUHUIHZACFGZUJDEZUKULUIUNUHUIIUIBJKZAJKZUIUNLZUHUICJKZUOCBJJDM
|
|
RNUHURUPCAJJDMRNZURUOUPUQOCBAPQSTULUPUNUKLUHUPUIUSUAUPUMAUJDAUBUCUDTCAUJDJU
|
|
EMUFUG $.
|
|
|
|
$( Adding one to a signed real yields a larger signed real. (Contributed by
|
|
Jim Kingdon, 7-Jul-2021.) $)
|
|
ltadd1sr $p |- ( A e. R. -> A <R ( A +R 1R ) ) $=
|
|
( cnr wcel c0r cplr co c1r cltr 0idsr wbr 0lt1sr wb 0r ltasrg mp3an12 mpbii
|
|
1sr eqbrtrrd ) ABCZADEFZAAGEFZHAISDGHJZTUAHJZKDBCGBCSUBUCLMQDGANOPR $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w v u f g h $.
|
|
$( The product of two positive signed reals is positive. (Contributed by
|
|
NM, 13-May-1996.) $)
|
|
mulgt0sr $p |- ( ( 0R <R A /\ 0R <R B ) -> 0R <R ( A .R B ) ) $=
|
|
( vf vg vh cnr wcel wa c0r cltr wbr co cv cnp wceq cmp cpp adantl mulclpr
|
|
syl2anc vx vy vz vw vv vu cmr ltrelsr simprd anim12i cop cer cec wi df-nr
|
|
brel breq2 anbi1d oveq1 breq2d imbi12d oveq2 cltp gt0srpr anbi12i ltexpri
|
|
anbi2d wrex addclpr simplrr simplr ad2antrr simplrl caovcld adantr simprl
|
|
eqeltrrd ltaddpr simprr oveq12 oveq1d distrprg eqtr3d mulcomprg caovdir2d
|
|
syl3anc oveq12d addcomprg addassprg caov4d 3eqtr4d caov12d caov32d oveq2d
|
|
w3a ad2antlr 3eqtr3d addcanprg breqtrrd rexlimdvaa mulsrpr syl6bb sylibrd
|
|
mpd syl5 impd syl5bi 2ecoptocl mpcom ) AFGZBFGZHIAJKZIBJKZHZIABUGLZJKZXLX
|
|
JXMXKXLIFGZXJIAFFJUHUPUIXMXQXKIBFFJUHUPUIUJIUAMZUBMZUKULUMZJKZIUCMZUDMZUK
|
|
ULUMZJKZHZIXTYDUGLZJKZUNXLYEHZIAYDUGLZJKZUNXNXPUNUAUBUCUDABNNULFUOXTAOZYF
|
|
YIYHYKYLYAXLYEXTAIJUQURYLYGYJIJXTAYDUGUSUTVAYDBOZYIXNYKXPYMYEXMXLYDBIJUQV
|
|
GYMYJXOIJYDBAUGVBUTVAYFXSXRVCKZYCYBVCKZHZXRNGZXSNGZHZYBNGZYCNGZHZHZYHYAYN
|
|
YEYOXRXSVDYBYCVDVEUUCYPXRYCPLZXSYBPLZQLZXRYBPLZXSYCPLZQLZVCKZYHUUCYNYOUUJ
|
|
YNXSUEMZQLZXROZUENVHUUCYOUUJUNZUEXSXRVFUUCUUMUUNUENYOYCUFMZQLZYBOZUFNVHUU
|
|
CUUKNGZUUMHZHZUUJUFYCYBVFUUTUUQUUJUFNUUTUUONGZUUQHZHZUUFUUFUUKUUOPLZQLZUU
|
|
IVCUVCUUFNGUVDNGZUUFUVEVCKUVCCDUUDUUENNNQCMZNGZDMZNGZHZUVGUVIQLZNGUVCUVGU
|
|
VIVIRZUVCYQUUAUUDNGZUVCUULXRNUUCUURUUMUVBVJZUVCCDXSUUKNNNQUVMUUCYRUUSUVBY
|
|
QYRUUBVKVLZUUCUURUUMUVBVMZVNZVQZUUTUUAUVBYSYTUUAUUSVJVOZXRYCSTZUVCYRYTUUE
|
|
NGZUVPUUTYTUVBYSYTUUAUUSVMVOZXSYBSTZVNZUVCUURUVAUVFUVQUUTUVAUUQVPZUUKUUOS
|
|
TZUUFUVDVRTUVCUUKYCPLZUUIQLZUWHUVEQLZOZUUIUVEOZUVCUUGUUHUWHQLZQLZUWHUUEUV
|
|
DQLZQLZUUDQLZUWIUWJUVCUULUUPPLZUWMQLZUWNUWQUVCUUMUUQUWSUWNOUVOUUTUVAUUQVS
|
|
UUMUUQHUWRUUGUWMQUULXRUUPYBPVTWATUVCUWRUWPUWMUUDQUVCUUHXSUUOPLZQLZUWHUVDQ
|
|
LZQLZUUEUXBQLUWRUWPUVCUXAUUEUXBQUVCXSUUPPLZUXAUUEUVCYRUUAUVAUXDUXAOUVPUVT
|
|
UWFXSYCUUOWBWFUVBUXDUUEOZUUTUUQUXEUVAUUPYBXSPVBRRWCWAUVCUULYCPLZUULUUOPLZ
|
|
QLZUWMUWTUVDQLZQLUWRUXCUVCUXFUWMUXGUXIQUVCCDEXSUUKYCNQPUVHUVJEMZNGWOZUVGU
|
|
VIUXJQLZPLUVGUVIPLZUVGUXJPLQLOUVCUVGUVIUXJWBRZUVPUVQUVTUVMUVKUXMUVIUVGPLO
|
|
UVCUVGUVIWDRZWEZUVCCDEXSUUKUUONQPUXNUVPUVQUWFUVMUXOWEWGUVCUULNGUUAUVAUWRU
|
|
XHOUVRUVTUWFUULYCUUOWBWFUVCCDEUUHUWTUWHUVDNQUVCYRUUAUUHNGZUVPUVTXSYCSTZUV
|
|
CYRUVAUWTNGUVPUWFXSUUOSTUVCUURUUAUWHNGZUVQUVTUUKYCSTZUVKUVLUVIUVGQLOUVCUV
|
|
GUVIWHRZUXKUVLUXJQLUVGUXLQLOUVCUVGUVIUXJWIRZUWGUVMWJWKUVCCDEUWHUUEUVDNQUX
|
|
TUWDUWGUYAUYBWLWKUVCUXFUWMUUDUXPUUSUXFUUDOZUUCUVBUUMUYCUURUULXRYCPUSRWPWC
|
|
WGWCUVCUUIUWHQLZUWNUWIUVCUUGNGZUXQUXSUYDUWNOUVCYQYTUYEUVSUWCXRYBSTZUXRUXT
|
|
UUGUUHUWHWIWFUVCUUINGZUXSUYDUWIOUVCUYEUXQUYGUYFUXRUUGUUHVITZUXTUUIUWHWHTW
|
|
CUVCUWHUUDQLUWOQLZUWHUUDUWOQLZQLZUWQUWJUVCUXSUVNUWONGUYIUYKOUXTUWAUVCCDUU
|
|
EUVDNNNQUVMUWDUWGVNZUWHUUDUWOWIWFUVCCDEUWHUWOUUDNQUXTUYLUWAUYAUYBWMUVCUVE
|
|
UYJUWHQUVCUVNUWBUVFUVEUYJOUWAUWDUWGUUDUUEUVDWIWFWNWKWQUVCUXSUYGUVENGUWKUW
|
|
LUNUXTUYHUVCCDUUFUVDNNNQUVMUWEUWGVNUWHUUIUVEWRWFXDWSWTXEWTXEXFUUCYHIUUIUU
|
|
FUKULUMZJKUUJUUCYGUYMIJXRXSYBYCXAUTUUIUUFVDXBXCXGXHXI $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w z $.
|
|
$( Apartness of signed reals is tight. (Contributed by Jim Kingdon,
|
|
29-Jan-2020.) $)
|
|
aptisr $p |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) )
|
|
-> A = B ) $=
|
|
( vx cnr wcel cltr wbr wo wn wceq cv cer wi cnp orbi12d notbid wa co cltp
|
|
cpp vy vz cop cec df-nr breq1 breq2 imbi12d addcomprg ad2ant2lr ad2ant2rl
|
|
eqeq1 breq12d orbi2d addclpr aptipr 3expia syl2anc sylbird ltsrprg ancoms
|
|
vw eqeq2 wb enreceq 3imtr4d 2ecoptocl 3impia ) ADEBDEABFGZBAFGZHZIZABJZCK
|
|
ZUAKZUCLUDZUBKZVBKZUCLUDZFGZVSVPFGZHZIZVPVSJZMAVSFGZVSAFGZHZIZAVSJZMVLVMM
|
|
CUAUBVBABNNLDUEVPAJZWCWHWDWIWJWBWGWJVTWEWAWFVPAVSFUFVPAVSFUGOPVPAVSULUHVS
|
|
BJZWHVLWIVMWKWGVKWKWEVIWFVJVSBAFUGVSBAFUFOPVSBAVCUHVNNEZVONEZQZVQNEZVRNEZ
|
|
QZQZVNVRTRZVOVQTRZSGZVQVOTRZVRVNTRZSGZHZIZWSWTJZWCWDWRXFXAWTWSSGZHZIZXGWR
|
|
XIXEWRXHXDXAWRWTXBWSXCSWMWOWTXBJWLWPVOVQUIUJWLWPWSXCJWMWOVNVRUIUKUMUNPWRW
|
|
SNEZWTNEZXJXGMWLWPXKWMWOVNVRUOUKWMWOXLWLWPVOVQUOUJXKXLXJXGWSWTUPUQURUSWRW
|
|
BXEWRVTXAWAXDVNVOVQVRUTWQWNWAXDVDVQVRVNVOUTVAOPVNVOVQVRVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d U f g h $. $d V f g h $. $d W f g h $. $d X f g h $. $d Y f g h $.
|
|
$d Z f g h $.
|
|
$( Lemma for ~ mulextsr1 . (Contributed by Jim Kingdon, 17-Feb-2020.) $)
|
|
mulextsr1lem $p |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. )
|
|
/\ ( U e. P. /\ V e. P. ) ) -> (
|
|
( ( ( X .P. U ) +P. ( Y .P. V ) ) +P.
|
|
( ( Z .P. V ) +P. ( W .P. U ) ) )
|
|
<P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P.
|
|
( ( Z .P. U ) +P. ( W .P. V ) ) ) ->
|
|
( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) ) $=
|
|
( vf vg vh cnp wcel cmp co cpp cltp wbr wceq adantl mulclpr syl2anc wa wo
|
|
w3a addcomprg addclpr simp2l simp3r simp1r caovcld simp1l simp3l caovcomd
|
|
cv simp2r addassprg caov411d distrprg mulcomprg caovdir2d oveq12d 3eqtr4d
|
|
eqtr4d breq12d wi eqeltrd addextpr syl22anc 3adant2 3adant1 ltmprr sylbid
|
|
syl3anc orim12d syld breq2d orbi12d sylibd ) DJKZEJKZUAZFJKZCJKZUAZAJKZBJ
|
|
KZUAZUCZDALMZEBLMZNMFBLMZCALMZNMNMZDBLMZEALMZNMFALMZCBLMZNMNMZOPZDCNMZFEN
|
|
MZOPZWTWSOPZUBZWSEFNMZOPZWTCDNMZOPZUBWGWRWSALMZWTBLMZNMZWTALMZWSBLMZNMZOP
|
|
ZXCWGWLXJWQXMOWGWJWINMZWHWKNMZNMXPXONMWLXJWGGHXOXPJNGUMZJKZHUMZJKZUAZXQXS
|
|
NMZXSXQNMQWGXQXSUDRZWGGHWJWIJJJNYAYBJKWGXQXSUERZWGWAWEWJJKVTWAWBWFUFZVTWC
|
|
WDWEUGZFBSTZWGVSWEWIJKVRVSWCWFUHZYFEBSTZUIZWGGHWHWKJJJNYDWGVRWDWHJKVRVSWC
|
|
WFUJZVTWCWDWEUKZDASTZWGWBWDWKJKVTWAWBWFUNZYLCASTZUIZULWGGHIWHWIWJWKJNYMYI
|
|
YGYCXRXTIUMZJKUCZYBYQNMXQXSYQNMZNMQWGXQXSYQUORZYOYDUPWGXHXPXIXONWGGHIDCAJ
|
|
NLYRXQYSLMXQXSLMZXQYQLMNMQWGXQXSYQUQRZYKYNYLYDYAUUAXSXQLMQWGXQXSURRZUSZWG
|
|
GHIFEBJNLUUBYEYHYFYDUUCUSZUTVAWGWQWOWNNMZWMWPNMZNMXMWGGHIWMWNWOWPJNWGVRWE
|
|
WMJKYKYFDBSTWGVSWDWNJKYHYLEASTWGWAWDWOJKYEYLFASTYCYTWGWBWEWPJKYNYFCBSTYDU
|
|
PWGXKUUFXLUUGNWGGHIFEAJNLUUBYEYHYLYDUUCUSWGGHIDCBJNLUUBYKYNYFYDUUCUSUTVBV
|
|
CWGXNXHXKOPZXIXLOPZUBZXCWGXHJKXIJKXKJKZXLJKZXNUUJVDWGXHXPJUUDYPVEWGXIXOJU
|
|
UEYJVEWGWTJKZWDUUKWGWAVSUUMYEYHFEUETZYLWTASTWGWSJKZWEUULWGVRWBUUOYKYNDCUE
|
|
TZYFWSBSTXHXIXKXLVFVGWGUUHXAUUIXBWGUUOUUMWDUUHXAVDUUPUUNYLUUOUUMWDUCZUUHA
|
|
WSLMZAWTLMZOPXAUUQXHUURXKUUSOUUOWDXHUURQUUMWSAURVHUUMWDXKUUSQUUOWTAURVIVC
|
|
WSWTAVJVKVLWGUUIBWTLMZBWSLMZOPZXBWGXIUUTXLUVAOWGUUMWEXIUUTQUUNYFWTBURTWGU
|
|
UOWEXLUVAQUUPYFWSBURTVCWGUUMUUOWEUVBXBVDUUNUUPYFWTWSBVJVLVKVMVNVKWGXAXEXB
|
|
XGWGWTXDWSOWGWAVSWTXDQYEYHFEUDTVOWGWSXFWTOWGVRWBWSXFQYKYNDCUDTVOVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d A u v w x y z $. $d B u v w z $. $d C u v $.
|
|
$( Strong extensionality of multiplication of signed reals. (Contributed
|
|
by Jim Kingdon, 18-Feb-2020.) $)
|
|
mulextsr1 $p |- ( ( A e. R. /\ B e. R. /\ C e. R. )
|
|
-> ( ( A .R C ) <R ( B .R C ) -> ( A <R B \/ B <R A ) ) ) $=
|
|
( cv cop cer cec cmr co cltr wbr wo cnp wceq wcel cmp cpp mulclpr syl2anc
|
|
wi vx vy vu vv vz vw df-nr oveq1 breq1d breq1 breq2 orbi12d imbi12d oveq2
|
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cnr breq2d breq12d imbi1d wa cltp mulextsr1lem mulsrpr 3adant2 3adant1 wb
|
|
simp1l simp3l simp1r simp3r addclpr simp2l ltsrprg syl22anc bitrd 3adant3
|
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w3a simp2r ancoms 3imtr4d 3ecoptocl ) UADZUBDZEFGZUCDZUDDZEFGZHIZUEDZUFDZ
|
|
EFGZWFHIZJKZWCWJJKZWJWCJKZLZTAWFHIZWKJKZAWJJKZWJAJKZLZTWPBWFHIZJKZABJKZBA
|
|
JKZLZTACHIZBCHIZJKZXETUAUBUEUFUCUDABCMFUOUGWCANZWLWQWOWTXIWGWPWKJWCAWFHUH
|
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UIXIWMWRWNWSWCAWJJUJWCAWJJUKULUMWJBNZWQXBWTXEXJWKXAWPJWJBWFHUHUPXJWRXCWSX
|
|
DWJBAJUKWJBAJUJULUMWFCNZXBXHXEXKWPXFXAXGJWFCAHUNWFCBHUNUQURWAMOZWBMOZUSZW
|
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HMOZWIMOZUSZWDMOZWEMOZUSZVPZWAWDPIZWBWEPIZQIZWHWEPIZWIWDPIZQIZQIWAWEPIZWB
|
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WDPIZQIZWHWDPIZWIWEPIZQIZQIUTKZWAWIQIWBWHQIUTKZWHWBQIWIWAQIUTKZLWLWOWDWEW
|
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|
|
KYRNXNWHWIWDWEVBVDUQYAYDMOZYJMOZYMMOZYGMOZYSYNVEYAYBMOZYCMOZYTYAXLXRUUDXL
|
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XMXQXTVFZXNXQXRXSVGZWAWDRSYAXMXSUUEXLXMXQXTVHZXNXQXRXSVIZWBWERSYBYCVJSYAY
|
|
HMOZYIMOZUUAYAXLXSUUJUUFUUIWAWERSYAXMXRUUKUUHUUGWBWDRSYHYIVJSYAYKMOZYLMOZ
|
|
UUBYAXOXRUULXNXOXPXTVKZUUGWHWDRSYAXPXSUUMXNXOXPXTVQZUUIWIWERSYKYLVJSYAYEM
|
|
OZYFMOZUUCYAXOXSUUPUUNUUIWHWERSYAXPXRUUQUUOUUGWIWDRSYEYFVJSYDYJYMYGVLVMVN
|
|
YAWMYOWNYPXNXQWMYOVEXTWAWBWHWIVLVOXNXQWNYPVEZXTXQXNUURWHWIWAWBVLVRVOULVSV
|
|
T $.
|
|
$}
|
|
|
|
${
|
|
$d A l u w x z $.
|
|
$( For any signed real, there is an integer that is greater than it. This
|
|
is also known as the "archimedean property". The expression
|
|
` [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } ` ,
|
|
` { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ` is the
|
|
embedding of the positive integer ` x ` into the signed reals.
|
|
(Contributed by Jim Kingdon, 23-Apr-2020.) $)
|
|
archsr $p |- ( A e. R. -> E. x e. N. A <R
|
|
[ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } ,
|
|
{ u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) $=
|
|
( vz cv cop cer cec wbr c1p cpp co cltr cnpi wrex cnp wcel cltp 1pr df-nr
|
|
vw c1o ceq cltq cab cnr wceq breq1 rexbidv wa addclpr mpan2 archpr adantr
|
|
nnprlu adantl sylancl simplr ltaddpr syl2anc addcomprg breqtrrd wi ltsopr
|
|
syl ltrelpr sotri expcom sylsyld reximdva mpd simpl a1i syl12anc rexbidva
|
|
wb ltsrprg mpbird ecoptocl ) EFZUBFZGHIZDFAFZUCGUDIZUEJDUFWEBFUEJBUFGZKLM
|
|
ZKGHIZNJZAOPZCWHNJZAOPEUBCQQHUGUAWCCUHWIWKAOWCCWHNUIUJWAQRZWBQRZUKZWJWAKL
|
|
MZWBWGLMZSJZAOPZWNWOWFSJZAOPZWRWLWTWMWLWOQRZWTWLKQRZXATWAKULUMABWODUNVFUO
|
|
WNWSWQAOWNWDORZUKZWGWPSJZWSWOWGSJZWQXDWGWGWBLMZWPSXDWGQRZWMWGXGSJXDWFQRZX
|
|
BXHXCXIWNBWDDUPUQZTWFKULURZWLWMXCUSZWGWBUTVAXDWMXHWPXGUHXLXKWBWGVBVAVCXDW
|
|
FWGSJZWSXFVDXDXIXBXMXJTWFKUTURWSXMXFWOWFWGSQVEVGVHVIVFXFXEWQWOWGWPSQVEVGV
|
|
HVIVJVKVLWNWIWQAOXDWNXHXBWIWQVQWNXCVMXKXBXDTVNWAWBWGKVRVOVPVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d A a b x $.
|
|
$( Mapping from a signed real greater than zero to a positive real.
|
|
(Contributed by Jim Kingdon, 25-Jun-2021.) $)
|
|
srpospr $p |- ( ( A e. R. /\ 0R <R A ) ->
|
|
E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) $=
|
|
( va vb cnr wcel c0r cltr wbr cv c1p cpp co cer wceq cnp wreu wi syl2anc
|
|
wa cop cec df-nr breq2 reubidv imbi12d cltp gt0srpr biimpi adantl lteupri
|
|
eqeq2 syl wb simpr 1pr addclpr simplll simpllr enreceq syl22anc addcomprg
|
|
a1i oveq1d addassprg syl3anc eqtrd eqeq1d addcanprg oveq2 impbid1 3bitrrd
|
|
bitrd reubidva mpbid ex ecoptocl imp ) BEFGBHIZAJZKLMZKUANUBZBOZAPQZGCJZD
|
|
JZUANUBZHIZWBWGOZAPQZRVSWDRCDBPPNEUCWGBOZWHVSWJWDWGBGHUDWKWIWCAPWGBWBULUE
|
|
UFWEPFZWFPFZTZWHWJWNWHTZWFVTLMZWEOZAPQZWJWOWFWEUGIZWRWHWSWNWHWSWEWFUHUIUJ
|
|
AWFWEUKUMWOWQWIAPWOVTPFZTZWIWAWFLMZKWELMZOZVTWFLMZWEOZWQXAWAPFZKPFZWLWMWI
|
|
XDUNXAWTXHXGWOWTUOZXHXAUPVCZVTKUQSXJWLWMWHWTURZWLWMWHWTUSZWAKWEWFUTVAXAXD
|
|
KXELMZXCOZXFXAXBXMXCXAXBKVTLMZWFLMZXMXAWAXOWFLXAWTXHWAXOOXIXJVTKVBSVDXAXH
|
|
WTWMXPXMOXJXIXLKVTWFVEVFVGVHXAXNXFXAXHXEPFZWLXNXFRXJXAWTWMXQXIXLVTWFUQSXK
|
|
KXEWEVIVFXEWEKLVJVKVMXAXEWPWEXAWTWMXEWPOXIXLVTWFVBSVHVLVNVOVPVQVR $.
|
|
$}
|
|
|
|
$( Mapping from a positive real to a signed real. (Contributed by Jim
|
|
Kingdon, 25-Jun-2021.) $)
|
|
prsrcl $p |- ( A e. P. ->
|
|
[ <. ( A +P. 1P ) , 1P >. ] ~R e. R. ) $=
|
|
( cnp wcel c1p cpp co cop cer cec cxp cqs cnr addclpr mpan2 opelxpi ecelqsi
|
|
1pr enrex 3syl df-nr syl6eleqr ) ABCZADEFZDGZHIZBBJZHKZLUBUCBCZUDUFCZUEUGCU
|
|
BDBCZUHQADMNUHUJUIQUCDBBONUFUDHRPSTUA $.
|
|
|
|
$( Mapping from a positive real to a signed real yields a result greater than
|
|
zero. (Contributed by Jim Kingdon, 25-Jun-2021.) $)
|
|
prsrpos $p |- ( A e. P. -> 0R <R [ <. ( A +P. 1P ) , 1P >. ] ~R ) $=
|
|
( cnp wcel c1p cpp cltp wbr c0r cop cer cec cltr 1pr ltaddpr mpan addcomprg
|
|
co wceq breqtrd gt0srpr sylibr ) ABCZDADEQZFGHUCDIJKLGUBDDAEQZUCFDBCZUBDUDF
|
|
GMDANOUEUBUDUCRMDAPOSUCDTUA $.
|
|
|
|
${
|
|
$d A f g h $. $d B f g h $.
|
|
$( Mapping from positive real addition to signed real addition.
|
|
(Contributed by Jim Kingdon, 29-Jun-2021.) $)
|
|
prsradd $p |- ( ( A e. P. /\ B e. P. ) ->
|
|
[ <. ( ( A +P. B ) +P. 1P ) , 1P >. ] ~R =
|
|
( [ <. ( A +P. 1P ) , 1P >. ] ~R +R
|
|
[ <. ( B +P. 1P ) , 1P >. ] ~R ) ) $=
|
|
( vf vg vh cnp wcel wa c1p cpp co cop cer cec wceq 1pr addclpr cv syl2anc
|
|
adantl mpan2 addsrpr mpanl2 mpanr2 syl2an simpl simpr addcomprg addassprg
|
|
cplr a1i w3a caov4d mp2an eqtrd oveq1d syl3anc wb enreceq mpbird eqtr2d )
|
|
AFGZBFGZHZAIJKZILMNBIJKZILMNUJKZVEVFJKZIIJKZLMNZABJKZIJKZILMNZVBVEFGZVFFG
|
|
ZVGVJOZVCVBIFGZVNPAIQUAZVCVQVOPBIQUAZVNVOVQVPPVNVQVOVQHVPPVEIVFIUBUCUDUEV
|
|
DVJVMOZVHIJKZVIVLJKZOZVDWAVIVKJKZIJKZWBVDVHWDIJVDVHVKVIJKZWDVDCDEAIBIFJVB
|
|
VCUFVQVDPUKZVBVCUGCRZFGZDRZFGZHZWHWJJKZWJWHJKOVDWHWJUHTWIWKERZFGULWMWNJKW
|
|
HWJWNJKJKOVDWHWJWNUITWGWLWMFGVDWHWJQTUMVDVKFGZVIFGZWFWDOABQZWPVDVQVQWPPPI
|
|
IQUNZUKZVKVIUHSUOUPVDWPWOVQWEWBOWSWQWGVIVKIUIUQUOVDVHFGZVLFGZVTWCURZVBVNV
|
|
OWTVCVRVSVEVFQUEVDWOVQXAWQWGVKIQSWTWPXAXBWRWTWPHXAVQXBPVHVIVLIUSUDUCSUTVA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A f g h $. $d B f g h $.
|
|
$( Mapping from positive real ordering to signed real ordering.
|
|
(Contributed by Jim Kingdon, 29-Jun-2021.) $)
|
|
prsrlt $p |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <->
|
|
[ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R ) ) $=
|
|
( vf vg vh cnp wcel wa c1p cpp co cltp wbr cop wceq wb syl2anc cv addclpr
|
|
cer cec cltr 1pr a1i simpr addassprg breq2d simpl ltaprg addcomprg breq1d
|
|
syl3anc w3a adantl caovord2d 3bitr2d ltsrprg syl22anc 3bitr4d ) AFGZBFGZH
|
|
ZAIJKZIJKZIBJKZIJKZLMZVDIBIJKZJKZLMZABLMZVCINTUAVHINTUAUBMZVBVFVIVDLVBIFG
|
|
ZVAVMVFVIOVMVBUCUDZUTVAUEZVNIBIUFULUGVBVKIAJKZVELMZVCVELMVGVBUTVAVMVKVQPU
|
|
TVAUHZVOVNABIUIULVBVCVPVELVBUTVMVCVPOVRVNAIUJQUKVBCDEVCVEILFJCRZFGZDRZFGZ
|
|
ERZFGUMVSWALMWCVSJKWCWAJKLMPVBVSWAWCUIUNVBUTVMVCFGZVRVNAISQZVBVMVAVEFGVNV
|
|
OIBSQVNVTWBHVSWAJKWAVSJKOVBVSWAUJUNUOUPVBWDVMVHFGZVMVLVJPWEVNVBVAVMWFVOVN
|
|
BISQVNVCIVHIUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Mapping a restricted iota from a positive real to a signed real.
|
|
(Contributed by Jim Kingdon, 29-Jun-2021.) $)
|
|
prsrriota $p |- ( ( A e. R. /\ 0R <R A ) -> [ <. ( ( iota_ x e. P.
|
|
[ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A ) $=
|
|
( vy wcel wa cv c1p cpp co cop cer cec wceq cnp wreu srpospr oveq1 opeq1d
|
|
syl eceq1d cnr c0r cltr crio reurex simprr wb simprl adantr eqeq1d riota2
|
|
wbr wrex syl2anc mpbid eqtrd rexlimddv ) BUADUBBUCULEZCFZGHIZGJZKLZBMZAFZ
|
|
GHIZGJZKLZBMZANUDZGHIZGJZKLZBMCNURVCCNOVCCNUMCBPVCCNUESURUSNDZVCEZEZVLVBB
|
|
VOVIUSMZVLVBMVOVCVPURVMVCUFZVOVMVHANOZVCVPUGURVMVCUHURVRVNABPUIVHVCANUSVD
|
|
USMZVGVBBVSVFVAKVSVEUTGVDUSGHQRTUJUKUNUOVPVKVAKVPVJUTGVIUSGHQRTSVQUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d A y $. $d F m $. $d F y $.
|
|
caucvgsrlemcl.f $e |- ( ph -> F : N. --> R. ) $.
|
|
caucvgsrlemcl.gt1 $e |- ( ph -> A. m e. N. 1R <R ( F ` m ) ) $.
|
|
$( Lemma for ~ caucvgsr . Terms of the sequence from ~ caucvgsrlemgt1 can
|
|
be mapped to positive reals. (Contributed by Jim Kingdon,
|
|
2-Jul-2021.) $)
|
|
caucvgsrlemcl $p |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P.
|
|
( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. ) $=
|
|
( cnpi wcel cfv cv c1p wceq cnp wreu cnr c0r cltr wbr c1r cpp cop cer cec
|
|
wa co crio ffvelrnda 0lt1sr fveq2 breq2d rspcv mpan9 ltsosr ltrelsr sotri
|
|
wral sylancr srpospr syl2anc eqcom reubii sylib riotacl syl ) ACHIZUEZCEJ
|
|
ZBKLUAUFLUBUCUDZMZBNOZVJBNUGNIVGVIVHMZBNOZVKVGVHPIQVHRSZVMAHPCEFUHVGQTRST
|
|
VHRSZVNUIATDKZEJZRSZDHUQVFVOGVRVODCHVPCMVQVHTRVPCEUJUKULUMQTVHRPUNUOUPURB
|
|
VHUSUTVLVJBNVIVHVAVBVCVJBNVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d A m $.
|
|
caucvgsrlemasr.bnd $e |- ( ph -> A. m e. N. A <R ( F ` m ) ) $.
|
|
$( Lemma for ~ caucvgsr . The lower bound is a signed real. (Contributed
|
|
by Jim Kingdon, 4-Jul-2021.) $)
|
|
caucvgsrlemasr $p |- ( ph -> A e. R. ) $=
|
|
( vx cnr wcel cnpi wral cfv cltr wbr ltrelsr brel simpld ralimi syl c1o
|
|
cv wex wb 1pi elex2 r19.3rmv mp2b sylibr ) ABGHZCIJZUHABCTDKZLMZCIJUIEUKU
|
|
HCIUKUHUJGHBUJGGLNOPQRSIHFTIHFUAUHUIUBUCFSIUDUHCFIUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
caucvgsr.f $e |- ( ph -> F : N. --> R. ) $.
|
|
caucvgsr.cau $e |- ( ph ->
|
|
A. n e. N. A. k e. N. ( n <N k ->
|
|
( ( F ` n ) <R ( ( F ` k ) +R
|
|
[ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R
|
|
) /\
|
|
( F ` k ) <R ( ( F ` n ) +R
|
|
[ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R
|
|
) ) ) ) $.
|
|
|
|
${
|
|
caucvgsrlemgt1.gt1 $e |- ( ph -> A. m e. N. 1R <R ( F ` m ) ) $.
|
|
|
|
${
|
|
caucvgsrlemf.xfr $e |- G =
|
|
( x e. N. |-> ( iota_ y e. P.
|
|
( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) ) $.
|
|
${
|
|
$d A m $. $d A x y $. $d F m $. $d F x y $. $d ph x $.
|
|
$( Lemma for ~ caucvgsr . Coercing sequence value from a positive
|
|
real to a signed real. (Contributed by Jim Kingdon,
|
|
29-Jun-2021.) $)
|
|
caucvgsrlemfv $p |- ( ( ph /\ A e. N. ) ->
|
|
[ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) $=
|
|
( cnpi c1p cpp wceq cnp wcel wa cfv co cop cer cec cv crio cmpt a1i
|
|
fveq2 eqeq1d adantl simpr caucvgsrlemcl fvmptd oveq1d opeq1d eceq1d
|
|
riotabidv eqcom riotabiia oveq1i opeq1i eceq1 mp1i cnr c0r cltr wbr
|
|
ffvelrnda c1r 0lt1sr wral breq2d rspcv mpan9 ltsosr ltrelsr sylancr
|
|
wb sotri prsrriota syl2anc 3eqtrd ) AEPUAZUBZEJUCZQRUDZQUEZUFUGEIUC
|
|
ZCUHZQRUDQUEUFUGZSZCTUIZQRUDZQUEZUFUGZWNWLSZCTUIZQRUDZQUEZUFUGZWLWH
|
|
WKWRUFWHWJWQQWHWIWPQRWHBEBUHZIUCZWNSZCTUIZWPPJTJBPXHUJSWHOUKXEESZXH
|
|
WPSWHXIXGWOCTXIXFWLWNXEEIULUMVAUNAWGUOACEGILNUPUQURUSUTWRXCSWSXDSWH
|
|
WQXBQWPXAQRWOWTCTWOWTWBWMTUAWLWNVBUKVCVDVEWRXCUFVFVGWHWLVHUAVIWLVJV
|
|
KZXDWLSAPVHEILVLWHVIVMVJVKVMWLVJVKZXJVNAVMGUHZIUCZVJVKZGPVOWGXKNXNX
|
|
KGEPXLESXMWLVMVJXLEIULVPVQVRVIVMWLVJVHVSVTWCWACWLWDWEWF $.
|
|
$}
|
|
|
|
${
|
|
$d F m $. $d F y $. $d m x $. $d ph x $. $d x y $.
|
|
$( Lemma for ~ caucvgsr . Defining the sequence in terms of positive
|
|
reals. (Contributed by Jim Kingdon, 23-Jun-2021.) $)
|
|
caucvgsrlemf $p |- ( ph -> G : N. --> P. ) $=
|
|
( cnpi cv cfv c1p cpp cnp cop cer cec wceq crio caucvgsrlemcl fmptd
|
|
co ) ABOBPZHQCPRSUHRUAUBUCUDCTUETIACUIFHKMUFNUG $.
|
|
$}
|
|
|
|
${
|
|
$d F m x $. $d F x y $. $d k m n x $. $d k n ph x $.
|
|
$d k n x y $. $d l n u $.
|
|
$( Lemma for ~ caucvgsr . Defining the Cauchy condition in terms of
|
|
positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) $)
|
|
caucvgsrlemcau $p |- ( ph ->
|
|
A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\
|
|
( G ` k ) <P ( ( G ` n ) +P.
|
|
<. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) ) $=
|
|
( wbr cop co cnpi c1p cnp cv clti cfv c1o ceq cec crq cltq cab cltp
|
|
cpp wa wi wral cer cplr cltr wcel wb caucvgsrlemf ad2antrr ffvelrnd
|
|
wf simplr adantr ffvelrnda recnnpr syl addclpr prsrlt caucvgsrlemfv
|
|
syl2anc prsradd adantlr oveq1d eqtrd breq12d bitrd anbi12d ralbidva
|
|
wceq imbi2d mpbird ) AGUAZEUAZUBOZWDIUCZWEIUCZJUAWDUDPUEUFUGUCZUHOJ
|
|
UIWIDUAUHODUIPZUKQZUJOZWHWGWJUKQZUJOZULZUMZERUNZGRUNWFWDHUCZWEHUCZW
|
|
JSUKQSPUOUFZUPQZUQOZWSWRWTUPQZUQOZULZUMZERUNZGRUNLAWQXGGRAWDRURZULZ
|
|
WPXFERXIWERURZULZWOXEWFXKWLXBWNXDXKWLWGSUKQSPUOUFZWKSUKQSPUOUFZUQOZ
|
|
XBXKWGTURZWKTURZWLXNUSXKRTWDIARTIVCZXHXJABCDEFGHIJKLMNUTZVAAXHXJVDZ
|
|
VBZXKWHTURZWJTURZXPXIRTWEIAXQXHXRVEVFZXKXHYBXSDWDJVGVHZWHWJVIVLWGWK
|
|
VJVLXKXLWRXMXAUQXIXLWRWAXJABCDWDEFGHIJKLMNVKVEZXKXMWHSUKQSPUOUFZWTU
|
|
PQZXAXKYAYBXMYGWAYCYDWHWJVMVLXKYFWSWTUPAXJYFWSWAXHABCDWEEFGHIJKLMNV
|
|
KVNZVOVPVQVRXKWNYFWMSUKQSPUOUFZUQOZXDXKYAWMTURZWNYJUSYCXKXOYBYKXTYD
|
|
WGWJVIVLWHWMVJVLXKYFWSYIXCUQYHXKYIXLWTUPQZXCXKXOYBYIYLWAXTYDWGWJVMV
|
|
LXKXLWRWTUPYEVOVPVQVRVSWBVTVTWC $.
|
|
$}
|
|
|
|
${
|
|
$d F m x y w $. $d ph x w $. $d G m w $.
|
|
$( Lemma for ~ caucvgsr . Defining the boundedness condition in
|
|
terms of positive reals. (Contributed by Jim Kingdon,
|
|
25-Jun-2021.) $)
|
|
caucvgsrlembound $p |- ( ph -> A. m e. N. 1P <P ( G ` m ) ) $=
|
|
( vw c1p wbr cnpi cltr c1r cv cfv cltp wral wcel wa cpp cop cer cec
|
|
co weq fveq2 breq2d cbvralv sylib r19.21bi wceq df-1r caucvgsrlemfv
|
|
eqcomi a1i 3brtr4d cnp wb 1pr caucvgsrlemf ffvelrnda prsrlt sylancr
|
|
mpbird ralrimiva ) APOUAZIUBZUCQZORUDPFUAZIUBZUCQZFRUDAVOORAVMRUEUF
|
|
ZVOPPUGUKPUHUIUJZVNPUGUKPUHUIUJZSQZVSTVMHUBZVTWASATWCSQZORATVPHUBZS
|
|
QZFRUDWDORUDMWFWDFORFOULWEWCTSVPVMHUMUNUOUPUQVTTURVSTVTUSVAVBABCDVM
|
|
EFGHIJKLMNUTVCVSPVDUEVNVDUEVOWBVEVFARVDVMIABCDEFGHIJKLMNVGVHPVNVIVJ
|
|
VKVLVOVROFROFULVNVQPUCVMVPIUMUNUOUP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d F a b j k l u $. $d F a i j k x $. $d F m $. $d F w z $.
|
|
$d F k l n u $. $d F a i j x y $. $d a b j k ph x $. $d ph z $.
|
|
$d ph n $. $d a b j k l u w $. $d a b j k l u z $. $d b c j k x $.
|
|
$d m w $. $d m z $. $d n u w $. $d n u z $. $d w x $. $d x z $.
|
|
$d k m $. $d m n $.
|
|
$( Lemma for ~ caucvgsr . A Cauchy sequence whose terms are greater
|
|
than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) $)
|
|
caucvgsrlemgt1 $p |- ( ph ->
|
|
E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. (
|
|
j <N i ->
|
|
( ( F ` i ) <R ( y +R x ) /\
|
|
y <R ( ( F ` i ) +R x ) ) ) ) ) $=
|
|
( wbr cnpi c1p co cnp cltr vz vw va vb vc cv clti cfv cpp cop cer cec
|
|
wceq crio cmpt cltp wa wral wrex c0r cplr caucvgsrlemf caucvgsrlemcau
|
|
wi eqid caucvgsrlembound caucvgprpr wcel prsrcl ad2antrl wreu srpospr
|
|
cnr riotacl syl adantll simplrr adantr oveq2 breq2d imbi2d rexralbidv
|
|
anbi12d rspcva syl2anc nfv nfcv nfre1 nfralya nfan nfrexya wb ad4antr
|
|
nfra1 wf simpr ffvelrnd simplrl addclpr caucvgsrlemfv adantlr prsradd
|
|
prsrlt oveq2d eqtrd breq12d bitrd oveq12d 3bitrd ralbida rexbid mpbid
|
|
prsrriota breq2 fveq2 breq1d oveq1d imbi12d rexbii sylib ex ralrimiva
|
|
cbvralv oveq1 breq1 ralbidv rspcev rexlimddv ) AFUFZGUFZUGOZYJUAPUAUF
|
|
JUHUBUFQUIRQUJUKULUMUBSUNUOZUHZUCUFZUDUFZUIRZUPOZYNYMYOUIRZUPOZUQZVDZ
|
|
GPURZFPUSZUDSURZUTBUFZTOZYIEUFZUGOZUUGJUHZCUFZUUEVARZTOZUUJUUIUUEVARZ
|
|
TOZUQZVDZEPURFPUSZVDZBVMURZCVMUSZUCSAUDUCDQFGHIYLKAUAUBDGHIJYLKLMNYLV
|
|
EZVBZAUAUBDGHIJYLKLMNUVAVCAUAUBDGHIJYLKLMNUVAVFVGAYNSVHZUUDUQZUQZYNQU
|
|
IRQUJUKULZVMVHZUUFUUHUUIUVFUUEVARZTOZUVFUUMTOZUQZVDZEPURZFPUSZVDZBVMU
|
|
RZUUTUVCUVGAUUDYNVIVJUVEUVOBVMUVEUUEVMVHZUQZUUFUVNUVRUUFUQZYKYJJUHZUV
|
|
HTOZUVFUVTUUEVARZTOZUQZVDZGPURZFPUSZUVNUVSYKYMYNUEUFQUIRQUJUKULUUEUMZ
|
|
UESUNZUIRZUPOZYNYMUWIUIRZUPOZUQZVDZGPURZFPUSZUWGUVSUWISVHZUUDUWQUVQUU
|
|
FUWRUVEUVQUUFUQZUWHUESVKUWRUEUUEVLUWHUESVNVOVPZUVRUUDUUFAUVCUUDUVQVQV
|
|
RUUCUWQUDUWISYOUWIUMZUUAUWOFGPPUXAYTUWNYKUXAYQUWKYSUWMUXAYPUWJYMUPYOU
|
|
WIYNUIVSVTUXAYRUWLYNUPYOUWIYMUIVSVTWCWAWBWDWEUVSUWPUWFFPUVRUUFFUVEUVQ
|
|
FAUVDFAFWFUVCUUDFUVCFWFUUCFUDSFSWGUUBFPWHWIWJWJUVQFWFWJUUFFWFWJUVSUWO
|
|
UWEGPUVRUUFGUVEUVQGAUVDGAGWFUVCUUDGUVCGWFUUCGUDSGSWGUUBGFPGPWGUUAGPWN
|
|
WKWIWJWJUVQGWFWJUUFGWFWJUVSYJPVHZUQZUWNUWDYKUXCUWKUWAUWMUWCUXCUWKYMQU
|
|
IRQUJUKULZUWJQUIRQUJUKULZTOZUWAUXCYMSVHZUWJSVHZUWKUXFWLUXCPSYJYLAPSYL
|
|
WOUVDUVQUUFUXBUVBWMUVSUXBWPWQZUVSUXHUXBUVSUVCUWRUXHUVRUVCUUFAUVCUUDUV
|
|
QWRVRZUWTYNUWIWSWEVRYMUWJXCWEUXCUXDUVTUXEUVHTUVRUXBUXDUVTUMZUUFUVEUXB
|
|
UXKUVQAUXBUXKUVDAUAUBDYJGHIJYLKLMNUVAWTXAXAXAZUVSUXEUVHUMUXBUVSUXEUVF
|
|
UWIQUIRQUJUKULZVARZUVHUVSUVCUWRUXEUXNUMUXJUWTYNUWIXBWEUVQUUFUXNUVHUMU
|
|
VEUWSUXMUUEUVFVAUEUUEXMZXDVPXEVRXFXGUXCUWMUVFUWLQUIRQUJUKULZTOZUVFUXD
|
|
UXMVARZTOUWCUXCUVCUWLSVHZUWMUXQWLUVSUVCUXBUXJVRUXCUXGUWRUXSUXIUVSUWRU
|
|
XBUWTVRZYMUWIWSWEYNUWLXCWEUXCUXPUXRUVFTUXCUXGUWRUXPUXRUMUXIUXTYMUWIXB
|
|
WEVTUXCUXRUWBUVFTUXCUXDUVTUXMUUEVAUXLUVSUXMUUEUMZUXBUVQUUFUYAUVEUXOVP
|
|
VRXHVTXIWCWAXJXKXLUWFUVMFPUWEUVLGEPYJUUGUMZYKUUHUWDUVKYJUUGYIUGXNUYBU
|
|
WAUVIUWCUVJUYBUVTUUIUVHTYJUUGJXOZXPUYBUWBUUMUVFTUYBUVTUUIUUEVAUYCXQVT
|
|
WCXRYCXSXTYAYBUUSUVPCUVFVMUUJUVFUMZUURUVOBVMUYDUUQUVNUUFUYDUUPUVLFEPP
|
|
UYDUUOUVKUUHUYDUULUVIUUNUVJUYDUUKUVHUUITUUJUVFUUEVAYDVTUUJUVFUUMTYEWC
|
|
WAWBWAYFYGWEYH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
caucvgsrlembnd.bnd $e |- ( ph -> A. m e. N. A <R ( F ` m ) ) $.
|
|
${
|
|
caucvgsrlembnd.offset $e |- G = ( a e. N. |->
|
|
( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) ) $.
|
|
|
|
${
|
|
$d A a $. $d A m $. $d F a $. $d J a $. $d a ph $.
|
|
$( Lemma for ~ caucvgsr . Offsetting the values of the sequence so
|
|
they are greater than one. (Contributed by Jim Kingdon,
|
|
3-Jul-2021.) $)
|
|
caucvgsrlemoffval $p |- ( ( ph /\ J e. N. ) ->
|
|
( ( G ` J ) +R A ) = ( ( F ` J ) +R 1R ) ) $=
|
|
( wcel cplr co cnr wceq cnpi wa cfv c1r cm1r cmr cv cmpt a1i oveq1d
|
|
fveq2 adantl simpr ffvelrnda 1sr addclsr sylancl caucvgsrlemasr m1r
|
|
adantr mulclsr syl2anc fvmptd addasssrg syl3anc c0r addcomsrg pn0sr
|
|
syl eqtrd oveq2d 0idsr 3eqtrd ) AIUAPZUBZIHUCZCQRIGUCZUDQRZCUEUFRZQ
|
|
RZCQRZVRVSCQRZQRZVRVOVPVTCQVOJIJUGZGUCZUDQRZVSQRZVTUAHSHJUAWGUHTVOO
|
|
UIWDITZWGVTTVOWHWFVRVSQWHWEVQUDQWDIGUKUJUJULAVNUMVOVRSPZVSSPZVTSPVO
|
|
VQSPUDSPWIAUASIGLUNUOVQUDUPUQZVOCSPZUESPWJAWLVNACEGNURUTZUSCUEVAUQZ
|
|
VRVSUPVBVCUJVOWIWJWLWAWCTWKWNWMVRVSCVDVEVOWCVRVFQRZVRVOWBVFVRQVOWBC
|
|
VSQRZVFVOWJWLWBWPTWNWMVSCVGVBVOWLWPVFTWMCVHVIVJVKVOWIWOVRTWKVRVLVIV
|
|
JVM $.
|
|
$}
|
|
|
|
${
|
|
$d A m $. $d a ph $.
|
|
$( Lemma for ~ caucvgsr . Offsetting the values of the sequence so
|
|
they are greater than one. (Contributed by Jim Kingdon,
|
|
3-Jul-2021.) $)
|
|
caucvgsrlemofff $p |- ( ph -> G : N. --> R. ) $=
|
|
( cnpi c1r co cm1r cnr wcel cv cfv cmr wa ffvelrnda addclsr sylancl
|
|
cplr 1sr caucvgsrlemasr adantr m1r mulclsr syl2anc fmptd ) AIOIUAZG
|
|
UBZPUHQZCRUCQZUHQZSHAUPOTZUDZURSTZUSSTZUTSTVBUQSTPSTVCAOSUPGKUEUIUQ
|
|
PUFUGVBCSTZRSTVDAVEVAACEGMUJUKULCRUMUGURUSUFUNNUO $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d A f g h $. $d A m $. $d F a $. $d F f g h $.
|
|
$d G f g h $. $d a k n $. $d a k n ph $. $d f g h k n ph $.
|
|
$d f g h l n u $.
|
|
$( Lemma for ~ caucvgsr . Offsetting the values of the sequence so
|
|
they are greater than one. (Contributed by Jim Kingdon,
|
|
3-Jul-2021.) $)
|
|
caucvgsrlemoffcau $p |- ( ph ->
|
|
A. n e. N. A. k e. N. ( n <N k ->
|
|
( ( G ` n ) <R ( ( G ` k ) +R
|
|
[ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R
|
|
) /\
|
|
( G ` k ) <R ( ( G ` n ) +R
|
|
[ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R
|
|
) ) ) ) $=
|
|
( co cplr cltr cnpi wcel cnr vf vg clti wbr cfv c1o cop ceq cec crq
|
|
vh cv cltq cab c1p cpp cer wa wi wral wceq caucvgsrlemoffval adantr
|
|
c1r eqcomd wf ad2antrr simpr ffvelrnd cnp simplr recnnpr prsrcl 1sr
|
|
3syl a1i addcomsrg adantl addasssrg caov32d adantlr caucvgsrlemofff
|
|
w3a oveq1d caucvgsrlemasr 3eqtr2d breq12d wb ltasrg addclsr syl2anc
|
|
caovord2d 3bitr4d anbi12d biimpd imim2d ralimdva mpd ) AFULZDULZUCU
|
|
DZWSGUEZWTGUEZJULWSUFUGUHUIUJUEZUMUDJUNXDBULUMUDBUNUGZUOUPOUOUGUQUI
|
|
ZPOZQUDZXCXBXFPOZQUDZURZUSZDRUTZFRUTXAWSHUEZWTHUEZXFPOZQUDZXOXNXFPO
|
|
ZQUDZURZUSZDRUTZFRUTLAXMYBFRAWSRSZURZXLYADRYDWTRSZURZXKXTXAYFXKXTYF
|
|
XHXQXJXSYFXBVDPOZXGVDPOZQUDXNCPOZXPCPOZQUDXHXQYFYGYIYHYJQYFYIYGYDYI
|
|
YGVAYEABCDEFGHWSIJKLMNVBVCZVEYFYHXCVDPOZXFPOXOCPOZXFPOYJYFUAUBUKXCX
|
|
FVDTPYFRTWTGARTGVFYCYEKVGZYDYEVHZVIZYFYCXEVJSXFTSZAYCYEVKZBWSJVLXEV
|
|
MVOZVDTSYFVNVPZUAULZTSZUBULZTSZURUUAUUCPOZUUCUUAPOVAYFUUAUUCVQVRZUU
|
|
BUUDUKULZTSWCZUUEUUGPOUUAUUCUUGPOPOVAYFUUAUUCUUGVSVRZVTYFYMYLXFPAYE
|
|
YMYLVAYCABCDEFGHWTIJKLMNVBWAZWDYFUAUBUKXOCXFTPYFRTWTHARTHVFYCYEABCD
|
|
EFGHIJKLMNWBVGZYOVIZACTSYCYEACEGMWEVGZYSUUFUUIVTWFWGYFUAUBUKXBXGVDQ
|
|
TPUUHUUAUUCQUDUUGUUAPOUUGUUCPOQUDWHYFUUAUUCUUGWIVRZYFRTWSGYNYRVIZYF
|
|
XCTSYQXGTSYPYSXCXFWJWKYTUUFWLYFUAUBUKXNXPCQTPUUNYFRTWSHUUKYRVIZYFXO
|
|
TSYQXPTSUULYSXOXFWJWKUUMUUFWLWMYFYLXIVDPOZQUDYMXRCPOZQUDXJXSYFYLYMU
|
|
UQUURQYFYMYLUUJVEYFUUQYGXFPOYIXFPOUURYFUAUBUKXBXFVDTPUUOYSYTUUFUUIV
|
|
TYFYIYGXFPYKWDYFUAUBUKXNCXFTPUUPUUMYSUUFUUIVTWFWGYFUAUBUKXCXIVDQTPU
|
|
UNYPYFXBTSYQXITSUUOYSXBXFWJWKYTUUFWLYFUAUBUKXOXRCQTPUUNUULYFXNTSYQX
|
|
RTSUUPYSXNXFWJWKUUMUUFWLWMWNWOWPWQWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d A a m $. $d A f g h m $. $d F a $. $d F f g h $. $d a m ph $.
|
|
$d f g h m ph $.
|
|
$( Lemma for ~ caucvgsr . Offsetting the values of the sequence so
|
|
they are greater than one. (Contributed by Jim Kingdon,
|
|
3-Jul-2021.) $)
|
|
caucvgsrlemoffgt1 $p |- ( ph -> A. m e. N. 1R <R ( G ` m ) ) $=
|
|
( c1r cltr wcel cplr co cnr vf vg vh cv cfv wbr cnpi wa r19.21bi wb
|
|
w3a ltasrg adantl caucvgsrlemasr adantr ffvelrnda 1sr a1i addcomsrg
|
|
caovord2d caucvgsrlemoffval caucvgsrlemofff syl2anc breqtrd syl3anc
|
|
wceq mpbid breqtrrd mpbird ralrimiva ) AOEUDZHUEZPUFZEUGAVKUGQZUHZV
|
|
MCORSZCVLRSZPUFZVOVPVLCRSZVQPVOVPVKGUEZORSZVSPVOCVTPUFZVPWAPUFAWBEU
|
|
GMUIVOUAUBUCCVTOPTRUAUDZTQZUBUDZTQZUCUDZTQUKWCWEPUFWGWCRSWGWERSPUFU
|
|
JVOWCWEWGULUMACTQZVNACEGMUNUOZAUGTVKGKUPOTQZVOUQURZWDWFUHWCWERSWEWC
|
|
RSVFVOWCWEUSUMUTVGABCDEFGHVKIJKLMNVAVHVOVLTQZWHVSVQVFAUGTVKHABCDEFG
|
|
HIJKLMNVBUPZWIVLCUSVCVDVOWJWLWHVMVRUJWKWMWIOVLCULVEVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d A a i k $. $d A f g h i x $. $d A i j k x $. $d A k m $.
|
|
$d A j k x y $. $d F a i k $. $d F f g h i z $. $d F k y z $.
|
|
$d G f g h i x z $. $d G i j k x z $. $d G j k l u $. $d G k m $.
|
|
$d G k l n u $. $d a k n ph $. $d f g h i ph x z $.
|
|
$d j k ph x z $. $d ph m $. $d m n $. $d a m $.
|
|
$( Lemma for ~ caucvgsr . Offsetting the values of the sequence so
|
|
they are greater than one. (Contributed by Jim Kingdon,
|
|
3-Jul-2021.) $)
|
|
caucvgsrlemoffres $p |- ( ph ->
|
|
E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. (
|
|
j <N k -> ( ( F ` k ) <R ( y +R x ) /\
|
|
y <R ( ( F ` k ) +R x ) ) ) ) ) $=
|
|
( cplr co cnr vi vz vf vg vh c0r cv cltr wbr clti wa cnpi wral wrex
|
|
cfv wi caucvgsrlemoffcau caucvgsrlemoffgt1 caucvgsrlemgt1 wcel cm1r
|
|
caucvgsrlemofff simprl caucvgsrlemasr adantr addclsr syl2anc m1r wb
|
|
sylancl c1r w3a ltasrg adantl ad3antrrr simpr ffvelrnd simpllr wceq
|
|
simplr addcomsrg caovord2d caucvgsrlemoffval breq1d bitrd addasssrg
|
|
wf adantlr caov32d breq2d ad2antrr ffvelrnda 1sr a1i syl3anc m1p1sr
|
|
mp2an eqtri oveq2i 0idsr syl5eq breq12d 3bitrd biimpd oveq1d syl6eq
|
|
syl eqtrd anim12d imim2d breq2 fveq2 anbi12d imbi12d cbvralv syl6ib
|
|
reximdv impr oveq1 breq1 imbi2d rexralbidv ralbidv rspcev rexlimddv
|
|
ralimdva ) AUFBUGZUHUIZFUGZUAUGZUJUIZYJKUOZUBUGZYGRSZUHUIZYMYLYGRSZ
|
|
UHUIZUKZUPZUAULUMZFULUNZUPZBTUMZYHYIGUGZUJUIZUUDJUOZCUGZYGRSZUHUIZU
|
|
UGUUFYGRSZUHUIZUKZUPZGULUMFULUNZUPZBTUMZCTUNZUBTABUBDUAFGHIKMADEGHI
|
|
JKLMNOPQVBZADEGHIJKLMNOPQUQADEGHIJKLMNOPQURUSAYMTUTZUUCUKZUKZYMERSZ
|
|
VARSZTUTZYHUUEUUFUVCYGRSZUHUIZUVCUUJUHUIZUKZUPZGULUMZFULUNZUPZBTUMZ
|
|
UUQUVAUVBTUTZVATUTZUVDUVAUUSETUTZUVNAUUSUUCVCAUVPUUTAEHJPVDZVEYMEVF
|
|
ZVGVHUVBVAVFVJAUUSUUCUVMAUUSUKZUUBUVLBTUVSYGTUTZUKZUUAUVKYHUWAYTUVJ
|
|
FULUWAYTYKYJJUOZUVEUHUIZUVCUWBYGRSZUHUIZUKZUPZUAULUMUVJUWAYSUWGUAUL
|
|
UWAYJULUTZUKZYRUWFYKUWIYOUWCYQUWEUWIYOUWCUWIYOUWBVKRSZYNERSZUHUIZUW
|
|
JUVBYGRSZUHUIZUWCUWIYOYLERSZUWKUHUIUWLUWIUCUDUEYLYNEUHTRUCUGZTUTZUD
|
|
UGZTUTZUEUGZTUTVLZUWPUWRUHUIUWTUWPRSUWTUWRRSUHUIVIUWIUWPUWRUWTVMVNZ
|
|
UWIULTYJKAULTKWGUUSUVTUWHUURVOUWAUWHVPVQZUWIUUSUVTYNTUTAUUSUVTUWHVR
|
|
ZUVSUVTUWHVTZYMYGVFVGAUVPUUSUVTUWHUVQVOZUWQUWSUKUWPUWRRSZUWRUWPRSVS
|
|
UWIUWPUWRWAVNZWBUWIUWOUWJUWKUHUVSUWHUWOUWJVSZUVTAUWHUXIUUSADEGHIJKY
|
|
JLMNOPQWCWHWHZWDWEUWIUWKUWMUWJUHUWIUCUDUEYMYGETRUXDUXEUXFUXHUXAUXGU
|
|
WTRSUWPUWRUWTRSRSVSUWIUWPUWRUWTWFVNZWIWJUWIUWNUWJVARSZUWMVARSZUHUIU
|
|
WCUWIUCUDUEUWJUWMVAUHTRUXBUWIUWBTUTZVKTUTZUWJTUTZUWAULTYJJAULTJWGUU
|
|
SUVTNWKWLZWMUWBVKVFVJZUWIUVNUVTUWMTUTUWIUUSUVPUVNUXDUXFUVRVGZUXEUVB
|
|
YGVFVGUVOUWIVHWNZUXHWBUWIUXLUWBUXMUVEUHUWIUXLUWBVKVARSZRSZUWBUWIUXN
|
|
UXOUVOUXLUYBVSUXQUXOUWIWMWNZUXTUWBVKVAWFWOUWIUYBUWBUFRSZUWBUYAUFUWB
|
|
RUYAVAVKRSZUFUXOUVOUYAUYEVSWMVHVKVAWAWQWPWRZWSUWIUXNUYDUWBVSUXQUWBW
|
|
TXGXAXHUWIUCUDUEUVBYGVATRUXSUXEUXTUXHUXKWIXBWEXCXDUWIYQUWEUWIYQUVBU
|
|
WJYGRSZUHUIZUVCUYGVARSZUHUIUWEUWIYQUVBYPERSZUHUIUYHUWIUCUDUEYMYPEUH
|
|
TRUXBUXDUWIYLTUTUVTYPTUTUXCUXEYLYGVFVGUXFUXHWBUWIUYJUYGUVBUHUWIUYJU
|
|
WOYGRSUYGUWIUCUDUEYLYGETRUXCUXEUXFUXHUXKWIUWIUWOUWJYGRUXJXEXHWJWEUW
|
|
IUCUDUEUVBUYGVAUHTRUXBUXSUWIUXPUVTUYGTUTUXRUXEUWJYGVFVGUXTUXHWBUWIU
|
|
YIUWDUVCUHUWIUYIUWDUFRSZUWDUWIUYIUWDUYARSZUYKUWIUYIUWDVKRSZVARSZUYL
|
|
UWIUYGUYMVARUWIUCUDUEUWBVKYGTRUXQUYCUXEUXHUXKWIXEUWIUWDTUTZUXOUVOUY
|
|
NUYLVSUWIUXNUVTUYOUXQUXEUWBYGVFVGZUYCUXTUWDVKVAWFWOXHUYAUFUWDRUYFWS
|
|
XFUWIUYOUYKUWDVSUYPUWDWTXGXHWJXCXDXIXJYFUWGUVIUAGULYJUUDVSZYKUUEUWF
|
|
UVHYJUUDYIUJXKUYQUWCUVFUWEUVGUYQUWBUUFUVEUHYJUUDJXLZWDUYQUWDUUJUVCU
|
|
HUYQUWBUUFYGRUYRXEWJXMXNXOXPXQXJYFXRUUPUVMCUVCTUUGUVCVSZUUOUVLBTUYS
|
|
UUNUVKYHUYSUUMUVIFGULULUYSUULUVHUUEUYSUUIUVFUUKUVGUYSUUHUVEUUFUHUUG
|
|
UVCYGRXSWJUUGUVCUUJUHXTXMYAYBYAYCYDVGYE $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A a b k n $. $d A a j k l u $. $d A a k m $. $d A a j k x $.
|
|
$d A j k x y $. $d F a b k n $. $d F a j k l u $. $d F a k m $.
|
|
$d F a j k x $. $d F j k x y $. $d b k n ph $. $d j k ph x $.
|
|
$d ph m $. $d l n u $. $d m n $. $d b m $.
|
|
$( Lemma for ~ caucvgsr . A Cauchy sequence with a lower bound
|
|
converges. (Contributed by Jim Kingdon, 19-Jun-2021.) $)
|
|
caucvgsrlembnd $p |- ( ph ->
|
|
E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. (
|
|
j <N k -> ( ( F ` k ) <R ( y +R x ) /\
|
|
y <R ( ( F ` k ) +R x ) ) ) ) ) $=
|
|
( va vb cnpi c1r cplr co cv cfv cm1r cmr cmpt fveq2 caucvgsrlemoffres
|
|
weq oveq1d cbvmptv ) ABCDEFGHIJOQOUAZJUBZRSTZEUCUDTZSTZUEPKLMNOPQUOPU
|
|
AZJUBZRSTZUNSTOPUHZUMURUNSUSULUQRSUKUPJUFUIUIUJUG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d F f g h m $. $d F j k l u $. $d F k l n u $. $d F j k x y $.
|
|
$d f g h m ph $. $d j k ph x $. $d k m ph $. $d n ph $. $d m n $.
|
|
$( A Cauchy sequence of signed reals with a modulus of convergence
|
|
converges to a signed real. This is basically Corollary 11.2.13 of
|
|
[HoTT], p. (varies). The HoTT book theorem has a modulus of
|
|
convergence (that is, a rate of convergence) specified by (11.2.9) in
|
|
HoTT whereas this theorem fixes the rate of convergence to say that
|
|
all terms after the nth term must be within ` 1 / n ` of the nth term
|
|
(it should later be able to prove versions of this theorem with a
|
|
different fixed rate or a modulus of convergence supplied as a
|
|
hypothesis).
|
|
|
|
This is similar to ~ caucvgprpr but is for signed reals rather than
|
|
positive reals.
|
|
|
|
Here is an outline of how we prove it:
|
|
|
|
1. Choose a lower bound for the sequence (see ~ caucvgsrlembnd ).
|
|
|
|
2. Offset each element of the sequence so that each element of the
|
|
resulting sequence is greater than one (greater than zero would not
|
|
suffice, because the limit as well as the elements of the sequence
|
|
need to be positive) (see ~ caucvgsrlemofff ).
|
|
|
|
3. Since a signed real (element of ` R. ` ) which is greater than
|
|
zero can be mapped to a positive real (element of ` P. ` ), perform
|
|
that mapping on each element of the sequence and invoke ~ caucvgprpr
|
|
to get a limit (see ~ caucvgsrlemgt1 ).
|
|
|
|
4. Map the resulting limit from positive reals back to signed reals
|
|
(see ~ caucvgsrlemgt1 ).
|
|
|
|
5. Offset that limit so that we get the limit of the original
|
|
sequence rather than the limit of the offsetted sequence (see
|
|
~ caucvgsrlemoffres ). (Contributed by Jim Kingdon, 20-Jun-2021.) $)
|
|
caucvgsr $p |- ( ph ->
|
|
E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k ->
|
|
( ( F ` k ) <R ( y +R x ) /\
|
|
y <R ( ( F ` k ) +R x ) ) ) ) ) $=
|
|
( c1o cplr co cltr wbr cnpi c1r c1p cnr vm vf vg vh cfv cm1r cv wcel wa
|
|
clti wceq cop ceq cec crq cltq cab cpp wi wral breq1 fveq2 opeq1 eceq1d
|
|
cer fveq2d breq2d abbidv breq1d opeq12d breq12d oveq12d anbi12d imbi12d
|
|
1pi oveq1d opeq1d oveq2d ralbidv rspcv mpsyl simpl imim2i syl breq2 c1q
|
|
ralimi mpan9 df-1nqqs fveq2i rec1nq eqtr3i breq2i breq1i opeq12i df-i1p
|
|
abbii eqtr4i oveq1i opeq1i eceq1 ax-mp df-1r oveq2i syl6ib imp ffvelrnd
|
|
wf adantr a1i ltadd1sr adantl breqtrd nlt1pig pm2.21d w3o pitri3or mpan
|
|
wn mpjao3dan w3a ltasrg ffvelrnda 1sr addclsr addcomsrg caovord2d mpbid
|
|
sylancl m1r addasssrg syl3anc c0r mp2an m1p1sr eqtri 0idsr syl5eq eqtrd
|
|
wb ralrimiva caucvgsrlembnd ) ABCDLHUEZUFMNZEFUAGHIJKAUUDUAUGZHUEZOPUAQ
|
|
AUUEQUHZUIZUUDUUFRMNZUFMNZUUFOUUHUUCUUIOPZUUDUUJOPUUHLUUEUJPZUUKLUUEUKZ
|
|
UUELUJPZUUHUULUUKUUHUULUUCUUFIUGZLLULZUMUNZUOUEZUPPZIUQZUURDUGZUPPZDUQZ
|
|
ULZSURNZSULZVEUNZMNZOPZUUKALFUGZUJPZUUCUVJHUEZUVGMNZOPZUSZFQUTZUUGUULUV
|
|
IUSZAUVKUVNUVLUUCUVGMNZOPZUIZUSZFQUTZUVPLQUHZAGUGZUVJUJPZUWDHUEZUVLUUOU
|
|
WDLULZUMUNZUOUEZUPPZIUQZUWIUVAUPPZDUQZULZSURNZSULZVEUNZMNZOPZUVLUWFUWQM
|
|
NZOPZUIZUSZFQUTZGQUTUWBVOKUXDUWBGLQUWDLUKZUXCUWAFQUXEUWEUVKUXBUVTUWDLUV
|
|
JUJVAUXEUWSUVNUXAUVSUXEUWFUUCUWRUVMOUWDLHVBZUXEUWQUVGUVLMUXEUWPUVFVEUXE
|
|
UWOUVESUXEUWNUVDSURUXEUWKUUTUWMUVCUXEUWJUUSIUXEUWIUURUUOUPUXEUWHUUQUOUX
|
|
EUWGUUPUMUWDLLVCVDVFZVGVHUXEUWLUVBDUXEUWIUURUVAUPUXGVIVHVJVPVQVDZVRVKUX
|
|
EUWTUVRUVLOUXEUWFUUCUWQUVGMUXFUXHVLVGVMVNVSVTWAUWAUVOFQUVTUVNUVKUVNUVSW
|
|
BWCWGWDUVOUVQFUUEQUVJUUEUKZUVKUULUVNUVIUVJUUELUJWEUXIUVMUVHUUCOUXIUVLUU
|
|
FUVGMUVJUUEHVBVPVGVNVTWHUVHUUIUUCOUVGRUUFMUVGSSURNZSULZVEUNZRUVFUXKUKUV
|
|
GUXLUKUVEUXJSUVDSSURUVDUUOWFUPPZIUQZWFUVAUPPZDUQZULSUUTUXNUVCUXPUUSUXMI
|
|
UURWFUUOUPWFUOUEUURWFWFUUQUOWIWJWKWLZWMWQUVBUXODUURWFUVAUPUXQWNWQWODIWP
|
|
WRWSWTUVFUXKVEXAXBXCWRXDWMXEXFUUHUUMUIUUCUUCRMNZUUIOUUHUUCUXROPZUUMUUHU
|
|
UCTUHUXSUUHQTLHAQTHXHUUGJXIUWCUUHVOXJXGZUUCXKWDXIUUMUXRUUIUKUUHUUMUUCUU
|
|
FRMLUUEHVBVPXLXMUUHUUNUUKUUHUUNUUKUUGUUNXSAUUEXNXLXOXFUUGUULUUMUUNXPZAU
|
|
WCUUGUYAVOLUUEXQXRXLXTUUHUBUCUDUUCUUIUFOTMUBUGZTUHZUCUGZTUHZUDUGZTUHYAU
|
|
YBUYDOPUYFUYBMNUYFUYDMNOPYTUUHUYBUYDUYFYBXLUXTUUHUUFTUHZRTUHZUUITUHAQTU
|
|
UEHJYCZYDUUFRYEYIUFTUHZUUHYJXJZUYCUYEUIUYBUYDMNUYDUYBMNUKUUHUYBUYDYFXLY
|
|
GYHUUHUUJUUFRUFMNZMNZUUFUUHUYGUYHUYJUUJUYMUKUYIUYHUUHYDXJUYKUUFRUFYKYLU
|
|
UHUYMUUFYMMNZUUFUYLYMUUFMUYLUFRMNZYMUYHUYJUYLUYOUKYDYJRUFYFYNYOYPXDUUHU
|
|
YGUYNUUFUKUYIUUFYQWDYRYSXMUUAUUB $.
|
|
$}
|
|
$}
|
|
|
|
$( Class of complex numbers. $)
|
|
cc $a class CC $.
|
|
$( Class of real numbers. $)
|
|
cr $a class RR $.
|
|
$( Extend class notation to include the complex number 0. $)
|
|
cc0 $a class 0 $.
|
|
$( Extend class notation to include the complex number 1. $)
|
|
c1 $a class 1 $.
|
|
$( Extend class notation to include the complex number i. $)
|
|
ci $a class _i $.
|
|
$( Addition on complex numbers. $)
|
|
caddc $a class + $.
|
|
$( 'Less than' predicate (defined over real subset of complex numbers). $)
|
|
cltrr $a class <RR $.
|
|
$( Multiplication on complex numbers. The token ` x. ` is a center dot. $)
|
|
cmul $a class x. $.
|
|
|
|
$( Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) $)
|
|
df-c $a |- CC = ( R. X. R. ) $.
|
|
|
|
$( Define the complex number 0. (Contributed by NM, 22-Feb-1996.) $)
|
|
df-0 $a |- 0 = <. 0R , 0R >. $.
|
|
|
|
$( Define the complex number 1. (Contributed by NM, 22-Feb-1996.) $)
|
|
df-1 $a |- 1 = <. 1R , 0R >. $.
|
|
|
|
$( Define the complex number ` _i ` (the imaginary unit). (Contributed by
|
|
NM, 22-Feb-1996.) $)
|
|
df-i $a |- _i = <. 0R , 1R >. $.
|
|
|
|
$( Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) $)
|
|
df-r $a |- RR = ( R. X. { 0R } ) $.
|
|
|
|
${
|
|
$d x y z w v u f $.
|
|
$( Define addition over complex numbers. (Contributed by NM,
|
|
28-May-1995.) $)
|
|
df-add $a |- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. )
|
|
/\ z = <. ( w +R u ) , ( v +R f ) >. ) ) } $.
|
|
|
|
$( Define multiplication over complex numbers. (Contributed by NM,
|
|
9-Aug-1995.) $)
|
|
df-mul $a |- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. )
|
|
/\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) ,
|
|
( ( v .R u ) +R ( w .R f ) ) >. ) ) } $.
|
|
|
|
$( Define 'less than' on the real subset of complex numbers. (Contributed
|
|
by NM, 22-Feb-1996.) $)
|
|
df-lt $a |- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\
|
|
E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) } $.
|
|
$}
|
|
|
|
$( Ordered pair membership in the class of complex numbers. (Contributed by
|
|
NM, 14-May-1996.) $)
|
|
opelcn $p |- ( <. A , B >. e. CC <-> ( A e. R. /\ B e. R. ) ) $=
|
|
( cop cc wcel cnr cxp wa df-c eleq2i opelxp bitri ) ABCZDEMFFGZEAFEBFEHDNMI
|
|
JABFFKL $.
|
|
|
|
$( Ordered pair membership in class of real subset of complex numbers.
|
|
(Contributed by NM, 22-Feb-1996.) $)
|
|
opelreal $p |- ( <. A , 0R >. e. RR <-> A e. R. ) $=
|
|
( c0r cop cr wcel cnr wceq eqid csn cxp wa df-r eleq2i opelxp 0r elexi elsn
|
|
anbi2i 3bitri mpbiran2 ) ABCZDEZAFEZBBGZBHUBUAFBIZJZEUCBUEEZKUCUDKDUFUALMAB
|
|
FUENUGUDUCBBBFOPQRST $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Membership in class of real numbers. (Contributed by NM,
|
|
31-Mar-1996.) $)
|
|
elreal $p |- ( A e. RR <-> E. x e. R. <. x , 0R >. = A ) $=
|
|
( vy cr wcel cnr c0r csn cxp cv cop wceq wrex eleq2i elxp2 0r elexi opeq2
|
|
df-r bitri eqeq2d rexsn eqcom rexbii ) BDEBFGHZIZEZAJZGKZBLZAFMZDUFBSNUGB
|
|
UHCJZKZLZCUEMZAFMUKACBFUEOUOUJAFUOBUILZUJUNUPCGGFPQULGLUMUIBULGUHRUAUBBUI
|
|
UCTUDTT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The real number mapping in ~ elreal is unique. (Contributed by Jim
|
|
Kingdon, 11-Jul-2021.) $)
|
|
elrealeu $p |- ( A e. RR <-> E! x e. R. <. x , 0R >. = A ) $=
|
|
( vy cr wcel cv c0r cop wceq wreu wrex wrmo elreal biimpi wa wi ralrimiva
|
|
cnr wral sylibr eqtr3 wb 0r opthg mpan2 ad2antlr syl5ib simpl syl6 eqeq1d
|
|
opeq1 rmo4 reu5 sylanbrc reurex impbii ) BDEZAFZGHZBIZARJZUQUTARKZUTARLZV
|
|
AUQVBABMZNUQUTCFZGHZBIZOZURVEIZPZCRSZARSVCUQVKARUQURREZOZVJCRVMVEREZOZVHV
|
|
IGGIZOZVIVHUSVFIZVOVQUSVFBUAVLVRVQUBZUQVNVLGREVSUCURGVEGRRUDUEUFUGVIVPUHU
|
|
IQQUTVGACRVIUSVFBURVEGUKUJULTUTARUMUNVAVBUQUTARUOVDTUP $.
|
|
$}
|
|
|
|
$( Ordered pair membership in the class of complex numbers. (Contributed by
|
|
Mario Carneiro, 15-Jun-2013.) $)
|
|
elreal2 $p |- ( A e. RR <->
|
|
( ( 1st ` A ) e. R. /\ A = <. ( 1st ` A ) , 0R >. ) ) $=
|
|
( cr wcel cnr c0r csn cxp c1st cfv cop wceq wa df-r eleq2i xp1st c2nd xp2nd
|
|
1st2nd2 elsni syl opeq2d eqtrd eleq1 0r elexi snid opelxp mpbiran2 biimparc
|
|
jca syl6bb impbii bitri ) ABCADEFZGZCZAHIZDCZAUQEJZKZLZBUOAMNUPVAUPURUTADUN
|
|
OUPAUQAPIZJUSADUNRUPVBEUQUPVBUNCVBEKADUNQVBESTUAUBUJUTUPURUTUPUSUOCZURAUSUO
|
|
UCVCUREUNCEEDUDUEUFUQEDUNUGUHUKUIULUM $.
|
|
|
|
$( The empty set is not a complex number. Note: do not use this after the
|
|
real number axioms are developed, since it is a construction-dependent
|
|
property. (Contributed by NM, 2-May-1996.) $)
|
|
0ncn $p |- -. (/) e. CC $=
|
|
( c0 cc wcel cnr cxp 0nelxp df-c eleq2i mtbir ) ABCADDEZCDDFBJAGHI $.
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( 'Less than' is a relation on real numbers. (Contributed by NM,
|
|
22-Feb-1996.) $)
|
|
ltrelre $p |- <RR C_ ( RR X. RR ) $=
|
|
( vx vy vz vw cltrr cv cr wcel wa c0r cop wceq wbr wex copab cxp opabssxp
|
|
cltr df-lt eqsstri ) EAFZGHBFZGHIUACFZJKLUBDFZJKLIUCUDRMIDNCNZIABOGGPABCD
|
|
SUEABGGQT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u f A $. $d x y z w v u f B $. $d x y z w v u f C $.
|
|
$d x y z w v u f D $.
|
|
$( Addition of complex numbers in terms of signed reals. (Contributed by
|
|
NM, 28-May-1995.) $)
|
|
addcnsr $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
|
|
( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. ) $=
|
|
( vx vy vz vw vv vu vf cv cplr co cop cnr wcel wa wceq cc addclsr opelxpi
|
|
cxp ad2ant2r ad2ant2l syl2anc simpll simprl oveq12d simplr simprr opeq12d
|
|
caddc wex coprab df-add df-c eleq2i anbi12i anbi1i oprabbii eqtri ovi3 )
|
|
EFGHIJABCDHLZJLZMNZILZKLZMNZOZACMNZBDMNZOZKUMPAPQZBPQZRCPQZDPQZRRVKPQZVLP
|
|
QZVMPPUCZQVNVPVRVOVQACUAUDVOVQVSVNVPBDUAUEVKVLPPUBUFVDASZVGBSZRZVECSZVHDS
|
|
ZRZRZVFVKVIVLWGVDAVECMWAWBWFUGWCWDWEUHUIWGVGBVHDMWAWBWFUJWCWDWEUKUIULUMEL
|
|
ZTQZFLZTQZRZWHVDVGOSWJVEVHOSRGLVJSRKUNJUNIUNHUNZRZEFGUOWHVTQZWJVTQZRZWMRZ
|
|
EFGUOEFGHIJKUPWNWREFGWLWQWMWIWOWKWPTVTWHUQURTVTWJUQURUSUTVAVBVC $.
|
|
|
|
$( Multiplication of complex numbers in terms of signed reals.
|
|
(Contributed by NM, 9-Aug-1995.) $)
|
|
mulcnsr $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
|
|
( <. A , B >. x. <. C , D >. ) =
|
|
<. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) ,
|
|
( ( B .R C ) +R ( A .R D ) ) >. ) $=
|
|
( vx vy vz vw vv cv cmr co cm1r cplr cnr wcel wa mulclsr wceq oveq12d cop
|
|
vu cmul cxp ad2ant2r ad2ant2l sylancr addclsr syl2anc ad2ant2lr ad2ant2rl
|
|
vf m1r opelxpi simpll simprl simplr simprr oveq2d opeq12d wex coprab df-c
|
|
cc df-mul eleq2i anbi12i anbi1i oprabbii eqtri ovi3 ) EFGHIUBABCDHJZUBJZK
|
|
LZMIJZULJZKLZKLZNLZVOVMKLZVLVPKLZNLZUAZACKLZMBDKLZKLZNLZBCKLZADKLZNLZUAZU
|
|
LUCOAOPZBOPZQCOPZDOPZQQZWGOPZWJOPZWKOOUDZPWPWDOPZWFOPZWQWLWNWTWMWOACRUEWP
|
|
MOPWEOPZXAUMWMWOXBWLWNBDRUFMWERUGWDWFUHUIWPWHOPZWIOPZWRWMWNXCWLWOBCRUJWLW
|
|
OXDWMWNADRUKWHWIUHUIWGWJOOUNUIVLASZVOBSZQZVMCSZVPDSZQZQZVSWGWBWJXKVNWDVRW
|
|
FNXKVLAVMCKXEXFXJUOZXGXHXIUPZTXKVQWEMKXKVOBVPDKXEXFXJUQZXGXHXIURZTUSTXKVT
|
|
WHWAWINXKVOBVMCKXNXMTXKVLAVPDKXLXOTTUTUCEJZVDPZFJZVDPZQZXPVLVOUASXRVMVPUA
|
|
SQGJWCSQULVAUBVAIVAHVAZQZEFGVBXPWSPZXRWSPZQZYAQZEFGVBEFGHIUBULVEYBYFEFGXT
|
|
YEYAXQYCXSYDVDWSXPVCVFVDWSXRVCVFVGVHVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
eqresr.1 $e |- A e. _V $.
|
|
$( Equality of real numbers in terms of intermediate signed reals.
|
|
(Contributed by NM, 10-May-1996.) $)
|
|
eqresr $p |- ( <. A , 0R >. = <. B , 0R >. <-> A = B ) $=
|
|
( c0r cop wceq eqid cnr 0r elexi opth mpbiran2 ) ADEBDEFABFDDFDGADBDCDHIJ
|
|
KL $.
|
|
$}
|
|
|
|
$( Addition of real numbers in terms of intermediate signed reals.
|
|
(Contributed by NM, 10-May-1996.) $)
|
|
addresr $p |- ( ( A e. R. /\ B e. R. ) ->
|
|
( <. A , 0R >. + <. B , 0R >. ) = <. ( A +R B ) , 0R >. ) $=
|
|
( cnr wcel wa c0r cop caddc co cplr wceq 0r addcnsr an4s 0idsr ax-mp opeq2i
|
|
mpanr12 syl6eq ) ACDZBCDZEZAFGBFGHIZABJIZFFJIZGZUDFGUBFCDZUGUCUFKZLLTUGUAUG
|
|
UHAFBFMNRUEFUDUGUEFKLFOPQS $.
|
|
|
|
$( Multiplication of real numbers in terms of intermediate signed reals.
|
|
(Contributed by NM, 10-May-1996.) $)
|
|
mulresr $p |- ( ( A e. R. /\ B e. R. ) ->
|
|
( <. A , 0R >. x. <. B , 0R >. ) = <. ( A .R B ) , 0R >. ) $=
|
|
( cnr wcel wa c0r cop cmul co cmr cm1r cplr wceq mulcnsr ax-mp oveq2i 0idsr
|
|
0r 00sr eqtrd an4s mpanr12 m1r mulclsr syl syl5eq mulcomsrg mpan oveqan12rd
|
|
eqtri syl6eq opeq12d ) ACDZBCDZEZAFGBFGHIZABJIZKFFJIZJIZLIZFBJIZAFJIZLIZGZU
|
|
QFGUOFCDZVEUPVDMZRRUMVEUNVEVFAFBFNUAUBUOUTUQVCFUOUTUQFLIZUQUSFUQLUSKFJIZFUR
|
|
FKJVEURFMRFSOPKCDVHFMUCKSOUJPUOUQCDVGUQMABUDUQQUEUFUOVCFFLIZFUNUMVAFVBFLUNV
|
|
ABFJIZFVEUNVAVJMRFBUGUHBSTASUIVEVIFMRFQOUKULT $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Ordering of real subset of complex numbers in terms of signed reals.
|
|
(Contributed by NM, 22-Feb-1996.) $)
|
|
ltresr $p |- ( <. A , 0R >. <RR <. B , 0R >. <-> A <R B ) $=
|
|
( vz vw vx vy c0r cop cltrr wbr cnr wcel wa cltr cr wceq wex anbi1d eqcom
|
|
cv ltrelre brel opelreal anbi12i sylib ltrelsr eleq1 eqeq1 2exbidv anbi2d
|
|
anbi12d df-lt brabg bianabs vex eqresr 3bitr4i opth2 bitr4i anbi1i 2exbii
|
|
wb syl6bb syl2anbr breq12 copsex2g bitrd pm5.21nii ) AGHZBGHZIJZAKLZBKLZM
|
|
ZABNJZVKVIOLZVJOLZMZVNVIVJOOIUAUBVPVLVQVMAUCZBUCZUDUEABKKNUFUBVNVKABHCTZD
|
|
TZHPZWAWBNJZMZDQCQZVOVLVPVQVKWFVBVMVSVTVRVKVIWAGHZPZVJWBGHZPZMZWDMZDQCQZW
|
|
FVRVKWMETZOLZFTZOLZMZWNWGPZWPWIPZMZWDMZDQCQZMVPWQMZWHWTMZWDMZDQCQZMVRWMME
|
|
FVIVJOOIWNVIPZWRXDXCXGXHWOVPWQWNVIOUGRXHXBXFCDXHXAXEWDXHWSWHWTWNVIWGUHRRU
|
|
IUKWPVJPZXDVRXGWMXIWQVQVPWPVJOUGUJXIXFWLCDXIXEWKWDXIWTWJWHWPVJWIUHUJRUIUK
|
|
EFCDULUMUNWLWECDWKWCWDWKAWAPZBWBPZMWCWHXJWJXKWGVIPWAAPWHXJWAACUOZUPVIWGSA
|
|
WASUQWIVJPWBBPWJXKWBBDUOZUPVJWISBWBSUQUDABWAWBXLXMURUSUTVAVCVDWDVOCDABKKW
|
|
AAWBBNVEVFVGVH $.
|
|
$}
|
|
|
|
$( Ordering of real subset of complex numbers in terms of signed reals.
|
|
(Contributed by NM, 22-Feb-1996.) $)
|
|
ltresr2 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A <RR B <-> ( 1st ` A ) <R ( 1st ` B ) ) ) $=
|
|
( cr wcel wa cltrr wbr c1st cfv c0r cop cltr wceq elreal2 simprbi breqan12d
|
|
cnr ltresr syl6bb ) ACDZBCDZEABFGAHIZJKZBHIZJKZFGUBUDLGTUAAUCBUEFTUBQDAUCMA
|
|
NOUAUDQDBUEMBNOPUBUDRS $.
|
|
|
|
$( Technical trick to permit reuse of previous lemmas to prove arithmetic
|
|
operation laws in ` CC ` from those in ` R. ` . The trick involves
|
|
~ qsid , which shows that the coset of the converse epsilon relation
|
|
(which is not an equivalence relation) acts as an identity divisor for the
|
|
quotient set operation. This lets us "pretend" that ` CC ` is a quotient
|
|
set, even though it is not (compare ~ df-c ), and allows us to reuse some
|
|
of the equivalence class lemmas we developed for the transition from
|
|
positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) $)
|
|
dfcnqs $p |- CC = ( ( R. X. R. ) /. `' _E ) $=
|
|
( cc cnr cxp cep ccnv cqs df-c qsid eqtr4i ) ABBCZJDEFGJHI $.
|
|
|
|
$( Technical trick to permit re-use of some equivalence class lemmas for
|
|
operation laws. See ~ dfcnqs and ~ mulcnsrec . (Contributed by NM,
|
|
13-Aug-1995.) $)
|
|
addcnsrec $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
|
|
( [ <. A , B >. ] `' _E + [ <. C , D >. ] `' _E ) =
|
|
[ <. ( A +R C ) , ( B +R D ) >. ] `' _E ) $=
|
|
( cnr wcel wa cop caddc co cplr cep ccnv cec addcnsr wceq opelxpi ecidg syl
|
|
addclsr cxp oveqan12d ad2ant2r ad2ant2l syl2anc 3eqtr4d ) AEFZBEFZGZCEFZDEF
|
|
ZGZGZABHZCDHZIJACKJZBDKJZHZUNLMZNZUOUSNZIJURUSNZABCDOUIULUTUNVAUOIUIUNEEUAZ
|
|
FUTUNPABEEQUNVCRSULUOVCFVAUOPCDEEQUOVCRSUBUMURVCFZVBURPUMUPEFZUQEFZVDUGUJVE
|
|
UHUKACTUCUHUKVFUGUJBDTUDUPUQEEQUEURVCRSUF $.
|
|
|
|
$( Technical trick to permit re-use of some equivalence class lemmas for
|
|
operation laws. The trick involves ~ ecidg , which shows that the coset
|
|
of the converse epsilon relation (which is not an equivalence relation)
|
|
leaves a set unchanged. See also ~ dfcnqs . (Contributed by NM,
|
|
13-Aug-1995.) $)
|
|
mulcnsrec $p |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) ->
|
|
( [ <. A , B >. ] `' _E x. [ <. C , D >. ] `' _E ) =
|
|
[ <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) ,
|
|
( ( B .R C ) +R ( A .R D ) ) >. ] `' _E ) $=
|
|
( cnr wcel wa cop cmul cmr cm1r cplr cec wceq opelxpi ecidg mulclsr syl2anc
|
|
co syl cep ccnv mulcnsr cxp simpll simprl m1r simplr simprr sylancr addclsr
|
|
oveqan12d 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZGZABHZCDHZISACJSZKBDJSZJSZLSZBCJSZA
|
|
DJSZLSZHZVAUAUBZMZVBVKMZISVJVKMZABCDUCUPUSVLVAVMVBIUPVAEEUDZFVLVANABEEOVAVO
|
|
PTUSVBVOFVMVBNCDEEOVBVOPTULUTVJVOFZVNVJNUTVFEFZVIEFZVPUTVCEFZVEEFZVQUTUNUQV
|
|
SUNUOUSUEZUPUQURUFZACQRUTKEFVDEFZVTUGUTUOURWCUNUOUSUHZUPUQURUIZBDQRKVDQUJVC
|
|
VEUKRUTVGEFZVHEFZVRUTUOUQWFWDWBBCQRUTUNURWGWAWEADQRVGVHUKRVFVIEEORVJVOPTUM
|
|
$.
|
|
|
|
${
|
|
$d f u v w x y z $.
|
|
$( Existence of a sum. This is dependent on how we define ` + ` so once we
|
|
proceed to real number axioms we will replace it with theorems such as
|
|
~ addcl . (Contributed by Jim Kingdon, 14-Jul-2021.) $)
|
|
addvalex $p |- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V ) $=
|
|
( vx vy vz vw vv vu vf wcel wa caddc cop cvv cv cnr cnp cc cfv df-ov cplr
|
|
co cxp cer cqs df-nr npex xpex qsex eqeltri wceq wex coprab df-add eleq2i
|
|
df-c anbi12i anbi1i oprabbii eqtri oprabex3 opexg fvexg sylancr syl5eqel
|
|
) ACLBDLMZABNUDABOZNUAZPABNUBVHNPLVIPLVJPLEFGHIJHQZJQZUCUDIQZKQZUCUDOZKNR
|
|
RSSUEZUFUGPUHVPUFSSUIUIUJUKULNEQZTLZFQZTLZMZVQVKVMOUMVSVLVNOUMMGQVOUMMKUN
|
|
JUNIUNHUNZMZEFGUOVQRRUEZLZVSWDLZMZWBMZEFGUOEFGHIJKUPWCWHEFGWAWGWBVRWEVTWF
|
|
TWDVQURUQTWDVSURUQUSUTVAVBVCABCDVDVINPPVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d K l u $.
|
|
$( Lemma for ~ pitonn . Two ways to write the number one. (Contributed by
|
|
Jim Kingdon, 24-Apr-2020.) $)
|
|
pitonnlem1 $p |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } ,
|
|
{ u | [ <. 1o , 1o >. ] ~Q <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R
|
|
, 0R >. = 1 $=
|
|
( c1r c0r cop cv c1o cec cltq wbr cab c1p cpp cer wceq c1q df-1nqqs abbii
|
|
co eqtri c1 ceq df-1 df-1r df-i1p breq2i breq1i oveq1i opeq1i eceq1 ax-mp
|
|
opeq12i eqtr2i ) UACDEBFZGGEUBHZIJZBKZUOAFZIJZAKZEZLMSZLEZNHZDEUCCVDDCLLM
|
|
SZLEZNHZVDUDVFVCOVGVDOVEVBLLVALMLUNPIJZBKZPURIJZAKZEVAABUEVIUQVKUTVHUPBPU
|
|
OUNIQUFRVJUSAPUOURIQUGRULTUHUIVFVCNUJUKTUIUM $.
|
|
|
|
$( Lemma for ~ pitonn . Simplifying an expression involving signed reals.
|
|
(Contributed by Jim Kingdon, 26-Apr-2020.) $)
|
|
pitonnlem1p1 $p |- ( A e. P. ->
|
|
[ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R =
|
|
[ <. ( A +P. 1P ) , 1P >. ] ~R ) $=
|
|
( cnp wcel c1p cpp co cop cer cec wceq 1pr addclpr mp2an addcomprg oveq1d
|
|
mpan2 addassprg mp3an13 eqtrd a1i wb enreceq syl22anc mpbird ) ABCZADDEFZ
|
|
EFZUFGHIADEFZDGHIJZUGDEFZUFUHEFZJZUEUJUFAEFZDEFZUKUEUGUMDEUEUFBCZUGUMJDBC
|
|
ZUPUOKKDDLMZAUFNPOUOUEUPUNUKJUQKUFADQRSUEUGBCZUOUHBCZUPUIULUAUEUOURUQAUFL
|
|
PUOUEUQTUEUPUSKADLPUPUEKTUGUFUHDUBUCUD $.
|
|
|
|
$( Lemma for ~ pitonn . Two ways to add one to a number. (Contributed by
|
|
Jim Kingdon, 24-Apr-2020.) $)
|
|
pitonnlem2 $p |- ( K e. N. ->
|
|
( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
|
|
{ u | [ <. K , 1o >. ] ~Q <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R
|
|
, 0R >. + 1 ) =
|
|
<. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } ,
|
|
{ u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R
|
|
, 0R >. ) $=
|
|
( wcel c1o cop cec cltq wbr cab c1p cpp co cer c0r c1r wceq cnp 1pr syl
|
|
cnpi cv ceq c1 caddc cplr cpli df-1 oveq2i cnr cxp nnprlu addclpr sylancl
|
|
cqs opelxpi enrex ecelqsi df-nr syl6eleqr 1sr addresr syl5eq pitonnlem1p1
|
|
df-1r mp2an addsrpr mpanl2 mpanr12 c1q cplq addpinq1 breq2d abbidv breq1d
|
|
wa opeq12d cnq nnnq addnqpr1 eqtrd oveq1d opeq1d eceq1d 3eqtr4d ) BUADZCU
|
|
BZBEFUCGZHICJWHAUBZHIAJFZKLMZKFZNGZOFZUDUEMZWMPUFMZOFZWGBEUGMEFUCGZHIZCJZ
|
|
WRWIHIZAJZFZKLMZKFZNGZOFWFWOWNPOFZUEMZWQUDXGWNUEUHUIWFWMUJDPUJDXHWQQWFWMR
|
|
RUKZNUOZUJWFWLXIDZWMXJDWFWKRDZKRDZXKWFWJRDXMXLABCULSWJKUMUNZSWKKRRUPUNXIW
|
|
LNUQURTUSUTVAWMPVBUNVCWFWPXFOWFWKKKLMZLMXOFNGZWKKLMZKFZNGZWPXFWFXLXPXSQXN
|
|
WKVDTWFWPWMXOKFNGZUFMZXPPXTWMUFVEUIWFXLYAXPQZXNXLXORDZXMYBXMXMYCSSKKUMVFS
|
|
XLXMYCXMVPYBSWKKXOKVGVHVITVCWFXEXRNWFXDXQKWFXCWKKLWFXCWGWHVJVKMZHIZCJZYDW
|
|
IHIZAJZFZWKWFWTYFXBYHWFWSYECWFWRYDWGHBVLZVMVNWFXAYGAWFWRYDWIHYJVOVNVQWFWH
|
|
VRDYIWKQBVSAWHCVTTWAWBWCWDWEWCWA $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d N l u w z $. $d k l u w y z $. $d x y z $.
|
|
$( Mapping from ` N. ` to ` NN ` . (Contributed by Jim Kingdon,
|
|
22-Apr-2020.) $)
|
|
pitonn $p |- ( N e. N. ->
|
|
<. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e.
|
|
|^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } ) $=
|
|
( wcel c1o cop ceq cec cltq wbr cab c1p cpp co cer c0r eceq1d abbidv cnpi
|
|
vz vw vk cv c1 caddc wral wa cint wi wal cpli opeq1 breq2d breq1d opeq12d
|
|
oveq1d opeq1d eleq1d imbi2d pitonnlem1 eleq1i biimpri adantr oveq1 rspccv
|
|
wceq ad2antll wb pitonnlem2 sylibd indpi alrimiv eleq2 raleqbi1dv anbi12d
|
|
a2d ralab sylibr cnp cxp cqs cnr nnprlu 1pr addclpr sylancl opelxpi enrex
|
|
ex ecelqsi syl 0r elintg mpbird ) DUAFZEUEZDGHZIJZKLZEMZWTCUEZKLZCMZHZNOP
|
|
ZNHZQJZRHZUFAUEZFZBUEZUFUGPZXKFZBXKUHZUIZAMZUJFZXJUBUEZFZUBXRUHZWQUFXTFZX
|
|
NXTFZBXTUHZUIZYAUKZUBULYBWQYGUBYFWRUCUEZGHZIJZKLZEMZYJXCKLZCMZHZNOPZNHZQJ
|
|
ZRHZXTFZUKYFWRGGHZIJZKLZEMZUUBXCKLZCMZHZNOPZNHZQJZRHZXTFZUKYFWRUDUEZGHZIJ
|
|
ZKLZEMZUUOXCKLZCMZHZNOPZNHZQJZRHZXTFZUKYFWRUUMGUMPZGHZIJZKLZEMZUVHXCKLZCM
|
|
ZHZNOPZNHZQJZRHZXTFZUKYGUCUDDYHGVHZYTUULYFUVSYSUUKXTUVSYRUUJRUVSYQUUIQUVS
|
|
YPUUHNUVSYOUUGNOUVSYLUUDYNUUFUVSYKUUCEUVSYJUUBWRKUVSYIUUAIYHGGUNSZUOTUVSY
|
|
MUUECUVSYJUUBXCKUVTUPTUQURUSSUSUTVAYHUUMVHZYTUVEYFUWAYSUVDXTUWAYRUVCRUWAY
|
|
QUVBQUWAYPUVANUWAYOUUTNOUWAYLUUQYNUUSUWAYKUUPEUWAYJUUOWRKUWAYIUUNIYHUUMGU
|
|
NSZUOTUWAYMUURCUWAYJUUOXCKUWBUPTUQURUSSUSUTVAYHUVFVHZYTUVRYFUWCYSUVQXTUWC
|
|
YRUVPRUWCYQUVOQUWCYPUVNNUWCYOUVMNOUWCYLUVJYNUVLUWCYKUVIEUWCYJUVHWRKUWCYIU
|
|
VGIYHUVFGUNSZUOTUWCYMUVKCUWCYJUVHXCKUWDUPTUQURUSSUSUTVAYHDVHZYTYAYFUWEYSX
|
|
JXTUWEYRXIRUWEYQXHQUWEYPXGNUWEYOXFNOUWEYLXBYNXEUWEYKXAEUWEYJWTWRKUWEYIWSI
|
|
YHDGUNSZUOTUWEYMXDCUWEYJWTXCKUWFUPTUQURUSSUSUTVAYCUULYEUULYCUUKUFXTCEVBVC
|
|
VDVEUUMUAFZYFUVEUVRUWGYFUVEUVRUKUWGYFUIUVEUVDUFUGPZXTFZUVRYEUVEUWIUKUWGYC
|
|
YDUWIBUVDXTXMUVDVHXNUWHXTXMUVDUFUGVFUTVGVIUWGUWIUVRVJYFUWGUWHUVQXTCUUMEVK
|
|
UTVEVLWKVRVMVNXQYFYAUBAXKXTVHXLYCXPYEXKXTUFVOXOYDBXKXTXKXTXNVOVPVQVSVTWQX
|
|
JWAWAWBZQWCZWDWBZFZXSYBVJWQXIUWKFZRWDFUWMWQXHUWJFZUWNWQXGWAFZNWAFZUWOWQXF
|
|
WAFUWQUWPCDEWEWFXFNWGWHWFXGNWAWAWIWHUWJXHQWJWLWMWNXIRUWKWDWIWHUBXJXRUWLWO
|
|
WMWP $.
|
|
$}
|
|
|
|
${
|
|
$d N l u $.
|
|
$( Embedding from ` N. ` to ` RR ` yields a number greater than zero.
|
|
(Contributed by Jim Kingdon, 15-Jul-2021.) $)
|
|
pitoregt0 $p |- ( N e. N. ->
|
|
0 <RR <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
) $=
|
|
( wcel c1p cop cer cec cv cltq wbr cab cpp co cltr c0r cltrr cltp cnp 1pr
|
|
cnpi c1o ceq cc0 addclpr mp2an ltaddpr sylancr wceq a1i addassprg syl3anc
|
|
nnprlu addcomprg oveq2d eqtrd breqtrd sylancl ltsrprg mpanl12 mpbird df-0
|
|
wb wa breq1i ltresr df-0r 3bitri sylibr ) BUADZEEFGHZCIBUBFUCHZJKCLVLAIJK
|
|
ALFZEMNZEFGHZOKZUDVOPFZQKZVJVPEEMNZEVNMNZRKZVJVSVSVMMNZVTRVJVSSDZVMSDZVSW
|
|
BRKESDZWEWCTTEEUEUFABCUMZVSVMUGUHVJWBEEVMMNZMNZVTVJWEWEWDWBWHUIWEVJTUJZWI
|
|
WFEEVMUKULVJWGVNEMVJWEWDWGVNUITWFEVMUNUHUOUPUQVJVNSDZWEVPWAVCZVJWDWEWJWFT
|
|
VMEUEURTWEWEWJWEVDWKTTEEVNEUSUTURVAVRPPFZVQQKPVOOKVPUDWLVQQVBVEPVOVFPVKVO
|
|
OVGVEVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d N l u $.
|
|
$( Embedding from ` N. ` to ` RR ` . Similar to ~ pitonn but separate in
|
|
the sense that we have not proved ~ nnssre yet. (Contributed by Jim
|
|
Kingdon, 15-Jul-2021.) $)
|
|
pitore $p |- ( N e. N. ->
|
|
<. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e.
|
|
RR ) $=
|
|
( cnpi wcel c1o cop ceq cec cltq wbr cab c1p cpp cer cnr cnp 1pr sylancl
|
|
cv co c0r cr cxp cqs nnprlu addclpr opelxpi enrex ecelqsi df-nr syl6eleqr
|
|
syl opelreal sylibr ) BDEZCTBFGHIZJKCLUQATJKALGZMNUAZMGZOIZPEVAUBGUCEUPVA
|
|
QQUDZOUEZPUPUTVBEZVAVCEUPUSQEZMQEZVDUPURQEVFVEABCUFRURMUGSRUSMQQUHSVBUTOU
|
|
IUJUMUKULVAUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d N l u $.
|
|
$( Embedding the reciprocal of a natural number into ` RR ` . (Contributed
|
|
by Jim Kingdon, 15-Jul-2021.) $)
|
|
recnnre $p |- ( N e. N. ->
|
|
<. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ,
|
|
0R >. e. RR ) $=
|
|
( cnpi wcel c1o cop ceq cec crq cltq wbr cab c1p cer cnr cnp 1pr sylancl
|
|
cv cfv cpp co c0r cxp cqs recnnpr addclpr opelxpi enrex ecelqsi syl df-nr
|
|
cr syl6eleqr opelreal sylibr ) BDEZCTBFGHIJUAZKLCMUSATKLAMGZNUBUCZNGZOIZP
|
|
EVCUDGUNEURVCQQUEZOUFZPURVBVDEZVCVEEURVAQEZNQEZVFURUTQEVHVGABCUGRUTNUHSRV
|
|
ANQQUISVDVBOUJUKULUMUOVCUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
peano1nnnn.n $e |- N =
|
|
|^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( One is an element of ` NN ` . This is a counterpart to ~ 1nn designed
|
|
for real number axioms which involve natural numbers (notably,
|
|
~ axcaucvg ). (Contributed by Jim Kingdon, 14-Jul-2021.)
|
|
(New usage is discouraged.) $)
|
|
peano1nnnn $p |- 1 e. N $=
|
|
( vz c1 wcel cv caddc co wral wa cab cint eleq2i cr wb c1r c0r eleq2
|
|
cop df-1 cnr 1sr opelreal mpbir eqeltri elintg ax-mp vex weq raleqbi1dv
|
|
bitri anbi12d elab simplbi mprgbir ) FCGZFEHZGZEFAHZGZBHFIJZVAGZBVAKZLZ
|
|
AMZURFVGNZGZUTEVGKZCVHFDOFPGVIVJQFRSUAZPUBVKPGRUCGUDRUEUFUGEFVGPUHUIUMU
|
|
SVGGUTVCUSGZBUSKZVFUTVMLAUSEUJAEUKVBUTVEVMVAUSFTVDVLBVAUSVAUSVCTULUNUOU
|
|
PUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d x y z $.
|
|
$( A successor of a positive integer is a positive integer. This is a
|
|
counterpart to ~ peano2nn designed for real number axioms which
|
|
involve to natural numbers (notably, ~ axcaucvg ). (Contributed by
|
|
Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) $)
|
|
peano2nnnn $p |- ( A e. N -> ( A + 1 ) e. N ) $=
|
|
( vz wcel c1 caddc co cv wral wa eleq2i elintg syl5bb cvv c1r c0r cnr
|
|
cab cint ibi vex wceq eleq2 raleqbi1dv anbi12d elab oveq1 eleq1d rspcva
|
|
simprbi sylan2 expcom ralimia syl wb cop df-1 0r opexg eqeltri addvalex
|
|
1sr mp2an mpan2 mpbird ) CDGZCHIJZDGZVJFKZGZFHAKZGZBKZHIJZVNGZBVNLZMZAU
|
|
AZLZVICVLGZFWALZWBVIWDVICWAUBZGVIWDDWECENFCWADOPUCWCVMFWAWCVLWAGZVMWFWC
|
|
VQVLGZBVLLZVMWFHVLGZWHVTWIWHMAVLFUDVNVLUEVOWIVSWHVNVLHUFVRWGBVNVLVNVLVQ
|
|
UFUGUHUIUMWGVMBCVLVPCUEVQVJVLVPCHIUJUKULUNUOUPUQVIVJQGZVKWBURVIHQGWJHRS
|
|
USZQUTRTGSTGWKQGVEVARSTTVBVFVCCHDQVDVGVKVJWEGWJWBDWEVJENFVJWAQOPUQVH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d J l $. $d J u $. $d K l $. $d K u $.
|
|
|
|
$( Ordering of natural numbers with ` <N ` or ` <RR ` . (Contributed by
|
|
Jim Kingdon, 13-Jul-2021.) $)
|
|
ltrennb $p |- ( ( J e. N. /\ K e. N. ) ->
|
|
( J <N K <-> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
|
|
{ u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
<RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
|
|
{ u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
) ) $=
|
|
( cnpi wcel wbr cv c1o cop ceq cec cltq cab c1p cpp co cer c0r cnq ltnnnq
|
|
wa clti cltr cltrr cltp wb adantr adantl ltnqpr syl2anc cnp nqprlu prsrlt
|
|
nnnq syl 3bitrd ltresr syl6bbr ) BEFZCEFZUBZBCUCGZDHZBIJKLZMGDNVEAHZMGANJ
|
|
ZOPQOJRLZVDCIJKLZMGDNVIVFMGANJZOPQOJRLZUDGZVHSJVKSJUEGVBVCVEVIMGZVGVJUFGZ
|
|
VLBCUAVBVETFZVITFZVMVNUGUTVOVABUOUHZVAVPUTCUOUIZAVEVIDUJUKVBVGULFZVJULFZV
|
|
NVLUGVBVOVSVQAVEDUMUPVBVPVTVRAVIDUMUPVGVJUNUKUQVHVKURUS $.
|
|
$}
|
|
|
|
${
|
|
$d J l $. $d J u $. $d K l $. $d K u $.
|
|
|
|
$( Ordering of natural numbers with ` <N ` or ` <RR ` . (Contributed by
|
|
Jim Kingdon, 12-Jul-2021.) $)
|
|
ltrenn $p |- ( J <N K -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
|
|
{ u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
<RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } ,
|
|
{ u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
) $=
|
|
( cnpi wcel clti wbr cv c1o cop ceq cec cltq cab c1p cpp co cer c0r cltrr
|
|
wa ltrelpi brel ltrennb biimpd mpcom ) BEFCEFUBZBCGHZDIZBJKLMZNHDOUKAIZNH
|
|
AOKPQRPKSMTKUJCJKLMZNHDOUMULNHAOKPQRPKSMTKUAHZBCEEGUCUDUHUIUNABCDUEUFUG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d N l u $.
|
|
$( Another way of saying that a number times its reciprocal is one.
|
|
(Contributed by Jim Kingdon, 17-Jul-2021.) $)
|
|
recidpipr $p |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. .P.
|
|
<. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P ) $=
|
|
( cnpi wcel cv c1o cop ceq cec cltq wbr cab c1p cnq wceq syl c1q abbidv
|
|
co crq cfv cmq nnnq recclnq mulnqpr syl2anc recidnq breq2d breq1d opeq12d
|
|
cmp df-i1p syl6eqr eqtr3d ) BDEZCFZBGHIJZURUAUBZUCTZKLZCMZUTAFZKLZAMZHZUQ
|
|
URKLCMURVCKLAMHUQUSKLCMUSVCKLAMHULTZNUPUROEZUSOEZVFVGPBUDZUPVHVIVJURUEQAU
|
|
RUSCUFUGUPVFUQRKLZCMZRVCKLZAMZHNUPVBVLVEVNUPVAVKCUPUTRUQKUPVHUTRPVJURUHQZ
|
|
UISUPVDVMAUPUTRVCKVOUJSUKACUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
recidpirqlemcalc.a $e |- ( ph -> A e. P. ) $.
|
|
recidpirqlemcalc.b $e |- ( ph -> B e. P. ) $.
|
|
recidpirqlemcalc.rec $e |- ( ph -> ( A .P. B ) = 1P ) $.
|
|
$( Lemma for ~ recidpirq . Rearranging some of the expressions.
|
|
(Contributed by Jim Kingdon, 17-Jul-2021.) $)
|
|
recidpirqlemcalc $p |- ( ph ->
|
|
( ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) +P. 1P ) =
|
|
( ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) +P.
|
|
( 1P +P. 1P ) ) ) $=
|
|
( c1p cpp co cmp cnp wcel wceq addclpr syl2anc syl3anc 1idpr eqtrd oveq1d
|
|
syl 1pr addassprg distrprg oveq2d mulcomprg oveq12d 3eqtrd mp1i addcomprg
|
|
a1i 3eqtr4d ) ABGHIZCGHIZHIZGHIZGHIZUNGGHIZHIZULUMJIZGGJIZHIZGHIULGJIZGUM
|
|
JIZHIZUQHIAUNKLZGKLZVFUPURMAULKLZUMKLZVEABKLZVFVGDVFAUAUJZBGNOZACKLZVFVHE
|
|
VJCGNOZULUMNOVJVJUNGGUBPAVAUOGHAVAGCHIZULHIZGHIUOAUSVOUTGHAUSULCJIZVBHIZV
|
|
PULHIVOAVGVLVFUSVQMVKEVJULCGUCPAVBULVPHAVGVBULMVKULQTZUDAVPVNULHAVPCULJIZ
|
|
CBJIZCGJIZHIZVNAVGVLVPVSMVKEULCUEOAVLVIVFVSWBMEDVJCBGUCPAVTGWACHAVTBCJIZG
|
|
AVLVIVTWCMEDCBUEOFRAVLWACMECQTUFUGSUGVFUTGMAUAGQUHUFAVOUNGHAVOUMULHIZUNAV
|
|
NUMULHAVFVLVNUMMVJEGCUIOSAVHVGWDUNMVMVKUMULUIORSRSAVDUNUQHAVBULVCUMHVRAVC
|
|
UMGJIZUMAVFVHVCWEMVJVMGUMUEOAVHWEUMMVMUMQTRUFSUK $.
|
|
$}
|
|
|
|
${
|
|
$d N l u $.
|
|
$( A real number times its reciprocal is one, where reciprocal is expressed
|
|
with ` *Q ` . (Contributed by Jim Kingdon, 15-Jul-2021.) $)
|
|
recidpirq $p |- ( N e. N. -> (
|
|
<. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
x. <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >.
|
|
] ~R , 0R >. ) = 1 ) $=
|
|
( wcel cop cec cltq wbr cab c1p cpp cer c0r wceq cnp syl2anc c1r addclpr
|
|
co cmp cnpi cv c1o ceq crq cfv cmul cmr cnr nnprlu prsrcl recnnpr mulresr
|
|
c1 syl 1pr a1i mulsrpr syl22anc recidpipr recidpirqlemcalc eqeq2i mulclpr
|
|
df-1r wb wa pm3.2i mp1i enreceq syl5bb mpbird eqtrd opeq1d df-1 syl6eqr )
|
|
BUADZCUBZBUCEUDFZGHCIVRAUBZGHAIEZJKSZJELFZMEVQVRUEUFZGHCIWCVSGHAIEZJKSZJE
|
|
LFZMEUGSZWBWFUHSZMEZUNVPWBUIDZWFUIDZWGWINVPVTODZWJABCUJZVTUKUOVPWDODZWKAB
|
|
CULZWDUKUOWBWFUMPVPWIQMEUNVPWHQMVPWHWAWETSZJJTSZKSZWAJTSZJWETSZKSZELFZQVP
|
|
WAODZJODZWEODZXDWHXBNVPWLXDXCWMXDVPUPUQZVTJRPZXFVPWNXDXEWOXFWDJRPZXFWAJWE
|
|
JURUSVPXBQNZWRJKSXAJJKSZKSNZVPVTWDWMWOABCUTVAXIXBXJJELFZNZVPXKQXLXBVDVBVP
|
|
WRODZXAODZXJODZXDXMXKVEVPWPODZWQODZXNVPXCXEXQXGXHWAWEVCPXDXDVFZXRVPXDXDUP
|
|
UPVGZJJVCVHWPWQRPVPWSODZWTODZXOVPXCXDYAXGXFWAJVCPVPXDXEYBXFXHJWEVCPWSWTRP
|
|
XSXPVPXTJJRVHXFWRXAXJJVIUSVJVKVLVMVNVOVL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Final derivation of real and complex number postulates
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The complex numbers form a set. Use ~ cnex instead. (Contributed by
|
|
Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) $)
|
|
axcnex $p |- CC e. _V $=
|
|
( cc cnr cxp cvv df-c cnp cer cqs df-nr cpw npex xpex pwex wss wtru wer a1i
|
|
enrer qsss eqeltri trud ssexi ) ABBCDEBBBFFCZGHZDIUDUCJZUCFFKKLMUDUENOUCGUC
|
|
GPORQSUAUBTZUFLT $.
|
|
|
|
$( The real numbers are a subset of the complex numbers. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly; instead, use ~ ax-resscn .
|
|
(Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon,
|
|
12-Aug-2011.) (New usage is discouraged.) $)
|
|
axresscn $p |- RR C_ CC $=
|
|
( cnr c0r csn cxp cr cc wcel wss 0r snssi xpss2 mp2b df-r df-c 3sstr4i ) AB
|
|
CZDZAADZEFBAGPAHQRHIBAJPAAKLMNO $.
|
|
|
|
$( 1 is a complex number. Axiom for real and complex numbers, derived from
|
|
set theory. This construction-dependent theorem should not be referenced
|
|
directly; instead, use ~ ax-1cn . (Contributed by NM, 12-Apr-2007.)
|
|
(New usage is discouraged.) $)
|
|
ax1cn $p |- 1 e. CC $=
|
|
( cr cc c1 axresscn c1r c0r cop df-1 wcel cnr opelreal mpbir eqeltri sselii
|
|
1sr ) ABCDCEFGZAHPAIEJIOEKLMN $.
|
|
|
|
$( 1 is a real number. Axiom for real and complex numbers, derived from set
|
|
theory. This construction-dependent theorem should not be referenced
|
|
directly; instead, use ~ ax-1re .
|
|
|
|
In the Metamath Proof Explorer, this is not a complex number axiom but is
|
|
proved from ~ ax-1cn and the other axioms. It is not known whether we can
|
|
do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020)
|
|
uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
ax1re $p |- 1 e. RR $=
|
|
( c1 c1r c0r cop cr df-1 wcel cnr 1sr opelreal mpbir eqeltri ) ABCDZEFMEGBH
|
|
GIBJKL $.
|
|
|
|
$( ` _i ` is a complex number. Axiom for real and complex numbers, derived
|
|
from set theory. This construction-dependent theorem should not be
|
|
referenced directly; instead, use ~ ax-icn . (Contributed by NM,
|
|
23-Feb-1996.) (New usage is discouraged.) $)
|
|
axicn $p |- _i e. CC $=
|
|
( ci cc wcel c0r cnr c1r 0r 1sr cop wa df-i eleq1i opelcn bitri mpbir2an )
|
|
ABCZDECZFECZGHPDFIZBCQRJASBKLDFMNO $.
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $.
|
|
$( Closure law for addition of complex numbers. Axiom for real and complex
|
|
numbers, derived from set theory. This construction-dependent theorem
|
|
should not be referenced directly, nor should the proven axiom
|
|
~ ax-addcl be used later. Instead, in most cases use ~ addcl .
|
|
(Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) $)
|
|
axaddcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $=
|
|
( vx vy vz vw cc wcel wa cv cop wceq cnr wex caddc co elxpi df-c ad2ant2l
|
|
eleq2s cxp anim12i ee4anv sylibr cplr simpll simprl oveq12d addcnsr eqtrd
|
|
addclsr ad2ant2r opelxpi syl2anc syl6eleqr eqeltrd exlimivv syl ) AGHZBGH
|
|
ZIZACJZDJZKZLZVBMHZVCMHZIZIZBEJZFJZKZLZVJMHZVKMHZIZIZIZFNENZDNCNZABOPZGHZ
|
|
VAVIDNCNZVQFNENZIVTUSWCUTWDWCAMMUAZGCDAMMQRTWDBWEGEFBMMQRTUBVIVQCDEFUCUDV
|
|
SWBCDVRWBEFVRWAVBVJUEPZVCVKUEPZKZGVRWAVDVLOPZWHVRAVDBVLOVEVHVQUFVIVMVPUGU
|
|
HVHVPWIWHLVEVMVBVCVJVKUISUJVHVPWHGHVEVMVHVPIZWHWEGWJWFMHZWGMHZWHWEHVFVNWK
|
|
VGVOVBVJUKULVGVOWLVFVNVCVKUKSWFWGMMUMUNRUOSUPUQUQUR $.
|
|
|
|
$( Closure law for addition in the real subfield of complex numbers. Axiom
|
|
for real and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly, nor
|
|
should the proven axiom ~ ax-addrcl be used later. Instead, in most
|
|
cases use ~ readdcl . (Contributed by NM, 31-Mar-1996.)
|
|
(New usage is discouraged.) $)
|
|
axaddrcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $=
|
|
( vx vy cv c0r cop caddc co cr wcel cnr elreal wceq oveq1 eleq1d oveq2 wa
|
|
cplr addresr addclsr opelreal sylibr eqeltrd 2gencl ) CEZFGZDEZFGZHIZJKAU
|
|
IHIZJKABHIZJKCDUGUIABLJCAMDBMUGANUJUKJUGAUIHOPUIBNUKULJUIBAHQPUFLKUHLKRZU
|
|
JUFUHSIZFGZJUFUHTUMUNLKUOJKUFUHUAUNUBUCUDUE $.
|
|
|
|
$( Closure law for multiplication of complex numbers. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly, nor should the proven axiom
|
|
~ ax-mulcl be used later. Instead, in most cases use ~ mulcl .
|
|
(Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) $)
|
|
axmulcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $=
|
|
( vx vy vz vw cc wcel wa cv cop wceq cnr wex cmul co df-c mulclsr syl2anc
|
|
cmr elxpi eleq2s anim12i ee4anv sylibr cm1r simpll simprl oveq12d mulcnsr
|
|
cxp cplr ad2ant2l simplrl simprrl m1r a1i simplrr simprrr addclsr opelxpi
|
|
eqtrd syl6eleqr eqeltrd exlimivv syl ) AGHZBGHZIZACJZDJZKZLZVJMHZVKMHZIZI
|
|
ZBEJZFJZKZLZVRMHZVSMHZIZIZIZFNENZDNCNZABOPZGHZVIVQDNCNZWEFNENZIWHVGWKVHWL
|
|
WKAMMUKZGCDAMMUAQUBWLBWMGEFBMMUAQUBUCVQWECDEFUDUEWGWJCDWFWJEFWFWIVJVRTPZU
|
|
FVKVSTPZTPZULPZVKVRTPZVJVSTPZULPZKZGWFWIVLVTOPZXAWFAVLBVTOVMVPWEUGVQWAWDU
|
|
HUIVPWDXBXALVMWAVJVKVRVSUJUMVBWFXAWMGWFWQMHZWTMHZXAWMHWFWNMHZWPMHZXCWFVNW
|
|
BXEVMVNVOWEUNZVQWAWBWCUOZVJVRRSWFUFMHZWOMHZXFXIWFUPUQWFVOWCXJVMVNVOWEURZV
|
|
QWAWBWCUSZVKVSRSUFWORSWNWPUTSWFWRMHZWSMHZXDWFVOWBXMXKXHVKVRRSWFVNWCXNXGXL
|
|
VJVSRSWRWSUTSWQWTMMVASQVCVDVEVEVF $.
|
|
|
|
$( Closure law for multiplication in the real subfield of complex numbers.
|
|
Axiom for real and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly, nor
|
|
should the proven axiom ~ ax-mulrcl be used later. Instead, in most
|
|
cases use ~ remulcl . (New usage is discouraged.) (Contributed by NM,
|
|
31-Mar-1996.) $)
|
|
axmulrcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $=
|
|
( vx vy cv c0r cop cmul co cr wcel cnr elreal wceq oveq1 eleq1d oveq2 cmr
|
|
wa mulresr mulclsr opelreal sylibr eqeltrd 2gencl ) CEZFGZDEZFGZHIZJKAUIH
|
|
IZJKABHIZJKCDUGUIABLJCAMDBMUGANUJUKJUGAUIHOPUIBNUKULJUIBAHQPUFLKUHLKSZUJU
|
|
FUHRIZFGZJUFUHTUMUNLKUOJKUFUHUAUNUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w z $.
|
|
$( Addition commutes. Axiom for real and complex numbers, derived from set
|
|
theory. This construction-dependent theorem should not be referenced
|
|
directly, nor should the proven axiom ~ ax-addcom be used later.
|
|
Instead, use ~ addcom .
|
|
|
|
In the Metamath Proof Explorer this is not a complex number axiom but is
|
|
instead proved from other axioms. That proof relies on real number
|
|
trichotomy and it is not known whether it is possible to prove this from
|
|
the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
axaddcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $=
|
|
( vx vy vz vw cv cop caddc co wceq cnr oveq1 oveq2 eqeq12d wcel addcomsrg
|
|
wa cplr addcnsr cc df-c ad2ant2r ad2ant2l opeq12d ancoms 3eqtr4d 2optocl
|
|
) CGZDGZHZEGZFGZHZIJZUNUKIJZKAUNIJZUNAIJZKABIJZBAIJZKCDEFABLLUAUBUKAKUOUQ
|
|
UPURUKAUNIMUKAUNINOUNBKUQUSURUTUNBAINUNBAIMOUILPZUJLPZRZULLPZUMLPZRZRZUIU
|
|
LSJZUJUMSJZHULUISJZUMUJSJZHZUOUPVGVHVJVIVKVAVDVHVJKVBVEUIULQUCVBVEVIVKKVA
|
|
VDUJUMQUDUEUIUJULUMTVFVCUPVLKULUMUIUJTUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( Multiplication of complex numbers is commutative. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly, nor should the proven axiom
|
|
~ ax-mulcom be used later. Instead, use ~ mulcom . (Contributed by NM,
|
|
31-Aug-1995.) (New usage is discouraged.) $)
|
|
axmulcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $=
|
|
( vx vy vz vw cv cmr co cm1r cplr cnr mulcnsrec wcel wa mulcomsrg syl2anc
|
|
wceq oveq12d mulclsr cc cmul dfcnqs simpll simprl simplr simprr addcomsrg
|
|
cep ccnv oveq2d eqtrd ecovicom ) CDEFABUACGZEGZHIZJDGZFGZHIZHIZKIUBUIUJLU
|
|
QUOHIZUNURHIZKIZUOUNHIZJURUQHIZHIZKIURUNHIZUOUQHIZKIZUCUNUQUOURMUOURUNUQM
|
|
UNLNZUQLNZOZUOLNZURLNZOZOZUPVDUTVFKVPVJVMUPVDRVJVKVOUDZVLVMVNUEZUNUOPQVPU
|
|
SVEJHVPVKVNUSVERVJVKVOUFZVLVMVNUGZUQURPQUKSVPVCVHVGKIZVIVPVAVHVBVGKVPVKVM
|
|
VAVHRVSVRUQUOPQVPVJVNVBVGRVQVTUNURPQSVPVHLNZVGLNZWAVIRVPVMVKWBVRVSUOUQTQV
|
|
PVNVJWCVTVQURUNTQVHVGUHQULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$( Addition of complex numbers is associative. This theorem transfers the
|
|
associative laws for the real and imaginary signed real components of
|
|
complex number pairs, to complex number addition itself. Axiom for real
|
|
and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly, nor
|
|
should the proven axiom ~ ax-addass be used later. Instead, use
|
|
~ addass . (Contributed by NM, 2-Sep-1995.)
|
|
(New usage is discouraged.) $)
|
|
axaddass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
|
|
( vx vy vz vw vv vu cv cplr co cnr addcnsrec wcel wa addclsr anim12i an4s
|
|
wceq cc caddc ccnv addasssrg 3adant3r 3adant2r 3adant1r 3adant3l 3adant2l
|
|
cep dfcnqs 3adant1l ecoviass ) DEFGHIABCUAUBGJZIJZKLZUJUCMDJZFJZKLZEJZUNK
|
|
LZUSHJZKLZVAUOKLZUQURVBKLZKLZUTUPKLZVEUKUQUTURUNNURUNVBUONUSVAVBUONUQUTVE
|
|
UPNUQMOZURMOZUTMOZUNMOZUSMOZVAMOZPVHVIPVLVJVKPVMUQURQUTUNQRSVIVBMOZVKUOMO
|
|
ZVEMOZUPMOZPVIVNPVPVKVOPVQURVBQUNUOQRSVHVIVKPZVNVOPZVCVFTZVJVHVIVSVTVKVHV
|
|
IVNVTVOUQURVBUDUEUFUGVJVRVSVDVGTZVHVJVKVSWAVIVJVKVOWAVNUTUNUOUDUHUIULUM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u f g h $.
|
|
$( Multiplication of complex numbers is associative. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly; instead, use ~ ax-mulass .
|
|
(Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) $)
|
|
axmulass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
|
|
( vf vg vh cv cmr cplr cnr cm1r wcel mulclsr syl2anc wceq adantl distrsrg
|
|
co wa syl3anc vx vy vz vw vv vu cc cmul cep ccnv dfcnqs mulcnsrec sylancr
|
|
m1r addclsr syl2an an4s syl2anr an42s jca w3a simp1l simp2l simp3l simp2r
|
|
simp3r simp1r addcomsrg addasssrg caov42d a1i eqtrd mulcomsrg 3coml simp3
|
|
oveq2d oveq12d 3adant3 simp1 3eqtr3d mulasssrg caovdilemd caovcld caov12d
|
|
simp2 3eqtrd 3eqtr4rd ecoviass ) UAUBUCUDUEUFABCUGUHUDGZUEGZHRZUCGZUFGZHR
|
|
ZIRZUIUJJUAGZWLHRZKUBGZWIHRZHRZIRZWRWLHRZWPWIHRZIRZXAWJHRZKXDWMHRZHRZIRZX
|
|
DWJHRZXAWMHRZIRZWPWLWJHRZKWIWMHRZHRZIRZHRZKWRWOHRZHRZIRZWRXOHRZWPWOHRZIRZ
|
|
XOUKWPWRWLWIULWLWIWJWMULXAXDWJWMULWPWRXOWOULWPJLZWRJLZSZWLJLZWIJLZSZSXAJL
|
|
ZXDJLZYCYFYDYGYIYCYFSWQJLWTJLZYIYDYGSZWPWLMYLKJLZWSJLZYKUNWRWIMZKWSMUMWQW
|
|
TUOUPUQYCYGYDYFYJYDYFSXBJLXCJLYJYCYGSWRWLMWPWIMXBXCUOURUSUTYHWJJLZWMJLZSZ
|
|
SXOJLZWOJLZYFYPYGYQYSYFYPSXLJLZXNJLZYSYGYQSZWLWJMZUUCYMXMJLZUUBUNWIWMMZKX
|
|
MMZUMXLXNUOUPUQYFYQYGYPYTYGYPSWKJLZWNJLZYTYFYQSWIWJMZWLWMMZWKWNUOURUSUTYE
|
|
YHYRVAZWPXLHRZWPXNHRZIRZKWRWKHRZHRZKWRWNHRZHRZIRZIRUUMUUQIRZUUSUUNIRZIRXS
|
|
XHUULDEFUUMUUNUUQUUSJIUULYCUUAUUMJLYCYDYHYRVBZUULYFYPUUAYEYFYGYRVCZYEYHYP
|
|
YQVDZUUDNZWPXLMNUULYCUUBUUNJLUVCUULYMUUEUUBUNUULYGYQUUEYEYFYGYRVEZYEYHYPY
|
|
QVFZUUFNZUUGUMZWPXNMNUULYMUUPJLZUUQJLUNUULYDUUHUVKYCYDYHYRVGZUULYGYPUUHUV
|
|
GUVEUUJNZWRWKMNZKUUPMUMDGZJLZEGZJLZSZUVOUVQIRZUVQUVOIROUULUVOUVQVHPZUVPUV
|
|
RFGZJLZVAZUVTUWBIRUVOUVQUWBIRIROUULUVOUVQUWBVIPZUULYMUURJLZUUSJLUNUULYDUU
|
|
IUWFUVLUULYFYQUUIUVDUVHUUKNZWRWNMNZKUURMUMUVSUVTJLZUULUVOUVQUOZPZVJUULXPU
|
|
UOXRUUTIUULYCUUAUUBXPUUOOUVCUVFUVJWPXLXNQTUULXRKUUPUURIRZHRZUUTUULXQUWLKH
|
|
UULYDUUHUUIXQUWLOUVLUVMUWGWRWKWNQTVPUULYMUVKUWFUWMUUTOYMUULUNVKZUVNUWHKUU
|
|
PUURQTVLVQUULXEUVAXGUVBIUULXEUUMKWSWJHRZHRZIRUVAUULDEFWPKWLWSJIHWJUVSUVOU
|
|
VQHRZUVQUVOHROUULUVOUVQVMPZUWDUVTUWBHRZUVOUWBHRZUVQUWBHRZIRZOUULUWDUWBUVT
|
|
HRZUWBUVOHRZUWBUVQHRZIRZUWSUXBUWCUVPUVRUXCUXFOUWBUVOUVQQVNUWDUWCUWIUXCUWS
|
|
OUVPUVRUWCVOZUVPUVRUWIUWCUWJVRUWBUVTVMNUWDUXDUWTUXEUXAIUWDUWCUVPUXDUWTOUX
|
|
GUVPUVRUWCVSUWBUVOVMNUWDUWCUVRUXEUXAOUXGUVPUVRUWCWEUWBUVQVMNVQVTPZUWDUWQU
|
|
WBHRUVOUXAHROUULUVOUVQUWBWAPZUVSUWQJLUULUVOUVQMPZUVCUWNUVDUULYDYGYNUVLUVG
|
|
YONZUVEWBUULUWPUUQUUMIUULUWOUUPKHUULYDYGYPUWOUUPOUVLUVGUVEWRWIWJWATVPVPVL
|
|
UULXGKUURWPXMHRZIRZHRZUUSKUXLHRZIRZUVBUULXFUXMKHUULDEFWRWPWLWIJIHWMUWRUXH
|
|
UXIUXJUVLUVCUVDUVGUVHWBVPUULYMUWFUXLJLUXNUXPOUWNUWHUULDEWPXMJJJHUXJUVCUVI
|
|
WCKUURUXLQTUULUXOUUNUUSIUULDEFKWPXMJHUWNUVCUVIUWRUXIWDVPWFVQWGUULWRXLHRZW
|
|
RXNHRZIRZWPWKHRZWPWNHRZIRZIRUXQUXTIRZUYAUXRIRZIRYBXKUULDEFUXQUXRUXTUYAJIU
|
|
ULDEWRXLJJJHUXJUVLUVFWCUULDEWRXNJJJHUXJUVLUVJWCUULDEWPWKJJJHUXJUVCUVMWCUW
|
|
AUWEUULDEWPWNJJJHUXJUVCUWGWCUWKVJUULXTUXSYAUYBIUULYDUUAUUBXTUXSOUVLUVFUVJ
|
|
WRXLXNQTUULYCUUHUUIYAUYBOUVCUVMUWGWPWKWNQTVQUULXIUYCXJUYDIUULDEFWRWPWLWIJ
|
|
IHWJUWRUXHUXIUXJUVLUVCUVDUVGUVEWBUULXJUYAKWSWMHRZHRZIRUYDUULDEFWPKWLWSJIH
|
|
WMUWRUXHUXIUXJUVCUWNUVDUXKUVHWBUULUYFUXRUYAIUULUYFKWRXMHRZHRUXRUULUYEUYGK
|
|
HUULYDYGYQUYEUYGOUVLUVGUVHWRWIWMWATVPUULDEFKWRXMJHUWNUVLUVIUWRUXIWDVLVPVL
|
|
VQWGWH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u f g h $.
|
|
$( Distributive law for complex numbers (left-distributivity). Axiom for
|
|
real and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly, nor
|
|
should the proven axiom ~ ax-distr be used later. Instead, use
|
|
~ adddi . (Contributed by NM, 2-Sep-1995.)
|
|
(New usage is discouraged.) $)
|
|
axdistr $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
|
|
( vf cnr cv cplr co cmr cm1r wcel wa addclsr mulclsr m1r sylancr distrsrg
|
|
wceq syl3anc syl2anc vx vy vz vw vv vu vg vh cc caddc ccnv cmul addcnsrec
|
|
cep dfcnqs mulcnsrec anim12i an4s syl2an syl2anr jca simp1l simp2l simp3l
|
|
an42s w3a simp1r simp2r simp3r oveq2d a1i oveq12d addcomsrg adantl caov4d
|
|
eqtrd addasssrg ecovidi ) UAUBUCUDUEUFABCUIUJUNUKEULUAFZUCFZUEFZGHZIHZJUB
|
|
FZUDFZUFFZGHZIHZIHZGHZWDWBIHZVSWGIHZGHZVSVTIHZJWDWEIHZIHZGHZVSWAIHZJWDWFI
|
|
HZIHZGHZGHZWDVTIHZVSWEIHZGHZWDWAIHZVSWFIHZGHZGHZWBWGWQXEXAXHUOVTWEWAWFUMV
|
|
SWDWBWGUPVSWDVTWEUPVSWDWAWFUPWQXEXAXHUMVTEKZWAEKZWEEKZWFEKZWBEKZWGEKZLXJX
|
|
KLXNXLXMLXOVTWAMWEWFMUQURVSEKZWDEKZLZXJXLLZLWQEKZXEEKZXPXJXQXLXTXPXJLWNEK
|
|
ZWPEKZXTXQXLLZVSVTNZYDJEKZWOEKZYCOWDWENZJWONZPWNWPMUSURXPXLXQXJYAXQXJLXCE
|
|
KZXDEKZYAXPXLLWDVTNZVSWENZXCXDMUTVEVAXRXKXMLZLXAEKZXHEKZXPXKXQXMYOXPXKLWR
|
|
EKZWTEKZYOXQXMLZVSWANZYSYFWSEKZYROWDWFNZJWSNZPWRWTMUSURXPXMXQXKYPXQXKLXFE
|
|
KZXGEKZYPXPXMLWDWANZVSWFNZXFXGMUTVEVAXRXSYNVFZWJWNWRGHZWPWTGHZGHXBUUHWCUU
|
|
IWIUUJGUUHXPXJXKWCUUIRXPXQXSYNVBZXRXJXLYNVCZXRXSXKXMVDZVSVTWAQSUUHWIJWOWS
|
|
GHZIHZUUJUUHWHUUNJIUUHXQXLXMWHUUNRXPXQXSYNVGZXRXJXLYNVHZXRXSXKXMVIZWDWEWF
|
|
QSVJUUHYFYGUUAUUOUUJRYFUUHOVKUUHXQXLYGUUPUUQYHTZUUHXQXMUUAUUPUURUUBTZJWOW
|
|
SQSVPVLUUHDUGUHWNWRWPWTEGUUHXPXJYBUUKUULYETUUHXPXKYQUUKUUMYTTUUHYFYGYCOUU
|
|
SYIPDFZEKZUGFZEKZLZUVAUVCGHZUVCUVAGHRUUHUVAUVCVMVNZUVBUVDUHFZEKVFUVFUVHGH
|
|
UVAUVCUVHGHGHRUUHUVAUVCUVHVQVNZUUHYFUUAYROUUTUUCPUVEUVFEKUUHUVAUVCMVNZVOV
|
|
PUUHWMXCXFGHZXDXGGHZGHXIUUHWKUVKWLUVLGUUHXQXJXKWKUVKRUUPUULUUMWDVTWAQSUUH
|
|
XPXLXMWLUVLRUUKUUQUURVSWEWFQSVLUUHDUGUHXCXFXDXGEGUUHXQXJYJUUPUULYLTUUHXQX
|
|
KUUDUUPUUMUUFTUUHXPXLYKUUKUUQYMTUVGUVIUUHXPXMUUEUUKUURUUGTUVJVOVPVR $.
|
|
$}
|
|
|
|
$( i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real
|
|
and complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly; instead, use ~ ax-i2m1 .
|
|
(Contributed by NM, 5-May-1996.) (New usage is discouraged.) $)
|
|
axi2m1 $p |- ( ( _i x. _i ) + 1 ) = 0 $=
|
|
( c0r c1r cop cmul co caddc ci cm1r cplr cmr cnr wcel wceq 0r 1sr ax-mp m1r
|
|
1idsr eqtri oveq12i c1 cc0 mulcnsr mp4an 00sr oveq2i addcomsrg mp2an 3eqtri
|
|
0idsr opeq12i oveq1i addresr m1p1sr opeq1i df-i df-1 df-0 3eqtr4i ) ABCZUTD
|
|
EZBACZFEZAACZGGDEZUAFEUBVCHACZVBFEZHBIEZACZVDVAVFVBFVAAAJEZHBBJEZJEZIEZBAJE
|
|
ZABJEZIEZCZVFAKLZBKLZVRVSVAVQMNONOABABUCUDVMHVPAVMAHIEZHAIEZHVJAVLHIVRVJAMN
|
|
AUEPVLHBJEZHVKBHJVSVKBMOBRPUFHKLZWBHMQHRPSTVRWCVTWAMNQAHUGUHWCWAHMQHUJPUIVP
|
|
AAIEZAVNAVOAIVSVNAMOBUEPVRVOAMNARPTVRWDAMNAUJPSUKSULWCVSVGVIMQOHBUMUHVHAAUN
|
|
UOUIVEVAUAVBFGUTGUTDUPUPTUQTURUS $.
|
|
|
|
$( 0 is less than 1. Axiom for real and complex numbers, derived from set
|
|
theory. This construction-dependent theorem should not be referenced
|
|
directly; instead, use ~ ax-0lt1 .
|
|
|
|
The version of this axiom in the Metamath Proof Explorer reads
|
|
` 1 =/= 0 ` ; here we change it to ` 0 <RR 1 ` . The proof of ` 0 <RR 1 `
|
|
from ` 1 =/= 0 ` in the Metamath Proof Explorer (accessed 12-Jan-2020)
|
|
relies on real number trichotomy. (Contributed by Jim Kingdon,
|
|
12-Jan-2020.) (New usage is discouraged.) $)
|
|
ax0lt1 $p |- 0 <RR 1 $=
|
|
( c0r cop c1r cc0 c1 cltrr wbr cltr 0lt1sr ltresr mpbir df-0 df-1 3brtr4i )
|
|
AABZCABZDEFOPFGACHGIACJKLMN $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( ` 1 ` is an identity element for real multiplication. Axiom for real
|
|
and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly;
|
|
instead, use ~ ax-1rid . (Contributed by Scott Fenton, 3-Jan-2013.)
|
|
(New usage is discouraged.) $)
|
|
ax1rid $p |- ( A e. RR -> ( A x. 1 ) = A ) $=
|
|
( vx vy cv cop c1 cmul co wceq cnr c0r csn cr df-r oveq1 id eqeq12d elsni
|
|
wcel c1r df-1 oveq2i cmr 1sr mpan2 1idsr opeq1d eqtrd syl5eq opeq2 oveq1d
|
|
mulresr syl5ibr impcom sylan2 optocl ) BDZCDZEZFGHZUSIZAFGHZAIBCAJKLZMNUS
|
|
AIZUTVBUSAUSAFGOVDPQURVCSUQJSZURKIZVAURKRVFVEVAVEVAVFUQKEZFGHZVGIVEVHVGTK
|
|
EZGHZVGFVIVGGUAUBVEVJUQTUCHZKEZVGVETJSVJVLIUDUQTULUEVEVKUQKUQUFUGUHUIVFUT
|
|
VHUSVGVFUSVGFGURKUQUJZUKVMQUMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( ` 0 ` is an identity element for real addition. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly; instead, use ~ ax-0id .
|
|
|
|
In the Metamath Proof Explorer this is not a complex number axiom but is
|
|
instead proved from other axioms. That proof relies on excluded middle
|
|
and it is not known whether it is possible to prove this from the other
|
|
axioms without excluded middle. (Contributed by Jim Kingdon,
|
|
16-Jan-2020.) (New usage is discouraged.) $)
|
|
ax0id $p |- ( A e. CC -> ( A + 0 ) = A ) $=
|
|
( vx vy cv cop cc0 caddc co wceq cnr cc df-c oveq1 id eqeq12d wcel c0r 0r
|
|
cplr 0idsr wa addcnsr mpanr12 eqcomi oveq2d adantr adantl opeq12d 3eqtr3d
|
|
df-0 a1i optocl ) BDZCDZEZFGHZUOIAFGHZAIBCAJJKLUOAIZUPUQUOAUOAFGMURNOUMJP
|
|
ZUNJPZUAZUOQQEZGHZUMQSHZUNQSHZEZUPUOVAQJPZVGVCVFIRRUMUNQQUBUCVAVBFUOGVBFI
|
|
VAFVBUJUDUKUEVAVDUMVEUNUSVDUMIUTUMTUFUTVEUNIUSUNTUGUHUIUL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Existence of negative of real number. Axiom for real and complex
|
|
numbers, derived from set theory. This construction-dependent theorem
|
|
should not be referenced directly; instead, use ~ ax-rnegex .
|
|
(Contributed by NM, 15-May-1996.) (New usage is discouraged.) $)
|
|
axrnegex $p |- ( A e. RR -> E. x e. RR ( A + x ) = 0 ) $=
|
|
( cr wcel c1st cfv cm1r cmr co c0r cop caddc cc0 wceq cv wrex cnr elreal2
|
|
simplbi syl2anc mulclsr sylancl opelreal sylibr cplr simprbi oveq1d pn0sr
|
|
m1r addresr opeq1d df-0 syl6eqr syl 3eqtrd oveq2 eqeq1d rspcev ) BCDZBEFZ
|
|
GHIZJKZCDZBVBLIZMNZBAOZLIZMNZACPUSVAQDZVCUSUTQDZGQDVIUSVJBUTJKZNZBRZSZUIU
|
|
TGUAUBZVAUCUDUSVDVKVBLIZUTVAUEIZJKZMUSBVKVBLUSVJVLVMUFUGUSVJVIVPVRNVNVOUT
|
|
VAUJTUSVJVRMNVNVJVRJJKMVJVQJJUTUHUKULUMUNUOVHVEAVBCVFVBNVGVDMVFVBBLUPUQUR
|
|
T $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y z $.
|
|
$( Existence of positive reciprocal of positive real number. Axiom for
|
|
real and complex numbers, derived from set theory. This
|
|
construction-dependent theorem should not be referenced directly;
|
|
instead, use ~ ax-precex .
|
|
|
|
In treatments which assume excluded middle, the ` 0 <RR A ` condition is
|
|
generally replaced by ` A =/= 0 ` , and it may not be necessary to state
|
|
that the reciproacal is positive. (Contributed by Jim Kingdon,
|
|
6-Feb-2020.) (New usage is discouraged.) $)
|
|
axprecex $p |- ( ( A e. RR /\ 0 <RR A ) ->
|
|
E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) ) $=
|
|
( vy vz cr wcel cc0 cltrr wbr cmul co c1 wceq wrex c0r cop cnr bitri c1r
|
|
wa cv wi wex elreal df-rex breq2 oveq1 eqeq1d anbi2d rexbidv imbi12d cltr
|
|
df-0 breq1i ltresr cmr recexgt0sr opelreal anbi1i a1i mulresr df-1 eqeq2i
|
|
wb eqid 1sr 0r opthg2 mp2an mpbiran2 syl6bb anbi12d pm5.32da syl5bb oveq2
|
|
rspcev syl6bir expd rexlimdv syl5 syl5bi gencl imp ) BEFZGBHIZGAUAZHIZBWF
|
|
JKZLMZTZAENZGCUAZOPZHIZWGWMWFJKZLMZTZAENZUBWEWKUBWLQFZWDCWMBWDWMBMZCQNWSW
|
|
TTCUCCBUDWTCQUERWTWNWEWRWKWMBGHUFWTWQWJAEWTWPWIWGWTWOWHLWMBWFJUGUHUIUJUKW
|
|
NOWLULIZWSWRWNOOPZWMHIXAGXBWMHUMUNOWLUORXAODUAZULIZWLXCUPKZSMZTZDQNWSWRDW
|
|
LUQWSXGWRDQWSXCQFZXGWRWSXHXGTZXCOPZEFZGXJHIZWMXJJKZLMZTZTZWRXPXHXOTWSXIXK
|
|
XHXOXCURUSWSXHXOXGWSXHTZXLXDXNXFXLXDVDXQXLXBXJHIXDGXBXJHUMUNOXCUORUTXQXNX
|
|
EOPZLMZXFXQXMXRLWLXCVAUHXSXRSOPZMZXFLXTXRVBVCYAXFOOMZOVESQFOQFYAXFYBTVDVF
|
|
VGXEOSOQQVHVIVJRVKVLVMVNWQXOAXJEWFXJMZWGXLWPXNWFXJGHUFYCWOXMLWFXJWMJVOUHV
|
|
LVPVQVRVSVTWAWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $.
|
|
$( A complex number can be expressed in terms of two reals. Definition
|
|
10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers,
|
|
derived from set theory. This construction-dependent theorem should not
|
|
be referenced directly; instead, use ~ ax-cnre . (Contributed by NM,
|
|
13-May-1996.) (New usage is discouraged.) $)
|
|
axcnre $p |- ( A e. CC ->
|
|
E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
|
|
( cop cmul co caddc wceq cr cnr wcel wa c0r cplr c1r cmr cm1r eqtrd cvv
|
|
0r vz vw cv ci wrex df-c eqeq1 2rexbidv wex opelreal anbi12i biimpri df-i
|
|
cc oveq1i 1sr mulcnsr mpanl12 mpan2 mulcomsrg mpan 00sr oveq1d oveq2i m1r
|
|
ax-mp eqtri 0idsr syl6eq 1idsr syl5eq opeq12d oveq2d adantl mpanl2 mpanr1
|
|
addcnsr addcomsrg opeq12 syl2an 3eqtrrd vex opexg mp2an eleq1 oveq1 oveq2
|
|
bi2anan9 sylan9eq eqeq2d anbi12d spc2ev syl2anc r2ex sylibr optocl ) UAUC
|
|
ZUBUCZDZAUCZUDBUCZEFZGFZHZBIUEAIUEZCXCHZBIUEAIUEUAUBCJJUNUFWSCHXDXFABIIWS
|
|
CXCUGUHWQJKZWRJKZLZWTIKZXAIKZLZXDLZBUIAUIZXEXIWQMDZIKZWRMDZIKZLZWSXOUDXQE
|
|
FZGFZHZXNXSXIXPXGXRXHWQUJWRUJUKULXIYAXOMWRDZGFZWQMNFZMWRNFZDZWSXHYAYDHXGX
|
|
HXTYCXOGXHXTMODZXQEFZYCUDYHXQEUMUOXHYIMWRPFZQOMPFZPFZNFZOWRPFZMMPFZNFZDZY
|
|
CXHMJKZYIYQHZTYROJKZXHYRLYSTUPMOWRMUQURUSXHYMMYPWRXHYMMYLNFZMXHYJMYLNXHYJ
|
|
WRMPFZMYRXHYJUUBHTMWRUTVAWRVBRVCUUAMMNFZMYLMMNYLQMPFZMYKMQPYTYKMHUPOVBVFV
|
|
DQJKUUDMHVEQVBVFVGVDYRUUCMHTMVHVFVGVIXHYPWRYONFZWRXHYNWRYONXHYNWROPFZWRYT
|
|
XHYNUUFHUPOWRUTVAWRVJRVCXHUUEWRMNFZWRYOMWRNYRYOMHTMVBVFVDWRVHZVKRVLRVKVMV
|
|
NXGYRXHYDYGHZTXGYRYRXHLUUITWQMMWRVQVOVPXGYEWQHYFWRHYGWSHXHWQVHXHYFUUGWRYR
|
|
XHYFUUGHTMWRVRVAUUHRYEYFWQWRVSVTWAXMXSYBLABXOXQWQSKYRXOSKUAWBTWQMSJWCWDWR
|
|
SKYRXQSKUBWBTWRMSJWCWDWTXOHZXAXQHZLZXLXSXDYBUUJXJXPUUKXKXRWTXOIWEXAXQIWEW
|
|
HUULXCYAWSUUJUUKXCXOXBGFYAWTXOXBGWFUUKXBXTXOGXAXQUDEWGVMWIWJWKWLWMXDABIIW
|
|
NWOWP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Real number less-than is irreflexive. Axiom for real and complex
|
|
numbers, derived from set theory. This construction-dependent theorem
|
|
should not be referenced directly; instead, use ~ ax-pre-ltirr .
|
|
(Contributed by Jim Kingdon, 12-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
axpre-ltirr $p |- ( A e. RR -> -. A <RR A ) $=
|
|
( vx cv c0r cop cltrr wbr wn cnr wcel cr wceq wrex wa elreal df-rex bitri
|
|
wex id cltr breq12d notbid ltsosr ltrelsr soirri ltresr mtbir a1i gencl )
|
|
BCZDEZUKFGZHZAAFGZHUJIJZAKJZBUKAUPUKALZBIMUOUQNBRBAOUQBIPQUQULUNUQUKAUKAF
|
|
UQSZURUAUBUMUOULUJUJTGUJTIUCUDUEUJUJUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Real number less-than is weakly linear. Axiom for real and complex
|
|
numbers, derived from set theory. This construction-dependent theorem
|
|
should not be referenced directly; instead, use ~ ax-pre-ltwlin .
|
|
(Contributed by Jim Kingdon, 12-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
axpre-ltwlin $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <RR B -> ( A <RR C \/ C <RR B ) ) ) $=
|
|
( vx vy vz cv c0r cop cltrr wbr wo cnr elreal wceq breq1 breq2 wcel cltr
|
|
wi orbi1d imbi12d orbi2d orbi12d imbi2d w3a wor ltsosr sowlin mpan ltresr
|
|
cr orbi12i 3imtr4g 3gencl ) DGZHIZEGZHIZJKZUQFGZHIZJKZVBUSJKZLZTAUSJKZAVB
|
|
JKZVDLZTABJKZVGVBBJKZLZTVIACJKZCBJKZLZTDEFUQUSVBAMULBCDANEBNFCNUQAOZUTVFV
|
|
EVHUQAUSJPVOVCVGVDUQAVBJPUAUBUSBOZVFVIVHVKUSBAJQVPVDVJVGUSBVBJQUCUBVBCOZV
|
|
KVNVIVQVGVLVJVMVBCAJQVBCBJPUDUEUPMRURMRVAMRUFZUPURSKZUPVASKZVAURSKZLZUTVE
|
|
MSUGVRVSWBTUHMUPURVASUIUJUPURUKVCVTVDWAUPVAUKVAURUKUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $.
|
|
$( Ordering on reals is transitive. Axiom for real and complex numbers,
|
|
derived from set theory. This construction-dependent theorem should not
|
|
be referenced directly; instead, use ~ ax-pre-lttrn . (Contributed by
|
|
NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.)
|
|
(New usage is discouraged.) $)
|
|
axpre-lttrn $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A <RR B /\ B <RR C ) -> A <RR C ) ) $=
|
|
( vx vy vz cv c0r cop cltrr wbr wa cnr elreal wceq breq1 breq2 wcel cltr
|
|
wi anbi1d imbi12d anbi12d imbi1d anbi2d w3a ltresr ltsosr ltrelsr syl2anb
|
|
cr sotri sylibr a1i 3gencl ) DGZHIZEGZHIZJKZUSFGZHIZJKZLZUQVBJKZTZAUSJKZV
|
|
CLZAVBJKZTABJKZBVBJKZLZVITVJBCJKZLZACJKZTDEFUQUSVBAMUKBCDANEBNFCNUQAOZVDV
|
|
HVEVIVPUTVGVCUQAUSJPUAUQAVBJPUBUSBOZVHVLVIVQVGVJVCVKUSBAJQUSBVBJPUCUDVBCO
|
|
ZVLVNVIVOVRVKVMVJVBCBJQUEVBCAJQUBVFUPMRURMRVAMRUFVDUPVASKZVEUTUPURSKURVAS
|
|
KVSVCUPURUGURVAUGUPURVASMUHUIULUJUPVAUGUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Apartness of reals is tight. Axiom for real and complex numbers,
|
|
derived from set theory. This construction-dependent theorem should not
|
|
be referenced directly; instead, use ~ ax-pre-apti .
|
|
|
|
(Contributed by Jim Kingdon, 29-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
axpre-apti $p |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) ->
|
|
A = B ) $=
|
|
( vx vy cr wcel cltrr wbr wo wn wceq cv c0r cop wi cnr elreal breq1 breq2
|
|
orbi12d notbid eqeq1 imbi12d eqeq2 wa aptisr 3expia ltresr orbi12i notbii
|
|
cltr vex eqresr 3imtr4g 2gencl 3impia ) AEFBEFABGHZBAGHZIZJZABKZCLZMNZDLZ
|
|
MNZGHZVEVCGHZIZJZVCVEKZOAVEGHZVEAGHZIZJZAVEKZOUTVAOCDVCVEABPECAQDBQVCAKZV
|
|
IVNVJVOVPVHVMVPVFVKVGVLVCAVEGRVCAVEGSTUAVCAVEUBUCVEBKZVNUTVOVAVQVMUSVQVKU
|
|
QVLURVEBAGSVEBAGRTUAVEBAUDUCVBPFZVDPFZUEVBVDUKHZVDVBUKHZIZJZVBVDKZVIVJVRV
|
|
SWCWDVBVDUFUGVHWBVFVTVGWAVBVDUHVDVBUHUIUJVBVDCULUMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Ordering property of addition on reals. Axiom for real and complex
|
|
numbers, derived from set theory. This construction-dependent theorem
|
|
should not be referenced directly; instead, use ~ ax-pre-ltadd .
|
|
(Contributed by NM, 11-May-1996.) (New usage is discouraged.) $)
|
|
axpre-ltadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <RR B -> ( C + A ) <RR ( C + B ) ) ) $=
|
|
( vx vy vz cr wcel w3a cltrr wbr caddc co cv c0r cop wb cnr elreal wceq
|
|
breq1 oveq2 breq1d bibi12d breq2 breq2d breq12d bibi2d cltr ltasrg ltresr
|
|
oveq1 cplr simp3 simp1 simp2 wa addresr breqan12d syl22anc syl6bb 3bitr4d
|
|
a1i 3gencl biimpd ) AGHBGHCGHIABJKZCALMZCBLMZJKZDNZOPZENZOPZJKZFNZOPZVKLM
|
|
ZVPVMLMZJKZQAVMJKZVPALMZVRJKZQVFWAVPBLMZJKZQVFVIQDEFVKVMVPARGBCDASEBSFCSV
|
|
KATZVNVTVSWBVKAVMJUAWEVQWAVRJVKAVPLUBUCUDVMBTZVTVFWBWDVMBAJUEWFVRWCWAJVMB
|
|
VPLUBUFUDVPCTZWDVIVFWGWAVGWCVHJVPCALULVPCBLULUGUHVJRHZVLRHZVORHZIZVJVLUIK
|
|
ZVOVJUMMZVOVLUMMZUIKZVNVSVJVLVOUJVNWLQWKVJVLUKVCWKVSWMOPZWNOPZJKZWOWKWJWH
|
|
WJWIVSWRQWHWIWJUNZWHWIWJUOWSWHWIWJUPWJWHUQWJWIUQVQWPVRWQJVOVJURVOVLURUSUT
|
|
WMWNUKVAVBVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The product of two positive reals is positive. Axiom for real and
|
|
complex numbers, derived from set theory. This construction-dependent
|
|
theorem should not be referenced directly; instead, use
|
|
~ ax-pre-mulgt0 . (Contributed by NM, 13-May-1996.)
|
|
(New usage is discouraged.) $)
|
|
axpre-mulgt0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) ) $=
|
|
( vx vy cc0 cv c0r cop cltrr wbr wa cmul cnr elreal wceq cltr df-0 ltresr
|
|
co wi cr breq2 anbi1d oveq1 breq2d imbi12d anbi2d oveq2 wcel breq1i bitri
|
|
cmr mulgt0sr syl2anb a1i mulresr breq12d syl6bb syl5ibr 2gencl ) ECFZGHZI
|
|
JZEDFZGHZIJZKZEVBVELSZIJZTEAIJZVFKZEAVELSZIJZTVJEBIJZKZEABLSZIJZTCDVBVEAB
|
|
MUACANDBNVBAOZVGVKVIVMVRVCVJVFVBAEIUBUCVRVHVLEIVBAVELUDUEUFVEBOZVKVOVMVQV
|
|
SVFVNVJVEBEIUBUGVSVLVPEIVEBALUHUEUFVGVIVAMUIVDMUIKZGVAVDULSZPJZVCGVAPJZGV
|
|
DPJZWBVFVCGGHZVBIJWCEWEVBIQUJGVARUKVFWEVEIJWDEWEVEIQUJGVDRUKVAVDUMUNVTVIW
|
|
EWAGHZIJWBVTEWEVHWFIEWEOVTQUOVAVDUPUQGWARURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d z C $.
|
|
$( Strong extensionality of multiplication (expressed in terms of
|
|
` <RR ` ). Axiom for real and complex numbers, derived from set theory.
|
|
This construction-dependent theorem should not be referenced directly;
|
|
instead, use ~ ax-pre-mulext .
|
|
|
|
(Contributed by Jim Kingdon, 18-Feb-2020.)
|
|
(New usage is discouraged.) $)
|
|
axpre-mulext $p |- ( ( A e. RR /\ B e. RR /\ C e. RR )
|
|
-> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) ) $=
|
|
( vx vz vy cv c0r cop cmul co cltrr wbr wo wi cnr elreal wceq wcel cltr
|
|
cr oveq1 breq1d breq1 breq2 orbi12d imbi12d breq2d breq12d imbi1d w3a cmr
|
|
oveq2 mulextsr1 mulresr 3adant2 3adant1 ltresr syl6bb orbi12i a1i 3imtr4d
|
|
wb 3gencl ) DGZHIZEGZHIZJKZFGZHIZVHJKZLMZVFVKLMZVKVFLMZNZOAVHJKZVLLMZAVKL
|
|
MZVKALMZNZOVQBVHJKZLMZABLMZBALMZNZOACJKZBCJKZLMZWFODFEVFVKVHAPUABCDAQFBQE
|
|
CQVFARZVMVRVPWAWJVIVQVLLVFAVHJUBUCWJVNVSVOVTVFAVKLUDVFAVKLUEUFUGVKBRZVRWC
|
|
WAWFWKVLWBVQLVKBVHJUBUHWKVSWDVTWEVKBALUEVKBALUDUFUGVHCRZWCWIWFWLVQWGWBWHL
|
|
VHCAJUMVHCBJUMUIUJVEPSZVJPSZVGPSZUKZVEVGULKZVJVGULKZTMZVEVJTMZVJVETMZNZVM
|
|
VPVEVJVGUNWPVMWQHIZWRHIZLMWSWPVIXCVLXDLWMWOVIXCRWNVEVGUOUPWNWOVLXDRWMVJVG
|
|
UOUQUIWQWRURUSVPXBVCWPVNWTVOXAVEVJURVJVEURUTVAVBVD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( The reciprocal from ~ axprecex is unique. (Contributed by Jim Kingdon,
|
|
15-Jul-2021.) $)
|
|
rereceu $p |- ( ( A e. RR /\ 0 <RR A ) ->
|
|
E! x e. RR ( A x. x ) = 1 ) $=
|
|
( vy vz cr wcel cc0 cltrr wbr wa cv cmul co c1 wceq wrex adantr cc sseldi
|
|
axresscn wrmo wreu axprecex simpr reximi wi wral eqtr3 wb simpll axmulcom
|
|
syl simprl syl2anc simprr eqeq12d syl6bi axmulass syl3anc sylibd ad2antll
|
|
oveq1 oveq2 ax1rid sylan9eqr rexlimddv syl5 ralrimivva eqeq1d rmo4 sylibr
|
|
reu5 sylanbrc ) BEFZGBHIZJZBAKZLMZNOZAEPZVSAEUAZVSAEUBVPGVQHIZVSJZAEPVTAB
|
|
UCWCVSAEWBVSUDUEULVPVSBCKZLMZNOZJZVQWDOZUFZCEUGAEUGWAVPWIACEEWGVRWEOZVPVQ
|
|
EFZWDEFZJZJZWHVRWENUHWNGDKZHIZBWOLMZNOZJZWJWHUFDEVPWSDEPWMDBUCQWNWOEFZWSJ
|
|
ZJZWJVQWQLMZWDWQLMZOZWHXBWJVQBLMZWOLMZWDBLMZWOLMZOZXEXBWJXFXHOZXJWNWJXKUI
|
|
XAWNVRXFWEXHWNBRFZVQRFZVRXFOWNERBTVNVOWMUJSZWNERVQTVPWKWLUMZSZBVQUKUNWNXL
|
|
WDRFZWEXHOXNWNERWDTVPWKWLUOSZBWDUKUNUPQXFXHWOLVBUQXBXGXCXIXDXBXMXLWORFZXG
|
|
XCOWNXMXAXPQWNXLXAXNQZXBERWOTWNWTWSUMSZVQBWOURUSXBXQXLXSXIXDOWNXQXAXRQXTY
|
|
AWDBWOURUSUPUTXBXCVQXDWDXAWNXCVQNLMZVQWRXCYBOWTWPWQNVQLVCVAWNWKYBVQOXOVQV
|
|
DULVEXAWNXDWDNLMZWDWRXDYCOWTWPWQNWDLVCVAWLYCWDOVPWKWDVDVAVEUPUTVFVGVHVSWF
|
|
ACEWHVRWENVQWDBLVCVIVJVKVSAEVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d N l r u y $.
|
|
$( Two ways to express the reciprocal of a natural number. (Contributed by
|
|
Jim Kingdon, 11-Jul-2021.) $)
|
|
recriota $p |- ( N e. N. -> ( iota_ r e. RR ( <. [ <. ( <.
|
|
{ l | l <Q [ <. N , 1o >. ] ~Q } ,
|
|
{ u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R
|
|
>. x. r ) = 1 ) =
|
|
<. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >.
|
|
] ~R , 0R >. ) $=
|
|
( vy wcel cv wbr cop cltq cab c1p co cmul c1 wceq cr syl2anc adantr cc wa
|
|
cnpi cc0 cltrr c1o ceq cec cpp cer c0r crio crq cfv wrex pitore pitoregt0
|
|
axprecex simprrr wreu wb simprl oveq2 eqeq1d riota2 mpbid oveq2d axresscn
|
|
rereceu sseldi recnnre axmulcom recidpirq eqtr3d axmulass syl3anc sylancr
|
|
oveq1d ax1cn 3eqtr3d ax1rid syl eqtrd rexlimddv ) BUBFZUCEGZUDHZDGZBUEIUF
|
|
UGZJHDKWHAGZJHAKILUHMLIUIUGUJIZWENMZOPZUAZWJCGZNMZOPZCQUKZWGWHULUMZJHDKWR
|
|
WIJHAKILUHMLIUIUGUJIZPEQWDWJQFZUCWJUDHZWMEQUNABDUOZABDUPZEWJUQRWDWEQFZWMU
|
|
AZUAZWQWEWSXFWLWQWEPZWDXDWFWLURZXFXDWPCQUSZWLXGUTWDXDWMVAZXFWTXAXIWDWTXEX
|
|
BSZWDXAXEXCSCWJVHRWPWLCQWEWNWEPWOWKOWNWEWJNVBVCVDRVEXFWSWKNMZWSONMZWEWSXF
|
|
WKOWSNXHVFXFXLWEONMZWEXFWSWJNMZWENMZOWENMZXLXNXFXOOWENXFWJWSNMZXOOXFWJTFZ
|
|
WSTFZXRXOPXFQTWJVGXKVIZXFQTWSVGWDWSQFZXEABDVJSZVIZWJWSVKRWDXROPXEABDVLSVM
|
|
VQXFXTXSWETFZXPXLPYDYAXFQTWEVGXJVIZWSWJWEVNVOXFOTFYEXQXNPVRYFOWEVKVPVSXFX
|
|
DXNWEPXJWEVTWAWBXFYBXMWSPYCWSVTWAVSWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d A l n u w x y z $.
|
|
|
|
$( Archimedean axiom. The Archimedean property is more naturally stated
|
|
once we have defined ` NN ` . Unless we find another way to state it,
|
|
we'll just use the right hand side of ~ dfnn2 in stating what we mean by
|
|
"natural number" in the context of this axiom.
|
|
|
|
This construction-dependent theorem should not be referenced directly;
|
|
instead, use ~ ax-arch . (Contributed by Jim Kingdon, 22-Apr-2020.)
|
|
(New usage is discouraged.) $)
|
|
axarch $p |- ( A e. RR -> E. n e.
|
|
|^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n ) $=
|
|
( vz vl vw vu wcel cv c0r cop wceq cltrr wbr wa cab wrex cnr cnpi cr wral
|
|
c1 caddc co cint elreal biimpi c1o ceq cec cltq c1p cpp cer cltr ad2antrl
|
|
archsr simplrr simprr ltresr sylibr pitonn simpr breq2d rspcedv rexlimddv
|
|
eqbrtrrd mpd ) CUAIZEJZKLZCMZCDJZNOZDUCAJZIBJUCUDUEVPIBVPUBPAQUFZRZESVJVM
|
|
ESRECUGUHVJVKSIZVMPPZVKFJGJZUILUJUKZULOFQWBHJULOHQLUMUNUEUMLUOUKZUPOZVRGT
|
|
VSWDGTRVJVMGHVKFURUQVTWATIZWDPZPZCWCKLZNOZVRWGVLCWHNVJVSVMWFUSWGWDVLWHNOV
|
|
TWEWDUTVKWCVAVBVHWGVOWIDWHVQWEWHVQIVTWDABHWAFVCUQWGVNWHMZPVNWHCNWGWJVDVEV
|
|
FVIVGVG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
nntopi.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
|
|
${
|
|
$d A x y $. $d A y z $.
|
|
$( Peano's inductive postulate. This is a counterpart to ~ peano5nni
|
|
designed for real number axioms which involve natural numbers
|
|
(notably, ~ axcaucvg ). (Contributed by Jim Kingdon, 14-Jul-2021.)
|
|
(New usage is discouraged.) $)
|
|
peano5nnnn $p |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A ) $=
|
|
( cv c1 caddc co wcel wral wss wceq wa cr ax1re elin mpan2 wi oveq1 cin
|
|
eleq1d cbvralv inss1 ssralv ax-mp inss2 sseli axaddrcl simplbi2com 3syl
|
|
biimpri ralimia syl cvv cc axcnex axresscn ssexi inex2 eleq2 raleqbi1dv
|
|
cab anbi12d elabg cint intss1 syl5eqss syl6bir syl2an syl6ss sylan2br )
|
|
CGZHIJZDKZCDLHDKZBGZHIJZDKZBDLZEDMVTVPBCDVRVNNVSVODVRVNHIUAUCUDVQWAOEDP
|
|
UBZDVQHWBKZVSWBKZBWBLZEWBMZWAVQHPKZWCQWCVQWGOHDPRUMSWAVTBWBLZWEWBDMWAWH
|
|
TDPUEZVTBWBDUFUGVTWDBWBVRWBKVRPKZVSPKZVTWDTWBPVRDPUHUIWJWGWKQVRHUJSWDVT
|
|
WKVSDPRUKULUNUOWBUPKZWCWEOZWFTPDPUQURUSUTVAWLWMWBHAGZKZVSWNKZBWNLZOZAVD
|
|
ZKZWFWRWMAWBUPWNWBNWOWCWQWEWNWBHVBWPWDBWNWBWNWBVSVBVCVEVFWTEWSVGWBFWBWS
|
|
VHVIVJUGVKWIVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d z k $. $d z A $. $d z ps $. $d z ch $. $d z th $. $d z ta $.
|
|
$d k ph $. $d N k y z $. $d N x y z $. $d ph x y $.
|
|
nnindnn.1 $e |- ( z = 1 -> ( ph <-> ps ) ) $.
|
|
nnindnn.y $e |- ( z = k -> ( ph <-> ch ) ) $.
|
|
nnindnn.y1 $e |- ( z = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
nnindnn.a $e |- ( z = A -> ( ph <-> ta ) ) $.
|
|
nnindnn.basis $e |- ps $.
|
|
nnindnn.step $e |- ( k e. N -> ( ch -> th ) ) $.
|
|
$( Principle of Mathematical Induction (inference schema). This is a
|
|
counterpart to ~ nnind designed for real number axioms which involve
|
|
natural numbers (notably, ~ axcaucvg ). (Contributed by Jim Kingdon,
|
|
14-Jul-2021.) (New usage is discouraged.) $)
|
|
nnindnn $p |- ( A e. N -> ta ) $=
|
|
( wcel c1 crab wa cv caddc co wral wss peano1nnnn elrab mpbir2an elrabi
|
|
peano2nnnn a1d anim12d 3imtr4g mpcom rgen peano5nnnn mp2an sseli simprd
|
|
sylib ) IKSZVCEVCIAHKUAZSVCEUBKVDITVDSZJUCZTUDUEZVDSZJVDUFKVDUGVETKSBFG
|
|
KLUHQABHTKMUIUJVHJVDVFKSZVFVDSZVHAHVFKUKVIVICUBVGKSZDUBVJVHVIVIVKCDVIVK
|
|
VIFGVFKLULUMRUNACHVFKNUIADHVGKOUIUOUPUQFGJVDKLURUSUTAEHIKPUIVBVA $.
|
|
$}
|
|
|
|
${
|
|
$d A w z $. $d N z $. $d N k y $. $d N w y $. $d N x $.
|
|
$d k l u v z $. $d k l u w z $. $d l u y $. $d x y z $. $d w x $.
|
|
$d l x $. $d u x $.
|
|
$( Mapping from ` NN ` to ` N. ` . (Contributed by Jim Kingdon,
|
|
13-Jul-2021.) $)
|
|
nntopi $p |- ( A e. N -> E. z e. N.
|
|
<. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } ,
|
|
{ u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
|
|
= A ) $=
|
|
( c1o cop ceq cec cltq wbr cab c1p cpp wceq cnpi c1 vw vk vv cv cer c0r
|
|
co wrex caddc eqeq2 rexbidv wcel 1pi eqid c1r c1q opeq1 eceq1d df-1nqqs
|
|
syl6eqr breq2d abbidv breq1d opeq12d df-i1p oveq1d opeq1d eqeq1d rspcev
|
|
df-1r df-1 mp2an wa cpli simplr addclpi sylancl pitonnlem2 simpr eqtr3d
|
|
syl syl2anc ex rexlimdva cbvrexv syl6ib nnindnn ) GUDZCUDZIJZKLZMNZGOZW
|
|
KDUDZMNZDOZJZPQUGZPJZUELZUFJZUAUDZRZCSUHXATRZCSUHZXAUBUDZRZCSUHZXAXFTUI
|
|
UGZRZCSUHZXAERZCSUHABUAEUBFHXBTRXCXDCSXBTXAUJUKXBXFRXCXGCSXBXFXAUJUKXBX
|
|
IRXCXJCSXBXIXAUJUKXBERXCXLCSXBEXAUJUKISULZTTRZXEUMTUNXDXNCISWIIRZXATTXO
|
|
XAUOUFJTXOWTUOUFXOWTPPQUGZPJZUELUOXOWSXQUEXOWRXPPXOWQPPQXOWQWHUPMNZGOZU
|
|
PWNMNZDOZJPXOWMXSWPYAXOWLXRGXOWKUPWHMXOWKIIJZKLUPXOWJYBKWIIIUQURUSUTZVA
|
|
VBXOWOXTDXOWKUPWNMYCVCVBVDDGVEUTVFVGURVJUTVGVKUTVHVIVLXFFULZXHWHUCUDZIJ
|
|
ZKLZMNZGOZYGWNMNZDOZJZPQUGZPJZUELZUFJZXIRZUCSUHZXKYDXGYRCSYDWISULZVMZXG
|
|
YRYTXGVMZWIIVNUGZSULZWHUUBIJZKLZMNZGOZUUEWNMNZDOZJZPQUGZPJZUELZUFJZXIRZ
|
|
YRUUAYSXMUUCYDYSXGVOZUMWIIVPVQUUAXATUIUGZUUNXIUUAYSUUQUUNRUUPDWIGVRWAUU
|
|
AXAXFTUIYTXGVSVFVTYQUUOUCUUBSYEUUBRZYPUUNXIUURYOUUMUFUURYNUULUEUURYMUUK
|
|
PUURYLUUJPQUURYIUUGYKUUIUURYHUUFGUURYGUUEWHMUURYFUUDKYEUUBIUQURZVAVBUUR
|
|
YJUUHDUURYGUUEWNMUUSVCVBVDVFVGURVGVHVIWBWCWDYQXJUCCSYEWIRZYPXAXIUUTYOWT
|
|
UFUUTYNWSUEUUTYMWRPUUTYLWQPQUUTYIWMYKWPUUTYHWLGUUTYGWKWHMUUTYFWJKYEWIIU
|
|
QURZVAVBUUTYJWODUUTYGWKWNMUVAVCVBVDVFVGURVGVHWEWFWG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d F z $. $d J l u z $. $d l u y $. $d x y $.
|
|
axcaucvglemcl.n $e |- N =
|
|
|^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
axcaucvglemcl.f $e |- ( ph -> F : N --> RR ) $.
|
|
$( Lemma for ~ axcaucvg . Mapping to ` N. ` and ` R. ` . (Contributed by
|
|
Jim Kingdon, 10-Jul-2021.) $)
|
|
axcaucvglemcl $p |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R.
|
|
( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
|
|
{ u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) =
|
|
<. z , 0R >. ) e. R. ) $=
|
|
( wcel wa cv cop cec cltq wbr cab cnr cnpi c1o ceq c1p cpp co cer c0r cfv
|
|
wceq wreu crio cr c1 wral cint pitonn syl6eleqr ffvelrnda sylan2 elrealeu
|
|
caddc sylib eqcom reubii riotacl syl ) AGUALZMZINGUBOUCPZQRISVJENQRESOUDU
|
|
EUFUDOUGPUHOZFUIZDNUHOZUJZDTUKZVNDTULTLVIVMVLUJZDTUKZVOVIVLUMLZVQVHAVKHLV
|
|
RVHVKUNBNZLCNUNVBUFVSLCVSUOMBSUPHBCEGIUQJURAHUMVKFKUSUTDVLVAVCVPVNDTVMVLV
|
|
DVEVCVNDTVFVG $.
|
|
$}
|
|
|
|
${
|
|
axcaucvg.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
axcaucvg.f $e |- ( ph -> F : N --> RR ) $.
|
|
axcaucvg.cau $e |- ( ph ->
|
|
A. n e. N A. k e. N ( n <RR k ->
|
|
( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\
|
|
( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) ) $.
|
|
|
|
${
|
|
axcaucvg.g $e |- G = ( j e. N. |-> ( iota_ z e. R.
|
|
( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } ,
|
|
{ u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) =
|
|
<. z , 0R >. ) ) $.
|
|
|
|
${
|
|
$d j ph $. $d F z $. $d j l u z $. $d l u y $. $d x y $.
|
|
$( Lemma for ~ axcaucvg . Mapping to ` N. ` and ` R. ` yields a
|
|
sequence. (Contributed by Jim Kingdon, 9-Jul-2021.) $)
|
|
axcaucvglemf $p |- ( ph -> G : N. --> R. ) $=
|
|
( cv cop cec cnpi c1o ceq cltq wbr cab c1p cpp cer wceq axcaucvglemcl
|
|
co c0r cfv cnr crio fmptd ) AFUAMRFRZUBSUCTZUDUEMUFUSERUDUEEUFSUGUHUL
|
|
UGSUITUMSIUNDRUMSUJDUOUPUOJABCDEIURKMNOUKQUQ $.
|
|
$}
|
|
|
|
${
|
|
$d F j z $. $d G z $. $d J j l u z $. $d j ph $. $d l u y $.
|
|
$d x y $.
|
|
$( Lemma for ~ axcaucvg . Value of sequence when mapping to ` N. ` and
|
|
` R. ` . (Contributed by Jim Kingdon, 10-Jul-2021.) $)
|
|
axcaucvglemval $p |- ( ( ph /\ J e. N. ) ->
|
|
( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } ,
|
|
{ u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
|
|
] ~R , 0R >. ) =
|
|
<. ( G ` J ) , 0R >. ) $=
|
|
( cop cnr cnpi wcel wa cv c1o ceq cec cltq wbr cab c1p cpp co cer c0r
|
|
cfv wceq crio opeq1 eceq1d breq2d abbidv breq1d opeq12d oveq1d opeq1d
|
|
cmpt fveq2d eqeq1d riotabidv adantl simpr axcaucvglemcl fvmptd eqcomd
|
|
a1i wreu wb eqeltrd cr wf adantr caddc wral pitonn syl6eleqr ffvelrnd
|
|
c1 cint elrealeu sylib eqcom reubii eqeq2d riota2 syl2anc mpbird ) AK
|
|
UAUBZUCZNUDZKUESZUFUGZUHUIZNUJZXBEUDZUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZI
|
|
UPZKJUPZUOSZUQZXMDUDZUOSZUQZDTURZXNUQZWSXNXTWSFKWTFUDZUESZUFUGZUHUIZN
|
|
UJZYDXEUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZIUPZXRUQZDTURZXTUAJTJFUAYPVGUQW
|
|
SRVPYBKUQZYPXTUQWSYQYOXSDTYQYNXMXRYQYMXLIYQYLXKUOYQYKXJUNYQYJXIUKYQYI
|
|
XHUKULYQYFXDYHXGYQYEXCNYQYDXBWTUHYQYCXAUFYBKUEUSUTZVAVBYQYGXFEYQYDXBX
|
|
EUHYRVCVBVDVEVFUTVFVHVIVJVKAWRVLABCDEIKLNOPVMZVNZVOWSXNTUBXSDTVQZXPYA
|
|
VRWSXNXTTYTYSVSWSXRXMUQZDTVQZUUAWSXMVTUBUUCWSLVTXLIALVTIWAWRPWBWRXLLU
|
|
BAWRXLWHBUDZUBCUDWHWCUMUUDUBCUUDWDUCBUJWILBCEKNWEOWFVKWGDXMWJWKUUBXSD
|
|
TXRXMWLWMWKXSXPDTXNXQXNUQXRXOXMXQXNUOUSWNWOWPWQ $.
|
|
$}
|
|
|
|
${
|
|
$d F a b k n $. $d F z $. $d F j $. $d N a b k n $. $d G z $.
|
|
$d a b k l n r u $. $d j l u z $. $d j ph $. $d k n ph $.
|
|
$d l u y $. $d x y $. $d j n $. $d z n $. $d j k $. $d k z $.
|
|
$( Lemma for ~ axcaucvg . The result of mapping to ` N. ` and ` R. `
|
|
satisfies the Cauchy condition. (Contributed by Jim Kingdon,
|
|
9-Jul-2021.) $)
|
|
axcaucvglemcau $p |- ( ph -> A. n e. N. A. k e. N. ( n <N k ->
|
|
( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <.
|
|
{ l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R )
|
|
/\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <.
|
|
{ l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } ,
|
|
{ u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >.
|
|
+P. 1P ) , 1P >. ] ~R )
|
|
) ) ) $=
|
|
( wbr co cltrr vb va cv clti cfv c1o cop ceq cec crq cltq cab c1p cpp
|
|
cer cplr cltr wa wi cnpi wral wcel c0r cmul c1 wceq crio caddc ltrenn
|
|
cr adantl pitonn syl6eleqr ad2antlr ad3antlr breq1 fveq2 oveq1 eqeq1d
|
|
riotabidv oveq2d breq12d oveq12d breq2d anbi12d imbi12d oveq1d breq1d
|
|
cint breq2 cbvral2v sylib ad3antrrr ralbidv rspcva syl2anc mpd simpld
|
|
axcaucvglemval ad2antrr adantlr adantr recriota axcaucvglemf ffvelrnd
|
|
cnr simplr cnp recnnpr prsrcl syl addresr eqtrd 3brtr3d ltresr simprd
|
|
wf simpllr jca ex ralrimiva ) AHUCZGUCZUDRZYBJUEZYCJUEZMUCZYBUFUGUHUI
|
|
ZUJUEZUKRMULYIEUCZUKREULUGZUMUNSUMUGUOUIZUPSZUQRZYFYEYLUPSZUQRZURZUSZ
|
|
GUTVAHUTAYBUTVBZURZYRGUTYTYCUTVBZURZYDYQUUBYDURZYNYPUUCYEVCUGZYMVCUGZ
|
|
TRYNUUCYGYHUKRMULYHYJUKREULUGUMUNSUMUGUOUIVCUGZIUEZYGYCUFUGUHUIZUKRMU
|
|
LUUHYJUKREULUGUMUNSUMUGUOUIVCUGZIUEZUUFLUCZVDSZVEVFZLVJVGZVHSZUUDUUET
|
|
UUCUUGUUOTRZUUJUUGUUNVHSZTRZUUCUUFUUITRZUUPUURURZYDUUSUUBEYBYCMVIVKUU
|
|
CUUIKVBZUUFUAUCZTRZUUGUVBIUEZUUNVHSZTRZUVDUUQTRZURZUSZUAKVAZUUSUUTUSZ
|
|
UUAUVAYTYDUUAUUIVEBUCZVBCUCVEVHSUVLVBCUVLVAURBULWIZKBCEYCMVLNVMVNUUCU
|
|
UFKVBZUBUCZUVBTRZUVOIUEZUVDUVOUUKVDSZVEVFZLVJVGZVHSZTRZUVDUVQUVTVHSZT
|
|
RZURZUSZUAKVAZUBKVAZUVJYSUVNAUUAYDYSUUFUVMKBCEYBMVLNVMVOAUWHYSUUAYDAY
|
|
BYCTRZYBIUEZYCIUEZYBUUKVDSZVEVFZLVJVGZVHSZTRZUWKUWJUWNVHSZTRZURZUSZGK
|
|
VAHKVAUWHPUWTUWFUVOYCTRZUVQUWKUVTVHSZTRZUWKUWCTRZURZUSHGUBUAKKYBUVOVF
|
|
ZUWIUXAUWSUXEYBUVOYCTVPUXFUWPUXCUWRUXDUXFUWJUVQUWOUXBTYBUVOIVQZUXFUWN
|
|
UVTUWKVHUXFUWMUVSLVJUXFUWLUVRVEYBUVOUUKVDVRVSVTZWAWBUXFUWQUWCUWKTUXFU
|
|
WJUVQUWNUVTVHUXGUXHWCWDWEWFYCUVBVFZUXAUVPUXEUWEYCUVBUVOTWJUXIUXCUWBUX
|
|
DUWDUXIUXBUWAUVQTUXIUWKUVDUVTVHYCUVBIVQZWGWDUXIUWKUVDUWCTUXJWHWEWFWKW
|
|
LWMUWGUVJUBUUFKUVOUUFVFZUWFUVIUAKUXKUVPUVCUWEUVHUVOUUFUVBTVPUXKUWBUVF
|
|
UWDUVGUXKUVQUUGUWAUVETUVOUUFIVQZUXKUVTUUNUVDVHUXKUVSUUMLVJUXKUVRUULVE
|
|
UVOUUFUUKVDVRVSVTZWAWBUXKUWCUUQUVDTUXKUVQUUGUVTUUNVHUXLUXMWCWDWEWFWNW
|
|
OWPUVIUVKUAUUIKUVBUUIVFZUVCUUSUVHUUTUVBUUIUUFTWJUXNUVFUUPUVGUURUXNUVE
|
|
UUOUUGTUXNUVDUUJUUNVHUVBUUIIVQZWGWDUXNUVDUUJUUQTUXOWHWEWFWOWPWQZWRYTU
|
|
UGUUDVFUUAYDABCDEFGHIJYBKLMNOPQWSWTZUUCUUOYFVCUGZYLVCUGZVHSZUUEUUCUUJ
|
|
UXRUUNUXSVHUUBUUJUXRVFZYDAUUAUYAYSABCDEFGHIJYCKLMNOPQWSXAXBZYSUUNUXSV
|
|
FAUUAYDEYBLMXCVOZWCUUCYFXFVBYLXFVBZUXTUUEVFUUCUTXFYCJAUTXFJXQYSUUAYDA
|
|
BCDEFGHIJKLMNOPQXDWMZYTUUAYDXGXEYSUYDAUUAYDYSYKXHVBUYDEYBMXIYKXJXKVOZ
|
|
YFYLXLWPXMXNYEYMXOWLUUCUXRYOVCUGZTRYPUUCUUJUUQUXRUYGTUUCUUPUURUXPXPUY
|
|
BUUCUUQUUDUXSVHSZUYGUUCUUGUUDUUNUXSVHUXQUYCWCUUCYEXFVBUYDUYHUYGVFUUCU
|
|
TXFYBJUYEAYSUUAYDXRXEUYFYEYLXLWPXMXNYFYOXOWLXSXTYAYA $.
|
|
$}
|
|
|
|
${
|
|
$d F b e f g k $. $d F b e f j k $. $d F k n $. $d F b j k y $.
|
|
$d F j z $. $d G a b c d e f g k x $. $d G c f g k l u $.
|
|
$d G k l n u $. $d G z $. $d N b e f g k $. $d N b e f j k $.
|
|
$d N k n $. $d N b j k y $. $d N x $. $d a b c e f g k ph x $.
|
|
$d j k l u y $. $d j k ph x $. $d k l n r u $. $d l u z $.
|
|
$d n ph $. $d x y $. $d j n $. $d n z $. $d k z $. $d g j $.
|
|
$d g z $. $d g y $. $d l x $. $d u x $.
|
|
$( Lemma for ~ axcaucvg . Mapping the limit from ` N. ` and ` R. ` .
|
|
(Contributed by Jim Kingdon, 10-Jul-2021.) $)
|
|
axcaucvglemres $p |- ( ph ->
|
|
E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
|
|
( ( F ` k ) <RR ( y + x ) /\
|
|
y <RR ( ( F ` k ) + x ) ) ) ) ) $=
|
|
( wbr wa cnpi va vc vb vd ve vf vg c0r cv cltr clti cplr co wral wrex
|
|
cfv cnr cc0 cltrr caddc axcaucvglemf axcaucvglemcau caucvgsr wcel cop
|
|
wi cr opelreal biimpri ad2antrl weq breq2 fveq2 breq1d oveq1d anbi12d
|
|
breq2d imbi12d cbvralv rexbii imbi2i ralbii anbi2i wceq elreal biimpi
|
|
ad2antlr simplrr ad2antrr simprr simplr breq1i ltresr syl5rbbr biimpa
|
|
df-0 bitri syl2anc oveq2 imbi2d rexralbidv rspcv breq1 imbi1d ralbidv
|
|
mp2d cbvrexv sylib c1o ceq cec cltq cab c1p cpp cer c1 cint syl6eleqr
|
|
pitonn nntopi adantl simprl adantr sylc simplrl ltrennb bitrd simplll
|
|
wb ad4antr axcaucvglemval fveq2d eqtr3d ad3antrrr rexlimddv ralrimiva
|
|
addresr syl5bbr rspcev ad5antr oveq2d breq12d wf syl ffvelrnd oveq12d
|
|
3imtr3d ex sylan2br oveq1 ) AUHUAUIZUJRZUBUIZGUIZUKRZUUOJUPZUCUIZUULU
|
|
LUMZUJRZUURUUQUULULUMZUJRZSZVFZGTUNZUBTUOZVFZUAUQUNZURBUIZUSRZFUIZUUO
|
|
USRZUUOIUPZCUIZUVIUTUMZUSRZUVNUVMUVIUTUMZUSRZSZVFZGKUNFKUOZVFZBVGUNZC
|
|
VGUOZUCUQAUAUCEUBGHJMABCDEFGHIJKLMNOPQVAZABCDEFGHIJKLMNOPQVBVCAUURUQV
|
|
DZUVHSZSUURUHVEZVGVDZUVJUVLUVMUWHUVIUTUMZUSRZUWHUVQUSRZSZVFZGKUNZFKUO
|
|
ZVFZBVGUNZUWDUWFUWIAUVHUWIUWFUURVHVIVJUWGAUWFUUMUUNUDUIZUKRZUWSJUPZUU
|
|
SUJRZUURUXAUULULUMZUJRZSZVFZUDTUNZUBTUOZVFZUAUQUNZSZUWRUXJUVHUWFUXIUV
|
|
GUAUQUXHUVFUUMUXGUVEUBTUXFUVDUDGTUDGVKZUWTUUPUXEUVCUWSUUOUUNUKVLUXLUX
|
|
BUUTUXDUVBUXLUXAUUQUUSUJUWSUUOJVMZVNUXLUXCUVAUURUJUXLUXAUUQUULULUXMVO
|
|
VQVPVRVSVTWAWBWCAUXKSZUWQBVGUXNUVIVGVDZSZUVJUWPUXPUVJSZUEUIZUHVEZUVIW
|
|
DZUWPUEUQUXOUXTUEUQUOZUXNUVJUXOUYAUEUVIWEWFWGUXQUXRUQVDZUXTSZSZUFUIZU
|
|
WSUKRZUXAUURUXRULUMZUJRZUURUXAUXRULUMZUJRZSZVFZUDTUNZUWPUFTUYDUWTUYKV
|
|
FZUDTUNZUBTUOZUYMUFTUOUYDUXJUHUXRUJRZUYPUXPUXJUVJUYCAUWFUXJUXOWHWIUYD
|
|
UXTUVJUYQUXQUYBUXTWJZUXPUVJUYCWKUXTUVJUYQUYQURUXSUSRZUXTUVJUYSUHUHVEZ
|
|
UXSUSRUYQURUYTUXSUSWPWLUHUXRWMWQUXSUVIURUSVLWNWOWRUYBUXJUYQUYPVFZVFUX
|
|
QUXTUXIVUAUAUXRUQUAUEVKZUUMUYQUXHUYPUULUXRUHUJVLVUBUXFUYNUBUDTTVUBUXE
|
|
UYKUWTVUBUXBUYHUXDUYJVUBUUSUYGUXAUJUULUXRUURULWSVQVUBUXCUYIUURUJUULUX
|
|
RUXAULWSVQVPWTXAVRXBVJXFUYOUYMUBUFTUBUFVKZUYNUYLUDTVUCUWTUYFUYKUUNUYE
|
|
UWSUKXCXDXEXGXHUYDUYETVDZUYMSZSZMUIZUYEXIVEXJXKZXLRMXMVUHEUIZXLREXMVE
|
|
XNXOUMXNVEXPXKUHVEZKVDZVUJUUOUSRZUWMVFZGKUNZUWPVUDVUKUYDUYMVUDVUJXQUV
|
|
IVDUVNXQUTUMUVIVDCUVIUNSBXMXRKBCEUYEMXTNXSVJVUFVUMGKVUFUUOKVDZSZVUGUG
|
|
UIZXIVEXJXKZXLRMXMVURVUIXLREXMVEXNXOUMXNVEXPXKUHVEZUUOWDZVUMUGTVUOVUT
|
|
UGTUOVUFBCUGEUUOKMNYAYBVUPVUQTVDZVUTSZSZUYEVUQUKRZVUQJUPZUYGUJRZUURVV
|
|
EUXRULUMZUJRZSZVULUWMVVCVVAUYMVVDVVIVFZVUPVVAVUTYCZVUPUYMVVBUYDVUDUYM
|
|
VUOWHYDUYLVVJUDVUQTUDUGVKZUYFVVDUYKVVIUWSVUQUYEUKVLVVLUYHVVFUYJVVHVVL
|
|
UXAVVEUYGUJUWSVUQJVMZVNVVLUYIVVGUURUJVVLUXAVVEUXRULVVMVOVQVPVRXBYEVVC
|
|
VVDVUJVUSUSRZVULVVCVUDVVAVVDVVNYJVUPVUDVVBUYDVUDUYMVUOYFYDVVKEUYEVUQM
|
|
YGWRVVCVUSUUOVUJUSVUPVVAVUTWJZVQYHVVCVVFUWKVVHUWLVVFVVEUHVEZUYGUHVEZU
|
|
SRVVCUWKVVEUYGWMVVCVVPUVMVVQUWJUSVVCVUSIUPZVVPUVMVVCAVVAVVRVVPWDUXQAU
|
|
YCVUEVUOVVBAUXKUXOUVJYIYKZVVKABCDEFGHIJVUQKLMNOPQYLWRVVCVUSUUOIVVOYMY
|
|
NZVVCUWHUXSUTUMZVVQUWJVVCUWFUYBVWAVVQWDUXPUWFUVJUYCVUEVUOVVBAUWFUXJUX
|
|
OYFUUAVUFUYBVUOVVBUXQUYBUXTVUEYFWIZUURUXRYRWRUYDVWAUWJWDVUEVUOVVBUYDU
|
|
XSUVIUWHUTUYRUUBYOYNUUCYSVVHUWHVVGUHVEZUSRVVCUWLUURVVGWMVVCVWCUVQUWHU
|
|
SVVCVVPUXSUTUMZVWCUVQVVCVVEUQVDUYBVWDVWCWDVVCTUQVUQJVVCATUQJUUDVVSUWE
|
|
UUEVVKUUFVWBVVEUXRYRWRVVCVVPUVMUXSUVIUTVVTUYDUXTVUEVUOVVBUYRYOUUGYNVQ
|
|
YSVPUUHYPYQUWOVUNFVUJKUVKVUJWDZUWNVUMGKVWEUVLVULUWMUVKVUJUUOUSXCXDXEY
|
|
TWRYPYPUUIYQUUJUWCUWRCUWHVGUVNUWHWDZUWBUWQBVGVWFUWAUWPUVJVWFUVTUWNFGK
|
|
KVWFUVSUWMUVLVWFUVPUWKUVRUWLVWFUVOUWJUVMUSUVNUWHUVIUTUUKVQUVNUWHUVQUS
|
|
XCVPWTXAWTXEYTWRYP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d F a j k l n u z $. $d F a b c j k l u x y $. $d N j k n $.
|
|
$d N x y $. $d b c j k l n u z $. $d j k n ph $. $d k l n r u $.
|
|
$d ph x $. $d x y $. $d l x $. $d u x $.
|
|
$( Real number completeness axiom. A Cauchy sequence with a modulus of
|
|
convergence converges. This is basically Corollary 11.2.13 of [HoTT],
|
|
p. (varies). The HoTT book theorem has a modulus of convergence
|
|
(that is, a rate of convergence) specified by (11.2.9) in HoTT whereas
|
|
this theorem fixes the rate of convergence to say that all terms after
|
|
the nth term must be within ` 1 / n ` of the nth term (it should later
|
|
be able to prove versions of this theorem with a different fixed rate
|
|
or a modulus of convergence supplied as a hypothesis).
|
|
|
|
Because we are stating this axiom before we have introduced notations
|
|
for ` NN ` or division, we use ` N ` for the natural numbers and
|
|
express a reciprocal in terms of ` iota_ ` .
|
|
|
|
This construction-dependent theorem should not be referenced directly;
|
|
instead, use ~ ax-caucvg . (Contributed by Jim Kingdon, 8-Jul-2021.)
|
|
(New usage is discouraged.) $)
|
|
axcaucvg $p |- ( ph ->
|
|
E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
|
|
( ( F ` k ) <RR ( y + x ) /\
|
|
y <RR ( ( F ` k ) + x ) ) ) ) ) $=
|
|
( vz vu cv cop cltq c1p c0r wceq vb vc va vl c1o ceq cec wbr cab cpp co
|
|
cnpi cer cfv cnr crio cmpt breq1 cbvabv breq2 oveq1i opeq1i eceq1 ax-mp
|
|
opeq12i fveq2i a1i opeq1 eqeq12d cbvriotav mpteq2i axcaucvglemres ) ABC
|
|
MNDEFGDULUAOZDOUEPUFUGZQUHZUAUIZVNUBOZQUHZUBUIZPZRUJUKZRPZUMUGZSPZGUNZU
|
|
COZSPZTZUCUOUPZUQHIUDJKLDULWIUDOZVNQUHZUDUIZVNNOZQUHZNUIZPZRUJUKZRPZUMU
|
|
GZSPZGUNZMOZSPZTZMUOUPWHXDUCMUOWFXBTZWEXAWGXCWEXATXEWDWTGWCWSSWBWRTWCWS
|
|
TWAWQRVTWPRUJVPWLVSWOVOWKUAUDVMWJVNQURUSVRWNUBNVQWMVNQUTUSVEVAVBWBWRUMV
|
|
CVDVBVFVGWFXBSVHVIVJVKVL $.
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Real and complex number postulates restated as axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The complex numbers form a set. Proofs should normally use ~ cnex
|
|
instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) $)
|
|
ax-cnex $a |- CC e. _V $.
|
|
$( $j restatement 'ax-cnex' of 'axcnex'; $)
|
|
|
|
$( The real numbers are a subset of the complex numbers. Axiom for real and
|
|
complex numbers, justified by theorem ~ axresscn . (Contributed by NM,
|
|
1-Mar-1995.) $)
|
|
ax-resscn $a |- RR C_ CC $.
|
|
$( $j restatement 'ax-resscn' of 'axresscn'; $)
|
|
|
|
$( 1 is a complex number. Axiom for real and complex numbers, justified by
|
|
theorem ~ ax1cn . (Contributed by NM, 1-Mar-1995.) $)
|
|
ax-1cn $a |- 1 e. CC $.
|
|
$( $j restatement 'ax-1cn' of 'ax1cn'; $)
|
|
|
|
$( 1 is a real number. Axiom for real and complex numbers, justified by
|
|
theorem ~ ax1re . Proofs should use ~ 1re instead. (Contributed by Jim
|
|
Kingdon, 13-Jan-2020.) (New usage is discouraged.) $)
|
|
ax-1re $a |- 1 e. RR $.
|
|
$( $j restatement 'ax-1re' of 'ax1re'; $)
|
|
|
|
$( ` _i ` is a complex number. Axiom for real and complex numbers, justified
|
|
by theorem ~ axicn . (Contributed by NM, 1-Mar-1995.) $)
|
|
ax-icn $a |- _i e. CC $.
|
|
$( $j restatement 'ax-icn' of 'axicn'; $)
|
|
|
|
$( Closure law for addition of complex numbers. Axiom for real and complex
|
|
numbers, justified by theorem ~ axaddcl . Proofs should normally use
|
|
~ addcl instead, which asserts the same thing but follows our naming
|
|
conventions for closures. (New usage is discouraged.) (Contributed by
|
|
NM, 22-Nov-1994.) $)
|
|
ax-addcl $a |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $.
|
|
$( $j restatement 'ax-addcl' of 'axaddcl'; $)
|
|
|
|
$( Closure law for addition in the real subfield of complex numbers. Axiom
|
|
for real and complex numbers, justified by theorem ~ axaddrcl . Proofs
|
|
should normally use ~ readdcl instead. (New usage is discouraged.)
|
|
(Contributed by NM, 22-Nov-1994.) $)
|
|
ax-addrcl $a |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $.
|
|
$( $j restatement 'ax-addrcl' of 'axaddrcl'; $)
|
|
|
|
$( Closure law for multiplication of complex numbers. Axiom for real and
|
|
complex numbers, justified by theorem ~ axmulcl . Proofs should normally
|
|
use ~ mulcl instead. (New usage is discouraged.) (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
ax-mulcl $a |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $.
|
|
$( $j restatement 'ax-mulcl' of 'axmulcl'; $)
|
|
|
|
$( Closure law for multiplication in the real subfield of complex numbers.
|
|
Axiom for real and complex numbers, justified by theorem ~ axmulrcl .
|
|
Proofs should normally use ~ remulcl instead. (New usage is discouraged.)
|
|
(Contributed by NM, 22-Nov-1994.) $)
|
|
ax-mulrcl $a |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $.
|
|
$( $j restatement 'ax-mulrcl' of 'axmulrcl'; $)
|
|
|
|
$( Addition commutes. Axiom for real and complex numbers, justified by
|
|
theorem ~ axaddcom . Proofs should normally use ~ addcom instead.
|
|
(New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) $)
|
|
ax-addcom $a |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $.
|
|
$( $j restatement 'ax-addcom' of 'axaddcom'; $)
|
|
|
|
$( Multiplication of complex numbers is commutative. Axiom for real and
|
|
complex numbers, justified by theorem ~ axmulcom . Proofs should normally
|
|
use ~ mulcom instead. (New usage is discouraged.) (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
ax-mulcom $a |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $.
|
|
$( $j restatement 'ax-mulcom' of 'axmulcom'; $)
|
|
|
|
$( Addition of complex numbers is associative. Axiom for real and complex
|
|
numbers, justified by theorem ~ axaddass . Proofs should normally use
|
|
~ addass instead. (New usage is discouraged.) (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
ax-addass $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) + C ) = ( A + ( B + C ) ) ) $.
|
|
$( $j restatement 'ax-addass' of 'axaddass'; $)
|
|
|
|
$( Multiplication of complex numbers is associative. Axiom for real and
|
|
complex numbers, justified by theorem ~ axmulass . Proofs should normally
|
|
use ~ mulass instead. (New usage is discouraged.) (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
ax-mulass $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $.
|
|
$( $j restatement 'ax-mulass' of 'axmulass'; $)
|
|
|
|
$( Distributive law for complex numbers (left-distributivity). Axiom for
|
|
real and complex numbers, justified by theorem ~ axdistr . Proofs should
|
|
normally use ~ adddi instead. (New usage is discouraged.) (Contributed
|
|
by NM, 22-Nov-1994.) $)
|
|
ax-distr $a |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $.
|
|
$( $j restatement 'ax-distr' of 'axdistr'; $)
|
|
|
|
$( i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real
|
|
and complex numbers, justified by theorem ~ axi2m1 . (Contributed by NM,
|
|
29-Jan-1995.) $)
|
|
ax-i2m1 $a |- ( ( _i x. _i ) + 1 ) = 0 $.
|
|
$( $j restatement 'ax-i2m1' of 'axi2m1'; $)
|
|
|
|
$( 0 is less than 1. Axiom for real and complex numbers, justified by
|
|
theorem ~ ax0lt1 . Proofs should normally use ~ 0lt1 instead.
|
|
(New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) $)
|
|
ax-0lt1 $a |- 0 <RR 1 $.
|
|
$( $j restatement 'ax-0lt1' of 'ax0lt1'; $)
|
|
|
|
$( ` 1 ` is an identity element for real multiplication. Axiom for real and
|
|
complex numbers, justified by theorem ~ ax1rid . (Contributed by NM,
|
|
29-Jan-1995.) $)
|
|
ax-1rid $a |- ( A e. RR -> ( A x. 1 ) = A ) $.
|
|
$( $j restatement 'ax-1rid' of 'ax1rid'; $)
|
|
|
|
$( ` 0 ` is an identity element for real addition. Axiom for real and
|
|
complex numbers, justified by theorem ~ ax0id .
|
|
|
|
Proofs should normally use ~ addid1 instead. (New usage is discouraged.)
|
|
(Contributed by Jim Kingdon, 16-Jan-2020.) $)
|
|
ax-0id $a |- ( A e. CC -> ( A + 0 ) = A ) $.
|
|
$( $j restatement 'ax-0id' of 'ax0id'; $)
|
|
|
|
${
|
|
$d x A $.
|
|
$( Existence of negative of real number. Axiom for real and complex
|
|
numbers, justified by theorem ~ axrnegex . (Contributed by Eric
|
|
Schmidt, 21-May-2007.) $)
|
|
ax-rnegex $a |- ( A e. RR -> E. x e. RR ( A + x ) = 0 ) $.
|
|
$( $j restatement 'ax-rnegex' of 'axrnegex'; $)
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Existence of reciprocal of positive real number. Axiom for real and
|
|
complex numbers, justified by theorem ~ axprecex . (Contributed by Jim
|
|
Kingdon, 6-Feb-2020.) $)
|
|
ax-precex $a |- ( ( A e. RR /\ 0 <RR A ) ->
|
|
E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) ) $.
|
|
$( $j restatement 'ax-precex' of 'axprecex'; $)
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A complex number can be expressed in terms of two reals. Definition
|
|
10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers,
|
|
justified by theorem ~ axcnre . For naming consistency, use ~ cnre for
|
|
new proofs. (New usage is discouraged.) (Contributed by NM,
|
|
9-May-1999.) $)
|
|
ax-cnre $a |- ( A e. CC ->
|
|
E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $.
|
|
$( $j restatement 'ax-cnre' of 'axcnre'; $)
|
|
$}
|
|
|
|
$( Real number less-than is irreflexive. Axiom for real and complex numbers,
|
|
justified by theorem ~ ax-pre-ltirr . (Contributed by Jim Kingdon,
|
|
12-Jan-2020.) $)
|
|
ax-pre-ltirr $a |- ( A e. RR -> -. A <RR A ) $.
|
|
$( $j restatement 'ax-pre-ltirr' of 'axpre-ltirr'; $)
|
|
|
|
$( Real number less-than is weakly linear. Axiom for real and complex
|
|
numbers, justified by theorem ~ axpre-ltwlin . (Contributed by Jim
|
|
Kingdon, 12-Jan-2020.) $)
|
|
ax-pre-ltwlin $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <RR B -> ( A <RR C \/ C <RR B ) ) ) $.
|
|
$( $j restatement 'ax-pre-ltwlin' of 'axpre-ltwlin'; $)
|
|
|
|
$( Ordering on reals is transitive. Axiom for real and complex numbers,
|
|
justified by theorem ~ axpre-lttrn . (Contributed by NM, 13-Oct-2005.) $)
|
|
ax-pre-lttrn $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A <RR B /\ B <RR C ) -> A <RR C ) ) $.
|
|
$( $j restatement 'ax-pre-lttrn' of 'axpre-lttrn'; $)
|
|
|
|
$( Apartness of reals is tight. Axiom for real and complex numbers,
|
|
justified by theorem ~ axpre-apti . (Contributed by Jim Kingdon,
|
|
29-Jan-2020.) $)
|
|
ax-pre-apti $a |- ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) ->
|
|
A = B ) $.
|
|
$( $j restatement 'ax-pre-apti' of 'axpre-apti'; $)
|
|
|
|
$( Ordering property of addition on reals. Axiom for real and complex
|
|
numbers, justified by theorem ~ axpre-ltadd . (Contributed by NM,
|
|
13-Oct-2005.) $)
|
|
ax-pre-ltadd $a |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <RR B -> ( C + A ) <RR ( C + B ) ) ) $.
|
|
$( $j restatement 'ax-pre-ltadd' of 'axpre-ltadd'; $)
|
|
|
|
$( The product of two positive reals is positive. Axiom for real and complex
|
|
numbers, justified by theorem ~ axpre-mulgt0 . (Contributed by NM,
|
|
13-Oct-2005.) $)
|
|
ax-pre-mulgt0 $a |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) ) $.
|
|
$( $j restatement 'ax-pre-mulgt0' of 'axpre-mulgt0'; $)
|
|
|
|
$( Strong extensionality of multiplication (expressed in terms of ` <RR ` ).
|
|
Axiom for real and complex numbers, justified by theorem ~ axpre-mulext
|
|
|
|
(Contributed by Jim Kingdon, 18-Feb-2020.) $)
|
|
ax-pre-mulext $a |- ( ( A e. RR /\ B e. RR /\ C e. RR )
|
|
-> ( ( A x. C ) <RR ( B x. C ) -> ( A <RR B \/ B <RR A ) ) ) $.
|
|
$( $j restatement 'ax-pre-mulext' of 'axpre-mulext'; $)
|
|
|
|
${
|
|
$d A n x y $.
|
|
|
|
$( Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
|
|
real and complex numbers, justified by theorem ~ axarch .
|
|
|
|
This axiom should not be used directly; instead use ~ arch (which is the
|
|
same, but stated in terms of ` NN ` and ` < ` ). (Contributed by Jim
|
|
Kingdon, 2-May-2020.) (New usage is discouraged.) $)
|
|
ax-arch $a |- ( A e. RR -> E. n e.
|
|
|^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n ) $.
|
|
$( $j restatement 'ax-arch' of 'axarch'; $)
|
|
$}
|
|
|
|
${
|
|
ax-caucvg.n $e |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
ax-caucvg.f $e |- ( ph -> F : N --> RR ) $.
|
|
ax-caucvg.cau $e |- ( ph ->
|
|
A. n e. N A. k e. N ( n <RR k ->
|
|
( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\
|
|
( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) ) $.
|
|
|
|
$d F j k n $. $d F j k x y $. $d N j k n $. $d N x y $. $d j k n $.
|
|
$d j k n ph $. $d k n r $. $d ph x $. $d x y $.
|
|
$( Completeness. Axiom for real and complex numbers, justified by theorem
|
|
~ axcaucvg .
|
|
|
|
A Cauchy sequence (as defined here, which has a rate convergence built
|
|
in) of real numbers converges to a real number. Specifically on rate of
|
|
convergence, all terms after the nth term must be within ` 1 / n ` of
|
|
the nth term.
|
|
|
|
This axiom should not be used directly; instead use ~ caucvgre (which is
|
|
the same, but stated in terms of the ` NN ` and ` 1 / n ` notations).
|
|
(Contributed by Jim Kingdon, 19-Jul-2021.)
|
|
(New usage is discouraged.) $)
|
|
ax-caucvg $a |- ( ph ->
|
|
E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k ->
|
|
( ( F ` k ) <RR ( y + x ) /\
|
|
y <RR ( ( F ` k ) + x ) ) ) ) ) $.
|
|
|
|
$( $j restatement 'ax-caucvg' of 'axcaucvg'; $)
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Derive the basic properties from the field axioms
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Some deductions from the field axioms for complex numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Alias for ~ ax-cnex . (Contributed by Mario Carneiro, 17-Nov-2014.) $)
|
|
cnex $p |- CC e. _V $=
|
|
( ax-cnex ) A $.
|
|
|
|
$( Alias for ~ ax-addcl , for naming consistency with ~ addcli . Use this
|
|
theorem instead of ~ ax-addcl or ~ axaddcl . (Contributed by NM,
|
|
10-Mar-2008.) $)
|
|
addcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) $=
|
|
( ax-addcl ) ABC $.
|
|
|
|
$( Alias for ~ ax-addrcl , for naming consistency with ~ readdcli .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
readdcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) $=
|
|
( ax-addrcl ) ABC $.
|
|
|
|
$( Alias for ~ ax-mulcl , for naming consistency with ~ mulcli .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
mulcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) $=
|
|
( ax-mulcl ) ABC $.
|
|
|
|
$( Alias for ~ ax-mulrcl , for naming consistency with ~ remulcli .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
remulcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) $=
|
|
( ax-mulrcl ) ABC $.
|
|
|
|
$( Alias for ~ ax-mulcom , for naming consistency with ~ mulcomi .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
mulcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) ) $=
|
|
( ax-mulcom ) ABC $.
|
|
|
|
$( Alias for ~ ax-addass , for naming consistency with ~ addassi .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
addass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
|
|
( ax-addass ) ABCD $.
|
|
|
|
$( Alias for ~ ax-mulass , for naming consistency with ~ mulassi .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
mulass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
|
|
( ax-mulass ) ABCD $.
|
|
|
|
$( Alias for ~ ax-distr , for naming consistency with ~ adddii .
|
|
(Contributed by NM, 10-Mar-2008.) $)
|
|
adddi $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
|
|
( ax-distr ) ABCD $.
|
|
|
|
$( A real number is a complex number. (Contributed by NM, 10-Aug-1999.) $)
|
|
recn $p |- ( A e. RR -> A e. CC ) $=
|
|
( cr cc ax-resscn sseli ) BCADE $.
|
|
|
|
$( The real numbers form a set. (Contributed by Mario Carneiro,
|
|
17-Nov-2014.) $)
|
|
reex $p |- RR e. _V $=
|
|
( cr cc cnex ax-resscn ssexi ) ABCDE $.
|
|
|
|
$( Reals are a subset of the pair of real and complex numbers (common case).
|
|
(Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
reelprrecn $p |- RR e. { RR , CC } $=
|
|
( cr cc reex prid1 ) ABCD $.
|
|
|
|
$( Complex numbers are a subset of the pair of real and complex numbers
|
|
(common case). (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
cnelprrecn $p |- CC e. { RR , CC } $=
|
|
( cr cc cnex prid2 ) ABCD $.
|
|
|
|
$( Distributive law for complex numbers (right-distributivity). (Contributed
|
|
by NM, 10-Oct-2004.) $)
|
|
adddir $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) ) $=
|
|
( cc wcel w3a caddc co cmul adddi 3coml wa addcl mulcom sylan 3impa 3adant2
|
|
wceq 3adant1 oveq12d 3eqtr4d ) ADEZBDEZCDEZFZCABGHZIHZCAIHZCBIHZGHZUFCIHZAC
|
|
IHZBCIHZGHUDUBUCUGUJRCABJKUBUCUDUKUGRZUBUCLUFDEUDUNABMUFCNOPUEULUHUMUIGUBUD
|
|
ULUHRUCACNQUCUDUMUIRUBBCNSTUA $.
|
|
|
|
$( 0 is a complex number. (Contributed by NM, 19-Feb-2005.) $)
|
|
0cn $p |- 0 e. CC $=
|
|
( ci cmul co c1 caddc cc0 cc ax-i2m1 wcel ax-icn mulcl mp2an addcl eqeltrri
|
|
ax-1cn ) AABCZDECZFGHPGIZDGIQGIAGIZSRJJAAKLOPDMLN $.
|
|
|
|
$( 0 is a complex number, deductive form. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
0cnd $p |- ( ph -> 0 e. CC ) $=
|
|
( cc0 cc wcel 0cn a1i ) BCDAEF $.
|
|
|
|
$( 0 is a set (common case). (Contributed by David A. Wheeler,
|
|
7-Jul-2016.) $)
|
|
c0ex $p |- 0 e. _V $=
|
|
( cc0 cc 0cn elexi ) ABCD $.
|
|
|
|
$( 1 is a set. Common special case. (Contributed by David A. Wheeler,
|
|
7-Jul-2016.) $)
|
|
1ex $p |- 1 e. _V $=
|
|
( c1 cc ax-1cn elexi ) ABCD $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Alias for ~ ax-cnre , for naming consistency. (Contributed by NM,
|
|
3-Jan-2013.) $)
|
|
cnre $p |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
|
|
( ax-cnre ) ABCD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( ` 1 ` is an identity element for multiplication. Based on ideas by Eric
|
|
Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) $)
|
|
mulid1 $p |- ( A e. CC -> ( A x. 1 ) = A ) $=
|
|
( vx vy cc wcel cv ci cmul co caddc wceq cr wrex c1 ax-icn ax-1cn ax-1rid
|
|
recn syl eqtrd cnre wa sylancr adddir mp3an3 syl2an mulass mp3an13 oveq2d
|
|
mulcl oveqan12d oveq1 id eqeq12d syl5ibrcom rexlimivv ) ADEABFZGCFZHIZJIZ
|
|
KZCLMBLMANHIZAKZBCAUAVAVCBCLLUQLEZURLEZUBZVCVAUTNHIZUTKVFVGUQNHIZUSNHIZJI
|
|
ZUTVDUQDEZUSDEZVGVJKZVEUQRVEGDEZURDEZVLOURRZGURUJUCVKVLNDEZVMPUQUSNUDUEUF
|
|
VDVEVHUQVIUSJUQQVEVIGURNHIZHIZUSVEVOVIVSKZVPVNVOVQVTOPGURNUGUHSVEVRURGHUR
|
|
QUITUKTVAVBVGAUTAUTNHULVAUMUNUOUPS $.
|
|
$}
|
|
|
|
$( Identity law for multiplication. Note: see ~ mulid1 for commuted version.
|
|
(Contributed by NM, 8-Oct-1999.) $)
|
|
mulid2 $p |- ( A e. CC -> ( 1 x. A ) = A ) $=
|
|
( cc wcel c1 cmul co wceq ax-1cn mulcom mpan mulid1 eqtrd ) ABCZDAEFZADEFZA
|
|
DBCMNOGHDAIJAKL $.
|
|
|
|
$( ` 1 ` is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) $)
|
|
1re $p |- 1 e. RR $=
|
|
( ax-1re ) A $.
|
|
|
|
$( ` 0 ` is a real number. (Contributed by Eric Schmidt, 21-May-2007.)
|
|
(Revised by Scott Fenton, 3-Jan-2013.) $)
|
|
0re $p |- 0 e. RR $=
|
|
( vx c1 cr wcel cv caddc cc0 wceq wrex 1re ax-rnegex readdcl mpan syl5ibcom
|
|
co eleq1 rexlimiv mp2b ) BCDZBAEZFOZGHZACIGCDZJABKUBUCACTCDZUACDZUBUCSUDUEJ
|
|
BTLMUAGCPNQR $.
|
|
|
|
$( ` 0 ` is a real number, deductive form. (Contributed by David A. Wheeler,
|
|
6-Dec-2018.) $)
|
|
0red $p |- ( ph -> 0 e. RR ) $=
|
|
( cc0 cr wcel 0re a1i ) BCDAEF $.
|
|
|
|
${
|
|
axi.1 $e |- A e. CC $.
|
|
$( Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) $)
|
|
mulid1i $p |- ( A x. 1 ) = A $=
|
|
( cc wcel c1 cmul co wceq mulid1 ax-mp ) ACDAEFGAHBAIJ $.
|
|
|
|
$( Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) $)
|
|
mulid2i $p |- ( 1 x. A ) = A $=
|
|
( cc wcel c1 cmul co wceq mulid2 ax-mp ) ACDEAFGAHBAIJ $.
|
|
|
|
axi.2 $e |- B e. CC $.
|
|
$( Closure law for addition. (Contributed by NM, 23-Nov-1994.) $)
|
|
addcli $p |- ( A + B ) e. CC $=
|
|
( cc wcel caddc co addcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) $)
|
|
mulcli $p |- ( A x. B ) e. CC $=
|
|
( cc wcel cmul co mulcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( Commutative law for multiplication. (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
mulcomi $p |- ( A x. B ) = ( B x. A ) $=
|
|
( cc wcel cmul co wceq mulcom mp2an ) AEFBEFABGHBAGHICDABJK $.
|
|
|
|
${
|
|
mulcomli.3 $e |- ( A x. B ) = C $.
|
|
$( Commutative law for multiplication. (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
mulcomli $p |- ( B x. A ) = C $=
|
|
( cmul co mulcomi eqtri ) BAGHABGHCBAEDIFJ $.
|
|
$}
|
|
|
|
axi.3 $e |- C e. CC $.
|
|
$( Associative law for addition. (Contributed by NM, 23-Nov-1994.) $)
|
|
addassi $p |- ( ( A + B ) + C ) = ( A + ( B + C ) ) $=
|
|
( cc wcel caddc co wceq addass mp3an ) AGHBGHCGHABIJCIJABCIJIJKDEFABCLM
|
|
$.
|
|
|
|
$( Associative law for multiplication. (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
mulassi $p |- ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) $=
|
|
( cc wcel cmul co wceq mulass mp3an ) AGHBGHCGHABIJCIJABCIJIJKDEFABCLM $.
|
|
|
|
$( Distributive law (left-distributivity). (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
adddii $p |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) $=
|
|
( cc wcel caddc co cmul wceq adddi mp3an ) AGHBGHCGHABCIJKJABKJACKJIJLDEF
|
|
ABCMN $.
|
|
|
|
$( Distributive law (right-distributivity). (Contributed by NM,
|
|
16-Feb-1995.) $)
|
|
adddiri $p |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) $=
|
|
( cc wcel caddc co cmul wceq adddir mp3an ) AGHBGHCGHABIJCKJACKJBCKJIJLDE
|
|
FABCMN $.
|
|
$}
|
|
|
|
${
|
|
recni.1 $e |- A e. RR $.
|
|
$( A real number is a complex number. (Contributed by NM, 1-Mar-1995.) $)
|
|
recni $p |- A e. CC $=
|
|
( cr cc ax-resscn sselii ) CDAEBF $.
|
|
|
|
axri.2 $e |- B e. RR $.
|
|
$( Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) $)
|
|
readdcli $p |- ( A + B ) e. RR $=
|
|
( cr wcel caddc co readdcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( Closure law for multiplication of reals. (Contributed by NM,
|
|
17-Jan-1997.) $)
|
|
remulcli $p |- ( A x. B ) e. RR $=
|
|
( cr wcel cmul co remulcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
$}
|
|
|
|
$( 1 is an real number, deductive form (common case). (Contributed by David
|
|
A. Wheeler, 6-Dec-2018.) $)
|
|
1red $p |- ( ph -> 1 e. RR ) $=
|
|
( c1 cr wcel 1re a1i ) BCDAEF $.
|
|
|
|
$( 1 is a complex number, deductive form (common case). (Contributed by
|
|
David A. Wheeler, 6-Dec-2018.) $)
|
|
1cnd $p |- ( ph -> 1 e. CC ) $=
|
|
( c1 cc wcel ax-1cn a1i ) BCDAEF $.
|
|
|
|
${
|
|
addcld.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Identity law for multiplication. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulid1d $p |- ( ph -> ( A x. 1 ) = A ) $=
|
|
( cc wcel c1 cmul co wceq mulid1 syl ) ABDEBFGHBICBJK $.
|
|
|
|
$( Identity law for multiplication. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulid2d $p |- ( ph -> ( 1 x. A ) = A ) $=
|
|
( cc wcel c1 cmul co wceq mulid2 syl ) ABDEFBGHBICBJK $.
|
|
|
|
addcld.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Closure law for addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addcld $p |- ( ph -> ( A + B ) e. CC ) $=
|
|
( cc wcel caddc co addcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure law for multiplication. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulcld $p |- ( ph -> ( A x. B ) e. CC ) $=
|
|
( cc wcel cmul co mulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Commutative law for multiplication. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulcomd $p |- ( ph -> ( A x. B ) = ( B x. A ) ) $=
|
|
( cc wcel cmul co wceq mulcom syl2anc ) ABFGCFGBCHICBHIJDEBCKL $.
|
|
|
|
addassd.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Associative law for addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addassd $p |- ( ph -> ( ( A + B ) + C ) = ( A + ( B + C ) ) ) $=
|
|
( cc wcel caddc co wceq addass syl3anc ) ABHICHIDHIBCJKDJKBCDJKJKLEFGBCDM
|
|
N $.
|
|
|
|
$( Associative law for multiplication. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulassd $p |- ( ph -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) ) $=
|
|
( cc wcel cmul co wceq mulass syl3anc ) ABHICHIDHIBCJKDJKBCDJKJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Distributive law (left-distributivity). (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
adddid $p |- ( ph -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) ) $=
|
|
( cc wcel caddc co cmul wceq adddi syl3anc ) ABHICHIDHIBCDJKLKBCLKBDLKJKM
|
|
EFGBCDNO $.
|
|
|
|
$( Distributive law (right-distributivity). (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
adddird $p |- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) ) $=
|
|
( cc wcel caddc co cmul wceq adddir syl3anc ) ABHICHIDHIBCJKDLKBDLKCDLKJK
|
|
MEFGBCDNO $.
|
|
$}
|
|
|
|
${
|
|
adddirp1d.a $e |- ( ph -> A e. CC ) $.
|
|
adddirp1d.b $e |- ( ph -> B e. CC ) $.
|
|
$( Distributive law, plus 1 version. (Contributed by Glauco Siliprandi,
|
|
11-Dec-2019.) $)
|
|
adddirp1d $p |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) ) $=
|
|
( c1 caddc co cmul 1cnd adddird mulid2d oveq2d eqtrd ) ABFGHCIHBCIHZFCIHZ
|
|
GHOCGHABFCDAJEKAPCOGACELMN $.
|
|
$}
|
|
|
|
${
|
|
joinlmuladdmuld.1 $e |- ( ph -> A e. CC ) $.
|
|
joinlmuladdmuld.2 $e |- ( ph -> B e. CC ) $.
|
|
joinlmuladdmuld.3 $e |- ( ph -> C e. CC ) $.
|
|
joinlmuladdmuld.4 $e |- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D ) $.
|
|
$( Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler,
|
|
26-Oct-2019.) $)
|
|
joinlmuladdmuld $p |- ( ph -> ( ( A + C ) x. B ) = D ) $=
|
|
( caddc co cmul adddird eqtrd ) ABDJKCLKBCLKDCLKJKEABDCFHGMIN $.
|
|
$}
|
|
|
|
${
|
|
recnd.1 $e |- ( ph -> A e. RR ) $.
|
|
$( Deduction from real number to complex number. (Contributed by NM,
|
|
26-Oct-1999.) $)
|
|
recnd $p |- ( ph -> A e. CC ) $=
|
|
( cr wcel cc recn syl ) ABDEBFECBGH $.
|
|
|
|
readdcld.2 $e |- ( ph -> B e. RR ) $.
|
|
$( Closure law for addition of reals. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
readdcld $p |- ( ph -> ( A + B ) e. RR ) $=
|
|
( cr wcel caddc co readdcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure law for multiplication of reals. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
remulcld $p |- ( ph -> ( A x. B ) e. RR ) $=
|
|
( cr wcel cmul co remulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Infinity and the extended real number system
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c <_ $. $( 'Less than or equal to' relation. $)
|
|
$c +oo $. $( Plus infinity $)
|
|
$c -oo $. $( Minus infinity $)
|
|
$c RR* $. $( The set of extended reals $)
|
|
$c < $. $( 'Less than' relation (over extended reals) $)
|
|
|
|
$( Plus infinity. $)
|
|
cpnf $a class +oo $.
|
|
$( Minus infinity. $)
|
|
cmnf $a class -oo $.
|
|
$( The set of extended reals (includes plus and minus infinity). $)
|
|
cxr $a class RR* $.
|
|
$( 'Less than' predicate (extended to include the extended reals). $)
|
|
clt $a class < $.
|
|
$( Extend wff notation to include the 'less than or equal to' relation. $)
|
|
cle $a class <_ $.
|
|
|
|
$( Define plus infinity. Note that the definition is arbitrary, requiring
|
|
only that ` +oo ` be a set not in ` RR ` and different from ` -oo `
|
|
( ~ df-mnf ). We use ` ~P U. CC ` to make it independent of the
|
|
construction of ` CC ` , and Cantor's Theorem will show that it is
|
|
different from any member of ` CC ` and therefore ` RR ` . See ~ pnfnre
|
|
and ~ mnfnre , and we'll also be able to prove ` +oo =/= -oo ` .
|
|
|
|
A simpler possibility is to define ` +oo ` as ` CC ` and ` -oo ` as
|
|
` { CC } ` , but that approach requires the Axiom of Regularity to show
|
|
that ` +oo ` and ` -oo ` are different from each other and from all
|
|
members of ` RR ` . (Contributed by NM, 13-Oct-2005.)
|
|
(New usage is discouraged.) $)
|
|
df-pnf $a |- +oo = ~P U. CC $.
|
|
|
|
$( Define minus infinity as the power set of plus infinity. Note that the
|
|
definition is arbitrary, requiring only that ` -oo ` be a set not in
|
|
` RR ` and different from ` +oo ` (see ~ mnfnre ). (Contributed by NM,
|
|
13-Oct-2005.) (New usage is discouraged.) $)
|
|
df-mnf $a |- -oo = ~P +oo $.
|
|
|
|
$( Define the set of extended reals that includes plus and minus infinity.
|
|
Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM,
|
|
13-Oct-2005.) $)
|
|
df-xr $a |- RR* = ( RR u. { +oo , -oo } ) $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define 'less than' on the set of extended reals. Definition 12-3.1 of
|
|
[Gleason] p. 173. Note that in our postulates for complex numbers,
|
|
` <RR ` is primitive and not necessarily a relation on ` RR ` .
|
|
(Contributed by NM, 13-Oct-2005.) $)
|
|
df-ltxr $a |- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) }
|
|
u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) $.
|
|
$}
|
|
|
|
$( Define 'less than or equal to' on the extended real subset of complex
|
|
numbers. (Contributed by NM, 13-Oct-2005.) $)
|
|
df-le $a |- <_ = ( ( RR* X. RR* ) \ `' < ) $.
|
|
|
|
$( Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) $)
|
|
pnfnre $p |- +oo e/ RR $=
|
|
( cpnf cr wcel cc cuni cpw cvv wn uniex pwuninel2 ax-mp df-pnf eleq1i mtbir
|
|
cnex recn mto nelir ) ABABCADCZSDEZFZDCZTGCUBHDOIDGJKAUADLMNAPQR $.
|
|
|
|
$( Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) $)
|
|
mnfnre $p |- -oo e/ RR $=
|
|
( cmnf cr wcel cc cuni cpw cvv wn cnex 2pwuninelg ax-mp df-mnf df-pnf pweqi
|
|
cpnf eqtri eleq1i mtbir recn mto nelir ) ABABCADCZUBDEFZFZDCZDGCUEHIDGJKAUD
|
|
DAOFUDLOUCMNPQRASTUA $.
|
|
|
|
$( The standard reals are a subset of the extended reals. (Contributed by
|
|
NM, 14-Oct-2005.) $)
|
|
ressxr $p |- RR C_ RR* $=
|
|
( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtr4i ) AABCDZEFAJGHI $.
|
|
|
|
$( The Cartesian product of standard reals are a subset of the Cartesian
|
|
product of extended reals (common case). (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
rexpssxrxp $p |- ( RR X. RR ) C_ ( RR* X. RR* ) $=
|
|
( cr cxr wss cxp ressxr xpss12 mp2an ) ABCZHAADBBDCEEABABFG $.
|
|
|
|
$( A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) $)
|
|
rexr $p |- ( A e. RR -> A e. RR* ) $=
|
|
( cr cxr ressxr sseli ) BCADE $.
|
|
|
|
$( Zero is an extended real. (Contributed by Mario Carneiro,
|
|
15-Jun-2014.) $)
|
|
0xr $p |- 0 e. RR* $=
|
|
( cr cxr cc0 ressxr 0re sselii ) ABCDEF $.
|
|
|
|
$( No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.)
|
|
(Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
renepnf $p |- ( A e. RR -> A =/= +oo ) $=
|
|
( cr wcel cpnf wceq pnfnre neli eleq1 mtbiri necon2ai ) ABCZADADEKDBCDBFGAD
|
|
BHIJ $.
|
|
|
|
$( No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof
|
|
shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
renemnf $p |- ( A e. RR -> A =/= -oo ) $=
|
|
( cr wcel cmnf wceq mnfnre neli eleq1 mtbiri necon2ai ) ABCZADADEKDBCDBFGAD
|
|
BHIJ $.
|
|
|
|
${
|
|
rexrd.1 $e |- ( ph -> A e. RR ) $.
|
|
$( A standard real is an extended real. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rexrd $p |- ( ph -> A e. RR* ) $=
|
|
( cr cxr ressxr sseldi ) ADEBFCG $.
|
|
|
|
$( No (finite) real equals plus infinity. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
renepnfd $p |- ( ph -> A =/= +oo ) $=
|
|
( cr wcel cpnf wne renepnf syl ) ABDEBFGCBHI $.
|
|
|
|
$( No real equals minus infinity. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
renemnfd $p |- ( ph -> A =/= -oo ) $=
|
|
( cr wcel cmnf wne renemnf syl ) ABDEBFGCBHI $.
|
|
$}
|
|
|
|
$( Plus infinity belongs to the set of extended reals. (Contributed by NM,
|
|
13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) $)
|
|
pnfxr $p |- +oo e. RR* $=
|
|
( cpnf cr cmnf cpr cun cxr ssun2 cc cuni cpw df-pnf cnex uniex pwex eqeltri
|
|
cvv prid1 sselii df-xr eleqtrri ) ABACDZEZFUAUBAUABGACAHIZJPKUCHLMNOQRST $.
|
|
|
|
$( Plus infinity exists (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
pnfex $p |- +oo e. _V $=
|
|
( cpnf cxr pnfxr elexi ) ABCD $.
|
|
|
|
$( Plus and minus infinity are different elements of ` RR* ` . (Contributed
|
|
by NM, 14-Oct-2005.) $)
|
|
pnfnemnf $p |- +oo =/= -oo $=
|
|
( cpnf cpw cmnf cxr wcel wne pnfxr pwne ax-mp necomi df-mnf neeqtrri ) AABZ
|
|
CMAADEMAFGADHIJKL $.
|
|
|
|
$( Minus and plus infinity are different (common case). (Contributed by
|
|
David A. Wheeler, 8-Dec-2018.) $)
|
|
mnfnepnf $p |- -oo =/= +oo $=
|
|
( cpnf cmnf pnfnemnf necomi ) ABCD $.
|
|
|
|
$( Minus infinity belongs to the set of extended reals. (Contributed by NM,
|
|
13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof
|
|
shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
mnfxr $p |- -oo e. RR* $=
|
|
( cmnf cr cpnf cpr cun cxr wcel cpw cvv df-mnf pnfex pwex prid2 elun2 ax-mp
|
|
eqeltri df-xr eleqtrri ) ABCADZEZFASGATGCAACHIJCKLPMASBNOQR $.
|
|
|
|
${
|
|
rexri.1 $e |- A e. RR $.
|
|
$( A standard real is an extended real (inference form.) (Contributed by
|
|
David Moews, 28-Feb-2017.) $)
|
|
rexri $p |- A e. RR* $=
|
|
( cr wcel cxr rexr ax-mp ) ACDAEDBAFG $.
|
|
$}
|
|
|
|
$( The reals and the infinities are disjoint. (Contributed by NM,
|
|
25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
renfdisj $p |- ( RR i^i { +oo , -oo } ) = (/) $=
|
|
( vx cr cpnf cmnf cpr cin c0 wceq cv wcel wn disj vex elpr renepnf necon2bi
|
|
wo renemnf jaoi sylbi con2i mprgbir ) BCDEZFGHAIZUCJZKABABUCLUEUDBJZUEUDCHZ
|
|
UDDHZQUFKZUDCDAMNUGUIUHUFUDCUDOPUFUDDUDRPSTUAUB $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( 'Less than' is a relation on extended reals. (Contributed by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
ltrelxr $p |- < C_ ( RR* X. RR* ) $=
|
|
( vx vy cv cr wcel copab cmnf csn cun cxp cxr wa eqsstri sstri wss ressxr
|
|
cpnf unssi xpss12 mp2an clt cltrr wbr w3a df-ltxr df-3an opabbii opabssxp
|
|
rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtr4i snsspr1 ) UAACZDEZBCZDEZUPURU
|
|
BUCZUDZABFZDGHZIZQHZJZVCDJZIZIKKJZABUEVBVHVIVBDDJZVIVBUQUSLUTLZABFVJVAVKA
|
|
BUQUSUTUFUGUTABDDUHMUINVFVGVIVDKOVEKOVFVIODVCKPVCQGUJZKQGUKVLDVLIKVLDULUM
|
|
UNZNZRVEVLKQGUOVMNVDKVEKSTVCKODKOVGVIOVNPVCKDKSTRRM $.
|
|
$}
|
|
|
|
$( 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) $)
|
|
ltrel $p |- Rel < $=
|
|
( clt cxr cxp wss wrel ltrelxr relxp relss mp2 ) ABBCZDJEAEFBBGAJHI $.
|
|
|
|
$( 'Less than or equal' is a relation on extended reals. (Contributed by
|
|
Mario Carneiro, 28-Apr-2015.) $)
|
|
lerelxr $p |- <_ C_ ( RR* X. RR* ) $=
|
|
( cle cxr cxp clt ccnv cdif df-le difss eqsstri ) ABBCZDEZFJGJKHI $.
|
|
|
|
$( 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.)
|
|
(Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
lerel $p |- Rel <_ $=
|
|
( cle cxr cxp wss wrel lerelxr relxp relss mp2 ) ABBCZDJEAEFBBGAJHI $.
|
|
|
|
$( 'Less than or equal to' expressed in terms of 'less than', for extended
|
|
reals. (Contributed by NM, 14-Oct-2005.) $)
|
|
xrlenlt $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) ) $=
|
|
( cxr wcel wa cle wbr cop clt ccnv wn df-br cxp wb opelxpi cdif df-le eldif
|
|
eleq2i bitri baib syl syl5bb opelcnvg syl6rbbr notbid bitr4d ) ACDBCDEZABFG
|
|
ZABHZIJZDZKZBAIGZKUIUJFDZUHUMABFLUHUJCCMZDZUOUMNABCCOUOUQUMUOUJUPUKPZDUQUME
|
|
FURUJQSUJUPUKRTUAUBUCUHUNULUHULBAHIDUNABCCIUDBAILUEUFUG $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The standard less-than ` <RR ` and the extended real less-than ` < ` are
|
|
identical when restricted to the non-extended reals ` RR ` .
|
|
(Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
ltxrlt $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) ) $=
|
|
( vx vy cr wcel clt wbr cltrr cv w3a cmnf csn cun cpnf cxp wo brun wceq
|
|
wi wa copab df-ltxr breqi bitri eleq1 breq1 3anbi13d breq2 3anbi23d brabg
|
|
eqid simp3 syl6bi brxp simprbi elsni syl a1i renepnf neneqd pm2.24 syl6ci
|
|
adantl simplbi renemnf adantr jaod syl5bi 3adant3 ibir orcd sylibr 3expia
|
|
wn wb impbid ) AEFZBEFZUAZABGHZABIHZWAABCJZEFZDJZEFZWCWEIHZKZCDUBZHZABELM
|
|
ZNZOMZPZWKEPZNZHZQZVTWBWAABWIWPNZHWRABGWSCDUCUDABWIWPRUEZVTWJWBWQVTWJVRVS
|
|
WBKZWBWHVRWFAWEIHZKXACDABEEWIWCASWDVRWGXBWFWCAEUFWCAWEIUGUHWEBSWFVSXBWBVR
|
|
WEBEUFWEBAIUIUJWIULUKZVRVSWBUMUNWQABWNHZABWOHZQVTWBABWNWORVTXDWBXEVSXDWBT
|
|
VRVSXDBOSZXFVOWBXDXFTVSXDBWMFZXFXDAWLFXGABWLWMUOUPBOUQURUSVSBOBUTVAXFWBVB
|
|
VCVDVRXEWBTVSVRXEALSZXHVOWBXEXHTVRXEAWKFZXHXEXIVSABWKEUOVEALUQURUSVRALAVF
|
|
VAXHWBVBVCVGVHVIVHVIVRVSWBWAXAWRWAXAWJWQXAWJVRVSWJXAVPWBXCVJVKVLWTVMVNVQ
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Restate the ordering postulates with extended real "less than"
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Real number less-than is irreflexive. Axiom for real and complex numbers,
|
|
derived from set theory. This restates ~ ax-pre-ltirr with ordering on
|
|
the extended reals. New proofs should use ~ ltnr instead for naming
|
|
consistency. (New usage is discouraged.) (Contributed by Jim Kingdon,
|
|
15-Jan-2020.) $)
|
|
axltirr $p |- ( A e. RR -> -. A < A ) $=
|
|
( cr wcel clt wbr cltrr ax-pre-ltirr wb ltxrlt anidms mtbird ) ABCZAADEZAAF
|
|
EZAGLMNHAAIJK $.
|
|
|
|
$( Real number less-than is weakly linear. Axiom for real and complex
|
|
numbers, derived from set theory. This restates ~ ax-pre-ltwlin with
|
|
ordering on the extended reals. (Contributed by Jim Kingdon,
|
|
15-Jan-2020.) $)
|
|
axltwlin $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B -> ( A < C \/ C < B ) ) ) $=
|
|
( cr wcel w3a cltrr wbr wo clt ax-pre-ltwlin ltxrlt 3adant3 3adant2 3adant1
|
|
wb ancoms orbi12d 3imtr4d ) ADEZBDEZCDEZFZABGHZACGHZCBGHZIABJHZACJHZCBJHZIA
|
|
BCKTUAUGUDPUBABLMUCUHUEUIUFTUBUHUEPUAACLNUAUBUIUFPZTUBUAUJCBLQORS $.
|
|
|
|
$( Ordering on reals is transitive. Axiom for real and complex numbers,
|
|
derived from set theory. This restates ~ ax-pre-lttrn with ordering on
|
|
the extended reals. New proofs should use ~ lttr instead for naming
|
|
consistency. (New usage is discouraged.) (Contributed by NM,
|
|
13-Oct-2005.) $)
|
|
axlttrn $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A < B /\ B < C ) -> A < C ) ) $=
|
|
( cr wcel w3a cltrr wbr wa clt ax-pre-lttrn 3adant3 3adant1 anbi12d 3adant2
|
|
wb ltxrlt 3imtr4d ) ADEZBDEZCDEZFZABGHZBCGHZIACGHZABJHZBCJHZIACJHZABCKUBUFU
|
|
CUGUDSTUFUCPUAABQLTUAUGUDPSBCQMNSUAUHUEPTACQOR $.
|
|
|
|
$( Ordering property of addition on reals. Axiom for real and complex
|
|
numbers, derived from set theory. (This restates ~ ax-pre-ltadd with
|
|
ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) $)
|
|
axltadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B -> ( C + A ) < ( C + B ) ) ) $=
|
|
( cr wcel w3a cltrr wbr caddc co clt ax-pre-ltadd wb ltxrlt 3adant3 readdcl
|
|
wa syl2an 3impdi 3coml 3imtr4d ) ADEZBDEZCDEZFABGHZCAIJZCBIJZGHZABKHZUFUGKH
|
|
ZABCLUBUCUIUEMUDABNOUDUBUCUJUHMZUDUBUCUKUDUBQUFDEUGDEUKUDUCQCAPCBPUFUGNRSTU
|
|
A $.
|
|
|
|
$( Apartness of reals is tight. Axiom for real and complex numbers, derived
|
|
from set theory. (This restates ~ ax-pre-apti with ordering on the
|
|
extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) $)
|
|
axapti $p |- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B ) $=
|
|
( cr wcel clt wbr wo wn wceq cltrr ltxrlt ancoms orbi12d notbid ax-pre-apti
|
|
wa wb 3expia sylbid 3impia ) ACDZBCDZABEFZBAEFZGZHZABIZUAUBPZUFABJFZBAJFZGZ
|
|
HZUGUHUEUKUHUCUIUDUJABKUBUAUDUJQBAKLMNUAUBULUGABORST $.
|
|
|
|
$( The product of two positive reals is positive. Axiom for real and complex
|
|
numbers, derived from set theory. (This restates ~ ax-pre-mulgt0 with
|
|
ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) $)
|
|
axmulgt0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) ) $=
|
|
( cr wcel wa cc0 cltrr wbr cmul co clt ax-pre-mulgt0 wb 0re ltxrlt bi2anan9
|
|
mpan remulcl sylancr 3imtr4d ) ACDZBCDZEZFAGHZFBGHZEFABIJZGHZFAKHZFBKHZEFUF
|
|
KHZABLUAUHUDUBUIUEFCDZUAUHUDMNFAOQUKUBUIUEMNFBOQPUCUKUFCDUJUGMNABRFUFOST $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordering on reals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Alias for ~ axlttrn , for naming consistency with ~ lttri . New proofs
|
|
should generally use this instead of ~ ax-pre-lttrn . (Contributed by NM,
|
|
10-Mar-2008.) $)
|
|
lttr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A < B /\ B < C ) -> A < C ) ) $=
|
|
( axlttrn ) ABCD $.
|
|
|
|
$( The product of two positive numbers is positive. (Contributed by NM,
|
|
10-Mar-2008.) $)
|
|
mulgt0 $p |- ( ( ( A e. RR /\ 0 < A ) /\
|
|
( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) ) $=
|
|
( cr wcel cc0 clt wbr cmul co wa axmulgt0 imp an4s ) ACDZBCDZEAFGZEBFGZEABH
|
|
IFGZNOJPQJRABKLM $.
|
|
|
|
$( 'Less than or equal to' expressed in terms of 'less than'. Part of
|
|
definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM,
|
|
13-May-1999.) $)
|
|
lenlt $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) ) $=
|
|
( cr wcel cxr cle wbr clt wn wb rexr xrlenlt syl2an ) ACDAEDBEDABFGBAHGIJBC
|
|
DAKBKABLM $.
|
|
|
|
$( 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) $)
|
|
ltnr $p |- ( A e. RR -> -. A < A ) $=
|
|
( axltirr ) AB $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) $)
|
|
ltso $p |- < Or RR $=
|
|
( vx vy vz cr clt wor wpo cv wbr wo wi wral wtru wcel wn ltnr adantl lttr
|
|
w3a wa ispod trud axltwlin rgen3 df-iso mpbir2an ) DEFDEGZAHZBHZEIZUHCHZE
|
|
IZUKUIEIJKZCDLBDLADLUGMABCDEUHDNZUHUHEIOMUHPQUNUIDNUKDNSUJUIUKEITULKMUHUI
|
|
UKRQUAUBUMABCDDDUHUIUKUCUDABCDEUEUF $.
|
|
$}
|
|
|
|
$( 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) $)
|
|
gtso $p |- `' < Or RR $=
|
|
( vx cr clt wor ccnv ltso cc0 wcel cv wex wb 0re elex2 cnvsom mp2b mpbi ) B
|
|
CDZBCEDZFGBHAIBHAJQRKLAGBMABCNOP $.
|
|
|
|
$( Tightness of real apartness. (Contributed by NM, 5-May-1999.) $)
|
|
lttri3 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
|
|
( cr wcel wa wceq clt wbr wn wi ltnr breq2 notbid syl5ibcom breq1 adantr wo
|
|
jcad ioran axapti 3expia syl5bir impbid ) ACDZBCDZEZABFZABGHZIZBAGHZIZEZUDU
|
|
GULJUEUDUGUIUKUDAAGHZIZUGUIAKZUGUMUHABAGLMNUDUNUGUKUOUGUMUJABAGOMNRPULUHUJQ
|
|
IZUFUGUHUJSUDUEUPUGABTUAUBUC $.
|
|
|
|
$( Tightness of real apartness. (Contributed by NM, 14-May-1999.) $)
|
|
letri3 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
|
|
( cr wcel wa wceq clt wbr wn cle lttri3 ancom syl6bbr ancoms anbi12d bitr4d
|
|
lenlt wb ) ACDZBCDZEZABFZBAGHIZABGHIZEZABJHZBAJHZEUAUBUDUCEUEABKUCUDLMUAUFU
|
|
CUGUDABQTSUGUDRBAQNOP $.
|
|
|
|
$( Transitive law, weaker form of ` ( A < B /\ B <_ C ) -> A < C ` .
|
|
(Contributed by AV, 14-Oct-2018.) $)
|
|
ltleletr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A < B /\ B <_ C ) -> A <_ C ) ) $=
|
|
( cr wcel w3a clt wbr cle wn wi wa lttr 3coml expcomd con3 wb lenlt 3adant1
|
|
syl6 3adant2 imbi12d sylibrd impd ) ADEZBDEZCDEZFZABGHZBCIHZACIHZUHUICBGHZJ
|
|
ZCAGHZJZKZUJUKKUHUIUNULKUPUHUNUIULUGUEUFUNUILULKCABMNOUNULPTUHUJUMUKUOUFUGU
|
|
JUMQUEBCRSUEUGUKUOQUFACRUAUBUCUD $.
|
|
|
|
$( Transitive law. (Contributed by NM, 12-Nov-1999.) $)
|
|
letr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A <_ B /\ B <_ C ) -> A <_ C ) ) $=
|
|
( cr w3a clt wbr wo wn cle wa wi axltwlin 3coml orcom syl6ib con3d wb lenlt
|
|
wcel 3adant3 3adant1 anbi12d ioran syl6bbr 3adant2 3imtr4d ) ADTZBDTZCDTZEZ
|
|
BAFGZCBFGZHZIZCAFGZIZABJGZBCJGZKZACJGZUKUPUNUKUPUMULHZUNUJUHUIUPVBLCABMNUMU
|
|
LOPQUKUTULIZUMIZKUOUKURVCUSVDUHUIURVCRUJABSUAUIUJUSVDRUHBCSUBUCULUMUDUEUHUJ
|
|
VAUQRUIACSUFUG $.
|
|
|
|
$( 'Less than or equal to' is reflexive. (Contributed by NM,
|
|
18-Aug-1999.) $)
|
|
leid $p |- ( A e. RR -> A <_ A ) $=
|
|
( cr wcel cle wbr clt wn ltnr wb lenlt anidms mpbird ) ABCZAADEZAAFEGZAHMNO
|
|
IAAJKL $.
|
|
|
|
$( 'Less than' implies not equal. See also ~ ltap which is the same but for
|
|
apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro,
|
|
16-Sep-2015.) $)
|
|
ltne $p |- ( ( A e. RR /\ A < B ) -> B =/= A ) $=
|
|
( cr wcel clt wbr wne wn wceq ltnr breq2 notbid syl5ibrcom necon2ad imp ) A
|
|
CDZABEFZBAGPQBAPQHBAIZAAEFZHAJRQSBAAEKLMNO $.
|
|
|
|
$( 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) $)
|
|
ltnsym $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) ) $=
|
|
( cr wcel wa clt wbr wi wn lttr 3anidm13 expd ltnr adantr con3 syl6ci ) ACD
|
|
ZBCDZEZABFGZBAFGZAAFGZHUBIZUAISTUAUBQRTUAEUBHABAJKLQUCRAMNUAUBOP $.
|
|
|
|
$( 'Less than' implies 'less than or equal to'. (Contributed by NM,
|
|
25-Aug-1999.) $)
|
|
ltle $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A < B -> A <_ B ) ) $=
|
|
( cr wcel wa clt wbr wn cle ltnsym lenlt sylibrd ) ACDBCDEABFGBAFGHABIGABJA
|
|
BKL $.
|
|
|
|
$( Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies).
|
|
(Contributed by NM, 23-May-1999.) $)
|
|
lelttr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A <_ B /\ B < C ) -> A < C ) ) $=
|
|
( cr wcel w3a cle wbr clt wa wn simprl wb simpl1 simpl2 lenlt syl2anc mpbid
|
|
pm2.21d idd wo simprr wi simpl3 axltwlin syl3anc mpd mpjaod ex ) ADEZBDEZCD
|
|
EZFZABGHZBCIHZJZACIHZUMUPJZBAIHZUQUQURUSUQURUNUSKZUMUNUOLURUJUKUNUTMUJUKULU
|
|
PNZUJUKULUPOZABPQRSURUQTURUOUSUQUAZUMUNUOUBURUKULUJUOVCUCVBUJUKULUPUDVABCAU
|
|
EUFUGUHUI $.
|
|
|
|
$( Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies).
|
|
(Contributed by NM, 25-Aug-1999.) $)
|
|
ltletr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A < B /\ B <_ C ) -> A < C ) ) $=
|
|
( cr wcel w3a clt wbr cle wa wn simprr wb simpl2 simpl3 lenlt syl2anc mpbid
|
|
wo simprl wi axltwlin adantr mpd ecased ex ) ADEZBDEZCDEZFZABGHZBCIHZJZACGH
|
|
ZUJUMJZUNCBGHZUOULUPKZUJUKULLUOUHUIULUQMUGUHUIUMNUGUHUIUMOBCPQRUOUKUNUPSZUJ
|
|
UKULTUJUKURUAUMABCUBUCUDUEUF $.
|
|
|
|
$( 'Less than' is antisymmetric and irreflexive. (Contributed by NM,
|
|
13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
ltnsym2 $p |- ( ( A e. RR /\ B e. RR ) -> -. ( A < B /\ B < A ) ) $=
|
|
( cr clt wor wcel wa wbr wn ltso so2nr mpan ) CDEACFBCFGABDHBADHGIJCABDKL
|
|
$.
|
|
|
|
$( Equality implies 'less than or equal to'. (Contributed by NM,
|
|
4-Apr-2005.) $)
|
|
eqle $p |- ( ( A e. RR /\ A = B ) -> A <_ B ) $=
|
|
( cr wcel cle wbr wceq leid breq2 biimpac sylan ) ACDAAEFZABGZABEFZAHMLNABA
|
|
EIJK $.
|
|
|
|
${
|
|
lt.1 $e |- A e. RR $.
|
|
$( 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) $)
|
|
ltnri $p |- -. A < A $=
|
|
( cr wcel clt wbr wn ltnr ax-mp ) ACDAAEFGBAHI $.
|
|
|
|
$( Equality implies 'less than or equal to'. (Contributed by NM,
|
|
23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) $)
|
|
eqlei $p |- ( A = B -> A <_ B ) $=
|
|
( cr wcel wceq cle wbr eqle mpan ) ADEABFABGHCABIJ $.
|
|
|
|
$( Equality implies 'less than or equal to'. (Contributed by Alexander van
|
|
der Vekens, 20-Mar-2018.) $)
|
|
eqlei2 $p |- ( B = A -> B <_ A ) $=
|
|
( cr wcel wceq cle wbr eleq1 mpbiri eqle mpancom ) BDEZBAFZBAGHNMADECBADI
|
|
JBAKL $.
|
|
|
|
${
|
|
ltneii.2 $e |- A < B $.
|
|
$( 'Less than' implies not equal. See also ~ gtapii which is the same
|
|
for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.) $)
|
|
gtneii $p |- B =/= A $=
|
|
( cr wcel clt wbr wne ltne mp2an ) AEFABGHBAICDABJK $.
|
|
|
|
$( 'Greater than' implies not equal. (Contributed by Mario Carneiro,
|
|
16-Sep-2015.) $)
|
|
ltneii $p |- A =/= B $=
|
|
( gtneii necomi ) BAABCDEF $.
|
|
$}
|
|
|
|
lt.2 $e |- B e. RR $.
|
|
$( Tightness of real apartness. (Contributed by NM, 14-May-1999.) $)
|
|
lttri3i $p |- ( A = B <-> ( -. A < B /\ -. B < A ) ) $=
|
|
( cr wcel wceq clt wbr wn wa wb lttri3 mp2an ) AEFBEFABGABHIJBAHIJKLCDABM
|
|
N $.
|
|
|
|
$( Tightness of real apartness. (Contributed by NM, 14-May-1999.) $)
|
|
letri3i $p |- ( A = B <-> ( A <_ B /\ B <_ A ) ) $=
|
|
( cr wcel wceq cle wbr wa wb letri3 mp2an ) AEFBEFABGABHIBAHIJKCDABLM $.
|
|
|
|
$( 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) $)
|
|
ltnsymi $p |- ( A < B -> -. B < A ) $=
|
|
( cr wcel clt wbr wn wi ltnsym mp2an ) AEFBEFABGHBAGHIJCDABKL $.
|
|
|
|
$( 'Less than or equal to' in terms of 'less than'. (Contributed by NM,
|
|
24-May-1999.) $)
|
|
lenlti $p |- ( A <_ B <-> -. B < A ) $=
|
|
( cr wcel cle wbr clt wn wb lenlt mp2an ) AEFBEFABGHBAIHJKCDABLM $.
|
|
|
|
$( 'Less than' implies 'less than or equal to'. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
ltlei $p |- ( A < B -> A <_ B ) $=
|
|
( cr wcel clt wbr cle wi ltle mp2an ) AEFBEFABGHABIHJCDABKL $.
|
|
|
|
${
|
|
ltlei.1 $e |- A < B $.
|
|
$( 'Less than' implies 'less than or equal to' (inference). (Contributed
|
|
by NM, 22-Aug-1999.) $)
|
|
ltleii $p |- A <_ B $=
|
|
( clt wbr cle ltlei ax-mp ) ABFGABHGEABCDIJ $.
|
|
$}
|
|
|
|
$( 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) $)
|
|
ltnei $p |- ( A < B -> B =/= A ) $=
|
|
( cr wcel clt wbr wne ltne mpan ) AEFABGHBAICABJK $.
|
|
|
|
${
|
|
lt.3 $e |- C e. RR $.
|
|
$( 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20.
|
|
(Contributed by NM, 14-May-1999.) $)
|
|
lttri $p |- ( ( A < B /\ B < C ) -> A < C ) $=
|
|
( cr wcel clt wbr wa wi lttr mp3an ) AGHBGHCGHABIJBCIJKACIJLDEFABCMN $.
|
|
|
|
$( 'Less than or equal to', 'less than' transitive law. (Contributed by
|
|
NM, 14-May-1999.) $)
|
|
lelttri $p |- ( ( A <_ B /\ B < C ) -> A < C ) $=
|
|
( cr wcel cle wbr clt wa wi lelttr mp3an ) AGHBGHCGHABIJBCKJLACKJMDEFAB
|
|
CNO $.
|
|
|
|
$( 'Less than', 'less than or equal to' transitive law. (Contributed by
|
|
NM, 14-May-1999.) $)
|
|
ltletri $p |- ( ( A < B /\ B <_ C ) -> A < C ) $=
|
|
( cr wcel clt wbr cle wa wi ltletr mp3an ) AGHBGHCGHABIJBCKJLACIJMDEFAB
|
|
CNO $.
|
|
|
|
$( 'Less than or equal to' is transitive. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
letri $p |- ( ( A <_ B /\ B <_ C ) -> A <_ C ) $=
|
|
( cr wcel cle wbr wa wi letr mp3an ) AGHBGHCGHABIJBCIJKACIJLDEFABCMN $.
|
|
|
|
$( Extended trichotomy law for 'less than or equal to'. (Contributed by
|
|
NM, 14-Aug-2000.) $)
|
|
le2tri3i $p |- ( ( A <_ B /\ B <_ C /\ C <_ A ) <->
|
|
( A = B /\ B = C /\ C = A ) ) $=
|
|
( cle wbr wceq wa letri letri3i biimpri sylan2 3impb 3comr eqcomd sylan
|
|
w3a eqlei 3impa 3jca 3anim123i impbii ) ABGHZBCGHZCAGHZSZABIZBCIZCAIZSU
|
|
HUIUJUKUEUFUGUIUFUGJUEBAGHZUIBCAEFDKUIUEULJABDELMNOUFUGUEUJUFUGUEUJUGUE
|
|
JUFCBGHZUJCABFDEKUJUFUMJBCEFLMNOPUEUFUGUKUEUFJACGHZUGUKABCDEFKUNUGJZACA
|
|
CIUOACDFLMQRUAUBUIUEUJUFUKUGABDTBCETCAFTUCUD $.
|
|
$}
|
|
|
|
$( The product of two positive numbers is positive. (Contributed by NM,
|
|
16-May-1999.) $)
|
|
mulgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co wi axmulgt0 mp2an ) AEFBEFGAHIGBHIJGABKL
|
|
HIMCDABNO $.
|
|
|
|
mulgt0i.3 $e |- 0 < A $.
|
|
mulgt0i.4 $e |- 0 < B $.
|
|
$( The product of two positive numbers is positive. (Contributed by NM,
|
|
18-May-1999.) $)
|
|
mulgt0ii $p |- 0 < ( A x. B ) $=
|
|
( cc0 clt wbr cmul co mulgt0i mp2an ) GAHIGBHIGABJKHIEFABCDLM $.
|
|
$}
|
|
|
|
${
|
|
ltd.1 $e |- ( ph -> A e. RR ) $.
|
|
$( 'Less than' is irreflexive. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltnrd $p |- ( ph -> -. A < A ) $=
|
|
( cr wcel clt wbr wn ltnr syl ) ABDEBBFGHCBIJ $.
|
|
|
|
${
|
|
ltned.2 $e |- ( ph -> A < B ) $.
|
|
$( 'Less than' implies not equal. See also ~ gtapd which is the same but
|
|
for apartness. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
gtned $p |- ( ph -> B =/= A ) $=
|
|
( cr wcel clt wbr wne ltne syl2anc ) ABFGBCHICBJDEBCKL $.
|
|
|
|
$( 'Greater than' implies not equal. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltned $p |- ( ph -> A =/= B ) $=
|
|
( gtned necomd ) ACBABCDEFG $.
|
|
$}
|
|
|
|
ltd.2 $e |- ( ph -> B e. RR ) $.
|
|
$( Tightness of real apartness. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
lttri3d $p |- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
|
|
( cr wcel wceq clt wbr wn wa wb lttri3 syl2anc ) ABFGCFGBCHBCIJKCBIJKLMDE
|
|
BCNO $.
|
|
|
|
$( Tightness of real apartness. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
letri3d $p |- ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
|
|
( cr wcel wceq cle wbr wa wb letri3 syl2anc ) ABFGCFGBCHBCIJCBIJKLDEBCMN
|
|
$.
|
|
|
|
$( 'Less than or equal to' in terms of 'less than'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
lenltd $p |- ( ph -> ( A <_ B <-> -. B < A ) ) $=
|
|
( cr wcel cle wbr clt wn wb lenlt syl2anc ) ABFGCFGBCHICBJIKLDEBCMN $.
|
|
|
|
${
|
|
ltled.1 $e |- ( ph -> A < B ) $.
|
|
$( 'Less than' implies 'less than or equal to'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltled $p |- ( ph -> A <_ B ) $=
|
|
( clt wbr cle cr wcel wi ltle syl2anc mpd ) ABCGHZBCIHZFABJKCJKPQLDEBCM
|
|
NO $.
|
|
|
|
$( 'Less than' implies 'less than or equal to'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltnsymd $p |- ( ph -> -. B < A ) $=
|
|
( cle wbr clt wn ltled lenltd mpbid ) ABCGHCBIHJABCDEFKABCDELM $.
|
|
$}
|
|
|
|
${
|
|
nltled.1 $e |- ( ph -> -. B < A ) $.
|
|
$( 'Not less than ' implies 'less than or equal to'. (Contributed by
|
|
Glauco Siliprandi, 11-Dec-2019.) $)
|
|
nltled $p |- ( ph -> A <_ B ) $=
|
|
( cle wbr clt wn lenltd mpbird ) ABCGHCBIHJFABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
lensymd.3 $e |- ( ph -> A <_ B ) $.
|
|
$( 'Less than or equal to' implies 'not less than'. (Contributed by
|
|
Glauco Siliprandi, 11-Dec-2019.) $)
|
|
lensymd $p |- ( ph -> -. B < A ) $=
|
|
( cle wbr clt wn lenltd mpbid ) ABCGHCBIHJFABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
mulgt0d.3 $e |- ( ph -> 0 < A ) $.
|
|
mulgt0d.4 $e |- ( ph -> 0 < B ) $.
|
|
$( The product of two positive numbers is positive. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
mulgt0d $p |- ( ph -> 0 < ( A x. B ) ) $=
|
|
( cr wcel cc0 clt wbr cmul co mulgt0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLD
|
|
FEGBCOP $.
|
|
$}
|
|
|
|
letrd.3 $e |- ( ph -> C e. RR ) $.
|
|
${
|
|
letrd.4 $e |- ( ph -> A <_ B ) $.
|
|
letrd.5 $e |- ( ph -> B <_ C ) $.
|
|
$( Transitive law deduction for 'less than or equal to'. (Contributed by
|
|
NM, 20-May-2005.) $)
|
|
letrd $p |- ( ph -> A <_ C ) $=
|
|
( cle wbr cr wcel wa wi letr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLMCLM
|
|
DLMSTNUAOEFGBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
lelttrd.4 $e |- ( ph -> A <_ B ) $.
|
|
lelttrd.5 $e |- ( ph -> B < C ) $.
|
|
$( Transitive law deduction for 'less than or equal to', 'less than'.
|
|
(Contributed by NM, 8-Jan-2006.) $)
|
|
lelttrd $p |- ( ph -> A < C ) $=
|
|
( cle wbr clt cr wcel wa wi lelttr syl3anc mp2and ) ABCJKZCDLKZBDLKZHIA
|
|
BMNCMNDMNTUAOUBPEFGBCDQRS $.
|
|
$}
|
|
|
|
${
|
|
lttrd.4 $e |- ( ph -> A < B ) $.
|
|
lttrd.5 $e |- ( ph -> B < C ) $.
|
|
$( Transitive law deduction for 'less than'. (Contributed by NM,
|
|
9-Jan-2006.) $)
|
|
lttrd $p |- ( ph -> A < C ) $=
|
|
( clt wbr cr wcel wa wi lttr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLMCLM
|
|
DLMSTNUAOEFGBCDPQR $.
|
|
$}
|
|
$}
|
|
|
|
$( 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition
|
|
11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.) $)
|
|
0lt1 $p |- 0 < 1 $=
|
|
( cc0 c1 clt wbr cltrr ax-0lt1 cr wcel wb 0re 1re ltxrlt mp2an mpbir ) ABCD
|
|
ZABEDZFAGHBGHOPIJKABLMN $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Initial properties of the complex numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Commutative/associative law for multiplication. (Contributed by NM,
|
|
30-Apr-2005.) $)
|
|
mul12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) ) $=
|
|
( cc wcel w3a cmul co wceq wa mulcom oveq1d 3adant3 mulass 3com12 3eqtr3d )
|
|
ADEZBDEZCDEZFABGHZCGHZBAGHZCGHZABCGHGHBACGHGHZQRUAUCISQRJTUBCGABKLMABCNRQSU
|
|
CUDIBACNOP $.
|
|
|
|
$( Commutative/associative law. (Contributed by NM, 8-Oct-1999.) $)
|
|
mul32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) ) $=
|
|
( cc wcel w3a cmul co wceq wa mulcom oveq2d 3adant1 mulass 3com23 3eqtr4d )
|
|
ADEZBDEZCDEZFABCGHZGHZACBGHZGHZABGHCGHACGHBGHZRSUAUCIQRSJTUBAGBCKLMABCNQSRU
|
|
DUCIACBNOP $.
|
|
|
|
$( Commutative/associative law. (Contributed by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
mul31 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) ) $=
|
|
( cc wcel cmul co wceq wa mulcom oveq2d 3adant1 mulass mulcl ancoms mulcomd
|
|
w3a simp1 3eqtr4d ) ADEZBDEZCDEZQZABCFGZFGZACBFGZFGZABFGCFGUFAFGUAUBUEUGHTU
|
|
AUBIUDUFAFBCJKLABCMUCUFAUAUBUFDEZTUBUAUHCBNOLTUAUBRPS $.
|
|
|
|
$( Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.) $)
|
|
mul4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A x. B ) x. ( C x. D ) ) =
|
|
( ( A x. C ) x. ( B x. D ) ) ) $=
|
|
( cc wcel wa cmul co wceq w3a mul32 oveq1d 3expa adantrr mulcl mulass 3expb
|
|
sylan an4s 3eqtr3d ) AEFZBEFZGZCEFZDEFZGZGABHIZCHIZDHIZACHIZBHIZDHIZUHCDHIH
|
|
IZUKBDHIHIZUDUEUJUMJZUFUBUCUEUPUBUCUEKUIULDHABCLMNOUDUHEFZUGUJUNJZABPUQUEUF
|
|
URUHCDQRSUBUEUCUFUMUOJZUBUEGUKEFZUCUFGUSACPUTUCUFUSUKBDQRSTUA $.
|
|
|
|
$( A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.) $)
|
|
muladd11 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) =
|
|
( ( 1 + A ) + ( B + ( A x. B ) ) ) ) $=
|
|
( cc wcel wa c1 caddc cmul wceq ax-1cn addcl mpan adddi mp3an2 sylan adantr
|
|
co mulid1d adddir eqtrd mp3an1 mulid2 adantl oveq1d oveq12d ) ACDZBCDZEZFAG
|
|
QZFBGQHQZUIFHQZUIBHQZGQZUIBABHQZGQZGQUFUICDZUGUJUMIZFCDZUFUPJFAKLZUPURUGUQJ
|
|
UIFBMNOUHUKUIULUOGUFUKUIIUGUFUIUSRPUHULFBHQZUNGQZUOURUFUGULVAIJFABSUAUHUTBU
|
|
NGUGUTBIUFBUBUCUDTUET $.
|
|
|
|
$( Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
1p1times $p |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) ) $=
|
|
( cc wcel c1 caddc co cmul ax-1cn a1i id adddird mulid2 oveq12d eqtrd ) ABC
|
|
ZDDEFAGFDAGFZPEFAAEFODDADBCOHIZQOJKOPAPAEALZRMN $.
|
|
|
|
$( A theorem for complex numbers analogous the second Peano postulate
|
|
~ peano2 . (Contributed by NM, 17-Aug-2005.) $)
|
|
peano2cn $p |- ( A e. CC -> ( A + 1 ) e. CC ) $=
|
|
( cc wcel c1 caddc co ax-1cn addcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( A theorem for reals analogous the second Peano postulate ~ peano2 .
|
|
(Contributed by NM, 5-Jul-2005.) $)
|
|
peano2re $p |- ( A e. RR -> ( A + 1 ) e. RR ) $=
|
|
( cr wcel c1 caddc co 1re readdcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.) $)
|
|
addcom $p |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) $=
|
|
( ax-addcom ) ABC $.
|
|
|
|
$( ` 0 ` is an additive identity. (Contributed by Jim Kingdon,
|
|
16-Jan-2020.) $)
|
|
addid1 $p |- ( A e. CC -> ( A + 0 ) = A ) $=
|
|
( ax-0id ) AB $.
|
|
|
|
$( ` 0 ` is a left identity for addition. (Contributed by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
addid2 $p |- ( A e. CC -> ( 0 + A ) = A ) $=
|
|
( cc wcel cc0 caddc co wceq 0cn addcom mpan2 addid1 eqtr3d ) ABCZADEFZDAEFZ
|
|
AMDBCNOGHADIJAKL $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Cancellation law for addition over the reals. (Contributed by Scott
|
|
Fenton, 3-Jan-2013.) $)
|
|
readdcan $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( C + A ) = ( C + B ) <-> A = B ) ) $=
|
|
( vx cr wcel caddc co wceq wa oveq2 recnd addassd adantr cc oveq1d addid2
|
|
cc0 syl eqtrd w3a cv wi ax-rnegex 3ad2ant3 adantl wb simprl simpl3 simpl1
|
|
simpl2 eqeq12d mpbird addcom syl2anc simplrr 3eqtr3d ex rexlimddv impbid1
|
|
wrex eqtr3d ) AEFZBEFZCEFZUAZCAGHZCBGHZIZABIZVFCDUBZGHZRIZVIVJUCDEVEVCVMD
|
|
EVAVDDCUDUEVFVKEFZVMJZJZVIVJVPVIJZVKCGHZAGHZVRBGHZABVQVSVTIZVKVGGHZVKVHGH
|
|
ZIZVIWDVPVGVHVKGKUFVPWAWDUGVIVPVSWBVTWCVPVKCAVPVKVFVNVMUHLZVPCVCVDVEVOUIL
|
|
ZVPAVCVDVEVOUJLZMVPVKCBWEWFVPBVCVDVEVOUKLZMULNUMVQVSRAGHZAVQVRRAGVQVLVRRV
|
|
QCOFZVKOFZVLVRIVPWJVIWFNVPWKVIWENCVKUNUOVFVNVMVIUPVBZPVQAOFZWIAIVPWMVIWGN
|
|
AQSTVQVTRBGHZBVQVRRBGWLPVQBOFZWNBIVPWOVIWHNBQSTUQURUSABCGKUT $.
|
|
$}
|
|
|
|
$( ` 0 ` is its own additive identity. (Contributed by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
00id $p |- ( 0 + 0 ) = 0 $=
|
|
( cc0 cc wcel caddc co wceq 0cn addid1 ax-mp ) ABCAADEAFGAHI $.
|
|
|
|
${
|
|
mul.1 $e |- A e. CC $.
|
|
$( ` 0 ` is an additive identity. (Contributed by NM, 23-Nov-1994.)
|
|
(Revised by Scott Fenton, 3-Jan-2013.) $)
|
|
addid1i $p |- ( A + 0 ) = A $=
|
|
( cc wcel cc0 caddc co wceq addid1 ax-mp ) ACDAEFGAHBAIJ $.
|
|
|
|
$( ` 0 ` is a left identity for addition. (Contributed by NM,
|
|
3-Jan-2013.) $)
|
|
addid2i $p |- ( 0 + A ) = A $=
|
|
( cc wcel cc0 caddc co wceq addid2 ax-mp ) ACDEAFGAHBAIJ $.
|
|
|
|
mul.2 $e |- B e. CC $.
|
|
$( Addition commutes. Based on ideas by Eric Schmidt. (Contributed by
|
|
Scott Fenton, 3-Jan-2013.) $)
|
|
addcomi $p |- ( A + B ) = ( B + A ) $=
|
|
( cc wcel caddc co wceq addcom mp2an ) AEFBEFABGHBAGHICDABJK $.
|
|
|
|
${
|
|
addcomli.2 $e |- ( A + B ) = C $.
|
|
$( Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
addcomli $p |- ( B + A ) = C $=
|
|
( caddc co addcomi eqtri ) BAGHABGHCBAEDIFJ $.
|
|
$}
|
|
|
|
mul.3 $e |- C e. CC $.
|
|
$( Commutative/associative law that swaps the first two factors in a triple
|
|
product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew
|
|
Salmon, 19-Nov-2011.) $)
|
|
mul12i $p |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) $=
|
|
( cc wcel cmul co wceq mul12 mp3an ) AGHBGHCGHABCIJIJBACIJIJKDEFABCLM $.
|
|
|
|
$( Commutative/associative law that swaps the last two factors in a triple
|
|
product. (Contributed by NM, 11-May-1999.) $)
|
|
mul32i $p |- ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) $=
|
|
( cc wcel cmul co wceq mul32 mp3an ) AGHBGHCGHABIJCIJACIJBIJKDEFABCLM $.
|
|
|
|
mul4.4 $e |- D e. CC $.
|
|
$( Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.) $)
|
|
mul4i $p |- ( ( A x. B ) x. ( C x. D ) ) =
|
|
( ( A x. C ) x. ( B x. D ) ) $=
|
|
( cc wcel cmul co wceq mul4 mp4an ) AIJBIJCIJDIJABKLCDKLKLACKLBDKLKLMEFGH
|
|
ABCDNO $.
|
|
$}
|
|
|
|
${
|
|
muld.1 $e |- ( ph -> A e. CC ) $.
|
|
$( ` 0 ` is an additive identity. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addid1d $p |- ( ph -> ( A + 0 ) = A ) $=
|
|
( cc wcel cc0 caddc co wceq addid1 syl ) ABDEBFGHBICBJK $.
|
|
|
|
$( ` 0 ` is a left identity for addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addid2d $p |- ( ph -> ( 0 + A ) = A ) $=
|
|
( cc wcel cc0 caddc co wceq addid2 syl ) ABDEFBGHBICBJK $.
|
|
|
|
addcomd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Addition commutes. Based on ideas by Eric Schmidt. (Contributed by
|
|
Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
addcomd $p |- ( ph -> ( A + B ) = ( B + A ) ) $=
|
|
( cc wcel caddc co wceq addcom syl2anc ) ABFGCFGBCHICBHIJDEBCKL $.
|
|
|
|
mul12d.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Commutative/associative law that swaps the first two factors in a triple
|
|
product. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
mul12d $p |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) ) $=
|
|
( cc wcel cmul co wceq mul12 syl3anc ) ABHICHIDHIBCDJKJKCBDJKJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Commutative/associative law that swaps the last two factors in a triple
|
|
product. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
mul32d $p |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) ) $=
|
|
( cc wcel cmul co wceq mul32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Commutative/associative law. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mul31d $p |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) ) $=
|
|
( cc wcel cmul co wceq mul31 syl3anc ) ABHICHIDHIBCJKDJKDCJKBJKLEFGBCDMN
|
|
$.
|
|
|
|
mul4d.4 $e |- ( ph -> D e. CC ) $.
|
|
$( Rearrangement of 4 factors. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mul4d $p |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) =
|
|
( ( A x. C ) x. ( B x. D ) ) ) $=
|
|
( cc wcel cmul co wceq mul4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLMN
|
|
FGHIBCDEOP $.
|
|
$}
|
|
|
|
$( A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.) $)
|
|
muladd11r $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) =
|
|
( ( ( A x. B ) + ( A + B ) ) + 1 ) ) $=
|
|
( cc wcel wa c1 caddc cmul simpl 1cnd addcomd simpr oveq12d muladd11 addcld
|
|
co mulcl addassd addcl 3eqtrd eqtr3d oveq1d ) ACDZBCDZEZAFGPZBFGPZHPFAGPZFB
|
|
GPZHPUHBABHPZGPZGPZUJABGPZGPZFGPZUEUFUHUGUIHUEAFUCUDIZUEJZKUEBFUCUDLZUQKMAB
|
|
NUEULFAUKGPZGPUSFGPUOUEFAUKUQUPUEBUJURABQZOZRUEFUSUQUEAUKUPVAOKUEUSUNFGUEUM
|
|
UJGPUSUNUEABUJUPURUTRUEUMUJABSUTKUAUBTT $.
|
|
|
|
${
|
|
comraddd.1 $e |- ( ph -> B e. CC ) $.
|
|
comraddd.2 $e |- ( ph -> C e. CC ) $.
|
|
comraddd.3 $e |- ( ph -> A = ( B + C ) ) $.
|
|
$( Commute RHS addition, in deduction form. (Contributed by David A.
|
|
Wheeler, 11-Oct-2018.) $)
|
|
comraddd $p |- ( ph -> A = ( C + B ) ) $=
|
|
( caddc co addcomd eqtrd ) ABCDHIDCHIGACDEFJK $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Real and complex numbers - basic operations
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Addition
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Commutative/associative law that swaps the first two terms in a triple
|
|
sum. (Contributed by NM, 11-May-2004.) $)
|
|
add12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A + ( B + C ) ) = ( B + ( A + C ) ) ) $=
|
|
( cc wcel w3a caddc co wceq wa addcom oveq1d 3adant3 addass 3com12 3eqtr3d
|
|
) ADEZBDEZCDEZFABGHZCGHZBAGHZCGHZABCGHGHBACGHGHZQRUAUCISQRJTUBCGABKLMABCNRQ
|
|
SUCUDIBACNOP $.
|
|
|
|
$( Commutative/associative law that swaps the last two terms in a triple sum.
|
|
(Contributed by NM, 13-Nov-1999.) $)
|
|
add32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) + C ) = ( ( A + C ) + B ) ) $=
|
|
( cc wcel w3a caddc co wceq wa addcom oveq2d 3adant1 addass 3com23 3eqtr4d
|
|
) ADEZBDEZCDEZFABCGHZGHZACBGHZGHZABGHCGHACGHBGHZRSUAUCIQRSJTUBAGBCKLMABCNQS
|
|
RUDUCIACBNOP $.
|
|
|
|
$( Commutative/associative law that swaps the last two terms in a triple sum,
|
|
rearranging the parentheses. (Contributed by Paul Chapman,
|
|
18-May-2007.) $)
|
|
add32r $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A + ( B + C ) ) = ( ( A + C ) + B ) ) $=
|
|
( cc wcel w3a caddc co addass add32 eqtr3d ) ADEBDECDEFABGHCGHABCGHGHACGHBG
|
|
HABCIABCJK $.
|
|
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.)
|
|
(Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
add4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) ) $=
|
|
( cc wcel wa caddc wceq add12 3expb oveq2d adantll addcl addass sylan2 an4s
|
|
co 3expa 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZGABCDHRZHRZHRZACBDHRZHRZHRZABHRUGHRZ
|
|
ACHRUJHRZUBUFUIULIUAUBUFGUHUKAHUBUDUEUHUKIBCDJKLMUFUCUGEFZUMUIIZCDNUAUBUOUP
|
|
ABUGOSPUAUDUBUEUNULIZUBUEGUAUDGUJEFZUQBDNUAUDURUQACUJOSPQT $.
|
|
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.) $)
|
|
add42 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) ) $=
|
|
( cc wcel wa caddc co add4 wceq addcom ad2ant2l oveq2d eqtrd ) AEFZBEFZGCEF
|
|
ZDEFZGGZABHICDHIHIACHIZBDHIZHIUADBHIZHIABCDJTUBUCUAHQSUBUCKPRBDLMNO $.
|
|
|
|
${
|
|
add.1 $e |- A e. CC $.
|
|
add.2 $e |- B e. CC $.
|
|
add.3 $e |- C e. CC $.
|
|
$( Commutative/associative law that swaps the first two terms in a triple
|
|
sum. (Contributed by NM, 21-Jan-1997.) $)
|
|
add12i $p |- ( A + ( B + C ) ) = ( B + ( A + C ) ) $=
|
|
( cc wcel caddc co wceq add12 mp3an ) AGHBGHCGHABCIJIJBACIJIJKDEFABCLM $.
|
|
|
|
$( Commutative/associative law that swaps the last two terms in a triple
|
|
sum. (Contributed by NM, 21-Jan-1997.) $)
|
|
add32i $p |- ( ( A + B ) + C ) = ( ( A + C ) + B ) $=
|
|
( cc wcel caddc co wceq add32 mp3an ) AGHBGHCGHABIJCIJACIJBIJKDEFABCLM $.
|
|
|
|
add4.4 $e |- D e. CC $.
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.) $)
|
|
add4i $p |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) $=
|
|
( cc wcel caddc co wceq add4 mp4an ) AIJBIJCIJDIJABKLCDKLKLACKLBDKLKLMEFG
|
|
HABCDNO $.
|
|
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) $)
|
|
add42i $p |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) $=
|
|
( caddc co add4i addcomi oveq2i eqtri ) ABIJCDIJIJACIJZBDIJZIJODBIJZIJABC
|
|
DEFGHKPQOIBDFHLMN $.
|
|
$}
|
|
|
|
${
|
|
addd.1 $e |- ( ph -> A e. CC ) $.
|
|
addd.2 $e |- ( ph -> B e. CC ) $.
|
|
addd.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Commutative/associative law that swaps the first two terms in a triple
|
|
sum. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
add12d $p |- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) ) $=
|
|
( cc wcel caddc co wceq add12 syl3anc ) ABHICHIDHIBCDJKJKCBDJKJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Commutative/associative law that swaps the last two terms in a triple
|
|
sum. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
add32d $p |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) ) $=
|
|
( cc wcel caddc co wceq add32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
|
|
$.
|
|
|
|
add4d.4 $e |- ( ph -> D e. CC ) $.
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
add4d $p |- ( ph ->
|
|
( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) ) $=
|
|
( cc wcel caddc co wceq add4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLM
|
|
NFGHIBCDEOP $.
|
|
|
|
$( Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
add42d $p |- ( ph ->
|
|
( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) ) $=
|
|
( cc wcel caddc co wceq add42 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMECLML
|
|
MNFGHIBCDEOP $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Subtraction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare symbols needed for some defined terms. $)
|
|
$c - $. $( Subtraction (binary minus). $)
|
|
$c -u $. $( Unary minus sign. $)
|
|
|
|
$( Extend class notation to include subtraction. $)
|
|
cmin $a class - $.
|
|
|
|
$( Extend class notation to include unary minus. The symbol ` -u ` is not a
|
|
class by itself but part of a compound class definition. We do this
|
|
rather than making it a formal function since it is so commonly used.
|
|
Note: We use different symbols for unary minus ( ` -u ` ) and subtraction
|
|
~ cmin ( ` - ` ) to prevent syntax ambiguity. For example, looking at the
|
|
syntax definition ~ co , if we used the same symbol
|
|
then " ` ( - A - B ) ` " could mean either " ` - A ` " minus " ` B ` ", or
|
|
it could represent the (meaningless) operation of
|
|
classes " ` - ` " and " ` - B ` " connected with "operation" " ` A ` ".
|
|
On the other hand, " ` ( -u A - B ) ` " is unambiguous. $)
|
|
cneg $a class -u A $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define subtraction. Theorem ~ subval shows its value (and describes how
|
|
this definition works), theorem ~ subaddi relates it to addition, and
|
|
theorems ~ subcli and ~ resubcli prove its closure laws. (Contributed
|
|
by NM, 26-Nov-1994.) $)
|
|
df-sub $a |- - = ( x e. CC , y e. CC |->
|
|
( iota_ z e. CC ( y + z ) = x ) ) $.
|
|
$}
|
|
|
|
$( Define the negative of a number (unary minus). We use different symbols
|
|
for unary minus ( ` -u ` ) and subtraction ( ` - ` ) to prevent syntax
|
|
ambiguity. See ~ cneg for a discussion of this. (Contributed by NM,
|
|
10-Feb-1995.) $)
|
|
df-neg $a |- -u A = ( 0 - A ) $.
|
|
|
|
${
|
|
$d a b c d x y z A $. $d x B $. $d x C $.
|
|
$( Addition cancellation of a real number from two complex numbers. Lemma
|
|
for ~ cnegex . (Contributed by Eric Schmidt, 22-May-2007.) $)
|
|
cnegexlem1 $p |- ( ( A e. RR /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) = ( A + C ) <-> B = C ) ) $=
|
|
( vx cr wcel cc caddc co wceq cc0 wi recn wa oveq2 addassd eqeq12d adantl
|
|
wb oveq1 w3a wrex ax-rnegex 3ad2ant1 simpr simpll simplrl simplrr syl5ibr
|
|
cv adantr addcom eqeq1d adantlr addid2 eqeqan12d ad2antrr bitrd ex sylbid
|
|
imp sylibd sylan2 rexlimdva 3impb syl3an1 mpd impbid1 ) AEFZBGFZCGFZUAZAB
|
|
HIZACHIZJZBCJZVLADUJZHIZKJZDEUBZVOVPLZVIVJVTVKDAUCUDVIAGFZVJVKVTWALZAMWBV
|
|
JVKWCWBVJVKNZNZVSWADEVQEFWEVQGFZVSWALVQMWEWFNZVSWAWGVSNVOVQAHIZBHIZWHCHIZ
|
|
JZVPWGVOWKLVSVOWKWGVQVMHIZVQVNHIZJVMVNVQHOWGWIWLWJWMWGVQABWEWFUEZWBWDWFUF
|
|
ZWBVJVKWFUGPWGVQACWNWOWBVJVKWFUHPQUIUKWGVSWKVPSZWGVSWHKJZWPWBWFVSWQSWDWBW
|
|
FNVRWHKAVQULUMUNWGWQWPWGWQNWKKBHIZKCHIZJZVPWQWKWTSWGWQWIWRWJWSWHKBHTWHKCH
|
|
TQRWEWTVPSZWFWQWDXAWBVJVKWRBWSCBUOCUOUPRUQURUSUTVAVBUSVCVDVEVFVGBCAHOVH
|
|
$.
|
|
|
|
$( Existence of a real number which produces a real number when multiplied
|
|
by ` _i ` . (Hint: zero is such a number, although we don't need to
|
|
prove that yet). Lemma for ~ cnegex . (Contributed by Eric Schmidt,
|
|
22-May-2007.) $)
|
|
cnegexlem2 $p |- E. y e. RR ( _i x. y ) e. RR $=
|
|
( vx vz cc wcel cv ci cmul co caddc wceq cr wrex wa wi adantr recn addid2
|
|
cc0 oveq1 0cn cnre ax-rnegex ax-icn mulcl sylancr 3ad2ant3 ad2antrl add32
|
|
w3a 3com23 eqcomd sylan9eq adantrl 3ad2ant2 eqtr3d ex syl3an 3expa simplr
|
|
3eqtr3d imp eqeltrrd exp32 rexlimdva mpd reximdva rexlimiv mp2b ) SDESBFZ
|
|
GAFZHIZJIZKZALMZBLMVLLEZALMZUABASUBVOVQBLVJLEZVNVPALVRVKLEZNZVJCFZJIZSKZC
|
|
LMZVNVPOZVRWDVSCVJUCPVTWCWECLVTWALEZNZWCVNVPWGWCVNNZNWAVLLWGWHWAVLKZVRVSW
|
|
FWHWIOZVRVJDEZVSVLDEZWFWADEZWJVJQVSGDEVKDEWLUDVKQGVKUEUFWAQWKWLWMUJZWHWIW
|
|
NWHNZSWAJIZWAVLWNWPWAKZWHWMWKWQWLWARUGPWOWBVLJIZSVLJIZWPVLWCWRWSKWNVNWBSV
|
|
LJTUHWNVNWRWPKWCWNVNWRVMWAJIZWPWKWMWLWRWTKVJWAVLUIUKVNWPWTSVMWAJTULUMUNWN
|
|
WSVLKZWHWLWKXAWMVLRUOPVAUPUQURUSVBVTWFWHUTVCVDVEVFVGVHVI $.
|
|
|
|
$( Existence of real number difference. Lemma for ~ cnegex . (Contributed
|
|
by Eric Schmidt, 22-May-2007.) $)
|
|
cnegexlem3 $p |- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y ) $=
|
|
( vx cv cr wcel wa caddc co wceq wrex ax-rnegex adantlr adantr cc wb recn
|
|
cc0 anim1i readdcl anim12i add32 3expa addcl addcom sylan eqtr2d adantllr
|
|
syl sylanl2 ancoms sylan2 id sylan9eq adantlll eqeq12d simpllr cnegexlem1
|
|
an32s simplr syl3anc bitr3d syl2an rexbidva mpbid adantl r19.29a ) BEZFGZ
|
|
AEZFGZHZVKDEZIJZSKZVICEZIJZVKKZCFLZDFVMVNFGZHZVPHZVIVNIJZVQIJZSKZCFLZVTWB
|
|
WGVPVJWAWGVLVJWAHWDFGWGVIVNUACWDMUJNOWCWFVSCFWCVIPGZVKPGZHZWAHZVPHZVQPGZW
|
|
FVSQVQFGWBWKVPVMWJWAVJWHVLWIVIRVKRUBTTVQRWLWMHZVNVRIJZVNVKIJZKZWFVSWNWOWE
|
|
WPSWKWMWOWEKZVPWHWAWMWRWIWAWHVNPGZWMWRVNRZWHWSHWMHWEVRVNIJZWOWHWSWMWEXAKV
|
|
IVNVQUCUDWHWMWSXAWOKZWHWMHVRPGZWSXBVIVQUEZVRVNUFUGUTUHUKUINWLWPSKZWMWIWAV
|
|
PXEWHWIWAHVPWPVOSWAWIWSWPVOKZWTWSWIXFVNVKUFULUMVPUNUOUPOUQWKWMWQVSQZVPWKW
|
|
MHWAXCWIXGWJWAWMVAWJWMXCWAWHWMXCWIXDNNWHWIWAWMURVNVRVKUSVBNVCVDVEVFVLVPDF
|
|
LVJDVKMVGVH $.
|
|
|
|
$( Existence of the negative of a complex number. (Contributed by Eric
|
|
Schmidt, 21-May-2007.) $)
|
|
cnegex $p |- ( A e. CC -> E. x e. CC ( A + x ) = 0 ) $=
|
|
( va vb vy vc vd cc wcel cv ci cmul co caddc wceq cr wrex cc0 wa ax-icn
|
|
cnre cnegexlem2 cnegexlem3 ad2ant2lr mulcl sylancr syl2an adantlr adantll
|
|
addcl adantr anim12i sylan2 ad2antrr ad2antlr simpr addassd simpll adantl
|
|
recn mpan adddi mp3an1 oveq2d eqtr4d oveq1d eqtr3d adantlrr eqtr2d eqeq1d
|
|
oveq2 biimpa rspcev syl2anc readdcl ax-rnegex ad2ant2rl r19.29a rexlimddv
|
|
adantllr syl rexlimdvaa mpi oveq1 rexbidv syl5ibrcom rexlimivv ) BHIBCJZK
|
|
DJZLMZNMZOZDPQCPQBAJZNMZROZAHQZCDBUAWLWPCDPPWHPIZWIPIZSZWPWLWKWMNMZROZAHQ
|
|
ZWSKEJZLMZPIZEPQXBEUBWSXEXBEPWSXCPIZXESZSZWIFJZNMZXCOZXBFPWRXFXKFPQWQXEED
|
|
FUCUDXHXIPIZXKSZSZWHXDNMZGJZNMZROZXBGPXNXPPIZSZXRSKXILMZXPNMZHIZWKYBNMZRO
|
|
ZXBXTYCXRXMXSYCXHXLXSYCXKXLYAHIZXPHIZYCXSXLKHIZXIHIZYFTXIUTZKXIUEZUFXPUTZ
|
|
YAXPUJUGUHUIUKXTXRYEXTXQYDRWSXMXSXQYDOXGWSXMSXSSYDWHKXJLMZNMZXPNMZXQWSXLX
|
|
SYDYOOZXKWSXLSWHHIZWIHIZSZYISZYGYPXSWSYSXLYIWQYQWRYRWHUTWIUTULYJULYLYTYGS
|
|
ZWKYANMZXPNMYDYOUUAWKYAXPYSWKHIZYIYGYRYQWJHIZUUCYHYRUUDTKWIUEVAZWHWJUJUMU
|
|
NYIYFYSYGYHYIYFTYKVAZUOYTYGUPUQUUAUUBYNXPNYTUUBYNOYGYTUUBWHWJYANMZNMYNYTW
|
|
HWJYAYQYRYIURYRUUDYQYIUUEUOYIYFYSUUFUSUQYTYMUUGWHNYRYIYMUUGOZYQYHYRYIUUHT
|
|
KWIXIVBVCUIVDVEUKVFVGUGVHXMYOXQOZWSXSXKUUIXLXKYNXOXPNXKYMXDWHNXJXCKLVKVDV
|
|
FUSUOVIVTVJVLXAYEAYBHWMYBOWTYDRWMYBWKNVKVJVMVNXHXRGPQZXMWQXEUUJWRXFWQXESX
|
|
OPIUUJWHXDVOGXOVPWAVQUKVRVSWBWCWLWOXAAHWLWNWTRBWKWMNWDVJWEWFWGWA $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Existence of a left inverse for addition. (Contributed by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
cnegex2 $p |- ( A e. CC -> E. x e. CC ( x + A ) = 0 ) $=
|
|
( cc wcel cv caddc co cc0 wceq wrex cnegex addcom eqeq1d rexbidva mpbid
|
|
wa ) BCDZBAEZFGZHIZACJRBFGZHIZACJABKQTUBACQRCDPSUAHBRLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|
|
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) = ( A + C ) <-> B = C ) ) $=
|
|
( vx cc wcel w3a cv caddc co cc0 wceq wb wa oveq1d addassd addid2 3eqtr3d
|
|
oveq2 syl wrex cnegex2 3ad2ant1 simprr simprl simpl1 simpl2 simpl3 syl5ib
|
|
eqeq12d impbid1 rexlimddv ) AEFZBEFZCEFZGZDHZAIJZKLZABIJZACIJZLZBCLZMDEUM
|
|
UNUSDEUAUODAUBUCUPUQEFZUSNZNZVBVCVBUQUTIJZUQVAIJZLVFVCUTVAUQISVFVGBVHCVFU
|
|
RBIJKBIJZVGBVFURKBIUPVDUSUDZOVFUQABUPVDUSUEZUMUNUOVEUFZUMUNUOVEUGZPVFUNVI
|
|
BLVMBQTRVFURCIJKCIJZVHCVFURKCIVJOVFUQACVKVLUMUNUOVEUHZPVFUOVNCLVOCQTRUJUI
|
|
BCAISUKUL $.
|
|
|
|
$( Cancellation law for addition. (Contributed by NM, 30-Jul-2004.)
|
|
(Revised by Scott Fenton, 3-Jan-2013.) $)
|
|
addcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + C ) = ( B + C ) <-> A = B ) ) $=
|
|
( vx cc wcel w3a cv caddc co cc0 wb wa oveq1 addassd oveq2d addid1 3eqtrd
|
|
wceq syl wrex cnegex 3ad2ant3 simpl1 simpl3 simprl simprr eqeq12d impbid1
|
|
simpl2 syl5ib rexlimddv ) AEFZBEFZCEFZGZCDHZIJZKSZACIJZBCIJZSZABSZLDEUOUM
|
|
USDEUAUNDCUBUCUPUQEFZUSMZMZVBVCVBUTUQIJZVAUQIJZSVFVCUTVAUQINVFVGAVHBVFVGA
|
|
URIJAKIJZAVFACUQUMUNUOVEUDZUMUNUOVEUEZUPVDUSUFZOVFURKAIUPVDUSUGZPVFUMVIAS
|
|
VJAQTRVFVHBURIJBKIJZBVFBCUQUMUNUOVEUJZVKVLOVFURKBIVMPVFUNVNBSVOBQTRUHUKAB
|
|
CINUIUL $.
|
|
$}
|
|
|
|
${
|
|
addcani.1 $e |- A e. CC $.
|
|
addcani.2 $e |- B e. CC $.
|
|
addcani.3 $e |- C e. CC $.
|
|
$( Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|
|
(Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
addcani $p |- ( ( A + B ) = ( A + C ) <-> B = C ) $=
|
|
( cc wcel caddc co wceq wb addcan mp3an ) AGHBGHCGHABIJACIJKBCKLDEFABCMN
|
|
$.
|
|
|
|
$( Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|
|
(Contributed by NM, 14-May-2003.) (Revised by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
addcan2i $p |- ( ( A + C ) = ( B + C ) <-> A = B ) $=
|
|
( cc wcel caddc co wceq wb addcan2 mp3an ) AGHBGHCGHACIJBCIJKABKLDEFABCMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
addcand.1 $e |- ( ph -> A e. CC ) $.
|
|
addcand.2 $e |- ( ph -> B e. CC ) $.
|
|
addcand.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addcand $p |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) ) $=
|
|
( cc wcel caddc co wceq wb addcan syl3anc ) ABHICHIDHIBCJKBDJKLCDLMEFGBCD
|
|
NO $.
|
|
|
|
$( Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addcan2d $p |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) ) $=
|
|
( cc wcel caddc co wceq wb addcan2 syl3anc ) ABHICHIDHIBDJKCDJKLBCLMEFGBC
|
|
DNO $.
|
|
|
|
${
|
|
addcanad.4 $e |- ( ph -> ( A + B ) = ( A + C ) ) $.
|
|
$( Cancelling a term on the left-hand side of a sum in an equality.
|
|
Consequence of ~ addcand . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
addcanad $p |- ( ph -> B = C ) $=
|
|
( caddc co wceq addcand mpbid ) ABCIJBDIJKCDKHABCDEFGLM $.
|
|
$}
|
|
|
|
${
|
|
addcan2ad.4 $e |- ( ph -> ( A + C ) = ( B + C ) ) $.
|
|
$( Cancelling a term on the right-hand side of a sum in an equality.
|
|
Consequence of ~ addcan2d . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
addcan2ad $p |- ( ph -> A = B ) $=
|
|
( caddc co wceq addcan2d mpbid ) ABDIJCDIJKBCKHABCDEFGLM $.
|
|
$}
|
|
|
|
${
|
|
addneintrd.4 $e |- ( ph -> B =/= C ) $.
|
|
$( Introducing a term on the left-hand side of a sum in a negated
|
|
equality. Contrapositive of ~ addcanad . Consequence of ~ addcand .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
addneintrd $p |- ( ph -> ( A + B ) =/= ( A + C ) ) $=
|
|
( caddc co wne addcand necon3bid mpbird ) ABCIJZBDIJZKCDKHAOPCDABCDEFGL
|
|
MN $.
|
|
$}
|
|
|
|
${
|
|
addneintr2d.4 $e |- ( ph -> A =/= B ) $.
|
|
$( Introducing a term on the right-hand side of a sum in a negated
|
|
equality. Contrapositive of ~ addcan2ad . Consequence of
|
|
~ addcan2d . (Contributed by David Moews, 28-Feb-2017.) $)
|
|
addneintr2d $p |- ( ph -> ( A + C ) =/= ( B + C ) ) $=
|
|
( caddc co wne addcan2d necon3bid mpbird ) ABDIJZCDIJZKBCKHAOPBCABCDEFG
|
|
LMN $.
|
|
$}
|
|
$}
|
|
|
|
$( Alternate proof of ~ 0cn . (Contributed by NM, 19-Feb-2005.) (Revised by
|
|
Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
0cnALT $p |- 0 e. CC $=
|
|
( vx ci cv caddc co cc0 wceq wrex wcel ax-icn cnegex ax-mp addcl mpan eleq1
|
|
cc syl5ibcom rexlimiv ) BACZDEZFGZAPHZFPIZBPIZUBJABKLUAUCAPSPIZTPIZUAUCUDUE
|
|
UFJBSMNTFPOQRL $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18.
|
|
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
negeu $p |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B ) $=
|
|
( vy cc wcel wa cv caddc co wceq wreu wrex cnegex adantr wral simpl simpr
|
|
cc0 wb addcl syl2anr simplrr oveq1d simplll simplrl simpllr eqeq2d addcld
|
|
addassd addid2d 3eqtr3rd addcand bitrd ralrimiva reu6i syl2anc rexlimddv
|
|
) BEFZCEFZGZBDHZIJZSKZBAHZIJZCKZAELZDEUSVDDEMUTDBNOVAVBEFZVDGZGZVBCIJZEFZ
|
|
VGVEVLKZTZAEPVHVJVIUTVMVAVIVDQUSUTRVBCUAUBVKVOAEVKVEEFZGZVGVFBVLIJZKVNVQC
|
|
VRVFVQVCCIJSCIJVRCVQVCSCIVAVIVDVPUCUDVQBVBCUSUTVJVPUEZVAVIVDVPUFZUSUTVJVP
|
|
UGZUJVQCWAUKULUHVQBVEVLVSVKVPRVQVBCVTWAUIUMUNUOVGAEVLUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Value of subtraction, which is the (unique) element ` x ` such that
|
|
` B + x = A ` . (Contributed by NM, 4-Aug-2007.) (Revised by Mario
|
|
Carneiro, 2-Nov-2013.) $)
|
|
subval $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( A - B ) = ( iota_ x e. CC ( B + x ) = A ) ) $=
|
|
( vy vz cc wcel cv caddc co wceq crio cmin wa negeu riotacl syl riotabidv
|
|
wreu ancoms eqeq2 oveq1 eqeq1d df-sub ovmpt2g mpd3an3 ) BFGZCFGZCAHZIJZBK
|
|
ZAFLZFGZBCMJULKUHUGUMUHUGNUKAFSUMACBOUKAFPQTDEBCFFEHZUIIJZDHZKZAFLULMUOBK
|
|
ZAFLFUPBKUQURAFUPBUOUARUNCKZURUKAFUSUOUJBUNCUIIUBUCRDEAUDUEUF $.
|
|
$}
|
|
|
|
$( Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.) $)
|
|
negeq $p |- ( A = B -> -u A = -u B ) $=
|
|
( wceq cc0 cmin co cneg oveq2 df-neg 3eqtr4g ) ABCDAEFDBEFAGBGABDEHAIBIJ $.
|
|
|
|
${
|
|
negeqi.1 $e |- A = B $.
|
|
$( Equality inference for negatives. (Contributed by NM, 14-Feb-1995.) $)
|
|
negeqi $p |- -u A = -u B $=
|
|
( wceq cneg negeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
negeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for negatives. (Contributed by NM, 14-May-1999.) $)
|
|
negeqd $p |- ( ph -> -u A = -u B ) $=
|
|
( wceq cneg negeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
nfnegd.1 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfneg . (Contributed by NM, 29-Feb-2008.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfnegd $p |- ( ph -> F/_ x -u A ) $=
|
|
( cneg cc0 cmin co df-neg nfcvd nfovd nfcxfrd ) ABCEFCGHCIABFCGABFJABGJDK
|
|
L $.
|
|
$}
|
|
|
|
${
|
|
nfneg.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for the negative of a complex number.
|
|
(Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfneg $p |- F/_ x -u A $=
|
|
( cneg wnfc wtru a1i nfnegd trud ) ABDEFABABEFCGHI $.
|
|
$}
|
|
|
|
$( Move class substitution in and out of the negative of a number.
|
|
(Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
csbnegg $p |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B ) $=
|
|
( wcel cc0 cmin co csb cneg csbov2g df-neg csbeq2i 3eqtr4g ) BDEABFCGHZIFAB
|
|
CIZGHABCJZIPJABFCGDKABQOCLMPLN $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Closure law for subtraction. (Contributed by NM, 10-May-1999.)
|
|
(Revised by Mario Carneiro, 21-Dec-2013.) $)
|
|
subcl $p |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC ) $=
|
|
( vx cc wcel wa cmin co cv caddc wceq crio subval wreu ancoms riotacl syl
|
|
negeu eqeltrd ) ADEZBDEZFZABGHBCIJHAKZCDLZDCABMUBUCCDNZUDDEUATUECBAROUCCD
|
|
PQS $.
|
|
$}
|
|
|
|
$( Closure law for negative. (Contributed by NM, 6-Aug-2003.) $)
|
|
negcl $p |- ( A e. CC -> -u A e. CC ) $=
|
|
( cc wcel cneg cc0 cmin co df-neg 0cn subcl mpan syl5eqel ) ABCZADEAFGZBAHE
|
|
BCMNBCIEAJKL $.
|
|
|
|
$( ` -u _i ` is a complex number (common case). (Contributed by David A.
|
|
Wheeler, 7-Dec-2018.) $)
|
|
negicn $p |- -u _i e. CC $=
|
|
( ci cc wcel cneg ax-icn negcl ax-mp ) ABCADBCEAFG $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Subtraction is an operation on the complex numbers. (Contributed by NM,
|
|
4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) $)
|
|
subf $p |- - : ( CC X. CC ) --> CC $=
|
|
( vy vz vx cv caddc co wceq cc crio wcel wral cxp cmin wf wa subval subcl
|
|
eqeltrrd rgen2a df-sub fmpt2 mpbi ) ADZBDEFCDZGBHIZHJZAHKCHKHHLHMNUFCAHUD
|
|
HJUCHJOUDUCMFUEHBUDUCPUDUCQRSCAHHUEHMCABTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Relationship between subtraction and addition. (Contributed by NM,
|
|
20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.) $)
|
|
subadd $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) = C <-> ( B + C ) = A ) ) $=
|
|
( vx cc wcel w3a cmin co wceq cv caddc crio wb subval eqeq1d 3adant3 wreu
|
|
wa negeu oveq2 riota2 sylan2 3impb 3com13 bitr4d ) AEFZBEFZCEFZGABHIZCJZB
|
|
DKZLIZAJZDEMZCJZBCLIZAJZUGUHUKUPNUIUGUHSUJUOCDABOPQUIUHUGURUPNZUIUHUGUSUH
|
|
UGSUIUNDERUSDBATUNURDECULCJUMUQAULCBLUAPUBUCUDUEUF $.
|
|
$}
|
|
|
|
$( Relationship between subtraction and addition. (Contributed by Scott
|
|
Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
subadd2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) = C <-> ( C + B ) = A ) ) $=
|
|
( cc wcel w3a cmin co wceq caddc subadd simp2 simp3 addcomd eqeq1d bitrd )
|
|
ADEZBDEZCDEZFZABGHCIBCJHZAICBJHZAIABCKTUAUBATBCQRSLQRSMNOP $.
|
|
|
|
$( Swap subtrahend and result of subtraction. (Contributed by NM,
|
|
14-Dec-2007.) $)
|
|
subsub23 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) = C <-> ( A - C ) = B ) ) $=
|
|
( cc wcel caddc co wceq cmin addcom 3adant1 eqeq1d subadd wb 3com23 3bitr4d
|
|
w3a ) ADEZBDEZCDEZQZBCFGZAHCBFGZAHZABIGCHACIGBHZUAUBUCASTUBUCHRBCJKLABCMRTS
|
|
UEUDNACBMOP $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 10-May-2004.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
pncan $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A ) $=
|
|
( cc wcel wa caddc co cmin wceq simpr simpl addcomd wb addcl subadd syl3anc
|
|
mpbird ) ACDZBCDZEZABFGZBHGAIZBAFGUAIZTBARSJZRSKZLTUACDSRUBUCMABNUDUEUABAOP
|
|
Q $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.) $)
|
|
pncan2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B ) $=
|
|
( cc wcel caddc co cmin wceq wa addcom oveq1d pncan eqtr3d ancoms ) BCDZACD
|
|
ZABEFZAGFZBHOPIZBAEFZAGFRBSTQAGBAJKBALMN $.
|
|
|
|
$( Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) $)
|
|
pncan3 $p |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B ) $=
|
|
( cc wcel wa cmin co wceq caddc eqid simpr simpl subcl ancoms syl3anc mpbii
|
|
wb subadd ) ACDZBCDZEZBAFGZUBHZAUBIGBHZUBJUATSUBCDZUCUDQSTKSTLTSUEBAMNBAUBR
|
|
OP $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 10-May-2004.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
npcan $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A ) $=
|
|
( cc wcel wa cmin co caddc subcl simpr addcomd wceq pncan3 ancoms eqtrd ) A
|
|
CDZBCDZEZABFGZBHGBSHGZARSBABIPQJKQPTALBAMNO $.
|
|
|
|
$( Associative-type law for addition and subtraction. (Contributed by NM,
|
|
6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
addsubass $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) - C ) = ( A + ( B - C ) ) ) $=
|
|
( cc wcel cmin co caddc simp1 subcl 3adant1 simp3 addassd wceq npcan oveq2d
|
|
w3a eqtrd oveq1d addcld pncan syl2anc eqtr3d ) ADEZBDEZCDEZQZABCFGZHGZCHGZC
|
|
FGZABHGZCFGUIUGUJULCFUGUJAUHCHGZHGULUGAUHCUDUEUFIZUEUFUHDEUDBCJKZUDUEUFLZMU
|
|
GUMBAHUEUFUMBNUDBCOKPRSUGUIDEUFUKUINUGAUHUNUOTUPUICUAUBUC $.
|
|
|
|
$( Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.)
|
|
(Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
addsub $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) - C ) = ( ( A - C ) + B ) ) $=
|
|
( cc wcel w3a caddc co cmin wa addcom oveq1d 3adant3 addsubass 3com12 subcl
|
|
wceq sylan2 3impb 3eqtrd ) ADEZBDEZCDEZFABGHZCIHZBAGHZCIHZBACIHZGHZUHBGHZUA
|
|
UBUEUGQUCUAUBJUDUFCIABKLMUBUAUCUGUIQBACNOUBUAUCUIUJQZUBUAUCUKUAUCJUBUHDEUKA
|
|
CPBUHKRSOT $.
|
|
|
|
$( Commutative/associative law for addition and subtraction. (Contributed by
|
|
NM, 1-Feb-2007.) $)
|
|
subadd23 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) + C ) = ( A + ( C - B ) ) ) $=
|
|
( cc wcel cmin co caddc wceq w3a addsub addsubass eqtr3d 3com23 ) ADEZCDEZB
|
|
DEZABFGCHGZACBFGHGZIOPQJACHGBFGRSACBKACBLMN $.
|
|
|
|
$( Commutative/associative law for addition and subtraction. (Contributed by
|
|
NM, 8-Feb-2005.) $)
|
|
addsub12 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A + ( B - C ) ) = ( B + ( A - C ) ) ) $=
|
|
( cc wcel cmin co caddc w3a subadd23 subcl addcom sylan 3impa eqtr3d 3com23
|
|
wceq wa ) ADEZCDEZBDEZABCFGHGZBACFGZHGZQSTUAIUCBHGZUBUDACBJSTUAUEUDQZSTRUCD
|
|
EUAUFACKUCBLMNOP $.
|
|
|
|
$( Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.) $)
|
|
2addsub $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) ) $=
|
|
( cc wcel wa caddc co cmin wceq add32 3expa oveq1d addcl addsub 3expb sylan
|
|
adantrr an4s eqtrd ) AEFZBEFZGZCEFZDEFZGGZABHICHIZDJIACHIZBHIZDJIZUIDJIBHIZ
|
|
UGUHUJDJUDUEUHUJKZUFUBUCUEUMABCLMSNUBUEUCUFUKULKZUBUEGUIEFZUCUFGUNACOUOUCUF
|
|
UNUIBDPQRTUA $.
|
|
|
|
$( Relation between sums and differences. (Contributed by Jeff Madsen,
|
|
17-Jun-2010.) $)
|
|
addsubeq4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) ) $=
|
|
( cc wcel wa cmin co wceq caddc eqcom subcl ancoms subadd 3expa an4s syl5bb
|
|
wb sylan addcom adantl oveq1d addsubass 3com12 eqtrd adantlr addcl 3bitr2rd
|
|
eqeq1d 3expb sylan2 ) AEFZBEFZGZCEFZDEFZGZGZCAHIZBDHIZJZDUTKIZBJZCDKIZAHIZB
|
|
JZABKIVEJZVBVAUTJZUSVDUTVALUMUPUNUQVIVDSZUMUPGUTEFZUNUQGZVJUPUMVKCAMNVLVKVJ
|
|
UNUQVKVJBDUTOPNTQRUSVFVCBUMURVFVCJUNUMURGZVFDCKIZAHIZVCVMVEVNAHURVEVNJUMCDU
|
|
AUBUCURUMVOVCJZUPUQUMVPUQUPUMVPDCAUDUEPNUFUGUJURUOVEEFZVGVHSZCDUHVQUOVRVQUM
|
|
UNVRVEABOUKNULUI $.
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${
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pncan3oi.1 $e |- A e. CC $.
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pncan3oi.2 $e |- B e. CC $.
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$( Subtraction and addition of equals. Almost but not exactly the same as
|
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~ pncan3i and ~ pncan , this order happens often when applying
|
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"operations to both sides" so create a theorem specifically for it. A
|
|
deduction version of this is available as ~ pncand . (Contributed by
|
|
David A. Wheeler, 11-Oct-2018.) $)
|
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pncan3oi $p |- ( ( A + B ) - B ) = A $=
|
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( cc wcel caddc co cmin wceq pncan mp2an ) AEFBEFABGHBIHAJCDABKL $.
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$}
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${
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mvrraddi.1 $e |- B e. CC $.
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mvrraddi.2 $e |- C e. CC $.
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mvrraddi.3 $e |- A = ( B + C ) $.
|
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$( Move RHS right addition to LHS. (Contributed by David A. Wheeler,
|
|
11-Oct-2018.) $)
|
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mvrraddi $p |- ( A - C ) = B $=
|
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( cmin co caddc oveq1i pncan3oi eqtri ) ACGHBCIHZCGHBAMCGFJBCDEKL $.
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$}
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${
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mvlladdi.1 $e |- A e. CC $.
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mvlladdi.2 $e |- B e. CC $.
|
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mvlladdi.3 $e |- ( A + B ) = C $.
|
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$( Move LHS left addition to RHS. (Contributed by David A. Wheeler,
|
|
11-Oct-2018.) $)
|
|
mvlladdi $p |- B = ( C - A ) $=
|
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( caddc co cmin pncan3oi addcomi eqtr3i oveq1i ) BAGHZAIHBCAIHBAEDJNCAIAB
|
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GHNCABDEKFLML $.
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$}
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$( Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
subid $p |- ( A e. CC -> ( A - A ) = 0 ) $=
|
|
( cc wcel cc0 caddc co cmin addid1 oveq1d wceq 0cn pncan2 mpan2 eqtr3d ) AB
|
|
CZADEFZAGFZAAGFDOPAAGAHIODBCQDJKADLMN $.
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|
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$( Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised
|
|
by Mario Carneiro, 27-May-2016.) $)
|
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subid1 $p |- ( A e. CC -> ( A - 0 ) = A ) $=
|
|
( cc wcel cc0 caddc co cmin addid1 oveq1d wceq 0cn pncan mpan2 eqtr3d ) ABC
|
|
ZADEFZDGFZADGFAOPADGAHIODBCQAJKADLMN $.
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|
|
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$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) $)
|
|
npncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) + ( B - C ) ) = ( A - C ) ) $=
|
|
( cc wcel w3a cmin co caddc wceq subcl 3adant3 addsubass syld3an1 wa oveq1d
|
|
npcan eqtr3d ) ADEZBDEZCDEZFABGHZBIHZCGHZUBBCGHIHZACGHZUBDEZTSUAUDUEJSTUGUA
|
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ABKLUBBCMNSTUDUFJUASTOUCACGABQPLR $.
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$( Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.) $)
|
|
nppcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( ( A - B ) + C ) + B ) = ( A + C ) ) $=
|
|
( cc wcel w3a cmin co caddc subcl 3adant3 simp3 simp2 add32d wceq wa oveq1d
|
|
npcan eqtrd ) ADEZBDEZCDEZFZABGHZCIHBIHUDBIHZCIHZACIHZUCUDCBTUAUDDEUBABJKTU
|
|
AUBLTUAUBMNTUAUFUGOUBTUAPUEACIABRQKS $.
|
|
|
|
$( Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex
|
|
numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.) $)
|
|
nnpcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC )
|
|
-> ( ( ( A - B ) - C ) + B ) = ( A - C ) ) $=
|
|
( cc wcel w3a cmin co caddc wceq subcl 3adant3 addsub eqcomd syld3an1 npcan
|
|
oveq1d eqtrd ) ADEZBDEZCDEZFZABGHZCGHBIHZUCBIHZCGHZACGHUCDEZTSUAUDUFJSTUGUA
|
|
ABKLUGTUAFUFUDUCBCMNOUBUEACGSTUEAJUAABPLQR $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
14-Sep-2015.) $)
|
|
nppcan3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) + ( C + B ) ) = ( A + C ) ) $=
|
|
( cc wcel w3a cmin co caddc subcl 3adant3 simp3 simp2 addassd nppcan eqtr3d
|
|
) ADEZBDEZCDEZFZABGHZCIHBIHUACBIHIHACIHTUACBQRUADESABJKQRSLQRSMNABCOP $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) $)
|
|
subcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - C ) = ( B - C ) <-> A = B ) ) $=
|
|
( cc wcel w3a cmin co wceq caddc wb simp1 simp3 subcl 3adant1 subadd2 npcan
|
|
syl3anc eqeq1d eqcom syl6bb bitrd ) ADEZBDEZCDEZFZACGHBCGHZIZUGCJHZAIZABIZU
|
|
FUCUEUGDEZUHUJKUCUDUELUCUDUEMUDUEULUCBCNOACUGPRUFUJBAIUKUFUIBAUDUEUIBIUCBCQ
|
|
OSBATUAUB $.
|
|
|
|
$( If the difference between two numbers is zero, they are equal.
|
|
(Contributed by NM, 16-Nov-1999.) $)
|
|
subeq0 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) ) $=
|
|
( cc wcel wa cmin co wceq cc0 subid adantl eqeq2d subcan2 3anidm23 bitr3d
|
|
wb ) ACDZBCDZEZABFGZBBFGZHZTIHABHZSUAITRUAIHQBJKLQRUBUCPABBMNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Scott Fenton,
|
|
21-Jun-2013.) $)
|
|
npncan2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( B - A ) ) = 0 ) $=
|
|
( cc wcel wa cmin co caddc cc0 wceq npncan 3anidm13 subid adantr eqtrd ) AC
|
|
DZBCDZEABFGBAFGHGZAAFGZIPQRSJABAKLPSIJQAMNO $.
|
|
|
|
$( Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
subsub2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A - ( B - C ) ) = ( A + ( C - B ) ) ) $=
|
|
( cc wcel w3a cmin co caddc wceq cc0 subcl 3adant1 simp1 simp3 simp2 add12d
|
|
syl2anc npncan2 oveq2d addid1d 3eqtrd wb addcld subadd syl3anc mpbird ) ADE
|
|
ZBDEZCDEZFZABCGHZGHACBGHZIHZJZULUNIHZAJZUKUPAULUMIHZIHAKIHAUKULAUMUIUJULDEZ
|
|
UHBCLMZUHUIUJNZUKUJUIUMDEUHUIUJOUHUIUJPCBLRZQUKURKAIUIUJURKJUHBCSMTUKAVAUAU
|
|
BUKUHUSUNDEUOUQUCVAUTUKAUMVAVBUDAULUNUEUFUG $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.)
|
|
(Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
nncan $p |- ( ( A e. CC /\ B e. CC ) -> ( A - ( A - B ) ) = B ) $=
|
|
( cc wcel wa cmin co caddc wceq subsub2 3anidm12 pncan3 eqtrd ) ACDZBCDZEAA
|
|
BFGFGZABAFGHGZBNOPQIAABJKABLM $.
|
|
|
|
$( Law for double subtraction. (Contributed by NM, 13-May-2004.) $)
|
|
subsub $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A - ( B - C ) ) = ( ( A - B ) + C ) ) $=
|
|
( cc wcel w3a cmin caddc subsub2 wceq addsubass addsub eqtr3d 3com23 eqtrd
|
|
co ) ADEZBDEZCDEZFABCGPGPACBGPHPZABGPCHPZABCIQSRTUAJQSRFACHPBGPTUAACBKACBLM
|
|
NO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.) $)
|
|
nppcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - ( B + C ) ) + C ) = ( A - B ) ) $=
|
|
( cc wcel w3a caddc co cmin wceq addcl 3adant1 subsub syld3an2 pncan oveq2d
|
|
eqtr3d ) ADEZBDEZCDEZFZABCGHZCIHZIHZAUBIHCGHZABIHRUBDEZSTUDUEJSTUFRBCKLAUBC
|
|
MNUAUCBAISTUCBJRBCOLPQ $.
|
|
|
|
$( Law for double subtraction. (Contributed by NM, 27-Jul-2005.) $)
|
|
subsub3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A - ( B - C ) ) = ( ( A + C ) - B ) ) $=
|
|
( cc wcel w3a cmin co caddc subsub2 wceq addsubass 3com23 eqtr4d ) ADEZBDEZ
|
|
CDEZFABCGHGHACBGHIHZACIHBGHZABCJOQPSRKACBLMN $.
|
|
|
|
$( Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
subsub4 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) - C ) = ( A - ( B + C ) ) ) $=
|
|
( cc wcel w3a cmin co caddc wceq nppcan2 wb simp1 simp2 subcl syl2anc simp3
|
|
addcld subadd2 syl3anc mpbird ) ADEZBDEZCDEZFZABGHZCGHABCIHZGHZJZUHCIHUFJZA
|
|
BCKUEUFDEZUDUHDEZUIUJLUEUBUCUKUBUCUDMZUBUCUDNZABOPUBUCUDQZUEUBUGDEULUMUEBCU
|
|
NUORAUGOPUFCUHSTUA $.
|
|
|
|
$( Swap the second and third terms in a double subtraction. (Contributed by
|
|
NM, 19-Aug-2005.) $)
|
|
sub32 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) - C ) = ( ( A - C ) - B ) ) $=
|
|
( cc wcel w3a caddc cmin wceq addcom 3adant1 oveq2d subsub4 3com23 3eqtr4d
|
|
co ) ADEZBDEZCDEZFZABCGPZHPACBGPZHPZABHPCHPACHPBHPZTUAUBAHRSUAUBIQBCJKLABCM
|
|
QSRUDUCIACBMNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.) $)
|
|
nnncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - ( B - C ) ) - C ) = ( A - B ) ) $=
|
|
( cc wcel cmin co caddc wceq subcl 3adant1 subsub4 syld3an2 wa npcan oveq2d
|
|
w3a eqtrd ) ADEZBDEZCDEZQABCFGZFGCFGZAUBCHGZFGZABFGZSUBDEZTUAUCUEITUAUGSBCJ
|
|
KAUBCLMTUAUEUFISTUANUDBAFBCOPKR $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
|
(Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
nnncan1 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) - ( A - C ) ) = ( C - B ) ) $=
|
|
( cc wcel w3a cmin co wceq subcl 3adant2 sub32 syld3an3 nncan oveq1d eqtrd
|
|
) ADEZBDEZCDEZFZABGHACGHZGHZAUAGHZBGHZCBGHQRSUADEZUBUDIQSUERACJKABUALMTUCCB
|
|
GQSUCCIRACNKOP $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.) $)
|
|
nnncan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - C ) - ( B - C ) ) = ( A - B ) ) $=
|
|
( cc wcel w3a cmin co wceq subcl 3adant1 sub32 syld3an2 nnncan eqtr3d ) ADE
|
|
ZBDEZCDEZFABCGHZGHCGHZACGHSGHZABGHPSDEZQRTUAIQRUBPBCJKASCLMABCNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Scott Fenton,
|
|
23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
npncan3 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) + ( C - A ) ) = ( C - B ) ) $=
|
|
( cc wcel w3a cmin co caddc simp1 subcl ancoms 3adant2 simp2 addsub syl3anc
|
|
wceq pncan3 oveq1d eqtr3d ) ADEZBDEZCDEZFZACAGHZIHZBGHZABGHUEIHZCBGHUDUAUED
|
|
EZUBUGUHQUAUBUCJUAUCUIUBUCUAUICAKLMUAUBUCNAUEBOPUDUFCBGUAUCUFCQUBACRMST $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by NM,
|
|
4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
pnpcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) - ( A + C ) ) = ( B - C ) ) $=
|
|
( cc wcel w3a caddc co cmin simp1 simp2 addcld simp3 subsub4 syl3anc pncan2
|
|
wceq 3adant3 oveq1d eqtr3d ) ADEZBDEZCDEZFZABGHZAIHZCIHZUEACGHIHZBCIHUDUEDE
|
|
UAUCUGUHQUDABUAUBUCJZUAUBUCKLUIUAUBUCMUEACNOUDUFBCIUAUBUFBQUCABPRST $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
Scott Fenton, 9-Jun-2006.) $)
|
|
pnpcan2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + C ) - ( B + C ) ) = ( A - B ) ) $=
|
|
( cc wcel w3a caddc co cmin wceq addcom 3adant2 3adant1 oveq12d 3coml eqtrd
|
|
pnpcan ) ADEZBDEZCDEZFZACGHZBCGHZIHCAGHZCBGHZIHZABIHZUAUBUDUCUEIRTUBUDJSACK
|
|
LSTUCUEJRBCKMNTRSUFUGJCABQOP $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by NM,
|
|
30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
pnncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) - ( A - C ) ) = ( B + C ) ) $=
|
|
( cc wcel caddc co cmin wceq simp1 simp2 addcld simp3 subsub syl3anc pncan2
|
|
w3a 3adant3 oveq1d eqtrd ) ADEZBDEZCDEZQZABFGZACHGHGZUEAHGZCFGZBCFGUDUEDEUA
|
|
UCUFUHIUDABUAUBUCJZUAUBUCKLUIUAUBUCMUEACNOUDUGBCFUAUBUGBIUCABPRST $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by NM,
|
|
30-Jun-2005.) $)
|
|
ppncan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A + B ) + ( C - B ) ) = ( A + C ) ) $=
|
|
( cc wcel w3a caddc co cmin wceq addcom 3adant3 oveq1d addcl subsub2 pnncan
|
|
syld3an1 3com12 3eqtr3d ) ADEZBDEZCDEZFZABGHZBCIHZIHZBAGHZUEIHZUDCBIHGHZACG
|
|
HZUCUDUGUEITUAUDUGJUBABKLMUDDEZUATUBUFUIJTUAUKUBABNLUDBCOQUATUBUHUJJBACPRS
|
|
$.
|
|
|
|
$( Rearrangement of 4 terms in a mixed addition and subtraction.
|
|
(Contributed by NM, 4-Mar-2005.) $)
|
|
addsub4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) ) $=
|
|
( cc wcel caddc cmin wceq simpll simplr simprl addsub syl3anc oveq1d addcld
|
|
wa co simprr subsub4 subcl ad2ant2r addsubass 3eqtr3d ) AEFZBEFZQZCEFZDEFZQ
|
|
ZQZABGRZCHRZDHRZACHRZBGRZDHRZULCDGRHRZUOBDHRGRZUKUMUPDHUKUEUFUHUMUPIUEUFUJJ
|
|
ZUEUFUJKZUGUHUILZABCMNOUKULEFUHUIUNURIUKABUTVAPVBUGUHUISZULCDTNUKUOEFZUFUIU
|
|
QUSIUEUHVDUFUIACUAUBVAVCUOBDUCNUD $.
|
|
|
|
$( Rearrangement of 4 terms in a mixed addition and subtraction.
|
|
(Contributed by NM, 24-Aug-2006.) $)
|
|
subadd4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) ) $=
|
|
( cc wcel wa cmin caddc wceq subcl subsub2 3expb sylan addsub4 an42s eqtr4d
|
|
co ) AEFZBEFZGZCEFZDEFZGZGABHRZCDHRHRZUEDCHRIRZADIRBCIRHRZUAUEEFZUDUFUGJZAB
|
|
KUIUBUCUJUECDLMNSUCTUBUHUGJADBCOPQ $.
|
|
|
|
$( Rearrangement of 4 terms in a subtraction. (Contributed by NM,
|
|
23-Nov-2007.) $)
|
|
sub4 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) ) $=
|
|
( cc wcel wa caddc cmin wceq addcom ad2ant2lr oveq2d subadd4 an4s 3eqtr4d
|
|
co ) AEFZBEFZGCEFZDEFZGGZADHQZBCHQZIQUCCBHQZIQZABIQCDIQIQACIQBDIQIQZUBUDUEU
|
|
CISTUDUEJRUABCKLMABCDNRTSUAUGUFJACBDNOP $.
|
|
|
|
$( Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.) $)
|
|
neg0 $p |- -u 0 = 0 $=
|
|
( cc0 cneg cmin co df-neg cc wcel wceq 0cn subid ax-mp eqtri ) ABAACDZAAEAF
|
|
GMAHIAJKL $.
|
|
|
|
$( Addition of a number and its negative. (Contributed by NM,
|
|
14-Mar-2005.) $)
|
|
negid $p |- ( A e. CC -> ( A + -u A ) = 0 ) $=
|
|
( cc wcel cneg caddc co cc0 cmin df-neg oveq2i wceq 0cn pncan3 mpan2 syl5eq
|
|
) ABCZAADZEFAGAHFZEFZGQRAEAIJPGBCSGKLAGMNO $.
|
|
|
|
$( Relationship between subtraction and negative. Theorem I.3 of [Apostol]
|
|
p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negsub $p |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) ) $=
|
|
( cc wcel wa cneg caddc co cc0 cmin wceq df-neg oveq2i a1i addsubass mp3an2
|
|
0cn simpl addid1d oveq1d 3eqtr2d ) ACDZBCDZEZABFZGHZAIBJHZGHZAIGHZBJHZABJHU
|
|
FUHKUDUEUGAGBLMNUBICDUCUJUHKQAIBOPUDUIABJUDAUBUCRSTUA $.
|
|
|
|
$( Relationship between subtraction and negative. (Contributed by NM,
|
|
10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
subneg $p |- ( ( A e. CC /\ B e. CC ) -> ( A - -u B ) = ( A + B ) ) $=
|
|
( cc wcel wa cneg cmin co cc0 caddc df-neg oveq2i wceq subsub mp3an2 syl5eq
|
|
0cn subid1 adantr oveq1d eqtrd ) ACDZBCDZEZABFZGHZAIGHZBJHZABJHUDUFAIBGHZGH
|
|
ZUHUEUIAGBKLUBICDUCUJUHMQAIBNOPUDUGABJUBUGAMUCARSTUA $.
|
|
|
|
$( A number is equal to the negative of its negative. Theorem I.4 of
|
|
[Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negneg $p |- ( A e. CC -> -u -u A = A ) $=
|
|
( cc wcel cneg cc0 caddc co cmin df-neg wceq 0cn subneg syl5eq addid2 eqtrd
|
|
mpan ) ABCZADZDZEAFGZAQSERHGZTRIEBCQUATJKEALPMANO $.
|
|
|
|
$( Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
neg11 $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A = -u B <-> A = B ) ) $=
|
|
( cc wcel wa cneg wceq negeq negneg eqeqan12d syl5ib impbid1 ) ACDZBCDZEZAF
|
|
ZBFZGZABGZRPFZQFZGOSPQHMNTAUABAIBIJKABHL $.
|
|
|
|
$( Negative contraposition law. (Contributed by NM, 9-May-2004.) $)
|
|
negcon1 $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) ) $=
|
|
( cc wcel wa cneg wceq negcl neg11 sylan negneg adantr eqeq1d bitr3d syl6bb
|
|
wb eqcom ) ACDZBCDZEZAFZBGZABFZGZUCAGTUAFZUCGZUBUDRUACDSUFUBPAHUABIJTUEAUCR
|
|
UEAGSAKLMNAUCQO $.
|
|
|
|
$( Negative contraposition law. (Contributed by NM, 14-Nov-2004.) $)
|
|
negcon2 $p |- ( ( A e. CC /\ B e. CC ) -> ( A = -u B <-> B = -u A ) ) $=
|
|
( cc wcel wa cneg wceq negcon1 eqcom syl6rbbr syl6bb ) ACDBCDEZABFZGZAFZBGZ
|
|
BOGLPMAGNABHAMIJOBIK $.
|
|
|
|
$( A number is zero iff its negative is zero. (Contributed by NM,
|
|
12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
negeq0 $p |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) ) $=
|
|
( cneg cc0 wceq cc wcel neg0 eqeq2i wb 0cn neg11 mpan2 syl5rbbr ) ABZCDNCBZ
|
|
DZAEFZACDZOCNGHQCEFPRIJACKLM $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
subcan $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) = ( A - C ) <-> B = C ) ) $=
|
|
( cc wcel w3a caddc co wceq cmin simp2 simp1 addcomd eqeq1d simp3 addsubeq4
|
|
wb syl22anc addcan 3bitr3d ) ADEZBDEZCDEZFZBAGHZACGHZIZABGHZUFIABJHACJHIZBC
|
|
IUDUEUHUFUDBAUAUBUCKZUAUBUCLZMNUDUBUAUAUCUGUIQUJUKUKUAUBUCOBAACPRABCST $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by NM,
|
|
15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
negsubdi $p |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) ) $=
|
|
( cc wcel wa cc0 cmin co caddc cneg 0cn subsub mp3an1 df-neg oveq1i 3eqtr4g
|
|
wceq ) ACDZBCDZEFABGHZGHZFAGHZBIHZTJAJZBIHFCDRSUAUCQKFABLMTNUDUBBIANOP $.
|
|
|
|
$( Distribution of negative over addition. (Contributed by NM, 10-May-2004.)
|
|
(Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
negdi $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
-u ( A + B ) = ( -u A + -u B ) ) $=
|
|
( cc wcel wa cneg cmin co caddc subneg negeqd negcl negsubdi sylan2 eqtr3d
|
|
wceq ) ACDZBCDZEZABFZGHZFZABIHZFAFTIHZSUAUCABJKRQTCDUBUDPBLATMNO $.
|
|
|
|
$( Distribution of negative over addition. (Contributed by NM,
|
|
1-Jan-2006.) $)
|
|
negdi2 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
-u ( A + B ) = ( -u A - B ) ) $=
|
|
( cc wcel wa caddc co cneg cmin negdi wceq negcl negsub sylan eqtrd ) ACDZB
|
|
CDZEABFGHAHZBHFGZRBIGZABJPRCDQSTKALRBMNO $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by NM,
|
|
4-Oct-1999.) $)
|
|
negsubdi2 $p |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) ) $=
|
|
( cc wcel wa cmin cneg caddc negsubdi wceq negcl addcom sylan negsub ancoms
|
|
co 3eqtrd ) ACDZBCDZEABFPGAGZBHPZBTHPZBAFPZABIRTCDSUAUBJAKTBLMSRUBUCJBANOQ
|
|
$.
|
|
|
|
$( Relationship between subtraction and negative. (Contributed by Paul
|
|
Chapman, 8-Oct-2007.) $)
|
|
neg2sub $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( B - A ) ) $=
|
|
( cc wcel wa cneg cmin co caddc wceq negcl sylan negsubdi negsubdi2 3eqtr2d
|
|
subneg ) ACDZBCDZEAFZBFGHZSBIHZABGHFBAGHQSCDRTUAJAKSBPLABMABNO $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.) $)
|
|
renegcl $p |- ( A e. RR -> -u A e. RR ) $=
|
|
( vx cr wcel cv caddc co cc0 wceq wrex cneg ax-rnegex wa recn cmin df-neg
|
|
cc wb eqeq1i 0cn subadd mp3an1 syl5bb sylan2 wi eleq1a adantl sylbird mpd
|
|
sylan rexlimdva ) ACDZABEZFGHIZBCJAKZCDZBALULUNUPBCULUMCDZMUNUOUMIZUPUQUL
|
|
UMQDZURUNRUMNURHAOGZUMIZULUSMUNUOUTUMAPSULAQDZUSVAUNRZANHQDVBUSVCTHAUMUAU
|
|
BUJUCUDUQURUPUEULUMCUOUFUGUHUKUI $.
|
|
$}
|
|
|
|
${
|
|
renegcl.1 $e |- A e. RR $.
|
|
$( Closure law for negative of reals. (Note: this inference proof style
|
|
and the deduction theorem usage in ~ renegcl is deprecated, but is
|
|
retained for its demonstration value.) (Contributed by NM,
|
|
17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
renegcli $p |- -u A e. RR $=
|
|
( cr wcel cneg renegcl ax-mp ) ACDAECDBAFG $.
|
|
|
|
resubcl.2 $e |- B e. RR $.
|
|
$( Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
resubcli $p |- ( A - B ) e. RR $=
|
|
( cneg caddc co cmin cr cc wcel wceq recni negsub mp2an renegcli readdcli
|
|
eqeltrri ) ABEZFGZABHGZIAJKBJKTUALACMBDMABNOASCBDPQR $.
|
|
$}
|
|
|
|
$( Closure law for subtraction of reals. (Contributed by NM,
|
|
20-Jan-1997.) $)
|
|
resubcl $p |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR ) $=
|
|
( cr wcel wa cneg caddc cmin wceq recn negsub syl2an renegcl readdcl sylan2
|
|
co cc eqeltrrd ) ACDZBCDZEABFZGPZABHPZCSAQDBQDUBUCITAJBJABKLTSUACDUBCDBMAUA
|
|
NOR $.
|
|
|
|
$( The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
negreb $p |- ( A e. CC -> ( -u A e. RR <-> A e. RR ) ) $=
|
|
( cc wcel cneg cr renegcl negneg eleq1d syl5ib impbid1 ) ABCZADZECZAECZMLDZ
|
|
ECKNLFKOAEAGHIAFJ $.
|
|
|
|
$( "Reverse" second Peano postulate analog for complex numbers: A complex
|
|
number minus 1 is a complex number. (Contributed by Alexander van der
|
|
Vekens, 18-Mar-2018.) $)
|
|
peano2cnm $p |- ( N e. CC -> ( N - 1 ) e. CC ) $=
|
|
( cc wcel c1 cmin co ax-1cn subcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( "Reverse" second Peano postulate analog for reals. (Contributed by NM,
|
|
6-Feb-2007.) $)
|
|
peano2rem $p |- ( N e. RR -> ( N - 1 ) e. RR ) $=
|
|
( cr wcel c1 cmin co 1re resubcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
${
|
|
negidi.1 $e |- A e. CC $.
|
|
$( Closure law for negative. (Contributed by NM, 26-Nov-1994.) $)
|
|
negcli $p |- -u A e. CC $=
|
|
( cc wcel cneg negcl ax-mp ) ACDAECDBAFG $.
|
|
|
|
$( Addition of a number and its negative. (Contributed by NM,
|
|
26-Nov-1994.) $)
|
|
negidi $p |- ( A + -u A ) = 0 $=
|
|
( cc wcel cneg caddc co cc0 wceq negid ax-mp ) ACDAAEFGHIBAJK $.
|
|
|
|
$( A number is equal to the negative of its negative. Theorem I.4 of
|
|
[Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by
|
|
Andrew Salmon, 22-Oct-2011.) $)
|
|
negnegi $p |- -u -u A = A $=
|
|
( cc wcel cneg wceq negneg ax-mp ) ACDAEEAFBAGH $.
|
|
|
|
$( Subtraction of a number from itself. (Contributed by NM,
|
|
26-Nov-1994.) $)
|
|
subidi $p |- ( A - A ) = 0 $=
|
|
( cc wcel cmin co cc0 wceq subid ax-mp ) ACDAAEFGHBAIJ $.
|
|
|
|
$( Identity law for subtraction. (Contributed by NM, 29-May-1999.) $)
|
|
subid1i $p |- ( A - 0 ) = A $=
|
|
( cc wcel cc0 cmin co wceq subid1 ax-mp ) ACDAEFGAHBAIJ $.
|
|
|
|
$( A number is nonzero iff its negative is nonzero. (Contributed by NM,
|
|
10-Aug-1999.) $)
|
|
negne0bi $p |- ( A =/= 0 <-> -u A =/= 0 ) $=
|
|
( cc0 cneg cc wcel wceq wb negeq0 ax-mp necon3bii ) ACADZCAEFACGLCGHBAIJK
|
|
$.
|
|
|
|
$( The negative of a real is real. (Contributed by NM, 11-Aug-1999.) $)
|
|
negrebi $p |- ( -u A e. RR <-> A e. RR ) $=
|
|
( cc wcel cneg cr wb negreb ax-mp ) ACDAEFDAFDGBAHI $.
|
|
|
|
${
|
|
negne0i.2 $e |- A =/= 0 $.
|
|
$( The negative of a nonzero number is nonzero. (Contributed by NM,
|
|
30-Jul-2004.) $)
|
|
negne0i $p |- -u A =/= 0 $=
|
|
( cc0 wne cneg negne0bi mpbi ) ADEAFDECABGH $.
|
|
$}
|
|
|
|
pncan3i.2 $e |- B e. CC $.
|
|
$( Closure law for subtraction. (Contributed by NM, 26-Nov-1994.)
|
|
(Revised by Mario Carneiro, 21-Dec-2013.) $)
|
|
subcli $p |- ( A - B ) e. CC $=
|
|
( cc wcel cmin co subcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( Subtraction and addition of equals. (Contributed by NM,
|
|
26-Nov-1994.) $)
|
|
pncan3i $p |- ( A + ( B - A ) ) = B $=
|
|
( cc wcel cmin co caddc wceq pncan3 mp2an ) AEFBEFABAGHIHBJCDABKL $.
|
|
|
|
$( Relationship between subtraction and negative. Theorem I.3 of [Apostol]
|
|
p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew
|
|
Salmon, 22-Oct-2011.) $)
|
|
negsubi $p |- ( A + -u B ) = ( A - B ) $=
|
|
( cc wcel cneg caddc co cmin wceq negsub mp2an ) AEFBEFABGHIABJIKCDABLM
|
|
$.
|
|
|
|
$( Relationship between subtraction and negative. (Contributed by NM,
|
|
1-Dec-2005.) $)
|
|
subnegi $p |- ( A - -u B ) = ( A + B ) $=
|
|
( cc wcel cneg cmin co caddc wceq subneg mp2an ) AEFBEFABGHIABJIKCDABLM
|
|
$.
|
|
|
|
$( If the difference between two numbers is zero, they are equal.
|
|
(Contributed by NM, 8-May-1999.) $)
|
|
subeq0i $p |- ( ( A - B ) = 0 <-> A = B ) $=
|
|
( cc wcel cmin co cc0 wceq wb subeq0 mp2an ) AEFBEFABGHIJABJKCDABLM $.
|
|
|
|
$( Negative is one-to-one. (Contributed by NM, 1-Aug-1999.) $)
|
|
neg11i $p |- ( -u A = -u B <-> A = B ) $=
|
|
( cc wcel cneg wceq wb neg11 mp2an ) AEFBEFAGBGHABHICDABJK $.
|
|
|
|
$( Negative contraposition law. (Contributed by NM, 25-Aug-1999.) $)
|
|
negcon1i $p |- ( -u A = B <-> -u B = A ) $=
|
|
( cc wcel cneg wceq wb negcon1 mp2an ) AEFBEFAGBHBGAHICDABJK $.
|
|
|
|
$( Negative contraposition law. (Contributed by NM, 25-Aug-1999.) $)
|
|
negcon2i $p |- ( A = -u B <-> B = -u A ) $=
|
|
( cc wcel cneg wceq wb negcon2 mp2an ) AEFBEFABGHBAGHICDABJK $.
|
|
|
|
$( Distribution of negative over addition. (Contributed by NM,
|
|
28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
negdii $p |- -u ( A + B ) = ( -u A + -u B ) $=
|
|
( caddc co cneg wceq cc0 addcli negidi oveq12i eqtri negcli add4i 3eqtr2i
|
|
00id addcani mpbi ) ABEFZTGZEFZTAGZBGZEFZEFZHUAUEHUBIAUCEFZBUDEFZEFZUFTAB
|
|
CDJZKUIIIEFIUGIUHIEACKBDKLQMAUCBUDCACNZDBDNZOPTUAUEUJTUJNUCUDUKULJRS $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by NM,
|
|
6-Aug-1999.) $)
|
|
negsubdii $p |- -u ( A - B ) = ( -u A + B ) $=
|
|
( cneg caddc co cmin negcli negdii negsubi negeqi negnegi oveq2i 3eqtr3i
|
|
) ABEZFGZEAEZPEZFGABHGZERBFGAPCBDIJQTABCDKLSBRFBDMNO $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by NM,
|
|
1-Oct-1999.) $)
|
|
negsubdi2i $p |- -u ( A - B ) = ( B - A ) $=
|
|
( cmin co cneg caddc negsubdii negcli negsubi addcomli eqtri ) ABEFGAGZBH
|
|
FBAEFZABCDIBNODACJBADCKLM $.
|
|
|
|
subadd.3 $e |- C e. CC $.
|
|
$( Relationship between subtraction and addition. (Contributed by NM,
|
|
26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.) $)
|
|
subaddi $p |- ( ( A - B ) = C <-> ( B + C ) = A ) $=
|
|
( cc wcel cmin co wceq caddc wb subadd mp3an ) AGHBGHCGHABIJCKBCLJAKMDEFA
|
|
BCNO $.
|
|
|
|
$( Relationship between subtraction and addition. (Contributed by NM,
|
|
15-Dec-2006.) $)
|
|
subadd2i $p |- ( ( A - B ) = C <-> ( C + B ) = A ) $=
|
|
( cc wcel cmin co wceq caddc wb subadd2 mp3an ) AGHBGHCGHABIJCKCBLJAKMDEF
|
|
ABCNO $.
|
|
|
|
${
|
|
subaddri.4 $e |- ( B + C ) = A $.
|
|
$( Relationship between subtraction and addition. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
subaddrii $p |- ( A - B ) = C $=
|
|
( cmin co wceq caddc subaddi mpbir ) ABHICJBCKIAJGABCDEFLM $.
|
|
$}
|
|
|
|
$( Swap subtrahend and result of subtraction. (Contributed by NM,
|
|
7-Oct-1999.) $)
|
|
subsub23i $p |- ( ( A - B ) = C <-> ( A - C ) = B ) $=
|
|
( cc wcel cmin co wceq wb subsub23 mp3an ) AGHBGHCGHABIJCKACIJBKLDEFABCMN
|
|
$.
|
|
|
|
$( Associative-type law for subtraction and addition. (Contributed by NM,
|
|
16-Sep-1999.) $)
|
|
addsubassi $p |- ( ( A + B ) - C ) = ( A + ( B - C ) ) $=
|
|
( cc wcel caddc co cmin wceq addsubass mp3an ) AGHBGHCGHABIJCKJABCKJIJLDE
|
|
FABCMN $.
|
|
|
|
$( Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.) $)
|
|
addsubi $p |- ( ( A + B ) - C ) = ( ( A - C ) + B ) $=
|
|
( cc wcel caddc co cmin wceq addsub mp3an ) AGHBGHCGHABIJCKJACKJBIJLDEFAB
|
|
CMN $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) $)
|
|
subcani $p |- ( ( A - B ) = ( A - C ) <-> B = C ) $=
|
|
( cc wcel cmin co wceq wb subcan mp3an ) AGHBGHCGHABIJACIJKBCKLDEFABCMN
|
|
$.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) $)
|
|
subcan2i $p |- ( ( A - C ) = ( B - C ) <-> A = B ) $=
|
|
( cc wcel cmin co wceq wb subcan2 mp3an ) AGHBGHCGHACIJBCIJKABKLDEFABCMN
|
|
$.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
NM, 14-Jan-2006.) $)
|
|
pnncani $p |- ( ( A + B ) - ( A - C ) ) = ( B + C ) $=
|
|
( cc wcel caddc co cmin wceq pnncan mp3an ) AGHBGHCGHABIJACKJKJBCIJLDEFAB
|
|
CMN $.
|
|
|
|
addsub4i.4 $e |- D e. CC $.
|
|
$( Rearrangement of 4 terms in a mixed addition and subtraction.
|
|
(Contributed by NM, 17-Oct-1999.) $)
|
|
addsub4i $p |- ( ( A + B ) - ( C + D ) ) =
|
|
( ( A - C ) + ( B - D ) ) $=
|
|
( cc wcel caddc co cmin wceq addsub4 mp4an ) AIJBIJCIJDIJABKLCDKLMLACMLBD
|
|
MLKLNEFGHABCDOP $.
|
|
$}
|
|
|
|
$( Alternate proof of ~ 0re . (Contributed by NM, 19-Feb-2005.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
0reALT $p |- 0 e. RR $=
|
|
( c1 cmin co cc0 cr ax-1cn subidi 1re resubcli eqeltrri ) AABCDEAFGAAHHIJ
|
|
$.
|
|
|
|
${
|
|
negidd.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Closure law for negative. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
negcld $p |- ( ph -> -u A e. CC ) $=
|
|
( cc wcel cneg negcl syl ) ABDEBFDECBGH $.
|
|
|
|
$( Subtraction of a number from itself. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subidd $p |- ( ph -> ( A - A ) = 0 ) $=
|
|
( cc wcel cmin co cc0 wceq subid syl ) ABDEBBFGHICBJK $.
|
|
|
|
$( Identity law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subid1d $p |- ( ph -> ( A - 0 ) = A ) $=
|
|
( cc wcel cc0 cmin co wceq subid1 syl ) ABDEBFGHBICBJK $.
|
|
|
|
$( Addition of a number and its negative. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
negidd $p |- ( ph -> ( A + -u A ) = 0 ) $=
|
|
( cc wcel cneg caddc co cc0 wceq negid syl ) ABDEBBFGHIJCBKL $.
|
|
|
|
$( A number is equal to the negative of its negative. Theorem I.4 of
|
|
[Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
negnegd $p |- ( ph -> -u -u A = A ) $=
|
|
( cc wcel cneg wceq negneg syl ) ABDEBFFBGCBHI $.
|
|
|
|
$( A number is zero iff its negative is zero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negeq0d $p |- ( ph -> ( A = 0 <-> -u A = 0 ) ) $=
|
|
( cc wcel cc0 wceq cneg wb negeq0 syl ) ABDEBFGBHFGICBJK $.
|
|
|
|
$( A number is nonzero iff its negative is nonzero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negne0bd $p |- ( ph -> ( A =/= 0 <-> -u A =/= 0 ) ) $=
|
|
( cc0 cneg negeq0d necon3bid ) ABDBEDABCFG $.
|
|
|
|
${
|
|
negcon1d.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Contraposition law for unary minus. Deduction form of ~ negcon1 .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
negcon1d $p |- ( ph -> ( -u A = B <-> -u B = A ) ) $=
|
|
( cc wcel cneg wceq wb negcon1 syl2anc ) ABFGCFGBHCICHBIJDEBCKL $.
|
|
$}
|
|
|
|
${
|
|
negcon1ad.2 $e |- ( ph -> -u A = B ) $.
|
|
$( Contraposition law for unary minus. One-way deduction form of
|
|
~ negcon1 . (Contributed by David Moews, 28-Feb-2017.) $)
|
|
negcon1ad $p |- ( ph -> -u B = A ) $=
|
|
( cneg wceq cc negcld eqeltrrd negcon1d mpbid ) ABFZCGCFBGEABCDAMCHEABD
|
|
IJKL $.
|
|
$}
|
|
|
|
${
|
|
neg11ad.2 $e |- ( ph -> B e. CC ) $.
|
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$( The negatives of two complex numbers are equal iff they are equal.
|
|
Deduction form of ~ neg11 . Generalization of ~ neg11d .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
neg11ad $p |- ( ph -> ( -u A = -u B <-> A = B ) ) $=
|
|
( cc wcel cneg wceq wb neg11 syl2anc ) ABFGCFGBHCHIBCIJDEBCKL $.
|
|
$}
|
|
|
|
${
|
|
negned.2 $e |- ( ph -> B e. CC ) $.
|
|
negned.3 $e |- ( ph -> A =/= B ) $.
|
|
$( If two complex numbers are unequal, so are their negatives.
|
|
Contrapositive of ~ neg11d . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
negned $p |- ( ph -> -u A =/= -u B ) $=
|
|
( cneg wne neg11ad necon3bid mpbird ) ABGZCGZHBCHFALMBCABCDEIJK $.
|
|
$}
|
|
|
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${
|
|
negne0d.2 $e |- ( ph -> A =/= 0 ) $.
|
|
$( The negative of a nonzero number is nonzero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negne0d $p |- ( ph -> -u A =/= 0 ) $=
|
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( cc0 wne cneg negne0bd mpbid ) ABEFBGEFDABCHI $.
|
|
$}
|
|
|
|
${
|
|
negrebd.2 $e |- ( ph -> -u A e. RR ) $.
|
|
$( The negative of a real is real. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
negrebd $p |- ( ph -> A e. RR ) $=
|
|
( cneg cr wcel cc wb negreb syl mpbid ) ABEFGZBFGZDABHGMNICBJKL $.
|
|
$}
|
|
|
|
pncand.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Closure law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subcld $p |- ( ph -> ( A - B ) e. CC ) $=
|
|
( cc wcel cmin co subcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
pncand $p |- ( ph -> ( ( A + B ) - B ) = A ) $=
|
|
( cc wcel caddc co cmin wceq pncan syl2anc ) ABFGCFGBCHICJIBKDEBCLM $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
pncan2d $p |- ( ph -> ( ( A + B ) - A ) = B ) $=
|
|
( cc wcel caddc co cmin wceq pncan2 syl2anc ) ABFGCFGBCHIBJICKDEBCLM $.
|
|
|
|
$( Subtraction and addition of equals. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
pncan3d $p |- ( ph -> ( A + ( B - A ) ) = B ) $=
|
|
( cc wcel cmin co caddc wceq pncan3 syl2anc ) ABFGCFGBCBHIJICKDEBCLM $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
npcand $p |- ( ph -> ( ( A - B ) + B ) = A ) $=
|
|
( cc wcel cmin co caddc wceq npcan syl2anc ) ABFGCFGBCHICJIBKDEBCLM $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nncand $p |- ( ph -> ( A - ( A - B ) ) = B ) $=
|
|
( cc wcel cmin co wceq nncan syl2anc ) ABFGCFGBBCHIHICJDEBCKL $.
|
|
|
|
$( Relationship between subtraction and negative. Theorem I.3 of [Apostol]
|
|
p. 18. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
negsubd $p |- ( ph -> ( A + -u B ) = ( A - B ) ) $=
|
|
( cc wcel cneg caddc co cmin wceq negsub syl2anc ) ABFGCFGBCHIJBCKJLDEBCM
|
|
N $.
|
|
|
|
$( Relationship between subtraction and negative. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
subnegd $p |- ( ph -> ( A - -u B ) = ( A + B ) ) $=
|
|
( cc wcel cneg cmin co caddc wceq subneg syl2anc ) ABFGCFGBCHIJBCKJLDEBCM
|
|
N $.
|
|
|
|
${
|
|
subeq0d.3 $e |- ( ph -> ( A - B ) = 0 ) $.
|
|
$( If the difference between two numbers is zero, they are equal.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
subeq0d $p |- ( ph -> A = B ) $=
|
|
( cmin co cc0 wceq cc wcel wb subeq0 syl2anc mpbid ) ABCGHIJZBCJZFABKLC
|
|
KLQRMDEBCNOP $.
|
|
$}
|
|
|
|
${
|
|
subne0d.3 $e |- ( ph -> A =/= B ) $.
|
|
$( Two unequal numbers have nonzero difference. See also ~ subap0d which
|
|
is the same thing for apartness rather than negated equality.
|
|
(Contributed by Mario Carneiro, 1-Jan-2017.) $)
|
|
subne0d $p |- ( ph -> ( A - B ) =/= 0 ) $=
|
|
( cmin co cc0 wne cc wcel wceq wb subeq0 syl2anc necon3bid mpbird ) ABC
|
|
GHZIJBCJFASIBCABKLCKLSIMBCMNDEBCOPQR $.
|
|
$}
|
|
|
|
$( The difference of two complex numbers is zero iff they are equal.
|
|
Deduction form of ~ subeq0 . Generalization of ~ subeq0d .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
subeq0ad $p |- ( ph -> ( ( A - B ) = 0 <-> A = B ) ) $=
|
|
( cc wcel cmin co cc0 wceq wb subeq0 syl2anc ) ABFGCFGBCHIJKBCKLDEBCMN $.
|
|
|
|
${
|
|
subne0ad.3 $e |- ( ph -> ( A - B ) =/= 0 ) $.
|
|
$( If the difference of two complex numbers is nonzero, they are unequal.
|
|
Converse of ~ subne0d . Contrapositive of ~ subeq0bd . (Contributed
|
|
by David Moews, 28-Feb-2017.) $)
|
|
subne0ad $p |- ( ph -> A =/= B ) $=
|
|
( cmin co cc0 wne subeq0ad necon3bid mpbid ) ABCGHZIJBCJFANIBCABCDEKLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
neg11d.3 $e |- ( ph -> -u A = -u B ) $.
|
|
$( If the difference between two numbers is zero, they are equal.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
neg11d $p |- ( ph -> A = B ) $=
|
|
( cneg wceq neg11ad mpbid ) ABGCGHBCHFABCDEIJ $.
|
|
$}
|
|
|
|
$( Distribution of negative over addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
negdid $p |- ( ph -> -u ( A + B ) = ( -u A + -u B ) ) $=
|
|
( cc wcel caddc co cneg wceq negdi syl2anc ) ABFGCFGBCHIJBJCJHIKDEBCLM $.
|
|
|
|
$( Distribution of negative over addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
negdi2d $p |- ( ph -> -u ( A + B ) = ( -u A - B ) ) $=
|
|
( cc wcel caddc co cneg cmin wceq negdi2 syl2anc ) ABFGCFGBCHIJBJCKILDEBC
|
|
MN $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negsubdid $p |- ( ph -> -u ( A - B ) = ( -u A + B ) ) $=
|
|
( cc wcel cmin co cneg caddc wceq negsubdi syl2anc ) ABFGCFGBCHIJBJCKILDE
|
|
BCMN $.
|
|
|
|
$( Distribution of negative over subtraction. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
negsubdi2d $p |- ( ph -> -u ( A - B ) = ( B - A ) ) $=
|
|
( cc wcel cmin co cneg wceq negsubdi2 syl2anc ) ABFGCFGBCHIJCBHIKDEBCLM
|
|
$.
|
|
|
|
$( Relationship between subtraction and negative. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
neg2subd $p |- ( ph -> ( -u A - -u B ) = ( B - A ) ) $=
|
|
( cc wcel cneg cmin co wceq neg2sub syl2anc ) ABFGCFGBHCHIJCBIJKDEBCLM $.
|
|
|
|
subaddd.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Relationship between subtraction and addition. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
subaddd $p |- ( ph -> ( ( A - B ) = C <-> ( B + C ) = A ) ) $=
|
|
( cc wcel cmin co wceq caddc wb subadd syl3anc ) ABHICHIDHIBCJKDLCDMKBLNE
|
|
FGBCDOP $.
|
|
|
|
$( Relationship between subtraction and addition. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
subadd2d $p |- ( ph -> ( ( A - B ) = C <-> ( C + B ) = A ) ) $=
|
|
( cc wcel cmin co wceq caddc wb subadd2 syl3anc ) ABHICHIDHIBCJKDLDCMKBLN
|
|
EFGBCDOP $.
|
|
|
|
$( Associative-type law for subtraction and addition. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
addsubassd $p |- ( ph -> ( ( A + B ) - C ) = ( A + ( B - C ) ) ) $=
|
|
( cc wcel caddc co cmin wceq addsubass syl3anc ) ABHICHIDHIBCJKDLKBCDLKJK
|
|
MEFGBCDNO $.
|
|
|
|
$( Law for subtraction and addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addsubd $p |- ( ph -> ( ( A + B ) - C ) = ( ( A - C ) + B ) ) $=
|
|
( cc wcel caddc co cmin wceq addsub syl3anc ) ABHICHIDHIBCJKDLKBDLKCJKMEF
|
|
GBCDNO $.
|
|
|
|
$( Commutative/associative law for addition and subtraction. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
subadd23d $p |- ( ph -> ( ( A - B ) + C ) = ( A + ( C - B ) ) ) $=
|
|
( cc wcel cmin co caddc wceq subadd23 syl3anc ) ABHICHIDHIBCJKDLKBDCJKLKM
|
|
EFGBCDNO $.
|
|
|
|
$( Commutative/associative law for addition and subtraction. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
addsub12d $p |- ( ph -> ( A + ( B - C ) ) = ( B + ( A - C ) ) ) $=
|
|
( cc wcel cmin co caddc wceq addsub12 syl3anc ) ABHICHIDHIBCDJKLKCBDJKLKM
|
|
EFGBCDNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
npncand $p |- ( ph -> ( ( A - B ) + ( B - C ) ) = ( A - C ) ) $=
|
|
( cc wcel cmin co caddc wceq npncan syl3anc ) ABHICHIDHIBCJKCDJKLKBDJKMEF
|
|
GBCDNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nppcand $p |- ( ph -> ( ( ( A - B ) + C ) + B ) = ( A + C ) ) $=
|
|
( cc wcel cmin co caddc wceq nppcan syl3anc ) ABHICHIDHIBCJKDLKCLKBDLKMEF
|
|
GBCDNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nppcan2d $p |- ( ph -> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) $=
|
|
( cc wcel caddc co cmin wceq nppcan2 syl3anc ) ABHICHIDHIBCDJKLKDJKBCLKME
|
|
FGBCDNO $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nppcan3d $p |- ( ph -> ( ( A - B ) + ( C + B ) ) = ( A + C ) ) $=
|
|
( cc wcel cmin co caddc wceq nppcan3 syl3anc ) ABHICHIDHIBCJKDCLKLKBDLKME
|
|
FGBCDNO $.
|
|
|
|
$( Law for double subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subsubd $p |- ( ph -> ( A - ( B - C ) ) = ( ( A - B ) + C ) ) $=
|
|
( cc wcel cmin co caddc wceq subsub syl3anc ) ABHICHIDHIBCDJKJKBCJKDLKMEF
|
|
GBCDNO $.
|
|
|
|
$( Law for double subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subsub2d $p |- ( ph -> ( A - ( B - C ) ) = ( A + ( C - B ) ) ) $=
|
|
( cc wcel cmin co caddc wceq subsub2 syl3anc ) ABHICHIDHIBCDJKJKBDCJKLKME
|
|
FGBCDNO $.
|
|
|
|
$( Law for double subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subsub3d $p |- ( ph -> ( A - ( B - C ) ) = ( ( A + C ) - B ) ) $=
|
|
( cc wcel cmin co caddc wceq subsub3 syl3anc ) ABHICHIDHIBCDJKJKBDLKCJKME
|
|
FGBCDNO $.
|
|
|
|
$( Law for double subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subsub4d $p |- ( ph -> ( ( A - B ) - C ) = ( A - ( B + C ) ) ) $=
|
|
( cc wcel cmin co caddc wceq subsub4 syl3anc ) ABHICHIDHIBCJKDJKBCDLKJKME
|
|
FGBCDNO $.
|
|
|
|
$( Swap the second and third terms in a double subtraction. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
sub32d $p |- ( ph -> ( ( A - B ) - C ) = ( ( A - C ) - B ) ) $=
|
|
( cc wcel cmin co wceq sub32 syl3anc ) ABHICHIDHIBCJKDJKBDJKCJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnncand $p |- ( ph -> ( ( A - ( B - C ) ) - C ) = ( A - B ) ) $=
|
|
( cc wcel cmin co wceq nnncan syl3anc ) ABHICHIDHIBCDJKJKDJKBCJKLEFGBCDMN
|
|
$.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnncan1d $p |- ( ph -> ( ( A - B ) - ( A - C ) ) = ( C - B ) ) $=
|
|
( cc wcel cmin co wceq nnncan1 syl3anc ) ABHICHIDHIBCJKBDJKJKDCJKLEFGBCDM
|
|
N $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnncan2d $p |- ( ph -> ( ( A - C ) - ( B - C ) ) = ( A - B ) ) $=
|
|
( cc wcel cmin co wceq nnncan2 syl3anc ) ABHICHIDHIBDJKCDJKJKBCJKLEFGBCDM
|
|
N $.
|
|
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
npncan3d $p |- ( ph -> ( ( A - B ) + ( C - A ) ) = ( C - B ) ) $=
|
|
( cc wcel cmin co caddc wceq npncan3 syl3anc ) ABHICHIDHIBCJKDBJKLKDCJKME
|
|
FGBCDNO $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
pnpcand $p |- ( ph -> ( ( A + B ) - ( A + C ) ) = ( B - C ) ) $=
|
|
( cc wcel caddc co cmin wceq pnpcan syl3anc ) ABHICHIDHIBCJKBDJKLKCDLKMEF
|
|
GBCDNO $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
pnpcan2d $p |- ( ph -> ( ( A + C ) - ( B + C ) ) = ( A - B ) ) $=
|
|
( cc wcel caddc co cmin wceq pnpcan2 syl3anc ) ABHICHIDHIBDJKCDJKLKBCLKME
|
|
FGBCDNO $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
pnncand $p |- ( ph -> ( ( A + B ) - ( A - C ) ) = ( B + C ) ) $=
|
|
( cc wcel caddc co cmin wceq pnncan syl3anc ) ABHICHIDHIBCJKBDLKLKCDJKMEF
|
|
GBCDNO $.
|
|
|
|
$( Cancellation law for mixed addition and subtraction. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
ppncand $p |- ( ph -> ( ( A + B ) + ( C - B ) ) = ( A + C ) ) $=
|
|
( cc wcel caddc co cmin wceq ppncan syl3anc ) ABHICHIDHIBCJKDCLKJKBDJKMEF
|
|
GBCDNO $.
|
|
|
|
${
|
|
subcand.4 $e |- ( ph -> ( A - B ) = ( A - C ) ) $.
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subcand $p |- ( ph -> B = C ) $=
|
|
( cmin co wceq cc wcel wb subcan syl3anc mpbid ) ABCIJBDIJKZCDKZHABLMCL
|
|
MDLMRSNEFGBCDOPQ $.
|
|
$}
|
|
|
|
${
|
|
subcan2d.4 $e |- ( ph -> ( A - C ) = ( B - C ) ) $.
|
|
$( Cancellation law for subtraction. (Contributed by Mario Carneiro,
|
|
22-Sep-2016.) $)
|
|
subcan2d $p |- ( ph -> A = B ) $=
|
|
( cmin co wceq cc wcel wb subcan2 syl3anc mpbid ) ABDIJCDIJKZBCKZHABLMC
|
|
LMDLMRSNEFGBCDOPQ $.
|
|
$}
|
|
|
|
$( Cancellation law for subtraction. Deduction form of ~ subcan .
|
|
Generalization of ~ subcand . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
subcanad $p |- ( ph -> ( ( A - B ) = ( A - C ) <-> B = C ) ) $=
|
|
( cc wcel cmin co wceq wb subcan syl3anc ) ABHICHIDHIBCJKBDJKLCDLMEFGBCDN
|
|
O $.
|
|
|
|
${
|
|
subneintrd.4 $e |- ( ph -> B =/= C ) $.
|
|
$( Introducing subtraction on both sides of a statement of inequality.
|
|
Contrapositive of ~ subcand . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
subneintrd $p |- ( ph -> ( A - B ) =/= ( A - C ) ) $=
|
|
( cmin co wne subcanad necon3bid mpbird ) ABCIJZBDIJZKCDKHAOPCDABCDEFGL
|
|
MN $.
|
|
$}
|
|
|
|
$( Cancellation law for subtraction. Deduction form of ~ subcan2 .
|
|
Generalization of ~ subcan2d . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
subcan2ad $p |- ( ph -> ( ( A - C ) = ( B - C ) <-> A = B ) ) $=
|
|
( cc wcel cmin co wceq wb subcan2 syl3anc ) ABHICHIDHIBDJKCDJKLBCLMEFGBCD
|
|
NO $.
|
|
|
|
${
|
|
subneintr2d.4 $e |- ( ph -> A =/= B ) $.
|
|
$( Introducing subtraction on both sides of a statement of inequality.
|
|
Contrapositive of ~ subcan2d . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
subneintr2d $p |- ( ph -> ( A - C ) =/= ( B - C ) ) $=
|
|
( cmin co wne subcan2ad necon3bid mpbird ) ABDIJZCDIJZKBCKHAOPBCABCDEFG
|
|
LMN $.
|
|
$}
|
|
|
|
addsub4d.4 $e |- ( ph -> D e. CC ) $.
|
|
$( Rearrangement of 4 terms in a mixed addition and subtraction.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addsub4d $p |- ( ph ->
|
|
( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) ) $=
|
|
( cc wcel caddc co cmin wceq addsub4 syl22anc ) ABJKCJKDJKEJKBCLMDELMNMBD
|
|
NMCENMLMOFGHIBCDEPQ $.
|
|
|
|
$( Rearrangement of 4 terms in a mixed addition and subtraction.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
subadd4d $p |- ( ph ->
|
|
( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) ) $=
|
|
( cc wcel cmin co caddc wceq subadd4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBE
|
|
NMCDNMLMOFGHIBCDEPQ $.
|
|
|
|
$( Rearrangement of 4 terms in a subtraction. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
sub4d $p |- ( ph ->
|
|
( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) ) $=
|
|
( cc wcel cmin co wceq sub4 syl22anc ) ABJKCJKDJKEJKBCLMDELMLMBDLMCELMLMN
|
|
FGHIBCDEOP $.
|
|
|
|
$( Law for subtraction and addition. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
2addsubd $p |- ( ph ->
|
|
( ( ( A + B ) + C ) - D ) = ( ( ( A + C ) - D ) + B ) ) $=
|
|
( cc wcel caddc co cmin wceq 2addsub syl22anc ) ABJKCJKDJKEJKBCLMDLMENMBD
|
|
LMENMCLMOFGHIBCDEPQ $.
|
|
|
|
$( Relation between sums and differences. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
addsubeq4d $p |- ( ph ->
|
|
( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) ) $=
|
|
( cc wcel caddc co wceq cmin wb addsubeq4 syl22anc ) ABJKCJKDJKEJKBCLMDEL
|
|
MNDBOMCEOMNPFGHIBCDEQR $.
|
|
$}
|
|
|
|
${
|
|
mvlraddd.1 $e |- ( ph -> A e. CC ) $.
|
|
mvlraddd.2 $e |- ( ph -> B e. CC ) $.
|
|
mvlraddd.3 $e |- ( ph -> ( A + B ) = C ) $.
|
|
$( Move LHS right addition to RHS. (Contributed by David A. Wheeler,
|
|
15-Oct-2018.) $)
|
|
mvlraddd $p |- ( ph -> A = ( C - B ) ) $=
|
|
( caddc co cmin pncand oveq1d eqtr3d ) ABCHIZCJIBDCJIABCEFKANDCJGLM $.
|
|
$}
|
|
|
|
${
|
|
mvrraddd.1 $e |- ( ph -> B e. CC ) $.
|
|
mvrraddd.2 $e |- ( ph -> C e. CC ) $.
|
|
mvrraddd.3 $e |- ( ph -> A = ( B + C ) ) $.
|
|
$( Move RHS right addition to LHS. (Contributed by David A. Wheeler,
|
|
15-Oct-2018.) $)
|
|
mvrraddd $p |- ( ph -> ( A - C ) = B ) $=
|
|
( cmin co caddc oveq1d pncand eqtrd ) ABDHICDJIZDHICABNDHGKACDEFLM $.
|
|
$}
|
|
|
|
${
|
|
subaddeqd.a $e |- ( ph -> A e. CC ) $.
|
|
subaddeqd.b $e |- ( ph -> B e. CC ) $.
|
|
subaddeqd.c $e |- ( ph -> C e. CC ) $.
|
|
subaddeqd.d $e |- ( ph -> D e. CC ) $.
|
|
subaddeqd.1 $e |- ( ph -> ( A + B ) = ( C + D ) ) $.
|
|
$( Transfer two terms of a subtraction to an addition in an equality.
|
|
(Contributed by Thierry Arnoux, 2-Feb-2020.) $)
|
|
subaddeqd $p |- ( ph -> ( A - D ) = ( C - B ) ) $=
|
|
( caddc co cmin oveq1d addcomd eqtrd pnpcan2d pnpcand 3eqtr3d ) ABCKLZECK
|
|
LZMLZEDKLZUAMLZBEMLDCMLAUBDEKLZUAMLUDATUEUAMJNAUEUCUAMADEHIONPABECFIGQAED
|
|
CIHGRS $.
|
|
$}
|
|
|
|
${
|
|
addlsub.a $e |- ( ph -> A e. CC ) $.
|
|
addlsub.b $e |- ( ph -> B e. CC ) $.
|
|
addlsub.c $e |- ( ph -> C e. CC ) $.
|
|
$( Left-subtraction: Subtraction of the left summand from the result of an
|
|
addition. (Contributed by BJ, 6-Jun-2019.) $)
|
|
addlsub $p |- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) ) $=
|
|
( caddc co wceq cmin oveq1 pncand wa wi eqtr2 eqcomd a1i mpan2d syl5 eqtr
|
|
npcand impbid ) ABCHIZDJZBDCKIZJZUEUDCKIZUFJZAUGUDDCKLAUIUHBJZUGABCEFMUIU
|
|
JNZUGOAUKUFBUHUFBPQRSTUGUDUFCHIZJZAUEBUFCHLAUMULDJZUEADCGFUBUMUNNUEOAUDUL
|
|
DUARSTUC $.
|
|
|
|
$( Right-subtraction: Subtraction of the right summand from the result of
|
|
an addition. (Contributed by BJ, 6-Jun-2019.) $)
|
|
addrsub $p |- ( ph -> ( ( A + B ) = C <-> B = ( C - A ) ) ) $=
|
|
( caddc co wceq cmin addcomd eqeq1d addlsub bitrd ) ABCHIZDJCBHIZDJCDBKIJ
|
|
APQDABCEFLMACBDFEGNO $.
|
|
|
|
$( A subtraction law: Exchanging the subtrahend and the result of the
|
|
subtraction. (Contributed by BJ, 6-Jun-2019.) $)
|
|
subexsub $p |- ( ph -> ( A = ( C - B ) <-> B = ( C - A ) ) ) $=
|
|
( caddc co wceq cmin addlsub addrsub bitr3d ) ABCHIDJBDCKIJCDBKIJABCDEFGL
|
|
ABCDEFGMN $.
|
|
$}
|
|
|
|
$( If adding a number to a another number yields the other number, the added
|
|
number must be ` 0 ` . This shows that ` 0 ` is the unique (right)
|
|
identity of the complex numbers. (Contributed by AV, 17-Jan-2021.) $)
|
|
addid0 $p |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) ) $=
|
|
( cc wcel wa caddc co wceq cc0 simpl simpr subaddd eqcom subid adantr eqtrd
|
|
cmin wi ex syl5bi sylbird oveq2 addid1 sylan9eqr impbid ) ACDZBCDZEZABFGZAH
|
|
ZBIHZUHUJAAQGZBHZUKUHAABUFUGJZUNUFUGKLUFUMUKRUGUMBULHZUFUKULBMUFUOUKUFUOEBU
|
|
LIUFUOKUFULIHUOANOPSTOUAUFUKUJRUGUFUKUJUKUFUIAIFGABIAFUBAUCUDSOUE $.
|
|
|
|
$( Adding a nonzero number to a complex number does not yield the complex
|
|
number. (Contributed by AV, 17-Jan-2021.) $)
|
|
addn0nid $p |- ( ( X e. CC /\ Y e. CC /\ Y =/= 0 ) -> ( X + Y ) =/= X ) $=
|
|
( cc wcel cc0 wne caddc co wa wceq addid0 biimpd necon3d 3impia ) ACDZBCDZB
|
|
EFABGHZAFOPIZQABERQAJBEJABKLMN $.
|
|
|
|
${
|
|
pnpncand.1 $e |- ( ph -> A e. CC ) $.
|
|
pnpncand.2 $e |- ( ph -> B e. CC ) $.
|
|
pnpncand.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Addition/subtraction cancellation law. (Contributed by Scott Fenton,
|
|
14-Dec-2017.) $)
|
|
pnpncand $p |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A ) $=
|
|
( cmin co caddc subcld addcld subsub2d pncand eqtr3d ) ABCDHIZJIZPHIQDCHI
|
|
JIBAQCDABPEACDFGKZLFGMABPERNO $.
|
|
$}
|
|
|
|
$( Reverse the order of subtraction in an equality. (Contributed by Scott
|
|
Fenton, 8-Jul-2013.) $)
|
|
subeqrev $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) ) $=
|
|
( cc wcel wa cmin co cneg wb subcl neg11 syl2an negsubdi2 eqeqan12d bitr3d
|
|
wceq ) AEFBEFGZCEFDEFGZGABHIZJZCDHIZJZRZUAUCRZBAHIZDCHIZRSUAEFUCEFUEUFKTABL
|
|
CDLUAUCMNSTUBUGUDUHABOCDOPQ $.
|
|
|
|
$( Cancellation law for addition and subtraction with 1. (Contributed by
|
|
Alexander van der Vekens, 3-Oct-2018.) $)
|
|
pncan1 $p |- ( A e. CC -> ( ( A + 1 ) - 1 ) = A ) $=
|
|
( cc wcel c1 id ax-1cn a1i pncand ) ABCZADIEDBCIFGH $.
|
|
|
|
$( Cancellation law for subtraction and addition with 1. (Contributed by
|
|
Alexander van der Vekens, 5-Oct-2018.) $)
|
|
npcan1 $p |- ( A e. CC -> ( ( A - 1 ) + 1 ) = A ) $=
|
|
( cc wcel c1 id ax-1cn a1i npcand ) ABCZADIEDBCIFGH $.
|
|
|
|
${
|
|
subeq0bd.1 $e |- ( ph -> A e. CC ) $.
|
|
subeq0bd.2 $e |- ( ph -> A = B ) $.
|
|
$( If two complex numbers are equal, their difference is zero. Consequence
|
|
of ~ subeq0ad . Converse of ~ subeq0d . Contrapositive of ~ subne0ad .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
subeq0bd $p |- ( ph -> ( A - B ) = 0 ) $=
|
|
( cmin co cc0 wceq cc eqeltrrd subeq0ad mpbird ) ABCFGHIBCIEABCDABCJEDKLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
renegcld.1 $e |- ( ph -> A e. RR ) $.
|
|
$( Closure law for negative of reals. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
renegcld $p |- ( ph -> -u A e. RR ) $=
|
|
( cr wcel cneg renegcl syl ) ABDEBFDECBGH $.
|
|
|
|
resubcld.2 $e |- ( ph -> B e. RR ) $.
|
|
$( Closure law for subtraction of reals. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
resubcld $p |- ( ph -> ( A - B ) e. RR ) $=
|
|
( cr wcel cmin co resubcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
${
|
|
$d A n x y $.
|
|
negf1o.1 $e |- F = ( x e. A |-> -u x ) $.
|
|
$( Negation is an isomorphism of a subset of the real numbers to the
|
|
negated elements of the subset. (Contributed by AV, 9-Aug-2020.) $)
|
|
negf1o $p |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } ) $=
|
|
( vy cr cv cneg wcel wceq wa imp wi cc recn eleq1d adantl negeq elrab wss
|
|
crab wf1o ccnv cmpt ssel renegcl syl6 negneg syl biimpcd mpd sylanbrc weq
|
|
eqcomd simpr a1i syl5bi wb syl6com ad3antrrr negcon2 syl2anc exp31 impcom
|
|
sylbi f1ocnv2d simpld ) BGUAZBCHZIZBJZCGUBZDUCDUDFVMFHZIZUEKVIAFBVMAHZIZV
|
|
ODEVIVPBJZLZVQGJZVQIZBJZVQVMJVIVRVTVIVRVPGJZVTBGVPUFZVPUGUHMVSWCWBVIVRWCW
|
|
DMVRWCWBNVIWCVRWBWCVPWABWCVPOJZVPWAKVPPZWEWAVPVPUIUOUJQUKRULVLWBCVQGVJVQK
|
|
VKWABVJVQSQTUMVIVNVMJZVOBJZWGVNGJZWHLZVIWHVLWHCVNGCFUNVKVOBVJVNSQTZWJWHNV
|
|
IWIWHUPUQURMVRWGLVIVPVOKVNVQKUSZWGVRVIWLNZWGWJVRWMNWKWJVRVIWLWJVRLZVILWEV
|
|
NOJZWLWNVIWEVRVIWENWJVIVRWCWEWDWFUTRMWIWOWHVRVIVNPVAVPVNVBVCVDVFVEVEVGVH
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Multiplication
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( k times k minus 1 is a complex number if k is a complex number.
|
|
(Contributed by Alexander van der Vekens, 11-Mar-2018.) $)
|
|
kcnktkm1cn $p |- ( K e. CC -> ( K x. ( K - 1 ) ) e. CC ) $=
|
|
( cc wcel c1 cmin co id ax-1cn a1i subcld mulcld ) ABCZAADEFLGZLADMDBCLHIJK
|
|
$.
|
|
|
|
$( Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened
|
|
by Andrew Salmon, 19-Nov-2011.) $)
|
|
muladd $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A + B ) x. ( C + D ) ) =
|
|
( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) ) $=
|
|
( cc wcel wa caddc co cmul wceq addcl 3expa adantrl oveq12d mulcl ad2ant2lr
|
|
adddir syl2an ad2ant2l adddi adantrr ad2ant2r anandirs add32d mulcom oveq2d
|
|
3expb sylan ad2ant2rl addassd ancoms 3eqtr3d an4s 3eqtrd ) AEFZBEFZGZCEFZDE
|
|
FZGZGZABHIZCDHIJIZVCCJIZVCDJIZHIZACJIZBCJIZHIZADJIZBDJIZHIZHIZVHDBJIZHIZVKC
|
|
BJIZHIHIZURVCEFZVAVDVGKZABLVSUSUTVTVCCDUAUHUIVBVEVJVFVMHURUSVEVJKZUTUPUQUSW
|
|
AABCRMUBURUTVFVMKZUSUPUQUTWBABDRMNOVBVNVHVMHIZVIHIVPVKHIZVQHIVRVBVHVIVMUPUS
|
|
VHEFZUQUTACPZUCZUQUSVIEFUPUTBCPQURUTVMEFZUSUPUQUTWHUPUTGVKEFZVLEFZWHUQUTGZA
|
|
DPZBDPZVKVLLSUDNUEVBWCWDVIVQHVBVHVKHIZVLHIWNVOHIWCWDVBVLVOWNHUQUTVLVOKUPUSB
|
|
DUFTUGVBVHVKVLWGUPUTWIUQUSWLUJZUQUTWJUPUSWMTUKVBVHVKVOWGWOUQUTVOEFZUPUSUTUQ
|
|
WPDBPULZTUEUMUQUSVIVQKUPUTBCUFQOVBVPVKVQUPUSUQUTVPEFZUPUSGWEWPWRWKWFWQVHVOL
|
|
SUNWOUQUSVQEFZUPUTUSUQWSCBPULQUKUOUO $.
|
|
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of [Apostol]
|
|
p. 18. (Contributed by NM, 18-Nov-2004.) $)
|
|
subdi $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) ) $=
|
|
( cc wcel w3a cmul cmin wceq caddc simp1 simp3 3adant1 adddid pncan3 ancoms
|
|
co subcl oveq2d mulcl eqtr3d 3adant3 3adant2 wa sylan2 3impb subaddd mpbird
|
|
eqcomd ) ADEZBDEZCDEZFZABGQZACGQZHQZABCHQZGQZUMUPURIUOURJQZUNIUMACUQJQZGQUS
|
|
UNUMACUQUJUKULKUJUKULLUKULUQDEZUJBCRZMNUMUTBAGUKULUTBIZUJULUKVCCBOPMSUAUMUN
|
|
UOURUJUKUNDEULABTUBUJULUODEUKACTUCUJUKULURDEZUKULUDUJVAVDVBAUQTUEUFUGUHUI
|
|
$.
|
|
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of [Apostol]
|
|
p. 18. (Contributed by NM, 30-Dec-2005.) $)
|
|
subdir $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) ) $=
|
|
( cc wcel cmin co cmul wceq subdi 3coml wa subcl mulcom sylan 3impa 3adant2
|
|
w3a 3adant1 oveq12d 3eqtr4d ) ADEZBDEZCDEZRZCABFGZHGZCAHGZCBHGZFGZUFCHGZACH
|
|
GZBCHGZFGUDUBUCUGUJICABJKUBUCUDUKUGIZUBUCLUFDEUDUNABMUFCNOPUEULUHUMUIFUBUDU
|
|
LUHIUCACNQUCUDUMUIIUBBCNSTUA $.
|
|
|
|
$( Multiplication by ` 0 ` . Theorem I.6 of [Apostol] p. 18. (Contributed
|
|
by NM, 10-Aug-1999.) $)
|
|
mul02 $p |- ( A e. CC -> ( 0 x. A ) = 0 ) $=
|
|
( cc wcel cc0 cmul co cmin 0cn subidi oveq1i subdir mp3an12 wa mulcl subidd
|
|
wceq mpan eqtrd syl5eqr ) ABCZDAEFZDDGFZAEFZDUBDAEDHIJTUCUAUAGFZDDBCZUETUCU
|
|
DPHHDDAKLUETUDDPHUETMUADANOQRS $.
|
|
|
|
$( Zero times a real is zero. Although we prove it as a corollary of
|
|
~ mul02 , the name is for consistency with the Metamath Proof Explorer
|
|
which proves it before ~ mul02 . (Contributed by Scott Fenton,
|
|
3-Jan-2013.) $)
|
|
mul02lem2 $p |- ( A e. RR -> ( 0 x. A ) = 0 ) $=
|
|
( cr wcel cc cc0 cmul co wceq recn mul02 syl ) ABCADCEAFGEHAIAJK $.
|
|
|
|
$( Multiplication by ` 0 ` . Theorem I.6 of [Apostol] p. 18. (Contributed
|
|
by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.) $)
|
|
mul01 $p |- ( A e. CC -> ( A x. 0 ) = 0 ) $=
|
|
( cc wcel cc0 cmul co wceq 0cn mulcom mpan2 mul02 eqtrd ) ABCZADEFZDAEFZDMD
|
|
BCNOGHADIJAKL $.
|
|
|
|
${
|
|
mul01i.1 $e |- A e. CC $.
|
|
$( Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by
|
|
NM, 23-Nov-1994.) $)
|
|
mul02i $p |- ( 0 x. A ) = 0 $=
|
|
( cc wcel cc0 cmul co wceq mul02 ax-mp ) ACDEAFGEHBAIJ $.
|
|
|
|
$( Multiplication by ` 0 ` . Theorem I.6 of [Apostol] p. 18. (Contributed
|
|
by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) $)
|
|
mul01i $p |- ( A x. 0 ) = 0 $=
|
|
( cc wcel cc0 cmul co wceq mul01 ax-mp ) ACDAEFGEHBAIJ $.
|
|
$}
|
|
|
|
${
|
|
mul01d.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
mul02d $p |- ( ph -> ( 0 x. A ) = 0 ) $=
|
|
( cc wcel cc0 cmul co wceq mul02 syl ) ABDEFBGHFICBJK $.
|
|
|
|
$( Multiplication by ` 0 ` . Theorem I.6 of [Apostol] p. 18. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
mul01d $p |- ( ph -> ( A x. 0 ) = 0 ) $=
|
|
( cc wcel cc0 cmul co wceq mul01 syl ) ABDEBFGHFICBJK $.
|
|
$}
|
|
|
|
$( The imaginary unit ` _i ` is not zero. (Contributed by NM,
|
|
6-May-1999.) $)
|
|
ine0 $p |- _i =/= 0 $=
|
|
( ci cc0 wceq c1 0lt1 gtneii neii caddc co cmul oveq2 ax-icn mul01i syl6req
|
|
0re oveq1d ax-1cn addid2i ax-i2m1 3eqtr3g mto neir ) ABABCZDBCDBBDOEFGUCBDH
|
|
IAAJIZDHIDBUCBUDDHUCUDABJIBABAJKALMNPDQRSTUAUB $.
|
|
|
|
$( Product with negative is negative of product. Theorem I.12 of [Apostol]
|
|
p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
mulneg1 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( -u A x. B ) = -u ( A x. B ) ) $=
|
|
( cc wcel wa cmin co cmul cneg wceq subdir mp3an1 simpr mul02d oveq1d eqtrd
|
|
cc0 0cn df-neg oveq1i 3eqtr4g ) ACDZBCDZEZQAFGZBHGZQABHGZFGZAIZBHGUGIUDUFQB
|
|
HGZUGFGZUHQCDUBUCUFUKJRQABKLUDUJQUGFUDBUBUCMNOPUIUEBHASTUGSUA $.
|
|
|
|
$( The product with a negative is the negative of the product. (Contributed
|
|
by NM, 30-Jul-2004.) $)
|
|
mulneg2 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( A x. -u B ) = -u ( A x. B ) ) $=
|
|
( cc wcel wa cneg cmul co mulneg1 ancoms negcl mulcom sylan2 negeqd 3eqtr4d
|
|
wceq ) ACDZBCDZEZBFZAGHZBAGHZFZATGHZABGHZFRQUAUCPBAIJRQTCDUDUAPBKATLMSUEUBA
|
|
BLNO $.
|
|
|
|
$( Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.) $)
|
|
mulneg12 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( -u A x. B ) = ( A x. -u B ) ) $=
|
|
( cc wcel wa cneg cmul co mulneg1 mulneg2 eqtr4d ) ACDBCDEAFBGHABGHFABFGHAB
|
|
IABJK $.
|
|
|
|
$( Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed
|
|
by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
mul2neg $p |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) ) $=
|
|
( cc wcel wa cneg cmul co negcl mulneg12 sylan2 negneg adantl oveq2d eqtrd
|
|
wceq ) ACDZBCDZEZAFBFZGHZATFZGHZABGHRQTCDUAUCPBIATJKSUBBAGRUBBPQBLMNO $.
|
|
|
|
$( Convert a subtraction to addition using multiplication by a negative.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
submul2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A - ( B x. C ) ) = ( A + ( B x. -u C ) ) ) $=
|
|
( cc wcel cmul co cmin cneg caddc wceq wa adantl oveq2d mulcl negsub sylan2
|
|
mulneg2 eqtr2d 3impb ) ADEZBDEZCDEZABCFGZHGZABCIFGZJGZKUAUBUCLZLZUGAUDIZJGZ
|
|
UEUIUFUJAJUHUFUJKUABCRMNUHUAUDDEUKUEKBCOAUDPQST $.
|
|
|
|
$( Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) $)
|
|
mulm1 $p |- ( A e. CC -> ( -u 1 x. A ) = -u A ) $=
|
|
( cc wcel c1 cneg cmul co wceq ax-1cn mulneg1 mpan mulid2 negeqd eqtrd ) AB
|
|
CZDEAFGZDAFGZEZAEDBCOPRHIDAJKOQAALMN $.
|
|
|
|
$( Product of two differences. (Contributed by NM, 14-Jan-2006.) $)
|
|
mulsub $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A - B ) x. ( C - D ) ) =
|
|
( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) ) $=
|
|
( cc wcel wa cneg caddc cmul cmin negsub oveqan12d wceq negcl mulneg2 mulcl
|
|
co ancoms syl2an muladd sylanr2 sylanl2 mul2neg oveq2d negdi eqtr4d ancom2s
|
|
ad2ant2l an42s oveq12d addcl an4s negsubd 3eqtrd eqtr3d ) AEFZBEFZGZCEFZDEF
|
|
ZGZGZABHZIRZCDHZIRZJRZABKRZCDKRZJRACJRZDBJRZIRZADJRZCBJRZIRZKRZUSVBVEVIVGVJ
|
|
JABLCDLMVCVHVKVFVDJRZIRZAVFJRZCVDJRZIRZIRZVMVPHZIRVQURUQVDEFZVBVHWCNZBOVAUQ
|
|
WEGUTVFEFWFDOAVDCVFUAUBUCVCVSVMWBWDIURVAVSVMNUQUTURVAGZVRVLVKIVAURVRVLNDBUD
|
|
SUEUIUQVAURUTWBWDNZUQVAGZUTURWHWIUTURGZGWBVNHZVOHZIRZWDWIWJVTWKWAWLIADPCBPM
|
|
WIVNEFZVOEFZWDWMNWJADQZCBQZVNVOUFTUGUHUJUKVCVMVPUQUTURVAVMEFZUQUTGVKEFVLEFZ
|
|
WRWGACQVAURWSDBQSVKVLULTUMUQVAURUTVPEFZWIWNWOWTURUTGWPUTURWOWQSVNVOULTUJUNU
|
|
OUP $.
|
|
|
|
$( Swap the order of subtraction in a multiplication. (Contributed by Scott
|
|
Fenton, 24-Jun-2013.) $)
|
|
mulsub2 $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) ->
|
|
( ( A - B ) x. ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) ) $=
|
|
( cc wcel wa cmin co cneg cmul wceq subcl syl2an negsubdi2 oveqan12d eqtr3d
|
|
mul2neg ) AEFBEFGZCEFDEFGZGABHIZJZCDHIZJZKIZUAUCKIZBAHIZDCHIZKISUAEFUCEFUEU
|
|
FLTABMCDMUAUCRNSTUBUGUDUHKABOCDOPQ $.
|
|
|
|
${
|
|
mulm1.1 $e |- A e. CC $.
|
|
$( Product with minus one is negative. (Contributed by NM,
|
|
31-Jul-1999.) $)
|
|
mulm1i $p |- ( -u 1 x. A ) = -u A $=
|
|
( cc wcel c1 cneg cmul co wceq mulm1 ax-mp ) ACDEFAGHAFIBAJK $.
|
|
|
|
mulneg.2 $e |- B e. CC $.
|
|
$( Product with negative is negative of product. Theorem I.12 of [Apostol]
|
|
p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulneg1i $p |- ( -u A x. B ) = -u ( A x. B ) $=
|
|
( cc wcel cneg cmul co wceq mulneg1 mp2an ) AEFBEFAGBHIABHIGJCDABKL $.
|
|
|
|
$( Product with negative is negative of product. (Contributed by NM,
|
|
31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
mulneg2i $p |- ( A x. -u B ) = -u ( A x. B ) $=
|
|
( cc wcel cneg cmul co wceq mulneg2 mp2an ) AEFBEFABGHIABHIGJCDABKL $.
|
|
|
|
$( Product of two negatives. Theorem I.12 of [Apostol] p. 18.
|
|
(Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mul2negi $p |- ( -u A x. -u B ) = ( A x. B ) $=
|
|
( cc wcel cneg cmul co wceq mul2neg mp2an ) AEFBEFAGBGHIABHIJCDABKL $.
|
|
|
|
subdi.3 $e |- C e. CC $.
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of
|
|
[Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) $)
|
|
subdii $p |- ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) $=
|
|
( cc wcel cmin co cmul wceq subdi mp3an ) AGHBGHCGHABCIJKJABKJACKJIJLDEFA
|
|
BCMN $.
|
|
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of
|
|
[Apostol] p. 18. (Contributed by NM, 8-May-1999.) $)
|
|
subdiri $p |- ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) $=
|
|
( cc wcel cmin co cmul wceq subdir mp3an ) AGHBGHCGHABIJCKJACKJBCKJIJLDEF
|
|
ABCMN $.
|
|
|
|
muladdi.4 $e |- D e. CC $.
|
|
$( Product of two sums. (Contributed by NM, 17-May-1999.) $)
|
|
muladdi $p |- ( ( A + B ) x. ( C + D ) ) =
|
|
( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) $=
|
|
( cc wcel caddc co cmul wceq muladd mp4an ) AIJBIJCIJDIJABKLCDKLMLACMLDBM
|
|
LKLADMLCBMLKLKLNEFGHABCDOP $.
|
|
$}
|
|
|
|
${
|
|
mulm1d.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Product with minus one is negative. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulm1d $p |- ( ph -> ( -u 1 x. A ) = -u A ) $=
|
|
( cc wcel c1 cneg cmul co wceq mulm1 syl ) ABDEFGBHIBGJCBKL $.
|
|
|
|
mulnegd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Product with negative is negative of product. Theorem I.12 of [Apostol]
|
|
p. 18. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
mulneg1d $p |- ( ph -> ( -u A x. B ) = -u ( A x. B ) ) $=
|
|
( cc wcel cneg cmul co wceq mulneg1 syl2anc ) ABFGCFGBHCIJBCIJHKDEBCLM $.
|
|
|
|
$( Product with negative is negative of product. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
mulneg2d $p |- ( ph -> ( A x. -u B ) = -u ( A x. B ) ) $=
|
|
( cc wcel cneg cmul co wceq mulneg2 syl2anc ) ABFGCFGBCHIJBCIJHKDEBCLM $.
|
|
|
|
$( Product of two negatives. Theorem I.12 of [Apostol] p. 18.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
mul2negd $p |- ( ph -> ( -u A x. -u B ) = ( A x. B ) ) $=
|
|
( cc wcel cneg cmul co wceq mul2neg syl2anc ) ABFGCFGBHCHIJBCIJKDEBCLM $.
|
|
|
|
subdid.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of
|
|
[Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
subdid $p |- ( ph -> ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) ) $=
|
|
( cc wcel cmin co cmul wceq subdi syl3anc ) ABHICHIDHIBCDJKLKBCLKBDLKJKME
|
|
FGBCDNO $.
|
|
|
|
$( Distribution of multiplication over subtraction. Theorem I.5 of
|
|
[Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
subdird $p |- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) ) $=
|
|
( cc wcel cmin co cmul wceq subdir syl3anc ) ABHICHIDHIBCJKDLKBDLKCDLKJKM
|
|
EFGBCDNO $.
|
|
|
|
muladdd.4 $e |- ( ph -> D e. CC ) $.
|
|
$( Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
muladdd $p |- ( ph -> ( ( A + B ) x. ( C + D ) ) =
|
|
( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) ) $=
|
|
( cc wcel caddc co cmul wceq muladd syl22anc ) ABJKCJKDJKEJKBCLMDELMNMBDN
|
|
MECNMLMBENMDCNMLMLMOFGHIBCDEPQ $.
|
|
|
|
$( Product of two differences. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
mulsubd $p |- ( ph -> ( ( A - B ) x. ( C - D ) ) =
|
|
( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) ) $=
|
|
( cc wcel cmin co cmul caddc wceq mulsub syl22anc ) ABJKCJKDJKEJKBCLMDELM
|
|
NMBDNMECNMOMBENMDCNMOMLMPFGHIBCDEQR $.
|
|
$}
|
|
|
|
${
|
|
mulsubfacd.1 $e |- ( ph -> A e. CC ) $.
|
|
mulsubfacd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Multiplication followed by the subtraction of a factor. (Contributed by
|
|
Alexander van der Vekens, 28-Aug-2018.) $)
|
|
mulsubfacd $p |- ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) ) $=
|
|
( c1 cmin co cmul cc wcel ax-1cn a1i subdird mulid2d oveq2d eqtr2d ) ABFG
|
|
HCIHBCIHZFCIHZGHRCGHABFCDFJKALMENASCRGACEOPQ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordering on reals (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Addition to both sides of 'less than'. (Contributed by NM,
|
|
12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
ltadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B <-> ( C + A ) < ( C + B ) ) ) $=
|
|
( vx cr wcel clt wbr caddc co axltadd wceq wi wa readdcld addassd breq12d
|
|
cc0 recnd cc w3a cv wrex ax-rnegex 3ad2ant3 simpl3 simpl1 syl3anc sylibrd
|
|
simpl2 simprl simprr wb addcom eqeq1d syl2anc mpbid oveq1d addid2d sylibd
|
|
eqtrd rexlimddv impbid ) AEFZBEFZCEFZUAZABGHZCAIJZCBIJZGHZABCKVGCDUBZIJZR
|
|
LZVKVHMDEVFVDVNDEUCVEDCUDUEVGVLEFZVNNZNZVKVLCIJZAIJZVRBIJZGHZVHVQVKVLVIIJ
|
|
ZVLVJIJZGHZWAVQVIEFVJEFVOVKWDMVQCAVDVEVFVPUFZVDVEVFVPUGZOVQCBWEVDVEVFVPUJ
|
|
ZOVGVOVNUKZVIVJVLKUHVQVSWBVTWCGVQVLCAVQVLWHSZVQCWESZVQAWFSZPVQVLCBWIWJVQB
|
|
WGSZPQUIVQVSAVTBGVQVSRAIJAVQVRRAIVQVNVRRLZVGVOVNULVQCTFZVLTFZVNWMUMWJWIWN
|
|
WONVMVRRCVLUNUOUPUQZURVQAWKUSVAVQVTRBIJBVQVRRBIWPURVQBWLUSVAQUTVBVC $.
|
|
$}
|
|
|
|
${
|
|
ltadd2i.1 $e |- A e. RR $.
|
|
ltadd2i.2 $e |- B e. RR $.
|
|
ltadd2i.3 $e |- C e. RR $.
|
|
$( Addition to both sides of 'less than'. (Contributed by NM,
|
|
21-Jan-1997.) $)
|
|
ltadd2i $p |- ( A < B <-> ( C + A ) < ( C + B ) ) $=
|
|
( cr wcel clt wbr caddc co wb ltadd2 mp3an ) AGHBGHCGHABIJCAKLCBKLIJMDEFA
|
|
BCNO $.
|
|
$}
|
|
|
|
${
|
|
ltadd2d.1 $e |- ( ph -> A e. RR ) $.
|
|
ltadd2d.2 $e |- ( ph -> B e. RR ) $.
|
|
ltadd2d.3 $e |- ( ph -> C e. RR ) $.
|
|
$( Addition to both sides of 'less than'. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltadd2d $p |- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) ) $=
|
|
( cr wcel clt wbr caddc co wb ltadd2 syl3anc ) ABHICHIDHIBCJKDBLMDCLMJKNE
|
|
FGBCDOP $.
|
|
|
|
${
|
|
ltletrd.4 $e |- ( ph -> A < B ) $.
|
|
$( Addition to both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 30-May-2016.) $)
|
|
ltadd2dd $p |- ( ph -> ( C + A ) < ( C + B ) ) $=
|
|
( clt wbr caddc co ltadd2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.
|
|
|
|
ltletrd.5 $e |- ( ph -> B <_ C ) $.
|
|
$( Transitive law deduction for 'less than', 'less than or equal to'.
|
|
(Contributed by NM, 9-Jan-2006.) $)
|
|
ltletrd $p |- ( ph -> A < C ) $=
|
|
( clt wbr cle cr wcel wa wi ltletr syl3anc mp2and ) ABCJKZCDLKZBDJKZHIA
|
|
BMNCMNDMNTUAOUBPEFGBCDQRS $.
|
|
$}
|
|
$}
|
|
|
|
$( Adding a negative number to another number decreases it. (Contributed by
|
|
Glauco Siliprandi, 11-Dec-2019.) $)
|
|
ltaddneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an2 wceq recn addid1d adantl
|
|
0re breq2d bitrd ) ACDZBCDZEZAFGHZBAIJZBFIJZGHZUEBGHUAFCDUBUDUGKRAFBLMUCUFB
|
|
UEGUBUFBNUAUBBBOPQST $.
|
|
|
|
$( Adding a negative number to another number decreases it. (Contributed by
|
|
AV, 19-Mar-2021.) $)
|
|
ltaddnegr $p |- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( A + B ) < B ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co ltaddneg cc wceq recn addcom breq1d bitrd
|
|
syl2anr ) ACDZBCDZEZAFGHBAIJZBGHABIJZBGHABKUAUBUCBGTBLDALDUBUCMSBNANBAORPQ
|
|
$.
|
|
|
|
${
|
|
lelttrdi.r $e |- ( ph -> ( A e. RR /\ B e. RR /\ C e. RR ) ) $.
|
|
lelttrdi.l $e |- ( ph -> B <_ C ) $.
|
|
$( If a number is less than another number, and the other number is less
|
|
than or equal to a third number, the first number is less than the third
|
|
number. (Contributed by Alexander van der Vekens, 24-Mar-2018.) $)
|
|
lelttrdi $p |- ( ph -> ( A < B -> A < C ) ) $=
|
|
( clt wbr wa cr wcel simp1d adantr simp2d simp3d simpr cle ltletrd ex ) A
|
|
BCGHZBDGHATIBCDABJKZTAUACJKZDJKZELMAUBTAUAUBUCENMAUCTAUAUBUCEOMATPACDQHTF
|
|
MRS $.
|
|
$}
|
|
|
|
$( Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof
|
|
shortened by Mario Carneiro, 27-May-2016.) $)
|
|
gt0ne0 $p |- ( ( A e. RR /\ 0 < A ) -> A =/= 0 ) $=
|
|
( cr wcel cc0 clt wbr wne 0red ltne sylan ) ABCZDBCDAEFADGKHDAIJ $.
|
|
|
|
$( A number which is less than zero is not zero. (Contributed by Stefan
|
|
O'Rear, 13-Sep-2014.) $)
|
|
lt0ne0 $p |- ( ( A e. RR /\ A < 0 ) -> A =/= 0 ) $=
|
|
( cr wcel cc0 clt wbr wa ltne necomd ) ABCADEFGDAADHI $.
|
|
|
|
$( Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of
|
|
[HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B <-> ( A + C ) < ( B + C ) ) ) $=
|
|
( cr wcel w3a clt wbr caddc co ltadd2 simp3 recnd simp1 addcomd simp2 bitrd
|
|
breq12d ) ADEZBDEZCDEZFZABGHCAIJZCBIJZGHACIJZBCIJZGHABCKUBUCUEUDUFGUBCAUBCS
|
|
TUALMZUBASTUANMOUBCBUGUBBSTUAPMORQ $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. Part of definition
|
|
11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.)
|
|
(Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
leadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) $=
|
|
( cr w3a clt wbr wn caddc co cle wb ltadd1 3com12 notbid simp1 simp2 lenltd
|
|
wcel readdcld simp3 3bitr4d ) ADSZBDSZCDSZEZBAFGZHBCIJZACIJZFGZHABKGUIUHKGU
|
|
FUGUJUDUCUEUGUJLBACMNOUFABUCUDUEPZUCUDUEQZRUFUIUHUFACUKUCUDUEUAZTUFBCULUMTR
|
|
UB $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by NM,
|
|
26-Oct-1999.) $)
|
|
leadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ B <-> ( C + A ) <_ ( C + B ) ) ) $=
|
|
( cr wcel w3a cle wbr caddc co leadd1 simp1 recnd simp3 addcomd simp2 bitrd
|
|
breq12d ) ADEZBDEZCDEZFZABGHACIJZBCIJZGHCAIJZCBIJZGHABCKUBUCUEUDUFGUBACUBAS
|
|
TUALMUBCSTUANMZOUBBCUBBSTUAPMUGORQ $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
ltsubadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) < C <-> A < ( C + B ) ) ) $=
|
|
( cr wcel w3a co clt wbr caddc wb simp1 simp2 resubcld simp3 ltadd1 syl3anc
|
|
cmin recnd npcand breq1d bitrd ) ADEZBDEZCDEZFZABRGZCHIZUGBJGZCBJGZHIZAUJHI
|
|
UFUGDEUEUDUHUKKUFABUCUDUELZUCUDUEMZNUCUDUEOUMUGCBPQUFUIAUJHUFABUFAULSUFBUMS
|
|
TUAUB $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 21-Jan-1997.) $)
|
|
ltsubadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) < C <-> A < ( B + C ) ) ) $=
|
|
( cr wcel w3a cmin clt wbr caddc ltsubadd simp2 recnd addcomd breq2d bitr4d
|
|
co simp3 ) ADEZBDEZCDEZFZABGQCHIACBJQZHIABCJQZHIABCKUBUDUCAHUBBCUBBSTUALMUB
|
|
CSTUARMNOP $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
lesubadd $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) <_ C <-> A <_ ( C + B ) ) ) $=
|
|
( cr wcel w3a co cle wbr caddc wb simp1 simp2 resubcld simp3 leadd1 syl3anc
|
|
cmin recnd npcand breq1d bitrd ) ADEZBDEZCDEZFZABRGZCHIZUGBJGZCBJGZHIZAUJHI
|
|
UFUGDEUEUDUHUKKUFABUCUDUELZUCUDUEMZNUCUDUEOUMUGCBPQUFUIAUJHUFABUFAULSUFBUMS
|
|
TUAUB $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by NM, 10-Aug-1999.) $)
|
|
lesubadd2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) <_ C <-> A <_ ( B + C ) ) ) $=
|
|
( cr wcel w3a cmin cle wbr caddc lesubadd simp2 recnd addcomd breq2d bitr4d
|
|
co simp3 ) ADEZBDEZCDEZFZABGQCHIACBJQZHIABCJQZHIABCKUBUDUCAHUBBCUBBSTUALMUB
|
|
CSTUARMNOP $.
|
|
|
|
$( 'Less than' relationship between addition and subtraction. (Contributed
|
|
by NM, 17-Nov-2004.) $)
|
|
ltaddsub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A + B ) < C <-> A < ( C - B ) ) ) $=
|
|
( cr wcel w3a co clt wbr caddc wb simp1 simp3 simp2 resubcld ltadd1 syl3anc
|
|
cmin recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACBRGZHIZABJGZUGBJGZHIZUICH
|
|
IUFUCUGDEUDUHUKKUCUDUELUFCBUCUDUEMZUCUDUENZOUMAUGBPQUFUJCUIHUFCBUFCULSUFBUM
|
|
STUAUB $.
|
|
|
|
$( 'Less than' relationship between addition and subtraction. (Contributed
|
|
by NM, 17-Nov-2004.) $)
|
|
ltaddsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A + B ) < C <-> B < ( C - A ) ) ) $=
|
|
( cr wcel w3a caddc co clt wbr cmin wceq cc addcom syl2an 3adant3 breq1d wb
|
|
recn ltaddsub 3com12 bitrd ) ADEZBDEZCDEZFZABGHZCIJBAGHZCIJZBCAKHIJZUFUGUHC
|
|
IUCUDUGUHLZUEUCAMEBMEUKUDASBSABNOPQUDUCUEUIUJRBACTUAUB $.
|
|
|
|
$( 'Less than or equal to' relationship between addition and subtraction.
|
|
(Contributed by NM, 6-Apr-2005.) $)
|
|
leaddsub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A + B ) <_ C <-> A <_ ( C - B ) ) ) $=
|
|
( cr wcel w3a co cle wbr caddc wb simp1 simp3 simp2 resubcld leadd1 syl3anc
|
|
cmin recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACBRGZHIZABJGZUGBJGZHIZUICH
|
|
IUFUCUGDEUDUHUKKUCUDUELUFCBUCUDUEMZUCUDUENZOUMAUGBPQUFUJCUIHUFCBUFCULSUFBUM
|
|
STUAUB $.
|
|
|
|
$( 'Less than or equal to' relationship between and addition and subtraction.
|
|
(Contributed by NM, 6-Apr-2005.) $)
|
|
leaddsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A + B ) <_ C <-> B <_ ( C - A ) ) ) $=
|
|
( cr wcel w3a caddc co cle wbr cmin wceq cc addcom syl2an 3adant3 breq1d wb
|
|
recn leaddsub 3com12 bitrd ) ADEZBDEZCDEZFZABGHZCIJBAGHZCIJZBCAKHIJZUFUGUHC
|
|
IUCUDUGUHLZUEUCAMEBMEUKUDASBSABNOPQUDUCUEUIUJRBACTUAUB $.
|
|
|
|
$( Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) $)
|
|
suble $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) $=
|
|
( cr wcel w3a cmin co cle wbr caddc lesubadd wb lesubadd2 3com23 bitr4d ) A
|
|
DEZBDEZCDEZFABGHCIJACBKHIJZACGHBIJZABCLQSRUATMACBNOP $.
|
|
|
|
$( Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
|
|
(Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
lesub $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ ( B - C ) <-> C <_ ( B - A ) ) ) $=
|
|
( cr wcel cmin co cle wbr wb w3a caddc leaddsub leaddsub2 bitr3d 3com23 ) A
|
|
DEZCDEZBDEZABCFGHIZCBAFGHIZJQRSKACLGBHITUAACBMACBNOP $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 4-Oct-1999.) $)
|
|
ltsub23 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A - B ) < C <-> ( A - C ) < B ) ) $=
|
|
( cr wcel w3a cmin co clt wbr caddc ltsubadd wb ltsubadd2 3com23 bitr4d ) A
|
|
DEZBDEZCDEZFABGHCIJACBKHIJZACGHBIJZABCLQSRUATMACBNOP $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 17-Nov-2004.) $)
|
|
ltsub13 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < ( B - C ) <-> C < ( B - A ) ) ) $=
|
|
( cr wcel cmin co clt wbr wb w3a caddc ltaddsub ltaddsub2 bitr3d 3com23 ) A
|
|
DEZCDEZBDEZABCFGHIZCBAFGHIZJQRSKACLGBHITUAACBMACBNOP $.
|
|
|
|
$( Adding both sides of two 'less than or equal to' relations. (Contributed
|
|
by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
le2add $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) ) $=
|
|
( cr wcel wa cle wbr caddc co wb simpll simprl simplr leadd1 syl3anc simprr
|
|
leadd2 readdcld anbi12d wi letr sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDHIZGA
|
|
BJKZCBJKZHIZUOCDJKZHIZGZUNUQHIZUKULUPUMURUKUEUHUFULUPLUEUFUJMZUGUHUINZUEUFU
|
|
JOZACBPQUKUFUIUHUMURLVCUGUHUIRZVBBDCSQUAUKUNEFUOEFUQEFUSUTUBUKABVAVCTUKCBVB
|
|
VCTUKCDVBVDTUNUOUQUCQUD $.
|
|
|
|
$( Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol]
|
|
p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
lt2add $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) ) $=
|
|
( cr wcel wa clt wbr caddc co wb simpll simprl simplr ltadd1 syl3anc simprr
|
|
ltadd2d readdcld anbi12d wi lttr sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDHIZG
|
|
ABJKZCBJKZHIZUOCDJKZHIZGZUNUQHIZUKULUPUMURUKUEUHUFULUPLUEUFUJMZUGUHUINZUEUF
|
|
UJOZACBPQUKBDCVCUGUHUIRZVBSUAUKUNEFUOEFUQEFUSUTUBUKABVAVCTUKCBVBVCTUKCDVBVD
|
|
TUNUOUQUCQUD $.
|
|
|
|
$( Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.) $)
|
|
ltleadd $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A < C /\ B <_ D ) -> ( A + B ) < ( C + D ) ) ) $=
|
|
( cr wcel wa clt wbr caddc co wb ltadd1 3com23 3expa adantrr leadd2 readdcl
|
|
cle 3expb adantll anbi12d wi adantr ancoms ad2ant2lr adantl ltletr syl3anc
|
|
sylbid ) AEFZBEFZGZCEFZDEFZGZGZACHIZBDSIZGABJKZCBJKZHIZVACDJKZSIZGZUTVCHIZU
|
|
QURVBUSVDUMUNURVBLZUOUKULUNVGUKUNULVGACBMNOPULUPUSVDLZUKULUNUOVHULUOUNVHBDC
|
|
QNTUAUBUQUTEFZVAEFZVCEFZVEVFUCUMVIUPABRUDULUNVJUKUOUNULVJCBRUEUFUPVKUMCDRUG
|
|
UTVAVCUHUIUJ $.
|
|
|
|
$( Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.) $)
|
|
leltadd $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A <_ C /\ B < D ) -> ( A + B ) < ( C + D ) ) ) $=
|
|
( cr wcel wa cle wbr caddc co wi ltleadd ancomsd cc wceq recn addcom syl2an
|
|
clt ancom2s ancom1s breqan12d sylibrd ) AEFZBEFZGZCEFZDEFZGZGACHIZBDTIZGZBA
|
|
JKZDCJKZTIZABJKZCDJKZTIUFUEUJUMUPLZUFUEGZUIUHUSUTUIUHGGULUKUPBADCMNUAUBUGUJ
|
|
UQUNURUOTUEAOFBOFUQUNPUFAQBQABRSUHCOFDOFURUOPUICQDQCDRSUCUD $.
|
|
|
|
$( The sum of 2 positive numbers is positive. (Contributed by NM,
|
|
1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addgt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) ->
|
|
0 < ( A + B ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co 00id wi 0re lt2add mpanl12 imp syl5eqbrr
|
|
) ACDBCDEZFAGHFBGHEZEFFFIJZABIJZGKRSTUAGHZFCDZUCRSUBLMMFFABNOPQ $.
|
|
|
|
$( The sum of nonnegative and positive numbers is positive. (Contributed by
|
|
NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addgegt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < B ) ) ->
|
|
0 < ( A + B ) ) $=
|
|
( cr wcel wa cc0 cle wbr clt caddc co 00id wi 0re leltadd mpanl12 syl5eqbrr
|
|
imp ) ACDBCDEZFAGHFBIHEZEFFFJKZABJKZILSTUAUBIHZFCDZUDSTUCMNNFFABOPRQ $.
|
|
|
|
$( The sum of nonnegative and positive numbers is positive. (Contributed by
|
|
NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addgtge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) ->
|
|
0 < ( A + B ) ) $=
|
|
( cr wcel wa cc0 clt wbr cle caddc co 00id wi 0re ltleadd mpanl12 syl5eqbrr
|
|
imp ) ACDBCDEZFAGHFBIHEZEFFFJKZABJKZGLSTUAUBGHZFCDZUDSTUCMNNFFABOPRQ $.
|
|
|
|
$( The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM,
|
|
17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\
|
|
( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A + B ) ) $=
|
|
( cr wcel wa cc0 cle wbr caddc co 00id wi 0re le2add mpanl12 imp syl5eqbrr
|
|
) ACDBCDEZFAGHFBGHEZEFFFIJZABIJZGKRSTUAGHZFCDZUCRSUBLMMFFABNOPQ $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed by
|
|
NM, 17-Nov-2004.) $)
|
|
ltaddpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co wb ltadd2 mp3an1 wceq recn addid1d adantl
|
|
0re breq1d bitrd ) ACDZBCDZEZFAGHZBFIJZBAIJZGHZBUFGHFCDUAUBUDUGKRFABLMUCUEB
|
|
UFGUBUEBNUAUBBBOPQST $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed by
|
|
NM, 8-Apr-2005.) $)
|
|
ltaddpos2 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co ltaddpos wceq addcom syl2an breq2d bitr4d
|
|
cc recn ) ACDZBCDZEZFAGHBBAIJZGHBABIJZGHABKUAUCUBBGSAQDBQDUCUBLTARBRABMNOP
|
|
$.
|
|
|
|
$( Subtracting a positive number from another number decreases it.
|
|
(Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon,
|
|
19-Nov-2011.) $)
|
|
ltsubpos $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B - A ) < B ) ) $=
|
|
( cr wcel wa cc0 clt wbr caddc co cmin ltaddpos wb ltsubadd 3anidm13 ancoms
|
|
bitr4d ) ACDZBCDZEFAGHBBAIJGHZBAKJBGHZABLSRUATMZSRUBBABNOPQ $.
|
|
|
|
$( Comparison of two numbers whose difference is positive. (Contributed by
|
|
NM, 17-Nov-2004.) $)
|
|
posdif $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) ) $=
|
|
( cr wcel wa cc0 cmin co clt caddc wb resubcl ancoms simpl ltaddpos syl2anc
|
|
wbr cc wceq recn pncan3 syl2an breq2d bitr2d ) ACDZBCDZEZFBAGHZIQZAAUHJHZIQ
|
|
ZABIQUGUHCDZUEUIUKKUFUEULBALMUEUFNUHAOPUGUJBAIUEARDBRDUJBSUFATBTABUAUBUCUD
|
|
$.
|
|
|
|
$( Subtraction from both sides of 'less than or equal to'. (Contributed by
|
|
NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
lesub1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ B <-> ( A - C ) <_ ( B - C ) ) ) $=
|
|
( cr wcel w3a cmin co cle caddc simp1 simp3 simp2 resubcld lesubadd syl3anc
|
|
wbr wb recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACGHBCGHZIQZAUGCJHZIQZABI
|
|
QUFUCUEUGDEUHUJRUCUDUEKUCUDUELZUFBCUCUDUEMZUKNACUGOPUFUIBAIUFBCUFBULSUFCUKS
|
|
TUAUB $.
|
|
|
|
$( Subtraction of both sides of 'less than or equal to'. (Contributed by NM,
|
|
29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
lesub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ B <-> ( C - B ) <_ ( C - A ) ) ) $=
|
|
( cr wcel w3a cle wbr cmin co caddc leadd2 wb simp3 simp1 readdcld lesubadd
|
|
simp2 syl3anc recnd addsubd breq1d 3bitr2d resubcld leaddsub bitrd ) ADEZBD
|
|
EZCDEZFZABGHZCBIJZAKJZCGHZULCAIJGHZUJUKCAKJZCBKJGHZUPBIJZCGHZUNABCLUJUPDEUH
|
|
UIUSUQMUJCAUGUHUINZUGUHUIOZPUGUHUIRZUTUPBCQSUJURUMCGUJCABUJCUTTUJAVATUJBVBT
|
|
UAUBUCUJULDEUGUIUNUOMUJCBUTVBUDVAUTULACUESUF $.
|
|
|
|
$( Subtraction from both sides of 'less than'. (Contributed by FL,
|
|
3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
ltsub1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B <-> ( A - C ) < ( B - C ) ) ) $=
|
|
( cr wcel w3a cmin co clt caddc simp1 simp3 simp2 resubcld ltsubadd syl3anc
|
|
wbr wb recnd npcand breq2d bitr2d ) ADEZBDEZCDEZFZACGHBCGHZIQZAUGCJHZIQZABI
|
|
QUFUCUEUGDEUHUJRUCUDUEKUCUDUELZUFBCUCUDUEMZUKNACUGOPUFUIBAIUFBCUFBULSUFCUKS
|
|
TUAUB $.
|
|
|
|
$( Subtraction of both sides of 'less than'. (Contributed by NM,
|
|
29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
ltsub2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < B <-> ( C - B ) < ( C - A ) ) ) $=
|
|
( cr wcel w3a clt wbr cmin co caddc ltadd2 wb simp3 simp1 readdcld ltsubadd
|
|
simp2 syl3anc recnd addsubd breq1d 3bitr2d resubcld ltaddsub bitrd ) ADEZBD
|
|
EZCDEZFZABGHZCBIJZAKJZCGHZULCAIJGHZUJUKCAKJZCBKJGHZUPBIJZCGHZUNABCLUJUPDEUH
|
|
UIUSUQMUJCAUGUHUINZUGUHUIOZPUGUHUIRZUTUPBCQSUJURUMCGUJCABUJCUTTUJAVATUJBVBT
|
|
UAUBUCUJULDEUGUIUNUOMUJCBUTVBUDVAUTULACUESUF $.
|
|
|
|
$( Subtracting both sides of two 'less than' relations. (Contributed by
|
|
Mario Carneiro, 14-Apr-2016.) $)
|
|
lt2sub $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A < C /\ D < B ) -> ( A - B ) < ( C - D ) ) ) $=
|
|
( cr wcel wa clt wbr cmin simpll simprl simplr ltsub1 syl3anc simprr ltsub2
|
|
co wb resubcl anbi12d wi adantr resubcld adantl lttr sylbid ) AEFZBEFZGZCEF
|
|
ZDEFZGZGZACHIZDBHIZGABJRZCBJRZHIZURCDJRZHIZGZUQUTHIZUNUOUSUPVAUNUHUKUIUOUSS
|
|
UHUIUMKUJUKULLZUHUIUMMZACBNOUNULUIUKUPVASUJUKULPVEVDDBCQOUAUNUQEFZUREFUTEFZ
|
|
VBVCUBUJVFUMABTUCUNCBVDVEUDUMVGUJCDTUEUQURUTUFOUG $.
|
|
|
|
$( Subtracting both sides of two 'less than or equal to' relations.
|
|
(Contributed by Mario Carneiro, 14-Apr-2016.) $)
|
|
le2sub $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) ) $=
|
|
( cr wcel wa cle wbr cmin simpll simprl simplr lesub1 syl3anc simprr lesub2
|
|
co wb resubcl anbi12d wi adantr resubcld adantl letr sylbid ) AEFZBEFZGZCEF
|
|
ZDEFZGZGZACHIZDBHIZGABJRZCBJRZHIZURCDJRZHIZGZUQUTHIZUNUOUSUPVAUNUHUKUIUOUSS
|
|
UHUIUMKUJUKULLZUHUIUMMZACBNOUNULUIUKUPVASUJUKULPVEVDDBCQOUAUNUQEFZUREFUTEFZ
|
|
VBVCUBUJVFUMABTUCUNCBVDVEUDUMVGUJCDTUEUQURUTUFOUG $.
|
|
|
|
$( Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
|
|
(Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) ) $=
|
|
( cr wcel wa clt wbr cc0 cmin cneg 0re ltsub2 mp3an3 df-neg breq12i syl6bbr
|
|
co wb ) ACDZBCDZEABFGZHBIQZHAIQZFGZBJZAJZFGSTHCDUAUDRKABHLMUEUBUFUCFBNANOP
|
|
$.
|
|
|
|
$( Contraposition of negative in 'less than'. (Contributed by NM,
|
|
8-Nov-2004.) $)
|
|
ltnegcon1 $p |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) ) $=
|
|
( cr wcel wa cneg clt wbr wb renegcl ltneg sylan simpl recnd negnegd breq2d
|
|
bitrd ) ACDZBCDZEZAFZBGHZBFZUAFZGHZUCAGHRUACDSUBUEIAJUABKLTUDAUCGTATARSMNOP
|
|
Q $.
|
|
|
|
$( Contraposition of negative in 'less than'. (Contributed by Mario
|
|
Carneiro, 25-Feb-2015.) $)
|
|
ltnegcon2 $p |- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) ) $=
|
|
( cr wcel wa cneg clt wbr wb renegcl ltneg sylan2 simpr recnd negnegd bitrd
|
|
breq1d ) ACDZBCDZEZABFZGHZUAFZAFZGHZBUDGHSRUACDUBUEIBJAUAKLTUCBUDGTBTBRSMNO
|
|
QP $.
|
|
|
|
$( Negative of both sides of 'less than or equal to'. (Contributed by NM,
|
|
12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
leneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) ) $=
|
|
( cr wcel wa cle wbr cc0 cmin cneg 0re lesub2 mp3an3 df-neg breq12i syl6bbr
|
|
co wb ) ACDZBCDZEABFGZHBIQZHAIQZFGZBJZAJZFGSTHCDUAUDRKABHLMUEUBUFUCFBNANOP
|
|
$.
|
|
|
|
$( Contraposition of negative in 'less than or equal to'. (Contributed by
|
|
NM, 10-May-2004.) $)
|
|
lenegcon1 $p |- ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ A ) ) $=
|
|
( cr wcel wa cneg cle wbr wb renegcl leneg sylan recn negnegd breq2d adantr
|
|
bitrd ) ACDZBCDZEAFZBGHZBFZTFZGHZUBAGHZRTCDSUAUDIAJTBKLRUDUEISRUCAUBGRAAMNO
|
|
PQ $.
|
|
|
|
$( Contraposition of negative in 'less than or equal to'. (Contributed by
|
|
NM, 8-Oct-2005.) $)
|
|
lenegcon2 $p |- ( ( A e. RR /\ B e. RR ) -> ( A <_ -u B <-> B <_ -u A ) ) $=
|
|
( cr wcel wa cneg cle wbr wb renegcl sylan2 wceq recn negnegd adantl breq1d
|
|
leneg bitrd ) ACDZBCDZEZABFZGHZUBFZAFZGHZBUEGHTSUBCDUCUFIBJAUBQKUAUDBUEGTUD
|
|
BLSTBBMNOPR $.
|
|
|
|
$( Comparison of a number and its negative to zero. Theorem I.23 of
|
|
[Apostol] p. 20. (Contributed by NM, 14-May-1999.) $)
|
|
lt0neg1 $p |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) ) $=
|
|
( cr wcel cc0 clt wbr cneg wb 0re ltneg mpan2 neg0 breq1i syl6bb ) ABCZADEF
|
|
ZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by NM,
|
|
10-May-2004.) $)
|
|
lt0neg2 $p |- ( A e. RR -> ( 0 < A <-> -u A < 0 ) ) $=
|
|
( cr wcel cc0 clt wbr cneg wb 0re ltneg mpan neg0 breq2i syl6bb ) ABCZDAEFZ
|
|
AGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by NM,
|
|
10-May-2004.) $)
|
|
le0neg1 $p |- ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) ) $=
|
|
( cr wcel cc0 cle wbr cneg wb 0re leneg mpan2 neg0 breq1i syl6bb ) ABCZADEF
|
|
ZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by NM,
|
|
24-Aug-1999.) $)
|
|
le0neg2 $p |- ( A e. RR -> ( 0 <_ A <-> -u A <_ 0 ) ) $=
|
|
( cr wcel cc0 cle wbr cneg wb 0re leneg mpan neg0 breq2i syl6bb ) ABCZDAEFZ
|
|
AGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative number.
|
|
(Contributed by NM, 21-Feb-2005.) $)
|
|
addge01 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( A + B ) ) ) $=
|
|
( cr wcel wa cc0 cle wbr caddc co wb leadd2 mp3an1 ancoms wceq recn addid1d
|
|
0re adantr breq1d bitrd ) ACDZBCDZEZFBGHZAFIJZABIJZGHZAUGGHUCUBUEUHKZFCDUCU
|
|
BUIRFBALMNUDUFAUGGUBUFAOUCUBAAPQSTUA $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative number.
|
|
(Contributed by NM, 27-Jul-2005.) $)
|
|
addge02 $p |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( B + A ) ) ) $=
|
|
( cr wcel wa cc0 cle wbr caddc co addge01 cc wceq recn addcom syl2an breq2d
|
|
bitrd ) ACDZBCDZEZFBGHAABIJZGHABAIJZGHABKUAUBUCAGSALDBLDUBUCMTANBNABOPQR $.
|
|
|
|
$( Two nonnegative numbers are zero iff their sum is zero. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
add20 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) )
|
|
-> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa caddc wceq simpllr simplrl simplll addge02 syl2anc
|
|
co wb mpbid simpr breqtrd simplrr letri3d mpbir2and oveq2d addid1d 3eqtr3rd
|
|
0red recnd jca ex oveq12 00id syl6eq impbid1 ) ACDZEAFGZHZBCDZEBFGZHZHZABIP
|
|
ZEJZAEJZBEJZHZVAVCVFVAVCHZVDVEVGVBAEIPEAVGBEAIVGVEBEFGUSVGBVBEFVGUPBVBFGZUO
|
|
UPUTVCKVGURUOUPVHQUQURUSVCLZUOUPUTVCMZBANORVAVCSZTUQURUSVCUAVGBEVIVGUGUBUCZ
|
|
UDVKVGAVGAVJUHUEUFVLUIUJVFVBEEIPEAEBEIUKULUMUN $.
|
|
|
|
$( Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof
|
|
shortened by Mario Carneiro, 27-May-2016.) $)
|
|
subge0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( 0 <_ ( A - B ) <-> B <_ A ) ) $=
|
|
( cr wcel wa cc0 caddc co cle wbr cmin wb 0red simpr simpl leaddsub syl3anc
|
|
recnd addid2d breq1d bitr3d ) ACDZBCDZEZFBGHZAIJZFABKHIJZBAIJUDFCDUCUBUFUGL
|
|
UDMUBUCNZUBUCOFBAPQUDUEBAIUDBUDBUHRSTUA $.
|
|
|
|
$( Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof
|
|
shortened by Mario Carneiro, 27-May-2016.) $)
|
|
suble0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( A - B ) <_ 0 <-> A <_ B ) ) $=
|
|
( cr wcel wa cmin co cc0 cle wbr wb suble mp3an3 simpl recnd subid1d breq1d
|
|
0re bitrd ) ACDZBCDZEZABFGHIJZAHFGZBIJZABIJTUAHCDUCUEKRABHLMUBUDABIUBAUBATU
|
|
ANOPQS $.
|
|
|
|
$( The sum of a real number and a second real number is less then the real
|
|
number iff the second real number is negative. (Contributed by Alexander
|
|
van der Vekens, 30-May-2018.) $)
|
|
leaddle0 $p |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) <_ A <-> B <_ 0 ) ) $=
|
|
( cr wcel wa caddc co cle wbr cmin wb leaddsub2 3anidm13 wceq subidd adantr
|
|
cc0 recn breq2d bitrd ) ACDZBCDZEZABFGAHIZBAAJGZHIZBQHIUAUBUDUFKABALMUCUEQB
|
|
HUAUEQNUBUAAAROPST $.
|
|
|
|
$( Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.) $)
|
|
subge02 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( 0 <_ B <-> ( A - B ) <_ A ) ) $=
|
|
( cr wcel wa cc0 cle wbr caddc co cmin addge01 wb lesubadd 3anidm13 bitr4d
|
|
) ACDZBCDZEFBGHAABIJGHZABKJAGHZABLQRTSMABANOP $.
|
|
|
|
$( Lemma to show a nonnegative number is zero. (Contributed by NM,
|
|
8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
lesub0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) ) $=
|
|
( cr wcel wa cc0 wceq cle wbr cmin co 0red letri3 sylan2 ancom simpr lesub2
|
|
wb simpl syl3anc recnd subid1d breq1d bitrd ancoms anbi2d syl5bb bitr2d ) A
|
|
CDZBCDZEZAFGZAFHIZFAHIZEZUNBBAJKZHIZEZUJUIFCDZULUORUJLAFMNUOUNUMEUKURUMUNOU
|
|
KUMUQUNUJUIUMUQRUJUIEZUMBFJKZUPHIZUQUTUIUSUJUMVBRUJUIPUTLUJUISZAFBQTUTVABUP
|
|
HUTBUTBVCUAUBUCUDUEUFUGUH $.
|
|
|
|
$( The product of two negative numbers is positive. (Contributed by Jeff
|
|
Hankins, 8-Jun-2009.) $)
|
|
mullt0 $p |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) )
|
|
-> 0 < ( A x. B ) ) $=
|
|
( cr wcel cc0 clt wbr wa cneg cmul renegcl adantr lt0neg1 biimpa jca mulgt0
|
|
co syl2an cc recn wceq mul2neg ad2ant2r breqtrd ) ACDZAEFGZHZBCDZBEFGZHZHEA
|
|
IZBIZJQZABJQZFUGUKCDZEUKFGZHULCDZEULFGZHEUMFGUJUGUOUPUEUOUFAKLUEUFUPAMNOUJU
|
|
QURUHUQUIBKLUHUIURBMNOUKULPRUEUHUMUNUAZUFUIUEASDBSDUSUHATBTABUBRUCUD $.
|
|
|
|
$( 0 is less than or equal to 1. (Contributed by Mario Carneiro,
|
|
29-Apr-2015.) $)
|
|
0le1 $p |- 0 <_ 1 $=
|
|
( cc0 c1 0re 1re 0lt1 ltleii ) ABCDEF $.
|
|
|
|
${
|
|
lt2.1 $e |- A e. RR $.
|
|
$( 'Less than or equal to' is reflexive. (Contributed by NM,
|
|
18-Aug-1999.) $)
|
|
leidi $p |- A <_ A $=
|
|
( cr wcel cle wbr leid ax-mp ) ACDAAEFBAGH $.
|
|
|
|
$( Positive means nonzero (useful for ordering theorems involving
|
|
division). (Contributed by NM, 16-Sep-1999.) $)
|
|
gt0ne0i $p |- ( 0 < A -> A =/= 0 ) $=
|
|
( cc0 0re ltnei ) CADBE $.
|
|
|
|
${
|
|
gt0ne0i.2 $e |- 0 < A $.
|
|
$( Positive implies nonzero. (Contributed by NM, 15-May-1999.) $)
|
|
gt0ne0ii $p |- A =/= 0 $=
|
|
( cc0 clt wbr wne gt0ne0i ax-mp ) DAEFADGCABHI $.
|
|
$}
|
|
|
|
lt2.2 $e |- B e. RR $.
|
|
$( Addition of 2 positive numbers is positive. (Contributed by NM,
|
|
16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A + B ) ) $=
|
|
( cr wcel cc0 clt wbr wa caddc co addgt0 mpanl12 ) AEFBEFGAHIGBHIJGABKLHI
|
|
CDABMN $.
|
|
|
|
$( Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM,
|
|
28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
addge0i $p |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A + B ) ) $=
|
|
( cr wcel cc0 cle wbr wa caddc co addge0 mpanl12 ) AEFBEFGAHIGBHIJGABKLHI
|
|
CDABMN $.
|
|
|
|
$( Addition of nonnegative and positive numbers is positive. (Contributed
|
|
by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
addgegt0i $p |- ( ( 0 <_ A /\ 0 < B ) -> 0 < ( A + B ) ) $=
|
|
( cr wcel cc0 cle wbr clt wa caddc co addgegt0 mpanl12 ) AEFBEFGAHIGBJIKG
|
|
ABLMJICDABNO $.
|
|
|
|
${
|
|
addgt0i.3 $e |- 0 < A $.
|
|
addgt0i.4 $e |- 0 < B $.
|
|
$( Addition of 2 positive numbers is positive. (Contributed by NM,
|
|
18-May-1999.) $)
|
|
addgt0ii $p |- 0 < ( A + B ) $=
|
|
( cc0 clt wbr caddc co addgt0i mp2an ) GAHIGBHIGABJKHIEFABCDLM $.
|
|
$}
|
|
|
|
$( Two nonnegative numbers are zero iff their sum is zero. (Contributed by
|
|
NM, 28-Jul-1999.) $)
|
|
add20i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa caddc co wceq wb add20 an4s mpanl12 ) AEFZBEFZGA
|
|
HIZGBHIZJABKLGMAGMBGMJNZCDRTSUAUBABOPQ $.
|
|
|
|
$( Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
ltnegi $p |- ( A < B <-> -u B < -u A ) $=
|
|
( cr wcel clt wbr cneg wb ltneg mp2an ) AEFBEFABGHBIAIGHJCDABKL $.
|
|
|
|
$( Negative of both sides of 'less than or equal to'. (Contributed by NM,
|
|
1-Aug-1999.) $)
|
|
lenegi $p |- ( A <_ B <-> -u B <_ -u A ) $=
|
|
( cr wcel cle wbr cneg wb leneg mp2an ) AEFBEFABGHBIAIGHJCDABKL $.
|
|
|
|
$( Contraposition of negative in 'less than'. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
ltnegcon2i $p |- ( A < -u B <-> B < -u A ) $=
|
|
( cr wcel cneg clt wbr wb ltnegcon2 mp2an ) AEFBEFABGHIBAGHIJCDABKL $.
|
|
|
|
$( Lemma to show a nonnegative number is zero. (Contributed by NM,
|
|
8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
lesub0i $p |- ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) $=
|
|
( cr wcel cc0 cle wbr cmin co wa wceq wb lesub0 mp2an ) AEFBEFGAHIBBAJKHI
|
|
LAGMNCDABOP $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed
|
|
by NM, 25-Aug-1999.) $)
|
|
ltaddposi $p |- ( 0 < A <-> B < ( B + A ) ) $=
|
|
( cr wcel cc0 clt wbr caddc co wb ltaddpos mp2an ) AEFBEFGAHIBBAJKHILCDAB
|
|
MN $.
|
|
|
|
$( Comparison of two numbers whose difference is positive. (Contributed by
|
|
NM, 19-Aug-2001.) $)
|
|
posdifi $p |- ( A < B <-> 0 < ( B - A ) ) $=
|
|
( cr wcel clt wbr cc0 cmin co wb posdif mp2an ) AEFBEFABGHIBAJKGHLCDABMN
|
|
$.
|
|
|
|
$( Contraposition of negative in 'less than'. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
ltnegcon1i $p |- ( -u A < B <-> -u B < A ) $=
|
|
( cr wcel cneg clt wbr wb ltnegcon1 mp2an ) AEFBEFAGBHIBGAHIJCDABKL $.
|
|
|
|
$( Contraposition of negative in 'less than or equal to'. (Contributed by
|
|
NM, 6-Apr-2005.) $)
|
|
lenegcon1i $p |- ( -u A <_ B <-> -u B <_ A ) $=
|
|
( cr wcel cneg cle wbr wb lenegcon1 mp2an ) AEFBEFAGBHIBGAHIJCDABKL $.
|
|
|
|
$( Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.) $)
|
|
subge0i $p |- ( 0 <_ ( A - B ) <-> B <_ A ) $=
|
|
( cr wcel cc0 cmin co cle wbr wb subge0 mp2an ) AEFBEFGABHIJKBAJKLCDABMN
|
|
$.
|
|
|
|
lt2.3 $e |- C e. RR $.
|
|
$( Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
ltadd1i $p |- ( A < B <-> ( A + C ) < ( B + C ) ) $=
|
|
( cr wcel clt wbr caddc co wb ltadd1 mp3an ) AGHBGHCGHABIJACKLBCKLIJMDEFA
|
|
BCNO $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by NM,
|
|
11-Aug-1999.) $)
|
|
leadd1i $p |- ( A <_ B <-> ( A + C ) <_ ( B + C ) ) $=
|
|
( cr wcel cle wbr caddc co wb leadd1 mp3an ) AGHBGHCGHABIJACKLBCKLIJMDEFA
|
|
BCNO $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by NM,
|
|
11-Aug-1999.) $)
|
|
leadd2i $p |- ( A <_ B <-> ( C + A ) <_ ( C + B ) ) $=
|
|
( cr wcel cle wbr caddc co wb leadd2 mp3an ) AGHBGHCGHABIJCAKLCBKLIJMDEFA
|
|
BCNO $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon,
|
|
19-Nov-2011.) $)
|
|
ltsubaddi $p |- ( ( A - B ) < C <-> A < ( C + B ) ) $=
|
|
( cr wcel cmin co clt wbr caddc wb ltsubadd mp3an ) AGHBGHCGHABIJCKLACBMJ
|
|
KLNDEFABCOP $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon,
|
|
19-Nov-2011.) $)
|
|
lesubaddi $p |- ( ( A - B ) <_ C <-> A <_ ( C + B ) ) $=
|
|
( cr wcel cmin co cle wbr caddc wb lesubadd mp3an ) AGHBGHCGHABIJCKLACBMJ
|
|
KLNDEFABCOP $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 21-Jan-1997.) $)
|
|
ltsubadd2i $p |- ( ( A - B ) < C <-> A < ( B + C ) ) $=
|
|
( cr wcel cmin co clt wbr caddc wb ltsubadd2 mp3an ) AGHBGHCGHABIJCKLABCM
|
|
JKLNDEFABCOP $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by NM, 3-Aug-1999.) $)
|
|
lesubadd2i $p |- ( ( A - B ) <_ C <-> A <_ ( B + C ) ) $=
|
|
( cr wcel cmin co cle wbr caddc wb lesubadd2 mp3an ) AGHBGHCGHABIJCKLABCM
|
|
JKLNDEFABCOP $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by NM, 14-May-1999.) $)
|
|
ltaddsubi $p |- ( ( A + B ) < C <-> A < ( C - B ) ) $=
|
|
( cr wcel caddc co clt wbr cmin wb ltaddsub mp3an ) AGHBGHCGHABIJCKLACBMJ
|
|
KLNDEFABCOP $.
|
|
|
|
lt.4 $e |- D e. RR $.
|
|
$( Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20.
|
|
(Contributed by NM, 14-May-1999.) $)
|
|
lt2addi $p |- ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) $=
|
|
( cr wcel clt wbr wa caddc co wi lt2add mp4an ) AIJBIJCIJDIJACKLBDKLMABNO
|
|
CDNOKLPEFGHABCDQR $.
|
|
|
|
$( Adding both side of two inequalities. (Contributed by NM,
|
|
16-Sep-1999.) $)
|
|
le2addi $p |- ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) $=
|
|
( cr wcel cle wbr wa caddc co wi le2add mp4an ) AIJBIJCIJDIJACKLBDKLMABNO
|
|
CDNOKLPEFGHABCDQR $.
|
|
$}
|
|
|
|
${
|
|
gt0ne0d.1 $e |- ( ph -> 0 < A ) $.
|
|
$( Positive implies nonzero. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
gt0ne0d $p |- ( ph -> A =/= 0 ) $=
|
|
( cc0 cr wcel clt wbr wne 0re ltne sylancr ) ADEFDBGHBDIJCDBKL $.
|
|
$}
|
|
|
|
${
|
|
lt0ne0d.1 $e |- ( ph -> A < 0 ) $.
|
|
$( Something less than zero is not zero. Deduction form. (Contributed by
|
|
David Moews, 28-Feb-2017.) $)
|
|
lt0ne0d $p |- ( ph -> A =/= 0 ) $=
|
|
( cc0 clt wbr wne wceq 0re ltnri breq1 mtbiri necon2ai syl ) ABDEFZBDGCOB
|
|
DBDHODDEFDIJBDDEKLMN $.
|
|
$}
|
|
|
|
${
|
|
leidd.1 $e |- ( ph -> A e. RR ) $.
|
|
$( 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
leidd $p |- ( ph -> A <_ A ) $=
|
|
( cr wcel cle wbr leid syl ) ABDEBBFGCBHI $.
|
|
|
|
$( Comparison of a number and its negative to zero. Theorem I.23 of
|
|
[Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lt0neg1d $p |- ( ph -> ( A < 0 <-> 0 < -u A ) ) $=
|
|
( cr wcel cc0 clt wbr cneg wb lt0neg1 syl ) ABDEBFGHFBIGHJCBKL $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
lt0neg2d $p |- ( ph -> ( 0 < A <-> -u A < 0 ) ) $=
|
|
( cr wcel cc0 clt wbr cneg wb lt0neg2 syl ) ABDEFBGHBIFGHJCBKL $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
le0neg1d $p |- ( ph -> ( A <_ 0 <-> 0 <_ -u A ) ) $=
|
|
( cr wcel cc0 cle wbr cneg wb le0neg1 syl ) ABDEBFGHFBIGHJCBKL $.
|
|
|
|
$( Comparison of a number and its negative to zero. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
le0neg2d $p |- ( ph -> ( 0 <_ A <-> -u A <_ 0 ) ) $=
|
|
( cr wcel cc0 cle wbr cneg wb le0neg2 syl ) ABDEFBGHBIFGHJCBKL $.
|
|
|
|
ltnegd.2 $e |- ( ph -> B e. RR ) $.
|
|
${
|
|
addgegt0d.3 $e |- ( ph -> 0 <_ A ) $.
|
|
addgegt0d.4 $e |- ( ph -> 0 < B ) $.
|
|
$( Addition of nonnegative and positive numbers is positive.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addgegt0d $p |- ( ph -> 0 < ( A + B ) ) $=
|
|
( cr wcel cc0 cle wbr clt caddc co addgegt0 syl22anc ) ABHICHIJBKLJCMLJ
|
|
BCNOMLDEFGBCPQ $.
|
|
$}
|
|
|
|
${
|
|
addgt0d.3 $e |- ( ph -> 0 < A ) $.
|
|
addgt0d.4 $e |- ( ph -> 0 < B ) $.
|
|
$( Addition of 2 positive numbers is positive. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
addgt0d $p |- ( ph -> 0 < ( A + B ) ) $=
|
|
( cc0 0red ltled addgegt0d ) ABCDEAHBAIDFJGK $.
|
|
$}
|
|
|
|
${
|
|
addge0d.3 $e |- ( ph -> 0 <_ A ) $.
|
|
addge0d.4 $e |- ( ph -> 0 <_ B ) $.
|
|
$( Addition of 2 nonnegative numbers is nonnegative. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
addge0d $p |- ( ph -> 0 <_ ( A + B ) ) $=
|
|
( cr wcel cc0 cle wbr caddc co addge0 syl22anc ) ABHICHIJBKLJCKLJBCMNKL
|
|
DEFGBCOP $.
|
|
$}
|
|
|
|
$( Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
ltnegd $p |- ( ph -> ( A < B <-> -u B < -u A ) ) $=
|
|
( cr wcel clt wbr cneg wb ltneg syl2anc ) ABFGCFGBCHICJBJHIKDEBCLM $.
|
|
|
|
$( Negative of both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
lenegd $p |- ( ph -> ( A <_ B <-> -u B <_ -u A ) ) $=
|
|
( cr wcel cle wbr cneg wb leneg syl2anc ) ABFGCFGBCHICJBJHIKDEBCLM $.
|
|
|
|
${
|
|
ltnegcon1d.3 $e |- ( ph -> -u A < B ) $.
|
|
$( Contraposition of negative in 'less than'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltnegcon1d $p |- ( ph -> -u B < A ) $=
|
|
( cneg clt wbr cr wcel wb ltnegcon1 syl2anc mpbid ) ABGCHIZCGBHIZFABJKC
|
|
JKPQLDEBCMNO $.
|
|
$}
|
|
|
|
${
|
|
ltnegcon2d.3 $e |- ( ph -> A < -u B ) $.
|
|
$( Contraposition of negative in 'less than'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltnegcon2d $p |- ( ph -> B < -u A ) $=
|
|
( cneg clt wbr cr wcel wb ltnegcon2 syl2anc mpbid ) ABCGHIZCBGHIZFABJKC
|
|
JKPQLDEBCMNO $.
|
|
|
|
$}
|
|
|
|
${
|
|
lenegcon1d.3 $e |- ( ph -> -u A <_ B ) $.
|
|
$( Contraposition of negative in 'less than or equal to'. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
lenegcon1d $p |- ( ph -> -u B <_ A ) $=
|
|
( cneg cle wbr cr wcel wb lenegcon1 syl2anc mpbid ) ABGCHIZCGBHIZFABJKC
|
|
JKPQLDEBCMNO $.
|
|
|
|
$}
|
|
|
|
${
|
|
lenegcon2d.3 $e |- ( ph -> A <_ -u B ) $.
|
|
$( Contraposition of negative in 'less than or equal to'. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
lenegcon2d $p |- ( ph -> B <_ -u A ) $=
|
|
( cneg cle wbr cr wcel wb lenegcon2 syl2anc mpbid ) ABCGHIZCBGHIZFABJKC
|
|
JKPQLDEBCMNO $.
|
|
$}
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltaddposd $p |- ( ph -> ( 0 < A <-> B < ( B + A ) ) ) $=
|
|
( cr wcel cc0 clt wbr caddc co wb ltaddpos syl2anc ) ABFGCFGHBIJCCBKLIJMD
|
|
EBCNO $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltaddpos2d $p |- ( ph -> ( 0 < A <-> B < ( A + B ) ) ) $=
|
|
( cr wcel cc0 clt wbr caddc co wb ltaddpos2 syl2anc ) ABFGCFGHBIJCBCKLIJM
|
|
DEBCNO $.
|
|
|
|
$( Subtracting a positive number from another number decreases it.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
ltsubposd $p |- ( ph -> ( 0 < A <-> ( B - A ) < B ) ) $=
|
|
( cr wcel cc0 clt wbr cmin co wb ltsubpos syl2anc ) ABFGCFGHBIJCBKLCIJMDE
|
|
BCNO $.
|
|
|
|
$( Comparison of two numbers whose difference is positive. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
posdifd $p |- ( ph -> ( A < B <-> 0 < ( B - A ) ) ) $=
|
|
( cr wcel clt wbr cc0 cmin co wb posdif syl2anc ) ABFGCFGBCHIJCBKLHIMDEBC
|
|
NO $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative number.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addge01d $p |- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) ) $=
|
|
( cr wcel cc0 cle wbr caddc co wb addge01 syl2anc ) ABFGCFGHCIJBBCKLIJMDE
|
|
BCNO $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative number.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
addge02d $p |- ( ph -> ( 0 <_ B <-> A <_ ( B + A ) ) ) $=
|
|
( cr wcel cc0 cle wbr caddc co wb addge02 syl2anc ) ABFGCFGHCIJBCBKLIJMDE
|
|
BCNO $.
|
|
|
|
$( Nonnegative subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subge0d $p |- ( ph -> ( 0 <_ ( A - B ) <-> B <_ A ) ) $=
|
|
( cr wcel cc0 cmin co cle wbr wb subge0 syl2anc ) ABFGCFGHBCIJKLCBKLMDEBC
|
|
NO $.
|
|
|
|
$( Nonpositive subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
suble0d $p |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) ) $=
|
|
( cr wcel cmin co cc0 cle wbr wb suble0 syl2anc ) ABFGCFGBCHIJKLBCKLMDEBC
|
|
NO $.
|
|
|
|
$( Nonnegative subtraction. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subge02d $p |- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) ) $=
|
|
( cr wcel cc0 cle wbr cmin co wb subge02 syl2anc ) ABFGCFGHCIJBCKLBIJMDEB
|
|
CNO $.
|
|
|
|
ltadd1d.3 $e |- ( ph -> C e. RR ) $.
|
|
$( Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
ltadd1d $p |- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) ) $=
|
|
( cr wcel clt wbr caddc co wb ltadd1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNE
|
|
FGBCDOP $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
leadd1d $p |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) $=
|
|
( cr wcel cle wbr caddc co wb leadd1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNE
|
|
FGBCDOP $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
leadd2d $p |- ( ph -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) ) $=
|
|
( cr wcel cle wbr caddc co wb leadd2 syl3anc ) ABHICHIDHIBCJKDBLMDCLMJKNE
|
|
FGBCDOP $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltsubaddd $p |- ( ph -> ( ( A - B ) < C <-> A < ( C + B ) ) ) $=
|
|
( cr wcel cmin co clt wbr caddc wb ltsubadd syl3anc ) ABHICHIDHIBCJKDLMBD
|
|
CNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lesubaddd $p |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) ) $=
|
|
( cr wcel cmin co cle wbr caddc wb lesubadd syl3anc ) ABHICHIDHIBCJKDLMBD
|
|
CNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltsubadd2d $p |- ( ph -> ( ( A - B ) < C <-> A < ( B + C ) ) ) $=
|
|
( cr wcel cmin co clt wbr caddc wb ltsubadd2 syl3anc ) ABHICHIDHIBCJKDLMB
|
|
CDNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than or equal to' relationship between subtraction and addition.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lesubadd2d $p |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) ) $=
|
|
( cr wcel cmin co cle wbr caddc wb lesubadd2 syl3anc ) ABHICHIDHIBCJKDLMB
|
|
CDNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by Mario Carneiro, 27-May-2016.) $)
|
|
ltaddsubd $p |- ( ph -> ( ( A + B ) < C <-> A < ( C - B ) ) ) $=
|
|
( cr wcel caddc co clt wbr cmin wb ltaddsub syl3anc ) ABHICHIDHIBCJKDLMBD
|
|
CNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than' relationship between subtraction and addition. (Contributed
|
|
by Mario Carneiro, 29-Dec-2016.) $)
|
|
ltaddsub2d $p |- ( ph -> ( ( A + B ) < C <-> B < ( C - A ) ) ) $=
|
|
( cr wcel caddc co clt wbr cmin wb ltaddsub2 syl3anc ) ABHICHIDHIBCJKDLMC
|
|
DBNKLMOEFGBCDPQ $.
|
|
|
|
$( 'Less than or equal to' relationship between and addition and
|
|
subtraction. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
leaddsub2d $p |- ( ph -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) ) $=
|
|
( cr wcel caddc co cle wbr cmin wb leaddsub2 syl3anc ) ABHICHIDHIBCJKDLMC
|
|
DBNKLMOEFGBCDPQ $.
|
|
|
|
${
|
|
subled.4 $e |- ( ph -> ( A - B ) <_ C ) $.
|
|
$( Swap subtrahends in an inequality. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
subled $p |- ( ph -> ( A - C ) <_ B ) $=
|
|
( cmin co cle wbr cr wcel wb suble syl3anc mpbid ) ABCIJDKLZBDIJCKLZHAB
|
|
MNCMNDMNSTOEFGBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
lesubd.4 $e |- ( ph -> A <_ ( B - C ) ) $.
|
|
$( Swap subtrahends in an inequality. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
lesubd $p |- ( ph -> C <_ ( B - A ) ) $=
|
|
( cmin co cle wbr cr wcel wb lesub syl3anc mpbid ) ABCDIJKLZDCBIJKLZHAB
|
|
MNCMNDMNSTOEFGBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
ltsub23d.4 $e |- ( ph -> ( A - B ) < C ) $.
|
|
$( 'Less than' relationship between subtraction and addition.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
ltsub23d $p |- ( ph -> ( A - C ) < B ) $=
|
|
( cmin co clt wbr cr wcel wb ltsub23 syl3anc mpbid ) ABCIJDKLZBDIJCKLZH
|
|
ABMNCMNDMNSTOEFGBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
ltsub13d.4 $e |- ( ph -> A < ( B - C ) ) $.
|
|
$( 'Less than' relationship between subtraction and addition.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
ltsub13d $p |- ( ph -> C < ( B - A ) ) $=
|
|
( cmin co clt wbr cr wcel wb ltsub13 syl3anc mpbid ) ABCDIJKLZDCBIJKLZH
|
|
ABMNCMNDMNSTOEFGBCDPQR $.
|
|
$}
|
|
|
|
$( Subtraction from both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
lesub1d $p |- ( ph -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) ) $=
|
|
( cr wcel cle wbr cmin co wb lesub1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNEF
|
|
GBCDOP $.
|
|
|
|
$( Subtraction of both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
lesub2d $p |- ( ph -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) ) $=
|
|
( cr wcel cle wbr cmin co wb lesub2 syl3anc ) ABHICHIDHIBCJKDCLMDBLMJKNEF
|
|
GBCDOP $.
|
|
|
|
$( Subtraction from both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltsub1d $p |- ( ph -> ( A < B <-> ( A - C ) < ( B - C ) ) ) $=
|
|
( cr wcel clt wbr cmin co wb ltsub1 syl3anc ) ABHICHIDHIBCJKBDLMCDLMJKNEF
|
|
GBCDOP $.
|
|
|
|
$( Subtraction of both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
ltsub2d $p |- ( ph -> ( A < B <-> ( C - B ) < ( C - A ) ) ) $=
|
|
( cr wcel clt wbr cmin co wb ltsub2 syl3anc ) ABHICHIDHIBCJKDCLMDBLMJKNEF
|
|
GBCDOP $.
|
|
|
|
${
|
|
ltadd1dd.4 $e |- ( ph -> A < B ) $.
|
|
$( Addition to both sides of 'less than'. Theorem I.18 of [Apostol]
|
|
p. 20. (Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
ltadd1dd $p |- ( ph -> ( A + C ) < ( B + C ) ) $=
|
|
( clt wbr caddc co ltadd1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
|
|
$( Subtraction from both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 30-May-2016.) $)
|
|
ltsub1dd $p |- ( ph -> ( A - C ) < ( B - C ) ) $=
|
|
( clt wbr cmin co ltsub1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
|
|
$( Subtraction of both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 30-May-2016.) $)
|
|
ltsub2dd $p |- ( ph -> ( C - B ) < ( C - A ) ) $=
|
|
( clt wbr cmin co ltsub2d mpbid ) ABCIJDCKLDBKLIJHABCDEFGMN $.
|
|
$}
|
|
|
|
${
|
|
leadd1dd.4 $e |- ( ph -> A <_ B ) $.
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 30-May-2016.) $)
|
|
leadd1dd $p |- ( ph -> ( A + C ) <_ ( B + C ) ) $=
|
|
( cle wbr caddc co leadd1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
|
|
$( Addition to both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 30-May-2016.) $)
|
|
leadd2dd $p |- ( ph -> ( C + A ) <_ ( C + B ) ) $=
|
|
( cle wbr caddc co leadd2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.
|
|
|
|
$( Subtraction from both sides of 'less than or equal to'. (Contributed
|
|
by Mario Carneiro, 30-May-2016.) $)
|
|
lesub1dd $p |- ( ph -> ( A - C ) <_ ( B - C ) ) $=
|
|
( cle wbr cmin co lesub1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
|
|
$( Subtraction of both sides of 'less than or equal to'. (Contributed by
|
|
Mario Carneiro, 30-May-2016.) $)
|
|
lesub2dd $p |- ( ph -> ( C - B ) <_ ( C - A ) ) $=
|
|
( cle wbr cmin co lesub2d mpbid ) ABCIJDCKLDBKLIJHABCDEFGMN $.
|
|
$}
|
|
|
|
lt2addd.4 $e |- ( ph -> D e. RR ) $.
|
|
${
|
|
le2addd.5 $e |- ( ph -> A <_ C ) $.
|
|
le2addd.6 $e |- ( ph -> B <_ D ) $.
|
|
$( Adding both side of two inequalities. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
le2addd $p |- ( ph -> ( A + B ) <_ ( C + D ) ) $=
|
|
( cle wbr caddc co cr wcel wa wi le2add syl22anc mp2and ) ABDLMZCELMZBC
|
|
NODENOLMZJKABPQCPQDPQEPQUCUDRUESFGHIBCDETUAUB $.
|
|
|
|
$( Subtracting both sides of two 'less than or equal to' relations.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
le2subd $p |- ( ph -> ( A - D ) <_ ( C - B ) ) $=
|
|
( cle wbr cmin co cr wcel wa wi le2sub syl22anc mp2and ) ABDLMZCELMZBEN
|
|
ODCNOLMZJKABPQEPQDPQCPQUCUDRUESFIHGBEDCTUAUB $.
|
|
$}
|
|
|
|
${
|
|
ltleaddd.5 $e |- ( ph -> A < C ) $.
|
|
ltleaddd.6 $e |- ( ph -> B <_ D ) $.
|
|
$( Adding both sides of two orderings. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltleaddd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
|
|
( clt wbr cle caddc co cr wcel wa wi ltleadd syl22anc mp2and ) ABDLMZCE
|
|
NMZBCOPDEOPLMZJKABQRCQRDQREQRUDUESUFTFGHIBCDEUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
leltaddd.5 $e |- ( ph -> A <_ C ) $.
|
|
leltaddd.6 $e |- ( ph -> B < D ) $.
|
|
$( Adding both sides of two orderings. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
leltaddd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
|
|
( cle wbr clt caddc co cr wcel wa wi leltadd syl22anc mp2and ) ABDLMZCE
|
|
NMZBCOPDEOPNMZJKABQRCQRDQREQRUDUESUFTFGHIBCDEUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
lt2addd.5 $e |- ( ph -> A < C ) $.
|
|
lt2addd.6 $e |- ( ph -> B < D ) $.
|
|
$( Adding both side of two inequalities. Theorem I.25 of [Apostol]
|
|
p. 20. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lt2addd $p |- ( ph -> ( A + B ) < ( C + D ) ) $=
|
|
( ltled ltleaddd ) ABCDEFGHIJACEGIKLM $.
|
|
|
|
$( Subtracting both sides of two 'less than' relations. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
lt2subd $p |- ( ph -> ( A - D ) < ( C - B ) ) $=
|
|
( clt wbr cmin co cr wcel wa wi lt2sub syl22anc mp2and ) ABDLMZCELMZBEN
|
|
ODCNOLMZJKABPQEPQDPQCPQUCUDRUESFIHGBEDCTUAUB $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
possumd.1 $e |- ( ph -> A e. RR ) $.
|
|
possumd.2 $e |- ( ph -> B e. RR ) $.
|
|
$( Condition for a positive sum. (Contributed by Scott Fenton,
|
|
16-Dec-2017.) $)
|
|
possumd $p |- ( ph -> ( 0 < ( A + B ) <-> -u B < A ) ) $=
|
|
( cneg clt wbr cmin co caddc renegcld posdifd recnd subnegd breq2d bitr2d
|
|
cc0 ) ACFZBGHRBSIJZGHRBCKJZGHASBACELDMATUARGABCABDNACENOPQ $.
|
|
$}
|
|
|
|
${
|
|
sublt0d.1 $e |- ( ph -> A e. RR ) $.
|
|
sublt0d.2 $e |- ( ph -> B e. RR ) $.
|
|
$( When a subtraction gives a negative result. (Contributed by Glauco
|
|
Siliprandi, 11-Dec-2019.) $)
|
|
sublt0d $p |- ( ph -> ( ( A - B ) < 0 <-> A < B ) ) $=
|
|
( cmin co cc0 clt wbr caddc 0red ltsubaddd recnd addid2d breq2d bitrd ) A
|
|
BCFGHIJBHCKGZIJBCIJABCHDEALMARCBIACACENOPQ $.
|
|
$}
|
|
|
|
$( Addition and subtraction on one side of 'less than'. (Contributed by AV,
|
|
24-Nov-2018.) $)
|
|
ltaddsublt $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( B < C <-> ( ( A + B ) - C ) < A ) ) $=
|
|
( cr wcel w3a clt caddc co cmin wb ltadd2 3comr readdcl 3adant3 simp3 simp1
|
|
wbr ltsubaddd bitr4d ) ADEZBDEZCDEZFZBCGRZABHIZACHIGRZUFCJIAGRUBUCUAUEUGKBC
|
|
ALMUDUFCAUAUBUFDEUCABNOUAUBUCPUAUBUCQST $.
|
|
|
|
$( ` 1 <_ 1 ` . Common special case. (Contributed by David A. Wheeler,
|
|
16-Jul-2016.) $)
|
|
1le1 $p |- 1 <_ 1 $=
|
|
( c1 1re leidi ) ABC $.
|
|
|
|
$( A positive sum must have a positive addend. Part of Definition 11.2.7(vi)
|
|
of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) $)
|
|
gt0add $p |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) ->
|
|
( 0 < A \/ 0 < B ) ) $=
|
|
( cr wcel cc0 caddc co clt wbr w3a wo simp3 wi 0red simp1 readdcld axltwlin
|
|
simp2 syl3anc mpd ltaddposd orbi2d mpbird ) ACDZBCDZEABFGZHIZJZEAHIZEBHIZKU
|
|
IAUFHIZKZUHUGULUDUEUGLUHECDUFCDUDUGULMUHNUHABUDUEUGOZUDUEUGRZPUMEUFAQSTUHUJ
|
|
UKUIUHBAUNUMUAUBUC $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Real Apartness
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c #RR $. $( 'Apart' relation (over real numbers) $)
|
|
|
|
$( Class of real apartness relation. $)
|
|
creap $a class #RR $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define real apartness. Definition in Section 11.2.1 of [HoTT], p.
|
|
(varies). Although ` #RR ` is an apartness relation on the reals (see
|
|
~ df-ap for more discussion of apartness relations), for our purposes it
|
|
is just a stepping stone to defining ` # ` which is an apartness
|
|
relation on complex numbers. On the reals, ` #RR ` and ` # ` agree
|
|
( ~ apreap ). (Contributed by Jim Kingdon, 26-Jan-2020.) $)
|
|
df-reap $a |- #RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\
|
|
( x < y \/ y < x ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Real apartness in terms of classes. Beyond the development of ` # `
|
|
itself, proofs should use ~ reaplt instead. (New usage is discouraged.)
|
|
(Contributed by Jim Kingdon, 29-Jan-2020.) $)
|
|
reapval $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A #RR B <-> ( A < B \/ B < A ) ) ) $=
|
|
( vx vy creap wbr cr wcel wa clt wo cv breq12 simpr simpl breq12d orbi12d
|
|
wceq df-reap brab2ga baib ) ABEFAGHBGHIABJFZBAJFZKZCLZDLZJFZUFUEJFZKUDCDA
|
|
BGGEUEARZUFBRZIZUGUBUHUCUEAUFBJMUKUFBUEAJUIUJNUIUJOPQCDSTUA $.
|
|
$}
|
|
|
|
$( Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT],
|
|
p. (varies). Beyond the development of ` # ` itself, proofs should use
|
|
~ apirr instead. (Contributed by Jim Kingdon, 26-Jan-2020.) $)
|
|
reapirr $p |- ( A e. RR -> -. A #RR A ) $=
|
|
( cr wcel creap wbr clt ltnr wo wb reapval anidms oridm syl6bb mtbird ) ABC
|
|
ZAADEZAAFEZAGOPQQHZQOPRIAAJKQLMN $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Existence of reciprocal of real number. (Contributed by Jim Kingdon,
|
|
29-Jan-2020.) $)
|
|
recexre $p |- ( ( A e. RR /\ A #RR 0 ) -> E. x e. RR ( A x. x ) = 1 ) $=
|
|
( vy cr wcel cc0 wbr cmul co c1 wceq wrex clt wb 0re wa cneg pm5.32i recn
|
|
cltrr creap cv reapval mpan2 lt0neg1 renegcl ltxrlt bitrd ax-precex simpr
|
|
sylancr reximi syl sylan sylbi negnegd oveq2d eqeq1d rspcev sylbir adantl
|
|
wo negeq rexlimddv cc mul2neg syl2an rexbidva adantr mpbid ex mpan sylbid
|
|
jaod imp ) BDEZBFUAGZBAUBZHIZJKZADLZVPVQBFMGZFBMGZVBZWAVPFDEZVQWDNOBFUCUD
|
|
VPWBWAWCVPWBWAVPWBPZBQZVRQZHIZJKZADLZWAWFWGCUBZHIZJKZWKCDWFVPFWGTGZPWNCDL
|
|
ZVPWBWOVPWBFWGMGZWOBUEVPWEWGDEZWQWONOBUFZFWGUGUKUHRVPWRWOWPWSWRWOPFWLTGZW
|
|
NPZCDLWPCWGUIXAWNCDWTWNUJULUMUNUOWLDEZWNPZWKWFXCXBWGWLQZQZHIZJKZPWKXBXGWN
|
|
XBXFWMJXBXEWLWGHXBWLWLSUPUQURRXBXDDEXGWKWLUFWJXGAXDDVRXDKZWIXFJXHWHXEWGHV
|
|
RXDVCUQURUSUNUTVAVDVPWKWANWBVPWJVTADVPVRDEZPWIVSJVPBVEEVRVEEWIVSKXIBSVRSB
|
|
VRVFVGURVHVIVJVKVPWCWAVPWCPVPFBTGZPZWAVPWCXJWEVPWCXJNOFBUGVLRXKFVRTGZVTPZ
|
|
ADLWAABUIXMVTADXLVTUJULUMUOVKVNVMVO $.
|
|
$}
|
|
|
|
$( Real apartness is tight. Beyond the development of apartness itself,
|
|
proofs should use ~ apti . (Contributed by Jim Kingdon, 30-Jan-2020.)
|
|
(New usage is discouraged.) $)
|
|
reapti $p |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #RR B ) ) $=
|
|
( cr wcel wa wceq creap wbr wn clt wo ltnr adantr oridm breq2 breq1 orbi12d
|
|
syl5bbr notbid syl5ibcom reapval sylibrd axapti 3expia sylbid impbid ) ACDZ
|
|
BCDZEZABFZABGHZIZUIUJABJHZBAJHZKZIZULUIAAJHZIZUJUPUGURUHALMUJUQUOUQUQUQKUJU
|
|
OUQNUJUQUMUQUNABAJOABAJPQRSTUIUKUOABUASZUBUIULUPUJUSUGUHUPUJABUCUDUEUF $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Existence of reciprocal of positive real number. (Contributed by Jim
|
|
Kingdon, 6-Feb-2020.) $)
|
|
recexgt0 $p |- ( ( A e. RR /\ 0 < A ) ->
|
|
E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) ) $=
|
|
( cr wcel cc0 cltrr wbr wa cv cmul co c1 wceq clt ax-precex wb 0re ltxrlt
|
|
wrex mpan pm5.32i anbi1d rexbiia 3imtr4i ) BCDZEBFGZHEAIZFGZBUGJKLMZHZACS
|
|
UEEBNGZHEUGNGZUIHZACSABOUEUKUFECDZUEUKUFPQEBRTUAUMUJACUGCDZULUHUIUNUOULUH
|
|
PQEUGRTUBUCUD $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Complex Apartness
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c # $. $( 'Apart' relation (over complex numbers) $)
|
|
|
|
$( Class of complex apartness relation. $)
|
|
cap $a class # $.
|
|
|
|
${
|
|
$d r s t u x y $.
|
|
$( Define complex apartness. Definition 6.1 of [Geuvers], p. 17.
|
|
|
|
Two numbers are considered apart if it is possible to separate them.
|
|
One common usage is that we can divide by a number if it is apart from
|
|
zero (see for example ~ recclap which says that a number apart from zero
|
|
has a reciprocal).
|
|
|
|
The defining characteristics of an apartness are irreflexivity
|
|
( ~ apirr ), symmetry ( ~ apsym ), and cotransitivity ( ~ apcotr ).
|
|
Apartness implies negated equality, as seen at ~ apne , and the converse
|
|
would also follow if we assumed excluded middle.
|
|
|
|
In addition, apartness of complex numbers is tight, which means that two
|
|
numbers which are not apart are equal ( ~ apti ).
|
|
|
|
(Contributed by Jim Kingdon, 26-Jan-2020.) $)
|
|
df-ap $a |- # = { <. x , y >. | E. r e. RR E. s e. RR E. t e. RR E. u e. RR
|
|
( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\
|
|
( r #RR t \/ s #RR u ) ) } $.
|
|
$}
|
|
|
|
$( ` _i ` times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof
|
|
shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
ixi $p |- ( _i x. _i ) = -u 1 $=
|
|
( c1 cneg cc0 cmin co ci cmul df-neg caddc ax-i2m1 0cn ax-1cn ax-icn mulcli
|
|
wceq subadd2i mpbir eqtr2i ) ABCADEZFFGEZAHSTOTAIECOJCATKLFFMMNPQR $.
|
|
|
|
$( The imaginary unit ` _i ` is not a real number. (Contributed by NM,
|
|
6-May-1999.) $)
|
|
inelr $p |- -. _i e. RR $=
|
|
( ci cr wcel cc0 wceq ine0 neii clt wbr wn co c1 0re 1re caddc wa anidms ex
|
|
cmul mtoi 0lt1 ltnsymi cneg renegcli eqeltri ltadd1i ax-1cn addid2i ax-i2m1
|
|
ax-mp ixi breq12i bitri mtbir mullt0 mulgt0 wb lttri3 mpan2 mpbir2and mto )
|
|
ABCZADEZADFGVBVCADHIZJZDAHIZJZVBVDDAASKZHIZVILDHIZDLHIVJJUADLMNUBUJVIDLOKZV
|
|
HLOKZHIVJDVHLMVHLUCBUKLNUDUENUFVKLVLDHLUGUHUIULUMUNZVBVDVIVBVDPVIAAUOQRTVBV
|
|
FVIVMVBVFVIVBVFPVIAAUPQRTVBDBCVCVEVGPUQMADURUSUTVA $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A real number times the imaginary unit is real only if the number is 0.
|
|
(Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
rimul $p |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> A = 0 ) $=
|
|
( vx cr wcel ci cmul co wa cc0 wceq creap wbr wn inelr cv c1 recnd ax-icn
|
|
wrex cc recexre adantlr simplll simprl mulass mp3an1 syl2anc oveq2 syl6eq
|
|
mulid1i ad2antll eqtrd simpllr remulcld eqeltrrd rexlimddv ex mtoi wb 0re
|
|
reapti mpan2 adantr mpbird ) ACDZEAFGZCDZHZAIJZAIKLZMZVHVJECDZNVHVJVLVHVJ
|
|
HZABOZFGZPJZVLBCVEVJVPBCSVGBAUAUBVMVNCDZVPHZHZVFVNFGZECVSVTEVOFGZEVSATDZV
|
|
NTDZVTWAJZVSAVEVGVJVRUCQVSVNVMVQVPUDZQETDWBWCWDREAVNUEUFUGVPWAEJVMVQVPWAE
|
|
PFGEVOPEFUHERUJUIUKULVSVFVNVEVGVJVRUMWEUNUOUPUQURVEVIVKUSZVGVEICDWFUTAIVA
|
|
VBVCVD $.
|
|
$}
|
|
|
|
$( Decomposition of a real number into real part (itself) and imaginary part
|
|
(zero). (Contributed by Jim Kingdon, 30-Jan-2020.) $)
|
|
rereim $p |- ( ( ( A e. RR /\ B e. RR ) /\
|
|
( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) ) $=
|
|
( cr wcel wa ci cmul co caddc wceq cc0 simpll recnd simplr simprr eqcomd cc
|
|
cmin ax-icn a1i simprl mulcld subaddd mpbird resubcld eqeltrrd rimul oveq2d
|
|
syl2anc mul01i syl6eq eqtrd subeq0d jca ) ADEZBDEZFZCDEZABGCHIZJIZKZFZFZBAK
|
|
CLKZVDABVDABVDAUPUQVCMZNZVDBUPUQVCOZNZVDABSIZUTLVDVJUTKVAAKVDAVAURUSVBPQVDA
|
|
BUTVGVIVDGCGREVDTUAVDCURUSVBUBZNUCUDUEZVDUTGLHILVDCLGHVDUSUTDEVEVKVDVJUTDVL
|
|
VDABVFVHUFUGCUHUJZUIGTUKULUMUNQVMUO $.
|
|
|
|
${
|
|
$d A r s t u x y $. $d B r s t u x y $.
|
|
|
|
$( Complex apartness and real apartness agree on the real numbers.
|
|
(Contributed by Jim Kingdon, 31-Jan-2020.) $)
|
|
apreap $p |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> A #RR B ) ) $=
|
|
( vr vs vt vu cr wcel wa wbr creap cv ci co caddc wceq wrex 2rexbidv cc0
|
|
wo vx vy cap cmul eqeq1 anbi1d anbi2d df-ap brabg simplll simplrl simplrr
|
|
wn adantr simprll rereim syl22anc simprd simpllr simprlr eqtr4d wb reapti
|
|
syl2anc mpbid simprr ecased simpld ex rexlimdvva sylbid w3a ax-icn mul01i
|
|
3brtr3d oveq2i simp1 recnd addid1d syl5req simp2 olc 3ad2ant3 orcomd 0red
|
|
jca31 simpr oveq2d eqeq2d breq2d orbi2d anbi12d oveq1d orbi1d syld breq1d
|
|
rspcedv rexbidv 3syld 3adant3 sylibrd mpd 3expia impbid ) AGHZBGHZIZABUCJ
|
|
ZABKJZXGXHACLZMDLZUDNZONZPZBELZMFLZUDNZONZPZIZXJXOKJZXKXPKJZTZIZFGQZEGQZD
|
|
GQZCGQZXIUALZXMPZUBLZXRPZIZYCIZFGQEGQZDGQCGQXNYLIZYCIZFGQEGQZDGQCGQYHUAUB
|
|
ABGGUCYIAPZYOYRCDGGYSYNYQEFGGYSYMYPYCYSYJXNYLYIAXMUEUFUFRRYKBPZYRYFCDGGYT
|
|
YQYDEFGGYTYPXTYCYTYLXSXNYKBXRUEUGUFRRUAUBFEDCUHUIZXGYFXICDGGXGXJGHZXKGHZI
|
|
ZIZYDXIEFGGUUEXOGHZXPGHZIZIZYDXIUUIYDIZXJXOABKUUJYAYBUUJXKXPPZYBUMZUUJXKS
|
|
XPUUJXJAPZXKSPZUUJXEUUBUUCXNUUMUUNIUUIXEYDXEXFUUDUUHUJUNUUIUUBYDXGUUBUUCU
|
|
UHUKUNUUIUUCYDXGUUBUUCUUHULUNZUUIXNXSYCUOAXJXKUPUQZURUUJXOBPZXPSPZUUJXFUU
|
|
FUUGXSUUQUURIUUIXFYDXEXFUUDUUHUSUNUUEUUFUUGYDUKUUEUUFUUGYDULZUUIXNXSYCUTB
|
|
XOXPUPUQZURVAUUJUUCUUGUUKUULVBUUOUUSXKXPVCVDVEUUIXTYCVFVGUUJUUMUUNUUPVHUU
|
|
JUUQUURUUTVHVOVIVJVJVKXEXFXIXHXEXFXIVLZAAMSUDNZONZPZBBUVBONZPZIZXISSKJZTZ
|
|
IZXHUVAUVDUVFUVIUVAUVCASONAUVBSAOMVMVNZVPUVAAUVAAXEXFXIVQZVRVSVTUVAUVEBSO
|
|
NBUVBSBOUVKVPUVABUVABXEXFXIWAZVRVSVTUVAUVHXIXIXEUVHXITXFXIUVHWBWCWDWFUVAU
|
|
VJYHXHUVAUVJUVDXSIZAXOKJZSXPKJZTZIZFGQZEGQZAAXLONZPZXSIZUVOYBTZIZFGQZEGQZ
|
|
DGQZYHUVAUVJUVDBBXQONZPZIZXIUVPTZIZFGQZUVTUVAUWMUVJFSGUVAWEZUVAUURIZUWKUV
|
|
GUWLUVIUWPUWJUVFUVDUWPUWIUVEBUWPXQUVBBOUWPXPSMUDUVAUURWGZWHWHWIUGUWPUVPUV
|
|
HXIUWPXPSSKUWQWJWKWLWQUVAUVSUWNEBGUVMUVAUUQIZUVRUWMFGUWRUVNUWKUVQUWLUWRXS
|
|
UWJUVDUWRXRUWIBUWRXOBXQOUVAUUQWGZWMWIUGUWRUVOXIUVPUWRXOBAKUWSWJWNWLWRWQWO
|
|
UVAUWGUVTDSGUWOUVAUUNIZUWEUVREFGGUWTUWCUVNUWDUVQUWTUWBUVDXSUWTUWAUVCAUWTX
|
|
LUVBAOUWTXKSMUDUVAUUNWGZWHWHWIUFUWTYBUVPUVOUWTXKSXPKUXAWPWKWLRWQUVAYGUWHC
|
|
AGUVLUVAUUMIZYEUWFDEGGUXBYDUWEFGUXBXTUWCYCUWDUXBXNUWBXSUXBXMUWAAUXBXJAXLO
|
|
UVAUUMWGZWMWIUFUXBYAUVOYBUXBXJAXOKUXCWPWNWLWRRWQWSXEXFXHYHVBXIUUAWTXAXBXC
|
|
XD $.
|
|
$}
|
|
|
|
$( Real apartness in terms of less than. Part of Definition 11.2.7(vi) of
|
|
[HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) $)
|
|
reaplt $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A # B <-> ( A < B \/ B < A ) ) ) $=
|
|
( cr wcel wa cap wbr creap clt wo apreap reapval bitrd ) ACDBCDEABFGABHGABI
|
|
GBAIGJABKABLM $.
|
|
|
|
$( Real apartness in terms of less than (exclusive-or version). (Contributed
|
|
by Jim Kingdon, 23-Mar-2020.) $)
|
|
reapltxor $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A # B <-> ( A < B \/_ B < A ) ) ) $=
|
|
( cr wcel wa cap wbr clt wo reaplt wn ltnsym2 biantrud df-xor syl6bbr bitrd
|
|
wxo ) ACDBCDEZABFGABHGZBAHGZIZSTQZABJRUAUASTEKZEUBRUCUAABLMSTNOP $.
|
|
|
|
$( One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
1ap0 $p |- 1 # 0 $=
|
|
( c1 cc0 cap wbr clt wo 0lt1 olci cr wcel wb 1re 0re reaplt mp2an mpbir ) A
|
|
BCDZABEDZBAEDZFZSRGHAIJBIJQTKLMABNOP $.
|
|
|
|
$( Multiplication of both sides of 'less than' by a positive number. Theorem
|
|
I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
ltmul1a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ A < B )
|
|
-> ( A x. C ) < ( B x. C ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co simpl2 simpl1 resubcld simpl3l posdifd
|
|
cmin recnd remulcld simpr mpbid simpl3r mulgt0d subdird breqtrd mpbird ) AD
|
|
EZBDEZCDEZFCGHZIZJZABGHZIZACKLZBCKLZGHFUQUPRLZGHUOFBARLZCKLURGUOUSCUOBAUHUI
|
|
ULUNMZUHUIULUNNZOUJUKUHUIUNPZUOUNFUSGHUMUNUAUOABVAUTQUBUJUKUHUIUNUCUDUOBACU
|
|
OBUTSUOAVASUOCVBSUEUFUOUPUQUOACVAVBTUOBCUTVBTQUG $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of
|
|
[HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
ltmul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
|
|
( vx cr wcel cc0 clt wbr wa cmul co ltmul1a ex c1 cc recnd adantr 3brtr3d
|
|
w3a cv wceq wrex recexgt0 3ad2ant3 simpl1 simpl3l remulcld simpl2 simprrl
|
|
wi simprl jca 3jca sylan mulassd simprrr oveq2d mulid1d rexlimddv impbid
|
|
) AEFZBEFZCEFZGCHIZJZTZABHIZACKLZBCKLZHIZVGVHVKABCMNVGGDUAZHIZCVLKLZOUBZJ
|
|
ZVKVHUKDEVFVBVPDEUCVCDCUDUEVGVLEFZVPJZJZVKVHVSVKJZAOKLZBOKLZABHVTAVNKLZBV
|
|
NKLZWAWBHVTVIVLKLZVJVLKLZWCWDHVSVIEFZVJEFZVQVMJZTVKWEWFHIVSWGWHWIVSACVBVC
|
|
VFVRUFZVDVEVBVCVRUGZUHVSBCVBVCVFVRUIZWKUHVSVQVMVGVQVPULZVGVQVMVOUJUMUNVIV
|
|
JVLMUOVTACVLVSAPFVKVSAWJQRZVSCPFVKVSCWKQRZVSVLPFVKVSVLWMQRZUPVTBCVLVSBPFV
|
|
KVSBWLQRZWOWPUPSVTVNOAKVSVOVKVGVQVMVOUQRZURVTVNOBKWRURSVTAWNUSVTBWQUSSNUT
|
|
VA $.
|
|
$}
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by NM, 21-Feb-2005.) $)
|
|
lemul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a wn cmul co cle wb ltmul1 notbid lenlt remulcld
|
|
3com12 3adant3 simp1 simp3l simp2 lenltd 3bitr4d ) ADEZBDEZCDEZFCGHZIZJZBAG
|
|
HZKZBCLMZACLMZGHZKZABNHZUPUONHUHUGUKUNUROUHUGUKJUMUQBACPQTUGUHUSUNOUKABRUAU
|
|
LUPUOULACUGUHUKUBUGUHUIUJUCZSULBCUGUHUKUDUTSUEUF $.
|
|
|
|
$( Lemma for ~ reapmul1 . (Contributed by Jim Kingdon, 8-Feb-2020.) $)
|
|
reapmul1lem $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( A # B <-> ( A x. C ) # ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a wo cmul co cap ltmul1 wb 3com12 orbi12d reaplt
|
|
remulcld 3adant3 simp1 simp3l simp2 syl2anc 3bitr4d ) ADEZBDEZCDEZFCGHZIZJZ
|
|
ABGHZBAGHZKZACLMZBCLMZGHZUQUPGHZKZABNHZUPUQNHZULUMURUNUSABCOUHUGUKUNUSPBACO
|
|
QRUGUHVAUOPUKABSUAULUPDEUQDEVBUTPULACUGUHUKUBUGUHUIUJUCZTULBCUGUHUKUDVCTUPU
|
|
QSUEUF $.
|
|
|
|
$( Multiplication of both sides of real apartness by a real number apart from
|
|
zero. Special case of ~ apmul1 . (Contributed by Jim Kingdon,
|
|
8-Feb-2020.) $)
|
|
reapmul1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C # 0 ) ) ->
|
|
( A # B <-> ( A x. C ) # ( B x. C ) ) ) $=
|
|
( cr wcel cc0 cap wbr wa cmul co wb clt reaplt cneg recnd mulneg2d renegcld
|
|
wo reapmul1lem 0re mpan2 pm5.32i simp1 simp3l simp2 breq12d simp3r lt0neg1d
|
|
mpbid syl112anc remulcld ltnegd orbi12d syl2anc orcom 3bitr4d 3expa anassrs
|
|
w3a syl6bb jaodan anasss sylan2b 3impa ) ADEZBDEZCDEZCFGHZIZABGHZACJKZBCJKZ
|
|
GHZLZVJVFVGIZVHCFMHZFCMHZSZIVOVHVIVSVHFDEVIVSLUACFNUBUCVPVHVSVOVPVHIVQVOVRV
|
|
PVHVQVOVFVGVHVQIZVOVFVGVTUTZACOZJKZBWBJKZGHZVLOZVMOZGHZVKVNWAWCWFWDWGGWAACW
|
|
AAVFVGVTUDZPWACVFVGVHVQUEZPZQWABCWABVFVGVTUFZPWKQUGWAVFVGWBDEFWBMHZVKWELWIW
|
|
LWACWJRWAVQWMVFVGVHVQUHWACWJUIUJABWBTUKWAVLVMMHZVMVLMHZSZWGWFMHZWFWGMHZSZVN
|
|
WHWAWNWQWOWRWAVLVMWAACWIWJULZWABCWLWJULZUMWAVMVLXAWTUMUNWAVLDEVMDEVNWPLWTXA
|
|
VLVMNUOWAWHWRWQSZWSWAWFDEWGDEWHXBLWAVLWTRWAVMXARWFWGNUOWRWQUPVAUQUQURUSVPVH
|
|
VRVOVFVGVHVRIVOABCTURUSVBVCVDVE $.
|
|
|
|
$( Real addition respects apartness. (Contributed by Jim Kingdon,
|
|
13-Feb-2020.) $)
|
|
reapadd1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A # B <-> ( A + C ) # ( B + C ) ) ) $=
|
|
( cr wcel w3a clt wbr wo caddc co cap ltadd1 wb 3com12 orbi12d reaplt simp1
|
|
3adant3 readdcld simp3 simp2 syl2anc 3bitr4d ) ADEZBDEZCDEZFZABGHZBAGHZIZAC
|
|
JKZBCJKZGHZUMULGHZIZABLHZULUMLHZUHUIUNUJUOABCMUFUEUGUJUONBACMOPUEUFUQUKNUGA
|
|
BQSUHULDEUMDEURUPNUHACUEUFUGRUEUFUGUAZTUHBCUEUFUGUBUSTULUMQUCUD $.
|
|
|
|
$( Real negation respects apartness. (Contributed by Jim Kingdon,
|
|
13-Feb-2020.) $)
|
|
reapneg $p |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) ) $=
|
|
( cr wcel wa cap wbr cneg clt wo reaplt ltneg wb ancoms orbi12d bitrd orcom
|
|
syl6bb simpl renegcld simpr syl2anc bitr4d ) ACDZBCDZEZABFGZAHZBHZIGZUIUHIG
|
|
ZJZUHUIFGZUFUGUKUJJZULUFUGABIGZBAIGZJUNABKUFUOUKUPUJABLUEUDUPUJMBALNOPUKUJQ
|
|
RUFUHCDUICDUMULMUFAUDUESTUFBUDUEUATUHUIKUBUC $.
|
|
|
|
$( Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT],
|
|
p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) $)
|
|
reapcotr $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A # B -> ( A # C \/ B # C ) ) ) $=
|
|
( cr w3a cap wbr clt wo wb reaplt 3adant3 axltwlin wi 3com12 orim12d sylbid
|
|
wcel orcom orbi2i or42 bitri syl6ib 3adant2 3adant1 orbi12d sylibrd ) ADRZB
|
|
DRZCDRZEZABFGZACHGZCAHGZIZBCHGZCBHGZIZIZACFGZBCFGZIUKULUMUQIZUPUNIZIZUSUKUL
|
|
ABHGZBAHGZIZVDUHUIULVGJUJABKLUKVEVBVFVCABCMUIUHUJVFVCNBACMOPQVDVBUNUPIZIUSV
|
|
CVHVBUPUNSTUMUQUNUPUAUBUCUKUTUOVAURUHUJUTUOJUIACKUDUIUJVAURJUHBCKUEUFUG $.
|
|
|
|
$( Left extensionality for multiplication. (Contributed by Jim Kingdon,
|
|
19-Feb-2020.) $)
|
|
remulext1 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> (
|
|
( A x. C ) # ( B x. C ) -> A # B ) ) $=
|
|
( cr wcel w3a cmul co cap wbr clt wo wb remulcld reaplt cltrr ax-pre-mulext
|
|
syl2anc ltxrlt 3imtr4d simp1 simp3 simp2 orbi12d bitrd wi 3com12 orcom jaod
|
|
syl6bb sylbid ) ADEZBDEZCDEZFZACGHZBCGHZIJZUPUQKJZUQUPKJZLZABIJZUOUPDEZUQDE
|
|
ZURVAMUOACULUMUNUAZULUMUNUBZNZUOBCULUMUNUCZVFNZUPUQORUOUSVBUTUOUPUQPJZABPJZ
|
|
BAPJZLZUSVBABCQUOVCVDUSVJMVGVIUPUQSRUOVBABKJZBAKJZLZVMUOULUMVBVPMVEVHABORUO
|
|
VNVKVOVLUOULUMVNVKMVEVHABSRUOUMULVOVLMVHVEBASRUDUEZTUOUQUPPJZVLVKLZUTVBUMUL
|
|
UNVRVSUFBACQUGUOVDVCUTVRMVIVGUQUPSRUOVBVMVSVQVKVLUHUJTUIUK $.
|
|
|
|
$( Right extensionality for real multiplication. (Contributed by Jim
|
|
Kingdon, 22-Feb-2020.) $)
|
|
remulext2 $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> (
|
|
( C x. A ) # ( C x. B ) -> A # B ) ) $=
|
|
( cr wcel w3a cmul co cap simp1 recnd simp3 mulcomd simp2 breq12d remulext1
|
|
wbr sylbird ) ADEZBDEZCDEZFZCAGHZCBGHZIQACGHZBCGHZIQABIQUBUEUCUFUDIUBACUBAS
|
|
TUAJKUBCSTUALKZMUBBCUBBSTUANKUGMOABCPR $.
|
|
|
|
$( The square of a real number apart from zero is positive. (Contributed by
|
|
Jim Kingdon, 7-Feb-2020.) $)
|
|
apsqgt0 $p |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) ) $=
|
|
( cr wcel cc0 cap wbr wa clt wo cmul co wb 0re reaplt pm5.32i mullt0 anidms
|
|
mpan2 mulgt0 jaodan sylbi ) ABCZADEFZGUBADHFZDAHFZIZGDAAJKHFZUBUCUFUBDBCUCU
|
|
FLMADNROUBUDUGUEUBUDGUGAAPQUBUEGUGAASQTUA $.
|
|
|
|
$( The representation of complex numbers in terms of real and imaginary parts
|
|
is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM,
|
|
9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
cru $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc wceq simplrl simplll cmin cc0 mulcld resubcld
|
|
recnd subeq0d simpr cc ax-icn a1i simpllr simplrr addsubeq4d subdid eqeltrd
|
|
mpbid eqtr4d rimul syl2anc oveq2d oveq1d subidd 3eqtrd eqcomd jca ex oveq12
|
|
oveq2 sylan2 impbid1 ) AEFZBEFZGZCEFZDEFZGZGZAHBIJZKJCHDIJZKJLZACLZBDLZGZVK
|
|
VNVQVKVNGZVOVPVRCAVRCAVRCVGVHVIVNMZSZVRAVEVFVJVNNZSZVRCAOJZVLVMOJZVMVMOJPVR
|
|
VNWCWDLVKVNUAVRAVLCVMWBVRHBHUBFVRUCUDZVRBVEVFVJVNUEZSZQVTVRHDWEVRDVGVHVIVNU
|
|
FZSZQZUGUJZVRVLVMVMOVRBDHIVRBDWGWIVRBDOJZEFHWLIJZEFWLPLVRBDWFWHRVRWMWCEVRWM
|
|
WDWCVRHBDWEWGWIUHWKUKVRCAVSWARUIWLULUMTZUNUOVRVMWJUPUQTURWNUSUTVPVOVLVMLVNB
|
|
DHIVBACVLVMKVAVCVD $.
|
|
|
|
${
|
|
$d A r s t u x y $. $d B r s t u x y $. $d C r s t u x y $.
|
|
$d D r s t u x y $.
|
|
$( Complex apartness in terms of real and imaginary parts. (Contributed by
|
|
Jim Kingdon, 12-Feb-2020.) $)
|
|
apreim $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> ( A # C \/ B # D ) )
|
|
) $=
|
|
( vr vs vt vu cr wcel wa ci cmul co caddc wbr wo wceq creap wrex vx vy cv
|
|
cc wb simpll recnd ax-icn simplr mulcld addcld simprl simprr eqeq1 anbi1d
|
|
cap 2rexbidv anbi2d df-ap brabg syl2anc ad3antrrr orbi12d simprll simpllr
|
|
a1i apreap syl21anc mpbid simpld simprlr breq12d simprd mpbird rexlimdvva
|
|
cru bitrd ex sylbid ad2ant2r ad2ant2l pm5.32i eqid pm3.2i biantrur eqeq2d
|
|
oveq1 breq2 orbi1d anbi12d oveq2d orbi2d rspc2ev syl3an3b syl3an3 anassrs
|
|
oveq2 3expa breq1 adantr sylbi impbid ) AIJZBIJZKZCIJZDIJZKZKZALBMNZONZCL
|
|
DMNZONZUPPZACUPPZBDUPPZQZXIXNXKEUCZLFUCZMNZONZRZXMGUCZLHUCZMNZONZRZKZXRYC
|
|
SPZXSYDSPZQZKZHITGITZFITEITZXQXIXKUDJXMUDJXNYNUEZXIAXJXIAXCXDXHUFZUGXILBL
|
|
UDJXIUHVFZXIBXCXDXHUIZUGUJUKXICXLXICXEXFXGULZUGXILDYQXIDXEXFXGUMZUGUJUKUA
|
|
UCZYARZUBUCZYFRZKZYKKZHITGITZFITEITYBUUDKZYKKZHITGITZFITEITYNUAUBXKXMUDUD
|
|
UPUUAXKRZUUGUUJEFIIUUKUUFUUIGHIIUUKUUEUUHYKUUKUUBYBUUDUUAXKYAUNUOUOUQUQUU
|
|
CXMRZUUJYMEFIIUULUUIYLGHIIUULUUHYHYKUULUUDYGYBUUCXMYFUNURUOUQUQUAUBHGFEUS
|
|
UTVAZXIYMXQEFIIXIXRIJXSIJKZKZYLXQGHIIUUOYCIJYDIJKZKZYLXQUUQYLKZXQYKUUQYHY
|
|
KUMUURXQACSPZBDSPZQZYKUURXOUUSXPUUTUURXCXFXOUUSUEZXIXCUUNUUPYLYPVBZXIXFUU
|
|
NUUPYLYSVBZACVGZVAUURXDXGXPUUTUEZXIXDUUNUUPYLYRVBZXIXGUUNUUPYLYTVBZBDVGZV
|
|
AVCUURUUSYIUUTYJUURAXRCYCSUURAXRRZBXSRZUURYBUVJUVKKZUUQYBYGYKVDUURXCXDUUN
|
|
YBUVLUEUVCUVGXIUUNUUPYLVEABXRXSVPVHVIZVJUURCYCRZDYDRZUURYGUVNUVOKZUUQYBYG
|
|
YKVKUURXFXGUUPYGUVPUEUVDUVHUUOUUPYLUICDYCYDVPVHVIZVJVLUURBXSDYDSUURUVJUVK
|
|
UVMVMUURUVNUVOUVQVMVLVCVQVNVRVOVOVSXIXQXNXIXQKXIUVAKZXNXIXQUVAXIXOUUSXPUU
|
|
TXCXFUVBXDXGUVEVTXDXGUVFXCXFUVIWAVCWBUVRXNYNXEXHUVAYNXCXDXHUVAKZYNUVSXCXD
|
|
XKXKRZYGKZAYCSPZBYDSPZQZKZHITGITZYNXFXGUVAUWFUVAXFXGUVTXMXMRZKZUVAKZUWFUW
|
|
HUVAUVTUWGXKWCXMWCWDWEUWEUWIUVTXMCYEONZRZKZUUSUWCQZKGHCDIIYCCRZUWAUWLUWDU
|
|
WMUWNYGUWKUVTUWNYFUWJXMYCCYEOWGWFURUWNUWBUUSUWCYCCASWHWIWJYDDRZUWLUWHUWMU
|
|
VAUWOUWKUWGUVTUWOUWJXMXMUWOYEXLCOYDDLMWQWKWFURUWOUWCUUTUUSYDDBSWHWLWJWMWN
|
|
WRYMUWFXKAXTONZRZYGKZUWBYJQZKZHITGITEFABIIXRARZYLUWTGHIIUXAYHUWRYKUWSUXAY
|
|
BUWQYGUXAYAUWPXKXRAXTOWGWFUOUXAYIUWBYJXRAYCSWSWIWJUQXSBRZUWTUWEGHIIUXBUWR
|
|
UWAUWSUWDUXBUWQUVTYGUXBUWPXKXKUXBXTXJAOXSBLMWQWKWFUOUXBYJUWCUWBXSBYDSWSWL
|
|
WJUQWMWOWRWPXIYOUVAUUMWTVNXAVRXB $.
|
|
$}
|
|
|
|
$( Complex multiplication in terms of real and imaginary parts. (Contributed
|
|
by Jim Kingdon, 23-Feb-2020.) $)
|
|
mulreim $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) ->
|
|
( ( A + ( _i x. B ) ) x. ( C + ( _i x. D ) ) ) =
|
|
( ( ( A x. C ) + -u ( B x. D ) ) +
|
|
( _i x. ( ( C x. B ) + ( D x. A ) ) ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc cneg simpll recnd mulcld mulcomd oveq2d eqtrd
|
|
mul12d oveq12d cc ax-icn a1i simplr simprl simprr muladdd ixi oveq1i syl6eq
|
|
c1 mul4d mulm1d negeqd 3eqtrd addcomd adddid 3eqtr4d ) AEFZBEFZGZCEFZDEFZGZ
|
|
GZAHBIJZKJCHDIJZKJIJACIJZVGVFIJZKJZAVGIJZCVFIJZKJZKJVHBDIJZLZKJZHCBIJZDAIJZ
|
|
KJIJZKJVEAVFCVGVEAUSUTVDMNZVEHBHUAFVEUBUCZVEBUSUTVDUDNZOVECVAVBVCUENZVEHDWA
|
|
VEDVAVBVCUFNZOUGVEVJVPVMVSKVEVIVOVHKVEVIUKLZDBIJZIJZWFLVOVEVIHHIJZWFIJWGVEH
|
|
DHBWAWDWAWBULWHWEWFIUHUIUJVEWFVEDBWDWBOUMVEWFVNVEDBWDWBPUNUOQVEHVRIJZHVQIJZ
|
|
KJWJWIKJVMVSVEWIWJVEHVRWAVEDAWDVTOZOVEHVQWAVECBWCWBOZOUPVEVKWIVLWJKVEVKHADI
|
|
JZIJWIVEAHDVTWAWDSVEWMVRHIVEADVTWDPQRVECHBWCWAWBSTVEHVQVRWAWLWKUQURTR $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) $)
|
|
apirr $p |- ( A e. CC -> -. A # A ) $=
|
|
( vx vy cc wcel cv ci co cr wrex cap wn wa creap reapirr wb apreap anidms
|
|
wbr mtbird cmul caddc wceq cnre wo anim12i sylibr apreim ad2antlr breq12d
|
|
ioran id notbid adantl mpbird ex rexlimdvva mpd ) ADEZABFZGCFZUAHUBHZUCZC
|
|
IJBIJAAKSZLZBCAUDUSVCVEBCIIUSUTIEZVAIEZMZMZVCVEVIVCMVEVBVBKSZLZVHVKUSVCVH
|
|
VJUTUTKSZVAVAKSZUEZVHVLLZVMLZMVNLVFVOVGVPVFVLUTUTNSZUTOVFVLVQPUTUTQRTVGVM
|
|
VAVANSZVAOVGVMVRPVAVAQRTUFVLVMUKUGVHVJVNPUTVAUTVAUHRTUIVCVEVKPVIVCVDVJVCA
|
|
VBAVBKVCULZVSUJUMUNUOUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $.
|
|
$( Apartness is symmetric. This theorem for real numbers is part of
|
|
Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim
|
|
Kingdon, 16-Feb-2020.) $)
|
|
apsym $p |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) ) $=
|
|
( vz vw vx vy cc wcel wa cv co cr wrex cap wbr wb wo clt reaplt syl2anc
|
|
ci cmul caddc wceq adantl ad3antrrr simplrl ad2antrr orcom syl6bbr bitr4d
|
|
cnre simplrr orbi12d syl22anc 3bitr4d simpr simpllr breq12d ex rexlimdvva
|
|
apreim mpd ) AGHZBGHZIZBCJZUADJZUBKUCKZUDZDLMCLMZABNOZBANOZPZVEVKVDCDBULU
|
|
EVFVJVNCDLLVFVGLHZVHLHZIZIZVJVNVRVJIZAEJZUAFJZUBKUCKZUDZFLMELMZVNVDWDVEVQ
|
|
VJEFAULUFVSWCVNEFLLVSVTLHZWALHZIZIZWCVNWHWCIZWBVINOZVIWBNOZVLVMWIVTVGNOZW
|
|
AVHNOZQZVGVTNOZVHWANOZQZWJWKWIWLWOWMWPWIWLVTVGROZVGVTROZQZWOWIWEVOWLWTPVS
|
|
WEWFWCUGZVSVOWGWCVFVOVPVJUGUHZVTVGSTWIWOWSWRQZWTWIVOWEWOXCPXBXAVGVTSTWRWS
|
|
UIUJUKWIWMWAVHROZVHWAROZQZWPWIWFVPWMXFPVSWEWFWCUMZVSVPWGWCVFVOVPVJUMUHZWA
|
|
VHSTWIWPXEXDQZXFWIVPWFWPXIPXHXGVHWASTXDXEUIUJUKUNWIWEWFVOVPWJWNPXAXGXBXHV
|
|
TWAVGVHVBUOWIVOVPWEWFWKWQPXBXHXAXGVGVHVTWAVBUOUPWIAWBBVINWHWCUQZVRVJWGWCU
|
|
RZUSWIBVIAWBNXKXJUSUPUTVAVCUTVAVC $.
|
|
$}
|
|
|
|
${
|
|
$d A u v w x y z $. $d B u v w x y z $. $d C u v w x y z $.
|
|
$( Apartness is cotransitive. (Contributed by Jim Kingdon,
|
|
16-Feb-2020.) $)
|
|
apcotr $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( A # B -> ( A # C \/ B # C ) ) ) $=
|
|
( vu vv vz vw vx vy cc wcel cv co cr wrex cap wbr wo wa ad3antrrr ci cmul
|
|
w3a caddc wceq wi cnre 3ad2ant3 3ad2ant2 ad2antrr 3ad2ant1 adantr simpllr
|
|
simpr breq12d simplrl simplrr simprl simprr apreim syl22anc bitrd syl3anc
|
|
reapcotr orim12d sylbid or4 syl6ib simplr orbi12d sylibrd rexlimdvva mpd
|
|
wb ex ) AJKZBJKZCJKZUCZCDLZUAELZUBMUDMZUEZENODNOZABPQZACPQZBCPQZRZUFZVRVP
|
|
WDVQDECUGUHVSWCWIDENNVSVTNKZWANKZSZSZWCWIWMWCSZBFLZUAGLZUBMUDMZUEZGNOFNOZ
|
|
WIVSWSWLWCVQVPWSVRFGBUGUIUJWNWRWIFGNNWNWONKZWPNKZSZSZWRWIXCWRSZAHLZUAILZU
|
|
BMUDMZUEZINOHNOZWIWMXIWCXBWRVSXIWLVPVQXIVRHIAUGUKULTXDXHWIHINNXDXENKZXFNK
|
|
ZSZSZXHWIXMXHSZWEXEVTPQZXFWAPQZRZWOVTPQZWPWAPQZRZRZWHXNWEXOXRRZXPXSRZRZYA
|
|
XNWEXEWOPQZXFWPPQZRZYDXNWEXGWQPQZYGXNAXGBWQPXMXHUNZXCWRXLXHUMZUOXNXJXKWTX
|
|
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|
|
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|
|
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|
|
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YNYOYPWOWPVTWAUTVAVBVJVKVOVLVMVOVLVMVOVLVM $.
|
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$}
|
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|
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(Contributed by Jim Kingdon, 13-Feb-2020.) $)
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|
|
ZBUAGZHUBGZIJZKJZUFZUBLMUALMZWMWBXDWPWGVTVSXDWAUAUBBUGUIRWRXCWMUAUBLLWRWS
|
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|
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|
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|
|
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|
|
JZWSWCKJZHWTWDKJZIJZKJZNOZYKYONOZYLYPNOZUNZWLYDXTYKLFYLLFYOLFYPLFYSUUBPXT
|
|
XJWCYEXHWNXCXRXNWQWNWGXGWBWNWOUORSZVAXTXKWDYFXHWOXCXRXNWQWOWGXGWBWNWOUPRS
|
|
ZVAXTWSWCYGUUCVAXTWTWDYHUUDVAYKYLYOYPUQURXTWJYNWKYRNXTXMWFKJYKXLWEKJZKJWJ
|
|
YNXTXJXLWCWEXTXJYETXTHXKHUEFXTVBVEZXTXKYFTZVCXTWCUUCTZXTHWDUUFXTWDUUDTZVC
|
|
ZVNXTAXMCWFKYIXHWGXCXRXNWQWGXGVFSZVGXTYMUUEYKKXTHXKWDUUFUUGUUIVHVIVJXTXBW
|
|
FKJYOXAWEKJZKJWKYRXTWSXAWCWEXTWSYGTXTHWTUUFXTWTYHTZVCUUHUUJVNXTBXBCWFKYJU
|
|
UKVGXTYQUULYOKXTHWTWDUUFUUMUUIVHVIVJVDXTYBYTYCUUAXTXPXEWNYBYTPYEYGUUCXJWS
|
|
WCVKVLXTXQXFWOYCUUAPYFYHUUDXKWTWDVKVLVMVOVOVPVQVRVPVQVRVPVQVR $.
|
|
$}
|
|
|
|
$( Addition respects apartness. (Contributed by Jim Kingdon,
|
|
16-Feb-2020.) $)
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apadd2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
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( A # B <-> ( C + A ) # ( C + B ) ) ) $=
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( cc wcel w3a cap wbr caddc apadd1 simp1 simp3 addcomd simp2 breq12d bitrd
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co ) ADEZBDEZCDEZFZABGHACIQZBCIQZGHCAIQZCBIQZGHABCJUAUBUDUCUEGUAACRSTKRSTLZ
|
|
MUABCRSTNUFMOP $.
|
|
|
|
$( Strong extensionality for addition. Given excluded middle, apartness
|
|
would be equivalent to negated equality and this would follow readily (for
|
|
all operations) from ~ oveq12 . For us, it is proved a different way.
|
|
(Contributed by Jim Kingdon, 15-Feb-2020.) $)
|
|
addext $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> (
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( A + B ) # ( C + D ) -> ( A # C \/ B # D ) ) ) $=
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( cc wcel wa caddc co cap wbr wo simpll simplr addcld simprl simprr syl3anc
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|
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|
|
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|
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|
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${
|
|
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|
|
14-Feb-2020.) $)
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|
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|
|
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|
|
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|
|
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|
|
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|
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$}
|
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|
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|
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Kingdon, 22-Feb-2020.) $)
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|
|
KULXIXMXDUCUDJJXIXJJEZXKJEZOZOZXMXDXRXMOZAUEFZUGUFFZGHIHZUHZUFJKUEJKZXDXH
|
|
YDWRXQXMWNYDXGWKWLYDWMUEUFAUIUNUOPXSYCXDUEUFJJXSXTJEZYAJEZOZOZYCXDYHYCOZX
|
|
BYBXLLMZXCYIXTWOGHZYAWPGHZUPZIHZXJWOGHZXKWPGHZUPZIHZLMZWOYAGHZWPXTGHZIHZW
|
|
OXKGHZWPXJGHZIHZLMZUQZXTXJLMZYAXKLMZUQZXBYJYIYSUUJUUFYIYSYKYOLMZYMYQLMZUQ
|
|
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|
|
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|
|
LPZQZSZVAYIXJWOYIXJXRXOXMYGYCXIXOXPUSPZQZUUQSYIYPYIXKWPYIXKXRXPXMYGYCXIXO
|
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XPUTPZQZUVASZVAYKYMYOYQVBRYIUUKUUHUULUUIYIYEXOXEUUKUUHNUUNUVCUUPXTXJWOVCV
|
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|
|
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|
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|
|
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|
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NUVCUUTXTXJWPVKVEVIVJUUIUUHVLVMVNYIXBYBWQGHZXLWQGHZLMYNUGUUBGHIHZYRUGUUEG
|
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HIHZLMZUUGYIWTUVMXAUVNLYIAYBCWQGYHYCVOZXRWRXMYGYCXHWRXQVPPZVQYIBXLCWQGXRX
|
|
MYGYCVRZUVSVQVSYIUVMUVOUVNUVPLYIYEYFXEXFUVMUVOUHUUNUURUUPUUTXTYAWOWPWERYI
|
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XOXPXEXFUVNUVPUHUVCUVEUUPUUTXJXKWOWPWERVSYIYNJEUUBJEYRJEUUEJEUVQUUGWDYIYK
|
|
YMYIXTWOUUNUUPTYIYLYIYAWPUURUUTTVTWAYIYTUUAYIWOYAUUPUURTYIWPXTUUTUUNTWAYI
|
|
YOYQYIXJWOUVCUUPTYIYPYIXKWPUVEUUTTVTWAYIUUCUUDYIWOXKUUPUVETYIWPXJUUTUVCTW
|
|
AYNUUBYRUUEWBRWCYIYEYFXOXPYJUUJWDUUNUURUVCUVEXTYAXJXKWBRWFYIAYBBXLLUVRUVT
|
|
VSWGWHWIWJWHWIWJWHWIWJ $.
|
|
$}
|
|
|
|
$( Right extensionality for complex multiplication. (Contributed by Jim
|
|
Kingdon, 22-Feb-2020.) $)
|
|
mulext2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> (
|
|
( C x. A ) # ( C x. B ) -> A # B ) ) $=
|
|
( cc wcel w3a cmul cap wbr simp3 simp1 mulcomd simp2 breq12d mulext1 sylbid
|
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co ) ADEZBDEZCDEZFZCAGQZCBGQZHIACGQZBCGQZHIABHIUAUBUDUCUEHUACARSTJZRSTKLUAC
|
|
BUFRSTMLNABCOP $.
|
|
|
|
$( Strong extensionality for multiplication. Given excluded middle,
|
|
apartness would be equivalent to negated equality and this would follow
|
|
readily (for all operations) from ~ oveq12 . For us, it is proved a
|
|
different way. (Contributed by Jim Kingdon, 23-Feb-2020.) $)
|
|
mulext $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> (
|
|
( A x. B ) # ( C x. D ) -> ( A # C \/ B # D ) ) ) $=
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( cc wcel wa cmul cap wbr simpll simplr mulcld simprl simprr apcotr syl3anc
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co wo wi mulext1 mulext2 wb apsym syl2anc sylibd orim12d syld ) AEFZBEFZGZC
|
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EFZDEFZGZGZABHRZCDHRZIJZUPCBHRZIJZUQUSIJZSZACIJZBDIJZSUOUPEFUQEFUSEFURVBTUO
|
|
ABUIUJUNKZUIUJUNLZMUOCDUKULUMNZUKULUMOZMUOCBVGVFMUPUQUSPQUOUTVCVAVDUOUIULUJ
|
|
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|
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UFUGUH $.
|
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|
|
$( A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed
|
|
by Jim Kingdon, 24-Feb-2020.) $)
|
|
mulap0r $p |- ( ( A e. CC /\ B e. CC /\ ( A x. B ) # 0 ) ->
|
|
( A # 0 /\ B # 0 ) ) $=
|
|
( cc wcel cmul co cc0 cap wbr wo breqtrrd wi mulext syl22anc wn apirr biorf
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|
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|
|
$( A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by
|
|
NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
msqge0 $p |- ( A e. RR -> 0 <_ ( A x. A ) ) $=
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wa wo orc wb reaplt syl5ibr wi w3a cc mulap0r syl3an1 syl3an2 simpld 3expia
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CVERZVCVESUTVGVHVJVLTVIMVADUAOUBUTVJVKUCUTUTVJVKUTUTVJUDVKVKUTUTAUECZVJVKVK
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|
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|
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${
|
|
msqge0i.1 $e |- A e. RR $.
|
|
$( A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof
|
|
shortened by Andrew Salmon, 19-Nov-2011.) $)
|
|
msqge0i $p |- 0 <_ ( A x. A ) $=
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( cr wcel cc0 cmul co cle wbr msqge0 ax-mp ) ACDEAAFGHIBAJK $.
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$}
|
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|
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${
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msqge0d.1 $e |- ( ph -> A e. RR ) $.
|
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$( A square is nonnegative. (Contributed by Mario Carneiro,
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27-May-2016.) $)
|
|
msqge0d $p |- ( ph -> 0 <_ ( A x. A ) ) $=
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( cr wcel cc0 cmul co cle wbr msqge0 syl ) ABDEFBBGHIJCBKL $.
|
|
$}
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|
|
$( The product of two nonnegative numbers is nonnegative. (Contributed by
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|
NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
mulge0 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\
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|
( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) ) $=
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( cr wcel cc0 cle wbr wa clt wn ad2ant2r 0re sylancl cap wo wb reaplt lenlt
|
|
cc mpbird co remulcl ltnsym2 orc syl5ibr simplll simplrl wi mulap0r syl3an1
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cmul recn syl3an2 3expia imp simpld mpan2 ad3antrrr mpbid mpan biimpa biorf
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ad2antrr simprd ad2antrl adantr ad2antlr mulgt0d ex syld ancld mtod sylancr
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syl ) ACDZEAFGZHZBCDZEBFGZHZHZEABUKUAZFGZWBEIGZJZWAWDWDEWBIGZHZWAWBCDZECDZW
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WBEQMUEWAWKWFWAWKHZABVOVPVTWKUFVQVRVSWKUGWMEAIGZAEIGZWNOZWMAENGZWPWMWQBENGZ
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CWOWNVBVNTWMEBIGZBEIGZXDOZWMWRXFWMWQWRXAVDWAWRXFPZWKVRXGVQVSVRWIXGLBEQUQVEV
|
|
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|
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|
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|
|
${
|
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mulge0i.1 $e |- A e. RR $.
|
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mulge0i.2 $e |- B e. RR $.
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$( The product of two nonnegative numbers is nonnegative. (Contributed by
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|
NM, 30-Jul-1999.) $)
|
|
mulge0i $p |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A x. B ) ) $=
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( cr wcel cc0 cle wbr wa cmul co mulge0 an4s mpanl12 ) AEFZBEFZGAHIZGBHIZ
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$}
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mulge0d.2 $e |- ( ph -> B e. RR ) $.
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mulge0d.3 $e |- ( ph -> 0 <_ A ) $.
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mulge0d.4 $e |- ( ph -> 0 <_ B ) $.
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|
$( The product of two nonnegative numbers is nonnegative. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
mulge0d $p |- ( ph -> 0 <_ ( A x. B ) ) $=
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|
( cr wcel cc0 cle wbr cmul co mulge0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLDFE
|
|
GBCOP $.
|
|
$}
|
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|
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${
|
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$d A w x y z $. $d B w x y z $.
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|
$( Complex apartness is tight. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
apti $p |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) ) $=
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( vx vy vz vw cc wcel wa cv co wceq cr wrex cap wbr wn wb syl2anc notbid
|
|
cmul caddc cnre adantr adantl ad2antrr simpr ad3antrrr simplr cru simpllr
|
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ci eqeq12d apreim ioran syl6bb breq12d simpld reapti apreap bitr4d simprd
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wo creap anbi12d 3bitr4d ex rexlimdvva mpd ) AGHZBGHZIZACJZULDJZUAKUBKZLZ
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DMNCMNZABLZABOPZQZRZVJVQVKCDAUCUDVLVPWACDMMVLVMMHZVNMHZIZIZVPWAWEVPIZBEJZ
|
|
ULFJZUAKUBKZLZFMNEMNZWAVLWKWDVPVKWKVJEFBUCUEUFWFWJWAEFMMWFWGMHZWHMHZIZIZW
|
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JWAWOWJIZVOWILZVMWGLZVNWHLZIZVRVTWPWDWNWQWTRWEWDVPWNWJVLWDUGUHZWFWNWJUIZV
|
|
MVNWGWHUJSWPAVOBWIWEVPWNWJUKZWOWJUGZUMWPVOWIOPZQZVMWGOPZQZVNWHOPZQZIZVTWT
|
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WPWDWNXFXKRXAXBWDWNIZXFXGXIVCZQXKXLXEXMVMVNWGWHUNTXGXIUOUPSWPVSXEWPAVOBWI
|
|
OXCXDUQTWPWRXHWSXJWPWBWLWRXHRWPWBWCXAURWPWLWMXBURWBWLIZWRVMWGVDPZQXHVMWGU
|
|
SXNXGXOVMWGUTTVASWPWCWMWSXJRWPWBWCXAVBWPWLWMXBVBWCWMIZWSVNWHVDPZQXJVNWHUS
|
|
XPXIXQVNWHUTTVASVEVFVFVGVHVIVGVHVI $.
|
|
$}
|
|
|
|
$( Apartness implies negated equality. We cannot in general prove the
|
|
converse, which is the whole point of having separate notations for
|
|
apartness and negated equality. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
apne $p |- ( ( A e. CC /\ B e. CC ) -> ( A # B -> A =/= B ) ) $=
|
|
( cc wcel wa cap wbr wceq wn apti biimpd necon2ad ) ACDBCDEZABFGZABMABHNIAB
|
|
JKL $.
|
|
|
|
$( '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon,
|
|
13-Aug-2021.) $)
|
|
leltap $p |- ( ( A e. RR /\ B e. RR /\ A <_ B ) ->
|
|
( A < B <-> B # A ) ) $=
|
|
( cr wcel cle wbr w3a clt wo cap wn wb simp3 simp1 simp2 lenltd mpbid biorf
|
|
syl reaplt syl2anc bitr4d ) ACDZBCDZABEFZGZABHFZBAHFZUGIZBAJFZUFUHKZUGUILUF
|
|
UEUKUCUDUEMUFABUCUDUENZUCUDUEOZPQUHUGRSUFUDUCUJUILUMULBATUAUB $.
|
|
|
|
$( Positive implies apart from zero. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
gt0ap0 $p |- ( ( A e. RR /\ 0 < A ) -> A # 0 ) $=
|
|
( cr wcel cc0 clt wbr wa cap wo simpr olcd wb 0red reaplt syldan mpbird ) A
|
|
BCZDAEFZGZADHFZADEFZRIZSRUAQRJKQRDBCTUBLSMADNOP $.
|
|
|
|
${
|
|
gt0ap0i.1 $e |- A e. RR $.
|
|
$( Positive means apart from zero (useful for ordering theorems involving
|
|
division). (Contributed by Jim Kingdon, 27-Feb-2020.) $)
|
|
gt0ap0i $p |- ( 0 < A -> A # 0 ) $=
|
|
( cr wcel cc0 clt wbr cap gt0ap0 mpan ) ACDEAFGAEHGBAIJ $.
|
|
|
|
${
|
|
gt0ap0i.2 $e |- 0 < A $.
|
|
$( Positive implies apart from zero. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
gt0ap0ii $p |- A # 0 $=
|
|
( cc0 clt wbr cap gt0ap0i ax-mp ) DAEFADGFCABHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
gt0ap0d.1 $e |- ( ph -> A e. RR ) $.
|
|
gt0ap0d.2 $e |- ( ph -> 0 < A ) $.
|
|
$( Positive implies apart from zero. Because of the way we define ` # ` ,
|
|
` A ` must be an element of ` RR ` , not just ` RR* ` . (Contributed by
|
|
Jim Kingdon, 27-Feb-2020.) $)
|
|
gt0ap0d $p |- ( ph -> A # 0 ) $=
|
|
( cr wcel cc0 clt wbr cap gt0ap0 syl2anc ) ABEFGBHIBGJICDBKL $.
|
|
$}
|
|
|
|
$( A number is apart from zero iff its negative is apart from zero.
|
|
(Contributed by Jim Kingdon, 27-Feb-2020.) $)
|
|
negap0 $p |- ( A e. CC -> ( A # 0 <-> -u A # 0 ) ) $=
|
|
( cc wcel cc0 cap wbr cneg wb 0cn apneg mpan2 neg0 breq2i syl6bb ) ABCZADEF
|
|
ZAGZDGZEFZQDEFODBCPSHIADJKRDQELMN $.
|
|
|
|
$( Less than in terms of non-strict order and apartness. (Contributed by Jim
|
|
Kingdon, 28-Feb-2020.) $)
|
|
ltleap $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A < B <-> ( A <_ B /\ A # B ) ) ) $=
|
|
( cr wcel wa clt wbr cle cap ltle wo orc reaplt syl5ibr wn simprl wb adantr
|
|
jcad mpbid lenlt simprr ecased ex impbid ) ACDBCDEZABFGZABHGZABIGZEZUFUGUHU
|
|
IABJUGUIUFUGBAFGZKZUGUKLABMZNSUFUJUGUFUJEZUGUKUNUHUKOZUFUHUIPUFUHUOQUJABUAR
|
|
TUNUIULUFUHUIUBUFUIULQUJUMRTUCUDUE $.
|
|
|
|
$( 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) $)
|
|
ltap $p |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A ) $=
|
|
( cr wcel clt wbr w3a cap wo simp3 olcd simp2 simp1 reaplt syl2anc mpbird
|
|
wb ) ACDZBCDZABEFZGZBAHFZBAEFZTIZUATUCRSTJKUASRUBUDQRSTLRSTMBANOP $.
|
|
|
|
${
|
|
ltapii.a $e |- A e. RR $.
|
|
ltapii.b $e |- B e. RR $.
|
|
${
|
|
ltapii.lt $e |- A < B $.
|
|
$( 'Greater than' implies apart. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
gtapii $p |- B # A $=
|
|
( cr wcel clt wbr cap ltap mp3an ) AFGBFGABHIBAJICDEABKL $.
|
|
|
|
$( 'Less than' implies apart. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
ltapii $p |- A # B $=
|
|
( cap wbr gtapii cc wcel wb recni apsym mp2an mpbi ) BAFGZABFGZABCDEHBI
|
|
JAIJPQKBDLACLBAMNO $.
|
|
$}
|
|
|
|
$( 'Less than' implies apart. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
ltapi $p |- ( A < B -> B # A ) $=
|
|
( cr wcel clt wbr cap ltap mp3an12 ) AEFBEFABGHBAIHCDABJK $.
|
|
$}
|
|
|
|
${
|
|
ltapd.a $e |- ( ph -> A e. RR ) $.
|
|
ltapd.b $e |- ( ph -> B e. RR ) $.
|
|
|
|
${
|
|
ltapd.lt $e |- ( ph -> A < B ) $.
|
|
$( 'Greater than' implies apart. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
gtapd $p |- ( ph -> B # A ) $=
|
|
( cr wcel clt wbr cap ltap syl3anc ) ABGHCGHBCIJCBKJDEFBCLM $.
|
|
|
|
$( 'Less than' implies apart. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
ltapd $p |- ( ph -> A # B ) $=
|
|
( cap wbr gtapd cc wcel wb recnd apsym syl2anc mpbid ) ACBGHZBCGHZABCDE
|
|
FIACJKBJKQRLACEMABDMCBNOP $.
|
|
$}
|
|
|
|
${
|
|
leltapd.3 $e |- ( ph -> A <_ B ) $.
|
|
$( '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon,
|
|
13-Aug-2021.) $)
|
|
leltapd $p |- ( ph -> ( A < B <-> B # A ) ) $=
|
|
( cr wcel cle wbr clt cap wb leltap syl3anc ) ABGHCGHBCIJBCKJCBLJMDEFBC
|
|
NO $.
|
|
$}
|
|
$}
|
|
|
|
$( A nonnegative number is apart from zero if and only if it is positive.
|
|
(Contributed by Jim Kingdon, 11-Aug-2021.) $)
|
|
ap0gt0 $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A # 0 <-> 0 < A ) ) $=
|
|
( cr wcel cc0 cle wbr wa cap wo wb 0red reaplt syldan wn simpr simpl lenltd
|
|
clt mpbid biorf syl bitr4d ) ABCZDAEFZGZADHFZADRFZDARFZIZUHUCUDDBCUFUIJUEKZ
|
|
ADLMUEUGNZUHUIJUEUDUKUCUDOUEDAUJUCUDPQSUGUHTUAUB $.
|
|
|
|
${
|
|
ap0gt0d.1 $e |- ( ph -> A e. RR ) $.
|
|
${
|
|
ap0gt0d.2 $e |- ( ph -> 0 <_ A ) $.
|
|
ap0gt0d.3 $e |- ( ph -> A # 0 ) $.
|
|
$( A nonzero nonnegative number is positive. (Contributed by Jim
|
|
Kingdon, 11-Aug-2021.) $)
|
|
ap0gt0d $p |- ( ph -> 0 < A ) $=
|
|
( cc0 cap wbr clt cr wcel cle wb ap0gt0 syl2anc mpbid ) ABFGHZFBIHZEABJ
|
|
KFBLHQRMCDBNOP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
subap0d.a $e |- ( ph -> A e. CC ) $.
|
|
subap0d.b $e |- ( ph -> B e. CC ) $.
|
|
${
|
|
subap0d.ap $e |- ( ph -> A # B ) $.
|
|
$( Two numbers apart from each other have difference apart from zero.
|
|
(Contributed by Jim Kingdon, 12-Aug-2021.) $)
|
|
subap0d $p |- ( ph -> ( A - B ) # 0 ) $=
|
|
( cneg caddc co cmin cc0 cap wbr cc wcel wb negcld apadd1 syl3anc mpbid
|
|
negsubd negidd 3brtr3d ) ABCGZHIZCUDHIZBCJIKLABCLMZUEUFLMZFABNOCNOUDNOU
|
|
GUHPDEACEQBCUDRSTABCDEUAACEUBUC $.
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Reciprocals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Lemma for ~ recexap . (Contributed by Eric Schmidt, 23-May-2007.) $)
|
|
recextlem1 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) x.
|
|
( A - ( _i x. B ) ) ) = ( ( A x. A ) + ( B x. B ) ) ) $=
|
|
( cc wcel wa ci cmul caddc cmin cneg ax-icn mulcl adantl sylan2 subdid wceq
|
|
co eqtrd anidms oveq12d simpl subcl adddird mulcom c1 oveq1i mulm1d syl5req
|
|
mpan ixi mul4 mpanl12 eqtr4d adantr negcld npncand subneg syl2an 3eqtrd ) A
|
|
CDZBCDZEZAFBGQZHQAVCIQZGQAVDGQZVCVDGQZHQAAGQZAVCGQZIQZVHBBGQZJZIQZHQZVGVJHQ
|
|
ZVBAVCVDUTVAUAZVAVCCDZUTFCDZVAVPKFBLUIZMZVAUTVPVDCDVRAVCUBNUCVBVEVIVFVLHVBA
|
|
AVCVOVOVSOVBVFVCAGQZVCVCGQZIQVLVBVCAVCVSVOVSOVBVHVTVKWAIVAUTVPVHVTPVRAVCUDN
|
|
VAVKWAPZUTVAWBVAVAEZVKFFGQZVJGQZWAWCWEUEJZVJGQVKWDWFVJGUJUFWCVJBBLZUGUHVQVQ
|
|
WCWEWAPKKFFBBUKULRSMTUMTVBVMVGVKIQZVNVBVGVHVKUTVGCDZVAUTWIAALSZUNVAUTVPVHCD
|
|
VRAVCLNVAVKCDZUTVAWKWCVJWGUOSMUPUTWIVJCDZWHVNPVAWJVAWLWGSVGVJUQURRUS $.
|
|
|
|
$( Lemma for ~ recexap . (Contributed by Jim Kingdon, 20-Feb-2020.) $)
|
|
recexaplem2 $p |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) ->
|
|
( ( A x. A ) + ( B x. B ) ) # 0 ) $=
|
|
( cr wcel ci cmul co caddc cc0 cap wbr clt wo wa wb 0re cle remulcl anim12i
|
|
anidms w3a ax-icn mul01i oveq2i eqtr2i breq2i apreim mpanr12 syl5bb pm5.32i
|
|
00id adantr apsqgt0 msqge0 an32s addgtge0 syl2anc anassrs jaodan sylbi olcd
|
|
addgegt0 3impa simp1 remulcld simp2 readdcld reaplt sylancl mpbird ) ACDZBC
|
|
DZAEBFGHGZIJKZUAZAAFGZBBFGZHGZIJKZVRILKZIVRLKZMZVOWAVTVKVLVNWAVKVLNZVNNWCAI
|
|
JKZBIJKZMZNWAWCVNWFVNVMIEIFGZHGZJKZWCWFIWHVMJWHIIHGIWGIIHEUBUCUDUKUEUFWCICD
|
|
ZWJWIWFOPPABIIUGUHUIUJWCWDWAWEWCWDNVPCDZVQCDZNZIVPLKZIVQQKZNZWAWCWMWDVKWKVL
|
|
WLVKWKAARTVLWLBBRTSZULVKWDVLWPVKWDNWNVLWOAUMBUNSUOVPVQUPUQWCWENWMIVPQKZIVQL
|
|
KZNZWAWCWMWEWQULVKVLWEWTVKWRVLWENWSAUNBUMSURVPVQVBUQUSUTVCVAVOVRCDWJVSWBOVO
|
|
VPVQVOAAVKVLVNVDZXAVEVOBBVKVLVNVFZXBVEVGPVRIVHVIVJ $.
|
|
|
|
${
|
|
$d x y a b A $.
|
|
$( Existence of reciprocal of nonzero complex number. (Contributed by Jim
|
|
Kingdon, 20-Feb-2020.) $)
|
|
recexap $p |- ( ( A e. CC /\ A # 0 ) -> E. x e. CC ( A x. x ) = 1 ) $=
|
|
( va vb vy cc wcel cc0 cap wbr cv cmul co c1 wceq wrex cr wi wa adantr ci
|
|
caddc cnre recexaplem2 3expia wb remulcl anidms readdcl syl2an 0re apreap
|
|
creap sylancl recexre sylan recn cmin ax-icn mulcl mpan subcl addcl simpr
|
|
sylan2 mulassd recextlem1 oveq1d eqtr3d id sylan9eq eqeq1d rspcev syl2anc
|
|
oveq2 exp31 syl5 rexlimdv mpd ex sylbid syld breq1 adantl rexbidv 3imtr4d
|
|
oveq1 rexlimivv syl imp ) BFGZBHIJZBAKZLMZNOZAFPZWKBCKZUADKZLMZUBMZOZDQPC
|
|
QPWLWPRZCDBUCXAXBCDQQWQQGZWRQGZSZXAXBXEXASWTHIJZWTWMLMZNOZAFPZWLWPXEXFXIR
|
|
XAXEXFWQWQLMZWRWRLMZUBMZHIJZXIXCXDXFXMWQWRUDUEXEXMXLHUMJZXIXEXLQGZHQGXMXN
|
|
UFXCXJQGZXKQGZXOXDXCXPWQWQUGUHXDXQWRWRUGUHXJXKUIUJZUKXLHULUNXEXNXIXEXNSXL
|
|
EKZLMZNOZEQPZXIXEXOXNYBXREXLUOUPXEYBXIRZXNXCWQFGZWRFGZYCXDWQUQWRUQYDYESZY
|
|
AXIEQXSQGXSFGZYFYAXIRXSUQYFYGYAXIYFYGSZYASWQWSURMZXSLMZFGZWTYJLMZNOZXIYHY
|
|
KYAYFYIFGZYGYKYEYDWSFGZYNUAFGYEYOUSUAWRUTVAZWQWSVBVEZYIXSUTUPTYHYAYLXTNYH
|
|
WTYILMZXSLMYLXTYHWTYIXSYFWTFGZYGYEYDYOYSYPWQWSVCVETYFYNYGYQTYFYGVDVFYHYRX
|
|
LXSLYFYRXLOYGWQWRVGTVHVIYAVJVKXHYMAYJFWMYJOXGYLNWMYJWTLVOVLVMVNVPVQVRUJTV
|
|
SVTWAWBTXAWLXFUFXEBWTHIWCWDXAWPXIUFXEXAWOXHAFXAWNXGNBWTWMLWGVLWEWDWFVTWHW
|
|
IWJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( The product of two numbers apart from zero is apart from zero. Lemma
|
|
2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.) $)
|
|
mulap0 $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( A x. B ) # 0 ) $=
|
|
( vx cc wcel cc0 cap wa cv cmul co c1 wceq recexap adantl simpllr simplll
|
|
wbr wrex simplrl simprl mulassd simprr oveq2d mulid1d 3eqtrd 3brtr4d 0cnd
|
|
mul02d wi mulcld mulext1 syl3anc mpd rexlimddv ) ADEZAFGRZHZBDEZBFGRZHZHZ
|
|
BCIZJKZLMZABJKZFGRZCDVAVECDSURCBNOVBVCDEZVEHZHZVFVCJKZFVCJKZGRZVGVJAFVKVL
|
|
GUPUQVAVIPVJVKAVDJKALJKAVJABVCUPUQVAVIQZURUSUTVITZVBVHVEUAZUBVJVDLAJVBVHV
|
|
EUCUDVJAVNUEUFVJVCVPUIUGVJVFDEFDEVHVMVGUJVJABVNVOUKVJUHVPVFFVCULUMUNUO $.
|
|
$}
|
|
|
|
$( The product of two numbers apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
mulap0b $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <->
|
|
( A x. B ) # 0 ) ) $=
|
|
( cc wcel wa cc0 cap wbr cmul co mulap0 an4s ex mulap0r 3expia impbid ) ACD
|
|
ZBCDZEZAFGHZBFGHZEZABIJFGHZSUBUCQTRUAUCABKLMQRUCUBABNOP $.
|
|
|
|
${
|
|
mulap0i.1 $e |- A e. CC $.
|
|
mulap0i.2 $e |- B e. CC $.
|
|
mulap0i.3 $e |- A # 0 $.
|
|
mulap0i.4 $e |- B # 0 $.
|
|
$( The product of two numbers apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 23-Feb-2020.) $)
|
|
mulap0i $p |- ( A x. B ) # 0 $=
|
|
( cc wcel cc0 cap wbr cmul co mulap0 mp4an ) AGHAIJKBGHBIJKABLMIJKCEDFABN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
mulap0d.1 $e |- ( ph -> A e. CC ) $.
|
|
mulap0d.2 $e |- ( ph -> B e. CC ) $.
|
|
$( The product of two numbers apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
mulap0bd $p |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) ) $=
|
|
( cc wcel cc0 cap wbr wa cmul co wb mulap0b syl2anc ) ABFGCFGBHIJCHIJKBCL
|
|
MHIJNDEBCOP $.
|
|
|
|
mulap0d.3 $e |- ( ph -> A # 0 ) $.
|
|
mulap0d.4 $e |- ( ph -> B # 0 ) $.
|
|
$( The product of two numbers apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 23-Feb-2020.) $)
|
|
mulap0d $p |- ( ph -> ( A x. B ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr cmul co mulap0 syl22anc ) ABHIBJKLCHICJKLBCMNJKLDFE
|
|
GBCOP $.
|
|
$}
|
|
|
|
${
|
|
mulap0bad.1 $e |- ( ph -> A e. CC ) $.
|
|
mulap0bad.2 $e |- ( ph -> B e. CC ) $.
|
|
mulap0bad.3 $e |- ( ph -> ( A x. B ) # 0 ) $.
|
|
$( A factor of a complex number apart from zero is apart from zero.
|
|
Partial converse of ~ mulap0d and consequence of ~ mulap0bd .
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
mulap0bad $p |- ( ph -> A # 0 ) $=
|
|
( cc0 cap wbr wa cmul co mulap0bd mpbird simpld ) ABGHIZCGHIZAPQJBCKLGHIF
|
|
ABCDEMNO $.
|
|
|
|
$( A factor of a complex number apart from zero is apart from zero.
|
|
Partial converse of ~ mulap0d and consequence of ~ mulap0bd .
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
mulap0bbd $p |- ( ph -> B # 0 ) $=
|
|
( cc0 cap wbr wa cmul co mulap0bd mpbird simprd ) ABGHIZCGHIZAPQJBCKLGHIF
|
|
ABCDEMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ph $.
|
|
mulcand.1 $e |- ( ph -> A e. CC ) $.
|
|
mulcand.2 $e |- ( ph -> B e. CC ) $.
|
|
mulcand.3 $e |- ( ph -> C e. CC ) $.
|
|
mulcand.4 $e |- ( ph -> C # 0 ) $.
|
|
$( Cancellation law for multiplication. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanapd $p |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) $=
|
|
( vx cmul co wceq c1 cc wcel wa oveq2 adantr oveq1d mulassd cv wi cc0 cap
|
|
wrex recexap syl2anc simprl mulcomd simprr mulid2d 3eqtr3d eqeq12d syl5ib
|
|
wbr eqtrd rexlimddv impbid1 ) ADBJKZDCJKZLZBCLZADIUAZJKZMLZVAVBUBINADNOZD
|
|
UCUDUOVEINUEGHIDUFUGVAVCUSJKZVCUTJKZLAVCNOZVEPZPZVBUSUTVCJQVKVGBVHCVKVCDJ
|
|
KZBJKMBJKVGBVKVLMBJVKVLVDMVKVCDAVIVEUHZAVFVJGRZUIAVIVEUJUPZSVKVCDBVMVNABN
|
|
OVJERZTVKBVPUKULVKVLCJKMCJKVHCVKVLMCJVOSVKVCDCVMVNACNOVJFRZTVKCVQUKULUMUN
|
|
UQBCDJQUR $.
|
|
|
|
$( Cancellation law for multiplication. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanap2d $p |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) ) $=
|
|
( cmul co wceq mulcomd eqeq12d mulcanapd bitrd ) ABDIJZCDIJZKDBIJZDCIJZKB
|
|
CKAPRQSABDEGLACDFGLMABCDEFGHNO $.
|
|
$}
|
|
|
|
${
|
|
mulcanapad.1 $e |- ( ph -> A e. CC ) $.
|
|
mulcanapad.2 $e |- ( ph -> B e. CC ) $.
|
|
mulcanapad.3 $e |- ( ph -> C e. CC ) $.
|
|
mulcanapad.4 $e |- ( ph -> C # 0 ) $.
|
|
${
|
|
mulcanapad.5 $e |- ( ph -> ( C x. A ) = ( C x. B ) ) $.
|
|
$( Cancellation of a nonzero factor on the left in an equation. One-way
|
|
deduction form of ~ mulcanapd . (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanapad $p |- ( ph -> A = B ) $=
|
|
( cmul co wceq mulcanapd mpbid ) ADBJKDCJKLBCLIABCDEFGHMN $.
|
|
$}
|
|
|
|
${
|
|
mulcanap2ad.5 $e |- ( ph -> ( A x. C ) = ( B x. C ) ) $.
|
|
$( Cancellation of a nonzero factor on the right in an equation. One-way
|
|
deduction form of ~ mulcanap2d . (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanap2ad $p |- ( ph -> A = B ) $=
|
|
( cmul co wceq mulcanap2d mpbid ) ABDJKCDJKLBCLIABCDEFGHMN $.
|
|
$}
|
|
$}
|
|
|
|
$( Cancellation law for multiplication (full theorem form). (Contributed by
|
|
Jim Kingdon, 21-Feb-2020.) $)
|
|
mulcanap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) ->
|
|
( ( C x. A ) = ( C x. B ) <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a simp1 simp2 simp3l simp3r mulcanapd ) ADEZBDEZ
|
|
CDEZCFGHZIZJABCPQTKPQTLPQRSMPQRSNO $.
|
|
|
|
$( Cancellation law for multiplication. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanap2 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) ->
|
|
( ( A x. C ) = ( B x. C ) <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a simp1 simp2 simp3l simp3r mulcanap2d ) ADEZBDE
|
|
ZCDEZCFGHZIZJABCPQTKPQTLPQRSMPQRSNO $.
|
|
|
|
${
|
|
mulcanapi.1 $e |- A e. CC $.
|
|
mulcanapi.2 $e |- B e. CC $.
|
|
mulcanapi.3 $e |- C e. CC $.
|
|
mulcanapi.4 $e |- C # 0 $.
|
|
$( Cancellation law for multiplication. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
mulcanapi $p |- ( ( C x. A ) = ( C x. B ) <-> A = B ) $=
|
|
( cc wcel cc0 cap wbr wa cmul co wceq wb pm3.2i mulcanap mp3an ) AHIBHICH
|
|
IZCJKLZMCANOCBNOPABPQDEUAUBFGRABCST $.
|
|
$}
|
|
|
|
$( Property of numbers whose product equals their sum. Equation 5 of
|
|
[Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.) $)
|
|
muleqadd $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) ) $=
|
|
( cc wcel wa c1 cmin co cmul wceq caddc ax-1cn mulsub mpanr2 mpanl2 mulid1i
|
|
cc0 oveq2i a1i mulid1 oveqan12d oveq12d addsub mp3an2 syl2anc 3eqtrd eqeq1d
|
|
mulcl addcl addid2i eqeq2i subcld 0cn addcan2 mp3an23 syl syl5rbbr subeq0ad
|
|
wb 3bitr2rd ) ACDZBCDZEZAFGHBFGHIHZFJABIHZABKHZGHZFKHZFJZVGQJZVEVFJVCVDVHFV
|
|
CVDVEFFIHZKHZAFIHZBFIHZKHZGHZVEFKHZVFGHZVHVAFCDZVBVDVPJZLVAVSEVBVSVTLAFBFMN
|
|
OVCVLVQVOVFGVLVQJVCVKFVEKFLPRSVAVBVMAVNBKATBTUAUBVCVECDZVFCDZVRVHJZABUHZABU
|
|
IZWAVSWBWCLVEFVFUCUDUEUFUGVIVHQFKHZJZVCVJWFFVHFLUJUKVCVGCDZWGVJUSZVCVEVFWDW
|
|
EULWHQCDVSWIUMLVGQFUNUOUPUQVCVEVFWDWEURUT $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Existential uniqueness of reciprocals. (Contributed by Jim Kingdon,
|
|
21-Feb-2020.) $)
|
|
receuap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
E! x e. CC ( B x. x ) = A ) $=
|
|
( vy cc wcel cc0 w3a cv cmul co wceq wrex wa wi wral 3adant1 oveq2 eqeq1d
|
|
c1 cap wbr wreu recexap simprl simpll mulcld oveq1 simplr mulassd mulid2d
|
|
ad2antll 3eqtr3d rspcev syl2anc rexlimdvaa 3adant3 mpd eqtr3 syl5ib 3expa
|
|
mulcanap expcom ralrimivv reu4 sylanbrc ) BEFZCEFZCGUAUBZHZCAIZJKZBLZAEMZ
|
|
VMCDIZJKZBLZNZVKVOLZOZDEPAEPVMAEUCVJVPTLZDEMZVNVHVIWBVGDCUDQVGVHWBVNOVIVG
|
|
VHNZWAVNDEWCVOEFZWANZNZVOBJKZEFCWGJKZBLZVNWFVOBWCWDWAUEZVGVHWEUFZUGWFVPBJ
|
|
KZTBJKZWHBWAWLWMLWCWDVPTBJUHULWFCVOBVGVHWEUIWJWKUJWFBWKUKUMVMWIAWGEVKWGLV
|
|
LWHBVKWGCJRSUNUOUPUQURVJVTADEEVHVIVKEFZWDNZVTOVGWOVHVINZVTWNWDWPVTVRVLVPL
|
|
WNWDWPHVSVLVPBUSVKVOCVBUTVAVCQVDVMVQADEVSVLVPBVKVOCJRSVEVF $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Division
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend class notation to include division. $)
|
|
cdiv $a class / $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define division. Theorem ~ divmulap relates it to multiplication, and
|
|
~ divclap and ~ redivclap prove its closure laws. (Contributed by NM,
|
|
2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.)
|
|
(New usage is discouraged.) $)
|
|
df-div $a |- / = ( x e. CC , y e. ( CC \ { 0 } ) |->
|
|
( iota_ z e. CC ( y x. z ) = x ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y C $.
|
|
$( Value of division: the (unique) element ` x ` such that
|
|
` ( B x. x ) = A ` . This is meaningful only when ` B ` is apart from
|
|
zero. (Contributed by Jim Kingdon, 21-Feb-2020.) $)
|
|
divvalap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) ) $=
|
|
( vz vy cc wcel cc0 cap wbr w3a csn cdif cv cmul wceq crio cdiv riotabidv
|
|
co simp1 wne simp2 wi 0cn apne mpan2 adantl eldifsn sylanbrc wreu receuap
|
|
3impia riotacl syl eqeq2 oveq1 eqeq1d df-div ovmpt2g syl3anc ) BFGZCFGZCH
|
|
IJZKZVBCFHLMZGZCANZOTZBPZAFQZFGZBCRTVKPVBVCVDUAVEVCCHUBZVGVBVCVDUCVBVCVDV
|
|
MVCVDVMUDZVBVCHFGVNUECHUFUGUHUMCFHUIUJVEVJAFUKVLABCULVJAFUNUODEBCFVFENZVH
|
|
OTZDNZPZAFQVKRVPBPZAFQFVQBPVRVSAFVQBVPUPSVOCPZVSVJAFVTVPVIBVOCVHOUQURSDEA
|
|
USUTVA $.
|
|
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 22-Feb-2020.) $)
|
|
divmulap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A / C ) = B <-> ( C x. B ) = A ) ) $=
|
|
( vx cc wcel cc0 cap wbr wa w3a cdiv co wceq cv cmul 3expb 3adant2 eqeq1d
|
|
crio divvalap wreu wb simp2 receuap oveq2 riota2 syl2anc bitr4d ) AEFZBEF
|
|
ZCEFZCGHIZJZKZACLMZBNCDOZPMZANZDETZBNZCBPMZANZUOUPUTBUJUNUPUTNZUKUJULUMVD
|
|
DACUAQRSUOUKUSDEUBZVCVAUCUJUKUNUDUJUNVEUKUJULUMVEDACUEQRUSVCDEBUQBNURVBAU
|
|
QBCPUFSUGUHUI $.
|
|
$}
|
|
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 22-Feb-2020.) $)
|
|
divmulap2 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A / C ) = B <-> A = ( C x. B ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co wceq cmul divmulap eqcom syl6bb ) ADEB
|
|
DECDECFGHIJACKLBMCBNLZAMARMABCORAPQ $.
|
|
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 22-Feb-2020.) $)
|
|
divmulap3 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A / C ) = B <-> A = ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co wceq cmul divmulap2 mulcom adantrr 3adant1
|
|
w3a eqeq2d bitr4d ) ADEZBDEZCDEZCFGHZIZRZACJKBLACBMKZLABCMKZLABCNUFUHUGAUBU
|
|
EUHUGLZUAUBUCUIUDBCOPQST $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) $)
|
|
divclap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC ) $=
|
|
( vx cc wcel cc0 cap wbr w3a cdiv co cmul wceq crio divvalap wreu receuap
|
|
cv riotacl syl eqeltrd ) ADEBDEBFGHIZABJKBCRLKAMZCDNZDCABOUBUCCDPUDDECABQ
|
|
UCCDSTUA $.
|
|
$}
|
|
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.) $)
|
|
recclap $p |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) e. CC ) $=
|
|
( c1 cc wcel cc0 cap wbr cdiv co ax-1cn divclap mp3an1 ) BCDACDAEFGBAHICDJB
|
|
AKL $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
22-Feb-2020.) $)
|
|
divcanap2 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( B x. ( A / B ) ) = A ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co wceq cmul eqid wa wb simp1 divclap 3simpc
|
|
divmulap syl3anc mpbii ) ACDZBCDZBEFGZHZABIJZUFKZBUFLJAKZUFMUEUBUFCDUCUDNUG
|
|
UHOUBUCUDPABQUBUCUDRAUFBSTUA $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
22-Feb-2020.) $)
|
|
divcanap1 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 )
|
|
-> ( ( A / B ) x. B ) = A ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv cmul divclap simp2 mulcomd divcanap2 eqtrd
|
|
co ) ACDZBCDZBEFGZHZABIPZBJPBUAJPATUABABKQRSLMABNO $.
|
|
|
|
$( A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon,
|
|
22-Feb-2020.) $)
|
|
diveqap0 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( ( A / B ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co wceq cmul wa wb simp1 0cnd 3simpc syl3anc
|
|
divmulap2 simp2 mul01d eqeq2d bitrd ) ACDZBCDZBEFGZHZABIJEKZABELJZKZAEKUGUD
|
|
ECDUEUFMUHUJNUDUEUFOUGPUDUEUFQAEBSRUGUIEAUGBUDUEUFTUAUBUC $.
|
|
|
|
$( The ratio of numbers apart from zero is apart from zero. (Contributed by
|
|
Jim Kingdon, 22-Feb-2020.) $)
|
|
divap0b $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( A # 0 <-> ( A / B ) # 0 ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wa cmul simp3 biantrud divclap simp2 mulap0bd
|
|
w3a divcanap1 breq1d 3bitrrd ) ACDZBCDZBEFGZQZABHIZEFGZUFUCJUEBKIZEFGAEFGUD
|
|
UCUFUAUBUCLMUDUEBABNUAUBUCOPUDUGAEFABRST $.
|
|
|
|
$( The ratio of numbers apart from zero is apart from zero. (Contributed by
|
|
Jim Kingdon, 22-Feb-2020.) $)
|
|
divap0 $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( A / B ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co wb divap0b 3expb biimpa an32s ) ACDZBCDZBE
|
|
FGZHZAEFGZABIJEFGZPSHTUAPQRTUAKABLMNO $.
|
|
|
|
$( The reciprocal of a number apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
recap0 $p |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) # 0 ) $=
|
|
( c1 cc wcel cc0 cap wbr wa cdiv co ax-1cn 1ap0 divap0 mpanl12 ) BCDBEFGACD
|
|
AEFGHBAIJEFGKLBAMN $.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by Jim
|
|
Kingdon, 24-Feb-2020.) $)
|
|
recidap $p |- ( ( A e. CC /\ A # 0 ) -> ( A x. ( 1 / A ) ) = 1 ) $=
|
|
( c1 cc wcel cc0 cap wbr cdiv co cmul wceq ax-1cn divcanap2 mp3an1 ) BCDACD
|
|
AEFGABAHIJIBKLBAMN $.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by Jim
|
|
Kingdon, 24-Feb-2020.) $)
|
|
recidap2 $p |- ( ( A e. CC /\ A # 0 ) -> ( ( 1 / A ) x. A ) = 1 ) $=
|
|
( c1 cc wcel cc0 cap wbr cdiv co cmul wceq ax-1cn divcanap1 mp3an1 ) BCDACD
|
|
AEFGBAHIAJIBKLBAMN $.
|
|
|
|
$( Relationship between division and reciprocal. (Contributed by Jim
|
|
Kingdon, 24-Feb-2020.) $)
|
|
divrecap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( A / B ) = ( A x. ( 1 / B ) ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co c1 cmul wceq simp2 recclap 3adant1 mul12d
|
|
simp1 recidap oveq2d mulid1d 3eqtrd mulcld 3simpc divmulap syl3anc mpbird
|
|
wa wb ) ACDZBCDZBEFGZHZABIJAKBIJZLJZMZBUOLJZAMZUMUQABUNLJZLJAKLJAUMBAUNUJUK
|
|
ULNUJUKULRZUKULUNCDUJBOPZQUMUSKALUKULUSKMUJBSPTUMAUTUAUBUMUJUOCDUKULUHUPURU
|
|
IUTUMAUNUTVAUCUJUKULUDAUOBUEUFUG $.
|
|
|
|
$( Relationship between division and reciprocal. (Contributed by Jim
|
|
Kingdon, 25-Feb-2020.) $)
|
|
divrecap2 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( A / B ) = ( ( 1 / B ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co c1 divrecap simp1 recclap 3adant1 mulcomd
|
|
cmul eqtrd ) ACDZBCDZBEFGZHZABIJAKBIJZQJUCAQJABLUBAUCSTUAMTUAUCCDSBNOPR $.
|
|
|
|
$( An associative law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divassap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) ->
|
|
( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cmul cdiv wceq recclap mulass syl3an3 divrecap
|
|
co c1 syl3anc mulcl 3adant3 simp3l simp3r simp2 oveq2d 3eqtr4d ) ADEZBDEZCD
|
|
EZCFGHZIZJZABKRZSCLRZKRZABUOKRZKRZUNCLRZABCLRZKRULUHUIUODEUPURMCNABUOOPUMUN
|
|
DEZUJUKUSUPMUHUIVAULABUAUBUHUIUJUKUCZUHUIUJUKUDZUNCQTUMUTUQAKUMUIUJUKUTUQMU
|
|
HUIULUEVBVCBCQTUFUG $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
div23ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cmul co cdiv wceq mulcom oveq1d 3adant3 3com12
|
|
divassap simp2 divclap 3expb 3adant2 mulcomd 3eqtrd ) ADEZBDEZCDEZCFGHZIZJZ
|
|
ABKLZCMLZBAKLZCMLZBACMLZKLZUPBKLUFUGUMUONUJUFUGIULUNCMABOPQUGUFUJUOUQNBACSR
|
|
UKBUPUFUGUJTUFUJUPDEZUGUFUHUIURACUAUBUCUDUE $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
div32ap $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ C e. CC )
|
|
-> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv cmul wceq w3a div23ap divassap eqtr3d 3com23
|
|
co ) ADEZCDEZBDEBFGHIZABJRCKRZACBJRKRZLSTUAMACKRBJRUBUCACBNACBOPQ $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
div13ap $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ C e. CC )
|
|
-> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cmul co cdiv wceq mulcom oveq1d 3adant2 3com23
|
|
div23ap 3coml 3eqtr3d ) ADEZBDEBFGHIZCDEZJACKLZBMLZCAKLZBMLZABMLCKLZCBMLAKL
|
|
ZUBUDUFUHNUCUBUDIUEUGBMACOPQUBUDUCUFUINACBSRUDUBUCUHUJNCABSTUA $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
div12ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co cmul divclap 3expb mulcom sylan2 3impb
|
|
wceq div13ap 3comr sylan 3impa 3com23 3eqtrd ) ADEZBDEZCDEZCFGHZIZJABCKLZML
|
|
ZUKAMLZACKLZBMLZBUNMLZUFUGUJULUMSZUGUJIUFUKDEZUQUGUHUIURBCNOAUKPQRUGUJUFUMU
|
|
OSBCATUAUFUJUGUOUPSZUFUJUGUSUFUJIUNDEZUGUSUFUHUIUTACNOUNBPUBUCUDUE $.
|
|
|
|
$( An associative law for division and multiplication. (Contributed by Jim
|
|
Kingdon, 24-Jan-2022.) $)
|
|
divmulassap $p |- ( ( ( A e. CC /\ B e. CC /\ C e. CC )
|
|
/\ ( D e. CC /\ D # 0 ) )
|
|
-> ( ( A x. ( B / D ) ) x. C ) = ( ( A x. B ) x. ( C / D ) ) ) $=
|
|
( cc wcel w3a cc0 cap wbr wa cdiv co cmul wceq simpl1 simpl2 simpr divassap
|
|
syl3anc eqcomd oveq1d mulcl 3adant3 adantr simpl3 div32ap eqtrd ) AEFZBEFZC
|
|
EFZGZDEFDHIJKZKZABDLMNMZCNMABNMZDLMZCNMZUPCDLMNMZUNUOUQCNUNUQUOUNUIUJUMUQUO
|
|
OUIUJUKUMPUIUJUKUMQULUMRZABDSTUAUBUNUPEFZUMUKURUSOULVAUMUIUJVAUKABUCUDUEUTU
|
|
IUJUKUMUFUPDCUGTUH $.
|
|
|
|
$( An associative/commutative law for division and multiplication.
|
|
(Contributed by Jim Kingdon, 24-Jan-2022.) $)
|
|
divmulasscomap $p |- ( ( ( A e. CC /\ B e. CC /\ C e. CC )
|
|
/\ ( D e. CC /\ D # 0 ) )
|
|
-> ( ( A x. ( B / D ) ) x. C ) = ( B x. ( ( A x. C ) / D ) ) ) $=
|
|
( cc wcel w3a cc0 cap wbr wa cdiv co cmul divmulassap mulcom 3adant3 adantr
|
|
wceq oveq1d simpl2 simpl1 anim1i 3anass sylibr divclap syl mulassd divassap
|
|
simp3 simpr syl3anc eqcomd oveq2d eqtrd 3eqtrd ) AEFZBEFZCEFZGZDEFZDHIJZKZK
|
|
ZABDLMNMCNMABNMZCDLMZNMBANMZVFNMZBACNMDLMZNMZABCDOVDVEVGVFNUTVEVGSZVCUQURVK
|
|
USABPQRTVDVHBAVFNMZNMVJVDBAVFUQURUSVCUAUQURUSVCUBZVDUSVAVBGZVFEFVDUSVCKVNUT
|
|
USVCUQURUSUJZUCUSVAVBUDUECDUFUGUHVDVLVIBNVDVIVLVDUQUSVCVIVLSVMUTUSVCVORUTVC
|
|
UKACDUIULUMUNUOUP $.
|
|
|
|
$( Distribution of division over addition. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a caddc co c1 cdiv cmul simp1 simp2 wceq syl3anc
|
|
divrecap recclap 3ad2ant3 adddird addcld simp3l simp3r oveq12d 3eqtr4d ) AD
|
|
EZBDEZCDEZCFGHZIZJZABKLZMCNLZOLZAUPOLZBUPOLZKLUOCNLZACNLZBCNLZKLUNABUPUIUJU
|
|
MPZUIUJUMQZUMUIUPDEUJCUAUBUCUNUODEUKULUTUQRUNABVCVDUDUIUJUKULUEZUIUJUKULUFZ
|
|
UOCTSUNVAURVBUSKUNUIUKULVAURRVCVEVFACTSUNUJUKULVBUSRVDVEVFBCTSUGUH $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divcanap3 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( ( B x. A ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr w3a cmul co cdiv wceq wa wb simp2 simp1 mulcld 3simpc
|
|
eqid divmulap syl3anc mpbiri ) ACDZBCDZBEFGZHZBAIJZBKJALZUGUGLZUGSUFUGCDUCU
|
|
DUEMUHUINUFBAUCUDUEOUCUDUEPZQUJUCUDUERUGABTUAUB $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divcanap4 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( ( A x. B ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr w3a cmul co cdiv wceq mulcom 3adant3 oveq1d divcanap3
|
|
eqtrd ) ACDZBCDZBEFGZHZABIJZBKJBAIJZBKJAUAUBUCBKRSUBUCLTABMNOABPQ $.
|
|
|
|
$( One-to-one relationship for division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
div11ap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) ->
|
|
( ( A / C ) = ( B / C ) <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co cmul wceq simp1 simp3l divclap syl3anc
|
|
simp3r divcanap2 simp2 mulcanapd eqeq12d bitr3d ) ADEZBDEZCDEZCFGHZIZJZCACK
|
|
LZMLZCBCKLZMLZNUKUMNABNUJUKUMCUJUEUGUHUKDEUEUFUIOZUEUFUGUHPZUEUFUGUHSZACQRU
|
|
JUFUGUHUMDEUEUFUIUAZUPUQBCQRUPUQUBUJULAUNBUJUEUGUHULANUOUPUQACTRUJUFUGUHUNB
|
|
NURUPUQBCTRUCUD $.
|
|
|
|
$( A number divided by itself is one. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
dividap $p |- ( ( A e. CC /\ A # 0 ) -> ( A / A ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul wceq divrecap 3anidm12 recidap eqtrd
|
|
c1 ) ABCZADEFZGAAHIZAPAHIJIZPQRSTKAALMANO $.
|
|
|
|
$( Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.) $)
|
|
div0ap $p |- ( ( A e. CC /\ A # 0 ) -> ( 0 / A ) = 0 ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co wceq cmul simpl mul01d wb divmulap mp3an12
|
|
0cn mpbird ) ABCZADEFZGZDAHIDJZADKIDJZUAASTLMDBCZUDUAUBUCNQQDDAOPR $.
|
|
|
|
$( A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof
|
|
shortened by Mario Carneiro, 27-May-2016.) $)
|
|
div1 $p |- ( A e. CC -> ( A / 1 ) = A ) $=
|
|
( cc wcel c1 cdiv co wceq cmul mulid2 wb cc0 cap wbr ax-1cn pm3.2i divmulap
|
|
wa 1ap0 mp3an3 anidms mpbird ) ABCZADEFAGZDAHFAGZAIUBUCUDJZUBUBDBCZDKLMZQUE
|
|
UFUGNROAADPSTUA $.
|
|
|
|
$( 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
|
|
7-Dec-2018.) $)
|
|
1div1e1 $p |- ( 1 / 1 ) = 1 $=
|
|
( c1 cc wcel cdiv co wceq ax-1cn div1 ax-mp ) ABCAADEAFGAHI $.
|
|
|
|
$( Equality in terms of unit ratio. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
diveqap1 $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( ( A / B ) = 1 <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co c1 wceq cmul wb wa ax-1cn divmulap2 3impb
|
|
mp3an2 simp2 mulid1d eqeq2d bitrd ) ACDZBCDZBEFGZHZABIJKLZABKMJZLZABLUDUEUF
|
|
UHUJNZUDKCDUEUFOUKPAKBQSRUGUIBAUGBUDUEUFTUAUBUC $.
|
|
|
|
$( Move negative sign inside of a division. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divnegap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
-u ( A / B ) = ( -u A / B ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cneg c1 cdiv co cmul wceq wa recclap sylan2 3impb
|
|
mulneg1 divrecap negcl syl3an1 negeqd 3eqtr4rd ) ACDZBCDZBEFGZHZAIZJBKLZMLZ
|
|
AUJMLZIZUIBKLZABKLZIUEUFUGUKUMNZUFUGOUEUJCDUPBPAUJSQRUEUICDUFUGUNUKNAUAUIBT
|
|
UBUHUOULABTUCUD $.
|
|
|
|
$( Distribution of division over addition with a multiplication.
|
|
(Contributed by Jim Kingdon, 11-Nov-2021.) $)
|
|
muldivdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( ( C x. A ) + B ) / C ) = ( A + ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cmul co caddc cdiv wceq simp3l mulcld divdirap
|
|
simp1 syld3an1 3anass biimpri 3adant2 divcanap3 syl oveq1d eqtrd ) ADEZBDEZ
|
|
CDEZCFGHZIZJZCAKLZBMLCNLZUNCNLZBCNLZMLZAUQMLUNDEUIUHULUOUROUMCAUHUIUJUKPUHU
|
|
IULSQUNBCRTUMUPAUQMUMUHUJUKJZUPAOUHULUSUIUSUHULIUHUJUKUAUBUCACUDUEUFUG $.
|
|
|
|
$( Distribution of division over subtraction. (Contributed by NM,
|
|
4-Mar-2005.) $)
|
|
divsubdirap $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv cneg caddc cmin wceq eqtr3d 3expb 3adant1
|
|
co divclap divdirap syl3an2 negsub oveq1d 3adant3 divnegap 3adant2 negsubd
|
|
negcl oveq2d ) ADEZBDEZCDEZCFGHZIZJZACKSZBLZCKSZMSZABNSZCKSZUQBCKSZNSZUPAUR
|
|
MSZCKSZUTVBULUKURDEUOVFUTOBUIAURCUAUBUKULVFVBOUOUKULIVEVACKABUCUDUEPUPUQVCL
|
|
ZMSUTVDUPVGUSUQMULUOVGUSOZUKULUMUNVHBCUFQRUJUPUQVCUKUOUQDEZULUKUMUNVIACTQUG
|
|
ULUOVCDEZUKULUMUNVJBCTQRUHPP $.
|
|
|
|
$( A number is equal to the reciprocal of its reciprocal. (Contributed by
|
|
Jim Kingdon, 25-Feb-2020.) $)
|
|
recrecap $p |- ( ( A e. CC /\ A # 0 ) -> ( 1 / ( 1 / A ) ) = A ) $=
|
|
( cc wcel cc0 cap wbr wa c1 cdiv co wceq cmul recidap2 wb 1cnd simpl recap0
|
|
recclap divmulap syl112anc mpbird ) ABCZADEFZGZHHAIJZIJAKZUEALJHKZAMUDHBCUB
|
|
UEBCUEDEFUFUGNUDOUBUCPARAQHAUESTUA $.
|
|
|
|
$( Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.) $)
|
|
rec11ap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) )
|
|
-> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr wa c1 cdiv co wceq cmul wb 1cnd adantl simpl divmulap
|
|
recclap syl3anc simpll simprl simprr divrecap eqeq1d diveqap1 3bitr2d ) ACD
|
|
ZAEFGZHZBCDZBEFGZHZHZIAJKIBJKZLZAUOMKZILZABJKZILZABLZUNICDUOCDZUJUPURNUNOUM
|
|
VBUJBSPUJUMQIUOARTUNUSUQIUNUHUKULUSUQLUHUIUMUAZUJUKULUBZUJUKULUCZABUDTUEUNU
|
|
HUKULUTVANVCVDVEABUFTUG $.
|
|
|
|
$( Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.) $)
|
|
rec11rap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) )
|
|
-> ( ( 1 / A ) = B <-> ( 1 / B ) = A ) ) $=
|
|
( cc wcel cc0 cap wa c1 cdiv co wceq cmul wb simprl simpll simplr divmulap2
|
|
wbr 1cnd syl112anc simprr divmulap3 bitr4d ) ACDZAEFRZGZBCDZBEFRZGZGZHAIJBK
|
|
ZHABLJKZHBIJAKZUJHCDZUGUDUEUKULMUJSZUFUGUHNZUDUEUIOZUDUEUIPHBAQTUJUNUDUGUHU
|
|
MULMUOUQUPUFUGUHUAHABUBTUC $.
|
|
|
|
$( Multiplication of two ratios. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divmuldivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 )
|
|
/\ ( D e. CC /\ D # 0 ) ) )
|
|
-> ( ( A / C ) x. ( B / D ) ) = ( ( A x. B ) / ( C x. D ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul wceq w3a 3anass mulcl syl2an 3adantr1
|
|
divclap ad2ant2r 3adantl1 mulap0 divcanap3 syl3anc simp2 jca mul4 divcanap2
|
|
oveqan12d eqtr3d oveq1d syl2anbr an4s ) AEFZCEFZCGHIZJZBEFZDEFZDGHIZJZACKLZ
|
|
BDKLZMLZABMLZCDMLZKLZNZUOURJUOUPUQOZUSUTVAOZVIUSVBJUOUPUQPUSUTVAPVJVKJZVGVE
|
|
MLZVGKLZVEVHVLVEEFZVGEFZVGGHIZVNVENVJVCEFZVDEFZVOVKACTZBDTZVCVDQRUPUQVKVPUO
|
|
URUTVAVPUSUPUTVPUQVACDQUASUBUPUQVKVQUOURUTVAVQUSCDUCSUBVEVGUDUEVLVMVFVGKVLC
|
|
VCMLZDVDMLZMLZVMVFVJUPVRJUTVSJWDVMNVKVJUPVRUOUPUQUFVTUGVKUTVSUSUTVAUFWAUGCV
|
|
CDVDUHRVJVKWBAWCBMACUIBDUIUJUKULUKUMUN $.
|
|
|
|
$( Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by
|
|
Jim Kingdon, 25-Feb-2020.) $)
|
|
divdivdivap $p |- ( ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) /\
|
|
( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) )
|
|
-> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul wceq divclap syl3anc mulcomd eqtrd c1
|
|
mulcld simprrl simprll simprlr simpll simplrl simplr simprl syl22anc oveq2d
|
|
simplrr divmuldivap simprrr mulap0d dividap syl2anc mulassd mulid2d 3eqtr3d
|
|
simprr oveq1d eqtr3d wb mulap0 ad2ant2lr divap0 adantl divmulap syl112anc
|
|
mpbird ) AEFZBEFZBGHIZJZJZCEFZCGHIZJZDEFZDGHIZJZJZJZABKLZCDKLZKLADMLZBCMLZK
|
|
LZNZWDWGMLZWCNZWBWDDCKLZWCMLZMLZWIWCWBWLWGWDMWBWLWCWKMLZWGWBWKWCWBVRVOVPWKE
|
|
FVNVQVRVSUAZVNVOVPVTUBZVNVOVPVTUCZDCOPZWBVJVKVLWCEFZVJVMWAUDZVJVKVLWAUEZVJV
|
|
KVLWAUJABOPZQWBVJVRVMVQWNWGNWTWOVJVMWAUFVNVQVTUGZADBCUKUHRUIWBWDWKMLZWCMLSW
|
|
CMLWMWCWBXDSWCMWBXDCDMLZDCMLZKLZSWBVOVRVTVQXDXGNWPWOVNVQVTUSXCCDDCUKUHWBXGX
|
|
FXFKLZSWBXEXFXFKWBCDWPWOQUTWBXFEFXFGHIXHSNWBDCWOWPTWBDCWOWPVNVQVRVSULZWQUMX
|
|
FUNUORRUTWBWDWKWCWBVOVRVSWDEFZWPWOXICDOPZWRXBUPWBWCXBUQURVAWBWSWGEFZXJWDGHI
|
|
ZWHWJVBXBWBWEEFWFEFWFGHIZXLWBADWTWOTWBBCXAWPTVMVQXNVJVTBCVCVDWEWFOPXKWAXMVN
|
|
CDVEVFWCWGWDVGVHVI $.
|
|
|
|
$( Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
|
|
25-Feb-2020.) $)
|
|
divcanap5 $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\
|
|
( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co cmul c1 dividap oveq1d 3ad2ant3 simp3l
|
|
wceq simp1 simp3 simp2 divmuldivap syl22anc divclap mulid2d 3adant3 3eqtr3d
|
|
3expb ) ADEZBDEZBFGHZIZCDEZCFGHZIZJZCCKLZABKLZMLZNUSMLZCAMLCBMLKLZUSUPUJUTV
|
|
ASUMUPURNUSMCOPQUQUNUJUPUMUTVBSUJUMUNUORUJUMUPTUJUMUPUAUJUMUPUBCACBUCUDUJUM
|
|
VAUSSUPUJUMIUSUJUKULUSDEABUEUIUFUGUH $.
|
|
|
|
$( Swap the denominators in the product of two ratios. (Contributed by Jim
|
|
Kingdon, 26-Feb-2020.) $)
|
|
divmul13ap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 )
|
|
/\ ( D e. CC /\ D # 0 ) ) )
|
|
-> ( ( A / C ) x. ( B / D ) ) = ( ( B / C ) x. ( A / D ) ) ) $=
|
|
( cc wcel wa cc0 cap wbr cmul co cdiv wceq mulcom adantr oveq1d divmuldivap
|
|
ancom1s 3eqtr4d ) AEFZBEFZGZCEFCHIJGDEFDHIJGGZGZABKLZCDKLZMLBAKLZUGMLZACMLB
|
|
DMLKLBCMLADMLKLZUEUFUHUGMUCUFUHNUDABOPQABCDRUBUAUDUJUINBACDRST $.
|
|
|
|
$( Swap the numerators in the product of two ratios. (Contributed by Jim
|
|
Kingdon, 26-Feb-2020.) $)
|
|
divmul24ap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 )
|
|
/\ ( D e. CC /\ D # 0 ) ) )
|
|
-> ( ( A / C ) x. ( B / D ) ) = ( ( A / D ) x. ( B / C ) ) ) $=
|
|
( cc wcel wa cc0 cap wbr cmul co cdiv wceq mulcom adantl oveq2d divmuldivap
|
|
ad2ant2r ancom2s 3eqtr4d ) AEFBEFGZCEFZCHIJZGZDEFZDHIJZGZGZGZABKLZCDKLZMLUK
|
|
DCKLZMLZACMLBDMLKLADMLBCMLKLZUJULUMUKMUIULUMNZUBUCUFUPUDUGCDOSPQABCDRUBUHUE
|
|
UOUNNABDCRTUA $.
|
|
|
|
$( Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
divmuleqap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\
|
|
( D e. CC /\ D # 0 ) ) ) ->
|
|
( ( A / C ) = ( B / D ) <-> ( A x. D ) = ( B x. C ) ) ) $=
|
|
( cc wcel wa cc0 cap cdiv cmul wceq divclap 3expb ad2ant2r ad2ant2l mulassd
|
|
wbr co divcanap1 mulap0 jca adantl mulcanap2 syl3anc simprll simprrl oveq1d
|
|
wb mulcl eqtr3d mulcomd oveq2d 3eqtr2d eqeq12d bitr3d ) AEFZBEFZGZCEFZCHIRZ
|
|
GZDEFZDHIRZGZGZGZACJSZCDKSZKSZBDJSZVIKSZLZVHVKLZADKSZBCKSZLVGVHEFZVKEFZVIEF
|
|
ZVIHIRZGZVMVNUIUQVBVQURVEUQUTVAVQACMNOZURVEVRUQVBURVCVDVRBDMNPZVFWAUSVFVSVT
|
|
UTVCVSVAVDCDUJOCDUAUBUCVHVKVIUDUEVGVJVOVLVPVGVHCKSZDKSVJVOVGVHCDWBUSUTVAVEU
|
|
FZUSVBVCVDUGZQVGWDADKUQVBWDALZURVEUQUTVAWGACTNOUHUKVGVLVKDCKSZKSVKDKSZCKSVP
|
|
VGVIWHVKKVGCDWEWFULUMVGVKDCWCWFWEQVGWIBCKURVEWIBLZUQVBURVCVDWJBDTNPUHUNUOUP
|
|
$.
|
|
|
|
$( The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
recdivap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( 1 / ( A / B ) ) = ( B / A ) ) $=
|
|
( cc wcel cc0 cap wa c1 cdiv co cmul 1div1e1 oveq1i wceq ax-1cn 1ap0 pm3.2i
|
|
wbr divdivdivap mulid2 mpanl12 syl5eqr oveqan12rd ad2ant2r eqtrd ) ACDZAEFR
|
|
ZGBCDZBEFRZGGZHABIJZIJZHBKJZHAKJZIJZBAIJZUJULHHIJZUKIJZUOUQHUKILMHCDZUSHEFR
|
|
ZGUJURUONOUSUTOPQHHABSUAUBUFUHUOUPNUGUIUHUFUMBUNAIBTATUCUDUE $.
|
|
|
|
$( Cancellation of inverted fractions. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
divcanap6 $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( ( A / B ) x. ( B / A ) ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co c1 cmul recdivap oveq2d wceq divclap 3expb
|
|
adantlr divap0 recidap syl2anc eqtr3d ) ACDZAEFGZHBCDZBEFGZHZHZABIJZKUIIJZL
|
|
JZUIBAIJZLJKUHUJULUILABMNUHUICDZUIEFGUKKOUCUGUMUDUCUEUFUMABPQRABSUITUAUB $.
|
|
|
|
$( Swap denominators in a division. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
divdiv32ap $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 )
|
|
/\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
|
|
( cc wcel cc0 cap wbr wa c1 cdiv cmul wceq recclap div23ap syl3an2 divrecap
|
|
w3a co 3expb 3adant3 oveq1d divclap syl3an1 3impa 3com23 3eqtr4d ) ADEZBDEZ
|
|
BFGHZIZCDEZCFGHZIZRZAJBKSZLSZCKSZACKSZUPLSZABKSZCKSUSBKSZUKUHUPDEUNURUTMBNA
|
|
UPCOPUOVAUQCKUHUKVAUQMZUNUHUIUJVCABQTUAUBUHUNUKVBUTMZUHUNUKVDUHUNIZUIUJVDVE
|
|
USDEZUIUJVDUHULUMVFACUCTUSBQUDTUEUFUG $.
|
|
|
|
$( Cancel equal divisors in a division. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
divcanap7 $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\
|
|
( C e. CC /\ C # 0 ) ) -> ( ( A / C ) / ( B / C ) ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv cmul wceq divdivdivap 3impdir mulcom adantrr
|
|
wa co 3adant2 oveq1d divcanap5 3eqtrd ) ADEZBDEBFGHQZCDEZCFGHZQZIZACJRBCJRJ
|
|
RZACKRZCBKRZJRZCAKRZUKJRABJRUCUGUDUIULLACBCMNUHUJUMUKJUCUGUJUMLZUDUCUEUNUFA
|
|
COPSTABCUAUB $.
|
|
|
|
$( Cancellation law for division and multiplication. (Contributed by Jim
|
|
Kingdon, 26-Feb-2020.) $)
|
|
dmdcanap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\
|
|
C e. CC ) ->
|
|
( ( A / B ) x. ( C / A ) ) = ( C / B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co cmul wceq simp1l simp3 divclap syl3anc
|
|
simp1r simp2l simp2r div23ap syl112anc divcanap2 oveq1d eqtr3d ) ADEZAFGHZI
|
|
ZBDEZBFGHZIZCDEZJZACAKLZMLZBKLZABKLUOMLZCBKLUNUGUODEZUJUKUQURNUGUHULUMOZUNU
|
|
MUGUHUSUIULUMPZUTUGUHULUMSZCAQRUIUJUKUMTUIUJUKUMUAAUOBUBUCUNUPCBKUNUMUGUHUP
|
|
CNVAUTVBCAUDRUEUF $.
|
|
|
|
$( Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
divdivap1 $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\
|
|
( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a cdiv co c1 cmul wceq ax-1cn pm3.2i divdivdivap
|
|
1ap0 mpanr2 3impa oveq2d ad2antrl 3adant1 mulid1 oveq1d 3ad2ant1 3eqtr3d
|
|
div1 ) ADEZBDEBFGHIZCDEZCFGHZIZJABKLZCMKLZKLZAMNLZBCNLZKLZUOCKLZAUSKLZUJUKU
|
|
NUQUTOZUJUKIUNMDEZMFGHZIVCVDVEPSQABCMRTUAUKUNUQVAOZUJULVFUKUMULUPCUOKCUIUBU
|
|
CUDUJUKUTVBOUNUJURAUSKAUEUFUGUH $.
|
|
|
|
$( Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
divdivap2 $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 )
|
|
) -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) ) $=
|
|
( cc wcel cc0 cap wbr wa w3a c1 cdiv co cmul wceq ax-1cn pm3.2i divdivdivap
|
|
1ap0 mpanl2 3impb 3ad2ant1 oveq1d mulid2 ad2antrl 3adant3 oveq2d 3eqtr3d
|
|
div1 ) ADEZBDEZBFGHZIZCDECFGHIZJZAKLMZBCLMZLMZACNMZKBNMZLMZAUQLMUSBLMUJUMUN
|
|
URVAOZUJKDEZKFGHZIUMUNIVBVCVDPSQAKBCRTUAUOUPAUQLUJUMUPAOUNAUIUBUCUOUTBUSLUJ
|
|
UMUTBOZUNUKVEUJULBUDUEUFUGUH $.
|
|
|
|
$( Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
recdivap2 $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( ( 1 / A ) / B ) = ( 1 / ( A x. B ) ) ) $=
|
|
( c1 cc wcel cc0 cap wbr wa cdiv co cmul wceq ax-1cn divdivap1 mp3an1 ) CDE
|
|
ADEAFGHIBDEBFGHICAJKBJKCABLKJKMNCABOP $.
|
|
|
|
$( Cancellation in a double division. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
ddcanap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ->
|
|
( A / ( A / B ) ) = B ) $=
|
|
( cc wcel cc0 cap wbr cdiv wceq cmul simpll simprl simprr divcanap1 syl3anc
|
|
wa co wb divclap divap0 divmulap syl112anc mpbird ) ACDZAEFGZPZBCDZBEFGZPZP
|
|
ZAABHQZHQBIZUKBJQAIZUJUDUGUHUMUDUEUIKZUFUGUHLZUFUGUHMZABNOUJUDUGUKCDZUKEFGU
|
|
LUMRUNUOUJUDUGUHUQUNUOUPABSOABTABUKUAUBUC $.
|
|
|
|
$( Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
divadddivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 )
|
|
/\ ( D e. CC /\ D # 0 ) ) ) ->
|
|
( ( A / C ) + ( B / D ) ) = ( ( ( A x. D ) + ( B x. C ) ) / ( C x. D ) ) ) $=
|
|
( cc wcel wa cc0 cap cmul co caddc cdiv wceq mulcl ad2ant2r syl3anc mulcomd
|
|
wbr oveq12d adantrl adantrr mulap0 jca adantl divdirap simpll simprr simpld
|
|
ad2ant2lr simprll simprl divcanap5 eqtrd simplr oveq1d eqtr2d ) AEFZBEFZGZC
|
|
EFZCHISZGZDEFZDHISZGZGZGZADJKZBCJKZLKCDJKZMKZVIVKMKZVJVKMKZLKZACMKZBDMKZLKV
|
|
HVIEFZVJEFZVKEFZVKHISZGZVLVONUTVFVRVCURVDVRUSVEADOPUAUSVCVSURVFUSVAVSVBBCOU
|
|
BUJVGWBUTVGVTWAVAVDVTVBVECDOPCDUCUDUEVIVJVKUFQVHVMVPVNVQLVHVMDAJKZDCJKZMKZV
|
|
PVHVIWCVKWDMVHADURUSVGUGZVHVDVEUTVCVFUHZUIZRVHCDUTVAVBVFUKZWHRTVHURVCVFWEVP
|
|
NWFUTVCVFULZWGACDUMQUNVHVNCBJKZVKMKZVQVHVJWKVKMVHBCURUSVGUOZWIRUPVHUSVFVCWL
|
|
VQNWMWGWJBDCUMQUNTUQ $.
|
|
|
|
$( Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
divsubdivap $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 )
|
|
/\ ( D e. CC /\ D # 0 ) ) ) ->
|
|
( ( A / C ) - ( B / D ) ) = ( ( ( A x. D ) - ( B x. C ) ) / ( C x. D ) ) ) $=
|
|
( cc wcel wa cc0 cap wbr cmul co cneg caddc cdiv cmin syl3anc oveq2d eqtr3d
|
|
wceq sylanl2 simplr simprrl simprrr divnegap simpll simprll simprlr divclap
|
|
negcl divadddivap negsubd mulneg1d mulcld eqtrd oveq1d ) AEFZBEFZGZCEFZCHIJ
|
|
ZGZDEFZDHIJZGZGZGZADKLZBMZCKLZNLZCDKLZOLZACOLZBDOLZPLZVHBCKLZPLZVLOLVGVNVID
|
|
OLZNLZVMVPURUQVIEFVFVTVMTBUJAVICDUKUAVGVNVOMZNLVTVPVGWAVSVNNVGURVCVDWAVSTUQ
|
|
URVFUBZUSVBVCVDUCZUSVBVCVDUDZBDUEQRVGVNVOVGUQUTVAVNEFUQURVFUFZUSUTVAVEUGZUS
|
|
UTVAVEUHACUIQVGURVCVDVOEFWBWCWDBDUIQULSSVGVKVRVLOVGVKVHVQMZNLVRVGVJWGVHNVGB
|
|
CWBWFUMRVGVHVQVGADWEWCUNVGBCWBWFUNULUOUPS $.
|
|
|
|
$( Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy
|
|
this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.) $)
|
|
conjmulap $p |- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) ->
|
|
( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) ) $=
|
|
( cc wcel cc0 cap wa cmul co c1 cdiv caddc wceq cmin recclap adantr recidap
|
|
wbr 3eqtrd ad2ant2r simpll simprl mul32d oveq1d mulid2 adantl oveq2d mulid1
|
|
ad2antrl mulassd ad2antrr oveq12d mulcl adddid addcom 3eqtr4d mulid1d addcl
|
|
eqeq12d syl2an mulap0 ax-1cn mulcanap mp3an2 syl12anc eqcom muleqadd syl5bb
|
|
wb 3bitr3d ) ACDZAEFRZGZBCDZBEFRZGZGZABHIZJAKIZJBKIZLIZHIZVRJHIZMZABLIZVRMZ
|
|
WAJMZAJNIBJNIHIJMZVQWBWEWCVRVQVRVSHIZVRVTHIZLIBALIZWBWEVQWIBWJALVQWIAVSHIZB
|
|
HIZJBHIZBVQABVSVKVLVPUAZVMVNVOUBZVMVSCDZVPAOZPZUCVMWMWNMVPVMWLJBHAQUDPVNWNB
|
|
MVMVOBUEUISVQWJABVTHIZHIZAJHIZAVQABVTWOWPVPVTCDZVMBOZUFZUJVPXAXBMVMVPWTJAHB
|
|
QUGUFVKXBAMVLVPAUHUKSULVQVRVSVTVKVNVRCDZVLVOABUMZTZWSXEUNVKVNWEWKMVLVOABUOT
|
|
UPVKVNWCVRMVLVOVKVNGZVRXGUQTUSVQWACDZXFVREFRZWDWGVIZVMWQXCXJVPWRXDVSVTURUTX
|
|
HABVAXJJCDXFXKGXLVBWAJVRVCVDVEVKVNWFWHVIVLVOWFVRWEMXIWHWEVRVFABVGVHTVJ $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
rerecclap $p |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR ) $=
|
|
( vx cr wcel cc0 cap wbr wa cv c1 cdiv co wceq wrex creap wb 0re cc recnd
|
|
cmul apreap mpan2 pm5.32i recexre sylbi eqcom 1cnd simpll simplr divmulap
|
|
simpr syl112anc syl5bb rexbidva mpbird risset sylibr ) ACDZAEFGZHZBIZJAKL
|
|
ZMZBCNZVBCDUTVDAVATLJMZBCNZUTURAEOGZHVFURUSVGURECDUSVGPQAEUAUBUCBAUDUEUTV
|
|
CVEBCVCVBVAMZUTVACDZHZVEVAVBUFVJJRDVARDARDUSVHVEPVJUGVJVAUTVIUKSVJAURUSVI
|
|
UHSURUSVIUIJVAAUJULUMUNUOBVBCUPUQ $.
|
|
$}
|
|
|
|
$( Closure law for division of reals. (Contributed by Jim Kingdon,
|
|
26-Feb-2020.) $)
|
|
redivclap $p |- ( ( A e. RR /\ B e. RR /\ B # 0 ) -> ( A / B ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr cdiv co c1 cmul wceq simp1 recnd simp2 simp3 divrecap
|
|
w3a cc syl3anc rerecclap 3adant1 remulcld eqeltrd ) ACDZBCDZBEFGZRZABHIZAJB
|
|
HIZKIZCUHASDBSDUGUIUKLUHAUEUFUGMZNUHBUEUFUGONUEUFUGPABQTUHAUJULUFUGUJCDUEBU
|
|
AUBUCUD $.
|
|
|
|
$( A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.)
|
|
(Revised by Mario Carneiro, 27-May-2016.) $)
|
|
eqneg $p |- ( A e. CC -> ( A = -u A <-> A = 0 ) ) $=
|
|
( cc wcel c1 caddc cmul cc0 wceq cneg 1p1times negid ax-1cn addcli syl6reqr
|
|
co mul01i eqeq12d a1i 1re 0lt1 id 0cnd readdcli addgt0ii gt0ap0ii mulcanapd
|
|
cap wbr negcl addcand 3bitr3rd ) ABCZDDEOZAFOZUMGFOZHAAEOZAAIZEOZHAGHAUQHUL
|
|
UNUPUOURAJULURGUOAKUMDDLLMZPNQULAGUMULUAZULUBUMBCULUSRUMGUGUHULUMDDSSUCDDSS
|
|
TTUDUERUFULAAUQUTUTAUIUJUK $.
|
|
|
|
${
|
|
eqnegd.1 $e |- ( ph -> A e. CC ) $.
|
|
$( A complex number equals its negative iff it is zero. Deduction form of
|
|
~ eqneg . (Contributed by David Moews, 28-Feb-2017.) $)
|
|
eqnegd $p |- ( ph -> ( A = -u A <-> A = 0 ) ) $=
|
|
( cc wcel cneg wceq cc0 wb eqneg syl ) ABDEBBFGBHGICBJK $.
|
|
$}
|
|
|
|
${
|
|
eqnegad.1 $e |- ( ph -> A e. CC ) $.
|
|
eqnegad.2 $e |- ( ph -> A = -u A ) $.
|
|
$( If a complex number equals its own negative, it is zero. One-way
|
|
deduction form of ~ eqneg . (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
eqnegad $p |- ( ph -> A = 0 ) $=
|
|
( cneg wceq cc0 eqnegd mpbid ) ABBEFBGFDABCHI $.
|
|
$}
|
|
|
|
$( Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) $)
|
|
div2negap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( -u A / -u B ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cneg cdiv co wceq cmul 3ad2ant2 syl112anc 3adant1
|
|
negcl c1 eqtrd 3ad2ant1 simp1 simp2 simp3 div12ap divnegap syld3an1 dividap
|
|
negeqd eqtr3d oveq2d ax-1cn negcli mulcom mpan2 mulm1 divclap negap0 biimpa
|
|
wb divmulap mpbird ) ACDZBCDZBEFGZHZAIZBIZJKABJKZLZVGVHMKZVFLZVEVJAVGBJKZMK
|
|
ZVFVEVGCDZVBVCVDVJVMLVCVBVNVDBQNZVBVCVDUAVBVCVDUBZVBVCVDUCVGABUDOVEVMARIZMK
|
|
ZVFVEVLVQAMVEBBJKZIZVLVQVCVCVBVDVTVLLVPBBUEUFVEVSRVCVDVSRLVBBUGPUHUIUJVBVCV
|
|
RVFLVDVBVRVQAMKZVFVBVQCDVRWALRUKULAVQUMUNAUOSTSSVEVFCDZVHCDVNVGEFGZVIVKUSVB
|
|
VCWBVDAQTABUPVOVCVDWCVBVCVDWCBUQURPVFVHVGUTOVA $.
|
|
|
|
$( Move negative sign inside of a division. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
divneg2ap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
-u ( A / B ) = ( A / -u B ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co cneg divnegap wceq negcl div2negap negneg
|
|
syl3an1 3ad2ant1 oveq1d 3eqtr2d ) ACDZBCDZBEFGZHZABIJKAKZBIJZUEKZBKZIJZAUHI
|
|
JABLUAUECDUBUCUIUFMANUEBOQUDUGAUHIUAUBUGAMUCAPRST $.
|
|
|
|
${
|
|
divclz.1 $e |- A e. CC $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
recclapzi $p |- ( A # 0 -> ( 1 / A ) e. CC ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co recclap mpan ) ACDAEFGHAIJCDBAKL $.
|
|
|
|
$( The reciprocal of a number apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 27-Feb-2020.) $)
|
|
recap0apzi $p |- ( A # 0 -> ( 1 / A ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co recap0 mpan ) ACDAEFGHAIJEFGBAKL $.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by Jim
|
|
Kingdon, 27-Feb-2020.) $)
|
|
recidapzi $p |- ( A # 0 -> ( A x. ( 1 / A ) ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap mpan ) ACDAEFGAHAIJKJH
|
|
LBAMN $.
|
|
|
|
$( A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) $)
|
|
div1i $p |- ( A / 1 ) = A $=
|
|
( cc wcel c1 cdiv co wceq div1 ax-mp ) ACDAEFGAHBAIJ $.
|
|
|
|
$( A number equal to its negative is zero. (Contributed by NM,
|
|
29-May-1999.) $)
|
|
eqnegi $p |- ( A = -u A <-> A = 0 ) $=
|
|
( cc wcel cneg wceq cc0 wb eqneg ax-mp ) ACDAAEFAGFHBAIJ $.
|
|
|
|
${
|
|
recclap.2 $e |- A # 0 $.
|
|
$( Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) $)
|
|
recclapi $p |- ( 1 / A ) e. CC $=
|
|
( cc0 cap wbr c1 cdiv co cc wcel recclapzi ax-mp ) ADEFGAHIJKCABLM $.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by NM,
|
|
9-Feb-1995.) $)
|
|
recidapi $p |- ( A x. ( 1 / A ) ) = 1 $=
|
|
( cc0 cap wbr c1 cdiv co cmul wceq recidapzi ax-mp ) ADEFAGAHIJIGKCABLM
|
|
$.
|
|
|
|
$( A number is equal to the reciprocal of its reciprocal. Theorem I.10
|
|
of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.) $)
|
|
recrecapi $p |- ( 1 / ( 1 / A ) ) = A $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co wceq recrecap mp2an ) ADEAFGHIIAJKJKAL
|
|
BCAMN $.
|
|
|
|
$( A number divided by itself is one. (Contributed by NM,
|
|
9-Feb-1995.) $)
|
|
dividapi $p |- ( A / A ) = 1 $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 wceq dividap mp2an ) ADEAFGHAAIJKLBCAM
|
|
N $.
|
|
|
|
$( Division into zero is zero. (Contributed by NM, 12-Aug-1999.) $)
|
|
div0api $p |- ( 0 / A ) = 0 $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq div0ap mp2an ) ADEAFGHFAIJFKBCALM $.
|
|
$}
|
|
|
|
divclz.2 $e |- B e. CC $.
|
|
$( Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) $)
|
|
divclapzi $p |- ( B # 0 -> ( A / B ) e. CC ) $=
|
|
( cc wcel cc0 cap wbr cdiv co divclap mp3an12 ) AEFBEFBGHIABJKEFCDABLM $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
divcanap1zi $p |- ( B # 0 -> ( ( A / B ) x. B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 mp3an12 ) AEFBEFBGHIABJ
|
|
KBLKAMCDABNO $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
divcanap2zi $p |- ( B # 0 -> ( B x. ( A / B ) ) = A ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 mp3an12 ) AEFBEFBGHIBAB
|
|
JKLKAMCDABNO $.
|
|
|
|
$( Relationship between division and reciprocal. (Contributed by Jim
|
|
Kingdon, 27-Feb-2020.) $)
|
|
divrecapzi $p |- ( B # 0 -> ( A / B ) = ( A x. ( 1 / B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap mp3an12 ) AEFBEFBGHIA
|
|
BJKALBJKMKNCDABOP $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
divcanap3zi $p |- ( B # 0 -> ( ( B x. A ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 mp3an12 ) AEFBEFBGHIBAJ
|
|
KBLKAMCDABNO $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
27-Feb-2020.) $)
|
|
divcanap4zi $p |- ( B # 0 -> ( ( A x. B ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 mp3an12 ) AEFBEFBGHIABJ
|
|
KBLKAMCDABNO $.
|
|
|
|
$( Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.) $)
|
|
rec11api $p |- ( ( A # 0 /\ B # 0 ) ->
|
|
( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) $=
|
|
( cc0 cap wbr cc wcel c1 cdiv co wceq wb wa rec11ap mpanl1 mpanr1 ) AEFGZ
|
|
BHIZBEFGZJAKLJBKLMABMNZDAHISTUAOUBCABPQR $.
|
|
|
|
${
|
|
divclap.3 $e |- B # 0 $.
|
|
$( Closure law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divclapi $p |- ( A / B ) e. CC $=
|
|
( cc wcel cc0 cap wbr cdiv co divclap mp3an ) AFGBFGBHIJABKLFGCDEABMN
|
|
$.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divcanap2i $p |- ( B x. ( A / B ) ) = A $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 mp3an ) AFGBFGBHIJBAB
|
|
KLMLANCDEABOP $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divcanap1i $p |- ( ( A / B ) x. B ) = A $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 mp3an ) AFGBFGBHIJABK
|
|
LBMLANCDEABOP $.
|
|
|
|
$( Relationship between division and reciprocal. (Contributed by Jim
|
|
Kingdon, 28-Feb-2020.) $)
|
|
divrecapi $p |- ( A / B ) = ( A x. ( 1 / B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap mp3an ) AFGBFGBHIJA
|
|
BKLAMBKLNLOCDEABPQ $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divcanap3i $p |- ( ( B x. A ) / B ) = A $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 mp3an ) AFGBFGBHIJBAK
|
|
LBMLANCDEABOP $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divcanap4i $p |- ( ( A x. B ) / B ) = A $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 mp3an ) AFGBFGBHIJABK
|
|
LBMLANCDEABOP $.
|
|
$}
|
|
|
|
${
|
|
divap0i.3 $e |- A # 0 $.
|
|
divap0.i4 $e |- B # 0 $.
|
|
$( The ratio of numbers apart from zero is apart from zero. (Contributed
|
|
by Jim Kingdon, 28-Feb-2020.) $)
|
|
divap0i $p |- ( A / B ) # 0 $=
|
|
( cc wcel cc0 cap wbr cdiv co divap0 mp4an ) AGHAIJKBGHBIJKABLMIJKCEDFA
|
|
BNO $.
|
|
|
|
$( Reciprocal is one-to-one. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
rec11apii $p |- ( ( 1 / A ) = ( 1 / B ) <-> A = B ) $=
|
|
( cc0 cap wbr c1 cdiv co wceq wb rec11api mp2an ) AGHIBGHIJAKLJBKLMABMN
|
|
EFABCDOP $.
|
|
$}
|
|
|
|
divmulz.3 $e |- C e. CC $.
|
|
$( An associative law for division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divassapzi $p |- ( C # 0 -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq wa divassap mp3an12 mpan ) CGHZCI
|
|
JKZABLMCNMABCNMLMOZFAGHBGHTUAPUBDEABCQRS $.
|
|
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 28-Feb-2020.) $)
|
|
divmulapzi $p |- ( B # 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq cmul wb wa divmulap mp3an12 mpan ) BGH
|
|
ZBIJKZABLMCNBCOMANPZEAGHCGHUAUBQUCDFACBRST $.
|
|
|
|
$( Distribution of division over addition. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divdirapzi $p |- ( C # 0 ->
|
|
( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr caddc co cdiv wceq wa divdirap mp3an12 mpan ) CGHZC
|
|
IJKZABLMCNMACNMBCNMLMOZFAGHBGHTUAPUBDEABCQRS $.
|
|
|
|
$( Swap denominators in a division. (Contributed by Jim Kingdon,
|
|
28-Feb-2020.) $)
|
|
divdiv23apzi $p |- ( ( B # 0 /\ C # 0 ) ->
|
|
( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
|
|
( cc0 cap wbr cc wcel cdiv co wceq wa divdiv32ap mp3an1 mpanl1 mpanr1 ) B
|
|
GHIZCJKZCGHIZABLMCLMACLMBLMNZFBJKZTUAUBOZUCEAJKUDTOUEUCDABCPQRS $.
|
|
|
|
${
|
|
divmulapi.4 $e |- B # 0 $.
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 29-Feb-2020.) $)
|
|
divmulapi $p |- ( ( A / B ) = C <-> ( B x. C ) = A ) $=
|
|
( cc0 cap wbr cdiv co wceq cmul wb divmulapzi ax-mp ) BHIJABKLCMBCNLAMO
|
|
GABCDEFPQ $.
|
|
|
|
divdiv32api.5 $e |- C # 0 $.
|
|
$( Swap denominators in a division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divdiv32api $p |- ( ( A / B ) / C ) = ( ( A / C ) / B ) $=
|
|
( cc0 cap wbr cdiv co wceq divdiv23apzi mp2an ) BIJKCIJKABLMCLMACLMBLMN
|
|
GHABCDEFOP $.
|
|
$}
|
|
|
|
${
|
|
divassap.4 $e |- C # 0 $.
|
|
$( An associative law for division. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
divassapi $p |- ( ( A x. B ) / C ) = ( A x. ( B / C ) ) $=
|
|
( cc wcel cc0 cap wbr wa cmul co cdiv wceq pm3.2i divassap mp3an ) AHIB
|
|
HICHIZCJKLZMABNOCPOABCPONOQDEUAUBFGRABCST $.
|
|
|
|
$( Distribution of division over addition. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
divdirapi $p |- ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) $=
|
|
( cc wcel cc0 cap wbr wa caddc co cdiv wceq pm3.2i divdirap mp3an ) AHI
|
|
BHICHIZCJKLZMABNOCPOACPOBCPONOQDEUAUBFGRABCST $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim
|
|
Kingdon, 9-Mar-2020.) $)
|
|
div23api $p |- ( ( A x. B ) / C ) = ( ( A / C ) x. B ) $=
|
|
( cc wcel cc0 cap wbr wa cmul co cdiv wceq pm3.2i div23ap mp3an ) AHIBH
|
|
ICHIZCJKLZMABNOCPOACPOBNOQDEUAUBFGRABCST $.
|
|
|
|
$( One-to-one relationship for division. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
div11api $p |- ( ( A / C ) = ( B / C ) <-> A = B ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co wceq wb pm3.2i div11ap mp3an ) AHIBHIC
|
|
HIZCJKLZMACNOBCNOPABPQDEUAUBFGRABCST $.
|
|
$}
|
|
|
|
divmuldivap.4 $e |- D e. CC $.
|
|
divmuldivap.5 $e |- B # 0 $.
|
|
divmuldivap.6 $e |- D # 0 $.
|
|
$( Multiplication of two ratios. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
divmuldivapi $p |- ( ( A / B ) x. ( C / D ) ) =
|
|
( ( A x. C ) / ( B x. D ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul pm3.2i wceq divmuldivap mp4an ) AKL
|
|
CKLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRACSRBDSRQRUAEGUDUEFITUFUGHJTACBDUBUC $.
|
|
|
|
$( Swap denominators of two ratios. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
divmul13api $p |- ( ( A / B ) x. ( C / D ) ) =
|
|
( ( C / B ) x. ( A / D ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul pm3.2i wceq divmul13ap mp4an ) AKLC
|
|
KLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRCBQRADQRSRUAEGUDUEFITUFUGHJTACBDUBUC $.
|
|
|
|
$( Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) $)
|
|
divadddivapi $p |- ( ( A / B ) + ( C / D ) ) =
|
|
( ( ( A x. D ) + ( C x. B ) ) / ( B x. D ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co caddc cmul wceq pm3.2i divadddivap mp4an
|
|
) AKLCKLBKLZBMNOZPDKLZDMNOZPABQRCDQRSRADTRCBTRSRBDTRQRUAEGUEUFFIUBUGUHHJU
|
|
BACBDUCUD $.
|
|
|
|
divdivdivap.7 $e |- C # 0 $.
|
|
$( Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) $)
|
|
divdivdivapi $p |- ( ( A / B ) / ( C / D ) )
|
|
= ( ( A x. D ) / ( B x. C ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co pm3.2i cmul wceq divdivdivap mp4an ) ALM
|
|
BLMZBNOPZQCLMZCNOPZQDLMZDNOPZQABRSCDRSRSADUASBCUASRSUBEUEUFFITUGUHGKTUIUJ
|
|
HJTABCDUCUD $.
|
|
$}
|
|
|
|
${
|
|
redivclap.1 $e |- A e. RR $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
rerecclapzi $p |- ( A # 0 -> ( 1 / A ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr c1 cdiv co rerecclap mpan ) ACDAEFGHAIJCDBAKL $.
|
|
|
|
${
|
|
rerecclap.2 $e |- A # 0 $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
rerecclapi $p |- ( 1 / A ) e. RR $=
|
|
( cc0 cap wbr c1 cdiv co cr wcel rerecclapzi ax-mp ) ADEFGAHIJKCABLM $.
|
|
$}
|
|
|
|
redivclap.2 $e |- B e. RR $.
|
|
$( Closure law for division of reals. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
redivclapzi $p |- ( B # 0 -> ( A / B ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr cdiv co redivclap mp3an12 ) AEFBEFBGHIABJKEFCDABLM
|
|
$.
|
|
|
|
redivclap.3 $e |- B # 0 $.
|
|
$( Closure law for division of reals. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
redivclapi $p |- ( A / B ) e. RR $=
|
|
( cc0 cap wbr cdiv co cr wcel redivclapzi ax-mp ) BFGHABIJKLEABCDMN $.
|
|
$}
|
|
|
|
${
|
|
div1d.1 $e |- ( ph -> A e. CC ) $.
|
|
$( A number divided by 1 is itself. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
div1d $p |- ( ph -> ( A / 1 ) = A ) $=
|
|
( cc wcel c1 cdiv co wceq div1 syl ) ABDEBFGHBICBJK $.
|
|
|
|
${
|
|
reccld.2 $e |- ( ph -> A # 0 ) $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
recclapd $p |- ( ph -> ( 1 / A ) e. CC ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co recclap syl2anc ) ABEFBGHIJBKLEFCDBMN
|
|
$.
|
|
|
|
$( The reciprocal of a number apart from zero is apart from zero.
|
|
(Contributed by Jim Kingdon, 3-Mar-2020.) $)
|
|
recap0d $p |- ( ph -> ( 1 / A ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co recap0 syl2anc ) ABEFBGHIJBKLGHICDBMN
|
|
$.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by Jim
|
|
Kingdon, 3-Mar-2020.) $)
|
|
recidapd $p |- ( ph -> ( A x. ( 1 / A ) ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap syl2anc ) ABEFBGHIBJ
|
|
BKLMLJNCDBOP $.
|
|
|
|
$( Multiplication of a number and its reciprocal. (Contributed by Jim
|
|
Kingdon, 3-Mar-2020.) $)
|
|
recidap2d $p |- ( ph -> ( ( 1 / A ) x. A ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recidap2 syl2anc ) ABEFBGHIJ
|
|
BKLBMLJNCDBOP $.
|
|
|
|
$( A number is equal to the reciprocal of its reciprocal. (Contributed
|
|
by Jim Kingdon, 3-Mar-2020.) $)
|
|
recrecapd $p |- ( ph -> ( 1 / ( 1 / A ) ) = A ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co wceq recrecap syl2anc ) ABEFBGHIJJBKLK
|
|
LBMCDBNO $.
|
|
|
|
$( A number divided by itself is one. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
dividapd $p |- ( ph -> ( A / A ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 wceq dividap syl2anc ) ABEFBGHIBBJKLMC
|
|
DBNO $.
|
|
|
|
$( Division into zero is zero. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
div0apd $p |- ( ph -> ( 0 / A ) = 0 ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq div0ap syl2anc ) ABEFBGHIGBJKGLCDBMN
|
|
$.
|
|
$}
|
|
$}
|
|
|
|
$( Multiplication of both sides of complex apartness by a complex number
|
|
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.) $)
|
|
apmul1 $p |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) )
|
|
-> ( A # B <-> ( A x. C ) # ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cmul co c1 mulassd oveq2d mulid1d 3eqtrd wi mulcld
|
|
w3a mulext1 cdiv simp1 simp3l simp3r recclapd simp2 breq12d syl3anc sylbird
|
|
recidapd 3adant3r impbid ) ADEZBDEZCDEZCFGHZIZSZABGHZACJKZBCJKZGHZURUSUTLCU
|
|
AKZJKZVAVCJKZGHZVBURVDAVEBGURVDACVCJKZJKALJKAURACVCUMUNUQUBZUMUNUOUPUCZURCV
|
|
IUMUNUOUPUDZUEZMURVGLAJURCVIVJUJZNURAVHOPURVEBVGJKBLJKBURBCVCUMUNUQUFZVIVKM
|
|
URVGLBJVLNURBVMOPUGURUTDEVADEVCDEVFVBQURACVHVIRURBCVMVIRVKUTVAVCTUHUIUMUNUO
|
|
VBUSQUPABCTUKUL $.
|
|
|
|
${
|
|
divcld.1 $e |- ( ph -> A e. CC ) $.
|
|
divcld.2 $e |- ( ph -> B e. CC ) $.
|
|
${
|
|
divclapd.3 $e |- ( ph -> B # 0 ) $.
|
|
$( Closure law for division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divclapd $p |- ( ph -> ( A / B ) e. CC ) $=
|
|
( cc wcel cc0 cap wbr cdiv co divclap syl3anc ) ABGHCGHCIJKBCLMGHDEFBCN
|
|
O $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divcanap1d $p |- ( ph -> ( ( A / B ) x. B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap1 syl3anc ) ABGHCGHCIJK
|
|
BCLMCNMBODEFBCPQ $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divcanap2d $p |- ( ph -> ( B x. ( A / B ) ) = A ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divcanap2 syl3anc ) ABGHCGHCIJK
|
|
CBCLMNMBODEFBCPQ $.
|
|
|
|
$( Relationship between division and reciprocal. Theorem I.9 of
|
|
[Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.) $)
|
|
divrecapd $p |- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap syl3anc ) ABGHCGHCI
|
|
JKBCLMBNCLMOMPDEFBCQR $.
|
|
|
|
$( Relationship between division and reciprocal. (Contributed by Jim
|
|
Kingdon, 29-Feb-2020.) $)
|
|
divrecap2d $p |- ( ph -> ( A / B ) = ( ( 1 / B ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 cmul wceq divrecap2 syl3anc ) ABGHCGHC
|
|
IJKBCLMNCLMBOMPDEFBCQR $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divcanap3d $p |- ( ph -> ( ( B x. A ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap3 syl3anc ) ABGHCGHCIJK
|
|
CBLMCNMBODEFBCPQ $.
|
|
|
|
$( A cancellation law for division. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
divcanap4d $p |- ( ph -> ( ( A x. B ) / B ) = A ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap4 syl3anc ) ABGHCGHCIJK
|
|
BCLMCNMBODEFBCPQ $.
|
|
|
|
${
|
|
diveqap0d.4 $e |- ( ph -> ( A / B ) = 0 ) $.
|
|
$( If a ratio is zero, the numerator is zero. (Contributed by Jim
|
|
Kingdon, 19-Mar-2020.) $)
|
|
diveqap0d $p |- ( ph -> A = 0 ) $=
|
|
( cdiv co cc0 wceq cc wcel cap wbr wb diveqap0 syl3anc mpbid ) ABCHIJ
|
|
KZBJKZGABLMCLMCJNOTUAPDEFBCQRS $.
|
|
$}
|
|
|
|
${
|
|
diveqap1d.4 $e |- ( ph -> ( A / B ) = 1 ) $.
|
|
$( Equality in terms of unit ratio. (Contributed by Jim Kingdon,
|
|
19-Mar-2020.) $)
|
|
diveqap1d $p |- ( ph -> A = B ) $=
|
|
( cdiv co c1 wceq cc wcel cc0 cap wbr wb diveqap1 syl3anc mpbid ) ABC
|
|
HIJKZBCKZGABLMCLMCNOPUAUBQDEFBCRST $.
|
|
$}
|
|
|
|
$( The quotient of two complex numbers is one iff they are equal.
|
|
Deduction form of ~ diveqap1 . Generalization of ~ diveqap1d .
|
|
(Contributed by Jim Kingdon, 19-Mar-2020.) $)
|
|
diveqap1ad $p |- ( ph -> ( ( A / B ) = 1 <-> A = B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c1 wceq wb diveqap1 syl3anc ) ABGHCGHCIJK
|
|
BCLMNOBCOPDEFBCQR $.
|
|
|
|
$( A fraction of complex numbers is zero iff its numerator is. Deduction
|
|
form of ~ diveqap0 . (Contributed by Jim Kingdon, 19-Mar-2020.) $)
|
|
diveqap0ad $p |- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq wb diveqap0 syl3anc ) ABGHCGHCIJKBCL
|
|
MINBINODEFBCPQ $.
|
|
|
|
${
|
|
divap1d.4 $e |- ( ph -> A # B ) $.
|
|
$( If two complex numbers are apart, their quotient is apart from one.
|
|
(Contributed by Jim Kingdon, 20-Mar-2020.) $)
|
|
divap1d $p |- ( ph -> ( A / B ) # 1 ) $=
|
|
( c1 cdiv co cmul cap wbr cc wcel cc0 wb recclapd recap0d apmul1
|
|
syl112anc mpbid divrecapd eqcomd recidapd 3brtr3d ) ABHCIJZKJZCUGKJZB
|
|
CIJZHLABCLMZUHUILMZGABNOCNOUGNOUGPLMUKULQDEACEFRACEFSBCUGTUAUBAUJUHAB
|
|
CDEFUCUDACEFUEUF $.
|
|
$}
|
|
|
|
$( A ratio is zero iff the numerator is zero. (Contributed by Jim
|
|
Kingdon, 19-Mar-2020.) $)
|
|
divap0bd $p |- ( ph -> ( A # 0 <-> ( A / B ) # 0 ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wb divap0b syl3anc ) ABGHCGHCIJKBIJKBCLMI
|
|
JKNDEFBCOP $.
|
|
|
|
$( Move negative sign inside of a division. (Contributed by Jim Kingdon,
|
|
19-Mar-2020.) $)
|
|
divnegapd $p |- ( ph -> -u ( A / B ) = ( -u A / B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cneg wceq divnegap syl3anc ) ABGHCGHCIJKB
|
|
CLMNBNCLMODEFBCPQ $.
|
|
|
|
$( Move negative sign inside of a division. (Contributed by Jim Kingdon,
|
|
19-Mar-2020.) $)
|
|
divneg2apd $p |- ( ph -> -u ( A / B ) = ( A / -u B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cneg wceq divneg2ap syl3anc ) ABGHCGHCIJK
|
|
BCLMNBCNLMODEFBCPQ $.
|
|
|
|
$( Quotient of two negatives. (Contributed by Jim Kingdon,
|
|
19-Mar-2020.) $)
|
|
div2negapd $p |- ( ph -> ( -u A / -u B ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr cneg cdiv co wceq div2negap syl3anc ) ABGHCGHCIJK
|
|
BLCLMNBCMNODEFBCPQ $.
|
|
$}
|
|
|
|
${
|
|
divap0d.3 $e |- ( ph -> A # 0 ) $.
|
|
divap0d.4 $e |- ( ph -> B # 0 ) $.
|
|
$( The ratio of numbers apart from zero is apart from zero. (Contributed
|
|
by Jim Kingdon, 3-Mar-2020.) $)
|
|
divap0d $p |- ( ph -> ( A / B ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr cdiv co divap0 syl22anc ) ABHIBJKLCHICJKLBCMNJKLD
|
|
FEGBCOP $.
|
|
|
|
$( The reciprocal of a ratio. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
recdivapd $p |- ( ph -> ( 1 / ( A / B ) ) = ( B / A ) ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co wceq recdivap syl22anc ) ABHIBJKLCHICJ
|
|
KLMBCNONOCBNOPDFEGBCQR $.
|
|
|
|
$( Division into a reciprocal. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
recdivap2d $p |- ( ph -> ( ( 1 / A ) / B ) = ( 1 / ( A x. B ) ) ) $=
|
|
( cc wcel cc0 cap wbr c1 cdiv co cmul wceq recdivap2 syl22anc ) ABHIBJK
|
|
LCHICJKLMBNOCNOMBCPONOQDFEGBCRS $.
|
|
|
|
$( Cancellation of inverted fractions. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
divcanap6d $p |- ( ph -> ( ( A / B ) x. ( B / A ) ) = 1 ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul c1 wceq divcanap6 syl22anc ) ABHIBJK
|
|
LCHICJKLBCMNCBMNONPQDFEGBCRS $.
|
|
|
|
$( Cancellation in a double division. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
ddcanapd $p |- ( ph -> ( A / ( A / B ) ) = B ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq ddcanap syl22anc ) ABHIBJKLCHICJKLBB
|
|
CMNMNCODFEGBCPQ $.
|
|
|
|
rec11d.5 $e |- ( ph -> ( 1 / A ) = ( 1 / B ) ) $.
|
|
$( Reciprocal is one-to-one. (Contributed by Jim Kingdon,
|
|
3-Mar-2020.) $)
|
|
rec11apd $p |- ( ph -> A = B ) $=
|
|
( c1 cdiv co wceq cc wcel cc0 cap wbr wb rec11ap syl22anc mpbid ) AIBJK
|
|
ICJKLZBCLZHABMNBOPQCMNCOPQUBUCRDFEGBCSTUA $.
|
|
$}
|
|
|
|
divmuld.3 $e |- ( ph -> C e. CC ) $.
|
|
${
|
|
divmulapd.4 $e |- ( ph -> B # 0 ) $.
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
divmulapd $p |- ( ph -> ( ( A / B ) = C <-> ( B x. C ) = A ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap syl112anc ) ABIJDIJ
|
|
CIJCKLMBCNODPCDQOBPREGFHBDCST $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
div32apd $p |- ( ph -> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq div32ap syl121anc ) ABIJCIJCKLM
|
|
DIJBCNODPOBDCNOPOQEFHGBCDRS $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
div13apd $p |- ( ph -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq div13ap syl121anc ) ABIJCIJCKLM
|
|
DIJBCNODPODCNOBPOQEFHGBCDRS $.
|
|
|
|
divdiv23apd.5 $e |- ( ph -> C # 0 ) $.
|
|
$( Swap denominators in a division. (Contributed by Jim Kingdon,
|
|
8-Mar-2020.) $)
|
|
divdiv32apd $p |- ( ph -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq divdiv32ap syl122anc ) ABJKCJKCLMNDJ
|
|
KDLMNBCOPDOPBDOPCOPQEFHGIBCDRS $.
|
|
|
|
$( Cancellation of common factor in a ratio. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
divcanap5d $p |- ( ph -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divcanap5 syl122anc ) ABJKCJKCL
|
|
MNDJKDLMNDBOPDCOPQPBCQPREFHGIBCDST $.
|
|
|
|
$( Cancellation of common factor in a ratio. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
divcanap5rd $p |- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( A / B ) ) $=
|
|
( cmul co cdiv mulcomd oveq12d divcanap5d eqtrd ) ABDJKZCDJKZLKDBJKZDCJ
|
|
KZLKBCLKAQSRTLABDEGMACDFGMNABCDEFGHIOP $.
|
|
|
|
$( Cancel equal divisors in a division. (Contributed by Jim Kingdon,
|
|
8-Mar-2020.) $)
|
|
divcanap7d $p |- ( ph -> ( ( A / C ) / ( B / C ) ) = ( A / B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq divcanap7 syl122anc ) ABJKCJKCLMNDJK
|
|
DLMNBDOPCDOPOPBCOPQEFHGIBCDRS $.
|
|
|
|
$( Cancellation law for division and multiplication. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
dmdcanapd $p |- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq dmdcanap syl221anc ) ACJKCLMNDJ
|
|
KDLMNBJKCDOPBCOPQPBDOPRFHGIECDBST $.
|
|
|
|
$( Cancellation law for division and multiplication. (Contributed by Jim
|
|
Kingdon, 8-Mar-2020.) $)
|
|
dmdcanap2d $p |- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( A / C ) ) $=
|
|
( cdiv co cmul divclapd mulcomd dmdcanapd eqtrd ) ABCJKZCDJKZLKRQLKBDJK
|
|
AQRABCEFHMACDFGIMNABCDEFGHIOP $.
|
|
|
|
$( Division into a fraction. (Contributed by Jim Kingdon,
|
|
8-Mar-2020.) $)
|
|
divdivap1d $p |- ( ph -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divdivap1 syl122anc ) ABJKCJKCL
|
|
MNDJKDLMNBCOPDOPBCDQPOPREFHGIBCDST $.
|
|
|
|
$( Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.) $)
|
|
divdivap2d $p |- ( ph -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq divdivap2 syl122anc ) ABJKCJKCL
|
|
MNDJKDLMNBCDOPOPBDQPCOPREFHGIBCDST $.
|
|
$}
|
|
|
|
${
|
|
divassapd.4 $e |- ( ph -> C # 0 ) $.
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 2-Mar-2020.) $)
|
|
divmulap2d $p |- ( ph -> ( ( A / C ) = B <-> A = ( C x. B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap2 syl112anc ) ABIJCI
|
|
JDIJDKLMBDNOCPBDCQOPREFGHBCDST $.
|
|
|
|
$( Relationship between division and multiplication. (Contributed by Jim
|
|
Kingdon, 2-Mar-2020.) $)
|
|
divmulap3d $p |- ( ph -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co wceq cmul wb divmulap3 syl112anc ) ABIJCI
|
|
JDIJDKLMBDNOCPBCDQOPREFGHBCDST $.
|
|
|
|
$( An associative law for division. (Contributed by Jim Kingdon,
|
|
2-Mar-2020.) $)
|
|
divassapd $p |- ( ph -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq divassap syl112anc ) ABIJCIJDIJ
|
|
DKLMBCNODPOBCDPONOQEFGHBCDRS $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim
|
|
Kingdon, 2-Mar-2020.) $)
|
|
div12apd $p |- ( ph -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul wceq div12ap syl112anc ) ABIJCIJDIJD
|
|
KLMBCDNOPOCBDNOPOQEFGHBCDRS $.
|
|
|
|
$( A commutative/associative law for division. (Contributed by Jim
|
|
Kingdon, 2-Mar-2020.) $)
|
|
div23apd $p |- ( ph -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) ) $=
|
|
( cc wcel cc0 cap wbr cmul co cdiv wceq div23ap syl112anc ) ABIJCIJDIJD
|
|
KLMBCNODPOBDPOCNOQEFGHBCDRS $.
|
|
|
|
$( Distribution of division over addition. (Contributed by Jim Kingdon,
|
|
2-Mar-2020.) $)
|
|
divdirapd $p |- ( ph ->
|
|
( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr caddc co cdiv wceq divdirap syl112anc ) ABIJCIJDI
|
|
JDKLMBCNODPOBDPOCDPONOQEFGHBCDRS $.
|
|
|
|
$( Distribution of division over subtraction. (Contributed by Jim
|
|
Kingdon, 2-Mar-2020.) $)
|
|
divsubdirapd $p |- ( ph ->
|
|
( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) ) $=
|
|
( cc wcel cc0 cap wbr cmin co cdiv wceq divsubdirap syl112anc ) ABIJCIJ
|
|
DIJDKLMBCNODPOBDPOCDPONOQEFGHBCDRS $.
|
|
|
|
div11apd.5 $e |- ( ph -> ( A / C ) = ( B / C ) ) $.
|
|
$( One-to-one relationship for division. (Contributed by Jim Kingdon,
|
|
2-Mar-2020.) $)
|
|
div11apd $p |- ( ph -> A = B ) $=
|
|
( cdiv co wceq cc wcel cc0 cap wbr wb div11ap syl112anc mpbid ) ABDJKCD
|
|
JKLZBCLZIABMNCMNDMNDOPQUBUCREFGHBCDSTUA $.
|
|
$}
|
|
|
|
divmuldivapd.4 $e |- ( ph -> D e. CC ) $.
|
|
divmuldivapd.5 $e |- ( ph -> B # 0 ) $.
|
|
divmuldivapd.6 $e |- ( ph -> D # 0 ) $.
|
|
$( Multiplication of two ratios. (Contributed by Jim Kingdon,
|
|
30-Jul-2021.) $)
|
|
divmuldivapd $p |- ( ph -> ( ( A / B ) x. ( C / D ) ) =
|
|
( ( A x. C ) / ( B x. D ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cdiv co cmul wceq jca divmuldivap syl22anc ) ABL
|
|
MDLMCLMZCNOPZQELMZENOPZQBCRSDERSTSBDTSCETSRSUAFHAUEUFGJUBAUGUHIKUBBDCEUCU
|
|
D $.
|
|
$}
|
|
|
|
${
|
|
redivclapd.1 $e |- ( ph -> A e. RR ) $.
|
|
${
|
|
rerecclapd.2 $e |- ( ph -> A # 0 ) $.
|
|
$( Closure law for reciprocal. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
rerecclapd $p |- ( ph -> ( 1 / A ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr c1 cdiv co rerecclap syl2anc ) ABEFBGHIJBKLEFCDBM
|
|
N $.
|
|
$}
|
|
|
|
redivclapd.2 $e |- ( ph -> B e. RR ) $.
|
|
redivclapd.3 $e |- ( ph -> B # 0 ) $.
|
|
$( Closure law for division of reals. (Contributed by Jim Kingdon,
|
|
29-Feb-2020.) $)
|
|
redivclapd $p |- ( ph -> ( A / B ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr cdiv co redivclap syl3anc ) ABGHCGHCIJKBCLMGHDEFBCN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
mvllmulapd.1 $e |- ( ph -> A e. CC ) $.
|
|
mvllmulapd.2 $e |- ( ph -> B e. CC ) $.
|
|
mvllmulapd.3 $e |- ( ph -> A # 0 ) $.
|
|
mvllmulapd.4 $e |- ( ph -> ( A x. B ) = C ) $.
|
|
$( Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
|
|
10-Jun-2020.) $)
|
|
mvllmulapd $p |- ( ph -> B = ( C / A ) ) $=
|
|
( cmul co cdiv divcanap4d mulcomd eqtr3d oveq1d ) ACBIJZBKJCDBKJACBFEGLAP
|
|
DBKABCIJPDABCEFMHNON $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordering on reals (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) $)
|
|
ltp1 $p |- ( A e. RR -> A < ( A + 1 ) ) $=
|
|
( c1 cr wcel caddc co clt wbr 1re wa cc0 0lt1 ltaddpos mpbii mpan ) BCDZACD
|
|
ZAABEFGHZIPQJKBGHRLBAMNO $.
|
|
|
|
$( A number is less than or equal to itself plus 1. (Contributed by NM,
|
|
5-Jan-2006.) $)
|
|
lep1 $p |- ( A e. RR -> A <_ ( A + 1 ) ) $=
|
|
( cr wcel c1 caddc co clt wbr cle ltp1 wi peano2re ltle mpdan mpd ) ABCZAAD
|
|
EFZGHZAQIHZAJPQBCRSKALAQMNO $.
|
|
|
|
$( A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) $)
|
|
ltm1 $p |- ( A e. RR -> ( A - 1 ) < A ) $=
|
|
( cr wcel c1 cmin co cc0 clt wbr 0lt1 wb 0re 1re mp3an12 mpbii recn subid1d
|
|
ltsub2 breqtrd ) ABCZADEFZAGEFZAHTGDHIZUAUBHIZJGBCDBCTUCUDKLMGDARNOTAAPQS
|
|
$.
|
|
|
|
$( A number minus 1 is less than or equal to itself. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
lem1 $p |- ( A e. RR -> ( A - 1 ) <_ A ) $=
|
|
( cr wcel c1 cmin co clt wbr cle ltm1 wi peano2rem ltle mpancom mpd ) ABCZA
|
|
DEFZAGHZQAIHZAJQBCPRSKALQAMNO $.
|
|
|
|
$( A transitive property of 'less than or equal' and plus 1. (Contributed by
|
|
NM, 5-Aug-2005.) $)
|
|
letrp1 $p |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ ( B + 1 ) ) $=
|
|
( cr wcel cle wbr w3a c1 caddc co clt wa adantl peano2re ancli lelttr 3expb
|
|
ltp1 wi sylan2 mpan2d 3impia ltle 3adant3 mpd ) ACDZBCDZABEFZGABHIJZKFZAUIE
|
|
FZUFUGUHUJUFUGLUHBUIKFZUJUGULUFBRMUGUFUGUICDZLUHULLUJSZUGUMBNZOUFUGUMUNABUI
|
|
PQTUAUBUFUGUJUKSZUHUGUFUMUPUOAUIUCTUDUE $.
|
|
|
|
$( A transitive property of plus 1 and 'less than or equal'. (Contributed by
|
|
NM, 16-Aug-2005.) $)
|
|
p1le $p |- ( ( A e. RR /\ B e. RR /\ ( A + 1 ) <_ B ) -> A <_ B ) $=
|
|
( cr wcel c1 caddc co cle wa lep1 adantr wi peano2re ancli letr 3expa sylan
|
|
wbr mpand 3impia ) ACDZBCDZAEFGZBHRZABHRZUAUBIAUCHRZUDUEUAUFUBAJKUAUAUCCDZI
|
|
UBUFUDIUELZUAUGAMNUAUGUBUHAUCBOPQST $.
|
|
|
|
$( The reciprocal of a positive number is positive. Exercise 4 of [Apostol]
|
|
p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
recgt0 $p |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 co cap 0re cneg cmul adantr rerecclapd lt0neg1d
|
|
wn cc syl2anc wb cdiv cle 0lt1 1re ax-mp simpll gt0ap0 renegcld simpr simpl
|
|
ltnsymi mpbid simplr mulgt0d recnd recclap mulneg1d recidap2 negeqd breqtrd
|
|
wceq eqtrd 1red mpbird ex mtoi lenlt sylancr recap0 apsym sylancl mpbir2and
|
|
0cn ltleap ) ABCZDAEFZGZDHAUAIZEFZDVRUBFZDVRJFZVQVTVRDEFZQZVQWBHDEFZDHEFWDQ
|
|
UCDHKUDUKUEVQWBWDVQWBGZWDDHLZEFWEDVRLZAMIZWFEWEWGAWEVRWEAVOVPWBUFZVQADJFZWB
|
|
AUGZNZOUHWIWEWBDWGEFVQWBUIWEVRVQVRBCZWBVQAVOVPUJZWKOZNPULVOVPWBUMUNWEWHVRAM
|
|
IZLWFWEVRAWEARCZWJVRRCZVQWQWBVQAWNUOZNZWLAUPZSWTUQWEWPHWEWQWJWPHVAWTWLAURSU
|
|
SVBUTWEHWEVCPVDVEVFVQDBCZWMVTWCTKWODVRVGVHVDVQVRDJFZWAVQWQWJXCWSWKAVISVQWRD
|
|
RCXCWATVQWQWJWRWSWKXASVMVRDVJVKULVQXBWMVSVTWAGTKWODVRVNVHVL $.
|
|
|
|
$( Infer that a multiplicand is positive from a positive multiplier and
|
|
positive product. See ~ prodgt0 for the same theorem with ` 0 < A `
|
|
replaced by the weaker condition ` 0 <_ A ` . (Contributed by Jim
|
|
Kingdon, 29-Feb-2020.) $)
|
|
prodgt0gt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) )
|
|
-> 0 < B ) $=
|
|
( cr wcel wa cc0 clt wbr cmul co cdiv simpll simplr remulcld simprl gt0ap0d
|
|
c1 rerecclapd simprr recnd recgt0 mulgt0d divrecapd simpr adantr divcanap3d
|
|
ad2ant2r cc eqtr3d breqtrd ) ACDZBCDZEZFAGHZFABIJZGHZEZEZFUOQAKJZIJZBGURUOU
|
|
SURABUKULUQLZUKULUQMNZURAVAURAVAUMUNUPOPZRUMUNUPSUKUNFUSGHULUPAUAUGUBURUOAK
|
|
JUTBURUOAURUOVBTURAVATZVCUCURBAUMBUHDUQUMBUKULUDTUEVDVCUFUIUJ $.
|
|
|
|
$( Infer that a multiplicand is positive from a nonnegative multiplier and
|
|
positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
prodgt0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) )
|
|
-> 0 < B ) $=
|
|
( cr wcel wa cc0 cle wbr cmul co clt wn cneg renegcld lt0neg1d recnd negcld
|
|
cap mpbird 0red simpllr simplll simplr biimpa simprr simpll mulcomd breqtrd
|
|
mul2negd breqtrrd adantr prodgt0gt0 syl22anc lenltd mpbid pm2.65da remulcld
|
|
simplrl gt0ap0d mulap0bbd cc wb 0cnd apsym syl2anc ltleap mpbir2and ) ACDZB
|
|
CDZEZFAGHZFABIJZKHZEZEZFBKHZFBGHZFBRHZVOVQBFKHZLVOVSAFKHZVOVSEZVTFAMZKHZWAB
|
|
MZCDWBCDFWDKHZFWDWBIJZKHZWCWABVHVIVNVSUANWAAVHVIVNVSUBZNVOVSWEVOBVHVIVNUCZO
|
|
UDVOWGVSVOFWBWDIJZWFKVOFVLWJKVJVKVMUEZVOABVOAVHVIVNUFZPZVOBWIPZUIUJVOWBWDVO
|
|
AWMQVOBWNQUGUHUKWDWBULUMWAAWHOSWAVKVTLVJVKVMVSURWAFAWATWHUNUOUPVOFBVOTZWIUN
|
|
SVOBFRHZVRVOABWMWNVOVLVOABWLWIUQWKUSUTVOBVADFVADWPVRVBWNVOVCBFVDVEUOVOFCDVI
|
|
VPVQVREVBWOWIFBVFVEVG $.
|
|
|
|
$( Infer that a multiplier is positive from a nonnegative multiplicand and
|
|
positive product. (Contributed by NM, 24-Apr-2005.) $)
|
|
prodgt02 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ B /\ 0 < ( A x. B ) ) )
|
|
-> 0 < A ) $=
|
|
( cr wcel wa cc0 cle wbr cmul co clt cc wceq mulcom syl2an breq2d biimpd wi
|
|
recn prodgt0 ex ancoms sylan2d imp ) ACDZBCDZEZFBGHZFABIJZKHZEFAKHZUGUJFBAI
|
|
JZKHZUHUKUGUJUMUGUIULFKUEALDBLDUIULMUFASBSABNOPQUFUEUHUMEZUKRUFUEEUNUKBATUA
|
|
UBUCUD $.
|
|
|
|
$( Infer that a multiplicand is nonnegative from a positive multiplier and
|
|
nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
prodge0 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ ( A x. B ) ) )
|
|
-> 0 <_ B ) $=
|
|
( cr wcel wa cc0 clt wbr cmul co cle wn cneg simpll renegcld recnd lt0neg1d
|
|
simplr 3imtr4d lenltd simprl simprr mulgt0d mulneg2d breqtrd remulcld con3d
|
|
expr 0red impr ) ACDZBCDZEZFAGHZFABIJZKHZFBKHZUMUNEZUOFGHZLBFGHZLUPUQURUTUS
|
|
URFBMZGHZFUOMZGHZUTUSUMUNVBVDUMUNVBEZEZFAVAIJVCGVFAVAUKULVENZVFBUKULVERZOUM
|
|
UNVBUAUMUNVBUBUCVFABVFAVGPVFBVHPUDUEUHURBUKULUNRZQURUOURABUKULUNNVIUFZQSUGU
|
|
RFUOURUIZVJTURFBVKVITSUJ $.
|
|
|
|
$( Infer that a multiplier is nonnegative from a positive multiplicand and
|
|
nonnegative product. (Contributed by NM, 2-Jul-2005.) $)
|
|
prodge02 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < B /\ 0 <_ ( A x. B ) ) )
|
|
-> 0 <_ A ) $=
|
|
( cr wcel wa cc0 clt wbr cmul co cle cc wceq mulcom syl2an breq2d biimpd wi
|
|
recn prodge0 ex ancoms sylan2d imp ) ACDZBCDZEZFBGHZFABIJZKHZEFAKHZUGUJFBAI
|
|
JZKHZUHUKUGUJUMUGUIULFKUEALDBLDUIULMUFASBSABNOPQUFUEUHUMEZUKRUFUEEUNUKBATUA
|
|
UBUCUD $.
|
|
|
|
$( Multiplication of both sides of 'less than' by a positive number. Theorem
|
|
I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) $)
|
|
ltmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co ltmul1 wb cc recn mulcom sylan 3adant2
|
|
wceq 3adant1 breq12d syl3an3 3adant3r bitrd ) ADEZBDEZCDEZFCGHZIJABGHACKLZB
|
|
CKLZGHZCAKLZCBKLZGHZABCMUFUGUHULUONZUIUHUFUGCOEZUPCPUFUGUQJUJUMUKUNGUFUQUJU
|
|
MTZUGUFAOEUQURAPACQRSUGUQUKUNTZUFUGBOEUQUSBPBCQRUAUBUCUDUE $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by NM, 16-Mar-2005.) $)
|
|
lemul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cle cmul co lemul1 wb wceq recn mulcom syl2an
|
|
cc 3adant2 3adant1 breq12d 3adant3r bitrd ) ADEZBDEZCDEZFCGHZIJABKHACLMZBCL
|
|
MZKHZCALMZCBLMZKHZABCNUFUGUHULUOOUIUFUGUHJUJUMUKUNKUFUHUJUMPZUGUFATECTEZUPU
|
|
HAQCQZACRSUAUGUHUKUNPZUFUGBTEUQUSUHBQURBCRSUBUCUDUE $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a nonnegative
|
|
number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies).
|
|
(Contributed by NM, 21-Feb-2005.) $)
|
|
lemul1a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\
|
|
A <_ B ) -> ( A x. C ) <_ ( B x. C ) ) $=
|
|
( cr wcel cc0 cle wbr wa w3a cmul co simpl2 simpl1 resubcld simpl3l subge0d
|
|
cmin recnd remulcld simpr mpbird simpl3r mulge0d subdird breqtrd mpbid ) AD
|
|
EZBDEZCDEZFCGHZIZJZABGHZIZFBCKLZACKLZRLZGHUQUPGHUOFBARLZCKLURGUOUSCUOBAUHUI
|
|
ULUNMZUHUIULUNNZOUJUKUHUIUNPZUOFUSGHUNUMUNUAUOBAUTVAQUBUJUKUHUIUNUCUDUOBACU
|
|
OBUTSUOAVASUOCVBSUEUFUOUPUQUOBCUTVBTUOACVAVBTQUG $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a nonnegative
|
|
number. (Contributed by Paul Chapman, 7-Sep-2007.) $)
|
|
lemul2a $p |- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\
|
|
A <_ B ) -> ( C x. A ) <_ ( C x. B ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co lemul1a wceq cc recn mulcom syl2an adantrr
|
|
w3a adantr 3adant2 3adant1 3brtr3d ) ADEZBDEZCDEZFCGHZIZSZABGHZIACJKZBCJKZC
|
|
AJKZCBJKZGABCLUIUKUMMZUJUDUHUOUEUDUFUOUGUDANECNEZUOUFAOCOZACPQRUATUIULUNMZU
|
|
JUEUHURUDUEUFURUGUEBNEUPURUFBOUQBCPQRUBTUC $.
|
|
|
|
$( Comparison of product of two positive numbers. (Contributed by NM,
|
|
30-Dec-2005.) $)
|
|
ltmul12a $p |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\
|
|
( ( C e. RR /\ D e. RR ) /\ ( 0 <_ C /\ C < D ) ) ) ->
|
|
( A x. C ) < ( B x. D ) ) $=
|
|
( cr wcel cc0 cle wbr clt cmul simpllr ad2ant2l imp ad2ant2r lelttr remulcl
|
|
wa co wi simplll simpll simprl jca ltle adantrl lemul1a syl31anc wb simplrl
|
|
simplrr 0re mp3an1 adantlr ltmul2 syl112anc biimpa anasss ad2ant2lr syl3anc
|
|
adantrrl adantr mp2and an4s ) AEFZBEFZRZCEFZDEFZRZGAHIZABJIZRZGCHIZCDJIZRZA
|
|
CKSZBDKSZJIZVGVJRZVMVPRZRZVQBCKSZHIZWCVRJIZVSWBVEVFVHVNRZABHIZWDVEVFVJWAUAV
|
|
EVFVJWALVJVPWFVGVMVJVPRVHVNVHVIVPUBVJVNVOUCUDMVGVMWGVJVPVGVLWGVKVGVLWGABUEN
|
|
UFOABCUGUHVTVMVOWEVNVTVMVOWEVTVMRZVOWEWHVHVIVFGBJIZVOWEUIVGVHVIVMUJVGVHVIVM
|
|
UKVEVFVJVMLVGVMWIVJVGVMWIGEFVEVFVMWITULGABPUMNUNCDBUOUPUQURVAVTWDWERVSTZWAV
|
|
TVQEFZWCEFZVREFZWJVEVHWKVFVIACQOVFVHWLVEVIBCQUSVFVIWMVEVHBDQMVQWCVRPUTVBVCV
|
|
D $.
|
|
|
|
$( Comparison of product of two nonnegative numbers. (Contributed by NM,
|
|
22-Feb-2008.) $)
|
|
lemul12b $p |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\
|
|
( C e. RR /\ ( D e. RR /\ 0 <_ D ) ) ) -> ( ( A <_ B /\ C <_ D ) ->
|
|
( A x. C ) <_ ( B x. D ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co wi w3a lemul2a ex adantlr remulcl ad2ant2r
|
|
3comr 3expb adantrrr lemul1a 3expa adantllr adantrl anim12d ancomsd adantrr
|
|
ad2ant2rl ad2ant2l letr syl3anc syld ) AEFZGAHIZJZBEFZJZCEFZDEFZGDHIZJZJZJZ
|
|
ABHIZCDHIZJACKLZADKLZHIZVIBDKLZHIZJZVHVKHIZVEVGVFVMVEVGVJVFVLUQVDVGVJMZURUQ
|
|
UTVAVOVBUQUTVAVOUTVAUQVOUTVAUQNVGVJCDAOPTUAUBQUSVCVFVLMZUTUOURVCVPUPUOURVCV
|
|
PUOURVCNVFVLABDUCPUDUEUFUGUHVEVHEFZVIEFZVKEFZVMVNMUQUTVQURVCUOUTVQUPACRQSUQ
|
|
VCVRURUTUOVAVRUPVBADRSUJURVCVSUQUTURVAVSVBBDRUIUKVHVIVKULUMUN $.
|
|
|
|
$( Comparison of product of two nonnegative numbers. (Contributed by NM,
|
|
22-Feb-2008.) $)
|
|
lemul12a $p |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\
|
|
( ( C e. RR /\ 0 <_ C ) /\ D e. RR ) ) -> ( ( A <_ B /\ C <_ D ) ->
|
|
( A x. C ) <_ ( B x. D ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co simpll ad2antlr simplrr wi 0re letr mp3an1
|
|
exp4b com23 imp41 ad2ant2l jca jca32 simpr lemul12b sylc ex ) AEFGAHIJBEFJZ
|
|
CEFZGCHIZJZDEFZJZJZABHIZCDHIZJZACKLBDKLHIZUPUSJZUJUKUNGDHIZJZJJUSUTVAUJUKVC
|
|
UJUOUSMUOUKUJUSUKULUNMNVAUNVBUJUMUNUSOUOURVBUJUQUKULUNURVBUKUNULURVBPUKUNUL
|
|
URVBGEFUKUNULURJVBPQGCDRSTUAUBUCUDUEUPUSUFABCDUGUHUI $.
|
|
|
|
$( The product of two numbers greater than 1 is greater than 1. (Contributed
|
|
by NM, 13-Feb-2005.) $)
|
|
mulgt1 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) ->
|
|
1 < ( A x. B ) ) $=
|
|
( cr wcel wa c1 clt wbr cmul co wi a1i cc0 0lt1 1re lttr adantr mp3an1 syld
|
|
simpl 0re mp3an12 mpani ltmul2 biimpd exp32 impcom impd wceq ax-1rid breq1d
|
|
w3a sylibd jcad remulcl syldan imp ) ACDZBCDZEZFAGHZFBGHZEZFABIJZGHZUTVCVAA
|
|
VDGHZEZVEUTVCVAVFVCVAKUTVAVBTLUTVCAFIJZVDGHZVFUTVAVBVIUTVAMAGHZVBVIKZURVAVJ
|
|
KUSURMFGHZVAVJNMCDFCDZURVLVAEVJKUAOMFAPUBUCQUSURVJVKKUSURVJVKVMUSURVJEZVKOV
|
|
MUSVNULVBVIFBAUDUERUFUGSUHUTVHAVDGURVHAUIUSAUJQUKUMUNURUSVDCDZVGVEKZABUOVMU
|
|
RVOVPOFAVDPRUPSUQ $.
|
|
|
|
$( Multiplication by a number greater than 1. (Contributed by NM,
|
|
24-Dec-2005.) $)
|
|
ltmulgt11 $p |- ( ( A e. RR /\ B e. RR /\ 0 < A ) ->
|
|
( 1 < B <-> A < ( A x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr w3a c1 cmul co wb 1re ltmul2 mp3an1 3impb 3com12 wceq
|
|
wa ax-1rid 3ad2ant1 breq1d bitrd ) ACDZBCDZEAFGZHZIBFGZAIJKZABJKZFGZAUJFGUE
|
|
UDUFUHUKLZUEUDUFULICDUEUDUFSULMIBANOPQUGUIAUJFUDUEUIARUFATUAUBUC $.
|
|
|
|
$( Multiplication by a number greater than 1. (Contributed by NM,
|
|
24-Dec-2005.) $)
|
|
ltmulgt12 $p |- ( ( A e. RR /\ B e. RR /\ 0 < A ) ->
|
|
( 1 < B <-> A < ( B x. A ) ) ) $=
|
|
( cr wcel cc0 clt wbr w3a c1 cmul ltmulgt11 wceq recn mulcom syl2an 3adant3
|
|
co cc breq2d bitrd ) ACDZBCDZEAFGZHZIBFGAABJQZFGABAJQZFGABKUDUEUFAFUAUBUEUF
|
|
LZUCUAARDBRDUGUBAMBMABNOPST $.
|
|
|
|
$( Multiplication by a number greater than or equal to 1. (Contributed by
|
|
NM, 17-Dec-2005.) $)
|
|
lemulge11 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) ->
|
|
A <_ ( A x. B ) ) $=
|
|
( cr wcel wa cc0 cle wbr c1 cmul co wceq ax-1rid ad2antrr simpll simprl jca
|
|
simplr 1re 0le1 pm3.2i jctil jca31 leid simprr lemul12a sylc eqbrtrrd ) ACD
|
|
ZBCDZEZFAGHZIBGHZEZEZAIJKZAABJKZGUIUPALUJUNAMNUOUIULEZUIEICDZFIGHZEZUJEZEAA
|
|
GHZUMEUPUQGHUOURUIVBUOUIULUIUJUNOZUKULUMPQVDUOUJVAUIUJUNRUSUTSTUAUBUCUOVCUM
|
|
UIVCUJUNAUDNUKULUMUEQAAIBUFUGUH $.
|
|
|
|
$( Multiplication by a number greater than or equal to 1. (Contributed by
|
|
Paul Chapman, 21-Mar-2011.) $)
|
|
lemulge12 $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) ->
|
|
A <_ ( B x. A ) ) $=
|
|
( cr wcel wa cc0 cle wbr c1 cmul co lemulge11 wb cc wceq recn mulcom syl2an
|
|
breq2d adantr mpbid ) ACDZBCDZEZFAGHIBGHEZEAABJKZGHZABAJKZGHZABLUDUGUIMUEUD
|
|
UFUHAGUBANDBNDUFUHOUCAPBPABQRSTUA $.
|
|
|
|
$( Division of both sides of 'less than' by a positive number. (Contributed
|
|
by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
ltdiv1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( A < B <-> ( A / C ) < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a c1 cdiv co cmul wb simp1 simp2 recnd divrecapd
|
|
simp3l simp3r gt0ap0d rerecclapd recgt0 3ad2ant3 ltmul1 syl112anc breq12d
|
|
bitr4d ) ADEZBDEZCDEZFCGHZIZJZABGHZAKCLMZNMZBUQNMZGHZACLMZBCLMZGHUOUJUKUQDE
|
|
FUQGHZUPUTOUJUKUNPZUJUKUNQZUOCUJUKULUMTZUOCVFUJUKULUMUAUBZUCUNUJVCUKCUDUEAB
|
|
UQUFUGUOVAURVBUSGUOACUOAVDRUOCVFRZVGSUOBCUOBVERVHVGSUHUI $.
|
|
|
|
$( Division of both sides of a less than or equal to relation by a positive
|
|
number. (Contributed by NM, 18-Nov-2004.) $)
|
|
lediv1 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( A <_ B <-> ( A / C ) <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a wn co cle wb ltdiv1 3adant1 redivclap syld3an3
|
|
cdiv 3expb 3com12 notbid lenlt 3adant3 cap gt0ap0 3adant2 lenltd 3bitr4d )
|
|
ADEZBDEZCDEZFCGHZIZJZBAGHZKZBCSLZACSLZGHZKABMHZUSURMHUOUPUTUKUJUNUPUTNBACOU
|
|
AUBUJUKVAUQNUNABUCUDUOUSURUJUNUSDEZUKUJULUMVBUJULUMCFUEHZVBULUMVCUJCUFZPACQ
|
|
RTUGUKUNURDEZUJUKULUMVEUKULUMVCVEULUMVCUKVDPBCQRTPUHUI $.
|
|
|
|
$( Division of a positive number by a positive number. (Contributed by NM,
|
|
28-Sep-2005.) $)
|
|
gt0div $p |- ( ( A e. RR /\ B e. RR /\ 0 < B ) ->
|
|
( 0 < A <-> 0 < ( A / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr w3a cdiv co wb wa 0re ltdiv1 mp3an1 3impb wceq gt0ap0
|
|
cap cc recn div0ap sylan syldan breq1d 3adant1 bitrd ) ACDZBCDZEBFGZHEAFGZE
|
|
BIJZABIJZFGZEUMFGZUHUIUJUKUNKZECDUHUIUJLZUPMEABNOPUIUJUNUOKUHUQULEUMFUIUJBE
|
|
SGZULEQZBRUIBTDURUSBUABUBUCUDUEUFUG $.
|
|
|
|
$( Division of a nonnegative number by a positive number. (Contributed by
|
|
NM, 28-Sep-2005.) $)
|
|
ge0div $p |- ( ( A e. RR /\ B e. RR /\ 0 < B ) ->
|
|
( 0 <_ A <-> 0 <_ ( A / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr w3a cle cdiv co wb 0re lediv1 mp3an1 3impb cap gt0ap0
|
|
wa wceq cc recn div0ap sylan syldan breq1d 3adant1 bitrd ) ACDZBCDZEBFGZHEA
|
|
IGZEBJKZABJKZIGZEUNIGZUIUJUKULUOLZECDUIUJUKSZUQMEABNOPUJUKUOUPLUIURUMEUNIUJ
|
|
UKBEQGZUMETZBRUJBUADUSUTBUBBUCUDUEUFUGUH $.
|
|
|
|
$( The ratio of two positive numbers is positive. (Contributed by NM,
|
|
12-Oct-1999.) $)
|
|
divgt0 $p |- ( ( ( A e. RR /\ 0 < A ) /\
|
|
( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) ) $=
|
|
( cr wcel cc0 clt wbr cdiv co wi w3a gt0div biimpd 3exp com34 com23 imp43 )
|
|
ACDZEAFGZBCDZEBFGZEABHIFGZRTSUAUBJRTUASUBRTUASUBJRTUAKSUBABLMNOPQ $.
|
|
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by NM, 27-Sep-1999.) $)
|
|
divge0 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\
|
|
( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) ) $=
|
|
( cr wcel cc0 cle wbr clt cdiv co w3a ge0div biimpd 3exp com34 com23 imp43
|
|
wi ) ACDZEAFGZBCDZEBHGZEABIJFGZSUATUBUCRSUAUBTUCSUAUBTUCRSUAUBKTUCABLMNOPQ
|
|
$.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
ltmuldiv $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co cdiv wb simp1 simp3l remulcld syld3an1
|
|
ltdiv1 recnd simp3r gt0ap0d divcanap4d breq1d bitrd ) ADEZBDEZCDEZFCGHZIZJZ
|
|
ACKLZBGHZULCMLZBCMLZGHZAUOGHULDEUGUFUJUMUPNUKACUFUGUJOZUFUGUHUIPZQULBCSRUKU
|
|
NAUOGUKACUKAUQTUKCURTUKCURUFUGUHUIUAUBUCUDUE $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 18-Nov-2004.) $)
|
|
ltmuldiv2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( C x. A ) < B <-> A < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cdiv wceq cc mulcom syl2an adantrr 3adant2
|
|
w3a recn breq1d ltmuldiv bitr3d ) ADEZBDEZCDEZFCGHZIZSZACJKZBGHCAJKZBGHABCL
|
|
KGHUIUJUKBGUDUHUJUKMZUEUDUFULUGUDANECNEULUFATCTACOPQRUAABCUBUC $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 18-Nov-2004.) $)
|
|
ltdivmul $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( A / C ) < B <-> A < ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co cdiv wb remulcl ancoms adantrr 3adant1
|
|
cc recn ltdiv1 syld3an2 adantr ad2antrl cap gt0ap0 adantl divcanap3d breq2d
|
|
wceq bitr2d ) ADEZBDEZCDEZFCGHZIZJZACBKLZGHZACMLZURCMLZGHZUTBGHULURDEZUMUPU
|
|
SVBNUMUPVCULUMUNVCUOUNUMVCCBOPQRAURCUAUBUQVABUTGUMUPVABUJULUMUPIBCUMBSEUPBT
|
|
UCUNCSEUMUOCTUDUPCFUEHUMCUFUGUHRUIUK $.
|
|
|
|
$( 'Less than or equal to' relationship between division and multiplication.
|
|
(Contributed by NM, 9-Dec-2005.) $)
|
|
ledivmul $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( ( A / C ) <_ B <-> A <_ ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co cle simp3l simp2 remulcld lediv1 recnd
|
|
cdiv wb syld3an2 simp3r gt0ap0d divcanap3d breq2d bitr2d ) ADEZBDEZCDEZFCGH
|
|
ZIZJZACBKLZMHZACSLZUMCSLZMHZUOBMHUGUMDEUHUKUNUQTULCBUGUHUIUJNZUGUHUKOZPAUMC
|
|
QUAULUPBUOMULBCULBUSRULCURRULCURUGUHUIUJUBUCUDUEUF $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 24-Feb-2005.) $)
|
|
ltdivmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( A / C ) < B <-> A < ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cdiv co cmul ltdivmul wceq recn mulcom adantrr
|
|
cc syl2an 3adant1 breq2d bitr4d ) ADEZBDEZCDEZFCGHZIZJZACKLBGHACBMLZGHABCML
|
|
ZGHABCNUIUKUJAGUEUHUKUJOZUDUEUFULUGUEBSECSEULUFBPCPBCQTRUAUBUC $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 8-Jan-2006.) $)
|
|
lt2mul2div $p |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) )
|
|
/\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) ->
|
|
( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cdiv simprl simprrl mulcomd oveq1d syl3anc
|
|
recnd wb simplrl simplrr gt0ap0d divassapd eqtrd breq2d remulcld redivclapd
|
|
simpll simplr ltmuldiv simprr ltdivmul 3bitr4d ) AEFZBEFZGBHIZJZJZCEFZDEFZG
|
|
DHIZJZJZJZACDKLZBMLZHIZADCBMLZKLZHIZABKLVFHIZADMLVIHIZVEVGVJAHVEVGDCKLZBMLV
|
|
JVEVFVNBMVECDVECUSUTVCNZSZVEDUSUTVAVBOZSZPQVEDCBVRVPVEBUOUPUQVDUAZSVEBVSUOU
|
|
PUQVDUBUCZUDUEUFVEUOVFEFURVLVHTUOURVDUIZVECDVOVQUGUOURVDUJAVFBUKRVEUOVIEFVC
|
|
VMVKTWAVECBVOVSVTUHUSUTVCULAVIDUMRUN $.
|
|
|
|
$( 'Less than or equal to' relationship between division and multiplication.
|
|
(Contributed by NM, 9-Dec-2005.) $)
|
|
ledivmul2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( ( A / C ) <_ B <-> A <_ ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cdiv cle cmul ledivmul wceq recn mulcom syl2an
|
|
co cc adantrr 3adant1 breq2d bitr4d ) ADEZBDEZCDEZFCGHZIZJZACKSBLHACBMSZLHA
|
|
BCMSZLHABCNUJULUKALUFUIULUKOZUEUFUGUMUHUFBTECTEUMUGBPCPBCQRUAUBUCUD $.
|
|
|
|
$( 'Less than or equal' relationship between division and multiplication.
|
|
(Contributed by NM, 10-Mar-2006.) $)
|
|
lemuldiv $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( ( A x. C ) <_ B <-> A <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cdiv co wn cmul cle wb ltdivmul2 3com12 notbid
|
|
lenltd simp1 simp2 simp3l simp3r gt0ap0d redivclapd remulcld 3bitr4rd ) ADE
|
|
ZBDEZCDEZFCGHZIZJZBCKLZAGHZMBACNLZGHZMAUOOHUQBOHUNUPURUJUIUMUPURPBACQRSUNAU
|
|
OUIUJUMUAZUNBCUIUJUMUBZUIUJUKULUCZUNCVAUIUJUKULUDUEUFTUNUQBUNACUSVAUGUTTUH
|
|
$.
|
|
|
|
$( 'Less than or equal' relationship between division and multiplication.
|
|
(Contributed by NM, 10-Mar-2006.) $)
|
|
lemuldiv2 $p |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) ->
|
|
( ( C x. A ) <_ B <-> A <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cmul co cle cdiv wceq cc mulcom syl2an adantrr
|
|
recn 3adant2 breq1d lemuldiv bitr3d ) ADEZBDEZCDEZFCGHZIZJZACKLZBMHCAKLZBMH
|
|
ABCNLMHUJUKULBMUEUIUKULOZUFUEUGUMUHUEAPECPEUMUGATCTACQRSUAUBABCUCUD $.
|
|
|
|
$( The reciprocal of both sides of 'less than'. (Contributed by NM,
|
|
26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
ltrec $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
|
|
( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co cmul 1red simprl simpll simplr ltmuldiv
|
|
wb syl112anc recnd mulid2d breq1d gt0ap0d divrecapd breq2d 3bitr3d ltdivmul
|
|
rerecclapd simprr bitr4d ) ACDZEAFGZHZBCDZEBFGZHZHZABFGZIBIAJKZLKZFGZIBJKUS
|
|
FGZUQIALKZBFGZIBAJKZFGZURVAUQICDZUNUKULVDVFRUQMZUMUNUONZUKULUPOZUKULUPPZIBA
|
|
QSUQVCABFUQAUQAVJTZUAUBUQVEUTIFUQBAUQBVITVLUQAVJVKUCZUDUEUFUQVGUSCDUNUOVBVA
|
|
RVHUQAVJVMUHVIUMUNUOUIIUSBUGSUJ $.
|
|
|
|
$( The reciprocal of both sides of 'less than or equal to'. (Contributed by
|
|
NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
lerec $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
|
|
( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa wn c1 co cle wb ltrec ancoms notbid lenltd gt0ap0d
|
|
cdiv rerecclapd simpll simprl simprr simplr 3bitr4d ) ACDZEAFGZHZBCDZEBFGZH
|
|
ZHZBAFGZIJASKZJBSKZFGZIABLGUOUNLGULUMUPUKUHUMUPMBANOPULABUFUGUKUAZUHUIUJUBZ
|
|
QULUOUNULBURULBURUHUIUJUCRTULAUQULAUQUFUGUKUDRTQUE $.
|
|
|
|
$( Lemma for ~ lt2msq . (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lt2msq1 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) ->
|
|
( A x. A ) < ( B x. B ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt w3a co simp1l remulcld simp2 simp1 simp3 ltled
|
|
cmul lemul1a lelttrd syl31anc wb 0red simp1r ltmul2 syl112anc mpbid ) ACDZE
|
|
AFGZHZBCDZABIGZJZAARKZBARKZBBRKZUMAAUHUIUKULLZUQMUMBAUJUKULNZUQMUMBBURURMUM
|
|
UHUKUJABFGUNUOFGUQURUJUKULOUMABUQURUJUKULPZQABASUAUMULUOUPIGZUSUMUHUKUKEBIG
|
|
ULUTUBUQURURUMEABUMUCUQURUHUIUKULUDUSTABBUEUFUGT $.
|
|
|
|
$( Two nonnegative numbers compare the same as their squares. (Contributed
|
|
by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
lt2msq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A < B <-> ( A x. A ) < ( B x. B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt cmul co wi lt2msq1 3expia cap syl2anc remulcld
|
|
wn lenltd 3imtr4d adantrr simpr simpll con3d simprl wo recnd syl22anc oridm
|
|
cc mulext syl6ib anim12d wb ltleap impbid ) ACDZEAFGZHZBCDZEBFGZHZHZABIGZAA
|
|
JKZBBJKZIGZUSUTVDVGLVAUSUTVDVGABMNUAVCVEVFFGZVEVFOGZHZABFGZABOGZHZVGVDVCVHV
|
|
KVIVLVCVFVEIGZRBAIGZRVHVKVCVOVNVCVBUQVOVNLUSVBUBUQURVBUCZVBUQVOVNBAMNPUDVCV
|
|
EVFVCAAVPVPQZVCBBUSUTVAUEZVRQZSVCABVPVRSTVCVIVLVLUFZVLVCAUJDZWABUJDZWBVIVTL
|
|
VCAVPUGZWCVCBVRUGZWDAABBUKUHVLUIULUMVCVECDVFCDVGVJUNVQVSVEVFUOPVCUQUTVDVMUN
|
|
VPVRABUOPTUP $.
|
|
|
|
$( Division of a positive number by both sides of 'less than'. (Contributed
|
|
by NM, 27-Apr-2005.) $)
|
|
ltdiv2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
|
|
( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co cmul wb gt0ap0 3ad2ant1 recnd divrecapd
|
|
w3a cap simp2l 3ad2ant2 rerecclap syldan simp3 ltmul2 syl3anc ltrec 3adant3
|
|
rerecclapd simp3l simp1l breq12d 3bitr4d ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCGHZI
|
|
ZSZJBKLZJAKLZGHZCVEMLZCVFMLZGHZABGHZCBKLZCAKLZGHVDVEDEVFDEZVCVGVJNVDBUQURUS
|
|
VCUAZUTUQBFTHVCBOUBZUJUQUTVNVCUOUPAFTHZVNAOZAUCUDPUQUTVCUEVEVFCUFUGUQUTVKVG
|
|
NVCABUHUIVDVLVHVMVIGVDCBVDCUQUTVAVBUKQZVDBVOQVPRVDCAVSVDAUOUPUTVCULQUQUTVQV
|
|
CVRPRUMUN $.
|
|
|
|
$( Reciprocal swap in a 'less than' relation. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
ltrec1 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
|
|
( ( 1 / A ) < B <-> ( 1 / B ) < A ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co simpll simplr gt0ap0d rerecclapd recgt0
|
|
wb adantr simpr ltrec syl21anc recnd recrecapd breq2d bitrd ) ACDZEAFGZHZBC
|
|
DEBFGHZHZIAJKZBFGZIBJKZIUKJKZFGZUMAFGUJUKCDEUKFGZUIULUOQUJAUFUGUILZUJAUQUFU
|
|
GUIMNZOUHUPUIAPRUHUISUKBTUAUJUNAUMFUJAUJAUQUBURUCUDUE $.
|
|
|
|
$( Reciprocal swap in a 'less than or equal to' relation. (Contributed by
|
|
NM, 24-Feb-2005.) $)
|
|
lerec2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) ->
|
|
( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co cle wb simprl simprr gt0ap0d rerecclapd
|
|
simpl recgt0 adantl lerec syl12anc recnd recrecapd breq1d bitrd ) ACDEAFGHZ
|
|
BCDZEBFGZHZHZAIBJKZLGZIULJKZIAJKZLGZBUOLGUKUGULCDEULFGZUMUPMUGUJRUKBUGUHUIN
|
|
ZUKBURUGUHUIOPZQUJUQUGBSTAULUAUBUKUNBUOLUKBUKBURUCUSUDUEUF $.
|
|
|
|
$( Invert ratios of positive numbers and swap their ordering. (Contributed
|
|
by NM, 9-Jan-2006.) $)
|
|
ledivdiv $p |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\
|
|
( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) ->
|
|
( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co cle c1 gt0ap0d redivclapd divgt0 recdivapd
|
|
wb recnd simplll simplrl simplrr adantr simprll simprrl simprrr lerec bitrd
|
|
adantl syl22anc simprlr simpllr breq12d ) AEFZGAHIZJZBEFZGBHIZJZJZCEFZGCHIZ
|
|
JZDEFZGDHIZJZJZJZABKLZCDKLZMIZNVKKLZNVJKLZMIZDCKLZBAKLZMIVIVJEFGVJHIZVKEFGV
|
|
KHIZVLVOSVIABUOUPUTVHUAZUQURUSVHUBZVIBWAUQURUSVHUCOZPVAVRVHABQUDVICDVAVBVCV
|
|
GUEZVAVDVEVFUFZVIDWDVAVDVEVFUGOZPVHVSVACDQUJVJVKUHUKVIVMVPVNVQMVICDVICWCTVI
|
|
DWDTVICWCVAVBVCVGULOWERVIABVIAVTTVIBWATVIAVTUOUPUTVHUMOWBRUNUI $.
|
|
|
|
$( Division of a positive number by both sides of 'less than or equal to'.
|
|
(Contributed by NM, 10-Jan-2006.) $)
|
|
lediv2 $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
|
|
( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a c1 cdiv co cle wb gt0ap0d rerecclapd divrecapd
|
|
cmul recnd simp2l simp2r simp1l simp1r simp3l simp3r lemul2 syl112anc lerec
|
|
3adant3 breq12d 3bitr4d ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCGHZIZJZKBLMZKALMZNHZC
|
|
VCSMZCVDSMZNHZABNHZCBLMZCALMZNHVBVCDEVDDEUSUTVEVHOVBBUOUPUQVAUAZVBBVLUOUPUQ
|
|
VAUBPZQVBAUMUNURVAUCZVBAVNUMUNURVAUDPZQUOURUSUTUEZUOURUSUTUFVCVDCUGUHUOURVI
|
|
VEOVAABUIUJVBVJVFVKVGNVBCBVBCVPTZVBBVLTVMRVBCAVQVBAVNTVORUKUL $.
|
|
|
|
$( Swap denominator with other side of 'less than'. (Contributed by NM,
|
|
3-Oct-1999.) $)
|
|
ltdiv23 $p |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( A / B ) < C <-> ( A / C ) < B ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cdiv co cmul wb simp1 simp2l simp2r redivclapd
|
|
gt0ap0d recnd simp3l simp2 ltmul1 syl3anc divcanap1d breq1d remulcld ltdiv1
|
|
syld3an2 simp3r divcanap3d breq2d bitrd 3bitrd ) ADEZBDEZFBGHZIZCDEZFCGHZIZ
|
|
JZABKLZCGHZVCBMLZCBMLZGHZAVFGHZACKLZBGHZVBVCDEUSURVDVGNVBABUOURVAOZUOUPUQVA
|
|
PZVBBVLUOUPUQVAQSZRUOURUSUTUAZUOURVAUBVCCBUCUDVBVEAVFGVBABVBAVKTVBBVLTZVMUE
|
|
UFVBVHVIVFCKLZGHZVJUOVFDEURVAVHVQNVBCBVNVLUGAVFCUHUIVBVPBVIGVBBCVOVBCVNTVBC
|
|
VNUOURUSUTUJSUKULUMUN $.
|
|
|
|
$( Swap denominator with other side of 'less than or equal to'. (Contributed
|
|
by NM, 30-May-2005.) $)
|
|
lediv23 $p |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) )
|
|
-> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) ) $=
|
|
( cr wcel cc0 clt wbr wa w3a cdiv co cle cmul wb simp1 simp2l gt0ap0d recnd
|
|
simp2r redivclapd simp3l lemul1 syl112anc divcanap1d breq1d remulcld lediv1
|
|
syld3an2 simp3r divcanap3d breq2d bitrd 3bitrd ) ADEZBDEZFBGHZIZCDEZFCGHZIZ
|
|
JZABKLZCMHZVCBNLZCBNLZMHZAVFMHZACKLZBMHZVBVCDEUSUPUQVDVGOVBABUOURVAPZUOUPUQ
|
|
VAQZVBBVLUOUPUQVATZRZUAUOURUSUTUBZVLVMVCCBUCUDVBVEAVFMVBABVBAVKSVBBVLSZVNUE
|
|
UFVBVHVIVFCKLZMHZVJUOVFDEURVAVHVROVBCBVOVLUGAVFCUHUIVBVQBVIMVBBCVPVBCVOSVBC
|
|
VOUOURUSUTUJRUKULUMUN $.
|
|
|
|
$( Comparison of ratio of two nonnegative numbers. (Contributed by NM,
|
|
31-Dec-2005.) $)
|
|
lediv12a $p |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A <_ B ) ) /\
|
|
( ( C e. RR /\ D e. RR ) /\ ( 0 < C /\ C <_ D ) ) ) ->
|
|
( A / D ) <_ ( B / C ) ) $=
|
|
( cr wcel wa cc0 cle wbr clt c1 cdiv cmul syl22anc gt0ap0d rerecclapd recnd
|
|
co jca simplrr simprrr simprll simprrl simprlr 0red ltletrd simplll simplrl
|
|
wb lerec mpbid wi simpllr recgt0 syl2anc lemul12a mp2and divrecapd 3brtr4d
|
|
ltled ) AEFZBEFZGZHAIJZABIJZGZGZCEFZDEFZGZHCKJZCDIJZGZGZGZALDMSZNSZBLCMSZNS
|
|
ZADMSBCMSIVPVFVQVSIJZVRVTIJZVDVEVFVOUAVPVMWAVHVKVLVMUBZVPVIVLVJHDKJZVMWAUJV
|
|
HVIVJVNUCZVHVKVLVMUDZVHVIVJVNUEZVPHCDVPUFZWEWGWFWCUGZCDUKOULVPVBVEGVCVQEFZH
|
|
VQIJZGVSEFVFWAGWBUMVPVBVEVBVCVGVOUHZVDVEVFVOUITVBVCVGVOUNZVPWJWKVPDWGVPDWGW
|
|
IPZQZVPHVQWHWOVPVJWDHVQKJWGWIDUOUPVATVPCWEVPCWEWFPZQABVQVSUQOURVPADVPAWLRVP
|
|
DWGRWNUSVPBCVPBWMRVPCWERWPUSUT $.
|
|
|
|
$( Division of both sides of 'less than or equal to' into a nonnegative
|
|
number. (Contributed by Paul Chapman, 7-Sep-2007.) $)
|
|
lediv2a $p |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\
|
|
( C e. RR /\ 0 <_ C ) ) /\
|
|
A <_ B ) -> ( C / B ) <_ ( C / A ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle w3a cdiv co pm3.2 pm2.43i adantr anim2i ancoms
|
|
leid jca ad2antlr 3adantl2 ad2ant2r simplr anim1i 3adantl3 lediv12a syl2anc
|
|
id ) ADEZFAGHZIZBDEZFBGHZIZCDEZFCJHZIZKABJHZIUPUPIZUQCCJHZIZIZUJUMIZUKUSIZI
|
|
ZCBLMCALMJHULURUSVCUOURVCULUSURUTVBUPUTUQUPUTUPUPNOPUQUPVBUPVAUQCSQRTUAUBUL
|
|
UOUSVFURULUOIZUSIVDVEVGVDUSUJUMVDUKUNVDUIUCPVGUKUSUJUKUOUDUETUFCCABUGUH $.
|
|
|
|
$( The reciprocal of a positive number less than 1 is greater than 1.
|
|
(Contributed by NM, 23-Feb-2005.) $)
|
|
reclt1 $p |- ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> 1 < ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co wb 1re 0lt1 ltrec mpanr12 breq1i syl6bb
|
|
1div1e1 ) ABCDAEFGZAHEFZHHIJZHAIJZEFZHUBEFSHBCDHEFTUCKLMAHNOUAHUBERPQ $.
|
|
|
|
$( The reciprocal of a positive number greater than 1 is less than 1.
|
|
(Contributed by NM, 28-Dec-2005.) $)
|
|
recgt1 $p |- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co wb 1re 0lt1 ltrec mpanl12 breq2i syl6bb
|
|
1div1e1 ) ABCDAEFGZHAEFZHAIJZHHIJZEFZUAHEFHBCDHEFSTUCKLMHANOUBHUAERPQ $.
|
|
|
|
$( The reciprocal of a number greater than 1 is positive and less than 1.
|
|
(Contributed by NM, 23-Feb-2005.) $)
|
|
recgt1i $p |- ( ( A e. RR /\ 1 < A ) ->
|
|
( 0 < ( 1 / A ) /\ ( 1 / A ) < 1 ) ) $=
|
|
( cr wcel c1 clt wbr wa cc0 cdiv co 0lt1 wi 0re 1re mp3an12 mpani imdistani
|
|
lttr recgt0 syl recgt1 biimpa sylancom jca ) ABCZDAEFZGZHDAIJZEFZUHDEFZUGUE
|
|
HAEFZGZUIUEUFUKUEHDEFZUFUKKHBCDBCUEUMUFGUKLMNHDAROPQZASTUEUFULUJUNULUFUJAUA
|
|
UBUCUD $.
|
|
|
|
$( Construct a number less than 1 from any nonnegative number. (Contributed
|
|
by NM, 30-Dec-2005.) $)
|
|
recp1lt1 $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 ) $=
|
|
( cr wcel cc0 cle wbr wa c1 caddc co cdiv clt cmul simpl ltp1 recnd addcomd
|
|
syl 1cnd breqtrd addcld 1red readdcld 1re addgtge0 mpanr1 mpanl1 divcanap1d
|
|
0lt1 gt0ap0d mulid2d 3brtr4d wb redivclapd ltmul1 syl112anc mpbird ) ABCZDA
|
|
EFZGZAHAIJZKJZHLFZVBVAMJZHVAMJZLFZUTAVAVDVELUTAAHIJZVALUTURAVGLFURUSNZAORUT
|
|
AHUTAVHPZUTSZQTUTAVAVIUTHAVJVIUAZUTVAUTHAUTUBZVHUCZHBCZURUSDVALFZUDVNURGDHL
|
|
FUSVOUIHAUEUFUGZUJZUHUTVAVKUKULUTVBBCVNVABCVOVCVFUMUTAVAVHVMVQUNVLVMVPVBHVA
|
|
UOUPUQ $.
|
|
|
|
$( Given a positive number ` A ` , construct a new positive number less than
|
|
both ` A ` and 1. (Contributed by NM, 28-Dec-2005.) $)
|
|
recreclt $p |- ( ( A e. RR /\ 0 < A ) ->
|
|
( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co caddc recgt0 1re ltaddpos sylancl mpbid
|
|
wb sylancr 0lt1 cc simpl gt0ap0 rerecclapd readdcl wi 0re mp3an12 syl mpani
|
|
lttr mpd recgt1 syl2anc mpbii wceq recnd ax-1cn addcom breqtrd simpr ltrec1
|
|
syl22anc jca ) ABCZDAEFZGZHHHAIJZKJZIJZHEFZVIAEFZVFHVHEFZVJVFDVGEFZVLALVFVG
|
|
BCZHBCZVMVLQVFAVDVEUAZAUBUCZMVGHNOPZVFVHBCZDVHEFZVLVJQVFVOVNVSMVQHVGUDRZVFV
|
|
LVTVRVFDHEFZVLVTSVFVSWBVLGVTUEZWADBCVOVSWCUFMDHVHUJUGUHUIUKZVHULUMPVFVGVHEF
|
|
ZVKVFVGVGHKJZVHEVFWBVGWFEFZSVFVOVNWBWGQMVQHVGNRUNVFVGTCHTCWFVHUOVFVGVQUPUQV
|
|
GHUROUSVFVDVEVSVTWEVKQVPVDVEUTWAWDAVHVAVBPVC $.
|
|
|
|
$( The square function on nonnegative reals is monotonic. (Contributed by
|
|
NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
le2msq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt cmul lt2msq ancoms notbid simpll simprl lenltd
|
|
wn co wb remulcld 3bitr4d ) ACDZEAFGZHZBCDZEBFGZHZHZBAIGZQBBJRZAAJRZIGZQABF
|
|
GUKUJFGUHUIULUGUDUIULSBAKLMUHABUBUCUGNZUDUEUFOZPUHUKUJUHAAUMUMTUHBBUNUNTPUA
|
|
$.
|
|
|
|
$( The square of a nonnegative number is a one-to-one function. (Contributed
|
|
by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
msq11 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( A x. A ) = ( B x. B ) <-> A = B ) ) $=
|
|
( cr wcel cc0 cle wbr cmul wceq le2msq ancoms anbi12d simpll simprl letri3d
|
|
wa co wb remulcld 3bitr4rd ) ACDZEAFGZPZBCDZEBFGZPZPZABFGZBAFGZPAAHQZBBHQZF
|
|
GZUKUJFGZPABIUJUKIUGUHULUIUMABJUFUCUIUMRBAJKLUGABUAUBUFMZUCUDUENZOUGUJUKUGA
|
|
AUNUNSUGBBUOUOSOT $.
|
|
|
|
$( Less-than-or-equal-to and division relation. (Lemma for computing upper
|
|
bounds of products. The "+ 1" prevents division by zero.) (Contributed
|
|
by NM, 28-Sep-2005.) $)
|
|
ledivp1 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( A / ( B + 1 ) ) x. B ) <_ A ) $=
|
|
( cr wcel cc0 cle wbr wa c1 caddc cdiv cmul simprl peano2re syl simpll 0red
|
|
co clt recnd simprr ltp1 lelttrd redivclapd simpl divge0 syl12anc jca ltled
|
|
gt0ap0d lemul2a syl31anc divcanap1d breqtrd ) ACDZEAFGZHZBCDZEBFGZHZHZABIJR
|
|
ZKRZBLRZVCVBLRZAFVAURVBCDZVCCDZEVCFGZHBVBFGVDVEFGUQURUSMZVAURVFVIBNOZVAVGVH
|
|
VAAVBUOUPUTPZVJVAVBVJVAEBVBVAQVIVJUQURUSUAVAURBVBSGVIBUBOZUCZUJZUDVAUQVFEVB
|
|
SGVHUQUTUEVJVMAVBUFUGUHVABVBVIVJVLUIBVBVCUKULVAAVBVAAVKTVAVBVJTVNUMUN $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( If a nonnegative number is less than any positive number, it is zero.
|
|
(Contributed by NM, 11-Feb-2006.) $)
|
|
squeeze0 $p |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) ->
|
|
A = 0 ) $=
|
|
( cr wcel cc0 cle wbr cv clt wi wral w3a wceq ltnr 3ad2ant1 breq2 imbi12d
|
|
wn rspcva 3adant2 mtod simp1 0red lenltd mpbird simp2 letri3d mpbir2and )
|
|
BCDZEBFGZEAHZIGZBUKIGZJZACKZLZBEMBEFGZUJUPUQEBIGZRUPURBBIGZUIUJUSRUOBNOUI
|
|
UOURUSJZUJUNUTABCUKBMULURUMUSUKBEIPUKBBIPQSTUAUPBEUIUJUOUBZUPUCZUDUEUIUJU
|
|
OUFUPBEVAVBUGUH $.
|
|
$}
|
|
|
|
${
|
|
ltplus1.1 $e |- A e. RR $.
|
|
$( A number is less than itself plus 1. (Contributed by NM,
|
|
20-Aug-2001.) $)
|
|
ltp1i $p |- A < ( A + 1 ) $=
|
|
( cr wcel c1 caddc co clt wbr ltp1 ax-mp ) ACDAAEFGHIBAJK $.
|
|
|
|
$( The reciprocal of a positive number is positive. Exercise 4 of
|
|
[Apostol] p. 21. (Contributed by NM, 15-May-1999.) $)
|
|
recgt0i $p |- ( 0 < A -> 0 < ( 1 / A ) ) $=
|
|
( cr wcel cc0 clt wbr c1 cdiv co recgt0 mpan ) ACDEAFGEHAIJFGBAKL $.
|
|
|
|
${
|
|
recgt0i.2 $e |- 0 < A $.
|
|
$( The reciprocal of a positive number is positive. Exercise 4 of
|
|
[Apostol] p. 21. (Contributed by NM, 15-May-1999.) $)
|
|
recgt0ii $p |- 0 < ( 1 / A ) $=
|
|
( cr wcel cc0 clt wbr c1 cdiv co recgt0 mp2an ) ADEFAGHFIAJKGHBCALM $.
|
|
$}
|
|
|
|
prodgt0.2 $e |- B e. RR $.
|
|
$( Infer that a multiplicand is positive from a nonnegative multiplier and
|
|
positive product. (Contributed by NM, 15-May-1999.) $)
|
|
prodgt0i $p |- ( ( 0 <_ A /\ 0 < ( A x. B ) ) -> 0 < B ) $=
|
|
( cr wcel cc0 cle wbr cmul co clt wa prodgt0 mpanl12 ) AEFBEFGAHIGABJKLIM
|
|
GBLICDABNO $.
|
|
|
|
$( Infer that a multiplicand is nonnegative from a positive multiplier and
|
|
nonnegative product. (Contributed by NM, 2-Jul-2005.) $)
|
|
prodge0i $p |- ( ( 0 < A /\ 0 <_ ( A x. B ) ) -> 0 <_ B ) $=
|
|
( cr wcel cc0 clt wbr cmul co cle wa prodge0 mpanl12 ) AEFBEFGAHIGABJKLIM
|
|
GBLICDABNO $.
|
|
|
|
$( The ratio of two positive numbers is positive. (Contributed by NM,
|
|
16-May-1999.) $)
|
|
divgt0i $p |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A / B ) ) $=
|
|
( cr wcel cc0 clt wbr cdiv co wa divgt0 mpanr1 mpanl1 ) AEFZGAHIZGBHIZGAB
|
|
JKHIZCPQLBEFRSDABMNO $.
|
|
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by NM, 12-Aug-1999.) $)
|
|
divge0i $p |- ( ( 0 <_ A /\ 0 < B ) -> 0 <_ ( A / B ) ) $=
|
|
( cr wcel cc0 cle wbr clt cdiv co wa divge0 mpanr1 mpanl1 ) AEFZGAHIZGBJI
|
|
ZGABKLHIZCQRMBEFSTDABNOP $.
|
|
|
|
$( The reciprocal of both sides of 'less than'. (Contributed by NM,
|
|
15-Sep-1999.) $)
|
|
ltreci $p |- ( ( 0 < A /\ 0 < B ) ->
|
|
( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr c1 cdiv co wb wa ltrec mpanr1 mpanl1 ) AEFZGAHIZGBH
|
|
IZABHIJBKLJAKLHIMZCRSNBEFTUADABOPQ $.
|
|
|
|
$( The reciprocal of both sides of 'less than or equal to'. (Contributed
|
|
by NM, 16-Sep-1999.) $)
|
|
lereci $p |- ( ( 0 < A /\ 0 < B ) ->
|
|
( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr cle c1 cdiv co wb wa lerec mpanr1 mpanl1 ) AEFZGAHI
|
|
ZGBHIZABJIKBLMKALMJINZCSTOBEFUAUBDABPQR $.
|
|
|
|
$( The square function on nonnegative reals is strictly monotonic.
|
|
(Contributed by NM, 3-Aug-1999.) $)
|
|
lt2msqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( A < B <-> ( A x. A ) < ( B x. B ) ) ) $=
|
|
( cr wcel cc0 cle wbr clt cmul co wb wa lt2msq mpanr1 mpanl1 ) AEFZGAHIZG
|
|
BHIZABJIAAKLBBKLJIMZCRSNBEFTUADABOPQ $.
|
|
|
|
$( The square function on nonnegative reals is monotonic. (Contributed by
|
|
NM, 2-Aug-1999.) $)
|
|
le2msqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) ) $=
|
|
( cr wcel cc0 cle wbr cmul co wb wa le2msq mpanr1 mpanl1 ) AEFZGAHIZGBHIZ
|
|
ABHIAAJKBBJKHILZCQRMBEFSTDABNOP $.
|
|
|
|
$( The square of a nonnegative number is a one-to-one function.
|
|
(Contributed by NM, 29-Jul-1999.) $)
|
|
msq11i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( ( A x. A ) = ( B x. B ) <-> A = B ) ) $=
|
|
( cr wcel cc0 cle wbr cmul co wceq wb wa msq11 mpanr1 mpanl1 ) AEFZGAHIZG
|
|
BHIZAAJKBBJKLABLMZCRSNBEFTUADABOPQ $.
|
|
|
|
${
|
|
divgt0i2.3 $e |- 0 < B $.
|
|
$( The ratio of two positive numbers is positive. (Contributed by NM,
|
|
16-May-1999.) $)
|
|
divgt0i2i $p |- ( 0 < A -> 0 < ( A / B ) ) $=
|
|
( cc0 clt wbr cdiv co divgt0i mpan2 ) FAGHFBGHFABIJGHEABCDKL $.
|
|
$}
|
|
|
|
${
|
|
ltreci.3 $e |- 0 < A $.
|
|
ltreci.4 $e |- 0 < B $.
|
|
$( The reciprocal of both sides of 'less than'. (Contributed by NM,
|
|
15-Sep-1999.) $)
|
|
ltrecii $p |- ( A < B <-> ( 1 / B ) < ( 1 / A ) ) $=
|
|
( cc0 clt wbr c1 cdiv co wb ltreci mp2an ) GAHIGBHIABHIJBKLJAKLHIMEFABC
|
|
DNO $.
|
|
|
|
$( The ratio of two positive numbers is positive. (Contributed by NM,
|
|
18-May-1999.) $)
|
|
divgt0ii $p |- 0 < ( A / B ) $=
|
|
( cc0 clt wbr cdiv co divgt0i2i ax-mp ) GAHIGABJKHIEABCDFLM $.
|
|
$}
|
|
|
|
ltmul1.3 $e |- C e. RR $.
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) $)
|
|
ltmul1i $p |- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr cmul co wb wa ltmul1 mp3an12 mpan ) CGHZICJKZABJKAC
|
|
LMBCLMJKNZFAGHBGHSTOUADEABCPQR $.
|
|
|
|
$( Division of both sides of 'less than' by a positive number.
|
|
(Contributed by NM, 16-May-1999.) $)
|
|
ltdiv1i $p |- ( 0 < C -> ( A < B <-> ( A / C ) < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr cdiv co wb wa ltdiv1 mp3an12 mpan ) CGHZICJKZABJKAC
|
|
LMBCLMJKNZFAGHBGHSTOUADEABCPQR $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by NM, 12-Oct-1999.) $)
|
|
ltmuldivi $p |- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr cmul co cdiv wb wa ltmuldiv mp3an12 mpan ) CGHZICJK
|
|
ZACLMBJKABCNMJKOZFAGHBGHTUAPUBDEABCQRS $.
|
|
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) $)
|
|
ltmul2i $p |- ( 0 < C -> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr cmul co wb wa ltmul2 mp3an12 mpan ) CGHZICJKZABJKCA
|
|
LMCBLMJKNZFAGHBGHSTOUADEABCPQR $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by NM, 2-Aug-1999.) $)
|
|
lemul1i $p |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr cle cmul co wb wa lemul1 mp3an12 mpan ) CGHZICJKZAB
|
|
LKACMNBCMNLKOZFAGHBGHTUAPUBDEABCQRS $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by NM, 1-Aug-1999.) $)
|
|
lemul2i $p |- ( 0 < C -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr cle cmul co wb wa lemul2 mp3an12 mpan ) CGHZICJKZAB
|
|
LKCAMNCBMNLKOZFAGHBGHTUAPUBDEABCQRS $.
|
|
|
|
$( Swap denominator with other side of 'less than'. (Contributed by NM,
|
|
26-Sep-1999.) $)
|
|
ltdiv23i $p |- ( ( 0 < B /\ 0 < C ) ->
|
|
( ( A / B ) < C <-> ( A / C ) < B ) ) $=
|
|
( cc0 clt wbr cr wcel cdiv co wb wa ltdiv23 mp3an1 mpanl1 mpanr1 ) GBHIZC
|
|
JKZGCHIZABLMCHIACLMBHINZFBJKZTUAUBOZUCEAJKUDTOUEUCDABCPQRS $.
|
|
|
|
${
|
|
ltdiv23i.4 $e |- 0 < B $.
|
|
ltdiv23i.5 $e |- 0 < C $.
|
|
$( Swap denominator with other side of 'less than'. (Contributed by NM,
|
|
26-Sep-1999.) $)
|
|
ltdiv23ii $p |- ( ( A / B ) < C <-> ( A / C ) < B ) $=
|
|
( cc0 clt wbr cdiv co wb ltdiv23i mp2an ) IBJKICJKABLMCJKACLMBJKNGHABCD
|
|
EFOP $.
|
|
$}
|
|
|
|
${
|
|
ltmul1i.4 $e |- 0 < C $.
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
|
|
(Proof shortened by Paul Chapman, 25-Jan-2008.) $)
|
|
ltmul1ii $p |- ( A < B <-> ( A x. C ) < ( B x. C ) ) $=
|
|
( cc0 clt wbr cmul co wb ltmul1i ax-mp ) HCIJABIJACKLBCKLIJMGABCDEFNO
|
|
$.
|
|
|
|
$( Division of both sides of 'less than' by a positive number.
|
|
(Contributed by NM, 16-May-1999.) $)
|
|
ltdiv1ii $p |- ( A < B <-> ( A / C ) < ( B / C ) ) $=
|
|
( cc0 clt wbr cdiv co wb ltdiv1i ax-mp ) HCIJABIJACKLBCKLIJMGABCDEFNO
|
|
$.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ltp1d.1 $e |- ( ph -> A e. RR ) $.
|
|
$( A number is less than itself plus 1. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
ltp1d $p |- ( ph -> A < ( A + 1 ) ) $=
|
|
( cr wcel c1 caddc co clt wbr ltp1 syl ) ABDEBBFGHIJCBKL $.
|
|
|
|
$( A number is less than or equal to itself plus 1. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
lep1d $p |- ( ph -> A <_ ( A + 1 ) ) $=
|
|
( cr wcel c1 caddc co cle wbr lep1 syl ) ABDEBBFGHIJCBKL $.
|
|
|
|
$( A number minus 1 is less than itself. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
ltm1d $p |- ( ph -> ( A - 1 ) < A ) $=
|
|
( cr wcel c1 cmin co clt wbr ltm1 syl ) ABDEBFGHBIJCBKL $.
|
|
|
|
$( A number minus 1 is less than or equal to itself. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
lem1d $p |- ( ph -> ( A - 1 ) <_ A ) $=
|
|
( cr wcel c1 cmin co cle wbr lem1 syl ) ABDEBFGHBIJCBKL $.
|
|
|
|
${
|
|
recgt0d.2 $e |- ( ph -> 0 < A ) $.
|
|
$( The reciprocal of a positive number is positive. Exercise 4 of
|
|
[Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
recgt0d $p |- ( ph -> 0 < ( 1 / A ) ) $=
|
|
( cr wcel cc0 clt wbr c1 cdiv co recgt0 syl2anc ) ABEFGBHIGJBKLHICDBMN
|
|
$.
|
|
$}
|
|
|
|
divgt0d.2 $e |- ( ph -> B e. RR ) $.
|
|
${
|
|
divgt0d.3 $e |- ( ph -> 0 < A ) $.
|
|
divgt0d.4 $e |- ( ph -> 0 < B ) $.
|
|
$( The ratio of two positive numbers is positive. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
divgt0d $p |- ( ph -> 0 < ( A / B ) ) $=
|
|
( cr wcel cc0 clt wbr cdiv co divgt0 syl22anc ) ABHIJBKLCHIJCKLJBCMNKLD
|
|
FEGBCOP $.
|
|
$}
|
|
|
|
${
|
|
mulgt1d.3 $e |- ( ph -> 1 < A ) $.
|
|
mulgt1d.4 $e |- ( ph -> 1 < B ) $.
|
|
$( The product of two numbers greater than 1 is greater than 1.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
mulgt1d $p |- ( ph -> 1 < ( A x. B ) ) $=
|
|
( cr wcel c1 clt wbr cmul co mulgt1 syl22anc ) ABHICHIJBKLJCKLJBCMNKLDE
|
|
FGBCOP $.
|
|
$}
|
|
|
|
${
|
|
lemulge11d.3 $e |- ( ph -> 0 <_ A ) $.
|
|
lemulge11d.4 $e |- ( ph -> 1 <_ B ) $.
|
|
$( Multiplication by a number greater than or equal to 1. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
lemulge11d $p |- ( ph -> A <_ ( A x. B ) ) $=
|
|
( cr wcel cc0 cle wbr c1 cmul co lemulge11 syl22anc ) ABHICHIJBKLMCKLBB
|
|
CNOKLDEFGBCPQ $.
|
|
|
|
$( Multiplication by a number greater than or equal to 1. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
lemulge12d $p |- ( ph -> A <_ ( B x. A ) ) $=
|
|
( cr wcel cc0 cle wbr c1 cmul co lemulge12 syl22anc ) ABHICHIJBKLMCKLBC
|
|
BNOKLDEFGBCPQ $.
|
|
$}
|
|
|
|
lemul1ad.3 $e |- ( ph -> C e. RR ) $.
|
|
${
|
|
lemul1ad.4 $e |- ( ph -> 0 <_ C ) $.
|
|
lemul1ad.5 $e |- ( ph -> A <_ B ) $.
|
|
$( Multiplication of both sides of 'less than or equal to' by a
|
|
nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lemul1ad $p |- ( ph -> ( A x. C ) <_ ( B x. C ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co jca lemul1a syl31anc ) ABJKCJKDJKZLDMN
|
|
ZOBCMNBDPQCDPQMNEFAUAUBGHRIBCDST $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a
|
|
nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lemul2ad $p |- ( ph -> ( C x. A ) <_ ( C x. B ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co jca lemul2a syl31anc ) ABJKCJKDJKZLDMN
|
|
ZOBCMNDBPQDCPQMNEFAUAUBGHRIBCDST $.
|
|
$}
|
|
|
|
ltmul12ad.3 $e |- ( ph -> D e. RR ) $.
|
|
${
|
|
ltmul12ad.4 $e |- ( ph -> 0 <_ A ) $.
|
|
ltmul12ad.5 $e |- ( ph -> A < B ) $.
|
|
ltmul12ad.6 $e |- ( ph -> 0 <_ C ) $.
|
|
ltmul12ad.7 $e |- ( ph -> C < D ) $.
|
|
$( Comparison of product of two positive numbers. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
ltmul12ad $p |- ( ph -> ( A x. C ) < ( B x. D ) ) $=
|
|
( cr wcel wa cc0 wbr clt jca cle cmul co ltmul12a syl22anc ) ABNOZCNOZP
|
|
QBUARZBCSRZPDNOZENOZPQDUARZDESRZPBDUBUCCEUBUCSRAUFUGFGTAUHUIJKTAUJUKHIT
|
|
AULUMLMTBCDEUDUE $.
|
|
$}
|
|
|
|
${
|
|
lemul12ad.4 $e |- ( ph -> 0 <_ A ) $.
|
|
lemul12ad.5 $e |- ( ph -> 0 <_ C ) $.
|
|
lemul12ad.6 $e |- ( ph -> A <_ B ) $.
|
|
lemul12ad.7 $e |- ( ph -> C <_ D ) $.
|
|
$( Comparison of product of two nonnegative numbers. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
lemul12ad $p |- ( ph -> ( A x. C ) <_ ( B x. D ) ) $=
|
|
( cle wbr cmul co cr wcel wa cc0 wi jca lemul12a syl22anc mp2and ) ABCN
|
|
OZDENOZBDPQCEPQNOZLMABRSZUABNOZTCRSDRSZUADNOZTERSUGUHTUIUBAUJUKFJUCGAUL
|
|
UMHKUCIBCDEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
lemul12bd.4 $e |- ( ph -> 0 <_ A ) $.
|
|
lemul12bd.5 $e |- ( ph -> 0 <_ D ) $.
|
|
lemul12bd.6 $e |- ( ph -> A <_ B ) $.
|
|
lemul12bd.7 $e |- ( ph -> C <_ D ) $.
|
|
$( Comparison of product of two nonnegative numbers. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
lemul12bd $p |- ( ph -> ( A x. C ) <_ ( B x. D ) ) $=
|
|
( cle wbr cmul co cr wcel wa cc0 wi jca lemul12b syl22anc mp2and ) ABCN
|
|
OZDENOZBDPQCEPQNOZLMABRSZUABNOZTCRSDRSERSZUAENOZTUGUHTUIUBAUJUKFJUCGHAU
|
|
LUMIKUCBCDEUDUEUF $.
|
|
$}
|
|
$}
|
|
|
|
$( Multiplying a nonnegative number by a nonpositive number yields a
|
|
nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) $)
|
|
mulle0r $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ 0 /\ 0 <_ B ) ) ->
|
|
( A x. B ) <_ 0 ) $=
|
|
( cr wcel wa cc0 cle cmul co simpll recnd simplr mulcomd 0red simprr simprl
|
|
wbr lemul2ad mul01d breqtrd eqbrtrd ) ACDZBCDZEZAFGQZFBGQZEZEZABHIBAHIZFGUH
|
|
ABUHAUBUCUGJZKUHBUBUCUGLZKZMUHUIBFHIFGUHAFBUJUHNUKUDUEUFOUDUEUFPRUHBULSTUA
|
|
$.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Suprema
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d w x y S $. $d w y A $.
|
|
$( If a set of reals contains a lower bound, it contains a unique lower
|
|
bound. (Contributed by NM, 9-Oct-2005.) $)
|
|
lbreu $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
|
|
E! x e. S A. y e. S x <_ y ) $=
|
|
( vw cr wss cv cle wbr wral wrex wa weq wi wreu wcel breq2 im2anan9r ssel
|
|
rspcv anim12d impcom letri3 syl exbiri com23 syld ralrimivv anim2i ancoms
|
|
wb com3r breq1 ralbidv reu4 sylibr ) CEFZAGZBGZHIZBCJZACKZLVBVADGZUSHIZBC
|
|
JZLZADMZNZDCJACJZLZVAACOVBUQVJUQVIVBUQVHADCCURCPZVCCPZLZVFUQVGVMVFURVCHIZ
|
|
VCURHIZLZUQVGNVLVAVNVKVEVOUTVNBVCCUSVCURHQTVDVOBURCUSURVCHQTRVMUQVPVGVMUQ
|
|
VGVPVMUQLUREPZVCEPZLZVGVPUKUQVMVSUQVKVQVLVRCEURSCEVCSUAUBURVCUCUDUEUFUGUL
|
|
UHUIUJVAVEADCVGUTVDBCURVCUSHUMUNUOUP $.
|
|
|
|
$( If a set of reals contains a lower bound, it contains a unique lower
|
|
bound that belongs to the set. (Contributed by NM, 9-Oct-2005.)
|
|
(Revised by Mario Carneiro, 24-Dec-2016.) $)
|
|
lbcl $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
|
|
( iota_ x e. S A. y e. S x <_ y ) e. S ) $=
|
|
( cr wss cv cle wbr wral wrex wa wreu crio wcel lbreu riotacl syl ) CDEAF
|
|
BFGHBCIZACJKRACLRACMCNABCORACPQ $.
|
|
|
|
$( If a set of reals contains a lower bound, the lower bound is less than
|
|
or equal to all members of the set. (Contributed by NM, 9-Oct-2005.)
|
|
(Proof shortened by Mario Carneiro, 24-Dec-2016.) $)
|
|
lble $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) ->
|
|
( iota_ x e. S A. y e. S x <_ y ) <_ A ) $=
|
|
( cr wss cv cle wral wrex wcel crio wreu lbreu nfcv nfriota1 nfbr nfralxy
|
|
wbr wa eqid wceq nfra1 nfeq2 breq1 ralbid riotaprop syl simprd breq2 rspc
|
|
nfriota mpan9 3impa ) DEFZAGZBGZHSZBDIZADJZCDKZUSADLZCHSZUOUTTZVBUQHSZBDI
|
|
ZVAVCVDVBDKZVFVDUSADMVGVFTABDNUSVFADVBVEABDADOAVBUQHUSADPAHOAUQOQRVBUAUPV
|
|
BUBURVEBDBUPVBUSBADURBDUCBDOULZUDUPVBUQHUEUFUGUHUIVEVCBCDBVBCHVHBHOBCOQUQ
|
|
CVBHUJUKUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d S f g x y z $.
|
|
$( If a set of reals contains a lower bound, the lower bound is its
|
|
infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV,
|
|
4-Sep-2020.) $)
|
|
lbinf $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) ->
|
|
inf ( S , RR , < ) = ( iota_ x e. S A. y e. S x <_ y ) ) $=
|
|
( vz vg vf cr wss cv cle wbr wral wrex wa crio clt wcel weq wn adantr mpd
|
|
wb lttri3 adantl lbcl wi ssel ssel2 adantlr lble 3expa lensymd infminti )
|
|
CGHZAIBIJKBCLZACMZNZDEFGCUOACOZPFIZGQEIZGQNFERUSUTPKSUTUSPKSNUBUQUSUTUCUD
|
|
UQURCQZURGQZABCUEZUNVAVBUFUPCGURUGTUAZVCUQDIZCQZNURVEUQVBVFVDTUNVFVEGQUPC
|
|
GVEUHUIUNUPVFURVEJKABVECUJUKULUM $.
|
|
|
|
$( If a set of reals contains a lower bound, it contains its infimum.
|
|
(Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) $)
|
|
lbinfcl $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y )
|
|
-> inf ( S , RR , < ) e. S ) $=
|
|
( cr wss cv cle wbr wral wrex wa clt cinf crio lbinf lbcl eqeltrd ) CDEAF
|
|
BFGHBCIZACJKCDLMRACNCABCOABCPQ $.
|
|
|
|
$d A y $.
|
|
$( If a set of reals contains a lower bound, its infimum is less than or
|
|
equal to all members of the set. (Contributed by NM, 11-Oct-2005.)
|
|
(Revised by AV, 4-Sep-2020.) $)
|
|
lbinfle $p |- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S )
|
|
-> inf ( S , RR , < ) <_ A ) $=
|
|
( cr wss cle wbr wral wrex wcel w3a clt cinf crio wceq lbinf 3adant3 lble
|
|
cv eqbrtrd ) DEFZATBTGHBDIZADJZCDKZLDEMNZUCADOZCGUBUDUFUGPUEABDQRABCDSUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d f g ph x $.
|
|
suprubex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
|
|
/\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
|
|
suprubex.ss $e |- ( ph -> A C_ RR ) $.
|
|
|
|
${
|
|
suprubex.b $e |- ( ph -> B e. A ) $.
|
|
$( A member of a nonempty bounded set of reals is less than or equal to
|
|
the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.) $)
|
|
suprubex $p |- ( ph -> B <_ sup ( A , RR , < ) ) $=
|
|
( vg vf cr clt csup sseldd cv wcel wa wbr wn weq lttri3 supclti supubti
|
|
wb adantl mpd nltled ) AFELMNZAELFHIOABCDJKLEMKPZLQJPZLQRKJUAUJUKMSTUKU
|
|
JMSTRUEAUJUKUBUFZGUCAFEQUIFMSTIABCDJKLEFMULGUDUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y $. $d A w z $. $d B w z $.
|
|
suprlubex.b $e |- ( ph -> B e. RR ) $.
|
|
$( The supremum of a nonempty bounded set of reals is the least upper
|
|
bound. (Contributed by Jim Kingdon, 19-Jan-2022.) $)
|
|
suprlubex $p |- ( ph
|
|
-> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) ) $=
|
|
( vg vf cr wcel clt csup wbr cv wb wa wn wrex weq lttri3 adantl wor a1i
|
|
ltso suplub2ti mpdan ) AFLMFELNONPFDQNPDEUARIABCDJKLEFNKQZLMJQZLMSKJUBU
|
|
JUKNPTUKUJNPTSRAUJUKUCUDGLNUEAUGUFHUHUI $.
|
|
|
|
$( An upper bound is not less than the supremum of a nonempty bounded set
|
|
of reals. (Contributed by Jim Kingdon, 19-Jan-2022.) $)
|
|
suprnubex $p |- ( ph
|
|
-> ( -. B < sup ( A , RR , < ) <-> A. z e. A -. B < z ) ) $=
|
|
( cr clt csup wbr wn cv wrex wral suprlubex notbid ralnex syl6bbr ) AFE
|
|
JKLKMZNFDOKMZDEPZNUCNDEQAUBUDABCDEFGHIRSUCDETUA $.
|
|
|
|
$d A w x y z $. $d f g ph x $. $d ph w x $.
|
|
$( The supremum of a nonempty bounded set of reals is less than or equal
|
|
to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by
|
|
Mario Carneiro, 6-Sep-2014.) $)
|
|
suprleubex $p |- ( ph
|
|
-> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) ) $=
|
|
( vw vg vf cr clt cle wbr cv wral wn wcel csup wa weq wb lttri3 supclti
|
|
adantl lenltd bitrd breq2 notbid cbvralv syl6bbr sselda adantr ralbidva
|
|
suprnubex bitr4d breq1 syl6bb ) AEMNUAZFOPZJQZFOPZJERZDQZFOPZDERAVBFVCN
|
|
PZSZJERZVEAVBFVFNPZSZDERZVJAVBFVANPSVMAVAFABCDKLMENLQZMTKQZMTUBLKUCVNVO
|
|
NPSVOVNNPSUBUDAVNVOUEUGGUFIUHABCDEFGHIUQUIVIVLJDEJDUCVHVKVCVFFNUJUKULUM
|
|
AVDVIJEAVCETZUBVCFAEMVCHUNAFMTVPIUOUHUPURVDVGJDEVCVFFOUSULUT $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z F $.
|
|
negiso.1 $e |- F = ( x e. RR |-> -u x ) $.
|
|
$( Negation is an order anti-isomorphism of the real numbers, which is its
|
|
own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) $)
|
|
negiso $p |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F ) $=
|
|
( vz vy cr clt ccnv wceq cv wbr wb cneg wa wtru wcel simpr adantl negeq
|
|
cc wiso wf1o wral cmpt renegcld recn negcon2 syl2an f1ocnv2d simpli simpl
|
|
cfv trud recnd negcld brcnvg syl2anc fvmptd breq12d ltneg 3bitr4rd rgen2a
|
|
a1i df-isom mpbir2an cbvmptv simpri 3eqtr4i pm3.2i ) FFGGHZBUAZBHZBIVKFFB
|
|
UBZDJZEJZGKZVNBULZVOBULZVJKZLZEFUCDFUCVMVLEFVOMZUDZIZVMWCNOAEFFAJZMZWABCO
|
|
WDFPZNWDOWFQUEOVOFPZNVOOWGQUEWFWGNWDWAIVOWEILZOWFWDTPVOTPZWHWGWDUFVOUFZWD
|
|
VOUGUHRUIUMZUJVTDEFVNFPZWGNZVNMZWAVJKZWAWNGKZVSVPWMWNTPWATPWOWPLWMVNWMVNW
|
|
LWGUKZUNUOZWMVOWGWIWLWJRUOZWNWATTGUPUQWMVQWNVRWAVJWMAVNWEWNFBTBAFWEUDZIWM
|
|
CVCZWDVNIWEWNIWMWDVNSRWQWRURWMAVOWEWAFBTXAWDVOIWEWAIWMWDVOSRWLWGQWSURUSVN
|
|
VOUTVAVBDEFFGVJBVDVEWBWTVLBEAFWAWEVOWDSVFVMWCWKVGCVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( The infimum of a set of reals ` A ` . (Contributed by NM, 9-Oct-2005.)
|
|
(Revised by AV, 4-Sep-2020.) $)
|
|
dfinfre $p |- ( A C_ RR -> inf ( A , RR , < ) = U. { x e. RR |
|
|
( A. y e. A x <_ y /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) } ) $=
|
|
( cr clt cv wbr wral wrex wi wa crab cuni wn wcel wb vex brcnv syl5eq wss
|
|
cinf ccnv csup cle df-inf df-sup ssel2 lenlt notbii syl6rbbr sylan2 an32s
|
|
ancoms ralbidva rexbii imbi12i ralbii a1i anbi12d rabbidva unieqd ) DEUAZ
|
|
DEFUBDEFUCZUDZAGZBGZUEHZBDIZVFVGFHZCGZVGFHZCDJZKZBEIZLZAEMZNZDEFUFVCVEVFV
|
|
GVDHZOZBDIZVGVFVDHZVGVKVDHZCDJZKZBEIZLZAEMZNVRABCDEVDUGVCWHVQVCWGVPAEVCVF
|
|
EPZLZWAVIWFVOWJVTVHBDVCVGDPZWIVTVHQZWIVCWKLZWLWMWIVGEPZWLDEVGUHWIWNLVHVGV
|
|
FFHZOVTVFVGUIVSWOVFVGFARZBRZSUJUKULUNUMUOWFVOQWJWEVNBEWBVJWDVMVGVFFWQWPSW
|
|
CVLCDVGVKFWQCRSUPUQURUSUTVAVBTT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Imaginary and complex number properties
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The real representation of complex numbers is apart from zero iff one of
|
|
its terms is apart from zero. (Contributed by Jim Kingdon,
|
|
5-Mar-2020.) $)
|
|
crap0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( A # 0 \/ B # 0 ) <-> ( A + ( _i x. B ) ) # 0 ) ) $=
|
|
( ci cmul co caddc cc0 cap wbr cr wcel wa ax-icn mul01i oveq2i eqtri breq2i
|
|
wo 00id 0re wb apreim mpanr12 syl5rbbr ) ACBDEFEZGHIUEGCGDEZFEZHIZAJKBJKLZA
|
|
GHIBGHIRZUGGUEHUGGGFEGUFGGFCMNOSPQUIGJKZUKUHUJUATTABGGUBUCUD $.
|
|
|
|
${
|
|
$d x y z w A $.
|
|
$( The real part of a complex number is unique. Proposition 10-1.3 of
|
|
[Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
creur $p |- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) $=
|
|
( vz vw cc wcel cv ci cmul co caddc wceq cr wrex wreu cnre wa wb wral cru
|
|
ancoms eqcom ancom 3bitr4g anassrs rexbidva biidd ceqsrexv ad2antlr bitrd
|
|
ralrimiva reu6i syldan eqeq1 rexbidv reubidv syl5ibrcom rexlimivv syl ) C
|
|
FGCDHZIEHZJKLKZMZENODNOCAHZIBHZJKLKZMZBNOZANPZDECQVDVJDENNVANGZVBNGZRZVJV
|
|
DVCVGMZBNOZANPZVKVLVOVEVAMZSZANTVPVMVRANVMVENGZRZVOVFVBMZVQRZBNOZVQVTVNWB
|
|
BNVMVSVFNGZVNWBSVMVSWDRZRVGVCMZVQWARZVNWBWEVMWFWGSVEVFVAVBUAUBVCVGUCWAVQU
|
|
DUEUFUGVLWCVQSVKVSVQVQBVBNWAVQUHUIUJUKULVOANVAUMUNVDVIVOANVDVHVNBNCVCVGUO
|
|
UPUQURUSUT $.
|
|
|
|
$( The imaginary part of a complex number is unique. Proposition 10-1.3 of
|
|
[Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
creui $p |- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) ) $=
|
|
( vz vw cc wcel cv ci cmul co caddc wceq cr wrex wreu cnre wa wb wral cru
|
|
simpr eqcom ancoms syl5bb anass1rs rexbidva biidd ceqsrexv ad2antrr bitrd
|
|
ralrimiva reu6i syl2anc eqeq1 rexbidv reubidv syl5ibrcom rexlimivv syl )
|
|
CFGCDHZIEHZJKLKZMZENODNOCAHZIBHZJKLKZMZANOZBNPZDECQVDVJDENNVANGZVBNGZRZVJ
|
|
VDVCVGMZANOZBNPZVMVLVOVFVBMZSZBNTVPVKVLUBVMVRBNVMVFNGZRZVOVEVAMZVQRZANOZV
|
|
QVTVNWBANVMVENGZVSVNWBSVNVGVCMZVMWDVSRZRWBVCVGUCWFVMWEWBSVEVFVAVBUAUDUEUF
|
|
UGVKWCVQSVLVSVQVQAVANWAVQUHUIUJUKULVOBNVBUMUNVDVIVOBNVDVHVNANCVCVGUOUPUQU
|
|
RUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The complex conjugate of a complex number is unique. (Contributed by
|
|
Mario Carneiro, 6-Nov-2013.) $)
|
|
cju $p |- ( A e. CC -> E! x e. CC
|
|
( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) $=
|
|
( vy vz cc wcel cv caddc co cr ci cmin cmul wrex wceq ax-icn oveq2d oveq2
|
|
wa eleq1d wi wral wreu cnre recn mulcl sylancr subcl syl2an adantr adantl
|
|
ppncand readdcl anidms eqeltrd pnncand a1i adddid eqtr4d addcld mulass c1
|
|
mp3an12 syl cneg ixi 1re renegcli eqeltri readdcld remulcl anbi12d rspcev
|
|
simpr syl12anc rexbidv syl5ibrcom rexlimivv an4 cc0 resubcl pnpcan syl5ib
|
|
ancoms adantrl adantrr subdid nnncan1 3com23 eqtr3d anim12d subeq0 biimpd
|
|
oveq1 3expb rimul 3syld syl5bi ralrimivva reu4 sylanbrc ) BEFZBAGZHIZJFZK
|
|
BXCLIZMIZJFZSZAENZXIBCGZHIZJFZKBXKLIZMIZJFZSZSZXCXKOZUAZCEUBAEUBXIAEUCXBB
|
|
XKKDGZMIZHIZOZDJNCJNXJCDBUDYDXJCDJJXKJFZYAJFZSZXJYDYCXCHIZJFZKYCXCLIZMIZJ
|
|
FZSZAENZYGXKYBLIZEFZYCYOHIZJFZKYCYOLIZMIZJFZYNYEXKEFZYBEFZYPYFXKUEZYFKEFZ
|
|
YAEFZUUCPYAUEZKYAUFUGZXKYBUHUIYGYQXKXKHIZJYGXKYBXKYEUUBYFUUDUJZYFUUCYEUUH
|
|
UKZUUJULYEUUIJFZYFYEUULXKXKUMUNUJUOYGYTKKMIZYAYAHIZMIZJYGYTKKUUNMIZMIZUUO
|
|
YGYSUUPKMYGYSYBYBHIUUPYGXKYBYBUUJUUKUUKUPYGKYAYAUUEYGPUQYFUUFYEUUGUKZUURU
|
|
RUSQYGUUNEFZUUOUUQOZYGYAYAUURUURUTUUEUUEUUSUUTPPKKUUNVAVCVDUSYGUUMJFUUNJF
|
|
UUOJFUUMVBVEJVFVBVGVHVIYGYAYAYEYFVNZUVAVJUUMUUNVKUGUOYMYRUUASAYOEXCYOOZYI
|
|
YRYLUUAUVBYHYQJXCYOYCHRTUVBYKYTJUVBYJYSKMXCYOYCLRQTVLVMVOYDXIYMAEYDXEYIXH
|
|
YLYDXDYHJBYCXCHWNTYDXGYKJYDXFYJKMBYCXCLWNQTVLVPVQVRVDXBXTACEEXRXEXMSZXHXP
|
|
SZSZXBXCEFZUUBSZSZXSXEXHXMXPVSUVHUVEXCXKLIZJFZKUVIMIZJFZSZUVIVTOZXSUVHUVC
|
|
UVJUVDUVLUVCXDXLLIZJFUVHUVJXDXLWAUVHUVOUVIJXBUVFUUBUVOUVIOBXCXKWBWOTWCUVD
|
|
XOXGLIZJFZUVHUVLXPXHUVQXOXGWAWDUVHUVPUVKJUVHKXNXFLIZMIUVPUVKUVHKXNXFUUEUV
|
|
HPUQXBUUBXNEFUVFBXKUHWEXBUVFXFEFUUBBXCUHWFWGUVHUVRUVIKMXBUVFUUBUVRUVIOZXB
|
|
UUBUVFUVSBXKXCWHWIWOQWJTWCWKUVMUVNUAUVHUVIWPUQUVGUVNXSUAXBUVGUVNXSXCXKWLW
|
|
MUKWQWRWSXIXQACEXSXEXMXHXPXSXDXLJXCXKBHRTXSXGXOJXSXFXNKMXCXKBLRQTVLWTXA
|
|
$.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Integer sets
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Positive integers (as a subset of complex numbers)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c NN $. $( The set of positive integers (blackboard bold N). $)
|
|
|
|
$( Extend class notation to include the class of positive integers. $)
|
|
cn $a class NN $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Definition of the set of positive integers. For naming consistency with
|
|
the Metamath Proof Explorer usages should refer to ~ dfnn2 instead.
|
|
(Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro,
|
|
3-May-2014.) (New usage is discouraged.) $)
|
|
df-inn $a |- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $.
|
|
|
|
$( Definition of the set of positive integers. Another name for ~ df-inn .
|
|
(Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro,
|
|
3-May-2014.) $)
|
|
dfnn2 $p |- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
|
|
( df-inn ) ABC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Peano's inductive postulate. Theorem I.36 (principle of mathematical
|
|
induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.)
|
|
(Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
peano5nni $p |- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A ) $=
|
|
( vy c1 wcel cv caddc co wral wa cn cr cin wss elin ax-mp syl cvv eleq2
|
|
wi 1re biimpri mpan2 inss1 ssralv inss2 1red readdcld simplbi2com ralimia
|
|
sseli reex inex2 wceq raleqbi1dv anbi12d elabg cint dfnn2 intss1 syl5eqss
|
|
cab syl6bir syl2an syl6ss ) DBEZAFZDGHZBEZABIZJKBLMZBVFDVKEZVHVKEZAVKIZKV
|
|
KNZVJVFDLEZVLUAVLVFVPJDBLOUBUCVJVIAVKIZVNVKBNVJVQTBLUDZVIAVKBUEPVIVMAVKVG
|
|
VKEZVHLEZVIVMTVSVGDVKLVGBLUFUKVSUGUHVMVIVTVHBLOUIQUJQVKREZVLVNJZVOTLBULUM
|
|
WAWBVKDCFZEZVHWCEZAWCIZJZCVBZEZVOWGWBCVKRWCVKUNWDVLWFVNWCVKDSWEVMAWCVKWCV
|
|
KVHSUOUPUQWIKWHURVKCAUSVKWHUTVAVCPVDVRVE $.
|
|
$}
|
|
|
|
$( The positive integers are a subset of the reals. (Contributed by NM,
|
|
10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) $)
|
|
nnssre $p |- NN C_ RR $=
|
|
( vx c1 cr wcel cv caddc co wral cn wss 1re peano2re rgen peano5nni mp2an )
|
|
BCDAEZBFGCDZACHICJKQACPLMACNO $.
|
|
|
|
$( The positive integers are a subset of the complex numbers. (Contributed
|
|
by NM, 2-Aug-2004.) $)
|
|
nnsscn $p |- NN C_ CC $=
|
|
( cn cr cc nnssre ax-resscn sstri ) ABCDEF $.
|
|
|
|
$( The set of positive integers exists. (Contributed by NM, 3-Oct-1999.)
|
|
(Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
nnex $p |- NN e. _V $=
|
|
( cn cc cnex nnsscn ssexi ) ABCDE $.
|
|
|
|
$( A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) $)
|
|
nnre $p |- ( A e. NN -> A e. RR ) $=
|
|
( cn cr nnssre sseli ) BCADE $.
|
|
|
|
$( A positive integer is a complex number. (Contributed by NM,
|
|
18-Aug-1999.) $)
|
|
nncn $p |- ( A e. NN -> A e. CC ) $=
|
|
( cn cc nnsscn sseli ) BCADE $.
|
|
|
|
${
|
|
nnre.1 $e |- A e. NN $.
|
|
$( A positive integer is a real number. (Contributed by NM,
|
|
18-Aug-1999.) $)
|
|
nnrei $p |- A e. RR $=
|
|
( cn wcel cr nnre ax-mp ) ACDAEDBAFG $.
|
|
|
|
$( A positive integer is a complex number. (Contributed by NM,
|
|
18-Aug-1999.) $)
|
|
nncni $p |- A e. CC $=
|
|
( nnrei recni ) AABCD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Peano postulate: 1 is a positive integer. (Contributed by NM,
|
|
11-Jan-1997.) $)
|
|
1nn $p |- 1 e. NN $=
|
|
( vz vx vy c1 cn wcel cv caddc co wral wa cab cint dfnn2 eleq2i cr wb 1re
|
|
elintg eleq2 ax-mp bitri vex weq raleqbi1dv anbi12d elab simplbi mprgbir
|
|
) DEFZDAGZFZADBGZFZCGDHIZUMFZCUMJZKZBLZUJDUSMZFZULAUSJZEUTDBCNODPFVAVBQRA
|
|
DUSPSUAUBUKUSFULUOUKFZCUKJZURULVDKBUKAUCBAUDUNULUQVDUMUKDTUPVCCUMUKUMUKUO
|
|
TUEUFUGUHUI $.
|
|
|
|
$( Peano postulate: a successor of a positive integer is a positive
|
|
integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro,
|
|
17-Nov-2014.) $)
|
|
peano2nn $p |- ( A e. NN -> ( A + 1 ) e. NN ) $=
|
|
( vz vx vy cn wcel c1 caddc co cv wral wa eleq2i elintg syl5bb wceq eleq2
|
|
cab syl cr cint dfnn2 ibi vex raleqbi1dv anbi12d elab oveq1 eleq1d rspcva
|
|
simprbi sylan2 expcom ralimia wb nnre 1red readdcld mpbird ) AEFZAGHIZEFZ
|
|
VABJZFZBGCJZFZDJZGHIZVEFZDVEKZLZCRZKZUTAVCFZBVLKZVMUTVOUTAVLUAZFUTVOEVPAC
|
|
DUBZMBAVLENOUCVNVDBVLVNVCVLFZVDVRVNVHVCFZDVCKZVDVRGVCFZVTVKWAVTLCVCBUDVEV
|
|
CPVFWAVJVTVEVCGQVIVSDVEVCVEVCVHQUEUFUGUKVSVDDAVCVGAPVHVAVCVGAGHUHUIUJULUM
|
|
UNSUTVATFZVBVMUOUTAGAUPUTUQURVBVAVPFWBVMEVPVAVQMBVAVLTNOSUS $.
|
|
$}
|
|
|
|
${
|
|
nnred.1 $e |- ( ph -> A e. NN ) $.
|
|
$( A positive integer is a real number. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnred $p |- ( ph -> A e. RR ) $=
|
|
( cn cr nnssre sseldi ) ADEBFCG $.
|
|
|
|
$( A positive integer is a complex number. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nncnd $p |- ( ph -> A e. CC ) $=
|
|
( cn cc nnsscn sseldi ) ADEBFCG $.
|
|
|
|
$( Peano postulate: a successor of a positive integer is a positive
|
|
integer. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
peano2nnd $p |- ( ph -> ( A + 1 ) e. NN ) $=
|
|
( cn wcel c1 caddc co peano2nn syl ) ABDEBFGHDECBIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Principle of mathematical induction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ps $. $d x ch $. $d x th $. $d x ta $.
|
|
$d y ph $.
|
|
$( Substitutions. $)
|
|
nnind.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
|
|
nnind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
nnind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
nnind.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
nnind.5 $e |- ps $.
|
|
$( Induction step. $)
|
|
nnind.6 $e |- ( y e. NN -> ( ch -> th ) ) $.
|
|
$( Principle of Mathematical Induction (inference schema). The first four
|
|
hypotheses give us the substitution instances we need; the last two are
|
|
the basis and the induction step. See ~ nnaddcl for an example of its
|
|
use. This is an alternative for Metamath 100 proof #74. (Contributed
|
|
by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) $)
|
|
nnind $p |- ( A e. NN -> ta ) $=
|
|
( cn wcel crab wa c1 elrab cv caddc wral wss 1nn mpbir2an elrabi peano2nn
|
|
co a1d anim12d 3imtr4g mpcom rgen peano5nni mp2an sseli sylib simprd ) HO
|
|
PZUTEUTHAFOQZPUTEROVAHSVAPZGUAZSUBUIZVAPZGVAUCOVAUDVBSOPBUEMABFSOITUFVEGV
|
|
AVCOPZVCVAPZVEAFVCOUGVFVFCRVDOPZDRVGVEVFVFVHCDVFVHVFVCUHUJNUKACFVCOJTADFV
|
|
DOKTULUMUNGVAUOUPUQAEFHOLTURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ps $. $d x ch $. $d x th $. $d x ta $.
|
|
$d y ph $.
|
|
$( Induction step. $)
|
|
nnindALT.6 $e |- ( y e. NN -> ( ch -> th ) ) $.
|
|
$( Basis. $)
|
|
nnindALT.5 $e |- ps $.
|
|
$( Substitutions. $)
|
|
nnindALT.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
|
|
nnindALT.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
nnindALT.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
nnindALT.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
$( Principle of Mathematical Induction (inference schema). The last four
|
|
hypotheses give us the substitution instances we need; the first two are
|
|
the induction step and the basis.
|
|
|
|
This ALT version of ~ nnind has a different hypothesis order. It may be
|
|
easier to use with the metamath program's Proof Assistant, because
|
|
"MM-PA> assign last" will be applied to the substitution instances
|
|
first. We may eventually use this one as the official version. You may
|
|
use either version. After the proof is complete, the ALT version can be
|
|
changed to the non-ALT version with "MM-PA> minimize nnind /allow".
|
|
(Contributed by NM, 7-Dec-2005.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
nnindALT $p |- ( A e. NN -> ta ) $=
|
|
( nnind ) ABCDEFGHKLMNJIO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Every positive integer is one or a successor. (Contributed by Mario
|
|
Carneiro, 16-May-2014.) $)
|
|
nn1m1nn $p |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) ) $=
|
|
( vx vy cv c1 wceq cmin co cn wcel wo caddc orc 1cnd eqeq1 eleq1d orbi12d
|
|
cc oveq1 ax-1cn 2thd nncn pncan sylancl id eqeltrd olcd a1d nnind ) BDZEF
|
|
ZUJEGHZIJZKZERJZCDZEFZUPEGHZIJZKZUPELHZEFZVAEGHZIJZKZAEFZAEGHZIJZKBCAUKUN
|
|
UOUKUMMUKNUAUJUPFZUKUQUMUSUJUPEOVIULURIUJUPEGSPQUJVAFZUKVBUMVDUJVAEOVJULV
|
|
CIUJVAEGSPQUJAFZUKVFUMVHUJAEOVKULVGIUJAEGSPQTUPIJZVEUTVLVDVBVLVCUPIVLUPRJ
|
|
UOVCUPFUPUBTUPEUCUDVLUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x ps $. $d x ch $. $d x th $. $d y ph $.
|
|
$( Substitutions. $)
|
|
nn1suc.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
|
|
nn1suc.3 $e |- ( x = ( y + 1 ) -> ( ph <-> ch ) ) $.
|
|
nn1suc.4 $e |- ( x = A -> ( ph <-> th ) ) $.
|
|
$( Hypothesis showing it holds for 1. $)
|
|
nn1suc.5 $e |- ps $.
|
|
$( Hypothesis showing it holds for a successor. $)
|
|
nn1suc.6 $e |- ( y e. NN -> ch ) $.
|
|
$( If a statement holds for 1 and also holds for a successor, it holds for
|
|
all positive integers. The first three hypotheses give us the
|
|
substitution instances we need; the last two show that it holds for 1
|
|
and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by
|
|
Mario Carneiro, 16-May-2014.) $)
|
|
nn1suc $p |- ( A e. NN -> th ) $=
|
|
( cn wcel c1 wceq co wsbc wb caddc cmin 1ex sbcie mpbir 1nn mpbiri sbcieg
|
|
wi eleq1 syl dfsbcq bitr3d cv elisset wa pm5.32ri adantr cr nnre peano2re
|
|
a1i 3syl oveq1 sbceq1d adantl mpbid sylbir exlimddv cc nncn npcan sylancl
|
|
ax-1cn bitrd syl5ib nn1m1nn mpjaod ) GMNZGOPZDGOUAQZMNZVSDUHVRVSDAEORZWBB
|
|
KABEOUBHUCUDVSAEGRZDWBVSVRWCDSVSVROMNUEGOMUIUFADEGMJUGZUJAEGOUKULUFVAWAAE
|
|
VTOTQZRZVRDWAFUMZVTPZWFFFVTMUNWAWHUOWGMNZWHUOZWFWHWIWAWGVTMUIUPWJCWFWICWH
|
|
LUQWJAEWGOTQZRZCWFWIWLCSZWHWIWGURNWKURNWMWGUSWGUTACEWKURIUGVBUQWHWLWFSWIW
|
|
HAEWKWEWGVTOTVCVDVEULVFVGVHVRWFWCDVRAEWEGVRGVINOVINWEGPGVJVMGOVKVLVDWDVNV
|
|
OGVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Closure of addition of positive integers, proved by induction on the
|
|
second addend. (Contributed by NM, 12-Jan-1997.) $)
|
|
nnaddcl $p |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN ) $=
|
|
( vx vy cn wcel caddc co cv wi c1 wceq oveq2 eleq1d imbi2d peano2nn wa cc
|
|
nncn ax-1cn addass mp3an3 syl2an syl5ib expcom a2d nnind impcom ) BEFAEFZ
|
|
ABGHZEFZUIACIZGHZEFZJUIAKGHZEFZJUIADIZGHZEFZJUIAUQKGHZGHZEFZJUIUKJCDBULKL
|
|
ZUNUPUIVCUMUOEULKAGMNOULUQLZUNUSUIVDUMUREULUQAGMNOULUTLZUNVBUIVEUMVAEULUT
|
|
AGMNOULBLZUNUKUIVFUMUJEULBAGMNOAPUQEFZUIUSVBUIVGUSVBJUSURKGHZEFUIVGQZVBUR
|
|
PVIVHVAEUIARFZUQRFZVHVALZVGASUQSVJVKKRFVLTAUQKUAUBUCNUDUEUFUGUH $.
|
|
|
|
$( Closure of multiplication of positive integers. (Contributed by NM,
|
|
12-Jan-1997.) $)
|
|
nnmulcl $p |- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) e. NN ) $=
|
|
( vx vy cn wcel cmul co cv wi c1 caddc wceq oveq2 eleq1d imbi2d cc mulid1
|
|
nncn wa biimprd nnaddcl ancoms ax-1cn mp3an3 oveq2d adantr syl2an syl5ibr
|
|
mpcom adddi eqtrd exp4b pm2.43b a2d nnind impcom ) BEFAEFZABGHZEFZURACIZG
|
|
HZEFZJURAKGHZEFZJURADIZGHZEFZJURAVFKLHZGHZEFZJURUTJCDBVAKMZVCVEURVLVBVDEV
|
|
AKAGNOPVAVFMZVCVHURVMVBVGEVAVFAGNOPVAVIMZVCVKURVNVBVJEVAVIAGNOPVABMZVCUTU
|
|
RVOVBUSEVABAGNOPAQFZURVEASZVPVEURVPVDAEARZOUAUJVFEFZURVHVKVSURVHVKJURVSUR
|
|
VHVKURVHTVKURVSTZVGALHZEFZVHURWBVGAUBUCVTVJWAEURVPVFQFZVJWAMVSVQVFSVPWCTV
|
|
JVGVDLHZWAVPWCKQFVJWDMUDAVFKUKUEVPWDWAMWCVPVDAVGLVRUFUGULUHOUIUMUNUOUPUQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nnmulcli.1 $e |- A e. NN $.
|
|
nnmulcli.2 $e |- B e. NN $.
|
|
$( Closure of multiplication of positive integers. (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
nnmulcli $p |- ( A x. B ) e. NN $=
|
|
( cn wcel cmul co nnmulcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A positive integer is one or greater. (Contributed by NM,
|
|
25-Aug-1999.) $)
|
|
nnge1 $p |- ( A e. NN -> 1 <_ A ) $=
|
|
( vx vy c1 cv cle wbr caddc co breq2 wcel cr wi cc0 clt wn 0re 1re lenlt
|
|
wb 1le1 cn nnre recn addid1d breq2d 0lt1 axltadd mp3an12 wa readdcl mpan2
|
|
mpi peano2re mp3an3 syl2anc mpand con3d sylancr 3imtr4d sylbird syl nnind
|
|
lttr ) DBEZFGDDFGDCEZFGZDVFDHIZFGZDAFGBCAVEDDFJVEVFDFJVEVHDFJVEADFJUAVFUB
|
|
KVFLKZVGVIMVFUCVJVGDVFNHIZFGZVIVJVKVFDFVJVFVFUDUEUFVJVKDOGZPZVHDOGZPZVLVI
|
|
VJVOVMVJVKVHOGZVOVMVJNDOGZVQUGNLKZDLKZVJVRVQMQRNDVFUHUIUMVJVKLKZVHLKZVQVO
|
|
UJVMMZVJVSWAQVFNUKULZVFUNZWAWBVTWCRVKVHDVDUOUPUQURVJVTWAVLVNTRWDDVKSUSVJV
|
|
TWBVIVPTRWEDVHSUSUTVAVBVC $.
|
|
$}
|
|
|
|
$( A positive integer is less than or equal to one iff it is equal to one.
|
|
(Contributed by NM, 3-Apr-2005.) $)
|
|
nnle1eq1 $p |- ( A e. NN -> ( A <_ 1 <-> A = 1 ) ) $=
|
|
( cn wcel c1 cle wbr wa wceq nnge1 biantrud cr wb 1re letri3 sylancl bitr4d
|
|
nnre ) ABCZADEFZSDAEFZGZADHZRTSAIJRAKCDKCUBUALAQMADNOP $.
|
|
|
|
$( A positive integer is positive. (Contributed by NM, 26-Sep-1999.) $)
|
|
nngt0 $p |- ( A e. NN -> 0 < A ) $=
|
|
( cn wcel cr c1 cle wbr cc0 clt nnre nnge1 0lt1 wa wi 0re 1re mp3an12 mpani
|
|
ltletr sylc ) ABCADCZEAFGZHAIGZAJAKUAHEIGZUBUCLHDCEDCUAUDUBMUCNOPHEASQRT $.
|
|
|
|
$( A positive integer is not less than one. (Contributed by NM,
|
|
18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
nnnlt1 $p |- ( A e. NN -> -. A < 1 ) $=
|
|
( cn wcel c1 cle wbr clt wn nnge1 cr wb 1re nnre lenlt sylancr mpbid ) ABCZ
|
|
DAEFZADGFHZAIQDJCAJCRSKLAMDANOP $.
|
|
|
|
$( Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) $)
|
|
0nnn $p |- -. 0 e. NN $=
|
|
( cc0 cn wcel c1 clt wbr 0lt1 nnnlt1 mt2 ) ABCADEFGAHI $.
|
|
|
|
$( A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) $)
|
|
nnne0 $p |- ( A e. NN -> A =/= 0 ) $=
|
|
( cn wcel cc0 wceq 0nnn eleq1 mtbiri necon2ai ) ABCZADADEJDBCFADBGHI $.
|
|
|
|
$( A positive integer is apart from zero. (Contributed by Jim Kingdon,
|
|
8-Mar-2020.) $)
|
|
nnap0 $p |- ( A e. NN -> A # 0 ) $=
|
|
( cn wcel nnre nngt0 gt0ap0d ) ABCAADAEF $.
|
|
|
|
${
|
|
nngt0.1 $e |- A e. NN $.
|
|
$( A positive integer is positive (inference version). (Contributed by NM,
|
|
17-Sep-1999.) $)
|
|
nngt0i $p |- 0 < A $=
|
|
( cn wcel cc0 clt wbr nngt0 ax-mp ) ACDEAFGBAHI $.
|
|
|
|
$( A positive integer is nonzero (inference version). (Contributed by NM,
|
|
25-Aug-1999.) $)
|
|
nnne0i $p |- A =/= 0 $=
|
|
( nnrei nngt0i gt0ne0ii ) AABCABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( There exists a positive integer greater than or equal to any two others.
|
|
(Contributed by NM, 18-Aug-1999.) $)
|
|
nn2ge $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
E. x e. NN ( A <_ x /\ B <_ x ) ) $=
|
|
( cn wcel wa caddc co cle wbr cc0 cr nnre adantl nngt0 ltled adantr mpbid
|
|
clt breq2 cv wrex nnaddcl 0red addge01d addge02d anbi12d rspcev syl12anc
|
|
wceq ) BDEZCDEZFZBCGHZDEBUNIJZCUNIJZBAUAZIJZCUQIJZFZADUBBCUCUMKCIJUOUMKCU
|
|
MUDZULCLEUKCMNZULKCSJUKCONPUMBCUKBLEULBMQZVBUERUMKBIJUPUMKBVAVCUKKBSJULBO
|
|
QPUMCBVBVCUFRUTUOUPFAUNDUQUNUJURUOUSUPUQUNBITUQUNCITUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A positive integer is either one or greater than one. This is for
|
|
` NN ` ; ~ 0elnn is a similar theorem for ` _om ` (the natural numbers
|
|
as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) $)
|
|
nn1gt1 $p |- ( A e. NN -> ( A = 1 \/ 1 < A ) ) $=
|
|
( vx vy cv c1 wceq clt wbr wo caddc co eqeq1 breq2 orbi12d eqid orci wcel
|
|
cn cc0 cr nngt0 wb nnre 1re ltaddpos2 sylancl mpbid olcd a1d nnind ) BDZE
|
|
FZEUKGHZIEEFZEEGHZICDZEFZEUPGHZIZUPEJKZEFZEUTGHZIZAEFZEAGHZIBCAULULUNUMUO
|
|
UKEELUKEEGMNUKUPFULUQUMURUKUPELUKUPEGMNUKUTFULVAUMVBUKUTELUKUTEGMNUKAFULV
|
|
DUMVEUKAELUKAEGMNUNUOEOPUPRQZVCUSVFVBVAVFSUPGHZVBUPUAVFUPTQETQVGVBUBUPUCU
|
|
DUPEUEUFUGUHUIUJ $.
|
|
|
|
$}
|
|
|
|
$( A positive integer is greater than one iff it is not equal to one.
|
|
(Contributed by NM, 7-Oct-2004.) $)
|
|
nngt1ne1 $p |- ( A e. NN -> ( 1 < A <-> A =/= 1 ) ) $=
|
|
( cn wcel c1 clt wbr wne cr 1re ltne mpan wceq wn nn1gt1 ord syl5bi impbid2
|
|
df-ne ) ABCZDAEFZADGZDHCTUAIDAJKUAADLZMSTADRSUBTANOPQ $.
|
|
|
|
$( The quotient of a real and a positive integer is real. (Contributed by
|
|
NM, 28-Nov-2008.) $)
|
|
nndivre $p |- ( ( A e. RR /\ N e. NN ) -> ( A / N ) e. RR ) $=
|
|
( cn wcel cr cc0 cap wbr wa cdiv co nnre nnap0 jca redivclap 3expb sylan2 )
|
|
BCDZAEDZBEDZBFGHZIABJKEDZRTUABLBMNSTUAUBABOPQ $.
|
|
|
|
$( The reciprocal of a positive integer is real. (Contributed by NM,
|
|
8-Feb-2008.) $)
|
|
nnrecre $p |- ( N e. NN -> ( 1 / N ) e. RR ) $=
|
|
( c1 cr wcel cn cdiv co 1re nndivre mpan ) BCDAEDBAFGCDHBAIJ $.
|
|
|
|
$( The reciprocal of a positive integer is positive. (Contributed by NM,
|
|
25-Aug-1999.) $)
|
|
nnrecgt0 $p |- ( A e. NN -> 0 < ( 1 / A ) ) $=
|
|
( cn wcel c1 cle wbr cc0 cdiv co clt nnge1 0lt1 cr wa wi 0re ltletr mp3an12
|
|
nnre 1re recgt0 ex syld syl mpani mpd ) ABCZDAEFZGDAHIJFZAKUGGDJFZUHUILUGAM
|
|
CZUJUHNZUIOASUKULGAJFZUIGMCDMCUKULUMOPTGDAQRUKUMUIAUAUBUCUDUEUF $.
|
|
|
|
${
|
|
$d z A $. $d x y z B $.
|
|
$( Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.)
|
|
(Revised by Mario Carneiro, 16-May-2014.) $)
|
|
nnsub $p |- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( B - A ) e. NN ) ) $=
|
|
( vz vx cn wcel clt wbr cmin co wi wral wceq breq2 eleq1d imbi12d ralbidv
|
|
c1 oveq1 cr vy wa cv caddc nnnlt1 pm2.21d rgen breq1 oveq2 cbvralv adantr
|
|
cc nncn ax-1cn pncan sylancl simpl eqeltrd syl5ibrcom a1dd rspcv nnre 1re
|
|
ltsubadd mp3an2 syl2anr subsub3 mp3an3 syl2an biimpd syl9r nn1m1nn adantl
|
|
wb wo mpjaod ralrimdva syl5bi nnind rspcva sylan2 cc0 nngt0 posdif impbid
|
|
syl5ibr ) AEFZBEFZUBZABGHZBAIJZEFZWHWGCUCZBGHZBWMIJZEFZKZCELZWJWLKZWMDUCZ
|
|
GHZWTWMIJZEFZKZCELWMRGHZRWMIJZEFZKZCELWMUAUCZGHZXIWMIJZEFZKZCELZWMXIRUDJZ
|
|
GHZXOWMIJZEFZKZCELZWRDUABWTRMZXDXHCEYAXAXEXCXGWTRWMGNYAXBXFEWTRWMISOPQWTX
|
|
IMZXDXMCEYBXAXJXCXLWTXIWMGNYBXBXKEWTXIWMISOPQWTXOMZXDXSCEYCXAXPXCXRWTXOWM
|
|
GNYCXBXQEWTXOWMISOPQWTBMZXDWQCEYDXAWNXCWPWTBWMGNYDXBWOEWTBWMISOPQXHCEWMEF
|
|
ZXEXGWMUEUFUGXNWTXIGHZXIWTIJZEFZKZDELZXIEFZXTXMYICDEWMWTMZXJYFXLYHWMWTXIG
|
|
UHYLXKYGEWMWTXIIUIOPUJYKYJXSCEYKYEUBZWMRMZYJXSKWMRIJZEFZYMYNXSYJYMYNXRXPY
|
|
MXRYNXORIJZEFYMYQXIEYMXIULFZRULFZYQXIMYKYRYEXIUMZUKUNXIRUOUPYKYEUQURYNXQY
|
|
QEWMRXOIUIOUSUTUTYPYJYOXIGHZXIYOIJZEFZKZYMXSYIUUDDYOEWTYOMZYFUUAYHUUCWTYO
|
|
XIGUHUUEYGUUBEWTYOXIIUIOPVAYMUUDXSYMUUAXPUUCXRYEWMTFZXITFZUUAXPVNZYKWMVBX
|
|
IVBUUFRTFUUGUUHVCWMRXIVDVEVFYMUUBXQEYKYRWMULFZUUBXQMZYEYTWMUMYRUUIYSUUJUN
|
|
XIWMRVGVHVIOPVJVKYEYNYPVOYKWMVLVMVPVQVRVSWQWSCAEWMAMZWNWJWPWLWMABGUHUUKWO
|
|
WKEWMABIUIOPVTWAWLWJWIWBWKGHZWKWCWGATFBTFWJUULVNWHAVBBVBABWDVIWFWE $.
|
|
|
|
nnsub.1 $e |- A e. NN $.
|
|
nnsub.2 $e |- B e. NN $.
|
|
$( Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) $)
|
|
nnsubi $p |- ( A < B <-> ( B - A ) e. NN ) $=
|
|
( cn wcel clt wbr cmin co wb nnsub mp2an ) AEFBEFABGHBAIJEFKCDABLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Two ways to express " ` A ` divides ` B ` " for positive integers.
|
|
(Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro,
|
|
16-May-2014.) $)
|
|
nndiv $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( E. x e. NN ( A x. x ) = B <-> ( B / A ) e. NN ) ) $=
|
|
( cdiv co cn wcel cv wceq wrex cmul risset eqcom ad2antlr ad2antrr adantl
|
|
wa cc nncn cc0 cap wbr nnap0 divmulapd syl5bb rexbidva syl5rbb ) CBDEZFGA
|
|
HZUHIZAFJBFGZCFGZQZBUIKECIZAFJAUHFLUMUJUNAFUJUHUIIUMUIFGZQZUNUIUHMUPCBUIU
|
|
LCRGUKUOCSNUKBRGULUOBSOUOUIRGUMUISPUKBTUAUBULUOBUCOUDUEUFUG $.
|
|
$}
|
|
|
|
$( Transitive property of divisibility: if ` A ` divides ` B ` and ` B `
|
|
divides ` C ` , then ` A ` divides ` C ` . Typically, ` C ` would be an
|
|
integer, although the theorem holds for complex ` C ` . (Contributed by
|
|
NM, 3-May-2005.) $)
|
|
nndivtr $p |- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\
|
|
( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN ) $=
|
|
( cn wcel cc w3a cdiv co wa cmul cc0 cap wbr wceq nncn 3ad2ant2 nnap0 jca
|
|
c1 nnmulcl simp3 3ad2ant1 divmul24ap syl22anc dividapd oveq1d divclap 3expb
|
|
sylan2 ancoms mulid2d 3adant2 3eqtrd eleq1d syl5ib imp ) ADEZBDEZCFEZGZBAHI
|
|
ZDECBHIZDEJZCAHIZDEZVDVBVCKIZDEVAVFVBVCUAVAVGVEDVAVGBBHIZVEKIZTVEKIZVEVABFE
|
|
ZUTAFEZALMNZJZVKBLMNZJZVGVIOUSURVKUTBPZQURUSUTUBURUSVNUTURVLVMAPARSZUCUSURV
|
|
PUTUSVKVOVQBRZSQBCABUDUEUSURVIVJOUTUSVHTVEKUSBVQVSUFUGQURUTVJVEOUSURUTJVEUT
|
|
URVEFEZURUTVNVTVRUTVLVMVTCAUHUIUJUKULUMUNUOUPUQ $.
|
|
|
|
${
|
|
nnge1d.1 $e |- ( ph -> A e. NN ) $.
|
|
$( A positive integer is one or greater. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnge1d $p |- ( ph -> 1 <_ A ) $=
|
|
( cn wcel c1 cle wbr nnge1 syl ) ABDEFBGHCBIJ $.
|
|
|
|
$( A positive integer is positive. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nngt0d $p |- ( ph -> 0 < A ) $=
|
|
( cn wcel cc0 clt wbr nngt0 syl ) ABDEFBGHCBIJ $.
|
|
|
|
$( A positive integer is nonzero. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nnne0d $p |- ( ph -> A =/= 0 ) $=
|
|
( cn wcel cc0 wne nnne0 syl ) ABDEBFGCBHI $.
|
|
|
|
$( A positive integer is apart from zero. (Contributed by Jim Kingdon,
|
|
25-Aug-2021.) $)
|
|
nnap0d $p |- ( ph -> A # 0 ) $=
|
|
( cn wcel cc0 cap wbr nnap0 syl ) ABDEBFGHCBIJ $.
|
|
|
|
$( The reciprocal of a positive integer is real. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
nnrecred $p |- ( ph -> ( 1 / A ) e. RR ) $=
|
|
( cn wcel c1 cdiv co cr nnrecre syl ) ABDEFBGHIECBJK $.
|
|
|
|
nnmulcld.2 $e |- ( ph -> B e. NN ) $.
|
|
$( Closure of addition of positive integers. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
nnaddcld $p |- ( ph -> ( A + B ) e. NN ) $=
|
|
( cn wcel caddc co nnaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure of multiplication of positive integers. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
nnmulcld $p |- ( ph -> ( A x. B ) e. NN ) $=
|
|
( cn wcel cmul co nnmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
${
|
|
nndivred.1 $e |- ( ph -> A e. RR ) $.
|
|
nndivred.2 $e |- ( ph -> B e. NN ) $.
|
|
$( A positive integer is one or greater. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nndivred $p |- ( ph -> ( A / B ) e. RR ) $=
|
|
( cr wcel cn cdiv co nndivre syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Decimal representation of numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The decimal representation of numbers/integers is based on the decimal digits
|
|
0 through 9 ( ~ df-0 through ~ df-9 ), which are explicitly defined in the
|
|
following. Note that the numbers 0 and 1 are constants defined as primitives
|
|
of the complex number axiom system (see ~ df-0 and ~ df-1 ).
|
|
|
|
Integers can also be exhibited as sums of powers of 10 (e.g. the number 103
|
|
can be expressed as ` ( ( ; 1 0 ^ 2 ) + 3 ) `) or as some other expression
|
|
built from operations on the numbers 0 through 9. For example, the prime
|
|
number 823541 can be expressed as ` ( 7 ^ 7 ) - 2 ` .
|
|
|
|
Most abstract math rarely requires numbers larger than 4. Even in Wiles'
|
|
proof of Fermat's Last Theorem, the largest number used appears to be 12.
|
|
|
|
$)
|
|
|
|
$c 2 $. $( The decimal number 2 $)
|
|
$c 3 $. $( The decimal number 3 $)
|
|
$c 4 $. $( The decimal number 4 $)
|
|
$c 5 $. $( The decimal number 5 $)
|
|
$c 6 $. $( The decimal number 6 $)
|
|
$c 7 $. $( The decimal number 7 $)
|
|
$c 8 $. $( The decimal number 8 $)
|
|
$c 9 $. $( The decimal number 9 $)
|
|
$c 10 $. $( The decimal number 10 $)
|
|
|
|
$( Extend class notation to include the number 2. $)
|
|
c2 $a class 2 $.
|
|
$( Extend class notation to include the number 3. $)
|
|
c3 $a class 3 $.
|
|
$( Extend class notation to include the number 4. $)
|
|
c4 $a class 4 $.
|
|
$( Extend class notation to include the number 5. $)
|
|
c5 $a class 5 $.
|
|
$( Extend class notation to include the number 6. $)
|
|
c6 $a class 6 $.
|
|
$( Extend class notation to include the number 7. $)
|
|
c7 $a class 7 $.
|
|
$( Extend class notation to include the number 8. $)
|
|
c8 $a class 8 $.
|
|
$( Extend class notation to include the number 9. $)
|
|
c9 $a class 9 $.
|
|
$( Extend class notation to include the number 10. $)
|
|
c10 $a class 10 $.
|
|
|
|
$( Define the number 2. (Contributed by NM, 27-May-1999.) $)
|
|
df-2 $a |- 2 = ( 1 + 1 ) $.
|
|
|
|
$( Define the number 3. (Contributed by NM, 27-May-1999.) $)
|
|
df-3 $a |- 3 = ( 2 + 1 ) $.
|
|
$( Define the number 4. (Contributed by NM, 27-May-1999.) $)
|
|
df-4 $a |- 4 = ( 3 + 1 ) $.
|
|
$( Define the number 5. (Contributed by NM, 27-May-1999.) $)
|
|
df-5 $a |- 5 = ( 4 + 1 ) $.
|
|
$( Define the number 6. (Contributed by NM, 27-May-1999.) $)
|
|
df-6 $a |- 6 = ( 5 + 1 ) $.
|
|
$( Define the number 7. (Contributed by NM, 27-May-1999.) $)
|
|
df-7 $a |- 7 = ( 6 + 1 ) $.
|
|
$( Define the number 8. (Contributed by NM, 27-May-1999.) $)
|
|
df-8 $a |- 8 = ( 7 + 1 ) $.
|
|
$( Define the number 9. (Contributed by NM, 27-May-1999.) $)
|
|
df-9 $a |- 9 = ( 8 + 1 ) $.
|
|
|
|
$( ` 0 =/= 1 ` (common case). See aso ~ 1ap0 . (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
0ne1 $p |- 0 =/= 1 $=
|
|
( cc0 c1 0re 0lt1 ltneii ) ABCDE $.
|
|
|
|
$( ` 1 =/= 0 ` . See aso ~ 1ap0 . (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
1ne0 $p |- 1 =/= 0 $=
|
|
( cc0 c1 0ne1 necomi ) ABCD $.
|
|
|
|
$( ` ( 1 - 1 ) = 0 ` (common case). (Contributed by David A. Wheeler,
|
|
7-Jul-2016.) $)
|
|
1m1e0 $p |- ( 1 - 1 ) = 0 $=
|
|
( c1 ax-1cn subidi ) ABC $.
|
|
|
|
$( The number 2 is real. (Contributed by NM, 27-May-1999.) $)
|
|
2re $p |- 2 e. RR $=
|
|
( c2 c1 caddc co cr df-2 1re readdcli eqeltri ) ABBCDEFBBGGHI $.
|
|
|
|
$( The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) $)
|
|
2cn $p |- 2 e. CC $=
|
|
( c2 2re recni ) ABC $.
|
|
|
|
$( 2 is a set (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
2ex $p |- 2 e. _V $=
|
|
( c2 cc 2cn elexi ) ABCD $.
|
|
|
|
$( 2 is a complex number, deductive form (common case). (Contributed by
|
|
David A. Wheeler, 8-Dec-2018.) $)
|
|
2cnd $p |- ( ph -> 2 e. CC ) $=
|
|
( c2 cc wcel 2cn a1i ) BCDAEF $.
|
|
|
|
$( The number 3 is real. (Contributed by NM, 27-May-1999.) $)
|
|
3re $p |- 3 e. RR $=
|
|
( c3 c2 c1 caddc co cr df-3 2re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) $)
|
|
3cn $p |- 3 e. CC $=
|
|
( c3 3re recni ) ABC $.
|
|
|
|
$( 3 is a set (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
3ex $p |- 3 e. _V $=
|
|
( c3 cc 3cn elexi ) ABCD $.
|
|
|
|
$( The number 4 is real. (Contributed by NM, 27-May-1999.) $)
|
|
4re $p |- 4 e. RR $=
|
|
( c4 c3 c1 caddc co cr df-4 3re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 4 is a complex number. (Contributed by David A. Wheeler,
|
|
7-Jul-2016.) $)
|
|
4cn $p |- 4 e. CC $=
|
|
( c4 4re recni ) ABC $.
|
|
|
|
$( The number 5 is real. (Contributed by NM, 27-May-1999.) $)
|
|
5re $p |- 5 e. RR $=
|
|
( c5 c4 c1 caddc co cr df-5 4re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 5 is complex. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
5cn $p |- 5 e. CC $=
|
|
( c5 5re recni ) ABC $.
|
|
|
|
$( The number 6 is real. (Contributed by NM, 27-May-1999.) $)
|
|
6re $p |- 6 e. RR $=
|
|
( c6 c5 c1 caddc co cr df-6 5re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 6 is complex. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
6cn $p |- 6 e. CC $=
|
|
( c6 6re recni ) ABC $.
|
|
|
|
$( The number 7 is real. (Contributed by NM, 27-May-1999.) $)
|
|
7re $p |- 7 e. RR $=
|
|
( c7 c6 c1 caddc co cr df-7 6re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 7 is complex. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
7cn $p |- 7 e. CC $=
|
|
( c7 7re recni ) ABC $.
|
|
|
|
$( The number 8 is real. (Contributed by NM, 27-May-1999.) $)
|
|
8re $p |- 8 e. RR $=
|
|
( c8 c7 c1 caddc co cr df-8 7re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 8 is complex. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
8cn $p |- 8 e. CC $=
|
|
( c8 8re recni ) ABC $.
|
|
|
|
$( The number 9 is real. (Contributed by NM, 27-May-1999.) $)
|
|
9re $p |- 9 e. RR $=
|
|
( c9 c8 c1 caddc co cr df-9 8re 1re readdcli eqeltri ) ABCDEFGBCHIJK $.
|
|
|
|
$( The number 9 is complex. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
9cn $p |- 9 e. CC $=
|
|
( c9 9re recni ) ABC $.
|
|
|
|
$( Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) $)
|
|
0le0 $p |- 0 <_ 0 $=
|
|
( cc0 0re leidi ) ABC $.
|
|
|
|
$( 0 is less than or equal to 2. (Contributed by David A. Wheeler,
|
|
7-Dec-2018.) $)
|
|
0le2 $p |- 0 <_ 2 $=
|
|
( cc0 c1 caddc co c2 cle wbr 0le1 1re addge0i mp2an df-2 breqtrri ) ABBCDZE
|
|
FABFGZOANFGHHBBIIJKLM $.
|
|
|
|
$( The number 2 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
2pos $p |- 0 < 2 $=
|
|
( cc0 c1 caddc co c2 clt 1re 0lt1 addgt0ii df-2 breqtrri ) ABBCDEFBBGGHHIJK
|
|
$.
|
|
|
|
$( The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) $)
|
|
2ne0 $p |- 2 =/= 0 $=
|
|
( c2 2re 2pos gt0ne0ii ) ABCD $.
|
|
|
|
$( The number 2 is apart from zero. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
2ap0 $p |- 2 # 0 $=
|
|
( c2 2re 2pos gt0ap0ii ) ABCD $.
|
|
|
|
$( The number 3 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
3pos $p |- 0 < 3 $=
|
|
( cc0 c2 c1 caddc co c3 clt 2re 1re 2pos 0lt1 addgt0ii df-3 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 7-May-2011.) $)
|
|
3ne0 $p |- 3 =/= 0 $=
|
|
( c3 3re 3pos gt0ne0ii ) ABCD $.
|
|
|
|
$( The number 3 is apart from zero. (Contributed by Jim Kingdon,
|
|
10-Oct-2021.) $)
|
|
3ap0 $p |- 3 # 0 $=
|
|
( c3 3re 3pos gt0ap0ii ) ABCD $.
|
|
|
|
$( The number 4 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
4pos $p |- 0 < 4 $=
|
|
( cc0 c3 c1 caddc co c4 clt 3re 1re 3pos 0lt1 addgt0ii df-4 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 4 is nonzero. (Contributed by David A. Wheeler,
|
|
5-Dec-2018.) $)
|
|
4ne0 $p |- 4 =/= 0 $=
|
|
( c4 4re 4pos gt0ne0ii ) ABCD $.
|
|
|
|
$( The number 4 is apart from zero. (Contributed by Jim Kingdon,
|
|
10-Oct-2021.) $)
|
|
4ap0 $p |- 4 # 0 $=
|
|
( c4 4re 4pos gt0ap0ii ) ABCD $.
|
|
|
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$( The number 5 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
5pos $p |- 0 < 5 $=
|
|
( cc0 c4 c1 caddc co c5 clt 4re 1re 4pos 0lt1 addgt0ii df-5 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 6 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
6pos $p |- 0 < 6 $=
|
|
( cc0 c5 c1 caddc co c6 clt 5re 1re 5pos 0lt1 addgt0ii df-6 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 7 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
7pos $p |- 0 < 7 $=
|
|
( cc0 c6 c1 caddc co c7 clt 6re 1re 6pos 0lt1 addgt0ii df-7 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 8 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
8pos $p |- 0 < 8 $=
|
|
( cc0 c7 c1 caddc co c8 clt 7re 1re 7pos 0lt1 addgt0ii df-8 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
$( The number 9 is positive. (Contributed by NM, 27-May-1999.) $)
|
|
9pos $p |- 0 < 9 $=
|
|
( cc0 c8 c1 caddc co c9 clt 8re 1re 8pos 0lt1 addgt0ii df-9 breqtrri ) ABCD
|
|
EFGBCHIJKLMN $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Some properties of specific numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
This includes adding two pairs of values 1..10
|
|
(where the right is less than the left)
|
|
and where the left is less than the right for the values 1..10.
|
|
|
|
$)
|
|
|
|
$( -1 is a complex number (common case). (Contributed by David A. Wheeler,
|
|
7-Jul-2016.) $)
|
|
neg1cn $p |- -u 1 e. CC $=
|
|
( c1 ax-1cn negcli ) ABC $.
|
|
|
|
$( -1 is a real number (common case). (Contributed by David A. Wheeler,
|
|
5-Dec-2018.) $)
|
|
neg1rr $p |- -u 1 e. RR $=
|
|
( c1 1re renegcli ) ABC $.
|
|
|
|
$( -1 is nonzero (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
neg1ne0 $p |- -u 1 =/= 0 $=
|
|
( c1 ax-1cn 1ne0 negne0i ) ABCD $.
|
|
|
|
$( -1 is less than 0 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
neg1lt0 $p |- -u 1 < 0 $=
|
|
( c1 cneg cc0 clt wbr neg0 0lt1 eqbrtri 1re 0re ltnegcon1i mpbir ) ABCDECBZ
|
|
ADEMCADFGHACIJKL $.
|
|
|
|
$( -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) $)
|
|
neg1ap0 $p |- -u 1 # 0 $=
|
|
( c1 cc0 cap wbr cneg 1ap0 cc wcel wb ax-1cn negap0 ax-mp mpbi ) ABCDZAEBCD
|
|
ZFAGHNOIJAKLM $.
|
|
|
|
$( ` -u -u 1 ` is 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
negneg1e1 $p |- -u -u 1 = 1 $=
|
|
( c1 ax-1cn negnegi ) ABC $.
|
|
|
|
$( ` 1 + -u 1 ` is 0 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
1pneg1e0 $p |- ( 1 + -u 1 ) = 0 $=
|
|
( c1 ax-1cn negidi ) ABC $.
|
|
|
|
$( 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
0m0e0 $p |- ( 0 - 0 ) = 0 $=
|
|
( cc0 0cn subidi ) ABC $.
|
|
|
|
$( 1 - 0 = 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
1m0e1 $p |- ( 1 - 0 ) = 1 $=
|
|
( c1 ax-1cn subid1i ) ABC $.
|
|
|
|
$( 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) $)
|
|
0p1e1 $p |- ( 0 + 1 ) = 1 $=
|
|
( c1 ax-1cn addid2i ) ABC $.
|
|
|
|
$( 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
1p0e1 $p |- ( 1 + 0 ) = 1 $=
|
|
( c1 ax-1cn addid1i ) ABC $.
|
|
|
|
$( 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) $)
|
|
1p1e2 $p |- ( 1 + 1 ) = 2 $=
|
|
( c2 c1 caddc co df-2 eqcomi ) ABBCDEF $.
|
|
|
|
$( 2 - 1 = 1. The result is on the right-hand-side to be consistent with
|
|
similar proofs like ~ 4p4e8 . (Contributed by David A. Wheeler,
|
|
4-Jan-2017.) $)
|
|
2m1e1 $p |- ( 2 - 1 ) = 1 $=
|
|
( c2 c1 2cn ax-1cn 1p1e2 subaddrii ) ABBCDDEF $.
|
|
|
|
$( 1 = 2 - 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
1e2m1 $p |- 1 = ( 2 - 1 ) $=
|
|
( c2 c1 cmin co 2m1e1 eqcomi ) ABCDBEF $.
|
|
|
|
$( 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM,
|
|
10-Dec-2017.) $)
|
|
3m1e2 $p |- ( 3 - 1 ) = 2 $=
|
|
( c3 c1 c2 3cn ax-1cn 2cn caddc co addcomi df-3 eqtr4i subaddrii ) ABCDEFBC
|
|
GHCBGHABCEFIJKL $.
|
|
|
|
$( Two plus two equals four. For more information, see "2+2=4 Trivia" on the
|
|
Metamath Proof Explorer Home Page:
|
|
~ http://us.metamath.org/mpeuni/mmset.html#trivia . (Contributed by NM,
|
|
27-May-1999.) $)
|
|
2p2e4 $p |- ( 2 + 2 ) = 4 $=
|
|
( c2 caddc co c1 c4 df-2 oveq2i c3 df-4 oveq1i ax-1cn addassi 3eqtri eqtr4i
|
|
df-3 2cn ) AABCADDBCZBCZEAQABFGEHDBCADBCZDBCRIHSDBOJADDPKKLMN $.
|
|
|
|
$( Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario
|
|
Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.) $)
|
|
2times $p |- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) $=
|
|
( cc wcel c2 cmul co c1 caddc df-2 oveq1i 1p1times syl5eq ) ABCDAEFGGHFZAEF
|
|
AAHFDMAEIJAKL $.
|
|
|
|
$( A number times 2. (Contributed by NM, 16-Oct-2007.) $)
|
|
times2 $p |- ( A e. CC -> ( A x. 2 ) = ( A + A ) ) $=
|
|
( cc wcel c2 cmul co caddc wceq 2cn mulcom mpan2 2times eqtrd ) ABCZADEFZDA
|
|
EFZAAGFNDBCOPHIADJKALM $.
|
|
|
|
${
|
|
2times.1 $e |- A e. CC $.
|
|
$( Two times a number. (Contributed by NM, 1-Aug-1999.) $)
|
|
2timesi $p |- ( 2 x. A ) = ( A + A ) $=
|
|
( cc wcel c2 cmul co caddc wceq 2times ax-mp ) ACDEAFGAAHGIBAJK $.
|
|
|
|
$( A number times 2. (Contributed by NM, 11-May-2004.) $)
|
|
times2i $p |- ( A x. 2 ) = ( A + A ) $=
|
|
( cc wcel c2 cmul co caddc wceq times2 ax-mp ) ACDAEFGAAHGIBAJK $.
|
|
$}
|
|
|
|
$( 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
2div2e1 $p |- ( 2 / 2 ) = 1 $=
|
|
( c2 2cn 2ap0 dividapi ) ABCD $.
|
|
|
|
$( 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
2p1e3 $p |- ( 2 + 1 ) = 3 $=
|
|
( c3 c2 c1 caddc co df-3 eqcomi ) ABCDEFG $.
|
|
|
|
$( 1 + 2 = 3 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
1p2e3 $p |- ( 1 + 2 ) = 3 $=
|
|
( c2 c1 c3 2cn ax-1cn 2p1e3 addcomli ) ABCDEFG $.
|
|
|
|
$( 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
3p1e4 $p |- ( 3 + 1 ) = 4 $=
|
|
( c4 c3 c1 caddc co df-4 eqcomi ) ABCDEFG $.
|
|
|
|
$( 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
4p1e5 $p |- ( 4 + 1 ) = 5 $=
|
|
( c5 c4 c1 caddc co df-5 eqcomi ) ABCDEFG $.
|
|
|
|
$( 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
5p1e6 $p |- ( 5 + 1 ) = 6 $=
|
|
( c6 c5 c1 caddc co df-6 eqcomi ) ABCDEFG $.
|
|
|
|
$( 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
6p1e7 $p |- ( 6 + 1 ) = 7 $=
|
|
( c7 c6 c1 caddc co df-7 eqcomi ) ABCDEFG $.
|
|
|
|
$( 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
7p1e8 $p |- ( 7 + 1 ) = 8 $=
|
|
( c8 c7 c1 caddc co df-8 eqcomi ) ABCDEFG $.
|
|
|
|
$( 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.) $)
|
|
8p1e9 $p |- ( 8 + 1 ) = 9 $=
|
|
( c9 c8 c1 caddc co df-9 eqcomi ) ABCDEFG $.
|
|
|
|
$( 3 + 2 = 5. (Contributed by NM, 11-May-2004.) $)
|
|
3p2e5 $p |- ( 3 + 2 ) = 5 $=
|
|
( c3 c2 caddc co c4 c1 c5 df-2 oveq2i 3cn ax-1cn addassi eqtr4i df-4 oveq1i
|
|
df-5 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.
|
|
|
|
$( 3 + 3 = 6. (Contributed by NM, 11-May-2004.) $)
|
|
3p3e6 $p |- ( 3 + 3 ) = 6 $=
|
|
( c3 caddc co c2 c1 c6 df-3 oveq2i 3cn 2cn ax-1cn addassi eqtr4i df-6 3p2e5
|
|
c5 oveq1i ) AABCZADBCZEBCZFRADEBCZBCTAUAABGHADEIJKLMFPEBCTNSPEBOQMM $.
|
|
|
|
$( 4 + 2 = 6. (Contributed by NM, 11-May-2004.) $)
|
|
4p2e6 $p |- ( 4 + 2 ) = 6 $=
|
|
( c4 c2 caddc co c5 c1 c6 df-2 oveq2i 4cn ax-1cn addassi eqtr4i df-5 oveq1i
|
|
df-6 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.
|
|
|
|
$( 4 + 3 = 7. (Contributed by NM, 11-May-2004.) $)
|
|
4p3e7 $p |- ( 4 + 3 ) = 7 $=
|
|
( c4 c3 caddc co c2 c1 c7 df-3 oveq2i 4cn 2cn ax-1cn addassi eqtr4i c6 df-7
|
|
4p2e6 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.
|
|
|
|
$( 4 + 4 = 8. (Contributed by NM, 11-May-2004.) $)
|
|
4p4e8 $p |- ( 4 + 4 ) = 8 $=
|
|
( c4 caddc co c3 c1 c8 df-4 oveq2i 4cn 3cn ax-1cn addassi eqtr4i df-8 4p3e7
|
|
c7 oveq1i ) AABCZADBCZEBCZFRADEBCZBCTAUAABGHADEIJKLMFPEBCTNSPEBOQMM $.
|
|
|
|
$( 5 + 2 = 7. (Contributed by NM, 11-May-2004.) $)
|
|
5p2e7 $p |- ( 5 + 2 ) = 7 $=
|
|
( c5 c2 caddc co c6 c1 c7 df-2 oveq2i 5cn ax-1cn addassi eqtr4i df-6 oveq1i
|
|
df-7 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.
|
|
|
|
$( 5 + 3 = 8. (Contributed by NM, 11-May-2004.) $)
|
|
5p3e8 $p |- ( 5 + 3 ) = 8 $=
|
|
( c5 c3 caddc co c2 c1 c8 df-3 oveq2i 5cn 2cn ax-1cn addassi eqtr4i c7 df-8
|
|
5p2e7 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.
|
|
|
|
$( 5 + 4 = 9. (Contributed by NM, 11-May-2004.) $)
|
|
5p4e9 $p |- ( 5 + 4 ) = 9 $=
|
|
( c5 c4 caddc co c3 c1 c9 df-4 oveq2i 5cn 3cn ax-1cn addassi eqtr4i c8 df-9
|
|
5p3e8 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.
|
|
|
|
$( 6 + 2 = 8. (Contributed by NM, 11-May-2004.) $)
|
|
6p2e8 $p |- ( 6 + 2 ) = 8 $=
|
|
( c6 c2 caddc co c7 c1 c8 df-2 oveq2i 6cn ax-1cn addassi eqtr4i df-7 oveq1i
|
|
df-8 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.
|
|
|
|
$( 6 + 3 = 9. (Contributed by NM, 11-May-2004.) $)
|
|
6p3e9 $p |- ( 6 + 3 ) = 9 $=
|
|
( c6 c3 caddc co c2 c1 c9 df-3 oveq2i 6cn 2cn ax-1cn addassi eqtr4i c8 df-9
|
|
6p2e8 oveq1i ) ABCDZAECDZFCDZGSAEFCDZCDUABUBACHIAEFJKLMNGOFCDUAPTOFCQRNN $.
|
|
|
|
$( 7 + 2 = 9. (Contributed by NM, 11-May-2004.) $)
|
|
7p2e9 $p |- ( 7 + 2 ) = 9 $=
|
|
( c7 c2 caddc co c8 c1 c9 df-2 oveq2i 7cn ax-1cn addassi eqtr4i df-8 oveq1i
|
|
df-9 ) ABCDZEFCDZGQAFCDZFCDZRQAFFCDZCDTBUAACHIAFFJKKLMESFCNOMPM $.
|
|
|
|
$( 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) $)
|
|
1t1e1 $p |- ( 1 x. 1 ) = 1 $=
|
|
( c1 ax-1cn mulid1i ) ABC $.
|
|
|
|
$( 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) $)
|
|
2t1e2 $p |- ( 2 x. 1 ) = 2 $=
|
|
( c2 2cn mulid1i ) ABC $.
|
|
|
|
$( 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) $)
|
|
2t2e4 $p |- ( 2 x. 2 ) = 4 $=
|
|
( c2 cmul co caddc c4 2cn 2timesi 2p2e4 eqtri ) AABCAADCEAFGHI $.
|
|
|
|
$( 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
3t1e3 $p |- ( 3 x. 1 ) = 3 $=
|
|
( c3 3cn mulid1i ) ABC $.
|
|
|
|
$( 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) $)
|
|
3t2e6 $p |- ( 3 x. 2 ) = 6 $=
|
|
( c3 c2 cmul co caddc c6 3cn times2i 3p3e6 eqtri ) ABCDAAEDFAGHIJ $.
|
|
|
|
$( 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) $)
|
|
3t3e9 $p |- ( 3 x. 3 ) = 9 $=
|
|
( c3 cmul co c2 c1 caddc c9 df-3 oveq2i 3cn 2cn ax-1cn adddii 3t2e6 oveq12i
|
|
c6 3t1e3 eqtri 6p3e9 ) AABCADEFCZBCZGATABHIUAPAFCZGUAADBCZAEBCZFCUBADEJKLMU
|
|
CPUDAFNQORSRR $.
|
|
|
|
$( 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.) $)
|
|
4t2e8 $p |- ( 4 x. 2 ) = 8 $=
|
|
( c4 c2 cmul co caddc c8 4cn times2i 4p4e8 eqtri ) ABCDAAEDFAGHIJ $.
|
|
|
|
$( 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
2t0e0 $p |- ( 2 x. 0 ) = 0 $=
|
|
( c2 2cn mul01i ) ABC $.
|
|
|
|
$( One half of four is two. (Contributed by NM, 3-Sep-1999.) $)
|
|
4d2e2 $p |- ( 4 / 2 ) = 2 $=
|
|
( c4 c2 cdiv co wceq cmul 2t2e4 4cn 2cn 2ap0 divmulapi mpbir ) ABCDBEBBFDAE
|
|
GABBHIIJKL $.
|
|
|
|
$( 2 is a positive integer. (Contributed by NM, 20-Aug-2001.) $)
|
|
2nn $p |- 2 e. NN $=
|
|
( c2 c1 caddc co cn df-2 wcel 1nn peano2nn ax-mp eqeltri ) ABBCDZEFBEGLEGHB
|
|
IJK $.
|
|
|
|
$( 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) $)
|
|
3nn $p |- 3 e. NN $=
|
|
( c3 c2 c1 caddc co cn df-3 wcel 2nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) $)
|
|
4nn $p |- 4 e. NN $=
|
|
( c4 c3 c1 caddc co cn df-4 wcel 3nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
5nn $p |- 5 e. NN $=
|
|
( c5 c4 c1 caddc co cn df-5 wcel 4nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
6nn $p |- 6 e. NN $=
|
|
( c6 c5 c1 caddc co cn df-6 wcel 5nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
7nn $p |- 7 e. NN $=
|
|
( c7 c6 c1 caddc co cn df-7 wcel 6nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
8nn $p |- 8 e. NN $=
|
|
( c8 c7 c1 caddc co cn df-8 wcel 7nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) $)
|
|
9nn $p |- 9 e. NN $=
|
|
( c9 c8 c1 caddc co cn df-9 wcel 8nn peano2nn ax-mp eqeltri ) ABCDEZFGBFHMF
|
|
HIBJKL $.
|
|
|
|
$( 1 is less than 2. (Contributed by NM, 24-Feb-2005.) $)
|
|
1lt2 $p |- 1 < 2 $=
|
|
( c1 caddc co c2 clt 1re ltp1i df-2 breqtrri ) AAABCDEAFGHI $.
|
|
|
|
$( 2 is less than 3. (Contributed by NM, 26-Sep-2010.) $)
|
|
2lt3 $p |- 2 < 3 $=
|
|
( c2 c1 caddc co c3 clt 2re ltp1i df-3 breqtrri ) AABCDEFAGHIJ $.
|
|
|
|
$( 1 is less than 3. (Contributed by NM, 26-Sep-2010.) $)
|
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1lt3 $p |- 1 < 3 $=
|
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( c1 c2 clt wbr c3 1lt2 2lt3 1re 2re 3re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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$( 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
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3lt4 $p |- 3 < 4 $=
|
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( c3 c1 caddc co c4 clt 3re ltp1i df-4 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
2lt4 $p |- 2 < 4 $=
|
|
( c2 c3 clt wbr c4 2lt3 3lt4 2re 3re 4re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
1lt4 $p |- 1 < 4 $=
|
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( c1 c2 clt wbr c4 1lt2 2lt4 1re 2re 4re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
4lt5 $p |- 4 < 5 $=
|
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( c4 c1 caddc co c5 clt 4re ltp1i df-5 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
3lt5 $p |- 3 < 5 $=
|
|
( c3 c4 clt wbr c5 3lt4 4lt5 3re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
2lt5 $p |- 2 < 5 $=
|
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( c2 c4 clt wbr c5 2lt4 4lt5 2re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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$( 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
1lt5 $p |- 1 < 5 $=
|
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( c1 c4 clt wbr c5 1lt4 4lt5 1re 4re 5re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
5lt6 $p |- 5 < 6 $=
|
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( c5 c1 caddc co c6 clt 5re ltp1i df-6 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
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4lt6 $p |- 4 < 6 $=
|
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( c4 c5 clt wbr c6 4lt5 5lt6 4re 5re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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$( 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
3lt6 $p |- 3 < 6 $=
|
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( c3 c4 clt wbr c6 3lt4 4lt6 3re 4re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
2lt6 $p |- 2 < 6 $=
|
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( c2 c3 clt wbr c6 2lt3 3lt6 2re 3re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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$( 1 is less than 6. (Contributed by NM, 19-Oct-2012.) $)
|
|
1lt6 $p |- 1 < 6 $=
|
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( c1 c2 clt wbr c6 1lt2 2lt6 1re 2re 6re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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$( 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
6lt7 $p |- 6 < 7 $=
|
|
( c6 c1 caddc co c7 clt 6re ltp1i df-7 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
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5lt7 $p |- 5 < 7 $=
|
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( c5 c6 clt wbr c7 5lt6 6lt7 5re 6re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
4lt7 $p |- 4 < 7 $=
|
|
( c4 c5 clt wbr c7 4lt5 5lt7 4re 5re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
3lt7 $p |- 3 < 7 $=
|
|
( c3 c4 clt wbr c7 3lt4 4lt7 3re 4re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
2lt7 $p |- 2 < 7 $=
|
|
( c2 c3 clt wbr c7 2lt3 3lt7 2re 3re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
1lt7 $p |- 1 < 7 $=
|
|
( c1 c2 clt wbr c7 1lt2 2lt7 1re 2re 7re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
7lt8 $p |- 7 < 8 $=
|
|
( c7 c1 caddc co c8 clt 7re ltp1i df-8 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
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6lt8 $p |- 6 < 8 $=
|
|
( c6 c7 clt wbr c8 6lt7 7lt8 6re 7re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
5lt8 $p |- 5 < 8 $=
|
|
( c5 c6 clt wbr c8 5lt6 6lt8 5re 6re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
4lt8 $p |- 4 < 8 $=
|
|
( c4 c5 clt wbr c8 4lt5 5lt8 4re 5re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
3lt8 $p |- 3 < 8 $=
|
|
( c3 c4 clt wbr c8 3lt4 4lt8 3re 4re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
2lt8 $p |- 2 < 8 $=
|
|
( c2 c3 clt wbr c8 2lt3 3lt8 2re 3re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
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$( 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) $)
|
|
1lt8 $p |- 1 < 8 $=
|
|
( c1 c2 clt wbr c8 1lt2 2lt8 1re 2re 8re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
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|
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$( 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.) $)
|
|
8lt9 $p |- 8 < 9 $=
|
|
( c8 c1 caddc co c9 clt 8re ltp1i df-9 breqtrri ) AABCDEFAGHIJ $.
|
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|
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$( 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
7lt9 $p |- 7 < 9 $=
|
|
( c7 c8 clt wbr c9 7lt8 8lt9 7re 8re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
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$( 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
6lt9 $p |- 6 < 9 $=
|
|
( c6 c7 clt wbr c9 6lt7 7lt9 6re 7re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
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$( 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
5lt9 $p |- 5 < 9 $=
|
|
( c5 c6 clt wbr c9 5lt6 6lt9 5re 6re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
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$( 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
4lt9 $p |- 4 < 9 $=
|
|
( c4 c5 clt wbr c9 4lt5 5lt9 4re 5re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
|
$( 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
3lt9 $p |- 3 < 9 $=
|
|
( c3 c4 clt wbr c9 3lt4 4lt9 3re 4re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
|
$( 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
|
|
2lt9 $p |- 2 < 9 $=
|
|
( c2 c3 clt wbr c9 2lt3 3lt9 2re 3re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
|
$( 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario
|
|
Carneiro, 9-Mar-2015.) $)
|
|
1lt9 $p |- 1 < 9 $=
|
|
( c1 c2 clt wbr c9 1lt2 2lt9 1re 2re 9re lttri mp2an ) ABCDBECDAECDFGABEHIJ
|
|
KL $.
|
|
|
|
$( 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
0ne2 $p |- 0 =/= 2 $=
|
|
( c2 cc0 2ne0 necomi ) ABCD $.
|
|
|
|
$( 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.) $)
|
|
1ne2 $p |- 1 =/= 2 $=
|
|
( c1 c2 1re 1lt2 ltneii ) ABCDE $.
|
|
|
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$( 1 is less than or equal to 2 (common case). (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
1le2 $p |- 1 <_ 2 $=
|
|
( c1 c2 1re 2re 1lt2 ltleii ) ABCDEF $.
|
|
|
|
$( 2 is a nonzero complex number (common case). (Contributed by David A.
|
|
Wheeler, 7-Dec-2018.) $)
|
|
2cnne0 $p |- ( 2 e. CC /\ 2 =/= 0 ) $=
|
|
( c2 cc wcel cc0 wne 2cn 2ne0 pm3.2i ) ABCADEFGH $.
|
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|
|
$( 2 is a nonzero real number (common case). (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
2rene0 $p |- ( 2 e. RR /\ 2 =/= 0 ) $=
|
|
( c2 cr wcel cc0 wne 2re 2ne0 pm3.2i ) ABCADEFGH $.
|
|
|
|
$( 1 is less than or equal to 3. (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
1le3 $p |- 1 <_ 3 $=
|
|
( c1 c3 1re 3re 1lt3 ltleii ) ABCDEF $.
|
|
|
|
$( ` -u 1 x. -u 1 ` is 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
neg1mulneg1e1 $p |- ( -u 1 x. -u 1 ) = 1 $=
|
|
( c1 cneg cmul co ax-1cn mul2negi 1t1e1 eqtri ) ABZICDAACDAAAEEFGH $.
|
|
|
|
$( One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
halfre $p |- ( 1 / 2 ) e. RR $=
|
|
( c2 2re 2ap0 rerecclapi ) ABCD $.
|
|
|
|
$( One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
halfcn $p |- ( 1 / 2 ) e. CC $=
|
|
( c1 c2 cdiv co halfre recni ) ABCDEF $.
|
|
|
|
$( One-half is greater than zero. (Contributed by NM, 24-Feb-2005.) $)
|
|
halfgt0 $p |- 0 < ( 1 / 2 ) $=
|
|
( c2 2re 2pos recgt0ii ) ABCD $.
|
|
|
|
$( One-half is not negative. (Contributed by AV, 7-Jun-2020.) $)
|
|
halfge0 $p |- 0 <_ ( 1 / 2 ) $=
|
|
( cc0 c1 c2 cdiv co 0re halfre halfgt0 ltleii ) ABCDEFGHI $.
|
|
|
|
$( One-half is less than one. (Contributed by NM, 24-Feb-2005.) $)
|
|
halflt1 $p |- ( 1 / 2 ) < 1 $=
|
|
( c1 cdiv co clt wbr 1div1e1 1lt2 eqbrtri 1re 2re 0lt1 2pos ltdiv23ii mpbi
|
|
c2 ) AABCZODEAOBCADEPAODFGHAAOIIJKLMN $.
|
|
|
|
$( Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler,
|
|
4-Jan-2017.) $)
|
|
1mhlfehlf $p |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 ) $=
|
|
( c2 c1 cmin co cdiv cc wcel cc0 cap wbr wceq 2cn ax-1cn pm3.2i divsubdirap
|
|
wa 2ap0 mp3an 2m1e1 oveq1i 2div2e1 3eqtr3ri ) ABCDZAEDZAAEDZBAEDZCDZUFBUFCD
|
|
AFGZBFGUHAHIJZPUDUGKLMUHUILQNABAORUCBAESTUEBUFCUATUB $.
|
|
|
|
$( An eighth of four thirds is a sixth. (Contributed by Paul Chapman,
|
|
24-Nov-2007.) $)
|
|
8th4div3 $p |- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 ) $=
|
|
( c1 c8 cdiv co c4 c3 cmul c6 ax-1cn 8re recni 4cn gt0ap0ii c2 eqtr3i eqtri
|
|
3cn 2cn cc wcel 8pos divmuldivapi mulcomi mul32i 4t2e8 oveq1i mulassi 3t2e6
|
|
3re 3pos oveq2i oveq12i cc0 cap wbr wceq 6re 6pos 4re 4pos divcanap5 mp3an1
|
|
wa mp4an ) ABCDEFCDGDZEAGDZEHGDZCDZAHCDZVEAEGDZBFGDZCDVHABEFIBJKLQBJUAMFUIU
|
|
JMUBVJVFVKVGCAEILUCVKEFNGDZGDZVGEFGDNGDZVKVMENGDZFGDVNVKENFLRQUDVOBFGUEUFOE
|
|
FNLQRUGOVLHEGUHUKPULPHSTZHUMUNUOZESTZEUMUNUOZVHVIUPZHUQKHUQURMLEUSUTMASTVPV
|
|
QVCVRVSVCVTIAHEVAVBVDP $.
|
|
|
|
$( One half plus or minus one sixth. (Contributed by Paul Chapman,
|
|
17-Jan-2008.) $)
|
|
halfpm6th $p |- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\
|
|
( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) ) $=
|
|
( c1 c2 cdiv co c6 cmin c3 wceq caddc cmul 3cn ax-1cn 2cn 2ap0 divmuldivapi
|
|
oveq1i mulid1i 3t2e6 oveq12i 3eqtr3i 3re 3pos gt0ap0ii halfcn mulid2i eqtri
|
|
dividapi cc wcel cc0 cap wbr wa 6cn 6re 6pos pm3.2i divsubdirap mp3an 3m1e2
|
|
oveq2i recclapi 3eqtr2i c4 divdirapi df-4 3eqtr4ri 2t2e4 divclapi ) ABCDZAE
|
|
CDZFDZAGCDZHVJVKIDZBGCDZHVLGECDZVKFDZGAFDZECDZVMVJVPVKFGGCDZVJJDZGAJDZGBJDZ
|
|
CDVJVPGGABKKLMGUAUBUCZNOWAAVJJDVJVTAVJJGKWDUGPVJUDUEUFWBGWCECGKQRSTZPGUHUIA
|
|
UHUIEUHUIZEUJUKULZUMVSVQHKLWFWGUNEUOUPUCZUQGAEURUSVSBECDABJDZWCCDZVMVRBECUT
|
|
PWIBWCECBMUERSVMBBCDZJDVMAJDWJVMWKAVMJBMNUGZVAAGBBLKMMWDNOVMGKWDVBQTVCVCVNV
|
|
DECDZBBJDZWCCDZVOGAIDZECDVPVKIDWMVNGAEKLUNWHVEVDWPECVFPVJVPVKIWEPVGWNVDWCEC
|
|
VHRSVOWKJDVOAJDWOVOWKAVOJWLVABGBBMKMMWDNOVOBGMKWDVIQTVCUQ $.
|
|
|
|
$( i times 0 equals 0 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
it0e0 $p |- ( _i x. 0 ) = 0 $=
|
|
( ci ax-icn mul01i ) ABC $.
|
|
|
|
$( ` ( 2 x. _i ) e. CC ` (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
2mulicn $p |- ( 2 x. _i ) e. CC $=
|
|
( c2 ci 2cn ax-icn mulcli ) ABCDE $.
|
|
|
|
$( The imaginary unit ` _i ` is apart from zero. (Contributed by Jim
|
|
Kingdon, 9-Mar-2020.) $)
|
|
iap0 $p |- _i # 0 $=
|
|
( cc0 ci c1 cmul co caddc cap wbr wo 1ap0 olci cr wcel wb 0re apreim ax-icn
|
|
1re oveq2i eqtri mp4an mpbir mulid1i addid2i it0e0 00id 3brtr3i ) ABCDEZFEZ
|
|
ABADEZFEZBAGUIUKGHZAAGHZCAGHZIZUNUMJKALMZCLMUPUPULUONOROOACAAPUAUBUIABFEBUH
|
|
BAFBQUCSBQUDTUKAAFEAUJAAFUESUFTUG $.
|
|
|
|
$( ` 2 x. _i ` is apart from zero. (Contributed by Jim Kingdon,
|
|
9-Mar-2020.) $)
|
|
2muliap0 $p |- ( 2 x. _i ) # 0 $=
|
|
( c2 ci 2cn ax-icn 2ap0 iap0 mulap0i ) ABCDEFG $.
|
|
|
|
$( ` ( 2 x. _i ) =/= 0 ` . See also ~ 2muliap0 . (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
2muline0 $p |- ( 2 x. _i ) =/= 0 $=
|
|
( c2 ci cmul co cc0 cap wbr wne 2muliap0 cc wcel wi 2mulicn 0cn mp2an ax-mp
|
|
apne ) ABCDZEFGZREHZIRJKEJKSTLMNREQOP $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Simple number properties
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Closure of half of a number (common case). (Contributed by NM,
|
|
1-Jan-2006.) $)
|
|
halfcl $p |- ( A e. CC -> ( A / 2 ) e. CC ) $=
|
|
( cc wcel c2 cc0 cap wbr cdiv co 2cn 2ap0 divclap mp3an23 ) ABCDBCDEFGADHIB
|
|
CJKADLM $.
|
|
|
|
$( Real closure of half. (Contributed by NM, 1-Jan-2006.) $)
|
|
rehalfcl $p |- ( A e. RR -> ( A / 2 ) e. RR ) $=
|
|
( cr wcel c2 cc0 cap wbr cdiv co 2re 2ap0 redivclap mp3an23 ) ABCDBCDEFGADH
|
|
IBCJKADLM $.
|
|
|
|
$( Half of a number is zero iff the number is zero. (Contributed by NM,
|
|
20-Apr-2006.) $)
|
|
half0 $p |- ( A e. CC -> ( ( A / 2 ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel c2 cc0 cap wbr cdiv co wceq wb 2cn 2ap0 diveqap0 mp3an23 ) ABCDBC
|
|
DEFGADHIEJAEJKLMADNO $.
|
|
|
|
$( Two halves make a whole. (Contributed by NM, 11-Apr-2005.) $)
|
|
2halves $p |- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A ) $=
|
|
( cc wcel c2 cmul co cdiv 2times oveq1d cc0 cap wbr wceq 2cn 2ap0 divcanap3
|
|
caddc mp3an23 wa pm3.2i divdirap mp3an3 anidms 3eqtr3rd ) ABCZDAEFZDGFZAAQF
|
|
ZDGFZAADGFZUJQFZUEUFUHDGAHIUEDBCZDJKLZUGAMNOADPRUEUIUKMZUEUEULUMSUNULUMNOTA
|
|
ADUAUBUCUD $.
|
|
|
|
$( A number is positive iff its half is positive. (Contributed by NM,
|
|
10-Apr-2005.) $)
|
|
halfpos2 $p |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) ) $=
|
|
( cr wcel c2 cc0 clt wbr cdiv co wb 2re 2pos gt0div mp3an23 ) ABCDBCEDFGEAF
|
|
GEADHIFGJKLADMN $.
|
|
|
|
$( A positive number is greater than its half. (Contributed by NM,
|
|
28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) $)
|
|
halfpos $p |- ( A e. RR -> ( 0 < A <-> ( A / 2 ) < A ) ) $=
|
|
( cr wcel cc0 clt c2 cdiv co caddc halfpos2 rehalfcl ltaddposd cc wceq recn
|
|
wbr 2halves syl breq2d 3bitrd ) ABCZDAEPDAFGHZEPUBUBUBIHZEPUBAEPAJUAUBUBAKZ
|
|
UDLUAUCAUBEUAAMCUCANAOAQRST $.
|
|
|
|
$( A number is nonnegative iff its half is nonnegative. (Contributed by NM,
|
|
9-Dec-2005.) $)
|
|
halfnneg2 $p |- ( A e. RR -> ( 0 <_ A <-> 0 <_ ( A / 2 ) ) ) $=
|
|
( cr wcel c2 cc0 clt wbr cle cdiv co wb 2re 2pos ge0div mp3an23 ) ABCDBCEDF
|
|
GEAHGEADIJHGKLMADNO $.
|
|
|
|
$( Closure of half-sum and half-difference. (Contributed by Paul Chapman,
|
|
12-Oct-2007.) $)
|
|
halfaddsubcl $p |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) e. CC /\
|
|
( ( A - B ) / 2 ) e. CC ) ) $=
|
|
( cc wcel wa caddc co c2 cdiv cmin addcl halfcl syl subcl jca ) ACDBCDEZABF
|
|
GZHIGCDZABJGZHIGCDZPQCDRABKQLMPSCDTABNSLMO $.
|
|
|
|
$( Sum and difference of half-sum and half-difference. (Contributed by Paul
|
|
Chapman, 12-Oct-2007.) $)
|
|
halfaddsub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A /\
|
|
( ( ( A + B ) / 2 ) - ( ( A - B ) / 2 ) ) = B ) ) $=
|
|
( cc wcel wa caddc cdiv cmin wceq cmul 2times adantr eqtr4d oveq1d 2cn 2ap0
|
|
co c2 mp3an3 syl2anc ppncan 3anidm13 addcl subcl cc0 cap divdirap divcanap3
|
|
wbr pm3.2i mp3an23 3eqtr3d pnncan 3anidm23 adantl divsubdirap jca ) ACDZBCD
|
|
ZEZABFQZRGQZABHQZRGQZFQZAIVBVDHQZBIUTVAVCFQZRGQZRAJQZRGQZVEAUTVGVIRGUTVGAAF
|
|
QZVIURUSVGVKIABAUAUBURVIVKIUSAKLMNUTVACDZVCCDZVHVEIZABUCZABUDZVLVMRCDZRUEUF
|
|
UIZEZVNVQVROPUJZVAVCRUGSTURVJAIZUSURVQVRWAOPARUHUKLULUTVAVCHQZRGQZRBJQZRGQZ
|
|
VFBUTWBWDRGUTWBBBFQZWDURUSWBWFIABBUMUNUSWDWFIURBKUOMNUTVLVMWCVFIZVOVPVLVMVS
|
|
WGVTVAVCRUPSTUSWEBIZURUSVQVRWHOPBRUHUKUOULUQ $.
|
|
|
|
$( TODO - use this to reduce:
|
|
Statement "lt2add" is directly referenced in the proofs of 9 statements:
|
|
2climnn 2climnn0 climaddlem3 climmullem5 climcaui ser1f0i
|
|
MM> sh us 2halves
|
|
Statement "2halves" is directly referenced in the proofs of 10 statements:
|
|
2climnn 2climnn0 climmullem5 climcaui caucvgi ser1f0i
|
|
MM>
|
|
$)
|
|
$( A sum is less than the whole if each term is less than half. (Contributed
|
|
by NM, 13-Dec-2006.) $)
|
|
lt2halves $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( A < ( C / 2 ) /\ B < ( C / 2 ) ) -> ( A + B ) < C ) ) $=
|
|
( cr wcel w3a c2 co clt wbr wa caddc wi 3simpa rehalfcl jca 3ad2ant3 lt2add
|
|
cdiv syl2anc wb cc wceq recn 2halves syl breq2d sylibd ) ADEZBDEZCDEZFZACGS
|
|
HZIJBUMIJKZABLHZUMUMLHZIJZUOCIJZULUIUJKUMDEZUSKZUNUQMUIUJUKNUKUIUTUJUKUSUSC
|
|
OZVAPQABUMUMRTUKUIUQURUAUJUKUPCUOIUKCUBEUPCUCCUDCUEUFUGQUH $.
|
|
|
|
$( Sum is less than product for numbers greater than 2. (Contributed by
|
|
Stefan Allan, 24-Sep-2010.) $)
|
|
addltmul $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) )
|
|
-> ( A + B ) < ( A x. B ) ) $=
|
|
( cr wcel wa c2 clt wbr caddc co cmul c1 cmin 2re 1re ltsub1 syl2an remulcl
|
|
wb cc mp3an13 2m1e1 breq1i syl6bb bi2anan9 wi peano2rem mulgt1 ex wceq recn
|
|
sylbid ax-1cn mulsub mpanl2 mpanr2 breq2d 1t1e1 oveq2i breq2i mpan2 readdcl
|
|
remulcli sylancl ltaddsub2 mp3an2 syl2anc syl5rbbr ltadd1 ax-1rid oveqan12d
|
|
mp3an3 breq1d bitr3d 3bitrd sylibd imp ) ACDZBCDZEZFAGHZFBGHZEZABIJZABKJZGH
|
|
ZVTWCLALMJZBLMJZKJZGHZWFVTWCLWGGHZLWHGHZEZWJVRWAWKVSWBWLVRWAFLMJZWGGHZWKFCD
|
|
ZVRLCDZWAWOSNOFALPUAWNLWGGUBUCUDVSWBWNWHGHZWLWPVSWQWBWRSNOFBLPUAWNLWHGUBUCU
|
|
DUEVRWGCDZWHCDZWMWJUFVSAUGBUGWSWTEWMWJWGWHUHUIQULVTWJLWELLKJZIJZALKJZBLKJZI
|
|
JZMJZGHZXELIJZWELIJZGHZWFVTWIXFLGVRATDZBTDZWIXFUJZVSAUKBUKXKXLLTDZXMUMXKXNX
|
|
LXNEXMUMALBLUNUOUPQUQXJXHXBGHZVTXGXBXIXHGXALWEIURUSUTVTXECDZXBCDZXOXGSZVRXC
|
|
CDZXDCDZXPVSVRWQXSOALRVAVSWQXTOBLRVAXCXDVBQZVTWECDZXACDXQABRZLLOOVCWEXAVBVD
|
|
XPWQXQXROXELXBVEVFVGVHVTXEWEGHZXJWFVTXPYBYDXJSZYAYCXPYBWQYEOXEWELVIVLVGVTXE
|
|
WDWEGVRVSXCAXDBIAVJBVJVKVMVNVOVPVQ $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( There is no smallest positive real number. (Contributed by NM,
|
|
28-Oct-2004.) $)
|
|
nominpos $p |- -. E. x e. RR ( 0 < x /\
|
|
-. E. y e. RR ( 0 < y /\ y < x ) ) $=
|
|
( cc0 cv clt wbr wa cr wrex wn wcel wi c2 cdiv co rehalfcl divgt0 mpanr12
|
|
2re 2pos ex halfpos biimpd jcad wceq breq2 breq1 rspcev syl6an imanim syl
|
|
anbi12d nrex ) CADZEFZCBDZEFZUPUNEFZGZBHIZJGZAHUNHKZUOUTLVAJVBUNMNOZHKUOC
|
|
VCEFZVCUNEFZGZUTUNPVBUOVDVEVBUOVDVBUOGMHKCMEFVDSTUNMQRUAVBUOVEUNUBUCUDUSV
|
|
FBVCHUPVCUEUQVDURVEUPVCCEUFUPVCUNEUGULUHUIUOUTUJUKUM $.
|
|
$}
|
|
|
|
$( Ordering property for average. (Contributed by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
avglt1 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A < B <-> A < ( ( A + B ) / 2 ) ) ) $=
|
|
( cr wcel wa clt wbr caddc co c2 cmul cdiv wb ltadd2 3anidm13 cc wceq simpl
|
|
recnd times2 syl breq1d cc0 readdcl 2re pm3.2i a1i ltmuldiv syl3anc 3bitr2d
|
|
2pos ) ACDZBCDZEZABFGZAAHIZABHIZFGZAJKIZUQFGZAUQJLIFGZULUMUOURMABANOUNUSUPU
|
|
QFUNAPDUSUPQUNAULUMRZSATUAUBUNULUQCDJCDZUCJFGZEZUTVAMVBABUDVEUNVCVDUEUKUFUG
|
|
AUQJUHUIUJ $.
|
|
|
|
$( Ordering property for average. (Contributed by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
avglt2 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A < B <-> ( ( A + B ) / 2 ) < B ) ) $=
|
|
( cr wcel wa caddc co c2 cmul clt wbr cdiv cc simpr recnd 2times syl breq2d
|
|
wceq wb cc0 readdcl 2re 2pos pm3.2i a1i ltdivmul syl3anc 3anidm23 3bitr4rd
|
|
ltadd1 ) ACDZBCDZEZABFGZHBIGZJKZUOBBFGZJKZUOHLGBJKZABJKZUNUPURUOJUNBMDUPURS
|
|
UNBULUMNZOBPQRUNUOCDUMHCDZUAHJKZEZUTUQTABUBVBVEUNVCVDUCUDUEUFUOBHUGUHULUMVA
|
|
USTABBUKUIUJ $.
|
|
|
|
$( Ordering property for average. (Contributed by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
avgle1 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A <_ B <-> A <_ ( ( A + B ) / 2 ) ) ) $=
|
|
( cr wcel wa clt wbr wn caddc co c2 cdiv cle wb avglt2 ancoms cc wceq lenlt
|
|
recn addcom syl2an oveq1d breq1d bitr4d notbid readdcl rehalfcl syl 3bitr4d
|
|
syldan ) ACDZBCDZEZBAFGZHABIJZKLJZAFGZHZABMGAUQMGZUNUOURUNUOBAIJZKLJZAFGZUR
|
|
UMULUOVCNBAOPUNUQVBAFUNUPVAKLULAQDBQDUPVARUMATBTABUAUBUCUDUEUFABSULUMUQCDZU
|
|
TUSNUNUPCDVDABUGUPUHUIAUQSUKUJ $.
|
|
|
|
$( Ordering property for average. (Contributed by Jeff Hankins,
|
|
15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.) $)
|
|
avgle2 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A <_ B <-> ( ( A + B ) / 2 ) <_ B ) ) $=
|
|
( cr wcel wa clt wbr wn caddc co c2 cdiv cle wb avglt1 ancoms cc wceq lenlt
|
|
recn addcom syl2an oveq1d breq2d bitr4d notbid readdcl rehalfcl syl 3bitr4d
|
|
sylancom ) ACDZBCDZEZBAFGZHBABIJZKLJZFGZHZABMGUQBMGZUNUOURUNUOBBAIJZKLJZFGZ
|
|
URUMULUOVCNBAOPUNUQVBBFUNUPVAKLULAQDBQDUPVARUMATBTABUAUBUCUDUEUFABSULUMUQCD
|
|
ZUTUSNUNUPCDVDABUGUPUHUIUQBSUKUJ $.
|
|
|
|
${
|
|
2timesd.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
2timesd $p |- ( ph -> ( 2 x. A ) = ( A + A ) ) $=
|
|
( cc wcel c2 cmul co caddc wceq 2times syl ) ABDEFBGHBBIHJCBKL $.
|
|
|
|
$( A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
times2d $p |- ( ph -> ( A x. 2 ) = ( A + A ) ) $=
|
|
( cc wcel c2 cmul co caddc wceq times2 syl ) ABDEBFGHBBIHJCBKL $.
|
|
|
|
$( Closure of half of a number (frequently used special case).
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
halfcld $p |- ( ph -> ( A / 2 ) e. CC ) $=
|
|
( cc wcel c2 cdiv co halfcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
$( Two halves make a whole. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
2halvesd $p |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A ) $=
|
|
( cc wcel c2 cdiv co caddc wceq 2halves syl ) ABDEBFGHZMIHBJCBKL $.
|
|
$}
|
|
|
|
${
|
|
rehalfcld.1 $e |- ( ph -> A e. RR ) $.
|
|
$( Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
rehalfcld $p |- ( ph -> ( A / 2 ) e. RR ) $=
|
|
( cr wcel c2 cdiv co rehalfcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
lt2halvesd.2 $e |- ( ph -> B e. RR ) $.
|
|
lt2halvesd.3 $e |- ( ph -> C e. RR ) $.
|
|
lt2halvesd.4 $e |- ( ph -> A < ( C / 2 ) ) $.
|
|
lt2halvesd.5 $e |- ( ph -> B < ( C / 2 ) ) $.
|
|
$( A sum is less than the whole if each term is less than half.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
lt2halvesd $p |- ( ph -> ( A + B ) < C ) $=
|
|
( c2 cdiv co clt wbr caddc cr wcel wa wi lt2halves syl3anc mp2and ) ABDJK
|
|
LZMNZCUCMNZBCOLDMNZHIABPQCPQDPQUDUERUFSEFGBCDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
rehalfcli.1 $e |- A e. RR $.
|
|
$( Half a real number is real. Inference form. (Contributed by David
|
|
Moews, 28-Feb-2017.) $)
|
|
rehalfcli $p |- ( A / 2 ) e. RR $=
|
|
( cr wcel c2 cdiv co rehalfcl ax-mp ) ACDAEFGCDBAHI $.
|
|
$}
|
|
|
|
$( Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) $)
|
|
add1p1 $p |- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) ) $=
|
|
( cc wcel c1 caddc co c2 id 1cnd addassd wceq 1p1e2 a1i oveq2d eqtrd ) ABCZ
|
|
ADEFDEFADDEFZEFAGEFPADDPHPIZRJPQGAEQGKPLMNO $.
|
|
|
|
$( Subtracting two times 1 from a number. (Contributed by AV,
|
|
23-Oct-2018.) $)
|
|
sub1m1 $p |- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) ) $=
|
|
( cc wcel c1 cmin co caddc c2 id 1cnd subsub4d wceq 1p1e2 a1i oveq2d eqtrd
|
|
) ABCZADEFDEFADDGFZEFAHEFQADDQIQJZSKQRHAERHLQMNOP $.
|
|
|
|
$( Subtracting 2 and afterwards 1 from a number results in the difference
|
|
between the number and 3. (Contributed by Alexander van der Vekens,
|
|
16-Sep-2018.) $)
|
|
cnm2m1cnm3 $p |- ( A e. CC -> ( ( A - 2 ) - 1 ) = ( A - 3 ) ) $=
|
|
( cc wcel c2 cmin co c1 caddc c3 id 2cnd 1cnd subsub4d wceq 2p1e3 a1i eqtrd
|
|
oveq2d ) ABCZADEFGEFADGHFZEFAIEFSADGSJSKSLMSTIAETINSOPRQ $.
|
|
|
|
$( A complex number increased by 1, then divided by 2, then decreased by 1
|
|
equals the complex number decreased by 1 and then divided by 2.
|
|
(Contributed by AV, 24-May-2020.) $)
|
|
xp1d2m1eqxm1d2 $p |- ( X e. CC
|
|
-> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) ) $=
|
|
( cc wcel c1 caddc co cdiv cmin peano2cn halfcld peano2cnm syl 2cnd cc0 cap
|
|
c2 wbr a1i cmul divcanap1d 2ap0 1cnd subdird mulid2d oveq12d wceq oveq2d id
|
|
2m1e1 subsub3d 3eqtr2rd 3eqtrd mulcanap2ad ) ABCZADEFZPGFZDHFZADHFZPGFZPUNU
|
|
PBCUQBCUNUOAIZJZUPKLUNURAKZJUNMZPNOQUNUARZUNUQPSFUPPSFZDPSFZHFUOPHFZUSPSFZU
|
|
NUPDPVAUNUBZVCUCUNVEUOVFPHUNUOPUTVCVDTUNPVCUDUEUNVHURAPDHFZHFVGUNURPVBVCVDT
|
|
UNVJDAHVJDUFUNUIRUGUNAPDUNUHVCVIUJUKULUM $.
|
|
|
|
$( An integer greater than 5, divided by 4 and increased by 1, is less than
|
|
or equal to the half of the integer minus 1. (Contributed by AV,
|
|
8-Jul-2021.) $)
|
|
div4p1lem1div2 $p |- ( ( N e. RR /\ 6 <_ N )
|
|
-> ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) ) $=
|
|
( cr wcel c6 cle wbr c4 co c1 caddc cmin c2 cmul a1i wceq adantr wb syl3anc
|
|
cc cc0 wa cdiv 6re id leadd2d biimpa times2d breqtrrd 4cn 2cn addassd 4p2e6
|
|
recn oveq2i syl6eq breq1d mpbird clt 4re 4ap0 redivclapd peano2re peano2rem
|
|
cap syl rehalfcld 4pos pm3.2i recnd 1cnd divcanap1d mulid2i joinlmuladdmuld
|
|
lemul1 oveq12d 2t2e4 eqcomi oveq2d mulass eqcomd 2ap0 oveq1d subdird 3eqtrd
|
|
w3a breq12d readdcld 2re remulcld leaddsub bicomd 3bitrd ) ABCZDAEFZUAZAGUB
|
|
HZIJHZAIKHZLUBHZEFZAGJHZLJHZALMHZEFZWOXDADJHZXCEFZWOXEAAJHZXCEWMWNXEXGEFWMD
|
|
AADBCWMUCNWMUDZXHUEUFWMXCXGOWNWMAAUMZUGPUHWMXDXFQWNWMXBXEXCEWMXBAGLJHZJHXEW
|
|
MAGLXIGSCWMUINZLSCZWMUJNZUKXJDAJULUNUOUPPUQWMWTXDQWNWMWTWQGMHZWSGMHZEFZXAXC
|
|
LKHZEFZXDWMWQBCZWSBCGBCZTGURFZUAZWTXPQWMWPBCXSWMAGXHXTWMUSNZGTVDFWMUTNZVAZW
|
|
PVBVEWMWRAVCZVFZYBWMXTYAUSVGVHNWQWSGVNRWMXNXAXOXQEWMWPGIXAWMWPYEVIXKWMVJZWM
|
|
WPGMHAIGMHZGJWMAGXIXKYDVKYIGOWMGUIVLNVOVMWMXOWSLLMHZMHZWSLMHZLMHZXQWMGYJWSM
|
|
GYJOWMYJGVPVQNVRWMWSSCZXLXLYKYMOWMWSYGVIXMXMYNXLXLWEYMYKWSLLVSVTRWMYMWRLMHX
|
|
CILMHZKHXQWMYLWRLMWMWRLWMWRYFVIXMLTVDFWMWANVKWBWMAILXIYHXMWCWMYOLXCKYOLOWML
|
|
UJVLNVRWDWDWFWMXABCZLBCZXCBCZXRXDQWMAGXHYCWGYQWMWHNZWMALXHYSWIYPYQYRWEXDXRX
|
|
ALXCWJWKRWLPUQ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The Archimedean property
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d n x y A $.
|
|
$( Archimedean property of real numbers. For any real number, there is an
|
|
integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed
|
|
by NM, 21-Jan-1997.) $)
|
|
arch $p |- ( A e. RR -> E. n e. NN A < n ) $=
|
|
( vx vy cr wcel cv clt wbr cn wrex cltrr c1 caddc co wral wa cint ax-arch
|
|
cab dfnn2 rexeqi sylibr wb nnre ltxrlt sylan2 rexbidva mpbird ) AEFZABGZH
|
|
IZBJKAUKLIZBJKZUJUMBMCGZFDGMNOUOFDUOPQCTRZKUNCDABSUMBJUPCDUAUBUCUJULUMBJU
|
|
KJFUJUKEFULUMUDUKUEAUKUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d n A $.
|
|
$( There exists a positive integer whose reciprocal is less than a given
|
|
positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM,
|
|
8-Nov-2004.) $)
|
|
nnrecl $p |- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co cv cn wrex simpl gt0ap0 rerecclapd wb
|
|
jca adantr arch syl recgt0 nnre nngt0 ltrec syl2an recrecapd breq2d bitrd
|
|
cc recn rexbidva mpbid ) ACDZEAFGZHZIAJKZBLZFGZBMNZIUSJKZAFGZBMNUQURCDZVA
|
|
UQAUOUPOAPZQZURBUAUBUQUTVCBMUQUSMDZHUTVBIURJKZFGZVCUQVDEURFGZHUSCDZEUSFGZ
|
|
HUTVIRVGUQVDVJVFAUCSVGVKVLUSUDUSUESURUSUFUGUQVIVCRVGUQVHAVBFUQAUOAUKDUPAU
|
|
LTVEUHUITUJUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x k $.
|
|
$( A bounded real sequence ` A ( k ) ` is less than or equal to at least
|
|
one of its indices. (Contributed by NM, 18-Jan-2008.) $)
|
|
bndndx $p |- ( E. x e. RR A. k e. NN ( A e. RR /\ A <_ x ) ->
|
|
E. k e. NN A <_ k ) $=
|
|
( cr wcel cv cle wbr wa cn wral wrex wi clt arch nnre lelttr ltle 3adant2
|
|
w3a syld exp5o com3l imp4b com23 sylan2 reximdva mpd r19.35-1 rexlimiv
|
|
syl ) BDEZBAFZGHZIZCJKZBCFZGHZCJLZADUMDEZUOURMZCJLZUPUSMUTUMUQNHZCJLVBUMC
|
|
OUTVCVACJUQJEUTUQDEZVCVAMUQPUTVDIUOVCURUTVDULUNVCURMZULUTVDUNVEMULUTVDUNV
|
|
CURULUTVDTUNVCIBUQNHZURBUMUQQULVDVFURMUTBUQRSUAUBUCUDUEUFUGUHUOURCJUIUKUJ
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Nonnegative integers (as a subset of complex numbers)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c NN0 $. $( The set of nonnegative integers (blackboard bold N_0). $)
|
|
|
|
$( Extend class notation to include the class of nonnegative integers. $)
|
|
cn0 $a class NN0 $.
|
|
|
|
$( Define the set of nonnegative integers. (Contributed by Raph Levien,
|
|
10-Dec-2002.) $)
|
|
df-n0 $a |- NN0 = ( NN u. { 0 } ) $.
|
|
|
|
$( Nonnegative integers expressed in terms of naturals and zero.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
elnn0 $p |- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) ) $=
|
|
( cn0 wcel cn cc0 csn cun wceq df-n0 eleq2i elun c0ex elsn2 orbi2i 3bitri
|
|
wo ) ABCADEFZGZCADCZAQCZPSAEHZPBRAIJADQKTUASAELMNO $.
|
|
|
|
$( Positive naturals are a subset of nonnegative integers. (Contributed by
|
|
Raph Levien, 10-Dec-2002.) $)
|
|
nnssnn0 $p |- NN C_ NN0 $=
|
|
( cn cc0 csn cun cn0 ssun1 df-n0 sseqtr4i ) AABCZDEAIFGH $.
|
|
|
|
$( Nonnegative integers are a subset of the reals. (Contributed by Raph
|
|
Levien, 10-Dec-2002.) $)
|
|
nn0ssre $p |- NN0 C_ RR $=
|
|
( cn0 cn cc0 csn cun cr df-n0 nnssre wcel wss 0re snssi ax-mp unssi eqsstri
|
|
) ABCDZEFGBPFHCFIPFJKCFLMNO $.
|
|
|
|
$( Nonnegative integers are a subset of the complex numbers.) (Contributed
|
|
by NM, 9-May-2004.) $)
|
|
nn0sscn $p |- NN0 C_ CC $=
|
|
( cn0 cr cc nn0ssre ax-resscn sstri ) ABCDEF $.
|
|
|
|
$( The set of nonnegative integers exists. (Contributed by NM,
|
|
18-Jul-2004.) $)
|
|
nn0ex $p |- NN0 e. _V $=
|
|
( cn0 cn cc0 csn cun cvv df-n0 nnex c0ex snex unex eqeltri ) ABCDZEFGBMHCIJ
|
|
KL $.
|
|
|
|
$( A positive integer is a nonnegative integer. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
nnnn0 $p |- ( A e. NN -> A e. NN0 ) $=
|
|
( cn cn0 nnssnn0 sseli ) BCADE $.
|
|
|
|
${
|
|
nnnn0.1 $e |- N e. NN $.
|
|
$( A positive integer is a nonnegative integer. (Contributed by NM,
|
|
20-Jun-2005.) $)
|
|
nnnn0i $p |- N e. NN0 $=
|
|
( cn wcel cn0 nnnn0 ax-mp ) ACDAEDBAFG $.
|
|
$}
|
|
|
|
$( A nonnegative integer is a real number. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
nn0re $p |- ( A e. NN0 -> A e. RR ) $=
|
|
( cn0 cr nn0ssre sseli ) BCADE $.
|
|
|
|
$( A nonnegative integer is a complex number. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
nn0cn $p |- ( A e. NN0 -> A e. CC ) $=
|
|
( cn0 cc nn0sscn sseli ) BCADE $.
|
|
|
|
${
|
|
nn0re.1 $e |- A e. NN0 $.
|
|
$( A nonnegative integer is a real number. (Contributed by NM,
|
|
14-May-2003.) $)
|
|
nn0rei $p |- A e. RR $=
|
|
( cn0 cr nn0ssre sselii ) CDAEBF $.
|
|
|
|
$( A nonnegative integer is a complex number. (Contributed by NM,
|
|
14-May-2003.) $)
|
|
nn0cni $p |- A e. CC $=
|
|
( nn0rei recni ) AABCD $.
|
|
$}
|
|
|
|
$( The set of positive integers defined in terms of nonnegative integers.
|
|
(Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro,
|
|
13-Feb-2013.) $)
|
|
dfn2 $p |- NN = ( NN0 \ { 0 } ) $=
|
|
( cn0 cc0 csn cdif cn cun df-n0 difeq1i difun2 wcel wceq 0nnn difsn 3eqtrri
|
|
wn ax-mp ) ABCZDEQFZQDEQDZEARQGHEQIBEJOSEKLBEMPN $.
|
|
|
|
$( The positive integer property expressed in terms of difference from zero.
|
|
(Contributed by Stefan O'Rear, 12-Sep-2015.) $)
|
|
elnnne0 $p |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) ) $=
|
|
( cn wcel cn0 cc0 csn cdif wne wa dfn2 eleq2i eldifsn bitri ) ABCADEFGZCADC
|
|
AEHIBNAJKADELM $.
|
|
|
|
$( 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
0nn0 $p |- 0 e. NN0 $=
|
|
( cc0 wceq cn0 wcel eqid cn wo elnn0 biimpri olcs ax-mp ) AABZACDZAEAFDZLMM
|
|
NLGAHIJK $.
|
|
|
|
$( 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
1nn0 $p |- 1 e. NN0 $=
|
|
( c1 1nn nnnn0i ) ABC $.
|
|
|
|
$( 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
2nn0 $p |- 2 e. NN0 $=
|
|
( c2 2nn nnnn0i ) ABC $.
|
|
|
|
$( 3 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
3nn0 $p |- 3 e. NN0 $=
|
|
( c3 3nn nnnn0i ) ABC $.
|
|
|
|
$( 4 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
4nn0 $p |- 4 e. NN0 $=
|
|
( c4 4nn nnnn0i ) ABC $.
|
|
|
|
$( 5 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) $)
|
|
5nn0 $p |- 5 e. NN0 $=
|
|
( c5 5nn nnnn0i ) ABC $.
|
|
|
|
$( 6 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) $)
|
|
6nn0 $p |- 6 e. NN0 $=
|
|
( c6 6nn nnnn0i ) ABC $.
|
|
|
|
$( 7 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) $)
|
|
7nn0 $p |- 7 e. NN0 $=
|
|
( c7 7nn nnnn0i ) ABC $.
|
|
|
|
$( 8 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) $)
|
|
8nn0 $p |- 8 e. NN0 $=
|
|
( c8 8nn nnnn0i ) ABC $.
|
|
|
|
$( 9 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) $)
|
|
9nn0 $p |- 9 e. NN0 $=
|
|
( c9 9nn nnnn0i ) ABC $.
|
|
|
|
$( A nonnegative integer is greater than or equal to zero. (Contributed by
|
|
NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) $)
|
|
nn0ge0 $p |- ( N e. NN0 -> 0 <_ N ) $=
|
|
( cn0 wcel cn cc0 wceq wo cle wbr elnn0 cr clt nnre nngt0 wi ltle mpan sylc
|
|
0re 0le0 breq2 mpbiri jaoi sylbi ) ABCADCZAEFZGEAHIZAJUEUGUFUEAKCZEALIZUGAM
|
|
ANEKCUHUIUGOSEAPQRUFUGEEHITAEEHUAUBUCUD $.
|
|
|
|
$( A nonnegative integer is not less than zero. (Contributed by NM,
|
|
9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) $)
|
|
nn0nlt0 $p |- ( A e. NN0 -> -. A < 0 ) $=
|
|
( cn0 wcel cc0 cle wbr clt wn nn0ge0 cr wb 0re nn0re lenlt sylancr mpbid )
|
|
ABCZDAEFZADGFHZAIQDJCAJCRSKLAMDANOP $.
|
|
|
|
${
|
|
nn0ge0.1 $e |- N e. NN0 $.
|
|
$( Nonnegative integers are nonnegative. (Contributed by Raph Levien,
|
|
10-Dec-2002.) $)
|
|
nn0ge0i $p |- 0 <_ N $=
|
|
( cn0 wcel cc0 cle wbr nn0ge0 ax-mp ) ACDEAFGBAHI $.
|
|
$}
|
|
|
|
$( A nonnegative integer is less than or equal to zero iff it is equal to
|
|
zero. (Contributed by NM, 9-Dec-2005.) $)
|
|
nn0le0eq0 $p |- ( N e. NN0 -> ( N <_ 0 <-> N = 0 ) ) $=
|
|
( cn0 wcel cc0 cle wbr wa nn0ge0 biantrud cr wb nn0re letri3 sylancl bitr4d
|
|
wceq 0re ) ABCZADEFZSDAEFZGZADPZRTSAHIRAJCDJCUBUAKALQADMNO $.
|
|
|
|
$( A nonnegative integer increased by 1 is greater than 0. (Contributed by
|
|
Alexander van der Vekens, 3-Oct-2018.) $)
|
|
nn0p1gt0 $p |- ( N e. NN0 -> 0 < ( N + 1 ) ) $=
|
|
( cn0 wcel c1 nn0re 1red nn0ge0 cc0 clt wbr 0lt1 a1i addgegt0d ) ABCZADAENF
|
|
AGHDIJNKLM $.
|
|
|
|
$( A positive integer plus a nonnegative integer is a positive integer.
|
|
(Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro,
|
|
16-May-2014.) $)
|
|
nnnn0addcl $p |- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN ) $=
|
|
( cn0 wcel cn cc0 wceq wo caddc co elnn0 nnaddcl wa oveq2 addid1d sylan9eqr
|
|
nncn simpl eqeltrd jaodan sylan2b ) BCDAEDZBEDZBFGZHABIJZEDZBKUBUCUFUDABLUB
|
|
UDMUEAEUDUBUEAFIJABFAINUBAAQOPUBUDRSTUA $.
|
|
|
|
$( A nonnegative integer plus a positive integer is a positive integer.
|
|
(Contributed by NM, 22-Dec-2005.) $)
|
|
nn0nnaddcl $p |- ( ( M e. NN0 /\ N e. NN ) -> ( M + N ) e. NN ) $=
|
|
( cn wcel cn0 caddc co wa wceq nncn nn0cn addcom syl2an nnnn0addcl eqeltrrd
|
|
cc ancoms ) BCDZAEDZABFGZCDRSHBAFGZTCRBPDAPDUATISBJAKBALMBANOQ $.
|
|
|
|
$( The result of subtracting a positive integer from 0 is not a nonnegative
|
|
integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) $)
|
|
0mnnnnn0 $p |- ( N e. NN -> ( 0 - N ) e/ NN0 ) $=
|
|
( cn wcel cc0 cmin co cn0 wnel 0re cneg df-neg eqcomi eleq1i cle wbr nn0ge0
|
|
wn cr nnre le0neg1d clt nngt0 0red lenltd pm2.21 syl6bi mpid sylbird syl5bi
|
|
wi syl5 mt2i df-nel sylibr ) ABCZDAEFZGCZQUPGHUOUQDRCZIUQAJZGCZUOURQZUPUSGU
|
|
SUPAKLMUTDUSNOZUOVAUSPUOVBADNOZVAUOAASZTUOVCDAUAOZVAAUBUOVCVEQVEVAUJUOADVDU
|
|
OUCUDVEVAUEUFUGUHUKUIULUPGUMUN $.
|
|
|
|
${
|
|
un0addcl.1 $e |- ( ph -> S C_ CC ) $.
|
|
un0addcl.2 $e |- T = ( S u. { 0 } ) $.
|
|
${
|
|
un0addcl.3 $e |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M + N ) e. S ) $.
|
|
$( If ` S ` is closed under addition, then so is ` S u. { 0 } ` .
|
|
(Contributed by Mario Carneiro, 17-Jul-2014.) $)
|
|
un0addcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T ) $=
|
|
( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1
|
|
cun sseqtr4i sseldi expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom
|
|
wi impancom jaodan sylan2b 0cnd snssd unssd syl5eqss addid1d simpr jaod
|
|
oveq2d syl5bi impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZIV
|
|
OCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVSAV
|
|
RVLVKAVRVLNNBCVJBVQCBVMUBGUDZHUEUFAVLVSVKAVLNZVKVSLEJKZCIWCWDECWCEABREF
|
|
SUGABCEBCUHAWBUISTVSVJWDCVSDLEJDLUJUKULUMUOUPUQVPVKVNDLJKZCIVPWEDCVPDAC
|
|
RDACVQRGABVMRFALRAURUSUTVASVBAVHVCTVNVJWECVNELDJELUJVEULUMVDVFVG $.
|
|
$}
|
|
|
|
un0mulcl.3 $e |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S ) $.
|
|
$( If ` S ` is closed under multiplication, then so is ` S u. { 0 } ` .
|
|
(Contributed by Mario Carneiro, 17-Jul-2014.) $)
|
|
un0mulcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T ) $=
|
|
( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtr4i cc sselda csn wi expr
|
|
cun ssun1 sseldi mul02d wss ssun2 c0ex mpbir syl6eqel elsni oveq1d eleq1d
|
|
snss syl5ibrcom impancom jaodan sylan2b 0cnd snssd syl5eqss mul01d oveq2d
|
|
unssd jaod syl5bi impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOUDZ
|
|
IVQCVSEGOEBVOPQVRVNVMVPVJADBIZDVOIZMZVNVMUBZVJDVSIWBCVSDGODBVOPQAVTWCWAAV
|
|
TVNVMAVTVNNNBCVLBVSCBVOUEGRHUFUCAVNWAVMAVNNZVMWALEJKZCIWDWELCWDEABSEFTUGL
|
|
CIVOCUHVOVSCVOBUIGRLCUJUPUKZULWAVLWECWADLEJDLUMUNUOUQURUSUTVRVMVPDLJKZCIV
|
|
RWGLCVRDACSDACVSSGABVOSFALSAVAVBVFVCTVDWFULVPVLWGCVPELDJELUMVEUOUQVGVHVI
|
|
$.
|
|
$}
|
|
|
|
$( Closure of addition of nonnegative integers. (Contributed by Raph Levien,
|
|
10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) $)
|
|
nn0addcl $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 ) $=
|
|
( cn cc wss cn0 wcel wa caddc co nnsscn df-n0 nnaddcl adantl un0addcl mpan
|
|
id ) CDEZAFGBFGHABIJZFGKRCFABRQLACGBCGHSCGRABMNOP $.
|
|
|
|
$( Closure of multiplication of nonnegative integers. (Contributed by NM,
|
|
22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) $)
|
|
nn0mulcl $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 ) $=
|
|
( cn cc wss cn0 wcel wa cmul co nnsscn df-n0 nnmulcl adantl un0mulcl mpan
|
|
id ) CDEZAFGBFGHABIJZFGKRCFABRQLACGBCGHSCGRABMNOP $.
|
|
|
|
${
|
|
nn0addcl.1 $e |- M e. NN0 $.
|
|
nn0addcl.2 $e |- N e. NN0 $.
|
|
$( Closure of addition of nonnegative integers, inference form.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
nn0addcli $p |- ( M + N ) e. NN0 $=
|
|
( cn0 wcel caddc co nn0addcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
|
|
$( Closure of multiplication of nonnegative integers, inference form.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
nn0mulcli $p |- ( M x. N ) e. NN0 $=
|
|
( cn0 wcel cmul co nn0mulcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
$}
|
|
|
|
$( A nonnegative integer plus 1 is a positive integer. (Contributed by Raph
|
|
Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) $)
|
|
nn0p1nn $p |- ( N e. NN0 -> ( N + 1 ) e. NN ) $=
|
|
( cn0 wcel c1 cn caddc co 1nn nn0nnaddcl mpan2 ) ABCDECADFGECHADIJ $.
|
|
|
|
$( Second Peano postulate for nonnegative integers. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
peano2nn0 $p |- ( N e. NN0 -> ( N + 1 ) e. NN0 ) $=
|
|
( cn0 wcel c1 caddc co 1nn0 nn0addcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( A positive integer minus 1 is a nonnegative integer. (Contributed by
|
|
Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro,
|
|
16-May-2014.) $)
|
|
nnm1nn0 $p |- ( N e. NN -> ( N - 1 ) e. NN0 ) $=
|
|
( cn wcel c1 cmin co cc0 wceq wo cn0 nn1m1nn oveq1 syl6eq orim1i syl orcomd
|
|
1m1e0 elnn0 sylibr ) ABCZADEFZBCZUAGHZIUAJCTUCUBTADHZUBIUCUBIAKUDUCUBUDUADD
|
|
EFGADDELQMNOPUARS $.
|
|
|
|
$( The nonnegative integer property expressed in terms of positive integers.
|
|
(Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro,
|
|
16-May-2014.) $)
|
|
elnn0nn $p |- ( N e. NN0 <-> ( N e. CC /\ ( N + 1 ) e. NN ) ) $=
|
|
( cn0 wcel cc c1 caddc co cn nn0cn nn0p1nn jca cmin wceq simpl ax-1cn pncan
|
|
wa sylancl nnm1nn0 adantl eqeltrrd impbii ) ABCZADCZAEFGZHCZQZUCUDUFAIAJKUG
|
|
UEELGZABUGUDEDCUHAMUDUFNOAEPRUFUHBCUDUESTUAUB $.
|
|
|
|
$( The positive integer property expressed in terms of nonnegative integers.
|
|
(Contributed by NM, 10-May-2004.) $)
|
|
elnnnn0 $p |- ( N e. NN <-> ( N e. CC /\ ( N - 1 ) e. NN0 ) ) $=
|
|
( cn wcel cc c1 cmin co cn0 nncn caddc wa npcan1 eleq1d peano2cnm biantrurd
|
|
bitr3d elnn0nn syl6bbr biadan2 ) ABCZADCZAEFGZHCZAIUATUBDCZUBEJGZBCZKZUCUAU
|
|
FTUGUAUEABALMUAUDUFANOPUBQRS $.
|
|
|
|
$( The positive integer property expressed in terms of nonnegative integers.
|
|
(Contributed by NM, 1-Sep-2005.) $)
|
|
elnnnn0b $p |- ( N e. NN <-> ( N e. NN0 /\ 0 < N ) ) $=
|
|
( cn wcel cn0 cc0 clt wbr wa nnnn0 nngt0 jca wceq wo elnn0 ax-1 breq2 ltnri
|
|
wi 0re pm2.21i syl6bi jaoi sylbi imp impbii ) ABCZADCZEAFGZHUFUGUHAIAJKUGUH
|
|
UFUGUFAELZMUHUFRZANUFUJUIUFUHOUIUHEEFGZUFAEEFPUKUFESQTUAUBUCUDUE $.
|
|
|
|
$( The positive integer property expressed in terms of nonnegative integers.
|
|
(Contributed by NM, 10-Jan-2006.) $)
|
|
elnnnn0c $p |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) ) $=
|
|
( cn wcel cn0 c1 cle wbr wa nnnn0 nnge1 jca cc0 clt 0lt1 cr wi nn0re ltletr
|
|
0re 1re mp3an12 syl mpani imdistani elnnnn0b sylibr impbii ) ABCZADCZEAFGZH
|
|
ZUHUIUJAIAJKUKUILAMGZHUHUIUJULUILEMGZUJULNUIAOCZUMUJHULPZAQLOCEOCUNUOSTLEAR
|
|
UAUBUCUDAUEUFUG $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative integer.
|
|
(Contributed by NM, 10-Mar-2005.) $)
|
|
nn0addge1 $p |- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( A + N ) ) $=
|
|
( cn0 wcel cr cc0 cle wbr wa caddc co nn0re nn0ge0 jca addge01 3expb sylan2
|
|
biimp3a ) BCDZAEDZBEDZFBGHZIAABJKGHZSUAUBBLBMNTUAUBUCTUAUBUCABORPQ $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative integer.
|
|
(Contributed by NM, 10-Mar-2005.) $)
|
|
nn0addge2 $p |- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( N + A ) ) $=
|
|
( cn0 wcel cr cc0 cle wbr wa caddc co nn0re nn0ge0 jca addge02 3expb sylan2
|
|
biimp3a ) BCDZAEDZBEDZFBGHZIABAJKGHZSUAUBBLBMNTUAUBUCTUAUBUCABORPQ $.
|
|
|
|
${
|
|
nn0addge1.1 $e |- A e. RR $.
|
|
nn0addge1.2 $e |- N e. NN0 $.
|
|
$( A number is less than or equal to itself plus a nonnegative integer.
|
|
(Contributed by NM, 10-Mar-2005.) $)
|
|
nn0addge1i $p |- A <_ ( A + N ) $=
|
|
( cr wcel cn0 caddc co cle wbr nn0addge1 mp2an ) AEFBGFAABHIJKCDABLM $.
|
|
|
|
$( A number is less than or equal to itself plus a nonnegative integer.
|
|
(Contributed by NM, 10-Mar-2005.) $)
|
|
nn0addge2i $p |- A <_ ( N + A ) $=
|
|
( cr wcel cn0 caddc co cle wbr nn0addge2 mp2an ) AEFBGFABAHIJKCDABLM $.
|
|
$}
|
|
|
|
${
|
|
nn0le2x.1 $e |- N e. NN0 $.
|
|
$( A nonnegative integer is less than or equal to twice itself.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
nn0le2xi $p |- N <_ ( 2 x. N ) $=
|
|
( caddc co c2 cmul cle nn0rei nn0addge1i nn0cni 2timesi breqtrri ) AAACDE
|
|
AFDGAAABHBIAABJKL $.
|
|
$}
|
|
|
|
${
|
|
nn0lele2x.1 $e |- M e. NN0 $.
|
|
nn0lele2x.2 $e |- N e. NN0 $.
|
|
$( 'Less than or equal to' implies 'less than or equal to twice' for
|
|
nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
nn0lele2xi $p |- ( N <_ M -> N <_ ( 2 x. M ) ) $=
|
|
( cle wbr c2 cmul co nn0le2xi nn0rei 2re remulcli letri mpan2 ) BAEFAGAHI
|
|
ZEFBPEFACJBAPBDKACKZGALQMNO $.
|
|
$}
|
|
|
|
$( Two ways to write the support of a function on ` NN0 ` . (Contributed by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
nn0supp $p |- ( F : I --> NN0 ->
|
|
( `' F " ( _V \ { 0 } ) ) = ( `' F " NN ) ) $=
|
|
( cn0 wf ccnv cn cima cvv cc0 csn cdif dfn2 invdif eqtr4i imaeq2i wfun wceq
|
|
cin ffun inpreima syl wss cdm cnvimass fimacnv syl5sseq sseqin2 sylib eqtrd
|
|
fdm eqtr4d syl5req ) BCADZAEZFGUNCHIJZKZRZGZUNUPGZFUQUNFCUOKUQLCUOMNOUMURUN
|
|
CGZUSRZUSUMAPURVAQBCASCUPATUAUMUSUTUBVAUSQUMAUCZUSUTAUPUDUMVBBUTBCAUJBCAUEU
|
|
KUFUSUTUGUHUIUL $.
|
|
|
|
${
|
|
nnnn0d.1 $e |- ( ph -> A e. NN ) $.
|
|
$( A positive integer is a nonnegative integer. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
nnnn0d $p |- ( ph -> A e. NN0 ) $=
|
|
( cn cn0 nnssnn0 sseldi ) ADEBFCG $.
|
|
$}
|
|
|
|
${
|
|
nn0red.1 $e |- ( ph -> A e. NN0 ) $.
|
|
$( A nonnegative integer is a real number. (Contributed by Mario Carneiro,
|
|
27-May-2016.) $)
|
|
nn0red $p |- ( ph -> A e. RR ) $=
|
|
( cn0 cr nn0ssre sseldi ) ADEBFCG $.
|
|
|
|
$( A nonnegative integer is a complex number. (Contributed by Mario
|
|
Carneiro, 27-May-2016.) $)
|
|
nn0cnd $p |- ( ph -> A e. CC ) $=
|
|
( nn0red recnd ) ABABCDE $.
|
|
|
|
$( A nonnegative integer is greater than or equal to zero. (Contributed by
|
|
Mario Carneiro, 27-May-2016.) $)
|
|
nn0ge0d $p |- ( ph -> 0 <_ A ) $=
|
|
( cn0 wcel cc0 cle wbr nn0ge0 syl ) ABDEFBGHCBIJ $.
|
|
|
|
nn0addcld.2 $e |- ( ph -> B e. NN0 ) $.
|
|
$( Closure of addition of nonnegative integers, inference form.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
nn0addcld $p |- ( ph -> ( A + B ) e. NN0 ) $=
|
|
( cn0 wcel caddc co nn0addcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure of multiplication of nonnegative integers, inference form.
|
|
(Contributed by Mario Carneiro, 27-May-2016.) $)
|
|
nn0mulcld $p |- ( ph -> ( A x. B ) e. NN0 ) $=
|
|
( cn0 wcel cmul co nn0mulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
$( Closure law for addition of reals, restricted to nonnegative integers.
|
|
(Contributed by Alexander van der Vekens, 6-Apr-2018.) $)
|
|
nn0readdcl $p |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) e. RR ) $=
|
|
( cn0 wcel wa caddc co nn0addcl nn0red ) ACDBCDEABFGABHI $.
|
|
|
|
$( If a nonnegative integer is greater than or equal to two, the integer
|
|
decreased by 1 is a positive integer. (Contributed by Alexander van der
|
|
Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) $)
|
|
nn0ge2m1nn $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN ) $=
|
|
( cn0 wcel c2 cle wbr wa c1 cmin co cn wo wceq simpl w3a clt 1red 2re 1lt2
|
|
cr a1i nn0re 3jca adantr simpr jctil ltleletr elnnnn0c sylanbrc nn1m1nn syl
|
|
sylc wi wn 1re lenlti biimpi breq2 mtbiri pm2.21d com12 adantl orim1d oridm
|
|
mt2 mpd sylib ) ABCZDAEFZGZAHIJKCZVKLZVKVJAHMZVKLZVLVJAKCZVNVJVHHAEFZVOVHVI
|
|
NVJHTCZDTCZATCZOZHDPFZVIGVPVHVTVIVHVQVRVSVHQVRVHRUAAUBUCUDVJVIWAVHVIUESUFHD
|
|
AUGULAUHUIAUJUKVJVMVKVKVIVMVKUMVHVMVIVKVMVIVKVMVIDHEFZWBWASWBWAUNDHRUOUPUQV
|
|
EAHDEURUSUTVAVBVCVFVKVDVG $.
|
|
|
|
$( If a nonnegative integer is greater than or equal to two, the integer
|
|
decreased by 1 is also a nonnegative integer. (Contributed by Alexander
|
|
van der Vekens, 1-Aug-2018.) $)
|
|
nn0ge2m1nn0 $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN0 ) $=
|
|
( cn0 wcel c2 cle wbr wa c1 cmin co nn0ge2m1nn nnnn0d ) ABCDAEFGAHIJAKL $.
|
|
|
|
$( Closure law for dividing of a nonnegative integer by a positive integer.
|
|
(Contributed by Alexander van der Vekens, 14-Apr-2018.) $)
|
|
nn0nndivcl $p |- ( ( K e. NN0 /\ L e. NN ) -> ( K / L ) e. RR ) $=
|
|
( cn0 wcel cn wa cr nn0re adantr simpr nndivred ) ACDZBEDZFABLAGDMAHILMJK
|
|
$.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Extended nonnegative integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The function values of the hash (set size) function are either nonnegative
|
|
integers or positive infinity. To avoid the need to distinguish between
|
|
finite and infinite sets (and therefore if the set size is a nonnegative
|
|
integer or positive infinity), it is useful to provide a definition of the
|
|
set of nonnegative integers extended by positive infinity, analogously to
|
|
the extension of the real numbers ` RR* ` , see ~ df-xr .
|
|
|
|
$)
|
|
|
|
$c NN0* $. $( The set of extended nonnegative integers $)
|
|
|
|
$( The set of extended nonnegative integers. $)
|
|
cxnn0 $a class NN0* $.
|
|
|
|
$( Define the set of extended nonnegative integers that includes positive
|
|
infinity. Analogue of the extension of the real numbers ` RR* ` , see
|
|
~ df-xr . (Contributed by AV, 10-Dec-2020.) $)
|
|
df-xnn0 $a |- NN0* = ( NN0 u. { +oo } ) $.
|
|
|
|
$( An extended nonnegative integer is either a standard nonnegative integer
|
|
or positive infinity. (Contributed by AV, 10-Dec-2020.) $)
|
|
elxnn0 $p |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) ) $=
|
|
( cxnn0 wcel cn0 cpnf csn cun wceq df-xnn0 eleq2i pnfex elsn2 orbi2i 3bitri
|
|
wo elun ) ABCADEFZGZCADCZAQCZOSAEHZOBRAIJADQPTUASAEKLMN $.
|
|
|
|
$( The standard nonnegative integers are a subset of the extended nonnegative
|
|
integers. (Contributed by AV, 10-Dec-2020.) $)
|
|
nn0ssxnn0 $p |- NN0 C_ NN0* $=
|
|
( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtr4i ) AABCZDEAIFGH $.
|
|
|
|
$( A standard nonnegative integer is an extended nonnegative integer.
|
|
(Contributed by AV, 10-Dec-2020.) $)
|
|
nn0xnn0 $p |- ( A e. NN0 -> A e. NN0* ) $=
|
|
( cn0 cxnn0 nn0ssxnn0 sseli ) BCADE $.
|
|
|
|
$( An extended nonnegative integer is an extended real. (Contributed by AV,
|
|
10-Dec-2020.) $)
|
|
xnn0xr $p |- ( A e. NN0* -> A e. RR* ) $=
|
|
( cxnn0 wcel cn0 cpnf wceq wo cxr elxnn0 nn0re rexrd pnfxr eleq1 jaoi sylbi
|
|
mpbiri ) ABCADCZAEFZGAHCZAIQSRQAAJKRSEHCLAEHMPNO $.
|
|
|
|
$( Zero is an extended nonnegative integer. (Contributed by AV,
|
|
10-Dec-2020.) $)
|
|
0xnn0 $p |- 0 e. NN0* $=
|
|
( cn0 cxnn0 cc0 nn0ssxnn0 0nn0 sselii ) ABCDEF $.
|
|
|
|
$( Positive infinity is an extended nonnegative integer. (Contributed by AV,
|
|
10-Dec-2020.) $)
|
|
pnf0xnn0 $p |- +oo e. NN0* $=
|
|
( cpnf cxnn0 wcel cn0 wceq wo eqid olci elxnn0 mpbir ) ABCADCZAAEZFLKAGHAIJ
|
|
$.
|
|
|
|
$( No standard nonnegative integer equals positive infinity. (Contributed by
|
|
AV, 10-Dec-2020.) $)
|
|
nn0nepnf $p |- ( A e. NN0 -> A =/= +oo ) $=
|
|
( cn0 wcel cpnf wceq cr pnfnre neli nn0re mto eleq1 mtbiri necon2ai ) ABCZA
|
|
DADENDBCZODFCDFGHDIJADBKLM $.
|
|
|
|
${
|
|
nn0xnn0d.1 $e |- ( ph -> A e. NN0 ) $.
|
|
$( A standard nonnegative integer is an extended nonnegative integer,
|
|
deduction form. (Contributed by AV, 10-Dec-2020.) $)
|
|
nn0xnn0d $p |- ( ph -> A e. NN0* ) $=
|
|
( cn0 cxnn0 nn0ssxnn0 sseldi ) ADEBFCG $.
|
|
|
|
$( No standard nonnegative integer equals positive infinity, deduction
|
|
form. (Contributed by AV, 10-Dec-2020.) $)
|
|
nn0nepnfd $p |- ( ph -> A =/= +oo ) $=
|
|
( cn0 wcel cpnf wne nn0nepnf syl ) ABDEBFGCBHI $.
|
|
$}
|
|
|
|
$( No extended nonnegative integer equals negative infinity. (Contributed by
|
|
AV, 10-Dec-2020.) $)
|
|
xnn0nemnf $p |- ( A e. NN0* -> A =/= -oo ) $=
|
|
( cxnn0 wcel cn0 cpnf wceq wo cmnf wne elxnn0 nn0re renemnfd pnfnemnf neeq1
|
|
mpbiri jaoi sylbi ) ABCADCZAEFZGAHIZAJRTSRAAKLSTEHIMAEHNOPQ $.
|
|
|
|
$( The extended nonnegative integers are extended reals without negative
|
|
infinity. (Contributed by AV, 10-Dec-2020.) $)
|
|
xnn0xrnemnf $p |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) ) $=
|
|
( cxnn0 wcel cxr cmnf wne xnn0xr xnn0nemnf jca ) ABCADCAEFAGAHI $.
|
|
|
|
$( An extended nonnegative integer which is not a standard nonnegative
|
|
integer is positive infinity. (Contributed by AV, 10-Dec-2020.) $)
|
|
xnn0nnn0pnf $p |- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo ) $=
|
|
( cxnn0 wcel cn0 wn cpnf wceq wo wi elxnn0 pm2.53 sylbi imp ) ABCZADCZEZAFG
|
|
ZNOQHPQIAJOQKLM $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Integers (as a subset of complex numbers)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ZZ $. $( The set of integers (blackboard bold Z). $)
|
|
|
|
$( Extend class notation to include the class of integers. $)
|
|
cz $a class ZZ $.
|
|
|
|
$( Define the set of integers, which are the positive and negative integers
|
|
together with zero. Definition of integers in [Apostol] p. 22. The
|
|
letter Z abbreviates the German word Zahlen meaning "numbers."
|
|
(Contributed by NM, 8-Jan-2002.) $)
|
|
df-z $a |- ZZ = { n e. RR | ( n = 0 \/ n e. NN \/ -u n e. NN ) } $.
|
|
|
|
${
|
|
$d x N $.
|
|
$( Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) $)
|
|
elz $p |- ( N e. ZZ <->
|
|
( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) ) $=
|
|
( vx cv cc0 wceq cn wcel cneg w3o cr cz eqeq1 eleq1 eleq1d 3orbi123d df-z
|
|
negeq elrab2 ) BCZDEZSFGZSHZFGZIADEZAFGZAHZFGZIBAJKSAEZTUDUAUEUCUGSADLSAF
|
|
MUHUBUFFSAQNOBPR $.
|
|
$}
|
|
|
|
$( The negative of a positive integer is an integer. (Contributed by NM,
|
|
12-Jan-2002.) $)
|
|
nnnegz $p |- ( N e. NN -> -u N e. ZZ ) $=
|
|
( cn wcel cneg cr cc0 wceq w3o cz nnre renegcld cc nncn negneg eleq1d mpcom
|
|
biimprd 3mix3d elz sylanbrc ) ABCZADZECUBFGZUBBCZUBDZBCZHUBICUAAAJKUAUFUCUD
|
|
ALCZUAUFAMUGUFUAUGUEABANOQPRUBST $.
|
|
|
|
$( An integer is a real. (Contributed by NM, 8-Jan-2002.) $)
|
|
zre $p |- ( N e. ZZ -> N e. RR ) $=
|
|
( cz wcel cr cc0 wceq cn cneg w3o elz simplbi ) ABCADCAEFAGCAHGCIAJK $.
|
|
|
|
$( An integer is a complex number. (Contributed by NM, 9-May-2004.) $)
|
|
zcn $p |- ( N e. ZZ -> N e. CC ) $=
|
|
( cz wcel zre recnd ) ABCAADE $.
|
|
|
|
${
|
|
zre.1 $e |- A e. ZZ $.
|
|
$( An integer is a real number. (Contributed by NM, 14-Jul-2005.) $)
|
|
zrei $p |- A e. RR $=
|
|
( cz wcel cr zre ax-mp ) ACDAEDBAFG $.
|
|
$}
|
|
|
|
$( The integers are a subset of the reals. (Contributed by NM,
|
|
2-Aug-2004.) $)
|
|
zssre $p |- ZZ C_ RR $=
|
|
( vx cz cr cv zre ssriv ) ABCADEF $.
|
|
|
|
$( The integers are a subset of the complex numbers. (Contributed by NM,
|
|
2-Aug-2004.) $)
|
|
zsscn $p |- ZZ C_ CC $=
|
|
( vx cz cc cv zcn ssriv ) ABCADEF $.
|
|
|
|
$( The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised
|
|
by Mario Carneiro, 17-Nov-2014.) $)
|
|
zex $p |- ZZ e. _V $=
|
|
( cz cc cnex zsscn ssexi ) ABCDE $.
|
|
|
|
$( Positive integer property expressed in terms of integers. (Contributed by
|
|
NM, 8-Jan-2002.) $)
|
|
elnnz $p |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) ) $=
|
|
( cn wcel cr cneg cc0 wceq wo wa clt wbr cz orc nngt0 jca31 wi wn w3o bitri
|
|
nnre idd lt0neg2 renegcl 0re ltnsym sylancl sylbid nsyl gt0ne0 neneqd ioran
|
|
imp sylanbrc pm2.21d jaod ex com23 imp31 impbii 3orrot 3orass anbi2i anbi1i
|
|
elz bitr4i ) ABCZADCZVFAEZBCZAFGZHZHZIZFAJKZIZALCZVNIVFVOVFVGVLVNATVFVKMANO
|
|
VGVLVNVFVGVNVLVFVGVNVLVFPVGVNIZVFVFVKVQVFUAVQVKVFVQVIQVJQVKQVQFVHJKZVIVGVNV
|
|
RQZVGVNVHFJKZVSAUBVGVHDCFDCVTVSPAUCUDVHFUEUFUGULVHNUHVQAFAUIUJVIVJUKUMUNUOU
|
|
PUQURUSVPVMVNVPVGVJVFVIRZIVMAVDWAVLVGWAVFVIVJRVLVJVFVIUTVFVIVJVASVBSVCVE $.
|
|
|
|
$( Zero is an integer. (Contributed by NM, 12-Jan-2002.) $)
|
|
0z $p |- 0 e. ZZ $=
|
|
( cc0 cz wcel cr wceq cn cneg w3o 0re eqid 3mix1i elz mpbir2an ) ABCADCAAEZ
|
|
AFCZAGFCZHINOPAJKALM $.
|
|
|
|
$( Zero is an integer, deductive form (common case). (Contributed by David
|
|
A. Wheeler, 8-Dec-2018.) $)
|
|
0zd $p |- ( ph -> 0 e. ZZ ) $=
|
|
( cc0 cz wcel 0z a1i ) BCDAEF $.
|
|
|
|
$( Nonnegative integer property expressed in terms of integers. (Contributed
|
|
by NM, 9-May-2004.) $)
|
|
elnn0z $p |- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) ) $=
|
|
( cn0 wcel cz cc0 cle wbr wa cr wceq cn cneg w3o nn0re elnn0 adantr a1i clt
|
|
wo wi biimpi orcomd 3mix1 3mix2 jaoi syl elz sylanbrc nn0ge0 jca 0nn0 eleq1
|
|
simprbi mpbiri nnnn0 wn simpr 0red lenltd mpbid nngt0 lt0neg1d syl5ibr mtod
|
|
zre pm2.21d 3jaod mpd impbii ) ABCZADCZEAFGZHZVJVKVLVJAICZAEJZAKCZALZKCZMZV
|
|
KANVJVOVPSVSVJVPVOVJVPVOSAOUAUBVOVSVPVOVPVRUCVPVOVRUDUEUFAUGZUHAUIUJVMVSVJV
|
|
KVSVLVKVNVSVTUMPVMVOVJVPVRVOVJTVMVOVJEBCUKAEBULUNQVPVJTVMAUOQVMVRVJVMVRAERG
|
|
ZVMVLWAUPVKVLUQVMEAVMURVKVNVLAVEPZUSUTVRWAVMEVQRGVQVAVMAWBVBVCVDVFVGVHVI $.
|
|
|
|
$( Integer property expressed in terms nonnegative integers and positive
|
|
integers. (Contributed by NM, 10-May-2004.) $)
|
|
elznn0nn $p |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) $=
|
|
( cz wcel cr cc0 wceq cn cneg w3o wa cn0 wo elz andi df-3or anbi2i pm4.71ri
|
|
nn0re elnn0 bitri orcom orbi1i 3bitr4i ) ABCADCZAEFZAGCZAHGCZIZJZAKCZUDUGJZ
|
|
LZAMUDUEUFLZUGLZJUDUMJZUKLUIULUDUMUGNUHUNUDUEUFUGOPUJUOUKUJUDUJJUOUJUDARQUJ
|
|
UMUDUJUFUELUMASUFUEUATPTUBUCT $.
|
|
|
|
$( Integer property expressed in terms of nonnegative integers. (Contributed
|
|
by NM, 9-May-2004.) $)
|
|
elznn0 $p |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) ) $=
|
|
( cz wcel cr cc0 wceq cn cneg w3o wa cn0 wo elz wb elnn0 a1i recn 0cn bitri
|
|
cc negcon1 sylancl eqeq1i eqcom syl6bb orbi2d syl5bb orbi12d 3orass orordir
|
|
neg0 orcom 3bitrri syl6rbb pm5.32i ) ABCADCZAEFZAGCZAHZGCZIZJUPAKCZUSKCZLZJ
|
|
AMUPVAVDUPVDURUQLZUTUQLZLZVAUPVBVEVCVFVBVENUPAOPVCUTUSEFZLUPVFUSOUPVHUQUTUP
|
|
VHEHZAFZUQUPATCETCVHVJNAQRAEUAUBVJEAFUQVIEAUKUCEAUDSUEUFUGUHVAUQURUTLZLVKUQ
|
|
LVGUQURUTUIUQVKULURUTUQUJUMUNUOS $.
|
|
|
|
$( Integer property expressed in terms of positive integers and nonnegative
|
|
integers. (Contributed by NM, 12-Jul-2005.) $)
|
|
elznn $p |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) ) $=
|
|
( cz wcel cr cc0 wceq cn cneg w3o wa cn0 wo elz recn negeq0d elnn0 syl6rbbr
|
|
orbi2d 3orrot bitri 3orass pm5.32i ) ABCADCZAEFZAGCZAHZGCZIZJUCUEUFKCZLZJAM
|
|
UCUHUJUCUJUEUGUDLZLZUHUCUIUKUEUCUKUGUFEFZLUIUCUDUMUGUCAANORUFPQRUHUEUGUDIUL
|
|
UDUEUGSUEUGUDUATQUBT $.
|
|
|
|
$( Positive integers are a subset of integers. (Contributed by NM,
|
|
9-Jan-2002.) $)
|
|
nnssz $p |- NN C_ ZZ $=
|
|
( vx cn cz cv wcel cc0 clt wbr elnnz simplbi ssriv ) ABCADZBELCEFLGHLIJK $.
|
|
|
|
$( Nonnegative integers are a subset of the integers. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
nn0ssz $p |- NN0 C_ ZZ $=
|
|
( cn0 cn cc0 csn cun cz df-n0 nnssz wcel wss c0ex snss mpbi unssi eqsstri
|
|
0z ) ABCDZEFGBQFHCFIQFJPCFKLMNO $.
|
|
|
|
$( A positive integer is an integer. (Contributed by NM, 9-May-2004.) $)
|
|
nnz $p |- ( N e. NN -> N e. ZZ ) $=
|
|
( cn cz nnssz sseli ) BCADE $.
|
|
|
|
$( A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) $)
|
|
nn0z $p |- ( N e. NN0 -> N e. ZZ ) $=
|
|
( cn0 cz nn0ssz sseli ) BCADE $.
|
|
|
|
${
|
|
nnzi.1 $e |- N e. NN $.
|
|
$( A positive integer is an integer. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
nnzi $p |- N e. ZZ $=
|
|
( cn cz nnssz sselii ) CDAEBF $.
|
|
$}
|
|
|
|
${
|
|
nn0zi.1 $e |- N e. NN0 $.
|
|
$( A nonnegative integer is an integer. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
nn0zi $p |- N e. ZZ $=
|
|
( cn0 cz nn0ssz sselii ) CDAEBF $.
|
|
$}
|
|
|
|
$( Positive integer property expressed in terms of integers. (Contributed by
|
|
NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) $)
|
|
elnnz1 $p |- ( N e. NN <-> ( N e. ZZ /\ 1 <_ N ) ) $=
|
|
( cn wcel cz c1 cle wbr wa nnz nnge1 jca cc0 clt 0lt1 cr zre 0re 1re ltletr
|
|
wi mp3an12 syl mpani imdistani elnnz sylibr impbii ) ABCZADCZEAFGZHZUHUIUJA
|
|
IAJKUKUILAMGZHUHUIUJULUILEMGZUJULNUIAOCZUMUJHULTZAPLOCEOCUNUOQRLEASUAUBUCUD
|
|
AUEUFUG $.
|
|
|
|
$( Positive integers expressed as a subset of integers. (Contributed by NM,
|
|
3-Oct-2004.) $)
|
|
nnzrab $p |- NN = { x e. ZZ | 1 <_ x } $=
|
|
( cn cv cz wcel c1 cle wbr wa cab crab elnnz1 abbi2i df-rab eqtr4i ) BACZDE
|
|
FPGHZIZAJQADKRABPLMQADNO $.
|
|
|
|
$( Nonnegative integers expressed as a subset of integers. (Contributed by
|
|
NM, 3-Oct-2004.) $)
|
|
nn0zrab $p |- NN0 = { x e. ZZ | 0 <_ x } $=
|
|
( cn0 cv cz wcel cc0 cle wbr wa cab crab elnn0z abbi2i df-rab eqtr4i ) BACZ
|
|
DEFPGHZIZAJQADKRABPLMQADNO $.
|
|
|
|
$( One is an integer. (Contributed by NM, 10-May-2004.) $)
|
|
1z $p |- 1 e. ZZ $=
|
|
( c1 1nn nnzi ) ABC $.
|
|
|
|
$( 1 is an integer, deductive form (common case). (Contributed by David A.
|
|
Wheeler, 6-Dec-2018.) $)
|
|
1zzd $p |- ( ph -> 1 e. ZZ ) $=
|
|
( c1 cz wcel 1z a1i ) BCDAEF $.
|
|
|
|
$( Two is an integer. (Contributed by NM, 10-May-2004.) $)
|
|
2z $p |- 2 e. ZZ $=
|
|
( c2 2nn nnzi ) ABC $.
|
|
|
|
$( 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
3z $p |- 3 e. ZZ $=
|
|
( c3 3nn nnzi ) ABC $.
|
|
|
|
$( 4 is an integer. (Contributed by BJ, 26-Mar-2020.) $)
|
|
4z $p |- 4 e. ZZ $=
|
|
( c4 4nn nnzi ) ABC $.
|
|
|
|
$( Closure law for negative integers. (Contributed by NM, 9-May-2004.) $)
|
|
znegcl $p |- ( N e. ZZ -> -u N e. ZZ ) $=
|
|
( cz wcel cr cc0 wceq cn cneg w3o elz negeq neg0 syl6eq syl6eqel nnnegz nnz
|
|
wa 0z 3jaoi adantl sylbi ) ABCADCZAEFZAGCZAHZGCZIZQUEBCZAJUGUHUBUCUHUDUFUCU
|
|
EEBUCUEEHEAEKLMRNAOUEPSTUA $.
|
|
|
|
$( -1 is an integer (common case). (Contributed by David A. Wheeler,
|
|
5-Dec-2018.) $)
|
|
neg1z $p |- -u 1 e. ZZ $=
|
|
( c1 cn wcel cneg cz 1nn nnnegz ax-mp ) ABCADECFAGH $.
|
|
|
|
$( A number is an integer iff its negative is. (Contributed by Stefan
|
|
O'Rear, 13-Sep-2014.) $)
|
|
znegclb $p |- ( A e. CC -> ( A e. ZZ <-> -u A e. ZZ ) ) $=
|
|
( cc wcel cz cneg znegcl negneg eleq1d syl5ib impbid2 ) ABCZADCZAEZDCZAFNME
|
|
ZDCKLMFKOADAGHIJ $.
|
|
|
|
$( The negative of a nonnegative integer is an integer. (Contributed by NM,
|
|
9-May-2004.) $)
|
|
nn0negz $p |- ( N e. NN0 -> -u N e. ZZ ) $=
|
|
( cn0 wcel cz cneg nn0z znegcl syl ) ABCADCAEDCAFAGH $.
|
|
|
|
${
|
|
nn0negzi.1 $e |- N e. NN0 $.
|
|
$( The negative of a nonnegative integer is an integer. (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) $)
|
|
nn0negzi $p |- -u N e. ZZ $=
|
|
( cn0 wcel cneg cz nn0negz ax-mp ) ACDAEFDBAGH $.
|
|
$}
|
|
|
|
$( Second Peano postulate generalized to integers. (Contributed by NM,
|
|
13-Feb-2005.) $)
|
|
peano2z $p |- ( N e. ZZ -> ( N + 1 ) e. ZZ ) $=
|
|
( cz wcel c1 caddc co cr cn0 cneg wo readdcld cn wa a1i cc recnd cc0 neg1cn
|
|
1red eqtrd elznn0nn biimpi biantrurd orbi2d mpbird wi peano2nn0 adantr 1cnd
|
|
renegcld negdid oveq1d negcld addassd ax-1cn 1pneg1e0 oveq2i syl6eq addid1d
|
|
zre addcomli simpr eqeltrd elnn0nn sylanbrc ex orim12d mpd elznn0 ) ABCZADE
|
|
FZGCVKHCZVKIZHCZJZVKBCVJADAUTZVJSKVJAHCZAIZLCZJZVOVJVTVQAGCZVSMZJZVJWCAUAUB
|
|
VJVSWBVQVJWAVSVPUCUDUEVJVQVLVSVNVQVLUFVJAUGNVJVSVNVJVSMZVMOCVMDEFZLCVNWDVMW
|
|
DVKWDADVJWAVSVPUHZWDSKUJPWDWEVRLWDWEVRQEFZVRWDWEVRDIZDEFZEFZWGWDWEVRWHEFZDE
|
|
FWJWDVMWKDEWDADWDAWFPZWDUIZUKULWDVRWHDWDAWLUMZWHOCWDRNWMUNTWIQVREDWHQUORUPV
|
|
AUQURWDVRWNUSTVJVSVBVCVMVDVEVFVGVHVKVIVE $.
|
|
|
|
${
|
|
$d M x y $. $d N x $.
|
|
$( Lemma for ~ zaddcl . Special case in which ` N ` is a positive integer.
|
|
(Contributed by Jim Kingdon, 14-Mar-2020.) $)
|
|
zaddcllempos $p |- ( ( M e. ZZ /\ N e. NN ) -> ( M + N ) e. ZZ ) $=
|
|
( vx vy cn wcel cz caddc co cv wi c1 wceq oveq2 eleq1d imbi2d peano2z zcn
|
|
wa cc adantl nncn adantr 1cnd addassd syl5ib ex a2d nnind impcom ) BEFAGF
|
|
ZABHIZGFZUKACJZHIZGFZKUKALHIZGFZKUKADJZHIZGFZKUKAUSLHIZHIZGFZKUKUMKCDBUNL
|
|
MZUPURUKVEUOUQGUNLAHNOPUNUSMZUPVAUKVFUOUTGUNUSAHNOPUNVBMZUPVDUKVGUOVCGUNV
|
|
BAHNOPUNBMZUPUMUKVHUOULGUNBAHNOPAQUSEFZUKVAVDVIUKVAVDKVAUTLHIZGFVIUKSZVDU
|
|
TQVKVJVCGVKAUSLUKATFVIARUAVIUSTFUKUSUBUCVKUDUEOUFUGUHUIUJ $.
|
|
$}
|
|
|
|
$( "Reverse" second Peano postulate for integers. (Contributed by NM,
|
|
12-Sep-2005.) $)
|
|
peano2zm $p |- ( N e. ZZ -> ( N - 1 ) e. ZZ ) $=
|
|
( cz wcel c1 cmin co cneg caddc zcn negsubdid znegcl peano2z syl eqeltrd cc
|
|
1cnd wb subcld znegclb mpbird ) ABCZADEFZBCZUBGZBCZUAUDAGZDHFZBUAADAIZUAPZJ
|
|
UAUFBCUGBCAKUFLMNUAUBOCUCUEQUAADUHUIRUBSMT $.
|
|
|
|
${
|
|
$d M x y $. $d N x y $.
|
|
$( Lemma for ~ zaddcl . Special case in which ` -u N ` is a positive
|
|
integer. (Contributed by Jim Kingdon, 14-Mar-2020.) $)
|
|
zaddcllemneg $p |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) ->
|
|
( M + N ) e. ZZ ) $=
|
|
( vx vy cz wcel cneg cn caddc co oveq2d wa cv wi wceq negeq eleq1d imbi2d
|
|
c1 cmin cr w3a simp2 recnd negnegd cc zcn adantr negsubd peano2zm eqeltrd
|
|
1cnd nncn ad2antrr negdi2d ad2antlr negcld addsubassd adantl eqeltrrd a2d
|
|
exp31 nnind impcom 3impa ) AEFZBUAFZBGZHFZUBZAVHGZIJZABIJEVJVKBAIVJBVJBVF
|
|
VGVIUCUDUEKVFVGVIVLEFZVIVFVGLZVMVNACMZGZIJZEFZNVNASGZIJZEFZNVNADMZGZIJZEF
|
|
ZNVNAWBSIJZGZIJZEFZNVNVMNCDVHVOSOZVRWAVNWJVQVTEWJVPVSAIVOSPKQRVOWBOZVRWEV
|
|
NWKVQWDEWKVPWCAIVOWBPKQRVOWFOZVRWIVNWLVQWHEWLVPWGAIVOWFPKQRVOVHOZVRVMVNWM
|
|
VQVLEWMVPVKAIVOVHPKQRVNVTASTJZEVNASVFAUFFZVGAUGUHZVNULUIVFWNEFVGAUJUHUKWB
|
|
HFZVNWEWIWQVNWEWIWQVNLZWELZWHAWCSTJZIJZEWSWGWTAIWSWBSWQWBUFFVNWEWBUMUNZWS
|
|
ULZUOKWSWDSTJZXAEWSAWCSVNWOWQWEWPUPWSWBXBUQXCURWEXDEFWRWDUJUSUTUKVBVAVCVD
|
|
VEUT $.
|
|
$}
|
|
|
|
$( Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof
|
|
shortened by Mario Carneiro, 16-May-2014.) $)
|
|
zaddcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ ) $=
|
|
( cz wcel wa cc0 wceq cn cneg w3o caddc co cr elz simprbi adantl zcn adantr
|
|
cc wi addid1d simpl eqeltrd eleq1d syl5ibrcom zaddcllempos zre zaddcllemneg
|
|
oveq2 ex 3expia sylan2 3jaod mpd ) ACDZBCDZEZBFGZBHDZBIHDZJZABKLZCDZUPVAUOU
|
|
PBMDZVABNOPUQURVCUSUTUQVCURAFKLZCDUQVEACUQAUOASDUPAQRUAUOUPUBUCURVBVECBFAKU
|
|
IUDUEUOUSVCTUPUOUSVCABUFUJRUPUOVDUTVCTBUGUOVDUTVCABUHUKULUMUN $.
|
|
|
|
$( Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) $)
|
|
zsubcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ ) $=
|
|
( cz wcel wa cneg caddc co cmin wceq zcn negsub syl2an znegcl zaddcl sylan2
|
|
cc eqeltrrd ) ACDZBCDZEABFZGHZABIHZCSAQDBQDUBUCJTAKBKABLMTSUACDUBCDBNAUAOPR
|
|
$.
|
|
|
|
$( Integer trichotomy (with zero). (Contributed by Jim Kingdon,
|
|
14-Mar-2020.) $)
|
|
ztri3or0 $p |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) ) $=
|
|
( cz wcel cc0 wceq clt wbr w3o cn cneg cr elz simprbi idd wi nngt0 lt0neg1d
|
|
a1i zre syl5ibr 3orim123d mpd 3orrot sylibr ) ABCZADEZDAFGZADFGZHZUHUFUGHUE
|
|
UFAICZAJZICZHZUIUEAKCUMALMUEUFUFUJUGULUHUEUFNUJUGOUEAPRULUHUEDUKFGUKPUEAASQ
|
|
TUAUBUHUFUGUCUD $.
|
|
|
|
$( Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.) $)
|
|
ztri3or $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) ) $=
|
|
( cz wcel wa clt wbr wceq w3o cmin cc0 zsubcl ztri3or0 syl cneg zre posdifd
|
|
co cr recnd adantr adantl resubcld lt0neg2d negsubdi2d breq1d 3bitrd bicomd
|
|
subeq0ad 3orbi123d mpbird ) ACDZBCDZEZABFGZABHZBAFGZIABJRZKFGZURKHZKURFGZIZ
|
|
UNURCDVBABLURMNUNUOUSUPUTUQVAUNUOKBAJRZFGVCOZKFGUSUNABULASDUMAPUAZUMBSDULBP
|
|
UBZQUNVCUNBAVFVEUCUDUNVDURKFUNBAUNBVFTZUNAVETZUEUFUGUNUTUPUNABVHVGUIUHUNBAV
|
|
FVEQUJUK $.
|
|
|
|
$( Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.) $)
|
|
zletric $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) ) $=
|
|
( cz wcel wa cr clt wbr wceq w3o cle zre anim12i ztri3or ltle orc syl6 eqle
|
|
wo wi ex adantr ancoms olc 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJA
|
|
BKHZBAKHZSZUGUIUHUJALBLMABNUKULUQUMUNUKULUOUQABOUOUPPZQUKUMUOUQUIUMUOTUJUIU
|
|
MUOABRUAUBURQUKUNUPUQUJUIUNUPTBAOUCUPUOUDQUEUF $.
|
|
|
|
$( Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.) $)
|
|
zlelttric $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) ) $=
|
|
( cz wcel wa cr clt wbr wceq w3o cle zre anim12i ztri3or ltle orc syl6 eqle
|
|
wo wi ex adantr olc a1i 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJABKH
|
|
ZUNSZUGUIUHUJALBLMABNUKULUPUMUNUKULUOUPABOUOUNPZQUKUMUOUPUIUMUOTUJUIUMUOABR
|
|
UAUBUQQUNUPTUKUNUOUCUDUEUF $.
|
|
|
|
$( 'Less than' expressed in terms of 'less than or equal to'. (Contributed
|
|
by Jim Kingdon, 14-Mar-2020.) $)
|
|
zltnle $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> -. B <_ A ) ) $=
|
|
( cz wcel wa clt wbr cle wn cr zre lenlt syl2anr biimpd con2d wceq w3o syl6
|
|
wb wi ztri3or ax-1 a1i eqcom eqle sylan2b ex adantl sylan2 pm2.24 3jaod mpd
|
|
ltle impbid ) ACDZBCDZEZABFGZBAHGZIZUQUSURUQUSURIZUPBJDZAJDZUSVASUOBKZAKZBA
|
|
LMNOUQURABPZBAFGZQUTURTZABUAUQURVHVFVGURVHTUQURUTUBUCUQVFUSVHUPUOVBVFUSTZVD
|
|
VBVIUOVBVFUSVFVBBAPUSABUDBAUEUFUGUHUIUSURUJZRUQVGUSVHUPVBVCVGUSTUOVDVEBAUMM
|
|
VJRUKULUN $.
|
|
|
|
$( Integer 'Less than or equal to' expressed in terms of 'less than' or
|
|
'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.) $)
|
|
zleloe $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
( A <_ B <-> ( A < B \/ A = B ) ) ) $=
|
|
( cz wcel wa cle wbr clt wo wn cr wb zre lenlt syl2an w3o ztri3or df-3or wi
|
|
wceq sylib orcomd ord sylbid ltle eqle ex adantr jaod impbid ) ACDZBCDZEZAB
|
|
FGZABHGZABTZIZUMUNBAHGZJZUQUKAKDZBKDZUNUSLULAMZBMZABNOUMURUQUMUQURUMUOUPURP
|
|
UQURIABQUOUPURRUAUBUCUDUKUTVAUQUNSULVBVCUTVAEUOUNUPABUEUTUPUNSVAUTUPUNABUFU
|
|
GUHUIOUJ $.
|
|
|
|
$( An integer is not a positive integer iff it is less than one.
|
|
(Contributed by NM, 13-Jul-2005.) $)
|
|
znnnlt1 $p |- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) ) $=
|
|
( cz wcel cn wn c1 cle wbr clt elnnz1 baib notbid wb 1z zltnle mpan2 bitr4d
|
|
) ABCZADCZEFAGHZEZAFIHZRSTSRTAJKLRFBCUBUAMNAFOPQ $.
|
|
|
|
$( Transitive law of ordering for integers. (Contributed by Alexander van
|
|
der Vekens, 3-Apr-2018.) $)
|
|
zletr $p |- ( ( J e. ZZ /\ K e. ZZ /\ L e. ZZ )
|
|
-> ( ( J <_ K /\ K <_ L ) -> J <_ L ) ) $=
|
|
( cz wcel cr cle wbr wa wi zre letr syl3an ) ADEAFEBDEBFECDECFEABGHBCGHIACG
|
|
HJAKBKCKABCLM $.
|
|
|
|
$( Reverse closure law for addition of integers. (Contributed by NM,
|
|
11-May-2004.) $)
|
|
zrevaddcl $p |- ( N e. ZZ ->
|
|
( ( M e. CC /\ ( M + N ) e. ZZ ) <-> M e. ZZ ) ) $=
|
|
( cz wcel cc caddc co wa cmin zcn pncan sylan2 ancoms adantr zsubcl adantlr
|
|
wceq eqeltrrd ex wi zaddcl expcom impbid pm5.32da pm4.71ri syl6bbr ) BCDZAE
|
|
DZABFGZCDZHUHACDZHUKUGUHUJUKUGUHHZUJUKULUJUKULUJHUIBIGZACULUMAQZUJUHUGUNUGU
|
|
HBEDUNBJABKLMNUGUJUMCDZUHUJUGUOUIBOMPRSUGUKUJTUHUKUGUJABUAUBNUCUDUKUHAJUEUF
|
|
$.
|
|
|
|
$( The positive difference of unequal integers is a positive integer.
|
|
(Generalization of ~ nnsub .) (Contributed by NM, 11-May-2004.) $)
|
|
znnsub $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) ) $=
|
|
( cz wcel wa clt wbr cmin co cn cr wb posdif syl2an zsubcl ancoms biantrurd
|
|
cc0 zre bitrd elnnz syl6bbr ) ACDZBCDZEZABFGZBAHIZCDZRUGFGZEZUGJDUEUFUIUJUC
|
|
AKDBKDUFUILUDASBSABMNUEUHUIUDUCUHBAOPQTUGUAUB $.
|
|
|
|
$( The sum of a real number not being an integer and an integer is not an
|
|
integer. Note that "not being an integer" in this case means "the
|
|
negation of is an integer" rather than "is apart from any integer" (given
|
|
excluded middle, those two would be equivalent). (Contributed by AV,
|
|
19-Jul-2021.) $)
|
|
nzadd $p |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ )
|
|
-> ( A + B ) e. ( RR \ ZZ ) ) $=
|
|
( cr cz cdif wcel wa caddc co wn eldif zre readdcl sylan2 adantlr wi zsubcl
|
|
cmin expcom cc adantl wceq zcn pncan syl2an eleq1d sylibd con3d com23 imp31
|
|
recn ex jca sylanb sylibr ) ACDEZFZBDFZGABHIZCFZUSDFZJZGZUSUPFUQACFZADFZJZG
|
|
ZURVCACDKVGURGUTVBVDURUTVFURVDBCFUTBLABMNOVDVFURVBVDURVFVBVDURVFVBPVDURGZVA
|
|
VEVHVAUSBRIZDFZVEURVAVJPVDVAURVJUSBQSUAVHVIADVDATFBTFVIAUBURAUKBUCABUDUEUFU
|
|
GUHULUIUJUMUNUSCDKUO $.
|
|
|
|
$( Closure of multiplication of integers. (Contributed by NM,
|
|
30-Jul-2004.) $)
|
|
zmulcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ ) $=
|
|
( cz wcel cr cn0 cneg wo wa cmul co elznn0 nn0mulcl jctild cc syl2an eleq1d
|
|
wceq syl5ib syl6 wi orcd a1i remulcl mulneg1 olc mulneg2 mul2neg orc ccased
|
|
recn syl6ibr imp an4s syl2anb ) ACDAEDZAFDZAGZFDZHZIBEDZBFDZBGZFDZHZIABJKZC
|
|
DZBCDALBLUPVAUTVEVGUPVAIZUTVEIZVGVHVIVFEDZVFFDZVFGZFDZHZIZVGVHUQVBUSVDVOVHU
|
|
QVBIZVNVJVPVNUAVHVPVKVMABMUBUCABUDZNVHUSVBIZVNVJVHVRVMVNVRURBJKZFDVHVMURBMV
|
|
HVSVLFUPAODZBODZVSVLRVAAUKZBUKZABUEPQSVMVKUFZTVQNVHUQVDIZVNVJVHWEVMVNWEAVCJ
|
|
KZFDVHVMAVCMVHWFVLFUPVTWAWFVLRVAWBWCABUGPQSWDTVQNVHUSVDIZVNVJVHWGVKVNWGURVC
|
|
JKZFDVHVKURVCMVHWHVFFUPVTWAWHVFRVAWBWCABUHPQSVKVMUITVQNUJVFLULUMUNUO $.
|
|
|
|
$( Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof
|
|
shortened by Mario Carneiro, 16-May-2014.) $)
|
|
zltp1le $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) ) $=
|
|
( cz wcel wa clt wbr c1 caddc co cle cmin cn wi nnge1 a1i znnsub cr wb zre
|
|
1re leaddsub2 mp3an2 syl2an 3imtr4d adantr ltp1d peano2re syl adantl ltletr
|
|
syl3anc mpand impbid ) ACDZBCDZEZABFGZAHIJZBKGZUQBALJZMDZHVAKGZURUTVBVCNUQV
|
|
AOPABQUOARDZBRDZUTVCSZUPATZBTZVDHRDVEVFUAAHBUBUCUDUEUQAUSFGZUTURUQAUOVDUPVG
|
|
UFZUGUQVDUSRDZVEVIUTEURNVJUQVDVKVJAUHUIUPVEUOVHUJAUSBUKULUMUN $.
|
|
|
|
$( Integer ordering relation. (Contributed by NM, 10-May-2004.) $)
|
|
zleltp1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) ) $=
|
|
( cz wcel wa cle wbr c1 caddc co clt cr wb zre leadd1 mp3an3 syl2an peano2z
|
|
1re zltp1le sylan2 bitr4d ) ACDZBCDZEABFGZAHIJBHIJZFGZAUFKGZUCALDZBLDZUEUGM
|
|
ZUDANBNUIUJHLDUKSABHOPQUDUCUFCDUHUGMBRAUFTUAUB $.
|
|
|
|
$( Integer ordering relation. (Contributed by NM, 13-Nov-2004.) $)
|
|
zlem1lt $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) ) $=
|
|
( cz wcel wa c1 cmin co clt wbr caddc cle wb peano2zm zltp1le sylan wceq cc
|
|
zcn ax-1cn npcan sylancl adantr breq1d bitr2d ) ACDZBCDZEZAFGHZBIJZUIFKHZBL
|
|
JZABLJUFUICDUGUJULMANUIBOPUHUKABLUFUKAQZUGUFARDFRDUMASTAFUAUBUCUDUE $.
|
|
|
|
$( Integer ordering relation. (Contributed by NM, 13-Nov-2004.) $)
|
|
zltlem1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) ) $=
|
|
( cz wcel wa c1 cmin co cle wbr caddc clt wb peano2zm zleltp1 sylan2 cc zcn
|
|
wceq ax-1cn npcan sylancl adantl breq2d bitr2d ) ACDZBCDZEZABFGHZIJZAUIFKHZ
|
|
LJZABLJUGUFUICDUJULMBNAUIOPUHUKBALUGUKBSZUFUGBQDFQDUMBRTBFUAUBUCUDUE $.
|
|
|
|
$( An integer greater than ` 0 ` is greater than or equal to ` 1 ` .
|
|
(Contributed by AV, 14-Oct-2018.) $)
|
|
zgt0ge1 $p |- ( Z e. ZZ -> ( 0 < Z <-> 1 <_ Z ) ) $=
|
|
( cz wcel cc0 clt wbr c1 caddc co cle wb zltp1le mpan wceq 0p1e1 a1i breq1d
|
|
0z bitrd ) ABCZDAEFZDGHIZAJFZGAJFDBCTUAUCKRDALMTUBGAJUBGNTOPQS $.
|
|
|
|
$( TODO: The following 14 theorems do not contain ` ZZ ` - these theorems are
|
|
about positive or nonnegative integers which are defined earlier. But the
|
|
proofs of these theorems are based on theorems for ` ZZ `. It should be
|
|
clarified if these theorems could be proven without theorems for ` ZZ ` and
|
|
then could be moved up into the corresponding subsections. Alternately,
|
|
these theorems could be moved into a new seperate subsection "Positive and
|
|
nonnegative integers (cont.)". $)
|
|
|
|
$( Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.)
|
|
(Proof shortened by Mario Carneiro, 16-May-2014.) $)
|
|
nnleltp1 $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( A <_ B <-> A < ( B + 1 ) ) ) $=
|
|
( cn wcel cz cle wbr c1 caddc co clt wb nnz zleltp1 syl2an ) ACDAEDBEDABFGA
|
|
BHIJKGLBCDAMBMABNO $.
|
|
|
|
$( Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.) $)
|
|
nnltp1le $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( A < B <-> ( A + 1 ) <_ B ) ) $=
|
|
( cn wcel cz clt wbr c1 caddc co cle wb nnz zltp1le syl2an ) ACDAEDBEDABFGA
|
|
HIJBKGLBCDAMBMABNO $.
|
|
|
|
$( Closure of addition of positive integers minus one. (Contributed by NM,
|
|
6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.) $)
|
|
nnaddm1cl $p |- ( ( A e. NN /\ B e. NN ) -> ( ( A + B ) - 1 ) e. NN ) $=
|
|
( cn wcel wa caddc co c1 cmin wceq nncn ax-1cn addsub mp3an3 syl2an nnm1nn0
|
|
cc cn0 nn0nnaddcl sylan eqeltrd ) ACDZBCDZEABFGHIGZAHIGZBFGZCUBAQDZBQDZUDUF
|
|
JZUCAKBKUGUHHQDUILABHMNOUBUERDUCUFCDAPUEBSTUA $.
|
|
|
|
$( Nonnegative integer ordering relation. (Contributed by Raph Levien,
|
|
10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.) $)
|
|
nn0ltp1le $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M < N <-> ( M + 1 ) <_ N ) ) $=
|
|
( cn0 wcel cz clt wbr c1 caddc co cle wb nn0z zltp1le syl2an ) ACDAEDBEDABF
|
|
GAHIJBKGLBCDAMBMABNO $.
|
|
|
|
$( Nonnegative integer ordering relation. (Contributed by Raph Levien,
|
|
10-Apr-2004.) $)
|
|
nn0leltp1 $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M <_ N <-> M < ( N + 1 ) ) ) $=
|
|
( cn0 wcel cz cle wbr c1 caddc co clt wb nn0z zleltp1 syl2an ) ACDAEDBEDABF
|
|
GABHIJKGLBCDAMBMABNO $.
|
|
|
|
$( Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.)
|
|
(Proof shortened by Mario Carneiro, 16-May-2014.) $)
|
|
nn0ltlem1 $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M < N <-> M <_ ( N - 1 ) ) ) $=
|
|
( cn0 wcel cz clt wbr c1 cmin co cle wb nn0z zltlem1 syl2an ) ACDAEDBEDABFG
|
|
ABHIJKGLBCDAMBMABNO $.
|
|
|
|
$( The nonnegative difference of integers is a nonnegative integer.
|
|
(Generalization of ~ nn0sub .) (Contributed by NM, 14-Jul-2005.) $)
|
|
znn0sub $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M <_ N <-> ( N - M ) e. NN0 ) ) $=
|
|
( cz wcel wa cle wbr co cc0 cn0 wb cr subge0 syl2an zsubcl biantrurd bitr3d
|
|
cmin zre ancoms elnn0z syl6bbr ) ACDZBCDZEABFGZBARHZCDZIUFFGZEZUFJDUDUCUEUI
|
|
KUDUCEZUHUEUIUDBLDALDUHUEKUCBSASBAMNUJUGUHBAOPQTUFUAUB $.
|
|
|
|
$( Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) $)
|
|
nn0sub $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M <_ N <-> ( N - M ) e. NN0 ) ) $=
|
|
( cn0 wcel cz cle wbr cmin co wb nn0z znn0sub syl2an ) ACDAEDBEDABFGBAHICDJ
|
|
BCDAKBKABLM $.
|
|
|
|
$( A nonnegative integer which is neither 0 nor 1 is greater than or equal to
|
|
2. (Contributed by Alexander van der Vekens, 6-Dec-2017.) $)
|
|
nn0n0n1ge2 $p |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) $=
|
|
( cn0 wcel cc0 wne c1 w3a c2 cle wbr cmin co wceq caddc 3ad2ant1 cn elnnne0
|
|
wa nnm1nn0 syl nn0cn subsub4d oveq2i syl6req 3simpa sylibr subeq0ad necon3d
|
|
1cnd 1p1e2 biimpd imp 3adant2 sylanbrc eqeltrd wb 2nn0 jctl nn0sub mpbird )
|
|
ABCZADEZAFEZGZHAIJZAHKLZBCZVDVFAFKLZFKLZBVAVBVFVIMVCVAVIAFFNLZKLVFVAAFFAUAZ
|
|
VAUIZVLUBVJHAKUJUCUDOVDVHPCZVIBCVDVHBCZVHDEZVMVDAPCZVNVDVAVBRVPVAVBVCUEAQUF
|
|
ASTVAVCVOVBVAVCVOVAVHDAFVAVHDMAFMVAAFVKVLUGUKUHULUMVHQUNVHSTUOVDHBCZVARZVEV
|
|
GUPVAVBVRVCVAVQUQUROHAUSTUT $.
|
|
|
|
${
|
|
$d x y z N $.
|
|
$( Membership in the set of integers. Commonly used in constructions of
|
|
the integers as equivalence classes under subtraction of the positive
|
|
integers. (Contributed by Mario Carneiro, 16-May-2014.) $)
|
|
elz2 $p |- ( N e. ZZ <-> E. x e. NN E. y e. NN N = ( x - y ) ) $=
|
|
( cz wcel cr cn0 cneg wo wa cmin co wceq cn c1 1nn ax-1cn sylancr syl2an
|
|
cc wrex elznn0 caddc nn0p1nn adantl a1i recn adantr pncan sylancl rspceov
|
|
eqcomd syl3anc negsub simpr nnnn0addcl eqeltrrd nncan jaodan nnre resubcl
|
|
cv cle wbr nnz zletric syl2anr nnnn0 nn0sub nncn negsubdi2 eleq1d orbi12d
|
|
wb bitr4d mpbid jca eleq1 negeq anbi12d syl5ibrcom rexlimivv impbii bitri
|
|
) CDECFEZCGEZCHZGEZIZJZCAVBZBVBZKLZMZBNUAANUAZCUBWJWOWEWFWOWHWEWFJZCOUCLZ
|
|
NEZONEZCWQOKLZMWOWFWRWECUDUEWSWPPUFWPWTCWPCTEZOTEZWTCMWEXAWFCUGZUHQCOUIUJ
|
|
ULABNNWQOCKUKUMWEWHJZWSOCKLZNECOXEKLZMWOWSXDPUFXDOWGUCLZXENXDXBXAXGXEMQWE
|
|
XAWHXCUHZOCUNRXDWSWHXGNEPWEWHUOOWGUPRUQXDXFCXDXBXAXFCMQXHOCURRULABNNOXECK
|
|
UKUMUSWNWJABNNWKNEZWLNEZJZWJWNWMFEZWMGEZWMHZGEZIZJXKXLXPXIWKFEWLFEXLXJWKU
|
|
TWLUTWKWLVASXKWLWKVCVDZWKWLVCVDZIZXPXJWLDEWKDEXSXIWLVEWKVEWLWKVFVGXKXQXMX
|
|
RXOXJWLGEZWKGEZXQXMVNXIWLVHZWKVHZWLWKVIVGXKXRWLWKKLZGEZXOXIYAXTXRYEVNXJYC
|
|
YBWKWLVISXKXNYDGXIWKTEWLTEXNYDMXJWKVJWLVJWKWLVKSVLVOVMVPVQWNWEXLWIXPCWMFV
|
|
RWNWFXMWHXOCWMGVRWNWGXNGCWMVSVLVMVTWAWBWCWD $.
|
|
|
|
$( Alternate definition of the integers, based on ~ elz2 . (Contributed by
|
|
Mario Carneiro, 16-May-2014.) $)
|
|
dfz2 $p |- ZZ = ( - " ( NN X. NN ) ) $=
|
|
( vx vy vz cz cmin cn cxp cima cv wcel co wceq wrex elz2 cc wfn wb nnsscn
|
|
wss mp2an wf subf ffn ax-mp xpss12 ovelimab bitr4i eqriv ) ADEFFGZHZAIZDJ
|
|
UKBICIEKLCFMBFMZUKUJJZBCUKNEOOGZPZUIUNSZUMULQUNOEUAUOUBUNOEUCUDFOSZUQUPRR
|
|
FOFOUETBCUNFFUKEUFTUGUH $.
|
|
$}
|
|
|
|
$( Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.) $)
|
|
nn0sub2 $p |- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 ) $=
|
|
( cn0 wcel cle wbr cmin co nn0sub biimp3a ) ACDBCDABEFBAGHCDABIJ $.
|
|
|
|
$( Apartness is equivalent to not equal for integers. (Contributed by Jim
|
|
Kingdon, 14-Mar-2020.) $)
|
|
zapne $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M # N <-> M =/= N ) ) $=
|
|
( cz wcel wa cap wbr wne cc wi zcn apne syl2an wceq wn clt wo w3o cr zre wb
|
|
df-ne ztri3or 3orrot 3orass bitri sylib reaplt syl6bb sylibrd syl5bi impbid
|
|
ord orcom ) ACDZBCDZEZABFGZABHZUOAIDBIDURUSJUPAKBKABLMUSABNZOZUQURABUBUQVAB
|
|
APGZABPGZQZURUQUTVDUQVCUTVBRZUTVDQZABUCVEUTVBVCRVFVCUTVBUDUTVBVCUEUFUGUMUOA
|
|
SDZBSDZURVDUAUPATBTVGVHEURVCVBQVDABUHVCVBUNUIMUJUKUL $.
|
|
|
|
$( Equality of integers is decidable. (Contributed by Jim Kingdon,
|
|
14-Mar-2020.) $)
|
|
zdceq $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B ) $=
|
|
( cz wcel wa clt wbr wceq w3o wdc ztri3or cr zre wne ltne necomd sylibr syl
|
|
wi ex wo olc dcne adantr sylan orc a1i adantl 3jaod mpd ) ACDZBCDZEZABFGZAB
|
|
HZBAFGZIUOJZABKUMUNUQUOUPUKALDZULUNUQSZAMURUSULURUNUQURUNEZABNZUQUTBAABOPVA
|
|
UOVAUAZUQVAUOUBABUCZQZRTUDUEUOUQSUMUOVBUQUOVAUFVCQUGULUPUQSZUKULBLDZVEBMVFU
|
|
PUQVFUPEVAUQBAOVDRTRUHUIUJ $.
|
|
|
|
$( Integer ` <_ ` is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.) $)
|
|
zdcle $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B ) $=
|
|
( cz wcel wa clt wbr wceq w3o cle wdc ztri3or cr wi zre ltle wn sylibr syl6
|
|
wo orc df-dc eqle syl ex adantr lenlt biimpd con2d olc 3jaod syl2an mpd ) A
|
|
CDZBCDZEABFGZABHZBAFGZIZABJGZKZABLUNAMDZBMDZUSVANUOAOBOVBVCEZUPVAUQURVDUPUT
|
|
VAABPUTUTUTQZTZVAUTVEUAUTUBZRZSVBUQVANVCVBUQVAVBUQEUTVAABUCVHUDUEUFVDURVEVA
|
|
VDUTURVDUTURQABUGUHUIVEVFVAVEUTUJVGRSUKULUM $.
|
|
|
|
$( Integer ` < ` is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.) $)
|
|
zdclt $p |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B ) $=
|
|
( cz wcel wa clt wbr wceq w3o wdc ztri3or cr wi zre wn wo orc sylibr adantr
|
|
df-dc a1i ltnr wb breq2 adantl mtbid olc ex ltnsym ancoms syl6 3jaod syl2an
|
|
syl mpd ) ACDZBCDZEABFGZABHZBAFGZIZURJZABKUPALDZBLDZVAVBMUQANBNVCVDEZURVBUS
|
|
UTURVBMVEURURUROZPZVBURVFQURTZRUAVCUSVBMVDVCUSVBVCUSEZVFVBVIAAFGZURVCVJOUSA
|
|
UBSUSVJURUCVCABAFUDUEUFVFVGVBVFURUGVHRZUNUHSVEUTVFVBVDVCUTVFMBAUIUJVKUKULUM
|
|
UO $.
|
|
|
|
$( Integer 'Less than' expressed in terms of 'less than or equal to'. Also
|
|
see ~ ltleap which is a similar result for real numbers. (Contributed by
|
|
Jim Kingdon, 14-Mar-2020.) $)
|
|
zltlen $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
( A < B <-> ( A <_ B /\ B =/= A ) ) ) $=
|
|
( cz wcel wa clt wbr cle wne cap cr wb zre ltleap syl2an zapne anbi2d bitrd
|
|
necom anbi2i syl6bb ) ACDZBCDZEZABFGZABHGZABIZEZUFBAIZEUDUEUFABJGZEZUHUBAKD
|
|
BKDUEUKLUCAMBMABNOUDUJUGUFABPQRUGUIUFABSTUA $.
|
|
|
|
$( A nonnegative integer is neither 0 nor 1 if and only if it is greater than
|
|
or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.) $)
|
|
nn0n0n1ge2b $p |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) ) $=
|
|
( wcel cc0 wne c1 wa c2 wbr wdc wn wi wceq zdceq sylancl dcned sylc syl clt
|
|
cz wb cn0 cle nn0n0n1ge2 3expib nn0z 0z 1z dcan ianordc nnedc orbi12d bitrd
|
|
wo 2pos breq1 mpbiri a1d 1lt2 impcom 2z zltnle adantr mpbid ex sylbid condc
|
|
jaoi impbid ) AUABZACDZAEDZFZGAUBHZVIVJVKVMAUCUDVIVLIZVLJZVMJZKVMVLKVIVJIZV
|
|
KIVNVIACVIASBZCSBACLZIZAUEZUFACMNZOZVIAEVIVRESBAELZIZWAUGAEMNZOVJVKUHPVIVOV
|
|
SWDUMZVPVIVOVJJZVKJZUMZWGVIVQVOWJTWCVJVKUIQVIWHVSWIWDVIVTWHVSTWBACUJQVIWEWI
|
|
WDTWFAEUJQUKULVIWGVPVIWGFAGRHZVPWGVIWKVSVIWKKWDVSWKVIVSWKCGRHUNACGRUOUPUQWD
|
|
WKVIWDWKEGRHURAEGRUOUPUQVGUSVIWKVPTZWGVIVRGSBWLWAUTAGVANVBVCVDVEVLVMVFPVH
|
|
$.
|
|
|
|
$( A nonnegative integer less than ` 1 ` is ` 0 ` . (Contributed by Paul
|
|
Chapman, 22-Jun-2011.) $)
|
|
nn0lt10b $p |- ( N e. NN0 -> ( N < 1 <-> N = 0 ) ) $=
|
|
( cn0 wcel cc0 wceq cle wbr wa c1 clt cr wb nn0re 0re letri3 mpan2 biantrud
|
|
syl nn0ge0 cz nn0z caddc co 0z zleltp1 0p1e1 breq2i syl6bb 3bitr2rd ) ABCZA
|
|
DEZADFGZDAFGZHZULAIJGZUJAKCZUKUNLZAMUPDKCUQNADOPRUJUMULASQUJATCZULUOLAUAURU
|
|
LADIUBUCZJGZUOURDTCULUTLUDADUEPUSIAJUFUGUHRUI $.
|
|
|
|
$( A nonnegative integer less than 2 must be 0 or 1. (Contributed by
|
|
Alexander van der Vekens, 16-Sep-2018.) $)
|
|
nn0lt2 $p |- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) ) $=
|
|
( cn0 wcel c2 clt wbr wa c1 wceq cc0 wo wne wi olc a1i cle cz wb sylancl 1z
|
|
cmin co nn0z 2z zltlem1 2m1e1 breq2i syl6bb necom zltlen nn0lt10b biimpa ex
|
|
orcd sylbird expd syl7bi sylbid imp wdc zdceq adantr dcne sylib mpjaod ) AB
|
|
CZADEFZGZAHIZAJIZVIKZAHLZVIVKMVHVIVJNOVFVGVLVKMZVFVGAHPFZVMVFVGADHUAUBZPFZV
|
|
NVFAQCZDQCVGVPRAUCZUDADUESVOHAPUFUGUHVLHALZVFVNVKAHUIVFVNVSVKVFVNVSGZAHEFZV
|
|
KVFVQHQCZWAVTRVRTAHUJSVFWAVKVFWAGVJVIVFWAVJAUKULUNUMUOUPUQURUSVHVIUTZVIVLKV
|
|
FWCVGVFVQWBWCVRTAHVASVBAHVCVDVE $.
|
|
|
|
$( Nonnegative integer ordering relation. (Contributed by NM,
|
|
21-Jun-2005.) $)
|
|
nn0lem1lt $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M <_ N <-> ( M - 1 ) < N ) ) $=
|
|
( cn0 wcel cz cle wbr c1 cmin co clt wb nn0z zlem1lt syl2an ) ACDAEDBEDABFG
|
|
AHIJBKGLBCDAMBMABNO $.
|
|
|
|
$( Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) $)
|
|
nnlem1lt $p |- ( ( M e. NN /\ N e. NN ) -> ( M <_ N <-> ( M - 1 ) < N ) ) $=
|
|
( cn wcel cz cle wbr c1 cmin co clt wb nnz zlem1lt syl2an ) ACDAEDBEDABFGAH
|
|
IJBKGLBCDAMBMABNO $.
|
|
|
|
$( Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) $)
|
|
nnltlem1 $p |- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) ) $=
|
|
( cn wcel cz clt wbr c1 cmin co cle wb nnz zltlem1 syl2an ) ACDAEDBEDABFGAB
|
|
HIJKGLBCDAMBMABNO $.
|
|
|
|
$( A positive integer decreased by 1 is greater than or equal to 0.
|
|
(Contributed by AV, 30-Oct-2018.) $)
|
|
nnm1ge0 $p |- ( N e. NN -> 0 <_ ( N - 1 ) ) $=
|
|
( cn wcel cc0 clt wbr c1 cmin co cle nngt0 cz wb nnz zltlem1 sylancr mpbid
|
|
0z ) ABCZDAEFZDAGHIJFZAKSDLCALCTUAMRANDAOPQ $.
|
|
|
|
$( Division of a nonnegative integer by a positive number is not negative.
|
|
(Contributed by Alexander van der Vekens, 14-Apr-2018.) $)
|
|
nn0ge0div $p |- ( ( K e. NN0 /\ L e. NN ) -> 0 <_ ( K / L ) ) $=
|
|
( cn0 wcel cn wa cc0 cle wbr cdiv co nn0ge0 adantr cr clt wb cz elnnz nn0re
|
|
w3a zre ad2antrl simprr 3jca sylan2b ge0div syl mpbid ) ACDZBEDZFZGAHIZGABJ
|
|
KHIZUIULUJALMUKANDZBNDZGBOIZTZULUMPUJUIBQDZUPFZUQBRUIUSFUNUOUPUIUNUSASMURUO
|
|
UIUPBUAUBUIURUPUCUDUEABUFUGUH $.
|
|
|
|
${
|
|
$d k M $. $d k N $.
|
|
$( Two ways to express " ` M ` divides ` N ` . (Contributed by NM,
|
|
3-Oct-2008.) $)
|
|
zdiv $p |- ( ( M e. NN /\ N e. ZZ )
|
|
-> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) ) $=
|
|
( cn wcel cz wa cc0 cap wbr cv cmul co wceq wrex cdiv wb cc wi zcn adantr
|
|
nnap0 nncn divcanap3 3coml 3expa sylan2 3adantl2 oveq1 sylan9req eqeltrrd
|
|
w3a simplr exp31 rexlimdv divcanap2 3com12 oveq2 eqeq1d rspcev expcom syl
|
|
impbid 3expia syl2an mpd ) BDEZCFEZGBHIJZBAKZLMZCNZAFOZCBPMZFEZQZVGVIVHBU
|
|
BUAVGBREZCREZVIVPSVHBUCCTVQVRVIVPVQVRVIULZVMVOVSVLVOAFVSVJFEZVLVOVSVTGZVL
|
|
GVJVNFWAVLVJVKBPMZVNVQVIVTWBVJNZVRVTVQVIGVJREZWCVJTVQVIWDWCWDVQVIWCVJBUDU
|
|
EUFUGUHVKCBPUIUJVSVTVLUMUKUNUOVSBVNLMZCNZVOVMSVRVQVIWFCBUPUQVOWFVMVLWFAVN
|
|
FVJVNNVKWECVJVNBLURUSUTVAVBVCVDVEVF $.
|
|
$}
|
|
|
|
$( Property of divisibility: if ` D ` divides ` A ` and ` B ` then it divides
|
|
` A + B ` . (Contributed by NM, 3-Oct-2008.) $)
|
|
zdivadd $p |- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\
|
|
( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ ) $=
|
|
( cn wcel cz w3a cdiv co wa caddc wceq cc cc0 cap wbr zcn nncn nnap0 jca
|
|
divdirap syl3an 3comr adantr zaddcl adantl eqeltrd ) CDEZAFEZBFEZGZACHIZFEB
|
|
CHIZFEJZJABKICHIZULUMKIZFUKUOUPLZUNUIUJUHUQUIAMEUJBMEUHCMEZCNOPZJUQAQBQUHUR
|
|
USCRCSTABCUAUBUCUDUNUPFEUKULUMUEUFUG $.
|
|
|
|
$( Property of divisibility: if ` D ` divides ` A ` then it divides
|
|
` B x. A ` . (Contributed by NM, 3-Oct-2008.) $)
|
|
zdivmul $p |- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\
|
|
( A / D ) e. ZZ ) -> ( ( B x. A ) / D ) e. ZZ ) $=
|
|
( cn wcel cz w3a cdiv co wa cmul wceq cc cc0 cap wbr 3ad2ant2 3ad2ant1 nncn
|
|
zcn nnap0 3ad2ant3 divassap syl3anc 3comr adantr zmulcl 3ad2antl3 eqeltrd
|
|
jca ) CDEZAFEZBFEZGZACHIZFEZJBAKICHIZBUOKIZFUNUQURLZUPULUMUKUSULUMUKGBMEZAM
|
|
EZCMEZCNOPZJZUSUMULUTUKBTQULUMVAUKATRUKULVDUMUKVBVCCSCUAUJUBBACUCUDUEUFUMUK
|
|
UPURFEULBUOUGUHUI $.
|
|
|
|
${
|
|
$d k M $. $d k N $.
|
|
$( An extensionality-like property for integer ordering. (Contributed by
|
|
NM, 29-Oct-2005.) $)
|
|
zextle $p |- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k <_ M <-> k <_ N ) )
|
|
-> M = N ) $=
|
|
( cz wcel cv cle wbr wb wral wceq wa zre leidd adantr breq1 bibi12d mpbid
|
|
rspcva cr adantlr mpbird adantll jca ex letri3 syl2an sylibrd 3impia ) BD
|
|
EZCDEZAFZBGHZULCGHZIZADJZBCKZUJUKLZUPBCGHZCBGHZLZUQURUPVAURUPLUSUTUJUPUSU
|
|
KUJUPLBBGHZUSUJVBUPUJBBMZNOUOVBUSIABDULBKUMVBUNUSULBBGPULBCGPQSRUAUKUPUTU
|
|
JUKUPLUTCCGHZUKVDUPUKCCMZNOUOUTVDIACDULCKUMUTUNVDULCBGPULCCGPQSUBUCUDUEUJ
|
|
BTECTEUQVAIUKVCVEBCUFUGUHUI $.
|
|
|
|
$( An extensionality-like property for integer ordering. (Contributed by
|
|
NM, 29-Oct-2005.) $)
|
|
zextlt $p |- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k < M <-> k < N ) )
|
|
-> M = N ) $=
|
|
( cz wcel clt wbr wb wral wceq wa c1 cmin cle zltlem1 peano2zm syl2an zcn
|
|
co cc cv adantrr adantrl bibi12d ancoms ralbidva wi zextle 3expia subcan2
|
|
ax-1cn mp3an3 sylibd sylbid 3impia ) BDEZCDEZAUAZBFGZURCFGZHZADIZBCJZUPUQ
|
|
KZVBURBLMSZNGZURCLMSZNGZHZADIZVCVDVAVIADURDEZVDVAVIHVKVDKUSVFUTVHVKUPUSVF
|
|
HUQURBOUBVKUQUTVHHUPURCOUCUDUEUFVDVJVEVGJZVCUPVEDEZVGDEZVJVLUGUQBPCPVMVNV
|
|
JVLAVEVGUHUIQUPBTEZCTEZVLVCHZUQBRCRVOVPLTEVQUKBCLUJULQUMUNUO $.
|
|
$}
|
|
|
|
$( The reciprocal of a number greater than 1 is not an integer. (Contributed
|
|
by NM, 3-May-2005.) $)
|
|
recnz $p |- ( ( A e. RR /\ 1 < A ) -> -. ( 1 / A ) e. ZZ ) $=
|
|
( cr wcel c1 clt wbr wa cdiv co cz cc0 recgt1i simprd cle wn simpld zgt0ge1
|
|
syl5ibcom wb 1re cap 0lt1 0re lttr mp3an12 mpani imdistani gt0ap0 rerecclap
|
|
wi syl syldan lenlt sylancr sylibd mt2d ) ABCZDAEFZGZDAHIZJCZUTDEFZUSKUTEFZ
|
|
VBALZMUSVADUTNFZVBOZUSVCVAVEUSVCVBVDPUTQRUSDBCZUTBCZVEVFSTUQURAKUAFZVHUSUQK
|
|
AEFZGVIUQURVJUQKDEFZURVJUBKBCVGUQVKURGVJUJUCTKDAUDUEUFUGAUHUKAUIULDUTUMUNUO
|
|
UP $.
|
|
|
|
$( A number between an integer and its successor is not an integer.
|
|
(Contributed by NM, 3-May-2005.) $)
|
|
btwnnz $p |- ( ( A e. ZZ /\ A < B /\ B < ( A + 1 ) ) -> -. B e. ZZ ) $=
|
|
( cz wcel clt wbr c1 caddc co wn wa cle zltp1le cr wb peano2z zre syl lenlt
|
|
syl2an bitrd biimpd impancom con2d 3impia ) ACDZABEFZBAGHIZEFZBCDZJUFUGKUJU
|
|
IUFUJUGUIJZUFUJKZUGUKULUGUHBLFZUKABMUFUHNDZBNDUMUKOUJUFUHCDUNAPUHQRBQUHBSTU
|
|
AUBUCUDUE $.
|
|
|
|
$( A larger number does not divide a smaller positive integer. (Contributed
|
|
by NM, 3-May-2005.) $)
|
|
gtndiv $p |- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ ) $=
|
|
( cr wcel cn clt wbr w3a cc0 cdiv co c1 caddc cz wn nnre 3ad2ant2 wa mp3an1
|
|
wb simp1 nngt0 adantl 0re lttr sylan ancoms mpand 3impia divgt0d simp3 cmul
|
|
wi 1re ltdivmul2 mp3an2 syl12anc mulid2d breq2d 3ad2ant1 bitrd mpbird 0p1e1
|
|
recn syl6breqr 0z btwnnz syl2anc ) ACDZBEDZBAFGZHZIBAJKZFGZVMILMKZFGZVMNDOZ
|
|
VLBAVJVIBCDZVKBPZQZVIVJVKUAZVJVIIBFGZVKBUBZQVIVJVKIAFGZVIVJRWBVKWDVJWBVIWCU
|
|
CVJVIWBVKRWDUMZVJVRVIWEVSICDVRVIWEUDIBAUESUFUGUHUIZUJVLVMLVOFVLVMLFGZVKVIVJ
|
|
VKUKVLWGBLAULKZFGZVKVLVRVIWDWGWITZVTWAWFVRLCDVIWDRWJUNBLAUOUPUQVIVJWIVKTVKV
|
|
IWHABFVIAAVDURUSUTVAVBVCVEINDVNVPVQVFIVMVGSVH $.
|
|
|
|
$( One-half is not an integer. (Contributed by NM, 31-Jul-2004.) $)
|
|
halfnz $p |- -. ( 1 / 2 ) e. ZZ $=
|
|
( c2 cr wcel c1 clt wbr cdiv co cz wn 2re 1lt2 recnz mp2an ) ABCDAEFDAGHICJ
|
|
KLAMN $.
|
|
|
|
$( Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) $)
|
|
3halfnz $p |- -. ( 3 / 2 ) e. ZZ $=
|
|
( c1 cz wcel c3 c2 cdiv co clt wbr caddc wn cmul cr wb 3re 2re mp3an breq2i
|
|
c4 mpbir 1z 2cn mulid2i 2lt3 eqbrtri cc0 1re 2pos pm3.2i ltmuldiv mpbi 3lt4
|
|
wa 2t2e4 1p1e2 ltdivmul bitri btwnnz ) ABCADEFGZHIZUSAAJGZHIZUSBCKUAAELGZDH
|
|
IZUTVCEDHEUBUCUDUEAMCDMCZEMCZUFEHIZUMZVDUTNUGOVFVGPUHUIZADEUJQUKVBDEELGZHIZ
|
|
VKDSHIULVJSDHUNRTVBUSEHIZVKVAEUSHUORVEVFVHVLVKNOPVIDEEUPQUQTAUSURQ $.
|
|
|
|
${
|
|
$d A w x y z $. $d f g ph x $. $d ph w x z $.
|
|
suprzclex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\
|
|
A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
|
|
suprzclex.ss $e |- ( ph -> A C_ ZZ ) $.
|
|
$( The supremum of a set of integers is an element of the set.
|
|
(Contributed by Jim Kingdon, 20-Dec-2021.) $)
|
|
suprzclex $p |- ( ph -> sup ( A , RR , < ) e. A ) $=
|
|
( vw cr clt c1 cv wbr wcel wrex wa wn cz wral adantr vg vf csup cmin wceq
|
|
co lttri3 adantl supclti ltm1d zssre syl6ss peano2rem syl suprlubex mpbid
|
|
wb cle caddc sselda sseldi simprl sseldd peano2re ad2antrr simpr suprubex
|
|
wss zre wi simprr 1red ltsubaddd lelttrd zleltp1 syl2anc mpbird ralrimiva
|
|
breq2 cbvrexv imbi2i ralbii anbi2i rexbii sylib letri3d mpbir2and eqeltrd
|
|
suprleubex rexlimddv ) AEIJUCZKUDUFZDLZJMZWKENDEAWLWKJMWNDEOAWKABCDUAUBIE
|
|
JUBLZINUALZINPWOWPUEWOWPJMQWPWOJMQPUQAWOWPUGUHFUIZUJABCDEWLFAERIGUKULZAWK
|
|
INZWLINWQWKUMUNUOUPAWMENZWNPZPZWKWMEXBWKWMUEWKWMURMZWMWKURMXBXCHLZWMURMZH
|
|
ESXBXEHEXBXDENZPZXEXDWMKUSUFZJMZXGXDWKXHXGRIXDUKXBERXDAERVHXAGTZUTZVAXBWS
|
|
XFAWSXAWQTZTXBXHINZXFXBWMINZXMXBWMRNZXNXBERWMXJAWTWNVBZVCZWMVIUNZWMVDUNTX
|
|
GBCDEXDABLZCLZJMQCESZXTXSJMZXTWMJMZDEOZVJZCISZPZBIOZXAXFFVEAEIVHXAXFWRVEX
|
|
BXFVFVGXBWKXHJMZXFXBWNYIAWTWNVKXBWKKWMXLXBVLXRVMUPTVNXGXDRNXOXEXIUQXKXBXO
|
|
XFXQTXDWMVOVPVQVRXBBCHEWMAYAYBXTXDJMZHEOZVJZCISZPZBIOZXAAYHYOFYGYNBIYFYMY
|
|
AYEYLCIYDYKYBYCYJDHEWMXDXTJVSVTWAWBWCWDWETZXBERIXJUKULZXRWIVQXBBCHEWMYPYQ
|
|
XPVGXBWKWMXLXRWFWGXPWHWJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to express " ` A ` is a prime number (or 1)." (Contributed by
|
|
NM, 4-May-2005.) $)
|
|
prime $p |- ( A e. NN -> (
|
|
A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <->
|
|
A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) ) $=
|
|
( cn wcel cv cdiv c1 wceq wi clt wbr wa wb cz nnz wn syl6bbr adantl nnre
|
|
cr co wo cle w3a wne wdc 1z zdceq mpan2 dfordc df-ne imbi1i imbi2d impexp
|
|
3syl bi2.04 bitri nngt1ne1 anbi1d gtndiv 3expia sylan con2d lenlt sylibrd
|
|
syl2an ancoms syl5 pm4.71rd anbi2d 3anass bitr3d imbi1d bitrd ralbidva )
|
|
BCDZBAEZFUAZCDZVQGHZVQBHZUBZIZGVQJKZVQBUCKZVSUDZWAIZACVPVQCDZLZWCVQGUEZVS
|
|
LZWAIZWGWHWCWLMVPWHWCVSWJWAIZIZWLWHWBWMVSWHVQNDZVTUFZWBWMMVQOWOGNDWPUGVQG
|
|
UHUIWPWBVTPZWAIWMVTWAUJWJWQWAVQGUKULQUOUMWLWJVSWAIIWNWJVSWAUNWJVSWAUPUQQR
|
|
WIWKWFWAWIWDVSLZWKWFWIWDWJVSWHWDWJMVPVQURRUSWIWRWDWEVSLZLWFWIVSWSWDWIVSWE
|
|
VSVRNDZWIWEVROWHVPWTWEIWHVPLZWTBVQJKZPZWEXAXBWTWHVQTDZVPXBWTPZIVQSZXDVPXB
|
|
XEVQBUTVAVBVCWHXDBTDWEXCMVPXFBSVQBVDVFVEVGVHVIVJWDWEVSVKQVLVMVNVO $.
|
|
$}
|
|
|
|
$( The square of a nonzero integer is a positive integer. (Contributed by
|
|
NM, 2-Aug-2004.) $)
|
|
msqznn $p |- ( ( A e. ZZ /\ A =/= 0 ) -> ( A x. A ) e. NN ) $=
|
|
( cz wcel cc0 wne wa cmul co clt wbr cn zmulcl anidms adantr wb zapne mpan2
|
|
cap 0z pm5.32i cr zre apsqgt0 sylan sylbir elnnz sylanbrc ) ABCZADEZFZAAGHZ
|
|
BCZDUKIJZUKKCUHULUIUHULAALMNUJUHADRJZFUMUHUNUIUHDBCUNUIOSADPQTUHAUACUNUMAUB
|
|
AUCUDUEUKUFUG $.
|
|
|
|
$( No even integer equals an odd integer (i.e. no integer can be both even
|
|
and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM,
|
|
31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.) $)
|
|
zneo $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) ) $=
|
|
( cz wcel wa c2 cmul co c1 caddc cdiv wn wceq cmin cc 2cn zcn mulcl sylancr
|
|
a1i wne halfnz adantr adantl 1cnd subaddd subdid oveq1d zsubcl syl cc0 2ap0
|
|
cap wbr divcanap3d eqtr3d eqeltrd oveq1 eleq1d syl5ibcom sylbird mpi necomd
|
|
necon3bd ) ACDZBCDZEZFBGHZIJHZFAGHZVGIFKHZCDZLVIVJUAUBVGVLVIVJVGVIVJMVJVHNH
|
|
ZIMZVLVGVJVHIVGFODZAODZVJODPVEVPVFAQUCZFARSVGVOBODZVHODPVFVRVEBQUDZFBRSVGUE
|
|
UFVGVMFKHZCDVNVLVGVTABNHZCVGFWAGHZFKHVTWAVGWBVMFKVGFABVOVGPTZVQVSUGUHVGWAFV
|
|
GWACDWAODABUIZWAQUJWCFUKUMUNVGULTUOUPWDUQVNVTVKCVMIFKURUSUTVAVDVBVC $.
|
|
|
|
${
|
|
$d j k N $.
|
|
$( A positive integer is even or odd. (Contributed by Jim Kingdon,
|
|
15-Mar-2020.) $)
|
|
nneoor $p |- ( N e. NN ->
|
|
( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) ) $=
|
|
( vj vk cn wcel c1 caddc co c2 cdiv cv wceq oveq1d eleq1d orbi12d 2div2e1
|
|
wo oveq1 cc 2cn df-2 oveq1i eqtr3i 1nn eqeltri orci peano2nn nncn cc0 cap
|
|
add1p1 wbr wa 2ap0 pm3.2i divdirap mp3an23 oveq2i syl6eq eqtrd syl orim2d
|
|
syl5ibr orcom syl6ib nnind orcomd ) ADEAFGHZIJHZDEZAIJHZDEZBKZFGHZIJHZDEZ
|
|
VMIJHZDEZQFFGHZIJHZDEZFIJHZDEZQCKZFGHZIJHZDEZWDIJHZDEZQZWEFGHZIJHZDEZWGQZ
|
|
VJVLQBCAVMFLZVPWAVRWCWOVOVTDWOVNVSIJVMFFGRMNWOVQWBDVMFIJRNOVMWDLZVPWGVRWI
|
|
WPVOWFDWPVNWEIJVMWDFGRMNWPVQWHDVMWDIJRNOVMWELZVPWMVRWGWQVOWLDWQVNWKIJVMWE
|
|
FGRMNWQVQWFDVMWEIJRNOVMALZVPVJVRVLWRVOVIDWRVNVHIJVMAFGRMNWRVQVKDVMAIJRNOW
|
|
AWCVTFDIIJHZVTFIVSIJUAUBPUCUDUEUFWDDEZWJWGWMQWNWTWIWMWGWIWMWTWHFGHZDEWHUG
|
|
WTWLXADWTWDSEZWLXALWDUHXBWLWDIGHZIJHZXAXBWKXCIJWDUKMXBXDWHWSGHZXAXBISEZXF
|
|
IUIUJULZUMXDXELTXFXGTUNUOWDIIUPUQWSFWHGPURUSUTVANVCVBWGWMVDVEVFVG $.
|
|
|
|
$( A positive integer is even or odd but not both. (Contributed by NM,
|
|
1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.) $)
|
|
nneo $p |- ( N e. NN ->
|
|
( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) ) $=
|
|
( cn wcel c2 cdiv co c1 caddc cmul wceq nncn peano2cn syl 2cnd divcanap2d
|
|
wn cc cc0 cz nnz cap wbr 2ap0 a1i oveq1d eqtr4d wne syl2an expcom syl5com
|
|
zneo necon2bd nneoor orcomd ord impbid ) ABCZADEFZBCZAGHFZDEFZBCZPZUQDVAI
|
|
FZDURIFZGHFZJUSVCUQVDUTVFUQUTDUQAQCUTQCAKZALMUQNZDRUAUBUQUCUDZOUQVEAGHUQA
|
|
DVGVHVIOUEUFUSVBVDVFVBUSVDVFUGZVBVASCURSCVJUSVATURTVAURUKUHUIULUJUQVBUSUQ
|
|
USVBAUMUNUOUP $.
|
|
|
|
nneo.1 $e |- N e. NN $.
|
|
$( A positive integer is even or odd but not both. (Contributed by NM,
|
|
20-Aug-2001.) $)
|
|
nneoi $p |- ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) $=
|
|
( cn wcel c2 cdiv co c1 caddc wn wb nneo ax-mp ) ACDAEFGCDAHIGEFGCDJKBALM
|
|
$.
|
|
$}
|
|
|
|
$( An integer is even or odd. (Contributed by NM, 1-Jan-2006.) $)
|
|
zeo $p |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel cc0 wceq cn cneg c2 cdiv co c1 caddc 2cn 2ap0 syl cc mp3an23 cmin
|
|
eleq1d ax-1cn cr w3o wa wo elz oveq1 div0api eqeltri syl6eqel adantl nneoor
|
|
0z orcd nnz orim12i wi recn cap wbr divnegap nnnegz syl6bir halfcld negnegd
|
|
sylibd peano2zm negsubdi2i 2m1e1 eqtr2i negcon2i mpbi negcl addsubass negdi
|
|
subcli oveq2i mpan2 3eqtr4a oveq1d peano2cn pm3.2i 2div2e1 syl6reqr 3eqtr4d
|
|
divsubdirap eqcomi syl5ib znegcl syl6 peano2re recnd orim12d adantr 3jaodan
|
|
syl5 mpd sylbi ) ABCAUACZADEZAFCZAGZFCZUBUCAHIJZBCZAKLJZHIJZBCZUDZAUEWRWSXH
|
|
WTXBWSXHWRWSXDXGWSXCDHIJZBADHIUFXIDBHMNUGULUHUIUMUJWTXHWRWTXCFCZXFFCZUDXHAU
|
|
KXJXDXKXGXCUNXFUNUOOUJWRXBUCXAHIJZFCZXAKLJZHIJZFCZUDZXHXBXQWRXAUKUJWRXQXHUP
|
|
XBWRXMXDXPXGWRXMXCGZGZBCZXDWRXMXRFCXTWRXRXLFWRAPCZXRXLEZAUQZYAHPCZHDURUSZYB
|
|
MNAHUTQOSXRVAVBWRXSXCBWRXCWRAYCVCVDSVEXPXOBCZWRXGXOUNWRYFXFGZGZBCZXGWRYFYGB
|
|
CZYIYFXOKRJZBCWRYJXOVFWRYKYGBWRYAYKYGEYCYAXNHRJZHIJZXEGZHIJZYKYGYAYLYNHIYAX
|
|
AKHRJZLJZXAKGZLJZYLYNYPYRXALKYPGZEYPYREYTHKRJKKHTMVGVHVIKYPTKHTMVOVJVKVPYAX
|
|
APCZYLYQEZAVLZUUAKPCZYDUUBTMXAKHVMQOYAUUDYNYSETAKVNVQVRVSYAYMXOHHIJZRJZYKYA
|
|
XNPCZYMUUFEZYAUUAUUGUUCXAVTOUUGYDYDYEUCUUHMYDYEMNWAXNHHWEQOKUUEXORUUEKWBWFV
|
|
PWCYAXEPCZYGYOEZAVTUUIYDYEUUJMNXEHUTQOWDOSWGYGWHWIWRYHXFBWRXFWRXEWRXEAWJWKV
|
|
CVDSVEWOWLWMWPWNWQ $.
|
|
|
|
$( An integer is even or odd but not both. (Contributed by Mario Carneiro,
|
|
12-Sep-2015.) $)
|
|
zeo2 $p |- ( N e. ZZ ->
|
|
( ( N / 2 ) e. ZZ <-> -. ( ( N + 1 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel c2 cdiv co c1 caddc wn cmul wceq cc zcn peano2cn syl 2cnd cc0 cap
|
|
wbr divcanap2d a1i oveq1d eqtr4d wne expcom necon2bd syl5com zeo orcomd ord
|
|
2ap0 zneo impbid ) ABCZADEFZBCZAGHFZDEFZBCZIZUNDURJFZDUOJFZGHFZKUPUTUNVAUQV
|
|
CUNUQDUNALCUQLCAMZANOUNPZDQRSUNUKUAZTUNVBAGHUNADVDVEVFTUBUCUPUSVAVCUSUPVAVC
|
|
UDURUOULUEUFUGUNUSUPUNUPUSAUHUIUJUM $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Second Peano postulate for upper integers. (Contributed by NM,
|
|
3-Oct-2004.) $)
|
|
peano2uz2 $p |- ( ( A e. ZZ /\ B e. { x e. ZZ | A <_ x } ) ->
|
|
( B + 1 ) e. { x e. ZZ | A <_ x } ) $=
|
|
( cz wcel cle wbr wa c1 caddc co cv crab peano2z ad2antrl cr wi zre breq2
|
|
elrab lep1 adantl peano2re ancli letr 3expb sylan2 mpan2d syl2an impr jca
|
|
anbi2i 3imtr4i ) BDEZCDEZBCFGZHZHZCIJKZDEZBUSFGZHUNCBALZFGZADMZEZHUSVDEUR
|
|
UTVAUOUTUNUPCNOUNUOUPVAUNBPEZCPEZUPVAQUOBRCRVFVGHUPCUSFGZVAVGVHVFCUAUBVGV
|
|
FVGUSPEZHUPVHHVAQZVGVICUCUDVFVGVIVJBCUSUEUFUGUHUIUJUKVEUQUNVCUPACDVBCBFST
|
|
ULVCVAAUSDVBUSBFSTUM $.
|
|
$}
|
|
|
|
${
|
|
$d k n x A $. $d k n x N $.
|
|
$( Peano's inductive postulate for upper integers. (Contributed by NM,
|
|
6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.) $)
|
|
peano5uzti $p |- ( N e. ZZ -> ( ( N e. A /\ A. x e. A ( x + 1 ) e. A )
|
|
-> { k e. ZZ | N <_ k } C_ A ) ) $=
|
|
( vn cz wcel cv c1 caddc co wa cc wceq cn wi oveq1 eleq1d imbi2d ex breq2
|
|
wral cle wbr crab wss elrab anbi2i cmin ad2antrl 1cnd subcld npcan syl2an
|
|
zcn ax-1cn subsub mp3an3 cn0 znn0sub biimpa anasss ancoms adantll nn0p1nn
|
|
eqeltrd simpr simpll adantr pncan3d simprl rspccv ad2antll nncnd ad2antlr
|
|
syl add32d sylibd a2d nnind syl3c eqeltrrd sylanb expcom expdimp ssrdv )
|
|
DFGZDBGZAHZIJKZBGZABUBZLZDCHZUCUDZCFUEZBUFWGWMLZEWPBWGWMEHZWPGZWRBGZWMWSL
|
|
ZWGWTXAWMWRFGZDWRUCUDZLZLZWGWTWSXDWMWOXCCWRFWNWRDUCUAUGUHXEWGLZWRDIUIKZUI
|
|
KZXGJKZWRBXEWRMGZXGMGZXIWRNWGXBXJWMXCWRUOUJZWGDIDUOZWGUKULZWRXGUMUNXFXHOG
|
|
WGWMXIBGZXFXHWRDUIKZIJKZOXEXJDMGZXHXQNZWGXLXMXJXRIMGXSUPWRDIUQURUNXFXPUSG
|
|
ZXQOGXDWGXTWMWGXDXTWGXBXCXTWGXBLXCXTDWRUTVAVBVCVDXPVEVPVFXEWGVGWMXDWGVHWG
|
|
WMWNXGJKZBGZPZPWGWMIXGJKZBGZPZPWGWMWRXGJKZBGZPZPWGWMWRIJKZXGJKZBGZPZPWGWM
|
|
XOPZPCEXHWNINZYCYFWGYOYBYEWMYOYAYDBWNIXGJQRSSWNWRNZYCYIWGYPYBYHWMYPYAYGBW
|
|
NWRXGJQRSSWNYJNZYCYMWGYQYBYLWMYQYAYKBWNYJXGJQRSSWNXHNZYCYNWGYRYBXOWMYRYAX
|
|
IBWNXHXGJQRSSWGWMYEWQYDDBWQIDWQUKWGXRWMXMVIVJWGWHWLVKVFTWROGZWGYIYMYSWGYI
|
|
YMPYSWGLZWMYHYLYTWMYHYLPYTWMLZYHYGIJKZBGZYLWLYHUUCPYTWHWKUUCAYGBWIYGNWJUU
|
|
BBWIYGIJQRVLVMUUAUUBYKBUUAWRXGIUUAWRYSWGWMVHVNWGXKYSWMXNVOUUAUKVQRVRTVSTV
|
|
SVTWAWBWCWDWEWFT $.
|
|
|
|
${
|
|
peano5uz.1 $e |- N e. ZZ $.
|
|
$( Peano's inductive postulate for upper integers. (Contributed by NM,
|
|
6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.) $)
|
|
peano5uzi $p |- ( ( N e. A /\ A. x e. A ( x + 1 ) e. A ) ->
|
|
{ k e. ZZ | N <_ k } C_ A ) $=
|
|
( cz wcel cv c1 caddc co wral wa cle wbr crab wss wi peano5uzti ax-mp )
|
|
DFGDBGAHIJKBGABLMDCHNOCFPBQREABCDST $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z N $.
|
|
dfuz.1 $e |- N e. ZZ $.
|
|
$( An expression for the upper integers that start at ` N ` that is
|
|
analogous to ~ dfnn2 for positive integers. (Contributed by NM,
|
|
6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.) $)
|
|
dfuzi $p |- { z e. ZZ | N <_ z } =
|
|
|^| { x | ( N e. x /\ A. y e. x ( y + 1 ) e. x ) } $=
|
|
( cv cle wbr cz crab wcel c1 caddc co wral wa cab wss mpbir2an eleq2 cint
|
|
ssintab peano5uzi mpgbir zrei leidi breq2 elrab peano2uz2 mpan rgen rabex
|
|
wi zex wceq raleqbi1dv anbi12d elab intss1 ax-mp eqssi ) DCFZGHZCIJZDAFZK
|
|
ZBFZLMNZVEKZBVEOZPZAQZUAZVDVMRVKVDVERUMAVKAVDUBBVECDEUCUDVDVLKZVMVDRVNDVD
|
|
KZVHVDKZBVDOZVODIKZDDGHZEDDEUEUFVCVSCDIVBDDGUGUHSVPBVDVRVGVDKVPECDVGUIUJU
|
|
KVKVOVQPAVDVCCIUNULVEVDUOVFVOVJVQVEVDDTVIVPBVEVDVEVDVHTUPUQURSVDVLUSUTVA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d j w N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d k w ph $.
|
|
$d j k w M $.
|
|
$( Substitutions. $)
|
|
uzind.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
|
|
uzind.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
|
|
uzind.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
uzind.5 $e |- ( M e. ZZ -> ps ) $.
|
|
$( Induction step. $)
|
|
uzind.6 $e |- ( ( M e. ZZ /\ k e. ZZ /\ M <_ k ) -> ( ch -> th ) ) $.
|
|
$( Induction on the upper integers that start at ` M ` . The first four
|
|
hypotheses give us the substitution instances we need; the last two are
|
|
the basis and the induction step. (Contributed by NM, 5-Jul-2005.) $)
|
|
uzind $p |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ta ) $=
|
|
( cz wcel cle wbr wa vw w3a cv crab c1 caddc wral wss zre leidd jca ancli
|
|
co wceq breq2 anbi12d elrab sylibr wi peano2z a1i adantrd clt ltp1 adantl
|
|
cr peano2re lelttr 3expb sylan2 mpan2d ltle syld syl2an expimpd 3exp jcad
|
|
imp4d 3imtr4g ralrimiv peano5uzti mp2and sseld 3imtr3g 3impib simprd ) HP
|
|
QZIPQZHIRSZUBZWIEWJWHWIETZWGWHWIWHWKTZWGIHUAUCZRSZUAPUDZQIHFUCZRSZATZFPUD
|
|
ZQWHWITWLWGWOWSIWGHWSQZGUCZUEUFUMZWSQZGWSUGWOWSUHWGWGHHRSZBTZTWTWGXEWGXDB
|
|
WGHHUIZUJNUKULWRXEFHPWPHUNWQXDABWPHHRUOJUPUQURWGXCGWSWGXAPQZHXARSZCTZTZXB
|
|
PQZHXBRSZDTZTXAWSQXCWGXJXKXMWGXGXKXIXGXKUSWGXAUTVAVBWGXJXLDWGXGXIXLWGXGTX
|
|
HXLCWGHVFQZXAVFQZXHXLUSXGXFXAUIXNXOTZXHHXBVCSZXLXPXHXAXBVCSZXQXOXRXNXAVDV
|
|
EXOXNXOXBVFQZTXHXRTXQUSZXOXSXAVGZULXNXOXSXTHXAXBVHVIVJVKXOXNXSXQXLUSYAHXB
|
|
VLVJVMVNVBVOWGXGXHCDWGXGXHCDUSOVPVRVQVQWRXIFXAPWPXAUNWQXHACWPXAHRUOKUPUQW
|
|
RXMFXBPWPXBUNWQXLADWPXBHRUOLUPUQVSVTGWSUAHWAWBWCWNWIUAIPWMIHRUOUQWRWKFIPW
|
|
PIUNWQWIAEWPIHRUOMUPUQWDWEWFWF $.
|
|
$}
|
|
|
|
${
|
|
$d j N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d k ph $.
|
|
$d j k M $.
|
|
$( Substitutions. $)
|
|
uzind2.1 $e |- ( j = ( M + 1 ) -> ( ph <-> ps ) ) $.
|
|
uzind2.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
|
|
uzind2.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind2.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
uzind2.5 $e |- ( M e. ZZ -> ps ) $.
|
|
$( Induction step. $)
|
|
uzind2.6 $e |- ( ( M e. ZZ /\ k e. ZZ /\ M < k ) -> ( ch -> th ) ) $.
|
|
$( Induction on the upper integers that start _after_ an integer ` M ` .
|
|
The first four hypotheses give us the substitution instances we need;
|
|
the last two are the basis and the induction step. (Contributed by NM,
|
|
25-Jul-2005.) $)
|
|
uzind2 $p |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta ) $=
|
|
( cz wcel wbr wi wceq clt wa c1 caddc co zltp1le peano2z cv imbi2d 3expia
|
|
cle a1i w3a sylbird ex com3l imp 3adant1 a2d uzind 3exp syl com34 pm2.43a
|
|
sylbid 3impia ) HPQZIPQZHIUARZEVGVHUBVIHUCUDUEZIUKRZEHIUFVGVHVKESZVHVGVLV
|
|
GVHVKVGEVGVJPQZVHVKVGESZSSHUGVMVHVKVNVGASVGBSZVGCSVGDSVNFGVJIFUHZVJTABVGJ
|
|
UIVPGUHZTACVGKUIVPVQUCUDUETADVGLUIVPITAEVGMUIVOVMNULVMVQPQZVJVQUKRZUMVGCD
|
|
VRVSVGCDSZSZVMVRVSWAVGVRVSVTVGVRVSVTSVGVRUBVSHVQUARZVTHVQUFVGVRWBVTOUJUNU
|
|
OUPUQURUSUTVAVBVCVDUQVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d j k N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d m ph $.
|
|
$d j m k M $.
|
|
$( Substitutions. $)
|
|
uzind3.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
|
|
uzind3.2 $e |- ( j = m -> ( ph <-> ch ) ) $.
|
|
uzind3.3 $e |- ( j = ( m + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind3.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
uzind3.5 $e |- ( M e. ZZ -> ps ) $.
|
|
$( Induction step. $)
|
|
uzind3.6 $e |- ( ( M e. ZZ /\ m e. { k e. ZZ | M <_ k } ) ->
|
|
( ch -> th ) ) $.
|
|
$( Induction on the upper integers that start at an integer ` M ` . The
|
|
first four hypotheses give us the substitution instances we need, and
|
|
the last two are the basis and the induction step. (Contributed by NM,
|
|
26-Jul-2005.) $)
|
|
uzind3 $p |- ( ( M e. ZZ /\ N e. { k e. ZZ | M <_ k } ) -> ta ) $=
|
|
( cle wbr cz wcel cv wa breq2 elrab wi sylan2br 3impb uzind 3expb sylan2b
|
|
crab ) JIGUAZQRZGSUKZTISTZJSTZIJQRZUBEUMUQGJSULJIQUCUDUOUPUQEABCDEFHIJKLM
|
|
NOUOHUAZSTZIURQRZCDUEZUSUTUBUOURUNTVAUMUTGURSULURIQUCUDPUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A x $. $d ps x $. $d ch x $. $d th x $. $d ta x $.
|
|
$d ph y $.
|
|
nn0ind.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
|
|
nn0ind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
nn0ind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
nn0ind.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
nn0ind.5 $e |- ps $.
|
|
nn0ind.6 $e |- ( y e. NN0 -> ( ch -> th ) ) $.
|
|
$( Principle of Mathematical Induction (inference schema) on nonnegative
|
|
integers. The first four hypotheses give us the substitution instances
|
|
we need; the last two are the basis and the induction step.
|
|
(Contributed by NM, 13-May-2004.) $)
|
|
nn0ind $p |- ( A e. NN0 -> ta ) $=
|
|
( cn0 wcel cz cc0 cle wbr wa elnn0z 0z a1i cv sylbir 3adant1 uzind mp3an1
|
|
wi sylbi ) HOPHQPZRHSTZUAEHUBRQPZULUMEUCABCDEFGRHIJKLBUNMUDGUEZQPZRUOSTZC
|
|
DUJZUNUPUQUAUOOPURUOUBNUFUGUHUIUK $.
|
|
$}
|
|
|
|
${
|
|
$d K x $. $d M x y $. $d N x y $. $d ch x $. $d ph y $. $d ps x $.
|
|
$d ta x $. $d th x $.
|
|
fzind.1 $e |- ( x = M -> ( ph <-> ps ) ) $.
|
|
fzind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
fzind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
fzind.4 $e |- ( x = K -> ( ph <-> ta ) ) $.
|
|
fzind.5 $e |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ps ) $.
|
|
fzind.6 $e |- ( ( ( M e. ZZ /\ N e. ZZ ) /\
|
|
( y e. ZZ /\ M <_ y /\ y < N ) ) -> ( ch -> th ) ) $.
|
|
$( Induction on the integers from ` M ` to ` N ` inclusive . The first
|
|
four hypotheses give us the substitution instances we need; the last two
|
|
are the basis and the induction step. (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
fzind $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\
|
|
( K e. ZZ /\ M <_ K /\ K <_ N ) ) -> ta ) $=
|
|
( cle wbr wa wi cz wcel w3a cv c1 caddc co breq1 anbi2d imbi12d 3expib cr
|
|
wceq zre p1le 3expia syl2an imdistanda imim1d 3ad2ant2 wb zltp1le adantlr
|
|
clt expcom pm5.32d adantl 3expa com12 sylbird com23 expd 3impib impd syld
|
|
ex a2d uzind expcomd 3expb 3impia impcom ) HUAUBZIHQRZHJQRZUCZIUAUBZJUAUB
|
|
ZSZEWFWGWHEWCWDWEWGWHETZTWCWDSZWGWEWJWGWKWEWJTZWGWCWDWLWGWCWDUCWHWEEWHFUD
|
|
ZJQRZSZATWHIJQRZSZBTWHGUDZJQRZSZCTZWHWRUEUFUGZJQRZSZDTZWHWESZETFGIHWMIUMZ
|
|
WOWQABXGWNWPWHWMIJQUHUIKUJWMWRUMZWOWTACXHWNWSWHWMWRJQUHUILUJWMXBUMZWOXDAD
|
|
XIWNXCWHWMXBJQUHUIMUJWMHUMZWOXFAEXJWNWEWHWMHJQUHUINUJWGWHWPBOUKWGWRUAUBZI
|
|
WRQRZUCZXAXDCTZXEXKWGXAXNTXLXKXDWTCXKWHXCWSXKWRULUBZJULUBZXCWSTWHWRUNJUNX
|
|
OXPXCWSWRJUOUPUQURUSUTXMXDCDXMWHXCCDTZXMXCWHXQWGXKXLXCWHXQTZTWGXKXLSZXCXR
|
|
WGWHXSXCSZXQWGWHXTXQTWIXTXSWRJVDRZSZXQWHYBXTVAWGWHXSYAXCXSWHYAXCVAZXKWHYC
|
|
XLWRJVBVCVEVFVGYBWIXQXKXLYAWIXQTWIXKXLYAUCXQPVEVHVIVJVPVKVLVMVKVNVQVOVRVS
|
|
VTVEVKWAVNWB $.
|
|
$}
|
|
|
|
${
|
|
$d K x $. $d N x y $. $d ch x $. $d ph y $. $d ps x $. $d ta x $.
|
|
$d th x $.
|
|
fnn0ind.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
|
|
fnn0ind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
fnn0ind.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
fnn0ind.4 $e |- ( x = K -> ( ph <-> ta ) ) $.
|
|
fnn0ind.5 $e |- ( N e. NN0 -> ps ) $.
|
|
fnn0ind.6 $e |- ( ( N e. NN0 /\ y e. NN0 /\ y < N ) -> ( ch -> th ) ) $.
|
|
$( Induction on the integers from ` 0 ` to ` N ` inclusive . The first
|
|
four hypotheses give us the substitution instances we need; the last two
|
|
are the basis and the induction step. (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
fnn0ind $p |- ( ( N e. NN0 /\ K e. NN0 /\ K <_ N ) -> ta ) $=
|
|
( wcel wbr wa cc0 wi cn0 cle cz elnn0z w3a nn0z 0z sylbir 3adant1 clt zre
|
|
cv cr 0re lelttr ltle 3adant2 mp3an1 syl2an ex com23 3impib impcom anbi1i
|
|
syld 3expb syl2anbr expcom 3impa expd mpd adantll fzind mpanl1 syl5 3expa
|
|
sylanb 3impb ) IUAPZHUAPZHIUBQZEVTWARVSEVTHUCPZSHUBQZRWAVSETZHUDWBWCWAWDV
|
|
SIUCPZWBWCWAUEZEIUFWEWFESUCPZWEWFEUGABCDEFGHSIJKLMWESIUBQZBWGWEWHRZVSBIUD
|
|
ZNUHUIWEGULZUCPZSWKUBQZWKIUJQZUEZCDTZWGWEWORWHWPWOWEWHWLWMWNWEWHTWLWEWMWN
|
|
RZWHWLWEWQWHTZWLWKUMPZIUMPZWRWEWKUKIUKSUMPZWSWTWRUNXAWSWTUEWQSIUJQZWHSWKI
|
|
UOXAWTXBWHTWSSIUPUQVEURUSUTVAVBVCWOWEWHWPTWOWEWHWPWLWMWNWIWPTWIWLWMRZWNRZ
|
|
WPWIVSWKUAPZWNRWPXDWJXEXCWNWKUDVDVSXEWNWPOVFVGVHVIVJVCVKVLVMVNVHVOVPVQVCV
|
|
R $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A x z $. $d ps x z $. $d ch x z $. $d th x z $.
|
|
$d ta x z $. $d ph y z $.
|
|
nn0ind-raph.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
|
|
nn0ind-raph.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
nn0ind-raph.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
nn0ind-raph.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
nn0ind-raph.5 $e |- ps $.
|
|
nn0ind-raph.6 $e |- ( y e. NN0 -> ( ch -> th ) ) $.
|
|
$( Principle of Mathematical Induction (inference schema) on nonnegative
|
|
integers. The first four hypotheses give us the substitution instances
|
|
we need; the last two are the basis and the induction step. Raph Levien
|
|
remarks: "This seems a bit painful. I wonder if an explicit
|
|
substitution version would be easier." (Contributed by Raph Levien,
|
|
10-Apr-2004.) $)
|
|
nn0ind-raph $p |- ( A e. NN0 -> ta ) $=
|
|
( vz wcel cc0 wceq c1 wi cn0 cn wo elnn0 wsb wsbc dfsbcq2 cv sbhypf caddc
|
|
nfv co nfsbc1v 1ex c0ex wa 0nn0 eleq1a ax-mp mpbiri eqeq2 syl6bir pm5.74d
|
|
wb mpbii com12 vtocle sylc adantr oveq1 0p1e1 syl6eq eqeq2d imp mpbird ex
|
|
sbceq1a mpbid vtoclef nnnn0 syl nnind bicomd sylan9bb sylbird eqcoms jaoi
|
|
eqeq1 sylbi ) HUAPHUBPZHQRZUCEHUDWJEWKAFOUEAFSUFZCDEOGHAFOSUGACFOGUHZCFUK
|
|
JUIADFOWMSUJULZDFUKKUIAEFOHEFUKLUIWLFSAFSUMUNFUHZSRZAWLWPATGQUOWMQRZWPAWQ
|
|
WPUPADWQDWPWQWMUAPZCDQUAPWQWRTUQQUAWMURUSWQCTFQUOWQWOQRZCWQWSATWSCTWSABMI
|
|
UTWQWSACWQWSWOWMRACVDWMQWOVAJVBVCVEVFVGNVHVIWQWPADVDZWQWPWOWNRWTWQWNSWOWQ
|
|
WNQSUJULSWMQSUJVJVKVLVMKVBVNVOVPVGAFSVQVRVSWMUBPWRCDTWMVTNWAWBEQHQHRZETZF
|
|
QXBFUKUOWSXAWOHRZEWOQHWHWSXCEWSXCUPBEMWSBAXCEWSABIWCLWDVEVPWEVSWFWGWI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ch x $. $d et x $. $d ph y $. $d ps x $. $d ta x $.
|
|
$d th x $. $d x y ze $.
|
|
zindd.1 $e |- ( x = 0 -> ( ph <-> ps ) ) $.
|
|
zindd.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
zindd.3 $e |- ( x = ( y + 1 ) -> ( ph <-> ta ) ) $.
|
|
zindd.4 $e |- ( x = -u y -> ( ph <-> th ) ) $.
|
|
zindd.5 $e |- ( x = A -> ( ph <-> et ) ) $.
|
|
zindd.6 $e |- ( ze -> ps ) $.
|
|
zindd.7 $e |- ( ze -> ( y e. NN0 -> ( ch -> ta ) ) ) $.
|
|
zindd.8 $e |- ( ze -> ( y e. NN -> ( ch -> th ) ) ) $.
|
|
$( Principle of Mathematical Induction on all integers, deduction version.
|
|
The first five hypotheses give the substitutions; the last three are the
|
|
basis, the induction, and the extension to negative numbers.
|
|
(Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
zindd $p |- ( ze -> ( A e. ZZ -> et ) ) $=
|
|
( cz wcel wral wi cv cneg cn0 cn wo cr znegcl elznn0nn sylib simpr orim2i
|
|
wa syl zcn negnegd eleq1d orbi2d mpbid cc0 wceq imbi2d c1 caddc com12 a2d
|
|
co nn0ind nnnn0 mpdd jaod syl5 ralrimiv wb sylan9eqr eqcomd bicomd rspcdv
|
|
negeq rspccv 3syl ) GDISUAZAHSUAJSTFUBGDISIUCZSTZWDUDZUETZWDUFTZUGZGDWEWG
|
|
WFUDZUFTZUGZWIWEWGWFUHTZWKUNZUGZWLWEWFSTWOWDUIWFUJUKWNWKWGWMWKULUMUOWEWKW
|
|
HWGWEWJWDUFWEWDWDUPUQURUSUTGWGDWHWGGDGAUBZGBUBZGCUBZGEUBZGDUBHIWFHUCZVAVB
|
|
ABGKVCZWTWDVBACGLVCZWTWDVDVEVHVBAEGMVCZWTWFVBZADGNVCPWDUETZGCEGXECEUBQVFV
|
|
GZVIVFGWHCDWHGCWHXEWRWDVJWPWQWRWSWRHIWDXAXBXCXBPXFVIUOVFRVKVLVMVNWCAHSWTS
|
|
TZWCAXGDAIWTUDZSWTUIXGWDXHVBZUNZADXJXDADVOXJWFWTXIXGWFXHUDWTWDXHVTXGWTWTU
|
|
PUQVPVQNUOVRVSVFVNAFHJSOWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y A $.
|
|
$( Any real number can be sandwiched between two integers. Exercise 2 of
|
|
[Apostol] p. 28. (Contributed by NM, 10-Nov-2004.) $)
|
|
btwnz $p |- ( A e. RR -> ( E. x e. ZZ x < A /\ E. y e. ZZ A < y ) ) $=
|
|
( vz cr wcel cv clt wbr cz wrex cneg cn renegcl arch wa wb nnre ltnegcon1
|
|
syl ex syl5 pm5.32d nnnegz breq1 rspcev sylan syl6bi expd rexlimdv anim1i
|
|
mpd nnz reximi2 jca ) CEFZAGZCHIZAJKZCBGZHIZBJKZUPCLZDGZHIZDMKZUSUPVCEFVF
|
|
CNVCDOTUPVEUSDMUPVDMFZVEUSUPVGVEPVGVDLZCHIZPUSUPVGVEVIVGVDEFZUPVEVIQZVDRU
|
|
PVJVKCVDSUAUBUCVGVHJFVIUSVDUDURVIAVHJUQVHCHUEUFUGUHUIUJULUPVABMKVBCBOVAVA
|
|
BMJUTMFUTJFVAUTUMUKUNTUO $.
|
|
$}
|
|
|
|
${
|
|
nn0zd.1 $e |- ( ph -> A e. NN0 ) $.
|
|
$( A positive integer is an integer. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
nn0zd $p |- ( ph -> A e. ZZ ) $=
|
|
( cn0 cz nn0ssz sseldi ) ADEBFCG $.
|
|
$}
|
|
|
|
${
|
|
nnzd.1 $e |- ( ph -> A e. NN ) $.
|
|
$( A nonnegative integer is an integer. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
nnzd $p |- ( ph -> A e. ZZ ) $=
|
|
( nnnn0d nn0zd ) ABABCDE $.
|
|
$}
|
|
|
|
${
|
|
zred.1 $e |- ( ph -> A e. ZZ ) $.
|
|
$( An integer is a real number. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
zred $p |- ( ph -> A e. RR ) $=
|
|
( cz cr zssre sseldi ) ADEBFCG $.
|
|
|
|
$( An integer is a complex number. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
zcnd $p |- ( ph -> A e. CC ) $=
|
|
( zred recnd ) ABABCDE $.
|
|
|
|
$( Closure law for negative integers. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
znegcld $p |- ( ph -> -u A e. ZZ ) $=
|
|
( cz wcel cneg znegcl syl ) ABDEBFDECBGH $.
|
|
|
|
$( Deduction from second Peano postulate generalized to integers.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
peano2zd $p |- ( ph -> ( A + 1 ) e. ZZ ) $=
|
|
( cz wcel c1 caddc co peano2z syl ) ABDEBFGHDECBIJ $.
|
|
|
|
zaddcld.1 $e |- ( ph -> B e. ZZ ) $.
|
|
$( Closure of addition of integers. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
zaddcld $p |- ( ph -> ( A + B ) e. ZZ ) $=
|
|
( cz wcel caddc co zaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure of subtraction of integers. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
zsubcld $p |- ( ph -> ( A - B ) e. ZZ ) $=
|
|
( cz wcel cmin co zsubcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure of multiplication of integers. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
zmulcld $p |- ( ph -> ( A x. B ) e. ZZ ) $=
|
|
( cz wcel cmul co zmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
$( Increasing an integer by 2 results in an integer. (Contributed by
|
|
Alexander van der Vekens, 16-Sep-2018.) $)
|
|
zadd2cl $p |- ( N e. ZZ -> ( N + 2 ) e. ZZ ) $=
|
|
( cz wcel c2 id 2z a1i zaddcld ) ABCZADIEDBCIFGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Decimal arithmetic
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ; $.
|
|
$( Constant used for decimal constructor. $)
|
|
cdc $a class ; A B $.
|
|
|
|
$( Define the "decimal constructor", which is used to build up "decimal
|
|
integers" or "numeric terms" in base ` 1 0 ` . For example,
|
|
` ( ; ; ; 1 0 0 0 + ; ; ; 2 0 0 0 ) = ; ; ; 3 0 0 0 ` ~ 1kp2ke3k .
|
|
(Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV,
|
|
1-Aug-2021.) $)
|
|
df-dec $a |- ; A B = ( ( ( 9 + 1 ) x. A ) + B ) $.
|
|
|
|
$( 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by
|
|
Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) $)
|
|
9p1e10 $p |- ( 9 + 1 ) = ; 1 0 $=
|
|
( c1 cc0 cdc c9 caddc co cmul df-dec cn 9nn 1nn nnaddcl mp2an nncni mulid1i
|
|
wcel oveq1i addid1i 3eqtrri ) ABCDAEFZAGFZBEFTBEFTABHUATBETTDIPAIPTIPJKDALM
|
|
NZOQTUBRS $.
|
|
|
|
$( Version of the definition of the "decimal constructor" using ` ; 1 0 `
|
|
instead of the symbol 10. Of course, this statement cannot be used as
|
|
definition, because it uses the "decimal constructor". (Contributed by
|
|
AV, 1-Aug-2021.) $)
|
|
dfdec10 $p |- ; A B = ( ( ; 1 0 x. A ) + B ) $=
|
|
( cdc c9 c1 caddc co cmul cc0 df-dec 9p1e10 oveq1i eqtri ) ABCDEFGZAHGZBFGE
|
|
ICZAHGZBFGABJOQBFNPAHKLLM $.
|
|
|
|
$( Equality theorem for the decimal constructor. (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
deceq1 $p |- ( A = B -> ; A C = ; B C ) $=
|
|
( wceq c9 c1 caddc co cmul cdc oveq2 oveq1d df-dec 3eqtr4g ) ABDZEFGHZAIHZC
|
|
GHPBIHZCGHACJBCJOQRCGABPIKLACMBCMN $.
|
|
|
|
$( Equality theorem for the decimal constructor. (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
deceq2 $p |- ( A = B -> ; C A = ; C B ) $=
|
|
( wceq c9 c1 caddc co cmul cdc oveq2 df-dec 3eqtr4g ) ABDEFGHCIHZAGHNBGHCAJ
|
|
CBJABNGKCALCBLM $.
|
|
|
|
${
|
|
deceq1i.1 $e |- A = B $.
|
|
$( Equality theorem for the decimal constructor. (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) $)
|
|
deceq1i $p |- ; A C = ; B C $=
|
|
( wceq cdc deceq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality theorem for the decimal constructor. (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) $)
|
|
deceq2i $p |- ; C A = ; C B $=
|
|
( wceq cdc deceq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
deceq12i.2 $e |- C = D $.
|
|
$( Equality theorem for the decimal constructor. (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) $)
|
|
deceq12i $p |- ; A C = ; B D $=
|
|
( cdc deceq1i deceq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
numnncl.1 $e |- T e. NN0 $.
|
|
numnncl.2 $e |- A e. NN0 $.
|
|
${
|
|
numnncl.3 $e |- B e. NN $.
|
|
$( Closure for a numeral (with units place). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numnncl $p |- ( ( T x. A ) + B ) e. NN $=
|
|
( cmul co cn0 wcel cn caddc nn0mulcli nn0nnaddcl mp2an ) CAGHZIJBKJPBLH
|
|
KJCADEMFPBNO $.
|
|
$}
|
|
|
|
$( Add a zero in the units place. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
num0u $p |- ( T x. A ) = ( ( T x. A ) + 0 ) $=
|
|
( cmul co cc0 caddc nn0mulcli nn0cni addid1i eqcomi ) BAEFZGHFMMMBACDIJKL
|
|
$.
|
|
|
|
$( Add a zero in the higher places. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
num0h $p |- A = ( ( T x. 0 ) + A ) $=
|
|
( cc0 cmul co caddc nn0cni mul01i oveq1i addid2i eqtr2i ) BEFGZAHGEAHGANE
|
|
AHBBCIJKAADILM $.
|
|
|
|
numcl.2 $e |- B e. NN0 $.
|
|
$( Closure for a decimal integer (with units place). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numcl $p |- ( ( T x. A ) + B ) e. NN0 $=
|
|
( cmul co nn0mulcli nn0addcli ) CAGHBCADEIFJ $.
|
|
|
|
numsuc.4 $e |- ( B + 1 ) = C $.
|
|
numsuc.5 $e |- N = ( ( T x. A ) + B ) $.
|
|
$( The successor of a decimal integer (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numsuc $p |- ( N + 1 ) = ( ( T x. A ) + C ) $=
|
|
( c1 caddc co cmul oveq1i nn0mulcli nn0cni ax-1cn addassi oveq2i 3eqtri )
|
|
EKLMDANMZBLMZKLMUBBKLMZLMUBCLMEUCKLJOUBBKUBDAFGPQBHQRSUDCUBLITUA $.
|
|
$}
|
|
|
|
${
|
|
deccl.1 $e |- A e. NN0 $.
|
|
deccl.2 $e |- B e. NN0 $.
|
|
$( Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
deccl $p |- ; A B e. NN0 $=
|
|
( cdc c9 c1 caddc co cmul cn0 df-dec 9nn0 1nn0 nn0addcli numcl eqeltri )
|
|
ABEFGHIZAJIBHIKABLABRFGMNOCDPQ $.
|
|
$}
|
|
|
|
$( 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
10nn $p |- ; 1 0 e. NN $=
|
|
( c9 c1 caddc co cc0 cdc cn 9p1e10 wcel 9nn peano2nn ax-mp eqeltrri ) ABCDZ
|
|
BEFGHAGINGIJAKLM $.
|
|
|
|
$( The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by
|
|
AV, 8-Sep-2021.) $)
|
|
10pos $p |- 0 < ; 1 0 $=
|
|
( c1 cc0 cdc 10nn nngt0i ) ABCDE $.
|
|
|
|
$( 10 is a nonnegative integer. (Contributed by Mario Carneiro,
|
|
19-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
10nn0 $p |- ; 1 0 e. NN0 $=
|
|
( c1 cc0 1nn0 0nn0 deccl ) ABCDE $.
|
|
|
|
$( The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV,
|
|
8-Sep-2021.) $)
|
|
10re $p |- ; 1 0 e. RR $=
|
|
( c1 cc0 cdc 10nn nnrei ) ABCDE $.
|
|
|
|
${
|
|
decnncl.1 $e |- A e. NN0 $.
|
|
decnncl.2 $e |- B e. NN $.
|
|
$( Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
decnncl $p |- ; A B e. NN $=
|
|
( cdc c1 cc0 cmul co caddc cn dfdec10 10nn0 numnncl eqeltri ) ABEFGEZAHIB
|
|
JIKABLABPMCDNO $.
|
|
$}
|
|
|
|
${
|
|
dec0u.1 $e |- A e. NN0 $.
|
|
$( Add a zero in the units place. (Contributed by Mario Carneiro,
|
|
17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
dec0u $p |- ( ; 1 0 x. A ) = ; A 0 $=
|
|
( c1 cc0 cdc cmul co caddc 10nn0 num0u dfdec10 eqtr4i ) CDEZAFGZNDHGADEAM
|
|
IBJADKL $.
|
|
|
|
$( Add a zero in the higher places. (Contributed by Mario Carneiro,
|
|
17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
dec0h $p |- A = ; 0 A $=
|
|
( c1 cc0 cdc cmul co caddc 10nn0 num0h dfdec10 eqtr4i ) ACDEZDFGAHGDAEAMI
|
|
BJDAKL $.
|
|
$}
|
|
|
|
${
|
|
numnncl2.1 $e |- T e. NN $.
|
|
numnncl2.2 $e |- A e. NN $.
|
|
$( Closure for a decimal integer (zero units place). (Contributed by Mario
|
|
Carneiro, 9-Mar-2015.) $)
|
|
numnncl2 $p |- ( ( T x. A ) + 0 ) e. NN $=
|
|
( cmul co cc0 caddc cn nnmulcli nncni addid1i eqeltri ) BAEFZGHFNINNBACDJ
|
|
ZKLOM $.
|
|
$}
|
|
|
|
${
|
|
decnncl2.1 $e |- A e. NN $.
|
|
$( Closure for a decimal integer (zero units place). (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
decnncl2 $p |- ; A 0 e. NN $=
|
|
( cc0 cdc c1 cmul co caddc cn dfdec10 10nn numnncl2 eqeltri ) ACDECDZAFGC
|
|
HGIACJANKBLM $.
|
|
$}
|
|
|
|
${
|
|
numlt.1 $e |- T e. NN $.
|
|
numlt.2 $e |- A e. NN0 $.
|
|
numlt.3 $e |- B e. NN0 $.
|
|
${
|
|
numlt.4 $e |- C e. NN $.
|
|
numlt.5 $e |- B < C $.
|
|
$( Comparing two decimal integers (equal higher places). (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) $)
|
|
numlt $p |- ( ( T x. A ) + B ) < ( ( T x. A ) + C ) $=
|
|
( clt wbr cmul co caddc nn0rei nnrei nnnn0i nn0mulcli ltadd2i mpbi ) BC
|
|
JKDALMZBNMUACNMJKIBCUABGOCHPUADADEQFROST $.
|
|
$}
|
|
|
|
numltc.3 $e |- C e. NN0 $.
|
|
numltc.4 $e |- D e. NN0 $.
|
|
numltc.5 $e |- C < T $.
|
|
numltc.6 $e |- A < B $.
|
|
$( Comparing two decimal integers (unequal higher places). (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) $)
|
|
numltc $p |- ( ( T x. A ) + C ) < ( ( T x. B ) + D ) $=
|
|
( cmul co caddc clt wbr cle nn0rei wcel numlt nnrei ax-1cn adddii mulid1i
|
|
c1 recni oveq2i eqtri breqtrri cn0 wb nn0ltp1le mp2an cc0 nngt0i peano2re
|
|
mpbi cr ax-mp lemul2i remulcli readdcli ltletri nn0addge1i ) EAMNZCONZEBM
|
|
NZPQZVHVHDONZRQVGVJPQVGEAUFONZMNZPQVLVHRQZVIVGVFEONZVLPACEEFGIFKUAVLVFEUF
|
|
MNZONVNEAUFEEFUBZUGZAAGSZUGUCUDVOEVFOEVQUEUHUIUJVKBRQZVMABPQZVSLAUKTBUKTV
|
|
TVSULGHABUMUNURUOEPQVSVMULEFUPVKBEAUSTVKUSTVRAUQUTZBHSZVPVAUTURVGVLVHVFCE
|
|
AVPVRVBCISVCZEVKVPWAVBEBVPWBVBZVDUNVHDWDJVEVGVHVJWCWDVHDWDDJSVCVDUN $.
|
|
$}
|
|
|
|
${
|
|
le9lt10.c $e |- A e. NN0 $.
|
|
le9lt10.e $e |- A <_ 9 $.
|
|
$( A "decimal digit" (i.e. a nonnegative integer less than or equal to 9)
|
|
is less then 10. (Contributed by AV, 8-Sep-2021.) $)
|
|
le9lt10 $p |- A < ; 1 0 $=
|
|
( c9 c1 caddc co cc0 cdc clt cle cz wcel wb nn0zi 9nn0 zleltp1 mp2an mpbi
|
|
wbr 9p1e10 breqtri ) ADEFGZEHIJADKTZAUCJTZCALMDLMUDUENABODPOADQRSUAUB $.
|
|
$}
|
|
|
|
${
|
|
declt.a $e |- A e. NN0 $.
|
|
declt.b $e |- B e. NN0 $.
|
|
${
|
|
declt.c $e |- C e. NN $.
|
|
declt.l $e |- B < C $.
|
|
$( Comparing two decimal integers (equal higher places). (Contributed by
|
|
Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
declt $p |- ; A B < ; A C $=
|
|
( c1 cc0 cdc cmul co caddc clt 10nn numlt dfdec10 3brtr4i ) HIJZAKLZBML
|
|
TCMLABJACJNABCSODEFGPABQACQR $.
|
|
$}
|
|
|
|
${
|
|
decltc.c $e |- C e. NN0 $.
|
|
decltc.d $e |- D e. NN0 $.
|
|
decltc.s $e |- C < ; 1 0 $.
|
|
decltc.l $e |- A < B $.
|
|
$( Comparing two decimal integers (unequal higher places). (Contributed
|
|
by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decltc $p |- ; A C < ; B D $=
|
|
( c1 cc0 cdc cmul co caddc clt 10nn numltc dfdec10 3brtr4i ) KLMZANOCPO
|
|
UBBNODPOACMBDMQABCDUBREFGHIJSACTBDTUA $.
|
|
$}
|
|
|
|
${
|
|
declth.c $e |- C e. NN0 $.
|
|
declth.d $e |- D e. NN0 $.
|
|
declth.e $e |- C <_ 9 $.
|
|
declth.l $e |- A < B $.
|
|
$( Comparing two decimal integers (unequal higher places). (Contributed
|
|
by AV, 8-Sep-2021.) $)
|
|
declth $p |- ; A C < ; B D $=
|
|
( le9lt10 decltc ) ABCDEFGHCGIKJL $.
|
|
$}
|
|
|
|
decsuc.c $e |- ( B + 1 ) = C $.
|
|
decsuc.n $e |- N = ; A B $.
|
|
$( The successor of a decimal integer (no carry). (Contributed by Mario
|
|
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) $)
|
|
decsuc $p |- ( N + 1 ) = ; A C $=
|
|
( c1 caddc co cc0 cdc cmul 10nn0 dfdec10 eqtri numsuc eqtr4i ) DIJKILMZAN
|
|
KZCJKACMABCTDOEFGDABMUABJKHABPQRACPS $.
|
|
$}
|
|
|
|
${
|
|
3decltc.a $e |- A e. NN0 $.
|
|
3decltc.b $e |- B e. NN0 $.
|
|
3decltc.c $e |- C e. NN0 $.
|
|
3decltc.d $e |- D e. NN0 $.
|
|
3decltc.e $e |- E e. NN0 $.
|
|
3decltc.f $e |- F e. NN0 $.
|
|
3decltc.3 $e |- A < B $.
|
|
${
|
|
3declth.1 $e |- C <_ 9 $.
|
|
3declth.2 $e |- E <_ 9 $.
|
|
$( Comparing two decimal integers with three "digits" (unequal higher
|
|
places). (Contributed by AV, 8-Sep-2021.) $)
|
|
3declth $p |- ; ; A C E < ; ; B D F $=
|
|
( cdc deccl declth ) ACPBDPEFACGIQBDHJQKLOABCDGHIJNMRR $.
|
|
$}
|
|
|
|
3decltc.1 $e |- C < ; 1 0 $.
|
|
3decltc.2 $e |- E < ; 1 0 $.
|
|
$( Comparing two decimal integers with three "digits" (unequal higher
|
|
places). (Contributed by AV, 15-Jun-2021.) (Revised by AV,
|
|
6-Sep-2021.) $)
|
|
3decltc $p |- ; ; A C E < ; ; B D F $=
|
|
( cdc deccl decltc ) ACPBDPEFACGIQBDHJQKLOABCDGHIJNMRR $.
|
|
$}
|
|
|
|
${
|
|
decle.1 $e |- A e. NN0 $.
|
|
decle.2 $e |- B e. NN0 $.
|
|
decle.3 $e |- C e. NN0 $.
|
|
${
|
|
decle.4 $e |- B <_ C $.
|
|
$( Comparing two decimal integers (equal higher places). (Contributed by
|
|
AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) $)
|
|
decle $p |- ; A B <_ ; A C $=
|
|
( c1 cc0 cdc cmul caddc cle wbr nn0rei 10nn0 nn0mulcli leadd2i dfdec10
|
|
co mpbi 3brtr4i ) HIJZAKTZBLTZUDCLTZABJACJMBCMNUEUFMNGBCUDBEOCFOUDUCAPD
|
|
QORUAABSACSUB $.
|
|
$}
|
|
|
|
decleh.4 $e |- D e. NN0 $.
|
|
decleh.5 $e |- C <_ 9 $.
|
|
decleh.6 $e |- A < B $.
|
|
$( Comparing two decimal integers (unequal higher places). (Contributed by
|
|
AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) $)
|
|
decleh $p |- ; A C <_ ; B D $=
|
|
( cdc deccl nn0rei declth ltleii ) ACKZBDKZPACEGLMQBDFHLMABCDEFGHIJNO $.
|
|
$}
|
|
|
|
${
|
|
declei.1 $e |- A e. NN $.
|
|
declei.2 $e |- B e. NN0 $.
|
|
declei.3 $e |- C e. NN0 $.
|
|
declei.4 $e |- C <_ 9 $.
|
|
$( Comparing a digit to a decimal integer. (Contributed by AV,
|
|
17-Aug-2021.) $)
|
|
declei $p |- C <_ ; A B $=
|
|
( cc0 cdc cle dec0h 0nn0 nnnn0i nngt0i decleh eqbrtri ) CHCIABIJCFKHACBLA
|
|
DMFEGADNOP $.
|
|
$}
|
|
|
|
${
|
|
numlti.1 $e |- T e. NN $.
|
|
numlti.2 $e |- A e. NN $.
|
|
numlti.3 $e |- B e. NN0 $.
|
|
numlti.4 $e |- C e. NN0 $.
|
|
numlti.5 $e |- C < T $.
|
|
$( Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
numlti $p |- C < ( ( T x. A ) + B ) $=
|
|
( cc0 cmul co caddc clt nnnn0i num0h 0nn0 nngt0i numltc eqbrtri ) CDJKLCM
|
|
LDAKLBMLNCDDEOHPJACBDEQAFOHGIAFRST $.
|
|
$}
|
|
|
|
${
|
|
declti.a $e |- A e. NN $.
|
|
declti.b $e |- B e. NN0 $.
|
|
declti.c $e |- C e. NN0 $.
|
|
${
|
|
declti.l $e |- C < ; 1 0 $.
|
|
$( Comparing a digit to a decimal integer. (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
declti $p |- C < ; A B $=
|
|
( c1 cc0 cdc cmul co caddc clt 10nn numlti dfdec10 breqtrri ) CHIJZAKLB
|
|
MLABJNABCSODEFGPABQR $.
|
|
$}
|
|
|
|
decltdi.4 $e |- C <_ 9 $.
|
|
$( Comparing a digit to a decimal integer. (Contributed by AV,
|
|
8-Sep-2021.) $)
|
|
decltdi $p |- C < ; A B $=
|
|
( le9lt10 declti ) ABCDEFCFGHI $.
|
|
$}
|
|
|
|
${
|
|
numsucc.1 $e |- Y e. NN0 $.
|
|
numsucc.2 $e |- T = ( Y + 1 ) $.
|
|
numsucc.3 $e |- A e. NN0 $.
|
|
numsucc.4 $e |- ( A + 1 ) = B $.
|
|
numsucc.5 $e |- N = ( ( T x. A ) + Y ) $.
|
|
$( The successor of a decimal integer (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numsucc $p |- ( N + 1 ) = ( ( T x. B ) + 0 ) $=
|
|
( c1 caddc co cmul cc0 cn0 1nn0 nn0addcli nn0cni oveq2i 3eqtr4ri eqeltrri
|
|
eqeltri mulid1i ax-1cn adddii eqcomi numsuc num0u 3eqtri ) DKLMZCAKLMZNMZ
|
|
CBNMZUNOLMCANMZCKNMZLMUOCLMUMUKUPCUOLCCCEKLMZPGEKFQRUCZSZUDTCAKUSAHSUEUFA
|
|
ECCDURHFCUQGUGJUHUAULBCNITBCURULBPIAKHQRUBUIUJ $.
|
|
$}
|
|
|
|
${
|
|
decsucc.1 $e |- A e. NN0 $.
|
|
decsucc.2 $e |- ( A + 1 ) = B $.
|
|
decsucc.3 $e |- N = ; A 9 $.
|
|
$( The successor of a decimal integer (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decsucc $p |- ( N + 1 ) = ; B 0 $=
|
|
( c1 caddc co cc0 cmul c9 9nn0 9p1e10 eqcomi dfdec10 eqtri numsucc eqtr4i
|
|
cdc ) CGHIGJTZBKIJHIBJTABUACLMLGHIUANODECALTUAAKILHIFALPQRBJPS $.
|
|
$}
|
|
|
|
$( The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.) $)
|
|
1e0p1 $p |- 1 = ( 0 + 1 ) $=
|
|
( cc0 c1 caddc co 0p1e1 eqcomi ) ABCDBEF $.
|
|
|
|
$( Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
dec10p $p |- ( ; 1 0 + A ) = ; 1 A $=
|
|
( c1 cdc cc0 cmul co caddc dfdec10 10nn nncni mulid1i oveq1i eqtr2i ) BACBD
|
|
CZBEFZAGFNAGFBAHONAGNNIJKLM $.
|
|
|
|
${
|
|
numma.1 $e |- T e. NN0 $.
|
|
numma.2 $e |- A e. NN0 $.
|
|
numma.3 $e |- B e. NN0 $.
|
|
numma.4 $e |- C e. NN0 $.
|
|
numma.5 $e |- D e. NN0 $.
|
|
numma.6 $e |- M = ( ( T x. A ) + B ) $.
|
|
numma.7 $e |- N = ( ( T x. C ) + D ) $.
|
|
${
|
|
numma.8 $e |- P e. NN0 $.
|
|
numma.9 $e |- ( ( A x. P ) + C ) = E $.
|
|
numma.10 $e |- ( ( B x. P ) + D ) = F $.
|
|
$( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
|
|
a fixed multiplicand ` P ` (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numma $p |- ( ( M x. P ) + N ) = ( ( T x. E ) + F ) $=
|
|
( cmul caddc oveq1i oveq12i nn0cni mulcli adddii mulassi eqtr4i adddiri
|
|
co add4i oveq2i 3eqtr2i ) IEUAUKZJUBUKFAUAUKZBUBUKZEUAUKZFCUAUKZDUBUKZU
|
|
BUKZFAEUAUKZCUBUKZUAUKZBEUAUKZDUBUKZUBUKZFGUAUKZHUBUKUOURJUTUBIUQEUAPUC
|
|
QUDVGUPEUAUKZUSUBUKZVFUBUKZVAVDVJVFUBVDFVBUAUKZUSUBUKVJFVBCFKUEZAEALUEZ
|
|
ERUEZUFCNUEZUGVIVLUSUBFAEVMVNVOUHUCUIUCVAVIVEUBUKZUTUBUKVKURVQUTUBUPBEF
|
|
AVMVNUFZBMUEZVOUJUCVIUSVEDUPEVRVOUFFCVMVPUFBEVSVOUFDOUEULUIUIVDVHVFHUBV
|
|
CGFUASUMTUDUN $.
|
|
$}
|
|
|
|
${
|
|
nummac.8 $e |- P e. NN0 $.
|
|
nummac.9 $e |- F e. NN0 $.
|
|
nummac.10 $e |- G e. NN0 $.
|
|
nummac.11 $e |- ( ( A x. P ) + ( C + G ) ) = E $.
|
|
nummac.12 $e |- ( ( B x. P ) + D ) = ( ( T x. G ) + F ) $.
|
|
$( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
|
|
a fixed multiplicand ` P ` (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
nummac $p |- ( ( M x. P ) + N ) = ( ( T x. E ) + F ) $=
|
|
( cmin co cmul caddc nn0cni mulcli addassi eqtri addcli eqeltrri subdii
|
|
oveq1i wceq subadd2i mpbir eqcomi numma wcel npcan mp2an subcli 3eqtr4i
|
|
cc eqtr3i ) FGIUDUEZUFUEZFIUFUEZHUGUEZUGUEFGUFUEZVJUDUEZVKUGUEZJEUFUEKU
|
|
GUEVLHUGUEZVIVMVKUGFGIFLUHZAEUFUEZCUGUEZIUGUEZGVFVSVQCIUGUEUGUEGVQCIAEA
|
|
MUHESUHUIZCOUHZIUAUHZUJUBUKZVRIVQCVTWAULZWBULUMZWBUNUOABCDEFVHVKJKLMNOP
|
|
QRSVHVRVHVRUPVSGUPWCGIVRWEWBWDUQURUSUCUTVMVJUGUEZHUGUEVOVNWFVLHUGVLVFVA
|
|
VJVFVAWFVLUPFGVPWEUIZFIVPWBUIZVLVJVBVCUOVMVJHVLVJWGWHVDWHHTUHUJVGVE $.
|
|
$}
|
|
|
|
${
|
|
numma2c.8 $e |- P e. NN0 $.
|
|
numma2c.9 $e |- F e. NN0 $.
|
|
numma2c.10 $e |- G e. NN0 $.
|
|
numma2c.11 $e |- ( ( P x. A ) + ( C + G ) ) = E $.
|
|
numma2c.12 $e |- ( ( P x. B ) + D ) = ( ( T x. G ) + F ) $.
|
|
$( Perform a multiply-add of two decimal integers ` M ` and ` N ` against
|
|
a fixed multiplicand ` P ` (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
numma2c $p |- ( ( P x. M ) + N ) = ( ( T x. E ) + F ) $=
|
|
( cmul co caddc nn0cni cn0 numcl eqeltri mulcomi oveq1i eqtri nummac )
|
|
EJUDUEZKUFUEJEUDUEZKUFUEFGUDUEHUFUEUOUPKUFEJESUGZJJFAUDUEBUFUEUHQABFLMN
|
|
UIUJUGUKULABCDEFGHIJKLMNOPQRSTUAAEUDUEZCIUFUEZUFUEEAUDUEZUSUFUEGURUTUSU
|
|
FAEAMUGUQUKULUBUMBEUDUEZDUFUEEBUDUEZDUFUEFIUDUEHUFUEVAVBDUFBEBNUGUQUKUL
|
|
UCUMUNUM $.
|
|
$}
|
|
|
|
${
|
|
numadd.8 $e |- ( A + C ) = E $.
|
|
numadd.9 $e |- ( B + D ) = F $.
|
|
$( Add two decimal integers ` M ` and ` N ` (no carry). (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) $)
|
|
numadd $p |- ( M + N ) = ( ( T x. E ) + F ) $=
|
|
( co caddc c1 cmul numcl eqeltri nn0cni mulid1i oveq1i 1nn0 eqtri numma
|
|
cn0 eqtr3i ) HUAUBSZITSHITSEFUBSGTSUMHITHHHEAUBSBTSUKOABEJKLUCUDUEUFUGA
|
|
BCDUAEFGHIJKLMNOPUHAUAUBSZCTSACTSFUNACTAAKUEUFUGQUIBUAUBSZDTSBDTSGUOBDT
|
|
BBLUEUFUGRUIUJUL $.
|
|
$}
|
|
|
|
${
|
|
numaddc.8 $e |- F e. NN0 $.
|
|
numaddc.9 $e |- ( ( A + C ) + 1 ) = E $.
|
|
numaddc.10 $e |- ( B + D ) = ( ( T x. 1 ) + F ) $.
|
|
$( Add two decimal integers ` M ` and ` N ` (with carry). (Contributed
|
|
by Mario Carneiro, 18-Feb-2014.) $)
|
|
numaddc $p |- ( M + N ) = ( ( T x. E ) + F ) $=
|
|
( co c1 cmul caddc cn0 numcl eqeltri nn0cni mulid1i oveq1i 1nn0 addassi
|
|
ax-1cn 3eqtr2i eqtri nummac eqtr3i ) HUAUBTZIUCTHIUCTEFUBTGUCTUQHIUCHHH
|
|
EAUBTBUCTUDOABEJKLUEUFUGUHUIABCDUAEFGUAHIJKLMNOPUJQUJAUAUBTZCUAUCTZUCTA
|
|
USUCTACUCTUAUCTFURAUSUCAAKUGZUHUIACUAUTCMUGULUKRUMBUAUBTZDUCTBDUCTEUAUB
|
|
TGUCTVABDUCBBLUGUHUISUNUOUP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
nummul1c.1 $e |- T e. NN0 $.
|
|
nummul1c.2 $e |- P e. NN0 $.
|
|
nummul1c.3 $e |- A e. NN0 $.
|
|
nummul1c.4 $e |- B e. NN0 $.
|
|
nummul1c.5 $e |- N = ( ( T x. A ) + B ) $.
|
|
nummul1c.6 $e |- D e. NN0 $.
|
|
nummul1c.7 $e |- E e. NN0 $.
|
|
${
|
|
nummul1c.8 $e |- ( ( A x. P ) + E ) = C $.
|
|
nummul1c.9 $e |- ( B x. P ) = ( ( T x. E ) + D ) $.
|
|
$( The product of a decimal integer with a number. (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
nummul1c $p |- ( N x. P ) = ( ( T x. C ) + D ) $=
|
|
( co cc0 caddc cmul cn0 numcl eqeltri num0u num0h nn0cni addid2i oveq2i
|
|
0nn0 eqtri eqtr3i nummac ) HEUARZUNSTRFCUARDTREHHFAUARBTRUBMABFIKLUCUDJ
|
|
UEABSSEFCDGHSIKLUJUJMSFIUJUFJNOAEUARZSGTRZTRUOGTRCUPGUOTGGOUGUHUIPUKBEU
|
|
ARZUQSTRFGUARDTREBLJUEQULUMUK $.
|
|
$}
|
|
|
|
${
|
|
nummul2c.7 $e |- ( ( P x. A ) + E ) = C $.
|
|
nummul2c.8 $e |- ( P x. B ) = ( ( T x. E ) + D ) $.
|
|
$( The product of a decimal integer with a number (with carry).
|
|
(Contributed by Mario Carneiro, 18-Feb-2014.) $)
|
|
nummul2c $p |- ( P x. N ) = ( ( T x. C ) + D ) $=
|
|
( cmul co caddc cn0 numcl eqeltri nn0cni oveq1i eqtri mulcomli nummul1c
|
|
mulcomi ) HEFCRSDTSHHFARSBTSUAMABFIKLUBUCUDEJUDZABCDEFGHIJKLMNOAERSZGTS
|
|
EARSZGTSCUKULGTAEAKUDUJUIUEPUFEBFGRSDTSUJBLUDQUGUHUG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
decma.a $e |- A e. NN0 $.
|
|
decma.b $e |- B e. NN0 $.
|
|
decma.c $e |- C e. NN0 $.
|
|
decma.d $e |- D e. NN0 $.
|
|
decma.m $e |- M = ; A B $.
|
|
decma.n $e |- N = ; C D $.
|
|
${
|
|
decma.p $e |- P e. NN0 $.
|
|
decma.e $e |- ( ( A x. P ) + C ) = E $.
|
|
decma.f $e |- ( ( B x. P ) + D ) = F $.
|
|
$( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
|
|
multiplicand ` P ` (no carry). (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decma $p |- ( ( M x. P ) + N ) = ; E F $=
|
|
( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri numma eqtr4i ) HESTIUATU
|
|
BUCUDZFSTGUATFGUDABCDEUJFGHIUEJKLMHABUDUJASTBUATNABUFUGICDUDUJCSTDUATOC
|
|
DUFUGPQRUHFGUFUI $.
|
|
$}
|
|
|
|
${
|
|
decmac.p $e |- P e. NN0 $.
|
|
decmac.f $e |- F e. NN0 $.
|
|
decmac.g $e |- G e. NN0 $.
|
|
decmac.e $e |- ( ( A x. P ) + ( C + G ) ) = E $.
|
|
decmac.2 $e |- ( ( B x. P ) + D ) = ; G F $.
|
|
$( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
|
|
multiplicand ` P ` (with carry). (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decmac $p |- ( ( M x. P ) + N ) = ; E F $=
|
|
( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri nummac eqtr4i ) IEUBUCJU
|
|
DUCUEUFUGZFUBUCGUDUCFGUGABCDEUMFGHIJUHKLMNIABUGUMAUBUCBUDUCOABUIUJJCDUG
|
|
UMCUBUCDUDUCPCDUIUJQRSTBEUBUCDUDUCHGUGUMHUBUCGUDUCUAHGUIUJUKFGUIUL $.
|
|
$}
|
|
|
|
${
|
|
decma2c.p $e |- P e. NN0 $.
|
|
decma2c.f $e |- F e. NN0 $.
|
|
decma2c.g $e |- G e. NN0 $.
|
|
decma2c.e $e |- ( ( P x. A ) + ( C + G ) ) = E $.
|
|
decma2c.2 $e |- ( ( P x. B ) + D ) = ; G F $.
|
|
$( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
|
|
multiplier ` P ` (with carry). (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decma2c $p |- ( ( P x. M ) + N ) = ; E F $=
|
|
( cmul co caddc c1 cc0 cdc 10nn0 dfdec10 eqtri numma2c eqtr4i ) EIUBUCJ
|
|
UDUCUEUFUGZFUBUCGUDUCFGUGABCDEUMFGHIJUHKLMNIABUGUMAUBUCBUDUCOABUIUJJCDU
|
|
GUMCUBUCDUDUCPCDUIUJQRSTEBUBUCDUDUCHGUGUMHUBUCGUDUCUAHGUIUJUKFGUIUL $.
|
|
$}
|
|
|
|
${
|
|
decadd.e $e |- ( A + C ) = E $.
|
|
decadd.f $e |- ( B + D ) = F $.
|
|
$( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decadd $p |- ( M + N ) = ; E F $=
|
|
( caddc co cdc cmul c1 cc0 10nn0 dfdec10 eqtri numadd eqtr4i ) GHQRUAUB
|
|
SZETRFQREFSABCDUHEFGHUCIJKLGABSUHATRBQRMABUDUEHCDSUHCTRDQRNCDUDUEOPUFEF
|
|
UDUG $.
|
|
$}
|
|
|
|
decaddc.e $e |- ( ( A + C ) + 1 ) = E $.
|
|
${
|
|
decaddc.f $e |- F e. NN0 $.
|
|
decaddc.2 $e |- ( B + D ) = ; 1 F $.
|
|
$( Add two numerals ` M ` and ` N ` (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decaddc $p |- ( M + N ) = ; E F $=
|
|
( caddc co cdc c1 cc0 cmul 10nn0 dfdec10 eqtri numaddc eqtr4i ) GHRSUAU
|
|
BTZEUCSFRSEFTABCDUIEFGHUDIJKLGABTUIAUCSBRSMABUEUFHCDTUICUCSDRSNCDUEUFPO
|
|
BDRSUAFTUIUAUCSFRSQUAFUEUFUGEFUEUH $.
|
|
$}
|
|
|
|
decaddc2.t $e |- ( B + D ) = ; 1 0 $.
|
|
$( Add two numerals ` M ` and ` N ` (with carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decaddc2 $p |- ( M + N ) = ; E 0 $=
|
|
( cc0 0nn0 decaddc ) ABCDEPFGHIJKLMNQOR $.
|
|
$}
|
|
|
|
${
|
|
decrmanc.a $e |- A e. NN0 $.
|
|
decrmanc.b $e |- B e. NN0 $.
|
|
decrmanc.n $e |- N e. NN0 $.
|
|
decrmanc.m $e |- M = ; A B $.
|
|
decrmanc.p $e |- P e. NN0 $.
|
|
${
|
|
decrmanc.e $e |- ( A x. P ) = E $.
|
|
decrmanc.f $e |- ( ( B x. P ) + N ) = F $.
|
|
$( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
|
|
multiplicand ` P ` (no carry). (Contributed by AV, 16-Sep-2021.) $)
|
|
decrmanc $p |- ( ( M x. P ) + N ) = ; E F $=
|
|
( cc0 0nn0 dec0h cmul co caddc nn0mulcli nn0cni addid1i eqtri decma ) A
|
|
BOGCDEFGHIPJKGJQLACRSZOTSUFDUFUFACHLUAUBUCMUDNUE $.
|
|
$}
|
|
|
|
${
|
|
decrmac.f $e |- F e. NN0 $.
|
|
decrmac.g $e |- G e. NN0 $.
|
|
decrmac.e $e |- ( ( A x. P ) + G ) = E $.
|
|
decrmac.2 $e |- ( ( B x. P ) + N ) = ; G F $.
|
|
$( Perform a multiply-add of two numerals ` M ` and ` N ` against a fixed
|
|
multiplicand ` P ` (with carry). (Contributed by AV, 16-Sep-2021.) $)
|
|
decrmac $p |- ( ( M x. P ) + N ) = ; E F $=
|
|
( cc0 co caddc 0nn0 dec0h cmul nn0cni addid2i oveq2i eqtri decmac ) ABR
|
|
HCDEFGHIJUAKLHKUBMNOACUCSZRFTSZTSUIFTSDUJFUITFFOUDUEUFPUGQUH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
decaddm10.a $e |- A e. NN0 $.
|
|
decaddm10.b $e |- B e. NN0 $.
|
|
$( The sum of two multiples of 10 is a multiple of 10. (Contributed by AV,
|
|
30-Jul-2021.) $)
|
|
decaddm10 $p |- ( ; A 0 + ; B 0 ) = ; ( A + B ) 0 $=
|
|
( cc0 caddc co cdc 0nn0 eqid 00id decadd ) AEBEABFGZEAEHZBEHZCIDINJOJMJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
decaddi.1 $e |- A e. NN0 $.
|
|
decaddi.2 $e |- B e. NN0 $.
|
|
decaddi.3 $e |- N e. NN0 $.
|
|
decaddi.4 $e |- M = ; A B $.
|
|
${
|
|
decaddi.5 $e |- ( B + N ) = C $.
|
|
$( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
decaddi $p |- ( M + N ) = ; A C $=
|
|
( cc0 0nn0 dec0h nn0cni addid1i decadd ) ABKEACDEFGLHIEHMAAFNOJP $.
|
|
$}
|
|
|
|
decaddci.5 $e |- ( A + 1 ) = D $.
|
|
${
|
|
decaddci.6 $e |- C e. NN0 $.
|
|
decaddci.7 $e |- ( B + N ) = ; 1 C $.
|
|
$( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) $)
|
|
decaddci $p |- ( M + N ) = ; D C $=
|
|
( cc0 0nn0 dec0h caddc co c1 nn0cni addid1i oveq1i eqtri decaddc ) ABNF
|
|
DCEFGHOIJFIPANQRZSQRASQRDUEASQAAGTUAUBKUCLMUD $.
|
|
$}
|
|
|
|
${
|
|
decaddci2.6 $e |- ( B + N ) = ; 1 0 $.
|
|
$( Add two numerals ` M ` and ` N ` (no carry). (Contributed by Mario
|
|
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decaddci2 $p |- ( M + N ) = ; D 0 $=
|
|
( cc0 0nn0 decaddci ) ABLCDEFGHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
decsubi.5 $e |- ( B - N ) = C $.
|
|
$( Difference between a numeral ` M ` and a nonnegative integer ` N ` (no
|
|
underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV,
|
|
6-Sep-2021.) $)
|
|
decsubi $p |- ( M - N ) = ; A C $=
|
|
( c1 cdc co caddc cmin nn0cni dfdec10 eqtri cmul 10nn0 nn0mulcli oveq1i
|
|
cc0 addsubassi eqcomi oveq2i 3eqtr4i ) MUENZAUAOZBPOZFQOUKBFQOZPOZEFQOA
|
|
CNZUKBFUKUJAUBGUCRBHRFIRUFEULFQEABNULJABSTUDUOUKCPOUNACSCUMUKPUMCLUGUHT
|
|
UI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
decmul1.p $e |- P e. NN0 $.
|
|
decmul1.a $e |- A e. NN0 $.
|
|
decmul1.b $e |- B e. NN0 $.
|
|
decmul1.n $e |- N = ; A B $.
|
|
decmul1.0 $e |- D e. NN0 $.
|
|
${
|
|
decmul1.c $e |- ( A x. P ) = C $.
|
|
decmul1.d $e |- ( B x. P ) = D $.
|
|
$( The product of a numeral with a number (no carry). (Contributed by
|
|
AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) $)
|
|
decmul1 $p |- ( N x. P ) = ; C D $=
|
|
( cmul co cc0 cdc caddc 10nn0 nn0cni c1 dfdec10 eqtri nn0mulcli addid1i
|
|
0nn0 oveq2i addid2i mul01i eqcomi oveq1i 3eqtr3i nummul1c eqtr4i ) FENO
|
|
UAPQZCNODROCDQABCDEUOPFSGHIFABQUOANOBROJABUBUCKUFAENOZPROUPCUPUPAEHGUDT
|
|
UELUCPBENOZROPDROUQUOPNOZDROUQDPRMUGUQUQBEIGUDTUHPURDRURPUOUOSTUIUJUKUL
|
|
UMCDUBUN $.
|
|
$}
|
|
|
|
decmul1c.e $e |- E e. NN0 $.
|
|
${
|
|
decmul1c.c $e |- ( ( A x. P ) + E ) = C $.
|
|
decmul1c.2 $e |- ( B x. P ) = ; E D $.
|
|
$( The product of a numeral with a number (with carry). (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decmul1c $p |- ( N x. P ) = ; C D $=
|
|
( cmul co cdc caddc dfdec10 c1 cc0 10nn0 eqtri nummul1c eqtr4i ) GEPQUA
|
|
UBRZCPQDSQCDRABCDEUGFGUCHIJGABRUGAPQBSQKABTUDLMNBEPQFDRUGFPQDSQOFDTUDUE
|
|
CDTUF $.
|
|
$}
|
|
|
|
decmul2c.c $e |- ( ( P x. A ) + E ) = C $.
|
|
decmul2c.2 $e |- ( P x. B ) = ; E D $.
|
|
$( The product of a numeral with a number (with carry). (Contributed by
|
|
Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) $)
|
|
decmul2c $p |- ( P x. N ) = ; C D $=
|
|
( cmul co cdc caddc dfdec10 c1 cc0 10nn0 eqtri nummul2c eqtr4i ) EGPQUAUB
|
|
RZCPQDSQCDRABCDEUGFGUCHIJGABRUGAPQBSQKABTUDLMNEBPQFDRUGFPQDSQOFDTUDUECDTU
|
|
F $.
|
|
$}
|
|
|
|
${
|
|
decmulnc.n $e |- N e. NN0 $.
|
|
decmulnc.a $e |- A e. NN0 $.
|
|
decmulnc.b $e |- B e. NN0 $.
|
|
$( The product of a numeral with a number (no carry). (Contributed by AV,
|
|
15-Jun-2021.) $)
|
|
decmulnc $p |- ( N x. ; A B ) = ; ( N x. A ) ( N x. B ) $=
|
|
( cmul co cc0 cdc eqid nn0mulcli 0nn0 nn0cni addid1i dec0h decmul2c ) ABC
|
|
AGHZCBGHZCIABJZDEFTKCBDFLZMRRCADELNOSUAPQ $.
|
|
$}
|
|
|
|
${
|
|
11multnc.n $e |- N e. NN0 $.
|
|
$( The product of 11 (as numeral) with a number (no carry). (Contributed
|
|
by AV, 15-Jun-2021.) $)
|
|
11multnc $p |- ( N x. ; 1 1 ) = ; N N $=
|
|
( c1 cdc cmul co 1nn0 decmulnc nn0cni mulid1i deceq12i eqtri ) ACCDEFACEF
|
|
ZMDAADCCABGGHMAMAAABIJZNKL $.
|
|
$}
|
|
|
|
${
|
|
decmul10add.1 $e |- A e. NN0 $.
|
|
decmul10add.2 $e |- B e. NN0 $.
|
|
decmul10add.3 $e |- M e. NN0 $.
|
|
decmul10add.4 $e |- E = ( M x. A ) $.
|
|
decmul10add.5 $e |- F = ( M x. B ) $.
|
|
$( A multiplication of a number and a numeral expressed as addition with
|
|
first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
decmul10add $p |- ( M x. ; A B ) = ( ; E 0 + F ) $=
|
|
( cdc cmul co cc0 caddc nn0cni 10nn0 nn0mulcli eqcomi 3eqtri oveq2i dec0u
|
|
c1 dfdec10 adddii mul12i deceq1i oveq12i ) EABKZLMEUCNKZALMZBOMZLMEUKLMZE
|
|
BLMZOMCNKZDOMUIULELABUDUAEUKBEHPZUKUJAQFRPBGPUEUMUOUNDOUMUJEALMZLMUQNKUOE
|
|
UJAUPUJQPAFPUFUQEAHFRUBUQCNCUQISUGTDUNJSUHT $.
|
|
$}
|
|
|
|
${
|
|
6p5lem.1 $e |- A e. NN0 $.
|
|
6p5lem.2 $e |- D e. NN0 $.
|
|
6p5lem.3 $e |- E e. NN0 $.
|
|
6p5lem.4 $e |- B = ( D + 1 ) $.
|
|
6p5lem.5 $e |- C = ( E + 1 ) $.
|
|
6p5lem.6 $e |- ( A + D ) = ; 1 E $.
|
|
$( Lemma for ~ 6p5e11 and related theorems. (Contributed by Mario
|
|
Carneiro, 19-Apr-2015.) $)
|
|
6p5lem $p |- ( A + B ) = ; 1 C $=
|
|
( caddc co c1 cdc oveq2i nn0cni ax-1cn addassi 1nn0 eqcomi decsuc 3eqtr2i
|
|
) ABLMADNLMZLMADLMZNLMNCOBUDALIPADNAFQDGQRSNECUETHCENLMJUAKUBUC $.
|
|
$}
|
|
|
|
$( 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
|
|
7-Apr-2020.) (Revised by AV, 6-Sep-2021.) $)
|
|
5p5e10 $p |- ( 5 + 5 ) = ; 1 0 $=
|
|
( c5 caddc co c4 c1 cc0 cdc df-5 oveq2i 5cn 4cn ax-1cn addassi eqtr4i 5p4e9
|
|
c9 oveq1i 9p1e10 3eqtri ) AABCZADBCZEBCZPEBCEFGTADEBCZBCUBAUCABHIADEJKLMNUA
|
|
PEBOQRS $.
|
|
|
|
$( 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
|
|
7-Apr-2020.) (Revised by AV, 6-Sep-2021.) $)
|
|
6p4e10 $p |- ( 6 + 4 ) = ; 1 0 $=
|
|
( c6 c4 caddc co c3 c1 c9 cc0 cdc df-4 oveq2i 6cn 3cn ax-1cn addassi eqtr4i
|
|
6p3e9 oveq1i 9p1e10 3eqtri ) ABCDZAECDZFCDZGFCDFHIUAAEFCDZCDUCBUDACJKAEFLMN
|
|
OPUBGFCQRST $.
|
|
|
|
$( 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
6p5e11 $p |- ( 6 + 5 ) = ; 1 1 $=
|
|
( c6 c5 c1 c4 cc0 6nn0 4nn0 0nn0 df-5 1e0p1 6p4e10 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
6p6e12 $p |- ( 6 + 6 ) = ; 1 2 $=
|
|
( c6 c2 c5 c1 6nn0 5nn0 1nn0 df-6 df-2 6p5e11 6p5lem ) AABCDEFGHIJK $.
|
|
|
|
$( 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
|
|
7-Apr-2020.) (Revised by AV, 6-Sep-2021.) $)
|
|
7p3e10 $p |- ( 7 + 3 ) = ; 1 0 $=
|
|
( c7 c3 caddc co c2 c1 c9 cc0 cdc df-3 oveq2i 7cn 2cn ax-1cn addassi eqtr4i
|
|
7p2e9 oveq1i 9p1e10 3eqtri ) ABCDZAECDZFCDZGFCDFHIUAAEFCDZCDUCBUDACJKAEFLMN
|
|
OPUBGFCQRST $.
|
|
|
|
$( 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
7p4e11 $p |- ( 7 + 4 ) = ; 1 1 $=
|
|
( c7 c4 c1 c3 cc0 7nn0 3nn0 0nn0 df-4 1e0p1 7p3e10 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7p5e12 $p |- ( 7 + 5 ) = ; 1 2 $=
|
|
( c7 c5 c2 c4 c1 7nn0 4nn0 1nn0 df-5 df-2 7p4e11 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7p6e13 $p |- ( 7 + 6 ) = ; 1 3 $=
|
|
( c7 c6 c3 c5 c2 7nn0 5nn0 2nn0 df-6 df-3 7p5e12 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7p7e14 $p |- ( 7 + 7 ) = ; 1 4 $=
|
|
( c7 c4 c6 c3 7nn0 6nn0 3nn0 df-7 df-4 7p6e13 6p5lem ) AABCDEFGHIJK $.
|
|
|
|
$( 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
|
|
7-Apr-2020.) (Revised by AV, 6-Sep-2021.) $)
|
|
8p2e10 $p |- ( 8 + 2 ) = ; 1 0 $=
|
|
( c8 c2 caddc co c1 c9 cc0 cdc df-2 oveq2i 8cn ax-1cn addassi eqtr4i oveq1i
|
|
df-9 9p1e10 3eqtr2i ) ABCDZAECDZECDZFECDEGHSAEECDZCDUABUBACIJAEEKLLMNFTECPO
|
|
QR $.
|
|
|
|
$( 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
8p3e11 $p |- ( 8 + 3 ) = ; 1 1 $=
|
|
( c8 c3 c1 c2 cc0 8nn0 2nn0 0nn0 df-3 1e0p1 8p2e10 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8p4e12 $p |- ( 8 + 4 ) = ; 1 2 $=
|
|
( c8 c4 c2 c3 c1 8nn0 3nn0 1nn0 df-4 df-2 8p3e11 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8p5e13 $p |- ( 8 + 5 ) = ; 1 3 $=
|
|
( c8 c5 c3 c4 c2 8nn0 4nn0 2nn0 df-5 df-3 8p4e12 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8p6e14 $p |- ( 8 + 6 ) = ; 1 4 $=
|
|
( c8 c6 c4 c5 c3 8nn0 5nn0 3nn0 df-6 df-4 8p5e13 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8p7e15 $p |- ( 8 + 7 ) = ; 1 5 $=
|
|
( c8 c7 c5 c6 c4 8nn0 6nn0 4nn0 df-7 df-5 8p6e14 6p5lem ) ABCDEFGHIJKL $.
|
|
|
|
$( 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8p8e16 $p |- ( 8 + 8 ) = ; 1 6 $=
|
|
( c8 c6 c7 c5 8nn0 7nn0 5nn0 df-8 df-6 8p7e15 6p5lem ) AABCDEFGHIJK $.
|
|
|
|
$( 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
9p2e11 $p |- ( 9 + 2 ) = ; 1 1 $=
|
|
( c9 c2 c1 cc0 9nn0 1nn0 0nn0 df-2 1e0p1 9p1e10 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p3e12 $p |- ( 9 + 3 ) = ; 1 2 $=
|
|
( c9 c3 c2 c1 9nn0 2nn0 1nn0 df-3 df-2 9p2e11 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p4e13 $p |- ( 9 + 4 ) = ; 1 3 $=
|
|
( c9 c4 c3 c2 9nn0 3nn0 2nn0 df-4 df-3 9p3e12 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p5e14 $p |- ( 9 + 5 ) = ; 1 4 $=
|
|
( c9 c5 c4 c3 9nn0 4nn0 3nn0 df-5 df-4 9p4e13 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p6e15 $p |- ( 9 + 6 ) = ; 1 5 $=
|
|
( c9 c6 c5 c4 9nn0 5nn0 4nn0 df-6 df-5 9p5e14 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p7e16 $p |- ( 9 + 7 ) = ; 1 6 $=
|
|
( c9 c7 c6 c5 9nn0 6nn0 5nn0 df-7 df-6 9p6e15 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p8e17 $p |- ( 9 + 8 ) = ; 1 7 $=
|
|
( c9 c8 c7 c6 9nn0 7nn0 6nn0 df-8 df-7 9p7e16 6p5lem ) ABCCDEFGHIJK $.
|
|
|
|
$( 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9p9e18 $p |- ( 9 + 9 ) = ; 1 8 $=
|
|
( c9 c8 c7 9nn0 8nn0 7nn0 df-9 df-8 9p8e17 6p5lem ) AABBCDEFGHIJ $.
|
|
|
|
$( 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
|
|
AV, 6-Sep-2021.) $)
|
|
10p10e20 $p |- ( ; 1 0 + ; 1 0 ) = ; 2 0 $=
|
|
( c1 cc0 c2 cdc 1nn0 0nn0 eqid 1p1e2 00id decadd ) ABABCBABDZKEFEFKGZLHIJ
|
|
$.
|
|
|
|
$( 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.) $)
|
|
10m1e9 $p |- ( ; 1 0 - 1 ) = 9 $=
|
|
( c1 cc0 cdc c9 9cn ax-1cn caddc co 9p1e10 eqcomi mvrraddi ) ABCZDAEFDAGHLI
|
|
JK $.
|
|
|
|
${
|
|
4t3lem.1 $e |- A e. NN0 $.
|
|
4t3lem.2 $e |- B e. NN0 $.
|
|
4t3lem.3 $e |- C = ( B + 1 ) $.
|
|
4t3lem.4 $e |- ( A x. B ) = D $.
|
|
4t3lem.5 $e |- ( D + A ) = E $.
|
|
$( Lemma for ~ 4t3e12 and related theorems. (Contributed by Mario
|
|
Carneiro, 19-Apr-2015.) $)
|
|
4t3lem $p |- ( A x. C ) = E $=
|
|
( cmul co c1 caddc oveq2i nn0cni ax-1cn adddii mulid1i eqtri oveq12i ) AC
|
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KLABMNLZKLZECUBAKHOUCDANLZEUCABKLZAMKLZNLUDABMAFPZBGPQRUEDUFANIAUGSUATJTT
|
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$.
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$}
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$( 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
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4t3e12 $p |- ( 4 x. 3 ) = ; 1 2 $=
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( c4 c2 c3 c8 c1 cdc 4nn0 2nn0 df-3 4t2e8 8p4e12 4t3lem ) ABCDEBFGHIJKL $.
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$( 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
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4t4e16 $p |- ( 4 x. 4 ) = ; 1 6 $=
|
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( c4 c3 c1 c2 cdc c6 4nn0 3nn0 df-4 4t3e12 1nn0 2nn0 4cn 2cn 4p2e6 addcomli
|
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eqid decaddi 4t3lem ) ABACDEZCFEGHIJCDFTAKLGTQADFMNOPRS $.
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$( 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV,
|
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4-Sep-2021.) $)
|
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5t2e10 $p |- ( 5 x. 2 ) = ; 1 0 $=
|
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( c5 c1 c2 cc0 cdc 5nn0 1nn0 df-2 5cn mulid1i 5p5e10 4t3lem ) ABCABDEFGHAIJ
|
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KL $.
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$( 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
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(Revised by AV, 6-Sep-2021.) $)
|
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5t3e15 $p |- ( 5 x. 3 ) = ; 1 5 $=
|
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( c5 c2 c3 c1 cc0 cdc 5nn0 2nn0 df-3 5t2e10 dec10p 4t3lem ) ABCDEFDAFGHIJAK
|
|
L $.
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$( 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
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5t4e20 $p |- ( 5 x. 4 ) = ; 2 0 $=
|
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( c5 c3 c4 c1 cdc c2 5nn0 3nn0 df-4 5t3e15 1nn0 eqid 1p1e2 5p5e10 decaddci2
|
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cc0 4t3lem ) ABCDAEZFPEGHIJDAFRAKGGRLMNOQ $.
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$( 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
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5t5e25 $p |- ( 5 x. 5 ) = ; 2 5 $=
|
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( c5 c4 c1 cc0 cdc c2 cmul co 5nn0 4nn0 df-5 5t4e20 2nn0 dec0u eqtr4i caddc
|
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dfdec10 eqcomi 4t3lem ) ABACDEFGHZFAEZIJKABGHFDETLFMNOUATAPHFAQRS $.
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$( 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
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6t2e12 $p |- ( 6 x. 2 ) = ; 1 2 $=
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( c6 c2 cmul co caddc c1 cdc 6cn times2i 6p6e12 eqtri ) ABCDAAEDFBGAHIJK $.
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$( 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
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6t3e18 $p |- ( 6 x. 3 ) = ; 1 8 $=
|
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( c6 c2 c3 c1 cdc c8 6nn0 2nn0 df-3 6t2e12 1nn0 eqid 6cn 2cn 6p2e8 addcomli
|
|
decaddi 4t3lem ) ABCDBEZDFEGHIJDBFSAKHGSLABFMNOPQR $.
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$( 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
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6t4e24 $p |- ( 6 x. 4 ) = ; 2 4 $=
|
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( c6 c3 c4 c1 c8 cdc 6nn0 3nn0 df-4 6t3e18 1nn0 8nn0 eqid 1p1e2 4nn0 8p6e14
|
|
c2 decaddci 4t3lem ) ABCDEFZQCFGHIJDECQTAKLGTMNOPRS $.
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$( 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
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(Revised by AV, 6-Sep-2021.) $)
|
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6t5e30 $p |- ( 6 x. 5 ) = ; 3 0 $=
|
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( c6 c4 c5 c2 cdc c3 cc0 6nn0 4nn0 df-5 6t4e24 2nn0 2p1e3 c1 6cn 4cn 6p4e10
|
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eqid addcomli decaddci2 4t3lem ) ABCDBEZFGEHIJKDBFUBALIHUBRMABNGEOPQSTUA $.
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$( 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
6t6e36 $p |- ( 6 x. 6 ) = ; 3 6 $=
|
|
( c6 c5 c1 cc0 cdc c3 cmul co 6nn0 5nn0 df-6 6t5e30 3nn0 dec0u eqtr4i caddc
|
|
dfdec10 eqcomi 4t3lem ) ABACDEFGHZFAEZIJKABGHFDETLFMNOUATAPHFAQRS $.
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$( 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t2e14 $p |- ( 7 x. 2 ) = ; 1 4 $=
|
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( c7 c2 cmul co caddc c1 c4 cdc 7cn times2i 7p7e14 eqtri ) ABCDAAEDFGHAIJKL
|
|
$.
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$( 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t3e21 $p |- ( 7 x. 3 ) = ; 2 1 $=
|
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( c7 c2 c3 c1 c4 7nn0 2nn0 df-3 7t2e14 1nn0 4nn0 eqid 1p1e2 nn0cni addcomli
|
|
cdc 7p4e11 decaddci 4t3lem ) ABCDEPZBDPFGHIDEDBTAJKFTLMJAEDDPAFNEKNQORS $.
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$( 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t4e28 $p |- ( 7 x. 4 ) = ; 2 8 $=
|
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( c7 c3 c4 c2 c1 cdc c8 7nn0 3nn0 df-4 7t3e21 2nn0 1nn0 7cn ax-1cn addcomli
|
|
eqid 7p1e8 decaddi 4t3lem ) ABCDEFZDGFHIJKDEGUAALMHUAQAEGNORPST $.
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$( 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t5e35 $p |- ( 7 x. 5 ) = ; 3 5 $=
|
|
( c7 c4 c5 c2 c8 cdc 7nn0 4nn0 df-5 7t4e28 2nn0 8nn0 eqid 2p1e3 5nn0 8p7e15
|
|
c3 decaddci 4t3lem ) ABCDEFZQCFGHIJDECQTAKLGTMNOPRS $.
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$( 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t6e42 $p |- ( 7 x. 6 ) = ; 4 2 $=
|
|
( c7 c5 c6 c3 cdc c4 c2 7nn0 5nn0 df-6 7t5e35 3nn0 eqid 3p1e4 nn0cni 7p5e12
|
|
2nn0 c1 addcomli decaddci 4t3lem ) ABCDBEZFGEHIJKDBGFUBALIHUBMNQABRGEAHOBIO
|
|
PSTUA $.
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$( 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
7t7e49 $p |- ( 7 x. 7 ) = ; 4 9 $=
|
|
( c7 c6 c4 c2 cdc c9 7nn0 6nn0 df-7 7t6e42 4nn0 2nn0 7cn 2cn 7p2e9 addcomli
|
|
eqid decaddi 4t3lem ) ABACDEZCFEGHIJCDFTAKLGTQADFMNOPRS $.
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$( 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8t2e16 $p |- ( 8 x. 2 ) = ; 1 6 $=
|
|
( c8 c2 cmul co caddc c1 c6 cdc 8cn times2i 8p8e16 eqtri ) ABCDAAEDFGHAIJKL
|
|
$.
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$( 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8t3e24 $p |- ( 8 x. 3 ) = ; 2 4 $=
|
|
( c8 c2 c3 c1 c6 cdc 8nn0 2nn0 df-3 8t2e16 1nn0 6nn0 eqid 1p1e2 4nn0 nn0cni
|
|
c4 8p6e14 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUBAKLGUBMNOAEDQFAGPE
|
|
LPRSTUA $.
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$( 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8t4e32 $p |- ( 8 x. 4 ) = ; 3 2 $=
|
|
( c8 c3 c4 c2 8nn0 3nn0 df-4 8t3e24 2nn0 4nn0 eqid 2p1e3 c1 nn0cni addcomli
|
|
cdc 8p4e12 decaddci 4t3lem ) ABCDCPZBDPEFGHDCDBTAIJETKLIACMDPAENCJNQORS $.
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$( 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
8t5e40 $p |- ( 8 x. 5 ) = ; 4 0 $=
|
|
( c8 c4 c5 c3 c2 cdc cc0 8nn0 4nn0 df-5 8t4e32 3nn0 2nn0 eqid 3p1e4 8cn 2cn
|
|
c1 8p2e10 addcomli decaddci2 4t3lem ) ABCDEFZBGFHIJKDEBUCALMHUCNOAERGFPQSTU
|
|
AUB $.
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$( 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
|
(Revised by AV, 6-Sep-2021.) $)
|
|
8t6e48 $p |- ( 8 x. 6 ) = ; 4 8 $=
|
|
( c8 c5 c6 c1 cc0 cdc c4 cmul 8nn0 5nn0 df-6 8t5e40 4nn0 dec0u eqtr4i caddc
|
|
co dfdec10 eqcomi 4t3lem ) ABCDEFGHQZGAFZIJKABHQGEFUALGMNOUBUAAPQGARST $.
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|
$( 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8t7e56 $p |- ( 8 x. 7 ) = ; 5 6 $=
|
|
( c8 c6 c7 c4 cdc c5 8nn0 6nn0 df-7 8t6e48 4nn0 eqid 8p8e16 decaddci 4t3lem
|
|
4p1e5 ) ABCDAEZFBEGHIJDABFQAKGGQLPHMNO $.
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$( 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
8t8e64 $p |- ( 8 x. 8 ) = ; 6 4 $=
|
|
( c8 c7 c5 c6 cdc c4 8nn0 7nn0 df-8 8t7e56 5nn0 6nn0 eqid 5p1e6 4nn0 nn0cni
|
|
c1 8p6e14 addcomli decaddci 4t3lem ) ABACDEZDFEGHIJCDFDUBAKLGUBMNOADQFEAGPD
|
|
LPRSTUA $.
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|
|
$( 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t2e18 $p |- ( 9 x. 2 ) = ; 1 8 $=
|
|
( c9 c2 cmul co caddc c1 c8 cdc 9cn times2i 9p9e18 eqtri ) ABCDAAEDFGHAIJKL
|
|
$.
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$( 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t3e27 $p |- ( 9 x. 3 ) = ; 2 7 $=
|
|
( c9 c2 c3 c1 c8 cdc 9nn0 2nn0 df-3 9t2e18 1nn0 8nn0 eqid 1p1e2 7nn0 nn0cni
|
|
c7 9p8e17 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUBAKLGUBMNOAEDQFAGPE
|
|
LPRSTUA $.
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|
$( 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t4e36 $p |- ( 9 x. 4 ) = ; 3 6 $=
|
|
( c9 c3 c4 c2 c7 cdc 9nn0 3nn0 df-4 9t3e27 2nn0 7nn0 eqid 2p1e3 6nn0 nn0cni
|
|
c6 c1 9p7e16 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUCAKLGUCMNOAERQFA
|
|
GPELPSTUAUB $.
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|
$( 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t5e45 $p |- ( 9 x. 5 ) = ; 4 5 $=
|
|
( c9 c4 c5 c3 c6 cdc 9nn0 4nn0 df-5 9t4e36 3nn0 6nn0 eqid 3p1e4 5nn0 nn0cni
|
|
c1 9p6e15 addcomli decaddci 4t3lem ) ABCDEFZBCFGHIJDECBUBAKLGUBMNOAEQCFAGPE
|
|
LPRSTUA $.
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|
$( 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t6e54 $p |- ( 9 x. 6 ) = ; 5 4 $=
|
|
( c9 c5 c6 cdc 9nn0 5nn0 df-6 9t5e45 4nn0 eqid 4p1e5 nn0cni 9p5e14 addcomli
|
|
c4 c1 decaddci 4t3lem ) ABCOBDZBODEFGHOBOBSAIFESJKIABPODAELBFLMNQR $.
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$( 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t7e63 $p |- ( 9 x. 7 ) = ; 6 3 $=
|
|
( c9 c6 c7 c5 c4 cdc 9nn0 6nn0 df-7 9t6e54 5nn0 4nn0 eqid 5p1e6 3nn0 nn0cni
|
|
c3 c1 9p4e13 addcomli decaddci 4t3lem ) ABCDEFZBQFGHIJDEQBUCAKLGUCMNOAERQFA
|
|
GPELPSTUAUB $.
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$( 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t8e72 $p |- ( 9 x. 8 ) = ; 7 2 $=
|
|
( c9 c7 c8 c6 c3 cdc c2 9nn0 7nn0 df-8 9t7e63 6nn0 3nn0 eqid 6p1e7 2nn0 9cn
|
|
c1 3cn 9p3e12 addcomli decaddci 4t3lem ) ABCDEFZBGFHIJKDEGBUDALMHUDNOPAERGF
|
|
QSTUAUBUC $.
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|
|
$( 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) $)
|
|
9t9e81 $p |- ( 9 x. 9 ) = ; 8 1 $=
|
|
( c9 c8 c7 c2 cdc c1 9nn0 8nn0 df-9 9t8e72 7nn0 2nn0 eqid 7p1e8 1nn0 9p2e11
|
|
9cn 2cn addcomli decaddci 4t3lem ) ABACDEZBFEGHIJCDFBUBAKLGUBMNOADFFEQRPSTU
|
|
A $.
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|
$( 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV,
|
|
6-Sep-2021.) $)
|
|
9t11e99 $p |- ( 9 x. ; 1 1 ) = ; 9 9 $=
|
|
( c9 c1 cc0 cdc cmul co caddc 9cn 10nn0 nn0cni ax-1cn mulcli adddii mulid1i
|
|
oveq2i mulcomi eqtri oveq12i dfdec10 3eqtr4i ) ABCDZBEFZBGFZEFZUAAEFZAGFZAB
|
|
BDZEFAADUDAUBEFZABEFZGFUFAUBBHUABUAIJZKLKMUHUEUIAGUHAUAEFUEUBUAAEUAUJNOAUAH
|
|
UJPQAHNRQUGUCAEBBSOAAST $.
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|
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|
$( 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
|
|
by AV, 8-Sep-2021.) $)
|
|
9lt10 $p |- 9 < ; 1 0 $=
|
|
( c9 c1 caddc co cc0 cdc clt 9re ltp1i 9p1e10 breqtri ) AABCDBEFGAHIJK $.
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|
$( 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
|
|
by AV, 8-Sep-2021.) $)
|
|
8lt10 $p |- 8 < ; 1 0 $=
|
|
( c8 c9 clt wbr c1 cc0 cdc 8lt9 9lt10 8re 9re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
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|
|
|
$( 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
7lt10 $p |- 7 < ; 1 0 $=
|
|
( c7 c8 clt wbr c1 cc0 cdc 7lt8 8lt10 7re 8re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
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|
$( 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
6lt10 $p |- 6 < ; 1 0 $=
|
|
( c6 c7 clt wbr c1 cc0 cdc 6lt7 7lt10 6re 7re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
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|
$( 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
5lt10 $p |- 5 < ; 1 0 $=
|
|
( c5 c6 clt wbr c1 cc0 cdc 5lt6 6lt10 5re 6re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
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|
$( 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
4lt10 $p |- 4 < ; 1 0 $=
|
|
( c4 c5 clt wbr c1 cc0 cdc 4lt5 5lt10 4re 5re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
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|
$( 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
3lt10 $p |- 3 < ; 1 0 $=
|
|
( c3 c4 clt wbr c1 cc0 cdc 3lt4 4lt10 3re 4re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
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|
$( 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
2lt10 $p |- 2 < ; 1 0 $=
|
|
( c2 c3 clt wbr c1 cc0 cdc 2lt3 3lt10 2re 3re 10re lttri mp2an ) ABCDBEFGZC
|
|
DAOCDHIABOJKLMN $.
|
|
|
|
$( 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario
|
|
Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) $)
|
|
1lt10 $p |- 1 < ; 1 0 $=
|
|
( c1 c2 clt wbr cc0 cdc 1lt2 2lt10 1re 2re 10re lttri mp2an ) ABCDBAEFZCDAN
|
|
CDGHABNIJKLM $.
|
|
|
|
${
|
|
decbin.1 $e |- A e. NN0 $.
|
|
$( Decompose base 4 into base 2. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
decbin0 $p |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) ) $=
|
|
( c2 cmul co c4 2t2e4 oveq1i 2cn nn0cni mulassi eqtr3i ) CCDEZADEFADECCAD
|
|
EDEMFADGHCCAIIABJKL $.
|
|
|
|
$( Decompose base 4 into base 2. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
decbin2 $p |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) ) $=
|
|
( c2 cmul co c1 caddc c4 2t1e2 oveq2i nn0cni mulcli ax-1cn adddii decbin0
|
|
2cn oveq1i 3eqtr4ri ) CCADEZDEZCFDEZGETCGECSFGEDEHADEZCGEUACTGIJCSFPCAPAB
|
|
KLMNUBTCGABOQR $.
|
|
|
|
$( Decompose base 4 into base 2. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
decbin3 $p |- ( ( 4 x. A ) + 3 ) = ( ( 2 x. ( ( 2 x. A ) + 1 ) ) + 1 ) $=
|
|
( c2 cmul co c1 caddc c4 c3 4nn0 2nn0 2p1e3 decbin2 eqcomi numsuc ) CCADE
|
|
FGEDEZFGEHADEZIGEACIHPJBKLQCGEPABMNON $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Upper sets of integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ZZ>= $. $( ` ZZ ` with greater-than-or-equal-to subscript. $)
|
|
|
|
$( Extend class notation with the upper integer function.
|
|
Read " ` ZZ>= `` M ` " as "the set of integers greater than or equal to
|
|
` M ` ." $)
|
|
cuz $a class ZZ>= $.
|
|
|
|
${
|
|
$d j k N $. $d k M $.
|
|
$( Define a function whose value at ` j ` is the semi-infinite set of
|
|
contiguous integers starting at ` j ` , which we will also call the
|
|
upper integers starting at ` j ` . Read " ` ZZ>= `` M ` " as "the set
|
|
of integers greater than or equal to ` M ` ." See ~ uzval for its
|
|
value, ~ uzssz for its relationship to ` ZZ ` , ~ nnuz and ~ nn0uz for
|
|
its relationships to ` NN ` and ` NN0 ` , and ~ eluz1 and ~ eluz2 for
|
|
its membership relations. (Contributed by NM, 5-Sep-2005.) $)
|
|
df-uz $a |- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } ) $.
|
|
|
|
$( The value of the upper integers function. (Contributed by NM,
|
|
5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
uzval $p |- ( N e. ZZ -> ( ZZ>= ` N ) = { k e. ZZ | N <_ k } ) $=
|
|
( vj cv cle wbr cz crab cuz wceq breq1 rabbidv df-uz zex rabex fvmpt ) CB
|
|
CDZADZEFZAGHBREFZAGHGIQBJSTAGQBREKLCAMTAGNOP $.
|
|
|
|
$( The domain and range of the upper integers function. (Contributed by
|
|
Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
uzf $p |- ZZ>= : ZZ --> ~P ZZ $=
|
|
( vj vk cv cle wbr cz crab cpw wcel wral cuz wss ssrab2 elpw2 mpbir rgenw
|
|
wf zex df-uz fmpt mpbi ) ACBCDEZBFGZFHZIZAFJFUDKQUEAFUEUCFLUBBFMUCFRNOPAF
|
|
UDUCKABSTUA $.
|
|
|
|
$( Membership in the upper set of integers starting at ` M ` .
|
|
(Contributed by NM, 5-Sep-2005.) $)
|
|
eluz1 $p |- ( M e. ZZ ->
|
|
( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) ) $=
|
|
( vk cz wcel cuz cfv cv cle wbr crab wa uzval eleq2d breq2 elrab syl6bb )
|
|
ADEZBAFGZEBACHZIJZCDKZEBDEABIJZLRSUBBCAMNUAUCCBDTBAIOPQ $.
|
|
|
|
$( Implication of membership in an upper set of integers. (Contributed by
|
|
NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
eluzel2 $p |- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) $=
|
|
( cuz cfv wcel cdm cz wrel cpw wf frel ax-mp relelfvdm mpan fdmi syl6eleq
|
|
uzf ) BACDEZACFZGCHZRASEGGIZCJTQGUACKLBACMNGUACQOP $.
|
|
|
|
$( Membership in an upper set of integers. We use the fact that a
|
|
function's value (under our function value definition) is empty outside
|
|
of its domain to show ` M e. ZZ ` . (Contributed by NM, 5-Sep-2005.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
eluz2 $p |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) ) $=
|
|
( cuz cfv wcel cz cle wbr w3a eluzel2 simp1 wa eluz1 bitrd 3anass syl6bbr
|
|
ibar pm5.21nii ) BACDEZAFEZTBFEZABGHZIZABJTUAUBKTSTUAUBLZLZUCTSUDUEABMTUD
|
|
QNTUAUBOPR $.
|
|
$}
|
|
|
|
${
|
|
eluz.1 $e |- M e. ZZ $.
|
|
$( Membership in an upper set of integers. (Contributed by NM,
|
|
5-Sep-2005.) $)
|
|
eluz1i $p |- ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) $=
|
|
( cz wcel cuz cfv cle wbr wa wb eluz1 ax-mp ) ADEBAFGEBDEABHIJKCABLM $.
|
|
$}
|
|
|
|
$( An integer in an upper set of integers is an element of an upper set of
|
|
integers with a smaller bound. (Contributed by Alexander van der Vekens,
|
|
17-Jun-2018.) $)
|
|
eluzuzle $p |- ( ( B e. ZZ /\ B <_ A )
|
|
-> ( C e. ( ZZ>= ` A ) -> C e. ( ZZ>= ` B ) ) ) $=
|
|
( cuz cfv wcel cz cle wbr w3a wa eluz2 simpll simpr2 cr zre ad2antrr adantl
|
|
3ad2ant1 3ad2ant2 simplr simpr3 letrd syl3anbrc ex syl5bi ) CADEFAGFZCGFZAC
|
|
HIZJZBGFZBAHIZKZCBDEFZACLUMUJUNUMUJKZUKUHBCHIUNUKULUJMUMUGUHUINUOBACUKBOFUL
|
|
UJBPQUJAOFZUMUGUHUPUIAPSRUJCOFZUMUHUGUQUICPTRUKULUJUAUMUGUHUIUBUCBCLUDUEUF
|
|
$.
|
|
|
|
$( A member of an upper set of integers is an integer. (Contributed by NM,
|
|
6-Sep-2005.) $)
|
|
eluzelz $p |- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) $=
|
|
( cuz cfv wcel cz cle wbr eluz2 simp2bi ) BACDEAFEBFEABGHABIJ $.
|
|
|
|
$( A member of an upper set of integers is a real. (Contributed by Mario
|
|
Carneiro, 31-Aug-2013.) $)
|
|
eluzelre $p |- ( N e. ( ZZ>= ` M ) -> N e. RR ) $=
|
|
( cuz cfv wcel eluzelz zred ) BACDEBABFG $.
|
|
|
|
$( A member of an upper set of integers is a complex number. (Contributed by
|
|
Glauco Siliprandi, 29-Jun-2017.) $)
|
|
eluzelcn $p |- ( N e. ( ZZ>= ` M ) -> N e. CC ) $=
|
|
( cuz cfv wcel eluzelre recnd ) BACDEBABFG $.
|
|
|
|
$( Implication of membership in an upper set of integers. (Contributed by
|
|
NM, 6-Sep-2005.) $)
|
|
eluzle $p |- ( N e. ( ZZ>= ` M ) -> M <_ N ) $=
|
|
( cuz cfv wcel cz cle wbr eluz2 simp3bi ) BACDEAFEBFEABGHABIJ $.
|
|
|
|
$( Membership in an upper set of integers. (Contributed by NM,
|
|
2-Oct-2005.) $)
|
|
eluz $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( N e. ( ZZ>= ` M ) <-> M <_ N ) ) $=
|
|
( cz wcel cuz cfv cle wbr eluz1 baibd ) ACDBAEFDBCDABGHABIJ $.
|
|
|
|
$( Membership of the least member in an upper set of integers. (Contributed
|
|
by NM, 2-Sep-2005.) $)
|
|
uzid $p |- ( M e. ZZ -> M e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel cuz cfv cle wbr wa zre leidd ancli eluz1 mpbird ) ABCZAADECNAAFGZ
|
|
HNONAAIJKAALM $.
|
|
|
|
${
|
|
$d k M $.
|
|
$( The upper integers are all nonempty. (Contributed by Mario Carneiro,
|
|
16-Jan-2014.) $)
|
|
uzn0 $p |- ( M e. ran ZZ>= -> M =/= (/) ) $=
|
|
( vk cuz crn wcel cv cfv wceq cz wrex c0 wne cpw wfn uzf ffn fvelrnb mp2b
|
|
wf wb uzid ne0i syl neeq1 syl5ibcom rexlimiv sylbi ) ACDEZBFZCGZAHZBIJZAK
|
|
LZIIMZCSCINUHULTOIUNCPBIACQRUKUMBIUIIEZUJKLZUKUMUOUIUJEUPUIUAUJUIUBUCUJAK
|
|
UDUEUFUG $.
|
|
$}
|
|
|
|
$( Transitive law for sets of upper integers. (Contributed by NM,
|
|
20-Sep-2005.) $)
|
|
uztrn $p |- ( ( M e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` N ) ) ->
|
|
M e. ( ZZ>= ` N ) ) $=
|
|
( cuz cfv wcel wa cz cle eluzel2 adantl eluzelz adantr eluzle zletr syl3anc
|
|
wbr wi mp2and eluz2 syl3anbrc ) BADEFZACDEZFZGZCHFZBHFZCBIQZBUCFUDUFUBCAJKZ
|
|
UBUGUDABLMZUECAIQZABIQZUHUDUKUBCANKUBULUDABNMUEUFAHFZUGUKULGUHRUIUDUMUBCALK
|
|
UJCABOPSCBTUA $.
|
|
|
|
${
|
|
uztrn2.1 $e |- Z = ( ZZ>= ` K ) $.
|
|
$( Transitive law for sets of upper integers. (Contributed by Mario
|
|
Carneiro, 26-Dec-2013.) $)
|
|
uztrn2 $p |- ( ( N e. Z /\ M e. ( ZZ>= ` N ) ) -> M e. Z ) $=
|
|
( wcel cuz cfv wa eleq2i uztrn ancoms sylanb syl6eleqr ) CDFZBCGHFZIBAGHZ
|
|
DOCQFZPBQFZDQCEJPRSCBAKLMEN $.
|
|
$}
|
|
|
|
$( Contraposition law for upper integers. (Contributed by NM,
|
|
28-Nov-2005.) $)
|
|
uzneg $p |- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) ) $=
|
|
( cuz cfv wcel cneg cle wbr eluzle cz wb eluzel2 eluzelz zre syl2an syl2anc
|
|
cr leneg mpbid znegcl eluz mpbird ) BACDEZAFZBFZCDEZUEUDGHZUCABGHZUGABIUCAJ
|
|
EZBJEZUHUGKZABLZABMZUIAQEBQEUKUJANBNABROPSUCUJUIUFUGKZUMULUJUEJEUDJEUNUIBTA
|
|
TUEUDUAOPUB $.
|
|
|
|
${
|
|
$d k y M $. $d k N $.
|
|
$( An upper set of integers is a subset of all integers. (Contributed by
|
|
NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
uzssz $p |- ( ZZ>= ` M ) C_ ZZ $=
|
|
( vy cuz cfv cz cv eluzelz ssriv ) BACDEABFGH $.
|
|
|
|
$( Subset relationship for two sets of upper integers. (Contributed by NM,
|
|
5-Sep-2005.) $)
|
|
uzss $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( ZZ>= ` N ) C_ ( ZZ>= ` M ) ) $=
|
|
( vk cuz cfv wcel cv cz cle wbr wa eluzle adantr wi eluzel2 eluzelz eluz1
|
|
jca wb syl zletr 3expa sylan mpand imdistanda 3imtr4d ssrdv ) BADEZFZCBDE
|
|
ZUHUICGZHFZBUKIJZKZULAUKIJZKZUKUJFZUKUHFZUIULUMUOUIULKABIJZUMUOUIUSULABLM
|
|
UIAHFZBHFZKULUSUMKUONZUIUTVAABOZABPZRUTVAULVBABUKUAUBUCUDUEUIVAUQUNSVDBUK
|
|
QTUIUTURUPSVCAUKQTUFUG $.
|
|
$}
|
|
|
|
$( Trichotomy of the ordering relation on integers, stated in terms of upper
|
|
integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro,
|
|
25-Jun-2013.) $)
|
|
uztric $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) ) $=
|
|
( cz wcel wa cuz cfv wo cle wbr zletric eluz wb ancoms orbi12d mpbird ) ACD
|
|
ZBCDZEZBAFGDZABFGDZHABIJZBAIJZHABKSTUBUAUCABLRQUAUCMBALNOP $.
|
|
|
|
$( The upper integers function is one-to-one. (Contributed by NM,
|
|
12-Dec-2005.) $)
|
|
uz11 $p |- ( M e. ZZ ->
|
|
( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) ) $=
|
|
( cz wcel cuz cfv wceq wa eleq2 eluzel2 syl6bi mpan9 cle wbr syl5ibr eluzle
|
|
uzid syl6 cr zre syl5ib anim12d impl ancoms anassrs wb letri3 syl2an mpbird
|
|
adantlr mpdan ex fveq2 impbid1 ) ACDZAEFZBEFZGZABGZUOURUSUOURHZBCDZUSUOAUPD
|
|
ZURVAAQZURVBAUQDZVAUPUQAIZBAJKLUTVAHUSABMNZBAMNZHZUOURVAVHURVAHUOVHURVAUOVH
|
|
URVAVFUOVGURVABUPDZVFVAVIURBUQDBQUPUQBIOABPRURUOVDVGUOVBURVDVCVEUABAPRUBUCU
|
|
DUEUOVAUSVHUFZURUOASDBSDVJVAATBTABUGUHUJUIUKULABEUMUN $.
|
|
|
|
$( Membership in the next upper set of integers. (Contributed by NM,
|
|
12-Sep-2005.) $)
|
|
eluzp1m1 $p |- ( ( M e. ZZ /\
|
|
N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel c1 caddc co cuz cfv cmin cle wbr wa peano2zm ad2antrl cr wb eluz1
|
|
zre 1re leaddsub mp3an2 syl2an biimpa anasss jca ex peano2z syl 3imtr4d imp
|
|
) ACDZBAEFGZHIDZBEJGZAHIDZULBCDZUMBKLZMZUOCDZAUOKLZMZUNUPULUSVBULUSMUTVAUQU
|
|
TULURBNOULUQURVAULUQMURVAULAPDZBPDZURVAQZUQASBSVCEPDVDVETAEBUAUBUCUDUEUFUGU
|
|
LUMCDUNUSQAUHUMBRUIAUORUJUK $.
|
|
|
|
$( Strict ordering implied by membership in the next upper set of integers.
|
|
(Contributed by NM, 12-Sep-2005.) $)
|
|
eluzp1l $p |- ( ( M e. ZZ /\
|
|
N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N ) $=
|
|
( cz wcel c1 caddc cuz cfv clt wbr cle eluzle adantl eluzelz zltp1le sylan2
|
|
co wa wb mpbird ) ACDZBAEFQZGHDZRABIJZUBBKJZUCUEUAUBBLMUCUABCDUDUESUBBNABOP
|
|
T $.
|
|
|
|
$( Membership in the next upper set of integers. (Contributed by NM,
|
|
5-Oct-2005.) $)
|
|
eluzp1p1 $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) ) $=
|
|
( cz wcel cle wbr w3a c1 caddc co cuz cfv peano2z 3ad2ant1 3ad2ant2 zre 1re
|
|
cr wb eluz2 leadd1 mp3an3 syl2an biimp3a 3jca 3imtr4i ) ACDZBCDZABEFZGZAHIJ
|
|
ZCDZBHIJZCDZUKUMEFZGBAKLDUMUKKLDUJULUNUOUGUHULUIAMNUHUGUNUIBMOUGUHUIUOUGARD
|
|
ZBRDZUIUOSZUHAPBPUPUQHRDURQABHUAUBUCUDUEABTUKUMTUF $.
|
|
|
|
${
|
|
eluzaddi.1 $e |- M e. ZZ $.
|
|
eluzaddi.2 $e |- K e. ZZ $.
|
|
$( Membership in a later upper set of integers. (Contributed by Paul
|
|
Chapman, 22-Nov-2007.) $)
|
|
eluzaddi $p |- ( N e. ( ZZ>= ` M )
|
|
-> ( N + K ) e. ( ZZ>= ` ( M + K ) ) ) $=
|
|
( cuz cfv wcel caddc co cz cle wbr eluzelz zaddcl sylancl wa eluz1i zrei
|
|
cr wb zre leadd1 mp3an13 syl biimpa sylbi mp2an sylanbrc ) CBFGHZCAIJZKHZ
|
|
BAIJZUKLMZUKUMFGHUJCKHZAKHZULBCNECAOPUJUOBCLMZQUNBCDRUOUQUNUOCTHZUQUNUAZC
|
|
UBBTHURATHUSBDSAESBCAUCUDUEUFUGUMUKBKHUPUMKHDEBAOUHRUI $.
|
|
|
|
$( Membership in an earlier upper set of integers. (Contributed by Paul
|
|
Chapman, 22-Nov-2007.) $)
|
|
eluzsubi $p |- ( N e. ( ZZ>= ` ( M + K ) )
|
|
-> ( N - K ) e. ( ZZ>= ` M ) ) $=
|
|
( caddc co cuz cfv wcel cmin cz cle eluzelz zsubcl sylancl eluz1i cr zrei
|
|
wbr wa zaddcl mp2an wb zre leaddsub mp3an12 syl biimpa sylbi sylanbrc ) C
|
|
BAFGZHIJZCAKGZLJZBUNMTZUNBHIJUMCLJZALJZUOULCNECAOPUMUQULCMTZUAUPULCBLJURU
|
|
LLJDEBAUBUCQUQUSUPUQCRJZUSUPUDZCUEBRJARJUTVABDSAESBACUFUGUHUIUJBUNDQUK $.
|
|
$}
|
|
|
|
$( Membership in a later upper set of integers. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
eluzadd $p |- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) ->
|
|
( N + K ) e. ( ZZ>= ` ( M + K ) ) ) $=
|
|
( cuz cfv wcel cz wa caddc co cle eluzelz zaddcl sylan adantr zred wb eluz1
|
|
wbr syl eluzel2 simpr simpl mpbid simprd leadd1dd zaddcld mpbir2and ) CBDEF
|
|
ZAGFZHZCAIJZBAIJZDEFZULGFZUMULKSZUICGFZUJUOBCLZCAMNUKBCAUKBUIBGFZUJBCUAOZPU
|
|
KCUIUQUJUROPUKAUIUJUBZPUKUQBCKSZUKUIUQVBHZUIUJUCUKUSUIVCQUTBCRTUDUEUFUKUMGF
|
|
UNUOUPHQUKBAUTVAUGUMULRTUH $.
|
|
|
|
$( Membership in an earlier upper set of integers. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
eluzsub $p |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) )
|
|
-> ( N - K ) e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel caddc co cuz cfv w3a cmin cle wbr wa wb eluz1 syl mpbid cr zred
|
|
eluzelz 3ad2ant3 simp2 zsubcld simp3 simp1 zaddcld simprd syl3anc mpbir2and
|
|
leaddsub ) BDEZADEZCBAFGZHIEZJZCAKGZBHIEZUQDEZBUQLMZUPCAUOULCDEZUMUNCUAUBZU
|
|
LUMUOUCZUDUPUNCLMZUTUPVAVDUPUOVAVDNZULUMUOUEUPUNDEUOVEOUPBAULUMUOUFZVCUGUNC
|
|
PQRUHUPBSEASECSEVDUTOUPBVFTUPAVCTUPCVBTBACUKUIRUPULURUSUTNOVFBUQPQUJ $.
|
|
|
|
$( Choices for an element of an upper interval of integers. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
uzm1 $p |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) ) $=
|
|
( cuz cfv wcel wceq clt wbr wo c1 cmin co wn eluzle eluzel2 zred cz syl2anc
|
|
cle wb eluzelz lenltd mpbid w3o ztri3or df-3or sylib ecased orcomd wi eqcom
|
|
biimpi a1i zltlem1 1zzd zsubcld eluz bitr4d biimpd orim12d mpd ) BACDZEZABF
|
|
ZABGHZIBAFZBJKLZVBEZIVCVEVDVCVEVDIZBAGHZVCABSHVJMABNVCABVCAABOZPVCBABUAZPUB
|
|
UCVCVEVDVJUDZVIVJIVCAQEZBQEZVMVKVLABUERVEVDVJUFUGUHUIVCVDVFVEVHVDVFUJVCVDVF
|
|
ABUKULUMVCVEVHVCVEAVGSHZVHVCVNVOVEVPTVKVLABUNRVCVNVGQEVHVPTVKVCBJVLVCUOUPAV
|
|
GUQRURUSUTVA $.
|
|
|
|
$( The nonnegative difference of integers is a nonnegative integer.
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
uznn0sub $p |- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 ) $=
|
|
( cuz cfv wcel cz cle wbr w3a cmin co cn0 eluz2 znn0sub biimp3a sylbi ) BAC
|
|
DEAFEZBFEZABGHZIBAJKLEZABMQRSTABNOP $.
|
|
|
|
$( Intersection of two upper intervals of integers. (Contributed by Mario
|
|
Carneiro, 24-Dec-2013.) $)
|
|
uzin $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( ( ZZ>= ` M ) i^i ( ZZ>= ` N ) ) = ( ZZ>= ` if ( M <_ N , N , M ) ) ) $=
|
|
( cz wcel wa cuz cfv wo cle wbr wceq wss uzss sylib eluzle fveq2d eqtr4d cr
|
|
syl zre cin uztric sseqin2 iftrue df-ss wn wb eluzel2 eluzelz letri3 syl2an
|
|
cif syl2anc biantrurd bitr4d biimprcd eqeq1d sylibrd com12 wi iffalse zdcle
|
|
a1i wdc df-dc mpjaod jaoi ) ACDZBCDZEBAFGZDZABFGZDZHVJVLUAZABIJZBAULZFGZKZA
|
|
BUBVKVRVMVKVNVLVQVKVLVJLVNVLKABMVLVJUCNVKVPBFVKVOVPBKABOVOBAUDZSPQVMVNVJVQV
|
|
MVJVLLVNVJKBAMVJVLUENVMVPAFVMVOVPAKZVOUFZVOVMVTVOVMBAKZVTVMWBVOVMWBBAIJZVOE
|
|
ZVOVMVIVHWBWDUGZBAUHZBAUIZVIBRDARDWEVHBTATBAUJUKUMVMWCVOBAOUNUOUPVOVPBAVSUQ
|
|
URUSWAVTUTVMVOBAVAVCVMVOVDZVOWAHVMVHVIWHWGWFABVBUMVOVENVFPQVGS $.
|
|
|
|
$( Choices for an element of an upper interval of integers. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
uzp1 $p |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) ) $=
|
|
( cuz cfv wcel wceq c1 cmin co wo caddc uzm1 eluzp1p1 eluzelcn ax-1cn npcan
|
|
cc sylancl eleq1d syl5ib orim2d mpd ) BACDZEZBAFZBGHIZUCEZJUEBAGKICDZEZJABL
|
|
UDUGUIUEUGUFGKIZUHEUDUIAUFMUDUJBUHUDBQEGQEUJBFABNOBGPRSTUAUB $.
|
|
|
|
$( Nonnegative integers expressed as an upper set of integers. (Contributed
|
|
by NM, 2-Sep-2005.) $)
|
|
nn0uz $p |- NN0 = ( ZZ>= ` 0 ) $=
|
|
( vk cn0 cc0 cv cle wbr cz crab cuz nn0zrab wcel wceq 0z uzval ax-mp eqtr4i
|
|
cfv ) BCADEFAGHZCIQZAJCGKSRLMACNOP $.
|
|
|
|
$( Positive integers expressed as an upper set of integers. (Contributed by
|
|
NM, 2-Sep-2005.) $)
|
|
nnuz $p |- NN = ( ZZ>= ` 1 ) $=
|
|
( vk cn c1 cv cle wbr cz crab cuz cfv nnzrab wcel wceq uzval ax-mp eqtr4i
|
|
1z ) BCADEFAGHZCIJZAKCGLSRMQACNOP $.
|
|
|
|
$( A positive integer expressed as a member of an upper set of integers.
|
|
(Contributed by NM, 6-Jun-2006.) $)
|
|
elnnuz $p |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) ) $=
|
|
( cn c1 cuz cfv nnuz eleq2i ) BCDEAFG $.
|
|
|
|
$( A nonnegative integer expressed as a member an upper set of integers.
|
|
(Contributed by NM, 6-Jun-2006.) $)
|
|
elnn0uz $p |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) ) $=
|
|
( cn0 cc0 cuz cfv nn0uz eleq2i ) BCDEAFG $.
|
|
|
|
$( An integer is greater than or equal to 2 is a positive integer.
|
|
(Contributed by AV, 3-Nov-2018.) $)
|
|
eluz2nn $p |- ( A e. ( ZZ>= ` 2 ) -> A e. NN ) $=
|
|
( c2 cuz cfv wcel c1 cn cz cle wbr wi 1z 1le2 eluzuzle mp2an nnuz syl6eleqr
|
|
) ABCDEZAFCDZGFHEFBIJRASEKLMBFANOPQ $.
|
|
|
|
$( If an integer is greater than or equal to 2, then it is a nonnegative
|
|
integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV,
|
|
3-Nov-2018.) $)
|
|
eluzge2nn0 $p |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 ) $=
|
|
( c2 cuz cfv wcel eluz2nn nnnn0d ) ABCDEAAFG $.
|
|
|
|
$( An integer in the upper set of integers starting at 3 is element of the
|
|
upper set of integers starting at 2. (Contributed by Alexander van der
|
|
Vekens, 17-Sep-2018.) $)
|
|
uzuzle23 $p |- ( A e. ( ZZ>= ` 3 ) -> A e. ( ZZ>= ` 2 ) ) $=
|
|
( c2 cz wcel c3 cle wbr cuz cfv wi 2z 2re 3re 2lt3 ltleii eluzuzle mp2an )
|
|
BCDBEFGAEHIDABHIDJKBELMNOEBAPQ $.
|
|
|
|
$( If an integer is greater than 3, then it is a positive integer.
|
|
(Contributed by Alexander van der Vekens, 17-Sep-2018.) $)
|
|
eluzge3nn $p |- ( N e. ( ZZ>= ` 3 ) -> N e. NN ) $=
|
|
( c3 cuz cfv wcel c1 cn cz cle wbr wi 1z 1le3 eluzuzle mp2an elnnuz sylibr
|
|
) ABCDEZAFCDEZAGEFHEFBIJRSKLMBFANOAPQ $.
|
|
|
|
$( An integer greater than or equal to 3 decreased by 2 is a positive
|
|
integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) $)
|
|
uz3m2nn $p |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN ) $=
|
|
( c3 cuz cfv wcel c2 clt wbr cmin co cn cz cle w3a eluz2 2lt3 cr wa wi a1i
|
|
2re 3re zre ltletr syl3anc mpani imp 3adant1 sylbi wb 2nn eluzge3nn sylancr
|
|
nnsub mpbid ) ABCDEZFAGHZAFIJKEZUPBLEZALEZBAMHZNUQBAOUTVAUQUSUTVAUQUTFBGHZV
|
|
AUQPUTFQEZBQEZAQEVBVARUQSVCUTUATVDUTUBTAUCFBAUDUEUFUGUHUIUPFKEAKEUQURUJUKAU
|
|
LFAUNUMUO $.
|
|
|
|
$( 1 is an integer greater than or equal to 0. (Contributed by Alexander van
|
|
der Vekens, 8-Jun-2018.) $)
|
|
1eluzge0 $p |- 1 e. ( ZZ>= ` 0 ) $=
|
|
( c1 cc0 cuz cfv wcel cz cle wbr 0z 1z 0le1 eluz2 mpbir3an ) ABCDEBFEAFEBAG
|
|
HIJKBALM $.
|
|
|
|
$( 2 is an integer greater than or equal to 0. (Contributed by Alexander van
|
|
der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) $)
|
|
2eluzge0 $p |- 2 e. ( ZZ>= ` 0 ) $=
|
|
( c2 cn0 cc0 cuz cfv 2nn0 nn0uz eleqtri ) ABCDEFGH $.
|
|
|
|
$( 2 is an integer greater than or equal to 1. (Contributed by Alexander van
|
|
der Vekens, 8-Jun-2018.) $)
|
|
2eluzge1 $p |- 2 e. ( ZZ>= ` 1 ) $=
|
|
( c2 c1 cuz cfv wcel cz cle wbr 1z 2z 1le2 eluz2 mpbir3an ) ABCDEBFEAFEBAGH
|
|
IJKBALM $.
|
|
|
|
$( The upper integers starting from a natural are a subset of the naturals.
|
|
(Contributed by Scott Fenton, 29-Jun-2013.) $)
|
|
uznnssnn $p |- ( N e. NN -> ( ZZ>= ` N ) C_ NN ) $=
|
|
( cn wcel cuz cfv c1 wss elnnuz uzss sylbi nnuz syl6sseqr ) ABCZADEZFDEZBMA
|
|
OCNOGAHFAIJKL $.
|
|
|
|
${
|
|
$d m n M $.
|
|
$( Restricted universal quantification in an upper set of integers.
|
|
(Contributed by NM, 9-Sep-2005.) $)
|
|
raluz $p |- ( M e. ZZ -> ( A. n e. ( ZZ>= ` M ) ph <->
|
|
A. n e. ZZ ( M <_ n -> ph ) ) ) $=
|
|
( cz wcel cv cle wbr wi cuz cfv wa eluz1 imbi1d impexp syl6bb ralbidv2 )
|
|
CDEZACBFZGHZAIZBCJKZDRSUBEZAISDEZTLZAIUDUAIRUCUEACSMNUDTAOPQ $.
|
|
|
|
$( Restricted universal quantification in an upper set of integers.
|
|
(Contributed by NM, 9-Sep-2005.) $)
|
|
raluz2 $p |- ( A. n e. ( ZZ>= ` M ) ph <->
|
|
( M e. ZZ -> A. n e. ZZ ( M <_ n -> ph ) ) ) $=
|
|
( cuz cfv wral cz wcel cv cle wbr wi w3a eluz2 3anass bitri imbi1i impexp
|
|
wa imbi2i bi2.04 ralbii2 r19.21v ) ABCDEZFCGHZCBIZJKZALZLZBGFUEUHBGFLAUIB
|
|
UDGUFUDHZALUEUFGHZUGSZSZALZUKUILZUJUMAUJUEUKUGMUMCUFNUEUKUGOPQUNUEUKUHLZL
|
|
ZUOUNUEULALZLUQUEULARURUPUEUKUGARTPUEUKUHUAPPUBUEUHBGUCP $.
|
|
|
|
$( Restricted existential quantification in an upper set of integers.
|
|
(Contributed by NM, 9-Sep-2005.) $)
|
|
rexuz $p |- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ph <->
|
|
E. n e. ZZ ( M <_ n /\ ph ) ) ) $=
|
|
( cz wcel cv cle wbr wa cuz cfv eluz1 anbi1d anass syl6bb rexbidv2 ) CDEZ
|
|
ACBFZGHZAIZBCJKZDQRUAEZAIRDEZSIZAIUCTIQUBUDACRLMUCSANOP $.
|
|
|
|
$( Restricted existential quantification in an upper set of integers.
|
|
(Contributed by NM, 9-Sep-2005.) $)
|
|
rexuz2 $p |- ( E. n e. ( ZZ>= ` M ) ph <->
|
|
( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) ) $=
|
|
( cuz cfv wrex cz wcel cv cle wbr wa eluz2 df-3an bitri anbi1i anass an12
|
|
w3a rexbii2 r19.42v ) ABCDEZFCGHZCBIZJKZALZLZBGFUCUFBGFLAUGBUBGUDUBHZALUC
|
|
UDGHZLZUELZALZUIUGLZUHUKAUHUCUIUESUKCUDMUCUIUENOPULUJUFLZUMUJUEAQUNUCUIUF
|
|
LLUMUCUIUFQUCUIUFROOOTUCUFBGUAO $.
|
|
|
|
$( Double existential quantification in an upper set of integers.
|
|
(Contributed by NM, 3-Nov-2005.) $)
|
|
2rexuz $p |- ( E. m E. n e. ( ZZ>= ` m ) ph <->
|
|
E. m e. ZZ E. n e. ZZ ( m <_ n /\ ph ) ) $=
|
|
( cv cuz cfv wrex wex cz wcel cle wbr wa rexuz2 exbii df-rex bitr4i ) ACB
|
|
DZEFGZBHRIJRCDKLAMCIGZMZBHTBIGSUABACRNOTBIPQ $.
|
|
$}
|
|
|
|
$( Second Peano postulate for an upper set of integers. (Contributed by NM,
|
|
7-Sep-2005.) $)
|
|
peano2uz $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( N + 1 ) e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel cle wbr w3a c1 caddc co cuz cfv simp1 peano2z 3ad2ant2 zre letrp1
|
|
cr syl3an2 eluz2 syl3an1 3jca 3imtr4i ) ACDZBCDZABEFZGZUDBHIJZCDZAUHEFZGBAK
|
|
LZDUHUKDUGUDUIUJUDUEUFMUEUDUIUFBNOUDARDZUEUFUJAPUEULBRDUFUJBPABQSUAUBABTAUH
|
|
TUC $.
|
|
|
|
${
|
|
peano2uzs.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( Second Peano postulate for an upper set of integers. (Contributed by
|
|
Mario Carneiro, 26-Dec-2013.) $)
|
|
peano2uzs $p |- ( N e. Z -> ( N + 1 ) e. Z ) $=
|
|
( c1 caddc co wcel cuz cfv peano2uz syl6eleqr eleq2s ) BEFGZCHBAIJZCBOHNO
|
|
CABKDLDM $.
|
|
$}
|
|
|
|
$( Reversed second Peano axiom for upper integers. (Contributed by NM,
|
|
2-Jan-2006.) $)
|
|
peano2uzr $p |- ( ( M e. ZZ /\
|
|
N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel c1 caddc co cuz cfv wa cmin wceq cc eluzelcn ax-1cn npcan sylancl
|
|
adantl eluzp1m1 peano2uz syl eqeltrrd ) ACDZBAEFGZHIDZJZBEKGZEFGZBAHIZUEUHB
|
|
LZUCUEBMDEMDUJUDBNOBEPQRUFUGUIDUHUIDABSAUGTUAUB $.
|
|
|
|
${
|
|
$d j K $. $d j k M $. $d j k N $.
|
|
$( Addition closure law for an upper set of integers. (Contributed by NM,
|
|
4-Jun-2006.) $)
|
|
uzaddcl $p |- ( ( N e. ( ZZ>= ` M ) /\ K e. NN0 ) ->
|
|
( N + K ) e. ( ZZ>= ` M ) ) $=
|
|
( vj vk cn0 wcel cuz caddc co cv wi cc0 c1 wceq oveq2 eleq1d imbi2d wa cc
|
|
eluzelcn addid1d ibir ax-1cn addass mp3an3 syl2anr adantr peano2uz adantl
|
|
cfv nn0cn eqeltrrd exp31 a2d nn0ind impcom ) AFGCBHUKZGZCAIJZURGZUSCDKZIJ
|
|
ZURGZLUSCMIJZURGZLUSCEKZIJZURGZLUSCVGNIJZIJZURGZLUSVALDEAVBMOZVDVFUSVMVCV
|
|
EURVBMCIPQRVBVGOZVDVIUSVNVCVHURVBVGCIPQRVBVJOZVDVLUSVOVCVKURVBVJCIPQRVBAO
|
|
ZVDVAUSVPVCUTURVBACIPQRUSVFUSVECURUSCBCUAZUBQUCVGFGZUSVIVLVRUSVIVLVRUSSZV
|
|
ISVHNIJZVKURVSVTVKOZVIUSCTGZVGTGZWAVRVQVGULWBWCNTGWAUDCVGNUEUFUGUHVIVTURG
|
|
VSBVHUIUJUMUNUOUPUQ $.
|
|
$}
|
|
|
|
$( The sum of a nonnegative integer and an integer is an integer greater than
|
|
or equal to that integer. (Contributed by Alexander van der Vekens,
|
|
3-Oct-2018.) $)
|
|
nn0pzuz $p |- ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) ) $=
|
|
( cn0 wcel cz wa caddc co cle wbr cuz cfv simpr nn0z zaddcl sylan nn0addge2
|
|
cr zre ancoms eluz2 syl3anbrc ) ACDZBEDZFUDABGHZEDZBUEIJZUEBKLDUCUDMUCAEDUD
|
|
UFANABOPUDUCUGUDBRDUCUGBSBAQPTBUEUAUB $.
|
|
|
|
${
|
|
$d j m N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d k ph $.
|
|
$d j k m M $.
|
|
$( Substitutions. $)
|
|
uzind4.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
|
|
uzind4.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
|
|
uzind4.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind4.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
uzind4.5 $e |- ( M e. ZZ -> ps ) $.
|
|
$( Induction step. $)
|
|
uzind4.6 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
|
|
$( Induction on the upper set of integers that starts at an integer ` M ` .
|
|
The first four hypotheses give us the substitution instances we need,
|
|
and the last two are the basis and the induction step. (Contributed by
|
|
NM, 7-Sep-2005.) $)
|
|
uzind4 $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
|
|
( vm wcel cz cle wbr cuz cfv cv crab eluzel2 eluzelz breq2 elrab sylanbrc
|
|
eluzle wa wi w3a eluz2 biimpri 3expb sylan2b syl uzind3 syl2anc ) IHUAUBZ
|
|
QZHRQZIHPUCZSTZPRUDZQZEHIUEVBIRQHISTZVGHIUFHIUJVEVHPIRVDIHSUGUHUIABCDEFPG
|
|
HIJKLMNVCGUCZVFQZUKVIVAQZCDULVJVCVIRQZHVISTZUKVKVEVMPVIRVDVIHSUGUHVCVLVMV
|
|
KVKVCVLVMUMHVIUNUOUPUQOURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d j N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d k ph $.
|
|
$d j k M $.
|
|
$( Basis. $)
|
|
uzind4ALT.5 $e |- ( M e. ZZ -> ps ) $.
|
|
$( Induction step. $)
|
|
uzind4ALT.6 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
|
|
$( Substitutions. $)
|
|
uzind4ALT.1 $e |- ( j = M -> ( ph <-> ps ) ) $.
|
|
uzind4ALT.2 $e |- ( j = k -> ( ph <-> ch ) ) $.
|
|
uzind4ALT.3 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind4ALT.4 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Induction on the upper set of integers that starts at an integer ` M ` .
|
|
The last four hypotheses give us the substitution instances we need; the
|
|
first two are the basis and the induction step. Either ~ uzind4 or
|
|
~ uzind4ALT may be used; see comment for ~ nnind . (Contributed by NM,
|
|
7-Sep-2005.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
uzind4ALT $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
|
|
( uzind4 ) ABCDEFGHILMNOJKP $.
|
|
$}
|
|
|
|
${
|
|
$d m k j M $. $d j N $. $d j m ph $.
|
|
$( Basis. $)
|
|
uzind4s.1 $e |- ( M e. ZZ -> [. M / k ]. ph ) $.
|
|
$( Induction step. $)
|
|
uzind4s.2 $e |- ( k e. ( ZZ>= ` M )
|
|
-> ( ph -> [. ( k + 1 ) / k ]. ph ) ) $.
|
|
$( Induction on the upper set of integers that starts at an integer ` M ` ,
|
|
using explicit substitution. The hypotheses are the basis and the
|
|
induction step. (Contributed by NM, 4-Nov-2005.) $)
|
|
uzind4s $p |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph ) $=
|
|
( vj vm wsb wsbc cv c1 caddc co dfsbcq2 sbequ wcel wi nfim imbi12d uzind4
|
|
cuz cfv nfv nfs1v nfsbc1v weq eleq1 sbequ12 oveq1 sbceq1d chvar ) ABGIABC
|
|
JABHIZABHKZLMNZJZABDJGHCDABGCOAGHBPABGUOOABGDOEBKZCUBUCZQZAABUQLMNZJZRZRU
|
|
NURQZUMUPRZRBHVCVDBVCBUDUMUPBABHUEABUOUFSSBHUGZUSVCVBVDUQUNURUHVEAUMVAUPA
|
|
BHUIVEABUTUOUQUNLMUJUKTTFULUA $.
|
|
$}
|
|
|
|
${
|
|
$d k m n M $. $d m N $. $d k m n ph $. $d j k m n $.
|
|
$( Basis. $)
|
|
uzind4s2.1 $e |- ( M e. ZZ -> [. M / j ]. ph ) $.
|
|
$( Induction step. $)
|
|
uzind4s2.2 $e |- ( k e. ( ZZ>= ` M ) ->
|
|
( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) ) $.
|
|
$( Induction on the upper set of integers that starts at an integer ` M ` ,
|
|
using explicit substitution. The hypotheses are the basis and the
|
|
induction step. Use this instead of ~ uzind4s when ` j ` and ` k ` must
|
|
be distinct in ` [. ( k + 1 ) / j ]. ph ` . (Contributed by NM,
|
|
16-Nov-2005.) $)
|
|
uzind4s2 $p |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph ) $=
|
|
( vm vn cv wsbc c1 caddc co dfsbcq wi cuz cfv weq oveq1 sbceq1d imbi12d
|
|
vtoclga uzind4 ) ABHJZKABDKABIJZKZABUFLMNZKZABEKHIDEABUEDOABUEUFOABUEUHOA
|
|
BUEEOFABCJZKZABUJLMNZKZPUGUIPCUFDQRCISZUKUGUMUIABUJUFOUNABULUHUJUFLMTUAUB
|
|
GUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d j N $. $d j ps $. $d j ch $. $d j th $. $d j ta $. $d k ph $.
|
|
$d j k M $.
|
|
$( Lower bound. $)
|
|
uzind4i.1 $e |- M e. ZZ $.
|
|
$( Substitutions. $)
|
|
uzind4i.2 $e |- ( j = M -> ( ph <-> ps ) ) $.
|
|
uzind4i.3 $e |- ( j = k -> ( ph <-> ch ) ) $.
|
|
uzind4i.4 $e |- ( j = ( k + 1 ) -> ( ph <-> th ) ) $.
|
|
uzind4i.5 $e |- ( j = N -> ( ph <-> ta ) ) $.
|
|
$( Basis. $)
|
|
uzind4i.6 $e |- ps $.
|
|
$( Induction step. $)
|
|
uzind4i.7 $e |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) ) $.
|
|
$( Induction on the upper integers that start at ` M ` . The first
|
|
hypothesis specifies the lower bound, the next four give us the
|
|
substitution instances we need, and the last two are the basis and the
|
|
induction step. (Contributed by NM, 4-Sep-2005.) $)
|
|
uzind4i $p |- ( N e. ( ZZ>= ` M ) -> ta ) $=
|
|
( cz wcel a1i uzind4 ) ABCDEFGHIKLMNBHQROSPT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y w ph $. $d x w z ps $.
|
|
$( Substitutions. $)
|
|
indstr.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Induction step. $)
|
|
indstr.2 $e |- ( x e. NN -> ( A. y e. NN ( y < x -> ps ) -> ph ) ) $.
|
|
$( Strong Mathematical Induction for positive integers (inference schema).
|
|
(Contributed by NM, 17-Aug-2001.) $)
|
|
indstr $p |- ( x e. NN -> ph ) $=
|
|
( vz vw vu cv cn wcel clt wbr wi wral c1 breq2 imbi1d ralbidv co wceq weq
|
|
caddc nnnlt1 pm2.21d rgen wex wb 1nn elex2 nfra1 r19.3rm mp2b wa wo com12
|
|
rsp adantl wsbc nfv nfsbc1v nfim sbceq1a imbi12d cbvral mpbi rspec dfsbcq
|
|
vex sbcie syl5bbr biimprcd syl6 adantr jcad jaob syl6ibr cle nnleltp1 nnz
|
|
cz zleloe syl2an bitr3d ancoms sylibrd ralimdva syl5bi nnind mpd ) CJZKLD
|
|
JZWLMNZBOZDKPZAWMGJZMNZBOZDKPWMQMNZBOZDKPWMHJZMNZBOZDKPZWMXBQUDUAZMNZBOZD
|
|
KPZWPGHWLWQQUBZWSXADKXJWRWTBWQQWMMRSTGHUCZWSXDDKXKWRXCBWQXBWMMRSTWQXFUBZW
|
|
SXHDKXLWRXGBWQXFWMMRSTGCUCZWSWODKXMWRWNBWQWLWMMRSTXADKWMKLZWTBWMUEUFUGXEX
|
|
EDKPZXBKLZXIQKLIJKLIUHXEXOUIUJIQKUKXEDIKXDDKULUMUNXPXEXHDKXPXNUOZXEXCDHUC
|
|
ZUPZBOZXHXQXEXDXRBOZUOXTXQXEXDYAXNXEXDOXPXEXNXDXDDKURUQUSXPXEYAOXNXPXEACX
|
|
BUTZYAXEYBOZHKWPAOZCKPYCHKPYDCKFUGYDYCCHKYDHVAXEYBCXECVAACXBVBVCCHUCZWPXE
|
|
AYBYEWOXDDKYEWNXCBWLXBWMMRSTACXBVDVEVFVGVHXRBYBBACWMUTXRYBABCWMDVJEVKACWM
|
|
XBVIVLVMVNVOVPXCBXRVQVRXQXGXSBXNXPXGXSUIXNXPUOWMXBVSNZXGXSWMXBVTXNWMWBLXB
|
|
WBLYFXSUIXPWMWAXBWAWMXBWCWDWEWFSWGWHWIWJFWK $.
|
|
$}
|
|
|
|
${
|
|
$d A f g w x y z $. $d f g ph w x y z $.
|
|
infrenegsupex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. y < x
|
|
/\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) $.
|
|
infrenegsupex.ss $e |- ( ph -> A C_ RR ) $.
|
|
$( The infimum of a set of reals ` A ` is the negative of the supremum of
|
|
the negatives of its elements. (Contributed by Jim Kingdon,
|
|
14-Jan-2022.) $)
|
|
infrenegsupex $p |- ( ph -> inf ( A , RR , < )
|
|
= -u sup ( { z e. RR | -u z e. A } , RR , < ) ) $=
|
|
( vg vf vw cr clt cneg cv wcel csup wa wceq wbr wiso cinf crab wn infclti
|
|
wb lttri3 adantl recnd negnegd ccnv cmpt cfv cima negeq cbvmptv mptpreima
|
|
eqid negiso simpri imaeq1i eqtr3i supeq1i simpli isocnv ax-mp isoeq1 mpbi
|
|
a1i cnvinfex supisoti syl5eq df-inf eqcomi fveq2i cc negcld fvmptd eqtr2d
|
|
cnvti eqidd negeqd eqtr3d ) AEKLUAZMZMWCDNZMZEODKUBZKLPZMAWCAWCABCDHIKELI
|
|
NZKOHNZKOQWIWJRWIWJLSUCWJWILSUCQUEAWIWJUFUGZFUDZUHZUIAWDWHAWHEKLUJZPZJKJN
|
|
ZMZUKZULZWDAWHWREUMZKLPWSKWGWTLWRUJZEUMWGWTDKWFEWRJDKWQWFWPWEUNUOUPXAWREK
|
|
KLWNWRTZXAWRRZJWRWRUQURZUSZUTVAVBABCDHIKKEWNLWRKKWNLWRTZAKKWNLXATZXFXBXGX
|
|
BXCXDVCKKLWNWRVDVEXCXGXFUEXEKKWNLWRXAVFVEVGVHGABCDKELFVIAHIKLWKVSVJVKAWSW
|
|
CWRULWDWOWCWRWCWOEKLVLVMVNAJWCWQWDKWRVOAWRVTWPWCRWQWDRAWPWCUNUGWLAWCWMVPV
|
|
QVKVRWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c y z $. $d A a c w y z $. $d A a w x y z $. $d a c ph y $.
|
|
supinfneg.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
|
|
/\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
|
|
supinfneg.ss $e |- ( ph -> A C_ RR ) $.
|
|
$( If a set of real numbers has a least upper bound, the set of the
|
|
negation of those numbers has a greatest lower bound. For a theorem
|
|
which is similar but only for the boundedness part, see ~ ublbneg .
|
|
(Contributed by Jim Kingdon, 15-Jan-2022.) $)
|
|
supinfneg $p |- ( ph -> E. x e. RR (
|
|
A. y e. { w e. RR | -u w e. A } -. y < x /\
|
|
A. y e. RR ( x < y -> E. z e. { w e. RR | -u w e. A } z < y ) ) ) $=
|
|
( va vb vc cv clt wbr wral wrex wi cr wa wcel cneg crab weq breq1 ralbidv
|
|
notbid breq2 imbi1d anbi12d cbvrexv sylibr cbvralv imbi2i rexbidv imbi12d
|
|
wn ralbii bitri anbi12i rexbii ad2antlr simplr simprl elrabi negeq eleq1d
|
|
renegcl elrab3 biimpd mpcom wceq rspcv syl adantr ltnegcon2 sylan sylibrd
|
|
ancoms ralrimdva sylc nfcv nfre1 nfim nfralya nfan sseld ad6antr renegcld
|
|
wb nfv mpd recnd negnegd eqeltrd elrab sylanbrc simp-4r ltnegcon1d rspcev
|
|
simpr syl2anc simpllr jca31 ltnegcon1 adantl sylbid r19.29af ex ralrimiva
|
|
imp adantrl syl12anc rexlimdva ) AILZJLZMNZUPZJFOZXOXNMNZXOKLZMNZKFPZQZJR
|
|
OZSZIRPZCLZBLZMNZUPZCELZUAZFTZERUBZOZYHYGMNZDLZYGMNZDYNPZQZCROZSZBRPZAXNY
|
|
GMNZUPZCFOZYGXNMNZYGYQMNZDFPZQZCROZSZIRPZYFAYPUPZCFOZYIUUIQZCROZSZBRPUUMG
|
|
UULUURIBRIBUCZUUFUUOUUKUUQUUSUUEUUNCFUUSUUDYPXNYHYGMUDUFUEUUSUUJUUPCRUUSU
|
|
UGYIUUIXNYHYGMUGUHUEUIUJUKYEUULIRXRUUFYDUUKXQUUEJCFJCUCZXPUUDXOYGXNMUGUFU
|
|
LYDXSXOYQMNZDFPZQZJROUUKYCUVCJRYBUVBXSYAUVAKDFXTYQXOMUGUJUMUQUVCUUJJCRUUT
|
|
XSUUGUVBUUIXOYGXNMUDUUTUVAUUHDFXOYGYQMUDUNUOULURUSUTUKAYEUUCIRAXNRTZSZYEU
|
|
UCUVEYESZXNUAZRTZYGUVGMNZUPZCYNOZUVGYGMNZYSQZCROZUUCUVDUVHAYEXNVGVAUVFUVD
|
|
XRUVKAUVDYEVBUVEXRYDVCUVDXRUVJCYNYGYNTZUVDXRUVJQUVOUVDSXRXNYGUAZMNZUPZUVJ
|
|
UVOXRUVRQZUVDUVOUVPFTZUVSYGRTZUVOUVTYMEYGRVDZUWAUVOUVTYMUVTEYGRECUCYLUVPF
|
|
YKYGVEVFVHVIVJXQUVRJUVPFXOUVPVKZXPUVQXOUVPXNMUGUFVLVMVNUVOUWAUVDUVJUVRWIU
|
|
WBUWAUVDSUVIUVQYGXNVOUFVPVQVRVSVTUVEYDUVNXRUVEYDSZUVMCRUWDUWASZUVLYSUWEUV
|
|
LSZUVPXTMNZYSKFUWEUVLKUWDUWAKUVEYDKUVEKWJYCKJRKRWAXSYBKXSKWJYAKFWBWCWDWEU
|
|
WAKWJWEUVLKWJWEUWFXTFTZSZUWGSZXTUAZYNTZUWKYGMNZYSUWJUWKRTUWKUAZFTZUWLUWJX
|
|
TUWJUWHXTRTZUWFUWHUWGVBZAUWHUWPQUVDYDUWAUVLUWHUWGAFRXTHWFWGWKZWHUWJUWNXTF
|
|
UWJXTUWJXTUWRWLWMUWQWNYMUWOEUWKRYKUWKVKYLUWNFYKUWKVEVFWOWPUWJYGXTUWDUWAUV
|
|
LUWHUWGWQUWRUWIUWGWTWRYRUWMDUWKYNYQUWKYGMUDWSXAUWEUVDUWASZYDSZUVLUWGKFPZU
|
|
WEUVDUWAYDAUVDYDUWAXBUWDUWAWTUVEYDUWAVBXCUWTUVLUXAUWTUVLUVPXNMNZUXAUWSUVL
|
|
UXBWIYDXNYGXDVNUWSYDUXBUXAQZUWAYDUXCQZUVDUWAUVPRTUXDYGVGYCUXCJUVPRUWCXSUX
|
|
BYBUXAXOUVPXNMUDUWCYAUWGKFXOUVPXTMUDUNUOVLVMXEXJXFXJVPXGXHXIXKUUBUVKUVNSB
|
|
UVGRYHUVGVKZYOUVKUUAUVNUXEYJUVJCYNUXEYIUVIYHUVGYGMUGUFUEUXEYTUVMCRUXEYPUV
|
|
LYSYHUVGYGMUDUHUEUIWSXLXHXMWK $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c y z $. $d A a c w y z $. $d A a w x y z $. $d a c ph y $.
|
|
infsupneg.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. y < x
|
|
/\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) $.
|
|
infsupneg.ss $e |- ( ph -> A C_ RR ) $.
|
|
$( If a set of real numbers has a greatest lower bound, the set of the
|
|
negation of those numbers has a least upper bound. To go in the other
|
|
direction see ~ supinfneg . (Contributed by Jim Kingdon,
|
|
15-Jan-2022.) $)
|
|
infsupneg $p |- ( ph -> E. x e. RR (
|
|
A. y e. { w e. RR | -u w e. A } -. x < y /\
|
|
A. y e. RR ( y < x -> E. z e. { w e. RR | -u w e. A } y < z ) ) ) $=
|
|
( vb va vc cv clt wbr wral wrex wi cr wa wcel cneg crab weq breq2 ralbidv
|
|
notbid breq1 imbi1d anbi12d cbvrexv sylibr cbvralv imbi2i rexbidv imbi12d
|
|
wn ralbii bitri anbi12i rexbii ad2antlr simplr simprl elrabi negeq eleq1d
|
|
renegcl elrab3 biimpd mpcom wceq rspcv adantr wb ltnegcon1 ancoms sylibrd
|
|
syl sylan ralrimdva sylc nfv nfcv nfre1 nfim nfralya ad6antr mpd renegcld
|
|
nfan sseld recnd negnegd eqeltrd elrab sylanbrc simp-4r ltnegcon2d rspcev
|
|
simpr syl2anc simpllr jca31 ltnegcon2 adantl sylbid r19.29af ex ralrimiva
|
|
imp adantrl syl12anc rexlimdva ) AILZJLZMNZUPZIFOZXOXNMNZKLZXNMNZKFPZQZIR
|
|
OZSZJRPZBLZCLZMNZUPZCELZUAZFTZERUBZOZYHYGMNZYHDLZMNZDYNPZQZCROZSZBRPZAYHX
|
|
OMNZUPZCFOZXOYHMNZYQYHMNZDFPZQZCROZSZJRPZYFAYPUPZCFOZYIUUIQZCROZSZBRPUUMG
|
|
UULUURJBRJBUCZUUFUUOUUKUUQUUSUUEUUNCFUUSUUDYPXOYGYHMUDUFUEUUSUUJUUPCRUUSU
|
|
UGYIUUIXOYGYHMUGUHUEUIUJUKYEUULJRXRUUFYDUUKXQUUEICFICUCZXPUUDXNYHXOMUGUFU
|
|
LYDXSYQXNMNZDFPZQZIROUUKYCUVCIRYBUVBXSYAUVAKDFXTYQXNMUGUJUMUQUVCUUJICRUUT
|
|
XSUUGUVBUUIXNYHXOMUDUUTUVAUUHDFXNYHYQMUDUNUOULURUSUTUKAYEUUCJRAXORTZSZYEU
|
|
UCUVEYESZXOUAZRTZUVGYHMNZUPZCYNOZYHUVGMNZYSQZCROZUUCUVDUVHAYEXOVGVAUVFUVD
|
|
XRUVKAUVDYEVBUVEXRYDVCUVDXRUVJCYNYHYNTZUVDXRUVJQUVOUVDSXRYHUAZXOMNZUPZUVJ
|
|
UVOXRUVRQZUVDUVOUVPFTZUVSYHRTZUVOUVTYMEYHRVDZUWAUVOUVTYMUVTEYHRECUCYLUVPF
|
|
YKYHVEVFVHVIVJXQUVRIUVPFXNUVPVKZXPUVQXNUVPXOMUGUFVLVRVMUVOUWAUVDUVJUVRVNU
|
|
WBUWAUVDSUVIUVQUVDUWAUVIUVQVNXOYHVOVPUFVSVQVPVTWAUVEYDUVNXRUVEYDSZUVMCRUW
|
|
DUWASZUVLYSUWEUVLSZXTUVPMNZYSKFUWEUVLKUWDUWAKUVEYDKUVEKWBYCKIRKRWCXSYBKXS
|
|
KWBYAKFWDWEWFWJUWAKWBWJUVLKWBWJUWFXTFTZSZUWGSZXTUAZYNTZYHUWKMNZYSUWJUWKRT
|
|
UWKUAZFTZUWLUWJXTUWJUWHXTRTZUWFUWHUWGVBZAUWHUWPQUVDYDUWAUVLUWHUWGAFRXTHWK
|
|
WGWHZWIUWJUWNXTFUWJXTUWJXTUWRWLWMUWQWNYMUWOEUWKRYKUWKVKYLUWNFYKUWKVEVFWOW
|
|
PUWJXTYHUWRUWDUWAUVLUWHUWGWQUWIUWGWTWRYRUWMDUWKYNYQUWKYHMUDWSXAUWEUVDUWAS
|
|
ZYDSZUVLUWGKFPZUWEUVDUWAYDAUVDYDUWAXBUWDUWAWTUVEYDUWAVBXCUWTUVLUXAUWTUVLX
|
|
OUVPMNZUXAUWSUVLUXBVNZYDUWAUVDUXCYHXOXDVPVMUWSYDUXBUXAQZUWAYDUXDQZUVDUWAU
|
|
VPRTUXEYHVGYCUXDIUVPRUWCXSUXBYBUXAXNUVPXOMUDUWCYAUWGKFXNUVPXTMUDUNUOVLVRX
|
|
EXJXFXJVSXGXHXIXKUUBUVKUVNSBUVGRYGUVGVKZYOUVKUUAUVNUXFYJUVJCYNUXFYIUVIYGU
|
|
VGYHMUGUFUEUXFYTUVMCRUXFYPUVLYSYGUVGYHMUDUHUEUIWSXLXHXMWH $.
|
|
$}
|
|
|
|
${
|
|
$d A f g w x y z $. $d f g ph x y z $.
|
|
supminfex.ex $e |- ( ph -> E. x e. RR ( A. y e. A -. x < y
|
|
/\ A. y e. RR ( y < x -> E. z e. A y < z ) ) ) $.
|
|
supminfex.ss $e |- ( ph -> A C_ RR ) $.
|
|
$( A supremum is the negation of the infimum of that set's image under
|
|
negation. (Contributed by Jim Kingdon, 14-Jan-2022.) $)
|
|
supminfex $p |- ( ph ->
|
|
sup ( A , RR , < ) = -u inf ( { w e. RR | -u w e. A } , RR , < ) ) $=
|
|
( vg vf cv cneg wcel cr crab clt wceq adantl wb eleq1d cinf supinfneg wss
|
|
csup ssrab2 a1i infrenegsupex elrabi sselda negeq elrab3 renegcl syl recn
|
|
negnegd 3bitrd eqrdav supeq1d negeqd eqtrd cc wa wbr lttri3 infclti recnd
|
|
wn supclti negcon2 syl2anc mpbid ) AEKZLZFMZENOZNPUAZFNPUDZLZQZVQVPLQZAVP
|
|
DKZLZVOMZDNOZNPUDZLVRABCDVOABCDEFGHUBZVONUCAVNENUEUFUGAWEVQANWDFPABWDFNBK
|
|
ZWDMZWGNMZAWCDWGNUHRAFNWGHUIWIWHWGFMZSAWIWHWGLZVOMZWKLZFMZWJWCWLDWGNWAWGQ
|
|
WBWKVOWAWGUJTUKWIWKNMWLWNSWGULVNWNEWKNVLWKQVMWMFVLWKUJTUKUMWIWMWGFWIWGWGU
|
|
NUOTUPRUQURUSUTAVPVAMVQVAMVSVTSAVPABCDIJNVOPJKZNMIKZNMVBWOWPQWOWPPVCVGWPW
|
|
OPVCVGVBSAWOWPVDRZWFVEVFAVQABCDIJNFPWQGVHVFVPVQVIVJVK $.
|
|
$}
|
|
|
|
$( Membership in a nonnegative upper set of integers implies membership in
|
|
` NN0 ` . (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
eluznn0 $p |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 ) $=
|
|
( cc0 cn0 nn0uz uztrn2 ) CABDEF $.
|
|
|
|
$( Membership in a positive upper set of integers implies membership in
|
|
` NN ` . (Contributed by JJ, 1-Oct-2018.) $)
|
|
eluznn $p |- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN ) $=
|
|
( c1 cn nnuz uztrn2 ) CABDEF $.
|
|
|
|
$( Two ways to say "an integer greater than or equal to 2." (Contributed by
|
|
Paul Chapman, 23-Nov-2012.) $)
|
|
eluz2b1 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) $=
|
|
( c2 cuz cfv wcel cz cle wbr wa c1 clt 2z eluz1i caddc co zltp1le mpan df-2
|
|
wb 1z breq1i syl6bbr pm5.32i bitr4i ) ABCDEAFEZBAGHZIUEJAKHZIBALMUEUGUFUEUG
|
|
JJNOZAGHZUFJFEUEUGUISTJAPQBUHAGRUAUBUCUD $.
|
|
|
|
$( An integer greater than or equal to 2 is greater than 1. (Contributed by
|
|
AV, 24-May-2020.) $)
|
|
eluz2gt1 $p |- ( N e. ( ZZ>= ` 2 ) -> 1 < N ) $=
|
|
( c2 cuz cfv wcel cz c1 clt wbr eluz2b1 simprbi ) ABCDEAFEGAHIAJK $.
|
|
|
|
$( Two ways to say "an integer greater than or equal to 2." (Contributed by
|
|
Paul Chapman, 23-Nov-2012.) $)
|
|
eluz2b2 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ 1 < N ) ) $=
|
|
( c2 cuz cfv wcel cz c1 clt wbr wa cn eluz2b1 cle cr 1re zre ltle imdistani
|
|
wi sylancr elnnz1 sylibr simpr jca nnz anim1i impbii bitri ) ABCDEAFEZGAHIZ
|
|
JZAKEZUJJZALUKUMUKULUJUKUIGAMIZJULUIUJUNUIGNEANEUJUNSOAPGAQTRAUAUBUIUJUCUDU
|
|
LUIUJAUEUFUGUH $.
|
|
|
|
$( Two ways to say "an integer greater than or equal to 2." (Contributed by
|
|
Paul Chapman, 23-Nov-2012.) $)
|
|
eluz2b3 $p |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) ) $=
|
|
( c2 cuz cfv wcel cn c1 clt wbr wa wne eluz2b2 nngt1ne1 pm5.32i bitri ) ABC
|
|
DEAFEZGAHIZJPAGKZJALPQRAMNO $.
|
|
|
|
$( One less than an integer greater than or equal to 2 is a positive integer.
|
|
(Contributed by Paul Chapman, 17-Nov-2012.) $)
|
|
uz2m1nn $p |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN ) $=
|
|
( c2 cuz cfv wcel cz c1 clt wbr wa cmin co cn eluz2b1 wb znnsub mpan biimpa
|
|
1z sylbi ) ABCDEAFEZGAHIZJAGKLMEZANUAUBUCGFEUAUBUCOSGAPQRT $.
|
|
|
|
$( 1 is not in ` ( ZZ>= `` 2 ) ` . (Contributed by Paul Chapman,
|
|
21-Nov-2012.) $)
|
|
1nuz2 $p |- -. 1 e. ( ZZ>= ` 2 ) $=
|
|
( c1 c2 cuz cfv wcel wne neirr cn eluz2b3 simprbi mto ) ABCDEZAAFZAGLAHEMAI
|
|
JK $.
|
|
|
|
$( A positive integer is either 1 or greater than or equal to 2.
|
|
(Contributed by Paul Chapman, 17-Nov-2012.) $)
|
|
elnn1uz2 $p |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) ) $=
|
|
( cn wcel c1 wceq c2 cuz cfv wo wne wa olc cz nnz wn 1z zdceq orbi2i sylibr
|
|
wdc mpan2 df-dc sylib df-ne syl ordi sylanbrc eluz2b3 1nn eleq1 mpbiri jaoi
|
|
eluz2nn impbii ) ABCZADEZAFGHCZIZUOUPUOADJZKZIZURUOUPUOIUPUSIZVAUOUPLUOAMCZ
|
|
VBANVCUPUPOZIZVBVCUPTZVEVCDMCVFPADQUAUPUBUCUSVDUPADUDRSUEUPUOUSUFUGUQUTUPAU
|
|
HRSUPUOUQUPUODBCUIADBUJUKAUMULUN $.
|
|
|
|
$( Closure of multiplication of integers greater than or equal to 2.
|
|
(Contributed by Paul Chapman, 26-Oct-2012.) $)
|
|
uz2mulcl $p |- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) ->
|
|
( M x. N ) e. ( ZZ>= ` 2 ) ) $=
|
|
( c2 cuz cfv wcel wa cmul co cz c1 clt wbr eluzelz syl2an cr eluz2b1 anim1i
|
|
zre sylbi zmulcl mulgt1 an4s sylanbrc ) ACDEZFZBUEFZGABHIZJFZKUHLMZUHUEFUFA
|
|
JFZBJFZUIUGCANCBNABUAOUFAPFZKALMZGZBPFZKBLMZGZUJUGUFUKUNGUOAQUKUMUNASRTUGUL
|
|
UQGURBQULUPUQBSRTUMUPUNUQUJABUBUCOUHQUD $.
|
|
|
|
${
|
|
$d ph y $. $d ps x $. $d x y $.
|
|
indstr2.1 $e |- ( x = 1 -> ( ph <-> ch ) ) $.
|
|
indstr2.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
indstr2.3 $e |- ch $.
|
|
indstr2.4 $e |- ( x e. ( ZZ>= ` 2 ) ->
|
|
( A. y e. NN ( y < x -> ps ) -> ph ) ) $.
|
|
$( Strong Mathematical Induction for positive integers (inference schema).
|
|
The first two hypotheses give us the substitution instances we need; the
|
|
last two are the basis and the induction step. (Contributed by Paul
|
|
Chapman, 21-Nov-2012.) $)
|
|
indstr2 $p |- ( x e. NN -> ph ) $=
|
|
( cv cn wcel c1 wceq c2 cuz clt wbr wi wb cfv wral elnn1uz2 nnnlt1 adantl
|
|
wo wa wn breq2 adantr mtbird pm2.21d ralrimiva pm5.5 syl bitrd jaoi sylbi
|
|
mpbiri indstr ) ABDEGDJZKLVAMNZVAOPUALZUFEJZVAQRZBSZEKUBZASZVAUCVBVHVCVBV
|
|
HCHVBVHACVBVGVHATVBVFEKVBVDKLZUGZVEBVJVEVDMQRZVIVKUHVBVDUDUEVBVEVKTVIVAMV
|
|
DQUIUJUKULUMVGAUNUOFUPUSIUQURUT $.
|
|
$}
|
|
|
|
$( Membership of an integer in an upper set of integers is decidable.
|
|
(Contributed by Jim Kingdon, 18-Apr-2020.) $)
|
|
eluzdc $p |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID N e. ( ZZ>= ` M ) ) $=
|
|
( cz wcel wa cle wbr clt wo cuz cfv wdc zlelttric wn eluz biimprd wb zltnle
|
|
ancoms notbid sylbid orim12d df-dc syl6ibr mpd ) ACDZBCDZEZABFGZBAHGZIZBAJK
|
|
DZLZABMUHUKULULNZIUMUHUIULUJUNUHULUIABOZPUHUJUINZUNUGUFUJUPQBARSUHUNUPUHULU
|
|
IUOTPUAUBULUCUDUE $.
|
|
|
|
${
|
|
$d A a b x y z $.
|
|
$( The image under negation of a bounded-above set of reals is bounded
|
|
below. For a theorem which is similar but also adds that the bounds
|
|
need to be the tightest possible, see ~ supinfneg . (Contributed by
|
|
Paul Chapman, 21-Mar-2011.) $)
|
|
ublbneg $p |- ( E. x e. RR A. y e. A y <_ x ->
|
|
E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y ) $=
|
|
( vb va cv cle wbr wral cr wrex cneg wcel crab breq1 cbvralv wceq ralbidv
|
|
rexbii breq2 cbvrexv bitri renegcl wa wi elrabi negeq eleq1d elrab3 mpcom
|
|
biimpd rspcv adantl wb lenegcon1 sylan2 sylibrd ralrimdva rspcev rexlimiv
|
|
syl syl6an sylbir ) BGZAGZHIZBDJZAKLZEGZFGZHIZEDJZFKLZVFVEHIZBCGZMZDNZCKO
|
|
ZJZAKLZVNVEVKHIZBDJZFKLVIVMWCFKVLWBEBDVJVEVKHPQTWCVHFAKVKVFRWBVGBDVKVFVEH
|
|
UASUBUCVMWAFKVKKNZVKMZKNVMWEVEHIZBVSJZWAVKUDWDVMWFBVSWDVEVSNZUEVMVEMZVKHI
|
|
ZWFWHVMWJUFZWDWHWIDNZWKVEKNZWHWLVRCVEKUGZWMWHWLVRWLCVEKVPVERVQWIDVPVEUHUI
|
|
UJULUKVLWJEWIDVJWIVKHPUMVBUNWHWDWMWFWJUOWNVKVEUPUQURUSVTWGAWEKVFWERVOWFBV
|
|
SVFWEVEHPSUTVCVAVD $.
|
|
$}
|
|
|
|
${
|
|
$d A w z $.
|
|
$( Two ways to express the image under negation of a set of integers.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
eqreznegel $p |- ( A C_ ZZ ->
|
|
{ z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } ) $=
|
|
( vw cz wss cv cneg wcel cr crab wa wi ssel cc recn caddc cc0 negid elrab
|
|
co 0z syl6eqel pm4.71i zrevaddcl syl5bb syl5ib syl6 com23 impd simpr jcad
|
|
a1i zre anim1i impbid1 weq negeq eleq1d 3bitr4g eqrdv ) BDEZCAFZGZBHZAIJZ
|
|
VDADJZVACFZIHZVGGZBHZKZVGDHZVJKZVGVEHVGVFHVAVKVMVAVKVLVJVAVHVJVLVAVJVHVLV
|
|
AVJVIDHZVHVLLBDVIMVHVGNHZVNVLVGOVOVOVGVIPTZDHZKVNVLVOVQVOVPQDVGRUAUBUCVGV
|
|
IUDUEUFUGUHUIVKVJLVAVHVJUJULUKVLVHVJVGUMUNUOVDVJAVGIACUPVCVIBVBVGUQURZSVD
|
|
VJAVGDVRSUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( The image under negation of an inhabited set of reals is inhabited.
|
|
(Contributed by Jim Kingdon, 10-Apr-2020.) $)
|
|
negm $p |- ( ( A C_ RR /\ E. x x e. A ) ->
|
|
E. y y e. { z e. RR | -u z e. A } ) $=
|
|
( cr wss cv wcel wex cneg crab ssel renegcl wceq negeq eleq1d elrab3 recn
|
|
wb syl negnegd bitrd biimprd syli elex2 syl6 exlimdv imp ) DEFZAGZDHZAIBG
|
|
CGZJZDHZCEKZHBIZUIUKUPAUIUKUJJZUOHZUPUKUIUJEHZURDEUJLUSURUKUSURUQJZDHZUKU
|
|
SUQEHURVASUJMUNVACUQEULUQNUMUTDULUQOPQTUSUTUJDUSUJUJRUAPUBUCUDBUQUOUEUFUG
|
|
UH $.
|
|
|
|
$( If a set of reals is bounded below, it is bounded below by an integer.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
lbzbi $p |- ( A C_ RR -> ( E. x e. RR A. y e. A x <_ y <->
|
|
E. x e. ZZ A. y e. A x <_ y ) ) $=
|
|
( vz cr wss cv cle wbr wral wrex cz wcel wi wa clt expdimp com23 imp ex
|
|
nfv nfre1 btwnz simpld w3a zre ltleletr syl3an1 expd 3expia syl5 ralrimiv
|
|
ssel2 ralim syl anasss expcom imdistand breq1 ralbidv rspcev syl6 ancomsd
|
|
weq rexlimdv mpdi rexlimd zssre ssrexv ax-mp impbid1 ) CEFZAGZBGZHIZBCJZA
|
|
EKZVPALKZVLVPVRAEVLAUAVPALUBVLVPVMEMZVRVLVPVSVRNVLVPOZVSDGZVMPIZDLKZVRVSW
|
|
CVMWAPIDLKDDVMUCUDVSVTWCVRNZVSVLVPWDVSVLOZVPOWBVRDLWEVPWALMZWBVRNZWEWFVPW
|
|
GWEWBWFVPOZVRWEWBWHVRNWEWBOZWHWFWAVNHIZBCJZOVRWIWFVPWKWEWBWFVPWKNZNWEWFWB
|
|
WLWFWEWBWLNZWFVSVLWMWFVSOZVLOZWBWLWOWBOZVOWJNZBCJWLWPWQBCWOWBVNCMZWQNWOWR
|
|
WBWQWNVLWRWBWQNZVLWROVNEMZWNWSCEVNUMWFVSWTWSWFVSWTUEWBVOWJWFWAEMVSWTWBVOO
|
|
WJNWAUFWAVMVNUGUHUIUJUKQRSULVOWJBCUNUOTUPUQRSURVPWKAWALADVDVOWJBCVMWAVNHU
|
|
SUTVAVBTRVCQVEUPUQVFTRVGLEFVRVQNVHVPALEVIVJVK $.
|
|
$}
|
|
|
|
$( A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
|
|
by Alexander van der Vekens, 13-Sep-2018.) $)
|
|
nn01to3 $p |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 )
|
|
-> ( N = 1 \/ N = 2 \/ N = 3 ) ) $=
|
|
( cn0 wcel c1 cle wbr c3 c2 clt w3o wo wb cz zleloe sylancr syl mpbid caddc
|
|
wceq co w3a simp2 simp1 1z nn0z 1nn0 nn0ltp1le mpan breq1i syl6bbr 2z bitrd
|
|
orbi1d orcomd orcom orbi2i sylib 3orass sylibr wi 3mix1 eqcoms a1i 3mix2 wa
|
|
df-2 simp3 biantrurd 2nn0 cr nn0red 3re letri3 sylancl 3bitr4d 3mix3 syl6bi
|
|
df-3 3jaod mpd ) ABCZDAEFZAGEFZUAZDASZHASZHAIFZJZADSZAHSZAGSZJZWDWEWFWGKZKZ
|
|
WHWDWEWGWFKZKWNWDWOWEWDDAIFZWEKZWOWEKZWDWBWQWAWBWCUBWDWAWBWQLZWAWBWCUCZWADM
|
|
CAMCZWSUDAUEZDANOPQWDWAWQWRLWTWAWPWOWEWAWPHAEFZWOWAWPDDRTZAEFZXCDBCWAWPXELU
|
|
FDAUGUHHXDAEVFUIUJWAHMCXAXCWOLUKXBHANOULUMPQUNWOWMWEWGWFUOUPUQWEWFWGURUSWDW
|
|
EWLWFWGWEWLUTWDWLADWIWJWKVAVBVCWFWLUTWDWLAHWJWIWKVDVBVCWDWGWKWLWDGAEFZWCXFV
|
|
EZWGWKWDWCXFWAWBWCVGVHWDWAWGXFLWTWAWGHDRTZAEFZXFHBCWAWGXILVIHAUGUHGXHAEVRUI
|
|
UJPWDAVJCGVJCWKXGLWDAWTVKVLAGVMVNVOWKWIWJVPVQVSVT $.
|
|
|
|
$( Alternate proof of ~ nn0ge2m1nn : If a nonnegative integer is greater
|
|
than or equal to two, the integer decreased by 1 is a positive integer.
|
|
This version is proved using ~ eluz2 , a theorem for upper sets of
|
|
integers, which are defined later than the positive and nonnegative
|
|
integers. This proof is, however, much shorter than the proof of
|
|
~ nn0ge2m1nn . (Contributed by Alexander van der Vekens, 1-Aug-2018.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
nn0ge2m1nnALT $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN ) $=
|
|
( cn0 wcel c2 cle wbr wa cuz cfv c1 cmin co cn cz 2z a1i adantr simpr eluz2
|
|
nn0z syl3anbrc uz2m1nn syl ) ABCZDAEFZGZADHICZAJKLMCUFDNCZANCZUEUGUHUFOPUDU
|
|
IUEATQUDUERDASUAAUBUC $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Rational numbers (as a subset of complex numbers)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c QQ $. $( The set of rational numbers (blackboard bold Q). $)
|
|
|
|
$( Extend class notation to include the class of rationals. $)
|
|
cq $a class QQ $.
|
|
|
|
$( Define the set of rational numbers. Based on definition of rationals in
|
|
[Apostol] p. 22. See ~ elq for the relation "is rational." (Contributed
|
|
by NM, 8-Jan-2002.) $)
|
|
df-q $a |- QQ = ( / " ( ZZ X. NN ) ) $.
|
|
|
|
${
|
|
$d x y z $.
|
|
|
|
$( Division restricted to ` ZZ X. NN ` is a function. Given excluded
|
|
middle, it would be easy to prove this for ` CC X. ( CC \ { 0 } ) ` .
|
|
The key difference is that an element of ` NN ` is apart from zero,
|
|
whereas being an element of ` CC \ { 0 } ` implies being not equal to
|
|
zero. (Contributed by Jim Kingdon, 19-Mar-2020.) $)
|
|
divfnzn $p |- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN ) $=
|
|
( vy vz vx cv co wceq cc crio wcel cn wral cz cdiv cres nncn ad2antlr cc0
|
|
wa adantl eqeltrrd cmul cxp wfn zcn ad2antrr cap wbr divmulapd riotabidva
|
|
simpr nnap0 wb eqcom a1i riotabidv simpl divclapd reueq sylib riotacl syl
|
|
wreu sylan rgen2 csn cdif cmpt2 df-div reseq1i wss zsscn eldifsn sylanbrc
|
|
wne nnne0 ssriv resmpt2 mp2an eqtri fnmpt2 ax-mp ) ADZBDZUAECDZFZBGHZGIZA
|
|
JKCLKMLJUBZNZWHUCWGCALJWDLIZWBJIZRZWDWBMEZWCFZBGHZWFGWLWNWEBGWLWCGIZRWDWB
|
|
WCWJWDGIZWKWPWDUDZUEWKWBGIZWJWPWBOZPWLWPUJWKWBQUFUGZWJWPWBUKZPUHUIWLWCWMF
|
|
ZBGHZWOGWLXCWNBGXCWNULWLWCWMUMUNUOWJWQWKXDGIZWRWQWKRZXCBGVBZXEXFWMGIXGXFW
|
|
DWBWQWKUPWKWSWQWTSWKXAWQXBSUQBGWMURUSXCBGUTVAVCTTVDCALJWFWIGWICAGGQVEVFZW
|
|
FVGZWHNZCALJWFVGZMXIWHCABVHVILGVJJXHVJXJXKFVKCJXHWDJIWQWDQVNWDXHIWDOWDVOW
|
|
DGQVLVMVPCAGXHLJWFVQVRVSVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.)
|
|
(Revised by Mario Carneiro, 28-Jan-2014.) $)
|
|
elq $p |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) ) $=
|
|
( cq wcel cv cdiv cz cn cxp cres co wceq wrex cima df-q eleq2i resima wfn
|
|
wss wb divfnzn ssid ovelimab mp2an 3bitr2i wa ovres eqeq2d 2rexbiia bitri
|
|
) CDEZCAFZBFZGHIJZKZLZMZBINAHNZCUMUNGLZMZBINAHNULCGUOOZECUPUOOZEZUSDVBCPQ
|
|
VCVBCGUORQUPUOSUOUOTVDUSUAUBUOUCABUOHICUPUDUEUFURVAABHIUMHEUNIEUGUQUTCUMU
|
|
NHIGUHUIUJUK $.
|
|
|
|
$( If ` A ` is rational, then some integer multiple of it is an integer.
|
|
(Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro,
|
|
22-Jul-2014.) $)
|
|
qmulz $p |- ( A e. QQ -> E. x e. NN ( A x. x ) e. ZZ ) $=
|
|
( vy cq wcel cv cdiv co wceq cn wrex cz elq rexcom wa cc zcn adantr sylbi
|
|
cmul adantl nncn cc0 cap nnap0 divcanap1d simpr eqeltrd eleq1d syl5ibrcom
|
|
wbr oveq1 rexlimdva reximia ) BDEBCFZAFZGHZIZAJKCLKZBUPTHZLEZAJKZCABMUSUR
|
|
CLKZAJKVBURCALJNVCVAAJUPJEZURVACLVDUOLEZOZVAURUQUPTHZLEVFVGUOLVFUOUPVEUOP
|
|
EVDUOQUAVDUPPEVEUPUBRVDUPUCUDUKVEUPUERUFVDVEUGUHURUTVGLBUQUPTULUIUJUMUNSS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The ratio of an integer and a positive integer is a rational number.
|
|
(Contributed by NM, 12-Jan-2002.) $)
|
|
znq $p |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ ) $=
|
|
( vx vy cz wcel cn wa cdiv co cv wceq wrex eqid rspceov mp3an3 elq sylibr
|
|
cq ) AEFZBGFZHABIJZCKDKIJLDGMCEMZUBSFTUAUBUBLUCUBNCDEGABUBIOPCDUBQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A rational number is a real number. (Contributed by NM,
|
|
14-Nov-2002.) $)
|
|
qre $p |- ( A e. QQ -> A e. RR ) $=
|
|
( vx vy cq wcel cv cdiv co wceq cn wrex cz cr elq wa cc0 cap wbr zre nnre
|
|
nnap0 jca redivclap 3expb syl2an eleq1 syl5ibrcom rexlimivv sylbi ) ADEAB
|
|
FZCFZGHZIZCJKBLKAMEZBCANUMUNBCLJUJLEZUKJEZOUNUMULMEZUOUJMEZUKMEZUKPQRZOUQ
|
|
UPUJSUPUSUTUKTUKUAUBURUSUTUQUJUKUCUDUEAULMUFUGUHUI $.
|
|
|
|
$( An integer is a rational number. (Contributed by NM, 9-Jan-2002.) $)
|
|
zq $p |- ( A e. ZZ -> A e. QQ ) $=
|
|
( vx vy cv wceq cz wrex cdiv co cn wcel cq c1 div1d eqeq2d eqcom syl6rbbr
|
|
zcn 1nn oveq2 rspcev mpan syl6bi reximia risset elq 3imtr4i ) BDZAEZBFGAU
|
|
HCDZHIZEZCJGZBFGAFKALKUIUMBFUHFKZUIAUHMHIZEZUMUNUPAUHEUIUNUOUHAUNUHUHRNOU
|
|
HAPQMJKUPUMSULUPCMJUJMEUKUOAUJMUHHTOUAUBUCUDBAFUEBCAUFUG $.
|
|
$}
|
|
|
|
$( The integers are a subset of the rationals. (Contributed by NM,
|
|
9-Jan-2002.) $)
|
|
zssq $p |- ZZ C_ QQ $=
|
|
( vx cz cq cv zq ssriv ) ABCADEF $.
|
|
|
|
$( The nonnegative integers are a subset of the rationals. (Contributed by
|
|
NM, 31-Jul-2004.) $)
|
|
nn0ssq $p |- NN0 C_ QQ $=
|
|
( cn0 cz cq nn0ssz zssq sstri ) ABCDEF $.
|
|
|
|
$( The positive integers are a subset of the rationals. (Contributed by NM,
|
|
31-Jul-2004.) $)
|
|
nnssq $p |- NN C_ QQ $=
|
|
( cn cz cq nnssz zssq sstri ) ABCDEF $.
|
|
|
|
$( The rationals are a subset of the reals. (Contributed by NM,
|
|
9-Jan-2002.) $)
|
|
qssre $p |- QQ C_ RR $=
|
|
( vx cq cr cv qre ssriv ) ABCADEF $.
|
|
|
|
$( The rationals are a subset of the complex numbers. (Contributed by NM,
|
|
2-Aug-2004.) $)
|
|
qsscn $p |- QQ C_ CC $=
|
|
( cq cr cc qssre ax-resscn sstri ) ABCDEF $.
|
|
|
|
$( The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.)
|
|
(Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
qex $p |- QQ e. _V $=
|
|
( cq cc cnex qsscn ssexi ) ABCDE $.
|
|
|
|
$( A positive integer is rational. (Contributed by NM, 17-Nov-2004.) $)
|
|
nnq $p |- ( A e. NN -> A e. QQ ) $=
|
|
( cn cq nnssq sseli ) BCADE $.
|
|
|
|
$( A rational number is a complex number. (Contributed by NM,
|
|
2-Aug-2004.) $)
|
|
qcn $p |- ( A e. QQ -> A e. CC ) $=
|
|
( cq cc qsscn sseli ) BCADE $.
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $.
|
|
$( Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.) $)
|
|
qaddcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ ) $=
|
|
( vx vy vz vw vv vu cq wcel cv cdiv co wceq cn wrex cz caddc wa cc elq wi
|
|
cmul zmulcl sylan2 ad2ant2rl simpl adantl syl2anr zaddcld adantr ad2ant2l
|
|
nnz nnmulcl oveq12 cc0 cap wbr anim12i nncn nnap0 divadddivap syl2an an4s
|
|
zcn jca sylan9eqr w3a rspceov sylibr syl3anc rexlimivv rexlimdvv syl2anb
|
|
exp43 imp ) AIJACKZDKZLMZNZDOPCQPZBEKZFKZLMZNZFOPEQPZABRMZIJZBIJCDAUAEFBU
|
|
AWAWFWHWAWEWHEFQOVTWBQJZWCOJZSZWEWHUBUBCDQOVQQJZVROJZSZVTWKWEWHWNWKVTWEWH
|
|
WNWKSZVTWESZSVQWCUCMZWBVRUCMZRMZQJZVRWCUCMZOJZWGWSXALMZNZWHWOWTWPWOWQWRWL
|
|
WJWQQJZWMWIWJWLWCQJXEWCUMVQWCUDUEUFWKWIVRQJZWRQJWNWIWJUGWMXFWLVRUMUHWBVRU
|
|
DUIUJUKWOXBWPWMWJXBWLWIVRWCUNULUKWPWOWGVSWDRMZXCAVSBWDRUOWLWIWMWJXGXCNZWL
|
|
WISVQTJZWBTJZSVRTJZVRUPUQURZSZWCTJZWCUPUQURZSZSXHWMWJSWLXIWIXJVQVEWBVEUSW
|
|
MXMWJXPWMXKXLVRUTVRVAVFWJXNXOWCUTWCVAVFUSVQWBVRWCVBVCVDVGWTXBXDVHWGGKHKLM
|
|
NHOPGQPWHGHQOWSXAWGLVIGHWGUAVJVKVDVOVLVMVPVN $.
|
|
|
|
$( Closure law for the negative of a rational. (Contributed by NM,
|
|
2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.) $)
|
|
qnegcl $p |- ( A e. QQ -> -u A e. QQ ) $=
|
|
( vx vy cq wcel cv cdiv co wceq cn wrex cz cneg elq wa cc zcn adantr nncn
|
|
adantl cc0 cap wbr nnap0 divnegapd znegcl sylan eqeltrd eleq1d syl5ibrcom
|
|
znq negeq rexlimivv sylbi ) ADEABFZCFZGHZIZCJKBLKAMZDEZBCANURUTBCLJUOLEZU
|
|
PJEZOZUTURUQMZDEVCVDUOMZUPGHZDVCUOUPVAUOPEVBUOQRVBUPPEVAUPSTVBUPUAUBUCVAU
|
|
PUDTUEVAVELEVBVFDEUOUFVEUPUKUGUHURUSVDDAUQULUIUJUMUN $.
|
|
|
|
$( Closure of multiplication of rationals. (Contributed by NM,
|
|
1-Aug-2004.) $)
|
|
qmulcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ ) $=
|
|
( vx vy vz vw vv vu cq wcel cv cdiv co wceq cn wrex cz cmul wa cc nnmulcl
|
|
elq zmulcl anim12i an4s adantr oveq12 cc0 cap wbr zcn ad2ant2r nncn nnap0
|
|
jca ad2ant2l divmuldivap syl2anc sylan9eqr rspceov 3expa sylibr rexlimivv
|
|
wi exp43 rexlimdvv imp syl2anb ) AIJACKZDKZLMZNZDOPCQPZBEKZFKZLMZNZFOPEQP
|
|
ZABRMZIJZBIJCDAUBEFBUBVMVRVTVMVQVTEFQOVLVNQJZVOOJZSZVQVTVDVDCDQOVIQJZVJOJ
|
|
ZSZVLWCVQVTWFWCVLVQVTWFWCSZVLVQSZSVIVNRMZQJZVJVORMZOJZSZVSWIWKLMZNZVTWGWM
|
|
WHWDWAWEWBWMWDWASWJWEWBSWLVIVNUCVJVOUAUDUEUFWHWGVSVKVPRMZWNAVKBVPRUGWGVIT
|
|
JZVNTJZSZVJTJZVJUHUIUJZSZVOTJZVOUHUIUJZSZSZWPWNNWDWAWSWEWBWDWQWAWRVIUKVNU
|
|
KUDULWEWBXFWDWAWEXBWBXEWEWTXAVJUMVJUNUOWBXCXDVOUMVOUNUOUDUPVIVNVJVOUQURUS
|
|
WMWOSVSGKHKLMNHOPGQPZVTWJWLWOXGGHQOWIWKVSLUTVAGHVSUBVBURUEVEVCVFVGVH $.
|
|
$}
|
|
|
|
$( Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.) $)
|
|
qsubcl $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ ) $=
|
|
( cq wcel wa cneg caddc co cmin wceq qcn negsub syl2an qnegcl qaddcl sylan2
|
|
cc eqeltrrd ) ACDZBCDZEABFZGHZABIHZCSAQDBQDUBUCJTAKBKABLMTSUACDUBCDBNAUAOPR
|
|
$.
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $.
|
|
$( Apartness is equivalent to not equal for rationals. (Contributed by Jim
|
|
Kingdon, 20-Mar-2020.) $)
|
|
qapne $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A # B <-> A =/= B ) ) $=
|
|
( vz vw vx vy wcel wa cv cdiv co wceq cn cz cap wbr wb cmul c1 cc cq wrex
|
|
wne elq biimpi adantl simplll sylib simplrl simprl ad3antrrr simprr nncnd
|
|
cc0 zcnd syl divclapd simplrr mulcld recclapd recap0d syl112anc divrecapd
|
|
nnap0 apmul1 eqcomd mulassd recidapd oveq2d mulid1d breq12d bitrd zmulcld
|
|
3eqtrd breq1d mulcomd 3eqtr2d breq2d simpr simpllr 3bitr4d syl2anc bitr3d
|
|
nnzd zapne wn notbid apti qcn ad2antrr eqeltrd necon3bid rexlimdvva mpd
|
|
ex ) AUAGZBUAGZHZBCIZDIZJKZLZDMUBCNUBZABOPZABUCZQZWQXCWPWQXCCDBUDUEUFWRXB
|
|
XFCDNMWRWSNGZWTMGZHZHZXBXFXJXBHZAEIZFIZJKZLZFMUBENUBZXFXKWPXPWPWQXIXBUGEF
|
|
AUDUHXKXOXFEFNMXKXLNGZXMMGZHZHZXOXFXTXOHZXDXLWTRKZXMWSRKZUCZXEYAYBYCOPZXD
|
|
YDYAXLXAXMRKZOPZXNXAOPZYEXDYAYGXLSXMJKZRKZYFYIRKZOPZYHYAXLTGYFTGYITGYIUNO
|
|
PYGYLQYAXLXKXQXRXOUIZUOZYAXAXMYAWSWTYAWSXJXGXBXSXOWRXGXHUJUKZUOZYAWTXJXHX
|
|
BXSXOWRXGXHULUKZUMZYAXHWTUNOPYQWTVDUPZUQZYAXMXKXQXRXOURZUMZUSYAXMUUBYAXRX
|
|
MUNOPUUAXMVDUPZUTZYAXMUUBUUCVAXLYFYIVEVBYAYJXNYKXAOYAXNYJYAXLXMYNUUBUUCVC
|
|
VFYAYKXAXMYIRKZRKXASRKXAYAXAXMYIYTUUBUUDVGYAUUESXARYAXMUUBUUCVHVIYAXAYTVJ
|
|
VNVKVLYAYEXLYCSWTJKZRKZOPZYGYAYEYBUUFRKZUUGOPZUUHYAYBTGZYCTGZUUFTGUUFUNOP
|
|
YEUUJQYAYBYAXLWTYMYAWTYQWDVMZUOZYAYCYAXMWSYAXMUUAWDYOVMZUOZYAWTYRYSUTZYAW
|
|
TYRYSVAYBYCUUFVEVBYAUUIXLUUGOYAUUIXLWTUUFRKZRKXLSRKXLYAXLWTUUFYNYRUUQVGYA
|
|
UURSXLRYAWTYRYSVHVIYAXLYNVJVNVOVLYAUUGYFXLOYAUUGXMWSUUFRKZRKXMXARKYFYAXMW
|
|
SUUFUUBYPUUQVGYAXAUUSXMRYAWSWTYPYRYSVCVIYAXMXAUUBYTVPVQVRVLYAAXNBXAOXTXOV
|
|
SXJXBXSXOVTZVKWAZYAYBNGYCNGYEYDQUUMUUOYBYCWEWBWCYAYBYCABYAYEWFZXDWFZYBYCL
|
|
ZABLZYAYEXDUVAWGYAUUKUULUVDUVBQUUNUUPYBYCWHWBYAATGZBTGUVEUVCQXJUVFXBXSXOW
|
|
PUVFWQXIAWIWJUKYABXATUUTYTWKABWHWBWAWLVLWOWMWNWOWMWN $.
|
|
$}
|
|
|
|
$( Rational 'Less than' expressed in terms of 'less than or equal to'. Also
|
|
see ~ ltleap which is a similar result for real numbers. (Contributed by
|
|
Jim Kingdon, 11-Oct-2021.) $)
|
|
qltlen $p |- ( ( A e. QQ /\ B e. QQ ) ->
|
|
( A < B <-> ( A <_ B /\ B =/= A ) ) ) $=
|
|
( cq wcel wa clt wbr cle wne cap cr wb qre ltleap syl2an qapne anbi2d bitrd
|
|
necom anbi2i syl6bb ) ACDZBCDZEZABFGZABHGZABIZEZUFBAIZEUDUEUFABJGZEZUHUBAKD
|
|
BKDUEUKLUCAMBMABNOUDUJUGUFABPQRUGUIUFABSTUA $.
|
|
|
|
$( Apartness is equivalent to not equal for rationals. (Contributed by Jim
|
|
Kingdon, 9-Nov-2021.) $)
|
|
qlttri2 $p |- ( ( A e. QQ /\ B e. QQ ) ->
|
|
( A =/= B <-> ( A < B \/ B < A ) ) ) $=
|
|
( cq wcel wa cap wbr wne clt wo qapne cr wb qre reaplt syl2an bitr3d ) ACDZ
|
|
BCDZEABFGZABHABIGBAIGJZABKRALDBLDTUAMSANBNABOPQ $.
|
|
|
|
${
|
|
$d x y z w A $.
|
|
$( Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.) $)
|
|
qreccl $p |- ( ( A e. QQ /\ A =/= 0 ) -> ( 1 / A ) e. QQ ) $=
|
|
( vx vy vz vw cq wcel cc0 cap wbr c1 cdiv co wb cz cn cv wceq wrex wa wne
|
|
ax-1cn 1ap0 div0api 0z 1nn znq mp2an eqeltrri qapne mpan2 biimpar wi cmul
|
|
elq nnne0 ancli nnz zapne sylancl adantl pm5.32i anbi1i breq1 cc zcn nncn
|
|
anim12i divap0b 3expa sylan bicomd sylan9bbr sylbir simplll zmulcl sylan2
|
|
bitrd adantr msqznn adantlr bitri oveq2 dividap oveq1d simpll simpl simpr
|
|
jca divdivdivap syl22anc eqtr3d anass1rs sylan9eqr an32s ex sylbid anasss
|
|
an4s rspceov sylibr syl8 rexlimivv sylbi imp syldan ) AFGZAHUAZAHIJZKALMZ
|
|
FGZXGXIXHXGHFGXIXHNHKLMZHFKUBUCUDHOGZKPGXLFGUEUFHKUGUHUIAHUJUKULXGXIXKXGA
|
|
BQZCQZLMZRZCPSBOSXIXKUMZBCAUOXQXRBCOPXNOGZXOPGZTZXQXIXNXOUNMZOGZXNXNUNMZP
|
|
GZTZXJYBYDLMZRZTZXKXTXSXTXOHUAZTXQXIYIUMZUMZXTYJXOUPUQXSXTYJYLYAYJTZXQYKY
|
|
MXQTZXIXNHUAZYIYNXIXNHIJZYOYNYAXOHIJZTZXQTZXIYPNYRYMXQYAYQYJXTYQYJNZXSXTX
|
|
OOGZXMYTXOURZUEXOHUSUTVAVBVCZXQXIXPHIJZYRYPAXPHIVDYRYPUUDYAXNVEGZXOVEGZTZ
|
|
YQYPUUDNZXSUUEXTUUFXNVFXOVGVHZUUEUUFYQUUHXNXOVIVJVKVLVMVNYNXSXMYPYONXSXTY
|
|
JXQVOUEXNHUSUTZVRYNYOYIYNYOTZYFYHYMYOYFXQYAYOYFYJYAYOTYCYEYAYCYOXTXSUUAYC
|
|
UUBXNXOVPVQVSXSYOYEXTXNVTWAWIWAWAUUKYSYPTZYHUULYNYPTUUKYSYNYPUUCVCYNYPYOU
|
|
UJVBWBYRYPXQYHXQYRYPTXJKXPLMZYGAXPKLWCYAYPYQUUMYGRZYAUUGYPYQTUUNUUIUUEYPU
|
|
UFYQUUNUUEYPTZUUFYQTZTZXNXNLMZXPLMZUUMYGUUQUURKXPLUUOUURKRUUPXNWDVSWEUUQU
|
|
UEUUOUUOUUPUUSYGRUUEYPUUPWFUUOUUPWGZUUTUUOUUPWHXNXNXNXOWJWKWLWSVKWMWNWOVN
|
|
WIWPWQWPWRVQYIXJDQEQLMREPSDOSZXKYCYEYHUVADEOPYBYDXJLWTVJDEXJUOXAXBXCXDXEX
|
|
F $.
|
|
$}
|
|
|
|
$( Closure of division of rationals. (Contributed by NM, 3-Aug-2004.) $)
|
|
qdivcl $p |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ ) $=
|
|
( cq wcel cc0 wne w3a cdiv co c1 cmul cc qcn 3ad2ant1 3ad2ant2 cap simp3 wb
|
|
wbr cz 0z zq ax-mp qapne mpan2 mpbird divrecapd qreccl qmulcl 3impb eqeltrd
|
|
wa sylan2 ) ACDZBCDZBEFZGZABHIAJBHIZKIZCUQABUNUOALDUPAMNUOUNBLDUPBMOUQBEPSZ
|
|
UPUNUOUPQUOUNUTUPRZUPUOECDZVAETDVBUAEUBUCBEUDUEOUFUGUNUOUPUSCDZUOUPULUNURCD
|
|
VCBUHAURUIUMUJUK $.
|
|
|
|
$( Reverse closure law for addition of rationals. (Contributed by NM,
|
|
2-Aug-2004.) $)
|
|
qrevaddcl $p |- ( B e. QQ ->
|
|
( ( A e. CC /\ ( A + B ) e. QQ ) <-> A e. QQ ) ) $=
|
|
( cq wcel cc caddc co wa cmin qcn pncan sylan2 ancoms adantr qsubcl adantlr
|
|
wceq eqeltrrd ex wi qaddcl expcom impbid pm5.32da pm4.71ri syl6bbr ) BCDZAE
|
|
DZABFGZCDZHUHACDZHUKUGUHUJUKUGUHHZUJUKULUJUKULUJHUIBIGZACULUMAQZUJUHUGUNUGU
|
|
HBEDUNBJABKLMNUGUJUMCDZUHUJUGUOUIBOMPRSUGUKUJTUHUKUGUJABUAUBNUCUDUKUHAJUEUF
|
|
$.
|
|
|
|
$( The reciprocal of a positive integer is rational. (Contributed by NM,
|
|
17-Nov-2004.) $)
|
|
nnrecq $p |- ( A e. NN -> ( 1 / A ) e. QQ ) $=
|
|
( c1 cz wcel cn cdiv co cq 1z znq mpan ) BCDAEDBAFGHDIBAJK $.
|
|
|
|
$( The sum of an irrational number and a rational number is irrational.
|
|
(Contributed by NM, 7-Nov-2008.) $)
|
|
irradd $p |- ( ( A e. ( RR \ QQ ) /\ B e. QQ )
|
|
-> ( A + B ) e. ( RR \ QQ ) ) $=
|
|
( cr cq cdif wcel wa caddc co wn eldif qre readdcl sylan2 adantlr wi qsubcl
|
|
cmin expcom cc adantl wceq qcn pncan syl2an eleq1d sylibd con3d com23 imp31
|
|
recn ex jca sylanb sylibr ) ACDEZFZBDFZGABHIZCFZUSDFZJZGZUSUPFUQACFZADFZJZG
|
|
ZURVCACDKVGURGUTVBVDURUTVFURVDBCFUTBLABMNOVDVFURVBVDURVFVBVDURVFVBPVDURGZVA
|
|
VEVHVAUSBRIZDFZVEURVAVJPVDVAURVJUSBQSUAVHVIADVDATFBTFVIAUBURAUKBUCABUDUEUFU
|
|
GUHULUIUJUMUNUSCDKUO $.
|
|
|
|
$( The product of a real which is not rational with a nonzero rational is not
|
|
rational. Note that by "not rational" we mean the negation of "is
|
|
rational" (whereas "irrational" is often defined to mean apart from any
|
|
rational number - given excluded middle these two definitions would be
|
|
equivalent). (Contributed by NM, 7-Nov-2008.) $)
|
|
irrmul $p |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 )
|
|
-> ( A x. B ) e. ( RR \ QQ ) ) $=
|
|
( cr cq cdif wcel cc0 wne w3a cmul co wn wa eldif remulcl wi 3expb 3ad2ant2
|
|
qre cc sylan2 ad2ant2r cdiv qdivcl expcom adantl wceq recn 3ad2ant1 qcn cap
|
|
simp3 wb cz 0z zq ax-mp qapne mpan2 mpbird divcanap4d eleq1d con3d ex com23
|
|
wbr sylibd imp31 jca 3impb syl3an1b sylibr ) ACDEZFZBDFZBGHZIABJKZCFZVQDFZL
|
|
ZMZVQVMFVNACFZADFZLZMZVOVPWAACDNWEVOVPWAWEVOVPMZMVRVTWBVOVRWDVPVOWBBCFVRBSA
|
|
BOUAUBWBWDWFVTWBWFWDVTWBWFWDVTPWBWFMZVSWCWGVSVQBUCKZDFZWCWFVSWIPWBVSWFWIVSV
|
|
OVPWIVQBUDQUEUFWGWHADWBVOVPWHAUGWBVOVPIZABWBVOATFVPAUHUIVOWBBTFVPBUJRWJBGUK
|
|
VFZVPWBVOVPULVOWBWKVPUMZVPVOGDFZWLGUNFWMUOGUPUQBGURUSRUTVAQVBVGVCVDVEVHVIVJ
|
|
VKVQCDNVL $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Complex numbers as pairs of reals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d F u v w z $. $d x y u v w z $.
|
|
cnref1o.1 $e |- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) $.
|
|
$( There is a natural one-to-one mapping from ` ( RR X. RR ) ` to ` CC ` ,
|
|
where we map ` <. x , y >. ` to ` ( x + ( _i x. y ) ) ` . In our
|
|
construction of the complex numbers, this is in fact our _definition_ of
|
|
` CC ` (see ~ df-c ), but in the axiomatic treatment we can only show
|
|
that there is the expected mapping between these two sets. (Contributed
|
|
by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro,
|
|
17-Feb-2014.) $)
|
|
cnref1o $p |- F : ( RR X. RR ) -1-1-onto-> CC $=
|
|
( vz vw vu vv cr cc cv cfv wceq wral wcel ci cmul co caddc wa cxp wf1 wfo
|
|
wf1o wf wi wfn simpl recnd ax-icn simpr mulcld addcld rgen2a fnmpt2 ax-mp
|
|
a1i c1st c2nd 1st2nd2 fveq2d df-ov syl6eqr xp1st xp2nd oveq1 oveq2 oveq2d
|
|
cop ovmpt2g syl3anc eqtrd eqeltrd rgen ffnfv mpbir2an wb jca syl2an fveq2
|
|
cru oveq12d eqeq12d vtoclga eqeqan12d cvv vex 1stexg 2ndexg syl6bb biimpd
|
|
opth 3bitr4d dff13 wrex cnre eqeq2d 2rexbiia sylibr rexxp dffo3 df-f1o )
|
|
IIUAZJCUDXCJCUBZXCJCUCZXDXCJCUEZEKZCLZFKZCLZMZXGXIMZUFZFXCNEXCNXFCXCUGZXH
|
|
JOZEXCNAKZPBKZQRZSRZJOZBINAINXNXTABIXPIOZXQIOZTZXPXRYCXPYAYBUHUIYCPXQPJOZ
|
|
YCUJUQYCXQYAYBUKUIULUMUNABIIXSCJDUOUPXOEXCXGXCOZXHXGURLZPXGUSLZQRZSRZJYEX
|
|
HYFYGCRZYIYEXHYFYGVIZCLYJYEXGYKCXGIIUTZVAYFYGCVBVCYEYFIOZYGIOZYIJOYJYIMXG
|
|
IIVDZXGIIVEZYEYFYHYEYFYOUIYEPYGYDYEUJUQYEYGYPUIULUMZABYFYGIIXSYICYFXRSRJX
|
|
PYFXRSVFXQYGMXRYHYFSXQYGPQVGVHDVJVKVLZYQVMVNEXCJCVOVPZXMEFXCYEXIXCOZTZXKX
|
|
LUUAYIXIURLZPXIUSLZQRZSRZMZYFUUBMYGUUCMTZXKXLYEYMYNTUUBIOZUUCIOZTUUFUUGVQ
|
|
YTYEYMYNYOYPVRYTUUHUUIXIIIVDXIIIVEVRYFYGUUBUUCWAVSYEYTXHYIXJUUEYRXHYIMXJU
|
|
UEMEXIXCXLXHXJYIUUEXGXICVTXLYFUUBYHUUDSXGXIURVTXLYGUUCPQXGXIUSVTVHWBWCYRW
|
|
DWEUUAXLYKUUBUUCVIZMUUGYEYTXGYKXIUUJYLXIIIUTWEYFYGUUBUUCXGWFOZYFWFOEWGZXG
|
|
WFWHUPUUKYGWFOUULXGWFWIUPWLWJWMWKUNEFXCJCWNVPXEXFXIXHMZEXCWOZFJNYSUUNFJXI
|
|
JOZXIGKZHKZCRZMZHIWOGIWOZUUNUUOXIUUPPUUQQRZSRZMZHIWOGIWOUUTGHXIWPUUSUVCGH
|
|
IIUUPIOZUUQIOZTZUURUVBXIUVFUVDUVEUVBJOUURUVBMUVDUVEUHZUVDUVEUKZUVFUUPUVAU
|
|
VFUUPUVGUIUVFPUUQYDUVFUJUQUVFUUQUVHUIULUMABUUPUUQIIXSUVBCUUPXRSRJXPUUPXRS
|
|
VFXQUUQMXRUVAUUPSXQUUQPQVGVHDVJVKWQWRWSUUMUUSEGHIIXGUUPUUQVIZMZXHUURXIUVJ
|
|
XHUVICLUURXGUVICVTUUPUUQCVBVCWQWTWSVNEFXCJCXAVPXCJCXBVP $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Order sets
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Positive reals (as a subset of complex numbers)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c RR+ $. $( The set of positive reals (blackboard bold R^+). $)
|
|
|
|
$( Extend class notation to include the class of positive reals. $)
|
|
crp $a class RR+ $.
|
|
|
|
$( Define the set of positive reals. Definition of positive numbers in
|
|
[Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) $)
|
|
df-rp $a |- RR+ = { x e. RR | 0 < x } $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Membership in the set of positive reals. (Contributed by NM,
|
|
27-Oct-2007.) $)
|
|
elrp $p |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) ) $=
|
|
( vx cc0 cv clt wbr cr crp breq2 df-rp elrab2 ) CBDZEFCAEFBAGHLACEIBJK $.
|
|
$}
|
|
|
|
${
|
|
elrpi.1 $e |- A e. RR $.
|
|
elrpi.2 $e |- 0 < A $.
|
|
$( Membership in the set of positive reals. (Contributed by NM,
|
|
23-Feb-2008.) $)
|
|
elrpii $p |- A e. RR+ $=
|
|
( crp wcel cr cc0 clt wbr elrp mpbir2an ) ADEAFEGAHIBCAJK $.
|
|
$}
|
|
|
|
$( 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.) $)
|
|
1rp $p |- 1 e. RR+ $=
|
|
( c1 1re 0lt1 elrpii ) ABCD $.
|
|
|
|
$( 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
2rp $p |- 2 e. RR+ $=
|
|
( c2 2re 2pos elrpii ) ABCD $.
|
|
|
|
$( A positive real is a real. (Contributed by NM, 27-Oct-2007.) $)
|
|
rpre $p |- ( A e. RR+ -> A e. RR ) $=
|
|
( vx crp cr cc0 cv clt wbr crab df-rp ssrab2 eqsstri sseli ) CDACEBFGHZBDID
|
|
BJNBDKLM $.
|
|
|
|
$( A positive real is an extended real. (Contributed by Mario Carneiro,
|
|
21-Aug-2015.) $)
|
|
rpxr $p |- ( A e. RR+ -> A e. RR* ) $=
|
|
( crp wcel rpre rexrd ) ABCAADE $.
|
|
|
|
$( A positive real is a complex number. (Contributed by NM, 11-Nov-2008.) $)
|
|
rpcn $p |- ( A e. RR+ -> A e. CC ) $=
|
|
( crp wcel rpre recnd ) ABCAADE $.
|
|
|
|
$( A positive integer is a positive real. (Contributed by NM,
|
|
28-Nov-2008.) $)
|
|
nnrp $p |- ( A e. NN -> A e. RR+ ) $=
|
|
( cn wcel cr cc0 clt wbr crp nnre nngt0 elrp sylanbrc ) ABCADCEAFGAHCAIAJAK
|
|
L $.
|
|
|
|
$( The positive reals are a subset of the reals. (Contributed by NM,
|
|
24-Feb-2008.) $)
|
|
rpssre $p |- RR+ C_ RR $=
|
|
( vx crp cr cv rpre ssriv ) ABCADEF $.
|
|
|
|
$( A positive real is greater than zero. (Contributed by FL,
|
|
27-Dec-2007.) $)
|
|
rpgt0 $p |- ( A e. RR+ -> 0 < A ) $=
|
|
( crp wcel cr cc0 clt wbr elrp simprbi ) ABCADCEAFGAHI $.
|
|
|
|
$( A positive real is greater than or equal to zero. (Contributed by NM,
|
|
22-Feb-2008.) $)
|
|
rpge0 $p |- ( A e. RR+ -> 0 <_ A ) $=
|
|
( crp wcel cr cc0 clt wbr cle rpre rpgt0 wi 0re ltle mpan sylc ) ABCADCZEAF
|
|
GZEAHGZAIAJEDCPQRKLEAMNO $.
|
|
|
|
$( A positive real is a positive real number. (Contributed by NM,
|
|
11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
rpregt0 $p |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) ) $=
|
|
( crp wcel cr cc0 clt wbr wa elrp biimpi ) ABCADCEAFGHAIJ $.
|
|
|
|
$( A positive real is a nonnegative real number. (Contributed by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
rprege0 $p |- ( A e. RR+ -> ( A e. RR /\ 0 <_ A ) ) $=
|
|
( crp wcel cr cc0 cle wbr rpre rpge0 jca ) ABCADCEAFGAHAIJ $.
|
|
|
|
$( A positive real is nonzero. (Contributed by NM, 18-Jul-2008.) $)
|
|
rpne0 $p |- ( A e. RR+ -> A =/= 0 ) $=
|
|
( crp wcel cr cc0 clt wbr wa wne rpregt0 gt0ne0 syl ) ABCADCEAFGHAEIAJAKL
|
|
$.
|
|
|
|
$( A positive real is apart from zero. (Contributed by Jim Kingdon,
|
|
22-Mar-2020.) $)
|
|
rpap0 $p |- ( A e. RR+ -> A # 0 ) $=
|
|
( crp wcel rpre rpgt0 gt0ap0d ) ABCAADAEF $.
|
|
|
|
$( A positive real is a nonzero real number. (Contributed by NM,
|
|
11-Nov-2008.) $)
|
|
rprene0 $p |- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) ) $=
|
|
( crp wcel cr cc0 wne rpre rpne0 jca ) ABCADCAEFAGAHI $.
|
|
|
|
$( A positive real is a real number apart from zero. (Contributed by Jim
|
|
Kingdon, 22-Mar-2020.) $)
|
|
rpreap0 $p |- ( A e. RR+ -> ( A e. RR /\ A # 0 ) ) $=
|
|
( crp wcel cr cc0 cap wbr rpre rpap0 jca ) ABCADCAEFGAHAIJ $.
|
|
|
|
$( A positive real is a nonzero complex number. (Contributed by NM,
|
|
11-Nov-2008.) $)
|
|
rpcnne0 $p |- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) ) $=
|
|
( crp wcel cc cc0 wne rpcn rpne0 jca ) ABCADCAEFAGAHI $.
|
|
|
|
$( A positive real is a complex number apart from zero. (Contributed by Jim
|
|
Kingdon, 22-Mar-2020.) $)
|
|
rpcnap0 $p |- ( A e. RR+ -> ( A e. CC /\ A # 0 ) ) $=
|
|
( crp wcel cc cc0 cap wbr rpcn rpap0 jca ) ABCADCAEFGAHAIJ $.
|
|
|
|
$( Quantification over positive reals. (Contributed by NM, 12-Feb-2008.) $)
|
|
ralrp $p |- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) ) $=
|
|
( cc0 cv clt wbr wi crp cr wcel wa elrp imbi1i impexp bitri ralbii2 ) ACBDZ
|
|
EFZAGZBHIQHJZAGQIJZRKZAGUASGTUBAQLMUARANOP $.
|
|
|
|
$( Quantification over positive reals. (Contributed by Mario Carneiro,
|
|
21-May-2014.) $)
|
|
rexrp $p |- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) ) $=
|
|
( cc0 cv clt wbr wa crp cr wcel elrp anbi1i anass bitri rexbii2 ) ACBDZEFZA
|
|
GZBHIPHJZAGPIJZQGZAGTRGSUAAPKLTQAMNO $.
|
|
|
|
$( Closure law for addition of positive reals. Part of Axiom 7 of [Apostol]
|
|
p. 20. (Contributed by NM, 27-Oct-2007.) $)
|
|
rpaddcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A + B ) e. RR+ ) $=
|
|
( crp wcel wa caddc co cr cc0 clt wbr rpre readdcl syl2an elrp an4s syl2anb
|
|
addgt0 sylanbrc ) ACDZBCDZEABFGZHDZIUBJKZUBCDTAHDZBHDZUCUAALBLABMNTUEIAJKZE
|
|
UFIBJKZEUDUAAOBOUEUFUGUHUDABRPQUBOS $.
|
|
|
|
$( Closure law for multiplication of positive reals. Part of Axiom 7 of
|
|
[Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) $)
|
|
rpmulcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ ) $=
|
|
( crp wcel wa cmul co cc0 clt wbr rpre remulcl syl2an elrp syl2anb sylanbrc
|
|
cr mulgt0 ) ACDZBCDZEABFGZQDZHUAIJZUACDSAQDZBQDZUBTAKBKABLMSUDHAIJEUEHBIJEU
|
|
CTANBNABROUANP $.
|
|
|
|
$( Closure law for division of positive reals. (Contributed by FL,
|
|
27-Dec-2007.) $)
|
|
rpdivcl $p |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A / B ) e. RR+ ) $=
|
|
( crp wcel wa cdiv co cr cc0 clt wbr cap rpre rpreap0 redivclap syl2an elrp
|
|
3expb divgt0 syl2anb sylanbrc ) ACDZBCDZEABFGZHDZIUDJKZUDCDUBAHDZBHDZBILKZE
|
|
UEUCAMBNUGUHUIUEABORPUBUGIAJKEUHIBJKEUFUCAQBQABSTUDQUA $.
|
|
|
|
$( Closure law for reciprocation of positive reals. (Contributed by Jeff
|
|
Hankins, 23-Nov-2008.) $)
|
|
rpreccl $p |- ( A e. RR+ -> ( 1 / A ) e. RR+ ) $=
|
|
( c1 crp wcel cdiv co 1rp rpdivcl mpan ) BCDACDBAEFCDGBAHI $.
|
|
|
|
$( Closure law for half of a positive real. (Contributed by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
rphalfcl $p |- ( A e. RR+ -> ( A / 2 ) e. RR+ ) $=
|
|
( crp wcel c2 cdiv co 2rp rpdivcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( A number greater or equal to a positive real is positive real.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
rpgecl $p |- ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> B e. RR+ ) $=
|
|
( crp wcel cle wbr w3a cc0 clt simp2 0red rpre 3ad2ant1 rpgt0 simp3 ltletrd
|
|
cr elrp sylanbrc ) ACDZBQDZABEFZGZUAHBIFBCDTUAUBJZUCHABUCKTUAAQDUBALMUDTUAH
|
|
AIFUBANMTUAUBOPBRS $.
|
|
|
|
$( Half of a positive real is less than the original number. (Contributed by
|
|
Mario Carneiro, 21-May-2014.) $)
|
|
rphalflt $p |- ( A e. RR+ -> ( A / 2 ) < A ) $=
|
|
( crp wcel cr cc0 clt wbr wa c2 cdiv co elrp halfpos biimpa sylbi ) ABCADCZ
|
|
EAFGZHAIJKAFGZALPQRAMNO $.
|
|
|
|
$( Closure law for division of a real by a positive real. (Contributed by
|
|
NM, 10-Nov-2008.) $)
|
|
rerpdivcl $p |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR ) $=
|
|
( crp wcel cr cc0 cap wbr wa cdiv co rpreap0 redivclap 3expb sylan2 ) BCDAE
|
|
DZBEDZBFGHZIABJKEDZBLPQRSABMNO $.
|
|
|
|
$( A nonnegative number plus one is a positive number. (Contributed by Mario
|
|
Carneiro, 5-Oct-2015.) $)
|
|
ge0p1rp $p |- ( ( A e. RR /\ 0 <_ A ) -> ( A + 1 ) e. RR+ ) $=
|
|
( cr wcel cc0 cle wbr wa c1 caddc co clt crp peano2re 0red simpl simpr ltp1
|
|
adantr lelttrd elrp sylanbrc ) ABCZDAEFZGZAHIJZBCZDUEKFUELCUBUFUCAMRZUDDAUE
|
|
UDNUBUCOUGUBUCPUBAUEKFUCAQRSUETUA $.
|
|
|
|
$( Either a real apart from zero or its negation is a positive real, but not
|
|
both. (Contributed by Jim Kingdon, 23-Mar-2020.) $)
|
|
rpnegap $p |- ( ( A e. RR /\ A # 0 ) -> ( A e. RR+ \/_ -u A e. RR+ ) ) $=
|
|
( cr wcel cc0 cap wbr wa crp cneg wxo clt 0re reapltxor mpan2 xorcom syl6bb
|
|
wb pm5.32i anxordi elrp bitri biimpi a1i renegcl biantrurd syl6rbbr lt0neg1
|
|
ibar 3bitr2d adantr xorbi12d mpbird ) ABCZADEFZGZAHCZAIZHCZJUMDAKFZGZUMADKF
|
|
ZGZJZUOVCUOUMUSVAJZGVCUMUNVDUMUNVAUSJZVDUMDBCUNVEQLADMNVAUSOPRUMUSVASUAUBUO
|
|
UPUTURVBUPUTQUOATUCUMURVBQUNUMURDUQKFZVAVBUMVFUQBCZVFGURUMVGVFAUDUEUQTUFAUG
|
|
UMVAUHUIUJUKUL $.
|
|
|
|
$( Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by
|
|
NM, 27-Oct-2007.) $)
|
|
0nrp $p |- -. 0 e. RR+ $=
|
|
( cc0 crp wcel clt wbr 0re ltnri rpgt0 mto ) ABCAADEAFGAHI $.
|
|
|
|
$( Subtracting a positive real from another number decreases it.
|
|
(Contributed by FL, 27-Dec-2007.) $)
|
|
ltsubrp $p |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A ) $=
|
|
( crp wcel cr cc0 clt wbr wa cmin co elrp wi ltsubpos biimpd expcom sylan2b
|
|
imp32 ) BCDAEDZBEDZFBGHZIABJKAGHZBLSTUAUBTSUAUBMTSIUAUBBANOPRQ $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed by
|
|
FL, 27-Dec-2007.) $)
|
|
ltaddrp $p |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) ) $=
|
|
( crp wcel cr cc0 clt wbr wa caddc co elrp wi ltaddpos biimpd imp32 sylan2b
|
|
expcom ) BCDAEDZBEDZFBGHZIAABJKGHZBLSTUAUBTSUAUBMTSIUAUBBANORPQ $.
|
|
|
|
$( Two ways to say one number is less than another. (Contributed by Mario
|
|
Carneiro, 21-May-2014.) $)
|
|
difrp $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A < B <-> ( B - A ) e. RR+ ) ) $=
|
|
( cr wcel wa clt wbr cc0 cmin co crp posdif wb resubcl ancoms elrp baib syl
|
|
bitr4d ) ACDZBCDZEZABFGHBAIJZFGZUCKDZABLUBUCCDZUEUDMUATUFBANOUEUFUDUCPQRS
|
|
$.
|
|
|
|
${
|
|
elrpd.1 $e |- ( ph -> A e. RR ) $.
|
|
elrpd.2 $e |- ( ph -> 0 < A ) $.
|
|
$( Membership in the set of positive reals. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
elrpd $p |- ( ph -> A e. RR+ ) $=
|
|
( cr wcel cc0 clt wbr crp elrp sylanbrc ) ABEFGBHIBJFCDBKL $.
|
|
$}
|
|
|
|
${
|
|
nnrpd.1 $e |- ( ph -> A e. NN ) $.
|
|
$( A positive integer is a positive real. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
nnrpd $p |- ( ph -> A e. RR+ ) $=
|
|
( cn wcel crp nnrp syl ) ABDEBFECBGH $.
|
|
$}
|
|
|
|
${
|
|
rpred.1 $e |- ( ph -> A e. RR+ ) $.
|
|
$( A positive real is a real. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rpred $p |- ( ph -> A e. RR ) $=
|
|
( crp cr rpssre sseldi ) ADEBFCG $.
|
|
|
|
$( A positive real is an extended real. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rpxrd $p |- ( ph -> A e. RR* ) $=
|
|
( rpred rexrd ) ABABCDE $.
|
|
|
|
$( A positive real is a complex number. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rpcnd $p |- ( ph -> A e. CC ) $=
|
|
( rpred recnd ) ABABCDE $.
|
|
|
|
$( A positive real is greater than zero. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rpgt0d $p |- ( ph -> 0 < A ) $=
|
|
( crp wcel cc0 clt wbr rpgt0 syl ) ABDEFBGHCBIJ $.
|
|
|
|
$( A positive real is greater than or equal to zero. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpge0d $p |- ( ph -> 0 <_ A ) $=
|
|
( crp wcel cc0 cle wbr rpge0 syl ) ABDEFBGHCBIJ $.
|
|
|
|
$( A positive real is nonzero. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
rpne0d $p |- ( ph -> A =/= 0 ) $=
|
|
( crp wcel cc0 wne rpne0 syl ) ABDEBFGCBHI $.
|
|
|
|
$( A positive real is apart from zero. (Contributed by Jim Kingdon,
|
|
28-Jul-2021.) $)
|
|
rpap0d $p |- ( ph -> A # 0 ) $=
|
|
( crp wcel cc0 cap wbr rpap0 syl ) ABDEBFGHCBIJ $.
|
|
|
|
$( A positive real is real and greater than zero. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpregt0d $p |- ( ph -> ( A e. RR /\ 0 < A ) ) $=
|
|
( cr wcel cc0 clt wbr rpred rpgt0d jca ) ABDEFBGHABCIABCJK $.
|
|
|
|
$( A positive real is real and greater or equal to zero. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
rprege0d $p |- ( ph -> ( A e. RR /\ 0 <_ A ) ) $=
|
|
( cr wcel cc0 cle wbr rpred rpge0d jca ) ABDEFBGHABCIABCJK $.
|
|
|
|
$( A positive real is a nonzero real number. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rprene0d $p |- ( ph -> ( A e. RR /\ A =/= 0 ) ) $=
|
|
( cr wcel cc0 wne rpred rpne0d jca ) ABDEBFGABCHABCIJ $.
|
|
|
|
$( A positive real is a nonzero complex number. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpcnne0d $p |- ( ph -> ( A e. CC /\ A =/= 0 ) ) $=
|
|
( cc wcel cc0 wne rpcnd rpne0d jca ) ABDEBFGABCHABCIJ $.
|
|
|
|
$( Closure law for reciprocation of positive reals. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpreccld $p |- ( ph -> ( 1 / A ) e. RR+ ) $=
|
|
( crp wcel c1 cdiv co rpreccl syl ) ABDEFBGHDECBIJ $.
|
|
|
|
$( Closure law for reciprocation of positive reals. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rprecred $p |- ( ph -> ( 1 / A ) e. RR ) $=
|
|
( c1 cdiv co rpreccld rpred ) ADBEFABCGH $.
|
|
|
|
$( Closure law for half of a positive real. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rphalfcld $p |- ( ph -> ( A / 2 ) e. RR+ ) $=
|
|
( crp wcel c2 cdiv co rphalfcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
$( The reciprocal of a positive number less than 1 is greater than 1.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
reclt1d $p |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d reclt1 syl ) ABDEFBGHIBJG
|
|
HJJBKLGHMABCNBOP $.
|
|
|
|
$( The reciprocal of a positive number greater than 1 is less than 1.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
recgt1d $p |- ( ph -> ( 1 < A <-> ( 1 / A ) < 1 ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d recgt1 syl ) ABDEFBGHIJBG
|
|
HJBKLJGHMABCNBOP $.
|
|
|
|
rpaddcld.1 $e |- ( ph -> B e. RR+ ) $.
|
|
$( Closure law for addition of positive reals. Part of Axiom 7 of
|
|
[Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
rpaddcld $p |- ( ph -> ( A + B ) e. RR+ ) $=
|
|
( crp wcel caddc co rpaddcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure law for multiplication of positive reals. Part of Axiom 7 of
|
|
[Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
rpmulcld $p |- ( ph -> ( A x. B ) e. RR+ ) $=
|
|
( crp wcel cmul co rpmulcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( Closure law for division of positive reals. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpdivcld $p |- ( ph -> ( A / B ) e. RR+ ) $=
|
|
( crp wcel cdiv co rpdivcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
|
|
$( The reciprocal of both sides of 'less than'. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
ltrecd $p |- ( ph -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa c1 cdiv co wb rpregt0d ltrec syl2anc ) ABFGHBIJK
|
|
CFGHCIJKBCIJLCMNLBMNIJOABDPACEPBCQR $.
|
|
|
|
$( The reciprocal of both sides of 'less than or equal to'. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
lerecd $p |- ( ph -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle c1 cdiv co wb rpregt0d lerec syl2anc ) ABFGH
|
|
BIJKCFGHCIJKBCLJMCNOMBNOLJPABDQACEQBCRS $.
|
|
|
|
${
|
|
ltrec1d.2 $e |- ( ph -> ( 1 / A ) < B ) $.
|
|
$( Reciprocal swap in a 'less than' relation. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
ltrec1d $p |- ( ph -> ( 1 / B ) < A ) $=
|
|
( c1 cdiv co clt wbr cr wcel cc0 wa wb rpregt0d ltrec1 syl2anc mpbid )
|
|
AGBHICJKZGCHIBJKZFABLMNBJKOCLMNCJKOUAUBPABDQACEQBCRST $.
|
|
$}
|
|
|
|
${
|
|
lerec2d.2 $e |- ( ph -> A <_ ( 1 / B ) ) $.
|
|
$( Reciprocal swap in a 'less than or equal to' relation. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
lerec2d $p |- ( ph -> B <_ ( 1 / A ) ) $=
|
|
( c1 cdiv co cle wbr cr wcel cc0 clt wa wb rpregt0d lerec2 syl2anc
|
|
mpbid ) ABGCHIJKZCGBHIJKZFABLMNBOKPCLMNCOKPUBUCQABDRACERBCSTUA $.
|
|
$}
|
|
|
|
${
|
|
lediv2ad.3 $e |- ( ph -> C e. RR ) $.
|
|
lediv2ad.4 $e |- ( ph -> 0 <_ C ) $.
|
|
lediv2ad.5 $e |- ( ph -> A <_ B ) $.
|
|
$( Division of both sides of 'less than or equal to' into a nonnegative
|
|
number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lediv2ad $p |- ( ph -> ( C / B ) <_ ( C / A ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle cdiv co rpregt0d jca lediv2a syl31anc ) AB
|
|
JKLBMNOCJKLCMNODJKZLDPNZOBCPNDCQRDBQRPNABESACFSAUCUDGHTIBCDUAUB $.
|
|
$}
|
|
|
|
ltdiv2d.3 $e |- ( ph -> C e. RR+ ) $.
|
|
$( Division of a positive number by both sides of 'less than'.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltdiv2d $p |- ( ph -> ( A < B <-> ( C / B ) < ( C / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co wb rpregt0d ltdiv2 syl3anc ) ABHIJBKLMCH
|
|
IJCKLMDHIJDKLMBCKLDCNODBNOKLPABEQACFQADGQBCDRS $.
|
|
|
|
$( Division of a positive number by both sides of 'less than or equal to'.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lediv2d $p |- ( ph -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle cdiv co wb rpregt0d lediv2 syl3anc ) ABHIJBK
|
|
LMCHIJCKLMDHIJDKLMBCNLDCOPDBOPNLQABERACFRADGRBCDST $.
|
|
|
|
ledivdivd.4 $e |- ( ph -> D e. RR+ ) $.
|
|
${
|
|
ledivdivd.5 $e |- ( ph -> ( A / B ) <_ ( C / D ) ) $.
|
|
$( Invert ratios of positive numbers and swap their ordering.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ledivdivd $p |- ( ph -> ( D / C ) <_ ( B / A ) ) $=
|
|
( cdiv co cle wbr cr wcel cc0 clt wa rpregt0d ledivdiv syl22anc mpbid
|
|
wb ) ABCKLDEKLMNZEDKLCBKLMNZJABOPQBRNSCOPQCRNSDOPQDRNSEOPQERNSUEUFUDABF
|
|
TACGTADHTAEITBCDEUAUBUC $.
|
|
$}
|
|
$}
|
|
|
|
$( The ratio of a number over a smaller positive number is larger than 1.
|
|
(Contributed by Glauco Siliprandi, 5-Apr-2020.) $)
|
|
divge1 $p |- ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> 1 <_ ( B / A ) ) $=
|
|
( crp wcel cr cle wbr w3a c1 cdiv co wceq rpgecl rpcn rpap0 dividapd eqcomd
|
|
syl simp3 simp1 lediv2d mpbid eqbrtrd ) ACDZBEDZABFGZHZIBBJKZBAJKZFUGBCDZIU
|
|
HLABMZUJUHIUJBBNBOPQRUGUFUHUIFGUDUEUFSUGABBUDUEUFTUKUKUAUBUC $.
|
|
|
|
$( A real number divided by a positive real number is less than 1 iff the
|
|
real number is less than the positive real number. (Contributed by AV,
|
|
25-May-2020.) $)
|
|
divlt1lt $p |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> A < B ) ) $=
|
|
( cr wcel crp wa cdiv co c1 clt wbr cc0 wb simpl rpregt0 adantl 0lt1 pm3.2i
|
|
1re a1i ltdiv23 syl3anc wceq recn div1d adantr breq1d bitrd ) ACDZBEDZFZABG
|
|
HIJKZAIGHZBJKZABJKUKUIBCDLBJKFZICDZLIJKZFZULUNMUIUJNUJUOUIBOPURUKUPUQSQRTAB
|
|
IUAUBUKUMABJUIUMAUCUJUIAAUDUEUFUGUH $.
|
|
|
|
$( A real number divided by a positive real number is less than or equal to 1
|
|
iff the real number is less than or equal to the positive real number.
|
|
(Contributed by AV, 29-Jun-2021.) $)
|
|
divle1le $p |- ( ( A e. RR /\ B e. RR+ )
|
|
-> ( ( A / B ) <_ 1 <-> A <_ B ) ) $=
|
|
( cr wcel crp wa cdiv co c1 cle wbr cc0 clt simpl rpregt0 adantl 1re pm3.2i
|
|
wb 0lt1 a1i lediv23 syl3anc wceq recn div1d adantr breq1d bitrd ) ACDZBEDZF
|
|
ZABGHIJKZAIGHZBJKZABJKULUJBCDLBMKFZICDZLIMKZFZUMUOSUJUKNUKUPUJBOPUSULUQURQT
|
|
RUAABIUBUCULUNABJUJUNAUDUKUJAAUEUFUGUHUI $.
|
|
|
|
$( If a number is less than or equal to another number, the number divided by
|
|
a positive number greater than or equal to one is less than or equal to
|
|
the other number. (Contributed by AV, 29-Jun-2021.) $)
|
|
ledivge1le $p |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) )
|
|
-> ( A <_ B -> ( A / C ) <_ B ) ) $=
|
|
( cr wcel crp c1 cle wbr wa w3a cdiv co wi wb adantr 1red syl3anc cc0 clt
|
|
divle1le rerpdivcl rpre letr expd sylbird com23 expimpd ex 3imp1 simp1 0lt1
|
|
adantl 0red ltletr mpani imp jca 3ad2ant3 rpregt0 3ad2ant2 3jca lediv23 syl
|
|
mpbird ) ADEZBFEZCFEZGCHIZJZKZABHIZACLMBHIZVKVLJZVMABLMZCHIZVFVGVJVLVPVFVGV
|
|
JVLVPNZNVFVGJZVHVIVQVRVHJZVLVIVPVSVLVOGHIZVIVPNVRVTVLOVHABUAPVSVTVIVPVSVODE
|
|
ZGDEZCDEZVTVIJVPNVRWAVHABUBPVSQVHWCVRCUCZUMVOGCUDRUEUFUGUHUIUJVNVFWCSCTIZJZ
|
|
BDESBTIJZKZVMVPOVKWHVLVKVFWFWGVFVGVJUKVJVFWFVGVJWCWEVHWCVIWDPVHVIWEVHSGTIZV
|
|
IWEULVHSDEWBWCWIVIJWENVHUNVHQWDSGCUORUPUQURUSVGVFWGVJBUTVAVBPACBVCVDVEUI $.
|
|
|
|
${
|
|
rpgecld.1 $e |- ( ph -> A e. RR ) $.
|
|
${
|
|
ge0p1rp.2 $e |- ( ph -> 0 <_ A ) $.
|
|
$( A nonnegative number plus one is a positive number. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
ge0p1rpd $p |- ( ph -> ( A + 1 ) e. RR+ ) $=
|
|
( cr wcel cc0 cle wbr c1 caddc co crp ge0p1rp syl2anc ) ABEFGBHIBJKLMFC
|
|
DBNO $.
|
|
$}
|
|
|
|
rpgecld.2 $e |- ( ph -> B e. RR+ ) $.
|
|
$( Closure law for division of a real by a positive real. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
rerpdivcld $p |- ( ph -> ( A / B ) e. RR ) $=
|
|
( cr wcel crp cdiv co rerpdivcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
|
|
$( Subtracting a positive real from another number decreases it.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltsubrpd $p |- ( ph -> ( A - B ) < A ) $=
|
|
( cr wcel crp cmin co clt wbr ltsubrp syl2anc ) ABFGCHGBCIJBKLDEBCMN $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
ltaddrpd $p |- ( ph -> A < ( A + B ) ) $=
|
|
( cr wcel crp caddc co clt wbr ltaddrp syl2anc ) ABFGCHGBBCIJKLDEBCMN $.
|
|
|
|
$( Adding a positive number to another number increases it. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
ltaddrp2d $p |- ( ph -> A < ( B + A ) ) $=
|
|
( caddc co clt ltaddrpd recnd rpcnd addcomd breqtrd ) ABBCFGCBFGHABCDEIAB
|
|
CABDJACEKLM $.
|
|
|
|
$( Multiplication by a number greater than 1. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
ltmulgt11d $p |- ( ph -> ( 1 < A <-> B < ( B x. A ) ) ) $=
|
|
( cr wcel cc0 clt wbr c1 cmul co wb rpred rpgt0d ltmulgt11 syl3anc ) ACFG
|
|
BFGHCIJKBIJCCBLMIJNACEODACEPCBQR $.
|
|
|
|
$( Multiplication by a number greater than 1. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
ltmulgt12d $p |- ( ph -> ( 1 < A <-> B < ( A x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr c1 cmul co wb rpred rpgt0d ltmulgt12 syl3anc ) ACFG
|
|
BFGHCIJKBIJCBCLMIJNACEODACEPCBQR $.
|
|
|
|
$( Division of a positive number by a positive number. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
gt0divd $p |- ( ph -> ( 0 < A <-> 0 < ( A / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr cdiv co wb rpred rpgt0d gt0div syl3anc ) ABFGCFGHCI
|
|
JHBIJHBCKLIJMDACENACEOBCPQ $.
|
|
|
|
$( Division of a nonnegative number by a positive number. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
ge0divd $p |- ( ph -> ( 0 <_ A <-> 0 <_ ( A / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr cle cdiv co wb rpred rpgt0d ge0div syl3anc ) ABFGCF
|
|
GHCIJHBKJHBCLMKJNDACEOACEPBCQR $.
|
|
|
|
${
|
|
rpgecld.3 $e |- ( ph -> B <_ A ) $.
|
|
$( A number greater or equal to a positive real is positive real.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
rpgecld $p |- ( ph -> A e. RR+ ) $=
|
|
( crp wcel cr cle wbr rpgecl syl3anc ) ACGHBIHCBJKBGHEDFCBLM $.
|
|
$}
|
|
|
|
divge0d.3 $e |- ( ph -> 0 <_ A ) $.
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
divge0d $p |- ( ph -> 0 <_ ( A / B ) ) $=
|
|
( cr wcel cc0 cle wbr clt wa cdiv co rpregt0d divge0 syl21anc ) ABGHIBJKC
|
|
GHICLKMIBCNOJKDFACEPBCQR $.
|
|
$}
|
|
|
|
${
|
|
ltmul1d.1 $e |- ( ph -> A e. RR ) $.
|
|
ltmul1d.2 $e |- ( ph -> B e. RR ) $.
|
|
ltmul1d.3 $e |- ( ph -> C e. RR+ ) $.
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltmul1d $p |- ( ph -> ( A < B <-> ( A x. C ) < ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co wb rpregt0d ltmul1 syl3anc ) ABHICHIDHIJ
|
|
DKLMBCKLBDNOCDNOKLPEFADGQBCDRS $.
|
|
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
ltmul2d $p |- ( ph -> ( A < B <-> ( C x. A ) < ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co wb rpregt0d ltmul2 syl3anc ) ABHICHIDHIJ
|
|
DKLMBCKLDBNODCNOKLPEFADGQBCDRS $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lemul1d $p |- ( ph -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle cmul co wb rpregt0d lemul1 syl3anc ) ABHICHI
|
|
DHIJDKLMBCNLBDOPCDOPNLQEFADGRBCDST $.
|
|
|
|
$( Multiplication of both sides of 'less than or equal to' by a positive
|
|
number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lemul2d $p |- ( ph -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle cmul co wb rpregt0d lemul2 syl3anc ) ABHICHI
|
|
DHIJDKLMBCNLDBOPDCOPNLQEFADGRBCDST $.
|
|
|
|
$( Division of both sides of 'less than' by a positive number.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltdiv1d $p |- ( ph -> ( A < B <-> ( A / C ) < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co wb rpregt0d ltdiv1 syl3anc ) ABHICHIDHIJ
|
|
DKLMBCKLBDNOCDNOKLPEFADGQBCDRS $.
|
|
|
|
$( Division of both sides of a less than or equal to relation by a positive
|
|
number. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lediv1d $p |- ( ph -> ( A <_ B <-> ( A / C ) <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cle cdiv co wb rpregt0d lediv1 syl3anc ) ABHICHI
|
|
DHIJDKLMBCNLBDOPCDOPNLQEFADGRBCDST $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltmuldivd $p |- ( ph -> ( ( A x. C ) < B <-> A < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d ltmuldiv syl3anc ) ABHI
|
|
CHIDHIJDKLMBDNOCKLBCDPOKLQEFADGRBCDST $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltmuldiv2d $p |- ( ph -> ( ( C x. A ) < B <-> A < ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d ltmuldiv2 syl3anc ) ABH
|
|
ICHIDHIJDKLMDBNOCKLBCDPOKLQEFADGRBCDST $.
|
|
|
|
$( 'Less than or equal to' relationship between division and
|
|
multiplication. (Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
lemuldivd $p |- ( ph -> ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cle cdiv wb rpregt0d lemuldiv syl3anc )
|
|
ABHICHIDHIJDKLMBDNOCPLBCDQOPLREFADGSBCDTUA $.
|
|
|
|
$( 'Less than or equal to' relationship between division and
|
|
multiplication. (Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
lemuldiv2d $p |- ( ph -> ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cle cdiv wb rpregt0d lemuldiv2 syl3anc )
|
|
ABHICHIDHIJDKLMDBNOCPLBCDQOPLREFADGSBCDTUA $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltdivmuld $p |- ( ph -> ( ( A / C ) < B <-> A < ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co cmul wb rpregt0d ltdivmul syl3anc ) ABHI
|
|
CHIDHIJDKLMBDNOCKLBDCPOKLQEFADGRBCDST $.
|
|
|
|
$( 'Less than' relationship between division and multiplication.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ltdivmul2d $p |- ( ph -> ( ( A / C ) < B <-> A < ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co cmul wb rpregt0d ltdivmul2 syl3anc ) ABH
|
|
ICHIDHIJDKLMBDNOCKLBCDPOKLQEFADGRBCDST $.
|
|
|
|
$( 'Less than or equal to' relationship between division and
|
|
multiplication. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ledivmuld $p |- ( ph -> ( ( A / C ) <_ B <-> A <_ ( C x. B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co cle cmul wb rpregt0d ledivmul syl3anc )
|
|
ABHICHIDHIJDKLMBDNOCPLBDCQOPLREFADGSBCDTUA $.
|
|
|
|
$( 'Less than or equal to' relationship between division and
|
|
multiplication. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
ledivmul2d $p |- ( ph -> ( ( A / C ) <_ B <-> A <_ ( B x. C ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cdiv co cle cmul wb rpregt0d ledivmul2 syl3anc )
|
|
ABHICHIDHIJDKLMBDNOCPLBCDQOPLREFADGSBCDTUA $.
|
|
|
|
${
|
|
ltdiv1dd.4 $e |- ( ph -> A < B ) $.
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
ltmul1dd $p |- ( ph -> ( A x. C ) < ( B x. C ) ) $=
|
|
( clt wbr cmul co ltmul1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
|
|
$( Multiplication of both sides of 'less than' by a positive number.
|
|
Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro,
|
|
30-May-2016.) $)
|
|
ltmul2dd $p |- ( ph -> ( C x. A ) < ( C x. B ) ) $=
|
|
( clt wbr cmul co ltmul2d mpbid ) ABCIJDBKLDCKLIJHABCDEFGMN $.
|
|
|
|
$( Division of both sides of 'less than' by a positive number.
|
|
(Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
ltdiv1dd $p |- ( ph -> ( A / C ) < ( B / C ) ) $=
|
|
( clt wbr cdiv co ltdiv1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
$}
|
|
|
|
${
|
|
lediv1dd.4 $e |- ( ph -> A <_ B ) $.
|
|
$( Division of both sides of a less than or equal to relation by a
|
|
positive number. (Contributed by Mario Carneiro, 30-May-2016.) $)
|
|
lediv1dd $p |- ( ph -> ( A / C ) <_ ( B / C ) ) $=
|
|
( cle wbr cdiv co lediv1d mpbid ) ABCIJBDKLCDKLIJHABCDEFGMN $.
|
|
$}
|
|
|
|
${
|
|
lediv12ad.4 $e |- ( ph -> D e. RR ) $.
|
|
lediv12ad.5 $e |- ( ph -> 0 <_ A ) $.
|
|
lediv12ad.6 $e |- ( ph -> A <_ B ) $.
|
|
lediv12ad.7 $e |- ( ph -> C <_ D ) $.
|
|
$( Comparison of ratio of two nonnegative numbers. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
lediv12ad $p |- ( ph -> ( A / D ) <_ ( B / C ) ) $=
|
|
( cr wcel wa cc0 cle wbr cdiv jca clt co rpred rpgt0d lediv12a syl22anc
|
|
) ABMNZCMNZOPBQRZBCQRZODMNZEMNZOPDUARZDEQRZOBESUBCDSUBQRAUGUHFGTAUIUJJK
|
|
TAUKULADHUCITAUMUNADHUDLTBCDEUEUF $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ltdiv23d.1 $e |- ( ph -> A e. RR ) $.
|
|
ltdiv23d.2 $e |- ( ph -> B e. RR+ ) $.
|
|
ltdiv23d.3 $e |- ( ph -> C e. RR+ ) $.
|
|
${
|
|
ltdiv23d.4 $e |- ( ph -> ( A / B ) < C ) $.
|
|
$( Swap denominator with other side of 'less than'. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
ltdiv23d $p |- ( ph -> ( A / C ) < B ) $=
|
|
( cdiv co clt wbr cr wcel cc0 wa wb rpregt0d ltdiv23 syl3anc mpbid ) AB
|
|
CIJDKLZBDIJCKLZHABMNCMNOCKLPDMNODKLPUBUCQEACFRADGRBCDSTUA $.
|
|
$}
|
|
|
|
${
|
|
lediv23d.4 $e |- ( ph -> ( A / B ) <_ C ) $.
|
|
$( Swap denominator with other side of 'less than or equal to'.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lediv23d $p |- ( ph -> ( A / C ) <_ B ) $=
|
|
( cdiv co cle wbr cr wcel cc0 clt wa wb rpregt0d lediv23 syl3anc mpbid
|
|
) ABCIJDKLZBDIJCKLZHABMNCMNOCPLQDMNODPLQUCUDREACFSADGSBCDTUAUB $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
lt2mul2divd.1 $e |- ( ph -> A e. RR ) $.
|
|
lt2mul2divd.2 $e |- ( ph -> B e. RR+ ) $.
|
|
lt2mul2divd.3 $e |- ( ph -> C e. RR ) $.
|
|
lt2mul2divd.4 $e |- ( ph -> D e. RR+ ) $.
|
|
$( The ratio of nonnegative and positive numbers is nonnegative.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lt2mul2divd $p |- ( ph ->
|
|
( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) ) $=
|
|
( cr wcel cc0 clt wbr wa cmul co cdiv wb rpregt0d lt2mul2div syl22anc ) A
|
|
BJKCJKLCMNODJKEJKLEMNOBCPQDEPQMNBERQDCRQMNSFACGTHAEITBCDEUAUB $.
|
|
$}
|
|
|
|
$( Division of a positive integer by a positive number is less than or equal
|
|
to the integer iff the number is greater than or equal to 1. (Contributed
|
|
by AV, 19-Jun-2021.) $)
|
|
nnledivrp $p |- ( ( A e. NN /\ B e. RR+ )
|
|
-> ( 1 <_ B <-> ( A / B ) <_ A ) ) $=
|
|
( cn wcel crp wa c1 cle wbr cdiv co cr cc0 clt wb 1re pm3.2i rpregt0 adantr
|
|
0lt1 adantl nnre nngt0 jca lediv2 mp3an2i wceq nncn div1d breq2d bitrd ) AC
|
|
DZBEDZFZGBHIZABJKZAGJKZHIZUPAHIGLDZMGNIZFUNBLDMBNIFZALDZMANIZFZUOUROUSUTPTQ
|
|
UMVAULBRUAULVDUMULVBVCAUBAUCUDSGBAUEUFUNUQAUPHULUQAUGUMULAAUHUISUJUK $.
|
|
|
|
$( Division of a nonnegative integer by a positive integer is less than or
|
|
equal to the integer. (Contributed by AV, 19-Jun-2021.) $)
|
|
nn0ledivnn $p |- ( ( A e. NN0 /\ B e. NN ) -> ( A / B ) <_ A ) $=
|
|
( cn0 wcel cn cdiv co cle wbr cc0 wceq wo wi elnn0 wa c1 nnge1 adantl wb ex
|
|
crp nnrp nnledivrp sylan2 mpbid cap nncn nnap0 jca div0ap syl 0le0 syl6eqbr
|
|
cc oveq1 id breq12d adantr mpbird jaoi sylbi imp ) ACDZBEDZABFGZAHIZVCAEDZA
|
|
JKZLVDVFMZANVGVIVHVGVDVFVGVDOPBHIZVFVDVJVGBQRVDVGBUADVJVFSBUBABUCUDUETVHVDV
|
|
FVHVDOZVFJBFGZJHIZVKVLJJHVKBUNDZBJUFIZOZVLJKVDVPVHVDVNVOBUGBUHUIRBUJUKULUMV
|
|
HVFVMSVDVHVEVLAJHAJBFUOVHUPUQURUSTUTVAVB $.
|
|
|
|
|
|
$( If the sum of a real number and a positive real number is less than or
|
|
equal to a third real number, the first real number is less than the third
|
|
real number. (Contributed by AV, 1-Jul-2021.) $)
|
|
addlelt $p |- ( ( M e. RR /\ N e. RR /\ A e. RR+ )
|
|
-> ( ( M + A ) <_ N -> M < N ) ) $=
|
|
( cr wcel crp w3a caddc clt wbr cle cc0 rpgt0 3ad2ant3 rpre simp1 ltaddposd
|
|
co mpbid wa wi simpl adantl readdcld 3adant2 simp2 ltletr syl3anc mpand ) B
|
|
DEZCDEZAFEZGZBBAHRZIJZUNCKJZBCIJZUMLAIJZUOULUJURUKAMNUMABULUJADEZUKAOZNUJUK
|
|
ULPZQSUMUJUNDEZUKUOUPTUQUAVAUJULVBUKUJULTBAUJULUBULUSUJUTUCUDUEUJUKULUFBUNC
|
|
UGUHUI $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Infinity and the extended real number system (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
$c -e $. $( the negative of an extended real number. $)
|
|
$c +e $. $( Addition of extended real numbers. $)
|
|
$c *e $. $( Multiplication of extended real numbers. $)
|
|
|
|
$( Extend class notation to include the negative of an extended real. $)
|
|
cxne $a class -e A $.
|
|
|
|
$( Extend class notation to include addition of extended reals. $)
|
|
cxad $a class +e $.
|
|
|
|
$( Extend class notation to include multiplication of extended reals. $)
|
|
cxmu $a class *e $.
|
|
|
|
$( Define the negative of an extended real number. (Contributed by FL,
|
|
26-Dec-2011.) $)
|
|
df-xneg $a |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define addition over extended real numbers. (Contributed by Mario
|
|
Carneiro, 20-Aug-2015.) $)
|
|
df-xadd $a |- +e = ( x e. RR* , y e. RR* |-> if ( x = +oo ,
|
|
if ( y = -oo , 0 , +oo ) , if ( x = -oo ,
|
|
if ( y = +oo , 0 , -oo ) , if ( y = +oo , +oo ,
|
|
if ( y = -oo , -oo , ( x + y ) ) ) ) ) ) $.
|
|
|
|
$( Define multiplication over extended real numbers. (Contributed by Mario
|
|
Carneiro, 20-Aug-2015.) $)
|
|
df-xmul $a |- *e = ( x e. RR* , y e. RR* |-> if ( ( x = 0 \/ y = 0 ) , 0 ,
|
|
if ( ( ( ( 0 < y /\ x = +oo ) \/ ( y < 0 /\ x = -oo ) ) \/
|
|
( ( 0 < x /\ y = +oo ) \/ ( x < 0 /\ y = -oo ) ) ) , +oo ,
|
|
if ( ( ( ( 0 < y /\ x = -oo ) \/ ( y < 0 /\ x = +oo ) ) \/
|
|
( ( 0 < x /\ y = -oo ) \/ ( x < 0 /\ y = +oo ) ) ) , -oo ,
|
|
( x x. y ) ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The 'less than' binary relation on the set of extended reals.
|
|
Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
ltxr $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <->
|
|
( ( ( ( A e. RR /\ B e. RR ) /\ A <RR B ) \/
|
|
( A = -oo /\ B = +oo ) ) \/ ( ( A e. RR /\ B = +oo )
|
|
\/ ( A = -oo /\ B e. RR ) ) ) ) ) $=
|
|
( vx vy cxr wcel wa cv cr cltrr wbr cmnf csn cun cpnf cxp wceq clt syl5bb
|
|
wo w3a copab breq12 df-3an opabbii brab2ga a1i brun brxp elun orcom bitri
|
|
wb elsng orbi1d bi2anan9 andir syl6bb anbi1d adantr orbi12d orass df-ltxr
|
|
breqi 3bitr4g ) AEFZBEFZGZABCHZIFZDHZIFZVIVKJKZUAZCDUBZKZABILMZNZOMZPZVQI
|
|
PZNZKZTZAIFZBIFZGABJKZGZALQZBOQZGZWEWJGZWIWFGZTZTZTABRKZWHWKTWNTVHVPWHWCW
|
|
OVPWHUMVHVMWGCDABIIVOVIAVKBJUCVNVJVLGVMGCDVJVLVMUDUEUFUGWCABVTKZABWAKZTZV
|
|
HWOABVTWAUHVHWSWKWLTZWMTWOVHWQWTWRWMWQAVRFZBVSFZGZVHWTABVRVSUIVHXCWIWETZW
|
|
JGWTVFXAXDVGXBWJXAAVQFZWETZVFXDXAWEXETXFAIVQUJWEXEUKULVFXEWIWEALEUNZUOSBO
|
|
EUNUPWIWEWJUQURSWRXEWFGZVHWMABVQIUIVFXHWMUMVGVFXEWIWFXGUSUTSVAWKWLWMVBURS
|
|
VAWPABVOWBNZKWDABRXICDVCVDABVOWBUHULWHWKWNVBVE $.
|
|
$}
|
|
|
|
$( Membership in the set of extended reals. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
elxr $p |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) ) $=
|
|
( cxr wcel cr cpnf cmnf cpr cun wo wceq df-xr eleq2i elun pnfex mnfxr elexi
|
|
w3o elpr2 orbi2i 3orass bitr4i 3bitri ) ABCADEFGZHZCADCZAUCCZIZUEAEJZAFJZQZ
|
|
BUDAKLADUCMUGUEUHUIIZIUJUFUKUEAEFNFBOPRSUEUHUITUAUB $.
|
|
|
|
$( An extended real other than minus infinity is real or positive infinite.
|
|
(Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
xrnemnf $p |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) ) $=
|
|
( cr wcel cpnf wceq wo cmnf wn cxr wne pm5.61 w3o elxr df-3or bitri anbi12i
|
|
wa df-ne renemnf pnfnemnf neeq1 mpbiri jaoi neneqd pm4.71i 3bitr4i ) ABCZAD
|
|
EZFZAGEZFZUJHZQUIULQAICZAGJZQUIUIUJKUMUKUNULUMUGUHUJLUKAMUGUHUJNOAGRPUIULUI
|
|
AGUGUNUHASUHUNDGJTADGUAUBUCUDUEUF $.
|
|
|
|
$( An extended real other than plus infinity is real or negative infinite.
|
|
(Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
xrnepnf $p |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) ) $=
|
|
( cr wcel cmnf wceq wo cpnf wn wa cxr wne pm5.61 w3o elxr df-3or or32 df-ne
|
|
3bitri anbi12i renepnf mnfnepnf neeq1 mpbiri jaoi neneqd pm4.71i 3bitr4i )
|
|
ABCZADEZFZAGEZFZUKHZIUJUMIAJCZAGKZIUJUJUKLUNULUOUMUNUHUKUIMUHUKFUIFULANUHUK
|
|
UIOUHUKUIPRAGQSUJUMUJAGUHUOUIATUIUODGKUAADGUBUCUDUEUFUG $.
|
|
|
|
$( The extended real 'less than' is irreflexive. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
xrltnr $p |- ( A e. RR* -> -. A < A ) $=
|
|
( cxr wcel cr cpnf wceq cmnf clt wbr wa cltrr neli intnanr pnfnemnf pm3.2ni
|
|
wo intnan wb pnfxr ltxr w3o elxr ltnr pnfnre neii mp2an mtbir breq12 anidms
|
|
wn mtbiri mnfnre nesymi mnfxr 3jaoi sylbi ) ABCADCZAEFZAGFZUAAAHIZUJZAUBUQV
|
|
AURUSAUCURUTEEHIZVBEDCZVCJZEEKIZJZEGFZEEFZJZPZVCVHJZVGVCJZPZPZVJVMVFVIVDVEV
|
|
CVCEDUDLZQMVGVHEGNUEMOVKVLVCVHVOMVCVGVOQOOEBCZVPVBVNRSSEETUFUGURUTVBRAEAEHU
|
|
HUIUKUSUTGGHIZVQGDCZVRJZGGKIZJZGGFZGEFZJZPZVRWCJZWBVRJZPZPZWEWHWAWDVSVTVRVR
|
|
GDULLZQMWCWBEGNUMQOWFWGVRWCWJMVRWBWJQOOGBCZWKVQWIRUNUNGGTUFUGUSUTVQRAGAGHUH
|
|
UIUKUOUP $.
|
|
|
|
$( Any (finite) real is less than plus infinity. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
ltpnf $p |- ( A e. RR -> A < +oo ) $=
|
|
( cr wcel cpnf clt wbr wa cltrr cmnf wceq wo eqid orc mpan2 olcd rexr pnfxr
|
|
cxr wb ltxr sylancl mpbird ) ABCZADEFZUCDBCZGADHFGAIJZDDJZGKZUCUGGZUFUEGZKZ
|
|
KZUCUKUHUCUGUKDLUIUJMNOUCARCDRCUDULSAPQADTUAUB $.
|
|
|
|
$( Zero is less than plus infinity (common case). (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
0ltpnf $p |- 0 < +oo $=
|
|
( cc0 cr wcel cpnf clt wbr 0re ltpnf ax-mp ) ABCADEFGAHI $.
|
|
|
|
$( Minus infinity is less than any (finite) real. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
mnflt $p |- ( A e. RR -> -oo < A ) $=
|
|
( cr wcel cmnf clt wbr wa cltrr wceq cpnf wo eqid olc mpan olcd cxr wb rexr
|
|
mnfxr ltxr sylancr mpbird ) ABCZDAEFZDBCZUCGDAHFGDDIZAJIZGKZUEUGGZUFUCGZKZK
|
|
ZUCUKUHUFUCUKDLUJUIMNOUCDPCAPCUDULQSARDATUAUB $.
|
|
|
|
$( Minus infinity is less than 0 (common case). (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
mnflt0 $p |- -oo < 0 $=
|
|
( cc0 cr wcel cmnf clt wbr 0re mnflt ax-mp ) ABCDAEFGAHI $.
|
|
|
|
$( Minus infinity is less than plus infinity. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
mnfltpnf $p |- -oo < +oo $=
|
|
( cmnf cpnf clt wbr cr wcel wa cltrr wceq wo eqid olc mp2an cxr mnfxr pnfxr
|
|
orci wb ltxr mpbir ) ABCDZAEFZBEFZGABHDGZAAIZBBIZGZJZUBUFGUEUCGJZJZUHUIUEUF
|
|
UHAKBKUGUDLMQANFBNFUAUJROPABSMT $.
|
|
|
|
$( Minus infinity is less than an extended real that is either real or plus
|
|
infinity. (Contributed by NM, 2-Feb-2006.) $)
|
|
mnfltxr $p |- ( ( A e. RR \/ A = +oo ) -> -oo < A ) $=
|
|
( cr wcel cmnf clt wbr cpnf wceq mnflt mnfltpnf breq2 mpbiri jaoi ) ABCDAEF
|
|
ZAGHZAIONDGEFJAGDEKLM $.
|
|
|
|
$( No extended real is greater than plus infinity. (Contributed by NM,
|
|
15-Oct-2005.) $)
|
|
pnfnlt $p |- ( A e. RR* -> -. +oo < A ) $=
|
|
( cxr wcel cpnf clt wbr cr wa cltrr cmnf wceq wo neli intnanr pnfnemnf neii
|
|
pnfnre pm3.2ni wb pnfxr ltxr mpan mtbiri ) ABCZDAEFZDGCZAGCZHZDAIFZHZDJKZAD
|
|
KZHZLZUFULHZUKUGHZLZLZUNUQUJUMUHUIUFUGDGQMZNNUKULDJOPZNRUOUPUFULUSNUKUGUTNR
|
|
RDBCUDUEURSTDAUAUBUC $.
|
|
|
|
$( No extended real is less than minus infinity. (Contributed by NM,
|
|
15-Oct-2005.) $)
|
|
nltmnf $p |- ( A e. RR* -> -. A < -oo ) $=
|
|
( cxr wcel cmnf clt wbr cr wa cltrr wceq cpnf wo mnfnre neli intnan intnanr
|
|
pnfnemnf nesymi pm3.2ni wb mnfxr ltxr mpan2 mtbiri ) ABCZADEFZAGCZDGCZHZADI
|
|
FZHZADJZDKJZHZLZUGUMHZULUHHZLZLZUOURUKUNUIUJUHUGDGMNZOPUMULKDQRZOSUPUQUMUGV
|
|
AOUHULUTOSSUEDBCUFUSTUAADUBUCUD $.
|
|
|
|
$( Plus infinity is an upper bound for extended reals. (Contributed by NM,
|
|
30-Jan-2006.) $)
|
|
pnfge $p |- ( A e. RR* -> A <_ +oo ) $=
|
|
( cxr wcel cpnf cle wbr clt wn pnfnlt wb pnfxr xrlenlt mpan2 mpbird ) ABCZA
|
|
DEFZDAGFHZAIODBCPQJKADLMN $.
|
|
|
|
$( 0 less than or equal to positive infinity. (Contributed by David A.
|
|
Wheeler, 8-Dec-2018.) $)
|
|
0lepnf $p |- 0 <_ +oo $=
|
|
( cc0 cxr wcel cpnf cle wbr 0xr pnfge ax-mp ) ABCADEFGAHI $.
|
|
|
|
$( If a number is a nonnegative integer or positive infinity, it is greater
|
|
than or equal to 0. (Contributed by Alexander van der Vekens,
|
|
6-Jan-2018.) $)
|
|
nn0pnfge0 $p |- ( ( N e. NN0 \/ N = +oo ) -> 0 <_ N ) $=
|
|
( cn0 wcel cc0 cle wbr cpnf wceq nn0ge0 0lepnf breq2 mpbiri jaoi ) ABCDAEFZ
|
|
AGHZAIONDGEFJAGDEKLM $.
|
|
|
|
$( Minus infinity is less than or equal to any extended real. (Contributed
|
|
by NM, 19-Jan-2006.) $)
|
|
mnfle $p |- ( A e. RR* -> -oo <_ A ) $=
|
|
( cxr wcel cmnf cle wbr clt wn nltmnf wb mnfxr xrlenlt mpan mpbird ) ABCZDA
|
|
EFZADGFHZAIDBCOPQJKDALMN $.
|
|
|
|
$( Ordering on the extended reals is not symmetric. (Contributed by NM,
|
|
15-Oct-2005.) $)
|
|
xrltnsym $p |- ( ( A e. RR* /\ B e. RR* ) ->
|
|
( A < B -> -. B < A ) ) $=
|
|
( cxr wcel cr cpnf wceq cmnf clt wbr wn pnfnlt syl adantr adantl mtbird a1d
|
|
wa wb pm2.21d wi elxr ltnsym rexr breq1 nltmnf breq2 3jaodan sylan2br mnfxr
|
|
w3o ax-mp breq12 mtbiri ancoms xrltnr 3jaoian syl2anb ) ACDZAEDZAFGZAHGZUKB
|
|
EDZBFGZBHGZUKZABIJZBAIJZKZUAZBCDZAUBBUBZUTVFVJVAVBUTVCVJVDVEABUCUTVDRZVIVGV
|
|
MVHFAIJZUTVNKZVDUTUSVOAUDZALMNVDVHVNSUTBFAIUEOPQUTVERZVGVIVQVGAHIJZUTVRKZVE
|
|
UTUSVSVPAUFMNVEVGVRSUTBHAIUGOPTUHVFVAVKVJVLVAVKRZVGVIVTVGFBIJZVKWAKVABLOVAV
|
|
GWASVKAFBIUENPTUIVBVCVJVDVEVBVCRZVIVGWBVHBHIJZVCWCKZVBVCVKWDBUDBUFMOVBVHWCS
|
|
VCAHBIUGNPQVBVDRVIVGVDVBVIVDVBRVHFHIJZHCDZWEKUJHLULBFAHIUMUNUOQVBVERZVGVIWG
|
|
VGHHIJZWFWHKUJHUPULAHBHIUMUNTUHUQUR $.
|
|
|
|
$( 'Less than' is antisymmetric and irreflexive for extended reals.
|
|
(Contributed by NM, 6-Feb-2007.) $)
|
|
xrltnsym2 $p |- ( ( A e. RR* /\ B e. RR* ) -> -. ( A < B /\ B < A ) ) $=
|
|
( cxr wcel wa clt wbr wn wi xrltnsym imnan sylib ) ACDBCDEABFGZBAFGZHIMNEHA
|
|
BJMNKL $.
|
|
|
|
$( Ordering on the extended reals is transitive. (Contributed by NM,
|
|
15-Oct-2005.) $)
|
|
xrlttr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A < B /\ B < C ) -> A < C ) ) $=
|
|
( cxr wcel cr cpnf wceq cmnf w3o clt wbr wa wn adantr adantl mtbird pm2.21d
|
|
wb adantlr wi elxr lttr 3expa an32s pnfnlt syl breq1 adantll adantld nltmnf
|
|
rexr breq2 adantrd 3jaodan sylan2b ltpnf mpbird a1d anasss adantrr mnfltpnf
|
|
mnflt breq12 mpbiri 3jaoian 3impb syl3an3b syl3an1b ) ADEZAFEZAGHZAIHZJZBDE
|
|
ZCDEZABKLZBCKLZMZACKLZUAZAUBVPVNVOCFEZCGHZCIHZJZWACUBVNVOWEWAVKVOWEMWAVLVMV
|
|
KVOWEWAVKVOMZWBWAWCWDVKWBVOWAVOVKWBMZBFEZBGHZBIHZJWABUBWGWHWAWIWJVKWHWBWAVK
|
|
WHWBWAABCUCUDUEWGWIMVRVTVQWBWIVRVTUAVKWBWIMZVRVTWKVRGCKLZWBWLNZWIWBVPWMCULC
|
|
UFUGOWIVRWLSWBBGCKUHPQRUIUJWGWJMVQVTVRVKWJVQVTUAWBVKWJMZVQVTWNVQAIKLZVKWONZ
|
|
WJVKVJWPAULAUKUGOWJVQWOSVKBIAKUMPQRTUNUOUPUEWFWCMVTVSVKWCVTVOVKWCMVTAGKLZVK
|
|
WQWCAUQOWCVTWQSVKCGAKUMPURTUSVOWDWAVKVOWDMZVRVTVQWRVRVTWRVRBIKLZVOWSNWDBUKO
|
|
WDVRWSSVOCIBKUMPQRUJZUIUOUTVLVOWAWEVLVOMZVQVTVRXAVQVTXAVQGBKLZVOXBNVLBUFPVL
|
|
VQXBSVOAGBKUHOQRUNVAVMVOWEWAVMVOMWBWAWCWDVMWBWAVOVMWBMZVTVSXCVTICKLZWBXDVMC
|
|
VCPVMVTXDSWBAICKUHOURUSTVMWCWAVOVMWCMZVTVSXEVTIGKLVBAICGKVDVEUSTVOWDWAVMWTU
|
|
IUOUTVFVGVHVI $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( 'Less than' is a weakly linear ordering on the extended reals.
|
|
(Contributed by NM, 15-Oct-2005.) $)
|
|
xrltso $p |- < Or RR* $=
|
|
( vx vy vz cxr clt cv wbr wo wcel adantl wa cpnf wceq cmnf mpbird 3jaodan
|
|
wb a1d sylan2b ancoms wor wpo wi wral wtru wn xrltnr xrlttr ispod trud cr
|
|
w3a w3o elxr simplr simpll simpr axltwlin ltpnf ad2antlr breq2 orcd mnflt
|
|
syl3anc ad2antrr breq1 olcd anasss mnfltpnf breq12 mpbiri rexr nltmnf syl
|
|
adantlr mtbird pm2.21d pnfnlt df-3or bitri adantr eqtr3 breq1d olc syl6bi
|
|
mnfltxr jaodan 3impa syl3an3b 3com13 rgen3 df-iso mpbir2an ) DEUADEUBZAFZ
|
|
BFZEGZWOCFZEGZWRWPEGZHZUCZCDUDBDUDADUDWNUEABCDEWODIZWOWOEGUFUEWOUGJXCWPDI
|
|
ZWRDIZULWQWPWREGKWSUCUEWOWPWRUHJUIUJXBABCDDDXEXDXCXBXCXEXDWOUKIZWOLMZWONM
|
|
ZUMZXBWOUNXEXDXIXBXEXDKZXFXBXGXHXFXJXBXFXEXDXBXDXFXEKZWPUKIZWPLMZWPNMZUMX
|
|
BWPUNXKXLXBXMXNXLXKXBXLXFXEXBXEXLXFKZWRUKIZWRLMZWRNMZUMZXBWRUNZXOXPXBXQXR
|
|
XOXPKXFXLXPXBXLXFXPUOXLXFXPUPXOXPUQWOWPWRURVDXOXQKZXAWQYAWSWTYAWSWOLEGZXF
|
|
YBXLXQWOUSZUTXQWSYBQZXOWRLWOEVAZJOVBRXOXRKZXAWQYFWTWSYFWTNWPEGZXLYGXFXRWP
|
|
VCVEXRWTYGQXOWRNWPEVFJOVGRPSVHTXMXKXBXMXFXEXBXEXMXFKZXSXBXTYHXPXBXQXRYHXP
|
|
KZXAWQYIWTWSYIWTWRLEGZXPYJYHWRUSJXMWTYJQXFXPWPLWREVAVEOVGRYHXQKZXAWQYKWSW
|
|
TYKWSYBXFYBXMXQYCUTXQYDYHYEJOVBRYHXRKZXAWQYLWTWSXMXRWTXFXMXRKWTNLEGZVIXRX
|
|
MWTYMQWRNWPLEVJTVKVOVGRPSVHTXKXNKZWQXAYNWQWONEGZXFYOUFZXEXNXFXCYPWOVLWOVM
|
|
VNVEXNWQYOQXKWPNWOEVAJVPVQPSVHTXJXGKZWQXAYQWQLWPEGZXDYRUFXEXGWPVRUTXGWQYR
|
|
QXJWOLWPEVFJVPVQXEXHXBXDXHXEXBXEXHXPXQHZXRHZXBXEXSYTXTXPXQXRVSVTXHYSXBXRX
|
|
HYSKZXAWQUUAWSWTUUAWSNWREGZYSUUBXHWRWFJXHWSUUBQYSWONWREVFWAOVBRXHXRKZWQWT
|
|
XAUUCWOWRWPEWOWRNWBWCWTWSWDWEWGSTVOPWHWIWJWKABCDEWLWM $.
|
|
$}
|
|
|
|
$( Extended real version of ~ lttri3 . (Contributed by NM, 9-Feb-2006.) $)
|
|
xrlttri3 $p |- ( ( A e. RR* /\ B e. RR* ) ->
|
|
( A = B <-> ( -. A < B /\ -. B < A ) ) ) $=
|
|
( wcel cpnf wceq cmnf clt wbr wn wa ancoms wne adantr adantl mpbird 2falsed
|
|
wb neneqd breq12 mtbiri cxr cr elxr lttri3 renepnf neeq2 necomd ltpnf breq2
|
|
w3o wo notnot olcs ioran sylnib syl renemnf mnflt breq1 oranim 3syl 3jaodan
|
|
orc eqtr3 eqcomd pnfxr xrltnr ax-mp 2thd mnfnepnf eqeq12 necon3bid mnfltpnf
|
|
jca mpbiri neii neneqad mnfxr syl2anb ) AUACAUBCZADEZAFEZUJZBUBCZBDEZBFEZUJ
|
|
ABEZABGHZIZBAGHZIZJZQZBUACAUCBUCWCWDWMWEWFWDWCWMWDVTWMWAWBVTWDWMABUDKWDWAJZ
|
|
WGWLWNABWNBAWNBALZBDLZWDWPWABUEMWAWOWPQWDADBUFNOUGRWNWJWLIZWNWJBDGHZWDWRWAB
|
|
UHMWAWJWRQWDADBGUINOWJWHWJUKZIZWLWHWJWTIWSULUMWHWJUNUOZUPPWDWBJZWGWLXBABXBB
|
|
AXBWOBFLZWDXCWBBUQMWBWOXCQWDAFBUFNOUGRXBWHWSWQXBWHFBGHZWDXDWBBURMWBWHXDQWDA
|
|
FBGUSNOWHWJVCZWHWJUTZVAPVBKWEWCWMWEVTWMWAWBWEVTJZWGWLXGABXGABLZADLZVTXIWEAU
|
|
ENWEXHXIQVTBDAUFMORXGWHWSWQXGWHADGHZVTXJWEAUHNWEWHXJQVTBDAGUIMOXEXFVAPWEWAJ
|
|
ZWGWLXKBABADVDVEXKWIWKXKWHDDGHZDUACXLIVFDVGVHZWAWEWHXLQADBDGSKTXKWJXLXMBDAD
|
|
GSTVNVIWEWBJZWGWLXNABWBWEXHWBWEJZXHFDLVJXOABFDAFBDVKVLVOKRXNWHWSWQWBWEWHXOW
|
|
HFDGHZVMAFBDGSVOKXEXFVAPVBKWFWCWMWFVTWMWAWBWFVTJZWGWLXQABXQXHAFLZVTXRWFAUQN
|
|
WFXHXRQVTBFAUFMORXQWJWQXQWJFAGHZVTXSWFAURNWFWJXSQVTBFAGUSMOXAUPPWFWAJZWGWLX
|
|
TABXTBAXTBAXTBAEFDEFDVJVPBFADVKTVQUGRXTWJWQXTWJXPVMBFADGSVOXAUPPWFWBJZWGWLW
|
|
BWFWGABFVDKYAWIWKYAWHFFGHZFUACYBIVRFVGVHZWBWFWHYBQAFBFGSKTYAWJYBYCBFAFGSTVN
|
|
VIVBKVBVS $.
|
|
|
|
$( 'Less than' implies 'less than or equal' for extended reals. (Contributed
|
|
by NM, 19-Jan-2006.) $)
|
|
xrltle $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) ) $=
|
|
( cxr wcel wa clt wbr wn cle xrltnsym xrlenlt sylibrd ) ACDBCDEABFGBAFGHABI
|
|
GABJABKL $.
|
|
|
|
$( 'Less than or equal to' is reflexive for extended reals. (Contributed by
|
|
NM, 7-Feb-2007.) $)
|
|
xrleid $p |- ( A e. RR* -> A <_ A ) $=
|
|
( cxr wcel cle wbr clt wn xrltnr wb xrlenlt anidms mpbird ) ABCZAADEZAAFEGZ
|
|
AHMNOIAAJKL $.
|
|
|
|
$( Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.) $)
|
|
xrletri3 $p |- ( ( A e. RR* /\ B e. RR* ) ->
|
|
( A = B <-> ( A <_ B /\ B <_ A ) ) ) $=
|
|
( cxr wcel wa wceq clt wbr wn cle xrlttri3 ancom syl6bbr xrlenlt wb anbi12d
|
|
ancoms bitr4d ) ACDZBCDZEZABFZBAGHIZABGHIZEZABJHZBAJHZEUAUBUDUCEUEABKUCUDLM
|
|
UAUFUCUGUDABNTSUGUDOBANQPR $.
|
|
|
|
$( Transitive law for ordering on extended reals. (Contributed by NM,
|
|
19-Jan-2006.) $)
|
|
xrlelttr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A <_ B /\ B < C ) -> A < C ) ) $=
|
|
( cxr wcel w3a cle wbr clt wa wn simprl simpl1 simpl2 xrlenlt syl2anc mpbid
|
|
wb pm2.21d idd wo simprr wi simpl3 wor xrltso sowlin mpan syl3anc mpjaod ex
|
|
mpd ) ADEZBDEZCDEZFZABGHZBCIHZJZACIHZUPUSJZBAIHZUTUTVAVBUTVAUQVBKZUPUQURLVA
|
|
UMUNUQVCRUMUNUOUSMZUMUNUOUSNZABOPQSVAUTTVAURVBUTUAZUPUQURUBVAUNUOUMURVFUCZV
|
|
EUMUNUOUSUDVDDIUEUNUOUMFVGUFDBCAIUGUHUIULUJUK $.
|
|
|
|
$( Transitive law for ordering on extended reals. (Contributed by NM,
|
|
19-Jan-2006.) $)
|
|
xrltletr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A < B /\ B <_ C ) -> A < C ) ) $=
|
|
( cxr wcel w3a clt wbr cle wa wn simprr simpl2 simpl3 xrlenlt syl2anc mpbid
|
|
wb wo simprl wi wor xrltso sowlin mpan adantr mpd ecased ex ) ADEZBDEZCDEZF
|
|
ZABGHZBCIHZJZACGHZUMUPJZUQCBGHZURUOUSKZUMUNUOLURUKULUOUTRUJUKULUPMUJUKULUPN
|
|
BCOPQURUNUQUSSZUMUNUOTUMUNVAUAZUPDGUBUMVBUCDABCGUDUEUFUGUHUI $.
|
|
|
|
$( Transitive law for ordering on extended reals. (Contributed by NM,
|
|
9-Feb-2006.) $)
|
|
xrletr $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A <_ B /\ B <_ C ) -> A <_ C ) ) $=
|
|
( cxr wcel w3a clt wbr wo wn cle wa wi wor xrltso sowlin mpan 3coml xrlenlt
|
|
wb orcom syl6ib con3d 3adant3 3adant1 anbi12d ioran syl6bbr 3adant2 3imtr4d
|
|
) ADEZBDEZCDEZFZBAGHZCBGHZIZJZCAGHZJZABKHZBCKHZLZACKHZUNUSUQUNUSUPUOIZUQUMU
|
|
KULUSVEMZDGNUMUKULFVFODCABGPQRUPUOUAUBUCUNVCUOJZUPJZLURUNVAVGVBVHUKULVAVGTU
|
|
MABSUDULUMVBVHTUKBCSUEUFUOUPUGUHUKUMVDUTTULACSUIUJ $.
|
|
|
|
${
|
|
xrlttrd.1 $e |- ( ph -> A e. RR* ) $.
|
|
xrlttrd.2 $e |- ( ph -> B e. RR* ) $.
|
|
xrlttrd.3 $e |- ( ph -> C e. RR* ) $.
|
|
${
|
|
xrlttrd.4 $e |- ( ph -> A < B ) $.
|
|
xrlttrd.5 $e |- ( ph -> B < C ) $.
|
|
$( Transitive law for ordering on extended reals. (Contributed by Mario
|
|
Carneiro, 23-Aug-2015.) $)
|
|
xrlttrd $p |- ( ph -> A < C ) $=
|
|
( clt wbr cxr wcel wa wi xrlttr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLM
|
|
CLMDLMSTNUAOEFGBCDPQR $.
|
|
$}
|
|
|
|
${
|
|
xrlelttrd.4 $e |- ( ph -> A <_ B ) $.
|
|
xrlelttrd.5 $e |- ( ph -> B < C ) $.
|
|
$( Transitive law for ordering on extended reals. (Contributed by Mario
|
|
Carneiro, 23-Aug-2015.) $)
|
|
xrlelttrd $p |- ( ph -> A < C ) $=
|
|
( cle wbr clt cxr wcel wa wi xrlelttr syl3anc mp2and ) ABCJKZCDLKZBDLKZ
|
|
HIABMNCMNDMNTUAOUBPEFGBCDQRS $.
|
|
$}
|
|
|
|
${
|
|
xrltletrd.4 $e |- ( ph -> A < B ) $.
|
|
xrltletrd.5 $e |- ( ph -> B <_ C ) $.
|
|
$( Transitive law for ordering on extended reals. (Contributed by Mario
|
|
Carneiro, 23-Aug-2015.) $)
|
|
xrltletrd $p |- ( ph -> A < C ) $=
|
|
( clt wbr cle cxr wcel wa wi xrltletr syl3anc mp2and ) ABCJKZCDLKZBDJKZ
|
|
HIABMNCMNDMNTUAOUBPEFGBCDQRS $.
|
|
$}
|
|
|
|
${
|
|
xrletrd.4 $e |- ( ph -> A <_ B ) $.
|
|
xrletrd.5 $e |- ( ph -> B <_ C ) $.
|
|
$( Transitive law for ordering on extended reals. (Contributed by Mario
|
|
Carneiro, 23-Aug-2015.) $)
|
|
xrletrd $p |- ( ph -> A <_ C ) $=
|
|
( cle wbr cxr wcel wa wi xrletr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABLM
|
|
CLMDLMSTNUAOEFGBCDPQR $.
|
|
$}
|
|
$}
|
|
|
|
$( 'Less than' implies not equal for extended reals. (Contributed by NM,
|
|
20-Jan-2006.) $)
|
|
xrltne $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A ) $=
|
|
( cxr wcel clt wbr wne wceq xrltnr breq2 notbid syl5ibrcom necon2ad 3adant2
|
|
wn imp ) ACDZABEFZBAGZBCDQRSQRBAQROBAHZAAEFZOAITRUABAAEJKLMPN $.
|
|
|
|
$( An extended real is not less than plus infinity iff they are equal.
|
|
(Contributed by NM, 30-Jan-2006.) $)
|
|
nltpnft $p |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) ) $=
|
|
( cxr wcel cr cpnf wceq cmnf w3o clt wn wb elxr renepnf neneqd ltpnf notnot
|
|
wbr 2falsed breq1 mtbiri syl id pnfxr xrltnr ax-mp 2thd mnfnepnf neii eqeq1
|
|
mnfltpnf mpbiri necon3bi necon2bi 3jaoi sylbi ) ABCADCZAEFZAGFZHUQAEIQZJZKZ
|
|
ALUPVAUQURUPUQUTUPAEAMNUPUSUTJAOUSPUARUQUQUTUQUBUQUSEEIQZEBCVBJUCEUDUEAEEIS
|
|
TUFURUQUTURUQGEFGEUGUHAGEUITUTAGUSAGURUSGEIQUJAGEISUKULUMRUNUO $.
|
|
|
|
$( An extended real is not greater than minus infinity iff they are equal.
|
|
(Contributed by NM, 2-Feb-2006.) $)
|
|
ngtmnft $p |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) ) $=
|
|
( cxr wcel cr cpnf wceq cmnf w3o clt wbr wn wb renemnf neneqd mnflt 2falsed
|
|
elxr wne mpbiri breq2 notnot syl pnfnemnf mnfltpnf necon3bi necon2bi xrltnr
|
|
neeq1 id mnfxr ax-mp mtbiri 2thd 3jaoi sylbi ) ABCADCZAEFZAGFZHURGAIJZKZLZA
|
|
QUPVAUQURUPURUTUPAGAMNUPUSUTKAOUSUAUBPUQURUTUQAGUQAGREGRUCAEGUHSNUTAEUSAEUQ
|
|
USGEIJUDAEGITSUEUFPURURUTURUIURUSGGIJZGBCVBKUJGUGUKAGGITULUMUNUO $.
|
|
|
|
$( An extended real is real iff it is strictly bounded by infinities.
|
|
(Contributed by NM, 2-Feb-2006.) $)
|
|
xrrebnd $p |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) ) $=
|
|
( cxr wcel cr cpnf wceq cmnf w3o clt wa wb elxr id necon2bi wn xrltnr ax-mp
|
|
wbr mtbiri 2falsed mnflt ltpnf jca 2thd renepnf pnfxr breq1 intnand renemnf
|
|
mnfxr breq2 intnanrd 3jaoi sylbi ) ABCADCZAEFZAGFZHUOGAIRZAEIRZJZKZALUOVAUP
|
|
UQUOUOUTUOMUOURUSAUAAUBUCUDUPUOUTUOAEAUENUPUSURUPUSEEIRZEBCVBOUFEPQAEEIUGSU
|
|
HTUQUOUTUOAGAUINUQURUSUQURGGIRZGBCVCOUJGPQAGGIUKSULTUMUN $.
|
|
|
|
$( A way of proving that an extended real is real. (Contributed by NM,
|
|
9-Mar-2006.) $)
|
|
xrre $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( -oo < A /\ A <_ B ) ) ->
|
|
A e. RR ) $=
|
|
( cxr wcel cr wa cmnf clt wbr cle cpnf simprl ltpnf adantl wi rexr xrlelttr
|
|
pnfxr mp3an3 sylan2 mpan2d imp adantrl wb xrrebnd ad2antrr mpbir2and ) ACDZ
|
|
BEDZFZGAHIZABJIZFZFAEDZUKAKHIZUJUKULLUJULUOUKUJULUOUJULBKHIZUOUIUPUHBMNUIUH
|
|
BCDZULUPFUOOZBPUHUQKCDURRABKQSTUAUBUCUHUNUKUOFUDUIUMAUEUFUG $.
|
|
|
|
$( An extended real between two others is real. (Contributed by NM,
|
|
6-Feb-2007.) $)
|
|
xrre2 $p |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\
|
|
( A < B /\ B < C ) ) -> B e. RR ) $=
|
|
( cxr wcel w3a clt wbr wa cr cmnf wi cle mnfle adantr mnfxr xrlelttr mp3an1
|
|
cpnf mpand 3adant3 pnfge adantl pnfxr mp3an3 mpan2d 3adant1 anim12d xrrebnd
|
|
xrltletr wb 3ad2ant2 sylibrd imp ) ADEZBDEZCDEZFZABGHZBCGHZIZBJEZURVAKBGHZB
|
|
SGHZIZVBURUSVCUTVDUOUPUSVCLUQUOUPIKAMHZUSVCUOVFUPANOKDEUOUPVFUSIVCLPKABQRTU
|
|
AUPUQUTVDLUOUPUQIUTCSMHZVDUQVGUPCUBUCUPUQSDEUTVGIVDLUDBCSUJUEUFUGUHUPUOVBVE
|
|
UKUQBUIULUMUN $.
|
|
|
|
$( A way of proving that an extended real is real. (Contributed by FL,
|
|
29-May-2014.) $)
|
|
xrre3 $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) )
|
|
-> A e. RR ) $=
|
|
( cxr wcel cr wa cle wbr cpnf clt cmnf mnflt adantl wi mnfxr a1i rexr simpl
|
|
xrltletr syl3anc mpand imp adantrr simprr wb xrrebnd ad2antrr mpbir2and ) A
|
|
CDZBEDZFZBAGHZAIJHZFZFAEDZKAJHZUMUKULUPUMUKULUPUKKBJHZULUPUJUQUIBLMUKKCDZBC
|
|
DZUIUQULFUPNURUKOPUJUSUIBQMUIUJRKBASTUAUBUCUKULUMUDUIUOUPUMFUEUJUNAUFUGUH
|
|
$.
|
|
|
|
$( A nonnegative extended real is greater than negative infinity.
|
|
(Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
ge0gtmnf $p |- ( ( A e. RR* /\ 0 <_ A ) -> -oo < A ) $=
|
|
( cxr wcel cmnf cc0 clt wbr cle mnflt0 wa wi mnfxr 0xr xrltletr mp3an12 imp
|
|
mpanr1 ) ABCZDEFGZEAHGZDAFGZIRSTJZUADBCEBCRUBUAKLMDEANOPQ $.
|
|
|
|
$( A nonnegative extended real is greater than negative infinity.
|
|
(Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
ge0nemnf $p |- ( ( A e. RR* /\ 0 <_ A ) -> A =/= -oo ) $=
|
|
( cxr wcel cc0 cle wbr wa cmnf clt wne ge0gtmnf wn wb ngtmnft adantr biimpd
|
|
wceq necon2ad mpd ) ABCZDAEFZGZHAIFZAHJAKUBUCAHUBAHQZUCLZTUDUEMUAANOPRS $.
|
|
|
|
$( A nonnegative extended real that is less than a real bound is real.
|
|
(Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
xrrege0 $p |- ( ( ( A e. RR* /\ B e. RR ) /\ ( 0 <_ A /\ A <_ B ) ) ->
|
|
A e. RR ) $=
|
|
( cxr wcel cr cc0 cle wbr cmnf clt ge0gtmnf ad2ant2r simprr jca xrre syldan
|
|
wa ) ACDZBEDZQZFAGHZABGHZQZIAJHZUBQAEDTUCQUDUBRUAUDSUBAKLTUAUBMNABOP $.
|
|
|
|
${
|
|
$d k M $. $d k N $.
|
|
$( There exists an integer greater than or equal to any two others.
|
|
(Contributed by NM, 28-Aug-2005.) $)
|
|
z2ge $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
E. k e. ZZ ( M <_ k /\ N <_ k ) ) $=
|
|
( cz wcel wa cle wbr cv wrex simplr simpr zred leidd breq2 anbi12d rspcev
|
|
wceq syl12anc simpll zletric mpjaodan ) BDEZCDEZFZBCGHZBAIZGHZCUGGHZFZADJ
|
|
ZCBGHZUEUFFZUDUFCCGHZUKUCUDUFKZUEUFLUMCUMCUOMNUJUFUNFACDUGCRUHUFUIUNUGCBG
|
|
OUGCCGOPQSUEULFZUCBBGHZULUKUCUDULTZUPBUPBURMNUEULLUJUQULFABDUGBRUHUQUIULU
|
|
GBBGOUGBCGOPQSBCUAUB $.
|
|
$}
|
|
|
|
$( Equality of two extended numbers with ` -e ` in front of them.
|
|
(Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xnegeq $p |- ( A = B -> -e A = -e B ) $=
|
|
( wceq cpnf cmnf cneg cif cxne eqeq1 negeq ifbieq2d df-xneg 3eqtr4g ) ABCZA
|
|
DCZEAECZDAFZGZGBDCZEBECZDBFZGZGAHBHNOSRUBEABDINPTQUADABEIABJKKALBLM $.
|
|
|
|
$( Minus ` +oo ` . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL,
|
|
26-Dec-2011.) $)
|
|
xnegpnf $p |- -e +oo = -oo $=
|
|
( cpnf cxne wceq cmnf cneg cif df-xneg eqid iftruei eqtri ) ABAACZDADCAAEFZ
|
|
FDAGKDLAHIJ $.
|
|
|
|
$( Minus ` -oo ` . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL,
|
|
26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) $)
|
|
xnegmnf $p |- -e -oo = +oo $=
|
|
( cmnf cxne cpnf wceq cneg cif df-xneg wne mnfnepnf ifnefalse ax-mp iftruei
|
|
eqid 3eqtri ) ABACDAAADZCAEZFZFZQCAGACHRQDIACAQJKOCPAMLN $.
|
|
|
|
$( Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by
|
|
FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) $)
|
|
rexneg $p |- ( A e. RR -> -e A = -u A ) $=
|
|
( cr wcel cxne cpnf wceq cmnf cif df-xneg wne renepnf ifnefalse syl renemnf
|
|
cneg eqtrd syl5eq ) ABCZADAEFGAGFEAOZHZHZSAIRUATSRAEJUATFAKAEGTLMRAGJTSFANA
|
|
GESLMPQ $.
|
|
|
|
$( The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) $)
|
|
xneg0 $p |- -e 0 = 0 $=
|
|
( cc0 cxne cneg cr wcel wceq 0re rexneg ax-mp neg0 eqtri ) ABZACZAADELMFGAH
|
|
IJK $.
|
|
|
|
$( Closure of extended real negative. (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xnegcl $p |- ( A e. RR* -> -e A e. RR* ) $=
|
|
( cxr wcel cr cpnf wceq cmnf w3o cxne elxr cneg rexneg renegcl rexrd xnegeq
|
|
eqeltrd xnegpnf mnfxr eqeltri syl6eqel xnegmnf pnfxr 3jaoi sylbi ) ABCADCZA
|
|
EFZAGFZHAIZBCZAJUEUIUFUGUEUHUEUHAKDALAMPNUFUHEIZBAEOUJGBQRSTUGUHGIZBAGOUKEB
|
|
UAUBSTUCUD $.
|
|
|
|
$( Extended real version of ~ negneg . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xnegneg $p |- ( A e. RR* -> -e -e A = A ) $=
|
|
( cxr wcel cpnf wceq cmnf w3o cxne elxr cneg rexneg renegcl xnegmnf xnegpnf
|
|
cr xnegeq syl syl6eq id 3eqtr4a recn negnegd 3eqtrd 3jaoi sylbi ) ABCAOCZAD
|
|
EZAFEZGAHZHZAEZAIUFUKUGUHUFUJAJZHZULJZAUFUIULEUJUMEAKUIULPQUFULOCUMUNEALULK
|
|
QUFAAUAUBUCUGFHZDUJAMUGUIFEUJUOEUGUIDHZFADPNRUIFPQUGSTUHUPFUJANUHUIDEUJUPEU
|
|
HUIUODAFPMRUIDPQUHSTUDUE $.
|
|
|
|
$( Extended real version of ~ neg11 . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xneg11 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) ) $=
|
|
( cxr wcel wa cxne wceq xnegeq xnegneg eqeqan12d syl5ib impbid1 ) ACDZBCDZE
|
|
ZAFZBFZGZABGZRPFZQFZGOSPQHMNTAUABAIBIJKABHL $.
|
|
|
|
$( Forward direction of ~ xltneg . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xltnegi $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A ) $=
|
|
( cxr wcel clt wbr cxne cr cpnf wceq cmnf w3o wa wi elxr cneg syl adantr wn
|
|
sylbid rexneg breqan12rd bitr4d biimpd xnegeq xnegpnf syl6eq adantl renegcl
|
|
ltneg eqeltrd mnflt eqbrtrd a1d simpr breq2d nltmnf pm2.21d 3jaodan sylan2b
|
|
rexr expimpd simpl breq1d pnfnlt breq1 anbi2d ltpnf mnfltpnf syl6eqbr breq2
|
|
mnfxr ax-mp pm2.21i syl6bi imp 3jaoian sylanb xnegmnf syl5ibr 3jaoi 3impib
|
|
sylbi ) ACDZBCDZABEFZBGZAGZEFZWDAHDZAIJZAKJZLWEWFMZWINZAOWJWNWKWLWJWEWFWIWE
|
|
WJBHDZBIJZBKJZLZWFWINZBOZWJWOWSWPWQWJWOMZWFWIXAWFBPZAPZEFWIABUJWOWJWGXBWHXC
|
|
EBUAZAUAZUBUCUDWJWPMZWIWFXFWGKWHEWPWGKJZWJWPWGIGKBIUEUFUGZUHWJKWHEFZWPWJWHH
|
|
DXIWJWHXCHXEAUIUKWHULQRUMUNWJWQMZWFAKEFZWIXJBKAEWJWQUOUPXJXKWIWJXKSZWQWJWDX
|
|
LAVAAUQQRURTUSUTVBWKWEWFWIWKWEMZWFIBEFZWIXMAIBEWKWEVCVDXMXNWIWEXNSWKBVEUHUR
|
|
TVBWLWMWEKBEFZMZWIWLWFXOWEAKBEVFVGXPWIWLWGIEFZWEWRXOXQWTWOXOXQWPWQWOXOMWGHD
|
|
ZXQWOXRXOWOWGXBHXDBUIUKRWGVHQWPXOMWGKIEWPXGXOXHRVIVJWQXOXQWQXOKKEFZXQBKKEVK
|
|
XSXQKCDXSSVLKUQVMVNVOVPVQVRWLWHIWGEWLWHKGIAKUEVSUGUPVTTWAWCWB $.
|
|
|
|
$( Extended real version of ~ ltneg . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xltneg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) ) $=
|
|
( cxr wcel wa clt xltnegi 3expia wi xnegcl syl2anr xnegneg breqan12d sylibd
|
|
wbr cxne impbid ) ACDZBCDZEZABFOZBPZAPZFOZRSUAUDABGHTUDUCPZUBPZFOZUASUBCDZU
|
|
CCDZUDUGIRBJAJUHUIUDUGUBUCGHKRSUEAUFBFALBLMNQ $.
|
|
|
|
$( Extended real version of ~ leneg . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xleneg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) ) $=
|
|
( cxr wcel wa clt wbr wn cle wb xltneg ancoms notbid xrlenlt xnegcl syl2anr
|
|
cxne 3bitr4d ) ACDZBCDZEZBAFGZHAQZBQZFGZHZABIGUDUCIGZUAUBUETSUBUEJBAKLMABNT
|
|
UDCDUCCDUGUFJSBOAOUDUCNPR $.
|
|
|
|
$( Extended real version of ~ lt0neg1 . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xlt0neg1 $p |- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) ) $=
|
|
( cxr wcel cc0 clt wbr cxne wb 0xr xltneg mpan2 xneg0 breq1i syl6bb ) ABCZA
|
|
DEFZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.
|
|
|
|
$( Extended real version of ~ lt0neg2 . (Contributed by Mario Carneiro,
|
|
20-Aug-2015.) $)
|
|
xlt0neg2 $p |- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) ) $=
|
|
( cxr wcel cc0 clt wbr cxne wb 0xr xltneg mpan xneg0 breq2i syl6bb ) ABCZDA
|
|
EFZAGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.
|
|
|
|
$( Extended real version of ~ le0neg1 . (Contributed by Mario Carneiro,
|
|
9-Sep-2015.) $)
|
|
xle0neg1 $p |- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) ) $=
|
|
( cxr wcel cc0 cle wbr cxne wb 0xr xleneg mpan2 xneg0 breq1i syl6bb ) ABCZA
|
|
DEFZDGZAGZEFZDREFODBCPSHIADJKQDRELMN $.
|
|
|
|
$( Extended real version of ~ le0neg2 . (Contributed by Mario Carneiro,
|
|
9-Sep-2015.) $)
|
|
xle0neg2 $p |- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) ) $=
|
|
( cxr wcel cc0 cle wbr cxne wb 0xr xleneg mpan xneg0 breq2i syl6bb ) ABCZDA
|
|
EFZAGZDGZEFZQDEFDBCOPSHIDAJKRDQELMN $.
|
|
|
|
${
|
|
xnegcld.1 $e |- ( ph -> A e. RR* ) $.
|
|
$( Closure of extended real negative. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
xnegcld $p |- ( ph -> -e A e. RR* ) $=
|
|
( cxr wcel cxne xnegcl syl ) ABDEBFDECBGH $.
|
|
$}
|
|
|
|
$( The set of extended reals exists. (Contributed by NM, 24-Dec-2006.) $)
|
|
xrex $p |- RR* e. _V $=
|
|
( cxr cr cpnf cmnf cpr cun cvv df-xr reex wcel pnfxr mnfxr prexg mp2an unex
|
|
eqeltri ) ABCDEZFGHBQICAJDAJQGJKLCDAAMNOP $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Real number intervals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c (,) $.
|
|
$c (,] $.
|
|
$c [,) $.
|
|
$c [,] $.
|
|
|
|
$( Extend class notation with the set of open intervals of extended reals. $)
|
|
cioo $a class (,) $.
|
|
|
|
$( Extend class notation with the set of open-below, closed-above intervals
|
|
of extended reals. $)
|
|
cioc $a class (,] $.
|
|
|
|
$( Extend class notation with the set of closed-below, open-above intervals
|
|
of extended reals. $)
|
|
cico $a class [,) $.
|
|
|
|
$( Extend class notation with the set of closed intervals of extended
|
|
reals. $)
|
|
cicc $a class [,] $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define the set of open intervals of extended reals. (Contributed by NM,
|
|
24-Dec-2006.) $)
|
|
df-ioo $a |- (,) = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x < z /\ z < y ) } ) $.
|
|
|
|
$( Define the set of open-below, closed-above intervals of extended reals.
|
|
(Contributed by NM, 24-Dec-2006.) $)
|
|
df-ioc $a |- (,] = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x < z /\ z <_ y ) } ) $.
|
|
|
|
$( Define the set of closed-below, open-above intervals of extended reals.
|
|
(Contributed by NM, 24-Dec-2006.) $)
|
|
df-ico $a |- [,) = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x <_ z /\ z < y ) } ) $.
|
|
|
|
$( Define the set of closed intervals of extended reals. (Contributed by
|
|
NM, 24-Dec-2006.) $)
|
|
df-icc $a |- [,] = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x <_ z /\ z <_ y ) } ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z C $. $d x y z B $. $d x y z R $. $d x y z S $.
|
|
ixx.1 $e |- O = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x R z /\ z S y ) } ) $.
|
|
$( Value of the interval function. (Contributed by Mario Carneiro,
|
|
3-Nov-2013.) $)
|
|
ixxval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) =
|
|
{ z e. RR* | ( A R z /\ z S B ) } ) $=
|
|
( cxr cv wbr wa crab wceq breq1 anbi1d rabbidv breq2 anbi2d rabex ovmpt2
|
|
xrex ) ABDEJJAKZCKZFLZUEBKZGLZMZCJNDUEFLZUEEGLZMZCJNHUJUHMZCJNUDDOZUIUMCJ
|
|
UNUFUJUHUDDUEFPQRUGEOZUMULCJUOUHUKUJUGEUEGSTRIULCJUCUAUB $.
|
|
|
|
$( Membership in an interval of extended reals. (Contributed by Mario
|
|
Carneiro, 3-Nov-2013.) $)
|
|
elixx1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <->
|
|
( C e. RR* /\ A R C /\ C S B ) ) ) $=
|
|
( cxr wcel wa co cv wbr crab w3a ixxval eleq2d breq2 breq1 anbi12d 3anass
|
|
wceq elrab bitr4i syl6bb ) DKLEKLMZFDEINZLFDCOZGPZUKEHPZMZCKQZLZFKLZDFGPZ
|
|
FEHPZRZUIUJUOFABCDEGHIJSTUPUQURUSMZMUTUNVACFKUKFUEULURUMUSUKFDGUAUKFEHUBU
|
|
CUFUQURUSUDUGUH $.
|
|
|
|
$( The set of intervals of extended reals maps to subsets of extended
|
|
reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro,
|
|
16-Nov-2013.) $)
|
|
ixxf $p |- O : ( RR* X. RR* ) --> ~P RR* $=
|
|
( cv wbr wa cxr crab cpw wcel wral cxp wf wss ssrab2 xrex elpw2 rgen2w
|
|
mpbir fmpt2 mpbi ) AHCHZDIUFBHEIJZCKLZKMZNZBKOAKOKKPUIFQUJABKKUJUHKRUGCKS
|
|
UHKTUAUCUBABKKUHUIFGUDUE $.
|
|
|
|
$( The set of intervals of extended reals exists. (Contributed by Mario
|
|
Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) $)
|
|
ixxex $p |- O e. _V $=
|
|
( cxr cxp cpw xrex xpex pwex wf wss ixxf fssxp ax-mp ssexi ) FHHIZHJZIZTU
|
|
AHHKKLHKMLTUAFNFUBOABCDEFGPTUAFQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z S $. $d x y z A $. $d x y z B $. $d x y z C $.
|
|
ixxssxr.1 $e |- O = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x R z /\ z S y ) } ) $.
|
|
${
|
|
$d x y z O $.
|
|
$( The set of intervals of extended reals maps to subsets of extended
|
|
reals. (Contributed by Mario Carneiro, 4-Jul-2014.) $)
|
|
ixxssxr $p |- ( A O B ) C_ RR* $=
|
|
( co cxr wcel wa cv wbr crab elmpt2cl cpw ixxf fovcl elpwid sseld mpcom
|
|
ssriv ) ADEHJZKDKLEKLMZANZUELUGKLABKKUGCNZFOUHBNGOMCKPDEHUGIQUFUEKUGUFU
|
|
EKDEKRKKHABCFGHISTUAUBUCUD $.
|
|
$}
|
|
|
|
$( Membership in a set of open intervals of extended reals. We use the
|
|
fact that an operation's value is empty outside of its domain to show
|
|
` A e. RR* ` and ` B e. RR* ` . (Contributed by Mario Carneiro,
|
|
3-Nov-2013.) $)
|
|
elixx3g $p |- ( C e. ( A O B ) <->
|
|
( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) ) $=
|
|
( cxr wcel wa wbr w3a co anass df-3an anbi1i cv crab elixx1 3anass syl6bb
|
|
elmpt2cl biadan2 3bitr4ri ) DKLZEKLZMZFKLZMZDFGNZFEHNZMZMUJUKUOMZMUHUIUKO
|
|
ZUOMFDEIPLZUJUKUOQUQULUOUHUIUKRSURUJUPABKKATCTZGNUSBTHNMCKUADEIFJUEUJURUK
|
|
UMUNOUPABCDEFGHIJUBUKUMUNUCUDUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z C $. $d w x y z D $. $d w x O $.
|
|
$d w x y z B $. $d w P $. $d x y z R $. $d x y z S $. $d x y z T $.
|
|
$d x y z U $. $d w W $. $d w X $.
|
|
ixxssixx.1 $e |- O = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x R z /\ z S y ) } ) $.
|
|
${
|
|
ixx.2 $e |- P = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x T z /\ z U y ) } ) $.
|
|
ixx.3 $e |- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A T w ) ) $.
|
|
ixx.4 $e |- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w U B ) ) $.
|
|
$( An interval is a subset of its closure. (Contributed by Paul Chapman,
|
|
18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
ixxssixx $p |- ( A O B ) C_ ( A P B ) $=
|
|
( cxr wcel wa wbr co cv crab elmpt2cl w3a wi simp1 simpl 3simpa expimpd
|
|
a1i syl2im simpr 3simpb ancoms 3jcad elixx1 3imtr4d mpcom ssriv ) DEFLU
|
|
AZEFGUAZEQRZFQRZSZDUBZVARZVFVBRZABQQAUBCUBZHTVIBUBITSCQUCEFLVFMUDVEVFQR
|
|
ZEVFHTZVFFITZUEZVJEVFJTZVFFKTZUEVGVHVEVMVJVNVOVMVJUFVEVJVKVLUGUKVEVCVMV
|
|
JVKSVNVCVDUHVJVKVLUIVCVJVKVNOUJULVEVDVMVJVLSVOVCVDUMVJVKVLUNVDVJVLVOVJV
|
|
DVLVOUFPUOUJULUPABCEFVFHILMUQABCEFVFJKGNUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
ixxun.2 $e |- P = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x T z /\ z U y ) } ) $.
|
|
ixxun.3 $e |- ( ( B e. RR* /\ w e. RR* ) -> ( B T w <-> -. w S B ) ) $.
|
|
$( Split an interval into disjoint pieces. (Contributed by Mario
|
|
Carneiro, 16-Jun-2014.) $)
|
|
ixxdisj $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A O B ) i^i ( B P C ) ) = (/) ) $=
|
|
( cxr wcel c0 wbr w3a co cin wss wceq elin elixx1 3adant3 biimpa simp3d
|
|
cv wa wb adantrr wn 3adant1 simp2d simpl2 simp1d syl2anc mpbid pm2.65da
|
|
adantrl pm2.21d syl5bi ssrdv ss0 syl ) EQRZFQRZGQRZUAZEFMUBZFGHUBZUCZSU
|
|
DVOSUEVLDVOSDUKZVORVPVMRZVPVNRZULZVLVPSRZVPVMVNUFVLVSVTVLVSVPFJTZVLVQWA
|
|
VRVLVQULVPQRZEVPITZWAVLVQWBWCWAUAZVIVJVQWDUMVKABCEFVPIJMNUGUHUIUJUNVLVR
|
|
WAUOZVQVLVRULZFVPKTZWEWFWBWGVPGLTZVLVRWBWGWHUAZVJVKVRWIUMVIABCFGVPKLHOU
|
|
GUPUIZUQWFVJWBWGWEUMVIVJVKVRURWFWBWGWHWJUSPUTVAVCVBVDVEVFVOVGVH $.
|
|
$}
|
|
|
|
${
|
|
ixxss1.2 $e |- P = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x T z /\ z S y ) } ) $.
|
|
ixxss1.3 $e |- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) ->
|
|
( ( A W B /\ B T w ) -> A R w ) ) $.
|
|
$( Subset relationship for intervals of extended reals. (Contributed by
|
|
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
ixxss1 $p |- ( ( A e. RR* /\ A W B ) ->
|
|
( B P C ) C_ ( A O C ) ) $=
|
|
( cxr wcel wbr wa co cv w3a elixx3g simplbi adantl simp3d simplr simpld
|
|
simprbi wi simpll simp1d syl3anc mp2and simprd wb simp2d elixx1 syl2anc
|
|
mpbir3and ex ssrdv ) EQRZEFMSZTZDFGHUAZEGLUAZVFDUBZVGRZVIVHRZVFVJTZVKVI
|
|
QRZEVIISZVIGJSZVLFQRZGQRZVMVJVPVQVMUCZVFVJVRFVIKSZVOTZABCFGVIKJHOUDZUEU
|
|
FZUGZVLVEVSVNVDVEVJUHVLVSVOVJVTVFVJVRVTWAUJUFZUIVLVDVPVMVEVSTVNUKVDVEVJ
|
|
ULZVLVPVQVMWBUMWCPUNUOVLVSVOWDUPVLVDVQVKVMVNVOUCUQWEVLVPVQVMWBURABCEGVI
|
|
IJLNUSUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
ixxss2.2 $e |- P = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x R z /\ z T y ) } ) $.
|
|
ixxss2.3 $e |- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( w T B /\ B W C ) -> w S C ) ) $.
|
|
$( Subset relationship for intervals of extended reals. (Contributed by
|
|
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
ixxss2 $p |- ( ( C e. RR* /\ B W C ) ->
|
|
( A P B ) C_ ( A O C ) ) $=
|
|
( cxr wcel wbr wa co cv w3a elixx3g simplbi adantl simp3d simpld simprd
|
|
simprbi simplr wi simp2d simpll syl3anc mp2and wb simp1d elixx1 syl2anc
|
|
mpbir3and ex ssrdv ) GQRZFGMSZTZDEFHUAZEGLUAZVFDUBZVGRZVIVHRZVFVJTZVKVI
|
|
QRZEVIISZVIGJSZVLEQRZFQRZVMVJVPVQVMUCZVFVJVRVNVIFKSZTZABCEFVIIKHOUDZUEU
|
|
FZUGZVLVNVSVJVTVFVJVRVTWAUJUFZUHVLVSVEVOVLVNVSWDUIVDVEVJUKVLVMVQVDVSVET
|
|
VOULWCVLVPVQVMWBUMVDVEVJUNZPUOUPVLVPVDVKVMVNVOUCUQVLVPVQVMWBURWEABCEGVI
|
|
IJLNUSUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
ixxss12.2 $e |- P = ( x e. RR* , y e. RR* |->
|
|
{ z e. RR* | ( x T z /\ z U y ) } ) $.
|
|
ixxss12.3 $e |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) ->
|
|
( ( A W C /\ C T w ) -> A R w ) ) $.
|
|
ixxss12.4 $e |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) ->
|
|
( ( w U D /\ D X B ) -> w S B ) ) $.
|
|
$( Subset relationship for intervals of extended reals. (Contributed by
|
|
Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
ixxss12 $p |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) ->
|
|
( C P D ) C_ ( A O B ) ) $=
|
|
( cxr wa wbr co cv elixx3g simplbi adantl simp3d simplrl simprbi simpld
|
|
wcel w3a wi simplll simp1d syl3anc mp2and simprd simplrr simp2d simpllr
|
|
wb elixx1 ad2antrr mpbir3and ex ssrdv ) EUAUMZFUAUMZUBZEGOUCZHFPUCZUBZU
|
|
BZDGHIUDZEFNUDZVPDUEZVQUMZVSVRUMZVPVTUBZWAVSUAUMZEVSJUCZVSFKUCZWBGUAUMZ
|
|
HUAUMZWCVTWFWGWCUNZVPVTWHGVSLUCZVSHMUCZUBZABCGHVSLMIRUFZUGUHZUIZWBVMWIW
|
|
DVLVMVNVTUJWBWIWJVTWKVPVTWHWKWLUKUHZULWBVJWFWCVMWIUBWDUOVJVKVOVTUPWBWFW
|
|
GWCWMUQWNSURUSWBWJVNWEWBWIWJWOUTVLVMVNVTVAWBWCWGVKWJVNUBWEUOWNWBWFWGWCW
|
|
MVBVJVKVOVTVCTURUSVLWAWCWDWEUNVDVOVTABCEFVSJKNQVEVFVGVHVI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $.
|
|
$( The set of open intervals of extended reals exists. (Contributed by NM,
|
|
6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iooex $p |- (,) e. _V $=
|
|
( vx vy vz clt cioo df-ioo ixxex ) ABCDDEABCFG $.
|
|
|
|
$( Value of the open interval function. (Contributed by NM, 24-Dec-2006.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iooval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) =
|
|
{ x e. RR* | ( A < x /\ x < B ) } ) $=
|
|
( vy vz clt cioo df-ioo ixxval ) DEABCFFGDEAHI $.
|
|
|
|
$( An open interval with identical lower and upper bounds is empty.
|
|
(Contributed by Jim Kingdon, 29-Mar-2020.) $)
|
|
iooidg $p |- ( A e. RR* -> ( A (,) A ) = (/) ) $=
|
|
( vx cxr wcel cioo co cv clt wa crab c0 wceq iooval anidms wral xrltnsym2
|
|
wbr wn ralrimiva rabeq0 sylibr eqtrd ) ACDZAAEFZABGZHQUEAHQIZBCJZKUCUDUGL
|
|
BAAMNUCUFRZBCOUGKLUCUHBCAUEPSUFBCTUAUB $.
|
|
|
|
$( Membership in a set of open intervals of extended reals. We use the
|
|
fact that an operation's value is empty outside of its domain to show
|
|
` A e. RR* ` and ` B e. RR* ` . (Contributed by NM, 24-Dec-2006.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
elioo3g $p |- ( C e. ( A (,) B ) <->
|
|
( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ C < B ) ) ) $=
|
|
( vx vy vz clt cioo df-ioo elixx3g ) DEFABCGGHDEFIJ $.
|
|
|
|
$( Membership in an open interval of extended reals. (Contributed by NM,
|
|
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
elioo1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <->
|
|
( C e. RR* /\ A < C /\ C < B ) ) ) $=
|
|
( vx vy vz clt cioo df-ioo elixx1 ) DEFABCGGHDEFIJ $.
|
|
|
|
$( A member of an open interval of reals is a real. (Contributed by NM,
|
|
17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
elioore $p |- ( A e. ( B (,) C ) -> A e. RR ) $=
|
|
( cioo co wcel cxr w3a clt wbr wa cr elioo3g 3ancomb xrre2 sylanb sylbi )
|
|
ABCDEFBGFZCGFZAGFZHZBAIJACIJKZKALFZBCAMUARTSHUBUCRSTNBACOPQ $.
|
|
|
|
$( An open interval does not contain its left endpoint. (Contributed by
|
|
Jim Kingdon, 30-Mar-2020.) $)
|
|
lbioog $p |- ( ( A e. RR* /\ B e. RR* ) -> -. A e. ( A (,) B ) ) $=
|
|
( cxr wcel wa cioo co clt wbr w3a xrltnr simp2 nsyl adantr elioo1 mtbird
|
|
wn ) ACDZBCDZEAABFGDRAAHIZABHIZJZRUBQSRTUBAKRTUALMNABAOP $.
|
|
|
|
$( An open interval does not contain its right endpoint. (Contributed by
|
|
Jim Kingdon, 30-Mar-2020.) $)
|
|
ubioog $p |- ( ( A e. RR* /\ B e. RR* ) -> -. B e. ( A (,) B ) ) $=
|
|
( cxr wcel wa cioo co clt wbr w3a xrltnr simp3 nsyl adantl elioo1 mtbird
|
|
wn ) ACDZBCDZEBABFGDSABHIZBBHIZJZSUBQRSUAUBBKSTUALMNABBOP $.
|
|
|
|
$( Value of the open interval function. (Contributed by NM, 6-Feb-2007.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iooval2 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) =
|
|
{ x e. RR | ( A < x /\ x < B ) } ) $=
|
|
( cxr wcel wa cioo co cv clt wbr crab cr iooval cin inrab2 ressxr sseqin2
|
|
wceq wss mpbi rabeq ax-mp eqtri elioore ssriv syl6eqssr df-ss sylib eqtrd
|
|
syl5reqr ) BDECDEFZBCGHZBAIZJKUNCJKFZADLZUOAMLZABCNZULUQUPMOZUPUSUOADMOZL
|
|
ZUQUOADMPUTMSZVAUQSMDTVBQMDRUAUOAUTMUBUCUDULUPMTUSUPSULUPUMMURAUMMUNBCUEU
|
|
FUGUPMUHUIUKUJ $.
|
|
|
|
$( Subset relationship for open intervals of extended reals. (Contributed
|
|
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.) $)
|
|
iooss1 $p |- ( ( A e. RR* /\ A <_ B ) ->
|
|
( B (,) C ) C_ ( A (,) C ) ) $=
|
|
( vx vy vz vw cioo clt cle df-ioo cv xrlelttr ixxss1 ) DEFGABCHIIIHJDEFKZ
|
|
OABGLMN $.
|
|
|
|
$( Subset relationship for open intervals of extended reals. (Contributed
|
|
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iooss2 $p |- ( ( C e. RR* /\ B <_ C ) ->
|
|
( A (,) B ) C_ ( A (,) C ) ) $=
|
|
( vx vy vz vw cioo clt cle df-ioo cv xrltletr ixxss2 ) DEFGABCHIIIHJDEFKZ
|
|
OGLBCMN $.
|
|
|
|
$( Value of the open-below, closed-above interval function. (Contributed
|
|
by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iocval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) =
|
|
{ x e. RR* | ( A < x /\ x <_ B ) } ) $=
|
|
( vy vz clt cle cioc df-ioc ixxval ) DEABCFGHDEAIJ $.
|
|
|
|
$( Value of the closed-below, open-above interval function. (Contributed
|
|
by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
icoval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) =
|
|
{ x e. RR* | ( A <_ x /\ x < B ) } ) $=
|
|
( vy vz cle clt cico df-ico ixxval ) DEABCFGHDEAIJ $.
|
|
|
|
$( Value of the closed interval function. (Contributed by NM,
|
|
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
iccval $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,] B ) =
|
|
{ x e. RR* | ( A <_ x /\ x <_ B ) } ) $=
|
|
( vy vz cle cicc df-icc ixxval ) DEABCFFGDEAHI $.
|
|
|
|
$( Membership in an open interval of extended reals. (Contributed by NM,
|
|
6-Feb-2007.) $)
|
|
elioo2 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <->
|
|
( C e. RR /\ A < C /\ C < B ) ) ) $=
|
|
( vx cxr wcel wa cioo co cv clt wbr cr crab w3a iooval2 eleq2d wceq breq2
|
|
breq1 anbi12d elrab 3anass bitr4i syl6bb ) AEFBEFGZCABHIZFCADJZKLZUHBKLZG
|
|
ZDMNZFZCMFZACKLZCBKLZOZUFUGULCDABPQUMUNUOUPGZGUQUKURDCMUHCRUIUOUJUPUHCAKS
|
|
UHCBKTUAUBUNUOUPUCUDUE $.
|
|
|
|
$( Membership in an open-below, closed-above interval of extended reals.
|
|
(Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro,
|
|
3-Nov-2013.) $)
|
|
elioc1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <->
|
|
( C e. RR* /\ A < C /\ C <_ B ) ) ) $=
|
|
( vx vy vz clt cle cioc df-ioc elixx1 ) DEFABCGHIDEFJK $.
|
|
|
|
$( Membership in a closed-below, open-above interval of extended reals.
|
|
(Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro,
|
|
3-Nov-2013.) $)
|
|
elico1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <->
|
|
( C e. RR* /\ A <_ C /\ C < B ) ) ) $=
|
|
( vx vy vz cle clt cico df-ico elixx1 ) DEFABCGHIDEFJK $.
|
|
|
|
$( Membership in a closed interval of extended reals. (Contributed by NM,
|
|
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
elicc1 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <->
|
|
( C e. RR* /\ A <_ C /\ C <_ B ) ) ) $=
|
|
( vx vy vz cle cicc df-icc elixx1 ) DEFABCGGHDEFIJ $.
|
|
|
|
$( A closed interval with identical lower and upper bounds is a singleton.
|
|
(Contributed by Jeff Hankins, 13-Jul-2009.) $)
|
|
iccid $p |- ( A e. RR* -> ( A [,] A ) = { A } ) $=
|
|
( vx cxr wcel co csn cv cle wbr w3a wb wi wa clt wn xrlenlt ancoms sylbid
|
|
cicc syl5ibrcom elicc1 anidms xrlttri3 biimprd expcomd com23 3impd eleq1a
|
|
wceq ex xrleid breq2 breq1 3jcad impbid velsn syl6bbr bitrd eqrdv ) ACDZB
|
|
AASEZAFZUTBGZVADZVCCDZAVCHIZVCAHIZJZVCVBDZUTVDVHKAAVCUAUBUTVHVCAUIZVIUTVH
|
|
VJUTVEVFVGVJUTVEVFVGVJLZLUTVEMZVFVCANIOZVKAVCPVLVGVMVJVLVGAVCNIOZVMVJLVEU
|
|
TVGVNKVCAPQVLVMVNVJVEUTVMVNMZVJLVEUTMVJVOVCAUCUDQUERUFRUJUGUTVJVEVFVGACVC
|
|
UHUTVFVJAAHIZAUKZVCAAHULTUTVGVJVPVQVCAAHUMTUNUOBAUPUQURUS $.
|
|
|
|
$( An empty closed interval of extended reals. (Contributed by Jim
|
|
Kingdon, 30-Mar-2020.) $)
|
|
icc0r $p |- ( ( A e. RR* /\ B e. RR* ) ->
|
|
( B < A -> ( A [,] B ) = (/) ) ) $=
|
|
( vx cxr wcel wa clt wbr cv cle wrex wn cicc co c0 wi xrletr 3com23 3expa
|
|
wceq rexlimdva xrlenlt sylibd con2d crab iccval eqeq1d wral rabeq0 ralnex
|
|
bitri syl6bb sylibrd ) ADEZBDEZFZBAGHZACIZJHURBJHFZCDKZLZABMNZOTZUPUTUQUP
|
|
UTABJHZUQLUPUSVDCDUNUOURDEZUSVDPZUNVEUOVFAURBQRSUAABUBUCUDUPVCUSCDUEZOTZV
|
|
AUPVBVGOCABUFUGVHUSLCDUHVAUSCDUIUSCDUJUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( An inhabited open interval spans an interval of extended reals.
|
|
(Contributed by NM, 17-Aug-2008.) $)
|
|
eliooxr $p |- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) ) $=
|
|
( vx vy vz cxr cv clt wbr wa crab cioo df-ioo elmpt2cl ) DEGGDHFHZIJPEHIJ
|
|
KFGLBCMADEFNO $.
|
|
$}
|
|
|
|
$( Ordering implied by a member of an open interval of reals. (Contributed
|
|
by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.) $)
|
|
eliooord $p |- ( A e. ( B (,) C ) -> ( B < A /\ A < C ) ) $=
|
|
( cioo co wcel cr clt wbr w3a wa cxr wb eliooxr elioo2 syl ibi 3simpc ) ABC
|
|
DEFZAGFZBAHIZACHIZJZUAUBKSUCSBLFCLFKSUCMABCNBCAOPQTUAUBRP $.
|
|
|
|
$( The upper bound belongs to an open-below, closed-above interval. See
|
|
~ ubicc2 . (Contributed by FL, 29-May-2014.) $)
|
|
ubioc1 $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) ) $=
|
|
( cxr wcel clt wbr w3a co cle simp2 simp3 xrleid 3ad2ant2 wb elioc1 3adant3
|
|
cioc mpbir3and ) ACDZBCDZABEFZGBABQHDZTUABBIFZSTUAJSTUAKTSUCUABLMSTUBTUAUCG
|
|
NUAABBOPR $.
|
|
|
|
$( The lower bound belongs to a closed-below, open-above interval. See
|
|
~ lbicc2 . (Contributed by FL, 29-May-2014.) $)
|
|
lbico1 $p |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) ) $=
|
|
( cxr wcel clt wbr w3a co cle simp1 xrleid 3ad2ant1 simp3 wb elico1 3adant3
|
|
cico mpbir3and ) ACDZBCDZABEFZGAABQHDZSAAIFZUASTUAJSTUCUAAKLSTUAMSTUBSUCUAG
|
|
NUAABAOPR $.
|
|
|
|
$( An element of a closed interval is less than or equal to its upper bound.
|
|
(Contributed by Jeff Hankins, 14-Jul-2009.) $)
|
|
iccleub $p |- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> C <_ B ) $=
|
|
( cxr wcel cicc co cle wbr wa w3a elicc1 simp3 syl6bi 3impia ) ADEZBDEZCABF
|
|
GEZCBHIZPQJRCDEZACHIZSKSABCLTUASMNO $.
|
|
|
|
$( An element of a closed interval is more than or equal to its lower bound
|
|
(Contributed by Thierry Arnoux, 23-Dec-2016.) $)
|
|
iccgelb $p |- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C ) $=
|
|
( cxr wcel cicc co cle wbr wa w3a elicc1 biimpa simp2d 3impa ) ADEZBDEZCABF
|
|
GEZACHIZPQJZRJCDEZSCBHIZTRUASUBKABCLMNO $.
|
|
|
|
$( TODO - see if this shortens the uses of any other
|
|
elioo*'s $)
|
|
$( Membership in an open interval of extended reals. (Contributed by NM,
|
|
17-Aug-2008.) $)
|
|
elioo5 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* )
|
|
-> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) ) $=
|
|
( cxr wcel w3a co clt wbr wa wb elioo1 3adant3 3anass baibr 3ad2ant3 bitr4d
|
|
cioo ) ADEZBDEZCDEZFCABRGEZUAACHIZCBHIZFZUCUDJZSTUBUEKUAABCLMUASUFUEKTUEUAU
|
|
FUAUCUDNOPQ $.
|
|
|
|
$( Membership in an open interval of extended reals. (Contributed by NM,
|
|
8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
elioo4g $p |- ( C e. ( A (,) B ) <->
|
|
( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) ) $=
|
|
( cioo co wcel cxr cr w3a clt wbr wa eliooxr elioore df-3an sylibr eliooord
|
|
jca rexr 3anim3i anim1i elioo3g impbii ) CABDEFZAGFZBGFZCHFZIZACJKCBJKLZLZU
|
|
DUHUIUDUEUFLZUGLUHUDUKUGCABMCABNRUEUFUGOPCABQRUJUEUFCGFZIZUILUDUHUMUIUGULUE
|
|
UFCSTUAABCUBPUC $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An open interval is a set of reals. (Contributed by NM,
|
|
31-May-2007.) $)
|
|
ioossre $p |- ( A (,) B ) C_ RR $=
|
|
( vx cioo co cr cv elioore ssriv ) CABDEFCGABHI $.
|
|
$}
|
|
|
|
$( Membership in an open-below, closed-above real interval. (Contributed by
|
|
Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) $)
|
|
elioc2 $p |- ( ( A e. RR* /\ B e. RR ) ->
|
|
( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) ) $=
|
|
( cxr wcel cr wa cioc clt wbr cle w3a rexr cmnf cpnf a1i xrlelttrd ad2antlr
|
|
co wb elioc1 sylan2 mnfxr simpll simpr1 mnfle ad2antrr simpr2 pnfxr xrrebnd
|
|
simpr3 ltpnf syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBFEZGZCABHSE
|
|
ZCDEZACIJZCBKJZLZCFEZVEVFLZVAUTBDEZVCVGTBMZABCUAUBVBVGVIVBVGVIVBVGGZVHVEVFV
|
|
LVHNCIJZCOIJZVLNACNDEVLUCPUTVAVGUDVBVDVEVFUEZUTNAKJVAVGAUFUGVBVDVEVFUHZQVLC
|
|
BOVOVAVJUTVGVKRODEVLUIPVBVDVEVFUKZVABOIJUTVGBULRQVLVDVHVMVNGTVOCUJUMUNVPVQU
|
|
OUPVHVDVEVFCMUQURUS $.
|
|
|
|
$( Membership in a closed-below, open-above real interval. (Contributed by
|
|
Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.) $)
|
|
elico2 $p |- ( ( A e. RR /\ B e. RR* ) ->
|
|
( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) ) $=
|
|
( cr wcel cxr wa cico cle wbr clt w3a rexr cmnf cpnf a1i ad2antrr xrltletrd
|
|
co wb elico1 sylan mnfxr simpr1 mnflt simpr2 simplr simpr3 ad2antlr xrrebnd
|
|
pnfxr pnfge syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBFEZGZCABHSEZ
|
|
CFEZACIJZCBKJZLZCDEZVEVFLZUTAFEZVAVCVGTAMZABCUAUBVBVGVIVBVGVIVBVGGZVHVEVFVL
|
|
VHNCKJZCOKJZVLNACNFEVLUCPUTVJVAVGVKQVBVDVEVFUDZUTNAKJVAVGAUEQVBVDVEVFUFZRVL
|
|
CBOVOUTVAVGUGOFEVLUKPVBVDVEVFUHZVABOIJUTVGBULUIRVLVDVHVMVNGTVOCUJUMUNVPVQUO
|
|
UPVHVDVEVFCMUQURUS $.
|
|
|
|
$( Membership in a closed real interval. (Contributed by Paul Chapman,
|
|
21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) $)
|
|
elicc2 $p |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A [,] B ) <->
|
|
( C e. RR /\ A <_ C /\ C <_ B ) ) ) $=
|
|
( cr wcel wa cicc co cxr cle wbr w3a wb rexr cmnf clt a1i ad2antrr ad2antlr
|
|
cpnf elicc1 syl2an mnfxr simpr1 mnflt simpr2 xrltletrd pnfxr simpr3 xrrebnd
|
|
ltpnf xrlelttrd syl mpbir2and 3jca ex 3anim1i impbid1 bitrd ) ADEZBDEZFZCAB
|
|
GHEZCIEZACJKZCBJKZLZCDEZVEVFLZUTAIEZBIEZVCVGMVAANZBNZABCUAUBVBVGVIVBVGVIVBV
|
|
GFZVHVEVFVNVHOCPKZCTPKZVNOACOIEVNUCQUTVJVAVGVLRVBVDVEVFUDZUTOAPKVAVGAUERVBV
|
|
DVEVFUFZUGVNCBTVQVAVKUTVGVMSTIEVNUHQVBVDVEVFUIZVABTPKUTVGBUKSULVNVDVHVOVPFM
|
|
VQCUJUMUNVRVSUOUPVHVDVEVFCNUQURUS $.
|
|
|
|
${
|
|
elicc2i.1 $e |- A e. RR $.
|
|
elicc2i.2 $e |- B e. RR $.
|
|
$( Inference for membership in a closed interval. (Contributed by Scott
|
|
Fenton, 3-Jun-2013.) $)
|
|
elicc2i $p |- ( C e. ( A [,] B ) <-> ( C e. RR /\ A <_ C /\ C <_ B ) ) $=
|
|
( cr wcel cicc co cle wbr w3a wb elicc2 mp2an ) AFGBFGCABHIGCFGACJKCBJKLM
|
|
DEABCNO $.
|
|
$}
|
|
|
|
$( Membership in a closed real interval. (Contributed by Stefan O'Rear,
|
|
16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.) $)
|
|
elicc4 $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) $=
|
|
( cxr wcel cicc co cle wbr wa wb w3a elicc1 3anass syl6bb baibd 3impa ) ADE
|
|
ZBDEZCDEZCABFGEZACHIZCBHIZJZKRSJZUATUDUEUATUBUCLTUDJABCMTUBUCNOPQ $.
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $. $d w x y z D $.
|
|
$( Condition for a closed interval to be a subset of another closed
|
|
interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario
|
|
Carneiro, 20-Feb-2015.) $)
|
|
iccss $p |- ( ( ( A e. RR /\ B e. RR ) /\
|
|
( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) ) $=
|
|
( vx vy vz vw cr wcel wa cxr cle wbr cicc co wss rexr anim12i xrletr cv
|
|
df-icc ixxss12 sylan ) AIJZBIJZKALJZBLJZKACMNDBMNKCDOPABOPQUEUGUFUHARBRSE
|
|
FGHABCDOMMMMOMMEFGUBZUIACHUAZTUJDBTUCUD $.
|
|
|
|
$( Condition for a closed interval to be a subset of an open interval.
|
|
(Contributed by Mario Carneiro, 20-Feb-2015.) $)
|
|
iccssioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A < C /\ D < B ) ) -> ( C [,] D ) C_ ( A (,) B ) ) $=
|
|
( vx vy vz vw cicc clt cle cioo df-ioo df-icc xrltletr xrlelttr ixxss12
|
|
cv ) EFGHABCDIJJKKLJJEFGMEFGNACHRZOSDBPQ $.
|
|
|
|
$( Condition for a closed-below, open-above interval to be a subset of a
|
|
closed-below, open-above interval. (Contributed by Thierry Arnoux,
|
|
21-Sep-2017.) $)
|
|
icossico $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A <_ C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A [,) B ) ) $=
|
|
( vx vy vz vw cico cle clt df-ico cv xrletr xrltletr ixxss12 ) EFGHABCDIJ
|
|
KJKIJJEFGLZQACHMZNRDBOP $.
|
|
|
|
$( Condition for a closed interval to be a subset of another closed
|
|
interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
iccss2 $p |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ->
|
|
( C [,] D ) C_ ( A [,] B ) ) $=
|
|
( vx vy vz vw cicc co wcel cxr cle wbr w3a elixx3g adantr simprbi xrletr
|
|
wa wss df-icc simplbi simp1d simp2d simpld simprd adantl ixxss12 syl22anc
|
|
cv ) CABIJZKZDULKZTZALKZBLKZACMNZDBMNZCDIJULUAUOUPUQCLKZUMUPUQUTOZUNUMVAU
|
|
RCBMNZTZEFGABCMMIEFGUBZPZUCQZUDUOUPUQUTVFUEUOURVBUMVCUNUMVAVCVERQUFUNUSUM
|
|
UNADMNZUSUNUPUQDLKOVGUSTEFGABDMMIVDPRUGUHEFGHABCDIMMMMIMMVDVDACHUKZSVHDBS
|
|
UIUJ $.
|
|
|
|
$( Condition for a closed interval to be a subset of a half-open interval.
|
|
(Contributed by Mario Carneiro, 9-Sep-2015.) $)
|
|
iccssico $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A <_ C /\ D < B ) ) -> ( C [,] D ) C_ ( A [,) B ) ) $=
|
|
( vx vy vz vw cicc cle clt cico df-ico df-icc cv xrletr xrlelttr ixxss12
|
|
) EFGHABCDIJKJJLJKEFGMEFGNACHOZPSDBQR $.
|
|
|
|
$( Condition for a closed interval to be a subset of an open interval.
|
|
(Contributed by Mario Carneiro, 20-Feb-2015.) $)
|
|
iccssioo2 $p |- ( ( C e. ( A (,) B ) /\ D e. ( A (,) B ) ) ->
|
|
( C [,] D ) C_ ( A (,) B ) ) $=
|
|
( cioo co wcel cxr clt wbr cicc wss eliooxr adantr eliooord simpld adantl
|
|
wa simprd iccssioo syl12anc ) CABEFZGZDUBGZRZAHGBHGRZACIJZDBIJZCDKFUBLUCU
|
|
FUDCABMNUEUGCBIJZUCUGUIRUDCABONPUEADIJZUHUDUJUHRUCDABOQSABCDTUA $.
|
|
|
|
$( Condition for a closed interval to be a subset of a closed-below,
|
|
open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.) $)
|
|
iccssico2 $p |- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) ->
|
|
( C [,] D ) C_ ( A [,) B ) ) $=
|
|
( vx vy vz cico co wcel wa cxr cle wbr clt cv adantr w3a elixx3g simprbi
|
|
cicc wss crab df-ico elmpt2cl1 elmpt2cl2 simpld simprd iccssico syl22anc
|
|
adantl ) CABHIZJZDULJZKALJZBLJZACMNZDBONZCDUAIULUBUMUOUNEFLLEPGPZMNUSFPON
|
|
KGLUCZABHCEFGUDZUEQUMUPUNEFLLUTABHCVAUFQUMUQUNUMUQCBONZUMUOUPCLJRUQVBKEFG
|
|
ABCMOHVASTUGQUNURUMUNADMNZURUNUOUPDLJRVCURKEFGABDMOHVASTUHUKABCDUIUJ $.
|
|
$}
|
|
|
|
$( The open interval from minus to plus infinity. (Contributed by NM,
|
|
6-Feb-2007.) $)
|
|
ioomax $p |- ( -oo (,) +oo ) = RR $=
|
|
( vx cmnf cpnf cioo co cv clt wbr wa crab cxr wcel wceq mnfxr pnfxr iooval2
|
|
cr mp2an rabid2 mnflt ltpnf jca mprgbir eqtr4i ) BCDEZBAFZGHZUFCGHZIZAQJZQB
|
|
KLCKLUEUJMNOABCPRQUJMUIAQUIAQSUFQLUGUHUFTUFUAUBUCUD $.
|
|
|
|
$( The closed interval from minus to plus infinity. (Contributed by Mario
|
|
Carneiro, 4-Jul-2014.) $)
|
|
iccmax $p |- ( -oo [,] +oo ) = RR* $=
|
|
( vx cmnf cpnf cicc co cv cle wbr wa cxr crab wcel mnfxr pnfxr iccval mp2an
|
|
wceq rabid2 mnfle pnfge jca mprgbir eqtr4i ) BCDEZBAFZGHZUECGHZIZAJKZJBJLCJ
|
|
LUDUIQMNABCOPJUIQUHAJUHAJRUEJLUFUGUESUETUAUBUC $.
|
|
|
|
$( The set of positive reals expressed as an open interval. (Contributed by
|
|
NM, 7-May-2007.) $)
|
|
ioopos $p |- ( 0 (,) +oo ) = { x e. RR | 0 < x } $=
|
|
( cc0 cpnf cioo co cv clt wbr wa crab cxr wcel wceq 0xr pnfxr iooval2 mp2an
|
|
cr ltpnf biantrud rabbiia eqtr4i ) BCDEZBAFZGHZUDCGHZIZARJZUEARJBKLCKLUCUHM
|
|
NOABCPQUEUGARUDRLUFUEUDSTUAUB $.
|
|
|
|
$( The set of positive reals expressed as an open interval. (Contributed by
|
|
Steve Rodriguez, 25-Nov-2007.) $)
|
|
ioorp $p |- ( 0 (,) +oo ) = RR+ $=
|
|
( vx cc0 cpnf cioo co cv clt wbr cr crab crp ioopos df-rp eqtr4i ) BCDEBAFG
|
|
HAIJKALAMN $.
|
|
|
|
$( Shift the arguments of the open interval function. (Contributed by NM,
|
|
17-Aug-2008.) $)
|
|
iooshf $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) )
|
|
-> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) ) $=
|
|
( cr wcel wa caddc co clt wbr cioo wb 3expa cxr readdcl rexrd rexr ad2antrl
|
|
elioo5 ltaddsub 3com13 adantrr w3a ltsubadd bicomd adantrl anbi12d ad2ant2l
|
|
cmin ad2ant2rl syl3anc ancoms ad2antll resubcl adantr 3bitr4rd ) AEFZBEFZGZ
|
|
CEFZDEFZGZGZCBHIZAJKZADBHIZJKZGZCABUJIZJKZVJDJKZGZAVEVGLIFZVJCDLIFZVDVFVKVH
|
|
VLUTVAVFVKMZVBURUSVAVPVAUSURVPCBAUAUBNUCUTVBVHVLMZVAURUSVBVQURUSVBUDVLVHABD
|
|
UEUFNUGUHVCUTVNVIMZVCUTGVEOFZVGOFZAOFZVRVAUSVSVBURVAUSGVECBPQUKVBUSVTVAURVB
|
|
USGVGDBPQUIURWAVCUSARSVEVGATULUMVDCOFZDOFZVJOFZVOVMMVAWBUTVBCRSVBWCUTVADRUN
|
|
UTWDVCUTVJABUOQUPCDVJTULUQ $.
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $.
|
|
$( A closed-above interval with real upper bound is a set of reals.
|
|
(Contributed by FL, 29-May-2014.) $)
|
|
iocssre $p |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR ) $=
|
|
( vx cxr wcel cr wa cioc co cv w3a clt wbr cle elioc2 simp1d 3expia ssrdv
|
|
biimp3a ) ADEZBFEZGCABHIZFTUACJZUBEZUCFEZTUAUDKUEAUCLMZUCBNMZTUAUDUEUFUGK
|
|
ABUCOSPQR $.
|
|
|
|
$( A closed-below interval with real lower bound is a set of reals.
|
|
(Contributed by Mario Carneiro, 14-Jun-2014.) $)
|
|
icossre $p |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR ) $=
|
|
( vx cr wcel cxr wa cico co cv w3a cle wbr clt elico2 simp1d 3expia ssrdv
|
|
biimp3a ) ADEZBFEZGCABHIZDTUACJZUBEZUCDEZTUAUDKUEAUCLMZUCBNMZTUAUDUEUFUGK
|
|
ABUCOSPQR $.
|
|
|
|
$( A closed real interval is a set of reals. (Contributed by FL,
|
|
6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.) $)
|
|
iccssre $p |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR ) $=
|
|
( vx cr wcel wa cicc co cv w3a cle wbr elicc2 biimp3a simp1d 3expia ssrdv
|
|
) ADEZBDEZFCABGHZDRSCIZTEZUADEZRSUBJUCAUAKLZUABKLZRSUBUCUDUEJABUAMNOPQ $.
|
|
|
|
$( A closed interval is a set of extended reals. (Contributed by FL,
|
|
28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.) $)
|
|
iccssxr $p |- ( A [,] B ) C_ RR* $=
|
|
( vx vy vz cle cicc df-icc ixxssxr ) CDEABFFGCDEHI $.
|
|
|
|
$( An open-below, closed-above interval is a subset of the extended reals.
|
|
(Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro,
|
|
4-Jul-2014.) $)
|
|
iocssxr $p |- ( A (,] B ) C_ RR* $=
|
|
( vx vy vz clt cle cioc df-ioc ixxssxr ) CDEABFGHCDEIJ $.
|
|
|
|
$( A closed-below, open-above interval is a subset of the extended reals.
|
|
(Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro,
|
|
4-Jul-2014.) $)
|
|
icossxr $p |- ( A [,) B ) C_ RR* $=
|
|
( vx vy vz cle clt cico df-ico ixxssxr ) CDEABFGHCDEIJ $.
|
|
|
|
$( An open interval is a subset of its closure. (Contributed by Paul
|
|
Chapman, 18-Oct-2007.) $)
|
|
ioossicc $p |- ( A (,) B ) C_ ( A [,] B ) $=
|
|
( vx vy vz vw cicc clt cle cioo df-ioo df-icc cv xrltle ixxssixx ) CDEFAB
|
|
GHHIIJCDEKCDELAFMZNPBNO $.
|
|
$}
|
|
|
|
${
|
|
$d a b w x A $. $d a b w x B $. $d a b w x C $. $d a b w x D $.
|
|
$( A closed-below, open-above interval is a subset of its closure.
|
|
(Contributed by Thierry Arnoux, 25-Oct-2016.) $)
|
|
icossicc $p |- ( A [,) B ) C_ ( A [,] B ) $=
|
|
( va vb vx vw cicc cle clt cico df-ico df-icc cxr wcel cv wa wbr ixxssixx
|
|
idd xrltle ) CDEFABGHIHHJCDEKCDELAMNFOZMNPAUAHQSUABTR $.
|
|
|
|
$( A closed-above, open-below interval is a subset of its closure.
|
|
(Contributed by Thierry Arnoux, 1-Apr-2017.) $)
|
|
iocssicc $p |- ( A (,] B ) C_ ( A [,] B ) $=
|
|
( va vb vx vw cicc clt cle cioc df-ioc df-icc cv xrltle cxr wcel ixxssixx
|
|
wa wbr idd ) CDEFABGHIIIJCDEKCDELAFMZNUAOPBOPRUABISTQ $.
|
|
|
|
$( An open interval is a subset of its closure-below. (Contributed by
|
|
Thierry Arnoux, 3-Mar-2017.) $)
|
|
ioossico $p |- ( A (,) B ) C_ ( A [,) B ) $=
|
|
( va vb vx vw cico clt cle cioo df-ioo df-ico cv xrltle cxr wcel ixxssixx
|
|
wa wbr idd ) CDEFABGHHIHJCDEKCDELAFMZNUAOPBOPRUABHSTQ $.
|
|
|
|
$( Condition for a closed interval to be a subset of an open interval.
|
|
(Contributed by Thierry Arnoux, 29-Mar-2017.) $)
|
|
iocssioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A <_ C /\ D < B ) ) -> ( C (,] D ) C_ ( A (,) B ) ) $=
|
|
( va vb vx vw cioc clt cle cioo df-ioo df-ioc cv xrlelttr ixxss12 ) EFGHA
|
|
BCDIJJJKLKJEFGMEFGNACHOZPRDBPQ $.
|
|
|
|
$( Condition for a closed interval to be a subset of an open interval.
|
|
(Contributed by Thierry Arnoux, 29-Mar-2017.) $)
|
|
icossioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A < C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A (,) B ) ) $=
|
|
( va vb vx vw cico clt cle cioo df-ioo df-ico cv xrltletr ixxss12 ) EFGHA
|
|
BCDIJJKJLJKEFGMEFGNACHOZPRDBPQ $.
|
|
|
|
$( Condition for an open interval to be a subset of an open interval.
|
|
(Contributed by Thierry Arnoux, 26-Sep-2017.) $)
|
|
ioossioo $p |- ( ( ( A e. RR* /\ B e. RR* ) /\
|
|
( A <_ C /\ D <_ B ) ) -> ( C (,) D ) C_ ( A (,) B ) ) $=
|
|
( va vb vx vw cioo clt cle df-ioo cv xrlelttr xrltletr ixxss12 ) EFGHABCD
|
|
IJJJJIKKEFGLZQACHMZNRDBOP $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B x y $. $d S x y $.
|
|
$( A nonempty subset of a closed real interval satisfies the conditions for
|
|
the existence of its supremum. To be useful without excluded middle,
|
|
we'll probably need to change not equal to apart, and perhaps make other
|
|
changes, but the theorem does hold as stated here. (Contributed by Paul
|
|
Chapman, 21-Jan-2008.) $)
|
|
iccsupr $p |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) ->
|
|
( S C_ RR /\ S =/= (/) /\ E. x e. RR A. y e. S y <_ x ) ) $=
|
|
( cr wcel wa cicc co wss w3a c0 wne cv cle wbr wral 3adant3 iccssre sylan
|
|
wrex sstr ancoms ne0i 3ad2ant3 simplr elicc2 biimpd sylan9r imp ralrimiva
|
|
ssel simp3d wceq breq2 ralbidv rspcev syl2anc 3jca ) CGHZDGHZIZFCDJKZLZEF
|
|
HZMFGLZFNOZBPZAPZQRZBFSZAGUCZVDVFVHVGVDVEGLZVFVHCDUAVFVOVHFVEGUDUEUBTVGVD
|
|
VIVFFEUFUGVDVFVNVGVDVFIZVCVJDQRZBFSZVNVBVCVFUHVPVQBFVPVJFHZIVJGHZCVJQRZVQ
|
|
VPVSVTWAVQMZVFVSVJVEHZVDWBFVEVJUNVDWCWBCDVJUIUJUKULUOUMVMVRADGVKDUPVLVQBF
|
|
VKDVJQUQURUSUTTVA $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d x B $.
|
|
$( Membership in an unbounded interval of extended reals. (Contributed by
|
|
Mario Carneiro, 18-Jun-2014.) $)
|
|
elioopnf $p |- ( A e. RR* ->
|
|
( B e. ( A (,) +oo ) <-> ( B e. RR /\ A < B ) ) ) $=
|
|
( cxr wcel cpnf cioo co cr clt wbr w3a wa pnfxr elioo2 mpan2 df-3an ltpnf
|
|
wb adantr pm4.71i bitr4i syl6bb ) ACDZBAEFGDZBHDZABIJZBEIJZKZUEUFLZUCECDU
|
|
DUHRMAEBNOUHUIUGLUIUEUFUGPUIUGUEUGUFBQSTUAUB $.
|
|
|
|
$( Membership in an unbounded interval of extended reals. (Contributed by
|
|
Mario Carneiro, 18-Jun-2014.) $)
|
|
elioomnf $p |- ( A e. RR* ->
|
|
( B e. ( -oo (,) A ) <-> ( B e. RR /\ B < A ) ) ) $=
|
|
( cxr wcel cmnf cioo co cr clt wbr w3a wa wb mnfxr elioo2 mpan an32 mnflt
|
|
df-3an adantr pm4.71i 3bitr4i syl6bb ) ACDZBEAFGDZBHDZEBIJZBAIJZKZUFUHLZE
|
|
CDUDUEUIMNEABOPUFUGLUHLUJUGLUIUJUFUGUHQUFUGUHSUJUGUFUGUHBRTUAUBUC $.
|
|
|
|
$( Membership in a closed unbounded interval of reals. (Contributed by
|
|
Mario Carneiro, 16-Sep-2014.) $)
|
|
elicopnf $p |- ( A e. RR ->
|
|
( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B ) ) ) $=
|
|
( cr wcel cpnf cico co cle wbr clt w3a wa cxr wb pnfxr elico2 mpan2 ltpnf
|
|
adantr pm4.71i df-3an bitr4i syl6bbr ) ACDZBAEFGDZBCDZABHIZBEJIZKZUFUGLZU
|
|
DEMDUEUINOAEBPQUJUJUHLUIUJUHUFUHUGBRSTUFUGUHUAUBUC $.
|
|
|
|
$( Two ways of saying that a real number is positive. (Contributed by NM,
|
|
7-May-2007.) $)
|
|
repos $p |- ( A e. ( 0 (,) +oo ) <-> ( A e. RR /\ 0 < A ) ) $=
|
|
( vx cc0 cv clt wbr cr cpnf cioo co breq2 ioopos elrab2 ) CBDZEFCAEFBAGCH
|
|
IJNACEKBLM $.
|
|
|
|
$( The set of open intervals of extended reals maps to subsets of reals.
|
|
(Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro,
|
|
16-Nov-2013.) $)
|
|
ioof $p |- (,) : ( RR* X. RR* ) --> ~P RR $=
|
|
( vx vz vy cv clt wbr wa cxr crab cr cpw wcel wral cxp cioo wf iooval cvv
|
|
co vex wss ioossre cop df-ov iooex opex fvex eqeltri elpw mpbir syl6eqelr
|
|
cfv rgen2a df-ioo fmpt2 mpbi ) ADZBDZEFURCDZEFGBHIZJKZLZCHMAHMHHNVAOPVBAC
|
|
HUQHLUSHLGUTUQUSOSZVABUQUSQVCVALVCJUAUQUSUBVCJVCUQUSUCZOULRUQUSOUDVDORRUE
|
|
UQUSATCTUFUGUHUIUJUKUMACHHUTVAOACBUNUOUP $.
|
|
|
|
$( The set of closed intervals of extended reals maps to subsets of
|
|
extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario
|
|
Carneiro, 3-Nov-2013.) $)
|
|
iccf $p |- [,] : ( RR* X. RR* ) --> ~P RR* $=
|
|
( vx vy vz cle cicc df-icc ixxf ) ABCDDEABCFG $.
|
|
|
|
$( The union of the range of the open interval function. (Contributed by
|
|
NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) $)
|
|
unirnioo $p |- RR = U. ran (,) $=
|
|
( cr cioo crn cuni wcel wss cmnf cpnf co ioomax cxr cxp wfn wf ioof ax-mp
|
|
cpw ffn mnfxr pnfxr fnovrn mp3an eqeltrri elssuni frn sspwuni mpbi eqssi
|
|
) ABCZDZAUIEAUJFGHBIZAUIJBKKLZMZGKEHKEUKUIEULAQZBNZUMOULUNBRPSTKKGHBUAUBU
|
|
CAUIUDPUIUNFZUJAFUOUPOULUNBUEPUIAUFUGUH $.
|
|
|
|
$( Alternate definition of the set of open intervals of extended reals.
|
|
(Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
dfioo2 $p |- (,) = ( x e. RR* , y e. RR* |->
|
|
{ w e. RR | ( x < w /\ w < y ) } ) $=
|
|
( cioo cxr cv co cmpt2 clt wbr wa cr crab cxp cpw wf wfn wceq ioof ffn
|
|
fnovim mp2b iooval2 mpt2eq3ia eqtri ) DABEEAFZBFZDGZHZABEEUFCFZIJUJUGIJKC
|
|
LMZHEENZLOZDPDULQDUIRSULUMDTABEEDUAUBABEEUHUKCUFUGUCUDUE $.
|
|
|
|
$( Open intervals are elements of the set of all open intervals.
|
|
(Contributed by Jim Kingdon, 4-Apr-2020.) $)
|
|
ioorebasg $p |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ran (,) ) $=
|
|
( cioo cxr cxp wfn wcel co crn cr cpw wf ioof ffn ax-mp fnovrn mp3an1 ) C
|
|
DDEZFZADGBDGABCHCIGRJKZCLSMRTCNODDABCPQ $.
|
|
$}
|
|
|
|
$( The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) $)
|
|
elrege0 $p |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) ) $=
|
|
( cc0 cr wcel cpnf cico co cle wbr wa wb 0re elicopnf ax-mp ) BCDABEFGDACDB
|
|
AHIJKLBAMN $.
|
|
|
|
$( Nonnegative real numbers are real numbers. (Contributed by Thierry
|
|
Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.) $)
|
|
rge0ssre $p |- ( 0 [,) +oo ) C_ RR $=
|
|
( vx cc0 cpnf cico co cr cv wcel cle wbr elrege0 simplbi ssriv ) ABCDEZFAGZ
|
|
NHOFHBOIJOKLM $.
|
|
|
|
$( Elementhood in the set of nonnegative extended reals. (Contributed by
|
|
Mario Carneiro, 28-Jun-2014.) $)
|
|
elxrge0 $p |- ( A e. ( 0 [,] +oo ) <-> ( A e. RR* /\ 0 <_ A ) ) $=
|
|
( cxr wcel cc0 cle wbr cpnf w3a wa cicc co df-3an wb 0xr pnfxr elicc1 mp2an
|
|
pnfge adantr pm4.71i 3bitr4i ) ABCZDAEFZAGEFZHZUBUCIZUDIADGJKCZUFUBUCUDLDBC
|
|
GBCUGUEMNODGAPQUFUDUBUDUCARSTUA $.
|
|
|
|
$( 0 is a member of ` ( 0 [,) +oo ) ` (common case). (Contributed by David
|
|
A. Wheeler, 8-Dec-2018.) $)
|
|
0e0icopnf $p |- 0 e. ( 0 [,) +oo ) $=
|
|
( cc0 cpnf cico co wcel cr cle wbr 0re 0le0 elrege0 mpbir2an ) AABCDEAFEAAG
|
|
HIJAKL $.
|
|
|
|
$( 0 is a member of ` ( 0 [,] +oo ) ` (common case). (Contributed by David
|
|
A. Wheeler, 8-Dec-2018.) $)
|
|
0e0iccpnf $p |- 0 e. ( 0 [,] +oo ) $=
|
|
( cc0 cpnf cicc co wcel cxr cle wbr 0xr 0le0 elxrge0 mpbir2an ) AABCDEAFEAA
|
|
GHIJAKL $.
|
|
|
|
$( The nonnegative reals are closed under addition. (Contributed by Mario
|
|
Carneiro, 19-Jun-2014.) $)
|
|
ge0addcl $p |- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) ->
|
|
( A + B ) e. ( 0 [,) +oo ) ) $=
|
|
( cc0 cpnf cico co wcel cr cle wbr wa caddc elrege0 readdcl ad2ant2r addge0
|
|
an4s sylanbrc syl2anb ) ACDEFZGAHGZCAIJZKZBHGZCBIJZKZABLFZTGZBTGAMBMUCUFKUG
|
|
HGZCUGIJZUHUAUDUIUBUEABNOUAUDUBUEUJABPQUGMRS $.
|
|
|
|
$( The nonnegative reals are closed under multiplication. (Contributed by
|
|
Mario Carneiro, 19-Jun-2014.) $)
|
|
ge0mulcl $p |- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) ->
|
|
( A x. B ) e. ( 0 [,) +oo ) ) $=
|
|
( cc0 cpnf cico co wcel cr cle wbr wa cmul elrege0 ad2ant2r mulge0 sylanbrc
|
|
remulcl syl2anb ) ACDEFZGAHGZCAIJZKZBHGZCBIJZKZABLFZSGZBSGAMBMUBUEKUFHGZCUF
|
|
IJUGTUCUHUAUDABQNABOUFMPR $.
|
|
|
|
$( The lower bound of a closed interval is a member of it. (Contributed by
|
|
Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by
|
|
Mario Carneiro, 9-Sep-2015.) $)
|
|
lbicc2 $p |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) ) $=
|
|
( cxr wcel cle wbr w3a cicc co simp1 xrleid 3ad2ant1 simp3 elicc1 mpbir3and
|
|
wb 3adant3 ) ACDZBCDZABEFZGAABHIDZRAAEFZTRSTJRSUBTAKLRSTMRSUARUBTGPTABANQO
|
|
$.
|
|
|
|
$( The upper bound of a closed interval is a member of it. (Contributed by
|
|
Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) $)
|
|
ubicc2 $p |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) ) $=
|
|
( cxr wcel cle wbr w3a cicc co simp2 simp3 xrleid 3ad2ant2 elicc1 mpbir3and
|
|
wb 3adant3 ) ACDZBCDZABEFZGBABHIDZSTBBEFZRSTJRSTKSRUBTBLMRSUASTUBGPTABBNQO
|
|
$.
|
|
|
|
$( Zero is an element of the closed unit. (Contributed by Scott Fenton,
|
|
11-Jun-2013.) $)
|
|
0elunit $p |- 0 e. ( 0 [,] 1 ) $=
|
|
( cc0 c1 cicc co wcel cr cle wbr 0re 0le0 0le1 1re elicc2i mpbir3an ) AABCD
|
|
EAFEAAGHABGHIJKABAILMN $.
|
|
|
|
$( One is an element of the closed unit. (Contributed by Scott Fenton,
|
|
11-Jun-2013.) $)
|
|
1elunit $p |- 1 e. ( 0 [,] 1 ) $=
|
|
( c1 cc0 cicc co wcel cr cle wbr 1re 0le1 1le1 0re elicc2i mpbir3an ) ABACD
|
|
EAFEBAGHAAGHIJKBAALIMN $.
|
|
|
|
$( Membership in a negated open real interval. (Contributed by Paul Chapman,
|
|
26-Nov-2007.) $)
|
|
iooneg $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) ) $=
|
|
( cr wcel w3a clt wbr wa cneg cioo co ltneg 3adant2 cxr rexr elioo5 renegcl
|
|
wb syl3an ancoms 3adant1 anbi12d ancom syl6bb 3com12 3bitr4d ) ADEZBDEZCDEZ
|
|
FZACGHZCBGHZIZBJZCJZGHZUPAJZGHZIZCABKLEZUPUOURKLEZUKUNUSUQIUTUKULUSUMUQUHUJ
|
|
ULUSSUIACMNUIUJUMUQSZUHUJUIVCCBMUAUBUCUSUQUDUEUHAOEUIBOEUJCOEVAUNSAPBPCPABC
|
|
QTUIUHUJVBUTSZUIUODEZUHURDEZUJUPDEZVDBRARCRVEUOOEVFUROEVGUPOEVDUOPURPUPPUOU
|
|
RUPQTTUFUG $.
|
|
|
|
$( Membership in a negated closed real interval. (Contributed by Paul
|
|
Chapman, 26-Nov-2007.) $)
|
|
iccneg $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( C e. ( A [,] B ) <-> -u C e. ( -u B [,] -u A ) ) ) $=
|
|
( cr wcel w3a cle wbr wa cneg cicc co renegcl anbi12d elicc2 3adant3 3anass
|
|
wb leneg syl6bb ax-1 impbid2 3ad2ant3 ancom 3adant1 3adant2 syl5bbr syl2anr
|
|
ancoms 3bitr4d ) ADEZBDEZCDEZFZUMACGHZCBGHZIZIZCJZDEZBJZUSGHZUSAJZGHZIZIZCA
|
|
BKLEZUSVAVCKLEZUNUMUTUQVEUMUKUMUTRULUMUMUTCMUMUTUAUBUCUQUPUOIUNVEUPUOUDUNUP
|
|
VBUOVDULUMUPVBRZUKUMULVICBSUIUEUKUMUOVDRULACSUFNUGNUNVGUMUOUPFZURUKULVGVJRU
|
|
MABCOPUMUOUPQTUNVHUTVBVDFZVFUKULVHVKRZUMULVADEVCDEVLUKBMAMVAVCUSOUHPUTVBVDQ
|
|
TUJ $.
|
|
|
|
$( A shifted real is a member of a shifted, closed-below, open-above real
|
|
interval. (Contributed by Paul Chapman, 25-Mar-2008.) $)
|
|
icoshft $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) ->
|
|
( X + C ) e. ( ( A + C ) [,) ( B + C ) ) ) ) $=
|
|
( cr wcel w3a cico co cle wbr clt wa caddc wi cxr wb rexr elico2 readdcl
|
|
sylan2 biimpd 3adant3 3anass syl6ib leadd1 3com12 3expib 3adant2 imp ltadd1
|
|
com12 3adant1 anbi12d pm5.32da expcom syl6ibr 3ad2ant3 biimprd syl2anc syld
|
|
anim1d sylbid ) AEFZBEFZCEFZGZDABHIFZDEFZADJKZDBLKZMZMZDCNIZACNIZBCNIZHIFZV
|
|
GVHVIVJVKGZVMVDVEVHVROVFVDVEMVHVRVEVDBPFVHVRQBRABDSUAUBUCVIVJVKUDUEVGVMVIVO
|
|
VNJKZVNVPLKZMZMZVQVGVIVLWAVGVIMVJVSVKVTVGVIVJVSQZVDVFVIWCOVEVIVDVFMWCVIVDVF
|
|
WCVDVIVFWCADCUFUGUHULUIUJVGVIVKVTQZVEVFVIWDOVDVIVEVFMWDVIVEVFWDDBCUKUHULUMU
|
|
JUNUOVGWBVNEFZVSVTGZVQVFVDWBWFOVEVFWBWEWAMWFVFVIWEWAVIVFWEDCTUPVBWEVSVTUDUQ
|
|
URVGVOEFZVPEFZWFVQOVDVFWGVEACTUIVEVFWHVDBCTUMWGWHMVQWFWHWGVPPFVQWFQVPRVOVPV
|
|
NSUAUSUTVAVCVA $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d y F $.
|
|
icoshftf1o.1 $e |- F = ( x e. ( A [,) B ) |-> ( x + C ) ) $.
|
|
$( Shifting a closed-below, open-above interval is one-to-one onto.
|
|
(Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario
|
|
Carneiro, 1-Sep-2015.) $)
|
|
icoshftf1o $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) ) $=
|
|
( vy cr wcel cv caddc co cico wral wceq wreu icoshft cmin recnd negsubd
|
|
wf1o ralrimiv wa cneg wi readdcl 3adant2 3adant1 renegcl 3ad2ant3 syl3anc
|
|
w3a imp cxr rexrd icossre syl2anc sselda simpl3 simp3 simp1 eqtrd oveq12d
|
|
wss pncand simp2 adantr 3eltr3d reueq sylib simpll3 simpl1 subadd2d eqcom
|
|
simpl2 3bitr4g reubidva mpbid ralrimiva f1ompt sylanbrc ) BHIZCHIZDHIZULZ
|
|
AJZDKLZBDKLZCDKLZMLZIZABCMLZNGJZWGOZAWLPZGWJNWLWJEUAWEWKAWLBCDWFQUBWEWOGW
|
|
JWEWMWJIZUCZWFWMDRLZOZAWLPZWOWQWRWLIWTWQWMDUDZKLZWHXAKLZWIXAKLZMLZWRWLWEW
|
|
PXBXEIZWEWHHIZWIHIZXAHIZWPXFUEWBWDXGWCBDUFUGZWCWDXHWBCDUFUHZWDWBXIWCDUIUJ
|
|
WHWIXAWMQUKUMWQWMDWQWMWEWJHWMWEXGWIUNIWJHVDXJWEWIXKUOWHWIUPUQURZSWQDWBWCW
|
|
DWPUSSTWEXEWLOWPWEXCBXDCMWEXCWHDRLBWEWHDWEWHXJSWEDWBWCWDUTSZTWEBDWEBWBWCW
|
|
DVASXMVEVBWEXDWIDRLCWEWIDWEWIXKSXMTWECDWECWBWCWDVFSXMVEVBVCVGVHAWLWRVIVJW
|
|
QWSWNAWLWQWFWLIZUCZWRWFOWGWMOWSWNXOWMDWFXOWMWQWMHIXNXLVGSXODWBWCWDWPXNVKS
|
|
XOWFWQWLHWFWQWBCUNIWLHVDWBWCWDWPVLWQCWBWCWDWPVOUOBCUPUQURSVMWFWRVNWMWGVNV
|
|
PVQVRVSAGWLWJWGEFVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $. $d w x y z D $.
|
|
$( End-to-end closed-below, open-above real intervals are disjoint.
|
|
(Contributed by Mario Carneiro, 16-Jun-2014.) $)
|
|
icodisj $p |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) ->
|
|
( ( A [,) B ) i^i ( B [,) C ) ) = (/) ) $=
|
|
( vx cxr wcel w3a cico co cin c0 wss wceq wa clt wbr cle wb elico1 biimpa
|
|
cv 3adant3 simp3d adantrr wn 3adant1 simp2d simpl2 simp1d xrlenlt syl2anc
|
|
elin mpbid adantrl pm2.65da pm2.21d syl5bi ssrdv ss0 syl ) AEFZBEFZCEFZGZ
|
|
ABHIZBCHIZJZKLVGKMVDDVGKDUAZVGFVHVEFZVHVFFZNZVDVHKFZVHVEVFULVDVKVLVDVKVHB
|
|
OPZVDVIVMVJVDVINVHEFZAVHQPZVMVDVIVNVOVMGZVAVBVIVPRVCABVHSUBTUCUDVDVJVMUEZ
|
|
VIVDVJNZBVHQPZVQVRVNVSVHCOPZVDVJVNVSVTGZVBVCVJWARVABCVHSUFTZUGVRVBVNVSVQR
|
|
VAVBVCVJUHVRVNVSVTWBUIBVHUJUKUMUNUOUPUQURVGUSUT $.
|
|
|
|
$( If the upper bound of one open interval is less than or equal to the
|
|
lower bound of the other, the intervals are disjoint. (Contributed by
|
|
Jeff Hankins, 13-Jul-2009.) $)
|
|
ioodisj $p |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) )
|
|
/\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) ) $=
|
|
( vx vy vz vw cxr wcel wa cle cioo co cin c0 wss wceq cicc syl wbr iooss1
|
|
simpllr sylancom ioossicc syl6ss sslin simplll simplrr clt df-ioo xrlenlt
|
|
df-icc cv ixxdisj syl3anc sseqtrd ss0 ) AIJZBIJZKZCIJZDIJZKZKZBCLUAZKZABM
|
|
NZCDMNZOZPQVJPRVGVJVHBDSNZOZPVGVIVKQVJVLQVGVIBDMNZVKVEVFUTVIVMQUSUTVDVFUC
|
|
ZBCDUBUDBDUEUFVIVKVHUGTVGUSUTVCVLPRUSUTVDVFUHVNVAVBVCVFUIEFGHABDSUJUJLLME
|
|
FGUKEFGUMBHUNULUOUPUQVJURT $.
|
|
|
|
$}
|
|
|
|
${
|
|
iccshftr.1 $e |- ( A + R ) = C $.
|
|
iccshftr.2 $e |- ( B + R ) = D $.
|
|
$( Membership in a shifted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccshftr $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) )
|
|
-> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) ) $=
|
|
( cr wcel wa cle wbr w3a caddc co cicc wb readdcl leadd1 simpl 2thd 3expb
|
|
adantl adantlr breq1i syl6bb an12s adantll breq2i 3anbi123d elicc2 adantr
|
|
syl5eqelr syl2an anandirs adantrl 3bitr4d ) AIJZBIJZKZFIJZEIJZKZKZVBAFLMZ
|
|
FBLMZNZFEOPZIJZCVILMZVIDLMZNZFABQPJZVICDQPJZVEVBVJVFVKVGVLVDVBVJRVAVDVBVJ
|
|
VBVCUAFESUBUDVEVFAEOPZVILMZVKUSVDVFVQRZUTUSVBVCVRAFETUCUEVPCVILGUFUGVEVGV
|
|
IBEOPZLMZVLUTVDVGVTRZUSVBUTVCWAVBUTVCWAFBETUCUHUIVSDVILHUJUGUKVAVNVHRVDAB
|
|
FULUMVAVCVOVMRZVBUSUTVCWBUSVCKZCIJDIJWBUTVCKZWCCVPIGAESUNWDDVSIHBESUNCDVI
|
|
ULUOUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
iccshftri.1 $e |- A e. RR $.
|
|
iccshftri.2 $e |- B e. RR $.
|
|
iccshftri.3 $e |- R e. RR $.
|
|
iccshftri.4 $e |- ( A + R ) = C $.
|
|
iccshftri.5 $e |- ( B + R ) = D $.
|
|
$( Membership in a shifted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccshftri $p |- ( X e. ( A [,] B ) -> ( X + R ) e. ( C [,] D ) ) $=
|
|
( cr wcel cicc co caddc wss iccssre mp2an sseli wb iccshftr mpanl12 mpan2
|
|
wa biimpd mpcom ) FLMZFABNOZMZFEPOCDNOMZUILFALMZBLMZUILQGHABRSTUHUJUKUHEL
|
|
MZUJUKUAZIULUMUHUNUEUOGHABCDEFJKUBUCUDUFUG $.
|
|
$}
|
|
|
|
${
|
|
iccshftl.1 $e |- ( A - R ) = C $.
|
|
iccshftl.2 $e |- ( B - R ) = D $.
|
|
$( Membership in a shifted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccshftl $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) )
|
|
-> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) ) $=
|
|
( cr wcel wa cle wbr w3a cmin co cicc wb resubcl lesub1 simpl 2thd adantl
|
|
3expb adantlr breq1i syl6bb adantll breq2i 3anbi123d elicc2 adantr syl2an
|
|
an12s syl5eqelr anandirs adantrl 3bitr4d ) AIJZBIJZKZFIJZEIJZKZKZVBAFLMZF
|
|
BLMZNZFEOPZIJZCVILMZVIDLMZNZFABQPJZVICDQPJZVEVBVJVFVKVGVLVDVBVJRVAVDVBVJV
|
|
BVCUAFESUBUCVEVFAEOPZVILMZVKUSVDVFVQRZUTUSVBVCVRAFETUDUEVPCVILGUFUGVEVGVI
|
|
BEOPZLMZVLUTVDVGVTRZUSVBUTVCWAVBUTVCWAFBETUDUNUHVSDVILHUIUGUJVAVNVHRVDABF
|
|
UKULVAVCVOVMRZVBUSUTVCWBUSVCKZCIJDIJWBUTVCKZWCCVPIGAESUOWDDVSIHBESUOCDVIU
|
|
KUMUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
iccshftli.1 $e |- A e. RR $.
|
|
iccshftli.2 $e |- B e. RR $.
|
|
iccshftli.3 $e |- R e. RR $.
|
|
iccshftli.4 $e |- ( A - R ) = C $.
|
|
iccshftli.5 $e |- ( B - R ) = D $.
|
|
$( Membership in a shifted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccshftli $p |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) ) $=
|
|
( cr wcel cicc co cmin wss iccssre mp2an sseli wb iccshftl mpanl12 biimpd
|
|
wa mpan2 mpcom ) FLMZFABNOZMZFEPOCDNOMZUILFALMZBLMZUILQGHABRSTUHUJUKUHELM
|
|
ZUJUKUAZIULUMUHUNUEUOGHABCDEFJKUBUCUFUDUG $.
|
|
$}
|
|
|
|
${
|
|
iccdil.1 $e |- ( A x. R ) = C $.
|
|
iccdil.2 $e |- ( B x. R ) = D $.
|
|
$( Membership in a dilated interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccdil $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) )
|
|
-> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) ) $=
|
|
( cr wcel wa cle wbr w3a cmul co cicc wb remulcl sylan2 simpl rpre adantl
|
|
crp 2thd cc0 clt elrp lemul1 syl3an3b 3expb adantlr breq1i syl6bb adantll
|
|
breq2i 3anbi123d elicc2 adantr syl5eqelr syl2an anandirs adantrl 3bitr4d
|
|
an12s ) AIJZBIJZKZFIJZEUDJZKZKZVIAFLMZFBLMZNZFEOPZIJZCVPLMZVPDLMZNZFABQPJ
|
|
ZVPCDQPJZVLVIVQVMVRVNVSVKVIVQRVHVKVIVQVIVJUAVJVIEIJZVQEUBZFESTUEUCVLVMAEO
|
|
PZVPLMZVRVFVKVMWFRZVGVFVIVJWGVJVFVIWCUFEUGMKZWGEUHZAFEUIUJUKULWECVPLGUMUN
|
|
VLVNVPBEOPZLMZVSVGVKVNWKRZVFVIVGVJWLVIVGVJWLVJVIVGWHWLWIFBEUIUJUKVEUOWJDV
|
|
PLHUPUNUQVHWAVORVKABFURUSVHVJWBVTRZVIVJVHWCWMWDVFVGWCWMVFWCKZCIJDIJWMVGWC
|
|
KZWNCWEIGAESUTWODWJIHBESUTCDVPURVAVBTVCVD $.
|
|
$}
|
|
|
|
${
|
|
iccdili.1 $e |- A e. RR $.
|
|
iccdili.2 $e |- B e. RR $.
|
|
iccdili.3 $e |- R e. RR+ $.
|
|
iccdili.4 $e |- ( A x. R ) = C $.
|
|
iccdili.5 $e |- ( B x. R ) = D $.
|
|
$( Membership in a dilated interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
iccdili $p |- ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) ) $=
|
|
( cr wcel cicc co cmul wss iccssre mp2an sseli wb wa iccdil mpanl12 mpan2
|
|
crp biimpd mpcom ) FLMZFABNOZMZFEPOCDNOMZUJLFALMZBLMZUJLQGHABRSTUIUKULUIE
|
|
UFMZUKULUAZIUMUNUIUOUBUPGHABCDEFJKUCUDUEUGUH $.
|
|
$}
|
|
|
|
${
|
|
icccntr.1 $e |- ( A / R ) = C $.
|
|
icccntr.2 $e |- ( B / R ) = D $.
|
|
$( Membership in a contracted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
icccntr $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) )
|
|
-> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) ) $=
|
|
( cr wcel wa cle wbr w3a cdiv co cicc wb rerpdivcl lediv1 crp 2thd adantl
|
|
simpl cc0 clt elrp syl3an3b 3expb adantlr breq1i syl6bb adantll 3anbi123d
|
|
an12s breq2i elicc2 adantr syl5eqelr syl2an anandirs adantrl 3bitr4d ) AI
|
|
JZBIJZKZFIJZEUAJZKZKZVGAFLMZFBLMZNZFEOPZIJZCVNLMZVNDLMZNZFABQPJZVNCDQPJZV
|
|
JVGVOVKVPVLVQVIVGVORVFVIVGVOVGVHUDFESUBUCVJVKAEOPZVNLMZVPVDVIVKWBRZVEVDVG
|
|
VHWCVHVDVGEIJUEEUFMKZWCEUGZAFETUHUIUJWACVNLGUKULVJVLVNBEOPZLMZVQVEVIVLWGR
|
|
ZVDVGVEVHWHVGVEVHWHVHVGVEWDWHWEFBETUHUIUOUMWFDVNLHUPULUNVFVSVMRVIABFUQURV
|
|
FVHVTVRRZVGVDVEVHWIVDVHKZCIJDIJWIVEVHKZWJCWAIGAESUSWKDWFIHBESUSCDVNUQUTVA
|
|
VBVC $.
|
|
$}
|
|
|
|
${
|
|
icccntri.1 $e |- A e. RR $.
|
|
icccntri.2 $e |- B e. RR $.
|
|
icccntri.3 $e |- R e. RR+ $.
|
|
icccntri.4 $e |- ( A / R ) = C $.
|
|
icccntri.5 $e |- ( B / R ) = D $.
|
|
$( Membership in a contracted interval. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
icccntri $p |- ( X e. ( A [,] B ) -> ( X / R ) e. ( C [,] D ) ) $=
|
|
( cr wcel cicc co cdiv wss iccssre mp2an sseli crp icccntr mpanl12 biimpd
|
|
wb wa mpan2 mpcom ) FLMZFABNOZMZFEPOCDNOMZUJLFALMZBLMZUJLQGHABRSTUIUKULUI
|
|
EUAMZUKULUEZIUMUNUIUOUFUPGHABCDEFJKUBUCUGUDUH $.
|
|
$}
|
|
|
|
$( A condition for a ratio to be a member of the closed unit. (Contributed
|
|
by Scott Fenton, 11-Jun-2013.) $)
|
|
divelunit $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) ->
|
|
( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) ) $=
|
|
( cdiv co cc0 c1 cicc wcel cr cle wbr wa clt w3a 0re 1re elicc2i bitri cmul
|
|
df-3an wb ledivmul mp3an2 simpll simprl cap gt0ap0 adantl redivclapd divge0
|
|
adantlr jca biantrurd cc recn ad2antrl mulid1d breq2d 3bitr3d syl5bb ) ABCD
|
|
ZEFGDHZVAIHZEVAJKZLZVAFJKZLZAIHZEAJKZLZBIHZEBMKZLZLZABJKZVBVCVDVFNVGEFVAOPQ
|
|
VCVDVFTRVNVFABFSDZJKZVGVOVHVMVFVQUAZVIVHFIHVMVRPAFBUBUCUKVNVEVFVNVCVDVNABVH
|
|
VIVMUDVJVKVLUEVMBEUFKVJBUGUHUIABUJULUMVNVPBAJVNBVKBUNHVJVLBUOUPUQURUSUT $.
|
|
|
|
$( A linear combination of two reals which lies in the interval between them.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario
|
|
Carneiro, 8-Sep-2015.) $)
|
|
lincmb01cmp $p |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) )
|
|
-> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) ) $=
|
|
( cr wcel wbr cc0 c1 cicc co cmin cmul caddc wb 0re a1i 1re cle eqid recnd
|
|
clt w3a wa simpr elicc2i simp1bi adantl difrp biimp3a adantr syl22anc mpbid
|
|
crp iccdil simpl2 simpl1 resubcld mulid2d oveq12d eleqtrd remulcld iccshftr
|
|
mul02d mulcld subadd23d subdid oveq1d resubcl sylancr addcomd subdird eqtrd
|
|
1cnd oveq2d 3eqtr4d addid2d npcand 3eltr3d ) ADEZBDEZABUAFZUBZCGHIJEZUCZCBA
|
|
KJZLJZAMJZGAMJZWEAMJZIJZHCKJZALJZCBLJZMJZABIJWDWFGWEIJZEZWGWJEZWDWFGWELJZHW
|
|
ELJZIJZWOWDWCWFWTEZWBWCUDWDGDEZHDEZCDEZWEUMEZWCXANXBWDOPZXCWDQPWCXDWBWCXDGC
|
|
RFCHRFGHCOQUEUFUGZWBXEWCVSVTWAXEABUHUIUJGHWRWSWECWRSWSSUNUKULWDWRGWSWEIWDWE
|
|
WDWEWDBAVSVTWAWCUOZVSVTWAWCUPZUQZTZVCWDWEXKURUSUTWDXBWEDEWFDEVSWPWQNXFXJWDC
|
|
WEXGXJVAXIGWEWHWIAWFWHSWISVBUKULWDWMCALJZKJZAMJWMAXLKJZMJZWGWNWDWMXLAWDCBWD
|
|
CXGTZWDBXHTZVDZWDCAXPWDAXITZVDXSVEWDWFXMAMWDCBAXPXQXSVFVGWDWNWMWLMJXOWDWLWM
|
|
WDWLWDWKAWDXCXDWKDEQXGHCVHVIXIVATXRVJWDWLXNWMMWDWLHALJZXLKJXNWDHCAWDVMXPXSV
|
|
KWDXTAXLKWDAXSURVGVLVNVLVOWDWHAWIBIWDAXSVPWDBAXQXSVQUSVR $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
iccf1o.1 $e |- F = ( x e. ( 0 [,] 1 ) |->
|
|
( ( x x. B ) + ( ( 1 - x ) x. A ) ) ) $.
|
|
$( Describe a bijection from ` [ 0 , 1 ] ` to an arbitrary nontrivial
|
|
closed interval ` [ A , B ] ` . (Contributed by Mario Carneiro,
|
|
8-Sep-2015.) $)
|
|
iccf1o $p |- ( ( A e. RR /\ B e. RR /\ A < B ) ->
|
|
( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) /\ `' F =
|
|
( y e. ( A [,] B ) |-> ( ( y - A ) / ( B - A ) ) ) ) ) $=
|
|
( cr wcel wbr cc0 c1 cicc co cmul cmin caddc recnd cc wceq adantrl clt cv
|
|
w3a cdiv wa cle 0re 1re elicc2i simp1bi adantl simpl2 mulcld ax-1cn subcl
|
|
sylancr simpl1 addcomd lincmb01cmp eqeltrd simpr wb elicc2 3adant3 biimpa
|
|
simp1d eqid iccshftl syl22anc mpbid resubcld crp difrp biimp3a adantr cap
|
|
rpcnd rpap0 syl divcanap1d mul02d subidd eqtr4d mulid2d oveq12d 0red 1red
|
|
3eltr4d rerpdivcld iccdil mpbird eqcom adantrr divmulap3d syl5bb remulcld
|
|
subadd2d syl6bb subadd23d subdid oveq1d 1cnd subdird eqtrd oveq2d 3eqtr4d
|
|
eqeq2d 3bitrd f1ocnv2d ) CGHZDGHZCDUAIZUCZABJKLMZCDLMZAUBZDNMZKXPOMZCNMZP
|
|
MZBUBZCOMZDCOMZUDMZEFXMXPXNHZUEZXTXSXQPMXOYFXQXSYFXPDYFXPYEXPGHZXMYEYGJXP
|
|
UFIXPKUFIJKXPUGUHUIUJUKZQZYFDXJXKXLYEULZQZUMZYFXRCYFKRHXPRHZXRRHUNYIKXPUO
|
|
UPYFCXJXKXLYEUQZQZUMURCDXPUSUTXMYAXOHZUEZYDXNHZYDYCNMZJYCNMZKYCNMZLMZHZYQ
|
|
YBCCOMZYCLMZYSUUBYQYPYBUUEHZXMYPVAYQXJXKYAGHZXJYPUUFVBXJXKXLYPUQZXJXKXLYP
|
|
ULYQUUGCYAUFIZYADUFIZXMYPUUGUUIUUJUCZXJXKYPUUKVBXLCDYAVCVDVEVFZUUHCDUUDYC
|
|
CYAUUDVGYCVGVHVIVJYQYBYCYQYBYQYACUULUUHVKZQZYQYCXMYCVLHZYPXJXKXLUUOCDVMVN
|
|
VOZVQZYQUUOYCJVPIZUUPYCVRVSZVTYQYTUUDUUAYCLYQYTJUUDYQYCUUQWAYQCYQCUUHQZWB
|
|
WCYQYCUUQWDWEWHYQJGHKGHYDGHUUOYRUUCVBYQWFYQWGYQYBYCUUMUUPWIUUPJKYTUUAYCYD
|
|
YTVGUUAVGWJVIWKXMYEYPUEUEZXPYDSZYBXPYCNMZSZYAUVCCPMZSZYAXTSUVBYDXPSUVAUVD
|
|
XPYDWLUVAYBXPYCXMYPYBRHYEUUNTXMYEYMYPYIWMXMYPYCRHYEUUQTXMYPUURYEUUSTWNWOU
|
|
VAUVDUVEYASUVFUVAYACUVCUVAYAXMYPUUGYEUULTQXMYPCRHYEUUTTUVAUVCXMYEUVCGHYPY
|
|
FXPYCYHYFDCYJYNVKWPWMQWQUVEYAWLWRUVAUVEXTYAXMYEUVEXTSYPYFXQXPCNMZOMZCPMXQ
|
|
CUVGOMZPMUVEXTYFXQUVGCYLYFXPCYIYOUMYOWSYFUVCUVHCPYFXPDCYIYKYOWTXAYFXSUVIX
|
|
QPYFXSKCNMZUVGOMUVIYFKXPCYFXBYIYOXCYFUVJCUVGOYFCYOWDXAXDXEXFWMXGXHXI $.
|
|
$}
|
|
|
|
$( ` ( 0 [,] 1 ) ` is a subset of the reals. (Contributed by David Moews,
|
|
28-Feb-2017.) $)
|
|
unitssre $p |- ( 0 [,] 1 ) C_ RR $=
|
|
( cc0 cr wcel c1 cicc co wss 0re 1re iccssre mp2an ) ABCDBCADEFBGHIADJK $.
|
|
|
|
$( The sum of an integer and a real number between 0 and 1 is less than or
|
|
equal to a second integer iff the sum is less than the second integer.
|
|
(Contributed by AV, 1-Jul-2021.) $)
|
|
zltaddlt1le $p |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) )
|
|
-> ( ( M + A ) < N <-> ( M + A ) <_ N ) ) $=
|
|
( cz wcel cc0 c1 co w3a caddc clt wbr cle cr wi zre cxr simplbiim 3ad2ant3
|
|
wa cioo adantr elioore adantl readdcld 3adant2 3ad2ant2 syl2anc crp elioo3g
|
|
ltle simpl elrpd addlelt syl3an zltp1le 3adant3 1red 3ad2ant1 simpr peano2z
|
|
wb ltadd2dd zred ltletr syl3anc mpand sylbid syld impbid ) BDEZCDEZAFGUAHEZ
|
|
IZBAJHZCKLZVOCMLZVNVONEZCNEZVPVQOVKVMVRVLVKVMTBAVKBNEZVMBPZUBVMANEZVKAFGUCZ
|
|
UDUEUFZVLVKVSVMCPZUGZVOCUKUHVNVQBCKLZVPVKVTVLVSVMAUIEVQWGOWAWEVMAWCVMFQEGQE
|
|
AQEIZFAKLZAGKLZTZWIFGAUJZWIWJULRUMABCUNUOVNWGBGJHZCMLZVPVKVLWGWNVBVMBCUPUQV
|
|
NVOWMKLZWNVPVNAGBVMVKWBVLWCSVNURVKVLVTVMWAUSVMVKWJVLVMWHWKWJWLWIWJUTRSVCVNV
|
|
RWMNEZVSWOWNTVPOWDVKVLWPVMVKWMBVAVDUSWFVOWMCVEVFVGVHVIVJ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite intervals of integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ... $. $( Dots. $)
|
|
|
|
$( Extend class notation to include the notation for a contiguous finite set
|
|
of integers. Read " ` M ... N ` " as "the set of integers from ` M ` to
|
|
` N ` inclusive." $)
|
|
cfz $a class ... $.
|
|
|
|
${
|
|
$d m n k $.
|
|
$( Define an operation that produces a finite set of sequential integers.
|
|
Read " ` M ... N ` " as "the set of integers from ` M ` to ` N `
|
|
inclusive." See ~ fzval for its value and additional comments.
|
|
(Contributed by NM, 6-Sep-2005.) $)
|
|
df-fz $a |- ... = ( m e. ZZ , n e. ZZ |->
|
|
{ k e. ZZ | ( m <_ k /\ k <_ n ) } ) $.
|
|
$}
|
|
|
|
${
|
|
$d k m n M $. $d k m n N $.
|
|
$( The value of a finite set of sequential integers. E.g., ` 2 ... 5 `
|
|
means the set ` { 2 , 3 , 4 , 5 } ` . A special case of this definition
|
|
(starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where
|
|
` NN ` _k means our ` 1 ... k ` ; he calls these sets _segments_ of the
|
|
integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
|
|
3-Nov-2013.) $)
|
|
fzval $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) =
|
|
{ k e. ZZ | ( M <_ k /\ k <_ N ) } ) $=
|
|
( vm vn cz cv cle wbr wa crab cfz breq1 anbi1d rabbidv breq2 anbi2d df-fz
|
|
wceq zex rabex ovmpt2 ) DEBCFFDGZAGZHIZUDEGZHIZJZAFKBUDHIZUDCHIZJZAFKLUIU
|
|
GJZAFKUCBSZUHULAFUMUEUIUGUCBUDHMNOUFCSZULUKAFUNUGUJUIUFCUDHPQOADERUKAFTUA
|
|
UB $.
|
|
|
|
$( An alternate way of expressing a finite set of sequential integers.
|
|
(Contributed by Mario Carneiro, 3-Nov-2013.) $)
|
|
fzval2 $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M ... N ) = ( ( M [,] N ) i^i ZZ ) ) $=
|
|
( vk cz wcel wa cfz co cv cle wbr crab cicc cin fzval wceq cr zssre sseli
|
|
cxr ressxr sstri iccval syl2an ineq1d inrab2 wss sseqin2 mpbi rabeq ax-mp
|
|
eqtri syl6req eqtrd ) ADEZBDEZFZABGHACIZJKURBJKFZCDLZABMHZDNZCABOUQVBUSCT
|
|
LZDNZUTUQVAVCDUOATEBTEVAVCPUPDTADQTRUAUBZSDTBVESCABUCUDUEVDUSCTDNZLZUTUSC
|
|
TDUFVFDPZVGUTPDTUGVHVEDTUHUIUSCVFDUJUKULUMUN $.
|
|
|
|
$( Establish the domain and codomain of the finite integer sequence
|
|
function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario
|
|
Carneiro, 16-Nov-2013.) $)
|
|
fzf $p |- ... : ( ZZ X. ZZ ) --> ~P ZZ $=
|
|
( vm vk vn cv cle wbr wa cz crab cpw wcel cxp cfz wf wss ssrab2 zex elpw2
|
|
wral mpbir rgen2w df-fz fmpt2 mpbi ) ADBDZEFUECDEFGZBHIZHJZKZCHSAHSHHLUHM
|
|
NUIACHHUIUGHOUFBHPUGHQRTUAACHHUGUHMBACUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d j K $. $d j M $. $d j N $.
|
|
$( Membership in a finite set of sequential integers. (Contributed by NM,
|
|
21-Jul-2005.) $)
|
|
elfz1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <->
|
|
( K e. ZZ /\ M <_ K /\ K <_ N ) ) ) $=
|
|
( vj cz wcel wa cfz co cv cle wbr crab w3a fzval wceq breq2 breq1 anbi12d
|
|
eleq2d elrab 3anass bitr4i syl6bb ) BEFCEFGZABCHIZFABDJZKLZUGCKLZGZDEMZFZ
|
|
AEFZBAKLZACKLZNZUEUFUKADBCOTULUMUNUOGZGUPUJUQDAEUGAPUHUNUIUOUGABKQUGACKRS
|
|
UAUMUNUOUBUCUD $.
|
|
$}
|
|
|
|
$( Membership in a finite set of sequential integers. (Contributed by NM,
|
|
29-Sep-2005.) $)
|
|
elfz $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <->
|
|
( M <_ K /\ K <_ N ) ) ) $=
|
|
( cz wcel cfz co cle wbr wa wb w3a elfz1 3anass baib sylan9bb 3impa 3comr )
|
|
BDEZCDEZADEZABCFGEZBAHIZACHIZJZKZSTUAUFSTJUBUAUCUDLZUAUEABCMUGUAUEUAUCUDNOP
|
|
QR $.
|
|
|
|
${
|
|
$d x y z M $. $d z N $.
|
|
$( Membership in a finite set of sequential integers. We use the fact that
|
|
an operation's value is empty outside of its domain to show ` M e. ZZ `
|
|
and ` N e. ZZ ` . (Contributed by NM, 6-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfz2 $p |- ( K e. ( M ... N ) <->
|
|
( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) ) $=
|
|
( vx vy vz cz wcel wa cle wbr w3a cfz co anass df-3an anbi1i crab df-fz
|
|
cv elmpt2cl simpl elfz1 3anass ibar syl5bb bitrd pm5.21nii 3bitr4ri ) BGH
|
|
ZCGHZIZAGHZIZBAJKZACJKZIZIULUMUQIZIZUJUKUMLZUQIABCMNHZULUMUQOUTUNUQUJUKUM
|
|
PQVAULUSDEGGDTFTZJKVBETJKIFGRBCMAFDESUAULURUBULVAUMUOUPLZUSABCUCVCURULUSU
|
|
MUOUPUDULURUEUFUGUHUI $.
|
|
$}
|
|
|
|
$( Membership in a finite set of sequential integers. (Contributed by NM,
|
|
26-Dec-2005.) $)
|
|
elfz5 $p |- ( ( K e. ( ZZ>= ` M ) /\ N e. ZZ ) ->
|
|
( K e. ( M ... N ) <-> K <_ N ) ) $=
|
|
( cuz cfv wcel cz wa cfz co cle wbr wb eluzelz eluzel2 jca elfz 3expa sylan
|
|
eluzle biantrurd adantr bitr4d ) ABDEFZCGFZHABCIJFZBAKLZACKLZHZUHUDAGFZBGFZ
|
|
HUEUFUIMZUDUJUKBANBAOPUJUKUEULABCQRSUDUHUIMUEUDUGUHBATUAUBUC $.
|
|
|
|
$( Membership in a finite set of sequential integers. (Contributed by NM,
|
|
21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
elfz4 $p |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) )
|
|
-> K e. ( M ... N ) ) $=
|
|
( cfz co wcel cz w3a cle wbr wa elfz2 biimpri ) ABCDEFBGFCGFAGFHBAIJACIJKKA
|
|
BCLM $.
|
|
|
|
$( Membership in a finite set of sequential integers in terms of sets of
|
|
upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzuzb $p |- ( K e. ( M ... N ) <->
|
|
( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) ) $=
|
|
( cz wcel w3a cle wbr wa cfz cuz cfv df-3an an6 anandir ancom anbi2i 3bitri
|
|
co eluz2 anbi1i 3bitr4ri elfz2 anbi12i 3bitr4i ) BDEZCDEZADEZFZBAGHZACGHZIZ
|
|
IZUFUHUJFZUHUGUKFZIZABCJSEABKLEZCAKLEZIUFUHIZUHUGIZULFUSUTIZULIUPUMUSUTULMU
|
|
FUHUJUHUGUKNUIVAULUIUFUGIUHIUSUGUHIZIVAUFUGUHMUFUGUHOVBUTUSUGUHPQRUAUBABCUC
|
|
UQUNURUOBATACTUDUE $.
|
|
|
|
$( Membership in a finite set of sequential integers. (Contributed by NM,
|
|
4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
eluzfz $p |- ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) )
|
|
-> K e. ( M ... N ) ) $=
|
|
( cfz co wcel cuz cfv wa elfzuzb biimpri ) ABCDEFABGHFCAGHFIABCJK $.
|
|
|
|
$( A member of a finite set of sequential integers belongs to an upper set of
|
|
integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
elfzuz $p |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) ) $=
|
|
( cfz co wcel cuz cfv elfzuzb simplbi ) ABCDEFABGHFCAGHFABCIJ $.
|
|
|
|
$( Membership in a finite set of sequential integers implies membership in an
|
|
upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by
|
|
Mario Carneiro, 28-Apr-2015.) $)
|
|
elfzuz3 $p |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) ) $=
|
|
( cfz co wcel cuz cfv elfzuzb simprbi ) ABCDEFABGHFCAGHFABCIJ $.
|
|
|
|
$( Membership in a finite set of sequential integer implies the upper bound
|
|
is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzel2 $p |- ( K e. ( M ... N ) -> N e. ZZ ) $=
|
|
( cfz co wcel cuz cfv cz elfzuz3 eluzelz syl ) ABCDEFCAGHFCIFABCJACKL $.
|
|
|
|
$( Membership in a finite set of sequential integer implies the lower bound
|
|
is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzel1 $p |- ( K e. ( M ... N ) -> M e. ZZ ) $=
|
|
( cfz co wcel cuz cfv cz elfzuz eluzel2 syl ) ABCDEFABGHFBIFABCJBAKL $.
|
|
|
|
$( A member of a finite set of sequential integer is an integer.
|
|
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
elfzelz $p |- ( K e. ( M ... N ) -> K e. ZZ ) $=
|
|
( cfz co wcel cuz cfv cz elfzuz eluzelz syl ) ABCDEFABGHFAIFABCJBAKL $.
|
|
|
|
$( A member of a finite set of sequential integer is greater than or equal to
|
|
the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzle1 $p |- ( K e. ( M ... N ) -> M <_ K ) $=
|
|
( cfz co wcel cuz cfv cle wbr elfzuz eluzle syl ) ABCDEFABGHFBAIJABCKBALM
|
|
$.
|
|
|
|
$( A member of a finite set of sequential integer is less than or equal to
|
|
the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzle2 $p |- ( K e. ( M ... N ) -> K <_ N ) $=
|
|
( cfz co wcel cuz cfv cle wbr elfzuz3 eluzle syl ) ABCDEFCAGHFACIJABCKACLM
|
|
$.
|
|
|
|
$( Implication of membership in a finite set of sequential integers.
|
|
(Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
elfzuz2 $p |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) ) $=
|
|
( cfz co wcel cuz cfv wa elfzuzb eqid uztrn2 sylbi ) ABCDEFABGHZFCAGHFICNFA
|
|
BCJBCANNKLM $.
|
|
|
|
$( Membership in a finite set of sequential integer implies the bounds are
|
|
comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfzle3 $p |- ( K e. ( M ... N ) -> M <_ N ) $=
|
|
( cfz co wcel cuz cfv cle wbr elfzuz2 eluzle syl ) ABCDEFCBGHFBCIJABCKBCLM
|
|
$.
|
|
|
|
$( Membership in a finite set of sequential integers - special case.
|
|
(Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
eluzfz1 $p |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) ) $=
|
|
( cuz cfv wcel cfz co cz eluzel2 uzid syl eluzfz mpancom ) AACDZEZBNEZAABFG
|
|
EPAHEOABIAJKAABLM $.
|
|
|
|
$( Membership in a finite set of sequential integers - special case.
|
|
(Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
eluzfz2 $p |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) ) $=
|
|
( cuz cfv wcel cfz co cz eluzelz uzid syl eluzfz mpdan ) BACDEZBBCDEZBABFGE
|
|
NBHEOABIBJKBABLM $.
|
|
|
|
$( Membership in a finite set of sequential integers - special case.
|
|
(Contributed by NM, 14-Sep-2005.) $)
|
|
eluzfz2b $p |- ( N e. ( ZZ>= ` M ) <-> N e. ( M ... N ) ) $=
|
|
( cuz cfv wcel cfz co eluzfz2 elfzuz impbii ) BACDEBABFGEABHBABIJ $.
|
|
|
|
$( Membership in a finite set of sequential integers containing one integer.
|
|
(Contributed by NM, 21-Jul-2005.) $)
|
|
elfz3 $p |- ( N e. ZZ -> N e. ( N ... N ) ) $=
|
|
( cz wcel cuz cfv cfz co uzid eluzfz1 syl ) ABCAADECAAAFGCAHAAIJ $.
|
|
|
|
$( Membership in a finite set of sequential integers containing one integer.
|
|
(Contributed by NM, 19-Sep-2005.) $)
|
|
elfz1eq $p |- ( K e. ( N ... N ) -> K = N ) $=
|
|
( cfz co wcel wceq cle wbr elfzle2 elfzle1 cz wa elfzelz elfzel2 zre letri3
|
|
wb cr syl2an syl2anc mpbir2and ) ABBCDEZABFZABGHZBAGHZABBIABBJUBAKEZBKEZUCU
|
|
DUELQZABBMABBNUFAREBREUHUGAOBOABPSTUA $.
|
|
|
|
$( If there is a member in a finite set of sequential integers, the upper
|
|
bound is also a member of this finite set of sequential integers.
|
|
(Contributed by Alexander van der Vekens, 31-May-2018.) $)
|
|
elfzubelfz $p |- ( K e. ( M ... N ) -> N e. ( M ... N ) ) $=
|
|
( cfz co wcel cuz cfv elfzuz2 eluzfz2 syl ) ABCDEZFCBGHFCLFABCIBCJK $.
|
|
|
|
$( A Peano-postulate-like theorem for downward closure of a finite set of
|
|
sequential integers. (Contributed by Mario Carneiro, 27-May-2014.) $)
|
|
peano2fzr $p |- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M ... N ) )
|
|
-> K e. ( M ... N ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfz wa simpl cz eluzelz elfzuz3 peano2uzr syl2an
|
|
elfzuzb sylanbrc ) ABDEFZAGHIZBCJIZFZKTCADEFZAUBFTUCLTAMFCUADEFUDUCBANUABCO
|
|
ACPQABCRS $.
|
|
|
|
${
|
|
$d M x $. $d N x $.
|
|
$( Properties of a finite interval of integers which is inhabited.
|
|
(Contributed by Jim Kingdon, 15-Apr-2020.) $)
|
|
fzm $p |- ( E. x x e. ( M ... N ) <-> N e. ( ZZ>= ` M ) ) $=
|
|
( cv cfz co wcel wex cuz cfv elfzuz2 exlimiv eluzfz1 elex2 syl impbii ) A
|
|
DZBCEFZGZAHZCBIJGZSUAAQBCKLUABRGTBCMABRNOP $.
|
|
$}
|
|
|
|
$( Trichotomy in terms of a finite interval of integers. (Contributed by Jim
|
|
Kingdon, 1-Jun-2020.) $)
|
|
fztri3or $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( K < M \/ K e. ( M ... N ) \/ N < K ) ) $=
|
|
( cz wcel clt wbr wn adantl wa cle simpr zred lenltd mpbird adantr wo zdclt
|
|
wdc df-dc w3a cfz co 3mix1 3mix3 simpl2 simpl1 cr simpll3 wb elfz mpbir2and
|
|
w3o 3mix2d ancoms 3adant2 sylib mpjaodan 3adant3 ) ADEZBDEZCDEZUAZABFGZVDAB
|
|
CUBUCEZCAFGZUMZVDHZVDVGVCVDVEVFUDIVCVHJZVFVGVFHZVFVGVIVFVDVEUEIVIVJJZVEVDVF
|
|
VKVEBAKGZACKGZVIVLVJVIVLVHVCVHLVIBAVIBUTVAVBVHUFMVIAUTVAVBVHUGMZNOPVKVMVJVI
|
|
VJLVKACVIAUHEVJVNPVKCUTVAVBVHVJUIMNOVIVEVLVMJUJZVJVCVOVHABCUKPPULUNVIVFSZVF
|
|
VJQVCVPVHUTVBVPVAVBUTVPCARUOUPPVFTUQURVCVDSZVDVHQUTVAVQVBABRUSVDTUQUR $.
|
|
|
|
$( Decidability of membership in a finite interval of integers. (Contributed
|
|
by Jim Kingdon, 1-Jun-2020.) $)
|
|
fzdcel $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
DECID K e. ( M ... N ) ) $=
|
|
( cz wcel w3a clt wbr cfz co w3o wn cle wb zltnle con3i syl6bi nsyld sylibr
|
|
syl6 wdc fztri3or wa 3adant3 simpl elfz biimpd wo olc df-dc orc a1i 3adant2
|
|
wi ancoms simpr 3jaod mpd ) ADEZBDEZCDEZFZABGHZABCIJEZCAGHZKVDUAZABCUBVBVCV
|
|
FVDVEVBVCVDLZVFVBVCBAMHZACMHZUCZVDVBVCVHLZVJLZUSUTVCVKNVAABOUDVJVHVHVIUEPQV
|
|
BVDVJABCUFUGZRVGVDVGUHZVFVGVDUIVDUJZSZTVDVFUNVBVDVNVFVDVGUKVOSULVBVEVGVFVBV
|
|
EVJVDVBVEVILZVLUSVAVEVQNZUTVAUSVRCAOUOUMVJVIVHVIUPPQVMRVPTUQUR $.
|
|
|
|
${
|
|
$d K x k m $. $d M x k m $. $d N x k m $.
|
|
$( A finite set of sequential integers is empty if the bounds are reversed.
|
|
(Contributed by Jim Kingdon, 16-Apr-2020.) $)
|
|
fznlem $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( N < M -> ( M ... N ) = (/) ) ) $=
|
|
( vk cz wcel wa clt wbr cfz co c0 wceq cv cle wn cr wb zre adantr zred wi
|
|
crab wral lenlt syl2an biimpd con2d imp simplll simpllr letr syl3anc mtod
|
|
simpr ralrimiva rabeq0 sylibr fzval eqeq1d mpbird ex ) ADEZBDEZFZBAGHZABI
|
|
JZKLZVDVEFZVGACMZNHVIBNHFZCDUBZKLZVHVJOZCDUCVLVHVMCDVHVIDEZFZVJABNHZVHVPO
|
|
ZVNVDVEVQVDVPVEVDVPVEOZVBAPEZBPEZVPVRQVCARBRABUDUEUFUGUHSVOVSVIPEVTVJVPUA
|
|
VOAVBVCVEVNUITVOVIVHVNUNTVOBVBVCVEVNUJTAVIBUKULUMUOVJCDUPUQVDVGVLQVEVDVFV
|
|
KKCABURUSSUTVA $.
|
|
|
|
$( A finite set of sequential integers is empty if the bounds are reversed.
|
|
(Contributed by NM, 22-Aug-2005.) $)
|
|
fzn $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) ) $=
|
|
( vx cz wcel wa clt wbr cfz co c0 wceq fznlem cle wn cv wex neq0r adantr
|
|
wb simpr nsyl3 cuz cfv eluz syl5rbb mtbird zltnle ancoms mpbird ex impbid
|
|
fzm ) ADEZBDEZFZBAGHZABIJZKLZABMUPUSUQUPUSFZUQABNHZOZUTVACPURECQZVCUSUTCU
|
|
RRUPUSUAUBUPVAVCTUSVCBAUCUDEUPVACABUMABUEUFSUGUPUQVBTZUSUOUNVDBAUHUISUJUK
|
|
UL $.
|
|
|
|
$( A shifted finite set of sequential integers is equinumerous to the
|
|
original set. (Contributed by Paul Chapman, 11-Apr-2009.) $)
|
|
fzen $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) ->
|
|
( M ... N ) ~~ ( ( M + K ) ... ( N + K ) ) ) $=
|
|
( cz wcel w3a cfz co caddc cle wbr wi wa elfz1 biimpd cr zre syl3an wceq
|
|
wb vk vm cv cmin cvv cxp wfn cpw wf fzf ax-mp fnovex mp3an1 3adant3 simp1
|
|
simp3 zaddcld syl2anc zaddcl expcom 3ad2ant3 adantrd leadd1 3com23 3expia
|
|
simp2 impd 3adant2 adantld 3coml 3adant1 3jcad biimprd com12 3impb zsubcl
|
|
ffn syld leaddsub lesubadd ancoms imp simp1d ex zcn subadd 3bitr3g addcom
|
|
cc eqcom eqeq2d bitrd 3expib syl2and en3d ) BDEZCDEZADEZFZUAUBBCGHZBAIHZC
|
|
AIHZGHZUAUCZAIHZUBUCZAUDHZWPWQWTUEEZWRGDDUFZUGZWPWQXHXIDUHZGUIXJUJXIXKGVQ
|
|
UKZBCDDGULUMUNWSXADEZXBDEZXCUEEZWSBAWPWQWRUOWPWQWRUPZUQWSCAWPWQWRVFXPUQXJ
|
|
XMXNXOXLXAXBDDGULUMURWSXDWTEZXDDEZBXDJKZXDCJKZFZXEXCEZWPWQXQYALWRWPWQMZXQ
|
|
YAXDBCNOUNZYAWSYBXRXSXTWSYBLWSXRXSXTMZMZYBWSYFXEDEZXAXEJKZXEXBJKZFZYBWSYF
|
|
YGYHYIWSXRYGYEWRWPXRYGLWQXRWRYGXDAUSUTVAVBWPWRYFYHLWQWPWRMZXRYEYHWPWRXRYE
|
|
YHLZWPXRWRYLWPXRWRFZXSYHXTYMXSYHWPBPEZXRXDPEZWRAPEZXSYHTBQZXDQZAQZBXDAVCR
|
|
OVBVDVEVGVHWQWRYFYILWPWQWRMXRYEYIWQWRXRYEYILZXRWQWRYTXRWQWRFZXTYIXSUUAXTY
|
|
IXRYOWQCPEZWRYPXTYITYRCQZYSXDCAVCROVIVJVEVGVKVLWSYBYJWSXMXNYBYJTWPWRXMWQB
|
|
AUSVHZWQWRXNWPCAUSVKZXEXAXBNURVMVRVNVOVNVRWSXFXCEZXFDEZXAXFJKZXFXBJKZFZXG
|
|
WTEZWSUUFUUJWSXMXNUUFUUJTUUDUUEXFXAXBNUROZUUJWSUUKUUGUUHUUIWSUUKLWSUUGUUH
|
|
UUIMZMZUUKWSUUNXGDEZBXGJKZXGCJKZFZUUKWSUUNUUOUUPUUQWSUUGUUOUUMWRWPUUGUUOL
|
|
WQUUGWRUUOXFAVPUTVAVBWPWRUUNUUPLWQYKUUGUUMUUPWPWRUUGUUMUUPLWPWRUUGFZUUHUU
|
|
PUUIUUSUUHUUPWPYNWRYPUUGXFPEZUUHUUPTYQYSXFQZBAXFVSROVBVEVGVHWQWRUUNUUQLZW
|
|
PWRWQUVBWRWQMUUGUUMUUQWRWQUUGUUMUUQLZUUGWRWQUVCUUGWRWQFZUUIUUQUUHUVDUUQUU
|
|
IUUGUUTWRYPWQUUBUUQUUITUVAYSUUCXFACVTRVMVIVJVEVGWAVKVLWPWQUURUUKLWRYCUUKU
|
|
URXGBCNVMUNVRVNVOVNVRWSXQXRUUFUUGXDXGSZXFXESZTZWSXQXRWSXQMXRXSXTWSXQYAYDW
|
|
BWCWDWSUUFUUGWSUUFMUUGUUHUUIWSUUFUUJUULWBWCWDWRWPXRUUGMUVGLWQWRXRUUGUVGUU
|
|
GWRXRUVGUUGXFWIEZWRAWIEZXRXDWIEZUVGXFWEAWEXDWEUVHUVIUVJFZUVEXFAXDIHZSZUVF
|
|
UVKXGXDSUVLXFSUVEUVMXFAXDWFXGXDWJUVLXFWJWGUVKUVLXEXFUVIUVJUVLXESUVHAXDWHV
|
|
KWKWLRVJWMVAWNWO $.
|
|
$}
|
|
|
|
$( A 1-based finite set of sequential integers is empty iff it ends at index
|
|
` 0 ` . (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
fz1n $p |- ( N e. NN0 -> ( ( 1 ... N ) = (/) <-> N = 0 ) ) $=
|
|
( cn0 wcel c1 clt wbr cfz co c0 wceq cc0 cz wb 1z nn0z fzn sylancr nn0lt10b
|
|
bitr3d ) ABCZADEFZDAGHIJZAKJTDLCALCUAUBMNAODAPQARS $.
|
|
|
|
$( Two ways to say a finite 1-based sequence is empty. (Contributed by Paul
|
|
Chapman, 26-Oct-2012.) $)
|
|
0fz1 $p |- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( F = (/) <-> N = 0 ) ) $=
|
|
( c1 cfz co wfn c0 wceq cn0 wcel cc0 fndmu sylan2br ex fneq2 syl6bb biimpcd
|
|
fn0 impbid fz1n sylan9bbr ) ACBDEZFZAGHZUBGHZBIJBKHUCUDUEUCUDUEUDUCAGFZUEAR
|
|
ZUBGALMNUEUCUDUEUCUFUDUBGAOUGPQSBTUA $.
|
|
|
|
$( There are no integers between 1 and 0. (Contributed by Jeff Madsen,
|
|
16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) $)
|
|
fz10 $p |- ( 1 ... 0 ) = (/) $=
|
|
( cc0 c1 clt wbr cfz co c0 wceq 0lt1 cz wcel wb 1z 0z fzn mp2an mpbi ) ABCD
|
|
ZBAEFGHZIBJKAJKRSLMNBAOPQ $.
|
|
|
|
$( Membership of an integer greater than L decreased by ( L - M ) in an M
|
|
based finite set of sequential integers. (Contributed by Alexander van
|
|
der Vekens, 14-Sep-2018.) $)
|
|
uzsubsubfz $p |- ( ( L e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` L ) )
|
|
-> ( N - ( L - M ) ) e. ( M ... N ) ) $=
|
|
( cuz wcel cmin co cz cle wbr w3a wi wa adantr cr zre adantl syl2anr cc zcn
|
|
cfv cfz eluz2 simpr zsubcl adantlr zsubcld 3jca 3adant3 com12 imp caddc cc0
|
|
subge0d exbiri com23 3impia impcom resubcl addge02d mpbid 3ad2ant2 3ad2ant1
|
|
ex subsubd breqtrrd subge0 imp31 subge02d jca elfz2 sylanbrc 3adant2 syl5bi
|
|
wb sylbi ) ABDUAEZCADUAEZCABFGZFGZBCUBGEZVQBHEZAHEZBAIJZKZVRWALBAUCVRWCCHEZ
|
|
ACIJZKZWEWAACUCWBWDWHWALWCWBWDMZWHWAWIWHMZWBWFVTHEZKZBVTIJZVTCIJZMWAWIWHWLW
|
|
BWHWLLWDWHWBWLWCWFWBWLLWGWCWFMZWBWLWOWBMZWBWFWKWOWBUDWOWFWBWCWFUDNZWPCVSWQW
|
|
CWBVSHEWFABUEUFUGUHVDUIUJNUKWJWMWNWJBCAFGZBULGZVTIWJUMWRIJZBWSIJWHWIWTWCWFW
|
|
GWIWTLWOWIWGWTWOWIWTWGWOWIMCAWOCOEZWIWFXAWCCPZQNWOAOEZWIWCXCWFAPZNNUNUOUPUQ
|
|
URWJBWRWIBOEZWHWBXEWDBPZNZNWHWROEZWIWCWFXHWGWFXAXCXHWCXBXDCAUSRUIQUTVAWJCAB
|
|
WHCSEZWIWFWCXIWGCTVBQWHASEZWIWCWFXJWGATVCQWIBSEZWHWBXKWDBTNNVEVFWJUMVSIJZWN
|
|
WBWDWHXLWBWHWDXLWBWHXLWDWHXCXEXLWDVOWBWCWFXCWGXDVCZXFABVGRUOUPVHWJCVSWHXAWI
|
|
WFWCXAWGXBVBQWHXCXEVSOEWIXMXGABUSRVIVAVJVTBCVKVLVDVMVNVPUK $.
|
|
|
|
$( Membership of an integer greater than L decreased by ( L - 1 ) in a 1
|
|
based finite set of sequential integers. (Contributed by Alexander van
|
|
der Vekens, 14-Sep-2018.) $)
|
|
uzsubsubfz1 $p |- ( ( L e. NN /\ N e. ( ZZ>= ` L ) )
|
|
-> ( N - ( L - 1 ) ) e. ( 1 ... N ) ) $=
|
|
( cn wcel c1 cuz cfv cmin co cfz elnnuz uzsubsubfz sylanb ) ACDAEFGDBAFGDBA
|
|
EHIHIEBJIDAKAEBLM $.
|
|
|
|
$( Membership of an integer greater than 2 decreased by 2 in a 1 based finite
|
|
set of sequential integers. (Contributed by Alexander van der Vekens,
|
|
14-Sep-2018.) $)
|
|
ige3m2fz $p |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... N ) ) $=
|
|
( c3 cuz cfv wcel c2 cmin co c1 cfz wceq 3m1e2 eqcomi oveq2d cn uzsubsubfz1
|
|
a1i 3nn mpan eqeltrd ) ABCDEZAFGHABIGHZGHZIAJHZUAFUBAGFUBKUAUBFLMQNBOEUAUCU
|
|
DERBAPST $.
|
|
|
|
${
|
|
$d M x $. $d N x $. $d K x $.
|
|
$( Split a finite interval of integers into two parts. (Contributed by
|
|
Mario Carneiro, 13-Apr-2016.) $)
|
|
fzsplit2 $p |- ( ( ( K + 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) ->
|
|
( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) ) $=
|
|
( vx co cuz cfv wcel wa cfz wo cle wbr cz adantl syl2anr wb elfzuz syl2an
|
|
elfzuz3 c1 caddc cun cv clt elfzelz eluzel2 zlelttric elfz5 simpl eluzelz
|
|
syl eluz elfzuzb rbaib zltp1le 3bitr4d mpbird simpr uztrn sylanbrc jaodan
|
|
orbi12d impbida elun syl6bbr eqrdv ) AUAUBEZBFGZHZCAFGHZIZDBCJEZBAJEZVHCJ
|
|
EZUCZVLDUDZVMHZVQVNHZVQVOHZKZVQVPHVLVRWAVLVRIZWAVQALMZAVQUEMZKZVRVQNHZANH
|
|
ZWEVLVQBCUFZVKWGVJACUGOZVQAUHPWBVSWCVTWDVRVQVIHZWGVSWCQVLVQBCRWIVQBAUIPWB
|
|
VQVHFGHZVHVQLMZVTWDVLVHNHZWFWKWLQVRVLVJWMVJVKUJZBVHUKULWHVHVQUMSWBCVQFGZH
|
|
ZVTWKQVRWPVLVQBCTOVTWKWPVQVHCUNUOULVLWGWFWDWLQVRWIWHAVQUPSUQVCURVLVSVRVTV
|
|
LVSIWJWPVRVSWJVLVQBAROVLVKAWOHWPVSVJVKUSVQBATACVQUTSVQBCUNZVAVLVTIWJWPVRV
|
|
TWKVJWJVLVQVHCRWNVHVQBUTPVTWPVLVQVHCTOWQVAVBVDVQVNVOVEVFVG $.
|
|
|
|
$( Split a finite interval of integers into two parts. (Contributed by
|
|
Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.) $)
|
|
fzsplit $p |- ( K e. ( M ... N ) ->
|
|
( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) ) $=
|
|
( cfz co wcel caddc cuz cfv cun wceq elfzuz peano2uz syl elfzuz3 fzsplit2
|
|
c1 syl2anc ) ABCDEZFZAQGEZBHIZFZCAHIFSBADEUACDEJKTAUBFUCABCLBAMNABCOABCPR
|
|
$.
|
|
|
|
$d J x $.
|
|
$( Condition for two finite intervals of integers to be disjoint.
|
|
(Contributed by Jeff Madsen, 17-Jun-2010.) $)
|
|
fzdisj $p |- ( K < M -> ( ( J ... K ) i^i ( M ... N ) ) = (/) ) $=
|
|
( vx clt wbr cfz co cin cv wcel wa wn elin cle cz adantl zred adantr cr
|
|
elfzel1 elfzelz elfzel2 elfzle1 elfzle2 letrd lenltd mpbid sylbi eq0rdv
|
|
con2i ) BCFGZEABHIZCDHIZJZEKZUPLZUMURUQUNLZUQUOLZMZUMNZUQUNUOOVACBPGVBVAC
|
|
UQBVACUTCQLUSUQCDUBRSZUTUQUALUSUTUQUQCDUCSRVABUSBQLUTUQABUDTSZUTCUQPGUSUQ
|
|
CDUERUSUQBPGUTUQABUFTUGVACBVCVDUHUIUJULUK $.
|
|
$}
|
|
|
|
$( 0-based and 1-based finite sets of sequential integers are equinumerous.
|
|
(Contributed by Paul Chapman, 11-Apr-2009.) $)
|
|
fz01en $p |- ( N e. ZZ -> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) ) $=
|
|
( cz wcel cc0 c1 cmin co cfz caddc cen wbr peano2zm 0z 1z fzen mp3an13 wceq
|
|
syl 0p1e1 cc a1i zcn ax-1cn npcan sylancl oveq12d breqtrd ) ABCZDAEFGZHGZDE
|
|
IGZUIEIGZHGZEAHGJUHUIBCZUJUMJKZALDBCUNEBCUOMNEDUIOPRUHUKEULAHUKEQUHSUAUHATC
|
|
ETCULAQAUBUCAEUDUEUFUG $.
|
|
|
|
$( A member of a finite set of sequential integers starting at 1 is a
|
|
positive integer. (Contributed by NM, 24-Aug-2005.) $)
|
|
elfznn $p |- ( K e. ( 1 ... N ) -> K e. NN ) $=
|
|
( c1 cfz co wcel cz cle wbr cn elfzelz elfzle1 elnnz1 sylanbrc ) ACBDEFAGFC
|
|
AHIAJFACBKACBLAMN $.
|
|
|
|
$( A nonempty finite range of integers contains its end point. (Contributed
|
|
by Stefan O'Rear, 10-Oct-2014.) $)
|
|
elfz1end $p |- ( A e. NN <-> A e. ( 1 ... A ) ) $=
|
|
( cn wcel c1 cfz co cuz cfv elnnuz biimpi cz nnz uzid eluzfz syl2anc elfznn
|
|
syl impbii ) ABCZADAEFCZSADGHCZAAGHCZTSUAAIJSAKCUBALAMQADANOAAPR $.
|
|
|
|
$( Subtraction closure for a member of a finite set of sequential integers.
|
|
(Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fznn0sub $p |- ( K e. ( M ... N ) -> ( N - K ) e. NN0 ) $=
|
|
( cfz co wcel cuz cfv cmin cn0 elfzuz3 uznn0sub syl ) ABCDEFCAGHFCAIEJFABCK
|
|
ACLM $.
|
|
|
|
$( Subtracting the difference of a member of a finite range of integers and
|
|
the lower bound of the range from the difference of the upper bound and
|
|
the lower bound of the range results in the difference of the upper bound
|
|
of the range and the member. (Contributed by Alexander van der Vekens,
|
|
27-May-2018.) $)
|
|
fzmmmeqm $p |- ( M e. ( L ... N )
|
|
-> ( ( N - L ) - ( M - L ) ) = ( N - M ) ) $=
|
|
( cfz co wcel cc w3a cmin wceq cz cle wa elfz2 3anim123i 3comr adantr sylbi
|
|
wbr zcn nnncan2 syl ) BACDEFZCGFZBGFZAGFZHZCAIEBAIEIECBIEJUCAKFZCKFZBKFZHZA
|
|
BLSBCLSMZMUGBACNUKUGULUIUJUHUGUIUDUJUEUHUFCTBTATOPQRCBAUAUB $.
|
|
|
|
$( Membership of a sum in a finite set of sequential integers. (Contributed
|
|
by NM, 30-Jul-2005.) $)
|
|
fzaddel $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) ->
|
|
( J e. ( M ... N ) <-> ( J + K ) e. ( ( M + K ) ... ( N + K ) ) ) ) $=
|
|
( cz wcel wa cle wbr w3a caddc co cfz wb zaddcl cr zre leadd1 syl3an 3expb
|
|
simpl adantl adantlr 3com12 adantll 3anbi123d elfz1 adantr anandirs adantrl
|
|
2thd syl2an 3bitr4d ) CEFZDEFZGZAEFZBEFZGZGZUQCAHIZADHIZJZABKLZEFZCBKLZVDHI
|
|
ZVDDBKLZHIZJZACDMLFZVDVFVHMLFZUTUQVEVAVGVBVIUSUQVENUPUSUQVEUQURUAABOUKUBUNU
|
|
SVAVGNZUOUNUQURVMUNCPFUQAPFZURBPFZVMCQAQZBQZCABRSTUCUOUSVBVINZUNUOUQURVRUQU
|
|
OURVRUQVNUODPFURVOVRVPDQVQADBRSUDTUEUFUPVKVCNUSACDUGUHUPURVLVJNZUQUNUOURVSU
|
|
NURGVFEFVHEFVSUOURGCBODBOVDVFVHUGULUIUJUM $.
|
|
|
|
$( Membership of a difference in a finite set of sequential integers.
|
|
(Contributed by NM, 30-Jul-2005.) $)
|
|
fzsubel $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) ->
|
|
( J e. ( M ... N ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) ) $=
|
|
( cz wcel wa cfz co cneg caddc cmin wb znegcl fzaddel cc zcn anim12i negsub
|
|
wceq sylanr2 adantl oveqan12d anandirs adantrl eleq12d syl2an bitrd ) CEFZD
|
|
EFZGZAEFZBEFZGZGACDHIFZABJZKIZCUPKIZDUPKIZHIZFZABLIZCBLIZDBLIZHIZFZUMUKULUP
|
|
EFUOVAMBNAUPCDOUAUKCPFZDPFZGZAPFZBPFZGZVAVFMUNUIVGUJVHCQDQRULVJUMVKAQBQRVIV
|
|
LGUQVBUTVEVLUQVBTVIABSUBVIVKUTVETZVJVGVHVKVMVGVKGVHVKGURVCUSVDHCBSDBSUCUDUE
|
|
UFUGUH $.
|
|
|
|
$( A finite set of sequential integers can represent an ordered pair.
|
|
(Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fzopth $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) ) $=
|
|
( cuz cfv wcel cfz co wceq wa wss eluzfz1 adantr eleqtrd uzss 3syl eleqtrrd
|
|
elfzuz eqssd simpr elfzuz2 cz wb eluzel2 uz11 mpbid eluzfz2 elfzuz3 eluzelz
|
|
syl jca ex oveq12 impbid1 ) DCEFZGZCDHIZABHIZJZCAJZDBJZKZUQUTVCUQUTKZVAVBVD
|
|
UPAEFZJZVAVDUPVEVDCUSGZCVEGUPVELVDCURUSUQCURGUTCDMNUQUTUAZOZCABSACPQVDAURGA
|
|
UPGVEUPLVDAUSURVDVGBVEGZAUSGVICABUBZABMQVHRACDSCAPQTVDCUCGZVFVAUDUQVLUTCDUE
|
|
NCAUFUKUGVDDEFZBEFZJZVBVDVMVNVDBURGDVNGVMVNLVDBUSURVDVGVJBUSGVIVKABUHQVHRBC
|
|
DUIBDPQVDDUSGBVMGVNVMLVDDURUSUQDURGUTCDUHNVHODABUIDBPQTVDDUCGZVOVBUDUQVPUTC
|
|
DUJNDBUFUKUGULUMCADBHUNUO $.
|
|
|
|
$( Two ways to express a nondecreasing sequence of four integers.
|
|
(Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
fzass4 $p |- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <->
|
|
( B e. ( A ... C ) /\ C e. ( A ... D ) ) ) $=
|
|
( cuz cfv wcel wa cfz co simpll simprl uztrn ancoms ad2ant2r simprr elfzuzb
|
|
jca jca32 anbi12i ad2ant2l simplr impbii 3bitr4i ) BAEFZGZDBEFZGZHZCUGGZDCE
|
|
FGZHZHZUFUJHZCUEGZUKHZHZBADIJZGZCBDIJGZHBACIJGZCURGZHUMUQUMUNUOUKUMUFUJUFUH
|
|
ULKUIUJUKLRUFUJUOUHUKUJUFUOBCAMNOUIUJUKPSUQUIUJUKUQUFUHUFUJUPKUJUKUHUFUOUKU
|
|
JUHCDBMNUARUFUJUPUBUNUOUKPSUCUSUIUTULBADQCBDQTVAUNVBUPBACQCADQTUD $.
|
|
|
|
$(
|
|
@{
|
|
@d j K @. @d j M @. @d j N @.
|
|
@( Union of two adjacent finite sets of sequential integers. @)
|
|
fzunt @p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K <_ M /\ M <_ N ) )
|
|
-> ( ( K ... M ) u. ( M ... N ) ) = ( K ... N ) ) @=
|
|
( vj cz wcel w3a cle wbr wa cfz co cun cv wo cr wb wi letr expd com23
|
|
3exp com3l imp42 adantlll adantrl anim2d 3expa imp adantlrl adantrr
|
|
anim1d jaod impbid exp42 com34 ex 3imp zre syl3anOLD syl5OLD pm5.32d andi
|
|
bicomi syl5bbOLD elfz1 3anass syl6bb 3adant3 adantr 3adant1 orbi12d
|
|
3adant2 3bitr4d elun eqrdv ) AEFZBEFZCEFZGZABHIZBCHIZJZJZDABKLZBCKLZMZACK
|
|
LZWDDNZWEFZWIWFFZOZWIWHFZWIWGFWDWIEFZAWIHIZWIBHIZJZJZWNBWIHIZWICHIZJZJZOZ
|
|
WNWOWTJZJZWLWMWDWNWQXAOZJZXEXCWDWNXFXDWDWIPFZXFXDQZWNVTWCXHXIRZAPFZBPFZCP
|
|
FZWCXJRZVQVRVSXKXLXMXNXKXLXMXNRXKXLJZXMXHWCXIXOXMXHWCXIXOXMXHJZJZWCJZXFXD
|
|
XRWQXDXAXRWPWTWOXQWBWPWTRZWAXLXPWBXSXKXLXMXHWBXSXHXLXMWBXSRZXHXLXMXTXHXLX
|
|
MGZWPWBWTYAWPWBWTWIBCSTUAUBUCUDUEUFUGXRWSWOWTXQWAWSWORZWBXOXHWAYBXMXOXHJZ
|
|
WAYBYCWAWSWOXKXLXHWAWSJWORABWISUHTUIUJUKULUM?UNUOUPUQURAUSBUSCUSUTUIWIUSV
|
|
AVBXGXCWNWQXAVCVDVEWDWJWRWKXBVTWJWRQZWCVQVRYDVSVQVRJWJWNWOWPGWRWIABVFWNWO
|
|
WPVGVHVIVJVTWKXBQZWCVRVSYEVQVRVSJWKWNWSWTGXBWIBCVFWNWSWTVGVHVKVJVLVTWMXEQ
|
|
ZWCVQVSYFVRVQVSJWMWNWOWTGXEWIACVFWNWOWTVGVHVMVJVNWIWEWFVOVEVP @.
|
|
@( [ ?] @) @( [21-Jul-2005] @)
|
|
@}
|
|
$)
|
|
|
|
${
|
|
$d k M $. $d k N $. $d k K $.
|
|
$( Subset relationship for finite sets of sequential integers.
|
|
(Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fzss1 $p |- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) ) $=
|
|
( vk cuz cfv wcel co cv wa elfzuz id uztrn syl2anr elfzuz3 adantl elfzuzb
|
|
cfz sylanbrc ex ssrdv ) ABEFZGZDACRHZBCRHZUCDIZUDGZUFUEGZUCUGJUFUBGZCUFEF
|
|
GZUHUGUFAEFGUCUIUCUFACKUCLAUFBMNUGUJUCUFACOPUFBCQSTUA $.
|
|
|
|
$( Subset relationship for finite sets of sequential integers.
|
|
(Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro,
|
|
30-Apr-2015.) $)
|
|
fzss2 $p |- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) ) $=
|
|
( vk cuz cfv wcel cfz co cv wa elfzuz adantl elfzuz3 uztrn sylan2 elfzuzb
|
|
sylanbrc ex ssrdv ) CAEFGZDBAHIZBCHIZUADJZUBGZUDUCGZUAUEKUDBEFGZCUDEFZGZU
|
|
FUEUGUAUDBALMUEUAAUHGUIUDBANACUDOPUDBCQRST $.
|
|
|
|
$( A finite set of sequential integers is a subset of an upper set of
|
|
integers. (Contributed by NM, 28-Oct-2005.) $)
|
|
fzssuz $p |- ( M ... N ) C_ ( ZZ>= ` M ) $=
|
|
( vk cfz co cuz cfv cv elfzuz ssriv ) CABDEAFGCHABIJ $.
|
|
|
|
$( A finite interval of integers with one element. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
fzsn $p |- ( M e. ZZ -> ( M ... M ) = { M } ) $=
|
|
( vk cz wcel cfz co csn wceq elfz1eq elfz3 eleq1 syl5ibrcom impbid2 velsn
|
|
cv syl6bbr eqrdv ) ACDZBAAEFZAGZRBOZSDZUAAHZUATDRUBUCUAAIRUBUCASDAJUAASKL
|
|
MBANPQ $.
|
|
|
|
$( Subset relationship for finite sets of sequential integers.
|
|
(Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fzssp1 $p |- ( M ... N ) C_ ( M ... ( N + 1 ) ) $=
|
|
( vk cfz co c1 caddc cv wcel cuz cfv wss elfzel2 uzid peano2uz fzss2 4syl
|
|
cz id sseldd ssriv ) CABDEZABFGEZDEZCHZUBIZUBUDUEUFBRIBBJKZIUCUGIUBUDLUEA
|
|
BMBNBBOBAUCPQUFSTUA $.
|
|
$}
|
|
|
|
$( Join a successor to the end of a finite set of sequential integers.
|
|
(Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fzsuc $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfz cun csn peano2uz eluzfz2 syl peano2fzr mpdan
|
|
wceq fzsplit cz eluzelz fzsn 3syl uneq2d eqtrd ) BACDZEZABFGHZIHZABIHZUGUGI
|
|
HZJZUIUGKZJUFBUHEZUHUKQUFUGUHEZUMUFUGUEEZUNABLZAUGMNBAUGOPBAUGRNUFUJULUIUFU
|
|
OUGSEUJULQUPAUGTUGUAUBUCUD $.
|
|
|
|
$( Join a predecessor to the beginning of a finite set of sequential
|
|
integers. (Contributed by AV, 24-Aug-2019.) $)
|
|
fzpred $p |- ( N e. ( ZZ>= ` M )
|
|
-> ( M ... N ) = ( { M } u. ( ( M + 1 ) ... N ) ) ) $=
|
|
( cuz cfv wcel cfz co c1 caddc cun wceq eluzel2 uzid peano2uz 3syl fzsplit2
|
|
csn cz mpancom fzsn syl uneq1d eqtrd ) BACDZEZABFGZAAFGZAHIGZBFGZJZAQZUIJUH
|
|
UDEZUEUFUJKUEAREZAUDEULABLZAMAANOAABPSUEUGUKUIUEUMUGUKKUNATUAUBUC $.
|
|
|
|
$( A finite set of sequential integers is disjoint with its predecessor.
|
|
(Contributed by AV, 24-Aug-2019.) $)
|
|
fzpreddisj $p |- ( N e. ( ZZ>= ` M )
|
|
-> ( { M } i^i ( ( M + 1 ) ... N ) ) = (/) ) $=
|
|
( cuz cfv wcel csn c1 caddc co cfz cin c0 wn cz cle wbr wa cc0 wb cr w3a 0z
|
|
incom wceq clt 0lt1 1z zltnle mpbi eluzel2 zred 1re leaddle0 sylancl mtbiri
|
|
mp2an intnanrd intnand elfz2 sylnibr disjsn sylibr syl5eqr ) BACDEZAFZAGHIZ
|
|
BJIZKVGVEKZLVGVEUCVDAVGEZMVHLUDVDVFNEBNEANEUAZVFAOPZABOPZQZQVIVDVMVJVDVKVLV
|
|
DVKGROPZRGUEPZVNMZUFRNEGNEVOVPSUBUGRGUHUPUIVDATEGTEVKVNSVDAABUJUKULAGUMUNUO
|
|
UQURAVFBUSUTVGAVAVBVC $.
|
|
|
|
$( Append an element to a finite set of sequential integers. (Contributed by
|
|
NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) $)
|
|
elfzp1 $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( K e. ( M ... ( N + 1 ) ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfz csn cun wceq wo fzsuc eleq2d elun cz eluzelz
|
|
wb peano2zd elsn2g syl orbi2d syl5bb bitrd ) CBDEFZABCGHIZJIZFABCJIZUHKZLZF
|
|
ZAUJFZAUHMZNZUGUIULABCOPUMUNAUKFZNUGUPAUJUKQUGUQUOUNUGUHRFUQUOTUGCBCSUAAUHR
|
|
UBUCUDUEUF $.
|
|
|
|
$( Subset relationship for finite sets of sequential integers. (Contributed
|
|
by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
fzp1ss $p |- ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) ) $=
|
|
( cz wcel cuz cfv c1 caddc co cfz wss uzid peano2uz fzss1 3syl ) ACDAAEFZDA
|
|
GHIZPDQBJIABJIKALAAMQABNO $.
|
|
|
|
$( Membership in a set of sequential integers with an appended element.
|
|
(Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fzelp1 $p |- ( K e. ( M ... N ) -> K e. ( M ... ( N + 1 ) ) ) $=
|
|
( cfz co c1 caddc fzssp1 sseli ) BCDEBCFGEDEABCHI $.
|
|
|
|
$( Add one to an element of a finite set of integers. (Contributed by Jeff
|
|
Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
fzp1elp1 $p |- ( K e. ( M ... N ) -> ( K + 1 ) e. ( M ... ( N + 1 ) ) ) $=
|
|
( cfz co wcel c1 caddc cuz cfv elfzuz peano2uz syl elfzuz3 eluzp1p1 elfzuzb
|
|
sylanbrc ) ABCDEFZAGHEZBIJZFZCGHEZSIJFZSBUBDEFRATFUAABCKBALMRCAIJFUCABCNACO
|
|
MSBUBPQ $.
|
|
|
|
$( Shift membership in a finite sequence of naturals. (Contributed by Scott
|
|
Fenton, 17-Jul-2013.) $)
|
|
fznatpl1 $p |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) ->
|
|
( I + 1 ) e. ( 1 ... N ) ) $=
|
|
( wcel c1 co cfz wa caddc cle wbr cr zred adantl peano2re syl wb 1re mp3an2
|
|
cz syl2anc cn cmin elfzelz ltp1d elfzle1 leadd1 mp3an13 mpbid ltletrd ltled
|
|
1red elfzle2 nnz adantr leaddsub mpbird peano2zd 1z elfz mpbir2and ) BUACZA
|
|
DBDUBEZFECZGZADHEZDBFECZDVEIJZVEBIJZVDDVEVDUKZVDAKCZVEKCVCVJVAVCAADVBUCZLMZ
|
|
ANOZVDDDDHEZVEVIVDDKCZVNKCVIDNOVMVDDVIUDVDDAIJZVNVEIJZVCVPVAADVBUEMVDVJVPVQ
|
|
PZVLVOVJVOVRQQDADUFUGOUHUIUJVDVHAVBIJZVCVSVAADVBULMVDVJBKCZVHVSPZVLVDBVABSC
|
|
ZVCBUMUNZLVJVOVTWAQADBUORTUPVDVESCZWBVFVGVHGPZVCWDVAVCAVKUQMWCWDDSCWBWEURVE
|
|
DBUSRTUT $.
|
|
|
|
${
|
|
$d M m $.
|
|
|
|
$( A finite interval of integers with two elements. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
fzpr $p |- ( M e. ZZ -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } ) $=
|
|
( vm cz wcel c1 caddc co cfz cpr cv wceq cuz cfv uzid elfzp1 syl csn fzsn
|
|
wo wb eleq2d velsn syl6bb orbi1d bitrd vex elpr syl6bbr eqrdv ) ACDZBAAEF
|
|
GZHGZAUKIZUJBJZULDZUNAKZUNUKKZSZUNUMDUJUOUNAAHGZDZUQSZURUJAALMDUOVATANUNA
|
|
AOPUJUTUPUQUJUTUNAQZDUPUJUSVBUNARUABAUBUCUDUEUNAUKBUFUGUHUI $.
|
|
|
|
$( A finite interval of integers with three elements. (Contributed by NM,
|
|
13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.) $)
|
|
fztp $p |- ( M e. ZZ -> ( M ... ( M + 2 ) )
|
|
= { M , ( M + 1 ) , ( M + 2 ) } ) $=
|
|
( cz wcel c1 caddc co cfz csn cun c2 ctp cuz cfv wceq uzid peano2uz fzsuc
|
|
cc ax-1cn syl6eqr 3syl zcn addass mp3an23 syl df-2 oveq2i oveq2d cpr fzpr
|
|
sneqd uneq12d df-tp 3eqtr3d ) ABCZAADEFZDEFZGFZAUPGFZUQHZIZAAJEFZGFAUPVBK
|
|
ZUOAALMZCUPVDCURVANAOAAPAUPQUAUOUQVBAGUOUQADDEFZEFZVBUOARCZUQVFNZAUBVGDRC
|
|
ZVIVHSSADDUCUDUEJVEAEUFUGTZUHUOVAAUPUIZVBHZIVCUOUSVKUTVLAUJUOUQVBVJUKULAU
|
|
PVBUMTUN $.
|
|
$}
|
|
|
|
$( Join a successor to the end of a finite set of sequential integers.
|
|
(Contributed by Mario Carneiro, 7-Mar-2014.) $)
|
|
fzsuc2 $p |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) ->
|
|
( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) $=
|
|
( c1 cmin co cuz cfv wcel cz wceq caddc wo cfz csn cun cc oveq2d c0 uneq12d
|
|
sneqd uzp1 zcn ax-1cn npcan sylancl uncom un0 eqtri clt wbr zre wb peano2zm
|
|
ltm1d fzn mpdan mpbid fzsn 3eqtr4a eqtr4d oveq1 oveq2 eqeq12d syl5ibrcom wa
|
|
imp fveq2d eleq2d biimpa fzsuc syl jaodan sylan2 ) BACDEZFGHAIHZBVNJZBVNCKE
|
|
ZFGZHZLABCKEZMEZABMEZVTNZOZJZVNBUAVOVPWEVSVOVPWEVOWEVPAVQMEZAVNMEZVQNZOZJVO
|
|
WFAAMEZWIVOVQAAMVOAPHCPHVQAJAUBUCACUDUEZQVORANZOZWLWIWJWMWLROWLRWLUFWLUGUHV
|
|
OWGRWHWLVOVNAUIUJZWGRJZVOAAUKUNVOVNIHWNWOULAUMAVNUOUPUQVOVQAWKTSAURUSUTVPWA
|
|
WFWDWIVPVTVQAMBVNCKVAZQVPWBWGWCWHBVNAMVBVPVTVQWPTSVCVDVFVOVSVEBAFGZHZWEVOVS
|
|
WRVOVRWQBVOVQAFWKVGVHVIABVJVKVLVM $.
|
|
|
|
$( ` ( M ... ( N + 1 ) ) ` is the disjoint union of ` ( M ... N ) ` with
|
|
` { ( N + 1 ) } ` . (Contributed by Mario Carneiro, 7-Mar-2014.) $)
|
|
fzp1disj $p |- ( ( M ... N ) i^i { ( N + 1 ) } ) = (/) $=
|
|
( cfz co c1 caddc csn cin c0 wceq wcel wn cle wbr elfzle2 elfzel2 clt ltp1d
|
|
cz zre wb peano2z zltnle mpdan mpbid syl pm2.65i disjsn mpbir ) ABCDZBEFDZG
|
|
HIJUKUJKZLULUKBMNZUKABOULBSKZUMLZUKABPUNBUKQNZUOUNBBTRUNUKSKUPUOUABUBBUKUCU
|
|
DUEUFUGUJUKUHUI $.
|
|
|
|
${
|
|
$d M k $. $d N k $.
|
|
$( Remove a successor from the end of a finite set of sequential integers.
|
|
(Contributed by AV, 4-Sep-2019.) $)
|
|
fzdifsuc $p |- ( N e. ( ZZ>= ` M )
|
|
-> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) ) $=
|
|
( vk wcel cfz co cz elfzelz adantl wa cle wbr adantr elfz syl3anc syl2anc
|
|
wb wn anbi2d bitr4d cuz cfv c1 caddc csn cdif cv eldifi syl simpr eluzel2
|
|
eluzelz clt wne eldif peano2zd wceq velsn notbii nesym bitr4i a1i anbi12d
|
|
syl5bb anass syl6bb zltlen zleltp1 eqrdav ) BAUAUBDZCABEFZABUCUDFZEFZVLUE
|
|
ZUFZGCUGZVKDZVPGDZVJVPABHIVPVODZVRVJVSVPVMDZVRVPVMVNUHVPAVLHUIIVJVRJZVQAV
|
|
PKLZVPBKLZJZVSWAVRAGDZBGDZVQWDQVJVRUJZVJWEVRABUKMZVJWFVRABULMZVPABNOWAVSW
|
|
BVPVLUMLZJZWDWAVSWBVPVLKLZVLVPUNZJZJZWKWAVSWBWLJZWMJZWOVSVTVPVNDZRZJWAWQV
|
|
PVMVNUOWAVTWPWSWMWAVRWEVLGDZVTWPQWGWHWABWIUPZVPAVLNOWSWMQWAWSVPVLUQZRWMWR
|
|
XBCVLURUSVLVPUTVAVBVCVDWBWLWMVEVFWAWJWNWBWAVRWTWJWNQWGXAVPVLVGPSTWAWCWJWB
|
|
WAVRWFWCWJQWGWIVPBVHPSTTVI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( Two ways of defining the first two values of a sequence on ` NN ` .
|
|
(Contributed by NM, 5-Sep-2011.) $)
|
|
fzprval $p |- ( A. x e. ( 1 ... 2 ) ( F ` x ) = if ( x = 1 , A , B )
|
|
<-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B ) ) $=
|
|
( cv cfv c1 wceq cif c2 cfz co wral cpr wa caddc df-2 fveq2 eqeq12d wne
|
|
cz wcel 1z fzpr ax-mp oveq2i preq2i 3eqtr4i raleqi 1ex iftrue 1ne2 necomi
|
|
2ex pm13.181 mpan2 neneqd iffalsed ralpr bitri ) AEZDFZVAGHZBCIZHZAGJKLZM
|
|
VEAGJNZMGDFZBHZJDFZCHZOVEAVFVGGGGPLZKLZGVLNZVFVGGUAUBVMVNHUCGUDUEJVLGKQUF
|
|
JVLGQUGUHUIVEVIVKAGJUJUNVCVBVHVDBVAGDRVCBCUKSVAJHZVBVJVDCVAJDRVOVCBCVOVAG
|
|
VOJGTVAGTGJULUMVAJGUOUPUQURSUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( Two ways of defining the first three values of a sequence on ` NN ` .
|
|
(Contributed by NM, 13-Sep-2011.) $)
|
|
fztpval $p |- ( A. x e. ( 1 ... 3 ) ( F ` x )
|
|
= if ( x = 1 , A , if ( x = 2 , B , C ) )
|
|
<-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B /\ ( F ` 3 ) = C ) ) $=
|
|
( cfv c1 wceq c2 c3 cfz co ctp caddc ax-mp fveq2 eqeq12d wne gtneii neeq1
|
|
cv cif wral w3a cz wcel 1z fztp df-3 2cn ax-1cn addcomi eqtri oveq2i df-2
|
|
tpeq3 tpeq2 3eqtr4i raleqi 1ex 2ex 3ex iftrue 1lt2 mpbiri ifnefalse eqtrd
|
|
1re syl 1lt3 2re 2lt3 raltp bitri ) AUAZEFZVOGHZBVOIHZCDUBZUBZHZAGJKLZUCW
|
|
AAGIJMZUCGEFZBHZIEFZCHZJEFZDHZUDWAAWBWCGGINLZKLZGGGNLZWJMZWBWCGUEUFWKWMHU
|
|
GGUHOJWJGKJIGNLWJUIIGUJUKULUMZUNWCGIWJMZWMJWJHWCWOHWNJWJGIUPOIWLHWOWMHUOI
|
|
WLGWJUQOUMURUSWAWEWGWIAGIJUTVAVBVQVPWDVTBVOGEPVQBVSVCQVRVPWFVTCVOIEPVRVTV
|
|
SCVRVOGRZVTVSHZVRWPIGRGIVHVDSVOIGTVEVOGBVSVFZVIVRCDVCVGQVOJHZVPWHVTDVOJEP
|
|
WSVTVSDWSWPWQWSWPJGRGJVHVJSVOJGTVEWRVIWSVOIRZVSDHWSWTJIRIJVKVLSVOJITVEVOI
|
|
CDVFVIVGQVMVN $.
|
|
$}
|
|
|
|
$( Reversal of start and end of a finite set of sequential integers.
|
|
(Contributed by NM, 25-Nov-2005.) $)
|
|
fzrev $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) ->
|
|
( K e. ( ( J - N ) ... ( J - M ) ) <-> ( J - K ) e. ( M ... N ) ) ) $=
|
|
( cz wcel wa cmin co cle wbr cfz wb cr zre syl3an 3expb zsubcl ancoms elfz
|
|
ancom suble 3comr adantll lesub adantlr anbi12d syl5rbbr ad2ant2lr ad2ant2r
|
|
simprr syl3anc adantl simpll simplr 3bitr4d ) CEFZDEFZGZAEFZBEFZGZGZADHIZBJ
|
|
KZBACHIZJKZGZCABHIZJKZVIDJKZGZBVDVFLIFZVICDLIFZVLVKVJGVCVHVKVJUAVCVKVEVJVGU
|
|
RVBVKVEMZUQURUTVAVOUTVAURVOUTANFZVABNFZURDNFVOAOZBOZDOABDUBPUCQUDUQVBVJVGMZ
|
|
URUQUTVAVTUQCNFUTVPVAVQVTCOVRVSCABUEPQUFUGUHVCVAVDEFZVFEFZVMVHMUSUTVAUKURUT
|
|
WAUQVAUTURWAADRSUIUQUTWBURVAUTUQWBACRSUJBVDVFTULVCVIEFZUQURVNVLMVBWCUSABRUM
|
|
UQURVBUNUQURVBUOVICDTULUP $.
|
|
|
|
$( Reversal of start and end of a finite set of sequential integers.
|
|
(Contributed by NM, 25-Nov-2005.) $)
|
|
fzrev2 $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) ->
|
|
( K e. ( M ... N ) <-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) ) $=
|
|
( cz wcel wa cmin co cfz wb simpl zsubcl jca fzrev sylan2 wceq cc zcn nncan
|
|
syl2an adantl eleq1d bitr2d ) CEFDEFGZAEFZBEFZGZGZABHIZADHIACHIJIFZAUJHIZCD
|
|
JIZFZBUMFUHUEUFUJEFZGUKUNKUHUFUOUFUGLABMNAUJCDOPUIULBUMUHULBQZUEUFARFBRFUPU
|
|
GASBSABTUAUBUCUD $.
|
|
|
|
$( Reversal of start and end of a finite set of sequential integers.
|
|
(Contributed by NM, 25-Nov-2005.) $)
|
|
fzrev2i $p |- ( ( J e. ZZ /\ K e. ( M ... N ) )
|
|
-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) $=
|
|
( cz wcel co wa cmin simpr wb elfzel1 adantl elfzel2 simpl elfzelz syl22anc
|
|
cfz fzrev2 mpbid ) AEFZBCDRGFZHZUBABIGADIGACIGRGFZUAUBJUCCEFZDEFZUABEFZUBUD
|
|
KUBUEUABCDLMUBUFUABCDNMUAUBOUBUGUABCDPMABCDSQT $.
|
|
|
|
$( The "complement" of a member of a finite set of sequential integers.
|
|
(Contributed by NM, 20-Nov-2005.) $)
|
|
fzrev3 $p |- ( K e. ZZ -> ( K e. ( M ... N )
|
|
<-> ( ( M + N ) - K ) e. ( M ... N ) ) ) $=
|
|
( cz wcel cfz co caddc cmin w3a wa simpl elfzel1 adantl elfzel2 3jca wb zcn
|
|
cc 3adant1 wceq pncan pncan2 syl2an eleq2d 3simpc zaddcl simp1 fzrev bitr3d
|
|
oveq12d syl12anc pm5.21nd ) ADEZABCFGZEZBCHGZAIGZUOEZUNBDEZCDEZJZUNUPKUNUTV
|
|
AUNUPLUPUTUNABCMNUPVAUNABCONPUNUSKUNUTVAUNUSLUSUTUNURBCMNUSVAUNURBCONPVBAUQ
|
|
CIGZUQBIGZFGZEZUPUSUTVAVFUPQUNUTVAKZVEUOAUTBSEZCSEZVEUOUAVABRCRVHVIKVCBVDCF
|
|
BCUBBCUCUKUDUETVBVGUQDEZUNVFUSQUNUTVAUFUTVAVJUNBCUGTUNUTVAUHUQABCUIULUJUM
|
|
$.
|
|
|
|
$( The "complement" of a member of a finite set of sequential integers.
|
|
(Contributed by NM, 20-Nov-2005.) $)
|
|
fzrev3i $p |- ( K e. ( M ... N ) -> ( ( M + N ) - K ) e. ( M ... N ) ) $=
|
|
( cfz co wcel caddc cmin cz wb elfzelz fzrev3 syl ibi ) ABCDEZFZBCGEAHEOFZP
|
|
AIFPQJABCKABCLMN $.
|
|
|
|
$( Finite set of sequential integers starting at 1. (Contributed by NM,
|
|
31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) $)
|
|
fznn $p |- ( N e. ZZ
|
|
-> ( K e. ( 1 ... N ) <-> ( K e. NN /\ K <_ N ) ) ) $=
|
|
( c1 cfz co wcel cn cuz cfv wa cz cle wbr elfzuzb elnnuz anbi1i bitr4i eluz
|
|
wb nnz sylan ancoms pm5.32da syl5bb ) ACBDEFZAGFZBAHIFZJZBKFZUFABLMZJUEACHI
|
|
FZUGJUHACBNUFUKUGAOPQUIUFUGUJUFUIUGUJSZUFAKFUIULATABRUAUBUCUD $.
|
|
|
|
$( Membership in a 1 based finite set of sequential integers. (Contributed
|
|
by AV, 30-Oct-2018.) $)
|
|
elfz1b $p |- ( N e. ( 1 ... M ) <-> ( N e. NN /\ M e. NN /\ N <_ M ) ) $=
|
|
( c1 wcel cz w3a cle wbr wa cn wi cc0 clt simpl 0red 1red zre 3jca adantr
|
|
cr cfz co elfz2 a1i simpr ltletr imp syl12anc elnnz sylanbrc 3ad2ant3 com12
|
|
0lt1 impcom 3anim123i 3com23 letr syl ltletrd 3ad2ant2 syld simprr 1zzd nnz
|
|
ex 3ad2ant1 nnge1 simp3 jca32 impbii bitri ) BCAUAUBDCEDZAEDZBEDZFZCBGHZBAG
|
|
HZIZIZBJDZAJDZVQFZBCAUCVSWBVSVTWAVQVRVOVTVPVOVTKVQVOVPVTVNVLVPVTKVMVNVPVTVN
|
|
VPIZVNLBMHZVTVNVPNWCLTDZCTDZBTDZFZLCMHZVPWDVNWHVPVNWEWFWGVNOVNPBQZRSWIWCUMU
|
|
DVNVPUEWHWIVPIWDLCBUFUGUHBUIUJVEUKULSUNVOVRWAVOVRCAGHZWAVOWFWGATDZFZVRWKKVL
|
|
VNVMWMVLWFVNWGVMWLCQWJAQZUOUPCBAUQURVMVLWKWAKVNVMWKWAVMWKIZVMLAMHWAVMWKNWOL
|
|
CAWOOWOPVMWLWKWNSWIWOUMUDVMWKUEUSAUIUJVEUTVAUGVOVPVQVBRWBVOVPVQWBVLVMVNWBVC
|
|
WAVTVMVQAVDUTVTWAVNVQBVDVFRVTWAVPVQBVGVFVTWAVQVHVIVJVK $.
|
|
|
|
$( Membership in a finite set of sequential integers. (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
elfzm11 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... ( N - 1 ) ) <->
|
|
( K e. ZZ /\ M <_ K /\ K < N ) ) ) $=
|
|
( cz wcel wa c1 cmin co cfz cle wbr w3a clt wb peano2zm elfz1 sylan2 3anass
|
|
zltlem1 anbi2d expcom pm5.32d 3bitr4g adantl bitr4d ) BDEZCDEZFABCGHIZJIEZA
|
|
DEZBAKLZAUIKLZMZUKULACNLZMZUHUGUIDEUJUNOCPABUIQRUHUPUNOUGUHUKULUOFZFUKULUMF
|
|
ZFUPUNUHUKUQURUKUHUQUROUKUHFUOUMULACTUAUBUCUKULUOSUKULUMSUDUEUF $.
|
|
|
|
${
|
|
$d k M $. $d k N $.
|
|
$( Express an upper integer set as the disjoint (see ~ uzdisj ) union of
|
|
the first ` N ` values and the rest. (Contributed by Mario Carneiro,
|
|
24-Apr-2014.) $)
|
|
uzsplit $p |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` M ) =
|
|
( ( M ... ( N - 1 ) ) u. ( ZZ>= ` N ) ) ) $=
|
|
( vk cuz cfv wcel c1 cmin co cfz cun cv wo wa cle wbr cz eluzelz syl2an
|
|
wb clt zlelttric eluz eluzel2 elfzm11 df-3an syl6bb syl2anr eluzle adantl
|
|
w3a jca biantrurd bitr4d orbi12d mpbird orcomd ex elfzuz a1i uztrn expcom
|
|
wi jaod impbid elun syl6bbr eqrdv ) BADEZFZCVIABGHIZJIZBDEZKZVJCLZVIFZVOV
|
|
LFZVOVMFZMZVOVNFVJVPVSVJVPVSVJVPNZVRVQVTVRVQMBVOOPZVOBUAPZMZVJBQFZVOQFZWC
|
|
VPABRZAVORZBVOUBSVTVRWAVQWBVJWDWEVRWATVPWFWGBVOUCSVTVQWEAVOOPZNZWBNZWBVPA
|
|
QFZWDVQWJTVJAVOUDWFWKWDNVQWEWHWBUKWJVOABUEWEWHWBUFUGUHVTWIWBVPWIVJVPWEWHW
|
|
GAVOUIULUJUMUNUOUPUQURVJVQVPVRVQVPVCVJVOAVKUSUTVRVJVPBVOAVAVBVDVEVOVLVMVF
|
|
VGVH $.
|
|
|
|
$( The first ` N ` elements of an upper integer set are distinct from any
|
|
later members. (Contributed by Mario Carneiro, 24-Apr-2014.) $)
|
|
uzdisj $p |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/) $=
|
|
( vk c1 cmin co cfz cuz cfv cin c0 wss wceq wcel wbr cle syl mpbid zred
|
|
cz cv elin simprbi eluzle eluzel2 eluzelz zlem1lt syl2anc simplbi elfzle2
|
|
clt wb wn peano2zm lenltd pm2.21dd ssriv ss0 ax-mp ) ABDEFZGFZBHIZJZKLVCK
|
|
MCVCKCUAZVCNZUTVDUKOZVDKNVEBVDPOZVFVEVDVBNZVGVEVDVANZVHVDVAVBUBZUCZBVDUDQ
|
|
VEBTNZVDTNZVGVFULVEVHVLVKBVDUEQZVEVHVMVKBVDUFQZBVDUGUHRVEVDUTPOZVFUMVEVIV
|
|
PVEVIVHVJUIVDAUTUJQVEVDUTVEVDVOSVEUTVEVLUTTNVNBUNQSUORUPUQVCURUS $.
|
|
$}
|
|
|
|
${
|
|
fseq1p1m1.1 $e |- H = { <. ( N + 1 ) , B >. } $.
|
|
$( Add/remove an item to/from the end of a finite sequence. (Contributed
|
|
by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro,
|
|
7-Mar-2014.) $)
|
|
fseq1p1m1 $p |- ( N e. NN0 ->
|
|
( ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) <->
|
|
( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\
|
|
F = ( G |` ( 1 ... N ) ) ) ) ) $=
|
|
( cn0 wcel c1 co wf cun wceq cfv cres csn c0 cn 3syl cfz w3a caddc wa cin
|
|
simpr1 nn0p1nn adantr simpr2 fsng mpbiri syl2anc snssd fssd fzp1disj fun2
|
|
cop a1i syl21anc cz cmin cuz simpl cc0 nn0uz 1m1e0 fveq2i eqtr4i syl6eleq
|
|
1z fzsuc2 sylancr eqcomd feq2d mpbid simpr3 feq1d mpbird wfn wb fnresdisj
|
|
reseq1d ffn uneq1d resundir uncom un0 eqtr2i 3eqtr4g fnresdm 3eqtrd nn0zd
|
|
fveq1d peano2zd snidg fvres fveq1i fvsng syl5eq 3eqtr3d incom uneq2d 3jca
|
|
eqcomi 3eqtrrd fzssp1 fssres sylancl nnuz eluzfz2 ffvelrnd eqeltrrd opeq2
|
|
wss fnressn sneqd eqtrd syl6reqr uneq12d reseq2d resundi syl6req impbida
|
|
syl ) FHIZJFUAKZACLZBAIZDCEMZNZUBZJFJUCKZUAKZADLZYLDOZBNZCDYFPZNZUBZYEYKU
|
|
DZYNYPYRYTYNYMAYILZYTYFYLQZMZAYILZUUAYTYGUUBAELZYFUUBUEZRNZUUDYEYGYHYJUFZ
|
|
YTUUBBQZAEYTYLSIZYHUUBUUIELZYEUUJYKFUGZUHZYEYGYHYJUIZUUJYHUDZUUKEYLBUQZQZ
|
|
NGYLBSAEUJUKULZYTBAUUNUMUNZUUGYTJFUOURZYFUUBACEUPUSYTUUCYMAYIYTYMUUCYTJUT
|
|
IZFJJVAKZVBOZIZYMUUCNZVJYTFHUVCYEYKVCZHVDVBOUVCVEUVBVDVBVFVGVHZVIJFVKZVLV
|
|
MVNVOYTYMADYIYEYGYHYJVPZVQVRYTYLDUUBPZOZYLEOZYOBYTYLUVJEYTUVJYIUUBPZEUUBP
|
|
ZEYTDYIUUBUVIWBYTCUUBPZUVNMRUVNMZUVMUVNYTUVORUVNYTUUGUVORNZUUTYTYGCYFVSZU
|
|
UGUVQVTUUHYFACWCZYFUUBCWATVOWDCEUUBWEUVPUVNRMUVNRUVNWFUVNWGWHWIYTUUEEUUBV
|
|
SZUVNENUUSUUBAEWCUUBEWJTWKWMYTYLUTIYLUUBIUVKYONYTFYTFUVFWLWNYLUTWOYLUUBDW
|
|
PTYTUUJYHUVLBNUUMUUNUUOUVLYLUUQOBYLEUUQGWQYLBSAWRWSULWTYTYQYIYFPZCYFPZCYT
|
|
DYIYFUVIWBYTUWBEYFPZMUWBRMZUWAUWBYTUWCRUWBYTUUBYFUEZRNZUWCRNZYTUWEUUFRUUB
|
|
YFXAUUTWSYTUUKUVTUWFUWGVTUURUUBUUIEWCUUBYFEWATVOXBCEYFWEUWDUWBUWBWGXDWIYT
|
|
YGUVRUWBCNUUHUVSYFCWJTXEXCYEYSUDZYGYHYJUWHYGYFAYQLZUWHYNYFYMXNUWIYEYNYPYR
|
|
UFZJFXFYMAYFDXGXHUWHYFACYQYEYNYPYRVPZVQVRUWHYOBAYEYNYPYRUIZUWHYMAYLDUWJUW
|
|
HYLJVBOZIYLYMIZUWHYLSUWMYEUUJYSUULUHXIVIJYLXJYDZXKXLUWHYIYQUVJMZDYMPZDUWH
|
|
CYQEUVJUWKUWHUVJUUQEUWHUVJYLYOUQZQZUUQUWHDYMVSZUWNUVJUWSNUWHYNUWTUWJYMADW
|
|
CZYDUWOYMYLDXOULUWHYPUWSUUQNUWLYPUWRUUPYOBYLXMXPYDXQGXRXSUWHUWQDUUCPUWPUW
|
|
HYMUUCDUWHUVAUVDUVEVJUWHFHUVCYEYSVCUVGVIUVHVLXTDYFUUBYAYBUWHYNUWTUWQDNUWJ
|
|
UXAYMDWJTXEXCYC $.
|
|
$}
|
|
|
|
${
|
|
fseq1m1p1.1 $e |- H = { <. N , B >. } $.
|
|
$( Add/remove an item to/from the end of a finite sequence. (Contributed
|
|
by Paul Chapman, 17-Nov-2012.) $)
|
|
fseq1m1p1 $p |- ( N e. NN ->
|
|
( ( F : ( 1 ... ( N - 1 ) ) --> A /\ B e. A /\ G = ( F u. H ) ) <->
|
|
( G : ( 1 ... N ) --> A /\ ( G ` N ) = B /\
|
|
F = ( G |` ( 1 ... ( N - 1 ) ) ) ) ) ) $=
|
|
( cn wcel c1 co cfz wf cop csn cun wceq w3a cfv cc cmin caddc cres cn0 wb
|
|
nnm1nn0 eqid fseq1p1m1 syl nncn ax-1cn npcan sylancl opeq1d sneqd syl6eqr
|
|
uneq2d eqeq2d 3anbi3d oveq2d feq2d fveq2d eqeq1d 3anbi12d 3bitr3d ) FHIZJ
|
|
FJUAKZLKZACMZBAIZDCVGJUBKZBNZOZPZQZRZJVKLKZADMZVKDSZBQZCDVHUCQZRZVIVJDCEP
|
|
ZQZRJFLKZADMZFDSZBQZWARVFVGUDIVPWBUEFUFABCDVMVGVMUGUHUIVFVOWDVIVJVFVNWCDV
|
|
FVMECVFVMFBNZOEVFVLWIVFVKFBVFFTIJTIVKFQFUJUKFJULUMZUNUOGUPUQURUSVFVRWFVTW
|
|
HWAVFVQWEADVFVKFJLWJUTVAVFVSWGBVFVKFDWJVBVCVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d k N $.
|
|
$( Quantification over a one-member finite set of sequential integers in
|
|
terms of substitution. (Contributed by NM, 28-Nov-2005.) $)
|
|
fz1sbc $p |- ( N e. ZZ
|
|
-> ( A. k e. ( N ... N ) ph <-> [. N / k ]. ph ) ) $=
|
|
( cz wcel wsbc cv wceq wi wal cfz co wral sbc6g df-ral elfz1eq syl5ibrcom
|
|
elfz3 eleq1 impbid2 imbi1d albidv syl5rbb bitr2d ) CDEZABCFBGZCHZAIZBJZAB
|
|
CCKLZMZABCDNUKUFUJEZAIZBJUEUIABUJOUEUMUHBUEULUGAUEULUGUFCPUEULUGCUJECRUFC
|
|
UJSQTUAUBUCUD $.
|
|
$}
|
|
|
|
$( An integer is a member of a 0-based finite set of sequential integers iff
|
|
its successor is a member of the corresponding 1-based set. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
elfzp1b $p |- ( ( K e. ZZ /\ N e. ZZ ) ->
|
|
( K e. ( 0 ... ( N - 1 ) ) <-> ( K + 1 ) e. ( 1 ... N ) ) ) $=
|
|
( cz wcel wa c1 caddc co cfz cc0 wb peano2z 1z fzsubel mpanl1 mpanr2 sylan2
|
|
cmin cc wceq ancoms zcn ax-1cn pncan sylancl 1m1e0 oveq1i a1i adantr bitr2d
|
|
eleq12d ) ACDZBCDZEAFGHZFBIHDZUNFRHZFFRHZBFRHZIHZDZAJURIHZDZUMULUOUTKZULUMU
|
|
NCDZVCALUMVDFCDZVCMVEUMVDVEEVCMUNFFBNOPQUAULUTVBKUMULUPAUSVAULASDFSDUPATAUB
|
|
UCAFUDUEUSVATULUQJURIUFUGUHUKUIUJ $.
|
|
|
|
$( An integer is a member of a 1-based finite set of sequential integers iff
|
|
its predecessor is a member of the corresponding 0-based set.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
elfzm1b $p |- ( ( K e. ZZ /\ N e. ZZ ) ->
|
|
( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) ) $=
|
|
( cz wcel c1 cfz co cmin wb wa 1z fzsubel mpanl1 mpanr2 1m1e0 oveq1i eleq2i
|
|
cc0 syl6bb ancoms ) BCDZACDZAEBFGDZAEHGZRBEHGZFGZDZIUAUBJUCUDEEHGZUEFGZDZUG
|
|
UAUBECDZUCUJIZKUKUAUBUKJULKAEEBLMNUIUFUDUHRUEFOPQST $.
|
|
|
|
$( Options for membership in a finite interval of integers. (Contributed by
|
|
Jeff Madsen, 18-Jun-2010.) $)
|
|
elfzp12 $p |- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <->
|
|
( K = M \/ K e. ( ( M + 1 ) ... N ) ) ) ) $=
|
|
( cuz cfv wcel cfz co wceq c1 caddc wo cz elfzelz anim2i eluzel2 syl5ibrcom
|
|
wa eleq1 imdistani jaodan csn cun fzpred eleq2d elun syl6bb orbi1d sylan9bb
|
|
elsng pm5.21nd ) CBDEFZABCGHZFZABIZABJKHZCGHZFZLZULAMFZRZUNUTULABCNOULUOVAU
|
|
RULUOUTULUTUOBMFBCPABMSQTURUTULAUPCNOUAULUNABUBZFZURLZUTUSULUNAVBUQUCZFVDUL
|
|
UMVEABCUDUEAVBUQUFUGUTVCUOURABMUJUHUIUK $.
|
|
|
|
$( Choices for an element of a finite interval of integers. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
fzm1 $p |- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <->
|
|
( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) ) $=
|
|
( cuz wcel wceq cfz co c1 wo wb wa eleq2d adantl c0 clt wbr cz adantr mpbid
|
|
cfv cmin wi oveq1 elfz1eq syl6bir olc syl6 noel eluzelz ltm1d breq2 eluzel2
|
|
zred 1zzd zsubcld syl2anc mtbiri pm2.21d eluzfz2 ad2antrr eleq1 mpbird jaod
|
|
fzn ex impbid caddc elfzp1 cc zcnd npcan1 oveq2d eqeq2d orbi2d 3bitr3d uzm1
|
|
syl mpjaodan ) CBDUAZEZCBFZABCGHZEZABCIUBHZGHZEZACFZJZKWEVTEZWAWBLZWDWIWBWD
|
|
WIUCWAWBWDWHWIWBWDACCGHZEWHWBWLWCACBCGUDMACUEUFWHWGUGUHNWKWGWDWHWKWGWDWKWGA
|
|
OEAUIWKWFOAWKWEBPQZWFOFZWKWECPQZWMWKCWKCWACREZWBBCUJZSZUNUKWBWOWMKWACBWEPUL
|
|
NTWKBREZWEREWMWNKWAWSWBBCUMSWKCIWRWKUOUPBWEVEUQTMURUSWKWHWDWKWHLWDCWCEZWAWT
|
|
WBWHBCUTVAWHWDWTKWKACWCVBNVCVFVDVGWAWJLZABWEIVHHZGHZEZWGAXBFZJZWDWIWJXDXFKW
|
|
AABWEVINXAXCWCAXAXBCBGXACVJEXBCFXACWAWPWJWQSVKCVLVRZVMMXAXEWHWGXAXBCAXGVNVO
|
|
VPBCVQVS $.
|
|
|
|
$( No finite set of sequential integers equals an upper set of integers.
|
|
(Contributed by NM, 11-Dec-2005.) $)
|
|
fzneuz $p |- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ )
|
|
-> -. ( M ... N ) = ( ZZ>= ` K ) ) $=
|
|
( cuz cfv wcel cz wa cfz co wceq wn c1 peano2uz adantl wbr ad2antrr nelneq2
|
|
caddc syl2anc cle eluzelz clt zre ltp1d wb peano2z zltnle mpdan syl elfzle2
|
|
mpbid nsyl eqcom sylnib eluzfz2 sylancom wo simpr adantr eluzdc df-dc sylib
|
|
wdc mpjaodan ) CBDEFZAGFZHZCADEZFZBCIJZVIKZLZVJLZVHVJHZVIVKKZVLVOCMSJZVIFZV
|
|
QVKFZLZVPLVJVRVHACNOVFVTVGVJVFVQCUAPZVSVFCGFZWALZBCUBZWBCVQUCPZWCWBCCUDUEWB
|
|
VQGFWEWCUFCUGCVQUHUIULUJVQBCUKUMQVQVIVKRTVIVKUNUOVHVNCVKFZVMVFWFVGVNBCUPQCV
|
|
KVIRUQVHVJVDZVJVNURVHVGWBWGVFVGUSVFWBVGWDUTACVATVJVBVCVE $.
|
|
|
|
$( Disjointness of the upper integers and a finite sequence. (Contributed by
|
|
Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro,
|
|
24-Aug-2013.) $)
|
|
fznuz $p |- ( K e. ( M ... N ) -> -. K e. ( ZZ>= ` ( N + 1 ) ) ) $=
|
|
( cfz co wcel c1 caddc cuz cfv cle wbr elfzle2 clt wn cz wi elfzel2 eluzp1l
|
|
ex syl wb elfzelz zltnle syl2anc sylibd mt2d ) ABCDEFZACGHEIJFZACKLZABCMUHU
|
|
ICANLZUJOZUHCPFZUIUKQABCRZUMUIUKCASTUAUHUMAPFUKULUBUNABCUCCAUDUEUFUG $.
|
|
|
|
$( Disjointness of the upper integers and a finite sequence. (Contributed by
|
|
Mario Carneiro, 24-Aug-2013.) $)
|
|
uznfz $p |- ( K e. ( ZZ>= ` N ) -> -. K e. ( M ... ( N - 1 ) ) ) $=
|
|
( cuz cfv wcel c1 cmin co cfz cle wbr eluzle clt wn cz eluzel2 wi elfzel1
|
|
wa w3a elfzm11 simp3 syl6bi impancom mpancom syl5com eluzelz zltnle syl2anc
|
|
wb sylibd mt2d ) ACDEFZABCGHIZJIFZCAKLZCAMUNUPACNLZUQOZUNCPFZUPURCAQZBPFZUP
|
|
UTURRABUOSVBUTUPURVBUTTUPAPFZBAKLZURUAURABCUBVCVDURUCUDUEUFUGUNVCUTURUSUKCA
|
|
UHVAACUIUJULUM $.
|
|
|
|
$( One plus the upper bound of a finite set of integers is not a member of
|
|
that set. (Contributed by Scott Fenton, 16-Dec-2017.) $)
|
|
fzp1nel $p |- -. ( N + 1 ) e. ( M ... N ) $=
|
|
( c1 caddc co cfz wcel wn cz w3a cle wbr wa wi clt zre ltp1d peano2z zltnle
|
|
wb mpdan mpbid intnand 3ad2ant2 elfz2 notbii imnan bitr4i mpbir ) BCDEZABFE
|
|
GZHZAIGZBIGZUJIGZJZAUJKLZUJBKLZMZHZNZUNUMUTUOUNURUQUNBUJOLZURHZUNBBPQUNUOVB
|
|
VCTBRBUJSUAUBUCUDULUPUSMZHVAUKVDUJABUEUFUPUSUGUHUI $.
|
|
|
|
${
|
|
$d j k x K $. $d j k x M $. $d j k x N $. $d k x ph $.
|
|
$( Reversal of scanning order inside of a quantification over a finite set
|
|
of sequential integers. (Contributed by NM, 25-Nov-2005.) $)
|
|
fzrevral $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ )
|
|
-> ( A. j e. ( M ... N ) ph
|
|
<-> A. k e. ( ( K - N ) ... ( K - M ) ) [. ( K - k ) / j ]. ph ) ) $=
|
|
( cz wcel cfz co wral cv cmin wsbc wi wa wb elfzelz syl wceq simpr sylan2
|
|
w3a fzrev anassrs mpbid rspsbc ex 3impa ralrimdv nfv nfcv nfsbc1v nfralxy
|
|
com23 fzrev2i oveq2 sbceq1d rspcv cc zcn zcnd nncan syl2an eqcomd sbceq1a
|
|
sylibrd ralrimd 3ad2ant3 impbid ) EGHZFGHZDGHZUCZABEFIJZKZABDCLZMJZNZCDFM
|
|
JZDEMJZIJZKZVNVPVSCWBVNVQWBHZVPVSVKVLVMWDVPVSOZOVKVLPZVMPZWDWEWGWDPZVRVOH
|
|
ZWEWHWDWIWGWDUAWDWGVQGHZWDWIQZVQVTWARWFVMWJWKDVQEFUDUEUBUFABVRVOUGSUHUIUO
|
|
UJVMVKWCVPOVLVMWCABVOVMBUKVSBCWBBWBULABVRUMUNVMBLZVOHZWCAVMWMWCAOVMWMPZWC
|
|
ABDDWLMJZMJZNZAWNWOWBHWCWQODWLEFUPVSWQCWOWBVQWOTABVRWPVQWODMUQURUSSWNWLWP
|
|
TAWQQWNWPWLVMDUTHWLUTHWPWLTWMDVAWMWLWLEFRVBDWLVCVDVEABWPVFSVGUHUOVHVIVJ
|
|
$.
|
|
|
|
$( Reversal of scanning order inside of a quantification over a finite set
|
|
of sequential integers. (Contributed by NM, 25-Nov-2005.) $)
|
|
fzrevral2 $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ )
|
|
-> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph
|
|
<-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) ) $=
|
|
( cz wcel cmin co cfz wral wb w3a zsubcl 3adant2 3adant3 cc wceq zcn wsbc
|
|
cv simp1 fzrevral syl3anc nncan oveq12d syl3an raleqdv bitrd 3coml ) DGHZ
|
|
EGHZFGHZABDFIJZDEIJZKJLZABDCUBIJUAZCEFKJZLZMULUMUNNZUQURCDUPIJZDUOIJZKJZL
|
|
ZUTVAUOGHZUPGHZULUQVEMULUNVFUMDFOPULUMVGUNDEOQULUMUNUCABCDUOUPUDUEVAURCVD
|
|
USULDRHZUMERHZUNFRHZVDUSSDTETFTVHVIVJNVBEVCFKVHVIVBESVJDEUFQVHVJVCFSVIDFU
|
|
FPUGUHUIUJUK $.
|
|
|
|
$( Reversal of scanning order inside of a quantification over a finite set
|
|
of sequential integers. (Contributed by NM, 20-Nov-2005.) $)
|
|
fzrevral3 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( A. j e. ( M ... N ) ph
|
|
<-> A. k e. ( M ... N ) [. ( ( M + N ) - k ) / j ]. ph ) ) $=
|
|
( cz wcel wa cfz co wral caddc cv cmin wsbc wb zaddcl fzrevral cc zcn
|
|
mpd3an3 wceq pncan pncan2 oveq12d syl2an raleqdv bitrd ) DFGZEFGZHZABDEIJ
|
|
ZKZABDELJZCMNJOZCUNENJZUNDNJZIJZKZUOCULKUIUJUNFGUMUSPDEQABCUNDERUAUKUOCUR
|
|
ULUIDSGZESGZURULUBUJDTETUTVAHUPDUQEIDEUCDEUDUEUFUGUH $.
|
|
|
|
$( Shift the scanning order inside of a quantification over a finite set of
|
|
sequential integers. (Contributed by NM, 27-Nov-2005.) $)
|
|
fzshftral $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ )
|
|
-> ( A. j e. ( M ... N ) ph
|
|
<-> A. k e. ( ( M + K ) ... ( N + K ) ) [. ( k - K ) / j ]. ph ) ) $=
|
|
( vx cz wcel cfz co wral cc0 cmin wsbc caddc wb wa cc wceq cv 0z fzrevral
|
|
w3a mp3an3 3adant3 zsubcl mpan id syl3an 3com12 elfzelz oveq2 sbcco3g syl
|
|
sylan2 ralbidva 3ad2ant3 cneg df-neg oveq2i subneg addcom syl5eqr 3adant2
|
|
zcn eqtrd oveq12d 3coml raleqdv zcnd negsubdi2 syl2an sbceq1d 3bitrd ) EH
|
|
IZFHIZDHIZUDZABEFJKLZABMGUAZNKZOZGMFNKZMENKZJKLZWCGDCUAZNKZOZCDWENKZDWDNK
|
|
ZJKZLZABWGDNKZOZCEDPKZFDPKZJKZLZVPVQVTWFQZVRVPVQMHIZWTUBABGMEFUCUEUFVQVPV
|
|
RWFWMQZVQWDHIZVPWEHIZVRVRXBXAVQXCUBMFUGUHXAVPXDUBMEUGUHVRUIWCGCDWDWEUCUJU
|
|
KVSWMABMWHNKZOZCWLLZXFCWRLZWSVRVPWMXGQVQVRWIXFCWLWGWLIVRWGHIZWIXFQZWGWJWK
|
|
ULVRXIRWHHIXJDWGUGAGBWHWBXEHWAWHMNUMUNUOUPUQURVSXFCWLWRVPESIZVQFSIZVRDSIZ
|
|
WLWRTZEVFFVFDVFZXMXKXLXNXMXKXLUDWJWPWKWQJXMXKWJWPTXLXMXKRZWJDEUSZNKZWPXQW
|
|
EDNEUTVAXPXRDEPKWPDEVBDEVCVGVDUFXMXLWKWQTXKXMXLRZWKDFUSZNKZWQXTWDDNFUTVAX
|
|
SYADFPKWQDFVBDFVCVGVDVEVHVIUJVJVRVPXHWSQVQVRXFWOCWRVRWGWRIZRABXEWNVRXMWGS
|
|
IZXEWNTYBXOYBWGWGWPWQULVKXMYCRXEWHUSWNWHUTDWGVLVDVMVNUQURVOVO $.
|
|
$}
|
|
|
|
$( Membership of an integer greater than 1 decreased by 1 in a 1 based finite
|
|
set of sequential integers (Contributed by Alexander van der Vekens,
|
|
14-Sep-2018.) $)
|
|
ige2m1fz1 $p |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. ( 1 ... N ) ) $=
|
|
( c2 cuz cfv wcel c1 cmin co cfz wceq 1e2m1 a1i oveq2d 2nn uzsubsubfz1 mpan
|
|
cn eqeltrd ) ABCDEZAFGHABFGHZGHZFAIHZSFTAGFTJSKLMBQESUAUBENBAOPR $.
|
|
|
|
$( Membership in a 0 based finite set of sequential integers. (Contributed
|
|
by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander
|
|
van der Vekens, 15-Sep-2018.) $)
|
|
ige2m1fz $p |- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. ( 0 ... N ) ) $=
|
|
( cn0 wcel c2 cle wbr wa c1 cfz co cc0 cmin cuz cfv 1eluzge0 fzss1 ax-mp cz
|
|
wss 2z a1i nn0z adantr simpr eluz2 syl3anbrc ige2m1fz1 syl sseldi ) ABCZDAE
|
|
FZGZHAIJZKAIJZAHLJZHKMNCUMUNSOHKAPQULADMNCZUOUMCULDRCZARCZUKUPUQULTUAUJURUK
|
|
AUBUCUJUKUDDAUEUFAUGUHUI $.
|
|
|
|
$( An integer is in the integer range from zero to one iff it is either zero
|
|
or one. (Contributed by Jim Kingdon, 11-Nov-2021.) $)
|
|
fz01or $p |- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) ) $=
|
|
( cc0 c1 cfz co wcel caddc wo wceq cuz eluzfz1 ax-mp eleq2i elfz1eq eleqtri
|
|
cfv bitri eleq1 mpbiri impbii 1eluzge0 fzsplit elun 0nn0 nn0uz 0p1e1 oveq1i
|
|
cun cn0 cn 1nn nnuz orbi12i ) ABCDEZFZABBDEZFZABCGEZCDEZFZHZABIZACIZHUOAUPU
|
|
SUHZFVAUNVDABUNFZUNVDICBJPZFVEUABCKLBBCUBLMAUPUSUCQUQVBUTVCUQVBABNVBUQBUPFZ
|
|
BVFFVGBUIVFUDUEOBBKLABUPRSTUTACCDEZFZVCUSVHAURCCDUFUGMVIVCACNVCVICVHFZCCJPZ
|
|
FVJCUJVKUKULOCCKLACVHRSTQUMQ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite intervals of nonnegative integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite intervals of nonnegative integers (or "finite sets of sequential
|
|
nonnegative integers") are finite intervals of integers with 0 as lower bound:
|
|
` ( 0 ... N ) `, usually abbreviated by "fz0".
|
|
$)
|
|
|
|
$( Membership in a finite set of sequential nonnegative integers.
|
|
(Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
elfz2nn0 $p |- ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ N e. NN0 /\ K <_ N ) ) $=
|
|
( cc0 cuz cfv wcel cn0 cle wbr cfz w3a elnn0uz anbi1i eluznn0 eluzle adantl
|
|
wa co cz nn0z jca wb eluz syl2an biimprd impr impbida pm5.32i bitr3i 3anass
|
|
elfzuzb 3bitr4i ) ACDEFZBADEFZQZAGFZBGFZABHIZQZQZACBJRFUPUQURKUOUPUNQZUTUPU
|
|
MUNALMUPUNUSUPUNUSVAUQURBANUNURUPABOPUAUPUQURUNUPUQQUNURUPASFBSFUNURUBUQATB
|
|
TABUCUDUEUFUGUHUIACBUKUPUQURUJUL $.
|
|
|
|
$( Characterization of a finite set of sequential nonnegative integers.
|
|
(Contributed by NM, 1-Aug-2005.) $)
|
|
fznn0 $p |- ( N e. NN0
|
|
-> ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ K <_ N ) ) ) $=
|
|
( cn0 wcel cc0 cfz co cz cle wbr w3a wa wb nn0z elfz1 sylancr df-3an elnn0z
|
|
0z anbi1i bitr4i syl6bb ) BCDZAEBFGDZAHDZEAIJZABIJZKZACDZUGLZUCEHDBHDUDUHMS
|
|
BNAEBOPUHUEUFLZUGLUJUEUFUGQUIUKUGARTUAUB $.
|
|
|
|
$( A member of a finite set of sequential nonnegative integers is a
|
|
nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario
|
|
Carneiro, 28-Apr-2015.) $)
|
|
elfznn0 $p |- ( K e. ( 0 ... N ) -> K e. NN0 ) $=
|
|
( cc0 cfz co wcel cn0 cle wbr elfz2nn0 simp1bi ) ACBDEFAGFBGFABHIABJK $.
|
|
|
|
$( The upper bound of a nonempty finite set of sequential nonnegative
|
|
integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.)
|
|
(Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
elfz3nn0 $p |- ( K e. ( 0 ... N ) -> N e. NN0 ) $=
|
|
( cc0 cfz co wcel cn0 cle wbr elfz2nn0 simp2bi ) ACBDEFAGFBGFABHIABJK $.
|
|
|
|
$( 0 is an element of a finite set of sequential nonnegative integers with a
|
|
nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.) $)
|
|
0elfz $p |- ( N e. NN0 -> 0 e. ( 0 ... N ) ) $=
|
|
( cn0 wcel cc0 cle wbr cfz co 0nn0 a1i id nn0ge0 elfz2nn0 syl3anbrc ) ABCZD
|
|
BCZODAEFDDAGHCPOIJOKALDAMN $.
|
|
|
|
$( A nonnegative integer is always part of the finite set of sequential
|
|
nonnegative integers with this integer as upper bound. (Contributed by
|
|
Scott Fenton, 21-Mar-2018.) $)
|
|
nn0fz0 $p |- ( N e. NN0 <-> N e. ( 0 ... N ) ) $=
|
|
( cn0 wcel cc0 cfz co cle wbr nn0re leidd fznn0 mpbir2and elfz3nn0 impbii
|
|
id ) ABCZADAEFCZPQPAAGHPOPAAIJAAKLAAMN $.
|
|
|
|
$( An element of a finite set of sequential nonnegative integers is an
|
|
element of an extended finite set of sequential nonnegative integers.
|
|
(Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened
|
|
by OpenAI, 25-Mar-2020.) $)
|
|
elfz0add $p |- ( ( A e. NN0 /\ B e. NN0 )
|
|
-> ( N e. ( 0 ... A ) -> N e. ( 0 ... ( A + B ) ) ) ) $=
|
|
( cn0 wcel wa cc0 cfz co caddc cuz cfv wss cz nn0z uzid syl anim12i uzaddcl
|
|
id fzss2 sseld ) ADEZBDEZFZGAHIZGABJIZHIZCUEUGAKLZEZUFUHMUEAUIEZUDFUJUCUKUD
|
|
UDUCANEUKAOAPQUDTRBAASQAGUGUAQUB $.
|
|
|
|
$( An integer range from 0 to 2 is an unordered triple. (Contributed by
|
|
Alexander van der Vekens, 1-Feb-2018.) $)
|
|
fz0tp $p |- ( 0 ... 2 ) = { 0 , 1 , 2 } $=
|
|
( cc0 c2 cfz co caddc c1 ctp 2cn addid2i eqcomi oveq2i cz wcel wceq 0z fztp
|
|
ax-mp eqid id a1i 0p1e1 tpeq123d 3eqtri ) ABCDAABEDZCDZAAFEDZUDGZAFBGZBUDAC
|
|
UDBBHIZJKALMUEUGNOAPQAANZUGUHNARUJAAUFFUDBUJSUFFNUJUATUDBNUJUITUBQUC $.
|
|
|
|
$( An element of a finite set of sequential nonnegative integers is an
|
|
element of a finite set of sequential nonnegative integers with the upper
|
|
bound being an element of the finite set of sequential nonnegative
|
|
integers with the same lower bound as for the first interval and the
|
|
element under consideration as upper bound. (Contributed by Alexander van
|
|
der Vekens, 3-Apr-2018.) $)
|
|
elfz0ubfz0 $p |- ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) )
|
|
-> K e. ( 0 ... L ) ) $=
|
|
( cc0 cfz co wa cn0 cle wbr w3a wi elfz2nn0 cz elfz2 elnn0z sylbi com12 imp
|
|
wcel simpr1 simpr 0z zletr mp3an1 simplbi2 sylsyld impancom adantr 3ad2ant3
|
|
expd com13 3ad2ant1 impcom simplrl 3jca ex sylibr ) ADCEFTZBACEFTZGAHTZBHTZ
|
|
ABIJZKZADBEFTUSUTVDUSVACHTZACIJZKZUTVDLACMUTVGVDUTANTZCNTZBNTZKZVCBCIJZGZGZ
|
|
VGVDLBACOVNVGVDVNVGGVAVBVCVNVAVEVFUAVGVNVBVAVEVNVBLVFVNVAVBVKVMVAVBLZVJVHVM
|
|
VOLVIVMVJVOVCVJVOLVLVAVJVCVBVAVHDAIJZGVJVCVBLZLAPVHVJVPVQVHVJGZVPVCVBVRVJVP
|
|
VCGZDBIJZVBVHVJUBDNTVHVJVSVTLUCDABUDUEVBVJVTBPUFUGUKUHQULUIRUJSRUMUNVKVCVLV
|
|
GUOUPUQQRQSABMUR $.
|
|
|
|
$( A member of a finite set of sequential nonnegative integers is a member of
|
|
a finite set of sequential nonnegative integers with a member of a finite
|
|
set of sequential nonnegative integers starting at the upper bound of the
|
|
first interval. (Contributed by Alexander van der Vekens,
|
|
27-May-2018.) $)
|
|
elfz0fzfz0 $p |- ( ( M e. ( 0 ... L ) /\ N e. ( L ... X ) )
|
|
-> M e. ( 0 ... N ) ) $=
|
|
( cc0 cfz co wcel wa cn0 cle wbr w3a wi elfz2nn0 cz cr adantr imp sylbi zre
|
|
elfz2 nn0re 3anim123i 3expa letr syl simplll simpr elnn0z 0red adantl exp4b
|
|
syl3anc com23 sylanbrc 3jca ex syld 3impia com13 com12 3ad2ant3 sylibr ) BE
|
|
AFGHZCADFGHZIBJHZCJHZBCKLZMZBECFGHVEVFVJVEVGAJHZBAKLZMZVFVJNBAOVFVMVJVFAPHZ
|
|
DPHZCPHZMZACKLZCDKLZIZIVMVJNZCADUBVQVTWAVPVNVTWANVOVTVPWAVRVPWANVSVMVPVRVJV
|
|
GVKVLVPVRVJNZNVGVKIZVPVLWBWCVPVLVRVJWCVPIZVLVRIZVIVJWDBQHZAQHZCQHZMZWEVINVG
|
|
VKVPWIVGWFVKWGVPWHBUCAUCCUAZUDUEBACUFUGWDVIVJWDVIIZVGVHVIVGVKVPVIUHWKVPECKL
|
|
ZVHWDVPVIWCVPUIRWDVIWLWCVPVIWLNZVGVPWMNZVKVGBPHZEBKLZIWNBUJWOWPWNWOVPWPWMWO
|
|
VPWPVIWLWOVPIZEQHWFWHWPVIIWLNWQUKWOWFVPBUARVPWHWOWJULEBCUFUNUMUOSTRSSCUJUPW
|
|
DVIUIUQURUSUMUOUTVARVBVCSTVBTSBCOVD $.
|
|
|
|
$( If a member of a finite set of sequential integers with a lower bound
|
|
being a member of a finite set of sequential nonnegative integers with the
|
|
same upper bound, this member is also a member of the finite set of
|
|
sequential nonnegative integers. (Contributed by Alexander van der
|
|
Vekens, 21-Apr-2018.) $)
|
|
fz0fzelfz0 $p |- ( ( N e. ( 0 ... R ) /\ M e. ( N ... R ) )
|
|
-> M e. ( 0 ... R ) ) $=
|
|
( cc0 cfz co wcel wa cn0 cle wbr w3a wi elfz2nn0 cz adantr 3jca imp sylbi
|
|
cr elfz2 simplr 0red nn0re zre adantl nn0ge0 anim1i letr elnn0z exp31 com23
|
|
sylc sylanbrc 3ad2ant1 com13 adantrd 3ad2ant3 simpr2 simplrr com12 sylibr
|
|
ex ) CDAEFZGZBCAEFGZHBIGZAIGZBAJKZLZBVDGVEVFVJVECIGZVHCAJKZLZVFVJMCANVFVMVJ
|
|
VFCOGZAOGZBOGZLZCBJKZVIHZHZVMVJMBCAUAVTVMVJVTVMHVGVHVIVTVMVGVQVSVMVGMZVPVNV
|
|
SWAMVOVPVRWAVIVMVRVPVGVKVHVRVPVGMMVLVKVPVRVGVKVPVRVGVKVPHZVRHZVPDBJKZVGVKVP
|
|
VRUBWCDTGZCTGZBTGZLZDCJKZVRHWDWBWHVRWBWEWFWGWBUCVKWFVPCUDPVPWGVKBUEUFQPWBWI
|
|
VRVKWIVPCUGPUHDCBUIUMBUJUNUKULUOUPUQURRRVTVKVHVLUSVQVRVIVMUTQVCSVASRBANVB
|
|
$.
|
|
|
|
$( Subtraction closure for a member of a finite set of sequential nonnegative
|
|
integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
fznn0sub2 $p |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) ) $=
|
|
( cc0 cfz co wcel cmin cle wbr elfzle1 cz wb elfzel2 elfzelz cr zre subge02
|
|
syl2an syl2anc mpbid cuz cfv cn0 fznn0sub nn0uz syl6eleq elfz5 mpbird ) ACB
|
|
DEZFZBAGEZUIFZUKBHIZUJCAHIZUMACBJUJBKFZAKFZUNUMLZACBMZACBNUOBOFAOFUQUPBPAPB
|
|
AQRSTUJUKCUAUBZFUOULUMLUJUKUCUSACBUDUEUFURUKCBUGSUH $.
|
|
|
|
$( Membership of an integer greater than L decreased by L in a finite set of
|
|
sequential nonnegative integers. (Contributed by Alexander van der
|
|
Vekens, 16-Sep-2018.) $)
|
|
uzsubfz0 $p |- ( ( L e. NN0 /\ N e. ( ZZ>= ` L ) )
|
|
-> ( N - L ) e. ( 0 ... N ) ) $=
|
|
( cn0 wcel cuz cfv wa cc0 cfz co cmin cle wbr simpl eluznn0 eluzle elfz2nn0
|
|
adantl syl3anbrc fznn0sub2 syl ) ACDZBAEFDZGZAHBIJZDZBAKJUEDUDUBBCDABLMZUFU
|
|
BUCNBAOUCUGUBABPRABQSABTUA $.
|
|
|
|
$( The difference of an integer in a finite set of sequential nonnegative
|
|
integers and and an integer of a finite set of sequential integers with
|
|
the same upper bound and the nonnegative integer as lower bound is a
|
|
member of the finite set of sequential nonnegative integers. (Contributed
|
|
by Alexander van der Vekens, 6-Jun-2018.) $)
|
|
fz0fzdiffz0 $p |- ( ( M e. ( 0 ... N ) /\ K e. ( M ... N ) )
|
|
-> ( K - M ) e. ( 0 ... N ) ) $=
|
|
( cc0 cfz co wcel wa cn0 cle wbr w3a adantl wb elfznn0 adantr mpbid wi cz
|
|
cr cmin fz0fzelfz0 elfzle1 nn0sub syl2anr elfz3nn0 elfz2 zsubcl zred ancoms
|
|
elfz2nn0 3adant2 zre 3ad2ant3 3ad2ant2 3jca nn0ge0 nn0re syl2an anim1i letr
|
|
subge02 sylc exp31 a1i com14 impcom sylbi com13 3adant3 imp mpancom sylibr
|
|
) BDCEFZGZABCEFGZHZABUAFZIGZCIGZVRCJKZLZVRVNGAVNGZVQWBCABUBWCVQHZVSVTWAWDBA
|
|
JKZVSVQWEWCVPWEVOABCUCMMVQBIGZAIGWEVSNWCVOWFVPBCOPACOBAUDUEQWCVTVQACUFPVQWA
|
|
WCVOVPWAVOWFVTBCJKZLVPWARZBCUKWFVTWHWGVTWFWHVPWFVTWAVPBSGZCSGZASGZLZWEACJKZ
|
|
HZHWFVTWARRZABCUGWNWLWOWMWLWORWEVTWLWFWMWAWLWFWMWARRRVTWLWFWMWAWLWFHZWMHVRT
|
|
GZATGZCTGZLZVRAJKZWMHWAWPWTWMWLWTWFWLWQWRWSWIWKWQWJWKWIWQWKWIHVRABUHUIUJULW
|
|
KWIWRWJAUMUNZWJWIWSWKCUMUOUPPPWPXAWMWPDBJKZXAWFXCWLBUQMWLWRBTGXCXANWFXBBURA
|
|
BVBUSQUTVRACVAVCVDVEVFMVGVHVIVGVJVHVKMUPVLVRCUKVM $.
|
|
|
|
$( Subtracting the lower bound of a finite set of sequential integers from an
|
|
element of this set. (Contributed by Alexander van der Vekens,
|
|
29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) $)
|
|
elfzmlbm $p |- ( K e. ( M ... N ) -> ( K - M ) e. ( 0 ... ( N - M ) ) ) $=
|
|
( cfz wcel cmin cn0 cle wbr cc0 cuz cfv elfzuz uznn0sub syl elfzuz2 elfzelz
|
|
co zred elfzel2 elfzel1 elfzle2 lesub1dd elfz2nn0 syl3anbrc ) ABCDREZABFRZG
|
|
EZCBFRZGEZUGUIHIUGJUIDREUFABKLZEUHABCMBANOUFCUKEUJABCPBCNOUFACBUFAABCQSUFCA
|
|
BCTSUFBABCUASABCUBUCUGUIUDUE $.
|
|
|
|
$( Subtracting the lower bound of a finite set of sequential integers from an
|
|
element of this set. (Contributed by Alexander van der Vekens,
|
|
29-Mar-2018.) $)
|
|
elfzmlbp $p |- ( ( N e. ZZ /\ K e. ( M ... ( M + N ) ) )
|
|
-> ( K - M ) e. ( 0 ... N ) ) $=
|
|
( cz wcel co cfz wa cn0 cle wbr w3a cc0 wi wb adantr cr zre adantl imp cmin
|
|
caddc elfz2 znn0sub biimpcd impcom zaddcl adantlr zred letr syl3anc addge01
|
|
syl2an elnn0z simplbi2 sylbird syld df-3an bitr3i 3anim123i sylbi lesubadd2
|
|
3ancoma syl biimprcd 3jca exp31 com23 3adant2 com12 syl5bi elfz2nn0 sylibr
|
|
) CDEZABBCUBFZGFEZHABUAFZIEZCIEZVQCJKZLZVQMCGFEVNVPWAVPBDEZVODEZADEZLZBAJKZ
|
|
AVOJKZHZHZVNWAABVOUCWIVNWAWEWHVNWANZWBWDWHWJNWCWBWDHZVNWHWAWKVNWHWAWKVNHZWH
|
|
HVRVSVTWHWLVRWFWLVRNWGWLWFVRWKWFVROVNBAUDPUEPUFWLWHVSWLWHBVOJKZVSWLBQEZAQEZ
|
|
VOQEWHWMNWKWNVNWBWNWDBRZPZPWKWOVNWDWOWBARZSPWLVOWBVNWCWDBCUGUHUIBAVOUJUKWLW
|
|
MMCJKZVSWKWNCQEZWSWMOVNWQCRZBCULUMVNWSVSNWKVSVNWSCUNUOSUPUQTWHWLVTWGWLVTNWF
|
|
WLVTWGWLWOWNWTLZVTWGOWLWDWBVNLZXBWLWBWDVNLXCWBWDVNURWBWDVNVCUSWDWOWBWNVNWTW
|
|
RWPXAUTVAABCVBVDVESUFVFVGVHVITVJVKTVQCVLVM $.
|
|
|
|
$( Lemma for theorems about the central binomial coefficient. (Contributed
|
|
by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro,
|
|
2-Aug-2014.) $)
|
|
fzctr $p |- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) ) $=
|
|
( cn0 wcel cc0 c2 cmul co cfz cle wbr nn0ge0 caddc cr nn0re nn0addge1 nn0cn
|
|
mpancom 2timesd breqtrrd cz wa wb nn0z 0zd 2z zmulcl sylancr elfz mpbir2and
|
|
syl3anc ) ABCZADEAFGZHGCZDAIJZAULIJZAKUKAAALGZULIAMCUKAUPIJANAAOQUKAAPRSUKA
|
|
TCZDTCULTCZUMUNUOUAUBAUCZUKUDUKETCUQURUEUSEAUFUGADULUHUJUI $.
|
|
|
|
$( The difference of two integers from a finite set of sequential nonnegative
|
|
integers is also element of this finite set of sequential integers.
|
|
(Contributed by Alexander van der Vekens, 12-Jun-2018.) $)
|
|
difelfzle $p |- ( ( K e. ( 0 ... N ) /\ M e. ( 0 ... N ) /\ K <_ M )
|
|
-> ( M - K ) e. ( 0 ... N ) ) $=
|
|
( cc0 co wcel cle wbr w3a cn0 cz wa wi elfznn0 nn0z syl2anr adantr cr nn0re
|
|
wb cfz cmin zsubcl subge0 biimpar jca exp31 syl2im elnn0z elfz3nn0 3ad2ant1
|
|
3imp sylibr elfz2nn0 resubcl syl2an 3ad2ant2 nn0ge0 adantl subge02 mpbid ex
|
|
simpl3 letrd sylbi syl5com a1dd syl3anbrc ) ADCUAEZFZBVIFZABGHZIZBAUBEZJFZC
|
|
JFZVNCGHZVNVIFVMVNKFZDVNGHZLZVOVJVKVLVTVJAJFZVKBJFZVLVTMACNZBCNWAWBVLVTWAWB
|
|
LZVLLVRVSWDVRVLWBBKFAKFVRWABOAOBAUCPQWDVSVLWBBRFZARFZVSVLTWABSZASZBAUDPUEUF
|
|
UGUHULVNUIUMVJVKVPVLACUJUKVJVKVLVQVJVKVQVLVJWAVKVQWCVKWBVPBCGHZIZWAVQMBCUNW
|
|
JWAVQWJWALZVNBCWJWEWFVNRFWAWBVPWEWIWGUKZWHBAUOUPWJWEWAWLQWJCRFZWAVPWBWMWICS
|
|
UQQWKDAGHZVNBGHZWAWNWJAURUSWJWEWFWNWOTWAWLWHBAUTUPVAWBVPWIWAVCVDVBVEVFVGULV
|
|
NCUNVH $.
|
|
|
|
$( The difference of two integers from a finite set of sequential nonnegative
|
|
integers increased by the upper bound is also element of this finite set
|
|
of sequential integers. (Contributed by Alexander van der Vekens,
|
|
12-Jun-2018.) $)
|
|
difelfznle $p |- ( ( K e. ( 0 ... N ) /\ M e. ( 0 ... N ) /\ -. K <_ M )
|
|
-> ( ( M + N ) - K ) e. ( 0 ... N ) ) $=
|
|
( cc0 co wcel cle wbr w3a caddc cn0 cz elfz2nn0 wa 3adant3 sylbi cr zred wb
|
|
syl wn cmin nn0addcl nn0zd elfzelz zsubcl syl2anr adantr elfzel2 nn0readdcl
|
|
cfz adantl elfzle2 elfzle1 nn0re anim12ci addge02 anim12i letr imp syl31anc
|
|
mpbid zre readdcl sylan subge0 mpbird elnn0z sylanbrc elfz3nn0 3ad2ant1 clt
|
|
wi zltnle ancoms ltle sylbird syl2an 3impia leadd1d lesubadd2d syl3anbrc )
|
|
ADCUKEZFZBWCFZABGHUAZIZBCJEZAUBEZKFZCKFZWICGHZWIWCFWGWILFZDWIGHZWJWDWEWMWFW
|
|
EWHLFZALFZWMWDWEBKFZWKBCGHZIZWOBCMZWQWKWOWRWQWKNWHBCUCUDOPADCUEZWHAUFUGOWGW
|
|
NAWHGHZWDWEXBWFWDWENZAQFZCQFZWHQFZACGHZCWHGHZNZXBWDXDWEWDAXARUHZWDXEWEWDCAD
|
|
CUIRUHZWEXFWDWEWSXFWTWQWKXFWRBCUJOPULZWDXGWEXHADCUMWEDBGHZXHBDCUNWEXEBQFZNZ
|
|
XMXHSWEWSXOWTWQWKXOWRWQXNWKXEBUOZCUOZUPOPCBUQTVBURXDXEXFIXIXBACWHUSUTVAOWGX
|
|
FXDNZWNXBSWDWEXRWFWDWPWEXRXAWPXDWEXFAVCZWEXNXENZXFWEWSXTWTWQWKXTWRWQXNWKXEX
|
|
PXQUROPBCVDTUPVEOWHAVFTVGWIVHVIWDWEWKWFACVJVKWGWLWHACJEGHZWGBAGHZYAWDWEWFYB
|
|
WDWPBLFZWFYBVMWEXABDCUEZWPYCNWFBAVLHZYBYCWPYEWFSBAVNVOYCXNXDYEYBVMWPBVCXSBA
|
|
VPUGVQVRVSWDWEYBYASWFXCBACWEXNWDWEBYDRULXJXKVTOVBWDWEWLYASWFXCWHACXLXJXKWAO
|
|
VGWICMWB $.
|
|
|
|
$( Express the set of nonnegative integers as the disjoint (see ~ nn0disj )
|
|
union of the first ` N + 1 ` values and the rest. (Contributed by AV,
|
|
8-Nov-2019.) $)
|
|
nn0split $p |- ( N e. NN0
|
|
-> NN0 = ( ( 0 ... N ) u. ( ZZ>= ` ( N + 1 ) ) ) ) $=
|
|
( cn0 wcel cc0 cuz cfv c1 caddc co cmin cfz cun wceq a1i peano2nn0 syl6eleq
|
|
nn0uz uzsplit syl cc nn0cn pncan1 oveq2d uneq1d 3eqtrd ) ABCZBDEFZDAGHIZGJI
|
|
ZKIZUHEFZLZDAKIZUKLBUGMUFQNUFUHUGCUGULMUFUHBUGAOQPDUHRSUFUJUMUKUFUIADKUFATC
|
|
UIAMAUAAUBSUCUDUE $.
|
|
|
|
${
|
|
$d N k $.
|
|
$( The first ` N + 1 ` elements of the set of nonnegative integers are
|
|
distinct from any later members. (Contributed by AV, 8-Nov-2019.) $)
|
|
nn0disj $p |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) = (/) $=
|
|
( vk cc0 cfz co c1 caddc cuz cfv c0 wceq wcel clt wbr cle syl cz mpbid wn
|
|
zred cin wss cv cmin elin simprbi eluzel2 eluzelz zlem1lt syl2anc simplbi
|
|
eluzle wb elfzle2 elfzel2 adantr sylbi lenltd pncan1 eqcomd breq1d notbid
|
|
wa cc zcnd bitrd pm2.21dd ssriv ss0 ax-mp ) CADEZAFGEZHIZUAZJUBVNJKBVNJBU
|
|
CZVNLZVLFUDEZVOMNZVOJLVPVLVOONZVRVPVOVMLZVSVPVOVKLZVTVOVKVMUEZUFZVLVOULPV
|
|
PVLQLZVOQLZVSVRUMVPVTWDWCVLVOUGPVPVTWEWCVLVOUHPZVLVOUIUJRVPVOAONZVRSZVPWA
|
|
WGVPWAVTWBUKVOCAUNPVPWGAVOMNZSWHVPVOAVPVOWFTVPAVPWAVTVCAQLZWBWAWJVTVOCAUO
|
|
UPUQZTURVPWIVRVPAVQVOMVPVQAVPAVDLVQAKVPAWKVEAUSPUTVAVBVFRVGVHVNVIVJ $.
|
|
$}
|
|
|
|
$( A one value function. (Contributed by Alexander van der Vekens,
|
|
3-Dec-2017.) $)
|
|
1fv $p |- ( ( N e. V /\ P = { <. 0 , N >. } )
|
|
-> ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) ) $=
|
|
( wcel cc0 cop csn wceq wa cfz co wf cfv wf1o cz f1osng mpan biimpi adantr
|
|
0z wfo crn wi f1ofo dffo2 wss fzsn ax-mp eqcomi feq2i snssi fss syl2an 3syl
|
|
ex mpcom fvsng jca wb feq1 fveq1 eqeq1d anbi12d adantl mpbird ) BCDZAEBFGZH
|
|
ZIEEJKZCALZEAMZBHZIZVICVGLZEVGMZBHZIZVFVQVHVFVNVPEGZBGZVGNZVFVNEODZVFVTTEBO
|
|
CPQVTVRVSVGUAZVRVSVGLZVGUBVSHZIZVFVNUCZVRVSVGUDWBWEVRVSVGUERWCWFWDWCVFVNWCV
|
|
IVSVGLZVSCUFVNVFWCWGVRVIVSVGVIVRWAVIVRHTEUGUHUIUJRBCUKVIVSCVGULUMUOSUNUPWAV
|
|
FVPTEBOCUQQURSVHVMVQUSVFVHVJVNVLVPVICAVGUTVHVKVOBEAVGVAVBVCVDVE $.
|
|
|
|
${
|
|
$d P a b c d $. $d V a b c d $.
|
|
$( The first four function values of a word of length at least 4.
|
|
(Contributed by Alexander van der Vekens, 18-Nov-2017.) $)
|
|
4fvwrd4 $p |- ( ( L e. ( ZZ>= ` 3 ) /\ P : ( 0 ... L ) --> V )
|
|
-> E. a e. V E. b e. V E. c e. V E. d e. V
|
|
( ( ( P ` 0 ) = a /\ ( P ` 1 ) = b )
|
|
/\ ( ( P ` 2 ) = c /\ ( P ` 3 ) = d ) ) ) $=
|
|
( c3 cuz cfv wcel cc0 wa cv wceq wrex c1 c2 rexbii bitri cfz co simpr cn0
|
|
wf 0nn0 elnn0uz mpbi 3nn0 uzss ax-mp sseli eluzfz sylancr adantr ffvelrnd
|
|
wss risset eqcom sylib 1eluzge0 cz cle 1z 3z 1le3 eluz2 mpbir3an 2eluzge0
|
|
wbr jca uzuzle23 mpan r19.42v anbi2i 2rexbii r19.41v anbi1i 3bitri sylibr
|
|
jca32 ) BHIJZKZLBUAUBZCAUEZMZLAJZDNZOZDCPZQAJZENZOZECPZMZRAJZFNZOZFCPZHAJ
|
|
ZGNZOZGCPZMZMZWIWMMZWRXBMZMGCPZFCPZECPDCPZWFWOWSXCWFWJWNWFWGCKZWJWFWDCLAW
|
|
CWEUCZWCLWDKZWEWCLLIJZKZBXNKXMLUDKXOUFLUGUHWBXNBHXNKZWBXNUQHUDKXPUIHUGUHZ
|
|
LHUJUKULLLBUMUNUOUPXKWHWGOZDCPWJDWGCURXRWIDCWHWGUSSTUTWFWKCKZWNWFWDCQAXLW
|
|
CQWDKZWEWCQXNKBQIJZKXTVAWBYABHYAKZWBYAUQYBQVBKHVBKQHVCVJVDVEVFQHVGVHQHUJU
|
|
KULQLBUMUNUOUPXSWLWKOZECPWNEWKCURYCWMECWLWKUSSTUTVKWFWPCKZWSWFWDCRAXLWCRW
|
|
DKZWEWCRXNKBRIJKYEVIBVLRLBUMUNUOUPYDWQWPOZFCPWSFWPCURYFWRFCWQWPUSSTUTWFWT
|
|
CKZXCWFWDCHAXLWCHWDKZWEXPWCYHXQHLBUMVMUOUPYGXAWTOZGCPXCGWTCURYIXBGCXAWTUS
|
|
STUTWAXJXFWRXCMZMZFCPZECPDCPXFXDMZECPZDCPZXEXIYLDECCXHYKFCXHXFXGGCPZMYKXF
|
|
XGGCVNYPYJXFWRXBGCVNVOTSVPYLYMDECCYLXFYJFCPZMYMXFYJFCVNYQXDXFWRXCFCVQVOTV
|
|
PYOWIWNMZXDMZDCPYRDCPZXDMXEYNYSDCYNXFECPZXDMYSXFXDECVQUUAYRXDWIWMECVNVRTS
|
|
YRXDDCVQYTWOXDWIWNDCVQVRVSVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d F i $. $d M i $. $d P i $.
|
|
$( Two functions over 0 based finite set of sequential integers are equal
|
|
if and only if their domains have the same length and the function
|
|
values are the same at each position. (Contributed by Alexander van der
|
|
Vekens, 30-Jun-2018.) $)
|
|
2ffzeq $p |- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X
|
|
/\ P : ( 0 ... N ) --> Y ) -> ( F = P
|
|
<-> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) $=
|
|
( cn0 wcel cc0 cfz co wf w3a wceq cfv wa wfn wb ffn cv anim12i syl fzopth
|
|
wral 3adant1 eqfnfv2 cuz elnn0uz sylbi simpr syl6bi anim1d anim1i impbid1
|
|
oveq2 3ad2ant1 bitrd ) DHIZJDKLZFCMZJEKLZGAMZNZCAOZUTVBOZBUAZCPVGAPOBUTUE
|
|
ZQZDEOZVHQZVDCUTRZAVBRZQZVEVISVAVCVNUSVAVLVCVMUTFCTVBGATUBUFBUTVBCAUGUCUS
|
|
VAVIVKSVCUSVIVKUSVFVJVHUSVFJJOZVJQZVJUSDJUHPIVFVPSDUIJEJDUDUJVOVJUKULUMVJ
|
|
VFVHDEJKUPUNUOUQUR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Half-open integer ranges
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ..^ $.
|
|
$( Syntax for half-open integer ranges. $)
|
|
cfzo $a class ..^ $.
|
|
|
|
${
|
|
$d m n $.
|
|
$( Define a function generating sets of integers using a _half-open_ range.
|
|
Read ` ( M ..^ N ) ` as the integers from ` M ` up to, but not
|
|
including, ` N ` ; contrast with ` ( M ... N ) ` ~ df-fz , which
|
|
includes ` N ` . Not including the endpoint simplifies a number of
|
|
formulae related to cardinality and splitting; contrast ~ fzosplit with
|
|
~ fzsplit , for instance. (Contributed by Stefan O'Rear,
|
|
14-Aug-2015.) $)
|
|
df-fzo $a |- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) ) $.
|
|
|
|
$( Functionality of the half-open integer set function. (Contributed by
|
|
Stefan O'Rear, 14-Aug-2015.) $)
|
|
fzof $p |- ..^ : ( ZZ X. ZZ ) --> ~P ZZ $=
|
|
( vm vn cv c1 cmin co cfz cz cpw wcel wral cxp cfzo wf peano2zm fzf fovcl
|
|
sylan2 rgen2a df-fzo fmpt2 mpbi ) ACZBCZDEFZGFZHIZJZBHKAHKHHLUGMNUHABHUDH
|
|
JUCHJUEHJUHUDOUCUEUGHHGPQRSABHHUFUGMABTUAUB $.
|
|
|
|
$( Reverse closure for half-open integer sets. (Contributed by Stefan
|
|
O'Rear, 14-Aug-2015.) $)
|
|
elfzoel1 $p |- ( A e. ( B ..^ C ) -> B e. ZZ ) $=
|
|
( vm vn cz cv c1 cmin co cfz cfzo df-fzo elmpt2cl1 ) DEFFDGEGHIJKJBCLADEM
|
|
N $.
|
|
|
|
$( Reverse closure for half-open integer sets. (Contributed by Stefan
|
|
O'Rear, 14-Aug-2015.) $)
|
|
elfzoel2 $p |- ( A e. ( B ..^ C ) -> C e. ZZ ) $=
|
|
( vm vn cz cv c1 cmin co cfz cfzo df-fzo elmpt2cl2 ) DEFFDGEGHIJKJBCLADEM
|
|
N $.
|
|
|
|
$( Reverse closure for half-open integer sets. (Contributed by Stefan
|
|
O'Rear, 14-Aug-2015.) $)
|
|
elfzoelz $p |- ( A e. ( B ..^ C ) -> A e. ZZ ) $=
|
|
( cfzo co wcel cz cpw elfzoel1 elfzoel2 fzof fovcl syl2anc elpwid sseldd
|
|
id ) ABCDEZFZQGARQGRBGFCGFQGHZFABCIABCJBCSGGDKLMNRPO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z N $. $d x y z M $.
|
|
$( Value of the half-open integer set in terms of the closed integer set.
|
|
(Contributed by Stefan O'Rear, 14-Aug-2015.) $)
|
|
fzoval $p |- ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) ) $=
|
|
( vx vy vz cz wcel cfzo co c1 cmin cfz cv wi elfzoel1 a1i elfzel1 wb wceq
|
|
wa cpw peano2zm fzf fovcl sylan2 id oveq1 oveqan12d df-fzo mpd3an3 eleq2d
|
|
ovmpt2ga expcom pm5.21ndd eqrdv ) BFGZCABHIZABJKIZLIZUPAFGZCMZUQGZVAUSGZV
|
|
BUTNUPVAABOPVCUTNUPVAAURQPUTUPVBVCRUTUPTUQUSVAUTUPUSFUAZGZUQUSSUPUTURFGVE
|
|
BUBAURVDFFLUCUDUEDEABFFDMZEMZJKIZLIUSHVDVFASZVGBSVFAVHURLVIUFVGBJKUGUHDEU
|
|
IULUJUKUMUNUO $.
|
|
$}
|
|
|
|
$( Membership in a half-open finite set of integers. (Contributed by Stefan
|
|
O'Rear, 15-Aug-2015.) $)
|
|
elfzo $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ..^ N ) <->
|
|
( M <_ K /\ K < N ) ) ) $=
|
|
( cz wcel w3a c1 cmin co cfz cle wbr wa cfzo clt wb peano2zm syl3an3 fzoval
|
|
elfz eleq2d 3ad2ant3 zltlem1 3adant2 anbi2d 3bitr4d ) ADEZBDEZCDEZFZABCGHIZ
|
|
JIZEZBAKLZAUKKLZMZABCNIZEZUNACOLZMUIUGUHUKDEUMUPPCQABUKTRUIUGURUMPUHUIUQULA
|
|
BCSUAUBUJUSUOUNUGUIUSUOPUHACUCUDUEUF $.
|
|
|
|
$( Membership in a half-open integer interval. (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
elfzo2 $p |- ( K e. ( M ..^ N ) <->
|
|
( K e. ( ZZ>= ` M ) /\ N e. ZZ /\ K < N ) ) $=
|
|
( cz wcel w3a cle wbr clt wa cuz cfv cfzo an4 df-3an anbi1i 3ancoma 3bitr4i
|
|
co eluz2 3bitri elfzoelz elfzoel1 elfzoel2 3jca elfzo biadan2 3anass ) ADEZ
|
|
BDEZCDEZFZBAGHZACIHZJZJZABKLEZUKUNJZJZABCMSEZUQUKUNFUIUJJZUKJZUOJVAUMJZURJU
|
|
PUSVAUKUMUNNULVBUOUIUJUKOPUQVCURUQUJUIUMFUIUJUMFVCBATUJUIUMQUIUJUMOUAPRUTUL
|
|
UOUTUIUJUKABCUBABCUCABCUDUEABCUFUGUQUKUNUHR $.
|
|
|
|
$( Membership in a half-open integer interval. (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
elfzouz $p |- ( K e. ( M ..^ N ) -> K e. ( ZZ>= ` M ) ) $=
|
|
( cfzo co wcel cuz cfv cz clt wbr elfzo2 simp1bi ) ABCDEFABGHFCIFACJKABCLM
|
|
$.
|
|
|
|
$( The left endpoint of a half-open integer interval is in the set iff the
|
|
two arguments are integers with ` M < N ` . This provides an alternate
|
|
notation for the "strict upper integer" predicate by analogy to the "weak
|
|
upper integer" predicate ` M e. ( ZZ>= `` N ) ` . (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
fzolb $p |- ( M e. ( M ..^ N ) <-> ( M e. ZZ /\ N e. ZZ /\ M < N ) ) $=
|
|
( cfzo co wcel cuz cfv clt wbr w3a elfzo2 eluzel2 uzid impbii 3anbi1i bitri
|
|
cz ) AABCDEAAFGEZBQEZABHIZJAQEZSTJAABKRUASTRUAAALAMNOP $.
|
|
|
|
$( The left endpoint of a half-open integer interval is in the set iff the
|
|
two arguments are integers with ` M < N ` . This provides an alternate
|
|
notation for the "strict upper integer" predicate by analogy to the "weak
|
|
upper integer" predicate ` M e. ( ZZ>= `` N ) ` . (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
fzolb2 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ( M ..^ N ) <-> M < N ) ) $=
|
|
( cfzo co wcel cz wa clt wbr w3a fzolb df-3an bitri baib ) AABCDEZAFEZBFEZG
|
|
ZABHIZOPQSJRSGABKPQSLMN $.
|
|
|
|
$( A member in a half-open integer interval is greater than or equal to the
|
|
lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
elfzole1 $p |- ( K e. ( M ..^ N ) -> M <_ K ) $=
|
|
( cfzo co cle wbr clt wa cz wb elfzoelz elfzoel1 elfzoel2 elfzo syl3anc ibi
|
|
wcel simpld ) ABCDERZBAFGZACHGZTUAUBIZTAJRBJRCJRTUCKABCLABCMABCNABCOPQS $.
|
|
|
|
$( A member in a half-open integer interval is less than the upper bound.
|
|
(Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
elfzolt2 $p |- ( K e. ( M ..^ N ) -> K < N ) $=
|
|
( cfzo co cle wbr clt wa cz wb elfzoelz elfzoel1 elfzoel2 elfzo syl3anc ibi
|
|
wcel simprd ) ABCDERZBAFGZACHGZTUAUBIZTAJRBJRCJRTUCKABCLABCMABCNABCOPQS $.
|
|
|
|
$( Membership in a half-open integer interval implies that the bounds are
|
|
unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
elfzolt3 $p |- ( K e. ( M ..^ N ) -> M < N ) $=
|
|
( cfzo co wcel elfzoel1 zred elfzoelz elfzoel2 elfzole1 elfzolt2 lelttrd )
|
|
ABCDEFZBACNBABCGHNAABCIHNCABCJHABCKABCLM $.
|
|
|
|
$( A member in a half-open integer interval is less than the upper bound.
|
|
(Contributed by Mario Carneiro, 29-Sep-2015.) $)
|
|
elfzolt2b $p |- ( K e. ( M ..^ N ) -> K e. ( K ..^ N ) ) $=
|
|
( cfzo co wcel cz clt wbr elfzoelz elfzoel2 elfzolt2 fzolb syl3anbrc ) ABCD
|
|
EFAGFCGFACHIAACDEFABCJABCKABCLACMN $.
|
|
|
|
$( Membership in a half-open integer interval implies that the bounds are
|
|
unequal. (Contributed by Mario Carneiro, 29-Sep-2015.) $)
|
|
elfzolt3b $p |- ( K e. ( M ..^ N ) -> M e. ( M ..^ N ) ) $=
|
|
( cfzo co wcel cz clt wbr elfzoel1 elfzoel2 elfzolt3 fzolb syl3anbrc ) ABCD
|
|
EZFBGFCGFBCHIBOFABCJABCKABCLBCMN $.
|
|
|
|
$( A half-open range does not contain its right endpoint. (Contributed by
|
|
Stefan O'Rear, 25-Aug-2015.) $)
|
|
fzonel $p |- -. B e. ( A ..^ B ) $=
|
|
( cfzo co wcel clt wbr elfzolt2 elfzoel2 zred ltnrd pm2.65i ) BABCDEZBBFGBA
|
|
BHMBMBBABIJKL $.
|
|
|
|
$( The upper bound of a half-open range is greater or equal to an element of
|
|
the range. (Contributed by Mario Carneiro, 29-Sep-2015.) $)
|
|
elfzouz2 $p |- ( K e. ( M ..^ N ) -> N e. ( ZZ>= ` K ) ) $=
|
|
( cfzo co wcel cz cle wbr cuz cfv elfzoelz elfzoel2 clt elfzolt2 wi cr ltle
|
|
zre syl2an syl2anc mpd eluz2 syl3anbrc ) ABCDEFZAGFZCGFZACHIZCAJKFABCLZABCM
|
|
ZUEACNIZUHABCOUEUFUGUKUHPZUIUJUFAQFCQFULUGASCSACRTUAUBACUCUD $.
|
|
|
|
$( A half-open range is contained in the corresponding closed range.
|
|
(Contributed by Stefan O'Rear, 23-Aug-2015.) $)
|
|
elfzofz $p |- ( K e. ( M ..^ N ) -> K e. ( M ... N ) ) $=
|
|
( cfzo co wcel cuz cfv cfz elfzouz elfzouz2 elfzuzb sylanbrc ) ABCDEFABGHFC
|
|
AGHFABCIEFABCJABCKABCLM $.
|
|
|
|
$( Express membership in a half-open integer interval in terms of the "less
|
|
than or equal" and "less than" predicates on integers, resp.
|
|
` K e. ( ZZ>= `` M ) <-> M <_ K ` , ` K e. ( K ..^ N ) <-> K < N ` .
|
|
(Contributed by Mario Carneiro, 29-Sep-2015.) $)
|
|
elfzo3 $p |- ( K e. ( M ..^ N ) <->
|
|
( K e. ( ZZ>= ` M ) /\ K e. ( K ..^ N ) ) ) $=
|
|
( cuz cfv wcel cz clt wbr wa cfzo co 3anass elfzo2 eluzelz fzolb bitri baib
|
|
w3a wb syl pm5.32i 3bitr4i ) ABDEFZCGFZACHIZSUDUEUFJZJABCKLFUDAACKLFZJUDUEU
|
|
FMABCNUDUHUGUDAGFZUHUGTBAOUHUIUGUHUIUEUFSUIUGJACPUIUEUFMQRUAUBUC $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x M $. $d x N $.
|
|
$( A half-open integer interval is inhabited iff it contains its left
|
|
endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.) $)
|
|
fzom $p |- ( E. x x e. ( M ..^ N ) <-> M e. ( M ..^ N ) ) $=
|
|
( cv cfzo co wcel wex elfzolt3b exlimiv elex2 impbii ) ADZBCEFZGZAHBNGZOP
|
|
AMBCIJABNKL $.
|
|
|
|
$( A half-open range is contained in the corresponding closed range.
|
|
(Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
fzossfz $p |- ( A ..^ B ) C_ ( A ... B ) $=
|
|
( vx cfzo co cfz cv elfzofz ssriv ) CABDEABFECGABHI $.
|
|
$}
|
|
|
|
$( A half-open set of sequential integers is empty if the bounds are equal or
|
|
reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.) $)
|
|
fzon $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N <_ M <-> ( M ..^ N ) = (/) ) ) $=
|
|
( cz wcel wa c1 cmin co clt wbr cfz c0 wceq cle cfzo wb peano2zm fzn sylan2
|
|
zlem1lt ancoms fzoval adantl eqeq1d 3bitr4d ) ACDZBCDZEZBFGHZAIJZAUIKHZLMZB
|
|
ANJZABOHZLMUGUFUICDUJULPBQAUIRSUGUFUMUJPBATUAUHUNUKLUGUNUKMUFABUBUCUDUE $.
|
|
|
|
$( A half-open integer range is empty if the bounds are equal or reversed.
|
|
(Contributed by AV, 20-Oct-2018.) $)
|
|
fzonlt0 $p |- ( ( M e. ZZ /\ N e. ZZ )
|
|
-> ( -. M < N <-> ( M ..^ N ) = (/) ) ) $=
|
|
( cz wcel wa cle wbr clt wn cfzo co c0 wceq cr wb lenlt syl2anr fzon bitr3d
|
|
zre ) ACDZBCDZEBAFGZABHGIZABJKLMUBBNDANDUCUDOUABTATBAPQABRS $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( Half-open sets with equal endpoints are empty. (Contributed by Stefan
|
|
O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) $)
|
|
fzo0 $p |- ( A ..^ A ) = (/) $=
|
|
( vx cfzo co c0 wceq cv wcel wn wal wex fzonel fzom mtbir alnex mpbir eq0
|
|
) AACDZEFBGRHZIBJZTSBKZIUAARHAALBAAMNSBOPBRQP $.
|
|
$}
|
|
|
|
$( If ` K < N ` then ` N - K ` is a positive integer. (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.) $)
|
|
fzonnsub $p |- ( K e. ( M ..^ N ) -> ( N - K ) e. NN ) $=
|
|
( cfzo co wcel clt wbr cmin cn elfzolt2 cz elfzoelz elfzoel2 znnsub syl2anc
|
|
wb mpbid ) ABCDEFZACGHZCAIEJFZABCKSALFCLFTUAQABCMABCNACOPR $.
|
|
|
|
$( If ` M < N ` then ` N - M ` is a positive integer. (Contributed by Mario
|
|
Carneiro, 1-Jan-2017.) $)
|
|
fzonnsub2 $p |- ( K e. ( M ..^ N ) -> ( N - M ) e. NN ) $=
|
|
( cfzo co wcel cmin cn elfzolt3b fzonnsub syl ) ABCDEZFBLFCBGEHFABCIBBCJK
|
|
$.
|
|
|
|
${
|
|
$d K x $. $d M x $. $d N x $.
|
|
$( Subset relationship for half-open sequences of integers. (Contributed
|
|
by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro,
|
|
29-Sep-2015.) $)
|
|
fzoss1 $p |- ( K e. ( ZZ>= ` M ) -> ( K ..^ N ) C_ ( M ..^ N ) ) $=
|
|
( vx cuz cfv wcel cfzo co cv wa cz elfzoel2 adantl c1 cmin cfz wss fzoval
|
|
wceq fzss1 adantr 3sstr4d sseld impancom mpd ex ssrdv ) ABEFGZDACHIZBCHIZ
|
|
UIDJZUJGZULUKGZUIUMKCLGZUNUMUOUIULACMNUIUOUMUNUIUOKZUJUKULUPACOPIZQIZBUQQ
|
|
IZUJUKUIURUSRUOABUQUAUBUOUJURTUIACSNUOUKUSTUIBCSNUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
$( Subset relationship for half-open sequences of integers. (Contributed by
|
|
Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) $)
|
|
fzoss2 $p |- ( N e. ( ZZ>= ` K ) -> ( M ..^ K ) C_ ( M ..^ N ) ) $=
|
|
( cuz cfv wcel c1 cmin co cfz cfzo wss cz caddc eluzel2 peano2zm syl fzoval
|
|
cc wceq 1zzd id zcnd ax-1cn sylancl fveq2d eleqtrrd eluzsub syl3anc eluzelz
|
|
npcan fzss2 3sstr4d ) CADEZFZBAGHIZJIZBCGHIZJIZBAKIZBCKIZUOURUPDEFZUQUSLUOU
|
|
PMFZGMFCUPGNIZDEZFVBUOAMFZVCACOZAPQUOUAUOCUNVEUOUBUOVDADUOASFGSFVDATUOAVGUC
|
|
UDAGUKUEUFUGGUPCUHUIUPBURULQUOVFUTUQTVGBARQUOCMFVAUSTACUJBCRQUM $.
|
|
|
|
$( Subset of a half open range. (Contributed by Alexander van der Vekens,
|
|
1-Nov-2017.) $)
|
|
fzossrbm1 $p |- ( N e. ZZ -> ( 0 ..^ ( N - 1 ) ) C_ ( 0 ..^ N ) ) $=
|
|
( cz wcel c1 cmin co cuz cfv cc0 cfzo wss cle wbr peano2zm id zre syl3anbrc
|
|
lem1d eluz2 fzoss2 syl ) ABCZAADEFZGHCZIUCJFIAJFKUBUCBCUBUCALMUDANUBOUBAAPR
|
|
UCASQUCIATUA $.
|
|
|
|
$( Subset relationship for half-open integer ranges with lower bounds 0 and
|
|
1. (Contributed by Alexander van der Vekens, 18-Mar-2018.) $)
|
|
fzo0ss1 $p |- ( 1 ..^ N ) C_ ( 0 ..^ N ) $=
|
|
( c1 cc0 cuz cfv wcel cfzo co wss 1eluzge0 fzoss1 ax-mp ) BCDEFBAGHCAGHIJBC
|
|
AKL $.
|
|
|
|
$( A half-open integer range starting at a nonnegative integer is a subset of
|
|
the nonnegative integers. (Contributed by Alexander van der Vekens,
|
|
13-May-2018.) $)
|
|
fzossnn0 $p |- ( M e. NN0 -> ( M ..^ N ) C_ NN0 ) $=
|
|
( cn0 wcel cfzo co cc0 cfz fzossfz wss cuz fzss1 nn0uz eleq2s syl5ss fzssuz
|
|
cfv sseqtr4i syl6ss ) ACDZABEFZGBHFZCTUAABHFZUBABIUCUBJAGKQZCAGBLMNOUBUDCGB
|
|
PMRS $.
|
|
|
|
$( One direction of splitting a half-open integer range in half.
|
|
(Contributed by Stefan O'Rear, 14-Aug-2015.) $)
|
|
fzospliti $p |- ( ( A e. ( B ..^ C ) /\ D e. ZZ ) ->
|
|
( A e. ( B ..^ D ) \/ A e. ( D ..^ C ) ) ) $=
|
|
( cfzo co wcel cz wa wo cle wbr clt simpr elfzoelz adantr a1d elfzo syl3anc
|
|
wb zlelttric syl2anc orcomd elfzole1 ancrd elfzolt2 ancld elfzoel1 elfzoel2
|
|
orim12d mpd orbi12d mpbird ) ABCEFGZDHGZIZABDEFGZADCEFGZJBAKLZADMLZIZDAKLZA
|
|
CMLZIZJZUPUTVBJVEUPVBUTUPUOAHGZVBUTJUNUONZUNVFUOABCOPZDAUAUBUCUPUTVAVBVDUPU
|
|
TUSUPUSUTUNUSUOABCUDPQUEUPVBVCUPVCVBUNVCUOABCUFPQUGUJUKUPUQVAURVDUPVFBHGZUO
|
|
UQVATVHUNVIUOABCUHPVGABDRSUPVFUOCHGZURVDTVHVGUNVJUOABCUIPADCRSULUM $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
$( Split a half-open integer range in half. (Contributed by Stefan O'Rear,
|
|
14-Aug-2015.) $)
|
|
fzosplit $p |- ( D e. ( B ... C ) -> ( B ..^ C ) =
|
|
( ( B ..^ D ) u. ( D ..^ C ) ) ) $=
|
|
( vx cfz co wcel cfzo cun cv wa wo simpr elfzelz adantr cuz cfv wss syl
|
|
cz fzospliti syl2anc elun sylibr ssrdv elfzuz3 fzoss2 elfzuz fzoss1 unssd
|
|
ex eqssd ) CABEFGZABHFZACHFZCBHFZIZUMDUNUQUMDJZUNGZURUQGZUMUSKZURUOGURUPG
|
|
LZUTVAUSCTGZVBUMUSMUMVCUSCABNOURABCUAUBURUOUPUCUDUKUEUMUOUPUNUMBCPQGUOUNR
|
|
CABUFCABUGSUMCAPQGUPUNRCABUHCABUISUJUL $.
|
|
|
|
$( Abutting half-open integer ranges are disjoint. (Contributed by Stefan
|
|
O'Rear, 14-Aug-2015.) $)
|
|
fzodisj $p |- ( ( A ..^ B ) i^i ( B ..^ C ) ) = (/) $=
|
|
( vx cfzo co cin c0 wceq cv wcel wn wi disj1 cle wbr clt elfzolt2 cz wb
|
|
elfzoelz elfzoel2 zltnle syl2anc mpbid elfzole1 nsyl mpgbir ) ABEFZBCEFZG
|
|
HIDJZUIKZUKUJKZLMDDUIUJNULBUKOPZUMULUKBQPZUNLZUKABRULUKSKBSKUOUPTUKABUAUK
|
|
ABUBUKBUCUDUEUKBCUFUGUH $.
|
|
|
|
$( Split an upper integer set into a half-open integer range and another
|
|
upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.) $)
|
|
fzouzsplit $p |- ( B e. ( ZZ>= ` A ) ->
|
|
( ZZ>= ` A ) = ( ( A ..^ B ) u. ( ZZ>= ` B ) ) ) $=
|
|
( vx cuz cfv wcel co cun cv wo wa clt wbr cle cz eluzelz zlelttric syl2an
|
|
cfzo wb orcomd id elfzo2 df-3an bitri baib syl2anr eluz orbi12d mpbird ex
|
|
w3a elun syl6ibr ssrdv wss elfzouz ssriv a1i uzss unssd eqssd ) BADEZFZVC
|
|
ABSGZBDEZHZVDCVCVGVDCIZVCFZVHVEFZVHVFFZJZVHVGFVDVIVLVDVIKZVLVHBLMZBVHNMZJ
|
|
VMVOVNVDBOFZVHOFZVOVNJVIABPZAVHPZBVHQRUAVMVJVNVKVOVIVIVPVJVNTVDVIUBVRVJVI
|
|
VPKZVNVJVIVPVNULVTVNKVHABUCVIVPVNUDUEUFUGVDVPVQVKVOTVIVRVSBVHUHRUIUJUKVHV
|
|
EVFUMUNUOVDVEVFVCVEVCUPVDCVEVCVHABUQURUSABUTVAVB $.
|
|
|
|
$( A half-open integer range does not overlap the upper integer range
|
|
starting at the endpoint of the first range. (Contributed by Mario
|
|
Carneiro, 21-Sep-2016.) $)
|
|
fzouzdisj $p |- ( ( A ..^ B ) i^i ( ZZ>= ` B ) ) = (/) $=
|
|
( vx cfzo co cuz cfv cin c0 wceq cv wcel wn eq0 wa elfzolt2 adantr adantl
|
|
clt wbr cle eluzle cz eluzel2 zred cr eluzelre lenltd mpbid pm2.65i mtbir
|
|
elin mpgbir ) ABDEZBFGZHZIJCKZUPLZMCCUPNURUQUNLZUQUOLZOZVAUQBSTZUSVBUTUQA
|
|
BPQVABUQUATZVBMUTVCUSBUQUBRVABUQVABUTBUCLUSBUQUDRUEUTUQUFLUSBUQUGRUHUIUJU
|
|
QUNUOULUKUM $.
|
|
$}
|
|
|
|
$( An integer is strictly greater than zero iff it is a member of ` NN ` .
|
|
(Contributed by Mario Carneiro, 29-Sep-2015.) $)
|
|
lbfzo0 $p |- ( 0 e. ( 0 ..^ A ) <-> A e. NN ) $=
|
|
( cc0 cz wcel clt wbr w3a wa cfzo co cn 3anass mpbiran fzolb elnnz 3bitr4i
|
|
0z ) BCDZACDZBAEFZGZSTHZBBAIJDAKDUARUBQRSTLMBANAOP $.
|
|
|
|
$( Membership in a half-open integer range based at 0. (Contributed by
|
|
Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) $)
|
|
elfzo0 $p |- ( A e. ( 0 ..^ B ) <-> ( A e. NN0 /\ B e. NN /\ A < B ) ) $=
|
|
( cc0 cfzo co wcel cn0 clt wbr w3a cuz cfv elfzouz elnn0uz sylibr elfzolt3b
|
|
cn lbfzo0 sylib elfzolt2 3jca cz simp1 nnz 3ad2ant2 elfzo2 syl3anbrc impbii
|
|
simp3 ) ACBDEZFZAGFZBQFZABHIZJZUKULUMUNUKACKLFZULACBMANZOUKCUJFUMACBPBRSACB
|
|
TUAUOUPBUBFZUNUKUOULUPULUMUNUCUQSUMULURUNBUDUEULUMUNUIACBUFUGUH $.
|
|
|
|
$( An integer between 1 and an upper bound of a half-open integer range is
|
|
not 0 and between 0 and the upper bound of the half-open integer range.
|
|
(Contributed by Alexander van der Vekens, 21-Mar-2018.) $)
|
|
fzo1fzo0n0 $p |- ( K e. ( 1 ..^ N ) <-> ( K e. ( 0 ..^ N ) /\ K =/= 0 ) ) $=
|
|
( c1 cfzo co wcel cc0 wne wa cz clt wbr w3a cn wi adantr adantl 3jca sylbir
|
|
cr cuz cfv elfzo2 cn0 nnnn0 nngt0 0red nnre zre lttr syl3anc elnnz simplbi2
|
|
elnnuz syld exp4b com13 mpcom imp31 simpr exp31 3imp elfzo0 sylibr 3ad2ant1
|
|
nnne0 jca sylbi cle elnnne0 nnge1 3ad2antl1 simpl3 wb nn0z 1zzd nnz 3adant3
|
|
elfzo syl mpbir2and sylanb impbii ) ACBDEFZAGBDEFZAGHZIZWDACUAUBFZBJFZABKLZ
|
|
MZWGACBUCWKWEWFWKAUDFZBNFZWJMZWEWHWIWJWNWHANFZWIWJWNOOAUNZWOWIWJWNWOWIIZWJI
|
|
WLWMWJWQWLWJWOWLWIAUEPPWOWIWJWMGAKLZWOWIWJWMOZOAUFWIWOWRWSWIWOWRWJWMWIWOIZW
|
|
RWJIZGBKLZWMWTGTFATFZBTFZXAXBOWTUGWOXCWIAUHQWIXDWOBUIPGABUJUKWIXBWMOWOWMWIX
|
|
BBULUMPUOUPUQURUSWQWJUTRVASVBABVCZVDWHWIWFWJWHWOWFWPAVFSVEVGVHWEWNWFWDXEWNW
|
|
FIZWDCAVILZWJWLWMWFXGWJWLWFIWOXGAVJAVKSVLWLWMWJWFVMXFAJFZCJFZWIMZWDXGWJIVNW
|
|
NXJWFWLWMXJWJWLWMIZXHXIWIWLXHWMAVOPXKVPWMWIWLBVQQRVRPACBVSVTWAWBWC $.
|
|
|
|
$( Membership in a half-open range of nonnegative integers, generalization of
|
|
~ elfzo0 requiring the upper bound to be an integer only. (Contributed by
|
|
Alexander van der Vekens, 23-Sep-2018.) $)
|
|
elfzo0z $p |- ( A e. ( 0 ..^ B ) <-> ( A e. NN0 /\ B e. ZZ /\ A < B ) ) $=
|
|
( cc0 cfzo co wcel cn0 cn clt wbr w3a cz elfzo0 nnz 3anim2i wa wi cr adantl
|
|
zre simp1 cle elnn0z 0red adantr lelttr syl3anc simplbi2 syld expd impancom
|
|
elnnz sylbi 3imp simp3 3jca impbii bitri ) ACBDEFAGFZBHFZABIJZKZUSBLFZVAKZA
|
|
BMVBVDUTVCUSVABNOVDUSUTVAUSVCVAUAUSVCVAUTUSALFZCAUBJZPVCVAUTQZQAUCVEVCVFVGV
|
|
EVCPZVFVAUTVHVFVAPZCBIJZUTVHCRFARFZBRFZVIVJQVHUDVEVKVCATUEVCVLVEBTSCABUFUGV
|
|
CVJUTQVEUTVCVJBULUHSUIUJUKUMUNUSVCVAUOUPUQUR $.
|
|
|
|
$( A member in a half-open range of nonnegative integers is less than or
|
|
equal to the upper bound of the range. (Contributed by Alexander van der
|
|
Vekens, 23-Sep-2018.) $)
|
|
elfzo0le $p |- ( A e. ( 0 ..^ B ) -> A <_ B ) $=
|
|
( cc0 cfzo co wcel cn0 cn clt wbr w3a elfzo0 cr wi nn0re nnre syl2an 3impia
|
|
cle ltle sylbi ) ACBDEFAGFZBHFZABIJZKABSJZABLUBUCUDUEUBAMFBMFUDUENUCAOBPABT
|
|
QRUA $.
|
|
|
|
$( A member of a half-open range of nonnegative integers is a nonnegative
|
|
integer. (Contributed by Alexander van der Vekens, 21-May-2018.) $)
|
|
elfzonn0 $p |- ( K e. ( 0 ..^ N ) -> K e. NN0 ) $=
|
|
( cc0 cfzo co wcel cuz cfv cn0 elfzouz elnn0uz sylibr ) ACBDEFACGHFAIFACBJA
|
|
KL $.
|
|
|
|
$( The result of subtracting a nonnegative integer from a positive integer
|
|
and adding another nonnegative integer which is less than the first one is
|
|
less then the positive integer. (Contributed by Alexander van der Vekens,
|
|
19-May-2018.) $)
|
|
fzonmapblen $p |- ( ( A e. ( 0 ..^ N ) /\ B e. ( 0 ..^ N ) /\ B < A )
|
|
-> ( B + ( N - A ) ) < N ) $=
|
|
( cc0 cfzo co wcel clt wbr cmin caddc cr wa wi cn0 cn anim12i adantr recn
|
|
cc w3a elfzo0 nn0re nnre 3adant3 sylbi elfzoelz simpr simpll resubcl ancoms
|
|
zred ltadd1d biimpa wceq pncan3 syl breqtrd ex syl2an 3impia ) ADCEFZGZBVBG
|
|
ZBAHIZBCAJFZKFZCHIZVCALGZCLGZMZBLGZVEVHNVDVCAOGZCPGZACHIZUAVKACUBVMVNVKVOVM
|
|
VIVNVJAUCCUDQUEUFVDBBDCUGULVKVLMZVEVHVPVEMZVGAVFKFZCHVPVEVGVRHIVPBAVFVKVLUH
|
|
VIVJVLUIVKVFLGZVLVJVIVSCAUJUKRUMUNVQATGZCTGZMZVRCUOVPWBVEVKWBVLVIVTVJWAASCS
|
|
QRRACUPUQURUSUTVA $.
|
|
|
|
$( If a nonnegative integer in a finite interval of integers is not the upper
|
|
bound of the interval, it is contained in the corresponding half-open
|
|
integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.) $)
|
|
fzofzim $p |- ( ( K =/= M /\ K e. ( 0 ... M ) ) -> K e. ( 0 ..^ M ) ) $=
|
|
( wne cc0 co wcel wa cn0 clt wbr w3a cle wi cz nn0z adantl expd sylbi imp
|
|
cr cn cfzo elfz2nn0 simpl1 necom wb zltlen syl2an bicomd elnn0z 0red adantr
|
|
cfz zre nn0re lelttr syl3anc elnnz simplbi2 syl syld impancom sylbid syl7bi
|
|
3impia biimpd exp4b 3imp syl5bi 3jca ex impcom elfzo0 sylibr ) ABCZADBUMEFZ
|
|
GAHFZBUAFZABIJZKZADBUBEFVPVOVTVPVQBHFZABLJZKZVOVTMABUCWCVOVTWCVOGVQVRVSVQWA
|
|
WBVOUDWCVOVRVQWAWBVOVRMVOBACZVQWAGZWBVRABUEZWEWBWDVRWEWBWDGZVSVRWEVSWGVQANF
|
|
ZBNFZVSWGUFWAAOBOZABUGUHUIZVQWAVSVRMZVQWHDALJZGWAWLMAUJWHWAWMWLWHWAGZWMVSVR
|
|
WNWMVSGZDBIJZVRWNDTFATFZBTFZWOWPMWNUKWHWQWAAUNULWAWRWHBUOPDABUPUQWAWPVRMZWH
|
|
WAWIWSWJVRWIWPBURUSUTPVAQVBRSVCQVDVESWCVOVSVOWDWCVSWFVQWAWBWDVSMVQWAWBWDVSW
|
|
EWGVSWKVFVGVHVISVJVKRVLABVMVN $.
|
|
|
|
${
|
|
$d k N $.
|
|
$( Half-open integer ranges starting with 1 are subsets of NN. (Contributed
|
|
by Thierry Arnoux, 28-Dec-2016.) $)
|
|
fzossnn $p |- ( 1 ..^ N ) C_ NN $=
|
|
( vk c1 cfzo co cfz cn fzossfz cv elfznn ssriv sstri ) CADECAFEZGCAHBMGBI
|
|
AJKL $.
|
|
$}
|
|
|
|
$( Membership in a half-open integer range based at 1. (Contributed by
|
|
Thierry Arnoux, 14-Feb-2017.) $)
|
|
elfzo1 $p |- ( N e. ( 1 ..^ M ) <-> ( N e. NN /\ M e. NN /\ N < M ) ) $=
|
|
( c1 cfzo co wcel clt wbr w3a fzossnn sseli cuz cfv elfzouz2 eluznn syl2anc
|
|
cn elfzolt2 3jca cz nnuz eqimssi nnz id 3anim123i elfzo2 sylibr impbii ) BC
|
|
ADEZFZBQFZAQFZBAGHZIZUJUKULUMUIQBAJKZUJUKABLMFULUOBCANABOPBCARSUNBCLMZFZATF
|
|
ZUMIUJUKUQULURUMUMQUPBQUPUAUBKAUCUMUDUEBCAUFUGUH $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A half-open integer range based at 0 is inhabited precisely if the upper
|
|
bound is a positive integer. (Contributed by Jim Kingdon,
|
|
20-Apr-2020.) $)
|
|
fzo0m $p |- ( E. x x e. ( 0 ..^ A ) <-> A e. NN ) $=
|
|
( cv cc0 cfzo co wcel wex cn fzom lbfzo0 bitri ) ACDBEFZGAHDMGBIGADBJBKL
|
|
$.
|
|
$}
|
|
|
|
$( Translate membership in a half-open integer range. (Contributed by Stefan
|
|
O'Rear, 15-Aug-2015.) $)
|
|
fzoaddel $p |- ( ( A e. ( B ..^ C ) /\ D e. ZZ ) ->
|
|
( A + D ) e. ( ( B + D ) ..^ ( C + D ) ) ) $=
|
|
( cfzo co wcel cz wa caddc cle wbr clt elfzoel1 adantr zred elfzoelz zaddcl
|
|
simpr sylan elfzole1 leadd1dd elfzoel2 elfzolt2 ltadd1dd wb elfzo mpbir2and
|
|
syl3anc ) ABCEFGZDHGZIZADJFZBDJFZCDJFZEFGZUNUMKLZUMUOMLZULBADULBUJBHGZUKABC
|
|
NZOPULAUJAHGZUKABCQZOPZULDUJUKSPZUJBAKLUKABCUAOUBULACDVCULCUJCHGZUKABCUCZOP
|
|
VDUJACMLUKABCUDOUEULUMHGZUNHGZUOHGZUPUQURIUFUJVAUKVGVBADRTUJUSUKVHUTBDRTUJV
|
|
EUKVIVFCDRTUMUNUOUGUIUH $.
|
|
|
|
$( Translate membership in a shifted-down half-open integer range.
|
|
(Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
fzoaddel2 $p |- ( ( A e. ( 0 ..^ ( B - C ) ) /\ B e. ZZ /\ C e. ZZ ) ->
|
|
( A + C ) e. ( C ..^ B ) ) $=
|
|
( cc0 cmin co cfzo wcel cz w3a caddc fzoaddel 3adant2 wceq cc zcn wa addid2
|
|
adantl npcan oveq12d syl2an 3adant1 eleqtrd ) ADBCEFZGFHZBIHZCIHZJACKFZDCKF
|
|
ZUECKFZGFZCBGFZUFUHUIULHUGADUECLMUGUHULUMNZUFUGBOHZCOHZUNUHBPCPUOUPQUJCUKBG
|
|
UPUJCNUOCRSBCTUAUBUCUD $.
|
|
|
|
$( Translate membership in a half-open integer range. (Contributed by Stefan
|
|
O'Rear, 15-Aug-2015.) $)
|
|
fzosubel $p |- ( ( A e. ( B ..^ C ) /\ D e. ZZ ) ->
|
|
( A - D ) e. ( ( B - D ) ..^ ( C - D ) ) ) $=
|
|
( cfzo co wcel cz wa cneg caddc cmin znegcl fzoaddel sylan2 elfzoelz adantr
|
|
zcnd simpr negsubd elfzoel1 elfzoel2 oveq12d 3eltr3d ) ABCEFGZDHGZIZADJZKFZ
|
|
BUHKFZCUHKFZEFZADLFBDLFZCDLFZEFUFUEUHHGUIULGDMABCUHNOUGADUGAUEAHGUFABCPQRUG
|
|
DUEUFSRZTUGUJUMUKUNEUGBDUGBUEBHGUFABCUAQRUOTUGCDUGCUECHGUFABCUBQRUOTUCUD $.
|
|
|
|
$( Membership in a translated half-open integer range implies translated
|
|
membership in the original range. (Contributed by Stefan O'Rear,
|
|
15-Aug-2015.) $)
|
|
fzosubel2 $p |- ( ( A e. ( ( B + C ) ..^ ( B + D ) ) /\ ( B e. ZZ /\
|
|
C e. ZZ /\ D e. ZZ ) ) -> ( A - B ) e. ( C ..^ D ) ) $=
|
|
( caddc co cfzo wcel cz w3a cmin fzosubel 3ad2antr1 wceq zcn pncan2 3adant3
|
|
wa cc 3adant2 oveq12d syl3an adantl eleqtrd ) ABCEFZBDEFZGFHZBIHZCIHZDIHZJZ
|
|
RABKFZUEBKFZUFBKFZGFZCDGFZUGUIUHULUOHUJAUEUFBLMUKUOUPNZUGUHBSHZUICSHZUJDSHZ
|
|
UQBOCODOURUSUTJUMCUNDGURUSUMCNUTBCPQURUTUNDNUSBDPTUAUBUCUD $.
|
|
|
|
$( Membership in a translated half-open integer range when the original range
|
|
is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
fzosubel3 $p |- ( ( A e. ( B ..^ ( B + D ) ) /\ D e. ZZ ) ->
|
|
( A - B ) e. ( 0 ..^ D ) ) $=
|
|
( caddc co cfzo wcel cz wa cmin simpl elfzoel1 adantr zcnd addid1d eleqtrrd
|
|
cc0 oveq1d 0zd simpr fzosubel2 syl13anc ) ABBCDEZFEZGZCHGZIZABQDEZUCFEZGBHG
|
|
ZQHGUFABJEQCFEGUGAUDUIUEUFKUGUHBUCFUGBUGBUEUJUFABUCLMZNORPUKUGSUEUFTABQCUAU
|
|
B $.
|
|
|
|
$( Membership of the difference of integers in a half-open range of
|
|
nonnegative integers. (Contributed by Alexander van der Vekens,
|
|
17-Sep-2018.) $)
|
|
eluzgtdifelfzo $p |- ( ( A e. ZZ /\ B e. ZZ )
|
|
-> ( ( N e. ( ZZ>= ` A ) /\ B < A )
|
|
-> ( N - A ) e. ( 0 ..^ ( N - B ) ) ) ) $=
|
|
( cz wcel wa clt wbr cmin co cc0 cfzo caddc simpl adantl adantr ad2antrr cr
|
|
zre cc cuz cfv eluzelz simprr zsubcld ancoms zaddcld posdif syl2anr adantld
|
|
wi biimpd imp resubcl syl2an eluzelre ad2antrl ltaddposd mpbid zcn eluzelcn
|
|
wb w3a addsub12 breq2d syl3anc mpbird elfzo2 syl3anbrc fzosubel3 syl2anc ex
|
|
) ADEZBDEZFZCAUAUBEZBAGHZFZCAIJKCBIJZLJEZVOVRFZCAAVSMJZLJEZVSDEZVTWAVPWBDEC
|
|
WBGHZWCVRVPVOVPVQNOWAAVSVOVMVRVMVNNPVRVOWDVRVOFCBVPCDEVQVOACUCQVRVMVNUDUEUF
|
|
ZUGWAWECCABIJZMJZGHZWAKWGGHZWIVOVRWJVOVQWJVPVNBREZAREZVQWJUKVMBSZASZWKWLFVQ
|
|
WJBAUHULUIUJUMWAWGCVOWGREZVRVMWLWKWOVNWNWMABUNUOPVPCREVOVQACUPUQURUSWAATEZC
|
|
TEZBTEZWEWIVBVMWPVNVRAUTQVPWQVOVQACVAUQVOWRVRVNWRVMBUTOPWPWQWRVCWBWHCGACBVD
|
|
VEVFVGCAWBVHVIWFCAVSVJVKVL $.
|
|
|
|
$( Membership of an integer greater than 1 decreased by 2 in a half-open
|
|
range of nonnegative integers. (Contributed by Alexander van der Vekens,
|
|
3-Oct-2018.) $)
|
|
ige2m2fzo $p |- ( N e. ( ZZ>= ` 2 ) -> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) $=
|
|
( c2 cuz cfv wcel cz c1 wa clt wbr cmin co cc0 cfzo eluzel2 jctir 1lt2 jctr
|
|
1z eluzgtdifelfzo sylc ) ABCDEZBFEZGFEZHUBGBIJZHABKLMAGKLNLEUBUCUDBAOSPUBUE
|
|
QRBGATUA $.
|
|
|
|
$( Translate membership in a half-open integer range. (Contributed by
|
|
Thierry Arnoux, 28-Sep-2018.) $)
|
|
fzocatel $p |- ( ( ( A e. ( 0 ..^ ( B + C ) ) /\ -. A e. ( 0 ..^ B ) )
|
|
/\ ( B e. ZZ /\ C e. ZZ ) ) -> ( A - B ) e. ( 0 ..^ C ) ) $=
|
|
( cc0 caddc co cfzo wcel wn wa cz cmin simplr wo fzospliti ad2ant2r ord mpd
|
|
simprl zcnd fzosubel syl2anc wceq zcn subidd simprr pncan2d oveq12d eleqtrd
|
|
syl ) ADBCEFZGFHZADBGFHZIZJZBKHZCKHZJZJZABLFZBBLFZUKBLFZGFZDCGFUSABUKGFHZUP
|
|
UTVCHUSUNVDULUNURMUSUMVDULUPUMVDNUNUQADUKBOPQRUOUPUQSZABUKBUAUBUSVADVBCGUSU
|
|
PVADUCVEUPBBUDUEUJUSBCUSBVETUSCUOUPUQUFTUGUHUI $.
|
|
|
|
$( If an integer in a 1 based finite set of sequential integers is subtracted
|
|
from the upper bound of this finite set of sequential integers, the result
|
|
is contained in a half-open range of nonnegative integers with the same
|
|
upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV,
|
|
30-Oct-2018.) $)
|
|
ubmelfzo $p |- ( K e. ( 1 ... N ) -> ( N - K ) e. ( 0 ..^ N ) ) $=
|
|
( cn wcel cle wbr w3a co cn0 clt cc0 wa wb nnnn0 anim12i 3adant3 mpbid nnre
|
|
syl cr cmin cfz cfzo simp3 nn0sub simp2 nngt0 3ad2ant1 ltsubpos 3jca elfz1b
|
|
c1 elfzo0 3imtr4i ) ACDZBCDZABEFZGZBAUAHZIDZUPUSBJFZGAULBUBHDUSKBUCHDURUTUP
|
|
VAURUQUTUOUPUQUDURAIDZBIDZLZUQUTMUOUPVDUQUOVBUPVCANBNOPABUESQUOUPUQUFURKAJF
|
|
ZVAUOUPVEUQAUGUHURATDZBTDZLZVEVAMUOUPVHUQUOVFUPVGARBROPABUISQUJBAUKUSBUMUN
|
|
$.
|
|
|
|
$( If an integer is in a half-open range of nonnegative integers with a
|
|
difference as upper bound, the sum of the integer with the subtrahend of
|
|
the difference is in the a half-open range of nonnegative integers
|
|
containing the minuend of the difference. (Contributed by AV,
|
|
13-Nov-2018.) $)
|
|
elfzodifsumelfzo $p |- ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... P ) )
|
|
-> ( I e. ( 0 ..^ ( N - M ) )
|
|
-> ( I + M ) e. ( 0 ..^ P ) ) ) $=
|
|
( cc0 co wcel wi cn0 cle wbr w3a wa clt cz adantr cr nn0re adantl imp caddc
|
|
cfz cmin cfzo elfz2nn0 cn elfzo0 wb nn0z znnsub syl2an simpr simpll syl2anr
|
|
nn0addcl 0red 3jca nn0ge0 anim1i lelttr sylc ex ltletr syl3anc simplbi2 syl
|
|
elnnz syld exp4b com24 com13 readdcl ltaddsubd exbiri impd anasss syl3anbrc
|
|
com23 exp53 sylbird 3adant3 com14 3imp sylbi 3adant1 com12 ) CEDUBFGZDEAUBF
|
|
GZBEDCUCFZUDFGZBCUAFZEAUDFGZHZWGCIGZDIGZCDJKZLZWHWMHCDUEWHWQWMWHWOAIGZDAJKZ
|
|
LWQWMHZDAUEWRWSWTWOWJWQWRWSMZWLWJBIGZWIUFGZBWINKZLWQXAWLHZHZBWIUGXBXCXDXFWQ
|
|
XCXDXBXEWNWOXCXDXBXEHHZHWPWNWOMZXCCDNKZXGWNCOGDOGXIXCUHWOCUIDUICDUJUKXHXIXD
|
|
XBXAWLXHXIMZXDXBMZMZXAMWKIGZAUFGZWKANKZWLXLXMXAXKXBWNXMXJXDXBULWNWOXIUMBCUO
|
|
UNPXLXAXNXJXAXNHZXKXHXIXPXHXIEDNKZXPXHXIXQXJEQGZCQGZDQGZLZECJKZXIMXQXHYAXIX
|
|
HXRXSXTXHUPWNXSWOCRPZWOXTWNDRZSZUQPXHYBXIWNYBWOCURPUSECDUTVAVBWOXQXPHWNXAXQ
|
|
WOXNWRWSXQWOXNHHWRWOXQWSXNWRWOXQWSXNWRWOMZXQWSMZEANKZXNYFXRXTAQGZYGYHHYFUPW
|
|
OXTWRYDSWRYIWOARZPEDAVCVDWRYHXNHZWOWRAOGZYKAUIXNYLYHAVGVEVFPVHVIVJTVKSVHTPT
|
|
XLWRWSXOXLWRMZWSMWKQGZXTYILZWKDNKZWSMXOYMYOWSYMYNXTYIXLYNWRXKBQGZXSYNXJXBYQ
|
|
XDBRZSXHXSXIYCPBCVLUNPXLXTWRXJXTXKXHXTXIYEPPPWRYIXLYJSUQPYMYPWSXLYPWRXJXKYP
|
|
XHXKYPHXIXHXDXBYPXHXBXDYPXHXBYPXDXHXBMBCDXBYQXHYRSXHXSXBYCPXHXTXBYEPVMVNVRV
|
|
OPTPUSWKDAVCVAVPWKAUGVQVSVTWAWBWCWDVKWEWDWFWDT $.
|
|
|
|
$( Membership of an integer incremented by one in a half-open range of
|
|
nonnegative integers. (Contributed by Alexander van der Vekens,
|
|
24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.) $)
|
|
elfzom1elp1fzo $p |- ( ( N e. ZZ /\ I e. ( 0 ..^ ( N - 1 ) ) )
|
|
-> ( I + 1 ) e. ( 0 ..^ N ) ) $=
|
|
( cz wcel cc0 c1 cmin co cfzo cfz caddc cn0 cle wbr w3a sylibr 3syl wss jca
|
|
wa cn cuz cfv wi elfzofz elfzuz2 elnn0uz cc zcn anim1i expcom sylbir impcom
|
|
elnnnn0 1nn0 a1i nnnn0 nnge1 3jca syl fzossrbm1 adantr fzossfz syl6ss simpr
|
|
elfz2nn0 ssel2 elfzubelfz elfzodifsumelfzo sylc ) BCDZAEBFGHZIHZDZTZFEBJHZD
|
|
ZBVPDZTVNAFKHEBIHZDVOVQVRVOFLDZBLDZFBMNZOZVQVOBUADZWCVNVKWDVNAEVLJHDVLEUBUC
|
|
DZVKWDUDZAEVLUEAEVLUFWEVLLDZWFVLUGVKWGWDVKWGTBUHDZWGTWDVKWHWGBUIUJBUNPUKULQ
|
|
UMWDVTWAWBVTWDUOUPBUQBURUSUTFBVFPVOVMVPRZVNTAVPDVRVOWIVNVOVMVSVPVKVMVSRVNBV
|
|
AVBEBVCVDVKVNVEZSVMVPAVGAEBVHQSWJBAFBVIVJ $.
|
|
|
|
$( Membership in a half-open range of nonnegative integers. (Contributed by
|
|
Alexander van der Vekens, 18-Jun-2018.) $)
|
|
elfzom1elfzo $p |- ( ( N e. ZZ /\ I e. ( 0 ..^ ( N - 1 ) ) )
|
|
-> I e. ( 0 ..^ N ) ) $=
|
|
( cz wcel cc0 c1 cmin co cfzo fzossrbm1 sselda ) BCDEBFGHIHEBIHABJK $.
|
|
|
|
$( Expressing a closed integer range as a half-open integer range.
|
|
(Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
fzval3 $p |- ( N e. ZZ -> ( M ... N ) = ( M ..^ ( N + 1 ) ) ) $=
|
|
( cz wcel c1 caddc co cfzo cmin cfz wceq peano2z fzoval syl cc ax-1cn pncan
|
|
zcn sylancl oveq2d eqtr2d ) BCDZABEFGZHGZAUCEIGZJGZABJGUBUCCDUDUFKBLAUCMNUB
|
|
UEBAJUBBODEODUEBKBRPBEQSTUA $.
|
|
|
|
$( Expressing a singleton as a half-open range. (Contributed by Stefan
|
|
O'Rear, 23-Aug-2015.) $)
|
|
fzosn $p |- ( A e. ZZ -> ( A ..^ ( A + 1 ) ) = { A } ) $=
|
|
( cz wcel cfz co c1 caddc cfzo csn fzval3 fzsn eqtr3d ) ABCAADEAAFGEHEAIAAJ
|
|
AKL $.
|
|
|
|
$( Membership of an integer in the smallest open range of integers.
|
|
(Contributed by Alexander van der Vekens, 22-Sep-2018.) $)
|
|
elfzomin $p |- ( Z e. ZZ -> Z e. ( Z ..^ ( Z + 1 ) ) ) $=
|
|
( cz wcel csn c1 caddc co cfzo snidg fzosn eleqtrrd ) ABCAADAAEFGHGABIAJK
|
|
$.
|
|
|
|
$( Membership of an integer increased by a nonnegative integer in a half-
|
|
open integer range. (Contributed by Alexander van der Vekens,
|
|
22-Sep-2018.) $)
|
|
zpnn0elfzo $p |- ( ( Z e. ZZ /\ N e. NN0 )
|
|
-> ( Z + N ) e. ( Z ..^ ( ( Z + N ) + 1 ) ) ) $=
|
|
( cz wcel cn0 wa cuz cfv caddc cfzo uzid anim1i nn0z zaddcl sylan2 elfzomin
|
|
co c1 syl wss uzaddcl fzoss1 sselda syl2anc ) BCDZAEDZFZBBGHZDZUFFZBAIQZUKU
|
|
KRIQZJQZDZUKBULJQZDUEUIUFBKLUGUKCDZUNUFUEACDUPAMBANOUKPSUJUMUOUKUJUKUHDUMUO
|
|
TABBUAUKBULUBSUCUD $.
|
|
|
|
$( Membership of an integer increased by a nonnegative integer in a half-
|
|
open integer range. (Contributed by Alexander van der Vekens,
|
|
22-Sep-2018.) $)
|
|
zpnn0elfzo1 $p |- ( ( Z e. ZZ /\ N e. NN0 )
|
|
-> ( Z + N ) e. ( Z ..^ ( Z + ( N + 1 ) ) ) ) $=
|
|
( cz wcel cn0 wa caddc co c1 cfzo zpnn0elfzo cc adantr nn0cn adantl addassd
|
|
zcn 1cnd oveq2d eleqtrd ) BCDZAEDZFZBAGHZBUDIGHZJHBBAIGHGHZJHABKUCUEUFBJUCB
|
|
AIUABLDUBBQMUBALDUAANOUCRPST $.
|
|
|
|
$( Removing a singleton from a half-open integer range at the end.
|
|
(Contributed by Alexander van der Vekens, 23-Mar-2018.) $)
|
|
fzosplitsnm1 $p |- ( ( A e. ZZ /\ B e. ( ZZ>= ` ( A + 1 ) ) ) ->
|
|
( A ..^ B ) = ( ( A ..^ ( B - 1 ) ) u. { ( B - 1 ) } ) ) $=
|
|
( cz wcel c1 caddc co cuz cfv wa cfzo cmin cun csn wceq eluzelz zcnd adantl
|
|
cc syl ax-1cn npcan eqcomd sylancl cfz eluzp1m1 peano2zm uzid peano2uz 4syl
|
|
oveq2d elfzuzb sylanbrc fzosplit fzosn uneq2d 3eqtrd ) ACDZBAEFGZHIDZJZABKG
|
|
ABELGZEFGZKGZAVBKGZVBVCKGZMZVEVBNZMVABVCAKVABSDZESDZBVCOUTVIURUTBUSBPZQRUAV
|
|
IVJJVCBBEUBUCUDUKVAVBAVCUEGDZVDVGOVAVBAHIDVCVBHIZDZVLABUFVABCDZVBCDZVBVMDVN
|
|
UTVOURVKRBUGZVBUHVBVBUIUJVBAVCULUMAVCVBUNTVAVFVHVEVAVPVFVHOUTVPURUTVOVPVKVQ
|
|
TRVBUOTUPUQ $.
|
|
|
|
$( If an element of a half-open integer range is not less than the upper
|
|
bound of the range decreased by 1, it must be equal to the upper bound of
|
|
the range decreased by 1. (Contributed by AV, 3-Nov-2018.) $)
|
|
elfzonlteqm1 $p |- ( ( A e. ( 0 ..^ B ) /\ -. A < ( B - 1 ) )
|
|
-> A = ( B - 1 ) ) $=
|
|
( cc0 cfzo co wcel c1 cmin clt wbr wn wceq csn cun cuz cfv w3a elfzo0 sylbi
|
|
cn wi cz caddc 0z elnnuz biimpi 0p1e1 fveq2d eleqtrrd 3ad2ant2 fzosplitsnm1
|
|
cn0 a1i sylancr eleq2 elun pm2.24 3ad2ant3 elsni a1d jaoi syl6bi mpcom imp
|
|
wo ) ACBDEZFZABGHEZIJZKZAVHLZVFCVHDEZVHMZNZLZVGVJVKUAZVGCUBFBCGUCEZOPZFZVOU
|
|
DVGAULFZBTFZABIJZQVSABRWAVTVSWBWABGOPZVRWABWCFBUEUFWAVQGOVQGLWAUGUMUHUIUJSC
|
|
BUKUNVOVGAVNFZVPVFVNAUOWDAVLFZAVMFZVEVPAVLVMUPWEVPWFWEVTVHTFZVIQVPAVHRVIVTV
|
|
PWGVIVKUQURSWFVKVJAVHUSUTVASVBVCVD $.
|
|
|
|
$( A nonnegative integer is element of the half-open range of nonnegative
|
|
integers with the element increased by one as an upper bound.
|
|
(Contributed by Alexander van der Vekens, 5-Aug-2018.) $)
|
|
fzonn0p1 $p |- ( N e. NN0 -> N e. ( 0 ..^ ( N + 1 ) ) ) $=
|
|
( cn0 wcel c1 caddc co cn clt wbr cc0 cfzo id nn0p1nn nn0re ltp1d syl3anbrc
|
|
elfzo0 ) ABCZRADEFZGCASHIAJSKFCRLAMRAANOASQP $.
|
|
|
|
$( A half-open range of nonnegative integers is a subset of a half-open range
|
|
of nonnegative integers with the upper bound increased by one.
|
|
(Contributed by Alexander van der Vekens, 5-Aug-2018.) $)
|
|
fzossfzop1 $p |- ( N e. NN0 -> ( 0 ..^ N ) C_ ( 0 ..^ ( N + 1 ) ) ) $=
|
|
( cn0 wcel c1 caddc co cuz cfv cc0 cfzo wss cz cle wbr w3a nn0z peano2z zre
|
|
id syl lep1d 3jca eluz2 sylibr fzoss2 ) ABCZADEFZAGHCZIAJFIUGJFKUFALCZUGLCZ
|
|
AUGMNZOZUHUFUIULAPUIUIUJUKUISAQUIAARUAUBTAUGUCUDAIUGUET $.
|
|
|
|
$( If a nonnegative integer is element of a half-open range of nonnegative
|
|
integers, increasing this integer by one results in an element of a half-
|
|
open range of nonnegative integers with the upper bound increased by one.
|
|
(Contributed by Alexander van der Vekens, 5-Aug-2018.) $)
|
|
fzonn0p1p1 $p |- ( I e. ( 0 ..^ N ) -> ( I + 1 ) e. ( 0 ..^ ( N + 1 ) ) ) $=
|
|
( cc0 cfzo co wcel cn0 cn clt wbr w3a c1 elfzo0 peano2nn0 3ad2ant1 peano2nn
|
|
caddc 3ad2ant2 simp3 cr nn0re nnre 1red ltadd1 syl3an mpbid syl3anbrc sylbi
|
|
wb ) ACBDEFAGFZBHFZABIJZKZALQEZCBLQEZDEFZABMUMUNGFZUOHFZUNUOIJZUPUJUKUQULAN
|
|
OUKUJURULBPRUMULUSUJUKULSUJATFUKBTFULLTFULUSUIAUABUBULUCABLUDUEUFUNUOMUGUH
|
|
$.
|
|
|
|
$( Increasing an element of a half-open range of nonnegative integers by 1
|
|
results in an element of the half-open range of nonnegative integers with
|
|
an upper bound increased by 1. (Contributed by Alexander van der Vekens,
|
|
1-Aug-2018.) $)
|
|
elfzom1p1elfzo $p |- ( ( N e. NN /\ X e. ( 0 ..^ ( N - 1 ) ) )
|
|
-> ( X + 1 ) e. ( 0 ..^ N ) ) $=
|
|
( cc0 c1 cmin co cfzo wcel cn caddc cn0 clt wbr w3a elfzo0 peano2nn0 adantr
|
|
wi wa cr 3ad2ant1 simpr 1red nnre adantl ltaddsubd biimprd impancom 3adant2
|
|
nn0re imp syl3anbrc ex sylbi impcom ) BCADEFZGFHZAIHZBDJFZCAGFHZUQBKHZUPIHZ
|
|
BUPLMZNZURUTRBUPOVDURUTVDURSUSKHZURUSALMZUTVDVEURVAVBVEVCBPUAQVDURUBVDURVFV
|
|
AVCURVFRVBVAURVCVFVAURSZVFVCVGBDAVABTHURBUJQVGUCURATHVAAUDUEUFUGUHUIUKUSAOU
|
|
LUMUNUO $.
|
|
|
|
${
|
|
$d k N $.
|
|
$( Half-open integer ranges starting with 0 are subsets of NN0.
|
|
(Contributed by Thierry Arnoux, 8-Oct-2018.) $)
|
|
fzo0ssnn0 $p |- ( 0 ..^ N ) C_ NN0 $=
|
|
( vk cc0 cfzo co cfz cn0 fzossfz cv elfznn0 ssriv sstri ) CADECAFEZGCAHBM
|
|
GBIAJKL $.
|
|
$}
|
|
|
|
$( Expressing the singleton of ` 0 ` as a half-open integer range.
|
|
(Contributed by Stefan O'Rear, 15-Aug-2015.) $)
|
|
fzo01 $p |- ( 0 ..^ 1 ) = { 0 } $=
|
|
( cc0 c1 cfzo co caddc csn 1e0p1 oveq2i cz wcel wceq 0z fzosn ax-mp eqtri )
|
|
ABCDAABEDZCDZAFZBPACGHAIJQRKLAMNO $.
|
|
|
|
$( A 1-based half-open integer interval up to, but not including, 2 is a
|
|
singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.) $)
|
|
fzo12sn $p |- ( 1 ..^ 2 ) = { 1 } $=
|
|
( c1 c2 cfzo co caddc csn df-2 oveq2i cz wcel wceq 1z fzosn ax-mp eqtri ) A
|
|
BCDAAAEDZCDZAFZBPACGHAIJQRKLAMNO $.
|
|
|
|
$( A half-open integer range from 0 to 2 is an unordered pair. (Contributed
|
|
by Alexander van der Vekens, 4-Dec-2017.) $)
|
|
fzo0to2pr $p |- ( 0 ..^ 2 ) = { 0 , 1 } $=
|
|
( cc0 c2 cfzo co c1 cmin cfz caddc cz wcel wceq 2z fzoval ax-mp 2m1e1 0p1e1
|
|
cpr eqtr4i oveq2i 0z fzpr preq2i syl6eq 3eqtri ) ABCDZABEFDZGDZAAEHDZGDZAEQ
|
|
ZBIJUEUGKLABMNUFUHAGUFEUHOPRSAIJZUIUJKTUKUIAUHQUJAUAUHEAPUBUCNUD $.
|
|
|
|
$( A half-open integer range from 0 to 3 is an unordered triple.
|
|
(Contributed by Alexander van der Vekens, 9-Nov-2017.) $)
|
|
fzo0to3tp $p |- ( 0 ..^ 3 ) = { 0 , 1 , 2 } $=
|
|
( cc0 c3 cfzo co c1 cmin cfz c2 caddc ctp cz wcel 3z fzoval ax-mp 3m1e2 2cn
|
|
wceq addid2i a1i eqtr4i oveq2i 0z fztp eqidd 0p1e1 tpeq123d eqtrd 3eqtri )
|
|
ABCDZABEFDZGDZAAHIDZGDZAEHJZBKLUJULRMABNOUKUMAGUKHUMPHQSZUAUBAKLZUNUORUCUQU
|
|
NAAEIDZUMJUOAUDUQAAUREUMHUQAUEURERUQUFTUMHRUQUPTUGUHOUI $.
|
|
|
|
$( A half-open integer range from 0 to 4 is a union of two unordered pairs.
|
|
(Contributed by Alexander van der Vekens, 17-Nov-2017.) $)
|
|
fzo0to42pr $p |- ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } ) $=
|
|
( cc0 c4 cfzo co c2 cun c1 cpr c3 cfz wcel wceq ax-mp caddc cz ax-1cn eqtri
|
|
cn0 3cn eqcomi cle wbr 2nn0 4nn0 2re 2lt4 ltleii elfz2nn0 mpbir3an fzosplit
|
|
4re fzo0to2pr cmin 4z fzoval 4cn df-4 addcomi subaddrii df-3 oveq2i 2z fzpr
|
|
preq2i 3eqtri uneq12i ) ABCDZAECDZEBCDZFZAGHZEIHZFEABJDKZVGVJLVMERKBRKEBUAU
|
|
BUCUDEBUEUKUFUGEBUHUIABEUJMVHVKVIVLULVIEBGUMDZJDZEEGNDZHZVLBOKVIVOLUNEBUOMV
|
|
OEVPJDZVQVNVPEJVNIVPBGIUPPSBGINDZBIGNDVSUQIGSPURQTUSUTQVAEOKVRVQLVBEVCMQVPI
|
|
EIVPUTTVDVEVFQ $.
|
|
|
|
$( A half-open range of nonnegative integers is the union of the singleton
|
|
set containing 0 and a half-open range of positive integers. (Contributed
|
|
by Alexander van der Vekens, 18-May-2018.) $)
|
|
fzo0sn0fzo1 $p |- ( N e. NN -> ( 0 ..^ N ) = ( { 0 } u. ( 1 ..^ N ) ) ) $=
|
|
( cn wcel cc0 cfzo co c1 cun csn cfz wceq cn0 cle 1nn0 nnnn0 nnge1 elfz2nn0
|
|
wbr a1i syl3anbrc fzosplit syl fzo01 uneq1d eqtrd ) ABCZDAEFZDGEFZGAEFZHZDI
|
|
ZUIHUFGDAJFCZUGUJKUFGLCZALCGAMRULUMUFNSAOAPGAQTDAGUAUBUFUHUKUIUHUKKUFUCSUDU
|
|
E $.
|
|
|
|
$( The endpoint of a half-open integer range. (Contributed by Mario
|
|
Carneiro, 29-Sep-2015.) $)
|
|
fzoend $p |- ( A e. ( A ..^ B ) -> ( B - 1 ) e. ( A ..^ B ) ) $=
|
|
( cfzo co wcel c1 cmin cfz cuz cfv wceq elfzoel2 fzoval syl eleqtrd elfzuz3
|
|
id cz eluzfz2 eleqtrrd ) AABCDZEZBFGDZAUCHDZUAUBUCAIJEZUCUDEUBAUDEUEUBAUAUD
|
|
UBQUBBREUAUDKAABLABMNZOAAUCPNAUCSNUFT $.
|
|
|
|
$( The endpoint of a zero-based half-open range. (Contributed by Stefan
|
|
O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) $)
|
|
fzo0end $p |- ( B e. NN -> ( B - 1 ) e. ( 0 ..^ B ) ) $=
|
|
( cn wcel cc0 cfzo co c1 cmin lbfzo0 fzoend sylbir ) ABCDDAEFZCAGHFLCAIDAJK
|
|
$.
|
|
|
|
$( Subset relationship for half-open integer ranges. (Contributed by
|
|
Alexander van der Vekens, 16-Mar-2018.) $)
|
|
ssfzo12 $p |- ( ( K e. ZZ /\ L e. ZZ /\ K < L )
|
|
-> ( ( K ..^ L ) C_ ( M ..^ N ) -> ( M <_ K /\ L <_ N ) ) ) $=
|
|
( cfzo co wcel cz clt wbr w3a wss cle wa wi fzolb2 biimp3ar ssel2 ex mpcom
|
|
c1 cmin fzoend elfzolt2 wb simp2 elfzoel2 zlem1lt elfzole1 pm3.2 syl adantr
|
|
syl2anr sylbird com13 3syl com24 syl5com pm2.43a com14 ) AABEFZGZAHGZBHGZAB
|
|
IJZKZVACDEFZLZCAMJZBDMJZNZOZVCVDVBVEABPQBUAUBFZVAGZVBVFVLOABUCVHVBVFVNVKVBV
|
|
HVFVNVKOOZVHVBVHVOOVHVBNAVGGZVHVOVAVGARVHVNVFVPVKVHVNVFVPVKOOZVHVNNVMVGGVMD
|
|
IJZVQVAVGVMRVMCDUDVPVFVRVKVPVFVRVKOVPVFNVRVJVKVFVDDHGVJVRUEVPVCVDVEUFACDUGB
|
|
DUHUMVPVJVKOZVFVPVIVSACDUIVIVJUJUKULUNSUOUPSUQURSUSUTTT $.
|
|
|
|
${
|
|
$d K x $. $d L x $. $d M x $. $d N x $.
|
|
$( Subset relationship for half-open integer ranges. (Contributed by
|
|
Alexander van der Vekens, 5-Nov-2018.) $)
|
|
ssfzo12bi $p |- ( ( ( K e. ZZ /\ L e. ZZ ) /\ ( M e. ZZ /\ N e. ZZ )
|
|
/\ K < L )
|
|
-> ( ( K ..^ L ) C_ ( M ..^ N ) <-> ( M <_ K /\ L <_ N ) ) ) $=
|
|
( vx cz wcel wa clt wbr w3a cfzo cle wi adantr cr zre adantl imp com12 co
|
|
wss df-3an biimpri 3adant2 ssfzo12 syl cv cuz elfzo2 eluz2 simprrl simpll
|
|
cfv letr syl3anc exp31 com23 expdimp impancom com13 3adant3 impcom sylibr
|
|
simpl2r ad3antlr ltletr ex expcomd adantld syl3anbrc 3adant1 sylbi impbid
|
|
3jca ssrdv ) AFGZBFGZHZCFGZDFGZHZABIJZKZABLUAZCDLUAZUBZCAMJZBDMJZHZWDVQVR
|
|
WCKZWGWJNVSWCWKWBWKVSWCHVQVRWCUCUDUEABCDUFUGWDWJWGWDWJHZEWEWFEUHZWEGZWLWM
|
|
WFGZWNWMAUIUNGZVRWMBIJZKWLWONZWMABUJWPWQWRVRWPWQWRWPVQWMFGZAWMMJZKWQWRNZA
|
|
WMUKWSWTXAVQWSWTHZWQWLWOXBWQHZWLHZWMCUIUNGZWAWMDIJZWOXDVTWSCWMMJZKZXEXCWL
|
|
XHXBWLXHNWQWLXBXHWJWDXBXHNZWHWDXINWIWDWHXIVSWBWHXINWCXBWHVSWBHZXHWSWHWTXJ
|
|
XHNZWSWHWTXKWSXJWHWTHZXHWSXJXLXHWSXJHZXLHVTWSXGXMVTXLWSVSVTWAULOWSXJXLUMX
|
|
MXLXGXMCPGZAPGZWMPGZXLXGNXJXNWSWBXNVSVTXNWACQORRXJXOWSVSXOWBVQXOVRAQOORWS
|
|
XPXJWMQZOCAWMUOUPSVOUQURUSUTVAVBTOVCTOSCWMUKVDWLWAXCVTWAVSWCWJVERXCWLXFXB
|
|
WQWLXFNZWSWQXRNWTWLWQWSXFWDWJWQWSXFNZNZWDWIXTWHWDWQWIXSVSWBWQWIHZXSNWCXJW
|
|
SYAXFXJWSYAXFNZXJWSHXPBPGZDPGZYBWSXPXJXQRVRYCVQWBWSBQVFXJYDWSWBYDVSWAYDVT
|
|
DQRROWMBDVGUPVHURVBVIVJSVAOSSWMCDUJVKUQVLVMSUEVMTVPVHVN $.
|
|
$}
|
|
|
|
$( The result of subtracting 1 and an integer of a half-open range of
|
|
nonnegative integers from the upper bound of this range is contained in
|
|
this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV,
|
|
30-Oct-2018.) $)
|
|
ubmelm1fzo $p |- ( K e. ( 0 ..^ N ) -> ( ( N - K ) - 1 ) e. ( 0 ..^ N ) ) $=
|
|
( cc0 co wcel cn0 cn clt wbr cmin c1 elfzo0 cz wa adantr adantl syl 3adant3
|
|
wb cr cfzo w3a cle nnz zsubcld ancoms peano2zm simp3 anim12i znnsub nnm1ge0
|
|
nn0z mpbid elnn0z sylanbrc simp2 caddc cc nncn nn0cn 1cnd subsub4d nn0p1gt0
|
|
nn0re peano2re nnre ltsubpos syl2an eqbrtrd syl3anbrc sylbi ) ACBUADZEAFEZB
|
|
GEZABHIZUBZBAJDZKJDZVLEZABLVPVRFEZVNVRBHIZVSVPVRMEZCVRUCIZVTVMVNWBVOVMVNNZV
|
|
QMEZWBVNVMWEVNVMNBAVNBMEZVMBUDZOVMAMEZVNAULZPUEUFVQUGQRVPVQGEZWCVPVOWJVMVNV
|
|
OUHVPWHWFNZVOWJSVMVNWKVOVMWHVNWFWIWGUIRABUJQUMVQUKQVRUNUOVMVNVOUPVMVNWAVOWD
|
|
VRBAKUQDZJDZBHWDBAKVNBUREVMBUSPVMAUREVNAUTOWDVAVBWDCWLHIZWMBHIZVMWNVNAVCOVM
|
|
WLTEZBTEWNWOSVNVMATEWPAVDAVEQBVFWLBVGVHUMVIRVRBLVJVK $.
|
|
|
|
$( If a point is in a half-open range, the next point is in the closed range.
|
|
(Contributed by Stefan O'Rear, 23-Aug-2015.) $)
|
|
fzofzp1 $p |- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ... B ) ) $=
|
|
( cfzo co wcel c1 caddc cfz cz cuz cfv wss elfzoel1 uzid peano2uz fzoss1 1z
|
|
4syl fzoaddel mpan2 sseldd wceq elfzoel2 fzval3 syl eleqtrrd ) CABDEFZCGHEZ
|
|
ABGHEZDEZABIEZUHAGHEZUJDEZUKUIUHAJFAAKLZFUMUOFUNUKMCABNAOAAPUMAUJQSUHGJFUIU
|
|
NFRCABGTUAUBUHBJFULUKUCCABUDABUEUFUG $.
|
|
|
|
$( If a point is in a half-open range, the next point is in the closed range.
|
|
(Contributed by Mario Carneiro, 27-Sep-2015.) $)
|
|
fzofzp1b $p |- ( C e. ( ZZ>= ` A ) ->
|
|
( C e. ( A ..^ B ) <-> ( C + 1 ) e. ( A ... B ) ) ) $=
|
|
( cuz cfv wcel cfzo co c1 caddc fzofzp1 cmin simpl eluzelz elfzuz3 eluzp1m1
|
|
cfz wa cz syl2an elfzuzb sylanbrc wceq elfzel2 adantl fzoval syl ex impbid2
|
|
eleqtrrd ) CADEFZCABGHZFZCIJHZABQHFZABCKUKUOUMUKUORZCABILHZQHZULUPUKUQCDEFZ
|
|
CURFUKUOMUKCSFBUNDEFUSUOACNUNABOCBPTCAUQUAUBUPBSFZULURUCUOUTUKUNABUDUEABUFU
|
|
GUJUHUI $.
|
|
|
|
$( An integer is a member of a 1-based finite set of sequential integers iff
|
|
its predecessor is a member of the corresponding 0-based set.
|
|
(Contributed by Mario Carneiro, 27-Sep-2015.) $)
|
|
elfzom1b $p |- ( ( K e. ZZ /\ N e. ZZ ) ->
|
|
( K e. ( 1 ..^ N ) <-> ( K - 1 ) e. ( 0 ..^ ( N - 1 ) ) ) ) $=
|
|
( cz wcel wa c1 cmin co cfz cc0 cfzo wb peano2zm elfzm1b sylan2 wceq fzoval
|
|
adantl eleq2d syl 3bitr4d ) ACDZBCDZEZAFBFGHZIHZDZAFGHZJUEFGHIHZDZAFBKHZDUH
|
|
JUEKHZDUCUBUECDZUGUJLBMZAUENOUDUKUFAUCUKUFPUBFBQRSUDULUIUHUDUMULUIPUCUMUBUN
|
|
RJUEQTSUA $.
|
|
|
|
$( If an element of a half-open integer range is not contained in the lower
|
|
subrange, it must be in the upper subrange. (Contributed by Alexander van
|
|
der Vekens, 30-Mar-2018.) $)
|
|
elfzonelfzo $p |- ( N e. ZZ -> ( ( K e. ( M ..^ R ) /\ -. K e. ( M ..^ N ) )
|
|
-> K e. ( N ..^ R ) ) ) $=
|
|
( cfzo co wcel wn wa cz cuz cfv clt wbr wi elfzo2 cle simpr 3ad2ant1 adantr
|
|
w3a eluzelz ad2antrr wb eluzel2 syl3anc eluzle biantrurd bitr4d notbid zred
|
|
elfzo lenltd biimpd ex com23 imp31 eluz2 syl3anbrc simpll2 simpll3 sylanb
|
|
com12 ) BCAEFGZBCDEFGZHZIDJGZBDAEFGZVDBCKLGZAJGZBAMNZUAZVFVGVHOBCAPVLVFIZVG
|
|
VHVMVGIZBDKLGZVJVKVHVNVGBJGZDBQNZVOVMVGRVLVPVFVGVIVJVPVKCBUBZSUCVLVFVGVQVIV
|
|
JVFVGVQOOVKVIVGVFVQVIVGVFVQOVIVGIZVFVQVSVFBDMNZHVQVSVEVTVSVECBQNZVTIZVTVSVP
|
|
CJGZVGVEWBUDVIVPVGVRTZVIWCVGCBUETVIVGRZBCDULUFVSWAVTVIWAVGCBUGTUHUIUJVSDBVS
|
|
DWEUKVSBWDUKUMUIUNUOUPSUQDBURUSVIVJVKVFVGUTVIVJVKVFVGVABDAPUSUOVBVC $.
|
|
|
|
$( An integer increased by another integer is an element of a half-open
|
|
integer range if and only if the integer is contained in the half-open
|
|
integer range with bounds decreased by the other integer. (Contributed by
|
|
Alexander van der Vekens, 30-Mar-2018.) $)
|
|
elfzomelpfzo $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ L e. ZZ ) )
|
|
-> ( K e. ( ( M - L ) ..^ ( N - L ) ) <-> ( K + L ) e. ( M ..^ N ) ) ) $=
|
|
( cz wcel wa cmin co cuz cfv clt wbr w3a cfzo adantr 2thd adantl cr zre cle
|
|
caddc zsubcl ad2ant2rl simpl zaddcl lesubaddd eluz2 3bitr4g ad2ant2l simplr
|
|
3anbi123d ltaddsubd bicomd elfzo2 ) CEFZDEFZGZAEFZBEFZGZGZACBHIZJKFZDBHIZEF
|
|
ZAVELMZNABUBIZCJKFZUQVHDLMZNAVCVEOIFVHCDOIFVBVDVIVFUQVGVJVBVCEFZUSVCAUAMZNU
|
|
PVHEFZCVHUAMZNVDVIVBVKUPUSVMVLVNVBVKUPUPUTVKUQUSCBUCUDURUPVAUPUQUEPQVBUSVMV
|
|
AUSURUSUTUERVAVMURABUFRQVBCBAURCSFZVAUPVOUQCTPPVABSFZURUTVPUSBTRRZVAASFZURU
|
|
SVRUTATPRZUGULVCAUHCVHUHUIVBVFUQUQUTVFUPUSDBUCUJUPUQVAUKQVBVJVGVBABDVSVQURD
|
|
SFZVAUQVTUPDTRPUMUNULAVCVEUOVHCDUOUI $.
|
|
|
|
$( A Peano-postulate-like theorem for downward closure of a half-open integer
|
|
range. (Contributed by Mario Carneiro, 1-Oct-2015.) $)
|
|
peano2fzor $p |- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M ..^ N ) ) ->
|
|
K e. ( M ..^ N ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfzo wa cmin cfz simpr cz elfzoel2 adantl fzoval
|
|
wceq syl eleqtrd peano2fzr syldan eleqtrrd ) ABDEFZAGHIZBCJIZFZKZABCGLIZMIZ
|
|
UGUEUHUFUKFAUKFUIUFUGUKUEUHNUICOFZUGUKSUHULUEUFBCPQBCRTZUAABUJUBUCUMUD $.
|
|
|
|
$( Extending a half-open range by a singleton on the end. (Contributed by
|
|
Stefan O'Rear, 23-Aug-2015.) $)
|
|
fzosplitsn $p |- ( B e. ( ZZ>= ` A ) ->
|
|
( A ..^ ( B + 1 ) ) = ( ( A ..^ B ) u. { B } ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfzo cun csn cfz wceq eluzelz uzid peano2uz 3syl
|
|
id cz syl elfzuzb sylanbrc fzosplit fzosn uneq2d eqtrd ) BACDEZABFGHZIHZABI
|
|
HZBUHIHZJZUJBKZJUGBAUHLHEZUIULMUGUGUHBCDZEZUNUGRUGBSEZBUOEUPABNZBOBBPQBAUHU
|
|
AUBAUHBUCTUGUKUMUJUGUQUKUMMURBUDTUEUF $.
|
|
|
|
$( Extending a half-open integer range by an unordered pair at the end.
|
|
(Contributed by Alexander van der Vekens, 22-Sep-2018.) $)
|
|
fzosplitprm1 $p |- ( ( A e. ZZ /\ B e. ZZ /\ A < B ) ->
|
|
( A ..^ ( B + 1 ) ) = ( ( A ..^ ( B - 1 ) ) u. { ( B - 1 ) , B } ) ) $=
|
|
( cz wcel wbr c1 caddc co cfzo csn cun wceq cle cr zre syl3anbrc fzosplitsn
|
|
eluz2 syl cc clt w3a cmin cpr cuz cfv simp1 simp2 wi ltle syl2an 3impia zcn
|
|
ax-1cn npcan eqcomd sylancl 3ad2ant2 oveq2d peano2zm zltlem1 biimp3a uneq1d
|
|
wa eqtrd unass df-pr eqcomi a1i uneq2d syl5eq 3eqtrd ) ACDZBCDZABUAEZUBZABF
|
|
GHIHZABIHZBJZKZABFUCHZIHZWAJZKZVSKZWBWABUDZKZVPBAUEUFZDZVQVTLVPVMVNABMEZWIV
|
|
MVNVOUGZVMVNVOUHVMVNVOWJVMANDBNDVOWJUIVNAOBOABUJUKULABRPABQSVPVRWDVSVPVRAWA
|
|
FGHZIHZWDVPBWLAIVNVMBWLLZVOVNBTDZFTDZWNBUMUNWOWPVDWLBBFUOUPUQURUSVPWAWHDZWM
|
|
WDLVPVMWACDZAWAMEZWQWKVNVMWRVOBUTURVMVNVOWSABVAVBAWARPAWAQSVEVCVPWEWBWCVSKZ
|
|
KWGWBWCVSVFVPWTWFWBWTWFLVPWFWTWABVGVHVIVJVKVL $.
|
|
|
|
$( Membership in a half-open range extended by a singleton. (Contributed by
|
|
Stefan O'Rear, 23-Aug-2015.) $)
|
|
fzosplitsni $p |- ( B e. ( ZZ>= ` A ) -> ( C e. ( A ..^ ( B + 1 ) ) <->
|
|
( C e. ( A ..^ B ) \/ C = B ) ) ) $=
|
|
( cuz cfv wcel c1 caddc co cfzo csn wceq wo fzosplitsn eleq2d elsn2g orbi2d
|
|
cun elun syl5bb bitrd ) BADEZFZCABGHIJIZFCABJIZBKZRZFZCUEFZCBLZMZUCUDUGCABN
|
|
OUHUICUFFZMUCUKCUEUFSUCULUJUICBUBPQTUA $.
|
|
|
|
$( A finite interval of integers as union of a half-open integer range and a
|
|
singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.) $)
|
|
fzisfzounsn $p |- ( B e. ( ZZ>= ` A )
|
|
-> ( A ... B ) = ( ( A ..^ B ) u. { B } ) ) $=
|
|
( cuz cfv wcel cfz co caddc cfzo csn cun wceq eluzelz fzval3 syl fzosplitsn
|
|
c1 cz eqtrd ) BACDEZABFGZABQHGIGZABIGBJKTBREUAUBLABMABNOABPS $.
|
|
|
|
$( Two possibilities for a number one greater than a number in a half-open
|
|
range. (Contributed by Stefan O'Rear, 23-Aug-2015.) $)
|
|
fzostep1 $p |- ( A e. ( B ..^ C ) -> ( ( A + 1 ) e. ( B ..^ C ) \/
|
|
( A + 1 ) = C ) ) $=
|
|
( cfzo co wcel c1 caddc wceq wo cz cuz cfv wss elfzoel1 peano2uz wbr cr zre
|
|
uzid fzoss1 4syl 1z fzoaddel mpan2 sseldd wb cle elfzoel2 clt elfzolt3 ltle
|
|
wi syl2an syl2anc mpd eluz2 syl3anbrc fzosplitsni syl mpbid ) ABCDEZFZAGHEZ
|
|
BCGHEZDEZFZVDVBFVDCIJZVCBGHEZVEDEZVFVDVCBKFZBBLMZFVIVLFVJVFNABCOZBTBBPVIBVE
|
|
UAUBVCGKFVDVJFUCABCGUDUEUFVCCVLFZVGVHUGVCVKCKFZBCUHQZVNVMABCUIZVCBCUJQZVPAB
|
|
CUKVCVKVOVRVPUMZVMVQVKBRFCRFVSVOBSCSBCULUNUOUPBCUQURBCVDUSUTVA $.
|
|
|
|
${
|
|
$d K j k $. $d M j k $. $d N j k $. $d ph k $.
|
|
$( Shift the scanning order inside of a quantification over a half-open
|
|
integer range, analogous to ~ fzshftral . (Contributed by Alexander van
|
|
der Vekens, 23-Sep-2018.) $)
|
|
fzoshftral $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ )
|
|
-> ( A. j e. ( M ..^ N ) ph
|
|
<-> A. k e. ( ( M + K ) ..^ ( N + K ) ) [. ( k - K ) / j ]. ph ) ) $=
|
|
( cz wcel cfzo co wral c1 cmin cfz caddc wceq fzoval raleqdv 3adant1 cc
|
|
w3a cv 3ad2ant2 wb peano2zm fzshftral syl3an2 zaddcl syl wa adantr adantl
|
|
wsbc zcn 1cnd addsubd oveq2d eqtr2d 3bitrd ) EGHZFGHZDGHZUAZABEFIJZKABEFL
|
|
MJZNJZKZABCUBDMJUMZCEDOJZVEDOJZNJZKZVHCVIFDOJZIJZKVCABVDVFVAUTVDVFPVBEFQU
|
|
CRVAUTVEGHVBVGVLUDFUEABCDEVEUFUGVCVHCVKVNVCVNVIVMLMJZNJZVKVCVMGHZVNVPPVAV
|
|
BVQUTFDUHSVIVMQUIVCVOVJVINVAVBVOVJPUTVAVBUJZFDLVAFTHVBFUNUKVBDTHVADUNULVR
|
|
UOUPSUQURRUS $.
|
|
$}
|
|
|
|
${
|
|
$d K x $. $d M x y $. $d N x y $. $d ch x $. $d ph y $. $d ps x $.
|
|
$d ta x $. $d th x $.
|
|
fzind2.1 $e |- ( x = M -> ( ph <-> ps ) ) $.
|
|
fzind2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
fzind2.3 $e |- ( x = ( y + 1 ) -> ( ph <-> th ) ) $.
|
|
fzind2.4 $e |- ( x = K -> ( ph <-> ta ) ) $.
|
|
fzind2.5 $e |- ( N e. ( ZZ>= ` M ) -> ps ) $.
|
|
fzind2.6 $e |- ( y e. ( M ..^ N ) -> ( ch -> th ) ) $.
|
|
$( Induction on the integers from ` M ` to ` N ` inclusive. The first four
|
|
hypotheses give us the substitution instances we need; the last two are
|
|
the basis and the induction step. Version of ~ fzind using integer
|
|
range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) $)
|
|
fzind2 $p |- ( K e. ( M ... N ) -> ta ) $=
|
|
( wcel wa wbr w3a cfz co cz cle elfz2 df-3an anbi1i 3anass anbi2i 3bitr4i
|
|
anass bitri cuz cfv eluz2 sylbir cv clt wi cfzo elfzo syl6bir 3coml 3expa
|
|
impr sylan2b fzind sylbi ) HIJUAUBQZIUCQZJUCQZRZHUCQZIHUDSZHJUDSZTZRZEVIV
|
|
JVKVMTZVNVORZRZVQHIJUEVLVMRZVSRVLVMVSRZRVTVQVLVMVSUKVRWAVSVJVKVMUFUGVPWBV
|
|
LVMVNVOUHUIUJULABCDEFGHIJKLMNVJVKIJUDSTJIUMUNQBIJUOOUPGUQZUCQZIWCUDSZWCJU
|
|
RSZTVLWDWEWFRZRCDUSZWDWEWFUHVLWDWGWHVJVKWDWGWHUSZWDVJVKWIWDVJVKTWGWCIJUTU
|
|
BQWHWCIJVAPVBVCVDVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d M n w y $. $d N n w y $. $d n ph w y $. $d ps w y $.
|
|
exfzdc.1 $e |- ( ph -> M e. ZZ ) $.
|
|
exfzdc.2 $e |- ( ph -> N e. ZZ ) $.
|
|
exfzdc.3 $e |- ( ( ph /\ n e. ( M ... N ) ) -> DECID ps ) $.
|
|
$( Decidability of the existence of an integer defined by a decidable
|
|
proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) $)
|
|
exfzdc $p |- ( ph -> DECID E. n e. ( M ... N ) ps ) $=
|
|
( cfz co wrex wdc wa wcel syl2anc wi wceq oveq2 rexeqdv dcbid cle wbr cuz
|
|
vw vy wn cz wb eluz biimpar simpl eluzfz2 cv c1 caddc imbi2d wsbc eluzfz1
|
|
cfv wral adantl ralrimiva adantr nfsbc1v nfdc sbceq1a rspc csn syl rexsns
|
|
sylc fzsn syl6bb mpbird expcom cfzo simpr fzofzp1 ad2antrr ad2antlr dcbii
|
|
cun wo sylibr dcor rexun elfzouz fzsuc exp31 a2d fzind2 c0 clt zltnle fzn
|
|
rex0 mpbid mtbiri olcd df-dc zdcle exmiddc mpjaodan ) ADEUAUBZBCDEIJZKZLZ
|
|
XDUFZAXDMEDUCUSZNZAXGAXJXDADUGNZEUGNZXJXDUHFGDEUIOUJAXDUKXJEXENAXGPZDEULA
|
|
BCDUDUMZIJZKZLZPABCDDIJZKZLZPABCDUEUMZIJZKZLZPABCDYAUNUOJZIJZKZLZPXMUDUEE
|
|
DEXNDQZXQXTAYIXPXSYIBCXOXRXNDDIRSTUPXNYAQZXQYDAYJXPYCYJBCXOYBXNYADIRSTUPX
|
|
NYEQZXQYHAYKXPYGYKBCXOYFXNYEDIRSTUPXNEQZXQXGAYLXPXFYLBCXOXEXNEDIRSTUPAXJX
|
|
TAXJMZXTBCDUQZLZYMDXENZBLZCXEUTZYOXJYPADEURVAAYRXJAYQCXEHVBZVCYQYOCDXEYNC
|
|
BCDVDVECUMZDQBYNBCDVFTVGVKYMXSYNYMXSBCDVHZKYNYMBCXRUUAYMXKXRUUAQAXKXJFVCD
|
|
VLVISBCDVJVMTVNVOYADEVPJNZAYDYHUUBAYDYHUUBAMZYDMZYHBCYBYEVHZWBZKZLZUUDYCB
|
|
CUUEKZWCZLZUUHUUDYDUUILZUUKUUCYDVQUUDBCYEUQZLZUULUUDYEXENZYRUUNUUBUUOAYDD
|
|
EYAVRVSAYRUUBYDYSVTYQUUNCYEXEUUMCBCYEVDVEYTYEQBUUMBCYEVFTVGVKUUIUUMBCYEVJ
|
|
WAWDYCUUIWEVKUUGUUJBCYBUUEWFWAWDUUDYGUUGUUDBCYFUUFUUDYAXINZYFUUFQUUBUUPAY
|
|
DYADEWGVSDYAWHVISTVNWIWJWKVIVKAXHMZXFXFUFZWCXGUUQUURXFUUQXFBCWLKBCWPUUQBC
|
|
XEWLUUQEDWMUBZXEWLQZAUUSXHAXLXKUUSXHUHGFEDWNOUJAUUSUUTUHZXHAXKXLUVAFGDEWO
|
|
OVCWQSWRWSXFWTWDAXKXLXDXHWCZFGXKXLMXDLUVBDEXAXDXBVIOXC $.
|
|
$}
|
|
|
|
${
|
|
$d F v $. $d K v $.
|
|
$( The function values for the borders of a finite interval of integers,
|
|
which is the domain of the function, are not in the image of the
|
|
interior of the interval iff the intersection of the images of the
|
|
interior and the borders is empty. (Contributed by Alexander van der
|
|
Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.) $)
|
|
fvinim0ffz $p |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 )
|
|
-> ( ( ( F " { 0 , K } ) i^i ( F " ( 1 ..^ K ) ) ) = (/)
|
|
<-> ( ( F ` 0 ) e/ ( F " ( 1 ..^ K ) )
|
|
/\ ( F ` K ) e/ ( F " ( 1 ..^ K ) ) ) ) ) $=
|
|
( vv cc0 co cn0 wcel wa cpr cima cin c0 wceq cfv wnel cle wbr adantl wn
|
|
cfz wf c1 cfzo wfn ffn adantr 0nn0 a1i simpr nn0ge0 elfz2nn0 syl3anbrc id
|
|
nn0re leidd fnimapr syl3anc ineq1d eqeq1d cv wral wb simpl ffvelrnd eleq1
|
|
disj notbid df-nel syl6bbr ralprg syl2anc syl5bb bitrd ) EBUAFZCAUBZBGHZI
|
|
ZAEBJKZAUCBUDFKZLZMNEAOZBAOZJZVTLZMNZWBVTPZWCVTPZIZVRWAWEMVRVSWDVTVRAVOUE
|
|
ZEVOHZBVOHZVSWDNVPWJVQVOCAUFUGVREGHZVQEBQRZWKWMVRUHUIVPVQUJVQWNVPBUKSEBUL
|
|
UMZVQWLVPVQVQVQBBQRWLVQUNZWPVQBBUOUPBBULUMSZVOEBAUQURUSUTWFDVAZVTHZTZDWDV
|
|
BZVRWIDWDVTVGVRWBCHWCCHXAWIVCVRVOCEAVPVQVDZWOVEVRVOCBAXBWQVEWTWGWHDWBWCCC
|
|
WRWBNZWTWBVTHZTWGXCWSXDWRWBVTVFVHWBVTVIVJWRWCNZWTWCVTHZTWHXEWSXFWRWCVTVFV
|
|
HWCVTVIVJVKVLVMVN $.
|
|
$}
|
|
|
|
$( The difference between two elements in a half-open range of nonnegative
|
|
integers is greater than the negation of the upper bound and less than the
|
|
upper bound of the range. (Contributed by AV, 20-Mar-2021.) $)
|
|
subfzo0 $p |- ( ( I e. ( 0 ..^ N ) /\ J e. ( 0 ..^ N ) )
|
|
-> ( -u N < ( I - J ) /\ ( I - J ) < N ) ) $=
|
|
( cc0 co wcel cmin clt wbr wa caddc cr cle adantr syl2an 3adant3 anim12i cc
|
|
3ad2ant1 adantl cfzo cneg cn0 cn wi elfzo0 nn0re nnre resubcl ancoms nn0ge0
|
|
w3a wb posdif biimp3a addgegt0 syl2anc nn0cn nncn 3ad2ant2 breqtrrd possumd
|
|
subadd23d mpbid readdcl addge02 syl lelttrdi impancom imp ltsubadd2d mpbird
|
|
3jca jca ex syl5bi 3adant2 sylbi ) ADCUAEZFZBVSFZCUBABGEZHIZWBCHIZJZVTAUCFZ
|
|
CUDFZACHIZULWAWEUEZACUFWFWHWIWGWABUCFZWGBCHIZULZWFWHJZWEBCUFWMWLWEWMWLJZWCW
|
|
DWNDWBCKEZHIWCWNDACBGEZKEZWOHWNALFZWPLFZJDAMIZDWPHIZJDWQHIWMWRWLWSWFWRWHAUG
|
|
ZNZWJWGWSWKWGWJWSWGCLFZBLFZWSWJCUHZBUGZCBUIOUJPQWMWTWLXAWFWTWHAUKNWJWGWKXAW
|
|
JXEXDWKXAUMWGXGXFBCUNOUOQAWPUPUQWNABCWMARFZWLWFXHWHAURNNWLBRFZWMWJWGXIWKBUR
|
|
STWLCRFZWMWGWJXJWKCUSUTTVCVAWNWBCWMWRXEWBLFWLXCWJWGXEWKXGSZABUIOWLXDWMWGWJX
|
|
DWKXFUTZTZVBVDWNWDABCKEZHIZWMWLXOWFWLWHXOWFWLJZACXNXPWRXDXNLFZWFWRWLXBNWLXD
|
|
WFXLTWLXQWFWJWGXQWKWJXEXDXQWGXGXFBCVEOPTVMXPDBMIZCXNMIZWLXRWFWJWGXRWKBUKSTX
|
|
PXDXEJZXRXSUMWLXTWFWJWGXTWKWGWJXTWGXDWJXEXFXGQUJPTCBVFVGVDVHVIVJWNABCWMWRWL
|
|
XCNWLXEWMXKTXMVKVLVNVOVPVQVRVJ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Rational numbers (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d M w x y z $. $d N w x y z $.
|
|
$( Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) $)
|
|
qtri3or $p |- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) ) $=
|
|
( vz vw vx vy cq wcel wa cv cdiv co wceq cn wrex cz clt wbr w3o cc biimpi
|
|
elq adantl ad3antrrr cmul simplrl simplrr zmulcld ztri3or syl2anc simpllr
|
|
ad2antrr nnzd breq2d wb breq1 zred nnrpd lt2mul2divd 3bitr4rd eqeq12d cc0
|
|
simpr cap zcnd nncnd nnap0d jca divmuleqap syl22anc bitr2d breq12d bitr4d
|
|
3orbi123d mpbid ex rexlimdvva mpd ) AGHZBGHZIZBCJZDJZKLZMZDNOCPOZABQRZABM
|
|
ZBAQRZSZVTWFVSVTWFCDBUBUAUCWAWEWJCDPNWAWBPHZWCNHZIZIZWEWJWNWEIZAEJZFJZKLZ
|
|
MZFNOEPOZWJVSWTVTWMWEVSWTEFAUBUAUDWOWSWJEFPNWOWPPHZWQNHZIZIZWSWJXDWSIZWPW
|
|
CUELZWBWQUELZQRZXFXGMZXGXFQRZSZWJXEXFPHXGPHXKXEWPWCWOXAXBWSUFZXEWCWOWLXCW
|
|
SWAWKWLWEUGULZUMUHXEWBWQWOWKXCWSWAWKWLWEUFULZXEWQWOXAXBWSUGZUMUHXFXGUIUJX
|
|
EXHWGXIWHXJWIXEWRBQRZWRWDQRWGXHXEBWDWRQWNWEXCWSUKZUNWSWGXPUOXDAWRBQUPUCXE
|
|
WPWCWBWQXEWPXLUQZXEWCXMURZXEWBXNUQZXEWQXOURZUSUTXEWHWRWDMZXIXEAWRBWDXDWSV
|
|
CZXQVAXEWPTHWBTHWQTHZWQVBVDRZIWCTHZWCVBVDRZIYBXIUOXEWPXLVEXEWBXNVEXEYDYEX
|
|
EWQXOVFXEWQXOVGVHXEYFYGXEWCXMVFXEWCXMVGVHWPWBWQWCVIVJVKXEXJWDWRQRWIXEWBWQ
|
|
WPWCXTYAXRXSUSXEBWDAWRQXQYCVLVMVNVOVPVQVRVPVQVR $.
|
|
$}
|
|
|
|
$( Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) $)
|
|
qletric $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) ) $=
|
|
( cq wcel wa cr clt wbr wceq w3o cle qre anim12i qtri3or ltle orc syl6 eqle
|
|
wo wi ex adantr ancoms olc 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJA
|
|
BKHZBAKHZSZUGUIUHUJALBLMABNUKULUQUMUNUKULUOUQABOUOUPPZQUKUMUOUQUIUMUOTUJUIU
|
|
MUOABRUAUBURQUKUNUPUQUJUIUNUPTBAOUCUPUOUDQUEUF $.
|
|
|
|
$( Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) $)
|
|
qlelttric $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) ) $=
|
|
( cq wcel wa cr clt wbr wceq w3o cle qre anim12i qtri3or ltle orc syl6 eqle
|
|
wo wi ex adantr olc a1i 3jaod sylc ) ACDZBCDZEAFDZBFDZEZABGHZABIZBAGHZJABKH
|
|
ZUNSZUGUIUHUJALBLMABNUKULUPUMUNUKULUOUPABOUOUNPZQUKUMUOUPUIUMUOTUJUIUMUOABR
|
|
UAUBUQQUNUPTUKUNUOUCUDUEUF $.
|
|
|
|
$( 'Less than' expressed in terms of 'less than or equal to'. (Contributed
|
|
by Jim Kingdon, 8-Oct-2021.) $)
|
|
qltnle $p |- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> -. B <_ A ) ) $=
|
|
( cq wcel wa clt wbr cle wn cr qre lenlt syl2anr biimpd con2d wceq w3o syl6
|
|
wb wi qtri3or ax-1 a1i eqcom eqle sylan2b ex adantl sylan2 pm2.24 3jaod mpd
|
|
ltle impbid ) ACDZBCDZEZABFGZBAHGZIZUQUSURUQUSURIZUPBJDZAJDZUSVASUOBKZAKZBA
|
|
LMNOUQURABPZBAFGZQUTURTZABUAUQURVHVFVGURVHTUQURUTUBUCUQVFUSVHUPUOVBVFUSTZVD
|
|
VBVIUOVBVFUSVFVBBAPUSABUDBAUEUFUGUHUIUSURUJZRUQVGUSVHUPVBVCVGUSTUOVDVEBAUMM
|
|
VJRUKULUN $.
|
|
|
|
$( Equality of rationals is decidable. (Contributed by Jim Kingdon,
|
|
11-Oct-2021.) $)
|
|
qdceq $p |- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B ) $=
|
|
( cq wcel wa clt wbr wceq w3o wdc qtri3or cr qre wne ltne necomd sylibr syl
|
|
wi ex wo olc dcne adantr sylan orc a1i adantl 3jaod mpd ) ACDZBCDZEZABFGZAB
|
|
HZBAFGZIUOJZABKUMUNUQUOUPUKALDZULUNUQSZAMURUSULURUNUQURUNEZABNZUQUTBAABOPVA
|
|
UOVAUAZUQVAUOUBABUCZQZRTUDUEUOUQSUMUOVBUQUOVAUFVCQUGULUPUQSZUKULBLDZVEBMVFU
|
|
PUQVFUPEVAUQBAOVDRTRUHUIUJ $.
|
|
|
|
${
|
|
$d A j m $. $d K j m $.
|
|
$( Lemma for ~ qbtwnz . Induction step. (Contributed by Jim Kingdon,
|
|
8-Oct-2021.) $)
|
|
qbtwnzlemstep $p |- ( ( K e. NN /\ A e. QQ /\
|
|
E. m e. ZZ ( m <_ A /\ A < ( m + ( K + 1 ) ) ) ) ->
|
|
E. m e. ZZ ( m <_ A /\ A < ( m + K ) ) ) $=
|
|
( vj wcel cq cle wbr c1 caddc co clt wa wrex simpllr breq1 breq2d anbi12d
|
|
cz oveq1 cn w3a simpll ad2antrr nnzd zaddcld simpr qre ad4antlr zred 1red
|
|
cv readdcld nnred simplrr zcnd nncnd 1cnd addassd breqtrrd nnge1d ltletrd
|
|
cr leadd2dd wceq rspcev syl12anc simplrl weq wo zq ad2antlr nnq ad3antrrr
|
|
qaddcl syl2anc qlelttric mpjaodan ex rexlimdva 3impia cbvrexv sylibr ) CU
|
|
AEZAFEZBULZAGHZAWFCIJKJKZLHZMZBSNZUBDULZAGHZAWLCJKZLHZMZDSNZWGAWFCJKZLHZM
|
|
ZBSNWDWEWKWQWDWEMZWJWQBSXAWFSEZMZWJWQXCWJMZWRAGHZWQWSXDXEMZWRSEXEAWRCJKZL
|
|
HZWQXFWFCXAXBWJXEOZXFCXCWDWJXEWDWEXBUCUDZUEUFZXDXEUGXFAWRIJKZXGWEAVCEWDXB
|
|
WJXEAUHUIXFWRIXFWRXKUJZXFUKZUMXFWRCXMXFCXJUNZUMXFAWHXLLXCWGWIXEUOXFWFCIXF
|
|
WFXIUPXFCXJUQXFURUSUTXFICWRXNXOXMXFCXJVAVDVBWPXEXHMDWRSWLWRVEZWMXEWOXHWLW
|
|
RAGPXPWNXGALWLWRCJTQRVFVGXDWSMXBWGWSWQXAXBWJWSOXCWGWIWSVHXDWSUGWPWTDWFSDB
|
|
VIZWMWGWOWSWLWFAGPXQWNWRALWLWFCJTQRVFVGXDWRFEZWEXEWSVJXDWFFEZCFEZXRXBXSXA
|
|
WJWFVKVLWDXTWEXBWJCVMVNWFCVOVPWDWEXBWJOWRAVQVPVRVSVTWAWTWPBDSBDVIZWGWMWSW
|
|
OWFWLAGPYAWRWNALWFWLCJTQRWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d A k m w x $. $d J m w $.
|
|
$( Lemma for ~ qbtwnz . Shrinking the range around the given rational
|
|
number. (Contributed by Jim Kingdon, 8-Oct-2021.) $)
|
|
qbtwnzlemshrink $p |- ( ( A e. QQ /\ J e. NN /\
|
|
E. m e. ZZ ( m <_ A /\ A < ( m + J ) ) ) ->
|
|
E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) $=
|
|
( cv wbr caddc co clt wa cz wrex c1 wi oveq2 breq2d anbi2d rexbidv imbi1d
|
|
wceq vw vk cq wcel cn cle simp2 3simpb breq1 oveq1 anbi12d cbvrexv biimpi
|
|
w3a adantl qbtwnzlemstep 3expia imdistanda imim1d nnind sylc ) BUCUDZDUEU
|
|
DZCEZBUFFZBVDDGHZIFZJZCKLZUNVCVBVIJZAEZBUFFZBVKMGHZIFZJZAKLZVBVCVIUGVBVCV
|
|
IUHVBVEBVDUAEZGHZIFZJZCKLZJZVPNVBVEBVDMGHZIFZJZCKLZJZVPNVBVEBVDUBEZGHZIFZ
|
|
JZCKLZJZVPNVBVEBVDWHMGHZGHZIFZJZCKLZJZVPNVJVPNUAUBDVQMTZWBWGVPWTWAWFVBWTV
|
|
TWECKWTVSWDVEWTVRWCBIVQMVDGOPQRQSVQWHTZWBWMVPXAWAWLVBXAVTWKCKXAVSWJVEXAVR
|
|
WIBIVQWHVDGOPQRQSVQWNTZWBWSVPXBWAWRVBXBVTWQCKXBVSWPVEXBVRWOBIVQWNVDGOPQRQ
|
|
SVQDTZWBVJVPXCWAVIVBXCVTVHCKXCVSVGVEXCVRVFBIVQDVDGOPQRQSWFVPVBWFVPWEVOCAK
|
|
VDVKTZVEVLWDVNVDVKBUFUIXDWCVMBIVDVKMGUJPUKULUMUOWHUEUDZWSWMVPXEVBWRWLXEVB
|
|
WRWLBCWHUPUQURUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x y $.
|
|
$( Lemma for ~ qbtwnz . Existence of the integer.
|
|
|
|
The proof starts by finding two integers which are less than and greater
|
|
than the given rational number. Then this range can be shrunk by
|
|
choosing an integer in between the endpoints of the range and then
|
|
deciding which half of the range to keep based on rational number
|
|
trichotomy, and iterating until the range consists of two consecutive
|
|
integers. (Contributed by Jim Kingdon, 8-Oct-2021.) $)
|
|
qbtwnzlemex $p |- ( A e. QQ -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) $=
|
|
( vm vn vy cq wcel cv clt wbr wa cz wrex cle c1 caddc co cr zred zcnd qre
|
|
btwnz reeanv sylibr cmin cn simpll simplrl ad2antrr simplrr simprl simprr
|
|
syl lttrd wb znnsub ad2antlr mpbid ltled pncan3d breqtrrd weq breq1 oveq1
|
|
breq2d anbi12d rspcev syl12anc qbtwnzlemshrink syl3anc ex rexlimdvva mpd
|
|
) BFGZCHZBIJZBDHZIJZKZDLMCLMZAHZBNJBWAOPQIJKALMZVNVPCLMVRDLMKZVTVNBRGZWCB
|
|
UAZCDBUBUMVPVRCDLLUCUDVNVSWBCDLLVNVOLGZVQLGZKZKZVSWBWIVSKZVNVQVOUEQZUFGZE
|
|
HZBNJZBWMWKPQZIJZKZELMZWBVNWHVSUGWJVOVQIJZWLWJVOBVQWJVOVNWFWGVSUHZSZVNWDW
|
|
HVSWEUIZWJVQVNWFWGVSUJZSWIVPVRUKZWIVPVRULZUNWHWSWLUOVNVSVOVQUPUQURWJWFVOB
|
|
NJZBVOWKPQZIJZWRWTWJVOBXAXBXDUSWJBVQXGIXEWJVOVQWJVOWTTWJVQXCTUTVAWQXFXHKE
|
|
VOLECVBZWNXFWPXHWMVOBNVCXIWOXGBIWMVOWKPVDVEVFVGVHABEWKVIVJVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( There is a unique greatest integer less than or equal to a rational
|
|
number. (Contributed by Jim Kingdon, 8-Oct-2021.) $)
|
|
qbtwnz $p |- ( A e. QQ -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) $=
|
|
( vy wcel cv cle wbr c1 caddc co wa cz wral zred readdcld lelttrd zleltp1
|
|
clt wb syl2anc cq wrex wrmo wreu qbtwnzlemex weq simplrl ad2antrr simplrr
|
|
wi cr simprll simprrr mpbird simprrl simprlr letri3d mpbir2and ralrimivva
|
|
qre 1red ex breq1 oveq1 breq2d anbi12d rmo4 sylibr reu5 sylanbrc ) BUADZA
|
|
EZBFGZBVLHIJZRGZKZALUBVPALUCZVPALUDABUEVKVPCEZBFGZBVRHIJZRGZKZKZACUFZUJZC
|
|
LMALMVQVKWEACLLVKVLLDZVRLDZKZKZWCWDWIWCKZWDVLVRFGZVRVLFGZWJWKVLVTRGZWJVLB
|
|
VTWJVLVKWFWGWCUGZNZVKBUKDWHWCBUTUHZWJVRHWJVRVKWFWGWCUIZNZWJVAZOWIVMVOWBUL
|
|
WIVPVSWAUMPWJWFWGWKWMSWNWQVLVRQTUNWJWLVRVNRGZWJVRBVNWRWPWJVLHWOWSOWIVPVSW
|
|
AUOWIVMVOWBUPPWJWGWFWLWTSWQWNVRVLQTUNWJVLVRWOWRUQURVBUSVPWBACLWDVMVSVOWAV
|
|
LVRBFVCWDVNVTBRVLVRHIVDVEVFVGVHVPALVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d A j m $. $d K j m $.
|
|
$( Lemma for ~ rebtwn2z . Induction step. (Contributed by Jim Kingdon,
|
|
13-Oct-2021.) $)
|
|
rebtwn2zlemstep $p |- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\
|
|
E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) ->
|
|
E. m e. ZZ ( m < A /\ A < ( m + K ) ) ) $=
|
|
( vj c2 cuz wcel cr clt c1 caddc co wa cz wrex simpllr breq1 oveq1 breq2d
|
|
wbr cfv peano2z ad3antlr simpr simplrr zcnd 1cnd eluzelcn ad4antr addassd
|
|
cv w3a addcomd oveq2d eqtrd breqtrrd wceq anbi12d rspcev syl12anc simplrl
|
|
cc weq wo 1red eluzelre ad3antrrr simplr zred 1z eluzp1l mpan df-2 fveq2i
|
|
eleq2s ltadd2dd wi readdcld axltwlin syl3anc mpjaodan ex rexlimdva 3impia
|
|
mpd cbvrexv sylibr ) CEFUAZGZAHGZBUKZAITZAWKCJKLZKLZITZMZBNOZULDUKZAITZAW
|
|
RCKLZITZMZDNOZWLAWKCKLZITZMZBNOWIWJWQXCWIWJMZWPXCBNXGWKNGZMZWPXCXIWPMZWKJ
|
|
KLZAITZXCXEXJXLMZXKNGZXLAXKCKLZITZXCXHXNXGWPXLWKUBUCXJXLUDXMAWNXOIXIWLWOX
|
|
LUEXMXOWKJCKLZKLWNXMWKJCXMWKXGXHWPXLPUFXMUGZWICVBGWJXHWPXLECUHUIZUJXMXQWM
|
|
WKKXMJCXRXSUMUNUOUPXBXLXPMDXKNWRXKUQZWSXLXAXPWRXKAIQXTWTXOAIWRXKCKRSURUSU
|
|
TXJXEMXHWLXEXCXGXHWPXEPXIWLWOXEVAXJXEUDXBXFDWKNDBVCZWSWLXAXEWRWKAIQYAWTXD
|
|
AIWRWKCKRSURUSUTXJXKXDITZXLXEVDZXJJCWKXJVEZWICHGWJXHWPECVFVGZXJWKXGXHWPVH
|
|
VIZWIJCITZWJXHWPYGCJJKLZFUAZWHJNGCYIGYGVJJCVKVLEYHFVMVNVOVGVPXJXKHGXDHGWJ
|
|
YBYCVQXJWKJYFYDVRXJWKCYFYEVRWIWJXHWPPXKXDAVSVTWEWAWBWCWDXFXBBDNBDVCZWLWSX
|
|
EXAWKWRAIQYJXDWTAIWKWRCKRSURWFWG $.
|
|
$}
|
|
|
|
${
|
|
$d A k m w x $. $d J m w $.
|
|
$( Lemma for ~ rebtwn2z . Shrinking the range around the given real
|
|
number. (Contributed by Jim Kingdon, 13-Oct-2021.) $)
|
|
rebtwn2zlemshrink $p |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\
|
|
E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) ->
|
|
E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) ) $=
|
|
( c2 cv clt wbr caddc co wa cz wrex wi oveq2 breq2d anbi2d rexbidv imbi1d
|
|
wceq vw vk cr wcel cuz cfv simp2 3simpb c1 2z breq1 oveq1 anbi12d cbvrexv
|
|
w3a biimpi adantl rebtwn2zlemstep 3expia imdistanda imim1d uzind4i sylc )
|
|
BUCUDZDEUEUFZUDZCFZBGHZBVGDIJZGHZKZCLMZUOVFVDVLKZAFZBGHZBVNEIJZGHZKZALMZV
|
|
DVFVLUGVDVFVLUHVDVHBVGUAFZIJZGHZKZCLMZKZVSNVDVHBVGEIJZGHZKZCLMZKZVSNVDVHB
|
|
VGUBFZIJZGHZKZCLMZKZVSNVDVHBVGWKUIIJZIJZGHZKZCLMZKZVSNVMVSNUAUBEDUJVTETZW
|
|
EWJVSXCWDWIVDXCWCWHCLXCWBWGVHXCWAWFBGVTEVGIOPQRQSVTWKTZWEWPVSXDWDWOVDXDWC
|
|
WNCLXDWBWMVHXDWAWLBGVTWKVGIOPQRQSVTWQTZWEXBVSXEWDXAVDXEWCWTCLXEWBWSVHXEWA
|
|
WRBGVTWQVGIOPQRQSVTDTZWEVMVSXFWDVLVDXFWCVKCLXFWBVJVHXFWAVIBGVTDVGIOPQRQSW
|
|
IVSVDWIVSWHVRCALVGVNTZVHVOWGVQVGVNBGUKXGWFVPBGVGVNEIULPUMUNUPUQWKVEUDZXBW
|
|
PVSXHVDXAWOXHVDXAWOBCWKURUSUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x y $.
|
|
$( A real number can be bounded by integers above and below which are two
|
|
apart.
|
|
|
|
The proof starts by finding two integers which are less than and greater
|
|
than the given real number. Then this range can be shrunk by choosing
|
|
an integer in between the endpoints of the range and then deciding which
|
|
half of the range to keep based on weak linearity, and iterating until
|
|
the range consists of integers which are two apart. (Contributed by Jim
|
|
Kingdon, 13-Oct-2021.) $)
|
|
rebtwn2z $p |- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) ) $=
|
|
( vm vn vy cr wcel cv clt wbr wa cz wrex c2 caddc co c1 cuz cfv zred cmin
|
|
btwnz reeanv sylibr simpll cn simplrl simprl simprr lttrd znnsub ad2antlr
|
|
simplrr wb mpbid elnnuz eluzp1p1 sylbi df-2 fveq2i syl6eleqr zcnd pncan3d
|
|
eqeltrd resubcld 1red readdcld breqtrrd ltp1d ltadd2dd breq1 oveq1 breq2d
|
|
syl weq anbi12d rspcev syl12anc rebtwn2zlemshrink syl3anc rexlimdvva mpd
|
|
ex ) BFGZCHZBIJZBDHZIJZKZDLMCLMZAHZBIJBWKNOPIJKALMZWDWFCLMWHDLMKWJCDBUBWF
|
|
WHCDLLUCUDWDWIWLCDLLWDWELGZWGLGZKZKZWIWLWPWIKZWDWGWEUAPZQOPZNRSZGZEHZBIJZ
|
|
BXBWSOPZIJZKZELMZWLWDWOWIUEZWQWRUFGZXAWQWEWGIJZXIWQWEBWGWQWEWDWMWNWIUGZTZ
|
|
XHWQWGWDWMWNWIUMZTZWPWFWHUHZWPWFWHUIZUJWOXJXIUNWDWIWEWGUKULUOXIWSQQOPZRSZ
|
|
WTXIWRQRSGWSXRGWRUPQWRUQURNXQRUSUTVAVNWQWMWFBWEWSOPZIJZXGXKXOWQBWEWROPZXS
|
|
XHWQYAWGFWQWEWGWQWEXKVBWQWGXMVBVCZXNVDWQWEWSXLWQWRQWQWGWEXNXLVEZWQVFVGZVG
|
|
WQBWGYAIXPYBVHWQWRWSWEYCYDXLWQWRYCVIVJUJXFWFXTKEWELECVOZXCWFXEXTXBWEBIVKY
|
|
EXDXSBIXBWEWSOVLVMVPVQVRABEWSVSVTWCWAWB $.
|
|
$}
|
|
|
|
${
|
|
qbtwnrelemcalc.m $e |- ( ph -> M e. ZZ ) $.
|
|
qbtwnrelemcalc.n $e |- ( ph -> N e. NN ) $.
|
|
qbtwnrelemcalc.a $e |- ( ph -> A e. RR ) $.
|
|
qbtwnrelemcalc.b $e |- ( ph -> B e. RR ) $.
|
|
qbtwnrelemcalc.lt $e |- ( ph -> M < ( A x. ( 2 x. N ) ) ) $.
|
|
qbtwnrelemcalc.1n $e |- ( ph -> ( 1 / N ) < ( B - A ) ) $.
|
|
$( Lemma for ~ qbtwnre . Calculations involved in showing the constructed
|
|
rational number is less than ` B ` . (Contributed by Jim Kingdon,
|
|
14-Oct-2021.) $)
|
|
qbtwnrelemcalc $p |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B ) $=
|
|
( c2 co cmul cdiv clt wbr cmin a1i recnd caddc cr wcel 2re nnred remulcld
|
|
resubcld zred c1 2t1e2 oveq1i 1cnd nnap0d cc0 cap 2ap0 divcanap5d syl5eqr
|
|
eqbrtrd crp 2rp nnrpd rpmulcld ltdivmul2d mulcld subdird breqtrd ltsub2dd
|
|
mpbid lttrd ltaddsub2d mpbird readdcld ) ADLUAMZLENMZOMCPQVNCVONMZPQZAVQL
|
|
VPDRMZPQALVPBVONMZRMZVRLUBUCAUDSZAVPVSACVOIALEWAAEGUEZUFZUFZABVOHWCUFZUGA
|
|
VPDWDADFUHZUGALCBRMZVONMZVTPALVOOMZWGPQLWHPQAWIUIEOMZWGPAWILUINMZVOOMWJWK
|
|
LVOOUJUKAUIELAULAEWBTZALWATZAEGUMLUNUOQAUPSUQURKUSALWGVOWAACBIHUGALELUTUC
|
|
AVASAEGVBVCZVDVIACBVOACITABHTALEWMWLVEVFVGADVSVPWFWEWDJVHVJADLVPWFWAWDVKV
|
|
LAVNCVOADLWFWAVMIWNVDVL $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x $. $d B m n x $.
|
|
$( The rational numbers are dense in ` RR ` : any two real numbers have a
|
|
rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by
|
|
NM, 18-Nov-2004.) $)
|
|
qbtwnre $p |- ( ( A e. RR /\ B e. RR /\ A < B ) ->
|
|
E. x e. QQ ( A < x /\ x < B ) ) $=
|
|
( vn vm cr wcel clt wbr cv cdiv co wa cq wrex cn c2 cz adantr a1i c1 cmin
|
|
w3a cc0 simp2 simp1 resubcld simp3 posdifd mpbid nnrecl syl2anc caddc 2re
|
|
cmul simprl nnred remulcld rebtwn2z syl zaddcld 2nn nnmulcld simprrr zred
|
|
2z znq nnrpd ltmuldivd simpll2 simprrl simplrr qbtwnrelemcalc breq2 breq1
|
|
wceq anbi12d rspcev syl12anc rexlimddv ) BFGZCFGZBCHIZUCZUADJZKLCBUBLZHIZ
|
|
BAJZHIZWHCHIZMZANOZDPWDWFFGUDWFHIZWGDPOWDCBWAWBWCUEZWAWBWCUFZUGWDWCWMWAWB
|
|
WCUHWDBCWOWNUIUJWFDUKULWDWEPGZWGMZMZEJZBQWEUOLZUOLZHIZXAWSQUMLZHIZMZWLERW
|
|
RXAFGXEEROWRBWTWDWAWQWOSZWRQWEQFGWRUNTWRWEWDWPWGUPZUQURUREXAUSUTWRWSRGZXE
|
|
MZMZXCWTKLZNGZBXKHIZXKCHIZWLXJXCRGWTPGXLXJWSQWRXHXEUPZQRGXJVFTVAZXJQWEQPG
|
|
XJVBTWRWPXIXGSZVCZXCWTVGULXJXDXMWRXHXBXDVDXJBXCWTWRWAXIXFSZXJXCXPVEXJWTXR
|
|
VHVIUJXJBCWSWEXOXQXSWAWBWCWQXIVJWRXHXBXDVKWDWPWGXIVLVMWKXMXNMAXKNWHXKVPWI
|
|
XMWJXNWHXKBHVNWHXKCHVOVQVRVSVTVT $.
|
|
|
|
$( The rational numbers are dense in ` RR* ` : any two extended real
|
|
numbers have a rational between them. (Contributed by NM, 6-Feb-2007.)
|
|
(Proof shortened by Mario Carneiro, 23-Aug-2015.) $)
|
|
qbtwnxr $p |- ( ( A e. RR* /\ B e. RR* /\ A < B )
|
|
-> E. x e. QQ ( A < x /\ x < B ) ) $=
|
|
( wcel clt wbr wa cq wrex cr cpnf wceq qbtwnre c1 adantr adantl a1d breq2
|
|
cmnf breq1 cxr cv w3o wi elxr 3expia caddc co simpl peano2re ltp1 syl3anc
|
|
qre syl simplr breqtrrd anim2d reximdva mpd rexr wb nltmnf pm2.21d sylbid
|
|
ltpnf sylan 3jaodan sylan2b pnfnlt cmin peano2rem simpr ltm1 simpll mnflt
|
|
wn eqbrtrd anim1d 1re ax-mp mpbiri cz 1z anbi12d rspcev mpan syl2an 3mix3
|
|
zq sylibr 3jaoian sylanb 3impia ) BUADZCUADZBCEFZBAUBZEFZWQCEFZGZAHIZWNBJ
|
|
DZBKLZBSLZUCZWOWPXAUDZBUEZXBWOXFXCXDWOXBCJDZCKLZCSLZUCZXFCUEZXBXHXFXIXJXB
|
|
XHWPXAABCMUFXBXIGZXAWPXMWRWQBNUGUHZEFZGZAHIZXAXMXBXNJDZBXNEFZXQXBXIUIXBXR
|
|
XIBUJOXBXSXIBUKOABXNMULXMXPWTAHXMWQHDZGZXOWSWRYAWSXOYAWQKCEXTWQKEFZXMXTWQ
|
|
JDZYBWQUMZWQVEUNPXBXIXTUOUPQUQURUSQXBWNXJXFBUTWNXJGZWPBSEFZXAXJWPYFVAWNCS
|
|
BERPYEYFXAWNYFVPXJBVBOVCVDZVFVGVHXCWOGZWPKCEFZXAXCWPYIVAWOBKCETOYHYIXAWOY
|
|
IVPXCCVIPVCVDWOXDXKXFXLXDXHXFXIXJXDXHGZXAWPYJCNVJUHZWQEFZWSGZAHIZXAYJYKJD
|
|
ZXHYKCEFZYNXHYOXDCVKPXDXHVLXHYPXDCVMPAYKCMULYJYMWTAHYJXTGZYLWRWSYQWRYLYQB
|
|
SWQEXDXHXTVNYQYCSWQEFXTYCYJYDPWQVOUNVQQVRURUSQXDXIGXAWPXDBNEFZNCEFZXAXIXD
|
|
YRSNEFZNJDZYTVSNVOVTBSNETWAXIYSNKEFZUUAUUBVSNVEVTCKNERWANHDZYRYSGZXANWBDU
|
|
UCWCNWIVTWTUUDANHWQNLWRYRWSYSWQNBERWQNCETWDWEWFWGQXDWNXJXFXDXEWNXDXBXCWHX
|
|
GWJYGVFVGVHWKWLWM $.
|
|
$}
|
|
|
|
$( The average of two rational numbers is less than or equal to at least one
|
|
of them. (Contributed by Jim Kingdon, 3-Nov-2021.) $)
|
|
qavgle $p |- ( ( A e. QQ /\ B e. QQ ) ->
|
|
( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) ) $=
|
|
( cq wcel cle wbr wo caddc co c2 cdiv cr wb adantl adantr avgle2 syl2anc cc
|
|
qre qcn wa qletric orcomd addcomd oveq1d breq1d bitr4d orbi12d mpbid ) ACDZ
|
|
BCDZUAZBAEFZABEFZGABHIZJKIZAEFZUPBEFZGULUNUMABUBUCULUMUQUNURULUMBAHIZJKIZAE
|
|
FZUQULBLDZALDZUMVAMUKVBUJBSNZUJVCUKASOZBAPQULUPUTAEULUOUSJKULABUJARDUKATOUK
|
|
BRDUJBTNUDUEUFUGULVCVBUNURMVEVDABPQUHUI $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( An empty open interval of extended reals. (Contributed by NM,
|
|
6-Feb-2007.) $)
|
|
ioo0 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <->
|
|
B <_ A ) ) $=
|
|
( vx cxr wcel wa cioo co c0 wceq cv clt wbr crab cle iooval wrex wn cq wb
|
|
eqeq1d xrlttr 3com23 3expa rexlimdva w3a qbtwnxr qre rexrd anim1i reximi2
|
|
wi syl 3expia impbid notbid wral rabeq0 ralnex a1i xrlenlt ancoms 3bitr4d
|
|
bitri bitrd ) ADEZBDEZFZABGHZIJACKZLMVJBLMFZCDNZIJZBAOMZVHVIVLICABPUAVHVK
|
|
CDQZRZABLMZRZVMVNVHVOVQVHVOVQVHVKVQCDVFVGVJDEZVKVQULZVFVSVGVTAVJBUBUCUDUE
|
|
VFVGVQVOVFVGVQUFVKCSQVOCABUGVKVKCSDVJSEZVSVKWAVJVJUHUIUJUKUMUNUOUPVMVPTVH
|
|
VMVKRCDUQVPVKCDURVKCDUSVDUTVGVFVNVRTBAVAVBVCVE $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( An open interval of extended reals is inhabited iff the lower argument
|
|
is less than the upper argument. (Contributed by Jim Kingdon,
|
|
27-Nov-2021.) $)
|
|
ioom $p |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <->
|
|
A < B ) ) $=
|
|
( cxr wcel wa cv cioo wex clt wbr w3a elioo3g biimpi simpld simp1d simprd
|
|
co wi cq simp3d simp2d xrlttrd exlimdv qbtwnxr df-rex sylib simpl1 simpl2
|
|
a1i wrex cr qre ad2antrl rexrd simprrl simprrr biimpri syl32anc ex eximdv
|
|
mpd 3expia impbid ) BDEZCDEZFZAGZBCHREZAIZBCJKZVGVIVKAVIVKSVGVIBVHCVIVEVF
|
|
VHDEZVIVEVFVLLZBVHJKZVHCJKZFZVIVMVPFZBCVHMZNZOZPVIVEVFVLVTUAVIVEVFVLVTUBV
|
|
IVNVOVIVMVPVSQZOVIVNVOWAQUCUJUDVEVFVKVJVEVFVKLZVHTEZVPFZAIZVJWBVPATUKWEAB
|
|
CUEVPATUFUGWBWDVIAWBWDVIWBWDFZVEVFVLVNVOVIVEVFVKWDUHVEVFVKWDUIWFVHWCVHULE
|
|
WBVPVHUMUNUOWBWCVNVOUPWBWCVNVOUQVIVQVRURUSUTVAVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( An empty open interval of extended reals. (Contributed by FL,
|
|
30-May-2014.) $)
|
|
ico0 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <->
|
|
B <_ A ) ) $=
|
|
( vx cxr wcel wa cico co c0 wceq cv cle wbr clt wrex wn wi cq a1i wb crab
|
|
icoval eqeq1d xrlelttr 3com23 3expa rexlimdva w3a qbtwnxr qre rexrd simpl
|
|
simpr1 xrltle syl2anc anim1d anim12d ex syl adantr reximdv2 3expia impbid
|
|
pm2.43b mpd notbid wral rabeq0 ralnex bitri xrlenlt ancoms 3bitr4d bitrd
|
|
) ADEZBDEZFZABGHZIJACKZLMZVSBNMZFZCDUAZIJZBALMZVQVRWCICABUBUCVQWBCDOZPZAB
|
|
NMZPZWDWEVQWFWHVQWFWHVQWBWHCDVOVPVSDEZWBWHQZVOWJVPWKAVSBUDUEUFUGVOVPWHWFV
|
|
OVPWHUHZAVSNMZWAFZCROWFCABUIWLWNWBCRDWLVSREZWNFZWJWBFZWOWLWPWQQZQZWNWOWJW
|
|
SWOVSVSUJUKZWJWLWRWJWLFZWOWJWNWBWOWJQXAWTSXAWMVTWAXAVOWJWMVTQWJVOVPWHUMWJ
|
|
WLULAVSUNUOUPUQURUSUTVDVAVEVBVCVFWDWGTVQWDWBPCDVGWGWBCDVHWBCDVIVJSVPVOWEW
|
|
ITBAVKVLVMVN $.
|
|
|
|
$( An empty open interval of extended reals. (Contributed by FL,
|
|
30-May-2014.) $)
|
|
ioc0 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <->
|
|
B <_ A ) ) $=
|
|
( vx cxr wcel wa cioc co c0 wceq cv clt wbr cle wrex wn wi cq a1i wb crab
|
|
iocval eqeq1d xrltletr 3com23 3expa rexlimdva w3a qbtwnxr rexrd 3ad2antr2
|
|
qre xrltle anim2d anim12d ex adantr pm2.43b reximdv2 3expia impbid notbid
|
|
syl mpd wral rabeq0 ralnex bitri xrlenlt ancoms 3bitr4d bitrd ) ADEZBDEZF
|
|
ZABGHZIJACKZLMZVQBNMZFZCDUAZIJZBANMZVOVPWAICABUBUCVOVTCDOZPZABLMZPZWBWCVO
|
|
WDWFVOWDWFVOVTWFCDVMVNVQDEZVTWFQZVMWHVNWIAVQBUDUEUFUGVMVNWFWDVMVNWFUHZVRV
|
|
QBLMZFZCROWDCABUIWJWLVTCRDWJVQREZWLFZWHVTFZWMWJWNWOQZQZWLWMWHWQWMVQVQULUJ
|
|
ZWHWJWPWHWJFZWMWHWLVTWMWHQWSWRSWSWKVSVRWHVMVNWKVSQWFVQBUMUKUNUOUPVCUQURUS
|
|
VDUTVAVBWBWETVOWBVTPCDVEWEVTCDVFVTCDVGVHSVNVMWCWGTBAVIVJVKVL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Elementary integer functions
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The floor and ceiling functions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c |_ $. $( Floor function. $)
|
|
$c |^ $. $( Ceiling function. $)
|
|
|
|
$( Extend class notation with floor (greatest integer) function. $)
|
|
cfl $a class |_ $.
|
|
$( Extend class notation to include the ceiling function. $)
|
|
cceil $a class |^ $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the floor (greatest integer less than or equal to) function. See
|
|
~ flval for its value, ~ flqlelt for its basic property, and ~ flqcl for
|
|
its closure. For example, ` ( |_ `` ( 3 / 2 ) ) = 1 ` while
|
|
` ( |_ `` -u ( 3 / 2 ) ) = -u 2 ` ( ~ ex-fl ).
|
|
|
|
Although we define this on real numbers so that notations are similar to
|
|
the Metamath Proof Explorer, in the absence of excluded middle few
|
|
theorems will be possible beyond the rationals. Imagine a real number
|
|
which is around 2.99995 or 3.00001 . In order to determine whether its
|
|
floor is 2 or 3, it would be necessary to compute the number to
|
|
arbitrary precision.
|
|
|
|
The term "floor" was coined by Ken Iverson. He also invented a
|
|
mathematical notation for floor, consisting of an L-shaped left bracket
|
|
and its reflection as a right bracket. In APL, the left-bracket alone
|
|
is used, and we borrow this idea. (Thanks to Paul Chapman for this
|
|
information.) (Contributed by NM, 14-Nov-2004.) $)
|
|
df-fl $a |- |_ = ( x e. RR |->
|
|
( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) ) $.
|
|
$}
|
|
|
|
$( The ceiling (least integer greater than or equal to) function. Defined in
|
|
ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of
|
|
Mathematical Functions" , front introduction, "Common Notations and
|
|
Definitions" section at ~ http://dlmf.nist.gov/front/introduction#Sx4 .
|
|
See ~ ceilqval for its value, ~ ceilqge and ~ ceilqm1lt for its basic
|
|
properties, and ~ ceilqcl for its closure. For example,
|
|
` ( |^ `` ( 3 / 2 ) ) = 2 ` while ` ( |^ `` -u ( 3 / 2 ) ) = -u 1 `
|
|
( ~ ex-ceil ).
|
|
|
|
As described in ~ df-fl most theorems are only for rationals, not reals.
|
|
|
|
The symbol ` |^ ` is inspired by the gamma shaped left bracket of the
|
|
usual notation. (Contributed by David A. Wheeler, 19-May-2015.) $)
|
|
df-ceil $a |- |^ = ( x e. RR |-> -u ( |_ ` -u x ) ) $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Value of the floor (greatest integer) function. The floor of ` A ` is
|
|
the (unique) integer less than or equal to ` A ` whose successor is
|
|
strictly greater than ` A ` . (Contributed by NM, 14-Nov-2004.)
|
|
(Revised by Mario Carneiro, 2-Nov-2013.) $)
|
|
flval $p |- ( A e. RR ->
|
|
( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) ) $=
|
|
( vy cv cle wbr c1 caddc co clt wa cz crio cfl wceq breq2 breq1 cvv wcel
|
|
cr anbi12d riotabidv df-fl zex riotaexg ax-mp fvmpt3i ) CBADZCDZEFZUIUHGH
|
|
IZJFZKZALMZUHBEFZBUKJFZKZALMTNUIBOZUMUQALURUJUOULUPUIBUHEPUIBUKJQUAUBCAUC
|
|
LRSUNRSUDUMALRUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The floor (greatest integer) function yields an integer when applied to
|
|
a rational (closure law). It would presumably be possible to prove a
|
|
similar result for some real numbers (for example, those apart from any
|
|
integer), but not real numbers in general. (Contributed by Jim Kingdon,
|
|
8-Oct-2021.) $)
|
|
flqcl $p |- ( A e. QQ -> ( |_ ` A ) e. ZZ ) $=
|
|
( vx cq wcel cfl cfv cv cle wbr c1 caddc co clt wa cz crio wceq qre syl
|
|
cr flval wreu qbtwnz riotacl eqeltrd ) ACDZAEFZBGZAHIAUHJKLMINZBOPZOUFATD
|
|
UGUJQARBAUASUFUIBOUBUJODBAUCUIBOUDSUE $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A basic property of the floor (greatest integer) function. (Contributed
|
|
by Jim Kingdon, 8-Oct-2021.) $)
|
|
flqlelt $p |- ( A e. QQ ->
|
|
( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) ) $=
|
|
( vx cq wcel cfl cfv cle wbr c1 caddc co clt wa cv cz crio wceq qre flval
|
|
cr eqcomd syl wreu flqcl qbtwnz breq1 oveq1 breq2d anbi12d riota2 syl2anc
|
|
wb mpbird ) ACDZAEFZAGHZAUOIJKZLHZMZBNZAGHZAUTIJKZLHZMZBOPZUOQZUNATDZVFAR
|
|
VGUOVEBASUAUBUNUOODVDBOUCUSVFULAUDBAUEVDUSBOUOUTUOQZVAUPVCURUTUOAGUFVHVBU
|
|
QALUTUOIJUGUHUIUJUKUM $.
|
|
$}
|
|
|
|
${
|
|
flqcld.1 $e |- ( ph -> A e. QQ ) $.
|
|
$( The floor (greatest integer) function is an integer (closure law).
|
|
(Contributed by Jim Kingdon, 8-Oct-2021.) $)
|
|
flqcld $p |- ( ph -> ( |_ ` A ) e. ZZ ) $=
|
|
( cq wcel cfl cfv cz flqcl syl ) ABDEBFGHECBIJ $.
|
|
$}
|
|
|
|
$( A basic property of the floor (greatest integer) function. (Contributed
|
|
by Jim Kingdon, 8-Oct-2021.) $)
|
|
flqle $p |- ( A e. QQ -> ( |_ ` A ) <_ A ) $=
|
|
( cq wcel cfl cfv cle wbr c1 caddc co clt flqlelt simpld ) ABCADEZAFGANHIJK
|
|
GALM $.
|
|
|
|
$( A basic property of the floor (greatest integer) function. (Contributed
|
|
by Jim Kingdon, 8-Oct-2021.) $)
|
|
flqltp1 $p |- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) ) $=
|
|
( cq wcel cfl cfv cle wbr c1 caddc co clt flqlelt simprd ) ABCADEZAFGANHIJK
|
|
GALM $.
|
|
|
|
$( The fractional part of a rational number is less than one. (Contributed
|
|
by Jim Kingdon, 8-Oct-2021.) $)
|
|
qfraclt1 $p |- ( A e. QQ -> ( A - ( |_ ` A ) ) < 1 ) $=
|
|
( cq wcel cfl cfv cmin co c1 clt wbr caddc flqltp1 qre zred 1red ltsubadd2d
|
|
flqcl mpbird ) ABCZAADEZFGHIJATHKGIJALSATHAMSTAQNSOPR $.
|
|
|
|
$( The fractional part of a rational number is nonnegative. (Contributed by
|
|
Jim Kingdon, 8-Oct-2021.) $)
|
|
qfracge0 $p |- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) ) $=
|
|
( cq wcel cc0 cfl cfv cmin co cle wbr flqle qre flqcl zred subge0d mpbird )
|
|
ABCZDAAEFZGHIJRAIJAKQARALQRAMNOP $.
|
|
|
|
$( The floor function value is the greatest integer less than or equal to its
|
|
argument. (Contributed by Jim Kingdon, 8-Oct-2021.) $)
|
|
flqge $p |- ( ( A e. QQ /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) ) $=
|
|
( cq wcel cz wa cle wbr cfl cfv c1 caddc co clt adantr cr wi syl3anc mpan2d
|
|
zred flqltp1 simpr qre simpl flqcld peano2zd lelttr zleltp1 syl2anc sylibrd
|
|
wb flqle letr impbid ) ACDZBEDZFZBAGHZBAIJZGHZUQURBUSKLMZNHZUTUQURAVANHZVBU
|
|
OVCUPAUAOUQBPDZAPDZVAPDURVCFVBQUQBUOUPUBZTZUOVEUPAUCOZUQVAUQUSUQAUOUPUDUEZU
|
|
FTBAVAUGRSUQUPUSEDUTVBUKVFVIBUSUHUIUJUQUTUSAGHZURUOVJUPAULOUQVDUSPDVEUTVJFU
|
|
RQVGUQUSVITVHBUSAUMRSUN $.
|
|
|
|
$( The floor function value is less than the next integer. (Contributed by
|
|
Jim Kingdon, 8-Oct-2021.) $)
|
|
flqlt $p |- ( ( A e. QQ /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) ) $=
|
|
( cq wcel cz wa cle wbr wn cfl cfv clt flqge notbid wb qltnle sylan2 flqcld
|
|
zq simpl syl adantl syl2anc 3bitr4d ) ACDZBEDZFZBAGHZIZBAJKZGHZIZABLHZUJBLH
|
|
ZUGUHUKABMNUFUEBCDZUMUIOBSZABPQUGUJCDZUOUNULOUGUJEDUQUGAUEUFTRUJSUAUFUOUEUP
|
|
UBUJBPUCUD $.
|
|
|
|
$( An integer is its own floor. (Contributed by NM, 15-Nov-2004.) $)
|
|
flid $p |- ( A e. ZZ -> ( |_ ` A ) = A ) $=
|
|
( cz wcel cfl cfv wceq cle wbr cq zq flqle syl leidd wb flqge mpancom mpbid
|
|
zre flqcld zred letri3d mpbir2and ) ABCZADEZAFUDAGHZAUDGHZUCAICZUEAJZAKLUCA
|
|
AGHZUFUCAARZMUGUCUIUFNUHAAOPQUCUDAUCUDUCAUHSTUJUAUB $.
|
|
|
|
$( The floor function is idempotent. (Contributed by Jim Kingdon,
|
|
8-Oct-2021.) $)
|
|
flqidm $p |- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) ) $=
|
|
( cq wcel cfl cfv cz wceq flqcl flid syl ) ABCADEZFCKDEKGAHKIJ $.
|
|
|
|
$( A rational number equals its floor iff it is an integer. (Contributed by
|
|
Jim Kingdon, 9-Oct-2021.) $)
|
|
flqidz $p |- ( A e. QQ -> ( ( |_ ` A ) = A <-> A e. ZZ ) ) $=
|
|
( cq wcel cfl cfv wceq cz flqcl eleq1 syl5ibcom flid impbid1 ) ABCZADEZAFZA
|
|
GCZMNGCOPAHNAGIJAKL $.
|
|
|
|
$( If A is not an integer, then the floor of A is less than A. (Contributed
|
|
by Jim Kingdon, 9-Oct-2021.) $)
|
|
flqltnz $p |- ( ( A e. QQ /\ -. A e. ZZ ) -> ( |_ ` A ) < A ) $=
|
|
( cq wcel cz wn wa cfl cfv clt wbr cap wne wceq flqidz adantr mtbird neqned
|
|
simpr wb mpbird necomd simpl flqcld zq syl qapne syldan cr qre cle c1 caddc
|
|
zred co flqlelt simpld leltapd ) ABCZADCZEZFZAGHZAIJAVBKJZVAVCAVBLZVAVBAVAV
|
|
BAVAVBAMZUSURUTRURVEUSSUTANOPQUAURUTVBBCZVCVDSVAVBDCVFVAAURUTUBUCZVBUDUEAVB
|
|
UFUGTVAVBAVAVBVGUMURAUHCUTAUIOVAVBAUJJZAVBUKULUNIJZURVHVIFUTAUOOUPUQT $.
|
|
|
|
$( Ordering relationship for the greatest integer function. (Contributed by
|
|
Jim Kingdon, 9-Oct-2021.) $)
|
|
flqwordi $p |- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) ->
|
|
( |_ ` A ) <_ ( |_ ` B ) ) $=
|
|
( cq wcel cle wbr w3a cfl cfv simp1 flqcld zred qre simp2 flqle simp3 letrd
|
|
cr syl cz wb flqge syl2anc mpbid ) ACDZBCDZABEFZGZAHIZBEFZUIBHIEFZUHUIABUHU
|
|
IUHAUEUFUGJZKZLUHUEARDULAMSUHUFBRDUEUFUGNZBMSUHUEUIAEFULAOSUEUFUGPQUHUFUITD
|
|
UJUKUAUNUMBUIUBUCUD $.
|
|
|
|
$( Ordering relationship for the greatest integer function. (Contributed by
|
|
Jim Kingdon, 9-Oct-2021.) $)
|
|
flqword2 $p |- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) ->
|
|
( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) ) $=
|
|
( cq cle wbr w3a cfl cfv cz cuz simp1 flqcld simp2 flqwordi eluz2 syl3anbrc
|
|
wcel ) ACQZBCQZABDEZFZAGHZIQBGHZIQUBUCDEUCUBJHQUAARSTKLUABRSTMLABNUBUCOP $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( A condition equivalent to floor. (Contributed by Jim Kingdon,
|
|
9-Oct-2021.) $)
|
|
flqbi $p |- ( ( A e. QQ /\ B e. ZZ ) ->
|
|
( ( |_ ` A ) = B <-> ( B <_ A /\ A < ( B + 1 ) ) ) ) $=
|
|
( vx cq wcel cz wa cfl cfv wceq cv cle wbr c1 caddc co clt crio wb cr qre
|
|
flval eqeq1d adantr wreu qbtwnz breq1 breq2d anbi12d riota2 sylan2 ancoms
|
|
syl oveq1 bitr4d ) ADEZBFEZGAHIZBJZCKZALMZAUTNOPZQMZGZCFRZBJZBALMZABNOPZQ
|
|
MZGZUPUSVFSZUQUPATEZVKAUAVLURVEBCAUBUCUMUDUQUPVJVFSZUPUQVDCFUEVMCAUFVDVJC
|
|
FBUTBJZVAVGVCVIUTBALUGVNVBVHAQUTBNOUNUHUIUJUKULUO $.
|
|
$}
|
|
|
|
$( A condition equivalent to floor. (Contributed by Jim Kingdon,
|
|
9-Oct-2021.) $)
|
|
flqbi2 $p |- ( ( N e. ZZ /\ F e. QQ ) ->
|
|
( ( |_ ` ( N + F ) ) = N <-> ( 0 <_ F /\ F < 1 ) ) ) $=
|
|
( cz wcel cq wa caddc co cfl cfv wceq cle wbr c1 clt cc0 wb zq qaddcl cr
|
|
sylan simpl syl2anc zre qre addge01 1re ltadd2 mp3an2 ancoms anbi12d syl2an
|
|
flqbi bitr4d ) BCDZAEDZFZBAGHZIJBKZBURLMZURBNGHOMZFZPALMZANOMZFZUQUREDZUOUS
|
|
VBQUOBEDUPVFBRBASUAUOUPUBURBUMUCUOBTDZATDZVEVBQUPBUDAUEVGVHFVCUTVDVABAUFVHV
|
|
GVDVAQZVHNTDVGVIUGANBUHUIUJUKULUN $.
|
|
|
|
$( The floor of a sum of an integer and a fraction is equal to the integer
|
|
iff the denominator of the fraction is less than the numerator.
|
|
(Contributed by AV, 14-Jul-2021.) $)
|
|
adddivflid $p |- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN )
|
|
-> ( B < C <-> ( |_ ` ( A + ( B / C ) ) ) = A ) ) $=
|
|
( cz wcel cn0 cn w3a co cc0 cle wbr clt wa wb 3adant1 jca syl cr anim12i c1
|
|
cdiv caddc cfl cfv wceq simp1 nn0z znq sylan flqbi2 nn0re nn0ge0 nnre nngt0
|
|
cq divge0 biantrurd crp nnrp divlt1lt 3bitr2rd ) ADEZBFEZCGEZHZABCUBIZUCIUD
|
|
UEAUFZJVGKLZVGUAMLZNZVJBCMLZVFVCVGUPEZNVHVKOVFVCVMVCVDVEUGVDVEVMVCVDBDEVEVM
|
|
BUHBCUIUJPQVGAUKRVFVIVJVFBSEZJBKLZNZCSEZJCMLZNZNZVIVDVEVTVCVDVPVEVSVDVNVOBU
|
|
LZBUMQVEVQVRCUNCUOQTPBCUQRURVFVNCUSEZNZVJVLOVDVEWCVCVDVNVEWBWACUTTPBCVARVB
|
|
$.
|
|
|
|
$( The floor of a number greater than or equal to 0 is a nonnegative integer.
|
|
(Contributed by Jim Kingdon, 10-Oct-2021.) $)
|
|
flqge0nn0 $p |- ( ( A e. QQ /\ 0 <_ A ) -> ( |_ ` A ) e. NN0 ) $=
|
|
( cq wcel cc0 cle wbr wa cfl cfv cz cn0 flqcl adantr wb flqge biimpa elnn0z
|
|
0z mpan2 sylanbrc ) ABCZDAEFZGAHIZJCZDUCEFZUCKCUAUDUBALMUAUBUEUADJCUBUENRAD
|
|
OSPUCQT $.
|
|
|
|
$( The floor of a number greater than or equal to 1 is a positive integer.
|
|
(Contributed by Jim Kingdon, 10-Oct-2021.) $)
|
|
flqge1nn $p |- ( ( A e. QQ /\ 1 <_ A ) -> ( |_ ` A ) e. NN ) $=
|
|
( cq wcel c1 cle wbr wa cfl cfv cz cn flqcl adantr wb 1z flqge mpan2 biimpa
|
|
elnnz1 sylanbrc ) ABCZDAEFZGAHIZJCZDUCEFZUCKCUAUDUBALMUAUBUEUADJCUBUENOADPQ
|
|
RUCST $.
|
|
|
|
$( The floor function of a division of a nonnegative integer by a positive
|
|
integer is a nonnegative integer. (Contributed by Alexander van der
|
|
Vekens, 14-Apr-2018.) $)
|
|
fldivnn0 $p |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. NN0 ) $=
|
|
( cn0 wcel cn wa cdiv co cq cc0 cle wbr cfl cfv cz nn0z znq sylan nn0ge0div
|
|
flqge0nn0 syl2anc ) ACDZBEDZFABGHZIDZJUDKLUDMNCDUBAODUCUEAPABQRABSUDTUA $.
|
|
|
|
$( The floor of a fraction is 0 iff the denominator is less than the
|
|
numerator. (Contributed by AV, 8-Jul-2021.) $)
|
|
divfl0 $p |- ( ( A e. NN0 /\ B e. NN )
|
|
-> ( A < B <-> ( |_ ` ( A / B ) ) = 0 ) ) $=
|
|
( cn0 wcel cn wa cdiv co cfl cfv cc0 wceq caddc cle wbr c1 clt cq cz wb znq
|
|
cc nn0z sylan qcn addid2 eqcomd 3syl fveq2d eqeq1d flqbi2 sylancr nn0ge0div
|
|
0z biantrurd cr crp nn0re nnrp divlt1lt syl2an bitr3d 3bitrrd ) ACDZBEDZFZA
|
|
BGHZIJZKLKVGMHZIJZKLZKVGNOZVGPQOZFZABQOZVFVHVJKVFVGVIIVFVGRDZVGUBDZVGVILVDA
|
|
SDVEVPAUCABUAUDZVGUEVQVIVGVGUFUGUHUIUJVFKSDVPVKVNTUNVRVGKUKULVFVMVNVOVFVLVM
|
|
ABUMUOVDAUPDBUQDVMVOTVEAURBUSABUTVAVBVC $.
|
|
|
|
$( An integer can be moved in and out of the floor of a sum. (Contributed by
|
|
Jim Kingdon, 10-Oct-2021.) $)
|
|
flqaddz $p |- ( ( A e. QQ /\ N e. ZZ ) ->
|
|
( |_ ` ( A + N ) ) = ( ( |_ ` A ) + N ) ) $=
|
|
( cq wcel cz wa caddc co cfl cfv wceq cle wbr c1 clt flqcl adantr zred zcnd
|
|
cr qre simpr leadd1dd 1red readdcld flqltp1 ltadd1dd 1cnd add32d breqtrd wb
|
|
flqle zq qaddcl sylan2 simpl flqcld zaddcld flqbi syl2anc mpbir2and ) ACDZB
|
|
EDZFZABGHZIJAIJZBGHZKZVGVELMZVEVGNGHZOMZVDVFABVDVFVBVFEDVCAPQZRZVBATDVCAUAQ
|
|
ZVDBVBVCUBZRZVBVFALMVCAULQUCVDVEVFNGHZBGHVJOVDAVQBVNVDVFNVMVDUDUEVPVBAVQOMV
|
|
CAUFQUGVDVFNBVDVFVLSVDUHVDBVOSUIUJVDVECDZVGEDVHVIVKFUKVCVBBCDVRBUMABUNUOVDV
|
|
FBVDAVBVCUPUQVOURVEVGUSUTVA $.
|
|
|
|
$( An integer can be moved in and out of the floor of a sum. (Contributed by
|
|
Jim Kingdon, 10-Oct-2021.) $)
|
|
flqzadd $p |- ( ( N e. ZZ /\ A e. QQ ) ->
|
|
( |_ ` ( N + A ) ) = ( N + ( |_ ` A ) ) ) $=
|
|
( cq wcel cz caddc co cfl cfv wceq wa flqaddz cc addcom syl2an fveq2d flqcl
|
|
qcn zcn zcnd 3eqtr3d ancoms ) ACDZBEDZBAFGZHIZBAHIZFGZJUCUDKZABFGZHIUGBFGZU
|
|
FUHABLUIUJUEHUCAMDBMDZUJUEJUDARBSZABNOPUCUGMDULUKUHJUDUCUGAQTUMUGBNOUAUB $.
|
|
|
|
$( Move a nonnegative integer in and out of a floor. (Contributed by Jim
|
|
Kingdon, 10-Oct-2021.) $)
|
|
flqmulnn0 $p |- ( ( N e. NN0 /\ A e. QQ ) ->
|
|
( N x. ( |_ ` A ) ) <_ ( |_ ` ( N x. A ) ) ) $=
|
|
( cn0 wcel cq wa cfl cfv cmul co cle wbr cz flqcl adantl zred cr qre nn0red
|
|
simpl nn0ge0d flqle lemul2ad wb nn0z syl qmulcl sylan zmulcl syl2an syl2anc
|
|
zq flqge mpbid ) BCDZAEDZFZBAGHZIJZBAIJZKLZUSUTGHKLZUQURABUQURUPURMDZUOANZO
|
|
PUPAQDUOAROUQBUOUPTZSUQBVEUAUPURAKLUOAUBOUCUQUTEDZUSMDZVAVBUDUOBEDZUPVFUOBM
|
|
DZVHBUEZBULUFBAUGUHUOVIVCVGUPVJVDBURUIUJUTUSUMUKUN $.
|
|
|
|
$( A real bounded between an integer and its successor is nonnegative iff the
|
|
integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217.
|
|
(Contributed by NM, 12-Mar-2005.) $)
|
|
btwnzge0 $p |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) )
|
|
-> ( 0 <_ A <-> 0 <_ N ) ) $=
|
|
( cr wcel cz wa cle wbr c1 caddc co clt cc0 0red simplll simplr zred adantr
|
|
1red simpr readdcld simplrr lelttrd wb zleltp1 mpan ad3antlr mpbird simplrl
|
|
0z letrd impbida ) ACDZBEDZFZBAGHZABIJKZLHZFZFZMAGHZMBGHZUTVAFZVBMUQLHZVCMA
|
|
UQVCNUMUNUSVAOVCBIUTBCDZVAUTBUMUNUSPQZRVCSUAUTVATUOUPURVAUBUCUNVBVDUDZUMUSV
|
|
AMEDUNVGUJMBUEUFUGUHUTVBFZMBAVHNUTVEVBVFRUMUNUSVBOUTVBTUOUPURVBUIUKUL $.
|
|
|
|
$( Two times an integer plus one is not negative iff the integer is not
|
|
negative. (Contributed by AV, 19-Jun-2021.) $)
|
|
2tnp1ge0ge0 $p |- ( N e. ZZ -> ( 0 <_ ( ( 2 x. N ) + 1 ) <-> 0 <_ N ) ) $=
|
|
( cz wcel cc0 c2 cmul co c1 caddc cle wbr cdiv cr clt wb 2z syl3anc cc 2ap0
|
|
a1i id zmulcld peano2zd zred 2re 2pos ge0div cap wa wceq zcnd 1cnd divdirap
|
|
2cn pm3.2i zcn 2cnd divcanap3d oveq1d eqtrd breq2d halfre readdcld addge01d
|
|
zre halfge0 mpbii 1red halflt1 ltadd2dd btwnzge0 syl22anc 3bitrd ) ABCZDEAF
|
|
GZHIGZJKZDVPELGZJKZDAHELGZIGZJKZDAJKZVNVPMCEMCZDENKZVQVSOVNVPVNVOVNEAEBCVNP
|
|
TVNUAZUBZUCUDWDVNUETWEVNUFTVPEUGQVNVRWADJVNVRVOELGZVTIGZWAVNVORCHRCERCZEDUH
|
|
KZUIZVRWIUJVNVOWGUKVNULWLVNWJWKUNSUOTVOHEUMQVNWHAVTIVNAEAUPVNUQWKVNSTURUSUT
|
|
VAVNWAMCVNAWAJKZWAAHIGNKWBWCOVNAVTAVEZVTMCVNVBTZVCWFVNDVTJKWMVFVNAVTWNWOVDV
|
|
GVNVTHAWOVNVHWNVTHNKVNVITVJWAAVKVLVM $.
|
|
|
|
$( Ordering relation for the floor of half of an integer. (Contributed by
|
|
NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.) $)
|
|
flhalf $p |- ( N e. ZZ -> N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) ) $=
|
|
( cz wcel c2 c1 caddc co cdiv cfl cfv cle wbr clt syl cr readdcld a1i recnd
|
|
cmul mpbird cq cn peano2z 2nn znq sylancl flqltp1 peano2re flqcld zred 1red
|
|
zre crp 2rp ltdivmuld mpbid 2timesd oveq2d 2cnd adddid 2re remulcld addassd
|
|
3eqtr4d breqtrd ltadd1d wb 2z zmulcld zleltp1 mpdan ) ABCZADAEFGZDHGZIJZSGZ
|
|
KLZAVPEFGZMLZVLVSVMVREFGZMLVLVMDVOEFGZSGZVTMVLVNWAMLZVMWBMLVLVNUACZWCVLVMBC
|
|
DUBCWDAUCUDVMDUEUFZVNUGNVLVMWADVLAOCVMOCAULZAUHNVLVOEVLVOVLVNWEUIZUJZVLUKZP
|
|
DUMCVLUNQUOUPVLVPDESGZFGVPEEFGZFGWBVTVLWJWKVPFVLEVLEWIRZUQURVLDVOEVLUSVLVOW
|
|
HRWLUTVLVPEEVLVPVLDVODOCVLVAQWHVBZRWLWLVCVDVEVLAVREWFVLVPEWMWIPWIVFTVLVPBCV
|
|
QVSVGVLDVODBCVLVHQWGVIAVPVJVKT $.
|
|
|
|
$( The floor function of a division of a nonnegative integer by a positive
|
|
integer is less than or equal to the division. (Contributed by Alexander
|
|
van der Vekens, 14-Apr-2018.) $)
|
|
fldivnn0le $p |- ( ( K e. NN0 /\ L e. NN )
|
|
-> ( |_ ` ( K / L ) ) <_ ( K / L ) ) $=
|
|
( cn0 wcel cn wa cdiv co cq cfl cfv cle wbr cz nn0z znq sylan flqle syl ) A
|
|
CDZBEDZFABGHZIDZUBJKUBLMTANDUAUCAOABPQUBRS $.
|
|
|
|
$( The floor function of a division of a nonnegative integer by a positive
|
|
integer is less than the division of a greater dividend by the same
|
|
positive integer. (Contributed by Alexander van der Vekens,
|
|
14-Apr-2018.) $)
|
|
flltdivnn0lt $p |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN )
|
|
-> ( K < N -> ( |_ ` ( K / L ) ) < ( N / L ) ) ) $=
|
|
( cn0 wcel w3a clt wbr cdiv co wa cz nn0zd znq syl2anc adantr cr cq qre syl
|
|
cn cfl cfv simp1 simp3 flqcld zred simp2 cle fldivnn0le 3adant2 simpr nn0re
|
|
cc0 wb nnre nngt0 jca 3anim123i ltdiv1 mpbid lelttrd ex ) ADEZCDEZBUAEZFZAC
|
|
GHZABIJZUBUCZCBIJZGHVGVHKZVJVIVKVLVJVGVJLEZVHVGALEZVFVMVGAVDVEVFUDMZVDVEVFU
|
|
EZVNVFKZVIABNZUFOPUGVLVNVFVIQEZVGVNVHVOPVGVFVHVPPZVQVIREVSVRVISTOVLCLEZVFVK
|
|
QEZVGWAVHVGCVDVEVFUHMPVTWAVFKVKREWBCBNVKSTOVGVJVIUIHZVHVDVFWCVEABUJUKPVLVHV
|
|
IVKGHZVGVHULVLAQEZCQEZBQEZUNBGHZKZFZVHWDUOVGWJVHVDWEVEWFVFWIAUMCUMVFWGWHBUP
|
|
BUQURUSPACBUTTVAVBVC $.
|
|
|
|
$( The floor of an integer equal to 3 or greater than 4, increased by 1, is
|
|
less than or equal to the half of the integer minus 1. (Contributed by
|
|
AV, 8-Jul-2021.) $)
|
|
fldiv4p1lem1div2 $p |- ( ( N = 3 \/ N e. ( ZZ>= ` 5 ) )
|
|
-> ( ( |_ ` ( N / 4 ) ) + 1 ) <_ ( ( N - 1 ) / 2 ) ) $=
|
|
( c3 c4 cdiv co cfl cfv c1 caddc cmin c2 cle wbr c5 wcel a1i cc0 syl6eq syl
|
|
cr wceq cuz 1le1 oveq1 fveq2d clt 3lt4 cn0 cn wb 3nn0 4nn divfl0 mp2an mpbi
|
|
oveq1d 0p1e1 3m1e2 2div2e1 3brtr4d wo uzp1 2re leidi df-5 oveq1i 4cn ax-1cn
|
|
4ap0 divdirapi dividapi eqtri fveq2i 1re 0le1 4pos divge0 mp4an 1lt4 recgt1
|
|
4re cz cq wa 1z znq flqbi2 mpbir2an 1p1e2 mvrraddi 4d2e2 c6 w3a eluz2 mpan2
|
|
flqle adantr flqcld zred zre id redivclapd 3jca leadd1 mpbid div4p1lem1div2
|
|
cap sylan wi peano2re peano2rem rehalfcld mp2and 3adant1 sylbi 5p1e6 eleq2s
|
|
letr jaoi ) ABUAZACDEZFGZHIEZAHJEZKDEZLMZANUBGOZXTHHYCYELHHLMXTUCPXTYCQHIEH
|
|
XTYBQHIXTYBBCDEZFGZQXTYAYHFABCDUDUEBCUFMZYIQUAZUGBUHOCUIOZYJYKUJUKULBCUMUNU
|
|
ORUPUQRXTYEKKDEHXTYDKKDXTYDBHJEKABHJUDURRUPUSRUTYGANUAZANHIEZUBGZOZVAYFNAVB
|
|
YMYFYPYMKKYCYELKKLMYMKVCVDPYMYCHHIEKYMYBHHIYMYBNCDEZFGZHYMYAYQFANCDUDUEYRHH
|
|
CDEZIEZFGZHYQYTFYQCHIEZCDEZYTNUUBCDVEVFUUCCCDEZYSIEYTCHCVGVHVGVIVJUUDHYSICV
|
|
GVIVKVFVLVLVMUUAHUAZQYSLMZYSHUFMZHTOZQHLMCTOZQCUFMZUUFVNVOWAVPHCVQVRHCUFMZU
|
|
UGVSUUIUUJUUKUUGUJWAVPCVTUNUOHWBOZYSWCOZUUEUUFUUGWDUJWEUULYLUUMWEULHCWFUNYS
|
|
HWGUNWHVLRUPWIRYMYECKDEKYMYDCKDYMYDNHJECANHJUDNCHVGVHVEWJRUPWKRUTYFAWLUBGZY
|
|
OAUUNOWLWBOZAWBOZWLALMZWMYFWLAWNUUPUUQYFUUOUUPUUQWDZYCYAHIEZLMZUUSYELMZYFUU
|
|
RYBYALMZUUTUUPUVBUUQUUPYAWCOZUVBUUPYLUVCULACWFWOZYAWPSWQUURYBTOZYATOZUUHWMZ
|
|
UVBUUTUJUUPUVGUUQUUPUVEUVFUUHUUPYBUUPYAUVDWRWSZUUPATOZUVFAWTZUVIACUVIXAUUIU
|
|
VIWAPCQXGMUVIVIPXBSZUUHUUPVNPXCWQYBYAHXDSXEUUPUVIUUQUVAUVJAXFXHUURYCTOZUUST
|
|
OZYETOZWMZUUTUVAWDYFXIUUPUVOUUQUUPUVLUVMUVNUUPUVEUVLUVHYBXJSUUPUVFUVMUVKYAX
|
|
JSUUPUVIUVNUVJUVIYDAXKXLSXCWQYCUUSYEXRSXMXNXOYNWLUBXPVMXQXSSXS $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The value of the ceiling function. (Contributed by Jim Kingdon,
|
|
10-Oct-2021.) $)
|
|
ceilqval $p |- ( A e. QQ -> ( |^ ` A ) = -u ( |_ ` -u A ) ) $=
|
|
( vx cq wcel cr cneg cfl cfv cz cceil wceq qre qnegcl flqcl znegcld negeq
|
|
syl cv fveq2d negeqd df-ceil fvmptg syl2anc ) ACDZAEDAFZGHZFZIDZAJHUGKALU
|
|
DUECDZUHAMUIUFUENOQBABRZFZGHZFUGEIJUJAKZULUFUMUKUEGUJAPSTBUAUBUC $.
|
|
$}
|
|
|
|
$( The ceiling function returns an integer (closure law). (Contributed by
|
|
Jim Kingdon, 11-Oct-2021.) $)
|
|
ceiqcl $p |- ( A e. QQ -> -u ( |_ ` -u A ) e. ZZ ) $=
|
|
( cq wcel cneg cfl cfv qnegcl flqcld znegcld ) ABCZADZEFJKAGHI $.
|
|
|
|
$( Closure of the ceiling function. (Contributed by Jim Kingdon,
|
|
11-Oct-2021.) $)
|
|
ceilqcl $p |- ( A e. QQ -> ( |^ ` A ) e. ZZ ) $=
|
|
( cq wcel cceil cfv cneg cfl cz ceilqval ceiqcl eqeltrd ) ABCADEAFGEFHAIAJK
|
|
$.
|
|
|
|
$( The ceiling of a real number is greater than or equal to that number.
|
|
(Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceiqge $p |- ( A e. QQ -> A <_ -u ( |_ ` -u A ) ) $=
|
|
( cq wcel cneg cfl cfv qnegcl flqcld zred qre cle wbr flqle syl lenegcon2d
|
|
) ABCZADZEFZAPRPQAGZHIAJPQBCRQKLSQMNO $.
|
|
|
|
$( The ceiling of a real number is greater than or equal to that number.
|
|
(Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceilqge $p |- ( A e. QQ -> A <_ ( |^ ` A ) ) $=
|
|
( cq wcel cneg cfl cfv cceil cle ceiqge ceilqval breqtrrd ) ABCAADEFDAGFHAI
|
|
AJK $.
|
|
|
|
$( One less than the ceiling of a real number is strictly less than that
|
|
number. (Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceiqm1l $p |- ( A e. QQ -> ( -u ( |_ ` -u A ) - 1 ) < A ) $=
|
|
( cq wcel cneg cfl cfv c1 cmin co clt qnegcl flqcld zcnd 1cnd negdid negcld
|
|
caddc negsubd eqtr2d qre peano2zd zred wbr flqltp1 syl ltnegcon1d eqbrtrd )
|
|
ABCZADZEFZDZGHIZUJGQIZDZAJUHUNUKGDQIULUHUJGUHUJUHUIAKZLZMZUHNZOUHUKGUHUJUQP
|
|
URRSUHAUMATUHUMUHUJUPUAUBUHUIBCUIUMJUCUOUIUDUEUFUG $.
|
|
|
|
$( One less than the ceiling of a real number is strictly less than that
|
|
number. (Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceilqm1lt $p |- ( A e. QQ -> ( ( |^ ` A ) - 1 ) < A ) $=
|
|
( cq wcel cceil cfv c1 cmin co cneg cfl clt ceilqval oveq1d ceiqm1l eqbrtrd
|
|
) ABCZADEZFGHAIJEIZFGHAKPQRFGALMANO $.
|
|
|
|
$( The ceiling of a real number is the smallest integer greater than or equal
|
|
to it. (Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceiqle $p |- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> -u ( |_ ` -u A ) <_ B ) $=
|
|
( cq wcel cz cle wbr w3a cneg cfl cfv c1 cmin clt ceiqcl peano2rem 3ad2ant1
|
|
co cr zred syl qre zre 3ad2ant2 ceiqm1l simp3 ltletrd zlem1lt sylan 3adant3
|
|
wb mpbird ) ACDZBEDZABFGZHZAIJKIZBFGZUQLMRZBNGZUPUSABUMUNUSSDZUOUMUQSDVAUMU
|
|
QAOZTUQPUAQUMUNASDUOAUBQUNUMBSDUOBUCUDUMUNUSANGUOAUEQUMUNUOUFUGUMUNURUTUKZU
|
|
OUMUQEDUNVCVBUQBUHUIUJUL $.
|
|
|
|
$( The ceiling of a real number is the smallest integer greater than or equal
|
|
to it. (Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceilqle $p |- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> ( |^ ` A ) <_ B ) $=
|
|
( cq wcel cz cle wbr w3a cfv cneg cfl wceq ceilqval 3ad2ant1 ceiqle eqbrtrd
|
|
cceil ) ACDZBEDZABFGZHAQIZAJKIJZBFRSUAUBLTAMNABOP $.
|
|
|
|
$( An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) $)
|
|
ceilid $p |- ( A e. ZZ -> ( |^ ` A ) = A ) $=
|
|
( cz wcel cceil cfv cneg cfl cq wceq zq ceilqval syl znegcl flid negeqd zcn
|
|
negnegd 3eqtrd ) ABCZADEZAFZGEZFZUAFASAHCTUCIAJAKLSUBUASUABCUBUAIAMUANLOSAA
|
|
PQR $.
|
|
|
|
$( A rational number equals its ceiling iff it is an integer. (Contributed
|
|
by Jim Kingdon, 11-Oct-2021.) $)
|
|
ceilqidz $p |- ( A e. QQ -> ( A e. ZZ <-> ( |^ ` A ) = A ) ) $=
|
|
( cq wcel cz cceil cfv wceq ceilid ceilqcl eleq1 syl5ibcom impbid2 ) ABCZAD
|
|
CZAEFZAGZAHMODCPNAIOADJKL $.
|
|
|
|
$( The floor of a rational number is less than or equal to its ceiling.
|
|
(Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
flqleceil $p |- ( A e. QQ -> ( |_ ` A ) <_ ( |^ ` A ) ) $=
|
|
( cq wcel cfl cfv cceil flqcl zred qre ceilqcl flqle ceilqge letrd ) ABCZAD
|
|
EZAAFEZNOAGHAINPAJHAKALM $.
|
|
|
|
$( A rational number is an integer iff its floor equals its ceiling.
|
|
(Contributed by Jim Kingdon, 11-Oct-2021.) $)
|
|
flqeqceilz $p |- ( A e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = ( |^ ` A ) ) ) $=
|
|
( cq wcel cz cfv wceq wn wi syl mpancom wa adantr adantl sylbid cle wbr wne
|
|
wb ex clt cfl cceil flid ceilid eqtr4d wo wdc flqcl zq qdceq eqeq1 ceilqidz
|
|
exmiddc eqcom syl6bb biimprd flqle df-ne necom qltlen breq1 ceilqge ceilqcl
|
|
qre zred lenltd pm2.21 syl6bi mpd com23 sylbird expd com3r sylbir mpdi jaoi
|
|
sylbi mpcom impbid2 ) ABCZADCZAUAEZAUBEZFZWAWBAWCAUCAUDUEWBAFZWEGZUFZVTWDWA
|
|
HZVTWEUGZWGWBBCZVTWIVTWBDCWJAUHWBUIIZWBAUJJWEUMIWEVTWHHWFWEVTWHWEVTKWDAWCFZ
|
|
WAWEWDWLRVTWBAWCUKLVTWLWAHWEVTWAWLVTWAWCAFWLAULWCAUNUOUPMNSWFVTWBAOPZWHAUQW
|
|
FWBAQZVTWMWHHHZWBAURWNAWBQZWOWBAUSVTWMWPWHVTWMWPWHVTWMWPKZWBATPZWHWJVTWRWQR
|
|
WKWBAUTJVTWDWRWAVTWDWRWAHVTWDKWRWCATPZWAWDWRWSRVTWBWCATVAMVTWSWAHZWDVTAWCOP
|
|
ZWTAVBVTXAWSGWTVTAWCAVDVTWCAVCVEVFWSWAVGVHVILNSVJVKVLVMVQVNVOVPVRVS $.
|
|
|
|
${
|
|
intqfrac2.1 $e |- Z = ( |_ ` A ) $.
|
|
intqfrac2.2 $e |- F = ( A - Z ) $.
|
|
$( Decompose a real into integer and fractional parts. (Contributed by Jim
|
|
Kingdon, 18-Oct-2021.) $)
|
|
intqfrac2 $p |- ( A e. QQ -> ( 0 <_ F /\ F < 1 /\ A = ( Z + F ) ) ) $=
|
|
( cq wcel cc0 cle wbr c1 clt caddc co wceq cfl cfv cmin qfracge0 oveq2i
|
|
eqtri syl6breqr qfraclt1 syl5eqbr flqcl syl5eqel zcnd qcn pncan3d syl5req
|
|
cz 3jca ) AFGZHBIJBKLJACBMNZOUMHAAPQZRNZBIASBACRNZUPECUOARDTUAZUBUMBUPKLU
|
|
RAUCUDUMUNCUQMNABUQCMETUMCAUMCUMCUOUKDAUEUFUGAUHUIUJUL $.
|
|
$}
|
|
|
|
${
|
|
intfracq.1 $e |- Z = ( |_ ` ( M / N ) ) $.
|
|
intfracq.2 $e |- F = ( ( M / N ) - Z ) $.
|
|
$( Decompose a rational number, expressed as a ratio, into integer and
|
|
fractional parts. The fractional part has a tighter bound than that of
|
|
~ intqfrac2 . (Contributed by NM, 16-Aug-2008.) $)
|
|
intfracq $p |- ( ( M e. ZZ /\ N e. NN )
|
|
-> ( 0 <_ F /\ F <_ ( ( N - 1 ) / N ) /\ ( M / N ) = ( Z + F ) ) ) $=
|
|
( cz wcel cle wbr c1 cmin co cdiv clt syl cmul cr wb adantl cn caddc wceq
|
|
wa cc0 w3a znq intqfrac2 simp1d cfl cfv qfraclt1 oveq2i eqtri simpr nncnd
|
|
cq a1i nnap0d dividapd 3brtr4d qre flqcld syl5eqel zred resubcld nnre jca
|
|
nngt0 ltmuldiv2 syl3anc mpbird recnd zcnd subdid syl5eq cc zcn divcanap2d
|
|
adantr simpl eqeltrd nnz zmulcld zsubcld zltlem1 syl2anc peano2rem simp3d
|
|
mpbid lemuldiv2 3jca ) BGHZCUAHZUDZUEAIJZACKLMZCNMIJZBCNMZDAUBMUCZWOWPAKO
|
|
JZWTWOWSUQHZWPXAWTUFBCUGZWSADEFUHPZUIWOCAQMZWQIJZWRWOXECOJZXFWOXGACCNMZOJ
|
|
ZWOWSWSUJUKZLMZKAXHOWOXBXKKOJXCWSULPAXKUCWOAWSDLMZXKFDXJWSLEUMUNURWOCWOCW
|
|
MWNUOZUPZWOCXMUSZUTVAWOARHZCRHZXQUECOJZUDZXGXISWOAXLRFWOWSDWOXBWSRHXCWSVB
|
|
PZWODWODXJGEWOWSXCVCVDZVEVFVDZWNXQWMCVGZTWNXSWMWNXQXRYCCVIVHTZACCVJVKVLWO
|
|
XEGHCGHZXGXFSWOXECWSQMZCDQMZLMZGWOXECXLQMYHAXLCQFUMWOCWSDXNWOWSXTVMWODYAV
|
|
NVOVPWOYFYGWOYFBGWOBCWMBVQHWNBVRVTXNXOVSWMWNWAWBWOCDWNYEWMCWCTZYAWDWEWBYI
|
|
XECWFWGWJWOXPWQRHZXSXFWRSYBWNYJWMWNXQYJYCCWHPTYDAWQCWKVKWJWOWPXAWTXDWIWL
|
|
$.
|
|
$}
|
|
|
|
$( Cancellation of the embedded floor of a real divided by an integer.
|
|
(Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
flqdiv $p |- ( ( A e. QQ /\ N e. NN )
|
|
-> ( |_ ` ( ( |_ ` A ) / N ) ) = ( |_ ` ( A / N ) ) ) $=
|
|
( cq wcel wa cdiv co cfl cfv caddc wceq cc0 cle wbr c1 clt eqid cc syl cr
|
|
cn cmin intqfrac2 simp3d adantr oveq1d simpl flqcld cz zq qsubcl qcn syldan
|
|
simpr nncnd nnap0d divdirapd eqtrd flqcl intfracq sylan znq syl2anc wne nnq
|
|
zcnd adantl nnne0d qdivcl syl3anc addassd 3eqtrd fveq2d qre nndivred simp1d
|
|
nnrpd qfracge0 divge0d addge0d peano2rem nnap0 redivclapd nnrecred qfraclt1
|
|
nnre simp2d wb nnred nngt0d 1re ltdiv1 mp3an2 syl12anc leltaddd nncn npcan1
|
|
mpbid recnd ax-1cn divdirap dividapd 3eqtr3d qaddcl flqbi2 mpbir2and eqtr2d
|
|
cap breqtrd ) ACDZBUADZEZABFGZHIAHIZBFGZHIZXOXPUBGZAXNUBGZBFGZJGZJGZHIZXPXL
|
|
XMYAHXLXMXOXSJGZXPXQJGZXSJGYAXLXMXNXRJGZBFGYCXLAYEBFXJAYEKZXKXJLXRMNZXROPNZ
|
|
YFAXRXNXNQXRQUCUDUEUFXLXNXRBXLXNXLAXJXKUGUHZVFXJXKXNCDZXRRDZXLXNUIDZYJYIXNU
|
|
JSZXJYJEXRCDZYKAXNUKZXRULSUMXLBXJXKUNZUOXLBYPUPUQURXLXOYDXSJXJYLXKXOYDKZAUS
|
|
ZYLXKEZLXQMNZXQBOUBGZBFGZMNZYQXQXNBXPXPQXQQUTZUDVAUFXLXPXQXSXLXPXJYLXKXPUID
|
|
ZYRYSXOXNBVBZUHVAZVFXLXQCDZXQRDXLXOCDZXPCDZUUHXLYLXKUUIYIYPUUFVCZXLUUEUUJUU
|
|
GXPUJSXOXPUKVCZXQULSXLXSCDZXSRDXLYNBCDZBLVDUUMXJXKYJYNYMYOUMZXKUUNXJBVEVGXL
|
|
BYPVHXRBVIVJZXSULSVKVLVMXLYBXPKZLXTMNZXTOPNZXLXQXSXLUUHXQTDUULXQVNSZXLXRBXL
|
|
YNXRTDZUUOXRVNSZYPVOZXJYLXKYTYRYSYTUUCYQUUDVPVAXLXRBUVBXLBYPVQXJYGXKAVRUEVS
|
|
VTXLXTUUBOBFGZJGZOPXLXQXSUUBUVDUUTUVCXKUUBTDXJXKUUABXKBTDZUUATDBWFZBWASZUVG
|
|
BWBZWCVGXLBYPWDXJYLXKUUCYRYSYTUUCYQUUDWGVAXLYHXSUVDPNZXJYHXKAWEUEXLUVAUVFLB
|
|
PNZYHUVJWHZUVBXLBYPWIXLBYPWJUVAOTDUVFUVKEUVLWKXROBWLWMWNWRWOXKUVEOKXJXKUUAO
|
|
JGZBFGZBBFGUVEOXKUVMBBFXKBRDZUVMBKBWPZBWQSUFXKUUARDZUVOBLXHNZUVNUVEKZXKUUAU
|
|
VHWSUVPUVIUVQORDUVOUVREUVSWTUUAOBXAWMWNXKBUVPUVIXBXCVGXIXLUUEXTCDZUUQUURUUS
|
|
EWHXLXOUUKUHXLUUHUUMUVTUULUUPXQXSXDVCXTXPXEVCXFXG $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The modulo (remainder) operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c mod $. $( Modulo operation. $)
|
|
$c == $. $( Equivalence/congruence symbol
|
|
- currently used in comments only. The modulus should be
|
|
presented as follows: ` n == A ` (mod ` N ` ) $)
|
|
|
|
$( Extend class notation with the modulo operation. $)
|
|
cmo $a class mod $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the modulo (remainder) operation. See ~ modqval for its value.
|
|
For example, ` ( 5 mod 3 ) = 2 ` and ` ( -u 7 mod 2 ) = 1 ` . As with
|
|
~ df-fl we define this for first and second arguments which are real and
|
|
positive real, respectively, even though many theorems will need to be
|
|
more restricted (for example, specify rational arguments). (Contributed
|
|
by NM, 10-Nov-2008.) $)
|
|
df-mod $a |- mod = ( x e. RR , y e. RR+ |->
|
|
( x - ( y x. ( |_ ` ( x / y ) ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The value of the modulo operation. The modulo congruence notation of
|
|
number theory, ` J == K ` (modulo ` N ` ), can be expressed in our
|
|
notation as ` ( J mod N ) = ( K mod N ) ` . Definition 1 in Knuth, _The
|
|
Art of Computer Programming_, Vol. I (1972), p. 38. Knuth uses "mod"
|
|
for the operation and "modulo" for the congruence. Unlike Knuth, we
|
|
restrict the second argument to positive numbers to simplify certain
|
|
theorems. (This also gives us future flexibility to extend it to any
|
|
one of several different conventions for a zero or negative second
|
|
argument, should there be an advantage in doing so.) As with ~ flqcl we
|
|
only prove this for rationals although other particular kinds of real
|
|
numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.) $)
|
|
modqval $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B )
|
|
-> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) $=
|
|
( vx vy cq wcel cc0 cr crp cdiv co cfl cfv cmul cmin cmo wceq qre fveq2d
|
|
cv clt wbr w3a 3ad2ant1 3ad2ant2 simp3 wne gt0ne0d qdivcl syld3an3 flqcld
|
|
elrpd remulcld resubcld oveq1 oveq2d oveq12 mpdan df-mod ovmpt2g syl3anc
|
|
zred oveq2 ) AEFZBEFZGBUAUBZUCZAHFZBIFABABJKZLMZNKZOKZHFABPKVLQVDVEVHVFAR
|
|
UDZVGBVEVDBHFVFBRUEZVDVEVFUFZULVGAVKVMVGBVJVNVGVJVGVIVDVEVFBGUGVIEFVGBVOU
|
|
HABUIUJUKVBUMUNCDABHICTZDTZVPVQJKZLMZNKZOKZVLPAVQAVQJKZLMZNKZOKZHVPAQZVTW
|
|
DQWAWEQWFVSWCVQNWFVRWBLVPAVQJUOSUPVPAVTWDOUQURVQBQZWDVKAOWGWCVJQWDVKQWGWB
|
|
VILVQBAJVCSVQBWCVJNUQURUPCDUSUTVA $.
|
|
$}
|
|
|
|
$( The value of the modulo operation (multiplication in reversed order).
|
|
(Contributed by Jim Kingdon, 16-Oct-2021.) $)
|
|
modqvalr $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B )
|
|
-> ( A mod B ) = ( A - ( ( |_ ` ( A / B ) ) x. B ) ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cdiv cfl cfv cmul cmin modqval cc 3ad2ant2
|
|
qcn wne simp3 gt0ne0d qdivcl syld3an3 flqcld zcnd mulcomd oveq2d eqtrd ) AC
|
|
DZBCDZEBFGZHZABIJABABKJZLMZNJZOJAUOBNJZOJABPUMUPUQAOUMBUOUKUJBQDULBSRUMUOUM
|
|
UNUJUKULBETUNCDUMBUJUKULUAUBABUCUDUEUFUGUHUI $.
|
|
|
|
$( Closure law for the modulo operation. (Contributed by Jim Kingdon,
|
|
16-Oct-2021.) $)
|
|
modqcl $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. QQ ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cdiv cfl cfv cmul cmin modqval simp1 simp2
|
|
cz syl2anc wne gt0ne0d qdivcl syld3an3 flqcld zq syl qmulcl qsubcl eqeltrd
|
|
simp3 ) ACDZBCDZEBFGZHZABIJABABKJZLMZNJZOJZCABPUOULURCDZUSCDULUMUNQUOUMUQCD
|
|
ZUTULUMUNRUOUQSDVAUOUPULUMUNBEUAUPCDUOBULUMUNUKUBABUCUDUEUQUFUGBUQUHTAURUIT
|
|
UJ $.
|
|
|
|
$( Partition of a division into its integer part and the remainder.
|
|
(Contributed by Jim Kingdon, 16-Oct-2021.) $)
|
|
flqpmodeq $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B )
|
|
-> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A ) $=
|
|
( cq wcel cc0 clt wbr w3a cdiv co cfl cfv cmul cmin cmo wceq caddc modqvalr
|
|
cc qcn eqcomd 3ad2ant1 wne simp3 gt0ne0d qdivcl syld3an3 flqcld zcnd mulcld
|
|
3ad2ant2 modqcl syl subaddd mpbid ) ACDZBCDZEBFGZHZAABIJZKLZBMJZNJZABOJZPVB
|
|
VDQJAPUSVDVCABRUAUSAVBVDUPUQASDURATUBUSVABUSVAUSUTUPUQURBEUCUTCDUSBUPUQURUD
|
|
UEABUFUGUHUIUQUPBSDURBTUKUJUSVDCDVDSDABULVDTUMUNUO $.
|
|
|
|
${
|
|
modqcld.1 $e |- ( ph -> A e. QQ ) $.
|
|
modqcld.2 $e |- ( ph -> B e. QQ ) $.
|
|
modqcld.3 $e |- ( ph -> 0 < B ) $.
|
|
$( Closure law for the modulo operation. (Contributed by Jim Kingdon,
|
|
16-Oct-2021.) $)
|
|
modqcld $p |- ( ph -> ( A mod B ) e. QQ ) $=
|
|
( cq wcel cc0 clt wbr cmo co modqcl syl3anc ) ABGHCGHICJKBCLMGHDEFBCNO $.
|
|
$}
|
|
|
|
$( ` A mod B ` is zero iff ` A ` is evenly divisible by ` B ` . (Contributed
|
|
by Jim Kingdon, 17-Oct-2021.) $)
|
|
modq0 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co wceq cdiv cfl cfv cz cc qcn 3ad2ant2 bitrd
|
|
cmul cmin modqval eqeq1d 3ad2ant1 simp3 gt0ne0d qdivcl syld3an3 flqcld zcnd
|
|
wne mulcld subeq0ad eqcom cr qre gt0ap0d divmulap2d syl5rbbr wb flqidz syl
|
|
) ACDZBCDZEBFGZHZABIJZEKZABLJZMNZVIKZVIODZVFVHABVJTJZKZVKVFVHAVMUAJZEKVNVFV
|
|
GVOEABUBUCVFAVMVCVDAPDVEAQUDZVFBVJVDVCBPDVEBQRZVFVJVFVIVCVDVEBEUKVICDZVFBVC
|
|
VDVEUEZUFABUGUHZUIUJZULUMSVKVIVJKVFVNVIVJUNVFAVJBVPWAVQVFBVDVCBUODVEBUPRVSU
|
|
QURUSSVFVRVKVLUTVTVIVAVBS $.
|
|
|
|
$( The product of an integer and a positive rational number is 0 modulo the
|
|
positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
mulqmod0 $p |- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) ->
|
|
( ( A x. M ) mod M ) = 0 ) $=
|
|
( cz wcel cq cc0 clt wbr w3a cmul co cmo wceq cdiv simp1 cc qcn 3ad2ant2 cr
|
|
zcnd qre simp3 gt0ap0d divcanap4d eqeltrd wb syl simp2 qmulcl syl2anc modq0
|
|
zq syld3an1 mpbird ) ACDZBEDZFBGHZIZABJKZBLKFMZUSBNKZCDZURVAACURABURAUOUPUQ
|
|
OZTUPUOBPDUQBQRURBUPUOBSDUQBUARUOUPUQUBUCUDVCUEUSEDZUPUOUQUTVBUFURAEDZUPVDU
|
|
RUOVEVCAULUGUOUPUQUHABUIUJUSBUKUMUN $.
|
|
|
|
$( ` A ` is divisible by ` B ` iff its negative is. (Contributed by Jim
|
|
Kingdon, 18-Oct-2021.) $)
|
|
negqmod0 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( ( A mod B ) = 0 <-> ( -u A mod B ) = 0 ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cdiv co cz cneg cmo wceq cc qcn 3ad2ant1 3ad2ant2
|
|
wb modq0 cr qre simp3 gt0ap0d divclapd znegclb syl divnegapd eleq1d syl3an1
|
|
bitrd qnegcl 3bitr4d ) ACDZBCDZEBFGZHZABIJZKDZALZBIJZKDZABMJENUTBMJENZUQUSU
|
|
RLZKDZVBUQURODUSVESUQABUNUOAODUPAPQZUOUNBODUPBPRZUQBUOUNBUADUPBUBRUNUOUPUCU
|
|
DZUEURUFUGUQVDVAKUQABVFVGVHUHUIUKABTUNUTCDUOUPVCVBSAULUTBTUJUM $.
|
|
|
|
$( The modulo operation is nonnegative. (Contributed by Jim Kingdon,
|
|
18-Oct-2021.) $)
|
|
modqge0 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 <_ ( A mod B ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cdiv co cfl cfv cmul cmin cmo cle wne qre mpbird
|
|
cr simp3 gt0ne0d qdivcl syld3an3 flqle syl wb flqcld zred 3ad2ant1 3ad2ant2
|
|
lemuldiv2 syl112anc remulcld subge0d modqval breqtrrd ) ACDZBCDZEBFGZHZEABA
|
|
BIJZKLZMJZNJZABOJPVAEVEPGVDAPGZVAVFVCVBPGZVAVBCDZVGURUSUTBEQVHVABURUSUTUAZU
|
|
BABUCUDZVBUEUFVAVCTDATDZBTDZUTVFVGUGVAVCVAVBVJUHUIZURUSVKUTARUJZUSURVLUTBRU
|
|
KZVIVCABULUMSVAAVDVNVABVCVOVMUNUOSABUPUQ $.
|
|
|
|
$( The modulo operation is less than its second argument. (Contributed by
|
|
Jim Kingdon, 18-Oct-2021.) $)
|
|
modqlt $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cdiv cmin cmul cc qcn 3ad2ant2 cr qre syl
|
|
c1 cfl cfv 3ad2ant1 simp3 gt0ap0d divcanap2d oveq1d gt0ne0d qdivcl syld3an3
|
|
wne flqcld zcnd subdid modqval 3eqtr4rd qfraclt1 dividapd breqtrrd resubcld
|
|
wb zred ltmuldiv2 syl112anc mpbird eqbrtrd ) ACDZBCDZEBFGZHZABIJZBABKJZVLUA
|
|
UBZLJZMJZBFVJBVLMJZBVMMJZLJAVQLJVOVKVJVPAVQLVJABVGVHANDVIAOUCVHVGBNDVIBOPZV
|
|
JBVHVGBQDZVIBRPZVGVHVIUDZUEZUFUGVJBVLVMVRVJVLCDZVLNDVGVHVIBEUKWCVJBWAUHABUI
|
|
UJZVLOSVJVMVJVLWDULZUMUNABUOUPVJVOBFGZVNBBKJZFGZVJVNTWGFVJWCVNTFGWDVLUQSVJB
|
|
VRWBURUSVJVNQDVSVSVIWFWHVAVJVLVMVJWCVLQDWDVLRSVJVMWEVBUTVTVTWAVNBBVCVDVEVF
|
|
$.
|
|
|
|
$( Modular reduction produces a half-open interval. (Contributed by Jim
|
|
Kingdon, 18-Oct-2021.) $)
|
|
modqelico $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( A mod B ) e. ( 0 [,) B ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cico cle modqcl qre syl modqge0 modqlt cxr
|
|
cr wb 0re rexrd 3ad2ant2 elico2 sylancr mpbir3and ) ACDZBCDZEBFGZHZABIJZEBK
|
|
JDZUKSDZEUKLGZUKBFGZUJUKCDUMABMUKNOABPABQUJESDBRDZULUMUNUOHTUAUHUGUPUIUHBBN
|
|
UBUCEBUKUDUEUF $.
|
|
|
|
$( The modulo operation differs from ` A ` by an integer multiple of ` B ` .
|
|
(Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
modqdiffl $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cmin cdiv cfl cfv cmul modqval cc 3ad2ant2
|
|
qcn eqtrd oveq2d 3ad2ant1 gt0ne0d qdivcl syld3an3 flqcld zcnd mulcld nncand
|
|
wne simp3 oveq1d cr qre gt0ap0d divcanap3d ) ACDZBCDZEBFGZHZAABIJZKJZBLJBAB
|
|
LJZMNZOJZBLJVDUTVBVEBLUTVBAAVEKJZKJVEUTVAVFAKABPUAUTAVEUQURAQDUSASUBUTBVDUR
|
|
UQBQDUSBSRZUTVDUTVCUQURUSBEUJVCCDUTBUQURUSUKZUCABUDUEUFUGZUHUITULUTVDBVIVGU
|
|
TBURUQBUMDUSBUNRVHUOUPT $.
|
|
|
|
$( The modulo operation differs from ` A ` by an integer multiple of ` B ` .
|
|
(Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
modqdifz $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( ( A - ( A mod B ) ) / B ) e. ZZ ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co cmin cdiv cfl cfv cz modqdiffl wne gt0ne0d
|
|
simp3 qdivcl syld3an3 flqcld eqeltrd ) ACDZBCDZEBFGZHZAABIJKJBLJABLJZMNOABP
|
|
UGUHUDUEUFBEQUHCDUGBUDUEUFSRABTUAUBUC $.
|
|
|
|
$( The fractional part of a number is the number modulo 1. (Contributed by
|
|
Jim Kingdon, 18-Oct-2021.) $)
|
|
modqfrac $p |- ( A e. QQ -> ( A mod 1 ) = ( A - ( |_ ` A ) ) ) $=
|
|
( cq wcel c1 cmo co cdiv cfl cfv cmul cmin cc0 clt wbr wceq cz 1z zq oveq2d
|
|
eqtrd ax-mp 0lt1 modqval mp3an23 qcn div1d fveq2d flqcl zcnd mulid2d ) ABCZ
|
|
ADEFZADADGFZHIZJFZKFZAAHIZKFUKDBCZLDMNULUPODPCURQDRUAUBADUCUDUKUOUQAKUKUODU
|
|
QJFUQUKUNUQDJUKUMAHUKAAUEUFUGSUKUQUKUQAUHUIUJTST $.
|
|
|
|
$( The floor function expressed in terms of the modulo operation.
|
|
(Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
flqmod $p |- ( A e. QQ -> ( |_ ` A ) = ( A - ( A mod 1 ) ) ) $=
|
|
( cq wcel c1 cmo cmin cfl cfv modqfrac oveq2d qcn flqcl zcnd nncand eqtr2d
|
|
co ) ABCZAADEPZFPAAAGHZFPZFPSQRTAFAIJQASAKQSALMNO $.
|
|
|
|
$( Break a number into its integer part and its fractional part.
|
|
(Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
intqfrac $p |- ( A e. QQ -> A = ( ( |_ ` A ) + ( A mod 1 ) ) ) $=
|
|
( cq wcel cfl cfv c1 cmo co caddc cmin modqfrac oveq2d flqcl pncan3d eqtr2d
|
|
zcnd qcn ) ABCZADEZAFGHZIHSASJHZIHARTUASIAKLRSARSAMPAQNO $.
|
|
|
|
$( An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
zmod10 $p |- ( N e. ZZ -> ( N mod 1 ) = 0 ) $=
|
|
( cz wcel c1 cmo co cfl cfv cmin cc0 cq wceq zq modqfrac flid oveq2d subidd
|
|
syl zcn 3eqtrd ) ABCZADEFZAAGHZIFZAAIFJUAAKCUBUDLAMANRUAUCAAIAOPUAAASQT $.
|
|
|
|
$( Two arbitrary integers are congruent modulo 1, see example 4 in
|
|
[ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) $)
|
|
zmod1congr $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A mod 1 ) = ( B mod 1 ) ) $=
|
|
( cz wcel wa c1 cmo co cc0 wceq zmod10 adantr adantl eqtr4d ) ACDZBCDZEAFGH
|
|
ZIBFGHZOQIJPAKLPRIJOBKMN $.
|
|
|
|
$( Move a positive integer in and out of a floor in the first argument of a
|
|
modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.) $)
|
|
modqmulnn $p |- ( ( N e. NN /\ A e. QQ /\ M e. NN ) -> ( ( N x. ( |_ ` A ) )
|
|
mod ( N x. M ) ) <_ ( ( |_ ` ( N x. A ) ) mod ( N x. M ) ) ) $=
|
|
( cn wcel cq cfl cfv cmul co cdiv cmin cz syl syl2anc cc0 wbr syl3anc wceq
|
|
zq w3a cmo cle cr nnq 3ad2ant1 flqcl 3ad2ant2 qmulcl qre simp2 zred nnmulcl
|
|
flqcld wa 3adant2 wne cap simp1 nncnd simp3 nnap0d mulap0d wb ax-mp sylancl
|
|
0z qapne mpbid qdivcl remulcld cn0 nnnn0 flqmulnn0 sylan lesub1dd clt nnred
|
|
nngt0d mulgt0d modqval flqdiv 3adant1 zcnd divcanap5d fveq2d 3eqtr4rd eqtrd
|
|
cc qcn oveq2d 3brtr4d ) CDEZAFEZBDEZUAZCAGHZIJZCBIJZWRWSKJZGHZIJZLJZCAIJZGH
|
|
ZXBLJZWRWSUBJZXEWSUBJZUCWPWRXEXBWPWRFEZWRUDEWPCFEZWQFEZXIWMWNXJWOCUEUFZWNWM
|
|
XKWOWNWQMEXKAUGZWQTNUHCWQUIOZWRUJNWPXEWPXDWPXJWNXDFEZXLWMWNWOUKZCAUIOZUNZUL
|
|
WPWSXAWPWSFEZWSUDEWMWOXSWNWMWOUOWSDEZXSCBUMZWSUENUPZWSUJNWPXAWPWTWPXIXSWSPU
|
|
QZWTFEXNYBWPWSPURQZYCWPCBWPCWMWNWOUSZUTZWPBWMWNWOVAZUTZWPCYEVBZWPBYGVBZVCWP
|
|
XSPFEZYDYCVDYBPMEYKVGPTVEWSPVHVFVIWRWSVJRUNULVKWPWMWNWRXEUCQZYEXPWMCVLEWNYL
|
|
CVMACVNVOOVPWPXIXSPWSVQQZXGXCSXNYBWPCBWPCYEVRWPBYGVRWPCYEVSWPBYGVSVTZWRWSWA
|
|
RWPXHXEWSXEWSKJGHZIJZLJZXFWPXEFEZXSYMXHYQSWPXEMEYRXRXETNYBYNXEWSWARWPYPXBXE
|
|
LWPYOXAWSIWPYOXDWSKJZGHZXAWPXOXTYOYTSXQWMWOXTWNYAUPXDWSWBOWPWQBKJZGHZABKJZG
|
|
HZXAYTWNWOUUBUUDSWMABWBWCWPWTUUAGWPWQBCWPWNWQWIEXPWNWQXMWDNYHYFYJYIWEWFWPYS
|
|
UUCGWPABCWPWNAWIEXPAWJNYHYFYJYIWEWFWGWHWKWKWHWL $.
|
|
|
|
$( The value of the modulo operation (expressed with sum of denominator and
|
|
nominator). (Contributed by Jim Kingdon, 20-Oct-2021.) $)
|
|
modqvalp1 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B )
|
|
-> ( ( A + B ) - ( ( ( |_ ` ( A / B ) ) + 1 ) x. B ) ) = ( A mod B ) ) $=
|
|
( cq wcel cc0 clt wbr w3a caddc co cdiv cfl cfv cmul c1 cmo cc qcn 3ad2ant1
|
|
cmin wne simp3 gt0ne0d qdivcl syld3an3 flqcld zcnd 3ad2ant2 mulcld pnpcan2d
|
|
adddirp1d oveq2d modqvalr 3eqtr4d ) ACDZBCDZEBFGZHZABIJZABKJZLMZBNJZBIJZTJA
|
|
VBTJUSVAOIJBNJZTJABPJURAVBBUOUPAQDUQARSURVABURVAURUTUOUPUQBEUAUTCDURBUOUPUQ
|
|
UBUCABUDUEUFUGZUPUOBQDUQBRUHZUIVFUJURVDVCUSTURVABVEVFUKULABUMUN $.
|
|
|
|
$( Closure law for the modulo operation restricted to integers. (Contributed
|
|
by NM, 27-Nov-2008.) $)
|
|
zmodcl $p |- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. NN0 ) $=
|
|
( cz wcel cn wa cmo co cc0 cle wbr cn0 cdiv cfl cmul cmin cq adantl syl3anc
|
|
cfv clt wceq adantr nnq nngt0 modqval nnz znq flqcld zmulcld zsubcl eqeltrd
|
|
zq syldan modqge0 elnn0z sylanbrc ) ACDZBEDZFZABGHZCDIVAJKZVALDUTVAABABMHZN
|
|
TZOHZPHZCUTAQDZBQDZIBUAKZVAVFUBURVGUSAUMUCZUSVHURBUDRZUSVIURBUERZABUFSURUSV
|
|
ECDVFCDUTBVDUSBCDURBUGRUTVCABUHUIUJAVEUKUNULUTVGVHVIVBVJVKVLABUOSVAUPUQ $.
|
|
|
|
${
|
|
zmodcld.1 $e |- ( ph -> A e. ZZ ) $.
|
|
zmodcld.2 $e |- ( ph -> B e. NN ) $.
|
|
$( Closure law for the modulo operation restricted to integers.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
zmodcld $p |- ( ph -> ( A mod B ) e. NN0 ) $=
|
|
( cz wcel cn cmo co cn0 zmodcl syl2anc ) ABFGCHGBCIJKGDEBCLM $.
|
|
$}
|
|
|
|
$( An integer mod ` B ` lies in the first ` B ` nonnegative integers.
|
|
(Contributed by Jeff Madsen, 17-Jun-2010.) $)
|
|
zmodfz $p |- ( ( A e. ZZ /\ B e. NN ) ->
|
|
( A mod B ) e. ( 0 ... ( B - 1 ) ) ) $=
|
|
( cz wcel cn wa cmo co cc0 cmin cfz cle wbr clt zmodcl nn0zd nn0ge0d adantl
|
|
c1 cq zq adantr nnq nngt0 modqlt syl3anc w3a nnz elfzm11 sylancr mpbir3and
|
|
wb 0z ) ACDZBEDZFZABGHZIBSJHKHDZUQCDZIUQLMZUQBNMZUPUQABOZPUPUQVBQUPATDZBTDZ
|
|
IBNMZVAUNVCUOAUAUBUOVDUNBUCRUOVEUNBUDRABUEUFUPICDBCDZURUSUTVAUGULUMUOVFUNBU
|
|
HRUQIBUIUJUK $.
|
|
|
|
$( An integer mod ` B ` lies in the first ` B ` nonnegative integers.
|
|
(Contributed by Stefan O'Rear, 6-Sep-2015.) $)
|
|
zmodfzo $p |- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ..^ B ) ) $=
|
|
( cz wcel cn wa cmo co cc0 c1 cmin cfz cfzo zmodfz wceq nnz fzoval eleqtrrd
|
|
syl adantl ) ACDZBEDZFABGHIBJKHLHZIBMHZABNUBUDUCOZUAUBBCDUEBPIBQSTR $.
|
|
|
|
$( An integer mod ` B ` lies in the first ` B + 1 ` nonnegative integers.
|
|
(Contributed by AV, 27-Oct-2018.) $)
|
|
zmodfzp1 $p |- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ... B ) ) $=
|
|
( cz wcel cn wa cc0 cfzo co cfz cmo fzossfz zmodfzo sseldi ) ACDBEDFGBHIGBJ
|
|
IABKIGBLABMN $.
|
|
|
|
$( Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) $)
|
|
modqid $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) )
|
|
-> ( A mod B ) = A ) $=
|
|
( cq wcel wa cc0 cle wbr clt co cfl cfv cmul cmin wceq cr qre syl3anc eqtrd
|
|
c1 cmo cdiv simpll simplr 0red ad2antrr simprl simprr lelttrd modqval caddc
|
|
ad2antlr cc wne gt0ne0d qdivcl qcn addid2 fveq2d 3syl divge0 syl22anc recnd
|
|
mulid1d breqtrrd wb 1red ltdivmul syl112anc mpbird flqbi2 sylancr mpbir2and
|
|
cz 0z eqtr3d oveq2d mul01d subid1d ) ACDZBCDZEZFAGHZABIHZEZEZABUAJZABABUBJZ
|
|
KLZMJZNJZAWFVTWAFBIHZWGWKOVTWAWEUCZVTWAWEUDZWFFABWFUEVTAPDZWAWEAQUFZWABPDZV
|
|
TWEBQULZWBWCWDUGZWBWCWDUHZUIZABUJRWFWKAFNJAWFWJFANWFWJBFMJFWFWIFBMWFFWHUKJZ
|
|
KLZWIFWFWHCDZWHUMDZXCWIOWFVTWABFUNXDWMWNWFBXAUOABUPRZWHUQXEXBWHKWHURUSUTWFX
|
|
CFOZFWHGHZWHTIHZWFWOWCWQWLXHWPWSWRXAABVAVBWFXIABTMJZIHZWFABXJIWTWFBWFBWRVCZ
|
|
VDVEWFWOTPDWQWLXIXKVFWPWFVGWRXAATBVHVIVJWFFVNDXDXGXHXIEVFVOXFWHFVKVLVMVPVQW
|
|
FBXLVRSVQWFAWFAWPVCVSSS $.
|
|
|
|
$( A positive real number modulo itself is 0. (Contributed by Jim Kingdon,
|
|
21-Oct-2021.) $)
|
|
modqid0 $p |- ( ( N e. QQ /\ 0 < N ) -> ( N mod N ) = 0 ) $=
|
|
( cq wcel cc0 clt wbr wa cmo co wceq cdiv cz c1 cr qre adantr recnd gt0ap0d
|
|
simpr dividapd 1z syl6eqel wb modq0 3anidm12 mpbird ) ABCZDAEFZGZAAHIDJZAAK
|
|
IZLCZUIUKMLUIAUIAUGANCUHAOPZQUIAUMUGUHSRTUAUBUGUHUJULUCAAUDUEUF $.
|
|
|
|
$( Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) $)
|
|
modqid2 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) ->
|
|
( ( A mod B ) = A <-> ( 0 <_ A /\ A < B ) ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co wceq cle wa modqge0 modqlt jca breq2 breq1
|
|
anbi12d syl5ibcom wi modqid ex 3adant3 impbid ) ACDZBCDZEBFGZHZABIJZAKZEALG
|
|
ZABFGZMZUIEUJLGZUJBFGZMUKUNUIUOUPABNABOPUKUOULUPUMUJAELQUJABFRSTUFUGUNUKUAU
|
|
HUFUGMUNUKABUBUCUDUE $.
|
|
|
|
$( Identity law for modulo restricted to integers. (Contributed by Paul
|
|
Chapman, 22-Jun-2011.) $)
|
|
zmodid2 $p |- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) = M <->
|
|
M e. ( 0 ... ( N - 1 ) ) ) ) $=
|
|
( cz wcel cn wa cmo co wceq cc0 cle wbr clt c1 cmin cfz cq wb zq adantl nnq
|
|
adantr nngt0 modqid2 syl3anc nnz w3a 0z elfzm11 mpan 3anass syl6bb syl ibar
|
|
bicomd sylan9bbr bitr4d ) ACDZBEDZFZABGHAIZJAKLZABMLZFZAJBNOHPHDZUTAQDZBQDZ
|
|
JBMLZVAVDRURVFUSASUBUSVGURBUATUSVHURBUCTABUDUEUSVEURVDFZURVDUSBCDZVEVIRBUFV
|
|
JVEURVBVCUGZVIJCDVJVEVKRUHAJBUIUJURVBVCUKULUMURVDVIURVDUNUOUPUQ $.
|
|
|
|
$( Identity law for modulo restricted to integers. (Contributed by AV,
|
|
27-Oct-2018.) $)
|
|
zmodidfzo $p |- ( ( M e. ZZ /\ N e. NN )
|
|
-> ( ( M mod N ) = M <-> M e. ( 0 ..^ N ) ) ) $=
|
|
( cz wcel cn wa cmo co wceq cc0 cmin cfz cfzo zmodid2 nnz fzoval syl adantl
|
|
c1 eqcomd eleq2d bitrd ) ACDZBEDZFZABGHAIAJBSKHLHZDAJBMHZDABNUEUFUGAUEUGUFU
|
|
DUGUFIZUCUDBCDUHBOJBPQRTUAUB $.
|
|
|
|
$( Identity law for modulo restricted to integers. (Contributed by AV,
|
|
27-Oct-2018.) $)
|
|
zmodidfzoimp $p |- ( M e. ( 0 ..^ N ) -> ( M mod N ) = M ) $=
|
|
( cz wcel cn wa cc0 cfzo co cmo wceq cn0 clt wbr elfzo0 nn0z anim1i 3adant3
|
|
w3a sylbi zmodidfzo biimprd mpcom ) ACDZBEDZFZAGBHIDZABJIAKZUGALDZUEABMNZSU
|
|
FABOUIUEUFUJUIUDUEAPQRTUFUHUGABUAUBUC $.
|
|
|
|
$( Special case: 0 modulo a positive real number is 0. (Contributed by Jim
|
|
Kingdon, 21-Oct-2021.) $)
|
|
q0mod $p |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 ) $=
|
|
( cq wcel cc0 clt wbr wa cle cmo co wceq cz 0z zq mp1i simpl 0le0 a1i simpr
|
|
modqid syl22anc ) ABCZDAEFZGZDBCZUBDDHFZUCDAIJDKDLCUEUDMDNOUBUCPUFUDQRUBUCS
|
|
DATUA $.
|
|
|
|
$( Special case: 1 modulo a real number greater than 1 is 1. (Contributed by
|
|
Jim Kingdon, 21-Oct-2021.) $)
|
|
q1mod $p |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 ) $=
|
|
( cq wcel c1 clt wbr wa cc0 cle cmo co wceq cz 1z mp1i simpl 0le1 a1i simpr
|
|
zq modqid syl22anc ) ABCZDAEFZGZDBCZUCHDIFZUDDAJKDLDMCUFUENDTOUCUDPUGUEQRUC
|
|
UDSDAUAUB $.
|
|
|
|
${
|
|
modqabs.a $e |- ( ph -> A e. QQ ) $.
|
|
modqabs.b $e |- ( ph -> B e. QQ ) $.
|
|
modqabs.bgt0 $e |- ( ph -> 0 < B ) $.
|
|
modqabs.c $e |- ( ph -> C e. QQ ) $.
|
|
modqabs.bc $e |- ( ph -> B <_ C ) $.
|
|
$( Absorption law for modulo. (Contributed by Jim Kingdon,
|
|
21-Oct-2021.) $)
|
|
modqabs $p |- ( ph -> ( ( A mod B ) mod C ) = ( A mod B ) ) $=
|
|
( cmo co cq wcel cc0 wbr clt syl3anc cr qre syl cle modqcld modqlt modqid
|
|
wceq modqge0 ltletrd syl22anc ) ABCJKZLMZDLMZNUIUAOZUIDPOUIDJKUIUEABCEFGU
|
|
BZHABLMZCLMZNCPOZULEFGBCUFQAUICDAUJUIRMUMUISTAUOCRMFCSTAUKDRMHDSTAUNUOUPU
|
|
ICPOEFGBCUCQIUGUIDUDUH $.
|
|
$}
|
|
|
|
$( Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) $)
|
|
modqabs2 $p |- ( ( A e. QQ /\ B e. QQ /\ 0 < B )
|
|
-> ( ( A mod B ) mod B ) = ( A mod B ) ) $=
|
|
( cq wcel cc0 clt wbr w3a simp1 simp2 simp3 cr qre syl leidd modqabs ) ACDZ
|
|
BCDZEBFGZHZABBQRSIQRSJZQRSKUATBTRBLDUABMNOP $.
|
|
|
|
$( The modulo operation is periodic. (Contributed by Jim Kingdon,
|
|
21-Oct-2021.) $)
|
|
modqcyc $p |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) )
|
|
-> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) ) $=
|
|
( cq wcel wa cmul co caddc cdiv cfl cfv cmin wceq syl2anc syl3anc cc oveq2d
|
|
qcn syl cc0 clt wbr cmo simpll ad2antlr simprl qmulcl qaddcl simprr modqval
|
|
cz zq qre gt0ap0d divdirapd simplr zcnd divcanap4d eqtrd fveq2d wne gt0ne0d
|
|
cr qdivcl flqaddz flqcld adddid mulcomd 3eqtrd mulcld pnpcan2d eqtr4d ) ADE
|
|
ZCULEZFZBDEZUABUBUCZFZFZACBGHZIHZBUDHZABABJHZKLZGHZMHZABUDHZVTWCWBBWBBJHZKL
|
|
ZGHZMHZWBWFWAIHZMHWGVTWBDEZVQVRWCWLNVTVNWADEZWNVNVOVSUEZVTCDEZVQWOVOWQVNVSC
|
|
UMUFVPVQVRUGZCBUHOZAWAUIOWRVPVQVRUJZWBBUKPVTWKWMWBMVTWKBWECIHZGHWFBCGHZIHWM
|
|
VTWJXABGVTWJWDCIHZKLZXAVTWIXCKVTWIWDWABJHZIHXCVTAWABVTVNAQEWPASTZVTWOWAQEWS
|
|
WASTZVTVQBQEWRBSTZVTBVTVQBVDEWRBUNTWTUOZUPVTXECWDIVTCBVTCVNVOVSUQZURZXHXIUS
|
|
RUTVAVTWDDEZVOXDXANVTVNVQBUAVBXLWPWRVTBWTVCABVEPZXJWDCVFOUTRVTBWECXHVTWEVTW
|
|
DXMVGURZXKVHVTXBWAWFIVTBCXHXKVIRVJRVTAWFWAXFVTBWEXHXNVKXGVLVJVTVNVQVRWHWGNW
|
|
PWRWTABUKPVM $.
|
|
|
|
$( The modulo operation is periodic. (Contributed by Jim Kingdon,
|
|
21-Oct-2021.) $)
|
|
modqcyc2 $p |- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) )
|
|
-> ( ( A - ( B x. N ) ) mod B ) = ( A mod B ) ) $=
|
|
( cq wcel cz wa cc0 clt wbr cmul co cmin cmo cneg caddc simplr cc qcn wceq
|
|
zcnd ad2antrl mulneg1d mulcom negeqd syl2anc eqtr4d oveq2d ad2antrr negsubd
|
|
mulcld eqtr2d oveq1d znegcl modqcyc sylanl2 eqtrd ) ADEZCFEZGZBDEZHBIJZGZGZ
|
|
ABCKLZMLZBNLACOZBKLZPLZBNLZABNLZVDVFVIBNVDVIAVEOZPLVFVDVHVLAPVDVHCBKLZOZVLV
|
|
DCBVDCURUSVCQUAZVABREZUTVBBSUBZUCVDVPCREZVLVNTVQVOVPVRGVEVMBCUDUEUFUGUHVDAV
|
|
EURAREUSVCASUIVDBCVQVOUKUJULUMUSURVGFEVCVJVKTCUNABVGUOUPUQ $.
|
|
|
|
${
|
|
modqadd1.a $e |- ( ph -> A e. QQ ) $.
|
|
modqadd1.b $e |- ( ph -> B e. QQ ) $.
|
|
modqadd1.c $e |- ( ph -> C e. QQ ) $.
|
|
modqadd1.dq $e |- ( ph -> D e. QQ ) $.
|
|
modqadd1.dgt0 $e |- ( ph -> 0 < D ) $.
|
|
modqadd1.ab $e |- ( ph -> ( A mod D ) = ( B mod D ) ) $.
|
|
$( Addition property of the modulo operation. (Contributed by Jim Kingdon,
|
|
22-Oct-2021.) $)
|
|
modqadd1 $p |- ( ph -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) $=
|
|
( cmo co wceq caddc cmin cq wcel syl3anc cc cdiv cfl cfv cmul cc0 clt wbr
|
|
modqval eqeq12d oveq1 syl6bi qcn syl gt0ne0d qdivcl flqcld mulcld addsubd
|
|
wne zcnd sylibrd cz qaddcl syl2anc modqcyc2 syl22anc syl5ib syld mpd ) AB
|
|
ELMZCELMZNZBDOMZELMZCDOMZELMZNZKAVLVMEBEUAMZUBUCZUDMZPMZVOECEUAMZUBUCZUDM
|
|
ZPMZNZVQAVLBVTPMZDOMZCWDPMZDOMZNZWFAVLWGWINWKAVJWGVKWIABQRZEQRZUEEUFUGZVJ
|
|
WGNFIJBEUHSACQRZWMWNVKWINGIJCEUHSUIWGWIDOUJUKAWAWHWEWJABDVTAWLBTRFBULUMAD
|
|
QRZDTRHDULUMZAEVSAWMETRIEULUMZAVSAVRAWLWMEUEUSZVRQRFIAEJUNZBEUOSUPZUTUQUR
|
|
ACDWDAWOCTRGCULUMWQAEWCWRAWCAWBAWOWMWSWBQRGIWTCEUOSUPZUTUQURUIVAWFWAELMZW
|
|
EELMZNAVQWAWEELUJAXCVNXDVPAVMQRZVSVBRWMWNXCVNNAWLWPXEFHBDVCVDXAIJVMEVSVEV
|
|
FAVOQRZWCVBRWMWNXDVPNAWOWPXFGHCDVCVDXBIJVOEWCVEVFUIVGVHVI $.
|
|
$}
|
|
|
|
$( Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.) $)
|
|
modqaddabs $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) ->
|
|
( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) ) $=
|
|
( cq wa cc0 cmo co caddc cc modqcld qcn syl addcomd oveq1d modqabs2 syl3anc
|
|
wcel wceq modqadd1 clt wbr simpll simprl simprr simplr eqtr4d 3eqtrd ) ADRZ
|
|
BDRZEZCDRZFCUAUBZEZEZACGHZBCGHZIHZCGHUQUPIHZCGHZUPBIHZCGHZABIHCGHUOURUSCGUO
|
|
UPUQUOUPDRUPJRUOACUIUJUNUCZUKULUMUDZUKULUMUEZKZUPLMZUOUQDRUQJRUOBCUIUJUNUFZ
|
|
VDVEKZUQLMNOUOUTBUPIHZCGHVBUOUQBUPCVIVHVFVDVEUOUJULUMUQCGHUQSVHVDVEBCPQTUOV
|
|
AVJCGUOUPBVGUOUJBJRVHBLMNOUGUOUPABCVFVCVHVDVEUOUIULUMUPCGHUPSVCVDVEACPQTUH
|
|
$.
|
|
|
|
$( The sum of a number modulo a modulus and another number equals the sum of
|
|
the two numbers modulo the same modulus. (Contributed by Jim Kingdon,
|
|
23-Oct-2021.) $)
|
|
modqaddmod $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( ( A mod M ) + B ) mod M ) = ( ( A + B ) mod M ) ) $=
|
|
( cq wcel cc0 clt wbr cmo simpll simprl simprr modqcld simplr wceq modqabs2
|
|
wa co syl3anc modqadd1 ) ADEZBDEZQZCDEZFCGHZQZQZACIRZABCUGACUAUBUFJZUCUDUEK
|
|
ZUCUDUELZMUIUAUBUFNUJUKUGUAUDUEUHCIRUHOUIUJUKACPST $.
|
|
|
|
$( The sum of a positive rational number less than an upper bound and the
|
|
product of an integer and the upper bound is the positive rational number
|
|
modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.) $)
|
|
mulqaddmodid $p |- ( ( ( N e. ZZ /\ M e. QQ ) /\
|
|
( A e. QQ /\ A e. ( 0 [,) M ) ) ) ->
|
|
( ( ( N x. M ) + A ) mod M ) = A ) $=
|
|
( cz wcel cq wa cc0 cico co cmul caddc cmo simpll cr recnd wbr clt syl22anc
|
|
wceq zcnd qre ad2antlr mulcld cle w3a simprr cxr wb 0red rexrd elico2 mpbid
|
|
syl2anc simp1d addcomd oveq1d simprl simplr simp2d simp3d lelttrd modqcyc
|
|
modqid 3eqtrd ) CDEZBFEZGZAFEZAHBIJEZGZGZCBKJZALJZBMJAVMLJZBMJZABMJZAVLVNVO
|
|
BMVLVMAVLCBVLCVFVGVKNZUAVLBVGBOEVFVKBUBUCZPUDVLAVLAOEZHAUEQZABRQZVLVJVTWAWB
|
|
UFZVHVIVJUGVLHOEBUHEVJWCUIVLUJZVLBVSUKHBAULUNUMZUOZPUPUQVLVIVFVGHBRQVPVQTVH
|
|
VIVJURZVRVFVGVKUSZVLHABWDWFVSVLVTWAWBWEUTZVLVTWAWBWEVAZVBABCVCSVLVIVGWAWBVQ
|
|
ATWGWHWIWJABVDSVE $.
|
|
|
|
$( The product of an integer and an integer greater than 1 increased by 1 is
|
|
1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.) $)
|
|
mulp1mod1 $p |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) )
|
|
-> ( ( ( N x. A ) + 1 ) mod N ) = 1 ) $=
|
|
( cz wcel c2 cmul co cmo c1 caddc cc0 adantl oveq1d cq clt wbr wceq syl a1i
|
|
zq cuz cfv wa cc eluzelcn simpl zcnd mulcomd eluzelz 0red 2re zred 2pos cle
|
|
cr eluzle ltletrd mulqmod0 syl3anc eqtrd 0p1e1 syl6eq syl2anc 1z modqaddmod
|
|
qmulcl mp1i syl22anc eluz2gt1 q1mod 3eqtr3d ) ACDZBEUAUBDZUCZBAFGZBHGZIJGZB
|
|
HGZIBHGZVOIJGBHGZIVNVQIBHVNVQKIJGIVNVPKIJVNVPABFGZBHGZKVNVOWABHVNBAVMBUDDVL
|
|
EBUELVNAVLVMUFZUGUHMVNVLBNDZKBOPZWBKQWCVMWDVLVMBCDZWDEBUIZBTRLZVNKEBVNUJEUO
|
|
DVNUKSVNBVMWFVLWGLULKEOPVNUMSVMEBUNPVLEBUPLUQZABURUSUTMVAVBMVNVONDZINDZWDWE
|
|
VRVTQVNWDANDZWJWHVNVLWLWCATRBAVFVCICDWKVNVDITVGWHWIVOIBVEVHVNWDIBOPZVSIQWHV
|
|
MWMVLBVILBVJVCVK $.
|
|
|
|
${
|
|
$d A k $. $d B k $. $d M k $. $d ph k $.
|
|
modqmuladd.a $e |- ( ph -> A e. ZZ ) $.
|
|
modqmuladd.bq $e |- ( ph -> B e. QQ ) $.
|
|
modqmuladd.b $e |- ( ph -> B e. ( 0 [,) M ) ) $.
|
|
modqmuladd.m $e |- ( ph -> M e. QQ ) $.
|
|
modqmuladd.mgt0 $e |- ( ph -> 0 < M ) $.
|
|
$( Decomposition of an integer into a multiple of a modulus and a
|
|
remainder. (Contributed by Jim Kingdon, 23-Oct-2021.) $)
|
|
modqmuladd $p |- ( ph ->
|
|
( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) ) $=
|
|
( cmo co wceq cmul caddc cz cq wcel cc0 ad2antrr cv wrex cdiv cfl cfv wne
|
|
zq syl gt0ne0d qdivcl syl3anc flqcld oveq1 oveq1d eqeq2d adantl flqpmodeq
|
|
wb clt wbr eqcomd rspcedvd oveq2 eqcoms rexbidv syl5ibrcom wa cico simplr
|
|
mulqaddmodid syl22anc eqtrd ex rexlimdva impbid ) ABEKLZCMZBDUAZENLZCOLZM
|
|
ZDPUBZAWBVQBVSVPOLZMZDPUBAWDBBEUCLZUDUEZENLZVPOLZMZDWFPAWEABQRZEQRZESUFWE
|
|
QRABPRWJFBUGUHZIAEJUIBEUJUKULVRWFMZWDWIURAWMWCWHBWMVSWGVPOVRWFENUMUNUOUPA
|
|
WHBAWJWKSEUSUTWHBMWLIJBEUQUKVAVBVQWAWDDPWAWDURCVPCVPMVTWCBCVPVSOVCUOVDVEV
|
|
FAWAVQDPAVRPRZVGZWAVQWOWAVGZVPVTEKLZCWAVPWQMWOBVTEKUMUPWPWNWKCQRZCSEVHLRZ
|
|
WQCMAWNWAVIAWKWNWAITAWRWNWAGTAWSWNWAHTCEVRVJVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d A k $. $d B k $. $d M k $.
|
|
$( Implication of a decomposition of an integer into a multiple of a
|
|
modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.) $)
|
|
modqmuladdim $p |- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B
|
|
-> E. k e. ZZ A = ( ( k x. M ) + B ) ) ) $=
|
|
( cz wcel cq cc0 clt wbr w3a cmo co wceq cv syl cr cle qre syl3anc simpl1
|
|
cmul caddc wrex simpr simpl2 simpl3 modqcld eqeltrrd cico modqge0 breqtrd
|
|
wa zq modqlt eqbrtrrd cxr wb 0re rexr elico2 sylancr mpbir3and modqmuladd
|
|
3syl mpbid ex ) AEFZDGFZHDIJZKZADLMZBNZACODUBMBUCMNCEUDZVKVMUMZVMVNVKVMUE
|
|
ZVOABCDVHVIVJVMUAZVOVLBGVPVOADVOVHAGFZVQAUNPZVHVIVJVMUFZVHVIVJVMUGZUHUIZV
|
|
OBHDUJMFZBQFZHBRJZBDIJZVOBGFWDWBBSPVOHVLBRVOVRVIVJHVLRJVSVTWAADUKTVPULVOV
|
|
LBDIVPVOVRVIVJVLDIJVSVTWAADUOTUPVOHQFDUQFZWCWDWEWFKURUSVOVIDQFWGVTDSDUTVE
|
|
HDBVAVBVCVTWAVDVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d A i k $. $d B i k $. $d M i k $.
|
|
$( Implication of a decomposition of a nonnegative integer into a multiple
|
|
of a modulus and a remainder. (Contributed by Jim Kingdon,
|
|
23-Oct-2021.) $)
|
|
modqmuladdnn0 $p |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) ->
|
|
( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) ) $=
|
|
( vi cn0 wcel cq cc0 wbr co wceq cz cle adantr cdiv ad2antrr syl adantl
|
|
cc clt w3a cmo cv cmul caddc wrex wa simpr cfl cfv cmin nn0cn 3ad2ant1 zq
|
|
nn0z simpl2 simpl3 modqcld qcn wb eleq1 mpbid zcn subadd2d eqcom syl6rbbr
|
|
mulcld subcld qre 3ad2ant2 gt0ap0d divmulap3d oveq2 oveq1d eqcoms syl3an1
|
|
cr modqdiffl eqtrd eqeq1d 3bitr2d wi nn0ge0 simp3 divge0 syl22anc gt0ne0d
|
|
wne simp2 qdivcl syl3anc 0z flqge sylancl breq2 syl5ibcom sylbid sylanbrc
|
|
imp elnn0z oveq1 eqeq2d rspcedvd modqmuladdim r19.29a ex ) AFGZDHGZIDUAJZ
|
|
UBZADUCKZBLZACUDZDUEKZBUFKZLZCFUGZXKXMUHZAEUDZDUEKZBUFKZLZXREMXSXTMGZUHZY
|
|
CUHZXQYCCXTFYFYDIXTNJZXTFGYEYDYCXSYDUIOYEYCYGYEYCADPKZUJUKZXTLZYGYEYCABUL
|
|
KZYALZYKDPKZXTLYJYEYLYBALYCYEABYAXKATGZXMYDXHXIYNXJAUMUNZQXSBTGZYDXSXLTGZ
|
|
YPXSXLHGYQXSADXKAHGZXMXHXIYRXJXHAMGZYRAUPZAUORZUNZOXHXIXJXMUQZXHXIXJXMURZ
|
|
USXLUTRXMYQYPVAXKXLBTVBSVCZOYEXTDYDXTTGXSXTVDSZXSDTGZYDXSXIUUGUUCDUTROZVH
|
|
VEAYBVFVGYEYKXTDXSYKTGYDXSABXKYNXMYOOUUEVIOUUFUUHYEDXKDVRGZXMYDXIXHUUIXJD
|
|
VJVKZQXSXJYDUUDOVLVMYEYMYIXTYEYMAXLULKZDPKZYIXSYMUULLZYDXMUUMXKUUMBXLBXLL
|
|
YKUUKDPBXLAULVNVOVPSOXKUULYILZXMYDXHYRXIXJUUNUUAADVSVQQVTWAWBXKYJYGWCXMYD
|
|
XKIYINJZYJYGXKIYHNJZUUOXKAVRGZIANJZUUIXJUUPXKYRUUQUUBAVJRXHXIUURXJAWDUNUU
|
|
JXHXIXJWEZADWFWGXKYHHGZIMGUUPUUOVAXKYRXIDIWIUUTUUBXHXIXJWJXKDUUSWHADWKWLW
|
|
MYHIWNWOVCYIXTINWPWQQWRWTXTXAWSXNXTLZXQYCVAYFUVAXPYBAUVAXOYABUFXNXTDUEXBV
|
|
OXCSYEYCUIXDXKXMYCEMUGZXHYSXIXJXMUVBWCYTABEDXEVQWTXFXG $.
|
|
$}
|
|
|
|
$( The negation of a number modulo a positive number is equal to the
|
|
difference of the modulus and the number modulo the modulus. (Contributed
|
|
by Jim Kingdon, 24-Oct-2021.) $)
|
|
qnegmod $p |- ( ( A e. QQ /\ N e. QQ /\ 0 < N )
|
|
-> ( -u A mod N ) = ( ( N - A ) mod N ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmin co cmo cneg caddc cmul qcn 3ad2ant2 3ad2ant1
|
|
c1 cc oveq1d negsubd eqcomd mulid2d 1cnd mulcld qnegcl addcomd cz wceq 1zzd
|
|
syl simp2 simp3 modqcyc syl22anc eqtrd 3eqtr2rd ) ACDZBCDZEBFGZHZBAIJZBKJBA
|
|
LZMJZBKJRBNJZVCMJZBKJZVCBKJZVAVBVDBKVAVDVBVABAUSURBSDUTBOPZURUSASDUTAOQUAUB
|
|
TVAVFVDBKVAVEBVCMVABVIUCTTVAVGVCVEMJZBKJZVHVAVFVJBKVAVEVCVARBVAUDVIUEURUSVC
|
|
SDZUTURVCCDZVLAUFZVCOUKQUGTVAVMRUHDUSUTVKVHUIURUSVMUTVNQVAUJURUSUTULURUSUTU
|
|
MVCBRUNUOUPUQ $.
|
|
|
|
$( Minus one modulo a positive integer is equal to the integer minus one.
|
|
(Contributed by AV, 14-Jul-2021.) $)
|
|
m1modnnsub1 $p |- ( M e. NN -> ( -u 1 mod M ) = ( M - 1 ) ) $=
|
|
( cn wcel c1 cneg cmo co cmin cq cc0 clt wbr wceq cz 1z zq mp1i nnq qnegmod
|
|
nngt0 syl3anc cle qsubcl syl2anc nnm1ge0 nnre ltm1d modqid syl22anc eqtrd )
|
|
ABCZDEAFGZADHGZAFGZUMUKDICZAICZJAKLULUNMDNCUOUKODPQZARZATDASUAUKUMICZUPJUMU
|
|
BLUMAKLUNUMMUKUPUOUSURUQADUCUDURAUEUKAAUFUGUMAUHUIUJ $.
|
|
|
|
$( Minus one modulo an integer greater than two is greater than one.
|
|
(Contributed by AV, 14-Jul-2021.) $)
|
|
m1modge3gt1 $p |- ( M e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod M ) ) $=
|
|
( c3 cuz cfv wcel c1 cmin co cneg cmo clt caddc wbr c2 1p1e2 2p1e3 syl5eqbr
|
|
cle eluzle cz wb 2z eluzelz zltp1le sylancr mpbird 1red eluzelre ltaddsub2d
|
|
mpbid cn wceq eluzge3nn m1modnnsub1 syl breqtrrd ) ABCDEZFAFGHZFIAJHZKUQFFL
|
|
HZAKMFURKMUQUTNAKOUQNAKMZNFLHZARMZUQVBBARPBASQUQNTEATEVAVCUAUBBAUCNAUDUEUFQ
|
|
UQFFAUQUGZVDBAUHUIUJUQAUKEUSURULAUMAUNUOUP $.
|
|
|
|
$( The sum of a positive integer and a nonnegative integer less than the
|
|
positive integer is equal to the nonnegative integer modulo the positive
|
|
integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof
|
|
shortened by AV, 5-Jul-2020.) $)
|
|
addmodid $p |- ( ( A e. NN0 /\ M e. NN /\ A < M )
|
|
-> ( ( M + A ) mod M ) = A ) $=
|
|
( cn0 wcel cn clt wbr w3a caddc co cmo c1 cmul simp2 oveq1d cz cc0 3ad2ant2
|
|
cq cr nncnd mulid2d eqcomd cico wceq 1zzd nnq simp1 nn0zd zq nn0re 3ad2ant1
|
|
syl cle nn0ge0d simp3 cxr wb 0re nnre elico2 sylancr mpbir3and mulqaddmodid
|
|
rexrd syl22anc eqtrd ) ACDZBEDZABFGZHZBAIJZBKJLBMJZAIJZBKJZAVKVLVNBKVKBVMAI
|
|
VKVMBVKBVKBVHVIVJNUAUBUCOOVKLPDBSDZASDZAQBUDJDZVOAUEVKUFVIVHVPVJBUGRVKAPDVQ
|
|
VKAVHVIVJUHZUIAUJUMVKVRATDZQAUNGZVJVHVIVTVJAUKULVKAVSUOVHVIVJUPVKQTDBUQDZVR
|
|
VTWAVJHURUSVIVHWBVJVIBBUTVERQBAVAVBVCABLVDVFVG $.
|
|
|
|
$( The sum of a positive integer and a nonnegative integer less than the
|
|
positive integer is equal to the nonnegative integer modulo the positive
|
|
integer. (Contributed by AV, 19-Mar-2021.) $)
|
|
addmodidr $p |- ( ( A e. NN0 /\ M e. NN /\ A < M )
|
|
-> ( ( A + M ) mod M ) = A ) $=
|
|
( cn0 wcel cn clt wbr w3a caddc co wceq cc nn0cn nncn addcom syl2an 3adant3
|
|
cmo oveq1d addmodid eqtrd ) ACDZBEDZABFGZHZABIJZBRJBAIJZBRJAUEUFUGBRUBUCUFU
|
|
GKZUDUBALDBLDUHUCAMBNABOPQSABTUA $.
|
|
|
|
$( The sum of a number modulo a modulus and another number equals the sum of
|
|
the two numbers modulo the modulus. (Contributed by Jim Kingdon,
|
|
24-Oct-2021.) $)
|
|
modqadd2mod $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) ) $=
|
|
( cq wcel wa cc0 clt wbr cmo co caddc cc qcn ad2antlr simpll simprl addcomd
|
|
syl oveq1d simprr modqcld modqaddmod 3eqtrd ) ADEZBDEZFZCDEZGCHIZFZFZBACJKZ
|
|
LKZCJKULBLKZCJKABLKZCJKBALKZCJKUKUMUNCJUKBULUFBMEUEUJBNOZUKULDEULMEUKACUEUF
|
|
UJPZUGUHUIQUGUHUIUAUBULNSRTABCUCUKUOUPCJUKABUKUEAMEURANSUQRTUD $.
|
|
|
|
$( If a number modulo a modulus equals the modulus decreased by 1, the first
|
|
number increased by 1 modulo the modulus equals 0. (Contributed by Jim
|
|
Kingdon, 24-Oct-2021.) $)
|
|
modqm1p1mod0 $p |- ( ( A e. QQ /\ M e. QQ /\ 0 < M )
|
|
-> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmo co c1 cmin wceq caddc wa simpl1 adantr oveq1d
|
|
cz syl 1z zq mp1i simp2 simpl3 modqaddmod syl22anc oveq1 adantl qcn modqid0
|
|
cc npcan1 syl2anc 3eqtrd eqtr3d ex ) ACDZBCDZEBFGZHZABIJZBKLJZMZAKNJBIJZEMV
|
|
AVDOZVBKNJZBIJZVEEVFURKCDZUSUTVHVEMURUSUTVDPKSDVIVFUAKUBUCVAUSVDURUSUTUDZQZ
|
|
URUSUTVDUEZAKBUFUGVFVHVCKNJZBIJZBBIJZEVDVHVNMVAVDVGVMBIVBVCKNUHRUIVFVMBBIVF
|
|
BULDZVMBMVAVPVDVAUSVPVJBUJTQBUMTRVFUSUTVOEMVKVLBUKUNUOUPUQ $.
|
|
|
|
$( If a number modulo a modulus is less than the modulus decreased by 1, the
|
|
first number increased by 1 modulo the modulus equals the first number
|
|
modulo the modulus, increased by 1. (Contributed by Jim Kingdon,
|
|
24-Oct-2021.) $)
|
|
modqltm1p1mod $p |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\
|
|
( M e. QQ /\ 0 < M ) )
|
|
-> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) ) $=
|
|
( cq wcel cmo co c1 cmin clt wbr cc0 caddc wceq simpll syl22anc cle qre syl
|
|
wa cr cz mp1i simprl simprr modqaddmod modqcld qaddcl syl2anc 0red readdcld
|
|
1z 1red modqge0 syl3anc lep1d letrd simplr ltaddsubd mpbird modqid eqtr3d
|
|
zq ) ACDZABEFZBGHFIJZSZBCDZKBIJZSZSZVDGLFZBEFZAGLFBEFZVKVJVCGCDZVGVHVLVMMVC
|
|
VEVINZGUADVNVJUKGVBUBZVFVGVHUCZVFVGVHUDZAGBUEOVJVKCDZVGKVKPJVKBIJZVLVKMVJVD
|
|
CDZVNVSVJABVOVQVRUFZVPVDGUGUHVQVJKVDVKVJUIVJWAVDTDWBVDQRZVJVDGWCVJULZUJVJVC
|
|
VGVHKVDPJVOVQVRABUMUNVJVDWCUOUPVJVTVEVCVEVIUQVJVDGBWCWDVJVGBTDVQBQRURUSVKBU
|
|
TOVA $.
|
|
|
|
${
|
|
modqmul1.a $e |- ( ph -> A e. QQ ) $.
|
|
modqmul1.b $e |- ( ph -> B e. QQ ) $.
|
|
modqmul1.c $e |- ( ph -> C e. ZZ ) $.
|
|
modqmul1.d $e |- ( ph -> D e. QQ ) $.
|
|
modqmul1.dgt0 $e |- ( ph -> 0 < D ) $.
|
|
modqmul1.ab $e |- ( ph -> ( A mod D ) = ( B mod D ) ) $.
|
|
$( Multiplication property of the modulo operation. Note that the
|
|
multiplier ` C ` must be an integer. (Contributed by Jim Kingdon,
|
|
24-Oct-2021.) $)
|
|
modqmul1 $p |- ( ph
|
|
-> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) ) $=
|
|
( cmo co wceq cmul cmin cq wcel syl3anc syl cfl cfv cc0 clt modqval oveq1
|
|
cdiv wbr eqeq12d syl6bi qcn zcnd wne gt0ne0d qdivcl flqcld mulassd mul32d
|
|
cc eqtr3d oveq2d mulcld subdird eqtr4d sylibrd cz qmulcl syl2anc modqcyc2
|
|
zq zmulcld syl22anc syl5ib syld mpd ) ABELMZCELMZNZBDOMZELMZCDOMZELMZNZKA
|
|
VRVSEDBEUGMZUAUBZOMZOMZPMZWAEDCEUGMZUAUBZOMZOMZPMZNZWCAVRBEWEOMZPMZDOMZCE
|
|
WJOMZPMZDOMZNZWNAVRWPWSNXAAVPWPVQWSABQRZEQRZUCEUDUHZVPWPNFIJBEUESACQRZXCX
|
|
DVQWSNGIJCEUESUIWPWSDOUFUJAWHWQWMWTAWHVSWODOMZPMWQAWGXFVSPAEDOMZWEOMWGXFA
|
|
EDWEAXCEUSRIEUKTZADHULZAWEAWDAXBXCEUCUMZWDQRFIAEJUNZBEUOSUPZULZUQAEDWEXHX
|
|
IXMURUTVAABWODAXBBUSRFBUKTAEWEXHXMVBXIVCVDAWMWAWRDOMZPMWTAWLXNWAPAXGWJOMW
|
|
LXNAEDWJXHXIAWJAWIAXEXCXJWIQRGIXKCEUOSUPZULZUQAEDWJXHXIXPURUTVAACWRDAXECU
|
|
SRGCUKTAEWJXHXPVBXIVCVDUIVEWNWHELMZWMELMZNAWCWHWMELUFAXQVTXRWBAVSQRZWFVFR
|
|
XCXDXQVTNAXBDQRZXSFADVFRXTHDVJTZBDVGVHADWEHXLVKIJVSEWFVIVLAWAQRZWKVFRXCXD
|
|
XRWBNAXEXTYBGYACDVGVHADWJHXOVKIJWAEWKVIVLUIVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
modqmul12d.1 $e |- ( ph -> A e. ZZ ) $.
|
|
modqmul12d.2 $e |- ( ph -> B e. ZZ ) $.
|
|
modqmul12d.3 $e |- ( ph -> C e. ZZ ) $.
|
|
modqmul12d.4 $e |- ( ph -> D e. ZZ ) $.
|
|
modqmul12d.5 $e |- ( ph -> E e. QQ ) $.
|
|
modqmul12d.egt0 $e |- ( ph -> 0 < E ) $.
|
|
modqmul12d.6 $e |- ( ph -> ( A mod E ) = ( B mod E ) ) $.
|
|
modqmul12d.7 $e |- ( ph -> ( C mod E ) = ( D mod E ) ) $.
|
|
$( Multiplication property of the modulo operation, see theorem 5.2(b) in
|
|
[ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.) $)
|
|
modqmul12d $p |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) ) $=
|
|
( cmul co cmo cz wcel cq zq syl modqmul1 zcnd mulcomd oveq1d 3eqtrd eqtrd
|
|
) ABDOPFQPCDOPZFQPZCEOPZFQPZABCDFABRSBTSGBUAUBACRSCTSHCUAUBIKLMUCAUJDCOPZ
|
|
FQPECOPZFQPULAUIUMFQACDACHUDZADIUDUEUFADECFADRSDTSIDUAUBAERSETSJEUAUBHKLN
|
|
UCAUNUKFQAECAEJUDUOUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
modqnegd.1 $e |- ( ph -> A e. QQ ) $.
|
|
modqnegd.2 $e |- ( ph -> B e. QQ ) $.
|
|
modqnegd.3 $e |- ( ph -> C e. QQ ) $.
|
|
modqnegd.cgt0 $e |- ( ph -> 0 < C ) $.
|
|
modqnegd.4 $e |- ( ph -> ( A mod C ) = ( B mod C ) ) $.
|
|
$( Negation property of the modulo operation. (Contributed by Jim Kingdon,
|
|
24-Oct-2021.) $)
|
|
modqnegd $p |- ( ph -> ( -u A mod C ) = ( -u B mod C ) ) $=
|
|
( cneg cmul co cmo wcel cq cc qcn syl mulcomd mulm1d c1 cz neg1z modqmul1
|
|
a1i zcnd eqtrd oveq1d 3eqtr3d ) ABUAJZKLZDMLCUJKLZDMLBJZDMLCJZDMLABCUJDEF
|
|
UJUBNAUCUEZGHIUDAUKUMDMAUKUJBKLUMABUJABONBPNEBQRZAUJUOUFZSABUPTUGUHAULUND
|
|
MAULUJCKLUNACUJACONCPNFCQRZUQSACURTUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
modqadd12d.1 $e |- ( ph -> A e. QQ ) $.
|
|
modqadd12d.2 $e |- ( ph -> B e. QQ ) $.
|
|
modqadd12d.3 $e |- ( ph -> C e. QQ ) $.
|
|
modqadd12d.4 $e |- ( ph -> D e. QQ ) $.
|
|
modqadd12d.5 $e |- ( ph -> E e. QQ ) $.
|
|
modqadd12d.egt0 $e |- ( ph -> 0 < E ) $.
|
|
modqadd12d.6 $e |- ( ph -> ( A mod E ) = ( B mod E ) ) $.
|
|
modqadd12d.7 $e |- ( ph -> ( C mod E ) = ( D mod E ) ) $.
|
|
$( Additive property of the modulo operation. (Contributed by Jim Kingdon,
|
|
25-Oct-2021.) $)
|
|
modqadd12d $p |- ( ph -> ( ( A + C ) mod E ) = ( ( B + D ) mod E ) ) $=
|
|
( caddc co cmo cq wcel cc modqadd1 qcn syl addcomd oveq1d 3eqtrd eqtrd )
|
|
ABDOPFQPCDOPZFQPZCEOPZFQPZABCDFGHIKLMUAAUIDCOPZFQPECOPZFQPUKAUHULFQACDACR
|
|
SCTSHCUBUCZADRSDTSIDUBUCUDUEADECFIJHKLNUAAUMUJFQAECAERSETSJEUBUCUNUDUEUFU
|
|
G $.
|
|
|
|
$( Subtraction property of the modulo operation. (Contributed by Jim
|
|
Kingdon, 25-Oct-2021.) $)
|
|
modqsub12d $p |- ( ph -> ( ( A - C ) mod E ) = ( ( B - D ) mod E ) ) $=
|
|
( co cmo cq wcel syl cc cneg caddc qnegcl modqnegd modqadd12d qcn negsubd
|
|
cmin oveq1d 3eqtr3d ) ABDUAZUBOZFPOCEUAZUBOZFPOBDUHOZFPOCEUHOZFPOABCUKUMF
|
|
GHADQRZUKQRIDUCSAEQRZUMQRJEUCSKLMADEFIJKLNUDUEAULUOFPABDABQRBTRGBUFSAUQDT
|
|
RIDUFSUGUIAUNUPFPACEACQRCTRHCUFSAURETRJEUFSUGUIUJ $.
|
|
$}
|
|
|
|
$( The difference of a number modulo a modulus and another number equals the
|
|
difference of the two numbers modulo the modulus. (Contributed by Jim
|
|
Kingdon, 25-Oct-2021.) $)
|
|
modqsubmod $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) ) $=
|
|
( cq wcel cc0 clt wbr cmo simpll simprl simprr modqcld simplr wceq modqabs2
|
|
wa co syl3anc eqidd modqsub12d ) ADEZBDEZQZCDEZFCGHZQZQZACIRZABBCUHACUBUCUG
|
|
JZUDUEUFKZUDUEUFLZMUJUBUCUGNZUMUKULUHUBUEUFUICIRUIOUJUKULACPSUHBCIRTUA $.
|
|
|
|
$( The difference of a number modulo a modulus and another number modulo the
|
|
same modulus equals the difference of the two numbers modulo the modulus.
|
|
(Contributed by Jim Kingdon, 25-Oct-2021.) $)
|
|
modqsubmodmod $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( ( A mod M ) - ( B mod M ) ) mod M ) = ( ( A - B ) mod M ) ) $=
|
|
( cq wcel cc0 clt wbr cmo simpll simprl simprr modqcld simplr wceq modqabs2
|
|
wa co syl3anc modqsub12d ) ADEZBDEZQZCDEZFCGHZQZQZACIRZABCIRZBCUGACUAUBUFJZ
|
|
UCUDUEKZUCUDUELZMUJUGBCUAUBUFNZUKULMUMUKULUGUAUDUEUHCIRUHOUJUKULACPSUGUBUDU
|
|
EUICIRUIOUMUKULBCPST $.
|
|
|
|
$( Two times a positive number modulo the number is zero. (Contributed by
|
|
Jim Kingdon, 25-Oct-2021.) $)
|
|
q2txmodxeq0 $p |- ( ( X e. QQ /\ 0 < X ) -> ( ( 2 x. X ) mod X ) = 0 ) $=
|
|
( cq wcel cc0 clt wbr wa c2 cmul co cmo wceq cdiv cz 2cnd cc qcn adantr cr
|
|
2z qre simpr gt0ap0d divcanap4d syl6eqel wb ax-mp qmulcl mpan simpl syl3anc
|
|
zq modq0 mpbird ) ABCZDAEFZGZHAIJZAKJDLZURAMJZNCZUQUTHNUQHAUQOUOAPCUPAQRUQA
|
|
UOASCUPAUARUOUPUBZUCUDTUEUQURBCZUOUPUSVAUFUOVCUPHBCZUOVCHNCVDTHULUGHAUHUIRU
|
|
OUPUJVBURAUMUKUN $.
|
|
|
|
$( If a number is between a modulus and twice the modulus, the first number
|
|
modulo the modulus equals the first number minus the modulus.
|
|
(Contributed by Jim Kingdon, 25-Oct-2021.) $)
|
|
q2submod $p |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\
|
|
( B <_ A /\ A < ( 2 x. B ) ) ) ->
|
|
( A mod B ) = ( A - B ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cle c2 cmul co wa c1 cmin cmo wceq cr qre syl qcn
|
|
cc 3ad2ant2 adantr mulid1d oveq2d oveq1d simpl1 1zzd simpl2 simpl3 modqcyc2
|
|
syl22anc qsubcl syl2anc simpr subge0d bicomd caddc 2timesd breq2d ltsubaddd
|
|
cz bitr4d anbi12d mpbid modqid syl21anc 3eqtr3d ) ACDZBCDZEBFGZHZBAIGZAJBKL
|
|
ZFGZMZMZABNKLZOLZBPLZABOLZBPLZABPLZWBVRVTWBBPVRVSBAOVRBVMBUBDZVQVKVJWEVLBUA
|
|
UCUDZUEUFUGVRVJNVCDVKVLWAWDQVJVKVLVQUHZVRUIVJVKVLVQUJZVJVKVLVQUKABNULUMVRWB
|
|
CDZVKEWBIGZWBBFGZMZWCWBQVRVJVKWIWGWHABUNUOWHVRVQWLVMVQUPVRVNWJVPWKVRWJVNVRA
|
|
BVRVJARDWGASTZVRVKBRDWHBSTZUQURVRVPABBUSLZFGWKVRVOWOAFVRBWFUTVAVRABBWMWNWNV
|
|
BVDVEVFWBBVGVHVI $.
|
|
|
|
$( If a nonnegative integer is less than twice a positive integer, the
|
|
nonnegative integer modulo the positive integer equals the nonnegative
|
|
integer or the nonnegative integer minus the positive integer.
|
|
(Contributed by Alexander van der Vekens, 21-May-2018.) $)
|
|
modifeq2int $p |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) )
|
|
-> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) $=
|
|
( cn0 wcel cn c2 co clt wbr wa cq cc0 cle cz syl adantr eqcomd adantl eqtrd
|
|
wceq cmul w3a cmo cmin cif wn simp1 nn0z zq 3ad2ant2 nn0ge0d simpr syl22anc
|
|
nnq modqid iftrue simp2 nngt0d nnred nn0red lenltd simpl3 q2submod syl32anc
|
|
biimpar iffalse wo nnzd wdc zdclt exmiddc syl2anc mpjaodan ) ACDZBEDZAFBUAG
|
|
HIZUBZABHIZABUCGZVRAABUDGZUEZTVRUFZVQVRJZVSAWAWCAKDZBKDZLAMIZVRVSATVQWDVRVQ
|
|
VNWDVNVOVPUGZVNANDZWDAUHZAUIOOZPVQWEVRVOVNWEVPBUNUJZPVQWFVRVQAWGUKPVQVRULAB
|
|
UOUMVRAWATVQVRWAAVRAVTUPQRSVQWBJZVSVTWAWLWDWELBHIBAMIZVPVSVTTVQWDWBWJPVQWEW
|
|
BWKPWLBVQVOWBVNVOVPUQZPURVQWMWBVQBAVQBWNUSVQAWGUTVAVEVNVOVPWBVBABVCVDWLWAVT
|
|
WBWAVTTVQVRAVTVFRQSVQWHBNDZVRWBVGZVQVNWHWGWIOVQBWNVHWHWOJVRVIWPABVJVRVKOVLV
|
|
M $.
|
|
|
|
$( The sum of an integer modulo a positive integer and another integer minus
|
|
the positive integer equals the sum of the two integers modulo the
|
|
positive integer if the other integer is in the upper part of the range
|
|
between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.) $)
|
|
modaddmodup $p |- ( ( A e. ZZ /\ M e. NN )
|
|
-> ( B e. ( ( M - ( A mod M ) ) ..^ M )
|
|
-> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) ) ) $=
|
|
( cmo co cmin wcel cz wa caddc wceq cq clt wbr cle adantr adantl zq syl cr
|
|
cfzo cn cc0 c2 elfzoelz cn0 zmodcl nn0zd zaddcld simprr nnq nngt0d elfzole1
|
|
cmul nnred nn0red zred mpbid elfzolt2 ad2antrl modqlt syl3anc lt2addd nncnd
|
|
lesubaddd 2timesd breqtrrd q2submod syl32anc modqadd2mod syl22anc eqtr3d
|
|
expcom ) BCACDEZFEZCUAEGZAHGZCUBGZIZBVNJEZCFEZBAJECDEZKVPVSIZVTCDEZWAWBWCVT
|
|
LGZCLGZUCCMNZCVTONZVTUDCUNEZMNWDWAKWCVTHGWEWCBVNVPBHGZVSBVOCUEZPZWCVNVSVNUF
|
|
GVPACUGZQUHUIVTRSWCVRWFVPVQVRUJZCUKSZWCCWNULZWCVOBONZWHVPWQVSBVOCUMPWCCVNBW
|
|
CCWNUOZVSVNTGVPVSVNWMUPQZVPBTGVSVPBWKUQPZVEURWCVTCCJEWIMWCBVNCCWTWSWRWRVPBC
|
|
MNVSBVOCUSPWCALGZWFWGVNCMNVQXAVPVRARUTZWOWPACVAVBVCWCCWCCWNVDVFVGVTCVHVIWCX
|
|
ABLGZWFWGWDWBKXBWCWJXCWLBRSWOWPABCVJVKVLVM $.
|
|
|
|
$( The sum of an integer modulo a positive integer and another integer equals
|
|
the sum of the two integers modulo the positive integer if the other
|
|
integer is in the lower part of the range between 0 and the positive
|
|
integer. (Contributed by AV, 30-Oct-2018.) $)
|
|
modaddmodlo $p |- ( ( A e. ZZ /\ M e. NN )
|
|
-> ( B e. ( 0 ..^ ( M - ( A mod M ) ) )
|
|
-> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) ) ) $=
|
|
( cz wcel cn wa cc0 cmo co caddc wceq cq cle wbr clt adantl zq syl syl22anc
|
|
cmin cfzo elfzoelz cn0 zmodcl adantr nn0zd qaddcl syl2anc simplr nnq nn0red
|
|
zred elfzole1 nn0ge0d addge0d elfzolt2 nnred ltaddsubd mpbird modqid nngt0d
|
|
ad2antrr modqadd2mod eqtr3d ex ) ADEZCFEZGZBHCACIJZUAJZUBJEZBVJKJZBAKJCIJZL
|
|
VIVLGZVMCIJZVMVNVOVMMEZCMEZHVMNOVMCPOZVPVMLVOBMEZVJMEZVQVOBDEZVTVLWBVIBHVKU
|
|
CQZBRSZVOVJDEWAVOVJVIVJUDEVLACUEUFZUGVJRSBVJUHUIVOVHVRVGVHVLUJZCUKSZVOBVJVO
|
|
BWCUMZVOVJWEULZVLHBNOVIBHVKUNQVOVJWEUOUPVOVSBVKPOZVLWJVIBHVKUQQVOBVJCWHWIVO
|
|
CWFURUSUTVMCVATVOAMEZVTVRHCPOVPVNLVGWKVHVLARVCWDWGVOCWFVBABCVDTVEVF $.
|
|
|
|
$( The product of a rational number modulo a modulus and an integer equals
|
|
the product of the rational number and the integer modulo the modulus.
|
|
(Contributed by Jim Kingdon, 25-Oct-2021.) $)
|
|
modqmulmod $p |- ( ( ( A e. QQ /\ B e. ZZ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) ) $=
|
|
( cq wcel cz wa cc0 clt wbr cmo co simpll simprl simprr modqcld simplr wceq
|
|
modqabs2 syl3anc modqmul1 ) ADEZBFEZGZCDEZHCIJZGZGZACKLZABCUHACUBUCUGMZUDUE
|
|
UFNZUDUEUFOZPUJUBUCUGQUKULUHUBUEUFUICKLUIRUJUKULACSTUA $.
|
|
|
|
$( The product of an integer and a rational number modulo a modulus equals
|
|
the product of the integer and the rational number modulo the modulus.
|
|
(Contributed by Jim Kingdon, 26-Oct-2021.) $)
|
|
modqmulmodr $p |- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) )
|
|
-> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) ) $=
|
|
( cz wcel cq wa cc0 clt wbr cmo co cmul simpll zcnd qcn syl mulcomd oveq1d
|
|
cc simplr simprl simprr modqcld wceq modqmulmod ancom1s 3eqtrd ) ADEZBFEZGZ
|
|
CFEZHCIJZGZGZABCKLZMLZCKLUPAMLZCKLZBAMLZCKLZABMLZCKLUOUQURCKUOAUPUOAUIUJUNN
|
|
OZUOUPFEUPTEUOBCUIUJUNUAZUKULUMUBUKULUMUCUDUPPQRSUJUIUNUSVAUEBACUFUGUOUTVBC
|
|
KUOBAUOUJBTEVDBPQVCRSUH $.
|
|
|
|
$( The sum of a rational number and the product of a second rational number
|
|
modulo a modulus and an integer equals the sum of the rational number and
|
|
the product of the other rational number and the integer modulo the
|
|
modulus. (Contributed by Jim Kingdon, 26-Oct-2021.) $)
|
|
modqaddmulmod $p |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\
|
|
( M e. QQ /\ 0 < M ) )
|
|
-> ( ( A + ( ( B mod M ) x. C ) ) mod M )
|
|
= ( ( A + ( B x. C ) ) mod M ) ) $=
|
|
( cq wcel wa cmo co cmul caddc cc qcn modqcld qmulcl syl2anc addcomd oveq1d
|
|
syl wceq w3a cc0 clt wbr simpl1 simpl2 simprl simprr simpl3 3ad2ant3 adantr
|
|
cz modqmulmod 3adantl1 modqabs2 syl3anc eqtr4d modqadd1 modqaddmod syl22anc
|
|
zq eqtrd 3eqtrd ) AEFZBEFZCULFZUAZDEFZUBDUCUDZGZGZABDHIZCJIZKIZDHIVMAKIZDHI
|
|
BCJIZDHIZAKIDHIZAVPKIZDHIZVKVNVODHVKAVMVKVDALFVDVEVFVJUEZAMSZVKVMEFZVMLFVKV
|
|
LEFZCEFZWCVKBDVDVEVFVJUFZVGVHVIUGZVGVHVIUHZNZVKVFWEVDVEVFVJUICVAZSVLCOZPVMM
|
|
SQRVKVMVQADVKWDWEWCWIVGWEVJVFVDWEVEWJUJUKZWKPVKVPDVKVEWEVPEFZWFWLBCOPZWGWHN
|
|
WAWGWHVKVMDHIZVQVQDHIZVEVFVJWOVQTVDBCDUMUNVKWMVHVIWPVQTWNWGWHVPDUOUPUQURVKV
|
|
RVPAKIZDHIZVTVKWMVDVHVIVRWRTWNWAWGWHVPADUSUTVKWQVSDHVKVPAVKWMVPLFWNVPMSWBQR
|
|
VBVC $.
|
|
|
|
$( Distribute multiplication over a modulo operation. (Contributed by Jim
|
|
Kingdon, 26-Oct-2021.) $)
|
|
modqdi $p |- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) )
|
|
-> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) ) $=
|
|
( cq wcel cc0 clt wbr wa co cfl cmul cmin cc qcn syl syl3anc qmulcl syl2anc
|
|
oveq2d w3a cdiv cfv cmo simp1l simp2 simp3l cz simp3r gt0ne0d qdivcl flqcld
|
|
wne zq subdid cr gt0ap0d simp1r divcanap5d fveq2d zcnd mulassd eqtr2d eqtrd
|
|
qre wceq modqval mulgt0d 3eqtr4d ) ADEZFAGHZIZBDEZCDEZFCGHZIZUAZABCBCUBJZKU
|
|
CZLJZMJZLJZABLJZACLJZWCWDUBJZKUCZLJZMJZABCUDJZLJWCWDUDJZVQWBWCAVTLJZMJWHVQA
|
|
BVTVQVJANEVJVKVMVPUEZAOPZVQVMBNEVLVMVPUFZBOPZVQVTDEZVTNEVQVNVSDEZWPVLVMVNVO
|
|
UGZVQVSUHEWQVQVRVQVMVNCFUMVRDEWNWRVQCVLVMVNVOUIZUJBCUKQULZVSUNPCVSRSVTOPUOV
|
|
QWKWGWCMVQWGWDVSLJWKVQWFVSWDLVQWEVRKVQBCAWOVQVNCNEWRCOPZWMVQCVQVNCUPEWRCVEP
|
|
ZWSUQVQAVQVJAUPEWLAVEPZVJVKVMVPURZUQUSUTTVQACVSWMXAVQVSWTVAVBVCTVDVQWIWAALV
|
|
QVMVNVOWIWAVFWNWRWSBCVGQTVQWCDEZWDDEZFWDGHWJWHVFVQVJVMXEWLWNABRSVQVJVNXFWLW
|
|
RACRSVQACXCXBXDWSVHWCWDVGQVI $.
|
|
|
|
$( Distribute the modulo operation over a subtraction. (Contributed by Jim
|
|
Kingdon, 26-Oct-2021.) $)
|
|
modqsubdir $p |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) )
|
|
-> ( ( B mod C ) <_ ( A mod C )
|
|
<-> ( ( A - B ) mod C ) = ( ( A mod C ) - ( B mod C ) ) ) ) $=
|
|
( cq wcel wa cc0 clt wbr cmo co cmin cle wceq cr syl adantr syl3anc cc qcn
|
|
simpll simprl simprr modqcld qre simplr subge0d cdiv cfl cfv cmul cz qsubcl
|
|
wne gt0ne0d qdivcl flqcld zsubcld modqcyc2 syl22anc zq qmulcl syl2anc sub4d
|
|
subdid oveq2d modqval oveq12d 3eqtr4d oveq1d eqtr3d simpr resubcld subge02d
|
|
zcnd modqge0 mpbid modqlt lelttrd modqid eqtrd breqtrd impbida bitr3d ) ADE
|
|
ZBDEZFZCDEZGCHIZFZFZGACJKZBCJKZLKZMIZWMWLMIABLKZCJKZWNNZWKWLWMWKWLDEZWLOEWK
|
|
ACWEWFWJUAZWGWHWIUBZWGWHWIUCZUDZWLUEPZWKWMDEZWMOEWKBCWEWFWJUFZXAXBUDZWMUEPZ
|
|
UGWKWOWRWKWOFZWQWNCJKZWNWKWQXJNWOWKWPCACUHKZUIUJZBCUHKZUIUJZLKZUKKZLKZCJKZW
|
|
QXJWKWPDEZXOULEWHWIXRWQNWGXSWJABUMQZWKXLXNWKXKWKWEWHCGUNZXKDEWTXAWKCXBUOZAC
|
|
UPRUQZWKXMWKWFWHYAXMDEXFXAYBBCUPRUQZURXAXBWPCXOUSUTWKXQWNCJWKWPCXLUKKZCXNUK
|
|
KZLKZLKAYELKZBYFLKZLKXQWNWKABYEYFWKWEASEWTATPWKWFBSEXFBTPWKYEDEZYESEWKWHXLD
|
|
EZYJXAWKXLULEYKYCXLVAPCXLVBVCYETPWKYFDEZYFSEWKWHXNDEZYLXAWKXNULEYMYDXNVAPCX
|
|
NVBVCYFTPVDWKXPYGWPLWKCXLXNWKWHCSEXACTPWKXLYCVOWKXNYDVOVEVFWKWLYHWMYILWKWEW
|
|
HWIWLYHNWTXAXBACVGRWKWFWHWIWMYINXFXAXBBCVGRVHVIVJVKQXIWNDEZWHWOWNCHIZXJWNNW
|
|
KYNWOWKWSXEYNXCXGWLWMUMVCQWKWHWOXAQWKWOVLWKYOWOWKWNWLCWKWLWMXDXHVMXDWKWHCOE
|
|
XACUEPWKGWMMIZWNWLMIWKWFWHWIYPXFXAXBBCVPRWKWLWMXDXHVNVQWKWEWHWIWLCHIWTXAXBA
|
|
CVRRVSQWNCVTUTWAWKWRFGWQWNMWKGWQMIZWRWKXSWHWIYQXTXAXBWPCVPRQWKWRVLWBWCWD $.
|
|
|
|
$( A rational number equals the difference of the rational number and a
|
|
modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.) $)
|
|
modqeqmodmin $p |- ( ( A e. QQ /\ M e. QQ /\ 0 < M )
|
|
-> ( A mod M ) = ( ( A - M ) mod M ) ) $=
|
|
( cq wcel cc0 clt wbr w3a cmin cmo cle wceq modqid0 3adant1 modqge0 eqbrtrd
|
|
co wb simp1 simp2 simp3 modqsubdir syl22anc eqcomd oveq2d cc modqcl qcn syl
|
|
mpbid subid1d 3eqtr2rd ) ACDZBCDZEBFGZHZABIQBJQZABJQZBBJQZIQZUREIQURUPUSURK
|
|
GZUQUTLZUPUSEURKUNUOUSELUMBMNZABOPUPUMUNUNUOVAVBRUMUNUOSUMUNUOTZVDUMUNUOUAA
|
|
BBUBUCUJUPEUSURIUPUSEVCUDUEUPURUPURCDURUFDABUGURUHUIUKUL $.
|
|
|
|
${
|
|
$d J i $. $d K i $. $d N i $.
|
|
$( For a number within a half-open range of nonnegative integers with one
|
|
excluded integer there is a positive integer so that the number is equal
|
|
to the sum of the positive integer and the excluded integer modulo the
|
|
upper bound of the range. (Contributed by AV, 19-Mar-2021.) $)
|
|
modfzo0difsn $p |- ( ( J e. ( 0 ..^ N ) /\ K e. ( ( 0 ..^ N ) \ { J } ) )
|
|
-> E. i e. ( 1 ..^ N ) K = ( ( i + J ) mod N ) ) $=
|
|
( wbr co wcel wa caddc wceq syl syl2anr wi clt wb w3a adantr adantl com12
|
|
cn cle wn wo cc0 cfzo csn cdif cv cmo c1 wrex eldifi elfzoelz wdc exmiddc
|
|
cz zdcle zleloe cmin cn0 elfzo0 cc nn0cn 3ad2ant1 nncn 3ad2ant2 subadd23d
|
|
simpl nnz znnsub syl2an biimp3a nn0nnaddcl eqeltrd simp2 cr nn0re sublt0d
|
|
nn0z bicomd biimpa resubcl nnre jca ltaddnegr mpbid syl3anbrc exp31 sylbi
|
|
elfzo1 3adant2 impcom oveq1 zcnd 3jca 3adant3 imp nppcan sylan9eqr oveq1d
|
|
ex eqeq2d biimpi a1d addmodidr eqcomd rspcedvd wne eqneqall sylbid zltnle
|
|
eldifsn jaoi nn0ge0 subge02 ancoms simp3 lelttr mpand sylibr zmodidfzoimp
|
|
impancom npcan mpcom ) CBUAEZYEUBZUCZBUDDUEFZGZCYHBUFZUGGZHZCAUHZBIFZDUIF
|
|
ZJZAUJDUEFZUKZYKCUPGZBUPGZYGYIYKCYHGZYSCYHYJULZCUDDUMZKZBUDDUMZYSYTHYEUNY
|
|
GCBUQYEUOKLYEYLYRMZYFYLYEYRYLYECBNEZCBJZUCZYRYKYSYTYEUUIOYIUUDUUECBURLUUI
|
|
YLYRUUGUUFUUHUUGYLYRUUGYLHZYPCCDIFZDUIFZJZACBUSFZDIFZYQYLUUGUUOYQGZYKYIUU
|
|
GUUPMZYKUUAYIUUQMZUUBUUACUTGZDTGZCDNEZPZUURCDVAZUUSUVAUURUUTYIUUSUVAHZUUQ
|
|
YIBUTGZUUTBDNEZPZUVDUUQMBDVAZUVGUVDUUGUUPUVGUVDHZUUGHZUUOTGZUUTUUODNEZUUP
|
|
UVIUVKUUGUVIUUOCDBUSFZIFZTUVICBDUVDCVBGZUVGUUSUVOUVACVCQRUVGBVBGZUVDUVEUU
|
|
TUVPUVFBVCZVDQUVGDVBGZUVDUUTUVEUVRUVFDVEZVFQVGUVDUUSUVMTGZUVNTGUVGUUSUVAV
|
|
HUVEUUTUVFUVTUVEYTDUPGUVFUVTOUUTBVSZDVIBDVJVKVLCUVMVMLVNQUVIUUTUUGUVGUUTU
|
|
VDUVEUUTUVFVOZQQUVJUUNUDNEZUVLUVIUUGUWCUVIUWCUUGUVICBUVDCVPGZUVGUUSUWDUVA
|
|
CVQZQZRUVGBVPGZUVDUVEUUTUWGUVFBVQVDZQVRVTWAUVJUUNVPGZDVPGZHZUWCUVLOUVIUWK
|
|
UUGUVIUWIUWJUVDUWDUWGUWIUVGUWFUWHCBWBZLUVGUWJUVDUUTUVEUWJUVFDWCVFZQWDQUUN
|
|
DWEKWFDUUOWJWGWHWISWKWIKWLWLUUJYMUUOJZHZYOUULCUWOYNUUKDUIUWNUUJYNUUOBIFZU
|
|
UKYMUUOBIWMUUJUVOUVPUVRPZUWPUUKJYLUWQUUGYIYKUWQYIUVGYKUWQMZUVHUVEUUTUWRUV
|
|
FYKUVEUUTHZUWQYKUUAUWSUWQMUUBUUAUWSUWQUUAUWSHUVOUVPUVRUUAUVOUWSUUACUUCWNZ
|
|
QUWSUVPUUAUVEUVPUUTUVQQRUWSUVRUUAUUTUVRUVEUVSRRWOXAKSWPWIWQRCBDWRKWSWTXBU
|
|
UJUVBUUMYLUVBUUGYKYIUVBYKUUAYIUVBMUUBUUAUVBYIUUAUVBUVCXCXDKWLRUVBUULCCDXE
|
|
XFKXGXAYLUUHYRYKUUHYRMZYIYKUUACBXHZHUXACYHBXLUXBUXAUUAUUHUXBYRYRCBXISRWIR
|
|
SXMSXJSYFYLYRYFYLHZYPCCDUIFZJAUUNYQUXCUUNTGZUUTUUNDNEZPZUUNYQGYLYFUXGYLYF
|
|
BCNEZUXGYLUXHYFYIYTYSUXHYFOYKUUEUUDBCXKVKVTYIYKUXHUXGMZYIUVGYKUXIMUVHYKUV
|
|
GUXIYKUUAUVGUXIMZUUBUUAUVBUXJUVCUUSUVAUXJUUTUVDUVGUXHUXGUVDUVGHZUXHHUXEUU
|
|
TUXFUXKUXHUXEUVGYTYSUXHUXEOUVDUVEUUTYTUVFUWAVDUUSYSUVACVSQBCVJLWAUXKUUTUX
|
|
HUVGUUTUVDUWBRQUXKUXFUXHUVDUVGUXFUUSUVGUVAUXFUUSUVGHZUUNCUAEZUVAUXFUXLUDB
|
|
UAEZUXMUVGUXNUUSUVEUUTUXNUVFBXNVDRUUSUWDUWGUXNUXMOUVGUWEUWHCBXOVKWFUXLUWI
|
|
UWDUWJPZUXMUVAHUXFMUXLUWGUWDUWJPZUXOUXLUWGUWDUWJUVGUWGUUSUWHRUUSUWDUVGUWE
|
|
QUVGUWJUUSUWMRWOUXPUWIUWDUWJUWGUWDUWIUWJUWDUWGUWIUWLXPWPUWGUWDUWJVOUWGUWD
|
|
UWJXQWOKUUNCDXRKXSYBWQQWOWHWKWIKSWIWQXJWLDUUNWJXTUXCYMUUNJZHZYOUXDCUXRYNC
|
|
DUIUXQUXCYNUUNBIFZCYMUUNBIWMYLUXSCJZYFYKUVOUVPUXTYIYKUUAUVOUUBUWTKYIBUUEW
|
|
NCBYCLRWSWTXBUXCUXDCYLUXDCJZYFYKUYAYIYKUUAUYAUUBCDYAKRRXFXGXAXMYD $.
|
|
$}
|
|
|
|
$( The sum of a number within a half-open range of positive integers is an
|
|
element of the corresponding open range of nonnegative integers with one
|
|
excluded integer modulo the excluded integer. (Contributed by AV,
|
|
19-Mar-2021.) $)
|
|
modsumfzodifsn $p |- ( ( J e. ( 0 ..^ N ) /\ K e. ( 1 ..^ N ) )
|
|
-> ( ( K + J ) mod N ) e. ( ( 0 ..^ N ) \ { J } ) ) $=
|
|
( cc0 co wcel c1 wa caddc clt wbr cmo cq cle wceq cz syl cn0 elfzo0 adantr
|
|
cfzo csn cdif wn elfzoelz adantl zq w3a biimpi simp1d qaddcl syl2anc simp2d
|
|
cn nn0zd nnq elfzo1 nnnn0d nn0addcld nn0ge0d simpr modqid syl22anc wne zcnd
|
|
syl3anbrc 0cnd nn0cnd nnne0d addneintr2d addid2d neeqtrd sylanbrc cmin cneg
|
|
eldifsn eqeltrd cmul cc nncnd mulm1d oveq2d addcld negsubd eqtrd oveq1d a1i
|
|
neg1z nngt0d modqcyc qsubcl nnred nn0red lenltd mpbird subge0d zred lt2addd
|
|
cr simp3d ltsubaddd 3eqtr3d nnzd zsubcld elnn0z subcld ltned subne0d eqcomd
|
|
addsubd 3netr4d wo wdc zdclt exmiddc mpjaodan ) ADCUAEZFZBGCUAEFZHZBAIEZCJK
|
|
ZYACLEZXQAUBUCZFYBUDZXTYBHZYCYAYDYFYAMFZCMFZDYANKZYBYCYAOXTYGYBXTBMFZAMFZYG
|
|
XTBPFZYJXSYLXRBGCUEUFZBUGQXTAPFYKXTAXTARFZCUNFZACJKZXRYNYOYPUHZXSXRYQACSUIT
|
|
ZUJZUOAUGQBAUKULZTXTYHYBXTYOYHXTYNYOYPYRUMZCUPQZTXTYIYBXTYAXTBAXTBXTBUNFZYO
|
|
BCJKZXSUUCYOUUDUHZXRXSUUECBUQUIUFZUJZURYSUSZUTTXTYBVAZYACVBVCYFYAXQFZYAAVDZ
|
|
YAYDFYFYARFZYOYBUUJXTUULYBUUHTXTYOYBUUATUUIYACSVFXTUUKYBXTYADAIEZAXTBDAXTBY
|
|
MVEZXTVGZXTAYSVHZXTBUUGVIVJXTAUUPVKZVLTYAXQAVPVMVQXTYEHZYCYACVNEZYDUURYAGVO
|
|
ZCVREZIEZCLEZUUSCLEZYCUUSUURUVBUUSCLUURUVBYACVOZIEZUUSUURUVAUVEYAIUURCXTCVS
|
|
FYEXTCUUAVTZTWAWBXTUVFUUSOYEXTYACXTBAUUNUUPWCUVGWDTWEWFUURYGUUTPFZYHDCJKZUV
|
|
CYCOXTYGYEYTTUVHUURWHWGXTYHYEUUBTZXTUVIYEXTCUUAWITYACUUTWJVCUURUUSMFZYHDUUS
|
|
NKZUUSCJKZUVDUUSOXTUVKYEXTYGYHUVKYTUUBYACWKULTUVJUURUVLCYANKZUURUVNYEXTYEVA
|
|
UURCYAXTCWSFYEXTCUUAWLZTZXTYAWSFYEXTYAUUHWMZTZWNWOUURYACUVRUVPWPWOZXTUVMYEX
|
|
TUVMYACCIEJKXTBACCXTBYMWQZXTAYSWMUVOUVOXTUUCYOUUDUUFWTZXTYNYOYPYRWTWRXTYACC
|
|
UVQUVOUVOXAWOTZUUSCVBVCXBUURUUSXQFZUUSAVDZUUSYDFUURUUSRFZYOUVMUWCUURUUSPFZU
|
|
VLUWEXTUWFYEXTYACXTYAUUHUOZXTCUUAXCZXDTUVSUUSXEVMXTYOYEUUATUWBUUSCSVFXTUWDY
|
|
EXTBCVNEZAIEUUMUUSAXTUWIDAXTBCUUNUVGXFUUOUUPXTBCUUNUVGXTBCUVTUWAXGXHVJXTBAC
|
|
UUNUUPUVGXJXTUUMAUUQXIXKTUUSXQAVPVMVQXTYAPFZCPFZYBYEXLZUWGUWHUWJUWKHYBXMUWL
|
|
YACXNYBXOQULXP $.
|
|
|
|
$( Two nonnegative integers less than the modulus are equal iff they are
|
|
equal modulo the modulus. (Contributed by AV, 14-Mar-2021.) $)
|
|
modlteq $p |- ( ( I e. ( 0 ..^ N ) /\ J e. ( 0 ..^ N ) )
|
|
-> ( ( I mod N ) = ( J mod N ) <-> I = J ) ) $=
|
|
( cc0 cfzo co wcel cmo zmodidfzoimp eqeqan12d ) ADCEFZGBKGACHFABCHFBACIBCIJ
|
|
$.
|
|
|
|
${
|
|
$d I k $. $d J k $. $d N k $. $d S k $.
|
|
$( Two nonnegative integers less than the modulus are equal iff the sums of
|
|
these integer with another integer are equal modulo the modulus.
|
|
(Contributed by AV, 20-Mar-2021.) $)
|
|
addmodlteq $p |- ( ( I e. ( 0 ..^ N ) /\ J e. ( 0 ..^ N ) /\ S e. ZZ )
|
|
-> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) ) $=
|
|
( cc0 co wcel cz caddc cmo wceq cq clt wbr cmin nn0cnd ad2antrr ad3antrrr
|
|
c1 cle vk cfzo w3a elfzoelz 3ad2ant1 zq syl simp3 cn elfzo0 biimpi simp2d
|
|
cn0 nnq nngt0d modqaddmod syl22anc 3ad2ant2 eqeq12d zmodcld nn0zd zaddcld
|
|
eqcomd subeq0ad modqcld qaddcl syl2anc modqsubmodmod zcnd pnpcan2d oveq1d
|
|
oveq1 eqtrd q0mod zmodidfzoimp oveq12d eqeq1d cdiv wb qsubcl syl3anc cmul
|
|
modq0 cv wrex zsubcld zdiv wa cneg simpr oveq2d nncnd mul01d eqcom simp1d
|
|
biimpd syl5bi sylbid an32s subfzo0 3adant3 simprd mulid1d 3brtr4d simpllr
|
|
imp simplr zred crp nnrpd ltmul2d mpbird nnge1d lensymd pm2.21dd breqtrrd
|
|
1red simpld cr nnzd adantr zmulcld nnred possumd cc adddid eqtr4d breqtrd
|
|
1cnd peano2zd mulcomd zcn addcomd subnegd ad3antlr suble0d eqbrtrd nnnn0d
|
|
0red sylbird nn0ge0d mulle0r w3o elz simprbi ad2antlr mpjao3dan rexlimdva
|
|
ex syl5 impbid1 ) BEDUBFZGZCUULGZAHGZUCZBAIFZDJFZCAIFZDJFZKZBCKZUUPUVABDJ
|
|
FZAIFZDJFZCDJFZAIFZDJFZKZUVBUUPUURUVEUUTUVHUUPUVEUURUUPBLGZALGZDLGZEDMNZU
|
|
VEUURKUUPBHGZUVJUUMUUNUVNUUOBEDUDUEZBUFUGZUUPUUOUVKUUMUUNUUOUHZAUFUGZUUPD
|
|
UIGZUVLUUPBUMGZUVSBDMNZUUMUUNUVTUVSUWAUCZUUOUUMUWBBDUJUKUEZULZDUNUGZUUPDU
|
|
WDUOZBADUPUQVCUUPUVHUUTUUPCLGZUVKUVLUVMUVHUUTKUUPCHGZUWGUUNUUMUWHUUOCEDUD
|
|
URZCUFUGZUVRUWEUWFCADUPUQVCUSUUPUVIUVEUVHOFZEKZUVBUUPUVEUVHUUPUVEUUPUVDDU
|
|
UPUVCAUUPUVCUUPBDUVOUWDUTZVAUVQVBUWDUTPUUPUVHUUPUVGDUUPUVFAUUPUVFUUPCDUWI
|
|
UWDUTZVAUVQVBUWDUTPVDUWLUWKDJFZEDJFZKZUUPUVBUWKEDJVLUUPUWQUVCUVFOFZDJFZEK
|
|
ZUVBUUPUWOUWSUWPEUUPUWOUVDUVGOFZDJFZUWSUUPUVDLGZUVGLGZUVLUVMUWOUXBKUUPUVC
|
|
LGUVKUXCUUPBDUVPUWEUWFVEUVRUVCAVFVGUUPUVFLGUVKUXDUUPCDUWJUWEUWFVEUVRUVFAV
|
|
FVGUWEUWFUVDUVGDVHUQUUPUXAUWRDJUUPUVCUVFAUUPUVCUWMPUUPUVFUWNPUUPAUVQVIVJV
|
|
KVMUUPUVLUVMUWPEKUWEUWFDVNVGUSUUPUWTBCOFZDJFZEKZUVBUUPUWSUXFEUUPUWRUXEDJU
|
|
UPUVCBUVFCOUUMUUNUVCBKUUOBDVOUEUUNUUMUVFCKUUOCDVOURVPVKVQUUPUXGUXEDVRFHGZ
|
|
UVBUUPUXELGZUVLUVMUXGUXHVSUUPUVJUWGUXIUVPUWJBCVTVGUWEUWFUXEDWCWAUUPUXHDUA
|
|
WDZWBFZUXEKZUAHWEZUVBUUPUVSUXEHGUXMUXHVSUWDUUPBCUVOUWIWFUADUXEWGVGUUPUXLU
|
|
VBUAHUUPUXJHGZWHZUXLUVBUXOUXLWHZUXJEKZUVBUXJUIGZUXJWIZUIGZUXOUXQUXLUVBUXO
|
|
UXQWHZUXLUVBUYAUXLEUXEKZUVBUYAUXKEUXEUYAUXKDEWBFZEUYAUXJEDWBUXOUXQWJWKUUP
|
|
UYCEKUXNUXQUUPDUUPDUWDWLZWMQVMVQUYBUXEEKZUYAUVBEUXEWNUYAUYEUVBUYABCUYABUU
|
|
PUVTUXNUXQUUPUVTUVSUWAUWCWOQPUYACUUPCUMGZUXNUXQUUPUYFUVSCDMNZUUNUUMUYFUVS
|
|
UYGUCZUUOUUNUYHCDUJUKURWOQPVDWPWQWRXFWSUXPUXRWHZUXJSMNZUVBUYIUYJUXKDSWBFZ
|
|
MNUYIUXEDUXKUYKMUYIDWIZUXEMNZUXEDMNZUUPUYMUYNWHZUXNUXLUXRUUMUUNUYOUUOBCDW
|
|
TXAZRXBUXOUXLUXRXGUUPUYKDKUXNUXLUXRUUPDUYDXCZRXDUYIUXJSDUYIUXJUUPUXNUXLUX
|
|
RXEXHZUYIXQZUUPDXIGUXNUXLUXRUUPDUWDXJRXKXLUYISUXJUYSUYRUYIUXJUXPUXRWJXMXN
|
|
XOUXPUXTWHZEDUXJSIFZWBFZMNUVBUYTEUXKDIFZVUBMUYTEVUCMNUYLUXKMNUYTUYLUXEUXK
|
|
MUYTUYMUYNUUPUYOUXNUXLUXTUYPRXRUXOUXLUXTXGXPUYTUXKDUXOUXKXSGUXLUXTUXOUXKU
|
|
XODUXJUUPDHGUXNUUPDUWDXTYAZUUPUXNWJZYBXHQUUPDXSGZUXNUXLUXTUUPDUWDYCRZYDXL
|
|
UYTVUCUXKUYKIFZVUBUUPVUCVUHKUXNUXLUXTUUPDUYKUXKIUUPUYKDUYQVCWKRUYTDUXJSUU
|
|
PDYEGZUXNUXLUXTUYDRUXOUXJYEGUXLUXTUXOUXJVUEVIQUYTYIYFYGYHUYTVUBEUXOVUBXSG
|
|
UXLUXTUXOVUBUXODVUAVUDUXOUXJVUEYJZYBXHQUYTYSUYTVUBVUADWBFZETUXOVUBVUKKUXL
|
|
UXTUXODVUAUUPVUIUXNUYDYAUXOVUAVUJVIYKQUYTVUAXSGZVUFVUAETNEDTNZVUKETNUXOVU
|
|
LUXLUXTUXOVUAVUJXHQVUGUYTVUASUXSOFZETUXNVUAVUNKUUPUXLUXTUXNVUASUXJIFVUNUX
|
|
NUXJSUXJYLZUXNYIZYMUXNSUXJVUPVUOYNYGYOUYTVUNETNSUXSTNUYTUXSUXPUXTWJZXMUYT
|
|
SUXSUYTXQUYTUXSVUQYCYPXLYQUUPVUMUXNUXLUXTUUPDUUPDUWDYRUUARVUADUUBUQYQXNXO
|
|
UXNUXQUXRUXTUUCZUUPUXLUXNUXJXSGVURUXJUUDUUEUUFUUGUUIUUHYTWRWRWRUUJYTWRUVB
|
|
UUQUUSDJBCAIVLVKUUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Miscellaneous theorems about integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A w y z $. $d B w z $. $d C v w x y z $. $d G v w y z $.
|
|
frec2uz.1 $e |- ( ph -> C e. ZZ ) $.
|
|
frec2uz.2 $e |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) $.
|
|
$( The mapping ` G ` is a one-to-one mapping from ` _om ` onto upper
|
|
integers that will be used to construct a recursive definition
|
|
generator. Ordinal natural number 0 maps to complex number ` C `
|
|
(normally 0 for the upper integers ` NN0 ` or 1 for the upper integers
|
|
` NN ` ), 1 maps to ` C ` + 1, etc. This theorem shows the value of
|
|
` G ` at ordinal natural number zero. (Contributed by Jim Kingdon,
|
|
16-May-2020.) $)
|
|
frec2uz0d $p |- ( ph -> ( G ` (/) ) = C ) $=
|
|
( c0 cfv cz cv c1 caddc co cmpt cfrec fveq1i wcel wceq frec0g syl syl5eq
|
|
) AGDHGBIBJKLMNZCOZHZCGDUCFPACIQUDCRECUBISTUA $.
|
|
|
|
${
|
|
$d ph k y z $. $d C k $. $d k x $.
|
|
frec2uzzd.a $e |- ( ph -> A e. _om ) $.
|
|
$( The value of ` G ` (see ~ frec2uz0d ) is an integer. (Contributed by
|
|
Jim Kingdon, 16-May-2020.) $)
|
|
frec2uzzd $p |- ( ph -> ( G ` A ) e. ZZ ) $=
|
|
( vk cfv cz cv c1 caddc co cmpt cfrec fveq1i wcel wa wceq peano2zd eqid
|
|
simpr oveq1 fvmptg syl2anc eqeltrd freccl syl5eqel ) ACEJCBKBLZMNOZPZDQ
|
|
ZJKCEUNGRAIDCKUMFAILZKSZTZUOUMJZUOMNOZKUQUPUSKSURUSUAAUPUDZUQUOUTUBZBUO
|
|
ULUSKKUMUKUOMNUEUMUCUFUGVAUHHUIUJ $.
|
|
|
|
$( The value of ` G ` (see ~ frec2uz0d ) at a successor. (Contributed by
|
|
Jim Kingdon, 16-May-2020.) $)
|
|
frec2uzsucd $p |- ( ph -> ( G ` suc A ) = ( ( G ` A ) + 1 ) ) $=
|
|
( vy vz cfv cz cv c1 caddc co wcel wceq cvv fveq1i csuc cfrec com mptex
|
|
cmpt wal zex vex fvex ax-gen frecsucOLD mp3an1 syl2anc fveq2i frec2uzzd
|
|
3eqtr4g oveq1 cbvmptv peano2z fvmpt3 syl eqtrd ) ACUAZEKZCEKZBLBMZNOPZU
|
|
EZKZVENOPZAVCVHDUBZKZCVKKZVHKZVDVIADLQZCUCQZVLVNRZFHIMZVHKSQZIUFVOVPVQV
|
|
SIVRVHSSBLVGUGUDIUHUIUJIDCVHLUKULUMVCEVKGTVEVMVHCEVKGTUNUPAVELQVIVJRABC
|
|
DEFGHUOJVEJMZNOPZVJLVHLVTVENOUQBJLVGWAVFVTNOUQURVTUSUTVAVB $.
|
|
|
|
$( The value ` G ` (see ~ frec2uz0d ) at an ordinal natural number is in
|
|
the upper integers. (Contributed by Jim Kingdon, 16-May-2020.) $)
|
|
frec2uzuzd $p |- ( ph -> ( G ` A ) e. ( ZZ>= ` C ) ) $=
|
|
( vy vz com wcel cfv cv wi wceq wa eleq1d c0 fveq2 simpr fveq2d imbi12d
|
|
cuz csuc frec2uz0d cz uzid syl eqeltrd c1 caddc co peano2uz frec2uzsucd
|
|
adantl simpl syl5ibr ex finds2 com12 vtocld mpd ) ACKLZCEMZDUDMZLZHAINZ
|
|
KLZVHEMZVFLZOVDVGOICKHAVHCPZQZVIVDVKVGVMVHCKAVLUAZRVMVJVEVFVMVHCEVNUBRU
|
|
CVIAVKVKSEMZVFLJNZEMZVFLZVPUEZEMZVFLZAIJVHSPVJVOVFVHSETRVHVPPVJVQVFVHVP
|
|
ETRVHVSPVJVTVFVHVSETRAVODVFABDEFGUFADUGLZDVFLFDUHUIUJVPKLZAVRWAOVRWAWCA
|
|
QZVQUKULUMZVFLDVQUNWDVTWEVFWDBVPDEAWBWCFUPGWCAUQUORURUSUTVAVBVC $.
|
|
|
|
frec2uzltd.b $e |- ( ph -> B e. _om ) $.
|
|
$( Less-than relation for ` G ` (see ~ frec2uz0d ). (Contributed by Jim
|
|
Kingdon, 16-May-2020.) $)
|
|
frec2uzltd $p |- ( ph -> ( A e. B -> ( G ` A ) < ( G ` B ) ) ) $=
|
|
( wcel cfv clt wbr wi c0 wceq fveq2 breq2d imbi12d vz vy com csuc eleq2
|
|
cv imbi2d noel pm2.21i a1i wo wa id orim12d wb elsuc2g bicomd adantr c1
|
|
caddc co cle cz adantl simpl frec2uzsucd cuz frec2uzuzd eluzelz zleltp1
|
|
syl2an syl2anc frec2uzzd zleloe 3bitr2rd syl5ib ex a2d finds mpcom ) DU
|
|
CKACDKZCFLZDFLZMNZOZJACUAUFZKZWBWFFLZMNZOZOACPKZWBPFLZMNZOZOACUBUFZKZWB
|
|
WOFLZMNZOZOACWOUDZKZWBWTFLZMNZOZOAWEOUAUBDWFPQZWJWNAXEWGWKWIWMWFPCUEXEW
|
|
HWLWBMWFPFRSTUGWFWOQZWJWSAXFWGWPWIWRWFWOCUEXFWHWQWBMWFWOFRSTUGWFWTQZWJX
|
|
DAXGWGXAWIXCWFWTCUEXGWHXBWBMWFWTFRSTUGWFDQZWJWEAXHWGWAWIWDWFDCUEXHWHWCW
|
|
BMWFDFRSTUGWNAWKWMCUHUIUJWOUCKZAWSXDXIAWSXDOWSWPCWOQZUKZWRWBWQQZUKZOXIA
|
|
ULZXDWSWPWRXJXLWSUMXJXLOWSCWOFRUJUNXNXKXAXMXCXIXKXAUOAXIXAXKCWOUCUPUQUR
|
|
XNXCWBWQUSUTVAZMNZWBWQVBNZXMXNXBXOWBMXNBWOEFAEVCKXIGVDZHXIAVEZVFSXNWBEV
|
|
GLZKZWQXTKZXQXPUOZXNBCEFXRHACUCKXIIVDZVHXNBWOEFXRHXSVHYAWBVCKZWQVCKZYCY
|
|
BEWBVIEWQVIWBWQVJVKVLXNYEYFXQXMUOXNBCEFXRHYDVMXNBWOEFXRHXSVMWBWQVNVLVOT
|
|
VPVQVRVSVT $.
|
|
|
|
$( The mapping ` G ` (see ~ frec2uz0d ) preserves order. (Contributed by
|
|
Jim Kingdon, 16-May-2020.) $)
|
|
frec2uzlt2d $p |- ( ph -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) ) $=
|
|
( wcel cfv clt wbr frec2uzltd wceq wa adantr zred ex w3o wi com syl2anc
|
|
nntri3or ax-1 a1i fveq2 adantl breq2d biimpar cz frec2uzzd pm2.21dd imp
|
|
ltnrd ltnsymd pm2.21d 3jaod mpd impbid ) ACDKZCFLZDFLZMNZABCDEFGHIJOAVB
|
|
CDPZDCKZUAZVEVBUBZACUCKDUCKVHIJCDUEUDAVBVIVFVGVBVIUBAVBVEUFUGAVFVIAVFQZ
|
|
VEVBVJVEQZVCVCMNZVBVJVLVEVJVCVDVCMVFVCVDPACDFUHUIUJUKVKVCVKVCVJVCULKZVE
|
|
AVMVFABCEFGHIUMZRRSUPUNTTAVGVIAVGQZVEVBVOVDVCVOVDAVDULKVGABDEFGHJUMRSVO
|
|
VCAVMVGVNRSAVGVDVCMNABDCEFGHJIOUOUQURTUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $.
|
|
$( Range of ` G ` (see ~ frec2uz0d ). (Contributed by Jim Kingdon,
|
|
17-May-2020.) $)
|
|
frec2uzrand $p |- ( ph -> ran G = ( ZZ>= ` C ) ) $=
|
|
( vy vz cz wcel cfv cv com c1 caddc co cvv syl eleq1 c0 vw crn cuz wceq
|
|
wrex wfn wb cmpt cfrec wal zex mptex fvex ax-gen frecfnom fneq1i sylibr
|
|
vex fvelrnb simpl simpr frec2uzuzd syl5ibcom rexlimdva sylbid frec2uz0d
|
|
mpan wa id peano1 fnfvelrn sylancl eqeltrrd wi eluzel2 csuc frec2uzsucd
|
|
oveq1 sylan9eq peano2 syl2an adantr ex uzind4 impbid1 eqrdv ) ACIJZDUBZ
|
|
CUCKZUDEWGGWHWIWGGLZWHJZWJWIJZWGWKHLZDKZWJUDZHMUEZWLWGDMUFZWKWPUGWGBIBL
|
|
NOPZUHZCUIZMUFZWQWMWSKQJZHUJWGXAXBHWMWSQQBIWRUKULHURUMUNHCWSIUOVGMDWTFU
|
|
PUQZHMWJDUSRZWGWOWLHMWGWMMJZVHZWNWIJWOWLXFBWMCDWGXEUTZFWGXEVAZVBWNWJWIS
|
|
VCVDVEUALZWHJCWHJWKWJNOPZWHJZWKUAGCWJXICWHSXIWJWHSZXIXJWHSXLWGTDKZCWHWG
|
|
BCDWGVIFVFWGWQTMJXMWHJXCVJMTDVKVLVMWLWGWKXKVNCWJVOWGWKWPXKXDWGWOXKHMXFW
|
|
OXKXFWOVHWMVPZDKZXJWHXFWOXOWNNOPXJXFBWMCDXGFXHVQWNWJNOVRVSXFXOWHJZWOWGW
|
|
QXNMJXPXEXCWMVTMXNDVKWAWBVMWCVDVERWDWEWFR $.
|
|
|
|
$( ` G ` (see ~ frec2uz0d ) is a one-to-one onto mapping. (Contributed
|
|
by Jim Kingdon, 17-May-2020.) $)
|
|
frec2uzf1od $p |- ( ph -> G : _om -1-1-onto-> ( ZZ>= ` C ) ) $=
|
|
( vy vz com cfv wceq cv wi cz cvv wcel sylanbrc wn clt adantr wf1o wral
|
|
cuz wf1 crn wf wfn wss c1 caddc co cmpt cfrec wal zex mptex fvex ax-gen
|
|
vex frecfnom sylancr fneq1i sylibr frec2uzrand eqimss syl df-f wa simpr
|
|
w3a wbr frec2uzzd 3adant3 zred ltnrd mtbid simp2 simp3 frec2uzltd con3d
|
|
breq2d mpd breq1d wb nntri3 3adant1 mpbir2and ex 3expb ralrimivva dff13
|
|
mtod dff1o5 ) AICUCJZDUDZDUEZWNKZIWNDUAAIWNDUFZGLZDJZHLZDJZKZWSXAKZMZHI
|
|
UBGIUBWOADIUGZWPWNUHZWRABNBLUIUJUKZULZCUMZIUGZXFAXAXIJOPZHUNCNPZXKXLHXA
|
|
XIOOBNXHUOUPHUSUQUREHCXINUTVAIDXJFVBVCAWQXGABCDEFVDZWPWNVEVFIWNDVGQAXEG
|
|
HIIAWSIPZXAIPZXEAXOXPVJZXCXDXQXCVHZXDWSXAPZRZXAWSPZRZXRWTXBSVKZRZXTXRWT
|
|
WTSVKZYCXQYERXCXQWTXQWTAXOWTNPXPAXOVHBWSCDAXMXOETZFAXOVIVLVMVNVOTZXRWTX
|
|
BWTSXQXCVIZWAVPXQYDXTMXCXQXSYCXQBWSXACDAXOXMXPYFVMZFAXOXPVQZAXOXPVRZVSV
|
|
TTWBXRYAXBWTSVKZXRYEYLYGXRWTXBWTSYHWCVPXQYAYLMXCXQBXAWSCDYIFYKYJVSTWLXQ
|
|
XDXTYBVHWDZXCXOXPYMAWSXAWEWFTWGWHWIWJGHIWNDWKQXNIWNDWMQ $.
|
|
|
|
$( ` G ` (see ~ frec2uz0d ) is an isomorphism from natural ordinals to
|
|
upper integers. (Contributed by Jim Kingdon, 17-May-2020.) $)
|
|
frec2uzisod $p |- ( ph -> G Isom _E , < ( _om , ( ZZ>= ` C ) ) ) $=
|
|
( vy vz com cuz cfv wf1o cv cep wbr clt wb wral wcel wa wiso ralrimivva
|
|
frec2uzf1od cz adantr simprl simprr frec2uzlt2d syl5bb df-isom sylanbrc
|
|
epel ) AICJKZDLGMZHMZNOZUNDKUODKPOZQZHIRGIRIUMNPDUAABCDEFUCAURGHIIUPUNU
|
|
OSAUNISZUOISZTZTZUQGHULVBBUNUOCDACUDSVAEUEFAUSUTUFAUSUTUGUHUIUBGHIUMNPD
|
|
UJUK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z F $. $d A w y z $. $d C v w x y z $. $d G v w y z $.
|
|
$d R v w z $. $d S w x y z $. $d ph v w x y z $.
|
|
uzrdg.s $e |- ( ph -> S e. V ) $.
|
|
uzrdg.a $e |- ( ph -> A e. S ) $.
|
|
uzrdg.f $e |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) ->
|
|
( x F y ) e. S ) $.
|
|
uzrdg.2 $e |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |->
|
|
<. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) $.
|
|
|
|
${
|
|
$d D z $.
|
|
$( The function ` R ` (used in the definition of the recursive
|
|
definition generator on upper integers) yields ordered pairs of
|
|
integers and elements of ` S ` . New proofs should use
|
|
~ frecuzrdgrrn which is the same but with one fewer hypothesis.
|
|
(Contributed by Jim Kingdon, 27-May-2020.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
frecuzrdgrrnOLD $p |- ( ( ph /\ D e. _om ) ->
|
|
( R ` D ) e. ( ( ZZ>= ` C ) X. S ) ) $=
|
|
( wcel cfv cvv vz com wa cuz cv c1 caddc co cop cmpt2 cfrec fveq1i cz
|
|
cxp zex uzssz ssexi a1i adantr mpt2exga syl2anc vex fvexg alrimiv syl
|
|
uzid opelxp sylanbrc c1st c2nd wceq 1st2nd2 fveq2 df-ov syl6eqr xp1st
|
|
adantl xp2nd peano2uz wral ralrimivva ad2antrr wi oveq1 eleq1d rspc2v
|
|
oveq2 mpd opeq12d opeq2d eqid ovmpt2g syl3anc eqtrd eqeltrd frecclOLD
|
|
simpr syl5eqel ) AFUBRZUCZFGSFBCEUDSZHBUEZUFUGUHZXBCUEZIUHZUIZUJZEDUI
|
|
ZUKZSXAHUNZFGXIQULWTUAXHFXJXGWTUAUEZXGSZTRZUAWTXGTRZXKTRZXMWTXATRZHKR
|
|
ZXNXPWTXAUMUOEUPUQURAXQWSNUSBCXAHXFTKUTVAXOWTUAVBURXKXGTTVCVAVDAXHXJR
|
|
ZWSAEXARZDHRXRAEUMRXSLEVFVEOEDXAHVGVHUSWTXKXJRZUCZXLXKVISZUFUGUHZYBXK
|
|
VJSZIUHZUIZXJYAXLYBYDXGUHZYFXTXLYGVKZWTXTXKYBYDUIZVKZYHXKXAHVLYJXLYIX
|
|
GSYGXKYIXGVMYBYDXGVNVOVEVQYAYBXARZYDHRZYFXJRZYGYFVKXTYKWTXKXAHVPVQZXT
|
|
YLWTXKXAHVRVQZYAYCXARZYEHRZYMYAYKYPYNEYBVSVEYAXEHRZCHVTBXAVTZYQAYSWSX
|
|
TAYRBCXAHPWAWBYAYKYLYSYQWCYNYOYRYQYBXDIUHZHRBCYBYDXAHXBYBVKZXEYTHXBYB
|
|
XDIWDZWEXDYDVKZYTYEHXDYDYBIWGZWEWFVAWHYCYEXAHVGVHZBCYBYDXAHXFYFXGYCYT
|
|
UIXJUUAXCYCXEYTXBYBUFUGWDUUBWIUUCYTYEYCUUDWJXGWKWLWMWNUUEWOAWSWQWPWR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d B w z $.
|
|
uzrdg.b $e |- ( ph -> B e. _om ) $.
|
|
$( A helper lemma for the value of a recursive definition generator on
|
|
upper integers (typically either ` NN ` or ` NN0 ` ) with
|
|
characteristic function ` F ( x , y ) ` and initial value ` A ` .
|
|
This lemma shows that evaluating ` R ` at an element of ` _om `
|
|
gives an ordered pair whose first element is the index (translated
|
|
from ` _om ` to ` ( ZZ>= `` C ) ` ). See comment in ~ frec2uz0d
|
|
which describes ` G ` and the index translation. (Contributed by
|
|
Jim Kingdon, 24-May-2020.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
frec2uzrdgOLD $p |- ( ph -> ( R ` B ) =
|
|
<. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. ) $=
|
|
( wcel cfv vz vv vw com c2nd cop wceq cv fveq2 fveq2d opeq12d eqeq12d
|
|
wi imbi2d c0 csuc cuz c1 caddc co cmpt2 cfrec fveq1i cz opexg syl2anc
|
|
cvv frec0g syl syl5eq frec2uz0d uzid op2ndg eqtrd eqtr4d wa wal uzssz
|
|
zex ssexi a1i adantr mpt2exga fvexg alrimiv frecsucOLD syl3anc fveq2i
|
|
vex simpr 3eqtr4g df-ov cxp frec2uzuzd frecuzrdgrrnOLD xp2nd peano2uz
|
|
caovclg adantlr caovcld opelxp sylanbrc oveq1 opeq2d cbvmpt2v ovmpt2g
|
|
oveq2 syl5eqr sylan9eqr frec2uzsucd frec2uzzd peano2zd expcom vtoclga
|
|
ex finds2 mpcom ) EUDSAEGTZEJTZXRUETZUFZUGZRAUAUHZGTZYCJTZYDUETZUFZUG
|
|
ZUMAYBUMUAEUDYCEUGZYHYBAYIYDXRYGYAYCEGUIZYIYEXSYFXTYCEJUIYIYDXRUEYJUJ
|
|
UKULUNYHUOGTZUOJTZYKUETZUFZUGUBUHZGTZYOJTZYPUETZUFZUGZYOUPZGTZUUAJTZU
|
|
UBUETZUFZUGZAUAUBYCUOUGZYDYKYGYNYCUOGUIZUUGYEYLYFYMYCUOJUIUUGYDYKUEUU
|
|
HUJUKULYCYOUGZYDYPYGYSYCYOGUIZUUIYEYQYFYRYCYOJUIUUIYDYPUEUUJUJUKULYCU
|
|
UAUGZYDUUBYGUUEYCUUAGUIZUUKYEUUCYFUUDYCUUAJUIUUKYDUUBUEUULUJUKULAYKFD
|
|
UFZYNAYKUOBCFUQTZHBUHZURUSUTZUUOCUHZIUTZUFZVAZUUMVBZTZUUMUOGUVAQVCAUU
|
|
MVGSZUVBUUMUGAFVDSZDHSZUVCLOFDVDHVEVFZUUMUUTVGVHVIVJZAYLFYMDABFJLMVKA
|
|
YMUUMUETZDAYKUUMUEUVGUJAFUUNSZUVEUVHDUGAUVDUVILFVLVIOFDUUNHVMVFVNUKVO
|
|
AYOUDSZYTUUFUMAUVJVPZYTUUFUVKYTVPZUUBYQURUSUTZYQYRIUTZUFZUUEUVLUUBYPU
|
|
UTTZUVOUVKUUBUVPUGYTUVKUUAUVATZYOUVATZUUTTZUUBUVPUVKYCUUTTVGSZUAVQUVC
|
|
UVJUVQUVSUGUVKUVTUAUVKUUTVGSZYCVGSZUVTUVKUUNVGSZHKSZUWAUWCUVKUUNVDVSF
|
|
VRVTWAAUWDUVJNWBBCUUNHUUSVGKWCVFUWBUVKUAWIWAYCUUTVGVGWDVFWEAUVCUVJUVF
|
|
WBAUVJWJZUAUUMYOUUTVGWFWGUUAGUVAQVCYPUVRUUTYOGUVAQVCWHWKWBYTUVKUVPYSU
|
|
UTTZUVOYPYSUUTUIUVKUWFYQYRUUTUTZUVOYQYRUUTWLUVKYQUUNSZYRHSZUVOUUNHWMZ
|
|
SZUWGUVOUGUVKBYOFJAUVDUVJLWBZMUWEWNZUVKYPUWJSUWIABCDFYOGHIJKLMNOPQWOY
|
|
PUUNHWPVIZUVKUVMUUNSZUVNHSZUWKUVKUWHUWOUWMFYQWQVIUVKUAUCYQYRUUNHHIAYC
|
|
UUNSUCUHZHSVPYCUWQIUTHSUVJABCYCUWQUUNHHIPWRWSUWMUWNWTZUVMUVNUUNHXAXBU
|
|
CUAYQYRUUNHUWQURUSUTZUWQYCIUTZUFZUVOUUTUVMYQYCIUTZUFUWJUWQYQUGUWSUVMU
|
|
WTUXBUWQYQURUSXCUWQYQYCIXCUKYCYRUGUXBUVNUVMYCYRYQIXGXDBCUCUAUUNHUUSUX
|
|
AUWSUWQUUQIUTZUFUUOUWQUGUUPUWSUURUXCUUOUWQURUSXCUUOUWQUUQIXCUKUUQYCUG
|
|
UXCUWTUWSUUQYCUWQIXGXDXEXFWGXHXIVNZUVLUUCUVMUUDUVNUVKUUCUVMUGYTUVKBYO
|
|
FJUWLMUWEXJWBUVLUUDUVOUETZUVNUVLUUBUVOUEUXDUJUVLUVMVDSZUWPUXEUVNUGUVK
|
|
UXFYTUVKYQUVKBYOFJUWLMUWEXKXLWBUVKUWPYTUWRWBUVMUVNVDHVMVFVNUKVOXOXMXP
|
|
XNXQ $.
|
|
$}
|
|
|
|
$( The function ` R ` (used in the definition of the recursive definition
|
|
generator on upper integers) is a function defined for all natural
|
|
numbers. New proofs should use ~ frecuzrdgrcl instead. (Contributed
|
|
by Jim Kingdon, 26-May-2020.) (New usage is discouraged.) $)
|
|
frecuzrdgrom $p |- ( ph -> R Fn _om ) $=
|
|
( vz cv cvv wcel cuz cfv c1 caddc co cop cmpt2 cfrec com wfn wal cxp cz
|
|
zex uzssz ssexi mpt2exga sylancr vex fvexg sylancl alrimiv syl sylanbrc
|
|
uzid opelxp frecfnom syl2anc fneq1i sylibr ) ABCEUAUBZGBRZUCUDUEVLCRHUE
|
|
UFZUGZEDUFZUHZUIUJZFUIUJAQRZVNUBSTZQUKVOVKGULZTZVQAVSQAVNSTZVRSTVSAVKST
|
|
GJTWBVKUMUNEUOUPMBCVKGVMSJUQURQUSVRVNSSUTVAVBAEVKTZDGTWAAEUMTWCKEVEVCNE
|
|
DVKGVFVDQVOVNVTVGVHUIFVPPVIVJ $.
|
|
|
|
${
|
|
$d B x y $.
|
|
frecuzrdglemOLD.b $e |- ( ph -> B e. ( ZZ>= ` C ) ) $.
|
|
$( A helper lemma for the value of a recursive definition generator on
|
|
upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
frecuzrdglemOLD $p |- ( ph ->
|
|
<. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R ) $=
|
|
( cfv com ccnv cop crn cuz frec2uzf1od f1ocnvdm syl2anc frec2uzrdgOLD
|
|
c2nd wf1o wcel wceq f1ocnvfv2 opeq1d eqtrd wfn frecuzrdgrom fnfvelrn
|
|
eqeltrrd ) AEJUASZGSZEVAUISZUBZGUCZAVAUTJSZVBUBVCABCDUTFGHIJKLMNOPQAT
|
|
FUDSZJUJZEVFUKZUTTUKZABFJLMUEZRTVFEJUFUGZUHAVEEVBAVGVHVEEULVJRTVFEJUM
|
|
UGUNUOAGTUPVIVAVDUKABCDFGHIJKLMNOPQUQVKTUTGURUGUS $.
|
|
$}
|
|
|
|
${
|
|
$d v w z T $.
|
|
frecuzrdgfn.3 $e |- ( ph -> T = ran R ) $.
|
|
$( The recursive definition generator on upper integers is a function.
|
|
See comment in ~ frec2uz0d for the description of ` G ` as the
|
|
mapping from ` _om ` to ` ( ZZ>= `` C ) ` . New proofs should use
|
|
~ frecuzrdgtcl instead. (Contributed by Jim Kingdon, 26-May-2020.)
|
|
(New usage is discouraged.) $)
|
|
frecuzrdgfn $p |- ( ph -> T Fn ( ZZ>= ` C ) ) $=
|
|
( vz wcel vv vw wfun cdm cuz cfv wceq wfn wrel cv wbr wmo cvv cxp wss
|
|
wral com crn eleq2d wb frecuzrdgrom fvelrnb syl bitrd frecuzrdgrrnOLD
|
|
wrex eleq1 syl5ibcom rexlimdva sylbid ssrdv xpss syl6ss df-rel sylibr
|
|
wa cop wal wex ccnv c2nd wf1o frec2uzf1od f1ocnvdm sylan syldan xp2nd
|
|
wi cz adantr co adantlr simpr frec2uzrdgOLD eqeq1d vex simplbi syl6bi
|
|
opth2 f1ocnvfv syld fveq2 fveq2d syl6 op2ndd adantl eqtr2d ex alrimiv
|
|
imp eqeq2 imbi2d albidv spcegv sylc nfv dmxpss frecuzrdglemOLD mpbird
|
|
mo2r dmss opeldmg mpan eqssd pm5.32i df-br 3imtr4i ralrimiva sylanbrc
|
|
mobii dffun7 df-fn ) AHUCZHUDZEUEUFZUGHYOUHAHUIZUAUJZSUJZHUKZSULZUAYN
|
|
UPYMAHUMUMUNZUOYPAHYOGUNZUUAASHUUBAYRHTZUBUJZFUFZYRUGZUBUQVFZYRUUBTZA
|
|
UUCYRFURZTZUUGAHUUIYRRUSAFUQUHZUUJUUGUTABCDEFGIJKLMNOPQVAZUBUQYRFVBVC
|
|
VDAUUFUUHUBUQAUUDUQTZVPZUUEUUBTUUFUUHABCDEUUDFGIJKLMNOPQVEUUEYRUUBVGV
|
|
HVIVJVKZYOGVLVMHVNVOAYTUAYNAYQYOTZVPZYQYRVQZHTZSULZAYQYNTZVPYTUUQUUSY
|
|
RUUDUGZWHZSVRZUBVSZUUTUUQYQJVTUFZFUFZWAUFZGTZUUSYRUVHUGZWHZSVRZUVEUUQ
|
|
UVGUUBTZUVIAUUPUVFUQTZUVMAUQYOJWBZUUPUVNABEJLMWCZUQYOYQJWDWEABCDEUVFF
|
|
GIJKLMNOPQVEWFUVGYOGWGVCZAUVLUUPAUVKSAUUSUUEUURUGZUBUQVFZUVJAUUSUURUU
|
|
ITZUVSAHUUIUURRUSAUUKUVTUVSUTUULUBUQUURFVBVCVDAUVRUVJUBUQUUNUVRUVJUUN
|
|
UVRVPUVHUUEWAUFZYRUUNUVRUVHUWAUGZUUNUVRUVFUUDUGZUWBUUNUVRUUDJUFZYQUGZ
|
|
UWCUUNUVRUWDUWAVQZUURUGZUWEUUNUUEUWFUURUUNBCDUUDEFGIJKAEWITZUUMLWJMAG
|
|
KTZUUMNWJADGTZUUMOWJABUJZYOTCUJZGTVPZUWKUWLIWKGTZUUMPWLQAUUMWMWNWOUWG
|
|
UWEUWAYRUGZUWDUWAYQYRUAWPZSWPZWSWQWRAUVOUUMUWEUWCWHUVPUQYOUUDYQJWTWEX
|
|
AUWCUVGUUEWAUVFUUDFXBXCXDXJUVRUWOUUNYQYRUUEUWPUWQXEXFXGXHVIVJXIWJUVDU
|
|
VLUBUVHGUUDUVHUGZUVCUVKSUWRUVBUVJUUSUUDUVHYRXKXLXMXNXOUUSSUBUUSUBXPXT
|
|
VCAUVAUUPAYNYOYQAYNYOAYNUUBUDZYOAHUUBUOYNUWSUOUUOHUUBYAVCYOGXQVMAUAYO
|
|
YNAUUPUVAUUQUVIYQUVHVQZHTZUVAUVQUUQUXAUWTUUITZUUQBCDYQEFGIJKAUWHUUPLW
|
|
JMAUWIUUPNWJAUWJUUPOWJAUWMUWNUUPPWLQAUUPWMXRAUXAUXBUTUUPAHUUIUWTRUSWJ
|
|
XSYQUMTUVIUXAUVAWHUWPYQUVHHUMGYBYCXOXHVKYDZUSYEYSUUSSYQYRHYFYJYGYHUAS
|
|
HYKYIUXCHYOYLYI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d w x y z F $. $d A w y z $. $d C v w x y z $. $d G v w y z $.
|
|
$d R v w z $. $d S w x y z $. $d ph v w x y z $.
|
|
frecuzrdgrrn.a $e |- ( ph -> A e. S ) $.
|
|
frecuzrdgrrn.f $e |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) ->
|
|
( x F y ) e. S ) $.
|
|
frecuzrdgrrn.2 $e |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |->
|
|
<. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) $.
|
|
|
|
${
|
|
$d D z $.
|
|
$( The function ` R ` (used in the definition of the recursive
|
|
definition generator on upper integers) yields ordered pairs of
|
|
integers and elements of ` S ` . (Contributed by Jim Kingdon,
|
|
28-Mar-2022.) $)
|
|
frecuzrdgrrn $p |- ( ( ph /\ D e. _om ) ->
|
|
( R ` D ) e. ( ( ZZ>= ` C ) X. S ) ) $=
|
|
( wcel cfv co cop wceq vz com wa cuz cv c1 caddc cmpt2 cfrec cxp uzid
|
|
fveq1i cz syl opelxp sylanbrc adantr c1st 1st2nd2 fveq2 df-ov syl6eqr
|
|
c2nd adantl xp1st xp2nd peano2uz wral ralrimivva ad2antrr oveq1 oveq2
|
|
eleq1d rspc2v syl2anc mpd opeq12d opeq2d eqid ovmpt2g syl3anc eqeltrd
|
|
wi eqtrd simpr freccl syl5eqel ) AFUBPZUCZFGQFBCEUDQZHBUEZUFUGRZWKCUE
|
|
ZIRZSZUHZEDSZUIZQWJHUJZFGWROULWIUAWQFWSWPAWQWSPZWHAEWJPZDHPWTAEUMPXAK
|
|
EUKUNMEDWJHUOUPUQWIUAUEZWSPZUCZXBWPQZXBURQZUFUGRZXFXBVCQZIRZSZWSXDXEX
|
|
FXHWPRZXJXCXEXKTZWIXCXBXFXHSZTZXLXBWJHUSXNXEXMWPQXKXBXMWPUTXFXHWPVAVB
|
|
UNVDXDXFWJPZXHHPZXJWSPZXKXJTXCXOWIXBWJHVEVDZXCXPWIXBWJHVFVDZXDXGWJPZX
|
|
IHPZXQXDXOXTXREXFVGUNXDWNHPZCHVHBWJVHZYAAYCWHXCAYBBCWJHNVIVJXDXOXPYCY
|
|
AWCXRXSYBYAXFWMIRZHPBCXFXHWJHWKXFTZWNYDHWKXFWMIVKZVMWMXHTZYDXIHWMXHXF
|
|
IVLZVMVNVOVPXGXIWJHUOUPZBCXFXHWJHWOXJWPXGYDSWSYEWLXGWNYDWKXFUFUGVKYFV
|
|
QYGYDXIXGYHVRWPVSVTWAWDYIWBAWHWEWFWG $.
|
|
$}
|
|
|
|
${
|
|
$d B w z $.
|
|
frec2uzrdg.b $e |- ( ph -> B e. _om ) $.
|
|
$( A helper lemma for the value of a recursive definition generator on
|
|
upper integers (typically either ` NN ` or ` NN0 ` ) with
|
|
characteristic function ` F ( x , y ) ` and initial value ` A ` .
|
|
This lemma shows that evaluating ` R ` at an element of ` _om `
|
|
gives an ordered pair whose first element is the index (translated
|
|
from ` _om ` to ` ( ZZ>= `` C ) ` ). See comment in ~ frec2uz0d
|
|
which describes ` G ` and the index translation. (Contributed by
|
|
Jim Kingdon, 24-May-2020.) $)
|
|
frec2uzrdg $p |- ( ph -> ( R ` B ) =
|
|
<. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. ) $=
|
|
( wcel cfv c2nd wceq vz vv vw com cop cv fveq2 fveq2d opeq12d eqeq12d
|
|
wi imbi2d c0 csuc cuz c1 caddc co cmpt2 cfrec fveq1i cz opexg syl2anc
|
|
cvv frec0g syl syl5eq frec2uz0d uzid op2ndg eqtrd eqtr4d wa wral c1st
|
|
cxp 1st2nd2 adantl df-ov xp1st xp2nd peano2uz ralrimivva oveq1 eleq1d
|
|
ad2antrr oveq2 rspc2v mpd opelxp sylanbrc opeq2d eqid ovmpt2g syl3anc
|
|
syl5eqr eqeltrd ralrimiva adantr simpr fveq2i frec2uzuzd frecuzrdgrrn
|
|
frecsuc 3eqtr4g caovclg adantlr caovcld cbvmpt2v frec2uzsucd peano2zd
|
|
sylan9eqr frec2uzzd ex expcom finds2 vtoclga mpcom ) EUDQAEGRZEJRZXTS
|
|
RZUEZTZPAUAUFZGRZYEJRZYFSRZUEZTZUKAYDUKUAEUDYEETZYJYDAYKYFXTYIYCYEEGU
|
|
GZYKYGYAYHYBYEEJUGYKYFXTSYLUHUIUJULYJUMGRZUMJRZYMSRZUEZTUBUFZGRZYQJRZ
|
|
YRSRZUEZTZYQUNZGRZUUCJRZUUDSRZUEZTZAUAUBYEUMTZYFYMYIYPYEUMGUGZUUIYGYN
|
|
YHYOYEUMJUGUUIYFYMSUUJUHUIUJYEYQTZYFYRYIUUAYEYQGUGZUUKYGYSYHYTYEYQJUG
|
|
UUKYFYRSUULUHUIUJYEUUCTZYFUUDYIUUGYEUUCGUGZUUMYGUUEYHUUFYEUUCJUGUUMYF
|
|
UUDSUUNUHUIUJAYMFDUEZYPAYMUMBCFUORZHBUFZUPUQURZUUQCUFZIURZUEZUSZUUOUT
|
|
ZRZUUOUMGUVCOVAAUUOVEQZUVDUUOTAFVBQZDHQZUVEKMFDVBHVCVDUUOUVBVEVFVGVHZ
|
|
AYNFYODABFJKLVIAYOUUOSRZDAYMUUOSUVHUHAFUUPQZUVGUVIDTAUVFUVJKFVJVGZMFD
|
|
UUPHVKVDVLUIVMAYQUDQZUUBUUHUKAUVLVNZUUBUUHUVMUUBVNZUUDYSUPUQURZYSYTIU
|
|
RZUEZUUGUVNUUDYRUVBRZUVQUVMUUDUVRTUUBUVMUUCUVCRZYQUVCRZUVBRZUUDUVRUVM
|
|
YEUVBRZUUPHVQZQZUAUWCVOUUOUWCQZUVLUVSUWATUVMUWDUAUWCUVMYEUWCQZVNZUWBY
|
|
EVPRZYESRZUEZUVBRZUWCUWGYEUWJUVBUWFYEUWJTUVMYEUUPHVRVSUHUWGUWKUWHUPUQ
|
|
URZUWHUWIIURZUEZUWCUWGUWKUWHUWIUVBURZUWNUWHUWIUVBVTUWGUWHUUPQZUWIHQZU
|
|
WNUWCQZUWOUWNTUWFUWPUVMYEUUPHWAVSZUWFUWQUVMYEUUPHWBVSZUWGUWLUUPQZUWMH
|
|
QZUWRUWGUWPUXAUWSFUWHWCVGUWGUUTHQZCHVOBUUPVOZUXBAUXDUVLUWFAUXCBCUUPHN
|
|
WDWGUWGUWPUWQUXDUXBUKUWSUWTUXCUXBUWHUUSIURZHQBCUWHUWIUUPHUUQUWHTZUUTU
|
|
XEHUUQUWHUUSIWEZWFUUSUWITZUXEUWMHUUSUWIUWHIWHZWFWIVDWJUWLUWMUUPHWKWLZ
|
|
BCUWHUWIUUPHUVAUWNUVBUWLUXEUEUWCUXFUURUWLUUTUXEUUQUWHUPUQWEUXGUIUXHUX
|
|
EUWMUWLUXIWMUVBWNWOWPWQUXJWRWRWSUVMUVJUVGUWEAUVJUVLUVKWTAUVGUVLMWTFDU
|
|
UPHWKWLAUVLXAZUAUUOYQUWCUVBXEWPUUCGUVCOVAYRUVTUVBYQGUVCOVAXBXFWTUUBUV
|
|
MUVRUUAUVBRZUVQYRUUAUVBUGUVMUXLYSYTUVBURZUVQYSYTUVBVTUVMYSUUPQZYTHQZU
|
|
VQUWCQZUXMUVQTUVMBYQFJAUVFUVLKWTZLUXKXCZUVMYRUWCQUXOABCDFYQGHIJKLMNOX
|
|
DYRUUPHWBVGZUVMUVOUUPQZUVPHQZUXPUVMUXNUXTUXRFYSWCVGUVMUAUCYSYTUUPHHIA
|
|
YEUUPQUCUFZHQVNYEUYBIURHQUVLABCYEUYBUUPHHINXGXHUXRUXSXIZUVOUVPUUPHWKW
|
|
LUCUAYSYTUUPHUYBUPUQURZUYBYEIURZUEZUVQUVBUVOYSYEIURZUEUWCUYBYSTUYDUVO
|
|
UYEUYGUYBYSUPUQWEUYBYSYEIWEUIYEYTTUYGUVPUVOYEYTYSIWHWMBCUCUAUUPHUVAUY
|
|
FUYDUYBUUSIURZUEUUQUYBTUURUYDUUTUYHUUQUYBUPUQWEUUQUYBUUSIWEUIUUSYETUY
|
|
HUYEUYDUUSYEUYBIWHWMXJWOWPWQXMVLZUVNUUEUVOUUFUVPUVMUUEUVOTUUBUVMBYQFJ
|
|
UXQLUXKXKWTUVNUUFUVQSRZUVPUVNUUDUVQSUYIUHUVNUVOVBQZUYAUYJUVPTUVMUYKUU
|
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BUVMYSUVMBYQFJUXQLUXKXNXLWTUVMUYAUUBUYCWTUVOUVPVBHVKVDVLUIVMXOXPXQXRX
|
|
S $.
|
|
$}
|
|
|
|
$( The function ` R ` (used in the definition of the recursive definition
|
|
generator on upper integers) is a function defined for all natural
|
|
numbers. (Contributed by Jim Kingdon, 1-Apr-2022.) $)
|
|
frecuzrdgrcl $p |- ( ph -> R : _om --> ( ( ZZ>= ` C ) X. S ) ) $=
|
|
( vz cfv co cop wcel wceq com cuz cxp cv c1 caddc cmpt2 cfrec wral c1st
|
|
wf c2nd 1st2nd2 adantl fveq2d df-ov xp1st xp2nd peano2uz syl ralrimivva
|
|
wa adantr oveq1 eleq1d oveq2 rspc2v syl2anc mpd opelxp sylanbrc opeq12d
|
|
wi eqid ovmpt2g syl3anc syl5eqr eqeltrd ralrimiva cz uzid frecfcl feq1i
|
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opeq2d sylibr ) AUAEUBPZGUCZBCWFGBUDZUEUFQZWHCUDZHQZRZUGZEDRZUHZUKZUAWG
|
|
FUKAOUDZWMPZWGSZOWGUIWNWGSZWPAWSOWGAWQWGSZVBZWRWQUJPZWQULPZRZWMPZWGXBWQ
|
|
XEWMXAWQXETAWQWFGUMUNUOXBXFXCUEUFQZXCXDHQZRZWGXBXFXCXDWMQZXIXCXDWMUPXBX
|
|
CWFSZXDGSZXIWGSZXJXITXAXKAWQWFGUQUNZXAXLAWQWFGURUNZXBXGWFSZXHGSZXMXBXKX
|
|
PXNEXCUSUTXBWKGSZCGUIBWFUIZXQAXSXAAXRBCWFGMVAVCXBXKXLXSXQVMXNXOXRXQXCWJ
|
|
HQZGSBCXCXDWFGWHXCTZWKXTGWHXCWJHVDZVEWJXDTZXTXHGWJXDXCHVFZVEVGVHVIXGXHW
|
|
FGVJVKZBCXCXDWFGWLXIWMXGXTRWGYAWIXGWKXTWHXCUEUFVDYBVLYCXTXHXGYDWDWMVNVO
|
|
VPVQYEVRVRVSAEWFSZDGSWTAEVTSYFJEWAUTLEDWFGVJVKOWNWGWMWBVHUAWGFWONWCWE
|
|
$.
|
|
|
|
${
|
|
$d B x y $.
|
|
frecuzrdglem.b $e |- ( ph -> B e. ( ZZ>= ` C ) ) $.
|
|
$( A helper lemma for the value of a recursive definition generator on
|
|
upper integers. (Contributed by Jim Kingdon, 26-May-2020.) $)
|
|
frecuzrdglem $p |- ( ph ->
|
|
<. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R ) $=
|
|
( cfv com wcel syl2anc ccnv c2nd cop crn cuz wf1o f1ocnvdm frec2uzrdg
|
|
frec2uzf1od wceq f1ocnvfv2 opeq1d eqtrd wfn cxp frecuzrdgrcl fnfvelrn
|
|
wf ffn syl eqeltrrd ) AEJUAQZGQZEVCUBQZUCZGUDZAVCVBJQZVDUCVEABCDVBFGH
|
|
IJKLMNOARFUEQZJUFZEVHSZVBRSZABFJKLUIZPRVHEJUGTZUHAVGEVDAVIVJVGEUJVLPR
|
|
VHEJUKTULUMAGRUNZVKVCVFSARVHHUOZGURVNABCDFGHIJKLMNOUPRVOGUSUTVMRVBGUQ
|
|
TVA $.
|
|
$}
|
|
|
|
${
|
|
$d v w z T $.
|
|
frecuzrdgtcl.3 $e |- ( ph -> T = ran R ) $.
|
|
$( The recursive definition generator on upper integers is a function.
|
|
See comment in ~ frec2uz0d for the description of ` G ` as the
|
|
mapping from ` _om ` to ` ( ZZ>= `` C ) ` . (Contributed by Jim
|
|
Kingdon, 26-May-2020.) $)
|
|
frecuzrdgtcl $p |- ( ph -> T : ( ZZ>= ` C ) --> S ) $=
|
|
( vz vw wcel com vv cuz cfv wfn crn wss wf wfun cdm wceq wrel wbr wmo
|
|
cv wral cvv cxp wrex eleq2d wb frecuzrdgrcl ffn fvelrnb 3syl bitrd wa
|
|
frecuzrdgrrn eleq1 syl5ibcom rexlimdva sylbid ssrdv syl6ss df-rel cop
|
|
xpss sylibr wal wex ccnv c2nd frec2uzf1od f1ocnvdm sylan syldan xp2nd
|
|
wi syl cz adantr co adantlr simpr frec2uzrdg eqeq1d vex opth2 simplbi
|
|
wf1o syl6bi f1ocnvfv syld fveq2 fveq2d syl6 imp op2ndd adantl alrimiv
|
|
eqtr2d ex imbi2d albidv spcegv sylc nfv mo2r dmss dmxpss frecuzrdglem
|
|
eqeq2 mpbird opeldmg mpan eqssd pm5.32i df-br mobii 3imtr4i ralrimiva
|
|
dffun7 sylanbrc df-fn rnss rnxpss df-f ) AHEUBUCZUDZHUEZGUFYQGHUGAHUH
|
|
ZHUIZYQUJYRAHUKZUAUNZQUNZHULZQUMZUAUUAUOYTAHUPUPUQZUFUUBAHYQGUQZUUGAQ
|
|
HUUHAUUDHSZRUNZFUCZUUDUJZRTURZUUDUUHSZAUUIUUDFUEZSZUUMAHUUOUUDPUSATUU
|
|
HFUGZFTUDZUUPUUMUTABCDEFGIJKLMNOVAZTUUHFVBZRTUUDFVCVDVEAUULUUNRTAUUJT
|
|
SZVFZUUKUUHSUULUUNABCDEUUJFGIJKLMNOVGUUKUUDUUHVHVIVJVKVLZYQGVPVMHVNVQ
|
|
AUUFUAUUAAUUCYQSZVFZUUCUUDVOZHSZQUMZAUUCUUASZVFUUFUVEUVGUUDUUJUJZWGZQ
|
|
VRZRVSZUVHUVEUUCJVTUCZFUCZWAUCZGSZUVGUUDUVPUJZWGZQVRZUVMUVEUVOUUHSZUV
|
|
QAUVDUVNTSZUWAATYQJWSZUVDUWBABEJKLWBZTYQUUCJWCWDABCDEUVNFGIJKLMNOVGWE
|
|
UVOYQGWFWHZAUVTUVDAUVSQAUVGUUKUVFUJZRTURZUVRAUVGUVFUUOSZUWGAHUUOUVFPU
|
|
SAUUQUURUWHUWGUTUUSUUTRTUVFFVCVDVEAUWFUVRRTUVBUWFUVRUVBUWFVFUVPUUKWAU
|
|
CZUUDUVBUWFUVPUWIUJZUVBUWFUVNUUJUJZUWJUVBUWFUUJJUCZUUCUJZUWKUVBUWFUWL
|
|
UWIVOZUVFUJZUWMUVBUUKUWNUVFUVBBCDUUJEFGIJAEWISZUVAKWJLADGSZUVAMWJABUN
|
|
ZYQSCUNZGSVFZUWRUWSIWKGSZUVANWLOAUVAWMWNWOUWOUWMUWIUUDUJZUWLUWIUUCUUD
|
|
UAWPZQWPZWQWRWTAUWCUVAUWMUWKWGUWDTYQUUJUUCJXAWDXBUWKUVOUUKWAUVNUUJFXC
|
|
XDXEXFUWFUXBUVBUUCUUDUUKUXCUXDXGXHXJXKVJVKXIWJUVLUVTRUVPGUUJUVPUJZUVK
|
|
UVSQUXEUVJUVRUVGUUJUVPUUDYAXLXMXNXOUVGQRUVGRXPXQWHAUVIUVDAUUAYQUUCAUU
|
|
AYQAUUAUUHUIZYQAHUUHUFZUUAUXFUFUVCHUUHXRWHYQGXSVMAUAYQUUAAUVDUVIUVEUV
|
|
QUUCUVPVOZHSZUVIUWEUVEUXIUXHUUOSZUVEBCDUUCEFGIJAUWPUVDKWJLAUWQUVDMWJA
|
|
UWTUXAUVDNWLOAUVDWMXTAUXIUXJUTUVDAHUUOUXHPUSWJYBUUCUPSUVQUXIUVIWGUXCU
|
|
UCUVPHUPGYCYDXOXKVLYEZUSYFUUEUVGQUUCUUDHYGYHYIYJUAQHYKYLUXKHYQYMYLAYS
|
|
UUHUEZGAUXGYSUXLUFUVCHUUHYNWHYQGYOVMYQGHYPYL $.
|
|
|
|
$( Initial value of a recursive definition generator on upper integers.
|
|
See comment in ~ frec2uz0d for the description of ` G ` as the
|
|
mapping from ` _om ` to ` ( ZZ>= `` C ) ` . (Contributed by Jim
|
|
Kingdon, 27-May-2020.) $)
|
|
frecuzrdg0 $p |- ( ph -> ( T ` C ) = A ) $=
|
|
( wcel cfv c0 com wfun cop wceq cuz wf frecuzrdgtcl ffun syl cv caddc
|
|
crn c1 co cmpt2 cfrec fveq1i cvv cz opexg syl2anc frec0g frecuzrdgrcl
|
|
syl5eq wfn cxp peano1 fnfvelrn sylancl eqeltrrd eleqtrrd funopfv sylc
|
|
ffn ) AHUAZEDUBZHQEHRDUCAEUDRZGHUEVNABCDEFGHIJKLMNOPUFVPGHUGUHAVOFUKZ
|
|
HASFRZVOVQAVRSBCVPGBUIZULUJUMVSCUIIUMUBUNZVOUOZRZVOSFWAOUPAVOUQQZWBVO
|
|
UCAEURQDGQWCKMEDURGUSUTVOVTUQVAUHVCAFTVDZSTQVRVQQATVPGVEZFUEWDABCDEFG
|
|
IJKLMNOVBTWEFVMUHVFTSFVGVHVIPVJEDHVKVL $.
|
|
|
|
${
|
|
$d A w y z $. $d B w x y z $. $d C w x y z $. $d F w x y z $.
|
|
$d G w y z $. $d R w z $. $d S w x y z $. $d T w z $.
|
|
$d ph w x y z $.
|
|
$( Successor value of a recursive definition generator on upper
|
|
integers. See comment in ~ frec2uz0d for the description of ` G `
|
|
as the mapping from ` _om ` to ` ( ZZ>= `` C ) ` . (Contributed
|
|
by Jim Kingdon, 28-May-2020.) $)
|
|
frecuzrdgsuc $p |- ( ( ph /\ B e. ( ZZ>= ` C ) ) ->
|
|
( T ` ( B + 1 ) ) = ( B F ( T ` B ) ) ) $=
|
|
( cfv wcel wceq vz vw cuz wa c1 caddc ccnv csuc c2nd cop crn adantr
|
|
co cz cv adantlr peano2uz adantl frecuzrdglem eleqtrrd frecuzrdgtcl
|
|
wi wfun wf ffun syl funopfv mpd com wf1o frec2uzf1od f1ocnvdm sylan
|
|
frec2uzsucd f1ocnvfv2 oveq1d eqtrd peano2 f1ocnvfv fveq2d cmpt2 cxp
|
|
syl2an2r cfrec 1st2nd2 df-ov xp1st xp2nd oveq2 eleq1d oveq1 ralbidv
|
|
wral c1st ralrimivva ad2antrr rspcdva opelxp opeq12d opeq2d ovmpt2g
|
|
sylanbrc eqid syl3anc syl5eqr eqeltrd ralrimiva uzid frecsuc fveq1i
|
|
3eqtr4g frec2uzrdg syl6eqr frec2uzuzd frecuzrdgrrn caovclg cbvmpt2v
|
|
fveq2i mpdan caovcld 3eqtrd op2ndg syl2anc simpr eqcomd oveq12d ) A
|
|
EFUCRZSZUDZEUEUFUMZIRZEKUGZRZUHZGRZUIRZYMKRZYMGRZUIRZJUMZEEIRZJUMYI
|
|
YKYJYLRZGRZUIRZYPYIYJUUDUJZISZYKUUDTZYIUUEGUKZIYIBCDYJFGHJKAFUNSZYH
|
|
LULZMADHSZYHNULZABUOZYGSCUOZHSUDUUMUUNJUMZHSZYHOUPZPYHYJYGSAFEUQURZ
|
|
USAIUUHTYHQULZUTAUUFUUGVBZYHAIVCZUUTAYGHIVDUVAABCDFGHIJKLMNOPQVAYGH
|
|
IVEVFZYJUUDIVGVFULVHYIUUCYOUIYIUUBYNGYIYNKRZYJTZUUBYNTZYIUVCYQUEUFU
|
|
MZYJYIBYMFKUUJMAVIYGKVJZYHYMVISZABFKLMVKZVIYGEKVLVMZVNYIYQEUEUFAUVG
|
|
YHYQETUVIVIYGEKVOVMZVPZVQAUVGYHYNVISZUVDUVEVBUVIYIUVHUVMUVJYMVRVFVI
|
|
YGYNYJKVSWCVHVTVTVQYIYPUVFYTUJZUIRZYTYIYOUVNUIYIYOYRBCYGHUUMUEUFUMZ
|
|
UUOUJZWAZRZYQYSUVRUMZUVNYIYNUVRFDUJZWDZRZYMUWBRZUVRRZYOUVSYIUAUOZUV
|
|
RRZYGHWBZSZUAUWHWMUWAUWHSZUVHUWCUWETYIUWIUAUWHYIUWFUWHSZUDZUWGUWFWN
|
|
RZUWFUIRZUJZUVRRZUWHUWLUWFUWOUVRUWKUWFUWOTYIUWFYGHWEURVTUWLUWPUWMUE
|
|
UFUMZUWMUWNJUMZUJZUWHUWLUWPUWMUWNUVRUMZUWSUWMUWNUVRWFUWLUWMYGSZUWNH
|
|
SZUWSUWHSZUWTUWSTUWKUXAYIUWFYGHWGURZUWKUXBYIUWFYGHWHURZUWLUWQYGSZUW
|
|
RHSZUXCUWLUXAUXFUXDFUWMUQVFUWLUWMUUNJUMZHSZUXGCHUWNUUNUWNTZUXHUWRHU
|
|
UNUWNUWMJWIZWJUWLUUPCHWMZUXICHWMBYGUWMUUMUWMTZUUPUXICHUXMUUOUXHHUUM
|
|
UWMUUNJWKZWJWLAUXLBYGWMYHUWKAUUPBCYGHOWOWPUXDWQUXEWQUWQUWRYGHWRXBZB
|
|
CUWMUWNYGHUVQUWSUVRUWQUXHUJUWHUXMUVPUWQUUOUXHUUMUWMUEUFWKUXNWSUXJUX
|
|
HUWRUWQUXKWTUVRXCXAXDXEUXOXFXFXGYIFYGSZUUKUWJYIUUIUXPUUJFXHVFUULFDY
|
|
GHWRXBUVJUAUWAYMUWHUVRXIXDYNGUWBPXJYRUWDUVRYMGUWBPXJXRXKYIUVSYQYSUJ
|
|
ZUVRRUVTYIYRUXQUVRYIBCDYMFGHJKUUJMUULUUQPUVJXLVTYQYSUVRWFXMYIYQYGSY
|
|
SHSZUVNUWHSZUVTUVNTYIBYMFKUUJMUVJXNZYIYRUWHSZUXRYIUVHUYAUVJYIBCDFYM
|
|
GHJKUUJMUULUUQPXOXSYRYGHWHVFZYIUVFYGSZYTHSZUXSYIUVFYJYGUVLUURXFZYIU
|
|
AUBYQYSYGHHJYIBCUWFUBUOZYGHHJUUQXPUXTUYBXTZUVFYTYGHWRXBUAUBYQYSYGHU
|
|
WFUEUFUMZUWFUYFJUMZUJZUVNUVRUVFYQUYFJUMZUJUWHUWFYQTUYHUVFUYIUYKUWFY
|
|
QUEUFWKUWFYQUYFJWKWSUYFYSTUYKYTUVFUYFYSYQJWIWTBCUAUBYGHUVQUYJUYHUWF
|
|
UUNJUMZUJUUMUWFTUVPUYHUUOUYLUUMUWFUEUFWKUUMUWFUUNJWKWSUUNUYFTUYLUYI
|
|
UYHUUNUYFUWFJWIWTXQXAXDYAVTYIUYCUYDUVOYTTUYEUYGUVFYTYGHYBYCVQYIYQEY
|
|
SUUAJUVKYIUUAYSYIEYSUJZISZUUAYSTZYIUYMUUHIYIBCDEFGHJKUUJMUULUUQPAYH
|
|
YDUSUUSUTAUYNUYOVBZYHAUVAUYPUVBEYSIVGVFULVHYEYFYA $.
|
|
$}
|
|
$}
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x M $.
|
|
uzinf.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( An upper integer set is denumerable. (Contributed by Mario Carneiro,
|
|
15-Oct-2015.) $)
|
|
uzenom $p |- ( M e. ZZ -> Z ~~ _om ) $=
|
|
( vx cz wcel com cuz cfv cen cv c1 caddc co cmpt cfrec wf1o wbr id eqid
|
|
frec2uzf1od omex f1oen syl syl6breqr ensymd ) AEFZGBUGGAHIZBJUGGUHDEDKLMN
|
|
OAPZQGUHJRUGDAUIUGSUITUAGUHUIUBUCUDCUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d m n G $. $d n N $. $d x z $.
|
|
frecfzennn.1 $e |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) $.
|
|
$( The cardinality of a finite set of sequential integers. (See
|
|
~ frec2uz0d for a description of the hypothesis.) (Contributed by Jim
|
|
Kingdon, 18-May-2020.) $)
|
|
frecfzennn $p |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) ) $=
|
|
( vz c1 cv cfz co cfv cen wbr cc0 wceq c0 com wcel cn0 cvv cz vn vm caddc
|
|
ccnv oveq2 fveq2 breq12d 0ex enref fz10 cuz wf1o wa wtru frec2uzf1od trud
|
|
0zd peano1 frec2uz0d f1ocnvfv mp2 3brtr4i csn cun cin simpr peano2nn0 wfn
|
|
pm3.2i cmpt cfrec wal zex mptex vex ax-gen 0z frecfnom mp2an fneq1i mpbir
|
|
fvex omex fnex cnvex en2sn sylancl adantr fzp1disj a1i word f1ocnvdm mpan
|
|
wn nn0uz eleq2s nnord ordirr 3syl disjsn sylibr unen syl22anc cmin fveq2i
|
|
1z 1m1e0 eqtr4i eleq2i biimpi fzsuc2 sylancr peano2 syl jctil frec2uzsucd
|
|
csuc id f1ocnvfv2 oveq1d eqtrd sylc df-suc syl6eq 3brtr4d ex nn0ind ) FUA
|
|
GZHIZYHBUDZJZKLFMHIZMYJJZKLFUBGZHIZYNYJJZKLZFYNFUCIZHIZYRYJJZKLZFCHIZCYJJ
|
|
ZKLUAUBCYHMNYIYLYKYMKYHMFHUEYHMYJUFUGYHYNNYIYOYKYPKYHYNFHUEYHYNYJUFUGYHYR
|
|
NYIYSYKYTKYHYRFHUEYHYRYJUFUGYHCNYIUUBYKUUCKYHCFHUEYHCYJUFUGOOYLYMKOUHUIUJ
|
|
PMUKJZBULZOPQZUMOBJMNZYMONUUEUUFUUEUNAMBUNUQZDUOUPZURVIUUGUNAMBUUHDUSUPPU
|
|
UDOMBUTVAVBYNRQZYQUUAUUJYQUMZYOYRVCZVDZYPYPVCZVDZYSYTKUUKYQUULUUNKLZYOUUL
|
|
VEONZYPUUNVEONZUUMUUOKLUUJYQVFUUJUUPYQUUJYRRQYPSQUUPYNVGYNYJSSBBPVHZPSQBS
|
|
QUUSATAGFUCIZVJZMVKZPVHZEGZUVAJSQZEVLMTQUVCUVEEUVDUVASSATUUTVMVNEVOWBVPVQ
|
|
EMUVATVRVSPBUVBDVTWAWCPSBWDVSWEUBVOWBYRYPRSWFWGWHUUQUUKFYNWIWJUUKYPYPQWNZ
|
|
UURUUJUVFYQUUJYPPQZYPWKUVFUVGYNUUDRUUEYNUUDQZUVGUUIPUUDYNBWLWMWOWPZYPWQYP
|
|
WRWSWHYPYPWTXAYOYPUULUUNXBXCUUJYSUUMNZYQUUJFTQYNFFXDIZUKJZQZUVJXFUUJUVMRU
|
|
VLYNRUUDUVLWOUVKMUKXGXEXHXIXJFYNXKXLWHUUKYTYPXQZUUOUUJYTUVNNZYQUUJUUEUVNP
|
|
QZUMUVNBJZYRNUVOUUJUVPUUEUUJUVGUVPUVIYPXMXNUUIXOUUJUVQYPBJZFUCIZYRUUJUVGU
|
|
VQUVSNUVIUVGAYPMBUVGUQDUVGXRXPXNUUJUVRYNFUCUUJUUEUVHUVRYNNUUIUUJUVHRUUDYN
|
|
WOXIXJPUUDYNBXSXLXTYAPUUDUVNYRBUTYBWHYPYCYDYEYFYG $.
|
|
|
|
$( The cardinality of a finite set of sequential integers with arbitrary
|
|
endpoints. (Contributed by Jim Kingdon, 18-May-2020.) $)
|
|
frecfzen2 $p |- ( N e. ( ZZ>= ` M ) ->
|
|
( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) ) $=
|
|
( cfv wcel cfz co c1 caddc cmin cen wbr cz cc wceq ax-1cn zcn syl2anc cuz
|
|
ccnv eluzel2 eluzelz 1z zsubcl sylancr fzen syl3anc zcnd pncan3 addsubass
|
|
sylancl mp3an2 syl2an eqcomd oveq12d breqtrd peano2uz uznn0sub frecfzennn
|
|
cn0 3syl entr ) DCUAFZGZCDHIZJDJKIZCLIZHIZMNVJVIBUBFZMNZVGVKMNVFVGCJCLIZK
|
|
IZDVMKIZHIZVJMVFCOGZDOGZVMOGZVGVPMNCDUCZCDUDZVFJOGVQVSUEVTJCUFUGVMCDUHUIV
|
|
FVNJVOVIHVFCPGZJPGZVNJQVFCVTUJRCJUKUMVFVIVOVFVRVQVIVOQZWAVTVRDPGZWBWDVQDS
|
|
CSWEWCWBWDRDJCULUNUOTUPUQURVFVHVEGVIVBGVLCDUSCVHUTABVIEVAVCVGVJVKVDT $.
|
|
|
|
$( ` G ` maps ` _om ` one-to-one onto ` NN0 ` . (Contributed by Jim
|
|
Kingdon, 19-May-2020.) $)
|
|
frechashgf1o $p |- G : _om -1-1-onto-> NN0 $=
|
|
( com cn0 wf1o cc0 cuz cfv wtru 0zd frec2uzf1od trud wceq wb nn0uz f1oeq3
|
|
ax-mp mpbir ) DEBFZDGHIZBFZUBJAGBJKCLMEUANTUBOPEUADBQRS $.
|
|
$}
|
|
|
|
${
|
|
$d C x $. $d ph x $.
|
|
frec2uzled.1 $e |- ( ph -> C e. ZZ ) $.
|
|
frec2uzled.2 $e |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) $.
|
|
frec2uzled.a $e |- ( ph -> A e. _om ) $.
|
|
frec2uzled.b $e |- ( ph -> B e. _om ) $.
|
|
$( The mapping ` G ` (see ~ frec2uz0d ) preserves order. (Contributed by
|
|
Jim Kingdon, 24-Feb-2022.) $)
|
|
frec2uzled $p |- ( ph -> ( A C_ B <-> ( G ` A ) <_ ( G ` B ) ) ) $=
|
|
( wcel wceq wo cfv wbr com wb syl2anc cz frec2uzzd clt wss cle wf1o f1of1
|
|
frec2uzlt2d cuz wf1 frec2uzf1od syl f1fveq syl12anc bicomd orbi12d zleloe
|
|
nnsseleq 3bitr4d ) ACDKZCDLZMZCFNZDFNZUAOZVAVBLZMZCDUBZVAVBUCOZAURVCUSVDA
|
|
BCDEFGHIJUFAVDUSAPEUGNZFUHZCPKZDPKZVDUSQAPVHFUDVIABEFGHUIPVHFUEUJIJPVHCDF
|
|
UKULUMUNAVJVKVFUTQIJCDUPRAVASKVBSKVGVEQABCEFGHITABDEFGHJTVAVBUORUQ $.
|
|
$}
|
|
|
|
$( A finite interval of integers is finite. (Contributed by Jim Kingdon,
|
|
19-May-2020.) $)
|
|
fzfig $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) e. Fin ) $=
|
|
( vx cz wcel wa cle wbr cfz co cfn wn cuz cfv c1 caddc com cn0 syl c0 cfrec
|
|
eluz cmin cv cmpt cc0 ccnv cen wf1o frechashgf1o peano2uz uznn0sub f1ocnvdm
|
|
eqid sylancr nnfi frecfzen2 enfii syl2anc syl6bir wceq clt wb zltnle ancoms
|
|
fzn bitr3d 0fin eleq1 mpbiri syl6bi wdc wo zdcle df-dc sylib mpjaod ) ADEZB
|
|
DEZFZABGHZABIJZKEZWALZVTWABAMNZEZWCABUBWFBOPJZAUCJZCDCUDOPJUEUFUAZUGNZKEZWB
|
|
WJUHHWCWFWJQEZWKWFQRWIUIWHREZWLCWIWIUNZUJWFWGWEEWMABUKAWGULSQRWHWIUMUOWJUPS
|
|
CWIABWNUQWBWJURUSUTVTWDWBTVAZWCVTBAVBHZWDWOVSVRWPWDVCBAVDVEABVFVGWOWCTKEVHW
|
|
BTKVIVJVKVTWAVLWAWDVMABVNWAVOVPVQ $.
|
|
|
|
${
|
|
fzfigd.m $e |- ( ph -> M e. ZZ ) $.
|
|
fzfigd.n $e |- ( ph -> N e. ZZ ) $.
|
|
$( Deduction form of ~ fzfig . (Contributed by Jim Kingdon,
|
|
21-May-2020.) $)
|
|
fzfigd $p |- ( ph -> ( M ... N ) e. Fin ) $=
|
|
( cz wcel cfz co cfn fzfig syl2anc ) ABFGCFGBCHIJGDEBCKL $.
|
|
$}
|
|
|
|
$( Half-open integer sets are finite. (Contributed by Jim Kingdon,
|
|
21-May-2020.) $)
|
|
fzofig $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ..^ N ) e. Fin ) $=
|
|
( cz wcel wa cfzo co cmin cfz cfn wceq fzoval adantl peano2zm fzfig eqeltrd
|
|
c1 sylan2 ) ACDZBCDZEABFGZABQHGZIGZJTUAUCKSABLMTSUBCDUCJDBNAUBORP $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( The nonnegative integers are equinumerous to the positive integers.
|
|
(Contributed by NM, 19-Jul-2004.) $)
|
|
nn0ennn $p |- NN0 ~~ NN $=
|
|
( vx vy cn cv c1 caddc co cmin nn0ex nnex nn0p1nn nnm1nn0 wcel cc wceq wb
|
|
cn0 nncn ax-1cn eqcom nn0cn wa subadd mp3an2 3bitr4g addcom eqeq2d adantl
|
|
mpan bitrd syl2anr en3i ) ABQCADZEFGZBDZEHGZIJUMKUOLUOCMUONMZUMNMZUMUPOZU
|
|
OUNOZPUMQMUORUMUAUQURUBZUSUOEUMFGZOZUTVAUPUMOZVBUOOZUSVCUQENMZURVDVEPSUOE
|
|
UMUCUDUMUPTUOVBTUEURVCUTPUQURVBUNUOVFURVBUNOSEUMUFUIUGUHUJUKUL $.
|
|
|
|
$( The set of positive integers (as a subset of complex numbers) is
|
|
equinumerous to omega (the set of finite ordinal numbers). (Contributed
|
|
by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) $)
|
|
nnenom $p |- NN ~~ _om $=
|
|
( vx com cn0 cn cvv wcel cz cv c1 caddc cmpt cc0 cfrec wf1o cen wbr nn0ex
|
|
co omex eqid frechashgf1o f1oen2g mp3an nn0ennn entr2i ) BCDBEFCEFBCAGAHI
|
|
JRKLMZNBCOPSQAUFUFTUABCUFEEUBUCUDUE $.
|
|
$}
|
|
|
|
$( ` NN ` is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) $)
|
|
nnct $p |- NN ~<_ _om $=
|
|
( cn com cen wbr cdom nnenom endom ax-mp ) ABCDABEDFABGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Strong induction over upper sets of integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d ch x $. $d M k w x y $. $d N w x $. $d ph k w y $. $d ps x $.
|
|
uzsinds.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
uzsinds.2 $e |- ( x = N -> ( ph <-> ch ) ) $.
|
|
uzsinds.3 $e |- ( x e. ( ZZ>= ` M ) ->
|
|
( A. y e. ( M ... ( x - 1 ) ) ps -> ph ) ) $.
|
|
$( Strong (or "total") induction principle over an upper set of integers.
|
|
(Contributed by Scott Fenton, 16-May-2014.) $)
|
|
uzsinds $p |- ( N e. ( ZZ>= ` M ) -> ch ) $=
|
|
( wcel cfz co wral c1 wceq oveq2 raleqdv cz cmin vw vk cuz cfv caddc wsbc
|
|
cv csn c0 ral0 clt wbr zre ltm1d wb peano2zm mpdan mpbid mpbiri uzid rgen
|
|
fzn wi nfv nfsbc1v nfim oveq1 oveq2d sbceq1a imbi12d mpisyl ralsns mpbird
|
|
rspc mpd fzsn cun simpr cbvralv sylib eluzelz adantr zcnd pncand peano2uz
|
|
wa 1cnd sylbird peano2zd syl ralun syl2anc fzsuc uzind4 eluzfz2 rspcdva
|
|
ex ) GFUCUDZKACDFGLMZGIADFUAUGZLMZNADFFLMZNZADFUBUGZLMZNZADFXDOUEMZLMZNZA
|
|
DWSNUAUBFGWTFPADXAXBWTFFLQRWTXDPADXAXEWTXDFLQRWTXGPADXAXHWTXGFLQRWTGPADXA
|
|
WSWTGFLQRFSKZXCADFUHZNZXJXLADFUFZXJBEFFOTMZLMZNZXMXJXPBEUINBEUJXJBEXOUIXJ
|
|
XNFUKULZXOUIPZXJFFUMUNXJXNSKXQXRUOFUPFXNVBUQURRUSXJFWRKBEFDUGZOTMZLMZNZAV
|
|
CZDWRNZXPXMVCZFUTYCDWRJVAZYCYEDFWRXPXMDXPDVDADFVEVFXSFPZYBXPAXMYGBEYAXOYG
|
|
XTXNFLXSFOTVGVHRADFVIVJVNVKVOADFSVLVMXJADXBXKFVPRVMXDWRKZXFXIYHXFWFZXIADX
|
|
EXGUHZVQZNZYIXFADYJNZYLYHXFVRZYIYMADXGUFZYIBEXENZYOYIXFYPYNABDEXEHVSVTYIY
|
|
PBEFXGOTMZLMZNZYOYIBEYRXEYIYQXDFLYIXDOYIXDYHXDSKXFFXDWAZWBWCYIWGWDVHRYIXG
|
|
WRKZYDYSYOVCZYHUUAXFFXDWEWBYFYCUUBDXGWRYSYODYSDVDADXGVEVFXSXGPZYBYSAYOUUC
|
|
BEYAYRUUCXTYQFLXSXGOTVGVHRADXGVIVJVNVKWHVOYIXGSKZYMYOUOYHUUDXFYHXDYTWIWBA
|
|
DXGSVLWJVMADXEYJWKWLYHXIYLUOXFYHADXHYKFXDWMRWBVMWQWNFGWOWP $.
|
|
$}
|
|
|
|
${
|
|
$d ch x $. $d N x $. $d ph y $. $d ps x $. $d x y $.
|
|
nnsinds.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
nnsinds.2 $e |- ( x = N -> ( ph <-> ch ) ) $.
|
|
nnsinds.3 $e |- ( x e. NN ->
|
|
( A. y e. ( 1 ... ( x - 1 ) ) ps -> ph ) ) $.
|
|
$( Strong (or "total") induction principle over the naturals. (Contributed
|
|
by Scott Fenton, 16-May-2014.) $)
|
|
nnsinds $p |- ( N e. NN -> ch ) $=
|
|
( cn wcel c1 cuz cfv elnnuz cv cmin co cfz wral wi sylbir uzsinds sylbi )
|
|
FJKFLMNZKCFOABCDELFGHDPZUEKUFJKBELUFLQRSRTAUAUFOIUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d ch x $. $d N x $. $d ph y $. $d ps x $. $d x y $.
|
|
nn0sinds.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
nn0sinds.2 $e |- ( x = N -> ( ph <-> ch ) ) $.
|
|
nn0sinds.3 $e |- ( x e. NN0 ->
|
|
( A. y e. ( 0 ... ( x - 1 ) ) ps -> ph ) ) $.
|
|
$( Strong (or "total") induction principle over the nonnegative integers.
|
|
(Contributed by Scott Fenton, 16-May-2014.) $)
|
|
nn0sinds $p |- ( N e. NN0 -> ch ) $=
|
|
( cn0 wcel cc0 cuz cfv elnn0uz cv c1 cmin co cfz wi sylbir uzsinds sylbi
|
|
wral ) FJKFLMNZKCFOABCDELFGHDPZUFKUGJKBELUGQRSTSUEAUAUGOIUBUCUD $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The infinite sequence builder "seq"
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c seq $.
|
|
$( Extend class notation with recursive sequence builder. $)
|
|
cseq $a class seq M ( .+ , F , S ) $.
|
|
|
|
${
|
|
$d .+ x y $. $d F x y $. $d M x y $. $d S x y $.
|
|
$( Define a general-purpose operation that builds a recursive sequence
|
|
(i.e. a function on the positive integers ` NN ` or some other upper
|
|
integer set) whose value at an index is a function of its previous value
|
|
and the value of an input sequence at that index. This definition is
|
|
complicated, but fortunately it is not intended to be used directly.
|
|
Instead, the only purpose of this definition is to provide us with an
|
|
object that has the properties expressed by ~ iseq1 and ~ iseqp1 .
|
|
Typically, those are the main theorems that would be used in practice.
|
|
|
|
The first operand in the parentheses is the operation that is applied to
|
|
the previous value and the value of the input sequence (second operand).
|
|
The operand to the left of the parenthesis is the integer to start from.
|
|
For example, for the operation ` + ` , an input sequence ` F ` with
|
|
values 1, 1/2, 1/4, 1/8,... would be transformed into the output
|
|
sequence ` seq 1 ( + , F , QQ ) ` with values 1, 3/2, 7/4, 15/8,.., so
|
|
that ` ( seq 1 ( + , F , QQ ) `` 1 ) = 1 ` ,
|
|
` ( seq 1 ( + , F , QQ ) `` 2 ) = ` 3/2, etc. In other words,
|
|
` seq M ( + , F , QQ ) ` transforms a sequence ` F ` into an infinite
|
|
series.
|
|
|
|
Internally, the ` frec ` function generates as its values a set of
|
|
ordered pairs starting at ` <. M , ( F `` M ) >. ` , with the first
|
|
member of each pair incremented by one in each successive value. So,
|
|
the range of ` frec ` is exactly the sequence we want, and we just
|
|
extract the range and throw away the domain.
|
|
|
|
(Contributed by Jim Kingdon, 29-May-2020.) $)
|
|
df-iseq $a |- seq M ( .+ , F , S ) = ran frec (
|
|
( x e. ( ZZ>= ` M ) , y e. S |->
|
|
<. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) ,
|
|
<. M , ( F ` M ) >. ) $.
|
|
$}
|
|
|
|
${
|
|
$d .+ x y $. $d F x y $. $d M x y $. $d S x y $.
|
|
$( Existence of the sequence builder operation. (Contributed by Jim
|
|
Kingdon, 20-Aug-2021.) $)
|
|
iseqex $p |- seq M ( .+ , F , S ) e. _V $=
|
|
( vx vy cseq cuz cfv cv c1 caddc cop cmpt2 cfrec crn cvv df-iseq frecex
|
|
co rnex eqeltri ) ABCDGEFDHIBEJKLTZFJUCCIATMNZDDCIMZOZPQEFABCDRUFUEUDSUAU
|
|
B $.
|
|
$}
|
|
|
|
${
|
|
$d x y F $. $d x y .+ $. $d x y G $. $d x y M $. $d x y N $.
|
|
$d x y Q $. $d x y S $. $d x y T $.
|
|
$( Equality theorem for the sequence builder operation. (Contributed by
|
|
Jim Kingdon, 30-May-2020.) $)
|
|
iseqeq1 $p |- ( M = N -> seq M ( .+ , F , S ) = seq N ( .+ , F , S ) ) $=
|
|
( vx vy wceq cuz cfv cv co cop cmpt2 cfrec crn cseq fveq2 syl df-iseq c1
|
|
caddc opeq12d freceq2 eqid mpt2eq12 sylancl freceq1 eqtrd rneqd 3eqtr4g
|
|
id ) DEHZFGDIJZBFKUAUBLZGKUOCJALMZNZDDCJZMZOZPFGEIJZBUPNZEECJZMZOZPABCDQA
|
|
BCEQUMUTVEUMUTUQVDOZVEUMUSVDHUTVFHUMDEURVCUMULDECRUCUSVDUQUDSUMUQVBHZVFVE
|
|
HUMUNVAHBBHVGDEIRBUEFGUNBVABUPUFUGVDUQVBUHSUIUJFGABCDTFGABCETUK $.
|
|
|
|
$( Equality theorem for the sequence builder operation. (Contributed by
|
|
Jim Kingdon, 30-May-2020.) $)
|
|
iseqeq2 $p |- ( .+ = Q -> seq M ( .+ , F , S ) = seq M ( Q , F , S ) ) $=
|
|
( vx vy wceq cuz cfv cv c1 co cop cmpt2 cfrec crn cseq wcel df-iseq caddc
|
|
w3a simp1 oveqd opeq2d mpt2eq3dva freceq1 syl rneqd 3eqtr4g ) ABHZFGEIJZC
|
|
FKZLUAMZGKZUNDJZAMZNZOZEEDJNZPZQFGULCUNUOUPBMZNZOZUTPZQACDERBCDERUKVAVEUK
|
|
USVDHVAVEHUKFGULCURVCUKUMULSZUOCSZUBZUQVBUNVHABUOUPUKVFVGUCUDUEUFUTUSVDUG
|
|
UHUIFGACDETFGBCDETUJ $.
|
|
|
|
$( Equality theorem for the sequence builder operation. (Contributed by
|
|
Jim Kingdon, 30-May-2020.) $)
|
|
iseqeq3 $p |- ( F = G -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) ) $=
|
|
( vx vy wceq cuz cfv cv co cop cmpt2 cfrec crn cseq wcel opeq2d df-iseq
|
|
c1 caddc w3a simp1 fveq1d oveq2d mpt2eq3dva fveq1 freceq1 freceq2 syl2anc
|
|
sylan9eq rneqd 3eqtr4g ) CDHZFGEIJZBFKZUAUBLZGKZURCJZALZMZNZEECJZMZOZPFGU
|
|
PBURUSURDJZALZMZNZEEDJZMZOZPABCEQABDEQUOVFVMUOVCVJHZVEVLHZVFVMHUOFGUPBVBV
|
|
IUOUQUPRZUSBRZUCZVAVHURVRUTVGUSAVRURCDUOVPVQUDUEUFSUGUOVDVKEECDUHSVNVOVFV
|
|
JVEOVMVEVCVJUIVEVLVJUJULUKUMFGABCETFGABDETUN $.
|
|
|
|
$( Equality theorem for the sequence builder operation. (Contributed by
|
|
Jim Kingdon, 30-May-2020.) $)
|
|
iseqeq4 $p |- ( S = T -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) ) $=
|
|
( vx vy wceq cuz cfv cv c1 caddc co cop cmpt2 cfrec crn cseq df-iseq eqid
|
|
mpt2eq12 mpan freceq1 syl rneqd 3eqtr4g ) BCHZFGEIJZBFKLMNZGKUJDJANOZPZEE
|
|
DJOZQZRFGUICUKPZUMQZRABDESACDESUHUNUPUHULUOHZUNUPHUIUIHUHUQUIUAFGUIBUICUK
|
|
UBUCUMULUOUDUEUFFGABDETFGACDETUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z F $. $d y z .+ $. $d y z M $. $d y z S $.
|
|
nfiseq.1 $e |- F/_ x M $.
|
|
nfiseq.2 $e |- F/_ x .+ $.
|
|
nfiseq.3 $e |- F/_ x F $.
|
|
nfiseq.4 $e |- F/_ x S $.
|
|
$( Hypothesis builder for the sequence builder operation. (Contributed by
|
|
Jim Kingdon, 30-May-2020.) $)
|
|
nfiseq $p |- F/_ x seq M ( .+ , F , S ) $=
|
|
( vy vz cseq cuz cfv cv co cop nfcv nffv nfop c1 caddc cmpt2 df-iseq nfov
|
|
cfrec crn nfmpt2 nffrec nfrn nfcxfr ) ABCDELJKEMNZCJOUAUBPZKOZUMDNZBPZQZU
|
|
CZEEDNZQZUFZUGJKBCDEUDAVAAUTURJKAULCUQAEMAMRFSIAUMUPAUMRZAUNUOBAUNRGAUMDH
|
|
VBSUETUHAEUSFAEDHFSTUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d F w x y z $. $d .+ w x y z $. $d S w x y z $. $d w x y z ph $.
|
|
$d M w x z $.
|
|
iseqovex.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqovex.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Closure of a function used in proving sequence builder theorems. This
|
|
can be thought of as a lemma for the small number of sequence builder
|
|
theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.) $)
|
|
iseqovex $p |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x
|
|
( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) )
|
|
y ) e. S ) $=
|
|
( cv cfv wcel wa c1 caddc co weq simprr cmpt2 eqidd simprl oveq1d oveq12d
|
|
cuz fveq2d caovclg adantlr wral peano2uz syl ralrimiva fveq2 eleq1d sylib
|
|
cbvralv adantr wceq rspcv sylc caovcld ovmpt2d eqeltrd ) ABLZIUFMZNZCLZGN
|
|
ZOZOZVEVHDEVFGELZDLZPQRZHMZFRZUAZRVHVEPQRZHMZFRZGVKDEVEVHVFGVPVTVQGVKVQUB
|
|
VKDBSZECSZOOZVLVHVOVSFVKWAWBTWCVNVRHWCVMVEPQVKWAWBUCUDUGUEAVGVIUCZAVGVITZ
|
|
VKDEVHVSGGGFAVMGNVLGNOVMVLFRGNVJABCVMVLGGGFKUHUIWEVKVRVFNZVMHMZGNZDVFUJZV
|
|
SGNZVKVGWFWDIVEUKULAWIVJAVEHMZGNZBVFUJWIAWLBVFJUMWLWHBDVFBDSWKWGGVEVMHUNU
|
|
OUQUPURWHWJDVRVFVMVRUSWGVSGVMVRHUNUOUTVAVBZVCWMVD $.
|
|
$}
|
|
|
|
${
|
|
$d F a b w x y z $. $d .+ a b w x y z $. $d M a b w x y z $.
|
|
$d S a b w x y z $. $d a b x y ph $.
|
|
iseqval.1 $e |- R = frec (
|
|
( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) ,
|
|
( x ( z e. ( ZZ>= ` M ) , w e. S |->
|
|
( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) ,
|
|
<. M , ( F ` M ) >. ) $.
|
|
iseqval.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqval.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Value of the sequence builder function. (Contributed by Jim Kingdon,
|
|
30-May-2020.) $)
|
|
iseqval $p |- ( ph -> seq M ( .+ , F , S ) = ran R ) $=
|
|
( cfv cv c1 caddc co wceq wcel va crn cuz cop cmpt2 cfrec cseq w3a simprl
|
|
vb wa simprr caovclg adantlr wral ralrimiva fveq2 eleq1d cbvralv sylib wi
|
|
adantr peano2uz rspcv syl ad2antrl mpd caovcld fveq2d oveq2d eqid ovmpt2g
|
|
oveq1 syl3anc 3impb opeq2d mpt2eq3dva freceq1 syl5eq df-iseq syl6reqr
|
|
rneqd ) AGUBBCJUCNZHBOZPQRZCOZWEINZFRZUDZUEZJJINUDZUFZUBFHIJUGAGWLAGBCWCH
|
|
WEWDWFDEWCHEOZDOZPQRZINZFRZUEZRZUDZUEZWKUFZWLKAXAWJSXBWLSABCWCHWTWIAWDWCT
|
|
ZWFHTZUHWSWHWEAXCXDWSWHSZAXCXDUKZUKZXCXDWHHTXEAXCXDUIAXCXDULZXGUAUJWFWGHH
|
|
HFAUAOZHTUJOZHTUKXIXJFRHTXFABCXIXJHHHFMUMUNXHXGWFINZHTZCWCUOZWGHTZAXMXFAW
|
|
DINZHTZBWCUOXMAXPBWCLUPXPXLBCWCWDWFSXOXKHWDWFIUQURUSUTVBXCXMXNVAZAXDXCWEW
|
|
CTXQJWDVCXLXNCWEWCWFWESXKWGHWFWEIUQURVDVEVFVGVHDEWDWFWCHWQWHWRWMWGFRHWNWD
|
|
SZWPWGWMFXRWOWEIWNWDPQVMVIVJWMWFWGFVMWRVKVLVNVOVPVQWKXAWJVRVEVSWBBCFHIJVT
|
|
WA $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z F $. $d w x y z .+ $. $d w x y z M $. $d x y z w S $.
|
|
$d x y z w ph $.
|
|
iseq1.m $e |- ( ph -> M e. ZZ ) $.
|
|
iseq1.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseq1.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Value of the sequence builder function at its initial value.
|
|
(Contributed by Jim Kingdon, 31-May-2020.) $)
|
|
iseq1 $p |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) ) $=
|
|
( vz vw cfv cv c1 caddc co cmpt2 cop wcel cfrec cseq cmpt eqid wceq fveq2
|
|
cuz cz eleq1d ralrimiva uzid syl rspcdva iseqovex iseqval frecuzrdg0 ) AB
|
|
CGFMZGBCGUGMZEBNZOPQZUSCNKLURELNKNOPQFMDQRZQSRGUQSUAZEDEFGUBVABUHUTUCGUAZ
|
|
HVCUDAUSFMZETZUQETBURGUSGUEVDUQEUSGFUFUIAVEBURIUJAGUHTGURTHGUKULUMABCKLDE
|
|
FGIJUNVBUDZABCKLDVBEFGVFIJUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d .+ x y $. $d .+ w x y z $. $d F x y $. $d F w x y z $. $d M x y $.
|
|
$d M w x y z $. $d S x y $. $d S w x y z $. $d Z x $. $d ph x y $.
|
|
$d ph w x y z $.
|
|
iseqfcl.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
iseqfcl.2 $e |- ( ph -> M e. ZZ ) $.
|
|
iseqfcl.3 $e |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S ) $.
|
|
iseqfcl.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Range of the recursive sequence builder. (Contributed by Jim Kingdon,
|
|
11-Apr-2022.) $)
|
|
iseqfcl $p |- ( ph -> seq M ( .+ , F , S ) : Z --> S ) $=
|
|
( vz vw cfv wf cv c1 co wcel cuz cseq caddc cmpt2 cfrec cz cmpt eqid wceq
|
|
cop fveq2 eleq1d ralrimiva syl syl6eleqr rspcdva eleq2i sylan2br iseqovex
|
|
uzid iseqval frecuzrdgtcl feq2i sylibr ) AGUAOZEDEFGUBZPHEVFPABCGFOZGBCVE
|
|
EBQZRUCSZVHCQMNVEENQMQRUCSFODSUDZSUJUDGVGUJUEZEVFVJBUFVIUGGUEZJVLUHAVHFOZ
|
|
ETZVGETBHGVHGUIVMVGEVHGFUKULAVNBHKUMAGVEHAGUFTGVETJGUTUNIUOUPABCMNDEFGVHV
|
|
ETAVHHTVNHVEVHIUQKURZLUSVKUHZABCMNDVKEFGVPVOLVAVBHVEEVFIVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z F $. $d w x y z .+ $. $d w x y z M $. $d x y z w S $.
|
|
$d x y z w ph $.
|
|
iseqfn.m $e |- ( ph -> M e. ZZ ) $.
|
|
iseqfn.ex $e |- ( ph -> S e. V ) $.
|
|
iseqfn.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqfn.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( The sequence builder function is a function. Because this theorem
|
|
needlessly requires that ` S ` be a set, new proofs should use ~ iseqfcl
|
|
instead. (Contributed by Jim Kingdon, 30-May-2020.)
|
|
(New usage is discouraged.) $)
|
|
iseqfn $p |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) ) $=
|
|
( vz vw cfv cv c1 caddc co wcel cuz cmpt2 cop cseq cz cmpt eqid wral uzid
|
|
cfrec ralrimiva wceq fveq2 eleq1d rspcv sylc iseqovex iseqval frecuzrdgfn
|
|
syl ) ABCGFOZGBCGUAOZEBPZQRSZVCCPMNVBENPMPQRSFODSUBZSUCUBGVAUCUJZEDEFGUDV
|
|
EBUEVDUFGUJZHIVGUGJAGVBTZVCFOZETZBVBUHVAETZAGUETVHIGUIUTAVJBVBKUKVJVKBGVB
|
|
VCGULVIVAEVCGFUMUNUOUPABCMNDEFGKLUQVFUGZABCMNDVFEFGVLKLURUS $.
|
|
$}
|
|
|
|
${
|
|
$d .+ w x y z $. $d F w x y z $. $d M w x y z $. $d N x y $.
|
|
$d S w x y z $. $d ph w x y z $.
|
|
iseqcl.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqcl.2 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqcl.3 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Closure property of the recursive sequence builder. (Contributed by Jim
|
|
Kingdon, 1-Jun-2020.) $)
|
|
iseqcl $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) e. S ) $=
|
|
( vz vw cfv cv c1 caddc co cmpt2 wcel cuz cseq cop cfrec cmpt eluzel2 syl
|
|
cz eqid wceq fveq2 eleq1d ralrimiva rspcdva iseqovex iseqval frecuzrdgtcl
|
|
uzid ffvelrnd ) AGUANZEHDEFGUBZABCGFNZGBCUTEBOZPQRZVCCOLMUTEMOLOPQRFNDRSZ
|
|
RUCSGVBUCUDZEVAVEBUHVDUEGUDZAHUTTGUHTZIGHUFUGZVGUIAVCFNZETZVBETBUTGVCGUJV
|
|
JVBEVCGFUKULAVKBUTJUMAVHGUTTVIGURUGUNABCLMDEFGJKUOVFUIZABCLMDVFEFGVLJKUPU
|
|
QIUS $.
|
|
$}
|
|
|
|
${
|
|
$d k x y .+ $. $d k x y F $. $d k x y M $. $d k x y ph $. $d k x y S $.
|
|
$d x Z $.
|
|
iseqf.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
iseqf.ex $e |- ( ph -> S e. V ) $.
|
|
iseqf.2 $e |- ( ph -> M e. ZZ ) $.
|
|
iseqf.3 $e |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S ) $.
|
|
iseqf.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Range of the recursive sequence builder. Because this theorem
|
|
needlessly requires that ` S ` be a set, new proofs should use ~ iseqfcl
|
|
instead. (Contributed by Jim Kingdon, 23-Jul-2021.)
|
|
(New usage is discouraged.) $)
|
|
iseqf $p |- ( ph -> seq M ( .+ , F , S ) : Z --> S ) $=
|
|
( iseqfcl ) ABCDEFGIJLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d .+ w x y z $. $d F w x y z $. $d M w x y z $. $d N w x y z $.
|
|
$d S w x y z $. $d ph w x y z $.
|
|
iseqp1.m $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqp1.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqp1.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Value of the sequence builder function at a successor. (Contributed by
|
|
Jim Kingdon, 31-May-2020.) $)
|
|
iseqp1 $p |- ( ph ->
|
|
( seq M ( .+ , F , S ) ` ( N + 1 ) ) =
|
|
( ( seq M ( .+ , F , S ) ` N ) .+ ( F ` ( N + 1 ) ) ) ) $=
|
|
( vz vw c1 caddc co cfv cv wcel wceq cseq cuz cmpt2 cfrec cz cmpt eluzel2
|
|
cop syl eqid fveq2 eleq1d ralrimiva rspcdva iseqovex iseqval frecuzrdgsuc
|
|
mpdan iseqcl peano2uz caovcld oveq1 fveq2d oveq2d ovmpt2g syl3anc eqtrd
|
|
uzid ) AHNOPZDEFGUAZQZHHVJQZLMGUBQZEMRZLRZNOPZFQZDPZUCZPZVLVIFQZDPZAHVMSZ
|
|
VKVTTIABCGFQZHGBCVMEBRZNOPZWECRVSPUHUCGWDUHUDZEVJVSBUEWFUFGUDZAWCGUESZIGH
|
|
UGUIZWHUJAWEFQZESZWDESBVMGWEGTWKWDEWEGFUKULAWLBVMJUMZAWIGVMSWJGVHUIUNABCL
|
|
MDEFGJKUOWGUJZABCLMDWGEFGWNJKUPUQURAWCVLESWBESVTWBTIABCDEFGHIJKUSZABCVLWA
|
|
EEEDKWOAWLWAESBVMVIWEVITWKWAEWEVIFUKULWMAWCVIVMSIGHUTUIUNVALMHVLVMEVRWBVS
|
|
VNWADPEVOHTZVQWAVNDWPVPVIFVOHNOVBVCVDVNVLWADVBVSUJVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d .+ k n w x y $. $d F k n w x y $. $d M k n w x y $. $d S k n w x y $.
|
|
$d T k n w x y $. $d ph k n w x y $.
|
|
iseqss.m $e |- ( ph -> M e. ZZ ) $.
|
|
iseqss.t $e |- ( ph -> T e. V ) $.
|
|
iseqss.ss $e |- ( ph -> S C_ T ) $.
|
|
iseqss.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqss.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqss.plt $e |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T ) $.
|
|
$( Specifying a larger universe for ` seq ` . As long as ` F ` and ` .+ `
|
|
are closed over ` S ` , then any set which contains ` S ` can be used as
|
|
the last argument to ` seq ` . This theorem does not allow ` T ` to be
|
|
a proper class, however. It also currently requires that ` .+ ` be
|
|
closed over ` T ` (as well as ` S ` ). (Contributed by Jim Kingdon,
|
|
18-Aug-2021.) $)
|
|
iseqss $p |- ( ph ->
|
|
seq M ( .+ , F , S ) = seq M ( .+ , F , T ) ) $=
|
|
( cfv wcel wi wceq fveq2 vn vw vk cuz cseq wf wfn eqid iseqfcl ffn syl cv
|
|
wa sseld adantr mpd c1 caddc co eqeq12d imbi2d iseq1 eqtr4d oveq1 adantlr
|
|
cz a1i simpr iseqp1 syl5ibr expcom a2d uzind4 impcom eqfnfvd ) AUAHUDPZDE
|
|
GHUEZDFGHUEZAVPEVQUFVQVPUGABCDEGHVPVPUHZJMNUIVPEVQUJUKAVPFVRUFVRVPUGABCDF
|
|
GHVPVSJABULZVPQZUMVTGPZEQZWBFQZMAWCWDRWAAEFWBLUNUOUPZOUIVPFVRUJUKUAULZVPQ
|
|
AWFVQPZWFVRPZSZAUBULZVQPZWJVRPZSZRAHVQPZHVRPZSZRZAUCULZVQPZWRVRPZSZRAWRUQ
|
|
URUSZVQPZXBVRPZSZRAWIRUBUCHWFWJHSZWMWPAXFWKWNWLWOWJHVQTWJHVRTUTVAWJWRSZWM
|
|
XAAXGWKWSWLWTWJWRVQTWJWRVRTUTVAWJXBSZWMXEAXHWKXCWLXDWJXBVQTWJXBVRTUTVAWJW
|
|
FSZWMWIAXIWKWGWLWHWJWFVQTWJWFVRTUTVAWQHVFQAWNHGPWOABCDEGHJMNVBABCDFGHJWEO
|
|
VBVCVGWRVPQZAXAXEAXJXAXERXAXEAXJUMZWSXBGPZDUSZWTXLDUSZSWSWTXLDVDXKXCXMXDX
|
|
NXKBCDEGHWRAXJVHZAWAWCXJMVEAVTEQCULZEQUMVTXPDUSZEQXJNVEVIXKBCDFGHWRXOAWAW
|
|
DXJWEVEAVTFQXPFQUMXQFQXJOVEVIUTVJVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d .+ x y $. $d F x y $. $d M x y $. $d N x y $. $d S x y $.
|
|
$d ph x y $.
|
|
iseqm1.m $e |- ( ph -> M e. ZZ ) $.
|
|
iseqm1.n $e |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) ) $.
|
|
iseqm1.ex $e |- ( ph -> S e. V ) $.
|
|
iseqm1.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqm1.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Value of the sequence builder function at a successor. (Contributed by
|
|
Mario Carneiro, 24-Jun-2013.) $)
|
|
iseqm1 $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) =
|
|
( ( seq M ( .+ , F , S ) ` ( N - 1 ) ) .+ ( F ` N ) ) ) $=
|
|
( c1 co caddc cfv wcel cuz cmin cseq cz eluzp1m1 syl2anc wceq cc eluzelcn
|
|
iseqp1 ax-1cn npcan sylancl syl fveq2d oveq2d 3eqtr3d ) AHOUAPZOQPZDEFGUB
|
|
ZRUQUSRZURFRZDPHUSRUTHFRZDPABCDEFGUQAGUCSHGOQPZTRSZUQGTRSJKGHUDUEMNUIAURH
|
|
USAVDURHUFZKVDHUGSOUGSVEVCHUHUJHOUKULUMZUNAVAVBUTDAURHFVFUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d k x y z w F $. $d k x y z w G $. $d k x y z w K $. $d k x y z w N $.
|
|
$d k x y z w ph $. $d k x y z w M $. $d k x y z w .+ $.
|
|
$d k x y z w S $.
|
|
iseqfveq2.1 $e |- ( ph -> K e. ( ZZ>= ` M ) ) $.
|
|
iseqfveq2.2 $e |- ( ph -> ( seq M ( .+ , F , S ) ` K ) = ( G ` K ) ) $.
|
|
iseqfveq2.s $e |- ( ph -> S e. V ) $.
|
|
iseqfveq2.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqfveq2.g $e |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S ) $.
|
|
iseqfveq2.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x .+ y ) e. S ) $.
|
|
${
|
|
iseqfveq2.3 $e |- ( ph -> N e. ( ZZ>= ` K ) ) $.
|
|
iseqfveq2.4 $e |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) ->
|
|
( F ` k ) = ( G ` k ) ) $.
|
|
$( Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) $)
|
|
iseqfveq2 $p |- ( ph ->
|
|
( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) $=
|
|
( vz vw cfz co wcel cseq cfv cuz eluzfz2 syl wi cv c1 caddc eleq1 fveq2
|
|
wceq eqeq12d imbi12d imbi2d eluzelz iseq1 eqtr4d a1i13 peano2fzr adantl
|
|
cz wa expr imim1d oveq1 simprl adantr uztrn syl2anc adantlr iseqp1 wral
|
|
ralrimiva eluzp1p1 ad2antrl elfzuz3 ad2antll elfzuzb rspcdva oveq2d a2d
|
|
sylanbrc syl5ibr syld expcom uzind4 mpcom mpd ) AKIKUCUDZUEZKDEGJUFZUGZ
|
|
KDEHIUFZUGZUQZAKIUHUGZUEZWPSIKUIUJXCAWPXAUKZSAUAULZWOUEZXEWQUGZXEWSUGZU
|
|
QZUKZUKAIWOUEZIWQUGZIWSUGZUQZUKZUKAUBULZWOUEZXPWQUGZXPWSUGZUQZUKZUKAXPU
|
|
MUNUDZWOUEZYBWQUGZYBWSUGZUQZUKZUKAXDUKUAUBIKXEIUQZXJXOAYHXFXKXIXNXEIWOU
|
|
OYHXGXLXHXMXEIWQUPXEIWSUPURUSUTXEXPUQZXJYAAYIXFXQXIXTXEXPWOUOYIXGXRXHXS
|
|
XEXPWQUPXEXPWSUPURUSUTXEYBUQZXJYGAYJXFYCXIYFXEYBWOUOYJXGYDXHYEXEYBWQUPX
|
|
EYBWSUPURUSUTXEKUQZXJXDAYKXFWPXIXAXEKWOUOYKXGWRXHWTXEKWQUPXEKWSUPURUSUT
|
|
IVGUEZAXKXNAXLIHUGXMNABCDEHIAIJUHUGZUEZYLMJIVAUJQRVBVCVDXPXBUEZAYAYGAYO
|
|
YAYGUKAYOVHZYAYCXTUKYGYPYCXQXTAYOYCXQYOYCVHZXQAXPIKVEVFVIVJYPYCXTYFAYOY
|
|
CXTYFUKXTYFAYQVHZXRYBGUGZDUDZXSYSDUDZUQXRXSYSDVKYRYDYTYEUUAYRBCDEGJXPYR
|
|
YOYNXPYMUEAYOYCVLZAYNYQMVMIXPJVNVOABULZYMUEUUCGUGEUEYQPVPAUUCEUECULZEUE
|
|
VHUUCUUDDUDEUEYQRVPZVQYRYEXSYBHUGZDUDUUAYRBCDEHIXPUUBAUUCXBUEUUCHUGEUEY
|
|
QQVPUUEVQYRYSUUFXSDYRFULZGUGZUUGHUGZUQZYSUUFUQFIUMUNUDZKUCUDZYBUUGYBUQU
|
|
UHYSUUIUUFUUGYBGUPUUGYBHUPURAUUJFUULVRYQAUUJFUULTVSVMYRYBUUKUHUGUEZKYBU
|
|
HUGUEZYBUULUEYOUUMAYCIXPVTWAYCUUNAYOYBIKWBWCYBUUKKWDWHWEWFVCURWIVIWGWJW
|
|
KWGWLWMWN $.
|
|
$}
|
|
|
|
iseqfeq2.4 $e |- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) ->
|
|
( F ` k ) = ( G ` k ) ) $.
|
|
$( Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) $)
|
|
iseqfeq2 $p |- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` K ) ) =
|
|
seq K ( .+ , G , S ) ) $=
|
|
( cfv wcel vz cuz cseq cres wfn wss cz eluzel2 syl iseqfn fnssres syl2anc
|
|
uzss eluzelz cv wa wceq fvres adantl adantr adantlr co simpr c1 caddc cfz
|
|
elfzuz sylan2 iseqfveq2 eqtrd eqfnfvd ) AUAIUBSZDEGJUCZVLUDZDEHIUCZAVMJUB
|
|
SZUEVLVPUFZVNVLUEABCDEGJKAIVPTZJUGTLJIUHUINOQUJAVRVQLJIUMUIVPVLVMUKULABCD
|
|
EHIKAVRIUGTLJIUNUINPQUJAUAUOZVLTZUPZVSVNSZVSVMSZVSVOSVTWBWCUQAVSVLVMURUSW
|
|
ABCDEFGHIJVSKAVRVTLUTAIVMSIHSUQVTMUTAEKTVTNUTABUOZVPTWDGSETVTOVAAWDVLTWDH
|
|
SETVTPVAAWDETCUOZETUPWDWEDVBETVTQVAAVTVCAFUOZIVDVEVBZVSVFVBTZWFGSWFHSUQZV
|
|
TWHAWFWGUBSTWIWFWGVSVGRVHVAVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d k x y F $. $d k x y G $. $d k x y M $. $d k x y N $. $d k x y ph $.
|
|
$d k x y .+ $. $d k x y S $.
|
|
iseqfveq.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqfveq.2 $e |- ( ( ph /\ k e. ( M ... N ) ) ->
|
|
( F ` k ) = ( G ` k ) ) $.
|
|
iseqfveq.s $e |- ( ph -> S e. V ) $.
|
|
iseqfveq.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqfveq.g $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S ) $.
|
|
iseqfveq.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x .+ y ) e. S ) $.
|
|
$( Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) $)
|
|
iseqfveq $p |- ( ph ->
|
|
( seq M ( .+ , F , S ) ` N ) = ( seq M ( .+ , G , S ) ` N ) ) $=
|
|
( wcel cfv syl cz cuz eluzel2 uzid cseq iseq1 cv wceq cfz fveq2 ralrimiva
|
|
co eqeq12d eluzfz1 rspcdva eqtrd caddc wss fzp1ss sselda syldan iseqfveq2
|
|
c1 ) ABCDEFGHIIJKAIUARZIIUBSZRAJVERZVDLIJUCTZIUDTAIDEGIUESIGSZIHSZABCDEGI
|
|
VGOQUFAFUGZGSZVJHSZUHZVHVIUHFIJUIULZIVJIUHVKVHVLVIVJIGUJVJIHUJUMAVMFVNMUK
|
|
AVFIVNRLIJUNTUOUPNOPQLAVJIVCUQULJUIULZRVJVNRVMAVOVNVJAVDVOVNURVGIJUSTUTMV
|
|
AVB $.
|
|
$}
|
|
|
|
${
|
|
$d .+ k x y z $. $d F k x y z $. $d G k x y z $. $d M k x y z $.
|
|
$d S k x y z $. $d k ph x y z $.
|
|
iseqfeq.1 $e |- ( ph -> M e. ZZ ) $.
|
|
iseqfeq.s $e |- ( ph -> S e. V ) $.
|
|
iseqfeq.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqfeq.2 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) ) $.
|
|
iseqfeq.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x .+ y ) e. S ) $.
|
|
$( Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) $)
|
|
iseqfeq $p |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) ) $=
|
|
( cfv cv wcel wa adantlr vz cseq iseqfn wceq wral ralrimiva fveq2 eqeq12d
|
|
cuz rspcv mpan9 eqeltrrd simpr cfz elfzuz sylan2 adantr iseqfveq eqfnfvd
|
|
co ) AUAIUIPZDEGIUBDEHIUBABCDEGIJKLMOUCABCDEHIJKLABQZVARZSVBGPZVBHPZEAFQZ
|
|
GPZVFHPZUDZFVAUEVCVDVEUDZAVIFVANUFVIVJFVBVAVFVBUDVGVDVHVEVFVBGUGVFVBHUGUH
|
|
UJUKMULZOUCAUAQZVARZSBCDEFGHIVLJAVMUMAVFIVLUNUTRZVIVMVNAVFVARVIVFIVLUONUP
|
|
TAEJRVMLUQAVCVDERVMMTAVCVEERVMVKTAVBERCQZERSVBVODUTERVMOTURUS $.
|
|
$}
|
|
|
|
${
|
|
$d .+ n w x y $. $d F k n x w $. $d F n x y $. $d G k n x w $.
|
|
$d G n x y $. $d K k n x w $. $d K n x y $. $d M k n x w $.
|
|
$d M n x y $. $d N k n x w $. $d N n x y w $. $d S n x y w $.
|
|
$d k n ph x w $. $d ph x y $.
|
|
iseqshft2.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqshft2.2 $e |- ( ph -> K e. ZZ ) $.
|
|
iseqshft2.3 $e |- ( ( ph /\ k e. ( M ... N ) ) ->
|
|
( F ` k ) = ( G ` ( k + K ) ) ) $.
|
|
iseqshft2.s $e |- ( ph -> S e. V ) $.
|
|
iseqshft2.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqshft2.g $e |- ( ( ph /\ x e. ( ZZ>= ` ( M + K ) ) ) ->
|
|
( G ` x ) e. S ) $.
|
|
iseqshft2.pl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x .+ y ) e. S ) $.
|
|
$( Shifting the index set of a sequence. (Contributed by Jim Kingdon,
|
|
15-Aug-2021.) $)
|
|
iseqshft2 $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) =
|
|
( seq ( M + K ) ( .+ , G , S ) ` ( N + K ) ) ) $=
|
|
( cfv vw vn cfz co wcel cseq caddc wceq cuz eluzfz2 syl wi cv eleq1 fveq2
|
|
c1 oveq1 fveq2d eqeq12d imbi12d imbi2d cz ralrimiva eluzfz1 rspcdva iseq1
|
|
eluzel2 zaddcld 3eqtr4d a1i13 peano2fzr adantl expr imim1d simprl adantlr
|
|
wa iseqp1 adantr eluzadd syl2anc eluzelz cc zcn ax-1cn mp3an2 syl2an wral
|
|
add32 simprr eqtrd oveq2d syl5ibr a2d syld expcom uzind4 mpcom mpd ) AKJK
|
|
UCUDZUEZKDEGJUFZTZKIUGUDZDEHJIUGUDZUFZTZUHZAKJUITZUEZXAMJKUJUKXJAXAXHULZM
|
|
AUAUMZWTUEZXLXBTZXLIUGUDZXFTZUHZULZULAJWTUEZJXBTZXEXFTZUHZULZULAUBUMZWTUE
|
|
ZYDXBTZYDIUGUDZXFTZUHZULZULAYDUPUGUDZWTUEZYKXBTZYKIUGUDZXFTZUHZULZULAXKUL
|
|
UAUBJKXLJUHZXRYCAYRXMXSXQYBXLJWTUNYRXNXTXPYAXLJXBUOYRXOXEXFXLJIUGUQURUSUT
|
|
VAXLYDUHZXRYJAYSXMYEXQYIXLYDWTUNYSXNYFXPYHXLYDXBUOYSXOYGXFXLYDIUGUQURUSUT
|
|
VAXLYKUHZXRYQAYTXMYLXQYPXLYKWTUNYTXNYMXPYOXLYKXBUOYTXOYNXFXLYKIUGUQURUSUT
|
|
VAXLKUHZXRXKAUUAXMXAXQXHXLKWTUNUUAXNXCXPXGXLKXBUOUUAXOXDXFXLKIUGUQURUSUTV
|
|
AJVBUEZAXSYBAJGTZXEHTZXTYAAFUMZGTZUUEIUGUDZHTZUHZUUCUUDUHFWTJUUEJUHZUUFUU
|
|
CUUHUUDUUEJGUOUUJUUGXEHUUEJIUGUQURUSAUUIFWTOVCZAXJXSMJKVDUKVEABCDEGJAXJUU
|
|
BMJKVGUKZQSVFABCDEHXEAJIUULNVHRSVFVIVJYDXIUEZAYJYQAUUMYJYQULAUUMVQZYJYLYI
|
|
ULYQUUNYLYEYIAUUMYLYEUUMYLVQZYEAYDJKVKVLVMVNUUNYLYIYPAUUMYLYIYPULYIYPAUUO
|
|
VQZYFYKGTZDUDZYHUUQDUDZUHYFYHUUQDUQUUPYMUURYOUUSUUPBCDEGJYDAUUMYLVOZABUMZ
|
|
XIUEUVAGTEUEUUOQVPAUVAEUECUMZEUEVQUVAUVBDUDEUEUUOSVPZVRUUPYGUPUGUDZXFTYHU
|
|
VDHTZDUDYOUUSUUPBCDEHXEYGUUPUUMIVBUEZYGXEUITZUEUUTAUVFUUONVSZIJYDVTWAAUVA
|
|
UVGUEUVAHTEUEUUORVPUVCVRUUPYNUVDXFUUPYDVBUEZUVFYNUVDUHZUUPUUMUVIUUTJYDWBU
|
|
KUVHUVIYDWCUEZIWCUEZUVJUVFYDWDIWDUVKUPWCUEUVLUVJWEYDUPIWIWFWGWAZURUUPUUQU
|
|
VEYHDUUPUUQYNHTZUVEUUPUUIUUQUVNUHFWTYKUUEYKUHZUUFUUQUUHUVNUUEYKGUOUVOUUGY
|
|
NHUUEYKIUGUQURUSAUUIFWTWHUUOUUKVSAUUMYLWJVEUUPYNUVDHUVMURWKWLVIUSWMVMWNWO
|
|
WPWNWQWRWS $.
|
|
$}
|
|
|
|
${
|
|
$d k x F $. $d k x M $. $d k x ph $. $d k Z $.
|
|
iserf.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
iserf.2 $e |- ( ph -> M e. ZZ ) $.
|
|
${
|
|
iserf.3 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
$( An infinite series of complex terms is a function from ` NN ` to
|
|
` CC ` . (Contributed by Jim Kingdon, 15-Aug-2021.) $)
|
|
iserf $p |- ( ph -> seq M ( + , F , CC ) : Z --> CC ) $=
|
|
( vx caddc cc cvv wcel cnex a1i cv wa co addcl adantl iseqf ) ABIJKCDLE
|
|
FKLMANOGHBPZKMIPZKMQUBUCJRKMAUBUCSTUA $.
|
|
$}
|
|
|
|
iserfre.3 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
$( An infinite series of real numbers is a function from ` NN ` to ` RR ` .
|
|
(Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro,
|
|
27-May-2014.) $)
|
|
iserfre $p |- ( ph -> seq M ( + , F , RR ) : Z --> RR ) $=
|
|
( vx caddc cr cvv wcel reex a1i cv wa co readdcl adantl iseqf ) ABIJKCDLE
|
|
FKLMANOGHBPZKMIPZKMQUBUCJRKMAUBUCSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d k n x F $. $d k n x M $. $d k n x N $. $d k n x ph $.
|
|
monoord.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
monoord.2 $e |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) $.
|
|
monoord.3 $e |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) ->
|
|
( F ` k ) <_ ( F ` ( k + 1 ) ) ) $.
|
|
$( Ordering relation for a monotonic sequence, increasing case.
|
|
(Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro,
|
|
9-Feb-2014.) $)
|
|
monoord $p |- ( ph -> ( F ` M ) <_ ( F ` N ) ) $=
|
|
( co wcel cfv cle wbr syl wi c1 wceq eleq1 fveq2 cr vx vn cfz cuz eluzfz2
|
|
cv caddc breq2d imbi12d imbi2d cz wral eluzfz1 ralrimiva rspcv sylc leidd
|
|
eleq1d a1d wa simprl simprr peano2fzr syl2anc expr imim1d eluzelz elfzuz3
|
|
cmin eluzp1m1 elfzuzb sylanbrc adantr oveq1 fveq2d breq12d syl3anc mpan2d
|
|
a1i letr a2d syld expcom uzind4 mpcom mpd ) AEDEUCIZJZDCKZECKZLMZAEDUDKZJ
|
|
ZWHFDEUENWMAWHWKOZFAUAUFZWGJZWIWOCKZLMZOZOADWGJZWIWILMZOZOZAUBUFZWGJZWIXD
|
|
CKZLMZOZOAXDPUGIZWGJZWIXICKZLMZOZOAWNOUAUBDEWODQZWSXBAXNWPWTWRXAWODWGRXNW
|
|
QWIWILWODCSUHUIUJWOXDQZWSXHAXOWPXEWRXGWOXDWGRXOWQXFWILWOXDCSUHUIUJWOXIQZW
|
|
SXMAXPWPXJWRXLWOXIWGRXPWQXKWILWOXICSUHUIUJWOEQZWSWNAXQWPWHWRWKWOEWGRXQWQW
|
|
JWILWOECSUHUIUJXCDUKJAXAWTAWIAWTBUFZCKZTJZBWGULZWITJZAWMWTFDEUMNAXTBWGGUN
|
|
ZXTYBBDWGXRDQXSWITXRDCSURUOUPZUQUSVSXDWLJZAXHXMAYEXHXMOAYEUTZXHXJXGOXMYFX
|
|
JXEXGAYEXJXEAYEXJUTZUTZYEXJXEAYEXJVAZAYEXJVBZXDDEVCVDZVEVFYFXJXGXLAYEXJXG
|
|
XLOYHXGXFXKLMZXLYHXDDEPVIIZUCIZJZXSXRPUGIZCKZLMZBYNULZYLYHYEYMXDUDKJZYOYI
|
|
YHXDUKJZEXIUDKJZYTYHYEUUAYIDXDVGNYHXJUUBYJXIDEVHNXDEVJVDXDDYMVKVLAYSYGAYR
|
|
BYNHUNVMYRYLBXDYNXRXDQZXSXFYQXKLXRXDCSZUUCYPXICXRXDPUGVNVOVPUOUPYHYBXFTJZ
|
|
XKTJZXGYLUTXLOAYBYGYDVMYHXEYAUUEYKAYAYGYCVMZXTUUEBXDWGUUCXSXFTUUDURUOUPYH
|
|
XJYAUUFYJUUGXTUUFBXIWGXRXIQXSXKTXRXICSURUOUPWIXFXKVTVQVRVEWAWBWCWAWDWEWF
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d k n F $. $d k n M $. $d k n N $. $d k n ph $.
|
|
monoord2.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
monoord2.2 $e |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) $.
|
|
monoord2.3 $e |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) ->
|
|
( F ` ( k + 1 ) ) <_ ( F ` k ) ) $.
|
|
$( Ordering relation for a monotonic sequence, decreasing case.
|
|
(Contributed by Mario Carneiro, 18-Jul-2014.) $)
|
|
monoord2 $p |- ( ph -> ( F ` N ) <_ ( F ` M ) ) $=
|
|
( vn cfv cle wbr cneg cfz co cr wcel renegcld c1 wceq cmpt eqid ffvelrnda
|
|
cv wa fmptd cmin caddc ralrimiva oveq1 fveq2d fveq2 breq12d cbvralv sylib
|
|
wral r19.21bi fzp1elp1 adantl cc cuz cz eluzelz zcnd ax-1cn npcan sylancl
|
|
syl oveq2d adantr eleqtrd eleq1d rspcv sylc fzssp1 syl5sseq sselda lenegd
|
|
negeqd fvmptg syl2anc 3brtr4d monoord eluzfz1 eluzfz2 3brtr3d mpbird
|
|
mpbid ) AECJZDCJZKLWJMZWIMZKLADBDENOZBUDZCJZMZUAZJZEWQJZWKWLKAIWQDEFAWMPI
|
|
UDZWQABWMWPPWQAWNWMQUEWOGRWQUBZUFUCAWTDESUGOZNOZQZUEZWTCJZMZWTSUHOZCJZMZW
|
|
TWQJZXHWQJZKXEXIXFKLZXGXJKLAXMIXCAWNSUHOZCJZWOKLZBXCUPXMIXCUPAXPBXCHUIXPX
|
|
MBIXCWNWTTZXOXIWOXFKXQXNXHCWNWTSUHUJUKWNWTCULZUMUNUOUQXEXIXFXEXHWMQZWOPQZ
|
|
BWMUPZXIPQZXEXHDXBSUHOZNOZWMXDXHYDQAWTDXBURUSAYDWMTXDAYCEDNAEUTQSUTQYCETA
|
|
EAEDVAJQZEVBQFDEVCVHVDVEESVFVGVIZVJVKZAYAXDAXTBWMGUIZVJZXTYBBXHWMWNXHTZWO
|
|
XIPWNXHCULZVLVMVNZXEWTWMQZYAXFPQZAXCWMWTAYDXCWMDXBVOYFVPVQZYIXTYNBWTWMXQW
|
|
OXFPXRVLVMVNZVRWHXEYMXGPQXKXGTYOXEXFYPRBWTWPXGWMPWQXQWOXFXRVSXAVTWAXEXSXJ
|
|
PQXLXJTYGXEXIYLRBXHWPXJWMPWQYJWOXIYKVSXAVTWAWBWCADWMQZWKPQWRWKTAYEYQFDEWD
|
|
VHZAWJAYQYAWJPQZYRYHXTYSBDWMWNDTZWOWJPWNDCULZVLVMVNZRBDWPWKWMPWQYTWOWJUUA
|
|
VSXAVTWAAEWMQZWLPQWSWLTAYEUUCFDEWEVHZAWIAUUCYAWIPQZUUDYHXTUUEBEWMWNETZWOW
|
|
IPWNECULZVLVMVNZRBEWPWLWMPWQUUFWOWIUUGVSXAVTWAWFAWIWJUUHUUBVRWG $.
|
|
$}
|
|
|
|
${
|
|
$d k x y F $. $d k x y K $. $d k x y M $. $d k x y N $. $d k x y ph $.
|
|
isermono.1 $e |- ( ph -> K e. ( ZZ>= ` M ) ) $.
|
|
isermono.2 $e |- ( ph -> N e. ( ZZ>= ` K ) ) $.
|
|
isermono.3 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR ) $.
|
|
isermono.4 $e |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) ->
|
|
0 <_ ( F ` x ) ) $.
|
|
$( The partial sums in an infinite series of positive terms form a
|
|
monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) $)
|
|
isermono $p |- ( ph ->
|
|
( seq M ( + , F , RR ) ` K ) <_ ( seq M ( + , F , RR ) ` N ) ) $=
|
|
( caddc cr cfz co wcel cfv adantl c1 cle syl vk vy cv wa cuz elfzuz uztrn
|
|
cseq syl2anr adantlr readdcl iseqcl cmin wceq fveq2 breq2d wral ralrimiva
|
|
cc0 wbr adantr simpr cz wb eluzelz peano2zm elfzelz 1zzd fzaddel syl22anc
|
|
mpbid cc ax-1cn npcan sylancl oveq2d eleqtrd rspcdva fzelp1 syldan eleq1d
|
|
zcn wss fzss1 fzp1elp1 sseldd addge01d iseqp1 breqtrrd monoord ) AUAKLCEU
|
|
HZDFHAUAUCZDFMNZOZUDZBUBKLCEWLWNWLDUEPZODEUEPZOZWLWQOZAWLDFUFGDWLEUGUIZAB
|
|
UCZWQOZXACPZLOZWNIUJXALOUBUCZLOUDZXAXEKNLOZWOXAXEUKZQULZAWLDFRUMNZMNOZUDZ
|
|
WLWKPZXMWLRKNZCPZKNZXNWKPSXLUSXOSUTZXMXPSUTXLUSXCSUTZXQBDRKNZFMNZXNXAXNUN
|
|
ZXCXOUSSXAXNCUOZUPAXRBXTUQXKAXRBXTJURVAXLXNXSXJRKNZMNZXTXLXKXNYDOZAXKVBXL
|
|
DVCOZXJVCOZWLVCOZRVCOXKYEVDXLWRYFAWRXKGVAZEDVETXLFVCOZYGXLFWPOZYJAYKXKHVA
|
|
DFVETZFVFTXKYHAWLDXJVGQXLVHWLRDXJVIVJVKXLYCFXSMXLYJYCFUNZYLYJFVLORVLOYMFW
|
|
BVMFRVNVOTZVPVQVRXLXMXOAXKWNXMLOXLWLDYCMNZWMXKWLYOOAWLDXJVSQXLYCFDMYNVPZV
|
|
QZXIVTXLXDXOLOBWQXNYAXCXOLYBWAAXDBWQUQXKAXDBWQIURVAXLXNEFMNZOXNWQOXLWMYRX
|
|
NXLWRWMYRWCYIDEFWDTXLXNYOWMXKXNYOOAWLDXJWEQYPVQWFXNEFUFTVRWGVKXLBUBKLCEWL
|
|
AXKWNWSYQWTVTAXBXDXKIUJXFXGXLXHQWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d n x y z F $. $d n x y z K $. $d n x y z M $. $d n x y z ph $.
|
|
$d n x y z N $. $d n x y z .+ $. $d x y z S $.
|
|
iseqsplit.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqsplit.2 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) $.
|
|
iseqsplit.3 $e |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) ) $.
|
|
iseqsplit.s $e |- ( ph -> S e. V ) $.
|
|
${
|
|
$d .+ n x y z $. $d F n x y z $. $d K n x y z $. $d M n x y z $.
|
|
$d N n x y z $. $d S n x y z $. $d n ph x y z $.
|
|
iseqsplit.4 $e |- ( ph -> M e. ( ZZ>= ` K ) ) $.
|
|
iseqsplit.5 $e |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S ) $.
|
|
$( Split a sequence into two sequences. (Contributed by Jim Kingdon,
|
|
16-Aug-2021.) $)
|
|
iseqsplit $p |- ( ph -> ( seq K ( .+ , F , S ) ` N ) =
|
|
( ( seq K ( .+ , F , S ) ` M ) .+
|
|
( seq ( M + 1 ) ( .+ , F , S ) ` N ) ) ) $=
|
|
( co wcel cfv vn c1 caddc cfz cseq wceq cuz eluzfz2 syl wi eleq1 oveq2d
|
|
cv fveq2 eqeq12d imbi12d imbi2d iseqp1 eluzel2 eluzelz peano2uzr adantr
|
|
cz sylan uztrn syl2anc syldan iseq1 eqtr4d peano2fzr adantl expr imim1d
|
|
wa a1i13 oveq1 simprl peano2uz adantlr iseqcl eleq1d wral elfzuz sylan2
|
|
simpl ralrimiva fzss1 3syl simpr ssel2 syl2an rspcdva caovassg syl13anc
|
|
wss syl5ibr a2d syld expcom uzind4 mpcom mpd ) AJIUBUCRZJUDRZSZJEFGHUEZ
|
|
TZIXFTZJEFGXCUEZTZERZUFZAJXCUGTZSZXENXCJUHUIXNAXEXLUJZNABUMZXDSZXPXFTZX
|
|
HXPXITZERZUFZUJZUJAXCXDSZXCXFTZXHXCXITZERZUFZUJZUJAUAUMZXDSZYIXFTZXHYIX
|
|
ITZERZUFZUJZUJAYIUBUCRZXDSZYPXFTZXHYPXITZERZUFZUJZUJAXOUJBUAXCJXPXCUFZY
|
|
BYHAUUCXQYCYAYGXPXCXDUKUUCXRYDXTYFXPXCXFUNUUCXSYEXHEXPXCXIUNULUOUPUQXPY
|
|
IUFZYBYOAUUDXQYJYAYNXPYIXDUKUUDXRYKXTYMXPYIXFUNUUDXSYLXHEXPYIXIUNULUOUP
|
|
UQXPYPUFZYBUUBAUUEXQYQYAUUAXPYPXDUKUUEXRYRXTYTXPYPXFUNUUEXSYSXHEXPYPXIU
|
|
NULUOUPUQXPJUFZYBXOAUUFXQXEYAXLXPJXDUKUUFXRXGXTXKXPJXFUNUUFXSXJXHEXPJXI
|
|
UNULUOUPUQXCVCSZAYCYGAYDXHXCGTZERYFABCEFGHIPQLURAYEUUHXHEABCEFGXCAXNUUG
|
|
NXCJUSUIAXPXMSZXPHUGTZSZXPGTZFSZAUUIVNXPIUGTSZIUUJSZUUKAIVCSZUUIUUNAUUO
|
|
UUPPHIUTUIIXPVAVDAUUOUUIPVBIXPHVEVFQVGZLVHULVIVOYIXMSZAYOUUBAUURYOUUBUJ
|
|
AUURVNZYOYQYNUJUUBUUSYQYJYNAUURYQYJUURYQVNZYJAYIXCJVJVKVLVMUUSYQYNUUAAU
|
|
URYQYNUUAUJYNUUAAUUTVNZYKYPGTZERZYMUVBERZUFYKYMUVBEVPUVAYRUVCYTUVDUVABC
|
|
EFGHYIUVAUURXCUUJSZYIUUJSAUURYQVQZAUVEUUTAUUOUVEPHIVRZUIVBXCYIHVEVFAUUK
|
|
UUMUUTQVSAXPFSCUMZFSVNXPUVHERFSUUTLVSZURUVAYTXHYLUVBERZERZUVDUVAYSUVJXH
|
|
EUVABCEFGXCYIUVFAUUIUUMUUTUUQVSZUVIURULUVAAXHFSZYLFSUVBFSZUVDUVKUFAUUTW
|
|
EAUVMUUTABCEFGHIPQLVTVBUVABCEFGXCYIUVFUVLUVIVTUVAUUMUVNBHJUDRZYPUUEUULU
|
|
VBFXPYPGUNWAAUUMBUVOWBUUTAUUMBUVOXPUVOSAUUKUUMXPHJWCQWDWFVBAXDUVOWOZYQY
|
|
PUVOSUUTAUUOUVEUVPPUVGXCHJWGWHUURYQWIXDUVOYPWJWKWLABCDXHYLUVBFEMWMWNVIU
|
|
OWPVLWQWRWSWQWTXAXB $.
|
|
$}
|
|
|
|
iseq1p.4 $e |- ( ph -> M e. ZZ ) $.
|
|
iseq1p.5 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
$( Removing the first term from a sequence. (Contributed by Jim Kingdon,
|
|
16-Aug-2021.) $)
|
|
iseq1p $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) =
|
|
( ( F ` M ) .+ ( seq ( M + 1 ) ( .+ , F , S ) ` N ) ) ) $=
|
|
( cseq cfv co wcel c1 caddc cz cuz uzid syl iseqsplit iseq1 oveq1d eqtrd
|
|
) AIEFGHQZRHUKRZIEFGHUAUBSQRZESHGRZUMESABCDEFGHHIJKLMNAHUCTHHUDRTOHUEUFPU
|
|
GAULUNUMEABCEFGHOPKUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d .+ n x y z $. $d F k n x y z $. $d G k n x y z $. $d H k n x y z $.
|
|
$d M k n x y z $. $d N k n x y z $. $d Q k n x y z $. $d S k n x y z $.
|
|
$d k n ph x y z $.
|
|
iseqcaopr3.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqcaopr3.2 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S ) $.
|
|
iseqcaopr3.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqcaopr3.4 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S ) $.
|
|
iseqcaopr3.5 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S ) $.
|
|
iseqcaopr3.6 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) ->
|
|
( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) ) $.
|
|
iseqcaopr3.7 $e |- ( ( ph /\ n e. ( M ..^ N ) ) ->
|
|
( ( ( seq M ( .+ , F , S ) ` n ) Q ( seq M ( .+ , G , S ) ` n ) ) .+
|
|
( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) =
|
|
( ( ( seq M ( .+ , F , S ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q
|
|
( ( seq M ( .+ , G , S ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) ) $.
|
|
iseqcaopr3.s $e |- ( ph -> S e. V ) $.
|
|
$( Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.) $)
|
|
iseqcaopr3 $p |- ( ph -> ( seq M ( .+ , H , S ) ` N ) =
|
|
( ( seq M ( .+ , F , S ) ` N ) Q ( seq M ( .+ , G , S ) ` N ) ) ) $=
|
|
( vz cfz co wcel cseq cfv wceq cuz eluzfz2 cv wi c1 caddc oveq12d eqeq12d
|
|
syl fveq2 imbi2d ralrimiva cz eluzel2 uzid rspcdva wral ralrimivva adantr
|
|
wa oveq1 eleq1d oveq2 rspc2v syl2anc mpd eqeltrd rspcv mpan9 3eqtr4d cfzo
|
|
iseq1 elfzouz adantl adantlr iseqp1 fzofzp1 elfzuz oveq2d 3eqtr4rd expcom
|
|
a1i syl5ibr a2d fzind2 mpcom ) MLMUDUEZUFZAMDFKLUGZUHZMDFILUGZUHZMDFJLUGZ
|
|
UHZEUEZUIZAMLUJUHZUFZWQQLMUKURAUCULZWRUHZXHWTUHZXHXBUHZEUEZUIZUMALWRUHZLW
|
|
TUHZLXBUHZEUEZUIZUMZAHULZWRUHZXTWTUHZXTXBUHZEUEZUIZUMAXTUNUOUEZWRUHZYFWTU
|
|
HZYFXBUHZEUEZUIZUMAXEUMUCHMLMXHLUIZXMXRAYLXIXNXLXQXHLWRUSYLXJXOXKXPEXHLWT
|
|
USXHLXBUSUPUQUTXHXTUIZXMYEAYMXIYAXLYDXHXTWRUSYMXJYBXKYCEXHXTWTUSXHXTXBUSU
|
|
PUQUTXHYFUIZXMYKAYNXIYGXLYJXHYFWRUSYNXJYHXKYIEXHYFWTUSXHYFXBUSUPUQUTXHMUI
|
|
ZXMXEAYOXIWSXLXDXHMWRUSYOXJXAXKXCEXHMWTUSXHMXBUSUPUQUTXSXGALKUHZLIUHZLJUH
|
|
ZEUEZXNXQAGULZKUHZYTIUHZYTJUHZEUEZUIZYPYSUIGXFLYTLUIZUUAYPUUDYSYTLKUSUUFU
|
|
UBYQUUCYREYTLIUSYTLJUSUPUQAUUEGXFTVAZALVBUFZLXFUFAXGUUHQLMVCURZLVDURVEABC
|
|
DFKLUUIAUUAFUFZGXFVFBULZXFUFZUUKKUHZFUFZAUUJGXFAYTXFUFZVIZUUAUUDFTUUPUUKC
|
|
ULZEUEZFUFZCFVFBFVFZUUDFUFZAUUTUUOAUUSBCFFPVGVHUUPUUBFUFZUUCFUFZUUTUVAUMR
|
|
SUUSUVAUUBUUQEUEZFUFBCUUBUUCFFUUKUUBUIUURUVDFUUKUUBUUQEVJVKUUQUUCUIUVDUUD
|
|
FUUQUUCUUBEVLVKVMVNVOVPVAUUJUUNGUUKXFYTUUKUIZUUAUUMFYTUUKKUSVKVQVRZOWAAXO
|
|
YQXPYREABCDFILUUIAUVBGXFVFUULUUKIUHZFUFZAUVBGXFRVAUVBUVHGUUKXFUVEUUBUVGFY
|
|
TUUKIUSVKVQVRZOWAABCDFJLUUIAUVCGXFVFUULUUKJUHZFUFZAUVCGXFSVAUVCUVKGUUKXFU
|
|
VEUUCUVJFYTUUKJUSVKVQVRZOWAUPVSWKXTLMVTUEUFZAYEYKAUVMYEYKUMYEYKAUVMVIZYAY
|
|
FKUHZDUEZYDUVODUEZUIYAYDUVODVJUVNYGUVPYJUVQUVNBCDFKLXTUVMXTXFUFAXTLMWBWCZ
|
|
AUULUUNUVMUVFWDAUUKFUFUUQFUFVIUUKUUQDUEFUFUVMOWDZWEUVNYDYFIUHZYFJUHZEUEZD
|
|
UEYBUVTDUEZYCUWADUEZEUEUVQYJUAUVNUVOUWBYDDUVNUUEUVOUWBUIGXFYFYTYFUIZUUAUV
|
|
OUUDUWBYTYFKUSUWEUUBUVTUUCUWAEYTYFIUSYTYFJUSUPUQAUUEGXFVFUVMUUGVHUVMYFXFU
|
|
FZAUVMYFWPUFUWFLMXTWFYFLMWGURWCVEWHUVNYHUWCYIUWDEUVNBCDFILXTUVRAUULUVHUVM
|
|
UVIWDUVSWEUVNBCDFJLXTUVRAUULUVKUVMUVLWDUVSWEUPWIUQWLWJWMWNWO $.
|
|
$}
|
|
|
|
${
|
|
$d .+ n w x y z $. $d F k n w x y z $. $d G k n w x y z $.
|
|
$d H k n x y z $. $d M k n w x y z $. $d N k n x y z $.
|
|
$d Q k n w x y z $. $d S k n w x y z $. $d k n ph w x y z $.
|
|
iseqcaopr2.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqcaopr2.2 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S ) $.
|
|
iseqcaopr2.3 $e |- ( ( ph /\
|
|
( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) ->
|
|
( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) ) $.
|
|
iseqcaopr2.4 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqcaopr2.5 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S ) $.
|
|
iseqcaopr2.6 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S ) $.
|
|
iseqcaopr2.7 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) ->
|
|
( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) ) $.
|
|
iseqcaopr2.s $e |- ( ph -> S e. V ) $.
|
|
$( The sum of two infinite series (generalized to an arbitrary commutative
|
|
and associative operation). (Contributed by Mario Carneiro,
|
|
30-May-2014.) $)
|
|
iseqcaopr2 $p |- ( ph -> ( seq M ( .+ , H , S ) ` N ) =
|
|
( ( seq M ( .+ , F , S ) ` N ) Q ( seq M ( .+ , G , S ) ` N ) ) ) $=
|
|
( vn cv cfzo co wcel wa cseq cfv c1 caddc wceq wral cuz elfzouz ralrimiva
|
|
adantl adantr fveq2 eleq1d rspccva sylan adantlr iseqcl cfz fzssuz sseldi
|
|
fzofzp1 syl2an anassrs oveq1 oveq1d eqeq12d 2ralbidv oveq2d oveq2 rspc2va
|
|
ralrimivva syl21anc iseqcaopr3 ) ABCFGHIUDJKLMNOPQSTUAUBAUDUEZMNUFUGUHZUI
|
|
ZWCFHKMUJUKZHUHWCULUMUGZKUKZHUHZWCFHJMUJUKZDUEZGUGZWGJUKZEUEZGUGZFUGZWJWM
|
|
FUGZWKWNFUGZGUGZUNZEHUODHUOZWJWFGUGZWMWHGUGZFUGZWQWFWHFUGZGUGZUNZWEBCFHKM
|
|
WCWDWCMUPUKZUHAWCMNUQUSZWEIUEZKUKZHUHZIXHUOZBUEZXHUHZXNKUKZHUHZAXMWDAXLIX
|
|
HUAURZUTXLXQIXNXHXJXNUNZXKXPHXJXNKVAVBVCVDAXNHUHCUEZHUHUIZXNXTFUGZHUHWDPV
|
|
EZVFAXMWGXHUHZWIWDXRWDMNVGUGXHWGMNVHMNWCVJVIZXLWIIWGXHXJWGUNZXKWHHXJWGKVA
|
|
VBVCVKWEWJHUHWMHUHZXNWKGUGZXTWNGUGZFUGZYBWRGUGZUNZEHUODHUOZCHUOBHUOZXAWEB
|
|
CFHJMWCXIAXOXNJUKZHUHZWDAXJJUKZHUHZIXHUOZXOYPAYRIXHTURZYRYPIXNXHXSYQYOHXJ
|
|
XNJVAVBVCVDVEYCVFAYSYDYGWDYTYEYRYGIWGXHYFYQWMHXJWGJVAVBVCVKAYNWDAYMBCHHAY
|
|
AUIYLDEHHAYAWKHUHWNHUHUIYLRVLVTVTUTYMXAWLYIFUGZWJXTFUGZWRGUGZUNZEHUODHUOB
|
|
CWJWMHHXNWJUNZYLUUDDEHHUUEYJUUAYKUUCUUEYHWLYIFXNWJWKGVMVNUUEYBUUBWRGXNWJX
|
|
TFVMVNVOVPXTWMUNZUUDWTDEHHUUFUUAWPUUCWSUUFYIWOWLFXTWMWNGVMVQUUFUUBWQWRGXT
|
|
WMWJFVRVNVOVPVSWAWTXGXBWOFUGZWQWFWNFUGZGUGZUNDEWFWHHHWKWFUNZWPUUGWSUUIUUJ
|
|
WLXBWOFWKWFWJGVRVNUUJWRUUHWQGWKWFWNFVMVQVOWNWHUNZUUGXDUUIXFUUKWOXCXBFWNWH
|
|
WMGVRVQUUKUUHXEWQGWNWHWFFVRVQVOVSWAUCWB $.
|
|
$}
|
|
|
|
${
|
|
$d .+ a b c d k x y z $. $d F a b c d k $. $d G a b c d k $.
|
|
$d H a b c k $. $d M a b c d k $. $d N a b c k $.
|
|
$d S a b c d k x y z $. $d a b c d k ph x y z $.
|
|
iseqcaopr.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqcaopr.2 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x .+ y ) = ( y .+ x ) ) $.
|
|
iseqcaopr.3 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) $.
|
|
iseqcaopr.4 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqcaopr.5 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S ) $.
|
|
iseqcaopr.6 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S ) $.
|
|
iseqcaopr.7 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) ->
|
|
( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) ) $.
|
|
iseqcaopr.s $e |- ( ph -> S e. V ) $.
|
|
$( The sum of two infinite series (generalized to an arbitrary commutative
|
|
and associative operation). (Contributed by Jim Kingdon,
|
|
17-Aug-2021.) $)
|
|
iseqcaopr $p |- ( ph -> ( seq M ( .+ , H , S ) ` N ) =
|
|
( ( seq M ( .+ , F , S ) ` N ) .+ ( seq M ( .+ , G , S ) ` N ) ) ) $=
|
|
( va vb vc vd cv caovclg wcel wa co simpl simprrl simprlr caovcomg oveq1d
|
|
syl12anc simprrr caovassg syl13anc 3eqtr3d oveq2d simprll adantrl 3eqtr4d
|
|
wceq iseqcaopr2 ) AUBUCUDUEEEFGHIJKLMABCUBUFZUCUFZFFFENUGZVIAVGFUHZVHFUHZ
|
|
UIZUDUFZFUHZUEUFZFUHZUIZUIZUIZVGVMVHVOEUJZEUJZEUJZVGVHVMVOEUJZEUJZEUJZVGV
|
|
MEUJVTEUJZVGVHEUJWCEUJZVSWAWDVGEVSVMVHEUJZVOEUJZVHVMEUJZVOEUJZWAWDVSWHWJV
|
|
OEVSAVNVKWHWJVEAVRUKZAVLVNVPULZAVJVKVQUMZABCVMVHFEOUNUPUOVSAVNVKVPWIWAVEW
|
|
LWMWNAVLVNVPUQZABCDVMVHVOFEPURUSVSAVKVNVPWKWDVEWLWNWMWOABCDVHVMVOFEPURUSU
|
|
TVAVSAVJVNVTFUHZWFWBVEWLAVJVKVQVBZWMVSAVKVPWPWLWNWOABCVHVOFFFENUGUPABCDVG
|
|
VMVTFEPURUSVSAVJVKWCFUHZWGWEVEWLWQWNAVQWRVLABCVMVOFFFENUGVCABCDVGVHWCFEPU
|
|
RUSVDQRSTUAVF $.
|
|
$}
|
|
|
|
${
|
|
$d k x y z F $. $d k x y z G $. $d k x y z M $. $d k x y z ph $.
|
|
$d k z H $. $d k x y z N $.
|
|
iseradd.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseradd.2 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC ) $.
|
|
iseradd.3 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC ) $.
|
|
iseradd.4 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) ->
|
|
( H ` k ) = ( ( F ` k ) + ( G ` k ) ) ) $.
|
|
$( The sum of two infinite series. (Contributed by NM, 17-Mar-2005.)
|
|
(Revised by Mario Carneiro, 26-May-2014.) $)
|
|
iseradd $p |- ( ph -> ( seq M ( + , H , CC ) ` N ) =
|
|
( ( seq M ( + , F , CC ) ` N ) + ( seq M ( + , G , CC ) ` N ) ) ) $=
|
|
( vx vy vz caddc cc cv wcel co adantl cvv wa addcl wceq addcom w3a addass
|
|
cnex a1i iseqcaopr ) ALMNOPBCDEFGUALQZPRZMQZPRZUBZUKUMOSZPRAUKUMUCTUOUPUM
|
|
UKOSUDAUKUMUETULUNNQZPRUFUPUQOSUKUMUQOSOSUDAUKUMUQUGTHIJKPUARAUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d F k w x y z $. $d G k w x y z $. $d H k x y z $. $d M k w x y z $.
|
|
$d N k x y z $. $d k ph w x y z $.
|
|
isersub.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
isersub.2 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC ) $.
|
|
isersub.3 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC ) $.
|
|
isersub.4 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) ->
|
|
( H ` k ) = ( ( F ` k ) - ( G ` k ) ) ) $.
|
|
$( The difference of two infinite series. (Contributed by NM,
|
|
17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.) $)
|
|
isersub $p |- ( ph -> ( seq M ( + , H , CC ) ` N ) =
|
|
( ( seq M ( + , F , CC ) ` N ) - ( seq M ( + , G , CC ) ` N ) ) ) $=
|
|
( vx caddc cmin cc cv wcel wa co adantl vy vz vw addcl subcl wceq addsub4
|
|
cvv eqcomd cnex a1i iseqcaopr2 ) ALUAUBUCMNOBCDEFGUHLPZOQUAPZOQRZUMUNMSZO
|
|
QAUMUNUDTUOUMUNNSOQAUMUNUETUOUBPZOQUCPZOQRRZUMUQNSUNURNSMSZUPUQURMSNSZUFA
|
|
USVAUTUMUNUQURUGUITHIJKOUHQAUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y .+ $. $d x y F $. $d x y M $. $d x y ph $. $d x y Z $.
|
|
$d x y N $.
|
|
iseqid3.1 $e |- ( ph -> ( Z .+ Z ) = Z ) $.
|
|
iseqid3.2 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqid3.3 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = Z ) $.
|
|
iseqid3.z $e |- ( ph -> Z e. V ) $.
|
|
$( A sequence that consists entirely of zeroes (or whatever the identity
|
|
` Z ` is for operation ` .+ ` ) sums to zero. (Contributed by Jim
|
|
Kingdon, 18-Aug-2021.) $)
|
|
iseqid3 $p |- ( ph -> ( seq M ( .+ , F , { Z } ) ` N ) = Z ) $=
|
|
( vy cfv wcel wceq cv wa syl elsni csn cuz elsn2g adantr mpbird oveqan12d
|
|
cseq wb co eleq1d syl5ibrcom imp iseqcl ) AFCHUAZDEUGNZUNOUOHPABMCUNDEFJA
|
|
BQZEUBNOZRUPDNZUNOZURHPZKAUSUTUHZUQAHGOZVALURHGUCSUDUEAUPUNOZMQZUNOZRZUPV
|
|
DCUIZUNOZAVHVFHHCUIZUNOZAVJVIHPZIAVBVJVKUHLVIHGUCSUEVFVGVIUNVCVEUPHVDHCUP
|
|
HTVDHTUFUJUKULUMUOHTS $.
|
|
$}
|
|
|
|
${
|
|
$d x y k w .+ $. $d x y k w F $. $d x y k w M $. $d x y k w ph $.
|
|
$d x y k w Z $. $d x y k w N $. $d x y k w S $.
|
|
iseqid3s.1 $e |- ( ph -> ( Z .+ Z ) = Z ) $.
|
|
iseqid3s.2 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqid3s.3 $e |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z ) $.
|
|
iseqid3s.z $e |- ( ph -> Z e. S ) $.
|
|
iseqid3s.s $e |- ( ph -> S e. V ) $.
|
|
iseqid3s.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqid3s.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( A sequence that consists of zeroes up to ` N ` sums to zero at ` N ` .
|
|
In this case by "zero" we mean whatever the identity ` Z ` is for the
|
|
operation ` .+ ` ). (Contributed by Jim Kingdon, 18-Aug-2021.) $)
|
|
iseqid3s $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) $=
|
|
( cfv wceq wcel vw vk cseq cuz cfz co wi eluzfz2 cv c1 caddc fveq2 eqeq1d
|
|
imbi2d eluzel2 syl iseq1 wral ralrimiva eluzfz1 rspcv 3syl mpd eqtrd cfzo
|
|
cz a1i elfzouz adantl adantlr iseqp1 adantr simpr fzofzp1 rspcdva oveq12d
|
|
wa ad2antrr 3eqtrd ex expcom a2d fzind2 pm2.43i ) AHDEFGUCZRZJSZAHGUDRZTZ
|
|
HGHUEUFZTAWGUGZLGHUHAUAUIZWERZJSZUGAGWERZJSZUGZAUBUIZWERZJSZUGAWRUJUKUFZW
|
|
ERZJSZUGWKUAUBHGHWLGSZWNWPAXDWMWOJWLGWEULUMUNWLWRSZWNWTAXEWMWSJWLWRWEULUM
|
|
UNWLXASZWNXCAXFWMXBJWLXAWEULUMUNWLHSZWNWGAXGWMWFJWLHWEULUMUNWQWIAWOGFRZJA
|
|
BCDEFGAWIGVFTLGHUOUPPQUQABUIZFRZJSZBWJURZXHJSZAXKBWJMUSZAWIGWJTXLXMUGLGHU
|
|
TXKXMBGWJXIGSXJXHJXIGFULUMVAVBVCVDVGWRGHVEUFTZAWTXCAXOWTXCUGAXOVQZWTXCXPW
|
|
TVQZXBWSXAFRZDUFZJJDUFZJXPXBXSSWTXPBCDEFGWRXOWRWHTAWRGHVHVIAXIWHTXJETXOPV
|
|
JAXIETCUIZETVQXIYADUFETXOQVJVKVLXQWSJXRJDXPWTVMXPXRJSZWTXPXKYBBWJXAXIXASX
|
|
JXRJXIXAFULUMAXLXOXNVLXOXAWJTAGHWRVNVIVOVLVPAXTJSXOWTKVRVSVTWAWBWCVBWD $.
|
|
$}
|
|
|
|
${
|
|
$d .+ k x y $. $d F k x y $. $d M k x y $. $d N k x y $. $d S k x y $.
|
|
$d Z x y $. $d k ph x y $.
|
|
iseqid.1 $e |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x ) $.
|
|
iseqid.2 $e |- ( ph -> Z e. S ) $.
|
|
iseqid.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqid.4 $e |- ( ph -> ( F ` N ) e. S ) $.
|
|
iseqid.5 $e |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z ) $.
|
|
iseqid.s $e |- ( ph -> S e. V ) $.
|
|
iseqid.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqid.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( Discard the first few terms of a sequence that starts with all zeroes
|
|
(or whatever the identity ` Z ` is for operation ` .+ ` ). (Contributed
|
|
by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro,
|
|
27-May-2014.) $)
|
|
iseqid $p |- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` N ) ) =
|
|
seq N ( .+ , F , S ) ) $=
|
|
( cfv wcel vk wceq cseq c1 caddc co cuz cz eluzelz syl cv wa simpr adantr
|
|
uztrn syl2anc syldan iseqeq1 fveq1d eqeq1d syl5ibcom cmin eluzel2 adantlr
|
|
iseq1 iseqm1 oveq2 id eqeq12d ralrimiva rspcdva eluzp1m1 sylan cfz oveq1d
|
|
iseqid3s wral 3eqtrd ex wo uzp1 mpjaod eqidd iseqfeq2 ) ABCDEUAFFHGIMAHGU
|
|
BZHDEFGUCZSZHFSZUBZHGUDUEUFUGSTZAHDEFHUCZSZWHUBWEWIABCDEFHAHGUGSZTZHUHTMG
|
|
HUIUJABUKZHUGSTZWOWMTZWOFSZETZAWPULWPWNWQAWPUMAWNWPMUNHWOGUOUPQUQZRVEWEWL
|
|
WGWHWEHWKWFDEFHGURUSUTVAAWJWIAWJULZWGHUDVBUFZWFSZWHDUFJWHDUFZWHXABCDEFGHI
|
|
AGUHTZWJAWNXEMGHVCUJZUNAWJUMAEITWJPUNZAWQWSWJQVDZAWOETCUKZETULWOXIDUFETWJ
|
|
RVDZVFXAXCJWHDXABCDEFGXBIJAJJDUFZJUBZWJAJWODUFZWOUBZXLBEJWOJUBZXMXKWOJWOJ
|
|
JDVGXOVHVIAXNBEKVJZLVKUNAXEWJXBWMTXFGHVLVMAWOGXBVNUFTWRJUBWJOVDAJETWJLUNX
|
|
GXHXJVPVOXAXNXDWHUBBEWHWOWHUBZXMXDWOWHWOWHJDVGXQVHVIAXNBEVQWJXPUNAWHETWJN
|
|
UNVKVRVSAWNWEWJVTMGHWAUJWBPQWTRAUAUKZHUDUEUFUGSTULXRFSWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d .+ n x y $. $d F n x y $. $d K n x y $. $d M n x y $. $d N n x y $.
|
|
$d S n x y $. $d Z x $. $d n ph x y $.
|
|
iseqid2.1 $e |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x ) $.
|
|
iseqid2.2 $e |- ( ph -> K e. ( ZZ>= ` M ) ) $.
|
|
iseqid2.3 $e |- ( ph -> N e. ( ZZ>= ` K ) ) $.
|
|
iseqid2.4 $e |- ( ph -> ( seq M ( .+ , F , S ) ` K ) e. S ) $.
|
|
iseqid2.5 $e |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z ) $.
|
|
iseqid2.s $e |- ( ph -> S e. V ) $.
|
|
iseqid2.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqid2.cl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
$( The last few partial sums of a sequence that ends with all zeroes (or
|
|
any element which is a right-identity for ` .+ ` ) are all the same.
|
|
(Contributed by Jim Kingdon, 5-Mar-2022.) $)
|
|
iseqid2 $p |- ( ph -> ( seq M ( .+ , F , S ) ` K )
|
|
= ( seq M ( .+ , F , S ) ` N ) ) $=
|
|
( wcel vn cfz co cseq cfv wceq cuz eluzfz2 syl wi cv c1 caddc eleq1 fveq2
|
|
eqeq2d imbi12d imbi2d cz eqidd 2a1i wa peano2fzr adantl expr imim1d oveq1
|
|
eqeq1d wral ralrimiva eluzp1p1 ad2antrl elfzuz3 ad2antll elfzuzb sylanbrc
|
|
adantr rspcdva oveq2d eqeq12d eqtr2d simprl uztrn syl2anc adantlr syl5ibr
|
|
id iseqp1 a2d syld expcom uzind4 mpcom mpd ) AIGIUBUCZTZGDEFHUDZUEZIWQUEZ
|
|
UFZAIGUGUEZTZWPNGIUHUIXBAWPWTUJZNABUKZWOTZWRXDWQUEZUFZUJZUJAGWOTZWRWRUFZU
|
|
JZUJAUAUKZWOTZWRXLWQUEZUFZUJZUJAXLULUMUCZWOTZWRXQWQUEZUFZUJZUJAXCUJBUAGIX
|
|
DGUFZXHXKAYBXEXIXGXJXDGWOUNYBXFWRWRXDGWQUOUPUQURXDXLUFZXHXPAYCXEXMXGXOXDX
|
|
LWOUNYCXFXNWRXDXLWQUOUPUQURXDXQUFZXHYAAYDXEXRXGXTXDXQWOUNYDXFXSWRXDXQWQUO
|
|
UPUQURXDIUFZXHXCAYEXEWPXGWTXDIWOUNYEXFWSWRXDIWQUOUPUQURGUSTAXKXIWRUTVAXLX
|
|
ATZAXPYAAYFXPYAUJAYFVBZXPXRXOUJYAYGXRXMXOAYFXRXMYFXRVBZXMAXLGIVCVDVEVFYGX
|
|
RXOXTAYFXRXOXTUJXOXTAYHVBZWRXQFUEZDUCZXNYJDUCZUFWRXNYJDVGYIWRYKXSYLYIYKWR
|
|
KDUCZWRYIYJKWRDYIXDFUEZKUFZYJKUFBGULUMUCZIUBUCZXQYDYNYJKXDXQFUOVHAYOBYQVI
|
|
YHAYOBYQPVJVQYIXQYPUGUETZIXQUGUETZXQYQTYFYRAXRGXLVKVLXRYSAYFXQGIVMVNXQYPI
|
|
VOVPVRVSAYMWRUFZYHAXDKDUCZXDUFZYTBEWRXDWRUFZUUAYMXDWRXDWRKDVGUUCWGVTAUUBB
|
|
ELVJOVRVQWAYIBCDEFHXLYIYFGHUGUEZTZXLUUDTAYFXRWBAUUEYHMVQGXLHWCWDAXDUUDTYN
|
|
ETYHRWEAXDETCUKZETVBXDUUFDUCETYHSWEWHVTWFVEWIWJWKWIWLWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d n x y F $. $d n x y H $. $d n x y M $. $d n x y N $. $d n x y ph $.
|
|
$d n x G $. $d x y K $. $d n x y .+ $. $d n x y Q $. $d x y S $.
|
|
$d x y Z $.
|
|
iseqhomo.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqhomo.2 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqhomo.s $e |- ( ph -> S e. V ) $.
|
|
${
|
|
$d .+ n x y w $. $d F n x y w $. $d G n x y w $. $d H n x y w $.
|
|
$d M n x y w $. $d N n x y w $. $d Q n x y w $. $d S n x y w $.
|
|
$d ph x y w $.
|
|
iseqhomo.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqhomo.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) ) $.
|
|
iseqhomo.5 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) ->
|
|
( H ` ( F ` x ) ) = ( G ` x ) ) $.
|
|
iseqhomo.g $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S ) $.
|
|
iseqhomo.qcl $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x Q y ) e. S ) $.
|
|
$( Apply a homomorphism to a sequence. (Contributed by Jim Kingdon,
|
|
21-Aug-2021.) $)
|
|
iseqhomo $p |- ( ph -> ( H ` ( seq M ( .+ , F , S ) ` N ) ) =
|
|
( seq M ( Q , G , S ) ` N ) ) $=
|
|
( vw vn cuz cfv wcel cseq cv wi c1 caddc co fveq2 fveq2d eqeq12d imbi2d
|
|
wceq cz ralrimiva eluzel2 syl uzid rspcdva iseq1 3eqtr4d wa oveq1 simpr
|
|
a1i adantlr iseqp1 wral ralrimivva adantr iseqcl eleq1d peano2uz oveq1d
|
|
oveq2 oveq2d rspc2v syl2anc mpd 3eqtrd syl5ibr expcom a2d uzind4 mpcom
|
|
) KJUCUDZUEZAKDFGJUFZUDZIUDZKEFHJUFZUDZUPZPAUAUGZWKUDZIUDZWQWNUDZUPZUHA
|
|
JWKUDZIUDZJWNUDZUPZUHZAUBUGZWKUDZIUDZXGWNUDZUPZUHAXGUIUJUKZWKUDZIUDZXLW
|
|
NUDZUPZUHAWPUHUAUBJKWQJUPZXAXEAXQWSXCWTXDXQWRXBIWQJWKULUMWQJWNULUNUOWQX
|
|
GUPZXAXKAXRWSXIWTXJXRWRXHIWQXGWKULUMWQXGWNULUNUOWQXLUPZXAXPAXSWSXNWTXOX
|
|
SWRXMIWQXLWKULUMWQXLWNULUNUOWQKUPZXAWPAXTWSWMWTWOXTWRWLIWQKWKULUMWQKWNU
|
|
LUNUOXFJUQUEZAJGUDZIUDZJHUDZXCXDABUGZGUDZIUDZYEHUDZUPZYCYDUPBWIJYEJUPZY
|
|
GYCYHYDYJYFYBIYEJGULUMYEJHULUNAYIBWIRURZAYAJWIUEAWJYAPJKUSUTZJVAUTVBAXB
|
|
YBIABCDFGJYLNMVCUMABCEFHJYLSTVCVDVHXGWIUEZAXKXPAYMXKXPUHXKXPAYMVEZXIXLH
|
|
UDZEUKZXJYOEUKZUPXIXJYOEVFYNXNYPXOYQYNXNXHXLGUDZDUKZIUDZXIYRIUDZEUKZYPY
|
|
NXMYSIYNBCDFGJXGAYMVGZAYEWIUEZYFFUEZYMNVIZAYEFUECUGZFUEVEZYEUUGDUKZFUEY
|
|
MMVIZVJUMYNUUIIUDZYEIUDZUUGIUDZEUKZUPZCFVKBFVKZYTUUBUPZAUUPYMAUUOBCFFQV
|
|
LVMYNXHFUEYRFUEZUUPUUQUHYNBCDFGJXGUUCUUFUUJVNYNUUEUURBWIXLYEXLUPZYFYRFY
|
|
EXLGULZVOAUUEBWIVKYMAUUEBWINURVMYNYMXLWIUEUUCJXGVPUTZVBUUOUUQXHUUGDUKZI
|
|
UDZXIUUMEUKZUPBCXHYRFFYEXHUPZUUKUVCUUNUVDUVEUUIUVBIYEXHUUGDVFUMUVEUULXI
|
|
UUMEYEXHIULVQUNUUGYRUPZUVCYTUVDUUBUVFUVBYSIUUGYRXHDVRUMUVFUUMUUAXIEUUGY
|
|
RIULVSUNVTWAWBYNUUAYOXIEYNYIUUAYOUPBWIXLUUSYGUUAYHYOUUSYFYRIUUTUMYEXLHU
|
|
LUNAYIBWIVKYMYKVMUVAVBVSWCYNBCEFHJXGUUCAUUDYHFUEYMSVIAUUHYEUUGEUKFUEYMT
|
|
VIVJUNWDWEWFWGWH $.
|
|
$}
|
|
|
|
$d .+ k w $. $d .+ x y $. $d F k w $. $d F x y $. $d K k w $.
|
|
$d K k x y $. $d M k w $. $d M x y $. $d N k w $. $d N x y $.
|
|
$d S k w $. $d S k x y $. $d Z k w $. $d Z k x y $. $d k ph w $.
|
|
$d ph x y $.
|
|
iseqz.3 $e |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z ) $.
|
|
iseqz.4 $e |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z ) $.
|
|
iseqz.5 $e |- ( ph -> K e. ( M ... N ) ) $.
|
|
iseqz.6 $e |- ( ph -> N e. V ) $.
|
|
iseqz.7 $e |- ( ph -> ( F ` K ) = Z ) $.
|
|
$( If the operation ` .+ ` has an absorbing element ` Z ` (a.k.a. zero
|
|
element), then any sequence containing a ` Z ` evaluates to ` Z ` .
|
|
(Contributed by Mario Carneiro, 27-May-2014.) $)
|
|
iseqz $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) $=
|
|
( wceq vw vk cuz cfv wcel cseq cfz co elfzuz3 cv wi c1 caddc fveq2 eqeq1d
|
|
imbi2d cz elfzuz eluzelz wa simpr adantr uztrn syl2anc syldan iseq1 eqtrd
|
|
syl iseqeq1 fveq1d syl5ibcom cmin eluzel2 adantlr iseqm1 oveq2d ralrimiva
|
|
oveq1 wral eluzp1m1 sylan iseqcl rspcdva 3eqtrd ex uzp1 mpjaod a1i iseqp1
|
|
wo oveq1d oveq2 eleq1d peano2uz expcom a2d uzind4 mpcom ) IGUCUDZUEZAIDEF
|
|
HUFZUDZKTZAGHIUGUHUEZWTQGHIUIVHAUAUJZXAUDZKTZUKAGXAUDZKTZUKZAUBUJZXAUDZKT
|
|
ZUKAXKULUMUHZXAUDZKTZUKAXCUKUAUBGIXEGTZXGXIAXQXFXHKXEGXAUNUOUPXEXKTZXGXMA
|
|
XRXFXLKXEXKXAUNUOUPXEXNTZXGXPAXSXFXOKXEXNXAUNUOUPXEITZXGXCAXTXFXBKXEIXAUN
|
|
UOUPXJGUQUEZAGHTZXIGHULUMUHUCUDUEZAGDEFGUFZUDZKTYBXIAYEGFUDZKABCDEFGAGHUC
|
|
UDZUEZYAAXDYHQGHIURVHZHGUSVHABUJZWSUEZYJYGUEZYJFUDZEUEZAYKUTYKYHYLAYKVAAY
|
|
HYKYIVBGYJHVCVDMVELVFSVGYBYEXHKYBGYDXADEFGHVIVJUOVKAYCXIAYCUTZXHGULVLUHZX
|
|
AUDZYFDUHYQKDUHZKYOBCDEFHGJAHUQUEZYCAYHYSYIHGVMVHZVBAYCVAAEJUEYCNVBAYLYNY
|
|
CMVNZAYJEUECUJZEUEUTZYJUUBDUHEUEZYCLVNZVOYOYFKYQDAYFKTYCSVBVPYOYJKDUHZKTZ
|
|
YRKTBEYQYJYQTUUFYRKYJYQKDVRUOAUUGBEVSYCAUUGBEPVQVBYOBCDEFHYPAYSYCYPYGUEYT
|
|
HGVTWAUUAUUEWBWCWDWEAYHYBYCWJYIHGWFVHWGWHXKWSUEZAXMXPAUUHXMXPUKAUUHUTZXMX
|
|
PUUIXMUTZXOXLXNFUDZDUHZKUUKDUHZKUUIXOUULTXMUUIBCDEFHXKUUIUUHYHXKYGUEZAUUH
|
|
VAAYHUUHYIVBGXKHVCVDZAYLYNUUHMVNAUUCUUDUUHLVNWIVBUUJXLKUUKDUUIXMVAWKUUIUU
|
|
MKTZXMUUIKYJDUHZKTZUUPBEUUKYJUUKTUUQUUMKYJUUKKDWLUOAUURBEVSUUHAUURBEOVQVB
|
|
UUIYNUUKEUEBYGXNYJXNTYMUUKEYJXNFUNWMAYNBYGVSUUHAYNBYGMVQVBUUIUUNXNYGUEUUO
|
|
HXKWNVHWCWCVBWDWEWOWPWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d .+ b x y $. $d .+ b x y z $. $d C a b x y z $. $d F x y $.
|
|
$d G x y z $. $d M x y z $. $d N x y z $. $d S a b x y z $.
|
|
$d T a b x y z $. $d ph x y $.
|
|
iseqdistr.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) $.
|
|
iseqdistr.2 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) ) $.
|
|
iseqdistr.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
iseqdistr.4 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S ) $.
|
|
iseqdistr.5 $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) ->
|
|
( F ` x ) = ( C T ( G ` x ) ) ) $.
|
|
iseqdistr.s $e |- ( ph -> S e. V ) $.
|
|
iseqdistr.t $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x T y ) e. S ) $.
|
|
iseqdistr.f $e |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) $.
|
|
iseqdistr.c $e |- ( ph -> C e. S ) $.
|
|
$( The distributive property for series. (Contributed by Jim Kingdon,
|
|
21-Aug-2021.) $)
|
|
iseqdistr $p |- ( ph -> ( seq M ( .+ , F , S ) ` N ) =
|
|
( C T ( seq M ( .+ , G , S ) ` N ) ) ) $=
|
|
( vz va vb cseq cfv cv co cmpt wcel wceq wral adantr ralrimivva weq oveq1
|
|
wa eleq1d oveq2 cbvral2v sylib rspc2va syl21anc eqid fvmptg simprl simprr
|
|
syl2anc oveq12d 3eqtr4d eqeltrrd eqtr4d iseqhomo iseqcl caovcld eqtr3d
|
|
cuz ) AKEFIJUEUFZUBFDUBUGZGUHZUIZUFZKEFHJUEUFDVRGUHZABCEEFIHWAJKLMPROABUG
|
|
ZFUJZCUGZFUJZUQZUQZDWDWFEUHZGUHZDWDGUHZDWFGUHZEUHWJWAUFZWDWAUFZWFWAUFZEUH
|
|
NWIWJFUJZWKFUJZWNWKUKMWIDFUJZWQUCUGZUDUGZGUHZFUJZUDFULUCFULZWRAWSWHUAUMZM
|
|
AXDWHAWDWFGUHZFUJZCFULBFULXDAXGBCFFSUNXGXCWTWFGUHZFUJBCUCUDFFBUCUOXFXHFWD
|
|
WTWFGUPURCUDUOXHXBFWFXAWTGUSURUTVAUMZXCWRDXAGUHZFUJZUCUDDWJFFWTDUKXBXJFWT
|
|
DXAGUPURZXAWJUKXJWKFXAWJDGUSURVBVCUBWJVTWKFFWAVSWJDGUSWAVDZVEVHWIWOWLWPWM
|
|
EWIWEWLFUJZWOWLUKAWEWGVFZWIWSWEXDXNXEXOXIXCXNXKUCUDDWDFFXLUDBUOXJWLFXAWDD
|
|
GUSURVBVCUBWDVTWLFFWAVSWDDGUSXMVEVHWIWGWMFUJZWPWMUKAWEWGVGZWIWSWGXDXPXEXQ
|
|
XIXCXPXKUCUDDWFFFXLUDCUOXJWMFXAWFDGUSURVBVCUBWFVTWMFFWAVSWFDGUSXMVEVHVIVJ
|
|
AWDJVQUFUJUQZWDIUFZWAUFZDXSGUHZWDHUFZXRXSFUJYAFUJXTYAUKPXRYBYAFQTVKUBXSVT
|
|
YAFFWAVSXSDGUSXMVEVHQVLTMVMAVRFUJWCFUJWBWCUKABCEFIJKOPMVNZABCDVRFFFGSUAYC
|
|
VOUBVRVTWCFFWAVSVRDGUSXMVEVHVP $.
|
|
$}
|
|
|
|
${
|
|
$d M k v $. $d N k v $. $d Z k v $.
|
|
iser0.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( The value of the partial sums in a zero-valued infinite series.
|
|
(Contributed by Jim Kingdon, 19-Aug-2021.) $)
|
|
iser0 $p |- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) , CC ) ` N ) = 0 ) $=
|
|
( vk vv wcel caddc cc cc0 cvv co wceq a1i cfv eleq2i cv wa 0cn adantl csn
|
|
cxp 00id cuz biimpi elfzuz syl6eleqr fvconst2g sylancr 0cnd cnex syl6eqel
|
|
cfz biimpri addcl iseqid3s ) BCGZEFHICJUAUBZABKJJJHLJMUQUCNUQBAUDOZGCUSBD
|
|
PUEUQEQZABUMLGZRJIGZUTCGZUTUROZJMZSVAVCUQVAUTUSCUTABUFDUGTCJUTIUHZUIUQUJI
|
|
KGUQUKNUQUTUSGZRZVDJIVHVBVCVESVGVCUQVCVGCUSUTDPUNTVFUISULUTIGFQZIGRUTVIHL
|
|
IGUQUTVIUOTUP $.
|
|
|
|
$( A zero-valued infinite series is equal to the constant zero function.
|
|
(Contributed by Jim Kingdon, 19-Aug-2021.) $)
|
|
iser0f $p |- ( M e. ZZ ->
|
|
seq M ( + , ( Z X. { 0 } ) , CC ) = ( Z X. { 0 } ) ) $=
|
|
( vk vv cz wcel caddc cc cc0 csn cxp cseq wceq cv cfv c0ex wfn cvv adantl
|
|
wral iser0 fvconst2 eqtr4d rgen wb cuz id cnex eleq2i 0cnd eqeltrd sylbir
|
|
a1i wa co addcl iseqfn fneq2i sylibr wf fconst ffn eqfnfv sylancl mpbiri
|
|
ax-mp ) AFGZHIBJKZLZAMZVJNZDOZVKPZVMVJPZNZDBUAZVPDBVMBGZVNJVOAVMBCUBBJVMQ
|
|
UCZUDUEVHVKBRZVJBRZVLVQUFVHVKAUGPZRVTVHDEHIVJASVHUHISGVHUIUNVMWBGZVOIGZVH
|
|
WCVRWDBWBVMCUJVRVOJIVSVRUKULUMTVMIGEOZIGUOVMWEHUPIGVHVMWEUQTURBWBVKCUSUTB
|
|
VIVJVAWABJQVBBVIVJVCVGDBVKVJVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d k x y F $. $d k x G $. $d k x y M $. $d k x y N $. $d k x y ph $.
|
|
serige0.1 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
serige0.2 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR ) $.
|
|
${
|
|
serige0.3 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> 0 <_ ( F ` k ) ) $.
|
|
$( A finite sum of nonnegative terms is nonnegative. (Contributed by Jim
|
|
Kingdon, 22-Aug-2021.) $)
|
|
serige0 $p |- ( ph -> 0 <_ ( seq M ( + , F , CC ) ` N ) ) $=
|
|
( vx caddc cc cfv cc0 cle wbr cr wcel wa breq2 elrab vy cseq cv cvv cuz
|
|
crab cz eluzel2 syl cnex a1i ssrab2 ax-resscn sstri sylanbrc co readdcl
|
|
ad2ant2r addge0 an4s syl2anb adantl addcl iseqss fveq1d iseqcl eqeltrrd
|
|
wss simprbi ) AEJKCDUBZLZMIUCZNOZIPUFZQZMVKNOZAEJVNCDUBZLVKVNAEVQVJABUA
|
|
JVNKCDUDAEDUELZQDUGQFDEUHUIKUDQAUJUKVNKVHAVNPKVMIPULUMUNUKABUCZVRQRVSCL
|
|
ZPQMVTNOZVTVNQGHVMWAIVTPVLVTMNSTUOZVSVNQZUAUCZVNQZRVSWDJUPZVNQZAWCVSPQZ
|
|
MVSNOZRZWDPQZMWDNOZRZWGWEVMWIIVSPVLVSMNSTVMWLIWDPVLWDMNSTWJWMRWFPQZMWFN
|
|
OZWGWHWKWNWIWLVSWDUQURWHWKWIWLWOVSWDUSUTVMWOIWFPVLWFMNSTUOVAVBZVSKQWDKQ
|
|
RWFKQAVSWDVCVBVDVEABUAJVNCDEFWBWPVFVGVOVKPQVPVMVPIVKPVLVKMNSTVIUI $.
|
|
$}
|
|
|
|
serile.3 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR ) $.
|
|
serile.4 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) ) $.
|
|
$( Comparison of partial sums of two infinite series of reals.
|
|
(Contributed by Jim Kingdon, 22-Aug-2021.) $)
|
|
serile $p |- ( ph ->
|
|
( seq M ( + , F , CC ) ` N ) <_ ( seq M ( + , G , CC ) ` N ) ) $=
|
|
( vx cc0 caddc cc cseq cfv cle cvv wcel cr cmin co wbr cv cmpt cuz wa vex
|
|
wceq resubcld oveq12d eqid fvmptg sylancr eqeltrd subge0d mpbird breqtrrd
|
|
fveq2 serige0 recnd isersub breqtrd cz eluzel2 syl cnex a1i wss ax-resscn
|
|
readdcl adantl addcl iseqss fveq1d iseqcl eqeltrrd mpbid ) ALFMNDEOZPZFMN
|
|
CEOZPZUAUBZQUCWBVTQUCALFMNKRKUDZDPZWDCPZUAUBZUEZEOPWCQABWHEFGABUDZEUFPZSU
|
|
GZWIWHPZWIDPZWICPZUAUBZTWKWIRSWOTSWLWOUIBUHWKWMWNIHUJZKWIWGWORTWHWDWIUIWE
|
|
WMWFWNUAWDWIDUSWDWICUSUKWHULUMUNZWPUOWKLWOWLQWKLWOQUCWNWMQUCJWKWMWNIHUPUQ
|
|
WQURUTABDCWHEFGWKWMIVAWKWNHVAWQVBVCAVTWBAFMTDEOZPVTTAFWRVSABKMTNDERAFWJSE
|
|
VDSGEFVEVFZNRSAVGVHZTNVIAVJVHZIWITSWDTSUGWIWDMUBZTSAWIWDVKVLZWINSWDNSUGXB
|
|
NSAWIWDVMVLZVNVOABKMTDEFGIXCVPVQAFMTCEOZPWBTAFXEWAABKMTNCERWSWTXAHXCXDVNV
|
|
OABKMTCEFGHXCVPVQUPVR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Integer powers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ^ $. $( Exponentiation. $)
|
|
|
|
$( Extend class notation to include exponentiation of a complex number to an
|
|
integer power. $)
|
|
cexp $a class ^ $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define exponentiation to nonnegative integer powers. This definition is
|
|
not meant to be used directly; instead, ~ exp0 and ~ expp1 provide the
|
|
standard recursive definition. The up-arrow notation is used by Donald
|
|
Knuth for iterated exponentiation (_Science_ 194, 1235-1242, 1976) and
|
|
is convenient for us since we don't have superscripts. 10-Jun-2005: The
|
|
definition was extended to include zero exponents, so that ` 0 ^ 0 = 1 `
|
|
per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014:
|
|
The definition was extended to include negative integer exponents. The
|
|
case ` x = 0 , y < 0 ` gives the value ` ( 1 / 0 ) ` , so we will avoid
|
|
this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.) $)
|
|
df-iexp $a |- ^ = ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y ,
|
|
( seq 1 ( x. , ( NN X. { x } ) , CC ) ` y ) ,
|
|
( 1 / ( seq 1 ( x. , ( NN X. { x } ) , CC ) ` -u y ) ) ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d A k n x y $. $d N n $.
|
|
|
|
$( Lemma for ~ expival . If we take a complex number apart from zero and
|
|
raise it to a positive integer power, the result is apart from zero.
|
|
(Contributed by Jim Kingdon, 7-Jun-2020.) $)
|
|
expivallem $p |- ( ( A e. CC /\ A # 0 /\ N e. NN ) ->
|
|
( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) $=
|
|
( vx vy cc wcel cc0 cap wbr cn cmul c1 cfv wa cv wi co wceq fveq2 breq1d
|
|
vn vk csn cxp cseq caddc imbi2d simpr 1zzd cuz fvconst2g sylan2br adantlr
|
|
elnnuz simpll eqeltrd mulcl adantl iseq1 mpan2 adantr eqtrd simpl adantll
|
|
1nn mpbird iseqcl simplrl simplrr mulap0d iseqp1 simprl peano2nnd syl2anc
|
|
sylib wb oveq2d exp31 a2d nnind impcom 3impa ) AEFZAGHIZBJFZBKEJAUCUDZLUE
|
|
ZMZGHIZWEWCWDNZWIWJUAOZWGMZGHIZPWJLWGMZGHIZPWJUBOZWGMZGHIZPWJWPLUFQZWGMZG
|
|
HIZPWJWIPUAUBBWKLRZWMWOWJXBWLWNGHWKLWGSTUGWKWPRZWMWRWJXCWLWQGHWKWPWGSTUGW
|
|
KWSRZWMXAWJXDWLWTGHWKWSWGSTUGWKBRZWMWIWJXEWLWHGHWKBWGSTUGWJWOWDWCWDUHWJWN
|
|
AGHWJWNLWFMZAWJCDKEWFLWJUIWJCOZLUJMZFZNXGWFMZAEWCXIXJARZWDXIWCXGJFXKXGUNJ
|
|
AXGEUKULUMWCWDXIUOUPZXGEFDOZEFNZXGXMKQEFZWJXGXMUQZURUSWCXFARZWDWCLJFXQVEJ
|
|
ALEUKUTVAVBTVFWPJFZWJWRXAXRWJWRXAXRWJNZWRNZXAWQAKQZGHIZXTWQAXTCDKEWFLWPXS
|
|
WPXHFZWRXSXRYCXRWJVCZWPUNVOZVAXSXIXJEFZWRWJXIYFXRXLVDZUMXNXOXTXPURVGXRWCW
|
|
DWRVHXSWRUHXRWCWDWRVIVJXSXAYBVPWRXSWTYAGHXSWTWQWSWFMZKQYAXSCDKEWFLWPYEYGX
|
|
NXOXSXPURVKXSYHAWQKXSWCWSJFYHARXRWCWDVLXSWPYDVMJAWSEUKVNVQVBTVAVFVRVSVTWA
|
|
WB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w N $.
|
|
$( Value of exponentiation to integer powers. (Contributed by Jim Kingdon,
|
|
7-Jun-2020.) $)
|
|
expival $p |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) ->
|
|
( A ^ N ) =
|
|
if ( N = 0 , 1 , if ( 0 < N ,
|
|
( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ,
|
|
( 1 / ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` -u N ) ) ) ) ) $=
|
|
( cc wcel cz cc0 wbr wo wceq c1 clt cmul cn cfv cdiv co cif wa adantl wb
|
|
vz vw vx vy cap cle csn cxp cseq cneg cexp w3a iftrue ax-1cn syl6eqel a1i
|
|
wn wi cuz elnnz elnnuz sylbb1 adantll cv fvconst2g sylan2br eleq1d mpbird
|
|
simpl adantlr mulcl iseqcl adantr 3adantl3 simpll2 znegcld wne zred simpr
|
|
ex 0red nltled simplr neneqad necomd 0z mpan2 3ad2ant2 ad2antrr mpbir2and
|
|
zltlen lt0neg1d mpbid sylanbrc 3ad2antl1 simpll1 expivallem 3com23 3expia
|
|
sylib syl2anc neneqd ioran zleloe mtbird pm2.21d simpll3 recclapd iffalse
|
|
mpan mpjaod wdc zdclt df-dc sneq xpeq2d iseqeq3 syl fveq1d oveq2d ifeq12d
|
|
zdceq ifeq2d eqeq1 breq2 fveq2 fveq2d ifbieq12d ifbieq2d df-iexp syld3an3
|
|
negeq ovmpt2g ) ACDZBEDZAFUEGZFBUFGZHZBFIZJFBKGZBLCMAUGZUHZJUIZNZJBUJZUUC
|
|
NZOPZQZQZCDZABUKPUUIIYNYOYRULZYSUUJYSUQZYSUUJURUUKYSUUIJCYSJUUHUMUNUOUPUU
|
|
KUULUUJUUKUULRZUUJUUHCDZUUMYTUUNYTUQZYNYOUULYTUUNURZYRYNYORZUUPUULUUQYTUU
|
|
NUUQYTRZUUNUUDCDZUURUAUBLCUUBJBYOYTBJUSNZDZYNBMDYOYTRUVABUTBVAVBVCUUQUAVD
|
|
ZUUTDZUVBUUBNZCDZYTYNUVCUVEYOYNUVCRZUVEYNYNUVCVIUVFUVDACUVCYNUVBMDUVDAIUV
|
|
BVAMAUVBCVEVFVGVHZVJVJUVBCDUBVDZCDRZUVBUVHLPCDZUURUVBUVHVKZSVLYTUUNUUSTUU
|
|
QYTUUHUUDCYTUUDUUGUMVGSVHVTVMVNUUMUUOUUNUUMUUORZUUNUUGCDZUVLUUFUVLUAUBLCU
|
|
UBJUUEUVLUUEMDZUUEUUTDUVLUUEEDFUUEKGZUVNUVLBYNYOYRUULUUOVOZVPUVLBFKGZUVOU
|
|
VLUVQBFUFGZFBVQZUVLBFUVLBUVPVRZUVLWAUUMUUOVSZWBUVLBFUVLBFUUKUULUUOWCWDWEZ
|
|
UUKUVQUVRUVSRTZUULUUOYOYNUWCYRYOFEDZUWCWFBFWKWGWHWIWJUVLBUVTWLWMUUEUTWNZU
|
|
UEVAWTUUMUVCUVEUUOUUKUVCUVEUULYNYOUVCUVEYRUVGWOVJVJUVIUVJUVLUVKSVLUVLYPUU
|
|
FFUEGZYQUVLYNUVNYPUWFURYNYOYRUULUUOWPUWEYNUVNYPUWFYNYPUVNUWFAUUEWQWRWSXAU
|
|
VLYQUWFUVLYQYTFBIZHZUVLUUOUWGUQUWHUQUWAUVLFBUWBXBYTUWGXCWNUUKYQUWHTZUULUU
|
|
OYOYNUWIYRUWDYOUWIWFFBXDXJWHWIXEXFYNYOYRUULUUOXGXKXHUUOUUNUVMTUUMUUOUUHUU
|
|
GCYTUUDUUGXIVGSVHVTYNYOUULYTUUOHZYRUUQUWJUULYOUWJYNYOYTXLZUWJUWDYOUWKWFFB
|
|
XMXJYTXNWTSVMVNXKUULUUJUUNTUUKUULUUIUUHCYSJUUHXIVGSVHVTYOYNYSUULHZYRYOYSX
|
|
LZUWLYOUWDUWMWFBFYBWGYSXNWTWHXKUCUDABCEUDVDZFIZJFUWNKGZUWNLCMUCVDZUGZUHZJ
|
|
UIZNZJUWNUJZUWTNZOPZQZQUUIUKUWOJUWPUWNUUCNZJUXBUUCNZOPZQZQCUWQAIZUWOUXEUX
|
|
IJUXJUWPUXAUXFUXDUXHUXJUWNUWTUUCUXJUWSUUBIUWTUUCIUXJUWRUUAMUWQAXOXPLCUWSU
|
|
UBJXQXRZXSUXJUXCUXGJOUXJUXBUWTUUCUXKXSXTYAYCUWNBIZUWOYSUXIUUHJUWNBFYDUXLU
|
|
WPYTUXFUXHUUDUUGUWNBFKYEUWNBUUCYFUXLUXGUUFJOUXLUXBUUEUUCUWNBYLYGXTYHYIUCU
|
|
DYJYMYK $.
|
|
$}
|
|
|
|
$( Value of exponentiation to positive integer powers. (Contributed by Jim
|
|
Kingdon, 8-Jun-2020.) $)
|
|
expinnval $p |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) =
|
|
( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) ) $=
|
|
( cc wcel cn wa cexp co cc0 wceq clt wbr cmul csn cxp cfv cif adantl eqtrd
|
|
c1 cseq cneg cdiv cz cap cle wo simpl nnz simpr nnnn0d nn0ge0d olcd expival
|
|
syl3anc nnne0 neneqd iffalsed nngt0 iftrued ) ACDZBEDZFZABGHZBIJZTIBKLZBMCE
|
|
ANOTUAZPZTBUBVGPUCHZQZQZVHVCVABUDDZAIUELZIBUFLZUGVDVKJVAVBUHVBVLVABUIRVCVNV
|
|
MVCBVCBVAVBUJUKULUMABUNUOVBVKVHJVAVBVKVJVHVBVETVJVBBIBUPUQURVBVFVHVIBUSUTSR
|
|
S $.
|
|
|
|
$( Value of a complex number raised to the 0th power. Note that under our
|
|
definition, ` 0 ^ 0 = 1 ` , following the convention used by Gleason.
|
|
Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM,
|
|
20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.) $)
|
|
exp0 $p |- ( A e. CC -> ( A ^ 0 ) = 1 ) $=
|
|
( cc wcel cc0 cexp co wceq c1 clt wbr cmul cn csn cxp cseq cfv cneg cdiv cz
|
|
cif cap cle wo 0zd 0le0 a1i olcd expival mpd3an23 eqid iftruei syl6eq ) ABC
|
|
ZADEFZDDGZHDDIJDKBLAMNHOZPHDQUPPRFTZTZHUMDSCADUAJZDDUBJZUCUNURGUMUDUMUTUSUT
|
|
UMUEUFUGADUHUIUOHUQDUJUKUL $.
|
|
|
|
$( ` 0 ^ 0 = 1 ` (common case). This is our convention. It follows the
|
|
convention used by Gleason; see Part of Definition 10-4.1 of [Gleason]
|
|
p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.) $)
|
|
0exp0e1 $p |- ( 0 ^ 0 ) = 1 $=
|
|
( cc0 cc wcel cexp co c1 wceq 0cn exp0 ax-mp ) ABCAADEFGHAIJ $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Value of a complex number raised to the first power. (Contributed by
|
|
NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.) $)
|
|
exp1 $p |- ( A e. CC -> ( A ^ 1 ) = A ) $=
|
|
( vx vy cc wcel c1 cexp co cmul csn cxp cseq cfv wceq 1nn mpan2 fvconst2g
|
|
cn cv wa expinnval 1zzd elnnuz sylan2br simpl eqeltrd mulcl adantl 3eqtrd
|
|
cuz iseq1 ) ADEZAFGHZFIDRAJKZFLMZFUNMZAULFREZUMUONOAFUAPULBCIDUNFULUBULBS
|
|
ZFUJMEZTURUNMZADUSULURREUTANURUCRAURDQUDULUSUEUFURDECSZDETURVAIHDEULURVAU
|
|
GUHUKULUQUPANORAFDQPUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d N x y $.
|
|
$( Value of a complex number raised to a nonnegative integer power plus
|
|
one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by
|
|
NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.) $)
|
|
expp1 $p |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ ( N + 1 ) ) =
|
|
( ( A ^ N ) x. A ) ) $=
|
|
( vx vy wcel cc cn cc0 wceq c1 caddc co cexp cmul cfv simpr elnnuz oveq1d
|
|
wa cv cn0 wo elnn0 csn cxp cseq sylib simpll wb fvconst2g eleq1d sylan2br
|
|
adantlr mpbird mulcl adantl iseqp1 peano2nn sylan2 oveq2d eqtrd expinnval
|
|
cuz 3eqtr4d exp1 mulid2 eqtr4d adantr 0p1e1 syl6eq oveq2 sylan9eqr jaodan
|
|
exp0 sylan2b ) BUAEAFEZBGEZBHIZUBABJKLZMLZABMLZANLZIZBUCVPVQWCVRVPVQSZVSN
|
|
FGAUDUEZJUFZOZBWFOZANLZVTWBWDWGWHVSWEOZNLWIWDCDNFWEJBWDVQBJVCOZEVPVQPBQUG
|
|
WDCTZWKEZSWLWEOZFEZVPVPVQWMUHVPWMWOVPUIZVQWMVPWLGEZWPWLQVPWQSWNAFGAWLFUJU
|
|
KULUMUNWLFEDTZFESWLWRNLFEWDWLWRUOUPUQWDWJAWHNVQVPVSGEZWJAIBURZGAVSFUJUSUT
|
|
VAVQVPWSVTWGIWTAVSVBUSWDWAWHANABVBRVDVPVRSZAJMLZJANLZVTWBVPXBXCIVRVPXBAXC
|
|
AVEAVFVGVHXAVSJAMXAVSHJKLJXABHJKVPVRPRVIVJUTXAWAJANVRVPWAAHMLJBHAMVKAVNVL
|
|
RVDVMVO $.
|
|
$}
|
|
|
|
$( Value of a complex number raised to a negative integer power.
|
|
(Contributed by Jim Kingdon, 8-Jun-2020.) $)
|
|
expnegap0 $p |- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) ->
|
|
( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 wbr cneg cexp co c1 cdiv wceq wa cn wo clt cfv adantl adantlr
|
|
oveq2d cap cn0 elnn0 cmul csn cxp cseq cif wn nnne0 nncn negeq0d necon3abid
|
|
wne mpbid iffalsed nnnn0 nn0nlt0 syl nn0red mtbid negnegd fveq2d 3eqtrd w3a
|
|
lt0neg1d cz cle simp1 simp3 nnzd znegcld simp2 orcd expival 3expa expinnval
|
|
syl3anc 3eqtr4d 1div1e1 eqcomi negeq neg0 syl6eq exp0 oveq2 3eqtr4a sylan2b
|
|
sylan9eqr jaodan 3impa ) ACDZAEUAFZBUBDZABGZHIZJABHIZKIZLZWNWLWMMZBNDZBELZO
|
|
WSBUCWTXAWSXBWTXAMWOELZJEWOPFZWOUDCNAUEUFJUGZQZJWOGZXEQZKIZUHZUHZJBXEQZKIZW
|
|
PWRWLXAXKXMLWMWLXAMZXKXJXIXMXNXCJXJXNBEUNZXCUIXAXOWLBUJRXNXCBEXNBXABCDWLBUK
|
|
RZULUMUOUPXNXDXFXIXNBEPFZXDXNWNXQUIXAWNWLBUQRZBURUSXNBXNBXRUTVFVAUPXNXHXLJK
|
|
XNXGBXEXNBXPVBVCTVDSWLWMXAWPXKLZWLWMXAVEZWLWOVGDWMEWOVHFZOXSWLWMXAVIXTBXTBW
|
|
LWMXAVJVKVLXTWMYAWLWMXAVMVNAWOVOVRVPWLXAWRXMLWMXNWQXLJKABVQTSVSWLXBWSWMWLXB
|
|
MZJJJKIZWPWRYCJVTWAXBWLWPAEHIZJXBWOEAHXBWOEGEBEWBWCWDTAWEZWIYBWQJJKXBWLWQYD
|
|
JBEAHWFYEWITWGSWJWHWK $.
|
|
|
|
$( Value of a complex number raised to a negative integer power.
|
|
(Contributed by Jim Kingdon, 8-Jun-2020.) $)
|
|
expineg2 $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) ->
|
|
( A ^ N ) = ( 1 / ( A ^ -u N ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cneg cn0 cexp co c1 cdiv negneg ad2antrl expnegap0
|
|
wceq oveq2d 3expa adantrl eqtr3d ) ACDZAEFGZHZBCDZBIZJDZHHZAUGIZKLZABKLMAUG
|
|
KLNLZUIUJBAKUFUJBRUEUHBOPSUEUHUKULRZUFUCUDUHUMAUGQTUAUB $.
|
|
|
|
$( A number to the negative one power is the reciprocal. (Contributed by Jim
|
|
Kingdon, 8-Jun-2020.) $)
|
|
expn1ap0 $p |- ( ( A e. CC /\ A # 0 ) -> ( A ^ -u 1 ) = ( 1 / A ) ) $=
|
|
( cc wcel cc0 cap wbr wa cneg cexp cdiv cn0 wceq 1nn0 expnegap0 mp3an3 exp1
|
|
c1 co oveq2d adantr eqtrd ) ABCZADEFZGAQHIRZQAQIRZJRZQAJRZUBUCQKCUDUFLMAQNO
|
|
UBUFUGLUCUBUEAQJAPSTUA $.
|
|
|
|
${
|
|
$d x y z w A $. $d x z B $. $d x y z w F $.
|
|
expcllem.1 $e |- F C_ CC $.
|
|
expcllem.2 $e |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F ) $.
|
|
expcllem.3 $e |- 1 e. F $.
|
|
$( Lemma for proving nonnegative integer exponentiation closure laws.
|
|
(Contributed by NM, 14-Dec-2005.) $)
|
|
expcllem $p |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F ) $=
|
|
( vz wcel cc0 wceq cexp co wi c1 oveq2 eleq1d imbi2d wa vw cn wo elnn0 cv
|
|
cn0 caddc cc sseli exp1 syl ibir caovcl ancoms adantlr nnnn0 expp1 syl2an
|
|
cmul adantr mpbird exp31 com12 a2d nnind impcom sylan9eqr syl6eqel jaodan
|
|
wb exp0 sylan2b ) DUFJCEJZDUBJZDKLZUCCDMNZEJZDUDVMVNVQVOVNVMVQVMCIUEZMNZE
|
|
JZOVMCPMNZEJZOVMCUAUEZMNZEJZOVMCWCPUGNZMNZEJZOVMVQOIUADVRPLZVTWBVMWIVSWAE
|
|
VRPCMQRSVRWCLZVTWEVMWJVSWDEVRWCCMQRSVRWFLZVTWHVMWKVSWGEVRWFCMQRSVRDLZVTVQ
|
|
VMWLVSVPEVRDCMQRSVMWBVMWACEVMCUHJZWACLEUHCFUIZCUJUKRULWCUBJZVMWEWHVMWOWEW
|
|
HOVMWOWEWHVMWOTZWETWHWDCUSNZEJZVMWEWRWOWEVMWRABWDCEUSGUMUNUOWPWHWRVJWEWPW
|
|
GWQEVMWMWCUFJWGWQLWOWNWCUPCWCUQURRUTVAVBVCVDVEVFVMVOTVPPEVOVMVPCKMNZPDKCM
|
|
QVMWMWSPLWNCVKUKVGHVHVIVL $.
|
|
|
|
expcl2lemap.4 $e |- ( ( x e. F /\ x # 0 ) -> ( 1 / x ) e. F ) $.
|
|
$( Lemma for proving integer exponentiation closure laws. (Contributed by
|
|
Jim Kingdon, 8-Jun-2020.) $)
|
|
expcl2lemap $p |- ( ( A e. F /\ A # 0 /\ B e. ZZ ) -> ( A ^ B ) e. F ) $=
|
|
( vz wcel cc0 cap wbr co wa c1 cc breq1 elrab cz cexp cr cneg cn elznn0nn
|
|
cn0 wo wi expcllem ex adantr cdiv simpll sseldi simplr simprl recnd nnnn0
|
|
wceq ad2antll expineg2 syl22anc crab ssrab2 simpl sylibr sstri cmul sseli
|
|
cv syl2an anim1i sylbi mulap0 sylanbrc 1ap0 mpbir2an syl2anc sylib simprd
|
|
oveq2 eleq1d imbi12d vtoclga sylc eqeltrd jaod syl5bi 3impia ) CEKZCLMNZD
|
|
UAKZCDUBOZEKZWMDUGKZDUCKZDUDZUEKZPZUHWKWLPZWODUFXAWPWOWTWKWPWOUIWLWKWPWOA
|
|
BCDEFGHUJUKULXAWTWOXAWTPZWNQCWRUBOZUMOZEXBCRKWLDRKWRUGKZWNXDUTXBERCFWKWLW
|
|
TUNUOWKWLWTUPXBDXAWQWSUQURWSXEXAWQWRUSVAZCDVBVCXBXCEKZXCLMNZXDEKZXBJVKZLM
|
|
NZJEVDZEXCXKJEVEZXBCXLKZXEXCXLKZXBXAXNXAWTVFXKWLJCEXJCLMSTVGXFABCWRXLXLER
|
|
XMFVHAVKZXLKZBVKZXLKZPXPXRVIOZEKZXTLMNZXTXLKXQXPEKZXREKZYAXSXLEXPXMVJXLEX
|
|
RXMVJGVLXQXPRKZXPLMNZPZXRRKZXRLMNZPZYBXSXQYCYFPYGXKYFJXPEXJXPLMSTYCYEYFER
|
|
XPFVJVMVNXSYDYIPYJXKYIJXREXJXRLMSTYDYHYIERXRFVJVMVNXPXRVOVLXKYBJXTEXJXTLM
|
|
STVPQXLKQEKQLMNZHVQXKYKJQEXJQLMSTVRUJVSZUOXBXGXHXBXOXGXHPYLXKXHJXCEXJXCLM
|
|
STVTWAYFQXPUMOZEKZUIXHXIUIAXCEXPXCUTZYFXHYNXIXPXCLMSYOYMXDEXPXCQUMWBWCWDY
|
|
CYFYNIUKWEWFWGUKWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z N $.
|
|
$( Closure of exponentiation of nonnegative integers. (Contributed by NM,
|
|
16-Dec-2005.) $)
|
|
nnexpcl $p |- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN ) $=
|
|
( vx vy cn nnsscn cv nnmulcl 1nn expcllem ) CDABEFCGDGHIJ $.
|
|
|
|
$( Closure of exponentiation of nonnegative integers. (Contributed by NM,
|
|
14-Dec-2005.) $)
|
|
nn0expcl $p |- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 ) $=
|
|
( vx vy cn0 nn0sscn cv nn0mulcl 1nn0 expcllem ) CDABEFCGDGHIJ $.
|
|
|
|
$( Closure of exponentiation of integers. (Contributed by NM,
|
|
16-Dec-2005.) $)
|
|
zexpcl $p |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ ) $=
|
|
( vx vy cz zsscn cv zmulcl 1z expcllem ) CDABEFCGDGHIJ $.
|
|
|
|
$( Closure of exponentiation of rationals. (Contributed by NM,
|
|
16-Dec-2005.) $)
|
|
qexpcl $p |- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ ) $=
|
|
( vx vy cq qsscn cv qmulcl c1 cz wcel 1z zq ax-mp expcllem ) CDABEFCGDGHI
|
|
JKIEKLIMNO $.
|
|
|
|
$( Closure of exponentiation of reals. (Contributed by NM,
|
|
14-Dec-2005.) $)
|
|
reexpcl $p |- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR ) $=
|
|
( vx vy cr ax-resscn cv remulcl 1re expcllem ) CDABEFCGDGHIJ $.
|
|
|
|
$( Closure law for nonnegative integer exponentiation. (Contributed by NM,
|
|
26-May-2005.) $)
|
|
expcl $p |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC ) $=
|
|
( vx vy cc ssid cv mulcl ax-1cn expcllem ) CDABEEFCGDGHIJ $.
|
|
|
|
$( Closure law for exponentiation of positive reals. (Contributed by NM,
|
|
24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) $)
|
|
rpexpcl $p |- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ ) $=
|
|
( vx vy crp wcel cz wa cc0 cap wbr cexp co simpl rpap0 adantr simpr cr cc
|
|
cv rpssre ax-resscn sstri rpmulcl 1rp c1 cdiv rpreccl expcl2lemap syl3anc
|
|
) AEFZBGFZHUKAIJKZULABLMEFUKULNUKUMULAOPUKULQCDABEERSUAUBUCCTZDTUDUEUNEFU
|
|
FUNUGMEFUNIJKUNUHPUIUJ $.
|
|
|
|
$( Closure of exponentiation of reals. (Contributed by Jim Kingdon,
|
|
9-Jun-2020.) $)
|
|
reexpclzap $p |- ( ( A e. RR /\ A # 0 /\ N e. ZZ ) ->
|
|
( A ^ N ) e. RR ) $=
|
|
( vx vy cr ax-resscn cv remulcl 1re rerecclap expcl2lemap ) CDABEFCGZDGHI
|
|
LJK $.
|
|
|
|
$( Closure of exponentiation of rational numbers. (Contributed by Mario
|
|
Carneiro, 9-Sep-2014.) $)
|
|
qexpclz $p |- ( ( A e. QQ /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. QQ ) $=
|
|
( vx vy cq wcel cc0 wne cz w3a cap wbr co wb zq ax-mp qapne mpan2 cv c1
|
|
cexp 0z 3anbi2d 3ad2ant1 ibir qsscn qmulcl 1z wa cdiv pm5.32i expcl2lemap
|
|
qreccl sylbi syl ) AEFZAGHZBIFZJZUPAGKLZURJZABUAMEFUSVAUPUQVAUSNURUPUTUQU
|
|
PURUPGEFZUTUQNGIFVBUBGOPZAGQRUCUDUECDABEUFCSZDSUGTIFTEFUHTOPVDEFZVDGKLZUI
|
|
VEVDGHZUITVDUJMEFVEVFVGVEVBVFVGNVCVDGQRUKVDUMUNULUO $.
|
|
|
|
$( Closure of exponentiation of negative one. (Contributed by Mario
|
|
Carneiro, 18-Jun-2015.) $)
|
|
m1expcl2 $p |- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } ) $=
|
|
( vx vy c1 cneg wcel cc0 cap wbr neg1cn ax-1cn cmul wceq eqeltri syl6eqel
|
|
co cc jaoi syl cdiv cpr cz cexp prid1g ax-mp neg1ap0 prssi mp2an cv wo wi
|
|
wss elpri sseli mulm1d negeq negneg1e1 prid2 eqeltrd oveq1 eleq1d syl5ibr
|
|
1ex mulid2d id imp oveq2 1ap0 divneg2ap 1div1e1 negeqi eqtr3i expcl2lemap
|
|
mp3an adantr mp3an12 ) DEZVQDUAZFZVQGHIAUBFVQAUCPVRFVQQFZVSJVQDQUDUEZUFBC
|
|
VQAVRVTDQFZVRQULJKVQDQUGUHZBUIZVRFZCUIZVRFZWDWFLPZVRFZWEWDVQMZWDDMZUJZWGW
|
|
IUKZWDVQDUMZWJWMWKWGWIWJVQWFLPZVRFWGWOWFEZVRWGWFVRQWFWCUNZUOWGWFVQMZWFDMZ
|
|
UJWPVRFZWFVQDUMWRWTWSWRWPVQEZVRWFVQUPXADVRUQVQDVCURZNOWSWPVQVRWFDUPWAORSU
|
|
SWJWHWOVRWDVQWFLUTVAVBWGWIWKDWFLPZVRFWGXCWFVRWGWFWQVDWGVEUSWKWHXCVRWDDWFL
|
|
UTVAVBRSVFXBWEDWDTPZVRFZWDGHIWEWLXEWNWJXEWKWJXDDVQTPZVRWDVQDTVGXFVQVRDDTP
|
|
ZEZXFVQWBWBDGHIXHXFMKKVHDDVIVNXGDVJVKVLWANOWKXDXGVRWDDDTVGXGDVRVJXBNORSVO
|
|
VMVP $.
|
|
|
|
$( Closure of exponentiation of negative one. (Contributed by Mario
|
|
Carneiro, 18-Jun-2015.) $)
|
|
m1expcl $p |- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ ) $=
|
|
( cz wcel c1 cneg cpr cexp co wss neg1z 1z prssi mp2an m1expcl2 sseldi )
|
|
ABCDEZDFZBPAGHPBCDBCQBIJKPDBLMANO $.
|
|
|
|
$( Closure law for integer exponentiation. Lemma for ~ expclzap and
|
|
~ expap0i . (Contributed by Jim Kingdon, 9-Jun-2020.) $)
|
|
expclzaplem $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) ->
|
|
( A ^ N ) e. { z e. CC | z # 0 } ) $=
|
|
( vx vy cc wcel cc0 cap wbr cz cexp co cv crab wi wa breq1 elrab c1 mulcl
|
|
ssrab2 cmul ad2ant2r mulap0 sylanbrc syl2anb ax-1cn 1ap0 mpbir2an recclap
|
|
cdiv recap0 jca 3imtr4i adantr expcl2lemap 3expia sylanbr anabss3 3impia
|
|
) BFGZBHIJZCKGZBCLMANZHIJZAFOZGZVBVCVDVHPZVBVCQBVGGZVCVIVFVCABFVEBHIRSVJV
|
|
CVDVHDEBCVGVFAFUBDNZVGGZVKFGZVKHIJZQZENZFGZVPHIJZQZVKVPUCMZVGGZVPVGGVFVNA
|
|
VKFVEVKHIRSZVFVRAVPFVEVPHIRSVOVSQVTFGZVTHIJZWAVMVQWCVNVRVKVPUAUDVKVPUEVFW
|
|
DAVTFVEVTHIRSUFUGTVGGTFGTHIJZUHUIVFWEATFVETHIRSUJVLTVKULMZVGGZVNVOWFFGZWF
|
|
HIJZQVLWGVOWHWIVKUKVKUMUNWBVFWIAWFFVEWFHIRSUOUPUQURUSUTVA $.
|
|
|
|
$( Closure law for integer exponentiation. (Contributed by Jim Kingdon,
|
|
9-Jun-2020.) $)
|
|
expclzap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC ) $=
|
|
( vz cc wcel cc0 cap wbr cz cexp co cv crab expclzaplem breq1 elrab sylib
|
|
w3a wa simpld ) ADEAFGHBIERZABJKZDEZUBFGHZUAUBCLZFGHZCDMEUCUDSCABNUFUDCUB
|
|
DUEUBFGOPQT $.
|
|
$}
|
|
|
|
${
|
|
nn0expcli.1 $e |- A e. NN0 $.
|
|
nn0expcli.2 $e |- N e. NN0 $.
|
|
$( Closure of exponentiation of nonnegative integers. (Contributed by
|
|
Mario Carneiro, 17-Apr-2015.) $)
|
|
nn0expcli $p |- ( A ^ N ) e. NN0 $=
|
|
( cn0 wcel cexp co nn0expcl mp2an ) AEFBEFABGHEFCDABIJ $.
|
|
$}
|
|
|
|
$( The square of a nonnegative integer is a nonnegative integer.
|
|
(Contributed by Stefan O'Rear, 16-Oct-2014.) $)
|
|
nn0sqcl $p |- ( A e. NN0 -> ( A ^ 2 ) e. NN0 ) $=
|
|
( cn0 wcel c2 cexp co 2nn0 nn0expcl mpan2 ) ABCDBCADEFBCGADHI $.
|
|
|
|
$( Exponentiation in terms of predecessor exponent. (Contributed by NM,
|
|
19-Dec-2005.) $)
|
|
expm1t $p |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) =
|
|
( ( A ^ ( N - 1 ) ) x. A ) ) $=
|
|
( cc wcel cn wa c1 cmin co caddc cexp cmul wceq ax-1cn npcan sylancl oveq2d
|
|
nncn adantl cn0 nnm1nn0 expp1 sylan2 eqtr3d ) ACDZBEDZFABGHIZGJIZKIZABKIZAU
|
|
GKIALIZUFUIUJMUEUFUHBAKUFBCDGCDUHBMBRNBGOPQSUFUEUGTDUIUKMBUAAUGUBUCUD $.
|
|
|
|
${
|
|
$d x y N $.
|
|
$( Value of one raised to a nonnegative integer power. (Contributed by NM,
|
|
15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) $)
|
|
1exp $p |- ( N e. ZZ -> ( 1 ^ N ) = 1 ) $=
|
|
( vx vy wcel c1 co wceq cc0 cap wbr 1ex cc cv wa cmul syl6eq cvv syl cdiv
|
|
elsni cz cexp csn snid 1ap0 wss ax-1cn snssi ax-mp oveq12 1t1e1 syl2an wb
|
|
eleq1 mpbiri elsng ibir oveq2d 1div1e1 adantr expcl2lemap mp3an12 ) AUADZ
|
|
EAUBFZEUCZDZVDEGEVEDEHIJVCVFEKUDZUEBCEAVEELDVELUFUGELUHUIBMZVEDZCMZVEDZNV
|
|
HVJOFZEGZVLVEDZVIVHEGZVJEGZVMVKVHETZVJETVOVPNVLEEOFEVHEVJEOUJUKPULVMVNVMV
|
|
LQDZVNVMUMVMVREQDZKVLEQUNUOVLEQUPRUQRVGVIEVHSFZVEDZVHHIJVIVTEGZWAVIVTEESF
|
|
EVIVHEESVQURUSPWBWAWBVTQDZWAWBUMWBWCVSKVTEQUNUOVTEQUPRUQRUTVAVBVDETR $.
|
|
$}
|
|
|
|
${
|
|
$d A j k $. $d N j $.
|
|
$( Positive integer exponentiation is apart from zero iff its mantissa is
|
|
apart from zero. That it is easier to prove this first, and then prove
|
|
~ expeq0 in terms of it, rather than the other way around, is perhaps an
|
|
illustration of the maxim "In constructive analysis, the apartness is
|
|
more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim
|
|
Kingdon, 10-Jun-2020.) $)
|
|
expap0 $p |- ( ( A e. CC /\ N e. NN ) ->
|
|
( ( A ^ N ) # 0 <-> A # 0 ) ) $=
|
|
( vj vk cn wcel cexp co cc0 cap wbr wb wi wceq oveq2 breq1d bibi1d imbi2d
|
|
c1 wa cc cv caddc exp1 cmul cn0 nnnn0 expp1 ancoms adantr simplr ad2antrr
|
|
sylan expcl syl2anc mulap0bd anbi1 adantl 3bitr2d anidm syl6bb a2d impcom
|
|
exp31 nnind ) BEFAUAFZABGHZIJKZAIJKZLZVFACUBZGHZIJKZVILZMVFASGHZIJKZVILZM
|
|
VFADUBZGHZIJKZVILZMVFAVRSUCHZGHZIJKZVILZMVFVJMCDBVKSNZVNVQVFWFVMVPVIWFVLV
|
|
OIJVKSAGOPQRVKVRNZVNWAVFWGVMVTVIWGVLVSIJVKVRAGOPQRVKWBNZVNWEVFWHVMWDVIWHV
|
|
LWCIJVKWBAGOPQRVKBNZVNVJVFWIVMVHVIWIVLVGIJVKBAGOPQRVFVOAIJAUDPVREFZVFWAWE
|
|
WJVFWAWEWJVFTZWATZWDVIVITZVIWLWDVSAUEHZIJKZVTVITZWMWKWDWOLZWAWJVRUFFZVFWQ
|
|
VRUGZVFWRWQVFWRTWCWNIJAVRUHPUIUMUJWLVSAWLVFWRVSUAFWJVFWAUKZWJWRVFWAWSULAV
|
|
RUNUOWTUPWAWPWMLWKVTVIVIUQURUSVIUTVAVDVBVEVC $.
|
|
$}
|
|
|
|
$( Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed
|
|
by NM, 23-Feb-2005.) $)
|
|
expeq0 $p |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel cn wa cexp co cc0 cap wbr wn wceq expap0 notbid wb cn0 nnnn0 apti
|
|
0cn expcl sylan2 sylancl mpan2 adantr 3bitr4d ) ACDZBEDZFZABGHZIJKZLZAIJKZL
|
|
ZUJIMZAIMZUIUKUMABNOUIUJCDZICDZUOULPUHUGBQDUQBRABUAUBTUJISUCUGUPUNPZUHUGURU
|
|
STAISUDUEUF $.
|
|
|
|
${
|
|
$d A z $. $d N z $.
|
|
$( Integer exponentiation is apart from zero if its mantissa is apart from
|
|
zero. (Contributed by Jim Kingdon, 10-Jun-2020.) $)
|
|
expap0i $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 ) $=
|
|
( vz cc wcel cc0 cap wbr cz cexp co cv crab expclzaplem breq1 elrab sylib
|
|
w3a wa simprd ) ADEAFGHBIERZABJKZDEZUBFGHZUAUBCLZFGHZCDMEUCUDSCABNUFUDCUB
|
|
DUEUBFGOPQT $.
|
|
$}
|
|
|
|
$( Nonnegative integer exponentiation with a positive mantissa is positive.
|
|
(Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro,
|
|
4-Jun-2014.) $)
|
|
expgt0 $p |- ( ( A e. RR /\ N e. ZZ /\ 0 < A ) -> 0 < ( A ^ N ) ) $=
|
|
( cr wcel cc0 clt wbr cz cexp co wa crp rpexpcl rpgt0d sylanbr 3impa 3com23
|
|
elrp ) ACDZEAFGZBHDZEABIJZFGZSTUAUCSTKALDZUAUCARUDUAKUBABMNOPQ $.
|
|
|
|
$( Value of a complex number raised to a negative power. (Contributed by
|
|
Mario Carneiro, 4-Jun-2014.) $)
|
|
expnegzap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) ->
|
|
( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cz cneg cexp co c1 cdiv wceq cr cn0 wo wa elznn0 wi
|
|
expnegap0 3expia adantr simpl simprl recnd simprr expineg2 oveq2d ad2ant2rl
|
|
syl12anc expcl simpll simplr nn0zd expap0i syl3anc recrecapd eqtr2d expimpd
|
|
expr jaod syl5bi 3impia ) ACDZAEFGZBHDZABIZJKZLABJKZMKZNZVGBODZBPDZVHPDZQZR
|
|
VEVFRZVLBSVQVMVPVLVQVMRVNVLVOVQVNVLTVMVEVFVNVLABUAUBUCVQVMVOVLVQVMVORZRZVKL
|
|
LVIMKZMKVIVSVJVTLMVSVQBCDVOVJVTNVQVRUDVSBVQVMVOUEUFVQVMVOUGZABUHUKUIVSVIVEV
|
|
OVICDVFVMAVHULUJVSVEVFVHHDVIEFGVEVFVRUMVEVFVRUNVSVHWAUOAVHUPUQURUSVAVBUTVCV
|
|
D $.
|
|
|
|
$( Value of zero raised to a positive integer power. (Contributed by NM,
|
|
19-Aug-2004.) $)
|
|
0exp $p |- ( N e. NN -> ( 0 ^ N ) = 0 ) $=
|
|
( cn wcel cc0 cexp co wceq eqid cc wb 0cn expeq0 mpan mpbiri ) ABCZDAEFDGZD
|
|
DGZDHDICOPQJKDALMN $.
|
|
|
|
${
|
|
$d x y z A $. $d x z N $.
|
|
$( Nonnegative integer exponentiation with a nonnegative mantissa is
|
|
nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario
|
|
Carneiro, 4-Jun-2014.) $)
|
|
expge0 $p |- ( ( A e. RR /\ N e. NN0 /\ 0 <_ A ) -> 0 <_ ( A ^ N ) ) $=
|
|
( vz vx vy cr wcel cc0 cle wbr cn0 cexp co wa cv crab breq2 elrab cc c1
|
|
ssrab2 ax-resscn sstri cmul remulcl ad2ant2r mulge0 sylanbrc syl2anb 0le1
|
|
1re mpbir2an expcllem simprbi syl sylanbr 3impa 3com23 ) AFGZHAIJZBKGZHAB
|
|
LMZIJZUSUTVAVCUSUTNAHCOZIJZCFPZGZVAVCVEUTCAFVDAHIQRVGVANVBVFGZVCDEABVFVFF
|
|
SVECFUAUBUCDOZVFGVIFGZHVIIJZNZEOZFGZHVMIJZNZVIVMUDMZVFGZVMVFGVEVKCVIFVDVI
|
|
HIQRVEVOCVMFVDVMHIQRVLVPNVQFGZHVQIJZVRVJVNVSVKVOVIVMUEUFVIVMUGVEVTCVQFVDV
|
|
QHIQRUHUITVFGTFGHTIJZUKUJVEWACTFVDTHIQRULUMVHVBFGVCVEVCCVBFVDVBHIQRUNUOUP
|
|
UQUR $.
|
|
|
|
$( Nonnegative integer exponentiation with a mantissa greater than or equal
|
|
to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.)
|
|
(Revised by Mario Carneiro, 4-Jun-2014.) $)
|
|
expge1 $p |- ( ( A e. RR /\ N e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ N ) ) $=
|
|
( vz vx vy cr wcel cn0 c1 cle wbr w3a co cv wa breq2 elrab cmul 1re jctl
|
|
cexp crab cc ssrab2 ax-resscn sstri remulcl ad2ant2r 1t1e1 wi 0le1 pm3.2i
|
|
cc0 lemul12a syl2an imp syl5eqbrr an4s sylanbrc syl2anb mpbir2an expcllem
|
|
1le1 sylanbr 3impa 3com23 simprbi syl ) AFGZBHGZIAJKZLABUAMZICNZJKZCFUBZG
|
|
ZIVLJKZVIVKVJVPVIVKVJVPVIVKOAVOGVJVPVNVKCAFVMAIJPQDEABVOVOFUCVNCFUDUEUFDN
|
|
ZVOGVRFGZIVRJKZOZENZFGZIWBJKZOZVRWBRMZVOGZWBVOGVNVTCVRFVMVRIJPQVNWDCWBFVM
|
|
WBIJPQWAWEOWFFGZIWFJKZWGVSWCWHVTWDVRWBUGUHVSWCVTWDWIVSWCOZVTWDOZOIIIRMZWF
|
|
JUIWJWKWLWFJKZVSIFGZUMIJKZOZVSOWPWCOWKWMUJWCVSWPWNWOSUKULZTWCWPWQTIVRIWBU
|
|
NUOUPUQURVNWICWFFVMWFIJPQUSUTIVOGWNIIJKZSVCVNWRCIFVMIIJPQVAVBVDVEVFVPVLFG
|
|
VQVNVQCVLFVMVLIJPQVGVH $.
|
|
$}
|
|
|
|
$( Positive integer exponentiation with a mantissa greater than 1 is greater
|
|
than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro,
|
|
4-Jun-2014.) $)
|
|
expgt1 $p |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) ) $=
|
|
( cr wcel cn c1 clt wbr w3a cexp 1re a1i simp1 cn0 reexpcl syl2anc cmul cle
|
|
co cc0 simp2 nnnn0d simp3 cmin nnm1nn0 syl wi sylancr mpd expge1 syl3anc wb
|
|
ltle 0red 0lt1 lttrd lemul1 syl112anc mpbid cc recn 3ad2ant1 mulid2d eqcomd
|
|
wceq expm1t 3brtr4d ltletrd ) ACDZBEDZFAGHZIZFAABJSZFCDZVLKLZVIVJVKMZVLVIBN
|
|
DVMCDVPVLBVIVJVKUAZUBABOPVIVJVKUCZVLFAQSZABFUDSZJSZAQSZAVMRVLFWARHZVSWBRHZV
|
|
LVIVTNDZFARHZWCVPVLVJWEVQBUEUFZVLVKWFVRVLVNVIVKWFUGKVPFAUMUHUIAVTUJUKVLVNWA
|
|
CDZVITAGHWCWDULVOVLVIWEWHVPWGAVTOPVPVLTFAVLUNVOVPTFGHVLUOLVRUPFWAAUQURUSVLV
|
|
SAVLAVIVJAUTDZVKAVAVBZVCVDVLWIVJVMWBVEWJVQABVFPVGVH $.
|
|
|
|
${
|
|
$d j k A $. $d j k B $. $d j k M $. $d j N $.
|
|
$( Positive integer exponentiation of a product. Proposition 10-4.2(c) of
|
|
[Gleason] p. 135, restricted to nonnegative integer exponents.
|
|
(Contributed by NM, 13-Feb-2005.) $)
|
|
mulexp $p |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) ->
|
|
( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) $=
|
|
( vj cc wcel cmul co cexp wceq wa wi c1 oveq2 oveq12d eqeq12d imbi2d exp0
|
|
cc0 expp1 vk cn0 cv caddc mulcl oveqan12d 1t1e1 syl6eq eqtr4d sylan oveq1
|
|
syl adantr expcl anim12i anandirs simpl syl2anc adantlr adantll sylan9eqr
|
|
mul4 eqtrd exp31 com12 a2d nn0ind expdcom 3imp ) AEFZBEFZCUBFZABGHZCIHZAC
|
|
IHZBCIHZGHZJZVLVJVKVRVJVKKZVMDUCZIHZAVTIHZBVTIHZGHZJZLVSVMSIHZASIHZBSIHZG
|
|
HZJZLVSVMUAUCZIHZAWKIHZBWKIHZGHZJZLVSVMWKMUDHZIHZAWQIHZBWQIHZGHZJZLVSVRLD
|
|
UACVTSJZWEWJVSXCWAWFWDWIVTSVMINXCWBWGWCWHGVTSAINVTSBINOPQVTWKJZWEWPVSXDWA
|
|
WLWDWOVTWKVMINXDWBWMWCWNGVTWKAINVTWKBINOPQVTWQJZWEXBVSXEWAWRWDXAVTWQVMINX
|
|
EWBWSWCWTGVTWQAINVTWQBINOPQVTCJZWEVRVSXFWAVNWDVQVTCVMINXFWBVOWCVPGVTCAINV
|
|
TCBINOPQVSWFMWIVSVMEFZWFMJABUEZVMRULVSWIMMGHMVJVKWGMWHMGARBRUFUGUHUIWKUBF
|
|
ZVSWPXBVSXIWPXBLVSXIWPXBVSXIKZWPKWRWLVMGHZXAXJWRXKJZWPVSXGXIXLXHVMWKTUJUM
|
|
WPXJXKWOVMGHZXAWLWOVMGUKXJXMWMAGHZWNBGHZGHZXAXJWMEFZWNEFZKZVSXMXPJVJVKXIX
|
|
SVJXIKXQVKXIKXRAWKUNBWKUNUOUPVSXIUQWMWNABVBURXJWSXNWTXOGVJXIWSXNJVKAWKTUS
|
|
VKXIWTXOJVJBWKTUTOUIVAVCVDVEVFVGVHVI $.
|
|
|
|
$( Integer exponentiation of a product. (Contributed by Jim Kingdon,
|
|
10-Jun-2020.) $)
|
|
mulexpzap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\
|
|
N e. ZZ ) ->
|
|
( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cz cmul cexp wceq cdiv expineg2 syl22anc syl3anc
|
|
co c1 eqtr4d cn0 cr cneg cn wo elznn0nn simpl anim12i 3expa sylan simplll
|
|
mulexp simplrl mulcld simpllr simplrr mulap0d recn ad2antrl nnnn0 oveq12d
|
|
ad2antll oveq2d 1t1e1 oveq1i syl6eqr expcl syl2anc nnz ax-1cn divmuldivap
|
|
expap0i mpanl12 jaodan sylan2b 3impa ) ADEZAFGHZIZBDEZBFGHZIZCJEZABKRZCLR
|
|
ZACLRZBCLRZKRZMZWCVSWBIZCUAEZCUBEZCUCZUDEZIZUEWICUFWJWKWIWOWJVQVTIWKWIVSV
|
|
QWBVTVQVRUGVTWAUGUHVQVTWKWIABCULUIUJWJWOIZWESWDWMLRZNRZWHWPWDDEWDFGHCDEZW
|
|
MUAEZWEWRMWPABVQVRWBWOUKZVSVTWAWOUMZUNWPABXAXBVQVRWBWOUOZVSVTWAWOUPZUQWLW
|
|
SWJWNCURUSZWNWTWJWLWMUTVBZWDCOPWPWHSAWMLRZNRZSBWMLRZNRZKRZWRWPWFXHWGXJKWP
|
|
VQVRWSWTWFXHMXAXCXEXFACOPWPVTWAWSWTWGXJMXBXDXEXFBCOPVAWPWRSSKRZXGXIKRZNRZ
|
|
XKWPWRSXMNRXNWPWQXMSNWPVQVTWTWQXMMXAXBXFABWMULQVCXLSXMNVDVEVFWPXGDEZXGFGH
|
|
ZXIDEZXIFGHZXKXNMZWPVQWTXOXAXFAWMVGVHWPVQVRWMJEZXPXAXCWNXTWJWLWMVIVBZAWMV
|
|
LQWPVTWTXQXBXFBWMVGVHWPVTWAXTXRXBXDYABWMVLQSDEZYBXOXPIXQXRIIXSVJVJSSXGXIV
|
|
KVMPTTTVNVOVP $.
|
|
|
|
$( Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim
|
|
Kingdon, 10-Jun-2020.) $)
|
|
exprecap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) ->
|
|
( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cz w3a cexp co cdiv expclzap recclap 3adant3 recap0
|
|
c1 simp3 cmul wceq syl3anc expap0i simp1 simp2 oveq1d mulexpzap syl221anc
|
|
recidapd 1exp syl 3eqtr3d mvllmulapd ) ACDZAEFGZBHDZIZABJKZQALKZBJKZQABMU
|
|
PURCDZUREFGZUOUSCDUMUNUTUOANOZUMUNVAUOAPOZUMUNUORZURBMUAABUBUPAURSKZBJKZQ
|
|
BJKZUQUSSKZQUPVEQBJUPAUMUNUOUCZUMUNUOUDZUHUEUPUMUNUTVAUOVFVHTVIVJVBVCVDAU
|
|
RBUFUGUPUOVGQTVDBUIUJUKUL $.
|
|
|
|
$( Sum of exponents law for nonnegative integer exponentiation.
|
|
Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM,
|
|
30-Nov-2004.) $)
|
|
expadd $p |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) ->
|
|
( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) $=
|
|
( vj cc wcel cn0 caddc co cexp cmul wi cc0 c1 oveq2 oveq2d eqeq12d imbi2d
|
|
wceq eqtr4d vk wa cv nn0cn addid1d adantl expcl mulid1d exp0 adantr oveq1
|
|
ax-1cn addass mp3an3 syl2an adantll simpll nn0addcl expp1 syl2anc adantlr
|
|
eqtr3d mulassd syl5ibr expcom a2d nn0ind expdcom 3imp ) AEFZBGFZCGFZABCHI
|
|
ZJIZABJIZACJIZKIZSZVLVJVKVRVJVKUBZABDUCZHIZJIZVOAVTJIZKIZSZLVSABMHIZJIZVO
|
|
AMJIZKIZSZLVSABUAUCZHIZJIZVOAWKJIZKIZSZLVSABWKNHIZHIZJIZVOAWQJIZKIZSZLVSV
|
|
RLDUACVTMSZWEWJVSXCWBWGWDWIXCWAWFAJVTMBHOPXCWCWHVOKVTMAJOPQRVTWKSZWEWPVSX
|
|
DWBWMWDWOXDWAWLAJVTWKBHOPXDWCWNVOKVTWKAJOPQRVTWQSZWEXBVSXEWBWSWDXAXEWAWRA
|
|
JVTWQBHOPXEWCWTVOKVTWQAJOPQRVTCSZWEVRVSXFWBVNWDVQXFWAVMAJVTCBHOPXFWCVPVOK
|
|
VTCAJOPQRVSWGVONKIZWIVSWGVOXGVSWFBAJVKWFBSVJVKBBUDZUEUFPVSVOABUGZUHTVSWHN
|
|
VOKVJWHNSVKAUIUJPTWKGFZVSWPXBVSXJWPXBLWPXBVSXJUBZWMAKIZWOAKIZSWMWOAKUKXKW
|
|
SXLXAXMXKAWLNHIZJIZWSXLXKXNWRAJVKXJXNWRSZVJVKBEFZWKEFZXPXJXHWKUDXQXRNEFXP
|
|
ULBWKNUMUNUOUPPXKVJWLGFZXOXLSVJVKXJUQZVKXJXSVJBWKURUPAWLUSUTVBXKXAVOWNAKI
|
|
ZKIXMXKWTYAVOKVJXJWTYASVKAWKUSVAPXKVOWNAVSVOEFXJXIUJVJXJWNEFVKAWKUGVAXTVC
|
|
TQVDVEVFVGVHVI $.
|
|
|
|
$( Lemma for ~ expaddzap . (Contributed by Jim Kingdon, 10-Jun-2020.) $)
|
|
expaddzaplem $p |- ( ( ( A e. CC /\ A # 0 ) /\
|
|
( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) ->
|
|
( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) $=
|
|
( cc wcel wa cn0 cexp co cdiv c1 cmul caddc cz syl3anc wceq adantr eqtr3d
|
|
eqtrd oveq1d cc0 cap wbr cr cneg cn w3a simp1l simp3 expcl syl2anc simp2r
|
|
nnnn0d simp1r expap0i divrecap2d simp2l recnd negnegd nnnegz syl eqeltrrd
|
|
nnzd nn0zd zaddcld expclzap divcanap4d expadd cmin negsubd nn0cnd pncan2d
|
|
simpr oveq2d expineg2 syl22anc znegcld negdi2d negcld npcand recdivapd wo
|
|
zcnd elznn0 simprbi mpjaodan 3eqtr4d ) ADEZAUAUBUCZFZBUDEZBUEZUFEZFZCGEZU
|
|
GZACHIZAWLHIZJIZKWRJIZWQLIABCMIZHIZABHIZWQLIWPWQWRWPWHWOWQDEZWHWIWNWOUHZW
|
|
JWNWOUIZACUJUKZWPWHWLGEZWRDEZXEWPWLWJWKWMWOULZUMZAWLUJUKZWPWHWIWLNEWRUAUB
|
|
UCZXEWHWIWNWOUNZWPWLXJVCAWLUOOZUPWPXAGEZXBWSPXAUEZGEZWPXPFZXBWRLIZWRJIXBW
|
|
SXSXBWRWPXBDEZXPWPWHWIXANEZYAXEXNWPBCWPWLUEZBNWPBWPBWJWKWMWOUQURZUSWPWMYC
|
|
NEXJWLUTVAVBWPCXFVDZVEZAXAVFOQWPXIXPXLQWPXMXPXOQVGXSXTWQWRJXSAXAWLMIZHIZX
|
|
TWQXSWHXPXHYHXTPWPWHXPXEQWPXPVMWPXHXPXKQAXAWLVHOXSYGCAHWPYGCPXPWPYGXABVII
|
|
CWPXABWPXAYFWCZYDVJWPBCYDWPCXFVKZVLSQVNRTRWPXRFZXBKAXQHIZJIZWSYKWHWIXADEZ
|
|
XRXBYMPWPWHXRXEQZWPWIXRXNQWPYNXRYIQWPXRVMZAXAVOVPYKYMKWRWQJIZJIZWSYKYLYQK
|
|
JYKYLWQLIZWQJIYLYQYKYLWQWPYLDEZXRWPWHWIXQNEYTXEXNWPXAYFVQAXQVFOQWPXDXRXGQ
|
|
WPWQUAUBUCZXRWPWHWICNEUUAXEXNYEACUOOZQVGYKYSWRWQJYKAXQCMIZHIZYSWRYKWHXRWO
|
|
UUDYSPYOYPWPWOXRXFQAXQCVHOYKUUCWLAHWPUUCWLPXRWPUUCWLCVIIZCMIWLWPXQUUECMWP
|
|
BCYDYJVRTWPWLCWPBYDVSYJVTSQVNRTRVNWPYRWSPXRWPWRWQXLXGXOUUBWAQSSWPYBXPXRWB
|
|
ZYFYBXAUDEUUFXAWDWEVAWFWPXCWTWQLWPWHWIBDEXHXCWTPXEXNYDXKABVOVPTWG $.
|
|
|
|
$( Sum of exponents law for integer exponentiation. (Contributed by Jim
|
|
Kingdon, 10-Jun-2020.) $)
|
|
expaddzap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) ->
|
|
( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wa cz co cexp cmul wceq cn0 cneg 3expia syl3anc c1 cdiv
|
|
syl22anc wbr caddc cr cn wo elznn0nn wi expadd adantlr expaddzaplem simp3
|
|
jaodan w3a nn0cnd simp2l recnd addcomd oveq2d simp1l expcl syl2anc simp1r
|
|
negnegd simp2r nnnn0d nn0negz syl eqeltrrd expclzap mulcomd simp3l negdid
|
|
3eqtr4d impancom simp3r eqtrd 1t1e1 oveq1i syl6eqr expap0i ax-1cn mpanl12
|
|
divmuldivap eqtr4d addcld nn0addcld eqeltrd expineg2 oveq12d jaod sylan2b
|
|
nn0zd syl5bi impr ) ADEZAFGUAZHZBIEZCIEZABCUBJZKJZABKJZACKJZLJZMZWSCNEZCU
|
|
CEZCOZUDEZHZUEZWQWRHXECUFWRWQBNEZBUCEZBOZUDEZHZUEZXKXEUGBUFWQXQHXFXEXJWQX
|
|
LXFXEUGZXPWOXLXRWPWOXLXFXEABCUHPUIWQXPXFXEABCUJPULWQXLXJXEUGXPWQXJXLXEWQX
|
|
JXLXEWQXJXLUMZACBUBJZKJXCXBLJXAXDACBUJXSWTXTAKXSBCXSBWQXJXLUKZUNXSCWQXGXI
|
|
XLUOUPZUQURXSXBXCXSWOXLXBDEWOWPXJXLUSZYAABUTVAXSWOWPWSXCDEYCWOWPXJXLVBXSX
|
|
HOZCIXSCYBVCXSXHNEZYDIEXSXHWQXGXIXLVDVEXHVFVGVHACVIQVJVMPVNWQXPXJXEWQXPXJ
|
|
UMZRAWTOZKJZSJZRAXNKJZSJZRAXHKJZSJZLJZXAXDYFYIRRLJZYJYLLJZSJZYNYFYIRYPSJY
|
|
QYFYHYPRSYFYHAXNXHUBJZKJZYPYFYGYRAKYFBCYFBWQXMXOXJUOUPZYFCWQXPXGXIVKUPZVL
|
|
ZURYFWOXNNEZYEYSYPMWOWPXPXJUSZYFXNWQXMXOXJVDVEZYFXHWQXPXGXIVOVEZAXNXHUHQV
|
|
PURYORYPSVQVRVSYFYJDEZYJFGUAZYLDEZYLFGUAZYNYQMZYFWOUUCUUGUUDUUEAXNUTVAYFW
|
|
OWPXNIEUUHUUDWOWPXPXJVBZYFXNUUEWLAXNVTQYFWOYEUUIUUDUUFAXHUTVAYFWOWPXHIEUU
|
|
JUUDUULYFXHUUFWLAXHVTQRDEZUUMUUGUUHHUUIUUJHHUUKWAWARRYJYLWCWBTWDYFWOWPWTD
|
|
EYGNEXAYIMUUDUULYFBCYTUUAWEYFYGYRNUUBYFXNXHUUEUUFWFWGAWTWHTYFXBYKXCYMLYFW
|
|
OWPBDEUUCXBYKMUUDUULYTUUEABWHTYFWOWPCDEYEXCYMMUUDUULUUAUUFACWHTWIVMPULWJW
|
|
KWMWN $.
|
|
|
|
$( Product of exponents law for positive integer exponentiation.
|
|
Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative
|
|
integer exponents. (Contributed by NM, 4-Jan-2006.) $)
|
|
expmul $p |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) ->
|
|
( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) $=
|
|
( vj cc wcel cn0 cmul co cexp wceq wa wi cc0 c1 caddc oveq2 oveq2d imbi2d
|
|
eqeq12d vk cv nn0cn mul01d exp0 sylan9eqr expcl eqtr4d oveq1 ax-1cn adddi
|
|
mp3an3 mulid1 adantr syl2an adantll simpll nn0mulcl simplr expadd syl3anc
|
|
syl eqtrd expp1 sylan syl5ibr expcom a2d nn0ind expdcom 3imp ) AEFZBGFZCG
|
|
FZABCHIZJIZABJIZCJIZKZVNVLVMVSVLVMLZABDUBZHIZJIZVQWAJIZKZMVTABNHIZJIZVQNJ
|
|
IZKZMVTABUAUBZHIZJIZVQWJJIZKZMVTABWJOPIZHIZJIZVQWOJIZKZMVTVSMDUACWANKZWEW
|
|
IVTWTWCWGWDWHWTWBWFAJWANBHQRWANVQJQTSWAWJKZWEWNVTXAWCWLWDWMXAWBWKAJWAWJBH
|
|
QRWAWJVQJQTSWAWOKZWEWSVTXBWCWQWDWRXBWBWPAJWAWOBHQRWAWOVQJQTSWACKZWEVSVTXC
|
|
WCVPWDVRXCWBVOAJWACBHQRWACVQJQTSVTWGOWHVMVLWGANJIOVMWFNAJVMBBUCZUDRAUEUFV
|
|
TVQEFZWHOKABUGZVQUEVBUHWJGFZVTWNWSVTXGWNWSMWNWSVTXGLZWLVQHIZWMVQHIZKWLWMV
|
|
QHUIXHWQXIWRXJXHWQAWKBPIZJIZXIXHWPXKAJVMXGWPXKKZVLVMBEFZWJEFZXMXGXDWJUCXN
|
|
XOLZWPWKBOHIZPIZXKXNXOOEFWPXRKUJBWJOUKULXPXQBWKPXNXQBKXOBUMUNRVCUOUPRXHVL
|
|
WKGFZVMXLXIKVLVMXGUQVMXGXSVLBWJURUPVLVMXGUSAWKBUTVAVCVTXEXGWRXJKXFVQWJVDV
|
|
ETVFVGVHVIVJVK $.
|
|
|
|
$( Product of exponents law for integer exponentiation. (Contributed by
|
|
Jim Kingdon, 11-Jun-2020.) $)
|
|
expmulzap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) ->
|
|
( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) $=
|
|
( cc wcel cc0 cap wbr wa cz cmul co cexp wceq cn0 c1 cdiv oveq2d syl22anc
|
|
syl3anc cr cn wo elznn0nn wi expmul 3expia adantlr w3a simp2l recnd simp3
|
|
cneg nn0cnd simp1l simp2r nnnn0d eqtr3d expcl syl2anc simp1r nnzd expap0i
|
|
mulneg1d nn0zd exprecap eqtr4d mulcld nn0mulcld eqeltrrd expineg2 3eqtr4d
|
|
oveq1d jaodan simp3l mulneg2d recclapd recap0d recrecapd mul2negd 3eqtrrd
|
|
simp2 simp3r 3eqtrd jaod sylan2b syl5bi impr ) ADEZAFGHZIZBJEZCJEZABCKLZM
|
|
LZABMLZCMLZNZWMCOEZCUAEZCUMZUBEZIZUCZWKWLIWRCUDWLWKBOEZBUAEZBUMZUBEZIZUCZ
|
|
XDWRUEBUDWKXJIWSWRXCWKXEWSWRUEZXIWIXEXKWJWIXEWSWRABCUFUGUHWKXIWSWRWKXIWSU
|
|
IZPAWNUMZMLZQLZPAXGMLZQLZCMLZWOWQXLXOPXPCMLZQLZXRXLXNXSPQXLAXGCKLZMLZXNXS
|
|
XLYAXMAMXLBCXLBWKXFXHWSUJUKZXLCWKXIWSULZUNZVDZRXLWIXGOEZWSYBXSNWIWJXIWSUO
|
|
ZXLXGWKXFXHWSUPZUQZYDAXGCUFTURRXLXPDEZXPFGHZWMXRXTNXLWIYGYKYHYJAXGUSZUTXL
|
|
WIWJXGJEZYLYHWIWJXIWSVAZXLXGYIVBAXGVCZTXLCYDVEXPCVFTVGXLWIWJWNDEZXMOEZWOX
|
|
ONZYHYOXLBCYCYEVHXLYAXMOYFXLXGCYJYDVIVJAWNVKZSXLWPXQCMXLWIWJBDEZYGWPXQNZY
|
|
HYOYCYJABVKZSVMVLUGVNWKXEXCWRUEXIWKXEXCWRWKXEXCUIZXOPWPXAMLZQLZWOWQUUDXNU
|
|
UEPQUUDABXAKLZMLZXNUUEUUDUUGXMAMUUDBCUUDBWKXEXCWBZUNZUUDCWKXEWTXBVOUKZVPZ
|
|
RUUDWIXEXAOEZUUHUUENWIWJXEXCUOZUUIUUDXAWKXEWTXBWCUQZABXAUFTURRUUDWIWJYQYR
|
|
YSUUNWIWJXEXCVAZUUDBCUUJUUKVHUUDUUGXMOUULUUDBXAUUIUUOVIVJYTSUUDWPDEZWPFGH
|
|
ZCDEZUUMWQUUFNUUDWIXEUUQUUNUUIABUSUTUUDWIWJWLUURUUNUUPUUDBUUIVEABVCTUUKUU
|
|
OWPCVKSVLUGWKXIXCWRWKXIXCUIZWQXRPXQXAMLZQLZWOUUTWPXQCMUUTWIWJUUAYGUUBWIWJ
|
|
XIXCUOZWIWJXIXCVAZUUTBWKXFXHXCUJUKZUUTXGWKXFXHXCUPZUQZUUCSVMUUTXQDEXQFGHU
|
|
USUUMXRUVBNUUTXPUUTWIYGYKUVCUVGYMUTZUUTWIWJYNYLUVCUVDUUTXGUVFVBYPTZVQUUTX
|
|
PUVHUVIVRUUTCWKXIWTXBVOUKZUUTXAWKXIWTXBWCZUQZXQCVKSUUTUVBPPXPXAMLZQLZQLUV
|
|
MWOUUTUVAUVNPQUUTYKYLXAJEZUVAUVNNUVHUVIUUTXAUVKVBZXPXAVFTRUUTUVMUUTYKUUMU
|
|
VMDEUVHUVLXPXAUSUTUUTYKYLUVOUVMFGHUVHUVIUVPXPXAVCTVSUUTAXGXAKLZMLZUVMWOUU
|
|
TWIYGUUMUVRUVMNUVCUVGUVLAXGXAUFTUUTUVQWNAMUUTBCUVEUVJVTRURWDWAUGVNWEWFWGW
|
|
H $.
|
|
$}
|
|
|
|
$( Exponentiation of negative one to an even power. (Contributed by Scott
|
|
Fenton, 17-Jan-2018.) $)
|
|
m1expeven $p |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 ) $=
|
|
( cz wcel c1 cneg c2 cmul cexp caddc zcn 2timesd oveq2d wceq neg1ap0 anidms
|
|
co cc cr oveq12 syl6eq cc0 cap wbr wa neg1cn expaddzap mpanl12 cpr m1expcl2
|
|
wo wb neg1rr reexpclzap mp3an12 elprg syl neg1mulneg1e1 1t1e1 syl6bi 3eqtrd
|
|
jaoi mpd ) ABCZDEZFAGPZHPVDAAIPZHPZVDAHPZVHGPZDVCVEVFVDHVCAAJKLVCVGVIMZVDQC
|
|
VDUAUBUCZVCVCUDVJUENVDAAUFUGOVCVHVDDUHCZVIDMZAUIVCVLVHVDMZVHDMZUJZVMVCVHRCZ
|
|
VLVPUKVDRCVKVCVQULNVDAUMUNVHVDDRUOUPVNVMVOVNVIVDVDGPZDVNVIVRMVHVDVHVDGSOUQT
|
|
VOVIDDGPZDVOVIVSMVHDVHDGSOURTVAUSVBUT $.
|
|
|
|
$( Exponent subtraction law for nonnegative integer exponentiation.
|
|
(Contributed by Jim Kingdon, 11-Jun-2020.) $)
|
|
expsubap $p |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) ->
|
|
( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cz co cexp cmul cdiv wceq zcn oveq2d 3expa adantrl
|
|
expclzap cneg caddc znegcl expaddzap sylanr2 negsub syl2an adantl expnegzap
|
|
cmin c1 adantrr expap0i divrecapd eqtr4d 3eqtr3d ) ADEZAFGHZIZBJEZCJEZIZIZA
|
|
BCUAZUBKZLKZABLKZAVDLKZMKZABCUJKZLKVGACLKZNKZVAUSUTVDJEVFVIOCUCABVDUDUEVCVE
|
|
VJALVBVEVJOZUSUTBDECDEVMVABPCPBCUFUGUHQVCVIVGUKVKNKZMKVLVCVHVNVGMUSVAVHVNOZ
|
|
UTUQURVAVOACUIRSQVCVGVKUSUTVGDEZVAUQURUTVPABTRULUSVAVKDEZUTUQURVAVQACTRSUSV
|
|
AVKFGHZUTUQURVAVRACUMRSUNUOUP $.
|
|
|
|
$( Value of a nonzero complex number raised to an integer power plus one.
|
|
(Contributed by Jim Kingdon, 11-Jun-2020.) $)
|
|
expp1zap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ ( N + 1 ) ) =
|
|
( ( A ^ N ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr cz w3a c1 caddc co cexp cmul wceq wa expaddzap mpanr2
|
|
1z 3impa exp1 3ad2ant1 oveq2d eqtrd ) ACDZAEFGZBHDZIZABJKLMLZABMLZAJMLZNLZU
|
|
JANLUEUFUGUIULOZUEUFPUGJHDUMSABJQRTUHUKAUJNUEUFUKAOUGAUAUBUCUD $.
|
|
|
|
$( Value of a complex number raised to an integer power minus one.
|
|
(Contributed by Jim Kingdon, 11-Jun-2020.) $)
|
|
expm1ap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) =
|
|
( ( A ^ N ) / A ) ) $=
|
|
( cc wcel cc0 cap wbr cz w3a c1 cmin co cexp cdiv wceq wa 1z expsubap 3impa
|
|
mpanr2 exp1 3ad2ant1 oveq2d eqtrd ) ACDZAEFGZBHDZIZABJKLMLZABMLZAJMLZNLZUJA
|
|
NLUEUFUGUIULOZUEUFPUGJHDUMQABJRTSUHUKAUJNUEUFUKAOUGAUAUBUCUD $.
|
|
|
|
$( Nonnegative integer exponentiation of a quotient. (Contributed by Jim
|
|
Kingdon, 11-Jun-2020.) $)
|
|
expdivap $p |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) ->
|
|
( ( A / B ) ^ N ) = ( ( A ^ N ) / ( B ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa cn0 w3a cdiv co cexp c1 cmul wceq divrecap syl3anc
|
|
expcl 3expb 3adant3 oveq1d recclap mulexp syl3an2 cz simp2l simp2r 3ad2ant3
|
|
exprecap oveq2d 3adant2 adantlr 3adant1 expap0i divrecapd eqtr4d 3eqtrd
|
|
nn0z ) ADEZBDEZBFGHZIZCJEZKZABLMZCNMAOBLMZPMZCNMZACNMZVHCNMZPMZVKBCNMZLMZVF
|
|
VGVICNVAVDVGVIQZVEVAVBVCVPABRUAUBUCVDVAVHDEVEVJVMQBUDAVHCUEUFVFVMVKOVNLMZPM
|
|
VOVFVLVQVKPVFVBVCCUGEZVLVQQVAVBVCVEUHZVAVBVCVEUIZVEVAVRVDCUTUJZBCUKSULVFVKV
|
|
NVAVEVKDEVDACTUMVDVEVNDEZVAVBVEWBVCBCTUNUOVFVBVCVRVNFGHVSVTWABCUPSUQURUS $.
|
|
|
|
$( Ordering relationship for exponentiation. (Contributed by NM,
|
|
2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) $)
|
|
ltexp2a $p |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\
|
|
( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) ) $=
|
|
( cr wcel cz w3a c1 clt wbr wa cexp co cmul crp rpexpcl syl2anc rpred recnd
|
|
cc0 simpl1 0red 1red 0lt1 a1i simprl lttrd elrpd simpl2 mulid2d cdiv simprr
|
|
cmin cn wb simpl3 znnsub mpbid expgt1 syl3anc cc cap wceq expsubap syl22anc
|
|
gt0ap0d breqtrd ltmuldivd mpbird eqbrtrrd ) ADEZBFEZCFEZGZHAIJZBCIJZKZKZHAB
|
|
LMZNMZVSACLMZIVRVSVRVSVRVSVRAOEZVLVSOEVRAVKVLVMVQUAZVRTHAVRUBVRUCZWCTHIJVRU
|
|
DUEVNVOVPUFZUGZUHZVKVLVMVQUIZABPQZRSUJVRVTWAIJHWAVSUKMZIJVRHACBUMMZLMZWJIVR
|
|
VKWKUNEZVOHWLIJWCVRVPWMVNVOVPULVRVLVMVPWMUOWHVKVLVMVQUPZBCUQQURWEAWKUSUTVRA
|
|
VAEATVBJVMVLWLWJVCVRAWCSVRAWCWFVFWNWHACBVDVEVGVRHWAVSWDVRWAVRWBVMWAOEWGWNAC
|
|
PQRWIVHVIVJ $.
|
|
|
|
$( Weak ordering relationship for exponentiation. (Contributed by NM,
|
|
14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.) $)
|
|
leexp2a $p |- ( ( A e. RR /\ 1 <_ A /\ N e. ( ZZ>= ` M ) ) ->
|
|
( A ^ M ) <_ ( A ^ N ) ) $=
|
|
( cr wcel c1 cle wbr cuz cfv cexp co crp cc0 3ad2ant3 rpexpcl syl2anc rpred
|
|
cz recnd w3a cmul simp1 0red 1red clt 0lt1 simp2 ltletrd elrpd eluzel2 cdiv
|
|
a1i mulid2d cmin cn0 uznn0sub expge1 syl3anc wceq gt0ap0d expsubap syl22anc
|
|
cc cap eluzelz breqtrd lemuldivd mpbird eqbrtrrd ) ADEZFAGHZCBIJEZUAZFABKLZ
|
|
UBLZVOACKLZGVNVOVNVOVNVOVNAMEZBSEZVOMEVNAVKVLVMUCZVNNFAVNUDVNUEZVTNFUFHVNUG
|
|
UMVKVLVMUHZUIZUJZVMVKVSVLBCUKOZABPQZRTUNVNVPVQGHFVQVOULLZGHVNFACBUOLZKLZWGG
|
|
VNVKWHUPEZVLFWIGHVTVMVKWJVLBCUQOWBAWHURUSVNAVDEANVEHCSEZVSWIWGUTVNAVTTVNAVT
|
|
WCVAVMVKWKVLBCVFOZWEACBVBVCVGVNFVQVOWAVNVQVNVRWKVQMEWDWLACPQRWFVHVIVJ $.
|
|
|
|
${
|
|
$d j k A $. $d j k B $. $d j k M $. $d j k N $.
|
|
$( Weak ordering relationship for exponentiation. (Contributed by Paul
|
|
Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) $)
|
|
leexp2r $p |- ( ( ( A e. RR /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) /\
|
|
( 0 <_ A /\ A <_ 1 ) ) ->
|
|
( A ^ N ) <_ ( A ^ M ) ) $=
|
|
( vj cr wcel cn0 cle wbr c1 wa cexp co wceq breq1d imbi2d reexpcl syl2anc
|
|
wi oveq2 vk cuz cfv w3a cc0 cv caddc cz adantr leidd cmul simprll simprlr
|
|
1red simpl eluznn0 simprrl expge0 syl3anc simprrr lemul2ad cc recnd expp1
|
|
a1i mulid1d eqcomd 3brtr4d peano2nn0 syl ad2antrl letr ex a2d uzind4 expd
|
|
mpand com12 3impia imp ) AEFZBGFZCBUBUCZFZUDUEAHIZAJHIZKZACLMZABLMZHIZWAW
|
|
BWDWGWJSZWDWAWBKZWKWDWLWGWJWLWGKZADUFZLMZWIHIZSWMWIWIHIZSZWMAUAUFZLMZWIHI
|
|
ZSWMAWSJUGMZLMZWIHIZSWMWJSDUABCWNBNZWPWQWMXEWOWIWIHWNBALTOPWNWSNZWPXAWMXF
|
|
WOWTWIHWNWSALTOPWNXBNZWPXDWMXGWOXCWIHWNXBALTOPWNCNZWPWJWMXHWOWHWIHWNCALTO
|
|
PWRBUHFWMWIWLWIEFZWGABQZUIUJVEWSWCFZWMXAXDXKWMXAXDSXKWMKZXCWTHIZXAXDXLWTA
|
|
UKMZWTJUKMZXCWTHXLAJWTXKWAWBWGULZXLUNXLWAWSGFZWTEFZXPXLWBXKXQXKWAWBWGUMXK
|
|
WMUOWSBUPRZAWSQRZXLWAXQWEUEWTHIXPXSXKWLWEWFUQAWSURUSXKWLWEWFUTVAXLAVBFXQX
|
|
CXNNXLAXPVCXSAWSVDRXLXOWTXLWTXLWTXTVCVFVGVHXLXCEFZXRXIXMXAKXDSXLWAXBGFZYA
|
|
XPXLXQYBXSWSVIVJAXBQRXTWLXIXKWGXJVKXCWTWIVLUSVQVMVNVOVPVRVSVT $.
|
|
|
|
$( Weak mantissa ordering relationship for exponentiation. (Contributed by
|
|
NM, 18-Dec-2005.) $)
|
|
leexp1a $p |- ( ( ( A e. RR /\ B e. RR /\ N e. NN0 ) /\
|
|
( 0 <_ A /\ A <_ B ) ) -> ( A ^ N ) <_ ( B ^ N ) ) $=
|
|
( cr wcel cc0 cle wa cexp co wi c1 wceq oveq2 breq12d imbi2d adantr sylan
|
|
wbr adantlr vj vk cn0 cv caddc cc recn exp0 1le1 syl6eqbr adantl breqtrrd
|
|
syl2an simpll reexpcl simplll simpr simplrl expge0 syl3anc simplr anim12i
|
|
cmul jca31 simpl simpllr jca32 simplrr jca lemul12a expp1 adantll 3brtr4d
|
|
sylc ex expcom a2d nn0ind exp4c com3l 3imp1 ) ADEZBDEZCUCEZFAGSZABGSZHZAC
|
|
IJZBCIJZGSZWDWBWCWGWJKWDWBWCWGWJWBWCHZWGHZAUAUDZIJZBWMIJZGSZKWLAFIJZBFIJZ
|
|
GSZKWLAUBUDZIJZBWTIJZGSZKWLAWTLUEJZIJZBXDIJZGSZKWLWJKUAUBCWMFMZWPWSWLXHWN
|
|
WQWOWRGWMFAINWMFBINOPWMWTMZWPXCWLXIWNXAWOXBGWMWTAINWMWTBINOPWMXDMZWPXGWLX
|
|
JWNXEWOXFGWMXDAINWMXDBINOPWMCMZWPWJWLXKWNWHWOWIGWMCAINWMCBINOPWKWSWGWBAUF
|
|
EZBUFEZWSWCAUGZBUGZXLXMHZWQLWRGXPWQLLGXLWQLMXMAUHQUIUJXMWRLMXLBUHUKULUMQW
|
|
TUCEZWLXCXGWLXQXCXGKWLXQHZXCXGXRXCHZXAAVCJZXBBVCJZXEXFGXSXADEZFXAGSZHXBDE
|
|
ZHZWBWEHZWCHHZXCWFHXTYAGSXRYGXCXRYEYFWCXRYBYCYDWLWBXQYBWBWCWGUNAWTUORXRWB
|
|
XQWEYCWBWCWGXQUPWLXQUQWKWEWFXQURAWTUSUTWLWCXQYDWBWCWGVABWTUORVDWLYFXQWKWB
|
|
WGWEWBWCVEWEWFVEVBQWBWCWGXQVFVGQXSXCWFXRXCUQXRWFXCWKWEWFXQVHQVIXAXBABVJVN
|
|
XRXEXTMZXCWKXQYHWGWBXQYHWCWBXLXQYHXNAWTVKRTTQXRXFYAMZXCWKXQYIWGWCXQYIWBWC
|
|
XMXQYIXOBWTVKRVLTQVMVOVPVQVRVSVTWA $.
|
|
$}
|
|
|
|
$( Nonnegative integer exponentiation with a mantissa between 0 and 1
|
|
inclusive is less than or equal to 1. (Contributed by Paul Chapman,
|
|
29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.) $)
|
|
exple1 $p |- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) ->
|
|
( A ^ N ) <_ 1 ) $=
|
|
( cr wcel cc0 cle wbr c1 w3a cn0 wa cexp co cuz cfv simpl1 0nn0 simpr nn0uz
|
|
a1i syl6eleq simpl2 simpl3 leexp2r syl32anc cc wceq recnd exp0 syl breqtrd
|
|
) ACDZEAFGZAHFGZIZBJDZKZABLMZAELMZHFUQULEJDZBENOZDUMUNURUSFGULUMUNUPPZUTUQQ
|
|
TUQBJVAUOUPRSUAULUMUNUPUBULUMUNUPUCAEBUDUEUQAUFDUSHUGUQAVBUHAUIUJUK $.
|
|
|
|
$( An upper bound on ` A ^ N ` when ` 2 <_ A ` . (Contributed by NM,
|
|
19-Dec-2005.) $)
|
|
expubnd $p |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) ->
|
|
( A ^ N ) <_ ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) ) $=
|
|
( cr wcel c2 cle wbr cexp co c1 cmin cmul cc0 wa 2re caddc mp3an1 wceq 2cn
|
|
cc cn0 w3a simp1 peano2rem remulcl sylancr 3ad2ant1 simp2 0le2 letr mp3an12
|
|
wi 0re mpani wb resubcl mpan2 leadd2 mpdan biimpa recn npcan sylancl adantr
|
|
imp ax-1cn subdi mp3an13 2times 2t1e2 a1i oveq12d addsub mp3an3 3eqtrrd syl
|
|
anidms 3brtr3d 3adant2 leexp1a syl31anc recnd mulexp sylan 3adant3 breqtrd
|
|
jca ) ACDZBUADZEAFGZUBZABHIZEAJKIZLIZBHIZEBHIWMBHILIZFWKWHWNCDZWIMAFGZAWNFG
|
|
ZNZWLWOFGWHWIWJUCWHWIWQWJWHECDZWMCDWQOAUDZEWMUEUFUGWHWIWJUHWHWJWTWIWHWJNZWR
|
|
WSWHWJWRWHMEFGZWJWRUIMCDXAWHXDWJNWRULUMOMEAUJUKUNVEXCAEKIZEPIZXEAPIZAWNFWHW
|
|
JXFXGFGZWHXECDZWJXHUOZWHXAXIOAEUPUQXAWHXIXJOEAXEURQUSUTWHXFARZWJWHATDZETDZX
|
|
KAVAZSAEVBVCVDWHXGWNRZWJWHXLXOXNXLWNEALIZEJLIZKIZAAPIZEKIZXGXMXLJTDWNXRRSVF
|
|
EAJVGVHXLXPXSXQEKAVIXQERXLVJVKVLXLXTXGRZXLXLXMYASAAEVMVNVQVOVPVDVRWGVSAWNBV
|
|
TWAWHWIWOWPRZWJWHWMTDZWIYBWHWMXBWBXMYCWIYBSEWMBWCQWDWEWF $.
|
|
|
|
$( Value of the square of a complex number. (Contributed by Raph Levien,
|
|
10-Apr-2004.) $)
|
|
sqval $p |- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) ) $=
|
|
( cc wcel c2 cexp co c1 cmul caddc df-2 oveq2i wceq 1nn0 expp1 mpan2 syl5eq
|
|
cn0 exp1 oveq1d eqtrd ) ABCZADEFZAGEFZAHFZAAHFUAUBAGGIFZEFZUDDUEAEJKUAGQCUF
|
|
UDLMAGNOPUAUCAAHARST $.
|
|
|
|
$( The square of the negative of a number.) (Contributed by NM,
|
|
15-Jan-2006.) $)
|
|
sqneg $p |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) ) $=
|
|
( cc wcel cneg cmul co c2 cexp wceq mul2neg anidms negcl sqval syl 3eqtr4d
|
|
) ABCZADZQEFZAAEFZQGHFZAGHFPRSIAAJKPQBCTRIALQMNAMO $.
|
|
|
|
$( Swap the order of subtraction in a square. (Contributed by Scott Fenton,
|
|
10-Jun-2013.) $)
|
|
sqsubswap $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( A - B ) ^ 2 ) = ( ( B - A ) ^ 2 ) ) $=
|
|
( cc wcel wa cmin co cneg cexp wceq subcl sqneg syl negsubdi2 oveq1d eqtr3d
|
|
c2 ) ACDBCDEZABFGZHZQIGZSQIGZBAFGZQIGRSCDUAUBJABKSLMRTUCQIABNOP $.
|
|
|
|
$( Closure of square. (Contributed by NM, 10-Aug-1999.) $)
|
|
sqcl $p |- ( A e. CC -> ( A ^ 2 ) e. CC ) $=
|
|
( cc wcel c2 cexp co cmul sqval mulcl anidms eqeltrd ) ABCZADEFAAGFZBAHLMBC
|
|
AAIJK $.
|
|
|
|
$( Distribution of square over multiplication. (Contributed by NM,
|
|
21-Mar-2008.) $)
|
|
sqmul $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) $=
|
|
( cc wcel c2 cn0 cmul co cexp wceq 2nn0 mulexp mp3an3 ) ACDBCDEFDABGHEIHAEI
|
|
HBEIHGHJKABELM $.
|
|
|
|
$( A number is zero iff its square is zero. (Contributed by NM,
|
|
11-Mar-2006.) $)
|
|
sqeq0 $p |- ( A e. CC -> ( ( A ^ 2 ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel c2 cn cexp co cc0 wceq wb 2nn expeq0 mpan2 ) ABCDECADFGHIAHIJKADL
|
|
M $.
|
|
|
|
$( Distribution of square over division. (Contributed by Jim Kingdon,
|
|
11-Jun-2020.) $)
|
|
sqdivap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cmul c2 cexp wa wceq simp1 3simpc divmuldivap
|
|
w3a syl22anc sqval divclap syl oveqan12d 3adant3 3eqtr4d ) ACDZBCDZBEFGZRZA
|
|
BHIZUJJIZAAJIZBBJIZHIZUJKLIZAKLIZBKLIZHIZUIUFUFUGUHMZUSUKUNNUFUGUHOZUTUFUGU
|
|
HPZVAAABBQSUIUJCDUOUKNABUAUJTUBUFUGURUNNUHUFUGUPULUQUMHATBTUCUDUE $.
|
|
|
|
$( A number is nonzero iff its square is nonzero. See also ~ sqap0 which is
|
|
the same but with not equal changed to apart. (Contributed by NM,
|
|
11-Mar-2006.) $)
|
|
sqne0 $p |- ( A e. CC -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) ) $=
|
|
( cc wcel c2 cexp co cc0 sqeq0 necon3bid ) ABCADEFGAGAHI $.
|
|
|
|
$( A number is apart from zero iff its square is apart from zero.
|
|
(Contributed by Jim Kingdon, 13-Aug-2021.) $)
|
|
sqap0 $p |- ( A e. CC -> ( ( A ^ 2 ) # 0 <-> A # 0 ) ) $=
|
|
( cc wcel c2 cexp co cc0 cap wbr cmul sqval breq1d wa anidm mulap0b syl5bbr
|
|
wb anidms bitr4d ) ABCZADEFZGHIAAJFZGHIZAGHIZTUAUBGHAKLUDUDUDMZTUCUDNTUEUCQ
|
|
AAORPS $.
|
|
|
|
$( Closure of the square of a real number. (Contributed by NM,
|
|
18-Oct-1999.) $)
|
|
resqcl $p |- ( A e. RR -> ( A ^ 2 ) e. RR ) $=
|
|
( cr wcel c2 cn0 cexp co 2nn0 reexpcl mpan2 ) ABCDECADFGBCHADIJ $.
|
|
|
|
$( The square of a nonzero real is positive. (Contributed by Jim Kingdon,
|
|
11-Jun-2020.) $)
|
|
sqgt0ap $p |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A ^ 2 ) ) $=
|
|
( cr wcel cc0 cap wbr wa cmul co c2 cexp clt apsqgt0 wceq cc recn sqval syl
|
|
adantr breqtrrd ) ABCZADEFZGDAAHIZAJKIZLAMUAUDUCNZUBUAAOCUEAPAQRST $.
|
|
|
|
$( The naturals are closed under squaring. (Contributed by Scott Fenton,
|
|
29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
nnsqcl $p |- ( A e. NN -> ( A ^ 2 ) e. NN ) $=
|
|
( cn wcel c2 cexp co cmul cc wceq nncn sqval syl nnmulcl anidms eqeltrd ) A
|
|
BCZADEFZAAGFZBPAHCQRIAJAKLPRBCAAMNO $.
|
|
|
|
$( Integers are closed under squaring. (Contributed by Scott Fenton,
|
|
18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
zsqcl $p |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ ) $=
|
|
( cz wcel c2 cn0 cexp co 2nn0 zexpcl mpan2 ) ABCDECADFGBCHADIJ $.
|
|
|
|
$( The square of a rational is rational. (Contributed by Stefan O'Rear,
|
|
15-Sep-2014.) $)
|
|
qsqcl $p |- ( A e. QQ -> ( A ^ 2 ) e. QQ ) $=
|
|
( cq wcel c2 cexp co cmul cc wceq qcn sqval syl qmulcl anidms eqeltrd ) ABC
|
|
ZADEFZAAGFZBPAHCQRIAJAKLPRBCAAMNO $.
|
|
|
|
$( The square function is one-to-one for nonnegative reals. Also see
|
|
~ sq11ap which would easily follow from this given excluded middle, but
|
|
which for us is proved another way. (Contributed by NM, 8-Apr-2001.)
|
|
(Proof shortened by Mario Carneiro, 28-May-2016.) $)
|
|
sq11 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) ) $=
|
|
( cr wcel cc0 cle wbr wa c2 cexp co wceq cc simpl recnd sqval syl eqeqan12d
|
|
cmul msq11 bitrd ) ACDZEAFGZHZBCDZEBFGZHZHAIJKZBIJKZLAASKZBBSKZLABLUDUGUHUJ
|
|
UIUKUDAMDUHUJLUDAUBUCNOAPQUGBMDUIUKLUGBUEUFNOBPQRABTUA $.
|
|
|
|
$( The square function on nonnegative reals is strictly monotonic.
|
|
(Contributed by NM, 24-Feb-2006.) $)
|
|
lt2sq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt cmul co c2 cexp lt2msq wb recn sqval breqan12d
|
|
cc syl2an ad2ant2r bitr4d ) ACDZEAFGZHBCDZEBFGZHHABIGAAJKZBBJKZIGZALMKZBLMK
|
|
ZIGZABNUCUEULUIOZUDUFUCASDZBSDZUMUEAPBPUNUOUJUGUKUHIAQBQRTUAUB $.
|
|
|
|
$( The square function on nonnegative reals is monotonic. (Contributed by
|
|
NM, 18-Oct-1999.) $)
|
|
le2sq $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co c2 cexp le2msq wb cc recn breqan12d syl2an
|
|
sqval ad2ant2r bitr4d ) ACDZEAFGZHBCDZEBFGZHHABFGAAIJZBBIJZFGZAKLJZBKLJZFGZ
|
|
ABMUBUDUKUHNZUCUEUBAODZBODZULUDAPBPUMUNUIUFUJUGFASBSQRTUA $.
|
|
|
|
$( The square of a 'less than or equal to' ordering. (Contributed by NM,
|
|
21-Mar-2008.) $)
|
|
le2sq2 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ A <_ B ) ) ->
|
|
( A ^ 2 ) <_ ( B ^ 2 ) ) $=
|
|
( cr wcel cc0 cle wbr wa c2 cexp co simprr wb simprl 0re mp3an1 exp4b com23
|
|
wi letr imp43 jca le2sq syldan mpbid ) ACDZEAFGZHZBCDZABFGZHZHZUJAIJKBIJKFG
|
|
ZUHUIUJLUHUKUIEBFGZHUJUMMULUIUNUHUIUJNUFUGUIUJUNUFUIUGUJUNSUFUIUGUJUNECDUFU
|
|
IUGUJHUNSOEABTPQRUAUBABUCUDUE $.
|
|
|
|
$( A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.) $)
|
|
sqge0 $p |- ( A e. RR -> 0 <_ ( A ^ 2 ) ) $=
|
|
( cr wcel cc0 cmul co c2 cexp cle msqge0 cc wceq recn sqval syl breqtrrd )
|
|
ABCZDAAEFZAGHFZIAJQAKCSRLAMANOP $.
|
|
|
|
$( The square of an integer is a nonnegative integer. (Contributed by Mario
|
|
Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
zsqcl2 $p |- ( A e. ZZ -> ( A ^ 2 ) e. NN0 ) $=
|
|
( cz wcel c2 cexp co cc0 cle wbr cn0 zsqcl cr zre sqge0 syl elnn0z sylanbrc
|
|
) ABCZADEFZBCGSHIZSJCAKRALCTAMANOSPQ $.
|
|
|
|
$( Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed
|
|
by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof
|
|
shortened by Mario Carneiro, 28-May-2016.) $)
|
|
sumsqeq0 $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 ) ) $=
|
|
( cr wcel wa c2 cexp co cc0 wceq cle wbr wb resqcl sqge0 jca recn sqeq0 syl
|
|
cc caddc add20 syl2an bi2anan9 bitr2d ) ACDZBCDZEAFGHZBFGHZUAHIJZUHIJZUIIJZ
|
|
EZAIJZBIJZEUFUHCDZIUHKLZEUICDZIUIKLZEUJUMMUGUFUPUQANAOPUGURUSBNBOPUHUIUBUCU
|
|
FUKUNUGULUOUFATDUKUNMAQARSUGBTDULUOMBQBRSUDUE $.
|
|
|
|
${
|
|
sqval.1 $e |- A e. CC $.
|
|
$( Value of square. Inference version. (Contributed by NM,
|
|
1-Aug-1999.) $)
|
|
sqvali $p |- ( A ^ 2 ) = ( A x. A ) $=
|
|
( cc wcel c2 cexp co cmul wceq sqval ax-mp ) ACDAEFGAAHGIBAJK $.
|
|
|
|
$( Closure of square. (Contributed by NM, 2-Aug-1999.) $)
|
|
sqcli $p |- ( A ^ 2 ) e. CC $=
|
|
( cc wcel c2 cexp co sqcl ax-mp ) ACDAEFGCDBAHI $.
|
|
|
|
$( A number is zero iff its square is zero. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
sqeq0i $p |- ( ( A ^ 2 ) = 0 <-> A = 0 ) $=
|
|
( cc wcel c2 cexp co cc0 wceq wb sqeq0 ax-mp ) ACDAEFGHIAHIJBAKL $.
|
|
|
|
sqmul.2 $e |- B e. CC $.
|
|
$( Distribution of square over multiplication. (Contributed by NM,
|
|
3-Sep-1999.) $)
|
|
sqmuli $p |- ( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) $=
|
|
( cc wcel cmul co c2 cexp wceq sqmul mp2an ) AEFBEFABGHIJHAIJHBIJHGHKCDAB
|
|
LM $.
|
|
|
|
sqdivap.3 $e |- B # 0 $.
|
|
$( Distribution of square over division. (Contributed by Jim Kingdon,
|
|
12-Jun-2020.) $)
|
|
sqdivapi $p |- ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) $=
|
|
( cdiv co cmul c2 cexp divmuldivapi divclapi sqvali oveq12i 3eqtr4i ) ABF
|
|
GZPHGAAHGZBBHGZFGPIJGAIJGZBIJGZFGABABCDCDEEKPABCDELMSQTRFACMBDMNO $.
|
|
$}
|
|
|
|
${
|
|
resqcl.1 $e |- A e. RR $.
|
|
$( Closure of square in reals. (Contributed by NM, 2-Aug-1999.) $)
|
|
resqcli $p |- ( A ^ 2 ) e. RR $=
|
|
( cr wcel c2 cexp co resqcl ax-mp ) ACDAEFGCDBAHI $.
|
|
|
|
$( The square of a nonzero real is positive. (Contributed by Jim Kingdon,
|
|
12-Jun-2020.) $)
|
|
sqgt0api $p |- ( A # 0 -> 0 < ( A ^ 2 ) ) $=
|
|
( cr wcel cc0 cap wbr c2 cexp co clt sqgt0ap mpan ) ACDAEFGEAHIJKGBALM $.
|
|
|
|
$( A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.) $)
|
|
sqge0i $p |- 0 <_ ( A ^ 2 ) $=
|
|
( cc0 cmul co c2 cexp cle msqge0i recni sqvali breqtrri ) CAADEAFGEHABIAA
|
|
BJKL $.
|
|
|
|
lt2sq.2 $e |- B e. RR $.
|
|
$( The square function on nonnegative reals is strictly monotonic.
|
|
(Contributed by NM, 12-Sep-1999.) $)
|
|
lt2sqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) ) $=
|
|
( cc0 cle wbr wa clt cmul co c2 cexp lt2msqi recni sqvali breq12i syl6bbr
|
|
) EAFGEBFGHABIGAAJKZBBJKZIGALMKZBLMKZIGABCDNUASUBTIAACOPBBDOPQR $.
|
|
|
|
$( The square function on nonnegative reals is monotonic. (Contributed by
|
|
NM, 12-Sep-1999.) $)
|
|
le2sqi $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) ) $=
|
|
( cc0 cle wbr wa cmul co c2 cexp le2msqi recni sqvali breq12i syl6bbr ) E
|
|
AFGEBFGHABFGAAIJZBBIJZFGAKLJZBKLJZFGABCDMTRUASFAACNOBBDNOPQ $.
|
|
|
|
$( The square function is one-to-one for nonnegative reals. (Contributed
|
|
by NM, 27-Oct-1999.) $)
|
|
sq11i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) ) $=
|
|
( c2 cexp co wceq cmul cc0 cle wbr wa recni sqvali eqeq12i msq11i syl5bb
|
|
) AEFGZBEFGZHAAIGZBBIGZHJAKLJBKLMABHSUATUBAACNOBBDNOPABCDQR $.
|
|
$}
|
|
|
|
$( The square of 0 is 0. (Contributed by NM, 6-Jun-2006.) $)
|
|
sq0 $p |- ( 0 ^ 2 ) = 0 $=
|
|
( cc0 c2 cexp co wceq eqid 0cn sqeq0i mpbir ) ABCDAEAAEAFAGHI $.
|
|
|
|
$( If a number is zero, its square is zero. (Contributed by FL,
|
|
10-Dec-2006.) $)
|
|
sq0i $p |- ( A = 0 -> ( A ^ 2 ) = 0 ) $=
|
|
( cc0 wceq c2 cexp co oveq1 sq0 syl6eq ) ABCADEFBDEFBABDEGHI $.
|
|
|
|
${
|
|
sq0id.1 $e |- ( ph -> A = 0 ) $.
|
|
$( If a number is zero, its square is zero. Deduction form of ~ sq0i .
|
|
Converse of ~ sqeq0d . (Contributed by David Moews, 28-Feb-2017.) $)
|
|
sq0id $p |- ( ph -> ( A ^ 2 ) = 0 ) $=
|
|
( cc0 wceq c2 cexp co sq0i syl ) ABDEBFGHDECBIJ $.
|
|
$}
|
|
|
|
$( The square of 1 is 1. (Contributed by NM, 22-Aug-1999.) $)
|
|
sq1 $p |- ( 1 ^ 2 ) = 1 $=
|
|
( c2 cz wcel c1 cexp co wceq 2z 1exp ax-mp ) ABCDAEFDGHAIJ $.
|
|
|
|
$( ` -u 1 ` squared is 1 (common case). (Contributed by David A. Wheeler,
|
|
8-Dec-2018.) $)
|
|
neg1sqe1 $p |- ( -u 1 ^ 2 ) = 1 $=
|
|
( c1 cneg c2 cexp co cc wcel wceq ax-1cn sqneg ax-mp sq1 eqtri ) ABCDEZACDE
|
|
ZAAFGNOHIAJKLM $.
|
|
|
|
$( The square of 2 is 4. (Contributed by NM, 22-Aug-1999.) $)
|
|
sq2 $p |- ( 2 ^ 2 ) = 4 $=
|
|
( c2 cexp co cmul c4 2cn sqvali 2t2e4 eqtri ) AABCAADCEAFGHI $.
|
|
|
|
$( The square of 3 is 9. (Contributed by NM, 26-Apr-2006.) $)
|
|
sq3 $p |- ( 3 ^ 2 ) = 9 $=
|
|
( c3 c2 cexp co cmul c9 3cn sqvali 3t3e9 eqtri ) ABCDAAEDFAGHIJ $.
|
|
|
|
$( The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.) $)
|
|
cu2 $p |- ( 2 ^ 3 ) = 8 $=
|
|
( c2 c3 cexp co c1 caddc c8 df-3 oveq2i cmul wcel cn0 wceq 2nn0 expp1 mp2an
|
|
cc 2cn c4 eqtri sq2 oveq1i 4t2e8 ) ABCDAAEFDZCDZGBUDACHIUEAACDZAJDZGAQKALKU
|
|
EUGMRNAAOPUGSAJDGUFSAJUAUBUCTTT $.
|
|
|
|
$( The reciprocal of ` _i ` . (Contributed by NM, 11-Oct-1999.) $)
|
|
irec $p |- ( 1 / _i ) = -u _i $=
|
|
( c1 ci cdiv co cneg wceq cmul ax-icn mulneg2i ax-1cn mulcli negcon2i mpbir
|
|
ixi eqtr4i negicn iap0 divmulapi ) ABCDBEZFBSGDZAFTBBGDZEZABBHHIAUBFUAAEFNA
|
|
UAJBBHHKLMOABSJHPQRM $.
|
|
|
|
$( ` _i ` squared. (Contributed by NM, 6-May-1999.) $)
|
|
i2 $p |- ( _i ^ 2 ) = -u 1 $=
|
|
( ci c2 cexp co cmul c1 cneg ax-icn sqvali ixi eqtri ) ABCDAAEDFGAHIJK $.
|
|
|
|
$( ` _i ` cubed. (Contributed by NM, 31-Jan-2007.) $)
|
|
i3 $p |- ( _i ^ 3 ) = -u _i $=
|
|
( ci c3 cexp co c2 c1 caddc cneg df-3 oveq2i cmul wcel cn0 wceq ax-icn 2nn0
|
|
cc expp1 mp2an eqtri i2 oveq1i mulm1i ) ABCDAEFGDZCDZAHZBUDACIJUEAECDZAKDZU
|
|
FAQLEMLUEUHNOPAERSUHFHZAKDUFUGUIAKUAUBAOUCTTT $.
|
|
|
|
$( ` _i ` to the fourth power. (Contributed by NM, 31-Jan-2007.) $)
|
|
i4 $p |- ( _i ^ 4 ) = 1 $=
|
|
( ci c2 caddc co cexp cmul c4 c1 cc wcel cn0 wceq ax-icn expadd mp3an 2p2e4
|
|
2nn0 oveq2i i2 ax-1cn cneg oveq12i mul2negi 1t1e1 3eqtri 3eqtr3i ) ABBCDZED
|
|
ZABEDZUIFDZAGEDHAIJBKJZUKUHUJLMQQABBNOUGGAEPRUJHUAZULFDHHFDHUIULUIULFSSUBHH
|
|
TTUCUDUEUF $.
|
|
|
|
$( A positive integer is less than or equal to its square. (Contributed by
|
|
NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.) $)
|
|
nnlesq $p |- ( N e. NN -> N <_ ( N ^ 2 ) ) $=
|
|
( cn wcel cmul co c2 cexp cle c1 nncn mulid1d wbr nnge1 cr cc0 wb 1red nnre
|
|
clt nngt0 lemul2 syl112anc mpbid eqbrtrrd cc wceq sqval syl breqtrrd ) ABCZ
|
|
AAADEZAFGEZHUJAIDEZAUKHUJAAJZKUJIAHLZUMUKHLZAMUJINCANCZUQOASLUOUPPUJQARZURA
|
|
TIAAUAUBUCUDUJAUECULUKUFUNAUGUHUI $.
|
|
|
|
$( Taking ` _i ` to the ` K ` -th power is the same as using the ` K mod 4 `
|
|
-th power instead, by ~ i4 . (Contributed by Mario Carneiro,
|
|
7-Jul-2014.) $)
|
|
iexpcyc $p |- ( K e. ZZ -> ( _i ^ ( K mod 4 ) ) = ( _i ^ K ) ) $=
|
|
( cz wcel ci c4 co cexp cdiv cq wceq zq cc0 wbr 4z oveq2d ax-icn iap0 eqtrd
|
|
syl c1 cmo cfl cfv cmul cmin clt ax-mp 4pos modqval mp3an23 cn mpan2 flqcld
|
|
4nn znq zmulcl sylancr cc cap wa expsubap mpanl12 mpdan expmulzap i4 oveq1i
|
|
1exp syl5eq expclzap mp3an12 div1d ) ABCZDAEUAFZGFDAEAEHFZUBUCZUDFZUEFZGFZD
|
|
AGFZVLVMVQDGVLAICZVMVQJZAKVTEICZLEUFMWAEBCZWBNEKUGUHAEUIUJSOVLVRVSDVPGFZHFZ
|
|
VSVLVPBCZVRWEJZVLWCVOBCZWFNVLVNVLEUKCVNICUNAEUOULUMZEVOUPUQDURCZDLUSMZVLWFU
|
|
TWGPQDAVPVAVBVCVLWEVSTHFVSVLWDTVSHVLWDDEGFZVOGFZTVLWCWHWDWMJZNWIWJWKWCWHUTW
|
|
NPQDEVOVDVBUQVLWMTVOGFZTWLTVOGVEVFVLWHWOTJWIVOVGSVHROVLVSWJWKVLVSURCPQDAVIV
|
|
JVKRRR $.
|
|
|
|
$( A counterexample showing that exponentiation is not associative.
|
|
(Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.) $)
|
|
expnass $p |- ( ( 3 ^ 3 ) ^ 3 ) < ( 3 ^ ( 3 ^ 3 ) ) $=
|
|
( c3 cmul co cexp clt cc wcel 3cn 3nn0 mp3an wbr 3re nn0zi w3a 1lt3 ltexp2a
|
|
cz c2 mpanr12 eqbrtrri wceq expmul cr nn0mulcli nn0expcli c1 sqvali 2z 2lt3
|
|
cn0 3z ) AAABCZDCZAADCZADCZAUNDCZEAFGAUJGZUQUMUOUAHIIAAAUBJAUCGZULQGZUNQGZU
|
|
MUPEKZLULAAIIUDMUNAAIIUEMURUSUTNUFAEKZULUNEKVAOARDCZULUNEAHUGURRQGZAQGZVCUN
|
|
EKZLUHUKURVDVENVBRAEKVFOUIARAPSJTAULUNPSJT $.
|
|
|
|
$( Factor the difference of two squares. (Contributed by NM,
|
|
21-Feb-2008.) $)
|
|
subsq $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) ) $=
|
|
( cc wcel wa caddc co cmin cmul cexp simpl simpr sqval adantr eqtr4d adantl
|
|
c2 wceq oveq12d sqcl subcl adddird subdi oveq1d subdid mulcom mulcl npncand
|
|
3anidm12 3eqtrrd ) ACDZBCDZEZABFGABHGZIGAUNIGZBUNIGZFGAQJGZABIGZHGZURBQJGZH
|
|
GZFGUQUTHGUMABUNUKULKZUKULLZABUAUBUMUOUSUPVAFUMUOAAIGZURHGZUSUKULUOVERAABUC
|
|
UIUMUQVDURHUKUQVDRULAMNUDOUMUPBAIGZBBIGZHGVAUMBABVCVBVCUEUMURVFUTVGHABUFULU
|
|
TVGRUKBMPSOSUMUQURUTUKUQCDULATNABUGULUTCDUKBTPUHUJ $.
|
|
|
|
$( Express the difference of the squares of two numbers as a polynomial in
|
|
the difference of the numbers. (Contributed by NM, 21-Feb-2008.) $)
|
|
subsq2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) =
|
|
( ( ( A - B ) ^ 2 ) + ( ( 2 x. B ) x. ( A - B ) ) ) ) $=
|
|
( cc wcel wa caddc co cmin cmul c2 cexp wceq 2cn mulcl mpan adantl subadd23
|
|
mpd3an3 oveq1d eqtrd 2times pncan anidms oveq2d subcl adddird subsq 3eqtr4d
|
|
eqtr3d sqval syl ) ACDZBCDZEZABFGZABHGZIGZUPUPIGZJBIGZUPIGZFGZAJKGBJKGHGUPJ
|
|
KGZUTFGUNUPUSFGZUPIGUQVAUNVCUOUPIUNVCAUSBHGZFGZUOULUMUSCDZVCVELUMVFULJCDUMV
|
|
FMJBNOPZABUSQRUNVDBAFUMVDBLULUMVDBBFGZBHGZBUMUSVHBHBUASUMVIBLBBUBUCTPUDTSUN
|
|
UPUSUPABUEZVGVJUFUIABUGUNVBURUTFUNUPCDVBURLVJUPUJUKSUH $.
|
|
|
|
${
|
|
binom2.1 $e |- A e. CC $.
|
|
binom2.2 $e |- B e. CC $.
|
|
$( The square of a binomial. (Contributed by NM, 11-Aug-1999.) $)
|
|
binom2i $p |- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x.
|
|
( A x. B ) ) ) + ( B ^ 2 ) ) $=
|
|
( caddc co cmul c2 cexp addcli adddii adddiri mulcomi oveq2i eqtri mulcli
|
|
oveq12i addassi oveq1i sqvali 3eqtr2i 2timesi 3eqtr4i ) ABEFZUDGFZAAGFZAB
|
|
GFZUGEFZEFZBBGFZEFZUDHIFAHIFZHUGGFZEFZBHIFZEFUEUDAGFZUDBGFZEFZUKUDABABCDJ
|
|
ZCDKURUFUGEFZUGUJEFZEFUTUGEFZUJEFUKUPUTUQVAEUPUFBAGFZEFUTABACDCLVCUGUFEBA
|
|
DCMNOABBCDDLQUTUGUJUFUGAACCPZABCDPZJVEBBDDPRVBUIUJEUFUGUGVDVEVERSUAOUDUST
|
|
UNUIUOUJEULUFUMUHEACTUGVEUBQBDTQUC $.
|
|
|
|
$( Factor the difference of two squares. (Contributed by NM,
|
|
7-Feb-2005.) $)
|
|
subsqi $p |- ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) $=
|
|
( cc wcel c2 cexp co cmin caddc cmul wceq subsq mp2an ) AEFBEFAGHIBGHIJIA
|
|
BKIABJILIMCDABNO $.
|
|
$}
|
|
|
|
$( The square of a binomial. (Contributed by FL, 10-Dec-2006.) $)
|
|
binom2 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) =
|
|
( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) ) $=
|
|
( cc wcel wa caddc co cmul c2 cexp addcl simpl adddird eqtrd oveq12d mulcld
|
|
addassd wceq sqval syl adddid mulcomd oveq2d addcld 3eqtr2d 2timesd 3eqtr4d
|
|
simpr oveq1d ) ACDZBCDZEZABFGZUMHGZAAHGZABHGZUPFGZFGZBBHGZFGZUMIJGZAIJGZIUP
|
|
HGZFGZBIJGZFGULUNUMAHGZUMBHGZFGZUTULUMABABKZUJUKLZUJUKUHZUAULVHUOUPFGZUPUSF
|
|
GZFGVLUPFGZUSFGUTULVFVLVGVMFULVFUOBAHGZFGVLULABAVJVKVJMULVOUPUOFULBAVKVJUBU
|
|
CNULABBVJVKVKMOULVLUPUSULUOUPULAAVJVJPZULABVJVKPZUDVQULBBVKVKPQULVNURUSFULU
|
|
OUPUPVPVQVQQUIUENULUMCDVAUNRVIUMSTULVDURVEUSFULVBUOVCUQFULUJVBUORVJASTULUPV
|
|
QUFOULUKVEUSRVKBSTOUG $.
|
|
|
|
$( Special case of ~ binom2 where ` B = 1 ` . (Contributed by Scott Fenton,
|
|
11-May-2014.) $)
|
|
binom21 $p |- ( A e. CC -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2
|
|
) + ( 2 x. A ) ) + 1 ) ) $=
|
|
( cc wcel c1 caddc co c2 cexp cmul wceq ax-1cn binom2 mulid1 oveq2d sq1 a1i
|
|
mpan2 oveq12d eqtrd ) ABCZADEFGHFZAGHFZGADIFZIFZEFZDGHFZEFZUBGAIFZEFZDEFTDB
|
|
CUAUGJKADLQTUEUIUFDETUDUHUBETUCAGIAMNNUFDJTOPRS $.
|
|
|
|
$( Expand the square of a subtraction. (Contributed by Scott Fenton,
|
|
10-Jun-2013.) $)
|
|
binom2sub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( A - B ) ^ 2 ) =
|
|
( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) ) $=
|
|
( cc wcel wa c2 cexp co cneg cmul caddc cmin wceq eqtr3d mulneg2 oveq2d 2cn
|
|
negcl mulcl sylancr binom2 sylan2 negsub oveq1d eqtr2d adantr negsubd sqneg
|
|
sqcl adantl oveq12d ) ACDZBCDZEZAFGHZFABIZJHZJHZKHZUPFGHZKHZABLHZFGHZUOFABJ
|
|
HZJHZLHZBFGHZKHUNAUPKHZFGHZVAVCUMULUPCDVIVAMBRAUPUAUBUNVHVBFGABUCUDNUNUSVFU
|
|
TVGKUNUOVEIZKHUSVFUNVJURUOKUNURFVDIZJHZVJUNUQVKFJABOPUNFCDZVDCDZVLVJMQABSZF
|
|
VDOTUEPUNUOVEULUOCDUMAUIUFUNVMVNVECDQVOFVDSTUGNUMUTVGMULBUHUJUKN $.
|
|
|
|
${
|
|
binom2subi.1 $e |- A e. CC $.
|
|
binom2subi.2 $e |- B e. CC $.
|
|
$( Expand the square of a subtraction. (Contributed by Scott Fenton,
|
|
13-Jun-2013.) $)
|
|
binom2subi $p |- ( ( A - B ) ^ 2 ) =
|
|
( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) $=
|
|
( cc wcel cmin co c2 cexp cmul caddc wceq binom2sub mp2an ) AEFBEFABGHIJH
|
|
AIJHIABKHKHGHBIJHLHMCDABNO $.
|
|
$}
|
|
|
|
$( The square of a binomial with factor. (Contributed by AV,
|
|
19-Jul-2021.) $)
|
|
mulbinom2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC )
|
|
-> ( ( ( C x. A ) + B ) ^ 2 )
|
|
= ( ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) )
|
|
+ ( B ^ 2 ) ) ) $=
|
|
( cc wcel w3a cmul co caddc c2 cexp wceq mulcl ancoms 3adant2 simp2 syl2anc
|
|
binom2 mulass oveq2d 3coml 2cnd simp3 3adant3 mulassd eqtr4d oveq1d eqtrd )
|
|
ADEZBDEZCDEZFZCAGHZBIHJKHZUMJKHZJUMBGHZGHZIHZBJKHZIHZUOJCGHABGHZGHZIHZUSIHU
|
|
LUMDEZUJUNUTLUIUKVDUJUKUIVDCAMNOUIUJUKPUMBRQULURVCUSIULUQVBUOIULUQJCVAGHZGH
|
|
VBULUPVEJGUKUIUJUPVELCABSUATULJCVAULUBUIUJUKUCUIUJVADEUKABMUDUEUFTUGUH $.
|
|
|
|
$( The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.) $)
|
|
binom3 $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 3 ) =
|
|
( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) +
|
|
( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) $=
|
|
( cc wcel caddc co c3 cexp cmul df-3 wceq sylancl syl oveq1d adddird oveq2d
|
|
c2 c1 eqtr4d oveq12d wa oveq2i cn0 addcl 2nn0 expp1 syl5eq sqcl simpl simpr
|
|
adddid binom2 mulcl sylancr addcld sqval mul32d 2cnd mulassd mulcomd 3eqtrd
|
|
2cn eqtrd mulcld 3nn0 expcl addassd eqtr3d add4d 1cnd mulid2d 1p2e3 syl5eqr
|
|
oveq1i eqtr2d ) ACDZBCDZUAZABEFZGHFZVSQHFZVSIFZAGHFZQAQHFZBIFZIFZEFZWEEFZAB
|
|
QHFZIFZQWJIFZBGHFZEFZEFZEFZWCGWEIFZEFZGWJIFZWLEFZEFVRVTVSQREFZHFZWBGWTVSHJU
|
|
BVRVSCDZQUCDZXAWBKABUDZUEVSQUFLUGVRWBWAAIFZWABIFZEFWGWJEFZWEWMEFZEFWOVRWAAB
|
|
VRXBWACDXDVSUHMVPVQUIZVPVQUJZUKVRXEXGXFXHEVRXEWDQABIFZIFZEFZWIEFZAIFXMAIFZW
|
|
IAIFZEFXGVRWAXNAIABULZNVRXMWIAVRWDXLVRVPWDCDXIAUHMZVRQCDZXKCDXLCDVBABUMZQXK
|
|
UMUNZUOZVRVQWICDXJBUHMZXIOVRXOWGXPWJEVRXOWDAIFZXLAIFZEFWGVRWDXLAXRYAXIOVRWC
|
|
YDWFYEEVRWCAWTHFZYDGWTAHJUBVRVPXCYFYDKXIUEAQUFLUGVRWFQXKAIFZIFYEVRWEYGQIVRW
|
|
EAAIFZBIFYGVRWDYHBIVRVPWDYHKXIAUPMNVRAABXIXIXJUQVCPVRQXKAVRURZXTXIUSSTSVRWI
|
|
AYCXIUTTVAVRXFXNBIFXMBIFZWIBIFZEFZXHVRWAXNBIXQNVRXMWIBYBYCXJOVRWEWKEFZWLEFY
|
|
LXHVRYMYJWLYKEVRYMWEXLBIFZEFYJVRWKYNWEEVRWKQXKBIFZIFYNVRWJYOQIVRWJABBIFZIFY
|
|
OVRWIYPAIVRVQWIYPKXJBUPMPVRABBXIXJXJUSSPVRQXKBYIXTXJUSSPVRWDXLBXRYAXJOSVRWL
|
|
BWTHFZYKGWTBHJUBVRVQXCYQYKKXJUEBQUFLUGTVRWEWKWLVRWDBXRXJVDZVRXSWJCDWKCDVBVR
|
|
AWIXIYCVDZQWJUMUNZVRVQGUCDZWLCDXJVEBGVFLZVGVHVATVRWGWJWEWMVRWCWFVRVPUUAWCCD
|
|
XIVEAGVFLZVRXSWECDWFCDVBYRQWEUMUNZUOYSYRVRWKWLYTUUBUOVIVAVRWHWQWNWSEVRWHWCW
|
|
FWEEFZEFWQVRWCWFWEUUCUUDYRVGVRWPUUEWCEVRWPWFRWEIFZEFZUUEVRWPWTWEIFUUGGWTWEI
|
|
JVNVRQRWEYIVRVJZYROUGVRUUFWEWFEVRWEYRVKPVCPSVRWSWJWKEFZWLEFWNVRWRUUIWLEVRWR
|
|
RWJIFZWKEFZUUIVRWRRQEFZWJIFUUKUULGWJIVLVNVRRQWJUUHYIYSOVMVRUUJWJWKEVRWJYSVK
|
|
NVCNVRWJWKWLYSYTUUBVGVOTVA $.
|
|
|
|
$( An integer is even iff its square is even. (Contributed by Mario
|
|
Carneiro, 12-Sep-2015.) $)
|
|
zesq $p |- ( N e. ZZ ->
|
|
( ( N / 2 ) e. ZZ <-> ( ( N ^ 2 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel c2 cdiv co cexp wa cmul wceq sqval syl oveq1d 2cnd 2ap0 divassapd
|
|
cc a1i c1 caddc zcn cc0 cap wbr adantr zmulcl eqeltrd wn cmin sqcl peano2cn
|
|
eqtrd halfcld pncand binom21 mulcl sylancr 1cnd add32d divdirapd divcanap3d
|
|
2cn 3eqtr3d oveq2d peano2z sylan eqeltrrd simpl zsubcld ex con3d zsqcl zeo2
|
|
wb 3imtr4d imp impbida ) ABCZADEFZBCZADGFZDEFZBCZVRVTHWBAVSIFZBVRWBWDJVTVRW
|
|
BAAIFZDEFWDVRWAWEDEVRAQCZWAWEJAUAZAKLMVRAADWGWGVRNDUBUCUDZVRORPULUEAVSUFUGV
|
|
RWCVTVRWASTFZDEFZBCZUHZASTFZDEFZBCZUHWCVTVRWOWKVRWOWKVRWOHZWJATFZAUIFWJBWPW
|
|
JAWPWIWPWAQCZWIQCWPWFWRVRWFWOWGUEZAUJLZWAUKLZUMWSUNWPWQAWPWMWNIFZWQBWPWMWMI
|
|
FZDEFWIDAIFZTFZDEFZXBWQWPXCXEDEWPWMDGFZWAXDTFSTFZXCXEWPWFXGXHJWSAUOLWPWMQCZ
|
|
XGXCJWPWFXIWSAUKLZWMKLWPWAXDSWTWPDQCWFXDQCVBWSDAUPUQZWPURUSVCMWPWMWMDXJXJWP
|
|
NZWHWPORZPWPXFWJXDDEFZTFWQWPWIXDDXAXKXLXMUTWPXNAWJTWPADWSXLXMVAVDULVCVRWMBC
|
|
WOXBBCAVEWMWNUFVFVGVRWOVHVIVGVJVKVRWABCWCWLVNAVLWAVMLAVMVOVPVQ $.
|
|
|
|
$( A positive integer is even iff its square is even. (Contributed by NM,
|
|
20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) $)
|
|
nnesq $p |- ( N e. NN ->
|
|
( ( N / 2 ) e. NN <-> ( ( N ^ 2 ) / 2 ) e. NN ) ) $=
|
|
( cn wcel c2 cdiv co cz cc0 clt wbr cexp nnz zesq syl nnrp rphalfcld rpgt0d
|
|
wa wb elnnz nnsqcl nnrpd 2thd anbi12d 3bitr4g ) ABCZADEFZGCZHUGIJZRADKFZDEF
|
|
ZGCZHUKIJZRUGBCUKBCUFUHULUIUMUFAGCUHULSALAMNUFUIUMUFUGUFAAOPQUFUKUFUJUFUJAU
|
|
AUBPQUCUDUGTUKTUE $.
|
|
|
|
${
|
|
$d j k A $. $d j k N $.
|
|
$( Bernoulli's inequality, due to Johan Bernoulli (1667-1748).
|
|
(Contributed by NM, 21-Feb-2005.) $)
|
|
bernneq $p |- ( ( A e. RR /\ N e. NN0 /\ -u 1 <_ A ) ->
|
|
( 1 + ( A x. N ) ) <_ ( ( 1 + A ) ^ N ) ) $=
|
|
( cr wcel c1 cle wbr cmul co caddc cexp wi wa wceq oveq2 oveq2d cc ax-1cn
|
|
cc0 adantr vj vk cn0 cneg cv breq12d imbi2d recn mul01 1p0e1 syl6eq addcl
|
|
1le1 mpan exp0 syl syl5breqr eqbrtrd nn0re remulcl sylan2 readdcl sylancr
|
|
1re simpl syl2anc remulcld reexpcl anidms msqge0 jca nn0ge0 mulge0 syl2an
|
|
sylan nn0cn adantl mul32d breqtrd addge01d mulcld addassd muladd11 eqtr4d
|
|
mpbid mulcl wb neg1rr leadd2 mp3an13 1pneg1e0 breq1i syl6bb biimpa simprr
|
|
ad2ant2r letrd adddi mp3an3 mulid1 eqtrd addass expp1 3brtr4d exp43 com12
|
|
lemul1ad mp3an1 impd a2d nn0ind expd 3imp ) ACDZBUCDZEUDZAFGZEABHIZJIZEAJ
|
|
IZBKIZFGZXOXNXQYBLXOXNXQYBXNXQMZEAUAUEZHIZJIZXTYDKIZFGZLYCEASHIZJIZXTSKIZ
|
|
FGZLYCEAUBUEZHIZJIZXTYMKIZFGZLYCEAYMEJIZHIZJIZXTYRKIZFGZLYCYBLUAUBBYDSNZY
|
|
HYLYCUUCYFYJYGYKFUUCYEYIEJYDSAHOPYDSXTKOUFUGYDYMNZYHYQYCUUDYFYOYGYPFUUDYE
|
|
YNEJYDYMAHOPYDYMXTKOUFUGYDYRNZYHUUBYCUUEYFYTYGUUAFUUEYEYSEJYDYRAHOPYDYRXT
|
|
KOUFUGYDBNZYHYBYCUUFYFXSYGYAFUUFYEXREJYDBAHOPYDBXTKOUFUGXNYLXQXNAQDZYLAUH
|
|
ZUUGYJEYKFUUGYJESJIEUUGYISEJAUIPUJUKUUGEEYKFUMUUGXTQDZYKENEQDZUUGUUIREAUL
|
|
ZUNXTUOUPUQURUPTYMUCDZYCYQUUBUULXNXQYQUUBLZXNUULXQUUMLXNUULXQYQUUBXNUULMZ
|
|
XQYQMZMZYOAJIZYPXTHIZYTUUAFUUPUUQYOXTHIZUURUUNUUQCDZUUOUUNYOCDZXNUUTUUNEC
|
|
DZYNCDZUVAVDUULXNYMCDZUVCYMUSZAYMUTZVAEYNVBVCZXNUULVEYOAVBVFZTUUNUUSCDUUO
|
|
UUNYOXTUVGXNXTCDZUULUVBXNUVIVDEAVBUNZTZVGTUUNUURCDUUOUUNYPXTXNUVIUULYPCDZ
|
|
UVJXTYMVHVOZUVKVGTUUNUUQUUSFGUUOUUNUUQUUQYNAHIZJIZUUSFUUNSUVNFGUUQUVOFGUU
|
|
NSAAHIZYMHIZUVNFXNUVPCDZSUVPFGZMUVDSYMFGZMSUVQFGUULXNUVRUVSXNUVRAAUTVIAVJ
|
|
VKUULUVDUVTUVEYMVLVKUVPYMVMVNUUNAAYMXNUUGUULUUHTZUWAUULYMQDZXNYMVPZVQVRVS
|
|
UUNUUQUVNUVHUULXNUVDUVNCDUVEXNUVDMYNAUVFXNUVDVEVGVAVTWEXNUUGUWBUVOUUSNUUL
|
|
UUHUWCUUGUWBMZUVOYOAUVNJIJIZUUSUWDYOAUVNUWDUUJYNQDZYOQDRAYMWFZEYNULVCUUGU
|
|
WBVEZUWDYNAUWGUWHWAWBUWDUWFUUGUUSUWENUWGUWHYNAWCVFWDVNVSTUUPYOYPXTUUNUVAU
|
|
UOUVGTUUNUVLUUOUVMTUUNUVIUUOUVKTXNXQSXTFGZUULYQXNXQUWIXNXQEXPJIZXTFGZUWIX
|
|
PCDXNUVBXQUWKWGWHVDXPAEWIWJUWJSXTFWKWLWMWNWPUUNXQYQWOXGWQUUNYTUUQNZUUOXNU
|
|
UGUWBUWLUULUUHUWCUWDYTEYNAJIZJIZUUQUWDYSUWMEJUWDYSYNAEHIZJIZUWMUUGUWBUUJY
|
|
SUWPNRAYMEWRWSUWDUWOAYNJUUGUWOANUWBAWTTPXAPUWDUWFUUGUUQUWNNZUWGUWHUUJUWFU
|
|
UGUWQREYNAXBXHVFWDVNTUUNUUAUURNZUUOXNUUIUULUWRXNUUJUUGUUIRUUHUUKVCXTYMXCV
|
|
OTXDXEXFXIXJXKXLXFXM $.
|
|
$}
|
|
|
|
$( Variation of Bernoulli's inequality ~ bernneq . (Contributed by NM,
|
|
18-Oct-2007.) $)
|
|
bernneq2 $p |- ( ( A e. RR /\ N e. NN0 /\ 0 <_ A ) ->
|
|
( ( ( A - 1 ) x. N ) + 1 ) <_ ( A ^ N ) ) $=
|
|
( cr wcel cn0 cc0 cle wbr w3a c1 cmin co caddc cexp 3ad2ant1 wa wceq ax-1cn
|
|
cc sylancr cmul cneg peano2rem simp2 df-neg 0re 1re lesub1 mp3an13 syl5eqbr
|
|
biimpa 3adant2 bernneq syl3anc recnd nn0cn mulcl syl2an addcom 3adant3 recn
|
|
wb pncan3 oveq1d 3brtr3d ) ACDZBEDZFAGHZIZJAJKLZBUALZMLZJVJMLZBNLZVKJMLZABN
|
|
LZGVIVJCDZVGJUBZVJGHZVLVNGHVFVGVQVHAUCZOVFVGVHUDVFVHVSVGVFVHPVRFJKLZVJGJUEV
|
|
FVHWAVJGHZFCDVFJCDVHWBVBUFUGFAJUHUIUKUJULVJBUMUNVFVGVLVOQZVHVFVGPJSDZVKSDZW
|
|
CRVFVJSDBSDWEVGVFVJVTUOBUPVJBUQURJVKUSTUTVFVGVNVPQVHVFVMABNVFWDASDVMAQRAVAJ
|
|
AVCTVDOVE $.
|
|
|
|
$( A corollary of ~ bernneq . (Contributed by Mario Carneiro,
|
|
11-Mar-2014.) $)
|
|
bernneq3 $p |- ( ( P e. ( ZZ>= ` 2 ) /\ N e. NN0 ) ->
|
|
N < ( P ^ N ) ) $=
|
|
( c2 cuz cfv wcel cn0 wa c1 caddc co adantl peano2re syl adantr cc0 cle wbr
|
|
cr nn0ge0 cexp nn0re eluzelre reexpcl sylan ltp1d cn uz2m1nn nnred remulcld
|
|
cmin cmul 1red nnge1d lemulge12d leadd1dd simpr eluzge2nn0 bernneq2 syl3anc
|
|
letrd ltletrd ) ACDEFZBGFZHZBBIJKZABUAKZVDBSFZVCBUBLZVEVHVFSFVIBMNZVCASFZVD
|
|
VGSFCAUCZABUDUEZVEBVIUFVEVFAIUKKZBULKZIJKZVGVJVEVOSFVPSFVEVNBVEVNVCVNUGFVDA
|
|
UHOZUIZVIUJZVOMNVMVEBVOIVIVSVEUMVEBVNVIVRVDPBQRVCBTLVEVNVQUNUOUPVEVKVDPAQRZ
|
|
VPVGQRVCVKVDVLOVCVDUQVCVTVDVCAGFVTAURATNOABUSUTVAVB $.
|
|
|
|
${
|
|
$d j k A $. $d j k B $.
|
|
$( Exponentiation with a mantissa greater than 1 has no upper bound.
|
|
(Contributed by NM, 20-Oct-2007.) $)
|
|
expnbnd $p |- ( ( A e. RR /\ B e. RR /\ 1 < B ) ->
|
|
E. k e. NN A < ( B ^ k ) ) $=
|
|
( cr wcel c1 clt cexp co cn wrex wa adantr simpr cmin 1red resubcld mpbid
|
|
wbr cc0 w3a cv simp1 simp2 simp3 cdiv posdifd gt0ap0d redivclapd arch syl
|
|
3expa adantrl cmul caddc simplll simpllr remulcld readdcld nnnn0d reexpcl
|
|
nnred cn0 syl2anc wb simplr simplrr ltdivmul syl112anc ltsubaddd cle 0red
|
|
0lt1 wi 0re 1re lttr mp3an12 mpani sylc ltled bernneq2 syl3anc ltletrd ex
|
|
reximdva mpd syl22anc 1nn cc wceq simpl2 recnd exp1 breqtrrd oveq2 breq2d
|
|
rspcev sylancr wo axltwlin mp3an1 ancoms 3impia mpjaodan ) ADEZBDEZFBGSZU
|
|
AZFAGSZABCUBZHIZGSZCJKZABGSZXIXJLXFXGXJXHXNXIXFXJXFXGXHUCZMXIXGXJXFXGXHUD
|
|
ZMXIXJNXIXHXJXFXGXHUEZMXFXGLZXJXHLZLZAFOIZBFOIZUFIZXKGSZCJKZXNXSXHYFXJXFX
|
|
GXHYFXIYDDEYFXIYBYCXIAFXPXIPZQXIBFXQYGQZXIYCYHXIXHTYCGSZXRXIFBYGXQUGRUHUI
|
|
YDCUJUKULUMYAYEXMCJYAXKJEZLZYEXMYKYELZAYCXKUNIZFUOIZXLYKXFYEXFXGXTYJUPMZY
|
|
KYNDEYEYKYMFYKYCXKYKBFXFXGXTYJUQZYKPZQYKXKYAYJNZVBURYQUSMYKXLDEZYEYKXGXKV
|
|
CEZYSYPYKXKYRUTBXKVAVDMYLYBYMGSZAYNGSYLYEUUAYKYENYLYBDEXKDEYCDEYIYEUUAVEY
|
|
LAFYOYLPZQYLXKYAYJYEVFZVBZYLBFYKXGYEYPMZUUBQZYLXHYIYKXHYEXSXJXHYJVGZMYLFB
|
|
UUBUUEUGRYBXKYCVHVIRYLAFYMYOUUBYLYCXKUUFUUDURVJRYLXGYTTBVKSZYNXLVKSUUEYLX
|
|
KUUCUTYKUUHYEYKTBYKVLYPYKXGXHTBGSZYPUUGXGTFGSZXHUUIVMTDEFDEZXGUUJXHLUUIVN
|
|
VOVPTFBVQVRVSVTWAMBXKWBWCWDWEWFWGWHXIXOLZFJEABFHIZGSZXNWIUULABUUMGXIXONUU
|
|
LBWJEUUMBWKUULBXFXGXHXOWLWMBWNUKWOXMUUNCFJXKFWKXLUUMAGXKFBHWPWQWRWSXFXGXH
|
|
XJXOWTZXGXFXHUUOVNZUUKXGXFUUPVPFBAXAXBXCXDXE $.
|
|
|
|
$( The reciprocal of exponentiation with a mantissa greater than 1 has no
|
|
lower bound. (Contributed by NM, 18-Jul-2008.) $)
|
|
expnlbnd $p |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) ->
|
|
E. k e. NN ( 1 / ( B ^ k ) ) < A ) $=
|
|
( crp wcel cr c1 clt wbr w3a cdiv co cv cexp cn wrex rpre wa cc0 adantr
|
|
rpap0 rerecclapd expnbnd syl3an1 wb rpregt0 3ad2ant1 nnnn0 reexpcl sylan2
|
|
cn0 adantlr cz simpll nnz adantl 0lt1 wi 0re 1re mp3an12 mpani imp expgt0
|
|
lttr syl3anc jca 3adantl1 ltrec1 syl2anc rexbidva mpbid ) ADEZBFEZGBHIZJZ
|
|
GAKLZBCMZNLZHIZCOPZGVSKLAHIZCOPVMVQFEVNVOWAVMAAQAUAUBVQBCUCUDVPVTWBCOVPVR
|
|
OEZRAFESAHIRZVSFEZSVSHIZRZVTWBUEVPWDWCVMVNWDVOAUFUGTVNVOWCWGVMVNVORZWCRZW
|
|
EWFVNWCWEVOWCVNVRUKEWEVRUHBVRUIUJULWIVNVRUMEZSBHIZWFVNVOWCUNWCWJWHVRUOUPW
|
|
HWKWCVNVOWKVNSGHIZVOWKUQSFEGFEVNWLVORWKURUSUTSGBVEVAVBVCTBVRVDVFVGVHAVSVI
|
|
VJVKVL $.
|
|
|
|
$( The reciprocal of exponentiation with a mantissa greater than 1 has no
|
|
lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by
|
|
Mario Carneiro, 5-Jun-2014.) $)
|
|
expnlbnd2 $p |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) ->
|
|
E. j e. NN A. k e. ( ZZ>= ` j ) ( 1 / ( B ^ k ) ) < A ) $=
|
|
( crp wcel cr c1 clt wbr cv cexp co cdiv cn wrex wa wi cle mpd w3a simpl2
|
|
cuz cfv wral expnlbnd simpl3 1re ltle sylancr simprr leexp2a syl3anc 0red
|
|
cz cc0 1red a1i lttrd elrpd nnz ad2antrl rpexpcl syl2anc eluzelz ad2antll
|
|
0lt1 lerecd mpbid rprecred simpl1 rpred lelttr anassrs ralrimdva reximdva
|
|
mpand ) AEFZBGFZHBIJZUAZHBCKZLMZNMZAIJZCOPHBDKZLMZNMZAIJZDWBUCUDZUEZCOPAB
|
|
CUFWAWEWKCOWAWBOFZQWEWIDWJWAWLWFWJFZWEWIRWAWLWMQZQZWHWDSJZWEWIWOWCWGSJZWP
|
|
WOVSHBSJZWMWQVRVSVTWNUBZWOVTWRVRVSVTWNUGZWOHGFVSVTWRRUHWSHBUIUJTWAWLWMUKB
|
|
WBWFULUMWOWCWGWOBEFZWBUOFZWCEFWOBWSWOUPHBWOUNWOUQWSUPHIJWOVGURWTUSUTZWLXB
|
|
WAWMWBVAVBBWBVCVDZWOXAWFUOFZWGEFXCWMXEWAWLWBWFVEVFBWFVCVDZVHVIWOWHGFWDGFA
|
|
GFWPWEQWIRWOWGXFVJWOWCXDVJWOAVRVSVTWNVKVLWHWDAVMUMVQVNVOVPT $.
|
|
$}
|
|
|
|
${
|
|
expcld.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Value of a complex number raised to the 0th power. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
exp0d $p |- ( ph -> ( A ^ 0 ) = 1 ) $=
|
|
( cc wcel cc0 cexp co c1 wceq exp0 syl ) ABDEBFGHIJCBKL $.
|
|
|
|
$( Value of a complex number raised to the first power. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
exp1d $p |- ( ph -> ( A ^ 1 ) = A ) $=
|
|
( cc wcel c1 cexp co wceq exp1 syl ) ABDEBFGHBICBJK $.
|
|
|
|
${
|
|
expeq0d.2 $e |- ( ph -> N e. NN ) $.
|
|
expeq0d.3 $e |- ( ph -> ( A ^ N ) = 0 ) $.
|
|
$( Positive integer exponentiation is 0 iff its mantissa is 0.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
expeq0d $p |- ( ph -> A = 0 ) $=
|
|
( cexp co cc0 wceq cc wcel cn wb expeq0 syl2anc mpbid ) ABCGHIJZBIJZFAB
|
|
KLCMLRSNDEBCOPQ $.
|
|
$}
|
|
|
|
$( Value of square. Inference version. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
sqvald $p |- ( ph -> ( A ^ 2 ) = ( A x. A ) ) $=
|
|
( cc wcel c2 cexp co cmul wceq sqval syl ) ABDEBFGHBBIHJCBKL $.
|
|
|
|
$( Closure of square. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
sqcld $p |- ( ph -> ( A ^ 2 ) e. CC ) $=
|
|
( cc wcel c2 cexp co sqcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
${
|
|
sqeq0d.1 $e |- ( ph -> ( A ^ 2 ) = 0 ) $.
|
|
$( A number is zero iff its square is zero. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
sqeq0d $p |- ( ph -> A = 0 ) $=
|
|
( c2 cn wcel 2nn a1i expeq0d ) ABECEFGAHIDJ $.
|
|
$}
|
|
|
|
${
|
|
expcld.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Closure law for nonnegative integer exponentiation. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
expcld $p |- ( ph -> ( A ^ N ) e. CC ) $=
|
|
( cc wcel cn0 cexp co expcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
|
|
$( Value of a complex number raised to a nonnegative integer power plus
|
|
one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
expp1d $p |- ( ph -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) $=
|
|
( cc wcel cn0 c1 caddc co cexp cmul wceq expp1 syl2anc ) ABFGCHGBCIJKLK
|
|
BCLKBMKNDEBCOP $.
|
|
|
|
expaddd.2 $e |- ( ph -> M e. NN0 ) $.
|
|
$( Sum of exponents law for nonnegative integer exponentiation.
|
|
Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
expaddd $p |- ( ph -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) $=
|
|
( cc wcel cn0 caddc co cexp cmul wceq expadd syl3anc ) ABHICJIDJIBCDKLM
|
|
LBCMLBDMLNLOEGFBCDPQ $.
|
|
|
|
$( Product of exponents law for positive integer exponentiation.
|
|
Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative
|
|
integer exponents. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
expmuld $p |- ( ph -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) $=
|
|
( cc wcel cn0 cmul co cexp wceq expmul syl3anc ) ABHICJIDJIBCDKLMLBCMLD
|
|
MLNEGFBCDOP $.
|
|
$}
|
|
|
|
${
|
|
sqrecd.1 $e |- ( ph -> A # 0 ) $.
|
|
$( Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.) $)
|
|
sqrecapd $p |- ( ph -> ( ( 1 / A ) ^ 2 ) = ( 1 / ( A ^ 2 ) ) ) $=
|
|
( cc wcel cc0 cap c1 cdiv co c2 cexp wceq cz 2z exprecap mp3an3 syl2anc
|
|
wbr ) ABEFZBGHTZIBJKLMKIBLMKJKNZCDUAUBLOFUCPBLQRS $.
|
|
|
|
expclzd.3 $e |- ( ph -> N e. ZZ ) $.
|
|
$( Closure law for integer exponentiation. (Contributed by Jim Kingdon,
|
|
12-Jun-2020.) $)
|
|
expclzapd $p |- ( ph -> ( A ^ N ) e. CC ) $=
|
|
( cc wcel cc0 cap wbr cz cexp co expclzap syl3anc ) ABGHBIJKCLHBCMNGHDE
|
|
FBCOP $.
|
|
|
|
$( Nonnegative integer exponentiation is nonzero if its mantissa is
|
|
nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.) $)
|
|
expap0d $p |- ( ph -> ( A ^ N ) # 0 ) $=
|
|
( cc wcel cc0 cap wbr cz cexp co expap0i syl3anc ) ABGHBIJKCLHBCMNIJKDE
|
|
FBCOP $.
|
|
|
|
$( Value of a complex number raised to a negative power. (Contributed by
|
|
Jim Kingdon, 12-Jun-2020.) $)
|
|
expnegapd $p |- ( ph -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cz cneg cexp co c1 cdiv wceq expnegzap syl3anc )
|
|
ABGHBIJKCLHBCMNOPBCNOQORDEFBCST $.
|
|
|
|
$( Nonnegative integer exponentiation of a reciprocal. (Contributed by
|
|
Jim Kingdon, 12-Jun-2020.) $)
|
|
exprecapd $p |- ( ph -> ( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cz c1 cdiv co cexp wceq exprecap syl3anc ) ABGHBI
|
|
JKCLHMBNOCPOMBCPONOQDEFBCRS $.
|
|
|
|
$( Value of a nonzero complex number raised to an integer power plus one.
|
|
(Contributed by Jim Kingdon, 12-Jun-2020.) $)
|
|
expp1zapd $p |- ( ph -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) $=
|
|
( cc wcel cc0 cap wbr cz c1 caddc co cexp cmul wceq expp1zap syl3anc )
|
|
ABGHBIJKCLHBCMNOPOBCPOBQORDEFBCST $.
|
|
|
|
$( Value of a complex number raised to an integer power minus one.
|
|
(Contributed by Jim Kingdon, 12-Jun-2020.) $)
|
|
expm1apd $p |- ( ph -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / A ) ) $=
|
|
( cc wcel cc0 cap wbr cz c1 cmin co cexp cdiv wceq expm1ap syl3anc ) AB
|
|
GHBIJKCLHBCMNOPOBCPOBQORDEFBCST $.
|
|
|
|
expsubd.3 $e |- ( ph -> M e. ZZ ) $.
|
|
$( Exponent subtraction law for nonnegative integer exponentiation.
|
|
(Contributed by Jim Kingdon, 12-Jun-2020.) $)
|
|
expsubapd $p |- ( ph -> ( A ^ ( M - N ) ) =
|
|
( ( A ^ M ) / ( A ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cz cmin co cexp cdiv wceq expsubap syl22anc ) ABI
|
|
JBKLMCNJDNJBCDOPQPBCQPBDQPRPSEFHGBCDTUA $.
|
|
$}
|
|
|
|
mulexpd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Distribution of square over multiplication. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
sqmuld $p |- ( ph -> ( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) $=
|
|
( cc wcel cmul co c2 cexp wceq sqmul syl2anc ) ABFGCFGBCHIJKIBJKICJKIHILD
|
|
EBCMN $.
|
|
|
|
${
|
|
sqdivapd.3 $e |- ( ph -> B # 0 ) $.
|
|
$( Distribution of square over division. (Contributed by Jim Kingdon,
|
|
13-Jun-2020.) $)
|
|
sqdivapd $p |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co c2 cexp wceq sqdivap syl3anc ) ABGHCGHCIJ
|
|
KBCLMNOMBNOMCNOMLMPDEFBCQR $.
|
|
|
|
expdivapd.3 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Nonnegative integer exponentiation of a quotient. (Contributed by Jim
|
|
Kingdon, 13-Jun-2020.) $)
|
|
expdivapd $p |- ( ph -> ( ( A / B ) ^ N ) =
|
|
( ( A ^ N ) / ( B ^ N ) ) ) $=
|
|
( cc wcel cc0 cap wbr cn0 cdiv co cexp wceq expdivap syl121anc ) ABIJCI
|
|
JCKLMDNJBCOPDQPBDQPCDQPOPREFGHBCDST $.
|
|
$}
|
|
|
|
mulexpd.3 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Positive integer exponentiation of a product. Proposition 10-4.2(c) of
|
|
[Gleason] p. 135, restricted to nonnegative integer exponents.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
mulexpd $p |- ( ph -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) $=
|
|
( cc wcel cn0 cmul co cexp wceq mulexp syl3anc ) ABHICHIDJIBCKLDMLBDMLCDM
|
|
LKLNEFGBCDOP $.
|
|
$}
|
|
|
|
${
|
|
0exp.1 $e |- ( ph -> N e. NN ) $.
|
|
$( Value of zero raised to a positive integer power. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
0expd $p |- ( ph -> ( 0 ^ N ) = 0 ) $=
|
|
( cn wcel cc0 cexp co wceq 0exp syl ) ABDEFBGHFICBJK $.
|
|
$}
|
|
|
|
${
|
|
reexpcld.1 $e |- ( ph -> A e. RR ) $.
|
|
reexpcld.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Closure of exponentiation of reals. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
reexpcld $p |- ( ph -> ( A ^ N ) e. RR ) $=
|
|
( cr wcel cn0 cexp co reexpcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
|
|
${
|
|
expge0d.3 $e |- ( ph -> 0 <_ A ) $.
|
|
$( Nonnegative integer exponentiation with a nonnegative mantissa is
|
|
nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
expge0d $p |- ( ph -> 0 <_ ( A ^ N ) ) $=
|
|
( cr wcel cn0 cc0 cle wbr cexp co expge0 syl3anc ) ABGHCIHJBKLJBCMNKLDE
|
|
FBCOP $.
|
|
$}
|
|
|
|
${
|
|
expge1d.3 $e |- ( ph -> 1 <_ A ) $.
|
|
$( Nonnegative integer exponentiation with a nonnegative mantissa is
|
|
nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
expge1d $p |- ( ph -> 1 <_ ( A ^ N ) ) $=
|
|
( cr wcel cn0 c1 cle wbr cexp co expge1 syl3anc ) ABGHCIHJBKLJBCMNKLDEF
|
|
BCOP $.
|
|
$}
|
|
$}
|
|
|
|
$( A squared odd number minus 1 divided by 8 is the odd number multiplied
|
|
with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) $)
|
|
sqoddm1div8 $p |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) )
|
|
-> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) ) $=
|
|
( wcel c2 cmul co c1 caddc wceq cexp cmin c8 cdiv c4 a1i zcnd oveq1d eqcomd
|
|
cc eqtrd cz wa oveq1 2z id zmulcld binom21 syl sylan9eqr zcn sqmuld sq2 w3a
|
|
mulass syl3anc 2t2e4 oveq12d 4z zsqcl addcld pncan1 adantr 4cn adddid 4t2e8
|
|
2cnd oveq2d zaddcld cc0 cap 2ap0 4ap0 divcanap5d sqvald mulid1d 1cnd 3eqtrd
|
|
wbr adddi ) BUACZADBEFZGHFZIZUBZADJFZGKFZLMFNBDJFZEFZNBEFZHFZLMFZNWGBHFZEFZ
|
|
LMFZBBGHFEFZDMFZWDWFWJLMWDWFWADJFZDWAEFZHFZGHFZGKFZWJWDWEWTGKWCVTWEWBDJFZWT
|
|
AWBDJUCVTWASCXBWTIVTWAVTDBDUACVTUDOVTUEZUFPWAUGUHUIQVTXAWJIWCVTXAWJGHFZGKFZ
|
|
WJVTWTXDGKVTWSWJGHVTWQWHWRWIHVTWQDDJFZWGEFWHVTDBVTVFZBUJZUKVTXFNWGEXFNIVTUL
|
|
OQTVTWRDDEFZBEFZWIVTDSCZXKBSCZWRXJIXGXGXHXKXKXLUMXJWRDDBUNRUOVTXINBEXINIVTU
|
|
POQTUQQQVTWJSCXEWJIVTWHWIVTWHVTNWGNUACVTUROZBUSZUFPVTWIVTNBXMXCUFPUTWJVAUHT
|
|
VBTQVTWKWNIWCVTWJWMLMVTWMWJVTNWGBNSCVTVCOZVTWGXNPXHVDRQVBVTWNWPIWCVTWNWMNDE
|
|
FZMFWLDMFWPVTLXPWMMVTXPLXPLIVTVEORVGVTWLDNVTWLVTWGBXNXCVHPXGXODVIVJVRVTVKON
|
|
VIVJVRVTVLOVMVTWLWODMVTWLBBEFZBHFXQBGEFZHFZWOVTWGXQBHVTBXHVNQVTBXRXQHVTXRBV
|
|
TBXHVORVGVTXLXLGSCZXSWOIXHXHVTVPXLXLXTUMWOXSBBGVSRUOVQQVQVBVQ $.
|
|
|
|
${
|
|
nnexpcld.1 $e |- ( ph -> A e. NN ) $.
|
|
$( The naturals are closed under squaring. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
nnsqcld $p |- ( ph -> ( A ^ 2 ) e. NN ) $=
|
|
( cn wcel c2 cexp co nnsqcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
nnexpcld.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Closure of exponentiation of nonnegative integers. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
nnexpcld $p |- ( ph -> ( A ^ N ) e. NN ) $=
|
|
( cn wcel cn0 cexp co nnexpcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
$}
|
|
|
|
${
|
|
nn0expcld.1 $e |- ( ph -> A e. NN0 ) $.
|
|
nn0expcld.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Closure of exponentiation of nonnegative integers. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
nn0expcld $p |- ( ph -> ( A ^ N ) e. NN0 ) $=
|
|
( cn0 wcel cexp co nn0expcl syl2anc ) ABFGCFGBCHIFGDEBCJK $.
|
|
$}
|
|
|
|
${
|
|
rpexpcld.1 $e |- ( ph -> A e. RR+ ) $.
|
|
rpexpcld.2 $e |- ( ph -> N e. ZZ ) $.
|
|
$( Closure law for exponentiation of positive reals. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
rpexpcld $p |- ( ph -> ( A ^ N ) e. RR+ ) $=
|
|
( crp wcel cz cexp co rpexpcl syl2anc ) ABFGCHGBCIJFGDEBCKL $.
|
|
$}
|
|
|
|
${
|
|
reexpclzapd.1 $e |- ( ph -> A e. RR ) $.
|
|
reexpclzapd.2 $e |- ( ph -> A # 0 ) $.
|
|
reexpclzapd.3 $e |- ( ph -> N e. ZZ ) $.
|
|
$( Closure of exponentiation of reals. (Contributed by Jim Kingdon,
|
|
13-Jun-2020.) $)
|
|
reexpclzapd $p |- ( ph -> ( A ^ N ) e. RR ) $=
|
|
( cr wcel cc0 cap wbr cz cexp co reexpclzap syl3anc ) ABGHBIJKCLHBCMNGHDE
|
|
FBCOP $.
|
|
$}
|
|
|
|
${
|
|
resqcld.1 $e |- ( ph -> A e. RR ) $.
|
|
$( Closure of square in reals. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
resqcld $p |- ( ph -> ( A ^ 2 ) e. RR ) $=
|
|
( cr wcel c2 cexp co resqcl syl ) ABDEBFGHDECBIJ $.
|
|
|
|
$( A square of a real is nonnegative. (Contributed by Mario Carneiro,
|
|
28-May-2016.) $)
|
|
sqge0d $p |- ( ph -> 0 <_ ( A ^ 2 ) ) $=
|
|
( cr wcel cc0 c2 cexp co cle wbr sqge0 syl ) ABDEFBGHIJKCBLM $.
|
|
|
|
${
|
|
sqgt0apd.2 $e |- ( ph -> A # 0 ) $.
|
|
$( The square of a real apart from zero is positive. (Contributed by Jim
|
|
Kingdon, 13-Jun-2020.) $)
|
|
sqgt0apd $p |- ( ph -> 0 < ( A ^ 2 ) ) $=
|
|
( cr wcel cc0 cap wbr c2 cexp co clt sqgt0ap syl2anc ) ABEFBGHIGBJKLMIC
|
|
DBNO $.
|
|
$}
|
|
|
|
${
|
|
leexp2ad.2 $e |- ( ph -> 1 <_ A ) $.
|
|
leexp2ad.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
$( Ordering relationship for exponentiation. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
leexp2ad $p |- ( ph -> ( A ^ M ) <_ ( A ^ N ) ) $=
|
|
( cr wcel c1 cle wbr cuz cfv cexp co leexp2a syl3anc ) ABHIJBKLDCMNIBCO
|
|
PBDOPKLEFGBCDQR $.
|
|
$}
|
|
|
|
${
|
|
leexp2rd.2 $e |- ( ph -> M e. NN0 ) $.
|
|
leexp2rd.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
leexp2rd.4 $e |- ( ph -> 0 <_ A ) $.
|
|
leexp2rd.5 $e |- ( ph -> A <_ 1 ) $.
|
|
$( Ordering relationship for exponentiation. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
leexp2rd $p |- ( ph -> ( A ^ N ) <_ ( A ^ M ) ) $=
|
|
( cr wcel cn0 cuz cfv cc0 cle wbr c1 cexp co leexp2r syl32anc ) ABJKCLK
|
|
DCMNKOBPQBRPQBDSTBCSTPQEFGHIBCDUAUB $.
|
|
$}
|
|
|
|
lt2sqd.2 $e |- ( ph -> B e. RR ) $.
|
|
lt2sqd.3 $e |- ( ph -> 0 <_ A ) $.
|
|
lt2sqd.4 $e |- ( ph -> 0 <_ B ) $.
|
|
$( The square function on nonnegative reals is strictly monotonic.
|
|
(Contributed by Mario Carneiro, 28-May-2016.) $)
|
|
lt2sqd $p |- ( ph -> ( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr clt c2 cexp co wb lt2sq syl22anc ) ABHIJBKLCHIJCKLB
|
|
CMLBNOPCNOPMLQDFEGBCRS $.
|
|
|
|
$( The square function on nonnegative reals is monotonic. (Contributed by
|
|
Mario Carneiro, 28-May-2016.) $)
|
|
le2sqd $p |- ( ph -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr c2 cexp co wb le2sq syl22anc ) ABHIJBKLCHIJCKLBCKLB
|
|
MNOCMNOKLPDFEGBCQR $.
|
|
|
|
sq11d.5 $e |- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) ) $.
|
|
$( The square function is one-to-one for nonnegative reals. (Contributed
|
|
by Mario Carneiro, 28-May-2016.) $)
|
|
sq11d $p |- ( ph -> A = B ) $=
|
|
( c2 cexp co wceq cr wcel cc0 cle wbr wb sq11 syl22anc mpbid ) ABIJKCIJKL
|
|
ZBCLZHABMNOBPQCMNOCPQUBUCRDFEGBCSTUA $.
|
|
$}
|
|
|
|
$( Analogue to ~ sq11 but for apartness. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
sq11ap $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( A ^ 2 ) # ( B ^ 2 ) <-> A # B ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt wo c2 cexp co cap lt2sq wb ancoms reaplt simpl
|
|
resqcld orbi12d ad2ant2r syl2an 3bitr4rd ) ACDZEAFGZHZBCDZEBFGZHZHZABIGZBAI
|
|
GZJZAKLMZBKLMZIGZUPUOIGZJZABNGZUOUPNGZUKULUQUMURABOUJUGUMURPBAOQUAUEUHUTUNP
|
|
UFUIABRUBUGUOCDUPCDVAUSPUJUGAUEUFSTUJBUHUISTUOUPRUCUD $.
|
|
|
|
$( The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by
|
|
AV, 1-Aug-2021.) $)
|
|
sq10 $p |- ( ; 1 0 ^ 2 ) = ; ; 1 0 0 $=
|
|
( c1 cc0 cdc c2 cexp cmul 1nn0 0nn0 deccl nn0cni sqvali eqid mulid2i mul02i
|
|
co decmul1 eqtri ) ABCZDEORRFORBCRRABGHIZJZKABRBRRSGHRLHRTMRTNPQ $.
|
|
|
|
$( The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.)
|
|
(Revised by AV, 1-Aug-2021.) $)
|
|
sq10e99m1 $p |- ( ; 1 0 ^ 2 ) = ( ; 9 9 + 1 ) $=
|
|
( c1 cc0 cdc c2 cexp co c9 caddc sq10 9nn0 9p1e10 eqid decsucc eqtr4i ) ABC
|
|
ZDEFOBCGGCZAHFIGOPJKPLMN $.
|
|
|
|
${
|
|
3dec.a $e |- A e. NN0 $.
|
|
3dec.b $e |- B e. NN0 $.
|
|
$( A "decimal constructor" which is used to build up "decimal integers" or
|
|
"numeric terms" in base 10 with 3 "digits". (Contributed by AV,
|
|
14-Jun-2021.) (Revised by AV, 1-Aug-2021.) $)
|
|
3dec $p |- ; ; A B C = ( ( ( ( ; 1 0 ^ 2 ) x. A )
|
|
+ ( ; 1 0 x. B ) ) + C ) $=
|
|
( cdc c1 cc0 cmul co caddc c2 cexp dfdec10 oveq2i 1nn nn0cni eqtri eqcomi
|
|
oveq1i decnncl2 nncni mulcli adddii mulassi sqvali ) ABFZCFGHFZUGIJZCKJUH
|
|
LMJZAIJZUHBIJZKJZCKJUGCNUIUMCKUIUHUHAIJZIJZULKJZUMUIUHUNBKJZIJUPUGUQUHIAB
|
|
NOUHUNBUHGPUAUBZUHAURADQZUCBEQUDRUOUKULKUOUHUHIJZAIJZUKVAUOUHUHAURURUSUES
|
|
UTUJAIUJUTUHURUFSTRTRTR $.
|
|
$}
|
|
|
|
${
|
|
expcanlem.a $e |- ( ph -> A e. RR ) $.
|
|
expcanlem.m $e |- ( ph -> M e. ZZ ) $.
|
|
expcanlem.n $e |- ( ph -> N e. ZZ ) $.
|
|
expcanlem.gt1 $e |- ( ph -> 1 < A ) $.
|
|
$( Lemma for ~ expcan . Proving the order in one direction. (Contributed
|
|
by Jim Kingdon, 29-Jan-2022.) $)
|
|
expcanlem $p |- ( ph -> ( ( A ^ M ) <_ ( A ^ N ) -> M <_ N ) ) $=
|
|
( cexp co clt wbr wn cle wcel cz c1 cc0 reexpclzapd lenltd cr w3a ltexp2a
|
|
wi expr syl31anc con3d 0red 1red 0lt1 a1i lttrd gt0ap0d zred 3imtr4d ) AB
|
|
DIJZBCIJZKLZMDCKLZMUQUPNLCDNLAUSURABUAOZDPOZCPOZQBKLZUSURUDEGFHUTVAVBUBVC
|
|
USURBDCUCUEUFUGAUQUPABCEABEARQBAUHAUIERQKLAUJUKHULUMZFSABDEVDGSTACDACFUNA
|
|
DGUNTUO $.
|
|
$}
|
|
|
|
$( Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.)
|
|
(Revised by Mario Carneiro, 4-Jun-2014.) $)
|
|
expcan $p |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\
|
|
1 < A ) -> ( ( A ^ M ) = ( A ^ N ) <-> M = N ) ) $=
|
|
( cr wcel cz w3a c1 clt wbr cexp wceq cle expcanlem cc0 reexpclzapd letri3d
|
|
wa co zred simpl1 simpl2 simpl3 simpr anim12d 0red 1red a1i gt0ap0d 3imtr4d
|
|
0lt1 lttrd oveq2 impbid1 ) ADEZBFEZCFEZGZHAIJZRZABKSZACKSZLZBCLZUTVAVBMJZVB
|
|
VAMJZRBCMJZCBMJZRVCVDUTVEVGVFVHUTABCUOUPUQUSUAZUOUPUQUSUBZUOUPUQUSUCZURUSUD
|
|
ZNUTACBVIVKVJVLNUEUTVAVBUTABVIUTAVIUTOHAUTUFUTUGVIOHIJUTUKUHVLULUIZVJPUTACV
|
|
IVMVKPQUTBCUTBVJTUTCVKTQUJBCAKUMUN $.
|
|
|
|
${
|
|
expcand.1 $e |- ( ph -> A e. RR ) $.
|
|
expcand.2 $e |- ( ph -> M e. ZZ ) $.
|
|
expcand.3 $e |- ( ph -> N e. ZZ ) $.
|
|
expcand.4 $e |- ( ph -> 1 < A ) $.
|
|
expcand.5 $e |- ( ph -> ( A ^ M ) = ( A ^ N ) ) $.
|
|
$( Ordering relationship for exponentiation. (Contributed by Mario
|
|
Carneiro, 28-May-2016.) $)
|
|
expcand $p |- ( ph -> M = N ) $=
|
|
( cexp co wceq cr wcel cz c1 clt wbr wb expcan syl31anc mpbid ) ABCJKBDJK
|
|
LZCDLZIABMNCONDONPBQRUCUDSEFGHBCDTUAUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered pair theorem for nonnegative integers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
nn0le2msqd.1 $e |- ( ph -> A e. NN0 ) $.
|
|
nn0le2msqd.2 $e |- ( ph -> B e. NN0 ) $.
|
|
$( The square function on nonnegative integers is monotonic. (Contributed
|
|
by Jim Kingdon, 31-Oct-2021.) $)
|
|
nn0le2msqd $p |- ( ph -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) ) $=
|
|
( cle wbr c2 cexp cmul nn0red nn0ge0d le2sqd nn0cnd sqvald breq12d bitrd
|
|
co ) ABCFGBHIRZCHIRZFGBBJRZCCJRZFGABCABDKACEKABDLACELMASUATUBFABABDNOACAC
|
|
ENOPQ $.
|
|
$}
|
|
|
|
${
|
|
nn0opthlem1d.1 $e |- ( ph -> A e. NN0 ) $.
|
|
nn0opthlem1d.2 $e |- ( ph -> C e. NN0 ) $.
|
|
$( A rather pretty lemma for ~ nn0opth2 . (Contributed by Jim Kingdon,
|
|
31-Oct-2021.) $)
|
|
nn0opthlem1d $p |- ( ph ->
|
|
( A < C <-> ( ( A x. A ) + ( 2 x. A ) ) < ( C x. C ) ) ) $=
|
|
( c1 caddc co cle wbr cmul clt cn0 wcel a1i syl2anc nn0mulcld cexp sqvald
|
|
c2 1nn0 nn0addcld nn0le2msqd wb nn0ltp1le 2nn0 cc wceq nn0cnd 1cnd binom2
|
|
addcld oveq1d oveq12d 3eqtr3d mulid1d oveq2d eqtrd breq1d bitr4d 3bitr4d
|
|
) ABFGHZCIJZVBVBKHZCCKHZIJZBCLJZBBKHZTBKHZGHZVELJZAVBCABFDFMNAUAOUBEUCABM
|
|
NCMNVGVCUDDEBCUEPAVKVJFGHZVEIJZVFAVJMNVEMNVKVMUDAVHVIABBDDQATBTMNAUFODQUB
|
|
ACCEEQVJVEUEPAVDVLVEIAVDVHTBFKHZKHZGHZFFKHZGHZVLAVBTRHZBTRHZVOGHZFTRHZGHZ
|
|
VDVRABUGNFUGNVSWCUHABDUIZAUJZBFUKPAVBABFWDWEULSAWAVPWBVQGAVTVHVOGABWDSUMA
|
|
FWESUNUOAVPVJVQFGAVOVIVHGAVNBTKABWDUPUQUQAFWEUPUNURUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
nn0opthd.1 $e |- ( ph -> A e. NN0 ) $.
|
|
nn0opthd.2 $e |- ( ph -> B e. NN0 ) $.
|
|
nn0opthd.3 $e |- ( ph -> C e. NN0 ) $.
|
|
nn0opthd.4 $e |- ( ph -> D e. NN0 ) $.
|
|
$( Lemma for ~ nn0opth2 . (Contributed by Jim Kingdon, 31-Oct-2021.) $)
|
|
nn0opthlem2d $p |- ( ph -> ( ( A + B ) < C
|
|
-> ( ( C x. C ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) ) $=
|
|
( caddc co wbr cmul cr wcel nn0red remulcld readdcld adantr cle nn0addcld
|
|
clt wne wa 2re a1i cn0 nn0addge2 syl2anc nn0addge1 recnd 2timesd breqtrrd
|
|
c2 letrd leadd2dd nn0opthlem1d biimpa lelttrd ltletrd gtned ex ) ABCJKZDU
|
|
BLZDDMKZEJKZVCVCMKZCJKZUCAVDUDZVHVFAVHNOVDAVGCAVCVCAVCABCFGUAZPZVKQZACGPZ
|
|
RSZVIVHVEVFVNAVENOZVDADDADHPZVPQZSZAVFNOVDAVEEVQAEIPRSVIVHVGUNVCMKZJKZVEV
|
|
NAVTNOVDAVGVSVLAUNVCUNNOAUEUFVKQZRSVRAVHVTTLVDACVSVGVMWAVLACVCVSVMVKWAACN
|
|
OBUGOCVCTLVMFCBUHUIAVCVCVCJKZVSTAVCNOVCUGOVCWBTLVKVJVCVCUJUIAVCAVCVKUKULU
|
|
MUOUPSAVDVTVEUBLAVCDVJHUQURUSAVEVFTLZVDAVOEUGOWCVQIVEEUJUISUTVAVB $.
|
|
|
|
$( An ordered pair theorem for nonnegative integers. Theorem 17.3 of
|
|
[Quine] p. 124. We can represent an ordered pair of nonnegative
|
|
integers ` A ` and ` B ` by ` ( ( ( A + B ) x. ( A + B ) ) + B ) ` . If
|
|
two such ordered pairs are equal, their first elements are equal and
|
|
their second elements are equal. Contrast this ordered pair
|
|
representation with the standard one ~ df-op that works for any set.
|
|
(Contributed by Jim Kingdon, 31-Oct-2021.) $)
|
|
nn0opthd $p |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) =
|
|
( ( ( C + D ) x. ( C + D ) ) + D ) <-> ( A = C /\ B = D ) ) ) $=
|
|
( caddc co cmul wceq wbr nn0addcld wcel wb syl2anc cc nn0cnd wa cap wn wo
|
|
clt wne nn0opthlem2d imp necomd ex cr nn0red reaplt nn0mulcld nn0zd zapne
|
|
jaod 3imtr4d con3d apti simpr oveq12d oveq1d eqtr4d mulcld addcand adantr
|
|
cz mpbid oveq2d addcan2d jca oveq12 impbid1 ) ABCJKZVOLKZCJKZDEJKZVRLKZEJ
|
|
KZMZBDMZCEMZUAZAWAWDAWAUAZWBWCWEVODCJKZMZWBWEVOVRWFAWAVOVRMZAVQVTUBNZUCZV
|
|
OVRUBNZUCZWAWHAWKWIAVOVRUENZVRVOUENZUDZVQVTUFZWKWIAWMWPWNAWMWPAWMUAVTVQAW
|
|
MVTVQUFABCVREFGADEHIOZIUGUHUIUJADEVOCHIABCFGOZGUGUQAVOUKPVRUKPWKWOQAVOWRU
|
|
LAVRWQULVOVRUMRAVQVHPVTVHPWIWPQAVQAVPCAVOVOWRWRUNGOZUOAVTAVSEAVRVRWQWQUNI
|
|
OZUOVQVTUPRURUSAVQSPVTSPWAWJQAVQWSTAVTWTTVQVTUTRAVOSPVRSPWHWLQAVOWRTZAVRW
|
|
QTVOVRUTRURUHZWECEDJWEVQVPEJKZMZWCWEVQVTXCAWAVAWEVPVSEJWEVOVRVOVRLXBXBVBV
|
|
CVDAXDWCQWAAVPCEAVOVOXAXAVEACGTZAEITVFVGVIZVJVDAWGWBQWAABDCABFTADHTXEVKVG
|
|
VIXFVLUJWDVPVSCEJWDVOVRVOVRLBDCEJVMZXGVBWBWCVAVBVN $.
|
|
|
|
$( An ordered pair theorem for nonnegative integers. Theorem 17.3 of
|
|
[Quine] p. 124. See comments for ~ nn0opthd . (Contributed by Jim
|
|
Kingdon, 31-Oct-2021.) $)
|
|
nn0opth2d $p |- ( ph ->
|
|
( ( ( ( A + B ) ^ 2 ) + B ) = ( ( ( C + D ) ^ 2 ) + D ) <->
|
|
( A = C /\ B = D ) ) ) $=
|
|
( caddc co c2 cexp wceq cmul wa nn0addcld nn0cnd sqvald oveq1d nn0opthd
|
|
eqeq12d bitrd ) ABCJKZLMKZCJKZDEJKZLMKZEJKZNUDUDOKZCJKZUGUGOKZEJKZNBDNCEN
|
|
PAUFUKUIUMAUEUJCJAUDAUDABCFGQRSTAUHULEJAUGAUGADEHIQRSTUBABCDEFGHIUAUC $.
|
|
$}
|
|
|
|
$( An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine]
|
|
p. 124. See ~ nn0opthd . (Contributed by NM, 22-Jul-2004.) $)
|
|
nn0opth2 $p |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( C e. NN0 /\ D e. NN0 ) ) ->
|
|
( ( ( ( A + B ) ^ 2 ) + B ) = ( ( ( C + D ) ^ 2 ) + D ) <->
|
|
( A = C /\ B = D ) ) ) $=
|
|
( cn0 wcel wa simpll simplr simprl simprr nn0opth2d ) AEFZBEFZGZCEFZDEFZGZG
|
|
ABCDMNRHMNRIOPQJOPQKL $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Factorial function
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ! $. $( Factorial. $)
|
|
|
|
$( Extend class notation to include the factorial of nonnegative integers. $)
|
|
cfa $a class ! $.
|
|
|
|
$( Define the factorial function on nonnegative integers. For example,
|
|
` ( ! `` 5 ) = 1 2 0 ` because ` 1 x. 2 x. 3 x. 4 x. 5 = 1 2 0 `
|
|
( ~ ex-fac ). In the literature, the factorial function is written as a
|
|
postscript exclamation point. (Contributed by NM, 2-Dec-2004.) $)
|
|
df-fac $a |- ! = ( { <. 0 , 1 >. } u. seq 1 ( x. , _I , CC ) ) $.
|
|
|
|
${
|
|
$d f g $.
|
|
$( Value of the factorial function for positive integers. (Contributed by
|
|
NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) $)
|
|
facnn $p |- ( N e. NN ->
|
|
( ! ` N ) = ( seq 1 ( x. , _I , CC ) ` N ) ) $=
|
|
( vf vg cfa cfv cmul cc cid c1 wceq cn0 cc0 csn cun cres dfn2 eqtr3i wtru
|
|
cn wcel cseq cdif c0ex 1ex cop cuz nnuz reseq2i wfn cvv 1zzd cnex a1i fvi
|
|
df-fac cv eleq1d ibir eluzelcn syl adantl wa co mulcl iseqfn trud fnresdm
|
|
ax-mp uneq2i eqtr4i fvsnun2 eleq2s ) ADEAFGHIUAZEJAKLMUBZSLIKAVMDUCUDDLIU
|
|
EMZVMNVOVMVNOZNUOVPVMVOVMIUFEZOZVPVMVQVNVMSVQVNUGPQUHVMVQUIZVRVMJVSRBCFGH
|
|
IUJRUKGUJTRULUMBUPZVQTZVTHEZGTZRWAWBVQTZWCWAWDWAWBVTVQVTVQUNUQURIWBUSUTVA
|
|
VTGTCUPZGTVBVTWEFVCGTRVTWEVDVAVEVFVQVMVGVHQVIVJVKPVL $.
|
|
$}
|
|
|
|
${
|
|
$d f g $.
|
|
$( The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario
|
|
Carneiro, 13-Jul-2013.) $)
|
|
fac0 $p |- ( ! ` 0 ) = 1 $=
|
|
( vf vg cc0 c1 cn0 cmul cc cid cseq cfa csn cun cres cfv eqtr3i wtru wcel
|
|
cvv cv adantl c0ex 1ex cop cdif df-fac cuz cn nnuz dfn2 reseq2i wceq 1zzd
|
|
wfn cnex a1i fvi eleq1d ibir eluzelcn syl wa co mulcl iseqfn trud fnresdm
|
|
ax-mp uneq2i eqtr4i fvsnun1 ) CDEFGHDIZJUAUBJCDUCKZVKLVLVKECKUDZMZLUEVNVK
|
|
VLVKDUFNZMZVNVKVOVMVKUGVOVMUHUIOUJVKVOUMZVPVKUKVQPABFGHDRPULGRQPUNUOASZVO
|
|
QZVRHNZGQZPVSVTVOQZWAVSWBVSVTVRVOVRVOUPUQURDVTUSUTTVRGQBSZGQVAVRWCFVBGQPV
|
|
RWCVCTVDVEVOVKVFVGOVHVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d f g $.
|
|
$( The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario
|
|
Carneiro, 13-Jul-2013.) $)
|
|
fac1 $p |- ( ! ` 1 ) = 1 $=
|
|
( vf vg c1 cfa cfv cmul cc cid cseq cn wcel wceq facnn ax-mp wtru 1zzd cv
|
|
1nn fvi adantl cuz eleq1d ibir eluzelcn syl wa co mulcl iseq1 trud 3eqtri
|
|
) CDEZCFGHCIEZCHEZCCJKZULUMLRCMNUMUNLOABFGHCOPAQZCUAEZKZUPHEZGKZOURUSUQKZ
|
|
UTURVAURUSUPUQUPUQSUBUCCUSUDUETUPGKBQZGKUFUPVBFUGGKOUPVBUHTUIUJUOUNCLRCJS
|
|
NUK $.
|
|
$}
|
|
|
|
${
|
|
$d N f g $.
|
|
$( The factorial of a successor. (Contributed by NM, 2-Dec-2004.)
|
|
(Revised by Mario Carneiro, 13-Jul-2013.) $)
|
|
facp1 $p |- ( N e. NN0 ->
|
|
( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) ) $=
|
|
( vf vg wcel cn cc0 wceq c1 caddc co cfa cfv cmul cc cid fvi adantl facnn
|
|
cv syl cn0 wo elnn0 cseq elnnuz biimpi eluzelcn eqeltrd wa mulcl peano2nn
|
|
cuz iseqp1 oveq2d eqtrd oveq1d 0p1e1 fveq2i fac1 eqtri oveq1 fveq2d fveq2
|
|
3eqtr4d oveq12d fac0 oveq12i 1t1e1 syl6eq 3eqtr4a jaoi sylbi ) AUADAEDZAF
|
|
GZUBAHIJZKLZAKLZVOMJZGZAUCVMVSVNVMVOMNOHUDZLZAVTLZVOMJZVPVRVMWAWBVOOLZMJW
|
|
CVMBCMNOHAVMAHULLZDAUEUFBSZWEDZWFOLZNDVMWGWHWFNWFWEPHWFUGUHQWFNDCSZNDUIWF
|
|
WIMJNDVMWFWIUJQUMVMWDVOWBMVMVOEDZWDVOGAUKZVOEPTUNUOVMWJVPWAGWKVORTVMVQWBV
|
|
OMARUPVDVNFHIJZKLZHVPVRWMHKLHWLHKUQURUSUTVNVOWLKAFHIVAZVBVNVRFKLZWLMJZHVN
|
|
VQWOVOWLMAFKVCWNVEWPHHMJHWOHWLHMVFUQVGVHUTVIVJVKVL $.
|
|
$}
|
|
|
|
$( The factorial of 2. (Contributed by NM, 17-Mar-2005.) $)
|
|
fac2 $p |- ( ! ` 2 ) = 2 $=
|
|
( c2 cfa cfv c1 caddc df-2 fveq2i cmul cn0 wcel wceq 1nn0 facp1 ax-mp 1p1e2
|
|
co fac1 oveq12i 2cn eqtri mulid2i ) ABCDDEPZBCZAAUBBFGUCDBCZUBHPZADIJUCUEKL
|
|
DMNUEDAHPAUDDUBAHQORASUATTT $.
|
|
|
|
$( The factorial of 3. (Contributed by NM, 17-Mar-2005.) $)
|
|
fac3 $p |- ( ! ` 3 ) = 6 $=
|
|
( c3 cfa cfv c2 c1 caddc co cmul df-3 fveq2i cn0 wcel wceq 2nn0 facp1 ax-mp
|
|
c6 fac2 2p1e3 3eqtri oveq12i 2cn 3cn mulcomi 3t2e6 ) ABCDEFGZBCZDBCZUFHGZQA
|
|
UFBIJDKLUGUIMNDOPUIDAHGADHGQUHDUFAHRSUADAUBUCUDUETT $.
|
|
|
|
$( The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) $)
|
|
fac4 $p |- ( ! ` 4 ) = ; 2 4 $=
|
|
( c3 c1 caddc co cfa cfv cmul c4 cdc cn0 wcel wceq facp1 ax-mp 3p1e4 fveq2i
|
|
c2 3nn0 c6 fac3 oveq12i 6t4e24 eqtri 3eqtr3i ) ABCDZEFZAEFZUEGDZHEFQHIZAJKU
|
|
FUHLRAMNUEHEOPUHSHGDUIUGSUEHGTOUAUBUCUD $.
|
|
|
|
$( Value of the factorial function expressed recursively. (Contributed by
|
|
NM, 2-Dec-2004.) $)
|
|
facnn2 $p |- ( N e. NN ->
|
|
( ! ` N ) = ( ( ! ` ( N - 1 ) ) x. N ) ) $=
|
|
( cn wcel cc c1 cmin co cn0 wa cfa cfv cmul wceq elnnnn0 caddc facp1 adantl
|
|
npcan1 fveq2d adantr oveq2d 3eqtr3d sylbi ) ABCADCZAEFGZHCZIZAJKZUEJKZALGZM
|
|
ANUGUEEOGZJKZUIUKLGZUHUJUFULUMMUDUEPQUDULUHMUFUDUKAJARZSTUDUMUJMUFUDUKAUILU
|
|
NUATUBUC $.
|
|
|
|
${
|
|
$d j k N $.
|
|
$( Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) $)
|
|
faccl $p |- ( N e. NN0 -> ( ! ` N ) e. NN ) $=
|
|
( vj vk cv cfa cfv cn wcel cc0 c1 caddc co wceq fveq2 eleq1d fac0 eqeltri
|
|
1nn cn0 wa cmul facp1 adantl nn0p1nn nnmulcl sylan2 eqeltrd expcom nn0ind
|
|
) BDZEFZGHIEFZGHCDZEFZGHZUMJKLZEFZGHZAEFZGHBCAUJIMUKULGUJIENOUJUMMUKUNGUJ
|
|
UMENOUJUPMUKUQGUJUPENOUJAMUKUSGUJAENOULJGPRQUOUMSHZURUOUTTUQUNUPUALZGUTUQ
|
|
VAMUOUMUBUCUTUOUPGHVAGHUMUDUNUPUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
faccld.1 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Closure of the factorial function, deduction version of ~ faccl .
|
|
(Contributed by Glauco Siliprandi, 5-Apr-2020.) $)
|
|
faccld $p |- ( ph -> ( ! ` N ) e. NN ) $=
|
|
( cn0 wcel cfa cfv cn faccl syl ) ABDEBFGHECBIJ $.
|
|
$}
|
|
|
|
$( The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) $)
|
|
facne0 $p |- ( N e. NN0 -> ( ! ` N ) =/= 0 ) $=
|
|
( cn0 wcel cfa cfv faccl nnne0d ) ABCADEAFG $.
|
|
|
|
${
|
|
$d j M $. $d j k N $.
|
|
$( A positive integer divides the factorial of an equal or larger number.
|
|
(Contributed by NM, 2-May-2005.) $)
|
|
facdiv $p |- ( ( M e. NN0 /\ N e. NN /\ N <_ M ) ->
|
|
( ( ! ` M ) / N ) e. NN ) $=
|
|
( wcel cn cle wbr cfa cfv cdiv co wi cc0 wceq breq2 oveq1d eleq1d imbi12d
|
|
fveq2 imbi2d cmul vj vk cn0 cv c1 caddc clt wn nngt0 cz wb 0z nnz sylancr
|
|
zltnle mpbid pm2.21d wa wo peano2nn0 nn0zd zleloe nnnn0 nn0leltp1 nn0p1nn
|
|
syl2an sylan nnmulcl sylan2 expcom adantl cc faccl nn0cnd nncn adantr cap
|
|
nncnd nnap0 div23apd sylibrd imim2d com23 sylbird eqeltrd oveq2 syl5ibcom
|
|
divcanap4d a1dd jaod sylbid com34 com12 imp4d facp1 exp4d a2d nn0ind 3imp
|
|
ex ) AUCCBDCZBAEFZAGHZBIJZDCZXABUAUDZEFZXFGHZBIJZDCZKZKXABLEFZLGHZBIJZDCZ
|
|
KZKXABUBUDZEFZXQGHZBIJZDCZKZKXABXQUEUFJZEFZYCGHZBIJZDCZKZKXAXBXEKZKUAUBAX
|
|
FLMZXKXPXAYJXGXLXJXOXFLBENYJXIXNDYJXHXMBIXFLGROPQSXFXQMZXKYBXAYKXGXRXJYAX
|
|
FXQBENYKXIXTDYKXHXSBIXFXQGROPQSXFYCMZXKYHXAYLXGYDXJYGXFYCBENYLXIYFDYLXHYE
|
|
BIXFYCGROPQSXFAMZXKYIXAYMXGXBXJXEXFABENYMXIXDDYMXHXCBIXFAGROPQSXAXLXOXALB
|
|
UGFZXLUHZBUIXALUJCBUJCZYNYOUKULBUMZLBUOUNUPUQXQUCCZXAYBYHYRXAYBYDYGYRXAYB
|
|
YDURURXSYCTJZBIJZDCZYGYRXAYBYDUUAXAYRYBYDUUAKKXAYRYDYBUUAXAYRYDYBUUAKZKXA
|
|
YRURZYDBYCUGFZBYCMZUSZUUBXAYPYCUJCYDUUFUKYRYQYRYCXQUTZVABYCVBVFUUCUUDUUBU
|
|
UEUUCUUDXRUUBXABUCCYRXRUUDUKBVCBXQVDVGUUCYBXRUUAUUCYAUUAXRUUCYAXTYCTJZDCZ
|
|
UUAYRYAUUIKXAYAYRUUIYRYAYCDCUUIXQVEXTYCVHVIVJVKUUCYTUUHDUUCXSYCBYRXSVLCXA
|
|
YRXSXQVMZVRVKZYRYCVLCXAYRYCUUGVNVKXABVLCYRBVOVPZXABLVQFYRBVSVPZVTPWAWBWCW
|
|
DUUCUUEUUAYBUUCXSBTJZBIJZDCUUEUUAUUCUUOXSDUUCXSBUUKUULUUMWHYRXSDCXAUUJVKW
|
|
EUUEUUOYTDUUEUUNYSBIBYCXSTWFOPWGWIWJWKWTWLWMWNYRYFYTDYRYEYSBIXQWOOPWAWPWQ
|
|
WRWS $.
|
|
$}
|
|
|
|
$( No positive integer (greater than one) divides the factorial plus one of
|
|
an equal or larger number. (Contributed by NM, 3-May-2005.) $)
|
|
facndiv $p |- ( ( ( M e. NN0 /\ N e. NN ) /\ ( 1 < N /\ N <_ M ) ) ->
|
|
-. ( ( ( ! ` M ) + 1 ) / N ) e. ZZ ) $=
|
|
( cn0 wcel cn wa c1 clt wbr cle cfa cfv caddc co cdiv cz cmin ad2antrr wceq
|
|
cc wn cr nnre recnz sylan ad2ant2lr facdiv 3expa nnzd adantrl syl5com faccl
|
|
zsubcl nncnd peano2cn syl ad2antlr simplr nnap0d divsubdirapd ax-1cn pncan2
|
|
ex nncn sylancl oveq1d eqtr3d eleq1d sylibd mtod ) ACDZBEDZFZGBHIZBAJIZFZFZ
|
|
AKLZGMNZBONZPDZGBONZPDZVLVNWCUAZVKVOVLBUBDVNWDBUCBUDUEUFVQWAVTVRBONZQNZPDZW
|
|
CVQWEPDZWAWGVMVOWHVNVMVOFWEVKVLVOWEEDABUGUHUIUJWAWHWGVTWEUMVCUKVQWFWBPVQVSV
|
|
RQNZBONZWFWBVQVSVRBVKVSTDZVLVPVKVRTDZWKVKVRAULUNZVRUOUPRVKWLVLVPWMRVLBTDVKV
|
|
PBVDUQVQBVKVLVPURUSUTVKWJWBSVLVPVKWIGBOVKWLGTDWIGSWMVAVRGVBVEVFRVGVHVIVJ $.
|
|
|
|
${
|
|
$d j k M $. $d j N $.
|
|
$( Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) $)
|
|
facwordi $p |- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) ->
|
|
( ! ` M ) <_ ( ! ` N ) ) $=
|
|
( cn0 wcel cle wbr cfa cfv wa wi cc0 breq2 anbi2d fveq2 breq2d imbi12d cr
|
|
wceq adantl nnred vj vk cv caddc nn0le0eq0 biimpa fveq2d fac0 1re eqeltri
|
|
c1 co leidi syl6eqbr impexp wo cz wb simpl nn0zd peano2nn0 zleloe syl2anc
|
|
clt nn0leltp1 cmul faccl nn0re peano2re syl nnnn0d nn0ge0d nn0p1nn nnge1d
|
|
lemulge11d facp1 breqtrrd adantr faccld letr syl3anc mpan2d com23 sylbird
|
|
imim2d leidd syl5ibcom syl5 a1dd jaod sylbid com13 com4l a2d imp4a syl5bi
|
|
ex nn0ind 3impib 3com12 ) BCDZACDZABEFZAGHZBGHZEFZXAXBXCXFXBAUAUCZEFZIZXD
|
|
XGGHZEFZJXBAKEFZIZXDKGHZEFZJXBAUBUCZEFZIZXDXPGHZEFZJZXBAXPUKUDULZEFZIZXDY
|
|
BGHZEFZJZXBXCIZXFJUAUBBXGKRZXIXMXKXOYIXHXLXBXGKAELMYIXJXNXDEXGKGNOPXGXPRZ
|
|
XIXRXKXTYJXHXQXBXGXPAELMYJXJXSXDEXGXPGNOPXGYBRZXIYDXKYFYKXHYCXBXGYBAELMYK
|
|
XJYEXDEXGYBGNOPXGBRZXIYHXKXFYLXHXCXBXGBAELMYLXJXEXDEXGBGNOPXMXDXNXNEXMAKG
|
|
XBXLAKRAUEUFUGXNXNUKQUHUIUJUMUNYAXBXQXTJZJZXPCDZYGXBXQXTUOYOYNXBYCYFYOXBY
|
|
MYCYFJYCYOXBYMYFXBYOYCYMYFJZXBYOYCYPJXBYOIZYCAYBVDFZAYBRZUPZYPYQAUQDYBUQD
|
|
YCYTURYQAXBYOUSUTYQYBYOYBCDXBXPVAZSUTAYBVBVCYQYRYPYSYQYRXQYPAXPVEYQYMXQYF
|
|
YQXTYFXQYQXTXSYEEFZYFYOUUBXBYOXSXSYBVFULYEEYOXSYBYOXSXPVGZTZYOXPQDYBQDXPV
|
|
HXPVIVJYOXSYOXSUUCVKVLYOYBXPVMVNVOXPVPVQSYQXDQDZXSQDZYEQDZXTUUBIYFJXBUUEY
|
|
OXBXDAVGTZVRYOUUFXBUUDSYOUUGXBYOYEYOYBUUAVSTSXDXSYEVTWAWBWEWCWDYQYSYFYMXB
|
|
YSYFJYOYSXDYERZXBYFAYBGNXBXDXDEFUUIYFXBXDUUHWFXDYEXDELWGWHVRWIWJWKWQWLWMW
|
|
NWOWPWRWSWT $.
|
|
|
|
$( A lower bound for the factorial function. (Contributed by NM,
|
|
17-Dec-2005.) $)
|
|
faclbnd $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M ^ ( N + 1 ) ) <_ ( ( M ^ M ) x. ( ! ` N ) ) ) $=
|
|
( cn0 wcel cc0 wceq c1 caddc co cexp cfa cfv cmul cle wbr wi oveq2d wa cr
|
|
adantr vj vk cn wo elnn0 oveq1 fveq2 breq12d imbi2d nnre nnge1 cuz elnnuz
|
|
biimpi leexp2ad 0p1e1 oveq2i a1i fac0 nnnn0 reexpcld recnd mulid1d syl5eq
|
|
cv 3brtr4d clt ad3antrrr simpllr peano2nn0 faccld nnred remulcld peano2re
|
|
syl nn0re 3syl 0re ltle mpan sylc expge0d simplr simprr lemul12ad anandis
|
|
nngt0 cc nncn expp1 syl2an facp1 adantl faccl nncnd nn0cn peano2cn eqtr4d
|
|
mulassd exp32 com23 wb syl2anr reexpcl ad2antrr remulcl simpr cz ad2antlr
|
|
nn0ltp1le nn0zd nnz eluz syl2anc mpbird anim12i id nn0ge0 lemulge11 letrd
|
|
nnge1d ex sylbid zlelttric mpjaod expcom a2d nn0ind impcom nnnn0d nn0ge0d
|
|
a1dd nn0p1nn 0expd 0exp0e1 oveq1i mulid2d oveq12 anidms oveq1d imp jaoian
|
|
syl5ibr sylanb ) ACDZAUCDZAEFZUDBCDZABGHIZJIZAAJIZBKLZMIZNOZAUEUUFUUHUUNU
|
|
UGUUHUUFUUNUUFAUAVEZGHIZJIZUUKUUOKLZMIZNOZPUUFAEGHIZJIZUUKEKLZMIZNOZPUUFA
|
|
UBVEZGHIZJIZUUKUVFKLZMIZNOZPUUFAUVGGHIZJIZUUKUVGKLZMIZNOZPUUFUUNPUAUBBUUO
|
|
EFZUUTUVEUUFUVQUUQUVBUUSUVDNUVQUUPUVAAJUUOEGHUFQUVQUURUVCUUKMUUOEKUGQUHUI
|
|
UUOUVFFZUUTUVKUUFUVRUUQUVHUUSUVJNUVRUUPUVGAJUUOUVFGHUFQUVRUURUVIUUKMUUOUV
|
|
FKUGQUHUIUUOUVGFZUUTUVPUUFUVSUUQUVMUUSUVONUVSUUPUVLAJUUOUVGGHUFQUVSUURUVN
|
|
UUKMUUOUVGKUGQUHUIUUOBFZUUTUUNUUFUVTUUQUUJUUSUUMNUVTUUPUUIAJUUOBGHUFQUVTU
|
|
URUULUUKMUUOBKUGQUHUIUUFAGJIZUUKUVBUVDNUUFAGAAUJZAUKZUUFAGULLDAUMUNUOUVBU
|
|
WAFUUFUVAGAJUPUQURUUFUVDUUKGMIUUKUVCGUUKMUSUQUUFUUKUUFUUKUUFAAUWBAUTZVAZV
|
|
BZVCVDVFUVFCDZUUFUVKUVPUUFUWGUVKUVPPZUUFUWGRZAUVGNOZUWHUVGAVGOZUWIUVKUWJU
|
|
VPUWIUVKUWJUVPUWIUVKUWJRZRUVHAMIZUVJUVGMIZUVMUVONUWIUVKUWJUWMUWNNOUWIUVKR
|
|
ZUWIUWJRZRZUVHUVJAUVGUWQAUVGUUFASDZUWGUVKUWPUWBVHZUWQUWGUVGCDZUUFUWGUVKUW
|
|
PVIZUVFVJZVOZVAUWQUUKUVIUWQAAUWSUUFUUEUWGUVKUWPUWDVHVAUWQUVIUWQUVFUXAVKVL
|
|
VMUWSUWQUWGUVFSDUVGSDUXAUVFVPUVFVNVQUWQAUVGUWSUXCUWQUWREAVGOZEANOZUWSUUFU
|
|
XDUWGUVKUWPAWGVHESDUWRUXDUXEPVREAVSVTWAZWBUXFUWIUVKUWPWCUWOUWIUWJWDWEWFUW
|
|
IUVMUWMFZUWLUUFAWHDUWTUXGUWGAWIUXBAUVGWJWKTUWIUVOUWNFUWLUWIUVOUUKUVIUVGMI
|
|
ZMIUWNUWIUVNUXHUUKMUWGUVNUXHFUUFUVFWLWMQUWIUUKUVIUVGUUFUUKWHDUWGUWFTUWGUV
|
|
IWHDUUFUWGUVIUVFWNWOWMUWGUVGWHDZUUFUWGUVFWHDUXIUVFWPUVFWQVOWMWSWRTVFWTXAU
|
|
WIUWKUVPUVKUWIUWKUVLANOZUVPUWGUWTUUEUWKUXJXBUUFUXBUWDUVGAXJXCUWIUXJUVPUWI
|
|
UXJRZUVMUUKUVOUWIUVMSDZUXJUUFUWRUVLCDZUXLUWGUWBUWGUWTUXMUXBUVGVJVOZAUVLXD
|
|
WKTUUFUUKSDZUWGUXJUWEXEUWIUVOSDZUXJUUFUXOUVNSDZUXPUWGUWEUWGUVNUWGUVGUXBVK
|
|
ZVLZUUKUVNXFWKTUXKAUVLAUUFUWRUWGUXJUWBXEUUFGANOUWGUXJUWCXEUXKAUVLULLDZUXJ
|
|
UWIUXJXGUXKUVLXHDAXHDZUXTUXJXBUXKUVLUWGUXMUUFUXJUXNXIXKUUFUYAUWGUXJAXLZXE
|
|
UVLAXMXNXOUOUWIUUKUVONOZUXJUWIUXOUXQREUUKNOZGUVNNOZRUYCUUFUXOUWGUXQUWEUXS
|
|
XPUUFUYDUWGUYEUUFUUEUYDUWDUUEAAAVPUUEXQAXRWBVOUWGUVNUXRYAXPUUKUVNXSXNTXTY
|
|
BYCYLUWIUYAUVGXHDUWJUWKUDUUFUYAUWGUYBTUWIUVGUWGUWTUUFUXBWMXKAUVGYDXNYEYFY
|
|
GYHYIUUGUUHUUNUUHUUNUUGEUUIJIZEEJIZUULMIZNOUUHEUULUYFUYHNUUHUULUUHUULBWNZ
|
|
YJYKUUHUUIBYMYNUUHUYHGUULMIUULUYGGUULMYOYPUUHUULUUHUULUYIWOYQVDVFUUGUUJUY
|
|
FUUMUYHNAEUUIJUFUUGUUKUYGUULMUUGUUKUYGFAEAEJYRYSYTUHUUCUUAUUBUUD $.
|
|
$}
|
|
|
|
$( A lower bound for the factorial function. (Contributed by NM,
|
|
17-Dec-2005.) $)
|
|
faclbnd2 $p |- ( N e. NN0 -> ( ( 2 ^ N ) / 2 ) <_ ( ! ` N ) ) $=
|
|
( cn0 wcel c2 cexp co cdiv c1 cle cmul c4 sq2 oveq2i cc 2cn mpan syl5eq cc0
|
|
wbr cr caddc cfa cfv 2t2e4 eqtr4i wceq expp1 oveq1d expcl 2ap0 divmuldivapd
|
|
a1i cap 2div2e1 halfcld mulid1d 3eqtr2rd 2nn0 faclbnd 2re peano2nn0 reexpcl
|
|
wb sylancr faccl nnred clt 4re eqeltri 4pos breqtrri pm3.2i ledivmul mp3an3
|
|
wa syl2anc mpbird eqbrtrd ) ABCZDAEFZDGFZDAHUAFZEFZDDEFZGFZAUBUCZIVSWEVTDJF
|
|
ZDDJFZGFZWADDGFZJFZWAVSWEWCWHGFWIWDWHWCGWDKWHLUDUEMVSWCWGWHGDNCZVSWCWGUFODA
|
|
UGPUHQVSVTDDDWLVSVTNCODAUIPZWLVSOULZWNWNDRUMSVSUJULZWOUKVSWKWAHJFWAWJHWAJUN
|
|
MVSWAVSVTWMUOUPQUQVSWEWFISZWCWDWFJFISZDBCVSWQURDAUSPVSWCTCZWFTCZWPWQVCZVSDT
|
|
CWBBCWRUTAVADWBVBVDVSWFAVEVFWRWSWDTCZRWDVGSZVOWTXAXBWDKTLVHVIRKWDVGVJLVKVLW
|
|
CWFWDVMVNVPVQVR $.
|
|
|
|
$( A lower bound for the factorial function. (Contributed by NM,
|
|
19-Dec-2005.) $)
|
|
faclbnd3 $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( M ^ N ) <_ ( ( M ^ M ) x. ( ! ` N ) ) ) $=
|
|
( cn0 wcel cn cc0 wceq wo cexp co cfv cle wbr wa c1 cr adantr sylan reexpcl
|
|
cmul cfa elnn0 caddc nnre nnge1 cuz nn0z adantl uzid peano2uz 3syl leexp2ad
|
|
cz nnnn0 faclbnd wi nn0re peano2nn0 syl2an mpancom faccl nnred remulcl letr
|
|
syl3anc mp2and 0exp 0le1 syl6eqbr oveq2 0exp0e1 1le1 eqbrtri jaoi sylbi 1nn
|
|
nnmulcl sylancr nnge1d 0re 1re mp3an2 syl2anc wb oveq1 oveq12 anidms syl6eq
|
|
mpan oveq1d breq12d mpbird jaoian sylanb ) ACDZAEDZAFGZHBCDZABIJZAAIJZBUAKZ
|
|
TJZLMZAUBWPWRXCWQWPWRNZWSABOUCJZIJZLMZXFXBLMZXCXDABXEWPAPDZWRAUDQWPOALMWRAU
|
|
EQXDBUMDZBBUFKZDXEXKDWRXJWPBUGUHBUIBBUJUKULWPWOWRXHAUNZABUORWPWOWRXGXHNXCUP
|
|
ZXLWOWRNWSPDZXFPDZXBPDZXMWOXIWRXNAUQZABSRWOXIXECDXOWRXQBURAXESUSWOWTPDZXAPD
|
|
ZXPWRXIWOXRXQAASUTWRXABVAZVBZWTXAVCUSWSXFXBVDVERVFWQWRNXCFBIJZOXATJZLMZWRYD
|
|
WQWRYBOLMZOYCLMZYDWRBEDZBFGZHYEBUBYGYEYHYGYBFOLBVGVHVIYHYBFFIJZOLBFFIVJYIOO
|
|
LVKVLVMVIVNVOWRYCWROEDXAEDYCEDVPXTOXAVQVRVSWRYBPDZYCPDZYEYFNYDUPZFPDWRYJVTF
|
|
BSWIWROPDZXSYKWAYAOXAVCVRYJYMYKYLWAYBOYCVDWBWCVFUHWQXCYDWDWRWQWSYBXBYCLAFBI
|
|
WEWQWTOXATWQWTYIOWQWTYIGAFAFIWFWGVKWHWJWKQWLWMWN $.
|
|
|
|
${
|
|
$d M m $. $d N k m $.
|
|
$( Geometric lower bound for the factorial function, where N is usually
|
|
held constant. (Contributed by Paul Chapman, 28-Dec-2007.) $)
|
|
faclbnd6 $p |- ( ( N e. NN0 /\ M e. NN0 ) ->
|
|
( ( ! ` N ) x. ( ( N + 1 ) ^ M ) ) <_ ( ! ` ( N + M ) ) ) $=
|
|
( wcel cfa cfv c1 caddc co cexp cmul cle wbr cc0 wceq oveq2 oveq2d adantr
|
|
wi wa cr vm vk cn0 cv fveq2d breq12d imbi2d faccl nnred leidd cc peano2cn
|
|
weq nn0cn syl exp0d nncnd mulid1d eqtrd addid1d 3brtr4d peano2nn0 reexpcl
|
|
nn0red sylan remulcld cn nnnn0 nn0ge0d simpr expge0d mulge0d jca nn0addcl
|
|
nn0re readdcl syl2an jca31 nn0ge0 adantl 0re leadd2 mp3an1 mpbid eqbrtrrd
|
|
syl2anc 1re leadd1 mp3an3 ax-1cn addass breqtrd lemul12a sylc expp1 expcl
|
|
wb mulassd eqtr4d facp1 3eqtr3d ex expcom a2d nn0ind impcom ) AUCCBUCCZBD
|
|
EZBFGHZAIHZJHZBAGHZDEZKLZXGXHXIUAUDZIHZJHZBXOGHZDEZKLZRXGXHXIMIHZJHZBMGHZ
|
|
DEZKLZRXGXHXIUBUDZIHZJHZBYFGHZDEZKLZRXGXHXIYFFGHZIHZJHZBYLGHZDEZKLZRXGXNR
|
|
UAUBAXOMNZXTYEXGYRXQYBXSYDKYRXPYAXHJXOMXIIOPYRXRYCDXOMBGOUEUFUGUAUBUMZXTY
|
|
KXGYSXQYHXSYJKYSXPYGXHJXOYFXIIOPYSXRYIDXOYFBGOUEUFUGXOYLNZXTYQXGYTXQYNXSY
|
|
PKYTXPYMXHJXOYLXIIOPYTXRYODXOYLBGOUEUFUGXOANZXTXNXGUUAXQXKXSXMKUUAXPXJXHJ
|
|
XOAXIIOPUUAXRXLDXOABGOUEUFUGXGXHXHYBYDKXGXHXGXHBUHZUIZUJXGYBXHFJHXHXGYAFX
|
|
HJXGXIXGBUKCZXIUKCZBUNZBULUOZUPPXGXHXGXHUUBUQZURUSXGYCBDXGBUUFUTZUEVAYFUC
|
|
CZXGYKYQXGUUJYKYQRXGUUJSZYKYQUUKYKSZYHXIJHZYJYOJHZYNYPKUULYHTCZMYHKLZSZYJ
|
|
TCZSXITCZMXIKLZSYOTCZSZSZYKXIYOKLZSUUMUUNKLUUKUVCYKUUKUUQUURUVBUUKUUOUUPU
|
|
UKXHYGXGXHTCUUJUUCQZXGUUSUUJYGTCXGXIBVBZVDZXIYFVCVEZVFUUKXHYGUVEUVHXGMXHK
|
|
LZUUJXGXHVGCZUVIUUBUVJXHXHVHVIUOQUUKXIYFXGUUSUUJUVGQZXGUUJVJXGUUTUUJXGXIU
|
|
VFVIQZVKVLVMUUKYJUUKYIUCCZYJVGCBYFVNZYIUHUOUIUUKUUSUUTUVAUVKUVLXGBTCZYLTC
|
|
UVAUUJBVOZUUJYLYFVBVDBYLVPVQVRVRQUULYKUVDUUKYKVJUUKUVDYKUUKXIYIFGHZYOKUUK
|
|
BYIKLZXIUVQKLZUUKYCBYIKXGYCBNUUJUUIQUUKMYFKLZYCYIKLZUUJUVTXGYFVSVTUUKYFTC
|
|
ZUVOUVTUWAWQZUUJUWBXGYFVOVTXGUVOUUJUVPQZMTCUWBUVOUWCWAMYFBWBWCWFWDWEUUKUV
|
|
OYITCZUVRUVSWQZUWDUUKYIUVNVDUVOUWEFTCUWFWGBYIFWHWIWFWDXGUUDYFUKCZUVQYONZU
|
|
UJUUFYFUNUUDUWGFUKCUWHWJBYFFWKWIVQZWLQVMYHYJXIYOWMWNUUKYNUUMNYKUUKYNXHYGX
|
|
IJHZJHUUMUUKYMUWJXHJXGUUEUUJYMUWJNUUGXIYFWOVEPUUKXHYGXIXGXHUKCUUJUUHQXGUU
|
|
EUUJYGUKCUUGXIYFWPVEXGUUEUUJUUGQWRWSQUUKYPUUNNYKUUKUVQDEZYJUVQJHZYPUUNUUK
|
|
UVMUWKUWLNUVNYIWTUOUUKUVQYODUWIUEUUKUVQYOYJJUWIPXAQVAXBXCXDXEXF $.
|
|
|
|
$( An upper bound for the factorial function. (Contributed by Mario
|
|
Carneiro, 15-Apr-2016.) $)
|
|
facubnd $p |- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) ) $=
|
|
( vm vk cv cfa cfv cexp co cle wbr c1 cc0 wceq fveq2 oveq12d breq12d wcel
|
|
id adantr cr caddc fac0 syl6eq 0exp0e1 1le1 cn0 wa cmul faccl nnred nn0re
|
|
cn simpl reexpcld nn0p1nn simpr nn0ge0 lep1d syl32anc letrd clt wb nngt0d
|
|
leexp1a lemul1 syl112anc mpbid facp1 nncnd expp1d 3brtr4d ex nn0ind ) BDZ
|
|
EFZVNVNGHZIJKKIJCDZEFZVQVQGHZIJZVQKUAHZEFZWAWAGHZIJZAEFZAAGHZIJBCAVNLMZVO
|
|
KVPKIWGVOLEFKVNLENUBUCWGVPLLGHKWGVNLVNLGWGRZWHOUDUCPVNVQMZVOVRVPVSIVNVQEN
|
|
WIVNVQVNVQGWIRZWJOPVNWAMZVOWBVPWCIVNWAENWKVNWAVNWAGWKRZWLOPVNAMZVOWEVPWFI
|
|
VNAENWMVNAVNAGWMRZWNOPUEVQUFQZVTWDWOVTUGZVRWAUHHZWAVQGHZWAUHHZWBWCIWPVRWR
|
|
IJZWQWSIJZWPVRVSWRWPVRWOVRULQVTVQUISUJZWPVQVQWOVQTQZVTVQUKSZWOVTUMZUNWPWA
|
|
VQWPWAWOWAULQVTVQUOSZUJZXEUNZWOVTUPWPXCWATQZWOLVQIJZVQWAIJVSWRIJXDXGXEWOX
|
|
JVTVQUQSWPVQXDURVQWAVQVDUSUTWPVRTQWRTQXILWAVAJWTXAVBXBXHXGWPWAXFVCVRWRWAV
|
|
EVFVGWOWBWQMVTVQVHSWPWAVQWPWAXFVIXEVJVKVLVM $.
|
|
$}
|
|
|
|
$( The product of two factorials is greater than or equal to the factorial of
|
|
(the floor of) their average. (Contributed by NM, 9-Dec-2005.) $)
|
|
facavg $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( ! ` ( |_ ` ( ( M + N ) / 2 ) ) ) <_ ( ( ! ` M ) x. ( ! ` N ) ) ) $=
|
|
( cn0 wcel wa co c2 cle wbr cfv cfa syl cr wi adantr letr syl3anc c1 adantl
|
|
cc0 caddc cdiv cfl cmul cq cz cn nn0addcl nn0zd 2nn znq sylancl flqcld zred
|
|
flqle nn0readdcl rehalfcld nn0re mpand nn0ge0d wb halfnneg2 mpbid flqge0nn0
|
|
syl2anc simpl facwordi 3exp sylc wceq faccl nncnd mulid1d nnnn0d jca nnge1d
|
|
nnred lemul2a mp3anl1 syl21anc eqbrtrrd remulcl syl2an mpan2d 3syld mulid2d
|
|
1re simpr lemul1a wo zq qavgle mpjaod ) ACDZBCDZEZABUAFZGUBFZAHIZWRUCJZKJZA
|
|
KJZBKJZUDFZHIZWRBHIZWPWSWTAHIZXAXBHIZXEWPWTWRHIZWSXGWPWRUEDZXIWPWQUFDGUGDXJ
|
|
WPWQABUHZUIUJWQGUKULZWRUOLZWPWTMDZWRMDZAMDZXIWSEXGNWPWTWPWRXLUMUNZWPWQABUPZ
|
|
UQZWNXPWOAUROWTWRAPQUSWPWTCDZWNXGXHNWPXJTWRHIZXTXLWPTWQHIZYAWPWQXKUTWPWQMDY
|
|
BYAVAXRWQVBLVCWRVDVEZWNWOVFZXTWNXGXHWTAVGVHVIWPXHXBXDHIZXEWPXBRUDFZXBXDHWNY
|
|
FXBVJWOWNXBWNXBAVKZVLVMOWPXCMDZXBMDZTXBHIZEZRXCHIZYFXDHIZWOYHWNWOXCBVKZVQZS
|
|
ZWNYKWOWNYIYJWNXBYGVQZWNXBWNXBYGVNUTVOOWOYLWNWOXCYNVPSRMDZYHYKYLYMWGRXCXBVR
|
|
VSVTWAWPXAMDZYIXDMDZXHYEEXENWPXAWPXTXAUGDYCWTVKLVQZWNYIWOYQOZWNYIYHYTWOYQYO
|
|
XBXCWBWCZXAXBXDPQWDWEWPXFWTBHIZXAXCHIZXEWPXIXFUUDXMWPXNXOBMDZXIXFEUUDNXQXSW
|
|
OUUFWNBURSWTWRBPQUSWPXTWOUUDUUENYCWNWOWHZXTWOUUDUUEWTBVGVHVIWPUUEXCXDHIZXEW
|
|
PRXCUDFZXCXDHWOUUIXCVJWNWOXCWOXCYNVLWFSWPYIYHTXCHIZEZRXBHIZUUIXDHIZUUBWOUUK
|
|
WNWOYHUUJYOWOXCWOXCYNVNUTVOSWNUULWOWNXBYGVPOYRYIUUKUULUUMWGRXBXCWIVSVTWAWPY
|
|
SYHYTUUEUUHEXENUUAYPUUCXAXCXDPQWDWEWPAUEDZBUEDZWSXFWJWPAUFDUUNWPAYDUIAWKLWP
|
|
BUFDUUOWPBUUGUIBWKLABWLVEWM $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The binomial coefficient operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c _C $. $( Binomial coefficient operation. $)
|
|
|
|
$( Extend class notation to include the binomial coefficient operation
|
|
(combinatorial choose operation). $)
|
|
cbc $a class _C $.
|
|
|
|
${
|
|
$d n k $.
|
|
$( Define the binomial coefficient operation. For example,
|
|
` ( 5 _C 3 ) = 1 0 ` ( ~ ex-bc ).
|
|
|
|
In the literature, this function is often written as a column vector of
|
|
the two arguments, or with the arguments as subscripts before and after
|
|
the letter "C". ` ( N _C K ) ` is read " ` N ` choose ` K ` ."
|
|
Definition of binomial coefficient in [Gleason] p. 295. As suggested by
|
|
Gleason, we define it to be 0 when ` 0 <_ k <_ n ` does not hold.
|
|
(Contributed by NM, 10-Jul-2005.) $)
|
|
df-bc $a |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) ,
|
|
( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d n k N $. $d n k K $.
|
|
$( Value of the binomial coefficient, ` N ` choose ` K ` . Definition of
|
|
binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we
|
|
define it to be 0 when ` 0 <_ K <_ N ` does not hold. See ~ bcval2 for
|
|
the value in the standard domain. (Contributed by NM, 10-Jul-2005.)
|
|
(Revised by Mario Carneiro, 7-Nov-2013.) $)
|
|
bcval $p |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N )
|
|
, ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) ) $=
|
|
( vn vk cn0 wcel cz cc0 cfz cfa cfv cmin cmul cdiv cif wceq adantl faccld
|
|
co cq cbc wa iftrue simpll nnzd fznn0sub elfznn0 nnmulcld syl2anc eqeltrd
|
|
wn cn znq iffalse 0z zq ax-mp syl6eqel wdc simpr 0zd simpl fzdcel syl3anc
|
|
wo nn0zd exmiddc syl mpjaodan cv oveq2 eleq2d fveq2 fveq2d oveq1d oveq12d
|
|
oveq1 ifbieq1d eleq1 oveq2d df-bc ovmpt2g mpd3an3 ) BEFZAGFZAHBISZFZBJKZB
|
|
ALSZJKZAJKZMSZNSZHOZTFZBAUASWNPWDWEUBZWGWOWGUKZWPWGUBZWNWMTWGWNWMPWPWGWMH
|
|
UCQWRWHGFWLULFWMTFWRWHWRBWDWEWGUDRUEWRWJWKWRWIWGWIEFWPAHBUFQRWRAWGAEFWPAB
|
|
UGQRUHWHWLUMUIUJWQWOWPWQWNHTWGWMHUNHGFZHTFUOHUPUQURQWPWGUSZWGWQVEWPWEWSBG
|
|
FWTWDWEUTWPVAWPBWDWEVBVFAHBVCVDWGVGVHVICDBAEGDVJZHCVJZISZFZXBJKZXBXALSZJK
|
|
ZXAJKZMSZNSZHOWNUAXAWFFZWHBXALSZJKZXHMSZNSZHOTXBBPZXDXKXJXOHXPXCWFXAXBBHI
|
|
VKVLXPXEWHXIXNNXBBJVMXPXGXMXHMXPXFXLJXBBXALVQVNVOVPVRXAAPZXKWGXOWMHXAAWFV
|
|
SXQXNWLWHNXQXMWJXHWKMXQXLWIJXAABLVKVNXAAJVMVPVTVRDCWAWBWC $.
|
|
|
|
$( Value of the binomial coefficient, ` N ` choose ` K ` , in its standard
|
|
domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro,
|
|
7-Nov-2013.) $)
|
|
bcval2 $p |- ( K e. ( 0 ... N ) -> ( N _C K ) =
|
|
( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) ) $=
|
|
( cc0 cfz co wcel cbc cfa cfv cmin cmul cdiv cif cz wceq elfz3nn0 elfzelz
|
|
cn0 bcval syl2anc iftrue eqtrd ) ACBDEFZBAGEZUCBHIBAJEHIAHIKELEZCMZUEUCBR
|
|
FANFUDUFOABPACBQABSTUCUECUAUB $.
|
|
|
|
$( Value of the binomial coefficient, ` N ` choose ` K ` , outside of its
|
|
standard domain. Remark in [Gleason] p. 295. (Contributed by NM,
|
|
14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $)
|
|
bcval3 $p |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) ->
|
|
( N _C K ) = 0 ) $=
|
|
( cn0 wcel cz cc0 cfz co wn w3a cbc cfa cfv cmin cmul cdiv cif wceq bcval
|
|
3adant3 iffalse 3ad2ant3 eqtrd ) BCDZAEDZAFBGHDZIZJBAKHZUFBLMBANHLMALMOHP
|
|
HZFQZFUDUEUHUJRUGABSTUGUDUJFRUEUFUIFUAUBUC $.
|
|
|
|
$( Value of the binomial coefficient, ` N ` choose ` K ` , outside of its
|
|
standard domain. Remark in [Gleason] p. 295. (Contributed by NM,
|
|
14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) $)
|
|
bcval4 $p |- ( ( N e. NN0 /\ K e. ZZ /\ ( K < 0 \/ N < K ) ) ->
|
|
( N _C K ) = 0 ) $=
|
|
( cn0 wcel cz cc0 clt wbr wo cfz co wn cbc wa cle cr lenlt mpbid adantl
|
|
wb wceq elfzle1 elfzelz zred sylancr elfzle2 nn0re syl2anr ioran sylanbrc
|
|
wi 0re ex adantr con2d 3impia bcval3 syld3an3 ) BCDZAEDZAFGHZBAGHZIZAFBJK
|
|
DZLZBAMKFUAUSUTVCVEUSUTNVDVCUSVDVCLZUKUTUSVDVFUSVDNZVALZVBLZVFVDVHUSVDFAO
|
|
HZVHAFBUBVDFPDAPDZVJVHTULVDAAFBUCUDZFAQUERSVGABOHZVIVDVMUSAFBUFSVDVKBPDVM
|
|
VITUSVLBUGABQUHRVAVBUIUJUMUNUOUPABUQUR $.
|
|
$}
|
|
|
|
$( Closure of the binomial coefficient in the positive reals. (This is
|
|
mostly a lemma before we have ~ bccl2 .) (Contributed by Mario Carneiro,
|
|
10-Mar-2014.) $)
|
|
bcrpcl $p |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ ) $=
|
|
( cc0 cfz co wcel cbc cfa cfv cmin cmul cdiv crp bcval2 cn cn0 faccl syl2an
|
|
syl2anc nnrp elfz3nn0 syl fznn0sub elfznn0 nnmulcl rpdivcl eqeltrd ) ACBDEF
|
|
ZBAGEBHIZBAJEZHIZAHIZKEZLEZMABNUHUIOFZUMOFZUNMFZUHBPFUOABUABQUBUHUJPFZAPFZU
|
|
PACBUCABUDURUKOFULOFUPUSUJQAQUKULUERSUOUIMFUMMFUQUPUITUMTUIUMUFRSUG $.
|
|
|
|
$( "Complementing" its second argument doesn't change a binary coefficient.
|
|
(Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro,
|
|
5-Mar-2014.) $)
|
|
bccmpl $p |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) =
|
|
( N _C ( N - K ) ) ) $=
|
|
( wcel cz cc0 co cbc cmin wceq wn cfa cfv cmul cdiv bcval2 fznn0sub2 syl cc
|
|
elfznn0 eqtr4d cn0 wa cfz faccld nncnd mulcomd elfz3nn0 elfzelz nn0cn nncan
|
|
zcn syl2an syl2anc fveq2d oveq1d oveq2d adantl w3a bcval3 simp1 nn0z zsubcl
|
|
sylan 3adant3 eleq1d syl5ib con3d 3impia syl3anc 3expa wdc simpr 0zd adantr
|
|
wo fzdcel exmiddc mpjaodan ) BUACZADCZUBZAEBUCFZCZBAGFZBBAHFZGFZIZWCJZWCWGW
|
|
AWCWDBKLZWEKLZAKLZMFZNFZWFABOWCWFWIBWEHFZKLZWJMFZNFZWMWCWEWBCZWFWQIABPZWEBO
|
|
QWCWLWPWINWCWLWKWJMFWPWCWJWKWCWRWJRCWSWRWJWRWEWEBSUDUEQWCWKWCAABSUDUEUFWCWO
|
|
WKWJMWCWNAKWCVSVTWNAIZABUGAEBUHVSBRCARCWTVTBUIAUKBAUJULZUMUNUOTUPTTUQVSVTWH
|
|
WGVSVTWHURZWDEWFABUSXBVSWEDCZWRJZWFEIVSVTWHUTVSVTXCWHVSBDCZVTXCBVAZBAVBVCVD
|
|
VSVTWHXDWAWRWCWRWNWBCWAWCWEBPWAWNAWBXAVEVFVGVHWEBUSVITVJWAWCVKZWCWHVOWAVTED
|
|
CXEXGVSVTVLWAVMVSXEVTXFVNAEBVPVIWCVQQVR $.
|
|
|
|
$( ` N ` choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM,
|
|
17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $)
|
|
bcn0 $p |- ( N e. NN0 -> ( N _C 0 ) = 1 ) $=
|
|
( cn0 wcel cc0 cbc co cfa cfv cmin cmul cdiv c1 cfz wceq 0elfz bcval2 nn0cn
|
|
syl subid1d eqtrd fveq2d oveq12 sylancl faccl nncnd mulid1d oveq2d dividapd
|
|
fac0 nnap0d ) ABCZADEFZAGHZADIFZGHZDGHZJFZKFZLUKDDAMFCULURNAODAPRUKURUMUMKF
|
|
LUKUQUMUMKUKUQUMLJFZUMUKUOUMNUPLNUQUSNUKUNAGUKAAQSUAUIUOUMUPLJUBUCUKUMUKUMA
|
|
UDZUEZUFTUGUKUMVAUKUMUTUJUHTT $.
|
|
|
|
$( The binomial coefficient " 0 choose ` K ` " is 0 for a positive integer K.
|
|
Note that ` ( 0 _C 0 ) = 1 ` (see ~ bcn0 ). (Contributed by Alexander van
|
|
der Vekens, 1-Jan-2018.) $)
|
|
bc0k $p |- ( K e. NN -> ( 0 _C K ) = 0 ) $=
|
|
( cn wcel cc0 cn0 cz clt wbr wo cbc co wceq 0nn0 a1i nnz nngt0 olcd syl3anc
|
|
bcval4 ) ABCZDECZAFCADGHZDAGHZIDAJKDLUATMNAOTUCUBAPQADSR $.
|
|
|
|
$( ` N ` choose ` N ` is 1. Remark in [Gleason] p. 296. (Contributed by NM,
|
|
17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $)
|
|
bcnn $p |- ( N e. NN0 -> ( N _C N ) = 1 ) $=
|
|
( cn0 wcel cc0 co cmin c1 cz wceq 0z bccmpl mpan2 bcn0 nn0cn subid1d oveq2d
|
|
cbc 3eqtr3rd ) ABCZADQEZAADFEZQEZGAAQESDHCTUBIJDAKLAMSUAAAQSAANOPR $.
|
|
|
|
$( Binomial coefficient: ` N ` choose ` 1 ` . (Contributed by NM,
|
|
21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $)
|
|
bcn1 $p |- ( N e. NN0 -> ( N _C 1 ) = N ) $=
|
|
( cn0 wcel cn cc0 wceq wo c1 cbc co elnn0 cfa cfv cmin cmul cfz cuz clt wbr
|
|
cdiv 1eluzge0 a1i elnnuz biimpi elfzuzb sylanbrc bcval2 facnn2 fac1 nnm1nn0
|
|
oveq2i faccld nncnd mulid1d syl5eq oveq12d nncn nnap0d divcanap3d 3eqtrd cz
|
|
syl 0nn0 1z 0lt1 olci bcval4 mp3an oveq1 eqeq12 mpancom mpbiri jaoi sylbi
|
|
wb ) ABCADCZAEFZGAHIJZAFZAKVPVSVQVPVRALMZAHNJZLMZHLMZOJZTJZWBAOJZWBTJAVPHEA
|
|
PJCZVRWEFVPHEQMCZAHQMCZWGWHVPUAUBVPWIAUCUDHEAUEUFHAUGVBVPVTWFWDWBTAUHVPWDWB
|
|
HOJWBWCHWBOUIUKVPWBVPWBVPWAAUJULZUMZUNUOUPVPAWBAUQWKVPWBWJURUSUTVQVSEHIJZEF
|
|
ZEBCHVACHERSZEHRSZGWMVCVDWOWNVEVFHEVGVHVRWLFVQVSWMVOAEHIVIVRWLAEVJVKVLVMVN
|
|
$.
|
|
|
|
$( Binomial coefficient: ` N + 1 ` choose ` N ` . (Contributed by NM,
|
|
20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) $)
|
|
bcnp1n $p |- ( N e. NN0 -> ( ( N + 1 ) _C N ) = ( N + 1 ) ) $=
|
|
( cn0 wcel c1 caddc co cbc cmin cz wceq peano2nn0 nn0z bccmpl syl2anc nn0cn
|
|
cc ax-1cn pncan2 sylancl oveq2d bcn1 syl 3eqtrd ) ABCZADEFZAGFZUEUEAHFZGFZU
|
|
EDGFZUEUDUEBCZAICUFUHJAKZALAUEMNUDUGDUEGUDAPCDPCUGDJAOQADRSTUDUJUIUEJUKUEUA
|
|
UBUC $.
|
|
|
|
$( The proportion of one binomial coefficient to another with ` K ` decreased
|
|
by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) $)
|
|
bcm1k $p |- ( K e. ( 1 ... N ) -> ( N _C K ) =
|
|
( ( N _C ( K - 1 ) ) x. ( ( N - ( K - 1 ) ) / K ) ) ) $=
|
|
( c1 cfz co wcel cfa cfv cmin cmul cdiv caddc cn nnnn0d faccld nncnd nnap0d
|
|
syl wceq cc0 cbc cuz elfzuz2 nnuz syl6eleqr fznn0sub nn0p1nn elfznn nnm1nn0
|
|
cn0 faccl 3syl nnmulcld divmuldivapd elfzel2 zcnd 1cnd fveq2d oveq1d oveq2d
|
|
subsubd oveq12d facp1 eqcomd facnn2 mul32d mulassd 3eqtr4d divcanap5d 0p1e1
|
|
mulcomd oveq1i cz wss 0z fzp1ss ax-mp eqsstr3i sseli bcval2 cc ax-1cn npcan
|
|
3eqtrrd sylancl peano2zm uzid peano2uz 4syl eqeltrrd fzss2 elfzmlbm sseldd
|
|
) ACBDEZFZBGHZBAIEZGHZAGHZJEZKEZWPBACIEZIEZGHZXBGHZJEZKEZXCAKEZJEZBAUAEZBXB
|
|
UAEZXHJEWOXIWPWQCLEZJEZWTXLJEZKEZXLWPJEZXLWTJEZKEXAWOWPXLGHZXEJEZKEZXLAKEZJ
|
|
EXMXSAJEZKEXIXOWOWPXSXLAWOWPWOBWOBWOBCUBHMACBUCUDUENOPZWOXSWOXRXEWOXLWOXLWO
|
|
WQUJFZXLMFACBUFZWQUGRZNOZWOAMFZXBUJFXEMFABUHZAUIXBUKULZUMZPWOXLYFPZWOAYIPZW
|
|
OXSYKQWOAYIQUNWOXGXTXHYAJWOXFXSWPKWOXDXRXEJWOXCXLGWOBACWOBACBUOZUPZYMWOUQVA
|
|
ZURUSUTWOXCXLAKYPUSVBWOXNYBXMKWOWRXLJEZWSJEXRXEAJEZJEXNYBWOYQXRWSYRJWOXRYQW
|
|
OYDXRYQSYEWQVCRVDWOYHWSYRSYIAVERVBWOWRWSXLWOWRWOWQYEOZPWOWSWOAWOAYINOZPYLVF
|
|
WOXRXEAWOXRYGPWOXEYJPYMVGVHUTVHWOXMXPXNXQKWOWPXLYCYLVKWOWTXLWOWTWOWRWSYSYTU
|
|
MZPZYLVKVBWOWPWTXLYCUUBYLWOWTUUAQWOXLYFQVIWDWOATBDEZFXJXASWNUUCAWNTCLEZBDEZ
|
|
UUCUUDCBDVJVLTVMFUUEUUCVNVOTBVPVQVRVSABVTRWOXKXGXHJWOXBUUCFXKXGSWOTBCIEZDEZ
|
|
UUCXBWOBUUFUBHZFUUGUUCVNWOUUFCLEZBUUHWOBWAFCWAFUUIBSYOWBBCWCWEWOBVMFUUFVMFU
|
|
UFUUHFUUIUUHFYNBWFUUFWGUUFUUFWHWIWJUUFTBWKRACBWLWMXBBVTRUSVH $.
|
|
|
|
$( The proportion of one binomial coefficient to another with ` N ` increased
|
|
by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) $)
|
|
bcp1n $p |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) =
|
|
( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) ) $=
|
|
( cc0 cfz co wcel c1 caddc cfa cfv cmin cmul cdiv cbc cn0 wceq faccld nncnd
|
|
syl cn elfz3nn0 facp1 fznn0sub nn0cnd elfznn0 addsubd fveq2d oveq2d 3eqtr4d
|
|
1cnd oveq1d nn0p1nn eqeltrd mul32d eqtrd oveq12d nnap0d divmuldivapd eqtr4d
|
|
nnmulcld fzelp1 bcval2 ) ACBDEFZBGHEZIJZVDAKEZIJZAIJZLEZMEZBIJZBAKEZIJZVHLE
|
|
ZMEZVDVFMEZLEZVDANEZBANEZVPLEVCVJVKVDLEZVNVFLEZMEVQVCVEVTVIWAMVCBOFZVEVTPAB
|
|
UAZBUBSVCVIVMVFLEZVHLEWAVCVGWDVHLVCVLGHEZIJZVMWELEZVGWDVCVLOFZWFWGPACBUCZVL
|
|
UBSVCVFWEIVCBGAVCBWCUDVCUJVCAABUEZUDUFZUGVCVFWEVMLWKUHUIUKVCVMVFVHVCVMVCVLW
|
|
IQZRVCVFVCVFWETWKVCWHWETFWIVLULSUMZRZVCVHVCAWJQZRUNUOUPVCVKVNVDVFVCVKVCBWCQ
|
|
RVCVNVCVMVHWLWOUTZRVCVDVCWBVDTFWCBULSRWNVCVNWPUQVCVFWMUQURUSVCACVDDEFVRVJPA
|
|
CBVAAVDVBSVCVSVOVPLABVBUKUI $.
|
|
|
|
$( The proportion of one binomial coefficient to another with ` N ` and ` K `
|
|
increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.) $)
|
|
bcp1nk $p |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( K + 1 ) ) =
|
|
( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) ) $=
|
|
( cc0 cfz co wcel c1 caddc cbc cmin cdiv cmul wceq cz wb syl cc oveq2d zred
|
|
eqtrd elfzel1 elfzel2 elfzelz fzaddel syl22anc 1e0p1 oveq1i syl6eleqr bcm1k
|
|
1zzd ibi zcnd ax-1cn pncan sylancl bcp1n oveq1d oveq12d bcrpcl peano2zd clt
|
|
rpcnd wbr elfzle2 ltp1d lelttrd znnsub syl2anc mpbid nndivred recnd elfznn0
|
|
cn nnred cn0 nn0p1nn mulassd nncnd nnap0d dmdcanap2d ) ACBDEFZBGHEZAGHEZIEZ
|
|
WBWCGJEZIEZWBWEJEZWCKEZLEZBAIEZWBWCKEZLEZWAWCGWBDEZFWDWIMWAWCCGHEZWBDEZWMWA
|
|
WCWOFZWACNFBNFANFZGNFWAWPOACBUAACBUBZACBUCZWAUJAGCBUDUEUKGWNWBDUFUGUHWCWBUI
|
|
PWAWIWJWBWBAJEZKEZLEZWTWCKEZLEZWLWAWFXBWHXCLWAWFWBAIEXBWAWEAWBIWAAQFGQFWEAM
|
|
WAAWSULUMAGUNUOZRABUPTWAWGWTWCKWAWEAWBJXERUQURWAXDWJXAXCLEZLEWLWAWJXAXCWAWJ
|
|
ABUSVBWAXAWAWBWTWAWBWABWRUTZSZWAAWBVAVCZWTVMFZWAABWBWAAWSSWABWRSZXHACBVDWAB
|
|
XKVEVFWAWQWBNFXIXJOWSXGAWBVGVHVIZVJVKWAXCWAWTWCWAWTXLVNWAAVOFWCVMFABVLAVPPZ
|
|
VJVKVQWAXFWKWJLWAWBWTWCWAWBXGULWAWTXLVRWAWCXMVRWAWTXLVSWAWCXMVSVTRTTT $.
|
|
|
|
${
|
|
$d k x y K $. $d k x y N $. $d K f g $. $d N f g $.
|
|
$( Write out the top and bottom parts of the binomial coefficient
|
|
` ( N _C K ) = ( N x. ( N - 1 ) x. ... x. ( ( N - K ) + 1 ) ) / K ! `
|
|
explicitly. In this form, it is valid even for ` N < K ` , although it
|
|
is no longer valid for nonpositive ` K ` . (Contributed by Jim Kingdon,
|
|
6-Nov-2021.) $)
|
|
ibcval5 $p |- ( ( N e. NN0 /\ K e. NN ) -> ( N _C K ) =
|
|
( ( seq ( ( N - K ) + 1 ) ( x. , _I , CC ) ` N ) / ( ! ` K ) ) ) $=
|
|
( wcel wa cc0 co cmul cc cid c1 cfv cfa cdiv wceq adantl cvv cv cle wbr
|
|
cz vk vx vy vf vg cn0 cfz cbc cmin caddc cseq bcval2 simprl simprr mulcld
|
|
cn w3a simpr1 simpr2 simpr3 mulassd cuz simpll nn0zd simplr nnzd peano2zd
|
|
wn zsubcld 1red nnred nn0red nnge1d lesub2dd zred leaddsub syl3anc mpbird
|
|
cr wb eluz2 syl3anbrc adantrr cnex a1i syl6eleq vex fvi eluzelcn syl5eqel
|
|
nnuz ax-mp iseqsplit elfzuz3 eluznn syl2anc facnn syl oveq1d 3eqtr4d expr
|
|
faccld nncnd mulid2d oveq2d eqtr3d fveq2 fac0 syl6eq oveq1 iseqeq1 fveq1d
|
|
0p1e1 oveq12d eqeq2d syl5ibrcom fznn0sub elnn0 sylib mpjaod iseqcl nnnn0d
|
|
wo mulcl nnap0d divcanap5d 3eqtrd div0apd simpr mul02d mul01d nn0uz elfz5
|
|
clt nn0re ad2antrr nnre 0z sylancl 0zd ad2antlr bitr4d mtbid simpl adantr
|
|
subge0d zltnle zltp1le mpbid nn0ge0 elfz mpbir2and elex mp1i iseqz bcval3
|
|
0cn nnz syl3an2 3expa 3eqtr4rd wdc fzdcel exmiddc mpjaodan ) BUFCZAUPCZDZ
|
|
AEBUGFCZBAUHFZBGHIBAUIFZJUJFZUKZKZALKZMFZNUVIVHZUVHUVIDZUVJBLKZUVKLKZUVOG
|
|
FZMFZUVTUVNGFZUWAMFUVPUVIUVJUWBNUVHABULOUVRUVSUWCUWAMUVRUVKUPCZUVSUWCNZUV
|
|
KENZUVHUVIUWDUWEUVHUVIUWDDDZBGHIJUKZKZUVKUWHKZUVNGFUVSUWCUWGUAUBUCGHIJUVK
|
|
BPUWGUAQZHCZUBQZHCZDZDUWKUWMUWGUWLUWNUMUWGUWLUWNUNUOUWGUWLUWNUCQZHCZUQDUW
|
|
KUWMUWPUWGUWLUWNUWQURUWGUWLUWNUWQUSUWGUWLUWNUWQUTVAUVHUVIBUVLVBKZCZUWDUVR
|
|
UVLTCZBTCZUVLBRSZUWSUVRUVKUVRBAUVRBUVFUVGUVIVCZVDZUVRAUVFUVGUVIVEZVFVIZVG
|
|
UXDUVRUXBUVKBJUIFRSZUVRJABUVRVJZUVRAUXEVKUVRBUXCVLZUVRAUXEVMVNUVRUVKVSCJV
|
|
SCBVSCZUXBUXGVTUVRUVKUXFVOUXHUXIUVKJBVPVQVRUVLBWAWBZWCHPCZUWGWDWEUWGUVKUP
|
|
JVBKZUVHUVIUWDUNZWKWFUWGUWKUXMCZDUWKIKZUWKHUWKPCUXPUWKNUAWGUWKPWHWLZUXOUW
|
|
LUWGJUWKWIOWJWMUWGBUPCZUVSUWINZUVHUVIUXRUWDUVRUVGBAVBKCZUXRUXEUVIUXTUVHAE
|
|
BWNOBAWOWPZWCBWQZWRUWGUVTUWJUVNGUWGUWDUVTUWJNUXNUVKWQWRWSWTXAUVRUWEUWFUVS
|
|
JUWIGFZNUVRJUVSGFUVSUYCUVRUVSUVRUVSUVRBUXCXBXCXDUVRUVSUWIJGUVRUXRUXSUYAUY
|
|
BWRXEXFUWFUWCUYCUVSUWFUVTJUVNUWIGUWFUVTELKJUVKELXGXHXIUWFBUVMUWHUWFUVLJNU
|
|
VMUWHNUWFUVLEJUJFJUVKEJUJXJXMXIGHIUVLJXKWRXLXNXOXPUVRUVKUFCZUWDUWFYCUVIUY
|
|
DUVHAEBXQOZUVKXRXSXTWSUVRUVNUVOUVTUVRUDUEGHIUVLBUXKUVRUDQZUWRCZDUYFIKZUYF
|
|
HUYFPCUYHUYFNUDWGUYFPWHWLUYGUYFHCZUVRUVLUYFWIOWJUYIUEQZHCDUYFUYJGFHCUVRUY
|
|
FUYJYDOYAUVRUVOUVRAUVRAUXEYBXBZXCUVRUVTUVRUVKUYEXBZXCUVRUVOUYKYEUVRUVTUYL
|
|
YEYFYGUVHUVQDZEUVOMFEUVPUVJUYMUVOUYMUVOUYMAUYMAUVFUVGUVQVEYBZXBZXCUYMUVOU
|
|
YOYEYHUYMUVNEUVOMUYMUAUBGHIEUVLBPEUWOUWKUWMGFHCUYMUWKUWMYDOUYMUWKUWRCZDUX
|
|
PUWKHUXQUYPUWLUYMUVLUWKWIOWJUXLUYMWDWEUYMUWLDZUWKUYMUWLYIZYJUYQUWKUYRYKUY
|
|
MEUVLBUGFCZUVLERSZEBRSZUYMUVKEYNSZUYTUYMVUBEUVKRSZVHZUYMUVIVUCUVHUVQYIUYM
|
|
UVIABRSZVUCUYMAEVBKZCUXAUVIVUEVTUYMAUFVUFUYNYLWFUYMBUVFUVGUVQVCVDZAEBYMWP
|
|
UYMBAUVFUXJUVGUVQBYOYPUVGAVSCUVFUVQAYQUUAUUFUUBUUCUYMUVKTCZETCZVUBVUDVTUV
|
|
HVUHUVQUVHBAUVHBUVFUVGUUDVDZUVHAUVFUVGYIVFZVIUUEZYRUVKEUUGYSVRUYMVUHVUIVU
|
|
BUYTVTVULYRUVKEUUHYSUUIUVFVUAUVGUVQBUUJYPUYMVUIUWTUXAUYSUYTVUADVTUYMYTUYM
|
|
UVKVULVGVUGEUVLBUUKVQUULUVFBPCUVGUVQBUFUUMYPEHCEIKENUYMUUQEHWHUUNUUOWSUVF
|
|
UVGUVQUVJENZUVGUVFATCZUVQVUMAUURABUUPUUSUUTUVAUVHUVIUVBZUVIUVQYCUVHVUNVUI
|
|
UXAVUOVUKUVHYTVUJAEBUVCVQUVIUVDWRUVE $.
|
|
$}
|
|
|
|
${
|
|
$d N x y $.
|
|
$( Binomial coefficient: ` N ` choose ` 2 ` . (Contributed by Mario
|
|
Carneiro, 22-May-2014.) $)
|
|
bcn2 $p |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) ) $=
|
|
( vx vy wcel c2 co cmul cc cid cmin c1 caddc cfv cdiv wceq ax-1cn syl cuz
|
|
cz eqtrd cn0 cbc cseq cfa 2nn ibcval5 mpan2 2m1e1 oveq2i nn0cn 2cn npncan
|
|
cn mp3an23 syl5eqr iseqeq1 fveq1d nn0z peano2zm uzid npcan sylancl fveq2d
|
|
cvv eleqtrrd cnex a1i cv wa eluzelcn adantl fvi eleq1d mulcl iseqm1 iseq1
|
|
ibir oveq12d subcl mulcomd fac2 ) AUADZAEUBFZAGHIAEJFZKLFZUCZMZEUDMZNFZAA
|
|
KJFZGFZENFWBEUMDWCWIOUEEAUFUGWBWGWKWHENWBWGAGHIWJUCZMZWKWBAWFWLWBWEWJOWFW
|
|
LOWBWEWDEKJFZLFZWJWNKWDLUHUIWBAHDZWOWJOZAUJZWPEHDKHDZWQUKPAEKULUNQUOGHIWE
|
|
WJUPQUQWBWMWJAGFZWKWBWMWJWLMZAIMZGFWTWBBCGHIWJAVDWBASDZWJSDZAURZAUSQZWBAA
|
|
RMZWJKLFZRMWBXCAXGDXEAUTQWBXHARWBWPWSXHAOWRPAKVAVBVCVEHVDDWBVFVGWBBVHZWJR
|
|
MDZVIXIHDZXIIMZHDZXJXKWBWJXIVJVKXKXMXKXLXIHXIHVLVMVQQZXKCVHZHDVIXIXOGFHDW
|
|
BXIXOVNVKZVOWBXAWJXBAGWBXAWJIMZWJWBBCGHIWJXFXNXPVPWBXDXQWJOXFWJSVLQTAUAVL
|
|
VRTWBWJAWBWPWSWJHDWRPAKVSVBWRVTTTWHEOWBWAVGVRT $.
|
|
$}
|
|
|
|
$( Compute the binomial coefficient of ` ( N + 1 ) ` over ` ( N - 1 ) `
|
|
(Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro,
|
|
22-May-2014.) $)
|
|
bcp1m1 $p |- ( N e. NN0 -> ( ( N + 1 ) _C ( N - 1 ) ) =
|
|
( ( ( N + 1 ) x. N ) / 2 ) ) $=
|
|
( cn0 wcel c1 caddc co cmin cbc cmul c2 cdiv cz wceq peano2nn0 peano2zm syl
|
|
nn0z oveq2d cc eqtrd bccmpl syl2anc nn0cn 1cnd pnncand syl6eqr ax-1cn pncan
|
|
df-2 bcn2 sylancl oveq1d ) ABCZADEFZADGFZHFZUNUNUOGFZHFZUNAIFZJKFZUMUNBCZUO
|
|
LCZUPURMANZUMALCVBAQAOPUOUNUAUBUMURUNJHFZUTUMUQJUNHUMUQDDEFJUMADDAUCZUMUDZV
|
|
FUEUIUFRUMVDUNUNDGFZIFZJKFZUTUMVAVDVIMVCUNUJPUMVHUSJKUMVGAUNIUMASCDSCVGAMVE
|
|
UGADUHUKRULTTT $.
|
|
|
|
$( Pascal's rule for the binomial coefficient, generalized to all integers
|
|
` K ` . Equation 2 of [Gleason] p. 295. (Contributed by NM,
|
|
13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.) $)
|
|
bcpasc $p |- ( ( N e. NN0 /\ K e. ZZ ) ->
|
|
( ( N _C K ) + ( N _C ( K - 1 ) ) ) = ( ( N + 1 ) _C K ) ) $=
|
|
( cn0 wcel cz wa cc0 c1 caddc co cfz cbc cmin wceq wo syl oveq12d cdiv cmul
|
|
oveq2d wn wb peano2nn0 cuz cfv elfzp12 nn0uz eleq2s 1p0e1 clt wbr 0z zsubcl
|
|
bcn0 1z mp2an cr ltm1 ax-mp orci bcval4 mp3an23 3eqtr4a oveq2 oveq1 eqeq12d
|
|
syl5ibrcom simpr 0p1e1 oveq1i syl6eleq cn nn0p1nn nnuz fzm1 biimpa sylan cc
|
|
0re nn0cn ax-1cn pncan sylancl eleq2d wss 1eluzge0 fzss1 sseli bcp1n bcrpcl
|
|
crp rpcnd elfzuz2 syl6eleqr peano2nnd 1cnd elfzelz addsubd fznn0sub eqeltrd
|
|
nncnd nnap0d div12apd nnrpd rpdivcld mulcomd eqtrd npcand adddid divcanap1d
|
|
zcnd 3eqtr2d elfznn divdivap2d bcm1k subsub3d oveq1d fzelp1 elfzm1b syl2anc
|
|
nnzd mpbid eleqtrd divap0d divmulap3d div23apd 3eqtr3rd 3eqtrrd nn0re ltp1d
|
|
mpbird olcd mpd3an23 sylan9eqr bcnn adantr jaodan syldan bcval3 syl3anc imp
|
|
ex jaod sylbid adantlr 00id con3i 3expa sylan2 simpll simplr peano2zm nn0zd
|
|
syl2anr fzp1ss eqsstr3i syl6bir sylbird con3dimp 0zd w3a wdc fzdcel exmiddc
|
|
id mpjaodan ) BCDZAEDZFZAGBHIJZKJZDZBALJZBAHMJZLJZIJZUVJALJZNZUVLUAZUVGUVLU
|
|
VRUVHUVGUVLUVRUVGUVLAGNZAGHIJZUVJKJZDZOZUVRUVGUVJCDZUVLUWDUBZBUCZUWFUVJGUDU
|
|
EZCAGUVJUFUGUHPUVGUVTUVRUWCUVGUVRUVTBGLJZBGHMJZLJZIJZUVJGLJZNUVGHGIJHUWLUWM
|
|
UIUVGUWIHUWKGIBUNUVGUWJEDZUWJGUJUKZBUWJUJUKZOUWKGNGEDZHEDUWNULUOGHUMUPUWOUW
|
|
PGUQDUWOVSGURUSUTUWJBVAVBQUVGUWEUWMHNUWGUVJUNPVCUVTUVPUWLUVQUWMUVTUVMUWIUVO
|
|
UWKIAGBLVDUVTUVNUWJBLAGHMVETQAGUVJLVDVFVGUVGUWCUVRUVGUWCAHUVJKJZDZUVRUVGUWC
|
|
FAUWBUWRUVGUWCVHUWAHUVJKVIVJZVKUVGUWSAHUVJHMJZKJZDZAUVJNZOZUVRUVGUVJHUDUEZD
|
|
ZUWSUXEUVGUVJVLUXFBVMZVNVKUXGUWSUXEAHUVJVOVPVQUVGUXCUVRUXDUVGUXCFAHBKJZDZUV
|
|
RUVGUXCUXJUVGUXBUXIAUVGUXABHKUVGBVRDZHVRDZUXABNZBVTZWABHWBZWCZTWDVPUXJUVQUV
|
|
MUVJUVJAMJZRJSJZUVMUXQRJZUXQSJZUXSASJZIJZUVPUXJAGBKJZDZUVQUXRNUXIUYCAHUWHDU
|
|
XIUYCWEWFHGBWGUSWHZABWIPUXJUXRUXSUVJSJZUXSUXQAIJZSJUYBUXJUXRUVJUXSSJUYFUXJU
|
|
VMUVJUXQUXJUVMUXJUYDUVMWKDUYEABWJPZWLZUXJUVJUXJBUXJBUXFVLAHBWMVNWNZWOZXAZUX
|
|
JUXQUXJUXQBAMJZHIJZVLUXJBHAUXJBUYJXAZUXJWPZUXJAAHBWQZXKZWRUXJUYMCDUYNVLDAHB
|
|
WSUYMVMPWTZXAZUXJUXQUYSXBZXCUXJUVJUXSUYLUXJUXSUXJUVMUXQUYHUXJUXQUYSXDZXEWLZ
|
|
XFXGUXJUYGUVJUXSSUXJUVJAUYLUYRXHTUXJUXSUXQAVUCUYTUYRXIXLUXJUXTUVMUYAUVOIUXJ
|
|
UVMUXQUYIUYTVUAXJUXJUVMUXQARJZRJZUVMASJUXQRJUVOUYAUXJUVMUXQAUYIUYTUYRVUAUXJ
|
|
AABXMZXBZXNUXJVUEUVONUVMUVOVUDSJZNUXJUVMUVOBUVNMJZARJZSJVUHABXOUXJVUJVUDUVO
|
|
SUXJVUIUXQARUXJBAHUYOUYRUYPXPXQTXGUXJUVMUVOVUDUYIUXJUVOUXJUVNUYCDZUVOWKDUXJ
|
|
UVNGUXAKJZUYCUXJUWSUVNVULDZAHBXRUXJUVHUVJEDZUWSVUMUBZUYQUXJUVJUYKYAAUVJXSZX
|
|
TYBUXJUXABGKUXJUXKUXLUXMUYOWAUXOWCTYCUVNBWJPWLUXJVUDUXJUXQAVUBUXJAVUFXDXEWL
|
|
UXJUXQAUYTUYRVUAVUGYDYEYKUXJUVMAUXQUYIUYRUYTVUAYFYGQYHPUVGUXDFZUWAHUVPUVQVI
|
|
VUQUVMGUVOHIUXDUVGUVMBUVJLJZGAUVJBLVDUVGVUNUVJGUJUKZBUVJUJUKZOVURGNUVGUVJUX
|
|
HYAZUVGVUTVUSUVGBBYIYJYLUVJBVAYMYNVUQUVOBBLJZHVUQUVNBBLUXDUVGUVNUXABAUVJHMV
|
|
EUXPYNTUVGVVBHNUXDBYOYPXGQUXDUVGUVQUVJUVJLJZHAUVJUVJLVDUVGUWEVVCHNUWGUVJYOP
|
|
YNVCYQYRYRUUBUUCUUDUUAUUEUVIUVSFZGGIJGUVPUVQUUFVVDUVMGUVOGIUVSUVIUYDUAZUVMG
|
|
NZUYDUVLAGBXRUUGUVGUVHVVEVVFABYSUUHUUIVVDUVGUVNEDZVUKUAUVOGNUVGUVHUVSUUJZVV
|
|
DUVHVVGUVGUVHUVSUUKZAUULPUVIVUKUVLUVIVUKVUMUVLUVIVULUYCUVNUVIUXABGKUVIUXKUX
|
|
LUXMUVGUXKUVHUXNYPWAUXOWCTWDUVIVUMUWSUVLUVHUVHVUNVUOUVGUVHUVEUVGUVJUWGUUMVU
|
|
PUUNUWRUVKAUWRUWBUVKUWTUWQUWBUVKWEULGUVJUUOUSUUPWHUUQUURUUSUVNBYSYTQVVDUWEU
|
|
VHUVSUVQGNVVDUVGUWEVVHUWGPVVIUVIUVSVHAUVJYSYTVCUVIUVHUWQVUNUVLUVSOZUVGUVHVH
|
|
UVIUUTUVGVUNUVHVVAYPUVHUWQVUNUVAUVLUVBVVJAGUVJUVCUVLUVDPYTUVF $.
|
|
|
|
${
|
|
$d k m n N $. $d k m n K $.
|
|
$( A binomial coefficient, in its extended domain, is a nonnegative
|
|
integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro,
|
|
9-Nov-2013.) $)
|
|
bccl $p |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) e. NN0 ) $=
|
|
( vk vm cn0 wcel cv cbc co cz wral cc0 c1 wceq oveq1 eleq1d ralbidv oveq2
|
|
wa 0nn0 vn caddc cfz wn elfz1eq adantl bcn0 ax-mp eqeltri syl6eqel bcval3
|
|
1nn0 syl mp3an1 0zd w3a wdc fzdcel exmiddc mpd3an23 mpjaodan rgen cbvralv
|
|
wo cmin bcpasc adantlr rspccva peano2zm sylan2 nn0addcld adantll eqeltrrd
|
|
ralrimiva ex syl5bi nn0ind sylan ) BEFBCGZHIZEFZCJKZAJFBAHIZEFZDGZVSHIZEF
|
|
ZCJKLVSHIZEFZCJKUAGZVSHIZEFZCJKZWJMUBIZVSHIZEFZCJKZWBDUABWELNZWGWICJWRWFW
|
|
HEWELVSHOPQWEWJNZWGWLCJWSWFWKEWEWJVSHOPQWEWNNZWGWPCJWTWFWOEWEWNVSHOPQWEBN
|
|
ZWGWACJXAWFVTEWEBVSHOPQWICJVSJFZVSLLUCIFZWIXCUDZXBXCSVSLNZWIXCXEXBVSLUEUF
|
|
XEWHLLHIZEVSLLHRXFMELEFZXFMNTLUGUHULUIUJUMXBXDSWHLEXGXBXDWHLNTVSLUKUNTUJX
|
|
BLJFZXHXCXDVDZXBUOZXJXBXHXHUPXCUQXIVSLLURXCUSUMUTVAVBWMWJWEHIZEFZDJKZWJEF
|
|
ZWQWLXLCDJVSWENWKXKEVSWEWJHRPVCXNXMWQXNXMSZWPCJXOXBSWKWJVSMVEIZHIZUBIZWOE
|
|
XNXBXRWONXMVSWJVFVGXMXBXREFXNXMXBSWKXQXLWLDVSJWEVSNXKWKEWEVSWJHRPVHXBXMXP
|
|
JFXQEFZVSVIXLXSDXPJWEXPNXKXQEWEXPWJHRPVHVJVKVLVMVNVOVPVQWAWDCAJVSANVTWCEV
|
|
SABHRPVHVR $.
|
|
|
|
$( A binomial coefficient, in its standard domain, is a positive integer.
|
|
(Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro,
|
|
10-Mar-2014.) $)
|
|
bccl2 $p |- ( K e. ( 0 ... N ) -> ( N _C K ) e. NN ) $=
|
|
( cc0 cfz co wcel cbc cn0 clt wbr cn elfz3nn0 elfzelz bccl syl2anc bcrpcl
|
|
cz rpgt0d elnnnn0b sylanbrc ) ACBDEFZBAGEZHFZCUBIJUBKFUABHFAQFUCABLACBMAB
|
|
NOUAUBABPRUBST $.
|
|
$}
|
|
|
|
$( Compute the binomial coefficient " ` N ` choose 2 " from " ` ( N - 1 ) `
|
|
choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van
|
|
der Vekens, 7-Jan-2018.) $)
|
|
bcn2m1 $p |- ( N e. NN
|
|
-> ( ( N - 1 ) + ( ( N - 1 ) _C 2 ) ) = ( N _C 2 ) ) $=
|
|
( cn wcel c1 cmin co c2 cbc caddc nnm1nn0 nn0cnd cz 2z bccl sylancl addcomd
|
|
cn0 wceq oveq2d eqtrd bcn1 eqcomd syl 1e2m1 a1i bcpasc npcand oveq1d 3eqtrd
|
|
nncn 1cnd ) ABCZADEFZUMGHFZIFUNUMIFUNUMGDEFZHFZIFZAGHFZULUMUNULUMAJZKULUNUL
|
|
UMQCZGLCZUNQCUSMGUMNOKPULUMUPUNIULUMUMDHFZUPULUTUMVBRUSUTVBUMUMUAUBUCULDUOU
|
|
MHDUORULUDUESTSULUQUMDIFZGHFZURULUTVAUQVDRUSMGUMUFOULVCAGHULADAUJULUKUGUHTU
|
|
I $.
|
|
|
|
$( Compute the binomial coefficient " ` ( N + 1 ) ` choose 2 " from " ` N `
|
|
choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der
|
|
Vekens, 8-Jan-2018.) $)
|
|
bcn2p1 $p |- ( N e. NN0 -> ( N + ( N _C 2 ) ) = ( ( N + 1 ) _C 2 ) ) $=
|
|
( cn0 wcel c2 cbc co caddc c1 cmin nn0cn cz 2z bccl mpan2 addcomd bcn1 wceq
|
|
nn0cnd 1e2m1 oveq2d a1i eqtr3d bcpasc 3eqtrd ) ABCZAADEFZGFUFAGFUFADHIFZEFZ
|
|
GFZAHGFDEFZUEAUFAJUEUFUEDKCZUFBCLDAMNROUEAUHUFGUEAHEFAUHAPUEHUGAEHUGQUESUAT
|
|
UBTUEUKUIUJQLDAUCNUD $.
|
|
|
|
$( The number of permutations of ` N - R ` objects from a collection of ` N `
|
|
objects is a positive integer. (Contributed by Jason Orendorff,
|
|
24-Jan-2007.) $)
|
|
permnn $p |- ( R e. ( 0 ... N ) -> ( ( ! ` N ) / ( ! ` R ) ) e. NN ) $=
|
|
( cc0 cfz co wcel cfa cn cmin cmul cc cdiv elfznn0 faccld fznn0sub nnmulcld
|
|
cfv cn0 elfz3nn0 nncnd faccl syl nnap0d divcanap4d eqeltrd cbc bcval2 bccl2
|
|
eqeltrrd nndivtr syl32anc ) ACBDEFZAGQZHFBAIEZGQZUMJEZHFBGQZKFZUPUMLEZHFUQU
|
|
PLEZHFUQUMLEHFULAABMNZULUOUMULUNACBONZVAPULBRFZURABSVCUQBUATUBULUSUOHULUOUM
|
|
ULUOVBTULUMVATULUMVAUCUDVBUEULBAUFEUTHABUGABUHUIUMUPUQUJUK $.
|
|
|
|
$( The binomial coefficent of ` ( N - 1 ) ` is ` N ` . (Contributed by Scott
|
|
Fenton, 16-May-2014.) $)
|
|
bcnm1 $p |- ( N e. NN0 -> ( N _C ( N - 1 ) ) = N ) $=
|
|
( cn0 wcel c1 cbc co cmin cz wceq 1z bccmpl mpan2 bcn1 eqtr3d ) ABCZADEFZAA
|
|
DGFEFZAODHCPQIJDAKLAMN $.
|
|
|
|
$( The value of four choose three. (Contributed by Scott Fenton,
|
|
11-Jun-2016.) $)
|
|
4bc3eq4 $p |- ( 4 _C 3 ) = 4 $=
|
|
( c4 c1 cmin co cbc 4cn ax-1cn 3cn 3p1e4 addcomli subaddrii oveq2i cn0 wcel
|
|
c3 wceq 4nn0 bcnm1 ax-mp eqtr3i ) AABCDZEDZAOEDAUAOAEABOFGHOBAHGIJKLAMNUBAP
|
|
QARST $.
|
|
|
|
$( The value of four choose two. (Contributed by Scott Fenton,
|
|
9-Jan-2017.) $)
|
|
4bc2eq6 $p |- ( 4 _C 2 ) = 6 $=
|
|
( c4 c2 co cfa cfv cmul cdiv c6 cc0 wcel wceq cz cle wbr ax-mp c3 3nn0 df-4
|
|
fveq2i eqtri cbc cmin cfz w3a wa 0z 3pm3.2i 0le2 2re 4re 2lt4 ltleii pm3.2i
|
|
4z 2z elfz4 mp2an bcval2 caddc cn0 facp1 oveq2i 3eqtr4i 4cn 2p2e4 subaddrii
|
|
c1 2cn fac2 oveq12i 2t2e4 cn faccl nncni 4ap0 divcanap4i fac3 ) ABUACZADEZA
|
|
BUBCZDEZBDEZFCZGCZHBIAUCCJZVRWDKILJZALJZBLJZUDIBMNZBAMNZUEWEWFWGWHUFUNUOUGW
|
|
IWJUHBAUIUJUKULUMBIAUPUQBAUROWDPDEZAFCZAGCZHVSWLWCAGPVGUSCZDEZWKWNFCZVSWLPU
|
|
TJZWOWPKQPVAOAWNDRSAWNWKFRVBVCWCBBFCAWABWBBFWAWBBVTBDABBVDVHVHVEVFSVITVIVJV
|
|
KTVJWMWKHWKAWKWQWKVLJQPVMOVNVDVOVPVQTTT $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The ` size ` (set size) function
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c size $.
|
|
|
|
$( Extend the definition of a class to include the set size function. $)
|
|
csize $a class size $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the set size function ` size ` , which gives the cardinality of a
|
|
finite set as a member of ` NN0 ` , and assigns all infinite sets the
|
|
value ` +oo ` . For example, ` ( size `` { 0 , 1 , 2 } ) = 3 ` .
|
|
|
|
Note that we use the sharp sign ( ` size ` ) for this function and we
|
|
use the different character octothorpe ( ` # ` ) for the apartness
|
|
relation (see ~ df-ap ). We adopt the former notation from Corollary
|
|
8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for
|
|
finite sets).
|
|
|
|
This definition (in terms of ` U. ` and ` ~<_ ` ) is not taken directly
|
|
from the literature, but for finite sets should be equivalent to the
|
|
conventional definition that the size of a finite set is the unique
|
|
natural number which is equinumerous to the given set. (Contributed by
|
|
Jim Kingdon, 19-Feb-2022.) $)
|
|
df-size $a |- size = ( ( frec (
|
|
( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. _om , +oo >. } ) o.
|
|
( x e. _V |-> U. { y e. ( _om u. { _om } ) | y ~<_ x } ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $.
|
|
$( The ordinal size of an infinite set is ` _om ` . (Contributed by Jim
|
|
Kingdon, 20-Feb-2022.) $)
|
|
sizeinfuni $p |- ( _om ~<_ A
|
|
-> U. { y e. ( _om u. { _om } ) | y ~<_ A } = _om ) $=
|
|
( vz com cdom wbr cv csn cun crab cuni wcel wss wral wceq omex snid elun2
|
|
wb syl breq1 elrab3 mp2b biimpri wo elrabi elun sylib ordom ordelss elsni
|
|
word mpan eqimss jaoi adantl ralrimiva ssunieq syl2anc eqcomd ) DBEFZDAGZ
|
|
BEFZADDHZIZJZKZVADVFLZCGZDMZCVFNDVGOVHVADVDLDVELVHVASDPQDVDDRVCVAADVEVBDB
|
|
EUAUBUCUDVAVJCVFVIVFLZVJVAVKVIDLZVIVDLZUEZVJVKVIVELVNVCAVIVEUFVIDVDUGUHVL
|
|
VJVMDULVLVJUIDVIUJUMVMVIDOVJVIDUKVIDUNTUOTUPUQCDVFURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( The value of the ` size ` function on an infinite set. (Contributed by
|
|
Jim Kingdon, 20-Feb-2022.) $)
|
|
sizeinf $p |- ( _om ~<_ A -> ( size ` A ) = +oo ) $=
|
|
( vx vy vz com cdom wbr csize cfv cz cv cmpt cc0 cpnf csn wcel wceq ax-mp
|
|
cvv c0 c1 caddc co cfrec cop cun crab cuni ccom df-size fveq1i cdm funmpt
|
|
wfun wrel funrel peano1 reldom brrelex2i sizeinfuni omex syl6eqel rabbidv
|
|
breq2 unieqd eqid fvmptg syl2anc eqtrd syl5eleqr relelfvdm sylancr syl5eq
|
|
fvco fveq2d cxr pnfxr word ordom ordirr wfn wal zex mptex vex fvex ax-gen
|
|
wn 0z frecfnom mp2an fndm eleq2i mtbir fsnunfv mp3an syl6eq ) EAFGZAHIZEB
|
|
JBKZUAUBUCZLZMUDZENUEOUFZIZNWRWSABSCKZWTFGZCEEOUFZUGZUHZLZIZXDIZXEWRWSAXD
|
|
XKUIZIZXMAHXNBCUJUKWRXKUNZAXKULPZXOXMQBSXJUMZWRXKUOZTXLPXQXPXSXRXKUPRWRTE
|
|
XLUQWRXLXFAFGZCXHUGZUHZEWRASPYBSPXLYBQEAFURUSWRYBESCAUTZVAVBBAXJYBSSXKWTA
|
|
QZXIYAYDXGXTCXHWTAXFFVDVCVEXKVFVGVHYCVIZVJTAXKVKVLAXDXKVNVLVMWRXLEXDYEVOV
|
|
IESPNVPPEXCULZPZWHXENQVAVQYGEEPZEVRYHWHVSEVTRYFEEXCEWAZYFEQDKZXBISPZDWBMJ
|
|
PYIYKDYJXBSSBJXAWCWDDWEWFWGWIDMXBJWJWKEXCWLRWMWNXCSVPENWOWPWQ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d N y z $. $d A y z $.
|
|
$( The ordinal size of a set equinumerous to an element of ` _om ` is that
|
|
element of ` _om ` . (Contributed by Jim Kingdon, 20-Feb-2022.) $)
|
|
sizeennnuni $p |- ( ( N e. _om /\ N ~~ A )
|
|
-> U. { y e. ( _om u. { _om } ) | y ~<_ A } = N ) $=
|
|
( vz com wcel cen wbr wa cv cdom csn adantr adantl breq1 elrab syl2anc wb
|
|
mpbid cfn cun crab cuni wss wral wceq elun1 sylanbrc biimpi simprd simplr
|
|
endom ensymd domentr simpr simplll nndomo ad3antrrr enfii wn elsni breq1d
|
|
nnfi infnfi syl pm2.21dd wo simpld elun mpjaodan ralrimiva ssunieq eqcomd
|
|
sylib ) CEFZCBGHZIZCAJZBKHZAEELZUAZUBZUCZVQCWBFZDJZCUDZDWBUECWCUFVQCWAFZC
|
|
BKHZWDVOWGVPCEVTUGMVPWHVOCBULNVSWHACWAVRCBKOPUHVQWFDWBVQWEWBFZIZWEEFZWFWE
|
|
VTFZWJWKIZWECKHZWFWJWNWKWJWEBKHZBCGHZWNWJWEWAFZWOWIWQWOIZVQWIWRVSWOAWEWAV
|
|
RWEBKOPUINZUJZWJCBVOVPWIUKUMZWEBCUNQMWMWKVOWNWFRWJWKUOVOVPWIWKUPWECUQQSWJ
|
|
WLIZBTFZWFXBCTFZWPXCVOXDVPWIWLCVCURWJWPWLXAMBCUSQXBEBKHZXCUTXBWOXEWJWOWLW
|
|
TMWLWOXERWJWLWEEBKWEEVAVBNSBVDVEVFWJWQWKWLVGWJWQWOWSVHWEEVTVIVNVJVKDCWBVL
|
|
QVM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d N x y z $.
|
|
$( The size of a set equinumerous to an element of ` _om ` . (Contributed
|
|
by Jim Kingdon, 21-Feb-2022.) $)
|
|
sizeennn $p |- ( ( N e. _om /\ N ~~ A )
|
|
-> ( size ` A ) = ( frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` N ) ) $=
|
|
( vy vz com wcel cen wbr wa csize cfv cv cvv cdom wceq cfn adantr syl2anc
|
|
eqtrd cz c1 caddc co cmpt cc0 cfrec cpnf cop csn crab cuni df-size fveq1i
|
|
cun ccom wfun cdm funmpt copab sizeennnuni eqcomd nnfi simpr ensymd enfii
|
|
wb simpl breq2 rabbidv unieqd eqeq12d opelopabga mpbird mptv syl6eleqr wi
|
|
opeldmg mpd fvco sylancr syl5eq elexd eqeltrd eqid fvmptg fveq2d wne word
|
|
wn ordom ordirr ax-mp eleq1 mtbii necon2ai fvunsng mpdan ) CFGZCBHIZJZBKL
|
|
ZCAUAAMZUBUCUDUEUFUGZFUHUIUJUOZLZCXDLZXAXBBANDMZXCOIZDFFUJUOZUKZULZUEZLZX
|
|
ELZXFXAXBBXEXMUPZLZXOBKXPADUMUNXAXMUQBXMURGZXQXOPANXLUSXABCUIZXMGZXRXAXSE
|
|
MZXLPZAEUTZXMXAXSYCGZCXHBOIZDXJUKZULZPZXAYGCDBCVAZVBXABQGZWSYDYHVGXACQGZB
|
|
CHIYJWSYKWTCVCRXACBWSWTVDVEBCVFSZWSWTVHZYBYHAEBCQFXCBPZYACPZJZYACXLYGYNYO
|
|
VDYPXKYFYPXIYEDXJYNXIYEVGYOXCBXHOVIZRVJVKVLVMSVNAEXLVOVPXAYJWSXTXRVQYLYMB
|
|
CXMQFVRSVSBXEXMVTWAWBXAXNCXEXAXNYGCXABNGYGFGXNYGPXABQYLWCXAYGCFYIYMWDABXL
|
|
YGNFXMYNXKYFYNXIYEDXJYQVJVKXMWEWFSYITWGTWSXFXGPZWTWSFCWHYRWSFCFCPFFGZWSFW
|
|
IYSWJWKFWLWMFCFWNWOWPXDFUHCFWQWRRT $.
|
|
$}
|
|
|
|
${
|
|
$d A n x $.
|
|
$( Closure of the ` size ` function on an finite set. (Contributed by Jim
|
|
Kingdon, 20-Feb-2022.) $)
|
|
sizecl $p |- ( A e. Fin -> ( size ` A ) e. NN0 ) $=
|
|
( vn vx cfn wcel cv cen wbr csize cfv cn0 wrex isfi biimpi wa cz c1 caddc
|
|
com cc0 co cmpt cfrec wceq simprl simprr ensymd sizeennn syl2anc cuz eqid
|
|
0zd id frec2uzuzd nn0uz syl6eleqr syl eqeltrd rexlimddv ) ADEZABFZGHZAIJZ
|
|
KEBSUTVBBSLBAMNUTVASEZVBOOZVCVACPCFQRUAUBTUCZJZKVEVDVAAGHVCVGUDUTVDVBUEZV
|
|
EAVAUTVDVBUFUGCAVAUHUIVEVDVGKEVHVDVGTUJJKVDCVATVFVDULVFUKVDUMUNUOUPUQURUS
|
|
$.
|
|
$}
|
|
|
|
$( The size of a finite set is either 0 or 1 or greater than 1. (Contributed
|
|
by Jim Kingdon, 21-Feb-2022.) $)
|
|
sizefiv01gt1 $p |- ( M e. Fin
|
|
-> ( ( size ` M ) = 0 \/ ( size ` M ) = 1 \/ 1 < ( size ` M ) ) ) $=
|
|
( cfn wcel csize cfv cc0 clt wbr c1 w3o cfz co wa simpr wn cn0 wo adantl cz
|
|
wceq sizecl nn0nlt0 syl adantr pm2.21dd orc fz01or df-3or 3imtr4i 3mix3 0zd
|
|
nn0zd 1zzd fztri3or syl3anc mpjao3dan ) ABCZADEZFGHZURFTZURITZIURGHZJZURFIK
|
|
LCZVBUQUSMUSVCUQUSNUQUSOZUSUQURPCVEAUAZURUBUCUDUEVDVCUQUTVAQZVGVBQVDVCVGVBU
|
|
FURUGUTVAVBUHUIRVBVCUQVBUTVAUJRUQURSCFSCISCUSVDVBJUQURVFULUQUKUQUMURFIUNUOU
|
|
P $.
|
|
|
|
${
|
|
$d A m n x $. $d B m n x $. $d N x y $.
|
|
$( The set ` ( 1 ... N ) ` has ` N ` elements. (Contributed by Jim
|
|
Kingdon, 21-Feb-2022.) $)
|
|
sizefz1 $p |- ( N e. NN0 -> ( size ` ( 1 ... N ) ) = N ) $=
|
|
( vx vy cn0 wcel c1 cfz co csize cfv cz cv caddc cmpt cc0 cfrec wceq wf1o
|
|
com syl2anc ccnv cen wbr cuz wf eqid frec2uzf1od f1ocnv f1of 3syl elnn0uz
|
|
biimpi ffvelrnd frecfzennn ensymd sizeennn wa oveq1 cbvmptv freceq1 ax-mp
|
|
0zd fveq1i f1ocnvfv2 syl5eqr eqtrd ) ADEZFAGHZIJZABKBLZFMHZNZOPZUAZJZCKCL
|
|
ZFMHZNZOPZJZAVGVOSEVOVHUBUCVIVTQVGOUDJZSAVNVGSWAVMRZWASVNRWASVNUEVGBOVMVG
|
|
VBVMUFZUGZSWAVMUHWASVNUIUJVGAWAEZAUKULZUMVGVHVOBVMAWCUNUOCVHVOUPTVGWBWEVT
|
|
AQWDWFWBWEUQVTVOVMJAVOVMVSVLVRQVMVSQBCKVKVQVJVPFMURUSOVLVRUTVAVCSWAAVMVDV
|
|
ETVF $.
|
|
|
|
$( Two finite sets have the same number of elements iff they are
|
|
equinumerous. (Contributed by Jim Kingdon, 21-Feb-2022.) $)
|
|
sizeen $p |- ( ( A e. Fin /\ B e. Fin ) ->
|
|
( ( size ` A ) = ( size ` B ) <-> A ~~ B ) ) $=
|
|
( vn vm vx cfn wcel wa cv cen wbr csize cfv wceq com wrex syl2anc syl cc0
|
|
wb isfi biimpi adantr ad2antlr simplrl simprl nneneq simplrr enen1 simprr
|
|
enen2 bitrd cz c1 caddc co cmpt cfrec ensymd sizeennn eqeq12d cuz wf1 0zd
|
|
wf1o eqid frec2uzf1od f1of1 f1fveq syl12anc 3bitr4rd rexlimddv ) AFGZBFGZ
|
|
HZACIZJKZALMZBLMZNZABJKZTZCOVMVQCOPZVNVMWCCAUAUBUCVOVPOGZVQHZHZBDIZJKZWBD
|
|
OVNWHDOPZVMWEVNWIDBUAUBUDWFWGOGZWHHZHZVPWGJKZVPWGNZWAVTWLWDWJWMWNTVOWDVQW
|
|
KUEZWFWJWHUFZVPWGUGQWLWAVPBJKZWMWLVQWAWQTVOWDVQWKUHZAVPBUIRWLWHWQWMTWFWJW
|
|
HUJZBWGVPUKRULWLVTVPEUMEIUNUOUPUQSURZMZWGWTMZNZWNWLVRXAVSXBWLWDVPAJKVRXAN
|
|
WOWLAVPWRUSEAVPUTQWLWJWGBJKVSXBNWPWLBWGWSUSEBWGUTQVAWLOSVBMZWTVCZWDWJXCWN
|
|
TWLOXDWTVEXEWLESWTWLVDWTVFVGOXDWTVHRWOWPOXDVPWGWTVIVJULVKVLVL $.
|
|
$}
|
|
|
|
${
|
|
$d A f $. $d B f $.
|
|
$( The size of two finite sets is equal if and only if there is a bijection
|
|
mapping one of the sets onto the other. (Contributed by Alexander van
|
|
der Vekens, 17-Dec-2017.) $)
|
|
sizeeqf1o $p |- ( ( A e. Fin /\ B e. Fin )
|
|
-> ( ( size ` A ) = ( size ` B ) <-> E. f f : A -1-1-onto-> B ) ) $=
|
|
( cfn wcel wa csize cfv wceq cen wbr cv wf1o wex sizeen bren syl6bb ) ADE
|
|
BDEFAGHBGHIABJKABCLMCNABOABCPQ $.
|
|
|
|
$( There is no bijection between a finite set and an infinite set. By
|
|
~ infnfi the theorem would also hold if "infinite" were expressed as
|
|
` _om ~<_ B ` . (Contributed by Alexander van der Vekens,
|
|
25-Dec-2017.) $)
|
|
fiinfnf1o $p |- ( ( A e. Fin /\ -. B e. Fin )
|
|
-> -. E. f f : A -1-1-onto-> B ) $=
|
|
( cfn wcel cv wf1o wex f1ofi ex exlimdv con3dimp ) ADEZABCFZGZCHBDEZMOPCM
|
|
OPABNIJKL $.
|
|
$}
|
|
|
|
$( The codomain of an onto function is a set if its domain is a set.
|
|
(Contributed by AV, 4-May-2021.) $)
|
|
focdmex $p |- ( ( A e. V /\ F : A -onto-> B ) -> B e. _V ) $=
|
|
( wcel wfo wa crn cvv wf fof anim2i ancomd rnexg 3syl wb forn eleq1d adantl
|
|
fex mpbid ) ADEZABCFZGZCHZIEZBIEZUDABCJZUBGCIEUFUDUBUHUCUHUBABCKLMABDCTCINO
|
|
UCUFUGPUBUCUEBIABCQRSUA $.
|
|
|
|
${
|
|
$d A f $. $d B f $. $d F f $.
|
|
$( The size of two finite sets is equal if there is a bijection mapping one
|
|
of the sets onto the other. (Contributed by Jim Kingdon,
|
|
21-Feb-2022.) $)
|
|
sizeeqf1oi $p |- ( ( A e. Fin /\ F : A -1-1-onto-> B )
|
|
-> ( size ` A ) = ( size ` B ) ) $=
|
|
( vf cfn wcel wf1o wa csize cfv wceq cv wex f1odm eleq1d biimparc wfun wb
|
|
cdm mpbird f1ofun adantl fundmfibi syl simpr f1oeq1 spcegv sylc sizeeqf1o
|
|
f1ofi syldan ) AEFZABCGZHZAIJBIJKZABDLZGZDMZUNCEFZUMURUNUSCSZEFZUMVAULUMU
|
|
TAEABCNOPUNCQZUSVARUMVBULABCUAUBCUCUDTULUMUEUQUMDCEABUPCUFUGUHULUMBEFUOUR
|
|
RABCUJABDUIUKT $.
|
|
$}
|
|
|
|
${
|
|
$d A f $. $d B f $. $d F f $.
|
|
$( The size of a finite set which is a one-to-one function is equal to the
|
|
size of the function's range. (Contributed by Jim Kingdon,
|
|
21-Feb-2022.) $)
|
|
sizef1rn $p |- ( ( A e. Fin /\ F : A -1-1-> B )
|
|
-> ( size ` F ) = ( size ` ran F ) ) $=
|
|
( vf vx cfn wcel wf1 wa crn cv wf1o wex csize cfv wceq c2nd wi cvv ex wfn
|
|
f1fn simpl fnfi syl2an2 cres f1o2ndf1 wfun csn cuni df-2nd funmpt2 wf f1f
|
|
anim2i ancomd fex syl resfunexg sylancr f1oeq1 biimpd eqcoms adantl com13
|
|
spcimedv mpcom impcom sizeeqf1oi exlimdv sylc ) AFGZABCHZIZCFGZCCJZDKZLZD
|
|
MZCNOVPNOPZVMCAUAVLVLVOABCUBVLVMUCACUDUEVMVLVSCVPQCUFZLZVMVLVSRABCUGVLVMW
|
|
BVSVLVMWBVSRVNVRWBDWASVNQUHCSGZWASGESEKUIJUJQEUKULVNABCUMZVLIWCVNVLWDVMWD
|
|
VLABCUNUOUPABFCUQURQCSUSUTVQWAPWBVRRZVNWEWAVQWAVQPWBVRCVPWAVQVAVBVCVDVFTV
|
|
EVGVHVOVRVTDVOVRVTCVPVQVITVJVK $.
|
|
$}
|
|
|
|
${
|
|
sizeeqf1od.a $e |- ( ph -> A e. Fin ) $.
|
|
sizeeqf1od.f $e |- ( ph -> F : A -1-1-onto-> B ) $.
|
|
$( The size of two finite sets is equal if there is a bijection mapping one
|
|
of the sets onto the other. (Contributed by Jim Kingdon,
|
|
21-Feb-2022.) $)
|
|
sizeeqf1od $p |- ( ph -> ( size ` A ) = ( size ` B ) ) $=
|
|
( cfn wcel wf1o csize cfv wceq sizeeqf1oi syl2anc ) ABGHBCDIBJKCJKLEFBCDM
|
|
N $.
|
|
$}
|
|
|
|
$( Two possibly-empty 1-based finite sets of sequential integers are equal
|
|
iff their endpoints are equal. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.) $)
|
|
fz1eqb $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( ( 1 ... M ) = ( 1 ... N ) <-> M = N ) ) $=
|
|
( cn0 wcel wa c1 cfz co wceq csize cfv fveq2 sizefz1 eqeqan12d syl5ib oveq2
|
|
impbid1 ) ACDZBCDZEZFAGHZFBGHZIZABIZUCUAJKZUBJKZITUDUAUBJLRSUEAUFBAMBMNOABF
|
|
GPQ $.
|
|
|
|
$( The size of an infinite set is greater than the size of a finite set.
|
|
(Contributed by Jim Kingdon, 21-Feb-2022.) $)
|
|
filtinf $p |- ( ( A e. Fin /\ _om ~<_ B )
|
|
-> ( size ` A ) < ( size ` B ) ) $=
|
|
( cfn wcel com cdom wbr wa csize cfv cpnf clt cr sizecl nn0red ltpnf adantr
|
|
syl wceq sizeinf adantl breqtrrd ) ACDZEBFGZHAIJZKBIJZLUCUEKLGZUDUCUEMDUGUC
|
|
UEANOUEPRQUDUFKSUCBTUAUB $.
|
|
|
|
$( A finite set is equinumerous to the range of integers from one up to the
|
|
hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.) $)
|
|
isfinite4im $p |- ( A e. Fin -> ( 1 ... ( size ` A ) ) ~~ A ) $=
|
|
( cfn wcel c1 csize cfv cfz co wceq cen wbr cn0 sizecl sizefz1 syl wb cz 1z
|
|
nn0zd fzfig sylancr sizeen mpancom mpbid ) ABCZDAEFZGHZEFUFIZUGAJKZUEUFLCUH
|
|
AMZUFNOUGBCZUEUHUIPUEDQCUFQCUKRUEUFUJSDUFTUAUGAUBUCUD $.
|
|
|
|
$( Two ways of saying a finite set is empty. (Contributed by Paul Chapman,
|
|
26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized
|
|
by Jim Kingdon, 23-Feb-2022.) $)
|
|
sizeeq0 $p |- ( A e. Fin -> ( ( size ` A ) = 0 <-> A = (/) ) ) $=
|
|
( cfn wcel csize cfv c0 wceq cen wbr cc0 wb 0fin sizeen mpan2 c1 cfz fveq2i
|
|
co fz10 cn0 0nn0 sizefz1 ax-mp eqtr3i eqeq2i en0 3bitr3g ) ABCZADEZFDEZGZAF
|
|
HIZUIJGAFGUHFBCUKULKLAFMNUJJUIOJPRZDEZUJJUMFDSQJTCUNJGUAJUBUCUDUEAUFUG $.
|
|
|
|
$( Two ways of saying a finite set is not empty. Also, "A is inhabited"
|
|
would be equivalent by ~ fin0 . (Contributed by Alexander van der Vekens,
|
|
23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.) $)
|
|
sizeneq0 $p |- ( A e. Fin -> ( 0 < ( size ` A ) <-> A =/= (/) ) ) $=
|
|
( cfn csize cfv cc0 cap wbr wne clt c0 cz wb sizecl nn0zd 0zd zapne syl2anc
|
|
wcel cn0 cr cle nn0re nn0ge0 ap0gt0 syl sizeeq0 necon3bid 3bitr3d ) ABRZACD
|
|
ZEFGZUJEHZEUJIGZAJHUIUJKREKRUKULLUIUJAMZNUIOUJEPQUIUJSRZUKUMLZUNUOUJTREUJUA
|
|
GUPUJUBUJUCUJUDQUEUIUJEAJAUFUGUH $.
|
|
|
|
$( Positive natural closure of the set size function. (Contributed by Mario
|
|
Carneiro, 16-Jan-2015.) $)
|
|
sizenncl $p |- ( A e. Fin -> ( ( size ` A ) e. NN <-> A =/= (/) ) ) $=
|
|
( cfn wcel csize cfv cn c0 wne wb cc0 wceq wa simpr nnne0 sizeeq0 necon3bid
|
|
adantl adantr mpbid wn 2thd necon2bi biimpa syl 2falsed cn0 wo sizecl elnn0
|
|
nner sylib mpjaodan ) ABCZADEZFCZUOAGHZIUNJKZUMUOLZUOUPUMUOMURUNJHZUPUOUSUM
|
|
UNNZQUMUSUPIUOUMUNJAGAOZPRSUAUMUQLZUOUPUQUOTUMUOUNJUTUBQVBAGKZUPTUMUQVCVAUC
|
|
AGUJUDUEUMUNUFCUOUQUGAUHUNUIUKUL $.
|
|
|
|
$( The empty set has size zero. (Contributed by Mario Carneiro,
|
|
8-Jul-2014.) $)
|
|
size0 $p |- ( size ` (/) ) = 0 $=
|
|
( c0 csize cfv cc0 wceq eqid cfn wcel wb 0fin sizeeq0 ax-mp mpbir ) ABCDEZA
|
|
AEZAFAGHNOIJAKLM $.
|
|
|
|
$( The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.)
|
|
(Proof shortened by Mario Carneiro, 13-Feb-2013.) $)
|
|
sizesng $p |- ( A e. V -> ( size ` { A } ) = 1 ) $=
|
|
( wcel csn csize cfv c1 cen wbr cz 1z en2sn mpan2 cfn wb snfig ax-mp sizeen
|
|
wceq sylancl mpbird cfz co cn0 1nn0 sizefz1 fzsn fveq2d syl5reqr syl6eq ) A
|
|
BCZADZEFZGDZEFZGUKUMUOSZULUNHIZUKGJCZUQKAGBJLMUKULNCUNNCZUPUQOABPURUSKGJPQU
|
|
LUNRTUAURUOGSKURGGGUBUCZEFZUOGUDCVAGSUEGUFQURUTUNEGUGUHUIQUJ $.
|
|
|
|
$( A finite set has size 1 if and only if it is equinumerous to the ordinal
|
|
1. (Contributed by AV, 14-Apr-2019.) (Intuitoinized by Jim Kingdon,
|
|
23-Feb-2022.) $)
|
|
sizeen1 $p |- ( A e. Fin -> ( ( size ` A ) = 1 <-> A ~~ 1o ) ) $=
|
|
( cfn wcel csize cfv c1 c0 csn cen wbr c1o cvv 0ex sizesng ax-mp eqcomi a1i
|
|
wceq eqeq2d wb snfig sizeen mpan2 df1o2 breq2i 3bitrd ) ABCZADEZFRUHGHZDEZR
|
|
ZAUIIJZAKIJZUGFUJUHFUJRUGUJFGLCZUJFRMGLNOPQSUGUIBCZUKULTUNUOMGLUAOAUIUBUCUL
|
|
UMTUGUIKAIKUIUDPUEQUF $.
|
|
|
|
$( A function on a finite set is equinumerous to its domain. (Contributed by
|
|
Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon,
|
|
24-Feb-2022.) $)
|
|
sizefn $p |- ( ( F Fn A /\ A e. Fin ) -> ( size ` F ) = ( size ` A ) ) $=
|
|
( wfn cfn wcel wa csize cfv wceq cen fndmeng ensymd wb fnfi sizeen sylancom
|
|
wbr mpbird ) BACZADEZFZBGHAGHIZBAJQZUAABADBKLSTBDEUBUCMABNBAOPR $.
|
|
|
|
$( The value of the size function on a finite 1-based sequence. (Contributed
|
|
by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro,
|
|
12-Mar-2015.) $)
|
|
fseq1size $p |- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( size ` F ) = N ) $=
|
|
( cn0 wcel c1 cfz co wfn csize cfv cfn wceq simpr simpl nn0zd fzfigd sizefn
|
|
wa 1zzd syl2anc sizefz1 adantr eqtrd ) BCDZAEBFGZHZRZAIJZUEIJZBUGUFUEKDUHUI
|
|
LUDUFMUGEBUGSUGBUDUFNOPUEAQTUDUIBLUFBUAUBUC $.
|
|
|
|
${
|
|
$d A n z $. $d B n $. $d G n z $.
|
|
omgadd.1 $e |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) $.
|
|
$( Mapping ordinal addition to integer addition. (Contributed by Jim
|
|
Kingdon, 24-Feb-2022.) $)
|
|
omgadd $p |- ( ( A e. _om /\ B e. _om ) ->
|
|
( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) $=
|
|
( com wcel coa co cfv caddc wceq wi c0 oveq2 fveq2d oveq2d cc0 cn0 c1 0zd
|
|
vn vz cv csuc fveq2 eqeq12d imbi2d wf1o wf frechashgf1o f1of ax-mp nn0cnd
|
|
ffvelrni addid1d frec2uz0d 3eqtr4rd w3a wa nnasuc nnacl frec2uzsucd eqtrd
|
|
nna0 3adant3 3ad2ant1 3ad2ant2 1cnd addassd oveq1 3ad2ant3 3eqtr4d 3expia
|
|
cc id expcom a2d finds impcom ) CFGBFGZBCHIZDJZBDJZCDJZKIZLZWABUBUDZHIZDJ
|
|
ZWDWHDJZKIZLZMWABNHIZDJZWDNDJZKIZLZMWABUCUDZHIZDJZWDWSDJZKIZLZMWABWSUEZHI
|
|
ZDJZWDXEDJZKIZLZMWAWGMUBUCCWHNLZWMWRWAXKWJWOWLWQXKWIWNDWHNBHOPXKWKWPWDKWH
|
|
NDUFQUGUHWHWSLZWMXDWAXLWJXAWLXCXLWIWTDWHWSBHOPXLWKXBWDKWHWSDUFQUGUHWHXELZ
|
|
WMXJWAXMWJXGWLXIXMWIXFDWHXEBHOPXMWKXHWDKWHXEDUFQUGUHWHCLZWMWGWAXNWJWCWLWF
|
|
XNWIWBDWHCBHOPXNWKWEWDKWHCDUFQUGUHWAWDRKIWDWQWOWAWDWAWDFSBDFSDUIFSDUJADEU
|
|
KFSDULUMZUOUNZUPWAWPRWDKWAARDWAUAEUQQWAWNBDBVEPURWSFGZWAXDXJWAXQXDXJMWAXQ
|
|
XDXJWAXQXDUSZXGXATKIZXIWAXQXGXSLXDWAXQUTZXGWTUEZDJXSXTXFYADBWSVAPXTAWTRDX
|
|
TUAEBWSVBVCVDVFXRXCTKIZWDXBTKIZKIZXSXIXRWDXBTWAXQWDVOGXDXPVGXQWAXBVOGXDXQ
|
|
XBFSWSDXOUOUNVHXRVIVJXDWAXSYBLXQXAXCTKVKVLXQWAXIYDLXDXQXHYCWDKXQAWSRDXQUA
|
|
EXQVPVCQVHVMVDVNVQVRVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x $. $d B m n x $.
|
|
$( Dominance relation for the size function. (Contributed by Jim Kingdon,
|
|
24-Feb-2022.) $)
|
|
sizedom $p |- ( ( A e. Fin /\ B e. Fin )
|
|
-> ( ( size ` A ) <_ ( size ` B ) <-> A ~<_ B ) ) $=
|
|
( vn vm vx cfn wcel wa cv cen wbr csize cfv cle cdom wb wrex isfi syl2anc
|
|
com biimpi adantr ad2antlr simplrr domen1 syl simprr domen2 cz caddc cmpt
|
|
wss c1 co cc0 cfrec 0zd eqid simplrl simprl frec2uzled nndomo wceq ensymd
|
|
sizeennn breq12d 3bitr4rd rexlimddv ) AFGZBFGZHZACIZJKZALMZBLMZNKZABOKZPZ
|
|
CTVIVMCTQZVJVIVSCARUAUBVKVLTGZVMHZHZBDIZJKZVRDTVJWDDTQZVIWAVJWEDBRUAUCWBW
|
|
CTGZWDHZHZAWCOKZVLWCOKZVQVPWHVMWIWJPVKVTVMWGUDZAVLWCUEUFWHWDVQWIPWBWFWDUG
|
|
ZBWCAUHUFWHVLWCULZVLEUIEIUMUJUNUKUOUPZMZWCWNMZNKWJVPWHEVLWCUOWNWHUQWNURVK
|
|
VTVMWGUSZWBWFWDUTZVAWHVTWFWJWMPWQWRVLWCVBSWHVNWOVOWPNWHVTVLAJKVNWOVCWQWHA
|
|
VLWKVDEAVLVESWHWFWCBJKVOWPVCWRWHBWCWLVDEBWCVESVFVGVGVHVH $.
|
|
$}
|
|
|
|
${
|
|
$d A j w k y z $. $d B j w y z $. $d M j $. $d N j w k y z $.
|
|
$d j ph w y z $.
|
|
sizeunlem.a $e |- ( ph -> A e. Fin ) $.
|
|
sizeunlem.b $e |- ( ph -> B e. Fin ) $.
|
|
sizeunlem.disj $e |- ( ph -> ( A i^i B ) = (/) ) $.
|
|
sizeunlem.n $e |- ( ph -> N e. _om ) $.
|
|
sizeunlem.m $e |- ( ph -> M e. _om ) $.
|
|
sizeunlem.an $e |- ( ph -> A ~~ N ) $.
|
|
sizeunlem.bm $e |- ( ph -> B ~~ M ) $.
|
|
$( Lemma for ~ sizeun . Ordinal size of the union. (Contributed by Jim
|
|
Kingdon, 25-Feb-2022.) $)
|
|
sizeunlem $p |- ( ph -> ( A u. B ) ~~ ( N +o M ) ) $=
|
|
( vj cen wbr wa com c0 wceq wcel vw vy vz vk cv cun coa co wrex csn breq1
|
|
uneq2 breq1d anbi12d rexbidv peano1 a1i 0ex enref un0 3brtr4d breq2 oveq2
|
|
nna0 syl breq2d rspcev syl12anc cfn wss cdif csuc peano2 ad2antlr simp-4r
|
|
cvv vex simprr ad2antrr eldifbd eldifd simplr simprl fiunsnnn cin ad4antr
|
|
syl22anc incom eqeq1i sylan2b syl5eqr syl2anc unfidisj syl3anc wn eldifad
|
|
ssdisj minel wo ioran elun xchnxbir sylanbrc nnacl unass eqcomd nnasuc ex
|
|
rexlimdva cbvrexv syl6ibr findcard2sd simprrr ensymd entr syl2an2r nneneq
|
|
simprrl wb mpbid oveq2d breqtrrd rexlimddv ) ACMUEZNOZBCUFZEYDUGUHZNOZPZY
|
|
FEDUGUHZNOMQAUAUEZYDNOZBYKUFZYGNOZPZMQUIRYDNOZBRUFZYGNOZPZMQUIZUBUEZYDNOZ
|
|
BUUAUFZYGNOZPZMQUIZUUAUCUEZUJZUFZYDNOZBUUIUFZYGNOZPZMQUIZYIMQUIUAUBUCCYKR
|
|
SZYOYSMQUUOYLYPYNYRYKRYDNUKUUOYMYQYGNYKRBULUMUNUOYKUUASZYOUUEMQUUPYLUUBYN
|
|
UUDYKUUAYDNUKUUPYMUUCYGNYKUUABULUMUNUOYKUUISZYOUUMMQUUQYLUUJYNUULYKUUIYDN
|
|
UKUUQYMUUKYGNYKUUIBULUMUNUOYKCSZYOYIMQUURYLYEYNYHYKCYDNUKUURYMYFYGNYKCBUL
|
|
UMUNUOARQTZRRNOZYQERUGUHZNOZYTUUSAUPUQUUTARURUSUQABEYQUVANKYQBSABUTUQAEQT
|
|
ZUVAESIEVDVEVAYSUUTUVBPMRQYDRSZYPUUTYRUVBYDRRNVBUVDYGUVAYQNYDREUGVCVFUNVG
|
|
VHAUUAVITZPZUUACVJZUUGCUUAVKTZPZPZUUFUUIUDUEZNOZUUKEUVKUGUHZNOZPZUDQUIZUU
|
|
NUVJUUEUVPMQUVJYDQTZPZUUEUVPUVRUUEPZYDVLZQTZUUIUVTNOZUUKEUVTUGUHZNOZUVPUV
|
|
QUWAUVJUUEYDVMVNUVSUVEUUGVPUUAVKTUVQUUBUWBAUVEUVIUVQUUEVOZUVSUUGVPUUAUUGV
|
|
PTUVSUCVQUQZUVSUUGCUUAUVJUVHUVQUUEUVFUVGUVHVRVSZVTZWAUVJUVQUUEWBZUVRUUBUU
|
|
DWCUUAUUGYDWDWGUVSUUCUUHUFZYGVLZUUKUWCNUVSUUCVITZUUGVPUUCVKTYGQTZUUDUWJUW
|
|
KNOUVSBVITZUVEBUUAWEZRSZUWLAUWNUVEUVIUVQUUEFWFUWEUVSUVGBCWEZRSZUWPUVJUVGU
|
|
VQUUEUVFUVGUVHWCVSAUWRUVEUVIUVQUUEHWFZUVGUWRPUWOUUABWEZRUUABWHUWRUVGCBWEZ
|
|
RSUWTRSUWQUXARBCWHWIUUACBWQWJWKWLBUUAWMWNUVSUUGVPUUCUWFUVSUUGBTZWOZUUGUUA
|
|
TZWOZUUGUUCTZWOUVSUUGCTUWRUXCUVSUUGCUUAUWGWPUWSUUGCBWRWLUWHUXBUXDWSUXCUXE
|
|
PUXFUXBUXDWTUUGBUUAXAXBXCWAUVSUVCUVQUWMAUVCUVEUVIUVQUUEIWFZUWIEYDXDWLUVRU
|
|
UBUUDVRUUCUUGYGWDWGUVSUWJUUKUWJUUKSUVSBUUAUUHXEUQXFUVSUVCUVQUWCUWKSUXGUWI
|
|
EYDXGWLVAUVOUWBUWDPUDUVTQUVKUVTSZUVLUWBUVNUWDUVKUVTUUINVBUXHUVMUWCUUKNUVK
|
|
UVTEUGVCVFUNVGVHXHXIUUMUVOMUDQYDUVKSZUUJUVLUULUVNYDUVKUUINVBUXIYGUVMUUKNY
|
|
DUVKEUGVCVFUNXJXKGXLAUVQYIPZPZYFYGYJNAUVQYEYHXMUXKDYDEUGUXKDYDNOZDYDSZADC
|
|
NOUXJYEUXLACDLXNAUVQYEYHXRDCYDXOXPADQTUXJUVQUXLUXMXSJAUVQYIWCDYDXQXPXTYAY
|
|
BYC $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x $. $d B m n x $.
|
|
$( The size of the union of disjoint finite sets is the sum of their sizes.
|
|
(Contributed by Jim Kingdon, 25-Feb-2022.) $)
|
|
sizeun $p |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) ->
|
|
( size ` ( A u. B ) ) = ( ( size ` A ) + ( size ` B ) ) ) $=
|
|
( vn vm vx cfn wcel wceq cv cen wbr csize cfv caddc co com wa syl2anc syl
|
|
sizeennn cin c0 w3a cun wrex isfi biimpi 3ad2ant1 3ad2ant2 adantr cz cmpt
|
|
coa cc0 cfrec simplrl simprl eqid omgadd nnacl enrefg vex sylancl oveq12d
|
|
c1 enref 3eqtr4d simpll1 simpll2 simpll3 simplrr simprr unfidisj ad2antrr
|
|
sizeunlem wb nnfi sizeen mpbird rexlimddv ) AFGZBFGZABUAUBHZUCZACIZJKZABU
|
|
DZLMZALMZBLMZNOZHZCPWAWBWFCPUEZWCWAWMCAUFUGUHWDWEPGZWFQZQZBDIZJKZWLDPWDWR
|
|
DPUEZWOWBWAWSWCWBWSDBUFUGUIUJWPWQPGZWRQZQZWEWQUMOZLMZWELMZWQLMZNOZWHWKXBX
|
|
CEUKEIVENOULUNUOZMZWEXHMZWQXHMZNOZXDXGXBWNWTXIXLHWDWNWFXAUPZWPWTWRUQZEWEW
|
|
QXHXHURUSRXBXCPGZXCXCJKZXDXIHXBWNWTXOXMXNWEWQUTZRZXBXOXPXRXCPVASEXCXCTRXB
|
|
XEXJXFXKNXBWNWEWEJKXEXJHXMWECVBVFEWEWETVCXBWTWQWQJKXFXKHXNWQDVBVFEWQWQTVC
|
|
VDVGXBWHXDHZWGXCJKZXBABWQWEWAWBWCWOXAVHZWAWBWCWOXAVIZWAWBWCWOXAVJXMXNWDWN
|
|
WFXAVKZWPWTWRVLZVOXBWGFGZXCFGZXSXTVPWDYEWOXAABVMVNXBWNWTYFXMXNWNWTQXOYFXQ
|
|
XCVQSRWGXCVRRVSXBWIXEWJXFNXBWIXEHZWFYCXBWAWEFGZYGWFVPYAXBWNYHXMWEVQSAWEVR
|
|
RVSXBWJXFHZWRYDXBWBWQFGZYIWRVPYBXBWTYJXNWQVQSBWQVRRVSVDVGVTVT $.
|
|
$}
|
|
|
|
$( 1 is an element of the finite set of sequential nonnegative integers
|
|
bounded by the size of a nonempty finite set. (Contributed by AV,
|
|
9-May-2020.) $)
|
|
1elfz0size $p |- ( ( A e. Fin /\ A =/= (/) )
|
|
-> 1 e. ( 0 ... ( size ` A ) ) ) $=
|
|
( cfn wcel c0 wne wa c1 cn0 csize cfv cle wbr cc0 cfz 1nn0 sizenncl biimpar
|
|
co a1i cn nnnn0d nnge1d elfz2nn0 syl3anbrc ) ABCZADEZFZGHCZAIJZHCGUIKLGMUIN
|
|
RCUHUGOSUGUIUEUITCUFAPQZUAUGUIUJUBGUIUCUD $.
|
|
|
|
$( The size of the union of a finite set with a disjoint singleton is one
|
|
more than the size of the set. (Contributed by Paul Chapman,
|
|
30-Nov-2012.) $)
|
|
sizeunsng $p |- ( B e. V -> ( ( A e. Fin /\ -. B e. A ) ->
|
|
( size ` ( A u. { B } ) ) = ( ( size ` A ) + 1 ) ) ) $=
|
|
( cfn wcel wn wa csn cun csize cfv c1 caddc co wceq cin simpll snfig adantl
|
|
c0 simplr disjsn sylibr sizeun syl3anc sizesng oveq2d eqtrd expcom ) ADEZBA
|
|
EFZGZBCEZABHZIJKZAJKZLMNZOULUMGZUOUPUNJKZMNZUQURUJUNDEZAUNPTOZUOUTOUJUKUMQU
|
|
MVAULBCRSURUKVBUJUKUMUAABUBUCAUNUDUEUMUTUQOULUMUSLUPMBCUFUGSUHUI $.
|
|
|
|
$( The size of an unordered pair. (Contributed by Mario Carneiro,
|
|
27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV,
|
|
18-Sep-2021.) $)
|
|
sizeprg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A =/= B <-> ( size ` { A , B } ) = 2 ) ) $=
|
|
( wcel wa wne cpr csize cfv c2 wceq csn cun c1 caddc co cfn imp adantr df-2
|
|
simplr snfig ad2antrr elsni eqcomd necon3ai adantl sizeunsng sizesng oveq1d
|
|
wn syl12anc eqtrd df-pr fveq2i 3eqtr4g 1ne2 a1i eqnetrd dfsn2 preq2 syl5req
|
|
fveq2d neeq1d syl5ibrcom necon2d impbida ) ACEZBDEZFZABGZABHZIJZKLZVKVLFZAM
|
|
ZBMNZIJZOOPQZVNKVPVSVQIJZOPQZVTVPVJVQREZBVQEZULZVSWBLZVIVJVLUBVIWCVJVLACUCU
|
|
DVLWEVKWDABWDBABAUEUFUGUHVJWCWEFWFVQBDUISUMVPWAOOPVKWAOLZVLVIWGVJACUJTZTUKU
|
|
NVMVRIABUOUPUAUQVKVOVLVKABVNKVKVNKGABLZWAKGVKWAOKWHOKGVKURUSUTWIVNWAKWIVMVQ
|
|
IWIVQAAHVMAVAABAVBVCVDVEVFVGSVH $.
|
|
|
|
$( There is (at least) one set with two different elements: the unordered
|
|
pair containing ` 0 ` and ` 1 ` . In contrast to ~ pr0size2ex , numbers
|
|
are used instead of sets because their representation is shorter (and more
|
|
comprehensive). (Contributed by AV, 29-Jan-2020.) $)
|
|
prsize2ex $p |- ( size ` { 0 , 1 } ) = 2 $=
|
|
( cc0 c1 cpr csize cfv c2 wceq wne 0ne1 cvv wcel wb c0ex 1ex sizeprg bicomd
|
|
wa mp2an mpbir ) ABCDEFGZABHZIAJKZBJKZTUALMNUBUCQUATABJJOPRS $.
|
|
|
|
${
|
|
sizep1i.1 $e |- A e. _om $.
|
|
sizep1i.2 $e |- B = suc A $.
|
|
sizep1i.3 $e |- ( size ` A ) = M $.
|
|
sizep1i.4 $e |- ( M + 1 ) = N $.
|
|
$( Size of a finite ordinal. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
sizep1i $p |- ( size ` B ) = N $=
|
|
( csize cfv csn cun csuc eqtri c1 caddc co wcel com ax-mp df-suc cfn wceq
|
|
fveq2i wn nnfi word nnord ordirr mp2b wa wi sizeunsng mp2an oveq1i ) BIJA
|
|
AKLZIJZDBUPIBAMUPFAUANUDUQAIJZOPQZDAUBRZAARUEZUQUSUCZASRZUTEAUFTVCAUGVAEA
|
|
UHAUIUJVCUTVAUKVBULEAASUMTUNUSCOPQDURCOPGUOHNNN $.
|
|
$}
|
|
|
|
$( Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) $)
|
|
size1 $p |- ( size ` 1o ) = 1 $=
|
|
( c0 c1o cc0 c1 peano1 df-1o size0 0p1e1 sizep1i ) ABCDEFGHI $.
|
|
|
|
$( Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) $)
|
|
size2 $p |- ( size ` 2o ) = 2 $=
|
|
( c1o c2o c1 c2 1onn df-2o size1 1p1e2 sizep1i ) ABCDEFGHI $.
|
|
|
|
$( Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) $)
|
|
size3 $p |- ( size ` 3o ) = 3 $=
|
|
( c2o c3o c2 c3 2onn df-3o size2 2p1e3 sizep1i ) ABCDEFGHI $.
|
|
|
|
$( Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) $)
|
|
size4 $p |- ( size ` 4o ) = 4 $=
|
|
( c3o c4o c3 c4 3onn df-4o size3 3p1e4 sizep1i ) ABCDEFGHI $.
|
|
|
|
$( There is (at least) one set with two different elements: the unordered
|
|
pair containing the empty set and the singleton containing the empty set.
|
|
(Contributed by AV, 29-Jan-2020.) $)
|
|
pr0size2ex $p |- ( size ` { (/) , { (/) } } ) = 2 $=
|
|
( c0 csn cpr csize cfv c2o c2 df2o2 eqcomi fveq2i size2 eqtri ) AABCZDEFDEG
|
|
MFDFMHIJKL $.
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Elementary real and complex functions
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The "shift" operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c shift $. $( Symbol for function shifter. $)
|
|
|
|
$( Extend class notation with function shifter. $)
|
|
cshi $a class shift $.
|
|
|
|
${
|
|
$d x y z f $.
|
|
$( Define a function shifter. This operation offsets the value argument of
|
|
a function (ordinarily on a subset of ` CC ` ) and produces a new
|
|
function on ` CC ` . See ~ shftval for its value. (Contributed by NM,
|
|
20-Jul-2005.) $)
|
|
df-shft $a |- shift = ( f e. _V , x e. CC |->
|
|
{ <. y , z >. | ( y e. CC /\ ( y - x ) f z ) } ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Two ways to write a shifted set ` ( B + A ) ` . (Contributed by Mario
|
|
Carneiro, 3-Nov-2013.) $)
|
|
shftlem $p |- ( ( A e. CC /\ B C_ CC ) ->
|
|
{ x e. CC | ( x - A ) e. B } = { x | E. y e. B x = ( y + A ) } ) $=
|
|
( cc wcel wss wa cv cmin co crab cab caddc wceq wrex ancoms oveq1 sylan
|
|
wi df-rab npcan eqcomd eqeq2d rspcev expcom syl expimpd ssel2 addcl pncan
|
|
adantr simplr eqeltrd jca anassrs eleq1d anbi12d syl5ibrcom impbid abbidv
|
|
eleq1 rexlimdva syl5eq ) CEFZDEGZHZAIZCJKZDFZAELVHEFZVJHZAMVHBIZCNKZOZBDP
|
|
ZAMVJAEUAVGVLVPAVGVLVPVEVLVPTVFVEVKVJVPVEVKHZVHVICNKZOZVJVPTVQVRVHVKVEVRV
|
|
HOVHCUBQUCVJVSVPVOVSBVIDVMVIOVNVRVHVMVICNRUDUEUFUGUHULVGVOVLBDVGVMDFZHVLV
|
|
OVNEFZVNCJKZDFZHZVEVFVTWDVFVTHZVEWDWEVEHZWAWCWEVMEFZVEWADEVMUIZVMCUJSWFWB
|
|
VMDWEWGVEWBVMOWHVMCUKSVFVTVEUMUNUOQUPVOVKWAVJWCVHVNEVBVOVIWBDVHVNCJRUQURU
|
|
SVCUTVAVD $.
|
|
|
|
$( A shift of the upper integers. (Contributed by Mario Carneiro,
|
|
5-Nov-2013.) $)
|
|
shftuz $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
{ x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) ) $=
|
|
( cz wcel wa cv cmin co cuz cfv cc crab cab caddc df-rab w3a simp2 ancoms
|
|
wi zcn 3ad2ant1 npcand eluzadd 3adant2 eqeltrrd 3expib adantr a1i eluzsub
|
|
eluzelcn 3expia jcad impbid abbi1dv syl5eq ) BDEZCDEZFZAGZBHIZCJKEZALMUTL
|
|
EZVBFZANCBOIZJKZVBALPUSVDAVFUSVDUTVFEZUQVDVGTURUQVCVBVGUQVCVBQZVABOIZUTVF
|
|
VHUTBUQVCVBRUQVCBLEVBBUAUBUCUQVBVIVFEZVCVBUQVJBCVAUDSUEUFUGUHUSVGVCVBVGVC
|
|
TUSVEUTUKUIURUQVGVBTURUQVGVBBCUTUJULSUMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z F $.
|
|
$( The value of the sequence shifter operation is a function on ` CC ` .
|
|
` A ` is ordinarily an integer. (Contributed by NM, 20-Jul-2005.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
shftfvalg $p |- ( ( A e. CC /\ F e. V ) -> ( F shift A ) =
|
|
{ <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } ) $=
|
|
( vz vw cc wcel wa cv cmin co wbr copab cvv cshi wceq caddc wrex sylancom
|
|
simpr simpl cdm cab crn cxp wss simplr simpll subcld breldmg mp3an2 npcan
|
|
vex eqcomd ancoms adantr oveq1 eqeq2d rspcev syl2anc eqeq1 rexbidv sylibr
|
|
elab brelrng jca ssopab2dv df-xp syl6sseqr dmexg abrexexg syl rnexg xpexg
|
|
expl ssexg syl2an elex breq anbi2d opabbidv oveq2 df-shft ovmpt2g syl3an1
|
|
breq1d syl3anc ) CHIZDEIZJWKWJAKZHIZWLCLMZBKZDNZJZABOZPIZDCQMWRRZWJWKUBWJ
|
|
WKUCWJWRFKZGKZCSMZRZGDUDZTZFUEZDUFZUGZUHXIPIZWSWKWJWRWLXGIZWOXHIZJZABOXIW
|
|
JWQXMABWJWMWPXMWJWMJZWPJZXKXLXOWLXCRZGXETZXKXOWNXEIZWLWNCSMZRZXQXNWPWNHIZ
|
|
XRXOWLCWJWMWPUIWJWMWPUJUKZYAWOPIZWPXRBUOZWNWOHPDULUMUAXNXTWPWMWJXTWMWJJXS
|
|
WLWLCUNUPUQURXPXTGWNXEXBWNRXCXSWLXBWNCSUSUTVAVBXFXQFWLAUOXAWLRXDXPGXEXAWL
|
|
XCVCVDVFVEXNWPYAXLYBYAYCWPXLYDWNWODHPVGUMUAVHVQVIABXGXHVJVKWKXGPIZXHPIXJW
|
|
KXEPIYEDEVLGFXEXCPVMVNDEVOXGXHPPVPVBWRXIPVRVSWKDPIWJWSWTDEVTFGDCPHWMWLXBL
|
|
MZWOXANZJZABOWRQWMYFWODNZJZABOPXADRZYHYJABYKYGYIWMYFWOXADWAWBWCXBCRZYJWQA
|
|
BYLYIWPWMYLYFWNWODXBCWLLWDWHWBWCGABFWEWFWGWI $.
|
|
$}
|
|
|
|
${
|
|
$d A u w z $. $d F u w z $. $d V z $.
|
|
$( Existence of the result of applying shift. (Contributed by Jim Kingdon,
|
|
15-Aug-2021.) $)
|
|
ovshftex $p |- ( ( F e. V /\ A e. CC ) -> ( F shift A ) e. _V ) $=
|
|
( vz vw vu wcel cc wa cshi co cv cmin wbr copab cvv wceq shftfvalg ancoms
|
|
cnex a1i cab crn rnexg ad2antrr wss breq2 elab simpr simpl subcld brelrng
|
|
vex mp3an2 sylan ex syl5bi ssrdv adantll ssexd opabex3d eqeltrd ) BCGZAHG
|
|
ZIZBAJKZDLZHGZVGAMKZELZBNZIDEOZPVDVCVFVLQDEABCRSVEVKDEHHPGVETUAVEVHIVKEUB
|
|
ZBUCZPVCVNPGVDVHBCUDUEVDVHVMVNUFVCVDVHIZFVMVNFLZVMGVIVPBNZVOVPVNGZVKVQEVP
|
|
FUMZVJVPVIBUGUHVOVQVRVOVIHGZVQVRVOVGAVDVHUIVDVHUJUKVTVPPGVQVRVSVIVPBHPULU
|
|
NUOUPUQURUSUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d F x y z $. $d V z $.
|
|
$( Value of a fiber of the relation ` F ` . (Contributed by Jim Kingdon,
|
|
15-Aug-2021.) $)
|
|
shftfibg $p |- ( ( F e. V /\ A e. CC /\ B e. CC ) ->
|
|
( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) ) $=
|
|
( vz vx vy wcel cc cv co wbr cab cmin csn cima wa wb cvv wceq simp2 simp1
|
|
w3a cshi simp3 copab shftfvalg breqd vex eleq1 oveq1 breq1d anbi12d breq2
|
|
anbi2d eqid brabg sylan9bb syl21anc 3anibar abbidv imasng 3ad2ant3 subcld
|
|
mpan2 syl 3eqtr4d ) CDHZAIHZBIHZUCZBEJZCAUDKZLZEMZBANKZVLCLZEMZVMBOPZCVPO
|
|
PZVKVNVQEVHVIVJVNVQVKVIVHVJVNVJVQQZRVHVIVJUAZVHVIVJUBVHVIVJUEZVIVHQZVNBVL
|
|
FJZIHZWEANKZGJZCLZQZFGUFZLZVJWAWDVMWKBVLFGACDUGUHVJVLSHWLWAREUIWJVJVPWHCL
|
|
ZQWAFGBVLISWKWEBTZWFVJWIWMWEBIUJWNWGVPWHCWEBANUKULUMWHVLTWMVQVJWHVLVPCUNU
|
|
OWKUPUQVEURUSUTVAVJVHVSVOTVIEBIVMVBVCVKVPIHVTVRTVKBAWCWBVDEVPICVBVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z F $. $d w x y z B $.
|
|
shftfval.1 $e |- F e. _V $.
|
|
$( The value of the sequence shifter operation is a function on ` CC ` .
|
|
` A ` is ordinarily an integer. (Contributed by NM, 20-Jul-2005.)
|
|
(Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
shftfval $p |- ( A e. CC -> ( F shift A ) =
|
|
{ <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } ) $=
|
|
( vz vw cc wcel cv cmin co wbr wa copab cvv cshi wceq caddc wrex sylancom
|
|
cdm cab crn cxp wss simplr simpll subcld vex breldmg mp3an2 eqcomd ancoms
|
|
npcan adantr oveq1 eqeq2d rspcev syl2anc eqeq1 rexbidv sylibr brelrng jca
|
|
elab expl ssopab2dv df-xp syl6sseqr dmex abrexex rnex xpex sylancl anbi2d
|
|
ssexg breq opabbidv oveq2 breq1d df-shft ovmpt2g mp3an1 mpdan ) CHIZAJZHI
|
|
ZWGCKLZBJZDMZNZABOZPIZDCQLWMRZWFWMFJZGJZCSLZRZGDUBZTZFUCZDUDZUEZUFXDPIWNW
|
|
FWMWGXBIZWJXCIZNZABOXDWFWLXGABWFWHWKXGWFWHNZWKNZXEXFXIWGWRRZGWTTZXEXIWIWT
|
|
IZWGWICSLZRZXKXHWKWIHIZXLXIWGCWFWHWKUGWFWHWKUHUIZXOWJPIZWKXLBUJZWIWJHPDUK
|
|
ULUAXHXNWKWHWFXNWHWFNXMWGWGCUOUMUNUPXJXNGWIWTWQWIRWRXMWGWQWICSUQURUSUTXAX
|
|
KFWGAUJWPWGRWSXJGWTWPWGWRVAVBVFVCXHWKXOXFXPXOXQWKXFXRWIWJDHPVDULUAVEVGVHA
|
|
BXBXCVIVJXBXCGFWTWRDEVKVLDEVMVNWMXDPVQVODPIWFWNWOEFGDCPHWHWGWQKLZWJWPMZNZ
|
|
ABOWMQWHXSWJDMZNZABOPWPDRZYAYCABYDXTYBWHXSWJWPDVRVPVSWQCRZYCWLABYEYBWKWHY
|
|
EXSWIWJDWQCWGKVTWAVPVSGABFWBWCWDWE $.
|
|
|
|
$( Domain of a relation shifted by ` A ` . The set on the right is more
|
|
commonly notated as ` ( dom F + A ) ` (meaning add ` A ` to every
|
|
element of ` dom F ` ). (Contributed by Mario Carneiro, 3-Nov-2013.) $)
|
|
shftdm $p |- ( A e. CC -> dom ( F shift A ) =
|
|
{ x e. CC | ( x - A ) e. dom F } ) $=
|
|
( vy cc wcel cshi co cdm cv cmin wbr wa copab crab shftfval dmeqd wex cab
|
|
wb simpr simpl subcld eldmg pm5.32da 19.42v syl6rbbr abbidv dmopab df-rab
|
|
syl 3eqtr4g eqtrd ) BFGZCBHIZJAKZFGZUQBLIZEKCMZNZAEOZJZUSCJGZAFPZUOUPVBAE
|
|
BCDQRUOVAESZATURVDNZATVCVEUOVFVGAUOVGURUTESZNVFUOURVDVHUOURNZUSFGVDVHUAVI
|
|
UQBUOURUBUOURUCUDEUSCFUEULUFURUTEUGUHUIVAAEUJVDAFUKUMUN $.
|
|
|
|
$( Value of a fiber of the relation ` F ` . (Contributed by Mario
|
|
Carneiro, 4-Nov-2013.) $)
|
|
shftfib $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) ) $=
|
|
( vz vx vy cc wcel wa cv co wbr cab cmin csn cima cvv wb wceq copab breqd
|
|
cshi shftfval eleq1 oveq1 breq1d anbi12d breq2 anbi2d eqid brabg sylan9bb
|
|
vex mpan2 ibar adantl bitr4d abbidv imasng simpr simpl subcld syl 3eqtr4d
|
|
) AHIZBHIZJZBEKZCAUCLZMZENZBAOLZVICMZENZVJBPQZCVMPQZVHVKVNEVHVKVGVNJZVNVF
|
|
VKBVIFKZHIZVSAOLZGKZCMZJZFGUAZMZVGVRVFVJWEBVIFGACDUDUBVGVIRIWFVRSEUNWDVGV
|
|
MWBCMZJVRFGBVIHRWEVSBTZVTVGWCWGVSBHUEWHWAVMWBCVSBAOUFUGUHWBVITWGVNVGWBVIV
|
|
MCUIUJWEUKULUOUMVGVNVRSVFVGVNUPUQURUSVGVPVLTVFEBHVJUTUQVHVMHIVQVOTVHBAVFV
|
|
GVAVFVGVBVCEVMHCUTVDVE $.
|
|
|
|
$( Functionality and domain of a sequence shifted by ` A ` . (Contributed
|
|
by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) $)
|
|
shftfn $p |- ( ( F Fn B /\ A e. CC ) ->
|
|
( F shift A ) Fn { x e. CC | ( x - A ) e. B } ) $=
|
|
( vy vz vw wfn cc wcel wa co wfun cdm cv cmin crab wceq wbr cshi wrel wmo
|
|
copab wal relopab a1i fnfun adantr funmo eleq1 oveq1 breq1d anbi12d breq2
|
|
vex anbi2d eqid simprbi moimi syl alrimiv dffun6 sylanbrc shftfval adantl
|
|
brab funeqd mpbird shftdm fndm eleq2d rabbidv sylan9eqr df-fn ) DCIZBJKZL
|
|
ZDBUAMZNZVSOZAPZBQMZCKZAJRZSVSWEIVRVTWBJKZWCFPZDTZLZAFUDZNZVRWJUBZGPZHPZW
|
|
JTZHUCZGUEZWKWLVRWIAFUFUGVRDNZWQVPWRVQCDUHUIWRWPGWRWMBQMZWNDTZHUCWPHWSDUJ
|
|
WOWTHWOWMJKZWTWIXAWSWGDTZLXAWTLAFWMWNWJGUPHUPWBWMSZWFXAWHXBWBWMJUKXCWCWSW
|
|
GDWBWMBQULUMUNWGWNSXBWTXAWGWNWSDUOUQWJURVGUSUTVAVBVAGHWJVCVDVRVSWJVQVSWJS
|
|
VPAFBDEVEVFVHVIVQVPWAWCDOZKZAJRWEABDEVJVPXEWDAJVPXDCWCCDVKVLVMVNVSWEVOVD
|
|
$.
|
|
|
|
$( Value of a sequence shifted by ` A ` . (Contributed by NM,
|
|
20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.) $)
|
|
shftval $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( F shift A ) ` B ) = ( F ` ( B - A ) ) ) $=
|
|
( vx cc wcel wa cv cshi co csn cima cio cmin cfv shftfib wceq dffv3g syl
|
|
eleq2d iotabidv simpr simpl subcld 3eqtr4d ) AFGZBFGZHZEIZCAJKZBLMZGZENZU
|
|
JCBAOKZLMZGZENZBUKPZUOCPZUIUMUQEUIULUPUJABCDQUAUBUIUHUSUNRUGUHUCZEBUKFSTU
|
|
IUOFGUTURRUIBAVAUGUHUDUEEUOCFSTUF $.
|
|
|
|
$( Value of a sequence shifted by ` A - B ` . (Contributed by NM,
|
|
20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftval2 $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) ) $=
|
|
( cc wcel w3a caddc co cmin cshi wceq subcl 3adant3 addcl 3adant2 shftval
|
|
cfv syl2anc pnncan 3com23 addcom 3adant1 eqtr4d fveq2d eqtrd ) AFGZBFGZCF
|
|
GZHZACIJZDABKJZLJSZULUMKJZDSZBCIJZDSUKUMFGZULFGZUNUPMUHUIURUJABNOUHUJUSUI
|
|
ACPQUMULDERTUKUOUQDUKUOCBIJZUQUHUJUIUOUTMACBUAUBUIUJUQUTMUHBCUCUDUEUFUG
|
|
$.
|
|
|
|
$( Value of a sequence shifted by ` A - B ` . (Contributed by NM,
|
|
20-Jul-2005.) $)
|
|
shftval3 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( F shift ( A - B ) ) ` A ) = ( F ` B ) ) $=
|
|
( cc wcel wa cc0 caddc co cmin cshi cfv 0cn shftval2 mp3an3 addid1 adantr
|
|
wceq fveq2d adantl 3eqtr3d ) AEFZBEFZGZAHIJZCABKJLJZMZBHIJZCMZAUGMBCMUCUD
|
|
HEFUHUJSNABHCDOPUEUFAUGUCUFASUDAQRTUEUIBCUDUIBSUCBQUATUB $.
|
|
|
|
$( Value of a sequence shifted by ` -u A ` . (Contributed by NM,
|
|
18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftval4 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) ) $=
|
|
( cc wcel wa cneg cshi co cfv cmin caddc wceq negcl shftval subneg ancoms
|
|
sylan addcom eqtr4d fveq2d eqtrd ) AEFZBEFZGZBCAHZIJKZBUGLJZCKZABMJZCKUDU
|
|
GEFUEUHUJNAOUGBCDPSUFUIUKCUFUIBAMJZUKUEUDUIULNBAQRABTUAUBUC $.
|
|
|
|
$( Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
|
|
(Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftval5 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( F shift A ) ` ( B + A ) ) = ( F ` B ) ) $=
|
|
( cc wcel caddc co cshi cfv wceq wa cmin simpr addcl shftval pncan fveq2d
|
|
syl2anc eqtrd ancoms ) BEFZAEFZBAGHZCAIHJZBCJZKUBUCLZUEUDAMHZCJZUFUGUCUDE
|
|
FUEUIKUBUCNBAOAUDCDPSUGUHBCBAQRTUA $.
|
|
|
|
$d y C $.
|
|
$( Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
|
|
(Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftf $p |- ( ( F : B --> C /\ A e. CC ) ->
|
|
( F shift A ) : { x e. CC | ( x - A ) e. B } --> C ) $=
|
|
( vy wf cc wcel wa co cv cmin wfn cfv wceq simpr simpl syl2an cshi shftfn
|
|
crab wral ffn sylan oveq1 elrab shftval ffvelrn eqeltrd sylan2b ralrimiva
|
|
eleq1d ffnfv sylanbrc ) CDEHZBIJZKZEBUALZAMZBNLZCJZAIUCZOZGMZUTPZDJZGVDUD
|
|
VDDUTHUQECOURVECDEUEABCEFUBUFUSVHGVDVFVDJUSVFIJZVFBNLZCJZKZVHVCVKAVFIVAVF
|
|
QVBVJCVAVFBNUGUNUHUSVLKVGVJEPZDUSURVIVGVMQVLUQURRVIVKSBVFEFUITUSUQVKVMDJV
|
|
LUQURSVIVKRCDVJEUJTUKULUMGVDDUTUOUP $.
|
|
|
|
$( Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised
|
|
by Mario Carneiro, 5-Nov-2013.) $)
|
|
2shfti $p |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) shift B ) =
|
|
( F shift ( A + B ) ) ) $=
|
|
( vx vy vz vw cc wcel wa cv cmin co cshi wbr copab wb cvv wceq simplr vex
|
|
caddc shftfval breqd simpr subcld eleq1 oveq1 breq1d anbi12d breq2 anbi2d
|
|
ad2antrr brabg sylancl bitrd subcl biantrurd ancoms adantll sub32 subsub4
|
|
eqid w3a eqtr3d 3expb 3bitr2d pm5.32da opabbidv ovshftex shftfvalg sylan2
|
|
mpan addcl syl 3eqtr4d ) AIJZBIJZKZELZIJZWABMNZFLZCAONZPZKZEFQZWBWAABUCNZ
|
|
MNZWDCPZKZEFQZWEBONZCWIONZVTWGWLEFVTWBWFWKVTWBKZWFWCIJZWCAMNZWDCPZKZWSWKW
|
|
PWFWCWDGLZIJZXAAMNZHLZCPZKZGHQZPZWTVRWFXHRVSWBVRWEXGWCWDGHACDUDUEUNWPWQWD
|
|
SJXHWTRWPWABVTWBUFVRVSWBUAUGFUBXFWQWRXDCPZKWTGHWCWDISXGXAWCTZXBWQXEXIXAWC
|
|
IUHXJXCWRXDCXAWCAMUIUJUKXDWDTXIWSWQXDWDWRCULUMXGVDUOUPUQVSWBWSWTRZVRWBVSX
|
|
KWBVSKWQWSWABURUSUTVAWPWRWJWDCWBVTWRWJTZWBVRVSXLWBVRVSVEWAAMNBMNWRWJWAABV
|
|
BWAABVCVFVGUTUJVHVIVJVSVRWNWHTZVRVSWESJZXMCSJVRXNDACSVKVNEFBWESVLVMUTVTWI
|
|
IJWOWMTABVOEFWICDUDVPVQ $.
|
|
|
|
$( Identity law for the shift operation. (Contributed by Mario Carneiro,
|
|
5-Nov-2013.) $)
|
|
shftidt2 $p |- ( F shift 0 ) = ( F |` CC ) $=
|
|
( vx vy cv cc wcel cc0 cmin co wbr copab cshi cres subid1 pm5.32i opabbii
|
|
wa breq1d wceq 0cn shftfval ax-mp dfres2 3eqtr4i ) CEZFGZUFHIJZDEZAKZRZCD
|
|
LZUGUFUIAKZRZCDLAHMJZAFNUKUNCDUGUJUMUGUHUFUIAUFOSPQHFGUOULTUACDHABUBUCCDF
|
|
AUDUE $.
|
|
|
|
$( Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.)
|
|
(Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftidt $p |- ( A e. CC -> ( ( F shift 0 ) ` A ) = ( F ` A ) ) $=
|
|
( cc wcel cc0 cshi co cfv cres shftidt2 fveq1i fvres syl5eq ) ADEABFGHZIA
|
|
BDJZIABIAOPBCKLADBMN $.
|
|
|
|
$( Cancellation law for the shift operation. (Contributed by NM,
|
|
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftcan1 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( ( F shift A ) shift -u A ) ` B ) = ( F ` B ) ) $=
|
|
( cc wcel cshi co cneg cfv cc0 caddc wceq negcl 2shfti mpdan negid oveq2d
|
|
eqtrd fveq1d shftidt sylan9eq ) AEFZBEFBCAGHAIZGHZJBCKGHZJBCJUCBUEUFUCUEC
|
|
AUDLHZGHZUFUCUDEFUEUHMANAUDCDOPUCUGKCGAQRSTBCDUAUB $.
|
|
|
|
$( Cancellation law for the shift operation. (Contributed by NM,
|
|
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) $)
|
|
shftcan2 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( ( F shift -u A ) shift A ) ` B ) = ( F ` B ) ) $=
|
|
( cc wcel wa cneg cshi co wceq negneg adantr oveq2d fveq1d negcl shftcan1
|
|
cfv sylan eqtr3d ) AEFZBEFZGZBCAHZIJZUDHZIJZRZBUEAIJZRBCRZUCBUGUIUCUFAUEI
|
|
UAUFAKUBALMNOUAUDEFUBUHUJKAPUDBCDQST $.
|
|
$}
|
|
|
|
${
|
|
$d F f $. $d A f $. $d B f $.
|
|
$( Value of a sequence shifted by ` A ` . (Contributed by Scott Fenton,
|
|
16-Dec-2017.) $)
|
|
shftvalg $p |- ( ( F e. V /\ A e. CC /\ B e. CC ) ->
|
|
( ( F shift A ) ` B ) = ( F ` ( B - A ) ) ) $=
|
|
( vf wcel cc cshi co cmin wceq wa cv wi oveq1 fveq1d fveq1 eqeq12d imbi2d
|
|
cfv vex shftval vtoclg 3impib ) CDFAGFZBGFZBCAHIZTZBAJIZCTZKZUEUFLZBEMZAH
|
|
IZTZUIUMTZKZNULUKNECDUMCKZUQUKULURUOUHUPUJURBUNUGUMCAHOPUIUMCQRSABUMEUAUB
|
|
UCUD $.
|
|
|
|
$( Value of a sequence shifted by ` -u A ` . (Contributed by Jim Kingdon,
|
|
19-Aug-2021.) $)
|
|
shftval4g $p |- ( ( F e. V /\ A e. CC /\ B e. CC ) ->
|
|
( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) ) $=
|
|
( vf wcel cc cneg cshi co cfv caddc wceq wa cv oveq1 fveq1d fveq1 eqeq12d
|
|
wi imbi2d vex shftval4 vtoclg 3impib ) CDFAGFZBGFZBCAHZIJZKZABLJZCKZMZUFU
|
|
GNZBEOZUHIJZKZUKUOKZMZTUNUMTECDUOCMZUSUMUNUTUQUJURULUTBUPUIUOCUHIPQUKUOCR
|
|
SUAABUOEUBUCUDUE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Real and imaginary parts; conjugate
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c Re $. $( Real part of a complex number. $)
|
|
$c Im $. $( Imaginary part of a complex number. $)
|
|
$c * $. $( Complex conjugate function. $)
|
|
|
|
$( Extend class notation to include complex conjugate function. $)
|
|
ccj $a class * $.
|
|
|
|
$( Extend class notation to include real part of a complex number. $)
|
|
cre $a class Re $.
|
|
|
|
$( Extend class notation to include imaginary part of a complex number. $)
|
|
cim $a class Im $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the complex conjugate function. See ~ cjcli for its closure and
|
|
~ cjval for its value. (Contributed by NM, 9-May-1999.) (Revised by
|
|
Mario Carneiro, 6-Nov-2013.) $)
|
|
df-cj $a |- * = ( x e. CC |-> ( iota_ y e. CC
|
|
( ( x + y ) e. RR /\ ( _i x. ( x - y ) ) e. RR ) ) ) $.
|
|
|
|
$( Define a function whose value is the real part of a complex number. See
|
|
~ reval for its value, ~ recli for its closure, and ~ replim for its use
|
|
in decomposing a complex number. (Contributed by NM, 9-May-1999.) $)
|
|
df-re $a |- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) ) $.
|
|
|
|
$( Define a function whose value is the imaginary part of a complex number.
|
|
See ~ imval for its value, ~ imcli for its closure, and ~ replim for its
|
|
use in decomposing a complex number. (Contributed by NM,
|
|
9-May-1999.) $)
|
|
df-im $a |- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The value of the conjugate of a complex number. (Contributed by Mario
|
|
Carneiro, 6-Nov-2013.) $)
|
|
cjval $p |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC
|
|
( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) ) $=
|
|
( vy cc wcel cv caddc co cr ci cmin cmul wa crio ccj cfv wceq wreu eleq1d
|
|
oveq1 cju riotacl syl oveq2d anbi12d riotabidv df-cj fvmptg mpdan ) BDEZB
|
|
AFZGHZIEZJBUKKHZLHZIEZMZADNZDEZBOPURQUJUQADRUSABUAUQADUBUCCBCFZUKGHZIEZJU
|
|
TUKKHZLHZIEZMZADNURDDOUTBQZVFUQADVGVBUMVEUPVGVAULIUTBUKGTSVGVDUOIVGVCUNJL
|
|
UTBUKKTUDSUEUFCAUGUHUI $.
|
|
|
|
$( The defining property of the complex conjugate. (Contributed by Mario
|
|
Carneiro, 6-Nov-2013.) $)
|
|
cjth $p |- ( A e. CC ->
|
|
( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) ) $=
|
|
( vx cc wcel cv caddc co cr ci cmin cmul ccj cfv wsbc crio wreu syl oveq2
|
|
wa eleq1d riotasbc cjval sbceq1d mpbird wb riotacl eqeltrd oveq2d anbi12d
|
|
cju wceq sbcieg mpbid ) ACDZABEZFGZHDZIAUOJGZKGZHDZSZBALMZNZAVBFGZHDZIAVB
|
|
JGZKGZHDZSZUNVCVABVABCOZNZUNVABCPZVKBAUJZVABCUAQUNVABVBVJBAUBZUCUDUNVBCDV
|
|
CVIUEUNVBVJCVNUNVLVJCDVMVABCUFQUGVAVIBVBCUOVBUKZUQVEUTVHVOUPVDHUOVBAFRTVO
|
|
USVGHVOURVFIKUOVBAJRUHTUIULQUM $.
|
|
|
|
$( Domain and codomain of the conjugate function. (Contributed by Mario
|
|
Carneiro, 6-Nov-2013.) $)
|
|
cjf $p |- * : CC --> CC $=
|
|
( vx vy cc cv caddc co cr wcel ci cmin cmul wa crio ccj df-cj cju riotacl
|
|
wreu syl fmpti ) ACCADZBDZEFGHIUAUBJFKFGHLZBCMZNABOUACHUCBCRUDCHBUAPUCBCQ
|
|
ST $.
|
|
|
|
$( The conjugate of a complex number is a complex number (closure law).
|
|
(Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
|
|
6-Nov-2013.) $)
|
|
cjcl $p |- ( A e. CC -> ( * ` A ) e. CC ) $=
|
|
( cc ccj cjf ffvelrni ) BBACDE $.
|
|
|
|
$( The value of the real part of a complex number. (Contributed by NM,
|
|
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
reval $p |- ( A e. CC -> ( Re ` A ) = ( ( A + ( * ` A ) ) / 2 ) ) $=
|
|
( vx cc wcel ccj cfv caddc co c2 cdiv cre wceq id addcld halfcld cv fveq2
|
|
cjcl oveq12 mpdan oveq1d df-re fvmptg ) ACDZAAEFZGHZIJHZCDAKFUGLUDUFUDAUE
|
|
UDMARNOBABPZUHEFZGHZIJHUGCCKUHALZUJUFIJUKUIUELUJUFLUHAEQUHAUIUEGSTUABUBUC
|
|
T $.
|
|
|
|
$( The value of the imaginary part of a complex number. (Contributed by
|
|
NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
imval $p |- ( A e. CC -> ( Im ` A ) = ( Re ` ( A / _i ) ) ) $=
|
|
( vx cc wcel ci cdiv co cre cfv cim wceq ccj caddc ax-icn a1i cc0 cap syl
|
|
c2 id wbr iap0 divclapd reval cjcl addcld halfcld eqeltrd cv oveq1 fveq2d
|
|
df-im fvmptg mpdan ) ACDZAEFGZHIZCDAJIUQKUOUQUPUPLIZMGZSFGZCUOUPCDZUQUTKU
|
|
OAEUOTECDUONOEPQUAUOUBOUCZUPUDRUOUSUOUPURVBUOVAURCDVBUPUERUFUGUHBABUIZEFG
|
|
ZHIUQCCJVCAKVDUPHVCAEFUJUKBULUMUN $.
|
|
|
|
$( The imaginary part of a complex number in terms of the real part
|
|
function. (Contributed by NM, 12-May-2005.) (Revised by Mario
|
|
Carneiro, 6-Nov-2013.) $)
|
|
imre $p |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) ) $=
|
|
( cc wcel cim cfv ci cdiv co cre cneg cmul imval c1 cc0 cap wbr wceq iap0
|
|
ax-icn divrecap2 mp3an23 irec oveq1i syl6eq fveq2d eqtrd ) ABCZADEAFGHZIE
|
|
FJZAKHZIEALUGUHUJIUGUHMFGHZAKHZUJUGFBCFNOPUHULQSRAFTUAUKUIAKUBUCUDUEUF $.
|
|
|
|
$( The real part of a complex number in terms of the imaginary part
|
|
function. (Contributed by Mario Carneiro, 31-Mar-2015.) $)
|
|
reim $p |- ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) $=
|
|
( cc wcel ci cmul co cim cfv cdiv cre wceq ax-icn mulcl imval syl cc0 cap
|
|
mpan wbr iap0 divcanap3 mp3an23 fveq2d eqtr2d ) ABCZDAEFZGHZUFDIFZJHZAJHU
|
|
EUFBCZUGUIKDBCZUEUJLDAMRUFNOUEUHAJUEUKDPQSUHAKLTADUAUBUCUD $.
|
|
|
|
$( The real part of a complex number is real. (Contributed by NM,
|
|
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
recl $p |- ( A e. CC -> ( Re ` A ) e. RR ) $=
|
|
( cc wcel cre cfv caddc co c2 cdiv cr reval ci cmin cmul simpld rehalfcld
|
|
ccj cjth eqeltrd ) ABCZADEAAQEZFGZHIGJAKTUBTUBJCLAUAMGNGJCAROPS $.
|
|
|
|
$( The imaginary part of a complex number is real. (Contributed by NM,
|
|
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
imcl $p |- ( A e. CC -> ( Im ` A ) e. RR ) $=
|
|
( cc wcel cim cfv ci cneg cmul co cre imre negicn mulcl mpan recl eqeltrd
|
|
cr syl ) ABCZADEFGZAHIZJEZQAKSUABCZUBQCTBCSUCLTAMNUAORP $.
|
|
|
|
$( Domain and codomain of the real part function. (Contributed by Paul
|
|
Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
ref $p |- Re : CC --> RR $=
|
|
( vx cc cr cv ccj cfv caddc co c2 cdiv cre df-re wcel reval recl eqeltrrd
|
|
fmpti ) ABCADZREFGHIJHZKALRBMRKFSCRNROPQ $.
|
|
|
|
$( Domain and codomain of the imaginary part function. (Contributed by
|
|
Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) $)
|
|
imf $p |- Im : CC --> RR $=
|
|
( vx cc cr cv ci cdiv co cre cfv cim df-im wcel imval imcl eqeltrrd fmpti
|
|
) ABCADZEFGHIZJAKQBLQJIRCQMQNOP $.
|
|
|
|
$( The real part of a complex number representation. Definition 10-3.1 of
|
|
[Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario
|
|
Carneiro, 7-Nov-2013.) $)
|
|
crre $p |- ( ( A e. RR /\ B e. RR )
|
|
-> ( Re ` ( A + ( _i x. B ) ) ) = A ) $=
|
|
( cr wcel ci cmul co caddc c2 cdiv cc ax-icn mulcl sylancr syl addcld a1i
|
|
cmin oveq1d 3eqtr4d cre cfv ccj wceq recn addcl syl2an reval cjcl halfcld
|
|
wa adantr cc0 recl eqeltrrd simpl resubcld adantl subcld pnpcand pnpcan2d
|
|
subdid eqtr4d addsubd subsubd 2timesd oveq2d 2cn cap divsubdirapd 3eqtr3d
|
|
wbr 2ap0 divcanap3d mulassd divassapd oveq12d c1 ixi neg1rr eqeltri simpr
|
|
cneg remulcl cjth simprd rehalfcld eqeltrd rimul syl2anc subeq0d eqtrd )
|
|
ACDZBCDZUKZAEBFGZHGZUAUBZWQWQUCUBZHGZIJGZAWOWQKDZWRXAUDWMAKDZWPKDZXBWNAUE
|
|
ZWNEKDZBKDZXDLBUEZEBMZNAWPUFUGZWQUHOZWOXAAWOWTWOWQWSXJWOXBWSKDXJWQUIOZPZU
|
|
JWMXCWNXEULZWOXAARGZCDEXOFGZCDXOUMUDWOXAAWOWRXACXKWOXBWRCDXJWQUNOUOWMWNUP
|
|
UQWOXPEEFGZBFGZEWQWSRGZFGZIJGZRGZCWOEWPXSIJGZRGZFGEWPFGZEYCFGZRGXPYBWOEWP
|
|
YCXFWOLQZWOXFXGXDLWNXGWMXHURZXINZWOXSWOWQWSXJXLUSZUJVBWOXOYDEFWOXAIAFGZIJ
|
|
GZRGZIWPFGZIJGZYCRGZXOYDWOWTYKRGZIJGYNXSRGZIJGYMYPWOYQYRIJWOWTAAHGZRGZWPW
|
|
PHGZXSRGZYQYRWOWQYSRGZWSHGUUAWQRGZWSHGYTUUBWOUUCUUDWSHWOUUCWPARGUUDWOAWPA
|
|
XNYIXNUTWOWPAWPYIXNYIVAVCSWOWQWSYSXJXLWOAAXNXNPVDWOUUAWQWSWOWPWPYIYIPXJXL
|
|
VETWOYKYSWTRWOAXNVFVGWOYNUUAXSRWOWPYIVFSTSWOWTYKIXMWOIKDZXCYKKDVHXNIAMNUU
|
|
EWOVHQZIUMVIVLWOVMQZVJWOYNXSIWOUUEXDYNKDVHYIIWPMNYJUUFUUGVJVKWOYLAXARWOAI
|
|
XNUUFUUGVNVGWOYOWPYCRWOWPIYIUUFUUGVNSVKVGWOXRYEYAYFRWOEEBYGYGYHVOWOEXSIYG
|
|
YJUUFUUGVPVQTWOXRYAWOXQCDWNXRCDXQVRWCCVSVTWAWMWNWBXQBWDNWOXTWOXBXTCDZXJXB
|
|
WTCDUUHWQWEWFOWGUQWHXOWIWJWKWL $.
|
|
|
|
$( The real part of a complex number representation. Definition 10-3.1 of
|
|
[Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario
|
|
Carneiro, 7-Nov-2013.) $)
|
|
crim $p |- ( ( A e. RR /\ B e. RR )
|
|
-> ( Im ` ( A + ( _i x. B ) ) ) = B ) $=
|
|
( cr wcel wa ci cmul co caddc cfv cdiv cre cneg cc wceq recn ax-icn mulcl
|
|
sylancr iap0 cim addcl syl2an imval syl mpan cc0 cap wbr divdirap mpanr12
|
|
3expa sylan2 c1 divrecap2 mp3an23 oveq1i a1i mulneg12 divcanap3 oveqan12d
|
|
irec 3eqtrd negcl addcom sylan 3eqtrrd fveq2d id renegcl syl2anr 3eqtr2d
|
|
crre ) ACDZBCDZEZAFBGHZIHZUAJZVRFKHZLJZBFAMZGHZIHZLJZBVPVRNDZVSWAOVNANDZV
|
|
QNDZWFVOAPZVOFNDZBNDZWHQBPZFBRZSAVQUBUCVRUDUEVPWDVTLVNWGWKWDVTOVOWIWLWGWK
|
|
EVTAFKHZVQFKHZIHZWCBIHZWDWKWGWHVTWPOZWJWKWHQWMUFWGWHEWJFUGUHUIZWRQTWGWHWJ
|
|
WSEWRAVQFUJULUKUMWGWKWNWCWOBIWGWNUNFKHZAGHZFMZAGHZWCWGWJWSWNXAOQTAFUOUPXA
|
|
XCOWGWTXBAGVBUQURWJWGXCWCOQFAUSUFVCWKWJWSWOBOQTBFUTUPVAWGWCNDZWKWQWDOWGWJ
|
|
WBNDXDQAVDFWBRSWCBVEVFVGUCVHVOVOWBCDWEBOVNVOVIAVJBWBVMVKVL $.
|
|
|
|
$( Reconstruct a complex number from its real and imaginary parts.
|
|
(Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
|
|
7-Nov-2013.) $)
|
|
replim $p |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) $=
|
|
( vx vy cc wcel cv ci cmul co caddc wceq cr wrex cre cfv cim cnre oveq12d
|
|
oveq2d fveq2 wa crre crim eqcomd id eqeq12d syl5ibrcom rexlimivv syl ) AD
|
|
EABFZGCFZHIZJIZKZCLMBLMAANOZGAPOZHIZJIZKZBCAQUNUSBCLLUJLEUKLEUAZUSUNUMUMN
|
|
OZGUMPOZHIZJIZKUTVDUMUTVAUJVCULJUJUKUBUTVBUKGHUJUKUCSRUDUNAUMURVDUNUEUNUO
|
|
VAUQVCJAUMNTUNUPVBGHAUMPTSRUFUGUHUI $.
|
|
|
|
$( Value of the conjugate of a complex number. The value is the real part
|
|
minus ` _i ` times the imaginary part. Definition 10-3.2 of [Gleason]
|
|
p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
|
|
7-Nov-2013.) $)
|
|
remim $p |- ( A e. CC ->
|
|
( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) ) $=
|
|
( vx cc wcel caddc co cr ci cmin cmul wa wceq oveq1d recnd ax-icn sylancr
|
|
cfv eqtrd readdcld eqeltrd ccj cv crio cre cjval replim recl imcl ppncand
|
|
cim mulcl pnncand a1i adddid 3eqtr4d oveq2d mulass mp3an12 eqtr4d c1 cneg
|
|
syl ixi neg1rr eqeltri remulcl wreu wb subcld oveq2 eleq1d anbi12d riota2
|
|
cju syl2anc mpbi2and ) ACDZAUAQABUBZEFZGDZHAVRIFZJFZGDZKZBCUCZAUDQZHAUJQZ
|
|
JFZIFZBAUEVQAWIEFZGDZHAWIIFZJFZGDZWEWILZVQWJWFWFEFZGVQWJWFWHEFZWIEFWPVQAW
|
|
QWIEAUFZMVQWFWHWFVQWFAUGZNZVQHCDZWGCDWHCDOVQWGAUHZNZHWGUKPZWTUIRVQWFWFWSW
|
|
SSTVQWMHHJFZWGWGEFZJFZGVQWMHHXFJFZJFZXGVQWLXHHJVQWQWIIFWHWHEFWLXHVQWFWHWH
|
|
WTXDXDULVQAWQWIIWRMVQHWGWGXAVQOUMXCXCUNUOUPVQXFCDZXGXILZVQXFVQWGWGXBXBSZN
|
|
XAXAXJXKOOHHXFUQURVBUSVQXEGDXFGDXGGDXEUTVAGVCVDVEXLXEXFVFPTVQWICDWDBCVGWK
|
|
WNKZWOVHVQWFWHWTXDVIBAVNWDXMBCWIVRWILZVTWKWCWNXNVSWJGVRWIAEVJVKXNWBWMGXNW
|
|
AWLHJVRWIAIVJUPVKVLVMVOVPR $.
|
|
$}
|
|
|
|
$( The imaginary part of a real number is 0. (Contributed by NM,
|
|
18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) $)
|
|
reim0 $p |- ( A e. RR -> ( Im ` A ) = 0 ) $=
|
|
( cr wcel ci cc0 cmul co caddc cim cfv wceq recn it0e0 oveq2i addid1 syl5eq
|
|
cc syl fveq2d 0re crim mpan2 eqtr3d ) ABCZADEFGZHGZIJZAIJEUDUFAIUDAQCZUFAKA
|
|
LUHUFAEHGAUEEAHMNAOPRSUDEBCUGEKTAEUAUBUC $.
|
|
|
|
$( A number is real iff its imaginary part is 0. (Contributed by NM,
|
|
26-Sep-2005.) $)
|
|
reim0b $p |- ( A e. CC -> ( A e. RR <-> ( Im ` A ) = 0 ) ) $=
|
|
( cc wcel cr cim cfv cc0 wceq reim0 wa ci cmul co caddc replim adantr oveq2
|
|
cre it0e0 syl6eq oveq2d recl recnd addid1d sylan9eqr eqtrd eqeltrd impbid2
|
|
ex ) ABCZADCZAEFZGHZAIUJUMUKUJUMJZAARFZDUNAUOKULLMZNMZUOUJAUQHUMAOPUMUJUQUO
|
|
GNMUOUMUPGUONUMUPKGLMGULGKLQSTUAUJUOUJUOAUBZUCUDUEUFUJUODCUMURPUGUIUH $.
|
|
|
|
$( A number is real iff it equals its real part. Proposition 10-3.4(f) of
|
|
[Gleason] p. 133. (Contributed by NM, 20-Aug-2008.) $)
|
|
rereb $p |- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) ) $=
|
|
( cc wcel cr cre cfv wceq wa ci cim co caddc cc0 replim adantr reim0 oveq2d
|
|
cmul it0e0 syl6eq adantl recl recnd addid1d 3eqtrrd simpr eqeltrrd impbida
|
|
) ABCZADCZAEFZAGZUIUJHZAUKIAJFZRKZLKZUKMLKZUKUIAUPGUJANOUMUOMUKLUJUOMGUIUJU
|
|
OIMRKMUJUNMIRAPQSTUAQUIUQUKGUJUIUKUIUKAUBZUCUDOUEUIULHUKADUIULUFUIUKDCULURO
|
|
UGUH $.
|
|
|
|
$( A product with a real multiplier apart from zero is real iff the
|
|
multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.) $)
|
|
mulreap $p |- ( ( A e. CC /\ B e. RR /\ B # 0 )
|
|
-> ( A e. RR <-> ( B x. A ) e. RR ) ) $=
|
|
( cc wcel cr cre cfv wceq co wb rereb 3ad2ant1 wa recnd caddc adantl ax-icn
|
|
cmul ci mulcl cc0 cap wbr w3a recl recn anim1i 3adant1 mulcanap syl3anc cim
|
|
simp1 adantr imcl sylancr adddid replim oveq2d mp3an1 syl2an 3eqtr4d fveq2d
|
|
mul12 remulcl sylan2 crre syl2anc eqtr2d eqeq1d sylan bitr4d ancoms 3adant3
|
|
syl 3bitr2d ) ACDZBEDZBUAUBUCZUDZAEDZAFGZAHZBWARIZBARIZHZWDEDZVPVQVTWBJVRAK
|
|
LVSWACDZVPBCDZVRMZWEWBJVPVQWGVRVPWAAUEZNZLVPVQVRULVQVRWIVPVQWHVRBUFZUGUHWAA
|
|
BUIUJVPVQWEWFJZVRVQVPWMVQVPMZWEWDFGZWDHZWFWNWCWOWDWNWOWCSBAUKGZRIZRIZOIZFGZ
|
|
WCWNWDWTFWNBWASWQRIZOIZRIWCBXBRIZOIWDWTWNBWAXBVQWHVPWLUMVPWGVQWKPVPXBCDZVQV
|
|
PSCDZWQCDZXEQVPWQAUNZNZSWQTUOPUPWNAXCBRVPAXCHVQAUQPURWNWSXDWCOVQWHXGWSXDHZV
|
|
PWLXIXFWHXGXJQSBWQVCUSUTURVAVBWNWCEDZWREDZXAWCHVPVQWAEDXKWJBWAVDVEVPVQWQEDX
|
|
LXHBWQVDVEWCWRVFVGVHVIWNWDCDZWFWPJVQWHVPXMWLBATVJWDKVNVKVLVMVO $.
|
|
|
|
$( A real number equals its real part. One direction of Proposition
|
|
10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman,
|
|
7-Sep-2007.) $)
|
|
rere $p |- ( A e. RR -> ( Re ` A ) = A ) $=
|
|
( cr wcel cre cfv wceq cc wb recn rereb syl ibi ) ABCZADEAFZMAGCMNHAIAJKL
|
|
$.
|
|
|
|
$( A number is real iff it equals its complex conjugate. Proposition
|
|
10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised
|
|
by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjreb $p |- ( A e. CC -> ( A e. RR <-> ( * ` A ) = A ) ) $=
|
|
( cc wcel ccj cfv wceq cre ci cneg cmul co caddc cr cmin recnd ax-icn mulcl
|
|
cim sylancr cc0 negsubd mulneg2 oveq2d remim 3eqtr4rd replim eqeq12d negcld
|
|
recl imcl addcand eqcom eqnegd syl5bb cap wbr wa wb iap0 pm3.2i a1i syl3anc
|
|
mulcanap reim0b 3bitr4d 3bitrrd ) ABCZADEZAFAGEZHAREZIZJKZLKZVIHVJJKZLKZFVL
|
|
VNFZAMCZVGVHVMAVOVGVIVNIZLKVIVNNKVMVHVGVIVNVGVIAUIOZVGHBCZVJBCZVNBCPVGVJAUJ
|
|
OZHVJQSZUAVGVLVRVILVGVTWAVLVRFPWBHVJUBSUCAUDUEAUFUGVGVIVLVNVSVGVTVKBCZVLBCP
|
|
VGVJWBUHZHVKQSWCUKVGVKVJFZVJTFZVPVQWFVJVKFVGWGVKVJULVGVJWBUMUNVGWDWAVTHTUOU
|
|
PZUQZVPWFURWEWBWIVGVTWHPUSUTVAVKVJHVCVBAVDVEVF $.
|
|
|
|
$( Real part of a complex conjugate. (Contributed by Mario Carneiro,
|
|
14-Jul-2014.) $)
|
|
recj $p |- ( A e. CC -> ( Re ` ( * ` A ) ) = ( Re ` A ) ) $=
|
|
( cc wcel ccj cfv cre ci cim cneg cmul co caddc cmin recl recnd ax-icn imcl
|
|
sylancr wceq cr mulcl negsubd mulneg2 oveq2d remim 3eqtr4rd fveq2d renegcld
|
|
crre syl2anc eqtrd ) ABCZADEZFEAFEZGAHEZIZJKZLKZFEZUNULUMURFULUNGUOJKZIZLKU
|
|
NUTMKURUMULUNUTULUNANZOULGBCZUOBCZUTBCPULUOAQZOZGUOUARUBULUQVAUNLULVCVDUQVA
|
|
SPVFGUOUCRUDAUEUFUGULUNTCUPTCUSUNSVBULUOVEUHUNUPUIUJUK $.
|
|
|
|
$( Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by
|
|
Mario Carneiro, 14-Jul-2014.) $)
|
|
reneg $p |- ( A e. CC -> ( Re ` -u A ) = -u ( Re ` A ) ) $=
|
|
( cc wcel cneg cre cfv ci cim cmul co caddc recl recnd ax-icn mulcl sylancr
|
|
imcl wceq cr renegcld negdid replim negeqd mulneg2 oveq2d fveq2d crre eqtrd
|
|
3eqtr4d syl2anc ) ABCZADZEFAEFZDZGAHFZDZIJZKJZEFZUNUKULUREUKUMGUOIJZKJZDUNU
|
|
TDZKJULURUKUMUTUKUMALZMUKGBCZUOBCZUTBCNUKUOAQZMZGUOOPUAUKAVAAUBUCUKUQVBUNKU
|
|
KVDVEUQVBRNVGGUOUDPUEUIUFUKUNSCUPSCUSUNRUKUMVCTUKUOVFTUNUPUGUJUH $.
|
|
|
|
$( Real part distributes over addition. (Contributed by NM, 17-Mar-2005.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
readd $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) ) $=
|
|
( cc wcel caddc co cre cfv ci cmul cr recl adantr recnd ax-icn imcl sylancr
|
|
cim mulcl adantl wa add4d replim oveqan12d a1i adddid oveq2d 3eqtr4d fveq2d
|
|
wceq readdcl syl2an crre syl2anc eqtrd ) ACDZBCDZUAZABEFZGHAGHZBGHZEFZIARHZ
|
|
BRHZEFZJFZEFZGHZVBURUSVGGURUTIVCJFZEFZVAIVDJFZEFZEFVBVIVKEFZEFUSVGURUTVIVAV
|
|
KURUTUPUTKDZUQALZMNURICDZVCCDVICDOURVCUPVCKDZUQAPZMNZIVCSQURVAUQVAKDZUPBLZT
|
|
NURVPVDCDVKCDOURVDUQVDKDZUPBPZTNZIVDSQUBUPUQAVJBVLEAUCBUCUDURVFVMVBEURIVCVD
|
|
VPUROUEVSWDUFUGUHUIURVBKDZVEKDZVHVBUJUPVNVTWEUQVOWAUTVAUKULUPVQWBWFUQVRWCVC
|
|
VDUKULVBVEUMUNUO $.
|
|
|
|
$( Real part distributes over subtraction. (Contributed by NM,
|
|
17-Mar-2005.) $)
|
|
resub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) ) $=
|
|
( cc wcel wa cneg caddc cre cfv cmin negcl readd sylan2 reneg adantl negsub
|
|
co wceq recl recnd oveq2d eqtrd fveq2d syl2an 3eqtr3d ) ACDZBCDZEZABFZGQZHI
|
|
ZAHIZBHIZFZGQZABJQZHIULUMJQZUHUKULUIHIZGQZUOUGUFUICDUKUSRBKAUILMUHURUNULGUG
|
|
URUNRUFBNOUAUBUHUJUPHABPUCUFULCDUMCDUOUQRUGUFULASTUGUMBSTULUMPUDUE $.
|
|
|
|
$( Lemma for ~ remul , ~ immul , and ~ cjmul . (Contributed by NM,
|
|
28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
remullem $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( Re ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) /\
|
|
( Im ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) /\
|
|
( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) ) ) $=
|
|
( cc wcel cmul co cre cfv cim cmin wceq caddc ci recnd ax-icn mulcl sylancr
|
|
cr oveq12d remulcld wa ccj replim oveqan12d recl adantr imcl addcld adddird
|
|
adantl adddid mulcld add42d cneg mul4d c1 oveq1i mulm1d syl5eq eqtrd oveq2d
|
|
a1i negsubd addcomd mulassd mul12d 3eqtr4d 3eqtr2d 3eqtrd resubcld readdcld
|
|
ixi fveq2d crre syl2anc crim syl subcld subdird subadd4d subdid eqtr4d 3jca
|
|
remim ) ACDZBCDZUAZABEFZGHZAGHZBGHZEFZAIHZBIHZEFZJFZKWHIHZWJWNEFZWMWKEFZLFZ
|
|
KWHUBHZAUBHZBUBHZEFZKWGWIWPMWTEFZLFZGHZWPWGWHXFGWGWHWJMWMEFZLFZWKMWNEFZLFZE
|
|
FXIWKEFZXIXJEFZLFZXFWEWFAXIBXKEAUCBUCUDWGXIWKXJWGWJXHWGWJWEWJRDWFAUEUFZNZWG
|
|
MCDZWMCDXHCDOWGWMWEWMRDWFAUGUFZNZMWMPQZUHWGWKWFWKRDWEBUEUJZNZWGXQWNCDXJCDOW
|
|
GWNWFWNRDWEBUGUJZNZMWNPQZUKWGXNWLXHWKEFZLFZWJXJEFZXHXJEFZLFZLFWLYILFZYFYHLF
|
|
ZLFXFWGXLYGXMYJLWGWJXHWKXPXTYBUIWGWJXHXJXPXTYEUISWGWLYIYFYHWGWLWGWJWKXOYATZ
|
|
NZWGXHXJXTYEULZWGXHWKXTYBULZWGWJXJXPYEULZUMWGYKWPYLXELWGYKWLWOUNZLFWPWGYIYR
|
|
WLLWGYIMMEFZWOEFZYRWGMWMMWNXQWGOVBZXSUUAYDUOWGYTUPUNZWOEFYRYSUUBWOEVLUQWGWO
|
|
WGWOWGWMWNXRYCTZNZURUSUTVAWGWLWOYNUUDVCUTZWGMWSEFZMWREFZLFUUGUUFLFZYLXEWGUU
|
|
FUUGWGXQWSCDUUFCDOWGWSWGWMWKXRYATZNZMWSPQWGXQWRCDUUGCDOWGWRWGWJWNXOYCTZNZMW
|
|
RPQVDWGYFUUFYHUUGLWGMWMWKUUAXSYBVEZWGWJMWNXPUUAYDVFZSWGMWRWSUUAUULUUJUKZVGS
|
|
VHVIZVMZWGWPRDZWTRDZXGWPKWGWLWOYMUUCVJZWGWRWSUUKUUIVKZWPWTVNVOZUTWGWQXFIHZW
|
|
TWGWHXFIUUPVMWGUURUUSUVCWTKUUTUVAWPWTVPVOUTZWGXAWIMWQEFZJFZXDWGWHCDXAUVFKAB
|
|
PWHWDVQWGXDWJXHJFZWKXJJFZEFWJUVHEFZXHUVHEFZJFZUVFWEWFXBUVGXCUVHEAWDBWDUDWGW
|
|
JXHUVHXPXTWGWKXJYBYEVRVSWGWLYHJFZYFYIJFZJFYKYHYFLFZJFUVKUVFWGWLYHYFYIYNYQYP
|
|
YOVTWGUVIUVLUVJUVMJWGWJWKXJXPYBYEWAWGXHWKXJXTYBYEWASWGWIYKUVEUVNJWGXGWPWIYK
|
|
UVBUUQUUEVGWGXEUUHUVEUVNUUOWGWQWTMEUVDVAWGYHUUGYFUUFLUUNUUMSVGSVGVIWBWC $.
|
|
|
|
$( Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by
|
|
Mario Carneiro, 14-Jul-2014.) $)
|
|
remul $p |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) ) $=
|
|
( cc wcel wa cmul co cre cfv cim cmin wceq caddc ccj remullem simp1d ) ACDB
|
|
CDEABFGZHIAHIZBHIZFGAJIZBJIZFGKGLQJIRUAFGTSFGMGLQNIANIBNIFGLABOP $.
|
|
|
|
$( Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) $)
|
|
remul2 $p |- ( ( A e. RR /\ B e. CC ) ->
|
|
( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) ) $=
|
|
( cr wcel cc wa cmul cre cfv cim cmin cc0 wceq recn remul sylan rere oveq1d
|
|
co recnd adantr reim0 imcl mul02d sylan9eq oveq12d recl mulcl syl2an 3eqtrd
|
|
subid1d ) ACDZBEDZFZABGSHIZAHIZBHIZGSZAJIZBJIZGSZKSZAUQGSZLKSVCULAEDZUMUOVB
|
|
MANZABOPUNURVCVALKUNUPAUQGULUPAMUMAQUARULUMVALUTGSLULUSLUTGAUBRUMUTUMUTBUCT
|
|
UDUEUFUNVCULVDUQEDVCEDUMVEUMUQBUGTAUQUHUIUKUJ $.
|
|
|
|
$( Real part of a division. Related to ~ remul2 . (Contributed by Jim
|
|
Kingdon, 14-Jun-2020.) $)
|
|
redivap $p |- ( ( A e. CC /\ B e. RR /\ B # 0 ) ->
|
|
( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) ) $=
|
|
( cc wcel cr cc0 cap wbr w3a c1 cdiv co cmul cre cfv wa ancom 3anass bitr4i
|
|
wceq rerecclap anim1i sylbir remul2 syl recn divrecap2 fveq2d syl3an2 recnd
|
|
recl 3ad2ant1 3ad2ant2 simp3 divrecap2d 3eqtr4d ) ACDZBEDZBFGHZIZJBKLZAMLZN
|
|
OZVAANOZMLZABKLZNOZVDBKLUTVAEDZUQPZVCVETUTURUSPZUQPZVIVKUQVJPUTVJUQQUQURUSR
|
|
SVJVHUQBUAUBUCVAAUDUEURUQBCDZUSVGVCTBUFZUQVLUSIVFVBNABUGUHUIUTVDBUQURVDCDUS
|
|
UQVDAUKUJULURUQVLUSVMUMUQURUSUNUOUP $.
|
|
|
|
$( Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
imcj $p |- ( A e. CC -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) ) $=
|
|
( cc wcel ccj cfv cim cre ci cneg cmul co caddc cmin recl recnd ax-icn imcl
|
|
sylancr wceq cr mulcl negsubd mulneg2 oveq2d remim 3eqtr4rd fveq2d renegcld
|
|
crim syl2anc eqtrd ) ABCZADEZFEAGEZHAFEZIZJKZLKZFEZUPULUMURFULUNHUOJKZIZLKU
|
|
NUTMKURUMULUNUTULUNANZOULHBCZUOBCZUTBCPULUOAQZOZHUOUARUBULUQVAUNLULVCVDUQVA
|
|
SPVFHUOUCRUDAUEUFUGULUNTCUPTCUSUPSVBULUOVEUHUNUPUIUJUK $.
|
|
|
|
$( The imaginary part of a negative number. (Contributed by NM,
|
|
18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
imneg $p |- ( A e. CC -> ( Im ` -u A ) = -u ( Im ` A ) ) $=
|
|
( cc wcel cneg cim cfv cre ci cmul co caddc recl recnd ax-icn mulcl sylancr
|
|
imcl wceq cr renegcld negdid replim negeqd mulneg2 oveq2d fveq2d crim eqtrd
|
|
3eqtr4d syl2anc ) ABCZADZEFAGFZDZHAEFZDZIJZKJZEFZUPUKULUREUKUMHUOIJZKJZDUNU
|
|
TDZKJULURUKUMUTUKUMALZMUKHBCZUOBCZUTBCNUKUOAQZMZHUOOPUAUKAVAAUBUCUKUQVBUNKU
|
|
KVDVEUQVBRNVGHUOUDPUEUIUFUKUNSCUPSCUSUPRUKUMVCTUKUOVFTUNUPUGUJUH $.
|
|
|
|
$( Imaginary part distributes over addition. (Contributed by NM,
|
|
18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
imadd $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) ) $=
|
|
( cc wcel caddc co cim cfv ci cmul cr recl adantr recnd ax-icn imcl sylancr
|
|
cre mulcl adantl wa add4d replim oveqan12d a1i adddid oveq2d 3eqtr4d fveq2d
|
|
wceq readdcl syl2an crim syl2anc eqtrd ) ACDZBCDZUAZABEFZGHARHZBRHZEFZIAGHZ
|
|
BGHZEFZJFZEFZGHZVEURUSVGGURUTIVCJFZEFZVAIVDJFZEFZEFVBVIVKEFZEFUSVGURUTVIVAV
|
|
KURUTUPUTKDZUQALZMNURICDZVCCDVICDOURVCUPVCKDZUQAPZMNZIVCSQURVAUQVAKDZUPBLZT
|
|
NURVPVDCDVKCDOURVDUQVDKDZUPBPZTNZIVDSQUBUPUQAVJBVLEAUCBUCUDURVFVMVBEURIVCVD
|
|
VPUROUEVSWDUFUGUHUIURVBKDZVEKDZVHVEUJUPVNVTWEUQVOWAUTVAUKULUPVQWBWFUQVRWCVC
|
|
VDUKULVBVEUMUNUO $.
|
|
|
|
$( Imaginary part distributes over subtraction. (Contributed by NM,
|
|
18-Mar-2005.) $)
|
|
imsub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) ) $=
|
|
( cc wcel wa cneg caddc cim cfv cmin negcl imadd sylan2 imneg adantl negsub
|
|
co wceq imcl recnd oveq2d eqtrd fveq2d syl2an 3eqtr3d ) ACDZBCDZEZABFZGQZHI
|
|
ZAHIZBHIZFZGQZABJQZHIULUMJQZUHUKULUIHIZGQZUOUGUFUICDUKUSRBKAUILMUHURUNULGUG
|
|
URUNRUFBNOUAUBUHUJUPHABPUCUFULCDUMCDUOUQRUGUFULASTUGUMBSTULUMPUDUE $.
|
|
|
|
$( Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised
|
|
by Mario Carneiro, 14-Jul-2014.) $)
|
|
immul $p |- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) ) $=
|
|
( cc wcel wa cmul co cre cfv cim cmin wceq caddc ccj remullem simp2d ) ACDB
|
|
CDEABFGZHIAHIZBHIZFGAJIZBJIZFGKGLQJIRUAFGTSFGMGLQNIANIBNIFGLABOP $.
|
|
|
|
$( Imaginary part of a product. (Contributed by Mario Carneiro,
|
|
2-Aug-2014.) $)
|
|
immul2 $p |- ( ( A e. RR /\ B e. CC ) ->
|
|
( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) ) $=
|
|
( cr wcel cc wa cmul co cim cfv cre caddc wceq recn immul sylan rere oveq1d
|
|
cc0 recnd adantr reim0 recl mul02d oveq12d imcl mulcl syl2an addid1d 3eqtrd
|
|
sylan9eq ) ACDZBEDZFZABGHIJZAKJZBIJZGHZAIJZBKJZGHZLHZAUQGHZSLHVCULAEDZUMUOV
|
|
BMANZABOPUNURVCVASLUNUPAUQGULUPAMUMAQUARULUMVASUTGHSULUSSUTGAUBRUMUTUMUTBUC
|
|
TUDUKUEUNVCULVDUQEDVCEDUMVEUMUQBUFTAUQUGUHUIUJ $.
|
|
|
|
$( Imaginary part of a division. Related to ~ immul2 . (Contributed by Jim
|
|
Kingdon, 14-Jun-2020.) $)
|
|
imdivap $p |- ( ( A e. CC /\ B e. RR /\ B # 0 ) ->
|
|
( Im ` ( A / B ) ) = ( ( Im ` A ) / B ) ) $=
|
|
( cc wcel cr cc0 cap wbr w3a c1 cdiv co cmul cim cfv wa ancom 3anass bitr4i
|
|
wceq rerecclap anim1i sylbir immul2 syl recn divrecap2 fveq2d syl3an2 recnd
|
|
imcl 3ad2ant1 3ad2ant2 simp3 divrecap2d 3eqtr4d ) ACDZBEDZBFGHZIZJBKLZAMLZN
|
|
OZVAANOZMLZABKLZNOZVDBKLUTVAEDZUQPZVCVETUTURUSPZUQPZVIVKUQVJPUTVJUQQUQURUSR
|
|
SVJVHUQBUAUBUCVAAUDUEURUQBCDZUSVGVCTBUFZUQVLUSIVFVBNABUGUHUIUTVDBUQURVDCDUS
|
|
UQVDAUKUJULURUQVLUSVMUMUQURUSUNUOUP $.
|
|
|
|
$( A real number equals its complex conjugate. Proposition 10-3.4(f) of
|
|
[Gleason] p. 133. (Contributed by NM, 8-Oct-1999.) $)
|
|
cjre $p |- ( A e. RR -> ( * ` A ) = A ) $=
|
|
( cc wcel cr ccj cfv wceq recn cjreb biimpd mpcom ) ABCZADCZAEFAGZAHLMNAIJK
|
|
$.
|
|
|
|
$( The conjugate of the conjugate is the original complex number.
|
|
Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM,
|
|
29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjcj $p |- ( A e. CC -> ( * ` ( * ` A ) ) = A ) $=
|
|
( cc wcel ccj cfv cre ci cim cmul co caddc wceq cjcl recj eqtrd cneg negeqd
|
|
syl imcj replim imcl recnd negnegd oveq2d oveq12d 3syl 3eqtr4d ) ABCZADEZDE
|
|
ZFEZGUJHEZIJZKJZAFEZGAHEZIJZKJUJAUHUKUOUMUQKUHUKUIFEZUOUHUIBCZUKURLAMZUINRA
|
|
NOUHULUPGIUHULUIHEZPZUPUHUSULVBLUTUISRUHVBUPPZPUPUHVAVCASQUHUPUHUPAUAUBUCOO
|
|
UDUEUHUSUJBCUJUNLUTUIMUJTUFATUG $.
|
|
|
|
$( Complex conjugate distributes over addition. Proposition 10-3.4(a) of
|
|
[Gleason] p. 133. (Contributed by NM, 31-Jul-1999.) (Revised by Mario
|
|
Carneiro, 14-Jul-2014.) $)
|
|
cjadd $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) ) $=
|
|
( cc wcel caddc co cre cfv ci cim cmul cmin ccj ax-icn cr imcl adantr recnd
|
|
adantl remim readd imadd oveq2d a1i adddid eqtrd oveq12d recl mulcl sylancr
|
|
wa addsub4d wceq addcl syl oveqan12d 3eqtr4d ) ACDZBCDZUKZABEFZGHZIVAJHZKFZ
|
|
LFZAGHZIAJHZKFZLFZBGHZIBJHZKFZLFZEFZVAMHZAMHZBMHZEFUTVEVFVJEFZVHVLEFZLFVNUT
|
|
VBVRVDVSLABUAUTVDIVGVKEFZKFVSUTVCVTIKABUBUCUTIVGVKICDZUTNUDUTVGURVGODUSAPQR
|
|
ZUTVKUSVKODURBPSRZUEUFUGUTVFVJVHVLUTVFURVFODUSAUHQRUTVJUSVJODURBUHSRUTWAVGC
|
|
DVHCDNWBIVGUIUJUTWAVKCDVLCDNWCIVKUIUJULUFUTVACDVOVEUMABUNVATUOURUSVPVIVQVME
|
|
ATBTUPUQ $.
|
|
|
|
$( Complex conjugate distributes over multiplication. Proposition 10-3.4(c)
|
|
of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened
|
|
by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjmul $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) ) $=
|
|
( cc wcel wa cmul co cre cfv cim cmin wceq caddc ccj remullem simp3d ) ACDB
|
|
CDEABFGZHIAHIZBHIZFGAJIZBJIZFGKGLQJIRUAFGTSFGMGLQNIANIBNIFGLABOP $.
|
|
|
|
$( Standard inner product on complex numbers. (Contributed by NM,
|
|
29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
ipcnval $p |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) ) $=
|
|
( cc wcel wa cfv cmul co cre cmin cneg wceq adantl oveq2d imcl recnd syl2an
|
|
cim recl mulcl ccj caddc cjcl remul recj imcj mulneg2 eqtrd oveq12d subnegd
|
|
sylan2 3eqtrd ) ACDZBCDZEZABUAFZGHIFZAIFZUPIFZGHZARFZUPRFZGHZJHZURBIFZGHZVA
|
|
BRFZGHZKZJHVFVHUBHUNUMUPCDUQVDLBUCAUPUDUKUOUTVFVCVIJUOUSVEURGUNUSVELUMBUEMN
|
|
UOVCVAVGKZGHZVIUOVBVJVAGUNVBVJLUMBUFMNUMVACDZVGCDZVKVILUNUMVAAOPZUNVGBOPZVA
|
|
VGUGQUHUIUOVFVHUMURCDVECDVFCDUNUMURASPUNVEBSPURVETQUMVLVMVHCDUNVNVOVAVGTQUJ
|
|
UL $.
|
|
|
|
$( A complex number times its conjugate is real. (Contributed by NM,
|
|
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjmulrcl $p |- ( A e. CC -> ( A x. ( * ` A ) ) e. RR ) $=
|
|
( cc wcel ccj cfv cmul co cr wceq cjcj oveq2d cjmul mpdan mulcom 3eqtr4d wb
|
|
cjcl mulcl cjreb syl mpbird ) ABCZAADEZFGZHCZUDDEZUDIZUBUCUCDEZFGZUCAFGZUFU
|
|
DUBUHAUCFAJKUBUCBCZUFUIIAQZAUCLMUBUKUDUJIULAUCNMOUBUDBCZUEUGPUBUKUMULAUCRMU
|
|
DSTUA $.
|
|
|
|
$( A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjmulval $p |- ( A e. CC -> ( A x. ( * ` A ) ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel cre cfv c2 cexp cim caddc cmul ccj recl recnd sqvald imcl oveq12d
|
|
co wceq ipcnval anidms cr cjmulrcl rere syl 3eqtr2rd ) ABCZADEZFGQZAHEZFGQZ
|
|
IQUGUGJQZUIUIJQZIQZAAKEJQZDEZUNUFUHUKUJULIUFUGUFUGALMNUFUIUFUIAOMNPUFUOUMRA
|
|
ASTUFUNUACUOUNRAUBUNUCUDUE $.
|
|
|
|
$( A complex number times its conjugate is nonnegative. (Contributed by NM,
|
|
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjmulge0 $p |- ( A e. CC -> 0 <_ ( A x. ( * ` A ) ) ) $=
|
|
( cc wcel cc0 cre cfv c2 cexp co cim caddc ccj cmul cle recl resqcld sqge0d
|
|
imcl addge0d cjmulval breqtrrd ) ABCZDAEFZGHIZAJFZGHIZKIAALFMINUBUDUFUBUCAO
|
|
ZPUBUEARZPUBUCUGQUBUEUHQSATUA $.
|
|
|
|
$( Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
cjneg $p |- ( A e. CC -> ( * ` -u A ) = -u ( * ` A ) ) $=
|
|
( cc wcel cneg cre cfv ci cim cmul co cmin ccj recl recnd imcl sylancr wceq
|
|
ax-icn 3eqtr4d remim mulcl neg2subd reneg oveq2d mulneg2 oveq12d negsubdi2d
|
|
imneg eqtrd negcl syl negeqd ) ABCZADZEFZGUNHFZIJZKJZAEFZGAHFZIJZKJZDZUNLFZ
|
|
ALFZDUMUSDZVADZKJVAUSKJURVCUMUSVAUMUSAMNZUMGBCZUTBCZVABCRUMUTAONZGUTUAPZUBU
|
|
MUOVFUQVGKAUCUMUQGUTDZIJZVGUMUPVMGIAUHUDUMVIVJVNVGQRVKGUTUEPUIUFUMUSVAVHVLU
|
|
GSUMUNBCVDURQAUJUNTUKUMVEVBATULS $.
|
|
|
|
$( A number plus its conjugate is twice its real part. Compare Proposition
|
|
10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
|
|
(Revised by Mario Carneiro, 14-Jul-2014.) $)
|
|
addcj $p |- ( A e. CC -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) ) $=
|
|
( cc wcel c2 cre cfv cmul ccj caddc cdiv reval oveq2d wceq cjcl addcl mpdan
|
|
co cc0 cap wbr 2cn 2ap0 divcanap2 mp3an23 syl eqtr2d ) ABCZDAEFZGQDAAHFZIQZ
|
|
DJQZGQZUJUGUHUKDGAKLUGUJBCZULUJMZUGUIBCUMANAUIOPUMDBCDRSTUNUAUBUJDUCUDUEUF
|
|
$.
|
|
|
|
$( Complex conjugate distributes over subtraction. (Contributed by NM,
|
|
28-Apr-2005.) $)
|
|
cjsub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) ) $=
|
|
( cc wcel wa cneg caddc ccj cfv cmin negcl cjadd sylan2 negsub fveq2d cjneg
|
|
co wceq adantl cjcl oveq2d syl2an eqtrd 3eqtr3d ) ACDZBCDZEZABFZGQZHIZAHIZU
|
|
HHIZGQZABJQZHIUKBHIZJQZUFUEUHCDUJUMRBKAUHLMUGUIUNHABNOUGUMUKUOFZGQZUPUGULUQ
|
|
UKGUFULUQRUEBPSUAUEUKCDUOCDURUPRUFATBTUKUONUBUCUD $.
|
|
|
|
${
|
|
$d j k A $. $d j k N $.
|
|
$( Complex conjugate of positive integer exponentiation. (Contributed by
|
|
NM, 7-Jun-2006.) $)
|
|
cjexp $p |- ( ( A e. CC /\ N e. NN0 ) ->
|
|
( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) $=
|
|
( vj vk cn0 wcel cc cexp co ccj cfv wi cc0 c1 oveq2 fveq2d eqeq12d imbi2d
|
|
wceq cmul cv caddc exp0 cjcl cr 1re cjre ax-mp syl6eqr eqtr4d expp1 expcl
|
|
syl wa simpl cjmul syl2anc eqtrd adantr oveq1 sylan sylan9eqr exp31 com12
|
|
eqcomd a2d nn0ind impcom ) BEFAGFZABHIZJKZAJKZBHIZSZVIACUAZHIZJKZVLVOHIZS
|
|
ZLVIAMHIZJKZVLMHIZSZLVIADUAZHIZJKZVLWDHIZSZLVIAWDNUBIZHIZJKZVLWIHIZSZLVIV
|
|
NLCDBVOMSZVSWCVIWNVQWAVRWBWNVPVTJVOMAHOPVOMVLHOQRVOWDSZVSWHVIWOVQWFVRWGWO
|
|
VPWEJVOWDAHOPVOWDVLHOQRVOWISZVSWMVIWPVQWKVRWLWPVPWJJVOWIAHOPVOWIVLHOQRVOB
|
|
SZVSVNVIWQVQVKVRVMWQVPVJJVOBAHOPVOBVLHOQRVIWANJKZWBVIVTNJAUCPVIVLGFZWBWRS
|
|
AUDZWSWBNWRVLUCNUEFWRNSUFNUGUHUIUMUJWDEFZVIWHWMVIXAWHWMLVIXAWHWMVIXAUNZWH
|
|
UNWKWFVLTIZWLXBWKXCSWHXBWKWEATIZJKZXCXBWJXDJAWDUKPXBWEGFVIXEXCSAWDULVIXAU
|
|
OWEAUPUQURUSWHXBXCWGVLTIZWLWFWGVLTUTXBWLXFVIWSXAWLXFSWTVLWDUKVAVEVBURVCVD
|
|
VFVGVH $.
|
|
$}
|
|
|
|
$( The imaginary part of a number in terms of complex conjugate.
|
|
(Contributed by NM, 30-Apr-2005.) $)
|
|
imval2 $p |- ( A e. CC ->
|
|
( Im ` A ) = ( ( A - ( * ` A ) ) / ( 2 x. _i ) ) ) $=
|
|
( cc wcel cim cfv c2 ci cmul co cdiv ccj cmin wceq imcl recnd 2mulicn caddc
|
|
syl ax-icn oveq1d cc0 cap wbr 2muliap0 divcanap4 mp3an23 recl mulcl sylancr
|
|
cre addcld subsubd replim remim oveq12d 2timesd mulcom mpan2 mulass mp3an12
|
|
2cn eqtrd pncan2d 3eqtr4d 3eqtr4rd eqtr3d ) ABCZADEZFGHIZHIZVIJIZVHAAKEZLIZ
|
|
VIJIVGVHBCZVKVHMZVGVHANOZVNVIBCZVIUAUBUCVOPUDVHVIUEUFRVGVJVMVIJVGAUJEZGVHHI
|
|
ZQIZVRVSLIZLIVTVRLIZVSQIZVMVJVGVTVRVSVGVRVSVGVRAUGOZVGGBCZVNVSBCSVPGVHUHUIZ
|
|
UKWDWFULVGAVTVLWALAUMAUNUOVGFVSHIZVSVSQIVJWCVGVSWFUPVGVNVJWGMVPVNVJVIVHHIZW
|
|
GVNVQVJWHMPVHVIUQURFBCWEVNWHWGMVASFGVHUSUTVBRVGWBVSVSQVGVRVSWDWFVCTVDVETVF
|
|
$.
|
|
|
|
$( The real part of zero. (Contributed by NM, 27-Jul-1999.) $)
|
|
re0 $p |- ( Re ` 0 ) = 0 $=
|
|
( cc0 cr wcel cre cfv wceq 0re rere ax-mp ) ABCADEAFGAHI $.
|
|
|
|
$( The imaginary part of zero. (Contributed by NM, 27-Jul-1999.) $)
|
|
im0 $p |- ( Im ` 0 ) = 0 $=
|
|
( cc0 cr wcel cim cfv wceq 0re reim0 ax-mp ) ABCADEAFGAHI $.
|
|
|
|
$( The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.) $)
|
|
re1 $p |- ( Re ` 1 ) = 1 $=
|
|
( c1 cr wcel cre cfv wceq 1re rere ax-mp ) ABCADEAFGAHI $.
|
|
|
|
$( The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.) $)
|
|
im1 $p |- ( Im ` 1 ) = 0 $=
|
|
( c1 cr wcel cim cfv cc0 wceq 1re reim0 ax-mp ) ABCADEFGHAIJ $.
|
|
|
|
$( The real part of ` _i ` . (Contributed by Scott Fenton, 9-Jun-2006.) $)
|
|
rei $p |- ( Re ` _i ) = 0 $=
|
|
( cc0 ci c1 cmul co caddc cre ax-icn ax-1cn mulcli addid2i fveq2i wcel wceq
|
|
cfv cr 0re 1re crre mp2an mulid1i 3eqtr3ri ) ABCDEZFEZGOZUCGOABGOUDUCGUCBCH
|
|
IJKLAPMCPMUEANQRACSTUCBGBHUALUB $.
|
|
|
|
$( The imaginary part of ` _i ` . (Contributed by Scott Fenton,
|
|
9-Jun-2006.) $)
|
|
imi $p |- ( Im ` _i ) = 1 $=
|
|
( ci c1 cmul cim cfv cc0 ax-icn ax-1cn mulcli addid2i eqcomi fveq2i mulid1i
|
|
co caddc cr wcel wceq 0re 1re crim mp2an 3eqtr3i ) ABCNZDEFUDONZDEZADEBUDUE
|
|
DUEUDUDABGHIJKLUDADAGMLFPQBPQUFBRSTFBUAUBUC $.
|
|
|
|
$( The conjugate of zero. (Contributed by NM, 27-Jul-1999.) $)
|
|
cj0 $p |- ( * ` 0 ) = 0 $=
|
|
( cc0 cr wcel ccj cfv wceq 0re cjre ax-mp ) ABCADEAFGAHI $.
|
|
|
|
$( The complex conjugate of the imaginary unit. (Contributed by NM,
|
|
26-Mar-2005.) $)
|
|
cji $p |- ( * ` _i ) = -u _i $=
|
|
( ci cre cfv cim cmul co cmin cc0 ccj cneg rei c1 imi oveq2i ax-icn mulid1i
|
|
eqtri oveq12i cc wcel wceq remim ax-mp df-neg 3eqtr4i ) ABCZAADCZEFZGFZHAGF
|
|
AICZAJUFHUHAGKUHALEFAUGLAEMNAOPQRASTUJUIUAOAUBUCAUDUE $.
|
|
|
|
$( The conjugate of a representation of a complex number in terms of real and
|
|
imaginary parts. (Contributed by NM, 1-Jul-2005.) $)
|
|
cjreim $p |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) =
|
|
( A - ( _i x. B ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc ccj cfv cneg cc wceq recn ax-icn sylancr cjre
|
|
syl2an 3eqtrd cmin mulcl cjadd cjmul cji oveq12d mulneg1 oveqan12d negsub
|
|
a1i ) ACDZBCDZEAFBGHZIHJKZAJKZUMJKZIHZAUMLZIHZAUMUAHZUKAMDZUMMDZUNUQNULAOZU
|
|
LFMDZBMDZVBPBOZFBUBQZAUMUCSUKULUOAUPURIARULUPFJKZBJKZGHZFLZBGHZURULVDVEUPVJ
|
|
NPVFFBUDQULVHVKVIBGVHVKNULUEUJBRUFULVDVEVLURNPVFFBUGQTUHUKVAVBUSUTNULVCVGAU
|
|
MUIST $.
|
|
|
|
$( The conjugate of the representation of a complex number in terms of real
|
|
and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened
|
|
by Mario Carneiro, 29-May-2016.) $)
|
|
cjreim2 $p |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A - ( _i x. B ) ) ) =
|
|
( A + ( _i x. B ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc ccj cfv cmin cjreim fveq2d simpl recnd ax-icn
|
|
cc wceq a1i simpr mulcld addcld cjcj syl eqtr3d ) ACDZBCDZEZAFBGHZIHZJKZJKZ
|
|
AUJLHZJKUKUIULUNJABMNUIUKRDUMUKSUIAUJUIAUGUHOPUIFBFRDUIQTUIBUGUHUAPUBUCUKUD
|
|
UEUF $.
|
|
|
|
$( Complex conjugate is a one-to-one function. (Contributed by NM,
|
|
29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.) $)
|
|
cj11 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( * ` A ) = ( * ` B ) <-> A = B ) ) $=
|
|
( cc wcel wa ccj cfv wceq fveq2 cjcj eqeqan12d syl5ib impbid1 ) ACDZBCDZEZA
|
|
FGZBFGZHZABHZSQFGZRFGZHPTQRFINOUAAUBBAJBJKLABFIM $.
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $.
|
|
$( Complex conjugate and apartness. (Contributed by Jim Kingdon,
|
|
14-Jun-2020.) $)
|
|
cjap $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( * ` A ) # ( * ` B ) <-> A # B ) ) $=
|
|
( vx vy vz vw cc wcel wa cv ci cmul co caddc wceq cr wrex ccj cap wbr cfv
|
|
wb cnre adantr ad3antlr cneg simplrr ad2antrr recnd apneg syl2anc simpllr
|
|
wo orbi2d simpr breq12d simplrl apreim syl22anc bitrd fveq2d cjreim eqtrd
|
|
cmin ax-icn a1i submul2 syl3anc renegcld 3bitrd 3bitr4rd rexlimdvva mpd
|
|
ex ) AGHZBGHZIZACJZKDJZLMZNMZOZDPQCPQZARUAZBRUAZSTZABSTZUBZVOWCVPCDAUCUDV
|
|
QWBWHCDPPVQVRPHZVSPHZIZIZWBWHWLWBIZBEJZKFJZLMZNMZOZFPQEPQZWHVPWSVOWKWBEFB
|
|
UCUEWMWRWHEFPPWMWNPHZWOPHZIZIZWRWHXCWRIZVRWNSTZVSWOSTZUMZXEVSUFZWOUFZSTZU
|
|
MZWGWFXDXFXJXEXDVSGHZWOGHZXFXJUBXDVSWMWJXBWRVQWIWJWBUGUHZUIZXDWOWMWTXAWRU
|
|
GZUIZVSWOUJUKUNXDWGWAWQSTZXGXDAWABWQSWLWBXBWRULZXCWRUOZUPXDWIWJWTXAXRXGUB
|
|
WMWIXBWRVQWIWJWBUQUHZXNWMWTXAWRUQZXPVRVSWNWOURUSUTXDWFVRVTVDMZWNWPVDMZSTV
|
|
RKXHLMNMZWNKXILMNMZSTZXKXDWDYCWEYDSXDWDWARUAZYCXDAWARXSVAXDWIWJYHYCOYAXNV
|
|
RVSVBUKVCXDWEWQRUAZYDXDBWQRXTVAXDWTXAYIYDOYBXPWNWOVBUKVCUPXDYCYEYDYFSXDVR
|
|
GHKGHZXLYCYEOXDVRYAUIYJXDVEVFZXOVRKVSVGVHXDWNGHYJXMYDYFOXDWNYBUIYKXQWNKWO
|
|
VGVHUPXDWIXHPHWTXIPHYGXKUBYAXDVSXNVIYBXDWOXPVIVRXHWNXIURUSVJVKVNVLVMVNVLV
|
|
M $.
|
|
$}
|
|
|
|
$( A number is apart from zero iff its complex conjugate is apart from zero.
|
|
(Contributed by Jim Kingdon, 14-Jun-2020.) $)
|
|
cjap0 $p |- ( A e. CC -> ( A # 0 <-> ( * ` A ) # 0 ) ) $=
|
|
( ccj cfv cc0 cap wbr cc wcel cj0 breq2i wb 0cn cjap mpan2 syl5rbbr ) ABCZD
|
|
EFPDBCZEFZAGHZADEFZQDPEIJSDGHRTKLADMNO $.
|
|
|
|
$( A number is nonzero iff its complex conjugate is nonzero. Also see
|
|
~ cjap0 which is similar but for apartness. (Contributed by NM,
|
|
29-Apr-2005.) $)
|
|
cjne0 $p |- ( A e. CC -> ( A =/= 0 <-> ( * ` A ) =/= 0 ) ) $=
|
|
( cc wcel cc0 ccj cfv wceq cj0 eqeq2i wb 0cn cj11 mpan2 syl5rbbr necon3bid
|
|
) ABCZADAEFZDQDGQDEFZGZPADGZRDQHIPDBCSTJKADLMNO $.
|
|
|
|
$( Complex conjugate distributes over division. (Contributed by Jim Kingdon,
|
|
14-Jun-2020.) $)
|
|
cjdivap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co ccj cfv cmul divclap cjcl syl simp2 simp3
|
|
wb eqtr3d cjap0 mpbid divcanap4d wceq cjmul syl2anc divcanap1 fveq2d oveq1d
|
|
) ACDZBCDZBEFGZHZABIJZKLZBKLZMJZUPIJUOAKLZUPIJUMUOUPUMUNCDZUOCDABNZUNOPUMUK
|
|
UPCDUJUKULQZBOPUMULUPEFGZUJUKULRUMUKULVBSVABUAPUBUCUMUQURUPIUMUNBMJZKLZUQUR
|
|
UMUSUKVDUQUDUTVAUNBUEUFUMVCAKABUGUHTUIT $.
|
|
|
|
${
|
|
$d F z $. $d x y z $.
|
|
cnrecnv.1 $e |- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) $.
|
|
$( The inverse to the canonical bijection from ` ( RR X. RR ) ` to ` CC `
|
|
from ~ cnref1o . (Contributed by Mario Carneiro, 25-Aug-2014.) $)
|
|
cnrecnv $p |- `' F = ( z e. CC |-> <. ( Re ` z ) , ( Im ` z ) >. ) $=
|
|
( cc cv cfv cmpt wceq wtru cr wf1o a1i wcel ci cmul co caddc recnd cre wf
|
|
ccnv cim cop cnref1o f1ocnv f1of mp2b feqmptd trud df-ov recl imcl ax-icn
|
|
cxp mulcld addcld oveq1 oveq2 oveq2d ovmpt2g syl3anc replim eqtr4d fveq2d
|
|
syl5eqr opelxpi syl2anc f1ocnvfv1 sylancr eqtr3d mpteq2ia eqtri ) DUCZCFC
|
|
GZVOHZIZCFVPUAHZVPUDHZUEZIVOVRJKCFLLUPZVOFWBVOUBZKWBFDMZFWBVOMWCABDEUFZWB
|
|
FDUGFWBVOUHUINUJUKCFVQWAVPFOZWADHZVOHZVQWAWFWGVPVOWFWGVSPVTQRZSRZVPWFWGVS
|
|
VTDRZWJVSVTDULWFVSLOZVTLOZWJFOWKWJJVPUMZVPUNZWFVSWIWFVSWNTWFPVTPFOWFUONWF
|
|
VTWOTUQURABVSVTLLAGZPBGZQRZSRWJDVSWRSRFWPVSWRSUSWQVTJWRWIVSSWQVTPQUTVAEVB
|
|
VCVGVPVDVEVFWFWDWAWBOZWHWAJWEWFWLWMWSWNWOVSVTLLVHVIWBFWADVJVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
recl.1 $e |- A e. CC $.
|
|
$( The real part of a complex number is real (closure law). (Contributed
|
|
by NM, 11-May-1999.) $)
|
|
recli $p |- ( Re ` A ) e. RR $=
|
|
( cc wcel cre cfv cr recl ax-mp ) ACDAEFGDBAHI $.
|
|
|
|
$( The imaginary part of a complex number is real (closure law).
|
|
(Contributed by NM, 11-May-1999.) $)
|
|
imcli $p |- ( Im ` A ) e. RR $=
|
|
( cc wcel cim cfv cr imcl ax-mp ) ACDAEFGDBAHI $.
|
|
|
|
$( Closure law for complex conjugate. (Contributed by NM, 11-May-1999.) $)
|
|
cjcli $p |- ( * ` A ) e. CC $=
|
|
( cc wcel ccj cfv cjcl ax-mp ) ACDAEFCDBAGH $.
|
|
|
|
$( Construct a complex number from its real and imaginary parts.
|
|
(Contributed by NM, 1-Oct-1999.) $)
|
|
replimi $p |- A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) $=
|
|
( cc wcel cre cfv ci cim cmul co caddc wceq replim ax-mp ) ACDAAEFGAHFIJK
|
|
JLBAMN $.
|
|
|
|
$( The conjugate of the conjugate is the original complex number.
|
|
Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM,
|
|
11-May-1999.) $)
|
|
cjcji $p |- ( * ` ( * ` A ) ) = A $=
|
|
( cc wcel ccj cfv wceq cjcj ax-mp ) ACDAEFEFAGBAHI $.
|
|
|
|
$( A number is real iff its imaginary part is 0. (Contributed by NM,
|
|
29-May-1999.) $)
|
|
reim0bi $p |- ( A e. RR <-> ( Im ` A ) = 0 ) $=
|
|
( cc wcel cr cim cfv cc0 wceq wb reim0b ax-mp ) ACDAEDAFGHIJBAKL $.
|
|
|
|
$( A real number equals its real part. Proposition 10-3.4(f) of [Gleason]
|
|
p. 133. (Contributed by NM, 27-Oct-1999.) $)
|
|
rerebi $p |- ( A e. RR <-> ( Re ` A ) = A ) $=
|
|
( cc wcel cr cre cfv wceq wb rereb ax-mp ) ACDAEDAFGAHIBAJK $.
|
|
|
|
$( A number is real iff it equals its complex conjugate. Proposition
|
|
10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) $)
|
|
cjrebi $p |- ( A e. RR <-> ( * ` A ) = A ) $=
|
|
( cc wcel cr ccj cfv wceq wb cjreb ax-mp ) ACDAEDAFGAHIBAJK $.
|
|
|
|
$( Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) $)
|
|
recji $p |- ( Re ` ( * ` A ) ) = ( Re ` A ) $=
|
|
( cc wcel ccj cfv cre wceq recj ax-mp ) ACDAEFGFAGFHBAIJ $.
|
|
|
|
$( Imaginary part of a complex conjugate. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
imcji $p |- ( Im ` ( * ` A ) ) = -u ( Im ` A ) $=
|
|
( cc wcel ccj cfv cim cneg wceq imcj ax-mp ) ACDAEFGFAGFHIBAJK $.
|
|
|
|
$( A complex number times its conjugate is real. (Contributed by NM,
|
|
11-May-1999.) $)
|
|
cjmulrcli $p |- ( A x. ( * ` A ) ) e. RR $=
|
|
( cc wcel ccj cfv cmul co cr cjmulrcl ax-mp ) ACDAAEFGHIDBAJK $.
|
|
|
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$( A complex number times its conjugate. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
cjmulvali $p |- ( A x. ( * ` A ) ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) $=
|
|
( cc wcel ccj cfv cmul co cre c2 cexp cim caddc wceq cjmulval ax-mp ) ACD
|
|
AAEFGHAIFJKHALFJKHMHNBAOP $.
|
|
|
|
$( A complex number times its conjugate is nonnegative. (Contributed by
|
|
NM, 28-May-1999.) $)
|
|
cjmulge0i $p |- 0 <_ ( A x. ( * ` A ) ) $=
|
|
( cc wcel cc0 ccj cfv cmul co cle wbr cjmulge0 ax-mp ) ACDEAAFGHIJKBALM
|
|
$.
|
|
|
|
$( Real part of negative. (Contributed by NM, 2-Aug-1999.) $)
|
|
renegi $p |- ( Re ` -u A ) = -u ( Re ` A ) $=
|
|
( cc wcel cneg cre cfv wceq reneg ax-mp ) ACDAEFGAFGEHBAIJ $.
|
|
|
|
$( Imaginary part of negative. (Contributed by NM, 2-Aug-1999.) $)
|
|
imnegi $p |- ( Im ` -u A ) = -u ( Im ` A ) $=
|
|
( cc wcel cneg cim cfv wceq imneg ax-mp ) ACDAEFGAFGEHBAIJ $.
|
|
|
|
$( Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.) $)
|
|
cjnegi $p |- ( * ` -u A ) = -u ( * ` A ) $=
|
|
( cc wcel cneg ccj cfv wceq cjneg ax-mp ) ACDAEFGAFGEHBAIJ $.
|
|
|
|
$( A number plus its conjugate is twice its real part. Compare Proposition
|
|
10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) $)
|
|
addcji $p |- ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) $=
|
|
( cc wcel ccj cfv caddc co c2 cre cmul wceq addcj ax-mp ) ACDAAEFGHIAJFKH
|
|
LBAMN $.
|
|
|
|
readdi.2 $e |- B e. CC $.
|
|
$( Real part distributes over addition. (Contributed by NM,
|
|
28-Jul-1999.) $)
|
|
readdi $p |- ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) $=
|
|
( cc wcel caddc co cre cfv wceq readd mp2an ) AEFBEFABGHIJAIJBIJGHKCDABLM
|
|
$.
|
|
|
|
$( Imaginary part distributes over addition. (Contributed by NM,
|
|
28-Jul-1999.) $)
|
|
imaddi $p |- ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) $=
|
|
( cc wcel caddc co cim cfv wceq imadd mp2an ) AEFBEFABGHIJAIJBIJGHKCDABLM
|
|
$.
|
|
|
|
$( Real part of a product. (Contributed by NM, 28-Jul-1999.) $)
|
|
remuli $p |- ( Re ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) $=
|
|
( cc wcel cmul co cre cfv cim cmin wceq remul mp2an ) AEFBEFABGHIJAIJBIJG
|
|
HAKJBKJGHLHMCDABNO $.
|
|
|
|
$( Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) $)
|
|
immuli $p |- ( Im ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) $=
|
|
( cc wcel cmul co cim cfv cre caddc wceq immul mp2an ) AEFBEFABGHIJAKJBIJ
|
|
GHAIJBKJGHLHMCDABNO $.
|
|
|
|
$( Complex conjugate distributes over addition. Proposition 10-3.4(a) of
|
|
[Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) $)
|
|
cjaddi $p |- ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) $=
|
|
( cc wcel caddc co ccj cfv wceq cjadd mp2an ) AEFBEFABGHIJAIJBIJGHKCDABLM
|
|
$.
|
|
|
|
$( Complex conjugate distributes over multiplication. Proposition
|
|
10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) $)
|
|
cjmuli $p |- ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) $=
|
|
( cc wcel cmul co ccj cfv wceq cjmul mp2an ) AEFBEFABGHIJAIJBIJGHKCDABLM
|
|
$.
|
|
|
|
$( Standard inner product on complex numbers. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
ipcni $p |- ( Re ` ( A x. ( * ` B ) ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) $=
|
|
( cc wcel ccj cfv cmul co cre cim caddc wceq ipcnval mp2an ) AEFBEFABGHIJ
|
|
KHAKHBKHIJALHBLHIJMJNCDABOP $.
|
|
|
|
$( Complex conjugate distributes over division. (Contributed by Jim
|
|
Kingdon, 14-Jun-2020.) $)
|
|
cjdivapi $p |- ( B # 0 ->
|
|
( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co ccj cfv wceq cjdivap mp3an12 ) AEFBEFBGHIAB
|
|
JKLMALMBLMJKNCDABOP $.
|
|
$}
|
|
|
|
${
|
|
crre.1 $e |- A e. RR $.
|
|
crre.2 $e |- B e. RR $.
|
|
$( The real part of a complex number representation. Definition 10-3.1 of
|
|
[Gleason] p. 132. (Contributed by NM, 10-May-1999.) $)
|
|
crrei $p |- ( Re ` ( A + ( _i x. B ) ) ) = A $=
|
|
( cr wcel ci cmul co caddc cre cfv wceq crre mp2an ) AEFBEFAGBHIJIKLAMCDA
|
|
BNO $.
|
|
|
|
$( The imaginary part of a complex number representation. Definition
|
|
10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) $)
|
|
crimi $p |- ( Im ` ( A + ( _i x. B ) ) ) = B $=
|
|
( cr wcel ci cmul co caddc cim cfv wceq crim mp2an ) AEFBEFAGBHIJIKLBMCDA
|
|
BNO $.
|
|
$}
|
|
|
|
${
|
|
recld.1 $e |- ( ph -> A e. CC ) $.
|
|
$( The real part of a complex number is real (closure law). (Contributed
|
|
by Mario Carneiro, 29-May-2016.) $)
|
|
recld $p |- ( ph -> ( Re ` A ) e. RR ) $=
|
|
( cc wcel cre cfv cr recl syl ) ABDEBFGHECBIJ $.
|
|
|
|
$( The imaginary part of a complex number is real (closure law).
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
imcld $p |- ( ph -> ( Im ` A ) e. RR ) $=
|
|
( cc wcel cim cfv cr imcl syl ) ABDEBFGHECBIJ $.
|
|
|
|
$( Closure law for complex conjugate. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
cjcld $p |- ( ph -> ( * ` A ) e. CC ) $=
|
|
( cc wcel ccj cfv cjcl syl ) ABDEBFGDECBHI $.
|
|
|
|
$( Construct a complex number from its real and imaginary parts.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
replimd $p |- ( ph -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) $=
|
|
( cc wcel cre cfv ci cim cmul co caddc wceq replim syl ) ABDEBBFGHBIGJKLK
|
|
MCBNO $.
|
|
|
|
$( Value of the conjugate of a complex number. The value is the real part
|
|
minus ` _i ` times the imaginary part. Definition 10-3.2 of [Gleason]
|
|
p. 132. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
remimd $p |- ( ph -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) ) $=
|
|
( cc wcel ccj cfv cre ci cim cmul co cmin wceq remim syl ) ABDEBFGBHGIBJG
|
|
KLMLNCBOP $.
|
|
|
|
$( The conjugate of the conjugate is the original complex number.
|
|
Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
cjcjd $p |- ( ph -> ( * ` ( * ` A ) ) = A ) $=
|
|
( cc wcel ccj cfv wceq cjcj syl ) ABDEBFGFGBHCBIJ $.
|
|
|
|
${
|
|
reim0bd.2 $e |- ( ph -> ( Im ` A ) = 0 ) $.
|
|
$( A number is real iff its imaginary part is 0. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
reim0bd $p |- ( ph -> A e. RR ) $=
|
|
( cr wcel cim cfv cc0 wceq cc wb reim0b syl mpbird ) ABEFZBGHIJZDABKFPQ
|
|
LCBMNO $.
|
|
$}
|
|
|
|
${
|
|
rerebd.2 $e |- ( ph -> ( Re ` A ) = A ) $.
|
|
$( A real number equals its real part. Proposition 10-3.4(f) of
|
|
[Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
rerebd $p |- ( ph -> A e. RR ) $=
|
|
( cr wcel cre cfv wceq cc wb rereb syl mpbird ) ABEFZBGHBIZDABJFOPKCBLM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
cjrebd.2 $e |- ( ph -> ( * ` A ) = A ) $.
|
|
$( A number is real iff it equals its complex conjugate. Proposition
|
|
10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
cjrebd $p |- ( ph -> A e. RR ) $=
|
|
( cr wcel ccj cfv wceq cc wb cjreb syl mpbird ) ABEFZBGHBIZDABJFOPKCBLM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
cjne0d.2 $e |- ( ph -> A =/= 0 ) $.
|
|
$( A number which is nonzero has a complex conjugate which is nonzero.
|
|
Also see ~ cjap0d which is similar but for apartness. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
cjne0d $p |- ( ph -> ( * ` A ) =/= 0 ) $=
|
|
( cc0 wne ccj cfv cc wcel wb cjne0 syl mpbid ) ABEFZBGHEFZDABIJOPKCBLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
cjap0d.2 $e |- ( ph -> A # 0 ) $.
|
|
$( A number which is apart from zero has a complex conjugate which is
|
|
apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.) $)
|
|
cjap0d $p |- ( ph -> ( * ` A ) # 0 ) $=
|
|
( cc0 cap wbr ccj cfv cc wcel wb cjap0 syl mpbid ) ABEFGZBHIEFGZDABJKPQ
|
|
LCBMNO $.
|
|
$}
|
|
|
|
$( Real part of a complex conjugate. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
recjd $p |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) ) $=
|
|
( cc wcel ccj cfv cre wceq recj syl ) ABDEBFGHGBHGICBJK $.
|
|
|
|
$( Imaginary part of a complex conjugate. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
imcjd $p |- ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) ) $=
|
|
( cc wcel ccj cfv cim cneg wceq imcj syl ) ABDEBFGHGBHGIJCBKL $.
|
|
|
|
$( A complex number times its conjugate is real. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
cjmulrcld $p |- ( ph -> ( A x. ( * ` A ) ) e. RR ) $=
|
|
( cc wcel ccj cfv cmul co cr cjmulrcl syl ) ABDEBBFGHIJECBKL $.
|
|
|
|
$( A complex number times its conjugate. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
cjmulvald $p |- ( ph -> ( A x. ( * ` A ) ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel ccj cfv cmul co cre c2 cexp cim caddc wceq cjmulval syl ) ABDEB
|
|
BFGHIBJGKLIBMGKLINIOCBPQ $.
|
|
|
|
$( A complex number times its conjugate is nonnegative. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
cjmulge0d $p |- ( ph -> 0 <_ ( A x. ( * ` A ) ) ) $=
|
|
( cc wcel cc0 ccj cfv cmul co cle wbr cjmulge0 syl ) ABDEFBBGHIJKLCBMN $.
|
|
|
|
$( Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
renegd $p |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) ) $=
|
|
( cc wcel cneg cre cfv wceq reneg syl ) ABDEBFGHBGHFICBJK $.
|
|
|
|
$( Imaginary part of negative. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
imnegd $p |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) ) $=
|
|
( cc wcel cneg cim cfv wceq imneg syl ) ABDEBFGHBGHFICBJK $.
|
|
|
|
$( Complex conjugate of negative. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
cjnegd $p |- ( ph -> ( * ` -u A ) = -u ( * ` A ) ) $=
|
|
( cc wcel cneg ccj cfv wceq cjneg syl ) ABDEBFGHBGHFICBJK $.
|
|
|
|
$( A number plus its conjugate is twice its real part. Compare Proposition
|
|
10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
addcjd $p |- ( ph -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) ) $=
|
|
( cc wcel ccj cfv caddc co c2 cre cmul wceq addcj syl ) ABDEBBFGHIJBKGLIM
|
|
CBNO $.
|
|
|
|
${
|
|
cjexpd.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Complex conjugate of positive integer exponentiation. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
cjexpd $p |- ( ph -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) $=
|
|
( cc wcel cn0 cexp co ccj cfv wceq cjexp syl2anc ) ABFGCHGBCIJKLBKLCIJM
|
|
DEBCNO $.
|
|
$}
|
|
|
|
readdd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Real part distributes over addition. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
readdd $p |- ( ph -> ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) ) $=
|
|
( cc wcel caddc co cre cfv wceq readd syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEB
|
|
CMN $.
|
|
|
|
$( Imaginary part distributes over addition. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
imaddd $p |- ( ph -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) ) $=
|
|
( cc wcel caddc co cim cfv wceq imadd syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEB
|
|
CMN $.
|
|
|
|
$( Real part distributes over subtraction. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
resubd $p |- ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) ) $=
|
|
( cc wcel cmin co cre cfv wceq resub syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEBC
|
|
MN $.
|
|
|
|
$( Imaginary part distributes over subtraction. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
imsubd $p |- ( ph -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) ) $=
|
|
( cc wcel cmin co cim cfv wceq imsub syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEBC
|
|
MN $.
|
|
|
|
$( Real part of a product. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
remuld $p |- ( ph -> ( Re ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) ) $=
|
|
( cc wcel cmul co cre cfv cim cmin wceq remul syl2anc ) ABFGCFGBCHIJKBJKC
|
|
JKHIBLKCLKHIMINDEBCOP $.
|
|
|
|
$( Imaginary part of a product. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
immuld $p |- ( ph -> ( Im ` ( A x. B ) ) =
|
|
( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) ) $=
|
|
( cc wcel cmul co cim cfv cre caddc wceq immul syl2anc ) ABFGCFGBCHIJKBLK
|
|
CJKHIBJKCLKHIMINDEBCOP $.
|
|
|
|
$( Complex conjugate distributes over addition. Proposition 10-3.4(a) of
|
|
[Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
cjaddd $p |- ( ph -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) ) $=
|
|
( cc wcel caddc co ccj cfv wceq cjadd syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEB
|
|
CMN $.
|
|
|
|
$( Complex conjugate distributes over multiplication. Proposition
|
|
10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
cjmuld $p |- ( ph -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) ) $=
|
|
( cc wcel cmul co ccj cfv wceq cjmul syl2anc ) ABFGCFGBCHIJKBJKCJKHILDEBC
|
|
MN $.
|
|
|
|
$( Standard inner product on complex numbers. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
ipcnd $p |- ( ph -> ( Re ` ( A x. ( * ` B ) ) ) =
|
|
( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) ) $=
|
|
( cc wcel ccj cfv cmul co cre cim caddc wceq ipcnval syl2anc ) ABFGCFGBCH
|
|
IJKLIBLICLIJKBMICMIJKNKODEBCPQ $.
|
|
|
|
cjdivapd.2 $e |- ( ph -> B # 0 ) $.
|
|
$( Complex conjugate distributes over division. (Contributed by Jim
|
|
Kingdon, 15-Jun-2020.) $)
|
|
cjdivapd $p |- ( ph ->
|
|
( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co ccj cfv wceq cjdivap syl3anc ) ABGHCGHCIJKB
|
|
CLMNOBNOCNOLMPDEFBCQR $.
|
|
$}
|
|
|
|
${
|
|
crred.1 $e |- ( ph -> A e. RR ) $.
|
|
$( A real number equals its real part. One direction of Proposition
|
|
10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
rered $p |- ( ph -> ( Re ` A ) = A ) $=
|
|
( cr wcel cre cfv wceq rere syl ) ABDEBFGBHCBIJ $.
|
|
|
|
$( The imaginary part of a real number is 0. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
reim0d $p |- ( ph -> ( Im ` A ) = 0 ) $=
|
|
( cr wcel cim cfv cc0 wceq reim0 syl ) ABDEBFGHICBJK $.
|
|
|
|
$( A real number equals its complex conjugate. Proposition 10-3.4(f) of
|
|
[Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
cjred $p |- ( ph -> ( * ` A ) = A ) $=
|
|
( cr wcel ccj cfv wceq cjre syl ) ABDEBFGBHCBIJ $.
|
|
|
|
${
|
|
remul2d.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Real part of a product. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
remul2d $p |- ( ph -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) ) $=
|
|
( cr wcel cc cmul co cre cfv wceq remul2 syl2anc ) ABFGCHGBCIJKLBCKLIJM
|
|
DEBCNO $.
|
|
|
|
$( Imaginary part of a product. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
immul2d $p |- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) ) $=
|
|
( cr wcel cc cmul co cim cfv wceq immul2 syl2anc ) ABFGCHGBCIJKLBCKLIJM
|
|
DEBCNO $.
|
|
|
|
redivapd.2 $e |- ( ph -> A # 0 ) $.
|
|
$( Real part of a division. Related to ~ remul2 . (Contributed by Jim
|
|
Kingdon, 15-Jun-2020.) $)
|
|
redivapd $p |- ( ph -> ( Re ` ( B / A ) ) = ( ( Re ` B ) / A ) ) $=
|
|
( cc wcel cr cc0 cap wbr cdiv co cre cfv wceq redivap syl3anc ) ACGHBIH
|
|
BJKLCBMNOPCOPBMNQEDFCBRS $.
|
|
|
|
$( Imaginary part of a division. Related to ~ remul2 . (Contributed by
|
|
Jim Kingdon, 15-Jun-2020.) $)
|
|
imdivapd $p |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) ) $=
|
|
( cc wcel cr cc0 cap wbr cdiv co cim cfv wceq imdivap syl3anc ) ACGHBIH
|
|
BJKLCBMNOPCOPBMNQEDFCBRS $.
|
|
$}
|
|
|
|
crred.2 $e |- ( ph -> B e. RR ) $.
|
|
$( The real part of a complex number representation. Definition 10-3.1 of
|
|
[Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
crred $p |- ( ph -> ( Re ` ( A + ( _i x. B ) ) ) = A ) $=
|
|
( cr wcel ci cmul co caddc cre cfv wceq crre syl2anc ) ABFGCFGBHCIJKJLMBN
|
|
DEBCOP $.
|
|
|
|
$( The imaginary part of a complex number representation. Definition
|
|
10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
crimd $p |- ( ph -> ( Im ` ( A + ( _i x. B ) ) ) = B ) $=
|
|
( cr wcel ci cmul co caddc cim cfv wceq crim syl2anc ) ABFGCFGBHCIJKJLMCN
|
|
DEBCOP $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Sequence convergence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A r $.
|
|
$( Two ways to express a reciprocal. (Contributed by Jim Kingdon,
|
|
20-Jul-2021.) $)
|
|
caucvgrelemrec $p |- ( ( A e. RR /\ A # 0 ) ->
|
|
( iota_ r e. RR ( A x. r ) = 1 ) = ( 1 / A ) ) $=
|
|
( cr wcel cc0 cap wbr wa cv cmul co c1 wceq cdiv rerecclap eqcom cc simpr
|
|
wb recnd simpll simplr ax-1cn divmulap mp3an1 syl12anc syl5rbbr riota5 )
|
|
ACDZAEFGZHZABIZJKLMZBCLANKZAOULUNMUNULMZUKULCDZHZUMUNULPUQULQDZAQDZUJUOUM
|
|
SZUQULUKUPRTUQAUIUJUPUATUIUJUPUBLQDURUSUJHUTUCLULAUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d F i j m x y $. $d F i k m x y $. $d F k m n $. $d k m n ph $.
|
|
$d k n r $. $d ph x y $.
|
|
caucvgre.f $e |- ( ph -> F : NN --> RR ) $.
|
|
caucvgre.cau $e |- ( ph ->
|
|
A. n e. NN A. k e. ( ZZ>= ` n )
|
|
( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\
|
|
( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) ) $.
|
|
${
|
|
$d k n ph $. $d n r $.
|
|
|
|
$( Lemma for ~ caucvgre . Converting the Cauchy condition. (Contributed
|
|
by Jim Kingdon, 20-Jul-2021.) $)
|
|
caucvgrelemcau $p |- ( ph -> A. n e. NN A. k e. NN ( n <RR k ->
|
|
( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\
|
|
( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) )
|
|
) ) ) $=
|
|
( cv cltrr wbr cfv co cr caddc wa cn wcel clt syl2anc wb cmul wceq crio
|
|
c1 wi wral cdiv cle simplr nnred simpr ltle cuz eluznn ex nnz eluz1 syl
|
|
syl6bi jcad anim1i syl5ibr impbid adantl biimpar r19.21bi syldan ltxrlt
|
|
cz expr syld ad2antrr ffvelrnd nnrecred readdcld cc0 cap caucvgrelemrec
|
|
wf nnap0 oveq2d breq2d bitr4d anbi12d 3imtr3d ralrimiva ) ACHZBHZIJZWGD
|
|
KZWHDKZWGEHUALUDUBEMUCZNLZIJZWKWJWLNLZIJZOZUEZBPUFCPAWGPQZOZWRBPWTWHPQZ
|
|
OZWGWHRJZWJWKUDWGUGLZNLZRJZWKWJXDNLZRJZOZWIWQXBXCWGWHUHJZXIXBWGMQZWHMQZ
|
|
XCXJUEXBWGAWSXAUIZUJZXBWHWTXAUKZUJZWGWHULSWTXAXJXIWTXAXJOZWHWGUMKZQZXIW
|
|
TXSXQWSXSXQTAWSXSXQWSXSXAXJWSXSXAWHWGUNUOWSXSWHVIQZXJOZXJWSWGVIQXSYATWG
|
|
UPWGWHUQURZXTXJUKUSUTXQXSWSYAXAXTXJWHUPVAYBVBVCVDVEWTXIBXRAXIBXRUFCPGVF
|
|
VFVGVJVKXBXKXLXCWITXNXPWGWHVHSXBXFWNXHWPXBXFWJXEIJZWNXBWJMQXEMQXFYCTXBP
|
|
MWGDAPMDVSWSXAFVLZXMVMZXBWKXDXBPMWHDYDXOVMZXBWGXMVNZVOWJXEVHSXBWMXEWJIX
|
|
BWLXDWKNXBXKWGVPVQJZWLXDUBXNXBWSYHXMWGVTURWGEVRSZWAWBWCXBXHWKXGIJZWPXBW
|
|
KMQXGMQXHYJTYFXBWJXDYEYGVOWKXGVHSXBWOXGWKIXBWLXDWJNYIWAWBWCWDWEWFWF $.
|
|
$}
|
|
|
|
$( Convergence of real sequences.
|
|
|
|
A Cauchy sequence (as defined here, which has a rate of convergence
|
|
built in) of real numbers converges to a real number. Specifically on
|
|
rate of convergence, all terms after the nth term must be within
|
|
` 1 / n ` of the nth term.
|
|
|
|
(Contributed by Jim Kingdon, 19-Jul-2021.) $)
|
|
caucvgre $p |- ( ph ->
|
|
E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j )
|
|
( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) ) $=
|
|
( vm cv wbr wa cn wral cr clt wcel wb vr cltrr cfv caddc wrex crp cuz cc0
|
|
co wi dfnn2 caucvgrelemcau ax-caucvg ralrp 0re ltxrlt mpan imbi1d ralbiia
|
|
bitri rexbii sylibr c1 wss simpr peano2nnd uznnssnn ssralv 3syl cz eluznn
|
|
cle sylan simplr nnzd eluz1 syl biimpd impancom mpd simprd ad2antlr nnred
|
|
1re ltadd1 mp3an3 syl2anc nnleltp1 bitr4d syldan mpbird syl2an adantll wf
|
|
ad4antr ffvelrnd simpllr adantr ad3antlr readdcld anbi12d imbi12d biimprd
|
|
nnre rpre mpid ralimdva syld breq1d oveq1d breq2d cbvralv syl6ib reximdva
|
|
wceq fveq2 raleqdv rspcev ex rexlimdva ) AKLZFLZUBMZYBHUCZCLZBLZUDUIZUBMZ
|
|
YEYDYFUDUIZUBMZNZUJZFOPZKOUEZBUFPZCQUEZDLZHUCZYGRMZYEYRYFUDUIZRMZNZDELZUG
|
|
UCZPZEOUEZBUFPZCQUEAUHYFUBMZYNUJZBQPZCQUEYPABCKFGHOUABCUKIAFGHUAIJULUMYOU
|
|
UJCQYOUHYFRMZYNUJZBQPUUJYNBUNUULUUIBQYFQSZUUKUUHYNUHQSUUMUUKUUHTUOUHYFUPU
|
|
QURUSUTVAVBAYOUUGCQAYEQSZNZYNUUFBUFUUOYFUFSZNZYNUUBDYAVCUDUIZUGUCZPZKOUEU
|
|
UFUUQYMUUTKOUUQYAOSZNZYMYDYGRMZYEYIRMZNZFUUSPZUUTUVBYMYLFUUSPZUVFUVBUUROS
|
|
ZUUSOVDYMUVGUJUVBYAUUQUVAVEVFZUURVGYLFUUSOVHVIUVBYLUVEFUUSUVBYBUUSSZNZYLY
|
|
AYBRMZUVEUVKUVLUURYBVLMZUVKYBVJSZUVMUVKYBOSZUVNUVMNZUVBUVHUVJUVOUVIYBUURV
|
|
KVMZUVBUVOUVJUVPUVBUVONZUVJUVPUVRUURVJSUVJUVPTUVRUURUVRYAUUQUVAUVOVNVFZVO
|
|
UURYBVPVQVRVSVTWAUVBUVJUVOUVLUVMTUVQUVRUVLUURYBVCUDUIRMZUVMUVRYAQSZYBQSZU
|
|
VLUVTTZUVAUWAUUQUVOYAXDZWBUVRYBUVBUVOVEZWCUWAUWBVCQSUWCWDYAYBVCWEWFWGUVRU
|
|
VHUVOUVMUVTTUVSUWEUURYBWHWGWIWJWKUVBUVJUVOYLUVLUVEUJZUJUVQUVRUWFYLUVRUVLY
|
|
CUVEYKUVAUVOUVLYCTZUUQUVAUWAUWBUWGUVOUWDYBXDYAYBUPWLWMUVRUVCYHUVDYJUVRYDQ
|
|
SYGQSUVCYHTUVROQYBHAOQHWNUUNUUPUVAUVOIWOUWEWPZUVRYEYFUVBUUNUVOAUUNUUPUVAW
|
|
QWRZUUPUUMUUOUVAUVOYFXEWSZWTYDYGUPWGUVRUUNYIQSUVDYJTUWIUVRYDYFUWHUWJWTYEY
|
|
IUPWGXAXBXCWJXFXGXHUVEUUBFDUUSYBYQXOZUVCYSUVDUUAUWKYDYRYGRYBYQHXPZXIUWKYI
|
|
YTYERUWKYDYRYFUDUWLXJXKXAXLXMXNUUQUUTUUFKOUVBUUTUUFUVBUVHUUTUUFUVIUUEUUTE
|
|
UUROUUCUURXOUUBDUUDUUSUUCUURUGXPXQXRVMXSXTXHXGXNVT $.
|
|
$}
|
|
|
|
${
|
|
cvg1nlemcxze.c $e |- ( ph -> C e. RR+ ) $.
|
|
cvg1nlemcxze.x $e |- ( ph -> X e. RR+ ) $.
|
|
cvg1nlemcxze.z $e |- ( ph -> Z e. NN ) $.
|
|
cvg1nlemcxze.e $e |- ( ph -> E e. NN ) $.
|
|
cvg1nlemcxze.a $e |- ( ph -> A e. NN ) $.
|
|
cvg1nlemcxze.1 $e |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) < E ) $.
|
|
$( Lemma for ~ cvg1n . Rearranging an expression related to the rate of
|
|
convergence. (Contributed by Jim Kingdon, 6-Aug-2021.) $)
|
|
cvg1nlemcxze $p |- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) ) $=
|
|
( c2 cmul co clt wbr cdiv nnrpd rpred rpcnd 2cnd rpap0d div23apd crp wcel
|
|
caddc rpmulcld rpdivcld nnred readdcld ltaddrpd lttrd ltdivmul2d eqbrtrrd
|
|
2rp mpbid rerpdivcld nnmulcld ltmuldivd lt2mul2divd nncnd mulcomd breqtrd
|
|
a1i mpbird ) ACMNOZEDFNOZNOZPQCVHROEMROPQAVGVHENOZVIPAVGVJPQCEROZVHMROPQZ
|
|
AVKMNOZVHPQVLAVGEROZVMVHPACMEACGUAAUBAEHUAZAEHUCUDAVNFROZDPQVNVHPQAVPVPBU
|
|
GODAVPAVNFAVGEACMGMUEUFAUPVEZUHHUIZAFISZUITZAVPBVTABKUJUKADJUJZAVPBVTABKS
|
|
ULLUMAVNDFAVNVRTWAVSUNUQUOAVKVHMACEACGTZHURAVHADFJIUSZUJZVQUTUQACMVHEWBVQ
|
|
WDHVAVFAVHEAVHWCVBVOVCVDACMEVHWBVQAEHTAVHWCSVAUQ $.
|
|
$}
|
|
|
|
${
|
|
cvg1n.f $e |- ( ph -> F : NN --> RR ) $.
|
|
cvg1n.c $e |- ( ph -> C e. RR+ ) $.
|
|
cvg1n.cau $e |- ( ph ->
|
|
A. n e. NN A. k e. ( ZZ>= ` n )
|
|
( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\
|
|
( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) ) $.
|
|
|
|
${
|
|
cvg1nlem.g $e |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) ) $.
|
|
cvg1nlem.z $e |- ( ph -> Z e. NN ) $.
|
|
cvg1nlem.start $e |- ( ph -> C < Z ) $.
|
|
|
|
${
|
|
$d j ph $.
|
|
$( Lemma for ~ cvg1n . The modified sequence ` G ` is a sequence.
|
|
(Contributed by Jim Kingdon, 1-Aug-2021.) $)
|
|
cvg1nlemf $p |- ( ph -> G : NN --> RR ) $=
|
|
( cn cv cmul cr wcel adantr co cfv wa simpr nnmulcld ffvelrnd fmptd
|
|
wf ) ACOCPZHQUAZFUBRGAUIOSZUCZORUJFAORFUHUKITULUIHAUKUDAHOSUKMTUEUFLU
|
|
G $.
|
|
$}
|
|
|
|
${
|
|
$d C a b n $. $d C b k n $. $d F a b n $. $d F j k n $. $d Z a b $.
|
|
$d Z j $. $d k n ph $.
|
|
$( Lemma for ~ cvg1n . By selecting spaced out terms for the modified
|
|
sequence ` G ` , the terms are within ` 1 / n ` (without the
|
|
constant ` C ` ). (Contributed by Jim Kingdon, 1-Aug-2021.) $)
|
|
cvg1nlemcau $p |- ( ph ->
|
|
A. n e. NN A. k e. ( ZZ>= ` n )
|
|
( ( G ` n ) < ( ( G ` k ) + ( 1 / n ) ) /\
|
|
( G ` k ) < ( ( G ` n ) + ( 1 / n ) ) ) ) $=
|
|
( cfv co caddc clt wbr cn vb va cv c1 cdiv wa cuz wral wcel cmul wceq
|
|
cr simplr wf ad2antrr nnmulcld ffvelrnd fveq2d fvmptg syl2anc eqeltrd
|
|
oveq1 eluznn rpred nndivred readdcld nnrecred cle eluzle adantl nnred
|
|
adantll nnrpd lemul1d mpbid cz nnzd mpbird fveq2 oveq1d breq2d breq1d
|
|
wb eluz anbi12d cbvralv ralbii oveq2 oveq2d breq12d oveq12d raleqbidv
|
|
bitri sylib rspcv sylc simpld ltmul1dd nncnd mulid2d 1red lt2mul2divd
|
|
mulcomd 3bitr3d ltadd2dd lttrd simprd jca ralrimiva ) AEUCZGOZDUCZGOZ
|
|
UDXJUEPZQPZRSZXMXKXNQPZRSZUFZDXJUGOZUHETAXJTUIZUFZXSDXTYBXLXTUIZUFZXP
|
|
XRYDXKXMBXJHUJPZUEPZQPZXOYDXKYEFOZULYDYAYHULUIXKYHUKAYAYCUMZYDTULYEFA
|
|
TULFUNYAYCIUOZYDXJHYIAHTUIYAYCMUOZUPZUQZCXJCUCZHUJPZFOZYHTULGYNXJUKYO
|
|
YEFYNXJHUJVBURLUSUTZYMVAZYDXMYFYDXMXLHUJPZFOZULYDXLTUIZYTULUIXMYTUKYA
|
|
YCUUAAXLXJVCVLZYDTULYSFYJYDXLHUUBYKUPZUQZCXLYPYTTULGYNXLUKYOYSFYNXLHU
|
|
JVBURLUSUTZUUDVAZYDBYEABULUIYAYCABJVDUOZYLVEZVFYDXMXNUUFYDXJYIVGZVFYD
|
|
XKYGRSZXMXKYFQPZRSZYDUUJUULUFYHYGRSZXMYHYFQPZRSZUFZYDUUPYHYTYFQPZRSZY
|
|
TUUNRSZUFZYDYSYEUGOZUIZYHUAUCZFOZYFQPZRSZUVDUUNRSZUFZUAUVAUHZUUTYDUVB
|
|
YEYSVHSZYDXJXLVHSZUVJYCUVKYBXJXLVIVJYDXJXLHYDXJYIVKYDXLUUBVKYDHYKVMVN
|
|
VOYDYEVPUIYSVPUIUVBUVJWCYDYEYLVQYDYSUUCVQYEYSWDUTVRYDYETUIUBUCZFOZUVD
|
|
BUVLUEPZQPZRSZUVDUVMUVNQPZRSZUFZUAUVLUGOZUHZUBTUHZUVIYLAUWBYAYCAXJFOZ
|
|
XLFOZBXJUEPZQPZRSZUWDUWCUWEQPZRSZUFZDXTUHZETUHZUWBKUWLUWCUVDUWEQPZRSZ
|
|
UVDUWHRSZUFZUAXTUHZETUHUWBUWKUWQETUWJUWPDUAXTXLUVCUKZUWGUWNUWIUWOUWRU
|
|
WFUWMUWCRUWRUWDUVDUWEQXLUVCFVSZVTWAUWRUWDUVDUWHRUWSWBWEWFWGUWQUWAEUBT
|
|
XJUVLUKZUWPUVSUAXTUVTXJUVLUGVSUWTUWNUVPUWOUVRUWTUWCUVMUWMUVORXJUVLFVS
|
|
ZUWTUWEUVNUVDQXJUVLBUEWHZWIWJUWTUWHUVQUVDRUWTUWCUVMUWEUVNQUXAUXBWKWAW
|
|
EWLWFWMWNUOUWAUVIUBYETUVLYEUKZUVSUVHUAUVTUVAUVLYEUGVSUXCUVPUVFUVRUVGU
|
|
XCUVMYHUVOUVERUVLYEFVSZUXCUVNYFUVDQUVLYEBUEWHZWIWJUXCUVQUUNUVDRUXCUVM
|
|
YHUVNYFQUXDUXEWKWAWEWLWOWPUVHUUTUAYSUVAUVCYSUKZUVFUURUVGUUSUXFUVEUUQY
|
|
HRUXFUVDYTYFQUVCYSFVSZVTWAUXFUVDYTUUNRUXGWBWEWOWPYDUUMUURUUOUUSYDYGUU
|
|
QYHRYDXMYTYFQUUEVTWAYDXMYTUUNRUUEWBWEVRYDUUJUUMUULUUOYDXKYHYGRYQWBYDU
|
|
UKUUNXMRYDXKYHYFQYQVTWAWEVRZWQYDYFXNXMUUHUUIUUFYDYFXNRSZBXJUJPZHXJUJP
|
|
ZRSZYDBHXJUUGYDHYKVKYDXJYIVMZABHRSYAYCNUOWRYDUXJUDYEUJPZRSUXJYERSUXIU
|
|
XLYDUXNYEUXJRYDYEYDYEYLWSWTWAYDBXJUDYEUUGUXMYDXAYDYEYLVMXBYDYEUXKUXJR
|
|
YDXJHYDXJYIWSYDHYKWSXCWAXDVRZXEXFYDXMUUKXQUUFYDXKYFYRUUHVFYDXKXNYRUUI
|
|
VFYDUUJUULUXHXGYDYFXNXKUUHUUIYRUXOXEXFXHXIXI $.
|
|
$}
|
|
|
|
${
|
|
$d C e i k $. $d C e k n $. $d F a e j $. $d F e j k n $.
|
|
$d G a b c e i w y $. $d G b c e i k y $. $d G e k n $.
|
|
$d G a b c e i x y $. $d Z e i j k $. $d Z e j k n $.
|
|
$d a c e i ph x y $. $d j k n ph $. $d j ph x y $.
|
|
$( Lemma for ~ cvg1n . The original sequence ` F ` has a limit (turns
|
|
out it is the same as the limit of the modified sequence ` G ` ).
|
|
(Contributed by Jim Kingdon, 1-Aug-2021.) $)
|
|
cvg1nlemres $p |- ( ph ->
|
|
E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j )
|
|
( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) ) $=
|
|
( caddc co clt vb vc va vw ve cv cfv wbr wa cuz wral cn crp cvg1nlemf
|
|
wrex cr cvg1nlemcau caucvgre wcel fveq2 raleqdv cbvrexv ralbii anbi2i
|
|
weq anbi1i cdiv simpr rphalfcld simplr wceq breq2d anbi12d rexralbidv
|
|
c2 oveq2 rspcv sylc sylbir cmul rpred ad4antr 2re remulcld rerpdivcld
|
|
a1i nndivred simprl readdcld arch syl adantr nnmulcld ad6antr simplrl
|
|
nnred wf eluznn sylancom ffvelrnd ad2antrr rehalfcld ad3antrrr oveq2d
|
|
simpllr breq12d oveq12d raleqbidv oveq1d breq1d cvg1nlemcxze leadd2dd
|
|
simplrr ltled ltletrd cle 2rp rpmulcld rpdivcld nnrpd ltaddrp2d lttrd
|
|
simprd cz wb nnzd syl2anc mpbird simprr fveq2d fvmptg simpld ltadd1dd
|
|
eluz oveq1 recnd addassd breqtrd ralrimiva rexlimddv mpbid jca rspcev
|
|
rpcnd 2halvesd sylbi ex reximdva mpd ) AUAUFZJUGZCUFZUBUFZRSZTUHZUULU
|
|
UKUUMRSZTUHZUIZUAUCUFZUJUGZUKZUCULUOZUBUMUKZCUPUOEUFZIUGZUULBUFZRSZTU
|
|
HZUULUVEUVFRSZTUHZUIZEFUFZUJUGZUKZFULUOZBUMUKZCUPUOAUBCUAUCGHJADFGHIJ
|
|
KLMNOPQUNADFGHIJKLMNOPQUQURAUVCUVPCUPAUULUPUSZUIZUVCUVPUVRUVCUIZUVOBU
|
|
MUVSUVFUMUSZUIZUVRUURUAUDUFZUJUGZUKZUDULUOZUBUMUKZUIZUVTUIZUVOUVSUWGU
|
|
VTUVCUWFUVRUVBUWEUBUMUVAUWDUCUDULUCUDVEUURUAUUTUWCUUSUWBUJUTVAVBVCVDV
|
|
FZUWHUUKUULUVFVOVGSZRSZTUHZUULUUKUWJRSZTUHZUIZUAUUTUKZUVOUCULUWHUWAUW
|
|
PUCULUOZUWIUWAUWJUMUSUVCUWQUWAUVFUVSUVTVHVIUVRUVCUVTVJUVBUWQUBUWJUMUU
|
|
MUWJVKZUURUWOUCUAULUUTUWRUUOUWLUUQUWNUWRUUNUWKUUKTUUMUWJUULRVPVLUWRUU
|
|
PUWMUULTUUMUWJUUKRVPVLVMVNVQVRVSUWHUUSULUSZUWPUIZUIZDVOVTSZUVFVGSZKVG
|
|
SZUUSRSZUEUFZTUHZUVOUEULUXAUXEUPUSUXGUEULUOUXAUXDUUSUXAUXCKUXAUXBUVFU
|
|
XADVOADUPUSZUVQUWFUVTUWTADMWAZWBVOUPUSUXAWCWFWDUWGUVTUWTVJZWEAKULUSZU
|
|
VQUWFUVTUWTPWBZWGUXAUUSUWHUWSUWPWHZWPWIUXEUEWJWKUXAUXFULUSZUXGUIZUIZU
|
|
XFKVTSZULUSZUVKEUXQUJUGZUKZUVOUXPUXFKUXAUXNUXGWHUXAUXKUXOUXLWLWMUXPUV
|
|
KEUXSUXPUVDUXSUSZUIZUVHUVJUYBUVEUULUWJUWJRSZRSZUVGTUYBUVEUWKUWJRSZUYD
|
|
TUYBUVEUXQIUGZUWJRSZUYEUYBULUPUVDIAULUPIWQUVQUWFUVTUWTUXOUYALWNZUXPUY
|
|
AUXRUVDULUSUYBUXFKUXAUXNUXGUYAWOZAUXKUVQUWFUVTUWTUXOUYAPWNZWMZUVDUXQW
|
|
RWSWTZUYBUYFUWJUYBULUPUXQIUYHUYKWTZUYBUVFUYBUVFUXAUVTUXOUYAUXJXAZWAXB
|
|
ZWIZUYBUWKUWJUYBUULUWJUWHUVQUWTUXOUYAAUVQUWFUVTXEXCZUYOWIZUYOWIUYBUVE
|
|
UYFDUXQVGSZRSZUYGUYLUYBUYFUYSUYMUYBDUXQAUXHUVQUWFUVTUWTUXOUYAUXIWNUYK
|
|
WGZWIUYPUYBUYFUVEUYSRSZTUHZUVEUYTTUHZUYBUYAUYFGUFZIUGZUYSRSZTUHZVUFUY
|
|
TTUHZUIZGUXSUKZVUCVUDUIZUXPUYAVHUYBUXRHUFZIUGZVUFDVUMVGSZRSZTUHZVUFVU
|
|
NVUORSZTUHZUIZGVUMUJUGZUKZHULUKZVUKUYKAVVCUVQUWFUVTUWTUXOUYANWNVVBVUK
|
|
HUXQULVUMUXQVKZVUTVUJGVVAUXSVUMUXQUJUTVVDVUQVUHVUSVUIVVDVUNUYFVUPVUGT
|
|
VUMUXQIUTZVVDVUOUYSVUFRVUMUXQDVGVPZXDXFVVDVURUYTVUFTVVDVUNUYFVUOUYSRV
|
|
VEVVFXGVLVMXHVQVRVUJVULGUVDUXSGEVEZVUHVUCVUIVUDVVGVUGVUBUYFTVVGVUFUVE
|
|
UYSRVUEUVDIUTZXIVLVVGVUFUVEUYTTVVHXJVMVQVRZYCUYBUYSUWJUYFVUAUYOUYMUYB
|
|
UYSUWJVUAUYBUVFUYBUVFUWHUVTUWTUXOUYAUWGUVTVHXCZWAXBUYBUUSDUXFUVFKADUM
|
|
USUVQUWFUVTUWTUXOUYAMWNZVVJUYJUYIUXAUWSUXOUYAUXMXAZUXAUXNUXGUYAXMZXKX
|
|
NZXLXOUYBUYFUWKUWJUYMUYRUYOUYBUYFUWKTUHZUULUYGTUHZUYBUXFJUGZUWKTUHZUU
|
|
LVVQUWJRSZTUHZUIZVVOVVPUIUYBUXFUUTUSZUWPVWAUYBVWBUUSUXFXPUHZUYBUUSUXF
|
|
UYBUUSVVLWPZUYBUXFUYIWPZUYBUUSUXEUXFVWDUYBUXDUUSUYBUXDUYBUXCKUYBUXBUV
|
|
FUYBDVOVVKVOUMUSUYBXQWFXRVVJXSUYBKUYJXTXSZWAVWDWIVWEUYBUUSUXDVWDVWFYA
|
|
VVMYBXNUYBUUSYDUSUXFYDUSVWBVWCYEUYBUUSVVLYFUYBUXFUYIYFUUSUXFYNYGYHUXA
|
|
UWPUXOUYAUWHUWSUWPYIXAUWOVWAUAUXFUUTUAUEVEZUWLVVRUWNVVTVWGUUKVVQUWKTU
|
|
UJUXFJUTZXJVWGUWMVVSUULTVWGUUKVVQUWJRVWHXIVLVMVQVRUYBVVRVVOVVTVVPUYBV
|
|
VQUYFUWKTUYBUXNUYFUPUSVVQUYFVKUYIUYMFUXFUVLKVTSZIUGUYFULUPJFUEVEVWIUX
|
|
QIUVLUXFKVTYOYJOYKYGZXJUYBVVSUYGUULTUYBVVQUYFUWJRVWJXIVLVMUUAZYLYMYBU
|
|
YBUULUWJUWJUYBUULUYQYPUYBUWJUYOYPZVWLYQYRUYBUYCUVFUULRUYBUVFUYBUVFUYN
|
|
UUDUUEZXDYRUYBUULUVEUYCRSZUVITUYBUULUVEUWJRSZUWJRSZVWNTUYBUULUYGVWPUY
|
|
QUYPUYBVWOUWJUYBUVEUWJUYLUYOWIZUYOWIUYBVVOVVPVWKYCUYBUYFVWOUWJUYMVWQU
|
|
YOUYBUYFVUBVWOUYMUYBUVEUYSUYLVUAWIVWQUYBVUCVUDVVIYLUYBUYSUWJUVEVUAUYO
|
|
UYLVVNXLXOYMYBUYBUVEUWJUWJUYBUVEUYLYPVWLVWLYQYRUYBUYCUVFUVERVWMXDYRUU
|
|
BYSUVNUXTFUXQULUVLUXQVKUVKEUVMUXSUVLUXQUJUTVAUUCYGYTYTUUFYSUUGUUHUUI
|
|
$.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d C k n z $. $d C i j $. $d C x y $. $d F x y z $. $d F k n $.
|
|
$d F i j $. $d ph k n z $. $d ph j $. $d ph i x y $. $d i z $.
|
|
$d j z $. $d j x y $. $d j x $. $d k n z $. $d j n $. $d k y $.
|
|
$d j k $. $d i k $.
|
|
$( Convergence of real sequences.
|
|
|
|
This is a version of ~ caucvgre with a constant multiplier ` C ` on
|
|
the rate of convergence. That is, all terms after the nth term must
|
|
be within ` C / n ` of the nth term.
|
|
|
|
(Contributed by Jim Kingdon, 1-Aug-2021.) $)
|
|
cvg1n $p |- ( ph ->
|
|
E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j )
|
|
( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) ) $=
|
|
( vz cv clt wbr cfv caddc co cn wa cuz wral wrex crp cr wcel rpred arch
|
|
syl cmul cmpt wf adantr cdiv eqid simprl simprr cvg1nlemres rexlimddv )
|
|
ADMNZOPZENIQZCNZBNZRSOPVDVCVERSOPUAEFNZUBQUCFTUDBUEUCCUFUDMTADUFUGVBMTU
|
|
DADKUHDMUIUJAVATUGZVBUAZUABCDEFGHIFTVFVAUKSIQULZVAATUFIUMVHJUNADUEUGVHK
|
|
UNAHNZIQZGNIQZDVJUOSZRSOPVLVKVMRSOPUAGVJUBQUCHTUCVHLUNVIUPAVGVBUQAVGVBU
|
|
RUSUT $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The upper integers are closed under intersection. (Contributed by Mario
|
|
Carneiro, 24-Dec-2013.) $)
|
|
uzin2 $p |- ( ( A e. ran ZZ>= /\ B e. ran ZZ>= ) ->
|
|
( A i^i B ) e. ran ZZ>= ) $=
|
|
( vx vy cv cuz cfv cin crn wcel cz wfn wceq wrex wb cpw wf fvelrnb eleq1d
|
|
ax-mp uzf ffn ineq1 ineq2 cle wbr uzin simpr simpl zdcle ifcldcd fnfvelrn
|
|
wa cif sylancr eqeltrd 2gencl ) CEZFGZDEZFGZHZFIZJAVAHZVCJABHZVCJCDUSVAAB
|
|
KVCFKLZAVCJUSAMZCKNOKKPZFQVFUAKVHFUBTZCKAFRTVFBVCJVABMZDKNOVIDKBFRTVGVBVD
|
|
VCUSAVAUCSVJVDVEVCVABAUDSURKJZUTKJZUMZVBURUTUEUFZUTURUNZFGZVCURUTUGVMVFVO
|
|
KJVPVCJVIVMVNUTURKVKVLUHVKVLUIURUTUJUKKVOFULUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d j k x y z $. $d j x y z ph $. $d j x y z ps $.
|
|
$( Combine two different upper integer properties into one. (Contributed
|
|
by Mario Carneiro, 25-Dec-2013.) $)
|
|
rexanuz $p |- ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <->
|
|
( E. j e. ZZ A. k e. ( ZZ>= ` j ) ph /\
|
|
E. j e. ZZ A. k e. ( ZZ>= ` j ) ps ) ) $=
|
|
( vx vy vz wa cv cuz wral cz wrex wb uzf raleq rexrn mp2b wi wcel cfv crn
|
|
r19.26 rexbii r19.40 sylbi cpw wf wfn ffn cin uzin2 wss inss1 ax-mp inss2
|
|
ssralv anim12i sylibr rspcev an4s rexlimdvaa rexlimiva imp sylib syl2anbr
|
|
syl2an impbii ) ABHZDCIJUAZKZCLMZADVJKZCLMZBDVJKZCLMZHZVLVMVOHZCLMVQVKVRC
|
|
LABDVJUCUDVMVOCLUEUFVNADEIZKZEJUBZMZBDFIZKZFWAMZVLVPLLUGZJUHZJLUIZWBVNNOL
|
|
WFJUJZVTVMECLJADVSVJPQRWGWHWEVPNOWIWDVOFCLJBDWCVJPQRWBWEHVIDGIZKZGWAMZVLW
|
|
BWEWLVTWEWLSEWAVSWATZVTHWDWLFWAWMWCWATZVTWDWLWMWNHVSWCUKZWATVIDWOKZWLVTWD
|
|
HZVSWCULWQADWOKZBDWOKZHWPVTWRWDWSWOVSUMVTWRSVSWCUNADWOVSUQUOWOWCUMWDWSSVS
|
|
WCUPBDWOWCUQUOURABDWOUCUSWKWPGWOWAVIDWJWOPUTVGVAVBVCVDWGWHWLVLNOWIWKVKGCL
|
|
JVIDWJVJPQRVEVFVH $.
|
|
$}
|
|
|
|
${
|
|
$d j k n x y z A $. $d j x y z ph $.
|
|
$( Combine finitely many different upper integer properties into one.
|
|
(Contributed by Mario Carneiro, 6-Jun-2014.) $)
|
|
rexfiuz $p |- ( A e. Fin ->
|
|
( E. j e. ZZ A. k e. ( ZZ>= ` j ) A. n e. A ph <->
|
|
A. n e. A E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) ) $=
|
|
( vx vz cv wral cz wrex wb c0 wceq raleq rexralbidv bibi12d wcel wa cvv
|
|
vy cuz cfv csn cun wex cc0 0z elex2 ax-mp ral0 rgen2w r19.2m mp2an 2th wi
|
|
cfn anbi1 rexanuz ralunb ralbii rexbii wsbc ralsnsg ralcom syl5bb rexbidv
|
|
vex sbcrex syl6rbbr bitrd anbi2i 3bitr4i 3bitr4g a1i findcard2 ) AEFHZIZD
|
|
CHZUBUCZICJKZADVTIZCJKZEVQIZLAEMIZDVTIZCJKZWCEMIZLAEUAHZIZDVTICJKZWCEWIIZ
|
|
LZAEWIGHZUDZUEZIZDVTIZCJKZWCEWPIZLZAEBIZDVTICJKZWCEBIZLFUAGBVQMNZWAWGWDWH
|
|
XEVRWECDJVTAEVQMOPWCEVQMOQVQWINZWAWKWDWLXFVRWJCDJVTAEVQWIOPWCEVQWIOQVQWPN
|
|
ZWAWSWDWTXGVRWQCDJVTAEVQWPOPWCEVQWPOQVQBNZWAXCWDXDXHVRXBCDJVTAEVQBOPWCEVQ
|
|
BOQWGWHVSJRCUFZWFCJIWGUGJRXIUHCUGJUIUJWECDJVTAEUKULWFCJUMUNWCEUKUOWMXAUPW
|
|
IUQRWMWKWCEWOIZSZWLXJSWSWTWKWLXJURWJAEWOIZSZDVTIZCJKWKXLDVTIZCJKZSWSXKWJX
|
|
LCDUSWRXNCJWQXMDVTAEWIWOUTVAVBXJXPWKWNTRZXJXPLGVHXQXJWCEWNVCZXPWCEWNTVDXQ
|
|
XPWBEWNVCZCJKXRXQXOXSCJXOWBEWOIXQXSADEVTWOVEWBEWNTVDVFVGWBECWNJVIVJVKUJVL
|
|
VMWCEWIWOUTVNVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d j M $. $d j ph $. $d j k Z $.
|
|
rexuz3.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( Restrict the base of the upper integers set to another upper integers
|
|
set. (Contributed by Mario Carneiro, 26-Dec-2013.) $)
|
|
rexuz3 $p |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph <->
|
|
E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) ) $=
|
|
( cz wcel cv cuz cfv wral wrex wa id rgen wceq fveq2 syl6eqr ralimi mpan2
|
|
raleqdv rspcev biantrurd uztrn2 ancrd ralimdva eluzelz eleq2s jctild uzid
|
|
a1d imp simpl eleq1 rspcva syl2an simpr adantl jca impbii rexbii2 rexanuz
|
|
bitr2i syl6rbb ) DGHZACBIZJKZLZBGMZCIZEHZCVHLZBGMZVJNZVIBEMZVFVNVJVFVLCEL
|
|
ZVNVLCEVLOPVMVQBDGVGDQZVLCVHEVRVHDJKZEVGDJRFSUBUCUAUDVPVLANZCVHLZBGMVOVIW
|
|
ABEGVGEHZVINVGGHZWANZWBVIWDWBVIWAWCWBAVTCVHWBVKVHHNZAVLWEVLADVKVGEFUEULUF
|
|
UGWCVGVSEDVGUHFUIUJUMWDWBVIWCVGVHHVMWBWAVGUKVTVLCVHVLAUNTVLWBCVGVHVKVGEUO
|
|
UPUQWAVIWCVTACVHVLAURTUSUTVAVBVLABCVCVDVE $.
|
|
|
|
${
|
|
$d j ps $.
|
|
$( Combine two different upper integer properties into one. (Contributed
|
|
by Mario Carneiro, 26-Dec-2013.) $)
|
|
rexanuz2 $p |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <->
|
|
( E. j e. Z A. k e. ( ZZ>= ` j ) ph /\
|
|
E. j e. Z A. k e. ( ZZ>= ` j ) ps ) ) $=
|
|
( wa cv cuz cfv wral wrex cz wcel eluzel2 eleq2s a1d rexlimiv rexuz3
|
|
adantr anbi12d rexanuz syl6rbbr bitrd pm5.21nii ) ABHZDCIZJKZLZCFMZENOZ
|
|
ADUILZCFMZBDUILZCFMZHZUJULCFUHFOZULUJULUHEJKFEUHPGQZRSUNULUPUMULCFURULU
|
|
MUSRSUAULUKUJCNMZUQUGCDEFGTULUQUMCNMZUOCNMZHUTULUNVAUPVBACDEFGTBCDEFGTU
|
|
BABCDUCUDUEUF $.
|
|
$}
|
|
|
|
$( A version of ~ 19.29 for upper integer quantifiers. (Contributed by
|
|
Mario Carneiro, 10-Feb-2014.) $)
|
|
r19.29uz $p |- ( ( A. k e. Z ph /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) ->
|
|
E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) ) $=
|
|
( wral cv cuz cfv wrex wa wcel wi uztrn2 ex pm3.2 a1i imim12d impcom syl
|
|
ralimdv2 ralim reximdva imp ) ADFHZBDCIZJKZHZCFLABMZDUIHZCFLUGUJULCFUGUHF
|
|
NZMBUKOZDUIHZUJULOUMUGUOUMAUNDFUIUMDIZUINZUPFNZAUNUMUQUREUPUHFGPQAUNOUMAB
|
|
RSTUCUABUKDUIUDUBUEUF $.
|
|
|
|
$( A version of ~ r19.2m for upper integer quantifiers. (Contributed by
|
|
Mario Carneiro, 15-Feb-2014.) $)
|
|
r19.2uz $p |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ph -> E. k e. Z ph ) $=
|
|
( cv cuz cfv wral wrex wcel wex cz eluzelz uzid elex2 3syl eleq2s r19.2m
|
|
sylan uztrn2 ex anim1d reximdv2 imp syldan rexlimiva ) ACBGZHIZJZACEKZBEU
|
|
IELZUKACUJKZULUMCGZUJLZCMZUKUNUQUIDHIZEUIURLUINLUIUJLUQDUIOUIPCUIUJQRFSAC
|
|
UJTUAUMUNULUMAACUJEUMUPUOELZAUMUPUSDUOUIEFUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
recvguniqlem.f $e |- ( ph -> F : NN --> RR ) $.
|
|
recvguniqlem.a $e |- ( ph -> A e. RR ) $.
|
|
recvguniqlem.b $e |- ( ph -> B e. RR ) $.
|
|
recvguniqlem.k $e |- ( ph -> K e. NN ) $.
|
|
recvguniqlem.lt1 $e |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) ) $.
|
|
recvguniqlem.lt2 $e |- ( ph -> ( F ` K ) < ( B + ( ( A - B ) / 2 ) ) ) $.
|
|
$( Lemma for ~ recvguniq . Some of the rearrangements of the expressions.
|
|
(Contributed by Jim Kingdon, 8-Aug-2021.) $)
|
|
recvguniqlem $p |- ( ph -> F. ) $=
|
|
( clt wbr cfv cmin co c2 caddc readdcld recnd ffvelrnd resubcld rehalfcld
|
|
cn cr ltadd1dd addassd 2halvesd oveq2d pncan3d 3eqtrd breqtrd lttrd ltnrd
|
|
cdiv pm2.21fal ) ABBLMABEDNZBCOPZQUOPZRPZBGAUQUSAUDUEEDFIUAZAURABCGHUBZUC
|
|
ZSGJAUTCUSRPZUSRPZBLAUQVDUSVAACUSHVCSVCKUFAVECUSUSRPZRPCURRPBACUSUSACHTZA
|
|
USVCTZVHUGAVFURCRAURAURVBTUHUIACBVGABGTUJUKULUMABGUNUP $.
|
|
$}
|
|
|
|
${
|
|
$d F j x $. $d L j k x $. $d M j k x $. $d k ph $.
|
|
recvguniq.f $e |- ( ph -> F : NN --> RR ) $.
|
|
recvguniq.lre $e |- ( ph -> L e. RR ) $.
|
|
recvguniq.l $e |- ( ph ->
|
|
A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) < ( L + x ) /\ L < ( ( F ` k ) + x ) ) ) $.
|
|
recvguniq.mre $e |- ( ph -> M e. RR ) $.
|
|
recvguniq.m $e |- ( ph ->
|
|
A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) < ( M + x ) /\ M < ( ( F ` k ) + x ) ) ) $.
|
|
$( Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.) $)
|
|
recvguniq $p |- ( ph -> L = M ) $=
|
|
( wbr clt wcel wa co caddc cn crp wceq wn wfal wi wo cr wb reaplt syl2anc
|
|
cap cv cfv cmin c2 cdiv wrex wral simpr adantr difrp rphalfcld cuz r19.26
|
|
mpbid sylanbrc c1 nnuz rexanuz2 ralbii sylibr r19.2uz ralimi oveq2 breq2d
|
|
anbi12d rexbidv rspcv sylc wf ad2antrr simprl simprrr adantl recvguniqlem
|
|
syl simprll rexlimddv ex biimpa simprlr simprrl jaod sylbid dfnot cc apti
|
|
recnd mpbird ) AFGUAZFGUJMZUBZAWTUCUDXAAWTFGNMZGFNMZUEZUCAFUFOZGUFOZWTXDU
|
|
GIKFGUHUIAXBUCXCAXBUCAXBPZDUKZEULZFGFUMQZUNUOQZRQZNMZFXIXKRQZNMZPZXIGXKRQ
|
|
ZNMZGXNNMZPZPZUCDSXGXKTOXIFBUKZRQZNMZFXIYBRQZNMZPZXIGYBRQZNMZGYENMZPZPZDS
|
|
UPZBTUQZYADSUPZXGXJXGXBXJTOZAXBURXGXEXFXBYPUGAXEXBIUSAXFXBKUSFGUTUIVDVAAY
|
|
NXBAYLDCUKVBULZUQCSUPZBTUQZYNAYGDYQUQCSUPZYKDYQUQCSUPZPZBTUQZYSAYTBTUQUUA
|
|
BTUQUUCJLYTUUABTVCVEYRUUBBTYGYKCDVFSVGVHVIVJYRYMBTYLCDVFSVGVKVLWEZUSYMYOB
|
|
XKTYBXKUAZYLYADSUUEYGXPYKXTUUEYDXMYFXOUUEYCXLXINYBXKFRVMVNUUEYEXNFNYBXKXI
|
|
RVMZVNVOUUEYIXRYJXSUUEYHXQXINYBXKGRVMVNUUEYEXNGNUUFVNVOVOVPVQVRXGXHSOZYAP
|
|
ZPGFEXHASUFEVSZXBUUHHVTAXFXBUUHKVTAXEXBUUHIVTXGUUGYAWAUUHXSXGUUGXPXRXSWBW
|
|
CUUHXMXGUUGXMXOXTWFWCWDWGWHAXCUCAXCPZXIFFGUMQZUNUOQZRQZNMZFXIUULRQZNMZPZX
|
|
IGUULRQZNMZGUUONMZPZPZUCDSUUJUULTOYNUVBDSUPZUUJUUKAXCUUKTOZAXFXEXCUVDUGKI
|
|
GFUTUIWIVAAYNXCUUDUSYMUVCBUULTYBUULUAZYLUVBDSUVEYGUUQYKUVAUVEYDUUNYFUUPUV
|
|
EYCUUMXINYBUULFRVMVNUVEYEUUOFNYBUULXIRVMZVNVOUVEYIUUSYJUUTUVEYHUURXINYBUU
|
|
LGRVMVNUVEYEUUOGNUVFVNVOVOVPVQVRUUJUUGUVBPZPFGEXHAUUIXCUVGHVTAXEXCUVGIVTA
|
|
XFXCUVGKVTUUJUUGUVBWAUVGUUPUUJUUGUUNUUPUVAWJWCUVGUUSUUJUUGUUQUUSUUTWKWCWD
|
|
WGWHWLWMWTWNVJAFWOOGWOOWSXAUGAFIWQAGKWQFGWPUIWR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Square root; absolute value
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c sqrt $. $( Positive square root of a positive real number. $)
|
|
$c abs $. $( Absolute value function. $)
|
|
$c +- $. $( Plus-or-minus symbol (for comments only) $)
|
|
|
|
$( Extend class notation to include square root of a complex number. $)
|
|
csqrt $a class sqrt $.
|
|
|
|
$( Extend class notation to include a function for the absolute value
|
|
(modulus) of a complex number. $)
|
|
cabs $a class abs $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Define a function whose value is the square root of a nonnegative real
|
|
number.
|
|
|
|
Defining the square root for complex numbers has one difficult part:
|
|
choosing between the two roots. The usual way to define a principal
|
|
square root for all complex numbers relies on excluded middle or
|
|
something similar. But in the case of a nonnegative real number, we
|
|
don't have the complications presented for general complex numbers, and
|
|
we can choose the nonnegative root.
|
|
|
|
(Contributed by Jim Kingdon, 23-Aug-2020.) $)
|
|
df-rsqrt $a |- sqrt = ( x e. RR |-> ( iota_ y e. RR ( ( y ^ 2 ) = x /\
|
|
0 <_ y ) ) ) $.
|
|
|
|
$( Define the function for the absolute value (modulus) of a complex
|
|
number. (Contributed by NM, 27-Jul-1999.) $)
|
|
df-abs $a |- abs = ( x e. CC |-> ( sqrt ` ( x x. ( * ` x ) ) ) ) $.
|
|
|
|
$( Value of square root function. (Contributed by Jim Kingdon,
|
|
23-Aug-2020.) $)
|
|
sqrtrval $p |- ( A e. RR -> ( sqrt ` A ) =
|
|
( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) ) $=
|
|
( vy cv c2 cexp co wceq cc0 cle wbr wa cr crio csqrt anbi1d riotabidv cvv
|
|
eqeq2 wcel df-rsqrt reex riotaexg ax-mp fvmpt ) CBADZEFGZCDZHZIUFJKZLZAMN
|
|
UGBHZUJLZAMNZMOUHBHZUKUMAMUOUIULUJUHBUGSPQCAUAMRTUNRTUBUMAMRUCUDUE $.
|
|
|
|
$( The absolute value (modulus) of a complex number. Proposition 10-3.7(a)
|
|
of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by
|
|
Mario Carneiro, 7-Nov-2013.) $)
|
|
absval $p |- ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) ) $=
|
|
( vx vy cc wcel ccj cfv cmul co csqrt cvv cabs wceq cr cv c2 cexp cc0 cle
|
|
mpdan wbr wa crio cmpt df-rsqrt reex mptex eqeltri id cjcl mulcld sylancr
|
|
fvexg fveq2 oveq12 fveq2d df-abs fvmptg ) ADEZAAFGZHIZJGZKEZALGVBMUSJKEVA
|
|
DEVCJBNCOZPQIBOZMRVDSUAUBCNUCZUDKBCUEBNVFUFUGUHUSAUTUSUIAUJUKVAJKDUMULBAV
|
|
EVEFGZHIZJGVBDKLVEAMZVHVAJVIVGUTMVHVAMVEAFUNVEAVGUTHUOTUPBUQURT $.
|
|
$}
|
|
|
|
$( A real number does not lie on the negative imaginary axis. (Contributed
|
|
by Mario Carneiro, 8-Jul-2013.) $)
|
|
rennim $p |- ( A e. RR -> ( _i x. A ) e/ RR+ ) $=
|
|
( cr wcel ci cmul co crp wn wnel cc0 wceq cre cfv cc wi ax-icn recn sylancr
|
|
mulcl rpre rereb syl5ib caddc addid2d fveq2d crre mpan eqtr3d eqeq1d sylibd
|
|
syl 0re rpne0 necon2bi eqcoms syl6 pm2.01d df-nel sylibr ) ABCZDAEFZGCZHZVA
|
|
GIUTVBUTVBJVAKZVCUTVBVALMZVAKZVDUTVANCZVBVFOUTDNCANCVGPAQDASRZVBVABCVGVFVAT
|
|
VAUAUBUKUTVEJVAUTJVAUCFZLMZVEJUTVIVALUTVAVHUDUEJBCUTVJJKULJAUFUGUHUIUJVCVAJ
|
|
VBVAJVAUMUNUOUPUQVAGURUS $.
|
|
|
|
$( Lemma for ~ sqrt0 . (Contributed by Jim Kingdon, 26-Aug-2020.) $)
|
|
sqrt0rlem $p |- ( ( A e. RR /\ ( ( A ^ 2 ) = 0 /\ 0 <_ A ) ) <-> A = 0 ) $=
|
|
( cr wcel c2 cexp co cc0 wceq cle wbr wa cc wb sqeq0 syl biimpa adantrr 0re
|
|
recn mpbiri eleq1 sq0i 0le0 breq2 jca32 impbii ) ABCZADEFGHZGAIJZKKAGHZUGUH
|
|
UJUIUGUHUJUGALCUHUJMASANOPQUJUGUHUIUJUGGBCRAGBUATAUBUJUIGGIJUCAGGIUDTUEUF
|
|
$.
|
|
|
|
$( Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) $)
|
|
sqrt0 $p |- ( sqrt ` 0 ) = 0 $=
|
|
( vx cc0 csqrt cfv cv c2 cexp co wceq cle wbr wa cr crio 0re sqrtrval ax-mp
|
|
wcel id wb sqrt0rlem biimpi ex simpr sylbir impbid1 adantl riota5 eqtri ) B
|
|
CDZAEZFGHBIBUKJKLZAMNZBBMRZUJUMIOABPQUNUMBIOUNULAMBUNSUKMRZULUKBIZTUNUOULUP
|
|
UOULUPUOULLZUPUKUAZUBUCUPUQULURUOULUDUEUFUGUHQUI $.
|
|
|
|
${
|
|
resqrexlemex.seq $e |- F = seq 1 (
|
|
( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) ,
|
|
( NN X. { ( 1 + A ) } ) ,
|
|
RR+ ) $.
|
|
resqrexlemex.a $e |- ( ph -> A e. RR ) $.
|
|
resqrexlemex.agt0 $e |- ( ph -> 0 <_ A ) $.
|
|
|
|
${
|
|
$d A y z $. $d ph y z $.
|
|
$( Lemma for ~ resqrex . ` 1 + A ` is a positive real (expressed in a way
|
|
that will help apply ~ iseqf and similar theorems). (Contributed by
|
|
Jim Kingdon, 28-Jul-2021.) $)
|
|
resqrexlem1arp $p |- ( ( ph /\ N e. NN ) ->
|
|
( ( NN X. { ( 1 + A ) } ) ` N ) e. RR+ ) $=
|
|
( cn wcel wa c1 caddc crp cr adantr cc0 clt wbr co csn cxp cfv readdcld
|
|
wceq 1red cle 0lt1 addgtge0 syl22anc elrpd fvconst2g sylancom eqeltrd
|
|
a1i ) AFJKZLZFJMDNUAZUBUCUDZUSOAUQUSOKUTUSUFURUSURMDURUGZADPKZUQHQZUEUR
|
|
MPKVBRMSTZRDUHTZRUSSTVAVCVDURUIUPAVEUQIQMDUJUKULZJUSFOUMUNVFUO $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d ph y z $. $d B y z $. $d C y z $.
|
|
$( Lemma for ~ resqrex . Applying the recursion rule yields a positive
|
|
real (expressed in a way that will help apply ~ iseqf and similar
|
|
theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) $)
|
|
resqrexlemp1rp $p |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) ->
|
|
( B ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) C )
|
|
e. RR+ ) $=
|
|
( crp wcel cdiv co caddc c2 cr wceq cc0 wbr wa cv cmpt2 eqidd id oveq1d
|
|
oveq2 ad2antrl simprl simprr rpred adantr rerpdivcld readdcld rehalfcld
|
|
oveq12d ovmpt2d clt cle rpgt0d divge0d addgtge0 elrpd rphalfcld eqeltrd
|
|
syl22anc ) AEKLZFKLZUAZUAZEFBCKKBUBZDVKMNZONZPMNZUCZNEDEMNZONZPMNZKVJBC
|
|
EFKKVNVRVOQVJVOUDVKERZVNVRRVJCUBFRVSVMVQPMVSVKEVLVPOVSUEVKEDMUGUPUFUHAV
|
|
GVHUIZAVGVHUJVJVQVJEVPVJEVTUKZVJDEADQLVIIULZVTUMZUNZUOUQVJVQVJVQWDVJEQL
|
|
VPQLSEURTSVPUSTSVQURTWAWCVJEVTUTVJDEWBVTASDUSTVIJULVAEVPVBVFVCVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d A a b y z $. $d a b ph y z $.
|
|
$( Lemma for ~ resqrex . The sequence is a function. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 27-Jul-2021.) $)
|
|
resqrexlemf $p |- ( ph -> F : NN --> RR+ ) $=
|
|
( va vb cn crp cv cdiv co caddc c2 c1 wf cvv cmpt2 csn cxp cseq nnuz cr
|
|
wcel reex rpssre ssexi resqrexlem1arp resqrexlemp1rp iseqf feq1i sylibr
|
|
a1i 1zzd ) AKLBCLLBMZDURNOPOQNOUAZLKRDPOUBUCZRUDZSKLESAIJUSLUTRTKUELTUG
|
|
ALUFUHUIUJUPAUQABCDEIMZFGHUKABCDVBJMEFGHULUMKLEVAFUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d A a b y z $. $d a b ph y z $.
|
|
$( Lemma for ~ resqrex . Initial value. Although this sequence
|
|
converges to the square root with any positive initial value, this
|
|
choice makes various steps in the proof of convergence easier.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) $)
|
|
resqrexlemf1 $p |- ( ph -> ( F ` 1 ) = ( 1 + A ) ) $=
|
|
( va vb c1 cfv cn caddc co crp cv cdiv wcel cr csn c2 cmpt2 cseq fveq1i
|
|
cxp 1zzd cuz elnnuz resqrexlem1arp sylan2br resqrexlemp1rp iseq1 syl5eq
|
|
wceq 1red readdcld 1nn fvconst2g sylancl eqtrd ) AKELZKMKDNOZUAUFZLZVCA
|
|
VBKBCPPBQZDVFRONOUBROUCZPVDKUDZLVEKEVHFUEAIJVGPVDKAUGIQZKUHLSAVIMSVIVDL
|
|
PSVIUIABCDEVIFGHUJUKABCDVIJQEFGHULUMUNAVCTSKMSVEVCUOAKDAUPGUQURMVCKTUSU
|
|
TVA $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c d y z $. $d F c d $. $d N a b c d $. $d ph a b c d y z $.
|
|
$( Lemma for ~ resqrex . Recursion rule. This sequence is the ancient
|
|
method for computing square roots, often known as the babylonian
|
|
method, although known to many ancient cultures. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 27-Jul-2021.) $)
|
|
resqrexlemfp1 $p |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) =
|
|
( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) ) $=
|
|
( cn wcel c1 caddc co cfv crp cv cdiv c2 wceq va vb vc vd csn cxp cmpt2
|
|
wa cseq cuz elnnuz biimpi adantl resqrexlem1arp sylan2br resqrexlemp1rp
|
|
adantlr iseqp1 fveq1i oveq1i 3eqtr4g cr id oveq2 oveq12d eqidd cbvmpt2v
|
|
oveq1d ad2antrl resqrexlemf ffvelrnda peano2nn sylan2 adantr rerpdivcld
|
|
a1i rpred readdcld rehalfcld ovmpt2d eqtrd ) AFJKZUHZFLMNZEOZFEOZWDJLDM
|
|
NUEUFZOZBCPPBQZDWIRNZMNZSRNZUGZNZWFDWFRNZMNZSRNZWCWDWMPWGLUIZOFWROZWHWM
|
|
NWEWNWCUAUBWMPWGLFWBFLUJOZKZAWBXAFUKULUMAUAQZWTKZXBWGOPKZWBXCAXBJKXDXBU
|
|
KABCDEXBGHIUNUOUQAXBPKUBQZPKUHXBXEWMNPKWBABCDXBXEEGHIUPUQURWDEWRGUSWFWS
|
|
WHWMFEWRGUSUTVAWCUCUDWFWHPPUCQZDXFRNZMNZSRNZWQWMVBWMUCUDPPXIUGTWCBCUCUD
|
|
PPWLXIXIWIXFTZWKXHSRXJWIXFWJXGMXJVCWIXFDRVDVEVHCQUDQZTXIVFVGVPXFWFTZXIW
|
|
QTWCXKWHTXLXHWPSRXLXFWFXGWOMXLVCXFWFDRVDVEVHVIAJPFEABCDEGHIVJVKZWBAWDJK
|
|
WHPKFVLABCDEWDGHIUNVMWCWPWCWFWOWCWFXMVQWCDWFADVBKWBHVNXMVOVRVSVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d A f g h k w y z $. $d F f g h k $. $d F k w $. $d N w $.
|
|
$d ph f g h k w y z $.
|
|
$( Lemma for ~ resqrex . Each element of the sequence is an
|
|
overestimate. (Contributed by Mario Carneiro and Jim Kingdon,
|
|
27-Jul-2021.) $)
|
|
resqrexlemover $p |- ( ( ph /\ N e. NN ) -> A < ( ( F ` N ) ^ 2 ) ) $=
|
|
( wcel c2 cexp co clt wbr c1 caddc oveq1d cc0 c4 vw vk vf vg vh cn wceq
|
|
cfv cv wi fveq2 breq2d imbi2d cmul resqcld cr 2re a1i remulcld readdcld
|
|
1red cle recnd mulid2d w3a 1le2 lemul1a mpan2 syl112anc eqbrtrrd sqge0d
|
|
wa addge02d mpbid letrd ltp1d lelttrd resqrexlemf1 1cnd addcomd binom21
|
|
eqtrd syl breqtrrd cdiv cmin crp resqrexlemf ffvelrnda rpred rerpdivcld
|
|
cc adantr resubcld 4re posdifd biimpa rpgt0d divgt0d sqcld divsubdirapd
|
|
cap rpap0d divcanap3d breqtrd gt0ap0d sqgt0apd 4pos gt0ap0ii redivclapd
|
|
sqvald ltaddpos2d wb resqrexlemfp1 2cnd 2ap0 sqdivapd sq2 oveq2i syl6eq
|
|
divcanap2d oveq2d syl2anc binom2 cneg mulcld negcld 4cn addassd negsubd
|
|
binom2sub 2cn negcli addassi subidi negeqi oveq1i recn adantl 3eqtrd ex
|
|
negsubdii neg0 3eqtr3i addid2i eqtri 3eqtr3ri adddird mulneg1d syl5reqr
|
|
2p2e4 3eqtr3rd addcom syl2an addass syl3an caov32d subcld mpbird expcom
|
|
3eqtr4rd divdirapd a2d nnind impcom ) FUFJADFEUHZKLMZNOZADUAUIZEUHZKLMZ
|
|
NOZUJADPEUHZKLMZNOZUJADUBUIZEUHZKLMZNOZUJADUVPPQMZEUHZKLMZNOZUJAUVHUJUA
|
|
UBFUVIPUGZUVLUVOAUWDUVKUVNDNUWDUVJUVMKLUVIPEUKRULUMUVIUVPUGZUVLUVSAUWEU
|
|
VKUVRDNUWEUVJUVQKLUVIUVPEUKRULUMUVIUVTUGZUVLUWCAUWFUVKUWBDNUWFUVJUWAKLU
|
|
VIUVTEUKRULUMUVIFUGZUVLUVHAUWGUVKUVGDNUWGUVJUVFKLUVIFEUKRULUMADDKLMZKDU
|
|
NMZQMZPQMZUVNNADUWJUWKHAUWHUWIADHUOZAKDKUPJZAUQURZHUSZUTZAUWJPUWPAVAZUT
|
|
ADUWIUWJHUWOUWPAPDUNMZDUWIVBADADHVCZVDAPUPJZUWMDUPJZSDVBOZUWRUWIVBOZUWQ
|
|
UWNHIUWTUWMUXAUXBVLVEPKVBOUXCVFPKDVGVHVIVJASUWHVBOUWIUWJVBOADHVKAUWIUWH
|
|
UWOUWLVMVNVOAUWJUWPVPVQAUVNDPQMZKLMZUWKAUVMUXDKLAUVMPDQMUXDABCDEGHIVRAP
|
|
DAVSUWSVTWBRADWLJZUXEUWKUGUWSDWAWCWBWDUVPUFJZAUVSUWCAUXGUVSUWCUJAUXGVLZ
|
|
UVSUWCUXHUVSVLZUWCDUVQDUVQWEMZWFMZKLMZTWEMZDQMZNOZUXISUXMNOUXOUXIUXLTUX
|
|
IUXKUXHUXKUPJUVSUXHUVQUXJUXHUVQAUFWGUVPEABCDEGHIWHWIZWJZUXHDUVQAUXAUXGH
|
|
WMZUXPWKZWNWMZUOZTUPJZUXIWOURZUXIUXKUXTUXIUXKUXTUXISUVRDWFMZUVQWEMZUXKN
|
|
UXIUYDUVQUXHUYDUPJUVSUXHUVRDUXHUVQUXQUOZUXRWNWMUXHUVQUPJUVSUXQWMUXHUVSS
|
|
UYDNOUXHDUVRUXRUYFWPWQUXHSUVQNOUVSUXHUVQUXPWRWMWSUXIUYEUVRUVQWEMZUXJWFM
|
|
UXKUXIUVRDUVQUXHUVRWLJUVSUXHUVQUXHUVQUXQVCZWTZWMUXHUXFUVSAUXFUXGUWSWMZW
|
|
MUXHUVQWLJZUVSUYHWMZUXHUVQSXBOUVSUXHUVQUXPXCZWMZXAUXIUYGUVQUXJWFUXIUYGU
|
|
VQUVQUNMZUVQWEMUVQUXIUVRUYOUVQWEUXIUVQUYLXKRUXIUVQUVQUYLUYLUYNXDWBRWBXE
|
|
XFXGSTNOUXIXHURWSUXIUXMDUXIUXLTUYAUYCTSXBOZUXITWOXHXIZURXJUXHUXAUVSUXRW
|
|
MXLVNUXHUWCUXOXMUVSUXHUWBUXNDNUXHUWBUXLTDUNMZQMZTWEMZUXMUYRTWEMZQMUXNUX
|
|
HUWBUVQUXJQMZKLMZTWEMZUYTUXHUWBVUCKKLMZWEMZVUDUXHUWBVUBKWEMZKLMVUFUXHUW
|
|
AVUGKLABCDEUVPGHIXNRUXHVUBKUXHVUBUXHUVQUXJUXQUXSUTVCUXHXOZKSXBOUXHXPURX
|
|
QWBVUETVUCWEXRXSXTUXHVUCUYSTWEUXHUVRKUVQUXJUNMZUNMZWFMZUXJKLMZQMZUYRQMU
|
|
VRUWIWFMZVULQMZUYRQMZUYSVUCUXHVUMVUOUYRQUXHVUKVUNVULQUXHVUJUWIUVRWFUXHV
|
|
UIDKUNUXHDUVQUYJUYHUYMYAYBZYBRRUXHUXLVUMUYRQUXHUYKUXJWLJZUXLVUMUGUYHUXH
|
|
UXJUXSVCZUVQUXJYKYCRUXHVUCUVRUWIQMZVULQMZVUNUYRQMZVULQMVUPUXHVUCUVRVUJQ
|
|
MZVULQMZVVAUXHUYKVURVUCVVDUGUYHVUSUVQUXJYDYCUXHVVCVUTVULQUXHVUJUWIUVRQV
|
|
UQYBRWBUXHVUTVVBVULQUXHUVRUWIYEZQMZUYRQMUVRVVEUYRQMZQMVVBVUTUXHUVRVVEUY
|
|
RUYIUXHUWIUXHKDVUHUYJYFZYGUXHTDTWLJUXHYHURZUYJYFZYIUXHVVFVUNUYRQUXHUVRU
|
|
WIUYIVVHYJRUXHVVGUWIUVRQUXHUWIKYEZTQMZDUNMZVVGVVLKDUNVVKKQMZKQMZVVKKKQM
|
|
ZQMKVVLVVKKKKYLYMZYLYLYNVVOSKQMKVVNSKQKKWFMZYESYEVVNSVVRSKYLYOYPKKYLYLU
|
|
UBUUCUUDYQKYLUUEUUFVVPTVVKQUUKXSUUGYQUXHVVMVVKDUNMZUYRQMVVGUXHVVKTDVVKW
|
|
LJUXHVVQURVVIUYJUUHUXHVVSVVEUYRQUXHKDVUHUYJUUIRWBUUJYBUULRUXHUCUDUEVUNU
|
|
YRVULUPQUXHUVRUWIUYFUXHKDUWMUXHUQURUXRUSWNUXHTDUYBUXHWOURUXRUSUXHUXJUXS
|
|
UOUCUIZUPJZUDUIZUPJZVLVVTVWBQMZVWBVVTQMUGZUXHVWAVVTWLJZVWBWLJZVWEVWCVVT
|
|
YRZVWBYRZVVTVWBUUMUUNYSVWAVWCUEUIZUPJZVEVWDVWJQMVVTVWBVWJQMQMUGZUXHVWAV
|
|
WFVWCVWGVWKVWJWLJVWLVWHVWIVWJYRVVTVWBVWJUUOUUPYSUUQYTUVARWBUXHUXLUYRTUX
|
|
HUXKUXHUVQUXJUYHVUSUURWTVVJVVIUYPUXHUYQURZUVBUXHVUADUXMQUXHDTUYJVVIVWMX
|
|
DYBYTULWMUUSUUAUUTUVCUVDUVE $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d ph y z $.
|
|
$( Lemma for ~ resqrex . The sequence is decreasing. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 29-Jul-2021.) $)
|
|
resqrexlemdec $p |- ( ( ph /\ N e. NN ) ->
|
|
( F ` ( N + 1 ) ) < ( F ` N ) ) $=
|
|
( cn wcel caddc co cfv cdiv c2 clt wbr cmul crp wa resqrexlemfp1 adantr
|
|
c1 cr resqrexlemf ffvelrnda rerpdivcld rpred cexp resqrexlemover sqvald
|
|
breqtrd ltdivmuld mpbird ltadd2dd 2timesd breqtrrd readdcld 2rp eqbrtrd
|
|
rpcnd a1i ) AFJKZUAZFUDLMENFENZDVFOMZLMZPOMZVFQABCDEFGHIUBVEVIVFQRVHPVF
|
|
SMZQRVEVHVFVFLMVJQVEVGVFVFVEDVFADUEKVDHUCZAJTFEABCDEGHIUFUGZUHZVEVFVLUI
|
|
ZVNVEVGVFQRDVFVFSMZQRVEDVFPUJMVOQABCDEFGHIUKVEVFVEVFVLVBZULUMVEDVFVFVKV
|
|
NVLUNUOUPVEVFVPUQURVEVHVFPVEVFVGVNVMUSVNPTKVEUTVCUNUOVA $.
|
|
$}
|
|
|
|
${
|
|
resqrexlemdecn.n $e |- ( ph -> N e. NN ) $.
|
|
resqrexlemdecn.m $e |- ( ph -> M e. NN ) $.
|
|
resqrexlemdecn.nm $e |- ( ph -> N < M ) $.
|
|
$d A y z $. $d ph y z k w $. $d F k w $. $d M w $. $d N k w $.
|
|
|
|
$( Lemma for ~ resqrex . The sequence is decreasing. (Contributed by
|
|
Jim Kingdon, 31-Jul-2021.) $)
|
|
resqrexlemdecn $p |- ( ph -> ( F ` M ) < ( F ` N ) ) $=
|
|
( cfv clt wbr c1 wcel wi cn vw vk caddc co cz nnzd peano2zd wb nnltp1le
|
|
cle syl2anc mpbid cv fveq2 breq1d imbi2d resqrexlemdec mpdan a1i w3a wa
|
|
wceq crp wf resqrexlemf ad2antrr simplr2 1red zred cc0 nnred nngt0d 0re
|
|
cr ltle mpan sylc addge02d simplr3 letrd elnnz1 sylanbrc ffvelrnd rpred
|
|
peano2nnd simpll simpr lttrd ex expcom a2d uzind syl3anc pm2.43i ) AFEN
|
|
ZGENZOPZAGQUCUDZUERZFUERWRFUJPZAWQSZAGAGKUFUGZAFLUFAGFOPZWTMAGTRZFTRXCW
|
|
TUHKLGFUIUKULAUAUMZENZWPOPZSAWRENZWPOPZSZAUBUMZENZWPOPZSAXKQUCUDZENZWPO
|
|
PZSXAUAUBWRFXEWRVBZXGXIAXQXFXHWPOXEWREUNUOUPXEXKVBZXGXMAXRXFXLWPOXEXKEU
|
|
NUOUPXEXNVBZXGXPAXSXFXOWPOXEXNEUNUOUPXEFVBZXGWQAXTXFWOWPOXEFEUNUOUPXJWS
|
|
AXDXIKABCDEGHIJUQURUSWSXKUERZWRXKUJPZUTZAXMXPAYCXMXPSAYCVAZXMXPYDXMVAZX
|
|
OXLWPYEXOYETVCXNEATVCEVDYCXMABCDEHIJVEVFZYEXKYEYAQXKUJPXKTRZWSYAYBAXMVG
|
|
ZYEQWRXKYEVHYEWRAWSYCXMXBVFVIYEXKYHVIAQWRUJPZYCXMAVJGUJPZYIAGVNRZVJGOPZ
|
|
YJAGKVKZAGKVLVJVNRYKYLYJSVMVJGVOVPVQAQGAVHYMVRULVFWSYAYBAXMVSVTXKWAWBZW
|
|
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NABCDEXKHIJUQUKYDXMWGWHWIWJWKWLWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d ph y z $. $d F k w $. $d N w $. $d k ph w $.
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|
$( Lemma for ~ resqrex . A (variable) lower bound for each term of the
|
|
sequence. (Contributed by Mario Carneiro and Jim Kingdon,
|
|
29-Jul-2021.) $)
|
|
resqrexlemlo $p |- ( ( ph /\ N e. NN ) ->
|
|
( 1 / ( 2 ^ N ) ) < ( F ` N ) ) $=
|
|
( wcel c1 c2 cexp co cdiv clt wbr wi oveq2d adantr vw vk cn cfv cv wceq
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|
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|
imbi2d 1red halflt1 syl6eqbr cc0 addge01d ltletrd resqrexlemf1 breqtrrd
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|
cle mpbid wa cmul a1i nnz ad2antlr rpexpcld rpcnd rpap0d recdivap2d cn0
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|
cz nnnn0 expp1d cr resqrexlemf ffvelrnda rpred rerpdivcld simpr divge0d
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|
|
JAKLFMNZONZFEUDZPQZAKLUAUEZMNZONZXBEUDZPQZRAKLKMNZONZKEUDZPQZRAKLUBUEZM
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NZONZXKEUDZPQZRAKLXKKUGNZMNZONZXPEUDZPQZRAXARUAUBFXBKUFZXFXJAYAXDXHXEXI
|
|
PYAXCXGKOXBKLMUHSXBKEUIUJURXBXKUFZXFXOAYBXDXMXEXNPYBXCXLKOXBXKLMUHSXBXK
|
|
EUIUJURXBXPUFZXFXTAYCXDXRXEXSPYCXCXQKOXBXPLMUHSXBXPEUIUJURXBFUFZXFXAAYD
|
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XDWSXEWTPYDXCWRKOXBFLMUHSXBFEUIUJURAXHKDUGNZXIPAXHKYEAXGAXGLUKALAULUMZU
|
|
NUOUPAUSZAKDYGHUQAXHKLONKPAXGLKOYFSUTVAAVBDVGQZKYEVGQIAKDYGHVCVHVDABCDE
|
|
GHIVEVFXKUCJZAXOXTAYIXOXTRAYIVIZXOXTYJXOVIZXMLONZXRXSPYKYLKXLLVJNZONXRY
|
|
KXLLYKXLYKLXKLUKJYKUNVKZYIXKVSJAXOXKVLVMVNZVOYKULZYKXLYOVPYKLYNVPVQYKXQ
|
|
YMKOYKLXKYPYIXKVRJAXOXKVTVMWASWIYKYLXNDXNONZUGNZLONZXSPYKXMYRLYKXLYOUPZ
|
|
YKXNYQYJXNWBJXOYJXNAUCUKXKEABCDEGHIWCWDZWEZTZYJYQWBJXOYJDXNADWBJYIHTZUU
|
|
AWFZTUQZYNYKXMXNYRYTUUCUUFYJXOWGYJXNYRVGQZXOYJVBYQVGQUUGYJDXNUUDUUAAYHY
|
|
IITWHYJXNYQUUBUUEVCVHTVDWJYJXSYSUFXOABCDEXKGHIWKTVFWLWMWNWOWPWQ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d ph y z $.
|
|
$( Lemma for ~ resqrex . Some of the calculations involved in showing
|
|
that the sequence converges. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 29-Jul-2021.) $)
|
|
resqrexlemcalc1 $p |- ( ( ph /\ N e. NN ) ->
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|
( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) =
|
|
( ( ( ( ( F ` N ) ^ 2 ) - A ) ^ 2 ) /
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|
( 4 x. ( ( F ` N ) ^ 2 ) ) ) ) $=
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|
( wcel caddc co c2 cexp cmin cdiv c4 cmul oveq1d eqtrd cn resqrexlemfp1
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|
wa c1 cfv crp resqrexlemf ffvelrnda rpred cr adantr rerpdivcld readdcld
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|
recnd 2cnd cc0 cap wbr 2ap0 a1i sqdivapd sq2 oveq2i syl6eq 4cn 4re 4pos
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|
cc clt gt0ap0d divcanap3d eqcomd oveq12d sqcld mulcld divsubdirapd cneg
|
|
eqtr4d addsubassd 2cn negsubdi2i oveq1i pncan3oi eqtr3i negeqi mulneg1d
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subdird 3eqtr3a oveq2d negsubd 3eqtrd wceq binom2 syl2anc rpap0d addcld
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|
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|
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WPUVAXPYGXPXTMYSMXAJXPXBUTXCWOZXDXGXPUUMYIYGPXPUUJYGRLZUUKYGRLZKLYGMNLZ
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MYGDRLRLZOLZDMNLZKLZUUMYIXPUVQUWAUVRUWBKXPUVQYGYGRLZUUIYGRLZOLUWAXPYGUU
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IYGUVAUVBUVAWGXPUVSUWDUVTUWEOXPYGUVAXEXPUWEMYGRLDRLUVTXPMDYGUUDUUEUVAXF
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XPMYGDUUDUVAUUEXHXIVMVRXPUVRUWBYGPLZYGRLUWBXPUUKUWFYGRXPDXTUUEUUTUVMVAS
|
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XPUWBYGXPDUUEVNUVAUVPXJTVMXPUUJUUKYGUVOUVNUVAXKXPYGVHJDVHJYIUWCWLUVAUUE
|
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YGDXLWNXGSTSXPYKYIYGQRLZPLYMXPYIYGQXPYHXPYGDUVAUUEWTVNUVAUUFUVPUUGXMXPU
|
|
WGYLYIPXPYGQUVAUUFXNWITWK $.
|
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$}
|
|
|
|
${
|
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$d A y z $. $d ph y z $.
|
|
$( Lemma for ~ resqrex . Some of the calculations involved in showing
|
|
that the sequence converges. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 29-Jul-2021.) $)
|
|
resqrexlemcalc2 $p |- ( ( ph /\ N e. NN ) ->
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|
( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) <_
|
|
( ( ( ( F ` N ) ^ 2 ) - A ) / 4 ) ) $=
|
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( wcel c1 co c2 cexp c4 cmul cdiv cle wbr recnd cn cmin resqrexlemcalc1
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wa caddc cfv crp resqrexlemf ffvelrnda rpred resqcld cr adantr resubcld
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|
rpap0d sqgt0apd elrpd readdcld addge01d mpbid lesub1dd breqtrd lediv1dd
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cc0 pncand gt0ap0d dividapd redivclapd 1red clt resqrexlemover wb difrp
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syl2anc 4re a1i 4pos lemul1d divmuldivapd sqvald mulcomd oveq12d eqtr4d
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NLZDUBLZMNLZOWLPLZQLZWMOQLZRABCDEFGHIUCWJWMWLQLZWQPLZKWQPLZWPWQRWJWRKRS
|
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WSWTRSWJWRWLWLQLKRWJWMWLWLWJWLDWJWKWJWKAUAUGFEABCDEGHIUHUIZUJZUKZADULJZ
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WIHUMZUNZXCWJWLXCWJWKXBWJWKXAUOUPZUQWJWMWLDUELZDUBLWLRWJWLXHDXCWJWLDXCX
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EURXEWJVDDRSZWLXHRSAXIWIIUMWJWLDXCXEUSUTVAWJWLDWJWLXCTZWJDXETVEVBVCWJWL
|
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XJWJWLXCXGVFZVGVBWJWRKWQWJWMWLXFXCXKVHWJVIWJWMOWJDWLVJSZWMUGJZABCDEFGHI
|
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VKWJXDWLULJXLXMVLXEXCDWLVMVNUTWJOOULJWJVOVPZVDOVJSWJVQVPZUQWDZVRUTWJWSW
|
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MWMPLZWLOPLZQLWPWJWMWLWMOWJWMXFTZXJXSWJOXNTZXKWJOXNXOVFVSWJWNXQWOXRQWJW
|
|
MXSVTWJOWLXTXJWAWBWCWJWQWJWQXPWEWFWGWH $.
|
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$}
|
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|
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${
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$d A y z k w $. $d ph y z k w $. $d F k w $. $d N w $.
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|
$( Lemma for ~ resqrex . Some of the calculations involved in showing
|
|
that the sequence converges. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 29-Jul-2021.) $)
|
|
resqrexlemcalc3 $p |- ( ( ph /\ N e. NN ) ->
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|
( ( ( F ` N ) ^ 2 ) - A ) <_
|
|
( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) ) $=
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|
( wcel cexp co cmin c1 c4 cdiv cle wbr oveq2d adantr vw vk cn cfv c2 cv
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|
caddc wceq fveq2 oveq1d oveq1 breq12d imbi2d cneg cc0 renegcld 0red crp
|
|
wi resqrexlemf 1nn ffvelrnd rpred resqcld le0neg2d mpbid leadd2dd recnd
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a1i negsubd addid1d 3brtr3d 1m1e0 oveq2i cc 4cn exp0 ax-mp eqtri syl5eq
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div1d breqtrrd wa cr wf peano2nn adantl resubcld ffvelrnda 4re clt 4pos
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elrpd rerpdivcld cz nnz peano2zm rpexpcld resqrexlemcalc2 lediv1d letrd
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biimpa cmul ad2antrr rpcnd rpap0d divdivap1d simpr pncan1 simplr expm1t
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syl nncnd sylancr eqtrd eqtr4d breqtrd ex expcom a2d nnind impcom ) FUC
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JAFEUDZUEKLZDMLZNEUDZUEKLZOFNMLZKLZPLZQRZAUAUFZEUDZUEKLZDMLZYGOYLNMLZKL
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ZPLZQRZUSAYGDMLZYGONNMLZKLZPLZQRZUSAUBUFZEUDZUEKLZDMLZYGOUUENMLZKLZPLZQ
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RZUSAUUENUGLZEUDZUEKLZDMLZYGOUUMNMLZKLZPLZQRZUSAYKUSUAUBFYLNUHZYSUUDAUV
|
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AYOYTYRUUCQUVAYNYGDMUVAYMYFUEKYLNEUIUJUJUVAYQUUBYGPUVAYPUUAOKYLNNMUKSSU
|
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LUMYLUUEUHZYSUULAUVBYOUUHYRUUKQUVBYNUUGDMUVBYMUUFUEKYLUUEEUIUJUJUVBYQUU
|
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JYGPUVBYPUUIOKYLUUENMUKSSULUMYLUUMUHZYSUUTAUVCYOUUPYRUUSQUVCYNUUODMUVCY
|
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MUUNUEKYLUUMEUIUJUJUVCYQUURYGPUVCYPUUQOKYLUUMNMUKSSULUMYLFUHZYSYKAUVDYO
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YEYRYJQUVDYNYDDMUVDYMYCUEKYLFEUIUJUJUVDYQYIYGPUVDYPYHOKYLFNMUKSSULUMAYT
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YGUUCQAYGDUNZUGLYGUOUGLYTYGQAUVEUOYGADHUPAUQAYFAYFAUCURNEABCDEGHIUTZNUC
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JAVAVIVBVCVDZAUODQRUVEUOQRIADHVEVFVGAYGDAYGUVGVHZADHVHVJAYGUVHVKVLAUUCY
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GNPLYGUUBNYGPUUBOUOKLZNUUAUOOKVMVNOVOJZUVINUHVPOVQVRVSVNAYGUVHWAVTWBUUE
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UCJZAUULUUTAUVKUULUUTUSAUVKWCZUULUUTUVLUULWCZUUPUUKOPLZUUSQUVMUUPUUHOPL
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UUEWFWGVBVCVDADWDJUVKHTZWHTUVLUVOWDJUULUVLUUHOUVLUUGDUVLUUFUVLUUFAUCURU
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UVLYGUUJAYGWDJUVKUVGTUVLOUUIUVRUVKUUIWOJZAUVKUUEWOJUVSUUEWPUUEWQXLWGWRZ
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WNZUVRWNTUVLUUPUVOQRUULABCDEUUEGHIWSTUVLUULUVOUVNQRUVLUUHUUKOUVQUWAUVRW
|
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TXBXAUVMUVNYGUUJOXCLZPLUUSUVMYGUUJOAYGVOJUVKUULUVHXDUVMUUJUVLUUJURJUULU
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VTTZXEUVJUVMVPVIUVMUUJUWCXFUVMOUVLOURJUULUVRTXFXGUVMUURUWBYGPUVMUUROUUE
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KLZUWBUVLUURUWDUHUULUVLUUQUUEOKUVLUUEVOJUUQUUEUHUVLUUEAUVKXHXMUUEXIXLST
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UVMUVJUVKUWDUWBUHVPAUVKUULXJOUUEXKXNXOSXPXQXRXSXTYAYB $.
|
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$}
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${
|
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$d A y z $. $d ph y z $. $d M y z $. $d N y z $.
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|
resqrexlemnmsq.n $e |- ( ph -> N e. NN ) $.
|
|
resqrexlemnmsq.m $e |- ( ph -> M e. NN ) $.
|
|
resqrexlemnmsq.nm $e |- ( ph -> N <_ M ) $.
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$( Lemma for ~ resqrex . The difference between the squares of two terms
|
|
of the sequence. (Contributed by Mario Carneiro and Jim Kingdon,
|
|
30-Jul-2021.) $)
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|
resqrexlemnmsq $p |- ( ph ->
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|
( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) <
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|
( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) ) $=
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( c2 cexp co cmin c1 cn wcel cfv c4 cdiv clt resqrexlemf ffvelrnd rpred
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crp resqcld recnd nnncan2d resubcld 1nn a1i cz rpexpcld nnrpd nnzd 1zzd
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wb ltsubrpd cle resqrexlemcalc3 ltletrd eqbrtrrd ) AGEUAZNOPZDQPZFEUAZN
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OPZDQPZQPZVRWAQPREUAZNOPZUBGRQPZOPZUCPZUDAVRWADAVRAVQAVQASUHGEABCDEHIJU
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EZKUFUGUIZUJAWAAVTAVTASUHFEWILUFUGUIZUJADIUJUKAWCVSWHAVSWBAVRDWJIULZAWA
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DWKIULULWLAWHAWEWGAWDNASUHREWIRSTAUMUNUFNUOTAUTUNUPAUBWFAUBUBSTAVAUNUQA
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GRAGKURAUSVBUPVCUGAVSWBWLADWAUDVDZWBUHTZAFSTWMLABCDEFHIJVEVFADVGTWAVGTW
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$( Lemma for ~ resqrex . The difference between two terms of the
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sequence. (Contributed by Mario Carneiro and Jim Kingdon,
|
|
31-Jul-2021.) $)
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|
resqrexlemnm $p |- ( ph -> ( ( F ` N ) - ( F ` M ) ) <
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( ( ( ( F ` 1 ) ^ 2 ) x. 2 ) / ( 2 ^ ( N - 1 ) ) ) ) $=
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( co c1 c2 cexp cdiv cmul wcel cfv cmin c4 clt crp resqrexlemf ffvelrnd
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ltmul1dd lelttrd rpap0d div32apd 4d2e2 oveq1i nnm1nn0 expdivapd syl5eqr
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|
ZUBNZUUHSNZUUIAYSYTAYSAUHUEGEABCDEHIJUFZKUGZUIZAYTAUHUEFEUUPLUGZUIZUJZA
|
|
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|
|
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|
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|
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|
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|
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|
|
UTAUVDUWOYSUDWBKABCDEGHIJXCXEAYTUUSXFXGXHAOUVRUVCUWMAYSYTUURUUTXIZUVHXJ
|
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|
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|
|
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|
|
VQWIAUUGAPUUDAPUVFVAZUVOVFZXNZAYKZXOAUXFUXDPSAUXFUUCUUGUUERNZSNUXDAUUCU
|
|
UEUUGAUUCUVKXNZAUUEUVPXNZUXIAUUEUVPYDZYEAUXBUXKUUCSAUXBOUUEUUGRNZRNUXKA
|
|
UUGUXOORAUUGUCPRNZUUDQNUXOUXPPUUDQYFYGAUCPUUDAUCUVLXNUXJAPUXGYDAUVDUUDY
|
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LTKGYHVLYIYJWLAUUEUUGUXMUXIUXNAUUGUXHYDZYMYNWLXRXSXTAUUCUXBPUXLAUUGUXIU
|
|
XQYOUXJYPYNAUUJUUGAUUCPUXLUXJYQUXIUXQYRXRXM $.
|
|
$}
|
|
|
|
${
|
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$d A y z $. $d F i j k r $. $d F j k n r $. $d F i j x r $.
|
|
$d i j k ph r $. $d k n ph y z $. $d ph x r $.
|
|
$( Lemma for ~ resqrex . The sequence has a limit. (Contributed by Jim
|
|
Kingdon, 6-Aug-2021.) $)
|
|
resqrexlemcvg $p |- ( ph ->
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E. r e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j )
|
|
( ( F ` i ) < ( r + x ) /\ r < ( ( F ` i ) + x ) ) ) $=
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|
( c2 co cn crp a1i wcel clt wbr vk vn c1 cfv cexp cr resqrexlemf rpssre
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|
cmul wss fssd 1nn ffvelrnd cz 2z rpexpcld 2rp rpmulcld cv cdiv caddc wa
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cuz wral cmin wf ad2antrr simplr rpred eluznn adantll resubcld rpdivcld
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nnzd nnrpd cc0 cle eluzle adantl resqrexlemnm cc wceq 2cn expm1t oveq2d
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|
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Jim Kingdon, 8-Aug-2021.) $)
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|
(Contributed by Jim Kingdon, 7-Aug-2021.) $)
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|
|
CXNFUCWDZXRXIXTXKYGWNXFXHTWMXEHVOZVRYGXSXJITYGWNXFXGSYHVSVJVKVMVTVQWA
|
|
WBZWLWCKAUBBCDOUCPUAHWJIJKLMYIWLWEWF $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A r x y z $. $d A n $. $d A e $. $d F a b c e r $.
|
|
$d F b c e k n r $. $d F e k n r y z $. $d b c e k n ph r $.
|
|
$d b e k n ph r z $. $d ph r y z $.
|
|
|
|
$( Lemma for ~ resqrex . Existence of square root given a sequence which
|
|
converges to the square root. (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 27-Jul-2021.) $)
|
|
resqrexlemex $p |- ( ph ->
|
|
E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) ) $=
|
|
( ve vn vc vb cv caddc co clt wbr wa wral vk vr va cfv cuz wrex crp cc0
|
|
cn c2 cexp wceq cr resqrexlemcvg wcel simprl adantr simprr fveq2 breq1d
|
|
cle weq oveq1d breq2d anbi12d cbvralv rexbii raleqdv bitri ralbii oveq2
|
|
cbvrexv rexralbidv sylib resqrexlemgt0 resqrexlemsqa breq2 oveq1 eqeq1d
|
|
rspcev syl12anc rexlimddv ) AUANZFUDZUBNZJNZOPZQRZWEWDWFOPZQRZSZUAKNZUE
|
|
UDZTZKUIUFZJUGTZUHBNZVARZWQUJUKPZEULZSZBUMUFZUBUMAJCDEUAKFUBGHIUNAWEUMU
|
|
OZWPSZSZXCUHWEVARZWEUJUKPZEULZXBAXCWPUPZXECDEUCLMFWEGAEUMUOXDHUQZAUHEVA
|
|
RXDIUQZXIXEWPLNZFUDZWEUCNZOPZQRZWEXMXNOPZQRZSZLMNZUEUDZTMUIUFZUCUGTZAXC
|
|
WPURZWPXMWGQRZWEXMWFOPZQRZSZLYATZMUIUFZJUGTYCWOYJJUGWOYHLWMTZKUIUFYJWNY
|
|
KKUIWKYHUALWMUALVBZWHYEWJYGYLWDXMWGQWCXLFUSZUTYLWIYFWEQYLWDXMWFOYMVCVDV
|
|
EVFVGYKYIKMUIKMVBYHLWMYAWLXTUEUSVHVLVIVJYJYBJUCUGJUCVBZYHXSMLUIYAYNYEXP
|
|
YGXRYNWGXOXMQWFXNWEOVKVDYNYFXQWEQWFXNXMOVKVDVEVMVFVIVNVOXECDEJUAKFWEGXJ
|
|
XKXIYDVPXAXFXHSBWEUMBUBVBZWRXFWTXHWQWEUHVAVQYOWSXGEWQWEUJUKVRVSVEVTWAWB
|
|
$.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A a b x y z $.
|
|
$( Existence of a square root for positive reals. (Contributed by Mario
|
|
Carneiro, 9-Jul-2013.) $)
|
|
resqrex $p |- ( ( A e. RR /\ 0 <_ A ) ->
|
|
E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) ) $=
|
|
( va vb vy vz cr wcel cc0 cle crp cv cdiv co caddc c2 cmpt2 c1 cseq wceq
|
|
wbr wa cn csn cxp oveq2 oveq12d oveq1d eqidd cbvmpt2v iseqeq2 ax-mp simpl
|
|
id simpr resqrexlemex ) BGHZIBJUAZUBACDBEFKKELZBUSMNZONZPMNZQZKUCRBONUDUE
|
|
ZRSZVCCDKKCLZBVFMNZONZPMNZQZTVEVJKVDRSTEFCDKKVBVIVIUSVFTZVAVHPMVKUSVFUTVG
|
|
OVKUNUSVFBMUFUGUHFLDLTVIUIUJVCVJKVDRUKULUQURUMUQURUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Uniqueness for the square root function. (Contributed by Jim Kingdon,
|
|
10-Aug-2021.) $)
|
|
rsqrmo $p |- ( ( A e. RR /\ 0 <_ A ) -> E* x e. RR ( ( x ^ 2 ) = A /\
|
|
0 <_ x ) ) $=
|
|
( vy cr wcel cc0 cle wbr wa cv c2 cexp co wceq wi simplrl simplrr simprlr
|
|
wral wrmo simprrr simprll simprrl eqtr4d sq11d ex ralrimivva oveq1 eqeq1d
|
|
breq2 anbi12d rmo4 sylibr ) BDEFBGHIZAJZKLMZBNZFUOGHZIZCJZKLMZBNZFUTGHZIZ
|
|
IZUOUTNZOZCDSADSUSADTUNVGACDDUNUODEZUTDEZIIZVEVFVJVEIZUOUTUNVHVIVEPUNVHVI
|
|
VEQVJUQURVDRVJUSVBVCUAVKUPBVAVJUQURVDUBVJUSVBVCUCUDUEUFUGUSVDACDVFUQVBURV
|
|
CVFUPVABUOUTKLUHUIUOUTFGUJUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Existence and uniqueness for the real square root function.
|
|
(Contributed by Jim Kingdon, 10-Aug-2021.) $)
|
|
rersqreu $p |- ( ( A e. RR /\ 0 <_ A ) -> E! x e. RR ( ( x ^ 2 ) = A /\
|
|
0 <_ x ) ) $=
|
|
( cr wcel cc0 cle wbr wa cv c2 cexp co wceq wrex wrmo wreu resqrex rexbii
|
|
ancom sylib rsqrmo reu5 sylanbrc ) BCDEBFGHZAIZJKLBMZEUEFGZHZACNZUHACOUHA
|
|
CPUDUGUFHZACNUIABQUJUHACUGUFSRTABUAUHACUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Closure of the square root function. (Contributed by Mario Carneiro,
|
|
9-Jul-2013.) $)
|
|
resqrtcl $p |- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR ) $=
|
|
( vy vx cr wcel cc0 cle wbr wa cv c2 cexp wceq wrex csqrt cfv resqrex w3a
|
|
co crio simp1l sqrtrval syl simp3r simp3l wreu wb simp2 rersqreu 3ad2ant1
|
|
oveq1 eqeq1d breq2 anbi12d riota2 syl2anc mpbi2and eqeltrd rexlimdv3a mpd
|
|
eqtrd ) ADEZFAGHZIZFBJZGHZVEKLSZAMZIZBDNAOPZDEZBAQVDVIVKBDVDVEDEZVIRZVJVE
|
|
DVMVJCJZKLSZAMZFVNGHZIZCDTZVEVMVBVJVSMVBVCVLVIUACAUBUCVMVHVFVSVEMZVDVLVFV
|
|
HUDVDVLVFVHUEVMVLVRCDUFZVHVFIZVTUGVDVLVIUHZVDVLWAVICAUIUJVRWBCDVEVNVEMZVP
|
|
VHVQVFWDVOVGAVNVEKLUKULVNVEFGUMUNUOUPUQVAWCURUSUT $.
|
|
|
|
$( Lemma for ~ resqrtth . (Contributed by Jim Kingdon, 10-Aug-2021.) $)
|
|
rersqrtthlem $p |- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqrt ` A ) ^ 2 ) = A /\
|
|
0 <_ ( sqrt ` A ) ) ) $=
|
|
( vx cr wcel cc0 cle wbr wa csqrt c2 cexp co wceq cv crio sqrtrval eqcomd
|
|
cfv adantr wreu wb resqrtcl rersqreu eqeq1d anbi12d riota2 syl2anc mpbird
|
|
oveq1 breq2 ) ACDZEAFGZHZAIRZJKLZAMZEUNFGZHZBNZJKLZAMZEUSFGZHZBCOZUNMZUKV
|
|
EULUKUNVDBAPQSUMUNCDVCBCTURVEUAAUBBAUCVCURBCUNUSUNMZVAUPVBUQVFUTUOAUSUNJK
|
|
UIUDUSUNEFUJUEUFUGUH $.
|
|
$}
|
|
|
|
$( Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29.
|
|
(Contributed by Mario Carneiro, 9-Jul-2013.) $)
|
|
resqrtth $p |- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) ^ 2 ) = A ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co wceq rersqrtthlem simpld ) AB
|
|
CDAEFGAHIZJKLAMDPEFANO $.
|
|
|
|
$( Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.) $)
|
|
remsqsqrt $p |- ( ( A e. RR /\ 0 <_ A ) ->
|
|
( ( sqrt ` A ) x. ( sqrt ` A ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co cmul resqrtcl sqvald resqrtth
|
|
recnd eqtr3d ) ABCDAEFGZAHIZJKLTTMLASTSTANQOAPR $.
|
|
|
|
$( The square root function is nonnegative for nonnegative input.
|
|
(Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro,
|
|
9-Jul-2013.) $)
|
|
sqrtge0 $p |- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqrt ` A ) ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co wceq rersqrtthlem simprd ) AB
|
|
CDAEFGAHIZJKLAMDPEFANO $.
|
|
|
|
$( The square root function is positive for positive input. (Contributed by
|
|
Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtgt0 $p |- ( ( A e. RR /\ 0 < A ) -> 0 < ( sqrt ` A ) ) $=
|
|
( cr wcel cc0 clt wbr wa csqrt cfv c2 cexp co simpr wceq sq0 a1i 0red simpl
|
|
cle syldan ltled resqrtth 3brtr4d resqrtcl leidd sqrtge0 lt2sqd mpbird ) AB
|
|
CZDAEFZGZDAHIZEFDJKLZULJKLZEFUKDAUMUNEUIUJMZUMDNUKOPUIUJDASFZUNANUKDAUKQZUI
|
|
UJRUOUAZAUBTUCUKDULUQUIUJUPULBCURAUDTUKDUQUEUIUJUPDULSFURAUFTUGUH $.
|
|
|
|
$( Square root distributes over multiplication. (Contributed by NM,
|
|
30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) $)
|
|
sqrtmul $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( sqrt ` ( A x. B ) ) = ( ( sqrt ` A ) x. ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa cmul co csqrt cfv remulcld resqrtcl syl2anc adantr
|
|
sqrtge0 c2 cexp resqrtth simpll simprl mulge0 adantl oveqan12d recnd sqmuld
|
|
mulge0d wceq 3eqtr4rd sq11d ) ACDZEAFGZHZBCDZEBFGZHZHZABIJZKLZAKLZBKLZIJZUR
|
|
USCDZEUSFGZUTCDURABULUMUQUAUNUOUPUBMZABUCZUSNOURVAVBUNVACDUQANPZUQVBCDUNBNU
|
|
DZMURVDVEEUTFGVFVGUSQOURVAVBVHVIUNEVAFGUQAQPUQEVBFGUNBQUDUHURVARSJZVBRSJZIJ
|
|
USVCRSJUTRSJZUNUQVJAVKBIATBTUEURVAVBURVAVHUFURVBVIUFUGURVDVEVLUSUIVFVGUSTOU
|
|
JUK $.
|
|
|
|
$( Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof
|
|
shortened by Mario Carneiro, 29-May-2016.) $)
|
|
sqrtle $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A <_ B <-> ( sqrt ` A ) <_ ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co wb resqrtcl sqrtge0 jca le2sq
|
|
syl2an resqrtth breqan12d bitr2d ) ACDEAFGHZBCDEBFGHZHAIJZBIJZFGZUEKLMZUFKL
|
|
MZFGZABFGUCUECDZEUEFGZHUFCDZEUFFGZHUGUJNUDUCUKULAOAPQUDUMUNBOBPQUEUFRSUCUDU
|
|
HAUIBFATBTUAUB $.
|
|
|
|
$( Square root is strictly monotonic. Closed form of ~ sqrtlti .
|
|
(Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
sqrtlt $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( A < B <-> ( sqrt ` A ) < ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv clt c2 cexp co wb resqrtcl sqrtge0 lt2sq
|
|
jca resqrtth syl2an breqan12d bitr2d ) ACDEAFGHZBCDEBFGHZHAIJZBIJZKGZUFLMNZ
|
|
UGLMNZKGZABKGUDUFCDZEUFFGZHUGCDZEUGFGZHUHUKOUEUDULUMAPAQSUEUNUOBPBQSUFUGRUA
|
|
UDUEUIAUJBKATBTUBUC $.
|
|
|
|
$( Analogue to ~ sqrt11 but for apartness. (Contributed by Jim Kingdon,
|
|
11-Aug-2021.) $)
|
|
sqrt11ap $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( sqrt ` A ) # ( sqrt ` B ) <-> A # B ) ) $=
|
|
( cr wcel cc0 cle wbr wa clt wo csqrt cfv sqrtlt wb ancoms orbi12d ad2ant2r
|
|
cap reaplt resqrtcl syl2an 3bitr4rd ) ACDZEAFGZHZBCDZEBFGZHZHZABIGZBAIGZJZA
|
|
KLZBKLZIGZUNUMIGZJZABRGZUMUNRGZUIUJUOUKUPABMUHUEUKUPNBAMOPUCUFURULNUDUGABSQ
|
|
UEUMCDUNCDUSUQNUHATBTUMUNSUAUB $.
|
|
|
|
$( The square root function is one-to-one. Also see ~ sqrt11ap which would
|
|
follow easily from this given excluded middle, but which is proved another
|
|
way without it. (Contributed by Scott Fenton, 11-Jun-2013.) $)
|
|
sqrt11 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co wceq wb resqrtcl sqrtge0 sq11
|
|
jca resqrtth syl2an eqeqan12d bitr3d ) ACDEAFGHZBCDEBFGHZHAIJZKLMZBIJZKLMZN
|
|
ZUFUHNZABNUDUFCDZEUFFGZHUHCDZEUHFGZHUJUKOUEUDULUMAPAQSUEUNUOBPBQSUFUHRUAUDU
|
|
EUGAUIBATBTUBUC $.
|
|
|
|
$( A square root is zero iff its argument is 0. (Contributed by NM,
|
|
27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.) $)
|
|
sqrt00 $p |- ( ( A e. RR /\ 0 <_ A ) ->
|
|
( ( sqrt ` A ) = 0 <-> A = 0 ) ) $=
|
|
( csqrt cfv cc0 wceq cr wcel cle wbr wa sqrt0 eqeq2i wb 0le0 sqrt11 mpanr12
|
|
0re syl5bbr ) ABCZDESDBCZEZAFGDAHIJZADEZTDSKLUBDFGDDHIUAUCMQNADOPR $.
|
|
|
|
$( The square root of a positive real is a positive real. (Contributed by
|
|
NM, 22-Feb-2008.) $)
|
|
rpsqrtcl $p |- ( A e. RR+ -> ( sqrt ` A ) e. RR+ ) $=
|
|
( crp wcel csqrt cfv cr cc0 cle wbr rpre rpge0 resqrtcl syl2anc clt sqrtgt0
|
|
rpgt0 elrpd ) ABCZADEZRAFCZGAHISFCAJZAKALMRTGANIGSNIUAAPAOMQ $.
|
|
|
|
$( Square root distributes over division. (Contributed by Mario Carneiro,
|
|
5-May-2016.) $)
|
|
sqrtdiv $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) ->
|
|
( sqrt ` ( A / B ) ) = ( ( sqrt ` A ) / ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa crp cdiv co csqrt cfv cmul rerpdivcl adantlr recnd
|
|
clt adantl eqtr3d divge0 sylan2b resqrtcl syl2anc rpsqrtcl rpcnd divcanap4d
|
|
elrp rpap0d wceq rprege0 sqrtmul syl21anc simpll rpcn cap divcanap1d fveq2d
|
|
cc rpap0 oveq1d ) ACDZEAFGZHZBIDZHZABJKZLMZBLMZNKZVIJKVHALMZVIJKVFVHVIVFVHV
|
|
FVGCDZEVGFGZVHCDVBVEVLVCABOPZVEVDBCDZEBRGHVMBUHABUAUBZVGUCUDQVFVIVEVIIDVDBU
|
|
ESZUFVFVIVQUIUGVFVJVKVIJVFVGBNKZLMZVJVKVFVLVMVOEBFGHZVSVJUJVNVPVEVTVDBUKSVG
|
|
BULUMVFVRALVFABVFAVBVCVEUNQVEBUSDVDBUOSVEBEUPGVDBUTSUQURTVAT $.
|
|
|
|
$( Relationship between square root and squares. (Contributed by NM,
|
|
31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) $)
|
|
sqrtsq2 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr wa csqrt cfv c2 cexp co wceq wb resqrtcl sqrtge0 sq11
|
|
jca sylan resqrtth adantr eqeq1d bitr3d ) ACDEAFGHZBCDEBFGHZHZAIJZKLMZBKLMZ
|
|
NZUHBNZAUJNUEUHCDZEUHFGZHUFUKULOUEUMUNAPAQSUHBRTUGUIAUJUEUIANUFAUAUBUCUD $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
sqrtsq $p |- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A ^ 2 ) ) = A ) $=
|
|
( vx cr wcel cc0 cle wbr wa c2 cexp co csqrt cfv wceq crio simpl sqrtrval
|
|
cv resqcld simplr syl simplll simprr simpllr simprl sq11d ex wi oveq1 a1i
|
|
breq2 syl5ibrcom jcad impbid riota5 eqtrd ) ACDZEAFGZHZAIJKZLMZBRZIJKUTNZ
|
|
EVBFGZHZBCOZAUSUTCDVAVFNUSAUQURPZSBUTQUAUSVEBCAVGUSVBCDZHZVEVBANZVIVEVJVI
|
|
VEHVBAUSVHVETUQURVHVEUBVIVCVDUCUQURVHVEUDVIVCVDUEUFUGVIVJVCVDVJVCUHVIVBAI
|
|
JUIUJVIVDVJURUQURVHTVBAEFUKULUMUNUOUP $.
|
|
$}
|
|
|
|
$( Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
sqrtmsq $p |- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A x. A ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr wa c2 cexp csqrt cfv simpl recnd sqvald fveq2d sqrtsq
|
|
co cmul eqtr3d ) ABCZDAEFZGZAHIQZJKAARQZJKAUBUCUDJUBAUBATUALMNOAPS $.
|
|
|
|
$( The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.) $)
|
|
sqrt1 $p |- ( sqrt ` 1 ) = 1 $=
|
|
( c1 c2 cexp co csqrt cfv sq1 fveq2i wcel cc0 cle wbr wceq 1re sqrtsq mp2an
|
|
cr 0le1 eqtr3i ) ABCDZEFZAEFATAEGHAQIJAKLUAAMNRAOPS $.
|
|
|
|
$( The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.) $)
|
|
sqrt4 $p |- ( sqrt ` 4 ) = 2 $=
|
|
( c2 cexp co csqrt cfv c4 sq2 fveq2i wcel cc0 cle wbr wceq 2re sqrtsq mp2an
|
|
cr 0le2 eqtr3i ) AABCZDEZFDEATFDGHAQIJAKLUAAMNRAOPS $.
|
|
|
|
$( The square root of 9 is 3. (Contributed by NM, 11-May-2004.) $)
|
|
sqrt9 $p |- ( sqrt ` 9 ) = 3 $=
|
|
( c3 c2 cexp co csqrt cfv c9 sq3 fveq2i cr wcel cc0 cle wbr wceq 3re ltleii
|
|
0re 3pos sqrtsq mp2an eqtr3i ) ABCDZEFZGEFAUCGEHIAJKLAMNUDAOPLARPSQATUAUB
|
|
$.
|
|
|
|
$( The square root of 2 is bounded by 1 and 2. (Contributed by Roy F.
|
|
Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrt2gt1lt2 $p |- ( 1 < ( sqrt ` 2 ) /\ ( sqrt ` 2 ) < 2 ) $=
|
|
( c1 c2 csqrt cfv clt wbr sqrt1 1lt2 cr wcel cc0 cle wb 2re 0le2 mp4an mpbi
|
|
sqrtlt c4 4re 1re 0le1 eqbrtrri 2lt4 0re 4pos ltleii sqrt4 breqtri pm3.2i )
|
|
ABCDZEFUKBEFACDZAUKEGABEFZULUKEFZHAIJKALFBIJZKBLFZUMUNMUAUBNOABRPQUCUKSCDZB
|
|
EBSEFZUKUQEFZUDUOUPSIJKSLFURUSMNOTKSUETUFUGBSRPQUHUIUJ $.
|
|
|
|
$( Absolute value of negative. (Contributed by NM, 27-Feb-2005.) $)
|
|
absneg $p |- ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) ) $=
|
|
( cc wcel cneg ccj cfv cmul csqrt cabs cjneg oveq2d wceq cjcl mul2neg mpdan
|
|
co eqtrd fveq2d negcl absval syl 3eqtr4d ) ABCZADZUDEFZGPZHFZAAEFZGPZHFUDIF
|
|
ZAIFUCUFUIHUCUFUDUHDZGPZUIUCUEUKUDGAJKUCUHBCULUILAMAUHNOQRUCUDBCUJUGLASUDTU
|
|
AATUB $.
|
|
|
|
$( Real closure of absolute value. (Contributed by NM, 3-Oct-1999.) $)
|
|
abscl $p |- ( A e. CC -> ( abs ` A ) e. RR ) $=
|
|
( cc wcel cabs cfv ccj cmul csqrt absval cc0 cle cjmulrcl cjmulge0 resqrtcl
|
|
co cr wbr syl2anc eqeltrd ) ABCZADEAAFEGOZHEZPAITUAPCJUAKQUBPCALAMUANRS $.
|
|
|
|
$( The absolute value of a number and its conjugate are the same.
|
|
Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM,
|
|
28-Apr-2005.) $)
|
|
abscj $p |- ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) ) $=
|
|
( cc wcel ccj cfv cabs cmul co csqrt wceq cjcl absval syl mulcom mpdan cjcj
|
|
oveq2d eqtr4d fveq2d ) ABCZADEZFEZAUAGHZIEZAFETUBUAUADEZGHZIEZUDTUABCZUBUGJ
|
|
AKZUALMTUCUFITUCUAAGHZUFTUHUCUJJUIAUANOTUEAUAGAPQRSRALR $.
|
|
|
|
$( Square of value of absolute value function. (Contributed by NM,
|
|
16-Jan-2006.) $)
|
|
absvalsq $p |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) ) $=
|
|
( cc wcel cabs cfv c2 cexp co ccj cmul csqrt absval oveq1d cc0 cle wbr wceq
|
|
cr cjmulrcl cjmulge0 resqrtth syl2anc eqtrd ) ABCZADEZFGHAAIEJHZKEZFGHZUFUD
|
|
UEUGFGALMUDUFRCNUFOPUHUFQASATUFUAUBUC $.
|
|
|
|
$( Square of value of absolute value function. (Contributed by NM,
|
|
1-Feb-2007.) $)
|
|
absvalsq2 $p |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel cabs cfv c2 cexp ccj cmul cre cim caddc absvalsq cjmulval eqtrd
|
|
co ) ABCADEFGPAAHEIPAJEFGPAKEFGPLPAMANO $.
|
|
|
|
$( Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason]
|
|
p. 133. (Contributed by NM, 21-Jan-2007.) $)
|
|
sqabsadd $p |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) =
|
|
( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) )
|
|
+ ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) ) $=
|
|
( cc wcel wa caddc co ccj cmul cabs c2 cexp oveq2d wceq cjcl mpdan absvalsq
|
|
cfv eqtrd sylan2 cre cjadd anim12i muladd addcl syl mulcom oveqan12d addcjd
|
|
mulcl cjmul cjcj adantl eqtr3d oveq12d 3eqtr4d ) ACDZBCDZEZABFGZUTHRZIGZAAH
|
|
RZIGZBHRZBIGZFGZAVEIGZVCBIGZFGZFGZUTJRKLGZAJRKLGZBJRKLGZFGZKVHUARIGZFGUSVBU
|
|
TVCVEFGZIGZVKUSVAVQUTIABUBMUSVCCDZVECDZEVRVKNUQVSURVTAOBOZUCABVCVEUDPSUSUTC
|
|
DVLVBNABUEUTQUFUSVOVGVPVJFUQURVMVDVNVFFAQURVNBVEIGZVFBQURVTWBVFNWABVEUGPSUH
|
|
USVHVHHRZFGVPVJUSVHURUQVTVHCDWAAVEUJTUIUSWCVIVHFUSWCVCVEHRZIGZVIURUQVTWCWEN
|
|
WAAVEUKTUSWDBVCIURWDBNUQBULUMMSMUNUOUP $.
|
|
|
|
$( Square of absolute value of difference. (Contributed by NM,
|
|
21-Jan-2007.) $)
|
|
sqabssub $p |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) =
|
|
( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) )
|
|
- ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) ) $=
|
|
( cc wcel wa cmin co ccj cfv cmul caddc cabs c2 cexp oveq2d wceq cjcl mpdan
|
|
eqtrd absvalsq cre cjsub anim12i mulsub subcl mulcom oveqan12d mulcl sylan2
|
|
syl addcjd cjmul cjcj adantl eqtr3d oveq12d 3eqtr4d ) ACDZBCDZEZABFGZVAHIZJ
|
|
GZAAHIZJGZBHIZBJGZKGZAVFJGZVDBJGZKGZFGZVALIMNGZALIMNGZBLIMNGZKGZMVIUAIJGZFG
|
|
UTVCVAVDVFFGZJGZVLUTVBVRVAJABUBOUTVDCDZVFCDZEVSVLPURVTUSWAAQBQZUCABVDVFUDRS
|
|
UTVACDVMVCPABUEVATUJUTVPVHVQVKFURUSVNVEVOVGKATUSVOBVFJGZVGBTUSWAWCVGPWBBVFU
|
|
FRSUGUTVIVIHIZKGVQVKUTVIUSURWAVICDWBAVFUHUIUKUTWDVJVIKUTWDVDVFHIZJGZVJUSURW
|
|
AWDWFPWBAVFULUIUTWEBVDJUSWEBPURBUMUNOSOUOUPUQ $.
|
|
|
|
$( Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
|
|
(Contributed by NM, 17-Mar-2005.) $)
|
|
absval2 $p |- ( A e. CC -> ( abs ` A ) =
|
|
( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) ) $=
|
|
( cc wcel cabs cfv ccj cmul co csqrt cre c2 cexp cim absval cjmulval fveq2d
|
|
caddc eqtrd ) ABCZADEAAFEGHZIEAJEKLHAMEKLHQHZIEANSTUAIAOPR $.
|
|
|
|
$( The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
abs0 $p |- ( abs ` 0 ) = 0 $=
|
|
( cc0 cabs cfv ccj cmul co csqrt cc wcel wceq 0cn absval ax-mp cjcli mul02i
|
|
fveq2i sqrt0 3eqtri ) ABCZAADCZEFZGCZAGCAAHISUBJKALMUAAGTAKNOPQR $.
|
|
|
|
$( The absolute value of the imaginary unit. (Contributed by NM,
|
|
26-Mar-2005.) $)
|
|
absi $p |- ( abs ` _i ) = 1 $=
|
|
( ci cabs cfv cmul co csqrt c1 cc wcel wceq ax-icn absval ax-mp cneg oveq2i
|
|
ccj cji mulneg2i ixi 3eqtri negeqi negneg1e1 eqtri fveq2i sqrt1 ) ABCZAAPCZ
|
|
DEZFCZGFCGAHIUFUIJKALMUHGFUHAANZDEAADEZNZGUGUJADQOAAKKRULGNZNGUKUMSUAUBUCTU
|
|
DUET $.
|
|
|
|
$( Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.)
|
|
(Revised by Mario Carneiro, 29-May-2016.) $)
|
|
absge0 $p |- ( A e. CC -> 0 <_ ( abs ` A ) ) $=
|
|
( cc wcel cc0 ccj cfv cmul csqrt cabs cle cjmulrcl cjmulge0 sqrtge0 syl2anc
|
|
co cr wbr absval breqtrrd ) ABCZDAAEFGOZHFZAIFJTUAPCDUAJQDUBJQAKALUAMNARS
|
|
$.
|
|
|
|
$( The absolute value of a number apart from zero is a positive real.
|
|
(Contributed by Jim Kingdon, 11-Aug-2021.) $)
|
|
absrpclap $p |- ( ( A e. CC /\ A # 0 ) -> ( abs ` A ) e. RR+ ) $=
|
|
( cc wcel cc0 cap wbr wa cabs cfv ccj cmul co csqrt crp absval adantr simpl
|
|
wceq cjmulrcld cjmulge0d cjcld simpr mulap0d ap0gt0d elrpd rpsqrtcl eqeltrd
|
|
cjap0d syl ) ABCZADEFZGZAHIZAAJIZKLZMIZNUJUMUPRUKAOPULUONCUPNCULUOULAUJUKQZ
|
|
SZULUOURULAUQTULAUNUQULAUQUAUJUKUBZULAUQUSUHUCUDUEUOUFUIUG $.
|
|
|
|
$( The absolute value of a number is apart from zero iff the number is apart
|
|
from zero. (Contributed by Jim Kingdon, 11-Aug-2021.) $)
|
|
abs00ap $p |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) ) $=
|
|
( cc wcel cfv cc0 cap wbr co caddc wo c2 csqrt breq1d breq2i syl6bbr cr cle
|
|
cexp wb recnd cabs cre ci cim cmul absval2 sqrt0 recl resqcld imcl readdcld
|
|
sqge0d addge0d 0red leidd sqrt11ap syl22anc bitrd 00id wi addext sylbid 2nn
|
|
0cnd cn expap0 sylancl orbi12d sylibd crap0 syl2anc replim absrpclap rpap0d
|
|
sylibrd wa ex impbid ) ABCZAUADZEFGZAEFGZVSWAAUBDZUCAUDDZUEHIHZEFGZWBVSWAWC
|
|
EFGZWDEFGZJZWFVSWAWCKRHZEFGZWDKRHZEFGZJZWIVSWAWJWLIHZEEIHZFGZWNVSWAWOEFGZWQ
|
|
VSWAWOLDZELDZFGZWRVSWAWSEFGXAVSVTWSEFAUFMWTEWSFUGNOVSWOPCEWOQGEPCEEQGXAWRSV
|
|
SWJWLVSWCAUHZUIZVSWDAUJZUIZUKVSWJWLXCXEVSWCXBULVSWDXDULUMVSUNZVSEXFUOWOEUPU
|
|
QURWPEWOFUSNOVSWJBCWLBCEBCZXGWQWNUTVSWJXCTVSWLXETVSVDZXHWJWLEEVAUQVBVSWKWGW
|
|
MWHVSWCBCKVECZWKWGSVSWCXBTVCWCKVFVGVSWDBCXIWMWHSVSWDXDTVCWDKVFVGVHVIVSWCPCW
|
|
DPCWIWFSXBXDWCWDVJVKVIVSAWEEFAVLMVOVSWBWAVSWBVPVTAVMVNVQVR $.
|
|
|
|
$( Strong extensionality for absolute value. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
absext $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( abs ` A ) # ( abs ` B ) -> A # B ) ) $=
|
|
( cc wcel cfv cap cmul co caddc wo c2 cexp cr resqcld sqge0d syl22anc recnd
|
|
wbr wi sqvald wa cabs cre ci cim csqrt absval2 breqan12d cc0 wb simpl recld
|
|
cle imcld readdcld addge0d simpr sqrt11ap bitrd addext sylbid breq12d oridm
|
|
mulext syl6ib orim12d syld apreim sylibrd replimd ) ACDZBCDZUAZAUBEZBUBEZFR
|
|
ZAUCEZUDAUEEZGHIHZBUCEZUDBUEEZGHIHZFRZABFRVMVPVQVTFRZVRWAFRZJZWCVMVPVQKLHZV
|
|
TKLHZFRZVRKLHZWAKLHZFRZJZWFVMVPWGWJIHZWHWKIHZFRZWMVMVPWNUFEZWOUFEZFRZWPVKVL
|
|
VNWQVOWRFAUGBUGUHVMWNMDUIWNUMRWOMDUIWOUMRWSWPUJVMWGWJVMVQVMAVKVLUKZULZNZVMV
|
|
RVMAWTUNZNZUOVMWGWJXBXDVMVQXAOVMVRXCOUPVMWHWKVMVTVMBVKVLUQZULZNZVMWAVMBXEUN
|
|
ZNZUOVMWHWKXGXIVMVTXFOVMWAXHOUPWNWOURPUSVMWGCDWJCDWHCDWKCDWPWMSVMWGXBQVMWJX
|
|
DQVMWHXGQVMWKXIQWGWJWHWKUTPVAVMWIWDWLWEVMWIWDWDJZWDVMWIVQVQGHZVTVTGHZFRZXJV
|
|
MWGXKWHXLFVMVQVMVQXAQZTVMVTVMVTXFQZTVBVMVQCDZXPVTCDZXQXMXJSXNXNXOXOVQVQVTVT
|
|
VDPVAWDVCVEVMWLWEWEJZWEVMWLVRVRGHZWAWAGHZFRZXRVMWJXSWKXTFVMVRVMVRXCQZTVMWAV
|
|
MWAXHQZTVBVMVRCDZYDWACDZYEYAXRSYBYBYCYCVRVRWAWAVDPVAWEVCVEVFVGVMVQMDVRMDVTM
|
|
DWAMDWCWFUJXAXCXFXHVQVRVTWAVHPVIVMAVSBWBFVMAWTVJVMBXEVJVBVI $.
|
|
|
|
$( The absolute value of a number is zero iff the number is zero. Also see
|
|
~ abs00ap which is similar but for apartness. Proposition 10-3.7(c) of
|
|
[Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
abs00 $p |- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel cabs cfv cc0 cap wbr wn abs00ap notbid wb abscl recnd 0cn sylancl
|
|
wceq apti mpan2 3bitr4d ) ABCZADEZFGHZIZAFGHZIZUBFQZAFQZUAUCUEAJKUAUBBCFBCZ
|
|
UGUDLUAUBAMNOUBFRPUAUIUHUFLOAFRST $.
|
|
|
|
${
|
|
abs00ad.1 $e |- ( ph -> A e. CC ) $.
|
|
$( A complex number is zero iff its absolute value is zero. Deduction form
|
|
of ~ abs00 . (Contributed by David Moews, 28-Feb-2017.) $)
|
|
abs00ad $p |- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) ) $=
|
|
( cc wcel cabs cfv cc0 wceq wb abs00 syl ) ABDEBFGHIBHIJCBKL $.
|
|
$}
|
|
|
|
${
|
|
abs00bd.1 $e |- ( ph -> A = 0 ) $.
|
|
$( If a complex number is zero, its absolute value is zero. (Contributed
|
|
by David Moews, 28-Feb-2017.) $)
|
|
abs00bd $p |- ( ph -> ( abs ` A ) = 0 ) $=
|
|
( cc0 wceq cabs cfv fveq2 abs0 syl6eq syl ) ABDEZBFGZDECLMDFGDBDFHIJK $.
|
|
$}
|
|
|
|
$( Square of the absolute value of a number that has been decomposed into
|
|
real and imaginary parts. (Contributed by NM, 1-Feb-2007.) $)
|
|
absreimsq $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( abs ` ( A + ( _i x. B ) ) ) ^ 2 ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc cabs cfv cexp cre cim wceq recn ax-icn oveq1d
|
|
c2 cc mulcl sylancr addcl syl2an absvalsq2 syl crre crim oveq12d eqtrd ) AC
|
|
DZBCDZEZAFBGHZIHZJKSLHZUOMKZSLHZUONKZSLHZIHZASLHZBSLHZIHUMUOTDZUPVAOUKATDUN
|
|
TDZVDULAPULFTDBTDVEQBPFBUAUBAUNUCUDUOUEUFUMURVBUTVCIUMUQASLABUGRUMUSBSLABUH
|
|
RUIUJ $.
|
|
|
|
$( Absolute value of a number that has been decomposed into real and
|
|
imaginary parts. (Contributed by NM, 14-Jan-2006.) $)
|
|
absreim $p |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A + ( _i x. B ) ) ) =
|
|
( sqrt ` ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) $=
|
|
( cr wcel wa ci cmul co caddc cabs cfv c2 cexp csqrt cc0 cle wbr recn syl
|
|
cc ax-icn mulcl sylancr addcl syl2an absge0 sqrtsq syl2anc absreimsq fveq2d
|
|
wceq abscl eqtr3d ) ACDZBCDZEZAFBGHZIHZJKZLMHZNKZUSALMHBLMHIHZNKUPUSCDZOUSP
|
|
QZVAUSUKUPURTDZVCUNATDUQTDZVEUOARUOFTDBTDVFUABRFBUBUCAUQUDUEZURULSUPVEVDVGU
|
|
RUFSUSUGUHUPUTVBNABUIUJUM $.
|
|
|
|
$( Absolute value distributes over multiplication. Proposition 10-3.7(f) of
|
|
[Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absmul $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) ) $=
|
|
( cc wcel wa cmul co ccj cfv csqrt cabs cjcld eqtrd cr cc0 cle wbr cjmulrcl
|
|
wceq absval cjmul oveq2d simpl simpr mul4d fveq2d cjmulge0 jca syl2an mulcl
|
|
sqrtmul syl oveqan12d 3eqtr4d ) ACDZBCDZEZABFGZURHIZFGZJIZAAHIZFGZJIZBBHIZF
|
|
GZJIZFGZURKIZAKIZBKIZFGUQVAVCVFFGZJIZVHUQUTVLJUQUTURVBVEFGZFGVLUQUSVNURFABU
|
|
AUBUQABVBVEUOUPUCZUOUPUDZUQAVOLUQBVPLUEMUFUOVCNDZOVCPQZEVFNDZOVFPQZEVMVHSUP
|
|
UOVQVRARAUGUHUPVSVTBRBUGUHVCVFUKUIMUQURCDVIVASABUJURTULUOUPVJVDVKVGFATBTUMU
|
|
N $.
|
|
|
|
$( Absolute value distributes over division. (Contributed by Jim Kingdon,
|
|
11-Aug-2021.) $)
|
|
absdivap $p |- ( ( A e. CC /\ B e. CC /\ B # 0 ) ->
|
|
( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr w3a cdiv co cabs cfv cmul divclap abscl syl recnd crp
|
|
cr eqtr3d absrpclap 3adant1 rpcnd rpap0d divcanap4d simp2 syl2anc divcanap1
|
|
wceq absmul fveq2d oveq1d ) ACDZBCDZBEFGZHZABIJZKLZBKLZMJZUSIJURAKLZUSIJUPU
|
|
RUSUPURUPUQCDZURSDABNZUQOPQUPUSUNUOUSRDUMBUAUBZUCUPUSVDUDUEUPUTVAUSIUPUQBMJ
|
|
ZKLZUTVAUPVBUNVFUTUIVCUMUNUOUFUQBUJUGUPVEAKABUHUKTULT $.
|
|
|
|
$( A nonnegative number is its own absolute value. (Contributed by NM,
|
|
11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) $)
|
|
absid $p |- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A ) $=
|
|
( cr wcel cc0 cle wbr wa cabs cfv cmul co csqrt c2 cexp cc wceq simpl recnd
|
|
ccj absval syl cjred oveq2d sqvald eqtr4d fveq2d sqrtsq 3eqtrd ) ABCZDAEFZG
|
|
ZAHIZAASIZJKZLIZAMNKZLIAUKAOCULUOPUKAUIUJQZRZATUAUKUNUPLUKUNAAJKUPUKUMAAJUK
|
|
AUQUBUCUKAURUDUEUFAUGUH $.
|
|
|
|
$( The absolute value of 1. Common special case. (Contributed by David A.
|
|
Wheeler, 16-Jul-2016.) $)
|
|
abs1 $p |- ( abs ` 1 ) = 1 $=
|
|
( c1 cr wcel cc0 cle wbr cabs cfv wceq 1re 0le1 absid mp2an ) ABCDAEFAGHAIJ
|
|
KALM $.
|
|
|
|
$( A negative number is the negative of its own absolute value. (Contributed
|
|
by NM, 27-Feb-2005.) $)
|
|
absnid $p |- ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A ) $=
|
|
( cr wcel cc0 cle wbr cabs cfv cneg wceq le0neg1 wa recn absneg syl renegcl
|
|
cc adantr absid sylan eqtr3d ex sylbid imp ) ABCZADEFZAGHZAIZJZUEUFDUHEFZUI
|
|
AKUEUJUIUEUJLUHGHZUGUHUEUKUGJZUJUEAQCULAMANORUEUHBCUJUKUHJAPUHSTUAUBUCUD $.
|
|
|
|
$( A real number is less than or equal to its absolute value. (Contributed
|
|
by NM, 27-Feb-2005.) $)
|
|
leabs $p |- ( A e. RR -> A <_ ( abs ` A ) ) $=
|
|
( cr wcel cabs cfv cle wbr clt wn wa cc0 wfal simpr cc recn ad2antrr lenltd
|
|
syl 0red pm2.21fal absge0 abscl mpbid wceq simpll ltled absid syl2anc ltnrd
|
|
simplr eqbrtrrd wo wi 0re axltwlin mp3an3 mpancom imp mpjaodan inegd mpbird
|
|
id ) ABCZAADEZFGVDAHGZIVCVEVCVEJZVDKHGZLKAHGZVFVGJZVGVFVGMVIKVDFGZVGIVCVJVE
|
|
VGVCANCZVJAOZAUARPVIKVDVISVCVDBCZVEVGVCVKVMVLAUBRZPQUCTVFVHJZAAHGVOVDAAHVOV
|
|
CKAFGVDAUDVCVEVHUEZVOKAVOSVPVFVHMUFAUGUHVCVEVHUJUKVOAVPUITVCVEVGVHULZVMVCVE
|
|
VQUMZVNVMVCKBCVRUNVDAKUOUPUQURUSUTVCAVDVCVBVNQVA $.
|
|
|
|
$( The absolute value of a rational number is either that number or its
|
|
negative. (Contributed by Jim Kingdon, 8-Nov-2021.) $)
|
|
qabsor $p |- ( A e. QQ -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) ) $=
|
|
( cq wcel cr cc0 cle wbr wo cabs cfv wceq cneg qre cz 0z ax-mp qletric mpan
|
|
zq ex absid absnid orim12d sylc ) ABCZADCZEAFGZAEFGZHZAIJZAKZUJALKZHAMEBCZU
|
|
EUIENCUMOESPEAQRUFUGUKUHULUFUGUKAUATUFUHULAUBTUCUD $.
|
|
|
|
${
|
|
qabsord.1 $e |- ( ph -> A e. QQ ) $.
|
|
$( The absolute value of a rational number is either that number or its
|
|
negative. (Contributed by Jim Kingdon, 8-Nov-2021.) $)
|
|
qabsord $p |- ( ph -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) ) $=
|
|
( cq wcel cabs cfv wceq cneg wo qabsor syl ) ABDEBFGZBHMBIHJCBKL $.
|
|
$}
|
|
|
|
$( Absolute value of a real number. (Contributed by NM, 17-Mar-2005.) $)
|
|
absre $p |- ( A e. RR -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) ) $=
|
|
( cr wcel cabs cfv ccj cmul co csqrt c2 cexp cc wceq recn absval syl sqvald
|
|
cjre oveq2d eqtr4d fveq2d ) ABCZADEZAAFEZGHZIEZAJKHZIEUBALCUCUFMANZAOPUBUGU
|
|
EIUBUGAAGHUEUBAUHQUBUDAAGARSTUAT $.
|
|
|
|
$( Square of the absolute value of a real number. (Contributed by NM,
|
|
16-Jan-2006.) $)
|
|
absresq $p |- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) ) $=
|
|
( cr wcel ccj cmul co cabs c2 cexp cjre oveq2d cc wceq recn absvalsq sqvald
|
|
cfv syl 3eqtr4d ) ABCZAADQZEFZAAEFAGQHIFZAHIFTUAAAEAJKTALCUCUBMANZAORTAUDPS
|
|
$.
|
|
|
|
${
|
|
$d j k A $. $d j k N $.
|
|
$( Absolute value of positive integer exponentiation. (Contributed by NM,
|
|
5-Jan-2006.) $)
|
|
absexp $p |- ( ( A e. CC /\ N e. NN0 ) ->
|
|
( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) $=
|
|
( vj vk cn0 wcel cc cexp co cabs cfv wceq wi cc0 c1 fveq2d eqeq12d imbi2d
|
|
oveq2 cmul cv caddc abs1 exp0 abscl recnd exp0d 3eqtr4a oveq1 expp1 expcl
|
|
wa adantl simpl absmul syl2anc eqtrd adantr sylan 3eqtr4d exp31 com12 a2d
|
|
nn0ind impcom ) BEFAGFZABHIZJKZAJKZBHIZLZVFACUAZHIZJKZVIVLHIZLZMVFANHIZJK
|
|
ZVINHIZLZMVFADUAZHIZJKZVIWAHIZLZMVFAWAOUBIZHIZJKZVIWFHIZLZMVFVKMCDBVLNLZV
|
|
PVTVFWKVNVRVOVSWKVMVQJVLNAHSPVLNVIHSQRVLWALZVPWEVFWLVNWCVOWDWLVMWBJVLWAAH
|
|
SPVLWAVIHSQRVLWFLZVPWJVFWMVNWHVOWIWMVMWGJVLWFAHSPVLWFVIHSQRVLBLZVPVKVFWNV
|
|
NVHVOVJWNVMVGJVLBAHSPVLBVIHSQRVFOJKOVRVSUCVFVQOJAUDPVFVIVFVIAUEUFZUGUHWAE
|
|
FZVFWEWJVFWPWEWJMVFWPWEWJVFWPULZWEULWCVITIZWDVITIZWHWIWEWRWSLWQWCWDVITUIU
|
|
MWQWHWRLWEWQWHWBATIZJKZWRWQWGWTJAWAUJPWQWBGFVFXAWRLAWAUKVFWPUNWBAUOUPUQUR
|
|
WQWIWSLZWEVFVIGFWPXBWOVIWAUJUSURUTVAVBVCVDVE $.
|
|
$}
|
|
|
|
$( Absolute value of integer exponentiation. (Contributed by Jim Kingdon,
|
|
11-Aug-2021.) $)
|
|
absexpzap $p |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) ->
|
|
( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) $=
|
|
( cz wcel cc cc0 cap wbr cn0 cr wa cexp co cabs cfv wceq absexp ex c1 cdiv
|
|
cneg cn wo elznn0nn adantr 1cnd simpll nnnn0 ad2antll expcld simplr expap0d
|
|
wi absdivap syl3anc abs1 oveq1i syl2anc oveq2d syl5eq eqtrd simprl expineg2
|
|
nnz recnd syl22anc fveq2d abscl ad2antrr abs00ap mpbird 3eqtr4d jaod 3impia
|
|
wb syl3an3b ) BCDAEDZAFGHZBIDZBJDZBUAZUBDZKZUCZABLMZNOZANOZBLMZPZBUDVQVRWDW
|
|
IVQVRKZVSWIWCVQVSWIUMVRVQVSWIABQRUEWJWCWIWJWCKZSAWALMZTMZNOZSWGWALMZTMZWFWH
|
|
WKWNSNOZWLNOZTMZWPWKSEDWLEDWLFGHWNWSPWKUFWKAWAVQVRWCUGZWBWAIDZWJVTWAUHUIZUJ
|
|
WKAWAWTVQVRWCUKZWBWACDWJVTWAVDUIULSWLUNUOWKWSSWRTMWPWQSWRTUPUQWKWRWOSTWKVQX
|
|
AWRWOPWTXBAWAQURUSUTVAWKWEWMNWKVQVRBEDZXAWEWMPWTXCWKBWJVTWBVBVEZXBABVCVFVGW
|
|
KWGEDWGFGHZXDXAWHWPPWKWGVQWGJDVRWCAVHVIVEWKXFVRXCVQXFVRVOVRWCAVJVIVKXEXBWGB
|
|
VCVFVLRVMVNVP $.
|
|
|
|
$( Square can be moved in and out of absolute value. (Contributed by Scott
|
|
Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abssq $p |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( abs ` ( A ^ 2 ) ) ) $=
|
|
( cc wcel c2 cexp co cabs cfv cn0 wceq 2nn0 absexp mpan2 eqcomd ) ABCZADEFG
|
|
HZAGHDEFZODICPQJKADLMN $.
|
|
|
|
$( The squares of two reals are equal iff their absolute values are equal.
|
|
(Contributed by NM, 6-Mar-2009.) $)
|
|
sqabs $p |- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <->
|
|
( abs ` A ) = ( abs ` B ) ) ) $=
|
|
( cr wcel wa c2 cexp co wceq cabs cfv cc0 cle resqcl sqge0 absid syl2anc cc
|
|
wbr recn cn0 2nn0 absexp sylancl eqtr3d eqeqan12d wb absge0 jca sq11 syl2an
|
|
abscl bitrd ) ACDZBCDZEAFGHZBFGHZIAJKZFGHZBJKZFGHZIZURUTIZUNUOUPUSUQVAUNUPJ
|
|
KZUPUSUNUPCDLUPMSVDUPIANAOUPPQUNARDZFUADZVDUSIATZUBAFUCUDUEUOUQJKZUQVAUOUQC
|
|
DLUQMSVHUQIBNBOUQPQUOBRDZVFVHVAIBTZUBBFUCUDUEUFUNVEVIVBVCUGZUOVGVJVEURCDZLU
|
|
RMSZEUTCDZLUTMSZEVKVIVEVLVMAULAUHUIVIVNVOBULBUHUIURUTUJUKUKUM $.
|
|
|
|
$( The absolute value of a complex number is greater than or equal to the
|
|
absolute value of its real part. (Contributed by NM, 1-Apr-2005.) $)
|
|
absrele $p |- ( A e. CC -> ( abs ` ( Re ` A ) ) <_ ( abs ` A ) ) $=
|
|
( cc wcel cre cfv c2 cexp co csqrt cim cabs cle wbr cc0 imcl sqge0d resqcld
|
|
caddc mpbid cr recl addge01d wb readdcld addge0d sqrtle syl22anc wceq absre
|
|
syl absval2 3brtr4d ) ABCZADEZFGHZIEZUOAJEZFGHZRHZIEZUNKEZAKELUMUOUSLMZUPUT
|
|
LMZUMNURLMVBUMUQAOZPZUMUOURUMUNAUAZQZUMUQVDQZUBSUMUOTCNUOLMUSTCNUSLMVBVCUCV
|
|
GUMUNVFPZUMUOURVGVHUDUMUOURVGVHVIVEUEUOUSUFUGSUMUNTCVAUPUHVFUNUIUJAUKUL $.
|
|
|
|
$( The absolute value of a complex number is greater than or equal to the
|
|
absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.)
|
|
(Proof shortened by Mario Carneiro, 29-May-2016.) $)
|
|
absimle $p |- ( A e. CC -> ( abs ` ( Im ` A ) ) <_ ( abs ` A ) ) $=
|
|
( cc wcel ci cneg cmul co cre cfv cabs cim cle wbr negicn id mulcld absrele
|
|
a1i wceq c1 syl imre fveq2d absmul mpan ax-icn ax-mp absi eqtri abscl recnd
|
|
absneg oveq1i mulid2d syl5eq eqtr2d 3brtr4d ) ABCZDEZAFGZHIZJIZUTJIZAKIZJIA
|
|
JIZLURUTBCVBVCLMURUSAUSBCZURNRUROPUTQUAURVDVAJAUBUCURVCUSJIZVEFGZVEVFURVCVH
|
|
SNUSAUDUEURVHTVEFGVEVGTVEFVGDJIZTDBCVGVISUFDULUGUHUIUMURVEURVEAUJUKUNUOUPUQ
|
|
$.
|
|
|
|
$( The absolute value of an integer is a nonnegative integer. (Contributed
|
|
by NM, 27-Feb-2005.) $)
|
|
nn0abscl $p |- ( A e. ZZ -> ( abs ` A ) e. NN0 ) $=
|
|
( cz wcel cc0 cle wbr cabs cfv cn0 wa cneg cr wceq zre absnid sylan znegcld
|
|
simpl elnn0z eqeltrd simpr le0neg1d mpbid sylanbrc absid biimpri wo zletric
|
|
adantr 0z mpan2 mpjaodan ) ABCZADEFZAGHZICDAEFZUMUNJZUOAKZIUMALCZUNUOURMANZ
|
|
AOPUQURBCDUREFZURICUQAUMUNRQUQUNVAUMUNUAUQAUMUSUNUTUIUBUCURSUDTUMUPJZUOAIUM
|
|
USUPUOAMUTAUEPAICVBASUFTUMDBCUNUPUGUJADUHUKUL $.
|
|
|
|
$( The absolute value of an integer is an integer. (Contributed by Stefan
|
|
O'Rear, 24-Sep-2014.) $)
|
|
zabscl $p |- ( A e. ZZ -> ( abs ` A ) e. ZZ ) $=
|
|
( cz wcel cabs cfv nn0abscl nn0zd ) ABCADEAFG $.
|
|
|
|
$( A number which is less than its absolute value is negative. (Contributed
|
|
by Jim Kingdon, 12-Aug-2021.) $)
|
|
ltabs $p |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 ) $=
|
|
( cr wcel cabs cfv clt wbr wa cc0 simpr simpllr cle simpll adantr cap wo wb
|
|
wceq mpbid 0re 0red ltled absid syl2anc breqtrd ltnrd pm2.65da cc abscl syl
|
|
recn ad2antrr gt0ap0d abs00ap 3syl reaplt sylancl ecased wi axltwlin mp3an3
|
|
mpdan imp mpjaodan ) ABCZAADEZFGZHZAIFGZVIIVFFGZVHVIJVHVJHZVIIAFGZVKVLAAFGV
|
|
KVLHZAVFAFVEVGVJVLKVMVEIALGVFARVKVEVLVEVGVJMZNZVMIAVMUAVOVKVLJUBAUCUDUEVMAV
|
|
OUFUGVKAIOGZVIVLPZVKVFIOGZVPVKVFVEVFBCZVGVJVEAUHCZVSAUKZAUIUJZULVHVJJUMVKVE
|
|
VTVRVPQVNWAAUNUOSVKVEIBCZVPVQQVNTAIUPUQSURVEVGVIVJPZVEVSVGWDUSZWBVEVSWCWETA
|
|
VFIUTVAVBVCVD $.
|
|
|
|
$( Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
|
|
(Revised by Mario Carneiro, 29-May-2016.) $)
|
|
abslt $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) ) $=
|
|
( cr wcel cabs cfv clt wbr cneg simpll renegcld recnd syl simplr leabs wceq
|
|
wa cle simpr cc0 abscl absneg breqtrd lelttrd ad2antrr jca simpl 0red ltabs
|
|
cc ltled le0neg1d mpbid absid syl2anc eqtr3d sylan simplrl eqbrtrd axltwlin
|
|
wo simprr wi syl3anc mpd mpjaodan impbida ltnegcon1 anbi1d bitrd ) ACDZBCDZ
|
|
QZAEFZBGHZAIZBGHZABGHZQZBIAGHZVRQVMVOVSVMVOQZVQVRWAVPVNBWAAVKVLVOJZKZWAAUJD
|
|
ZVNCDZWAAWBLZAUAZMZVKVLVONZWAVPVPEFZVNRWAVPCDZVPWJRHWCVPOMWAWDWJVNPZWFAUBZM
|
|
UCVMVOSZUDWAAVNBWBWHWIVKAVNRHVLVOAOUEWNUDUFVMVSQZAVNGHZVOVOWOWPQVNVPBGWOVKW
|
|
PVNVPPVKVLVSJZVKWPQZWJVNVPWRWDWLWRAVKWPUGZLWMMWRWKTVPRHZWJVPPWRAWSKWRATRHWT
|
|
WRATWSWRUHAUIUKWRAWSULUMVPUNUOUPUQVMVQVRWPURUSWOVOSWOVRWPVOVAZVMVQVRVBWOVKV
|
|
LWEVRXAVCWQVKVLVSNWOWDWEWOAWQLWGMABVNUTVDVEVFVGVMVQVTVRABVHVIVJ $.
|
|
|
|
$( Absolute value and 'less than or equal to' relation. (Contributed by NM,
|
|
6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) $)
|
|
absle $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) ) $=
|
|
( cr wcel cabs cfv cle wbr cneg simpll renegcld recnd syl simplr leabs wceq
|
|
wa letrd clt cc0 cc abscl absneg breqtrd simpr ad2antrr wn axltwlin syl3anc
|
|
jca wo wi simprr lenltd mpbid pm2.53 syl6ci simpl 0red ltabs ltled le0neg1d
|
|
absid syl2anc eqtr3d syl6an simprl syl5ibrcom sylibd pm2.01d mpbird impbida
|
|
breq1 syld lenegcon1 anbi1d bitrd ) ACDZBCDZQZAEFZBGHZAIZBGHZABGHZQZBIAGHZW
|
|
EQVTWBWFVTWBQZWDWEWHWCWABWHAVRVSWBJZKZWHAUADZWACDZWHAWILZAUBZMZVRVSWBNZWHWC
|
|
WCEFZWAGWHWCCDZWCWQGHWJWCOMWHWKWQWAPZWMAUCZMUDVTWBUEZRWHAWABWIWOWPVRAWAGHVS
|
|
WBAOUFXARUJVTWFQZWBBWASHZUGZXBXCXBXCWBXDXBXCWAWCPZWBXBVRXCAWASHZXEVRVSWFJZX
|
|
BXCBASHZXFUKZXHUGZXFXBVSWLVRXCXIULVRVSWFNZXBWKWLXBAXGLWNMZXGBWAAUHUIXBWEXJV
|
|
TWDWEUMXBABXGXKUNUOXHXFUPUQVRXFQZWQWAWCXMWKWSXMAVRXFURZLWTMXMWRTWCGHZWQWCPX
|
|
MAXNKXMATGHXOXMATXNXMUSAUTVAXMAXNVBUOWCVCVDVEVFXBWBXEWDVTWDWEVGWAWCBGVMVHVN
|
|
XBWABXLXKUNZVIVJXPVKVLVTWDWGWEABVOVPVQ $.
|
|
|
|
$( If the absolute value of a complex number is less than a real, its
|
|
difference from the real is apart from zero. (Contributed by Jim Kingdon,
|
|
12-Aug-2021.) $)
|
|
abssubap0 $p |- ( ( A e. CC /\ B e. RR /\ ( abs ` A ) < B ) ->
|
|
( B - A ) # 0 ) $=
|
|
( cc wcel cr cabs cfv clt wbr cmin co cc0 cap simplr recnd simpll abscl syl
|
|
wa simpr cle leabs ltletrd gtapd wi absext syl2anc mpd subap0d 3impa ) ACDZ
|
|
BEDZAFGZBHIZBAJKLMIUKULSZUNSZBAUPBUKULUNNZOZUKULUNPZUPBFGZUMMIZBAMIZUPUMUTU
|
|
PUKUMEDUSAQRZUPBCDZUTEDURBQRZUPUMBUTVCUQVEUOUNTUPULBUTUAIUQBUBRUCUDUPVDUKVA
|
|
VBUEURUSBAUFUGUHUIUJ $.
|
|
|
|
$( If the absolute value of a complex number is less than a real, its
|
|
difference from the real is nonzero. See also ~ abssubap0 which is the
|
|
same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.) $)
|
|
abssubne0 $p |- ( ( A e. CC /\ B e. RR /\ ( abs ` A ) < B ) ->
|
|
( B - A ) =/= 0 ) $=
|
|
( cc wcel cr cabs cfv clt wbr w3a cmin co cc0 cap wne abssubap0 simp2 recnd
|
|
wi simp1 subcld 0cnd apne syl2anc mpd ) ACDZBEDZAFGBHIZJZBAKLZMNIZUJMOZABPU
|
|
IUJCDMCDUKULSUIBAUIBUFUGUHQRUFUGUHTUAUIUBUJMUCUDUE $.
|
|
|
|
$( The absolute value of a difference and 'less than' relation. (Contributed
|
|
by Paul Chapman, 18-Sep-2007.) $)
|
|
absdiflt $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( abs ` ( A - B ) ) < C <->
|
|
( ( B - C ) < A /\ A < ( B + C ) ) ) ) $=
|
|
( cr wcel w3a cmin co cabs cfv clt wbr cneg wa caddc wb resubcl abslt recn
|
|
cc stoic3 renegcl ltaddsub2 syl3an2 3comr wceq negsub syl2an 3adant1 breq1d
|
|
bitr3d ltsubadd2 anbi12d bitrd ) ADEZBDEZCDEZFZABGHZIJCKLZCMZUSKLZUSCKLZNZB
|
|
CGHZAKLZABCOHKLZNUOUPUSDEUQUTVDPABQUSCRUAURVBVFVCVGURBVAOHZAKLZVBVFUPUQUOVI
|
|
VBPZUQUPVADEUOVJCUBBVAAUCUDUEURVHVEAKUPUQVHVEUFZUOUPBTECTEVKUQBSCSBCUGUHUIU
|
|
JUKABCULUMUN $.
|
|
|
|
$( The absolute value of a difference and 'less than or equal to' relation.
|
|
(Contributed by Paul Chapman, 18-Sep-2007.) $)
|
|
absdifle $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( ( abs ` ( A - B ) ) <_ C <->
|
|
( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) ) $=
|
|
( cr wcel w3a cmin co cabs cfv cle wbr cneg wa caddc wb resubcl absle recn
|
|
cc stoic3 renegcl leaddsub2 syl3an2 3comr wceq negsub syl2an 3adant1 breq1d
|
|
bitr3d lesubadd2 anbi12d bitrd ) ADEZBDEZCDEZFZABGHZIJCKLZCMZUSKLZUSCKLZNZB
|
|
CGHZAKLZABCOHKLZNUOUPUSDEUQUTVDPABQUSCRUAURVBVFVCVGURBVAOHZAKLZVBVFUPUQUOVI
|
|
VBPZUQUPVADEUOVJCUBBVAAUCUDUEURVHVEAKUPUQVHVEUFZUOUPBTECTEVKUQBSCSBCUGUHUIU
|
|
JUKABCULUMUN $.
|
|
|
|
$( Membership in a symmetric closed real interval. (Contributed by Stefan
|
|
O'Rear, 16-Nov-2014.) $)
|
|
elicc4abs $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( C e. ( ( A - B ) [,] ( A + B ) ) <-> ( abs ` ( C - A ) ) <_ B ) ) $=
|
|
( cr wcel w3a cmin co caddc cicc cle wbr wa cabs cfv cxr wb resubcl 3adant3
|
|
rexrd readdcl rexr 3ad2ant3 elicc4 syl3anc absdifle 3coml bitr4d ) ADEZBDEZ
|
|
CDEZFZCABGHZABIHZJHEZUMCKLCUNKLMZCAGHNOBKLZULUMPEUNPECPEZUOUPQULUMUIUJUMDEU
|
|
KABRSTULUNUIUJUNDEUKABUASTUKUIURUJCUBUCUMUNCUDUEUKUIUJUQUPQCABUFUGUH $.
|
|
|
|
$( Comparison to a nonnegative number based on comparison to squares.
|
|
(Contributed by NM, 16-Jan-2006.) $)
|
|
lenegsq $p |- ( ( A e. RR /\ B e. RR /\ 0 <_ B ) ->
|
|
( ( A <_ B /\ -u A <_ B ) <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr cneg wa c2 cexp co wb cabs cfv recn abscl absge0 jca
|
|
cc le2sq sylan absle ancom lenegcon1 anbi1d syl5rbbr adantrr absresq breq1d
|
|
syl bitrd adantr 3bitr3d 3impb ) ACDZBCDZEBFGZABFGZAHBFGZIZAJKLZBJKLZFGZMUP
|
|
UQURIZIANOZBFGZVFJKLZVCFGZVAVDUPVFCDZEVFFGZIZVEVGVIMUPATDZVLAPVMVJVKAQARSUK
|
|
VFBUAUBUPUQVGVAMURUPUQIZVGBHAFGZUSIZVAABUCVAUTUSIVNVPUTUSUDVNUTVOUSABUEUFUG
|
|
ULUHUPVIVDMVEUPVHVBVCFAUIUJUMUNUO $.
|
|
|
|
$( The real part of a number is less than or equal to its absolute value.
|
|
Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM,
|
|
1-Apr-2005.) $)
|
|
releabs $p |- ( A e. CC -> ( Re ` A ) <_ ( abs ` A ) ) $=
|
|
( cc wcel cre cfv cabs recl cr recnd abscl syl cle wbr leabs absrele letrd
|
|
) ABCZADEZRFEZAFEAGZQRBCSHCQRTIRJKAJQRHCRSLMTRNKAOP $.
|
|
|
|
$( Reciprocal expressed with a real denominator. (Contributed by Jim
|
|
Kingdon, 13-Aug-2021.) $)
|
|
recvalap $p |- ( ( A e. CC /\ A # 0 ) ->
|
|
( 1 / A ) = ( ( * ` A ) / ( ( abs ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel cc0 cap wbr wa c1 cabs cfv c2 cexp ccj cdiv wceq cmul cjcl adantr
|
|
co mpbird simpl absvalsq eqtr4d cr abscl recnd sqcld cjap0 biimpa divmulapd
|
|
mulcomd oveq2d abs00ap biimpar wb sqap0 syl recdivapd eqtr3d ) ABCZADEFZGZH
|
|
AIJZKLSZAMJZNSZNSHANSVEVDNSVBVFAHNVBVFAOVEAPSZVDOVBVGAVEPSZVDVBVEAUTVEBCVAA
|
|
QRZUTVAUAZUKUTVDVHOVAAUBRUCVBVDVEAVBVCVBVCUTVCUDCVAAUERUFZUGZVIVJUTVAVEDEFA
|
|
UHUIZUJTULVBVDVEVLVIVBVDDEFZVCDEFZUTVOVAAUMUNVBVCBCVNVOUOVKVCUPUQTVMURUS $.
|
|
|
|
$( The absolute value function is idempotent. (Contributed by NM,
|
|
20-Nov-2004.) $)
|
|
absidm $p |- ( A e. CC -> ( abs ` ( abs ` A ) ) = ( abs ` A ) ) $=
|
|
( cc wcel cabs cfv cr cc0 cle wbr wceq abscl absge0 absid syl2anc ) ABCADEZ
|
|
FCGOHIODEOJAKALOMN $.
|
|
|
|
$( The absolute value of a number apart from zero is positive. (Contributed
|
|
by Jim Kingdon, 13-Aug-2021.) $)
|
|
absgt0ap $p |- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) ) $=
|
|
( cc wcel cc0 cabs cfv clt wbr cap 0red abscl absge0 leltapd abs00ap bitr2d
|
|
) ABCZDAEFZGHQDIHADIHPDQPJAKALMANO $.
|
|
|
|
$( The absolute value of a nonzero integer is a positive integer.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew
|
|
Salmon, 25-May-2011.) $)
|
|
nnabscl $p |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN ) $=
|
|
( cz wcel cc0 wne wa cabs cfv clt wbr cn zabscl adantr wb 0z zapne mpan2 cc
|
|
cap zcn absgt0ap syl bitr3d biimpa elnnz sylanbrc ) ABCZADEZFAGHZBCZDUIIJZU
|
|
IKCUGUJUHALMUGUHUKUGADSJZUHUKUGDBCULUHNOADPQUGARCULUKNATAUAUBUCUDUIUEUF $.
|
|
|
|
$( Swapping order of subtraction doesn't change the absolute value.
|
|
(Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abssub $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) ) $=
|
|
( cc wcel wa cmin co cneg cabs cfv subcl absneg syl negsubdi2 fveq2d eqtr3d
|
|
wceq ) ACDBCDEZABFGZHZIJZSIJZBAFGZIJRSCDUAUBQABKSLMRTUCIABNOP $.
|
|
|
|
$( Absolute value of a nonnegative difference. (Contributed by NM,
|
|
14-Feb-2008.) $)
|
|
abssubge0 $p |- ( ( A e. RR /\ B e. RR /\ A <_ B )
|
|
-> ( abs ` ( B - A ) ) = ( B - A ) ) $=
|
|
( cr wcel cle wbr cmin co cabs cfv wceq w3a resubcl 3adant3 subge0 biimp3ar
|
|
cc0 absid syl2anc 3com12 ) BCDZACDZABEFZBAGHZIJUDKZUAUBUCLUDCDZQUDEFZUEUAUB
|
|
UFUCBAMNUAUBUGUCBAOPUDRST $.
|
|
|
|
$( Absolute value of a nonpositive difference. (Contributed by FL,
|
|
3-Jan-2008.) $)
|
|
abssuble0 $p |- ( ( A e. RR /\ B e. RR /\ A <_ B )
|
|
-> ( abs ` ( A - B ) ) = ( B - A ) ) $=
|
|
( cr wcel cle wbr w3a cmin co cabs cfv wceq cc recn abssub syl2an abssubge0
|
|
3adant3 eqtrd ) ACDZBCDZABEFZGABHIJKZBAHIZJKZUDTUAUCUELZUBTAMDBMDUFUAANBNAB
|
|
OPRABQS $.
|
|
|
|
$( Triangle inequality for absolute value. Proposition 10-3.7(h) of
|
|
[Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
abstri $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) $=
|
|
( cc wcel caddc co cabs cfv cle wbr c2 cexp cmul cr remulcld abscl syl wceq
|
|
recnd cc0 wa ccj cre 2re a1i simpl simpr cjcld mulcld recld resqcld releabs
|
|
readdcld absmul syl2anc abscj oveq2d eqtrd breqtrd crp 2rp lemul2d leadd2dd
|
|
mpbid sqabsadd binom2 add32d 3brtr4d addcl absge0 addge0d le2sqd mpbird ) A
|
|
CDZBCDZUAZABEFZGHZAGHZBGHZEFZIJVRKLFZWAKLFZIJVPVSKLFZVTKLFZEFZKABUBHZMFZUCH
|
|
ZMFZEFWFKVSVTMFZMFZEFZWBWCIVPWJWLWFVPKWIKNDVPUDUEZVPWHVPAWGVNVOUFZVPBVNVOUG
|
|
ZUHZUIZUJZOVPKWKWNVPVSVTVPVNVSNDWOAPQZVPVOVTNDWPBPQZOZOZVPWDWEVPVSWTUKZVPVT
|
|
XAUKZUMVPWIWKIJWJWLIJVPWIWHGHZWKIVPWHCDWIXFIJWRWHULQVPXFVSWGGHZMFZWKVPVNWGC
|
|
DXFXHRWOWQAWGUNUOVPXGVTVSMVPVOXGVTRWPBUPQUQURUSVPWIWKKWSXBKUTDVPVAUEVBVDVCA
|
|
BVEVPWCWDWLEFWEEFZWMVPVSCDVTCDWCXIRVPVSWTSVPVTXASVSVTVFUOVPWDWLWEVPWDXDSVPW
|
|
LXCSVPWEXESVGURVHVPVRWAVPVQCDZVRNDABVIZVQPQVPVSVTWTXAUMVPXJTVRIJXKVQVJQVPVS
|
|
VTWTXAVPVNTVSIJWOAVJQVPVOTVTIJWPBVJQVKVLVM $.
|
|
|
|
$( Absolute value of differences around common element. (Contributed by FL,
|
|
9-Oct-2006.) $)
|
|
abs3dif $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) ->
|
|
( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) +
|
|
( abs ` ( C - B ) ) ) ) $=
|
|
( cc wcel w3a cmin caddc cabs cfv cle wceq npncan 3com23 fveq2d wbr 3adant2
|
|
co subcl ancoms 3adant1 abstri syl2anc eqbrtrrd ) ADEZBDEZCDEZFZACGRZCBGRZH
|
|
RZIJZABGRZIJUIIJUJIJHRZKUHUKUMIUEUGUFUKUMLACBMNOUHUIDEZUJDEZULUNKPUEUGUOUFA
|
|
CSQUFUGUPUEUGUFUPCBSTUAUIUJUBUCUD $.
|
|
|
|
$( Difference of absolute values. (Contributed by Paul Chapman,
|
|
7-Sep-2007.) $)
|
|
abs2dif $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) $=
|
|
( cc wcel wa cc0 cmin co cfv cle subid1 fveq2d wbr 0cn cr subcl mpan2 abscl
|
|
cabs syl oveqan12d caddc abs3dif mp3an2 wb anim12i df-3an sylanbrc lesubadd
|
|
w3a mpbird eqbrtrrd ) ACDZBCDZEZAFGHZSIZBFGHZSIZGHZASIZBSIZGHABGHZSIZJUMUNU
|
|
QVAUSVBGUMUPASAKLUNURBSBKLUAUOUTVDJMZUQVDUSUBHJMZUMFCDZUNVFNAFBUCUDUOUQODZU
|
|
SODZVDODZUJZVEVFUEUOVHVIEVJVKUMVHUNVIUMUPCDZVHUMVGVLNAFPQUPRTUNURCDZVIUNVGV
|
|
MNBFPQURRTUFUOVCCDVJABPVCRTVHVIVJUGUHUQUSVDUITUKUL $.
|
|
|
|
$( Difference of absolute values. (Contributed by Mario Carneiro,
|
|
14-Apr-2016.) $)
|
|
abs2dif2 $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( abs ` ( A - B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) $=
|
|
( cc wcel wa cneg caddc co cabs cfv cmin cle wbr negcl abstri sylan2 negsub
|
|
fveq2d wceq absneg adantl oveq2d 3brtr3d ) ACDZBCDZEZABFZGHZIJZAIJZUGIJZGHZ
|
|
ABKHZIJUJBIJZGHLUEUDUGCDUIULLMBNAUGOPUFUHUMIABQRUFUKUNUJGUEUKUNSUDBTUAUBUC
|
|
$.
|
|
|
|
$( Absolute value of difference of absolute values. (Contributed by Paul
|
|
Chapman, 7-Sep-2007.) $)
|
|
abs2difabs $p |- ( ( A e. CC /\ B e. CC ) ->
|
|
( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) ) $=
|
|
( cc wcel wa cabs cfv cmin co cle cneg abs2dif ancoms abscl recnd syl2an cr
|
|
wbr wb syl2anc wceq negsubdi2 abssub 3brtr4d resubcl subcl syl absle anbi1d
|
|
lenegcon1 bitr4d mpbir2and ) ACDZBCDZEZAFGZBFGZHIZFGABHIZFGZJRZURKZUTJRZURU
|
|
TJRZUOUQUPHIZBAHIFGZVBUTJUNUMVEVFJRBALMUMUPCDUQCDVBVEUAUNUMUPANZOUNUQBNZOUP
|
|
UQUBPABUCUDABLUOVAUTKURJRZVDEZVCVDEUOURQDZUTQDZVAVJSUMUPQDUQQDVKUNVGVHUPUQU
|
|
EPZUOUSCDVLABUFUSNUGZURUTUHTUOVCVIVDUOVKVLVCVISVMVNURUTUJTUIUKUL $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Cancellation law involving the real part of a complex number.
|
|
(Contributed by NM, 12-May-2005.) $)
|
|
recan $p |- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC
|
|
( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> A = B ) ) $=
|
|
( cc wcel cmul co cre cfv wceq c1 ci caddc wi oveq1 fveq2d eqeq12d oveq2d
|
|
rspcv oveq12d wa cv wral cneg ax-1cn negicn cim replim mulid2 eqcomd imre
|
|
ax-mp eqtrd eqeqan12d syl5ibr oveq2 ralrimivw impbid1 ) BDEZCDEZUAZAUBZBF
|
|
GZHIZVBCFGZHIZJZADUCZBCJZVHVIVAKBFGZHIZLLUDZBFGZHIZFGZMGZKCFGZHIZLVLCFGZH
|
|
IZFGZMGZJVHVKVRVOWAMKDEVHVKVRJZNUEVGWCAKDVBKJZVDVKVFVRWDVCVJHVBKBFOPWDVEV
|
|
QHVBKCFOPQSULVHVNVTLFVLDEVHVNVTJZNUFVGWEAVLDVBVLJZVDVNVFVTWFVCVMHVBVLBFOP
|
|
WFVEVSHVBVLCFOPQSULRTUSUTBVPCWBUSBBHIZLBUGIZFGZMGVPBUHUSWGVKWIVOMUSBVJHUS
|
|
VJBBUIUJPUSWHVNLFBUKRTUMUTCCHIZLCUGIZFGZMGWBCUHUTWJVRWLWAMUTCVQHUTVQCCUIU
|
|
JPUTWKVTLFCUKRTUMUNUOVIVGADVIVCVEHBCVBFUPPUQUR $.
|
|
|
|
$( Mapping domain and codomain of the absolute value function.
|
|
(Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro,
|
|
7-Nov-2013.) $)
|
|
absf $p |- abs : CC --> RR $=
|
|
( vx cc cr cv ccj cfv cmul co csqrt cabs df-abs wcel abscl eqeltrrd fmpti
|
|
absval ) ABCADZQEFGHIFZJAKQBLQJFRCQPQMNO $.
|
|
$}
|
|
|
|
$( Lemma involving absolute value of differences. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
abs3lem $p |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\
|
|
D e. RR ) ) -> (
|
|
( ( abs ` ( A - C ) ) < ( D / 2 ) /\
|
|
( abs ` ( C - B ) ) < ( D / 2 ) ) ->
|
|
( abs ` ( A - B ) ) < D ) ) $=
|
|
( cc wcel wa cr cmin co cabs cfv c2 cdiv clt wbr caddc subcld abscl syl cle
|
|
simplll simpllr simplrl readdcld simplrr abs3dif syl3anc lt2halvesd lelttrd
|
|
simprl simprr ex ) AEFZBEFZGZCEFZDHFZGZGZACIJZKLZDMNJZOPZCBIJZKLZVCOPZGZABI
|
|
JZKLZDOPUTVHGZVJVBVFQJZDVKVIEFVJHFVKABUNUOUSVHUBZUNUOUSVHUCZRVISTVKVBVFVKVA
|
|
EFVBHFVKACVMUPUQURVHUDZRVASTZVKVEEFVFHFVKCBVOVNRVESTZUEUPUQURVHUFZVKUNUOUQV
|
|
JVLUAPVMVNVOABCUGUHVKVBVFDVPVQVRUTVDVGUKUTVDVGULUIUJUM $.
|
|
|
|
$( Lemma for ~ fzomaxdif . (Contributed by Stefan O'Rear, 6-Sep-2015.) $)
|
|
fzomaxdiflem $p |- ( ( ( A e. ( C ..^ D ) /\ B e. ( C ..^ D ) ) /\
|
|
A <_ B ) -> ( abs ` ( B - A ) ) e. ( 0 ..^ ( D - C ) ) ) $=
|
|
( cfzo co wcel wa cle wbr cmin cabs cc0 elfzoelz adantl adantr zsubcld zred
|
|
cz clt cfv cr wceq subge0d biimpar absid syl2anc elfzoel1 resubcld elfzoel2
|
|
elfzole1 lesub2dd elfzolt2 ltsub1dd lelttrd elfzo syl3anc mpbir2and eqeltrd
|
|
wb 0zd ) ACDEFZGZBVBGZHZABIJZHZBAKFZLUAZVHMDCKFZEFZVGVHUBGZMVHIJZVIVHUCVEVL
|
|
VFVEVHVEBAVDBSGVCBCDNOZVCASGVDACDNPZQZRZPVEVMVFVEBAVEBVNRZVEAVORZUDUEZVHUFU
|
|
GVGVHVKGZVMVHVJTJZVTVEWBVFVEVHBCKFVJVQVEBCVRVECVDCSGVCBCDUHOZRZUIVEVJVEDCVD
|
|
DSGVCBCDUJOZWCQZRVECABWDVSVRVCCAIJVDACDUKPULVEBDCVRVEDWERWDVDBDTJVCBCDUMOUN
|
|
UOPVEWAVMWBHUTZVFVEVHSGMSGVJSGWGVPVEVAWFVHMVJUPUQPURUS $.
|
|
|
|
$( A bound on the separation of two points in a half-open range.
|
|
(Contributed by Stefan O'Rear, 6-Sep-2015.) $)
|
|
fzomaxdif $p |- ( ( A e. ( C ..^ D ) /\ B e. ( C ..^ D ) ) ->
|
|
( abs ` ( A - B ) ) e. ( 0 ..^ ( D - C ) ) ) $=
|
|
( cfzo co wcel wa cle wbr cmin cabs cfv cc0 cc elfzoelz syl2an fzomaxdiflem
|
|
zcnd cz wceq abssub adantr eqeltrd ancom1s wo zletric mpjaodan ) ACDEFZGZBU
|
|
IGZHZABIJZABKFLMZNDCKFEFZGZBAIJZULUMHUNBAKFLMZUOULUNURUAZUMUJAOGBOGUSUKUJAA
|
|
CDPZSUKBBCDPZSABUBQUCABCDRUDUKUJUQUPBACDRUEUJATGBTGUMUQUFUKUTVAABUGQUH $.
|
|
|
|
${
|
|
$d j z $. $d k m ch $. $d k m x z D $. $d k m x z F $. $d j k m x ph $.
|
|
$d k m x z G $. $d m x z ps $. $d x z ta $. $d k th $. $d x z Z $.
|
|
cau3lem.1 $e |- Z C_ ZZ $.
|
|
cau3lem.2 $e |- ( ta -> ps ) $.
|
|
cau3lem.3 $e |- ( ( F ` k ) = ( F ` j ) -> ( ps <-> ch ) ) $.
|
|
cau3lem.4 $e |- ( ( F ` k ) = ( F ` m ) -> ( ps <-> th ) ) $.
|
|
cau3lem.5 $e |- ( ( ph /\ ch /\ ps ) ->
|
|
( G ` ( ( F ` j ) D ( F ` k ) ) ) = ( G ` ( ( F ` k ) D ( F ` j ) ) ) ) $.
|
|
cau3lem.6 $e |- ( ( ph /\ th /\ ch ) ->
|
|
( G ` ( ( F ` m ) D ( F ` j ) ) ) = ( G ` ( ( F ` j ) D ( F ` m ) ) ) ) $.
|
|
cau3lem.7 $e |- ( ( ph /\ ( ps /\ th ) /\ ( ch /\ x e. RR ) ) ->
|
|
( ( ( G ` ( ( F ` k ) D ( F ` j ) ) ) < ( x / 2 ) /\
|
|
( G ` ( ( F ` j ) D ( F ` m ) ) ) < ( x / 2 ) ) ->
|
|
( G ` ( ( F ` k ) D ( F ` m ) ) ) < x ) ) $.
|
|
$( Lemma for ~ cau3 . (Contributed by Mario Carneiro, 15-Feb-2014.)
|
|
(Revised by Mario Carneiro, 1-May-2014.) $)
|
|
cau3lem $p |- ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ta /\
|
|
( G ` ( ( F ` k ) D ( F ` j ) ) ) < x ) <->
|
|
A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ta /\
|
|
A. m e. ( ZZ>= ` k ) ( G ` ( ( F ` k ) D ( F ` m ) ) ) < x ) ) ) $=
|
|
( vz cv cfv co clt wbr wa cuz wral wrex crp weq anbi2d rexralbidv cbvralv
|
|
breq2 wcel c2 cdiv wi rphalfcl wceq rspcv syl adantl ralimi r19.26 oveq1d
|
|
wb fveq2 fveq2d breq1d anbi12d biimpi a1i syl5bir expdimp cz sseli rspcva
|
|
uzid sylan adantll simplll simplrr simplrl syl3anc cr simpr simpllr rpred
|
|
jctild syl122anc sylbid expd impr an32s anassrs expimpd ralimdv expr uzss
|
|
wss ssralv sylan9 ralimdva ex com23 adantr mpdd sylan2 imdistanda 3imtr4g
|
|
reximdva syld ralrimdva syl5bi raleqbidv ad2antlr oveq2d anim2i w3a ralbi
|
|
3expia sylc syl5bb sylibd impbid ) AEIUBZKUCZHUBZKUCZGUDZLUCZFUBZUEUFZUGZ
|
|
IYKUHUCZUIZHMUJZFUKUIZEYJJUBZKUCZGUDZLUCZYOUEUFZJYIUHUCZUIZUGIYRUIZHMUJZF
|
|
UKUIZUUAEYNUAUBZUEUFZUGZIYRUIHMUJZUAUKUIZAUUKYTUUOFUAUKFUAULZYQUUNHIMYRUU
|
|
QYPUUMEYOUULYNUEUPUMUNUOAUUPUUJFUKAYOUKUQZUGZUUPEYNYOURUSUDZUEUFZUGZIYRUI
|
|
ZHMUJZUUJUURUUPUVDUTZAUURUUTUKUQUVEYOVAUUOUVDUAUUTUKUULUUTVBZUUNUVBHIMYRU
|
|
VFUUMUVAEUULUUTYNUEUPUMUNVCVDVEUUSUVCUUIHMUUSYKMUQZUGZEIYRUIZUVAIYRUIZUGU
|
|
VIUUHIYRUIZUGZUVCUUIUVHUVIUVJUVKUVIUVHBIYRUIZUVJUVKUTEBIYROVFZUVHUVMUGZUV
|
|
JCDUUCYLGUDZLUCZUUTUEUFZUGZJYRUIZUGZUVKUVOUVJUVTCUVHUVMUVJUVTUVMUVJUGZBUV
|
|
AUGZIYRUIZUVHUVTBUVAIYRVGZUWDUVTUTUVHUWDUVTUWCUVSIJYRIJULZBDUVAUVRUWFYJUU
|
|
CVBBDVIYIUUBKVJZQVDUWFYNUVQUUTUEUWFYMUVPLUWFYJUUCYLGUWGVHVKVLVMUOVNVOVPVQ
|
|
UVGUVMCUUSUVGYKYRUQZUVMCUVGYKVRUQUWHMVRYKNVSYKWAVDZBCIYKYRIHULZYJYLVBBCVI
|
|
YIYKKVJZPVDVTZWBWCWLUVHUVMUVJUWAUVKUTZUWBUWDUVHUWMUWEUUSUWDUWMUTUVGUUSUWA
|
|
UWDUVKUUSUWAUWDUVKUTUUSUWAUGZUWCUUHIYRUWNYIYRUQZUGBUVAUUHUWNBUWOUVAUUHUTU
|
|
WNBUGUVAUUFJYRUIZUWOUUHUWNBUVAUWPUUSUWCUWAUWPUUSUWCUGZCUVTUWPUWQCUGZUVSUU
|
|
FJYRUWRDUVRUUFUWQCDUVRUUFUTZUUSCDUGZUWCUWSUUSUWTUGZBUVAUWSUXABUGZUVAUVRUU
|
|
FUXBUVAUVRUGUVAYLUUCGUDZLUCZUUTUEUFZUGZUUFUXBUVRUXEUVAUXBUVQUXDUUTUEUXBAD
|
|
CUVQUXDVBAUURUWTBWDZUUSCDBWEZUUSCDBWFZSWGVLUMUXBABDCYOWHUQUXFUUFUTUXGUXAB
|
|
WIUXHUXIUXBYOAUURUWTBWJWKTWMWNWOWPWQWRWSWTWPWQXAUWOUUGYRXCUWPUUHUTYKYIXBU
|
|
UFJUUGYRXDVDXEWQWSXFXGXHXIVPVQXJXKXLEUVAIYRVGEUUHIYRVGZXMXNXOXPXQAUUJYTFU
|
|
KAUUIYSHMAUVGUGUVLUVIYPIYRUIZUGZUUIYSUVGAUWHUVLUXLUTUWIAUWHUGZUVIUVKUXKUV
|
|
IUXMUVMUVKUXKUTUVNUXMUVMUGZUVKUXDYOUEUFZJYRUIZUXKUWHUVKUXPUTAUVMUUHUXPIYK
|
|
YRUWJUUFUXOJUUGYRYIYKUHVJUWJUUEUXDYOUEUWJUUDUXCLUWJYJYLUUCGUWKVHVKVLXRVCX
|
|
SUXPYLYJGUDZLUCZYOUEUFZIYRUIZUXNUXKUXOUXSJIYRJIULZUXDUXRYOUEUYAUXCUXQLUYA
|
|
UUCYJYLGUUBYIKVJXTVKVLUOUXNUXSYPVIZIYRUIZUXTUXKVIUXNACUGZUVMUYCAUWHUVMUYD
|
|
UWHUVMUGCAUWLYAWRUXMUVMWIUYDBUYBIYRACBUYBACBYBUXRYNYOUERVLYDWTYEUXSYPIYRY
|
|
CVDYFYGXKXLXKUXJEYPIYRVGXMXNWTYH $.
|
|
$}
|
|
|
|
${
|
|
$d j k m x y F $. $d j k x M $. $d j k x N $. $d j k x y Z $.
|
|
$d j k x W $.
|
|
cau3.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( Convert between three-quantifier and four-quantifier versions of the
|
|
Cauchy criterion. (In particular, the four-quantifier version has no
|
|
occurrence of ` j ` in the assertion, so it can be used with ~ rexanuz
|
|
and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.) $)
|
|
cau3 $p |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <->
|
|
A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\
|
|
A. m e. ( ZZ>= ` k ) ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) ) $=
|
|
( cv cfv cc wcel cmin co cabs clt wbr wa wral wtru cuz wrex wb cz eqsstri
|
|
crp uzssz id eleq1 wceq abssub 3adant1 cr c2 cdiv wi abs3lem cau3lem trud
|
|
) CIZEJZKLZVABIZEJZMNOJZAIZPQRCVCUAJZSBGUBAUFSVBVADIEJZMNOJVFPQZDUTUAJSRC
|
|
VGSBGUBAUFSUCTVBVDKLZVHKLZVBAMBCDEOGGFUAJUDHFUGUEVBUHVAVDKUIVAVHKUIVJVBVD
|
|
VAMNOJVEUJTVDVAUKULVKVJVHVDMNOJVDVHMNOJZUJTVHVDUKULVBVKRVJVFUMLRVEVFUNUON
|
|
ZPQVLVMPQRVIUPTVAVHVDVFUQULURUS $.
|
|
|
|
${
|
|
cau4.2 $e |- W = ( ZZ>= ` N ) $.
|
|
$( Change the base of a Cauchy criterion. (Contributed by Mario
|
|
Carneiro, 18-Mar-2014.) $)
|
|
cau4 $p |- ( N e. Z -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x )
|
|
<-> A. x e. RR+ E. j e. W A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) ) ) $=
|
|
( vy wcel cv cfv cuz wral wrex crp wb cz cc cmin co cabs clt wa eluzel2
|
|
wbr rexuz3 syl eluzelz bitr4d eleq2s ralbidv cau3 3bitr4g ) FHLZCMZDNZU
|
|
ALZUSKMDNUBUCUDNAMZUEUHKURONPUFZCBMZONZPZBHQZARPVEBGQZARPUTUSVCDNUBUCUD
|
|
NVAUEUHUFCVDPZBHQARPVHBGQARPUQVFVGARVFVGSFEONZHFVILZVFVEBTQZVGVJETLVFVK
|
|
SEFUGVBBCEHIUIUJVJFTLVGVKSEFUKVBBCFGJUIUJULIUMUNABCKDEHIUOABCKDFGJUOUP
|
|
$.
|
|
$}
|
|
|
|
$( A Cauchy sequence of complex numbers is eventually bounded.
|
|
(Contributed by Mario Carneiro, 14-Feb-2014.) $)
|
|
caubnd2 $p |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) ->
|
|
E. y e. RR E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < y ) $=
|
|
( cv cfv cc wcel clt wbr wa wral wrex c1 cr syl cmin co cabs cuz crp wceq
|
|
wi breq2 anbi2d rexralbidv rspcv ax-mp caddc cz eluzelz eleq2s uzid simpl
|
|
1rp ralimi fveq2 eleq1d rspcva syl2an abscl 1re readdcl sylancl cle simpr
|
|
simplr abs2dif syl2anc resubcld subcld lelttr mp3an3 wb ltsubadd2 expimpd
|
|
mpand sylibd ralimdv impancom mpd ralbidv rspcev ex reximia rexcom sylib
|
|
) DIZEJZKLZWMCIZEJZUAUBZUCJZAIZMNZOZDWOUDJZPCGQZAUEPZWNWRRMNZOZDXBPZCGQZW
|
|
MUCJZBIZMNZDXBPZCGQBSQZRUELXDXHUGUSXCXHARUEWSRUFZXAXFCDGXBXNWTXEWNWSRWRMU
|
|
HUIUJUKULXHXLBSQZCGQXMXGXOCGWOGLZXGXOXPXGOZWPUCJZRUMUBZSLZXIXSMNZDXBPZXOX
|
|
QXRSLZRSLZXTXQWPKLZYCXPWOXBLZWNDXBPYEXGXPWOUNLZYFYGWOFUDJGFWOUOHUPWOUQTXF
|
|
WNDXBWNXEURUTWNYEDWOXBWLWOUFWMWPKWLWOEVAVBVCVDZWPVEZTVFXRRVGVHXQYEYBYHXPY
|
|
EXGYBXPYEOZXFYADXBYJWNXEYAYJWNOZXEXIXRUAUBZRMNZYAYKYLWRVINZXEYMYKWNYEYNYJ
|
|
WNVJZXPYEWNVKZWMWPVLVMYKYLSLZWRSLZYNXEOYMUGZYKXIXRYKWNXISLZYOWMVETZYKYEYC
|
|
YPYITZVNYKWQKLYRYKWMWPYOYPVOWQVETYQYRYDYSVFYLWRRVPVQVMWAYKYTYCYMYAVRZUUAU
|
|
UBYTYCYDUUCVFXIXRRVSVQVMWBVTWCWDWEXLYBBXSSXJXSUFXKYADXBXJXSXIMUHWFWGVMWHW
|
|
IXLCBGSWJWKT $.
|
|
$}
|
|
|
|
$( Arithmetic-geometric mean inequality for ` n = 2 ` . (Contributed by
|
|
Mario Carneiro, 2-Jul-2014.) $)
|
|
amgm2 $p |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) ->
|
|
( sqrt ` ( A x. B ) ) <_ ( ( A + B ) / 2 ) ) $=
|
|
( cr wcel cc0 cle wa c2 cmul co caddc cexp c4 cc wceq remulcl syl2anc recnd
|
|
wbr cmin csqrt cfv cdiv 2cn simpll simprl mulge0 resqrtcl sqmul sylancr sq2
|
|
oveq1i resqrtth oveq2d syl5eq eqtrd sqge0d binom2 binom2sub oveq12d resqcld
|
|
resubcld 2re readdcld pnpcan2d 2timesd 2t2e4 mulassd syl5eqr pnncand 3eqtrd
|
|
2cnd 3eqtr4rd wb 4re subsub23 syl3anc breqtrrd subge0d eqbrtrd sqrtge0 0le2
|
|
mpbid mpanl12 addge0 an4s le2sqd mpbird clt 2pos pm3.2i a1i lemuldiv2 ) ACD
|
|
ZEAFSZGZBCDZEBFSZGZGZHABIJZUAUBZIJZABKJZFSZXBXDHUCJFSZWTXEXCHLJZXDHLJZFSWTX
|
|
GMXAIJZXHFWTXGHHLJZXBHLJZIJZXIWTHNDXBNDXGXLOUDWTXBWTXACDZEXAFSZXBCDZWTWNWQX
|
|
MWNWOWSUEZWPWQWRUFZABPQZABUGZXAUHQZRHXBUIUJWTXLMXKIJXIXJMXKIUKULWTXKXAMIWTX
|
|
MXNXKXAOXRXSXAUMQUNUOUPWTEXHXITJZFSXIXHFSWTEABTJZHLJZYAFWTYBWTABXPXQVBZUQWT
|
|
XHYCTJZXIOZYAYCOZWTYEAHLJZHXAIJZKJZBHLJZKJZYHYITJZYKKJZTJYJYMTJZXIWTXHYLYCY
|
|
NTWTANDZBNDZXHYLOWTAXPRZWTBXQRZABURQWTYPYQYCYNOYRYSABUSQUTWTYJYMYKWTYJWTYHY
|
|
IWTAXPVAZWTHCDZXMYICDVCXRHXAPUJZVDRWTYMWTYHYIYTUUBVBRWTYKWTBXQVARVEWTHYIIJZ
|
|
YIYIKJXIYOWTYIWTYIUUBRZVFWTXIHHIJZXAIJUUCUUEMXAIVGULWTHHXAWTVLZUUFWTXAXRRVH
|
|
VIWTYHYIYIWTYHYTRUUDUUDVJVMVKWTXHNDYCNDXINDYFYGVNWTXHWTXDWTABXPXQVDZVAZRWTY
|
|
CWTYBYDVARWTXIWTMCDXMXICDVOXRMXAPUJZRXHYCXIVPVQWCVRWTXHXIUUHUUIVSWCVTWTXCXD
|
|
WTUUAXOXCCDVCXTHXBPUJUUGWTXOEXBFSZEXCFSZXTWTXMXNUUJXRXSXAWAQUUAEHFSXOUUJGUU
|
|
KVCWBHXBUGWDQWNWQWOWREXDFSABWEWFWGWHWTXOXDCDUUAEHWISZGZXEXFVNXTUUGUUMWTUUAU
|
|
ULVCWJWKWLXBXDHWMVQWC $.
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|
|
|
${
|
|
sqrth.1 $e |- A e. RR $.
|
|
$( Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by
|
|
NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtthi $p |- ( 0 <_ A -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv cmul co wceq remsqsqrt mpan ) ACDEAFGAHIZ
|
|
OJKALBAMN $.
|
|
|
|
$( The square root of a nonnegative real is a real. (Contributed by NM,
|
|
26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtcli $p |- ( 0 <_ A -> ( sqrt ` A ) e. RR ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv resqrtcl mpan ) ACDEAFGAHICDBAJK $.
|
|
|
|
$( The square root of a positive real is positive. (Contributed by NM,
|
|
26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtgt0i $p |- ( 0 < A -> 0 < ( sqrt ` A ) ) $=
|
|
( cr wcel cc0 clt wbr csqrt cfv sqrtgt0 mpan ) ACDEAFGEAHIFGBAJK $.
|
|
|
|
$( Square root of square. (Contributed by NM, 2-Aug-1999.) $)
|
|
sqrtmsqi $p |- ( 0 <_ A -> ( sqrt ` ( A x. A ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr cmul co csqrt cfv wceq sqrtmsq mpan ) ACDEAFGAAHIJK
|
|
ALBAMN $.
|
|
|
|
$( Square root of square. (Contributed by NM, 11-Aug-1999.) $)
|
|
sqrtsqi $p |- ( 0 <_ A -> ( sqrt ` ( A ^ 2 ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr c2 cexp co csqrt cfv wceq sqrtsq mpan ) ACDEAFGAHIJ
|
|
KLAMBANO $.
|
|
|
|
$( Square of square root. (Contributed by NM, 11-Aug-1999.) $)
|
|
sqsqrti $p |- ( 0 <_ A -> ( ( sqrt ` A ) ^ 2 ) = A ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv c2 cexp co wceq resqrtth mpan ) ACDEAFGAH
|
|
IJKLAMBANO $.
|
|
|
|
$( The square root of a nonnegative real is nonnegative. (Contributed by
|
|
NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtge0i $p |- ( 0 <_ A -> 0 <_ ( sqrt ` A ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv sqrtge0 mpan ) ACDEAFGEAHIFGBAJK $.
|
|
|
|
$( A nonnegative number is its own absolute value. (Contributed by NM,
|
|
2-Aug-1999.) $)
|
|
absidi $p |- ( 0 <_ A -> ( abs ` A ) = A ) $=
|
|
( cr wcel cc0 cle wbr cabs cfv wceq absid mpan ) ACDEAFGAHIAJBAKL $.
|
|
|
|
$( A negative number is the negative of its own absolute value.
|
|
(Contributed by NM, 2-Aug-1999.) $)
|
|
absnidi $p |- ( A <_ 0 -> ( abs ` A ) = -u A ) $=
|
|
( cr wcel cc0 cle wbr cabs cfv cneg wceq absnid mpan ) ACDAEFGAHIAJKBALM
|
|
$.
|
|
|
|
$( A real number is less than or equal to its absolute value. (Contributed
|
|
by NM, 2-Aug-1999.) $)
|
|
leabsi $p |- A <_ ( abs ` A ) $=
|
|
( cr wcel cabs cfv cle wbr leabs ax-mp ) ACDAAEFGHBAIJ $.
|
|
|
|
$( Absolute value of a real number. (Contributed by NM, 3-Aug-1999.) $)
|
|
absrei $p |- ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) $=
|
|
( cr wcel cabs cfv c2 cexp co csqrt wceq absre ax-mp ) ACDAEFAGHIJFKBALM
|
|
$.
|
|
|
|
${
|
|
sqrpclii.2 $e |- 0 < A $.
|
|
$( The square root of a positive real is a real. (Contributed by Mario
|
|
Carneiro, 6-Sep-2013.) $)
|
|
sqrtpclii $p |- ( sqrt ` A ) e. RR $=
|
|
( cc0 cle wbr csqrt cfv cr wcel 0re ltleii sqrtcli ax-mp ) DAEFAGHIJDAK
|
|
BCLABMN $.
|
|
|
|
$( The square root of a positive real is positive. (Contributed by NM,
|
|
26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) $)
|
|
sqrtgt0ii $p |- 0 < ( sqrt ` A ) $=
|
|
( cc0 clt wbr csqrt cfv sqrtgt0i ax-mp ) DAEFDAGHEFCABIJ $.
|
|
$}
|
|
|
|
sqr11.1 $e |- B e. RR $.
|
|
$( The square root function is one-to-one. (Contributed by NM,
|
|
27-Jul-1999.) $)
|
|
sqrt11i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv wceq wb wa sqrt11 mpanr1 mpanl1 ) AEFZGAH
|
|
IZGBHIZAJKBJKLABLMZCRSNBEFTUADABOPQ $.
|
|
|
|
$( Square root distributes over multiplication. (Contributed by NM,
|
|
30-Jul-1999.) $)
|
|
sqrtmuli $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( sqrt ` ( A x. B ) ) = ( ( sqrt ` A ) x. ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr cmul co csqrt cfv wceq wa sqrtmul mpanr1 mpanl1 ) A
|
|
EFZGAHIZGBHIZABJKLMALMBLMJKNZCSTOBEFUAUBDABPQR $.
|
|
|
|
${
|
|
sqrmuli.1 $e |- 0 <_ A $.
|
|
sqrmuli.2 $e |- 0 <_ B $.
|
|
$( Square root distributes over multiplication. (Contributed by NM,
|
|
30-Jul-1999.) $)
|
|
sqrtmulii $p |- ( sqrt ` ( A x. B ) )
|
|
= ( ( sqrt ` A ) x. ( sqrt ` B ) ) $=
|
|
( cc0 cle wbr cmul co csqrt cfv wceq sqrtmuli mp2an ) GAHIGBHIABJKLMALM
|
|
BLMJKNEFABCDOP $.
|
|
$}
|
|
|
|
$( Relationship between square root and squares. (Contributed by NM,
|
|
31-Jul-1999.) $)
|
|
sqrtmsq2i $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( ( sqrt ` A ) = B <-> A = ( B x. B ) ) ) $=
|
|
( cc0 cle wbr wa csqrt cfv wceq c2 cexp co cmul cr wcel wb sqrtsq2 mpanr1
|
|
mpanl1 recni sqvali eqeq2i syl6bb ) EAFGZEBFGZHAIJBKZABLMNZKZABBONZKAPQZU
|
|
FUGUHUJRZCULUFHBPQUGUMDABSTUAUIUKABBDUBUCUDUE $.
|
|
|
|
$( Square root is monotonic. (Contributed by NM, 3-Aug-1999.) $)
|
|
sqrtlei $p |- ( ( 0 <_ A /\ 0 <_ B ) ->
|
|
( A <_ B <-> ( sqrt ` A ) <_ ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv wb wa sqrtle mpanr1 mpanl1 ) AEFZGAHIZGBH
|
|
IZABHIAJKBJKHILZCQRMBEFSTDABNOP $.
|
|
|
|
$( Square root is strictly monotonic. (Contributed by Roy F. Longton,
|
|
8-Aug-2005.) $)
|
|
sqrtlti $p |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( sqrt ` A ) <
|
|
( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr clt csqrt cfv wb wa sqrtlt mpanr1 mpanl1 ) AEFZGAHI
|
|
ZGBHIZABJIAKLBKLJIMZCRSNBEFTUADABOPQ $.
|
|
|
|
$( Absolute value and 'less than' relation. (Contributed by NM,
|
|
6-Apr-2005.) $)
|
|
abslti $p |- ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) $=
|
|
( cr wcel cabs cfv clt wbr cneg wa wb abslt mp2an ) AEFBEFAGHBIJBKAIJABIJ
|
|
LMCDABNO $.
|
|
|
|
$( Absolute value and 'less than or equal to' relation. (Contributed by
|
|
NM, 6-Apr-2005.) $)
|
|
abslei $p |- ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) $=
|
|
( cr wcel cabs cfv cle wbr cneg wa wb absle mp2an ) AEFBEFAGHBIJBKAIJABIJ
|
|
LMCDABNO $.
|
|
$}
|
|
|
|
${
|
|
absvalsqi.1 $e |- A e. CC $.
|
|
$( Square of value of absolute value function. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
absvalsqi $p |- ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) $=
|
|
( cc wcel cabs cfv c2 cexp co ccj cmul wceq absvalsq ax-mp ) ACDAEFGHIAAJ
|
|
FKILBAMN $.
|
|
|
|
$( Square of value of absolute value function. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
absvalsq2i $p |- ( ( abs ` A ) ^ 2 ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) $=
|
|
( cc wcel cabs cfv c2 cexp co cre cim caddc wceq absvalsq2 ax-mp ) ACDAEF
|
|
GHIAJFGHIAKFGHILIMBANO $.
|
|
|
|
$( Real closure of absolute value. (Contributed by NM, 2-Aug-1999.) $)
|
|
abscli $p |- ( abs ` A ) e. RR $=
|
|
( cc wcel cabs cfv cr abscl ax-mp ) ACDAEFGDBAHI $.
|
|
|
|
$( Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.) $)
|
|
absge0i $p |- 0 <_ ( abs ` A ) $=
|
|
( cc wcel cc0 cabs cfv cle wbr absge0 ax-mp ) ACDEAFGHIBAJK $.
|
|
|
|
$( Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
|
|
(Contributed by NM, 2-Oct-1999.) $)
|
|
absval2i $p |- ( abs ` A ) =
|
|
( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel cabs cfv cre c2 cexp co cim caddc csqrt wceq absval2 ax-mp ) AC
|
|
DAEFAGFHIJAKFHIJLJMFNBAOP $.
|
|
|
|
$( The absolute value of a number is zero iff the number is zero.
|
|
Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM,
|
|
28-Jul-1999.) $)
|
|
abs00i $p |- ( ( abs ` A ) = 0 <-> A = 0 ) $=
|
|
( cc wcel cabs cfv cc0 wceq wb abs00 ax-mp ) ACDAEFGHAGHIBAJK $.
|
|
|
|
$( The absolute value of a nonzero number is positive. Remark in [Apostol]
|
|
p. 363. (Contributed by NM, 1-Oct-1999.) $)
|
|
absgt0api $p |- ( A # 0 <-> 0 < ( abs ` A ) ) $=
|
|
( cc wcel cc0 cap wbr cabs cfv clt wb absgt0ap ax-mp ) ACDAEFGEAHIJGKBALM
|
|
$.
|
|
|
|
$( Absolute value of negative. (Contributed by NM, 2-Aug-1999.) $)
|
|
absnegi $p |- ( abs ` -u A ) = ( abs ` A ) $=
|
|
( cc wcel cneg cabs cfv wceq absneg ax-mp ) ACDAEFGAFGHBAIJ $.
|
|
|
|
$( The absolute value of a number and its conjugate are the same.
|
|
Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
abscji $p |- ( abs ` ( * ` A ) ) = ( abs ` A ) $=
|
|
( cc wcel ccj cfv cabs wceq abscj ax-mp ) ACDAEFGFAGFHBAIJ $.
|
|
|
|
$( The real part of a number is less than or equal to its absolute value.
|
|
Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
releabsi $p |- ( Re ` A ) <_ ( abs ` A ) $=
|
|
( cc wcel cre cfv cabs cle wbr releabs ax-mp ) ACDAEFAGFHIBAJK $.
|
|
|
|
abssub.2 $e |- B e. CC $.
|
|
$( Swapping order of subtraction doesn't change the absolute value.
|
|
Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.) $)
|
|
abssubi $p |- ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) $=
|
|
( cc wcel cmin co cabs cfv wceq abssub mp2an ) AEFBEFABGHIJBAGHIJKCDABLM
|
|
$.
|
|
|
|
$( Absolute value distributes over multiplication. Proposition 10-3.7(f)
|
|
of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.) $)
|
|
absmuli $p |- ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) $=
|
|
( cc wcel cmul co cabs cfv wceq absmul mp2an ) AEFBEFABGHIJAIJBIJGHKCDABL
|
|
M $.
|
|
|
|
$( Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason]
|
|
p. 133. (Contributed by NM, 2-Oct-1999.) $)
|
|
sqabsaddi $p |- ( ( abs ` ( A + B ) ) ^ 2 ) =
|
|
( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) +
|
|
( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) $=
|
|
( cc wcel caddc co cabs cfv c2 cexp ccj cmul cre wceq sqabsadd mp2an ) AE
|
|
FBEFABGHIJKLHAIJKLHBIJKLHGHKABMJNHOJNHGHPCDABQR $.
|
|
|
|
$( Square of absolute value of difference. (Contributed by Steve
|
|
Rodriguez, 20-Jan-2007.) $)
|
|
sqabssubi $p |- ( ( abs ` ( A - B ) ) ^ 2 ) =
|
|
( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) )
|
|
- ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) $=
|
|
( cc wcel cmin co cabs cfv c2 cexp caddc ccj cmul cre wceq sqabssub mp2an
|
|
) AEFBEFABGHIJKLHAIJKLHBIJKLHMHKABNJOHPJOHGHQCDABRS $.
|
|
|
|
$( Absolute value distributes over division. (Contributed by Jim Kingdon,
|
|
13-Aug-2021.) $)
|
|
absdivapzi $p |- ( B # 0 ->
|
|
( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cabs cfv wceq absdivap mp3an12 ) AEFBEFBGHI
|
|
ABJKLMALMBLMJKNCDABOP $.
|
|
|
|
$( Triangle inequality for absolute value. Proposition 10-3.7(h) of
|
|
[Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
abstrii $p |- ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) $=
|
|
( cc wcel caddc co cabs cfv cle wbr abstri mp2an ) AEFBEFABGHIJAIJBIJGHKL
|
|
CDABMN $.
|
|
|
|
abs3dif.3 $e |- C e. CC $.
|
|
$( Absolute value of differences around common element. (Contributed by
|
|
NM, 2-Oct-1999.) $)
|
|
abs3difi $p |- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) +
|
|
( abs ` ( C - B ) ) ) $=
|
|
( cc wcel cmin co cabs cfv caddc cle wbr abs3dif mp3an ) AGHBGHCGHABIJKLA
|
|
CIJKLCBIJKLMJNODEFABCPQ $.
|
|
|
|
abs3lem.4 $e |- D e. RR $.
|
|
$( Lemma involving absolute value of differences. (Contributed by NM,
|
|
2-Oct-1999.) $)
|
|
abs3lemi $p |- ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\
|
|
( abs ` ( C - B ) ) < ( D / 2 ) ) ->
|
|
( abs ` ( A - B ) ) < D ) $=
|
|
( cc wcel cr cmin co cabs cfv c2 cdiv clt wbr wa wi abs3lem mp4an ) AIJBI
|
|
JCIJDKJACLMNODPQMZRSCBLMNOUDRSTABLMNODRSUAEFGHABCDUBUC $.
|
|
$}
|
|
|
|
${
|
|
sqrgt0d.1 $e |- ( ph -> A e. RR+ ) $.
|
|
$( The square root of a positive real is positive. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
rpsqrtcld $p |- ( ph -> ( sqrt ` A ) e. RR+ ) $=
|
|
( crp wcel csqrt cfv rpsqrtcl syl ) ABDEBFGDECBHI $.
|
|
|
|
$( The square root of a positive real is positive. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
sqrtgt0d $p |- ( ph -> 0 < ( sqrt ` A ) ) $=
|
|
( csqrt cfv rpsqrtcld rpgt0d ) ABDEABCFG $.
|
|
$}
|
|
|
|
${
|
|
resqrcld.1 $e |- ( ph -> A e. RR ) $.
|
|
${
|
|
absnidd.2 $e |- ( ph -> A <_ 0 ) $.
|
|
$( A negative number is the negative of its own absolute value.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
absnidd $p |- ( ph -> ( abs ` A ) = -u A ) $=
|
|
( cr wcel cc0 cle wbr cabs cfv cneg wceq absnid syl2anc ) ABEFBGHIBJKBL
|
|
MCDBNO $.
|
|
$}
|
|
|
|
$( A real number is less than or equal to its absolute value. (Contributed
|
|
by Mario Carneiro, 29-May-2016.) $)
|
|
leabsd $p |- ( ph -> A <_ ( abs ` A ) ) $=
|
|
( cr wcel cabs cfv cle wbr leabs syl ) ABDEBBFGHICBJK $.
|
|
|
|
$( Absolute value of a real number. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
absred $p |- ( ph -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) ) $=
|
|
( cr wcel cabs cfv c2 cexp co csqrt wceq absre syl ) ABDEBFGBHIJKGLCBMN
|
|
$.
|
|
|
|
resqrcld.2 $e |- ( ph -> 0 <_ A ) $.
|
|
$( The square root of a nonnegative real is a real. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
resqrtcld $p |- ( ph -> ( sqrt ` A ) e. RR ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv resqrtcl syl2anc ) ABEFGBHIBJKEFCDBLM $.
|
|
|
|
$( Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
sqrtmsqd $p |- ( ph -> ( sqrt ` ( A x. A ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr cmul co csqrt cfv wceq sqrtmsq syl2anc ) ABEFGBHIBB
|
|
JKLMBNCDBOP $.
|
|
|
|
$( Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
sqrtsqd $p |- ( ph -> ( sqrt ` ( A ^ 2 ) ) = A ) $=
|
|
( cr wcel cc0 cle wbr c2 cexp co csqrt cfv wceq sqrtsq syl2anc ) ABEFGBHI
|
|
BJKLMNBOCDBPQ $.
|
|
|
|
$( The square root of a nonnegative real is nonnegative. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
sqrtge0d $p |- ( ph -> 0 <_ ( sqrt ` A ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv sqrtge0 syl2anc ) ABEFGBHIGBJKHICDBLM $.
|
|
|
|
$( A nonnegative number is its own absolute value. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absidd $p |- ( ph -> ( abs ` A ) = A ) $=
|
|
( cr wcel cc0 cle wbr cabs cfv wceq absid syl2anc ) ABEFGBHIBJKBLCDBMN $.
|
|
|
|
${
|
|
sqrdivd.3 $e |- ( ph -> B e. RR+ ) $.
|
|
$( Square root distributes over division. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
sqrtdivd $p |- ( ph ->
|
|
( sqrt ` ( A / B ) ) = ( ( sqrt ` A ) / ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr crp cdiv co csqrt cfv wceq sqrtdiv syl21anc ) ABG
|
|
HIBJKCLHBCMNOPBOPCOPMNQDEFBCRS $.
|
|
$}
|
|
|
|
sqr11d.3 $e |- ( ph -> B e. RR ) $.
|
|
sqr11d.4 $e |- ( ph -> 0 <_ B ) $.
|
|
$( Square root distributes over multiplication. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
sqrtmuld $p |- ( ph ->
|
|
( sqrt ` ( A x. B ) ) = ( ( sqrt ` A ) x. ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr cmul co csqrt cfv wceq sqrtmul syl22anc ) ABHIJBKLC
|
|
HIJCKLBCMNOPBOPCOPMNQDEFGBCRS $.
|
|
|
|
$( Relationship between square root and squares. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
sqrtsq2d $p |- ( ph -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv wceq c2 cexp co wb sqrtsq2 syl22anc ) ABH
|
|
IJBKLCHIJCKLBMNCOBCPQROSDEFGBCTUA $.
|
|
|
|
$( Square root is monotonic. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
sqrtled $p |- ( ph -> ( A <_ B <-> ( sqrt ` A ) <_ ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr csqrt cfv wb sqrtle syl22anc ) ABHIJBKLCHIJCKLBCKLB
|
|
MNCMNKLODEFGBCPQ $.
|
|
|
|
$( Square root is strictly monotonic. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
sqrtltd $p |- ( ph -> ( A < B <-> ( sqrt ` A ) < ( sqrt ` B ) ) ) $=
|
|
( cr wcel cc0 cle wbr clt csqrt cfv wb sqrtlt syl22anc ) ABHIJBKLCHIJCKLB
|
|
CMLBNOCNOMLPDEFGBCQR $.
|
|
|
|
sqrt11d.5 $e |- ( ph -> ( sqrt ` A ) = ( sqrt ` B ) ) $.
|
|
$( The square root function is one-to-one. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
sqr11d $p |- ( ph -> A = B ) $=
|
|
( csqrt cfv wceq cr wcel cc0 cle wbr wb sqrt11 syl22anc mpbid ) ABIJCIJKZ
|
|
BCKZHABLMNBOPCLMNCOPUAUBQDEFGBCRST $.
|
|
$}
|
|
|
|
${
|
|
absltd.1 $e |- ( ph -> A e. RR ) $.
|
|
absltd.2 $e |- ( ph -> B e. RR ) $.
|
|
$( Absolute value and 'less than' relation. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absltd $p |- ( ph -> ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) ) $=
|
|
( cr wcel cabs cfv clt wbr cneg wa wb abslt syl2anc ) ABFGCFGBHICJKCLBJKB
|
|
CJKMNDEBCOP $.
|
|
|
|
$( Absolute value and 'less than or equal to' relation. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
absled $p |- ( ph -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) ) $=
|
|
( cr wcel cabs cfv cle wbr cneg wa wb absle syl2anc ) ABFGCFGBHICJKCLBJKB
|
|
CJKMNDEBCOP $.
|
|
|
|
${
|
|
abssubge0d.2 $e |- ( ph -> A <_ B ) $.
|
|
$( Absolute value of a nonnegative difference. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abssubge0d $p |- ( ph -> ( abs ` ( B - A ) ) = ( B - A ) ) $=
|
|
( cr wcel cle wbr cmin co cabs cfv wceq abssubge0 syl3anc ) ABGHCGHBCIJ
|
|
CBKLZMNRODEFBCPQ $.
|
|
|
|
$( Absolute value of a nonpositive difference. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abssuble0d $p |- ( ph -> ( abs ` ( A - B ) ) = ( B - A ) ) $=
|
|
( cr wcel cle wbr cmin co cabs cfv wceq abssuble0 syl3anc ) ABGHCGHBCIJ
|
|
BCKLMNCBKLODEFBCPQ $.
|
|
$}
|
|
|
|
absltd.3 $e |- ( ph -> C e. RR ) $.
|
|
$( The absolute value of a difference and 'less than' relation.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
absdifltd $p |- ( ph -> ( ( abs ` ( A - B ) ) < C <->
|
|
( ( B - C ) < A /\ A < ( B + C ) ) ) ) $=
|
|
( cr wcel cmin co cabs cfv clt wbr caddc wa wb absdiflt syl3anc ) ABHICHI
|
|
DHIBCJKLMDNOCDJKBNOBCDPKNOQREFGBCDST $.
|
|
|
|
$( The absolute value of a difference and 'less than or equal to' relation.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
absdifled $p |- ( ph -> ( ( abs ` ( A - B ) ) <_ C <->
|
|
( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) ) $=
|
|
( cr wcel cmin co cabs cfv cle wbr caddc wa wb absdifle syl3anc ) ABHICHI
|
|
DHIBCJKLMDNOCDJKBNOBCDPKNOQREFGBCDST $.
|
|
$}
|
|
|
|
$( Two elements in a half-open interval have separation _strictly_ less than
|
|
the difference between the endpoints. (Contributed by Stefan O'Rear,
|
|
12-Sep-2014.) $)
|
|
icodiamlt $p |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,) B ) /\ D e. (
|
|
A [,) B ) ) ) -> ( abs ` ( C - D ) ) < ( B - A ) ) $=
|
|
( cr wcel wa cico cmin cabs cfv clt wbr cle w3a cxr elico2 recnd resubcld
|
|
co rexr anbi12d biimpd sylan2 cneg simplr simpll negsubdi2d simprl1 simprr1
|
|
simprl2 lesub1dd simprr3 ltsub2dd lelttrd eqbrtrd simprl3 ltsub1dd lesub2dd
|
|
wi simprr2 ltletrd absltd mpbir2and ex syld imp ) AEFZBEFZGZCABHTZFZDVKFZGZ
|
|
CDITZJKBAITZLMZVJVNCEFZACNMZCBLMZOZDEFZADNMZDBLMZOZGZVQVIVHBPFZVNWFUTBUAVHW
|
|
GGZVNWFWHVLWAVMWEABCQABDQUBUCUDVJWFVQVJWFGZVQVPUEZVOLMVOVPLMWIWJABITZVOLWIB
|
|
AWIBVHVIWFUFZRWIAVHVIWFUGZRUHWIWKCBITVOWIABWMWLSWICBVRVSVTWEVJUIZWLSWICDWNW
|
|
BWCWDWAVJUJZSZWIACBWMWNWLVRVSVTWEVJUKULWIDBCWOWLWNWBWCWDWAVJUMUNUOUPWIVOBDI
|
|
TVPWPWIBDWLWOSWIBAWLWMSZWICBDWNWLWOVRVSVTWEVJUQURWIADBWMWOWLWBWCWDWAVJVAUSV
|
|
BWIVOVPWPWQVCVDVEVFVG $.
|
|
|
|
${
|
|
abscld.1 $e |- ( ph -> A e. CC ) $.
|
|
$( Real closure of absolute value. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abscld $p |- ( ph -> ( abs ` A ) e. RR ) $=
|
|
( cc wcel cabs cfv cr abscl syl ) ABDEBFGHECBIJ $.
|
|
|
|
$( Square of value of absolute value function. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absvalsqd $p |- ( ph -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) ) $=
|
|
( cc wcel cabs cfv c2 cexp co ccj cmul wceq absvalsq syl ) ABDEBFGHIJBBKG
|
|
LJMCBNO $.
|
|
|
|
$( Square of value of absolute value function. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absvalsq2d $p |- ( ph -> ( ( abs ` A ) ^ 2 ) =
|
|
( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) $=
|
|
( cc wcel cabs cfv c2 cexp co cre cim caddc wceq absvalsq2 syl ) ABDEBFGH
|
|
IJBKGHIJBLGHIJMJNCBOP $.
|
|
|
|
$( Absolute value is nonnegative. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
absge0d $p |- ( ph -> 0 <_ ( abs ` A ) ) $=
|
|
( cc wcel cc0 cabs cfv cle wbr absge0 syl ) ABDEFBGHIJCBKL $.
|
|
|
|
$( Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
absval2d $p |- ( ph -> ( abs ` A ) =
|
|
( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) ) $=
|
|
( cc wcel cabs cfv cre c2 cexp co cim caddc csqrt wceq absval2 syl ) ABDE
|
|
BFGBHGIJKBLGIJKMKNGOCBPQ $.
|
|
|
|
${
|
|
abs00d.2 $e |- ( ph -> ( abs ` A ) = 0 ) $.
|
|
$( The absolute value of a number is zero iff the number is zero.
|
|
Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abs00d $p |- ( ph -> A = 0 ) $=
|
|
( cabs cfv cc0 wceq abs00ad mpbid ) ABEFGHBGHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
absne0d.2 $e |- ( ph -> A =/= 0 ) $.
|
|
$( The absolute value of a number is zero iff the number is zero.
|
|
Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
absne0d $p |- ( ph -> ( abs ` A ) =/= 0 ) $=
|
|
( cabs cfv cc0 wne abs00ad necon3bid mpbird ) ABEFZGHBGHDALGBGABCIJK $.
|
|
$}
|
|
|
|
${
|
|
absrpclapd.2 $e |- ( ph -> A # 0 ) $.
|
|
$( The absolute value of a complex number apart from zero is a positive
|
|
real. (Contributed by Jim Kingdon, 13-Aug-2021.) $)
|
|
absrpclapd $p |- ( ph -> ( abs ` A ) e. RR+ ) $=
|
|
( cc wcel cc0 cap wbr cabs cfv crp absrpclap syl2anc ) ABEFBGHIBJKLFCDB
|
|
MN $.
|
|
$}
|
|
|
|
$( Absolute value of negative. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
absnegd $p |- ( ph -> ( abs ` -u A ) = ( abs ` A ) ) $=
|
|
( cc wcel cneg cabs cfv wceq absneg syl ) ABDEBFGHBGHICBJK $.
|
|
|
|
$( The absolute value of a number and its conjugate are the same.
|
|
Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abscjd $p |- ( ph -> ( abs ` ( * ` A ) ) = ( abs ` A ) ) $=
|
|
( cc wcel ccj cfv cabs wceq abscj syl ) ABDEBFGHGBHGICBJK $.
|
|
|
|
$( The real part of a number is less than or equal to its absolute value.
|
|
Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
releabsd $p |- ( ph -> ( Re ` A ) <_ ( abs ` A ) ) $=
|
|
( cc wcel cre cfv cabs cle wbr releabs syl ) ABDEBFGBHGIJCBKL $.
|
|
|
|
${
|
|
absexpd.2 $e |- ( ph -> N e. NN0 ) $.
|
|
$( Absolute value of positive integer exponentiation. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
absexpd $p |- ( ph ->
|
|
( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) $=
|
|
( cc wcel cn0 cexp co cabs cfv wceq absexp syl2anc ) ABFGCHGBCIJKLBKLCI
|
|
JMDEBCNO $.
|
|
$}
|
|
|
|
abssubd.2 $e |- ( ph -> B e. CC ) $.
|
|
$( Swapping order of subtraction doesn't change the absolute value.
|
|
Example of [Apostol] p. 363. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abssubd $p |- ( ph -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) ) $=
|
|
( cc wcel cmin co cabs cfv wceq abssub syl2anc ) ABFGCFGBCHIJKCBHIJKLDEBC
|
|
MN $.
|
|
|
|
$( Absolute value distributes over multiplication. Proposition 10-3.7(f)
|
|
of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
absmuld $p |- ( ph ->
|
|
( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) ) $=
|
|
( cc wcel cmul co cabs cfv wceq absmul syl2anc ) ABFGCFGBCHIJKBJKCJKHILDE
|
|
BCMN $.
|
|
|
|
${
|
|
absdivapd.2 $e |- ( ph -> B # 0 ) $.
|
|
$( Absolute value distributes over division. (Contributed by Jim
|
|
Kingdon, 13-Aug-2021.) $)
|
|
absdivapd $p |- ( ph ->
|
|
( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) ) $=
|
|
( cc wcel cc0 cap wbr cdiv co cabs cfv wceq absdivap syl3anc ) ABGHCGHC
|
|
IJKBCLMNOBNOCNOLMPDEFBCQR $.
|
|
$}
|
|
|
|
$( Triangle inequality for absolute value. Proposition 10-3.7(h) of
|
|
[Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
abstrid $p |- ( ph ->
|
|
( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) $=
|
|
( cc wcel caddc co cabs cfv cle wbr abstri syl2anc ) ABFGCFGBCHIJKBJKCJKH
|
|
ILMDEBCNO $.
|
|
|
|
$( Difference of absolute values. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abs2difd $p |- ( ph ->
|
|
( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) $=
|
|
( cc wcel cabs cfv cmin co cle wbr abs2dif syl2anc ) ABFGCFGBHICHIJKBCJKH
|
|
ILMDEBCNO $.
|
|
|
|
$( Difference of absolute values. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
abs2dif2d $p |- ( ph ->
|
|
( abs ` ( A - B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) $=
|
|
( cc wcel cmin co cabs cfv caddc cle wbr abs2dif2 syl2anc ) ABFGCFGBCHIJK
|
|
BJKCJKLIMNDEBCOP $.
|
|
|
|
$( Absolute value of difference of absolute values. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abs2difabsd $p |- ( ph ->
|
|
( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) ) $=
|
|
( cc wcel cabs cfv cmin co cle wbr abs2difabs syl2anc ) ABFGCFGBHICHIJKHI
|
|
BCJKHILMDEBCNO $.
|
|
|
|
abs3difd.3 $e |- ( ph -> C e. CC ) $.
|
|
$( Absolute value of differences around common element. (Contributed by
|
|
Mario Carneiro, 29-May-2016.) $)
|
|
abs3difd $p |- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) +
|
|
( abs ` ( C - B ) ) ) ) $=
|
|
( cc wcel cmin co cabs cfv caddc cle wbr abs3dif syl3anc ) ABHICHIDHIBCJK
|
|
LMBDJKLMDCJKLMNKOPEFGBCDQR $.
|
|
|
|
abs3lemd.4 $e |- ( ph -> D e. RR ) $.
|
|
abs3lemd.5 $e |- ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) ) $.
|
|
abs3lemd.6 $e |- ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) ) $.
|
|
$( Lemma involving absolute value of differences. (Contributed by Mario
|
|
Carneiro, 29-May-2016.) $)
|
|
abs3lemd $p |- ( ph -> ( abs ` ( A - B ) ) < D ) $=
|
|
( cmin co cabs cfv c2 clt wbr cc wcel cdiv cr wa abs3lem syl22anc mp2and
|
|
wi ) ABDLMNOEPUAMZQRZDCLMNOUHQRZBCLMNOEQRZJKABSTCSTDSTEUBTUIUJUCUKUGFGHIB
|
|
CDEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( The rational numbers are dense in ` RR ` : any real number can be
|
|
approximated with arbitrary precision by a rational number. For order
|
|
theoretic density, see ~ qbtwnre . (Contributed by BJ, 15-Oct-2021.) $)
|
|
qdenre $p |-
|
|
( ( A e. RR /\ B e. RR+ ) -> E. x e. QQ ( abs ` ( x - A ) ) < B ) $=
|
|
( cr wcel crp wa cmin co caddc clt wbr w3a cv cabs cfv wrex simpl adantl
|
|
cq rpre resubcld readdcld ltsubrp ltaddrp lttrd 3jca qbtwnre qre ad2antlr
|
|
wi simpll absdiflt biimprd syl3anc reximdva syl5 mpd ) BDEZCFEZGZBCHIZDEZ
|
|
BCJIZDEZVBVDKLZMZANZBHIOPCKLZATQZVAVCVEVFVABCUSUTRZUTCDEZUSCUAZSZUBZVABCV
|
|
KVNUCZVAVBBVDVOVKVPBCUDBCUEUFUGVGVBVHKLVHVDKLGZATQVAVJAVBVDUHVAVQVIATVAVH
|
|
TEZGVHDEZUSVLVQVIUKVRVSVAVHUISUSUTVRULUTVLUSVRVMUJVSUSVLMVIVQVHBCUMUNUOUP
|
|
UQUR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The maximum of two real numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
|
|
(Contributed by Jim Kingdon, 21-Dec-2021.) $)
|
|
maxcom $p |- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < ) $=
|
|
( cr cpr clt prcom supeq1i ) CABDBADEABFG $.
|
|
|
|
$( An upper bound for ` { A , B } ` . (Contributed by Jim Kingdon,
|
|
20-Dec-2021.) $)
|
|
maxabsle $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) $=
|
|
( cr wcel wa c2 cmul caddc cmin cabs cfv cle cdiv simpl recnd 2timesd simpr
|
|
co wbr ppncand eqtr4d resubcl readdcl leadd2dd eqbrtrd readdcld crp 2rp a1i
|
|
abscld leabsd lemuldiv2d mpbid ) ACDZBCDZEZFAGRZABHRZABIRZJKZHRZLSAVAFMRLSU
|
|
PUQURUSHRZVALUPUQAAHRVBUPAUPAUNUONZOZPUPABAVDUPBUNUOQOVDTUAUPUSUTURABUBZUPU
|
|
SUPUSVEOUJZABUCZUPUSVEUKUDUEUPAVAFVCUPURUTVGVFUFFUGDUPUHUIULUM $.
|
|
|
|
${
|
|
$d A f g y $. $d B f g y $.
|
|
$( Value of maximum when we know which number is larger. (Contributed by
|
|
Jim Kingdon, 21-Dec-2021.) $)
|
|
maxleim $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A <_ B -> sup ( { A , B } , RR , < ) = B ) ) $=
|
|
( vy vg vf cr wcel wa cle wbr clt wceq cv wn adantl ad2antrr breq2 notbid
|
|
wb mpbird cpr lttri3 simplr prid2g syl simpll simpllr lensymd ltnrd elpri
|
|
csup wo mpjaodan supmaxti ex ) AFGZBFGZHZABIJZABUAZFKUKBLURUSHZCDEFUTBKEM
|
|
ZFGDMZFGHVBVCLVBVCKJNVCVBKJNHSVAVBVCUBOUPUQUSUCZVAUQBUTGVDABFUDUEVACMZUTG
|
|
ZHZVEALZBVEKJZNZVEBLZVGVHHZVJBAKJZNZVLABVAUPVFVHUPUQUSUFPVAUQVFVHVDPURUSV
|
|
FVHUGUHVHVJVNSVGVHVIVMVEABKQROTVGVKHZVJBBKJZNZVOBVAUQVFVKVDPUIVKVJVQSVGVK
|
|
VIVPVEBBKQROTVFVHVKULVAVEABUJOUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
maxabslemab.a $e |- ( ph -> A e. RR ) $.
|
|
maxabslemab.b $e |- ( ph -> B e. RR ) $.
|
|
maxabslemab.ab $e |- ( ph -> A < B ) $.
|
|
$( Lemma for ~ maxabs . A variation of ~ maxleim - that is, if we know
|
|
which of two real numbers is larger, we know the maximum of the two.
|
|
(Contributed by Jim Kingdon, 21-Dec-2021.) $)
|
|
maxabslemab $p |- ( ph ->
|
|
( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) = B ) $=
|
|
( caddc co cmin cabs c2 cdiv wceq cmul recnd ppncand addcomd ltled addcld
|
|
cfv abssuble0d oveq12d 2timesd 3eqtr4rd resubcld eqeltrd 2cnd cc0 cap wbr
|
|
cr 2ap0 a1i divmulapd mpbird ) ABCGHZBCIHJTZGHZKLHCMKCNHZURMACBGHZCBIHZGH
|
|
CCGHURUSACBCACEOZABDOZVBPAUPUTUQVAGABCVCVBQABCDEABCDEFRUAZUBACVBUCUDAURKC
|
|
AUPUQABCVCVBSAUQAUQVAUKVDACBEDUEUFOSAUGVBKUHUIUJAULUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
maxabslemlub.a $e |- ( ph -> A e. RR ) $.
|
|
maxabslemlub.b $e |- ( ph -> B e. RR ) $.
|
|
maxabslemlub.c $e |- ( ph -> C e. RR ) $.
|
|
maxabslemlub.clt $e |- ( ph ->
|
|
C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) $.
|
|
$( Lemma for ~ maxabs . A least upper bound for ` { A , B } ` .
|
|
(Contributed by Jim Kingdon, 20-Dec-2021.) $)
|
|
maxabslemlub $p |- ( ph -> ( C < A \/ C < B ) ) $=
|
|
( clt wbr caddc co cmin c2 cr wcel recnd adantr cmul c1 cabs cfv readdcld
|
|
cdiv wo subcld abscld rehalfcld axltwlin syl3anc mpd cc0 resubcld 2re a1i
|
|
wi wa remulcld subsub4d 2cnd mulsubfacd 2m1e1 oveq1i syl6eq mulid2d eqtrd
|
|
oveq1d eqtr3d simpr crp 2rp ltmuldiv2d mpbird ltsubadd2d eqbrtrrd syl2anc
|
|
cc ltabs sublt0d mpbid maxabslemab breqtrd ex orim2d ) ADBIJZBBCKLZBCMLZU
|
|
AUBZKLZNUDLZIJZUEZWEDCIJZUEADWJIJZWLHADOPWJOPBOPZWNWLUPGAWIAWFWHABCEFUCZA
|
|
WGABCABEQZACFQZUFUGZUCZUHEDWJBUIUJUKAWKWMWEAWKWMAWKUQZDWJCIAWNWKHRXABCAWO
|
|
WKERZACOPWKFRZXAWGULIJZBCIJXAWGOPWGWHIJXDXABCXBXCUMXANBSLZWFMLZWGWHIXAXEB
|
|
MLZCMLXFWGXAXEBCXAXEXANBNOPXAUNUOXBURZQABVQPWKWQRZACVQPWKWRRUSXAXGBCMXAXG
|
|
TBSLZBXAXGNTMLZBSLXJXANBXAUTXIVAXKTBSVBVCVDXABXIVEVFVGVHXAXFWHIJXEWIIJZXA
|
|
XLWKAWKVIXABWINXBAWIOPWKWTRNVJPXAVKUOVLVMXAXEWFWHXHAWFOPWKWPRAWHOPWKWSRVN
|
|
VMVOWGVRVPXABCXBXCVSVTWAWBWCWDUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x z $. $d B x z $.
|
|
$( Lemma for ~ maxabs . Value of the supremum. (Contributed by Jim
|
|
Kingdon, 22-Dec-2021.) $)
|
|
maxabslemval $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\
|
|
A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\
|
|
A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ->
|
|
E. z e. { A , B } x < z ) ) ) $=
|
|
( cr wcel wa caddc co cmin cabs cfv c2 cdiv cv clt wbr wn wral breq2 wrex
|
|
cpr wi readdcl simpl recnd simpr subcld abscld readdcld rehalfcld wceq wo
|
|
vex maxabsle lensymd notbid syl5ibrcom cle ancoms addcomd abssubd oveq12d
|
|
elpr oveq1d breqtrd syl5bi ralrimiv prid1g ad4antr rspcev sylancom prid2g
|
|
jaod ad4antlr ad2antrr simplr maxabslemlub mpjaodan ex ralrimiva 3jca ) C
|
|
EFZDEFZGZCDHIZCDJIZKLZHIZMNIZEFWJAOZPQZRZACDUBZSWKWJPQZWKBOZPQZBWNUAZUCZA
|
|
ESWEWIWEWFWHCDUDWEWGWECDWECWCWDUEZUFZWEDWCWDUGZUFZUHUIUJUKZWEWMAWNWKWNFWK
|
|
CULZWKDULZUMWEWMWKCDAUNVDWEXEWMXFWEWMXEWJCPQZRWECWJWTXDCDUOUPXEWLXGWKCWJP
|
|
TUQURWEWMXFWJDPQZRWEDWJXBXDWEDDCHIZDCJIKLZHIZMNIZWJUSWDWCDXLUSQDCUOUTWEXK
|
|
WIMNWEXIWFXJWHHWEDCXCXAVAWEDCXCXAVBVCVEVFUPXFWLXHWKDWJPTUQURVNVGVHWEWSAEW
|
|
EWKEFZGZWOWRXNWOGZWKCPQZWRWKDPQZXOXPCWNFZWRWCXRWDXMWOXPCDEVIVJWQXPBCWNWPC
|
|
WKPTVKVLXOXQDWNFZWRWDXSWCXMWOXQCDEVMVOWQXQBDWNWPDWKPTVKVLXOCDWKWEWCXMWOWT
|
|
VPWEWDXMWOXBVPWEXMWOVQXNWOUGVRVSVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d A u v x z $. $d B u v x z $.
|
|
$( Maximum of two real numbers in terms of absolute value. (Contributed by
|
|
Jim Kingdon, 20-Dec-2021.) $)
|
|
maxabs $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
sup ( { A , B } , RR , < ) =
|
|
( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) $=
|
|
( vx vz vv vu cr wcel wa caddc co cmin cabs cfv cv clt wbr wn wral wceq
|
|
c2 cdiv cpr wrex wi w3a csup maxabslemval wb lttri3 adantl eqsupti mpd )
|
|
AGHBGHIZABJKABLKMNJKUAUBKZGHUOCOZPQRCABUCZSUPUOPQUPDOPQDUQUDUECGSUFUQGPUG
|
|
UOTCDABUHUNCDEFGUQUOPFOZGHEOZGHIURUSTURUSPQRUSURPQRIUIUNURUSUJUKULUM $.
|
|
$}
|
|
|
|
$( The maximum of two real numbers is a real number. (Contributed by Jim
|
|
Kingdon, 22-Dec-2021.) $)
|
|
maxcl $p |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR ) $=
|
|
( cr wcel wa cpr clt csup caddc co cmin cabs cfv c2 cdiv maxabs simpl recnd
|
|
readdcl simpr subcld abscld readdcld rehalfcld eqeltrd ) ACDZBCDZEZABFCGHAB
|
|
IJZABKJZLMZIJZNOJCABPUHULUHUIUKABSUHUJUHABUHAUFUGQRUHBUFUGTRUAUBUCUDUE $.
|
|
|
|
$( The maximum of two reals is no smaller than the first real. Lemma 3.10 of
|
|
[Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.) $)
|
|
maxle1 $p |- ( ( A e. RR /\ B e. RR ) -> A <_ sup ( { A , B } , RR , < ) ) $=
|
|
( cr wcel wa caddc co cmin cabs cfv cdiv cpr clt csup cle maxabsle breqtrrd
|
|
c2 maxabs ) ACDBCDEAABFGABHGIJFGRKGABLCMNOABPABSQ $.
|
|
|
|
$( The maximum of two reals is no smaller than the second real. Lemma 3.10
|
|
of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.) $)
|
|
maxle2 $p |- ( ( A e. RR /\ B e. RR ) -> B <_ sup ( { A , B } , RR , < ) ) $=
|
|
( cr wcel wa cpr clt csup cle wbr maxle1 ancoms maxcom syl6breq ) ACDZBCDZE
|
|
BBAFCGHZABFCGHIPOBQIJBAKLBAMN $.
|
|
|
|
${
|
|
$d A f g x $. $d A x y z $. $d B f g x $. $d B x y z $. $d C f g x $.
|
|
$d C x z $.
|
|
$( The maximum of two reals is a least upper bound. Lemma 3.11 of
|
|
[Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.) $)
|
|
maxleast $p |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\
|
|
( A <_ C /\ B <_ C ) ) ->
|
|
sup ( { A , B } , RR , < ) <_ C ) $=
|
|
( vz vx vy vg vf cr wcel cle wbr wa clt wn cv wceq wral wi co w3a csup wo
|
|
cpr ioran wrex simp3 wb lttri3 adantl caddc cmin cabs cfv c2 maxabslemval
|
|
3anass sylib breq1 notbid ralbidv breq2 imbi1d anbi12d rspcev syl 3adant3
|
|
cdiv suplubti mpand elpri adantr biimpcd orim12d mpd rexlimiva syl6 con3d
|
|
syl5bir simp1 lenltd simp2 supclti 3imtr4d imp ) AIJZBIJZCIJZUAZACKLZBCKL
|
|
ZMZABUDZINUBZCKLZWICANLZOZCBNLZOZMZCWNNLZOZWLWOWTWPWRUCZOWIXBWPWRUEWIXAXC
|
|
WIXACDPZNLZDWMUFZXCWIWHXAXFWFWGWHUGZWIEFDGHIWMCNHPZIJGPZIJMXHXIQXHXINLOXI
|
|
XHNLOMUHWIXHXIUIUJZWFWGEPZFPZNLZOZFWMRZXLXKNLZXLXDNLDWMUFZSZFIRZMZEIUFZWH
|
|
WFWGMZABUKTABULTUMUNUKTUOVHTZIJZYCXLNLZOZFWMRZXLYCNLZXQSZFIRZMZMZYAYBYDYG
|
|
YJUAYLFDABUPYDYGYJUQURXTYKEYCIXKYCQZXOYGXSYJYMXNYFFWMYMXMYEXKYCXLNUSUTVAY
|
|
MXRYIFIYMXPYHXQXKYCXLNVBVCVAVDVEVFVGZVIVJXEXCDWMXDWMJZXEMZXDAQZXDBQZUCZXC
|
|
YOYSXEXDABVKVLYPYQWPYRWRXEYQWPSYOYQXEWPXDACNVBVMUJXEYRWRSYOYRXEWRXDBCNVBV
|
|
MUJVNVOVPVQVRVSWIWJWQWKWSWIACWFWGWHVTXGWAWIBCWFWGWHWBXGWAVDWIWNCWIEFDGHIW
|
|
MNXJYNWCXGWAWDWE $.
|
|
$}
|
|
|
|
$( Two ways of saying the maximum of two numbers is less than or equal to a
|
|
third. (Contributed by Jim Kingdon, 31-Jan-2022.) $)
|
|
maxleastb $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( sup ( { A , B } , RR , < ) <_ C <-> ( A <_ C /\ B <_ C ) ) ) $=
|
|
( cr wcel w3a cpr clt csup cle wa simpl1 simpl2 maxcl syl2anc simpl3 maxle1
|
|
wbr simpr letrd maxle2 jca maxleast impbida ) ADEZBDEZCDEZFZABGDHIZCJRZACJR
|
|
ZBCJRZKUHUJKZUKULUMAUICUEUFUGUJLZUMUEUFUIDEUNUEUFUGUJMZABNOZUEUFUGUJPZUMUEU
|
|
FAUIJRUNUOABQOUHUJSZTUMBUICUOUPUQUMUEUFBUIJRUNUOABUAOURTUBABCUCUD $.
|
|
|
|
$( The maximum as a least upper bound, in terms of less than. (Contributed
|
|
by Jim Kingdon, 9-Feb-2022.) $)
|
|
maxleastlt $p |- ( ( ( A e. RR /\ B e. RR )
|
|
/\ ( C e. RR /\ C < sup ( { A , B } , RR , < ) ) )
|
|
-> ( C < A \/ C < B ) ) $=
|
|
( cr wcel wa cpr clt csup wbr simpll simplr simprl caddc cmin cabs cfv cdiv
|
|
co c2 simprr wb maxabs breq2d adantr mpbid maxabslemlub ) ADEZBDEZFZCDEZCAB
|
|
GDHIZHJZFZFZABCUHUIUNKUHUIUNLUJUKUMMUOUMCABNSABOSPQNSTRSZHJZUJUKUMUAUJUMUQU
|
|
BUNUJULUPCHABUCUDUEUFUG $.
|
|
|
|
$( Equivalence of ` <_ ` and being equal to the maximum of two reals. Lemma
|
|
3.12 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.) $)
|
|
maxleb $p |- ( ( A e. RR /\ B e. RR ) ->
|
|
( A <_ B <-> sup ( { A , B } , RR , < ) = B ) ) $=
|
|
( cr wcel cle wbr cpr clt csup wceq maxleim maxle1 breq2 syl5ibcom impbid
|
|
wa ) ACDBCDPZABEFZABGCHIZBJZABKQASEFTRABLSBAEMNO $.
|
|
|
|
$( Absolute value of a real number in terms of maximum. Definition 3.13 of
|
|
[Geuvers], p. 11. (Contributed by BJ and Jim Kingdon, 21-Dec-2021.) $)
|
|
dfabsmax $p |- ( A e. RR -> ( abs ` A ) = sup ( { A , -u A } , RR , < ) ) $=
|
|
( cr wcel cabs cfv cneg cpr clt csup wceq cle id renegcl maxcl mpdan maxle2
|
|
wbr lenegcon1d maxle1 mpbir2and wa absle recn abscld leabsd absnegd breqtrd
|
|
wb leabs maxleast syl32anc letri3d ) ABCZADEZAAFZGBHIZJUNUPKQZUPUNKQZUMUQUP
|
|
FAKQZAUPKQZUMAUPUMLZUMUOBCZUPBCZAMZAUONOZUMVBUOUPKQVDAUOPORUMVBUTVDAUOSOUMV
|
|
CUQUSUTUAUHVEAUPUBOTUMUMVBUNBCAUNKQUOUNKQURVAVDUMAAUCZUDZAUIUMUOUODEUNKUMUO
|
|
VDUEUMAVFUFUGAUOUNUJUKUMUNUPVGVEULT $.
|
|
|
|
$( Two ways of saying the maximum of two numbers is less than a third.
|
|
(Contributed by Jim Kingdon, 10-Feb-2022.) $)
|
|
maxltsup $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) ) $=
|
|
( cr wcel clt wa simpl1 simpl2 syl2anc simpl3 caddc co cmin c2 cmul eqbrtrd
|
|
wbr recnd mpbird w3a cpr csup maxcl cle maxle1 3adant3 adantr simpr lelttrd
|
|
maxle2 jca cabs cfv cdiv maxabs 2re a1i remulcld addcld negsubdi2d readdcld
|
|
wceq cneg resubcld 2timesd pnncand eqtr4d simprr ltmul2dd eqbrtrrd ltsub23d
|
|
crp 2rp nppcan3d simprl ltaddsubd mpbid absltd ltaddsub2d ltdivmuld impbida
|
|
abscld ) ADEZBDEZCDEZUAZABUBDFUCZCFRZACFRZBCFRZGZWGWIGZWJWKWMAWHCWDWEWFWIHZ
|
|
WMWDWEWHDEWNWDWEWFWIIZABUDJZWDWEWFWIKZWGAWHUERZWIWDWEWRWFABUFUGUHWGWIUIZUJW
|
|
MBWHCWOWPWQWMWDWEBWHUERWNWOABUKJWSUJULWGWLGZWHABLMZABNMZUMUNZLMZOUOMZCFWGWH
|
|
XEVCZWLWDWEXFWFABUPUGUHWTXECFRXDOCPMZFRZWTXHXCXGXANMZFRZWTXJXIVDZXBFRZXBXIF
|
|
RZGWTXLXMWTXKXAXGNMXBFWTXGXAWTXGWTOCODEWTUQURWDWEWFWLKZUSZSWTABWTAWDWEWFWLH
|
|
ZSZWTBWDWEWFWLIZSZUTVAWTXAXBXGWTABXPXRVBZWTABXPXRVEZXOWTOBPMZXAXBNMZXGFWTYB
|
|
BBLMYCWTBXSVFWTABBXQXSXSVGVHWTBCOXRXNOVMEWTVNURZWGWJWKVIVJVKVLQWTXBXALMZXGF
|
|
RXMWTYEOAPMZXGFWTYEAALMYFWTABAXQXSXQVOWTAXQVFVHWTACOXPXNYDWGWJWKVPVJQWTXBXA
|
|
XGYAXTXOVQVRULWTXBXIYAWTXGXAXOXTVEVSTWTXAXCXGXTWTXBWTXBYASWCZXOVTTWTXDCOWTX
|
|
AXCXTYGVBXNYDWATQWB $.
|
|
|
|
$( The sum of the positive and negative part functions is the absolute value
|
|
function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.) $)
|
|
max0addsup $p |- ( A e. RR ->
|
|
( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) )
|
|
= ( abs ` A ) ) $=
|
|
( cr wcel cc0 cpr clt csup caddc cabs cfv cdiv cmin wceq 0re oveq12d oveq1d
|
|
co c2 eqtrd addcld cneg maxabs mpan2 addid1d subid1d fveq2d renegcl sylancl
|
|
cmul recn recnd absnegd abscld 2cnd cap wbr 2ap0 a1i divdirapd add4d negidd
|
|
addid2d 3eqtrd 3eqtr2d 2timesd divcanap3d ) ABCZADEBFGZAUAZDEBFGZHQZAIJZVLH
|
|
QZRKQZRVLUIQZRKQVLVGVKAVLHQZRKQZVIVLHQZRKQZHQVPVRHQZRKQVNVGVHVQVJVSHVGVHADH
|
|
QZADLQZIJZHQZRKQZVQVGDBCZVHWEMNADUBUCVGWDVPRKVGWAAWCVLHVGAAUJZUDVGWBAIVGAWG
|
|
UEUFOPSVGVJVIDHQZVIDLQZIJZHQZRKQZVSVGVIBCWFVJWLMAUGZNVIDUBUHVGWKVRRKVGWHVIW
|
|
JVLHVGVIVGVIWMUKZUDVGWJVIIJVLVGWIVIIVGVIWNUEUFVGAWGULSOPSOVGVPVRRVGAVLWGVGV
|
|
LVGAWGUMUKZTVGVIVLWNWOTVGUNZRDUOUPVGUQURZUSVGVTVMRKVGVTAVIHQZVMHQDVMHQVMVGA
|
|
VLVIVLWGWOWNWOUTVGWRDVMHVGAWGVAPVGVMVGVLVLWOWOTVBVCPVDVGVOVMRKVGVLWOVEPVGVL
|
|
RWOWPWQVFVD $.
|
|
|
|
${
|
|
$d j k x y A $. $d j x y ph $. $d j x y ps $.
|
|
$( Combine two different upper real properties into one. (Contributed by
|
|
Mario Carneiro, 8-May-2016.) $)
|
|
rexanre $p |- ( A C_ RR ->
|
|
( E. j e. RR A. k e. A ( j <_ k -> ( ph /\ ps ) ) <->
|
|
( E. j e. RR A. k e. A ( j <_ k -> ph ) /\
|
|
E. j e. RR A. k e. A ( j <_ k -> ps ) ) ) ) $=
|
|
( vx vy cr cv cle wbr wa wi wral wrex wceq breq1 imbi1d ralbidv wcel csup
|
|
wss simpl imim2i ralimi reximi simpr jca cbvrexv anbi12i reeanv cpr maxcl
|
|
bitr4i clt adantl r19.26 prth wb simplrl simplrr sselda maxleastb syl3anc
|
|
syl5ibr ralimdva syl5bir rspcev syl6an rexlimdvva syl5bi impbid2 ) CHUBZD
|
|
IZEIZJKZABLZMZECNZDHOZVPAMZECNZDHOZVPBMZECNZDHOZLZVTWCWFVSWBDHVRWAECVQAVP
|
|
ABUCUDUEUFVSWEDHVRWDECVQBVPABUGUDUEUFUHWGFIZVOJKZAMZECNZGIZVOJKZBMZECNZLZ
|
|
GHOFHOZVMVTWGWKFHOZWOGHOZLWQWCWRWFWSWBWKDFHVNWHPZWAWJECWTVPWIAVNWHVOJQRSU
|
|
IWEWODGHVNWLPZWDWNECXAVPWMBVNWLVOJQRSUIUJWKWOFGHHUKUNVMWPVTFGHHVMWHHTZWLH
|
|
TZLZLZWHWLULHUOUAZHTZWPXFVOJKZVQMZECNZVTXDXGVMWHWLUMUPWPWJWNLZECNXEXJWJWN
|
|
ECUQXEXKXIECXKXIXEVOCTZLZWIWMLZVQMWIAWMBURXMXHXNVQXMXBXCVOHTXHXNUSVMXBXCX
|
|
LUTVMXBXCXLVAXECHVOVMXDUCVBWHWLVOVCVDRVEVFVGVSXJDXFHVNXFPZVRXIECXOVPXHVQV
|
|
NXFVOJQRSVHVIVJVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d j k n A $. $d j k n B $. $d j n ph $.
|
|
$( Restrict the base of an upper real quantifier to an upper real set.
|
|
(Contributed by Mario Carneiro, 12-May-2016.) $)
|
|
rexico $p |- ( ( A C_ RR /\ B e. RR ) ->
|
|
( E. j e. ( B [,) +oo ) A. k e. A ( j <_ k -> ph ) <->
|
|
E. j e. RR A. k e. A ( j <_ k -> ph ) ) ) $=
|
|
( vn cr wss wcel wa cv cle wbr wi wral cpnf wrex simpr adantll wb cico co
|
|
cxr pnfxr icossre sylancl ssrexv syl cpr clt csup maxcl elicopnf ad2antlr
|
|
maxle1 mpbir2and simpllr simplr simpll sselda syl6bi imim1d ralimdva wceq
|
|
maxleastb syl3anc imbi1d ralbidv rspcev syl6an rexlimdva cbvrexv syl6ib
|
|
breq1 impbid ) BGHZCGIZJZDKZEKZLMZANZEBOZDCPUAUBZQZWCDGQZVRWDGHZWEWFNVRVQ
|
|
PUCIWGVPVQRUDCPUEUFWCDWDGUGUHVRWFFKZVTLMZANZEBOZFWDQZWEVRWCWLDGVRVSGIZJZC
|
|
VSUIGUJUKZWDIZWCWOVTLMZANZEBOZWLWNWPWOGIZCWOLMZVQWMWTVPCVSULSVQWMXAVPCVSU
|
|
OSVQWPWTXAJTVPWMCWOUMUNUPWNWBWREBWNVTBIZJZWQWAAXCWQCVTLMZWAJZWAXCVQWMVTGI
|
|
WQXETVPVQWMXBUQVRWMXBURWNBGVTVPVQWMUSUTCVSVTVEVFXDWARVAVBVCWKWSFWOWDWHWOV
|
|
DZWJWREBXFWIWQAWHWOVTLVNVGVHVIVJVKWKWCFDWDWHVSVDZWJWBEBXGWIWAAWHVSVTLVNVG
|
|
VHVLVMVO $.
|
|
$}
|
|
|
|
$( The maximum of two real numbers is one of those numbers if and only if
|
|
dichotomy ( ` A <_ B \/ B <_ A ` ) holds. For example, this can be
|
|
combined with ~ zletric if one is dealing with integers, but real number
|
|
dichotomy in general does not follow from our axioms. (Contributed by Jim
|
|
Kingdon, 1-Feb-2022.) $)
|
|
maxclpr $p |- ( ( A e. RR /\ B e. RR ) -> (
|
|
sup ( { A , B } , RR , < ) e. { A , B }
|
|
<-> ( A <_ B \/ B <_ A ) ) ) $=
|
|
( cr wcel wa cpr clt csup cle wbr wo wceq wb maxcl syl maxleb maxcom eqeq1i
|
|
elprg syl6bb ancoms orbi12d bitr4d orcom ) ACDZBCDZEZABFZCGHZUHDZBAIJZABIJZ
|
|
KZULUKKUGUJUIALZUIBLZKZUMUGUICDUJUPMABNUIABCSOUGUKUNULUOUFUEUKUNMUFUEEUKBAF
|
|
CGHZALUNBAPUQUIABAQRTUAABPUBUCUKULUDT $.
|
|
|
|
${
|
|
$d A x y s u v w $.
|
|
$( A nonempty finite set of real numbers has an upper bound. (Contributed
|
|
by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro,
|
|
13-Feb-2014.) $)
|
|
fimaxre2 $p |- ( ( A C_ RR /\ A e. Fin ) ->
|
|
E. x e. RR A. y e. A y <_ x ) $=
|
|
( vs wcel cr wss cv cle wbr wral wrex wi c0 wceq sseq1 raleq cc0 syl2anc
|
|
wa vw vu vv cfn csn cun rexbidv imbi12d 0re ral0 breq2 ralbidv rspcev a1i
|
|
mp2an unss biimpri simpld adantl simplr mpd cbvrexv sylib cpr csup simprl
|
|
clt simprd vex snss sylibr ad2antlr maxcl nfv nfcv nfra1 nfim nfan simprr
|
|
nfrexxy maxle1 r19.27av r19.21bi simpr sseldd adantr letr syl3anc ralrimi
|
|
ad2antrr ex maxle2 wb breq1 ralsng mpbird ralun rexlimddv exp31 findcard2
|
|
syl impcom ) CUDECFGZBHZAHZIJZBCKZAFLZUAHZFGZXFBXIKZAFLZMNFGZXFBNKZAFLZMU
|
|
BHZFGZXFBXPKZAFLZMZXPUCHZUEZUFZFGZXFBYCKZAFLZMXCXHMUAUBUCCXINOZXJXMXLXOXI
|
|
NFPYGXKXNAFXFBXINQUGUHXIXPOZXJXQXLXSXIXPFPYHXKXRAFXFBXIXPQUGUHXIYCOZXJYDX
|
|
LYFXIYCFPYIXKYEAFXFBXIYCQUGUHXICOZXJXCXLXHXICFPYJXKXGAFXFBXICQUGUHXOXMRFE
|
|
XDRIJZBNKZXOUIYKBUJXNYLARFXEROXFYKBNXERXDIUKULUMUOUNXPUDEZXTYDYFYMXTTZYDT
|
|
ZXDDHZIJZBXPKZYFDFYOXSYRDFLYOXQXSYDXQYNYDXQYBFGZXQYSTYDXPYBFUPUQZURUSZYMX
|
|
TYDUTVAXRYRADFXEYPOXFYQBXPXEYPXDIUKULVBVCYOYPFEZYRTZTZYPYAVDFVGVEZFEZXDUU
|
|
EIJZBYCKZYFUUDUUBYAFEZUUFYOUUBYRVFZYDUUIYNUUCYDYSUUIYDXQYSYTVHYAFUCVIVJVK
|
|
VLZYPYAVMSZUUDUUGBXPKUUGBYBKZUUHUUDUUGBXPYOUUCBYNYDBYMXTBYMBVNXQXSBXQBVNX
|
|
RBAFBFVOXFBXPVPVTVQVRYDBVNVRUUBYRBUUBBVNYQBXPVPVRVRUUDXDXPEZUUGUUDUUNTZYQ
|
|
YPUUEIJZTZUUGUUDUUQBXPUUDYRUUPUUQBXPKYOUUBYRVSUUDUUBUUIUUPUUJUUKYPYAWASYQ
|
|
UUPBXPWBSWCUUOXDFEUUBUUFUUQUUGMUUOXPFXDYOXQUUCUUNUUAWJUUDUUNWDWEUUDUUBUUN
|
|
UUJWFUUDUUFUUNUULWFXDYPUUEWGWHVAWKWIUUDUUMYAUUEIJZUUDUUBUUIUURUUJUUKYPYAW
|
|
LSUUDUUIUUMUURWMUUKUUGUURBYAFXDYAUUEIWNWOXAWPUUGBXPYBWQSYEUUHAUUEFXEUUEOX
|
|
FUUGBYCXEUUEXDIUKULUMSWRWSWTXB $.
|
|
$}
|
|
|
|
${
|
|
$d A a n x $.
|
|
$( The negation of a finite set of real numbers is finite. (Contributed by
|
|
AV, 9-Aug-2020.) $)
|
|
negfi $p |- ( ( A C_ RR /\ A e. Fin )
|
|
-> { n e. RR | -u n e. A } e. Fin ) $=
|
|
( va vx cr cfn wcel cv cneg crab wceq imp syl eqcomd eleq1d wb cvv cab wa
|
|
adantl wss cmpt crn cdm wral ssel renegcl syl6 ralrimiva dmmptg fundmfibi
|
|
wfun funmpt mp1i bitr4d reex ssex mptexg wf1o eqid negf1o f1of1 f1vrnfibi
|
|
wf1 syl2anc wrex wi recn negnegd biimpcd jca mpdan eleq1 negeq syl5ibrcom
|
|
anbi12d rexlimdva simprr eqeq2d cc negneg ad2antrl rspcedvd impbid abbidv
|
|
ex rnmpt df-rab 3eqtr4g 3bitrd biimpa ) AEUAZAFGZBHZIZAGZBEJZFGZWLWMCACHZ
|
|
IZUBZFGZXAUCZFGZWRWLWMXAUDZFGZXBWLAXEFWLXEAWLWTEGZCAUEXEAKWLXGCAWLWSAGZXG
|
|
WLXHWSEGZXGAEWSUFZWSUGZUHLUICAWTEUJMNOXAULXBXFPWLCAWTUMXAUKUNUOWLXAQGZADH
|
|
IAGDEJZXAVDZXBXDPWLAQGXLAEUPUQCAWTQURMWLAXMXAUSXNCADXAXAUTZVAAXMXAVBMAXMX
|
|
AQVCVEWLXCWQFWLWNWTKZCAVFZBRWNEGZWPSZBRXCWQWLXQXSBWLXQXSWLXPXSCAWLXHSZXSX
|
|
PXGWTIZAGZSZXTXIYCWLXHXIXJLXTXISXGYBXIXGXTXKTXTXIYBXHXIYBVGWLXIXHYBXIWSYA
|
|
AXIYAWSXIWSWSVHVINOVJTLVKVLXPXRXGWPYBWNWTEVMXPWOYAAWNWTVNOVPVOVQWLXSXQWLX
|
|
SSZXPWNWOIZKZCWOAWLXRWPVRWSWOKZXPYFPYDYGWTYEWNWSWOVNVSTXRYFWLWPXRWNVTGZYF
|
|
WNVHYHYEWNWNWANMWBWCWFWDWECBAWTXAXOWGWPBEWHWIOWJWK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The minimum of two real numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The minimum of two reals is commutative. (Contributed by Jim Kingdon,
|
|
8-Feb-2021.) $)
|
|
mincom $p |- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < ) $=
|
|
( cr cpr clt prcom infeq1i ) CABDBADEABFG $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
|
|
8-Feb-2021.) $)
|
|
minmax $p |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < )
|
|
= -u sup ( { -u A , -u B } , RR , < ) ) $=
|
|
( vz vx vy cr wcel wa clt cneg wbr wral wceq wo simpr syl2an cle ad2antrr
|
|
wb mpbird cpr cinf cv crab csup wn wrex wi renegcl elprg syl adantl recnd
|
|
simpll negcon1d eqcom syl6bb simplr orbi12d bitrd rabbidva cin cab dfrab2
|
|
dfpr2 ineq1i eqtr4i prssi df-ss sylib syl5eq eqtrd supeq1d maxcl renegcld
|
|
eqeltrd negeqd maxle1 eqbrtrd breq2d notbid adantr lenltd lenegcon1 sylan
|
|
simplll 3bitr2d syl2anc maxle2 simpllr elpri mpjaodan ralrimiva ad3antrrr
|
|
wss ad3antlr breq1d mpbid ltnegcon1d maxleastlt syl22anc ltnegd rexprg ex
|
|
breq1 breq2 ralbidv imbi1d anbi12d rspcev syl12anc infrenegsupex ) AFGZBF
|
|
GZHZABUAZFIUBCUCZJZXPGZCFUDZFIUEZJZAJZBJZUAZFIUEZJZXODECXPXOYBFGEUCZYBIKZ
|
|
UFZEXPLZYBYHIKZXQYHIKZCXPUGZUHZEFLZYHDUCZIKZUFZEXPLZYQYHIKZYNUHZEFLZHZDFU
|
|
GXOYAXOYAYFFXOFXTYEIXOXTXQYCMZXQYDMZNZCFUDZYEXOXSUUGCFXOXQFGZHZXSXRAMZXRB
|
|
MZNZUUGUUIXSUUMSZXOUUIXRFGUUNXQUIXRABFUJUKULUUJUUKUUEUULUUFUUJUUKYCXQMUUE
|
|
UUJXQAUUJXQXOUUIOUMZUUJAXMXNUUIUNUMUOYCXQUPUQUUJUULYDXQMUUFUUJXQBUUOUUJBX
|
|
MXNUUIURUMUOYDXQUPUQUSUTVAXOUUHYEFVBZYEUUHUUGCVCZFVBUUPUUGCFVDYEUUQFCYCYD
|
|
VEVFVGXOYEFWOZUUPYEMXMYCFGZYDFGZUURXNAUIZBUIZYCYDFVHPYEFVIVJVKVLVMZXMUUSU
|
|
UTYFFGZXNUVAUVBYCYDVNPZVPVOXOYJEXPXOYHXPGZHZYHAMZYJYHBMZUVGUVHHZYJYHJZYFQ
|
|
KZUVJUVKYCYFQUVJYHAUVGUVHOZVQXOYCYFQKZUVFUVHXMUUSUUTUVNXNUVAUVBYCYDVRPRVS
|
|
UVJXOYHFGZYJUVLSZXOUVFUVHUNUVJYHAFUVMXMXNUVFUVHWFVPXOUVOHZYJYHYGIKZUFZYGY
|
|
HQKZUVLXOYJUVSSUVOXOYIUVRXOYBYGYHIXOYAYFUVCVQZVTWAWBUVQYGYHUVQYFXOUVDUVOU
|
|
VEWBVOXOUVOOWCXOUVDUVOUVTUVLSUVEYFYHWDWEWGZWHTUVGUVIHZYJUVLUWCUVKYDYFQUWC
|
|
YHBUVGUVIOZVQXOYDYFQKZUVFUVIXMUUSUUTUWEXNUVAUVBYCYDWIPRVSUWCXOUVOUVPXOUVF
|
|
UVIUNUWCYHBFUWDXMXNUVFUVIWJVPUWBWHTUVFUVHUVINXOYHABWKULWLWMXOYOEFUVQYLYNU
|
|
VQYLHZYNAYHIKZBYHIKZNZUWFUWIUVKYCIKZUVKYDIKZNZUWFUUSUUTUVKFGUVKYFIKUWLXMU
|
|
USXNUVOYLUVAWNXNUUTXMUVOYLUVBWPUWFYHXOUVOYLURZVOUWFYFYHXOUVDUVOYLUVERUWMU
|
|
WFYLYGYHIKZUVQYLOXOYLUWNSUVOYLXOYBYGYHIUWAWQRWRWSYCYDUVKWTXAUWFUWGUWJUWHU
|
|
WKUWFAYHXMXNUVOYLWFUWMXBUWFBYHXMXNUVOYLWJUWMXBUSTXOYNUWISUVOYLYMUWGUWHCAB
|
|
FFXQAYHIXEXQBYHIXEXCRTXDWMUUDYKYPHDYBFYQYBMZYTYKUUCYPUWOYSYJEXPUWOYRYIYQY
|
|
BYHIXFWAXGUWOUUBYOEFUWOUUAYLYNYQYBYHIXEXHXGXIXJXKABFVHXLUWAVL $.
|
|
$}
|
|
|
|
$( The minimum of two numbers is less than or equal to the first.
|
|
(Contributed by Jim Kingdon, 8-Feb-2021.) $)
|
|
min1inf $p |- ( ( A e. RR /\ B e. RR )
|
|
-> inf ( { A , B } , RR , < ) <_ A ) $=
|
|
( cr wcel wa cpr clt cinf cneg cle minmax simpl renegcl maxcl syl2an maxle1
|
|
csup wbr lenegcon1d eqbrtrd ) ACDZBCDZEZABFCGHAIZBIZFCGQZIAJABKUCAUFUAUBLUA
|
|
UDCDZUECDZUFCDUBAMZBMZUDUENOUAUGUHUDUFJRUBUIUJUDUEPOST $.
|
|
|
|
$( The minimum of two numbers is less than or equal to the second.
|
|
(Contributed by Jim Kingdon, 9-Feb-2021.) $)
|
|
min2inf $p |- ( ( A e. RR /\ B e. RR )
|
|
-> inf ( { A , B } , RR , < ) <_ B ) $=
|
|
( cr wcel wa cpr clt cinf cneg cle minmax simpr renegcl maxcl syl2an maxle2
|
|
csup wbr lenegcon1d eqbrtrd ) ACDZBCDZEZABFCGHAIZBIZFCGQZIBJABKUCBUFUAUBLUA
|
|
UDCDZUECDZUFCDUBAMZBMZUDUENOUAUGUHUEUFJRUBUIUJUDUEPOST $.
|
|
|
|
$( Two ways of saying a number is less than or equal to the minimum of two
|
|
others. (Contributed by NM, 3-Aug-2007.) $)
|
|
lemininf $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A <_ inf ( { B , C } , RR , < ) <-> ( A <_ B /\ A <_ C ) ) ) $=
|
|
( cr wcel w3a cpr clt cinf cle wbr cneg csup wa wceq simp2 syl2anc renegcld
|
|
wb lenegd simp3 minmax breq2d maxcl simp1 lenegcon2 syl3anc anbi12d 3bitr2d
|
|
maxleastb bitr4d ) ADEZBDEZCDEZFZABCGDHIZJKABLZCLZGDHMZLZJKZUSALZJKZABJKZAC
|
|
JKZNZUOUPUTAJUOUMUNUPUTOULUMUNPZULUMUNUAZBCUBQUCUOUSDEZULVCVASUOUQDEZURDEZV
|
|
IUOBVGRZUOCVHRZUQURUDQULUMUNUEZUSAUFQUOVCUQVBJKZURVBJKZNZVFUOVJVKVBDEVCVQSV
|
|
LVMUOAVNRUQURVBUJUGUOVDVOVEVPUOABVNVGTUOACVNVHTUHUKUI $.
|
|
|
|
$( Two ways of saying a number is less than the minimum of two others.
|
|
(Contributed by Jim Kingdon, 10-Feb-2022.) $)
|
|
ltmininf $p |- ( ( A e. RR /\ B e. RR /\ C e. RR ) ->
|
|
( A < inf ( { B , C } , RR , < ) <-> ( A < B /\ A < C ) ) ) $=
|
|
( cr wcel w3a cneg cpr clt csup wbr wa cinf wb simp2 renegcld simp3 syl2anc
|
|
simp1 ltnegd maxltsup syl3anc minmax breq2d 3adant1 maxcl ltnegcon2 anbi12d
|
|
bitrd 3bitr4d ) ADEZBDEZCDEZFZBGZCGZHDIJZAGZIKZUOURIKZUPURIKZLZABCHDIMZIKZA
|
|
BIKZACIKZLUNUODEZUPDEZURDEUSVBNUNBUKULUMOZPZUNCUKULUMQZPZUNAUKULUMSZPUOUPUR
|
|
UAUBUNVDAUQGZIKZUSULUMVDVONUKULUMLVCVNAIBCUCUDUEUNUKUQDEZVOUSNVMUNVGVHVPVJV
|
|
LUOUPUFRAUQUGRUIUNVEUTVFVAUNABVMVITUNACVMVKTUHUJ $.
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Elementary limits and convergence
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Limits
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c ~~> $. $( Limit arrow. $)
|
|
|
|
$( Extend class notation with convergence relation for limits. $)
|
|
cli $a class ~~> $.
|
|
|
|
${
|
|
$d j k x y f $.
|
|
$( Define the limit relation for complex number sequences. See ~ clim for
|
|
its relational expression. (Contributed by NM, 28-Aug-2005.) $)
|
|
df-clim $a |- ~~> = { <. f , y >. | ( y e. CC /\
|
|
A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j )
|
|
( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) } $.
|
|
|
|
$( The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
|
|
(Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climrel $p |- Rel ~~> $=
|
|
( vy vk vf vx vj cv cc wcel cfv cmin co cabs clt wbr wa cuz wral wrex crp
|
|
cz cli df-clim relopabi ) AFZGHBFCFIZGHUEUDJKLIDFMNOBEFPIQETRDSQOCAUADACE
|
|
BUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d f j k x y A $. $d f j k x y F $. $d j k x ph $.
|
|
clim.1 $e |- ( ph -> F e. V ) $.
|
|
clim.3 $e |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B ) $.
|
|
$( Express the predicate: The limit of complex number sequence ` F ` is
|
|
` A ` , or ` F ` converges to ` A ` . This means that for any real
|
|
` x ` , no matter how small, there always exists an integer ` j ` such
|
|
that the absolute difference of any later complex number in the sequence
|
|
and the limit is less than ` x ` . (Contributed by NM, 28-Aug-2005.)
|
|
(Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
clim $p |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ
|
|
A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) ) $=
|
|
( cc wcel cv cfv cmin cabs clt wa wral cz vy vf cli wbr co cuz crp cvv wi
|
|
wrex climrel brrelex2i a1i elex adantr wb simpr eleq1d fveq1 oveq12 sylan
|
|
fveq2d breq1d anbi12d ralbidv rexbidv df-clim brabga ex pm5.21ndd eluzelz
|
|
wceq syl oveq1d sylan2 ralbidva anbi2d bitrd ) AGCUCUDZCKLZFMZGNZKLZWBCOU
|
|
EZPNZBMZQUDZRZFEMZUFNZSZETUJZBUGSZRZVTDKLZDCOUEZPNZWFQUDZRZFWJSZETUJZBUGS
|
|
ZRACUHLZVSWNVSXCUIAGCUCUKULUMWNXCUIAVTXCWMCKUNUOUMAGHLZXCVSWNUPZUIIXDXCXE
|
|
UAMZKLZWAUBMZNZKLZXIXFOUEZPNZWFQUDZRZFWJSZETUJZBUGSZRWNUBUAGCUCHUHXHGVLZX
|
|
FCVLZRZXGVTXQWMXTXFCKXRXSUQURXTXPWLBUGXTXOWKETXTXNWHFWJXTXJWCXMWGXTXIWBKX
|
|
RXIWBVLZXSWAXHGUSZUOURXTXLWEWFQXTXKWDPXRYAXSXKWDVLYBXIWBXFCOUTVAVBVCVDVEV
|
|
FVEVDBUAUBEFVGVHVIVMVJAWMXBVTAWLXABUGAWKWTETAWHWSFWJWAWJLAWATLZWHWSUPWIWA
|
|
VKAYCRZWCWOWGWRYDWBDKJURYDWEWQWFQYDWDWPPYDWBDCOJVNVBVCVDVOVPVFVEVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d j k x A $. $d j k x F $.
|
|
$( Closure of the limit of a sequence of complex numbers. (Contributed by
|
|
NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
climcl $p |- ( F ~~> A -> A e. CC ) $=
|
|
( vk vx vj cli wbr cc wcel cv cfv cmin co cabs clt wa cuz wral cz wrex
|
|
crp cvv climrel brrelexi eqidd clim ibi simpld ) BAFGZAHIZCJZBKZHIULALMNK
|
|
DJOGPCEJQKRESTDUARZUIUJUMPUIDAULECBUBBAFUCUDUIUKSIPULUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d j k x A $. $d j k x F $. $d j M $. $d j k x ph $. $d j k Z $.
|
|
clim2.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
clim2.2 $e |- ( ph -> M e. ZZ ) $.
|
|
clim2.3 $e |- ( ph -> F e. V ) $.
|
|
clim2.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B ) $.
|
|
$( Express the predicate: The limit of complex number sequence ` F ` is
|
|
` A ` , or ` F ` converges to ` A ` , with more general quantifier
|
|
restrictions than ~ clim . (Contributed by NM, 6-Jan-2007.) (Revised
|
|
by Mario Carneiro, 31-Jan-2014.) $)
|
|
clim2 $p |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z
|
|
A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) ) $=
|
|
( wbr cc wcel cfv wa wral cli cv cmin co cabs clt cuz wrex crp eqidd clim
|
|
cz wb uztrn2 eleq1d oveq1d fveq2d breq1d anbi12d sylan2 ralbidva rexbidva
|
|
anassrs rexuz3 syl bitr3d ralbidv anbi2d bitr4d ) AGCUAOCPQZFUBZGRZPQZVLC
|
|
UCUDZUERZBUBZUFOZSZFEUBZUGRZTZEULUHZBUITZSVJDPQZDCUCUDZUERZVPUFOZSZFVTTZE
|
|
JUHZBUITZSABCVLEFGIMAVKULQSVLUJUKAWKWCVJAWJWBBUIAWAEJUHZWJWBAWAWIEJAVSJQZ
|
|
SVRWHFVTAWMVKVTQZVRWHUMZWMWNSAVKJQZWOHVKVSJKUNAWPSZVMWDVQWGWQVLDPNUOWQVOW
|
|
FVPUFWQVNWEUEWQVLDCUCNUPUQURUSUTVCVAVBAHULQWLWBUMLVREFHJKVDVEVFVGVHVI $.
|
|
|
|
clim2c.5 $e |- ( ph -> A e. CC ) $.
|
|
clim2c.6 $e |- ( ( ph /\ k e. Z ) -> B e. CC ) $.
|
|
$( Express the predicate ` F ` converges to ` A ` . (Contributed by NM,
|
|
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
clim2c $p |- ( ph -> ( F ~~> A <->
|
|
A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) ) $=
|
|
( wcel cv wa wral cmin cabs cfv clt wbr cuz wrex crp cli biantrurd uztrn2
|
|
cc co wb sylan2 anassrs ralbidva rexbidva ralbidv clim2 3bitr4rd ) ADULQZ
|
|
DCUAUMUBUCBRUDUEZSZFERZUFUCZTZEJUGZBUHTZCULQZVISVCFVFTZEJUGZBUHTGCUIUEAVJ
|
|
VIOUJAVLVHBUHAVKVGEJAVEJQZSVCVDFVFAVMFRZVFQZVCVDUNZVMVOSAVNJQZVPHVNVEJKUK
|
|
AVQSVBVCPUJUOUPUQURUSABCDEFGHIJKLMNUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d j k x F $. $d j M $. $d j k x ph $. $d j k Z $.
|
|
clim0.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
clim0.2 $e |- ( ph -> M e. ZZ ) $.
|
|
clim0.3 $e |- ( ph -> F e. V ) $.
|
|
clim0.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B ) $.
|
|
$( Express the predicate ` F ` converges to ` 0 ` . (Contributed by NM,
|
|
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
clim0 $p |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z
|
|
A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) ) $=
|
|
( cc0 wbr cabs cfv clt wa wral cli cc wcel cmin co cuz wrex crp clim2 0cn
|
|
cv biantrur subid1 fveq2d breq1d pm5.32i ralbii rexbii bitr3i syl6bb ) AF
|
|
NUAONUBUCZCUBUCZCNUDUEZPQZBUKZROZSZEDUKUFQZTZDIUGZBUHTZSZVBCPQZVEROZSZEVH
|
|
TZDIUGZBUHTZABNCDEFGHIJKLMUIVLVKVRVAVKUJULVJVQBUHVIVPDIVGVOEVHVBVFVNVBVDV
|
|
MVERVBVCCPCUMUNUOUPUQURUQUSUT $.
|
|
|
|
clim0c.6 $e |- ( ( ph /\ k e. Z ) -> B e. CC ) $.
|
|
$( Express the predicate ` F ` converges to ` 0 ` . (Contributed by NM,
|
|
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
clim0c $p |- ( ph -> ( F ~~> 0 <->
|
|
A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) ) $=
|
|
( cc0 wbr cabs cfv cv wral cli cmin co clt cuz wrex crp 0cnd clim2c wa wb
|
|
wcel uztrn2 subid1d fveq2d breq1d anassrs ralbidva rexbidva ralbidv bitrd
|
|
sylan2 ) AFOUAPCOUBUCZQRZBSZUDPZEDSZUERZTZDIUFZBUGTCQRZVEUDPZEVHTZDIUFZBU
|
|
GTABOCDEFGHIJKLMAUHNUIAVJVNBUGAVIVMDIAVGIULZUJVFVLEVHAVOESZVHULZVFVLUKZVO
|
|
VQUJAVPIULZVRGVPVGIJUMAVSUJZVDVKVEUDVTVCCQVTCNUNUOUPVBUQURUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d j k x A $. $d j k x C $. $d j k x F $. $d j k x ph $. $d j k x Z $.
|
|
$d x B $. $d j M $.
|
|
climi.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climi.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climi.3 $e |- ( ph -> C e. RR+ ) $.
|
|
climi.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B ) $.
|
|
${
|
|
climi.5 $e |- ( ph -> F ~~> A ) $.
|
|
$( Convergence of a sequence of complex numbers. (Contributed by NM,
|
|
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climi $p |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( B e. CC /\ ( abs ` ( B - A ) ) < C ) ) $=
|
|
( vx crp wcel clt wbr wa cc cmin cabs cfv cuz wral wrex cli cvv climrel
|
|
co cv brrelexi syl clim2 mpbid simprd wceq breq2 anbi2d rexralbidv sylc
|
|
rspcv ) ADPQCUAQZCBUBUKUCUDZOULZRSZTZFEULUEUDZUFEIUGZOPUFZVDVEDRSZTZFVI
|
|
UFEIUGZLABUAQZVKAGBUHSZVOVKTNAOBCEFGHUIIJKAVPGUIQNGBUHUJUMUNMUOUPUQVJVN
|
|
ODPVFDURZVHVMEFIVIVQVGVLVDVFDVERUSUTVAVCVB $.
|
|
|
|
$( Convergence of a sequence of complex numbers. (Contributed by NM,
|
|
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climi2 $p |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( abs ` ( B - A ) ) < C ) $=
|
|
( cc wcel cmin cfv wral wrex co cabs clt wbr wa cuz climi ralimi reximi
|
|
cv simpr syl ) ACOPZCBQUAUBRDUCUDZUEZFEUJUFRZSZEITUNFUPSZEITABCDEFGHIJK
|
|
LMNUGUQUREIUOUNFUPUMUNUKUHUIUL $.
|
|
$}
|
|
|
|
climi0.5 $e |- ( ph -> F ~~> 0 ) $.
|
|
$( Convergence of a sequence of complex numbers to zero. (Contributed by
|
|
NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climi0 $p |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C ) $=
|
|
( cc0 cabs cfv clt wbr wral wrex cc wcel co wa cv cuz climi subid1 fveq2d
|
|
cmin breq1d biimpa ralimi reximi syl ) ABUAUBZBNUJUCZOPZCQRZUDZEDUEUFPZSZ
|
|
DHTBOPZCQRZEVASZDHTANBCDEFGHIJKLMUGVBVEDHUTVDEVAUPUSVDUPURVCCQUPUQBOBUHUI
|
|
UKULUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d j k x A $. $d j k x F $. $d j M $. $d j k x ph $. $d j k Z $.
|
|
climconst.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climconst.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climconst.3 $e |- ( ph -> F e. V ) $.
|
|
climconst.4 $e |- ( ph -> A e. CC ) $.
|
|
climconst.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A ) $.
|
|
$( An (eventually) constant sequence converges to its value. (Contributed
|
|
by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climconst $p |- ( ph -> F ~~> A ) $=
|
|
( vx vj wbr cabs cfv cv wcel cc0 cli cmin co clt cuz wral wrex wa cz uzid
|
|
crp syl syl6eleqr adantr wceq subidd fveq2d syl6eq rpgt0 adantl ralrimivw
|
|
abs0 eqbrtrd fveq2 syl6eqr raleqdv rspcev syl2anc ralrimiva clim2c mpbird
|
|
cc ) ADBUAOBBUBUCZPQZMRZUDOZCNRZUEQZUFZNGUGZMUKUFAVTMUKAVOUKSZUHZEGSZVPCG
|
|
UFZVTAWCWAAEEUEQZGAEUISEWESIEUJULHUMUNWBVPCGWBVNTVOUDAVNTUOWAAVNTPQTAVMTP
|
|
ABKUPUQVBURUNWATVOUDOAVOUSUTVCVAVSWDNEGVQEUOZVPCVRGWFVRWEGVQEUEVDHVEVFVGV
|
|
HVIAMBBNCDEFGHIJLKABVLSCRGSKUNVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d k A $. $d k M $. $d k Z $.
|
|
climconst2.1 $e |- ( ZZ>= ` M ) C_ Z $.
|
|
climconst2.2 $e |- Z e. _V $.
|
|
$( A constant sequence converges to its value. (Contributed by NM,
|
|
6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climconst2 $p |- ( ( A e. CC /\ M e. ZZ ) -> ( Z X. { A } ) ~~> A ) $=
|
|
( vk cc wcel cz wa csn cxp cvv cuz cfv eqid simpr snexg adantr xpexg wceq
|
|
sylancr simpl cv sseli fvconst2g syl2an climconst ) AGHZBIHZJZAFCAKZLZBMB
|
|
NOZUNPUIUJQUKCMHULMHZUMMHEUIUOUJAGRSCULMMTUBUIUJUCZUKUIFUDZCHUQUMOAUAUQUN
|
|
HUPUNCUQDUECAUQGUFUGUH $.
|
|
$}
|
|
|
|
$( The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
|
|
(Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climz $p |- ( ZZ X. { 0 } ) ~~> 0 $=
|
|
( cc0 cc wcel cz csn cxp cli wbr 0cn 0z uzssz zex climconst2 mp2an ) ABCADC
|
|
DAEFAGHIJAADAKLMN $.
|
|
|
|
${
|
|
$d j k A $. $d j k B $. $d j k F $.
|
|
$( An infinite sequence of complex numbers converges to at most one limit.
|
|
(Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
climuni $p |- ( ( F ~~> A /\ F ~~> B ) -> A = B ) $=
|
|
( vk vj wbr wa c1 wcel cfv cc cmin co cabs clt wral cn wrex nnuz wi wn cz
|
|
cli wceq cap 1z w3a cv c2 cdiv 1zzd climcl 3ad2ant1 3ad2ant2 subcld simp3
|
|
cuz subap0d absrpclapd rphalfcld eqidd simp1 climi simp2 rexanuz2 wex nnz
|
|
sylanbrc uzid elex2 r19.2m ex simpr simpll abssubd breq1d cr simplr subcl
|
|
4syl adantr abscld abs3lem syl22anc pm2.21d syld expd sylbid impr adantld
|
|
ltnrd expimpd rexlimdvw sylan9r rexlimdva syl2anc mpd 3expia mt2i wb apti
|
|
syl2an mpbird ) CAUCFZCBUCFZGZABUDZABUEFZUAZXFXHHUBIZUFXDXEXHXJUAZXDXEXHU
|
|
GZDUHZCJZKIZXNALMNJZABLMZNJZUIUJMZOFZGZXOXNBLMNJXSOFZGZGZDEUHZUQJZPZEQRZX
|
|
KXLYADYFPEQRYCDYFPEQRYHXLAXNXSEDCHQSXLUKZXLXRXLXQXLABXDXEAKIZXHACULZUMZXE
|
|
XDBKIZXHBCULZUNZUOXLABYLYOXDXEXHUPURUSUTZXLXMQIGXNVAZXDXEXHVBVCXLBXNXSEDC
|
|
HQSYIYPYQXDXEXHVDVCYAYCEDHQSVEVHXLYJYMYHXKTYLYOYJYMGZYGXKEQYEQIZYGYDDYFRZ
|
|
YRXKYSYEUBIYEYFIXMYFIDVFZYGYTTYEVGYEVIDYEYFVJUUAYGYTYDDYFVKVLVTYRYDXKDYFY
|
|
RYAYCXKYRYAGYBXKXOYRXOXTYBXKTZYRXOGZXTAXNLMNJZXSOFZUUBUUCXPUUDXSOUUCXNAYR
|
|
XOVMZYJYMXOVNZVOVPUUCUUEYBXKUUCUUEYBGZXRXROFZXKUUCYJYMXOXRVQIUUHUUITUUGYJ
|
|
YMXOVRUUFUUCXQYRXQKIXOABVSWAWBZABXNXRWCWDUUCUUIXKUUCXRUUJWKWEWFWGWHWIWJWL
|
|
WMWNWOWPWQWRWSXDYJYMXGXIWTXEYKYNABXAXBXC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The limit relation is function-like, and with range the complex numbers.
|
|
(Contributed by Mario Carneiro, 31-Jan-2014.) $)
|
|
fclim $p |- ~~> : dom ~~> --> CC $=
|
|
( vx vy vz cli cdm cc wf wfn crn wss wfun wrel cv wbr wa weq wal mpbir2an
|
|
ax-gen wcel climrel climuni dffun2 funfn mpbi wex vex elrn climcl exlimiv
|
|
wi sylbi ssriv df-f ) DEZFDGDUOHZDIZFJDKZUPURDLAMZBMZDNZUSCMZDNOBCPUKZCQZ
|
|
BQZAQUAVEAVDBVCCUTVBUSUBSSSABCDUCRDUDUEBUQFUTUQTVAAUFUTFTZAUTDBUGUHVAVFAU
|
|
TUSUIUJULUMUOFDUNR $.
|
|
$}
|
|
|
|
$( Two ways to express that a function has a limit. (The expression
|
|
` ( ~~> `` F ) ` is sometimes useful as a shorthand for "the unique limit
|
|
of the function ` F ` "). (Contributed by Mario Carneiro,
|
|
18-Mar-2014.) $)
|
|
climdm $p |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) ) $=
|
|
( cli cdm cc wf wfun wcel cfv wbr wb fclim ffun funfvbrb mp2b ) BCZDBEBFAOG
|
|
AABHBIJKODBLABMN $.
|
|
|
|
${
|
|
$d y A $. $d x y F $.
|
|
$( An infinite sequence of complex numbers converges to at most one limit.
|
|
(Contributed by NM, 25-Dec-2005.) $)
|
|
climeu $p |- ( F ~~> A -> E! x F ~~> x ) $=
|
|
( vy cli wbr cv wex wa weq wi wal weu wcel climcl breq2 spcegv mpcom nfv
|
|
cc climuni gen2 jctir cbveu eu4 bitri sylibr ) CBEFZCDGZEFZDHZUJCAGZEFZID
|
|
AJKZALDLZIZUMAMZUHUKUOBTNUHUKBCOUJUHDBTUIBCEPQRUNDAUIULCUAUBUCUQUJDMUPUMU
|
|
JADUMDSUJASULUICEPUDUJUMDAUIULCEPUEUFUG $.
|
|
|
|
$( An infinite sequence of complex numbers converges to at most one limit.
|
|
(Contributed by NM, 25-Dec-2005.) $)
|
|
climreu $p |- ( F ~~> A -> E! x e. CC F ~~> x ) $=
|
|
( cli wbr cv weu cc wreu climeu climcl pm4.71ri eubii df-reu bitr4i sylib
|
|
wcel wa ) CBDECAFZDEZAGZTAHIZABCJUASHQZTRZAGUBTUDATUCSCKLMTAHNOP $.
|
|
|
|
$( An infinite sequence of complex numbers converges to at most one limit.
|
|
(Contributed by Mario Carneiro, 13-Jul-2013.) $)
|
|
climmo $p |- E* x F ~~> x $=
|
|
( vy cv cli wbr wmo wex weu breq2 cbvexv climeu exlimiv sylbi df-mo mpbir
|
|
wi ) BADZEFZAGSAHZSAIZQTBCDZEFZCHUASUCACRUBBEJKUCUACAUBBLMNSAOP $.
|
|
$}
|
|
|
|
${
|
|
$d k x y A $. $d k x y F $. $d k x y G $. $d k x y ph $. $d k y Z $.
|
|
$d y M $.
|
|
climeq.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climeq.2 $e |- ( ph -> F e. V ) $.
|
|
climeq.3 $e |- ( ph -> G e. W ) $.
|
|
climeq.5 $e |- ( ph -> M e. ZZ ) $.
|
|
climeq.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) ) $.
|
|
$( Two functions that are eventually equal to one another have the same
|
|
limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
climeq $p |- ( ph -> ( F ~~> A <-> G ~~> A ) ) $=
|
|
( vx vy wbr wcel cv cfv cli cc cmin co cabs clt cuz wral wrex clim2 eqidd
|
|
wa crp bitr4d ) ADBUAQBUBRCSZETZUBRUPBUCUDUETOSUFQULCPSUGTUHPIUIOUMUHULEB
|
|
UAQAOBUPPCDFGIJMKNUJAOBUPPCEFHIJMLAUOIRULUPUKUJUN $.
|
|
$}
|
|
|
|
${
|
|
$d j k m y A $. $d j k m x F $. $d j m x y G $. $d j m M $. $d m V $.
|
|
$d j k y ph $. $d j k m x Z $.
|
|
2clim.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
${
|
|
climmpt.2 $e |- G = ( k e. Z |-> ( F ` k ) ) $.
|
|
$( Exhibit a function ` G ` with the same convergence properties as the
|
|
not-quite-function ` F ` . (Contributed by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
climmpt $p |- ( ( M e. ZZ /\ F e. V ) ->
|
|
( F ~~> A <-> G ~~> A ) ) $=
|
|
( vm cz wcel wa cvv simpr cv cfv cuz syl syl5eqel cmpt cpw uzf ffvelrni
|
|
elex mptexg adantr simpl wceq fvexg adantll fveq2 fvmptg syl2anc eqcomd
|
|
climeq ) EKLZCFLZMZAJCDEFNGHUQUROUQDNLURUQDBGBPZCQZUAZNIUQGNLVBNLUQGERQ
|
|
ZNHUQVCKUBZLVCNLKVDERUCUDVCVDUESTBGVANUFSTUGUQURUHUSJPZGLZMZVEDQZVECQZV
|
|
GVFVINLZVHVIUIUSVFOURVFVJUQVECFGUJUKBVEVAVIGNDUTVECULIUMUNUOUP $.
|
|
$}
|
|
|
|
$d k G $.
|
|
2clim.2 $e |- ( ph -> M e. ZZ ) $.
|
|
2clim.3 $e |- ( ph -> G e. V ) $.
|
|
2clim.5 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) $.
|
|
2clim.6 $e |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j )
|
|
( abs ` ( ( F ` k ) - ( G ` k ) ) ) < x ) $.
|
|
2clim.7 $e |- ( ph -> F ~~> A ) $.
|
|
$( If two sequences converge to each other, they converge to the same
|
|
limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
2clim $p |- ( ph -> G ~~> A ) $=
|
|
( wbr cfv wcel wa vy cli cv cmin co cabs clt cuz wral wrex crp c2 cdiv cc
|
|
rphalfcl wceq breq2 rexralbidv rspccva syl2an adantr eqidd climi rexanuz2
|
|
cz adantl sylanbrc wi uztrn2 simprr ad2ant2r abssubd breq1d anbi1d climcl
|
|
an12 cr syl ad2antrr rpre ad2antlr abs3lem syl22anc sylbid anassrs syl5bi
|
|
expimpd sylan2 ralimdva reximdva mpd ralrimiva clim2c mpbird ) AGCUBQEUCZ
|
|
GRZCUDUEUFRUAUCZUGQZEDUCZUHRZUIZDJUJZUAUKUIAXBUAUKAWQUKSZTZWOFRZWPUDUEUFR
|
|
ZWQULUMUEZUGQZXEUNSZXECUDUEUFRXGUGQZTZTZEWTUIZDJUJZXBXDXHEWTUIDJUJZXKEWTU
|
|
IDJUJXNAXFBUCZUGQZEWTUIDJUJZBUKUIXGUKSZXOXCOWQUOZXRXOBXGUKXPXGUPXQXHDEJWT
|
|
XPXGXFUGUQURUSUTXDCXEXGDEFHJKAHVESXCLVAXCXSAXTVFXDWOJSZTZXEVBAFCUBQZXCPVA
|
|
VCXHXKDEHJKVDVGXDXMXADJXDWSJSZTXLWREWTXDYDWOWTSZXLWRVHZYDYETXDYAYFHWOWSJK
|
|
VIXLXIXHXJTZTYBWRXHXIXJVPYBXIYGWRXDYAXIYGWRVHXDYAXITZTZYGWPXEUDUEUFRZXGUG
|
|
QZXJTZWRYIXHYKXJYIXFYJXGUGYIXEWPXDYAXIVJZAYAWPUNSZXCXINVKZVLVMVNYIYNCUNSZ
|
|
XIWQVQSZYLWRVHYOAYPXCYHAYCYPPCFVOVRZVSYMXCYQAYHWQVTWAWPCXEWQWBWCWDWEWGWFW
|
|
HWEWIWJWKWLAUACWPDEGHIJKLMAYATWPVBYRNWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d f k m n x A $. $d f k m n x F $. $d f k m n x M $. $d k m n x V $.
|
|
$( A shifted function converges if the original function converges.
|
|
(Contributed by Mario Carneiro, 5-Nov-2013.) $)
|
|
climshftlemg $p |- ( ( M e. ZZ /\ F e. V ) ->
|
|
( F ~~> A -> ( F shift M ) ~~> A ) ) $=
|
|
( vm vx vk vn cz wcel wa cc cv cfv cmin co cabs clt wbr wral cuz wrex crp
|
|
cshi cli caddc zaddcl ancoms adantlr wi eluzsub 3com12 3expa fveq2 eleq1d
|
|
wceq oveq1d fveq2d breq1d anbi12d rspcv syl adantllr wb ad2antrr eluzelcn
|
|
simplr zcn adantl w3a shftvalg syl3anc sylibrd ralrimdva rspcev rexlimdva
|
|
raleqdv syl6an ralimdv anim2d simpr eqidd clim cvv ovshftex sylan 3imtr4d
|
|
) CIJZBDJZKZALJZEMZBNZLJZWMAOPZQNZFMZRSZKZEGMZUANZTZGIUBZFUCTZKWKHMZBCUDP
|
|
ZNZLJZXGAOPZQNZWQRSZKZHWLUANZTZEIUBZFUCTZKBAUESXFAUESWJXDXPWKWJXCXOFUCWJX
|
|
BXOGIWJWTIJZKZWTCUFPZIJZXBXLHXSUANZTZXOWHXQXTWIXQWHXTWTCUGUHUIXRXBXLHYAXR
|
|
XEYAJZKXBXECOPZBNZLJZYEAOPZQNZWQRSZKZXLWHXQYCXBYJUJZWIWHXQKYCKYDXAJZYKWHX
|
|
QYCYLXQWHYCYLCWTXEUKULUMWSYJEYDXAWLYDUPZWNYFWRYIYMWMYELWLYDBUNZUOYMWPYHWQ
|
|
RYMWOYGQYMWMYEAOYNUQURUSUTVAVBVCWJYCXLYJVDZXQWJYCKWICLJZXELJZYOWHWIYCVGWH
|
|
YPWIYCCVHZVEYCYQWJXSXEVFVIWIYPYQVJZXHYFXKYIYSXGYELCXEBDVKZUOYSXJYHWQRYSXI
|
|
YGQYSXGYEAOYTUQURUSUTVLUIVMVNXNYBEXSIWLXSUPXLHXMYAWLXSUAUNVQVOVRVPVSVTWJF
|
|
AWMGEBDWHWIWAWJWLIJKWMWBWCWJFAXGEHXFWDWHYPWIXFWDJZYRWIYPUUACBDWEUHWFWJXEI
|
|
JKXGWBWCWG $.
|
|
|
|
$( A function restricted to upper integers converges iff the original
|
|
function converges. (Contributed by Mario Carneiro, 13-Jul-2013.)
|
|
(Revised by Mario Carneiro, 31-Jan-2014.) $)
|
|
climres $p |- ( ( M e. ZZ /\ F e. V ) ->
|
|
( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) ) $=
|
|
( vk cz wcel wa cuz cfv cres eqid resexg adantl simpr simpl cv wceq fvres
|
|
cvv climeq ) CFGZBDGZHZAEBCIJZKZBCTDUEUELUCUFTGUBBUEDMNUBUCOUBUCPEQZUEGUG
|
|
UFJUGBJRUDUGUEBSNUA $.
|
|
|
|
$( A shifted function converges iff the original function converges.
|
|
(Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
climshft $p |- ( ( M e. ZZ /\ F e. V ) ->
|
|
( ( F shift M ) ~~> A <-> F ~~> A ) ) $=
|
|
( vf vk wcel cz cshi co cli wbr wb cv wi wceq cvv cc ovshftex cfv bibi12d
|
|
oveq1 breq1d breq1 imbi2d znegcl vex zcn sylancr climshftlemg syl2anc cuz
|
|
cneg eqid negcld a1i eluzelcn shftcan1 syl2an climeq sylibd impbid vtoclg
|
|
id mpan2 impcom ) BDGCHGZBCIJZAKLZBAKLZMZVGENZCIJZAKLZVLAKLZMZOVGVKOEBDVL
|
|
BPZVPVKVGVQVNVIVOVJVQVMVHAKVLBCIUBUCVLBAKUDUAUEVGVNVOVGVNVMCUMZIJZAKLZVOV
|
|
GVRHGVMQGZVNVTOCUFVGVLQGZCRGZWAEUGZCUHZCVLQSUIZAVMVRQUJUKVGAFVSVLCQQCULTZ
|
|
WGUNVGWAVRRGVSQGWFVGCWEUOVRVMQSUKWBVGWDUPVGVDVGWCFNZRGWHVSTWHVLTPWHWGGWEC
|
|
WHUQCWHVLWDURUSUTVAVGWBVOVNOWDAVLCQUJVEVBVCVF $.
|
|
$}
|
|
|
|
$( The zero series converges to zero. (Contributed by Jim Kingdon,
|
|
19-Aug-2021.) $)
|
|
iserclim0 $p |- ( M e. ZZ ->
|
|
seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) , CC ) ~~> 0 ) $=
|
|
( cz wcel caddc cuz cfv cc0 csn cxp cseq cli eqid iser0f wbr 0cn ssid uzssz
|
|
cc zex ssexi climconst2 mpan eqbrtrd ) ABCZDRAEFZGHIZAJUFGKAUEUELMGRCUDUFGK
|
|
NOGAUEUEPUEBSAQTUAUBUC $.
|
|
|
|
${
|
|
$d k F $. $d k G $. $d k K $. $d k M $. $d k ph $. $d k Z $.
|
|
$d k A $.
|
|
climshft2.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climshft2.2 $e |- ( ph -> M e. ZZ ) $.
|
|
${
|
|
climshft2.3 $e |- ( ph -> K e. ZZ ) $.
|
|
climshft2.5 $e |- ( ph -> F e. W ) $.
|
|
climshft2.6 $e |- ( ph -> G e. X ) $.
|
|
climshft2.7 $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` ( k + K ) ) = ( F ` k ) ) $.
|
|
$( A shifted function converges iff the original function converges.
|
|
(Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario
|
|
Carneiro, 6-Feb-2014.) $)
|
|
climshft2 $p |- ( ph -> ( F ~~> A <-> G ~~> A ) ) $=
|
|
( co cvv wcel cfv cneg cshi cli wbr cc negcld ovshftex syl2anc cv caddc
|
|
zcnd wa cid wceq wfun cdm funi syl dmi syl6eleqr funfvex sylancr adantr
|
|
cz cuz eluzelz eleq2s adantl shftval4g syl3anc fvi oveq1d fveq1d addcom
|
|
elex syl2an fveq12d 3eqtr3d eqtrd climeq wb znegcld climshft bitr3d ) A
|
|
EFUAZUBQZBUCUDZDBUCUDEBUCUDZABCWFDGRHJKAEISZWEUESWFRSOAFAFMUKZUFWEEIUGU
|
|
HNLACUIZJSZULZWKWFTZWKFUJQZETZWKDTWMWKEUMTZWEUBQZTZFWKUJQZWQTZWNWPWMWQR
|
|
SZFUESZWKUESZWSXAUNAXBWLAUMUOEUMUPZSXBUQAERXEAWIERSOEIVOURUSUTEUMVAVBVC
|
|
AXCWLWJVCWLXDAWLWKWKVDSWKGVETJGWKVFKVGUKZVHFWKWQRVIVJWMWKWRWFWMWQEWEUBA
|
|
WQEUNZWLAWIXGOEIVKURVCZVLVMWMWTWOWQEXHAXCXDWTWOUNWLWJXFFWKVNVPVQVRPVSVT
|
|
AWEVDSWIWGWHWAAFMWBOBEWEIWCUHWD $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d j k x F $. $d j k x G $. $d j M $. $d j k x ph $. $d j k Z $.
|
|
climabs0.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climabs0.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climabs0.3 $e |- ( ph -> F e. V ) $.
|
|
climabs0.4 $e |- ( ph -> G e. W ) $.
|
|
climabs0.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
climabs0.6 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) ) $.
|
|
$( Convergence to zero of the absolute value is equivalent to convergence
|
|
to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
climabs0 $p |- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) ) $=
|
|
( vx vj cfv wbr wral wcel cv cabs clt cuz wrex crp cc0 cli wa uztrn2 wceq
|
|
wb cc absidm breq1d sylan2 anassrs ralbidva rexbidva ralbidv abscld recnd
|
|
syl clim0c eqidd 3bitr4rd ) ABUAZCQZUBQZUBQZOUAZUCRZBPUAZUDQZSZPHUEZOUFSV
|
|
IVKUCRZBVNSZPHUEZOUFSDUGUHRCUGUHRAVPVSOUFAVOVRPHAVMHTZUIVLVQBVNAVTVGVNTZV
|
|
LVQULZVTWAUIAVGHTZWBEVGVMHIUJAWCUIZVJVIVKUCWDVHUMTVJVIUKMVHUNVCUOUPUQURUS
|
|
UTAOVIPBDEGHIJLNWDVIWDVHMVAVBVDAOVHPBCEFHIJKWDVHVEMVDVF $.
|
|
$}
|
|
|
|
${
|
|
$d j k x y z A $. $d j k z B $. $d j k y z G $. $d j k x H $. $d j M $.
|
|
$d j k x y z F $. $d j k x y z ph $. $d j k y Z $.
|
|
climcn1.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climcn1.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climcn1.3 $e |- ( ph -> A e. B ) $.
|
|
climcn1.4 $e |- ( ( ph /\ z e. B ) -> ( F ` z ) e. CC ) $.
|
|
climcn1.5 $e |- ( ph -> G ~~> A ) $.
|
|
climcn1.6 $e |- ( ph -> H e. W ) $.
|
|
climcn1.7 $e |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ A. z e. B
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) $.
|
|
climcn1.8 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. B ) $.
|
|
climcn1.9 $e |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( F ` ( G ` k ) ) ) $.
|
|
$( Image of a limit under a continuous map. (Contributed by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
climcn1 $p |- ( ph -> H ~~> ( F ` A ) ) $=
|
|
( vj cfv cli wbr cv cmin co cabs clt cuz wral wrex crp wa wi adantr simpr
|
|
wcel cz eqidd climi2 uztrn2 adantlr wceq oveq1 fveq2d breq1d fveq2 oveq1d
|
|
imbi12d rspcva sylan an32s sylan2 anassrs ralimdva reximdva rexlimdva mpd
|
|
ex mpid ralrimiva cc eleq1d rspcv sylc clim2c mpbird ) AJEHUDZUEUFGUGZIUD
|
|
ZHUDZWKUHUIZUJUDZBUGZUKUFZGUCUGZULUDZUMZUCMUNZBUOUMAXBBUOAWQUOUTZUPDUGZEU
|
|
HUIZUJUDZCUGZUKUFZXDHUDZWKUHUIZUJUDZWQUKUFZUQZDFUMZCUOUNZXBTAXOXBUQXCAXNX
|
|
BCUOAXGUOUTZUPZXNWMEUHUIZUJUDZXGUKUFZGWTUMZUCMUNZXBXQEWMXGUCGIKMNAKVAUTXP
|
|
OURAXPUSXQWLMUTZUPZWMVBAIEUEUFXPRURVCXQXNYBXBUQXQXNUPZYAXAUCMYEWSMUTZUPXT
|
|
WRGWTYEYFWLWTUTZXTWRUQZYFYGUPYEYCYHKWLWSMNVDXQYCXNYHYDWMFUTZXNYHAYCYIXPUA
|
|
VEXMYHDWMFXDWMVFZXHXTXLWRYJXFXSXGUKYJXEXRUJXDWMEUHVGVHVIYJXKWPWQUKYJXJWOU
|
|
JYJXIWNWKUHXDWMHVJZVKVHVIVLVMVNVOVPVQVRVSWBWCVTURWAWDABWKWNUCGJKLMNOSUBAE
|
|
FUTXIWEUTZDFUMZWKWEUTZPAYLDFQWDZYLYNDEFXDEVFXIWKWEXDEHVJWFWGWHAYCUPYIYMWN
|
|
WEUTZUAAYMYCYOURYLYPDWMFYJXIWNWEYKWFWGWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d j k u v C $. $d j k u v D $. $d j k v y z H $. $d j k u v x y z ph $.
|
|
$d j k u v x y z A $. $d j k u v y z G $. $d j k x K $. $d j k y z Z $.
|
|
$d j k u v x y z B $. $d j k u v x y z F $. $d j M $.
|
|
climcn2.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climcn2.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climcn2.3a $e |- ( ph -> A e. C ) $.
|
|
climcn2.3b $e |- ( ph -> B e. D ) $.
|
|
climcn2.4 $e |- ( ( ph /\ ( u e. C /\ v e. D ) ) -> ( u F v ) e. CC ) $.
|
|
climcn2.5a $e |- ( ph -> G ~~> A ) $.
|
|
climcn2.5b $e |- ( ph -> H ~~> B ) $.
|
|
climcn2.6 $e |- ( ph -> K e. W ) $.
|
|
climcn2.7 $e |- ( ( ph /\ x e. RR+ ) ->
|
|
E. y e. RR+ E. z e. RR+ A. u e. C A. v e. D
|
|
( ( ( abs ` ( u - A ) ) < y /\ ( abs ` ( v - B ) ) < z ) ->
|
|
( abs ` ( ( u F v ) - ( A F B ) ) ) < x ) ) $.
|
|
climcn2.8a $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. C ) $.
|
|
climcn2.8b $e |- ( ( ph /\ k e. Z ) -> ( H ` k ) e. D ) $.
|
|
climcn2.9 $e |- ( ( ph /\ k e. Z ) ->
|
|
( K ` k ) = ( ( G ` k ) F ( H ` k ) ) ) $.
|
|
$( Image of a limit under a continuous map, two-arg version. (Contributed
|
|
by Mario Carneiro, 31-Jan-2014.) $)
|
|
climcn2 $p |- ( ph -> K ~~> ( A F B ) ) $=
|
|
( vj co cli wbr cv cfv cmin cabs clt cuz wral wrex crp wcel adantr simprl
|
|
wa wi cz eqidd climi2 simprr rexanuz2 sylanbrc uztrn2 oveq1 fveq2d breq1d
|
|
wceq anbi1d oveq1d imbi12d anbi2d oveq2 rspc2v syl2anc imp sylan2 anassrs
|
|
an32s ralimdva reximdva ex rexlimdvva mpd ralrimiva cc caovcld ralrimivva
|
|
mpid jca eleq1d sylc clim2c mpbird ) AOGHLULZUMUNKUOZMUPZXGNUPZLULZXFUQUL
|
|
ZURUPZBUOZUSUNZKUKUOZUTUPZVAZUKRVBZBVCVAAXRBVCAXMVCVDZVGFUOZGUQULZURUPZCU
|
|
OZUSUNZEUOZHUQULZURUPZDUOZUSUNZVGZXTYELULZXFUQULZURUPZXMUSUNZVHZEJVAFIVAZ
|
|
DVCVBCVCVBZXRUGAYQXRVHXSAYPXRCDVCVCAYCVCVDZYHVCVDZVGZVGZYPXHGUQULZURUPZYC
|
|
USUNZXIHUQULZURUPZYHUSUNZVGZKXPVAZUKRVBZXRUUAUUDKXPVAUKRVBUUGKXPVAUKRVBUU
|
|
JUUAGXHYCUKKMPRSAPVIVDYTTVEZAYRYSVFUUAXGRVDZVGZXHVJAMGUMUNYTUDVEVKUUAHXIY
|
|
HUKKNPRSUUKAYRYSVLUUMXIVJANHUMUNYTUEVEVKUUDUUGUKKPRSVMVNAYPUUJXRVHZVHYTAY
|
|
PUUNAYPVGZUUIXQUKRUUOXORVDZVGUUHXNKXPUUOUUPXGXPVDZUUHXNVHZUUPUUQVGUUOUULU
|
|
URPXGXORSVOAUULYPUURAUULVGZYPUURUUSXHIVDZXIJVDZYPUURVHUHUIYOUURUUDYIVGZXH
|
|
YELULZXFUQULZURUPZXMUSUNZVHFEXHXIIJXTXHVSZYJUVBYNUVFUVGYDUUDYIUVGYBUUCYCU
|
|
SUVGYAUUBURXTXHGUQVPVQVRVTUVGYMUVEXMUSUVGYLUVDURUVGYKUVCXFUQXTXHYELVPZWAV
|
|
QVRWBYEXIVSZUVBUUHUVFXNUVIYIUUGUUDUVIYGUUFYHUSUVIYFUUEURYEXIHUQVPVQVRWCUV
|
|
IUVEXLXMUSUVIUVDXKURUVIUVCXJXFUQYEXIXHLWDZWAVQVRWBWEWFWGWJWHWIWKWLWMVEWTW
|
|
NVEWOWPABXFXJUKKOPQRSTUFUJAFEGHIJWQLUCUAUBWRUUSUUTUVAVGYKWQVDZEJVAFIVAZXJ
|
|
WQVDZUUSUUTUVAUHUIXAAUVLUULAUVKFEIJUCWSVEUVKUVMUVCWQVDFEXHXIIJUVGYKUVCWQU
|
|
VHXBUVIUVCXJWQUVJXBWEXCXDXE $.
|
|
$}
|
|
|
|
${
|
|
$d u v w y z A $. $d u v w y z B $. $d u v w y z C $.
|
|
$( Complex number addition is a continuous function. Part of Proposition
|
|
14-4.16 of [Gleason] p. 243. (We write out the definition directly
|
|
because df-cn and df-cncf are not yet available to us. See addcn for
|
|
the abbreviated version.) (Contributed by Mario Carneiro,
|
|
31-Jan-2014.) $)
|
|
addcn2 $p |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) ->
|
|
E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC
|
|
( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) ->
|
|
( abs ` ( ( u + v ) - ( B + C ) ) ) < A ) ) $=
|
|
( crp wcel cc co cv cmin cabs cfv clt wbr wa wi wral w3a c2 cdiv rphalfcl
|
|
caddc 3ad2ant1 simprl simpl2 simprr pnpcan2d fveq2d breq1d simpl3 pnpcand
|
|
wrex anbi12d cr addcl adantl addcld simpl1 rpred abs3lem syl22anc sylbird
|
|
ralrimivva wceq breq2 anbi1d imbi1d 2ralbidv anbi2d rspc2ev syl3anc ) EHI
|
|
ZFJIZGJIZUAZEUBUCKZHIZVTDLZFMKZNOZVSPQZCLZGMKZNOZVSPQZRZWAWEUEKZFGUEKZMKN
|
|
OEPQZSZCJTDJTZWCALZPQZWGBLZPQZRZWLSZCJTDJTZBHUOAHUOVOVPVTVQEUDUFZXBVRWMDC
|
|
JJVRWAJIZWEJIZRZRZWIWJFWEUEKZMKZNOZVSPQZXGWKMKZNOZVSPQZRZWLXFXJWDXMWHXFXI
|
|
WCVSPXFXHWBNXFWAFWEVRXCXDUGVOVPVQXEUHZVRXCXDUIZUJUKULXFXLWGVSPXFXKWFNXFFW
|
|
EGXOXPVOVPVQXEUMZUNUKULUPXFWJJIZWKJIXGJIEUQIXNWLSXEXRVRWAWEURUSXFFGXOXQUT
|
|
XFFWEXOXPUTXFEVOVPVQXEVAVBWJWKXGEVCVDVEVFXAWNWDWRRZWLSZCJTDJTABVSVSHHWOVS
|
|
VGZWTXTDCJJYAWSXSWLYAWPWDWRWOVSWCPVHVIVJVKWQVSVGZXTWMDCJJYBXSWIWLYBWRWHWD
|
|
WQVSWGPVHVLVJVKVMVN $.
|
|
|
|
$( Complex number subtraction is a continuous function. Part of
|
|
Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
subcn2 $p |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) ->
|
|
E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC
|
|
( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) ->
|
|
( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) $=
|
|
( vw crp wcel cc cv cmin co cabs cfv clt wbr wa caddc w3a cneg wral negcl
|
|
wi wrex addcn2 syl3an3 wceq oveq1 fveq2d breq1d anbi2d oveq2 oveq1d rspcv
|
|
imbi12d syl adantl simpr simpll3 neg2subd abssubd adantll simpll2 negsubd
|
|
eqtrd negsub oveq12d sylibd ralrimdva ralimdva reximdv mpd ) EIJZFKJZGKJZ
|
|
UAZDLZFMNOPALQRZHLZGUBZMNZOPZBLZQRZSZVSWATNZFWBTNZMNZOPZEQRZUEZHKUCZDKUCZ
|
|
BIUFZAIUFZVTCLZGMNOPZWEQRZSZVSWRMNZFGMNZMNZOPZEQRZUEZCKUCZDKUCZBIUFZAIUFV
|
|
QVOVPWBKJWQGUDABHDEFWBUGUHVRWPXJAIVRWOXIBIVRWNXHDKVRVSKJZSZWNXGCKXLWRKJZS
|
|
ZWNVTWRUBZWBMNZOPZWEQRZSZVSXOTNZWIMNZOPZEQRZUEZXGXMWNYDUEZXLXMXOKJYEWRUDW
|
|
MYDHXOKWAXOUIZWGXSWLYCYFWFXRVTYFWDXQWEQYFWCXPOWAXOWBMUJUKULUMYFWKYBEQYFWJ
|
|
YAOYFWHXTWIMWAXOVSTUNUOUKULUQUPURUSXNXSXAYCXFXNXRWTVTXNXQWSWEQXNXQGWRMNZO
|
|
PWSXNXPYGOXNWRGXLXMUTZVOVPVQXKXMVAZVBUKXNGWRYIYHVCVGULUMXNYBXEEQXNYAXDOXN
|
|
XTXBWIXCMXKXMXTXBUIVRVSWRVHVDXNFGVOVPVQXKXMVEYIVFVIUKULUQVJVKVLVMVMVN $.
|
|
|
|
$( Complex number multiplication is a continuous function. Part of
|
|
Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario
|
|
Carneiro, 31-Jan-2014.) $)
|
|
mulcn2 $p |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) ->
|
|
E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC
|
|
( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) ->
|
|
( abs ` ( ( u x. v ) - ( B x. C ) ) ) < A ) ) $=
|
|
( crp wcel cc co cabs cfv clt wbr wa cmul wi cr cle c2 cdiv c1 caddc cmin
|
|
w3a cv wral rphalfcl 3ad2ant1 abscl 3ad2ant3 3ad2ant2 1re readdcl sylancl
|
|
wrex cc0 0lt1 addgegt0 an4s mpanr12 syl2anc elrpd rpdivcld rpred readdcld
|
|
absge0 elrp sylan2b syl21anc simprl simpl2 subcld abscld adantr ltmuldivd
|
|
simprr simpl3 abs2difd resubcld lelttr syl3anc mpand ltsubadd2d ltle syld
|
|
sylibd absge0d lemul2a syl112anc remulcld expd 3syld sylbird impd absmuld
|
|
ex com23 subdird fveq2d eqtr3d breq1d ltmuldiv2d subdid lep1d jca lemul1a
|
|
syl3an3 mpd eqbrtrd mulcld adantld mulcl adantl simpl1 abs3lem ralrimivva
|
|
jcad syl22anc wceq breq2 anbi1d imbi1d 2ralbidv anbi2d rspc2ev ) EHIZFJIZ
|
|
GJIZUFZEUAUBKZGLMZYLFLMZUCUDKZUBKZUDKZUBKZHIYPHIZDUGZFUEKZLMZYRNOZCUGZGUE
|
|
KZLMZYPNOZPZYTUUDQKZFGQKZUEKLMENOZRZCJUHDJUHZUUBAUGZNOZUUFBUGZNOZPZUUKRZC
|
|
JUHDJUHZBHUQAHUQYKYLYQYHYIYLHIZYJEUIUJZYKYQYKYMYPYJYHYMSIZYIGUKULZYKYPYKY
|
|
LYOUVBYKYOYKYNSIZUCSIZYOSIZYIYHUVEYJFUKZUMUNYNUCUOUPZYIYHURYONOZYJYIUVEUR
|
|
YNTOZUVJUVHFVHUVEUVKPUVFURUCNOZUVJUNUSUVEUVFUVKUVLUVJYNUCUTVAVBVCUMVDZVEZ
|
|
VFZVGZYKUVCURYMTOZYSURYQNOZUVDYJYHUVQYIGVHULUVNYSUVCUVQPYPSIZURYPNOZPUVRY
|
|
PVIUVCUVSUVQUVTUVRYMYPUTVAVJVKVDZVEUVNYKUULDCJJYKYTJIZUUDJIZPZPZUUHUUIFUU
|
|
DQKZUEKZLMZYLNOZUWFUUJUEKZLMZYLNOZPZUUKUWEUUHUWIUWLUWEUUHUUBUUDLMZQKZYLNO
|
|
ZUWIUWEUUCUUGUWPUWEUUCUUBYQQKZYLNOZUUGUWPRUWEUUBYLYQUWEUUAUWEYTFYKUWBUWCV
|
|
LZYHYIYJUWDVMZVNZVOZUWEYLYKUVAUWDUVBVPVFZYKYQHIUWDUWAVPVQUWEUUGUWRUWPUWEU
|
|
UGUWNYQTOZUWOUWQTOZUWRUWPRUWEUUGUWNYQNOZUXDUWEUUGUWNYMUEKZYPNOZUXFUWEUXGU
|
|
UFTOZUUGUXHUWEUUDGYKUWBUWCVRZYHYIYJUWDVSZVTUWEUXGSIUUFSIZUVSUXIUUGPUXHRUW
|
|
EUWNYMUWEUUDUXJVOZYKUVCUWDUVDVPZWAUWEUUEUWEUUDGUXJUXKVNZVOZYKUVSUWDUVOVPZ
|
|
UXGUUFYPWBWCWDUWEUWNYMYPUXMUXNUXQWEWHUWEUWNSIZYQSIZUXFUXDRUXMYKUXSUWDUVPV
|
|
PZUWNYQWFVCWGUWEUXRUXSUUBSIZURUUBTOZUXDUXERUXMUXTUXBUWEUUAUXAWIUXRUXSUYAU
|
|
YBPUFUXDUXEUWNYQUUBWJWRWKUWEUXEUWRUWPUWEUWOSIUWQSIYLSIZUXEUWRPUWPRUWEUUBU
|
|
WNUXBUXMWLUWEUUBYQUXBUXTWLUXCUWOUWQYLWBWCWMWNWSWOWPUWEUWOUWHYLNUWEUUAUUDQ
|
|
KZLMUWOUWHUWEUUAUUDUXAUXJWQUWEUYDUWGLUWEYTFUUDUWSUWTUXJWTXAXBXCWHUWEUUGUW
|
|
LUUCUWEUUGYOUUFQKZYLNOZUWLUWEUUFYLYOUXPUXCYKYOHIUWDUVMVPXDUWEUWKUYETOZUYF
|
|
UWLUWEUWKYNUUFQKZUYETUWEFUUEQKZLMUWKUYHUWEUYIUWJLUWEFUUDGUWTUXJUXKXEXAUWE
|
|
FUUEUWTUXOWQXBUWEYNYOTOZUYHUYETOZUWEYNUWEFUWTVOZXFUWEUVEUVGUUEJIZUYJUYKRZ
|
|
UYLYKUVGUWDUVIVPZUXOUYMUVEUVGUXLURUUFTOZPZUYNUYMUXLUYPUUEUKUUEVHXGUVEUVGU
|
|
YQUFUYJUYKYNYOUUFXHWRXIWCXJXKUWEUWKSIUYESIUYCUYGUYFPUWLRUWEUWJUWEUWFUUJUW
|
|
EFUUDUWTUXJXLZUWEFGUWTUXKXLZVNVOUWEYOUUFUYOUXPWLUXCUWKUYEYLWBWCWDWOXMXSUW
|
|
EUUIJIZUUJJIUWFJIESIUWMUUKRUWDUYTYKYTUUDXNXOUYSUYRUWEEYHYIYJUWDXPVFUUIUUJ
|
|
UWFEXQXTWGXRUUTUUMUUCUUQPZUUKRZCJUHDJUHABYRYPHHUUNYRYAZUUSVUBDCJJVUCUURVU
|
|
AUUKVUCUUOUUCUUQUUNYRUUBNYBYCYDYEUUPYPYAZVUBUULDCJJVUDVUAUUHUUKVUDUUQUUGU
|
|
UCUUPYPUUFNYBYFYDYEYGWC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d y F $.
|
|
cn1lem.1 $e |- F : CC --> CC $.
|
|
cn1lem.2 $e |- ( ( z e. CC /\ A e. CC ) ->
|
|
( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) ) $.
|
|
$( A sufficient condition for a function to be continuous. (Contributed by
|
|
Mario Carneiro, 9-Feb-2014.) $)
|
|
cn1lem $p |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) $=
|
|
( cc wcel cv crp wa cmin co cabs cfv clt wbr wi cr wral wrex simpr simpll
|
|
cle syl2anc ffvelrni syl subcld abscld rpre ad2antlr lelttr syl3anc mpand
|
|
ralrimiva weq breq2 imbi1d ralbidv rspcev ) DHIZAJZKIZLZVDCJZDMNZOPZVCQRZ
|
|
VFEPZDEPZMNZOPZVCQRZSZCHUAZVHBJZQRZVNSZCHUAZBKUBVBVDUCVEVOCHVEVFHIZLZVMVH
|
|
UERZVIVNWBWAVBWCVEWAUCZVBVDWAUDZGUFWBVMTIVHTIVCTIZWCVILVNSWBVLWBVJVKWBWAV
|
|
JHIWDHHVFEFUGUHWBVBVKHIWEHHDEFUGUHUIUJWBVGWBVFDWDWEUIUJVDWFVBWAVCUKULVMVH
|
|
VCUMUNUOUPVTVPBVCKBAUQZVSVOCHWGVRVIVNVQVCVHQURUSUTVAUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $.
|
|
$( The absolute value function is continuous. (Contributed by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
abscn2 $p |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( abs ` z ) - ( abs ` A ) ) ) < x ) ) $=
|
|
( cabs cc cr wf wss absf ax-resscn fss mp2an cv abs2difabs cn1lem ) ABCDE
|
|
FGEHGFIFFEHJKFGFELMCNDOP $.
|
|
|
|
$( The complex conjugate function is continuous. (Contributed by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
cjcn2 $p |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( * ` z ) - ( * ` A ) ) ) < x ) ) $=
|
|
( ccj cjf cv cc wcel wa cfv cmin co cabs cr wceq cle wbr cjcl subcl cjsub
|
|
syl2an abscld fveq2d abscjd eqtr3d eqle syl2anc cn1lem ) ABCDEFCGZHIZDHIZ
|
|
JZUJEKZDEKZLMZNKZOIUQUJDLMZNKZPUQUSQRUMUPUKUNHIUOHIUPHIULUJSDSUNUOTUBUCUM
|
|
UREKZNKUQUSUMUTUPNUJDUAUDUMURUJDTUEUFUQUSUGUHUI $.
|
|
|
|
$( The real part function is continuous. (Contributed by Mario Carneiro,
|
|
9-Feb-2014.) $)
|
|
recn2 $p |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( Re ` z ) - ( Re ` A ) ) ) < x ) ) $=
|
|
( cre cc cr wf wss ref ax-resscn fss mp2an cv wcel cmin co cfv cabs cle
|
|
wa resub fveq2d wbr subcl absrele syl eqbrtrrd cn1lem ) ABCDEFGEHGFIFFEHJ
|
|
KFGFELMCNZFODFOUAZUJDPQZERZSRZUJERDERPQZSRULSRZTUKUMUOSUJDUBUCUKULFOUNUPT
|
|
UDUJDUEULUFUGUHUI $.
|
|
|
|
$( The imaginary part function is continuous. (Contributed by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
imcn2 $p |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( Im ` z ) - ( Im ` A ) ) ) < x ) ) $=
|
|
( cim cc cr wf wss imf ax-resscn fss mp2an cv wcel cmin co cfv cabs cle
|
|
wa imsub fveq2d wbr subcl absimle syl eqbrtrrd cn1lem ) ABCDEFGEHGFIFFEHJ
|
|
KFGFELMCNZFODFOUAZUJDPQZERZSRZUJERDERPQZSRULSRZTUKUMUOSUJDUBUCUKULFOUNUPT
|
|
UDUJDUEULUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d k x y z A $. $d k y z F $. $d k x G $. $d k x y z ph $. $d k y Z $.
|
|
$d k x y z H $. $d k M $.
|
|
climcn1lem.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climcn1lem.2 $e |- ( ph -> F ~~> A ) $.
|
|
climcn1lem.4 $e |- ( ph -> G e. W ) $.
|
|
climcn1lem.5 $e |- ( ph -> M e. ZZ ) $.
|
|
climcn1lem.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
${
|
|
climcn1lem.7 $e |- H : CC --> CC $.
|
|
climcn1lem.8 $e |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC
|
|
( ( abs ` ( z - A ) ) < y ->
|
|
( abs ` ( ( H ` z ) - ( H ` A ) ) ) < x ) ) $.
|
|
climcn1lem.9 $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( H ` ( F ` k ) ) ) $.
|
|
$( The limit of a continuous function, theorem form. (Contributed by
|
|
Mario Carneiro, 9-Feb-2014.) $)
|
|
climcn1lem $p |- ( ph -> G ~~> ( H ` A ) ) $=
|
|
( cc cli wbr wcel climcl syl cv cfv ffvelrni adantl crp cmin co cabs wi
|
|
clt wral wrex sylan climcn1 ) ABCDEUAFIGHJKLMPAGEUBUCEUAUDZNEGUEUFZDUGZ
|
|
UAUDVCIUHZUAUDAUAUAVCIRUIUJNOAVABUGZUKUDVCEULUMUNUHCUGUPUCVDEIUHULUMUNU
|
|
HVEUPUCUODUAUQCUKURVBSUSQTUT $.
|
|
$}
|
|
|
|
${
|
|
climabs.7 $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( abs ` ( F ` k ) ) ) $.
|
|
$( Limit of the absolute value of a sequence. Proposition 12-2.4(c) of
|
|
[Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
climabs $p |- ( ph -> G ~~> ( abs ` A ) ) $=
|
|
( vx vy vz cabs cc cr wf wss absf ax-resscn fss mp2an abscn2 climcn1lem
|
|
) AOPQBCDERFGHIJKLMSTRUATSUBSSRUAUCUDSTSRUEUFOPQBUGNUH $.
|
|
$}
|
|
|
|
${
|
|
climcj.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( * ` ( F ` k ) ) ) $.
|
|
$( Limit of the complex conjugate of a sequence. Proposition 12-2.4(c)
|
|
of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by
|
|
Mario Carneiro, 9-Feb-2014.) $)
|
|
climcj $p |- ( ph -> G ~~> ( * ` A ) ) $=
|
|
( vx vy vz ccj cjf cjcn2 climcn1lem ) AOPQBCDERFGHIJKLMSOPQBTNUA $.
|
|
$}
|
|
|
|
${
|
|
climre.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( Re ` ( F ` k ) ) ) $.
|
|
$( Limit of the real part of a sequence. Proposition 12-2.4(c) of
|
|
[Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
climre $p |- ( ph -> G ~~> ( Re ` A ) ) $=
|
|
( vx vy vz cre cc cr wf wss ref ax-resscn fss mp2an recn2 climcn1lem )
|
|
AOPQBCDERFGHIJKLMSTRUATSUBSSRUAUCUDSTSRUEUFOPQBUGNUH $.
|
|
$}
|
|
|
|
${
|
|
climim.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( Im ` ( F ` k ) ) ) $.
|
|
$( Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of
|
|
[Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario
|
|
Carneiro, 9-Feb-2014.) $)
|
|
climim $p |- ( ph -> G ~~> ( Im ` A ) ) $=
|
|
( vx vy vz cim cc cr wf wss imf ax-resscn fss mp2an imcn2 climcn1lem )
|
|
AOPQBCDERFGHIJKLMSTRUATSUBSSRUAUCUDSTSRUEUFOPQBUGNUH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d j k F $. $d j k M $. $d j k ph $. $d j k Z $. $d j k A $.
|
|
climrecl.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climrecl.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climrecl.3 $e |- ( ph -> F ~~> A ) $.
|
|
climrecl.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
$( The limit of a convergent real sequence is real. Corollary 12-2.5 of
|
|
[Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) $)
|
|
climrecl $p |- ( ph -> A e. RR ) $=
|
|
( cli wbr cc wcel syl cre cfv wceq cvv eqcomd climcl brrelexi cv wa recnd
|
|
climrel cr rere climre climuni syl2anc rerebd ) ABADBKLZBMNIBDUAOABBPQZAU
|
|
MDUNKLBUNRIABCDDESFGIAUMDSNIDBKUFUBOHACUCZFNUDZUODQZJUEUPUQUGNZUQUQPQZRJU
|
|
RUSUQUQUHTOUIBUNDUJUKTUL $.
|
|
|
|
climge0.5 $e |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) ) $.
|
|
$( A nonnegative sequence converges to a nonnegative number. (Contributed
|
|
by NM, 11-Sep-2005.) $)
|
|
climge0 $p |- ( ph -> 0 <_ A ) $=
|
|
( vj cc0 wbr clt wa cfv wcel adantr cr cle wn cv cmin cabs cneg wfal wral
|
|
co cuz wrex cz climrecl renegcld lt0neg1d biimpa elrpd cli climi2 r19.2uz
|
|
eqidd syl caddc simprr ad2ant2r absdifltd mpbid simprd recnd breqtrd 0red
|
|
negidd lenltd pm2.21fal rexlimddv inegd wb 0re lenlt sylancr mpbird ) AMB
|
|
UANZBMONZUBZAWCAWCPZCUCZDQZBUDUIUEQBUFZONZUGCFWEWICLUCUJQUHLFUKWICFUKWEBW
|
|
GWHLCDEFGAEULRWCHSWEWHWEBABTRZWCABCDEFGHIJUMZSZUNZAWCMWHONABWKUOUPUQWEWFF
|
|
RZPWGVAADBURNWCISUSWILCEFGUTVBWEWNWIPZPZWGMONZWPWGBWHVCUIZMOWPBWHUDUIWGON
|
|
ZWGWRONZWPWIWSWTPWEWNWIVDWPWGBWHAWNWGTRWCWIJVEZWEWJWOWLSZWEWHTRWOWMSVFVGV
|
|
HWPBWPBXBVIVLVJWPMWGUANZWQUBAWNXCWCWIKVEWPMWGWPVKXAVMVGVNVOVPAMTRWJWBWDVQ
|
|
VRWKMBVSVTWA $.
|
|
$}
|
|
|
|
${
|
|
$d k u v x y z B $. $d k C $. $d j k u v y z F $. $d j k u v x y z ph $.
|
|
$d j k u v x y z A $. $d j k v x y z G $. $d k x y z H $. $d j k x M $.
|
|
$d j k x y z Z $.
|
|
climadd.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
climadd.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climadd.4 $e |- ( ph -> F ~~> A ) $.
|
|
${
|
|
climadd.6 $e |- ( ph -> H e. X ) $.
|
|
climadd.7 $e |- ( ph -> G ~~> B ) $.
|
|
climadd.8 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
climadd.9 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) $.
|
|
${
|
|
climadd.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( H ` k ) = ( ( F ` k ) + ( G ` k ) ) ) $.
|
|
$( Limit of the sum of two converging sequences. Proposition 12-2.1(a)
|
|
of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof
|
|
shortened by Mario Carneiro, 31-Jan-2014.) $)
|
|
climadd $p |- ( ph -> H ~~> ( A + B ) ) $=
|
|
( cc wcel vx vy vz vv vu caddc cli wbr climcl syl cv wa co adantl crp
|
|
addcl cmin cabs cfv clt wral wrex simpr adantr addcn2 syl3anc climcn2
|
|
wi ) AUAUBUCUDUEBCSSDUFEFGHIJKLAEBUGUHBSTZMBEUIUJZAFCUGUHCSTZOCFUIUJZ
|
|
UEUKZSTUDUKZSTULVMVNUFUMZSTAVMVNUPUNMONAUAUKZUOTZULVQVIVKVMBUQUMURUSU
|
|
BUKUTUHVNCUQUMURUSUCUKUTUHULVOBCUFUMUQUMURUSVPUTUHVHUDSVAUESVAUCUOVBU
|
|
BUOVBAVQVCAVIVQVJVDAVKVQVLVDUBUCUDUEVPBCVEVFPQRVG $.
|
|
$}
|
|
|
|
${
|
|
climmul.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) ) $.
|
|
$( Limit of the product of two converging sequences. Proposition
|
|
12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.)
|
|
(Proof shortened by Mario Carneiro, 1-Feb-2014.) $)
|
|
climmul $p |- ( ph -> H ~~> ( A x. B ) ) $=
|
|
( cc wcel vx vy vz vv vu cli wbr climcl syl cv wa co mulcl adantl crp
|
|
cmul cmin cabs cfv clt wral wrex simpr adantr mulcn2 syl3anc climcn2
|
|
wi ) AUAUBUCUDUEBCSSDUPEFGHIJKLAEBUFUGBSTZMBEUHUIZAFCUFUGCSTZOCFUHUIZ
|
|
UEUJZSTUDUJZSTUKVMVNUPULZSTAVMVNUMUNMONAUAUJZUOTZUKVQVIVKVMBUQULURUSU
|
|
BUJUTUGVNCUQULURUSUCUJUTUGUKVOBCUPULUQULURUSVPUTUGVHUDSVAUESVAUCUOVBU
|
|
BUOVBAVQVCAVIVQVJVDAVKVQVLVDUBUCUDUEVPBCVEVFPQRVG $.
|
|
$}
|
|
|
|
${
|
|
climsub.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( H ` k ) = ( ( F ` k ) - ( G ` k ) ) ) $.
|
|
$( Limit of the difference of two converging sequences. Proposition
|
|
12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.)
|
|
(Proof shortened by Mario Carneiro, 1-Feb-2014.) $)
|
|
climsub $p |- ( ph -> H ~~> ( A - B ) ) $=
|
|
( cc cmin vx vy vz vv vu cli wbr climcl syl cv wa co subcl adantl crp
|
|
wcel cabs cfv clt wi wral wrex simpr adantr subcn2 syl3anc climcn2 )
|
|
AUAUBUCUDUEBCSSDTEFGHIJKLAEBUFUGBSUPZMBEUHUIZAFCUFUGCSUPZOCFUHUIZUEUJ
|
|
ZSUPUDUJZSUPUKVLVMTULZSUPAVLVMUMUNMONAUAUJZUOUPZUKVPVHVJVLBTULUQURUBU
|
|
JUSUGVMCTULUQURUCUJUSUGUKVNBCTULTULUQURVOUSUGUTUDSVAUESVAUCUOVBUBUOVB
|
|
AVPVCAVHVPVIVDAVJVPVKVDUBUCUDUEVOBCVEVFPQRVG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
climaddc1.5 $e |- ( ph -> C e. CC ) $.
|
|
climaddc1.6 $e |- ( ph -> G e. W ) $.
|
|
climaddc1.7 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
${
|
|
climaddc1.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( ( F ` k ) + C ) ) $.
|
|
$( Limit of a constant ` C ` added to each term of a sequence.
|
|
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
|
|
3-Feb-2014.) $)
|
|
climaddc1 $p |- ( ph -> G ~~> ( A + C ) ) $=
|
|
( cz cc wcel cfv csn cxp cc0 cli wbr 0z uzssz climconst2 sylancl wceq
|
|
zex cv wa cuz eluzelz eleq2s fvconst2g syl2an adantr eqeltrd caddc co
|
|
oveq2d eqtr4d climadd ) ABCDEQCUAUBZFGHIJKLNACRSZUCQSVFCUDUEMUFCUCQUC
|
|
UGUKUHUIOADULZISZUMZVHVFTZCRAVGVHQSZVKCUJVIMVLVHGUNTIGVHUOJUPQCVHRUQU
|
|
RZAVGVIMUSUTVJVHFTVHETZCVAVBVNVKVAVBPVJVKCVNVAVMVCVDVE $.
|
|
$}
|
|
|
|
${
|
|
climaddc2.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( C + ( F ` k ) ) ) $.
|
|
$( Limit of a constant ` C ` added to each term of a sequence.
|
|
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
|
|
3-Feb-2014.) $)
|
|
climaddc2 $p |- ( ph -> G ~~> ( C + A ) ) $=
|
|
( caddc co cli wcel cv wa cfv adantr addcomd climaddc1 wbr climcl syl
|
|
cc eqtrd breqtrd ) AFBCQRCBQRSABCDEFGHIJKLMNOADUAZITZUBZUMFUCCUMEUCZQ
|
|
RUPCQRPUOCUPACUJTUNMUDOUEUKUFABCAEBSUGBUJTLBEUHUIMUEUL $.
|
|
$}
|
|
|
|
${
|
|
climmulc2.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( C x. ( F ` k ) ) ) $.
|
|
$( Limit of a sequence multiplied by a constant ` C ` . Corollary
|
|
12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.)
|
|
(Revised by Mario Carneiro, 3-Feb-2014.) $)
|
|
climmulc2 $p |- ( ph -> G ~~> ( C x. A ) ) $=
|
|
( cz cc wcel cfv csn cxp cc0 cli wbr 0z uzssz climconst2 sylancl wceq
|
|
zex cv cuz eluzelz eleq2s fvconst2g syl2an adantr eqeltrd cmul oveq1d
|
|
wa co eqtr4d climmul ) ACBDQCUAUBZEFGHIJKACRSZUCQSVFCUDUEMUFCUCQUCUGU
|
|
KUHUINLADULZISZVBZVHVFTZCRAVGVHQSZVKCUJVIMVLVHGUMTIGVHUNJUOQCVHRUPUQZ
|
|
AVGVIMURUSOVJVHFTCVHETZUTVCVKVNUTVCPVJVKCVNUTVMVAVDVE $.
|
|
$}
|
|
|
|
${
|
|
climsubc1.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( ( F ` k ) - C ) ) $.
|
|
$( Limit of a constant ` C ` subtracted from each term of a sequence.
|
|
(Contributed by Mario Carneiro, 9-Feb-2014.) $)
|
|
climsubc1 $p |- ( ph -> G ~~> ( A - C ) ) $=
|
|
( cz cc wcel cfv csn cxp cc0 cli wbr 0z uzssz climconst2 sylancl wceq
|
|
zex cv cuz eluzelz eleq2s fvconst2g syl2an adantr eqeltrd cmin oveq2d
|
|
wa co eqtr4d climsub ) ABCDEQCUAUBZFGHIJKLNACRSZUCQSVFCUDUEMUFCUCQUCU
|
|
GUKUHUIOADULZISZVBZVHVFTZCRAVGVHQSZVKCUJVIMVLVHGUMTIGVHUNJUOQCVHRUPUQ
|
|
ZAVGVIMURUSVJVHFTVHETZCUTVCVNVKUTVCPVJVKCVNUTVMVAVDVE $.
|
|
$}
|
|
|
|
${
|
|
climsubc2.h $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( C - ( F ` k ) ) ) $.
|
|
$( Limit of a constant ` C ` minus each term of a sequence.
|
|
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
|
|
9-Feb-2014.) $)
|
|
climsubc2 $p |- ( ph -> G ~~> ( C - A ) ) $=
|
|
( cz cc wcel cfv csn cxp cc0 cli wbr 0z uzssz climconst2 sylancl wceq
|
|
zex cv cuz eluzelz eleq2s fvconst2g syl2an adantr eqeltrd cmin oveq1d
|
|
wa co eqtr4d climsub ) ACBDQCUAUBZEFGHIJKACRSZUCQSVFCUDUEMUFCUCQUCUGU
|
|
KUHUINLADULZISZVBZVHVFTZCRAVGVHQSZVKCUJVIMVLVHGUMTIGVHUNJUOQCVHRUPUQZ
|
|
AVGVIMURUSOVJVHFTCVHETZUTVCVKVNUTVCPVJVKCVNUTVMVAVDVE $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
climle.5 $e |- ( ph -> G ~~> B ) $.
|
|
climle.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
climle.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR ) $.
|
|
climle.8 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) ) $.
|
|
$( Comparison of the limits of two sequences. (Contributed by Paul
|
|
Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) $)
|
|
climle $p |- ( ph -> A <_ B ) $=
|
|
( vj cmin cle wbr cfv cc0 co cv cmpt cvv cuz cz zex uzssz ssexi eqeltri
|
|
wcel mptex wa recnd cr wceq simpr resubcld fveq2 oveq12d fvmptg syl2anc
|
|
a1i eqid climsub eqeltrd subge0d mpbird breqtrrd climge0 climrecl mpbid
|
|
) AUACBQUBZRSBCRSAVNDPHPUCZFTZVOETZQUBZUDZGHIJACBDFEVSGUEHIJLVSUEULAPHV
|
|
RHGUFTZUEIVTUGUHGUIUJUKUMVDKADUCZHULZUNZWAFTZNUOWCWAETZMUOWCWBWDWEQUBZU
|
|
PULWAVSTZWFUQAWBURWCWDWENMUSZPWAVRWFHUPVSVOWAUQVPWDVQWEQVOWAFUTVOWAEUTV
|
|
AVSVEVBVCZVFWCWGWFUPWIWHVGWCUAWFWGRWCUAWFRSWEWDRSOWCWDWENMVHVIWIVJVKACB
|
|
ACDFGHIJLNVLABDEGHIJKMVLVHVM $.
|
|
$}
|
|
|
|
climsqz.5 $e |- ( ph -> G e. W ) $.
|
|
climsqz.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
climsqz.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR ) $.
|
|
${
|
|
climsqz.8 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) ) $.
|
|
climsqz.9 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ A ) $.
|
|
$( Convergence of a sequence sandwiched between another converging
|
|
sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by
|
|
Mario Carneiro, 3-Feb-2014.) $)
|
|
climsqz $p |- ( ph -> G ~~> A ) $=
|
|
( vj wcel wa cr vx cli wbr cv cfv cmin co cabs clt cuz wral wrex crp cz
|
|
adantr simpr eqidd climi2 uztrn2 cle climrecl lesub2dd abssuble0d letrd
|
|
wi 3brtr4d adantlr ad2antrr resubcld recnd abscld rpre ad2antlr syl3anc
|
|
lelttr mpand sylan2 anassrs ralimdva reximdva ralrimiva clim2c mpbird
|
|
mpd ) AEBUBUCCUDZEUEZBUFUGZUHUEZUAUDZUIUCZCQUDZUJUEZUKZQHULZUAUMUKAWNUA
|
|
UMAWIUMRZSZWEDUEZBUFUGZUHUEZWIUIUCZCWLUKZQHULWNWPBWQWIQCDFHIAFUNRWOJUOA
|
|
WOUPWPWEHRZSZWQUQADBUBUCWOKUOURWPXAWMQHWPWKHRZSWTWJCWLWPXDWEWLRZWTWJVEZ
|
|
XDXESWPXBXFFWEWKHIUSXCWHWSUTUCZWTWJAXBXGWOAXBSZBWFUFUGBWQUFUGWHWSUTXHWQ
|
|
WFBMNABTRZXBABCDFHIJKMVAZUOZOVBXHWFBNXKPVCXHWQBMXKXHWQWFBMNXKOPVDVCVFVG
|
|
XCWHTRWSTRWITRZXGWTSWJVEXCWGXCWGXCWFBAXBWFTRWONVGAXIWOXBXJVHZVIVJVKXCWR
|
|
XCWRXCWQBAXBWQTRWOMVGXMVIVJVKWOXLAXBWIVLVMWHWSWIVOVNVPVQVRVSVTWDWAAUABW
|
|
FQCEFGHIJLXHWFUQABXJVJXHWFNVJWBWC $.
|
|
$}
|
|
|
|
${
|
|
climsqz2.8 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ ( F ` k ) ) $.
|
|
climsqz2.9 $e |- ( ( ph /\ k e. Z ) -> A <_ ( G ` k ) ) $.
|
|
$( Convergence of a sequence sandwiched between another converging
|
|
sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised
|
|
by Mario Carneiro, 3-Feb-2014.) $)
|
|
climsqz2 $p |- ( ph -> G ~~> A ) $=
|
|
( vj wcel wa cr vx cli wbr cv cfv cmin co cabs clt cuz wral wrex crp cz
|
|
adantr simpr eqidd climi2 uztrn2 cle climrecl lesub1dd abssubge0d letrd
|
|
wi 3brtr4d adantlr ad2antrr resubcld recnd abscld rpre ad2antlr syl3anc
|
|
lelttr mpand sylan2 anassrs ralimdva reximdva ralrimiva clim2c mpbird
|
|
mpd ) AEBUBUCCUDZEUEZBUFUGZUHUEZUAUDZUIUCZCQUDZUJUEZUKZQHULZUAUMUKAWNUA
|
|
UMAWIUMRZSZWEDUEZBUFUGZUHUEZWIUIUCZCWLUKZQHULWNWPBWQWIQCDFHIAFUNRWOJUOA
|
|
WOUPWPWEHRZSZWQUQADBUBUCWOKUOURWPXAWMQHWPWKHRZSWTWJCWLWPXDWEWLRZWTWJVEZ
|
|
XDXESWPXBXFFWEWKHIUSXCWHWSUTUCZWTWJAXBXGWOAXBSZWGWRWHWSUTXHWFWQBNMABTRZ
|
|
XBABCDFHIJKMVAZUOZOVBXHBWFXKNPVCXHBWQXKMXHBWFWQXKNMPOVDVCVFVGXCWHTRWSTR
|
|
WITRZXGWTSWJVEXCWGXCWGXCWFBAXBWFTRWONVGAXIWOXBXJVHZVIVJVKXCWRXCWRXCWQBA
|
|
XBWQTRWOMVGXMVIVJVKWOXLAXBWIVLVMWHWSWIVOVNVPVQVRVSVTWDWAAUABWFQCEFGHIJL
|
|
XHWFUQABXJVJXHWFNVJWBWC $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d j k A $. $d j k B $. $d j k x y F $. $d j k x y M $. $d j k x y N $.
|
|
$d j k x C $. $d j k G $. $d j k x y ph $. $d j k x Z $.
|
|
clim2ser.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
${
|
|
clim2ser.2 $e |- ( ph -> N e. Z ) $.
|
|
clim2ser.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
${
|
|
clim2ser.5 $e |- ( ph -> seq M ( + , F , CC ) ~~> A ) $.
|
|
$( The limit of an infinite series with an initial segment removed.
|
|
(Contributed by Jim Kingdon, 20-Aug-2021.) $)
|
|
clim2iser $p |- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~>
|
|
( A - ( seq M ( + , F , CC ) ` N ) ) ) $=
|
|
( caddc cc cfv co cvv wcel syl cv adantr vj vx vy cseq c1 cuz eqid cz
|
|
syl6eleq peano2uz eluzelz eluzel2 iserf ffvelrnd iseqex a1i syl6eleqr
|
|
wa uztrn2 sylan cmin addcl adantl w3a wceq addass simpr cnex sylan2br
|
|
wf eleq2i adantlr iseqsplit oveq1d syldan ffvelrnda pncan2d climsubc1
|
|
eqtr2d ) ABFLMDEUDZNZUAVTLMDFUELOZUDZWBPWBUFNZWDUGZAWBEUFNZQZWBUHQAFW
|
|
FQZWGAFGWFIHUIZEFUJRZEWBUKRZKAGMFVTACDEGHAWHEUHQWIEFULRJUMZIUNZWCPQAL
|
|
MDWBUOUPAUASZWDQZURZGMWNVTAGMVTVJWOWLTAWBGQZWOWNGQAWBWFGWJHUQZEWNWBGH
|
|
USUTUNWPWNVTNZWAVAOWAWNWCNZLOZWAVAOWTWPWSXAWAVAWPCUBUCLMDEFWNPCSZMQZU
|
|
BSZMQZURXBXDLOZMQWPXBXDVBVCXCXEUCSZMQVDXFXGLOXBXDXGLOLOVEWPXBXDXGVFVC
|
|
AWOVGMPQWPVHUPAWHWOWITAXBWFQZXBDNMQZWOXHAXBGQZXIGWFXBHVKJVIVLVMVNWPWA
|
|
WTAWAMQWOWMTAWDMWNWCACDWBWDWEWKAXBWDQZXJXIAWQXKXJWREXBWBGHUSUTJVOUMVP
|
|
VQVSVR $.
|
|
$}
|
|
|
|
${
|
|
clim2ser2.5 $e |- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~> A ) $.
|
|
$( The limit of an infinite series with an initial segment added.
|
|
(Contributed by Jim Kingdon, 21-Aug-2021.) $)
|
|
clim2iser2 $p |- ( ph -> seq M ( + , F , CC ) ~~>
|
|
( A + ( seq M ( + , F , CC ) ` N ) ) ) $=
|
|
( vj caddc cc cfv co cvv wcel syl cv vx vy cseq cuz syl6eleq peano2uz
|
|
c1 eqid cz eluzelz eluzel2 iserf ffvelrnd iseqex a1i syl6eleqr uztrn2
|
|
sylan syldan ffvelrnda addcl adantl w3a wceq addass simpr cnex adantr
|
|
wa eleq2i sylan2br adantlr iseqsplit addcomd eqtrd climaddc1 ) ABFMND
|
|
EUCZOZLMNDFUGMPZUCZVQVSQVSUDOZWAUHZAVSEUDOZRZVSUIRAFWCRZWDAFGWCIHUEZE
|
|
FUFSZEVSUJSZKAGNFVQACDEGHAWEEUIRWFEFUKSJULIUMZVQQRAMNDEUNUOAWANLTZVTA
|
|
CDVSWAWBWHACTZWARZWKGRZWKDONRZAVSGRWLWMAVSWCGWGHUPEWKVSGHUQURJUSULUTZ
|
|
AWJWARZVIZWJVQOVRWJVTOZMPWRVRMPWQCUAUBMNDEFWJQWKNRZUATZNRZVIWKWTMPZNR
|
|
WQWKWTVAVBWSXAUBTZNRVCXBXCMPWKWTXCMPMPVDWQWKWTXCVEVBAWPVFNQRWQVGUOAWE
|
|
WPWFVHAWKWCRZWNWPXDAWMWNGWCWKHVJJVKVLVMWQVRWRAVRNRWPWIVHWOVNVOVP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
iserex.2 $e |- ( ph -> N e. Z ) $.
|
|
iserex.3 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
$( An infinite series converges, if and only if the series does with
|
|
initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.)
|
|
(Revised by Mario Carneiro, 27-Apr-2014.) $)
|
|
iiserex $p |- ( ph -> ( seq M ( + , F , CC ) e. dom ~~> <->
|
|
seq N ( + , F , CC ) e. dom ~~> ) ) $=
|
|
( wceq caddc cc cseq cli wcel c1 co cfv wbr syl cdm wb cmin cuz iseqeq1
|
|
wi eleq1d bicomd a1i wa simpll cz syl6eleq eluzelz ax-1cn npcan sylancl
|
|
simplr syl6eleqr cv sylan simpr climdm sylib clim2iser eqbrtrrd climrel
|
|
zcnd releldmi adantr eqbrtrd clim2iser2 impbida ex wo uzm1 mpjaod ) AED
|
|
JZKLCDMZNUAZOZKLCEMZVTOZUBZEPUCQZDUDRZOZVRWDUFAVRWCWAVRWBVSVTKLCEDUEUGU
|
|
HUIAWGWDAWGUJZWAWCWHWAUJZWBVSNRZWEVSRZUCQZNSWCWIKLCWEPKQZMZWBWLNWIAWNWB
|
|
JZAWGWAUKZAWMEJZWOAELOPLOWQAEAEWFOZEULOAEFWFHGUMZDEUNTVHUOEPUPUQKLCWMEU
|
|
ETZTWIWJBCDWEFGWIWEWFFAWGWAURGUSWIABUTZFOZXACRLOZWPIVAWIWAVSWJNSWHWAVBV
|
|
SVCVDVEVFWBWLNVGVITWHWCUJZVSWBNRZWKKQZNSWAXDXEBCDWEFGWHWEFOWCWHWEWFFAWG
|
|
VBGUSVJXDAXBXCAWGWCUKZIVAXDWNWBXENXDAWOXGWTTXDWCWBXENSWHWCVBWBVCVDVKVLV
|
|
SXFNVGVITVMVNAWRVRWGVOWSDEVPTVQ $.
|
|
$}
|
|
|
|
${
|
|
$d G x $.
|
|
isermulc2.2 $e |- ( ph -> M e. ZZ ) $.
|
|
isermulc2.4 $e |- ( ph -> C e. CC ) $.
|
|
isermulc2.5 $e |- ( ph -> seq M ( + , F , CC ) ~~> A ) $.
|
|
isermulc2.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
isermulc2.7 $e |- ( ( ph /\ k e. Z ) ->
|
|
( G ` k ) = ( C x. ( F ` k ) ) ) $.
|
|
$( Multiplication of an infinite series by a constant. (Contributed by
|
|
Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro,
|
|
1-Feb-2014.) $)
|
|
iisermulc2 $p |- ( ph -> seq M ( + , G , CC ) ~~> ( C x. A ) ) $=
|
|
( caddc cc cvv wcel cmul co vj vx cseq iseqex a1i cv iserf ffvelrnda wa
|
|
addcl adantl wceq adantr adddi 3expb sylan cuz syl6eleq eleq2i sylan2br
|
|
cfv simpr adantlr cnex mulcl mulcld eqeltrd iseqdistr climmulc2 ) ABCUA
|
|
OPEGUCZOPFGUCZGQHIJLKVKQRAOPFGUDUEAHPUAUFZVJADEGHIJMUGUHAVLHRZUIZDUBCOP
|
|
SFEGVLQDUFZPRZUBUFZPRZUIZVOVQOTZPRVNVOVQUJUKVNCPRZVSCVTSTCVOSTCVQSTOTUL
|
|
ZAWAVMKUMZWAVPVRWBCVOVQUNUOUPVNVLHGUQVAZAVMVBIURAVOWDRZVOEVAZPRZVMWEAVO
|
|
HRZWGHWDVOIUSZMUTVCAWEVOFVAZCWFSTZULZVMWEAWHWLWINUTVCPQRVNVDUEVSVOVQSTP
|
|
RVNVOVQVEUKAWEWJPRZVMWEAWHWMWIAWHUIZWJWKPNWNCWFAWAWHKUMMVFVGUTVCWCVHVI
|
|
$.
|
|
$}
|
|
|
|
${
|
|
climlec2.2 $e |- ( ph -> M e. ZZ ) $.
|
|
climlec2.3 $e |- ( ph -> A e. RR ) $.
|
|
climlec2.4 $e |- ( ph -> F ~~> B ) $.
|
|
climlec2.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
climlec2.6 $e |- ( ( ph /\ k e. Z ) -> A <_ ( F ` k ) ) $.
|
|
$( Comparison of a constant to the limit of a sequence. (Contributed by
|
|
NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) $)
|
|
climlec2 $p |- ( ph -> A <_ B ) $=
|
|
( cz csn cxp wcel cc0 cfv cr cc cli wbr recnd 0z zex climconst2 sylancl
|
|
uzssz cv wa wceq cuz eluzelz eleq2s fvconst2g syl2an adantr eqeltrd cle
|
|
eqbrtrd climle ) ABCDNBOPZEFGHIABUAQRNQVCBUBUCABJUDUEBRNRUIUFUGUHKADUJZ
|
|
GQZUKZVDVCSZBTABTQZVDNQZVGBULVEJVIVDFUMSGFVDUNHUONBVDTUPUQZAVHVEJURUSLV
|
|
FVGBVDESUTVJMVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d G x $.
|
|
iserile.2 $e |- ( ph -> M e. ZZ ) $.
|
|
iserile.4 $e |- ( ph -> seq M ( + , F , CC ) ~~> A ) $.
|
|
iserile.5 $e |- ( ph -> seq M ( + , G , CC ) ~~> B ) $.
|
|
iserile.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
iserile.7 $e |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR ) $.
|
|
iserile.8 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) ) $.
|
|
$( Comparison of the limits of two infinite series. (Contributed by Jim
|
|
Kingdon, 22-Aug-2021.) $)
|
|
iserile $p |- ( ph -> A <_ B ) $=
|
|
( caddc cc wcel cr cfv vj vx cseq cv wa wceq cvv cnex a1i wss ax-resscn
|
|
cuz eleq2i sylan2br readdcl adantl addcl iseqss adantr fveq1d ffvelrnda
|
|
co iserfre eqeltrrd simpr adantlr cle wbr simpll biimpri syl2anc serile
|
|
syl6eleq climle ) ABCUAPQEGUCZPQFGUCZGHIJKLAUAUDZHRZUEZVQPSEGUCZTVQVOTS
|
|
VSVQVTVOAVTVOUFVRADUBPSQEGUGJQUGRAUHUIZSQUJAUKUIZDUDZGULTZRZAWCHRZWCETZ
|
|
SRZHWDWCIUMZMUNZWCSRUBUDZSRUEWCWKPVBZSRAWCWKUOUPZWCQRWKQRUEWLQRAWCWKUQU
|
|
PZURUSUTAHSVQVTADEGHIJMVCVAVDVSVQPSFGUCZTVQVPTSVSVQWOVPAWOVPUFVRADUBPSQ
|
|
FGUGJWAWBWEAWFWCFTZSRZWINUNZWMWNURUSUTAHSVQWOADFGHIJNVCVAVDVSDEFGVQVSVQ
|
|
HWDAVRVEIVMAWEWHVRWJVFAWEWQVRWRVFVSWEUEAWFWGWPVGVHAVRWEVIWEWFVSWFWEWIVJ
|
|
UPOVKVLVN $.
|
|
$}
|
|
|
|
${
|
|
iserige0.2 $e |- ( ph -> M e. ZZ ) $.
|
|
iserige0.3 $e |- ( ph -> seq M ( + , F , CC ) ~~> A ) $.
|
|
iserige0.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
iserige0.5 $e |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) ) $.
|
|
$( The limit of an infinite series of nonnegative reals is nonnegative.
|
|
(Contributed by Jim Kingdon, 22-Aug-2021.) $)
|
|
iserige0 $p |- ( ph -> 0 <_ A ) $=
|
|
( cc0 cuz cfv csn cxp cz wcel caddc syl cc cseq cli wbr iserclim0 cv wa
|
|
cr wceq simpr syl6eleq c0ex fvconst2 0re syl6eqel cle eqbrtrd iserile )
|
|
ALBCEMNZLOPZDEFGHAEQRSUAUTEUBLUCUDHEUETIACUFZFRZUGZVAUTNZLUHVCVAUSRVDLU
|
|
IVCVAFUSAVBUJGUKUSLVAULUMTZUNUOJVCVDLVADNUPVEKUQUR $.
|
|
$}
|
|
|
|
${
|
|
climub.2 $e |- ( ph -> N e. Z ) $.
|
|
climub.3 $e |- ( ph -> F ~~> A ) $.
|
|
climub.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
climub.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) ) $.
|
|
$( The limit of a monotonic sequence is an upper bound. (Contributed by
|
|
NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) $)
|
|
climub $p |- ( ph -> ( F ` N ) <_ A ) $=
|
|
( vj cfv cuz wcel cr wi syldan co eqid cz syl6eleq eluzelz syl cv fveq2
|
|
eleq1d imbi2d expcom vtoclga mpcom uztrn2 sylan impcom simpr cfz elfzuz
|
|
wceq wa sylan2 adantlr c1 cmin caddc cle wbr monoord climlec2 ) AFDNZBM
|
|
DFFONZVKUAAFEONZPFUBPAFGVLIHUCEFUDUEFGPZAVJQPZIACUFZDNZQPZRZAVNRCFGVOFU
|
|
SZVQVNAVSVPVJQVOFDUGUHUIAVOGPZVQKUJZUKULJAMUFZVKPZWBGPZWBDNZQPZAVMWCWDI
|
|
EWBFGHUMUNWDAWFVRAWFRCWBGVOWBUSZVQWFAWGVPWEQVOWBDUGUHUIWAUKUOSAWCUTCDFW
|
|
BAWCUPAVOFWBUQTPZVQWCWHAVOVKPZVQVOFWBURAWIVTVQAVMWIVTIEVOFGHUMUNZKSVAVB
|
|
AVOFWBVCVDTZUQTPZVPVOVCVETDNVFVGZWCWLAWIWMVOFWKURAWIVTWMWJLSVAVBVHVI $.
|
|
$}
|
|
|
|
${
|
|
climserile.2 $e |- ( ph -> N e. Z ) $.
|
|
climserile.3 $e |- ( ph -> seq M ( + , F , CC ) ~~> A ) $.
|
|
climserile.4 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $.
|
|
climserile.5 $e |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) ) $.
|
|
$( The partial sums of a converging infinite series with nonnegative
|
|
terms are bounded by its limit. (Contributed by Jim Kingdon,
|
|
22-Aug-2021.) $)
|
|
climserile $p |- ( ph -> ( seq M ( + , F , CC ) ` N ) <_ A ) $=
|
|
( caddc cc cr cfv wcel wa cle wi vj vx cseq cv cuz syl6eleq eluzel2 syl
|
|
wf cz iserfre cvv cnex a1i wss ax-resscn eleq2i sylan2br readdcl adantl
|
|
co addcl iseqss feq1d mpbid ffvelrnda c1 cc0 wbr peano2uzs fveq2 breq2d
|
|
imbi2d expcom vtoclga impcom sylan2 eleq1d addge01d simpr adantlr recnd
|
|
wceq iseqp1 breqtrrd climub ) ABUAMNDEUCZEFGHIJAGOUAUDZWGAGOMODEUCZUIGO
|
|
WGUIACDEGHAFEUEPZQEUJQAFGWJIHUFEFUGUHZKUKAGOWIWGACUBMONDEULWKNULQAUMUNO
|
|
NUOAUPUNCUDZWJQZAWLGQZWLDPZOQZGWJWLHUQKURZWLOQUBUDZOQRWLWRMVAZOQAWLWRUS
|
|
UTWLNQWRNQRZWSNQZAWLWRVBZUTVCVDVEVFZAWHGQZRZWHWGPZXFWHVGMVAZDPZMVAZXGWG
|
|
PSXEVHXHSVIZXFXISVIXDAXGGQZXJEWHGHVJZXKAXJAVHWOSVIZTAXJTCXGGWLXGWCZXMXJ
|
|
AXNWOXHVHSWLXGDVKZVLVMAWNXMLVNVOVPVQXEXFXHXCXDAXKXHOQZXLXKAXPAWPTAXPTCX
|
|
GGXNWPXPAXNWOXHOXOVRVMAWNWPKVNVOVPVQVSVEXECUBMNDEWHXEWHGWJAXDVTHUFXEWMR
|
|
WOAWMWPXDWQWAWBWTXAXEXBUTWDWEWF $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d j k x y F $. $d j k x y M $. $d j k x y Z $.
|
|
climcau.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( A converging sequence of complex numbers is a Cauchy sequence. The
|
|
converse would require excluded middle or a different definition of
|
|
Cauchy sequence (for example, fixing a rate of convergence as in
|
|
~ climcvg1n ). Theorem 12-5.3 of [Gleason] p. 180 (necessity part).
|
|
(Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro,
|
|
26-Apr-2014.) $)
|
|
climcau $p |- ( ( M e. ZZ /\ F e. dom ~~> ) -> A. x e. RR+ E. j e. Z
|
|
A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) $=
|
|
( vy wcel cli cv cfv cmin co cabs clt wbr crp wa cc cdm cuz wral wrex cop
|
|
cz wex eldm2g ibi df-br c2 cdiv simpll rphalfcl adantl eqidd simplr climi
|
|
wi eluzelz syl eleq2s weq fveq2 eleq1d oveq1d fveq2d breq1d anbi12d rspcv
|
|
uzid cr rpre simpllr climcl simprl simplrl simplll simprr abssubd simplrr
|
|
ad2antlr eqbrtrd abs3lemd ex ralimdv com23 syl2anc reximdva mpd ralrimiva
|
|
mpdd syl5bir exlimdv syl5 imp ) EUFIZDJUAZIZCKZDLZBKZDLZMNOLAKZPQZCXBUBLZ
|
|
UCZBFUDZARUCZWSDHKZUEJIZHUGZWQXIWSXLHDJWRUHUIWQXKXIHXKDXJJQZWQXIDXJJUJWQX
|
|
MXIWQXMSZXHARXNXDRIZSZXATIZXAXJMNZOLZXDUKULNZPQZSZCXFUCZBFUDXHXPXJXAXTBCD
|
|
EFGWQXMXOUMXOXTRIXNXDUNUOXPWTFISXAUPWQXMXOUQURXPYCXGBFXPXBFIZSZYCXCTIZXCX
|
|
JMNZOLZXTPQZSZXGYEXBXFIZYCYJUSYDYKXPYKXBEUBLZFXBYLIXBUFIYKEXBUTXBVKVAGVBU
|
|
OYBYJCXBXFCBVCZXQYFYAYIYMXAXCTWTXBDVDZVEYMXSYHXTPYMXRYGOYMXAXCXJMYNVFVGVH
|
|
VIVJVAYEXDVLIZXJTIZYCYJXGUSUSXOYOXNYDXDVMWBYEXMYPWQXMXOYDVNXJDVOVAYOYPSZY
|
|
JYCXGYQYJYCXGUSYQYJSZYBXECXFYRYBXEYRYBSZXAXCXJXDYRXQYAVPYQYFYIYBVQZYOYPYJ
|
|
YBVNZYOYPYJYBVRYRXQYAVSYSXJXCMNOLYHXTPYSXJXCUUAYTVTYQYFYIYBWAWCWDWEWFWEWG
|
|
WHWLWIWJWKWEWMWNWOWP $.
|
|
$}
|
|
|
|
${
|
|
$d C e i j y $. $d C i j k y $. $d C i k n $. $d F e i j y $.
|
|
$d F i j k y $. $d F i k n $. $d e i j ph y $. $d k n ph $.
|
|
climrecvg1n.f $e |- ( ph -> F : NN --> RR ) $.
|
|
climrecvg1n.c $e |- ( ph -> C e. RR+ ) $.
|
|
climrecvg1n.cau $e |- ( ph ->
|
|
A. n e. NN A. k e. ( ZZ>= ` n )
|
|
( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) ) $.
|
|
$( A Cauchy sequence of real numbers converges, existence version. The
|
|
rate of convergence is fixed: all terms after the nth term must be
|
|
within ` C / n ` of the nth term, where ` C ` is a constant multiplier.
|
|
(Contributed by Jim Kingdon, 23-Aug-2021.) $)
|
|
climrecvg1n $p |- ( ph -> F e. dom ~~> ) $=
|
|
( vj ve vi cv cfv co clt wbr wa cn wcel cr vy caddc cuz wral wrex crp cli
|
|
cdm cdiv cmin cabs r19.21bi ad2antrr eluznn adantll ffvelrnd simplr nnrpd
|
|
wf rpdivcld rpred absdifltd mpbid ltsubaddd anbi1d ralrimiva cvg1n adantr
|
|
ad3antrrr simpr simpllr bitrd ancom syl6bb ralbidva rexbidva c1 nnuz 1zzd
|
|
cvv nnex reex fex2 syl3anc eqidd recnd ffvelrnda climrel releldmi syl6bir
|
|
a1i clim2c sylbird impr rexlimddv ) AILZEMZUALZJLZUBNOPZWRWQWSUBNOPZQZIKL
|
|
ZUCMZUDZKRUEZJUFUDZEUGUHSZUATAJUABIKCDEFGADLZEMZCLZEMZBXIUINZUBNOPZXLXJXM
|
|
UBNOPZQZCXIUCMZUDDRAXIRSZQZXPCXQXSXKXQSZQZXJXMUJNXLOPZXOQZXPYAXLXJUJNUKMX
|
|
MOPZYCXSYDCXQAYDCXQUDDRHULULYAXLXJXMYARTXKEARTEUSZXRXTFUMZXRXTXKRSAXKXIUN
|
|
UOUPZYARTXIEYFAXRXTUQZUPZYAXMYABXIABUFSXRXTGUMYAXIYHURUTVAZVBVCYAYBXNXOYA
|
|
XJXMXLYIYJYGVDVEVCVFVFVGAWRTSZXGXHAYKQZXGWQWRUJNUKMWSOPZIXDUDZKRUEZJUFUDZ
|
|
XHYLYOXFJUFYLWSUFSZQZYNXEKRYRXCRSZQZYMXBIXDYTWPXDSZQZYMXAWTQZXBUUBYMWRWSU
|
|
JNWQOPZWTQUUCUUBWQWRWSUUBRTWPEYLYEYQYSUUAAYEYKFVHZVIYSUUAWPRSZYRWPXCUNUOU
|
|
PZYLYKYQYSUUAAYKVJZVIZUUBWSYLYQYSUUAVKVAZVBUUBUUDXAWTUUBWRWSWQUUIUUJUUGVD
|
|
VEVLXAWTVMVNVOVPVOYLYPEWRUGPXHYLJWRWQKIEVQVTRVRYLVSYLYERVTSZTVTSZEVTSUUEU
|
|
UKYLWAWKUULYLWBWKRTEVTVTWCWDYLUUFQZWQWEYLWRUUHWFUUMWQYLRTWPEUUEWGWFWLEWRU
|
|
GWHWIWJWMWNWO $.
|
|
$}
|
|
|
|
${
|
|
climcvg1n.f $e |- ( ph -> F : NN --> CC ) $.
|
|
climcvg1n.c $e |- ( ph -> C e. RR+ ) $.
|
|
climcvg1n.cau $e |- ( ph ->
|
|
A. n e. NN A. k e. ( ZZ>= ` n )
|
|
( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) ) $.
|
|
${
|
|
$d C k n $. $d F k x $. $d G k n $. $d H k n x $. $d J k $.
|
|
$d k n ph x $.
|
|
climcvg1nlem.g $e |- G = ( x e. NN |-> ( Re ` ( F ` x ) ) ) $.
|
|
climcvg1nlem.h $e |- H = ( x e. NN |-> ( Im ` ( F ` x ) ) ) $.
|
|
climcvg1nlem.j $e |- J = ( x e. NN |-> ( _i x. ( H ` x ) ) ) $.
|
|
$( Lemma for ~ climcvg1n . We construct sequences of the real and
|
|
imaginary parts of each term of ` F ` , show those converge, and use
|
|
that to show that ` F ` converges. (Contributed by Jim Kingdon,
|
|
24-Aug-2021.) $)
|
|
climcvg1nlem $p |- ( ph -> F e. dom ~~> ) $=
|
|
( cfv wcel cn cr cc cli ci cmul co caddc wbr cdm c1 cvv nnuz 1zzd cv wa
|
|
cre ffvelrnda recld cmin cabs cdiv clt cuz wral cle wceq eluznn adantll
|
|
wf ad2antrr ffvelrnd fveq2d fvmptg syl2anc simplr oveq12d resubd eqtr4d
|
|
fmptd fveq2 subcld absrele syl eqbrtrd wi recnd eqeltrd abscld rpdivcld
|
|
nnrpd rpred lelttr syl3anc mpand ralimdva climrecvg1n climdm sylib nnex
|
|
crp mpd fex sylancl cim imcld imsubd absimle resubcld ax-icn cmpt mptex
|
|
a1i eqeltri ax-resscn fssd oveq2d adantl simpr mulcld climmulc2 replimd
|
|
wss fvmptd eqtrd climadd climrel releldmi ) AFGUAPZUBHUAPZUCUDZUEUDZUAU
|
|
FFUAUGZQAYFYHDGIFUHUIRUJAUKZAGYJQGYFUAUFACDEGABRBULZFPZUNPZSGAYLRQUMZYM
|
|
ARTYLFJUOZUPMVQZKADULZFPZEULZFPZUQUDZURPZCYTUSUDZUTUFZDYTVAPZVBZERVBZYR
|
|
GPZYTGPZUQUDZURPZUUDUTUFZDUUFVBZERVBLAUUGUUNERAYTRQZUMZUUEUUMDUUFUUPYRU
|
|
UFQZUMZUULUUCVCUFZUUEUUMUURUULUUBUNPZURPZUUCVCUURUUKUUTURUURUUKYSUNPZUU
|
|
AUNPZUQUDUUTUURUUIUVBUUJUVCUQUURYRRQZUVBSQZUUIUVBVDZUUOUUQUVDAYRYTVEVFZ
|
|
UURYSUURRTYRFARTFVGZUUOUUQJVHZUVGVIZUPBYRYNUVBRSGYLYRVDZYMYSUNYLYRFVRZV
|
|
JMVKZVLUURUUOUVCSQUUJUVCVDAUUOUUQVMZUURUUAUURRTYTFUVIUVNVIZUPBYTYNUVCRS
|
|
GYLYTVDZYMUUAUNYLYTFVRZVJMVKVLVNUURYSUUAUVJUVOVOVPZVJUURUUBTQZUVAUUCVCU
|
|
FUURYSUUAUVJUVOVSZUUBVTWAWBUURUULSQUUCSQZUUDSQZUUSUUEUMUUMWCUURUUKUURUU
|
|
KUUTTUVRUURUUTUURUUBUVTUPWDWEWFUURUUBUVTWFZUURUUDUURCYTACWRQUUOUUQKVHUU
|
|
RYTUVNWHWGWIZUULUUCUUDWJWKWLWMWMWSWNGWOWPAUVHRUIQFUIQJWQRTUIFWTXAAYGUBD
|
|
HIUHUIRUJYKAHYJQHYGUAUFACDEHABRYMXBPZSHYOYMYPXCNVQZKAUUHYRHPZYTHPZUQUDZ
|
|
URPZUUDUTUFZDUUFVBZERVBLAUUGUWLERUUPUUEUWKDUUFUURUWJUUCVCUFZUUEUWKUURUW
|
|
JUUBXBPZURPZUUCVCUURUWIUWNURUURUWIYSXBPZUUAXBPZUQUDUWNUURUWGUWPUWHUWQUQ
|
|
UURUVDUWPSQZUWGUWPVDZUVGUURYSUVJXCZBYRUWEUWPRSHUVKYMYSXBUVLVJNVKZVLZUUR
|
|
UUOUWQSQUWHUWQVDUVNUURUUAUVOXCZBYTUWEUWQRSHUVPYMUUAXBUVQVJNVKVLZVNUURYS
|
|
UUAUVJUVOXDVPVJUURUVSUWOUUCVCUFUVTUUBXEWAWBUURUWJSQUWAUWBUWMUUEUMUWKWCU
|
|
URUWIUURUWIUURUWGUWHUURUWGUWPSUXBUWTWEUURUWHUWQSUXDUXCWEXFWDWFUWCUWDUWJ
|
|
UUCUUDWJWKWLWMWMWSWNHWOWPUBTQZAXGXJIUIQAIBRUBYLHPZUCUDZXHZUIOBRUXGWQXIX
|
|
KXJARTYRHARSTHUWFSTXTAXLXJXMUOZAUVDUMZBYRUXGUBUWGUCUDZRITIUXHVDUXJOXJUV
|
|
KUXGUXKVDUXJUVKUXFUWGUBUCYLYRHVRXNXOAUVDXPZUXJUBUWGUXEUXJXGXJUXIXQZYAZX
|
|
RUXJUUIARSYRGYQUOWDUXJYRIPZUXKTUXNUXMWEUXJYSUVBUBUWPUCUDZUEUDUUIUXOUEUD
|
|
UXJYSARTYRFJUOZXSUXJUUIUVBUXOUXPUEUXJUVDUVEUVFUXLUXJYSUXQUPUVMVLUXJUXOU
|
|
XKUXPUXNUXJUWGUWPUBUCUXJUVDUWRUWSUXLUXJYSUXQXCUXAVLXNYBVNVPYCFYIUAYDYEW
|
|
A $.
|
|
$}
|
|
|
|
${
|
|
$d C k n $. $d F k n x y $. $d k n ph x $.
|
|
$( A Cauchy sequence of complex numbers converges, existence version.
|
|
The rate of convergence is fixed: all terms after the nth term must be
|
|
within ` C / n ` of the nth term, where ` C ` is a constant
|
|
multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.) $)
|
|
climcvg1n $p |- ( ph -> F e. dom ~~> ) $=
|
|
( vx vy cn cv cfv cre cmpt cim ci cmul co eqid weq fveq2 fveq2d cbvmptv
|
|
climcvg1nlem ) AIBCDEIKILZEMZNMOZJKJLZEMZPMZOZIKQUFULMRSOZFGHUHTJIKUKUG
|
|
PMJIUAUJUGPUIUFEUBUCUDUMTUE $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d j k m n x F $. $d j k m n x M $. $d j k m n x Z $. $d j k n x ph $.
|
|
$d k m n V $.
|
|
climcauc.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
$( A converging sequence of complex numbers is a Cauchy sequence. This is
|
|
like ~ climcau but adds the part that ` ( F `` k ) ` is complex.
|
|
(Contributed by Jim Kingdon, 24-Aug-2021.) $)
|
|
climcaucn $p |- ( ( M e. ZZ /\ F e. dom ~~> ) ->
|
|
A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\
|
|
( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) ) $=
|
|
( vn cz wcel cli wa cv cfv cuz wral wrex wbr crp c1 cc cmin co cabs simpl
|
|
cdm clt 1rp a1i eqidd climdm biimpi adantl climi ralimi reximi wi eluzelz
|
|
eleq2s eqid climcau sylan r19.29uz ex ralimdv mpan9 an32s adantll wb cau4
|
|
syl ad2antrl sylibrd rexlimdvaa com23 imp mpd ) EIJZDKUFJZLZCMZDNZUAJZCHM
|
|
ZONZPZHFQZWCWBBMZDNUBUCUDNAMUGRZLCWHONZPZBFQASPZVTWCWBDKNZUBUCUDNTUGRZLZC
|
|
WEPZHFQWGVTWMWBTHCDEFGVRVSUETSJVTUHUIVTWAFJLWBUJVSDWMKRZVRVSWQDUKULUMUNWP
|
|
WFHFWOWCCWEWCWNUEUOUPVKVRVSWGWLUQVRWGVSWLVRWFVSWLUQHFVRWDFJZWFLZLZVSWKBWE
|
|
QZASPZWLWTVSXBWSVSXBVRWRVSWFXBWRVSLWICWJPBWEQZASPZWFXBWRWDIJZVSXDXEWDEONF
|
|
EWDURGUSABCDWDWEWEUTZVAVBWFXCXAASWFXCXAWCWIBCWDWEXFVCVDVEVFVGVHVDWRWLXBVI
|
|
VRWFABCDEWDWEFGXFVJVLVMVNVOVPVQ $.
|
|
|
|
$d F a b j k $. $d M a b j k $. $d Z a b j k $. $d a b j k ph $.
|
|
serif0.2 $e |- ( ph -> M e. ZZ ) $.
|
|
serif0.3 $e |- ( ph -> F e. V ) $.
|
|
serif0.4 $e |- ( ph -> seq M ( + , F , CC ) e. dom ~~> ) $.
|
|
serif0.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) $.
|
|
$( If an infinite series converges, its underlying sequence converges to
|
|
zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro,
|
|
16-Feb-2014.) $)
|
|
serif0 $p |- ( ph -> F ~~> 0 ) $=
|
|
( vx cfv cabs wral caddc cc wcel cmin co vn vm vj va cc0 cli wbr clt wrex
|
|
vb cv cuz crp cseq wa cz climcaucn syl2anc cau3 sylib c1 peano2uzs adantl
|
|
cdm wi eluzelz uzid peano2uz wceq fveq2 oveq2d fveq2d breq1d 4syl adantld
|
|
rspcv ralimia simpr syl6eleq syl eluzp1m1 sylan oveq1 oveq12d wf ad2antrr
|
|
iserf uztrn2 syldan ffvelrnd abssubd zcnd npcan sylancl cvv eluzp1p1 eqid
|
|
ax-1cn cnex a1i syl6eleqr ralrimiva ad3antrrr eleq1d rspcva iseqm1 oveq1d
|
|
addcl adantlr pncan2d eqtr2d 3eqtr4d sylibd ralrimdva syl5 raleqdv rspcev
|
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syl6an rexlimdva ralimdv mpd eqidd clim0c mpbird ) ACUEUFUGBUKZCMZNMZLUKZ
|
|
UHUGZBUAUKZULMZOZUAFUIZLUMOZAUBUKZPQCDUNZMZQRZYQYEYPMZSTZNMZYHUHUGZBYOULM
|
|
ZOZUOZUBUCUKZULMZOZUCFUIZLUMOZYNAYRYQUUFYPMSTNMYHUHUGUOUBUUGOUCFUILUMOZUU
|
|
JADUPRZYPUFVDRUUKHJLUCUBYPDFGUQURLUCUBBYPDFGUSUTAUUIYMLUMAUUHYMUCFAUUFFRZ
|
|
UOZUUFVAPTZFRZUUHYIBUUOULMZOZYMUUMUUPADUUFFGVBVCZUUHYQYOVAPTZYPMZSTZNMZYH
|
|
UHUGZUBUUGOZUUNUURUUEUVDUBUUGYOUUGRZUUDUVDYRUVFYOUPRYOUUCRUUTUUCRUUDUVDVE
|
|
UUFYOVFYOVGYOYOVHUUBUVDBUUTUUCYEUUTVIZUUAUVCYHUHUVGYTUVBNUVGYSUVAYQSYEUUT
|
|
YPVJVKVLVMVPVNVOVQUUNUVEYIBUUQUUNYEUUQRZUOZUVEYEVASTZYPMZUVJVAPTZYPMZSTZN
|
|
MZYHUHUGZYIUVIUVJUUGRZUVEUVPVEUUNUUFUPRZUVHUVQUUNUUFDULMZRZUVRUUNUUFFUVSA
|
|
UUMVRZGVSZDUUFVFVTUUFYEWAWBZUVDUVPUBUVJUUGYOUVJVIZUVCUVOYHUHUWDUVBUVNNUWD
|
|
YQUVKUVAUVMSYOUVJYPVJUWDUUTUVLYPYOUVJVAPWCVLWDVLVMVPVTUVIUVOYGYHUHUVIUVKY
|
|
SSTZNMYSUVKSTZNMUVOYGUVIUVKYSUVIFQUVJYPAFQYPWEUUMUVHABCDFGHKWGWFZUUNUVHUV
|
|
QUVJFRZUWCUUNUUMUVQUWHUWADUVJUUFFGWHWBWIWJZUVIFQYEYPUWGUUNUUPUVHYEFRZUUSD
|
|
YEUUOFGWHWBZWJWKUVIUVNUWENUVIUVMYSUVKSUVIUVLYEYPUVIYEQRVAQRUVLYEVIUVIYEUV
|
|
HYEUPRUUNUUOYEVFVCWLWRYEVAWMWNVLVKVLUVIYFUWFNUVIUWFUVKYFPTZUVKSTYFUVIYSUW
|
|
LUVKSUVIUDUJPQCDYEWOAUULUUMUVHHWFUUNUUODVAPTZULMZRZUVHYEUWNRUUNUVTUWOUWBD
|
|
UUFWPVTUWMYEUUOUWNUWNWQWHWBQWORUVIWSWTUVIUDUKZUVSRZUOZUWPFRYFQRZBFOZUWPCM
|
|
ZQRZUWRUWPUVSFUVIUWQVRGXAAUWTUUMUVHUWQAUWSBFKXBXCUWSUXBBUWPFYEUWPVIYFUXAQ
|
|
YEUWPCVJXDXEURUWPQRUJUKZQRUOUWPUXCPTQRUVIUWPUXCXHVCXFXGUVIUVKYFUWIUUNUVHU
|
|
WJUWSUWKAUWJUWSUUMKXIWIXJXKVLXLVMXMXNXOYLUURUAUUOFYJUUOVIYIBYKUUQYJUUOULV
|
|
JXPXQXRXSXTYAALYFUABCDEFGHIAUWJUOYFYBKYCYD $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite and infinite sums
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c sum_ $. $( Summation sign. $)
|
|
|
|
$( Extend class notation to include finite summations. (An underscore was
|
|
added to the ASCII token in order to facilitate set.mm text searches,
|
|
since "sum" is a commonly used word in comments.) $)
|
|
csu $a class sum_ k e. A B $.
|
|
|
|
$( Indicate that "k" is not bound in "A" in the definition of csu. $)
|
|
$( $j free_var_in 'csu' with 'A' 'k'; $)
|
|
|
|
${
|
|
$d k f m n x j $. $d A f m n x j $. $d B f m n x j $.
|
|
$( Define the sum of a series with an index set of integers ` A ` . ` k `
|
|
is normally a free variable in ` B ` , i.e. ` B ` can be thought of as
|
|
` B ( k ) ` . This definition is the result of a collection of
|
|
discussions over the most general definition for a sum that does not
|
|
need the index set to have a specified ordering. This definition is in
|
|
two parts, one for finite sums and one for subsets of the upper
|
|
integers. When summing over a subset of the upper integers, we extend
|
|
the index set to the upper integers by adding zero outside the domain,
|
|
and then sum the set in order, setting the result to the limit of the
|
|
partial sums, if it exists. This means that conditionally convergent
|
|
sums can be evaluated meaningfully. For finite sums, we are explicitly
|
|
order-independent, by picking any bijection to a 1-based finite sequence
|
|
and summing in the induced order. In both cases we have an ` if `
|
|
expression so that we only need ` B ` to be defined where ` k e. A ` .
|
|
In the infinite case, we also require that the indexing set be a
|
|
decidable subset of an upperset of integers (that is, membership of
|
|
integers in it is decidable). These two methods of summation produce
|
|
the same result on their common region of definition (i.e. finite sets
|
|
of integers). Examples: ` sum_ k e. { 1 , 2 , 4 } ` ` k ` means
|
|
` 1 + 2 + 4 = 7 ` , and ` sum_ k e. NN ` ` ( 1 / ( 2 ^ k ) ) = 1 ` means
|
|
1/2 + 1/4 + 1/8 + ... = 1. (Contributed by Jim Kingdon,
|
|
15-Feb-2022.) $)
|
|
df-isum $a |- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\
|
|
A. j e. ( ZZ>= ` m ) DECID j e. A /\
|
|
seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
|
|
~~> x ) \/
|
|
E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\
|
|
x = ( seq 1 ( + ,
|
|
( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ,
|
|
CC ) ` m ) ) ) ) $.
|
|
|
|
$d A f j m n x $. $d B f j m n x $. $d C f j m n x $. $d f j k m n x $.
|
|
$d C f m n x $.
|
|
$( Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised
|
|
by Mario Carneiro, 13-Jun-2019.) $)
|
|
sumeq1 $p |- ( A = B -> sum_ k e. A C = sum_ k e. B C ) $=
|
|
( vm vj vn vx vf wceq cv cfv wcel caddc cc cz cc0 cif wrex cn cuz wss wdc
|
|
wral csb cmpt cseq cli wbr w3a c1 cfz co wf1o cle wa wex wo cio csu sseq1
|
|
eleq2 dcbid ralbidv simpl eleq2d ifbid mpteq2dva iseqeq3 breq1d 3anbi123d
|
|
syl rexbidv f1oeq3 anbi1d exbidv orbi12d iotabidv df-isum 3eqtr4g ) ABJZA
|
|
EKZUALZUBZFKZAMZUCZFWCUDZNOGPGKZAMZDWICUEZQRZUFZWBUGZHKZUHUIZUJZEPSZUKWBU
|
|
LUMZAIKZUNZWOWBNOGTWIWBUOUIDWIWTLCUEQRUFUKUGLJZUPZIUQZETSZURZHUSBWCUBZWEB
|
|
MZUCZFWCUDZNOGPWIBMZWKQRZUFZWBUGZWOUHUIZUJZEPSZWSBWTUNZXBUPZIUQZETSZURZHU
|
|
SACDUTBCDUTWAXFYBHWAWRXQXEYAWAWQXPEPWAWDXGWHXJWPXOABWCVAWAWGXIFWCWAWFXHAB
|
|
WEVBVCVDWAWNXNWOUHWAWMXMJWNXNJWAGPWLXLWAWIPMZUPZWJXKWKQYDABWIWAYCVEVFVGVH
|
|
NOWMXMWBVIVLVJVKVMWAXDXTETWAXCXSIWAXAXRXBABWSWTVNVOVPVMVQVRHACIFDEGVSHBCI
|
|
FDEGVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d j f m n x k $. $d j f m n x A $. $d j f m n x B $.
|
|
nfsum1.1 $e |- F/_ k A $.
|
|
$( Bound-variable hypothesis builder for sum. (Contributed by NM,
|
|
11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) $)
|
|
nfsum1 $p |- F/_ k sum_ k e. A B $=
|
|
( vm vj vn vx vf cv cfv wcel caddc cc cz cc0 cli c1 cn nfcv csu wral cmpt
|
|
cuz wss wdc csb cif cseq wbr w3a wrex cfz co wf1o cle wceq wa wex df-isum
|
|
wo cio nfss nfcri nfdc nfralxy nfcsb1v nfif nfmpt nfiseq nfbr nf3an nff1o
|
|
nfrexya nfv nffv nfeq2 nfan nfex nfor nfiotaxy nfcxfr ) CABCUAAEJZUDKZUEZ
|
|
FJALZUFZFWDUBZMNGOGJZALZCWIBUGZPUHZUCZWCUIZHJZQUJZUKZEOULZRWCUMUNZAIJZUOZ
|
|
WOWCMNGSWIWCUPUJZCWIWTKZBUGZPUHZUCZRUIZKZUQZURZIUSZESULZVAZHVBHABIFCEGUTX
|
|
MCHWRXLCWQCEOCOTZWEWHWPCCAWDDCWDTZVCWGCFWDXOWFCCFADVDVEVFCWNWOQCMNWMWCCWC
|
|
TZCMTZCGOWLXNWJCWKPCGADVDCWIBVGCPTZVHVICNTZVJCQTCWOTVKVLVNXKCESCSTZXJCIXA
|
|
XICCWSAWTCWTTCWSTDVMCWOXHCWCXGCMNXFRCRTXQCGSXEXTXBCXDPXBCVOCXCBVGXRVHVIXS
|
|
VJXPVPVQVRVSVNVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d j f m n z k $. $d j f m n z x $. $d j f m n z A $. $d j f m n z B $.
|
|
nfsum.1 $e |- F/_ x A $.
|
|
nfsum.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for sum: if ` x ` is (effectively) not
|
|
free in ` A ` and ` B ` , it is not free in ` sum_ k e. A B ` .
|
|
(Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro,
|
|
13-Jun-2019.) $)
|
|
nfsum $p |- F/_ x sum_ k e. A B $=
|
|
( vm vj vn vz vf cv cfv caddc cc cz cc0 c1 cn nfcv csu cuz wcel wral cmpt
|
|
wss wdc csb cif cseq cli wbr w3a wrex cfz co wf1o cle wceq wa wex df-isum
|
|
wo cio nfss nfcri nfdc nfralxy nfcsb nfif nfmpt nfiseq nfbr nf3an nfrexxy
|
|
nff1o nfv nffv nfeq2 nfan nfex nfor nfiotaxy nfcxfr ) ABCDUABGLZUBMZUFZHL
|
|
BUCZUGZHWFUDZNOIPILZBUCZDWKCUHZQUIZUEZWEUJZJLZUKULZUMZGPUNZRWEUOUPZBKLZUQ
|
|
ZWQWENOISWKWEURULZDWKXBMZCUHZQUIZUEZRUJZMZUSZUTZKVAZGSUNZVCZJVDJBCKHDGIVB
|
|
XOAJWTXNAWSAGPAPTZWGWJWRAABWFEAWFTZVEWIAHWFXQWHAAHBEVFVGVHAWPWQUKANOWOWEA
|
|
WETZANTZAIPWNXPWLAWMQAIBEVFADWKCAWKTFVIAQTZVJVKAOTZVLAUKTAWQTVMVNVOXMAGSA
|
|
STZXLAKXCXKAAXABXBAXBTAXATEVPAWQXJAWEXIANOXHRARTXSAISXGYBXDAXFQXDAVQADXEC
|
|
AXETFVIXTVJVKYAVLXRVRVSVTWAVOWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d M x $. $d N x $.
|
|
sumdc.m $e |- ( ph -> M e. ZZ ) $.
|
|
sumdc.ss $e |- ( ph -> A C_ ( ZZ>= ` M ) ) $.
|
|
sumdc.dc $e |- ( ph -> A. x e. ( ZZ>= ` M ) DECID x e. A ) $.
|
|
sumdc.n $e |- ( ph -> N e. ZZ ) $.
|
|
$( Decidability of a subset of upper integers. (Contributed by Jim
|
|
Kingdon, 1-Jan-2022.) $)
|
|
sumdc $p |- ( ph -> DECID N e. A ) $=
|
|
( cuz cfv wcel wdc wn cv wral wceq eleq1 wo cz dcbid rspcv wa ssneld olcd
|
|
mpan9 imp df-dc sylibr eluzdc syl2anc exmiddc syl mpjaodan ) AEDJKZLZECLZ
|
|
MZUPNZABOZCLZMZBUOPUPURHVBURBEUOUTEQVAUQUTECRUAUBUFAUSUCZUQUQNZSURVCVDUQA
|
|
USVDACUOEGUDUGUEUQUHUIAUPMZUPUSSADTLETLVEFIDEUJUKUPULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A f j k m n x $. $d B f j m n x $. $d C f j m n x $.
|
|
$d f k m n ph x $.
|
|
sumeq2d.bc $e |- ( ( ph /\ k e. A ) -> B = C ) $.
|
|
$( Equality theorem for sum. (Contributed by Jim Kingdon, 11-Feb-2022.) $)
|
|
sumeq2d $p |- ( ph -> sum_ k e. A B = sum_ k e. A C ) $=
|
|
( vm vj vn vx vf cv wcel caddc cc cz cc0 cn wceq wa cuz cfv wss wral cmpt
|
|
wdc csb cif cseq cli wbr w3a wrex c1 cfz co wf1o cle wex wo cio csu simpr
|
|
ralrimiva ad4antr nfcsb1v nfeq csbeq1a eqeq12d rspc simpllr simplrl sumdc
|
|
sylc simplrr ifeq1dadc mpteq2dva iseqeq3 breq1d pm5.32da 3bitr4g rexbidva
|
|
df-3an wf f1of ad3antlr simplr simp-4r nnzd fznn mpbir2and ffvelrnd zdcle
|
|
syl wb syl2anc fveq1d eqeq2d exbidv orbi12d iotabidv df-isum 3eqtr4g ) AB
|
|
GLZUAUBZUCZHLBMUFHXEUDZNOIPILZBMZEXHCUGZQUHZUEZXDUIZJLZUJUKZULZGPUMZUNXDU
|
|
OUPZBKLZUQZXNXDNOIRXHXDURUKZEXHXSUBZCUGZQUHZUEZUNUIZUBZSZTZKUSZGRUMZUTZJV
|
|
AXFXGNOIPXIEXHDUGZQUHZUEZXDUIZXNUJUKZULZGPUMZXTXNXDNOIRYAEYBDUGZQUHZUEZUN
|
|
UIZUBZSZTZKUSZGRUMZUTZJVABCEVBBDEVBAYLUUIJAXQYSYKUUHAXPYRGPAXDPMZTZXFXGTZ
|
|
XOTUULYQTXPYRUUKUULXOYQUUKUULTZXMYPXNUJUUMXLYOSXMYPSUUMIPXKYNUUMXHPMZTZXI
|
|
XJYMQUUOXITXICDSZEBUDZXJYMSZUUOXIVCAUUQUUJUULUUNXIAUUPEBFVDZVEUUPUUREXHBE
|
|
XJYMEXHCVFEXHDVFVGELZXHSCXJDYMEXHCVHEXHDVHVIVJVNUUOHBXDXHAUUJUULUUNVKUUKX
|
|
FXGUUNVLUUKXFXGUUNVOUUMUUNVCVMVPVQNOXLYOXDVRWNVSVTXFXGXOWCXFXGYQWCWAWBAYJ
|
|
UUGGRAXDRMZTZYIUUFKUVBXTYHUUEUVBXTTZYGUUDXNUVCXDYFUUCUVCYEUUBSYFUUCSUVCIR
|
|
YDUUAUVCXHRMZTZYAYCYTQUVEYATZYBBMUUQYCYTSZUVFXRBXHXSXTXRBXSWDUVBUVDYAXRBX
|
|
SWEWFUVFXHXRMZUVDYAUVCUVDYAWGUVEYAVCUVFUUJUVHUVDYATWOUVFXDAUVAXTUVDYAWHWI
|
|
XHXDWJWNWKWLAUUQUVAXTUVDYAUUSVEUUPUVGEYBBEYCYTEYBCVFEYBDVFVGUUTYBSCYCDYTE
|
|
YBCVHEYBDVHVIVJVNUVEUUNUUJYAUFUVEXHUVCUVDVCWIUVEXDAUVAXTUVDVKWIXHXDWMWPVP
|
|
VQNOYEUUBUNVRWNWQWRVTWSWBWTXAJBCKHEGIXBJBDKHEGIXBXC $.
|
|
$}
|
|
|
|
${
|
|
$d A f j k m n x $. $d B f j m n x $. $d C f j m n x $.
|
|
$( Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised
|
|
by Mario Carneiro, 13-Jul-2013.) $)
|
|
sumeq2 $p |- ( A. k e. A B = C -> sum_ k e. A B = sum_ k e. A C ) $=
|
|
( vm vj vn vx vf wceq cv cfv wcel caddc cc cz csb cc0 cn wa wral cuz cmpt
|
|
wss wdc cif cseq cli wbr w3a wrex c1 cfz co wf1o cle wex wo cio csu simpr
|
|
simp-4l nfcsb1v nfeq csbeq1a eqeq12d rspc simpllr simplrl sumdc ifeq1dadc
|
|
sylc simplrr mpteq2dva iseqeq3 breq1d pm5.32da df-3an 3bitr4g rexbidva wf
|
|
syl f1of ad3antlr simplr wb simp-4r nnzd mpbir2and ffvelrnd zdcle syl2anc
|
|
fznn fveq1d eqeq2d exbidv orbi12d iotabidv df-isum 3eqtr4g ) BCJZDAUAZAEK
|
|
ZUBLZUDZFKAMUEFXDUAZNOGPGKZAMZDXGBQZRUFZUCZXCUGZHKZUHUIZUJZEPUKZULXCUMUNZ
|
|
AIKZUOZXMXCNOGSXGXCUPUIZDXGXRLZBQZRUFZUCZULUGZLZJZTZIUQZESUKZURZHUSXEXFNO
|
|
GPXHDXGCQZRUFZUCZXCUGZXMUHUIZUJZEPUKZXSXMXCNOGSXTDYACQZRUFZUCZULUGZLZJZTZ
|
|
IUQZESUKZURZHUSABDUTACDUTXBYKUUHHXBXPYRYJUUGXBXOYQEPXBXCPMZTZXEXFTZXNTUUK
|
|
YPTXOYQUUJUUKXNYPUUJUUKTZXLYOXMUHUULXKYNJXLYOJUULGPXJYMUULXGPMZTZXHXIYLRU
|
|
UNXHTXHXBXIYLJZUUNXHVAXBUUIUUKUUMXHVBXAUUODXGADXIYLDXGBVCDXGCVCVDDKZXGJBX
|
|
ICYLDXGBVEDXGCVEVFVGVLUUNFAXCXGXBUUIUUKUUMVHUUJXEXFUUMVIUUJXEXFUUMVMUULUU
|
|
MVAVJVKVNNOXKYNXCVOWBVPVQXEXFXNVRXEXFYPVRVSVTXBYIUUFESXBXCSMZTZYHUUEIUURX
|
|
SYGUUDUURXSTZYFUUCXMUUSXCYEUUBUUSYDUUAJYEUUBJUUSGSYCYTUUSXGSMZTZXTYBYSRUV
|
|
AXTTZYAAMXBYBYSJZUVBXQAXGXRXSXQAXRWAUURUUTXTXQAXRWCWDUVBXGXQMZUUTXTUUSUUT
|
|
XTWEUVAXTVAUVBUUIUVDUUTXTTWFUVBXCXBUUQXSUUTXTWGWHXGXCWMWBWIWJXBUUQXSUUTXT
|
|
VBXAUVCDYAADYBYSDYABVCDYACVCVDUUPYAJBYBCYSDYABVEDYACVEVFVGVLUVAUUMUUIXTUE
|
|
UVAXGUUSUUTVAWHUVAXCXBUUQXSUUTVHWHXGXCWKWLVKVNNOYDUUAULVOWBWNWOVQWPVTWQWR
|
|
HABIFDEGWSHACIFDEGWSWT $.
|
|
$}
|
|
|
|
${
|
|
$d f A $.
|
|
$( A lemma for working with finite sums. (Contributed by Mario Carneiro,
|
|
22-Apr-2014.) $)
|
|
fz1f1o $p |- ( A e. Fin -> ( A = (/) \/ ( ( size ` A ) e. NN /\
|
|
E. f f : ( 1 ... ( size ` A ) ) -1-1-onto-> A ) ) ) $=
|
|
( cfn wcel csize cfv cc0 wceq cn wo c0 c1 cfz co cv wf1o wex wa cn0 sylib
|
|
sizecl elnn0 orcomd sizeeq0 cen isfinite4im bren biantrud orbi12d mpbid
|
|
wbr ) ACDZAEFZGHZUMIDZJAKHZUOLUMMNZABOPBQZRZJULUOUNULUMSDUOUNJAUAUMUBTUCU
|
|
LUNUPUOUSAUDULURUOULUQAUEUKURAUFUQABUGTUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d k n A $. $d k B $. $d n F $. $d k n N $. $d k n ph $. $d k n M $.
|
|
isummo.1 $e |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) ) $.
|
|
isummo.2 $e |- ( ( ph /\ k e. A ) -> B e. CC ) $.
|
|
${
|
|
$d A k n z $. $d F n z $. $d M k n z $. $d N k n z $. $d ph k n z $.
|
|
isummo.dc $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A ) $.
|
|
isumrb.3 $e |- ( ph -> N e. ( ZZ>= ` M ) ) $.
|
|
$( Lemma for ~ isumrb . (Contributed by Mario Carneiro, 12-Aug-2013.) $)
|
|
isumrblem $p |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) ->
|
|
( seq M ( + , F , CC ) |` ( ZZ>= ` N ) ) = seq N ( + , F , CC ) ) $=
|
|
( cfv wa cc cc0 wcel co wceq adantl cz vn cuz wss caddc cvv addid2 0cnd
|
|
vz cv adantr cif eluzelz syl eleq1 ifbid eleq1d wn wo exmiddc wi iftrue
|
|
wdc eqeltrd iffalse 0cn syl6eqel a1i jaod mpd ralrimiva rspcdva syl2anc
|
|
ex fvmptg c1 cmin elfzelz wral ad2antrr elfzuz syl2an2 uznfz con2i ssel
|
|
cfz ad2antlr mtod iffalsed eqtrd cnex simpr addcl iseqid ) ABGUBLZUCZMZ
|
|
UAUHUDNEFGUEOUAUIZNPZOWQUDQWQRWPWQUFSWPUGAGFUBLZPZWOKUJZWPGELZGBPZCOUKZ
|
|
NWPGTPZXDNPZXBXDRWPWTXEXAFGULUMAXFWOADUIZBPZCOUKZNPZXFDWSGXGGRZXIXDNXKX
|
|
HXCCOXGGBUNUOZUPAXJDWSAXGWSPZMZXHXHUQZURZXJXNXHVBXPJXHUSUMAXPXJUTXMAXHX
|
|
JXOAXHXJAXHMXICNXHXICRAXHCOVASIVCVMXOXJUTAXOXIONXHCOVDVEVFVGVHUJVIVJZKV
|
|
KUJZDGXIXDTNEXLHVNVLXRVCWPWQFGVOVPQZWEQPZMZWQELZWQBPZCOUKZOXTWQTPZWPYDN
|
|
PZYBYDRZWQFXSVQYAXJYFDWSWQXGWQRZXIYDNYHXHYCCOXGWQBUNUOZUPZAXJDWSVRZWOXT
|
|
XQVSXTWQWSPZWPWQFXSVTSVKDWQXIYDTNEYIHVNZWAYAYCCOYAYCWQWNPZXTYNUQWPYNXTW
|
|
QFGWBWCSWOYCYNUTAXTBWNWQWDWFWGWHWINUEPWPWJVGWPYLMZYBYDNYLYEWPYFYGFWQULY
|
|
OXJYFDWSWQYJAYKWOYLXQVSWPYLWKVKZYMWAYPVCWRUHUIZNPMWQYQUDQNPWPWQYQWLSWM
|
|
$.
|
|
|
|
$d F k m n z $. $d M k m n z $. $d N k m n z $. $d k m n ph z $.
|
|
fisumcvg.4 $e |- ( ph -> A C_ ( M ... N ) ) $.
|
|
$( The sequence of partial sums of a finite sum converges to the whole
|
|
sum. (Contributed by Mario Carneiro, 20-Apr-2014.) $)
|
|
fisumcvg $p |- ( ph -> seq M ( + , F , CC )
|
|
~~> ( seq M ( + , F , CC ) ` N ) ) $=
|
|
( caddc cc cfv wcel cz wa cc0 wceq vn vm vz cseq cvv cuz eluzelz iseqex
|
|
eqid syl a1i eluzel2 cv cif adantl wn iftrue eqeltrd adantr iffalse 0cn
|
|
wi ex syl6eqel wo exmiddc mpjaod fvmpt2 syl2anc iserf ffvelrnd co simpr
|
|
wdc addid1 c1 cfz elfzuz cdif sseld fznuz con2d imp eldifd fveq2 eqeq1d
|
|
syl6 eldifi eldifn eqtrd vtoclga sylan2 adantlr cnex ralrimiva ad2antrr
|
|
eleq1d wral rspcdva addcl iseqid2 eqcomd climconst ) AGMNEFUDZOZUAXDGUE
|
|
GUFOZXFUIAGFUFOZPZGQPKFGUGUJXDUEPAMNEFUHUKAXGNGXDADEFXGXGUIAXHFQPKFGULU
|
|
JADUMZXGPZRZXIEOZXIBPZCSUNZNXKXIQPZXNNPZXLXNTZXJXOAFXIUGUOXKXMXPXMUPZAX
|
|
MXPVBXJAXMXPAXMRXNCNXMXNCTAXMCSUQUOIURVCUSXRXPVBXKXRXNSNXMCSUTZVAVDUKXK
|
|
XMVNXMXRVEJXMVFUJVGZDQXNNEHVHZVIXTURZVJKVKZAUAUMZXFPZRZXEYDXDOYFUBUCMNE
|
|
GFYDUESUBUMZNPZYGSMVLYGTYFYGVOUOAXHYEKUSAYEVMAXENPYEYCUSAYGGVPMVLZYDVQV
|
|
LPZYGEOZSTZYEYJAYGYIUFOPZYLYGYIYDVRAYMRZYGQBVSZPYLYNYGQBYMYGQPAYIYGUGUO
|
|
AYMYGBPZUPAYPYMAYPYGFGVQVLZPYMUPABYQYGLVTYGFGWAWGWBWCWDXLSTYLDYGYOXIYGT
|
|
ZXLYKSXIYGEWEZWFXIYOPZXLXNSYTXOXPXQXIQBWHYTXNSNYTXRXNSTXIQBWIXSUJZVAVDY
|
|
AVIUUAWJWKUJWLWMNUEPYFWNUKYFYGXGPZRXLNPZYKNPDXGYGYRXLYKNYSWQAUUCDXGWRYE
|
|
UUBAUUCDXGYBWOWPYFUUBVMWSYHUCUMZNPRYGUUDMVLNPYFYGUUDWTUOXAXBXC $.
|
|
$}
|
|
|
|
${
|
|
isumrb.4 $e |- ( ph -> M e. ZZ ) $.
|
|
isumrb.5 $e |- ( ph -> N e. ZZ ) $.
|
|
isumrb.6 $e |- ( ph -> A C_ ( ZZ>= ` M ) ) $.
|
|
isumrb.7 $e |- ( ph -> A C_ ( ZZ>= ` N ) ) $.
|
|
isumrb.mdc $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A ) $.
|
|
isumrb.ndc $e |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A ) $.
|
|
$( Rebase the starting point of a sum. (Contributed by Jim Kingdon,
|
|
5-Mar-2022.) $)
|
|
isumrb $p |- ( ph ->
|
|
( seq M ( + , F , CC ) ~~> C <-> seq N ( + , F , CC ) ~~> C ) ) $=
|
|
( wcel caddc cc cli cuz cfv cseq wbr wb wa cz cvv adantr iseqex climres
|
|
cres sylancl wss cv adantlr wdc simpr isumrblem mpidan breq1d bitr3d wo
|
|
wceq uztric syl2anc mpjaodan ) AHGUAUBZQZRSFGUCZDTUDZRSFHUCZDTUDZUEGHUA
|
|
UBZQZAVIUFZVJVNULZDTUDZVKVMVPHUGQZVJUHQVRVKUEAVSVILUIRSFGUJDVJHUHUKUMVP
|
|
VQVLDTAVIBVNUNVQVLVDNVPBCEFGHIAEUOZBQZCSQZVIJUPAVTVHQWAUQZVIOUPAVIURUSU
|
|
TVAVBAVOUFZVLVHULZDTUDZVKVMWDWEVJDTAVOBVHUNWEVJVDMWDBCEFHGIAWAWBVOJUPAV
|
|
TVNQWCVOPUPAVOURUSUTVAWDGUGQZVLUHQWFVMUEAWGVOKUIRSFHUJDVLGUHUKUMVBAWGVS
|
|
VIVOVCKLGHVEVFVG $.
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
ELEMENTARY NUMBER THEORY
|
|
###############################################################################
|
|
|
|
Here we introduce elementary number theory, in particular the
|
|
elementary properties of divisibility and elementary prime number theory.
|
|
$)
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Elementary properties of divisibility
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The divides relation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c || $.
|
|
|
|
$( Extend the definition of a class to include the divides relation. See
|
|
~ df-dvds . $)
|
|
cdvds $a class || $.
|
|
|
|
${
|
|
$d n x y $.
|
|
$( Define the divides relation, see definition in [ApostolNT] p. 14.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
df-dvds $a |- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\
|
|
E. n e. ZZ ( n x. x ) = y ) } $.
|
|
$}
|
|
|
|
${
|
|
$d M n x y $. $d N n x y $.
|
|
$( Define the divides relation. ` M || N ` means ` M ` divides into ` N `
|
|
with no remainder. For example, ` 3 || 6 ` ( ~ ex-dvds ). As proven in
|
|
~ dvdsval3 , ` M || N <-> ( N mod M ) = 0 ` . See ~ divides and
|
|
~ dvdsval2 for other equivalent expressions. (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
divides $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M || N <-> E. n e. ZZ ( n x. M ) = N ) ) $=
|
|
( vx vy cdvds wbr cop cv cz wcel wa cmul co wceq wrex copab df-br df-dvds
|
|
rexbidv eleq2i bitri oveq2 eqeq1d eqeq2 opelopab2 syl5bb ) BCFGZBCHZDIZJK
|
|
EIZJKLAIZUJMNZUKOZAJPZLDEQZKZBJKCJKLULBMNZCOZAJPZUHUIFKUQBCFRFUPUIDEASUAU
|
|
BUOURUKOZAJPUTDEBCJJUJBOZUNVAAJVBUMURUKUJBULMUCUDTUKCOVAUSAJUKCURUETUFUG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d k M $. $d k N $.
|
|
$( One nonzero integer divides another integer if and only if their
|
|
quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) $)
|
|
dvdsval2 $p |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) ->
|
|
( M || N <-> ( N / M ) e. ZZ ) ) $=
|
|
( vk cz wcel cc0 wne w3a cdvds wbr cmul co wceq wb wa zcn adantr 3ad2ant1
|
|
cc mpbird cv wrex cdiv divides 3adant2 3ad2ant3 adantl simpl2 zapne mpan2
|
|
cap 0z divmulap3d eqcom syl6bb biimprd impr simprl rexlimdvaa simpr simp2
|
|
eqeltrd divcanap1d oveq1 eqeq1d rspcev syl2anc ex impbid bitrd ) ADEZAFGZ
|
|
BDEZHZABIJZCUAZAKLZBMZCDUBZBAUCLZDEZVKVMVOVSNVLCABUDUEVNVSWAVNVRWACDVNVPD
|
|
EZVROOVTVPDVNWBVRVTVPMZVNWBOZWCVRWDWCBVQMVRWDBVPAVNBSEZWBVMVKWEVLBPUFZQWB
|
|
VPSEVNVPPUGVNASEZWBVKVLWGVMAPRZQWDAFUKJZVLVKVLVMWBUHVNWIVLNZWBVKVLWJVMVKF
|
|
DEWJULAFUIUJRZQTUMBVQUNUOUPUQVNWBVRURVBUSVNWAVSVNWAOWAVTAKLZBMZVSVNWAUTVN
|
|
WMWAVNBAWFWHVNWIVLVKVLVMVAWKTVCQVRWMCVTDVPVTMVQWLBVPVTAKVDVEVFVGVHVIVJ $.
|
|
$}
|
|
|
|
$( One nonzero integer divides another integer if and only if the remainder
|
|
upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by
|
|
Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro,
|
|
15-Jul-2014.) $)
|
|
dvdsval3 $p |- ( ( M e. NN /\ N e. ZZ ) ->
|
|
( M || N <-> ( N mod M ) = 0 ) ) $=
|
|
( cn wcel cz wa cdvds wbr cdiv co cmo cc0 wceq wne wb nnz nnne0 jca adantr
|
|
cq dvdsval2 3expa sylan clt zq adantl nnq nngt0 modq0 syl3anc bitr4d ) ACDZ
|
|
BEDZFZABGHZBAIJEDZBAKJLMZULAEDZALNZFUMUOUPOZULURUSAPAQRURUSUMUTABUAUBUCUNBT
|
|
DZATDZLAUDHZUQUPOUMVAULBUEUFULVBUMAUGSULVCUMAUHSBAUIUJUK $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Reverse closure for the divisibility relation. (Contributed by Stefan
|
|
O'Rear, 5-Sep-2015.) $)
|
|
dvdszrcl $p |- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) ) $=
|
|
( vx vy vz cz cdvds cv wcel wa cmul co wceq wrex df-dvds opabssxp eqsstri
|
|
copab cxp brel ) ABFFGGCHZFIDHZFIJEHUAKLUBMEFNZJCDRFFSCDEOUCCDFFPQT $.
|
|
$}
|
|
|
|
$( Strong form of ~ dvdsval2 for positive integers. (Contributed by Stefan
|
|
O'Rear, 13-Sep-2014.) $)
|
|
nndivdvds $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( B || A <-> ( A / B ) e. NN ) ) $=
|
|
( cn wcel wa cdvds wbr cc0 cdiv co clt cz wne wb nnz adantl adantr cr nngt0
|
|
nnre nnne0 dvdsval2 syl3anc anbi1d divgt0d biantrud elnnz a1i 3bitr4d ) ACD
|
|
ZBCDZEZBAFGZHABIJZKGZEUNLDZUOEZUMUNCDZULUMUPUOULBLDZBHMZALDZUMUPNUKUSUJBOPU
|
|
KUTUJBUAPUJVAUKAOQBAUBUCUDULUOUMULABUJARDUKATQUKBRDUJBTPUJHAKGUKASQUKHBKGUJ
|
|
BSPUEUFURUQNULUNUGUHUI $.
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( Definition of the divides relation for positive integers. (Contributed
|
|
by AV, 26-Jul-2021.) $)
|
|
nndivides $p |- ( ( M e. NN /\ N e. NN )
|
|
-> ( M || N <-> E. n e. NN ( n x. M ) = N ) ) $=
|
|
( cn wcel wa cv cmul co wceq wrex cdiv cdvds wbr nndiv cc adantl ad2antrr
|
|
nncn mulcomd eqeq1d rexbidva wb nndivdvds ancoms 3bitr4rd ) BDEZCDEZFZBAG
|
|
ZHIZCJZADKCBLIDEZUJBHIZCJZADKBCMNZABCOUIUOULADUIUJDEZFZUNUKCURUJBUQUJPEUI
|
|
UJSQUGBPEUHUQBSRTUAUBUHUGUPUMUCCBUDUEUF $.
|
|
$}
|
|
|
|
$( Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) $)
|
|
dvdsdc $p |- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N ) $=
|
|
( cn wcel cz wa cdvds wbr wdc cmo co cc0 simpr simpl zmodcld nn0zd 0z zdceq
|
|
wceq sylancl dvdsval3 dcbid mpbird ) ACDZBEDZFZABGHZIBAJKZLSZIZUFUHEDLEDUJU
|
|
FUHUFBAUDUEMUDUENOPQUHLRTUFUGUIABUAUBUC $.
|
|
|
|
$( Two ways to say ` A == B ` (mod ` N ` ), see also definition in
|
|
[ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.) $)
|
|
moddvds $p |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) ->
|
|
( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) ) $=
|
|
( wcel cz cmo co wceq wbr wb wa cc0 cq adantr syl2anc caddc zq simpr oveq1d
|
|
modqadd1 cn cmin cdvds clt nngt0 q0mod eqeq2d cneg ad2antrl ad2antll qnegcl
|
|
nnq syl simprl zcnd simprr negsubd negidd eqeq12d sylibd qsubcl mp1i npcand
|
|
ex 0z addid2d impbid zsubcl dvdsval3 sylan2 3bitr4d 3impb ) CUADZAEDZBEDZAC
|
|
FGZBCFGZHZCABUBGZUCIZJVMVNVOKZKZVSCFGZLCFGZHZWCLHZVRVTWBWDLWCWBCMDZLCUDIZWD
|
|
LHVMWGWACULNZVMWHWACUENZCUFOUGWBVRWEWBVRABUHZPGZCFGZBWKPGZCFGZHZWEWBVRWPWBV
|
|
RKZABWKCWBAMDZVRVNWRVMVOAQUIZNWBBMDZVRVOWTVMVNBQUJZNZWQWTWKMDXBBUKUMWBWGVRW
|
|
INWBWHVRWJNWBVRRTVDWBWMWCWOWDWBWLVSCFWBABWBAVMVNVOUNUOZWBBVMVNVOUPUOZUQSWBW
|
|
NLCFWBBXDURSUSUTWBWEVSBPGZCFGZLBPGZCFGZHZVRWBWEXIWBWEKZVSLBCXJWRWTVSMDWBWRW
|
|
EWSNWBWTWEXANZABVAOLEDLMDXJVELQVBXKWBWGWEWINWBWHWEWJNWBWERTVDWBXFVPXHVQWBXE
|
|
ACFWBABXCXDVCSWBXGBCFWBBXDVFSUSUTVGWAVMVSEDVTWFJABVHCVSVIVJVKVL $.
|
|
|
|
${
|
|
$d K x $. $d M x $. $d N x $.
|
|
$( A lemma to assist theorems of ` || ` with no antecedents. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds0lem $p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) ->
|
|
M || N ) $=
|
|
( vx cz wcel w3a cmul co wceq cdvds wbr wi wa cv wrex oveq1 eqeq1d rspcev
|
|
adantl wb divides adantr mpbird expr 3impa 3comr imp ) AEFZBEFZCEFZGABHIZ
|
|
CJZBCKLZUJUKUIUMUNMZUJUKUIUOUJUKNZUIUMUNUPUIUMNZNUNDOZBHIZCJZDEPZUQVAUPUT
|
|
UMDAEURAJUSULCURABHQRSTUPUNVAUAUQDBCUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d J x $. $d K x $. $d M x z $. $d N x z $. $d Z z $. $d ph x $.
|
|
dvds1lem.1 $e |- ( ph -> ( J e. ZZ /\ K e. ZZ ) ) $.
|
|
dvds1lem.2 $e |- ( ph -> ( M e. ZZ /\ N e. ZZ ) ) $.
|
|
dvds1lem.3 $e |- ( ( ph /\ x e. ZZ ) -> Z e. ZZ ) $.
|
|
dvds1lem.4 $e |- ( ( ph /\ x e. ZZ ) ->
|
|
( ( x x. J ) = K -> ( Z x. M ) = N ) ) $.
|
|
$( A lemma to assist theorems of ` || ` with one antecedent. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds1lem $p |- ( ph -> ( J || K -> M || N ) ) $=
|
|
( vz cv cmul co wceq cz wrex wcel wa cdvds wbr oveq1 eqeq1d rspcev syl6an
|
|
rexlimdva wb divides syl 3imtr4d ) ABMZCNODPZBQRZLMZENOZFPZLQRZCDUAUBZEFU
|
|
AUBZAUMURBQAULQSTGQSUMGENOZFPZURJKUQVBLGQUOGPUPVAFUOGENUCUDUEUFUGACQSDQST
|
|
USUNUHHBCDUIUJAEQSFQSTUTURUHILEFUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d I x y $. $d J x y $. $d K x y $. $d L x y $. $d M x y z $.
|
|
$d N x y z $. $d Z z $. $d ph x y $.
|
|
dvds2lem.1 $e |- ( ph -> ( I e. ZZ /\ J e. ZZ ) ) $.
|
|
dvds2lem.2 $e |- ( ph -> ( K e. ZZ /\ L e. ZZ ) ) $.
|
|
dvds2lem.3 $e |- ( ph -> ( M e. ZZ /\ N e. ZZ ) ) $.
|
|
dvds2lem.4 $e |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> Z e. ZZ ) $.
|
|
dvds2lem.5 $e |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) ->
|
|
( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> ( Z x. M ) = N ) ) $.
|
|
$( A lemma to assist theorems of ` || ` with two antecedents. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds2lem $p |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) ) $=
|
|
( wa cmul wceq cz wcel vz cdvds wbr cv co wrex wb divides bi2anan9 biimpd
|
|
syl2anc reeanv syl6ibr oveq1 eqeq1d rspcev syl6an rexlimdvva syld sylibrd
|
|
syl ) ADEUBUCZFGUBUCZPZUAUDZHQUEZIRZUASUFZHIUBUCZAVDBUDZDQUEERZCUDZFQUEGR
|
|
ZPZCSUFBSUFZVHAVDVKBSUFZVMCSUFZPZVOAVDVRADSTESTPZFSTGSTPZVDVRUGKLVSVBVPVT
|
|
VCVQBDEUHCFGUHUIUKUJVKVMBCSSULUMAVNVHBCSSAVJSTVLSTPPJSTVNJHQUEZIRZVHNOVGW
|
|
BUAJSVEJRVFWAIVEJHQUNUOUPUQURUSAHSTISTPVIVHUGMUAHIUHVAUT $.
|
|
$}
|
|
|
|
$( An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive
|
|
property of the divides relation). (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
iddvds $p |- ( N e. ZZ -> N || N ) $=
|
|
( cz wcel c1 cmul co wceq cdvds wbr zcn mulid2d 1z dvds0lem mp3anl1 anabsan
|
|
mpdan ) ABCZDAEFAGZAAHIZQAAJKQRSDBCQQRSLDAAMNOP $.
|
|
|
|
$( 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
1dvds $p |- ( N e. ZZ -> 1 || N ) $=
|
|
( cz wcel c1 cmul co wceq cdvds wbr zcn mulid1d 1z dvds0lem mp3anl2 anabsan
|
|
mpdan ) ABCZADEFAGZDAHIZQAAJKQRSQDBCQRSLADAMNOP $.
|
|
|
|
$( Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds0 $p |- ( N e. ZZ -> N || 0 ) $=
|
|
( cz wcel cc0 cmul co wceq cdvds wbr zcn mul02d wi w3a dvds0lem mp3an13 mpd
|
|
0z ex ) ABCZDAEFDGZADHIZSAAJKDBCZSUBTUALQQUBSUBMTUADADNROP $.
|
|
|
|
${
|
|
$d M x $. $d N x $.
|
|
$( An integer divides another iff its negation does. (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
negdvdsb $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) ) $=
|
|
( vx cz wcel wa cdvds wbr cneg znegcl cmul co wceq cc zcn syl2anr adantlr
|
|
eqeq1d biimprd dvds1lem cv id anim1i adantl mul2neg mulneg12 impbid ) ADE
|
|
ZBDEZFZABGHAIZBGHUJCABUKBCUAZIZUJUBZUHUKDEUIAJUCZULDEZUMDEUJULJUDZUJUPFZU
|
|
MUKKLZBMULAKLZBMURUSUTBUHUPUSUTMZUIUPULNEZANEZVAUHULOZAOZULAUEPQRSTUJCUKB
|
|
ABUMUOUNUQURUMAKLZBMULUKKLZBMURVFVGBUHUPVFVGMZUIUPVBVCVHUHVDVEULAUFPQRSTU
|
|
G $.
|
|
|
|
$( An integer divides another iff it divides its negation. (Contributed by
|
|
Paul Chapman, 21-Mar-2011.) $)
|
|
dvdsnegb $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || -u N ) ) $=
|
|
( vx cz wcel wa cdvds cneg cv id znegcl cmul co wceq wi cc negeq dvds1lem
|
|
wbr zcn anim2i adantl mulneg1 eqeq2d syl5ibcom syl2anr sylan9eqr sylan9eq
|
|
adantlr negneg expr 3impa syl3an 3coml 3expa impbid ) ADEZBDEZFZABGSABHZG
|
|
SUSCABAUTCIZHZUSJZURUTDEUQBKUAZVADEZVBDEUSVAKUBZUQVEVAALMZBNZVBALMZUTNZOZ
|
|
URVEVAPEZAPEZVKUQVATZATZVLVMFZVIVGHZNVHVJVAAUCZVHVQUTVIVGBQUDUEUFUIRUSCAU
|
|
TABVBVDVCVFUQURVEVGUTNZVIBNZOZVEUQURWAVEVLUQVMURBPEZWAVNVOBTVLVMWBWAVPWBV
|
|
SVTVPWBVSFVIVQBVRVSWBVQUTHBVGUTQBUJUGUHUKULUMUNUORUP $.
|
|
$}
|
|
|
|
$( An integer divides another iff its absolute value does. (Contributed by
|
|
Paul Chapman, 21-Mar-2011.) $)
|
|
absdvdsb $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M || N <-> ( abs ` M ) || N ) ) $=
|
|
( cz wcel wa cabs cfv wceq cdvds wbr wb cneg breq1 bicomd negdvdsb sylan9bb
|
|
wi a1i ex wo zq qabsord adantr mpjaod ) ACDZBCDZEZAFGZAHZABIJZUHBIJZKZUHALZ
|
|
HZUIULQUGUIUKUJUHABIMNRUGUNULUGUJUMBIJZUNUKABOUNUKUOUHUMBIMNPSUEUIUNTUFUEAA
|
|
UAUBUCUD $.
|
|
|
|
$( An integer divides another iff it divides its absolute value.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdsabsb $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M || N <-> M || ( abs ` N ) ) ) $=
|
|
( cz wcel wa cabs cfv wceq cdvds wbr wb cneg breq2 bicomd dvdsnegb sylan9bb
|
|
wi a1i ex wo zq qabsord adantl mpjaod ) ACDZBCDZEZBFGZBHZABIJZAUHIJZKZUHBLZ
|
|
HZUIULQUGUIUKUJUHBAIMNRUGUNULUGUJAUMIJZUNUKABOUNUKUOUHUMAIMNPSUFUIUNTUEUFBB
|
|
UAUBUCUD $.
|
|
|
|
${
|
|
$d N n $.
|
|
$( Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
0dvds $p |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) ) $=
|
|
( vn cz wcel cc0 cdvds wbr wceq cv cmul co wrex wb 0z divides mpan mul01d
|
|
zcn eqtr2 sylan2 ancoms rexlimiva syl6bi dvds0 ax-mp breq2 mpbiri impbid1
|
|
) ACDZEAFGZAEHZUIUJBIZEJKZAHZBCLZUKECDZUIUJUOMNBEAOPUNUKBCUNULCDZUKUQUNUM
|
|
EHUKUQULULRQUMAESTUAUBUCUKUJEEFGZUPURNEUDUEAEEFUFUGUH $.
|
|
$}
|
|
|
|
$( Divisibility of integers is decidable. (Contributed by Jim Kingdon,
|
|
17-Jan-2022.) $)
|
|
zdvdsdc $p |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N ) $=
|
|
( cz wcel wa cc0 clt wbr cdvds wdc wceq cn simpll simpr elnnz simplr dvdsdc
|
|
sylanbrc syl2anc wb cneg znegcld zred lt0neg1d mpbid negdvdsb adantr mpbird
|
|
dcbid zdceq mpan2 ad2antlr breq1 adantl 0dvds bitrd w3o ztri3or0 mpjao3dan
|
|
0z ) ACDZBCDZEZAFGHZABIHZJZAFKZFAGHZVCVDEZVFAUAZBIHZJZVIVJLDZVBVLVIVJCDFVJG
|
|
HZVMVIAVAVBVDMZUBVIVDVNVCVDNVIAVIAVOUCUDUEVJORVAVBVDPVJBQSVIVEVKVCVEVKTVDAB
|
|
UFUGUIUHVCVGEZVFBFKZJZVBVRVAVGVBFCDVRUTBFUJUKULVPVEVQVPVEFBIHZVQVGVEVSTVCAF
|
|
BIUMUNVBVSVQTVAVGBUOULUPUIUHVCVHEZALDZVBVFVTVAVHWAVAVBVHMVCVHNAORVAVBVHPABQ
|
|
SVAVDVGVHUQVBAURUGUS $.
|
|
|
|
$( An integer divides a multiple of itself. (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
dvdsmul1 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) ) $=
|
|
( cz wcel wa cmul co cdvds wbr cc zcn mulcom syl2anr wi zmulcl w3a dvds0lem
|
|
wceq ex 3com12 mpd3an3 mpd ) ACDZBCDZEBAFGABFGZRZAUEHIZUDBJDAJDUFUCBKAKBALM
|
|
UCUDUECDZUFUGNZABOUDUCUHUIUDUCUHPUFUGBAUEQSTUAUB $.
|
|
|
|
$( An integer divides a multiple of itself. (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
dvdsmul2 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) ) $=
|
|
( cz wcel cmul co cdvds wbr zmulcl w3a wceq eqid dvds0lem mpan2 mpd3an3 ) A
|
|
CDZBCDZABEFZCDZBRGHZABIPQSJRRKTRLABRMNO $.
|
|
|
|
$( An integer divides a positive integer power of itself. (Contributed by
|
|
Paul Chapman, 26-Oct-2012.) $)
|
|
iddvdsexp $p |- ( ( M e. ZZ /\ N e. NN ) -> M || ( M ^ N ) ) $=
|
|
( cz wcel cn wa c1 cmin cexp cmul cdvds wbr cn0 nnm1nn0 zexpcl sylan2 simpl
|
|
co dvdsmul2 syl2anc cc wceq zcn expm1t sylan breqtrrd ) ACDZBEDZFZAABGHRZIR
|
|
ZAJRZABIRZKUIUKCDZUGAULKLUHUGUJMDUNBNAUJOPUGUHQUKASTUGAUADUHUMULUBAUCABUDUE
|
|
UF $.
|
|
|
|
${
|
|
$d K x $. $d M x $. $d N x $.
|
|
$( If a product divides an integer, so does one of its factors.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
muldvds1 $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K x. M ) || N -> K || N ) ) $=
|
|
( vx cz wcel w3a cmul co cv wa zmulcl anim1i 3simpb ancoms 3ad2antl2 wceq
|
|
3impa cc zcn mulass mul32 eqtr3d syl3an 3adantl3 eqeq1d biimpd dvds1lem
|
|
3coml 3expa ) AEFZBEFZCEFZGZDABHIZCACDJZBHIZUKULUMUOEFZUMKUKULKURUMABLMRU
|
|
KULUMNULUKUPEFZUQEFZUMUSULUTUPBLOPUNUSKZUPUOHIZCQUQAHIZCQVAVBVCCUKULUSVBV
|
|
CQZUMUKULUSVDUSUKULVDUSUPSFZUKASFZULBSFZVDUPTATBTVEVFVGGUPAHIBHIVBVCUPABU
|
|
AUPABUBUCUDUIUJUEUFUGUH $.
|
|
|
|
$( If a product divides an integer, so does one of its factors.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
muldvds2 $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K x. M ) || N -> M || N ) ) $=
|
|
( vx cz wcel w3a cmul co cv wa zmulcl anim1i 3simpc ancoms 3ad2antl1 wceq
|
|
3impa cc zcn mulass syl3an 3coml 3expa 3adantl3 eqeq1d biimprd dvds1lem )
|
|
AEFZBEFZCEFZGZDABHIZCBCDJZAHIZUIUJUKUMEFZUKKUIUJKUPUKABLMRUIUJUKNUIUJUNEF
|
|
ZUOEFZUKUQUIURUNALOPULUQKZUOBHIZCQUNUMHIZCQUSUTVACUIUJUQUTVAQZUKUIUJUQVBU
|
|
QUIUJVBUQUNSFUIASFUJBSFVBUNTATBTUNABUAUBUCUDUEUFUGUH $.
|
|
|
|
$( Multiplication by a constant maintains the divides relation. Theorem
|
|
1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides
|
|
relation). (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdscmul $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) ->
|
|
( M || N -> ( K x. M ) || ( K x. N ) ) ) $=
|
|
( vx cz wcel cdvds wbr cmul co wi w3a cv 3simpc zmulcl 3adant3 wceq 3coml
|
|
cc zcn 3adant2 jca simpr wa mul12 syl3an 3expa 3adantl3 oveq2 sylan9eq ex
|
|
dvds1lem ) AEFZBEFZCEFZBCGHABIJZACIJZGHKUMUNUOLZDBCUPUQDMZUMUNUONURUPEFZU
|
|
QEFZUMUNUTUOABOPUMUOVAUNACOUAUBURUSEFZUCURVBUDZUSBIJZCQZUSUPIJZUQQVCVEVFA
|
|
VDIJZUQUMUNVBVFVGQZUOUMUNVBVHVBUMUNVHVBUSSFUMASFUNBSFVHUSTATBTUSABUEUFRUG
|
|
UHVDCAIUIUJUKULR $.
|
|
|
|
$( Multiplication by a constant maintains the divides relation.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdsmulc $p |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) ->
|
|
( M || N -> ( M x. K ) || ( N x. K ) ) ) $=
|
|
( vx cz wcel cdvds wbr cmul co wi w3a cv 3simpc wa zmulcl 3adant2 wceq cc
|
|
zcn 3adant1 jca 3comr simpr mulass syl3an 3com13 3expa 3adantl3 sylan9req
|
|
oveq1 ex dvds1lem 3coml ) AEFZBEFZCEFZBCGHBAIJZCAIJZGHKUOUPUQLZDBCURUSDMZ
|
|
UOUPUQNUPUQUOUREFZUSEFZOUPUQUOLVBVCUPUOVBUQBAPQUQUOVCUPCAPUAUBUCUTVAEFZUD
|
|
UTVDOZVABIJZCRZVAURIJZUSRVEVGVHVFAIJZUSUOUPVDVIVHRZUQUOUPVDVJVDUPUOVJVDVA
|
|
SFUPBSFUOASFVJVATBTATVABAUEUFUGUHUIVFCAIUKUJULUMUN $.
|
|
|
|
$( Cancellation law for the divides relation. Theorem 1.1(e) in
|
|
[ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdscmulr $p |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) ->
|
|
( ( K x. M ) || ( K x. N ) <-> M || N ) ) $=
|
|
( vx cz wcel cc0 wne wa w3a cmul co cdvds wbr cv zmulcld wceq adantr zcnd
|
|
jca simp3l simp1 simp2 simpr mul12d eqeq1d mulcld cap simpl3r wb 0z zapne
|
|
sylancl mpbird mulcanapd bitr3d biimpd dvds1lem dvdscmul 3adant3r impbid
|
|
wi ) BEFZCEFZAEFZAGHZIZJZABKLZACKLZMNZBCMNZVHDVIVJBCDOZVHVIEFVJEFVHABVCVD
|
|
VEVFUAZVCVDVGUBZPVHACVNVCVDVGUCZPTVHVCVDVOVPTVHVMEFZUDZVHVQIZVMVIKLZVJQZV
|
|
MBKLZCQZVSAWBKLZVJQWAWCVSWDVTVJVSAVMBVSAVHVEVQVNRZSZVSVMVRSZVSBVHVCVQVORS
|
|
ZUEUFVSWBCAVSVMBWGWHUGVSCVHVDVQVPRSWFVSAGUHNZVFVEVFVCVDVQUIVSVEGEFWIVFUJW
|
|
EUKAGULUMUNUOUPUQURVCVDVEVLVKVBVFABCUSUTVA $.
|
|
|
|
$( Cancellation law for the divides relation. (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
dvdsmulcr $p |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) ->
|
|
( ( M x. K ) || ( N x. K ) <-> M || N ) ) $=
|
|
( vx cz wcel cc0 wne wa w3a cmul co cdvds wbr cv zmulcld wceq zcnd adantr
|
|
simp1 simp3l simp2 jca 3simpa simpr cc mulassd eqeq1d mulcld cap wb zapne
|
|
simpl3r 0z mpan2 3ad2ant3 mpbird bitr3d biimpd dvds1lem dvdsmulc 3adant3r
|
|
mulcanap2d wi impbid ) BEFZCEFZAEFZAGHZIZJZBAKLZCAKLZMNZBCMNZVKDVLVMBCDOZ
|
|
VKVLEFVMEFVKBAVFVGVJTZVFVGVHVIUAZPVKCAVFVGVJUBZVRPUCVFVGVJUDVKVPEFZUEZVKV
|
|
TIZVPVLKLZVMQZVPBKLZCQZWBWEAKLZVMQWDWFWBWGWCVMWBVPBAWBVPWARZVKBUFFVTVKBVQ
|
|
RSZWBAVKVHVTVRSRZUGUHWBWECAWBVPBWHWIUIWBCVKVGVTVSSRWJWBAGUJNZVIVHVIVFVGVT
|
|
UMVKWKVIUKZVTVJVFWLVGVHWLVIVHGEFWLUNAGULUOSUPSUQVCURUSUTVFVGVHVOVNVDVIABC
|
|
VAVBVE $.
|
|
$}
|
|
|
|
$( The sum of two integers modulo a positive integer equals zero iff the
|
|
first of the two integers equals the negative of the other integer modulo
|
|
the positive integer. (Contributed by AV, 25-Jul-2021.) $)
|
|
summodnegmod $p |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN )
|
|
-> ( ( ( A + B ) mod N ) = 0 <-> ( A mod N ) = ( -u B mod N ) ) ) $=
|
|
( cz wcel cn w3a cmo co cneg wceq cmin cdvds wbr caddc cc0 wb zcn 3adant3
|
|
cc simp3 simp1 znegcl 3ad2ant2 moddvds syl3anc wa anim12i subneg eqcomd syl
|
|
breq2d zaddcl dvdsval3 syl2anc 3bitr2rd ) ADEZBDEZCFEZGZACHIBJZCHIKZCAVALIZ
|
|
MNZCABOIZMNZVECHIPKZUTUSUQVADEZVBVDQUQURUSUAZUQURUSUBURUQVHUSBUCUDAVACUEUFU
|
|
TVEVCCMUTATEZBTEZUGZVEVCKUQURVLUSUQVJURVKARBRUHSVLVCVEABUIUJUKULUTUSVEDEZVF
|
|
VGQVIUQURVMUSABUMSCVEUNUOUP $.
|
|
|
|
$( Constant multiplication in a modulo operation, see theorem 5.3 in
|
|
[ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) $)
|
|
modmulconst $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN )
|
|
-> ( ( A mod M ) = ( B mod M )
|
|
<-> ( ( C x. A ) mod ( C x. M ) )
|
|
= ( ( C x. B ) mod ( C x. M ) ) ) ) $=
|
|
( cz wcel cn w3a wa cmin co cdvds wbr cmul cmo wceq wb adantr syl3anc cc
|
|
cc0 wne nnz adantl zsubcl 3adant3 nnne0 3ad2ant3 dvdscmulr bicomd 3anim123i
|
|
jca zcn nncn 3anrot sylibr subdi syl breq2d bitrd simpr simp1 simp2 moddvds
|
|
simpl3 nnmulcld zmulcld 3bitr4d ) AEFZBEFZCGFZHZDGFZIZDABJKZLMZCDNKZCANKZCB
|
|
NKZJKZLMZADOKBDOKPZVRVQOKVSVQOKPZVNVPVQCVONKZLMZWAVNDEFZVOEFZCEFZCUAUBZIZVP
|
|
WEQVMWFVLDUCUDVLWGVMVIVJWGVKABUEUFRVLWJVMVKVIWJVJVKWHWICUCZCUGULUHRWFWGWJHW
|
|
EVPCDVOUIUJSVNWDVTVQLVLWDVTPZVMVLCTFZATFZBTFZHZWLVLWNWOWMHWPVIWNVJWOVKWMAUM
|
|
BUMCUNUKWMWNWOUOUPCABUQURRUSUTVNVMVIVJWBVPQVLVMVAZVLVIVMVIVJVKVBZRVLVJVMVIV
|
|
JVKVCZRABDVDSVNVQGFVREFZVSEFZWCWAQVNCDVIVJVKVMVEWQVFVLWTVMVLCAVKVIWHVJWKUHZ
|
|
WRVGRVLXAVMVLCBXBWSVGRVRVSVQVDSVH $.
|
|
|
|
${
|
|
$d I x y $. $d J x y $. $d K x y $. $d M x y $. $d N x y $.
|
|
$( If an integer divides each of two other integers, it divides any linear
|
|
combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity
|
|
property of the divides relation). (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
dvds2ln $p |- ( ( ( I e. ZZ /\ J e. ZZ ) /\
|
|
( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ) ->
|
|
( ( K || M /\ K || N ) -> K || ( ( I x. M ) + ( J x. N ) ) ) ) $=
|
|
( vx vy cz wcel wa cmul co caddc cv jca adantr wceq cc zcn adantl zmulcld
|
|
w3a simpr1 simpr2 simpr3 simpll simplr zaddcld zmulcl anim12i an4s expcom
|
|
imp zaddcl syl zcnd adddir 3expa syl2anc ad3antrrr mul32d oveq12d mulcomd
|
|
wi mulcld 3eqtrd oveq2 oveqan12d sylan9eq ex dvds2lem ) AHIZBHIZJZCHIZDHI
|
|
ZEHIZUBZJZFGCDCECADKLZBEKLZMLZFNZAKLZGNZBKLZMLZVSVOVPVNVOVPVQUCZVNVOVPVQU
|
|
DZOVSVOVQWHVNVOVPVQUEZOVSVOWBHIWHVSVTWAVSADVLVMVRUFWIUAVSBEVLVMVRUGZWJUAU
|
|
HOVSWCHIZWEHIZJZJZWDHIZWFHIZJZWGHIVSWNWRVNWNWRVDVRWNVNWRWLVLWMVMWRWLVLJWP
|
|
WMVMJWQWCAUIWEBUIUJUKULPUMZWDWFUNUOWOWCCKLZDQZWECKLZEQZJZWGCKLZWBQWOXDXEA
|
|
WTKLZBXBKLZMLZWBWOXEWDCKLZWFCKLZMLZWTAKLZXBBKLZMLXHWOWDRIZWFRIZJZCRIZXEXK
|
|
QZWOWRXPWSWPXNWQXOWDSWFSUJUOVSXQWNVSCWHUPPZXNXOXQXRWDWFCUQURUSWOXIXLXJXMM
|
|
WOWCACWNWCRIZVSWLXTWMWCSPTZVLARIVMVRWNASUTZXSVAWOWEBCWNWERIZVSWMYCWLWESTT
|
|
ZVSBRIWNVSBWKUPPZXSVAVBWOXLXFXMXGMWOWTAWOWCCYAXSVEYBVCWOXBBWOWECYDXSVEYEV
|
|
CVBVFXAXCXFVTXGWAMWTDAKVGXBEBKVGVHVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d K x y $. $d M x y $. $d N x y $.
|
|
$( If an integer divides each of two other integers, it divides their sum.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds2add $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M /\ K || N ) -> K || ( M + N ) ) ) $=
|
|
( vx vy cz wcel w3a caddc co cv 3simpa 3simpb zaddcl anim2i cmul wceq zcn
|
|
wa cc 3impb adantl wi adddir syl3an 3comr 3expb oveq12 sylan9eq 3ad2antl1
|
|
ex dvds2lem ) AFGZBFGZCFGZHZDEABACABCIJZDKZEKZIJZUMUNUOLUMUNUOMUMUNUOUMUQ
|
|
FGZSUNUOSVAUMBCNOUAURFGZUSFGZSZUTFGUPURUSNUBUMUNVDURAPJZBQUSAPJZCQSZUTAPJ
|
|
ZUQQZUCUOUMVDSZVGVIVJVGVHVEVFIJZUQUMVBVCVHVKQZVBVCUMVLVBURTGVCUSTGUMATGVL
|
|
URRUSRARURUSAUDUEUFUGVEBVFCIUHUIUKUJUL $.
|
|
|
|
$( If an integer divides each of two other integers, it divides their
|
|
difference. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvds2sub $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M /\ K || N ) -> K || ( M - N ) ) ) $=
|
|
( vx vy cz wcel w3a cmin co cv 3simpa 3simpb wa zsubcl anim2i cmul cc zcn
|
|
wceq 3impb adantl wi subdir syl3an 3comr 3expb oveq12 sylan9eq 3ad2antl1
|
|
ex dvds2lem ) AFGZBFGZCFGZHZDEABACABCIJZDKZEKZIJZUMUNUOLUMUNUOMUMUNUOUMUQ
|
|
FGZNUNUONVAUMBCOPUAURFGZUSFGZNZUTFGUPURUSOUBUMUNVDURAQJZBTUSAQJZCTNZUTAQJ
|
|
ZUQTZUCUOUMVDNZVGVIVJVGVHVEVFIJZUQUMVBVCVHVKTZVBVCUMVLVBURRGVCUSRGUMARGVL
|
|
URSUSSASURUSAUDUEUFUGVEBVFCIUHUIUKUJUL $.
|
|
|
|
${
|
|
dvds2subd.1 $e |- ( ph -> K e. ZZ ) $.
|
|
dvds2subd.2 $e |- ( ph -> K || M ) $.
|
|
dvds2subd.3 $e |- ( ph -> K || N ) $.
|
|
dvds2subd.4 $e |- ( ph -> M e. ZZ ) $.
|
|
dvds2subd.5 $e |- ( ph -> N e. ZZ ) $.
|
|
$( Natural deduction form of ~ dvds2sub . (Contributed by Stanislas
|
|
Polu, 9-Mar-2020.) $)
|
|
dvds2subd $p |- ( ph -> K || ( M - N ) ) $=
|
|
( cdvds wbr cmin co cz wcel wa wi dvds2sub syl3anc mp2and ) ABCJKZBDJKZ
|
|
BCDLMJKZFGABNOCNODNOUAUBPUCQEHIBCDRST $.
|
|
$}
|
|
|
|
$( The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14
|
|
(transitive property of the divides relation). (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
dvdstr $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M /\ M || N ) -> K || N ) ) $=
|
|
( vx vy cz wcel w3a cv cmul co 3simpa 3simpc 3simpb wa zmulcl adantl wceq
|
|
cc zcn oveq2 adantr eqeq2 mpbid mulass mul12 eqtrd syl3an 3comr 3ad2antl1
|
|
wb 3expb eqeq1d syl5ibr dvds2lem ) AFGZBFGZCFGZHZDEABBCACDIZEIZJKZUPUQURL
|
|
UPUQURMUPUQURNUTFGZVAFGZOZVBFGUSUTVAPQUTAJKZBRZVABJKZCRZOZVBAJKZCRUSVEOZV
|
|
AVFJKZCRZVJVMVHRZVNVGVOVIVFBVAJUAUBVIVOVNUKVGVHCVMUCQUDVLVKVMCUPUQVEVKVMR
|
|
ZURUPVCVDVPVCVDUPVPVCUTSGZVDVASGZUPASGZVPUTTVATATVQVRVSHVKUTVAAJKJKVMUTVA
|
|
AUEUTVAAUFUGUHUIULUJUMUNUO $.
|
|
$}
|
|
|
|
$( If an integer divides another, it divides a multiple of it. (Contributed
|
|
by Paul Chapman, 17-Nov-2012.) $)
|
|
dvdsmultr1 $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( K || M -> K || ( M x. N ) ) ) $=
|
|
( cz wcel w3a cdvds wbr cmul co dvdsmul1 3adant1 wa wi zmulcl dvdstr mpan2d
|
|
syld3an3 ) ADEZBDEZCDEZFABGHZBBCIJZGHZAUCGHZTUAUDSBCKLSTUAUCDEZUBUDMUENTUAU
|
|
FSBCOLABUCPRQ $.
|
|
|
|
${
|
|
dvdsmultr1d.1 $e |- ( ph -> K e. ZZ ) $.
|
|
dvdsmultr1d.2 $e |- ( ph -> M e. ZZ ) $.
|
|
dvdsmultr1d.3 $e |- ( ph -> N e. ZZ ) $.
|
|
dvdsmultr1d.4 $e |- ( ph -> K || M ) $.
|
|
$( Natural deduction form of ~ dvdsmultr1 . (Contributed by Stanislas
|
|
Polu, 9-Mar-2020.) $)
|
|
dvdsmultr1d $p |- ( ph -> K || ( M x. N ) ) $=
|
|
( cdvds wbr cmul co cz wcel wi dvdsmultr1 syl3anc mpd ) ABCIJZBCDKLIJZHAB
|
|
MNCMNDMNSTOEFGBCDPQR $.
|
|
$}
|
|
|
|
$( If an integer divides another, it divides a multiple of it. (Contributed
|
|
by Paul Chapman, 17-Nov-2012.) $)
|
|
dvdsmultr2 $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( K || N -> K || ( M x. N ) ) ) $=
|
|
( cz wcel w3a cdvds wbr cmul co wa wb dvdsmul2 biantrud 3adant1 simp1 simp3
|
|
wi zmulcl dvdstr syl3anc sylbid ) ADEZBDEZCDEZFZACGHZUGCBCIJZGHZKZAUHGHZUDU
|
|
EUGUJLUCUDUEKUIUGBCMNOUFUCUEUHDEZUJUKRUCUDUEPUCUDUEQUDUEULUCBCSOACUHTUAUB
|
|
$.
|
|
|
|
$( If an integer divides either of two others, it divides their product.
|
|
(Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario
|
|
Carneiro, 17-Jul-2014.) $)
|
|
ordvdsmul $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M \/ K || N ) -> K || ( M x. N ) ) ) $=
|
|
( cz wcel w3a cdvds wbr cmul co dvdsmultr1 dvdsmultr2 jaod ) ADEBDECDEFABGH
|
|
ABCIJGHACGHABCKABCLM $.
|
|
|
|
$( If an integer divides a difference, then it divides one term iff it
|
|
divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.) $)
|
|
dvdssub2 $p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) ->
|
|
( K || M <-> K || N ) ) $=
|
|
( cz wcel cmin co cdvds wbr wa wi 3adant1 imp wceq cc syl2an adantr breqtrd
|
|
zcn expr w3a zsubcl dvds2sub syld3an3 ancomsd nncan caddc dvds2add syld3an2
|
|
npcan impbid ) ADEZBDEZCDEZUAZABCFGZHIZJABHIZACHIZUOUQURUSUOUQURJZJABUPFGZC
|
|
HUOUTAVAHIZUOURUQVBULUMUNUPDEZURUQJVBKUMUNVCULBCUBLZABUPUCUDUEMUOVACNZUTUMU
|
|
NVEULUMBOEZCOEZVEUNBSZCSZBCUFPLQRTUOUQUSURUOUQUSJZJAUPCUGGZBHUOVJAVKHIZULVC
|
|
UMUNVJVLKVDAUPCUHUIMUOVKBNZVJUMUNVMULUMVFVGVMUNVHVIBCUJPLQRTUK $.
|
|
|
|
$( An integer divides another iff it divides their sum. (Contributed by Paul
|
|
Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.) $)
|
|
dvdsadd $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M + N ) ) ) $=
|
|
( cz wcel wa caddc co cdvds wbr cmin wb simpl zaddcl simpr iddvds adantr cc
|
|
wceq zcn pncan syl2an breqtrrd dvdssub2 syl31anc bicomd ) ACDZBCDZEZAABFGZH
|
|
IZABHIZUHUFUICDUGAUIBJGZHIUJUKKUFUGLABMUFUGNUHAAULHUFAAHIUGAOPUFAQDBQDULARU
|
|
GASBSABTUAUBAUIBUCUDUE $.
|
|
|
|
$( An integer divides another iff it divides their sum. (Contributed by Paul
|
|
Chapman, 31-Mar-2011.) $)
|
|
dvdsaddr $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N + M ) ) ) $=
|
|
( cz wcel wa cdvds wbr caddc co dvdsadd wceq zcn addcom syl2an breq2d bitrd
|
|
cc ) ACDZBCDZEZABFGAABHIZFGABAHIZFGABJTUAUBAFRAQDBQDUAUBKSALBLABMNOP $.
|
|
|
|
$( An integer divides another iff it divides their difference. (Contributed
|
|
by Paul Chapman, 31-Mar-2011.) $)
|
|
dvdssub $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M - N ) ) ) $=
|
|
( cz wcel wa cdvds wbr cneg caddc co cmin dvdsnegb wb znegcl dvdsadd sylan2
|
|
cc wceq zcn negsub syl2an breq2d 3bitrd ) ACDZBCDZEZABFGABHZFGZAAUGIJZFGZAA
|
|
BKJZFGABLUEUDUGCDUHUJMBNAUGOPUFUIUKAFUDAQDBQDUIUKRUEASBSABTUAUBUC $.
|
|
|
|
$( An integer divides another iff it divides their difference. (Contributed
|
|
by Paul Chapman, 31-Mar-2011.) $)
|
|
dvdssubr $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N - M ) ) ) $=
|
|
( cz wcel wa cmin co cdvds wbr caddc wb zsubcl ancoms dvdsadd syldan cc zcn
|
|
wceq pncan3 syl2an breq2d bitr2d ) ACDZBCDZEZABAFGZHIZAAUFJGZHIZABHIUCUDUFC
|
|
DZUGUIKUDUCUJBALMAUFNOUEUHBAHUCAPDBPDUHBRUDAQBQABSTUAUB $.
|
|
|
|
$( Adding a multiple of the base does not affect divisibility. (Contributed
|
|
by Stefan O'Rear, 23-Sep-2014.) $)
|
|
dvdsadd2b $p |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) ->
|
|
( A || B <-> A || ( C + B ) ) ) $=
|
|
( cz cdvds wbr wa w3a caddc co simpl1 simpl3l simpl2 simpl3r simpr dvds2add
|
|
wcel imp syl2anc adantr syl32anc simp3l simp2 zaddcl znegcld dvdsnegb mpbid
|
|
cneg wb wceq cmin ancoms zcnd cc zcn adantl negsubd pncan2d breqtrd impbida
|
|
eqtrd ) ADQZBDQZCDQZACEFZGZHZABEFZACBIJZEFZVGVHGVBVDVCVEVHVJVBVCVFVHKVDVEVB
|
|
VCVHLVBVCVFVHMVDVEVBVCVHNVGVHOVBVDVCHVEVHGVJACBPRUAVGVJGZAVICUHZIJZBEVKVBVI
|
|
DQZVLDQZVJAVLEFZAVMEFZVBVCVFVJKZVGVNVJVGVDVCVNVBVCVDVEUBZVBVCVFUCCBUDZSTVGV
|
|
OVJVGCVSUETVGVJOVKVEVPVDVEVBVCVJNVKVBVDVEVPUIVRVDVEVBVCVJLZACUFSUGVBVNVOHVJ
|
|
VPGVQAVIVLPRUAVKVCVDVMBUJVBVCVFVJMWAVCVDGZVMVICUKJBWBVICWBVIVDVCVNVTULUMVDC
|
|
UNQVCCUOUPZUQWBCBWCVCBUNQVDBUOTURVASUSUT $.
|
|
|
|
${
|
|
dvdslelemd.1 $e |- ( ph -> M e. ZZ ) $.
|
|
dvdslelemd.2 $e |- ( ph -> N e. NN ) $.
|
|
dvdslelemd.3 $e |- ( ph -> K e. ZZ ) $.
|
|
dvdslelemd.lt $e |- ( ph -> N < M ) $.
|
|
$( Lemma for ~ dvdsle . (Contributed by Jim Kingdon, 8-Nov-2021.) $)
|
|
dvdslelemd $p |- ( ph -> ( K x. M ) =/= N ) $=
|
|
( cmul clt wbr wo cc0 cle cz wcel syl cr adantr cq co wne c1 0z zlelttric
|
|
sylancl wb zgt0ge1 orbi2d mpbid wa zred remulcld 0red nnred cneg renegcld
|
|
le0neg1d biimpa nngt0d lttrd ltled mulge0d mulneg1d breq2d bitr4d lelttrd
|
|
recnd mpbird ex simpr lemulge12d ltletrd orim12d mpd zq qmulcl syl2anc cn
|
|
nnq qlttri2 ) ABCIUAZDUBZWBDJKZDWBJKZLZABMNKZUCBNKZLZWFAWGMBJKZLZWIABOPZM
|
|
OPWKGUDBMUEUFAWJWHWGAWLWJWHUGGBUHQUIUJAWGWDWHWEAWGWDAWGUKZWBMDWMBCABRPZWG
|
|
ABGULZSZACRPZWGACEULZSZUMZWMUNADRPZWGADFUOZSWMWBMNKZMBUPZCIUAZNKZWMXDCWMB
|
|
WPUQWSAWGMXDNKABWOURUSAMCNKZWGAMCAUNZWRAMDCXHXBWRADFUTZHVAVBZSVCWMXCMWBUP
|
|
ZNKXFWMWBWTURWMXEXKMNWMBCWMBWPVHWMCWSVHVDVEVFVIAMDJKWGXISVGVJAWHWEAWHUKZD
|
|
CWBAXAWHXBSAWQWHWRSZXLBCAWNWHWOSZXMUMADCJKWHHSXLCBXMXNAXGWHXJSAWHVKVLVMVJ
|
|
VNVOAWBTPZDTPZWCWFUGABTPZCTPZXOAWLXQGBVPQACOPXRECVPQBCVQVRADVSPXPFDVTQWBD
|
|
WAVRVI $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( The divisors of a positive integer are bounded by it. The proof does
|
|
not use ` / ` . (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdsle $p |- ( ( M e. ZZ /\ N e. NN ) -> ( M || N -> M <_ N ) ) $=
|
|
( vn cz wcel cn wa cle wbr cdvds wi clt simpr cv cmul co wceq wrex simplr
|
|
a1d simplll simpllr dvdslelemd neneqd nrexdv nnzd divides syl2anc pm2.21d
|
|
wb simpll mtbird wo nnz zlelttric sylan2 mpjaodan ) ADEZBFEZGZABHIZABJIZV
|
|
AKBALIZUTVAGVAVBUTVAMTUTVCGZVBVAVDVBCNZAOPZBQZCDRZVDVGCDVDVEDEZGZVFBVJVEA
|
|
BURUSVCVIUAURUSVCVIUBVDVIMUTVCVISUCUDUEVDURBDEZVBVHUJURUSVCUKVDBURUSVCSUF
|
|
CABUGUHULUIUSURVKVAVCUMBUNABUOUPUQ $.
|
|
$}
|
|
|
|
$( The divisors of a nonzero integer are bounded by its absolute value.
|
|
Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides
|
|
relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened
|
|
by Fan Zheng, 3-Jul-2016.) $)
|
|
dvdsleabs $p |- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) ->
|
|
( M || N -> M <_ ( abs ` N ) ) ) $=
|
|
( cz wcel cc0 wne w3a cdvds wbr cabs cfv cle wb dvdsabsb 3adant3 wi nnabscl
|
|
wa cn dvdsle sylan2 3impb sylbid ) ACDZBCDZBEFZGABHIZABJKZHIZAUHLIZUDUEUGUI
|
|
MUFABNOUDUEUFUIUJPZUEUFRUDUHSDUKBQAUHTUAUBUC $.
|
|
|
|
$( Transfer divisibility to an order constraint on absolute values.
|
|
(Contributed by Stefan O'Rear, 24-Sep-2014.) $)
|
|
dvdsleabs2 $p |- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 )
|
|
-> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) ) $=
|
|
( cz wcel cc0 wne w3a cdvds wbr cabs cfv cle zabscl 3anim1i adantr absdvdsb
|
|
wa wb 3adant3 biimpa dvdsleabs sylc ex ) ACDZBCDZBEFZGZABHIZAJKZBJKLIZUGUHQ
|
|
UICDZUEUFGZUIBHIZUJUGULUHUDUKUEUFAMNOUGUHUMUDUEUHUMRUFABPSTUIBUAUBUC $.
|
|
|
|
$( If two integers divide each other, they must be equal, up to a difference
|
|
in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario
|
|
Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.) $)
|
|
dvdsabseq $p |- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) ) $=
|
|
( cdvds wbr cabs cfv wceq cz wcel wa wi cc0 wn simpr breq1 wb adantr expcom
|
|
0dvds adantl dvdszrcl zcn abs00ad bicomd bitrd sylan9bb fveq2 syl6eq eqeq2d
|
|
abs0 bitr4d syl5ib expd cle simprl neqne dvdsleabs2 syl3anc syl5rbbr eqeq1d
|
|
wne eqcom a1dd expcomd cr abscld letri3 syl2anr syl5bb biimprd syld a1d wdc
|
|
wo 0z zdceq mpan2 exmiddc syl mpjaod com34 mpdd mpcom imp ) ABCDZBACDZAEFZB
|
|
EFZGZAHIZBHIZJZWEWFWIKZABUAWLBLGZWEWMKZWNMZWNWLWOWNWLJZWEWFWIWEWFJWFWQWIWEW
|
|
FNWQWFWGLGZWIWNWFLACDZWLWRBLACOWLWSALGZWRWJWSWTPWKASQWJWTWRPWKWJWRWTWJAAUBZ
|
|
UCUDQUEUFWQWHLWGWNWHLGZWLWNWHLEFZLBLEUGUJUHQUIUKULUMRWPWLWOWPWLJZWEWGWHUNDZ
|
|
WMXDWJWKBLVAZWEXEKWPWJWKUOWLWKWPWJWKNZTWPXFWLBLUPQABUQURWLWEXEWMKKWPWLWEWFX
|
|
EWIWLWTWEWFXEWIKZKZKZWTMZWTWLXJWTWLJZWFWEXHXLWFWEJZWIXEXMWEXLWIWFWENXLWELWH
|
|
GZWIWTWELBCDZWLXNALBCOWKXOXNPWJWKXOWNXNBSXNXBWKWNWHLVBWKBBUBZUCUSUETUFXLWGL
|
|
WHWTWRWLWTWGXCLALEUGUJUHQUTUKULVCVDRXKWLXJXKWLJZXIWEXQWFWHWGUNDZXHXQWKWJALV
|
|
AZWFXRKWLWKXKXGTXKWJWKUOXKXSWLALUPQBAUQURWLXRXHKXKWLXRXEWIWLWIXRXEJZWIWHWGG
|
|
ZWLXTWGWHVBWKWHVEIWGVEIYAXTPWJWKBXPVFWJAXAVFWHWGVGVHVIVJUMTVKVLRWJWTXKVNZWK
|
|
WJWTVMZYBWJLHIZYCVOALVPVQWTVRVSQVTWATWBRWKWNWPVNZWJWKWNVMZYEWKYDYFVOBLVPVQW
|
|
NVRVSTVTWCWD $.
|
|
|
|
$( If two nonnegative integers divide each other, they must be equal.
|
|
(Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV,
|
|
7-Aug-2021.) $)
|
|
dvdseq $p |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( M || N /\ N || M ) ) ->
|
|
M = N ) $=
|
|
( cdvds wbr wa cn0 wcel cabs cfv wceq dvdsabseq nn0ge0 absidd adantr eqcomd
|
|
nn0re simpr ad2antlr 3eqtrd sylan2 ) ABCDBACDEAFGZBFGZEZAHIZBHIZJZABJABKUCU
|
|
FEAUDUEBUCAUDJUFUCUDAUAUDAJUBUAAAPALMNONUCUFQUBUEBJUAUFUBBBPBLMRST $.
|
|
|
|
${
|
|
$d M m $. $d N m $.
|
|
$( If a nonzero integer ` M ` divides another integer ` N ` , the other
|
|
integer ` N ` divided by the nonzero integer ` M ` (i.e. the _divisor
|
|
conjugate_ of ` N ` to ` M ` ) divides the other integer ` N ` .
|
|
Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV,
|
|
7-Aug-2021.) $)
|
|
divconjdvds $p |- ( ( M || N /\ M =/= 0 ) -> ( N / M ) || N ) $=
|
|
( vm cdvds wbr cc0 cdiv co cz wcel wa cmul wceq wb adantl cc adantr simpr
|
|
wne zcn wi dvdszrcl cv wrex simpll oveq1 eqeq1d 0z zapne mpan2 divcanap2d
|
|
cap biimpar rspcedvd w3a 3jca dvdsval2 mpbid divides syl2anc mpbird exp31
|
|
syl com3r mpd imp ) ABDEZAFSZBAGHZBDEZVGAIJZBIJZKZVHVJUAABUBVMVHVGVJVMVHV
|
|
GVJVMVHKZVGKZVJCUCZVILHZBMZCIUDZVNVSVGVNVRAVILHZBMZCAIVKVLVHUEZVPAMZVRWAN
|
|
VNWCVQVTBVPAVILUFUGOVNBAVMBPJZVHVLWDVKBTOQVMAPJZVHVKWEVLATQQVMAFULEZVHVKW
|
|
FVHNZVLVKFIJWGUHAFUIUJQUMUKUNQVOVIIJZVLVJVSNVOVGWHVNVGRVOVKVHVLUOZVGWHNVN
|
|
WIVGVNVKVHVLWBVMVHRVMVLVHVKVLRQZUPQABUQVCURVNVLVGWJQCVIBUSUTVAVBVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x N $.
|
|
$( The complement of a divisor of ` N ` is also a divisor of ` N ` .
|
|
(Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV,
|
|
9-Aug-2021.) $)
|
|
dvdsdivcl $p |- ( ( N e. NN /\ A e. { x e. NN | x || N } ) ->
|
|
( N / A ) e. { x e. NN | x || N } ) $=
|
|
( cn wcel cv cdvds crab wa cdiv co wi breq1 elrab nndivdvds biimpd expcom
|
|
wbr com23 imp cc0 wne nnne0 anim1i ancomd divconjdvds jctird sylbi impcom
|
|
syl sylibr ) CDEZBAFZCGRZADHZEZICBJKZDEZUQCGRZIZUQUOEUPULUTUPBDEZBCGRZIZU
|
|
LUTLUNVBABDUMBCGMNVCULURUSVAVBULURLVAULVBURULVAVBURLULVAIVBURCBOPQSTVCVBB
|
|
UAUBZIUSVCVDVBVAVDVBBUCUDUEBCUFUJUGUHUIUNUSAUQDUMUQCGMNUK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z N $.
|
|
dvdsflip.a $e |- A = { x e. NN | x || N } $.
|
|
dvdsflip.f $e |- F = ( y e. A |-> ( N / y ) ) $.
|
|
$( An involution of the divisors of a number. (Contributed by Stefan
|
|
O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro,
|
|
13-May-2016.) $)
|
|
dvdsflip $p |- ( N e. NN -> F : A -1-1-onto-> A ) $=
|
|
( vz cn wcel cv cdiv co wa eleq2i dvdsdivcl sylan2b wceq cc nncn wbr crab
|
|
cdvds syl6eleqr wb ssrab2 eqsstri anim12i adantr ad2antrl ad2antll simprr
|
|
sseli cmul nnap0d divmulap3d simprl divmulap2d bitr4d eqcom 3bitr4g f1o2d
|
|
sylan2 ) EIJZBHCCEBKZLMZEHKZLMZDGVDVECJZNVFAKEUCUAZAIUBZCVIVDVEVKJVFVKJCV
|
|
KVEFOAVEEPQFUDVDVGCJZNVHVKCVLVDVGVKJVHVKJCVKVGFOAVGEPQFUDVDVIVLNZNVHVERZV
|
|
FVGRZVEVHRVGVFRVMVDVEIJZVGIJZNZVNVOUEVIVPVLVQCIVECVKIFVJAIUFUGZUMCIVGVSUM
|
|
UHVDVRNZVNEVEVGUNMRVOVTEVEVGVDESJVRETUIZVPVESJVDVQVETUJZVQVGSJVDVPVGTUKZV
|
|
TVGVDVPVQULUOUPVTEVGVEWAWCWBVTVEVDVPVQUQUOURUSVCVEVHUTVGVFUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d A p $.
|
|
$( The set of divisors of a number is a subset of a finite set.
|
|
(Contributed by Mario Carneiro, 22-Sep-2014.) $)
|
|
dvdsssfz1 $p |- ( A e. NN -> { p e. NN | p || A } C_ ( 1 ... A ) ) $=
|
|
( cn wcel cv cdvds wbr c1 cfz co wi wral crab wss wa cle cz nnz id wb syl
|
|
dvdsle syl2anr ibar adantl adantr bitr4d sylibd ralrimiva rabss sylibr
|
|
fznn ) ACDZBEZAFGZUNHAIJZDZKZBCLUOBCMUPNUMURBCUMUNCDZOZUOUNAPGZUQUSUNQDUM
|
|
UOVAKUMUNRUMSUNAUBUCUTVAUSVAOZUQUSVAVBTUMUSVAUDUEUTAQDZUQVBTUMVCUSARUFUNA
|
|
ULUAUGUHUIUOBCUPUJUK $.
|
|
$}
|
|
|
|
$( The only nonnegative integer that divides 1 is 1. (Contributed by Mario
|
|
Carneiro, 2-Jul-2015.) $)
|
|
dvds1 $p |- ( M e. NN0 -> ( M || 1 <-> M = 1 ) ) $=
|
|
( cn0 wcel c1 cdvds wbr wceq wa simpl 1nn0 a1i simpr cz 1dvds adantr dvdseq
|
|
nn0z syl syl22anc ex id 1z iddvds ax-mp syl6eqbr impbid1 ) ABCZADEFZADGZUGU
|
|
HUIUGUHHZUGDBCZUHDAEFZUIUGUHIUKUJJKUGUHLUGULUHUGAMCULAQANROADPSTUIADDEUIUAD
|
|
MCDDEFUBDUCUDUEUF $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x N $.
|
|
$( Only 0 is divisible by all integers. (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
alzdvds $p |- ( N e. ZZ -> ( A. x e. ZZ x || N <-> N = 0 ) ) $=
|
|
( cz wcel cv cdvds wbr wral cc0 wceq wa wne wn wrex cn wb adantl ralrimiv
|
|
syl expcom cabs cfv cle clt wss nnssz zcn abscld arch ssrexv mpsyl zabscl
|
|
cr zltnle sylan rexbidva rexnalim syl6bi wi ralim dvdsleabs 3expb expdimp
|
|
mpd syl11 mtod wdc 0z zdceq mpan2 nnedc mpbid dvds0 breq2 syl5ibr impbid1
|
|
) BCDZAEZBFGZACHZBIJZVTVQWAVTVQKZBILZMZWAWBWCVRBUAUBZUCGZACHZVQWGMZVTVQWE
|
|
VRUDGZACNZWHOCUEVQWIAONZWJUFVQWEUMDWKVQBBUGUHWEAUISWIAOCUJUKVQWJWFMZACNWH
|
|
VQWIWLACVQWECDVRCDZWIWLPBULWEVRUNUOUPWFACUQURVDQVTVQWCWGVSWFUSZACHVTWGVQW
|
|
CKZVSWFACUTWOWNACWMWOWNWMVQWCWNVRBVAVBTRVEVCVFVQWDWAPZVTVQWAVGZWPVQICDWQV
|
|
HBIVIVJBIVKSQVLTWAVSACWMVSWAVRIFGVRVMBIVRFVNVORVP $.
|
|
|
|
$( Poset extensionality for division. (Contributed by Stefan O'Rear,
|
|
6-Sep-2015.) $)
|
|
dvdsext $p |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <->
|
|
A. x e. NN0 ( A || x <-> B || x ) ) ) $=
|
|
( cn0 wcel wa wceq cv cdvds wbr wb wral breq1 cz iddvds syl breq2 bibi12d
|
|
nn0z rspcva ralrimivw simpll simplr ad2antlr adantll mpbird adantlr mpbid
|
|
ad2antrr dvdseq syl22anc ex impbid2 ) BDEZCDEZFZBCGZBAHZIJZCURIJZKZADLZUQ
|
|
VAADBCURIMUAUPVBUQUPVBFZUNUOBCIJZCBIJZUQUNUOVBUBUNUOVBUCVCVDCCIJZUOVFUNVB
|
|
UOCNEVFCSCOPUDUOVBVDVFKZUNVAVGACDURCGUSVDUTVFURCBIQURCCIQRTUEUFVCBBIJZVEU
|
|
NVHUOVBUNBNEVHBSBOPUIUNVBVHVEKZUOVAVIABDURBGUSVHUTVEURBBIQURBCIQRTUGUHBCU
|
|
JUKULUM $.
|
|
$}
|
|
|
|
$( No number between ` 1 ` and ` M - 1 ` divides ` M ` . (Contributed by
|
|
Mario Carneiro, 24-Jan-2015.) $)
|
|
fzm1ndvds $p |- ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N ) $=
|
|
( cn wcel c1 cmin co cfz wa cdvds wbr cle wn elfzle2 adantl elfzelz syl2anc
|
|
clt cz wb nnz adantr zltlem1 mpbird elfznn nnzd zltnle mpbid wi dvdsle mtod
|
|
) ACDZBEAEFGZHGDZIZABJKZABLKZUOBARKZUQMZUOURBUMLKZUNUTULBEUMNOUOBSDZASDZURU
|
|
TTUNVAULBEUMPOULVBUNAUAUBZBAUCQUDUOVAVBURUSTUOBUNBCDZULBUMUEOZUFVCBAUGQUHUO
|
|
VBVDUPUQUIVCVEABUJQUK $.
|
|
|
|
$( Zero is the only one of the first ` A ` nonnegative integers that is
|
|
divisible by ` A ` . (Contributed by Stefan O'Rear, 6-Sep-2015.) $)
|
|
fzo0dvdseq $p |- ( B e. ( 0 ..^ A ) -> ( A || B <-> B = 0 ) ) $=
|
|
( cc0 cfzo co wcel cdvds wbr wceq wa wne wn cz syl2anc mpbid adantr cn0 syl
|
|
wb cn cle clt elfzolt2 elfzoelz elfzoel2 zltnle cdif elfzonn0 simpr eldifsn
|
|
wi csn sylanbrc dfn2 syl6eleqr dvdsle impancom mtod 0z zdceq mpan2 nnedc ex
|
|
wdc dvds0 breq2 syl5ibrcom impbid ) BCADEFZABGHZBCIZVIVJVKVIVJJZBCKZLZVKVLV
|
|
MABUAHZVIVOLZVJVIBAUBHZVPBCAUCVIBMFZAMFZVQVPSBCAUDZBCAUEZBAUFNOPVIVMVJVOVIV
|
|
MJZVSBTFVJVOUKVIVSVMWAPWBBQCULUGZTWBBQFZVMBWCFVIWDVMBAUHPVIVMUIBQCUJUMUNUOA
|
|
BUPNUQURVIVNVKSZVJVIVRWEVTVRVKVDZWEVRCMFWFUSBCUTVABCVBRRPOVCVIVJVKACGHZVIVS
|
|
WGWAAVERBCAGVFVGVH $.
|
|
|
|
$( Two different elements of a half-open range are not congruent mod its
|
|
length. (Contributed by Stefan O'Rear, 6-Sep-2015.) $)
|
|
fzocongeq $p |- ( ( A e. ( C ..^ D ) /\ B e. ( C ..^ D ) ) ->
|
|
( ( D - C ) || ( A - B ) <-> A = B ) ) $=
|
|
( cfzo co wcel wa cmin cdvds wbr cc0 wceq cz wb elfzoelz syl2an bitrd zcnd
|
|
cc cabs elfzoel2 elfzoel1 zsubcld adantl zsubcl dvdsabsb syl2anc fzo0dvdseq
|
|
cfv fzomaxdif syl subcl abs00ad subeq0 ) ACDEFZGZBUPGZHZDCIFZABIFZJKZVAUAUJ
|
|
ZLMZABMZUSVBUTVCJKZVDUSUTNGZVANGZVBVFOURVGUQURDCBCDUBBCDUCUDUEUQANGBNGVHURA
|
|
CDPZBCDPZABUFQUTVAUGUHUSVCLUTEFGVFVDOABCDUKUTVCUIULRUSVDVALMZVEUSVAUQATGZBT
|
|
GZVATGURUQAVISZURBVJSZABUMQUNUQVLVMVKVEOURVNVOABUOQRR $.
|
|
|
|
$( Two nonnegative integers less than the modulus are equal iff the sums of
|
|
these integer with another integer are equal modulo the modulus. Shorter
|
|
proof of ~ addmodlteq based on the "divides" relation. (Contributed by
|
|
AV, 14-Mar-2021.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
addmodlteqALT $p |- ( ( I e. ( 0 ..^ N ) /\ J e. ( 0 ..^ N ) /\ S e. ZZ )
|
|
-> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) ) $=
|
|
( cc0 co wcel cz w3a caddc cmo wceq cmin cdvds wbr wb wi wa syl cc cfzo cn0
|
|
cn clt elfzo0 elfzoelz simplrr nn0z ad2antrl sylan adantlr 3jca exp31 com12
|
|
zaddcl 3adant3 sylbi 3imp moddvds elfzoel2 zcn subid1d eqcomd 3ad2ant1 zcnd
|
|
pnpcan2 syl3an breq12d fzocongeq 3bitrd ) BEDUAFZGZCVKGZAHGZIZBAJFZDKFCAJFZ
|
|
DKFLZDVPVQMFZNOZDEMFZBCMFZNOZBCLZVODUCGZVPHGZVQHGZIZVRVTPVLVMVNWHVLBUBGZWEB
|
|
DUDOZIVMVNWHQZQZBDUEWIWEWLWJVMWIWERZWKVMCHGZWMWKQCEDUFZWNWMVNWHWNWMRZVNRWEW
|
|
FWGWNWIWEVNUGWPBHGZVNWFWIWQWNWEBUHUIBAUOUJWNVNWGWMCAUOUKULUMSUNUPUQURVPVQDU
|
|
SSVODWAVSWBNVLVMDWALZVNVLDHGZWRBEDUTWSWADWSDDVAVBVCSVDVLBTGVMCTGVNATGVSWBLV
|
|
LBBEDUFVEVMCWOVEAVABCAVFVGVHVLVMWCWDPVNBCEDVIUPVJ $.
|
|
|
|
${
|
|
$d K x y $. $d N x $.
|
|
$( A positive integer divides any greater factorial. (Contributed by Paul
|
|
Chapman, 28-Nov-2012.) $)
|
|
dvdsfac $p |- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) ) $=
|
|
( vx vy cfv wcel cn cfa cdvds wbr wi c1 wceq fveq2 breq2d imbi2d syl nnzd
|
|
co cz cuz caddc cmin cmul cn0 nnm1nn0 faccl nnz dvdsmul2 syl2anc breqtrrd
|
|
cv facnn2 wa adantl elnnuz uztrn sylan2b sylibr nnnn0d dvdsmultr1 syl3anc
|
|
a1i peano2zd facp1 sylibrd ex a2d uzind4 impcom ) BAUAEZFAGFZABHEZIJZVLAC
|
|
ULZHEZIJZKVLAAHEZIJZKZVLADULZHEZIJZKVLAWALUBSZHEZIJZKVLVNKCDABVOAMZVQVSVL
|
|
WGVPVRAIVOAHNOPVOWAMZVQWCVLWHVPWBAIVOWAHNOPVOWDMZVQWFVLWIVPWEAIVOWDHNOPVO
|
|
BMZVQVNVLWJVPVMAIVOBHNOPVTATFZVLAALUCSZHEZAUDSZVRIVLWMTFWKAWNIJVLWMVLWLUE
|
|
FWMGFAUFWLUGQRAUHZWMAUIUJAUMUKVCWAVKFZVLWCWFWPVLWCWFKWPVLUNZWCAWBWDUDSZIJ
|
|
ZWFWQWKWBTFWDTFWCWSKVLWKWPWOUOWQWBWQWAUEFZWBGFWQWAWQWALUAEZFZWAGFVLWPAXAF
|
|
XBAUPAWALUQURWAUPUSZUTZWAUGQRWQWAWQWAXCRVDAWBWDVAVBWQWEWRAIWQWTWEWRMXDWAV
|
|
EQOVFVGVHVIVJ $.
|
|
$}
|
|
|
|
$( A power divides a power with a greater exponent. (Contributed by Mario
|
|
Carneiro, 23-Feb-2014.) $)
|
|
dvdsexp $p |- ( ( A e. ZZ /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) ->
|
|
( A ^ M ) || ( A ^ N ) ) $=
|
|
( cz wcel cn0 cuz cfv w3a cexp co cmin cmul cdvds wbr simp1 uznn0sub zexpcl
|
|
3ad2ant3 syl2anc 3adant3 dvdsmul2 caddc zcnd simp2 expaddd cc nn0cnd npcand
|
|
eluzelcn oveq2d eqtr3d breqtrd ) ADEZBFEZCBGHEZIZABJKZACBLKZJKZURMKZACJKZNU
|
|
QUTDEZURDEZURVANOUQUNUSFEZVCUNUOUPPZUPUNVEUOBCQSZAUSRTUNUOVDUPABRUAUTURUBTU
|
|
QAUSBUCKZJKVAVBUQAUSBUQAVFUDUNUOUPUEZVGUFUQVHCAJUQCBUPUNCUGEUOBCUJSUQBVIUHU
|
|
IUKULUM $.
|
|
|
|
$( Any number ` K ` whose mod base ` N ` is divisible by a divisor ` P ` of
|
|
the base is also divisible by ` P ` . This means that primes will also be
|
|
relatively prime to the base when reduced ` mod N ` for any base.
|
|
(Contributed by Mario Carneiro, 13-Mar-2014.) $)
|
|
dvdsmod $p |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) ->
|
|
( P || ( K mod N ) <-> P || K ) ) $=
|
|
( cn wcel cz cdvds wbr cmo co cmin cc0 wceq syl syl3anc breq2d nnzd moddvds
|
|
cq wb w3a cdiv cfl cfv cmul clt simpl3 simpl2 nnq nngt0d modqval simpl1 znq
|
|
wa zq syl2anc flqcld simpr dvdsmultr1d zmulcld zcnd subid1d breqtrrd mpbird
|
|
0zd eqeq2d 3bitr3d 3bitrd ) ADEZCDEZBFEZUAZACGHZUNZABCIJZGHABCBCUBJZUCUDZUE
|
|
JZKJZGHZABLKJZGHZABGHVNVOVSAGVNBSEZCSEZLCUFHVOVSMVNVKWCVIVJVKVMUGZBUONVNVJW
|
|
DVIVJVKVMUHZCUINVNCWFUJBCUKOPVNBAIJZVRAIJZMZWGLAIJZMZVTWBVNWHWJWGVNWHWJMZAV
|
|
RLKJZGHZVNAVRWMGVNACVQVNAVIVJVKVMULZQVNCWFQZVNVPVNVKVJVPSEWEWFBCUMUPUQZVLVM
|
|
URUSVNVRVNVRVNCVQWPWQUTZVAVBVCVNVIVRFEZLFEZWLWNTWOWRVNVEZVRLAROVDVFVNVIVKWS
|
|
WIVTTWOWEWRBVRAROVNVIVKWTWKWBTWOWEXABLAROVGVNWABAGVNBVNBWEVAVBPVH $.
|
|
|
|
$( If an integer is divisible by a positive integer, the product of this
|
|
integer with another integer modulo the positive integer is 0.
|
|
(Contributed by Alexander van der Vekens, 30-Aug-2018.) $)
|
|
mulmoddvds $p |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A
|
|
-> ( ( A x. B ) mod N ) = 0 ) ) $=
|
|
( cn wcel cz w3a cdvds wbr cmul co cmo cc0 wceq wa syl adantr oveq1d eqtrd
|
|
cq clt simp2 zq simp3 simp1 nnq nngt0d modqmulmod syl22anc dvdsval3 3adant3
|
|
eqcomd wb biimpa zcnd mul02d q0mod syl2anc ex ) CDEZAFEZBFEZGZCAHIZABJKCLKZ
|
|
MNVCVDOZVEACLKZBJKZCLKZMVCVEVINVDVCVIVEVCATEZVBCTEZMCUAIZVIVENVCVAVJUTVAVBU
|
|
BAUCPUTVAVBUDZVCUTVKUTVAVBUEZCUFPZVCCVNUGZABCUHUIULQVFVIMBJKZCLKZMVFVHVQCLV
|
|
FVGMBJVCVDVGMNZUTVAVDVSUMVBCAUJUKUNRRVFVRMCLKZMVFVQMCLVFBVFBVCVBVDVMQUOUPRV
|
|
FVKVLVTMNVCVKVDVOQVCVLVDVPQCUQURSSSUS $.
|
|
|
|
${
|
|
3dvdsdec.a $e |- A e. NN0 $.
|
|
3dvdsdec.b $e |- B e. NN0 $.
|
|
$( A decimal number is divisible by three iff the sum of its two "digits"
|
|
is divisible by three. The term "digits" in its narrow sense is only
|
|
correct if ` A ` and ` B ` actually are digits (i.e. nonnegative
|
|
integers less than 10). However, this theorem holds for arbitrary
|
|
nonnegative integers ` A ` and ` B ` , especially if ` A ` is itself a
|
|
decimal number, e.g. ` A = ; C D ` . (Contributed by AV, 14-Jun-2021.)
|
|
(Revised by AV, 8-Sep-2021.) $)
|
|
3dvdsdec $p |- ( 3 || ; A B <-> 3 || ( A + B ) ) $=
|
|
( c3 cdc cdvds wbr c9 cmul co caddc c1 eqcomi oveq1i 9cn cz wcel 3z mp2an
|
|
dfdec10 9p1e10 ax-1cn nn0cni adddiri mulid2i oveq2i 3eqtri mulcli addassi
|
|
cc0 breq2i wa wb nn0zi zaddcl 9nn zmulcl dvdsmul1 3t3e9 3cn mulassi eqtri
|
|
nnzi breqtrri pm3.2i dvdsadd2b mp3an bitr4i ) EABFZGHEIAJKZABLKZLKZGHZEVL
|
|
GHZVJVMEGVJMUKFZAJKZBLKVKALKZBLKVMABUAVQVRBLVQIMLKZAJKVKMAJKZLKVRVPVSAJVS
|
|
VPUBNOIMAPUCACUDZUEVTAVKLAWAUFUGUHOVKABIAPWAUIWABDUDUJUHULEQRZVLQRZVKQRZE
|
|
VKGHZUMVOVNUNSAQRZBQRWCACUOZBDUOABUPTWDWEIQRWFWDIUQVDWGIAURTEEEAJKZJKZVKG
|
|
WBWHQRZEWIGHSWBWFWJSWGEAURTEWHUSTVKEEJKZAJKWIIWKAJWKIUTNOEEAVAVAWAVBVCVEV
|
|
FEVLVKVGVHVI $.
|
|
|
|
3dvds2dec.c $e |- C e. NN0 $.
|
|
$( A decimal number is divisible by three iff the sum of its three "digits"
|
|
is divisible by three. The term "digits" in its narrow sense is only
|
|
correct if ` A ` , ` B ` and ` C ` actually are digits (i.e. nonnegative
|
|
integers less than 10). However, this theorem holds for arbitrary
|
|
nonnegative integers ` A ` , ` B ` and ` C ` . (Contributed by AV,
|
|
14-Jun-2021.) (Revised by AV, 1-Aug-2021.) $)
|
|
3dvds2dec $p |- ( 3 || ; ; A B C <-> 3 || ( ( A + B ) + C ) ) $=
|
|
( c3 cdc cdvds c1 cmul co caddc c9 oveq1i nn0cni 3eqtri wcel cz mp2an wbr
|
|
cc0 c2 cexp 3dec sq10e99m1 9nn0 deccl ax-1cn adddiri oveq2i 9p1e10 eqcomi
|
|
mulid2i 9cn oveq12i cc wceq mulcli wa oveq1d mp4an addcli addassi 9t11e99
|
|
add4 1nn0 mulassi eqtri adddii 3t3e9 3cn breq2i wb 3z nn0zi zaddcl zmulcl
|
|
dvdsmul1 pm3.2i dvdsadd2b mp3an bitr4i ) GABHCHZIUAGGGJJHZAKLZBMLZKLZKLZA
|
|
BMLZCMLZMLZIUAZGWKIUAZWDWLGIWDJUBHZUCUDLZAKLZWOBKLZMLZCMLNNHZAKLZAMLZNBKL
|
|
ZBMLZMLZCMLZWLABCDEUEWSXECMWQXBWRXDMWQWTJMLZAKLXAJAKLZMLXBWPXGAKUFOWTJAWT
|
|
NNUGUGUHPZUIADPZUJXHAXAMAXJUNUKQWRNJMLZBKLXCJBKLZMLXDWOXKBKXKWOULUMONJBUO
|
|
UIBEPZUJXLBXCMBXMUNUKQUPOXFXAXCMLZWJMLZCMLZXNWKMLWLXAUQRZAUQRZXCUQRZBUQRZ
|
|
XFXPURWTAXIXJUSZXJNBUOXMUSZXMXQXRUTXSXTUTUTXEXOCMXAAXCBVFVAVBXNWJCXAXCYAY
|
|
BVCABXJXMVCCFPVDXNWIWKMXNNWFKLZXCMLZNWGKLZWIXAYCXCMXANWEKLZAKLYCWTYFAKYFW
|
|
TVEUMONWEAUOWEJJVGVGUHZPZXJVHVIOYEYDNWFBUOWEAYHXJUSZXMVJUMYEGGKLZWGKLWINY
|
|
JWGKYJNVKUMOGGWGVLVLWFBYIXMVCVHVIQOQQVMGSRZWKSRZWISRZGWIIUAZUTWNWMVNVOWJS
|
|
RZCSRYLASRZBSRZYOADVPZBEVPZABVQTCFVPWJCVQTYMYNYKWHSRZYMVOYKWGSRZYTVOWFSRZ
|
|
YQUUAWESRYPUUBWEYGVPYRWEAVRTYSWFBVQTGWGVRTZGWHVRTYKYTYNVOUUCGWHVSTVTGWKWI
|
|
WAWBWC $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Even and odd numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The set ` ZZ ` of integers can be partitioned into the set of even numbers and
|
|
the set of odd numbers, see ~ zeo4 . Instead of defining new class variables
|
|
Even and Odd to represent these sets, we use the idiom ` 2 || N ` to
|
|
say that "` N ` is _even_" (which implies ` N e. ZZ `, see ~ evenelz ) and
|
|
` -. 2 || N ` to say that "` N ` is _odd_" (under the assumption that
|
|
` N e. ZZ `). The previously proven theorems about even and odd numbers, like
|
|
~ zneo , ~ zeo , ~ zeo2 , etc. use different representations, which are
|
|
equivalent with the representations using the divides relation, see ~ evend2
|
|
and ~ oddp1d2 . The corresponding theorems are ~ zeneo , ~ zeo3 and ~ zeo4 .
|
|
$)
|
|
|
|
$( An even number is an integer. This follows immediately from the reverse
|
|
closure of the divides relation, see ~ dvdszrcl . (Contributed by AV,
|
|
22-Jun-2021.) $)
|
|
evenelz $p |- ( 2 || N -> N e. ZZ ) $=
|
|
( c2 cdvds wbr cz wcel dvdszrcl simprd ) BACDBEFAEFBAGH $.
|
|
|
|
$( An integer is even or odd. (Contributed by AV, 17-Jun-2021.) $)
|
|
zeo3 $p |- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) ) $=
|
|
( cz wcel c2 cdvds wbr wn c1 cmin co cdiv caddc wo peano2zm zeo syl wb zeo2
|
|
zcn 1cnd npcand oveq1d eleq1d cc0 wne 2z 2ne0 dvdsval2 mp3an12 bitr4d bitrd
|
|
notbid orbi12d mpbid orcomd ) ABCZDAEFZGZUQUPAHIJZDKJBCZUSHLJZDKJZBCZMZURUQ
|
|
MUPUSBCZVDANZUSOPUPUTURVCUQUPUTVCGZURUPVEUTVGQVFUSRPUPVCUQUPVCADKJZBCZUQUPV
|
|
BVHBUPVAADKUPAHASUPTUAUBUCDBCDUDUEUPUQVIQUFUGDAUHUIUJZULUKVJUMUNUO $.
|
|
|
|
$( An integer is even or odd but not both. (Contributed by Jim Kingdon,
|
|
10-Nov-2021.) $)
|
|
zeoxor $p |- ( N e. ZZ -> ( 2 || N \/_ -. 2 || N ) ) $=
|
|
( cz wcel c2 cdvds wbr wn wo wxo zeo3 wa pm3.24 df-xor mpbiran2 sylibr ) AB
|
|
CDAEFZPGZHZPQIZAJSRPQKGPLPQMNO $.
|
|
|
|
$( An integer is even or odd but not both. (Contributed by AV,
|
|
17-Jun-2021.) $)
|
|
zeo4 $p |- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) ) $=
|
|
( cz wcel c2 cdvds wbr wn wxo wb zeoxor xorbin syl ) ABCDAEFZMGZHMNGIAJMNKL
|
|
$.
|
|
|
|
$( No even integer equals an odd integer (i.e. no integer can be both even
|
|
and odd). Exercise 10(a) of [Apostol] p. 28. This variant of ~ zneo
|
|
follows immediately from the fact that a contradiction implies anything,
|
|
see ~ pm2.21i . (Contributed by AV, 22-Jun-2021.) $)
|
|
zeneo $p |- ( ( A e. ZZ /\ B e. ZZ )
|
|
-> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) ) $=
|
|
( c2 cdvds wbr wn wa wne wi cz wcel nbrne1 a1i ) CADECBDEFGABHIAJKBJKGCABDL
|
|
M $.
|
|
|
|
${
|
|
$d j k $. $d j m $. $d j n $. $d j x $. $d j y $. $d k m $. $d k y $.
|
|
$d m n $. $d m x $. $d m y $. $d N j $. $d N k $. $d N n $. $d n x $.
|
|
$d n y $. $d k x $.
|
|
$( Lemma for ~ odd2np1 . (Contributed by Scott Fenton, 3-Apr-2014.)
|
|
(Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
odd2np1lem $p |- ( N e. NN0 -> (
|
|
E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) ) $=
|
|
( vy c2 cv cmul co c1 caddc wceq cz wrex wo cc0 eqeq2 rexbidv eqeq1d wcel
|
|
2cn vj vx vm orbi12d weq oveq2 oveq1d syl6bb oveq1 0z mul02i rspcev mp2an
|
|
cbvrexv olci cn0 orcom wa cc zcn mulcom sylancl adantl wi mpan2 syl5ibcom
|
|
eqid eqeq2d sylbid rexlimdva peano2z mulid2i oveq12d oveq2i syl6eq ax-1cn
|
|
a1i df-2 adddir mp3an23 mpan addass 3eqtr4d syl2anc orim12d syl5bi nn0ind
|
|
mulcl syl ) EBFZGHZIJHZUAFZKZBLMZAFZEGHZWMKZALMZNWLOKZBLMZWQOKZALMZNEUBFZ
|
|
GHZIJHZUCFZKZUBLMZDFZEGHZXGKZDLMZNZWLXGIJHZKZBLMZWQXOKZALMZNZWLCKZBLMZWQC
|
|
KZALMZNUAUCCWMOKZWOXAWSXCYEWNWTBLWMOWLPQYEWRXBALWMOWQPQUDUAUCUEZWOXIWSXMY
|
|
FWOWLXGKZBLMXIYFWNYGBLWMXGWLPQYGXHBUBLBUBUEZWLXFXGYHWKXEIJWJXDEGUFUGRUNUH
|
|
YFWSWQXGKZALMXMYFWRYIALWMXGWQPQYIXLADLADUEWQXKXGWPXJEGUIRUNUHUDWMXOKZWOXQ
|
|
WSXSYJWNXPBLWMXOWLPQYJWRXRALWMXOWQPQUDWMCKZWOYBWSYDYKWNYABLWMCWLPQYKWRYCA
|
|
LWMCWQPQUDXCXAOLSOEGHZOKZXCUJETUKXBYMAOLWPOKWQYLOWPOEGUIRULUMUOXNXMXINXGU
|
|
PSZXTXIXMUQYNXMXQXIXSYNXLXQDLYNXJLSZURZXLEXJGHZXGKZXQYPXKYQXGYOXKYQKZYNYO
|
|
XJUSSEUSSZYSXJUTTXJEVAVBVCRYOYRXQVDYNYOWLYQIJHZKZBLMZYRXQYOUUAUUAKZUUCUUA
|
|
VGUUBUUDBXJLBDUEZWLUUAUUAUUEWKYQIJWJXJEGUFUGRULVEYRUUBXPBLYRUUAXOWLYQXGIJ
|
|
UIVHQVFVCVIVJYNXHXSUBLYNXDLSZURZWQXFIJHZKZALMZXHXSUUGXDIJHZLSZUUKEGHZUUHK
|
|
ZUUJUUFUULYNXDVKVCUUFUUNYNUUFXDUSSZUUNXDUTUUOXDEGHZIEGHZJHZXEIIJHZJHZUUMU
|
|
UHUUOUURXEEJHUUTUUOUUPXEUUQEJUUOYTUUPXEKTXDEVAVEUUQEKUUOETVLVQVMEUUSXEJVR
|
|
VNVOUUOIUSSZYTUUMUURKVPTXDIEVSVTUUOXEUSSZUUHUUTKZYTUUOUVBTEXDWHWAUVBUVAUV
|
|
AUVCVPVPXEIIWBVTWIWCWIVCUUIUUNAUUKLWPUUKKWQUUMUUHWPUUKEGUIRULWDXHUUIXRALX
|
|
HUUHXOWQXFXGIJUIVHQVFVJWEWFWG $.
|
|
$}
|
|
|
|
${
|
|
$d k n x y N $.
|
|
$( An integer is odd iff it is one plus twice another integer.
|
|
(Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro,
|
|
19-Apr-2014.) $)
|
|
odd2np1 $p |- ( N e. ZZ ->
|
|
( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) ) $=
|
|
( vk cz wcel c2 cmul co wceq wrex c1 caddc wb wa cneg syl 2cn cmin ax-1cn
|
|
cc vx vy cdvds wbr wn cv 2z divides notbid wo cr elznn0 odd2np1lem adantl
|
|
mpan cn0 peano2z znegcl ad2antlr zcn mulcl peano2cn simpl negcon2 syl2anc
|
|
recnd eqcom sylancr mulcli mp3an23 2t1e2 oveq1i 2m1e1 eqtri syl6req adddi
|
|
addsubass oveq2i mp3an13 oveq1d eqtr4d negeqd zcnd mulneg2 sylancl eqeq1d
|
|
negsubdi syl5bb bitrd biimpa oveq2 rspcev rexlimdva recn syl2anr syl6rbbr
|
|
ex mulneg1 oveq1 orim12d syl5 imp jaodan sylbi halfnz reeanv eqtr3 mulcom
|
|
cdiv eqeq2d subadd mp3an3 subcl cc0 cap 2ap0 pm3.2i divmulap ancoms subdi
|
|
w3a mp3an1 syl2an wi zsubcl syl5ibcom sylbird sylbid rexlimivv sylbir mto
|
|
eleq1 wxo df-xor xorbin bicomd ) BDEZFBUCUDZUECUFZFGHZBIZCDJZUEZFAUFZGHZK
|
|
LHZBIZADJZYQYRUUBFDEYQYRUUBMUGCFBUHUOUIYQUUHUUBUJZUUHUUBNZUEZUUCUUHMYQBUK
|
|
EZBUPEZBOZUPEZUJNUUIBULUULUUMUUIUUOUUMUUIUULCABUMUNUULUUOUUIUUOFUAUFZGHZK
|
|
LHZUUNIZUADJZUBUFZFGHZUUNIZUBDJZUJUULUUIUBUAUUNUMUULUUTUUHUVDUUBUULUUSUUH
|
|
UADUULUUPDEZNZUUSUUHUVFUUSNUUPKLHZOZDEZFUVHGHZKLHZBIZUUHUVEUVIUULUUSUVEUV
|
|
GDEUVIUUPUQZUVGURPUSUVFUUSUVLUVFUUSBUUROZIZUVLUVFUURTEZBTEZUUSUVOMUVEUVPU
|
|
ULUVEUUPTEZUVPUUPUTZUVRUUQTEZUVPFTEZUVRUVTQFUUPVAZUOUUQVBPPUNUVFBUULUVEVC
|
|
VFUURBVDVEUVOUVNBIUVFUVLBUVNVGUVFUVNUVKBUVEUVNUVKIUULUVEUVNFUVGGHZKRHZOZU
|
|
VKUVEUURUWDUVEUURUUQFKGHZLHZKRHZUWDUVEUWHUUQUWFKRHZLHZUURUVEUVTUWHUWJIZUV
|
|
EUWAUVRUVTQUVSUWBVHUVTUWFTEKTEZUWKFKQSVISUUQUWFKVQVJPUWIKUUQLUWIFKRHKUWFF
|
|
KRVKVLVMVNVRVOUVEUWCUWGKRUVEUVRUWCUWGIZUVSUWAUVRUWLUWMQSFUUPKVPVSPVTWAWBU
|
|
VEUVKUWCOZKLHZUWEUVEUVJUWNKLUVEUWAUVGTEZUVJUWNIQUVEUVGUVMWCZFUVGWDVHVTUVE
|
|
UWCTEZUWLUWEUWOIUVEUWAUWPUWRQUWQFUVGVAVHSUWCKWGWEWAWAUNWFWHWIWJUUGUVLAUVH
|
|
DUUDUVHIZUUFUVKBUWSUUEUVJKLUUDUVHFGWKVTWFWLVEWQWMUULUVCUUBUBDUULUVADEZNZU
|
|
VCUUBUXAUVCNUVAOZDEZUXBFGHZBIZUUBUWTUXCUULUVCUVAURUSUXAUVCUXEUXAUVCBUVBOZ
|
|
IZUXEUWTUVBTEZUVQUVCUXGMUULUWTUVATEZUWAUXHUVAUTZQUVAFVAWEBWNUVBBVDWOUXAUX
|
|
EUXFBIUXGUXAUXDUXFBUWTUXDUXFIZUULUWTUXIUWAUXKUXJQUVAFWRWEUNWFBUXFVGWPWIWJ
|
|
UUAUXECUXBDYSUXBIYTUXDBYSUXBFGWSWFWLVEWQWMWTXAXBXCXDUUJKFXIHZDEZXEUUJUUGU
|
|
UANZCDJADJUXMUUGUUAACDDXFUXNUXMACDDUXNUUFYTIZUUDDEZYSDEZNZUXMUUFYTBXGUXRU
|
|
XOUUFFYSGHZIZUXMUXQUXOUXTMUXPUXQYTUXSUUFUXQYSTEZUWAYTUXSIYSUTZQYSFXHWEXJU
|
|
NUXRUXTUXSUUERHZKIZUXMUXQUXSTEZUUETEZUYDUXTMZUXPUXQUWAUYAUYEQUYBFYSVAVHUX
|
|
PUWAUUDTEZUYFQUUDUTZFUUDVAVHUYEUYFUWLUYGSUXSUUEKXKXLWOUXRUYDYSUUDRHZUXLIZ
|
|
UXMUXPUYHUYAUYKUYDMUXQUYIUYBUYHUYANZUYKFUYJGHZKIZUYDUYAUYHUYKUYNMZUYAUYHN
|
|
UYJTEZUYOYSUUDXMUWLUYPUWAFXNXOUDZNZUYOSUWAUYQQXPXQUYKUXLUYJIUWLUYPUYRYAUY
|
|
NUYJUXLVGKUYJFXRWHVSPXSUYLUYMUYCKUYAUYHUYMUYCIZUWAUYAUYHUYSQFYSUUDXTYBXSW
|
|
FWIYCUXQUXPUYKUXMYDUXQUXPNUYJDEUYKUXMYSUUDYEUYJUXLDYLYFXSYGYGYHXAYIYJYKUU
|
|
IUUKNZUUHUUCUYTUUHUUBYMUUHUUCMUUHUUBYNUUHUUBYOYJYPWEWI $.
|
|
|
|
$( An integer is even iff it is twice another integer. (Contributed by AV,
|
|
25-Jun-2020.) $)
|
|
even2n $p |- ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N ) $=
|
|
( c2 cdvds wbr cz wcel cv cmul co wceq wrex evenelz wa a1i zmulcld adantr
|
|
2z id wb eleq1 adantl mpbid rexlimiva divides 2cnd mulcomd eqeq1d rexbiia
|
|
zcn syl6bb mpan pm5.21nii ) CBDEZBFGZCAHZIJZBKZAFLZBMURUOAFUPFGZURNUQFGZU
|
|
OUTVAURUTCUPCFGZUTROUTSPQURVAUOTUTUQBFUAUBUCUDVBUOUNUSTRVBUONUNUPCIJZBKZA
|
|
FLUSACBUEVDURAFUTVCUQBUTUPCUPUJUTUFUGUHUIUKULUM $.
|
|
|
|
$( An integer is odd iff its predecessor is even. (Contributed by Mario
|
|
Carneiro, 5-Sep-2016.) $)
|
|
oddm1even $p |- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N - 1 ) ) ) $=
|
|
( vn cz wcel c2 cv cmul co c1 caddc wceq wrex cmin cdvds wn wa simpl zcnd
|
|
wbr 1cnd 2cnd simpr mulcld subadd2d mulcomd eqeq1d syl5bb bitr3d rexbidva
|
|
eqcom odd2np1 wb 2z peano2zm divides sylancr 3bitr4d ) ACDZEBFZGHZIJHAKZB
|
|
CLUSEGHZAIMHZKZBCLZEANSOEVCNSZURVAVDBCURUSCDZPZVCUTKZVAVDVHAIUTVHAURVGQRV
|
|
HTVHEUSVHUAZVHUSURVGUBRZUCUDVIUTVCKVHVDVCUTUJVHUTVBVCVHEUSVJVKUEUFUGUHUIB
|
|
AUKURECDVCCDVFVEULUMAUNBEVCUOUPUQ $.
|
|
|
|
$( An integer is odd iff its successor is even. (Contributed by Mario
|
|
Carneiro, 5-Sep-2016.) $)
|
|
oddp1even $p |- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N + 1 ) ) ) $=
|
|
( cz wcel c2 cdvds wn c1 cmin co caddc oddm1even peano2zm dvdsadd sylancr
|
|
wbr wb 2z 2cnd zcn 1cnd addsub12d 2m1e1 oveq2i syl6eq breq2d 3bitrd ) ABC
|
|
ZDAEOFDAGHIZEOZDDUHJIZEOZDAGJIZEOAKUGDBCUHBCUIUKPQALDUHMNUGUJULDEUGUJADGH
|
|
IZJIULUGDAGUGRASUGTUAUMGAJUBUCUDUEUF $.
|
|
|
|
$d n A $.
|
|
$( The exponential of the negative of a number, when the exponent is odd.
|
|
(Contributed by Mario Carneiro, 25-Apr-2015.) $)
|
|
oexpneg $p |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) ->
|
|
( -u A ^ N ) = -u ( A ^ N ) ) $=
|
|
( vn wcel c2 wbr cmul co c1 wceq cneg cz syl wa cn0 oveq1d cc0 a1i eqtr3d
|
|
cexp cc cn cdvds wn w3a caddc wrex nnz odd2np1 biimpa 3adant1 simpl1 cmin
|
|
cv wb simprr simpl2 nncnd 1cnd simprl zmulcl sylancr zcnd subadd2d mpbird
|
|
2z nnm1nn0 eqeltrrd expcld mulneg2d sqneg negcld cle cr clt 2re zred 2pos
|
|
nn0ge0d prodge0 syl22anc elnn0z 2nn0 expmuld 3eqtr4d expp1d oveq2d negeqd
|
|
sylanbrc rexlimddv ) AUADZBUBDZEBUCFUDZUEZECUNZGHZIUFHZBJZAKZBTHZABTHZKZJ
|
|
CLWLWMWRCLUGZWKWLWMXCWLBLDWMXCUOBUHCBUIMUJUKWNWOLDZWRNZNZAWPTHZAGHZKZWTXB
|
|
XFXGWSGHZXIWTXFXGAXFAWPWKWLWMXEULZXFBIUMHZWPOXFXLWPJWRWNXDWRUPZXFBIWPXFBW
|
|
KWLWMXEUQZURXFUSXFWPXFELDXDWPLDVFWNXDWRUTZEWOVAVBVCVDVEXFWLXLODXNBVGMVHZV
|
|
IXKVJXFWSWPTHZWSGHZXJWTXFXQXGWSGXFWSETHZWOTHAETHZWOTHXQXGXFXSXTWOTXFWKXSX
|
|
TJXKAVKMPXFWSEWOXFAXKVLZXFXDQWOVMFZWOODXOXFEVNDZWOVNDQEVOFZQWPVMFYBYCXFVP
|
|
RXFWOXOVQYDXFVRRXFWPXPVSEWOVTWAWOWBWIZEODXFWCRZWDXFAEWOXKYEYFWDWEPXFWSWQT
|
|
HXRWTXFWSWPYAXPWFXFWQBWSTXMWGSSSXFXHXAXFAWQTHXHXAXFAWPXKXPWFXFWQBATXMWGSW
|
|
HSWJ $.
|
|
$}
|
|
|
|
$( An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example
|
|
2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) $)
|
|
mod2eq0even $p |- ( N e. ZZ -> ( ( N mod 2 ) = 0 <-> 2 || N ) ) $=
|
|
( cz wcel c2 cdvds wbr cmo co cc0 wceq cn wb 2nn dvdsval3 mpan bicomd ) ABC
|
|
ZDAEFZADGHIJZDKCQRSLMDANOP $.
|
|
|
|
$( An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see
|
|
example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) $)
|
|
mod2eq1n2dvds $p |- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) ) $=
|
|
( cz wcel c2 cmo co cc0 wceq c1 cdvds wn wb wa pm13.181 mpan2 neneqd adantl
|
|
wbr wne cfz 0ne1 mod2eq0even biimpa notnotd 2falsed simpr 1ne0 notbid mpbid
|
|
adantr 2thd wo cmin 2nn zmodfz 2m1e1 oveq2i syl6eleq fz01or sylib mpjaodan
|
|
cn ) ABCZADEFZGHZVDIHZDAJRZKZLVFVCVEMZVFVHVEVFKVCVEVDIVEGISVDISUAVDGINOPQVI
|
|
VGVCVEVGAUBZUCUDUEVCVFMZVFVHVCVFUFVKVEKZVHVFVLVCVFVDGVFIGSVDGSUGVDIGNOPQVCV
|
|
LVHLVFVCVEVGVJUHUJUIUKVCVDGITFZCVEVFULVCVDGDIUMFZTFZVMVCDVBCVDVOCUNADUOOVNI
|
|
GTUPUQURVDUSUTVA $.
|
|
|
|
${
|
|
$d N n $.
|
|
$( A nonnegative integer is odd iff it is one plus twice another
|
|
nonnegative integer. (Contributed by AV, 19-Jun-2021.) $)
|
|
oddnn02np1 $p |- ( N e. NN0 -> ( -. 2 || N
|
|
<-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = N ) ) $=
|
|
( cn0 wcel cv c2 cmul co c1 caddc wceq wa cz wrex wbr wi cc0 cle elnn0z
|
|
wb cdvds wn 2tnp1ge0ge0 biimpd imdistani expcom syl6ibr simplbiim syl6bir
|
|
eleq1 com13 impcom pm4.71rd bicomd rexbidva wss nn0ssz rexss mp1i odd2np1
|
|
nn0z syl 3bitr4rd ) BCDZAEZCDZFVEGHIJHZBKZLZAMNZVHAMNZVHACNZFBUAOUBZVDVIV
|
|
HAMVDVEMDZLZVHVIVOVHVFVNVDVHVFPVHVDVNVFVHVDVGCDZVNVFPZVGBCUJVPVGMDQVGROZV
|
|
QVGSVRVNVNQVEROZLZVFVNVRVTVNVRVSVNVRVSVEUCUDUEUFVESUGUHUIUKULUMUNUOCMUPVL
|
|
VJTVDUQVHACMURUSVDBMDVMVKTBVAABUTVBVC $.
|
|
|
|
$( An integer greater than one is odd iff it is one plus twice a positive
|
|
integer. (Contributed by AV, 16-Aug-2021.) $)
|
|
oddge22np1 $p |- ( N e. ( ZZ>= ` 2 ) -> ( -. 2 || N
|
|
<-> E. n e. NN ( ( 2 x. n ) + 1 ) = N ) ) $=
|
|
( c2 wcel cn co c1 wa cn0 wrex wbr wi cz cc0 clt cle cr 2re a1i wb cuz cv
|
|
cfv cmul caddc wceq cdvds eleq1 nn0z adantl w3a eluz2 cmin 1red nn0mulcld
|
|
wn 2nn0 id nn0red lesubaddd 2m1e1 breq1i cdiv nn0re 2pos ledivmul syl3anc
|
|
pm3.2i halfgt0 0red halfre ltletr mpani sylbird syl5bi com12 3ad2ant3 imp
|
|
sylbi elnnz sylanbrc ex syl6bir com13 impcom pm4.71rd bicomd rexbidva wss
|
|
nnssnn0 rexss mp1i eluzge2nn0 oddnn02np1 syl 3bitr4rd ) BCUAUCZDZAUBZEDZC
|
|
WSUDFZGUEFZBUFZHZAIJZXCAIJZXCAEJZCBUGKUPZWRXDXCAIWRWSIDZHZXCXDXJXCWTXIWRX
|
|
CWTLXCWRXIWTXCWRXBWQDZXIWTLXBBWQUHXKXIWTXKXIHWSMDZNWSOKZWTXIXLXKWSUIUJXKX
|
|
IXMXKCMDZXBMDZCXBPKZUKXIXMLZCXBULXPXNXQXOXIXPXMXIXPCGUMFZXAPKZXMXICGXACQD
|
|
ZXIRSXIUNZXIXAXICWSCIDXIUQSXIURUOUSUTXSGXAPKZXIXMXRGXAPVAVBXIYBGCVCFZWSPK
|
|
ZXMXIGQDWSQDZXTNCOKZHZYDYBTYAWSVDZYGXIXTYFRVEVHSGWSCVFVGXINYCOKZYDXMVIXIN
|
|
QDYCQDZYEYIYDHXMLXIVJYJXIVKSYHNYCWSVLVGVMVNVOVNVPVQVSVRWSVTWAWBWCWDWEWFWG
|
|
WHEIWIXGXETWRWJXCAEIWKWLWRBIDXHXFTBWMABWNWOWP $.
|
|
|
|
$( A nonnegative integer is even iff it is twice another nonnegative
|
|
integer. (Contributed by AV, 12-Aug-2021.) $)
|
|
evennn02n $p |- ( N e. NN0
|
|
-> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) ) $=
|
|
( cn0 wcel cv c2 cmul co wceq wa cz wrex cdvds wbr wi cc0 cle cr a1i wb
|
|
eleq1 simpr clt 2re zre adantl 2pos nn0ge0 adantr prodge0 syl22anc elnn0z
|
|
sylanbrc ex syl6bir com13 impcom pm4.71rd bicomd rexbidva wss nn0ssz mp1i
|
|
rexss even2n 3bitr4rd ) BCDZAEZCDZFVHGHZBIZJZAKLZVKAKLZVKACLZFBMNZVGVLVKA
|
|
KVGVHKDZJZVKVLVRVKVIVQVGVKVIOVKVGVQVIVKVGVJCDZVQVIOVJBCUAVSVQVIVSVQJZVQPV
|
|
HQNZVIVSVQUBVTFRDZVHRDZPFUCNZPVJQNZWAWBVTUDSVQWCVSVHUEUFWDVTUGSVSWEVQVJUH
|
|
UIFVHUJUKVHULUMUNUOUPUQURUSUTCKVAVOVMTVGVBVKACKVDVCVPVNTVGABVESVF $.
|
|
|
|
$( A positive integer is even iff it is twice another positive integer.
|
|
(Contributed by AV, 12-Aug-2021.) $)
|
|
evennn2n $p |- ( N e. NN -> ( 2 || N <-> E. n e. NN ( 2 x. n ) = N ) ) $=
|
|
( cn wcel cv c2 cmul co wceq wa cz wrex cdvds wbr wi cc0 clt cr a1i wb ex
|
|
eleq1 simpr cle 2re zre adantl 0le2 nngt0 prodgt0 syl22anc elnnz sylanbrc
|
|
adantr syl6bir com13 impcom pm4.71rd bicomd rexbidva wss nnssz rexss mp1i
|
|
even2n 3bitr4rd ) BCDZAEZCDZFVHGHZBIZJZAKLZVKAKLZVKACLZFBMNZVGVLVKAKVGVHK
|
|
DZJZVKVLVRVKVIVQVGVKVIOVKVGVQVIVKVGVJCDZVQVIOVJBCUBVSVQVIVSVQJZVQPVHQNZVI
|
|
VSVQUCVTFRDZVHRDZPFUDNZPVJQNZWAWBVTUESVQWCVSVHUFUGWDVTUHSVSWEVQVJUIUNFVHU
|
|
JUKVHULUMUAUOUPUQURUSUTCKVAVOVMTVGVBVKACKVCVDVPVNTVGABVESVF $.
|
|
$}
|
|
|
|
${
|
|
$d A k $.
|
|
$( A number which is twice an integer increased by 1 is odd. (Contributed
|
|
by AV, 16-Jul-2021.) $)
|
|
2tp1odd $p |- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B ) $=
|
|
( vk cz wcel c2 cmul co c1 caddc wceq wa cdvds wbr wn cv wrex id adantl
|
|
wb oveq2 oveq1d eqeq1d eqidd rspcedvd 2z a1i zmulcld peano2zd odd2np1 syl
|
|
mpbird adantr breq2 mtbird ) ADEZBFAGHZIJHZKZLFBMNZFURMNZUPVAOZUSUPVBFCPZ
|
|
GHZIJHZURKZCDQZUPVFURURKZCADUPRZVCAKZVFVHTUPVJVEURURVJVDUQIJVCAFGUAUBUCSU
|
|
PURUDUEUPURDEVBVGTUPUQUPFAFDEUPUFUGVIUHUICURUJUKULUMUSUTVATUPBURFMUNSUO
|
|
$.
|
|
$}
|
|
|
|
$( An integer multiplied with its successor divided by 2 yields an integer,
|
|
i.e. an integer multiplied with its successor is even. (Contributed by
|
|
AV, 19-Jul-2021.) $)
|
|
mulsucdiv2z $p |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) $=
|
|
( c2 cdiv co cz wcel c1 caddc wo cmul wi wa peano2z zmulcl wb eleq1d adantl
|
|
zeo mpbird ex sylan2 zcn zcnd 2cnd cc0 cap wbr 2ap0 div23apd divassapd jaoi
|
|
a1i ancoms mpcom ) ABCDZEFZAGHDZBCDZEFZIAEFZAUQJDBCDZEFZARUPUTVBKUSUPUTVBUP
|
|
UTLVBUOUQJDZEFZUTUPUQEFVDAMZUOUQNUAUTVBVDOUPUTVAVCEUTAUQBAUBZUTUQVEUCZUTUDZ
|
|
BUEUFUGUTUHULZUIPQSTUSUTVBUSUTLVBAURJDZEFZUTUSVKAURNUMUTVBVKOUSUTVAVJEUTAUQ
|
|
BVFVGVHVIUJPQSTUKUN $.
|
|
|
|
${
|
|
$d N k $.
|
|
$( A squared odd number minus 1 divided by 8 is an integer. (Contributed
|
|
by AV, 19-Jul-2021.) $)
|
|
sqoddm1div8z $p |- ( ( N e. ZZ /\ -. 2 || N )
|
|
-> ( ( ( N ^ 2 ) - 1 ) / 8 ) e. ZZ ) $=
|
|
( vk cz wcel c2 cdvds wbr wn wa cv cmul co caddc wceq wrex cexp cmin cdiv
|
|
c1 c8 odd2np1 biimpa eqcom sqoddm1div8 adantll mulsucdiv2z eqeltrd syl5bi
|
|
ad2antlr ex rexlimdva mpd ) ACDZEAFGHZIZEBJZKLSMLZANZBCOZAEPLSQLTRLZCDZUM
|
|
UNUSBAUAUBUOURVABCURAUQNZUOUPCDZIZVAUQAUCVDVBVAVDVBIUTUPUPSMLKLERLZCVCVBU
|
|
TVENUOAUPUDUEVCVECDUOVBUPUFUIUGUJUHUKUL $.
|
|
$}
|
|
|
|
$( A number which is twice an integer is even. (Contributed by AV,
|
|
16-Jul-2021.) $)
|
|
2teven $p |- ( ( A e. ZZ /\ B = ( 2 x. A ) ) -> 2 || B ) $=
|
|
( cz wcel c2 cmul co wceq wa cdvds wbr 2z dvdsmul1 mpan adantr breq2 adantl
|
|
wb mpbird ) ACDZBEAFGZHZIEBJKZEUAJKZTUDUBECDTUDLEAMNOUBUCUDRTBUAEJPQS $.
|
|
|
|
$( An integer is either even or odd, version of ~ zeo3 avoiding the negation
|
|
of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.)
|
|
(Contributed by AV, 26-Jun-2020.) $)
|
|
zeo5 $p |- ( N e. ZZ -> ( 2 || N \/ 2 || ( N + 1 ) ) ) $=
|
|
( cz wcel c2 cdvds wbr c1 caddc co wo zeo3 oddp1even bicomd orbi2d mpbird
|
|
wn ) ABCZDAEFZDAGHIEFZJRRPZJAKQSTRQTSALMNO $.
|
|
|
|
$( An integer is even iff its quotient with 2 is an integer. This is a
|
|
representation of even numbers without using the divides relation, see
|
|
~ zeo and ~ zeo2 . (Contributed by AV, 22-Jun-2021.) $)
|
|
evend2 $p |- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) ) $=
|
|
( c2 cz wcel cc0 wne cdvds wbr cdiv co wb 2z 2ne0 dvdsval2 mp3an12 ) BCDBEF
|
|
ACDBAGHABIJCDKLMBANO $.
|
|
|
|
$( An integer is odd iff its successor divided by 2 is an integer. This is a
|
|
representation of odd numbers without using the divides relation, see
|
|
~ zeo and ~ zeo2 . (Contributed by AV, 22-Jun-2021.) $)
|
|
oddp1d2 $p |- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel c2 cdvds wbr wn c1 caddc co cdiv oddp1even cc0 wb 2z 2ne0 peano2z
|
|
wne dvdsval2 mp3an12i bitrd ) ABCZDAEFGDAHIJZEFZUCDKJBCZALDBCDMRUBUCBCUDUEN
|
|
OPAQDUCSTUA $.
|
|
|
|
$( Alternate characterizations of an odd number. (Contributed by AV,
|
|
7-Jun-2020.) $)
|
|
zob $p |- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) e. ZZ
|
|
<-> ( ( N - 1 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel c1 caddc co c2 cdiv cmin peano2zm peano2z cc wceq zcnd npcan1 syl
|
|
halfcld eqcomd eleq1d syl5ibr impbid2 zcn xp1d2m1eqxm1d2 bitrd ) ABCZADEFZG
|
|
HFZBCZUGDIFZBCZADIFGHFZBCUEUHUJUGJUJUHUEUIDEFZBCUIKUEUGULBUEULUGUEUGLCULUGM
|
|
UEUFUEUFAKNQUGOPRSTUAUEUIUKBUEALCUIUKMAUBAUCPSUD $.
|
|
|
|
$( An integer is odd iff its predecessor divided by 2 is an integer. This is
|
|
another representation of odd numbers without using the divides relation.
|
|
(Contributed by AV, 18-Jun-2021.) (Proof shortened by AV,
|
|
22-Jun-2021.) $)
|
|
oddm1d2 $p |- ( N e. ZZ -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) ) $=
|
|
( cz wcel c2 cdvds wbr wn c1 caddc co cdiv cmin oddp1d2 zob bitrd ) ABCDAEF
|
|
GAHIJDKJBCAHLJDKJBCAMANO $.
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( An integer is less than half of an odd number iff it is less than or
|
|
equal to the half of the predecessor of the odd number (which is an even
|
|
number). (Contributed by AV, 29-Jun-2021.) $)
|
|
ltoddhalfle $p |- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ )
|
|
-> ( M < ( N / 2 ) <-> M <_ ( ( N - 1 ) / 2 ) ) ) $=
|
|
( vn cz wcel c2 wbr cdiv co clt c1 cle wb wceq wi wa cr a1i adantr cc w3a
|
|
cdvds wn cmin cv cmul caddc wrex odd2np1 halfre 1red 3jca halflt1 axltadd
|
|
mpisyl adantl readdcld peano2z zred syl3anc mpan2d zleltp1 ancoms sylibrd
|
|
zre lttr cc0 halfgt0 jca ltaddpos syl lelttr impbid cap zcn 1cnd 2cn 2ap0
|
|
mpbii pm3.2i muldivdirap breq2d 2z id zmulcld zcnd pncan1 2cnd divcanap3d
|
|
oveq1d eqtrd 3bitr4d oveq1 bibi12d syl5ibcom com23 rexlimdva sylbid 3imp
|
|
ex ) BDEZFBUBGUCZADEZABFHIZJGZABKUDIZFHIZLGZMZXAXBFCUEZUFIZKUGIZBNZCDUHXC
|
|
XIOZCBUIXAXMXNCDXAXJDEZPXCXMXIXOXCXMXIOZOXAXOXCXPXOXCPZAXLFHIZJGZAXLKUDIZ
|
|
FHIZLGZMXMXIXQAXJKFHIZUGIZJGZAXJLGZXSYBXQYEYFXQYEAXJKUGIZJGZYFXQYEYDYGJGZ
|
|
YHXQYCQEZKQEZXJQEZUAZYCKJGYIXOYMXCXOYJYKYLYJXOUJRZXOUKXJVEZULSUMYCKXJUNUO
|
|
XQAQEZYDQEZYGQEZYEYIPYHOXCYPXOAVEUPZXOYQXCXOXJYCYOYNUQSZXOYRXCXOYGXJURUSS
|
|
AYDYGVFUTVAXCXOYFYHMAXJVBVCVDXQYFXJYDJGZYEXQVGYCJGZUUAVHXQYJYLPZUUBUUAMXO
|
|
UUCXCXOYJYLYNYOVISYCXJVJVKVSXQYPYLYQYFUUAPYEOYSXOYLXCYOSYTAXJYDVLUTVAVMXO
|
|
XSYEMXCXOXRYDAJXOXJTEKTEFTEZFVGVNGZPZXRYDNXJVOZXOVPUUFXOUUDUUEVQVRVTRXJKF
|
|
WAUTWBSXQYAXJALXQYAXKFHIZXJXQXTXKFHXQXKTEZXTXKNXOUUIXCXOXKXOFXJFDEXOWCRXO
|
|
WDWEWFSXKWGVKWJXOUUHXJNXCXOXJFUUGXOWHUUEXOVRRWISWKWBWLXMXSXEYBXHXMXRXDAJX
|
|
LBFHWMWBXMYAXGALXMXTXFFHXLBKUDWMWJWBWNWOWTUPWPWQWRWS $.
|
|
|
|
$( An integer is greater than half of an odd number iff it is greater than
|
|
or equal to the half of the odd number. (Contributed by AV,
|
|
1-Jul-2021.) $)
|
|
halfleoddlt $p |- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ )
|
|
-> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) $=
|
|
( vn cz wcel c2 wbr cdiv co cle clt wb c1 caddc wi wa cc0 cxr cc breq1d
|
|
cdvds wn cv cmul wceq wrex odd2np1 cioo w3a 0xr 1re rexri 3pm3.2i halfgt0
|
|
halfre halflt1 pm3.2i elioo3g mpbir2an zltaddlt1le mp3an3 cap adantr 1cnd
|
|
zcn 2cn 2ap0 a1i muldivdirap syl3anc 3bitr4rd bibi12d syl5ibcom ex adantl
|
|
oveq1 com23 rexlimdva sylbid 3imp ) BDEZFBUAGUBZADEZBFHIZAJGZWDAKGZLZWAWB
|
|
FCUCZUDIMNIZBUEZCDUFWCWGOZCBUGWAWJWKCDWAWHDEZPWCWJWGWLWCWJWGOZOWAWLWCWMWL
|
|
WCPZWIFHIZAJGZWOAKGZLWJWGWNWHMFHIZNIZAKGZWSAJGZWQWPWLWCWRQMUHIEZWTXALXBQR
|
|
EZMREZWRREZUIQWRKGZWRMKGZPXCXDXEUJMUKULWRUOULUMXFXGUNUPUQQMWRURUSWRWHAUTV
|
|
AWNWOWSAKWNWHSEZMSEFSEZFQVBGZPZWOWSUEWLXHWCWHVEVCWNVDXKWNXIXJVFVGUQVHWHMF
|
|
VIVJZTWNWOWSAJXLTVKWJWPWEWQWFWJWOWDAJWIBFHVPZTWJWOWDAKXMTVLVMVNVOVQVRVSVT
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A a b c $. $d B a b c $.
|
|
$( The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.)
|
|
(Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
opoe $p |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) ->
|
|
2 || ( A + B ) ) $=
|
|
( va vb cz wcel c2 cdvds wbr wn caddc co wa cmul c1 wceq wrex 2cn ax-1cn
|
|
cc cv odd2np1 bi2anan9 reeanv 2z zaddcl peano2zd dvdsmul1 zcn addcl adddi
|
|
sylancr syl mp3an1 oveq1d eqtrd 2t1e2 df-2 eqtri oveq2i syl6eq mulcl mpan
|
|
mp3an13 add4 mpanr12 syl2an breqtrd oveq12 breq2d syl5ibcom sylbir syl6bi
|
|
rexlimivv imp an4s ) AEFZBEFZGAHIJZGBHIJZGABKLZHIZVQVRMZVSVTMZWBWCWDGCUAZ
|
|
NLZOKLZAPZCEQZGDUAZNLZOKLZBPZDEQZMZWBVQVSWIVRVTWNCAUBDBUBUCWOWHWMMZDEQCEQ
|
|
WBWHWMCDEEUDWPWBCDEEWEEFZWJEFZMZGWGWLKLZHIWPWBWSGGWEWJKLZOKLZNLZWTHWSGEFX
|
|
BEFGXCHIUEWSXAWEWJUFUGGXBUHULWQWETFZWJTFZXCWTPWRWEUIWJUIXDXEMZXCWFWKKLZOO
|
|
KLZKLZWTXFXCXGGONLZKLZXIXFXCGXANLZXJKLZXKXFXATFZXCXMPZWEWJUJGTFZXNOTFZXOR
|
|
SGXAOUKVDUMXFXLXGXJKXPXDXEXLXGPRGWEWJUKUNUOUPXJXHXGKXJGXHUQURUSUTVAXDWFTF
|
|
ZWKTFZXIWTPZXEXPXDXRRGWEVBVCXPXEXSRGWJVBVCXRXSMXQXQXTSSWFWKOOVEVFVGUPVGVH
|
|
WPWTWAGHWGAWLBKVIVJVKVNVLVMVOVP $.
|
|
|
|
$( The difference of two odds is even. (Contributed by Scott Fenton,
|
|
7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
omoe $p |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) ->
|
|
2 || ( A - B ) ) $=
|
|
( va vb cz wcel c2 cdvds wbr wn cmin co wa cv cmul c1 wceq wrex cc 2cn 2z
|
|
caddc odd2np1 bi2anan9 reeanv zsubcl dvdsmul1 sylancr mpan ax-1cn pnpcan2
|
|
zcn mulcl mp3an3 syl2an mp3an1 eqtr4d breqtrrd oveq12 syl5ibcom rexlimivv
|
|
subdi breq2d sylbir syl6bi imp an4s ) AEFZBEFZGAHIJZGBHIJZGABKLZHIZVHVIMZ
|
|
VJVKMZVMVNVOGCNZOLZPUBLZAQZCERZGDNZOLZPUBLZBQZDERZMZVMVHVJVTVIVKWECAUCDBU
|
|
CUDWFVSWDMZDERCERVMVSWDCDEEUEWGVMCDEEVPEFZWAEFZMZGVRWCKLZHIWGVMWJGGVPWAKL
|
|
ZOLZWKHWJGEFWLEFGWMHIUAVPWAUFGWLUGUHWHVPSFZWASFZWKWMQWIVPULWAULWNWOMWKVQW
|
|
BKLZWMWNVQSFZWBSFZWKWPQZWOGSFZWNWQTGVPUMUIWTWOWRTGWAUMUIWQWRPSFWSUJVQWBPU
|
|
KUNUOWTWNWOWMWPQTGVPWAVBUPUQUOURWGWKVLGHVRAWCBKUSVCUTVAVDVEVFVG $.
|
|
|
|
$( The sum of an odd and an even is odd. (Contributed by Scott Fenton,
|
|
7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
opeo $p |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) ->
|
|
-. 2 || ( A + B ) ) $=
|
|
( vc va vb cz wcel c2 cdvds wbr wa caddc co cmul c1 wceq wrex mpan cc 2cn
|
|
wn cv odd2np1 wb 2z divides bi2anan9 reeanv zaddcl zcn adddi mp3an1 mulcl
|
|
oveq1d ax-1cn mp3an3 syl2an mulcom adantl oveq2d 3eqtrd syl2anc syl5ibcom
|
|
add32 oveq2 eqeq1d rspcev oveq12 eqeq2d rexbidv rexlimivv sylbir imp an4s
|
|
syl6bi ad2ant2r syl mpbird ) AFGZHAIJUAZKBFGZHBIJZKKZHABLMZIJUAZHCUBZNMZO
|
|
LMZWDPZCFQZVSWAVTWBWJVSWAKZVTWBKZWJWKWLHDUBZNMZOLMZAPZDFQZEUBZHNMZBPZEFQZ
|
|
KZWJVSVTWQWAWBXADAUCHFGWAWBXAUDUEEHBUFRUGXBWPWTKZEFQDFQWJWPWTDEFFUHXCWJDE
|
|
FFWMFGZWRFGZKZWHWOWSLMZPZCFQZXCWJXFWMWRLMZFGHXJNMZOLMZXGPZXIWMWRUIXDWMSGZ
|
|
WRSGZXMXEWMUJWRUJXNXOKZXLWNHWRNMZLMZOLMZWOXQLMZXGXPXKXROLHSGZXNXOXKXRPTHW
|
|
MWRUKULUNXNWNSGZXQSGZXSXTPZXOYAXNYBTHWMUMRYAXOYCTHWRUMRYBYCOSGYDUOWNXQOVD
|
|
UPUQXPXQWSWOLXOXQWSPZXNYAXOYETHWRURRUSUTVAUQXHXMCXJFWFXJPZWHXLXGYFWGXKOLW
|
|
FXJHNVEUNVFVGVBXCXHWICFXCXGWDWHWOAWSBLVHVIVJVCVKVLVOVMVNWCWDFGZWEWJUDVSWA
|
|
YGVTWBABUIVPCWDUCVQVR $.
|
|
|
|
$( The difference of an odd and an even is odd. (Contributed by Scott
|
|
Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
omeo $p |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) ->
|
|
-. 2 || ( A - B ) ) $=
|
|
( vc va vb cz wcel c2 cdvds wa cmin co cmul c1 caddc wceq wrex mpan 2cn
|
|
cc wn cv odd2np1 wb 2z divides bi2anan9 reeanv zsubcl subdi mp3an1 oveq1d
|
|
wbr mulcl ax-1cn addsub mp3an2 syl2an mulcom oveq2d adantl 3eqtr2d eqeq1d
|
|
zcn oveq2 rspcev syl2anc oveq12 eqeq2d rexbidv syl5ibcom rexlimivv sylbir
|
|
syl6bi imp an4s ad2ant2r syl mpbird ) AFGZHAIUMUAZJBFGZHBIUMZJJZHABKLZIUM
|
|
UAZHCUBZMLZNOLZWEPZCFQZVTWBWAWCWKVTWBJZWAWCJZWKWLWMHDUBZMLZNOLZAPZDFQZEUB
|
|
ZHMLZBPZEFQZJZWKVTWAWRWBWCXBDAUCHFGWBWCXBUDUEEHBUFRUGXCWQXAJZEFQDFQWKWQXA
|
|
DEFFUHXDWKDEFFWNFGZWSFGZJZWIWPWTKLZPZCFQZXDWKXGWNWSKLZFGHXKMLZNOLZXHPZXJW
|
|
NWSUIXEWNTGZWSTGZXNXFWNVDWSVDXOXPJZXMWOHWSMLZKLZNOLZWPXRKLZXHXQXLXSNOHTGZ
|
|
XOXPXLXSPSHWNWSUJUKULXOWOTGZXRTGZYAXTPZXPYBXOYCSHWNUNRYBXPYDSHWSUNRYCNTGY
|
|
DYEUOWONXRUPUQURXPYAXHPXOXPXRWTWPKYBXPXRWTPSHWSUSRUTVAVBURXIXNCXKFWGXKPZW
|
|
IXMXHYFWHXLNOWGXKHMVEULVCVFVGXDXIWJCFXDXHWEWIWPAWTBKVHVIVJVKVLVMVNVOVPWDW
|
|
EFGZWFWKUDVTWBYGWAWCABUIVQCWEUCVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d N n $.
|
|
$( Exponentiation of -1 by an even power. Variant of ~ m1expeven .
|
|
(Contributed by AV, 25-Jun-2021.) $)
|
|
m1expe $p |- ( 2 || N -> ( -u 1 ^ N ) = 1 ) $=
|
|
( vn c2 cdvds wbr cv cmul co wceq cz wrex c1 cneg cexp even2n wcel eqcoms
|
|
oveq2 m1expeven sylan9eqr rexlimiva sylbi ) CADECBFZGHZAIZBJKLMZANHZLIZBA
|
|
OUEUHBJUEUCJPUGUFUDNHZLUGUIIAUDAUDUFNRQUCSTUAUB $.
|
|
|
|
$( Exponentiation of -1 by an odd power. (Contributed by AV,
|
|
26-Jun-2021.) $)
|
|
m1expo $p |- ( ( N e. ZZ /\ -. 2 || N ) -> ( -u 1 ^ N ) = -u 1 ) $=
|
|
( vn cz wcel c2 cdvds wbr wn c1 cneg cexp co wceq cmul caddc wrex odd2np1
|
|
cv neg1cn a1i wa oveq2 eqcoms cc0 cap neg1ap0 zmulcld expp1zapd m1expeven
|
|
cc 2z id oveq1d mulid2i syl6eq eqtrd adantl sylan9eqr ex rexlimdva sylbid
|
|
imp ) ACDZEAFGHZIJZAKLZVEMZVCVDEBRZNLZIOLZAMZBCPVGBAQVCVKVGBCVCVHCDZUAZVK
|
|
VGVKVMVFVEVJKLZVEVFVNMAVJAVJVEKUBUCVLVNVEMVCVLVNVEVIKLZVENLZVEVLVEVIVEUJD
|
|
VLSTVEUDUEGVLUFTVLEVHECDVLUKTVLULUGUHVLVPIVENLVEVLVOIVENVHUIUMVESUNUOUPUQ
|
|
URUSUTVAVB $.
|
|
|
|
$( Exponentiation of negative one is one iff the exponent is even.
|
|
(Contributed by AV, 20-Jun-2021.) $)
|
|
m1exp1 $p |- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) ) $=
|
|
( vn cz wcel c2 wbr c1 cexp co wceq wb wa cmul wrex 2z oveq2 eqcoms eqtrd
|
|
cc0 a1i cdvds cneg wn cv divides mpan zcn 2cnd oveq2d m1expeven sylan9eqr
|
|
mulcomd rexlimiva syl6bi impcom simpl 2thd expcom 1ne0 eqcom ax-1cn bitri
|
|
eqnegi nemtbir caddc odd2np1 neg1cn cap neg1ap0 zmulcld expp1zapd mulid2i
|
|
cc id oveq1d syl6eq eqeq1d mtbiri 2falsed zeo3 mpjaod ) ACDZEAUAFZGUBZAHI
|
|
ZGJZWCKZWCUCZWCWBWGWCWBLWFWCWBWCWFWBWCBUDZEMIZAJZBCNZWFECDZWBWCWLKOBEAUEU
|
|
FWKWFBCWKWICDZWEWDWJHIZGWEWOJAWJAWJWDHPQWNWOWDEWIMIZHIZGWNWJWPWDHWNWIEWIU
|
|
GWNUHULUIWIUJZRUKUMUNUOWCWBUPUQURWHWBWGWHWBLZWFWCWSWFWDGJZWTGSUSWTGWDJGSJ
|
|
WDGUTGVAVCVBVDWSWEWDGWBWHWEWDJZWBWHWPGVEIZAJZBCNXABAVFXCXABCXCWNWEWDXBHIZ
|
|
WDWEXDJAXBAXBWDHPQWNXDWQWDMIZWDWNWDWPWDVMDWNVGTWDSVHFWNVITWNEWIWMWNOTWNVN
|
|
VJVKWNXEGWDMIWDWNWQGWDMWRVOWDVGVLVPRUKUMUNUOVQVRWHWBUPVSURAVTWA $.
|
|
$}
|
|
|
|
$( A positive integer is an even nonnegative integer iff it is an even
|
|
positive integer. (Contributed by AV, 30-May-2020.) $)
|
|
nn0enne $p |- ( N e. NN -> ( ( N / 2 ) e. NN0 <-> ( N / 2 ) e. NN ) ) $=
|
|
( cn wcel c2 cdiv co cn0 cc0 wceq wo wi elnn0 nncn 2cnd cap 2ap0 diveqap0ad
|
|
wbr a1i com12 eleq1 0nnn pm2.21i syl6bi sylbid jao1i sylbi nnnn0 impbid1 )
|
|
ABCZADEFZGCZUKBCZULUJUMULUMUKHIZJUJUMKUKLUMUNUJUJUNUMUJUNAHIZUMUJADAMUJNDHO
|
|
RUJPSQUOUJUMUOUJHBCZUMAHBUAUPUMUBUCUDTUETUFUGTUKUHUI $.
|
|
|
|
$( The half of an even nonnegative integer is a nonnegative integer.
|
|
(Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) $)
|
|
nn0ehalf $p |- ( ( N e. NN0 /\ 2 || N ) -> ( N / 2 ) e. NN0 ) $=
|
|
( cn0 wcel c2 cdvds wbr cdiv co cz wb nn0z evend2 syl wa cc0 cle nn0ge0 clt
|
|
cr a1i nn0re 2re ge0div syl3anc mpbid anim1i ancomd elnn0z sylibr ex sylbid
|
|
2pos imp ) ABCZDAEFZADGHZBCZUNUOUPICZUQUNAICUOURJAKALMUNURUQUNURNZUROUPPFZN
|
|
UQUSUTURUNUTURUNOAPFZUTAQUNASCDSCZODRFZVAUTJAUAVBUNUBTVCUNULTADUCUDUEUFUGUP
|
|
UHUIUJUKUM $.
|
|
|
|
$( The half of an even positive integer is a positive integer. (Contributed
|
|
by AV, 28-Jun-2021.) $)
|
|
nnehalf $p |- ( ( N e. NN /\ 2 || N ) -> ( N / 2 ) e. NN ) $=
|
|
( cn wcel c2 cdvds wbr wa cdiv co cn0 nnnn0 nn0ehalf sylan wb nn0enne mpbid
|
|
adantr ) ABCZDAEFZGADHIZJCZTBCZRAJCSUAAKALMRUAUBNSAOQP $.
|
|
|
|
$( An odd nonnegative integer is either 1 or greater than 2. (Contributed by
|
|
AV, 2-Jun-2020.) $)
|
|
nn0o1gt2 $p |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 )
|
|
-> ( N = 1 \/ 2 < N ) ) $=
|
|
( cn0 wcel c1 caddc co c2 cdiv wceq clt wbr wo cc0 wi cz wb nn0z jaoi com12
|
|
cle cn elnn0 wa elnnnn0c 1z zleloe sylancr zltp1le syl2anc 1p1e2 breq1i a1i
|
|
1zzd 2z 3bitrd olc c3 oveq1 oveq1d eqcoms adantl 2p1e3 oveq1i syl6eq eleq1d
|
|
2a1d wn 3halfnz pm2.24d mpi syl6bi expcom sylbid orc imp sylbi 0p1e1 halfnz
|
|
) ABCZADEFZGHFZBCZADIZGAJKZLZVSAUACZAMIZLWBWENZAUBWFWHWGWFVSDATKZUCWHAUDVSW
|
|
IWHVSWIDAJKZDAIZLZWHVSDOCZAOCZWIWLPUEAQZDAUFUGWLVSWHWJVSWHNZWKVSWJWHVSWJWDG
|
|
AIZLZWHVSWJDDEFZATKZGATKZWRVSWMWNWJWTPVSUMWODAUHUIWTXAPVSWSGATUJUKULVSGOCWN
|
|
XAWRPUNWOGAUFUGUOWRVSWHWDWPWQWDWEVSWBWDWCUPVFVSWQWHVSWQUCZWBUQGHFZBCZWEXBWA
|
|
XCBXBWAGDEFZGHFZXCWQWAXFIZVSXGAGAGIVTXEGHAGDEURUSUTVAXEUQGHVBVCVDVEXDXCOCZV
|
|
GWEVHXDXHWEXCQVIVJVKVLRSVMSWKWEVSWBWEADWCWDVNUTVFRSVMVOVPWGWBDGHFZBCZWEWGWA
|
|
XIBWGVTDGHWGVTMDEFDAMDEURVQVDUSVEXJXIOCZVGWEVRXJXKWEXIQVIVJVKRVPVO $.
|
|
|
|
$( An alternate characterization of an odd integer greater than 1.
|
|
(Contributed by AV, 2-Jun-2020.) $)
|
|
nno $p |- ( ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 )
|
|
-> ( ( N - 1 ) / 2 ) e. NN ) $=
|
|
( c2 wcel c1 co cdiv cn0 cmin cn wa wi wceq clt wbr a1d cz cc0 adantr mpbid
|
|
cr cuz cfv caddc wne eluz2b3 wo nnnn0 nn0o1gt2 sylan eqneqall nn0z peano2zm
|
|
syl ad2antlr cmul 2cn mulid2i nnre ltp1d 2re peano2nn nnred syl3anc expdimp
|
|
a1i lttr mpd syl5eqbr wb 1red 2pos ltmuldiv rehalfcld posdifd adantlr elnnz
|
|
pm3.2i sylanbrc nncn xp1d2m1eqxm1d2 eleq1d expcom jaoi mpcom impancom sylbi
|
|
cc imp ) ABUAUBCZADUCEZBFEZGCZADHEBFEZICZWIAICZADUDZJWLWNKAUEWOWLWPWNADLZBA
|
|
MNZUFZWOWLJZWPWNKZWOAGCWLWSAUGAUHUIWQWTXAKWRWQXAWTWNADUJOWTWRXAWTWRJZWNWPXB
|
|
WKDHEZICZWNXBXCPCZQXCMNZXDWLXEWOWRWLWKPCXEWKUKWKULUMUNWOWRXFWLWOWRJZDWKMNZX
|
|
FXGDBUOEZWJMNZXHXGXIBWJMBUPUQXGAWJMNZBWJMNZWOXKWRWOAAURZUSRWOWRXKXLWOBTCZAT
|
|
CWJTCZWRXKJXLKXNWOUTVEXMWOWJAVAVBZBAWJVFVCVDVGVHXGDTCXOXNQBMNZJZXJXHVIXGVJZ
|
|
WOXOWRXPRXRXGXNXQUTVKVQVEDWJBVLVCSXGDWKXSWOWKTCWRWOWJXPVMRVNSVOXCVPVRWTXDWN
|
|
VIZWRWOXTWLWOXCWMIWOAWGCXCWMLAVSAVTUMWARRSOWBWCWDWEWFWH $.
|
|
|
|
$( An alternate characterization of an odd nonnegative integer. (Contributed
|
|
by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) $)
|
|
nn0o $p |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 )
|
|
-> ( ( N - 1 ) / 2 ) e. NN0 ) $=
|
|
( c1 wceq c2 clt wbr wo cn0 wcel caddc co cdiv wa cmin wi cc0 adantr ex cz
|
|
cr nn0o1gt2 1m1e0 oveq1i 2cn 2ap0 div0api eqtri eqeltri oveq1 oveq1d eleq1d
|
|
0nn0 wb mpbiri cuz cfv cn cle 2z a1i nn0z ad2antrl 2re nn0re sylancr impcom
|
|
ltle eluz2 syl3anbrc simprr jca nno nnnn0 3syl jaoi mpcom ) ABCZDAEFZGAHIZA
|
|
BJKDLKHIZMZABNKZDLKZHIZAUAVQWAWDOVRVQWAWDVQWAMWDBBNKZDLKZHIZWFPHWFPDLKPWEPD
|
|
LUBUCDUDUEUFUGULUHVQWDWGUMWAVQWCWFHVQWBWEDLABBNUIUJUKQUNRVRWAWDVRWAMZADUOUP
|
|
IZVTMWCUQIWDWHWIVTWHDSIZASIZDAURFZWIWJWHUSUTVSWKVRVTAVAVBWAVRWLVSVRWLOZVTVS
|
|
DTIATIWMVCAVDDAVGVEQVFDAVHVIVRVSVTVJVKAVLWCVMVNRVOVP $.
|
|
|
|
$( Alternate characterizations of an odd nonnegative integer. (Contributed
|
|
by AV, 4-Jun-2020.) $)
|
|
nn0ob $p |- ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0
|
|
<-> ( ( N - 1 ) / 2 ) e. NN0 ) ) $=
|
|
( wcel c1 caddc co c2 cdiv cmin nn0o wa wceq cc nn0cn xp1d2m1eqxm1d2 eqcomd
|
|
cn0 syl peano2cnm halfcld peano2nn0 nn0cnd addlsub mpbird eqeltrrd impbida
|
|
1cnd adantr adantl ) APBZACDEZFGEZPBACHEZFGEZPBZAIUIUNJUMCDEZUKPUIUOUKKZUNU
|
|
IUPUMUKCHEZKZUIALBZURAMZUSUQUMANOQUIUMCUKUIULUIUSULLBUTARQSUIUFUIUJUIUJATUA
|
|
SUBUCUGUNUOPBUIUMTUHUDUE $.
|
|
|
|
$( A positive integer is odd iff its predecessor divided by 2 is a positive
|
|
integer. (Contributed by AV, 28-Jun-2021.) $)
|
|
nn0oddm1d2 $p |- ( N e. NN0 -> ( -. 2 || N
|
|
<-> ( ( N - 1 ) / 2 ) e. NN0 ) ) $=
|
|
( cn0 wcel c2 cdvds wbr wn c1 caddc co cdiv cz cmin wb nn0z wa cc0 cle a1i
|
|
cr oddp1d2 syl nn0re 1red nn0ge0 0le1 addge0d clt peano2nn0 nn0red 2re 2pos
|
|
ge0div syl3anc mpbid anim1i ancomd elnn0z sylibr ex impbid1 nn0ob 3bitrd )
|
|
ABCZDAEFGZAHIJZDKJZLCZVGBCZAHMJDKJBCVDALCVEVHNAOAUAUBVDVHVIVDVHVIVDVHPZVHQV
|
|
GRFZPVIVJVKVHVDVKVHVDQVFRFZVKVDAHAUCVDUDAUEQHRFVDUFSUGVDVFTCDTCZQDUHFZVLVKN
|
|
VDVFAUIUJVMVDUKSVNVDULSVFDUMUNUOUPUQVGURUSUTVGOVAAVBVC $.
|
|
|
|
$( A positive integer is odd iff its successor divided by 2 is a positive
|
|
integer. (Contributed by AV, 28-Jun-2021.) $)
|
|
nnoddm1d2 $p |- ( N e. NN -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. NN ) ) $=
|
|
( cn wcel c2 cdvds wbr wn c1 caddc co cdiv cz wb nnz oddp1d2 syl wa cc0 clt
|
|
a1i peano2nn nnred cr 2re nnre 1red 0lt1 addgt0d 2pos divgt0d anim1i ancomd
|
|
nngt0 elnnz sylibr ex impbid1 bitrd ) ABCZDAEFGZAHIJZDKJZLCZVBBCZUSALCUTVCM
|
|
ANAOPUSVCVDUSVCVDUSVCQZVCRVBSFZQVDVEVFVCUSVFVCUSVADUSVAAUAUBDUCCUSUDTUSAHAU
|
|
EUSUFAUMRHSFUSUGTUHRDSFUSUITUJUKULVBUNUOUPVBNUQUR $.
|
|
|
|
$( 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV,
|
|
23-Jun-2021.) $)
|
|
z0even $p |- 2 || 0 $=
|
|
( c2 cz wcel cc0 cdvds wbr 2z dvds0 ax-mp ) ABCADEFGAHI $.
|
|
|
|
$( 2 does not divide 1 (common case). That means 1 is odd. (Contributed by
|
|
David A. Wheeler, 8-Dec-2018.) $)
|
|
n2dvds1 $p |- -. 2 || 1 $=
|
|
( c2 c1 cdvds wbr cle clt wn 1lt2 cz wcel wb 1z 2z zltnle mp2an mpbi cn 1nn
|
|
wi dvdsle mto ) ABCDZABEDZBAFDZUCGZHBIJAIJZUDUEKLMBANOPUFBQJUBUCSMRABTOUA
|
|
$.
|
|
|
|
$( 2 does not divide -1. That means -1 is odd. (Contributed by AV,
|
|
15-Aug-2021.) $)
|
|
n2dvdsm1 $p |- -. 2 || -u 1 $=
|
|
( c2 c1 cneg cdvds wbr wn caddc co z0even ax-1cn 1pneg1e0 addcomli breqtrri
|
|
cc0 neg1cn cz wcel wb neg1z oddp1even ax-mp mpbir ) ABCZDEFZAUCBGHZDEZANUED
|
|
IBUCNJOKLMUCPQUDUFRSUCTUAUB $.
|
|
|
|
$( 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV,
|
|
23-Jun-2021.) $)
|
|
z2even $p |- 2 || 2 $=
|
|
( c2 cz wcel cdvds wbr 2z iddvds ax-mp ) ABCAADEFAGH $.
|
|
|
|
$( 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV,
|
|
28-Feb-2021.) $)
|
|
n2dvds3 $p |- -. 2 || 3 $=
|
|
( c2 c3 cdvds wbr wn c1 cmin co cz 2z iddvds ax-mp 3m1e2 breqtrri oddm1even
|
|
wcel wb 3z mpbir ) ABCDEZABFGHZCDZAAUACAIPAACDJAKLMNBIPTUBQRBOLS $.
|
|
|
|
$( 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV,
|
|
4-Jul-2021.) $)
|
|
z4even $p |- 2 || 4 $=
|
|
( c2 cmul co c4 cdvds cz wcel wbr 2z dvdsmul1 mp2an 2t2e4 breqtri ) AAABCZD
|
|
EAFGZOANEHIIAAJKLM $.
|
|
|
|
$( An integer which is divisible by 4 is an even integer. (Contributed by
|
|
AV, 4-Jul-2021.) $)
|
|
4dvdseven $p |- ( 4 || N -> 2 || N ) $=
|
|
( c4 cdvds wbr c2 cz wcel w3a wa 2z a1i 4z dvdszrcl simprd 3jca z4even jctl
|
|
dvdstr sylc ) BACDZEFGZBFGZAFGZHEBCDZTIEACDTUAUBUCUATJKUBTLKTUBUCBAMNOTUDPQ
|
|
EBARS $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The division algorithm
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d D q r $. $d N q r $.
|
|
$( Lemma for ~ divalg . Existence for a positive denominator.
|
|
(Contributed by Jim Kingdon, 30-Nov-2021.) $)
|
|
divalglemnn $p |- ( ( N e. ZZ /\ D e. NN ) ->
|
|
E. r e. ZZ E. q e. ZZ
|
|
( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) $=
|
|
( cz wcel co cfv cc0 cle wbr clt cmul caddc wceq cv w3a wrex cq adantl cn
|
|
wa cmo cdiv cfl cabs zmodcl nn0zd znq flqcld nn0ge0d zq adantr nnq modqlt
|
|
nngt0 syl3anc cr nnre 0red ltled absidd breqtrrd nn0cnd zcnd simpr mulcld
|
|
nncnd cmin modqvalr oveq1d simpl npcand eqtr2d comraddd breq2 breq1 oveq2
|
|
eqeq2d 3anbi123d oveq1 3anbi3d rspc2ev syl113anc ) BEFZAUAFZUBZBAUCGZEFBA
|
|
UDGZUEHZEFIWHJKZWHAUFHZLKZBWJAMGZWHNGZOZICPZJKZWQWLLKZBDPZAMGZWQNGZOZQZDE
|
|
RCERWGWHBAUGZUHWGWIBAUIUJZWGWHXEUKWGWHAWLLWGBSFZASFZIALKZWHALKWEXGWFBULUM
|
|
ZWFXHWEAUNTZWFXIWEAUPTZBAUOUQWGAWFAURFWEAUSTZWGIAWGUTXMXLVAVBVCWGBWHWNWGW
|
|
HXEVDWGWJAWGWJXFVEWGAWEWFVFVHVGZWGWHWNNGBWNVIGZWNNGBWGWHXOWNNWGXGXHXIWHXO
|
|
OXJXKXLBAVJUQVKWGBWNWGBWEWFVLVEXNVMVNVOXDWKWMWPQWKWMBXAWHNGZOZQCDWHWJEEWQ
|
|
WHOZWRWKWSWMXCXQWQWHIJVPWQWHWLLVQXRXBXPBWQWHXANVRVSVTWTWJOZXQWPWKWMXSXPWO
|
|
BXSXAWNWHNWTWJAMWAVKVSWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
divalglemqt.d $e |- ( ph -> D e. ZZ ) $.
|
|
divalglemqt.r $e |- ( ph -> R e. ZZ ) $.
|
|
divalglemqt.s $e |- ( ph -> S e. ZZ ) $.
|
|
divalglemqt.q $e |- ( ph -> Q e. ZZ ) $.
|
|
divalglemqt.t $e |- ( ph -> T e. ZZ ) $.
|
|
divalglemqt.qt $e |- ( ph -> Q = T ) $.
|
|
divalglemqt.eq $e |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) ) $.
|
|
$( Lemma for ~ divalg . The ` Q = T ` case involved in showing uniqueness.
|
|
(Contributed by Jim Kingdon, 5-Dec-2021.) $)
|
|
divalglemqt $p |- ( ph -> R = S ) $=
|
|
( cmul co cc oveq1d zmulcld zcnd caddc eqeltrrd eqtr3d addcanad ) AFBNOZD
|
|
EACBNOZUDPACFBNLQZAUEACBJGRSUAADHSAEISAUEDTOUDDTOUDETOAUEUDDTUFQMUBUC $.
|
|
$}
|
|
|
|
${
|
|
divalglemnqt.d $e |- ( ph -> D e. NN ) $.
|
|
divalglemnqt.r $e |- ( ph -> R e. ZZ ) $.
|
|
divalglemnqt.s $e |- ( ph -> S e. ZZ ) $.
|
|
divalglemnqt.q $e |- ( ph -> Q e. ZZ ) $.
|
|
divalglemnqt.t $e |- ( ph -> T e. ZZ ) $.
|
|
divalglemnqt.0s $e |- ( ph -> 0 <_ S ) $.
|
|
divalglemnqt.rd $e |- ( ph -> R < D ) $.
|
|
divalglemnqt.eq $e |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) ) $.
|
|
$( Lemma for ~ divalg . The ` Q < T ` case involved in showing uniqueness.
|
|
(Contributed by Jim Kingdon, 4-Dec-2021.) $)
|
|
divalglemnqt $p |- ( ph -> -. Q < T ) $=
|
|
( wbr adantr wcel caddc co cle clt wa cn nnred zred readdcld cc0 addge01d
|
|
cz mpbid cmul recnd mulcld addassd remulcld adddirp1d peano2re syl nnnn0d
|
|
c1 nn0ge0d simpr zltp1le syl2an2r lemul1ad eqbrtrrd leadd1dd wceq leadd2d
|
|
cr wb breqtrrd mpbird letrd lensymd pm2.65da ) ACFUAOZDBUAOZAVRVQMPAVQUBZ
|
|
BDVSBABUCQVQGPZUDZVSDADUIQVQHPUEZVSBBERSZDWAVSBEWAVSEAEUIQVQIPUEZUFZWBVSU
|
|
GETOZBWCTOAWFVQLPVSBEWAWDUHUJVSWCDTOCBUKSZWCRSZWGDRSZTOVSWGBRSZERSZWHWITV
|
|
SWGBEVSCBVSCVSCACUIQZVQJPUEZULZVSBWAULZUMWOVSEWDULUNVSWKFBUKSZERSZWITVSWJ
|
|
WPEVSWGBVSCBWMWAUOZWAUFVSFBVSFAFUIQZVQKPZUEZWAUOWDVSCUTRSZBUKSWJWPTVSCBWN
|
|
WOUPVSXBFBVSCVJQXBVJQWMCUQURXAWAVSBVSBVTUSVAVSVQXBFTOZAVQVBAWLVQWSVQXCVKJ
|
|
WTCFVCVDUJVEVFVGAWIWQVHVQNPVLVFVSWCDWGWEWBWRVIVMVNVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d D q r s t $. $d N q r s t $.
|
|
$( Lemma for ~ divalg . Uniqueness for a positive denominator.
|
|
(Contributed by Jim Kingdon, 4-Dec-2021.) $)
|
|
divalglemeunn $p |- ( ( N e. ZZ /\ D e. NN ) ->
|
|
E! r e. ZZ E. q e. ZZ
|
|
( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) $=
|
|
( vs vt cz wcel wa cc0 cv cle wbr clt co caddc wceq wrex simpr adantr cfv
|
|
cn cabs cmul w3a wrmo wreu divalglemnn wral nfv nfre1 oveq1 oveq1d eqeq2d
|
|
wi nfim 3anbi3d cbvrexv wn simplr ad4antr simplrl ad3antrrr simpr1 simpr2
|
|
simplrr ad2antrr nnnn0d nn0ge0d absidd breqtrd simpr3 eqtr3d divalglemnqt
|
|
nnred pm2.21dd divalglemqt w3o ztri3or syl2anc mpjao3dan rexlimdva syl5bi
|
|
nnzd ex exp31 rexlimd impd ralrimivva breq2 breq1 oveq2 3anbi123d rexbidv
|
|
rmo4 sylibr reu5 sylanbrc ) BGHZAUBHZIZJCKZLMZXBAUCUAZNMZBDKZAUDOZXBPOZQZ
|
|
UEZDGRZCGRXKCGUFZXKCGUGABCDUHXAXKJEKZLMZXMXDNMZBXGXMPOZQZUEZDGRZIXBXMQZUO
|
|
ZEGUICGUIXLXAYACEGGXAXBGHZXMGHZIZIZXKXSXTYEXJXSXTUOZDGYEDUJXSXTDXRDGUKXTD
|
|
UJUPYEXFGHZXJYFXSXNXOBFKZAUDOZXMPOZQZUEZFGRYEYGIZXJIZXTXRYLDFGXFYHQZXQYKX
|
|
NXOYOXPYJBYOXGYIXMPXFYHAUDULUMUNUQURYNYLXTFGYNYHGHZIZYLXTYQYLIZXFYHNMZXTY
|
|
OYHXFNMZYRYSIYSXTYRYSSYRYSUSYSYRAXFXBXMYHYEWTYGXJYPYLWSWTYDUTVAZYMYBXJYPY
|
|
LXAYBYCYGVBVCZYMYCXJYPYLXAYBYCYGVFVCZYMYGXJYPYLYEYGSVCZYNYPYLUTZYQXNXOYKV
|
|
DYRXBXDANYNXEYPYLYMXCXEXIVEVGYRAYRAUUAVOYRAYRAUUAVHVIVJZVKYRBXHYJYNXIYPYL
|
|
YMXCXEXIVLVGZYQXNXOYKVLZVMZVNTVPYRYOIZAXFXBXMYHUUJAYRWTYOUUATWDYRYBYOUUBT
|
|
YRYCYOUUCTYRYGYOUUDTYRYPYOUUETYRYOSYRXHYJQYOUUITVQYRYTIYTXTYRYTSYRYTUSYTY
|
|
RAYHXMXBXFUUAUUCUUBUUEUUDYNXCYPYLYMXCXEXIVDVGYRXMXDANYQXNXOYKVEUUFVKYRBYJ
|
|
XHUUHUUGVMVNTVPYRYGYPYSYOYTVRUUDUUEXFYHVSVTWAWEWBWCWFWGWHWIXKXSCEGXTXJXRD
|
|
GXTXCXNXEXOXIXQXBXMJLWJXBXMXDNWKXTXHXPBXBXMXGPWLUNWMWNWOWPXKCGWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d D k q r $. $d N k q r $.
|
|
$( Lemma for ~ divalg . The quotient and remainder exist. (Contributed by
|
|
Jim Kingdon, 30-Nov-2021.) $)
|
|
divalglemex $p |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) ->
|
|
E. r e. ZZ E. q e. ZZ
|
|
( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) $=
|
|
( vk cz wcel cc0 w3a clt wbr cv cmul co caddc wceq wrex wa simpr syl2anc
|
|
wne cle cabs cfv cneg simpl1 simpl2 znegcld lt0neg1d mpbid elnnz sylanbrc
|
|
cn zred divalglemnn simplr simpr1 simpr2 simpll2 ad2antrr absnegd breqtrd
|
|
zcnd simpr3 mulneg12 oveq1d eqtr4d oveq1 eqeq2d rspcev syl13anc rexlimdva
|
|
cc 3anbi3d reximdva mpd simpl3 pm2.21ddne w3o ztri3or0 3ad2ant2 mpjao3dan
|
|
ex ) BFGZAFGZAHUAZIZAHJKZHCLZUBKZWIAUCUDZJKZBDLZAMNZWIONZPZIZDFQZCFQZAHPZ
|
|
HAJKZWGWHRZWJWIAUEZUCUDZJKZBELZXCMNZWIONZPZIZEFQZCFQZWSXBWDXCUMGZXLWDWEWF
|
|
WHUFXBXCFGHXCJKZXMXBAWDWEWFWHUGZUHXBWHXNWGWHSXBAXBAXOUNUIUJXCUKULXCBCEUOT
|
|
XBXKWRCFXBWIFGZRZXJWREFXQXFFGZRZXJWRXSXJRZXFUEZFGWJWLBYAAMNZWIONZPZWRXTXF
|
|
XQXRXJUPZUHXSWJXEXIUQXTWIXDWKJXSWJXEXIURXTAXTAXQWEXRXJWDWEWFWHXPUSUTVCZVA
|
|
VBXTBXHYCXSWJXEXIVDXTYBXGWIOXTXFVMGAVMGYBXGPXTXFYEVCYFXFAVETVFVGWQWJWLYDI
|
|
DYAFWMYAPZWPYDWJWLYGWOYCBYGWNYBWIOWMYAAMVHVFVIVNVJVKWCVLVOVPWGWTRWSAHWGWT
|
|
SWDWEWFWTVQVRWGXARZWDAUMGZWSWDWEWFXAUFYHWEXAYIWDWEWFXAUGWGXASAUKULABCDUOT
|
|
WEWDWHWTXAVSWFAVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d D q r s t $. $d N q r s t $.
|
|
$( Lemma for ~ divalg . Uniqueness for a negative denominator.
|
|
(Contributed by Jim Kingdon, 4-Dec-2021.) $)
|
|
divalglemeuneg $p |- ( ( N e. ZZ /\ D e. ZZ /\ D < 0 ) ->
|
|
E! r e. ZZ E. q e. ZZ
|
|
( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) $=
|
|
( vs vt cz wcel cc0 clt wbr w3a cv cmul co caddc wceq wa ad3antrrr adantr
|
|
cle cabs cfv wrex wrmo wreu wne simp3 lt0ne0d divalglemex syld3an3 wi nfv
|
|
wral nfre1 nfim oveq1 oveq1d eqeq2d 3anbi3d cbvrexv simpr wn cneg znegcld
|
|
cn simp2 zred lt0neg1d mpbid elnnz sylanbrc ad5antr simplrr simplr simpr1
|
|
simplrl ad2antrr simpr2 simpll2 0red simpll3 ltled absnidd breqtrd simpr3
|
|
mul2negd eqtr4d eqtr3d divalglemnqt ltnegd mtbird divalglemqt w3o ztri3or
|
|
zcnd pm2.21dd syl2anc mpjao3dan rexlimdva syl5bi exp31 rexlimd ralrimivva
|
|
ex impd breq2 breq1 oveq2 3anbi123d rexbidv rmo4 sylibr reu5 ) BGHZAGHZAI
|
|
JKZLZICMZUAKZXSAUBUCZJKZBDMZANOZXSPOZQZLZDGUDZCGUDZYHCGUEZYHCGUFXOXPXQAIU
|
|
GYIXRAXOXPXQUHZUIABCDUJUKXRYHIEMZUAKZYLYAJKZBYDYLPOZQZLZDGUDZRXSYLQZULZEG
|
|
UNCGUNYJXRYTCEGGXRXSGHZYLGHZRZRZYHYRYSUUDYGYRYSULZDGUUDDUMYRYSDYQDGUOYSDU
|
|
MUPUUDYCGHZYGUUEYRYMYNBFMZANOZYLPOZQZLZFGUDUUDUUFRZYGRZYSYQUUKDFGYCUUGQZY
|
|
PUUJYMYNUUNYOUUIBUUNYDUUHYLPYCUUGANUQURUSUTVAUUMUUKYSFGUUMUUGGHZRZUUKYSUU
|
|
PUUKRZYCUUGJKZYSUUNUUGYCJKZUUQUURRUURYSUUQUURVBUUQUURVCUURUUQUURUUGVDZYCV
|
|
DZJKUUQAVDZUUTYLXSUVAXRUVBVFHZUUCUUFYGUUOUUKXRUVBGHIUVBJKZUVCXRAXOXPXQVGZ
|
|
VEXRXQUVDYKXRAXRAUVEVHVIVJUVBVKVLVMZUULUUBYGUUOUUKXRUUAUUBUUFVNSZUULUUAYG
|
|
UUOUUKXRUUAUUBUUFVQSZUUQUUGUUMUUOUUKVOZVEZUUQYCUULUUFYGUUOUUKUUDUUFVBSZVE
|
|
ZUUMXTUUOUUKUULXTYBYFVPVRUUQYLYAUVBJUUPYMYNUUJVSUUQAUUQAUULXPYGUUOUUKXOXP
|
|
XQUUCUUFVTSZVHZUUQAIUVNUUQWAUULXQYGUUOUUKXOXPXQUUCUUFWBSWCWDZWEUUQBUUTUVB
|
|
NOZYLPOZUVAUVBNOZXSPOZUUQBUUIUVQUUPYMYNUUJWFZUUQUVPUUHYLPUUQUUGAUUQUUGUVI
|
|
WPUUQAUVMWPZWGURWHZUUQBYEUVSUUMYFUUOUUKUULXTYBYFWFVRZUUQUVRYDXSPUUQYCAUUQ
|
|
YCUVKWPUWAWGURWHZWIWJUUQYCUUGUUQYCUVKVHZUUQUUGUVIVHZWKWLTWQUUQUUNRZAYCXSY
|
|
LUUGUUQXPUUNUVMTUUQUUAUUNUVHTUUQUUBUUNUVGTUUQUUFUUNUVKTUUQUUOUUNUVITUUQUU
|
|
NVBUWGBYEUUIUUQYFUUNUWCTUUQUUJUUNUVTTWIWMUUQUUSRUUSYSUUQUUSVBUUQUUSVCUUSU
|
|
UQUUSUVAUUTJKUUQUVBUVAXSYLUUTUVFUVHUVGUVLUVJUUPYMYNUUJVPUUQXSYAUVBJUUMYBU
|
|
UOUUKUULXTYBYFVSVRUVOWEUUQBUVSUVQUWDUWBWIWJUUQUUGYCUWFUWEWKWLTWQUUQUUFUUO
|
|
UURUUNUUSWNUUMUUFUUOUUKUUDUUFYGVOVRUVIYCUUGWOWRWSXEWTXAXBXCXFXDYHYRCEGYSY
|
|
GYQDGYSXTYMYBYNYFYPXSYLIUAXGXSYLYAJXHYSYEYOBXSYLYDPXIUSXJXKXLXMYHCGXNVL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d D q r $. $d N q r $.
|
|
$( The division algorithm (theorem). Dividing an integer ` N ` by a
|
|
nonzero integer ` D ` produces a (unique) quotient ` q ` and a unique
|
|
remainder ` 0 <_ r < ( abs `` D ) ` . Theorem 1.14 in [ApostolNT]
|
|
p. 19. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
divalg $p |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> E! r e. ZZ E. q e. ZZ
|
|
( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) ) $=
|
|
( cz wcel cc0 wne w3a clt wbr cv cle cabs co wceq wa simpl1 simpl2 simpr
|
|
cmul caddc wrex wreu divalglemeuneg syl3anc simpl3 pm2.21ddne cn sylanbrc
|
|
cfv elnnz divalglemeunn syl2anc w3o ztri3or0 3ad2ant2 mpjao3dan ) BEFZAEF
|
|
ZAGHZIZAGJKZGCLZMKVDANUKJKBDLAUAOVDUBOPIDEUCCEUDZAGPZGAJKZVBVCQUSUTVCVEUS
|
|
UTVAVCRUSUTVAVCSVBVCTABCDUEUFVBVFQVEAGVBVFTUSUTVAVFUGUHVBVGQZUSAUIFZVEUSU
|
|
TVAVGRVHUTVGVIUSUTVAVGSVBVGTAULUJABCDUMUNUTUSVCVFVGUOVAAUPUQUR $.
|
|
|
|
$( Express the division algorithm as stated in ~ divalg in terms of
|
|
` || ` . (Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
divalgb $p |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) ->
|
|
( E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\
|
|
N = ( ( q x. D ) + r ) ) <->
|
|
E! r e. NN0 ( r < ( abs ` D ) /\ D || ( N - r ) ) ) ) $=
|
|
( cz wcel cc0 cv wbr co caddc wceq w3a wrex wreu wa cn0 wb cc zcn cle cfv
|
|
cabs clt cmul cmin cdvds wne zsubcl divides sylan2 3impb 3com12 wi zmulcl
|
|
zcnd subadd syl3an addcom syl2an 3adant1 eqeq1d bitrd eqcom 3expia 3impia
|
|
3bitr3g expcomd imp rexbidva 3com23 anbi2d df-3an rexbii r19.42v syl6rbbr
|
|
bitri anass syl6bb 3expa reubidva weu elnn0z anbi1i eubii df-reu 3bitr4ri
|
|
3adant3 ) BEFZAEFZGCHZUAIZWKAUCUBUDIZBDHZAUEJZWKKJZLZMZDENZCEOZWMABWKUFJZ
|
|
UGIZPZCQOZRAGUHWIWJPZWTWLXCPZCEOZXDXEWSXFCEWIWJWKEFZWSXFRWIWJXHMZWSWLWMPZ
|
|
XBPZXFXIXKXJWQDENZPZWSXIXBXLXJXIXBWOXALZDENZXLWJWIXHXBXORZWJWIXHXPWIXHPZW
|
|
JXAEFXPBWKUIDAXAUJUKULUMWIXHWJXOXLRWIXHWJMZXNWQDEXRWNEFZXNWQRZWIXHWJXSXTU
|
|
NXQXSWJXTWIXHXSWJPZXTWIXHYAMZXAWOLZWPBLZXNWQYBYCWKWOKJZBLZYDWIBSFXHWKSFZY
|
|
AWOSFZYCYFRBTWKTZYAWOWNAUOUPZBWKWOUQURYBYEWPBXHYAYEWPLZWIXHYGYHYKYAYIYJWK
|
|
WOUSUTVAVBVCXAWOVDWPBVDVGVEVHVFVIVJVKVCVLWSXJWQPZDENXMWRYLDEWLWMWQVMVNXJW
|
|
QDEVOVQVPWLWMXBVRVSVTWAWKQFZXCPZCWBXHXFPZCWBXDXGYNYOCYNXHWLPZXCPYOYMYPXCW
|
|
KWCWDXHWLXCVRVQWEXCCQWFXFCEWFWGVSWH $.
|
|
|
|
$( The division algorithm (theorem) for a positive divisor. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
divalg2 $p |- ( ( N e. ZZ /\ D e. NN ) ->
|
|
E! r e. NN0 ( r < D /\ D || ( N - r ) ) ) $=
|
|
( vq cz wcel cn wa cv cabs cfv clt wbr cmin co cn0 wreu cc0 w3a mpbid wne
|
|
cdvds nnz nnne0 jca cmul caddc wceq wrex divalg divalgb 3expb sylan2 nnre
|
|
cle wb nnnn0 nn0ge0d absidd breq2d anbi1d reubidv adantl ) BEFZAGFZHCIZAJ
|
|
KZLMZABVFNOUBMZHZCPQZVFALMZVIHZCPQZVEVDAEFZARUAZHVKVEVOVPAUCAUDUEVDVOVPVK
|
|
VDVOVPSRVFUOMVHBDIAUFOVFUGOUHSDEUICEQVKABCDUJABCDUKTULUMVEVKVNUPVDVEVJVMC
|
|
PVEVHVLVIVEVGAVFLVEAAUNVEAAUQURUSUTVAVBVCT $.
|
|
$}
|
|
|
|
${
|
|
$d D z $. $d N z $. $d R z $.
|
|
$( The result of the ` mod ` operator satisfies the requirements for the
|
|
remainder ` R ` in the division algorithm for a positive divisor
|
|
(compare ~ divalg2 and ~ divalgb ). This demonstration theorem
|
|
justifies the use of ` mod ` to yield an explicit remainder from this
|
|
point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by
|
|
AV, 21-Aug-2021.) $)
|
|
divalgmod $p |- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <->
|
|
( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) ) ) $=
|
|
( vz cz wcel wa wceq clt wbr cmin cdvds cn0 csn adantr adantl syl syl3anc
|
|
co cq cn cmo cv crab zq simpr nngt0d modqcld snidg eleq1 syl5ibrcom elsni
|
|
nnq impbid1 crio cc0 modqlt cdiv cfl cfv cmul znq flqcld nnz zmodcl nn0zd
|
|
zsubcl syldan cc nncn zcnd mulcomd modqval nn0cnd zmulcl syl2an2 subexsub
|
|
mpbid eqtr3d dvds0lem syl31anc wreu wb divalg2 breq1 oveq2 breq2d anbi12d
|
|
zcn riota2 syl2anc mpbi2and eqcomd sneqd eqtr4d eleq2d bitrd elrab syl6bb
|
|
snriota ) CEFZAUAFZGZBCAUBSZHZBDUCZAIJZACXFKSZLJZGZDMUDZFZBMFBAIJZACBKSZL
|
|
JZGZGXCXEBXDNZFZXLXCXEXRXCXRXEXDXQFZXCXDTFXSXCCAXACTFZXBCUEOZXBATFZXAAUMP
|
|
ZXCAXAXBUFUGZUHXDTUIQBXDXQUJUKBXDULUNXCXQXKBXCXQXJDMUOZNZXKXCXDYEXCYEXDXC
|
|
XDAIJZACXDKSZLJZYEXDHZXCXTYBUPAIJZYGYAYCYDCAUQRXCCAURSZUSUTZEFZAEFZYHEFZY
|
|
MAVASZYHHYIXCYLCAVBVCZXBYOXAAVDZPXAXBXDEFYPXCXDCAVEZVFCXDVGVHXCAYMVASZYQY
|
|
HXCAYMXBAVIFXAAVJPXCYMYRVKVLXCXDCUUAKSHZUUAYHHXCXTYBYKUUBYAYCYDCAVMRXCXDU
|
|
UACXCXDYTVNXCUUAXBYOXAYNUUAEFYSYRAYMVOVPVKXACVIFXBCWIOVQVRVSYMAYHVTWAXCXD
|
|
MFXJDMWBZYGYIGZYJWCYTACDWDZXJUUDDMXDXFXDHZXGYGXIYIXFXDAIWEUUFXHYHALXFXDCK
|
|
WFWGWHWJWKWLWMWNXCUUCXKYFHUUEXJDMWTQWOWPWQXJXPDBMXFBHZXGXMXIXOXFBAIWEUUGX
|
|
HXNALXFBCKWFWGWHWRWS $.
|
|
$}
|
|
|
|
$( The result of the ` mod ` operator satisfies the requirements for the
|
|
remainder ` R ` in the division algorithm for a positive divisor. Variant
|
|
of ~ divalgmod . (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof
|
|
shortened by AV, 21-Aug-2021.) $)
|
|
divalgmodcl $p |- ( ( N e. ZZ /\ D e. NN /\ R e. NN0 )
|
|
-> ( R = ( N mod D ) <-> ( R < D /\ D || ( N - R ) ) ) ) $=
|
|
( cz wcel cn cn0 cmo co wceq clt wbr cmin cdvds wa wb divalgmod baibd 3impa
|
|
) CDEZAFEZBGEZBCAHIJZBAKLACBMINLOZPTUAOUCUBUDABCQRS $.
|
|
|
|
${
|
|
$d D z $. $d N z $. $d R z $.
|
|
$( The result of the modulo operation is the remainder of the division
|
|
algorithm. (Contributed by AV, 19-Aug-2021.) $)
|
|
modremain $p |- ( ( N e. ZZ /\ D e. NN /\ ( R e. NN0 /\ R < D ) )
|
|
-> ( ( N mod D ) = R
|
|
<-> E. z e. ZZ ( ( z x. D ) + R ) = N ) ) $=
|
|
( cmo co wceq cz wcel cn cn0 wbr wa wrex eqcom wb 3ad2ant3 adantr syl5bb
|
|
cc clt cv cmul caddc cmin cdvds divalgmodcl 3adant3r ibar adantl 3ad2ant2
|
|
w3a nnz simp1 zsubcld divides syl2anc zcn 3ad2ant1 nn0cn zmulcld subadd2d
|
|
nn0z simpr zcnd rexbidva bitrd 3bitr2d ) DBEFZCGCVIGZDHIZBJIZCKIZCBUALZMZ
|
|
ULZAUBZBUCFZCUDFDGZAHNZVICOVPVJVNBDCUEFZUFLZMZWBVTVKVLVMVJWCPVNBCDUGUHVOV
|
|
KWBWCPZVLVNWDVMVNWBUIUJQVPWBVRWAGZAHNZVTVPBHIZWAHIWBWFPVLVKWGVOBUMUKZVPDC
|
|
VKVLVOUNVOVKCHIZVLVMWIVNCVCRQUOABWAUPUQVPWEVSAHWEWAVRGVPVQHIZMZVSVRWAOWKD
|
|
CVRVPDTIZWJVKVLWLVODURUSRVPCTIZWJVOVKWMVLVMWMVNCUTRQRWKVRWKVQBVPWJVDVPWGW
|
|
JWHRVAVEVBSVFVGVHS $.
|
|
$}
|
|
|
|
${
|
|
$d D r x $. $d K r $. $d N r x $.
|
|
$( Corollary of the division algorithm. If an integer ` D ` greater than
|
|
` 1 ` divides ` N ` , then it does not divide any of ` N - 1 ` ,
|
|
` N - 2 ` ... ` N - ( D - 1 ) ` . (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
ndvdssub $p |- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) ->
|
|
( D || N -> -. D || ( N - K ) ) ) $=
|
|
( vr vx wcel clt wbr wa cdvds cmin co wi cn0 wceq wral breq1 oveq2 breq2d
|
|
cc0 cz cn wn w3a wne nnnn0 nnne0 jca df-ne anbi2i cv wrex divalg2 anbi12d
|
|
wreu sylib nngt0 3ad2ant2 zcn subid1d biimpar 3adant2 3expa anim2i ancoms
|
|
reu4 0nn0 anbi2d eqeq2 imbi12d rspcv syl5 ralimi simpl2im r19.21v pm2.43i
|
|
ax-mp expd 3impia eqeq1 syl5com pm3.37 syl6 syl7bi exp4a com23 imp4a syl7
|
|
impd 3expia ) CUAFZAUBFZBUBFZBAGHZIZACJHZACBKLZJHZUCZMWKWLIZWPWOWSWKWLWPW
|
|
OWSMWKWLWPUDZWMWNWSXAWNWMWSWMBNFZBTUEZIXAWNWSWMXBXCBUFBUGUHXAWNXBXCWSXAXB
|
|
WNXCWSMXAXBWNXCWSWNXCIWNBTOZUCZIZXAXBWSXCXEWNBTUIUJXAXBWNWRIZXDMZXFWSMXAD
|
|
UKZAGHZACXIKLZJHZIZXITOZMZDNPZXBXHWKWLWPXPWTWPXPMWTWTWPXPWTWTWPIZXOMZDNPZ
|
|
XQXPMWTXMDNULZXMEUKZAGHZACYAKLZJHZIZIZXIYAOZMZENPZDNPZXSWTXMDNUOXTYJIACDU
|
|
MXMYEDENYGXJYBXLYDXIYAAGQYGXKYCAJXIYACKRSUNVFUPYIXRDNYIXQXMXNXQXMIXMTAGHZ
|
|
ACTKLZJHZIZIZYIXNXMXQYOXQYNXMWKWLWPYNXAYKYMWLWKYKWPAUQURWKWPYMWLWKYMWPWKY
|
|
LCAJWKCCUSUTSVAVBUHVCVDVETNFYIYOXNMZMVGYHYPETNYATOZYFYOYGXNYQYEYNXMYQYBYK
|
|
YDYMYATAGQYQYCYLAJYATCKRSUNVHYATXIVIVJVKVQVLVRVMVNXQXODNVOUPVRVPVSXOXHDBN
|
|
XIBOZXMXGXNXDYRXJWNXLWRXIBAGQYRXKWQAJXIBCKRSUNXIBTVTVJVKWAWNWRXDWBWCWDWEW
|
|
FWGWHWFWIWJWFVS $.
|
|
$}
|
|
|
|
$( Corollary of the division algorithm. If an integer ` D ` greater than
|
|
` 1 ` divides ` N ` , then it does not divide any of ` N + 1 ` ,
|
|
` N + 2 ` ... ` N + ( D - 1 ) ` . (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
ndvdsadd $p |- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) ->
|
|
( D || N -> -. D || ( N + K ) ) ) $=
|
|
( cz wcel cn clt wbr wa w3a cdvds cmin co wn wi cc0 cr wb syl2an cc syl2anr
|
|
caddc nnre posdif pm5.32i nnz zsubcl elnnz biimpri sylan sylbi anasss nngt0
|
|
ltsubpos biimpd expcom mpdi imp adantrr jca ndvdssub syld3an3 zaddcl sylan2
|
|
3adant1 dvdssubr an12s 3impb wceq zcn subsub3 syl3an breq2d bitr4d 3adant3r
|
|
nncn notbid sylibrd ) CDEZAFEZBFEZBAGHZIZJACKHZACABLMZLMZKHZNZACBUBMZKHZNZV
|
|
SVTWCWEFEZWEAGHZIZWDWHOVTWCWNVSVTWCIWLWMVTWAWBWLVTWAIZWBIWOPWEGHZIWLWOWBWPW
|
|
ABQEZAQEZWBWPRVTBUCZAUCZBAUDUAUEWOWEDEZWPWLVTADEZBDEZXAWAAUFZBUFZABUGSWLXAW
|
|
PIWEUHUIUJUKULVTWAWMWBVTWAWMVTWAPBGHZWMBUMWAVTXFWMOWAVTIXFWMWAWQWRXFWMRVTWS
|
|
WTBAUNSUOUPUQURUSUTVEAWECVAVBVSVTWAWKWHRWBVSVTWAJZWJWGXGWJAWIALMZKHZWGVSVTW
|
|
AWJXIRZVTVSWAXJVTXBWIDEZXJVSWAIXDWAVSXCXKXECBVCVDAWIVFSVGVHXGWFXHAKVSCTEVTA
|
|
TEWABTEWFXHVICVJAVPBVPCABVKVLVMVNVQVOVR $.
|
|
|
|
$( Special case of ~ ndvdsadd . If an integer ` D ` greater than ` 1 `
|
|
divides ` N ` , it does not divide ` N + 1 ` . (Contributed by Paul
|
|
Chapman, 31-Mar-2011.) $)
|
|
ndvdsp1 $p |- ( ( N e. ZZ /\ D e. NN /\ 1 < D ) ->
|
|
( D || N -> -. D || ( N + 1 ) ) ) $=
|
|
( c1 clt wbr cz wcel cn wa cdvds caddc co wn wi 1nn jctl ndvdsadd syl3an3 )
|
|
CADEZBFGAHGCHGZSIABJEABCKLJEMNSTOPACBQR $.
|
|
|
|
${
|
|
ndvdsi.1 $e |- A e. NN $.
|
|
ndvdsi.2 $e |- Q e. NN0 $.
|
|
ndvdsi.3 $e |- R e. NN $.
|
|
ndvdsi.4 $e |- ( ( A x. Q ) + R ) = B $.
|
|
ndvdsi.5 $e |- R < A $.
|
|
$( A quick test for non-divisibility. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) $)
|
|
ndvdsi $p |- -. A || B $=
|
|
( cmul co caddc cdvds wbr wn cz wcel nnzi mp2an cn dvdsmul1 clt wa zmulcl
|
|
nn0zi wi pm3.2i ndvdsadd mp3an ax-mp breq2i mtbi ) AACJKZDLKZMNZABMNAUMMN
|
|
ZUOOZAPQZCPQZUPAERZCFUEZACUASUMPQZATQDTQZDAUBNZUCUPUQUFURUSVBUTVAACUDSEVC
|
|
VDGIUGADUMUHUIUJUNBAMHUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( The floor of an odd integer divided by 4. (Contributed by AV,
|
|
17-Jun-2021.) $)
|
|
flodddiv4 $p |- ( ( M e. ZZ /\ N = ( ( 2 x. M ) + 1 ) )
|
|
-> ( |_ ` ( N / 4 ) )
|
|
= if ( 2 || M , ( M / 2 ) , ( ( M - 1 ) / 2 ) ) ) $=
|
|
( cz wcel c2 co c1 caddc wceq wa c4 cdiv cfl wbr cc a1i cc0 eqtrd adantr
|
|
c3 vx cmul cfv cdvds cmin cif 2cnd zcn mulcld 1cnd 4cn cap 4ap0 divdirapd
|
|
oveq1 eqcomi oveq2d 2ap0 divcanap5d oveq1d sylan9eqr fveq2d wn iftrue cle
|
|
2t2e4 clt cr 1re 0le1 4re 4pos divge0 mp4an 1lt4 recgt1 mp2an mpbi pm3.2i
|
|
wb cq evend2 biimpac 4nn nnrecq ax-mp flqbi2 sylancl mpbiri eqtr4d expcom
|
|
cn iffalse cv wrex odd2np1 wi ax-1cn divcanap5 mp3an 2t1e2 oveq12i eqtr3i
|
|
2cn oveq1i divdirapi 2p1e3 3eqtr2i 3re 0re 3pos ltleii 3lt4 nnrp divlt1lt
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|
crp mpbir 3z znq mpan2 eqcoms 2z zmulcld zcnd divcanap3d recclapi addassd
|
|
id halfcn pncan1 3eqtr4rd ex adantl rexlimdva sylbid impcom mpjaod eqcomd
|
|
syl zeo3 ) ACDZBEAUBFZGHFZIZJZBKLFZMUCAELFZGKLFZHFZMUCZEAUDNZUUGAGUEFZELF
|
|
ZUFZUUEUUFUUIMUUDUUAUUFUUCKLFZUUIBUUCKLUOUUAUUOUUBKLFZUUHHFUUIUUAUUBGKUUA
|
|
EAUUAUGZAUHZUIUUAUJKODUUAUKPKQULNUUAUMPUNUUAUUPUUGUUHHUUAUUPUUBEEUBFZLFUU
|
|
GUUAKUUSUUBLKUUSIUUAUUSKVFUPPUQUUAAEEUURUUQUUQEQULNZUUAURPZUVAUSRUTRVAVBU
|
|
UAUUJUUNIUUDUUAUUNUUJUUAUUKUUNUUJIZUUKVCZUUKUUAUVBUUKUUAJZUUNUUGUUJUUKUUN
|
|
UUGIUUAUUKUUGUUMVDSUVDUUJUUGIZQUUHVENZUUHGVGNZJZUVFUVGGVHDQGVENKVHDZQKVGN
|
|
ZUVFVIVJVKVLGKVMVNGKVGNZUVGVOUVIUVJUVKUVGVTVKVLKVPVQVRVSUVDUUGCDZUUHWADZU
|
|
VEUVHVTUUAUUKUVLAWBWCKWLDZUVMWDKWEWFUUHUUGWGWHWIWJWKUVCUUAUVBUVCUUAJUUNUU
|
|
MUUJUVCUUNUUMIUUAUUKUUGUUMWMSUUAUVCUUMUUJIZUUAUVCEUAWNZUBFZGHFZAIZUACWOUV
|
|
OUAAWPUUAUVSUVOUACUVPCDZUVSUVOWQUUAUVTUVSUVOUVTUVSJZUVPGELFZUUHHFZHFZMUCZ
|
|
UVPUUJUUMUVTUWEUVPIUVSUVTUWEUVPTKLFZHFZMUCZUVPUVTUWDUWGMUVTUWCUWFUVPHUWCU
|
|
WFIUVTUWCEKLFZUUHHFEGHFZKLFUWFUWBUWIUUHHEGUBFZUUSLFZUWBUWIGODEODZUUTJZUWN
|
|
UWLUWBIWRUWMUUTXDURVSZUWOGEEWSWTUWKEUUSKLXAVFXBXCXEEGKXDWRUKUMXFUWJTKLXGX
|
|
EXHPUQVBUVTUWHUVPIZQUWFVENZUWFGVGNZJZUWQUWRTVHDZQTVENUVIUVJUWQXIQTXJXIXKX
|
|
LVKVLTKVMVNUWRTKVGNZXMUWTKXPDZUWRUXAVTXIUVNUXBWDKXNWFTKXOVQXQVSUVTUWFWADZ
|
|
UWPUWSVTTCDUVNUXCXRWDTKXSVQUWFUVPWGXTWIRSUWAUUIUWDMUWAUUIUVPUWBHFZUUHHFZU
|
|
WDUWAUUGUXDUUHHUVSUVTUUGUVRELFZUXDUUGUXFIAUVRAUVRELUOYAUVTUXFUVQELFZUWBHF
|
|
UXDUVTUVQGEUVTUVQUVTEUVPECDUVTYBPUVTYHYCYDZUVTUJUVTUGZUUTUVTURPZUNUVTUXGU
|
|
VPUWBHUVTUVPEUVPUHZUXIUXJYEZUTRVAUTUVTUXEUWDIUVSUVTUVPUWBUUHUXKUWBODUVTYI
|
|
PUUHODUVTKUKUMYFPYGSRVBUWAUUMUXGUVPUWAUULUVQELUVSUVTUULUVRGUEFZUVQUULUXMI
|
|
AUVRAUVRGUEUOYAUVTUVQODUXMUVQIUXHUVQYJYSVAUTUVTUXGUVPIUVSUXLSRYKYLYMYNYOY
|
|
PRWKAYTYQYRSR $.
|
|
$}
|
|
|
|
$( The floor of an integer divided by a nonzero integer not dividing the
|
|
first integer is less than the integer divided by the positive integer.
|
|
(Contributed by AV, 4-Jul-2021.) $)
|
|
fldivndvdslt $p |- ( ( K e. ZZ /\ ( L e. ZZ /\ L =/= 0 ) /\ -. L || K )
|
|
-> ( |_ ` ( K / L ) ) < ( K / L ) ) $=
|
|
( cz wcel cc0 wne wa cdvds wbr wn w3a co cq cfl cfv clt zq 3ad2ant1 syl3anc
|
|
cdiv adantr 3ad2ant2 simp2r qdivcl wb simprl simprr dvdsval2 notbid biimp3a
|
|
simpl flqltnz syl2anc ) ACDZBCDZBEFZGZBAHIZJZKZABTLZMDZVACDZJZVANOVAPIUTAMD
|
|
ZBMDZUPVBUNUQVEUSAQRUQUNVFUSUOVFUPBQUAUBUNUOUPUSUCABUDSUNUQUSVDUNUQGZURVCVG
|
|
UOUPUNURVCUEUNUOUPUFUNUOUPUGUNUQUKBAUHSUIUJVAULUM $.
|
|
|
|
$( The floor of an odd number divided by 4 is less than the odd number
|
|
divided by 4. (Contributed by AV, 4-Jul-2021.) $)
|
|
flodddiv4lt $p |- ( ( N e. ZZ /\ -. 2 || N )
|
|
-> ( |_ ` ( N / 4 ) ) < ( N / 4 ) ) $=
|
|
( cz wcel c2 cdvds wbr wn wa c4 cc0 wne cdiv co cfl cfv clt simpl 4z pm3.2i
|
|
4ne0 a1i 4dvdseven con3i adantl fldivndvdslt syl3anc ) ABCZDAEFZGZHZUGIBCZI
|
|
JKZHZIAEFZGZAILMZNOUPPFUGUIQUMUJUKULRTSUAUIUOUGUNUHAUBUCUDAIUEUF $.
|
|
|
|
$( The floor of an odd number divided by 4, multiplied by 2 is less than the
|
|
half of the odd number. (Contributed by AV, 4-Jul-2021.) $)
|
|
flodddiv4t2lthalf $p |- ( ( N e. ZZ /\ -. 2 || N )
|
|
-> ( ( |_ ` ( N / 4 ) ) x. 2 ) < ( N / 2 ) ) $=
|
|
( cz wcel c2 cdvds wbr wn wa c4 cdiv co cfl cfv cmul clt cr cc0 adantr wceq
|
|
a1i flodddiv4lt wb cn cq 4nn znq flqcld zred qre syl 2re 2pos pm3.2i ltmul1
|
|
mpan2 syl3anc mpbid zcn halfcld 2cnd cap divcanap1d divdivap1d 2t2e4 oveq2d
|
|
2ap0 eqtrd oveq1d eqtr3d breqtrrd ) ABCZDAEFGZHZAIJKZLMZDNKZVNDNKZADJKZOVMV
|
|
OVNOFZVPVQOFZAUAVMVOPCZVNPCZDPCZQDOFZHZVSVTUBVKWAVLVKVOVKVNVKIUCCVNUDCZUEAI
|
|
UFUOZUGUHRVKWBVLVKWFWBWGVNUIUJRWEVMWCWDUKULUMTVOVNDUNUPUQVKVRVQSVLVKVRDJKZD
|
|
NKVRVQVKVRDVKAAURZUSVKUTZDQVAFVKVFTZVBVKWHVNDNVKWHADDNKZJKVNVKADDWIWJWJWKWK
|
|
VCVKWLIAJWLISVKVDTVEVGVHVIRVJ $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The greatest common divisor operator
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c gcd $.
|
|
|
|
$( Extend the definition of a class to include the greatest common divisor
|
|
operator. $)
|
|
cgcd $a class gcd $.
|
|
|
|
${
|
|
$d n x y $.
|
|
$( Define the ` gcd ` operator. For example, ` ( -u 6 gcd 9 ) = 3 `
|
|
( ~ ex-gcd ). (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
df-gcd $a |- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 ,
|
|
sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) ) $.
|
|
$}
|
|
|
|
$( Decidablity lemma used in various proofs related to ` gcd ` .
|
|
(Contributed by Jim Kingdon, 12-Dec-2021.) $)
|
|
gcdmndc $p |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) ) $=
|
|
( cz wcel wa cc0 wceq wdc 0z zdceq mpan2 adantr adantl dcan sylc ) ACDZBCDZ
|
|
EAFGZHZBFGZHZRTEHPSQPFCDZSIAFJKLQUAPQUBUAIBFJKMRTNO $.
|
|
|
|
${
|
|
$d K n x y z $. $d M n y $. $d n ph y $. $d ps x y z $.
|
|
zsupcllemstep.dc $e |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps ) $.
|
|
$( Lemma for ~ zsupcl . Induction step. (Contributed by Jim Kingdon,
|
|
7-Dec-2021.) $)
|
|
zsupcllemstep $p |- ( K e. ( ZZ>= ` M )
|
|
-> ( ( ( ph /\ A. n e. ( ZZ>= ` K ) -. ps )
|
|
-> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y
|
|
/\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
|
|
-> ( ( ph /\ A. n e. ( ZZ>= ` ( K + 1 ) ) -. ps )
|
|
-> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y
|
|
/\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
|
|
) ) $=
|
|
( wcel wn wral wa clt wbr cz wi cr wb wceq cuz cv crab wrex c1 caddc wsbc
|
|
cfv co eluzelz ad3antrrr nfv nfcv nfra1 nfan nfrexya nfim elrabsf simprbi
|
|
wsb sylibr ad2antlr cle elrabi zltp1le syl2an biimpa peano2zd eluz adantr
|
|
sbsbc mpbird simprr nfs1v nfn sbequ12 rspc sylc pm2.65da ralrimi ad2antrr
|
|
notbid simpllr sylanbrc breq2 rspcev sylancom exp31 breq1 ralbidv anbi12d
|
|
imbi1d syl12anc csn sbcng biimpar sbcsng mpbid simplrr cfzo uzid peano2uz
|
|
ex cun fzouzsplit 3syl fzosn uneq1d eqtrd raleqdv ralunb syl6bb mpbir2and
|
|
syl simprl simplr mpand mpd wdc wo ralrimiva nfsbc1v nfdc sbceq1a exmiddc
|
|
dcbid mpjaodan ) GHUAUHZJZABKZFGUAUHZLZMZCUBZDUBZNOZKZDBFPUCZLZYOYNNOZYOE
|
|
UBZNOZEYRUDZQZDRLZMZCPUDZQZAYJFGUEUFUIZUAUHZLZMZUUGYIUUHMZUULMZBFGUGZUUGU
|
|
UOKZUUNUUOMZGPJZGYONOZKZDYRLZYOGNOZUUCQZDRLZUUGYIUURUUHUULUUOHGUJZUKZUUQU
|
|
UTDYRUUNUUODUUMUULDYIUUHDYIDULYMUUGDYMDULUUFDCPDPUMYSUUEDYQDYRUNUUDDRUNUO
|
|
UPUQUOUULDULUOUUODULUOZUUQYOYRJZUUTUUQUVHMZUUSBFDUTZUVHUVJUUQUUSUVHBFYOUG
|
|
ZUVJUVHYOPJZUVKBFYOPFPUMZURUSBFDVKVAVBUVIUUSMZYOUUJJZUUKUVJKZUVNUVOUUIYOV
|
|
COZUVIUUSUVQUUQUURUVLUUSUVQSUVHUVFBFYOPVDZGYOVEVFVGUVIUVOUVQSZUUSUUQUUIPJ
|
|
UVLUVSUVHUUQGUVFVHUVRUUIYOVIVFVJVLUUNUUKUUOUVHUUSUUMAUUKVMUKYJUVPFYOUUJUV
|
|
JFBFDVNVOFUBZYOTBUVJBFDVPWBVQVRVSXCVTUUQUVCDRUVGUUQYORJZUVBUUCUUQUWAMZUVB
|
|
GYRJZUUCUWBUVBMUURUUOUWCUUQUURUWAUVBUVFWAUUNUUOUWAUVBWCBFGPUVMURWDUUBUVBE
|
|
GYRUUAGYONWEWFWGWHVTUUFUVAUVDMCGPYNGTZYSUVAUUEUVDUWDYQUUTDYRUWDYPUUSYNGYO
|
|
NWIWBWJUWDUUDUVCDRUWDYTUVBUUCYNGYONWEWLWJWKWFWMUUNUUPMZYLUUGUWEYLYJFGWNZL
|
|
ZUUKUWEYJFGUGZUWGUUNUWHUUPYIUWHUUPSUUHUULBFGYHWOWAWPYIUWHUWGSUUHUULUUPYJF
|
|
GYHWQUKWRUUMAUUKUUPWSYIYLUWGUUKMZSUUHUULUUPYIYLYJFUWFUUJXDZLUWIYIYJFYKUWJ
|
|
YIYKGUUIWTUIZUUJXDZUWJYIUURUUIYKJZYKUWLTUVEUURGYKJUWMGXAGGXBXNGUUIXEXFYIU
|
|
WKUWFUUJYIUURUWKUWFTUVEGXGXNXHXIXJYJFUWFUUJXKXLUKXMUUNYLUUGQUUPUUNAYLUUGU
|
|
UMAUUKXOZYIUUHUULXPXQVJXRUUNUUOXSZUUOUUPXTUUNBXSZFYHLZUWOUUNAUWQUWNAUWPFY
|
|
HIYAXNYIUWQUWOQUUHUULUWPUWOFGYHUUOFBFGYBYCUVTGTBUUOBFGYDYFVQWAXRUUOYEXNYG
|
|
WH $.
|
|
$}
|
|
|
|
${
|
|
$d M k n w y $. $d ch n $. $d j n ph w y $. $d j ps w x y z $.
|
|
$d k n ph w y $. $d k ps w x y z $. $d n w x y z $.
|
|
zsupcllemex.m $e |- ( ph -> M e. ZZ ) $.
|
|
zsupcllemex.sbm $e |- ( n = M -> ( ps <-> ch ) ) $.
|
|
zsupcllemex.mtru $e |- ( ph -> ch ) $.
|
|
zsupcllemex.dc $e |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps ) $.
|
|
zsupcllemex.bnd $e |- ( ph ->
|
|
E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps ) $.
|
|
$( Lemma for ~ zsupcl . Existence of the supremum. (Contributed by Jim
|
|
Kingdon, 7-Dec-2021.) $)
|
|
zsupcllemex $p |- ( ph -> E. x e. ZZ (
|
|
A. y e. { n e. ZZ | ps } -. x < y /\
|
|
A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) $=
|
|
( cv cuz cfv wral wi wa vw vk wn clt cz crab wrex cr wcel simpl simprr c1
|
|
wbr caddc co wceq fveq2 raleqdv anbi2d imbi1d adantr uzid notbid 3syl imp
|
|
rspcv pm2.21dd a1i zsupcllemstep uzind4 ad2antrl mp2and rexlimddv ) ABUCZ
|
|
HGOZPQZRZDOZEOZUDUMUCEBHUEUFZRVSVRUDUMVSFOUDUMFVTUGSEUHRTDUEUGZGIPQZNAVOW
|
|
BUIZVQTZTAVQWAAWDUJAWCVQUKWCAVQTZWASZAVQAVNHUAOZPQZRZTZWASAVNHWBRZTZWASZA
|
|
VNHUBOZPQZRZTZWASAVNHWNULUNUOZPQZRZTZWASWFUAUBIVOWGIUPZWJWLWAXBWIWKAXBVNH
|
|
WHWBWGIPUQURUSUTWGWNUPZWJWQWAXCWIWPAXCVNHWHWOWGWNPUQURUSUTWGWRUPZWJXAWAXD
|
|
WIWTAXDVNHWHWSWGWRPUQURUSUTWGVOUPZWJWEWAXEWIVQAXEVNHWHVPWGVOPUQURUSUTWMIU
|
|
EUIZWLCWAACWKLVAAWKCUCZAXFIWBUIWKXGSJIVBVNXGHIWBHOIUPBCKVCVFVDVEVGVHABDEF
|
|
HWNIMVIVJVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d ph j n x y z u v $. $d ps j x y z u v $. $d ch j n x y z u v $.
|
|
$d M j n x y z u v $.
|
|
zsupcl.m $e |- ( ph -> M e. ZZ ) $.
|
|
zsupcl.sbm $e |- ( n = M -> ( ps <-> ch ) ) $.
|
|
zsupcl.mtru $e |- ( ph -> ch ) $.
|
|
zsupcl.dc $e |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps ) $.
|
|
zsupcl.bnd $e |- ( ph ->
|
|
E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps ) $.
|
|
$( Closure of supremum for decidable integer properties. The property
|
|
which defines the set we are taking the supremum of must (a) be true at
|
|
` M ` (which corresponds to the non-empty condition of classical
|
|
supremum theorems), (b) decidable at each value after ` M ` , and (c) be
|
|
false after ` j ` (which corresponds to the upper bound condition found
|
|
in classical supremum theorems). (Contributed by Jim Kingdon,
|
|
7-Dec-2021.) $)
|
|
zsupcl $p |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) ) $=
|
|
( vx vy vz vv cz cr clt wcel wbr vu crab csup cuz cfv cle cv wa weq wn wb
|
|
zred lttri3 adantl wss wral wrex wi zssre zsupcllemex mpsyl supclti elrab
|
|
ssrexv sylanbrc supubti mpd nltled a1i supelti eluz syl2anc mpbird ) ABEP
|
|
UBZQRUCZFUDUESZFVOUFTZAFVOAFGULALMNOUAQVNRUAUGZQSOUGZQSUHUAOUIVRVSRTUJVSV
|
|
RRTUJUHUKAVRVSUMUNZPQUOZALUGZMUGZRTUJMVNUPWCWBRTWCNUGRTNVNUQURMQUPUHZLPUQ
|
|
WDLQUQUSABCLMNDEFGHIJKUTZWDLPQVDVAZVBAFVNSZVOFRTUJAFPSZCWGGIBCEFPHVCVEALM
|
|
NOUAQVNFRVTWFVFVGVHAWHVOPSVPVQUKGALMNOUAQVNPRVTWEWAAUSVIVJFVOVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d B f g x y z $. $d f g ph x y z $.
|
|
zssinfcl.ex $e |- ( ph -> E. x e. RR ( A. y e. B -. y < x /\
|
|
A. y e. RR ( x < y -> E. z e. B z < y ) ) ) $.
|
|
zssinfcl.ss $e |- ( ph -> B C_ ZZ ) $.
|
|
zssinfcl.zz $e |- ( ph -> inf ( B , RR , < ) e. ZZ ) $.
|
|
$( The infimum of a set of integers is an element of the set. (Contributed
|
|
by Jim Kingdon, 16-Jan-2022.) $)
|
|
zssinfcl $p |- ( ph -> inf ( B , RR , < ) e. B ) $=
|
|
( vg vf cv cr clt c1 wbr wcel wa wn adantr cz cinf caddc co wrex readdcld
|
|
zred 1red ltp1d wceq lttri3 adantl infglbti mp2and cle wss simprl inflbti
|
|
sseldd imp adantrr nltled simprr zleltp1 syl2anc mpbird letri3d mpbir2and
|
|
wb eqeltrd rexlimddv ) ADKZELMUAZNUBUCZMOZVLEPDEAVMLPVLVMMOVNDEUDAVLNAVLH
|
|
UFZAUGUEAVLVOUHABCDIJLEVMMJKZLPIKZLPQVPVQUIVPVQMORVQVPMORQVHAVPVQUJUKZFUL
|
|
UMAVKEPZVNQZQZVLVKEWAVLVKUIVLVKUNOVKVLUNOZWAVLVKAVLLPVTVOSZWAVKWAETVKAETU
|
|
OVTGSAVSVNUPZURZUFZAVSVKVLMORZVNAVSWGABCDIJLEVKMVRFUQUSUTVAWAWBVNAVSVNVBW
|
|
AVKTPVLTPZWBVNVHWEAWHVTHSVKVLVCVDVEWAVLVKWCWFVFVGWDVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d A j m y $. $d A m n $. $d M j m $. $d M m n $. $d S a m w $.
|
|
$d S b m $. $d S j m x y z $. $d a m n $. $d b m n ph $.
|
|
$d j m ph x y $. $d w x y z $.
|
|
infssuzledc.m $e |- ( ph -> M e. ZZ ) $.
|
|
infssuzledc.s $e |- S = { n e. ( ZZ>= ` M ) | ps } $.
|
|
infssuzledc.a $e |- ( ph -> A e. S ) $.
|
|
infssuzledc.dc $e |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps ) $.
|
|
$( Existence of the infimum of a subset of an upper set of integers.
|
|
(Contributed by Jim Kingdon, 13-Jan-2022.) $)
|
|
infssuzex $p |- ( ph -> E. x e. RR ( A. y e. S -. y < x
|
|
/\ A. y e. RR ( x < y -> E. z e. S z < y ) ) ) $=
|
|
( vm wbr wcel cr wa cz syl vw vj vb va cv wn cneg crab wral wrex wi zssre
|
|
clt wss cuz eleq2i sylib elrabi eluzelz znegcld wceq negeq eleq1d negnegd
|
|
cfv zcnd eqeltrd cle wdc wsbc cfz co adantr zred adantl eluzle lenegcon1d
|
|
simpr jca wb ad2antrr elfz syl3anc mpbird ralrimiva nfsbc1v sbceq1a dcbid
|
|
nfdc rspc sylc elfzuz biantrurd nfcv elrabsf syl6rbbr syl5bb wo nsyl olcd
|
|
eleq2s df-dc sylibr zdcle syl2anc mpjaodan c1 caddc lenegd mpbid peano2uz
|
|
exmiddc renegcld peano2re ad2antlr adantlr lenegcon2d letrd zltp1le ltnrd
|
|
eluz pm2.65da fveq2 raleqdv rspcev zsupcllemex zre anim1i cc recn znegclb
|
|
impbii weq elrab 3bitr4i a1i eqrdv rexeqdv imbi2d ralbidv anbi12d rexbidv
|
|
ssrexv mpsyl ssrab2 supinfneg eluzelre elrab3 renegcl 3bitrd eqriv raleqi
|
|
pm5.21nii rexeqi imbi2i ralbii anbi12i rexbii ) ADUEZCUEZUMOZUFZDUAUEZUGZ
|
|
NUEZUGZGPZNQUHZPZUAQUHZUIZUUTUUSUMOZEUEZUUSUMOZEUVJUJZUKZDQUIZRZCQUJUVBDG
|
|
UIZUVLUVNEGUJZUKZDQUIZRZCQUJACDEUAUVHSQUNAUVLUFZDUVHUIZUVAUUSUVMUMOZEUVHU
|
|
JZUKZDQUIZRZCSUJZUWJCQUJULAUWDDUVGNSUHZUIZUVAUWFEUWLUJZUKZDQUIZRZCSUJUWKA
|
|
UVGFUGZUGZGPCDEUBNUWRAFAFIUOVEZPZFSPZAFBHUWTUHZPZUXAAFGPUXDLGUXCFKUPUQBHF
|
|
UWTURTZIFUSTZUTZUVEUWRVAUVFUWSGUVEUWRVBVCAUWSFGAFAFUXFVFVDLVGAUVEUWRUOVEZ
|
|
PZRZIUVFVHOZUVGVIZUXKUFZUXJUXKRZUXLBHUVFVJZVIZUXNUVFIFVKVLZPZBVIZHUXQUIZU
|
|
XPUXNUXRUXKUVFFVHOZRZUXNUXKUYAUXJUXKVRUXJUYAUXKUXJFUVEUXJFAUXBUXIUXFVMVNU
|
|
XJUVEUXIUVESPZAUWRUVEUSVOZVNUXIUWRUVEVHOAUWRUVEVPVOVQVMVSUXNUVFSPZISPZUXB
|
|
UXRUYBVTUXJUYEUXKUXJUVEUYDUTZVMAUYFUXIUXKJWAAUXBUXIUXKUXFWAUVFIFWBWCWDZAU
|
|
XTUXIUXKAUXSHUXQMWEWAUXSUXPHUVFUXQUXOHBHUVFWFWIHUEUVFVABUXOBHUVFWGWHWJWKU
|
|
XNUVGUXOUVGUVFUXCPZUXNUXOGUXCUVFKUPUXNUXOUVFUWTPZUXORUYIUXNUYJUXOUXNUXRUY
|
|
JUYHUVFIFWLTWMBHUVFUWTHUWTWNWOWPWQWHWDUXJUXMRZUVGUVGUFZWRUXLUYKUYLUVGUYKU
|
|
XKUVGUXJUXMVRUXKUVFUXCGUYIUYJUXKBHUVFUWTURZIUVFVPZTKXAWSWTUVGXBXCUXJUXKVI
|
|
ZUXKUXMWRUXJUYFUYEUYOAUYFUXIJVMUYGIUVFXDXEUXKXLTXFAIUGZXGXHVLZUXHPZUYLNUY
|
|
QUOVEZUIZUYLNUBUEZUOVEZUIZUBUXHUJAUYPUXHPZUYRAVUDUWRUYPVHOZAIFVHOZVUEAUXA
|
|
VUFUXEIFVPTAIFAIJVNZAFUXFVNXIXJAUWRSPUYPSPZVUDVUEVTUXGAIJUTZUWRUYPYAXEWDU
|
|
WRUYPXKTAUYLNUYSAUVEUYSPZRZUVGUYPUYPUMOZVUKUVGRZVULUYQUYPVHOZVUMUYQUVEUYP
|
|
VUMUYPQPUYQQPVUMIAIQPVUJUVGVUGWAZXMZUYPXNTVUMUVEVUJUYCAUVGUYQUVEUSXOVNZVU
|
|
PVUJUYQUVEVHOAUVGUYQUVEVPXOVUMIUVEVUOVUQAUVGUXKVUJAUVGRUYJUXKUVGUYJAUYJUV
|
|
FUXCGUYMKXAVOUYNTXPXQXRVUMVUHVUHVULVUNVTAVUHVUJUVGVUIWAZVURUYPUYPXSXEWDVU
|
|
MUYPVUPXTYBWEVUCUYTUBUYQUXHVUAUYQVAUYLNVUBUYSVUAUYQUOYCYDYEXEYFAUWQUWJCSA
|
|
UWMUWEUWPUWIAUWDDUWLUVHAUCUWLUVHUCUEZUWLPZVUSUVHPZVTAVUSSPZVUSUGZGPZRZVUS
|
|
QPZVVDRZVUTVVAVVEVVGVVBVVFVVDVUSYGYHVVGVVBVVDVVGVVBVVCSPZVVDVVHVVFVVDVVCU
|
|
WTPZVVHVVIVVCUXCGBHVVCUWTURKXAIVVCUSTVOVVFVVBVVHVTZVVDVVFVUSYIPVVJVUSYJVU
|
|
SYKTVMWDVVFVVDVRVSYLUVGVVDNVUSSNUCYMUVFVVCGUVEVUSVBVCZYNUVGVVDNVUSQVVKYNY
|
|
OYPYQZYDAUWOUWHDQAUWNUWGUVAAUWFEUWLUVHVVLYRYSYTUUAUUBXJUWJCSQUUCUUDUVHQUN
|
|
AUVGNQUUEYPUUFUVRUWCCQUVKUVSUVQUWBUVBDUVJGUDUVJGUDUEZUVJPZVVMQPZVVMGPZUVI
|
|
UAVVMQURVVPVVMUWTPZVVOVVQVVMUXCGBHVVMUWTURKXAIVVMUUGTVVOVVNVVMUGZUVHPZVVR
|
|
UGZGPZVVPUVIVVSUAVVMQUAUDYMUVDVVRUVHUVCVVMVBVCUUHVVOVWAVVRQPZVWARVVSVVOVW
|
|
BVWAVVMUUIWMUVGVWANVVRQUVEVVRVAUVFVVTGUVEVVRVBVCYNWPVVOVVTVVMGVVOVVMVVMYJ
|
|
VDVCUUJUUMUUKZUULUVPUWADQUVOUVTUVLUVNEUVJGVWCUUNUUOUUPUUQUURUQ $.
|
|
|
|
$d S a b x y z $. $d a ph x y z $.
|
|
$( The infimum of a subset of an upper set of integers is less than or
|
|
equal to all members of the subset. (Contributed by Jim Kingdon,
|
|
13-Jan-2022.) $)
|
|
infssuzledc $p |- ( ph -> inf ( S , RR , < ) <_ A ) $=
|
|
( vx vy vz vb va cr clt wcel wbr wn cinf cv wa wb lttri3 adantl infssuzex
|
|
weq infclti cuz cfv crab elrabi eleq2s eluzelre 3syl inflbti mpd nltled )
|
|
ADPQUAZCAKLMNOPDQOUBZPRNUBZPRUCONUHVAVBQSTVBVAQSTUCUDAVAVBUEUFZABKLMCDEFG
|
|
HIJUGZUIACDRZCFUJUKZRZCPRIVGCBEVFULDBECVFUMHUNFCUOUPAVECUTQSTIAKLMNOPDCQV
|
|
CVDUQURUS $.
|
|
|
|
${
|
|
$d S u w $. $d ph w x y z $.
|
|
$( The infimum of a subset of an upper set of integers belongs to the
|
|
subset. (Contributed by Jim Kingdon, 20-Jan-2022.) $)
|
|
infssuzcldc $p |- ( ph -> inf ( S , RR , < ) e. S ) $=
|
|
( vw vx vy vz vu cr clt cneg wcel cz cinf cv crab infssuzex wss cuz cfv
|
|
csup ssrab2 eqsstri uzssz sstri zssre a1i infrenegsupex infsupneg negeq
|
|
wa eleq1d elrab sseli adantl cc wb simpl recnd znegclb syl mpbird sylbi
|
|
wceq ssriv suprzclex nfrab1 nfcv nfsup nfneg nfel1 elrabf sylib eqeltrd
|
|
simprd ) ADPQUAKUBZRZDSZKPUCZPQUHZRZDALMKDABLMKCDEFGHIJUDDPUEADTPDFUFUG
|
|
ZTDBEWIUCWIHBEWIUIUJFUKULZUMULUNZUOAWGPSZWHDSZAWGWFSWLWMURALMNWFALMNKDA
|
|
BLMNCDEFGHIJUDWKUPWFTUEAOWFTOUBZWFSWNPSZWNRZDSZURZWNTSZWEWQKWNPWCWNVKWD
|
|
WPDWCWNUQUSUTWRWSWPTSZWQWTWODTWPWJVAVBWRWNVCSWSWTVDWRWNWOWQVEVFWNVGVHVI
|
|
VJVLUNVMWEWMKWGPKWFPQWEKPVNKPVOZKQVOVPZXAKWHDKWGXBVQVRWCWGVKWDWHDWCWGUQ
|
|
USVSVTWBWA $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A m n $.
|
|
$( There is an upper bound to the divisors of a nonzero integer.
|
|
(Contributed by Jim Kingdon, 11-Dec-2021.) $)
|
|
dvdsbnd $p |- ( ( A e. ZZ /\ A =/= 0 ) ->
|
|
E. n e. NN A. m e. ( ZZ>= ` n ) -. m || A ) $=
|
|
( cz wcel cc0 wne wa cfv cv clt wbr cn wn cle ad3antrrr adantl simplr mpd
|
|
wrex cabs cdvds cuz wral simpl zcnd abscld arch syl simpllr nnred eluzelz
|
|
cr zred eluzle ltletrd wb zabscl ad4antr zltnle syl2anc mpbid wi ad2antrr
|
|
w3a dvdsleabs con3d syl3anc ralrimiva ex reximdva ) ADEZAFGZHZAUAIZCJZKLZ
|
|
CMTZBJZAUBLZNZBVPUCIZUDZCMTVNVOUMEZVRVNAVNAVLVMUEZUFUGZVOCUHUIVNVQWCCMVNV
|
|
PMEZHZVQWCWHVQHZWABWBWIVSWBEZHZVSVOOLZNZWAWKVOVSKLZWMWKVOVPVSVNWDWGVQWJWF
|
|
PWKVPVNWGVQWJUJUKWKVSWJVSDEZWIVPVSULQZUNWHVQWJRWJVPVSOLWIVPVSUOQUPWKVODEZ
|
|
WOWNWMUQVLWQVMWGVQWJAURUSWPVOVSUTVAVBWKWOVLVMWMWAVCWPVNVLWGVQWJWEPWHVMVQW
|
|
JVLVMWGRVDWOVLVMVEVTWLVSAVFVGVHSVIVJVKS $.
|
|
$}
|
|
|
|
${
|
|
$d X j n x y z $. $d Y j n x y z $.
|
|
$( Existence of the supremum used in defining ` gcd ` . (Contributed by
|
|
Jim Kingdon, 12-Dec-2021.) $)
|
|
gcdsupex $p |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) ->
|
|
E. x e. ZZ ( A. y e. { n e. ZZ | ( n || X /\ n || Y ) } -. x < y /\
|
|
A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || X /\ n || Y ) }
|
|
y < z ) ) ) $=
|
|
( vj cz wcel wa cc0 wceq wn cdvds wbr c1 wdc cn wral wrex cv 1zzd anbi12d
|
|
breq1 anim12i adantr cuz elnnuz biimpri simpll dvdsdc syl2an2 simplr dcan
|
|
1dvds cfv sylc adantlr wne simplll dvdsbnd nnuz rexeqi id intnanrd ralimi
|
|
sylib reximi syl sylancom simpllr intnand wo simpr wb zdceq sylancl mpbid
|
|
0z ianordc df-ne orbi12i sylibr mpjaodan zsupcllemex ) EHIZFHIZJZEKLZFKLZ
|
|
JMZJZDUAZENOZWMFNOZJZPENOZPFNOZJZABCGDPWLUBWMPLWNWQWOWRWMPENUDWMPFNUDUCWH
|
|
WSWKWFWQWGWREUOFUOUEUFWHWMPUGUPZIZWPQZWKWHXAJWNQZWOQZXBXAWMRIZWHWFXCXEXAW
|
|
MUHUIZWFWGXAUJWMEUKULXAXEWHWGXDXFWFWGXAUMWMFUKULWNWOUNUQURWLEKUSZWPMZDGUA
|
|
UGUPZSZGWTTZFKUSZWLXGWFXKWFWGWKXGUTWFXGJZWNMZDXISZGWTTZXKXMXOGRTXPEDGVAXO
|
|
GRWTVBVCVGXOXJGWTXNXHDXIXNWNWOXNVDVEVFVHVIVJWLXLWGXKWFWGWKXLVKWGXLJZWOMZD
|
|
XISZGWTTZXKXQXSGRTXTFDGVAXSGRWTVBVCVGXSXJGWTXRXHDXIXRWOWNXRVDVLVFVHVIVJWL
|
|
WIMZWJMZVMZXGXLVMWLWKYCWHWKVNWLWIQZWKYCVOWLWFKHIYDWFWGWKUJVSEKVPVQWIWJVTV
|
|
IVRXGYAXLYBEKWAFKWAWBWCWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d X j n $. $d Y j n $.
|
|
$( Closure of the supremum used in defining ` gcd ` . A lemma for ~ gcdval
|
|
and ~ gcdn0cl . (Contributed by Jim Kingdon, 11-Dec-2021.) $)
|
|
gcdsupcl $p |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) ->
|
|
sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN ) $=
|
|
( vj cz wcel wa cc0 wceq wn cv cdvds wbr c1 cn wdc wral wrex nnuz syl clt
|
|
crab csup cuz cfv 1zzd breq1 anbi12d anim12i adantr elnnuz biimpri simpll
|
|
cr 1dvds dvdsdc syl2an2 simplr dcan sylc adantlr wne simplll rexeqi sylib
|
|
dvdsbnd id intnanrd ralimi reximi sylancom simpllr intnand wo simpr wb 0z
|
|
zdceq sylancl ianordc mpbid orbi12i sylibr mpjaodan zsupcl syl6eleqr
|
|
df-ne ) BEFZCEFZGZBHIZCHIZGJZGZAKZBLMZWOCLMZGZAEUBUNUAUCNUDUEZOWNWRNBLMZN
|
|
CLMZGZDANWNUFWONIWPWTWQXAWONBLUGWONCLUGUHWJXBWMWHWTWIXABUOCUOUIUJWJWOWSFZ
|
|
WRPZWMWJXCGWPPZWQPZXDXCWOOFZWJWHXEXGXCWOUKULZWHWIXCUMWOBUPUQXCXGWJWIXFXHW
|
|
HWIXCURWOCUPUQWPWQUSUTVAWNBHVBZWRJZADKUDUEZQZDWSRZCHVBZWNXIWHXMWHWIWMXIVC
|
|
WHXIGZWPJZAXKQZDWSRZXMXOXQDORXRBADVFXQDOWSSVDVEXQXLDWSXPXJAXKXPWPWQXPVGVH
|
|
VIVJTVKWNXNWIXMWHWIWMXNVLWIXNGZWQJZAXKQZDWSRZXMXSYADORYBCADVFYADOWSSVDVEY
|
|
AXLDWSXTXJAXKXTWQWPXTVGVMVIVJTVKWNWKJZWLJZVNZXIXNVNWNWMYEWJWMVOWNWKPZWMYE
|
|
VPWNWHHEFYFWHWIWMUMVQBHVRVSWKWLVTTWAXIYCXNYDBHWGCHWGWBWCWDWESWF $.
|
|
$}
|
|
|
|
${
|
|
$d M n x y $. $d N n x y $.
|
|
$( The value of the ` gcd ` operator. ` ( M gcd N ) ` is the greatest
|
|
common divisor of ` M ` and ` N ` . If ` M ` and ` N ` are both ` 0 ` ,
|
|
the result is defined conventionally as ` 0 ` . (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.) $)
|
|
gcdval $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) =
|
|
if ( ( M = 0 /\ N = 0 ) , 0 ,
|
|
sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) ) $=
|
|
( vx vy cz wcel cc0 wceq wa cv cdvds wbr crab cr clt csup cif cn0 cgcd co
|
|
wn simpr iftrued 0nn0 syl6eqel cn iffalsed gcdsupcl eqeltrd nnnn0d wdc wo
|
|
gcdmndc exmiddc syl mpjaodan eqeq1 anbi1d rabbidv supeq1d ifbieq2d anbi2d
|
|
breq2 df-gcd ovmpt2g mpd3an3 ) BFGZCFGZBHIZCHIZJZHAKZBLMZVMCLMZJZAFNZOPQZ
|
|
RZSGZBCTUAVSIVHVIJZVLVTVLUBZWAVLJZVSHSWCVLHVRWAVLUCUDUEUFWAWBJZVSWDVSVRUG
|
|
WDVLHVRWAWBUCUHABCUIUJUKWAVLULVLWBUMBCUNVLUOUPUQDEBCFFDKZHIZEKZHIZJZHVMWE
|
|
LMZVMWGLMZJZAFNZOPQZRVSTVJWHJZHVNWKJZAFNZOPQZRSWEBIZWIWOWNWRHWSWFVJWHWEBH
|
|
URUSWSOWMWQPWSWLWPAFWSWJVNWKWEBVMLVDUSUTVAVBWGCIZWOVLWRVRHWTWHVKVJWGCHURV
|
|
CWTOWQVQPWTWPVPAFWTWKVOVNWGCVMLVDVCUTVAVBDEAVEVFVG $.
|
|
$}
|
|
|
|
$( The value, by convention, of the ` gcd ` operator when both operands are
|
|
0. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
gcd0val $p |- ( 0 gcd 0 ) = 0 $=
|
|
( vn cc0 cgcd co wceq wa cv cdvds wbr cz crab cr clt csup cif wcel 0z mp2an
|
|
gcdval eqid iftrue eqtri ) BBCDZBBEZUDFZBAGBHIZUFFAJKLMNZOZBBJPZUIUCUHEQQAB
|
|
BSRUDUDUHBEBTZUJUEBUGUARUB $.
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( The value of the ` gcd ` operator when at least one operand is nonzero.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
gcdn0val $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) ->
|
|
( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) $=
|
|
( cz wcel wa cc0 wceq wn cgcd co cv cdvds wbr crab cr clt csup cif gcdval
|
|
iffalse sylan9eq ) BDECDEFBGHCGHFZIBCJKUCGALZBMNUDCMNFADOPQRZSUEABCTUCGUE
|
|
UAUB $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( Closure of the ` gcd ` operator. (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
gcdn0cl $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) ->
|
|
( M gcd N ) e. NN ) $=
|
|
( vn cz wcel wa cc0 wceq wn cgcd co cv cdvds wbr crab cr csup cn gcdn0val
|
|
clt gcdsupcl eqeltrd ) ADEBDEFAGHBGHFIFABJKCLZAMNUCBMNFCDOPTQRCABSCABUAUB
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d M n x y z $. $d N n x y z $.
|
|
$( The gcd of two integers divides each of them. (Contributed by Paul
|
|
Chapman, 21-Mar-2011.) $)
|
|
gcddvds $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( ( M gcd N ) || M /\ ( M gcd N ) || N ) ) $=
|
|
( vn vx vy vz cz wcel wa cc0 wceq cgcd co cdvds wbr wn cv cr clt wrex wss
|
|
0z dvds0 ax-mp breq2 bi2anan9 syl6bb mpbiri oveq12 gcd0val syl6eq anbi12d
|
|
anidm breq1d mpbird adantl crab csup gcdn0val zssre gcdsupex ssrexv mpsyl
|
|
wral wi ssrab2 a1i suprzclex eqeltrd wb gcdn0cl nnzd breq1 elrab3 syl wdc
|
|
mpbid wo gcdmndc exmiddc mpjaodan ) AGHBGHIZAJKZBJKZIZABLMZANOZWFBNOZIZWE
|
|
PZWEWIWBWEWIJANOZJBNOZIZWEWMJJNOZJGHWNUBJUCUDWEWMWNWNIWNWCWKWNWDWLWNAJJNU
|
|
EBJJNUEUFWNUMUGUHWEWGWKWHWLWEWFJANWEWFJJLMJAJBJLUIUJUKZUNWEWFJBNWOUNULUOU
|
|
PWBWJIZWFCQZANOZWQBNOZIZCGUQZHZWIWPWFXARSURXACABUSWPDEFXAGRUAWPDQZEQZSOPE
|
|
XAVDXDXCSOXDFQSOFXATVEERVDIZDGTXEDRTUTDEFCABVAXEDGRVBVCXAGUAWPWTCGVFVGVHV
|
|
IWPWFGHXBWIVJWPWFABVKVLWTWICWFGWQWFKWRWGWSWHWQWFANVMWQWFBNVMULVNVOVQWBWEV
|
|
PWEWJVRABVSWEVTVOWA $.
|
|
$}
|
|
|
|
${
|
|
$d K n $. $d M f g x $. $d M n x y z $. $d N f g x $. $d N n x y z $.
|
|
$( An integer which divides both operands of the ` gcd ` operator is
|
|
bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
dvdslegcd $p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\
|
|
-. ( M = 0 /\ N = 0 ) ) ->
|
|
( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) ) $=
|
|
( vn vx vy vz vg vf cz wcel wceq wa wn cdvds wbr cv cr clt wrex cgcd crab
|
|
w3a cc0 co cle csup simpll1 zred simpll2 simpll3 simplr lttri3 adantl wss
|
|
wb wral zssre gcdsupex ssrexv mpsyl supclti syl21anc simpr anbi12d elrab3
|
|
wi breq1 syl mpbird supubti mpd nltled gcdn0val breqtrrd ex ) AJKZBJKZCJK
|
|
ZUCZBUDLCUDLMNZMZABOPZACOPZMZABCUAUEZUFPWBWEMZADQZBOPZWHCOPZMZDJUBZRSUGZW
|
|
FUFWGAWMWGAVQVRVSWAWEUHZUIWGVRVSWAWMRKVQVRVSWAWEUJZVQVRVSWAWEUKZVTWAWEULZ
|
|
VRVSMWAMZEFGHIRWLSIQZRKHQZRKMWSWTLWSWTSPNWTWSSPNMUPWRWSWTUMUNZJRUOWREQZFQ
|
|
ZSPNFWLUQXCXBSPXCGQSPGWLTVGFRUQMZEJTXDERTUREFGDBCUSXDEJRUTVAZVBVCWGAWLKZW
|
|
MASPNZWGXFWEWBWEVDWGVQXFWEUPWNWKWEDAJWHALWIWCWJWDWHABOVHWHACOVHVEVFVIVJWG
|
|
VRVSWAXFXGVGWOWPWQWREFGHIRWLASXAXEVKVCVLVMWGVRVSWAWFWMLWOWPWQDBCVNVCVOVP
|
|
$.
|
|
$}
|
|
|
|
$( A positive integer which divides both positive operands of the ` gcd `
|
|
operator is bounded by it. (Contributed by AV, 9-Aug-2020.) $)
|
|
nndvdslegcd $p |- ( ( K e. NN /\ M e. NN /\ N e. NN )
|
|
-> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) ) $=
|
|
( cn wcel w3a cz cc0 wceq wa wn cdvds wbr cgcd co cle nnz 3anim123i nnne0
|
|
wi neneqd 3ad2ant2 intnanrd dvdslegcd syl2anc ) ADEZBDEZCDEZFZAGEZBGEZCGEZF
|
|
BHIZCHIZJKABLMACLMJABCNOPMTUFUJUGUKUHULAQBQCQRUIUMUNUGUFUMKUHUGBHBSUAUBUCAB
|
|
CUDUE $.
|
|
|
|
$( Closure of the ` gcd ` operator. (Contributed by Paul Chapman,
|
|
21-Mar-2011.) $)
|
|
gcdcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 ) $=
|
|
( cz wcel wa cc0 wceq cgcd co wn oveq12 gcd0val syl6eq 0nn0 syl6eqel adantl
|
|
cn0 gcdn0cl nnnn0d wdc wo gcdmndc exmiddc syl mpjaodan ) ACDBCDEZAFGBFGEZAB
|
|
HIZQDZUGJZUGUIUFUGUHFQUGUHFFHIFAFBFHKLMNOPUFUJEUHABRSUFUGTUGUJUAABUBUGUCUDU
|
|
E $.
|
|
|
|
$( Closure of the ` gcd ` operator. (Contributed by Thierry Arnoux,
|
|
2-Feb-2020.) $)
|
|
gcdnncl $p |- ( ( M e. NN /\ N e. NN ) -> ( M gcd N ) e. NN ) $=
|
|
( cn wcel wa cz cc0 wceq wn cgcd co simpl nnzd simpr nnne0d intnand gcdn0cl
|
|
neneqd syl21anc ) ACDZBCDZEZAFDBFDAGHZBGHZEIABJKCDUBATUALMUBBTUANZMUBUDUCUB
|
|
BGUBBUEORPABQS $.
|
|
|
|
${
|
|
gcdcld.1 $e |- ( ph -> M e. ZZ ) $.
|
|
gcdcld.2 $e |- ( ph -> N e. ZZ ) $.
|
|
$( Closure of the ` gcd ` operator. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
gcdcld $p |- ( ph -> ( M gcd N ) e. NN0 ) $=
|
|
( cz wcel cgcd co cn0 gcdcl syl2anc ) ABFGCFGBCHIJGDEBCKL $.
|
|
$}
|
|
|
|
$( Closure of the ` gcd ` operator if the second operand is not 0.
|
|
(Contributed by AV, 10-Jul-2021.) $)
|
|
gcd2n0cl $p |- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M gcd N ) e. NN ) $=
|
|
( cz wcel cc0 wne w3a wa wceq wn cgcd co neneq intnand anim2i 3impa gcdn0cl
|
|
cn syl ) ACDZBCDZBEFZGTUAHZAEIZBEIZHJZHZABKLRDTUAUBUGUBUFUCUBUEUDBEMNOPABQS
|
|
$.
|
|
|
|
${
|
|
$d A n $. $d B n $.
|
|
$( An integer is the product of an integer and the gcd of it and another
|
|
integer. (Contributed by AV, 11-Jul-2021.) $)
|
|
zeqzmulgcd $p |- ( ( A e. ZZ /\ B e. ZZ )
|
|
-> E. n e. ZZ A = ( n x. ( A gcd B ) ) ) $=
|
|
( cz wcel wa cgcd co cdvds cv cmul wceq wrex gcddvds wb gcdcl nn0zd simpl
|
|
wbr divides syl2anc eqcom a1i rexbidv biimpd sylbid adantrd mpd ) ADEZBDE
|
|
ZFZABGHZAISZULBISZFACJULKHZLZCDMZABNUKUMUQUNUKUMUOALZCDMZUQUKULDEUIUMUSOU
|
|
KULABPQUIUJRCULATUAUKUSUQUKURUPCDURUPOUKUOAUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
$( An integer divided by the gcd of it and a nonzero integer is an integer.
|
|
(Contributed by AV, 11-Jul-2021.) $)
|
|
divgcdz $p |- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 )
|
|
-> ( A / ( A gcd B ) ) e. ZZ ) $=
|
|
( cz wcel cc0 wne w3a cgcd co cdvds cdiv wa gcddvds 3adant3 simpld gcd2n0cl
|
|
wbr wb cn syl nnz nnne0 jca simp1 df-3an sylanbrc dvdsval2 mpbid ) ACDZBCDZ
|
|
BEFZGZABHIZAJQZAUMKICDZULUNUMBJQZUIUJUNUPLUKABMNOULUMCDZUMEFZUIGZUNUORULUQU
|
|
RLZUIUSULUMSDZUTABPVAUQURUMUAUMUBUCTUIUJUKUDUQURUIUEUFUMAUGTUH $.
|
|
|
|
${
|
|
$d n x y $.
|
|
$( Domain and codomain of the ` gcd ` operator. (Contributed by Paul
|
|
Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.) $)
|
|
gcdf $p |- gcd : ( ZZ X. ZZ ) --> NN0 $=
|
|
( vx vy vn cv cc0 wceq wa cdvds wbr cz crab cr clt csup cif cn0 wcel wral
|
|
cxp cgcd wf co gcdval gcdcl eqeltrrd rgen2a df-gcd fmpt2 mpbi ) ADZEFBDZE
|
|
FGECDZUJHIULUKHIGCJKLMNOZPQZBJRAJRJJSPTUAUNABJUJJQUKJQGUJUKTUBUMPCUJUKUCU
|
|
JUKUDUEUFABJJUMPTABCUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d N n $.
|
|
$( The ` gcd ` operator is commutative. Theorem 1.4(a) in [ApostolNT]
|
|
p. 16. (Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
gcdcom $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( N gcd M ) ) $=
|
|
( vn cz wcel wa cc0 wceq cv cdvds wbr crab cr clt csup cif cgcd co gcdval
|
|
ancom wb a1i rabbiia supeq1i ifbieq2i ancoms 3eqtr4a ) ADEZBDEZFAGHZBGHZF
|
|
ZGCIZAJKZUMBJKZFZCDLZMNOZPUKUJFZGUOUNFZCDLZMNOZPZABQRBAQRZULUSURVBGUJUKTM
|
|
UQVANUPUTCDUPUTUAUMDEUNUOTUBUCUDUECABSUIUHVDVCHCBASUFUG $.
|
|
$}
|
|
|
|
$( A positive integer divided by the gcd of it and another integer is a
|
|
positive integer. (Contributed by AV, 10-Jul-2021.) $)
|
|
divgcdnn $p |- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( A gcd B ) ) e. NN ) $=
|
|
( cn wcel cz wa cgcd co cdvds wbr cdiv nnz anim1i gcddvds simpld syl wb cc0
|
|
wceq wn nnne0 neneqd adantr intnanrd gcdn0cl syl2anc nndivdvds syldan mpbid
|
|
) ACDZBEDZFZABGHZAIJZAUMKHCDZULAEDZUKFZUNUJUPUKALMZUQUNUMBIJABNOPUJUKUMCDZU
|
|
NUOQULUQARSZBRSZFTUSURULUTVAUJUTTUKUJARAUAUBUCUDABUEUFAUMUGUHUI $.
|
|
|
|
$( A positive integer divided by the gcd of it and another integer is a
|
|
positive integer. (Contributed by AV, 10-Jul-2021.) $)
|
|
divgcdnnr $p |- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( B gcd A ) ) e. NN ) $=
|
|
( cn wcel cz wa cgcd co cdiv wceq nnz gcdcom eqcomd oveq2d divgcdnn eqeltrd
|
|
sylan ) ACDZBEDZFZABAGHZIHAABGHZIHCTUAUBAITUBUARAEDSUBUAJAKABLQMNABOP $.
|
|
|
|
$( The gcd of two integers is zero iff they are both zero. (Contributed by
|
|
Paul Chapman, 21-Mar-2011.) $)
|
|
gcdeq0 $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( ( M gcd N ) = 0 <-> ( M = 0 /\ N = 0 ) ) ) $=
|
|
( cz wcel wa cgcd co cc0 wceq wdc wn wi gcdmndc wne gcdn0cl nnne0d ex df-ne
|
|
syl6ib condc sylc oveq12 gcd0val syl6eq impbid1 ) ACDBCDEZABFGZHIZAHIBHIEZU
|
|
FUIJUIKZUHKZLUHUILABMUFUJUGHNZUKUFUJULUFUJEUGABOPQUGHRSUIUHTUAUIUGHHFGHAHBH
|
|
FUBUCUDUE $.
|
|
|
|
$( The gcd of two integers is positive (nonzero) iff they are not both zero.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
gcdn0gt0 $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( -. ( M = 0 /\ N = 0 ) <-> 0 < ( M gcd N ) ) ) $=
|
|
( cz wcel wa cc0 cgcd co clt wbr wne wceq wn cn0 wb gcdcl cap cr cle 0re
|
|
nn0re nn0ge0 leltap mp3an2i nn0z 0zd zapne syl2anc gcdeq0 necon3abid bitr2d
|
|
bitrd syl ) ACDBCDEZFABGHZIJZUOFKZAFLBFLEZMUNUONDZUPUQOABPUSUPUOFQJZUQFRDUS
|
|
UORDFUOSJUPUTOTUOUAUOUBFUOUCUDUSUOCDFCDUTUQOUOUEUSUFUOFUGUHULUMUNURUOFABUIU
|
|
JUK $.
|
|
|
|
$( The gcd of 0 and an integer is the integer's absolute value. (Contributed
|
|
by Paul Chapman, 21-Mar-2011.) $)
|
|
gcd0id $p |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) ) $=
|
|
( cz wcel cc0 wceq cgcd co cabs cfv wn wa cle wbr cdvds mpan adantr mpd syl
|
|
0z wi gcd0val oveq2 fveq2 abs0 syl6eq 3eqtr4a adantl wne gcddvds simprd cn0
|
|
df-ne gcdcl nn0zd dvdsleabs 3anidm12 zabscl dvds0 iddvds wb absdvdsb anidms
|
|
syl3an1 jca eqid biantrur necon3abii w3a dvdslegcd ex mp3an2 mpancom syl5bi
|
|
mpbid imp zred letri3d mpbir2and sylan2br wdc zdceq mpan2 exmiddc mpjaodan
|
|
wo ) ABCZADEZDAFGZAHIZEZWGJZWGWJWFWGDDFGDWHWIUAADDFUBWGWIDHIDADHUCUDUEUFUGW
|
|
KWFADUHZWJADULWFWLKZWJWHWILMZWIWHLMZWMWHANMZWNWFWPWLWFWHDNMZWPDBCZWFWQWPKSD
|
|
AUIOUJPWFWLWPWNTZWFWHBCWFWLWSWFWHWRWFWHUKCSDAUMOUNZWHAUOVCUPQWMWIDNMZWIANMZ
|
|
KZWOWFXCWLWFXAXBWFWIBCZXAAUQZWIURRWFAANMZXBAUSWFXFXBUTAAVAVBVNVDPWFWLXCWOTZ
|
|
WLDDEZWGKZJZWFXGXIADXHWGDVEVFVGXDWFXJXGTZXEXDWRWFXKSXDWRWFVHXJXGWIDAVIVJVKV
|
|
LVMVOQWFWJWNWOKUTWLWFWHWIWFWHWTVPWFWIXEVPVQPVRVSWFWGVTZWGWKWEWFWRXLSADWAWBW
|
|
GWCRWD $.
|
|
|
|
$( The gcd of an integer and 0 is the integer's absolute value. Theorem
|
|
1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
gcdid0 $p |- ( N e. ZZ -> ( N gcd 0 ) = ( abs ` N ) ) $=
|
|
( cz wcel cc0 cgcd co cabs cfv wceq 0z gcdcom mpan gcd0id eqtr3d ) ABCZDAEF
|
|
ZADEFZAGHDBCOPQIJDAKLAMN $.
|
|
|
|
$( The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul
|
|
Chapman, 31-Mar-2011.) $)
|
|
nn0gcdid0 $p |- ( N e. NN0 -> ( N gcd 0 ) = N ) $=
|
|
( cn0 wcel cc0 cgcd co cabs cfv cz wceq nn0z gcdid0 syl nn0re nn0ge0 absidd
|
|
eqtrd ) ABCZADEFZAGHZARAICSTJAKALMRAANAOPQ $.
|
|
|
|
$( Negating one operand of the ` gcd ` operator does not alter the result.
|
|
(Contributed by Paul Chapman, 21-Mar-2011.) $)
|
|
gcdneg $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) ) $=
|
|
( cz wcel wa cgcd co cc0 wceq wn oveq12 adantl anbi2d cle wbr cdvds gcddvds
|
|
wb gcdcl wi cneg negeq0d syl6bi eqtr4d nn0zd dvdsnegb sylancom mpbid notbid
|
|
zcn imp simpl znegcl dvdslegcd ex syl3anc sylbid com12 impcom sylan2 mpbird
|
|
w3a mpdi simpr letri3d adantr mpbir2and wdc wo gcdmndc exmiddc syl mpjaodan
|
|
zred eqcomd ) ACDZBCDZEZABFGZABUAZFGZVRAHIZBHIZEZVSWAIZWDJZVRWDEVSHHFGZWAWD
|
|
VSWGIVRAHBHFKLVRWDWAWGIZVRWDWBVTHIZEZWHVQWDWJRVPVQWCWIWBVQBBUJUBMLZAHVTHFKU
|
|
CUKUDVRWFEWEVSWANOZWAVSNOZWFVRWLWFVRVSAPOZVSVTPOZEZWLVRWNVSBPOZEWPABQVRWQWO
|
|
WNVPVQVSCDZWQWORVRVSABSUEZVSBUFUGMUHVRWFWPWLTZVRWFWJJZWTVRWDWJWKUIVRWRVPVTC
|
|
DZXAWTTWSVPVQULZVQXBVPBUMZLWRVPXBVBXAWTVSAVTUNUOUPUQURVCUSWFVRWMWFVRWAAPOZW
|
|
ABPOZEZWMVRXGXEWAVTPOZEZVQVPXBXIXDAVTQUTVRXFXHXEVPVQWACDZXFXHRVQVPXBXJXDVPX
|
|
BEWAAVTSUEUTZWABUFUGMVAVRWFXGWMTZVRXJVPVQWFXLTXKXCVPVQVDXJVPVQVBWFXLWAABUNU
|
|
OUPURVCUSVRWEWLWMERWFVRVSWAVRVSWSVNVRWAXKVNVEVFVGVRWDVHWDWFVIABVJWDVKVLVMVO
|
|
$.
|
|
|
|
$( Negating one operand of the ` gcd ` operator does not alter the result.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
neggcd $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M gcd N ) = ( M gcd N ) ) $=
|
|
( cz wcel wa cneg cgcd co wceq gcdneg ancoms znegcl gcdcom sylan 3eqtr4d )
|
|
ACDZBCDZEBAFZGHZBAGHZRBGHZABGHQPSTIBAJKPRCDQUASIALRBMNABMO $.
|
|
|
|
$( Adding a multiple of one operand of the ` gcd ` operator to the other does
|
|
not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
gcdaddm $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( M gcd N ) = ( M gcd ( N + ( K x. M ) ) ) ) $=
|
|
( cz wcel cgcd co cmul caddc cc0 wceq wa wbr cdvds c1 wi zcnd syl2anc eqtrd
|
|
wdc w3a cle gcddvds 3adant1 simpld simp1 1zzd cn0 gcdcl nn0zd simp2 dvds2ln
|
|
simp3 syl23anc mpd mulid2d oveq2d breqtrd zmulcld zaddcld dvdslegcd syl3anc
|
|
wn jca ex mpid znegcld gcdcld mulneg1d oveq12d mulcld negcld addcomd negidd
|
|
eqtr3d oveq1d addassd addid2d 3eqtr3d anim12d nn0red letri3d sylibrd wo 0zd
|
|
cneg wb zdceq dcan orandc simpr mul01d adantr eqeq1d pm5.32da oveq12 adantl
|
|
sylc syl6bir imp eqtr4d sylbid jaod sylbird dcn syl exmiddc mpjaod ) ADEZBD
|
|
EZCDEZUAZBCFGZBABHGZCIGZFGZBCXNIGZFGXLBJKZXOJKZLZVCZXRCJKZLZVCZLZXMXPKZYEVC
|
|
ZXLYEXMXPUBMZXPXMUBMZLYFXLYAYHYDYIXLYAXMBNMZXMXONMZLZYHXLYJYKXLYJXMCNMZXJXK
|
|
YJYMLZXIBCUCUDZUEXLXMXNOCHGZIGZXONXLYNXMYQNMZYOXLXIODEZXMDEZXJXKYNYRPXIXJXK
|
|
UFZXLUGZXLXMXJXKXMUHEXIBCUIUDZUJZXIXJXKUKZXIXJXKUMZAOXMBCULUNUOXLYPCXNIXLCX
|
|
LCUUFQZUPUQURVDXLYTXJXODEZYAYLYHPZPUUDUUEXLXNCXLABUUAUUEUSUUFUTZYTXJUUHUAYA
|
|
UUIXMBXOVAVEVBVFXLYDXPBNMZXPCNMZLZYIXLUUKUULXLUUKXPXONMZXLXJUUHUUKUUNLZUUEU
|
|
UJBXOUCRZUEXLXPAWFZBHGZOXOHGZIGZCNXLUUOXPUUTNMZUUPXLUUQDEYSXPDEZXJUUHUUOUVA
|
|
PXLAUUAVGUUBXLXPXLBXOUUEUUJVHZUJZUUEUUJUUQOXPBXOULUNUOXLUUTXNWFZXOIGZCXLUUR
|
|
UVEUUSXOIXLABXLAUUAQZXLBUUEQZVIXLXOXLXOUUJQUPVJXLUVEXNIGZCIGJCIGZUVFCXLUVIJ
|
|
CIXLXNUVEIGUVIJXLXNUVEXLABUVGUVHVKZXLXNUVKVLZVMXLXNUVKVNVOVPXLUVEXNCUVLUVKU
|
|
UGVQXLCUUGVRZVSSURVDXLUVBXJXKYDUUMYIPZPUVDUUEUUFUVBXJXKUAYDUVNXPBCVAVEVBVFV
|
|
TXLXMXPXLXMUUCWAXLXPUVCWAWBWCXLYGXTYCWDZYFXLXTTZYCTZUVOYGWGXLXRTZXSTZUVPXLX
|
|
JJDEZUVRUUEXLWEZBJWHRZXLUUHUVTUVSUUJUWAXOJWHRXRXSWIWRZXLUVRYBTZUVQUWBXLXKUV
|
|
TUWDUUFUWACJWHRXRYBWIWRZXTYCWJRXLXTYFYCXLXTYCYFXLXRXSYBXLXRLZXOCJUWFXOUVJCU
|
|
WFXNJCIUWFXNAJHGZJUWFBJAHXLXRWKUQXLUWGJKXRXLAUVGWLWMSVPXLUVJCKXRUVMWMSWNWOZ
|
|
XLYCYFXLYCLXMJJFGZXPYCXMUWIKXLBJCJFWPWQXLYCXPUWIKZXLYCXTUWJUWHBJXOJFWPWSWTX
|
|
AVEZXBUWKXCXDXLYETZYEYGWDXLYATZYDTZUWLXLUVPUWMUWCXTXEXFXLUVQUWNUWEYCXEXFYAY
|
|
DWIWRYEXGXFXHXLXOXQBFXLXNCUVKUUGVMUQS $.
|
|
|
|
$( The GCD of two numbers is the same as the GCD of the left and their sum.
|
|
(Contributed by Scott Fenton, 20-Apr-2014.) $)
|
|
gcdadd $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M gcd N ) = ( M gcd ( N + M ) ) ) $=
|
|
( cz wcel wa cgcd co c1 cmul caddc wceq 1z gcdaddm mp3an1 zcn mulid2 oveq2d
|
|
cc syl adantr eqtrd ) ACDZBCDZEABFGZABHAIGZJGZFGZABAJGZFGZHCDUBUCUDUGKLHABM
|
|
NUBUGUIKZUCUBARDZUJAOUKUFUHAFUKUEABJAPQQSTUA $.
|
|
|
|
$( The gcd of a number and itself is its absolute value. (Contributed by
|
|
Paul Chapman, 31-Mar-2011.) $)
|
|
gcdid $p |- ( N e. ZZ -> ( N gcd N ) = ( abs ` N ) ) $=
|
|
( cz wcel cc0 cgcd co c1 cmul caddc cabs cfv wceq 1z gcdaddm mp3an13 gcdid0
|
|
0z cc zcn oveq2d mulid2 addid2 eqtrd syl 3eqtr3rd ) ABCZADEFZADGAHFZIFZEFZA
|
|
JKAAEFGBCUFDBCUGUJLMQGADNOAPUFUIAAEUFARCZUIALASUKUIDAIFAUKUHADIAUATAUBUCUDT
|
|
UE $.
|
|
|
|
$( The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16.
|
|
(Contributed by Mario Carneiro, 19-Feb-2014.) $)
|
|
gcd1 $p |- ( M e. ZZ -> ( M gcd 1 ) = 1 ) $=
|
|
( cz wcel c1 cgcd co cle wbr wceq cdvds wa 1z gcddvds mpan2 simprd cn wi wn
|
|
cc0 wne 1ne0 simpr necon3ai gcdn0cl nnzd 1nn dvdsle sylancl mpd wb nnle1eq1
|
|
ax-mp syl mpbid ) ABCZADEFZDGHZUPDIZUOUPDJHZUQUOUPAJHZUSUODBCZUTUSKLADMNOUO
|
|
UPBCDPCUSUQQUOUPUOVAUPPCZLUOVAKASIZDSIZKZRZVBDSTVFUAVEDSVCVDUBUCULADUDNNZUE
|
|
UFUPDUGUHUIUOVBUQURUJVGUPUKUMUN $.
|
|
|
|
$( The gcd of two integers is the same as that of their absolute values.
|
|
(Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
gcdabs $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) ) $=
|
|
( cz wcel wa cabs cfv wceq cneg wo co cq zq qabsor anim12i oveq12 sylan9eqr
|
|
cgcd ex gcdneg syl2an wi a1i neggcd znegcl sylan eqtrd ccased mpd ) ACDZBCD
|
|
ZEZAFGZAHZUMAIZHZJZBFGZBHZURBIZHZJZEZUMURRKZABRKZHZUJALDZBLDZVCUKAMBMVGUQVH
|
|
VBANBNOUAULUNUSUPVAVFUNUSEVFUBULUMAURBRPUCULUPUSEZVFVIULVDUOBRKZVEUMUOURBRP
|
|
ABUDZQSULUNVAEZVFVLULVDAUTRKVEUMAURUTRPABTQSULUPVAEZVFVMULVDUOUTRKZVEUMUOUR
|
|
UTRPULVNVJVEUJUOCDUKVNVJHAUEUOBTUFVKUGQSUHUI $.
|
|
|
|
$( ` gcd ` of the absolute value of the first operator. (Contributed by
|
|
Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
gcdabs1 $p |- ( ( N e. ZZ /\ M e. ZZ ) ->
|
|
( ( abs ` N ) gcd M ) = ( N gcd M ) ) $=
|
|
( cz wcel wa cabs cfv wceq cgcd co wi oveq1 a1i neggcd eqeq1d syl5ibrcom wo
|
|
cneg zq qabsord adantr mpjaod ) BCDZACDZEZBFGZBHZUFAIJZBAIJZHZUFBRZHZUGUJKU
|
|
EUFBAILMUEUJULUKAIJZUIHBANULUHUMUIUFUKAILOPUCUGULQUDUCBBSTUAUB $.
|
|
|
|
$( ` gcd ` of the absolute value of the second operator. (Contributed by
|
|
Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
gcdabs2 $p |- ( ( N e. ZZ /\ M e. ZZ ) ->
|
|
( N gcd ( abs ` M ) ) = ( N gcd M ) ) $=
|
|
( cz wcel wa cabs cfv cgcd wceq gcdabs1 ancoms zabscl gcdcom sylan2 3eqtr4d
|
|
co ) BCDZACDZEAFGZBHPZABHPZBSHPZBAHPRQTUAIBAJKRQSCDUBTIALBSMNBAMO $.
|
|
|
|
$( The gcd remains unchanged if one operand is replaced with its remainder
|
|
modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
modgcd $p |- ( ( M e. ZZ /\ N e. NN ) ->
|
|
( ( M mod N ) gcd N ) = ( M gcd N ) ) $=
|
|
( cz wcel wa co cgcd cneg cmul caddc cq wceq adantr adantl syl3anc cc eqtrd
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|
oveq2d eqtr4d gcdcom cn cmo cdiv cfl cfv cmin cc0 clt wbr nnq nngt0 modqval
|
|
zq zcn nncn znq flqcld zcnd w3a mulneg1 mulcom negeqd ancoms 3adant1 negsub
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|
mulcl sylan2 3impb znegcld nnz simpl gcdaddm zmodcl nn0zd syl2anc 3eqtr3d )
|
|
ACDZBUADZEZBABUBFZGFZBAGFZVTBGFZABGFZVSWABAABUCFZUDUEZHZBIFZJFZGFZWBVSVTWIB
|
|
GVSVTABWFIFZUFFZWIVSAKDZBKDZUGBUHUIZVTWLLVQWMVRAUMMVRWNVQBUJNVRWOVQBUKNABUL
|
|
OVSAPDZBPDZWFPDZWIWLLVQWPVRAUNMVRWQVQBUONVSWFVSWEABUPUQZURWPWQWRUSZWIAWKHZJ
|
|
FZWLWTWHXAAJWQWRWHXALZWPWRWQXCWRWQEZWHWFBIFZHXAWFBUTXDXEWKWFBVAVBQVCVDRWPWQ
|
|
WRXBWLLZWQWREWPWKPDXFBWFVFAWKVEVGVHQOSRVSWGCDBCDZVQWBWJLVSWFWSVIVRXGVQBVJNZ
|
|
VQVRVKZWGBAVLOSVSXGVTCDWAWCLXHVSVTABVMVNBVTTVOVSXGVQWBWDLXHXIBATVOVP $.
|
|
|
|
$( The GCD of one and an integer is one. (Contributed by Scott Fenton,
|
|
17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
1gcd $p |- ( M e. ZZ -> ( 1 gcd M ) = 1 ) $=
|
|
( cz wcel c1 cgcd co wceq 1z gcdcom mpan gcd1 eqtrd ) ABCZDAEFZADEFZDDBCMNO
|
|
GHDAIJAKL $.
|
|
|
|
$( The greatest common divisor of six and four is two. To calculate this
|
|
gcd, a simple form of Euclid's algorithm is used:
|
|
` ( 6 gcd 4 ) = ( ( 4 + 2 ) gcd 4 ) = ( 2 gcd 4 ) ` and
|
|
` ( 2 gcd 4 ) = ( 2 gcd ( 2 + 2 ) ) = ( 2 gcd 2 ) = 2 ` . (Contributed by
|
|
AV, 27-Aug-2020.) $)
|
|
6gcd4e2 $p |- ( 6 gcd 4 ) = 2 $=
|
|
( c6 c4 cgcd co c2 caddc cz wcel wceq 6nn 4z gcdcom mp2an 4cn 2cn oveq2i 2z
|
|
nnzi gcdadd eqtri 4p2e6 addcomli 2p2e4 cabs cfv gcdid ax-mp cc0 cle wbr 2re
|
|
cr 0le2 absid 3eqtr3ri 3eqtr2i ) ABCDZBACDZBEBFDZCDZEAGHBGHZUQURIAJRKABLMUS
|
|
ABCBEANOUAUBPEECDZBECDZEUTVBEEEFDZCDZVCEGHZVFVBVEIQQEESMVEEBCDZVCVDBECUCPVF
|
|
VAVGVCIQKEBLMTTVBEUDUEZEVFVBVHIQEUFUGEULHUHEUIUJVHEIUKUMEUNMTVAVFVCUTIKQBES
|
|
MUOUP $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Bézout's identity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
bezoutlemstep.is-bezout $e |- ( ph <-> E. s e. ZZ E. t e. ZZ
|
|
r = ( ( A x. s ) + ( B x. t ) ) ) $.
|
|
bezoutlemstep.a $e |- ( th -> A e. NN0 ) $.
|
|
bezoutlemstep.b $e |- ( th -> B e. NN0 ) $.
|
|
bezoutlemstep.w $e |- ( th -> W e. NN ) $.
|
|
bezoutlemstep.y-is-bezout $e |- ( th -> [ y / r ] ph ) $.
|
|
bezoutlemstep.y-nn0 $e |- ( th -> y e. NN0 ) $.
|
|
bezoutlemstep.w-is-bezout $e |- ( th -> [. W / r ]. ph ) $.
|
|
${
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$d A j k q s u $. $d A q r s t u v $. $d B j k q s u $.
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$d B q r s t u v $. $d W j k q s u $. $d W q r s t u v $.
|
|
$d j k q s u y $. $d k q s t u v y $. $d ph q s t u v $.
|
|
$d q s t th u v $. $d r s t u v y $.
|
|
$( Lemma for Bézout's identity. The is-bezout predicate holds for
|
|
` ( y mod W ) ` . (Contributed by Jim Kingdon, 6-Jan-2022.) $)
|
|
bezoutlemnewy $p |- ( th -> [. ( y mod W ) / r ]. ph ) $=
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|
( cmul co wceq cz vu vv vq vj vk cv caddc wrex cmo wsbc sbcbii cn eqeq1
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|
wcel wb 2rexbidv sbcieg syl syl5bb mpbid wsb oveq2 oveq1d eqeq2d oveq2d
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wa cbvrex2v bitri sbbii nfv sbie sylib ad2antrr cn0 clt ad4antr zmodcld
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|
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syl112anc simprl simplrl ad3antrrr zmulcld zsubcld simprr simplrr simpr
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|
cmin simplr ad5antr nn0cnd mulcld adddid mul12d oveq12d 3eqtrd nncnd cc
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adantr modqcld qcn subaddd mpbird addsub4d 3eqtr3d subdid eqtr4d rspcev
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|
zcnd syl2anc rexbidv sylibr rexlimddv ex rexlimdvva mpd ) BGEHUFZQRZFDU
|
|
FZQRZUGRZSZDTUHHTUHZAICUFZGUIRZUJZBAIGUJZYOPYSIUFZYMSZDTUHHTUHZIGUJZBYO
|
|
AUUBIGJUKBGULUNZUUCYOUOMUUBYOIGULYTGSUUAYNHDTTYTGYMUMUPUQURUSUTBYNYRHDT
|
|
TBYITUNZYKTUNZVFZVFZYNYRUUHYNVFZYPEUAUFZQRZFUBUFZQRZUGRZSZUBTUHUATUHZYR
|
|
BUUPUUGYNBAICVAZUUPNUUQYTUUNSZUBTUHUATUHZICVAUUPAUUSICAUUBUUSJUUAUURYTU
|
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UKYLUGRZSHDUAUBTTYIUUJSZYMUUTYTUVAYJUUKYLUGYIUUJEQVBVCVDYKUULSZUUTUUNYT
|
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UVBYLUUMUUKUGYKUULFQVBVEVDVGVHVIUUSUUPICUUPIVJYTYPSUURUUOUAUBTTYTYPUUNU
|
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MUPVKVHVLVMUUIUUOYRUAUBTTUUIUUJTUNZUULTUNZVFZVFZUUOYRUVFUUOVFZUCUFZGQRZ
|
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YQUGRYPSZYRUCTUVGYPTUNZUUDYQVNUNZYQGVOVRZUVJUCTUHZUVGYPBYPVNUNZUUGYNUVE
|
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UUOOVPVSZBUUDUUGYNUVEUUOMVPZUVGYPGUVPUVQVQZUVGYPVTUNZGVTUNZWAGVOVRZUVMU
|
|
VGUVKUVSUVPYPWBURZUVGGTUNUVTUVGGUVQWCGWBURZUVGGUVQWDZYPGWEWFUVKUUDUVLUV
|
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MVFWGYQYQSUVNYQWHUCGYQYPWIWJWKUVGUVHTUNZUVJVFZVFZUUBIYQUJZYRUWGUWHYQYMS
|
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ZDTUHHTUHZUWGYQEUDUFZQRZFUEUFZQRZUGRZSZUETUHZUDTUHZUWJUWGUUJUVHYIQRZWTR
|
|
ZTUNYQEUWTQRZUWNUGRZSZUETUHZUWRUWGUUJUWSUVFUVCUUOUWFUUIUVCUVDWLVMZUWGUV
|
|
HYIUVGUWEUVJWLZUUIUUEUVEUUOUWFBUUEUUFYNWMWNZWOWPUWGUULUVHYKQRZWTRZTUNYQ
|
|
UXAFUXIQRZUGRZSZUXDUWGUULUXHUVFUVDUUOUWFUUIUVCUVDWQVMZUWGUVHYKUXFUUIUUF
|
|
UVEUUOUWFBUUEUUFYNWRWNZWOWPUWGYQUUKEUWSQRZWTRZUUMFUXHQRZWTRZUGRZUXKUWGY
|
|
PUVIWTRZUUNUXOUXQUGRZWTRYQUXSUWGYPUUNUVIUYAWTUVFUUOUWFXAUWGUVIUVHYMQRUV
|
|
HYJQRZUVHYLQRZUGRUYAUWGGYMUVHQUUIYNUVEUUOUWFUUHYNWSWNVEUWGUVHYJYLUWGUVH
|
|
UXFYAZUWGEYIUWGEBEVNUNUUGYNUVEUUOUWFKXBXCZUWGYIUXGYAZXDUWGFYKUWGFBFVNUN
|
|
UUGYNUVEUUOUWFLXBXCZUWGYKUXNYAZXDXEUWGUYBUXOUYCUXQUGUWGUVHEYIUYDUYEUYFX
|
|
FUWGUVHFYKUYDUYGUYHXFXGXHXGUWGUXTYQSUVJUVGUWEUVJWQUWGYPUVIYQUWGYPBUVOUU
|
|
GYNUVEUUOUWFOXBXCUWGUVHGUYDUWGGUVGUUDUWFUVQXKXIXDUWGYQVTUNYQXJUNUWGYPGU
|
|
VGUVSUWFUWBXKUVGUVTUWFUWCXKUVGUWAUWFUWDXKXLYQXMURXNXOUWGUUKUUMUXOUXQUWG
|
|
EUUJUYEUWGUUJUXEYAZXDUWGFUULUYGUWGUULUXMYAZXDUWGEUWSUYEUWGUVHYIUYDUYFXD
|
|
ZXDUWGFUXHUYGUWGUVHYKUYDUYHXDZXDXPXQUWGUXAUXPUXJUXRUGUWGEUUJUWSUYEUYIUY
|
|
KXRUWGFUULUXHUYGUYJUYLXRXGXSUXCUXLUEUXITUWMUXISZUXBUXKYQUYMUWNUXJUXAUGU
|
|
WMUXIFQVBVEVDXTYBUWQUXDUDUWTTUWKUWTSZUWPUXCUETUYNUWOUXBYQUYNUWLUXAUWNUG
|
|
UWKUWTEQVBVCVDYCXTYBUWPUWIYQYJUWNUGRZSUDUEHDTTUWKYISZUWOUYOYQUYPUWLYJUW
|
|
NUGUWKYIEQVBVCVDUWMYKSZUYOYMYQUYQUWNYLYJUGUWMYKFQVBVEVDVGVLUWGUVLUWHUWJ
|
|
UOUVGUVLUWFUVRXKUUBUWJIYQVNYTYQSUUAUWIHDTTYTYQYMUMUPUQURXOAUUBIYQJUKYDY
|
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EYFYGYHYFYGYH $.
|
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$}
|
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|
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${
|
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$d A r s t $. $d B r s t $. $d W q r x y z $. $d W r s t y $.
|
|
$d ph q z $. $d ph s t $. $d ps q z $. $d q th z $. $d s t th $.
|
|
bezoutlemstep.sub-gcd $e |- ( ps <->
|
|
A. z e. NN0 ( z || r -> ( z || x /\ z || y ) ) ) $.
|
|
bezoutlemstep.hyp $e |- ( ( th /\ [. ( y mod W ) / r ]. ph )
|
|
-> E. r e. NN0 ( [. ( y mod W ) / x ]. [. W / y ]. ps /\ ph ) ) $.
|
|
bezoutlemstep.thx $e |- F/ x th $.
|
|
bezoutlemstep.thr $e |- F/ r th $.
|
|
$( Lemma for Bézout's identity. This is the induction step for
|
|
the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.) $)
|
|
bezoutlemstep $p |- ( th -> E. r e. NN0 ( [. W / x ]. ps /\ ph ) ) $=
|
|
( vq wsbc cv cmo co wa wrex bezoutlemnewy mpdan wcel cmul caddc wceq cz
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cn0 eqidd cn clt wbr wb nn0zd ad2antrr zmodcld cq cc0 syl nngt0d modqlt
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|
zq nnzd syl3anc modremain syl112anc mpbid cdvds wi simplrl sbcbii breq2
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wral anbi2d imbi2d sbcieg syl5bb sbcbid anbi1d bitrd ad3antrrr r19.21bi
|
|
ralbidv imp simprd simplr simprl dvdsmultr2 mpd simpld zmulcld dvds2add
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mp2and simprr breqtrd jca ex ralrimiva simplrr rexlimddv exp31 reximdai
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|
mpbird ) CBEJUEZDEUFZJUGUHZUEZAUIZLURUJZBDJUEZAUIZLURUJCALXPUEXSACEGHIJ
|
|
KLMNOPQRSUKUAULCXRYALURUCCLUFZURUMZXRYACYCUIZXRUIZUDUFZJUNUHZXPUOUHZXOU
|
|
PZYAUDUQYEXPXPUPZYIUDUQUJZYEXPUSYEXOUQUMZJUTUMZXPURUMZXPJVAVBZYJYKVCCYL
|
|
YCXRCXORVDZVEZCYMYCXRPVEZCYNYCXRCXOJYPPVFZVEZYEXOVGUMZJVGUMZVHJVAVBYOYE
|
|
YLUUAYQXOVLVIYEJUQUMZUUBYEJYRVMZJVLVIYEJYRVJXOJVKVNUDJXPXOVOVPVQYEYFUQU
|
|
MZYIUIZUIZXTAUUGXTFUFZYBVRVBZUUHJVRVBZUUHXOVRVBZUIZVSZFURWCZUUGUUMFURUU
|
|
GUUHURUMZUIZUUIUULUUPUUIUIZUUJUUKUUQUUHXPVRVBZUUJUUPUUIUURUUJUIZUUGUUIU
|
|
USVSZFURUUGXQUUTFURWCZYDXQAUUFVTCXQUVAVCYCXRUUFCXQUUIUUHDUFZVRVBZUUJUIZ
|
|
VSZFURWCZDXPUEZUVACXNUVFDXPUBXNUUIUVCUUKUIZVSZFURWCZEJUEZCUVFBUVJEJTWAC
|
|
YMUVKUVFVCPUVJUVFEJUTXOJUPZUVIUVEFURUVLUVHUVDUUIUVLUUKUUJUVCXOJUUHVRWBW
|
|
DWEWMWFVIWGWHCYNUVGUVAVCYSUVFUVADXPURUVBXPUPZUVEUUTFURUVMUVDUUSUUIUVMUV
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CUURUUJUVBXPUUHVRWBWIWEWMWFVIWJWKVQWLWNZWOZUUQUUHYHXOVRUUQUUHYGVRVBZUUR
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RYGUQUMXPUQUMUVPUURUIUVQVSUVSUUQYFJUVTUWAXAUUQXPYEYNUUFUUOUUIYTWKVDUUHY
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GXPXBVNXCUUGYIUUOUUIYEUUEYIXDVEXEXFXGXHCXTUUNVCYCXRUUFXTUVJDJUEZCUUNBUV
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$}
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$}
|
|
|
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${
|
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|
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$d th a r s t w z $. $d th x y $. $d A r s t $. $d B r s t $.
|
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|
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r = ( ( A x. s ) + ( B x. t ) ) ) $.
|
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bezout.sub-gcd $e |- ( ps <->
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bezout.a $e |- ( th -> A e. NN0 ) $.
|
|
bezout.b $e |- ( th -> B e. NN0 ) $.
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$( Lemma for Bézout's identity. This is the main result which we
|
|
prove by induction and which represents the application of the Extended
|
|
Euclidean algorithm. (Contributed by Jim Kingdon, 30-Dec-2021.) $)
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|
bezoutlemmain $p |- ( th -> A. x e. NN0 ( [ x / r ] ph -> A. y e. NN0 (
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LUKUMUNUOPDUHZUUTUVNUVEUUOUVPUUSUUJCAPDKUIUJUVPUVDUUNERUVPUVCUUMUUKUVPUVB
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|
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|
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XGUUDUUEWRUUFWTUUGUUBUUHUUI $.
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$}
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|
|
$d A r s t $. $d B r s t $.
|
|
bezoutlema.is-bezout $e |- ( ph <-> E. s e. ZZ E. t e. ZZ
|
|
r = ( ( A x. s ) + ( B x. t ) ) ) $.
|
|
bezoutlema.a $e |- ( th -> A e. NN0 ) $.
|
|
bezoutlema.b $e |- ( th -> B e. NN0 ) $.
|
|
$( Lemma for Bézout's identity. The is-bezout condition is
|
|
satisfied by ` A ` . (Contributed by Jim Kingdon, 30-Dec-2021.) $)
|
|
bezoutlema $p |- ( th -> [. A / r ]. ph ) $=
|
|
( cv cmul co caddc wceq cz wrex c1 wcel cc0 wsbc 1z 0z nn0cnd mul01d 1cnd
|
|
oveq2d mulcld addid1d mulid1d 3eqtrrd oveq2 oveq1d eqeq2d mp3an12i cn0 wb
|
|
rspc2ev eqeq1 2rexbidv syl5bb sbcieg syl mpbird ) BAGDUAZDDFKZLMZECKZLMZN
|
|
MZOZCPQFPQZRPSTPSBDDRLMZETLMZNMZOZVLUBUCBVOVMTNMVMDBVNTVMNBEBEJUDUEUGBVMB
|
|
DRBDIUDZBUFUHUIBDVQUJUKVKVPDVMVINMZOFCRTPPVFROZVJVRDVSVGVMVINVFRDLULUMUNV
|
|
HTOZVRVODVTVIVNVMNVHTELULUGUNURUOBDUPSVEVLUQIAVLGDUPAGKZVJOZCPQFPQWADOZVL
|
|
HWCWBVKFCPPWADVJUSUTVAVBVCVD $.
|
|
|
|
$( Lemma for Bézout's identity. The is-bezout condition is
|
|
satisfied by ` B ` . (Contributed by Jim Kingdon, 30-Dec-2021.) $)
|
|
bezoutlemb $p |- ( th -> [. B / r ]. ph ) $=
|
|
( cv cmul co caddc wceq cz wrex cc0 wcel c1 wsbc 0z 1z nn0cnd mul01d 1cnd
|
|
oveq1d mulcld addid2d mulid1d 3eqtrrd oveq2 eqeq2d oveq2d mp3an12i cn0 wb
|
|
rspc2ev eqeq1 2rexbidv syl5bb sbcieg syl mpbird ) BAGEUAZEDFKZLMZECKZLMZN
|
|
MZOZCPQFPQZRPSTPSBEDRLMZETLMZNMZOZVLUBUCBVORVNNMVNEBVMRVNNBDBDIUDUEUGBVNB
|
|
ETBEJUDZBUFUHUIBEVQUJUKVKVPEVMVINMZOFCRTPPVFROZVJVREVSVGVMVINVFRDLULUGUMV
|
|
HTOZVRVOEVTVIVNVMNVHTELULUNUMURUOBEUPSVEVLUQJAVLGEUPAGKZVJOZCPQFPQWAEOZVL
|
|
HWCWBVKFCPPWAEVJUSUTVAVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d A a b d s t w x y $. $d A a b d s t w z $. $d B a b d s t w x y $.
|
|
$d B a b d s t w z $.
|
|
$( Lemma for Bézout's identity. Existence of a number which we will
|
|
later show to be the greater common divisor and its decomposition into
|
|
cofactors. (Contributed by Mario Carneiro and Jim Kingdon,
|
|
3-Jan-2022.) $)
|
|
bezoutlemex $p |- ( ( A e. NN0 /\ B e. NN0 ) ->
|
|
E. d e. NN0 ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\
|
|
E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) $=
|
|
( vt vs vb cn0 wa cv cmul co wceq cz wrex cdvds wbr wi va wcel caddc wsbc
|
|
vw wral oveq2 oveq2d eqeq2d cbvrexv rexbii rexbidv bitri simpl bezoutlemb
|
|
oveq1d simpr wsb dfsbcq2 breq2 anbi2d imbi2d ralbidv anbi1d imbi12d breq1
|
|
bezoutlema anbi12d cbvralv bezoutlemmain rspcdva mpd ) DJUBZEJUBZKZFLZDAL
|
|
ZMNZEBLZMNZUCNZOZBPQZAPQZFEUDZCLZVPRSZWFDRSZWFERSZKZTZCJUFZWDKZFJQZWDVOGD
|
|
EHFWDVPVREGLZMNZUCNZOZGPQZAPQVPDHLZMNZWPUCNZOZGPQZHPQWCWSAPWBWRBGPVSWOOZW
|
|
AWQVPXEVTWPVRUCVSWOEMUGUHUIUJUKWSXDAHPVQWTOZWRXCGPXFWQXBVPXFVRXAWPUCVQWTD
|
|
MUGUPUIULUJUMZVMVNUNZVMVNUQZUOVOWDFIURZWGWHWFILZRSZKZTZCJUFZWDKZFJQZTZWEW
|
|
NTIJEXKEOZXJWEXQWNWDFIEUSXSXPWMFJXSXOWLWDXSXNWKCJXSXMWJWGXSXLWIWHXKEWFRUT
|
|
VAVBVCVDULVEVOWDFDUDZXRIJUFZWDVOGDEHFXGXHXIVGVOWDFUAURZXJWGWFUALZRSZXLKZT
|
|
ZCJUFZWDKZFJQZTZIJUFZTXTYATUAJDYCDOZYBXTYKYAWDFUADUSYLYJXRIJYLYIXQXJYLYHX
|
|
PFJYLYGXOWDYLYFXNCJYLYEXMWGYLYDWHXLYCDWFRUTVDVBVCVDULVBVCVEWDYGVOUAIUEGDE
|
|
HFXGYFUELZVPRSZYMYCRSZYMXKRSZKZTCUEJWFYMOZWGYNYEYQWFYMVPRVFYRYDYOXLYPWFYM
|
|
YCRVFWFYMXKRVFVHVEVIXHXIVJXHVKVLXIVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A d x y w $. $d B d x y w $. $d A z w d $. $d B w z $.
|
|
$( Lemma for Bézout's identity. Like ~ bezoutlemex but
|
|
where ' z ' is any integer, not just a nonnegative one. (Contributed by
|
|
Mario Carneiro and Jim Kingdon, 8-Jan-2022.) $)
|
|
bezoutlemzz $p |- ( ( A e. NN0 /\ B e. NN0 ) ->
|
|
E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\
|
|
E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) $=
|
|
( vw cn0 wcel wa cv cdvds wbr wi wral co cz wrex ad2antrr breq1 cmul wceq
|
|
caddc bezoutlemex nfv nfra1 nfan rsp mpd adantlll anbi12d imbi12d cbvralv
|
|
cneg simpr biimpi rspcdva wb simplr simpllr adantr nn0zd negdvdsb syl2anc
|
|
simplll 3imtr4d wo elznn0 simprbi adantl mpjaodan ralrimi anim1d reximdva
|
|
cr ex ) DHIZEHIZJZCKZFKZLMZVTDLMZVTELMZJZNZCHOZWADAKUAPEBKUAPUCPUBBQRAQRZ
|
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JZFHRWFCQOZWHJZFHRABCDEFUDVSWIWKFHVSWAHIZJZWGWJWHWMWGWJWMWGJZWFCQWMWGCWMC
|
|
UEWFCHUFUGWNVTQIZWFWNWOJZVTHIZWFVTUNZHIZWGWOWQWFWMWGWOJZWQJWQWFWTWQUOWGWQ
|
|
WFNWOWQWFCHUHSUIUJWPWSJZWRWALMZWRDLMZWRELMZJZWBWEWGWOWSXBXENZWMWTWSJGKZWA
|
|
LMZXGDLMZXGELMZJZNZXFGHWRXGWRUBZXHXBXKXEXGWRWALTXMXIXCXJXDXGWRDLTXGWRELTU
|
|
KULWGXLGHOZWOWSWGXNWFXLCGHVTXGUBZWBXHWEXKVTXGWALTXOWCXIWDXJVTXGDLTVTXGELT
|
|
UKULUMUPSWTWSUOUQUJXAWOWAQIWBXBURWNWOWSUSZXAWAWPWLWSVSWLWGWOUTVAVBVTWAVCV
|
|
DXAWCXCWDXDXAWODQIWCXCURXPXADWNVQWOWSVQVRWLWGVESVBVTDVCVDXAWOEQIWDXDURXPX
|
|
AEWNVRWOWSVQVRWLWGUTSVBVTEVCVDUKVFWOWQWSVGZWNWOVTVOIXQVTVHVIVJVKVPVLVPVMV
|
|
NUI $.
|
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$}
|
|
|
|
${
|
|
$d A d x y t $. $d B d x y t $. $d A z t d $. $d B t z $.
|
|
$( Lemma for Bézout's identity. Like ~ bezoutlemzz but
|
|
where ' A ' can be any integer, not just a nonnegative one.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) $)
|
|
bezoutlemaz $p |- ( ( A e. ZZ /\ B e. NN0 ) ->
|
|
E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\
|
|
E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) $=
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( vt cz wcel cn0 wa cv cdvds wbr wi cmul co caddc wceq wrex adantll simpr
|
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wral bezoutlemzz ancoms wb simpll ad2antrr dvdsnegb syl2anc anim1d imim2d
|
|
cneg biimprd ralimdva cc zcnd mulneg12 oveq1d eqeq2d rexbidv znegcl oveq2
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|
rspcev sylan ex adantl sylbid rexlimdva anim12d reximdva mpd wo cr elznn0
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simprbi adantr mpjaodan ) DHIZEJIZKZDJIZCLZFLZMNZWCDMNZWCEMNZKZOZCHUCZWDD
|
|
ALZPQZEBLPQZRQZSZBHTZAHTZKZFJTZDUMZJIZVTWBWSVSWBVTWSABCDEFUDUEUAWAXAKZWEW
|
|
CWTMNZWGKZOZCHUCZWDWTGLZPQZWMRQZSZBHTZGHTZKZFJTZWSVTXAXNVSXAVTXNGBCWTEFUD
|
|
UEUAXBXMWRFJXBWDJIZKZXFWJXLWQXPXEWICHXPWCHIZKZXDWHWEXRXCWFWGXRWFXCXRXQVSW
|
|
FXCUFXPXQUBXBVSXOXQVSVTXAUGZUHWCDUIUJUNUKULUOXPXKWQGHXPXGHIZKZXKWDDXGUMZP
|
|
QZWMRQZSZBHTZWQYAXJYEBHYAXIYDWDYAXHYCWMRYADUPIXGUPIXHYCSYADXBVSXOXTXSUHUQ
|
|
YAXGXPXTUBUQDXGURUJUSUTVAXTYFWQOXPXTYFWQXTYBHIYFWQXGVBWPYFAYBHWKYBSZWOYEB
|
|
HYGWNYDWDYGWLYCWMRWKYBDPVCUSUTVAVDVEVFVGVHVIVJVKVLVSWBXAVMZVTVSDVNIYHDVOV
|
|
PVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d A d x y t $. $d B d x y t $. $d A z t d $. $d B t z $.
|
|
$( Lemma for Bézout's identity. Like ~ bezoutlemaz but
|
|
where ' B ' can be any integer, not just a nonnegative one.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) $)
|
|
bezoutlembz $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
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E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\
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E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) $=
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|
( vt cz wcel wa cn0 cv cdvds wbr wi cmul co caddc wceq wrex adantlr simpr
|
|
wral cneg bezoutlemaz wb simplr ad2antrr dvdsnegb syl2anc anim2d ralimdva
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|
biimprd imim2d cc mulneg12 oveq2d eqeq2d znegcl oveq2 rspcev sylan adantl
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zcnd ex sylbid rexlimdva reximdv anim12d reximdva mpd wo simprbi mpjaodan
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cr elznn0 ) DHIZEHIZJZEKIZCLZFLZMNZWADMNZWAEMNZJZOZCHUCZWBDALPQZEBLZPQZRQ
|
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ZSZBHTZAHTZJZFKTZEUDZKIZVQVTWQVRABCDEFUEUAVSWSJZWCWDWAWRMNZJZOZCHUCZWBWIW
|
|
RGLZPQZRQZSZGHTZAHTZJZFKTZWQVQWSXLVRAGCDWRFUEUAWTXKWPFKWTWBKIZJZXDWHXJWOX
|
|
NXCWGCHXNWAHIZJZXBWFWCXPXAWEWDXPWEXAXPXOVRWEXAUFXNXOUBWTVRXMXOVQVRWSUGZUH
|
|
WAEUIUJUMUKUNULXNXIWNAHXNXHWNGHXNXEHIZJZXHWBWIEXEUDZPQZRQZSZWNXSXGYBWBXSX
|
|
FYAWIRXSEUOIXEUOIXFYASXSEWTVRXMXRXQUHVDXSXEXNXRUBVDEXEUPUJUQURXRYCWNOXNXR
|
|
YCWNXRXTHIYCWNXEUSWMYCBXTHWJXTSZWLYBWBYDWKYAWIRWJXTEPUTUQURVAVBVEVCVFVGVH
|
|
VIVJVKVRVTWSVLZVQVREVOIYEEVPVMVCVN $.
|
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$}
|
|
|
|
${
|
|
$d A d x y z $. $d B d x y z $.
|
|
$( Lemma for Bézout's identity. Like ~ bezoutlembz but the greatest
|
|
common divisor condition is a biconditional, not just an implication.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) $)
|
|
bezoutlembi $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
E. d e. NN0 ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\
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E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) ) $=
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|
( cz wcel wa cv cdvds wbr wi wral co wrex cn0 wb syl3anc ax-mp cmul caddc
|
|
wceq bezoutlembz simpllr simpll simplrl dvdsmultr1 simplr simplrr zmulcld
|
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ad3antrrr dvds2add syl2and simpr breq2d sylibrd bi3 syl5com ex rexlimdvva
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imdistan ancom imbi12i bitr4i sylib ralimdva wex cc0 0z r19.27mv reximdva
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elex2 3imtr3g mpd ) DGHZEGHZIZCJZFJZKLZVSDKLZVSEKLZIZMZCGNVTDAJZUAOZEBJZU
|
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AOZUBOZUCZBGPAGPZIZFQPWAWDRZCGNWLIZFQPABCDEFUDVRWMWOFQVRVTQHZIZWEWLIZCGNZ
|
|
WNWLIZCGNZWMWOWQWRWTCGWQVSGHZIZWLWEWNMZMZWRWTMZXCWKXDABGGXCWFGHZWHGHZIZIZ
|
|
WKXDXJWKIZWDWAMWEWNXKWDVSWJKLZWAXKWBVSWGKLZWCVSWIKLZXLXKXBVPXGWBXMMWQXBXI
|
|
WKUEZWQVPXBXIWKVPVQWPUFULZXCXGXHWKUGZVSDWFUHSXKXBVQXHWCXNMXOWQVQXBXIWKVPV
|
|
QWPUIULZXCXGXHWKUJZVSEWHUHSXKXBWGGHWIGHXMXNIXLMXOXKDWFXPXQUKXKEWHXRXSUKVS
|
|
WGWIUMSUNXKVTWJVSKXJWKUOUPUQWAWDURUSUTVAXEWLWEIZWLWNIZMXFWLWEWNVBWRXTWTYA
|
|
WEWLVCWNWLVCVDVEVFVGXBCVHZWSWMRVIGHYBVJCVIGVMTZWEWLCGVKTYBXAWORYCWNWLCGVK
|
|
TVNVLVO $.
|
|
$}
|
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|
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${
|
|
bezoutlemgcd.1 $e |- ( ph -> A e. ZZ ) $.
|
|
bezoutlemgcd.2 $e |- ( ph -> B e. ZZ ) $.
|
|
${
|
|
$d D z $.
|
|
bezoutlemgcd.3 $e |- ( ph -> D e. NN0 ) $.
|
|
bezoutlemgcd.4 $e |- ( ph ->
|
|
A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) ) $.
|
|
${
|
|
$d E z $. $d ph z $.
|
|
bezoutlemmo.5 $e |- ( ph -> E e. NN0 ) $.
|
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bezoutlemmo.6 $e |- ( ph ->
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A. z e. ZZ ( z || E <-> ( z || A /\ z || B ) ) ) $.
|
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$( Lemma for Bézout's identity. There is at most one
|
|
nonnegative integer meeting the greatest common divisor condition.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) $)
|
|
bezoutlemmo $p |- ( ph -> D = E ) $=
|
|
( wcel cdvds wbr wceq cz wb breq1 wral cn0 nn0zd iddvds cv bibi12d wa
|
|
r19.26 sylanbrc biantr ralimi rspcdva mpbid mpbird dvdseq syl22anc
|
|
syl ) AEUAMFUAMEFNOZFENOZEFPIKAEENOZUQAEQMUSAEIUBZEUCUPABUDZENOZVAFNO
|
|
ZRZUSUQRBQEVAEPVBUSVCUQVAEENSVAEFNSUEAVBVACNOVADNOUFZRZVCVERZUFZBQTZV
|
|
DBQTAVFBQTVGBQTVIJLVFVGBQUGUHVHVDBQVBVEVCUIUJUPZUTUKULAURFFNOZAFQMVKA
|
|
FKUBZFUCUPAVDURVKRBQFVAFPVBURVCVKVAFENSVAFFNSUEVJVLUKUMEFUNUO $.
|
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$}
|
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|
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${
|
|
$d A d e w z $. $d A d s t z $. $d B d e w z $. $d B d s t z $.
|
|
$d D z $. $d d e ph w $.
|
|
$( Lemma for Bézout's identity. There is exactly one
|
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nonnegative integer meeting the greatest common divisor condition.
|
|
(Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) $)
|
|
bezoutlemeu $p |- ( ph -> E! d e. NN0 A. z e. ZZ
|
|
( z || d <-> ( z || A /\ z || B ) ) ) $=
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( ve vw cv cdvds wbr wa wb cz wral cn0 vs vt wrex wrmo wreu wcel cmul
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co caddc bezoutlembi simpl reximi syl syl2anc weq wi ad2antrr simplrl
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simprl breq1 anbi12d bibi12d cbvralv sylib simplrr simprr bezoutlemmo
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wceq ex ralrimivva breq2 bibi1d ralbidv rmo4 sylibr reu5 sylanbrc ) A
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BMZFMZNOZVRCNOZVRDNOZPZQZBRSZFTUCZWEFTUDZWEFTUEACRUFZDRUFZWFGHWHWIPWE
|
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VSCUAMUGUHDUBMUGUHUIUHVHUBRUCUARUCZPZFTUCWFUAUBBCDFUJWKWEFTWEWJUKULUM
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UNAWEVRKMZNOZWCQZBRSZPZFKUOZUPZKTSFTSWGAWRFKTTAVSTUFZWLTUFZPZPZWPWQXB
|
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WPPZLCDVSWLAWHXAWPGUQAWIXAWPHUQAWSWTWPURXCWELMZVSNOZXDCNOZXDDNOZPZQZL
|
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RSXBWEWOUSWDXIBLRBLUOZVTXEWCXHVRXDVSNUTXJWAXFWBXGVRXDCNUTVRXDDNUTVAZV
|
|
BVCVDAWSWTWPVEXCWOXDWLNOZXHQZLRSXBWEWOVFWNXMBLRXJWMXLWCXHVRXDWLNUTXKV
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BVCVDVGVIVJWEWOFKTWQWDWNBRWQVTWMWCVSWLVRNVKVLVMVNVOWEFTVPVQ $.
|
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$}
|
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${
|
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$d A w z $. $d B w z $. $d D w z $. $d ph w z $.
|
|
bezoutlemgcd.5 $e |- ( ph -> -. ( A = 0 /\ B = 0 ) ) $.
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|
$( Lemma for Bézout's identity. The number satisfying the
|
|
greatest common divisor condition is the largest number which
|
|
divides both ` A ` and ` B ` . (Contributed by Mario Carneiro and
|
|
Jim Kingdon, 9-Jan-2022.) $)
|
|
bezoutlemle $p |- ( ph -> A. z e. ZZ
|
|
( ( z || A /\ z || B ) -> z <_ D ) ) $=
|
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( cdvds wbr wa cz wcel wb wceq breq1 ad2antrr cc0 vw cv simpr anbi12d
|
|
cle wi bibi12d equcom bicom 3imtr3i wral cbvralv sylib simplr rspcdva
|
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mpbird cn cn0 wne 0zd nn0zd 0dvds syl 3bitr3d mtbird elnnne0 sylanbrc
|
|
wn neqned dvdsle syl2anc mpd ex ralrimiva ) ABUBZCKLZVODKLZMZVOEUELZU
|
|
FBNAVONOZMZVRVSWAVRMZVOEKLZVSWBWCVRWAVRUCWBUAUBZEKLZWDCKLZWDDKLZMZPZW
|
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CVRPZUANVOVOWDQZWJWIPWDVOQWIWJPWKWCWEVRWHVOWDEKRWKVPWFVQWGVOWDCKRVOWD
|
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DKRUDUGZBUAUHWJWIUIUJAWIUANUKZVTVRAWJBNUKWMIWJWIBUANWLULUMSAVTVRUNZUO
|
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UPWBVTEUQOZWCVSUFWNWBEUROZETUSWOAWPVTVRHSZWBETWBETQZCTQZDTQZMZAXAVHVT
|
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VRJSWBTEKLZTCKLZTDKLZMZWRXAAXBXEPZVTVRAWJXFBNTVOTQZWCXBVRXEVOTEKRXGVP
|
|
XCVQXDVOTCKRVOTDKRUDUGIAUTUOSWBENOXBWRPWBEWQVAEVBVCWBXCWSXDWTWBCNOZXC
|
|
WSPAXHVTVRFSCVBVCWBDNOZXDWTPAXIVTVRGSDVBVCUDVDVEVIEVFVGVOEVJVKVLVMVN
|
|
$.
|
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|
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$d A f g u w z $. $d B f g u w z $. $d D f g u w z $.
|
|
$d f g ph w z $.
|
|
$( Lemma for Bézout's identity. The number satisfying the
|
|
greatest common divisor condition is the supremum of divisors of
|
|
both ` A ` and ` B ` . (Contributed by Mario Carneiro and Jim
|
|
Kingdon, 9-Jan-2022.) $)
|
|
bezoutlemsup $p |- ( ph -> D
|
|
= sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) ) $=
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( vw cv cdvds wbr wa cz cr clt wcel breq1 vu crab csup wral wrex wceq
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vg vf wn nn0red elrabi adantl zred adantr cle anbi12d simprbi imbi12d
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wi elrab bezoutlemle simpr rspcdva sylan2 mpd lensymd ralrimiva nn0zd
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iddvds syl wb bibi12d mpbid ad2antrr cn0 elrab3 mpbird breq2 sylancom
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rspcev ex lttri3 eqsupti mp3and eqcomd ) ABLZCMNZWFDMNZOZBPUBZQRUCZEA
|
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EQSZEKLZRNUIZKWJUDWMERNZWMUALZRNZUAWJUEZUSZKQUDWKEUFAEHUJZAWNKWJAWMWJ
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SZOZWMEXBWMXAWMPSZAWIBWMPUKZULUMAWLXAWTUNXBWMCMNZWMDMNZOZWMEUONZXAXGA
|
|
XAXCXGWIXGBWMPWFWMUFZWGXEWHXFWFWMCMTWFWMDMTUPZUTUQULXAAXCXGXHUSZXDAXC
|
|
OWIWFEUONZUSZXKBPWMXIWIXGXLXHXJWFWMEUOTURAXMBPUDXCABCDEFGHIJVAUNAXCVB
|
|
VCVDVEVFVGAWSKQAWMQSZOZWOWRXOWOEWJSZWRXOWOOZXPECMNZEDMNZOZAXTXNWOAEEM
|
|
NZXTAEPSZYAAEHVHZEVIVJAWFEMNZWIVKYAXTVKBPEWFEUFZYDYAWIXTWFEEMTYEWGXRW
|
|
HXSWFECMTWFEDMTUPZVLIYCVCVMVNXQYBXPXTVKXQEAEVOSXNWOHVNVHWIXTBEPYFVPVJ
|
|
VQWQWOUAEWJWPEWMRVRVTVSWAVGAKUAUGUHQWJERUHLZQSUGLZQSOYGYHUFYGYHRNUIYH
|
|
YGRNUIOVKAYGYHWBULWCWDWE $.
|
|
$}
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d M a b r w $. $d M d r w z $. $d N a b r w $. $d N d r w z $.
|
|
$( Alternate definition of the ` gcd ` operator. (Contributed by Jim
|
|
Kingdon, 31-Dec-2021.) $)
|
|
dfgcd3 $p |- ( ( M e. ZZ /\ N e. ZZ )
|
|
-> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ
|
|
( z || d <-> ( z || M /\ z || N ) ) ) ) $=
|
|
( vw vr cz wcel wa cc0 wceq co cv cdvds wbr wb wral cn0 mpbird breq1 cgcd
|
|
va vb crio gcd0val simprl simprr oveq12d 0nn0 a1i 0dvds ad2antrr ad2antlr
|
|
wn jca wi anbi12d bibi12d rspcv ax-mp adantl simplr nn0zd syl mpbid dvds0
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|
breq2 ad3antrrr breqtrrd 2thd ralrimiva impbida riota5 3eqtr4a wrex caddc
|
|
0z cmul bezoutlembi simpl reximi adantr crab csup simplll simpllr cbvralv
|
|
cr biimpi ad2antll bezoutlemsup riotabiia wreu bezoutlemeu bibi1d ralbidv
|
|
clt riota2 syl2anc syl5eqr gcdn0val 3eqtr4rd rexlimddv wo gcdmndc exmiddc
|
|
wdc mpjaodan ) BGHZCGHZIZBJKZCJKZIZBCUALZAMZDMZNOZXPBNOZXPCNOZIZPZAGQZDRU
|
|
DZKZXNUNZXKXNIZJJUALJXOYDUEYGBJCJUAXKXLXMUFZXKXLXMUGZUHYGYCDRJJRHYGUIUJYG
|
|
XQRHZIZYCXQJKZYKYCIZJXQNOZYLYMYNJBNOZJCNOZIZYGYQYJYCYGYOYPYGYOXLYHXIYOXLP
|
|
XJXNBUKULSYGYPXMYIXJYPXMPXIXNCUKUMSUOULYCYNYQPZYKJGHYCYRUPVQYBYRAJGXPJKZX
|
|
RYNYAYQXPJXQNTYSXSYOXTYPXPJBNTXPJCNTUQURUSUTVASYMXQGHYNYLPYMXQYGYJYCVBVCX
|
|
QUKVDVEYKYLIZYBAGYTXPGHZIZXRYAUUBXRXPJNOZUUAUUCYTXPVFVAZYLXRUUCPYKUUAXQJX
|
|
PNVGUMSUUBXSXTUUBXPJBNUUDYGXLYJYLUUAYHVHVIUUBXPJCNUUDYGXMYJYLUUAYIVHVIUOV
|
|
JVKVLVMVNXKYFIZEMZFMZNOZUUFBNOZUUFCNOZIZPZEGQZYEFRXKUUMFRVOZYFXKUUMUUGBUB
|
|
MVRLCUCMVRLVPLKUCGVOUBGVOZIZFRVOUUNUBUCEBCFVSUUPUUMFRUUMUUOVTWAVDWBUUEUUG
|
|
RHZUUMIZIZUUGYAAGWCWHWQWDZYDXOUUSABCUUGXIXJYFUURWEZXIXJYFUURWFZUUEUUQUUMU
|
|
FZUUMXPUUGNOZYAPZAGQZUUEUUQUUMUVFUULUVEEAGUUFXPKZUUHUVDUUKYAUUFXPUUGNTUVG
|
|
UUIXSUUJXTUUFXPBNTUUFXPCNTUQZURWGWIWJXKYFUURVBWKUUSYDUUFXQNOZUUKPZEGQZDRU
|
|
DZUUGUVKYCDRUVKYCPYJUVJYBEAGUVGUVIXRUUKYAUUFXPXQNTUVHURWGUJWLUUSUUMUVLUUG
|
|
KZUUEUUQUUMUGZUUSUUQUVKDRWMUUMUVMPUVCUUSEBCUUGDUVAUVBUVCUVNWNUVKUUMDRUUGX
|
|
QUUGKZUVJUULEGUVOUVIUUHUUKXQUUGUUFNVGWOWPWRWSVEWTUUEXOUUTKUURABCXAWBXBXCX
|
|
KXNXGXNYFXDBCXEXNXFVDXH $.
|
|
$}
|
|
|
|
${
|
|
$d A d w x y z $. $d B d w x y z $.
|
|
$( Bézout's identity: For any integers ` A ` and ` B ` , there are
|
|
integers ` x , y ` such that ` ( A gcd B ) = A x. x + B x. y ` . This
|
|
is Metamath 100 proof #60.
|
|
|
|
The proof is constructive, in the sense that it applies the Extended
|
|
Euclidian Algorithm to constuct a number which can be shown to be
|
|
` ( A gcd B ) ` and which satisfies the rest of the theorem. In the
|
|
presence of excluded middle, it is common to prove Bézout's
|
|
identity by taking the smallest number which satisfies the Bézout
|
|
condition, and showing it is the greatest common divisor. But we do not
|
|
have the ability to show that number exists other than by providing a
|
|
way to determine it. (Contributed by Mario Carneiro, 22-Feb-2014.) $)
|
|
bezout $p |- ( ( A e. ZZ /\ B e. ZZ ) ->
|
|
E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) ) $=
|
|
( vz vd vw cz wcel wa cv cdvds wbr wb co wceq wrex cn0 nfv nfan wral cmul
|
|
caddc cgcd bezoutlembi simprrr nfre1 nfcv nfrexya crio dfgcd3 adantr wreu
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|
simprrl simprl simpll simplr bezoutlemeu breq2 bibi1d ralbidv mpbid eqtrd
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|
riota2 syl2anc eqeq1d rexbid mpbird rexlimddv ) CHIZDHIZJZEKZFKZLMZVMCLMV
|
|
MDLMJZNZEHUAZVNCAKUBODBKUBOUCOZPZBHQZAHQZJZCDUDOZVSPZBHQZAHQZFRABECDFUEVL
|
|
VNRIZWCJZJZWGWBVLWHVRWBUFWJWFWAAHVLWIAVLASWHWCAWHASVRWBAVRASWAAHUGTTTWJWE
|
|
VTBHVLWIBVLBSWHWCBWHBSVRWBBVRBSWABAHBHUHVTBHUGUITTTWJWDVNVSWJWDVMGKZLMZVP
|
|
NZEHUAZGRUJZVNVLWDWOPWIECDGUKULWJVRWOVNPZVLWHVRWBUNZWJWHWNGRUMVRWPNVLWHWC
|
|
UOZWJECDVNGVJVKWIUPVJVKWIUQWRWQURWNVRGRVNWKVNPZWMVQEHWSWLVOVPWKVNVMLUSUTV
|
|
AVDVEVBVCVFVGVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d K x y $. $d M x y $. $d N x y $.
|
|
$( An integer which divides each of two others also divides their gcd.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro,
|
|
30-May-2014.) $)
|
|
dvdsgcd $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M /\ K || N ) -> K || ( M gcd N ) ) ) $=
|
|
( vx vy cz wcel w3a cdvds wbr wa co cv cmul caddc wceq wrex wi cc zcn ex
|
|
bezout 3adant1 dvds2ln 3impia simp3l simp12 mulcom syl2an syl2anc oveq12d
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cgcd 3coml simp3r simp13 breqtrd breq2 syl5ibrcom 3expia rexlimdvv mpid )
|
|
AFGZBFGZCFGZHZABIJACIJKZBCULLZBDMZNLZCEMZNLZOLZPZEFQDFQZAVGIJZVCVDVNVBDEB
|
|
CUBUCVEVFVNVORVEVFKVMVODEFFVEVFVHFGZVJFGZKZVMVORVEVFVRHZVOVMAVLIJVSAVHBNL
|
|
ZVJCNLZOLZVLIVRVEVFAWBIJZVRVEVFWCVHVJABCUDUEUMVSVTVIWAVKOVSVPVCVTVIPZVEVF
|
|
VPVQUFVBVCVDVFVRUGVPVHSGBSGWDVCVHTBTVHBUHUIUJVSVQVDWAVKPZVEVFVPVQUNVBVCVD
|
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VFVRUOVQVJSGCSGWEVDVJTCTVJCUHUIUJUKUPVGVLAIUQURUSUTUAVA $.
|
|
$}
|
|
|
|
$( Biconditional form of ~ dvdsgcd . (Contributed by Scott Fenton,
|
|
2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
dvdsgcdb $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || M /\ K || N ) <-> K || ( M gcd N ) ) ) $=
|
|
( cz wcel w3a cdvds wbr wa co dvdsgcd gcddvds simpld 3adant1 wi simp1 gcdcl
|
|
cgcd dvdstr mpan2d nn0zd simp2 syl3anc simprd syld3an2 jcad impbid ) ADEZBD
|
|
EZCDEZFZABGHZACGHZIABCRJZGHZABCKUKUOULUMUKUOUNBGHZULUIUJUPUHUIUJIZUPUNCGHZB
|
|
CLZMNUKUHUNDEZUIUOUPIULOUHUIUJPUIUJUTUHUQUNBCQUANZUHUIUJUBAUNBSUCTUKUOURUMU
|
|
IUJURUHUQUPURUSUDNUHUTUIUJUOURIUMOVAAUNCSUETUFUG $.
|
|
|
|
${
|
|
$d D e f g n y z $. $d M e f g n y z $. $d N e f g n y z $.
|
|
$( Alternate definition of the ` gcd ` operator, see definition in
|
|
[ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.) $)
|
|
dfgcd2 $p |- ( ( M e. ZZ /\ N e. ZZ )
|
|
-> ( D = ( M gcd N ) <-> ( 0 <_ D /\ ( D || M /\ D || N )
|
|
/\ A. e e. ZZ ( ( e || M /\ e || N ) -> e || D ) ) ) ) $=
|
|
( cz wcel wa wceq cc0 wbr cdvds cv wi adantr wb breq2 breq1 adantl cr clt
|
|
vn vy vz vg vf cgcd cle wral w3a gcdcl nn0ge0d gcddvds 3anass ancom bitri
|
|
dvdsgcd sylbir ralrimiva 3jca anbi12d imbi2d ralbidv 3anbi123d mpbird cif
|
|
co crab csup gcdval wn iftrue bi2anan9 imbi1d 3anbi23d dvdszrcl dvds0 jca
|
|
pm5.5 syl ralbidva 0z rspcv ax-mp 0dvds biimpd eqcom syl6ibr sylbid com12
|
|
syl5 ex 3imp syl6bi adantld imp eqtrd ancoms iffalse lttri3 zred 3ad2ant2
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|
simpld ad2antll elrab imbi12d com23 ad2antrr cn0 elnn0z simplbi2 ad2antrl
|
|
impcom zre ad3antlr cn simp-5l wo elnn0 2a1 anbi2d simplr wdc zdceq mpan2
|
|
ianordc mpbid pm2.24 mpcom jaoi sylbi dvdsle syl2anc exp31 lensymd mpan2d
|
|
com14 com13 syld 3impia expimpd sylibr rspcedvd eqsuptid gcdmndc mpjaodan
|
|
ancri simprr exmiddc eqtr2d impbida ) CEFZDEFZGZACDUFVFZHZIAUGJZACKJZADKJ
|
|
ZGZBLZCKJZUUTDKJZGZUUTAKJZMZBEUHZUIZUUMUUOGUVGIUUNUGJZUUNCKJZUUNDKJZGZUVC
|
|
UUTUUNKJZMZBEUHZUIZUUMUVOUUOUUMUVHUVKUVNUUMUUNCDUJUKCDULUUMUVMBEUUMUUTEFZ
|
|
GZUVPUUKUULUIZUVMUVRUVPUUMGUVQUVPUUKUULUMUVPUUMUNUOUUTCDUPUQURUSNUUOUVGUV
|
|
OOUUMUUOUUPUVHUUSUVKUVFUVNAUUNIUGPUUOUUQUVIUURUVJAUUNCKQAUUNDKQUTUUOUVEUV
|
|
MBEUUOUVDUVLUVCAUUNUUTKPVAVBVCRVDUUMUVGGZUUNCIHZDIHZGZIUALZCKJZUWCDKJZGZU
|
|
AEVGZSTVHZVEZAUUMUUNUWIHUVGUACDVINUVSUWBUWIAHZUWBVJZUWBUVSUWJUWBUVSGUWIIA
|
|
UWBUWIIHUVSUWBIUWHVKNUWBUVSIAHZUWBUVGUWLUUMUWBUVGUUPAIKJZUWMGZUUTIKJZUWOG
|
|
ZUVDMZBEUHZUIUWLUWBUUSUWNUVFUWRUUPUVTUUQUWMUWAUURUWMCIAKPDIAKPVLUWBUVEUWQ
|
|
BEUWBUVCUWPUVDUVTUVAUWOUWAUVBUWOCIUUTKPDIUUTKPVLVMVBVNUUPUWNUWRUWLUWNUUPU
|
|
WRUWLMZUWMUUPUWSMZUWMUWMAEFZIEFZGUWTAIVOUXAUWTUXBUXAUUPUWSUXAUUPGZUWRUVDB
|
|
EUHZUWLUXCUWQUVDBEUXCUVPGUWPUWQUVDOUVPUWPUXCUVPUWOUWOUUTVPZUXEVQRUWPUVDVR
|
|
VSVTUXAUXDUWLMUUPUXDIAKJZUXAUWLUXBUXDUXFMWAUVDUXFBIEUUTIAKQWBWCUXAUXFAIHZ
|
|
UWLUXAUXFUXGAWDWEIAWFWGWJNWHWKNVSNWIWLWMWNWOWPWQUWKUVSUWJUWKUVSGZUWIUWHAU
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WKUWIUWHHUVSUWBIUWHWRNUXHUBUCUDUESUWGATUELZSFUDLZSFGUXIUXJHUXIUXJTJVJUXJU
|
|
XITJVJGOUXHUXIUXJWSRUVGASFZUWKUUMUUSUUPUXKUVFUUQUXKUURUUQAUUQUXAUUKACVOZX
|
|
BZWTNZXAXCUBLZUWGFZUXHAUXOTJVJZUXPUXOEFZUXOCKJZUXODKJZGZGZUXHUXQMUWFUYAUA
|
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UXOEUWCUXOHUWDUXSUWEUXTUWCUXOCKQUWCUXODKQUTXDUYBUWKUVSUXQUYBUWKGZUUMUVGUX
|
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QUVGUYCUUMGZUXQUUPUUSUVFUYDUXQMZUYDUVFUUPUUSGZUXQUYDUVFUXOAKJZUYFUXQMUYBU
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VFUYGMZUWKUUMUXRUYAUYHUXRUVFUYAUYGUVEUYAUYGMBUXOEUUTUXOHZUVCUYAUVDUYGUYIU
|
|
VAUXSUVBUXTUUTUXOCKQUUTUXODKQUTUUTUXOAKQXEWBXFWOXGUYFUYGUYDUXQUYFUYGAXHFZ
|
|
UYEUUSUUPUYJUUQUUPUYJMZUURUUQUXAUUKGUYKUXLUXAUYKUUKUYJUXAUUPAXIXJNVSNXLUY
|
|
FUYGUYJGZUYDUXQUYFUYLGZUYDGUXOAUYCUXOSFZUYMUUMUXRUYNUYAUWKUXOXMXGXKUUSUXK
|
|
UUPUYLUYDUXNXNUYMUYDUXOAUGJZUYLUYFUYDUYOMZUYGUYJUYFUYPMUYDUYJUYFUYGUYOUYD
|
|
UYJUYFUYGUYOMZUYDUYJGZUYFGUXRAXOFZUYQUXRUYAUWKUUMUYJUYFXPUYRUYFUYSUYJUYDU
|
|
YFUYSMZUYJUYSUXGXQUYDUYTMZAXRUYSVUAUXGUYSUYDUYFXSUXGUYDUYTUXGUYDGUYFUUPIC
|
|
KJZIDKJZGZGZUYSUXGUYFVUEOUYDUXGUUSVUDUUPUXGUUQVUBUURVUCAICKQAIDKQUTXTNUYD
|
|
VUEUYSMUXGUYDVUDUYSUUPUYDUVTVJZUWAVJZXQZVUDUYSMZUYDUWKVUHUYBUWKUUMYAUUKUW
|
|
KVUHOZUYCUULUUKUVTYBZVUJUUKUXBVUKWACIYCYDUVTUWAYEVSXKYFVUFVUIVUGVUDVUFUYS
|
|
VUBVUFUYSMZVUCUXBUUKGVUBVULICVOUUKVUBVULMUXBUUKVUBUVTVULCWDUVTUYSYGWMRYHN
|
|
WIVUDVUGUYSVUCVUGUYSMZVUBUXBUULGVUCVUMIDVOUULVUCVUMMUXBUULVUCUWAVUMDWDUWA
|
|
UYSYGWMRYHRWIYIVSWNRWHWKYIYJXLWOUXOAYKYLYMYPWOXLWOYNYMYOYQYRYQYSWIYTYTYJX
|
|
LUXHUYNUXOATJZGZGZUXOUCLZTJZVUNUCAUWGVUPUXAUUSGZAUWGFUXHVUSVUOUVGVUSUWKUU
|
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MUUSUUPVUSUVFUUSUXAUUQUXAUURUXMNUUFXAXCNUWFUUSUAAEUWCAHUWDUUQUWEUURUWCACK
|
|
QUWCADKQUTXDUUAVUQAHVURVUNOVUPVUQAUXOTPRUXHUYNVUNUUGUUBUUCWPWQUVSUWBYBZUW
|
|
BUWKXQUUMVUTUVGCDUUDNUWBUUHVSUUEUUIUUJ $.
|
|
$}
|
|
|
|
${
|
|
$d N x $. $d M x $. $d P x $.
|
|
$( Associative law for ` gcd ` operator. Theorem 1.4(b) in [ApostolNT]
|
|
p. 16. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario
|
|
Carneiro, 19-Apr-2014.) $)
|
|
gcdass $p |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) ->
|
|
( ( N gcd M ) gcd P ) = ( N gcd ( M gcd P ) ) ) $=
|
|
( vx cz wcel cc0 wceq wa cdvds wbr crab cr clt csup cif cgcd co anass wb
|
|
w3a cv a1i rabbiia supeq1i ifbieq2i cn0 gcdcl 3adant3 nn0zd simp3 syl2anc
|
|
gcdval gcdeq0 anbi1d bicomd simpr simpl1 simpl2 dvdsgcdb syl3anc rabbidva
|
|
supeq1d ifbieq2d eqtr4d simp1 3adant1 anbi2d simpl3 3eqtr4a ) CEFZBEFZAEF
|
|
ZUAZCGHZBGHZIZAGHZIZGDUBZCJKZVTBJKZIZVTAJKZIZDELZMNOZPZVOVPVRIZIZGWAWBWDI
|
|
ZIZDELZMNOZPZCBQRZAQRZCBAQRZQRZVSWJWGWNGVOVPVRSMWFWMNWEWLDEWEWLTVTEFZWAWB
|
|
WDSUCUDUEUFVNWQWPGHZVRIZGVTWPJKZWDIZDELZMNOZPZWHVNWPEFVMWQXGHVNWPVKVLWPUG
|
|
FVMCBUHUIUJVKVLVMUKDWPAUMULVNVSXBWGXFGVNXBVSVNXAVQVRVKVLXAVQTVMCBUNUIUOUP
|
|
VNMWFXENVNWEXDDEVNWTIZWCXCWDXHWTVKVLWCXCTVNWTUQZVKVLVMWTURVKVLVMWTUSZVTCB
|
|
UTVAUOVBVCVDVEVNWSVOWRGHZIZGWAVTWRJKZIZDELZMNOZPZWOVNVKWREFWSXQHVKVLVMVFV
|
|
NWRVLVMWRUGFVKBAUHVGUJDCWRUMULVNWJXLWNXPGVNXLWJVNXKWIVOVLVMXKWITVKBAUNVGV
|
|
HUPVNMWMXONVNWLXNDEXHWKXMWAXHWTVLVMWKXMTXIXJVKVLVMWTVIVTBAUTVAVHVBVCVDVEV
|
|
J $.
|
|
$}
|
|
|
|
$( Distribute multiplication by a nonnegative integer over gcd. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro,
|
|
30-May-2014.) $)
|
|
mulgcd $p |- ( ( K e. NN0 /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) $=
|
|
( cn0 wcel cz cmul co cgcd cc0 wa cdvds wbr 3adant1 syl2anc dvdsgcd syl3anc
|
|
wi wb mp2and wceq cn elnn0 w3a simp1 nnzd simp2 zmulcld simp3 gcdcld nnnn0d
|
|
wo gcdcl nn0mulcld cdiv nn0cnd nnap0d divcanap2d gcddvds simpld eqbrtrd wne
|
|
nncnd dvdsmul1 nnne0d nn0zd dvdsval2 mpbid dvdscmulr syl112anc dvdscmul mpd
|
|
simprd eqbrtrrd dvdseq syl22anc 3expib mul02d gcd0val syl6reqr zcn 3ad2ant2
|
|
oveq1d cc eqtrd 3ad2ant3 oveq12d 3eqtr4d jaoi sylbi 3impib ) ADEZBFEZCFEZAB
|
|
GHZACGHZIHZABCIHZGHZUAZWLAUBEZAJUAZULWMWNKWTRZAUCXAXCXBXAWMWNWTXAWMWNUDZWQD
|
|
EWSDEWQWSLMWSWQLMZWTXDWOWPXDABXDAXAWMWNUEZUFZXAWMWNUGZUHZXDACXGXAWMWNUIZUHZ
|
|
UJZXDAWRXDAXFUKWMWNWRDEZXABCUMZNZUNZXDAWQAUOHZGHZWQWSLXDWQAXDWQXLUPXDAXFVCX
|
|
DAXFUQURZXDXQWRLMZXRWSLMZXDXQBLMZXQCLMZXTXDXRWOLMZYBXDXRWQWOLXSXDWQWOLMZWQW
|
|
PLMZXDWOFEZWPFEZYEYFKXIXKWOWPUSOZUTVAXDXQFEZWMAFEZAJVBZYDYBSXDAWQLMZYJXDAWO
|
|
LMZAWPLMZYMXDYKWMYNXGXHABVDOXDYKWNYOXGXJACVDOXDYKYGYHYNYOKYMRXGXIXKAWOWPPQT
|
|
XDYKYLWQFEYMYJSXGXDAXFVEZXDWQXLVFAWQVGQVHZXHXGYPAXQBVIVJVHXDXRWPLMZYCXDXRWQ
|
|
WPLXSXDYEYFYIVMVAXDYJWNYKYLYRYCSYQXJXGYPAXQCVIVJVHXDYJWMWNYBYCKXTRYQXHXJXQB
|
|
CPQTXDYJWRFEZYKXTYARYQXDWRXOVFZXGAXQWRVKQVLVNXDWSWOLMZWSWPLMZXEXDWRBLMZUUAX
|
|
DUUCWRCLMZWMWNUUCUUDKXABCUSNZUTXDYSWMYKUUCUUARYTXHXGAWRBVKQVLXDUUDUUBXDUUCU
|
|
UDUUEVMXDYSWNYKUUDUUBRYTXJXGAWRCVKQVLXDWSFEYGYHUUAUUBKXERXDWSXPVFXIXKWSWOWP
|
|
PQTWQWSVOVPVQXBWMWNWTXBWMWNUDZJJIHZJWRGHZWQWSUUFUUHJUUGUUFWRUUFWRWMWNXMXBXN
|
|
NUPVRVSVTUUFWOJWPJIUUFWOJBGHJUUFAJBGXBWMWNUEZWCUUFBWMXBBWDEWNBWAWBVRWEUUFWP
|
|
JCGHJUUFAJCGUUIWCUUFCWNXBCWDEWMCWAWFVRWEWGUUFAJWRGUUIWCWHVQWIWJWK $.
|
|
|
|
$( Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in
|
|
[ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
absmulgcd $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K x. M ) gcd ( K x. N ) ) = ( abs ` ( K x. ( M gcd N ) ) ) ) $=
|
|
( cz wcel cabs cfv cgcd co cmul wceq wa cn0 oveq2d 3adant1 cc absmul syl2an
|
|
zcn 3impdi w3a gcdcl nn0re nn0ge0 absidd syl nn0cnd oveqan12d syl3an zmulcl
|
|
3impb gcdabs nn0abscl zabscl mulgcd 3eqtr3d eqtrd 3eqtr4rd ) ADEZBDEZCDEZUA
|
|
ZAFGZBCHIZFGZJIZVCVDJIZAVDJIFGZABJIZACJIZHIZUTVAVFVGKUSUTVALZVEVDVCJVLVDMEZ
|
|
VEVDKBCUBZVMVDVDUCVDUDUEUFNOUSUTVAVHVFKZUSAPEZVDPEVOVLASZVLVDVNUGAVDQRUKVBV
|
|
KVCBFGZCFGZHIZJIZVGVBVIFGZVJFGZHIZVCVRJIZVCVSJIZHIZVKWAUSVPUTBPEZVACPEZWDWG
|
|
KZVQBSCSVPWHWIWJVPWHLVPWILWBWEWCWFHABQACQUHTUIUSUTVAWDVKKZUSUTLVIDEVJDEWKUS
|
|
VALABUJACUJVIVJULRTUSVCMEUTVRDEVAVSDEWGWAKAUMBUNCUNVCVRVSUOUIUPVBVTVDVCJUTV
|
|
AVTVDKUSBCULONUQUR $.
|
|
|
|
$( Reverse distribution law for the ` gcd ` operator. (Contributed by Scott
|
|
Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
mulgcdr $p |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN0 ) ->
|
|
( ( A x. C ) gcd ( B x. C ) ) = ( ( A gcd B ) x. C ) ) $=
|
|
( cz wcel cn0 w3a cmul co cgcd wceq mulgcd 3coml cc 3ad2ant1 nn0cn 3ad2ant3
|
|
zcn mulcomd 3ad2ant2 oveq12d gcdcl 3adant3 nn0cnd 3eqtr4d ) ADEZBDEZCFEZGZC
|
|
AHIZCBHIZJIZCABJIZHIZACHIZBCHIZJIUMCHIUHUFUGULUNKCABLMUIUOUJUPUKJUIACUFUGAN
|
|
EUHAROUHUFCNEUGCPQZSUIBCUGUFBNEUHBRTUQSUAUIUMCUIUMUFUGUMFEUHABUBUCUDUQSUE
|
|
$.
|
|
|
|
${
|
|
$d A a b $. $d B a b $. $d C a b $.
|
|
$( Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.)
|
|
(Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
gcddiv $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN )
|
|
/\ ( C || A /\ C || B ) ) ->
|
|
( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) $=
|
|
( va vb cz wcel w3a cdvds wa cgcd cdiv wceq cmul wrex 3ad2ant3 divcanap4d
|
|
wbr co cc cn cv wb nnz simp1 divides syl2anc simp2 anbi12d reeanv syl6bbr
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|
wi gcdcl nn0cnd 3adant3 simp3 nnap0d cn0 nnnn0 mulgcdr syl3an3 oveq1d zcn
|
|
nncn 3ad2ant1 3ad2ant2 oveq12d 3eqtr4d oveq12 oveqan12d eqeq12d syl5ibcom
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|
oveq1 3expa expcom rexlimdvv sylbid imp ) AFGZBFGZCUAGZHZCAIRZCBIRZJZABKS
|
|
ZCLSZACLSZBCLSZKSZMZWBWEDUBZCNSZAMZEUBZCNSZBMZJZEFODFOZWKWBWEWNDFOZWQEFOZ
|
|
JWSWBWCWTWDXAWBCFGZVSWCWTUCWAVSXBVTCUDPZVSVTWAUEDCAUFUGWBXBVTWDXAUCXCVSVT
|
|
WAUHECBUFUGUIWNWQDEFFUJUKWAVSWSWKULVTWAWRWKDEFFWLFGZWOFGZJZWAWRWKULZXDXEW
|
|
AXGXDXEWAHZWMWPKSZCLSZWMCLSZWPCLSZKSZMWRWKXHWLWOKSZCNSZCLSXNXJXMXHXNCXDXE
|
|
XNTGWAXFXNWLWOUMUNUOWAXDCTGXECVDPZXHCXDXEWAUPUQZQXHXIXOCLWAXDXECURGXIXOMC
|
|
USWLWOCUTVAVBXHXKWLXLWOKXHWLCXDXEWLTGWAWLVCVEXPXQQXHWOCXEXDWOTGWAWOVCVFXP
|
|
XQQVGVHWRXJWGXMWJWRXIWFCLWMAWPBKVIVBWNWQXKWHXLWIKWMACLVMWPBCLVMVJVKVLVNVO
|
|
VPPVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d M k n $. $d N k n $.
|
|
$( The GCD of a multiple of a number is the number itself. (Contributed by
|
|
Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro,
|
|
19-Apr-2014.) $)
|
|
gcdmultiple $p |- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M ) $=
|
|
( vk vn cn wcel cmul co cgcd wi c1 caddc oveq2 oveq2d eqeq1d imbi2d eqtrd
|
|
wceq cz cc cv weq nncn mulid1d cabs cfv nnz gcdid syl nnre nn0ge0d absidd
|
|
nnnn0 wa adantr zmulcl syl2an gcdaddm mp3an1 syl2anc ax-1cn mp3an3 mulcom
|
|
1z adddi mpan2 eqtr4d biimpd expcom a2d nnind impcom ) BEFAEFZAABGHZIHZAR
|
|
ZVMAACUAZGHZIHZARZJVMAAKGHZIHZARZJVMAADUAZGHZIHZARZJVMAAWDKLHZGHZIHZARZJV
|
|
MVPJCDBVQKRZVTWCVMWLVSWBAWLVRWAAIVQKAGMNOPCDUBZVTWGVMWMVSWFAWMVRWEAIVQWDA
|
|
GMNOPVQWHRZVTWKVMWNVSWJAWNVRWIAIVQWHAGMNOPVQBRZVTVPVMWOVSVOAWOVRVNAIVQBAG
|
|
MNOPVMWBAAIHZAVMWAAAIVMAAUCZUDNVMWPAUEUFZAVMASFZWPWRRAUGZAUHUIVMAAUJVMAAU
|
|
MUKULQQWDEFZVMWGWKVMXAWGWKJVMXAUNZWGWKXBWFWJAXBWFAWEKAGHZLHZIHZWJXBWSWESF
|
|
ZWFXERZVMWSXAWTUOVMWSWDSFXFXAWTWDUGAWDUPUQKSFWSXFXGVDKAWEURUSUTXBWIXDAIVM
|
|
ATFZWDTFZWIXDRXAWQWDUCXHXIUNZWIWEWALHZXDXHXIKTFZWIXKRVAAWDKVEVBXJWAXCWELX
|
|
HWAXCRZXIXHXLXMVAAKVCVFUONQUQNVGOVHVIVJVKVL $.
|
|
$}
|
|
|
|
$( Extend ~ gcdmultiple so ` N ` can be an integer. (Contributed by Scott
|
|
Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
gcdmultiplez $p |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) $=
|
|
( cn wcel cz cmul co cgcd wceq cc0 wdc wn wa cc oveq2d syl adantr eqtrd cfv
|
|
cabs wo wi zdceq mpan2 exmiddc nncn mul01 cn0 nnnn0 nn0gcdid0 oveq2 syl5ibr
|
|
0z eqeq1d wne df-ne zcn absmul syl2an nnre nn0ge0d absidd oveq1d simpll nnz
|
|
nnzd zmulcl sylan gcdabs2 syl2anc nnabscl gcdmultiple sylan2 anassrs expcom
|
|
3eqtr3d sylbir jaoi 3syl anabsi7 ) ACDZBEDZAABFGZHGZAIZWBBJIZKZWFWFLZUAWAWB
|
|
MZWEUBZWBJEDWGUMBJUCUDWFUEWFWJWHWIWEWFAAJFGZHGZAIWIWLAJHGZAWAWLWMIZWBWAANDZ
|
|
WNAUFZWOWKJAHAUGOPQWAWMAIZWBWAAUHDWQAUIZAUJPQRWFWDWLAWFWCWKAHBJAFUKOUNULWHB
|
|
JUOZWJBJUPWIWSWEWIWSMZAWCTSZHGZAABTSZFGZHGZWDAWIXBXEIWSWIXAXDAHWIXAATSZXCFG
|
|
ZXDWAWOBNDXAXGIWBWPBUQABURUSWAXGXDIWBWAXFAXCFWAAAUTWAAWRVAVBVCQROQWTAEDZWCE
|
|
DZXBWDIWTAWAWBWSVDVFWIXIWSWAXHWBXIAVEABVGVHQWCAVIVJWAWBWSXEAIZWBWSMWAXCCDXJ
|
|
BVKAXCVLVMVNVPVOVQVRVSVT $.
|
|
|
|
$( A positive integer ` A ` is equal to its gcd with an integer ` B ` if and
|
|
only if ` A ` divides ` B ` . Generalization of ~ gcdeq . (Contributed
|
|
by AV, 1-Jul-2020.) $)
|
|
gcdzeq $p |- ( ( A e. NN /\ B e. ZZ ) -> ( ( A gcd B ) = A <-> A || B ) ) $=
|
|
( cn wcel cz wa cgcd co wceq cdvds wbr nnz gcddvds sylan cle adantr syl cc0
|
|
wi simpl simprd breq1 syl5ibcom iddvds wn simpr wne nnne0 necon3ai syl31anc
|
|
dvdslegcd mpand simpld cn0 gcdcl nn0zd dvdsle syl2anc jctild nn0red cr nnre
|
|
mpd letri3d sylibrd impbid ) ACDZBEDZFZABGHZAIZABJKZVIVJBJKZVKVLVIVJAJKZVMV
|
|
GAEDZVHVNVMFALZABMNZUAVJABJUBUCVIVLVJAOKZAVJOKZFVKVIVLVSVRVIAAJKZVLVSVIVOVT
|
|
VGVOVHVPPZAUDQVIVOVOVHARIZBRIZFZUEZVTVLFVSSWAWAVGVHUFVGWEVHVGARUGWEAUHWDARW
|
|
BWCTUIQPAABUKUJULVIVNVRVIVNVMVQUMVIVJEDVGVNVRSVIVJVGVOVHVJUNDVPABUONZUPVGVH
|
|
TVJAUQURVCUSVIVJAVIVJWFUTVGAVADVHAVBPVDVEVF $.
|
|
|
|
|
|
$( ` A ` is equal to its gcd with ` B ` if and only if ` A ` divides ` B ` .
|
|
(Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV,
|
|
8-Aug-2021.) $)
|
|
gcdeq $p |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) = A <-> A || B ) ) $=
|
|
( cn wcel cz cgcd co wceq cdvds wbr wb nnz gcdzeq sylan2 ) BCDACDBEDABFGAHA
|
|
BIJKBLABMN $.
|
|
|
|
${
|
|
$d M k $. $d N k $.
|
|
$( Unidirectional form of ~ dvdssq . (Contributed by Scott Fenton,
|
|
19-Apr-2014.) $)
|
|
dvdssqim $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M || N -> ( M ^ 2 ) || ( N ^ 2 ) ) ) $=
|
|
( vk cz wcel wa cdvds wbr cv cmul co wceq wrex cexp divides zsqcl syl2anr
|
|
c2 cc zcn dvdsmul2 sqmul breqtrrd oveq1 breq2d syl5ibcom rexlimdva adantr
|
|
wi sylbid ) ADEZBDEZFABGHCIZAJKZBLZCDMZARNKZBRNKZGHZCABOUKUPUSUIULUKUOUSC
|
|
DUKUMDEZFZUQUNRNKZGHUOUSVAUQUMRNKZUQJKZVBGUTVCDEUQDEUQVDGHUKUMPAPVCUQUAQU
|
|
TUMSEASEVBVDLUKUMTATUMAUBQUCUOVBURUQGUNBRNUDUEUFUGUHUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Relationship between the order of an element and that of a multiple. (a
|
|
divisibility equivalent). (Contributed by Stefan O'Rear,
|
|
6-Sep-2015.) $)
|
|
dvdsmulgcd $p |- ( ( B e. ZZ /\ C e. ZZ ) ->
|
|
( A || ( B x. C ) <-> A || ( B x. ( C gcd A ) ) ) ) $=
|
|
( vx vy cz wcel wa cmul co cdvds wbr cv simplr simpld syl2anc zmulcld mpd
|
|
caddc zcnd cgcd wceq dvdszrcl adantl bezout adantr simplll simpllr simprl
|
|
simprr wi dvdsmultr1 syl3anc mulassd breqtrd dvdsmul1 mul12d breqtrrd w3a
|
|
wrex dvds2add imp adddid oveq2 breq2d syl5ibrcom rexlimdvva simprd zmulcl
|
|
syl32anc simpr gcddvds gcdcld nn0zd simpll dvdscmul dvdstr impbida ) BFGZ
|
|
CFGZHZABCIJZKLZABCAUAJZIJZKLZWAWCHZWDCDMZIJZAEMZIJZSJZUBZEFUTDFUTZWFWGVTA
|
|
FGZWNVSVTWCNWGWOWBFGZWCWOWPHWAAWBUCUDOZDECAUEPWGWMWFDEFFWGWHFGZWJFGZHZHZW
|
|
FWMABWLIJZKLXAABWIIJZBWKIJZSJZXBKXAWOXCFGZXDFGZAXCKLZAXDKLZAXEKLZWGWOWTWQ
|
|
UFZXABWIVSVTWCWTUGZXACWHVSVTWCWTUHZWGWRWSUIZQZQXABWKXLXAAWJXKWGWRWSUJZQZQ
|
|
XAAWBWHIJZXCKXAWCAXRKLZWAWCWTNXAWOWPWRWCXSUKXKXABCXLXMQXNAWBWHULUMRXABCWH
|
|
XABXLTZXACXMTXAWHXNTUNUOXAAABWJIJZIJZXDKXAWOYAFGAYBKLXKXABWJXLXPQAYAUPPXA
|
|
BAWJXTXAAXKTXAWJXPTUQURWOXFXGUSXHXIHXJAXCXDVAVBVJXABWIWKXTXAWIXOTXAWKXQTV
|
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CURWMWEXBAKWDWLBIVDVEVFVGRWAWFHZWOWEFGZWPWFWEWBKLZWCYCWOYDWFWOYDHWAAWEUCU
|
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DZOZYCWOYDYFVHWAWPWFBCVIUFWAWFVKYCWDCKLZYEYCYHWDAKLZYCVTWOYHYIHVSVTWFNZYG
|
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CAVLPOYCWDFGVTVSYHYEUKYCWDYCCAYJYGVMVNYJVSVTWFVOBWDCVPUMRWOYDWPUSWFYEHWCA
|
|
WEWBVQVBVJVR $.
|
|
$}
|
|
|
|
$( If ` K ` and ` M ` are relatively prime, then the GCD of ` K ` and
|
|
` M x. N ` is the GCD of ` K ` and ` N ` . (Contributed by Scott Fenton,
|
|
12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
rpmulgcd $p |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\
|
|
( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd N ) ) $=
|
|
( cn wcel w3a cgcd co c1 wceq wa cmul gcdmultiple 3adant2 nnz zmulcl syl2an
|
|
cz adantr eqtrd oveq1d 3ad2ant1 3adant1 gcdass syl3anc eqtr3d nnnn0 mulgcdr
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cn0 syl3an oveq1 sylan9eq cc nncn 3ad2ant3 mulid2d oveq2d ) ADEZBDEZCDEZFZA
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BGHZIJZKZABCLHZGHZAACLHZVEGHZGHZACGHVAVFVIJVCVAAVGGHZVEGHZVFVIVAVJAVEGURUTV
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JAJUSACMNUAVAAREZVGREZVEREZVKVIJURUSVLUTAOZUBURUTVMUSURVLCREZVMUTVOCOZACPQN
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|
USUTVNURUSBREZVPVNUTBOZVQBCPQUCVEVGAUDUEUFSVDVHCAGVDVHICLHZCVAVCVHVBCLHZVTU
|
|
RVLUSVRUTCUIEVHWAJVOVSCUGABCUHUJVBICLUKULVDCVACUMEZVCUTURWBUSCUNUOSUPTUQT
|
|
$.
|
|
|
|
${
|
|
$d A n k $. $d B n k $. $d N n k $.
|
|
$( If ` A ` and ` B ` are relatively prime, then so are ` A ^ N ` and
|
|
` B ` . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario
|
|
Carneiro, 19-Apr-2014.) $)
|
|
rplpwr $p |- ( ( A e. NN /\ B e. NN /\ N e. NN ) ->
|
|
( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) ) $=
|
|
( vk vn cn wcel cgcd co c1 wceq cexp wi oveq2 oveq1d eqeq1d imbi2d adantr
|
|
wa cz cv caddc weq nncn exp1d biimpar w3a df-3an cmul simpl1 nncnd simpl3
|
|
nnnn0d expp1d simp1 cn0 nnnn0 3ad2ant3 nnexpcld nnzd mulcomd eqtrd oveq2d
|
|
zcnd simpl2 nnz 3ad2ant1 3ad2ant2 gcdcom syl2anc biimpa rpmulgcd syl31anc
|
|
peano2nn 3eqtr4d biimprd sylanbr an32s expcom a2d nnind expd com12 3impia
|
|
) AFGZBFGZCFGZABHIZJKZACLIZBHIZJKZMZWGWEWFSZWMWGWNWIWLWNWISZADUAZLIZBHIZJ
|
|
KZMWOAJLIZBHIZJKZMWOAEUAZLIZBHIZJKZMWOAXCJUBIZLIZBHIZJKZMWOWLMDECWPJKZWSX
|
|
BWOXKWRXAJXKWQWTBHWPJALNOPQDEUCZWSXFWOXLWRXEJXLWQXDBHWPXCALNOPQWPXGKZWSXJ
|
|
WOXMWRXIJXMWQXHBHWPXGALNOPQWPCKZWSWLWOXNWRWKJXNWQWJBHWPCALNOPQWNXBWIWNXAW
|
|
HJWEXAWHKWFWEWTABHWEAAUDUEORPUFXCFGZWOXFXJWOXOXFXJMZWNXOWIXPWNXOSWEWFXOUG
|
|
ZWIXPWEWFXOUHXQWISZXJXFXRXIXEJXRBXHHIZBXDHIZXIXEXRXSBAXDUIIZHIZXTXRXHYABH
|
|
XRXHXDAUIIYAXRAXCXRAWEWFXOWIUJZUKZXRXCWEWFXOWIULUMUNXRXDAXRXDXQXDTGZWIXQX
|
|
DXQAXCWEWFXOUOXOWEXCUPGWFXCUQURUSZUTRZVDYDVAVBVCXRWFWEXDFGZBAHIZJKZYBXTKW
|
|
EWFXOWIVEYCXQYHWIYFRXQWIYJXQWHYIJXQATGZBTGZWHYIKWEWFYKXOAVFVGWFWEYLXOBVFV
|
|
HZABVIVJPVKBAXDVLVMVBXRXHTGYLXIXSKXRXHXRAXGYCXRXGXQXGFGZWIXOWEYNWFXCVNURR
|
|
UMUSUTXQYLWIYMRZXHBVIVJXRYEYLXEXTKYGYOXDBVIVJVOPVPVQVRVSVTWAWBWCWD $.
|
|
$}
|
|
|
|
$( If ` A ` and ` B ` are relatively prime, then so are ` A ^ N ` and
|
|
` B ^ N ` . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by
|
|
Mario Carneiro, 19-Apr-2014.) $)
|
|
rppwr $p |- ( ( A e. NN /\ B e. NN /\ N e. NN ) ->
|
|
( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 ) ) $=
|
|
( cn wcel w3a cgcd co c1 wceq cexp 3jca nnzd gcdcom syl2anc nnz rplpwr sylc
|
|
wa cz simpl1 simpl2 simpl3 nnnn0d nnexpcld 3ad2ant1 3ad2ant2 eqeq1d biimpa
|
|
eqtrd ex ) ADEZBDEZCDEZFZABGHZIJZACKHBCKHZGHIJZUOUQSZULURDEZUNFAURGHZIJUSUT
|
|
ULVAUNULUMUNUQUAZUTBCULUMUNUQUBZUTCULUMUNUQUCZUDUEZVELUTVBURAGHZIUTATEZURTE
|
|
VBVGJUTAVCMUTURVFMAURNOUTUMULUNFBAGHZIJZVGIJUTUMULUNVDVCVELUOUQVJUOUPVIIUOV
|
|
HBTEZUPVIJULUMVHUNAPUFUMULVKUNBPUGABNOUHUIBACQRUJAURCQRUK $.
|
|
|
|
$( Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.)
|
|
(Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
sqgcd $p |- ( ( M e. NN /\ N e. NN ) ->
|
|
( ( M gcd N ) ^ 2 ) = ( ( M ^ 2 ) gcd ( N ^ 2 ) ) ) $=
|
|
( cn wcel wa cgcd co c2 cexp c1 cdiv wceq cz cdvds adantr adantl syl2anc cc
|
|
wbr cc0 cmul gcdnncl nnsqcld nncnd mulid1d nnsqcl nnz gcddvds syl2an simpld
|
|
nnzd wi dvdssqim mpd simprd gcddiv syl32anc nncn sqdivapd syl31anc dividapd
|
|
nnap0d oveq12d eqtr3d clt wb nnne0d dvdsval2 syl3anc mpbid nnre nnred nngt0
|
|
wne cr nngt0d divgt0d elnnz sylanbrc 2nn rppwr mp3an3 3eqtr2d cap wn neneqd
|
|
anim12i intnanrd gcdn0cl ax-1cn divmulap mp3an2 syl12anc ) ACDZBCDZEZABFGZH
|
|
IGZJUAGZWRAHIGZBHIGZFGZWPWRWPWRWPWQABUBZUCZUDZUEWPXBWRKGZJLZWSXBLZWPXFWTWRK
|
|
GZXAWRKGZFGZAWQKGZHIGZBWQKGZHIGZFGZJWPWTMDZXAMDZWRCDWRWTNSZWRXANSZXFXKLWNXQ
|
|
WOWNWTAUFZUKZOWOXRWNWOXABUFUKZPXDWPWQANSZXSWPYDWQBNSZWNAMDZBMDZYDYEEZWOAUGZ
|
|
BUGZABUHUIZUJZWPWQMDZYFYDXSULWPWQXCUKZWNYFWOYIOZWQAUMQUNWPYEXTWPYDYEYKUOZWP
|
|
YMYGYEXTULYNWOYGWNYJPZWQBUMQUNWTXAWRUPUQWPXMXIXOXJFWPAWQWNARDWOAUROWPWQXCUD
|
|
ZWPWQXCVBZUSWPBWQWOBRDWNBURPYRYSUSVCWPXLXNFGZJLZXPJLZWPWQWQKGZYTJWPYFYGWQCD
|
|
YHUUCYTLYOYQXCYKABWQUPUTWPWQYRYSVAVDWPXLCDZXNCDZUUAUUBULZWPXLMDZTXLVESUUDWP
|
|
YDUUGYLWPYMWQTVNZYFYDUUGVFYNWPWQXCVGZYOWQAVHVIVJWPAWQWNAVODWOAVKOWPWQXCVLZW
|
|
NTAVESWOAVMOWPWQXCVPZVQXLVRVSWPXNMDZTXNVESUUEWPYEUULYPWPYMUUHYGYEUULVFYNUUI
|
|
YQWQBVHVIVJWPBWQWOBVODWNBVKPUUJWOTBVESWNBVMPUUKVQXNVRVSUUDUUEHCDUUFVTXLXNHW
|
|
AWBQUNWCWPXBRDZWRRDZWRTWDSZXGXHVFZWPXBWPXQXREWTTLZXATLZEWEZXBCDWNXQWOXRYBYC
|
|
WGWNUUSWOWNUUQUURWNWTTWNWTYAVGWFWHOWTXAWIQUDXEWPWRXDVBUUMJRDUUNUUOEUUPWJXBJ
|
|
WRWKWLWMVJVD $.
|
|
|
|
$( Lemma for ~ dvdssq . (Contributed by Scott Fenton, 18-Apr-2014.)
|
|
(Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
dvdssqlem $p |- ( ( M e. NN /\ N e. NN ) ->
|
|
( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) ) $=
|
|
( cn wcel wa cdvds wbr c2 cexp co cz nnz syl2an cgcd wceq adantr nnsqcl cc0
|
|
wb cr dvdssqim sqgcd gcdeq biimpar eqtrd cle cn0 gcdcl nn0red nn0ge0d nnnn0
|
|
wi nnre sq11 syl22anc mpbid gcddvds simprd eqbrtrrd ex impbid ) ACDZBCDZEZA
|
|
BFGZAHIJZBHIJZFGZVBAKDZBKDZVEVHULVCALZBLZABUAMVDVHVEVDVHEZABNJZABFVMVNHIJZV
|
|
FOZVNAOZVMVOVFVGNJZVFVDVOVROVHABUBPVDVRVFOZVHVBVFCDVGCDVSVHSVCAQBQVFVGUCMUD
|
|
UEVDVPVQSZVHVDVNTDRVNUFGATDZRAUFGZVTVDVNVBVIVJVNUGDVCVKVLABUHMZUIVDVNWCUJVB
|
|
WAVCAUMPVBWBVCVBAAUKUJPVNAUNUOPUPVMVNAFGZVNBFGZVDWDWEEZVHVBVIVJWFVCVKVLABUQ
|
|
MPURUSUTVA $.
|
|
|
|
$( Two numbers are divisible iff their squares are. (Contributed by Scott
|
|
Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $)
|
|
dvdssq $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) ) $=
|
|
( cz wcel cdvds wbr c2 cexp co wb cc0 wceq syl bitr4d zsqcl cabs cfv adantr
|
|
wa syl2an wdc wn wo wi 0z zdceq mpan2 exmiddc 0dvds cc zcn sqeq0 breq1 sq0i
|
|
adantl breq1d bibi12d syl5ibr wne df-ne nnabscl nnz dvds0 2thd breq2 breq2d
|
|
cn dvdssqlem sylan2 simpl dvdsabsb nnsqcl nnzd abssq 3bitr4d anassrs expcom
|
|
sylbir jaoi 3syl anabsi7 sylan absdvdsb adantlr eqcomd bitrd an32s anabsi5
|
|
) ACDZBCDZABEFZAGHIZBGHIZEFZJZWIAKLZUAZWPWPUBZUCWIWJSZWOUDZWIKCDZWQUEAKUFUG
|
|
WPUHWPWTWRWSWOWPKBEFZKWMEFZJZWJXDWIWJXBWMKLZXCWJXBBKLZXEBUIWJBUJDZXEXFJBUKZ
|
|
BULMNWJWMCDZXCXEJBOZWMUIMNUOWPWKXBWNXCAKBEUMWPWLKWMEAUNUPUQURWRAKUSZWTAKUTW
|
|
SXKWOWIXKWJWOWIXKSZWJSZAPQZBEFZXNGHIZWMEFZWKWNXLXNVGDZWJXOXQJZAVAXRWJXSWJXF
|
|
UAZXFXFUBZUCXRWJSZXSUDZWJXAXTUEBKUFUGXFUHXFYCYAYBXSXFXNKEFZXPKEFZJZXRYFWJXR
|
|
XNCDZYFXNVBZYGYDYEXNVCYGXPCDZYEXNOXPVCMVDMRXFXOYDXQYEBKXNEVEXFWMKXPEBUNVFUQ
|
|
URYABKUSZYCBKUTYBYJXSXRWJYJXSXRWJYJSZSZXNBPQZEFZXPYMGHIZEFZXOXQYKXRYMVGDYNY
|
|
PJBVAXNYMVHVIXRYGWJXOYNJYKYHWJYJVJXNBVKTYLXQXPWMPQZEFZYPXRYIXIXQYRJYKXRXPXN
|
|
VLVMWJXIYJXJRXPWMVKTYKYPYRJXRYKYOYQXPEYKXGYOYQLWJXGYJXHRBVNMVFUONVOVPVQVRVS
|
|
VTWAWBWIWJWKXOJXKABWCWDXMWNWLPQZWMEFZXQXLWLCDZXIWNYTJWJWIUUAXKAORXJWLWMWCTX
|
|
LYTXQJWJXLYSXPWMEWIYSXPLXKWIXPYSWIAUJDXPYSLAUKAVNMWERUPRWFVOWGVQVRVSVTWH $.
|
|
|
|
$( Partial converse to ~ bezout . Existence of a linear combination does not
|
|
set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear,
|
|
23-Sep-2014.) $)
|
|
bezoutr $p |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( A
|
|
gcd B ) || ( ( A x. X ) + ( B x. Y ) ) ) $=
|
|
( cz wcel wa cgcd co cmul cdvds wbr caddc adantr zmulcld w3a dvdsmultr1 imp
|
|
gcdcl syl31anc simpll simprl simplr simprr gcddvds simpld dvds2add syl32anc
|
|
nn0zd simprd ) AEFZBEFZGZCEFZDEFZGZGZABHIZEFZACJIZEFZBDJIZEFZURUTKLZURVBKLZ
|
|
URUTVBMIKLZUMUSUPUMURABSUINZUQACUKULUPUAZUMUNUOUBZOUQBDUKULUPUCZUMUNUOUDZOU
|
|
QUSUKUNURAKLZVDVGVHVIUQVLURBKLZUMVLVMGUPABUENZUFUSUKUNPVLVDURACQRTUQUSULUOV
|
|
MVEVGVJVKUQVLVMVNUJUSULUOPVMVEURBDQRTUSVAVCPVDVEGVFURUTVBUGRUH $.
|
|
|
|
$( Converse of ~ bezout for when the greater common divisor is one
|
|
(sufficient condition for relative primality). (Contributed by Stefan
|
|
O'Rear, 23-Sep-2014.) $)
|
|
bezoutr1 $p |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( (
|
|
( A x. X ) + ( B x. Y ) ) = 1 -> ( A gcd B ) = 1 ) ) $=
|
|
( cz wcel wa cmul co caddc c1 wceq wbr cdvds cn a1i syl2anc cc0 wne oveq1
|
|
cgcd cle bezoutr adantr simpr breqtrd wi gcdcl nn0zd ad2antrr 1nn dvdsle wb
|
|
mpd wn simpll oveqan12d zcn mul02d sylan9eqr 00id syl6eq adantll eqnetrd ex
|
|
0ne1 necon2bd imp gcdn0cl nnle1eq1 syl mpbid ) AEFBEFGZCEFZDEFZGZGZACHIZBDH
|
|
IZJIZKLZABUAIZKLZVQWAGZWBKUBMZWCWDWBKNMZWEWDWBVTKNVQWBVTNMWAABCDUCUDVQWAUEU
|
|
FWDWBEFZKOFZWFWEUGVMWGVPWAVMWBABUHUIUJWHWDUKPWBKULQUNWDWBOFZWEWCUMWDVMARLZB
|
|
RLZGZUOZWIVMVPWAUPVQWAWMVQWLVTKVQWLVTKSVQWLGZVTRKVPWLVTRLVMVPWLGVTRRJIZRWLV
|
|
PVTRCHIZRDHIZJIWOWJWKVRWPVSWQJARCHTBRDHTUQVNVOWPRWQRJVNCCURUSVODDURUSUQUTVA
|
|
VBVCRKSWNVFPVDVEVGVHABVIQWBVJVKVLVE $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Algorithms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d F k m $. $d N k m $. $d k m ph $.
|
|
nn0seqcvgd.1 $e |- ( ph -> F : NN0 --> NN0 ) $.
|
|
nn0seqcvgd.2 $e |- ( ph -> N = ( F ` 0 ) ) $.
|
|
nn0seqcvgd.3 $e |- ( ( ph /\ k e. NN0 ) ->
|
|
( ( F ` ( k + 1 ) ) =/= 0 -> ( F ` ( k + 1 ) ) < ( F ` k ) ) ) $.
|
|
$( A strictly-decreasing nonnegative integer sequence with initial term
|
|
` N ` reaches zero by the ` N ` th term. Deduction version.
|
|
(Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
nn0seqcvgd $p |- ( ph -> ( F ` N ) = 0 ) $=
|
|
( cfv cc0 wceq cle wbr cmin co cn0 wcel wi clt wa cr 0nn0 ffvelrn sylancl
|
|
vm wf eqeltrd nn0red leidd cv c1 caddc fveq2 oveq2 breq12d eqbrtrrd recnd
|
|
imbi2d subid1d breqtrrd a1i wne wb posdif syl2anr adantr adantl peano2nn0
|
|
nn0re breq1 cz syl2an nn0zd nn0z zsubcl zltlem1 syl2anc cc ax-1cn subsub4
|
|
nn0cn mp3an3 breq2d bitrd 3bitr2d biimpa an32s a1d ffvelrnda zred syl3anc
|
|
ltletr sylibd syland expdimp wdc wo 0zd zdceq dcne mpjaodan anasss expcom
|
|
a2d 3adant1 fnn0ind pm2.43i subidd breqtrd ffvelrnd nn0ge0d 0re mpbir2and
|
|
sylib letri3 ) ADCHZIJZXOIKLZIXOKLZAXODDMNZIKAXOXSKLZADOPZYADDKLAXTQZADIC
|
|
HZOFAOOCUEZIOPYCOPEUAOOICUBUCUFZYEADADYEUGZUHZAUDUIZCHZDYHMNZKLZQAYCDIMNZ
|
|
KLZQZABUIZCHZDYOMNZKLZQZAYOUJUKNZCHZDYTMNZKLZQZYBUDBDDYHIJZYKYMAUUEYIYCYJ
|
|
YLKYHICULYHIDMUMUNUQYHYOJZYKYRAUUFYIYPYJYQKYHYOCULYHYODMUMUNUQYHYTJZYKUUC
|
|
AUUGYIUUAYJUUBKYHYTCULYHYTDMUMUNUQYHDJZYKXTAUUHYIXOYJXSKYHDCULYHDDMUMUNUQ
|
|
YNYAAYCDYLKADYCDKFYGUOADADYFUPZURUSUTYOOPZYODRLZYSUUDQYAUUJUUKSZAYRUUCAUU
|
|
LYRUUCQZAUUJUUKUUMAUUJSZUUKSZUUAIJZUUMUUAIVAZUUOUUPSUUCYRUUNUUPUUKUUCUUNU
|
|
UPSZUUKUUCUURUUKIYQRLZUUAYQRLZUUCUUNUUKUUSVBZUUPUUJYOTPDTPUVAAYOVHYFYODVC
|
|
VDVEUUPUUTUUSVBUUNUUAIYQRVIVFUUNUUTUUCVBUUPUUNUUTUUAYQUJMNZKLZUUCUUNUUAVJ
|
|
PZYQVJPZUUTUVCVBUUNUUAAYDYTOPUUAOPUUJEYOVGOOYTCUBVKZVLZADVJPYOVJPUVEUUJAD
|
|
YEVLYOVMDYOVNVKZUUAYQVOVPUUNUVBUUBUUAKADVQPZYOVQPZUVBUUBJZUUJUUIYOVTUVIUV
|
|
JUJVQPUVKVRDYOUJVSWAVKWBWCZVEWDWEWFWGUUOUUQYRUUCUUNUUQYRSUUCQUUKUUNUUQUUA
|
|
YPRLZYRUUCGUUNUVMYRSZUUTUUCUUNUUATPYPTPYQTPUVNUUTQUUNUUAUVFUGUUNYPAOOYOCE
|
|
WHUGUUNYQUVHWIUUAYPYQWKWJUVLWLWMVEWNUUOUUPWOZUUPUUQWPUUOUVDIVJPUVOUUNUVDU
|
|
UKUVGVEUUOWQUUAIWRVPUUAIWSXMWTXAXBXCXDXEWJXFADUUIXGXHAXOAOODCEYEXIZXJAXOT
|
|
PITPXPXQXRSVBAXOUVPUGXKXOIXNUCXL $.
|
|
$}
|
|
|
|
${
|
|
ialgrlem1st.f $e |- ( ph -> F : S --> S ) $.
|
|
$( Lemma for ~ ialgr0 . Expressing ~ algrflemg in a form suitable for
|
|
theorems such as ~ iseq1 or ~ iseqfn . (Contributed by Jim Kingdon,
|
|
22-Jul-2021.) $)
|
|
ialgrlem1st $p |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x ( F o. 1st ) y ) e. S ) $=
|
|
( cv wcel wa c1st ccom co wceq algrflemg adantl wf adantr simprl ffvelrnd
|
|
cfv eqeltrd ) ABGZDHZCGZDHZIZIZUBUDEJKLZUBETZDUFUHUIMAUBUDEDDNOUGDDUBEADD
|
|
EPUFFQAUCUERSUA $.
|
|
$}
|
|
|
|
${
|
|
ialgrlemconst.z $e |- Z = ( ZZ>= ` M ) $.
|
|
ialgrlemconst.a $e |- ( ph -> A e. S ) $.
|
|
$( Lemma for ~ ialgr0 . Closure of a constant function, in a form suitable
|
|
for theorems such as ~ iseq1 or ~ iseqfn . (Contributed by Jim Kingdon,
|
|
22-Jul-2021.) $)
|
|
ialgrlemconst $p |- ( ( ph /\ x e. ( ZZ>= ` M ) ) ->
|
|
( ( Z X. { A } ) ` x ) e. S ) $=
|
|
( cv cuz cfv wcel wa csn cxp wceq eleq2i biimpri fvconst2g syl2an eqeltrd
|
|
adantr ) ABIZEJKZLZMUCFCNOKZCDACDLZUCFLZUFCPUEHUHUEFUDUCGQRFCUCDSTAUGUEHU
|
|
BUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $. $d x y M $. $d x y ph $. $d x y S $.
|
|
$d x y V $. $d x y Z $.
|
|
algrf.1 $e |- Z = ( ZZ>= ` M ) $.
|
|
algrf.2 $e |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S ) $.
|
|
algrf.3 $e |- ( ph -> M e. ZZ ) $.
|
|
algrf.4 $e |- ( ph -> A e. S ) $.
|
|
algrf.5 $e |- ( ph -> F : S --> S ) $.
|
|
algrf.s $e |- ( ph -> S e. V ) $.
|
|
$( The value of the algorithm iterator ` R ` at ` 0 ` is the initial state
|
|
` A ` . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario
|
|
Carneiro, 28-May-2014.) $)
|
|
ialgr0 $p |- ( ph -> ( R ` M ) = A ) $=
|
|
( vx vy cfv c1st ccom wcel csn cxp fveq1i ialgrlemconst ialgrlem1st iseq1
|
|
cseq wceq cuz cz uzid syl syl6eleqr fvconst2g syl2anc eqtrd syl5eq ) AFCQ
|
|
FERSZDHBUAUBZFUGZQZBFCUTJUCAVAFUSQZBAOPURDUSFKAOBDFHILUDAOPDEMUEUFABDTFHT
|
|
VBBUHLAFFUIQZHAFUJTFVCTKFUKULIUMHBFDUNUOUPUQ $.
|
|
|
|
$( An algorithm is a step function ` F : S --> S ` on a state space ` S ` .
|
|
An algorithm acts on an initial state ` A e. S ` by iteratively applying
|
|
` F ` to give ` A ` , ` ( F `` A ) ` , ` ( F `` ( F `` A ) ) ` and so
|
|
on. An algorithm is said to halt if a fixed point of ` F ` is reached
|
|
after a finite number of iterations.
|
|
|
|
The algorithm iterator ` R : NN0 --> S ` "runs" the algorithm ` F ` so
|
|
that ` ( R `` k ) ` is the state after ` k ` iterations of ` F ` on the
|
|
initial state ` A ` .
|
|
|
|
Domain and codomain of the algorithm iterator ` R ` . (Contributed by
|
|
Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
ialgrf $p |- ( ph -> R : Z --> S ) $=
|
|
( vx vy c1st wf wcel cfv ccom csn cxp cv cuz eleq2i ialgrlemconst sylan2b
|
|
cseq ialgrlem1st iseqf feq1i sylibr ) AHDEQUAZDHBUBUCZFUIZRHDCRAOPUNDUOFG
|
|
HINKOUDZHSAUQFUETZSUQUOTDSHURUQIUFAOBDFHILUGUHAOPDEMUJUKHDCUPJULUM $.
|
|
|
|
$d K x y $.
|
|
$( The value of the algorithm iterator ` R ` at ` ( K + 1 ) ` .
|
|
(Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro,
|
|
27-Dec-2014.) $)
|
|
ialgrp1 $p |- ( ( ph /\ K e. Z ) ->
|
|
( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) ) $=
|
|
( vx vy wcel co cfv wa caddc c1st ccom csn cxp cseq fveq1i simpr syl6eleq
|
|
c1 cuz adantr ialgrlemconst wf ialgrlem1st iseqp1 syl5eq iseqcl peano2uzs
|
|
wceq fvconst2g syl2an eqeltrd algrflemg syl2anc fveq2i syl6reqr eqtr4d )
|
|
AFIRZUAZFUKUBSZCTZFEUCUDZDIBUEUFZGUGZTZVLVOTZVNSZFCTZETZVKVMVLVPTVSVLCVPK
|
|
UHVKPQVNDVOGFVKFIGULTAVJUIJUJZVKPBDGIJABDRZVJMUMZUNZVKPQDEADDEUOVJNUMUPZU
|
|
QURVKVSVQETZWAVKVQDRVRDRVSWGVAVKPQVNDVOGFWBWEWFUSVKVRBDAWCVLIRVRBVAVJMGFI
|
|
JUTIBVLDVBVCWDVDVQVREDDVEVFVTVQEFCVPKUHVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d k z A $. $d x F $. $d k x z I $. $d k x z R $. $d k x z S $.
|
|
$d z K $.
|
|
alginv.1 $e |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S ) $.
|
|
alginv.2 $e |- F : S --> S $.
|
|
alginv.3 $e |- I Fn S $.
|
|
alginv.4 $e |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) ) $.
|
|
alginv.s $e |- S e. V $.
|
|
$( If ` I ` is an invariant of ` F ` , its value is unchanged after any
|
|
number of iterations of ` F ` . (Contributed by Paul Chapman,
|
|
31-Mar-2011.) $)
|
|
ialginv $p |- ( ( A e. S /\ K e. NN0 ) ->
|
|
( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) $=
|
|
( cn0 wcel cfv wceq wi fveq2 fveq2d vz vk cc0 cv c1 caddc co eqeq1d eqidd
|
|
imbi2d wa nn0uz 0zd id a1i ialgrp1 ialgrf ffvelrnda eqeq12d vtoclga eqtrd
|
|
wf syl biimprd expcom a2d nn0ind impcom ) GNOBDOZGCPZFPZUCCPZFPZQZVIUAUDZ
|
|
CPZFPZVMQZRVIVMVMQZRVIUBUDZCPZFPZVMQZRVIVTUEUFUGZCPZFPZVMQZRVIVNRUAUBGVOU
|
|
CQZVRVSVIWHVQVMVMWHVPVLFVOUCCSTUHUJVOVTQZVRWCVIWIVQWBVMWIVPWAFVOVTCSTUHUJ
|
|
VOWDQZVRWGVIWJVQWFVMWJVPWEFVOWDCSTUHUJVOGQZVRVNVIWKVQVKVMWKVPVJFVOGCSTUHU
|
|
JVIVMUIVTNOZVIWCWGVIWLWCWGRVIWLUKZWGWCWMWFWBVMWMWFWAEPZFPZWBWMWEWNFVIBCDE
|
|
VTUCHNULIVIUMZVIUNZDDEVBVIJUOZDHOVIMUOZUPTWMWADOWOWBQZVINDVTCVIBCDEUCHNUL
|
|
IWPWQWRWSUQURAUDZEPZFPZXAFPZQWTAWADXAWAQZXCWOXDWBXEXBWNFXAWAESTXAWAFSUSLU
|
|
TVCVAUHVDVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d A k $. $d C k z $. $d F z $. $d N k $. $d R k z $. $d S k z $.
|
|
algcvg.1 $e |- F : S --> S $.
|
|
algcvg.2 $e |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S ) $.
|
|
algcvg.3 $e |- C : S --> NN0 $.
|
|
algcvg.4 $e |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 ->
|
|
( C ` ( F ` z ) ) < ( C ` z ) ) ) $.
|
|
algcvg.5 $e |- N = ( C ` A ) $.
|
|
ialgcvg.s $e |- S e. V $.
|
|
$( One way to prove that an algorithm halts is to construct a countdown
|
|
function ` C : S --> NN0 ` whose value is guaranteed to decrease for
|
|
each iteration of ` F ` until it reaches ` 0 ` . That is, if ` X e. S `
|
|
is not a fixed point of ` F ` , then
|
|
` ( C `` ( F `` X ) ) < ( C `` X ) ` .
|
|
|
|
If ` C ` is a countdown function for algorithm ` F ` , the sequence
|
|
` ( C `` ( R `` k ) ) ` reaches ` 0 ` after at most ` N ` steps, where
|
|
` N ` is the value of ` C ` for the initial state ` A ` . (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
ialgcvg $p |- ( A e. S -> ( C ` ( R ` N ) ) = 0 ) $=
|
|
( wcel cfv cc0 cn0 wceq clt vk ccom wf nn0uz 0zd id a1i ffvelrni syl5eqel
|
|
ialgrf fvco3 syl2anc sylancr 0nn0 sylancl ialgr0 fveq2d eqtrd syl6reqr cv
|
|
fco wa wne wbr c1 caddc co ffvelrnda fveq2 neeq1d breq12d imbi12d vtoclga
|
|
wi syl peano2nn0 syl2an ialgrp1 sylan 3imtr4d nn0seqcvgd eqtr3d ) BEOZGCD
|
|
UBZPZGDPCPZQWCREDUCZGROWEWFSWCBDEFQHRUDJWCUEZWCUFZEEFUCWCIUGZEHOWCNUGZUJZ
|
|
WCGBCPZRMERBCKUHUIREGCDUKULWCUAWDGWCERCUCWGRRWDUCKWLRERCDVAUMWCQWDPZWMGWC
|
|
WNQDPZCPZWMWCWGQROWNWPSWLUNREQCDUKUOWCWOBCWCBDEFQHRUDJWHWIWJWKUPUQURMUSWC
|
|
UAUTZROZVBZWQDPZFPZCPZQVCZXBWTCPZTVDZWQVEVFVGZWDPZQVCXGWQWDPZTVDWSWTEOXCX
|
|
EVNZWCREWQDWLVHAUTZFPZCPZQVCZXLXJCPZTVDZVNXIAWTEXJWTSZXMXCXOXEXPXLXBQXPXK
|
|
XACXJWTFVIUQZVJXPXLXBXNXDTXQXJWTCVIVKVLLVMVOWSXGXBQWSXGXFDPZCPZXBWCWGXFRO
|
|
XGXSSWRWLWQVPREXFCDUKVQWSXRXACWCBDEFWQQHRUDJWHWIWJWKVRUQURZVJWSXGXBXHXDTX
|
|
TWCWGWRXHXDSWLREWQCDUKVSVKVTWAWB $.
|
|
$}
|
|
|
|
$( Lemma for ~ algcvgb . (Contributed by Paul Chapman, 31-Mar-2011.) $)
|
|
algcvgblem $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <->
|
|
( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) ) $=
|
|
( cn0 wcel wa cc0 wne clt wbr wi wceq wn wb wdc cz nn0z 0z syl adantl cr wo
|
|
zdceq sylancl dcned imordc cle zltnle sylancr adantr nn0le0eq0 notbid bitrd
|
|
df-ne syl6bbr anbi2d nnedc breq1 bi2 syl6 impd sylbird expd ax-1 jctir jaob
|
|
syl6bi sylibr sylbid nn0ge0 nn0re lelttr mp3an1 syl2anr mpand sylibd imim2d
|
|
0re jcad pm3.34 impbid1 con34bdc imbi12i bitr4d ) ACDZBCDZEZBFGZBAHIZJZAFGZ
|
|
WHJZWGWJJZEZWKAFKZBFKZJZEWFWIWMWFWIWKWLWFWIWGLZWHUAZWKWEWIWRMZWDWEWGNWSWEBF
|
|
WEBODZFODZWONZBPZQBFUBZUCUDWGWHUERSWFWQWKJZWHWKJZEWRWKJWFXEXFWFWQWJWHWFWQWJ
|
|
EWQFAHIZEWHWFXGWJWQWFXGWNLZWJWFXGAFUFIZLZXHWDXGXJMZWEWDXAAODXKQAPFAUGUHUIWD
|
|
XJXHMWEWDXIWNAUJUKUIULAFUMZUNZUOWFWQXGWHWFWQWHXGMZXGWHJWFWQWOXNWFXBWQWOMWFW
|
|
TXAXBWEWTWDXCSQXDUCZBFUPRBFAHUQVFWHXGURUSUTVAVBWHWJVCVDWQWKWHVEVGVHWFWHWJWG
|
|
WFWHXGWJWFFBUFIZWHXGWEXPWDBVISWEBTDZATDZXPWHEXGJZWDBVJAVJFTDXQXRXSVQFBAVKVL
|
|
VMVNXMVOVPVRWGWJWHVSVTWFWPWLWKWFWPWOLZXHJZWLWFXBWPYAMXOWNWOWARWGXTWJXHBFUMX
|
|
LWBUNUOWC $.
|
|
|
|
${
|
|
algcvgb.1 $e |- F : S --> S $.
|
|
algcvgb.2 $e |- C : S --> NN0 $.
|
|
$( Two ways of expressing that ` C ` is a countdown function for algorithm
|
|
` F ` . The first is used in these theorems. The second states the
|
|
condition more intuitively as a conjunction: if the countdown function's
|
|
value is currently nonzero, it must decrease at the next step; if it has
|
|
reached zero, it must remain zero at the next step. (Contributed by
|
|
Paul Chapman, 31-Mar-2011.) $)
|
|
algcvgb $p |- ( X e. S ->
|
|
( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <->
|
|
( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\
|
|
( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) ) $=
|
|
( wcel cfv cn0 cc0 wne clt wbr wi wceq wa wb ffvelrni syl algcvgblem
|
|
syl2anc ) DBGZDAHZIGDCHZAHZIGZUEJKUEUCLMZNUCJKUGNUCJOUEJONPQBIDAFRUBUDBGU
|
|
FBBDCERBIUDAFRSUCUETUA $.
|
|
$}
|
|
|
|
${
|
|
$d A k m $. $d C k m z $. $d F z $. $d K m $. $d N k m $.
|
|
$d R k m z $. $d S k m z $.
|
|
algcvga.1 $e |- F : S --> S $.
|
|
algcvga.2 $e |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S ) $.
|
|
algcvga.3 $e |- C : S --> NN0 $.
|
|
algcvga.4 $e |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 ->
|
|
( C ` ( F ` z ) ) < ( C ` z ) ) ) $.
|
|
algcvga.5 $e |- N = ( C ` A ) $.
|
|
ialgcvga.s $e |- S e. V $.
|
|
$( The countdown function ` C ` remains ` 0 ` after ` N ` steps.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
ialgcvga $p |- ( A e. S -> ( K e. ( ZZ>= ` N ) ->
|
|
( C ` ( R ` K ) ) = 0 ) ) $=
|
|
( wcel cfv cc0 wceq wi vm vk cn0 cuz ffvelrni syl5eqel cz nn0z cle wbr wa
|
|
eluz1 cv c1 caddc co fveq2 fveq2d eqeq1d imbi2d ialgcvg a1i nn0ge0 adantr
|
|
w3a cr nn0re zre 0re letr mp3an1 syl2an mpand elnn0z simplbi2 adantl syld
|
|
sylan impr expcom 3adant1 ancld nn0uz 0zd ialgrf ffvelrnda wne clt neeq1d
|
|
id wf breq12d imbi12d vtoclga algcvgb syl6bi mpd syl ialgrp1 sylibrd syl6
|
|
simpr a2d uzind 3expib sylbid com3r ) BEPZHUCPZGHUDQPZGDQZCQZRSZTXHHBCQUC
|
|
NEUCBCLUEUFZXIXJXHXMXIHUGPZXJXHXMTZTHUHXOXJGUGPZHGUIUJZUKXPHGULXOXQXRXPXH
|
|
UAUMZDQZCQZRSZTXHHDQZCQZRSZTZXHUBUMZDQZCQZRSZTXHYGUNUOUPZDQZCQZRSZTXPUAUB
|
|
HGXSHSZYBYEXHYOYAYDRYOXTYCCXSHDUQURUSUTXSYGSZYBYJXHYPYAYIRYPXTYHCXSYGDUQU
|
|
RUSUTXSYKSZYBYNXHYQYAYMRYQXTYLCXSYKDUQURUSUTXSGSZYBXMXHYRYAXLRYRXTXKCXSGD
|
|
UQURUSUTYFXOABCDEFHIJKLMNOVAVBXOYGUGPZHYGUIUJZVEZXHYJYNUUAXHXHYGUCPZUKZYJ
|
|
YNTUUAXHUUBYSYTXHUUBTXOXHYSYTUKUUBXHYSYTUUBXHXIYSYTUUBTXNXIYSUKZYTRYGUIUJ
|
|
ZUUBUUDRHUIUJZYTUUEXIUUFYSHVCVDXIHVFPZYGVFPZUUFYTUKUUETZYSHVGYGVHRVFPUUGU
|
|
UHUUIVIRHYGVJVKVLVMYSUUEUUBTXIUUBYSUUEYGVNVOVPVQVRVSVTWAWBUUCYJYHFQZCQZRS
|
|
ZYNUUCYHEPZYJUULTZXHUCEYGDXHBDEFRIUCWCKXHWDZXHWJZEEFWKXHJVBZEIPXHOVBZWEWF
|
|
UUMUUKRWGZUUKYIWHUJZTZUUNAUMZFQZCQZRWGZUVDUVBCQZWHUJZTUVAAYHEUVBYHSZUVEUU
|
|
SUVGUUTUVHUVDUUKRUVHUVCUUJCUVBYHFUQURZWIUVHUVDUUKUVFYIWHUVIUVBYHCUQWLWMMW
|
|
NUUMUVAYIRWGUUTTZUUNUKUUNCEFYHJLWOUVJUUNXBWPWQWRUUCYMUUKRUUCYLUUJCXHBDEFY
|
|
GRIUCWCKUUOUUPUUQUURWSURUSWTXAXCXDXEXFWRXGWQ $.
|
|
|
|
$d K z $. $d N z $.
|
|
algfx.6 $e |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) ) $.
|
|
$( If ` F ` reaches a fixed point when the countdown function ` C ` reaches
|
|
` 0 ` , ` F ` remains fixed after ` N ` steps. (Contributed by Paul
|
|
Chapman, 22-Jun-2011.) $)
|
|
ialgfx $p |- ( A e. S -> ( K e. ( ZZ>= ` N ) ->
|
|
( R ` K ) = ( R ` N ) ) ) $=
|
|
( wcel cfv wceq wi vm vk cz cuz cn0 ffvelrni syl5eqel nn0zd wa cv cle wbr
|
|
crab uzval eleq2d pm5.32i c1 caddc co fveq2 eqeq1d imbi2d eqidd a1i sylan
|
|
eluznn0 cc0 nn0uz 0zd id ialgrp1 syldan ialgrf ffvelrnda ialgcvga eqeq12d
|
|
wf imp imbi12d vtoclga sylc eqtrd biimprd expcom adantl sylbir a2d uzind3
|
|
sylbi ex com3r mpd ) BEQZHUCQZGHUDRZQZGDRZHDRZSZTWMHWMHBCRUENEUEBCLUFUGZU
|
|
HWNWPWMWSWNWPWMWSTZWNWPUIWNGHAUJZUKULAUCUMZQZUIXAWNWPXDWNWOXCGAHUNZUOUPWM
|
|
UAUJZDRZWRSZTWMWRWRSZTZWMUBUJZDRZWRSZTWMXKUQURUSZDRZWRSZTXAUAAUBHGXFHSZXH
|
|
XIWMXQXGWRWRXFHDUTVAVBXFXKSZXHXMWMXRXGXLWRXFXKDUTVAVBXFXNSZXHXPWMXSXGXOWR
|
|
XFXNDUTVAVBXFGSZXHWSWMXTXGWQWRXFGDUTVAVBXJWNWMWRVCVDWNXKXCQZUIZWMXMXPYBWN
|
|
XKWOQZUIWMXMXPTZTZWNYCYAWNWOXCXKXEUOUPYCYEWNWMYCYDWMYCUIZXPXMYFXOXLWRYFXO
|
|
XLFRZXLWMYCXKUEQZXOYGSWMHUEQYCYHWTXKHVFVEZWMBDEFXKVGIUEVHKWMVIZWMVJZEEFVQ
|
|
WMJVDZEIQWMOVDZVKVLYFXLEQZXLCRZVGSZYGXLSZWMYCYHYNYIWMUEEXKDWMBDEFVGIUEVHK
|
|
YJYKYLYMVMVNVLWMYCYPABCDEFXKHIJKLMNOVOVRXBCRZVGSZXBFRZXBSZTYPYQTAXLEXBXLS
|
|
ZYSYPUUAYQUUBYRYOVGXBXLCUTVAUUBYTYGXBXLXBXLFUTUUBVJVPVSPVTWAWBVAWCWDWEWFW
|
|
GWHWIWJWKWL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Euclid's Algorithm
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y M $. $d x y N $. $d x y X $. $d x y z A $. $d x z R $.
|
|
$d z E $.
|
|
eucalgval.1 $e |- E = ( x e. NN0 , y e. NN0 |->
|
|
if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) ) $.
|
|
$( The value of the step function ` E ` for Euclid's Algorithm on an
|
|
ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by
|
|
Mario Carneiro, 28-May-2014.) $)
|
|
eucalgval2 $p |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M E N ) =
|
|
if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) ) $=
|
|
( cn0 wcel cc0 wceq cop cmo co cif cvv wa opexg adantr simpr cz simpl wne
|
|
wn nn0zd cn neqned elnnne0 sylanbrc zmodcld syl2anc wdc zdceq ifcldadc cv
|
|
0zd eqeq1d opeq12 oveq12 opeq12d ifbieq12d ovmpt2ga mpd3an3 ) DGHZEGHZEIJ
|
|
ZDEKZEDELMZKZNZOHDECMVIJVCVDPZVEVFVHOVJVFOHVEDEGGQRVJVEUCZPZVDVGGHVHOHVJV
|
|
DVKVCVDSZRZVLDEVJDTHVKVJDVCVDUAUDRVLVDEIUBEUEHVNVLEIVJVKSUFEUGUHUIEVGGGQU
|
|
JVJETHITHVEUKVJEVMUDVJUOEIULUJUMABDEGGBUNZIJZAUNZVOKZVOVQVOLMZKZNVICOVQDJ
|
|
ZVOEJZPZVPVEVRVTVFVHWCVOEIWAWBSZUPVQVODEUQWCVOEVSVGWDVQDVOELURUSUTFVAVB
|
|
$.
|
|
|
|
$( Euclid's Algorithm ~ eucialg computes the greatest common divisor of two
|
|
nonnegative integers by repeatedly replacing the larger of them with its
|
|
remainder modulo the smaller until the remainder is 0.
|
|
|
|
The value of the step function ` E ` for Euclid's Algorithm.
|
|
(Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
eucalgval $p |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) =
|
|
if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) $=
|
|
( cn0 cxp wcel c1st cfv c2nd cop cc0 wceq cmo co cif df-ov xp1st fveq2d
|
|
xp2nd eucalgval2 syl2anc syl5eqr 1st2nd2 syl6eqr opeq2d ifeq12d 3eqtr4d )
|
|
DFFGHZDIJZDKJZLZCJZULMNZUMULUKULOPZLZQZDCJUODULDOJZLZQUJUNUKULCPZURUKULCR
|
|
UJUKFHULFHVAURNDFFSDFFUAABCUKULEUBUCUDUJDUMCDFFUEZTUJUODUMUTUQVBUJUSUPULU
|
|
JUSUMOJUPUJDUMOVBTUKULORUFUGUHUI $.
|
|
|
|
$( Domain and codomain of the step function ` E ` for Euclid's Algorithm.
|
|
(Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro,
|
|
28-May-2014.) $)
|
|
eucalgf $p |- E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 ) $=
|
|
( cv cc0 wceq cop cmo co cif cn0 cxp wcel wral wf adantl opelxpi eqeltrd
|
|
wa cn wne nnne0 neneqd iffalsed nnnn0 cz nn0z zmodcl sylan syl2anc iftrue
|
|
adantlr adantr wo simpr elnn0 sylib mpjaodan rgen2a fmpt2 mpbi ) BEZFGZAE
|
|
ZVCHZVCVEVCIJZHZKZLLMZNZBLOALOVJVJCPVKABLVELNZVCLNZTZVCUANZVKVDVLVOVKVMVL
|
|
VOTZVIVHVJVPVDVFVHVPVCFVOVCFUBVLVCUCQUDUEVPVMVGLNZVHVJNVOVMVLVCUFQVLVEUGN
|
|
VOVQVEUHVEVCUIUJVCVGLLRUKSUMVNVDTVIVFVJVDVIVFGVNVDVFVHULQVNVFVJNVDVEVCLLR
|
|
UNSVNVMVOVDUOVLVMUPVCUQURUSUTABLLVIVJCDVAVB $.
|
|
|
|
$( The invariant of the step function ` E ` for Euclid's Algorithm is the
|
|
` gcd ` operator applied to the state. (Contributed by Paul Chapman,
|
|
31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.) $)
|
|
eucalginv $p |- ( X e. ( NN0 X. NN0 ) ->
|
|
( gcd ` ( E ` X ) ) = ( gcd ` X ) ) $=
|
|
( cn0 wcel cfv cgcd cc0 wceq cmo fveq2d co adantr df-ov syl6eqr cz adantl
|
|
cop cxp c2nd cif eucalgval cn wa c1st 1st2nd2 oveq2d xp1st nn0zd sylancom
|
|
nnz zmodcl gcdcom syl2anc modgcd 3eqtrd wne nnne0 neneqd iffalsed 3eqtr4d
|
|
iftrue wo xp2nd elnn0 sylib mpjaodan eqtrd ) DFFUAGZDCHZIHDUBHZJKZDVMDLHZ
|
|
TZUCZIHZDIHZVKVLVQIABCDEUDMVKVMUEGZVRVSKZVNVKVTUFZVMVOINZDUGHZVMINZVRVSWB
|
|
WCVMWDVMLNZINZWFVMINZWEWBVOWFVMIWBVOWDVMTZLHWFWBDWILVKDWIKVTDFFUHOZMWDVML
|
|
PQUIWBVMRGZWFRGWGWHKVTWKVKVMUMSWBWFVKVTWDRGZWFFGWBWDVKWDFGVTDFFUJOUKZWDVM
|
|
UNULUKVMWFUOUPVKVTWLWHWEKWMWDVMUQULURWBVRVPIHWCWBVQVPIWBVNDVPWBVMJVTVMJUS
|
|
VKVMUTSVAVBMVMVOIPQWBVSWIIHWEWBDWIIWJMWDVMIPQVCVNWAVKVNVQDIVNDVPVDMSVKVMF
|
|
GVTVNVEDFFVFVMVGVHVIVJ $.
|
|
|
|
$( The second member of the state decreases with each iteration of the step
|
|
function ` E ` for Euclid's Algorithm. (Contributed by Paul Chapman,
|
|
31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.) $)
|
|
eucalglt $p |- ( X e. ( NN0 X. NN0 ) ->
|
|
( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) ) $=
|
|
( cn0 wcel cfv c2nd cc0 wne clt wbr cmo cop wceq adantr fveq2d nn0zd cq
|
|
cxp wa c1st co eucalgval wn simpr iftrue eqeq2d fveq2 syl6bi eqeq2 sylibd
|
|
cif syl5com necon3ad mpd iffalsed eqtrd xp2nd 1st2nd2 df-ov syl6eqr xp1st
|
|
cn neqned elnnne0 sylanbrc zmodcld eqeltrd op2ndg 3eqtrd cz zq syl nngt0d
|
|
syl2anc modqlt syl3anc eqbrtrd ex ) DFFUAGZDCHZIHZJKZWDDIHZLMWBWEUBZWDDUC
|
|
HZWFNUDZWFLWGWDWFDNHZOZIHZWJWIWGWCWKIWGWCWFJPZDWKUNZWKWBWCWNPZWEABCDEUEQZ
|
|
WGWMDWKWGWEWMUFWBWEUGWGWMWDJWGWOWMWDJPZWPWMWOWDWFPZWQWMWOWCDPWRWMWNDWCWMD
|
|
WKUHUIWCDIUJUKWFJWDULUMUOUPUQZURUSRWGWFFGZWJFGWLWJPWBWTWEDFFUTQZWGWJWIFWG
|
|
WJWHWFOZNHWIWGDXBNWBDXBPWEDFFVAQRWHWFNVBVCZWGWHWFWGWHWBWHFGWEDFFVDQSZWGWT
|
|
WFJKWFVEGXAWGWFJWSVFWFVGVHZVIVJWFWJFFVKVQXCVLWGWHTGZWFTGZJWFLMWIWFLMWGWHV
|
|
MGXFXDWHVNVOWGWFVMGXGWGWFXASWFVNVOWGWFXEVPWHWFVRVSVTWA $.
|
|
|
|
eucialg.2 $e |- R =
|
|
seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) , ( NN0 X. NN0 ) ) $.
|
|
${
|
|
eucialgcvga.3 $e |- N = ( 2nd ` A ) $.
|
|
$( Once Euclid's Algorithm halts after ` N ` steps, the second element of
|
|
the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.) $)
|
|
eucialgcvga $p |- ( A e. ( NN0 X. NN0 ) ->
|
|
( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) ) $=
|
|
( cn0 wcel cuz cfv c2nd cc0 wceq cvv fvres clt vz cxp wa xp2nd syl5eqel
|
|
cres eluznn0 sylan nn0uz 0zd id eucalgf a1i nn0ex xpex ialgrf ffvelrnda
|
|
syldan syl simpl syl6eqr fveq2d eleq2d biimpar f2ndres wne wbr eucalglt
|
|
wf cv ffvelrni neeq1d breq12d 3imtr4d eqid ialgcvga sylc eqtr3d ex ) CK
|
|
KUBZLZFGMNZLZFDNZONZPQWAWCUCZWDOVTUFZNZWEPWFWDVTLZWHWEQWAWCFKLZWIWAGKLW
|
|
CWJWAGCONZKJCKKUDUEFGUGUHWAKVTFDWACDVTEPRKUIIWAUJWAUKVTVTEVIWAABEHULZUM
|
|
VTRLWAKKUNUNUOZUMUPUQURWDVTOSUSWFWAFCWGNZMNZLZWHPQWAWCUTWAWPWCWAWOWBFWA
|
|
WNGMWAWNWKGCVTOSJVAVBVCVDUACWGDVTEFWNRWLIKKVEUAVJZVTLZWQENZONZPVFWTWQON
|
|
ZTVGWSWGNZPVFXBWQWGNZTVGABEWQHVHWRXBWTPWRWSVTLXBWTQVTVTWQEWLVKWSVTOSUSZ
|
|
VLWRXBWTXCXATXDWQVTOSVMVNWNVOWMVPVQVRVS $.
|
|
|
|
eucialg.3 $e |- A = <. M , N >. $.
|
|
$( Euclid's Algorithm computes the greatest common divisor of two
|
|
nonnegative integers by repeatedly replacing the larger of them with
|
|
its remainder modulo the smaller until the remainder is 0. Theorem
|
|
1.15 in [ApostolNT] p. 20.
|
|
|
|
Upon halting, the 1st member of the final state ` ( R `` N ) ` is
|
|
equal to the gcd of the values comprising the input state
|
|
` <. M , N >. ` . This is Metamath 100 proof #69 (greatest common
|
|
divisor algorithm). (Contributed by Jim Kingdon, 11-Jan-2022.) $)
|
|
eucialg $p |- ( ( M e. NN0 /\ N e. NN0 ) ->
|
|
( 1st ` ( R ` N ) ) = ( M gcd N ) ) $=
|
|
( cn0 wcel cfv cgcd cc0 c2nd wceq syl cz vz wa c1st co cop cxp wf nn0uz
|
|
cvv 0zd opelxpi syl5eqel eucalgf a1i nn0ex xpex ialgrf ffvelrn sylancom
|
|
1st2nd2 fveq2d df-ov syl6eqr fveq2i op2ndg syl5eq xp2nd nn0zd uzid eqid
|
|
cuz eucialgcvga mpd eqtr3d oveq2d xp1st nn0gcdid0 3syl 3eqtrrd cres wfn
|
|
wss gcdf ffn ax-mp nn0ssz xpss12 mp2an fnssres eucalginv ffvelrni fvres
|
|
cv 3eqtr4d ialginv 0nn0 sylancl 3eqtr3d ialgr0 syl6eq 3eqtrd ) FLMZGLMZ
|
|
UBZGDNZUCNZXEONZPDNZONZFGOUDZXDXGXFXEQNZOUDZXFPOUDZXFXDXGXFXKUEZONXLXDX
|
|
EXNOXDXELLUFZMZXEXNRXBXCLXODUGZXPXDCDXOEPUILUHIXDUJZXDCFGUEZXOKFGLLUKUL
|
|
ZXOXOEUGXDABEHUMZUNZXOUIMXDLLUOUOUPZUNZUQZLXOGDURUSZXELLUTSVAXFXKOVBVCX
|
|
DXKPXFOXDCQNZDNZQNZXKPXDYHXEQXDYGGDXDYGXSQNGCXSQKVDFGLLVEVFVAVAXDCXOMZY
|
|
IPRZXTYJYGYGVKNMZYKYJYGTMYLYJYGCLLVGVHYGVISABCDEYGYGHIYGVJVLVMSVNVOXDXP
|
|
XFLMXMXFRYFXELLVPXFVQVRVSXDXEOXOVTZNZXHYMNZXGXIXBXCYJYNYORXTUACDXOEYMGU
|
|
IIYAOTTUFZWAZXOYPWBZYMXOWAYPLOUGYQWCYPLOWDWELTWBZYSYRWFWFLTLTWGWHYPXOOW
|
|
IWHUAWMZXOMZYTENZONZYTONUUBYMNZYTYMNABEYTHWJUUAUUBXOMUUDUUCRXOXOYTEYAWK
|
|
UUBXOOWLSYTXOOWLWNYCWOUSXDXPYNXGRYFXEXOOWLSXDXHXOMZYOXIRXDXQPLMUUEYEWPL
|
|
XOPDURWQXHXOOWLSWRXDXIXSONXJXDXHXSOXDXHCXSXDCDXOEPUILUHIXRXTYBYDWSKWTVA
|
|
FGOVBVCXA $.
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The least common multiple
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
According to Wikipedia ("Least common multiple", 27-Aug-2020,
|
|
~ https://en.wikipedia.org/wiki/Least_common_multiple ): "In arithmetic and
|
|
number theory, the _least common multiple_, lowest common multiple, or smallest
|
|
common multiple of two integers a and b, usually denoted by lcm(a, b), is the
|
|
smallest positive integer that is divisible by both a and b. Since division of
|
|
integers by zero is undefined, this definition has meaning only if a and b are
|
|
both different from zero. However, some authors define lcm(a,0) as 0 for all
|
|
a, which is the result of taking the lcm to be the least upper bound in the
|
|
lattice of divisibility."
|
|
|
|
In this section, an operation calculating the least common multiple of two
|
|
integers ( ~ df-lcm ). The definition is valid for all integers, including
|
|
negative integers and 0, obeying the above mentioned convention.
|
|
$)
|
|
|
|
$c lcm $.
|
|
$( Extend the definition of a class to include the least common multiple
|
|
operator. $)
|
|
clcm $a class lcm $.
|
|
|
|
${
|
|
$d n x y $.
|
|
$( Define the ` lcm ` operator. For example, ` ( 6 lcm 9 ) = 1 8 ` .
|
|
(Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV,
|
|
16-Sep-2020.) $)
|
|
df-lcm $a |- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 ,
|
|
inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) ) $.
|
|
$}
|
|
|
|
$( Decidablity lemma used in various proofs related to ` lcm ` .
|
|
(Contributed by Jim Kingdon, 21-Jan-2022.) $)
|
|
lcmmndc $p |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) ) $=
|
|
( cz wcel wa cc0 wceq wdc wo 0z zdceq mpan2 adantr adantl dcor sylc ) ACDZB
|
|
CDZEAFGZHZBFGZHZSUAIHQTRQFCDZTJAFKLMRUBQRUCUBJBFKLNSUAOP $.
|
|
|
|
${
|
|
$d n x y M $. $d n x y N $.
|
|
$( Value of the ` lcm ` operator. ` ( M lcm N ) ` is the least common
|
|
multiple of ` M ` and ` N ` . If either ` M ` or ` N ` is ` 0 ` , the
|
|
result is defined conventionally as ` 0 ` . Contrast with ~ df-gcd and
|
|
~ gcdval . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by
|
|
AV, 16-Sep-2020.) $)
|
|
lcmval $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) =
|
|
if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) }
|
|
, RR , < ) ) ) $=
|
|
( vx vy cz wcel wa cv cc0 wceq cdvds wbr cn crab cr clt cfv c1 syl2anc wo
|
|
cinf cif clcm cvv cmpt2 df-lcm eqeq1 orbi1d breq1 anbi1d rabbidv ifbieq2d
|
|
a1i infeq1d orbi2d anbi2d sylan9eq adantl simpl simpr c0ex cmul cabs 1zzd
|
|
wn co cuz nnuz rabeq ax-mp dvdsmul1 adantr simpll simplr zmulcld dvdsabsb
|
|
wb mpbid dvdsmul2 zcnd absmuld ioran sylib simpld neqned nnabscl nnmulcld
|
|
wne simprd eqeltrd breq2 anbi12d elrab3 syl mpbir2and cfz elfzelz zdvdsdc
|
|
wdc syl2an dcan sylc infssuzcldc elexd lcmmndc ifcldadc ovmpt2d ) BFGZCFG
|
|
ZHZDEBCFFDIZJKZEIZJKZUAZJXLAIZLMZXNXQLMZHZANOZPQUBZUCZBJKZCJKZUAZJBXQLMZC
|
|
XQLMZHZANOZPQUBZUCZUDUEUDDEFFYCUFKXKDEAUGUNXLBKZXNCKZHYCYLKXKYMYNYCYDXOUA
|
|
ZJYGXSHZANOZPQUBZUCYLYMXPYOYBYRJYMXMYDXOXLBJUHUIYMPYAYQQYMXTYPANYMXRYGXSX
|
|
LBXQLUJUKULUOUMYNYOYFYRYKJYNXOYEYDXNCJUHUPYNPYQYJQYNYPYIANYNXSYHYGXNCXQLU
|
|
JUQULUOUMURUSXIXJUTXIXJVAXKYFJYKUEJUEGXKYFHVBUNXKYFVFZHZYKYJYTYIBCVCVGZVD
|
|
RZYJASYTVENSVHRZKYJYIAUUCOKVIYIANUUCVJVKYTUUBYJGZBUUBLMZCUUBLMZYTBUUALMZU
|
|
UEXKUUGYSBCVLVMYTXIUUAFGZUUGUUEVRXIXJYSVNZYTBCUUIXIXJYSVOZVPZBUUAVQTVSYTC
|
|
UUALMZUUFXKUULYSBCVTVMYTXJUUHUULUUFVRUUJUUKCUUAVQTVSYTUUBNGUUDUUEUUFHZVRY
|
|
TUUBBVDRZCVDRZVCVGNYTBCYTBUUIWAYTCUUJWAWBYTUUNUUOYTXIBJWIUUNNGUUIYTBJYTYD
|
|
VFZYEVFZYTYSUUPUUQHXKYSVAYDYEWCWDZWEWFBWGTYTXJCJWIUUONGUUJYTCJYTUUPUUQUUR
|
|
WJWFCWGTWHWKYIUUMAUUBNXQUUBKYGUUEYHUUFXQUUBBLWLXQUUBCLWLWMWNWOWPYTXQSUUBW
|
|
QVGGZHYGWTZYHWTZYIWTYTXIXQFGZUUTUUSUUIXQSUUBWRZBXQWSXAYTXJUVBUVAUUSUUJUVC
|
|
CXQWSXAYGYHXBXCXDXEBCXFXGXH $.
|
|
$}
|
|
|
|
${
|
|
$d n K $. $d n M $. $d n N $.
|
|
$( The ` lcm ` operator is commutative. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) $)
|
|
lcmcom $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) ) $=
|
|
( vn cz wcel wa cc0 wceq wo cdvds wbr cn crab cr clt cinf cif clcm lcmval
|
|
co cv orcom wb ancom a1i rabbiia infeq1i ifbieq2i ancoms 3eqtr4a ) ADEZBD
|
|
EZFAGHZBGHZIZGACUAZJKZBUPJKZFZCLMZNOPZQUNUMIZGURUQFZCLMZNOPZQZABRTBARTZUO
|
|
VBVAVEGUMUNUBNUTVDOUSVCCLUSVCUCUPLEUQURUDUEUFUGUHCABSULUKVGVFHCBASUIUJ $.
|
|
|
|
$( The value, by convention, of the ` lcm ` operator when either operand is
|
|
0. (Use ~ lcmcom for a left-hand 0.) (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) $)
|
|
lcm0val $p |- ( M e. ZZ -> ( M lcm 0 ) = 0 ) $=
|
|
( vn cz wcel cc0 clcm co wceq 0z wa wo cv cdvds wbr cn crab clt cinf cif
|
|
cr lcmval eqid olci iftruei syl6eq mpan2 ) ACDZECDZAEFGZEHIUGUHJUIAEHZEEH
|
|
ZKZEABLZMNEUMMNJBOPTQRZSEBAEUAULEUNUKUJEUBUCUDUEUF $.
|
|
|
|
$( The value of the ` lcm ` operator when both operands are nonzero.
|
|
(Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV,
|
|
16-Sep-2020.) $)
|
|
lcmn0val $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) )
|
|
-> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) $=
|
|
( cz wcel wa cc0 wceq wo wn clcm co cv cdvds wbr cn crab cr clt cinf cif
|
|
lcmval iffalse sylan9eq ) BDECDEFBGHCGHIZJBCKLUEGBAMZNOCUFNOFAPQRSTZUAUGA
|
|
BCUBUEGUGUCUD $.
|
|
|
|
$( Lemma for ~ lcmn0cl and ~ dvdslcm . (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) $)
|
|
lcmcllem $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) ->
|
|
( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } ) $=
|
|
( cz wcel wa cc0 wceq wn co cdvds wbr cn c1 wne cap zapne syl2anc mpbid
|
|
wb wo clcm cv crab cr clt cinf lcmn0val cmul cabs cfv 1zzd cuz nnuz rabeq
|
|
ax-mp simpll simplr zmulcld zcnd ioran biimpi adantl simpld neqned mpbird
|
|
0zd simprd mulap0d nnabscl dvdsmul1 zmulcl dvdsabsb syldan sylan2 anabss7
|
|
dvdsmul2 jca breq2 anbi12d elrab sylanbrc cfz wdc simplll elfzelz zdvdsdc
|
|
adantr simpllr dcan sylc infssuzcldc eqeltrd ) BDEZCDEZFZBGHZCGHZUAIZFZBC
|
|
UBJBAUCZKLZCXAKLZFZAMUDZUEUFUGXEABCUHWTXDBCUIJZUJUKZXEANWTULMNUMUKZHXEXDA
|
|
XHUDHUNXDAMXHUOUPWTXGMEZBXGKLZCXGKLZFZXGXEEWTXFDEZXFGOZXIWTBCWNWOWSUQZWNW
|
|
OWSURZUSZWTXFGPLZXNWTBCWTBXOUTWTCXPUTWTBGPLZBGOZWTBGWTWQIZWRIZWSYAYBFZWPW
|
|
SYCWQWRVAVBVCZVDVEWTWNGDEZXSXTTXOWTVGZBGQRVFWTCGPLZCGOZWTCGWTYAYBYDVHVEWT
|
|
WOYEYGYHTXPYFCGQRVFVIWTXMYEXRXNTXQYFXFGQRSXFVJRWPXLWSWPXJXKWPBXFKLZXJBCVK
|
|
WNWOXMYIXJTBCVLZBXFVMVNSWPCXFKLZXKBCVQWNWOYKXKTZWPWOXMYLYJCXFVMVOVPSVRWHX
|
|
DXLAXGMXAXGHXBXJXCXKXAXGBKVSXAXGCKVSVTWAWBWTXANXGWCJEZFZXBWDZXCWDZXDWDYNW
|
|
NXADEZYOWNWOWSYMWEYMYQWTXANXGWFVCZBXAWGRYNWOYQYPWNWOWSYMWIYRCXAWGRXBXCWJW
|
|
KWLWM $.
|
|
|
|
$( Closure of the ` lcm ` operator. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) $)
|
|
lcmn0cl $p |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) ->
|
|
( M lcm N ) e. NN ) $=
|
|
( vn cz wcel wa cc0 wceq wo wn cv cdvds wbr cn crab clcm co ssrab2 sseldi
|
|
lcmcllem ) ADEBDEFAGHBGHIJFACKZLMBUALMFZCNONABPQUBCNRCABTS $.
|
|
|
|
$( The lcm of two integers is divisible by each of them. (Contributed by
|
|
Steve Rodriguez, 20-Jan-2020.) $)
|
|
dvdslcm $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N )
|
|
/\ N || ( M lcm N ) ) ) $=
|
|
( vn cz wcel wa cc0 wceq wo co cdvds wbr dvds0 lcm0val sylan9eqr breqtrrd
|
|
clcm cn breq2 syl wn ad2antrr oveq1 0z lcmcom mpan2 adantll oveq2 adantlr
|
|
eqtr3d jaodan ad2antlr jca cv crab lcmcllem wb lcmn0cl anbi12d elrab3 wdc
|
|
mpbid lcmmndc exmiddc mpjaodan ) ADEZBDEZFZAGHZBGHZIZAABQJZKLZBVLKLZFZVKU
|
|
AZVHVKFZVMVNVQAGVLKVFAGKLVGVKAMUBVHVIVLGHZVJVGVIVRVFVIVGVLGBQJZGAGBQUCVGB
|
|
GQJZVSGVGGDEVTVSHUDBGUEUFBNUJOUGVFVJVRVGVJVFVLAGQJGBGAQUHANOUIUKZPVQBGVLK
|
|
VGBGKLVFVKBMULWAPUMVHVPFZVLACUNZKLZBWCKLZFZCRUOEZVOCABUPWBVLREWGVOUQABURW
|
|
FVOCVLRWCVLHWDVMWEVNWCVLAKSWCVLBKSUSUTTVBVHVKVAVKVPIABVCVKVDTVE $.
|
|
|
|
$( A positive integer which both operands of the ` lcm ` operator divide
|
|
bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof
|
|
shortened by AV, 16-Sep-2020.) $)
|
|
lcmledvds $p |- ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ )
|
|
/\ -. ( M = 0 \/ N = 0 ) )
|
|
-> ( ( M || K /\ N || K ) -> ( M lcm N ) <_ K ) ) $=
|
|
( vn cn wcel cz cc0 wceq wa cdvds wbr co cle c1 breq2 wdc zdvdsdc syl2anc
|
|
crab w3a wo wn clcm cv cr clt cinf lcmn0val 3adantl1 adantr 1zzd cuz nnuz
|
|
cfv rabeq ax-mp simpll1 simpr anbi12d sylanbrc cfz simpll2 elfzelz adantl
|
|
elrab simpll3 dcan sylc adantlr infssuzledc eqbrtrd ex ) AEFZBGFZCGFZUABH
|
|
ICHIUBUCZJZBAKLZCAKLZJZBCUDMZANLVRWAJZWBBDUEZKLZCWDKLZJZDETZUFUGUHZANVRWB
|
|
WIIZWAVOVPVQWJVNDBCUIUJUKWCWGAWHDOWCULEOUMUOZIWHWGDWKTIUNWGDEWKUPUQWCVNWA
|
|
AWHFVNVOVPVQWAURVRWAUSWGWADAEWDAIWEVSWFVTWDABKPWDACKPUTVFVAVRWDOAVBMFZWGQ
|
|
ZWAVRWLJZWEQZWFQZWMWNVOWDGFZWOVNVOVPVQWLVCWLWQVRWDOAVDVEZBWDRSWNVPWQWPVNV
|
|
OVPVQWLVGWRCWDRSWEWFVHVIVJVKVLVM $.
|
|
$}
|
|
|
|
$( The lcm of two integers is zero iff either is zero. (Contributed by Steve
|
|
Rodriguez, 20-Jan-2020.) $)
|
|
lcmeq0 $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = 0 <-> ( M = 0 \/
|
|
N = 0 ) ) ) $=
|
|
( cz wcel wa clcm co cc0 wceq wo wdc wn wi lcmmndc lcmn0cl nnne0d neneqd ex
|
|
lcm0val sylan9eqr condc sylc oveq1 lcmcom mpan2 eqtr3d adantll oveq2 jaodan
|
|
0z adantlr impbid ) ACDZBCDZEZABFGZHIZAHIZBHIZJZUOUTKUTLZUQLZMUQUTMABNUOVAV
|
|
BUOVAEZUPHVCUPABOPQRUTUQUAUBUOUTUQUOURUQUSUNURUQUMURUNUPHBFGZHAHBFUCUNBHFGZ
|
|
VDHUNHCDVEVDIUJBHUDUEBSUFTUGUMUSUQUNUSUMUPAHFGHBHAFUHASTUKUIRUL $.
|
|
|
|
$( Closure of the ` lcm ` operator. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) $)
|
|
lcmcl $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 ) $=
|
|
( cz wcel wa cc0 wceq wo clcm co cn0 wn lcmcom adantr oveq2 lcm0val adantll
|
|
sylan9eqr eqtrd adantlr jaodan 0nn0 syl6eqel lcmn0cl nnnn0d lcmmndc exmiddc
|
|
wdc syl mpjaodan ) ACDZBCDZEZAFGZBFGZHZABIJZKDUPLZUMUPEUQFKUMUNUQFGZUOUMUNE
|
|
UQBAIJZFUMUQUTGUNABMNULUNUTFGUKUNULUTBFIJFAFBIOBPRQSUKUOUSULUOUKUQAFIJFBFAI
|
|
OAPRTUAUBUCUMUREUQABUDUEUMUPUHUPURHABUFUPUGUIUJ $.
|
|
|
|
$( The greatest common divisor of two numbers divides their least common
|
|
multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.) $)
|
|
gcddvdslcm $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || ( M lcm N ) ) $=
|
|
( cz wcel wa cgcd co cdvds clcm gcddvds simpld dvdslcm wi gcdcl nn0zd simpl
|
|
wbr lcmcl dvdstr syl3anc mp2and ) ACDZBCDZEZABFGZAHQZAABIGZHQZUEUGHQZUDUFUE
|
|
BHQABJKUDUHBUGHQABLKUDUECDUBUGCDUFUHEUIMUDUEABNOUBUCPUDUGABROUEAUGSTUA $.
|
|
|
|
$( Negating one operand of the ` lcm ` operator does not alter the result.
|
|
(Contributed by Steve Rodriguez, 20-Jan-2020.) $)
|
|
lcmneg $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) ) $=
|
|
( cz wcel wa clcm co cneg cc0 wceq wo wb oveq2 mpbird sylan2 adantr syl2anc
|
|
wbr cdvds wi wn lcm0val znegcl eqtr4d ad2antlr eqeq12d adantl lcmcom oveq2i
|
|
syl eqcomi negeq oveq2d 3eqtr4a jaodan cle dvdslcm simpr cn0 lcmcl negdvdsb
|
|
neg0 anbi2d cn w3a zcn negeq0d orbi2d notbid biimpa adantll lcmn0cl sylanl2
|
|
nn0zd syldan simpl 3anass sylanbrc lcmledvds mpd simplr nnzd syl3an3 3expib
|
|
ex syl3c sylbid nn0red cr letri3d mpbir2and lcmmndc exmiddc mpjaodan eqcomd
|
|
wdc ) ACDZBCDZEZABFGZABHZFGZWSAIJZBIJZKZWTXBJZXEUAZWSXCXFXDWSXCEZXFBAFGZXAA
|
|
FGZJZXHXKBIFGZXAIFGZJZWRXNWQXCWRXLIXMBUBWRXACDZXMIJBUCZXAUBUJUDUEXCXKXNLWSX
|
|
CXIXLXJXMAIBFMAIXAFMUFUGNWSXFXKLXCWSWTXIXBXJABUHWRWQXOXBXJJXPAXAUHOUFPNXDXF
|
|
WSXDAIFGZAIHZFGZWTXBXSXQXRIAFVBUIUKBIAFMXDXAXRAFBIULUMUNUGUOWSXGEZXFWTXBUPR
|
|
ZXBWTUPRZXTAXBSRZBXBSRZEZYAWSYEXGWSYEYCXAXBSRZEZWRWQXOYGXPAXAUQOWSYDYFYCWSW
|
|
RXBCDYDYFLWQWRURWSXBWRWQXOXBUSDXPAXAUTZOVNBXBVAQVCNPXTXBVDDZWQWRVEZXGYEYATX
|
|
TYIWSYJWSXGXCXAIJZKZUAZYIWRXGYMWQWRXGYMWRXEYLWRXDYKXCWRBBVFVGVHVIVJVKZWRWQX
|
|
OYMYIXPAXAVLVMVOWSXGVPZYIWQWRVQVRWSXGURXBABVSQVTXTAWTSRZBWTSRZEZYBWSYRXGABU
|
|
QPXTYRYPXAWTSRZEZYBXTYQYSYPXTWRWTCDYQYSLWQWRXGWAXTWTABVLZWBBWTVAQVCXTWTVDDZ
|
|
WSYMYTYBTZUUAYOYNUUBWQWRYMUUCTZWRUUBWQXOUUDXPUUBWQXOVEYMUUCWTAXAVSWEWCWDWFW
|
|
GVTWSXFYAYBELXGWSWTXBWSWTABUTWHWRWQXOXBWIDXPWQXOEXBYHWHOWJPWKWSXEWPXEXGKABW
|
|
LXEWMUJWNWO $.
|
|
|
|
$( Negating one operand of the ` lcm ` operator does not alter the result.
|
|
(Contributed by Steve Rodriguez, 20-Jan-2020.) $)
|
|
neglcm $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) ) $=
|
|
( cz wcel wa cneg clcm co wceq lcmneg ancoms znegcl lcmcom sylan 3eqtr4d )
|
|
ACDZBCDZEBAFZGHZBAGHZRBGHZABGHQPSTIBAJKPRCDQUASIALRBMNABMO $.
|
|
|
|
$( The lcm of two integers is the same as that of their absolute values.
|
|
(Contributed by Steve Rodriguez, 20-Jan-2020.) $)
|
|
lcmabs $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) =
|
|
( M lcm N ) ) $=
|
|
( cz wcel wa cabs cfv wceq cneg wo co cq zq qabsor anim12i oveq12 sylan9eqr
|
|
clcm ex lcmneg syl2an wi a1i neglcm znegcl sylan eqtrd ccased mpd ) ACDZBCD
|
|
ZEZAFGZAHZUMAIZHZJZBFGZBHZURBIZHZJZEZUMURRKZABRKZHZUJALDZBLDZVCUKAMBMVGUQVH
|
|
VBANBNOUAULUNUSUPVAVFUNUSEVFUBULUMAURBRPUCULUPUSEZVFVIULVDUOBRKZVEUMUOURBRP
|
|
ABUDZQSULUNVAEZVFVLULVDAUTRKVEUMAURUTRPABTQSULUPVAEZVFVMULVDUOUTRKZVEUMUOUR
|
|
UTRPULVNVJVEUJUOCDUKVNVJHAUEUOBTUFVKUGQSUHUI $.
|
|
|
|
${
|
|
$d K n $. $d M f g n x $. $d M n x y $. $d N f g n x $. $d N n x y $.
|
|
$( Lemma for ~ lcmgcd and ~ lcmdvds . Prove them for positive ` M ` ,
|
|
` N ` , and ` K ` . (Contributed by Steve Rodriguez, 20-Jan-2020.)
|
|
(Proof shortened by AV, 16-Sep-2020.) $)
|
|
lcmgcdlem $p |- ( ( M e. NN /\ N e. NN ) ->
|
|
( ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) /\
|
|
( ( K e. NN /\ ( M || K /\ N || K ) ) -> ( M lcm N ) || K ) ) ) $=
|
|
( vx cn wcel wa co cmul wceq cdvds wbr wi cz adantr cc0 cdiv wb syl3anc
|
|
cc vn vg vf vy clcm cgcd cabs cfv nnmulcl nnred nnz zred adantl 0red nnre
|
|
vw cle nngt0 ltled mulge0d absidd cv crab cr clt cinf wo wn anim12i nnne0
|
|
neneqd ioran sylibr lcmn0val syl2anc lttri3 simpld gcdcl nn0zd dvdsmultr1
|
|
gcddvds 3expb mpancom mpd syl gcdnncl nndivdvds mpbid simprd wne dvdsval2
|
|
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|
|
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|
|
jca bezout simplll simpllr mulap0d cap divdivap2d oveq2 ad2antrl ad2antll
|
|
zcn mulcld mul12d divcanap5d 3eqtrd adantlrr simprl zmulcld sylibd 3impia
|
|
ex ad3antrrr an32s reximdva c1 1z r19.9rmv mp2b eqtr2d adddid oveq12d imp
|
|
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|
|
eqeltrd nnzd mpbird anassrs elex2 bitri sylan2b lensymd infminti eqeltrrd
|
|
wex dvdsle divmulap3d eleq1 anbi2d imbi12d breq1d vtoclg mpcom ) BEFZCEFZ
|
|
GZBCUEHZBCUFHZIHZBCIHZUGUHZJAEFZBAKLZCAKLZGZGZUVPAKLZMUVOUVTUVSUVRUVOUVSU
|
|
VOUVSBCUIZUJUVOBCUVOBUVMBNFZUVNBUKZOZULUVOCUVNCNFZUVMCUKZUMZULUVMPBUQLUVN
|
|
UVMPBUVMUNBUOBURUSOUVNPCUQLUVMUVNPCUVNUNCUOCURUSUMUTVAUVOUVSUVQQHZUVPJUVS
|
|
UVRJUVOUVPBDVBZKLZCUWOKLZGZDEVCZVDVEVFZUWNUVOUWHUWKGZBPJZCPJZVGVHZUVPUWTJ
|
|
UVMUWHUVNUWKUWIUWLVIZUVOUXBVHZUXCVHZGUXDUVMUXFUVNUXGUVMBPBVJZVKUVNCPCVJZV
|
|
KVIUXBUXCVLVMDBCVNVOUVOUAUBUCVDUWSUWNVEUCVBZVDFUBVBZVDFGUXJUXKJUXJUXKVELV
|
|
HUXKUXJVELVHGRUVOUXJUXKVPUMUVOUWNUVOUVQUVSKLZUWNEFZUVOUXAUXLUXEUXAUVQBKLZ
|
|
UXLUXAUXNUVQCKLZBCWAZVQZUVQNFZUXAUXNUXLMZUXAUVQBCVRVSZUXRUWHUWKUXSUVQBCVT
|
|
WBWCWDWEUVOUVSEFUVQEFUXLUXMRUWGBCWFZUVSUVQWGVOWHZUJZUVOUXMBUWNKLZCUWNKLZG
|
|
ZUWNUWSFUYBUVOUYDUYEUVOBBCUVQQHZIHZUWNKUVOUWHUYGNFZBUYHKLUWJUVOUXOUYIUVOU
|
|
XAUXOUXEUXAUXNUXOUXPWIWEUVOUXRUVQPWJZUWKUXOUYIRUVOUXAUXRUXEUXTWEZUVOUVQUY
|
|
AWLZUWMUVQCWKSWHBUYGWMVOUVOBCUVQUVMBTFZUVNBWNOZUVNCTFZUVMCWNZUMZUVOUVQUYA
|
|
WOZUVOUVQUYAWPZWQWRUVOCCBUVQQHZIHZUWNKUVOUWKUYTNFZCVUAKLUWMUVOUXNVUBUVOUX
|
|
AUXNUXEUXQWEUVOUXRUYJUWHUXNVUBRUYKUYLUWJUVQBWKSWHCUYTWMVOUVOUWNCBIHZUVQQH
|
|
VUAUVOUVSVUCUVQQUVOBCUYNUYQWSZWTUVOCBUVQUYQUYNUYRUYSWQXAWRXLUWRUYFDUWNEUW
|
|
OUWNJUWPUYDUWQUYEUWOUWNBKXBUWOUWNCKXBXCXDXEUVOUAVBZUWSFZGUWNVUEUVOUWNVDFV
|
|
UFUYCOVUFVUEVDFUVOVUFVUEUWRDVUEEXFUJUMVUFUVOVUEEFZBVUEKLZCVUEKLZGZGZUWNVU
|
|
EUQLZUWRVUJDVUEEUWOVUEJUWPVUHUWQVUIUWOVUEBKXBUWOVUECKXBXCXDUVOVUKGZUWNVUE
|
|
KLZVULVUMVUNUDNXGZDNXGZVUNVUMUVQBUWOIHZCUDVBZIHZXHHZJZUDNXGZDNXGZVUPUVOVV
|
|
CVUKUVOUXAVVCUXEDUDBCXMWEOVUMVVBVUODNVUMUWONFZGVVAVUNUDNVUMVVDVURNFZVVAVU
|
|
NMVUMVVDVVEGZGZVVAVUNVVGVVAGZVUNVUEUWNQHZNFZVVHVVIVUEUWOIHZCQHZVUEVURIHZB
|
|
QHZXHHZNVVGVVAVVIVVOJZUVOVUGVVFVVAVVPMVUJUVOVUGGZVVFGZVVAVVPVVRVVAGVVIVUE
|
|
UVQIHZUVSQHZVUEVUQIHZUVSQHZVUEVUSIHZUVSQHZXHHZVVOVVRVVIVVTJVVAVVRVUEUVSUV
|
|
QVUGVUETFUVOVVFVUEWNXIZUVOUVSTFVUGVVFUVOUVSUWGWOZXJZUVOUVQTFVUGVVFUYRXJVV
|
|
RBCUVOUYMVUGVVFUYNXJZUVNUYOUVMVUGVVFUYPXKZVVRBUVMUVNVUGVVFXNWPZVVRCUVMUVN
|
|
VUGVVFXOWPZXPZUVOUVQPXQLVUGVVFUYSXJXROVVAVVRVVTVUEVUTIHZUVSQHZVWEVVAVVSVW
|
|
NUVSQUVQVUTVUEIXSWTVVRVWOVWAVWCXHHZUVSQHVWEVVRVWNVWPUVSQVVRVUEVUQVUSVWFVV
|
|
RBUWOVWIVVDUWOTFVVQVVEUWOYBXTZYCZVVRCVURVWJVVEVURTFVVQVVDVURYBYAZYCZUUAWT
|
|
VVRVWAVWCUVSVVRVUEVUQVWFVWRYCVVRVUEVUSVWFVWTYCVWHVWMUUDXAUUEVVRVWEVVOJVVA
|
|
VVRVWBVVLVWDVVNXHVVRVWBBVVKIHZUVSQHVVLVVRVWAVXAUVSQVVRVUEBUWOVWFVWIVWQYDW
|
|
TVVRVVKCBVVRVUEUWOVWFVWQYCVWJVWIVWLVWKYEXAVVRVWDCVVMIHZUVSQHVXBVUCQHVVNVV
|
|
RVWCVXBUVSQVVRVUECVURVWFVWJVWSYDWTVVRUVSVUCVXBQUVOUVSVUCJVUGVVFVUDXJUUFVV
|
|
RVVMBCVVRVUEVURVWFVWSYCVWIVWJVWKVWLYEYFUUBOYFYLYGUUCVVGVVONFZVVAUVOVVFVUK
|
|
VXCUVOVVFGVUGVUJVXCUVOVUGVVFVUJVXCMVVQVVFVUJVXCVVQVVFVUJUUGVVLVVNVVQVVFVU
|
|
JVVLNFZVVRVUIVXDVUHVVRVUICVVKKLZVXDVVRUWKVUENFZVVDVUIVXEMUVNUWKUVMVUGVVFU
|
|
WLXKZVUGVXFUVOVVFVUEUKXIZVVQVVDVVEYHZCVUEUWOVTSVVRUWKCPWJZVVKNFVXEVXDRVXG
|
|
UVNVXJUVMVUGVVFUXIXKVVRVUEUWOVXHVXIYICVVKWKSYJUUHYKVVQVVFVUJVVNNFZVVRVUHV
|
|
XKVUIVVRVUHBVVMKLZVXKVVRUWHVXFVVEVUHVXLMUVMUWHUVNVUGVVFUWIYMZVXHVVQVVDVVE
|
|
UUIZBVUEVURVTSVVRUWHBPWJZVVMNFVXLVXKRVXMUVMVXOUVNVUGVVFUXHYMVVRVUEVURVXHV
|
|
XNYIBVVMWKSYJUUJYKUUKUULYNUUMYNOUUNVVGVUNVVJRZVVAVVGUWNNFZUWNPWJZVXFVXPUV
|
|
OVXQVUKVVFUVOUWNUYBUUOZXJUVOVXRVUKVVFUVOUWNUYBWLXJUVOVUGVVFVXFVUJVXHYGUWN
|
|
VUEWKSOUUPYLUUQYOYOWDVUNVUOVUPYPNFZUPVBNFUPUVDZVUNVUORYQUPYPNUURZVUNUDUPN
|
|
YRYSVXTVYAVUOVUPRYQVYBVUODUPNYRYSUUSVMZVUMVXQVUGVUNVULMUVOVXQVUKVXSOUVOVU
|
|
GVUJYHUWNVUEUVEVOWDUUTUVAUVBYTZUVOUVSUVPUVQVWGUVOUVPUVOUWNUVPEVYDUYBUVCWO
|
|
UYRUYSUVFWHYTUVOUWEUWFUWAUVOUWEGZUWFUVOUWAUWDYHVUMUVPVUEKLZMVYEUWFMUAAEVU
|
|
EAJZVUMVYEVYFUWFVYGVUKUWEUVOVYGVUGUWAVUJUWDVUEAEUVGVYGVUHUWBVUIUWCVUEABKX
|
|
BVUEACKXBXCXCUVHVUEAUVPKXBUVIVUMVUNVYFVYCUVOVUNVYFRVUKUVOUWNUVPVUEKVYDUVJ
|
|
OWHUVKUVLYLXL $.
|
|
$}
|
|
|
|
$( The product of two numbers' least common multiple and greatest common
|
|
divisor is the absolute value of the product of the two numbers. In
|
|
particular, that absolute value _is_ the least common multiple of two
|
|
coprime numbers, for which ` ( M gcd N ) = 1 ` .
|
|
|
|
Multiple methods exist for proving this, and it is often proven either as
|
|
a consequence of the fundamental theorem of arithmetic or of
|
|
Bézout's identity ~ bezout ; see e.g.
|
|
~ https://proofwiki.org/wiki/Product_of_GCD_and_LCM and
|
|
~ https://math.stackexchange.com/a/470827 . This proof uses the latter to
|
|
first confirm it for positive integers ` M ` and ` N ` (the "Second Proof"
|
|
in the above Stack Exchange page), then shows that implies it for all
|
|
nonzero integer inputs, then finally uses ~ lcm0val to show it applies
|
|
when either or both inputs are zero. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) $)
|
|
lcmgcd $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) x. ( M gcd N ) ) =
|
|
( abs ` ( M x. N ) ) ) $=
|
|
( cz wcel wa cc0 wceq clcm co cmul cabs cfv nn0cnd adantl oveq1d cc 3eqtr4d
|
|
adantr cn cdvds wo cgcd wn gcdcl mul02d lcmcom mpan2 lcm0val eqtr3d abs00bd
|
|
zcn simpr fveq2d mul01d oveq2d jaodan neanior nnabscl anim12i an4s sylan2br
|
|
0z wne wbr wi lcmgcdlem simpld syl lcmabs gcdabs oveq12d oveqan12d nn0abscl
|
|
absidm syl2an absmuld 3eqtr3d wdc lcmmndc exmiddc mpjaodan ) ACDZBCDZEZAFGZ
|
|
BFGZUAZABHIZABUBIZJIZABJIZKLZGZWGUCZWDWEWMWFWDWEEZFBHIZWIJIZFBJIZKLZWJWLWDW
|
|
QWSGWEWDFWIJIZFWQWSWDWIWDWIABUDMUEZWDWPFWIJWCWPFGWBWCBFHIZWPFWCFCDXBWPGVBBF
|
|
UFUGBUHUINOWDWRWDBWCBPDZWBBUKZNZUEUJQRWOWHWPWIJWOAFBHWDWEULZOOWOWKWRKWOAFBJ
|
|
XFOUMQWDWFEZAFHIZWIJIZAFJIZKLZWJWLWDXIXKGWFWDWTFXIXKXAWDXHFWIJWBXHFGWCAUHRO
|
|
WDXJWDAWBAPDZWCAUKZRZUNUJQRXGWHXHWIJXGBFAHWDWFULZUOOXGWKXJKXGBFAJXOUOUMQUPW
|
|
DWNEZAKLZBKLZHIZXQXRUBIZJIZXQXRJIZKLZWJWLXPXQSDZXRSDZEZYAYCGZWNWDAFVCZBFVCZ
|
|
EYFAFBFUQWBYHWCYIYFWBYHEYDWCYIEYEAURBURUSUTVAYFYGFSDXQFTVDXRFTVDEEXSFTVDVEF
|
|
XQXRVFVGVHWDYAWJGWNWDXSWHXTWIJABVIABVJVKRWDYCWLGWNWDXQKLZXRKLZJIZYBYCWLWBXL
|
|
XCYLYBGWCXMXDXLXCYJXQYKXRJAVNBVNVLVOWDXQXRWBXQPDWCWBXQAVMMRWCXRPDWBWCXRBVMM
|
|
NVPWDABXNXEVPQRVQWDWGVRWGWNUAABVSWGVTVHWA $.
|
|
|
|
$( The lcm of two integers divides any integer the two divide. (Contributed
|
|
by Steve Rodriguez, 20-Jan-2020.) $)
|
|
lcmdvds $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K )
|
|
-> ( M lcm N ) || K ) ) $=
|
|
( cz wcel cc0 wceq wo cdvds wbr wa clcm co wi wb adantl breq1d imbi12d cabs
|
|
cfv w3a wn id breq1 oveq1 0z lcmcom mpan lcm0val sylan9eqr mpbiri 3ad2antl3
|
|
eqtrd adantrd ex oveq2 3ad2antl2 adantld jaod wne neanior lcmcl nn0zd dvds0
|
|
syl a1d adantr breq2 anbi12d adantrl adantllr adantlrr anassrs nnabscl cgcd
|
|
mpbird cmul lcmgcdlem simprd sylani syl2an expdimp dvdsabsb zabscl absdvdsb
|
|
sylan2 bitrd adantlr adantll bicomd lcmabs sylan bitr4d adantrr mpbid zdceq
|
|
wdc mpan2 exmiddc df-ne orbi2i sylibr mpjaodan an4s sylan2br impancom 3impa
|
|
cn 3comr lcmmndc 3adant1 mpjaod ) ADEZBDEZCDEZUAZBFGZCFGZHZBAIJZCAIJZKZBCLM
|
|
ZAIJZNZXSUBZXPXQYEXRXPXQYEXPXQKXTYDYAXOXMXQXTYDNZXNXOXQKZYGFAIJZYINZYIUCZYH
|
|
XTYIYDYIXQXTYIOXOBFAIUDPYHYCFAIXQXOYCFCLMZFBFCLUEXOYLCFLMZFFDEZXOYLYMGUFFCU
|
|
GUHCUIUMUJQRUKULUNUOXPXRYEXPXRKYAYDXTXNXMXRYAYDNZXOXNXRKZYOYJYKYPYAYIYDYIXR
|
|
YAYIOXNCFAIUDPYPYCFAIXRXNYCBFLMFCFBLUPBUIUJQRUKUQURUOUSXNXOXMYFYENZXNXOXMYQ
|
|
XNXOKZYFXMYEYFYRBFUTZCFUTZKXMYENZBFCFVAXNYSXOYTUUAXNYSKZXOYTKZKZXMYEUUDXMKA
|
|
FGZYEAFUTZUUDXMUUEYEUUBXOXMUUEKZYEYTXNXOUUGYEYSYRUUEYEXMYRUUEKYEBFIJZCFIJZK
|
|
ZYCFIJZNZYRUULUUEYRUUKUUJYRYCDEZUUKYRYCBCVBVCZYCVDVEVFVGUUEYEUULOYRUUEYBUUJ
|
|
YDUUKUUEXTUUHYAUUIAFBIVHAFCIVHVIAFYCIVHRPVPVJVKVLVMUUDXMUUFYEUUDXMUUFKZKBST
|
|
ZASTZIJZCSTZUUQIJZKZUUPUUSLMZUUQIJZNZYEUUDUUOUVAUVCUUBUUPXHEZUUSXHEZUUOUVAK
|
|
UVCNUUCBVNCVNUUOUVEUVFKZUUQXHEZUVAUVCAVNUVGUVBUUPUUSVOMVQMUUPUUSVQMSTGUVHUV
|
|
AKUVCNUUQUUPUUSVRVSVTWAWBUUBXOUUOUVDYEOZYTXNXOUUOUVIYSYRXMUVIUUFYRXMKZUVAYB
|
|
UVCYDUVJYBUVAUVJXTUURYAUUTXNXMXTUUROXOXNXMKXTBUUQIJZUURBAWCXMXNUUQDEZUVKUUR
|
|
OAWDZBUUQWEWFWGWHXOXMYAUUTOXNXOXMKYACUUQIJZUUTCAWCXMXOUVLUVNUUTOUVMCUUQWEWF
|
|
WGWIVIWJUVJUVCYCUUQIJZYDYRUVCUVOOXMYRUVBYCUUQIBCWKQVGYRUUMXMYDUVOOUUNYCAWCW
|
|
LWMRWNVKVLWOVMXMUUEUUFHZUUDXMUUEUUEUBZHZUVPXMUUEWQZUVRXMYNUVSUFAFWPWRUUEWSV
|
|
EUUFUVQUUEAFWTXAXBPXCUOXDXEXFXGXIXNXOXSYFHZXMYRXSWQUVTBCXJXSWSVEXKXL $.
|
|
|
|
$( The lcm of an integer and itself is its absolute value. (Contributed by
|
|
Steve Rodriguez, 20-Jan-2020.) $)
|
|
lcmid $p |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) ) $=
|
|
( cz wcel cc0 wceq clcm co cabs cfv wn wa lcm0val adantr wb oveq2 mpbird cc
|
|
wne anidms cmul fveq2 abs0 syl6eq adantl df-ne lcmcl nn0cnd zabscl zcnd cap
|
|
eqeq12d wbr zcn simpr absne0d 0zd zapne syl2an2r cgcd lcmgcd oveq2d absmuld
|
|
gcdid 3eqtr3d mulcanap2ad sylan2br wdc wo zdceq mpan2 exmiddc syl mpjaodan
|
|
0z ) ABCZADEZAAFGZAHIZEZVPJZVOVPKVSADFGZDEZVOWBVPALMVPVSWBNVOVPVQWAVRDADAFO
|
|
VPVRDHIDADHUAUBUCUKUDPVTVOADRZVSADUEVOWCKZVQVRVRVOVQQCZWCVOWEVOVOKVQAAUFUGS
|
|
MVOVRQCWCVOVRAUHZUIMZWGWDVRDUJULZVRDRZWDAVOAQCWCAUMZMVOWCUNUOVOVRBCWCDBCZWH
|
|
WINWFWDUPVRDUQURPVOVQVRTGZVRVRTGZEWCVOVQAAUSGZTGZAATGHIZWLWMVOWOWPEAAUTSVOW
|
|
NVRVQTAVCVAVOAAWJWJVBVDMVEVFVOVPVGZVPVTVHVOWKWQVNADVIVJVPVKVLVM $.
|
|
|
|
$( The lcm of an integer and 1 is the absolute value of the integer.
|
|
(Contributed by AV, 23-Aug-2020.) $)
|
|
lcm1 $p |- ( M e. ZZ -> ( M lcm 1 ) = ( abs ` M ) ) $=
|
|
( cz wcel c1 clcm co cgcd cmul cabs cfv gcd1 oveq2d cn0 lcmcl mpan2 mulid1d
|
|
1z nn0cnd eqtr2d wceq lcmgcd zcn fveq2d 3eqtrd ) ABCZADEFZUFADGFZHFZADHFZIJ
|
|
ZAIJUEUHUFDHFUFUEUGDUFHAKLUEUFUEUFUEDBCZUFMCQADNORPSUEUKUHUJTQADUAOUEUIAIUE
|
|
AAUBPUCUD $.
|
|
|
|
$( The product of two positive integers' least common multiple and greatest
|
|
common divisor is the product of the two integers. (Contributed by AV,
|
|
27-Aug-2020.) $)
|
|
lcmgcdnn $p |- ( ( M e. NN /\ N e. NN )
|
|
-> ( ( M lcm N ) x. ( M gcd N ) ) = ( M x. N ) ) $=
|
|
( cn wcel wa clcm co cgcd cmul cabs cfv cz nnz lcmgcd syl2an cn0 cr cc0 cle
|
|
wceq wbr nnmulcl nnnn0d nn0re nn0ge0 jca absid 3syl eqtrd ) ACDZBCDZEZABFGA
|
|
BHGIGZABIGZJKZUNUJALDBLDUMUOTUKAMBMABNOULUNPDZUNQDZRUNSUAZEUOUNTULUNABUBUCU
|
|
PUQURUNUDUNUEUFUNUGUHUI $.
|
|
|
|
$( Two integers' absolute values are equal iff their least common multiple
|
|
and greatest common divisor are equal. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) $)
|
|
lcmgcdeq $p |- ( ( M e. ZZ /\ N e. ZZ ) ->
|
|
( ( M lcm N ) = ( M gcd N ) <-> ( abs ` M ) = ( abs ` N ) ) ) $=
|
|
( cz wcel wa clcm co cgcd wceq cabs cfv cdvds wbr adantr wi mp2and wb sylan
|
|
imp mpbid dvdslcm simpld gcddvds simprd breq1 syl5ibrcom lcmcl nn0zd dvdstr
|
|
syl3an2 3com12 3expb anidms absdvdsb zabscl dvdsabsb bitrd 3coml ancoms cn0
|
|
nn0abscl anim12i dvdseq ex lcmid syl gcdid eqtr4d eqeq12d syl5ibcom adantlr
|
|
oveq2 lcmabs gcdabs impbida ) ACDZBCDZEZABFGZABHGZIZAJKZBJKZIZVRWAEZWBWCLMZ
|
|
WCWBLMZWDWEABLMZWFWEAVSLMZVSBLMZWHVRWIWAVRWIBVSLMZABUAZUBNVRWAWJVRWJWAVTBLM
|
|
ZVRVTALMZWMABUCZUDVSVTBLUEUFSVRWIWJEWHOZWAVRWPVRVPVQWPVPVRVQWPVRVPVSCDZVQWP
|
|
VRVSABUGUHZAVSBUIUJUKULUMNPVRWHWFQWAVRWHWBBLMZWFABUNVPWBCDZVQWSWFQAUOZWBBUP
|
|
RUQNTWEBALMZWGWEWKVSALMZXBVRWKWAVRWIWKWLUDNVRWAXCVRXCWAWNVRWNWMWOUBVSVTALUE
|
|
UFSVRWKXCEXBOZWAVRXDVRVPVQXDVQVRVPXDVRVQWQVPXDWRBVSAUIUJURULUMNPVRXBWGQZWAV
|
|
QVPXEVQVPEXBWCALMZWGBAUNVQWCCDVPXFWGQBUOWCAUPRUQUSNTVRWFWGEZWDOWAVRXGWDVRWB
|
|
UTDZWCUTDZEXGWDVPXHVQXIAVABVAVBWBWCVCRVDNPVRWDEWBWCFGZWBWCHGZIZWAVPWDXLVQVP
|
|
WDXLVPWBWBFGZWBWBHGZIWDXLVPXMWBJKZXNVPWTXMXOIXAWBVEVFVPWTXNXOIXAWBVGVFVHWDX
|
|
MXJXNXKWBWCWBFVLWBWCWBHVLVIVJSVKVRXLWAQWDVRXJVSXKVTABVMABVNVINTVO $.
|
|
|
|
$( Biconditional form of ~ lcmdvds . (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) $)
|
|
lcmdvdsb $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K )
|
|
<-> ( M lcm N ) || K ) ) $=
|
|
( cz wcel w3a cdvds wbr wa co lcmdvds dvdslcm simpld 3adant1 wi simp2 lcmcl
|
|
clcm dvdstr mpand nn0zd simp1 syl3anc simprd 3com13 syld3an2 jcad impbid )
|
|
ADEZBDEZCDEZFZBAGHZCAGHZIBCRJZAGHZABCKULUPUMUNULBUOGHZUPUMUJUKUQUIUJUKIZUQC
|
|
UOGHZBCLZMNULUJUODEZUIUQUPIUMOUIUJUKPUJUKVAUIURUOBCQUANZUIUJUKUBBUOASUCTULU
|
|
SUPUNUJUKUSUIURUQUSUTUDNUIVAUJUKUSUPIUNOZVBUKVAUIVCCUOASUEUFTUGUH $.
|
|
|
|
${
|
|
$d x M $. $d x N $. $d x P $.
|
|
$( Associative law for ` lcm ` operator. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) $)
|
|
lcmass $p |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) =
|
|
( N lcm ( M lcm P ) ) ) $=
|
|
( vx cz wcel cc0 wceq wo cdvds wbr wa cn crab cr clt cinf cif clcm wb w3a
|
|
cv orass anass a1i rabbiia infeq1i ifbieq2i cn0 lcmcl 3adant3 nn0zd simp3
|
|
co lcmval syl2anc lcmeq0 orbi1d bicomd adantl simp1 adantr simpl2 syl3anc
|
|
nnz lcmdvdsb anbi1d rabbidva infeq1d eqtr4d 3adant1 orbi2d anbi2d 3eqtr4a
|
|
ifbieq2d ) CEFZBEFZAEFZUAZCGHZBGHZIZAGHZIZGCDUBZJKZBWEJKZLZAWEJKZLZDMNZOP
|
|
QZRZVTWAWCIZIZGWFWGWILZLZDMNZOPQZRZCBSUNZASUNZCBASUNZSUNZWDWOWLWSGVTWAWCU
|
|
COWKWRPWJWQDMWJWQTWEMFZWFWGWIUDUEUFUGUHVSXBXAGHZWCIZGXAWEJKZWILZDMNZOPQZR
|
|
ZWMVSXAEFVRXBXLHVSXAVPVQXAUIFVRCBUJUKULVPVQVRUMZDXAAUOUPVSWDXGWLXKGVSXGWD
|
|
VSXFWBWCVPVQXFWBTVRCBUQUKURUSVSOWKXJPVSWJXIDMVSXELZWHXHWIXNWEEFZVPVQWHXHT
|
|
XEXOVSWEVEUTZVSVPXEVPVQVRVAZVBVPVQVRXEVCZWECBVFVDVGVHVIVOVJVSXDVTXCGHZIZG
|
|
WFXCWEJKZLZDMNZOPQZRZWTVSVPXCEFXDYEHXQVSXCVQVRXCUIFVPBAUJVKULDCXCUOUPVSWO
|
|
XTWSYDGVSXTWOVSXSWNVTVQVRXSWNTVPBAUQVKVLUSVSOWRYCPVSWQYBDMXNWPYAWFXNXOVQV
|
|
RWPYATXPXRVSVRXEXMVBWEBAVFVDVMVHVIVOVJVN $.
|
|
$}
|
|
|
|
$( The least common multiple of three and two is six. This proof does not
|
|
use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez,
|
|
20-Jan-2020.) (Revised by AV, 27-Aug-2020.) $)
|
|
3lcm2e6woprm $p |- ( 3 lcm 2 ) = 6 $=
|
|
( c3 c2 co cgcd cdiv c1 c6 cc wcel cc0 wa wceq cz 3z 2z mp2an pm3.2i oveq2i
|
|
cn 1z clcm cmul cap wbr 3cn mulcli lcmcl nn0cnd wn 2ne0 neii intnan gcdn0cl
|
|
2cn nncnd wne nnne0i wb nnzi 0z zapne w3a 3nn 2nn lcmgcdnn eqcomd divmulap3
|
|
mpbir mp1i mpbird mp3an gcdcom caddc cabs cfv gcdid ax-mp abs1 eqtr2i 1p1e2
|
|
gcdadd 3eqtri 1p2e3 eqtri 3t2e6 oveq1i 6cn div1i ) ABUACZABUBCZABDCZECZWJFE
|
|
CZGWJHIZWIHIZWKHIZWKJUCUDZKZWIWLLABUEUNUFAMIZBMIZWONOWSWTKZWIABUGUHPWPWQXAA
|
|
JLZBJLZKUIZWPWSWTNOQZXCXBBJUJUKULZXAXDKWKABUMZUOPWQWKJUPZWKXAXDWKSIXEXFXGPZ
|
|
UQWKMIJMIWQXHURWKXIUSUTWKJVAPVHQWNWOWRVBZWLWIXJWLWILWJWIWKUBCZLZASIZBSIZKZX
|
|
LXJXMXNVCVDQXOXKWJABVEVFVIWJWIWKVGVJVFVKWKFWJEWKBADCZFWSWTWKXPLNOABVLPFBFBV
|
|
MCZDCZXPFFBDCZBFDCZXRFFFDCZFFFVMCZDCZXSYAFVNVOZFFMIZYAYDLTFVPVQVRVSYEYEYAYC
|
|
LTTFFWAPYBBFDVTRWBYEWTXSXTLTOFBVLPWTYEXTXRLOTBFWAPWBXQABDWCRVSWDRWMGFECGWJG
|
|
FEWEWFGWGWHWDWB $.
|
|
|
|
$( The least common multiple of six and four is twelve. (Contributed by AV,
|
|
27-Aug-2020.) $)
|
|
6lcm4e12 $p |- ( 6 lcm 4 ) = ; 1 2 $=
|
|
( c6 c4 co cmul cdiv c2 cc wcel cc0 wa 6cn 4cn cz 4z nn0cnd mp2an cn pm3.2i
|
|
wceq eqcomd clcm cgcd c1 cdc cap wbr mulcli 6nn0 nn0zi lcmcl gcdcl wne 4ne0
|
|
wn neii intnan gcdn0cl nnne0i wb nnzi zapne mpbir w3a 6nn 4nn lcmgcdnn mp1i
|
|
0z divmulap3 mpbird mp3an 6gcd4e2 oveq2i 2ap0 divassapi 4d2e2 6t2e12 3eqtri
|
|
2cn ) ABUACZABDCZABUBCZECZWAFECZUCFUDZWAGHZVTGHZWBGHZWBIUEUFZJZVTWCSABKLUGA
|
|
MHZBMHZWGAUHUIZNWKWLJZVTABUJOPWHWIWKWLWHWMNWNWBABUKOPWIWBIULZWBWNAISZBISZJU
|
|
NWBQHWKWLWMNRWQWPBIUMUOUPABUQPZURWBMHIMHWIWOUSWBWRUTVHWBIVAPVBRWFWGWJVCZWCV
|
|
TWSWCVTSWAVTWBDCZSWSWTWAAQHZBQHZJWTWASWSXAXBVDVERABVFVGTWAVTWBVIVJTVKWBFWAE
|
|
VLVMWDABFECZDCAFDCWEABFKLVSVNVOXCFADVPVMVQVRVR $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Coprimality and Euclid's lemma
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
According to Wikipedia "Coprime integers",
|
|
see ~ https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[[...] two
|
|
integers a and b are said to be _relatively prime_, mutually prime, or
|
|
_coprime_ [[...] if the only positive integer (factor) that divides both of
|
|
them is 1. Consequently, any prime number that divides one does not divide the
|
|
other. This is equivalent to their greatest common divisor (gcd) being 1.".
|
|
In the following, we use this equivalent characterization to say that
|
|
` A e. ZZ ` and ` B e. ZZ ` are coprime (or relatively prime) if
|
|
` ( A gcd B ) = 1 `. The equivalence of the definitions is shown by
|
|
~ coprmgcdb . The negation, i.e. two integers are not coprime, can be
|
|
expressed either by ` ( A gcd B ) =/= 1 `, see ~ ncoprmgcdne1b , or
|
|
equivalently by ` 1 < ( A gcd B ) `, see ~ ncoprmgcdgt1b .
|
|
|
|
A proof of Euclid's lemma based on coprimality is provided in ~ coprmdvds (as
|
|
opposed to Euclid's lemma for primes).
|
|
$)
|
|
|
|
${
|
|
$d A i $. $d B i $.
|
|
$( Two positive integers are coprime, i.e. the only positive integer that
|
|
divides both of them is 1, iff their greatest common divisor is 1.
|
|
(Contributed by AV, 9-Aug-2020.) $)
|
|
coprmgcdb $p |- ( ( A e. NN /\ B e. NN )
|
|
-> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 )
|
|
<-> ( A gcd B ) = 1 ) ) $=
|
|
( cn wcel wa cv cdvds wbr c1 wceq wi wral cgcd nnz adantr breq1 syl cle
|
|
cz co gcddvds syl2an simpr gcdnncl anbi12d eqeq1 imbi12d rspcv mpid mpdan
|
|
simpl anim1i ancomd 3anass sylibr nndvdslegcd wb breq2 nnge1 nnre letri3d
|
|
w3a 1red biimprd mpan2d adantl sylbid adantll syld ralrimiva ex impbid )
|
|
ADEZBDEZFZCGZAHIZVQBHIZFZVQJKZLZCDMZABNUAZJKZVPWDAHIZWDBHIZFZWCWELVNATEBT
|
|
EWHVOAOBOABUBUCVPWHFZWCWHWEVPWHUDWIWDDEZWCWHWELZLVPWJWHABUEPWBWKCWDDVQWDK
|
|
ZVTWHWAWEWLVRWFVSWGVQWDAHQVQWDBHQUFVQWDJUGUHUIRUJUKVPWEWCVPWEFZWBCDWMVQDE
|
|
ZFZVTVQWDSIZWAWOWNVNVOVCZVTWPLWOWNVPFWQWOVPWNWMVPWNVPWEULUMUNWNVNVOUOUPVQ
|
|
ABUQRWEWNWPWALVPWEWNFWPVQJSIZWAWEWPWRURWNWDJVQSUSPWNWRWALWEWNWRJVQSIZWAVQ
|
|
UTWNWAWRWSFWNVQJVQVAWNVDVBVEVFVGVHVIVJVKVLVM $.
|
|
|
|
$( Two positive integers are not coprime, i.e. there is an integer greater
|
|
than 1 which divides both integers, iff their greatest common divisor is
|
|
not 1. (Contributed by AV, 9-Aug-2020.) $)
|
|
ncoprmgcdne1b $p |- ( ( A e. NN /\ B e. NN )
|
|
-> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B )
|
|
<-> ( A gcd B ) =/= 1 ) ) $=
|
|
( cn wcel wa cdvds wbr c2 co c1 cle a1i cz ad2antlr syl2anc simpr wi nnzd
|
|
mpd cuz cfv wrex cgcd wne clt caddc df-2 2re eluzelz zred simplll simpllr
|
|
cv cr gcdnncl nnred eluzle dvdsgcd dvdsle letrd syl5eqbrr wb 1nn nnltp1le
|
|
syl3anc mpbird nngt1ne1 mpbid ex rexlimdva adantr eluz2b3 sylanbrc simpll
|
|
syl simplr gcddvds wceq breq1 anbi12d rspcev impbid ) ADEZBDEZFZCUNZAGHZW
|
|
GBGHZFZCIUAUBZUCZABUDJZKUEZWFWJWNCWKWFWGWKEZFZWJWNWPWJFZKWMUFHZWNWQWRKKUG
|
|
JZWMLHZWQWSIWMLUHWQIWGWMIUOEWQUIMWQWGWOWGNEZWFWJIWGUJOZUKWQWMWQWDWEWMDEZW
|
|
DWEWOWJULZWDWEWOWJUMZABUPZPZUQWOIWGLHWFWJIWGUROWQWGWMGHZWGWMLHZWQWJXHWPWJ
|
|
QWQXAANEZBNEZWJXHRXBWQAXDSWQBXESWGABUSVFTWQXAXCXHXIRXBXGWGWMUTPTVAVBWQKDE
|
|
ZXCWRWTVCXLWQVDMXGKWMVEPVGWQXCWRWNVCXGWMVHVPVIVJVKWFWNWLWFWNFZWMWKEZWMAGH
|
|
ZWMBGHZFZWLXMXCWNXNWFXCWNXFVLWFWNQWMVMVNXMXJXKXQXMAWDWEWNVOSXMBWDWEWNVQSA
|
|
BVRPWJXQCWMWKWGWMVSWHXOWIXPWGWMAGVTWGWMBGVTWAWBPVJWC $.
|
|
|
|
$( Two positive integers are not coprime, i.e. there is an integer greater
|
|
than 1 which divides both integers, iff their greatest common divisor is
|
|
greater than 1. (Contributed by AV, 9-Aug-2020.) $)
|
|
ncoprmgcdgt1b $p |- ( ( A e. NN /\ B e. NN )
|
|
-> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B )
|
|
<-> 1 < ( A gcd B ) ) ) $=
|
|
( cn wcel wa cv cdvds wbr c2 cuz cfv wrex cgcd co c1 wne ncoprmgcdne1b wb
|
|
clt gcdnncl nngt1ne1 syl bitr4d ) ADEBDEFZCGZAHIUFBHIFCJKLMABNOZPQZPUGTIZ
|
|
ABCRUEUGDEUIUHSABUAUGUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d F i $. $d G i $. $d I i $.
|
|
$( If two positive integers are coprime, i.e. their greatest common divisor
|
|
is 1, the only positive integer that divides both of them is 1.
|
|
(Contributed by AV, 4-Aug-2021.) $)
|
|
coprmdvds1 $p |- ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 )
|
|
-> ( ( I e. NN /\ I || F /\ I || G ) -> I = 1 ) ) $=
|
|
( vi cn wcel cgcd co c1 wceq cdvds wbr wi wa wral coprmgcdb breq1 anbi12d
|
|
w3a cv eqeq1 imbi12d rspcv com23 3impib com12 syl6bir 3impia ) AEFZBEFZAB
|
|
GHIJZCEFZCAKLZCBKLZSZCIJZMZUIUJNUKDTZAKLZURBKLZNZURIJZMZDEOZUQABDPUOVDUPU
|
|
LUMUNVDUPMULVDUMUNNZUPVCVEUPMDCEURCJZVAVEVBUPVFUSUMUTUNURCAKQURCBKQRURCIU
|
|
AUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
$( Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021,
|
|
~ https://proofwiki.org/wiki/Euclid's_Lemma ): If an integer divides the
|
|
product of two integers and is coprime to one of them, then it divides the
|
|
other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul
|
|
Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) $)
|
|
coprmdvds $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) ) $=
|
|
( cz wcel w3a cmul co cdvds wbr cgcd c1 wceq wa wb zcn breq2d bitrd adantr
|
|
cc mulcom syl2an dvdsmulgcd ancoms 3adant1 gcdcom 3adant3 eqeq1d syl6bi imp
|
|
wi oveq2 mulid1d 3ad2ant3 eqtrd biimpd ex com23 impd ) ADEZBDEZCDEZFZABCGHZ
|
|
IJZABKHZLMZACIJZVCVGVEVHVCVGVEVHUKVCVGNZVEVHVIVEACBAKHZGHZIJZVHVCVEVLOZVGVA
|
|
VBVMUTVAVBNZVEACBGHZIJZVLVNVDVOAIVABTECTEVDVOMVBBPCPZBCUAUBQVBVAVPVLOACBUCU
|
|
DRUESVIVKCAIVIVKCLGHZCVCVGVKVRMZVCVGVJLMVSVCVFVJLUTVAVFVJMVBABUFUGUHVJLCGUL
|
|
UIUJVCVRCMZVGVBUTVTVAVBCVQUMUNSUOQRUPUQURUS $.
|
|
|
|
${
|
|
$d x K $. $d x M $. $d x N $.
|
|
$( If an integer is divisible by two coprime integers, then it is divisible
|
|
by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) $)
|
|
coprmdvds2 $p |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\
|
|
( M gcd N ) = 1 ) ->
|
|
( ( M || K /\ N || K ) -> ( M x. N ) || K ) ) $=
|
|
( vx cz wcel w3a cgcd co wceq wa cdvds wbr wi cc zcn syl3anc sylbid breq2
|
|
cmul c1 cv wrex divides 3adant1 adantr simprr simpl2 mulcom syl2an breq2d
|
|
wb syl2anc simprl simpl1 coprmdvds mpan2d dvdsmulc syld imbi12d syl5ibcom
|
|
anassrs rexlimdva com23 impd ) BEFZCEFZAEFZGZBCHIUAJZKZBALMZCALMZBCTIZALM
|
|
ZVKVMVLVOVKVMDUBZCTIZAJZDEUCZVLVONZVIVMVSULZVJVGVHWAVFDCAUDUEUFVKVRVTDEVI
|
|
VJVPEFZVRVTNVIVJWBKZKZBVQLMZVNVQLMZNVRVTWDWEBVPLMZWFWDWEBCVPTIZLMZWGWDVQW
|
|
HBLWDWBVGVQWHJZVIVJWBUGZVFVGVHWCUHZWBVPOFCOFWJVGVPPCPVPCUIUJUMUKWDWIVJWGV
|
|
IVJWBUNWDVFVGWBWIVJKWGNVFVGVHWCUOZWLWKBCVPUPQUQRWDVFWBVGWGWFNWMWKWLCBVPUR
|
|
QUSVRWEVLWFVOVQABLSVQAVNLSUTVAVBVCRVDVE $.
|
|
$}
|
|
|
|
$( One half of ~ rpmulgcd2 , which does not need the coprimality assumption.
|
|
(Contributed by Mario Carneiro, 2-Jul-2015.) $)
|
|
mulgcddvds $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) ) $=
|
|
( cz wcel cgcd co cc0 cmul cdvds wbr wa zmulcld nn0zd syl adantr wb syl2anc
|
|
wi syl3anc w3a simp1 simp2 simp3 gcdcld dvds0 oveq2 nn0cnd mul01d sylan9eqr
|
|
wceq wne breqtrrd cdiv zcnd cap 0zd zapne biimpar divcanap1d gcddvds simpld
|
|
dvdsmultr1d eqbrtrd simprd dvdsmultr2 mpd dvdsgcd mp2and dvdsval2 dvdsmulcr
|
|
simpr mpbid syl112anc cabs cfv cn0 nn0abscl iddvds dvdsabsb dvdsmulc dvdstr
|
|
nn0red nn0ge0d absidd oveq2d absmuld mulgcd mpbird eqbrtrrd wdc wo zdceq wn
|
|
3eqtr4d exmiddc df-ne orbi2i sylibr mpjaodan ) ADEZBDEZCDEZUAZACFGZHUKZABCI
|
|
GZFGZABFGZXEIGZJKXEHULZXDXFLXHHXJJXDXHHJKZXFXDXHDEZXLXDXHXDAXGXAXBXCUBZXDBC
|
|
XAXBXCUCZXAXBXCUDZMZUENZXHUFOPXFXDXJXIHIGHXEHXIIUGXDXIXDXIXDABXNXOUEZUHUIUJ
|
|
UMXDXKLZXHXEUNGZXEIGZXHXJJXTXHXEXTXHXDXMXKXRPZUOXTXEXDXEDEZXKXDXEXDACXNXPUE
|
|
ZNZPZUOXDXEHUPKZXKXDYDHDEZYHXKQYFXDUQZXEHURRUSUTZXTYAXIJKZYBXJJKZXTYAAJKZYA
|
|
BJKZYLXTYBAXEIGZJKZYNXTYBXHYPJYKXDXHYPJKXKXDXHAXEXRXNYFXDXHAJKZXHXGJKZXDXAX
|
|
GDEZYRYSLXNXQAXGVARZVBZVCPVDXTYADEZXAYDXKYQYNQXTXEXHJKZUUCXDUUDXKXDXEAJKZXE
|
|
XGJKZUUDXDUUEXECJKZXDXAXCUUEUUGLXNXPACVARZVBXDUUGUUFXDUUEUUGUUHVEXDYDXBXCUU
|
|
GUUFSYFXOXPXEBCVFTVGXDYDXAYTUUEUUFLUUDSYFXNXQXEAXGVHTVIPXTYDXKXMUUDUUCQYGXD
|
|
XKVLZYCXEXHVJTVMZXDXAXKXNPZYGUUIXEYAAVKVNVMXTYBBXEIGZJKZYOXTYBXHUULJYKXDXHU
|
|
ULJKZXKXDUUNXHUULVOVPZJKZXDXHBVOVPZAIGZUUQCIGZFGZUUOJXDXHUURJKZXHUUSJKZXHUU
|
|
TJKZXDYRUVAUUBXDXMUUQDEZXAYRUVASXRXDUUQXDXBUUQVQEZXOBVROZNZXNXHUUQAVFTVGXDY
|
|
SXGUUSJKZUVBXDYRYSUUAVEXDBUUQJKZUVHXDBBJKZUVIXDXBUVJXOBVSOXDXBXBUVJUVIQXOXO
|
|
BBVTRVMXDXBUVDXCUVIUVHSXOUVGXPCBUUQWATVGXDXMYTUUSDEZYSUVHLUVBSXRXQXDUUQCUVG
|
|
XPMZXHXGUUSWBTVIXDXMUURDEUVKUVAUVBLUVCSXRXDUUQAUVGXNMUVLXHUURUUSVHTVIXDUUQX
|
|
EVOVPZIGUUQXEIGZUUOUUTXDUVMXEUUQIXDXEXDXEYEWCXDXEYEWDWEWFXDBXEXDBXOUOXDXEYE
|
|
UHWGXDUVEXAXCUUTUVNUKUVFXNXPUUQACWHTWOUMXDXMUULDEUUNUUPQXRXDBXEXOYFMXHUULVT
|
|
RWIPVDXTUUCXBYDXKUUMYOQUUJXDXBXKXOPZYGUUIXEYABVKVNVMXTUUCXAXBYNYOLYLSUUJUUK
|
|
UVOYAABVHTVIXTUUCXIDEZYDYLYMSUUJXDUVPXKXDXIXSNPYGXEYAXIWATVGWJXDXFWKZXFXKWL
|
|
ZXDYDYIUVQYFYJXEHWMRUVQXFXFWNZWLUVRXFWPXKUVSXFXEHWQWRWSOWT $.
|
|
|
|
$( If ` M ` is relatively prime to ` N ` , then the GCD of ` K ` with
|
|
` M x. N ` is the product of the GCDs with ` M ` and ` N ` respectively.
|
|
(Contributed by Mario Carneiro, 2-Jul-2015.) $)
|
|
rpmulgcd2 $p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\
|
|
( M gcd N ) = 1 ) ->
|
|
( K gcd ( M x. N ) ) = ( ( K gcd M ) x. ( K gcd N ) ) ) $=
|
|
( cz wcel cgcd co c1 wceq cmul cn0 cdvds wbr zmulcld gcdcld gcddvds syl3anc
|
|
wa wi mp2and simpl1 simpl2 simpl3 nn0mulcld mulgcddvds adantr syl2anc nn0zd
|
|
w3a simpld simprd dvdstr dvdsgcd simpr breqtrd wb dvds1 coprmdvds2 syl31anc
|
|
syl mpbid dvdscmul dvdsmulc syl2and dvdseq syl22anc ) ADEZBDEZCDEZUIZBCFGZH
|
|
IZRZABCJGZFGZKEABFGZACFGZJGZKEVOVRLMZVRVOLMZVOVRIVMAVNVGVHVIVLUAZVMBCVGVHVI
|
|
VLUBZVGVHVIVLUCZNZOVMVPVQVMABWAWBOZVMACWAWCOZUDVJVSVLABCUEUFVMVRALMZVRVNLMZ
|
|
VTVMVPALMZVQALMZWGVMWIVPBLMZVMVGVHWIWKRWAWBABPUGZUJVMWJVQCLMZVMVGVIWJWMRWAW
|
|
CACPUGZUJVMVPDEZVQDEZVGVPVQFGZHIZWIWJRWGSVMVPWEUHZVMVQWFUHZWAVMWQHLMZWRVMWQ
|
|
VKHLVMWQBLMZWQCLMZWQVKLMZVMWQVPLMZWKXBVMXEWQVQLMZVMWOWPXEXFRWSWTVPVQPUGZUJV
|
|
MWIWKWLUKZVMWQDEZWOVHXEWKRXBSVMWQVMVPVQWSWTOZUHZWSWBWQVPBULQTVMXFWMXCVMXEXF
|
|
XGUKVMWJWMWNUKZVMXIWPVIXFWMRXCSXKWTWCWQVQCULQTVMXIVHVIXBXCRXDSXKWBWCWQBCUMQ
|
|
TVJVLUNUOVMWQKEXAWRUPXJWQUQUTVAAVPVQURUSTVMWMWKWHXLXHVMWMVRVPCJGZLMZWKXMVNL
|
|
MZWHVMWPVIWOWMXNSWTWCWSVPVQCVBQVMWOVHVIWKXOSWSWBWCCVPBVCQVMVRDEZXMDEVNDEZXN
|
|
XORWHSVMVPVQWSWTNZVMVPCWSWCNWDVRXMVNULQVDTVMXPVGXQWGWHRVTSXRWAWDVRAVNUMQTVO
|
|
VRVEVF $.
|
|
|
|
$( Two equal reduced fractions have the same numerator and denominator.
|
|
(Contributed by Jeff Hankins, 29-Sep-2013.) $)
|
|
qredeq $p |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ
|
|
/\ Q e. NN /\ ( P gcd Q ) = 1 ) /\ ( M / N ) = ( P / Q ) ) -> ( M = P /\
|
|
N = Q ) ) $=
|
|
( cz wcel cgcd co c1 wceq wa cmul adantr adantl wbr 3adant3 3ad2ant2 syl2an
|
|
cc cdvds cn w3a cdiv zcn cc0 cap nnap0 divclapd mulcanapd divcanap2d eqeq1d
|
|
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|
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|
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|
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|
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|
|
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WMWNWPWQWRWSWT $.
|
|
|
|
${
|
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$d n x y z A $.
|
|
$( Every rational number has a unique reduced form. (Contributed by Jeff
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|
Hankins, 29-Sep-2013.) $)
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|
qredeu $p |- ( A e. QQ -> E! x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd `
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x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) $=
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( cdiv co wceq cn cz c1st cfv c2nd cgcd c1 wa wcel wbr cc0 adantl 3adant3
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cq syl2anc vz vn vy cv wrex cxp wi wral wreu w3a cop cdvds gcddvds simpld
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nngt0d jca elnnz sylibr opelxpi fveq2 znq op1stg sylan9eqr op2ndg oveq12d
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cr simp1 eqeq1d eqeq2d anbi12d gcdcld nn0cnd 1cnd nnap0d cmul mulid1d zcn
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cc adantr divcanap2d nncn mulgcd 3eqtr2rd mulcanapad cap nnap0 divcanap7d
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biimp3ar rspcedvd simprl ad2antrr simprr simprll ad2antlr simprrl simprlr
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reu4 ) BUAUDZUBUDZCDZEZUBFUEUAGUEAUDZHIZUUHJIZKDZLEZBUUIUUJCDZEZMZAGFUFZU
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EZUUOUCUDZHIZUURJIZKDZLEZBUUSUUTCDZEZMZMZUUHUUREZUGZUCUUPUHAUUPUHZMZBSNUU
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OAUUPUIUUGUVJUAUBGFUUDGNZUUEFNZUUGUVJUVKUVLUUGUJZUUQUVIUVMUUOUUDUUDUUEKDZ
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CDZUUEUVNCDZKDZLEZBUVOUVPCDZEZMAUVOUVPUKZUUPUVMUVOGNZUVPFNZUWAUUPNUVKUVLU
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WBUUGUVKUVLMZUVNUUDULOZUWBUVLUVKUUEGNZUWEUUEUOZUVKUWFMZUWEUVNUUEULOZUUDUU
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EUMZUNUPUWDUVNGNZUVNPUQZUVKUWEUWBURUWDUVNUVLUVKUWFUVNUSNZUWGUUDUUEUTUPZVA
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UWQUWPUVLUUEPUUEVDVHVEQUUDUUEVFVGZVIZUWTUVNUUDVJVKVLZRUVMUVPGNZPUVPVMOZMZ
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UWFUUGUXARUXNUUEUVNWETZUVOUVPSSWFTWGZUXIUVMUUJUWAJIZUVPUUHUWAJWDUVMUXLUXM
|
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UXRUVPEUXOUXPUVOUVPSSWHTWGZWIWLUXJUUMUVSBUXJUUIUVOUUJUVPCUXQUXSWIWMWNUVMU
|
|
VRUVTUVKUVLUVRUUGUWDUVQLUVNUWDUVQUWDUVOUVPUXDUXHWOWPUWDWQUWDUVNUWNWPZUWDU
|
|
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|
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UXTUYAXDWIUWDUWMUWBUXEUYDUYEEUWNUXDUXHUVNUVOUVPXFVKXGXHRUVKUVLUVTUUGUWDUV
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SUUFBUWDUUDUUEUVNUYFUYGUXTUVLUUEPXIOUVKUUEXJQUYAXKWMXLVTXMUVHAUCUUPUUHUUP
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HGFYAUURGFYAUYMUYSMZUVFUVGUYTUVFMZUYHUYNUUHUURVUAUUIUUSEUUJUUTEMZUYHUYNEV
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TUUOUVBUVDXSVUABUUMUVCUYTUULUUNUVEXTUYTUUOUVBUVDYBYCUUSUUTUUIUUJYDYEUUIUU
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YMUYOUYRUVFYMYNYOYPYQYRYSYTUAUBBUUAUUOUVEAUCUUPUVGUULUVBUUNUVDUVGUUKUVALU
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VGUUIUUSUUJUUTKUUHUURHWDZUUHUURJWDZWIWLUVGUUMUVCBUVGUUIUUSUUJUUTCVUEVUFWI
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WMWNUUCUUB $.
|
|
$}
|
|
|
|
$( If ` K ` is relatively prime to ` M ` and to ` N ` , it is also relatively
|
|
prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
|
|
(Proof shortened by Mario Carneiro, 2-Jul-2015.) $)
|
|
rpmul $p |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( ( ( K gcd M ) = 1 /\ ( K gcd N ) = 1 ) ->
|
|
( K gcd ( M x. N ) ) = 1 ) ) $=
|
|
( cz wcel w3a cgcd co c1 wceq cmul cdvds wbr mulgcddvds oveq12 1t1e1 syl6eq
|
|
wa breq2d syl5ibcom cn0 wb simp1 zmulcl 3adant1 gcdcld dvds1 syl sylibd ) A
|
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DEZBDEZCDEZFZABGHZIJACGHZIJRZABCKHZGHZILMZURIJZUMURUNUOKHZLMUPUSABCNUPVAIUR
|
|
LUPVAIIKHIUNIUOIKOPQSTUMURUAEUSUTUBUMAUQUJUKULUCUKULUQDEUJBCUDUEUFURUGUHUI
|
|
$.
|
|
|
|
$( If ` K ` is relatively prime to ` N ` then it is also relatively prime to
|
|
any divisor ` M ` of ` N ` . (Contributed by Mario Carneiro,
|
|
19-Jun-2015.) $)
|
|
rpdvds $p |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\
|
|
( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) = 1 ) $=
|
|
( cz wcel cgcd co c1 wceq cdvds wbr wa cle syl2anc simprr wi cc0 wn wne wb
|
|
w3a simpl1 simpl2 gcddvds simpld simprd cn 1ne0 simprl neeq1d mpbiri neneqd
|
|
simplrr eqbrtrrd simpll3 0dvds syl mpbid jca ex simpl3 sylibrd mtod gcdn0cl
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|
gcdeq0 syl21anc nnzd dvdstr syl3anc mp2and mtbid dvdslegcd syl31anc breqtrd
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|
nnle1eq1 ) ADEZBDEZCDEZUAZACFGZHIZBCJKZLZLZABFGZHMKZWEHIZWDWEVTHMWDWEAJKZWE
|
|
CJKZWEVTMKZWDWHWEBJKZWDVPVQWHWKLVPVQVRWCUBZVPVQVRWCUCZABUDNZUEWDWKWBWIWDWHW
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CJKZXCXEBQCJWDWPWQOVSWAWBWRUMUNXEVRXFXCTVPVQVRWCWRUOCUPUQURUSUTWDVPVRWTXDTW
|
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LVPVQVRWCVAZACVENZVBVCABVDVFZVGZWMXGWEBCVHVIVJWDWOVPVRXDRWHWILWJPXJWLXGWDWT
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XDXBXHVKWEACVLVMVJXAVNWDWSWFWGTXIWEVOUQUR $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Cancellability of congruences
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A n $. $d B n $. $d M n $.
|
|
$( Definition of congruence by integer multiple (see ProofWiki "Congruence
|
|
(Number Theory)", 11-Jul-2021,
|
|
~ https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory) ):
|
|
An integer ` A ` is congruent to an integer ` B ` modulo ` M ` if their
|
|
difference is a multiple of ` M ` . See also the definition in
|
|
[ApostolNT] p. 104: "... ` a ` is congruent to ` b ` modulo ` m ` , and
|
|
we write ` a == b ` (mod ` m ` ) if ` m ` divides the difference
|
|
` a - b ` ", or Wikipedia "Modular arithmetic - Congruence",
|
|
~ https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence ,
|
|
11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers
|
|
are said to be congruent modulo n, if n is a divisor of their difference
|
|
(i.e., if there is an integer k such that a-b = kn)". (Contributed by
|
|
AV, 11-Jul-2021.) $)
|
|
congr $p |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN )
|
|
-> ( ( A mod M ) = ( B mod M )
|
|
<-> E. n e. ZZ ( n x. M ) = ( A - B ) ) ) $=
|
|
( cz wcel cn w3a cmo co wceq cmin cdvds wbr cv cmul wrex wb moddvds 3coml
|
|
simp3 nnzd zsubcl 3adant3 divides syl2anc bitrd ) AEFZBEFZDGFZHZADIJBDIJK
|
|
ZDABLJZMNZCODPJUMKCEQZUJUHUIULUNRABDSTUKDEFUMEFZUNUORUKDUHUIUJUAUBUHUIUPU
|
|
JABUCUDCDUMUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A a b m n $. $d B a b m n $.
|
|
$( Integers divided by gcd are coprime. (Contributed by AV,
|
|
12-Jul-2021.) $)
|
|
divgcdcoprm0 $p |- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 )
|
|
-> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 ) $=
|
|
( va cz wcel co wa cdiv c1 wceq 3adant3 cmul wb wi caddc cc zcnd ad2antrr
|
|
syl eqcom vb vm vn cc0 wne w3a cgcd cdvds wbr gcddvds cv wrex gcdcl nn0zd
|
|
simpl divides simpr anbi12d bezout oveqan12rd eqeq2d bicomd adantl nn0cnd
|
|
jca oveq1 ad2antlr mul32d oveq12d cap zmulcld zaddcld gcd2n0cl nncn nnap0
|
|
cn0 cn div11ap syl3anc dividap divdirap simprd divcanap4d eqeq12d 3bitr2d
|
|
sylan9bbr anim1i ancomd bezoutr1 adantr simpll1 divmulap3 3bitr4g biimprd
|
|
eqtrd syl5bi imp32 simp2 a1dd eqeq1d sylibd sylbid exp32 com34 rexlimdvva
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|
a1d com23 ex mpd impl rexlimdva impd ) ADEZBDEZBUDUEZUFZABUGFZAUHUIZXQBUH
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|
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|
|
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|
|
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|
|
XPYEDEZGZYLYGYDYTYKYGYDNZUADXPYSYIDEZYKUUANZXPXQAUBUKZLFZBUCUKZLFZOFZJZUC
|
|
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|
|
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|
|
UIIYEUUDLFZYIUUFLFZOFZJZYDUURUUIXQYFUUDLFZYJUUFLFZOFZJZUUQUVCUURUVGUUIUUR
|
|
UVFUUHXQYGYKUVDUUEUVEUUGOYFAUUDLVFYJBUUFLVFUTVAVBUUQUVGXQUUTXQLFZUVAXQLFZ
|
|
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|
|
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|
|
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|
|
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|
|
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|
|
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|
|
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|
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|
|
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|
|
WFUUSUVCYEYIUGFZIJZYDUVCUVBIJZUUSUWTIUVBTUUQUXAUWTNZUURUUQUUKUUOGUXBUUQUU
|
|
OUUKUUPUUOUUKXPUUOUQWGWHYEYIUUDUUFWISWJWPUUSUWSYCIUUSYEYAYIYBUGUUQYKYGYEY
|
|
AJZUUQYGUXCNYKUUQUXCYGUUQYAYEJZAYFJZUXCYGUUQAPEUVOUWFUXDUXEMUUQAXMXNXOUUO
|
|
UUKWKQUVQUWNAYEXQWLVSYEYATYFATWMWNXFWQUUQYKYGYIYBJZUUQYKUXFYGUUQUXFYKUUQY
|
|
BYIJZBYJJZUXFYKUUQBPEZUWAUWFUXGUXHMXPUXIUUOUUKXPBXMXNXOWRQRUWCUWNBYIXQWLV
|
|
SYIYBTYJBTWMWNWSWQVIWTXAXBXCXDXGXHXGXEXIXJXKXGXKXLXBXI $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B a b $. $d M a b $.
|
|
$( Integers divided by gcd are coprime (see ProofWiki "Integers Divided by
|
|
GCD are Coprime", 11-Jul-2021,
|
|
~ https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime ): Any
|
|
pair of integers, not both zero, can be reduced to a pair of coprime
|
|
ones by dividing them by their gcd. (Contributed by AV,
|
|
12-Jul-2021.) $)
|
|
divgcdcoprmex $p |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 )
|
|
/\ M = ( A gcd B ) ) -> E. a e. ZZ E. b e. ZZ
|
|
( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) ) $=
|
|
( cz wcel cc0 wa cgcd co wceq cmul wrex syl 3adant3 zcn adantr ad2antrr
|
|
cc wne w3a cv c1 simpl anim2i zeqzmulgcd adantlr ancoms reeanv adantl cn0
|
|
gcdcl nn0cnd mulcomd simp3 eqcomd oveq1d eqtrd eqeq1 mpbird ancomd gcdcom
|
|
simpr oveq2d 3eqtrd sylan9eqr cdiv 3ad2ant1 simp1 3ad2ant2 gcdcld cap wbr
|
|
wb gcdeq0 syl6bi necon3d nn0zd 0zd zapne syl2anc divmulap3d bicomd eqeq2d
|
|
impr bitr2d anbi12d wi 3anass divgcdcoprm0 oveq12 eqeq1d syl5ibcom sylbid
|
|
biimpri imp 3jca ex reximdva syl5bir mp2and ) AFGZBFGZBHUAZIZCABJKZLZUBZA
|
|
DUCZXGMKZLZDFNZBEUCZBAJKZMKZLZEFNZACXJMKZLZBCXNMKZLZXJXNJKZUDLZUBZEFNZDFN
|
|
ZXCXFXMXHXCXFIZXCXDIZXMXFXDXCXDXEUEZUFZABDUGOPXCXFXRXHXFXCXRXDXCXRXEBAEUG
|
|
UHUIPXMXRIXLXQIZEFNZDFNXIYGXLXQDEFFUJXIYMYFDFXIXJFGZIZYLYEEFYOXNFGZIZYLYE
|
|
YQYLIZXTYBYDYRXTXKXSLZYOYSYPYLYOXKXGXJMKZXSYOXJXGYNXJTGZXIXJQUKZXIXGTGZYN
|
|
XCXFUUCXHYHXGYHYIXGULGZYKABUMOZUNPRUOXIYTXSLYNXIXGCXJMXICXGXCXFXHUPUQZURR
|
|
USSYLXTYSVOZYQXLUUGXQAXKXSUTRUKVAYLYQBXPYAXLXQVDXIYPXPYALYNXIYPIZXPXNXGMK
|
|
ZXGXNMKYAXIXPUUILYPXIXOXGXNMXCXFXOXGLZXHYHXDXCIUUJYHXCXDYKVBBAVCOPVERUUHX
|
|
NXGYPXNTGZXIXNQZUKUUHXGXIUUDYPXCXFUUDXHUUEPZRUNUOUUHXGCXNMXIXGCLYPUUFRURV
|
|
FUHVGYQYLYDYQYLAXGVHKZXJLZBXGVHKZXNLZIZYDYQXLUUOXQUUQYQUUOXLYQAXJXGXIATGZ
|
|
YNYPXCXFUUSXHAQVISYOUUAYPUUBRXIUUCYNYPXIXGXIABXCXFXHVJXFXCXDXHYJVKVLUNSZY
|
|
QXGHVMVNZXGHUAZXIUVBYNYPXCXFUVBXHXCXDXEUVBYIXGHBHYIXGHLAHLZBHLZIUVDABVPUV
|
|
CUVDVDVQVRWFPSYQXGFGHFGUVAUVBVOYQXGXIUUDYNYPUUMSVSYQVTXGHWAWBVAZWCWDYQUUQ
|
|
BUUILXQYQBXNXGXIBTGZYNYPXFXCUVFXHXDUVFXEBQRVKSYPUUKYOUULUKUUTUVEWCYQUUIXP
|
|
BYQXGXOXNMXIXGXOLZYNYPXIYIUVGXCXFYIXHYKPABVCOSVEWEWGWHXIUURYDWIYNYPXIUUNU
|
|
UPJKZUDLZUURYDXIXCXDXEUBZUVIXCXFUVJXHUVJYHXCXDXEWJWPPABWKOUURUVHYCUDUUNXJ
|
|
UUPXNJWLWMWNSWOWQWRWSWTWTXAXB $.
|
|
$}
|
|
|
|
${
|
|
$d A k r s $. $d B k r s $. $d C k r s $. $d M k r s $. $d N k r s $.
|
|
$( One direction of the bicondition in ~ cncongr . Theorem 5.4 in
|
|
[ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) $)
|
|
cncongr1 $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ )
|
|
/\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) )
|
|
-> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N )
|
|
-> ( A mod M ) = ( B mod M ) ) ) $=
|
|
( cz wcel co wceq wa cmul cmo cmin wb wi adantl adantr cc zcnd ad2antrr
|
|
vk vr vs w3a cn cgcd cdiv cv zmulcl 3adant2 3adant1 simpl congr syl2an3an
|
|
wrex c1 cc0 wne nnz nnne0 jca eqidd 3jca ex 3ad2ant3 impcom divgcdcoprmex
|
|
com12 oveq2 3ad2ant2 oveq12d 3ad2ant1 eqeq12d simpr simp3 ad2antrl gcdcld
|
|
syl nn0cnd mul12d simp1 ad3antrrr nnzd simp2 zmulcld subdid eqcomd eqeq2d
|
|
simprr ad2ant2r ad2ant2lr zsubcld 3adant3 imp cap gcd2n0cl syl3anc nnne0d
|
|
wbr cn0 nn0zd 0zd zapne syl2anc mpbird mulcanapd 3bitrd zcn df-3an sylibr
|
|
anim12i subdir nncnd divmulap2d simpll eleq1 eqcoms cdvds anim2i dvdsmul2
|
|
divgcdnnr breq2 zsubcl mulcomd breq2d gcdcom eqeq1d ancomd 3anass moddvds
|
|
ad2antll coprmdvds sylibrd expcomd sylbid com23 com3l syl6bi com14 mpd
|
|
impl eqtr2 sylbird a1i 3imp rexlimdvva rexlimdva ) AFGZBFGZCFGZUDZEUEGZDE
|
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|
|
UKUURFGZUUSFGZUUPUULUUTUVENUUHUUJUVIUUIACUIUJUUIUUJUVJUUHBCUIUKUULUUOULZU
|
|
URUUSUAEUMUNUUQUVDUVHUAFUUQUVAFGZJZCUUMUBUHZKHZIZEUUMUCUHZKHZIZUVNUVQUFHZ
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|
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|
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SUXDUWAUVPUURUWSUUSUWTMCUVOAKVICUVOBKVIVKVLPVMUWQUXBUVAUVQKHZAUVNKHZBUVNK
|
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|
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|
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UUQUUMUUQCEUUKUUJUUPUUHUUIUUJVOZQZUULUWEUUKUUOUWKVPVQVSTZUWOUVQRGUVMUWOUV
|
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|
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|
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|
|
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|
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|
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|
|
OUWOJZUXHUYPUXFUXGUUHUWMUXFFGUUIUWNAUVNUIWJUUIUWMUXGFGUUHUWNBUVNUIWKWLSVD
|
|
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|
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|
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|
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|
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|
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BVVDVVHUWAVUSVVDVVHUVQUVNUYTKHZXRWSZUWAVUSOVVDVUAVVIUVQXRVVDUYTUVNUYOUYTR
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GVVBUYOUYTABYCZSQVVBUYDUYOUWMUYDUVLVUKUVNXHVPPYDYEVVDUWAVVJVUSVVDUWAUVQUV
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NUFHZUPIZVVJVUSOVULUWAVVMNUYOUVLVULUVTVVLUPVULUWOUVTVVLIVUKUWNUWMVVGXSZUV
|
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|
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NUWMVVQXIXJUVQUVNUYTYLVRVVDVUKUUHUUIUDZVUSVVPNVVDVUKUYOJVVTVVDUYOVUKVVBVU
|
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KUYOUVLUWMVUKWIXSYHVUKUUHUUIYIXJABUVQYJVRYMYNYOYPYOYQYRYSYTVDWMQUUAQWNVUP
|
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UVHVUSNZUWAVUMVUGVWAUUQVUGVWAOZUVLVULUUOVWBUUKUULVWBUUNDUUNDIZVUGVWAVWCVU
|
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|
|
DUUCYQUUDUUEVFYOYOYOVDUUFYTUUGYO $.
|
|
$}
|
|
|
|
${
|
|
$d A k $. $d B k $. $d C k $. $d M k $. $d N k $.
|
|
$( The other direction of the bicondition in ~ cncongr . (Contributed by
|
|
AV, 11-Jul-2021.) $)
|
|
cncongr2 $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ )
|
|
/\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) )
|
|
-> ( ( A mod M ) = ( B mod M )
|
|
-> ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) ) ) $=
|
|
( cz wcel co wceq wa cmo cmul cc0 syl adantr cdvds wbr wb adantl cc vk cn
|
|
w3a cgcd cdiv simpll3 wne wo wi wn wdc 0z zdceq mpan2 df-ne orbi2i sylibr
|
|
exmiddc mul01d 3ad2ant1 3ad2ant2 eqtr4d oveq1d oveq2 eqeq12d syl5ibr cmin
|
|
zcn wrex simpl simp3 divgcdnnr syl2anr simpl1 simpl2 moddvds syl3anc nnzd
|
|
cv zsubcl 3adant3 divides syl2anc 3bitrd simpr zmulcld zcnd ad3antrrr cap
|
|
0zd zapne mpbird mulcanap2d subdir syl3an eqeq2d bitr3d nnz anim12i gcdcl
|
|
cn0 nn0cnd nnne0 neneqd intnand gcdeq0 necon3abid nn0zd divassapd divgcdz
|
|
jca eqeltrd dvdsmul1 syl2an2 nncnd divmulasscomap syl32anc breqtrrd exp32
|
|
3anass adantrd breq2 syl5ibcom sylbid rexlimdva zmulcl 3adant2 3adant1 ex
|
|
imp sylibrd com23 com12 jaoi mpcom ) AFGZBFGZCFGZUCZEUBGZDECEUDHZUEHZIZJZ
|
|
JZADKHZBDKHZIZACLHZEKHZBCLHZEKHZIZYRUUEUUHJZUUMYPYQYRUUDUUHUFYRCMIZCMUGZU
|
|
HZUUNUUMUIZYRUUOUUOUJZUHZUUQYRUUOUKZUUTYRMFGZUVAULCMUMUNUUOURNUUPUUSUUOCM
|
|
UOUPUQUUOUURUUPUUNUUMUUOAMLHZEKHZBMLHZEKHZIZUUEUVGUUHUUEUVCUVEEKYSUVCUVEI
|
|
UUDYSUVCMUVEYPYQUVCMIYRYPAAVHZUSUTYQYPUVEMIYRYQBBVHZUSVAVBOVCOUUOUUJUVDUU
|
|
LUVFUUOUUIUVCEKCMALVDVCUUOUUKUVEEKCMBLVDVCVEVFUUNUUPUUMUUEUUHUUPUUMUIZUUE
|
|
UUHUAVSZUUBLHZABVGHZIZUAFVIZUVJUUEUUHAUUBKHZBUUBKHZIZUUBUVMPQZUVOUUDUUHUV
|
|
RRZYSUUCUVTYTUUCUUFUVPUUGUVQDUUBAKVDDUUBBKVDVESSUUEUUBUBGZYPYQUVRUVSRUUDY
|
|
TYRUWAYSYTUUCVJZYPYQYRVKZECVLVMZYPYQYRUUDVNYPYQYRUUDVOABUUBVPVQUUEUUBFGZU
|
|
VMFGZUVSUVORUUEUUBUWDVRZYSUWFUUDYPYQUWFYRABVTZWAOUAUUBUVMWBWCWDUUEUUPUVOU
|
|
UMUUEUUPUVOUUMUIUUEUUPJZUVOEUUIUUKVGHZPQZUUMUWIUVNUWKUAFUWIUVKFGZJZUVNUVL
|
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CLHZUWJIZUWKUWMUWNUVMCLHZIUVNUWOUWMUVLUVMCUWMUVLUWMUVKUUBUWIUWLWEUWIUWEUW
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LUUEUWEUUPUWGOOWFWGYSUVMTGZUUDUUPUWLYPYQUWQYRYPYQJUVMUWHWGWAWHYSCTGZUUDUU
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PUWLYSCUWCWGZWHUWMCMWIQZUUPUWIUUPUWLUUEUUPWEOUWMYRUVBUWTUUPRYSYRUUDUUPUWL
|
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UWCWHUWMWJCMWKWCWLWMUWMUWPUWJUWNYSUWPUWJIZUUDUUPUWLYPATGYQBTGYRUWRUXAUVHU
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VICVHABCWNWOWHWPWQUWMEUWNPQZUWOUWKUWIUWLUXBUUEUWLUXBUIZUUPYSUUDUXCYSYTUXC
|
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UUCYSYTUWLUXBYSYTUWLJZJZEEUVKCLHUUAUEHZLHZUWNPUXDEFGZYSUXFFGEUXGPQYTUXHUW
|
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LEWRZOUXEUXFUVKCUUAUEHZLHFUXEUVKCUUAUXDUVKTGZYSUXDUVKYTUWLWEZWGSZYSUWRUXD
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UWSOZUXEUUAUXEYRUXHJZUUAXAGYSYRUXDUXHUWCUXDEYTUWLVJZVRWSZCEWTNZXBZUXEUUAM
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UYCUUAMUXEUXOUUAMIUYCRUXQCEXFNXGWLUXEUUAFGUVBUXTUYARUXEUUAUXRXHUXEWJUUAMW
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KWCWLZXIUXEUVKUXJUXDUWLYSUXLSUXEYRUXHEMUGZUCZUXJFGUXEYRUXHUYGJZJUYHYSYRUX
|
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DUYIUWCYTUYIUWLYTUXHUYGUXIUYEXKOWSYRUXHUYGXTUQCEXJNWFXLEUXFXMXNUXEUXKETGZ
|
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UWRUUATGUXTUWNUXGIUXMUXDUYJYSUXDEUXPXOSUXNUXSUYFUVKECUUAXPXQXRXSYAYJOYJUW
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NUWJEPYBYCYDYEUUEUUMUWKRZUUPUUEYTUUIFGZUUKFGZUYKUUDYTYSUWBSYSUYLUUDYPYRUY
|
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LYQACYFYGOYSUYMUUDYQYRUYMYPBCYFYHOUUIUUKEVPVQOYKYIYLYDYJYMYNNYOYI $.
|
|
$}
|
|
|
|
$( Cancellability of Congruences (see ProofWiki "Cancellability of
|
|
Congruences, ~ https://proofwiki.org/wiki/Cancellability_of_Congruences ,
|
|
10-Jul-2021): Two products with a common factor are congruent modulo a
|
|
positive integer iff the other factors are congruent modulo the integer
|
|
divided by the greates common divisor of the integer and the common
|
|
factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109.
|
|
(Contributed by AV, 13-Jul-2021.) $)
|
|
cncongr $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ )
|
|
/\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) )
|
|
-> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N )
|
|
<-> ( A mod M ) = ( B mod M ) ) ) $=
|
|
( cz wcel w3a cn cgcd co cdiv wceq wa cmul cmo cncongr1 cncongr2 impbid ) A
|
|
FGBFGCFGHEIGDECEJKLKMNNACOKEPKBCOKEPKMADPKBDPKMABCDEQABCDERS $.
|
|
|
|
$( Corollary 1 of Cancellability of Congruences: Two products with a common
|
|
factor are congruent modulo an integer being coprime to the common factor
|
|
iff the other factors are congruent modulo the integer. (Contributed by
|
|
AV, 13-Jul-2021.) $)
|
|
cncongrcoprm $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ )
|
|
/\ ( N e. NN /\ ( C gcd N ) = 1 ) )
|
|
-> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N )
|
|
<-> ( A mod N ) = ( B mod N ) ) ) $=
|
|
( cn wcel cgcd co c1 wceq wa cz w3a cdiv cmul cmo wb simpl nncn div1d oveq2
|
|
eqcomd sylan9req jca cncongr sylan2 ) DEFZCDGHZIJZKZALFBLFCLFMUGDDUHNHZJZKA
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|
COHDPHBCOHDPHJADPHBDPHJQUJUGULUGUIRUGUIDDINHZUKUGDDSTUIUKUMUHIDNUAUBUCUDABC
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DDUEUF $.
|
|
|
|
|
|
$(
|
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Elementary prime number theory
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Elementary properties
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Remark: to represent odd prime numbers, i.e., all prime numbers except ` 2 `,
|
|
the idiom ` P e. ( Prime \ { 2 } ) ` is used. It is a little bit shorter
|
|
than ` ( P e. Prime /\ P =/= 2 ) `. Both representations can be converted
|
|
into each other by ~ eldifsn .
|
|
|
|
$)
|
|
|
|
$c Prime $.
|
|
|
|
$( Extend the definition of a class to include the set of prime numbers. $)
|
|
cprime $a class Prime $.
|
|
|
|
${
|
|
$d n p $.
|
|
$( Define the set of prime numbers. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
df-prm $a |- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o } $.
|
|
$}
|
|
|
|
${
|
|
$d P n p z $.
|
|
$( The predicate "is a prime number". A prime number is a positive integer
|
|
with exactly two positive divisors. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
isprm $p |- ( P e. Prime <->
|
|
( P e. NN /\ { n e. NN | n || P } ~~ 2o ) ) $=
|
|
( vp cv cdvds wbr cn crab c2o cen cprime wceq breq2 rabbidv breq1d df-prm
|
|
elrab2 ) BDZCDZEFZBGHZIJFRAEFZBGHZIJFCAGKSALZUAUCIJUDTUBBGSAREMNOBCPQ $.
|
|
|
|
$( A prime number is a positive integer. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
prmnn $p |- ( P e. Prime -> P e. NN ) $=
|
|
( vz cprime wcel cn cv cdvds wbr crab c2o cen isprm simplbi ) ACDAEDBFAGH
|
|
BEIJKHABLM $.
|
|
|
|
$( A prime number is an integer. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) $)
|
|
prmz $p |- ( P e. Prime -> P e. ZZ ) $=
|
|
( cprime wcel prmnn nnzd ) ABCAADE $.
|
|
$}
|
|
|
|
$( The prime numbers are a subset of the positive integers. (Contributed by
|
|
AV, 22-Jul-2020.) $)
|
|
prmssnn $p |- Prime C_ NN $=
|
|
( vx cprime cn cv prmnn ssriv ) ABCADEF $.
|
|
|
|
$( The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) $)
|
|
prmex $p |- Prime e. _V $=
|
|
( cprime cn nnex prmssnn ssexi ) ABCDE $.
|
|
|
|
${
|
|
$d n z $.
|
|
$( 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|
|
(Proof shortened by Fan Zheng, 3-Jul-2016.) $)
|
|
1nprm $p |- -. 1 e. Prime $=
|
|
( vn vz c1 cprime wcel c1o c2o cen wbr 1nen2 cv cdvds cn crab csn wceq wa
|
|
1nn eleq1 mpbiri cn0 wb nnnn0 dvds1 syl bicomd velsn breq1 elrab 3bitr4ri
|
|
biadan2 eqriv 1ex ensn1 eqbrtri ensymi isprm simprbi entr sylancr mto ) C
|
|
DEZFGHIZJVBFAKZCLIZAMNZHIVFGHIZVCVFFVFCOZFHBVFVHBKZCPZVIMEZVICLIZQVIVHEVI
|
|
VFEVJVKVLVJVKCMEZRVICMSTVKVLVJVKVIUAEVLVJUBVIUCVIUDUEUFUKBCUGVEVLAVIMVDVI
|
|
CLUHUIUJULCUMUNUOUPVBVMVGCAUQURFVFGUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d N n $.
|
|
$( The positive divisors of a positive integer include 1 and itself.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
1idssfct $p |- ( N e. NN -> { 1 , N } C_ { n e. NN | n || N } ) $=
|
|
( cn wcel c1 cv cdvds wbr crab cpr wss 1nn cz nnz 1dvds syl breq1 biimpri
|
|
wa elrab sylancr iddvds mpdan prssi syl2anc ) BCDZEAFZBGHZACIZDZBUIDZEBJU
|
|
IKUFECDZEBGHZUJLUFBMDZUMBNZBOPUJULUMSUHUMAECUGEBGQTRUAUFBBGHZUKUFUNUPUOBU
|
|
BPUKUFUPSUHUPABCUGBBGQTRUCEBUIUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d n z P $.
|
|
$( Lemma for ~ isprm2 . (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
isprm2lem $p |- ( ( P e. NN /\ P =/= 1 ) ->
|
|
( { n e. NN | n || P } ~~ 2o <-> { n e. NN | n || P } = { 1 , P } ) ) $=
|
|
( cn wcel c1 wne wa cv cdvds wbr crab c2o cen cpr syl ad2antrr 1nn elrab3
|
|
wb breq1 wceq simplr necomd wi simpr nnz 1dvds ax-mp sylibr iddvds mpbird
|
|
cz en2eqpr syl3anc ex necom pr2ne mpan biimpar sylan2br syl5ibrcom impbid
|
|
mpd ) ACDZAEFZGZBHZAIJZBCKZLMJZVIEANZUAZVFVJVLVFVJGZEAFZVLVMAEVDVEVJUBUCV
|
|
MVJEVIDZAVIDZVNVLUDVFVJUEVMEAIJZVOVDVQVEVJVDAULDZVQAUFZAUGOPECDZVOVQSQVHV
|
|
QBECVGEAITRUHUIVMVPAAIJZVDWAVEVJVDVRWAVSAUJOPVDVPWASVEVJVHWABACVGAAITRPUK
|
|
EAVIUMUNVCUOVFVJVLVKLMJZVEVDVNWBEAUPVDWBVNVTVDWBVNSQEACCUQURUSUTVIVKLMTVA
|
|
VB $.
|
|
|
|
$( The predicate "is a prime number". A prime number is an integer greater
|
|
than or equal to 2 whose only positive divisors are 1 and itself.
|
|
Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman,
|
|
26-Oct-2012.) $)
|
|
isprm2 $p |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\
|
|
A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) ) $=
|
|
( vn cprime wcel c1 wa cn cv cdvds wbr wss wceq wi bitri ancom bitr4i wal
|
|
wb 3bitri wne crab cpr c2 cuz cfv wo wral 1nprm eleq1 biimpcd mtoi neqned
|
|
pm4.71i c2o cen isprm isprm2lem eqss imbi2i 1idssfct jcab mpbiran2 adantr
|
|
pm5.74ri bitrd expcom pm5.32d syl5bb anass eluz2b3 anbi1i dfss2 breq1 vex
|
|
pm5.32ri elrab elpr imbi12i impexp albii df-ral anbi2i ) BDEZWDBFUAZGBHEZ
|
|
CIZBJKZCHUBZFBUCZLZGZWEGZBUDUEUFEZAIZBJKZWOFMWOBMUGZNZAHUHZGZWDWEWDBFWDBF
|
|
MZFDEZUIXAWDXBBFDUJUKULUMUNWEWDWLWDWFWIUOUPKZGWEWLBCUQWEWFXCWKWFWEXCWKSWF
|
|
WEGZXCWIWJMZWKBCURWFXEWKSWEWFXEWKWFXENWFWKWJWILZGZNZWFWKNZXEXGWFWIWJUSUTX
|
|
HXIWFXFNCBVAWFWKXFVBVCOVEVDVFVGVHVIVPWMWEWFGZWKGZWNWKGWTWMWEWLGXKWLWEPWEW
|
|
FWKVJQXJWNWKXJXDWNWEWFPBVKQVLWKWSWNWKWOWIEZWOWJEZNZARZWSAWIWJVMXOWOHEZWRN
|
|
ZARWSXNXQAXNXPWPGZWQNXQXLXRXMWQWHWPCWOHWGWOBJVNVQWOFBAVOVRVSXPWPWQVTOWAWR
|
|
AHWBQOWCTT $.
|
|
|
|
$( The predicate "is a prime number". A prime number is an integer greater
|
|
than or equal to 2 with no divisors strictly between 1 and itself.
|
|
(Contributed by Paul Chapman, 26-Oct-2012.) $)
|
|
isprm3 $p |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\
|
|
A. z e. ( 2 ... ( P - 1 ) ) -. z || P ) ) $=
|
|
( wcel c2 cuz wbr c1 wi cn wa wn co wdc wb cz cle adantr sylan2 cr bitrd
|
|
cprime cfv cv cdvds wceq wo wral cmin isprm2 dvdszrcl simpld 1zzd syl2an2
|
|
cfz zdceq simprd adantl dcor sylc imandc syl clt eluz2nn nnz dvdsle sylan
|
|
nnge1 jctild wne cap zre 1re leltap mp3an1 zapne mpan2 3adant2 3expia w3a
|
|
1z syl3an1 syl3an2 ancoms 3adant3 anim12d pm4.38 df-ne nesym ioran bitr4i
|
|
anbi12i syl6bb syl6 syl2an syld eluzelz caddc zltp1le mpan breq1i syl6bbr
|
|
imp df-2 zltlem1 anbi12d peano2zm elfz mp3an2 bitr4d bitr3d anasss expcom
|
|
pm5.32d fzssuz wss 2eluzge1 uzss ax-mp sstri nnuz sseqtr4i sseli pm4.71ri
|
|
2z notbid pm5.74da bi2.04 con2b 3bitr3g ralbidv2 pm5.32i bitri ) BUACBDEU
|
|
BZCZAUCZBUDFZYOGUEZYOBUEZUFZHZAIUGZJYNYPKZADBGUHLZUNLZUGZJABUIYNUUAUUEYNY
|
|
TUUBAIUUDYNYPYOICZYSHZHYPYOUUDCZKZHUUFYTHUUHUUBHYNYPUUGUUIYNYPJZUUGUUFYSK
|
|
ZJZKZUUIUUJYSMZUUGUUMNUUJYQMZYRMZUUNYPYOOCZYNGOCZUUOYPUUQBOCZYOBUJZUKZUUJ
|
|
ULYOGUOUMYPUUQYNUUSUUPUVAYPUUSYNYPUUQUUSUUTUPUQYOBUOUMYQYRURUSUUFYSUTVAUU
|
|
JUULUUHUUJUULUUFUUHJUUHUUJUUFUUKUUHUUFUUJUUKUUHNZUUFYNYPUVBUUFYNJZYPJGYOV
|
|
BFZYOBVBFZJZUUKUUHUVCYPUVFUUKNZUVCYPGYOPFZYOBPFZJZUVGYNUUFBICZYPUVJHBVCZU
|
|
UFUVKJYPUVIUVHUUFUUQUVKYPUVIHYOVDZYOBVEVFUUFUVHUVKYOVGQVHRUUFUUQUVKUVJUVG
|
|
HYNUVMUVLUUQUVKJUVJUVDYOGVIZNZUVEBYOVIZNZJZUVGUVKUUQUUSUVJUVRHBVDUUQUUSJZ
|
|
UVHUVOUVIUVQUUQUUSUVHUVOUUQUVHUVOUUSUUQUVHJUVDYOGVJFZUVNUUQYOSCZUVHUVDUVT
|
|
NZYOVKZGSCUWAUVHUWBVLGYOVMVNVFUUQUVTUVNNZUVHUUQUURUWDVTYOGVOVPQTVQVRUUQUU
|
|
SUVIUVQUUQUUSUVIVSUVEBYOVJFZUVPUUSUUQBSCZUVIUVEUWENZBVKUUQUWAUWFUVIUWGUWC
|
|
YOBVMWAWBUUQUUSUWEUVPNZUVIUUSUUQUWHBYOVOWCWDTVRWERUVRUVFUVNUVPJZUUKUVDUVE
|
|
UVNUVPWFUWIYQKZYRKZJUUKUVNUWJUVPUWKYOGWGBYOWHWKYQYRWIWJWLWMWNWOXBUVCUVFUU
|
|
HNZYPUUFUUQUUSUWLYNUVMDBWPUVSUVFDYOPFZYOUUCPFZJZUUHUVSUVDUWMUVEUWNUUQUVDU
|
|
WMNUUSUUQUVDGGWQLZYOPFZUWMUURUUQUVDUWQNVTGYOWRWSDUWPYOPXCWTXAQYOBXDXEUUSU
|
|
UQUUCOCZUUHUWONZBXFUUQDOCUWRUWSYDYODUUCXGXHRXIWNQXJXKXLXMUUHUUFUUDIYOUUDG
|
|
EUBZIUUDYMUWTDUUCXNDUWTCYMUWTXOXPGDXQXRXSXTYAYBYCXAYETYFYPUUFYSYGYPUUHYHY
|
|
IYJYKYL $.
|
|
|
|
$( The predicate "is a prime number". A prime number is an integer greater
|
|
than or equal to 2 whose only divisor greater than or equal to 2 is
|
|
itself. (Contributed by Paul Chapman, 26-Oct-2012.) $)
|
|
isprm4 $p |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\
|
|
A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) ) $=
|
|
( cprime wcel c2 cuz cfv cv cdvds c1 wceq wi cn wral imbi1i impexp imbi2d
|
|
wa cz syl5bb wbr isprm2 eluz2nn pm4.71ri bitri wne eluz2b3 bi2.04 wdc nnz
|
|
wo wn 1zzd zdceq syl2anc dfordc syl df-ne syl6rbbr pm5.74i 3bitri ralbii2
|
|
wb pm5.4 anbi2i bitr4i ) BCDBEFGZDZAHZBIUAZVIJKZVIBKZUKZLZAMNZRVHVJVLLZAV
|
|
GNZRABUBVQVOVHVPVNAVGMVIVGDZVPLZVIMDZVSLZVTVTVNLZLWBVSVTVRRZVPLWAVRWCVPVR
|
|
VTVIUCUDOVTVRVPPUEVTVSWBVSVTVIJUFZRZVPLZVTWBVRWEVPVIUGOWFVTWDVPLZLVTWBVTW
|
|
DVPPVTWGVNVTWGVJWDVLLZLVTVNWDVJVLUHVTWHVMVJVTVMVKULZVLLZWHVTVKUIZVMWJVCVT
|
|
VISDJSDWKVIUJVTUMVIJUNUOVKVLUPUQWDWIVLVIJUROUSQTQTTUTVTVNVDVAVBVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d k x y $. $d n x A $. $d x z ch $. $d x et $. $d x ta $. $d x th $.
|
|
$d k n y z ph $.
|
|
prmind.1 $e |- ( x = 1 -> ( ph <-> ps ) ) $.
|
|
prmind.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
prmind.3 $e |- ( x = z -> ( ph <-> th ) ) $.
|
|
prmind.4 $e |- ( x = ( y x. z ) -> ( ph <-> ta ) ) $.
|
|
prmind.5 $e |- ( x = A -> ( ph <-> et ) ) $.
|
|
prmind.6 $e |- ps $.
|
|
${
|
|
prmind2.7 $e |- ( ( x e. Prime /\ A. y e. ( 1 ... ( x - 1 ) ) ch ) ->
|
|
ph ) $.
|
|
prmind2.8 $e |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ->
|
|
( ( ch /\ th ) -> ta ) ) $.
|
|
$( A variation on ~ prmind assuming complete induction for primes.
|
|
(Contributed by Mario Carneiro, 20-Jun-2015.) $)
|
|
prmind2 $p |- ( A e. NN -> et ) $=
|
|
( wcel c1 vn vk cn cfz co wral elfz1end biimpi caddc wceq oveq2 raleqdv
|
|
cv wb elfz1eq syl mpbiri rgen csn wa wsbc cdvds wbr cmin wrex cdiv cmul
|
|
c2 wn peano2nn ad2antrr nncnd cuz cfv elfzuz ad2antrl nnap0d divcanap2d
|
|
eluz2nn cz clt simprr cc0 wne nnne0d dvdsval2 syl3anc mpbid mulid2d cle
|
|
nnzd elfzle2 cc nncn ax-1cn sylancl breqtrd nnz zleltp1 syl2anc eqbrtrd
|
|
pncan cr 1red nngt0d ltmuldiv syl112anc eluz2b1 sylanbrc fznn mpbir2and
|
|
nnred simplr rspcv sylc nnrpd rpdivcld rpgt0d dividapd eluz2gt1 ltdiv23
|
|
elnnz syl122anc mpbird cbvralv sylib vex sbcie dfsbcq syl5bbr wi anbi2d
|
|
jca imbi12d eluzelz sylibrd ex cprime oveq2d wdc sbceq1d imbi2d zmulcld
|
|
ancoms adantl adantr sbcieg vtoclga sbceq1dd rexlimdvaa ralnex eluzp1p1
|
|
syl3c simpl elnnuz df-2 fveq2i syl6eleqr isprm3 baibr simpr nfv nfsbc1v
|
|
nfcv nfim oveq1 sbceq1a vtoclgaf syl5com sylbid syl5bir wo a1i peano2zd
|
|
2z zsubcld dvdsdc syl2anr exfzdc exmiddc mpjaod ralsnsg ancld cun fzsuc
|
|
1zzd sylbi ralunb syl6bb nnind ) JUCSZJTJUDUEZSZAGUWLUFZFUWKUWMJUGUHAGT
|
|
UAUMZUDUEZUFAGTTUDUEZUFAGTUBUMZUDUEZUFZAGTUWRTUIUEZUDUEZUFZUWNUAUBJUWOT
|
|
UJAGUWPUWQUWOTTUDUKULUWOUWRUJAGUWPUWSUWOUWRTUDUKULUWOUXAUJAGUWPUXBUWOUX
|
|
ATUDUKULUWOJUJAGUWPUWLUWOJTUDUKULAGUWQGUMZUWQSZABPUXEUXDTUJABUNUXDTUOKU
|
|
PUQURUWRUCSZUWTUWTAGUXAUSZUFZUTZUXCUXFUWTUXHUXFUWTAGUXAVAZUXHUXFUWTUXJU
|
|
XFUWTUTZHUMZUXAVBVCZHVHUXATVDUEZUDUEZVEZUXJUXPVIZUXKUXMUXJHUXOUXKUXLUXO
|
|
SZUXMUTZUTZAGUXLUXAUXLVFUEZVGUEZUXAUXTUXAUXLUXTUXAUXFUXAUCSZUWTUXSUWRVJ
|
|
ZVKZVLZUXTUXLUXTUXLVHVMVNZSZUXLUCSZUXRUYHUXKUXMUXLVHUXNVOZVPZUXLVSZUPZV
|
|
LZUXTUXLUYMVQVRUXTUYAUYGSZUYHCAGUYAVAZUTZAGUYBVAZUXTUYAVTSZTUYAWAVCZUYO
|
|
UXTUXMUYSUXKUXRUXMWBUXTUXLVTSZUXLWCWDUXAVTSZUXMUYSUNUXTUXLUYMWKZUXTUXLU
|
|
YMWEUXTUXAUYEWKUXLUXAWFWGWHZUXTTUXLVGUEZUXAWAVCZUYTUXTVUEUXLUXAWAUXTUXL
|
|
UYNWIUXTUXLUWRWJVCZUXLUXAWAVCZUXTUXLUXNUWRWJUXRUXLUXNWJVCUXKUXMUXLVHUXN
|
|
WLVPUXTUWRWMSZTWMSZUXNUWRUJZUXFVUIUWTUXSUWRWNVKWOUWRTXBZWPWQZUXTVUAUWRV
|
|
TSZVUGVUHUNVUCUXFVUNUWTUXSUWRWRVKZUXLUWRWSWTWHXAUXTTXCSUXAXCSZUXLXCSZWC
|
|
UXLWAVCZVUFUYTUNUXTXDUXTUXAUYEXLZUXTUXLUYMXLZUXTUXLUYMXEZTUXAUXLXFXGWHU
|
|
YAXHXIUYKUXTCUYPUXTUXLUWSSZUWTCUXTVVBUYIVUGUYMVUMUXTVUNVVBUYIVUGUTUNVUO
|
|
UXLUWRXJUPXKUXFUWTUXSXMZACGUXLUWSLXNXOUXTUYAUWSSZDIUWSUFZUYPUXTVVDUYAUC
|
|
SZUYAUWRWJVCZUXTUYSWCUYAWAVCVVFVUDUXTUYAUXTUXAUXLUXTUXAUYEXPUXTUXLUYMXP
|
|
XQXRUYAYBXIUXTVVGUYAUXAWAVCZUXTUXAUXAVFUEZUXLWAVCZVVHUXTVVITUXLWAUXTUXA
|
|
UYFUXTUXAUYEVQXSUXTUYHTUXLWAVCUYKUXLXTUPXAUXTVUPVUPWCUXAWAVCVUQVURVVJVV
|
|
HUNVUSVUSUXTUXAUYEXEVUTVVAUXAUXAUXLYAYCWHUXTUYSVUNVVGVVHUNVUDVUOUYAUWRW
|
|
SWTYDUXTVUNVVDVVFVVGUTUNVUOUYAUWRXJUPXKUXTUWTVVEVVCADGIUWSMYEYFDUYPIUYA
|
|
UWSDAGIUMZVAVVKUYAUJZUYPADGVVKIYGMYHAGVVKUYAYIYJZXNXOYMUYHCDUTZAGUXLVVK
|
|
VGUEZVAZYKZYKUYHUYQUYRYKZYKIUYAUYGVVLVVQVVRUYHVVLVVNUYQVVPUYRVVLDUYPCVV
|
|
MYLVVLAGVVOUYBVVKUYAUXLVGUKUUAYNUUBVVKUYGSZUYHVVQVVSUYHUTZVVNEVVPUYHVVS
|
|
VVNEYKRUUDVVTVVOVTSVVPEUNVVTUXLVVKUYHVUAVVSVHUXLYOUUEVVSVVKVTSUYHVHVVKY
|
|
OUUFUUCAEGVVOVTNUUGUPYPYQUUHUUMUUIUUJUXQUXMVIHUXOUFZUXKUXJUXMHUXOUUKUXK
|
|
VWAUXAYRSZUXJUXKUXAUYGSZVWAVWBUNUXKUXATTUIUEZVMVNZUYGUXKUWRTVMVNSZUXAVW
|
|
ESUXKUXFVWFUXFUWTUUNZUWRUUOZYFTUWRUULUPVHVWDVMUUPUUQUURVWBVWCVWAHUXAUUS
|
|
UUTUPUXKCHTUXNUDUEZUFZVWBUXJUXKVWJCHUWSUFZUXKUWTVWKUXFUWTUVAACGHUWSLYEY
|
|
FUXKCHVWIUWSUXKUXNUWRTUDUXKVUIVUJVUKUXKUWRVWGVLWOVULWPYSULYDCHTUXDTVDUE
|
|
ZUDUEZUFZAYKVWJUXJYKGUXAYRGUXAUVDVWJUXJGVWJGUVBAGUXAUVCUVEUXDUXAUJZVWNV
|
|
WJAUXJVWOCHVWMVWIVWOVWLUXNTUDUXDUXATVDUVFYSULAGUXAUVGYNUXDYRSVWNAQYQUVH
|
|
UVIUVJUVKUXKUXPYTUXPUXQUVLUXKUXMHVHUXNVHVTSUXKUVOUVMUXKUXATUXKUWRUXKUWR
|
|
VWGWKUVNZUXKUWFUVPUXRUYIVUBUXMYTUXKUXRUYHUYIUYJUYLUPVWPUXLUXAUVQUVRUVSU
|
|
XPUVTUPUWAYQUXFUYCUXHUXJUNUYDAGUXAUCUWBUPYPUWCUXFUXCAGUWSUXGUWDZUFUXIUX
|
|
FAGUXBVWQUXFVWFUXBVWQUJVWHTUWRUWEUWGULAGUWSUXGUWHUWIYPUWJAFGJUWLOXNXO
|
|
$.
|
|
$}
|
|
|
|
prmind.7 $e |- ( x e. Prime -> ph ) $.
|
|
prmind.8 $e |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ->
|
|
( ( ch /\ th ) -> ta ) ) $.
|
|
$( Perform induction over the multiplicative structure of ` NN ` . If a
|
|
property ` ph ( x ) ` holds for the primes and ` 1 ` and is preserved
|
|
under multiplication, then it holds for every positive integer.
|
|
(Contributed by Mario Carneiro, 20-Jun-2015.) $)
|
|
prmind $p |- ( A e. NN -> et ) $=
|
|
( c1 co cv cprime wcel cmin cfz wral adantr prmind2 ) ABCDEFGHIJKLMNOPGUA
|
|
ZUBUCACHSUISUDTUETUFQUGRUH $.
|
|
$}
|
|
|
|
${
|
|
$d M m $. $d P m $.
|
|
$( If ` M ` divides a prime, then ` M ` is either the prime or one.
|
|
(Contributed by Scott Fenton, 8-Apr-2014.) $)
|
|
dvdsprime $p |- ( ( P e. Prime /\ M e. NN ) ->
|
|
( M || P <-> ( M = P \/ M = 1 ) ) ) $=
|
|
( vm cprime wcel cn wa cdvds wbr wceq c1 wo c2 cuz breq1 eqeq1 syl adantr
|
|
wi syl5ibrcom cv wral isprm2 orbi12d orcom syl6bb imbi12d rspccva adantll
|
|
cfv sylanb cz prmz iddvds 1dvds jaod impbid ) ADEZBFEZGZBAHIZBAJZBKJZLZUR
|
|
AMNUJEZCUAZAHIZVFKJZVFAJZLZSZCFUBZGUSVAVDSZCAUCVLUSVMVEVKVMCBFVFBJZVGVAVJ
|
|
VDVFBAHOVNVJVCVBLVDVNVHVCVIVBVFBKPVFBAPUDVCVBUEUFUGUHUIUKUTVBVAVCUTVAVBAA
|
|
HIZURVOUSURAULEZVOAUMZAUNQRBAAHOTUTVAVCKAHIZURVRUSURVPVRVQAUOQRBKAHOTUPUQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A product of two integers greater than one is composite. (Contributed
|
|
by Mario Carneiro, 20-Jun-2015.) $)
|
|
nprm $p |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) ->
|
|
-. ( A x. B ) e. Prime ) $=
|
|
( vx c2 cuz cfv wcel cz eluzelz adantr clt wbr cn simprbi adantl cr cdvds
|
|
zred wceq wi wa cmul co wne cprime wn c1 eluz2b2 cc0 wb eluz2nn ltmulgt11
|
|
nngt0d syl3anc mpbid ltned dvdsmul1 syl2an wral breq1 eqeq1 imbi12d rspcv
|
|
cv isprm4 syl5 mpid necon3ad mpd ) ADEFZGZBVJGZUAZAABUBUCZUDVNUEGZUFVMAVN
|
|
VMAVKAHGZVLDAIZJRZVMUGBKLZAVNKLZVLVSVKVLBMGVSBUHNOVMAPGBPGUIAKLVSVTUJVRVM
|
|
BVLBHGZVKDBIZORVMAVKAMGVLAUKJUMABULUNUOUPVMVOAVNVMVOAVNQLZAVNSZVKVPWAWCVL
|
|
VQWBABUQURVKVOWCWDTZTVLVOCVDZVNQLZWFVNSZTZCVJUSZVKWEVOVNVJGWJCVNVENWIWECA
|
|
VJWFASWGWCWHWDWFAVNQUTWFAVNVAVBVCVFJVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
nprmi.1 $e |- A e. NN $.
|
|
nprmi.2 $e |- B e. NN $.
|
|
nprmi.3 $e |- 1 < A $.
|
|
nprmi.4 $e |- 1 < B $.
|
|
nprmi.5 $e |- ( A x. B ) = N $.
|
|
$( An inference for compositeness. (Contributed by Mario Carneiro,
|
|
18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) $)
|
|
nprmi $p |- -. N e. Prime $=
|
|
( cmul co cprime wcel cn c1 clt wbr wn wa c2 eluz2b2 nprm syl2anbr eleq1i
|
|
cuz cfv mp4an mtbi ) ABIJZKLZCKLAMLZNAOPZBMLZNBOPZUIQZDFEGUJUKRASUDUEZLBU
|
|
OLUNULUMRATBTABUAUBUFUHCKHUCUG $.
|
|
$}
|
|
|
|
${
|
|
$d A k $. $d N k $. $d ph k $.
|
|
dvdsnprmd.g $e |- ( ph -> 1 < A ) $.
|
|
dvdsnprmd.l $e |- ( ph -> A < N ) $.
|
|
dvdsnprmd.d $e |- ( ph -> A || N ) $.
|
|
$( If a number is divisible by an integer greater than 1 and less then the
|
|
number, the number is not prime. (Contributed by AV, 24-Jul-2021.) $)
|
|
dvdsnprmd $p |- ( ph -> -. N e. Prime ) $=
|
|
( vk wbr wcel cz wa wb c2 cle c1 clt adantr cr cc0 wi cdvds wn cv cmul co
|
|
cprime wceq wrex dvdszrcl divides 3syl cuz cfv 2z a1i simplr breq2 adantl
|
|
mpbird w3a zre 3ad2ant1 3ad2ant3 0lt1 0red 1red syl3anc mpani imp 3adant3
|
|
lttr 3jca 3exp mpd ltmulgt12 syl caddc df-2 breq1i zltp1le mpancom bicomd
|
|
1zzd syl5bb eluz2 syl3anbrc simpl biimpa sylibr nprm syl2anc eleq1 notbid
|
|
ex mpbid rexlimdva sylbid ) ABCUAHZCUFIZUBZFAWRGUCZBUDUEZCUGZGJUHZWTAWRBJ
|
|
IZCJIZKZWRXDLFBCUIZGBCUJUKAXCWTGJAXAJIZKZXCWTXJXCKZXBUFIZUBZWTXKXAMULUMZI
|
|
ZBXNIZXMXKMJIZXIMXANHZXOXQXKUNUOAXIXCUPXKXROXAPHZXKXSBXBPHZXKXTBCPHZXJYAX
|
|
CAYAXIEQQXCXTYALXJXBCBPUQURUSXKBRIZXARIZSBPHZUTZXSXTLXJYEXCAXIYEAOBPHZXIY
|
|
ETZDAWRXGYFYGTZFXHXEYHXFXEYFXIYEXEYFXIUTYBYCYDXEYFYBXIBVAZVBXIXEYCYFXAVAV
|
|
CXEYFYDXIXEYFYDXESOPHZYFYDVDXESRIORIYBYJYFKYDTXEVEXEVFYISOBVKVGVHVIVJVLVM
|
|
QUKVNVIQBXAVOVPUSXROOVQUEZXANHZXKXSMYKXANVRVSXJYLXSLZXCXIYMAXIXSYLOJIZXIX
|
|
SYLLXIWCOXAVTWAWBURQWDUSMXAWEWFXJXPXCAXPXIAXQXEMBNHZUTZXPAYFYPDAWRXGYFYPT
|
|
ZFXHXEYQXFXEYFYPXEYFKZXQXEYOXQYRUNUOXEYFWGYRYKBNHZYOXEYFYSYNXEYFYSLXEWCOB
|
|
VTWAWHMYKBNVRVSWIVLWNQUKVNMBWEWIQQXABWJWKXCXMWTLXJXCXLWSXBCUFWLWMURWOWNWP
|
|
WQVN $.
|
|
$}
|
|
|
|
$( A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) $)
|
|
prm2orodd $p |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) ) $=
|
|
( cprime wcel c2 wceq cdvds wbr wn wo wi c1 cn wb 2nn dvdsprime mpan2 eqcom
|
|
biimpi wne cz 1ne2 necom eqneqall com12 sylbi ax-mp jaoi syl6bi wdc prmz 2z
|
|
con3d zdceq dfordc 3syl mpbird ) ABCZADEZDAFGZHZIZURHUTJZUQUSURUQUSDAEZDKEZ
|
|
IZURUQDLCUSVEMNADOPVCURVDVCURDAQRKDSZVDURJZUAVFDKSZVGKDUBVDVHURURDKUCUDUEUF
|
|
UGUHULUQATCZURUIZVAVBMAUJVIDTCVJUKADUMPURUTUNUOUP $.
|
|
|
|
$( 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof
|
|
shortened by Fan Zheng, 16-Jun-2016.) $)
|
|
2prm $p |- 2 e. Prime $=
|
|
( vz c2 cprime wcel cuz cfv cv cdvds wbr wn c1 cmin co cfz wral cz mpbir2an
|
|
clt 2z c0 1lt2 eluz2b1 ral0 cin wceq fzssuz df-ss mpbi uzdisj eqtr3i raleqi
|
|
wss mpbir isprm3 ) BCDBBEFZDZAGBHIJZABBKLMZNMZOZUPBPDKBRISUABUBQUTUQATOUQAU
|
|
CUQAUSTUSUOUDZUSTUSUOULVAUSUEBURUFUSUOUGUHBBUIUJUKUMABUNQ $.
|
|
|
|
$( 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) $)
|
|
3prm $p |- 3 e. Prime $=
|
|
( vz c3 cprime wcel c2 cuz cfv cv cdvds wbr wn c1 cmin co cfz wral mpbir2an
|
|
cz clt 2z 3z 1lt3 eluz2b1 wceq elfz1eq caddc iddvds cn wi 2nn ndvdsp1 mp3an
|
|
1lt2 mp2b df-3 breq2i mtbir breq1 mtbiri 3m1e2 oveq2i eleq2s rgen isprm3
|
|
syl ) BCDBEFGDZAHZBIJZKZAEBLMNZONZPVFBRDLBSJUAUBBUCQVIAVKVIVGEEONZVKVGVLDVG
|
|
EUDZVIVGEUEVMVHEBIJZVNEELUFNZIJZERDZEEIJZVPKZTEUGVQEUHDLESJVRVSUITUJUMEEUKU
|
|
LUNBVOEIUOUPUQVGEBIURUSVEVJEEOUTVAVBVCABVDQ $.
|
|
|
|
$( 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|
|
(Proof shortened by Mario Carneiro, 18-Feb-2014.) $)
|
|
4nprm $p |- -. 4 e. Prime $=
|
|
( c2 c4 2nn 1lt2 2t2e4 nprmi ) AABCCDDEF $.
|
|
|
|
${
|
|
$d P x $.
|
|
$( A prime number is an integer greater than or equal to 2. (Contributed
|
|
by Paul Chapman, 17-Nov-2012.) $)
|
|
prmuz2 $p |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) ) $=
|
|
( vx cprime wcel c2 cuz cfv cv cdvds wbr wceq wi wral isprm4 simplbi ) AC
|
|
DAEFGZDBHZAIJQAKLBPMBANO $.
|
|
$}
|
|
|
|
$( A prime number is an integer greater than 1. (Contributed by Alexander
|
|
van der Vekens, 17-May-2018.) $)
|
|
prmgt1 $p |- ( P e. Prime -> 1 < P ) $=
|
|
( cprime wcel c2 cuz cfv c1 clt wbr prmuz2 cz cle w3a eluz2 caddc wceq df-2
|
|
co a1i breq1d wb 1z zltp1le mpan biimprd sylbid imp 3adant1 sylbi syl ) ABC
|
|
ADEFCZGAHIZAJUKDKCZAKCZDALIZMULDANUNUOULUMUNUOULUNUOGGORZALIZULUNDUPALDUPPU
|
|
NQSTUNULUQGKCUNULUQUAUBGAUCUDUEUFUGUHUIUJ $.
|
|
|
|
$( Subtracting 2 from a prime number results in a nonnegative integer.
|
|
(Contributed by Alexander van der Vekens, 30-Aug-2018.) $)
|
|
prmm2nn0 $p |- ( P e. Prime -> ( P - 2 ) e. NN0 ) $=
|
|
( cprime wcel c2 cuz cfv cmin co cn0 prmuz2 uznn0sub syl ) ABCADEFCADGHICAJ
|
|
DAKL $.
|
|
|
|
$( An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) $)
|
|
oddprmgt2 $p |- ( P e. ( Prime \ { 2 } ) -> 2 < P ) $=
|
|
( cprime c2 csn cdif wcel wne wa clt wbr eldifsn cuz cfv wi prmuz2 cz eluz2
|
|
cle w3a sylbi zltlen biimprd exp4b 3imp syl imp ) ABCDEFABFZACGZHCAIJZABCKU
|
|
GUHUIUGACLMFZUHUINZAOUJCPFZAPFZCARJZSUKCAQULUMUNUKULUMUNUHUIULUMHUIUNUHHCAU
|
|
AUBUCUDTUEUFT $.
|
|
|
|
$( An odd prime is greater than or equal to 3. (Contributed by Alexander van
|
|
der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.) $)
|
|
oddprmge3 $p |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) ) $=
|
|
( cprime c2 csn cdif wcel c3 cz cle wbr w3a cuz cfv clt eldifi oddprmgt2 wa
|
|
3z a1i prmz adantr c1 caddc co df-3 wb zltp1le sylancr biimpa syl5eqbr 3jca
|
|
2z syl2anc eluz2 sylibr ) ABCDZEFZGHFZAHFZGAIJZKZAGLMFUQABFZCANJZVAABUPOAPV
|
|
BVCQZURUSUTURVDRSVBUSVCATZUAVDGCUBUCUDZAIUEVBVCVFAIJZVBCHFUSVCVGUFULVECAUGU
|
|
HUIUJUKUMGAUNUO $.
|
|
|
|
$( A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) $)
|
|
sqnprm $p |- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime ) $=
|
|
( cz wcel c2 cexp co cprime cfv wa adantr syl recnd clt wbr eluz2b1 cc0 cle
|
|
cr c1 syl2anc cabs cmul zre absresq abscld sqvald eqtr3d simpr eqeltrrd cuz
|
|
wceq cn0 nn0abscl nn0zd sq1 prmuz2 adantl simprbi breqtrrd syl5eqbr absge0d
|
|
wn wb 1re 0le1 lt2sq mpanl12 mpbird sylanbrc nprm pm2.65da ) ABCZADEFZGCZAU
|
|
AHZVOUBFZGCZVLVNIZVMVPGVRVODEFZVMVPVRARCZVSVMUKVLVTVNAUCJZAUDKZVRVOVRVOVRAV
|
|
RAWALZUEZLUFUGVLVNUHUIVRVODUJHZCZWFVQVBVRVOBCSVOMNZWFVRVOVLVOULCVNAUMJUNVRW
|
|
GSDEFZVSMNZVRWHSVSMUOVRSVMVSMVRVMWECZSVMMNZVNWJVLVMUPUQWJVMBCWKVMOURKWBUSUT
|
|
VRVORCZPVOQNZWGWIVCZWDVRAWCVASRCPSQNWLWMIWNVDVESVOVFVGTVHVOOVIZWOVOVOVJTVK
|
|
$.
|
|
|
|
${
|
|
$d N z $. $d P z $.
|
|
$( An integer greater than or equal to 2 divides a prime number iff it is
|
|
equal to it. (Contributed by Paul Chapman, 26-Oct-2012.) $)
|
|
dvdsprm $p |- ( ( N e. ( ZZ>= ` 2 ) /\ P e. Prime ) ->
|
|
( N || P <-> N = P ) ) $=
|
|
( vz c2 cuz cfv wcel cprime wa cdvds wceq wi cv wral isprm4 simprbi breq1
|
|
wbr eqeq1 imbi12d rspcv mpan9 ancoms cz iddvds breq2 syl5ibcom syl adantr
|
|
eluzelz impbid ) BDEFZGZAHGZIBAJRZBAKZUNUMUOUPLZUNCMZAJRZURAKZLZCULNZUMUQ
|
|
UNAULGVBCAOPVAUQCBULURBKUSUOUTUPURBAJQURBASTUAUBUCUMUPUOLZUNUMBUDGZVCDBUJ
|
|
VDBBJRUPUOBUEBABJUFUGUHUIUK $.
|
|
$}
|
|
|
|
${
|
|
$d p x y z N $.
|
|
$( Every integer greater than or equal to 2 has a prime factor.
|
|
(Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario
|
|
Carneiro, 20-Jun-2015.) $)
|
|
exprmfct $p |- ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N ) $=
|
|
( vx cn wcel c2 cv cdvds wbr cprime wrex wi c1 wceq eleq1 rexbidv imbi12d
|
|
breq2 cz wa vy vz cuz cfv eluz2nn cmul co imbi1d cc0 cmin 1m1e0 syl5eqelr
|
|
uz2m1nn 0nnn pm2.21i syl prmz iddvds breq1 rspcev mpdan a1d simpl eluzelz
|
|
ad2antrr ad2antlr dvdsmul1 syl2anc adantl zmulcld dvdstr syl3anc reximdva
|
|
mpan2d embantd a1dd adantrd prmind mpcom ) ADEAFUCUDZEZBGZAHIZBJKZAUECGZV
|
|
TEZWBWEHIZBJKZLMVTEZWHLUAGZVTEZWBWJHIZBJKZLZUBGZVTEZWBWOHIZBJKZLZWJWOUFUG
|
|
ZVTEZWBWTHIZBJKZLZWAWDLCUAUBAWEMNWFWIWHWEMVTOUHWEWJNZWFWKWHWMWEWJVTOXEWGW
|
|
LBJWEWJWBHRPQWEWONZWFWPWHWRWEWOVTOXFWGWQBJWEWOWBHRPQWEWTNZWFXAWHXCWEWTVTO
|
|
XGWGXBBJWEWTWBHRPQWEANZWFWAWHWDWEAVTOXHWGWCBJWEAWBHRPQWIUIDEZWHWIUIMMUJUG
|
|
DUKMUMULXIWHUNUOUPWEJEZWHWFXJWEWEHIZWHXJWESEXKWEUQWEURUPWGXKBWEJWBWEWEHUS
|
|
UTVAVBWKWPTZWNXDWSXLWNXCXAXLWKWMXCWKWPVCXLWLXBBJXLWBJEZTZWLWJWTHIZXBXNWJS
|
|
EZWOSEZXOWKXPWPXMFWJVDVEZWPXQWKXMFWOVDVFZWJWOVGVHXNWBSEZXPWTSEWLXOTXBLXMX
|
|
TXLWBUQVIXRXNWJWOXRXSVJWBWJWTVKVLVNVMVOVPVQVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d I p $. $d N p $.
|
|
$( Each integer greater than 1 and less then or equal to a fixed number is
|
|
divisible by a prime less then or equal to this fixed number.
|
|
(Contributed by AV, 15-Aug-2020.) $)
|
|
prmdvdsfz $p |- ( ( N e. NN /\ I e. ( 2 ... N ) )
|
|
-> E. p e. Prime ( p <_ N /\ p || I ) ) $=
|
|
( cn wcel c2 cfz co wa cv wbr cprime wrex cle adantl syl wi ad2antlr zred
|
|
cr cuz cfv elfzuz exprmfct cz prmz eluz2nn dvdsle syl2anr elfzle2 elfzelz
|
|
cdvds nnre ad2antrr letr syl3anc mpan2d syld ancrd reximdva mpd ) BDEZAFB
|
|
GHEZIZCJZAULKZCLMZVEBNKZVFIZCLMVDAFUAUBEZVGVCVJVBAFBUCZOACUDPVDVFVICLVDVE
|
|
LEZIZVFVHVMVFVEANKZVHVLVEUEEADEZVFVNQVDVEUFZVCVOVBVCVJVOVKAUGPOVEAUHUIVMV
|
|
NABNKZVHVCVQVBVLAFBUJRVMVETEZATEZBTEZVNVQIVHQVLVRVDVLVEVPSOVCVSVBVLVCAAFB
|
|
UKSRVBVTVCVLBUMUNVEABUOUPUQURUSUTVA $.
|
|
$}
|
|
|
|
$( No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.)
|
|
(Proof shortened by Mario Carneiro, 2-Jul-2015.) $)
|
|
nprmdvds1 $p |- ( P e. Prime -> -. P || 1 ) $=
|
|
( cprime wcel c1 cdvds wbr 1nprm wceq cn0 wb prmnn nnnn0d dvds1 syl biimpcd
|
|
eleq1 sylbid mtoi ) ABCZADEFZDBCZGSTADHZUASAICTUBJSAAKLAMNUBSUAADBPOQR $.
|
|
|
|
$( Either ` A / ( A gcd B ) ` is odd or ` B / ( A gcd B ) ` is odd.
|
|
(Contributed by Scott Fenton, 19-Apr-2014.) $)
|
|
divgcdodd $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) ) $=
|
|
( cn wcel wa c2 cgcd co cdvds wbr wn wi c1 cz wb nnz dvdsval2 syl3anc mpbid
|
|
cdiv wo n2dvds1 2z gcddvds syl2an simpld cc0 wne gcdnncl nnzd nnne0d adantr
|
|
simprd adantl dvdsgcdb mp3an2i gcddiv syl31anc nncnd nnap0d dividapd eqtr3d
|
|
wceq breq2d biimpd sylbid expdimp mtoi ex wdc 2nn dvdsdc sylancr imordc syl
|
|
) ACDZBCDZEZFAABGHZTHZIJZFBVSTHZIJZKZLZWAKWDUAZVRWAWDVRWAEWCFMIJZUBVRWAWCWG
|
|
VRWAWCEZFVTWBGHZIJZWGFNDVRVTNDZWBNDZWHWJOUCVRVSAIJZWKVRWMVSBIJZVPANDZBNDZWM
|
|
WNEZVQAPZBPZABUDUEZUFVRVSNDZVSUGUHZWOWMWKOVRVSABUIZUJZVRVSXCUKZVPWOVQWRULZV
|
|
SAQRSZVRWNWLVRWMWNWTUMVRXAXBWPWNWLOXDXEVQWPVPWSUNZVSBQRSFVTWBUOUPVRWJWGVRWI
|
|
MFIVRVSVSTHZWIMVRWOWPVSCDWQXIWIVCXFXHXCWTABVSUQURVRVSVRVSXCUSVRVSXCUTVAVBVD
|
|
VEVFVGVHVIVRWAVJZWEWFOVRFCDWKXJVKXGFVTVLVMWAWDVNVOS $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Coprimality and Euclid's lemma (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
This section is about coprimality with respect to primes, and a special version
|
|
of Euclid's lemma for primes is provided, see ~ euclemma .
|
|
$)
|
|
|
|
${
|
|
$d N z $. $d P z $.
|
|
$( A prime number either divides an integer or is coprime to it, but not
|
|
both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
coprm $p |- ( ( P e. Prime /\ N e. ZZ ) ->
|
|
( -. P || N <-> ( P gcd N ) = 1 ) ) $=
|
|
( vz wcel cz wa cdvds wbr wn c1 sylan breq1 syl5ibcom wo cn cc0 adantr wi
|
|
wceq cr cprime cgcd co prmz gcddvds simprd con3d 0nnn prmnn mtoi intnanrd
|
|
eleq1 gcdn0cl ex mpd simpld cv wral c2 cuz isprm2 simprbi orbi12d imbi12d
|
|
cfv eqeq1 rspcv syl5com mp2d biorf orcom syl6bb syl5ibrcom cle clt iddvds
|
|
syld wne syl w3a dvdslegcd 3anidm12 mpand prmgt1 zred nnred ltletr mp3an1
|
|
1re syl2anc ltne mpan a1i 3syld necon2bd impbid ) AUADZBEDZFZABGHZIZABUBU
|
|
CZJSZWSXAXBASZIZXCWSXDWTWSXBBGHZXDWTWSXBAGHZXFWQAEDZWRXGXFFAUDZABUEKZUFXB
|
|
ABGLMUGWSXCXEXCXDNZWSXBODZXGXKWSAPSZBPSZFIZXLWQXOWRWQXMXNWQXMPODZUHWQAODX
|
|
MXPAUIAPOULMUJUKQZWQXHWRXOXLRXIXHWRFXOXLABUMUNKUOZWSXGXFXJUPWQXLXGXKRZRWR
|
|
WQCUQZAGHZXTJSZXTASZNZRZCOURZXLXSWQAUSUTVEDYFCAVAVBYEXSCXBOXTXBSZYAXGYDXK
|
|
XTXBAGLYGYBXCYCXDXTXBJVFXTXBAVFVCVDVGVHQVIXEXCXDXCNXKXDXCVJXDXCVKVLVMVQWS
|
|
WTXBJWSWTAXBVNHZJXBVOHZXBJVRZWSAAGHZWTYHWQYKWRWQXHYKXIAVPVSQWSXOYKWTFYHRZ
|
|
XQWQXHWRXOYLRZXIXHWRYMXHXHWRVTXOYLAABWAUNWBKUOWCWSJAVOHZYHYIWQYNWRAWDQWSA
|
|
TDZXBTDZYNYHFYIRZWQYOWRWQAXIWEQWSXBXRWFJTDZYOYPYQWIJAXBWGWHWJWCYIYJRWSYRY
|
|
IYJWIJXBWKWLWMWNWOWP $.
|
|
$}
|
|
|
|
$( Unequal prime numbers are relatively prime. (Contributed by Mario
|
|
Carneiro, 23-Feb-2014.) $)
|
|
prmrp $p |- ( ( P e. Prime /\ Q e. Prime ) ->
|
|
( ( P gcd Q ) = 1 <-> P =/= Q ) ) $=
|
|
( cprime wcel wa cdvds wbr wn cgcd co c1 wceq wne cz wb coprm sylan2 c2 cuz
|
|
prmz cfv prmuz2 dvdsprm sylan necon3bbid bitr3d ) ACDZBCDZEZABFGZHZABIJKLZA
|
|
BMUHUGBNDUKULOBTABPQUIUJABUGARSUADUHUJABLOAUBBAUCUDUEUF $.
|
|
|
|
|
|
$( Euclid's lemma. A prime number divides the product of two integers iff it
|
|
divides at least one of them. Theorem 1.9 in [ApostolNT] p. 17.
|
|
(Contributed by Paul Chapman, 17-Nov-2012.) $)
|
|
euclemma $p |- ( ( P e. Prime /\ M e. ZZ /\ N e. ZZ ) ->
|
|
( P || ( M x. N ) <-> ( P || M \/ P || N ) ) ) $=
|
|
( cprime wcel cz w3a cmul co cdvds wbr wo wn wi wa cgcd c1 wceq wb syl3an1
|
|
coprm 3adant3 anbi2d prmz coprmdvds sylbid expd wdc cn prmnn 3ad2ant1 simp2
|
|
dvdsdc syl2anc dfordc syl sylibrd ordvdsmul impbid ) ADEZBFEZCFEZGZABCHIJKZ
|
|
ABJKZACJKZLZVCVDVEMZVFNZVGVCVDVHVFVCVDVHOVDABPIQRZOZVFVCVHVJVDUTVAVHVJSVBAB
|
|
UAUBUCUTAFEZVAVBVKVFNAUDZABCUETUFUGVCVEUHZVGVISVCAUIEZVAVNUTVAVOVBAUJUKUTVA
|
|
VBULABUMUNVEVFUOUPUQUTVLVAVBVGVDNVMABCURTUS $.
|
|
|
|
${
|
|
$d x y z P $.
|
|
$( A number is prime iff it satisfies Euclid's lemma ~ euclemma .
|
|
(Contributed by Mario Carneiro, 6-Sep-2015.) $)
|
|
isprm6 $p |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. x e. ZZ
|
|
A. y e. ZZ ( P || ( x x. y ) -> ( P || x \/ P || y ) ) ) ) $=
|
|
( vz wcel cmul co cdvds wbr wo wi cz wa wb c1 wceq cn adantr syl ad2antrl
|
|
cprime c2 cuz cfv wral prmuz2 euclemma 3expb biimpd ralrimivva simpl cdiv
|
|
cv jca eluz2nn nnzd iddvds cc nncn cc0 cap nnap0 divcanap1d simprr simprl
|
|
breqtrrd nndivdvds syl2anc mpbid oveq1 breq2d breq2 orbi1d imbi12d orbi2d
|
|
nnz oveq2 rspc2va sylan mpd cle dvdsle div1d breq1d sylibrd nnrp rpregt0d
|
|
clt crp 1rp rpregt0 mp1i lediv2 syl3anc nnle1eq1 sylibd cn0 nnnn0 simplrr
|
|
cr simpr dvdseq syl22anc orim12d imp syldan an32s expr ralrimiva sylanbrc
|
|
ex isprm2 impbii ) CUAEZCUBUCUDEZCAUMZBUMZFGZHIZCXPHIZCXQHIZJZKZBLUEALUEZ
|
|
MZXNXOYDCUFXNYCABLLXNXPLEZXQLEZMMXSYBXNYFYGXSYBNCXPXQUGUHUIUJUNYEXODUMZCH
|
|
IZYHOPZYHCPZJZKZDQUEXNXOYDUKYEYMDQYEYHQEZYIYLXOYNYIMZYDYLXOYOMZYDCCYHULGZ
|
|
HIZCYHHIZJZYLYPYDMCYQYHFGZHIZYTYPUUBYDYPCCUUAHYPCLEZCCHIYPCXOCQEZYOCUORZU
|
|
PZCUQSYPCYHYPUUDCUREUUECUSSZYNYHUREXOYIYHUSTYNYHUTVAIXOYIYHVBTVCVFRYPYQLE
|
|
ZYHLEZMYDUUBYTKZYPUUHUUIYPYQYPYIYQQEZXOYNYIVDYPUUDYNYIUUKNUUEXOYNYIVECYHV
|
|
GVHVIZUPYNUUIXOYIYHVPTUNYCUUJCYQXQFGZHIZYRYAJZKABYQYHLLXPYQPZXSUUNYBUUOUU
|
|
PXRUUMCHXPYQXQFVJVKUUPXTYRYAXPYQCHVLVMVNXQYHPZUUNUUBUUOYTUUQUUMUUACHXQYHY
|
|
QFVQVKUUQYAYSYRXQYHCHVLVOVNVRVSVTYPYTYLYPYRYJYSYKYPYRYHOWAIZYJYPYRCOULGZY
|
|
QWAIZUURYPYRCYQWAIZUUTYPUUCUUKYRUVAKUUFUULCYQWBVHYPUUSCYQWAYPCUUGWCWDWEYP
|
|
YHWTEUTYHWHIMZOWTEUTOWHIMZCWTEUTCWHIMUURUUTNYNUVBXOYIYNYHYHWFWGTOWIEUVCYP
|
|
WJOWKWLYPCYPUUDCWIEUUECWFSWGYHOCWMWNWEYNUURYJNXOYIYHWOTWPYPYSYKYPYSMYHWQE
|
|
ZCWQEZYIYSYKYPUVDYSYNUVDXOYIYHWRTRYPUVEYSYPUUDUVEUUECWRSRXOYNYIYSWSYPYSXA
|
|
YHCXBXCXKXDXEXFXGXHXIDCXLXJXM $.
|
|
$}
|
|
|
|
${
|
|
$d N m $. $d P k m $. $d A k m $.
|
|
$( A prime divides a positive power of an integer iff it divides the
|
|
integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by
|
|
Mario Carneiro, 17-Jul-2014.) $)
|
|
prmdvdsexp $p |- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) ->
|
|
( P || ( A ^ N ) <-> P || A ) ) $=
|
|
( vm vk wcel cz cn cexp co cdvds wb wi c1 wceq oveq2 breq2d bibi1d imbi2d
|
|
wbr cprime wa cv caddc cc zcn adantl exp1d wo cmul cn0 nnnn0 expp1 syl2an
|
|
simpll simpr zexpcl simplr euclemma syl3anc bitrd syl6bb bibi2d syl5ibcom
|
|
orbi1 oridm expcom a2d nnind impcom 3impa ) BUAFZAGFZCHFZBACIJZKTZBAKTZLZ
|
|
VNVLVMUBZVRVSBADUCZIJZKTZVQLZMVSBANIJZKTZVQLZMVSBAEUCZIJZKTZVQLZMVSBAWGNU
|
|
DJZIJZKTZVQLZMVSVRMDECVTNOZWCWFVSWOWBWEVQWOWAWDBKVTNAIPQRSVTWGOZWCWJVSWPW
|
|
BWIVQWPWAWHBKVTWGAIPQRSVTWKOZWCWNVSWQWBWMVQWQWAWLBKVTWKAIPQRSVTCOZWCVRVSW
|
|
RWBVPVQWRWAVOBKVTCAIPQRSVSWDABKVSAVMAUEFZVLAUFUGZUHQWGHFZVSWJWNVSXAWJWNMV
|
|
SXAUBZWMWIVQUIZLWJWNXBWMBWHAUJJZKTZXCXBWLXDBKVSWSWGUKFZWLXDOXAWTWGULZAWGU
|
|
MUNQXBVLWHGFZVMXEXCLVLVMXAUOVSVMXFXHXAVLVMUPXGAWGUQUNVLVMXAURBWHAUSUTVAWJ
|
|
XCVQWMWJXCVQVQUIVQWIVQVQVEVQVFVBVCVDVGVHVIVJVK $.
|
|
$}
|
|
|
|
$( A prime divides a positive power of another iff they are equal.
|
|
(Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro,
|
|
24-Feb-2014.) $)
|
|
prmdvdsexpb $p |- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) ->
|
|
( P || ( Q ^ N ) <-> P = Q ) ) $=
|
|
( cprime wcel cn w3a cexp co cdvds wbr wceq cz wb prmdvdsexp syl3an2 c2 cuz
|
|
prmz cfv prmuz2 dvdsprm sylan 3adant3 bitrd ) ADEZBDEZCFEZGABCHIJKZABJKZABL
|
|
ZUGUFBMEUHUIUJNBSBACOPUFUGUJUKNZUHUFAQRTEUGULAUABAUBUCUDUE $.
|
|
|
|
$( If a prime divides a nonnegative power of another, then they are equal.
|
|
(Contributed by Mario Carneiro, 16-Jan-2015.) $)
|
|
prmdvdsexpr $p |- ( ( P e. Prime /\ Q e. Prime /\ N e. NN0 ) ->
|
|
( P || ( Q ^ N ) -> P = Q ) ) $=
|
|
( cprime wcel cn0 cexp co cdvds wbr wceq wi cn cc0 wo wa elnn0 w3a breq2d
|
|
c1 prmdvdsexpb biimpd 3expia prmnn adantl nncnd exp0d pm2.21d adantr sylbid
|
|
nprmdvds1 oveq2 imbi1d syl5ibrcom jaod syl5bi 3impia ) ADEZBDEZCFEZABCGHZIJ
|
|
ZABKZLZUTCMEZCNKZOURUSPZVDCQVGVEVDVFURUSVEVDURUSVERVBVCABCUAUBUCVGVDVFABNGH
|
|
ZIJZVCLVGVIATIJZVCVGVHTAIVGBVGBUSBMEURBUDUEUFUGSURVJVCLUSURVJVCAUKUHUIUJVFV
|
|
BVIVCVFVAVHAICNBGULSUMUNUOUPUQ $.
|
|
|
|
$( Two positive prime powers are equal iff the primes and the powers are
|
|
equal. (Contributed by Paul Chapman, 30-Nov-2012.) $)
|
|
prmexpb $p |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) ->
|
|
( ( P ^ M ) = ( Q ^ N ) <-> ( P = Q /\ M = N ) ) ) $=
|
|
( cprime wcel wa cn cexp co wceq w3a cdvds wbr cz prmz adantr 3ad2ant1 nnzd
|
|
wb simp2l iddvdsexp syl2anc breq2 3ad2ant3 simp1l simp1r simp2r prmdvdsexpb
|
|
syl3anc bitrd mpbid zred c1 clt prmgt1 ad2antrr 3adant3 simp3 oveq1d eqtr4d
|
|
expcand jca 3expia oveq12 impbid1 ) AEFZBEFZGZCHFZDHFZGZGACIJZBDIJZKZABKZCD
|
|
KZGZVIVLVOVRVIVLVOLZVPVQVSAVMMNZVPVSAOFZVJVTVIVLWAVOVGWAVHAPQRZVIVJVKVOUAZA
|
|
CUBUCVSVTAVNMNZVPVOVIVTWDTVLVMVNAMUDUEVSVGVHVKWDVPTVGVHVLVOUFVGVHVLVOUGVIVJ
|
|
VKVOUHZABDUIUJUKULZVSACDVSAWBUMVSCWCSVSDWESVIVLUNAUONZVOVGWGVHVLAUPUQURVSVM
|
|
VNADIJVIVLVOUSVSABDIWFUTVAVBVCVDABCDIVEVF $.
|
|
|
|
${
|
|
$d P x k $. $d N x $.
|
|
$( The factorial of a number only contains primes less than the base.
|
|
(Contributed by Mario Carneiro, 6-Mar-2014.) $)
|
|
prmfac1 $p |- ( ( N e. NN0 /\ P e. Prime /\ P || ( ! ` N ) ) -> P <_ N ) $=
|
|
( wcel cfa cfv cdvds wbr cle wi c1 wceq fveq2 breq2d breq2 imbi12d imbi2d
|
|
cc0 adantr cz cr vx vk cprime cv caddc co breq2i nprmdvds1 pm2.21d syl5bi
|
|
cn0 fac0 wa wo cmul facp1 wb simpr cn faccl nnzd nn0p1nn euclemma syl3anc
|
|
bitrd nn0re lep1d prmz adantl zred nnred letr mpan2d imim2d com23 syl2anc
|
|
dvdsle a1dd jaod sylbid ex a2d nn0ind 3imp ) BUKCAUCCZABDEZFGZABHGZWEAUAU
|
|
DZDEZFGZAWIHGZIZIWEAQDEZFGZAQHGZIZIWEAUBUDZDEZFGZAWRHGZIZIWEAWRJUEUFZDEZF
|
|
GZAXCHGZIZIWEWGWHIZIUAUBBWIQKZWMWQWEXIWKWOWLWPXIWJWNAFWIQDLMWIQAHNOPWIWRK
|
|
ZWMXBWEXJWKWTWLXAXJWJWSAFWIWRDLMWIWRAHNOPWIXCKZWMXGWEXKWKXEWLXFXKWJXDAFWI
|
|
XCDLMWIXCAHNOPWIBKZWMXHWEXLWKWGWLWHXLWJWFAFWIBDLMWIBAHNOPWOAJFGZWEWPWNJAF
|
|
ULUGWEXMWPAUHUIUJWRUKCZWEXBXGXNWEXBXGIXNWEUMZXEXBXFXOXEWTAXCFGZUNZXBXFIZX
|
|
OXEAWSXCUOUFZFGZXQXOXDXSAFXNXDXSKWEWRUPRMXOWEWSSCXCSCXTXQUQXNWEURXOWSXNWS
|
|
USCWEWRUTRVAXOXCXNXCUSCZWEWRVBRZVAAWSXCVCVDVEXOWTXRXPXOXBWTXFXOXAXFWTXOXA
|
|
WRXCHGZXFXOWRXNWRTCZWEWRVFRZVGXOATCYDXCTCXAYCUMXFIXOAWEASCZXNAVHVIZVJYEXO
|
|
XCYBVKAWRXCVLVDVMVNVOXOXPXFXBXOYFYAXPXFIYGYBAXCVQVPVRVSVTVOWAWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d p A $. $d p B $. $d p N $.
|
|
$( If two numbers ` A ` and ` B ` are relatively prime, then they are still
|
|
relatively prime if raised to a power. (Contributed by Mario Carneiro,
|
|
24-Feb-2014.) $)
|
|
rpexp $p |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) ->
|
|
( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) ) $=
|
|
( vp cz wcel cc0 wceq wa co cgcd c1 wb wn wi cdvds syl2anc adantr syl3anc
|
|
wbr cn w3a cexp 0exp oveq1d eqeq1d oveq1 oveq12 sylan syl5ibrcom 3ad2ant3
|
|
bibi12d wne c2 cuz cfv cv cprime exprmfct cn0 simpl1 simpl3 nnnn0d zexpcl
|
|
wrex simpl2 gcddvds simpld prmz adantl simpr cc zcnd expeq0 anbi1d mtbird
|
|
gcdn0cl syl21anc nnzd dvdstr mpan2d simpll1 prmdvdsexp sylibd simprd jcad
|
|
dvdsgcd nprmdvds1 breq2 notbid necon2ad 3syld rexlimdva 3adantl3 baib syl
|
|
eluz2b3 sylibrd syl5 iddvdsexp mp2and impbid wdc simp1 simp3 simp2 gcdcld
|
|
3bitr3d nn0zd 1zzd zdceq nebidc sylc mpbird ex wo gcdmndc exmiddc 3adant3
|
|
mpjaod ) AEFZBEFZCUAFZUBZAGHZBGHZIZACUCJZBKJZLHZABKJZLHZMZYGNZYCYAYGYMOYB
|
|
YCYMYGGCUCJZGKJZLHZGGKJZLHZMYCYPYRLYCYOGGKCUDUEUFYGYJYQYLYSYGYIYPLYEYHYOH
|
|
YFYIYPHAGCUCUGYHYOBGKUHUIUFYGYKYRLAGBGKUHUFULUJUKYDYNYMYDYNIZYMYILUMZYKLU
|
|
MZMZYTYIUNUOUPZFZYKUUDFZUUAUUBYTUUEUUFUUEDUQZYIPTZDURVEZYTUUFYIDUSYTUUIUU
|
|
BUUFYTUUHUUBDURYTUUGURFZIZUUHUUGAPTZUUGBPTZIZUUGYKPTZUUBUUKUUHUULUUMUUKUU
|
|
HUUGYHPTZUULUUKUUHYIYHPTZUUPUUKUUQYIBPTZUUKYHEFZYBUUQUURIYTUUSUUJYTYACUTF
|
|
ZUUSYAYBYCYNVAZYTCYAYBYCYNVBZVCACVDZQZRZYTYBUUJYAYBYCYNVFZRZYHBVGQZVHUUKU
|
|
UGEFZYIEFZUUSUUHUUQIUUPOUUJUVIYTUUGVIVJZYTUVJUUJYTYIYTUUSYBYHGHZYFIZNYIUA
|
|
FZUVDUVFYTUVMYGYDYNVKYTUVLYEYFYTAVLFYCUVLYEMYTAUVAVMUVBACVNQVOVPYHBVQVRZV
|
|
SRZUVEUUGYIYHVTSWAUUKUUJYAYCUUPUULMYTUUJVKYAYBYCYNUUJWBZYTYCUUJUVBRZAUUGC
|
|
WCSWDUUKUUHUURUUMUUKUUQUURUVHWEUUKUVIUVJYBUUHUURIUUMOUVKUVPUVGUUGYIBVTSWA
|
|
WFUUKUVIYAYBUUNUUOOUVKUVQUVGUUGABWGSUUJUUOUUBOYTUUJUUOYKLUUJUUONYLUUGLPTZ
|
|
NZUUGWHZYLUUOUVSYKLUUGPWIWJUJWKVJWLWMYTYKUAFZUUFUUBMYAYBYNUWBYCABVQWNZUUF
|
|
UWBUUBYKWQWOWPZWRWSUUFUUODURVEZYTUUEYKDUSYTUWEUUAUUEYTUUOUUADURUUKUUOUUPU
|
|
UMIZUUHUUAUUKUUOUUPUUMUUKUUOYKYHPTZUUPUUKYKAPTZAYHPTZUWGUUKUWHYKBPTZUUKYA
|
|
YBUWHUWJIUVQUVGABVGQZVHUUKYAYCUWIUVQUVRACWTQUUKYKEFZYAUUSUWHUWIIUWGOYTUWL
|
|
UUJYTYKUWCVSRZUVQUVEYKAYHVTSXAUUKUVIUWLUUSUUOUWGIUUPOUVKUWMUVEUUGYKYHVTSW
|
|
AUUKUUOUWJUUMUUKUWHUWJUWKWEUUKUVIUWLYBUUOUWJIUUMOUVKUWMUVGUUGYKBVTSWAWFUU
|
|
KUVIUUSYBUWFUUHOUVKUVEUVGUUGYHBWGSUUJUUHUUAOYTUUJUUHYILUUJUUHNYJUVTUWAYJU
|
|
UHUVSYILUUGPWIWJUJWKVJWLWMYTUVNUUEUUAMUVOUUEUVNUUAYIWQWOWPZWRWSXBUWNUWDXH
|
|
YDYMUUCMZYNYDYJXCZYLXCZUWOYDUVJLEFZUWPYDYIYDYHBYDYAUUTUUSYAYBYCXDZYDCYAYB
|
|
YCXEVCUVCQYAYBYCXFZXGXIYDXJZYILXKQYDUWLUWRUWQYDYKYDABUWSUWTXGXIUXAYKLXKQY
|
|
ILYKLXLXMRXNXOYAYBYGYNXPZYCYAYBIYGXCUXBABXQYGXRWPXSXT $.
|
|
$}
|
|
|
|
$( Relative primality passes to asymmetric powers. (Contributed by Stefan
|
|
O'Rear, 27-Sep-2014.) $)
|
|
rpexp1i $p |- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN0 ) -> ( ( A gcd B ) = 1 -> (
|
|
( A ^ M ) gcd B ) = 1 ) ) $=
|
|
( cz wcel cn0 cgcd co c1 wceq cexp wi wa cn cc0 wo elnn0 w3a rpexp eqtrd cc
|
|
biimprd 3expa simpr oveq2d zcn ad2antrr oveq1d 1gcd ad2antlr jaodan sylan2b
|
|
exp0d a1d 3impa ) ADEZBDEZCFEZABGHIJZACKHZBGHZIJZLZURUPUQMZCNEZCOJZPVCCQVDV
|
|
EVCVFUPUQVEVCUPUQVERVBUSABCSUBUCVDVFMZVBUSVGVAIBGHZIVGUTIBGVGUTAOKHIVGCOAKV
|
|
DVFUDUEVGAUPAUAEUQVFAUFUGUMTUHUQVHIJUPVFBUIUJTUNUKULUO $.
|
|
|
|
$( Relative primality passes to symmetric powers. (Contributed by Stefan
|
|
O'Rear, 27-Sep-2014.) $)
|
|
rpexp12i $p |- ( ( A e. ZZ /\ B e. ZZ /\ ( M e. NN0 /\ N e. NN0 ) ) -> ( ( A
|
|
gcd B ) = 1 -> ( ( A ^ M ) gcd ( B ^ N ) ) = 1 ) ) $=
|
|
( cz wcel cn0 wa w3a cgcd co wceq cexp rpexp1i zexpcl syl2anc gcdcom eqeq1d
|
|
c1 wi 3adant3r simp2 simp1 simp3l simp3r syl3anc 3imtr4d syld ) AEFZBEFZCGF
|
|
ZDGFZHZIZABJKSLZACMKZBJKZSLZUPBDMKZJKZSLZUIUJUKUOURTULABCNUAUNBUPJKZSLZUSUP
|
|
JKZSLZURVAUNUJUPEFZULVCVETUIUJUMUBZUNUIUKVFUIUJUMUCUIUJUKULUDACOPZUIUJUKULU
|
|
EZBUPDNUFUNUQVBSUNVFUJUQVBLVHVGUPBQPRUNUTVDSUNVFUSEFZUTVDLVHUNUJULVJVGVIBDO
|
|
PUPUSQPRUGUH $.
|
|
|
|
$( A prime number does not divide the factorial of a nonnegative integer less
|
|
than the prime number. (Contributed by AV, 13-Jul-2021.) $)
|
|
prmndvdsfaclt $p |- ( ( P e. Prime /\ N e. NN0 )
|
|
-> ( N < P -> -. P || ( ! ` N ) ) ) $=
|
|
( cprime wcel cn0 wa clt wbr cle wn cfa cfv cdvds cz wb nn0z zltnle syl2anr
|
|
prmz wi prmfac1 3exp impcom con3d sylbid ) ACDZBEDZFZBAGHZABIHZJZABKLMHZJUG
|
|
BNDANDUIUKOUFBPASBAQRUHULUJUGUFULUJTUGUFULUJABUAUBUCUDUE $.
|
|
|
|
$( Corollary 2 of Cancellability of Congruences: Two products with a common
|
|
factor are congruent modulo a prime number not dividing the common factor
|
|
iff the other factors are congruent modulo the prime number. (Contributed
|
|
by AV, 13-Jul-2021.) $)
|
|
cncongrprm $p |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ )
|
|
/\ ( P e. Prime /\ -. P || C ) )
|
|
-> ( ( ( A x. C ) mod P ) = ( ( B x. C ) mod P )
|
|
<-> ( A mod P ) = ( B mod P ) ) ) $=
|
|
( cz wcel w3a cprime cdvds wbr wn wa cn cgcd co c1 wceq cmul cmo wb wi prmz
|
|
prmnn ad2antrl coprm gcdcom sylan eqeq1d ancoms biimpd expimpd 3ad2ant3 imp
|
|
bitrd jca cncongrcoprm syldan ) AEFZBEFZCEFZGZDHFZDCIJKZLZDMFZCDNOZPQZLACRO
|
|
DSOBCRODSOQADSOBDSOQTVAVDLVEVGVBVEVAVCDUCUDVAVDVGUTURVDVGUAUSUTVBVCVGUTVBLV
|
|
CVGVBUTVCVGTVBUTLZVCDCNOZPQVGDCUEVHVIVFPVBDEFUTVIVFQDUBDCUFUGUHUNUIUJUKULUM
|
|
UOABCDUPUQ $.
|
|
|
|
$( The predicate "is an even number". An even number and 2 have 2 as
|
|
greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by
|
|
AV, 8-Aug-2021.) $)
|
|
isevengcd2 $p |- ( Z e. ZZ -> ( 2 || Z <-> ( 2 gcd Z ) = 2 ) ) $=
|
|
( cz wcel c2 cgcd co wceq cdvds wbr cn wb 2nn gcdzeq mpan bicomd ) ABCZDAEF
|
|
DGZDAHIZDJCPQRKLDAMNO $.
|
|
|
|
$( The predicate "is an odd number". An odd number and 2 have 1 as greatest
|
|
common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV,
|
|
8-Aug-2021.) $)
|
|
isoddgcd1 $p |- ( Z e. ZZ -> ( -. 2 || Z <-> ( 2 gcd Z ) = 1 ) ) $=
|
|
( c2 cprime wcel cz cdvds wbr wn cgcd co c1 wceq wb 2prm coprm mpan ) BCDAE
|
|
DBAFGHBAIJKLMNBAOP $.
|
|
|
|
$( The least common multiple of three and two is six. The operands are
|
|
unequal primes and thus coprime, so the result is (the absolute value of)
|
|
their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof
|
|
shortened by AV, 27-Aug-2020.) $)
|
|
3lcm2e6 $p |- ( 3 lcm 2 ) = 6 $=
|
|
( c3 c2 clcm co cmul c6 cgcd c1 wceq wne 2re 2lt3 cprime wcel mp2an 3nn 2nn
|
|
cn cz nnzi gtneii wb 3prm 2prm prmrp mpbir oveq2i lcmgcdnn cn0 lcmcl nn0cni
|
|
mulid1i 3eqtr3ri 3t2e6 eqtri ) ABCDZABEDZFUPABGDZEDZUPHEDUQUPURHUPEURHIZABJ
|
|
ZBAKLUAAMNBMNUTVAUBUCUDABUEOUFUGARNBRNUSUQIPQABUHOUPUPASNBSNUPUINAPTBQTABUJ
|
|
OUKULUMUNUO $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Non-rationality of square root of 2
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
sqrt2irrlem.1 $e |- ( ph -> A e. ZZ ) $.
|
|
sqrt2irrlem.2 $e |- ( ph -> B e. NN ) $.
|
|
sqrt2irrlem.3 $e |- ( ph -> ( sqrt ` 2 ) = ( A / B ) ) $.
|
|
$( Lemma for ~ sqrt2irr . This is the core of the proof: - if
|
|
` A / B = sqrt ( 2 ) ` , then ` A ` and ` B ` are even, so ` A / 2 ` and
|
|
` B / 2 ` are smaller representatives, which is absurd by the method of
|
|
infinite descent (here implemented by strong induction). (Contributed
|
|
by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) $)
|
|
sqrt2irrlem $p |- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) ) $=
|
|
( c2 cdiv co cz wcel cn cexp cmul cc0 wbr oveq1d nncnd eqtrd syl csqrt cr
|
|
cfv cle wceq 2re 0le2 resqrtth mp2an syl5eqr zcnd nnap0d sqdivapd nnsqcld
|
|
sqcld divcanap1d 2cnd cap 2ap0 a1i divcanap3d eqtr3d eqeltrd nnzd wb zesq
|
|
mpbird clt 2cn sqvali oveq2i divdivap1d 3eqtr4a eqeltrrd rphalfcld rpgt0d
|
|
zsqcl nnrpd elnnz sylanbrc nnesq jca ) ABGHIZJKZCGHILKZAWDBGMIZGHIZJKZAWG
|
|
AWGCGMIZLAGWINIZGHIWGWIAWJWFGHAWJWFWIHIZWINIWFAGWKWINAGBCHIZGMIZWKAGGUAUC
|
|
ZGMIZWMGUBKOGUDPWOGUEUFUGGUHUIAWNWLGMFQUJABCABDUKZACERACEULUMSQAWFWIABWPU
|
|
OZAWIACEUNZRZAWIWRULUPSQAWIGWSAUQZGOURPAUSUTZVAVBZWRVCVDABJKWDWHVEDBVFTVG
|
|
ZAWEWIGHIZLKZAXDJKOXDVHPXEAWCGMIZXDJAXFWGGHIZXDAWFGGMIZHIWFGGNIZHIXFXGXHX
|
|
IWFHGVIVJVKABGWPWTXAUMAWFGGWQWTWTXAXAVLVMAWGWIGHXBQSAWDXFJKXCWCVQTVNAXDAW
|
|
IAWIWRVRVOVPXDVSVTACLKWEXEVEECWATVGWB $.
|
|
$}
|
|
|
|
${
|
|
$d n x y z $.
|
|
$( The square root of 2 is not rational. That is, for any rational number,
|
|
` ( sqrt `` 2 ) ` does not equal it. However, if we were to say "the
|
|
square root of 2 is irrational" that would mean something stronger:
|
|
"for any rational number, ` ( sqrt `` 2 ) ` is apart from it" (the two
|
|
statements are equivalent given excluded middle). See ~ sqrt2irrap for
|
|
the proof that the square root of two is irrational.
|
|
|
|
The proof's core is proven in ~ sqrt2irrlem , which shows that if
|
|
` A / B = sqrt ( 2 ) ` , then ` A ` and ` B ` are even, so ` A / 2 ` and
|
|
` B / 2 ` are smaller representatives, which is absurd. (Contributed by
|
|
NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) $)
|
|
sqrt2irr $p |- ( sqrt ` 2 ) e/ QQ $=
|
|
( vx vy vz c2 wcel cv cdiv co wceq cz wrex cn c1 clt wbr wne wral ralbidv
|
|
wi wa vn csqrt cfv cq caddc peano2nn breq2 imbi1d nnnlt1 pm2.21d rgen crp
|
|
wn nnrp rphalflt breq1 oveq2 neeq2d imbi12d rspcv com13 simpr cc ad2antlr
|
|
syl zcn nncn ad2antrr 2cnd cc0 nnap0 2ap0 divcanap7d eqtr4d simplr simpll
|
|
cap a1i sqrt2irrlem simprd simpld oveq1 embantd necon2bd mpd ex ralrimdva
|
|
necon2ad cbvralv syl6ibr ceqsralv sylibrd ancld wo wb cle nnleltp1 zleloe
|
|
syld nnz syl2an bitr3d ancoms jaob syl6bb ralbidva nnind nnre ltp1d df-ne
|
|
r19.26 ralnex mp2d nrex elq rexcom bitri mtbir nelir ) DUBUCZUDXTUDEZXTAF
|
|
ZBFZGHZIZAJKZBLKZYFBLYCLEZCFZYCMUEHZNOZXTYBYIGHZPZAJQZSZCLQZYCYJNOZYFUMZY
|
|
HYJLEYPYCUFYIUAFZNOZYNSZCLQYIMNOZYNSZCLQYIYCNOZYNSZCLQZYPYPUABYJYSMIZUUAU
|
|
UCCLUUGYTUUBYNYSMYINUGUHRYSYCIZUUAUUECLUUHYTUUDYNYSYCYINUGUHRYSYJIZUUAYOC
|
|
LUUIYTYKYNYSYJYINUGUHRZUUJUUCCLYILEZUUBYNYIUIUJUKYHUUFUUFYIYCIZYNSZCLQZTZ
|
|
YPYHUUFUUNYHUUFXTYDPZAJQZUUNYHUUFXTYIYCGHZPZCJQZUUQYHUUFYCDGHZLEZXTYBUVAG
|
|
HZPZAJQZSZUUTYHUVAYCNOZUUFUVFSYHYCULEUVGYCUNYCUOVEUVBUUFUVGUVEUUEUVGUVESC
|
|
UVALYIUVAIZUUDUVGYNUVEYIUVAYCNUPUVHYMUVDAJUVHYLUVCXTYIUVAYBGUQURRUSUTVAVE
|
|
YHUVFUUSCJYHYIJEZTZUVFXTUURUVJXTUURIZUVFUMZUVJUVKTZXTYIDGHZUVAGHZIUVLUVMX
|
|
TUURUVOUVJUVKVBZUVMYIYCDUVIYIVCEYHUVKYIVFVDYHYCVCEUVIUVKYCVGVHUVMVIYHYCVJ
|
|
VQOUVIUVKYCVKVHDVJVQOUVMVLVRVMVNUVMUVFXTUVOUVMUVBUVEXTUVOPZUVMUVNJEZUVBUV
|
|
MYIYCYHUVIUVKVOYHUVIUVKVPUVPVSZVTUVMUVRUVEUVQSUVMUVRUVBUVSWAUVDUVQAUVNJYB
|
|
UVNIUVCUVOXTYBUVNUVAGWBURUTVEWCWDWEWFWHWGWSUUPUUSACJYBYIIYDUURXTYBYIYCGWB
|
|
URWIWJYNUUQCYCLUULYMUUPAJUULYLYDXTYIYCYBGUQURZRWKWLWMYHYPUUEUUMTZCLQUUOYH
|
|
YOUWACLYHUUKTZYOUUDUULWNZYNSUWAUWBYKUWCYNUUKYHYKUWCWOUUKYHTYIYCWPOZYKUWCY
|
|
IYCWQUUKUVIYCJEUWDUWCWOYHYIWTYCWTYIYCWRXAXBXCUHUUDYNUULXDXEXFUUEUUMCLXKXE
|
|
WLXGVEYHYCYCXHXIYOYQYRSCYCLUULYKYQYNYRYIYCYJNUPUULYNYEUMZAJQYRUULYMUWEAJU
|
|
ULYMUUPUWEUVTXTYDXJXERYEAJXLXEUSUTXMXNYAYEBLKAJKYGABXTXOYEABJLXPXQXRXS $.
|
|
$}
|
|
|
|
$( The square root of 2 exists and is a real number. (Contributed by NM,
|
|
3-Dec-2004.) $)
|
|
sqrt2re $p |- ( sqrt ` 2 ) e. RR $=
|
|
( c2 2re 2pos sqrtpclii ) ABCD $.
|
|
|
|
${
|
|
$d A w $. $d N k m w $.
|
|
$( Lemma for ~ pw2dvds . If a natural number has some power of two which
|
|
does not divide it, there is a highest power of two which does divide
|
|
it. (Contributed by Jim Kingdon, 14-Nov-2021.) $)
|
|
pw2dvdslemn $p |- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) ->
|
|
E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) $=
|
|
( cn wcel c2 cexp co cdvds wbr wn wa c1 caddc cn0 oveq2 breq1d notbid cc0
|
|
wceq vw vk w3a cv wrex 3simpb wi anbi2d imbi1d 0nn0 a1i wb oveq2d anbi12d
|
|
oveq1 adantl 2cnd exp0d cz simpl nnzd 1dvds syl simpr 0p1e1 oveq2i breq1i
|
|
eqbrtrd sylnibr jca rspcedvd simpll nnnn0d simplrr adantllr simprl anim1i
|
|
simpllr mpd wdc 2nn nnexpcl sylancr dvdsdc syl2anc exmiddc mpjaodan exp31
|
|
wo nnind 3ad2ant2 ) CDEZADEZFAGHZCIJZKZUCWLWPLZFBUDZGHZCIJZFWRMNHZGHZCIJZ
|
|
KZLZBOUEZWLWMWPUFWMWLWQXFUGZWPWLFUAUDZGHZCIJZKZLZXFUGWLFMGHZCIJZKZLZXFUGW
|
|
LFUBUDZGHZCIJZKZLZXFUGZWLFXQMNHZGHZCIJZKZLZXFUGXGUAUBAXHMTZXLXPXFYHXKXOWL
|
|
YHXJXNYHXIXMCIXHMFGPQRUHUIXHXQTZXLYAXFYIXKXTWLYIXJXSYIXIXRCIXHXQFGPQRUHUI
|
|
XHYCTZXLYGXFYJXKYFWLYJXJYEYJXIYDCIXHYCFGPQRUHUIXHATZXLWQXFYKXKWPWLYKXJWOY
|
|
KXIWNCIXHAFGPQRUHUIXPXEFSGHZCIJZFSMNHZGHZCIJZKZLZBSOSOEXPUJUKWRSTZXEYRULX
|
|
PYSWTYMXDYQYSWSYLCIWRSFGPQYSXCYPYSXBYOCIYSXAYNFGWRSMNUOUMQRUNUPXPYMYQXPYL
|
|
MCIXPFXPUQURXPCUSEZMCIJXPCWLXOUTVACVBVCVHXPXNYPWLXOVDYOXMCIYNMFGVEVFVGVIV
|
|
JVKXQDEZYBYGXFUUAYBLZYGLZXSXFXTUUAYGXSXFYBUUAYGLZXSLZXEXSYFLZBXQOUUEXQUUA
|
|
YGXSVLVMWRXQTZXEUUFULUUEUUGWTXSXDYFUUGWSXRCIWRXQFGPQUUGXCYEUUGXBYDCIUUGXA
|
|
YCFGWRXQMNUOUMQRUNUPUUEXSYFUUDXSVDUUAWLYFXSVNVJVKVOUUCXTLYAXFUUCWLXTUUBWL
|
|
YFVPZVQUUAYBYGXTVRVSUUCXSVTZXSXTWIUUCXRDEZYTUUIUUCFDEXQOEUUJWAUUCXQUUAYBY
|
|
GVLVMFXQWBWCUUCCUUHVAXRCWDWEXSWFVCWGWHWJWKVS $.
|
|
$}
|
|
|
|
${
|
|
$d N m $.
|
|
$( A natural number has a highest power of two which divides it.
|
|
(Contributed by Jim Kingdon, 14-Nov-2021.) $)
|
|
pw2dvds $p |- ( N e. NN ->
|
|
E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) $=
|
|
( cn wcel c2 co cdvds wbr wn cv c1 caddc wa cn0 sylancr cz cle 2z syl2anc
|
|
cexp wrex id cmin cfz 2nn nnnn0 nnexpcl 1zzd zexpcl zsubcld nnz nnge1 clt
|
|
cuz cfv uzid ax-mp bernneq3 wb mpbid elfz4 syl32anc fzm1ndvds pw2dvdslemn
|
|
zltlem1 mpd3an23 ) BCDZVGEBTFZBGHIZEAJZTFBGHEVJKLFTFBGHIMANUAVGUBVGVHCDZB
|
|
KVHKUCFZUDFDZVIVGECDBNDZVKUEBUFZEBUGOVGKPDVLPDBPDZKBQHBVLQHZVMVGUHZVGVHKV
|
|
GEPDZVNVHPDZRVOEBUIOZVRUJBUKZBULVGBVHUMHZVQVGEEUNUODZVNWCVSWDREUPUQVOEBUR
|
|
OVGVPVTWCVQUSWBWABVHVESUTBKVLVAVBVHBVCSBABVDVF $.
|
|
$}
|
|
|
|
${
|
|
pw2dvdseulemle.n $e |- ( ph -> N e. NN ) $.
|
|
pw2dvdseulemle.a $e |- ( ph -> A e. NN0 ) $.
|
|
pw2dvdseulemle.b $e |- ( ph -> B e. NN0 ) $.
|
|
pw2dvdseulemle.2a $e |- ( ph -> ( 2 ^ A ) || N ) $.
|
|
pw2dvdseulemle.n2b $e |- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N ) $.
|
|
$( Lemma for ~ pw2dvdseu . Powers of two which do and do not divide a
|
|
natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) $)
|
|
pw2dvdseulemle $p |- ( ph -> A <_ B ) $=
|
|
( nn0red wbr c2 co cexp cdvds cn0 wcel adantr cz nnzd clt caddc cmin cmul
|
|
c1 wa 2cnd cle peano2nn0 simpr wb nn0ltp1le syl2anc mpbid nn0sub2 syl3anc
|
|
syl expaddd nn0cnd pncan3d oveq2d eqbrtrd eqbrtrrd wi cn 2nn a1i nnexpcld
|
|
muldvds1 mpd mtand nltled ) ABCABFJACGJACBUAKZLCUEUBMZNMZDOKZIAVMUFZVOLBV
|
|
NUCMZNMZUDMZDOKZVPVQLVNVRUBMZNMZVTDOVQLVNVRVQUGVQVNPQZBPQZVNBUHKZVRPQVQCP
|
|
QZWDAWGVMGRZCUIUQZAWEVMFRZVQVMWFAVMUJVQWGWEVMWFUKWHWJCBULUMUNVNBUOUPZWIUR
|
|
VQWCLBNMZDOVQWBBLNVQVNBVQVNWIUSVQBWJUSUTVAAWLDOKVMHRVBVCVQVOSQVSSQDSQWAVP
|
|
VDVQVOVQLVNLVEQVQVFVGZWIVHTVQVSVQLVRWMWKVHTVQDADVEQVMERTVOVSDVIUPVJVKVL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d N m x $.
|
|
$( A natural number has a unique highest power of two which divides it.
|
|
(Contributed by Jim Kingdon, 16-Nov-2021.) $)
|
|
pw2dvdseu $p |- ( N e. NN ->
|
|
E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) $=
|
|
( vx wcel c2 cv cexp co cdvds wbr c1 caddc wn cn0 wral cle pw2dvdseulemle
|
|
wa nn0red breq1d cn wrex wrmo wreu pw2dvds weq wi simplrl simplrr simprll
|
|
simpll simprrr simprrl simprlr mpbir2and ex ralrimivva oveq2 oveq1 oveq2d
|
|
letri3d notbid anbi12d rmo4 sylibr reu5 sylanbrc ) BUADZEAFZGHZBIJZEVIKLH
|
|
ZGHZBIJZMZRZANUBVPANUCZVPANUDABUEVHVPECFZGHZBIJZEVRKLHZGHZBIJZMZRZRZACUFZ
|
|
UGZCNOANOVQVHWHACNNVHVINDZVRNDZRZRZWFWGWLWFRZWGVIVRPJVRVIPJWMVIVRBVHWKWFU
|
|
KZVHWIWJWFUHZVHWIWJWFUIZWLVKVOWEUJWLVPVTWDULQWMVRVIBWNWPWOWLVPVTWDUMWLVKV
|
|
OWEUNQWMVIVRWMVIWOSWMVRWPSVAUOUPUQVPWEACNWGVKVTVOWDWGVJVSBIVIVREGURTWGVNW
|
|
CWGVMWBBIWGVLWAEGVIVRKLUSUTTVBVCVDVEVPANVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d A z $. $d Y z $.
|
|
$( Lemma for ~ oddpwdc . Another way of stating that decomposing a natural
|
|
number into a power of two and an odd number is unique. (Contributed by
|
|
Jim Kingdon, 16-Nov-2021.) $)
|
|
oddpwdclemxy $p |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 )
|
|
/\ A = ( ( 2 ^ Y ) x. X ) ) ->
|
|
( X = ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A
|
|
/\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) )
|
|
/\ Y = ( iota_ z e. NN0 ( ( 2 ^ z ) || A
|
|
/\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) $=
|
|
( cn wcel c2 cdvds wbr wn wa cn0 cexp co cmul wceq c1 cz nnzd nncnd caddc
|
|
cv crio cdiv 2nn a1i simplll simplr nnexpcld simpr eqeltrd mulcomd eqtr2d
|
|
nnmulcld dvds0lem simpllr breq2d cc0 wne nnne0d dvdscmulr syl112anc bitrd
|
|
syl31anc wb mtbird expp1d breq1d wreu pw2dvdseu oveq2 oveq1 oveq2d notbid
|
|
syl anbi12d riota2 syl2anc mpbi2and nnap0d eqcomd oveq1d mvllmulapd jca )
|
|
CEFZGCHIZJZKZDLFZKZBGDMNZCONZPZKZCBGGAUBZMNZBHIZGWOQUANZMNZBHIZJZKZALUCZM
|
|
NZUDNPDXCPWNXDCBWNXDWNGXCGEFWNUEUFZWNXCDLWNWKBHIZGDQUANZMNZBHIZJZXCDPZWNC
|
|
RFZWKRFZBRFCWKONZBPXFWNCWEWGWIWMUGZSZWNWKWNGDXEWHWIWMUHZUIZSZWNBWNBWLEWJW
|
|
MUJZWNWKCXRXOUNUKZSWNBWLXNXTWNWKCWNWKXRTWNCXOTZULUMCWKBUOVDWNXIWKGONZBHIZ
|
|
WNYDWFWEWGWIWMUPWNYDYCWLHIZWFWNBWLYCHXTUQWNGRFXLXMWKURUSYEWFVEWNGXESXPXSW
|
|
NWKXRUTWKGCVAVBVCVFWNXHYCBHWNGDWNGXETXQVGVHVFWNWIXBALVIZXFXJKZXKVEXQWNBEF
|
|
YFYAABVJVOXBYGALDWODPZWQXFXAXJYHWPWKBHWODGMVKVHYHWTXIYHWSXHBHYHWRXGGMWODQ
|
|
UAVLVMVHVNVPVQVRVSZXQUKUIZTYBWNXDYJVTWNBWLXDCONXTWNWKXDCOWNDXCGMWNXCDYIWA
|
|
ZVMWBUMWCYKWD $.
|
|
$}
|
|
|
|
${
|
|
$d A z $.
|
|
$( Lemma for ~ oddpwdc . A natural number is divisible by the highest
|
|
power of two which divides it. (Contributed by Jim Kingdon,
|
|
17-Nov-2021.) $)
|
|
oddpwdclemdvds $p |- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0
|
|
( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A ) $=
|
|
( cn wcel c2 cv cexp co cdvds wbr c1 caddc cn0 wrex crio pw2dvds nfv nfcv
|
|
wn wa nfriota1 nfov nfbr wceq wreu wb pw2dvdseu riota1 syl biimpa simprrl
|
|
oveq2d eqbrtrd exp32 rexlimd mpd ) BCDZEAFZGHZBIJZEURKLHGHBIJSZTZAMNEVBAM
|
|
OZGHZBIJZABPUQVBVEAMUQAQAVDBIAEVCGAERAGRVBAMUAUBAIRABRUCUQURMDZVBVEUQVFVB
|
|
TZTZVDUSBIVHVCUREGUQVGVCURUDZUQVBAMUEVGVIUFABUGVBAMUHUIUJULUQVFUTVAUKUMUN
|
|
UOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A z $.
|
|
$( Lemma for ~ oddpwdc . A natural number is not divisible by one more
|
|
than the highest power of two which divides it. (Contributed by Jim
|
|
Kingdon, 17-Nov-2021.) $)
|
|
oddpwdclemndvds $p |- ( A e. NN -> -. ( 2 ^ ( ( iota_ z e. NN0
|
|
( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A ) $=
|
|
( cn wcel c2 cv cexp co cdvds wbr c1 caddc wn wrex crio pw2dvds nfcv nfov
|
|
wa cn0 nfv nfriota1 nfbr nfn simprrr wceq wreu wb pw2dvdseu riota1 biimpa
|
|
syl oveq1d oveq2d breq1d mtbird exp32 rexlimd mpd ) BCDZEAFZGHBIJZEVAKLHZ
|
|
GHZBIJZMZSZATNEVGATOZKLHZGHZBIJZMZABPUTVGVLATUTAUAVKAAVJBIAEVIGAEQAGQAVHK
|
|
LVGATUBALQAKQRRAIQABQUCUDUTVATDZVGVLUTVMVGSZSZVKVEUTVMVBVFUEVOVJVDBIVOVIV
|
|
CEGVOVHVAKLUTVNVHVAUFZUTVGATUGVNVPUHABUIVGATUJULUKUMUNUOUPUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d A z $.
|
|
$( Lemma for ~ oddpwdc . Removing the powers of two from a natural number
|
|
produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) $)
|
|
oddpwdclemodd $p |- ( A e. NN -> -. 2 ||
|
|
( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\
|
|
-. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) ) $=
|
|
( cn wcel c2 cv cexp co cdvds wbr c1 caddc wn wa cn0 crio cmul mtbid nnzd
|
|
cz cdiv oddpwdclemndvds wceq 2cn wreu pw2dvdseu riotacl syl expp1 sylancr
|
|
cc breq1d nncn 2nn a1i nnexpcld nncnd nnap0d divcanap2d eqcomd breq2d cc0
|
|
wb id oddpwdclemdvds nndivdvds biimpa syl21anc nnne0d dvdscmulr syl112anc
|
|
wne bitrd ) BCDZEEAFZGHBIJEVOKLHGHBIJMNZAOPZGHZEQHZBIJZEBVRUAHZIJZVNEVQKL
|
|
HGHZBIJVTABUBVNWCVSBIVNEUKDVQODZWCVSUCUDVNVPAOUEWDABUFVPAOUGUHZEVQUIUJULR
|
|
VNVTVSVRWAQHZIJZWBVNBWFVSIVNWFBVNBVRBUMVNVRVNEVQECDVNUNUOZWEUPZUQVNVRWIUR
|
|
USUTVAVNETDWATDVRTDVRVBVLWGWBVCVNEWHSVNWAVNVNVRCDZVRBIJZWACDZVNVDWIABVEVN
|
|
WJNWKWLBVRVFVGVHSVNVRWISVNVRWIVIVREWAVJVKVMR $.
|
|
$}
|
|
|
|
${
|
|
$d A z $. $d Y z $.
|
|
$( Lemma for ~ oddpwdc . Decomposing a number into odd and even parts.
|
|
(Contributed by Jim Kingdon, 16-Nov-2021.) $)
|
|
oddpwdclemdc $p |- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 )
|
|
/\ A = ( ( 2 ^ Y ) x. X ) )
|
|
<-> ( A e. NN /\ ( X = ( A / ( 2 ^
|
|
( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) )
|
|
) /\ Y =
|
|
( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) )
|
|
) ) $=
|
|
( cn wcel c2 cdvds wbr wn wa cn0 cexp cmul wceq simpr 2nn eqeltrd adantr
|
|
co cv c1 caddc crio a1i simplr nnexpcld simplll nnmulcld oddpwdclemxy jca
|
|
cdiv oddpwdclemdvds wb wreu pw2dvdseu riotacl syl nndivdvds mpdan adantrr
|
|
mpbid oddpwdclemodd breq2 notbid mpbird simprr oveq2d oveq12d simpl nncnd
|
|
ad2antrl nnap0d divcanap2d eqtr2d jca31 impbii ) CEFZGCHIZJZKZDLFZKZBGDMT
|
|
ZCNTZOZKZBEFZCBGGAUAZMTBHIGWIUBUCTMTBHIJKZALUDZMTZULTZOZDWKOZKZKZWGWHWPWG
|
|
BWEEWCWFPWGWDCWGGDGEFZWGQUEWAWBWFUFUGVRVTWBWFUHUIRABCDUJUKWQWAWBWFWQVRVTW
|
|
HWNVRWOWHWNKCWMEWHWNPZWHWMEFZWNWHWLBHIZWTABUMWHWLEFZXAWTUNWHGWKWRWHQUEWHW
|
|
JALUOWKLFZABUPWJALUQURZUGZBWLUSUTVBSRVAWQVTGWMHIZJZWHXGWPABVCSWNVTXGUNWHW
|
|
OWNVSXFCWMGHVDVEVLVFUKWQDWKLWHWNWOVGZWHXCWPXDSRWQWEWLWMNTBWQWDWLCWMNWQDWK
|
|
GMXHVHWHWNWNWOWSVAVIWQBWLWQBWHWPVJVKWQWLWHXBWPXESZVKWQWLXIVMVNVOVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d a x y z $. $d a F $. $d a x y J $.
|
|
oddpwdc.j $e |- J = { z e. NN | -. 2 || z } $.
|
|
oddpwdc.f $e |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) ) $.
|
|
$( The function ` F ` that decomposes a number into its "odd" and "even"
|
|
parts, which is to say the largest power of two and largest odd divisor
|
|
of a number, is a bijection from pairs of a nonnegative integer and an
|
|
odd number to positive integers. (Contributed by Thierry Arnoux,
|
|
15-Aug-2017.) $)
|
|
oddpwdc $p |- F : ( J X. NN0 ) -1-1-onto-> NN $=
|
|
( cn0 cn wtru c2 cv cexp co cdvds wbr wn wa wcel wceq va cxp wf1o cmul c1
|
|
caddc crio cdiv cc cr 2cnd simpr expcld breq2 notbid elrab2 simplbi nncnd
|
|
adantr mulcld adantl nnnn0 2nn wreu pw2dvdseu riotacl syl nnexpcl sylancr
|
|
nn0nndivcl syl2anc jca wb anbi1i oddpwdclemdc bitri a1i f1od2 trud ) EHUB
|
|
IDUCJABUAEHKBLZMNZALZUDNZIDUALZKKCLZMNWDOPKWEUEUFNMNWDOPQRZCHUGZMNZUHNZWG
|
|
UIUJHGWBESZVTHSZRZWCUISJWLWAWBWLKVTWLUKWJWKULUMWLWBWJWBISZWKWJWMKWBOPZQZK
|
|
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|
|
DHSWHISZWSWDVBWRKISWTXAVCWRWFCHVDWTCWDVEWFCHVFVGZKWGVHVIWDWHVJVKXBVLVAWLW
|
|
DWCTZRZWRWBWITVTWGTRRZVMJXDWMWORZWKRZXCRXEWLXGXCWJXFWKWQVNVNCWDWBVTVOVPVQ
|
|
VRVS $.
|
|
|
|
${
|
|
$d A x y z $. $d F x y z $. $d J x y $.
|
|
$( The greatest power of two dividing the square of an integer is an even
|
|
power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) $)
|
|
sqpweven $p |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) ) $=
|
|
( cn wcel c2 cfv cmul co cexp cdvds wbr cn0 syl wceq ccnv c2nd cxp wf1o
|
|
cz wf oddpwdc f1ocnv f1of ffvelrni xp2nd nn0zd 2nn a1i dvdsmul2 syl2anc
|
|
mp2b nnzd c1st cop wn xp1st breq2 notbid elrab2 simplbi nnsqcld simprbi
|
|
cv cprime wb 2prm w3a euclemma oridm syl6bb mp3an2i mtbird nncnd sqvald
|
|
breq2d sylanbrc nnnn0d nn0mulcld opelxp expmuld nnexpcld nnmulcld oveq2
|
|
wo oveq1d ovmpt2g syl3anc f1ocnvfv2 1st2nd2 fveq2d eqtr3d df-ov syl6eqr
|
|
mpan eqtrd sqmuld 3eqtr4rd syl6req wi f1ocnvfv sylc op2ndg breqtrrd ) D
|
|
IJZKDEUAZLZUBLZKMNZDKONZXKLZUBLZPXJXMUEJKUEJKXNPQXJXMXJXLFRUCZJZXMRJZIX
|
|
RDXKXRIEUDZIXRXKUDIXRXKUFABCEFGHUGZXRIEUHIXRXKUIUQUJZXLFRUKSZULXJKKIJXJ
|
|
UMUNZURXMKUOUPXJXQXLUSLZKONZXNUTZUBLZXNXJXPYHUBXJYHXRJZYHELZXOTZXPYHTZX
|
|
JYGFJZXNRJZYJXJYGIJKYGPQZVAZYNXJYFXJYFFJZYFIJZXJXSYRYCXLFRVBSZYRYSKYFPQ
|
|
ZVAZKCVIZPQZVAZUUBCYFIFUUCYFTUUDUUAUUCYFKPVCVDGVEZVFSZVGZXJYPKYFYFMNZPQ
|
|
ZXJUUJUUAXJYRUUBYTYRYSUUBUUFVHSKVJJZXJYFUEJZUULUUJUUAVKVLXJYFUUGURZUUMU
|
|
UKUULUULVMUUJUUAUUAWJUUAKYFYFVNUUAVOVPVQVRXJYGUUIKPXJYFXJYFUUGVSZVTWAVR
|
|
UUEYQCYGIFUUCYGTUUDYPUUCYGKPVCVDGVEWBZXJXMKYDXJKYEWCZWDZYGXNFRWEWBXJXOY
|
|
GXNENZYKXJKXNONZYGMNZKXMONZKONZYGMNZUURXOXJUUSUVBYGMXJKXMKXJKYEVSUUPYDW
|
|
FWKXJYNYOUUTIJUURUUTTUUOUUQXJUUSYGXJKXNYEUUQWGUUHWHABYGXNFRKBVIZONZAVIZ
|
|
MNZUUTEUVEYGMNIUVFYGUVEMWIUVDXNTUVEUUSYGMUVDXNKOWIWKHWLWMXJXOUVAYFMNZKO
|
|
NUVCXJDUVHKOXJDYFXMENZUVHXJDYFXMUTZELZUVIXJXLELZDUVKYAXJUVLDTYBXRIDEWNW
|
|
TXJXLUVJEXJXSXLUVJTYCXLFRWOSWPWQYFXMEWRWSXJYRXTUVHIJUVIUVHTYTYDXJUVAYFX
|
|
JKXMYEYDWGZUUGWHABYFXMFRUVGUVHEUVEYFMNIUVFYFUVEMWIUVDXMTUVEUVAYFMUVDXMK
|
|
OWIWKHWLWMXAWKXJUVAYFXJUVAUVMVSUUNXBXAXCYGXNEWRXDYAYJYLYMXEYBXRIYHXOEXF
|
|
WTXGWPXJYNYOYIXNTUUOUUQYGXNFRXHUPXAXI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d F x y z $. $d J x y $.
|
|
$( The greatest power of two dividing twice the square of an integer is
|
|
an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) $)
|
|
2sqpwodd $p |- ( A e. NN ->
|
|
-. 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) ) $=
|
|
( cn wcel c2 cexp co cmul cfv cdvds wbr cn0 syl wceq ccnv c2nd c1 caddc
|
|
cz wn cxp wf1o wf oddpwdc f1ocnv f1of mp2b ffvelrni xp2nd nn0zd 2nn a1i
|
|
nnzd zmulcld dvdsmul2 syl2anc oddp1even biimprd con2d sylc cop xp1st cv
|
|
c1st breq2 notbid elrab2 simplbi nnsqcld simprbi cprime wb w3a euclemma
|
|
2prm wo oridm syl6bb mp3an2i mtbird nncnd sqvald breq2d sylanbrc nnnn0d
|
|
nn0mulcld peano2nn0 opelxp expp1d expcld mulcomd oveq2d 3eqtrd nnexpcld
|
|
expmuld oveq1d nnmulcld ovmpt2g syl3anc f1ocnvfv2 1st2nd2 fveq2d eqtr3d
|
|
oveq2 mpan df-ov syl6eqr eqtrd sqmuld eqeltrrd mulassd 3eqtr4rd syl6req
|
|
cc eqtr4d wi f1ocnvfv op2ndg ) DIJZKKDKLMZNMZEUAZOZUBOZPQKDYHOZUBOZKNMZ
|
|
UCUDMZPQZYEYMUEJZKYMPQZYOUFYEYLKYEYLYEYKFRUGZJZYLRJZIYRDYHYRIEUHZIYRYHU
|
|
HIYRYHUIABCEFGHUJZYRIEUKIYRYHULUMUNZYKFRUOSZUPZYEKKIJYEUQURZUSZUTYEYLUE
|
|
JKUEJYQUUEUUGYLKVAVBYPYOYQYPYQUFYOYMVCVDVEVFYEYJYNKPYEYJYKVJOZKLMZYNVGZ
|
|
UBOZYNYEYIUUJUBYEUUJYRJZUUJEOZYGTZYIUUJTZYEUUIFJZYNRJZUULYEUUIIJKUUIPQZ
|
|
UFZUUPYEUUHYEUUHFJZUUHIJZYEYSUUTUUCYKFRVHSZUUTUVAKUUHPQZUFZKCVIZPQZUFZU
|
|
VDCUUHIFUVEUUHTUVFUVCUVEUUHKPVKVLGVMZVNSZVOZYEUURKUUHUUHNMZPQZYEUVLUVCY
|
|
EUUTUVDUVBUUTUVAUVDUVHVPSKVQJZYEUUHUEJZUVNUVLUVCVRWAYEUUHUVIUSZUVOUVMUV
|
|
NUVNVSUVLUVCUVCWBUVCKUUHUUHVTUVCWCWDWEWFYEUUIUVKKPYEUUHYEUUHUVIWGZWHWIW
|
|
FUVGUUSCUUIIFUVEUUITUVFUURUVEUUIKPVKVLGVMWJZYEYMRJUUQYEYLKUUDYEKUUFWKZW
|
|
LZYMWMSZUUIYNFRWNWJYEYGUUIYNEMZUUMYEKYNLMZUUINMZKKYLLMZKLMZNMZUUINMZUWA
|
|
YGYEUWBUWFUUINYEUWBKYMLMZKNMKUWHNMUWFYEKYMYEKUUFWGZUVSWOYEUWHKYEKYMUWIU
|
|
VSWPZUWIWQYEUWHUWEKNYEKYLKUWIUVRUUDXAZWRWSXBYEUUPUUQUWCIJUWAUWCTUVQUVTY
|
|
EUWBUUIYEKYNUUFUVTWTUVJXCABUUIYNFRKBVIZLMZAVIZNMZUWCEUWMUUINMIUWNUUIUWM
|
|
NXJUWLYNTUWMUWBUUINUWLYNKLXJXBHXDXEYEYGKUWEUUINMZNMUWGYEYFUWPKNYEYFUWDU
|
|
UHNMZKLMUWPYEDUWQKLYEDUUHYLEMZUWQYEDUUHYLVGZEOZUWRYEYKEOZDUWTUUAYEUXADT
|
|
UUBYRIDEXFXKYEYKUWSEYEYSYKUWSTUUCYKFRXGSXHXIUUHYLEXLXMYEUUTYTUWQIJUWRUW
|
|
QTUVBUUDYEUWDUUHYEKYLUUFUUDWTZUVIXCABUUHYLFRUWOUWQEUWMUUHNMIUWNUUHUWMNX
|
|
JUWLYLTUWMUWDUUHNUWLYLKLXJXBHXDXEXNXBYEUWDUUHYEUWDUXBWGUVPXOXNWRYEKUWEU
|
|
UIUWIYEUWHUWEXTUWKUWJXPYEUUIUVJWGXQYAXRUUIYNEXLXSUUAUULUUNUUOYBUUBYRIUU
|
|
JYGEYCXKVFXHYEUUPUUQUUKYNTUVQUVTUUIYNFRYDVBXNWIWF $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A a b c x y z $. $d B a b c x y z $.
|
|
$( The square of a natural number can never be equal to two times the
|
|
square of a natural number. (Contributed by Jim Kingdon,
|
|
17-Nov-2021.) $)
|
|
sqne2sq $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( A ^ 2 ) =/= ( 2 x. ( B ^ 2 ) ) ) $=
|
|
( va vb vc vx vy vz cn c2 cexp co cmul wceq cv cdvds wbr wn cfv c2nd wcel
|
|
wa crab cn0 cmpt2 ccnv breq2 notbid oveq2 oveq1d cbvmpt2v 2sqpwodd adantl
|
|
cbvrabv sqpweven ad2antrr wb fveq2 fveq2d breq2d mpbid mtand neqned ) AIU
|
|
AZBIUAZUBZAJKLZJBJKLMLZVFVGVHNZJVHCDJEOZPQZRZEIUCZUDJDOZKLZCOZMLZUEZUFZSZ
|
|
TSZPQZVEWBRVDFGHBVRVMVLJHOZPQZREHIVJWCNVKWDVJWCJPUGUHUNZCDFGVMUDVQJGOZKLZ
|
|
FOZMLVOWHMLVPWHVOMUIVNWFNVOWGWHMVNWFJKUIUJUKZULUMVFVIUBJVGVSSZTSZPQZWBVDW
|
|
LVEVIFGHAVRVMWEWIUOUPVIWLWBUQVFVIWKWAJPVIWJVTTVGVHVSURUSUTUMVAVBVC $.
|
|
$}
|
|
|
|
$( The absolute value of the difference between two unequal integers is at
|
|
least one. (Contributed by Jim Kingdon, 31-Jan-2022.) $)
|
|
znege1 $p |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B )
|
|
-> 1 <_ ( abs ` ( A - B ) ) ) $=
|
|
( cz wcel clt wbr c1 cmin co cle wa caddc zltp1le 3adant3 simpl1 zred mpbid
|
|
wb 1red simpr wne w3a cabs cfv wceq biimpa simpl2 leaddsub2d ltled breqtrrd
|
|
abssuble0d simpl3 pm2.21ddne syl2anc abssubge0d w3o ztri3or mpjao3dan ) ACD
|
|
ZBCDZABUAZUBZABEFZGABHIZUCUDZJFZABUEZBAEFZVBVCKZGBAHIZVEJVIAGLIBJFZGVJJFVBV
|
|
CVKUSUTVCVKRVAABMNUFVIAGBVIAUSUTVAVCOPZVISVIBUSUTVAVCUGPZUHQVIABVLVMVIABVLV
|
|
MVBVCTUIUKUJVBVGKVFABVBVGTUSUTVAVGULUMVBVHKZGVDVEJVNBGLIAJFZGVDJFVNVHVOVBVH
|
|
TZVNUTUSVHVORUSUTVAVHUGZUSUTVAVHOZBAMUNQVNBGAVNBVQPZVNSVNAVRPZUHQVNBAVSVTVN
|
|
BAVSVTVPUIUOUJUSUTVCVGVHUPVAABUQNUR $.
|
|
|
|
$( Lemma for ~ sqrt2irrap . The square root of 2 is apart from a positive
|
|
rational expressed as a numerator and denominator. (Contributed by Jim
|
|
Kingdon, 2-Oct-2021.) $)
|
|
sqrt2irraplemnn $p |- ( ( A e. NN /\ B e. NN ) ->
|
|
( sqrt ` 2 ) # ( A / B ) ) $=
|
|
( cn wcel cdiv co c2 csqrt cfv cap wbr nnred nngt0d ltled mpbird cq syl2anc
|
|
cc0 wb cz wa cexp simpl nnsqcld 0red simpr nnrpd sqrtdivd sqrtsqd eqtrd wne
|
|
oveq12d cmul sqne2sq nncnd 2cnd nnap0d divmulap3d necon3bid nnzd 2z zq mp1i
|
|
znq qapne cr cle qre syl divgt0d 2re a1i 0le2 sqrt11ap syl22anc eqbrtrrd cc
|
|
nnz sylan qcn sqrt2re recni apsym mpbid ) ACDZBCDZUAZABEFZGHIZJKZWIWHJKZWGA
|
|
GUBFZBGUBFZEFZHIZWHWIJWGWOWLHIZWMHIZEFWHWGWLWMWGWLWGAWEWFUCZUDZLZWGRWLWGUEZ
|
|
WTWGWLWSMZNWGWMWGBWEWFUFZUDZUGUHWGWPAWQBEWGAWGAWRLZWGRAXAXEWGAWRMNUIWGBWGBX
|
|
CLZWGRBXAXFWGBXCMNUIULUJWGWOWIJKZWNGJKZWGXHWNGUKZWGXIWLGWMUMFZUKABUNWGWNGWL
|
|
XJWGWLGWMWGWLWSUOWGUPWGWMXDUOWGWMXDUQURUSOWGWNPDZGPDZXHXISWGWLTDWMCDXKWGWLW
|
|
SUTXDWLWMVDQZGTDXLWGVAGVBVCWNGVEQOWGWNVFDZRWNVGKGVFDZRGVGKZXGXHSWGXKXNXMWNV
|
|
HVIZWGRWNXAXQWGWLWMWTWGWMXDLXBWGWMXDMVJNXOWGVKVLXPWGVMVLWNGVNVOOVPWGWHVQDZW
|
|
IVQDZWJWKSWGWHPDZXRWEATDWFXTAVRABVDVSWHVTVIXSWGWIWAWBVLWHWIWCQWD $.
|
|
|
|
${
|
|
$d Q a b $.
|
|
$( The square root of 2 is irrational. That is, for any rational number,
|
|
` ( sqrt `` 2 ) ` is apart from it. In the absence of excluded middle,
|
|
we can distinguish between this and "the square root of 2 is not
|
|
rational" which is ~ sqrt2irr . (Contributed by Jim Kingdon,
|
|
2-Oct-2021.) $)
|
|
sqrt2irrap $p |- ( Q e. QQ -> ( sqrt ` 2 ) # Q ) $=
|
|
( va vb cq wcel cv cdiv co cn cz cap wbr wa cc0 cle cr adantr a1i simpr
|
|
c1 wceq wrex csqrt cfv elq biimpi clt simplrl simplrr znq qre syl syl2anc
|
|
c2 sqrt2re 0red cmul zcnd nncnd nnap0d divrecapd zred nnrecred 1red nnrpd
|
|
0le1 divge0d mulle0r syl22anc eqbrtrd 2re 2pos sqrtgt0ii lelttrd sylanbrc
|
|
gtapd elnnz sqrt2irraplemnn wo zlelttric mpan2 ad2antrl mpjaodan breqtrrd
|
|
0z ex rexlimdvva mpd ) ADEZABFZCFZGHZUAZCIUBBJUBZUNUCUDZAKLZWIWNBCAUEUFWI
|
|
WMWPBCJIWIWJJEZWKIEZMZMZWMWPWTWMMZWOWLAKXAWJNOLZWOWLKLZNWJUGLZXAXBMZWLWOX
|
|
EWQWRWLPEZXAWQXBWIWQWRWMUHZQZXAWRXBWIWQWRWMUIZQZWSWLDEXFWJWKUJWLUKULUMZWO
|
|
PEXEUORZXEWLNWOXKXEUPXLXEWLWJTWKGHZUQHZNOXEWJWKXEWJXHURXEWKXJUSXEWKXJUTVA
|
|
XEWJPEXMPEXBNXMOLXNNOLXEWJXHVBXEWKXJVCXAXBSXETWKXEVDXEWKXJVENTOLXEVFRVGWJ
|
|
XMVHVIVJNWOUGLXEUNVKVLVMRVNVPXAXDMZWJIEZWRXCXOWQXDXPXAWQXDXGQXAXDSWJVQVOX
|
|
AWRXDXIQWJWKVRUMWTXBXDVSZWMWQXQWIWRWQNJEXQWEWJNVTWAWBQWCWTWMSWDWFWGWH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
GUIDES AND MISCELLANEA
|
|
###############################################################################
|
|
$)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Guides (conventions, explanations, and examples)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Conventions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
This section describes the conventions we use. However, these conventions
|
|
often refer to existing mathematical practices, which are discussed in more
|
|
detail in other references.
|
|
|
|
The following sources lay out how mathematics is developed without the law
|
|
of the excluded middle. Of course, there are a greater number of sources
|
|
which assume excluded middle and most of what is in them applies here too
|
|
(especially in a treatment such as ours which is built on first order logic
|
|
and set theory, rather than, say, type theory). Studying how a topic is
|
|
treated in the Metamath Proof Explorer and the references therein is often a
|
|
good place to start (and is easy to compare with the Intuitionistic Logic
|
|
Explorer).
|
|
|
|
The textbooks provide a motivation for what we are doing, whereas Metamath
|
|
lets you see in detail all hidden and implicit steps. Most standard theorems
|
|
are accompanied by citations. Some closely followed texts include the
|
|
following:
|
|
|
|
<HTML><ul>
|
|
<li>Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or
|
|
[Heyting].</li>
|
|
<li>Axioms of predicate calculus - our axioms are adapted from the ones in
|
|
the Metamath Proof Explorer.</li>
|
|
<li>Theorems of propositional calculus - [Heyting].</li>
|
|
<li>Theorems of pure predicate calculus - Metamath Proof Explorer.</li>
|
|
<li>Theorems of equality and substitution - Metamath Proof Explorer.</li>
|
|
<li>Axioms of set theory - [Crosilla].</li>
|
|
<li>Development of set theory - Chapter 10 of [HoTT].</li>
|
|
<li>Construction of real and complex numbers - Chapter 11 of [HoTT];
|
|
[BauerTaylor].</li>
|
|
<li>Theorems about real numbers - [Geuvers].</li></ul></HTML>
|
|
|
|
$)
|
|
|
|
${
|
|
$( Dummy premise for "conventions" $)
|
|
conventions.1 $e |- ph $.
|
|
$( Unless there is a reason to diverge, we follow the conventions of the
|
|
Metamath Proof Explorer (aka "set.mm"). This list of conventions is
|
|
intended to be read in conjunction with the corresponding conventions in
|
|
the Metamath Proof Explorer, and only the differences are described
|
|
below.
|
|
|
|
<HTML>
|
|
<br>
|
|
<ul>
|
|
|
|
<li><b>Minimizing axioms and the axiom of choice.</b>
|
|
We prefer proofs that depend on fewer and/or weaker axioms,
|
|
even if the proofs are longer. In particular, our choice of IZF
|
|
(Intuitionistic Zermelo-Fraenkel) over CZF (Constructive
|
|
Zermelo-Fraenkel, a weaker system) was just an expedient choice
|
|
because IZF is easier to formalize in Metamath. You can find some
|
|
development using CZF in BJ's mathbox starting at ~ wbd (and the
|
|
section header just above it).
|
|
As for the axiom of choice, the full axiom of choice implies
|
|
excluded middle as seen at ~ acexmid , although some authors will
|
|
use countable choice or dependent choice. For example, countable
|
|
choice or excluded middle is needed to show that the Cauchy reals
|
|
coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT],
|
|
p. (varies).</li>
|
|
|
|
<li><b>Junk/undefined results.</b>
|
|
Much of the discussion of this topic in the Metamath Proof
|
|
Explorer applies except that certain techniques are not
|
|
available to us. For example, the Metamath Proof Explorer will
|
|
often say "if a function is evaluated within its domain, a
|
|
certain result follows; if the function is evaluated outside
|
|
its domain, the same result follows. Since the function must
|
|
be evaluated within its domain or outside it, the result follows
|
|
unconditionally" (the use of excluded middle in this argument is
|
|
perhaps obvious when stated this way). For this reason, we
|
|
generally need to prove we are evaluating functions within
|
|
their domains and avoid the reverse closure theorems of the
|
|
Metamath Proof Explorer.</li>
|
|
|
|
<li><b>Bibliography references.</b>
|
|
The bibliography for the Intuitionistic Logic Explorer is
|
|
separate from the one for the Metamath Proof Explorer but feel
|
|
free to copy-paste a citation in either direction in order to
|
|
cite it.</li>
|
|
|
|
</ul></HTML>
|
|
|
|
<HTML><b>Label naming conventions</b></HTML>
|
|
|
|
Here are a few of the label naming conventions:
|
|
|
|
<HTML><ul>
|
|
|
|
<li><b>Suffixes.</b>
|
|
We follow the conventions of the Metamath Proof Explorer with a few
|
|
additions. A biconditional in set.mm which is an implication in
|
|
iset.mm should have a "r" (for the reverse direction), or "i"/"im"
|
|
(for the forward direction) appended. A theorem in set.mm which has
|
|
a decidability condition added should add "dc" to the theorem name.
|
|
A theorem in set.mm where "nonempty class" is changed to "inhabited
|
|
class" should add "m" (for member) to the theorem name.</li>
|
|
</ul></HTML>
|
|
|
|
<HTML>
|
|
The following table shows some commonly-used abbreviations in labels
|
|
which are not found in the Metamath Proof Explorer,
|
|
in alphabetical order.
|
|
For each abbreviation we provide a mnenomic to help you remember it,
|
|
the source theorem/assumption defining it, an expression showing
|
|
what it looks like, whether or not it is a "syntax fragment"
|
|
(an abbreviation that indicates a particular kind of syntax), and
|
|
hyperlinks to label examples that use the abbreviation.
|
|
The abbreviation is bolded if there is a df-NAME definition but
|
|
the label fragment is not NAME.
|
|
<table border="1" id="naming-abbreviation-table">
|
|
<tr><th>Abbreviation</th><th>Mnenomic</th><th>Source</th>
|
|
<th>Expression</th><th>Syntax?</th><th>Example(s)</th></tr>
|
|
<tr><td>ap</td><td>apart</td><td> ~ df-ap </td>
|
|
<td> </td><td>Yes</td><td> ~ apadd1 , ~ apne </td></tr>
|
|
</table>
|
|
</HTML>
|
|
|
|
<HTML><li><b>Community.</b>
|
|
The Metamath mailing list also covers the Intuitionistic Logic
|
|
Explorer and is at:
|
|
~ https://groups.google.com/forum/#!forum/metamath .
|
|
</li>
|
|
</ul></HTML>
|
|
|
|
(Contributed by Jim Kingdon, 24-Feb-2020.) $)
|
|
conventions $p |- ph $=
|
|
( ) B $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Definitional examples
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Example for ~ ax-io . Example by David A. Wheeler. (Contributed by Mario
|
|
Carneiro, 9-May-2015.) $)
|
|
ex-or $p |- ( 2 = 3 \/ 4 = 4 ) $=
|
|
( c4 wceq c2 c3 eqid olci ) AABCDBAEF $.
|
|
|
|
$( Example for ~ ax-ia1 . Example by David A. Wheeler. (Contributed by
|
|
Mario Carneiro, 9-May-2015.) $)
|
|
ex-an $p |- ( 2 = 2 /\ 3 = 3 ) $=
|
|
( c2 wceq c3 eqid pm3.2i ) AABCCBADCDE $.
|
|
|
|
$( Example for ~ df-dec , 1000 + 2000 = 3000.
|
|
|
|
This proof disproves (by counterexample) the assertion of Hao Wang, who
|
|
stated, "There is a theorem in the primitive notation of set theory that
|
|
corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula
|
|
would be forbiddingly long... even if (one) knows the definitions and is
|
|
asked to simplify the long formula according to them, chances are he will
|
|
make errors and arrive at some incorrect result." (Hao Wang, "Theory and
|
|
practice in mathematics" , In Thomas Tymoczko, editor, _New Directions in
|
|
the Philosophy of Mathematics_, pp 129-152, Birkauser Boston, Inc.,
|
|
Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)
|
|
|
|
This is noted in _Metamath: A Computer Language for Pure Mathematics_ by
|
|
Norman Megill (2007) section 1.1.3. Megill then states, "A number of
|
|
writers have conveyed the impression that the kind of absolute rigor
|
|
provided by Metamath is an impossible dream, suggesting that a complete,
|
|
formal verification of a typical theorem would take millions of steps in
|
|
untold volumes of books... These writers assume, however, that in order
|
|
to achieve the kind of complete formal verification they desire one must
|
|
break down a proof into individual primitive steps that make direct
|
|
reference to the axioms. This is not necessary. There is no reason not
|
|
to make use of previously proved theorems rather than proving them over
|
|
and over... A hierarchy of theorems and definitions permits an
|
|
exponential growth in the formula sizes and primitive proof steps to be
|
|
described with only a linear growth in the number of symbols used. Of
|
|
course, this is how ordinary informal mathematics is normally done anyway,
|
|
but with Metamath it can be done with absolute rigor and precision."
|
|
|
|
The proof here starts with ` ( 2 + 1 ) = 3 ` , commutes it, and repeatedly
|
|
multiplies both sides by ten. This is certainly longer than traditional
|
|
mathematical proofs, e.g., there are a number of steps explicitly shown
|
|
here to show that we're allowed to do operations such as multiplication.
|
|
However, while longer, the proof is clearly a manageable size - even
|
|
though every step is rigorously derived all the way back to the primitive
|
|
notions of set theory and logic. And while there's a risk of making
|
|
errors, the many independent verifiers make it much less likely that an
|
|
incorrect result will be accepted.
|
|
|
|
This proof heavily relies on the decimal constructor ~ df-dec developed by
|
|
Mario Carneiro in 2015. The underlying Metamath language has an
|
|
intentionally very small set of primitives; it doesn't even have a
|
|
built-in construct for numbers. Instead, the digits are defined using
|
|
these primitives, and the decimal constructor is used to make it easy to
|
|
express larger numbers as combinations of digits.
|
|
|
|
(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario
|
|
Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.) $)
|
|
1kp2ke3k $p |- ( ; ; ; 1 0 0 0 + ; ; ; 2 0 0 0 ) = ; ; ; 3 0 0 0 $=
|
|
( c1 cc0 cdc c2 c3 1nn0 0nn0 deccl 2nn0 eqid 1p2e3 00id decadd ) ABCZBCZBDB
|
|
CZBCZBEBCZBCBOBCZQBCZNBABFGHZGHGPBDBIGHZGHGSJTJNBPBRBOQUAGUBGOJQJABDBEBNPFG
|
|
IGNJPJKLMLMLM $.
|
|
|
|
$( Example for ~ df-fl . Example by David A. Wheeler. (Contributed by Mario
|
|
Carneiro, 18-Jun-2015.) $)
|
|
ex-fl $p |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 ) $=
|
|
( c3 c2 co c1 wceq cneg wbr clt 1re 3re 2cn eqbrtri wb 2re ax-mp mpbi wa cz
|
|
wcel mp2an cdiv cfl cfv cle caddc rehalfcli cmul mulid2i 2lt3 cc0 ltmuldivi
|
|
2pos ltleii c4 3lt4 2t2e4 breqtrri pm3.2i ltdivmul mp3an mpbir df-2 breqtri
|
|
cr cq cn 3z 2nn znq 1z flqbi mpbir2an renegcli ltnegi cmin cc negcli ax-1cn
|
|
negdi2 negnegi oveq1i eqtri 2m1e1 readdcli ltnegcon1i qnegcl 2z znegcl ) AB
|
|
UACZUBUCDEZWIFZUBUCBFZEZWJDWIUDGZWIDDUECZHGZDWIIAJUFZDBUGCZAHGZDWIHGZWRBAHB
|
|
KUHUILUJBHGZWSWTMULDABIJNUKOPZUMWIBWOHWIBHGZABBUGCZHGZAUNXDHUOUPUQAVDSBVDSZ
|
|
XFXAQXCXEMJNXFXANULURABBUSUTVAZVBVCWIVESZDRSWJWNWPQMARSBVFSXHVGVHABVITZVJWI
|
|
DVKTVLWMWLWKUDGZWKWLDUECZHGZWLWKBNVMZWIWQVMXCWLWKHGXGWIBWQNVNPUMXKFZWIHGXLX
|
|
NBDVOCZWIHXNWLFZDVOCZXOWLVPSDVPSXNXQEBKVQVRWLDVSTXPBDVOBKVTWAWBXODWIHWCXBLL
|
|
XKWIWLDXMIWDWQWEPWKVESZWLRSZWMXJXLQMXHXRXIWIWFOBRSXSWGBWHOWKWLVKTVLUR $.
|
|
|
|
$( Example for ~ df-ceil . (Contributed by AV, 4-Sep-2021.) $)
|
|
ex-ceil $p |- ( ( |^ ` ( 3 / 2 ) ) = 2 /\ ( |^ ` -u ( 3 / 2 ) ) = -u 1 ) $=
|
|
( c3 c2 cdiv co cfl cfv c1 wceq cneg wa cceil ex-fl cq cz cn ax-mp ceilqval
|
|
wcel negnegi syl5eq 3z 2nn znq mp2an qnegcl qcn eqcomi fveq2i eqeq1i biimpi
|
|
cc negeqd negeq 2cn syl6eq anim12ci ) ABCDZEFZGHZUQIZEFZBIZHZJUQKFZBHZUTKFZ
|
|
GIZHZJLUSVHVCVEUSVFUTIZEFZIZVGUTMRZVFVKHUQMRZVLANRBORVMUAUBABUCUDZUQUEPUTQP
|
|
USVJGUSVJGHURVJGUQVIEVIUQUQVMUQUKRVNUQUFPSUGUHUIUJULTVCVDVAIZBVMVDVOHVNUQQP
|
|
VCVOVBIBVAVBUMBUNSUOTUPP $.
|
|
|
|
$( Example for ~ df-fac . (Contributed by AV, 4-Sep-2021.) $)
|
|
ex-fac $p |- ( ! ` 5 ) = ; ; 1 2 0 $=
|
|
( c5 cfa cfv c4 c1 caddc co cmul c2 cdc cc0 df-5 fveq2i 4nn0 eqtri 2nn0 5cn
|
|
0nn0 2cn mulcomli wcel wceq facp1 ax-mp fac4 4p1e5 oveq12i 5nn0 eqid 5t2e10
|
|
cn0 1nn0 addid2i decaddi 4cn 5t4e20 decmul1c ) ABCZDBCZDEFGZHGZEIJZKJZURUTB
|
|
CZVAAUTBLMDUKUAVDVAUBNDUCUDOVAIDJZAHGVCUSVEUTAHUEUFUGIDVBKAIVEUHPNVEUIRPEKI
|
|
IAHGIULRPAIEKJQSUJTISUMUNADIKJQUOUPTUQOO $.
|
|
|
|
$( Example for ~ df-bc . (Contributed by AV, 4-Sep-2021.) $)
|
|
ex-bc $p |- ( 5 _C 3 ) = ; 1 0 $=
|
|
( c5 c3 cbc co c4 c1 caddc cc0 cdc df-5 oveq1i cmin c6 4bc3eq4 3m1e2 oveq2i
|
|
c2 4bc2eq6 eqtri wcel oveq12i cn0 cz wceq 4nn0 3z bcpasc mp2an 6cn addcomli
|
|
4cn 6p4e10 3eqtr3i ) ABCDEFGDZBCDZFHIZAUNBCJKEBCDZEBFLDZCDZGDZEMGDUOUPUQEUS
|
|
MGNUSEQCDMURQECOPRSUAEUBTBUCTUTUOUDUEUFBEUGUHMEUPUIUKULUJUMS $.
|
|
|
|
$( Example for ~ df-dvds : 3 divides into 6. (Contributed by David A.
|
|
Wheeler, 19-May-2015.) $)
|
|
ex-dvds $p |- 3 || 6 $=
|
|
( c2 cz wcel c3 c6 w3a cmul co wceq cdvds wbr 2z 6nn nnzi 3pm3.2i caddc 3cn
|
|
3z 2timesi 3p3e6 eqtri dvds0lem mp2an ) ABCZDBCZEBCZFADGHZEIDEJKUDUEUFLREMN
|
|
OUGDDPHEDQSTUAADEUBUC $.
|
|
|
|
$( Example for ~ df-gcd . (Contributed by AV, 5-Sep-2021.) $)
|
|
ex-gcd $p |- ( -u 6 gcd 9 ) = 3 $=
|
|
( c6 cneg c9 cgcd co c3 cz wcel wceq mp2an caddc eqcomi oveq2i eqtri gcdadd
|
|
nnzi 3z cc0 3re 3eqtr3i 6nn 9nn neggcd 6cn 6p3e9 addcomli gcdcom 3p3e6 cabs
|
|
3cn cfv gcdid ax-mp cr cle wbr 0re 3pos ltleii absid ) ABCDEZACDEZFAGHZCGHV
|
|
AVBIAUAPZCUBPACUCJVBAFAKEZDEZFCVEADVECAFCUDUJUEUFLMAFDEZFFFKEZDEZVFFVGFADEZ
|
|
VIVCFGHZVGVJIVDQAFUGJAVHFDVHAUHLMNVCVKVGVFIVDQAFOJFFDEZFUIUKZVIFVKVLVMIQFUL
|
|
UMVKVKVLVIIQQFFOJFUNHRFUOUPVMFISRFUQSURUSFUTJTTNN $.
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Appendix: Typesetting definitions for the tokens in this file
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$( $t
|
|
|
|
/* The '$ t' (no space between '$' and 't') token indicates the beginning
|
|
of the typesetting definition section, embedded in a Metamath
|
|
comment. There may only be one per source file, and the typesetting
|
|
section ends with the end of the Metamath comment. The typesetting
|
|
section uses C-style comment delimiters. Todo: Allow multiple
|
|
typesetting comments */
|
|
|
|
/* These are the LaTeX and HTML definitions in the order the tokens are
|
|
introduced in $c or $v statements. See HELP TEX or HELP HTML in the
|
|
Metamath program. */
|
|
|
|
|
|
/******* Web page format settings *******/
|
|
|
|
/* Custom CSS for Unicode fonts */
|
|
/* The woff font file was obtained from
|
|
http://fred-wang.github.io/MathFonts/XITS/xits-math.woff 28-Aug-2015 */
|
|
htmlcss '<STYLE TYPE="text/css">\n' +
|
|
'<!--\n' +
|
|
' .setvar { color: red; }\n' +
|
|
' .wff { color: blue; }\n' +
|
|
' .class { color: #C3C; }\n' +
|
|
' .symvar { border-bottom:1px dotted;color:#C3C}\n' +
|
|
' .typecode { color: gray }\n' +
|
|
' .hidden { color: gray }\n' +
|
|
' @font-face {\n' +
|
|
' font-family: XITSMath-Regular;\n' +
|
|
' src: url(xits-math.woff);\n' +
|
|
' }\n' +
|
|
' .math { font-family: XITSMath-Regular }\n' +
|
|
'-->\n' +
|
|
'</STYLE>\n' +
|
|
'<LINK href="mmset.css" title="mmset"\n' +
|
|
' rel="stylesheet" type="text/css">\n' +
|
|
'<LINK href="mmsetalt.css" title="mmsetalt"\n' +
|
|
' rel="alternate stylesheet" type="text/css">';
|
|
|
|
/* Tag(s) for the main SPAN surrounding all Unicode math */
|
|
htmlfont 'CLASS=math';
|
|
|
|
/* Page title, home page link */
|
|
htmltitle "Intuitionistic Logic Explorer";
|
|
htmlhome '<A HREF="mmil.html"><FONT SIZE=-2 FACE=sans-serif>' +
|
|
'<IMG SRC="_icon-il.gif" BORDER=0 ALT=' +
|
|
'"Home" HEIGHT=32 WIDTH=32 ALIGN=MIDDLE>' +
|
|
'Home</FONT></A>';
|
|
/* Optional file where bibliographic references are kept */
|
|
/* If specified, e.g. "mmset.html", Metamath will hyperlink all strings of the
|
|
form "[rrr]" (where "rrr" has no whitespace) to "mmset.html#rrr" */
|
|
/* A warning will be given if the file "mmset.html" with the bibliographical
|
|
references is not present. It is read in order to check correctness of
|
|
the references. */
|
|
htmlbibliography "mmil.html";
|
|
|
|
/* Variable color key at the bottom of each proof table */
|
|
htmlvarcolor '<FONT COLOR="#0000FF">wff</FONT> '
|
|
+ '<FONT COLOR="#FF0000">set</FONT> '
|
|
+ '<FONT COLOR="#CC33CC">class</FONT>';
|
|
|
|
/* GIF and Unicode HTML directories - these are used for the GIF version to
|
|
crosslink to the Unicode version and vice-versa */
|
|
htmldir "../ilegif/";
|
|
althtmldir "../ileuni/";
|
|
|
|
|
|
/******* Symbol definitions *******/
|
|
|
|
htmldef "(" as "<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT=' (' TITLE='('>";
|
|
althtmldef "(" as "(";
|
|
latexdef "(" as "(";
|
|
htmldef ")" as "<IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=' )' TITLE=')'>";
|
|
althtmldef ")" as ")";
|
|
latexdef ")" as ")";
|
|
htmldef "->" as
|
|
" <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT=' ->' TITLE='->'> ";
|
|
althtmldef "->" as " → ";
|
|
latexdef "->" as " \rightarrow ";
|
|
htmldef "&" as
|
|
" <IMG SRC='amp.gif' WIDTH=12 HEIGHT=19 ALT='&'> ";
|
|
althtmldef "&" as " & ";
|
|
latexdef "&" as " & ";
|
|
htmldef "=>" as
|
|
" <IMG SRC='bigto.gif' WIDTH=15 HEIGHT=19 ALT='=>'> ";
|
|
althtmldef "=>" as " ⇒ ";
|
|
latexdef "=>" as " \Rightarrow ";
|
|
htmldef "-." as
|
|
"<IMG SRC='lnot.gif' WIDTH=10 HEIGHT=19 ALT=' -.' TITLE='-.'> ";
|
|
althtmldef "-." as '¬ ';
|
|
latexdef "-." as "\lnot";
|
|
htmldef "wff" as
|
|
"<IMG SRC='_wff.gif' WIDTH=24 HEIGHT=19 ALT=' wff' TITLE='wff'> ";
|
|
althtmldef "wff" as
|
|
'<SPAN CLASS=typecode STYLE="color:gray">wff </SPAN>'; /* was #00CC00 */
|
|
latexdef "wff" as "{\rm wff}";
|
|
htmldef "|-" as
|
|
"<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT=' |-' TITLE='|-'> ";
|
|
althtmldef "|-" as
|
|
'<SPAN CLASS=hidden STYLE="color:gray">⊢ </SPAN>'; /* ⊢ */
|
|
/* Without sans-serif, way too big in FF3 */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "|-" as "\vdash";
|
|
htmldef "ph" as
|
|
"<IMG SRC='_varphi.gif' WIDTH=11 HEIGHT=19 ALT=' ph' TITLE='ph'>";
|
|
/* althtmldef "ph" as '<FONT COLOR="#0000FF">𝜑</SPAN>'; */
|
|
althtmldef "ph" as '<SPAN CLASS=wff STYLE="color:blue">𝜑</SPAN>';
|
|
latexdef "ph" as "\varphi";
|
|
htmldef "ps" as "<IMG SRC='_psi.gif' WIDTH=12 HEIGHT=19 ALT=' ps' TITLE='ps'>";
|
|
althtmldef "ps" as '<SPAN CLASS=wff STYLE="color:blue">𝜓</SPAN>';
|
|
latexdef "ps" as "\psi";
|
|
htmldef "ch" as "<IMG SRC='_chi.gif' WIDTH=12 HEIGHT=19 ALT=' ch' TITLE='ch'>";
|
|
althtmldef "ch" as '<SPAN CLASS=wff STYLE="color:blue">𝜒</SPAN>';
|
|
latexdef "ch" as "\chi";
|
|
htmldef "th" as
|
|
"<IMG SRC='_theta.gif' WIDTH=8 HEIGHT=19 ALT=' th' TITLE='th'>";
|
|
althtmldef "th" as
|
|
'<SPAN CLASS=wff STYLE="color:blue">𝜃</SPAN>';
|
|
latexdef "th" as "\theta";
|
|
htmldef "ta" as "<IMG SRC='_tau.gif' WIDTH=10 HEIGHT=19 ALT=' ta' TITLE='ta'>";
|
|
althtmldef "ta" as '<SPAN CLASS=wff STYLE="color:blue">𝜏</SPAN>';
|
|
latexdef "ta" as "\tau";
|
|
htmldef "et" as "<IMG SRC='_eta.gif' WIDTH=9 HEIGHT=19 ALT=' et' TITLE='et'>";
|
|
althtmldef "et" as '<SPAN CLASS=wff STYLE="color:blue">𝜂</SPAN>';
|
|
latexdef "et" as "\eta";
|
|
htmldef "ze" as "<IMG SRC='_zeta.gif' WIDTH=9 HEIGHT=19 ALT=' ze' TITLE='ze'>";
|
|
althtmldef "ze" as '<SPAN CLASS=wff STYLE="color:blue">𝜁</SPAN>';
|
|
latexdef "ze" as "\zeta";
|
|
htmldef "si" as
|
|
"<IMG SRC='_sigma.gif' WIDTH=10 HEIGHT=19 ALT=' si' TITLE='si'>";
|
|
althtmldef "si" as
|
|
'<SPAN CLASS=wff STYLE="color:blue">𝜎</SPAN>';
|
|
latexdef "si" as "\sigma";
|
|
htmldef "rh" as "<IMG SRC='_rho.gif' WIDTH=9 HEIGHT=19 ALT=' rh' TITLE='rh'>";
|
|
althtmldef "rh" as '<SPAN CLASS=wff STYLE="color:blue">𝜌</SPAN>';
|
|
latexdef "rh" as "\rho";
|
|
htmldef "mu" as "<IMG SRC='_mu.gif' WIDTH=10 HEIGHT=19 ALT=' mu' TITLE='mu'>";
|
|
althtmldef "mu" as '<SPAN CLASS=wff STYLE="color:blue">𝜇</SPAN>';
|
|
latexdef "mu" as "\mu";
|
|
htmldef "la" as
|
|
"<IMG SRC='_lambda.gif' WIDTH=9 HEIGHT=19 ALT=' la' TITLE='la'>";
|
|
althtmldef "la" as
|
|
'<SPAN CLASS=wff STYLE="color:blue">𝜆</SPAN>';
|
|
latexdef "la" as "\lambda";
|
|
htmldef "ka" as
|
|
"<IMG SRC='_kappa.gif' WIDTH=9 HEIGHT=19 ALT=' ka' TITLE='ka'>";
|
|
althtmldef "ka" as
|
|
'<SPAN CLASS=wff STYLE="color:blue">𝜅</SPAN>';
|
|
latexdef "ka" as "\kappa";
|
|
htmldef "<->" as " <IMG SRC='leftrightarrow.gif' WIDTH=15 HEIGHT=19 " +
|
|
"ALT=' <->' TITLE='<->'> ";
|
|
althtmldef "<->" as ' ↔ ';
|
|
latexdef "<->" as "\leftrightarrow";
|
|
htmldef "\/" as
|
|
" <IMG SRC='vee.gif' WIDTH=11 HEIGHT=19 ALT=' \/' TITLE='\/'> ";
|
|
althtmldef "\/" as ' ∨ ' ;
|
|
/* althtmldef "\/" as ' <FONT FACE=sans-serif>⋁</FONT> ' ; */
|
|
/* was ∨ - changed to match font of ∧ replacement */
|
|
/* Changed back 6-Mar-2012 NM */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "\/" as "\vee";
|
|
htmldef "/\" as
|
|
" <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT=' /\' TITLE='/\'> ";
|
|
althtmldef "/\" as ' ∧ ';
|
|
/* althtmldef "/\" as ' <FONT FACE=sans-serif>⋀</FONT> '; */
|
|
/* was ∧ which is circle in Mozilla on WinXP Pro (but not Home) */
|
|
/* Changed back 6-Mar-2012 NM */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "/\" as "\wedge";
|
|
htmldef "\/_" as
|
|
" <IMG SRC='veebar.gif' WIDTH=9 HEIGHT=19 ALT=' \/_' TITLE='\/_'> ";
|
|
althtmldef "\/_" as " ⊻ ";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "\/_" as "\veebar";
|
|
htmldef "T." as
|
|
" <IMG SRC='top.gif' WIDTH=11 HEIGHT=19 ALT='T.' TITLE='T.'> ";
|
|
althtmldef "T." as '⊤';
|
|
latexdef "T." as "\top";
|
|
htmldef "F." as
|
|
" <IMG SRC='perp.gif' WIDTH=11 HEIGHT=19 ALT='F.' TITLE='F.'> ";
|
|
althtmldef "F." as '⊥';
|
|
latexdef "F." as "\bot";
|
|
htmldef "A." as
|
|
"<IMG SRC='forall.gif' WIDTH=10 HEIGHT=19 ALT=' A.' TITLE='A.'>";
|
|
althtmldef "A." as '∀'; /* ∀ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "A." as "\forall";
|
|
htmldef "setvar" as
|
|
"<IMG SRC='_setvar.gif' WIDTH=40 HEIGHT=19 ALT=' setvar' TITLE='setvar'> ";
|
|
althtmldef
|
|
"setvar" as '<SPAN CLASS=typecode STYLE="color:gray">setvar </SPAN>';
|
|
latexdef "setvar" as "{\rm setvar}";
|
|
htmldef "x" as "<IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT=' x' TITLE='x'>";
|
|
/* http://www.unicode.org/charts/PDF/U1D400.pdf has math italic Unicode */
|
|
althtmldef "x" as '<SPAN CLASS=setvar STYLE="color:red">𝑥</SPAN>';
|
|
latexdef "x" as "x";
|
|
htmldef "y" as "<IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT=' y' TITLE='y'>";
|
|
althtmldef "y" as '<SPAN CLASS=setvar STYLE="color:red">𝑦</SPAN>';
|
|
latexdef "y" as "y";
|
|
htmldef "z" as "<IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 ALT=' z' TITLE='z'>";
|
|
althtmldef "z" as '<SPAN CLASS=setvar STYLE="color:red">𝑧</SPAN>';
|
|
latexdef "z" as "z";
|
|
htmldef "w" as "<IMG SRC='_w.gif' WIDTH=12 HEIGHT=19 ALT=' w' TITLE='w'>";
|
|
althtmldef "w" as '<SPAN CLASS=setvar STYLE="color:red">𝑤</SPAN>';
|
|
latexdef "w" as "w";
|
|
htmldef "v" as "<IMG SRC='_v.gif' WIDTH=9 HEIGHT=19 ALT=' v' TITLE='v'>";
|
|
althtmldef "v" as '<SPAN CLASS=setvar STYLE="color:red">𝑣</SPAN>';
|
|
latexdef "v" as "v";
|
|
htmldef "u" as "<IMG SRC='_u.gif' WIDTH=10 HEIGHT=19 ALT=' u' TITLE='u'>";
|
|
althtmldef "u" as '<SPAN CLASS=setvar STYLE="color:red">𝑢</SPAN>';
|
|
latexdef "u" as "u";
|
|
htmldef "t" as "<IMG SRC='_t.gif' WIDTH=7 HEIGHT=19 ALT=' t' TITLE='t'>";
|
|
althtmldef "t" as '<SPAN CLASS=setvar STYLE="color:red">𝑡</SPAN>';
|
|
latexdef "t" as "t";
|
|
htmldef "E." as
|
|
"<IMG SRC='exists.gif' WIDTH=9 HEIGHT=19 ALT=' E.' TITLE='E.'>";
|
|
althtmldef "E." as '∃'; /* ∃ */
|
|
/* Without sans-serif, bad in Opera and way too big in FF3 */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "E." as "\exists";
|
|
htmldef "F/" as
|
|
"<IMG SRC='finv.gif' WIDTH=9 HEIGHT=19 ALT=' F/' TITLE='F/'>";
|
|
althtmldef "F/" as "Ⅎ";
|
|
latexdef "F/" as "\Finv";
|
|
htmldef "class" as
|
|
"<IMG SRC='_class.gif' WIDTH=32 HEIGHT=19 ALT=' class' TITLE='class'> ";
|
|
althtmldef "class" as
|
|
'<SPAN CLASS=typecode STYLE="color:gray">class </SPAN>';
|
|
latexdef "class" as "{\rm class}";
|
|
htmldef "=" as " <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT=' =' TITLE='='> ";
|
|
althtmldef "=" as ' = '; /* = */
|
|
latexdef "=" as "=";
|
|
htmldef "e." as " <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT=' e.' TITLE='e.'> ";
|
|
althtmldef "e." as ' ∈ ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "e." as "\in";
|
|
htmldef "[" as "<IMG SRC='lbrack.gif' WIDTH=5 HEIGHT=19 ALT=' [' TITLE='['>";
|
|
althtmldef "[" as '['; /* [ */
|
|
latexdef "[" as "[";
|
|
htmldef "/" as
|
|
" <IMG SRC='solidus.gif' WIDTH=6 HEIGHT=19 ALT=' /' TITLE='/'> ";
|
|
althtmldef "/" as ' / '; /* / */
|
|
latexdef "/" as "/";
|
|
htmldef "]" as "<IMG SRC='rbrack.gif' WIDTH=5 HEIGHT=19 ALT=' ]' TITLE=']'>";
|
|
althtmldef "]" as ']'; /* ] */
|
|
latexdef "]" as "]";
|
|
htmldef "E!" as "<IMG SRC='_e1.gif' WIDTH=12 HEIGHT=19 ALT=' E!' TITLE='E!'>";
|
|
althtmldef "E!" as '∃!';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "E!" as "\exists{!}";
|
|
htmldef "E*" as "<IMG SRC='_em1.gif' WIDTH=15 HEIGHT=19 ALT=' E*' TITLE='E*'>";
|
|
althtmldef "E*" as '∃*';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "E*" as "\exists^\ast";
|
|
htmldef "{" as "<IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19 ALT=' {' TITLE='{'>";
|
|
althtmldef "{" as '{'; /* { */
|
|
latexdef "{" as "\{";
|
|
htmldef "|" as " <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT=' |' TITLE='|'> ";
|
|
althtmldef "|" as ' ∣ '; /* &vertbar; */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "|" as "|";
|
|
htmldef "}" as "<IMG SRC='rbrace.gif' WIDTH=6 HEIGHT=19 ALT=' }' TITLE='}'>";
|
|
althtmldef "}" as '}'; /* } */
|
|
latexdef "}" as "\}";
|
|
htmldef "F/_" as
|
|
"<IMG SRC='_finvbar.gif' WIDTH=9 HEIGHT=19 ALT=' F/_' TITLE='F/_'>";
|
|
althtmldef "F/_" as "<U>Ⅎ</U>";
|
|
latexdef "F/_" as "\underline{\Finv}";
|
|
htmldef "CondEq" as "CondEq";
|
|
althtmldef "CondEq" as "CondEq";
|
|
latexdef "CondEq" as "\mbox{CondEq}";
|
|
htmldef "./\" as
|
|
" <IMG SRC='_.wedge.gif' WIDTH=11 HEIGHT=19 ALT=' ./\' TITLE='./\'> ";
|
|
althtmldef "./\" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∧</SPAN> ';
|
|
latexdef "./\" as "\wedge";
|
|
htmldef ".\/" as
|
|
" <IMG SRC='_.vee.gif' WIDTH=11 HEIGHT=19 ALT=' .\/' TITLE='.\/'> ";
|
|
althtmldef ".\/" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∨</SPAN> ';
|
|
latexdef ".\/" as "\vee";
|
|
htmldef ".<_" as
|
|
" <IMG SRC='_.le.gif' WIDTH=11 HEIGHT=19 ALT=' .<_' TITLE='.<_'> ";
|
|
althtmldef ".<_" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'≤</SPAN> ';
|
|
latexdef ".<_" as "\le";
|
|
htmldef ".<" as /* Symbol as variable */
|
|
" <IMG SRC='_.lt.gif' WIDTH=11 HEIGHT=19 ALT=' .<' TITLE='.<'> ";
|
|
althtmldef ".<" as
|
|
/* This is how to put a dotted box around the symbol: */
|
|
/* border means box around symbol; border-bottom underlines symbol */
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'<</SPAN> ';
|
|
/* TODO: can this STYLE sequence be done with a CLASS? */
|
|
/* Move the underline down 3px so it isn't too close to symbol */
|
|
/*
|
|
' <SPAN STYLE="vertical-align:-3px">' +
|
|
'<SPAN CLASS=symvar STYLE="text-decoration:underline dotted;color:#C3C">' +
|
|
'<SPAN STYLE="vertical-align:3px"><</SPAN></SPAN></SPAN> ';
|
|
*/
|
|
latexdef ".<" as "<";
|
|
htmldef ".+" as
|
|
" <IMG SRC='_.plus.gif' WIDTH=13 HEIGHT=19 ALT=' .+' TITLE='.+'> ";
|
|
althtmldef ".+" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'+</SPAN> ';
|
|
latexdef ".+" as "+";
|
|
htmldef ".-" as
|
|
" <IMG SRC='_.minus.gif' WIDTH=11 HEIGHT=19 ALT=' .-' TITLE='.-'> ";
|
|
althtmldef ".-" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'−</SPAN> ';
|
|
latexdef ".-" as "-";
|
|
htmldef ".X." as
|
|
" <IMG SRC='_.times.gif' WIDTH=9 HEIGHT=19 ALT=' .X.' TITLE='.X.'> ";
|
|
althtmldef ".X." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'×</SPAN> ';
|
|
latexdef ".X." as "\times";
|
|
htmldef "./" as
|
|
" <IMG SRC='_.solidus.gif' WIDTH=8 HEIGHT=19 ALT=' ./' TITLE='./'> ";
|
|
althtmldef "./" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'/</SPAN> ';
|
|
latexdef "./" as "/";
|
|
htmldef ".^" as
|
|
" <IMG SRC='_.uparrow.gif' WIDTH=7 HEIGHT=19 ALT=' .^' TITLE='.^'> ";
|
|
althtmldef ".^" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'↑</SPAN> ';
|
|
latexdef ".^" as "\uparrow";
|
|
htmldef ".0." as
|
|
" <IMG SRC='_.0.gif' WIDTH=8 HEIGHT=19 ALT=' .0.' TITLE='.0.'> ";
|
|
althtmldef ".0." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'0</SPAN> ';
|
|
latexdef ".0." as "0";
|
|
htmldef ".1." as
|
|
" <IMG SRC='_.1.gif' WIDTH=7 HEIGHT=19 ALT=' .1.' TITLE='.1.'> ";
|
|
althtmldef ".1." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'1</SPAN> ';
|
|
latexdef ".1." as "1";
|
|
htmldef ".||" as
|
|
" <IMG SRC='_.parallel.gif' WIDTH=5 HEIGHT=19 ALT=' .||' TITLE='.||'> ";
|
|
althtmldef ".||" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∥</SPAN> ';
|
|
latexdef ".||" as "\parallel";
|
|
htmldef ".~" as
|
|
" <IMG SRC='_.sim.gif' WIDTH=13 HEIGHT=19 ALT=' .~' TITLE='.~'> ";
|
|
althtmldef ".~" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∼</SPAN> ';
|
|
latexdef ".~" as "\sim";
|
|
htmldef "._|_" as
|
|
" <IMG SRC='_.perp.gif' WIDTH=11 HEIGHT=19 ALT=' ._|_' TITLE='._|_'> ";
|
|
althtmldef "._|_" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊥</SPAN> ';
|
|
latexdef "._|_" as "\perp";
|
|
htmldef ".+^" as
|
|
" <IMG SRC='_.plushat.gif' WIDTH=11 HEIGHT=19 ALT=' .+^' TITLE='.+^'> ";
|
|
althtmldef ".+^" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⨣</SPAN> '; /* ⨣ */
|
|
latexdef ".+^" as "\hat{+}";
|
|
htmldef ".+b" as
|
|
" <IMG SRC='_.plusb.gif' WIDTH=14 HEIGHT=19 ALT=' .+b' TITLE='.+b'> ";
|
|
althtmldef ".+b" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'✚</SPAN> ';
|
|
latexdef ".+b" as "\boldsymbol{+}";
|
|
htmldef ".(+)" as
|
|
" <IMG SRC='_.oplus.gif' WIDTH=13 HEIGHT=19 ALT=' .(+)' TITLE='.(+)'> ";
|
|
althtmldef ".(+)" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊕</SPAN> ';
|
|
latexdef ".(+)" as "\oplus";
|
|
htmldef ".*" as
|
|
" <IMG SRC='_.ast.gif' WIDTH=7 HEIGHT=19 ALT=' .*' TITLE='.*'> ";
|
|
althtmldef ".*" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∗</SPAN> ';
|
|
latexdef ".*" as "\ast";
|
|
htmldef ".x." as
|
|
" <IMG SRC='_.cdot.gif' WIDTH=4 HEIGHT=19 ALT=' .x.' TITLE='.x.'> ";
|
|
althtmldef ".x." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'·</SPAN> ';
|
|
latexdef ".x." as "\cdot";
|
|
htmldef ".xb" as
|
|
" <IMG SRC='_.bullet.gif' WIDTH=8 HEIGHT=19 ALT=' .xb' TITLE='.xb'> ";
|
|
althtmldef ".xb" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∙</SPAN> ';
|
|
latexdef ".xb" as "\bullet";
|
|
htmldef ".," as
|
|
" <IMG SRC='_.comma.gif' WIDTH=4 HEIGHT=19 ALT=' .,' TITLE='.,'> ";
|
|
althtmldef ".," as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
',</SPAN> ';
|
|
latexdef ".," as ",";
|
|
htmldef ".(x)" as
|
|
" <IMG SRC='_.otimes.gif' WIDTH=13 HEIGHT=19 ALT=' .(x)' TITLE='.(x)'> ";
|
|
althtmldef ".(x)" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊗</SPAN> ';
|
|
latexdef ".(x)" as "\otimes";
|
|
htmldef ".0b" as
|
|
" <IMG SRC='_.bf0.gif' WIDTH=9 HEIGHT=19 ALT=' .0b' TITLE='.0b'> ";
|
|
althtmldef ".0b" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'𝟎</SPAN> ';
|
|
latexdef ".0b" as "\mbox{\boldmath$0$}";
|
|
htmldef "A" as "<IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT=' A' TITLE='A'>";
|
|
althtmldef "A" as '<SPAN CLASS=class STYLE="color:#C3C">𝐴</SPAN>';
|
|
latexdef "A" as "A";
|
|
htmldef "B" as "<IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT=' B' TITLE='B'>";
|
|
althtmldef "B" as '<SPAN CLASS=class STYLE="color:#C3C">𝐵</SPAN>';
|
|
latexdef "B" as "B";
|
|
htmldef "C" as "<IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT=' C' TITLE='C'>";
|
|
althtmldef "C" as '<SPAN CLASS=class STYLE="color:#C3C">𝐶</SPAN>';
|
|
latexdef "C" as "C";
|
|
htmldef "D" as "<IMG SRC='_cd.gif' WIDTH=12 HEIGHT=19 ALT=' D' TITLE='D'>";
|
|
althtmldef "D" as '<SPAN CLASS=class STYLE="color:#C3C">𝐷</SPAN>';
|
|
latexdef "D" as "D";
|
|
htmldef "P" as "<IMG SRC='_cp.gif' WIDTH=12 HEIGHT=19 ALT=' P' TITLE='P'>";
|
|
althtmldef "P" as '<SPAN CLASS=class STYLE="color:#C3C">𝑃</SPAN>';
|
|
latexdef "P" as "P";
|
|
htmldef "Q" as "<IMG SRC='_cq.gif' WIDTH=12 HEIGHT=19 ALT=' Q' TITLE='Q'>";
|
|
althtmldef "Q" as '<SPAN CLASS=class STYLE="color:#C3C">𝑄</SPAN>';
|
|
latexdef "Q" as "Q";
|
|
htmldef "R" as "<IMG SRC='_cr.gif' WIDTH=12 HEIGHT=19 ALT=' R' TITLE='R'>";
|
|
althtmldef "R" as '<SPAN CLASS=class STYLE="color:#C3C">𝑅</SPAN>';
|
|
latexdef "R" as "R";
|
|
htmldef "S" as "<IMG SRC='_cs.gif' WIDTH=11 HEIGHT=19 ALT=' S' TITLE='S'>";
|
|
althtmldef "S" as '<SPAN CLASS=class STYLE="color:#C3C">𝑆</SPAN>';
|
|
latexdef "S" as "S";
|
|
htmldef "T" as "<IMG SRC='_ct.gif' WIDTH=12 HEIGHT=19 ALT=' T' TITLE='T'>";
|
|
althtmldef "T" as '<SPAN CLASS=class STYLE="color:#C3C">𝑇</SPAN>';
|
|
latexdef "T" as "T";
|
|
htmldef "U" as "<IMG SRC='_cu.gif' WIDTH=12 HEIGHT=19 ALT=' U' TITLE='U'>";
|
|
althtmldef "U" as '<SPAN CLASS=class STYLE="color:#C3C">𝑈</SPAN>';
|
|
latexdef "U" as "U";
|
|
htmldef "e" as "<IMG SRC='_e.gif' WIDTH=8 HEIGHT=19 ALT=' e' TITLE='e'>";
|
|
althtmldef "e" as '<SPAN CLASS=setvar STYLE="color:red">𝑒</SPAN>';
|
|
latexdef "e" as "e";
|
|
htmldef "f" as "<IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 ALT=' f' TITLE='f'>";
|
|
althtmldef "f" as '<SPAN CLASS=setvar STYLE="color:red">𝑓</SPAN>';
|
|
latexdef "f" as "f";
|
|
htmldef "g" as "<IMG SRC='_g.gif' WIDTH=9 HEIGHT=19 ALT=' g' TITLE='g'>";
|
|
althtmldef "g" as '<SPAN CLASS=setvar STYLE="color:red">𝑔</SPAN>';
|
|
latexdef "g" as "g";
|
|
htmldef "h" as "<IMG SRC='_h.gif' WIDTH=10 HEIGHT=19 ALT=' h' TITLE='h'>";
|
|
althtmldef "h" as '<SPAN CLASS=setvar STYLE="color:red">ℎ</SPAN>';
|
|
latexdef "h" as "h";
|
|
htmldef "i" as "<IMG SRC='_i.gif' WIDTH=6 HEIGHT=19 ALT=' i' TITLE='i'>";
|
|
althtmldef "i" as '<SPAN CLASS=setvar STYLE="color:red">𝑖</SPAN>';
|
|
latexdef "i" as "i";
|
|
htmldef "j" as "<IMG SRC='_j.gif' WIDTH=7 HEIGHT=19 ALT=' j' TITLE='j'>";
|
|
althtmldef "j" as '<SPAN CLASS=setvar STYLE="color:red">𝑗</SPAN>';
|
|
latexdef "j" as "j";
|
|
htmldef "k" as "<IMG SRC='_k.gif' WIDTH=9 HEIGHT=19 ALT=' k' TITLE='k'>";
|
|
althtmldef "k" as '<SPAN CLASS=setvar STYLE="color:red">𝑘</SPAN>';
|
|
latexdef "k" as "k";
|
|
htmldef "m" as "<IMG SRC='_m.gif' WIDTH=14 HEIGHT=19 ALT=' m' TITLE='m'>";
|
|
althtmldef "m" as '<SPAN CLASS=setvar STYLE="color:red">𝑚</SPAN>';
|
|
latexdef "m" as "m";
|
|
htmldef "n" as "<IMG SRC='_n.gif' WIDTH=10 HEIGHT=19 ALT=' n' TITLE='n'>";
|
|
althtmldef "n" as '<SPAN CLASS=setvar STYLE="color:red">𝑛</SPAN>';
|
|
latexdef "n" as "n";
|
|
htmldef "o" as "<IMG SRC='_o.gif' WIDTH=8 HEIGHT=19 ALT=' o' TITLE='o'>";
|
|
althtmldef "o" as '<SPAN CLASS=setvar STYLE="color:red">𝑜</SPAN>';
|
|
latexdef "o" as "o";
|
|
htmldef "E" as "<IMG SRC='_ce.gif' WIDTH=13 HEIGHT=19 ALT=' E' TITLE='E'>";
|
|
althtmldef "E" as '<SPAN CLASS=class STYLE="color:#C3C">𝐸</SPAN>';
|
|
latexdef "E" as "E";
|
|
htmldef "F" as "<IMG SRC='_cf.gif' WIDTH=13 HEIGHT=19 ALT=' F' TITLE='F'>";
|
|
althtmldef "F" as '<SPAN CLASS=class STYLE="color:#C3C">𝐹</SPAN>';
|
|
latexdef "F" as "F";
|
|
htmldef "G" as "<IMG SRC='_cg.gif' WIDTH=12 HEIGHT=19 ALT=' G' TITLE='G'>";
|
|
althtmldef "G" as '<SPAN CLASS=class STYLE="color:#C3C">𝐺</SPAN>';
|
|
latexdef "G" as "G";
|
|
htmldef "H" as "<IMG SRC='_ch.gif' WIDTH=14 HEIGHT=19 ALT=' H' TITLE='H'>";
|
|
althtmldef "H" as '<SPAN CLASS=class STYLE="color:#C3C">𝐻</SPAN>';
|
|
latexdef "H" as "H";
|
|
htmldef "I" as "<IMG SRC='_ci.gif' WIDTH=8 HEIGHT=19 ALT=' I' TITLE='I'>";
|
|
althtmldef "I" as '<SPAN CLASS=class STYLE="color:#C3C">𝐼</SPAN>';
|
|
latexdef "I" as "I";
|
|
htmldef "J" as "<IMG SRC='_cj.gif' WIDTH=10 HEIGHT=19 ALT=' J' TITLE='J'>";
|
|
althtmldef "J" as '<SPAN CLASS=class STYLE="color:#C3C">𝐽</SPAN>';
|
|
latexdef "J" as "J";
|
|
htmldef "K" as "<IMG SRC='_ck.gif' WIDTH=14 HEIGHT=19 ALT=' K' TITLE='K'>";
|
|
althtmldef "K" as '<SPAN CLASS=class STYLE="color:#C3C">𝐾</SPAN>';
|
|
latexdef "K" as "K";
|
|
htmldef "L" as "<IMG SRC='_cl.gif' WIDTH=10 HEIGHT=19 ALT=' L' TITLE='L'>";
|
|
althtmldef "L" as '<SPAN CLASS=class STYLE="color:#C3C">𝐿</SPAN>';
|
|
latexdef "L" as "L";
|
|
htmldef "M" as "<IMG SRC='_cm.gif' WIDTH=15 HEIGHT=19 ALT=' M' TITLE='M'>";
|
|
althtmldef "M" as '<SPAN CLASS=class STYLE="color:#C3C">𝑀</SPAN>';
|
|
latexdef "M" as "M";
|
|
htmldef "N" as "<IMG SRC='_cn.gif' WIDTH=14 HEIGHT=19 ALT=' N' TITLE='N'>";
|
|
althtmldef "N" as '<SPAN CLASS=class STYLE="color:#C3C">𝑁</SPAN>';
|
|
latexdef "N" as "N";
|
|
htmldef "V" as "<IMG SRC='_cv.gif' WIDTH=12 HEIGHT=19 ALT=' V' TITLE='V'>";
|
|
althtmldef "V" as '<SPAN CLASS=class STYLE="color:#C3C">𝑉</SPAN>';
|
|
latexdef "V" as "V";
|
|
htmldef "W" as "<IMG SRC='_cw.gif' WIDTH=16 HEIGHT=19 ALT=' W' TITLE='W'>";
|
|
althtmldef "W" as '<SPAN CLASS=class STYLE="color:#C3C">𝑊</SPAN>';
|
|
latexdef "W" as "W";
|
|
htmldef "X" as "<IMG SRC='_cx.gif' WIDTH=13 HEIGHT=19 ALT=' X' TITLE='X'>";
|
|
althtmldef "X" as '<SPAN CLASS=class STYLE="color:#C3C">𝑋</SPAN>';
|
|
latexdef "X" as "X";
|
|
htmldef "Y" as "<IMG SRC='_cy.gif' WIDTH=12 HEIGHT=19 ALT=' Y' TITLE='Y'>";
|
|
althtmldef "Y" as '<SPAN CLASS=class STYLE="color:#C3C">𝑌</SPAN>';
|
|
latexdef "Y" as "Y";
|
|
htmldef "Z" as "<IMG SRC='_cz.gif' WIDTH=11 HEIGHT=19 ALT=' Z' TITLE='Z'>";
|
|
althtmldef "Z" as '<SPAN CLASS=class STYLE="color:#C3C">𝑍</SPAN>';
|
|
latexdef "Z" as "Z";
|
|
htmldef "O" as "<IMG SRC='_co.gif' WIDTH=12 HEIGHT=19 ALT=' O' TITLE='O'>";
|
|
althtmldef "O" as '<SPAN CLASS=class STYLE="color:#C3C">𝑂</SPAN>';
|
|
latexdef "O" as "O";
|
|
htmldef "s" as "<IMG SRC='_s.gif' WIDTH=7 HEIGHT=19 ALT=' s' TITLE='s'>";
|
|
althtmldef "s" as '<SPAN CLASS=setvar STYLE="color:red">𝑠</SPAN>';
|
|
latexdef "s" as "s";
|
|
htmldef "r" as "<IMG SRC='_r.gif' WIDTH=8 HEIGHT=19 ALT=' r' TITLE='r'>";
|
|
althtmldef "r" as '<SPAN CLASS=setvar STYLE="color:red">𝑟</SPAN>';
|
|
latexdef "r" as "r";
|
|
htmldef "q" as "<IMG SRC='_q.gif' WIDTH=8 HEIGHT=19 ALT=' q' TITLE='q'>";
|
|
althtmldef "q" as '<SPAN CLASS=setvar STYLE="color:red">𝑞</SPAN>';
|
|
latexdef "q" as "q";
|
|
htmldef "p" as "<IMG SRC='_p.gif' WIDTH=10 HEIGHT=19 ALT=' p' TITLE='p'>";
|
|
althtmldef "p" as '<SPAN CLASS=setvar STYLE="color:red">𝑝</SPAN>';
|
|
latexdef "p" as "p";
|
|
htmldef "a" as "<IMG SRC='_a.gif' WIDTH=9 HEIGHT=19 ALT=' a' TITLE='a'>";
|
|
althtmldef "a" as '<SPAN CLASS=setvar STYLE="color:red">𝑎</SPAN>';
|
|
latexdef "a" as "a";
|
|
htmldef "b" as "<IMG SRC='_b.gif' WIDTH=8 HEIGHT=19 ALT=' b' TITLE='b'>";
|
|
althtmldef "b" as '<SPAN CLASS=setvar STYLE="color:red">𝑏</SPAN>';
|
|
latexdef "b" as "b";
|
|
htmldef "c" as "<IMG SRC='_c.gif' WIDTH=7 HEIGHT=19 ALT=' c' TITLE='c'>";
|
|
althtmldef "c" as '<SPAN CLASS=setvar STYLE="color:red">𝑐</SPAN>';
|
|
latexdef "c" as "c";
|
|
htmldef "d" as "<IMG SRC='_d.gif' WIDTH=9 HEIGHT=19 ALT=' d' TITLE='d'>";
|
|
althtmldef "d" as '<SPAN CLASS=setvar STYLE="color:red">𝑑</SPAN>';
|
|
latexdef "d" as "d";
|
|
htmldef "l" as "<IMG SRC='_l.gif' WIDTH=6 HEIGHT=19 ALT=' l' TITLE='l'>";
|
|
althtmldef "l" as '<SPAN CLASS=setvar STYLE="color:red">𝑙</SPAN>';
|
|
latexdef "l" as "l";
|
|
htmldef "=/=" as
|
|
" <IMG SRC='ne.gif' WIDTH=12 HEIGHT=19 ALT=' =/=' TITLE='=/='> ";
|
|
althtmldef "=/=" as ' ≠ ';
|
|
latexdef "=/=" as "\ne";
|
|
htmldef "e/" as
|
|
" <IMG SRC='notin.gif' WIDTH=10 HEIGHT=19 ALT=' e/' TITLE='e/'> ";
|
|
althtmldef "e/" as ' ∉ ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "e/" as "\notin";
|
|
htmldef "_V" as "<IMG SRC='rmcv.gif' WIDTH=10 HEIGHT=19 ALT=' _V' TITLE='_V'>";
|
|
althtmldef "_V" as 'V';
|
|
latexdef "_V" as "{\rm V}";
|
|
htmldef "[." as
|
|
"<IMG SRC='_dlbrack.gif' WIDTH=7 HEIGHT=19 ALT=' [.' TITLE='[.'>";
|
|
/* althtmldef "[." as '⦏'; */ /* corner tick */
|
|
/* althtmldef "[." as '[̣'; */ /* combining dot below */
|
|
althtmldef "[." as '<B>[</B>'; /* 6-Aug-2018 nm */
|
|
/* \underaccent is in accents package */
|
|
latexdef "[." as "\underaccent{\dot}{[}";
|
|
htmldef "]." as
|
|
"<IMG SRC='_drbrack.gif' WIDTH=6 HEIGHT=19 ALT=' ].' TITLE='].'>";
|
|
/* althtmldef "]." as '⦎'; */ /* corner tick */
|
|
/* althtmldef "]." as ']̣'; */ /* combining dot below */
|
|
althtmldef "]." as '<B>]</B>'; /* 6-Aug-2018 nm */
|
|
latexdef "]." as "\underaccent{\dot}{]}";
|
|
htmldef "[_" as
|
|
"<IMG SRC='_ulbrack.gif' WIDTH=7 HEIGHT=19 ALT=' [_' TITLE='[_'>";
|
|
althtmldef "[_" as '<B>⦋</B>'; /* left sq brack w underbar */
|
|
latexdef "[_" as "\underline{[}";
|
|
htmldef "]_" as
|
|
"<IMG SRC='_urbrack.gif' WIDTH=6 HEIGHT=19 ALT=' ]_' TITLE=']_'>";
|
|
althtmldef "]_" as '<B>⦌</B>'; /* right sq brack w underbar */
|
|
latexdef "]_" as "\underline{]}";
|
|
htmldef "\" as
|
|
" <IMG SRC='setminus.gif' WIDTH=8 HEIGHT=19 ALT=' \' TITLE='\'> ";
|
|
althtmldef "\" as ' ∖ '; /* ∖ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "\" as "\setminus";
|
|
htmldef "u." as
|
|
" <IMG SRC='cup.gif' WIDTH=10 HEIGHT=19 ALT=' u.' TITLE='u.'> ";
|
|
althtmldef "u." as ' ∪ ';
|
|
latexdef "u." as "\cup";
|
|
htmldef "i^i" as
|
|
" <IMG SRC='cap.gif' WIDTH=10 HEIGHT=19 ALT=' i^i' TITLE='i^i'> ";
|
|
althtmldef "i^i" as ' ∩ ';
|
|
latexdef "i^i" as "\cap";
|
|
htmldef "C_" as
|
|
" <IMG SRC='subseteq.gif' WIDTH=12 HEIGHT=19 ALT=' C_' TITLE='C_'> ";
|
|
althtmldef "C_" as ' ⊆ '; /* ⫅ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "C_" as "\subseteq";
|
|
htmldef "(/)" as
|
|
"<IMG SRC='varnothing.gif' WIDTH=11 HEIGHT=19 ALT=' (/)' TITLE='(/)'>";
|
|
althtmldef "(/)" as '∅';
|
|
/*althtmldef "(/)" as '∅';*/ /* =∅ */ /* bad in Opera */
|
|
/*althtmldef "(/)" as '⌀';*/
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "(/)" as "\varnothing";
|
|
htmldef "if" as "<IMG SRC='_if.gif' WIDTH=11 HEIGHT=19 ALT=' if' TITLE='if'>";
|
|
althtmldef "if" as 'if';
|
|
latexdef "if" as "{\rm if}";
|
|
htmldef "," as "<IMG SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=' ,' TITLE=','> ";
|
|
althtmldef "," as ', ';
|
|
latexdef "," as ",";
|
|
htmldef "~P" as "<IMG SRC='scrp.gif' WIDTH=16 HEIGHT=19 ALT=' ~P' TITLE='~P'>";
|
|
/* althtmldef "~P" as '<FONT FACE=sans-serif>℘</FONT>'; */
|
|
/* althtmldef "~P" as '℘'; */ /* 2-Jan-2016 reverted sans-serif */
|
|
/* 14-Jan-2016 script P is now available in upper Unicode */
|
|
/* 4-Aug-2016 NM Put space after ~P, needed for e.g. ncanth where it
|
|
overlapped the _V */
|
|
althtmldef "~P" as '𝒫 ';
|
|
latexdef "~P" as "{\cal P}";
|
|
htmldef "<." as
|
|
"<IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT=' <.' TITLE='<.'>";
|
|
/* The Unicode below doesn't always work on Firefox and Chrome on Windows,
|
|
so revert to the image above */
|
|
althtmldef "<." as '⟨'; /* 〈 */
|
|
/* 2-Jan-2016 restored Unicode; reverted sans-serif */
|
|
latexdef "<." as "\langle";
|
|
htmldef ">." as
|
|
"<IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT=' >.' TITLE='>.'>";
|
|
/* The Unicode below doesn't always work on Firefox and Chrome on Windows,
|
|
so revert to the image above */
|
|
althtmldef ">." as '⟩'; /* 〉 */
|
|
/* 2-Jan-2016 restored Unicode; reverted sans-serif */
|
|
latexdef ">." as "\rangle";
|
|
htmldef "U." as
|
|
"<IMG SRC='bigcup.gif' WIDTH=13 HEIGHT=19 ALT=' U.' TITLE='U.'>";
|
|
/* 20-Sep-2017 nm Add space after U. in althtmldef to improve "U. ran" */
|
|
althtmldef "U." as '<FONT SIZE="+1">∪</FONT> '; /* ⋃ */
|
|
/* xcup does not render, and #8899 renders as a small bold cup, on
|
|
Mozilla 1.7.3 on Windows XP */
|
|
/*althtmldef "U." as '⋃';*/ /* ⋃ */
|
|
latexdef "U." as "\bigcup";
|
|
htmldef "|^|" as
|
|
"<IMG SRC='bigcap.gif' WIDTH=13 HEIGHT=19 ALT=' |^|' TITLE='|^|'>";
|
|
/* 20-Sep-2017 nm Add space after |^| in althtmldef to improve "|^| ran" */
|
|
althtmldef "|^|" as '<FONT SIZE="+1">∩</FONT> '; /* ⋂ */
|
|
/*althtmldef "|^|" as '⋂';*/ /* ⋂ */
|
|
latexdef "|^|" as "\bigcap";
|
|
htmldef "U_" as
|
|
"<IMG SRC='_cupbar.gif' WIDTH=13 HEIGHT=19 ALT=' U_' TITLE='U_'>";
|
|
/* 20-Sep-2017 nm Add space after U_ in althtmldef to improve "U_ ran" */
|
|
althtmldef "U_" as '<U><FONT SIZE="+1">∪</FONT></U> '; /* ⋃ */
|
|
latexdef "U_" as "\underline{\bigcup}";
|
|
htmldef "|^|_" as
|
|
"<IMG SRC='_capbar.gif' WIDTH=13 HEIGHT=19 ALT=' |^|_' TITLE='|^|_'>";
|
|
/* 20-Sep-2017 nm Add space after |^|_ in althtmldef to improve "|^|_ ran" */
|
|
althtmldef "|^|_" as '<U><FONT SIZE="+1">∩</FONT></U> '; /* ⋂ */
|
|
latexdef "|^|_" as "\underline{\bigcap}";
|
|
htmldef "Disj_" as "<u>Disj</u> ";
|
|
althtmldef "Disj_" as "<u>Disj</u> ";
|
|
latexdef "Disj_" as "\operatorname{\underline{Disj}}";
|
|
htmldef "|->" as " <IMG SRC='mapsto.gif' WIDTH=15 HEIGHT=19 ALT=' |->'" +
|
|
" TITLE='|->'> ";
|
|
althtmldef "|->" as ' ↦ ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "|->" as "\mapsto";
|
|
htmldef "Tr" as
|
|
"<IMG SRC='_ctr.gif' WIDTH=16 HEIGHT=19 ALT=' Tr' TITLE='Tr'> ";
|
|
althtmldef "Tr" as 'Tr ';
|
|
latexdef "Tr" as "{\rm Tr}";
|
|
htmldef "_E" as
|
|
" <IMG SRC='rmce.gif' WIDTH=9 HEIGHT=19 ALT=' _E' TITLE='_E'> ";
|
|
althtmldef "_E" as ' E ';
|
|
latexdef "_E" as "{\rm E}";
|
|
htmldef "_I" as
|
|
" <IMG SRC='rmci.gif' WIDTH=4 HEIGHT=19 ALT=' _I' TITLE='_I'> ";
|
|
althtmldef "_I" as ' I ';
|
|
latexdef "_I" as "{\rm I}";
|
|
htmldef "Po" as
|
|
" <IMG SRC='_po.gif' WIDTH=16 HEIGHT=19 ALT=' Po' TITLE='Po'> ";
|
|
althtmldef "Po" as ' Po ';
|
|
latexdef "Po" as "{\rm Po}";
|
|
htmldef "Or" as
|
|
" <IMG SRC='_or.gif' WIDTH=18 HEIGHT=19 ALT=' Or' TITLE='Or'> ";
|
|
althtmldef "Or" as ' Or ';
|
|
latexdef "Or" as "{\rm Or}";
|
|
htmldef "FrFor" as "FrFor ";
|
|
althtmldef "FrFor" as ' FrFor ';
|
|
latexdef "FrFor" as "{\rm FrFor}";
|
|
htmldef "Fr" as
|
|
" <IMG SRC='_fr.gif' WIDTH=15 HEIGHT=19 ALT=' Fr' TITLE='Fr'> ";
|
|
althtmldef "Fr" as ' Fr ';
|
|
latexdef "Fr" as "{\rm Fr}";
|
|
htmldef "Se" as ' Se ';
|
|
althtmldef "Se" as ' Se ';
|
|
latexdef "Se" as "{\rm Se}";
|
|
htmldef "We" as
|
|
" <IMG SRC='_we.gif' WIDTH=21 HEIGHT=19 ALT=' We' TITLE='We'> ";
|
|
althtmldef "We" as ' We ';
|
|
latexdef "We" as "{\rm We}";
|
|
htmldef "Ord" as
|
|
"<IMG SRC='_ord.gif' WIDTH=26 HEIGHT=19 ALT=' Ord' TITLE='Ord'> ";
|
|
althtmldef "Ord" as 'Ord ';
|
|
latexdef "Ord" as "{\rm Ord}";
|
|
htmldef "On" as "<IMG SRC='_on.gif' WIDTH=20 HEIGHT=19 ALT=' On' TITLE='On'>";
|
|
althtmldef "On" as 'On';
|
|
latexdef "On" as "{\rm On}";
|
|
htmldef "Lim" as
|
|
"<IMG SRC='_lim.gif' WIDTH=26 HEIGHT=19 ALT=' Lim' TITLE='Lim'> ";
|
|
althtmldef "Lim" as 'Lim ';
|
|
latexdef "Lim" as "{\rm Lim}";
|
|
htmldef "suc" as
|
|
"<IMG SRC='_suc.gif' WIDTH=22 HEIGHT=19 ALT=' suc' TITLE='suc'> ";
|
|
althtmldef "suc" as 'suc ';
|
|
latexdef "suc" as "{\rm suc}";
|
|
htmldef "_om" as
|
|
"<IMG SRC='omega.gif' WIDTH=11 HEIGHT=19 ALT=' om' TITLE='om'>";
|
|
/*althtmldef "_om" as 'ω';*/
|
|
althtmldef "_om" as 'ω'; /* upright lowercase Greek omega */
|
|
latexdef "_om" as "{\rm \omega}";
|
|
htmldef "X." as
|
|
" <IMG SRC='times.gif' WIDTH=9 HEIGHT=19 ALT=' X.' TITLE='X.'> ";
|
|
althtmldef "X." as ' × ';
|
|
latexdef "X." as "\times";
|
|
htmldef "`'" as "<IMG SRC='_cnv.gif' WIDTH=10 HEIGHT=19 ALT=" + '"' + " `'" +
|
|
'"' + "TITLE=" + '"' + "`'" + '"' + ">";
|
|
althtmldef "`'" as '<FONT SIZE="-1"><SUP>◡</SUP></FONT>'; /* or 8995 */
|
|
latexdef "`'" as "{}^{\smallsmile}";
|
|
htmldef "dom" as
|
|
"<IMG SRC='_dom.gif' WIDTH=26 HEIGHT=19 ALT=' dom' TITLE='dom'> ";
|
|
althtmldef "dom" as 'dom ';
|
|
latexdef "dom" as "{\rm dom}";
|
|
htmldef "ran" as
|
|
"<IMG SRC='_ran.gif' WIDTH=22 HEIGHT=19 ALT=' ran' TITLE='ran'> ";
|
|
althtmldef "ran" as 'ran ';
|
|
latexdef "ran" as "{\rm ran}";
|
|
htmldef "|`" as " <IMG SRC='restriction.gif' WIDTH=5 HEIGHT=19 ALT=' |`'" +
|
|
" TITLE='|`'> ";
|
|
althtmldef "|`" as ' ↾ '; /* ↾ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "|`" as "\restriction";
|
|
htmldef '"' as "<IMG SRC='backquote.gif' WIDTH=7 HEIGHT=19 ALT=' " + '"' +
|
|
"' TITLE='" + '"' + "'>";
|
|
althtmldef '"' as ' “ '; /* “ */
|
|
latexdef '"' as "``";
|
|
htmldef "o." as
|
|
" <IMG SRC='circ.gif' WIDTH=8 HEIGHT=19 ALT=' o.' TITLE='o.'> ";
|
|
althtmldef "o." as ' ∘ ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "o." as "\circ";
|
|
htmldef "Rel" as
|
|
"<IMG SRC='_rel.gif' WIDTH=22 HEIGHT=19 ALT=' Rel' TITLE='Rel'> ";
|
|
althtmldef "Rel" as 'Rel ';
|
|
latexdef "Rel" as "{\rm Rel}";
|
|
htmldef "Fun" as
|
|
"<IMG SRC='_fun.gif' WIDTH=25 HEIGHT=19 ALT=' Fun' TITLE='Fun'> ";
|
|
althtmldef "Fun" as 'Fun ';
|
|
latexdef "Fun" as "{\rm Fun}";
|
|
htmldef "Fn" as
|
|
" <IMG SRC='_fn.gif' WIDTH=17 HEIGHT=19 ALT=' Fn' TITLE='Fn'> ";
|
|
althtmldef "Fn" as ' Fn ';
|
|
latexdef "Fn" as "{\rm Fn}";
|
|
htmldef ":" as "<IMG SRC='colon.gif' WIDTH=4 HEIGHT=19 ALT=' :' TITLE=':'>";
|
|
althtmldef ":" as ':';
|
|
latexdef ":" as ":";
|
|
htmldef "-->" as
|
|
"<IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 " +
|
|
"ALT=' -->' TITLE='-->'>";
|
|
/* althtmldef "-->" as '–→'; */
|
|
althtmldef "-->" as '⟶';
|
|
/* ­‐–—− (possible symbols test) */
|
|
latexdef "-->" as "\longrightarrow";
|
|
htmldef "-1-1->" as "<IMG SRC='onetoone.gif' WIDTH=23 HEIGHT=19 " +
|
|
"ALT=' -1-1->' TITLE='-1-1->'>";
|
|
althtmldef "-1-1->" as
|
|
'–<FONT SIZE=-2 FACE=sans-serif>1-1</FONT>→';
|
|
latexdef "-1-1->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm 1" +
|
|
"\tt -\rm 1}}}\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm {\ }}}}}$}";
|
|
htmldef "-onto->" as
|
|
"<IMG SRC='onto.gif' WIDTH=23 HEIGHT=19 " +
|
|
"ALT=' -onto->' TITLE='-onto->'>";
|
|
althtmldef "-onto->" as
|
|
'–<FONT SIZE=-2 FACE=sans-serif>onto</FONT>→';
|
|
latexdef "-onto->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm {\ }}}}" +
|
|
"\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm onto}}}}$}";
|
|
htmldef "-1-1-onto->" as "<IMG SRC='onetooneonto.gif' WIDTH=23 HEIGHT=19 " +
|
|
"ALT=' -1-1-onto->' TITLE='-1-1-onto->'>";
|
|
althtmldef "-1-1-onto->" as '–<FONT SIZE=-2 '
|
|
+ 'FACE=sans-serif>1-1</FONT>-<FONT SIZE=-2 '
|
|
+ 'FACE=sans-serif>onto</FONT>→';
|
|
latexdef "-1-1-onto->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm 1" +
|
|
"\tt -\rm 1}}}\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm onto}}}}$}";
|
|
htmldef "`" as
|
|
"<IMG SRC='backtick.gif' WIDTH=4 HEIGHT=19 ALT=' ` ' TITLE='` '>";
|
|
/* Above, IE7 _printing_ is corrupted by '`'; use '` ' which works */
|
|
/*althtmldef "`" as ' ‘';*/
|
|
/* I took out the leading space to make e.g. ( log ` A ) look better.
|
|
I added the leading space a long time ago because the quote overlapped
|
|
the character to its left, making it sometimes hidden, but that seems
|
|
to be no longer a problem with the XITS font. - 29-Aug-2017 nm */
|
|
althtmldef "`" as '‘';
|
|
latexdef "`" as "`";
|
|
htmldef "Isom" as
|
|
" <IMG SRC='_isom.gif' WIDTH=30 HEIGHT=19 ALT=' Isom' TITLE='Isom'> ";
|
|
althtmldef "Isom" as ' Isom ';
|
|
latexdef "Isom" as "{\rm Isom}";
|
|
htmldef "oF" as
|
|
" <IMG SRC='circ.gif' WIDTH=8 HEIGHT=19 ALT=' o' TITLE='o'>" +
|
|
"<IMG SRC='subf.gif' WIDTH=6 HEIGHT=19 ALT='F' TITLE='F'>";
|
|
althtmldef "oF" as " ∘<SUB>𝑓</SUB> ";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "oF" as "\circ_f";
|
|
htmldef "oR" as
|
|
" <IMG SRC='circ.gif' WIDTH=8 HEIGHT=19 ALT=' o' TITLE='o'>" +
|
|
"<IMG SRC='subr.gif' WIDTH=5 HEIGHT=19 ALT='R' TITLE='R'>";
|
|
althtmldef "oR" as " ∘<SUB>𝑟</SUB> ";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "oR" as "\circ_r";
|
|
htmldef "1st" as
|
|
"<IMG SRC='_1st.gif' WIDTH=15 HEIGHT=19 ALT=' 1st' TITLE='1st'>";
|
|
althtmldef "1st" as '1<SUP>st</SUP> ';
|
|
latexdef "1st" as "1^{\rm st}";
|
|
htmldef "2nd" as
|
|
"<IMG SRC='_2nd.gif' WIDTH=21 HEIGHT=19 ALT=' 2nd' TITLE='2nd'>";
|
|
althtmldef "2nd" as '2<SUP>nd</SUP> ';
|
|
latexdef "2nd" as "2^{\rm nd}";
|
|
htmldef "tpos" as "tpos ";
|
|
althtmldef "tpos" as 'tpos ';
|
|
latexdef "tpos" as "{\rm tpos}";
|
|
/*
|
|
htmldef "iota" as
|
|
"<IMG SRC='iota.gif' WIDTH=6 HEIGHT=19 ALT=' iota' TITLE='iota'>";
|
|
althtmldef "iota" as '<FONT SIZE="+1">ι</FONT>';
|
|
latexdef "iota" as "\iota";
|
|
*/
|
|
/* 30-Nov-2013 changed to rotated iota */
|
|
htmldef "iota" as
|
|
"<IMG SRC='riota.gif' WIDTH=6 HEIGHT=19 ALT=' iota' TITLE='iota'>";
|
|
althtmldef "iota" as '℩';
|
|
latexdef "iota" as "\mathrm{\rotatebox[origin=C]{180}{$\iota$}}";
|
|
htmldef "Smo" as
|
|
"<IMG SRC='_smo.gif' WIDTH=27 HEIGHT=19 ALT=' Smo' TITLE='Smo'> ";
|
|
althtmldef "Smo" as "Smo ";
|
|
latexdef "Smo" as "{\rm Smo}";
|
|
htmldef "recs" as "recs";
|
|
althtmldef "recs" as "recs";
|
|
latexdef "recs" as "\mathrm{recs}";
|
|
htmldef "rec" as
|
|
"<IMG SRC='_rec.gif' WIDTH=21 HEIGHT=19 ALT=' rec' TITLE='rec'>";
|
|
althtmldef "rec" as 'rec';
|
|
latexdef "rec" as "{\rm rec}";
|
|
htmldef "frec" as "frec";
|
|
althtmldef "frec" as "frec";
|
|
latexdef "frec" as "{\rm frec}";
|
|
htmldef "1o" as "<IMG SRC='_1o.gif' WIDTH=13 HEIGHT=19 ALT=' 1o' TITLE='1o'>";
|
|
althtmldef "1o" as '1<SUB>𝑜</SUB>';
|
|
latexdef "1o" as "1_o";
|
|
htmldef "2o" as "<IMG SRC='_2o.gif' WIDTH=14 HEIGHT=19 ALT=' 2o' TITLE='2o'>";
|
|
althtmldef "2o" as '2<SUB>𝑜</SUB>';
|
|
latexdef "2o" as "2_o";
|
|
htmldef "3o" as "<IMG SRC='_3o.gif' WIDTH=14 HEIGHT=19 ALT=' 3o' TITLE='3o'>";
|
|
althtmldef "3o" as "3<SUB>𝑜</SUB>"; latexdef "3o" as "3_o";
|
|
htmldef "4o" as "<IMG SRC='_4o.gif' WIDTH=15 HEIGHT=19 ALT=' 4o' TITLE='4o'>";
|
|
althtmldef "4o" as "4<SUB>𝑜</SUB>"; latexdef "4o" as "4_o";
|
|
htmldef "+o" as
|
|
" <IMG SRC='_plo.gif' WIDTH=18 HEIGHT=19 ALT=' +o' TITLE='+o'> ";
|
|
althtmldef "+o" as ' +<SUB>𝑜</SUB> ';
|
|
latexdef "+o" as "+_o";
|
|
htmldef ".o" as
|
|
" <IMG SRC='_cdo.gif' WIDTH=10 HEIGHT=19 ALT=' .o' TITLE='.o'> ";
|
|
althtmldef ".o" as ' ·<SUB>𝑜</SUB> ';
|
|
latexdef ".o" as "\cdot_o";
|
|
htmldef "^oi" as ' ↑<SUB>𝑜</SUB> ';
|
|
althtmldef "^oi" as ' ↑<SUB>𝑜</SUB> ';
|
|
latexdef "^oi" as "\uparrow_{oi}";
|
|
htmldef "Er" as
|
|
" <IMG SRC='_er.gif' WIDTH=16 HEIGHT=19 ALT=' Er' TITLE='Er'> ";
|
|
althtmldef "Er" as ' Er ';
|
|
latexdef "Er" as "{\rm Er}";
|
|
htmldef "/." as
|
|
"<IMG SRC='diagup.gif' WIDTH=14 HEIGHT=19 ALT=' /.' TITLE='/.'>";
|
|
althtmldef "/." as ' <B>/</B> ';
|
|
latexdef "/." as "\diagup";
|
|
htmldef "~~" as
|
|
" <IMG SRC='approx.gif' WIDTH=13 HEIGHT=19 ALT=' ~~' TITLE='~~'> ";
|
|
althtmldef "~~" as ' ≈ '; /* ≈ */
|
|
latexdef "~~" as "\approx";
|
|
htmldef "~<_" as
|
|
" <IMG SRC='preccurlyeq.gif' WIDTH=11 HEIGHT=19 " +
|
|
"ALT=' ~<_' TITLE='~<_'> ";
|
|
althtmldef "~<_" as ' ≼ '; /* ≼ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "~<_" as "\preccurlyeq";
|
|
htmldef "Fin" as
|
|
"<IMG SRC='_fin.gif' WIDTH=21 HEIGHT=19 ALT=' Fin' TITLE='Fin'>";
|
|
althtmldef "Fin" as 'Fin';
|
|
latexdef "Fin" as "{\rm Fin}";
|
|
htmldef "iota_" as
|
|
"<IMG SRC='_riotabar.gif' WIDTH=6 HEIGHT=19 ALT=' iota_' TITLE='iota_'>";
|
|
althtmldef "iota_" as '<U>℩</U>';
|
|
latexdef "iota_" as
|
|
"\underline{\mathrm{\rotatebox[origin=C]{180}{$\iota$}}}";
|
|
htmldef "sup" as
|
|
"<IMG SRC='_sup.gif' WIDTH=22 HEIGHT=19 ALT=' sup' TITLE='sup'>";
|
|
althtmldef "sup" as 'sup';
|
|
latexdef "sup" as "{\rm sup}";
|
|
htmldef "inf" as 'inf';
|
|
althtmldef "inf" as 'inf';
|
|
latexdef "inf" as "{\rm inf}";
|
|
htmldef "card" as
|
|
"<IMG SRC='_card.gif' WIDTH=30 HEIGHT=19 ALT=' card' TITLE='card'>";
|
|
althtmldef "card" as 'card';
|
|
latexdef "card" as "{\rm card}";
|
|
htmldef "N." as "<IMG SRC='caln.gif' WIDTH=17 HEIGHT=19 ALT=' N.' TITLE='N.'>";
|
|
althtmldef "N." as '<I><B>N</B></I>';
|
|
latexdef "N." as "{\cal N}";
|
|
htmldef "+N" as
|
|
" <IMG SRC='_pln.gif' WIDTH=22 HEIGHT=19 ALT=' +N' TITLE='+N'> ";
|
|
althtmldef "+N" as ' +<I><SUB><B>N</B></SUB></I> ';
|
|
latexdef "+N" as "+_{\cal N}";
|
|
htmldef ".N" as
|
|
" <IMG SRC='_cdn.gif' WIDTH=14 HEIGHT=19 ALT=' .N' TITLE='.N'> ";
|
|
althtmldef ".N" as ' ·<I><SUB><B>N</B></SUB></I> ';
|
|
latexdef ".N" as "\cdot_{\cal N}";
|
|
htmldef "<N" as
|
|
" <IMG SRC='_ltn.gif' WIDTH=21 HEIGHT=19 ALT=' <N' TITLE='<N'> ";
|
|
althtmldef "<N" as ' <<I><SUB><B>N</B></SUB></I> ';
|
|
latexdef "<N" as "<_{\cal N}";
|
|
htmldef "+pQ" as
|
|
" <IMG SRC='_plpq.gif' WIDTH=28 HEIGHT=19 ALT=' +pQ' TITLE='+pQ'> ";
|
|
althtmldef "+pQ" as ' +<I><SUB>p<B>Q</B></SUB></I> ';
|
|
latexdef "+pQ" as "+_{p{\cal Q}}";
|
|
htmldef ".pQ" as
|
|
" <IMG SRC='_cdpq.gif' WIDTH=19 HEIGHT=19 ALT=' .pQ' TITLE='.pQ'> ";
|
|
althtmldef ".pQ" as ' ·<I><SUB>p<B>Q</B></SUB></I> ';
|
|
latexdef ".pQ" as "\cdot_{p{\cal Q}}";
|
|
htmldef "<pQ" as
|
|
" <IMG SRC='_ltpq.gif' WIDTH=27 HEIGHT=19 ALT=' <pQ' TITLE='<pQ'> ";
|
|
althtmldef "<pQ" as ' <<I><SUB>p<B>Q</B></SUB></I> ';
|
|
latexdef "<pQ" as "<_{p{\cal Q}}";
|
|
htmldef "~Q" as
|
|
" <IMG SRC='_simq.gif' WIDTH=21 HEIGHT=19 ALT=' ~Q' TITLE='~Q'> ";
|
|
althtmldef "~Q" as ' ~<I><SUB><B>Q</B></SUB></I> ';
|
|
latexdef "~Q" as "\sim_{\cal Q}";
|
|
htmldef "Q." as "<IMG SRC='calq.gif' WIDTH=12 HEIGHT=19 ALT=' Q.' TITLE='Q.'>";
|
|
althtmldef "Q." as '<I><B>Q</B></I>';
|
|
latexdef "Q." as "{\cal Q}";
|
|
htmldef "1Q" as "<IMG SRC='_1q.gif' WIDTH=16 HEIGHT=19 ALT=' 1Q' TITLE='1Q'>";
|
|
althtmldef "1Q" as '1<I><SUB><B>Q</B></SUB></I>';
|
|
latexdef "1Q" as "1_{\cal Q}";
|
|
htmldef "+Q" as
|
|
" <IMG SRC='_plq.gif' WIDTH=21 HEIGHT=19 ALT=' +Q' TITLE='+Q'> ";
|
|
althtmldef "+Q" as ' +<I><SUB><B>Q</B></SUB></I> ';
|
|
latexdef "+Q" as "+_{\cal Q}";
|
|
htmldef ".Q" as
|
|
" <IMG SRC='_cdq.gif' WIDTH=13 HEIGHT=19 ALT=' .Q' TITLE='.Q'> ";
|
|
althtmldef ".Q" as ' ·<I><SUB><B>Q</B></SUB></I> ';
|
|
latexdef ".Q" as "\cdot_{\cal Q}";
|
|
htmldef "*Q" as
|
|
"<IMG SRC='_astq.gif' WIDTH=16 HEIGHT=19 ALT=' *Q' TITLE='*Q'>";
|
|
althtmldef "*Q" as '*<I><SUB><B>Q</B></SUB></I>';
|
|
latexdef "*Q" as "\ast_{\cal Q}";
|
|
htmldef "<Q" as
|
|
" <IMG SRC='_ltq.gif' WIDTH=20 HEIGHT=19 ALT=' <Q' TITLE='<Q'> ";
|
|
althtmldef "<Q" as ' <<I><SUB><B>Q</B></SUB></I> ';
|
|
latexdef "<Q" as "<_{\cal Q}";
|
|
htmldef "~Q0" as ' ~<I><SUB><B>Q0</B></SUB></I> ';
|
|
althtmldef "~Q0" as ' ~<I><SUB><B>Q0</B></SUB></I> ';
|
|
latexdef "~Q0" as "\sim_{\cal Q0}";
|
|
htmldef "Q0." as '<I><B>Q<SUB>0</SUB></B></I>';
|
|
althtmldef "Q0." as '<I><B>Q<SUB>0</SUB></B></I>';
|
|
latexdef "Q0." as "{\cal Q}_0";
|
|
htmldef "0Q0" as '0<I><SUB><B>Q0</B></SUB></I>';
|
|
althtmldef "0Q0" as '0<I><SUB><B>Q0</B></SUB></I>';
|
|
latexdef "0Q0" as "0_{\cal Q0}";
|
|
htmldef "+Q0" as ' +<I><SUB><B>Q0</B></SUB></I> ';
|
|
althtmldef "+Q0" as ' +<I><SUB><B>Q0</B></SUB></I> ';
|
|
latexdef "+Q0" as "+_{\cal Q0}";
|
|
htmldef ".Q0" as ' ·<I><SUB><B>Q0</B></SUB></I> ';
|
|
althtmldef ".Q0" as ' ·<I><SUB><B>Q0</B></SUB></I> ';
|
|
latexdef ".Q0" as "\cdot_{\cal Q0}";
|
|
htmldef "P." as "<IMG SRC='calp.gif' WIDTH=13 HEIGHT=19 ALT=' P.' TITLE='P.'>";
|
|
althtmldef "P." as '<I><B>P</B></I>';
|
|
latexdef "P." as "{\cal P}";
|
|
htmldef "1P" as "<IMG SRC='_1p.gif' WIDTH=15 HEIGHT=19 ALT=' 1P' TITLE='1P'>";
|
|
althtmldef "1P" as '1<I><SUB><B>P</B></SUB></I>';
|
|
latexdef "1P" as "1_{\cal P}";
|
|
htmldef "+P." as
|
|
" <IMG SRC='_plp.gif' WIDTH=22 HEIGHT=19 ALT=' +P.' TITLE='+P.'> ";
|
|
althtmldef "+P." as ' +<I><SUB><B>P</B></SUB></I> ';
|
|
latexdef "+P." as "+_{\cal P}";
|
|
htmldef ".P." as
|
|
" <IMG SRC='_cdp.gif' WIDTH=13 HEIGHT=19 ALT=' .P.' TITLE='.P.'> ";
|
|
althtmldef ".P." as ' ·<I><SUB><B>P</B></SUB></I> ';
|
|
latexdef ".P." as "\cdot_{\cal P}";
|
|
htmldef "<P" as
|
|
" <IMG SRC='_ltp.gif' WIDTH=19 HEIGHT=19 ALT=' <P' TITLE='<P'> ";
|
|
althtmldef "<P" as '<<I><SUB><B>P</B></SUB></I> ';
|
|
latexdef "<P" as "<_{\cal P}";
|
|
htmldef "~R" as
|
|
" <IMG SRC='_simr.gif' WIDTH=23 HEIGHT=19 ALT=' ~R' TITLE='~R'> ";
|
|
althtmldef "~R" as ' ~<I><SUB><B>R</B></SUB></I> ';
|
|
latexdef "~R" as "\sim_{\cal R}";
|
|
htmldef "R." as "<IMG SRC='calr.gif' WIDTH=15 HEIGHT=19 ALT=' R.' TITLE='R.'>";
|
|
althtmldef "R." as '<I><B>R</B></I>';
|
|
latexdef "R." as "{\cal R}";
|
|
htmldef "0R" as "<IMG SRC='_0r.gif' WIDTH=18 HEIGHT=19 ALT=' 0R' TITLE='0R'>";
|
|
althtmldef "0R" as '0<I><SUB><B>R</B></SUB></I>';
|
|
latexdef "0R" as "0_{\cal R}";
|
|
htmldef "1R" as "<IMG SRC='_1cr.gif' WIDTH=16 HEIGHT=19 ALT=' 1R' TITLE='1R'>";
|
|
althtmldef "1R" as '1<I><SUB><B>R</B></SUB></I>';
|
|
latexdef "1R" as "1_{\cal R}";
|
|
htmldef "-1R" as
|
|
"<IMG SRC='_m1r.gif' WIDTH=22 HEIGHT=19 ALT=' -1R' TITLE='-1R'>";
|
|
althtmldef "-1R" as '-1<I><SUB><B>R</B></SUB></I>';
|
|
latexdef "-1R" as "-1_{\cal R}";
|
|
htmldef "+R" as
|
|
" <IMG SRC='_plr.gif' WIDTH=23 HEIGHT=19 ALT=' +R' TITLE='+R'> ";
|
|
althtmldef "+R" as ' +<I><SUB><B>R</B></SUB></I> ';
|
|
latexdef "+R" as "+_{\cal R}";
|
|
htmldef ".R" as
|
|
" <IMG SRC='_cdcr.gif' WIDTH=14 HEIGHT=19 ALT=' .R' TITLE='.R'> ";
|
|
althtmldef ".R" as ' ·<I><SUB><B>R</B></SUB></I> ';
|
|
latexdef ".R" as "\cdot_{\cal R}";
|
|
htmldef "<R" as
|
|
" <IMG SRC='_ltr.gif' WIDTH=20 HEIGHT=19 ALT=' <R' TITLE='<R'> ";
|
|
althtmldef "<R" as ' <<I><SUB><B>R</B></SUB></I> ';
|
|
latexdef "<R" as "<_{\cal R}";
|
|
htmldef "<RR" as
|
|
" <IMG SRC='_ltbbr.gif' WIDTH=20 HEIGHT=19 ALT=' <RR' TITLE='<RR'> ";
|
|
althtmldef "<RR" as ' <<SUB>ℝ</SUB> ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "<RR" as "<_\mathbb{R}";
|
|
htmldef "CC" as "<IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT=' CC' TITLE='CC'>";
|
|
althtmldef "CC" as 'ℂ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "CC" as "\mathbb{C}";
|
|
htmldef "RR" as "<IMG SRC='bbr.gif' WIDTH=13 HEIGHT=19 ALT=' RR' TITLE='RR'>";
|
|
althtmldef "RR" as 'ℝ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "RR" as "\mathbb{R}";
|
|
/*latexdef "" as "_{10}";*/
|
|
/*latexdef "" as "";*/
|
|
/* suppress base 10 suffix */
|
|
htmldef "0" as "<IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT=' 0' TITLE='0'>";
|
|
althtmldef "0" as '0';
|
|
latexdef "0" as "0";
|
|
htmldef "1" as "<IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT=' 1' TITLE='1'>";
|
|
althtmldef "1" as '1';
|
|
latexdef "1" as "1";
|
|
htmldef "_i" as "<IMG SRC='rmi.gif' WIDTH=4 HEIGHT=19 ALT=' _i' TITLE='_i'>";
|
|
althtmldef "_i" as 'i';
|
|
latexdef "_i" as "{\rm i}";
|
|
htmldef "+" as " <IMG SRC='plus.gif' WIDTH=13 HEIGHT=19 ALT=' +' TITLE='+'> ";
|
|
althtmldef "+" as ' + ';
|
|
latexdef "+" as "+";
|
|
htmldef "x." as
|
|
" <IMG SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT=' x.' TITLE='x.'> ";
|
|
althtmldef "x." as ' · '; /* unicode: · */
|
|
latexdef "x." as "\cdot";
|
|
htmldef "<_" as
|
|
" <IMG SRC='le.gif' WIDTH=11 HEIGHT=19 ALT=' <_' TITLE='<_'> ";
|
|
althtmldef "<_" as ' ≤ ';
|
|
latexdef "<_" as "\le";
|
|
htmldef "+oo" as " <IMG SRC='_pinf.gif' WIDTH=29 HEIGHT=19 ALT='+oo' " +
|
|
"TITLE='+oo'>";
|
|
althtmldef "+oo" as '+∞';
|
|
latexdef "+oo" as "+\infty";
|
|
htmldef "-oo" as " <IMG SRC='_minf.gif' WIDTH=24 HEIGHT=19 ALT='-oo' " +
|
|
"TITLE='-oo'>";
|
|
althtmldef "-oo" as '-∞';
|
|
latexdef "-oo" as "-\infty";
|
|
htmldef "RR*" as "<IMG SRC='_bbrast.gif' WIDTH=18 HEIGHT=19 ALT=' RR*' " +
|
|
"TITLE='RR*'>";
|
|
althtmldef "RR*" as 'ℝ<SUP>*</SUP>';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "RR*" as "\mathbb{R}^*";
|
|
htmldef "<" as
|
|
" <IMG SRC='lt.gif' WIDTH=11 HEIGHT=19 ALT=' <' TITLE='<'> ";
|
|
althtmldef "<" as ' < ';
|
|
latexdef "<" as "<";
|
|
htmldef "-" as
|
|
" <IMG SRC='minus.gif' WIDTH=11 HEIGHT=19 ALT=' -' TITLE='-'> ";
|
|
althtmldef "-" as ' − ';
|
|
latexdef "-" as "-";
|
|
htmldef "-u" as
|
|
"<IMG SRC='shortminus.gif' WIDTH=8 HEIGHT=19 ALT=' -u' TITLE='-u'>";
|
|
/* use standard minus sign */
|
|
althtmldef "-u" as '-';
|
|
latexdef "-u" as "\textrm{-}"; /* short minus */
|
|
/*latexdef "-u" as "-_u";*/
|
|
htmldef "#" as ' # ';
|
|
althtmldef "#" as ' # ';
|
|
latexdef "#" as "\apart";
|
|
htmldef "#RR" as ' #<SUB>ℝ</SUB> ';
|
|
althtmldef "#RR" as ' #<SUB>ℝ</SUB> ';
|
|
latexdef "#RR" as "\apart_\mathbb{R}";
|
|
htmldef "NN" as "<IMG SRC='bbn.gif' WIDTH=12 HEIGHT=19 ALT=' NN' TITLE='NN'>";
|
|
althtmldef "NN" as 'ℕ'; /* ℕ */
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "NN" as "\mathbb{N}";
|
|
htmldef "NN0" as
|
|
"<IMG SRC='_bbn0.gif' WIDTH=19 HEIGHT=19 ALT=' NN0' TITLE='NN0'>";
|
|
althtmldef "NN0" as 'ℕ<SUB>0</SUB>';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "NN0" as "\mathbb{N}_0";
|
|
htmldef "NN0*" as 'NN0*';
|
|
althtmldef "NN0*" as 'ℕ<SUB>0</SUB><SUP>*</SUP>';
|
|
latexdef "NN0*" as "\mathbb{N}_0^*";
|
|
htmldef "ZZ" as "<IMG SRC='bbz.gif' WIDTH=11 HEIGHT=19 ALT=' ZZ' TITLE='ZZ'>";
|
|
althtmldef "ZZ" as 'ℤ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "ZZ" as "\mathbb{Z}";
|
|
htmldef "QQ" as "<IMG SRC='bbq.gif' WIDTH=13 HEIGHT=19 ALT=' QQ' TITLE='QQ'>";
|
|
althtmldef "QQ" as 'ℚ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "QQ" as "\mathbb{Q}";
|
|
htmldef "RR+" as "<IMG SRC='_bbrplus.gif' WIDTH=20 HEIGHT=19 ALT=' RR+' " +
|
|
"TITLE='RR+'>";
|
|
althtmldef "RR+" as 'ℝ<SUP>+</SUP>';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "RR+" as "\mathbb{R}^+";
|
|
htmldef "2" as "<IMG SRC='2.gif' WIDTH=8 HEIGHT=19 ALT=' 2' TITLE='2'>";
|
|
althtmldef "2" as '2';
|
|
latexdef "2" as "2";
|
|
htmldef "3" as "<IMG SRC='3.gif' WIDTH=8 HEIGHT=19 ALT=' 3' TITLE='3'>";
|
|
althtmldef "3" as '3';
|
|
latexdef "3" as "3";
|
|
htmldef "4" as "<IMG SRC='4.gif' WIDTH=9 HEIGHT=19 ALT=' 4' TITLE='4'>";
|
|
althtmldef "4" as '4';
|
|
latexdef "4" as "4";
|
|
htmldef "5" as "<IMG SRC='5.gif' WIDTH=8 HEIGHT=19 ALT=' 5' TITLE='5'>";
|
|
althtmldef "5" as '5';
|
|
latexdef "5" as "5";
|
|
htmldef "6" as "<IMG SRC='6.gif' WIDTH=8 HEIGHT=19 ALT=' 6' TITLE='6'>";
|
|
althtmldef "6" as '6';
|
|
latexdef "6" as "6";
|
|
htmldef "7" as "<IMG SRC='7.gif' WIDTH=9 HEIGHT=19 ALT=' 7' TITLE='7'>";
|
|
althtmldef "7" as '7';
|
|
latexdef "7" as "7";
|
|
htmldef "8" as "<IMG SRC='8.gif' WIDTH=8 HEIGHT=19 ALT=' 8' TITLE='8'>";
|
|
althtmldef "8" as '8';
|
|
latexdef "8" as "8";
|
|
htmldef "9" as "<IMG SRC='9.gif' WIDTH=8 HEIGHT=19 ALT=' 9' TITLE='9'>";
|
|
althtmldef "9" as '9';
|
|
latexdef "9" as "9";
|
|
htmldef "10" as "<IMG SRC='_10.gif' WIDTH=14 HEIGHT=19 ALT=' 10' TITLE='10'>";
|
|
althtmldef "10" as '10';
|
|
latexdef "10" as "10";
|
|
htmldef ";" as '<FONT COLOR="#808080">;</FONT>';
|
|
althtmldef ";" as '<SPAN CLASS=hidden STYLE="color:gray">;</SPAN>';
|
|
latexdef ";" as "{\rm;}";
|
|
htmldef "ZZ>=" as "<IMG SRC='_bbzge.gif' WIDTH=20 HEIGHT=19 " +
|
|
"ALT=' ZZ>=' TITLE='ZZ>='>";
|
|
althtmldef "ZZ>=" as "ℤ<SUB>≥</SUB>";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "ZZ>=" as "\mathbb{Z}_\ge";
|
|
htmldef "-e" as " <IMG SRC='shortminus.gif' WIDTH=8 HEIGHT=19 ALT=' -' " +
|
|
"TITLE='-'><IMG SRC='sube.gif' WIDTH=6 HEIGHT=19 ALT='e' TITLE='e'>";
|
|
althtmldef "-e" as "-<SUB>𝑒</SUB>";
|
|
latexdef "-e" as "\textrm{-}_e";
|
|
htmldef "+e" as "<IMG SRC='plus.gif' WIDTH=13 HEIGHT=19 ALT=' +' TITLE='+'>" +
|
|
"<IMG SRC='sube.gif' WIDTH=6 HEIGHT=19 ALT='e' TITLE='e'>";
|
|
althtmldef "+e" as " +<SUB>𝑒</SUB> ";
|
|
latexdef "+e" as "+_e";
|
|
htmldef "*e" as "<IMG SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT=' x' TITLE='x'>" +
|
|
"<IMG SRC='sube.gif' WIDTH=6 HEIGHT=19 ALT='e' TITLE='e'>";
|
|
althtmldef "*e" as " ·<SUB>e</SUB> ";
|
|
latexdef "*e" as "\cdot_e";
|
|
htmldef "(,)" as
|
|
"<IMG SRC='_ioo.gif' WIDTH=13 HEIGHT=19 ALT=' (,)' TITLE='(,)'>";
|
|
althtmldef "(,)" as "(,)";
|
|
latexdef "(,)" as "(,)";
|
|
htmldef "(,]" as
|
|
"<IMG SRC='_ioc.gif' WIDTH=12 HEIGHT=19 ALT=' (,]' TITLE='(,]'>";
|
|
althtmldef "(,]" as "(,]";
|
|
latexdef "(,]" as "(,]";
|
|
htmldef "[,)" as
|
|
"<IMG SRC='_ico.gif' WIDTH=13 HEIGHT=19 ALT=' [,)' TITLE='[,)'>";
|
|
althtmldef "[,)" as "[,)";
|
|
latexdef "[,)" as "[,)";
|
|
htmldef "[,]" as
|
|
"<IMG SRC='_icc.gif' WIDTH=12 HEIGHT=19 ALT=' [,]' TITLE='[,]'>";
|
|
althtmldef "[,]" as "[,]";
|
|
latexdef "[,]" as "[,]";
|
|
htmldef "..." as "<IMG SRC='ldots.gif' WIDTH=18 HEIGHT=19 " +
|
|
"ALT=' ...' TITLE='...'>";
|
|
althtmldef "..." as "...";
|
|
latexdef "..." as "\ldots";
|
|
htmldef "..^" as "..^";
|
|
althtmldef "..^" as "..^";
|
|
latexdef "..^" as "..\^";
|
|
htmldef "|_" as "<IMG SRC='lfloor.gif' WIDTH=6 HEIGHT=19 ALT=' |_' " +
|
|
"TITLE='|_'>";
|
|
althtmldef "|_" as '⌊';
|
|
latexdef "|_" as "\lfloor";
|
|
htmldef "|^" as "⌈";
|
|
althtmldef "|^" as "⌈";
|
|
latexdef "|^" as "\lceil";
|
|
htmldef "mod" as " <IMG SRC='_mod.gif' WIDTH=29 HEIGHT=19 ALT=' mod' " +
|
|
"TITLE='mod'> ";
|
|
althtmldef "mod" as ' mod ';
|
|
latexdef "mod" as "{\rm mod}";
|
|
htmldef "==" as " <IMG SRC='equiv.gif' WIDTH=12 HEIGHT=19 ALT=' ==' " +
|
|
"TITLE='=='> ";
|
|
althtmldef "==" as "≡"; /* 2263 */
|
|
latexdef "==" as "\equiv";
|
|
htmldef "seq" as " <IMG SRC='_seq.gif' WIDTH=22 HEIGHT=19 ALT=' seq' " +
|
|
"TITLE='seq'>";
|
|
althtmldef "seq" as 'seq';
|
|
latexdef "seq" as "{\rm seq}";
|
|
htmldef "^" as "<IMG SRC='uparrow.gif' WIDTH=7 HEIGHT=19 ALT=' ^' TITLE='^'>";
|
|
althtmldef "^" as '↑';
|
|
latexdef "^" as "\uparrow"; /*
|
|
latexdef "^" as "\widehat{\ }"; */
|
|
htmldef "!" as "<IMG SRC='bang.gif' WIDTH=3 HEIGHT=19 ALT=' !' TITLE='!'>";
|
|
althtmldef "!" as '!';
|
|
latexdef "!" as "{!}";
|
|
htmldef "_C" as
|
|
" <IMG SRC='rmcc.gif' WIDTH=10 HEIGHT=19 ALT=' _C' TITLE='_C'> ";
|
|
althtmldef "_C" as 'C';
|
|
latexdef "_C" as "{\rm C}";
|
|
htmldef "size" as "♯";
|
|
althtmldef "size" as "♯";
|
|
latexdef "size" as "\sharp";
|
|
htmldef "shift" as " <IMG SRC='_shift.gif' WIDTH=30 HEIGHT=19 ALT=' shift' " +
|
|
"TITLE='shift'> ";
|
|
althtmldef "shift" as ' shift ';
|
|
latexdef "shift" as "{\rm shift}";
|
|
htmldef "Re" as "<IMG SRC='re.gif' WIDTH=12 HEIGHT=19 ALT=' Re' TITLE='Re'>";
|
|
althtmldef "Re" as 'ℜ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "Re" as "\Re";
|
|
htmldef "Im" as "<IMG SRC='im.gif' WIDTH=12 HEIGHT=19 ALT=' Im' TITLE='Im'>";
|
|
althtmldef "Im" as 'ℑ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "Im" as "\Im";
|
|
htmldef "*" as "<IMG SRC='ast.gif' WIDTH=7 HEIGHT=19 ALT=' *' TITLE='*'>";
|
|
althtmldef "*" as '∗';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "*" as "*";
|
|
htmldef "sqrt" as
|
|
"<IMG SRC='surd.gif' WIDTH=14 HEIGHT=19 ALT=' sqr' TITLE='sqr'>";
|
|
althtmldef "sqrt" as '√';
|
|
latexdef "sqrt" as "\surd";
|
|
htmldef "abs" as
|
|
"<IMG SRC='_abs.gif' WIDTH=22 HEIGHT=19 ALT=' abs' TITLE='abs'>";
|
|
althtmldef "abs" as 'abs';
|
|
latexdef "abs" as "{\rm abs}";
|
|
htmldef "+-" as
|
|
"<IMG SRC='pm.gif' WIDTH=14 HEIGHT=19 ALT=' pm' TITLE='pm'>";
|
|
althtmldef "+-" as '±';
|
|
latexdef "+-" as "\pm";
|
|
htmldef "~~>" as " <IMG SRC='rightsquigarrow.gif' WIDTH=15 HEIGHT=19 " +
|
|
"ALT=' ~~>' TITLE='~~>'> ";
|
|
althtmldef "~~>" as ' ⇝ ';
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "~~>" as '\rightsquigarrow';
|
|
htmldef "sum_" as "<IMG SRC='csigma.gif' WIDTH=11 HEIGHT=19 " +
|
|
"ALT=' sum_' TITLE='sum_'>";
|
|
althtmldef "sum_" as "Σ";
|
|
latexdef "sum_" as "\Sigma";
|
|
htmldef "||" as
|
|
" <IMG SRC='parallel.gif' WIDTH=5 HEIGHT=19 " + "ALT=' ||' TITLE='||'> ";
|
|
althtmldef "||" as ' ∥ ';
|
|
latexdef "||" as " \parallel ";
|
|
htmldef "gcd" as
|
|
" <IMG SRC='_gcd.gif' WIDTH=23 HEIGHT=19 ALT=' gcd' TITLE='gcd'> ";
|
|
althtmldef "gcd" as " gcd ";
|
|
latexdef "gcd" as "\,{\rm gcd}\,";
|
|
htmldef "lcm" as " lcm ";
|
|
althtmldef "lcm" as " lcm ";
|
|
latexdef "lcm" as "\,{\rm lcm}\,";
|
|
htmldef "Prime" as
|
|
"<IMG SRC='bbp.gif' WIDTH=11 HEIGHT=19 ALT=' Prime' TITLE='Prime'>";
|
|
althtmldef "Prime" as "ℙ";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "Prime" as "\mathbb{P}";
|
|
htmldef "STAB" as "<SMALL>STAB</SMALL> ";
|
|
althtmldef "STAB" as "<SMALL>STAB</SMALL> ";
|
|
latexdef "STAB" as "\mathrm{STAB} ";
|
|
htmldef "DECID" as "<SMALL>DECID</SMALL> ";
|
|
althtmldef "DECID" as "<SMALL>DECID</SMALL> ";
|
|
latexdef "DECID" as "\mathrm{DECID} ";
|
|
|
|
/* htmldef, althtmldef, latexdef for mathboxes */
|
|
/* Note the "Mathbox of" instead of "Mathbox for" to make searching easier. */
|
|
|
|
/* Mathbox of BJ */
|
|
htmldef "DECID_in" as " <SMALL><U>DECID</U><SUB>in</SUB></SMALL> ";
|
|
althtmldef "DECID_in" as " <SMALL><U>DECID</U><SUB>in</SUB></SMALL> ";
|
|
latexdef "DECID_in" as
|
|
" \normalfont\textsc{\underline{decid}$_\mathrm{in}$}} ";
|
|
htmldef "Delta0" as "Δ<sub>0</sub>";
|
|
althtmldef "Delta0" as "Δ<sub>0</sub>";
|
|
latexdef "Delta0" as "\Delta_0 ";
|
|
htmldef "Bdd" as "<SMALL>BOUNDED</SMALL> ";
|
|
althtmldef "Bdd" as "<SMALL>BOUNDED</SMALL> ";
|
|
latexdef "Bdd" as "\normalfont\textsc{bounded}} ";
|
|
htmldef "Bdd_" as "<SMALL><U>BOUNDED</U></SMALL> ";
|
|
althtmldef "Bdd_" as "<SMALL><U>BOUNDED</U></SMALL> ";
|
|
latexdef "Bdd_" as "\normalfont\textsc{\underline{bounded}}} ";
|
|
htmldef "Ind" as "Ind ";
|
|
althtmldef "Ind" as "Ind ";
|
|
latexdef "Ind" as "\mathrm{Ind} ";
|
|
/* End of BJ's mathbox */
|
|
|
|
/* Mathbox of David A. Wheeler */
|
|
htmldef "A!" as
|
|
"<IMG SRC='forall.gif' WIDTH=10 HEIGHT=19 ALT=' A.' TITLE='A.'>!";
|
|
althtmldef "A!" as '∀!'; /* ∀ */
|
|
latexdef "A!" as "\forall !";
|
|
/* End of David A. Wheeler's mathbox */
|
|
|
|
/* End of typesetting definition section */
|
|
$)
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
|
|
###############################################################################
|
|
$)
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Mathboxes for user contributions
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Mathbox guidelines
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
mathobox.1 $e |- ph $.
|
|
$( (_This theorem is a dummy placeholder for these guidelines. The name of
|
|
this theorem, "mathbox", is hard-coded into the Metamath program to
|
|
identify the start of the mathbox section for web page generation_.)
|
|
|
|
A "mathbox" is a user-contributed section that is maintained by its
|
|
contributor independently from the main part of set.mm.
|
|
|
|
_For contributors_:
|
|
|
|
By making a contribution, you agree to release it into the public
|
|
domain, according to the statement at the beginning of set.mm.
|
|
|
|
Mathboxes are provided to help keep your work synchronized with changes
|
|
in set.mm, but they shouldn't be depended on as a permanent archive. If
|
|
you want to preserve your original contribution, it is your
|
|
responsibility to keep your own copy of it along with the version of
|
|
set.mm that works with it.
|
|
|
|
_Guidelines_:
|
|
|
|
1. If at all possible, please use only nullary class constants for new
|
|
definitions.
|
|
|
|
2. Try to follow the style of the rest of set.mm. Each $p and $a
|
|
statement must be immediately preceded with the comment that will be
|
|
shown on its web page description. The metamath program command "write
|
|
source set.mm /rewrap" will take care of wrapping comment lines and
|
|
indentation conventions. All mathbox content will be on public display
|
|
and should hopefully reflect the overall quality of the website.
|
|
|
|
3. Before submitting a revised mathbox, please make sure it verifies
|
|
against the current set.mm.
|
|
|
|
4. Mathboxes should be independent i.e. the proofs should verify with
|
|
all other mathboxes removed. If you need a theorem from another
|
|
mathbox, that is fine (and encouraged), but let me know, so I can move
|
|
the theorem to the main section. One way avoid undesired accidental use
|
|
of other mathbox theorems is to develop your mathbox using a modified
|
|
set.mm that has mathboxes removed.
|
|
|
|
_Notes_:
|
|
|
|
1. We may decide to move some theorems to the main part of set.mm for
|
|
general use.
|
|
|
|
2. We may make changes to mathboxes to maintain the overall quality of
|
|
set.mm. Normally we will let you know if a change might impact what you
|
|
are working on.
|
|
|
|
3. If you use theorems from another user's mathbox, we don't provide
|
|
assurance that they are based on correct or consistent $a statements.
|
|
(If you find such a problem, please let us know so it can be corrected.)
|
|
(Contributed by NM, 20-Feb-2007.) (New usage is discouraged.) $)
|
|
mathbox $p |- ph $=
|
|
( ) B $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Mathbox for BJ
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Double negation of excluded middle. Intuitionistic logic refutes the
|
|
negation of excluded middle (but, of course, does not prove excluded
|
|
middle) for any formula. (Contributed by BJ, 9-Oct-2019.) $)
|
|
nnexmid $p |- -. -. ( ph \/ -. ph ) $=
|
|
( wn wo wa pm3.24 ioran mtbir ) AABZCBHHBDHEAHFG $.
|
|
|
|
$( Double negation of decidability of a formula. Intuitionistic logic
|
|
refutes undecidability (but, of course, does not prove decidability) of
|
|
any formula. (Contributed by BJ, 9-Oct-2019.) $)
|
|
nndc $p |- -. -. DECID ph $=
|
|
( wdc wn wo nnexmid df-dc notbii mtbir ) ABZCAACDZCAEIJAFGH $.
|
|
|
|
$( Decidability of a proposition is decidable if and only if that proposition
|
|
is decidable. ` DECID ` is idempotent. (Contributed by BJ,
|
|
9-Oct-2019.) $)
|
|
dcdc $p |- ( DECID DECID ph <-> DECID ph ) $=
|
|
( wdc wn wo df-dc nndc biorfi bitr4i ) ABZBIICZDIIEJIAFGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Predicate calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof
|
|
modification is discouraged because there are shorter proofs, but using
|
|
less basic results (like ~ exlimiv and ~ 19.9ht or ~ 19.23ht ).
|
|
(Proof modification is discouraged.) $)
|
|
bj-ex $p |- ( E. x ph -> ph ) $=
|
|
( wi wex wal wb ax-ie2 ax-17 mpg id mpgbi ) AACZABDACZBAABECLBEMFBAABGABH
|
|
IAJK $.
|
|
$}
|
|
|
|
$( Closed form of ~ hbal (copied from set.mm). (Contributed by BJ,
|
|
2-May-2019.) $)
|
|
bj-hbalt $p |- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) ) $=
|
|
( wal wi alim ax-7 syl6 ) AABDZECDACDZICDJBDAICFACBGH $.
|
|
|
|
$( Closed form of ~ nfal (copied from set.mm). (Contributed by BJ,
|
|
2-May-2019.) $)
|
|
bj-nfalt $p |- ( A. x F/ y ph -> F/ y A. x ph ) $=
|
|
( wnf wal wi df-nf albii bj-hbalt alimi alcoms sylbi sylibr ) ACDZBEZABEZPC
|
|
EFZCEZPCDOAACEFZCEZBERNTBACGHSRCBSBEQCACBIJKLPCGM $.
|
|
|
|
${
|
|
spimd.nf $e |- ( ph -> F/ x ch ) $.
|
|
spimd.1 $e |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( Deduction form of ~ spim . (Contributed by BJ, 17-Oct-2019.) $)
|
|
spimd $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( wnf weq wi wal spimt syl2anc ) ACDHDEIBCJJDKBDKCJFGBCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d x t $.
|
|
2spim.nfx $e |- F/ x ch $.
|
|
2spim.nfz $e |- F/ z ch $.
|
|
2spim.1 $e |- ( ( x = y /\ z = t ) -> ( ps -> ch ) ) $.
|
|
$( Double substitution, as in ~ spim . (Contributed by BJ,
|
|
17-Oct-2019.) $)
|
|
2spim $p |- ( A. z A. x ps -> ch ) $=
|
|
( wal weq wnf a1i wi expcom alrimiv spimd spim ) ACJBEFHEFKZABCDBCLSGMSCD
|
|
KZABNZNCTSUAIOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d x t $.
|
|
ch2var.nfx $e |- F/ x ps $.
|
|
ch2var.nfz $e |- F/ z ps $.
|
|
ch2var.maj $e |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) ) $.
|
|
ch2var.min $e |- ph $.
|
|
$( Implicit substitution of ` y ` for ` x ` and ` t ` for ` z ` into a
|
|
theorem. (Contributed by BJ, 17-Oct-2019.) $)
|
|
ch2var $p |- ps $=
|
|
( wal weq wa biimpd 2spim ax-gen mpg ) ACKBEABCDEFGHCDLEFLMABINOACJPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x z ps $. $d x t $.
|
|
ch2varv.maj $e |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) ) $.
|
|
ch2varv.min $e |- ph $.
|
|
$( Version of ~ ch2var with non-freeness hypotheses replaced by DV
|
|
conditions. (Contributed by BJ, 17-Oct-2019.) $)
|
|
ch2varv $p |- ps $=
|
|
( nfv ch2var ) ABCDEFBCIBEIGHJ $.
|
|
$}
|
|
|
|
${
|
|
bj-exlimmp.nf $e |- F/ x ps $.
|
|
bj-exlimmp.min $e |- ( ch -> ph ) $.
|
|
$( Lemma for ~ bj-vtoclgf . (Contributed by BJ, 21-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-exlimmp $p |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) ) $=
|
|
( wi wal nfa1 idd embantd a2i sps exlimd ) CABGZGZDHCBDPDIEPCBGDCOBCABBFC
|
|
BJKLMN $.
|
|
|
|
bj-exlimmpi.maj $e |- ( ch -> ( ph -> ps ) ) $.
|
|
$( Lemma for ~ bj-vtoclgf . (Contributed by BJ, 21-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-exlimmpi $p |- ( E. x ch -> ps ) $=
|
|
( mpd exlimi ) CBDECABFGHI $.
|
|
$}
|
|
|
|
${
|
|
bj-sbimedh.1 $e |- ( ph -> A. x ph ) $.
|
|
bj-sbimedh.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
bj-sbimedh.3 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( A strengthening of ~ sbiedh (same proof). (Contributed by BJ,
|
|
16-Dec-2019.) $)
|
|
bj-sbimedh $p |- ( ph -> ( [ y / x ] ps -> ch ) ) $=
|
|
( wsb wex weq wa sb1 impd eximdh syl5 19.9hd syld ) ABDEIZCDJZCSDEKZBLZDJ
|
|
ATBDEMAUBCDFAUABCHNOPCADFGQR $.
|
|
$}
|
|
|
|
${
|
|
bj-sbimeh.1 $e |- ( ps -> A. x ps ) $.
|
|
bj-sbimeh.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( A strengthening of ~ sbieh (same proof). (Contributed by BJ,
|
|
16-Dec-2019.) $)
|
|
bj-sbimeh $p |- ( [ y / x ] ph -> ps ) $=
|
|
( wsb wi wtru tru hbth wal a1i weq bj-sbimedh trud ) ACDGBHIABCDICJKBBCLH
|
|
IEMCDNABHHIFMOP $.
|
|
$}
|
|
|
|
${
|
|
bj-sbime.nf $e |- F/ x ps $.
|
|
bj-sbime.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( A strengthening of ~ sbie (same proof). (Contributed by BJ,
|
|
16-Dec-2019.) $)
|
|
bj-sbime $p |- ( [ y / x ] ph -> ps ) $=
|
|
( nfri bj-sbimeh ) ABCDBCEGFH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Extensionality
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Various utility theorems using FOL and extensionality.
|
|
|
|
$)
|
|
|
|
${
|
|
bj-vtoclgf.nf1 $e |- F/_ x A $.
|
|
bj-vtoclgf.nf2 $e |- F/ x ps $.
|
|
bj-vtoclgf.min $e |- ( x = A -> ph ) $.
|
|
$( Weakening two hypotheses of ~ vtoclgf . (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
bj-vtoclgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) ) $=
|
|
( wcel cvv cv wceq wi wal elex wex issetf bj-exlimmp syl5bi syl5 ) DEIDJI
|
|
ZCKDLZABMMCNZBDEOUAUBCPUCBCDFQABUBCGHRST $.
|
|
|
|
bj-vtoclgf.maj $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( Weakening two hypotheses of ~ vtoclgf . (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
bj-vtoclgf $p |- ( A e. V -> ps ) $=
|
|
( cv wceq wi wcel bj-vtoclgft mpg ) CJDKABLLDEMBLCABCDEFGHNIO $.
|
|
$}
|
|
|
|
$( Lemma for ~ elabgf . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabgf0 $p |- ( x = A -> ( A e. { x | ph } <-> ph ) ) $=
|
|
( cv cab wcel wceq abid eleq1 syl5rbbr ) ABDZABEZFKCGCLFABHKCLIJ $.
|
|
|
|
${
|
|
elabgf1.nf1 $e |- F/_ x A $.
|
|
elabgf1.nf2 $e |- F/ x ps $.
|
|
$( One implication of ~ elabgf , in closed form. (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
elabgft1 $p |-
|
|
( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) ) $=
|
|
( cv wceq wi wal cab wcel wb bi1 imim2 syl5 imim2i alimi nfab1 nfel nfim
|
|
elabgf0 bj-vtoclgft syl pm2.43d ) CGDHZABIZIZCJZDACKZLZBUIUFUKAMZUKBIZIZI
|
|
ZCJUKUMIUHUOCUGUNUFULUKAIUGUMUKANABUKOPQRULUMCDUJEUKBCCDUJEACSTFUAACDUBUC
|
|
UDUE $.
|
|
|
|
elabgf1.1 $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( One implication of ~ elabgf . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabgf1 $p |- ( A e. { x | ph } -> ps ) $=
|
|
( cv wceq wi cab wcel elabgft1 mpg ) CHDIABJJDACKLBJCABCDEFMGN $.
|
|
$}
|
|
|
|
${
|
|
elabgf2.nf1 $e |- F/_ x A $.
|
|
elabgf2.nf2 $e |- F/ x ps $.
|
|
elabgf2.1 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( One implication of ~ elabgf . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabgf2 $p |- ( A e. B -> ( ps -> A e. { x | ph } ) ) $=
|
|
( cab wcel wb wi nfab1 nfel nfim elabgf0 cv wceq bicom1 bi1 bj-vtoclgf
|
|
syl9 syl5 ) DACIZJZAKZBUELZCDEFBUECGCDUDFACMNOACDPUFAUEKZCQDRZUGUEASUIBAU
|
|
HUEHAUETUBUCUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
elabf1.nf $e |- F/ x ps $.
|
|
elabf1.1 $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( One implication of ~ elabf . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabf1 $p |- ( A e. { x | ph } -> ps ) $=
|
|
( nfcv elabgf1 ) ABCDCDGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
elabf2.nf $e |- F/ x ps $.
|
|
elabf2.s $e |- A e. _V $.
|
|
elabf2.1 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( One implication of ~ elabf . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabf2 $p |- ( ps -> A e. { x | ph } ) $=
|
|
( cvv wcel cab wi nfcv elabgf2 ax-mp ) DHIBDACJIKFABCDHCDLEGMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab1.1 $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( One implication of ~ elab . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elab1 $p |- ( A e. { x | ph } -> ps ) $=
|
|
( nfv elabf1 ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab2a.s $e |- A e. _V $.
|
|
elab2a.1 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( One implication of ~ elab . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elab2a $p |- ( ps -> A e. { x | ph } ) $=
|
|
( nfv elabf2 ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elabg2.1 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( One implication of ~ elabg . (Contributed by BJ, 21-Nov-2019.) $)
|
|
elabg2 $p |- ( A e. V -> ( ps -> A e. { x | ph } ) ) $=
|
|
( nfcv nfv elabgf2 ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
bj-rspg.nfa $e |- F/_ x A $.
|
|
bj-rspg.nfb $e |- F/_ x B $.
|
|
bj-rspg.nf2 $e |- F/ x ps $.
|
|
$( Restricted specialization, generalized. Weakens a hypothesis of
|
|
~ rspccv and seems to have a shorter proof. (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
bj-rspgt $p |- ( A. x ( x = A -> ( ph -> ps ) ) ->
|
|
( A. x e. B ph -> ( A e. B -> ps ) ) ) $=
|
|
( cv wceq wi wal wcel wral eleq1 imbi1d biimpd imim2 nfim com23 a2i alimi
|
|
imim2d syl9 nfel nfra1 rsp a1i bj-vtoclgft syl pm2.43d ) CIZDJZABKZKZCLZD
|
|
EMZACENZBUPUQURBKZUPUMULEMZURAKZKZUQUSKZKZKZCLUQVCKUOVECUMUNVDUMVBUQVAKZU
|
|
NVCUMVBVFUMUTUQVAULDEOPQUNVAUSUQABURRUCUDUAUBVBVCCDEFUQUSCCDEFGUEURBCACEU
|
|
FHSSUMURUTAURUTAKKUMACEUGUHTUIUJUKT $.
|
|
|
|
bj-rspg.is $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( Restricted specialization, generalized. Weakens a hypothesis of
|
|
~ rspccv and seems to have a shorter proof. (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
bj-rspg $p |- ( A. x e. B ph -> ( A e. B -> ps ) ) $=
|
|
( cv wceq wi wral wcel bj-rspgt mpg ) CJDKABLLACEMDENBLLCABCDEFGHOIP $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $. $d z ps $. $d z ch $.
|
|
cbvrald.nf0 $e |- F/ x ph $.
|
|
cbvrald.nf1 $e |- F/ y ph $.
|
|
cbvrald.nf2 $e |- ( ph -> F/ y ps ) $.
|
|
cbvrald.nf3 $e |- ( ph -> F/ x ch ) $.
|
|
cbvrald.is $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by BJ, 22-Nov-2019.) $)
|
|
cbvrald $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
|
|
( vz cv wcel wi nfv wnf a1i nfimd wb wal wral wsb nfs1v weq eleq1 sbequ12
|
|
wa adantl imbi12d cbv2 nfsbd sbequ sbied sylan9bbr bitrd df-ral 3bitr4g
|
|
ex ) ADMZFNZBOZDUAZEMZFNZCOZEUAZBDFUBCEFUBAVCLMZFNZBDLUCZOZLUAVGAVBVKDLGA
|
|
LPZAVABLVALQAVALPRBLQABLPRSAVIVJDVIDQAVIDPRVJDQABDLUDRSADLUEZVBVKTAVMUHVA
|
|
VIBVJVMVAVITAUTVHFUFUIVMBVJTABDLUGUIUJUSUKAVKVFLEVLHAVIVJEVIEQAVIEPRABDLE
|
|
GIULSAVECLVELQAVELPRCLQACLPRSALEUEZVKVFTAVNUHVIVEVJCVNVIVETAVHVDFUFUIVNVJ
|
|
BDEUCACBLEDUMABCDEGJKUNUOUJUSUKUPBDFUQCEFUQUR $.
|
|
$}
|
|
|
|
${
|
|
bj-intabssel.nf $e |- F/_ x A $.
|
|
$( Version of ~ intss1 using a class abstraction and explicit substitution.
|
|
(Contributed by BJ, 29-Nov-2019.) $)
|
|
bj-intabssel $p |-
|
|
( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) ) $=
|
|
( wcel wsbc cab cint wss nfsbc1 sbceq1a elabgf intss1 syl6bir ) CDFABCGZC
|
|
ABHZFQICJAPBCDEABCEKABCLMCQNO $.
|
|
$}
|
|
|
|
${
|
|
bj-intabssel1.nf $e |- F/_ x A $.
|
|
bj-intabssel1.nf2 $e |- F/ x ps $.
|
|
bj-intabssel1.is $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( Version of ~ intss1 using a class abstraction and implicit substitution.
|
|
Closed form of ~ intmin3 . (Contributed by BJ, 29-Nov-2019.) $)
|
|
bj-intabssel1 $p |-
|
|
( A e. V -> ( ps -> |^| { x | ph } C_ A ) ) $=
|
|
( wcel cab cint wss elabgf2 intss1 syl6 ) DEIBDACJZIPKDLABCDEFGHMDPNO $.
|
|
$}
|
|
|
|
${
|
|
bj-elssuniab.nf $e |- F/_ x A $.
|
|
$( Version of ~ elssuni using a class abstraction and explicit
|
|
substitution. (Contributed by BJ, 29-Nov-2019.) $)
|
|
bj-elssuniab $p |-
|
|
( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) ) $=
|
|
( wcel wsbc cab cuni wss sbc8g elssuni syl6bi ) CDFABCGCABHZFCNIJABCDKCNL
|
|
M $.
|
|
$}
|
|
|
|
${
|
|
bj-sseq.1 $e |- ( ph -> ( ps <-> A C_ B ) ) $.
|
|
bj-sseq.2 $e |- ( ph -> ( ch <-> B C_ A ) ) $.
|
|
$( If two converse inclusions are characterized each by a formula, then
|
|
equality is characterized by the conjunction of these formulas.
|
|
(Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-sseq $p |- ( ph -> ( ( ps /\ ch ) <-> A = B ) ) $=
|
|
( wa wss wceq anbi12d eqss syl6bbr ) ABCHDEIZEDIZHDEJABNCOFGKDELM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Dedidability of classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The question of decidability is essential in intuitionistic logic. In
|
|
intuitionistic set theories, it is natural to define decidability of a set
|
|
(or class) as decidability of membership in it. One can parameterize this
|
|
notion with another set (or class) since it is often important to assess
|
|
decidability of membership in one class among elements of another class.
|
|
Namely, one will say that " ` A ` is decidable in ` B ` " if
|
|
` A. x e. B DECID x e. A ` (see ~ df-dcin ).
|
|
|
|
Note the similarity with the definition of a bounded class as a class for
|
|
which membership in it is a bounded proposition ( ~ df-bdc ).
|
|
|
|
$)
|
|
|
|
$( Token for decidability of a class in another. $)
|
|
$c DECID_in $.
|
|
|
|
$( Syntax for decidability of a class in another. $)
|
|
wdcin $a wff A DECID_in B $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Define decidability of a class in another. (Contributed by BJ,
|
|
19-Feb-2022.) $)
|
|
df-dcin $a |- ( A DECID_in B <-> A. x e. B DECID x e. A ) $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d X x $.
|
|
$( Property of being decidable in another class. (Contributed by BJ,
|
|
19-Feb-2022.) $)
|
|
decidi $p |- ( A DECID_in B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) ) $=
|
|
( vx wdcin cv wcel wdc wral wn wo df-dcin df-dc ralbii wceq eleq1 orbi12d
|
|
wi notbid sylbi rspccv ) ABEDFZAGZHZDBIZCBGCAGZUFJZKZRZDABLUEUCUCJZKZDBIU
|
|
IUDUKDBUCMNUKUHDCBUBCOZUCUFUJUGUBCAPZULUCUFUMSQUATT $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $.
|
|
decidr.1 $e |- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) ) $.
|
|
$( Sufficient condition for being decidable in another class. (Contributed
|
|
by BJ, 19-Feb-2022.) $)
|
|
decidr $p |- ( ph -> A DECID_in B ) $=
|
|
( cv wcel wdc wi wal wdcin wn wo df-dc syl6ibr alrimiv wral df-dcin bitri
|
|
df-ral sylibr ) ABFZDGZUBCGZHZIZBJZCDKZAUFBAUCUDUDLMUEEUDNOPUHUEBDQUGBCDR
|
|
UEBDTSUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d C x $. $d ph x $.
|
|
decidin.ss $e |- ( ph -> A C_ B ) $.
|
|
decidin.a $e |- ( ph -> A DECID_in B ) $.
|
|
decidin.b $e |- ( ph -> B DECID_in C ) $.
|
|
$( If A is a decidable subclass of B (meaning: it is a subclass of B and it
|
|
is decidable in B), and B is decidable in C, then A is decidable in C.
|
|
(Contributed by BJ, 19-Feb-2022.) $)
|
|
decidin $p |- ( ph -> A DECID_in C ) $=
|
|
( vx cv wcel wn wo wdcin wi decidi syl ssneld olc syl6 jaod syld decidr )
|
|
AHBDAHIZDJZUCCJZUEKZLZUCBJZUHKZLZACDMUDUGNGCDUCOPAUEUJUFABCMUEUJNFBCUCOPA
|
|
UFUIUJABCUCEQUIUHRSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( An upperset of integers is decidable in the integers. Reformulation of
|
|
~ eluzdc . (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ,
|
|
19-Feb-2022.) $)
|
|
uzdcinzz $p |- ( M e. ZZ -> ( ZZ>= ` M ) DECID_in ZZ ) $=
|
|
( vx cz wcel cuz cfv cv wn wo wa cle wbr zlelttric eluz biimprd wb zltnle
|
|
clt ancoms notbid sylbid orim12d mpd ex decidr ) ACDZBAEFZCUFBGZCDZUHUGDZ
|
|
UJHZIZUFUIJZAUHKLZUHARLZIULAUHMUMUNUJUOUKUMUJUNAUHNZOUMUOUNHZUKUIUFUOUQPU
|
|
HAQSUMUKUQUMUJUNUPTOUAUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d M x y $. $d A x y $. $d ph y $. $d A z $. $d N z $.
|
|
sumdc2.m $e |- ( ph -> M e. ZZ ) $.
|
|
sumdc2.ss $e |- ( ph -> A C_ ( ZZ>= ` M ) ) $.
|
|
sumdc2.dc $e |- ( ph -> A. x e. ( ZZ>= ` M ) DECID x e. A ) $.
|
|
sumdc2.n $e |- ( ph -> N e. ZZ ) $.
|
|
$( Alternate proof of ~ sumdc , without DV condition on ` N , x ` (longer
|
|
because the statement is taylored to the proof ~ sumdc ). (Contributed
|
|
by BJ, 19-Feb-2022.) $)
|
|
sumdc2 $p |- ( ph -> DECID N e. A ) $=
|
|
( vy vz cz wdcin wcel wdc cv wral wi eleq1 dcbid cuz cfv wn wo weq rspccv
|
|
exmiddc syl6 syl decidr uzdcinzz decidin df-dcin wceq nfv rspct mpg com12
|
|
wb sylbi sylc ) ACLMZELNZECNZOZACDUAUBZLGAJCVFABPZCNZOZBVFQZJPZVFNZVKCNZV
|
|
MUCUDZRHVJVLVMOZVNVIVOBVKVFBJUEVHVMVGVKCSTUFVMUGUHUIUJADLNVFLMFDUKUIULIVB
|
|
KPZCNZOZKLQZVCVERKCLUMVCVSVEVPEUNZVRVEUSRVCVSVERRKVRVEKELVEKUOUPVTVQVDVPE
|
|
CSTUQURUTVA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
This section develops constructive Zermelo--Fraenkel set theory (CZF) on top
|
|
of intuitionistic logic. It is a constructive theory in the sense that its
|
|
logic is intuitionistic and it is predicative. "Predicative" means that new
|
|
sets can be constructed only from already constructed sets. In particular,
|
|
the axiom of separation ~ ax-sep is not predicative (because we cannot allow
|
|
all formulas to define a subset) and is replaced in CZF by bounded separation
|
|
~ ax-bdsep . Because this axiom is weaker than full separation, the
|
|
axiom of replacement or collection ~ ax-coll of ZF and IZF has to be
|
|
strengthened in CZF to the axiom of strong collection ~ ax-strcoll (which is
|
|
a theorem of IZF), and the axiom of infinity needs a more precise version,
|
|
the von Neumann axiom of infinity ~ ax-infvn . Similarly, the axiom of
|
|
powerset ~ ax-pow is not predicative (checking whether a set is included in
|
|
another requires to universally quantifier over that "not yet constructed"
|
|
set) and is replaced in CZF by the axiom of fullness or the axiom of subset
|
|
collection ~ ax-sscoll .
|
|
|
|
In an intuitionistic context, the axiom of regularity is stated in IZF as
|
|
well as in CZF as the axiom of set induction ~ ax-setind . It is sometimes
|
|
interesting to study the weakening of CZF where that axiom is replaced by
|
|
bounded set induction ~ ax-bdsetind .
|
|
|
|
For more details on CZF, a useful set of notes is
|
|
|
|
Peter Aczel and Michael Rathjen,
|
|
_CST Book draft_.
|
|
(available at ~ http://www1.maths.leeds.ac.uk/~~rathjen/book.pdf )
|
|
|
|
and an interesting article is
|
|
|
|
Michael Shulman,
|
|
_Comparing material and structural set theories_,
|
|
Annals of Pure and Applied Logic,
|
|
Volume 170, Issue 4 (Apr. 2019), 465--504.
|
|
(available at ~ https://arxiv.org/abs/1808.05204 )
|
|
|
|
I also thank Michael Rathjen and Michael Shulman for useful hints in the
|
|
formulation of some results.
|
|
|
|
$)
|
|
|
|
$( This declaration is simply to be able to display Delta0 in comments. $)
|
|
$c Delta0 $.
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Bounded formulas
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
The present definition of bounded formulas emerged from a discussion
|
|
on GitHub between Jim Kingdon, Mario Carneiro and I, started
|
|
23-Sept-2019 (see ~ https://github.com/metamath/set.mm/issues/1173 and links
|
|
therein).
|
|
|
|
In order to state certain axiom schemes of Constructive
|
|
Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or
|
|
restricted, or ` Delta0 ` ) separation, it is necessary to distinguish
|
|
certain formulas, called bounded (or restricted, or ` Delta0 ` ) formulas.
|
|
The necessity of considering bounded formulas also arises in several theories
|
|
of bounded arithmetic, both classical or intuitonistic, for instance to state
|
|
the axiom scheme of ` Delta0 ` -induction.
|
|
|
|
To formalize this in Metamath, there are several choices to make.
|
|
|
|
A first choice is to either create a new type for bounded formulas, or to
|
|
create a predicate on formulas that indicates whether they are bounded.
|
|
In the first case, one creates a new type "wff0" with a new set of
|
|
metavariables (ph_0 ...) and an axiom "$a wff ph_0 " ensuring that bounded
|
|
formulas are formulas, so that one can reuse existing theorems, and then
|
|
axioms take the form "$a wff0 ( ph_0 -> ps_0 )", etc.
|
|
In the second case, one introduces a predicate " ` Bdd ` " with the intended
|
|
meaning that " ` Bdd ph ` " is a formula meaning that ` ph ` is a bounded
|
|
formula.
|
|
We choose the second option, since the first would complicate the grammar,
|
|
risking to make it ambiguous.
|
|
(TODO: elaborate.)
|
|
|
|
A second choice is to view "bounded" either as a syntactic or a semantic
|
|
property.
|
|
For instance, ` A. x T. ` is not syntactically bounded since it has an
|
|
unbounded universal quantifier, but it is semantically bounded since it is
|
|
equivalent to ` T. ` which is bounded.
|
|
We choose the second option, so that formulas using defined symbols can be
|
|
proved bounded.
|
|
|
|
A third choice is in the form of the axioms, either in closed form or in
|
|
inference form.
|
|
One cannot state all the axioms in closed form, especially ~ ax-bd0 .
|
|
Indeed, if we posited it in closed form, then we could prove for instance
|
|
` |- ( ph -> Bdd ph ) ` and ` |- ( -. ph -> Bdd ph ) ` which is problematic
|
|
(with the law of excluded middle, this would entail that all formulas are
|
|
bounded, but even without it, too many formulas could be proved bounded...).
|
|
(TODO: elaborate.)
|
|
|
|
Having ~ ax-bd0 in inference form ensures that a formula can be proved
|
|
bounded only if it is equivalent *for all values of the free variables* to a
|
|
syntactically bounded one.
|
|
The other axioms ( ~ ax-bdim through ~ ax-bdsb ) can be written either in
|
|
closed or inference form. The fact that ~ ax-bd0 is an inference is enough
|
|
to ensure that the closed forms cannot be "exploited" to prove that some
|
|
unbounded formulas are bounded.
|
|
(TODO: check.)
|
|
However, we state all the axioms in inference form to make it clear that we
|
|
do not exploit any over-permissiveness.
|
|
|
|
Finally, note that our logic has no terms, only variables. Therefore, we
|
|
cannot prove for instance that ` x e. _om ` is a bounded formula. However,
|
|
since ` _om ` can be defined as "the ` y ` such that PHI" a proof using the
|
|
fact that ` x e. _om ` is bounded can be converted to a proof in iset.mm by
|
|
replacing ` _om ` with ` y ` everywhere and prepending the antecedent PHI,
|
|
since ` x e. y ` is bounded by ~ ax-bdel . For a similar method, see
|
|
~ bj-omtrans .
|
|
|
|
Note that one cannot add an axiom ` |- Bdd x e. A ` since by ~ bdph it would
|
|
imply that every formula is bounded.
|
|
|
|
$)
|
|
|
|
$( Symbol for the predicate ` Bdd ` . $)
|
|
$c Bdd $.
|
|
|
|
$( Syntax for the predicate ` Bdd ` . $)
|
|
wbd $a wff Bdd ph $.
|
|
|
|
${
|
|
ax-bd0.1 $e |- ( ph <-> ps ) $.
|
|
$( If two formulas are equivalent, then boundedness of one implies
|
|
boundedness of the other. (Contributed by BJ, 3-Oct-2019.) $)
|
|
ax-bd0 $a |- ( Bdd ph -> Bdd ps ) $.
|
|
$}
|
|
|
|
${
|
|
bdim.1 $e |- Bdd ph $.
|
|
bdim.2 $e |- Bdd ps $.
|
|
$( An implication between two bounded formulas is bounded. (Contributed by
|
|
BJ, 25-Sep-2019.) $)
|
|
ax-bdim $a |- Bdd ( ph -> ps ) $.
|
|
|
|
$( The conjunction of two bounded formulas is bounded. (Contributed by BJ,
|
|
25-Sep-2019.) $)
|
|
ax-bdan $a |- Bdd ( ph /\ ps ) $.
|
|
|
|
$( The disjunction of two bounded formulas is bounded. (Contributed by BJ,
|
|
25-Sep-2019.) $)
|
|
ax-bdor $a |- Bdd ( ph \/ ps ) $.
|
|
$}
|
|
|
|
${
|
|
bdn.1 $e |- Bdd ph $.
|
|
$( The negation of a bounded formula is bounded. (Contributed by BJ,
|
|
25-Sep-2019.) $)
|
|
ax-bdn $a |- Bdd -. ph $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
bdal.1 $e |- Bdd ph $.
|
|
$( A bounded universal quantification of a bounded formula is bounded.
|
|
Note the DV condition on ` x , y ` . (Contributed by BJ,
|
|
25-Sep-2019.) $)
|
|
ax-bdal $a |- Bdd A. x e. y ph $.
|
|
|
|
$( A bounded existential quantification of a bounded formula is bounded.
|
|
Note the DV condition on ` x , y ` . (Contributed by BJ,
|
|
25-Sep-2019.) $)
|
|
ax-bdex $a |- Bdd E. x e. y ph $.
|
|
$}
|
|
|
|
$( An atomic formula is bounded (equality predicate). (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
ax-bdeq $a |- Bdd x = y $.
|
|
|
|
$( An atomic formula is bounded (membership predicate). (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
ax-bdel $a |- Bdd x e. y $.
|
|
|
|
${
|
|
bdsb.1 $e |- Bdd ph $.
|
|
$( A formula resulting from proper substitution in a bounded formula is
|
|
bounded. This probably cannot be proved from the other axioms, since
|
|
neither the definiens in ~ df-sb , nor probably any other equivalent
|
|
formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) $)
|
|
ax-bdsb $a |- Bdd [ y / x ] ph $.
|
|
$}
|
|
|
|
${
|
|
bdeq.1 $e |- ( ph <-> ps ) $.
|
|
$( Equality property for the predicate ` Bdd ` . (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdeq $p |- ( Bdd ph <-> Bdd ps ) $=
|
|
( wbd ax-bd0 bicomi impbii ) ADBDABCEBAABCFEG $.
|
|
$}
|
|
|
|
${
|
|
bd0.min $e |- Bdd ph $.
|
|
bd0.maj $e |- ( ph <-> ps ) $.
|
|
$( A formula equivalent to a bounded one is bounded. See also ~ bd0r .
|
|
(Contributed by BJ, 3-Oct-2019.) $)
|
|
bd0 $p |- Bdd ps $=
|
|
( wbd ax-bd0 ax-mp ) AEBECABDFG $.
|
|
$}
|
|
|
|
${
|
|
bd0r.min $e |- Bdd ph $.
|
|
bd0r.maj $e |- ( ps <-> ph ) $.
|
|
$( A formula equivalent to a bounded one is bounded. Stated with a
|
|
commuted (compared with ~ bd0 ) biconditional in the hypothesis, to work
|
|
better with definitions ( ` ps ` is the definiendum that one wants to
|
|
prove bounded). (Contributed by BJ, 3-Oct-2019.) $)
|
|
bd0r $p |- Bdd ps $=
|
|
( bicomi bd0 ) ABCBADEF $.
|
|
$}
|
|
|
|
${
|
|
bdbi.1 $e |- Bdd ph $.
|
|
bdbi.2 $e |- Bdd ps $.
|
|
$( A biconditional between two bounded formulas is bounded. (Contributed
|
|
by BJ, 3-Oct-2019.) $)
|
|
bdbi $p |- Bdd ( ph <-> ps ) $=
|
|
( wi wa wb ax-bdim ax-bdan dfbi2 bd0r ) ABEZBAEZFABGLMABCDHBADCHIABJK $.
|
|
$}
|
|
|
|
${
|
|
bdstab.1 $e |- Bdd ph $.
|
|
$( Stability of a bounded formula is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdstab $p |- Bdd STAB ph $=
|
|
( wn wi wstab ax-bdn ax-bdim df-stab bd0r ) ACZCZADAEKAJABFFBGAHI $.
|
|
|
|
$( Decidability of a bounded formula is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bddc $p |- Bdd DECID ph $=
|
|
( wn wo wdc ax-bdn ax-bdor df-dc bd0r ) AACZDAEAJBABFGAHI $.
|
|
$}
|
|
|
|
${
|
|
bd3or.1 $e |- Bdd ph $.
|
|
bd3or.2 $e |- Bdd ps $.
|
|
bd3or.3 $e |- Bdd ch $.
|
|
$( A disjunction of three bounded formulas is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bd3or $p |- Bdd ( ph \/ ps \/ ch ) $=
|
|
( wo w3o ax-bdor df-3or bd0r ) ABGZCGABCHLCABDEIFIABCJK $.
|
|
|
|
$( A conjunction of three bounded formulas is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bd3an $p |- Bdd ( ph /\ ps /\ ch ) $=
|
|
( wa w3a ax-bdan df-3an bd0r ) ABGZCGABCHLCABDEIFIABCJK $.
|
|
$}
|
|
|
|
${
|
|
bdth.1 $e |- ph $.
|
|
$( A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ,
|
|
6-Oct-2019.) $)
|
|
bdth $p |- Bdd ph $=
|
|
( vx weq wi ax-bdeq ax-bdim id 2th bd0 ) CCDZKEZAKKCCFZMGLAKHBIJ $.
|
|
$}
|
|
|
|
$( The truth value ` T. ` is bounded. (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdtru $p |- Bdd T. $=
|
|
( wtru tru bdth ) ABC $.
|
|
|
|
$( The truth value ` F. ` is bounded. (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdfal $p |- Bdd F. $=
|
|
( wtru wn wfal bdtru ax-bdn df-fal bd0r ) ABCADEFG $.
|
|
|
|
${
|
|
bdnth.1 $e |- -. ph $.
|
|
$( A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) $)
|
|
bdnth $p |- Bdd ph $=
|
|
( wfal bdfal fal 2false bd0 ) CADCAEBFG $.
|
|
|
|
$( Alternate proof of ~ bdnth not using ~ bdfal . Then, ~ bdfal can be
|
|
proved from this theorem, using ~ fal . The total number of proof steps
|
|
would be 17 (for ~ bdnthALT ) + 3 = 20, which is more than 8 (for
|
|
~ bdfal ) + 9 (for ~ bdnth ) = 17. (Contributed by BJ, 6-Oct-2019.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bdnthALT $p |- Bdd ph $=
|
|
( wtru wn bdtru ax-bdn notnot trud 2false bd0 ) CDZACEFKAKDCGHBIJ $.
|
|
$}
|
|
|
|
${
|
|
bdxor.1 $e |- Bdd ph $.
|
|
bdxor.2 $e |- Bdd ps $.
|
|
$( The exclusive disjunction of two bounded formulas is bounded.
|
|
(Contributed by BJ, 3-Oct-2019.) $)
|
|
bdxor $p |- Bdd ( ph \/_ ps ) $=
|
|
( wo wa wn wxo ax-bdor ax-bdan ax-bdn df-xor bd0r ) ABEZABFZGZFABHNPABCDI
|
|
OABCDJKJABLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $.
|
|
bj-bdcel.bd $e |- Bdd y = A $.
|
|
$( Boundedness of a membership formula. (Contributed by BJ,
|
|
8-Dec-2019.) $)
|
|
bj-bdcel $p |- Bdd A e. x $=
|
|
( cv wceq wrex wcel ax-bdex risset bd0r ) BECFZBAEZGCMHLBADIBCMJK $.
|
|
$}
|
|
|
|
${
|
|
bdab.1 $e |- Bdd ph $.
|
|
$( Membership in a class defined by class abstraction using a bounded
|
|
formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdab $p |- Bdd x e. { y | ph } $=
|
|
( wsb cv cab wcel ax-bdsb df-clab bd0r ) ACBEBFACGHACBDIABCJK $.
|
|
$}
|
|
|
|
${
|
|
bdcdeq.1 $e |- Bdd ph $.
|
|
$( Conditional equality of a bounded formula is a bounded formula.
|
|
(Contributed by BJ, 16-Oct-2019.) $)
|
|
bdcdeq $p |- Bdd CondEq ( x = y -> ph ) $=
|
|
( weq wi wcdeq ax-bdeq ax-bdim df-cdeq bd0r ) BCEZAFABCGLABCHDIABCJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Bounded classes
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In line with our definitions of classes as extensions of predicates, it is
|
|
useful to define a predicate for bounded classes, which is done in ~ df-bdc .
|
|
Note that this notion is only a technical device which can be used to shorten
|
|
proofs of (semantic) boundedness of formulas.
|
|
|
|
As will be clear by the end of this subsection (see for instance ~ bdop ),
|
|
one can prove the boundedness of any concrete term using only setvars and
|
|
bounded formulas, for instance,
|
|
` |- Bdd ph => `
|
|
` |- Bdd_ <. { x | ph } , ( { y , suc z } X. <. t , (/) >. ) >. ` .
|
|
The proofs are long since one has to prove boundedness at each step of the
|
|
construction, without being able to prove general theorems like
|
|
` |- Bdd_ A => |- Bdd_ { A } ` .
|
|
|
|
$)
|
|
|
|
$( Symbol for the predicate ` Bdd_ ` . $)
|
|
$c Bdd_ $.
|
|
|
|
$( Syntax for the predicate ` Bdd_ ` . $)
|
|
wbdc $a wff Bdd_ A $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define a bounded class as one such that membership in this class is a
|
|
bounded formula. (Contributed by BJ, 3-Oct-2019.) $)
|
|
df-bdc $a |- ( Bdd_ A <-> A. x Bdd x e. A ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
bdceq.1 $e |- A = B $.
|
|
$( Equality property for the predicate ` Bdd_ ` . (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdceq $p |- ( Bdd_ A <-> Bdd_ B ) $=
|
|
( vx cv wcel wbd wal wbdc eleq2i bdeq albii df-bdc 3bitr4i ) DEZAFZGZDHOB
|
|
FZGZDHAIBIQSDPRABOCJKLDAMDBMN $.
|
|
$}
|
|
|
|
${
|
|
bdceqi.min $e |- Bdd_ A $.
|
|
bdceqi.maj $e |- A = B $.
|
|
$( A class equal to a bounded one is bounded. Note the use of ~ ax-ext .
|
|
See also ~ bdceqir . (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdceqi $p |- Bdd_ B $=
|
|
( wbdc bdceq mpbi ) AEBECABDFG $.
|
|
$}
|
|
|
|
${
|
|
bdceqir.min $e |- Bdd_ A $.
|
|
bdceqir.maj $e |- B = A $.
|
|
$( A class equal to a bounded one is bounded. Stated with a commuted
|
|
(compared with ~ bdceqi ) equality in the hypothesis, to work better
|
|
with definitions ( ` B ` is the definiendum that one wants to prove
|
|
bounded; see comment of ~ bd0r ). (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdceqir $p |- Bdd_ B $=
|
|
( eqcomi bdceqi ) ABCBADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The belonging of a setvar in a bounded class is a bounded formula.
|
|
(Contributed by BJ, 3-Oct-2019.) $)
|
|
bdel $p |- ( Bdd_ A -> Bdd x e. A ) $=
|
|
( wbdc cv wcel wbd wal df-bdc sp sylbi ) BCADBEFZAGKABHKAIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdeli.1 $e |- Bdd_ A $.
|
|
$( Inference associated with ~ bdel . Its converse is ~ bdelir .
|
|
(Contributed by BJ, 3-Oct-2019.) $)
|
|
bdeli $p |- Bdd x e. A $=
|
|
( wbdc cv wcel wbd bdel ax-mp ) BDAEBFGCABHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdelir.1 $e |- Bdd x e. A $.
|
|
$( Inference associated with ~ df-bdc . Its converse is ~ bdeli .
|
|
(Contributed by BJ, 3-Oct-2019.) $)
|
|
bdelir $p |- Bdd_ A $=
|
|
( wbdc cv wcel wbd df-bdc mpgbir ) BDAEBFGAABHCI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdcv $p |- Bdd_ x $=
|
|
( vy cv ax-bdel bdelir ) BACBADE $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
bdcab.1 $e |- Bdd ph $.
|
|
$( A class defined by class abstraction using a bounded formula is bounded.
|
|
(Contributed by BJ, 6-Oct-2019.) $)
|
|
bdcab $p |- Bdd_ { x | ph } $=
|
|
( vy cab bdab bdelir ) DABEADBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
bdph.1 $e |- Bdd_ { x | ph } $.
|
|
$( A formula which defines (by class abstraction) a bounded class is
|
|
bounded. (Contributed by BJ, 6-Oct-2019.) $)
|
|
bdph $p |- Bdd ph $=
|
|
( vy wsb cv cab wcel bdeli df-clab bd0 ax-bdsb sbid2v ) ABDEZDBEANDBDFABG
|
|
ZHNDOCIADBJKLADBMK $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $.
|
|
bds.bd $e |- Bdd ph $.
|
|
bds.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Boundedness of a formula resulting from implicit substitution in a
|
|
bounded formula. Note that the proof does not use ~ ax-bdsb ;
|
|
therefore, using implicit instead of explicit substitution when
|
|
boundedness is important, one might avoid using ~ ax-bdsb .
|
|
(Contributed by BJ, 19-Nov-2019.) $)
|
|
bds $p |- Bdd ps $=
|
|
( cab bdcab cbvabv bdceqi bdph ) BDACGBDGACEHABCDFIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdcrab.1 $e |- Bdd_ A $.
|
|
bdcrab.2 $e |- Bdd ph $.
|
|
$( A class defined by restricted abstraction from a bounded class and a
|
|
bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) $)
|
|
bdcrab $p |- Bdd_ { x e. A | ph } $=
|
|
( cv wcel wa cab crab bdeli ax-bdan bdcab df-rab bdceqir ) BFCGZAHZBIABCJ
|
|
QBPABCDKELMABCNO $.
|
|
$}
|
|
|
|
$( Inequality of two setvars is a bounded formula. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdne $p |- Bdd x =/= y $=
|
|
( weq wn cv wne ax-bdeq ax-bdn df-ne bd0r ) ABCZDAEZBEZFKABGHLMIJ $.
|
|
|
|
${
|
|
$d x A $.
|
|
bdnel.1 $e |- Bdd_ A $.
|
|
$( Non-membership of a setvar in a bounded formula is a bounded formula.
|
|
(Contributed by BJ, 16-Oct-2019.) $)
|
|
bdnel $p |- Bdd x e/ A $=
|
|
( cv wcel wn wnel bdeli ax-bdn df-nel bd0r ) ADZBEZFLBGMABCHILBJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
bdreu.1 $e |- Bdd ph $.
|
|
$( Boundedness of existential uniqueness.
|
|
|
|
Remark regarding restricted quantifiers: the formula ` A. x e. A ph `
|
|
need not be bounded even if ` A ` and ` ph ` are. Indeed, ` _V ` is
|
|
bounded by ~ bdcvv , and ` |- ( A. x e. _V ph <-> A. x ph ) ` (in
|
|
minimal propositional calculus), so by ~ bd0 , if ` A. x e. _V ph ` were
|
|
bounded when ` ph ` is bounded, then ` A. x ph ` would be bounded as
|
|
well when ` ph ` is bounded, which is not the case. The same remark
|
|
holds with ` E. , E! , E* ` . (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdreu $p |- Bdd E! x e. y ph $=
|
|
( vz cv wrex wi wral wa wreu ax-bdex ax-bdeq ax-bdim ax-bdal ax-bdan reu3
|
|
weq bd0r ) ABCFZGZABERZHZBTIZETGZJABTKUAUEABCDLUDECUCBCAUBDBEMNOLPABETQS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
bdrmo.1 $e |- Bdd ph $.
|
|
$( Boundedness of existential at-most-one. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdrmo $p |- Bdd E* x e. y ph $=
|
|
( cv wrex wreu wi wrmo ax-bdex bdreu ax-bdim rmo5 bd0r ) ABCEZFZABOGZHABO
|
|
IPQABCDJABCDKLABOMN $.
|
|
$}
|
|
|
|
$( The universal class is bounded. The formulation may sound strange, but
|
|
recall that here, "bounded" means " ` Delta0 ` ". (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcvv $p |- Bdd_ _V $=
|
|
( vx cvv cv wcel vex bdth bdelir ) ABACBDAEFG $.
|
|
|
|
${
|
|
bdcsbc.1 $e |- Bdd ph $.
|
|
$( A formula resulting from proper substitution of a setvar for a setvar in
|
|
a bounded formula is bounded. See also ~ bdsbcALT . (Contributed by
|
|
BJ, 16-Oct-2019.) $)
|
|
bdsbc $p |- Bdd [. y / x ]. ph $=
|
|
( wsb cv wsbc ax-bdsb sbsbc bd0 ) ABCEABCFGABCDHABCIJ $.
|
|
|
|
$( Alternate proof of ~ bdsbc . (Contributed by BJ, 16-Oct-2019.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bdsbcALT $p |- Bdd [. y / x ]. ph $=
|
|
( cv cab wcel wsbc bdab df-sbc bd0r ) CEZABFGABLHACBDIABLJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d A z $.
|
|
bdccsb.1 $e |- Bdd_ A $.
|
|
$( A class resulting from proper substitution of a setvar for a setvar in a
|
|
bounded class is bounded. (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdccsb $p |- Bdd_ [_ y / x ]_ A $=
|
|
( vz cv wcel wsbc cab csb bdeli bdsbc bdcab df-csb bdceqir ) EFCGZABFZHZE
|
|
IAQCJREPABECDKLMAEQCNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
bdcdif.1 $e |- Bdd_ A $.
|
|
bdcdif.2 $e |- Bdd_ B $.
|
|
$( The difference of two bounded classes is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcdif $p |- Bdd_ ( A \ B ) $=
|
|
( vx cv wcel wn wa cab cdif bdeli ax-bdn ax-bdan bdcab df-dif bdceqir ) E
|
|
FZAGZRBGZHZIZEJABKUBESUAEACLTEBDLMNOEABPQ $.
|
|
|
|
$( The union of two bounded classes is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcun $p |- Bdd_ ( A u. B ) $=
|
|
( vx cv wcel wo cab cun bdeli ax-bdor bdcab df-un bdceqir ) EFZAGZPBGZHZE
|
|
IABJSEQREACKEBDKLMEABNO $.
|
|
|
|
$( The intersection of two bounded classes is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcin $p |- Bdd_ ( A i^i B ) $=
|
|
( vx cv wcel wa cab cin bdeli ax-bdan bdcab df-in bdceqir ) EFZAGZPBGZHZE
|
|
IABJSEQREACKEBDKLMEABNO $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y A $.
|
|
bdss.1 $e |- Bdd_ A $.
|
|
$( The inclusion of a setvar in a bounded class is a bounded formula.
|
|
Note: apparently, we cannot prove from the present axioms that equality
|
|
of two bounded classes is a bounded formula. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdss $p |- Bdd x C_ A $=
|
|
( vy cv wcel wral wss bdeli ax-bdal dfss3 bd0r ) DEBFZDAEZGNBHMDADBCIJDNB
|
|
KL $.
|
|
$}
|
|
|
|
$( The empty class is bounded. See also ~ bdcnulALT . (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcnul $p |- Bdd_ (/) $=
|
|
( vx c0 cv wcel noel bdnth bdelir ) ABACZBDHEFG $.
|
|
|
|
$( Alternate proof of ~ bdcnul . Similarly, for the next few theorems
|
|
proving boundedness of a class, one can either use their definition
|
|
followed by ~ bdceqir , or use the corresponding characterizations of its
|
|
elements followed by ~ bdelir . (Contributed by BJ, 3-Oct-2019.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bdcnulALT $p |- Bdd_ (/) $=
|
|
( cvv cdif c0 bdcvv bdcdif df-nul bdceqir ) AABCAADDEFG $.
|
|
|
|
$( Boundedness of the formula expressing that a setvar is equal to the empty
|
|
class. (Contributed by BJ, 21-Nov-2019.) $)
|
|
bdeq0 $p |- Bdd x = (/) $=
|
|
( cv c0 wss wceq bdcnul bdss 0ss eqss mpbiran2 bd0r ) ABZCDZLCEZACFGNMCLDLH
|
|
LCIJK $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Boundedness of the formula "the empty set belongs to the setvar ` x ` ".
|
|
(Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-bd0el $p |- Bdd (/) e. x $=
|
|
( vy c0 bdeq0 bj-bdcel ) ABCBDE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdcpw.1 $e |- Bdd_ A $.
|
|
$( The power class of a bounded class is bounded. (Contributed by BJ,
|
|
3-Oct-2019.) $)
|
|
bdcpw $p |- Bdd_ ~P A $=
|
|
( vx cv wss cab cpw bdss bdcab df-pw bdceqir ) CDAEZCFAGLCCABHICAJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The singleton of a setvar is bounded. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdcsn $p |- Bdd_ { x } $=
|
|
( vy weq cab cv csn ax-bdeq bdcab df-sn bdceqir ) BACZBDAEZFKBBAGHBLIJ $.
|
|
$}
|
|
|
|
$( The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdcpr $p |- Bdd_ { x , y } $=
|
|
( cv csn cun cpr bdcsn bdcun df-pr bdceqir ) ACZDZBCZDZEKMFLNAGBGHKMIJ $.
|
|
|
|
$( The unordered triple of three setvars is bounded. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdctp $p |- Bdd_ { x , y , z } $=
|
|
( cv cpr csn cun ctp bdcpr bdcsn bdcun df-tp bdceqir ) ADZBDZEZCDZFZGNOQHPR
|
|
ABICJKNOQLM $.
|
|
|
|
${
|
|
$d x A $.
|
|
bdsnss.1 $e |- Bdd_ A $.
|
|
$( Inclusion of a singleton of a setvar in a bounded class is a bounded
|
|
formula. (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdsnss $p |- Bdd { x } C_ A $=
|
|
( cv wcel csn wss bdeli vex snss bd0 ) ADZBELFBGABCHLBAIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Equality of a setvar with a singleton of a setvar is a bounded formula.
|
|
(Contributed by BJ, 16-Oct-2019.) $)
|
|
bdvsn $p |- Bdd x = { y } $=
|
|
( cv csn wss wa wceq bdcsn bdss bdcv bdsnss ax-bdan eqss bd0r ) ACZBCDZEZ
|
|
POEZFOPGQRAPBHIBOAJKLOPMN $.
|
|
$}
|
|
|
|
${
|
|
$d z x $. $d z y $.
|
|
$( The ordered pair of two setvars is a bounded class. (Contributed by BJ,
|
|
21-Nov-2019.) $)
|
|
bdop $p |- Bdd_ <. x , y >. $=
|
|
( vz cv cop csn wceq cpr wo wcel bdvsn wss bdcpr ax-bdel ax-bdan vex ssel
|
|
wa mpi bd0r bdss prid1 prid2 jca prssi impbii eqss ax-bdor elop bdelir )
|
|
CADZBDZEZCDZUKFGZUNUKULHZGZIUNUMJUOUQCAKUNUPLZUPUNLZRUQURUSCUPABMUAUKUNJZ
|
|
ULUNJZRZUSUTVAACNBCNOUSVBUSUTVAUSUKUPJUTUKULAPZUBUPUNUKQSUSULUPJVAUKULBPZ
|
|
UCUPUNULQSUDUKULUNUEUFTOUNUPUGTUHUNUKULCPVCVDUITUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The union of a setvar is a bounded class. (Contributed by BJ,
|
|
15-Oct-2019.) $)
|
|
bdcuni $p |- Bdd_ U. x $=
|
|
( vy vz wel wa wex cab cv cuni ax-bdel ax-bdex bdcab df-rex exancom bitri
|
|
wrex abbii bdceqi df-uni bdceqir ) BCDZCADZECFZBGZAHZIUACUEPZBGUDUFBUACAB
|
|
CJKLUFUCBUFUBUAECFUCUACUEMUBUACNOQRBCUEST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The intersection of a setvar is a bounded class. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdcint $p |- Bdd_ |^| x $=
|
|
( vz vy wel wi wal cab cint wral ax-bdel ax-bdal df-ral bd0 bdcab bdceqir
|
|
cv df-int ) BADCBDZEBFZCGAPZHSCRBTISRBACBJKRBTLMNCBTQO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z A $.
|
|
bdciun.1 $e |- Bdd_ A $.
|
|
$( The indexed union of a bounded class with a setvar indexing set is a
|
|
bounded class. (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdciun $p |- Bdd_ U_ x e. y A $=
|
|
( vz cv wcel wrex cab ciun bdeli ax-bdex bdcab df-iun bdceqir ) EFCGZABFZ
|
|
HZEIAQCJREPABECDKLMAEQCNO $.
|
|
|
|
$( The indexed intersection of a bounded class with a setvar indexing set
|
|
is a bounded class. (Contributed by BJ, 16-Oct-2019.) $)
|
|
bdciin $p |- Bdd_ |^|_ x e. y A $=
|
|
( vz cv wcel wral cab ciin bdeli ax-bdal bdcab df-iin bdceqir ) EFCGZABFZ
|
|
HZEIAQCJREPABECDKLMAEQCNO $.
|
|
$}
|
|
|
|
$( The successor of a setvar is a bounded class. (Contributed by BJ,
|
|
16-Oct-2019.) $)
|
|
bdcsuc $p |- Bdd_ suc x $=
|
|
( cv csn cun csuc bdcv bdcsn bdcun df-suc bdceqir ) ABZKCZDKEKLAFAGHKIJ $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Boundedness of the formula expressing that a setvar is equal to the
|
|
successor of another. (Contributed by BJ, 21-Nov-2019.) $)
|
|
bdeqsuc $p |- Bdd x = suc y $=
|
|
( cv csuc wss wa wceq bdcsuc bdss csn cun bdcv bdsnss ax-bdan unss df-suc
|
|
bd0 sseq1i bd0r eqss ) ACZBCZDZEZUCUAEZFUAUCGUDUEAUCBHIUBUBJZKZUAEZUEUBUA
|
|
EZUFUAEZFUHUIUJBUAALZIBUAUKMNUBUFUAOQUCUGUAUBPRSNUAUCTS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Boundedness of the formula "the successor of the setvar ` x ` belongs to
|
|
the setvar ` y ` ". (Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-bdsucel $p |- Bdd suc x e. y $=
|
|
( vz cv csuc bdeqsuc bj-bdcel ) BCADECAFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z t $. $d z t ph $.
|
|
bdcriota.bd $e |- Bdd ph $.
|
|
bdcriota.ex $e |- E! x e. y ph $.
|
|
$( A class given by a restricted definition binder is bounded, under the
|
|
given hypotheses. (Contributed by BJ, 24-Nov-2019.) $)
|
|
bdcriota $p |- Bdd_ ( iota_ x e. y ph ) $=
|
|
( vz vt wel wa cio cv cab wi wal wsb bicomi albii bitri imbi1i bdceqir
|
|
crio cint wcel wral ax-bdsb ax-bdel ax-bdim ax-bdal df-ral impexp clelsb3
|
|
sban anbi1i df-clab bd0 df-int weu wceq wreu df-reu mpbi iotaint df-riota
|
|
bdcab ax-mp ) BCHZAIZBJZABCKZUAVGBLZUBZVHFKVJUCZGFHZMZFNZGLVKVOGABFOZVMMZ
|
|
FVIUDZVOVQFCVPVMABFDUEGFUFUGUHVRVGBFOZVMMZFNZVOVRFCHZVPIZVMMZFNZWAVRWBVQM
|
|
ZFNWEVQFVIUIWFWDFWDWFWBVPVMUJPQRWDVTFWCVSVMVSWCVSVFBFOZVPIWCVFABFULWGWBVP
|
|
FBVIUKUMRPSQRVTVNFVSVLVMVLVSVGFBUNPSQRUOVDGFVJUPTVGBUQZVHVKURABVIUSWHEABV
|
|
IUTVAVGBVBVETABVIVCT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
CZF: Bounded separation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In this section, we state the axiom scheme of bounded separation, which is
|
|
part of CZF set theory.
|
|
|
|
$)
|
|
|
|
${
|
|
$d a b x $. $d a b ph $.
|
|
ax-bdsep.1 $e |- Bdd ph $.
|
|
$( Axiom scheme of bounded (or restricted, or ` Delta0 ` ) separation. It
|
|
is stated with all possible disjoint variable conditions, to show that
|
|
this weak form is sufficient. For the full axiom of separation, see
|
|
~ ax-sep . (Contributed by BJ, 5-Oct-2019.) $)
|
|
ax-bdsep $a |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d a b x $. $d a b ph $.
|
|
bdsep1.1 $e |- Bdd ph $.
|
|
$( Version of ~ ax-bdsep without initial universal quantifier.
|
|
(Contributed by BJ, 5-Oct-2019.) $)
|
|
bdsep1 $p |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) $=
|
|
( wel wa wb wal wex ax-bdsep spi ) BDFBCFAGHBIDJCABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y $. $d b y ph $.
|
|
bdsep2.1 $e |- Bdd ph $.
|
|
$( Version of ~ ax-bdsep with one DV condition removed and without initial
|
|
universal quantifier. Use ~ bdsep1 when sufficient. (Contributed by
|
|
BJ, 5-Oct-2019.) $)
|
|
bdsep2 $p |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) $=
|
|
( vy wel wa wb wal wex cv eleq2 anbi1d bibi2d albidv exbidv bdsep1 chvarv
|
|
weq ) BDGZBFGZAHZIZBJZDKUABCGZAHZIZBJZDKFCFCTZUEUIDUJUDUHBUJUCUGUAUJUBUFA
|
|
FLCLBLMNOPQABFDERS $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y $. $d y ph $.
|
|
bdsepnft.1 $e |- Bdd ph $.
|
|
$( Closed form of ~ bdsepnf . Version of ~ ax-bdsep with one DV condition
|
|
removed, the other DV condition replaced by a non-freeness antecedent,
|
|
and without initial universal quantifier. Use ~ bdsep1 when sufficient.
|
|
(Contributed by BJ, 19-Oct-2019.) $)
|
|
bdsepnft $p |- ( A. x F/ b ph ->
|
|
E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) ) $=
|
|
( vy wnf wal wel wa wb wex bdsep2 nfnf1 nfal nfa1 nfvd nfv a1i sp nfan ex
|
|
nfand nfbid nfald weq elequ2 adantl bibi1d albid cbvexd mpbii ) ADGZBHZBF
|
|
IZBCIZAJZKZBHZFLBDIZUQKZBHZDLABCFEMUNUSVBFDUMDBADNOUNURDBUMBPZUNUOUQDUNUO
|
|
DQUNUPADUPDGUNUPDRSUMBTUCUDUEUNFDUFZUSVBKUNVDJZURVABUNVDBVCVDBRUAVEUOUTUQ
|
|
VDUOUTKUNFDBUGUHUIUJUBUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y $. $d y ph $.
|
|
bdsepnf.nf $e |- F/ b ph $.
|
|
bdsepnf.1 $e |- Bdd ph $.
|
|
$( Version of ~ ax-bdsep with one DV condition removed, the other DV
|
|
condition replaced by a non-freeness hypothesis, and without initial
|
|
universal quantifier. See also ~ bdsepnfALT . Use ~ bdsep1 when
|
|
sufficient. (Contributed by BJ, 5-Oct-2019.) $)
|
|
bdsepnf $p |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) $=
|
|
( wnf wel wa wb wal wex bdsepnft mpg ) ADGBDHBCHAIJBKDLBABCDFMEN $.
|
|
|
|
$( Alternate proof of ~ bdsepnf , not using ~ bdsepnft . (Contributed by
|
|
BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
bdsepnfALT $p |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) $=
|
|
( vy wel wa wb wal wex bdsep2 nfv nfan nfbi nfal weq elequ2 bibi1d albidv
|
|
cbvex mpbi ) BGHZBCHZAIZJZBKZGLBDHZUFJZBKZDLABCGFMUHUKGDUGDBUDUFDUDDNUEAD
|
|
UEDNEOPQUKGNGDRZUGUJBULUDUIUFGDBSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z ph $.
|
|
bdzfauscl.bd $e |- Bdd ph $.
|
|
$( Closed form of the version of ~ zfauscl for bounded formulas using
|
|
bounded separation. (Contributed by BJ, 13-Nov-2019.) $)
|
|
bdzfauscl $p |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) $=
|
|
( vz wel wa wb wal wex cv wcel wceq eleq2 anbi1d bibi2d albidv exbidv
|
|
bdsep1 vtoclg ) BCHZBGHZAIZJZBKZCLUCBMZDNZAIZJZBKZCLGDEGMZDOZUGULCUNUFUKB
|
|
UNUEUJUCUNUDUIAUMDUHPQRSTABGCFUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x ph z $. $d x y z $.
|
|
bdbm1.3ii.bd $e |- Bdd ph $.
|
|
bdbm1.3ii.1 $e |- E. x A. y ( ph -> y e. x ) $.
|
|
$( Bounded version of ~ bm1.3ii . (Contributed by BJ, 5-Oct-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdbm1.3ii $p |- E. x A. y ( y e. x <-> ph ) $=
|
|
( vz wel wi wal wa wex weq elequ2 imbi2d albidv cbvexv mpbi bdsep1 pm3.2i
|
|
wb exan 19.42v bimsc1 alanimi eximi sylbir exlimiv ax-mp ) ACFGZHZCIZCBGZ
|
|
UIAJTZCIZBKZJZFKULATZCIZBKZUKUOFUKFKZUOAULHZCIZBKUTEVBUKBFBFLZVAUJCVCULUI
|
|
ABFCMNOPQACFBDRSUAUPUSFUPUKUNJZBKUSUKUNBUBVDURBUJUMUQCAUIULUCUDUEUFUGUH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Lemma for ~ bj-axempty and ~ bj-axempty2 . (Contributed by BJ,
|
|
25-Oct-2020.) (Proof modification is discouraged.) Use ~ ax-nul
|
|
instead. (New usage is discouraged.) $)
|
|
bj-axemptylem $p |- E. x A. y ( y e. x -> F. ) $=
|
|
( vz wel wfal wa wb wal wi bdfal bdsep1 bi1 falimd syl6 alimi eximii ) BA
|
|
DZBCDZEFZGZBHQEIZBHAEBCAJKTUABTQSEQSLREMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of the empty set from bounded separation. It is provable from
|
|
bounded separation since the intuitionistic FOL used in iset.mm assumes
|
|
a non-empty universe. See ~ axnul . (Contributed by BJ, 25-Oct-2020.)
|
|
(Proof modification is discouraged.) Use ~ ax-nul instead.
|
|
(New usage is discouraged.) $)
|
|
bj-axempty $p |- E. x A. y e. x F. $=
|
|
( wfal cv wral wex wel wi wal bj-axemptylem df-ral exbii mpbir ) CBADZEZA
|
|
FBAGCHBIZAFABJOPACBNKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of the empty set from bounded separation, alternate version to
|
|
~ bj-axempty . (Contributed by BJ, 27-Oct-2020.)
|
|
(Proof modification is discouraged.) Use ~ ax-nul instead.
|
|
(New usage is discouraged.) $)
|
|
bj-axempty2 $p |- E. x A. y -. y e. x $=
|
|
( wel wn wal wex wfal wi bj-axemptylem dfnot albii exbii mpbir ) BACZDZBE
|
|
ZAFNGHZBEZAFABIPRAOQBNJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( ~ nalset from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nalset $p |- -. E. x A. y y e. x $=
|
|
( vz wel wn wex wal alexnim wa wb ax-bdel ax-bdn bdsep1 weq elequ1 elequ2
|
|
bitrd notbid anbi12d bibi12d spv pclem6 syl eximii mpg ) BADZEZBFUFBGAFEA
|
|
UFABHCBDZCADZCCDZEZIZJZCGZUGBUKCABUJCCKLMUNBBDZUFUOEZIZJZUGUMURCBCBNZUHUO
|
|
ULUQCBBOUSUIUFUKUPCBAOUSUJUOUSUJBCDUOCBCOCBBPQRSTUAUOUFUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( ~ vprc from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-vprc $p |- -. _V e. _V $=
|
|
( vx vy cvv wcel cv wceq wex wal bj-nalset wb vex tbt albii dfcleq bitr4i
|
|
exbii mtbi isset mtbir ) CCDAEZCFZAGZBEZTDZBHZAGUBABIUEUAAUEUDUCCDZJZBHUA
|
|
UDUGBUFUDBKLMBTCNOPQACRS $.
|
|
$}
|
|
|
|
$( ~ nvel from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nvel $p |- -. _V e. A $=
|
|
( cvv wcel bj-vprc elex mto ) BACBBCDBAEF $.
|
|
|
|
$( ~ vnex from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-vnex $p |- -. E. x x = _V $=
|
|
( cvv wcel cv wceq wex bj-vprc isset mtbi ) BBCADBEAFGABHI $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
bdinex1.bd $e |- Bdd_ B $.
|
|
bdinex1.1 $e |- A e. _V $.
|
|
$( Bounded version of ~ inex1 . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdinex1 $p |- ( A i^i B ) e. _V $=
|
|
( vx vy cin cv wceq wex wel wcel wa wal cvv bdeli bdzfauscl ax-mp dfcleq
|
|
wb elin bibi2i albii bitri exbii mpbir issetri ) EABGZEHZUHIZEJFEKZFHZALU
|
|
LBLZMZTZFNZEJZAOLUQDUMFEAOFBCPQRUJUPEUJUKULUHLZTZFNUPFUIUHSUSUOFURUNUKULA
|
|
BUAUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
bdinex2.bd $e |- Bdd_ B $.
|
|
bdinex2.1 $e |- A e. _V $.
|
|
$( Bounded version of ~ inex2 . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdinex2 $p |- ( B i^i A ) e. _V $=
|
|
( cin cvv incom bdinex1 eqeltri ) BAEABEFBAGABCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
bdinex1g.bd $e |- Bdd_ B $.
|
|
$( Bounded version of ~ inex1g . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdinex1g $p |- ( A e. V -> ( A i^i B ) e. _V ) $=
|
|
( vx cv cin cvv wcel wceq ineq1 eleq1d vex bdinex1 vtoclg ) EFZBGZHIABGZH
|
|
IEACPAJQRHPABKLPBDEMNO $.
|
|
$}
|
|
|
|
${
|
|
bdssex.bd $e |- Bdd_ A $.
|
|
bdssex.1 $e |- B e. _V $.
|
|
$( Bounded version of ~ ssex . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdssex $p |- ( A C_ B -> A e. _V ) $=
|
|
( wss cin wceq cvv wcel df-ss bdinex2 eleq1 mpbii sylbi ) ABEABFZAGZAHIZA
|
|
BJPOHIQBACDKOAHLMN $.
|
|
$}
|
|
|
|
${
|
|
bdssexi.bd $e |- Bdd_ A $.
|
|
bdssexi.1 $e |- B e. _V $.
|
|
bdssexi.2 $e |- A C_ B $.
|
|
$( Bounded version of ~ ssexi . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdssexi $p |- A e. _V $=
|
|
( wss cvv wcel bdssex ax-mp ) ABFAGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
bdssexg.bd $e |- Bdd_ A $.
|
|
$( Bounded version of ~ ssexg . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdssexg $p |- ( ( A C_ B /\ B e. C ) -> A e. _V ) $=
|
|
( vx wcel wss cvv cv wi wceq sseq2 imbi1d vex bdssex vtoclg impcom ) BCFA
|
|
BGZAHFZAEIZGZSJRSJEBCTBKUARSTBALMATDENOPQ $.
|
|
$}
|
|
|
|
${
|
|
bdssexd.1 $e |- ( ph -> B e. C ) $.
|
|
bdssexd.2 $e |- ( ph -> A C_ B ) $.
|
|
bdssexd.bd $e |- Bdd_ A $.
|
|
$( Bounded version of ~ ssexd . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdssexd $p |- ( ph -> A e. _V ) $=
|
|
( wss wcel cvv bdssexg syl2anc ) ABCHCDIBJIFEBCDGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdrabexg.bd $e |- Bdd ph $.
|
|
bdrabexg.bdc $e |- Bdd_ A $.
|
|
$( Bounded version of ~ rabexg . (Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdrabexg $p |- ( A e. V -> { x e. A | ph } e. _V ) $=
|
|
( crab wss wcel cvv ssrab2 bdcrab bdssexg mpan ) ABCGZCHCDIOJIABCKOCDABCF
|
|
ELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( The intersection of two sets is a set, from bounded separation.
|
|
(Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-inex $p |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V ) $=
|
|
( vx vy vz wcel cv wceq wex cin cvv elisset wal ax-17 19.29r sylan2 19.29
|
|
wa sylan eximi ineq12 2eximi crab dfin5 vex ax-bdel bdcv bdrabexg eqeltri
|
|
ax-mp eleq1 mpbii exlimivv syl 3syl syl2an ) ACHEIZAJZEKZFIZBJZFKZABLZMHZ
|
|
BDHEACNFBDNVAVDTUTVDTZEKZUTVCTZFKZEKZVFVDVAVDEOVHVDEPUTVDEQRVGVJEUTUTFOVD
|
|
VJUTFPUTVCFSUAUBVKUSVBLZVEJZFKEKVFVIVMEFUSAVBBUCUDVMVFEFVMVLMHVFVLGIVBHZG
|
|
USUEZMGUSVBUFUSMHVOMHEUGVNGUSMGFUHEUIUJULUKVLVEMUMUNUOUPUQUR $.
|
|
$}
|
|
|
|
$( ~ intexr from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-intexr $p |- ( |^| A e. _V -> A =/= (/) ) $=
|
|
( cint cvv wcel c0 wceq bj-vprc inteq int0 syl6eq eleq1d mtbiri necon2ai )
|
|
ABZCDZAEAEFZOCCDGPNCCPNEBCAEHIJKLM $.
|
|
|
|
$( ~ intnexr from bounded separation. (Contributed by BJ, 18-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-intnexr $p |- ( |^| A = _V -> -. |^| A e. _V ) $=
|
|
( cint cvv wceq wcel bj-vprc eleq1 mtbiri ) ABZCDICECCEFICCGH $.
|
|
|
|
${
|
|
$d x z w $. $d y z w $.
|
|
$( Proof of ~ zfpair2 using only bounded separation. (Contributed by BJ,
|
|
5-Oct-2019.) (Proof modification is discouraged.) $)
|
|
bj-zfpair2 $p |- { x , y } e. _V $=
|
|
( vz vw cv cpr wceq wex wel weq wo wb wal ax-bdeq ax-bdor ax-pr bdbm1.3ii
|
|
wcel dfcleq vex elpr bibi2i albii bitri exbii mpbir issetri ) CAEZBEZFZCE
|
|
ZUJGZCHDCIZDAJZDBJZKZLZDMZCHUPCDUNUODANDBNOABCDPQULURCULUMDEZUJRZLZDMURDU
|
|
KUJSVAUQDUTUPUMUSUHUIDTUAUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x W $.
|
|
$( Proof of ~ prexg using only bounded separation. (Contributed by BJ,
|
|
5-Oct-2019.) (Proof modification is discouraged.) $)
|
|
bj-prexg $p |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V ) $=
|
|
( vx vy wcel cpr cvv wi wceq preq2 eleq1d bj-zfpair2 vtoclg preq1 vtocleg
|
|
cv syl5ib imp ) ACGBDGZABHZIGZUAUCJEACUAERZBHZIGZUDAKZUCUDFRZHZIGUFFBDUHB
|
|
KUIUEIUHBUDLMEFNOUGUEUBIUDABPMSQT $.
|
|
$}
|
|
|
|
$( ~ snexg from bounded separation. (Contributed by BJ, 5-Oct-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-snexg $p |- ( A e. V -> { A } e. _V ) $=
|
|
( wcel csn cpr cvv dfsn2 bj-prexg anidms syl5eqel ) ABCZADAAEZFAGKLFCAABBHI
|
|
J $.
|
|
|
|
${
|
|
bj-snex.1 $e |- A e. _V $.
|
|
$( ~ snex from bounded separation. (Contributed by BJ, 5-Oct-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-snex $p |- { A } e. _V $=
|
|
( cvv wcel csn bj-snexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( If a class is a set, then it is a member of a set. (Copied from
|
|
set.mm.) (Contributed by BJ, 3-Apr-2019.) $)
|
|
bj-sels $p |- ( A e. V -> E. x A e. x ) $=
|
|
( wcel cv csn wsbc snidg csb cvv bj-snexg sbcel2g syl wceq csbvarg eleq2d
|
|
wb bitrd mpbird spesbcd ) BCDZBAEZDZABFZUAUCAUDGZBUDDZBCHUAUEBAUDUBIZDZUF
|
|
UAUDJDZUEUHQBCKZAUDBUBJLMUAUGUDBUAUIUGUDNUJAUDJOMPRST $.
|
|
$}
|
|
|
|
$(
|
|
${
|
|
$d x y $.
|
|
bdiinex.bd $e |- Bdd_ A $.
|
|
@( Bounded version of a strengthening of ~ iinexgm . (Contributed by BJ,
|
|
13-Nov-2019.) (Proof modification is discouraged.) @)
|
|
bdiinex $p |- ( E. x e. y A e. V -> |^|_ x e. y A e. _V ) $=
|
|
wcel.cA=cA wcel.cB=cV wph=wcel vx=vx vx.cv=vy cA=cv wph=wrex wps=? vx=vx
|
|
vx.cv=vy cA=cv cB=cA wcel.cA=ciin wcel.cB=cvv wch=wcel wph=? wps=? wch=?
|
|
wph=? wps=? wch=? syl.1=? vx=? cA=? cB=? cC=? syl.2=iinss jca.1=syl jca.2=?
|
|
syl.1=jca vx=vx vx.cv=vy cA=cv cB=cA cA=ciin cB=? cC=? vx=vx vy=vy cA=cA
|
|
bdciun.1=bdiinex.bd bdssexg.bd=bdciin syl.2=bdssexg bdiinex=syl $.
|
|
$}
|
|
$)
|
|
|
|
${
|
|
$d w x y z $.
|
|
$( ~ axun2 from bounded separation. (Contributed by BJ, 15-Oct-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-axun2 $p |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) ) $=
|
|
( wel wa wex cv wrex ax-bdel ax-bdex df-rex exancom bitri ax-un bdbm1.3ii
|
|
bd0 ) CDEZDAEZFDGZBCRDAHZIZTRDACDJKUBSRFDGTRDUALSRDMNQABCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( ~ uniex2 from bounded separation. (Contributed by BJ, 15-Oct-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-uniex2 $p |- E. y y = U. x $=
|
|
( vz cv cuni wceq wex wel wcel wb wal bdcuni bdeli wi wa zfun eluni exbii
|
|
imbi1i mpbir albii bdbm1.3ii dfcleq ) BDZADZEZFZBGCBHZCDZUFIZJCKZBGUJBCCU
|
|
FALMUJUHNZCKZBGUHBAHOBGZUHNZCKZBGBCAPUMUPBULUOCUJUNUHBUIUEQSUARTUBUGUKBCU
|
|
DUFUCRT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
bj-uniex.1 $e |- A e. _V $.
|
|
$( ~ uniex from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-uniex $p |- U. A e. _V $=
|
|
( vx vy cv cuni cvv wcel wceq unieq eleq1d bj-uniex2 issetri vtocl ) CEZF
|
|
ZGHAFZGHCABOAIPQGOAJKDPCDLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( ~ uniexg from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-uniexg $p |- ( A e. V -> U. A e. _V ) $=
|
|
( vx cv cuni cvv wcel wceq unieq eleq1d vex bj-uniex vtoclg ) CDZEZFGAEZF
|
|
GCABNAHOPFNAIJNCKLM $.
|
|
$}
|
|
|
|
${
|
|
bj-unex.1 $e |- A e. _V $.
|
|
bj-unex.2 $e |- B e. _V $.
|
|
$( ~ unex from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-unex $p |- ( A u. B ) e. _V $=
|
|
( cpr cuni cun cvv unipr wcel bj-prexg mp2an bj-uniex eqeltrri ) ABEZFABG
|
|
HABCDIOAHJBHJOHJCDABHHKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
bdunex.bd1 $e |- Bdd_ A $.
|
|
bdunex.bd2 $e |- Bdd_ B $.
|
|
$( Bounded version of ~ unexb . (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdunexb $p |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V ) $=
|
|
( vx vy cvv wcel wa cun cv wceq uneq1 eleq1d vex bj-unex wss bdssexg mpan
|
|
uneq2 vtocl2g ssun1 ssun2 jca impbii ) AGHZBGHZIABJZGHZEKZFKZJZGHAUKJZGHU
|
|
IEFABGGUJALULUMGUJAUKMNUKBLUMUHGUKBATNUJUKEOFOPUAUIUFUGAUHQUIUFABUBAUHGCR
|
|
SBUHQUIUGBAUCBUHGDRSUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( ~ unexg from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
|
|
( vx vy cv cun cvv wcel wceq wb uneq1 eleq1 syl uneq2 vex bj-unex vtocl2g
|
|
) EGZFGZHZIJZAUAHZIJZABHZIJZEFABCDTAKUBUDKUCUELTAUAMUBUDINOUABKUDUFKUEUGL
|
|
UABAPUDUFINOTUAEQFQRS $.
|
|
$}
|
|
|
|
$( ~ sucexg from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-sucexg $p |- ( A e. V -> suc A e. _V ) $=
|
|
( wcel csn cvv cun csuc bj-snexg pm4.71i biimpi bj-unexg df-suc eleq1i 3syl
|
|
wa biimpri ) ABCZQADZECZOZARFZECZAGZECZQTQSABHIJARBEKUDUBUCUAEALMPN $.
|
|
|
|
${
|
|
bj-sucex.1 $e |- A e. _V $.
|
|
$( ~ sucex from bounded separation. (Contributed by BJ, 13-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-sucex $p |- suc A e. _V $=
|
|
( cvv wcel csuc bj-sucexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Delta_0-classical logic
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
${
|
|
ax-bj-d0class.bd $e |- Bdd ph $.
|
|
$( Axiom for ` Delta0 ` -classical logic. (Contributed by BJ,
|
|
2-Jan-2020.) $)
|
|
ax-bj-d0cl $a |- DECID ph $.
|
|
$}
|
|
|
|
$( Equivalence property for negation. TODO: minimize all theorems using
|
|
~ notbid and ~ notbii . (Contributed by BJ, 27-Jan-2020.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-notbi $p |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) ) $=
|
|
( wb wn bi2 con3d bi1 impbid ) ABCZADBDIBAABEFIABABGFH $.
|
|
|
|
${
|
|
bj-notbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference associated with ~ bj-notbi . (Contributed by BJ,
|
|
27-Jan-2020.) (Proof modification is discouraged.) $)
|
|
bj-notbii $p |- ( -. ph <-> -. ps ) $=
|
|
( wb wn bj-notbi ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
bj-notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction form of ~ bj-notbi . (Contributed by BJ, 27-Jan-2020.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
|
|
( wb wn bj-notbi syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Equivalence property for ` DECID ` . TODO: solve conflict with ~ dcbi ;
|
|
minimize ~ dcbii and ~ dcbid with it, as well as theorems using those.
|
|
(Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) $)
|
|
bj-dcbi $p |- ( ( ph <-> ps ) -> ( DECID ph <-> DECID ps ) ) $=
|
|
( wb wn wo wdc id bj-notbi orbi12d df-dc 3bitr4g ) ABCZAADZEBBDZEAFBFLABMNL
|
|
GABHIAJBJK $.
|
|
|
|
${
|
|
$d a x ph $.
|
|
$( ` Delta0 ` -classical logic and separation implies classical logic.
|
|
(Contributed by BJ, 2-Jan-2020.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-d0clsepcl $p |- DECID ph $=
|
|
( va vx wdc wex c0 cv wcel wb csn wel wal 0ex bj-snex zfauscl wceq anbi1d
|
|
wa eleq1 eximii bibi12d spcv snid biantrur bicomi bibi2i exbii ax-bj-d0cl
|
|
mpbi bj-bd0el bj-dcbi mpbii bj-ex ax-mp ) ADZBEUOFBGZHZAIZUOBUQFFJZHZARZI
|
|
ZBEURBECBKZCGZUSHZARZIZCLVBBACBUSFMNOVGVBCFMVDFPZVCUQVFVAVDFUPSVHVEUTAVDF
|
|
USSQUAUBTVBURBVAAUQAVAUTAFMUCUDUEUFUGUIURUQDUOUQBUJUHUQAUKULTUOBUMUN $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Inductive classes and the class of natural numbers (finite ordinals)
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
$)
|
|
|
|
$( Introduce constant for inductive classes. $)
|
|
$c Ind $.
|
|
|
|
$( Syntax for inductive classes. $)
|
|
wind $a wff Ind A $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define the property of being an inductive class. (Contributed by BJ,
|
|
30-Nov-2019.) $)
|
|
df-bj-ind $a |- ( Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A direct consequence of the definition of ` Ind ` . (Contributed by BJ,
|
|
30-Nov-2019.) $)
|
|
bj-indsuc $p |- ( Ind A -> ( B e. A -> suc B e. A ) ) $=
|
|
( vx wind cv csuc wcel wral c0 df-bj-ind simprbi wceq suceq rspcv syl5com
|
|
eleq1d ) ADZCEZFZAGZCAHZBAGBFZAGZQIAGUACAJKTUCCBARBLSUBARBMPNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality property for ` Ind ` . (Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-indeq $p |- ( A = B -> ( Ind A <-> Ind B ) ) $=
|
|
( vx wind c0 wcel cv csuc wral wa wceq df-bj-ind eleq2 raleqbi1dv anbi12d
|
|
bicomd syl5rbb syl5bb ) ADEAFZCGHZAFZCAIZJZABKZBDZCALUEEBFZTBFZCBIZJUDUCC
|
|
BLUDUFSUHUBUDSUFABEMPUDUBUHUAUGCABABTMNPOQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Boundedness of the formula "the setvar ` x ` is an inductive class".
|
|
(Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-bdind $p |- Bdd Ind x $=
|
|
( vy c0 wcel csuc wral wind bj-bd0el bj-bdsucel ax-bdal ax-bdan df-bj-ind
|
|
cv wa bd0r ) CAMZDZBMEPDZBPFZNPGQSAHRBABAIJKBPLO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The property of being an inductive class is closed under intersections.
|
|
(Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-indint $p |- Ind |^| { x e. A | Ind x } $=
|
|
( vy cv wind crab cint c0 wcel csuc wral df-bj-ind simplbi rgenw elintrab
|
|
wi 0ex mpbir bj-indsuc a2i ralimi vex bj-sucex 3imtr4i rgen mpbir2an ) AD
|
|
ZEZABFGZEHUIIZCDZJZUIIZCUIKUJUHHUGIZPZABKUOABUHUNULUGIZCUGKCUGLMNUHAHBQOR
|
|
UMCUIUHUKUGIZPZABKUHUPPZABKUKUIIUMURUSABUHUQUPUGUKSTUAUHAUKBCUBZOUHAULBUK
|
|
UTUCOUDUECUILUF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( If ` A ` is inductive and ` B ` is "inductive in ` A ` ", then
|
|
` ( A i^i B ) ` is inductive. (Contributed by BJ, 25-Oct-2020.) $)
|
|
bj-indind $p |- ( ( Ind A /\ ( (/) e. B /\
|
|
A. x e. A ( x e. B -> suc x e. B ) ) ) -> Ind ( A i^i B ) ) $=
|
|
( wind c0 wcel cv csuc wi wral wa cin df-bj-ind id biimpri wal syl df-ral
|
|
elin sylibr sylanb r19.26 simpl simpr syl6an ralimi pm3.31 syl5bi anim12i
|
|
an4s alimi sylbi ) BDZECFZAGZCFZUOHZCFZIZABJZKZKZEBCLZFZUQVCFZAVCJZKZVCDV
|
|
BEBFZUNKZUQBFZABJZUTKZKZVGUMVHVKKVAVMABMVHUNVKUTVMVMNUJUAVIVDVLVFVDVIEBCS
|
|
OVLUOVCFZVEIZAPZVFVLUPVEIZABJZVPVLVJUSKZABJZVRVTVLVJUSABUBOVSVQABVSVJUPUR
|
|
VEVJUSUCVJUSUDVEVJURKUQBCSOUEUFQVRUOBFZVQIZAPVPVQABRWBVOAVNWAUPKWBVEUOBCS
|
|
WAUPVEUGUHUKULQVEAVCRTUIQAVCMT $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Alternate definition of ` _om ` , as the intersection of all the
|
|
inductive sets. Proposal: make this the definition. (Contributed by
|
|
BJ, 30-Nov-2019.) $)
|
|
bj-dfom $p |- _om = |^| { x | Ind x } $=
|
|
( vy com c0 cv wcel csuc wral cab cint wind dfom3 df-bj-ind bicomi inteqi
|
|
wa abbii eqtri ) CDAEZFBEGSFBSHPZAIZJSKZAIZJABLUAUCTUBAUBTBSMNQOR $.
|
|
$}
|
|
|
|
$( ` _om ` is an inductive class. (Contributed by BJ, 30-Nov-2019.) $)
|
|
bj-omind $p |- Ind _om $=
|
|
( vx com wind cv cvv crab cint bj-indint wb cab bj-dfom rabab inteqi eqtr4i
|
|
wceq bj-indeq ax-mp mpbir ) BCZADCZAEFZGZCZAEHBUBOSUCIBTAJZGUBAKUAUDTALMNBU
|
|
BPQR $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( ` _om ` is included in all the inductive sets (but for the moment, we
|
|
cannot prove that it is included in all the inductive classes).
|
|
(Contributed by BJ, 30-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-omssind $p |- ( A e. V -> ( Ind A -> _om C_ A ) ) $=
|
|
( wcel wind cab cint wss com nfcv nfv wceq bj-indeq biimprd bj-intabssel1
|
|
vx cv bj-dfom sseq1i syl6ibr ) ABCADZOPZDZOEFZAGHAGUBTOABOAITOJUAAKUBTUAA
|
|
LMNHUCAOQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A characterization of subclasses of ` _om ` . (Contributed by BJ,
|
|
30-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-ssom $p |- ( A. x ( Ind x -> A C_ x ) <-> A C_ _om ) $=
|
|
( vy cv wind wss wi wal cab cint com wral wcel ssint df-ral bj-indeq elab
|
|
vex imbi1i albii 3bitrri bj-dfom eqcomi sseq2i bitri ) ADZEZBUFFZGZAHZBCD
|
|
ZEZCIZJZFZBKFUOUHAUMLUFUMMZUHGZAHUJABUMNUHAUMOUQUIAUPUGUHULUGCUFARUKUFPQS
|
|
TUAUNKBKUNCUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A set is equal to ` _om ` if and only if it is the smallest inductive
|
|
set. (Contributed by BJ, 30-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-om $p |- ( A e. V ->
|
|
( A = _om <-> ( Ind A /\ A. x ( Ind x -> A C_ x ) ) ) ) $=
|
|
( wcel com wceq wind cv wss wi wal wa bj-omind bj-indeq mpbiri bj-omssind
|
|
cvv vex ax-mp sseq1 syl5ibr alrimiv jca bj-ssom biimpi adantl a1i adantrd
|
|
jcad eqss syl6ibr impbid2 ) BCDZBEFZBGZAHZGZBUPIZJZAKZLZUNUOUTUNUOEGMBENO
|
|
UNUSAUQURUNEUPIZUPQDUQVBJARUPQPSBEUPTUAUBUCUMVABEIZEBIZLUNUMVAVCVDVAVCJUM
|
|
UTVCUOUTVCABUDUEUFUGUMUOVDUTBCPUHUIBEUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Two formulations of the axiom of infinity (see ~ ax-infvn and
|
|
~ bj-omex ) . (Contributed by BJ, 30-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-2inf $p |- ( _om e. _V <->
|
|
E. x ( Ind x /\ A. y ( Ind y -> x C_ y ) ) ) $=
|
|
( com cvv wcel cv wind wss wi wal wa wceq eqid bj-om mpbii bj-indeq sseq1
|
|
wex imbi2d albidv anbi12d spcegv mpd vex ax-mp biimpri eximi isset sylibr
|
|
wb impbii ) CDEZAFZGZBFZGZUMUOHZIZBJZKZARZULCGZUPCUOHZIZBJZKZVAULCCLVFCMB
|
|
CDNOUTVFACDUMCLZUNVBUSVEUMCPVGURVDBVGUQVCUPUMCUOQSTUAUBUCVAVGARULUTVGAVGU
|
|
TUMDEVGUTUJAUDBUMDNUEUFUGACUHUIUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
The first three Peano postulates
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
The first three Peano postulates follow from constructive set theory
|
|
(actually, from its core axioms). The proofs ~ peano1 and ~ peano3 already
|
|
show this. In this section, we prove ~ bj-peano2 to complete this program.
|
|
We also prove a preliminary version of the fifth Peano postulate from the
|
|
core axioms.
|
|
|
|
$)
|
|
|
|
$( Constructive proof of ~ peano2 . Temporary note: another possibility is
|
|
to simply replace ~ sucexg with ~ bj-sucexg in the proof of ~ peano2 .
|
|
(Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-peano2 $p |- ( A e. _om -> suc A e. _om ) $=
|
|
( com wind wcel csuc wi bj-omind bj-indsuc ax-mp ) BCABDAEBDFGBAHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Version of ~ peano5 when ` _om i^i A ` is assumed to be a set, allowing
|
|
a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
peano5set $p |- ( ( _om i^i A ) e. V ->
|
|
( ( (/) e. A /\ A. x e. _om ( x e. A -> suc x e. A ) ) -> _om C_ A ) ) $=
|
|
( com cin wcel c0 cv csuc wi wral wind bj-omind bj-indind mpan bj-omssind
|
|
wa wss imp sylan2 inss2 syl6ss ex ) DBEZCFZGBFAHZBFUFIBFJADKQZDBRUEUGQDUD
|
|
BUGUEUDLZDUDRZDLUGUHMADBNOUEUHUIUDCPSTDBUAUBUC $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
CZF: Infinity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In the absence of full separation, the axiom of infinity has to be stated
|
|
more precisely, as the existence of the smallest class containing the empty
|
|
set and the successor of each of its elements.
|
|
|
|
$)
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
The set of natural numbers (finite ordinals)
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In this section, we introduce the axiom of infinity in a constructive setting
|
|
( ~ ax-infvn ) and deduce that the class ` _om ` of finite ordinals is a set
|
|
( ~ bj-omex ).
|
|
|
|
$)
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of infinity in a constructive setting. This asserts the existence
|
|
of the special set we want (the set of natural numbers), instead of the
|
|
existence of a set with some properties ( ~ ax-iinf ) from which one
|
|
then proves, using full separation, that the wanted set exists
|
|
( ~ omex ). "vn" is for "von Neumann". (Contributed by BJ,
|
|
14-Nov-2019.) $)
|
|
ax-infvn $a |- E. x ( Ind x /\ A. y ( Ind y -> x C_ y ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Proof of ~ omex from ~ ax-infvn . (Contributed by BJ, 14-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-omex $p |- _om e. _V $=
|
|
( vx vy com cvv wcel cv wind wss wi wal wa wex ax-infvn bj-2inf mpbir ) C
|
|
DEAFZGBFZGPQHIBJKALABMABNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Peano's fifth postulate
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In this section, we give constructive proofs of two versions of Peano's fifth
|
|
postulate.
|
|
|
|
$)
|
|
|
|
${
|
|
$d x A $.
|
|
bdpeano5.bd $e |- Bdd_ A $.
|
|
$( Bounded version of ~ peano5 . (Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdpeano5 $p |- ( ( (/) e. A /\ A. x e. _om ( x e. A -> suc x e. A ) ) ->
|
|
_om C_ A ) $=
|
|
( com cin cvv wcel c0 cv csuc wi wral wss bj-omex bdinex1 peano5set ax-mp
|
|
wa ) DBEFGHBGAIZBGSJBGKADLRDBMKDBCNOABFPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Version of ~ peano5 when ` A ` is assumed to be a set, allowing a proof
|
|
from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
speano5 $p |- ( ( A e. V /\ (/) e. A /\
|
|
A. x e. _om ( x e. A -> suc x e. A ) ) -> _om C_ A ) $=
|
|
( wcel c0 cv csuc com wral wss cin cvv bj-omex bj-inex mpan peano5set syl
|
|
wi wa 3impib ) BCDZEBDZAFZBDUCGBDRAHIZHBJZUAHBKLDZUBUDSUERHLDUAUFMHBLCNOA
|
|
BLPQT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Bounded induction and Peano's fourth postulate
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In this section, we prove various versions of bounded induction from
|
|
the basic axioms of CZF (in particular, without the axiom of set induction).
|
|
We also prove Peano's fourth postulate.
|
|
Together with the results from the previous sections, this proves from the
|
|
core axioms of CZF (with infinity) that the set of finite ordinals satisfies
|
|
the five Peano postulates and thus provides a model for the set of natural
|
|
numbers.
|
|
$)
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bounded induction (principle of induction when ` A ` is assumed to be a
|
|
set) allowing a proof from basic constructive axioms. See ~ find for a
|
|
nonconstructive proof of the general case. See ~ bdfind for a proof
|
|
when ` A ` is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
findset $p |- ( A e. V ->
|
|
( ( A C_ _om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = _om ) ) $=
|
|
( wcel com wss c0 cv csuc wral wceq wa simpr1 wi simp2 df-ral alral sylbi
|
|
w3a wal 3ad2ant3 jca 3anass biimpri sylan2 speano5 syl eqssd ex ) BCDZBEF
|
|
ZGBDZAHZIBDZABJZSZBEKUJUPLZBEUJUKULUOMUQUJULUMBDUNNZAEJZSZEBFUPUJULUSLZUT
|
|
UPULUSUKULUOOUOUKUSULUOURATUSUNABPURAEQRUAUBUTUJVALUJULUSUCUDUEABCUFUGUHU
|
|
I $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
bdfind.bd $e |- Bdd_ A $.
|
|
$( Bounded induction (principle of induction when ` A ` is assumed to be
|
|
bounded), proved from basic constructive axioms. See ~ find for a
|
|
nonconstructive proof of the general case. See ~ findset for a proof
|
|
when ` A ` is assumed to be a set. (Contributed by BJ, 22-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdfind $p |-
|
|
( ( A C_ _om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = _om ) $=
|
|
( cvv wcel com wss c0 cv csuc wral w3a wceq bj-omex bdssex 3ad2ant1 mpcom
|
|
findset ) BDEZBFGZHBEZAIJBEABKZLBFMTUASUBBFCNOPABDRQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
bj-bdfindis.bd $e |- Bdd ph $.
|
|
bj-bdfindis.nf0 $e |- F/ x ps $.
|
|
bj-bdfindis.nf1 $e |- F/ x ch $.
|
|
bj-bdfindis.nfsuc $e |- F/ x th $.
|
|
bj-bdfindis.0 $e |- ( x = (/) -> ( ps -> ph ) ) $.
|
|
bj-bdfindis.1 $e |- ( x = y -> ( ph -> ch ) ) $.
|
|
bj-bdfindis.suc $e |- ( x = suc y -> ( th -> ph ) ) $.
|
|
$( Bounded induction (principle of induction for bounded formulas), using
|
|
implicit substitutions (the biconditional versions of the hypotheses are
|
|
implicit substitutions, and we have weakened them to implications).
|
|
Constructive proof (from CZF). See ~ finds for a proof of full
|
|
induction in IZF. From this version, it is easy to prove bounded
|
|
versions of ~ finds , ~ finds2 , ~ finds1 . (Contributed by BJ,
|
|
21-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-bdfindis $p |-
|
|
( ( ps /\ A. y e. _om ( ch -> th ) ) -> A. x e. _om ph ) $=
|
|
( wi com wral wa c0 wcel elabf2 cab wss cv 0ex elabf1 vex bj-sucex ralimi
|
|
csuc imim12i bdcab bdpeano5 syl2an ssabral sylib ) BCDNZFOPZQOAEUAZUBZAEO
|
|
PBRURSFUCZURSZUTUIZURSZNZFOPUSUQABERHUDKTUPVDFOVACDVCACEUTILUEADEVBJUTFUF
|
|
UGMTUJUHFURAEGUKULUMAEOUNUO $.
|
|
|
|
bj-bdfindisg.nfa $e |- F/_ x A $.
|
|
bj-bdfindisg.nfterm $e |- F/ x ta $.
|
|
bj-bdfindisg.term $e |- ( x = A -> ( ph -> ta ) ) $.
|
|
$( Version of ~ bj-bdfindis using a class term in the consequent.
|
|
Constructive proof (from CZF). See the comment of ~ bj-bdfindis for
|
|
explanations. (Contributed by BJ, 21-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-bdfindisg $p |-
|
|
( ( ps /\ A. y e. _om ( ch -> th ) ) -> ( A e. _om -> ta ) ) $=
|
|
( wi com wral wa wcel bj-bdfindis nfcv bj-rspg syl ) BCDSGTUAUBAFTUAHTUCE
|
|
SABCDFGIJKLMNOUDAEFHTPFTUEQRUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
bj-bdfindes.bd $e |- Bdd ph $.
|
|
$( Bounded induction (principle of induction for bounded formulas), using
|
|
explicit substitutions. Constructive proof (from CZF). See the comment
|
|
of ~ bj-bdfindis for explanations. From this version, it is easy to
|
|
prove the bounded version of ~ findes . (Contributed by BJ,
|
|
21-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-bdfindes $p |- ( ( [. (/) / x ]. ph /\
|
|
A. x e. _om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. _om ph ) $=
|
|
( vy cv csuc wsbc wi com wral wsb nfv nfim nfs1v nfsbc1v weq wceq sbceq1a
|
|
c0 biimprd sbequ12 sbceq1d imbi12d cbvral sbequ1 bj-bdfindis sylan2b
|
|
suceq ) AABBEZFZGZHZBIJABSGZABDKZABDEZFZGZHZDIJABIJULURBDIAUKDADLUKDLMUNU
|
|
QBABDNZABUPOZMBDPZAUNUKUQABDUAVAABUJUPUIUOUHUBUCUDAUMUNUQBDCABSOUSUTUISQA
|
|
UMABSRTABDUEUIUPQAUQABUPRTUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Constructive proof of a variant of ~ nn0suc . For a constructive proof
|
|
of ~ nn0suc , see ~ bj-nn0suc . (Contributed by BJ, 19-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nn0suc0 $p |- ( A e. _om -> ( A = (/) \/ E. x e. A A = suc x ) ) $=
|
|
( vy vz cv c0 wceq csuc wrex wo com eqeq1 rexeqbi1dv wtru wi wral a1d weq
|
|
a1tru wn orbi12d tru rgenw bdeq0 bdeqsuc ax-bdex ax-bdor nfv orc expi wel
|
|
wcel vex sucid eleq2 mpbiri suceq eqeq2d rspcev mpancom bj-bdfindis mp2an
|
|
olcd vtoclri ) CEZFGZVEAEZHZGZAVEIZJZBFGZBVHGZABIZJCBKVEBGVFVLVJVNVEBFLVI
|
|
VMAVEBVEBVHLMUANNNOZDKPVKCKPUBVODKNSUCVKNNNCDVFVJCUDVIACCAUEUFUGNCUHZVPVP
|
|
VFVKNVFVJUIQCDRZVKNVQVKTOTSUJVEDEZHZGZVKNVTVJVFDCUKZVTVJVTWAVRVSULVRDUMUN
|
|
VEVSVRUOUPVIVTAVRVEADRVHVSVEVGVRUQURUSUTVCQVAVBVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $. $d x y z $.
|
|
$( A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nntrans $p |- ( A e. _om -> ( B e. A -> B C_ A ) ) $=
|
|
( vx vz vy com wcel cv wss wral c0 csuc wi syl5bi wceq raleqbi1dv biimprd
|
|
nfv sseq2 biimpd ral0 csn cun wel df-suc eleq2i wo elun sssucid sstr2 mpi
|
|
imim2i elsni syl6eqss a1i jaod ralimi2 bdcv bdss ax-bdal weq bj-bdfindisg
|
|
rgenw nfcv mp2an sseq1 rspc syl5com ) AFGZCHZAIZCAJZBAGBAIZVJKIZCKJZVJDHZ
|
|
IZCVPJZVJVPLZIZCVSJZMZDFJVIVLMVNCUAWBDFVQVTCVPVSVJVSGVJVPVPUBZUCZGZCDUDZV
|
|
QMZVTVSWDVJVPUEUFWEWFVJWCGZUGWGVTVJVPWCUHWGWFVTWHVQVTWFVQVPVSIVTVPUIZVJVP
|
|
VSUJUKULWHVTMWGWHVJVPVSVJVPUMWIUNUOUPNNUQVCVJEHZIZCWJJZVOVRWAVLEDAWKCECWJ
|
|
EURUSUTVOERVRERWAERWJKOWLVOWKVNCWJKWJKVJSPQEDVAWLVRWKVQCWJVPWJVPVJSPTWJVS
|
|
OWLWAWKVTCWJVSWJVSVJSPQEAVDVLERWJAOWLVLWKVKCWJAWJAVJSPTVBVEVKVMCBAVMCRVJB
|
|
AVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nntrans2 $p |- ( A e. _om -> Tr A ) $=
|
|
( vx com wcel cv wss wral wtr bj-nntrans ralrimiv dftr3 sylibr ) ACDZBEZA
|
|
FZBAGAHMOBAANIJBAKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y $.
|
|
$( A natural number does not belong to itself. Version of ~ elirr for
|
|
natural numbers, which does not require ~ ax-setind . (Contributed by
|
|
BJ, 24-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-nnelirr $p |- ( A e. _om -> -. A e. A ) $=
|
|
( vy vx c0 wcel wn cv csuc wi com wral syl5bi nfv wceq eleq1 eleq2 notbid
|
|
bitrd biimprd biimpd noel csn cun df-suc eleq2i wo wss bj-nntrans sucssel
|
|
elun syld vex sucid elsni syl5eleq a1i con3d ax-bdel ax-bdn elequ1 elequ2
|
|
jaod rgen nfcv bj-bdfindisg mp2an ) DDEZFZBGZVIEZFZVIHZVLEZFZIZBJKAJEAAEZ
|
|
FZIDUAVOBJVIJEZVMVJVMVLVIVIUBZUCZEZVRVJVLVTVLVIUDUEWAVLVIEZVLVSEZUFVRVJVL
|
|
VIVSUJVRWBVJWCVRWBVLVIUGVJVIVLUHVIVIJUIUKWCVJIVRWCVIVLVIVIBULUMVLVIUNUOUP
|
|
VBLLUQVCCGZWDEZFZVHVKVNVQCBAWECCURUSVHCMVKCMVNCMWDDNZWFVHWGWEVGWGWEDWDEVG
|
|
WDDWDOWDDDPRQSWDVINZWFVKWHWEVJWHWEVIWDEVJCBCUTCBBVARQTWDVLNZWFVNWIWEVMWIW
|
|
EVLWDEVMWDVLWDOWDVLVLPRQSCAVDVQCMWDANZWFVQWJWEVPWJWEAWDEVPWDAWDOWDAAPRQTV
|
|
EVF $.
|
|
$}
|
|
|
|
$( A version of ~ en2lp for natural numbers, which does not require
|
|
~ ax-setind .
|
|
|
|
Note: using this theorem and ~ bj-nnelirr , one can remove dependency on
|
|
~ ax-setind from ~ nntri2 and ~ nndcel ; one can actually remove more
|
|
dependencies from these. (Contributed by BJ, 28-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nnen2lp $p |- ( ( A e. _om /\ B e. _om ) -> -. ( A e. B /\ B e. A ) ) $=
|
|
( com wcel wa wn bj-nnelirr adantl wss wi bj-nntrans ssel syl6 impd mtod )
|
|
ACDZBCDZEZABDZBADZEBBDZQUAFPBGHRSTUARSABIZTUAJQSUBJPBAKHABBLMNO $.
|
|
|
|
$( Remove from ~ peano4 dependency on ~ ax-setind . Therefore, it only
|
|
requires core constructive axioms (albeit more of them). (Contributed by
|
|
BJ, 28-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-peano4 $p |-
|
|
( ( A e. _om /\ B e. _om ) -> ( suc A = suc B <-> A = B ) ) $=
|
|
( com wcel wa csuc wceq w3a 3simpa pm3.22 bj-nnen2lp 3syl sucidg syl5ibrcom
|
|
wn wo eleq2 elsucg sylibd imp 3adant1 syl5ibcom 3adant2 eqcom orbi2i anbi1i
|
|
jca sylib ordir sylibr ord mpd 3expia suceq impbid1 ) ACDZBCDZEZAFZBFZGZABG
|
|
ZUPUQVAVBUPUQVAHZBADZABDZEZOZVBVCURUQUPEVGUPUQVAIUPUQJBAKLVCVFVBVCVDVBPZVEV
|
|
BPZEZVFVBPVCVDBAGZPZVIEVJVCVLVIUQVAVLUPUQVAVLUQVABUSDZVLUQVMVABUTDBCMUSUTBQ
|
|
NBACRSTUAUPVAVIUQUPVAVIUPVAAUTDZVIUPAUSDVAVNACMUSUTAQUBABCRSTUCUGVLVHVIVKVB
|
|
VDBAUDUEUFUHVDVEVBUIUJUKULUMABUNUO $.
|
|
|
|
${
|
|
$d a x y A $.
|
|
$( The set ` _om ` is transitive. A natural number is included in
|
|
` _om ` . Constructive proof of ~ elnn .
|
|
|
|
The idea is to use bounded induction with the formula ` x C_ _om ` .
|
|
This formula, in a logic with terms, is bounded. So in our logic
|
|
without terms, we need to temporarily replace it with ` x C_ a ` and
|
|
then deduce the original claim. (Contributed by BJ, 29-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-omtrans $p |- ( A e. _om -> A C_ _om ) $=
|
|
( vy va vx cv com wss csuc wi wcel wral bj-omex wceq sseq2 imbi12d c0 nfv
|
|
sseq1 biimprd biimpd ralbidv imbi2d 0ss bdcv bdss nfcv bj-bdfindisg vtocl
|
|
mpan wa csn cun df-suc simpr simpl snssd unssd syl5eqss ex mprg ) BEZFGZV
|
|
AHZFGZIZAFJZAFGZIZBFVACEZGZVCVIGZIZBFKZVFAVIGZIZIVEBFKZVHICFLVIFMZVMVPVOV
|
|
HVQVLVEBFVQVJVBVKVDVIFVANVIFVCNOUAVQVNVGVFVIFANUBOPVIGZVMVOVIUCDEZVIGZVRV
|
|
JVKVNDBADVICUDUEVRDQVJDQVKDQVSPMVTVRVSPVIRSVSVAMVTVJVSVAVIRTVSVCMVTVKVSVC
|
|
VIRSDAUFVNDQVSAMVTVNVSAVIRTUGUIUHVAFJZVBVDWAVBUJZVCVAVAUKZULFVAUMWBVAWCFW
|
|
AVBUNWBVAFWAVBUOUPUQURUSUT $.
|
|
$}
|
|
|
|
$( The set ` _om ` is transitive. (Contributed by BJ, 29-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-omtrans2 $p |- Tr _om $=
|
|
( vx com wtr cv wss dftr3 bj-omtrans mprgbir ) BCADZBEABABFIGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( A natural number is an ordinal. Constructive proof of ~ nnord . Can
|
|
also be proved from ~ bj-nnelon if the latter is proved from
|
|
~ bj-omssonALT . (Contributed by BJ, 27-Oct-2020.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nnord $p |- ( A e. _om -> Ord A ) $=
|
|
( vx com wcel wtr cv wral word bj-nntrans2 wi wal bj-omtrans syl6 alrimiv
|
|
sseld df-ral sylibr dford3 sylanbrc ) ACDZAEBFZEZBAGZAHAITUAADZUBJZBKUCTU
|
|
EBTUDUACDUBTACUAALOUAIMNUBBAPQBARS $.
|
|
$}
|
|
|
|
$( A natural number is an ordinal. Constructive proof of ~ nnon . Can also
|
|
be proved from ~ bj-omssonALT . (Contributed by BJ, 27-Oct-2020.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-nnelon $p |- ( A e. _om -> A e. On ) $=
|
|
( com wcel con0 word bj-nnord elong mpbird ) ABCADCAEAFABGH $.
|
|
|
|
$( The set ` _om ` is an ordinal. Constructive proof of ~ ordom .
|
|
(Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) $)
|
|
bj-omord $p |- Ord _om $=
|
|
( vx com word wtr cv wral bj-omtrans2 bj-nntrans2 rgen dford3 mpbir2an ) BC
|
|
BDAEZDZABFGMABLHIABJK $.
|
|
|
|
$( The set ` _om ` is an ordinal. Constructive proof of ~ omelon .
|
|
(Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) $)
|
|
bj-omelon $p |- _om e. On $=
|
|
( com con0 wcel word bj-omord bj-omex elon mpbir ) ABCADEAFGH $.
|
|
|
|
$( Constructive proof of ~ omsson . See also ~ bj-omssonALT . (Contributed
|
|
by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is
|
|
discouraged. $)
|
|
bj-omsson $p |- _om C_ On $=
|
|
( vx com con0 cv bj-nnelon ssriv ) ABCADEF $.
|
|
|
|
$( Alternate proof of ~ bj-omsson . (Contributed by BJ, 27-Oct-2020.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bj-omssonALT $p |- _om C_ On $=
|
|
( com con0 wcel wss bj-omelon onss ax-mp ) ABCABDEAFG $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Proof of (biconditional form of) ~ nn0suc from the core axioms of CZF.
|
|
See also ~ bj-nn0sucALT . As a characterization of the elements of
|
|
` _om ` , this could be labeled "elom". (Contributed by BJ,
|
|
19-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-nn0suc $p |- ( A e. _om <-> ( A = (/) \/ E. x e. _om A = suc x ) ) $=
|
|
( com wcel c0 wceq cv csuc wrex wo bj-nn0suc0 wi bj-omtrans ssrexv orim2d
|
|
wss syl mpd peano1 eleq1 mpbiri bj-peano2 eleq1a imp sylan rexlimiva jaoi
|
|
impbii ) BCDZBEFZBAGZHZFZACIZJZUIUJUMABIZJUOABKUIUPUNUJUIBCPUPUNLBMUMABCN
|
|
QORUJUIUNUJUIECDSBECTUAUMUIACUKCDULCDZUMUIUKUBUQUMUIULCBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
CZF: Set induction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In this section, we add the axiom of set induction to the core axioms of CZF.
|
|
|
|
$)
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Set induction
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In this section, we prove some variants of the axiom of set induction.
|
|
|
|
$)
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
$( Axiom of set-induction with a DV condition replaced with a non-freeness
|
|
hypothesis (Contributed by BJ, 22-Nov-2019.) $)
|
|
setindft $p |- ( A. x F/ y ph ->
|
|
( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) ) $=
|
|
( vz wnf wal wsb cv wral wi nfa1 nfv nfnf1 nfsbt a1i weq wb sbequ cbvrald
|
|
nfal biimpd imim1d alimd ax-setind syl6 ) ACEZBFZABCGZCBHZIZAJZBFABDGZDUI
|
|
IZAJZBFABFUGUKUNBUFBKUGUMUJAUGUMUJUGULUHDCUIUGDLUFCBACMTABDCNUHDEUGUHDLOD
|
|
CPULUHQJUGADCBROSUAUBUCADBUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
setindf.nf $e |- F/ y ph $.
|
|
$( Axiom of set-induction with a DV condition replaced with a non-freeness
|
|
hypothesis (Contributed by BJ, 22-Nov-2019.) $)
|
|
setindf $p |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) $=
|
|
( wnf wsb cv wral wi wal setindft mpg ) ACEABCFCBGHAIBJABJIBABCKDL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
setindis.nf0 $e |- F/ x ps $.
|
|
setindis.nf1 $e |- F/ x ch $.
|
|
setindis.nf2 $e |- F/ y ph $.
|
|
setindis.nf3 $e |- F/ y ps $.
|
|
setindis.1 $e |- ( x = z -> ( ph -> ps ) ) $.
|
|
setindis.2 $e |- ( x = y -> ( ch -> ph ) ) $.
|
|
$( Axiom of set induction using implicit substitutions. (Contributed by
|
|
BJ, 22-Nov-2019.) $)
|
|
setindis $p |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph ) $=
|
|
( cv wral wi wal wsb nfcv nfralxy nfim raleq biimprd equcoms imim12d cbv3
|
|
weq bj-sbime ralimi imim1i alimi ax-setind 3syl ) BFEMZNZCOZEPBFDMZNZAOZD
|
|
PADFQZFUPNZAOZDPADPUOUREDUNCDBDFUMDUMRGSHTUQAEBEFUPEUPRJSITEDUFZUQUNCAVBU
|
|
NUQBFUMUPUAUBCAODELUCUDUEURVADUTUQAUSBFUPABDFGKUGUHUIUJAFDUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d y a $. $d ph y $.
|
|
ax-bdsetind.bd $e |- Bdd ph $.
|
|
$( Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) $)
|
|
ax-bdsetind $a |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
bdsetindis.bd $e |- Bdd ph $.
|
|
bdsetindis.nf0 $e |- F/ x ps $.
|
|
bdsetindis.nf1 $e |- F/ x ch $.
|
|
bdsetindis.nf2 $e |- F/ y ph $.
|
|
bdsetindis.nf3 $e |- F/ y ps $.
|
|
bdsetindis.1 $e |- ( x = z -> ( ph -> ps ) ) $.
|
|
bdsetindis.2 $e |- ( x = y -> ( ch -> ph ) ) $.
|
|
$( Axiom of bounded set induction using implicit substitutions.
|
|
(Contributed by BJ, 22-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bdsetindis $p |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph ) $=
|
|
( cv wral wi wal nfcv nfralxy nfim wsb raleq biimprd equcoms imim12d cbv3
|
|
weq bj-sbime ralimi imim1i alimi ax-bdsetind 3syl ) BFENZOZCPZEQBFDNZOZAP
|
|
ZDQADFUAZFUQOZAPZDQADQUPUSEDUOCDBDFUNDUNRHSITURAEBEFUQEUQRKSJTEDUGZURUOCA
|
|
VCUOURBFUNUQUBUCCAPDEMUDUEUFUSVBDVAURAUTBFUQABDFHLUHUIUJUKAFDGULUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Lemma for ~ bj-inf2vn . Remark: unoptimized proof (have to use more
|
|
deduction style). (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-inf2vnlem1 $p |-
|
|
( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A ) $=
|
|
( vz cv wcel c0 wceq csuc wrex wo wb wal wral wi wex eleq1 3syl alimi syl
|
|
wind wa bi2 jaob biimpi simpl mpbidi 0ex isseti pm2.27 ax-mp bj-ex simprd
|
|
exim eqid suceq eqeq2d rspcev mpan2 bj-sucex eqeq1 rexbidv imbi12d syl2im
|
|
vex spcv ralrimiv df-bj-ind sylanbrc ) AEZCFZVJGHZVJBEZIZHZBCJZKZLZAMZGCF
|
|
ZDEZIZCFZDCNCUAVSVLVTOZAMVLAPZVTAPZOZVTVRWDAVRVQVKOZVLVKOZVPVKOZUBZWDVKVQ
|
|
UCZWHWKVLVKVPUDUEZVLVKVTWKWIWJUFVJGCQUGRSVLVTAUNWGWFVTWEWGWFOAGUHUIWEWFUJ
|
|
UKVTAULTRVSWCDCVSWJAMWACFZWBVNHZBCJZWCVRWJAVRWHWJWLWHWIWJWMUMTSWNWBWBHZWP
|
|
WBUOWOWQBWACVMWAHVNWBWBVMWAUPUQURUSWJWPWCOAWBWADVEUTVJWBHZVPWPVKWCWRVOWOB
|
|
CVJWBVNVAVBVJWBCQVCVFVDVGDCVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y t u A $. $d x y t u v Z $.
|
|
$( Lemma for ~ bj-inf2vnlem3 and ~ bj-inf2vnlem4 . Remark: unoptimized
|
|
proof (have to use more deduction style). (Contributed by BJ,
|
|
8-Dec-2019.) (Proof modification is discouraged.) $)
|
|
bj-inf2vnlem2 $p |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) ->
|
|
( Ind Z ->
|
|
A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) ) $=
|
|
( vv cv c0 wceq csuc wrex wo wral wcel wi wal eqeq1 rspcv eleq1 df-bj-ind
|
|
wind rexbidv orbi12d simplbi syl5ibr a1dd vex sucid wb eleq2 eqcoms mpbii
|
|
imbi12d bj-indsuc eleq1a syl6com syl8 com13 com25 mpdi rexlimiv jaoi syl6
|
|
com3l alrimdv bi2.04 albii syl6ib ) AHZIJZVJBHZKZJZBELZMZAENZFUBZCHZEOZDH
|
|
ZEOZWAFOZPZDVSNZVSFOZPZPZCQWEVTWFPPZCQVQVRWHCVTVQVRWGVTVQVSIJZVSVMJZBELZM
|
|
ZVRWGPZVPWMAVSEVJVSJZVKWJVOWLVJVSIRWOVNWKBEVJVSVMRUCUDSWJWNWLWJVRWFWEVRWF
|
|
WJIFOZVRWPGHKFOGFNGFUAUEVSIFTUFUGWKWNBEVLEOZWKVLVSOZWNWKVLVMOZWRVLBUHUIWS
|
|
WRUJVMVSVMVSVLUKULUMWQWEWRVRWKWFWRWEWQVRWKWFPZPZWRWEWQVLFOZXAWDWQXBPDVLVS
|
|
WAVLJWBWQWCXBWAVLETWAVLFTUNSVRXBVMFOWTFVLUOVMFVSUPUQURUSUTVAVBVCVDVEVFWHW
|
|
ICVTWEWFVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z t u A $. $d x y z t u Z $.
|
|
bj-inf2vnlem3.bd1 $e |- Bdd_ A $.
|
|
bj-inf2vnlem3.bd2 $e |- Bdd_ Z $.
|
|
$( Lemma for ~ bj-inf2vn . (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-inf2vnlem3 $p |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) ->
|
|
( Ind Z -> A C_ Z ) ) $=
|
|
( vz vt vu cv wceq wral wcel wi wal bdeli nfv weq eleq1 imbi12d csuc wrex
|
|
c0 wo wind wss bj-inf2vnlem2 ax-bdim biimpd biimprd bdsetindis syl6 dfss2
|
|
syl6ibr ) AJZUCKUOBJUAKBCUBUDACLZDUEZGJZCMZURDMZNZGOZCDUFUPUQHJZCMZVCDMZN
|
|
ZHIJZLVGCMZVGDMZNZNIOVBABIHCDUGVAVFVJGIHUSUTGCEPGDFPUHVFGQVJGQVAIQVFIQGHR
|
|
ZVAVFVKUSVDUTVEURVCCSURVCDSTUIGIRZVAVJVLUSVHUTVIURVGCSURVGDSTUJUKULGCDUMU
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x y z t u A $. $d x y z t u Z $.
|
|
$( Lemma for ~ bj-inf2vn2 . (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-inf2vnlem4 $p |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) ->
|
|
( Ind Z -> A C_ Z ) ) $=
|
|
( vz vt vu cv c0 wceq csuc wrex wral wcel wi wal nfv weq eleq1 imbi12d wo
|
|
wind wss bj-inf2vnlem2 biimpd biimprd setindis syl6 dfss2 syl6ibr ) AHZIJ
|
|
UKBHKJBCLUAACMZDUBZEHZCNZUNDNZOZEPZCDUCULUMFHZCNZUSDNZOZFGHZMVCCNZVCDNZOZ
|
|
OGPURABGFCDUDUQVBVFEGFVBEQVFEQUQGQVBGQEFRZUQVBVGUOUTUPVAUNUSCSUNUSDSTUEEG
|
|
RZUQVFVHUOVDUPVEUNVCCSUNVCDSTUFUGUHECDUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
bj-inf2vn.1 $e |- Bdd_ A $.
|
|
$( A sufficient condition for ` _om ` to be a set. See ~ bj-inf2vn2 for
|
|
the unbounded version from full set induction. (Contributed by BJ,
|
|
8-Dec-2019.) (Proof modification is discouraged.) $)
|
|
bj-inf2vn $p |- ( A e. V -> ( A. x ( x e. A <->
|
|
( x = (/) \/ E. y e. A x = suc y ) ) -> A = _om ) ) $=
|
|
( vz cv wcel c0 wceq csuc wrex wo wb wal com wind wss wi wa bj-inf2vnlem1
|
|
wral bi1 alimi df-ral sylibr bdcv bj-inf2vnlem3 syl alrimiv bj-om syl5ibr
|
|
jca ) AGZCHZUNIJUNBGKJBCLMZNZAOZCPJCDHCQZFGZQCUTRSZFOZTURUSVBABCUAURVAFUR
|
|
UPACUBZVAURUOUPSZAOVCUQVDAUOUPUCUDUPACUEUFABCUTEFUGUHUIUJUMFCDUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( A sufficient condition for ` _om ` to be a set; unbounded version of
|
|
~ bj-inf2vn . (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-inf2vn2 $p |- ( A e. V -> ( A. x ( x e. A <->
|
|
( x = (/) \/ E. y e. A x = suc y ) ) -> A = _om ) ) $=
|
|
( vz cv wcel c0 wceq csuc wrex wo wb wal com wind wss wi wa bj-inf2vnlem1
|
|
wral bi1 alimi df-ral sylibr bj-inf2vnlem4 syl alrimiv jca bj-om syl5ibr
|
|
) AFZCGZULHIULBFJIBCKLZMZANZCOICDGCPZEFZPCURQRZENZSUPUQUTABCTUPUSEUPUNACU
|
|
AZUSUPUMUNRZANVAUOVBAUMUNUBUCUNACUDUEABCURUFUGUHUIECDUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Another axiom of infinity in a constructive setting (see ~ ax-infvn ).
|
|
(Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) $)
|
|
ax-inf2 $a |-
|
|
E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y a $.
|
|
$( Using bounded set induction and the strong axiom of infinity, ` _om ` is
|
|
a set, that is, we recover ~ ax-infvn (see ~ bj-2inf for the equivalence
|
|
of the latter with ~ bj-omex ). (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bj-omex2 $p |- _om e. _V $=
|
|
( va vx vy com wel cv c0 wceq csuc wrex wo wb wal ax-inf2 cvv wcel wi vex
|
|
bdcv bj-inf2vn ax-mp eximii issetri ) ADBAEBFZGHUDCFIHCAFZJKLBMZUEDHZABCA
|
|
NUEOPUFUGQARBCUEOASTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d a x y z A $.
|
|
$( Alternate proof of ~ bj-nn0suc , also constructive but from ~ ax-inf2 ,
|
|
hence requiring ~ ax-bdsetind . (Contributed by BJ, 8-Dec-2019.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
bj-nn0sucALT $p |- ( A e. _om <-> ( A = (/) \/ E. x e. _om A = suc x ) ) $=
|
|
( va vy vz com wcel c0 wceq cv csuc wrex wo wb wal wi ax-mp bibi12d eleq1
|
|
cvv wex ax-inf2 bdcv bj-inf2vn eleq2 rexeq orbi2d albidv nfcv eqeq1 suceq
|
|
vex nfv eqeq2d cbvrexv rexbidv syl5bb orbi12d bi1 syl6bi peano1 bj-peano2
|
|
spcimgf pm2.43b mpbiri eleq1a imp sylan rexlimiva jaoi mpcom eximii bj-ex
|
|
impbid1 ) BFGZBHIZBAJZKZIZAFLZMZNZCUAWBDJZCJZGZWCHIZWCEJZKZIZEWDLZMZNZDOZ
|
|
WBCDECUBWDFIZWMWBWDTGWMWNPCULDEWDTCUCUDQWNWMWCFGZWFWIEFLZMZNZDOZWBWNWLWRD
|
|
WNWEWOWKWQWDFWCUEWNWJWPWFWIEWDFUFUGRUHWSVOWAWSVOWAWRVOWAPZDBFDBUIWTDUMWCB
|
|
IZWRWBWTXAWOVOWQWAWCBFSXAWFVPWPVTWCBHUJWPWCVRIZAFLXAVTWIXBEAFWGVQIWHVRWCW
|
|
GVQUKUNUOXAXBVSAFWCBVRUJUPUQURRVOWAUSUTVCVDVPVOVTVPVOHFGVABHFSVEVSVOAFVQF
|
|
GVRFGZVSVOVQVBXCVSVOVRFBVFVGVHVIVJVNUTVKVLWBCVMQ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
Full induction
|
|
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
|
|
|
|
In this section, using the axiom of set induction, we prove full induction on
|
|
the set of natural numbers.
|
|
|
|
$)
|
|
|
|
${
|
|
$d x y z $. $d y z ph $. $d ps z $. $d ch z $. $d th z $.
|
|
bj-findis.nf0 $e |- F/ x ps $.
|
|
bj-findis.nf1 $e |- F/ x ch $.
|
|
bj-findis.nfsuc $e |- F/ x th $.
|
|
bj-findis.0 $e |- ( x = (/) -> ( ps -> ph ) ) $.
|
|
bj-findis.1 $e |- ( x = y -> ( ph -> ch ) ) $.
|
|
bj-findis.suc $e |- ( x = suc y -> ( th -> ph ) ) $.
|
|
$( Principle of induction, using implicit substitutions (the biconditional
|
|
versions of the hypotheses are implicit substitutions, and we have
|
|
weakened them to implications). Constructive proof (from CZF). See
|
|
~ bj-bdfindis for a bounded version not requiring ~ ax-setind . See
|
|
~ finds for a proof in IZF. From this version, it is easy to prove of
|
|
~ finds , ~ finds2 , ~ finds1 . (Contributed by BJ, 22-Dec-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-findis $p |- ( ( ps /\ A. y e. _om ( ch -> th ) ) -> A. x e. _om ph ) $=
|
|
( vz wi com wral wa wcel wceq nfv cv wal c0 csuc wrex wo bj-nn0suc pm3.21
|
|
ad2antrr pm2.04 ralimi2 imim2 ral2imi imp sylan2 r19.29 sucid mpbiri ax-1
|
|
vex eleq2 pm2.27 anim12ii mpdan impcom reximi syl adantll orim12d alrimiv
|
|
ex syl7bi nfim nfan nfcv nfrexxy nfor biimprd imim12d biimpd eqtr expimpd
|
|
eleq1 rexlimdvw jaod setindis df-ral sylibr ) BCDNZFOPZQZEUAZORZANZEUBZAE
|
|
OPWKFUAZORZCNZFMUAZPZWSORZWSUCSZBQZWSWPUDZSZDQZFOUEZUFZNZNZMUBWOWKXJMXAXB
|
|
XEFOUEZUFZWKWTXHFWSUGWKWTXLXHNWKWTQXBXCXKXGBXBXCNWJWTBXBUHUIWJWTXKXGNZBWJ
|
|
WTQWPWSRZDNZFOPZXMWTWJXNCNZFOPZXPWRXQFWSOXNWQCUJUKWJXRXPWIXQXOFOCDXNULUMU
|
|
NUOXPXKXGXPXKQXOXEQZFOUEXGXOXEFOUPXSXFFOXEXOXFXEXNXOXFNXEXNWPXDRWPFUTUQWS
|
|
XDWPVAURXEXOXEXNDXEXOUSXNDVBVCVDVEVFVGVKVGVHVIVKVLVJWNWRXIEMFWQCEWQETHVMX
|
|
AXHEXAETXCXGEXBBEXBETGVNXFEFOEOVOXEDEXEETIVNVPVQVMWNMTWRMTWLWPSZWQWMACXTW
|
|
MWQWLWPOWCVRKVSWLWSSZWMXAXHAYAWMXAWLWSOWCVTYAXCAXGYAXBBAYAXBQWLUCSBANWLWS
|
|
UCWAJVGWBYAXFAFOYAXEDAYAXEQWLXDSDANWLWSXDWALVGWBWDWEVSWFVGAEOWGWH $.
|
|
|
|
bj-findisg.nfa $e |- F/_ x A $.
|
|
bj-findisg.nfterm $e |- F/ x ta $.
|
|
bj-findisg.term $e |- ( x = A -> ( ph -> ta ) ) $.
|
|
$( Version of ~ bj-findis using a class term in the consequent.
|
|
Constructive proof (from CZF). See the comment of ~ bj-findis for
|
|
explanations. (Contributed by BJ, 21-Nov-2019.)
|
|
(Proof modification is discouraged.) $)
|
|
bj-findisg $p |-
|
|
( ( ps /\ A. y e. _om ( ch -> th ) ) -> ( A e. _om -> ta ) ) $=
|
|
( wi com wral wa wcel bj-findis nfcv bj-rspg syl ) BCDRGSTUAAFSTHSUBERABC
|
|
DFGIJKLMNUCAEFHSOFSUDPQUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Principle of induction, using explicit substitutions. Constructive
|
|
proof (from CZF). See the comment of ~ bj-findis for explanations.
|
|
From this version, it is easy to prove ~ findes . (Contributed by BJ,
|
|
21-Nov-2019.) (Proof modification is discouraged.) $)
|
|
bj-findes $p |- ( ( [. (/) / x ]. ph /\
|
|
A. x e. _om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. _om ph ) $=
|
|
( vy cv csuc wsbc wi com wral c0 wsb nfv nfim nfs1v nfsbc1v sbequ12 suceq
|
|
wceq sbceq1a biimprd sbceq1d imbi12d cbvral sbequ1 bj-findis sylan2b ) AA
|
|
BBDZEZFZGZBHIABJFZABCKZABCDZEZFZGZCHIABHIUJUPBCHAUICACLUICLMULUOBABCNZABU
|
|
NOZMUGUMRZAULUIUOABCPUSABUHUNUGUMQUAUBUCAUKULUOBCABJOUQURUGJRAUKABJSTABCU
|
|
DUGUNRAUOABUNSTUEUF $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
CZF: Strong collection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In this section, we state the axiom scheme of strong collection, which is
|
|
part of CZF set theory.
|
|
|
|
$)
|
|
|
|
${
|
|
$d a b x y $. $d a b ph $.
|
|
$( Axiom scheme of strong collection. It is stated with all possible
|
|
disjoint variable conditions, to show that this weak form is sufficient.
|
|
(Contributed by BJ, 5-Oct-2019.) $)
|
|
ax-strcoll $a |- A. a ( A. x e. a E. y ph ->
|
|
E. b A. y ( y e. b <-> E. x e. a ph ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y z $. $d b z ph $.
|
|
$( Version of ~ ax-strcoll with one DV condition removed and without
|
|
initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) $)
|
|
strcoll2 $p |- ( A. x e. a E. y ph ->
|
|
E. b A. y ( y e. b <-> E. x e. a ph ) ) $=
|
|
( vz wex cv wral wel wrex wb wal wi weq raleq rexeq bibi2d albidv exbidv
|
|
imbi12d ax-strcoll spi chvarv ) ACGZBFHZIZCEJZABUFKZLZCMZEGZNZUEBDHZIZUHA
|
|
BUNKZLZCMZEGZNFDFDOZUGUOULUSUEBUFUNPUTUKUREUTUJUQCUTUIUPUHABUFUNQRSTUAUMF
|
|
ABCFEUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y z $. $d z ph $.
|
|
$( Closed form of ~ strcollnf . Version of ~ ax-strcoll with one DV
|
|
condition removed, the other DV condition replaced by a non-freeness
|
|
antecedent, and without initial universal quantifier. (Contributed by
|
|
BJ, 21-Oct-2019.) $)
|
|
strcollnft $p |- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph ->
|
|
E. b A. y ( y e. b <-> E. x e. a ph ) ) ) $=
|
|
( vz wex cv wral wel wrex wb wal wnf strcoll2 nfnf1 nfal nfa2 nfvd nfa1
|
|
nfcvd sp nfrexdxy sps alcoms nfbid nfald weq wa nfan elequ2 adantl bibi1d
|
|
nfv albid ex cbvexd syl5ib ) ACGBDHZICFJZABUSKZLZCMZFGAENZCMZBMZCEJZVALZC
|
|
MZEGABCDFOVFVCVIFEVEEBVDECAEPQQVFVBECVDCBRZVFUTVAEVFUTESVDVAENZCBVDBMZVKC
|
|
VLAEBUSVDBTVLEUSUAVDBUBUCUDUEUFUGVFFEUHZVCVILVFVMUIZVBVHCVFVMCVJVMCUNUJVN
|
|
UTVGVAVMUTVGLVFFECUKULUMUOUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y z $. $d z ph $.
|
|
strcollnf.nf $e |- F/ b ph $.
|
|
$( Version of ~ ax-strcoll with one DV condition removed, the other DV
|
|
condition replaced by a non-freeness hypothesis, and without initial
|
|
universal quantifier. (Contributed by BJ, 21-Oct-2019.) $)
|
|
strcollnf $p |- ( A. x e. a E. y ph ->
|
|
E. b A. y ( y e. b <-> E. x e. a ph ) ) $=
|
|
( wnf wal wex cv wral wel wrex wb wi strcollnft ax-gen mpg ) AEGZCHACIBDJ
|
|
ZKCELABTMNCHEIOBABCDEPSCFQR $.
|
|
|
|
$( Alternate proof of ~ strcollnf , not using ~ strcollnft . (Contributed
|
|
by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
strcollnfALT $p |- ( A. x e. a E. y ph ->
|
|
E. b A. y ( y e. b <-> E. x e. a ph ) ) $=
|
|
( vz wex cv wral wel wrex wb wal strcoll2 nfv nfcv nfrexxy nfbi nfal weq
|
|
elequ2 bibi1d albidv cbvex sylib ) ACHBDIZJCGKZABUGLZMZCNZGHCEKZUIMZCNZEH
|
|
ABCDGOUKUNGEUJECUHUIEUHEPAEBUGEUGQFRSTUNGPGEUAZUJUMCUOUHULUIGECUBUCUDUEUF
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
CZF: Subset collection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In this section, we state the axiom scheme of subset collection, which is
|
|
part of CZF set theory.
|
|
|
|
$)
|
|
|
|
${
|
|
$d a b c d x y z $. $d a b c d ph $.
|
|
$( Axiom scheme of subset collection. It is stated with all possible
|
|
disjoint variable conditions, to show that this weak form is sufficient.
|
|
(Contributed by BJ, 5-Oct-2019.) $)
|
|
ax-sscoll $a |- A. a A. b E. c A. z ( A. x e. a E. y e. b ph ->
|
|
E. d e. c A. y ( y e. d <-> E. x e. a ph ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d a b c d x y z u v $. $d c d u v ph $.
|
|
$( Version of ~ ax-sscoll with two DV conditions removed and without
|
|
initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) $)
|
|
sscoll2 $p |- E. c A. z ( A. x e. a E. y e. b ph ->
|
|
E. d e. c A. y ( y e. d <-> E. x e. a ph ) ) $=
|
|
( vv vu cv wrex wral wb wal wi wex weq nfv rexeq wel adantl adantr bibi2d
|
|
wa simpl raleqbidv albid rexbid imbi12d exbid ax-sscoll spi ch2varv ) ACI
|
|
KZLZBJKZMZCHUAZABUQLZNZCOZHGKZLZPZDOZGQZACFKZLZBEKZMZUSABVJLZNZCOZHVCLZPZ
|
|
DOZGQJEIFJERZIFRZUEZVFVQGVTGSVTVEVPDVTDSVTURVKVDVOVTUPVIBUQVJVRVSUFVSUPVI
|
|
NVRACUOVHTUBUGVTVBVNHVCVTHSVTVAVMCVTCSVTUTVLUSVRUTVLNVSABUQVJTUCUDUHUIUJU
|
|
HUKVGIVGIOJABCDJIGHULUMUMUN $.
|
|
$}
|
|
|
|
$(
|
|
${
|
|
$d a b c d x y z u v $. $d u v ph $.
|
|
sscollnf.1 $e |- F/ c ph $.
|
|
sscollnf.2 $e |- F/ d ph $.
|
|
@( Version of ~ ax-sscoll with two DV conditions removed, the other two DV
|
|
conditions replaced by non-freeness hypotheses, and without initial
|
|
universal quantifiers. (Contributed by BJ, 5-Oct-2019.) @)
|
|
sscollnf $p |- E. c A. z ( A. x e. a E. y e. b ph ->
|
|
E. d e. c A. y ( y e. d <-> E. x e. a ph ) ) $=
|
|
? $.
|
|
$}
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Real numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Dedekind completeness for Dedekind real numbers: every inhabited
|
|
upper-bounded located set of reals has a real upper bound. Ideally, this
|
|
axiom should be "proved" as "axddkcomp" for the real numbers constructed
|
|
from IZF, and then the axiom ~ ax-ddkcomp should be used in place of
|
|
construction specific results. In particular, ~ axcaucvg should be proved
|
|
from it. (Contributed by BJ, 24-Oct-2021.) $)
|
|
ax-ddkcomp $a |- ( ( ( A C_ RR /\ E. x x e. A ) /\
|
|
E. x e. RR A. y e. A y < x /\
|
|
A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) )
|
|
) -> E. x e. RR (
|
|
A. y e. A y <_ x /\ ( ( B e. R /\ A. y e. A y <_ B ) -> x <_ B ) ) ) $.
|
|
|
|
$( (End of BJ's mathbox.) $)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Mathbox for Jim Kingdon
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
${
|
|
$d A u v x $. $d B u v x $. $d Q u v x $. $d u v z $.
|
|
qdencn.q $e |- Q =
|
|
{ z e. CC | ( ( Re ` z ) e. QQ /\ ( Im ` z ) e. QQ ) } $.
|
|
$( The set of complex numbers whose real and imaginary parts are rational
|
|
is dense in the complex plane. This is a two dimensional analogue to
|
|
~ qdenre (and also would hold for ` RR X. RR ` with the usual metric;
|
|
this is not about complex numbers in particular). (Contributed by Jim
|
|
Kingdon, 18-Oct-2021.) $)
|
|
qdencn $p |- ( ( A e. CC /\ B e. RR+ ) ->
|
|
E. x e. Q ( abs ` ( x - A ) ) < B ) $=
|
|
( vu cc wcel wa cfv cmin co cabs clt cq adantr ci cmul caddc vv crp cv c2
|
|
cre cdiv wbr wrex cr simpl recld simpr rphalfcld qdenre syl2anc cim imcld
|
|
simpll qcn ad2antrl ax-icn a1i mulcld addcld crred simplrl eqeltrd simprl
|
|
qre crimd jca fveq2 eleq1d anbi12d elrab2 sylanbrc subcld abscld ad2antrr
|
|
wceq recnd readdcld rpred cle replimd oveq2d eqtrd fveq2d abstrid eqbrtrd
|
|
addsub4d absmuld subdid absi oveq1i mulid2d syl5eq 3eqtr3d breqtrd simprr
|
|
c1 simplrr lt2halvesd lelttrd oveq1 breq1d rspcev rexlimddv ) CHIZDUBIZJZ
|
|
GUCZCUEKZLMZNKZDUDUFMZOUGZAUCZCLMZNKZDOUGZAEUHZGPXKXMUIIZXPUBIZXQGPUHXKCX
|
|
IXJUJZUKZXKDXIXJULZUMZGXMXPUNUOXKXLPIZXQJZJZUAUCZCUPKZLMZNKZXPOUGZYBUAPYK
|
|
YMUIIZYDYPUAPUHYKCXIXJYJURZUQZXKYDYJYHQUAYMXPUNUOYKYLPIZYPJZJZXLRYLSMZTMZ
|
|
EIZUUDCLMZNKZDOUGZYBUUBUUDHIUUDUEKZPIZUUDUPKZPIZJZUUEUUBXLUUCYKXLHIZUUAYI
|
|
UUNXKXQXLUSUTQZUUBRYLRHIUUBVAVBZYTYLHIYKYPYLUSUTZVCZVDZUUBUUJUULUUBUUIXLP
|
|
UUBXLYLYKXLUIIZUUAYIUUTXKXQXLVIUTQZYTYLUIIYKYPYLVIUTZVEXKYIXQUUAVFVGUUBUU
|
|
KYLPUUBXLYLUVAUVBVJYKYTYPVHVGVKBUCZUEKZPIZUVCUPKZPIZJUUMBUUDHEUVCUUDVTZUV
|
|
EUUJUVGUULUVHUVDUUIPUVCUUDUEVLVMUVHUVFUUKPUVCUUDUPVLVMVNFVOVPUUBUUGXOYOTM
|
|
ZDUUBUUFUUBUUDCUUSYKXIUUAYRQVQVRUUBXOYOUUBXNUUBXLXMUUOUUBXMXKYCYJUUAYFVSW
|
|
AZVQZVRZUUBYNUUBYLYMUUQUUBYMYKYQUUAYSQWAZVQZVRZWBUUBDXKXJYJUUAYGVSWCZUUBU
|
|
UGXOUUCRYMSMZLMZNKZTMZUVIWDUUBUUGXNUVRTMZNKUVTWDUUBUUFUWANUUBUUFUUDXMUVQT
|
|
MZLMZUWAXKUUFUWCVTYJUUAXKCUWBUUDLXKCYEWEWFVSUUBXLUUCXMUVQUUOUURUVJUUBRYMU
|
|
UPUVMVCZWKWGWHUUBXNUVRUVKUUBUUCUVQUURUWDVQWIWJUUBUVSYOXOTUUBRYNSMZNKRNKZY
|
|
OSMZUVSYOUUBRYNUUPUVNWLUUBUWEUVRNUUBRYLYMUUPUUQUVMWMWHUUBUWGXAYOSMYOUWFXA
|
|
YOSWNWOUUBYOUUBYOUVOWAWPWQWRWFWSUUBXOYODUVLUVOUVPXKYIXQUUAXBYKYTYPWTXCXDY
|
|
AUUHAUUDEXRUUDVTZXTUUGDOUWHXSUUFNXRUUDCLXEWHXFXGUOXHXH $.
|
|
$}
|
|
|
|
$( (End of Jim Kingdon's mathbox.) $)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Mathbox for Mykola Mostovenko
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$( Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by
|
|
MM, 11-Aug-2018.) $)
|
|
ax1hfs $p |- ( ph -> ( ph /\ ph ) ) $=
|
|
( wa ax-ia3 pm2.43i ) AAABAACD $.
|
|
|
|
$( (End of Mykola Mostovenko's mathbox.) $)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Mathbox for David A. Wheeler
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Allsome quantifier
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
These are definitions and proofs involving an experimental
|
|
"allsome" quantifier (aka "all some").
|
|
|
|
In informal language, statements like
|
|
"All Martians are green" imply that there is at least one Martian.
|
|
But it's easy to mistranslate informal language into formal notations
|
|
because similar statements like ` A. x ph -> ps ` do _not_
|
|
imply that ` ph ` is ever true, leading to vacuous truths.
|
|
Some systems include a mechanism to counter this, e.g., PVS allows
|
|
types to be appended with "+" to declare that they are nonempty.
|
|
This section presents a different solution to the same problem.
|
|
|
|
The "allsome" quantifier expressly includes the notion of both
|
|
"all" and "there exists at least one" (aka some), and is defined
|
|
to make it easier to more directly express both notions.
|
|
The hope is that if a quantifier more directly expresses this concept,
|
|
it will be used instead and reduce the risk of creating formal expressions
|
|
that look okay but in fact are mistranslations.
|
|
The term "allsome" was chosen because it's short, easy to say, and
|
|
clearly hints at the two concepts it combines.
|
|
|
|
I do not expect this to be used much in metamath, because in metamath
|
|
there's a general policy of avoiding the use of new definitions
|
|
unless there are very strong reasons to do so. Instead, my goal is to
|
|
rigorously define this quantifier and demonstrate a few
|
|
basic properties of it.
|
|
|
|
The syntax allows two forms that look like they would be problematic,
|
|
but they are fine. When applied to a top-level implication we allow
|
|
` A! x ( ph -> ps ) ` , and when restricted (applied to a class) we allow
|
|
` A! x e. A ph ` .
|
|
The first symbol after the setvar variable must
|
|
always be ` e. ` if it is the form applied to a class, and since
|
|
` e. ` cannot begin a wff, it is unambiguous.
|
|
The ` -> ` looks like it would be a problem because ` ph ` or ` ps `
|
|
might include implications, but any implication arrow
|
|
` -> ` within any wff must be surrounded by parentheses, so only the
|
|
implication arrow of ` A! ` can follow the wff.
|
|
The implication syntax would work fine without the parentheses, but
|
|
I added the parentheses because it makes things clearer inside
|
|
larger complex expressions, and it's also more consistent with
|
|
the rest of the syntax.
|
|
|
|
For more, see "The Allsome Quantifier" by David A. Wheeler at
|
|
https://dwheeler.com/essays/allsome.html
|
|
I hope that others will eventually agree that allsome is awesome.
|
|
$)
|
|
|
|
$c A! $. $( "inverted A" exclamation point (read: "all some"
|
|
or more briefly "allsome") $)
|
|
|
|
$( Extend wff definition to include "all some" applied to a top-level
|
|
implication, which means ` ps ` is true whenever ` ph ` is true, and there
|
|
is at least least one ` x ` where ` ph ` is true. (Contributed by David
|
|
A. Wheeler, 20-Oct-2018.) $)
|
|
walsi $a wff A! x ( ph -> ps ) $.
|
|
|
|
$( Extend wff definition to include "all some" applied to a class, which
|
|
means ` ph ` is true for all ` x ` in ` A ` , and there is at least one
|
|
` x ` in ` A ` . (Contributed by David A. Wheeler, 20-Oct-2018.) $)
|
|
walsc $a wff A! x e. A ph $.
|
|
|
|
$( Define "all some" applied to a top-level implication, which means ` ps `
|
|
is true whenever ` ph ` is true and there is at least one ` x ` where
|
|
` ph ` is true. (Contributed by David A. Wheeler, 20-Oct-2018.) $)
|
|
df-alsi $a |- ( A! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) ) $.
|
|
|
|
$( Define "all some" applied to a class, which means ` ph ` is true for all
|
|
` x ` in ` A ` and there is at least one ` x ` in ` A ` . (Contributed by
|
|
David A. Wheeler, 20-Oct-2018.) $)
|
|
df-alsc $a |- ( A! x e. A ph <-> ( A. x e. A ph /\ E. x x e. A ) ) $.
|
|
|
|
$( There is an equivalence between the two "all some" forms. (Contributed by
|
|
David A. Wheeler, 22-Oct-2018.) $)
|
|
alsconv $p |- ( A! x ( x e. A -> ph ) <-> A! x e. A ph ) $=
|
|
( wral cv wcel wex wa wi walsc walsi df-ral anbi1i df-alsc df-alsi 3bitr4ri
|
|
wal ) ABCDZBECFZBGZHSAIBQZTHABCJSABKRUATABCLMABCNSABOP $.
|
|
|
|
${
|
|
alsi1d.1 $e |- ( ph -> A! x ( ps -> ch ) ) $.
|
|
$( Deduction rule: Given "all some" applied to a top-level inference, you
|
|
can extract the "for all" part. (Contributed by David A. Wheeler,
|
|
20-Oct-2018.) $)
|
|
alsi1d $p |- ( ph -> A. x ( ps -> ch ) ) $=
|
|
( wi wal wex walsi wa df-alsi sylib simpld ) ABCFDGZBDHZABCDINOJEBCDKLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
alsi2d.1 $e |- ( ph -> A! x ( ps -> ch ) ) $.
|
|
$( Deduction rule: Given "all some" applied to a top-level inference, you
|
|
can extract the "exists" part. (Contributed by David A. Wheeler,
|
|
20-Oct-2018.) $)
|
|
alsi2d $p |- ( ph -> E. x ps ) $=
|
|
( wi wal wex walsi wa df-alsi sylib simprd ) ABCFDGZBDHZABCDINOJEBCDKLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
alsc1d.1 $e |- ( ph -> A! x e. A ps ) $.
|
|
$( Deduction rule: Given "all some" applied to a class, you can extract
|
|
the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) $)
|
|
alsc1d $p |- ( ph -> A. x e. A ps ) $=
|
|
( wral cv wcel wex walsc wa df-alsc sylib simpld ) ABCDFZCGDHCIZABCDJOPKE
|
|
BCDLMN $.
|
|
$}
|
|
|
|
${
|
|
alsc2d.1 $e |- ( ph -> A! x e. A ps ) $.
|
|
$( Deduction rule: Given "all some" applied to a class, you can extract
|
|
the "there exists" part. (Contributed by David A. Wheeler,
|
|
20-Oct-2018.) $)
|
|
alsc2d $p |- ( ph -> E. x x e. A ) $=
|
|
( wral cv wcel wex walsc wa df-alsc sylib simprd ) ABCDFZCGDHCIZABCDJOPKE
|
|
BCDLMN $.
|
|
$}
|
|
|
|
$( (End of David A. Wheeler's mathbox.) $)
|